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0911.4557
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hep-th/09114557
D-brane orbiting NS5-branes
Gyeong Yun Jun111E-mail: gyjun@ks.ac.kr and Pyung Seong Kwon222E-mail:
bskwon@ks.ac.kr
Department of Physics, Kyungsung University,
110-1 Daeyeon-dong, Nam-gu, Pusan 608-736, Korea
Abstract
We study real time dynamics of a Dp-brane orbiting a stack of NS5-branes. It
is generally known that a BPS D-brane moving in the vicinity of NS5-branes
becomes unstable due to the presence of tachyonic degree of freedom induced on
the D-brane. Indeed, the D-brane necessarily falls into the fivebranes due to
gravitational attraction and eventually collapses into a pressureless fluid.
Such a decay of the D-brane is known to be closely related to the rolling
tachyon problem. In this paper we show that in special cases the decay of
D-brane caused by gravitational attraction can be avoided. Namely for certain
values of energy and angular momentum the D-brane orbits around the
fivebranes, maintaining certain distance from the fivebranes all the time, and
the process of tachyon condensation is suppressed. We show that the tachyonic
degree of freedom induced on such a D-brane really disappears and the brane
returns to a stable D-brane.
KEYWORDS : D-brane, NS5-brane, tachyon
After the D-brane was found in 1995 [1], it has been generally conjectured
that our universe may be a stack of D-branes with standard model (SM) fields
living on it [2]. But recently there was an argument that true background
p-brane immanent in our spacetime may be an NS-NS type brane, rather than
D-brane [3]. Indeed, brane world models including NS-brane have been already
considered in the literature [4, 5] including ”Little String Theory”(LST) [6].
In these models the NS-branes usually appear as background branes near which
the D-brane is to be placed, and in particular in [5] it was argued that these
NS-branes play an important role in the context of the cosmological constant
problem. In [5] it was shown that in the presence of the background NS-branes
the disturbance of the bulk geometry due to quantum fluctuations of SM-fields
with support on the D-brane (SM-brane) is highly suppressed in the limit
$g_{s}\rightarrow 0$. So the bulk geometry, as well as the flat intrinsic
geometry of the brane, is practically insensitive to the quantum fluctuations
in this limit.
Apart from this, Kutasov noticed [7] that the real time dynamics of D-brane
near NS5-branes is closely related to the rolling tachyon problem of the
unstable D-brane. In [7] he considered a BPS Dp-brane propagating at some
distance from a stack of k parallel NS5-branes of the type II string theory.
In this configuration the supersymmetry of the system is completely broken and
the D-brane becomes unstable. Indeed, since the D-brane experiences an
attractive force it either escapes to infinity after deflected by the
fivebranes, or it moves towards the fivebranes and eventually decays into a
pressureless fluid. Such a decay of the Dp-brane is described by the rolling
tachyon solution where the role of the tachyon is played by the radial mode on
the D-brane. So, as the D-brane approaches the fivebranes tachyon condensation
occurs, and the D-brane turns into some ”tachyon matter” state which has an
equation of state of a pressureless fluid. Similar configurations have been
considered since then by some others [8], they all obtained basically the same
result. They did not find solutions corresponding to a D-brane in orbit around
the fivebranes.
If there does not really exist the solution in which the D-brane neither
escapes to infinity nor falls into the fivebranes, it would be unnatural to
consider the brane world models where D-branes are placed near background
NS5-branes, because in the former case the bound state of the D-brane and the
fivebranes can not form, while in the later case the D-brane will be absorbed
into the fivebranes and eventually lose most of its properties including
energy, charge and supersymmetry. So it would be interesting if we can find a
solution where the D-brane is in orbit around the five-branes. But with the
given configuration the solution with stable orbits does not exist (unless we
compactify one of the transverse directions), because in the 4d transverse
space the D-brane experiences a gravitational potential $V\sim-{1}/{r^{2}}$
and this potential does not allow for stable orbits. However, though the
solution with stable orbits does not exist, the solution with metastable
orbits surely exists. For certain values of energy and angular momentum of the
D-brane the attractive force between D-brane and five branes vanishes for all
$r$ just as in the case of two parallel BPS D-branes. In this paper we will
first show that the Dirac-Born-Infeld (DBI) action describing a Dp-brane
moving in the vicinity of NS5-branes really admits such a solution with
metastable orbits, then we will show that in this case the tachyon potential
becomes flat and the tachyon induced on the D-brane turns into a trivial
massless constant field. Namely the tachyonic degree of freedom disappears and
the D-brane returns to the stable brane.
Before we start we will briefly review the calculations in [7] which are
necessary to develop our discussion. In the presence of k coincident
NS5-branes, the metric, dilaton and NS-NS 3-form fields are respectively given
by
$\displaystyle ds^{2}=dx_{\mu}dx^{\mu}+H(x^{n})dx^{m}dx^{m}\equiv
G_{MN}dx^{M}dx^{N},$ $\displaystyle e^{2(\Phi-\Phi_{0})}=H(x^{n}),$ (1)
$\displaystyle H_{mnp}=-\epsilon^{q}_{mnp}\partial_{q}\Phi,$
where $x^{\mu}(\mu=0,1,...5)$ are the coordinates along the world volume of
the k coincident NS5-branes, while $x^{m}\;(m=6,7,8,9)$ the coordinates along
the transverse dimensions. Also $H(x^{n})$ is a harmonic function
$H=1+{kl^{2}_{s}}/{r^{2}},$ (2)
where $r^{2}=\sum_{n=6}^{9}x^{m}x_{m}$, and $l_{s}$ is the string length.
Now consider a Dp-brane stretched along the directions $(x_{1},...x_{p})$ with
$p\leq 5$, and label the world volume of the D-brane by $\xi^{\mu}$ ,
$\mu=0,1,...p$. Then in the static gauge we have $\xi^{\mu}=x^{\mu}$. The
dynamics of the world volume fields of the Dp-brane propagating in the above
background fields is governed by DBI (Dirac-Born-Infeld) action
$S_{p}=-\tau_{p}\int d^{p+1}\xi e^{-(\Phi-\Phi_{0})}\sqrt{-det\mid
G_{\mu\nu}+B_{\mu\nu}\mid}\;,$ (3)
where $G_{\mu\nu}$ and $B_{\mu\nu}$ are the pullbacks of $G_{MN}$ and
$B_{MN}$:
$G_{\mu\nu}=\frac{\partial x^{M}}{\partial\xi^{\mu}}\frac{\partial
x^{N}}{\partial\xi^{\nu}}G_{MN},\;\;\;\;B_{\mu\nu}=\frac{\partial{x^{M}}}{\partial\xi^{\mu}}\frac{\partial{x^{N}}}{\partial\xi^{\nu}}B_{MN}\;.$
(4)
In (4) $x^{M}=(\xi^{\mu},x^{m})$, and $x^{m}$ now represent the position of
the Dp-brane in the transverse space and give rise to world volume scalars
$X^{m}(\xi^{\mu})$. In this paper we will only consider the spatially
homogeneous solutions for which $X^{m}=X^{m}(t)$. $G_{\mu\nu}$ then reduces to
$G_{\mu\nu}=\eta_{\mu\nu}+\delta^{0}_{\mu}\delta^{0}_{\nu}\;H(X^{n})\dot{X}^{m}\dot{X}^{m}\;.$
(5)
We can also allow for nonzero values of $B_{\mu\nu}$ on the D-brane. But it
generally breaks the isotropy of the Dp-brane world volume, and generates off-
diagonal components of the metric and the stress tensor. In this paper we will
assume that the world volume components of the B-field vanish as in [7], which
implies that the induced B-field in (3) vanishes, i.e., $B_{\mu\nu}=0$ in the
given configuration.
With these values of world volume fields, the DBI action (3) becomes
$S_{p}=-\tau_{p}V_{p}\int dt\sqrt{H^{-1}(X^{n})-\dot{X}^{m}\dot{X}^{m}}\;,$
(6)
where $V_{p}$ is the volume of the Dp-brane. The action (6) admits a conserved
quantity $T_{\mu\nu}$, the stress-energy tensor of the scalar fields
$X^{m}(t)$. The nonzero components of $T_{\mu\nu}$ are given by
$\displaystyle
T_{00}=\tau_{p}\frac{1}{H\sqrt{H^{-1}(X^{n})-\dot{X}^{m}\dot{X}^{m}}}\;,\;\;\;T_{ij}=-\tau_{p}\;\delta_{ij}\;\sqrt{H^{-1}(X^{n})-\dot{X}^{m}\dot{X}^{m}}\;,$
(7)
where we have set $V_{p}$ equal to one. $T_{00}$ in (7) is a conserved energy
defined by
$E=P_{n}\dot{X}^{n}-\mathcal{L}\;\;,$ (8)
where the momentum $P_{n}$ is
$P_{n}=\frac{\delta\mathcal{L}}{\delta\dot{X}^{n}}=\tau_{p}\frac{\dot{X}^{n}}{\sqrt{H^{-1}(X^{n})-\dot{X}^{m}\dot{X}^{m}}}\;\;.$
(9)
One can check that substituting (9) into (8) gives $T_{00}$ in (7). There is
another conserved quantity. If we assume that the Dp-brane moves in the
$(x^{6},x^{7})$ plane it can be shown that the angular momentum defined by
$L=X^{6}P^{7}-X^{7}P^{6}$ is also conserved. For instance, see (24).
Let us introduce polar coordinates defined by $X^{6}=R\cos\theta$ and
$X^{7}=R\sin\theta$. In these coordinates the angular momentum and the
conserved energy take the forms
$L=\tau_{p}\frac{R^{2}\dot{\theta}}{\sqrt{H^{-1}(R)-(\dot{R}^{2}+R^{2}\dot{\theta}^{2})}},$
(10)
$E=\tau_{p}\frac{1}{H\sqrt{H^{-1}(R)-(\dot{R}^{2}+R^{2}\dot{\theta}^{2})}}\;.$
(11)
Solving these two equations in terms of $\dot{R}^{2}$ and $\dot{\theta}^{2}$
one obtains
$\dot{R}^{2}=\frac{1}{\epsilon^{2}H^{2}}\Big{[}\epsilon^{2}H-(1+\frac{l^{2}}{R^{2}})\Big{]}\;\;,$
(12)
and
$\dot{\theta}^{2}=\frac{1}{R^{4}H^{2}}\frac{l^{2}}{\epsilon^{2}}\;\;,$ (13)
where $l$ and $\epsilon$ are defined by $l\equiv L/\tau_{p}$ and
$\epsilon\equiv E/\tau_{p}$, respectively. Also from (7) and (11) (and using
$E/\tau_{p}=\epsilon$) one finds
$T_{ij}=-\frac{1}{H(R)}\;\frac{\tau_{p}}{\epsilon}\;\delta_{ij}\;.$ (14)
The radial equation of motion (12) describes a particle moving in a one
dimensional potential
$V_{eff}=-\frac{1}{\epsilon^{2}H^{2}}\Big{[}\epsilon^{2}H-(1+\frac{l^{2}}{R^{2}})\Big{]}$
(15)
with zero energy.
As discussed in [7], $V_{eff}$ in (15) does not allow for stable orbits, and
in general the D-brane escapes to infinity or falls into the fivebranes at
late times. Indeed the author considered two regimes $E>\tau_{p}$ and
$E<\tau_{p}$, and he found no solutions corresponding to a D-brane in orbit
around the fivebranes. In the intermediate regime, however, there exist
solutions in which the D-brane only orbits around the fivebranes without
escaping to infinity or falling into fivebranes. Note that $V_{eff}$ in (15)
vanishes for all $R$ if
$\epsilon=1\longleftrightarrow E=\tau_{p}\;,$ (16)
and
$l=\sqrt{k}l_{s}\longleftrightarrow L=\sqrt{k}l_{s}\tau_{p}\;\;.$ (17)
Since $V_{eff}$ vanishes, there is no force that pushes the D-branes into
infinity or pulls it to the fivebranes. The D-brane can maintain its orbit (or
the distance) around (from) the fivebranes all the time. This is very
reminiscent of the system consisting of two parallel BPS D-branes, where the
force between two BPS D-branes precisely vanishes. So in our system we can
place the D-brane at any distance from the fivebranes as we want as in the
case of the two parallel BPS D-branes.
Since the D-brane can maintain its orbit, certain amount of $T_{ij}$ is
preserved depending on the distance $R$ from the fivebranes. From (14) and
(16), $T_{ij}$ is now given by
$T_{ij}=-\frac{\tau_{p}}{H(R)}\;\delta_{ij}\;\;.$ (18)
Thus if $R\geq\sqrt{k}l_{s}$ for instance, then $\mid
T_{ij}\mid\geq\tau_{p}/2$, meaning that more than a half of $T_{ij}$ is
preserved if the radius of the orbit is greater than the string scale. This
suggests that in our case the D-brane does not decay into a pressureless
fluid. Rather, it will have nonzero (negative) pressure unlike the other
conventional cases. Besides this, one also finds from (13) that (16) and (17)
imply
$\dot{\theta}=\frac{\sqrt{k}l_{s}}{R^{2}H}\equiv\omega(R)\;\;.$ (19)
In (19) the angular velocity takes the value $\omega(R)\rightarrow 0$ as
$R\rightarrow\infty$, while $\omega(R)\rightarrow\frac{1}{\sqrt{k}l_{s}}$ as
$R\rightarrow 0$. This is rather unexpected result because it implies that the
tangential velocity $v_{t}(\equiv R\dot{\theta})$ of the D-brane becomes
$v_{t}\rightarrow 0$ as $R\rightarrow 0$. Typically the velocity of the
particle goes to infinity as the radius of the orbit goes to zero.
As mentioned already, real time dynamics of the D-brane near NS5-branes
necessarily leads to a decay of the D-brane in the ordinary circumstances. The
D-brane rolls down to the fivebrane throat due to gravitational attraction,
and as it approaches the fivebrane it loses most of its energy and finally
turns into a pressureless fluid. Such a decay of the D-brane is closely
related to tachyon condensation on the D-brane, which is described by the
rolling tachyon solutions. In our case the tachyonic degree of freedom arises
from the radial motion of the D-brane moving in the vicinity of NS5-branes. In
[7], it was noticed that $R$ is identified with geometrical tachyon $T$ by the
equation.
$dT=\sqrt{H(R)}dR\;\;\;,$ (20)
and the tachyon potential is given by $V(T)=\tau_{p}/\sqrt{H(R(T))}$. From
(20) one finds that as $R\rightarrow 0$,
$T(R)\sim\sqrt{k}l_{s}lnR/\sqrt{k}l_{s}$ or
$R(T)\sim\sqrt{k}l_{s}e^{T/\sqrt{k}l_{s}}$, and therefore $V(T)/\tau_{p}\sim
1/\sqrt{H(R(T))}\sim e^{T/\sqrt{k}l_{s}}$, indicating that the potential
$V(T)$ goes exponentially to zero as $R\rightarrow 0$. This precisely
coincides with the behavior exhibited by the tachyon potential relevant for
the rolling tachyon solutions. So the tachyon rolls towards the minimum of the
potential, and as a result the D-brane collapses into a pressureless fluid. In
our case, however, this is not to be the case anymore. Since the D-brane
maintains a certain distance from the fivebranes, the process of tachyon
condensation is expected to be suppressed. Indeed, $\dot{R}=0$ implies
$\dot{T}=0$ from (20), which means that tachyon does not roll anymore in our
case, i.e., when $\epsilon=1$ and $l=\sqrt{k}l_{s}$.
The fact that tachyon does not roll when $\epsilon=1$ and $l=\sqrt{k}l_{s}$
can be understood as follows. First, rewrite the DBI action (6) in terms of
$\dot{R}$ and $\dot{\theta}$ :
$S_{p}=-\tau_{p}\int
dt\sqrt{H^{-1}(R)-(\dot{R}^{2}+R^{2}\dot{\theta}^{2})}\;\;,$ (21)
where we have set $V_{p}=1$ as before. The equations of motion for $R$ and
$\theta$ are then respectively given by
$\frac{d}{dt}\Bigg{(}\frac{\dot{R}}{\Sigma(\dot{\theta})}\Bigg{)}=\frac{R}{\Sigma(\dot{\theta})}\;[\;\dot{\theta}^{2}-\omega^{2}(R)\;]\;,$
(22)
where
$\Sigma(\dot{\theta})\equiv\sqrt{H^{-1}(R)-(\dot{R}^{2}+R^{2}\dot{\theta}^{2})}\;,$
(23)
and
$\frac{dL}{dt}=0$ (24)
with $L$ given by (10). (Check that $\dot{R}=0$ with $\epsilon=1$ and
$l=\sqrt{k}l_{s}$ becomes a solution to (22). Also (24) ensures that $L$ in
(10) is a constant of motion.) As can be seen from (20) the geometrical
tachyon only arises from the radial mode $R$. So if we want to analyze the
tachyonic behavior it is only enough to consider the radial motion of the
D-brane. (22), however, contains both $\dot{R}$ and $\dot{\theta}$. To express
it only in terms of $\dot{R}$ and $R$, we solve (10) for $\dot{\theta}$ to get
$\dot{\theta}^{2}=\frac{l^{2}}{R^{4}H^{2}}\frac{H}{(1+\frac{l^{2}}{R^{2}})}-\frac{\dot{R}^{2}}{R^{4}}\frac{l^{2}}{(1+\frac{l^{2}}{R^{2}})}\;,$
(25)
and also using (25) we obtain
$\Sigma(\dot{\theta})=\frac{\Sigma(0)}{\sqrt{1+\frac{l^{2}}{R^{2}}}}\;,$ (26)
where
$\Sigma(0)=\sqrt{H^{-1}(R)-\dot{R}^{2}}\;.$ (27)
Substituting (25) and (26) into (22) one finally obtains a
$\dot{\theta}$-independent equation of motion
$\frac{d}{dt}\Bigg{[}\frac{\sqrt{1+\frac{l^{2}}{R^{2}}}}{\Sigma(0)}\;\dot{R}\Bigg{]}=\frac{1}{\sqrt{1+\frac{l^{2}}{R^{2}}}}\frac{R}{\Sigma(0)}\Bigg{[}\Bigg{(}\frac{l^{2}}{kl_{s}^{2}}-1\Bigg{)}\omega^{2}(R)-l^{2}\frac{\dot{R}^{2}}{R^{4}}\Bigg{]}\;\;.$
(28)
Now consider an action of the form
$\tilde{S_{p}}=-\tau_{p}\int
dt\;\sqrt{1+\frac{l^{2}}{R^{2}}}\;\sqrt{H^{-1}(R)-\dot{R}^{2}}\;\;.$ (29)
Since (29) does not contain $\dot{\theta}$ term it only gives an equation of
motion for $R$, and one can show that the equation of motion following from
(29) precisely coincides with (28). Besides this, the conserved energy
obtained from (29) is given by
$\tilde{E}=\tau_{p}\frac{\sqrt{1+\frac{l^{2}}{R^{2}}}}{H\sqrt{H^{-1}(R)-\dot{R}^{2}}},$
(30)
which is also equal to $E$ in (11) due to (26):
$\tilde{E}=E=\tau_{p}\epsilon\;\;.$ (31)
These things indicate that $\tilde{S_{p}}$ is classically equivalent to the
original action $S_{p}$ as far as the radial motion is concerned, which in
turn means that the tachyonic behavior described by $\tilde{S_{p}}$ is
equivalent to that described by the original action $S_{p}$ because the
tachyon $T$ is only a field redefinition of $R$ (Eq.(20)). Indeed the one
dimensional motion described by $\tilde{S}_{p}$ exactly coincides with the
radial motion described by $S_{p}$.
Let us now rewrite $\tilde{S}_{p}$ in terms of $T$:
$\tilde{S}_{p}=-\int dt\tilde{V}(T)\sqrt{1-\dot{T}^{2}}=\int dtL(t)\;\;\;,$
(32)
where $\tilde{V}(T)$ is given by
$\tilde{V}(T)=\tau_{p}\frac{\sqrt{1+\frac{l^{2}}{R^{2}}}}{\sqrt{H(R(T))}}\;\;.$
(33)
We observe that the tachyon potential has been changed from
$V(T)=\tau_{p}/\sqrt{H(R(T))}$ into (33). Such a change of the tachyon
potential is obviously due to the orbiting motion of the D-brane. Note that
the change has occurred in compensation for the elimination of the
$\dot{\theta}$ term. The tachyon potential $\tilde{V}(T)$ has a remarkable
feature. For $l=\sqrt{k}l_{s}$, it simply becomes a constant:
$\tilde{V}(T)=\tau_{p}\;\;.$ (34)
The decay of unstable D-brane essentially occurs as the tachyon rolls towards
the minimum of the tachyon potential. But in the case $l=\sqrt{k}l_{s}$, the
tachyon does not roll (provided the initial condition $\dot{T}=0$ is met)
because the potential $\tilde{V}(T)$ is flat, and the decay of the D-brane is
necessarily suppressed. To be more precise, in the case $l=\sqrt{k}l_{s}$ the
geometrical tachyon induced on the D-brane turns into a trivial massless
constant field, meaning that the tachyonic degree of freedom on the D-brane
disappears and the D-brane returns to the stable brane. Thus the whole issue
regarding rolling tachyon, including gravitational radiation, becomes
irrelevant to this case regardless of whether it is considered at the tree
level or quantum level.
The fact that the unstable D-brane returns to the stable brane when
$l=\sqrt{k}l_{s}$ and $\epsilon=1$ can be confirmed as follows. In the case of
the usual unstable D-brane (of the bosonic string theory) the spatially
homogeneous tachyon is often described by
$T(X^{0})=\lambda\cosh X^{0}$ (35)
and the corresponding tachyon potential
$V(T)=\frac{\tau_{p}}{\coth\frac{T}{2}}\;\;\;.$ (36)
The tree level analysis of the boundary conformal field theory (BCFT) shows
[9] that the energy-stress tensor $T_{\mu\nu}$ corresponding to the above
tachyon profile takes the form
$T_{00}=\frac{\tau_{p}}{2}(1+\cos
2\pi\lambda)\;,\;\;\;\;\;\;T_{ij}=-\tau_{p}\;f(t)\;\delta_{ij}\;,$ (37)
where $t=x^{0}$ and $f(t)$ is given by
$f(t)=\frac{1}{1+\sin(\lambda\pi)e^{t}}+\frac{1}{1+\sin(\lambda\pi)e^{-t}}-1\;\;.$
(38)
For positive $\lambda$ the function $f(t)$ goes to zero as
$t\rightarrow\infty$, showing that the pressure vanishes asymptotically and
the D-brane decays into a pressureless fluid. This is what happens as the
tachyon rolls towards the minimum of the potential $V(T)$.
In our case, however, the tachyon does not roll as mentioned already. Consider
the equation of motion following from (32):
$\dot{T}=\sqrt{1-\frac{\tilde{V}^{2}}{\tilde{E}^{2}}}\;\;.$ (39)
Using (31), (34) and (39) one finds that $\dot{T}$ really vanishes when
$l=\sqrt{k}l_{s}$ and $\epsilon=1$. Since the tachyon does not roll, the
function $f(t)$ is expected to be a constant. There is a simple way to find
$f(t)$ which does not use the BCFT analysis. On general grounds the function
$\tau_{p}f(t)$, which is a partition function on the disk in BCFT, can be
identified as the on-shell value of $-L(t)$ [10]. Substituting (34) together
with $\dot{T}=0$ into $L(t)$ one finds that the function $f(t)$ is just equal
to one:
$f(t)=1\;\;\;,$ (40)
and consequently the components of $\tilde{T}_{\mu\nu}$ following from
$\tilde{S}_{p}$ become
$\tilde{T}_{00}=\tilde{E}=\tau_{p}\;,\;\;\;\;\;\;\tilde{T}_{ij}=-\tau_{p}\delta_{ij}\;\;\;,$
(41)
which are typical of the BPS D-brane. So we expect that the action $S_{p}$
(with $\epsilon=1$ and $l=\sqrt{k}l_{s}$ ) also describes a BPS D-brane
because the tachyonic behavior described by $\tilde{S}_{p}$ is equivalent to
that described by $S_{p}$.
The D-brane described by $S_{p}$ really appears as a BPS brane to an observer
living on that D-brane. Notice that the conditions $\epsilon=1$ and
$l=\sqrt{k}l_{s}$ imply $R=$ const $\equiv R_{0}$ and
$\dot{\theta}=\sqrt{k}l_{s}/HR^{2}$, which then gives $G_{00}=-1/H(R_{0})$ and
$G_{ij}=\eta_{ij}$ from (5). Thus the observer on the D-brane finds the stress
tensor
$T_{\mu\nu}^{(brane)}=\;-\hat{\tau_{p}}\eta_{\mu\nu}\;,\;\;\;\;\;\hat{\tau_{p}}\equiv\frac{\tau_{p}}{H(R_{0})}\;,$
(42)
and sees the geometry $ds^{2}=-d\tau^{2}+d\vec{x}_{p}^{2}$, where the proper
time $\tau$ is related to $t$ by $d\tau^{2}=-G_{00}dt^{2}$. (42) is precisely
the stress tensor of the BPS D-brane with a tension $\hat{\tau}_{p}$. Thus the
D-brane orbiting around NS 5-branes with $\epsilon=1$ and $l=\sqrt{k}l_{s}$
appears as a BPS D-brane with an effective tension $\hat{\tau}_{p}$ (and with
no NS 5-branes nearby) to an observer on the D-brane. This suggests that the
effects of the NS 5-branes on the D-brane have disappeared due to the orbiting
motion of the D-brane. The presence of the NS5-branes only changes the tension
measured by an observer on the D-brane. The effective tension $\hat{\tau}_{p}$
goes to zero as $R_{0}\rightarrow 0$, while it approaches $\tau_{p}$ as
$R_{0}\rightarrow\infty$. In the brane world cosmology $\hat{\tau}_{p}$ serves
for a cosmological constant and makes a contribution to the dark energy.
The whole analysis of this paper has been made at the classical level. But
even at the quantum level we do not need to be concerned about the
gravitational or closed string radiations generated by tachyon because the
tachyon field coupled with graviton or closed string modes has been
disappeared already. The aim of this paper is to examine the possibility of
avoiding the Kutasov’s conjecture, which also has been made at the classical
level, that a D-brane moving around NS5-branes is necessarily absorbed into
the fivebranes and eventually decays into a ”pressureless fluid”. According to
the result of this paper the decay of the D-brane can be avoided if the brane
has specific values of energy and angular momentum. In that case the D-brane
orbits around the fivebranes, maintaining certain distance from the fivebranes
all the time, and consequently a stable bound state of D-brane and fivebranes
can be formed. But before concluding, it should be mentioned that the
discussion of this paper was based on the assumption that the radiation
emitted from the D-brane is entirely generated by the tachyonic degree of
freedom induced on the brane, as it was in other papers including [7]. In
general the D-brane is regarded as a source for the closed string modes. It
couples to the metric, dilaton and the $(p+1)$-form R-R gauge field of the
type II string theories. So if the D-brane is accelerated, or rotating for
instance, one would expect it to emit Larmor-type radiation of these fields.
Though this is not the main point of this paper it may be necessary to address
it for the completeness of our discussion.
In [11], the radiation emitted by accelerating D-branes has been studied in
the linear approximation to the supergravity limit of the string theory.
Assuming that the D-brane is moving in three uncompactified spatial dimensions
the authors found that the total radiated power per unit mass (energy) of the
D-brane is given by $P/M\sim\kappa^{2}M|\dot{\vec{v}}|^{2}$, where
$\kappa^{2}\equiv\kappa^{2}_{10}/V_{6}$ with $V_{6}$ the volume of the extra
dimensions is the 4d gravitational coupling, while $M$ and $\dot{\vec{v}}$ are
the mass and the acceleration of the Dp-brane respectively. This result,
however, may not be directly applicable to our case because in our case the
spatial dimensions of the spacetime in which the radiation propagates are
four, instead of three, and the green function is therefore proportional to
$\sim 1/r^{2}$, instead of $\sim 1/r$, where $r$ represents the distance from
the source (the rotating Dp-brane) to the observation point of the radiation
and it characterizes the scale of the transverse dimensions. Thus the Poynting
vector is expected to fall off as $\sim 1/r^{4}$, and we estimate $P/M$ to be
$P/M\sim\kappa^{2}M|\dot{\vec{v}}|^{2}/r$ where $\kappa^{2}$ is now given by
$\kappa^{2}=\kappa^{2}_{10}/V_{5}$. We see that $P/M$ goes to zero as
$r\rightarrow\infty$.
Apart from this, there is a good reason for neglecting the energy loss
resulting from the Larmor-type radiation. In order to see this, consider the
case $p=5$ for instance. For $p=5$, $\kappa^{2}$ and $M$ are respectively
given by $\kappa^{2}\sim g_{s}^{2}\alpha^{\prime 4}/V_{5}$, $M\sim
V_{5}/g_{s}\alpha^{\prime 3}$, and since $|\dot{\vec{v}}|\sim
R_{0}\dot{\theta}^{2}$, $P/M$ becomes111If the D5-brane is wrapped on a
2-cycle to become an effective D3-brane, $P/M$ gets even smaller than (43). In
this case $P/M$ acquires an extra factor $V^{(c)}_{2}/V_{2}$, where
$V_{2}^{(c)}$ is the volume of the 2-cycle, while $V_{2}$ the volume of the 2d
uncompactified space.
$\frac{P}{M}\sim\bigm{(}\frac{g_{s}}{k}\Bigm{)}\Bigm{(}\frac{1}{r}\Bigm{)}\frac{(\sqrt{k}l_{s}/R_{0})^{6}}{[1+(\sqrt{k}l_{s}/R_{0})^{2}]^{4}}\;\;.$
(43)
(43) is only valid for $R_{0}\gg\sqrt{k}l_{s}$ because it has been obtained by
assuming the situation in which the accelerated brane moves in a background
which is almost a flat spacetime. Also in our discussion it has been assumed
that the 4d transverse space is noncompact222In general the green function
describing radiation emitted from a single source does not exists in a compact
space, and therefore the Larmor-type radiation is automatically suppressed in
this case. as in [7], and we may take $r\sim\infty$ which characterizes the
scale of the transverse dimensions.
For these reasons we are only allowed to consider the case $r\gg
R_{0}\gg\sqrt{k}l_{s}$ as far as the Larmor-type radiation is concerned.
Clearly, the ratio $P/M$ in (43) goes to zero in the limit $g_{s}\rightarrow
0$, which suggests that the Larmor-type radiation could be ignored in our
case. Indeed, the time it takes for the whole energy of the D-brane to be
dissipated away by the Larmor-type radiation is given by $t=(1/c)M/P$, which
is expected to be very large for $g_{s}\rightarrow 0$ and $r\gg
R_{0}\gg\sqrt{k}l_{s}$ . In fact, as long as $r$ or the size of the transverse
dimensions is not so small, $t$ can be arbitrarily large in the limit
$g_{s}\rightarrow 0$ depending on how large the ratio $R_{0}/\sqrt{k}l_{s}$
is. In any case, however, if $r$ and $R_{0}$ are not sufficiently large, we
may have to resort to the quantum theory to avoid the Larmor-type radiation as
in the case of an electron orbiting the proton where the electron is not
absorbed into the proton due to orbit quantization.
Acknowledgements
This work is supported by Kyungsung University in 2009.
## References
* [1] J. Polchinski, Dirichlet branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 (1995) 4725 [arXiv:hep-th/9510017]
* [2] R. Sundrum, Effective field theory for a three-brane universe, Phys. Rew. D 59 (1999) 085009 [arXiv:hep-th/9805471];
L. Randall and R. Sundrum, An alternative to a compactification, Phys. Rew.
Lett 83 (1999) 4690 [arXiv:hep-th/9906064]
* [3] E. K. Park and P. S. Kwon, A comment on p-brane of $(p+3)$d string theory, JHEP 05(2009)057 [arXiv:hep-th/0812.0227]
* [4] S. Ribault, D3-branes in NS5-brane backgrounds , JHEP 0302 (2003) 044 [arXiv:hep-th/0301092];
S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and G. Sorkissian, D-branes
in the background of NS fivebranes, JHEP 0008 (2000) 046 [arXiv:hep-
th/0005052];
also see, for instance, O. Pelc, On the Quantization Constraints for a D3
Brane in the Geometry of NS5 Branes, JHEP 0008 (2000) 030 [arXiv:hep-
th/0007100];
O. Aharony, M. Berkooz, D. Kutasov and N. Seiberg, Linear Dilatons, NS5-branes
and Holography, JHEP 9810 (1998)004 [arXiv:hep-th/9808149]
* [5] E. K. Park and P. S. Kwon, A self-tuning mechanism in $(3+p)d$ gravity-scalar theory, JHEP 11(2007)051 [arXiv:hep-th/0702171]
* [6] O. Aharony, A brief review of ”little string theories”, Class. Quant. Grav. 17(2000) 929[arXiv:hep-th/9911147];
N. Seiberg, New theories in six dimensions and matrix description of M-theory
on $T^{5}$ and $T^{5}/Z_{2}$ Phys. Lett. B408 (1997) 98 [arXiv:hep-
th/9705221].
For the review of LST, see D. Kutasov, Introduction to Little String Theory,
Lectures given at the Spring School on Superstrings and Related Matters,
Trieste, 2-10 April 2001.
* [7] D. Kutasov, D-brane Dynamics near NS5-branes, [arXiv:hep-th/0405058]
* [8] Bin Chen and Bo Sun, Note on DBI dynamics of Dbrane near NS5-branes, Phys. Rew. D 72 (2005) 046005 [arXiv:hep-th/0501176];
Bin Chen, Miao Li and Bo Sun, Dbrane Near NS5-branes: with Electromagnetic
Field, JHEP 0412 (2004) 057 [arXiv:hep-th/0412022];
J. Kluson, Non-BPS D-brane near NS5-branes, JHEP 0411 (2004) 013 [arXiv:hep-
th/0409298]
David A. Sahakyan, Comments on D-brane dynamics near NS5-branes, JHEP 0410
(2004) 008 [arXiv:hep-th/0408070];
Y. Nakayama, Y. Sugawara and H. Takayanagi, Boundary states for the rolling
D-branes in NS5 background, JHEP 0407 (2004) 020 [arXiv:hep-th/0406173]
* [9] F. Larsen, A. Naqvi and S. Terashima, Rolling Tachyons and Decaying Branes, JHEP 0302(2003)039 [arXiv:hep-th/0212248];
A. Sen, Rolling Tachyon, JHEP 0204(2002) 048 [arXiv:hep-th/0203211]
* [10] N. Lambert, H. Liu and J. Maldacena, Closed Strings from Decaying D-branes, JHEP 0703(2007) 014 [arXiv:hep-th/0303139]
* [11] M. Abou-Zeid and M. S. Costa, Radiation from Accelerated Branes, Phys. Rew. D 61 (2000) 106007 [arXiv:hep-th/9909148]
|
arxiv-papers
| 2009-11-24T11:40:47 |
2024-09-04T02:49:06.650217
|
{
"license": "Public Domain",
"authors": "Gyeong Yun Jun, Pyung Seong Kwon",
"submitter": "PyungSeong Kwon",
"url": "https://arxiv.org/abs/0911.4557"
}
|
0911.4576
|
# Centers of symmetric cellular algebras ††thanks: keywords: symmetric
cellular algebras, center.
Yanbo Li
Department of Information and Computing Sciences,
Northeastern University at Qinhuangdao;
Qinhuangdao, 066004, P.R. China
School of Mathematics Sciences, Beijing Normal University;
Beijing, 100875, P.R. China
E-mail: liyanbo707@163.com
(November 18, 2009)
###### Abstract
Let $R$ be an integral domain and $A$ a symmetric cellular algebra over $R$
with a cellular basis $\\{C_{S,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$. We will construct an ideal $L(A)$ of the center of $A$ and
prove that $L(A)$ contains the so-called Higman ideal. When $R$ is a field, we
prove that the dimension of $L(A)$ is not less than the number of non-
isomorphic simple $A$-modules.
## 1 Introduction
In 1996, Graham and Lehrer [8] introduced cellular algebras in order to
provide a systematic framework for studying the representation theory of a
class of algebras. By the theory of cellular algebras, one can parameterize
simple modules for a finite dimensional cellular algebra by methods in linear
algebra. Many classes of algebras from mathematics and physics are found to be
cellular, including Hecke algebras of finite type, Ariki-Koike algebras,
$q$-Schur algebras, Brauer algebras, Temperley-Lieb algebras, cyclotomic
Temperley-Lieb algebras, partition algebras, Birman-Wenzl algebras and so on,
see [7], [8], [14], [15], [16] for details.
There are many papers on centers of Hecke algebras of finite type, which are
all cellular algebras [7]. In [11], Jones found bases for centers of Hecke
algebras of type A over $\mathbb{Q}[q,q^{-1}]$, where $q$ is an indeterminant.
This basis is an analog of conjugacy class sum in a group algebra. In [10],
Geck and Rouquier found bases for the centers of generic Hecke algebras over
$\mathbb{Z}[q,q^{-1}]$ with $q$ an indeterminant. However, it is not easy to
write the basis explicitly. Then one should ask, is there any basis which can
be written explicitly? In [3], Francis gave an integral minimal basis for the
center of a Hecke algebra. Then in [4], he used the minimal basis approach to
provide a way of describing and calculating elements of the minimal basis for
the center of an Iwahori-Hecke algebra which is entirely combinatorial. In
[6], Francis and Jones found an explicit non-recursive expression for the
coefficients appearing in these linear combinations for the Hecke algebras of
type A. The relations between the so-called Jucys-Murphy elements and centers
of Hecke algebras are also investigated widely. In [2], Dipper and James
conjectured that the center of a Hecke algebra of type A consists of symmetric
polynomials in the Jucys-Murphy elements. This conjecture was proved by
Francis and Graham [5] in 2006. An analogous conjecture for Ariki-Koike
algebras is still open.
The fact that Hecke algebras of finite type are all cellular leads us to
considering how to describe the centers of Hecke algebras by cellular bases.
Furthermore, how to describe the center of a cellular algebra in general?
Clearly, most of the approaches for studying Hecke algebras can not be used
directly for cellular algebras, since we have no Weyl group structure to use.
Then we must look for some new method. In fact, the symmetry of Hecke algebras
provides us a way. We will do some work on the centers of symmetric cellular
algebras in this paper.
In order to describe our result exactly, we fix some notations first. Let $A$
be a symmetric cellular $R$-algebra with a non-degenerate symmetric bilinear
form $f:A\times A\rightarrow R$. Then $f$ determines a map $\tau:A\rightarrow
R$ which is defined by $\tau(a)=f(a,1)$ for every $a\in A$. We call $\tau$ a
symmetrizing trace. Denote by $\\{D_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$ the dual basis determined by $\tau$. Let
$H(A)=\\{\sum\limits_{\lambda\in\Lambda,S,T\in
M(\lambda)}C_{S,T}^{\lambda}aD_{S,T}^{\lambda}\mid a\in A\\}$. It is the
Higman ideal of $Z(A)$. For any $\lambda\in\Lambda$ and some $T\in
M(\lambda)$, set $x_{\lambda}=\sum\limits_{S\in
M(\lambda)}C_{S,T}^{\lambda}D_{S,T}^{\lambda}$, where $x_{\lambda}$ is
independent of $T$, and
$L(A)=\\{\sum\limits_{\lambda\in\Lambda}r_{\lambda}x_{\lambda}\mid
r_{\lambda}\in R\\}$. Then the main result of this paper is as follows.
_Theorem. Let $A$ be a symmetric cellular algebra with a cellular basis
$\\{C_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$ and the dual
basis $\\{D_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$
determined by a symmetrizing trace $\tau$. Then_
(1) _$L(A)$ is an ideal of $Z(A)$ and contains the Higman ideal._
(2) _$L(A)$ is independent of the choice of $\tau$_.
(3) _If $R$ is a field, then the dimension of $L(A)$ is not less than the
number of non-isomorphic simple $A$-modules._
This theorem enlarge the well known Higman ideal to a new one for the center
of a symmetric cellular algebra.
## 2 Preliminaries
In this section, we first recall some basic results on symmetric algebras and
cellular algebras, which is needed in the following sections. The so-called
Higman ideal is also described. References for this section are the books [1]
and [9].
Let $R$ be a commutative ring with identity and $A$ an associative
$R$-algebra. As an $R$-module, $A$ is finitely generated and free. Suppose
that there exists an $R$-bilinear map $f:A\times A\rightarrow R$. We say that
$f$ is non-degenerate if the determinant of the matrix
$(f(a_{i},a_{j}))_{a_{i},a_{j}\in B}$ is a unit in $R$ for some $R$-basis $B$
of $A$. We say $f$ is associative if $f(ab,c)=f(a,bc)$ for all $a,b,c\in A$,
and symmetric if $f(a,b)=f(b,a)$ for all $a,b\in A$.
###### Definition 2.1.
An $R$-algebra $A$ is called symmetric if there is a non-degenerate
associative symmetric bilinear form $f$ on $A$. Define an $R$-linear map
$\tau:A\rightarrow R$ by $\tau(a)=f(a,1)$. We call $\tau$ a symmetrizing
trace.
Let $A$ be a symmetric algebra with a basis $B=\\{a_{i}\mid i=1,\ldots,n\\}$
and $\tau$ a symmetrizing trace. Denote by $D=\\{D_{i}\mid i=i,\ldots,n\\}$
the basis determined by the requirement that $\tau(D_{j}a_{i})=\delta_{ij}$
for all $i,j=1,\ldots,n$. We will call $D$ the dual basis of $B$. For
arbitrary $1\leq i,j\leq n$, we write
$a_{i}a_{j}=\sum\limits_{k}r_{ijk}a_{k}$, where $r_{ijk}\in R$. Fix a $\tau$
for $A$. Then in [13], we proved the following lemma.
###### Lemma 2.2.
Let $A$ be a symmetric algebra with a basis $B$ and the dual basis $D$. Then
the following hold:
$a_{i}D_{j}=\sum\limits_{k}r_{kij}D_{k};\,\,\,\,\,D_{i}a_{j}=\sum\limits_{k}r_{jki}D_{k}.$
$\Box$
We now consider the set $\\{\sum\limits_{i}D_{i}aa_{i}\mid a\in A\\}$. It is
well known that $\\{\sum\limits_{i}D_{i}aa_{i}\mid a\in A\\}$ is an ideal of
the center of $A$, see [1]. We here give a direct proof by the lemma above.
###### Proposition 2.3.
Let $A$ be a symmetric algebra with a basis $B$ and the dual basis $D$. Then
$\\{\sum\limits_{i}D_{i}aa_{i}\mid a\in A\\}$ is an ideal of the center of
$A$.
Proof: For arbitrary $a_{j}\in B$ and $a\in A$, we have
$\sum_{i}D_{i}aa_{i}a_{j}=\sum_{i,k}r_{ijk}D_{i}aa_{k},$
and
$\sum_{i}a_{j}D_{i}aa_{i}=\sum_{i,k}r_{kji}D_{k}aa_{i}.$
Obviously, the right sides of the above two equations are equal. Then
$\\{\sum\limits_{i}D_{i}aa_{i}\mid a\in A\\}\subseteq Z(A)$. It is clear that
the set is an ideal of the center of $A$. $\Box$
The ideal $H(A)=\\{\sum\limits_{i}D_{i}aa_{i}\mid a\in A\\}$ is called Higman
ideal of the center of the algebra $A$. It is independent the choice of the
dual bases.
The following proposition is also proved in [13].
###### Proposition 2.4.
Suppose that $A$ is a symmetric $R$-algebra with a basis $\\{a_{i}\mid
i=1,\cdots,n\\}$. Let $\tau,\tau^{{}^{\prime}}$ be two symmetrizing traces.
Denote by $\\{D_{i}\mid i=1,\cdots,n\\}$ and $\\{D_{i}^{{}^{\prime}}\mid
i=1,\cdots,n\\}$ the dual bases determined by $\tau$ and $\tau^{{}^{\prime}}$
respectively. Then for any $1\leq i\leq n$, we have
$D_{i}^{{}^{\prime}}=\sum\limits_{j=1}^{n}\tau(a_{j}D_{i}^{{}^{\prime}})D_{j}.$
$\Box$
We now recall the definition of cellular algebras introduced by Graham and
Lehrer [8] and some well known results.
###### Definition 2.5.
([8] 1.1) Let $R$ be a commutative ring with identity. An associative unital
$R$-algebra is called a cellular algebra with cell datum $(\Lambda,M,C,i)$ if
the following conditions are satisfied:
(C1) The finite set $\Lambda$ is a poset. Associated with each
$\lambda\in\Lambda$, there is a finite set $M(\lambda)$. The algebra $A$ has
an $R$-basis $\\{C_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$.
(C2) The map $i$ is an $R$-linear anti-automorphism of $A$ with $i^{2}=id$
which sends $C_{S,T}^{\lambda}$ to $C_{T,S}^{\lambda}$.
(C3) If $\lambda\in\Lambda$ and $S,T\in M(\lambda)$, then for any element
$a\in A$, we have
$aC_{S,T}^{\lambda}\equiv\sum_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{{}^{\prime}},S)C_{S^{{}^{\prime}},T}^{\lambda}\,\,\,\,(\rm{mod}\,\,\,A(<\lambda)),$
where $r_{a}(S^{{}^{\prime}},S)\in R$ is independent of $T$ and where
$A(<\lambda)$ is the $R$-submodule of $A$ generated by
$\\{C_{S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}}^{\mu}\mid
S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}\in M(\mu),\mu<\lambda\\}$.
Apply $i$ to the equation in (C3), we obtain
$(C3^{{}^{\prime}})\,\,C_{T,S}^{\lambda}i(a)\equiv\sum\limits_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{{}^{\prime}},S)C_{T,S^{{}^{\prime}}}^{\lambda}\,\,\,\,(\rm
mod\,\,\,A(<\lambda)).$
By Definition 2.5, it is easy to check that
$C_{S,S}^{\lambda}C_{T,T}^{\lambda}\equiv\Phi(S,T)C_{S,T}^{\lambda}\quad(\rm
mod\,\,\,A(<\lambda)),$
where $\Phi(S,T)\in R$ depends only on $S$ and $T$.
Let $A$ be a cellular algebra with cell datum $(\Lambda,M,C,i)$. We recall the
definition of cell modules.
###### Definition 2.6.
([8] 2.1) For each $\lambda\in\Lambda$, define the left $A$-module
$W(\lambda)$ as follows: $W(\lambda)$ is a free $R$-module with basis
$\\{C_{S}\mid S\in M(\lambda)\\}$ and $A$-action defined by
$aC_{S}=\sum_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{{}^{\prime}},S)C_{S^{{}^{\prime}}}\,\,\,\,(a\in A,S\in
M(\lambda)),$
where $r_{a}(S^{{}^{\prime}},S)$ is the element of $R$ defined in (C3).
For a cell module $W(\lambda)$, define a bilinear form
$\Phi_{\lambda}:\,\,W(\lambda)\times W(\lambda)\longrightarrow R$ by
$\Phi_{\lambda}(C_{S},C_{T})=\Phi(S,T)$, extended bilinearly and define
$\operatorname{rad}(\lambda):=\\{x\in
W(\lambda)\mid\Phi_{\lambda}(x,y)=0\,\,\,\text{for all}\,\,\,y\in
W(\lambda)\\}.$
Then Graham and Lehrer proved the following results in [8].
###### Theorem 2.7.
[8] Let $K$ be a field and $A$ a finite dimensional cellular algebra. Denote
the $A$-module $W(\lambda)/\operatorname{rad}\lambda$ by $L_{\lambda}$, where
$\lambda\in\Lambda$ with $\Phi_{\lambda}\neq 0$. Let
$\Lambda_{0}=\\{\lambda\in\Lambda\mid\Phi_{\lambda}\neq 0\\}$. Then the set
$\\{L_{\lambda}\mid\lambda\in\Lambda_{0}\\}$ is a complete set of
(representative of equivalence classes of ) absolutely simple $A$-modules.
$\Box$
## 3 Centers of symmetric cellular algebras
Let $A$ be a symmetric cellular algebra with a cell datum $(\Lambda,M,C,i)$.
Denote the dual basis by $D=\\{D_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$, which satisfies
$\tau(C_{S,T}^{\lambda}D_{U,V}^{\mu})=\begin{cases}1,&\text{$\lambda=\mu,\,\,\,S=U,\,\,\,T=V$;}\\\
0,\,&\text{otherwise.}\end{cases}$
For any $\lambda,\mu\in\Lambda$, $S,T\in M(\lambda)$, $U,V\in M(\mu)$, write
$C_{S,T}^{\lambda}C_{U,V}^{\mu}=\sum\limits_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(S,T,\lambda),(U,V,\mu),(X,Y,\epsilon)}C_{X,Y}^{\epsilon}.$
Then in [13], we proved the following lemma.
###### Lemma 3.1.
Let $A$ be a symmetric cellular algebra with a basis $B$. Let $D$ be the dual
basis determined by a given $\tau$. For arbitrary $\lambda,\mu\in\Lambda$ and
$S,T,P,Q\in M(\lambda)$, $U,V\in M(\mu)$, the following hold:
(1) $D_{U,V}^{\mu}C_{S,T}^{\lambda}=\sum\limits_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(S,T,\lambda),(X,Y,\epsilon),(U,V,\mu)}D_{X,Y}^{\epsilon}.$
(2) $C_{S,T}^{\lambda}D_{U,V}^{\mu}=\sum\limits_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(X,Y,\epsilon),(S,T,\lambda),(U,V,\mu)}D_{X,Y}^{\epsilon}.$
(3) $C_{S,T}^{\lambda}D_{S,T}^{\lambda}=C_{S,P}^{\lambda}D_{S,P}^{\lambda}.$
(4) $D_{S,T}^{\lambda}C_{S,T}^{\lambda}=D_{P,T}^{\lambda}C_{P,T}^{\lambda}.$
(5) $C_{S,T}^{\lambda}D_{P,Q}^{\lambda}=0\,\,if\,\,T\neq Q.$ (6)
$D_{P,Q}^{\lambda}C_{S,T}^{\lambda}=0\,\,if\,\,P\neq S.$
(7) $C_{S,T}^{\lambda}D_{U,V}^{\mu}=0\,\,\,\,if\,\,\,\mu\nleq\lambda.$ (8)
$D_{U,V}^{\mu}C_{S,T}^{\lambda}=0\,\,\,\,if\,\,\,\mu\nleq\lambda.$ $\Box$
Let $A$ be a symmetric cellular $R$-algebra with a symmetrizing trace $\tau$.
The dual basis $\\{D_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$ is determined by $\tau$. Then the Higman
ideal is $H(A)=\\{\sum\limits_{\lambda\in\Lambda,S,T\in
M(\lambda)}C_{S,T}^{\lambda}aD_{S,T}^{\lambda}\mid a\in A\\}$. For any
$\lambda\in\Lambda$ and $T\in M(\lambda)$, set $x_{\lambda}=\sum\limits_{S\in
M(\lambda)}C_{S,T}^{\lambda}D_{S,T}^{\lambda}$ and
$L(A)=\\{\sum\limits_{\lambda\in\Lambda}r_{\lambda}x_{\lambda}\mid
r_{\lambda}\in R\\}$. Now we are in a position to give the main result of this
paper.
###### Theorem 3.2.
Let $A$ be a symmetric cellular algebra with a cellular basis
$\\{C_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$ and the dual
basis $\\{D_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$
determined by a symmetrizing trace $\tau$. Then
(1) $L(A)$ is an ideal of $Z(A)$ and contains the Higman ideal $H(A)$.
(2) $L(A)$ is independent of the choice of $\tau$.
(3) If $R$ is a field, then the dimension of $L(A)$ is not less than the
number of non-isomorphic simple $A$-modules.
Proof: (1) Firstly, we show that $H(A)\subseteq L(A)$.
Clearly, we only need to show that $l:=\sum\limits_{S,T\in
M(\lambda),\lambda\in\Lambda}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}\in
L(A)$ for any $C_{U,V}^{\mu}\in B$, where $\mu\in\Lambda,U,V\in M(\mu)$. We
divide $l$ into three parts:
$l=l_{\lambda=\mu}+l_{\lambda<\mu}+l_{\lambda\nleq\mu}$, where
$l_{\lambda=\mu}:=\sum_{S,T\in
M(\lambda),\lambda=\mu}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}$
and the other two parts are defined similarly.
By Lemma 3.1 (7), $l_{\lambda\nleq\mu}=0$.
We now show that
$l_{\lambda=\mu}=\Phi_{\mu}(C_{U},C_{V})x_{\mu}.$
By Lemma 3.1 (5), $C_{U,V}^{\mu}D_{X,Y}^{\mu}=0$ if $V\neq Y$. Then
$l_{\lambda=\mu}=\sum\limits_{X\in
M(\mu)}C_{X,V}^{\mu}C_{U,V}^{\mu}D_{X,V}^{\mu}$. By Definition 2.5, we have
$l_{\lambda=\mu}=\sum_{X\in
M(\mu)}\Phi_{\mu}(C_{U},C_{V})C_{X,V}^{\mu}D_{X,V}^{\mu}+\sum_{\eta<\mu,P,Q\in
M(\eta)}r_{P,Q,\eta}C_{P,Q}^{\eta}D_{X,V}^{\mu},$
where $r_{P,Q,\eta}\in R$. Note that by Lemma 3.1 (7),
$\sum\limits_{\eta<\mu,P,Q\in
M(\mu)}r_{P,Q,\eta}C_{P,Q}^{\eta}D_{X,V}^{\mu}=0$, then
$l_{\lambda=\mu}=\Phi_{\mu}(C_{U},C_{V})x_{\mu}$. This implies that
$l_{\lambda=\mu}\in L(A)$.
Now let us consider $l_{\lambda<\mu}$. For arbitrary $\lambda<\mu$, we show
that
$\sum_{S,T\in
M(\lambda)}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}=\sum_{T\in
M(\lambda)}r_{(S,T,\lambda),(U,V,\mu),(S,T,\lambda)}x_{\lambda}.$
Note that
$\sum_{S,T\in
M(\lambda)}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}=\sum_{S,T\in
M(\lambda)}(\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(S,T,\lambda),(U,V,\mu),(X,Y,\epsilon)}C_{X,Y}^{\epsilon})D_{S,T}^{\lambda}.$
By $(C3)^{{}^{\prime}}$ of Definition 2.5, if $\epsilon\nleq\lambda$, then
$r_{(S,T,\lambda),(U,V,\mu),(X,Y,\epsilon)}=0$. By Lemma 3.1 (7), if
$\epsilon<\lambda$, then $C_{X,Y}^{\epsilon}D_{S,T}^{\lambda}=0$. Thus
$\sum_{S,T\in
M(\lambda)}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}=\sum_{S,T\in
M(\lambda)}\sum_{X,Y\in
M(\lambda)}r_{(S,T,\lambda),(U,V,\mu),(X,Y,\lambda)}C_{X,Y}^{\lambda}D_{S,T}^{\lambda}.$
By $(C3)^{{}^{\prime}}$ of Definition 2.5, if $X\neq S$, then
$r_{(S,T,\lambda),(U,V,\mu),(X,Y,\lambda)}=0$. By Lemma 3.1 (5), if $Y\neq T$,
then $C_{X,Y}^{\lambda}D_{S,T}^{\lambda}=0$. Hence,
$\sum_{S,T\in
M(\lambda)}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}=\sum_{S,T\in
M(\lambda)}r_{(S,T,\lambda),(U,V,\mu),(S,T,\lambda)}C_{S,T}^{\lambda}D_{S,T}^{\lambda}.$
Note that for arbitrary $S,S^{{}^{\prime}}\in M(\lambda)$, by
$(C3)^{{}^{\prime}}$ of Definition 2.5.
$r_{(S,T,\lambda),(U,V,\mu),(S,T,\lambda)}=r_{(S^{{}^{\prime}},T,\lambda),(U,V,\mu),(S^{{}^{\prime}},T,\lambda)}.$
We get
$\sum_{S,T\in
M(\lambda)}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}=\sum_{T\in
M(\lambda)}r_{(S,T,\lambda),(U,V,\mu),(S,T,\lambda)}x_{\lambda}.$
This implies $l_{\lambda<\mu}\in L(A)$. Then we obtain $l\in L(A)$.
Secondly, we show that $L(A)\subseteq Z(A).$
We only need to show that $x_{\lambda}C_{U,V}^{\mu}=C_{U,V}^{\mu}x_{\lambda}$
for arbitrary $\lambda\in\Lambda$ and $\mu\in\Lambda,U,V\in M(\mu)$.
On one hand, by Lemma 3.1 (1),
$\displaystyle x_{\lambda}C_{U,V}^{\mu}$ $\displaystyle=$
$\displaystyle\sum_{S\in
M(\lambda)}C_{S,T}^{\lambda}D_{S,T}^{\lambda}C_{U,V}^{\mu}$ $\displaystyle=$
$\displaystyle\sum_{S\in M(\lambda)}\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(U,V,\mu),(X,Y,\epsilon),(S,T,\lambda)}C_{S,T}^{\lambda}D_{X,Y}^{\epsilon}.$
By a similar method as in the first part, we get
$x_{\lambda}C_{U,V}^{\mu}=\sum_{S,X\in
M(\lambda)}r_{(U,V,\mu),(X,T,\lambda),(S,T,\lambda)}C_{S,T}^{\lambda}D_{X,T}^{\lambda}.$
On the other hand,
$\displaystyle C_{U,V}^{\mu}x_{\lambda}$ $\displaystyle=$
$\displaystyle\sum_{S\in M(\lambda)}\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(U,V,\mu),(S,T,\lambda),(X,Y,\epsilon)}C_{X,Y}^{\epsilon}D_{S,T}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{S,X\in
M(\lambda)}r_{(U,V,\mu),(S,T,\lambda),(X,T,\lambda)}C_{X,T}^{\lambda}D_{S,T}^{\lambda}.$
So $x_{\lambda}C_{U,V}^{\mu}=C_{U,V}^{\mu}x_{\lambda}$ for arbitrary
$\lambda,\mu\in\Lambda$, $U,V\in M(\mu)$, that is, $L(A)\subseteq Z(A)$.
Finally, we show that $L(A)$ is an ideal of $Z(A)$.
It suffices to show that for arbitrary $c\in Z(A)$ and $\lambda\in\Lambda$,
the element $cx_{\lambda}\in L(A)$, that is, $\sum\limits_{S\in
M(\lambda)}C_{S,T}^{\lambda}cD_{S,T}^{\lambda}\in L(A)$.
Since $c$ is $R$-linear combination of elements of $B$, then we only need to
prove that for arbitrary $C_{U,V}^{\mu}\in B$, the element $\sum\limits_{S\in
M(\lambda)}C_{S,T}^{\lambda}C_{U,V}^{\mu}D_{S,T}^{\lambda}\in L(A)$. Clearly,
this element is equal to
$\sum_{S\in M(\lambda)}\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(S,T,\lambda),(U,V,\mu),(X,Y,\epsilon)}C_{X,Y}^{\epsilon}D_{S,T}^{\lambda}.$
We know that it is equal to
$r_{(S,T,\lambda),(U,V,\mu),(X,Y,\epsilon)}x_{\lambda}$ by a similar way as in
the first part. This implies that $\sum\limits_{S\in
M(\lambda)}C_{S,T}^{\lambda}cD_{S,T}^{\lambda}\in L(A)$.
(2) $L(A)$ is independent of the choice of $\tau$.
Let $\tau$, $\tau^{{}^{\prime}}$ be two non-equal symmetrizing traces and $D$,
$d$ the dual bases determined by $\tau$ and $\tau^{{}^{\prime}}$ respectively.
For arbitrary $d_{S,T}^{\lambda}\in d$, by Proposition 2.4, we have
$d_{S,T}^{\lambda}=\sum_{\varepsilon\in\Lambda,X,Y\in
M(\varepsilon)}\tau(C_{X,Y}^{\varepsilon}d_{S,T}^{\lambda})D_{X,Y}^{\varepsilon}.$
Then by Lemma 3.1,
$\displaystyle\sum_{S\in M(\lambda)}C_{S,T}^{\lambda}d_{S,T}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{S\in
M(\lambda)}\sum_{\varepsilon\in\Lambda,X,Y\in
M(\varepsilon)}\tau(C_{X,Y}^{\varepsilon}d_{S,T}^{\lambda})C_{S,T}^{\lambda}D_{X,Y}^{\varepsilon}$
$\displaystyle=$ $\displaystyle\sum_{S\in M(\lambda)}\sum_{X\in
M(\lambda)}\tau(C_{X,T}^{\lambda}d_{S,T}^{\lambda})C_{S,T}^{\lambda}D_{X,T}^{\lambda}.$
By the definition of $\tau$, we have
$\tau(C_{X,T}^{\lambda}d_{S,T}^{\lambda})=\tau(d_{S,T}^{\lambda}C_{X,T}^{\lambda})$.
Then by Lemma 3.1, $\tau(d_{S,T}^{\lambda}C_{X,T}^{\lambda})=0$ if $X\neq S$,
that is,
$\sum_{S\in M(\lambda)}C_{S,T}^{\lambda}d_{S,T}^{\lambda}=\sum_{S\in
M(\lambda)}\tau(C_{S,T}^{\lambda}d_{S,T}^{\lambda})C_{S,T}^{\lambda}D_{S,T}^{\lambda}.$
We now need to show $\tau(C_{S,T}^{\lambda}d_{S,T}^{\lambda})$ is independent
of $S$. It is clear by the equations
$d_{S,T}^{\lambda}C_{S,T}^{\lambda}=d_{S^{{}^{\prime}},T}^{\lambda}C_{S^{{}^{\prime}},T}^{\lambda}$
for arbitrary $S^{{}^{\prime}}\in M(\lambda)$.
(3) We only need to find $|\Lambda_{0}|$ $R$-linear independent elements in
$L(A)$, where $|\Lambda_{0}|$ is the number of the elements in $\Lambda_{0}$.
By the definition of $\Lambda_{0}$, for each $\lambda\in\Lambda_{0}$, there
exist $S,T\in M(\lambda)$, such that $\Phi_{\lambda}(C_{S},C_{T})\neq 0$.
Write $x_{\lambda}=\sum\limits_{U\in
M(\lambda)}C_{U,T}^{\lambda}D_{U,T}^{\lambda}$. By Lemma 3.1, we know that the
coefficient of $D_{S,T}^{\lambda}$ in the expansion of
$C_{S,T}^{\lambda}D_{S,T}^{\lambda}$ is
$r_{(S,T,\lambda),(S,T,\lambda),(S,T,\lambda)}=\Phi_{\lambda}(C_{S},C_{T})\neq
0$ and is $0$ in the expansion of $C_{U,T}^{\lambda}D_{U,T}^{\lambda}$ for any
$U\neq S$. That is, the coefficient of $D_{S,T}^{\lambda}$ in the expansion of
$x_{\lambda}$ is not zero. We also know that the coefficient of
$D_{S,T}^{\lambda}$ in the expansion of $x_{\mu}$ is zero for any
$\mu\nleq\lambda$. Now let
$\sum\limits_{\lambda\in\Lambda_{0}}r_{\lambda}x_{\lambda}=0$ and $\mu$ a
minimal element in $\Lambda_{0}$. Then $r_{\mu}$ must be zero. By induction,
we know that $r_{\lambda}=0$ for each $\lambda\in\Lambda_{0}$. This implies
that $\\{x_{\lambda}\mid\lambda\in\Lambda_{0}\\}$ is $R$-linear independent.
That is, $\dim_{R}L(A)$ is not less than the number of (representatives of
equivalence classes of) simple $A$-modules. $\Box$
By a similar way, we obtain the following result.
###### Theorem 3.3.
Suppose that $R$ is a commutative ring with identity and $A$ a symmetric
cellular algebra over $R$ with a cellular basis $B=\\{C_{S,T}^{\lambda}\mid
S,T\in M(\lambda),\lambda\in\Lambda\\}$ and the dual basis
$D=\\{D_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$ . Denote
the set of the $R$-linear combination of the elements of the set
$\\{x_{\lambda}^{{}^{\prime}}=\sum\limits_{T\in
M(\lambda)}D_{S,T}^{\lambda}C_{S,T}^{\lambda}\mid\lambda\in\Lambda\\}$ by
$L(A)^{{}^{\prime}}$. Then the following hold:
(1) $L(A)^{{}^{\prime}}$ is an ideal of $Z(A)$ and contains the Higman ideal.
(2) $L(A)^{{}^{\prime}}$ is independent of the choice of $\tau$.
(3) If $R$ is a field, then the dimension of $L(A)^{{}^{\prime}}$ is not less
than the number of (representatives of equivalence classes of) simple
$A$-modules.
We now give some examples of $L(A)$.
Example Let $K$ be a field and $Q$ the following quiver
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{1}$$\scriptstyle{\alpha_{1}^{\prime}}$$\scriptstyle{2}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$$\scriptstyle{\alpha_{2}^{\prime}}$$\scriptstyle{\alpha_{n-1}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n-1}^{\prime}}$$\scriptstyle{n-1}$$\scriptstyle{n}$
with relation $\rho$ given as follows.
(1) all paths of length $\geq 3$;
(2)
$\alpha_{i}^{{}^{\prime}}\alpha_{i}-\alpha_{i+1}\alpha_{i+1}^{{}^{\prime}}$,
$i=1,\cdots,n-2$;
(3) $\alpha_{i}\alpha_{i+1}$,
$\alpha_{i+1}^{{}^{\prime}}\alpha_{i}^{{}^{\prime}}$, $i=1,\cdots,n-2$.
Let $A=K(Q,\rho)$. Define $\tau$ by
(1) $\tau(e_{1})=\cdots=\tau(e_{n})=1$;
(2)
$\tau(\alpha_{i}\alpha_{i}^{{}^{\prime}})=\tau(\alpha_{i}^{{}^{\prime}}\alpha_{i})=1$,
$i=1,\cdots,n-1$;
(3)$\tau(\alpha_{i})=\tau(\alpha_{i}^{{}^{\prime}})=0$.
Then $A$ is a symmetric cellular algebra with a cellular basis
$\begin{matrix}\begin{matrix}e_{1}\end{matrix};&\begin{matrix}\alpha_{1}\alpha_{1}^{{}^{\prime}}&\alpha_{1}\\\
\alpha_{1}^{{}^{\prime}}&e_{2}\end{matrix};&\begin{matrix}\alpha_{2}\alpha_{2}^{{}^{\prime}}&\alpha_{2}\\\
\alpha_{2}^{{}^{\prime}}&e_{3}\end{matrix};&\cdots;&\begin{matrix}\alpha_{n-1}\alpha_{n-1}^{{}^{\prime}}&\alpha_{n-1}\\\
\alpha_{n-1}^{{}^{\prime}}&e_{n}\end{matrix};&\begin{matrix}\alpha_{n-1}^{{}^{\prime}}\alpha_{n-1}\end{matrix}.\end{matrix}$
The dual basis is
$\begin{matrix}\begin{matrix}\alpha_{1}\alpha_{1}^{{}^{\prime}}\end{matrix};&\begin{matrix}e_{1}&\alpha_{1}^{{}^{\prime}}\\\
\alpha_{1}&\alpha_{1}^{{}^{\prime}}\alpha_{1}\end{matrix};&\begin{matrix}e_{2}&\alpha_{2}^{{}^{\prime}}\\\
\alpha_{2}&\alpha_{2}^{{}^{\prime}}\alpha_{2}\end{matrix};&\cdots;&\begin{matrix}e_{n-1}&\alpha_{n-1}^{{}^{\prime}}\\\
\alpha_{n-1}&\alpha_{n-1}^{{}^{\prime}}\alpha_{n-1}\end{matrix};&\begin{matrix}e_{n}\end{matrix}.\end{matrix}$
It is easy to know that $L(A)$ is an ideal of $Z(A)$ generated by
$\\{\alpha_{1}\alpha_{1}^{{}^{\prime}},\alpha_{1}\alpha_{1}^{{}^{\prime}}+\alpha_{2}\alpha_{2}^{{}^{\prime}},\alpha_{2}\alpha_{2}^{{}^{\prime}}+\alpha_{3}\alpha_{3}^{{}^{\prime}},\cdots,\alpha_{n-2}\alpha_{n-2}^{{}^{\prime}}+\alpha_{n-1}\alpha_{n-1}^{{}^{\prime}},\alpha_{n-1}^{{}^{\prime}}\alpha_{n-1}\\}$
and $H(A)$ is generated by
$\\{2\alpha_{1}\alpha_{1}^{{}^{\prime}}+\alpha_{2}\alpha_{2}^{{}^{\prime}},\alpha_{1}\alpha_{1}^{{}^{\prime}}+2\alpha_{2}\alpha_{2}^{{}^{\prime}}+\alpha_{3}\alpha_{3}^{{}^{\prime}},\alpha_{2}\alpha_{2}^{{}^{\prime}}+2\alpha_{3}\alpha_{3}^{{}^{\prime}}+\alpha_{4}\alpha_{4}^{{}^{\prime}},\cdots,\alpha_{n-3}\alpha_{n-3}^{{}^{\prime}}+2\alpha_{n-2}\alpha_{n-2}^{{}^{\prime}}+\alpha_{n-1}\alpha_{n-1}^{{}^{\prime}},\alpha_{n-2}\alpha_{n-2}^{{}^{\prime}}+2\alpha_{n-1}\alpha_{n-1}^{{}^{\prime}}\\}$.
Then $\dim_{K}L(A)=n$ since the rank of the matrix below is $n$.
$\begin{bmatrix}1&0&0&0&\cdots&0&0\\\ 1&1&0&0&\cdots&0&0\\\
0&1&1&0&\cdots&0&0\\\ &\cdots&&\cdots&&\cdots&&&\\\ 0&0&0&0&\cdots&1&1\\\
0&0&0&0&\cdots&0&1\end{bmatrix}_{(n+1)\times n}$
We know that $\dim_{K}H(A)<n$ if $CharK$ is a factor of $n+1$ and
$\dim_{K}H(A)=n$ otherwise, since the determinant of the matrix below is
$n+1$.
$\begin{bmatrix}2&1&0&0&\cdots&0&0\\\ 1&2&1&0&\cdots&0&0\\\
0&1&2&1&\cdots&0&0\\\ &\cdots&&\cdots&&\cdots&&&\\\ 0&0&0&0&\cdots&2&1\\\
0&0&0&0&\cdots&1&2\end{bmatrix}_{n\times n}$
Then $H(A)\subsetneq L(A)$ if $CharK$ is a factor of $n+1$ and $H(A)=L(A)$
otherwise.
Example Let $K$ be a field and $A=K[x]/(x^{n})$, where $n\in\mathbb{N}$.
Clearly, $A$ is a symmetric cellular algebra with a basis
$1,\bar{x},\ldots,\overline{x^{n-1}}$. It is easy to know that $L(A)$ has a
basis $\overline{x^{n-1}}$ and $Z(A)=A$. This example shows that
$\dim_{K}Z(A)-\dim_{K}L(A)$ may be very large.
###### Proposition 3.4.
Notations are as in Theorem A. Then $x_{\lambda}x_{\mu}=0$ for arbitrary
$\lambda,\mu\in\Lambda$ with $\lambda\neq\mu$.
Proof: For arbitrary $\lambda,\mu\in\Lambda$ with $x_{\lambda}x_{\mu}\neq 0$,
then there exist $S_{0}\in M(\lambda)$ and $U_{0}\in M(\mu)$ such that
$C_{S_{0},T}^{\lambda}D_{S_{0},T}^{\lambda}C_{U_{0},V}^{\mu}D_{U_{0},V}^{\mu}\neq
0.$
This implies $D_{S_{0},T}^{\lambda}C_{U_{0},V}^{\mu}\neq 0$. Then by Corollary
3.1, there exists some $C_{X,Y}^{\epsilon}$ such that
$r_{(U_{0},V,\mu),(X,Y,\epsilon),(S_{0},T,\lambda)}\neq 0.$
By (C3) of Definition 2.5, this implies that $\lambda\leq\mu$. By
$x_{\lambda}x_{\mu}=x_{\mu}x_{\lambda}$, we get $x_{\lambda}x_{\mu}\neq 0$
also implies that $\mu\leq\lambda$. Then $\lambda=\mu$ if
$x_{\lambda}x_{\mu}\neq 0$. $\Box$
## 4 Semisimple case
In this section, we consider the semisimple case. We will construct all the
central idempotents which are primitive in $Z(A)$ by elements $x_{\lambda}$
defined in Section 3 for a semisimple symmetric cellular algebra.
Firstly, let us recall the definition of Schur elements. For details, see [9].
Let $R$ be a commutative ring with identity and $A$ an $R$-algebra. Let $V$ be
an $A$-module which is finitely generated and free over $R$. The algebra
homomorphism
$\rho_{V}:A\rightarrow\rm
End_{R}(V),\,\,\,\rho_{V}(a)v=av,\,\,\,\,{\text{w}here}\,\,\,\,\,v\in
V,\,\,\,\,a\in A,$
is called the representation afforded by $V$. The corresponding character is
the $R$-linear map defined by
$\chi_{V}:A\rightarrow R,\,\,\,a\mapsto{\bf tr}(\rho_{V}(a)),$
where tr is the usual trace of a matrix.
Let $K$ be a field and $A$ a finite dimensional symmetric $K$-algebra with
symmetrizing trace $\tau$. Let $B=\\{B_{i}\mid i=1,\cdots,n\\}$ be a basis and
$D=\\{D_{i}\mid i=1,\cdots,n\\}$ the dual basis determined by $\tau$. If $V$
is a split simple $A$-module, denote the character by $\chi_{V}$, we have
$\sum_{i}\chi_{V}(b_{i})\chi_{V}(D_{i})=c_{V}\dim_{K}V,$
where $c_{V}\in K$ is the so-called Schur element associated with $V$. We also
denote it by $c_{\chi_{V}}$. It is non-zero if and only if $V$ is a split
simple projective $A$-module [9].
###### Lemma 4.1.
([9] 7.2.7) Let $A$ be a split semisimple $K$-algebra. Then
$\\{e_{V}:=c_{V}^{-1}\sum_{i}\chi_{V}(b_{i})D_{i}\mid V\,\,\text{is a simple
A-module}\\}$
is a complete set of central idempotents which are primitive in $Z(A)$.
Let $R$ be an integral domain and $A$ a symmetric cellular algebra with
cellular basis $\\{C_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$. Given a symmetrizing trace $\tau$, the dual
basis is $\\{D_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$.
Let $K$ be the field of fractions of $R$. Define $A_{K}:=A\bigotimes_{R}K$.
Consider $A$ as a subalgebra of $A_{K}$ and extend $\tau$ of $A$ to $A_{K}$.
Then we can construct all the central idempotents which are primitive in
$Z(A_{K})$ by $x_{\lambda}$.
###### Proposition 4.2.
If $A_{K}$ is split semisimple, then
$\\{c_{W(\lambda)}^{-1}x_{\lambda}\mid\lambda\in\Lambda\\}$ is a complete set
of central idempotents which are primitive in $Z(A_{K})$.
Proof: The left $A_{K}$-module $W(\lambda)$ is split simple since $A_{K}$ is
split semisimple. Then by Lemma 4.1, we have
$e_{W(\lambda)}=c_{W(\lambda)}^{-1}\sum_{\mu\in\Lambda,U,V\in
M(\mu)}\chi_{W(\lambda)}(C_{U,V}^{\mu})D_{U,V}^{\mu}.$
Note that the character afforded by $W(\lambda)$ is given by the following
formula
$\chi_{W(\lambda)}(a)=\sum_{S\in M(\lambda)}r_{a}(S,S)$
for all $a\in A$. Then we get
$\chi_{W(\lambda)}(C_{U,V}^{\mu})=\sum\limits_{S\in
M(\lambda)}r_{(U,V,\mu),(S,T,\lambda),(S,T,\lambda)}.$ Then
$\displaystyle e_{W(\lambda)}$ $\displaystyle=$ $\displaystyle
c_{W(\lambda)}^{-1}\sum_{\mu\in\Lambda,U,V\in M(\mu)}\sum_{S\in
M(\lambda)}r_{(U,V,\mu),(S,T,\lambda),(S,T,\lambda)}D_{U,V,}^{\mu}$
$\displaystyle=$ $\displaystyle c_{W(\lambda)}^{-1}\sum_{S\in
M(\lambda)}\sum_{\mu\in\Lambda,U,V\in
M(\mu)}r_{(U,V,\mu),(S,T,\lambda),(S,T,\lambda)}D_{U,V,}^{\mu}$
$\displaystyle=$ $\displaystyle c_{W(\lambda)}^{-1}\sum_{S\in
M(\lambda)}C_{S,T}^{\lambda}D_{S,T}^{\lambda}.$
$\Box$
Remark. Clearly, $\\{x_{\lambda}\mid\lambda\in\Lambda\\}$ is a basis of the
center of $A_{K}$ by this proposition. It is different from the one in [12]
when we consider Hecke algebras.
Note that the center of $A$ is equal to the intersection of $A$ and the center
of $A_{K}$. We now give a necessary condition for an element of the center of
$A_{K}$ being in $A$.
###### Corollary 4.3.
Let $a_{\lambda}\in K$ for all $\lambda\in\Lambda$ and
$a=\sum\limits_{\lambda\in\Lambda}a_{\lambda}x_{\lambda}\in A$. Then
$a_{\lambda}c_{W(\lambda)}n_{\lambda}\in R$ for arbitrary $\lambda\in\Lambda$,
where $n_{\lambda}$ is the number of elements in the set $M(\lambda)$.
Proof: For any $\lambda\in\Lambda$, we know $c_{W(\lambda)}^{-1}x_{\lambda}$
is a central idempotent of $A_{K}$ by Proposition 4.2, i.e.
$x_{\lambda}^{2}=c_{W(\lambda)}x_{\lambda}$. This implies that
$ax_{\lambda}=a_{\lambda}c_{W(\lambda)}x_{\lambda}$. Clearly, $ax_{\lambda}\in
A$ implies $\tau(ax_{\lambda})\in R$. By the definition of the dual basis,
$\tau(x_{\lambda})=m_{\lambda}$. This completes the proof. $\Box$
Acknowledgments The author acknowledges his supervisor Prof. C.C. Xi and the
support from the Research Fund of Doctor Program of Higher Education, Ministry
of Education of China. He also acknowledges Dr. Wei Hu for many helpful
conversations.
## References
* [1] C.W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Interscience, New York, 1964\.
* [2] R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 54, (1987), 57-82.
* [3] A. Francis, The minimal bases for the centre of an Iwahori-Hecke algebra, J. Algebra, 221, (1999), 1-28,
* [4] A. Francis, Centralizers of Iwahori-Hecke algebras, Tran. Amer. Math. Soc., 353, Number 7, (2001), 2725-2739.
* [5] A. Francis and J.J. Graham, Centers of Hecke algebras: The Dipper-James conjecture, J. Algebra, 306, (2006), 244-267.
* [6] A. Francis and L. Jones, On bases of centres of Iwahori-Hecke algebras of the symmetric group, J. Algebra, 289, (2005), 42-69.
* [7] M. Geck, Hecke algebras of finite type are cellular, Invent. math., 169, (2007), 501-517.
* [8] J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math., 123, (1996), 1-34.
* [9] M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Lond. Math. Soc. Monographs, New serries, vol. 21,Oxford University Press, New York (2000).
* [10] M. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke algebras, in: Finite Reductive Groups, Luminy, 1994, Birkhauser Boston, Boston, MA, 1997, 251-72.
* [11] L. Jones, Centers of generic Hecke algebras, Trans. Amer. Math. Soc., 317, (1990), 361-392.
* [12] Yeon-Kwan Jeong, In-Sok Lee, Hyekyung Oh and Kyung-Hwan Park, Cellular algebras and centers of Hecke algebras, Bull. Korean Math. Soc., 39, (2002), No.1, 71-79.
* [13] Yanbo Li, Radicals of symmetric cellular algebras, arXiv: 0911.3524v1 [math.RT], preprint (2009).
* [14] H.B. Rui and C.C. Xi, The representation theory of cyclotomic Temperley-Lieb algebras, Comment. Math. Helv., 79, no.2, (2004), 427-450.
* [15] C.C. Xi, Partition algebras are cellular Compositio math., 119, (1999), 99-109.
* [16] C.C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154, (2000), 280-298.
|
arxiv-papers
| 2009-11-24T08:42:32 |
2024-09-04T02:49:06.656076
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanbo Li",
"submitter": "Yanbo Li",
"url": "https://arxiv.org/abs/0911.4576"
}
|
0911.4701
|
# Wavelets Beyond Admissibility
Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds LS2 9JT,
UK
kisilv@maths.leeds.ac.uk
###### Abstract
The purpose of this paper is to articulate an observation that many
interesting type of _wavelets_ (or _coherent states_) arise from group
representations which _are not_ square integrable or vacuum vectors which _are
not_ admissible.
###### keywords:
Wavelets, coherent states, group representations, Hardy space, functional
calculus, Berezin calculus, Radon transform, Möbius map, maximal function,
affine group, special linear group, numerical range.
## 1 Covariant Transform
A general group-theoretical construction [1, 2, 3, 4, 5, 6] of _wavelets_ (or
_coherent states_) starts from an square integrable (_s.i._) representation.
However, such a setup is restrictive and is not necessary, in fact.
###### Definition 1
Let ${\rho}$ be a representation of a group $G$ in a space $V$ and $F$ be an
operator from $V$ to a space $U$. We define a _covariant transform_
$\mathcal{W}$ from $V$ to the space $L{}(G,U)$ of $U$-valued functions on $G$
by the formula:
$\mathcal{W}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v),\qquad v\in V,\ g\in G.$
(1)
###### Remark 1
We do not require that operator $F$ shall be linear.
###### Remark 2
Usefulness of the covariant transform is in the reverse proportion to the
dimensionality of the space $U$. The covariant transform encodes properties of
$v$ in a function $\mathcal{W}v$ on $G$. For a low dimensional $U$ this
function can be ultimately investigated by means of harmonic analysis. Thus
$\dim U=1$ is the ideal case, however, it is unattainable sometimes, see Ex.
2.4 below.
###### Theorem 1
The covariant transform $\mathcal{W}$ (1) intertwines ${\rho}$ and the left
regular representation $\Lambda$ on $L{}(G,U)$:
$\Lambda(g):f(h)\mapsto f(g^{-1}h).$ (2)
###### Proof 1.1.
We have a calculation similar to wavelet transform [3, Prop. 2.6]:
$[\mathcal{W}({\rho}(g)v)](h)=F({\rho}(h^{-1}){\rho}(g)v)=[\mathcal{W}v](g^{-1}h)=\Lambda(g)[\mathcal{W}v](h).$
###### Corollary 2.
The image space $\mathcal{W}(V)$ is invariant under the left shifts on $G$.
## 2 Examples of Covariant Transform
###### Example 2.1.
Let $V$ be a Hilbert space with an inner product
$\left\langle\cdot,\cdot\right\rangle$ and ${\rho}$ be a unitary
representation. Let $F:V\rightarrow\mathbb{C}{}$ be a functional
$v\mapsto\left\langle v,v_{0}\right\rangle$ defined by a vector $v_{0}\in V$.
Then the transformation (1) is the well-known expression for a _wavelet
transform_ [4, (7.48)] (or _representation coefficients_):
$\mathcal{W}:v\mapsto\hat{v}(g)=\left\langle{\rho}(g^{-1})v,v_{0}\right\rangle=\left\langle
v,{\rho}(g)v_{0}\right\rangle,\qquad v\in V,\ g\in G.$ (3)
The family of vectors $v_{g}={\rho}(g)v_{0}$ is called _wavelets_ or _coherent
states_. In this case we obtain scalar valued functions on $G$, thus the
fundamental rôle of this example is explained in Rem. 2.
This scheme is typically carried out for a s.i. representation ${\rho}$ and
$v_{0}$ being an admissible vector[1, 2, 4, 5, 6]. In this case the wavelet
(covariant) transform is a map into the s.i. functions [7] with respect to the
left Haar measure.
However s.i. representations and admissible vectors does not cover all
interesting cases.
###### Example 2.2.
Let $G$ be the “$ax+b$” (or _affine_) group [4, § 8.2]: the set of points
$(a,b)$, $a\in\mathbb{R}_{+}{}$, $b\in\mathbb{R}{}$ in the upper half-plane
with the group law:
$(a,b)*(a^{\prime},b^{\prime})=(aa^{\prime},ab^{\prime}+b)$ (4)
and left invariant measure $a^{-2}\,da\,db$. Its isometric representation on
$V=L_{p}{}(\mathbb{R}{})$ is given by the formula:
$[{\rho_{p}}(a,b)\,f](x)=a^{\frac{1}{p}}f\left(ax+b\right).$ (5)
We consider the operators
$F_{\pm}:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ defined by:
$F_{\pm}(f)=\frac{1}{2\pi
i}\int_{\mathbb{R}{}}\frac{f(t)\,dt}{t\mp\mathrm{i}}.$ (6)
Then the covariant transform (1) is the Cauchy integral from
$L_{2}{}(\mathbb{R}{})$ to the Hardy space in the upper/lower half-plane
$H_{2}{}(\mathbb{R}^{2}_{\pm}{})$. Although the representation (5) is s.i. for
$p=2$, the function $\frac{1}{t\pm\mathrm{i}}$ is not an admissible vacuum
vector. Thus the complex analysis become decoupled from the traditional
wavelets theory. As a result the application of wavelet theory shall relay on
an extraneous mother wavelets [8].
However many important objects in complex analysis are generated by
inadmissible mother wavelets like (6). For example, if
$F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ is defined by $F:f\mapsto
F_{+}f+F_{-}f$ then the covariant transform (1) is simply the _Poisson
integral_. If $F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}^{2}{}$ is defined
by $F:f\mapsto(F_{+}f,F_{-}f)$ then the covariant transform (1) represents a
function on the real line as a jump between functions analytic in the upper
and the lower half-planes. This makes a decomposition of
$L_{2}{}(\mathbb{R}{})$ into irreducible components of the representation (5).
Another interesting but non-admissible vector is the Gaussian $e^{-x^{2}}$.
###### Example 2.3.
For the group $G=SL_{2}{}(\mathbb{R}{})$ [14] let us consider the unitary
representation ${\rho}$ on the space of s.i. function
$L_{2}{}(\mathbb{R}^{2}_{+}{})$ on the upper half-plane through the Möbius
transformations:
${\rho}(g):f(z)\mapsto\frac{1}{(cz+d)^{2}}\,f\left(\frac{az+b}{cz+d}\right),\qquad
g^{-1}=\ \begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$
Let $F_{i}$ be the functional
$L_{2}{}(\mathbb{R}^{2}_{+}{})\rightarrow\mathbb{C}{}$ of pairing with the
lowest/highest $i$-weight vector in the corresponding irreducible component of
the discrete series [14, Ch. VI]. Then we can build an operator $F$ from
various $F_{i}$ similarly to the previous example, e.g. this generalises the
representation of an s.i. function as a sum of analytic ones from different
irreducible subspaces.
Covariant transform is also meaningful for principal and complementary series
of representations of the group $SL_{2}{}(\mathbb{R}{})$[9], which are not
s.i.
###### Example 2.4.
A straightforward generalisation of Ex.2.1 is obtained if $V$ is a Banach
space and $F:V\rightarrow\mathbb{C}{}$ is an element of $V^{*}$. Then the
covariant transform coincides with the construction of wavelets in Banach
spaces [3].
The next stage of generalisation is achieved if $V$ is a Banach space and
$F:V\rightarrow\mathbb{C}^{n}{}$ be a linear operator. Then the corresponding
covariant transform is a map $\mathcal{W}:V\rightarrow
L{}(G,\mathbb{C}^{n}{})$. This is closely related to M.G. Krein’s works on
_directing functionals_ [10], see also _multiresolution wavelet analysis_
[11], Clifford-valued Bargmann spaces [12] and [4, Thm. 7.3.1].
###### Example 2.5.
A step in a different direction is a consideration of non-linear operators.
Take again the “$ax+b$” group and its representation (5). We define $F$ to be
a homogeneous but non-linear functional $V\rightarrow\mathbb{R}_{+}{}$:
$F(f)=\frac{1}{2}\int\limits_{-1}^{1}\left|f(x)\right|\,dx.$
The covariant transform (1) becomes:
$\displaystyle[\mathcal{W}_{p}f](a,b)=\frac{1}{2}\int\limits_{-1}^{1}\left|a^{\frac{1}{p}}f\left(ax+b\right)\right|\,dx=a^{\frac{1}{p}}\frac{1}{2a}\int\limits^{b+a}_{b-a}\left|f\left(x\right)\right|\,dx.$
Obviously $M_{f}(b)=\max_{a}[\mathcal{W}_{\infty}f](a,b)$ coincides with the
Hardy _maximal function_ , which contains important information on the
original function $f$. However, the full covariant transform is even more
detailed. For example,
$\left\|f\right\|=\max_{b}[\mathcal{W}_{\infty}f](\frac{1}{2},b)$ is the shift
invariant norm [13].
From the Cor. 2 we deduce that the operator $M:f\mapsto M_{f}$ intertwines
${\rho_{p}}$ with itself ${\rho_{p}}M=M{\rho_{p}}$.
###### Example 2.6.
Let $V=L_{c}{}(\mathbb{R}^{2}{})$ be the space of compactly supported bounded
functions on the plane. We take $F$ be the linear operator
$V\rightarrow\mathbb{C}{}$ of integration over the real line:
$F:f(x,y)\mapsto F(f)=\int_{\mathbb{R}{}}f(x,0)\,dx.$
Let $G$ be the group of Euclidean motions of the plane represented by ${\rho}$
on $V$ by a change of variables. Then the wavelet transform $F({\rho}(g)f)$ is
the _Radon transform_.
###### Example 2.7.
Let a representation ${\rho}$ of a group $G$ act on a space $X$. Then there is
an associated representation ${\rho_{B}}$ of $G$ on a space $V=B{}(X,Y)$ of
linear operators $X\rightarrow Y$ defined by the identity:
$({\rho_{B}}(g)A)x=A({\rho}(g)x),\qquad x\in X,\ g\in G,\ A\in B{}(X,Y).$
Following the Remark 2 we take $F$ to be a functional
$V\rightarrow\mathbb{C}{}$, for example $F$ can be defined from a pair $x\in
X$, $l\in Y^{*}$ by the expression $F:A\mapsto\left\langle Ax,l\right\rangle$.
Then the covariant transform:
$\mathcal{W}:A\mapsto\hat{A}(g)=F({\rho_{B}}(g)A),\qquad$
this is an example of _covariant calculus_ [3, 15].
###### Example 2.8.
A modification of the previous construction is obtained if we have two groups
$G_{1}$ and $G_{2}$ represented by ${\rho_{1}}$ and ${\rho_{2}}$ on $X$ and
$Y^{*}$ respectively. Then we have a covariant transform $B{}(X,Y)\rightarrow
L{}(G_{1}\times G_{2},\mathbb{C}{})$ defined by the formula:
$\mathcal{W}:A\mapsto\hat{A}(g_{1},g_{2})=\left\langle
A{\rho_{1}}(g_{1})x,{\rho_{2}}(g_{2})l\right\rangle.$
This generalises _Berezin functional calculi_ [3].
###### Example 2.9.
Let us restrict the previous example to the case when $X=Y$ is a Hilbert
space, ${\rho_{1}}{}={\rho_{2}}{}={\rho}$ and $x=l$ with $\left\|x\right\|=1$.
Than the range of the covariant transform:
$\mathcal{W}:A\mapsto\hat{A}(g)=\left\langle
A{\rho}(g)x,{\rho}(g)x\right\rangle$
is a subset of the _numerical range_ of the operator $A$.
###### Example 2.10.
The group $SL_{2}{}(\mathbb{R}{})$ consists of $2\times 2$ matrices of the
form $\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}$
with the unit determinant [14, § IX.1]. Let $A$ be an operator with the
spectral radius less than $1$. Then the associated Möbius transformation
$g:A\mapsto g\cdot A=\frac{\alpha A+\beta
I}{\bar{\beta}A+\bar{\alpha}I},\qquad\text{where}\quad
g^{-1}=\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}\in
SL_{2}{}(\mathbb{R}{}),\ $
produces a well-defined operator with the spectral radius less than $1$ as
well. Thus we have a representation of $SL_{2}{}(\mathbb{R}{})$. A choise of
an operator $F$ will define the corresponding covariant transform. In this way
we obtain generalisations of _Riesz–Dunford functional calculus_ [15].
## 3 Inverse Covariant Transform
An object invariant under the left action $\Lambda$ (2) is called _left
invariant_. For example, let $L$ and $L^{\prime}$ be two left invariant spaces
of functions on $G$. We say that a pairing
$\left\langle\cdot,\cdot\right\rangle:L\times
L^{\prime}\rightarrow\mathbb{C}{}$ is _left invariant_ if
$\left\langle\Lambda(g)f,\Lambda(g)f^{\prime}\right\rangle=\left\langle
f,f^{\prime}\right\rangle,\quad\textrm{ for all }\quad f\in L,\ f^{\prime}\in
L^{\prime}.$ (7)
###### Remark 3.
1. 1.
We do not require the pairing to be linear in general.
2. 2.
If the pairing is invariant on space $L\times L^{\prime}$ it is not
necessarily invariant (or even defined) on the whole $C{}(G)\times C{}(G)$.
3. 3.
In a more general setting we shall study an invariant pairing on a homogeneous
spaces instead of the group. However due to length constraints we cannot
consider it here beyond the Example 3.2.
4. 4.
An invariant pairing on $G$ can be obtained from an invariant functional $l$
by the formula $\left\langle f_{1},f_{2}\right\rangle=l(f_{1}\bar{f}_{2})$.
For a representation ${\rho}$ of $G$ in $V$ and $v_{0}\in V$ we fix a function
$w(g)={\rho}(g)v_{0}$. We assume that the pairing can be extended in its
second component to this $V$-valued functions, say, in the weak sense.
###### Definition 4.
Let $\left\langle\cdot,\cdot\right\rangle$ be a left invariant pairing on
$L\times L^{\prime}$ as above, let ${\rho}$ be a representation of $G$ in a
space $V$, we define the function $w(g)={\rho}(g)v_{0}$ for $v_{0}\in V$. The
_inverse covariant transform_ $\mathcal{M}$ is a map $L\rightarrow V$ defined
by the pairing:
$\mathcal{M}:f\mapsto\left\langle f,w\right\rangle,\qquad\text{ where }f\in
L.$ (8)
###### Example 3.1.
Let $G$ be a group with a unitary s.i. representation $\rho$. An invariant
pairing of two s.i. functions is obviously done by the integration over the
Haar measure:
$\left\langle f_{1},f_{2}\right\rangle=\int_{G}f_{1}(g)\bar{f}_{2}(g)\,dg.$
For an admissible vector $v_{0}$ [7], [4, Chap. 8] the inverse covariant
transform is known in this setup as _reconstruction formula_.
###### Example 3.2.
Let $\rho$ be a s.i. representation of $G$ modulo a subgroup $H\subset G$ and
let $X=G/H$ be the corresponding homogeneous space with a quasi-invariant
measure $dx$. Then integration over $dx$ with an appropriate weight produces
an invariant pairing. The inverse covariant transform is a more general
version [4, (7.52)] of the _reconstruction formula_ mentioned in the previous
example.
Let $\rho$ be not a s.i. representation (even modulo a subgroup) or let
$v_{0}$ be inadmissible vector of a s.i. representation $\rho$. An invariant
pairing in this case is not associated with an integration over any non
singular invariant measure on $G$. In this case we have a _Hardy pairing_. The
following example explains the name.
###### Example 3.3.
Let $G$ be the “$ax+b$” group and its representation ${\rho}$ (5) from Ex.
2.2. An invariant pairing on $G$, which is not generated by the Haar measure
$a^{-2}da\,db$, is:
$\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b)\,\bar{f}_{2}(a,b)\,db.$ (9)
For this pairing we can consider functions $\frac{1}{2\pi i(x+i)}$ or
$e^{-x^{2}}$, which are not admissible vectors in the sense of s.i.
representations. Then the inverse covariant transform provides an _integral
resolutions_ of the identity.
Similar pairings can be defined for other semi-direct products of two groups.
We can also extend a Hardy pairing to a group, which has a subgroup with such
a pairing.
###### Example 3.4.
Let $G$ be the group $SL_{2}{}(\mathbb{R}{})$ from the Ex. 2.3. Then the
“$ax+b$” group is a subgroup of $SL_{2}{}(\mathbb{R}{})$, moreover we can
parametrise $SL_{2}{}(\mathbb{R}{})$ by triples $(a,b,\theta)$,
$\theta\in(-\pi,\pi]$ with the respective Haar measure [14, III.1(3)]. Then
the Hardy pairing
$\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b,\theta)\,\bar{f}_{2}(a,b,\theta)\,db\,d\theta.$
(10)
is invariant on $SL_{2}{}(\mathbb{R}{})$ as well. The corresponding inverse
covariant transform provides even a finer resolution of the identity which is
invariant under conformal mappings of the Lobachevsky half-plane.
A further study of covariant transform and its inverse shall be continued
elsewhere.
## References
* [1] A. Perelomov, Generalized coherent states and their applications (Springer-Verlag, Berlin, 1986).
* [2] Feichtinger, Hans G. and Groechenig, K.H., J. Funct. Anal. 86, 307 (1989).
* [3] V. V. Kisil, Acta Appl. Math. 59, 79 (1999), E-print: arXiv:math/9807141 .
* [4] S. T. Ali, J.-P. Antoine and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations (Springer-Verlag, New York, 2000).
* [5] H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Lecture Notes in Mathematics, Vol. 1863 (Springer-Verlag, Berlin, 2005).
* [6] J. G. Christensen and G. Ólafsson, Acta Appl. Math. 107, 25 (2009).
* [7] M. Duflo and C. C. Moore, J. Functional Analysis 21, 209 (1976).
* [8] O. Hutník, Integral Equations Operator Theory 63, 29 (2009).
* [9] V. V. Kisil, Complex Variables Theory Appl. 40, 93 (1999), E-print: arXiv:funct-an/9712003.
* [10] M. G. Kreĭn, Akad. Nauk Ukrain. RSR. Zbirnik Prac’ Inst. Mat. 1948, 83 (1948), MR#14:56c, reprinted in [16].
* [11] O. Bratteli and P. E. T. Jorgensen, Integral Equations Operator Theory 28, 382 (1997), E-print: arXiv:funct-an/9612003.
* [12] J. Cnops and V. V. Kisil, Math. Methods Appl. Sci. 22, 353 (1999), E-print: arXiv:math/9806150. Zbl 1005.22003.
* [13] A. Johansson, Systems Control Lett. 57, 105 (2008).
* [14] S. Lang, ${\rm SL}_{2}({\bf R})$ (Springer-Verlag, New York, 1985).
* [15] V. V. Kisil, Spectrum as the support of functional calculus, in Functional analysis and its applications, North-Holland Math. Stud. Vol. 197, pp. 133–141, (Elsevier, Amsterdam, 2004). E-print: arXiv:math.FA/0208249.
* [16] M. G. Kreĭn, Izbrannye Trudy. II (Akad. Nauk Ukrainy Inst. Mat., Kiev, 1997). MR#96m:01030.
|
arxiv-papers
| 2009-11-24T19:32:00 |
2024-09-04T02:49:06.663244
|
{
"license": "Public Domain",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/0911.4701"
}
|
0911.4726
|
KIAS-P09052
A Study of Wall-Crossing:
Flavored Kinks in $D=2$ QED
Sungjay Lee111sjlee@kias.re.kr and Piljin Yi222piljin@kias.re.kr
School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
We study spectrum of $D=2$ ${\cal N}=(2,2)$ QED with $N+1$ massive charged
chiral multiplets, with care given to precise supermultiplet countings. In the
infrared the theory flows to $\mathbb{CP}^{N}$ model with twisted masses,
where we construct generic flavored kink solitons for the large mass regime,
and study their quantum degeneracies. These kinks are qualitatively different
and far more numerous than those of small mass regime, with features
reminiscent of multi-pronged $(p,q)$ string web, complete with the wall-
crossing behavior. It has been also conjectured that spectrum of this theory
is equivalent to the hypermultiplet spectrum of a certain $D=4$ Seiberg-Witten
theory. We find that the correspondence actually extends beyond
hypermultiplets in $D=4$, and that many of the relevant indices match.
However, a $D=2$ BPS state is typically mapped to several different kind of
dyons whose individual supermultiplets are rather complicated; the match of
index comes about only after summing over indices of these different dyons. We
note general wall-crossing behavior of flavored BPS kink states, and compare
it to those of $D=4$ dyons.
###### Contents
1. 1 Introduction
2. 2 $\mathbb{CP}^{N}$ with twisted masses
1. 2.1 review on massive $\mathbb{CP}^{N}$-model
2. 2.2 BPS equations
3. 2.3 BPS (multi-)kinks
4. 2.4 zero modes
3. 3 Flavored kink solitons and marginal stability
1. 3.1 simple flavored kinks
2. 3.2 composite flavored kinks and marginal stability
4. 4 Quantum BPS states and wall-crossing
1. 4.1 low energy interactions of kinks
2. 4.2 counting generic BPS states
3. 4.3 wall-crossing
5. 5 $D=4$ ${\cal N}=2$ $SU(N+1)$ with flavors
1. 5.1 BPS dyons in pure $SU(N+1)$ and wall-crossing
2. 5.2 flavored dyons from wall-crossing formula
6. 6 Conclusion
7. A Miscellany
8. B Low energy dynamics of kinks
1. B.1 fermion zero mode counting with aligned masses
2. B.2 the two-kink moduli space metric
3. B.3 supersymmetric low energy dynamics with potential
## 1 Introduction
The wall-crossing in four-dimensional supersymmetric theory [1, 2, 3] has been
a subject of interests to many string theorists and mathematicians. This
phenomenon of discontinuity in the BPS spectrum across walls of marginal
stability, as one changes either parameters or vacuum expectation value of a
theory, has been a source of enormous difficulty in understanding the detailed
structure of theories like ${\cal N}=2$ Seiberg-Witten theories and Calabi-Yau
compactified type II string theories.
For BPS states in four-dimensional theories, this phenomenon has been
understood in various physical viewpoints,#1#1#1See Ref. [4] for a review in
the field theory side. such as from geometric realization of BPS states in
string theory [5, 6], from solitonic dynamics [7, 8] and quantum bound states
thereof [9, 10], from a classical soliton picture of the low energy effective
theory [11, 12], and also later from supergravity attractor flow [13]. From
the spacetime viewpoint, the wall-crossing occurs simply because the
wavefunction of the BPS state in question becomes so large (as one approaches
a wall of marginal stability) that the state in question cannot be regarded as
a one-particle state anymore [9, 10, 13]. Despite this simple and compelling
physical picture, a systematic and practical approach to the wall-crossing
phenomenon which can cover all part of the moduli space had not been
available.
Recently, there appeared a new remarkable development in this regard. It
states that such discontinuities of spectrum across walls of marginal
stability is actually necessary for the continuity of the vacuum moduli space
metric. According to Gaiotto, Moore and Neitzke (GMN) [14, 15], the continuity
of the vacuum moduli space metric of $S^{1}$-compactified Seiberg-Witten
theory implies the so-called Kontsevich-Soibelman relations [16] among BPS
dyons across any given wall of marginal stability, which in turn tells us how
the BPS spectra would change across such walls. Cecotti and Vafa [17] has
recently suggested another interesting explanation of Kontsevich-Soibelman’s
formulae with spin refinement [18], using the partition function of A-model
topological string.
While the derivation by GMN was intended for ${\cal N}=2$ Seiberg-Witten
theory, the idea itself must be applicable to all wall-crossing phenomena.
This new machinary is also important in that for the first time we have a
systematic and local prescription for computing BPS spectrum. Although there
were powerful methods which allowed explicit construction/counting of BPS
states in certain regions of the moduli space [9, 10, 19, 20], this new wall-
crossing formula is far more comprehensive in its potential applications.
This observation that discontinuity of BPS spectra is related to continuity of
some physical quantity has, on the other hand, a previously known analog in
the context of two-dimensional ${\cal N}=(2,2)$ theories. Cecotti and Vafa
[21, 22] noted some time ago that if one assumes continuity of a twisted
partition function
${\cal F}(\beta;m^{i})={\rm tr}(-1)^{R}Re^{-\beta H}$ (1.1)
throughout parameter space of the theory, this necessarily implies
(dis-)appearance of BPS topological kinks across walls of marginal stability.
Here $R$ is the fermion number, and $m^{i}$’s are the parameters of the
theory. The above twisted partition is in turn related to the natural metric
in the parameter space, and obeys the so-called $tt^{*}$ equation [23]. In
fact, GMN also noted that some of mathematical structures of $tt^{*}$ equation
is very closely mirrored by those that appear in their formulation of the
four-dimensional wall-crossing.
Independent of this, another interesting similarity between $D=4$ ${\cal N}=2$
and $D=2$ ${\cal N}=(2,2)$ theories was noted in the literature: It has been
conjectured [24, 25] that two-dimensional ${\cal N}=2$ QED with $N+1$ massive
chiral multiplets possesses a BPS spectrum which is related to that of
$SU(N+1)$ Seiberg-Witten theory with $N+1$ massive flavors at the root of the
baryonic branch. In view of the new development in the Seiberg-Witten theory
concerning the wall-crossing, and given its analog in $tt^{*}$ system, it is
of some interest to clarify the precise correspondence and potential
differences. In this article, we aim to study the two-dimensional theory with
care given to precise BPS multiplet countings, and compare their wall-crossing
phenomena against that of the Seiberg-Witten theory.
${\cal N}=(2,2)$ QED with $N+1$ chiral multiplets with twisted masses has been
studied much previously. Initial studies by Hanany and Hori [26] and also by
Dorey [24, 25] concentrated on implications of effective superpotential of the
gauge-multiplet and its similarity to certain Seiberg-Witten spectral curve of
$D=4$ theory. Later works [27, 28, 29, 30, 31] refined this relationship
further by giving physical reasonings, if somewhat sketchy, for the
correspondence and also looked at $D=2$ spectrum more closely by considering
massive excitations of simple kink solutions.
In this paper, we expand on these existing works and solve for all possible
flavored kinks. We give precise criteria for existence of such flavored kink
states, set up the low energy dynamics of kinks, count their degeneracies, and
provide wall-crossing formula. This allows a more refined look at the proposed
“equivalence” of the spectra. We also hope that it will provide a playground
for understanding wall-crossing phenomena in $D=2$ when conserved charged
other than the topological ones are present.
In section 2 and 3, we review the theory and search for all possible kink
soliton solutions. Although kinks are simple and well-known objects, global
charge allows the variety of kink solutions to increase greatly. Apart from
simple “dyonic” kinks whose flavor charge is proportional to the topological
charge, there are much more flavored kinks whose central charges and stability
criteria mimics those of the $(p,q)$ open string webs [6, 7]. In section 4, we
quantize these solitons, elevate them to quantum BPS states, and count their
degeneracy. These BPS states exhibit wall-crossing behavior, just as open
string web does, which we put in the context of general $D=2$ and ${\cal
N}=(2,2)$ theories following Cecotti and Vafa’s results. In section 5, we
compare this spectra to its conjectured counterpart in $D=4$ Seiberg-Witten
theory. Although, the two sides have some common features, essentially due to
the open string web analogy, absence of “angular momentum” in the $D=2$ theory
leads to quantitatively different spectra. However, a set of distinct dyons
with different quark contents are mapped to a single type of favored kink;
interestingly, if one sum over the relevant indices of the former, the result
matches precisely with the degeneracy of the flavored kink. We rely on the
four-dimensional wall-crossing formula to reach this conclusion. We close with
conclusion.
## 2 $\mathbb{CP}^{N}$ with twisted masses
Let us first summarize basic properties of ${\cal N}=(2,2)$ supersymmetric
theories in two dimensions.#2#2#2Please see Appendix A for fruther notations
and conventions. In particular we discuss the massive representation of
${\cal N}=(2,2)$ SUSY algebra and the CFIV index [21] which effectively counts
the short multiplets only.
#### supersymmetry algebra
The ${\cal N}=(2,2)$ superalgebra can read off from the four-dimensional
${\cal N}=1$ superalgebra via trivial dimensional reduction as
$\displaystyle\big{\\{}Q_{+},\bar{Q}_{+}\big{\\}}=2Z,$
$\displaystyle\big{\\{}Q_{+},\bar{Q}_{-}\big{\\}}=-2\big{(}P_{0}-P_{3}\big{)},$
$\displaystyle\big{\\{}Q_{-},\bar{Q}_{-}\big{\\}}=2\bar{Z},$
$\displaystyle\big{\\{}Q_{-},\bar{Q}_{+}\big{\\}}=-2\big{(}P_{0}+P_{3}\big{)}\
,$ (2.1)
where the central charge $Z$ is
$\displaystyle Z=P_{1}-iP_{2}\ .$ (2.2)
For later convenience, let us summarize the $U(1)_{\text{R}}\times
U(1)_{\text{A}}$ charges of supersymmetric generators
$\displaystyle\begin{array}[]{c|cccc}&Q_{+}&Q_{-}&\bar{Q}_{+}&\bar{Q}_{-}\\\
\hline\cr U(1)_{\text{R}}&+1&+1&-1&-1\\\
U(1)_{\text{A}}&+1&-1&+1&-1\end{array}\ .$ (2.6)
Here $U(1)_{\text{A}}$ symmetry comes from the rotational symmetry $SO(2)$ in
four dimensions.
In massive theories, one of the two $U(1)$ symmetries are explicitly broken,
and suppose we choose the following basis that preserveq $U(1)_{\text{R}}$
$\displaystyle{\cal A}=\frac{1}{\sqrt{2}}\big{(}Q_{+}+Q_{-}\big{)}\
,\qquad{\cal B}=\frac{1}{\sqrt{2}}\big{(}Q_{+}-Q_{-}\big{)}\ .$ (2.7)
Making the central charge $Z$ real via a suitable $U(1)_{\text{A}}$ rotation,
the supersymmetry algebra can be recast as
$\displaystyle\big{\\{}{\cal A},{\cal
A}^{\dagger}\big{\\}}=-2\big{(}M-Z\big{)}\ ,\qquad\big{\\{}{\cal B},{\cal
B}^{\dagger}\big{\\}}=-2\big{(}M+Z\big{)}\ ,\qquad\big{\\{}{\cal A},{\cal
B}^{\dagger}\big{\\}}=0\ ,$ (2.8)
One can therefore conclude that, for massive BPS multiplets, the algebra
eventually is reduced to that of a single fermion oscillator.
#### CFIV index
With this, the index that count BPS multiplets is
$\displaystyle\Omega=\text{tr}\Big{[}(-1)^{R}R\Big{]}\ .$ (2.9)
This is a proper index since for long multiplets in Fock vacuum of R-charge
$f$
$\displaystyle[{f}]\otimes\big{(}[{\bf 1}]\oplus[{\bf 0}]\big{)}^{2}\ \
\Longrightarrow\ \ [{f+2}]\oplus 2[{f+1}]\oplus[{f}]\ ,$
the index $\Omega$ identically vanishes
$\displaystyle\Omega=0\ .$ (2.10)
On the other hand, for generic BPS multiplets
$\displaystyle[{f}]\otimes\big{(}[{\bf 1}]\oplus[{\bf 0}]\big{)}\ \
\Longrightarrow\ \ [{f+1}]\oplus[{f}]\ ,$
one can have non-vanishing $\Omega$
$\displaystyle\Omega=(-1)^{f+1}\ .$ (2.11)
The simplicity of $D=2$ theory is such that we have only two types of BPS
multiplets, labeled by this sign, which is because of the small supersymmetry
compounded by absence of spin.#3#3#3 The mirror symmetry, or t-duality in two-
dimensional supersymmetric theory, exchanges those two R-symmetries
$\displaystyle U(1)_{\text{R}}\leftrightarrow U(1)_{\text{A}}\ ,\qquad
Q_{-}\leftrightarrow\bar{Q}_{+}\ .$ (2.12) In the mirror-symmetric dual, the
proper index now in turn is defined with $U(1)_{\text{A}}$ charge,
$\displaystyle\Omega=\text{tr}\Big{[}(-1)^{A}A\Big{]}\ .$
### 2.1 review on massive $\mathbb{CP}^{N}$-model
We consider a two-dimensional supersymmetric QED which flows down to a massive
$\mathbb{CP}^{N}$-model with twisted masses. It is well-known that the
massless $\mathbb{CP}^{N}$-model can be easily understood as IR limit of a
gauged linear sigma model (GLSM) with a photon field $V$ and $N+1$ chiral
matter fields $\phi^{i}$ of unit charge. Introducing the Fayet-Iliopoulos (FI)
parameter $r$ together with theta-angle $\theta$, the Lagrangian takes the
following form
$\displaystyle{\cal L}=\int d^{4}\theta\
\Big{[}\phi^{\dagger}_{i}e^{-2V}\phi^{i}-\frac{1}{4e^{2}}\bar{\Sigma}\Sigma\Big{]}-\text{Im}\Big{[}\tau\int
d^{2}\hat{\theta}\ \Sigma\Big{]}\ ,\qquad\tau=-ir+\frac{\theta}{2\pi}\ ,$
(2.13)
where $i$ run from $0,1,..,N$. Again, the notations and conventions used here
are introduced in appendix A. For a positive FI parameter $r>0$, the
supersymmetric vacuum can be described by
$\displaystyle\sum_{i}|\phi^{i}|^{2}=r\ ,\qquad\sigma=0\ ,$ (2.14)
which defines a projective space $\mathbb{CP}^{N}$. On the generic point of
vacuum moduli space, the $U(1)$ vector multiplet and chiral mode orthogonal to
$\mathbb{CP}^{N}$ are combined to a long multiplet of mass $\sqrt{r}e$ by the
Higgs mechanism. In the IR limit where $e^{2}$ diverges, these modes become
very heavy so that they decouple from the low-energy dynamics of the theory.
It leads to a ${\cal N}=(2,2)$ $\mathbb{CP}^{N}$ model.
We will present a simple way to obtain the effective Lagrangian for the above
low-energy theory, ${\cal N}=(2,2)$ $\mathbb{CP}^{N}$ model. For simplicity,
let us first turn off the theta-angle $\theta=0$ for a while. Note that we can
then rewrite the Fayet-Iliopoulos (FI) term as
$\displaystyle{\cal L}_{\text{FI}}=2r\int d^{4}\theta\ V\ .$ (2.15)
The decoupling phenomenon of massive modes in the Higgs phase can be realized
effectively as the vanishing Maxwell term in the limit of $e^{2}\to\infty$.
The low-energy theory at IR is now governed by the following Lagrangian
$\displaystyle{\cal L}\simeq\int d^{4}\theta\
\Big{[}\phi^{\dagger}_{i}e^{-2V}\phi^{i}+2rV\Big{]}\ .$ (2.16)
Here the vector multiplet becomes an auxiliary fields that one can solve out:
$\displaystyle\delta V\ :\ \ r=\phi^{\dagger}_{i}e^{-2V}\phi^{i}\ \Rightarrow\
V=-\frac{1}{2}\text{log}\big{(}\frac{r}{\phi_{i}^{\dagger}\phi^{i}}\big{)}\ .$
(2.17)
Componentwise, the gauge field, for examples, is determined by
$\displaystyle
A_{\mu}=\frac{1}{2i\phi_{i}^{\dagger}\phi^{i}}\Big{(}\phi_{i}^{\dagger}\partial_{\mu}\phi^{i}-\partial_{\mu}\phi_{i}^{\dagger}\phi^{i}-i\bar{\psi}_{i}\bar{\sigma}_{\mu}\psi^{i}\Big{)}\
,$ (2.18)
which implies that above procedure can be understood as supersymmetric version
of solving the Gauss law in GLSM. Inserting the result back into the
Lagrangian, one can finally obtain
$\displaystyle{\cal L}^{\text{IR}}=r\int d^{4}\theta\
\Big{[}\text{log}\big{(}\sum_{i}\phi_{i}^{\dagger}\phi^{i}\big{)}\Big{]}\ .$
(2.19)
Assuming one of matter fields, say $\phi^{0}$, does not vanish, one can
rewrite the above Lagrangian as
$\displaystyle{\cal L}^{\text{IR}}=r\int d^{4}\theta\
\Big{[}\text{log}\big{(}\phi_{0}^{\dagger}\phi^{0}\big{)}+\text{log}\big{(}1+Z_{m}^{\dagger}Z^{m}\big{)}\Big{]}=r\int
d^{4}\theta\ \Big{[}\text{log}\big{(}1+Z_{m}^{\dagger}Z^{m}\big{)}\Big{]}\ ,$
(2.20)
where we used for the last equality the chirality of $\phi^{0}$. (2.20) is
precisely the lagrangian for the ${\cal N}=(2,2)$ supersymmetric non-linear
sigma model with target space ${\mathbb{C}P}^{N}$. Here chiral superfields
$z^{m}$ ($m=1,2,...,N$) are defined as
$\displaystyle Z^{m}=\frac{\phi^{m}}{\phi^{0}}\ ,$ (2.21)
from which one can identify it bosonic and fermionic part as
$\displaystyle z^{m}=\frac{\phi^{m}}{\phi^{0}}\
,\qquad\chi^{m}=\frac{1}{(\phi^{0})^{2}}\big{(}\phi^{0}\psi^{m}-\psi^{0}\phi^{m}\big{)}\
.$ (2.22)
The model we are eventually interested in is a massive version of this theory.
The so-called twisted masses can be introduced by gauging the flavor symmetry
$U(N+1)$ and give expectation values to the corresponding twisted chiral field
$\hat{\Sigma}$ as
$\displaystyle\langle\hat{\Sigma}\rangle=\text{diag}\big{(}\langle\hat{\Sigma}_{0}\rangle,\langle\hat{\Sigma}_{1}\rangle,..,\langle\hat{\Sigma}_{N}\rangle\big{)}=\begin{pmatrix}m_{0}&&&\\\
&m_{1}&&\\\ &&\ddots&\\\ &&&m_{n}\end{pmatrix}\ .$ (2.23)
These vev acts as mass terms for the chiral multiplets, and can be
incorporated into the Lagrangian as
$\displaystyle{\cal L}=\int d^{4}\theta\
\Big{[}\phi^{\dagger}_{i}e^{-2V}\phi^{i}e^{2\langle\hat{V}_{i}\rangle}-\frac{1}{4e^{2}}\bar{\Sigma}\Sigma\Big{]}-\text{Im}\Big{[}\tau\int
d^{2}\hat{\theta}\ \Sigma\Big{]}\ .$ (2.24)
With these twisted masses, there are $N+1$ classical discrete vacua in this
theory. They correspond to
$\sigma=m_{i}\;,\;\;|\phi^{i}|^{2}=r\;\;{\rm and}\;\;\phi^{k}=0\;,\;\;k\neq i$
(2.25)
for each $i=0,1,\dots,N$. With such discrete set of vacua, various topological
kink solitons are present, which are the objects of our interest. One can show
that this massive theory flows down to
$\displaystyle{\cal L}^{\text{IR}}_{\text{mass}}=r\int d^{4}\theta\
\Big{[}\text{log}\big{(}1+z_{m}^{\dagger}e^{2\langle\hat{V}_{m}\rangle-2\langle\hat{V}_{0}\rangle}z^{m}\big{)}\Big{]}\
.$ (2.26)
In this article, we will be classifying and counting BPS multiplets of this
theory, with a care given to quantum degeneracy and wall-crossing in weak
coupling regime $r\gg 1$ of the sigma model.
The FI parameter $r$ indeed receives the quantum correction at one-loop level,
which leads to the RG running of renormalized FI parameter $r(\mu)$
$\displaystyle\mu\frac{\partial}{\partial\mu}r(\mu)=-\frac{N+1}{2\pi}\ \to\
r(\mu)\simeq\frac{N+1}{2\pi}\text{log}\Big{[}\frac{\mu}{\Lambda_{\sigma}}\Big{]}\
,$ (2.27)
where $\Lambda_{\sigma}$ denotes the RG-invariant dynamical scale where the
perturbative analysis breaks down. In order to rely on our analysis in the
article, we therefore have to introduce sufficiently large twisted masses
$m^{i}$
$\displaystyle e\gg|m^{i}-m^{j}|\gg\Lambda_{\sigma}\ ,$
such that the renormalized coupling $r(\mu)$ are frozen in the weak-coupling
regime. On the other hand, the low-energy theory of (2.13) in another
interesting parameter region $e\ll\Lambda$ have been explored in [26, 24] to
study the BPS states in $\mathbb{CP}^{N}$ model at strong coupling, which will
be briefly discussed in section 5. It has been shown that there is the
discrepancy between BPS spectra at weak and strong coupling of the theory,
which strongly implies the existence of curves of marginal stability somewhere
at strong coupling region. Quantum aspects of central charges and strong/weak
coupling marginal stability walls were also recently investigated in Ref. [32,
33].
As emphasized again, we will explore the curves of marginal stability and
wall-crossing phenomena not in strong-coupling regime but in weak-coupling
regime.
#### conserved charges
For later convenience, we summarize some conserved charges. The bosonic part
of energy functional of this theory takes the following simple form
$\displaystyle{\cal E}=\int dx^{3}\
\sum_{i}\Big{[}|D_{0}\phi^{i}|^{2}+|D_{3}\phi^{i}|^{2}+|\sigma-
m_{i}|^{2}|\phi^{i}|^{2}\Big{]}\ .$ (2.28)
In the infrared, one can express the energy functional in terms of sigma model
variables as
$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle r\int d{\bf x}^{3}\
\Big{[}\frac{(1+\bar{z}\cdot
z)\delta^{m}_{n}-\bar{z}_{n}z^{m}}{(1+\bar{z}\cdot
z)^{2}}\big{(}\dot{\bar{z}}_{m}\dot{z}^{n}+\partial_{3}\bar{z}_{m}\partial_{3}z^{n}\big{)}$
(2.29) $\displaystyle\hskip 8.5359pt+\frac{1}{(1+\bar{z}\cdot
z)^{2}}\sum_{n}|m_{n}-m_{0}|^{2}|z^{n}|^{2}$ $\displaystyle\hskip
8.5359pt+\frac{1}{(1+\bar{z}\cdot
z)^{3}}\sum_{n<p}(m_{n}-m_{p})^{2}|z_{n}|^{2}|z_{p}|^{2}\big{(}1+|z_{n}|^{2}+|z_{p}|^{2}\big{)}$
$\displaystyle\hskip 8.5359pt+\frac{1}{(1+\bar{z}\cdot z)^{3}}\sum_{n\neq
p\neq q}(m_{n}-m_{p})(m_{n}-m_{q})|z_{n}|^{2}|z_{p}|^{2}|z_{q}|^{2}\Big{]}\ .$
Introducing the twisted mass terms, flavor symmetry group $SU(N+1)$ of
$\mathbb{CP}^{N}$ model is spontaneously broken down to $U(1)^{N}$. Those
charges are defined by following: $N$ $U(1)$ charges can be parameterized by a
following $N+1$-vector
$\displaystyle\vec{Q}=\big{(}Q_{0},Q_{1},..,Q_{N}\big{)}\ ,$ (2.30)
where each component is given by
$\displaystyle Q_{0}$ $\displaystyle=$ $\displaystyle-i\int d{\bf x}^{3}\
\phi_{0}^{\dagger}D_{0}\phi^{0}+\text{c.c.}$ (2.31) $\displaystyle=$
$\displaystyle r\int d{\bf x}^{3}\
\frac{i\sum_{m}\big{(}\bar{z}_{m}\partial_{0}z^{m}-\partial_{0}z_{m}z^{m}\big{)}}{\big{(}1+\sum_{m}\bar{z}_{m}z^{m}\big{)}^{2}}\
,$ $\displaystyle Q_{n}$ $\displaystyle=$ $\displaystyle-i\int d{\bf x}^{3}\
\phi_{n}^{\dagger}D_{0}\phi^{n}+\text{c.c.}$ $\displaystyle=$ $\displaystyle
r\int d{\bf x}^{3}\
\frac{-i\big{(}\bar{z}_{n}\partial_{0}z^{n}-\partial_{0}\bar{z}_{n}z^{n}\big{)}}{1+\sum_{m}\bar{z}_{m}z^{m}}+\frac{\bar{z}_{n}z^{n}\cdot
i\sum_{m}\big{(}\bar{z}_{m}\partial_{0}z^{m}-\partial_{0}\bar{z}_{m}z^{m}\big{)}}{\big{(}1+\sum_{m}\bar{z}_{m}z^{m}\big{)}^{2}}\
.$
Note that the charge components $Q_{i}$ ($i=0,1,..N$) always satisfy the
traceless condition
$\displaystyle Q_{0}+Q_{1}+..+Q_{N}=0\ .$ (2.32)
#### central charge
Finally let us recall the expression of central charge $Z$ for ${\cal
N}=(2,2)$ massive $\mathbb{CP}^{N}$ model. Based on the two-dimensional Witten
effect and simple BPS spectra of (2.26) such as fundamental excitations and
kink solutions, central charge $Z$ takes the following form at weak coupling
limit $r\gg 1$
$\displaystyle Z=\sum_{i}m^{i}\big{(}Q_{i}+\tau T_{i}\big{)}\
,\qquad\tau=\frac{\theta}{2\pi}-ir\ ,$ (2.33)
as discussed in [24]. Here $T$ denotes the topological charge associated with
kinks. Because the theory possesses $N+1$ discrete vacua, $T$ naturally live
in the $SU(N+1)$ root lattice. For a topological kink from vacuum j to vacuum
i, our convention is such that $T_{j}=-1$, $T_{i}=1$, and $T_{k}=0$ for $k\neq
j,i$.
The exact expression for central charge $Z$ has also proposed in [26] as
$\displaystyle Z=\sum_{i}\big{(}m^{i}Q_{i}+m_{D}^{i}T_{i}\big{)}\ ,\qquad
m_{D}^{i}={\cal W}(e_{i})\ ,$ (2.34)
where $e_{i}$ are determined by roots of the polynomial equation
$\displaystyle\prod_{i}\big{(}x-m_{i}\big{)}-\Lambda^{N+1}_{\sigma}=\prod_{i}\big{(}x-e_{i}\big{)}=0\
,$ (2.35)
and ${\cal W}(e_{i})$ are given by
$\displaystyle{\cal
W}(e_{i})=\frac{N+1}{2\pi}e_{i}+\sum_{i}\frac{m_{i}}{2\pi}\text{log}\Big{[}\frac{e_{i}-m_{i}}{\mu}\Big{]}\
.$ (2.36)
We will discuss in Section 5 an interesting implication of the exact
expression of central charge $Z$ in relation to four-dimensional ${\cal N}=2$
supersymmetric gauge theories.
### 2.2 BPS equations
The supersymmetry transformation for $z^{m}$ can be read off from those of
GLSM fields: for examples, the variation rules for fermions $\chi^{m}$ are
given by
$\displaystyle\delta\chi^{m}=\frac{1}{(\phi^{0})^{2}}\big{(}\phi^{0}\delta\psi^{m}-\delta\psi^{0}\phi^{m}\big{)}+\cdots\
,$ (2.37)
where we suppressed the irrelevant terms in our discussion. The transformation
rules for GLSM fermion fields $\psi^{i}$ are given by
$\displaystyle\delta\psi^{i}=\tau^{3}\epsilon D_{3}\phi^{i}+\epsilon
D_{0}\phi^{i}-i\tau^{I}\epsilon\big{(}\sigma_{I}\phi^{i}-\phi^{i}m_{I}^{i}\big{)}\
,$ (2.38)
where $I$ run form $1,2$. Here we substitute (2.18) for the GLSM gauge fields:
$\displaystyle
A_{\mu}=\frac{\bar{z}_{m}\partial_{\mu}z^{m}-\partial_{\mu}\bar{z}_{m}\cdot
z^{m}}{2i\big{(}1+\bar{z}_{m}z^{m}\big{)}}+\cdots\ ,$ (2.39)
and also substitute the following for the GLSM vector scalar $\sigma$
$\displaystyle\sigma=\frac{m_{0}+m_{n}\bar{z}_{n}z^{n}}{1+\bar{z}_{m}z^{m}}+\cdots\
,$ (2.40)
with $m_{i}\equiv(m^{i})_{1}-i(m^{i})_{2}$. We dropped again the fermion
contribution here, which are irrelevant in our discussion below.
Inserting the above results (2.38) back into (2.37), BPS solitons of
$\mathbb{CP}^{N}$-model should satisfy the following condition
$\displaystyle\phi^{0}\big{(}\tau^{3}\epsilon D_{3}\phi^{n}+\epsilon
D_{0}\phi^{n}+i\hat{\tau}_{m_{n}}\epsilon\phi^{n}\big{)}-\phi^{n}\big{(}\tau^{3}\epsilon
D_{3}\phi^{0}+\epsilon D_{0}\phi^{0}\big{)}=0\ .$ (2.41)
where $\hat{\tau}_{m_{n}}$ is defined as
$\displaystyle\hat{\tau}_{m_{n}}\equiv\tau^{I}(m^{n}-m^{0})_{I}=\begin{pmatrix}&m_{n0}\\\
\bar{m}_{n0}&\end{pmatrix}\ ,\qquad m_{n0}=m_{n}-m_{0}\ .$ (2.42)
### 2.3 BPS (multi-)kinks
#### simple BPS kinks
Let us first review BPS kinks solutions. Since they are static particle, the
BPS equation (2.41) can be simplifies as
$\displaystyle\tau^{3}\big{(}D_{3}\phi^{n}-z^{n}D_{3}\phi^{0}+i\tau^{3}\hat{\tau}_{m_{n}}\phi^{n}\big{)}\epsilon=0\
.$ (2.43)
As referred to appendix for detailed computation, one can show that
$\displaystyle
D_{3}\phi^{n}-z^{n}D_{3}\phi^{0}=r\frac{\partial_{3}z^{n}}{\sqrt{1+\bar{z}_{n}z^{n}}}\
,$ (2.44)
from which one can massage the above BPS equation into
$\displaystyle\Big{[}\frac{\partial_{3}z^{n}}{\sqrt{1+\bar{z}_{n}z^{n}}}+i\tau^{3}\hat{\tau}_{m_{n}}\frac{z^{n}}{\sqrt{1+\bar{z}_{n}z^{n}}}\Big{]}\epsilon=0\
.$ (2.45)
Since $\big{(}\tau^{3}\hat{\tau}_{m_{n}}\big{)}^{2}=-|m_{n0}|^{2}$, the BPS
equation is finally given by
$\displaystyle\partial_{3}z^{n}\pm|m_{n0}|z^{n}=0\ ,\qquad z^{m}=0\ \ \text{
for }m\neq n\ ,$ (2.46)
provided that $m_{n}\neq m_{n}$. The solutions are therefore given by
$\displaystyle z^{n}=\text{exp}\Big{[}\pm|m_{n0}|({\bf x}^{3}-{\bf
x}_{0})\Big{]}\,.$ (2.47)
The energy of this configuration saturate a topological energy bound since
$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle r\int d{\bf x}^{3}\
\Big{[}\frac{1}{(1+\bar{z}_{n}z^{n})^{2}}\big{|}\partial_{3}z^{n}\mp|m_{n0}|z^{n}\big{|}^{2}\pm\frac{|m_{n0}|}{(1+\bar{z}z)^{2}}\partial_{3}\big{(}\bar{z}_{n}z_{n}\big{)}\Big{]}$
(2.48) $\displaystyle\geq$
$\displaystyle-r|m_{n0}|\Big{[}\frac{1}{1+\bar{z}_{n}z^{n}}\Big{]}^{{\bf
x}^{3}=+\infty}_{{\bf x}^{3}=-\infty}=r|m_{n0}|\,.$
#### composite kinks
Let us denote a BPS kink which interpolates from $m$th vacuum to $n$th vacuum
as $nm$-kink. Suppose that the phases of two mass-parameters $m_{10}$ and
$m_{20}$ are aligned as parallel. Without loss of generality, one can set
$|m_{20}|>|m_{10}|$. Then, the $20$-kink can be also understood as a bound
state of a $10$-kink and a $21$-kink: the BPS equations for $20$-kink are
$\displaystyle\partial_{3}z^{1}+i\tau^{3}\hat{\tau}_{m_{10}}z^{1}=0\
,\qquad\partial_{3}z^{1}+i\tau^{3}\hat{\tau}_{m_{20}}z^{1}=0\
,\qquad\big{[}\tau^{3}\hat{\tau}_{m_{10}},\tau^{3}\hat{\tau}_{m_{20}}\big{]}=0\
,$ (2.49)
or equivalently
$\displaystyle\partial_{3}z^{1}\mp|m_{10}|z^{1}=0\
,\qquad\partial_{3}z^{2}\mp|m_{20}|z^{2}=0\ .$ (2.50)
The solution then turns out to be
$\displaystyle z^{1}=\text{exp}\Big{[}\pm|m_{10}|{\bf x}^{3}\Big{]}\ ,\qquad
z^{2}=\text{exp}\Big{[}\pm|m_{20}|({\bf x}^{3}-{\bf x}_{0})\Big{]}\ ,$ (2.51)
after a suitable choice of the origin. Here ${\bf x}_{0}$ parameterizes the
relative distance between constituent BPS kinks. Note that the phase factor of
each $z^{m}$ describes one-parameter degeneracy of such kink solutions, so in
fact we can have an arbitrary complex number multiplying each of $z^{1,2}$’s.
The fact that they have the same energy can be directly checked. See Appendix
A.
Obviously, this can be repeated for other $z_{m}$’s straightforwardly. When
all $m_{n0}$’s are aligned in the complex plane, the general solution is
$\displaystyle z^{m}=\zeta^{m}\text{exp}\Big{[}\pm|m_{m0}|{\bf x}^{3}\Big{]}\
$ (2.52)
with arbitrary complex numbers $\zeta^{m}$’s which are moduli coordinates of
the soliton.
Figure 2.1: Configuration of the GLSM field $\sigma$. It implies that the
system is placed in $\sigma=m_{1}$ vacuum at ${\bf x}^{3}=-\infty$, and in
$\sigma=m_{2}$ vacuum at ${\bf x}^{3}=+\infty$. The size of the plateau near
$m_{1}$ is determined by how far 10-kink and 21-kink are separated, which is
in turn determined by certain ratio between $\zeta^{1}$ and $\zeta^{2}$.
The GLSM $\sigma$ field (2.40) is useful for describing the general behavior
of the kink solution, which is depicted in Figure 2.1 for this solution. For a
finite ${\bf x}_{0}$, $\sigma$ starts with the vacuum $\sigma=m_{0}$,
approaches the vacuum $\sigma=m_{1}$ (never touches it), and eventually goes
to the vacuum $\sigma=m_{2}$ as ${\bf x}^{3}$ increases. This shows that the
solution indeed a sequential sum of 10-kink and 21-kink.
### 2.4 zero modes
Here we briefly dwell on details of fermion zero mode counting. Bosonic ones
were already noted in previous section: there is one complex bosonic
collective coordinate for each $z^{n}$ kink, provided that all masses $m_{n0}$
are of the same phase. We will find below that for each $z_{n}$ kink there is
also one complex fermionic zero modes. The linearized fermion equations of
motion are given by
$\displaystyle\bar{\sigma}^{M}\big{(}D_{M}\chi^{n}+D_{M}z^{m}\Gamma^{n}_{\
ml}\chi^{l}\big{)}=0\ ,$ (2.53)
with
$\displaystyle\Gamma^{n}_{ml}=-\frac{\delta^{n}_{l}\bar{z}_{m}+\delta^{n}_{m}\bar{z}_{l}}{1+\bar{z}\cdot
z}\
,\qquad\Gamma^{\bar{n}}_{{\bar{m}}{\bar{l}}}=-\frac{\delta^{\bar{n}}_{\bar{l}}z_{\bar{m}}+\delta^{\bar{n}}_{\bar{m}}z_{\bar{l}}}{1+\bar{z}\cdot
z}\ ,$ (2.54)
where the covariant derivatives are defined as
$\displaystyle
D_{M}\chi^{n}=\partial_{M}\chi^{n}+i\big{(}\hat{A}^{n}_{M}-\hat{A}^{0}_{M}\big{)}\chi^{n}\
.$ (2.55)
Here $M$ run from $0,1,2,3$. The twisted mass terms are written as if it is
gauge field along $2,3$ directions, and contributes
$\displaystyle\bar{\sigma}^{M}\big{(}\hat{A}^{n}_{M}-\hat{A}^{0}_{M}\big{)}=-\hat{\tau}_{m_{n0}}=-\begin{pmatrix}0&m_{n}-m_{0}\\\
\bar{m}_{n}-\bar{m}_{0}&0\end{pmatrix}\ .$ (2.56)
Clearly the derivative $\partial_{M}$ runs only for $M=0,1$.
For simplicity let us again take the example of a double-kink with aligned
masses $|m_{20}|>|m_{10}|>0$. The BPS solution in this case was
$\displaystyle z^{1}=\zeta^{1}\text{exp}\Big{[}|m_{10}|{\bf x}^{3}\Big{]}\,,$
$\displaystyle z^{2}=\zeta^{2}\text{exp}\Big{[}|m_{20}|{\bf x}^{3}\Big{]}\,.$
Recall that, despite its deceptively simple appearance, the solution should be
viewed as a combination of two kinks, one from 0 to 1 and another from 1 to 2,
which will interact with each other when one begins to move them around. The
fermionic zero modes in this background are equally simple and deceptive.
There are exactly one zero mode for each $\chi$, and we find (in the limit of
$\zeta^{1}=0$)
$\displaystyle\chi^{1}_{0}=e^{|m_{10}|{\bf x}^{3}}\epsilon_{0}$
$\displaystyle\chi^{2}_{0}=e^{|m_{20}|{\bf x}^{3}}\epsilon_{0}\ .$ (2.57)
with the constant spinor obeying
$i\tau^{3}\hat{\tau}_{m_{20}}\epsilon_{0}=-|m_{20}|\epsilon_{0}$.
The Goldstino mode, in the limit $|\zeta^{1}|\ll 1$, is the combination
$\chi^{1,2}=\zeta^{1,2}e^{|m_{10,20}|{\bf x}^{3}}\epsilon_{0}$, quantization
of which endows the soliton with the basic BPS multiplet structure. The other
combination is more interesting. This is a superpartner to the nontrivial
bosonic moduli of the kinks that encodes relative separation and mutual
interaction of 10-kink and 21-kink. We will come back to them later when we
search for quantum spectrum of flavored kinks.
## 3 Flavored kink solitons and marginal stability
Since the theory has $U(1)^{N}$ flavor charges, BPS objects may carry both
topological and flavor charges. A kink with generic flavor charge will be
called flavored kinks. We present in this section the explicit construction of
flavored kink solitons together with preliminary discussion on their marginal
stability behavior. An important fact here is that these generic flavored
kinks appears only when the mass parameters of the theory is misaligned, i.e.,
when they are no longer lined up in the complex plain. This is analogous to
(dis-)appearance of generic dyons in $D=4$ ${\cal N}=2$ SYM and also of 1/4
BPS dyons in $D=4$ ${\cal N}=4$ SYM, depending on how the vacuum expectation
values of adjoint scalar fields are aligned or misaligned.
In order to investigate the dyonic spectrum of the two-dimensional
$\mathbb{CP}^{N}$ model, let us introduce the time-dependence on the phase
factor of sigma model fields $z^{m}$. Then, the BPS equation (2.41) can be
rewritten as
$\displaystyle\tau^{3}\big{(}D_{3}\phi^{n}-z^{n}D_{3}\phi^{0}+i\tau^{3}\hat{\tau}_{m_{n}}\phi^{n}\big{)}\epsilon+\big{(}D_{0}\phi^{0}-z^{n}D_{0}\phi^{n}\big{)}\epsilon=0\
.$ (3.1)
Inserting (2.39) into the above equation, one can show that flavored kinks
should satisfy the following
$\displaystyle\Big{[}\tau^{3}\frac{\partial_{3}z^{n}}{\sqrt{1+\sum_{m}\bar{z}_{m}z^{m}}}+i\hat{\tau}_{m_{n}}\frac{z^{n}}{\sqrt{1+\sum_{m}\bar{z}_{m}z^{m}}}+\frac{\partial_{0}z^{n}}{\sqrt{1+\sum_{m}\bar{z}_{m}z^{m}}}\Big{]}\epsilon=0\
.$ (3.2)
### 3.1 simple flavored kinks
Let us again review simple flavored kink solutions whose topological charge
and flavor charge are parallel [24]. In this case, without loss of generality,
one can turn off all complex field $z^{n}$ expect one, say $z^{1}$.
Then, the above BPS equations (3.2) can be simplified as
$\displaystyle\big{(}\partial_{0}z^{1}+i\hat{\tau}_{\text{E}}\big{)}\epsilon+\tau^{3}\big{(}\partial_{3}z^{1}+i\tau^{3}\hat{\tau}_{\text{M}}\big{)}\epsilon=0\
,$ (3.3)
where $\hat{\tau}_{\text{E,M}}$ are defined by
$\displaystyle\hat{\tau}_{\text{E}}+\hat{\tau}_{\text{M}}=\hat{\tau}_{m_{10}}\
.$ (3.4)
In order to have solutions to this equation, we have to demand the projectors
$\hat{\tau}_{\text{E,M}}$ to satisfy the following compatibility condition
$\displaystyle\big{[}\hat{\tau}_{\text{E}},\tau^{3}\hat{\tau}_{\text{M}}\big{]}=0\
.$ (3.5)
Figure 3.1: For a simple flavor kink, the mass parameter $\vec{m}_{10}$ can be
decomposed into arbitrary two orthogonal vectors ${\vec{m}}_{\text{M}}$ and
${\vec{m}}_{\text{E}}$. For (a), $m_{\text{M}}$ lies on the right hand side of
$m_{10}$ while for (b) $m_{\text{M}}$ lies on the left hand side of $m_{10}$.
One can easily find a family of solution, parameterized by
$\displaystyle\hat{\tau}_{\text{E}}=\vec{\tau}\cdot{\vec{m}}_{\text{E}}\
,\qquad\hat{\tau}_{\text{M}}=\vec{\tau}\cdot{\vec{m}}_{\text{M}}\ ,$ (3.6)
where vectors $\vec{m}_{\text{E}}$ and $\vec{m}_{\text{M}}$ are orthogonal
decomposition of $\vec{m}_{10}$ as depicted in figure 3.1.
For the case (a), the flavored kink solution is
$\displaystyle z^{1}=\text{exp}\Big{[}\pm|m_{\text{M}}|{\bf x}^{3}\pm
i|m_{\text{E}}|t\Big{]}\ ,$ (3.7)
For the case (b), the flavor kink solution is instead given by
$\displaystyle z^{1}=\text{exp}\Big{[}\pm|m_{\text{M}}|{\bf x}^{3}\mp
i|m_{\text{E}}|t\Big{]}\ .$ (3.8)
Without loss of generality, let us concentrate on the case (a). Some conserved
charges of the simple flavored kink solutions are in order.
#### flavor charge
For a simple flavored kink, the nonvanishing flavor charges (2.31) are
$\displaystyle Q_{1}=-Q_{0}=\pm r\int_{-\infty}^{+\infty}d{\bf x}^{3}\
\frac{|m_{\text{E}}|}{2\text{cosh}^{2}(|m_{\text{M}}|{\bf
x}^{3})}=r\frac{|m_{\text{E}}|}{|m_{\text{M}}|}\ .$ (3.9)
#### energy
For the simple flavored kinks, the energy functional (2.29) can be massaged
into a sum of complete squares like
$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle r\int d{\bf x}^{3}\
\frac{1}{(1+|z^{1}|^{2})^{2}}\bigg{[}\big{|}\partial_{3}z^{1}\mp|m_{\text{M}}|z^{1}\big{|}^{2}+\big{|}\partial_{0}z^{1}\mp
i|m_{\text{E}}|z^{1}\big{|}^{2}$ (3.10) $\displaystyle\hskip
128.0374pt\mp|m_{\text{E}}|i\big{(}\bar{z}_{1}\partial_{0}z^{1}-\partial_{0}\bar{z}_{1}z^{1}\big{)}\pm|m_{\text{M}}|\partial_{3}\big{(}\bar{z}_{n}z_{n}\big{)}\bigg{]}$
$\displaystyle\geq$ $\displaystyle\mp|m_{\text{E}}|Q_{0}\mp
r|m_{\text{M}}|\left.\frac{1}{1+\bar{z}_{n}z^{n}}\right|^{{\bf
x}^{3}=+\infty}_{{\bf x}^{3}=-\infty}=\pm\frac{r|m_{10}|^{2}}{|m_{\text{M}}|}\
,$
where we used $|m_{\text{M}}|^{2}+|m_{\text{E}}|^{2}=|m_{10}|^{2}$. Since
$\displaystyle
Z=-m_{10}Q_{0}+irm_{10}=r\frac{|m_{10}|}{|m_{\text{M}}|}e^{i\varphi_{m_{10}}}\big{(}-|m_{\text{E}}|+i|m_{\text{M}}|\big{)}=\frac{r|m_{10}|^{2}}{|m_{\text{M}}|}e^{i\varphi_{m_{\text{E}}}}\
,$ (3.11)
the solutions are indeed BPS with ${\cal E}=|Z|\ .$
### 3.2 composite flavored kinks and marginal stability
It has been noted previously that the solitonic sector of this $D=2$ QED has
some features reminiscent of certain $D=4$ Seiberg-Witten theory, where the
topological charge and the flavor charges are mapped to the magnetic charge
and the electric charges, respectively. On the other hand, dyonic solitons in
the ${\cal N}=2$ supersymmetric gauge theories in four dimensions are such
that magnetic and electric charges are generically not parallel [7, 6]. This
is in turn related to existence of multi-pronged strings in string theory.
These class of $D=4$ BPS states are useful in that one can study the issue of
marginal stability in weakly-coupled regime of the theory. In this subsection,
we will look for their analog in $D=2$ theory, considering flavored kinks
whose topological and flavor charge are not parallel misaligned, and discuss
their marginal stability briefly. In section 4, their quantum spectrum and
wall-crossing phenomena will be explored in more details.
For simplicity, let us first assume that
$\displaystyle z^{1}=z^{1}({\bf x}^{3},t)\ ,\qquad z^{2}=z^{2}({\bf x}^{3})\
,\qquad z^{m}=0\ \ \text{ for }m\neq 1,2\ .$ (3.12)
For this ansatz, the BPS equation (3.2) can be rewritten as
$\displaystyle\Big{[}\partial_{3}z^{2}+i\tau^{3}\hat{\tau}_{m_{20}}z^{2}\Big{]}\epsilon$
$\displaystyle=$ $\displaystyle 0\ ,$
$\displaystyle\Big{[}\tau^{3}\partial_{3}z^{1}+\partial_{0}z^{1}+i\hat{\tau}_{m_{10}}z^{1}\Big{]}\epsilon$
$\displaystyle=$ $\displaystyle 0\ .$ (3.13)
Guided by the previous example of simple flavored kink, let us rewrite the
second equation into the following form
$\displaystyle\tau^{3}\Big{[}\partial_{3}z^{1}+i\tau^{3}\hat{\tau}_{m_{\text{M}}}z^{1}\Big{]}\epsilon+\Big{[}\partial_{0}z^{1}+i\hat{\tau}_{m_{\text{E}}}z^{1}\Big{]}\epsilon=0\
,\qquad\hat{\tau}_{\text{E}}+\hat{\tau}_{\text{M}}=\hat{\tau}_{m_{10}}\ .$
(3.14)
In order to find out half-BPS solutions, we therefore have to demand three
projectors to commute to each other
$\displaystyle\big{[}\tau^{3}\hat{\tau}_{m_{20}},\tau^{3}\hat{\tau}_{m_{\text{M}}}\big{]}=0\
,\qquad\big{[}\tau^{3}\hat{\tau}_{m_{\text{M}}},\hat{\tau}_{m_{\text{E}}}\big{]}=0\
,\qquad\big{[}\tau^{3}\hat{\tau}_{m_{20}},\hat{\tau}_{m_{\text{E}}}\big{]}=0\
.$ (3.15)
One can again easily parameterize the solutions of the above relations as
$\displaystyle\hat{\tau}_{\text{E}}=\vec{\tau}\cdot{\vec{m}}_{\text{E}}\
,\qquad\hat{\tau}_{\text{M}}=\vec{\tau}\cdot{\vec{m}}_{\text{M}}\ ,$ (3.16)
where vectors $\vec{m}_{\text{E}}$ and $\vec{m}_{\text{M}}$ are depicted in
figure 3.2.
Figure 3.2: (a) Schematic diagram for decomposition of the mass parameter
$\vec{m}_{10}$. Let us denote the relative angle between two mass parameters
$m_{10}$ and $m_{20}$ by $\theta$. By definition, $\vec{m}_{\text{M}}$ is
parallel to $\vec{m}_{20}$. We are considering cases where
$|\vec{m}_{\text{M}}|<|m_{20}|$. (b) Each node denotes the vacuum of the
theory, i.e., grey for $\sigma=m_{0}$, red for $\sigma=m_{1}$ and green for
$\sigma=m_{2}$. The solid lines schematically describe the GLSM $\sigma$
field. It somehow parallels with the four-dimensional picture of pronged
strings where each node represents the D3-brane and solid line denotes the
(p,q)-string. In section 5, the parallel between $D=2$ sigma models and $D=4$
gauge theories will be discussed in more details.
The BPS solutions of interests are
$\displaystyle z^{1}=\text{exp}\Big{[}|m_{\text{M}}|{\bf
x}^{3}+i|m_{\text{E}}|t\Big{]}\ ,\qquad z^{2}=\text{exp}\Big{[}|m_{20}|({\bf
x}^{3}-{\bf x}_{0})\Big{]}\ ,$ (3.17)
after a suitable choice of origin of ${\bf x}^{3}$.
#### flavor charge and marginal stability
For the above solution, the flavor charges (2.31) are
$\displaystyle Q_{0}$ $\displaystyle=$ $\displaystyle-2r|m_{\text{E}}|\
\int_{-\infty}^{+\infty}d{\bf x}^{3}\
\frac{|z^{1}|^{2}}{\big{(}1+|z^{1}|^{2}+|z^{2}|^{2}\big{)}^{2}}\ ,$
$\displaystyle Q_{2}$ $\displaystyle=$ $\displaystyle-2r|m_{\text{E}}|\
\int_{-\infty}^{+\infty}d{\bf x}^{3}\
\frac{|z^{1}|^{2}|z^{2}|^{2}}{\big{(}1+|z^{1}|^{2}+|z^{2}|^{2}\big{)}^{2}}\ ,$
$\displaystyle Q_{1}$ $\displaystyle=$ $\displaystyle-Q_{0}-Q_{2}\ .$ (3.18)
When we place the mass parameter $m_{10}$ on a so-called wall of marginal
stability, as depicted in figure 3.2 (b), the relative distance ${\bf x}_{0}$
diverges such that the $20$-flavored kink decays into two constituent $10$\-
and $21$-flavored kinks. This is an underlying physical reason for the
phenomenon of wall-crossing. At wall-crossing, one can easily show that the
GLSM field $\sigma$ actually turn touches the vacuum $\sigma=m_{1}$, as
described in figure 3.2 (b).
For classical soliton whose flavored charges are not quantized, this can be
viewed backward as a process where the flavor charges are increased until the
kink solution decompose into two. This “maximal” or “critical” flavor charge
can can be read off from the solution as
$\displaystyle Q_{0}^{\text{cr}}\simeq-r\tan\theta\ ,\qquad
Q_{2}^{\text{cr}}\simeq-r\tan\tilde{\theta}\ ,\qquad
Q_{1}^{\text{cr}}\simeq+r\big{(}\tan\theta+\tan\tilde{\theta}\big{)}\ .$
(3.19)
With quantized (and thus fixed) flavor charges, we can use this formula to
determine the critical values of $\theta$ and $\tilde{\theta}$, which in turn
determine the marginal stability wall for breaking this soliton to a simple
flavored 10-kink and a simple flavored 21-kink.
#### central charge
As discussed before, the central charge of the present model can take the
following form
$\displaystyle Z=\sum_{n}m^{n}\big{(}Q_{n}+\tau T_{n}\big{)}\
,\qquad\tau=\frac{\theta}{2\pi}-ir\ .$ (3.20)
For the composite flavored kinks, the central charge $Z_{20}$ can be
decomposed into those of constituent particles, say
$\displaystyle Z_{20}=Z_{10}+Z_{21}\ ,\qquad Z_{10}=-m_{10}Q_{0}+\tau m_{10}\
,\ Z_{21}=+m_{21}Q_{2}+\tau m_{21}\ .$ (3.21)
On the wall of marginal stability where the flavor charges take their critical
values $\vec{Q}^{\text{cr}}$, the central charges of constituent particles
become
$\displaystyle Z_{10}$ $\displaystyle=$ $\displaystyle
m_{10}\big{(}+\tan\theta-i\big{)}\ ,$ $\displaystyle Z_{21}$ $\displaystyle=$
$\displaystyle m_{21}\big{(}-\tan{\tilde{\theta}}-i\big{)}\ .$ (3.22)
Note that, on the wall of marginal stability, the phases of two mass-
parameters satisfy the relations below
$\displaystyle\theta+\tilde{\theta}=\varphi_{m_{21}}-\varphi_{m_{10}}\ ,$
from which one can conclude that phase difference between $Z_{10}$ and
$Z_{21}$ is
$\displaystyle\text{arg}\big{(}Z_{21}\big{)}-\text{arg}\big{(}Z_{10}\big{)}=-\tilde{\theta}-\theta+\varphi_{m_{21}}-\varphi_{m_{10}}=0\
!$ (3.23)
As expected, we find that phases of the two central charges $Z_{10},Z_{21}$
coincides at the marginal stability wall.
## 4 Quantum BPS states and wall-crossing
### 4.1 low energy interactions of kinks
In this section, we construct and count quantum BPS states of topological
kinks with flavor charges, by studying the low energy interactions of simple
kinks. When $m_{i0}$ are all of same phase, each kink carries one complex
bosonic moduli, and their moduli space is naturally Kähler. The holomorphic
coordinates $\zeta^{i}$’s are defined in terms of the soliton solution as
$\displaystyle z^{i}=e^{m_{i0}{\bf x}^{3}}\cdot e^{m_{i0}{\bf
x}^{i}+i\theta^{i}}\equiv e^{m_{i0}{\bf x}^{3}}\zeta^{i}\ .$ (4.1)
The moduli space dynamics is obtained by taking time-dependence of the form
$\zeta^{i}(t)$ with small velocity as usual. The Kähler potential is found by
integrating the field theory kinetic term as [38]
$\displaystyle K\big{(}\bar{\zeta},\zeta\big{)}=\int d{\bf x}^{3}\ {\cal
K}\big{(}\bar{z},z\big{)}\ ,=r\int d{\bf x}^{3}\
\text{log}\bigg{[}1+\sum_{i}e^{2m_{i0}{\bf
x}^{3}}\bar{\zeta}_{i}\zeta^{i}\bigg{]}\ ,$ (4.2)
from which the moduli space metric follows
$\displaystyle g_{i{\bar{j}}}\big{(}\zeta^{i},\bar{\zeta}_{i}\big{)}=r\int
d{\bf x}^{3}\ \Bigg{[}\frac{e^{2m_{i0}{\bf
x}^{3}}\delta_{i}^{j}}{1+\sum_{k}e^{2m_{k0}{\bf
x}^{3}}\bar{\zeta}_{k}\zeta^{k}}-\frac{e^{2(m_{i0}+m_{j0}){\bf
x}^{3}}\bar{\zeta}_{i}\zeta^{j}}{\big{(}1+\sum_{k}e^{2m_{k0}{\bf
x}^{3}}\bar{\zeta}_{k}\zeta^{k}\big{)}^{2}}\Bigg{]}\,.$ (4.3)
Here let us first concentrate on $\mathbb{CP}^{2}$ model, from which we can
read off the indices of all BPS states following an argument of type found in
Ref. [19].
For the moment, let us further assume $m_{20}=2m_{10}$. This causes two
different restrictions on the mass parameters for our purpose. One is the
special ratio between the two absolute values, which is harmless in counting
supersymmetric states. The other, namely alignment of the two phases, pose a
physical restriction to the spectrum. We will shortly abandon the latter.
The moduli space metric is then compactly written as
$\displaystyle g=g_{\text{com}}+g_{\text{rel}}\ ,\qquad
g_{\text{com}}=\frac{r}{4m}\Big{|}d\text{log}\zeta^{2}\Big{|}^{2}\ ,\quad
g_{\text{rel}}=\frac{r}{4m}F({|\zeta^{1}|^{4}/|\zeta^{2}|^{2}})\Bigg{|}d\frac{\zeta^{2}}{{\zeta^{1}}^{2}}\Bigg{|}^{2}\
,$
with
$\displaystyle
F(1/w)=\frac{1}{w(1-4w)}+\frac{2}{(1-4w)^{3/2}}\text{log}\left(\frac{1-\sqrt{1-4w}}{1+\sqrt{1-4w}}\right)\,,$
(4.4)
for $4w<1$ and
$\displaystyle
F(1/w)=-\frac{1}{w(4w-1)}+\frac{4}{(4w-1)^{3/2}}\text{tan}^{-1}\left(\sqrt{4w-1}\right)\,,$
(4.5)
for $4w>1$. This shows that $\zeta^{2}$ plays the role of the center of mass
coordinates, while
$\zeta_{rel}\equiv\zeta^{1}/\sqrt{\zeta^{2}}$
plays the role of the relative coordinate. It is important for a later purpose
to note that in the limit of $|\zeta_{rel}|\to\infty$ $g_{\text{rel}}$ is
reduced simply to
$\displaystyle
g_{\text{rel}}\simeq\frac{r}{m}\biggr{|}d\zeta_{rel}/{\zeta_{rel}}\bigg{|}^{2}\
.$ (4.6)
On the other hand, in the limit of $\zeta_{rel}\to 0$, we have
$\displaystyle g_{\text{rel}}\sim\big{|}d\zeta_{rel}\big{|}^{2}\ .$ (4.7)
so $\zeta_{rel}$ is itself a good coordinate near origin where the two kinks
coincides in real space.
The phases $\theta^{1,2}$ of $\zeta^{1,2}$ are each $2\pi$-periodic and
turning on their (integral) momenta corresponds to turning on $U(1)$ flavor
charges of type $q^{i0}=q^{i}-q^{0}$; $q^{i}$ is the charge of $i$-th diagonal
unbroken favor group. Defining the phase of $\zeta_{cm}$ as $\theta_{cm}$ and
$\zeta_{rel}$ as $\varphi$, we find
$\theta_{cm}=\theta^{2},\qquad\varphi=\theta^{1}-\frac{\theta^{2}}{2}\,,$
(4.8)
and thus
$q^{10}=q,\qquad q^{20}=q_{cm}-\frac{q}{2}\,,$ (4.9)
where $q_{cm}$ and $q$ are conjugate momenta of $\theta_{cm}$ and $\varphi$.
The actual flavor charge for these are
$(q^{0},q^{1},q^{2},\dots)=(q_{cm}-q/2,q,-q_{cm}-q/2,0,0,\dots)\,.$ (4.10)
Note that $q$ is integral while $q_{cm}$ should be integral or half-integral
depending on whether $q$ is even or odd. Such a correlation between relative
and center of mass charges is common, and here due to the identification
$(\theta_{cm},\varphi)\sim(\theta_{cm}+2\pi,\varphi-\pi)\,.$ (4.11)
The total moduli space has the form
${\mathbb{R}}\times\frac{[0,4\pi]\times{\cal M}_{2}}{{\mathbb{Z}_{2}}}\,,$
(4.12)
where the relative moduli space ${\cal M}_{2}$ has a topology of $R^{2}$ and
where ${\mathbb{Z}}_{2}$ acts as (4.11). The center of mass phase and the
quotient action depends on the masses of individual kinks, in general.
Such a charge state, say with $q_{cm}=0$, precisely corresponds to the
classical solution we find in the previous section with $q=Q_{E}$. As we saw
there, however, a flavored kink states of this kind do not appear unless some
of the twisted masses are misaligned in the complex plane. On the other hand,
with such misaligned masses, the composite kink for which we obtained the
moduli dynamics is no longer a solution to the equation of motion unless
$\zeta_{rel}=0$. With $m_{20}=2m_{M}>0$ and
$m_{10}=m_{\text{M}}+im_{\text{E}}$, the relative moduli space makes sense
only if $m_{E}=0$ while the flavored kinks appears only if $m_{E}\neq 0$.
These two issues are in fact tied together. Whenever $m_{E}\neq 0$, unflavored
20-kink configuration costs more energy than the central charge bound and this
extra energy,
$\displaystyle\Delta{\cal E}=r|m_{\text{E}}|^{2}\int d{\bf x}^{3}\
\frac{|z^{1}|^{2}\big{(}1+|z^{2}|^{2}\big{)}}{\big{(}1+|z^{1}|^{2}+|z^{2}|^{2}\big{)}^{2}}\,\;,$
(4.13)
should be interpreted as a potential in the two-kink moduli space dynamics.
With $m_{20}=2m_{\text{M}}\equiv 2m$, we find
$\displaystyle\Delta{\cal
E}=\frac{r|m_{\text{E}}|^{2}}{m}\frac{|\zeta^{2}|^{2}}{|\zeta^{1}|^{4}}F(|\zeta^{1}|^{4}/|z^{2}|^{2})=\frac{m_{E}^{2}}{2}g_{rel}\left(\frac{\partial}{\partial\varphi},\frac{\partial}{\partial\varphi}\right)$
(4.14)
Thus, the bosonic part of relative moduli space dynamics must be modified to
$L_{rel}=\frac{1}{2}(g_{rel})_{\mu\nu}\dot{y}^{\mu}\dot{y}^{\nu}-\frac{1}{2}m_{E}^{2}(g_{rel})_{\mu\nu}K^{\mu}K^{\nu}$
(4.15)
where
$K=\frac{\partial}{\partial\varphi}\ .$ (4.16)
happens to be a holomorphic Killing vector field on the moduli space.
Figure 4.1: The profiles of attractive scalar potential in the moduli space
dynamics of two-kinks system, induced by tension of composite kinks.
This potential energy on the moduli space $\Delta{\cal E}$, depicted in figure
4.1, shows physical separations between simple kinks are no longer a moduli
degree of freedom, since it generates an attractive force between the two
kinks. On the other hand, when the conjugate momentum $q$ of $\varphi$ is
turned on, this induces a repulsive angular momentum barrier between the two
kinks. For finite relative charge $q$, then, one can generically expect
flavored two kinks states with the relative position determined by the balance
of these two forces. The amount of the flavor charge, the mass parameter
$m_{E}$, and the size of $\zeta_{rel}$ are all interrelated, which was shown
implicitly in the classical analysis of section 3.
More generally, we may consider $L0$-kink dynamics, regarded as a collection
of 10-kink, 21-kink,32-kink etc, with
$m_{p0}=m^{(p)}_{M}+im^{(p)}_{E}$
for $p=1,2,...,L-1$ and
$0<m^{(1)}_{M}<m^{(2)}_{M}<\cdots<m^{(L-1)}_{M}<m_{L0}$. The above Lagrangian
generalizes to
$L_{rel}=\frac{1}{2}(g_{rel})_{\mu\nu}\dot{y}^{\mu}\dot{y}^{\nu}-\frac{1}{2}(g_{rel})_{\mu\nu}(m_{E}^{(p)}K_{p}^{\mu})(m_{E}^{(q)}K_{q}^{\nu})\,,$
(4.17)
where$K_{p}$’s are linear combinations of holomorphic Killing vector fields,
induced by flavor $U(1)$ rotations on the soliton.
### 4.2 counting generic BPS states
This form of moduli dynamics with potential has well-known supersymmetric
extensions, provided that $K$ is a Killing vector field. Such massive
nonlinear sigma-model mechanics first appeared with complex supersymmetry in a
work by Freedman and Alvarez-Gaume [39], while the form of relevance for us
was found more recently in the context of BPS dyons of the Seiberg-Witten
theory [10, 40, 4]. In this subsection, let us outline this modified moduli
dynamics and solve for flavored BPS multi-kink states explicitly. See appendix
B for a short review.
Without the potential, the moduli dynamics of the kinks would be the ordinary
nonlinear sigma model where the real fermions match 1-1 with real bosons.
Therefore the supercharge in question can be understood geometrically as the
spinorial Dirac operator on the moduli space,
${\cal Q}=i\Gamma^{I}\nabla_{I}\,,$ (4.18)
where $\nabla_{I}$’s are the covariant derivative with ordinary spin
connection and $\Gamma^{I}$’s the Dirac matrices.
The addition potential energy shifts this supercharge. With general $L$-kink
case, the supercharge is shifted as
$\displaystyle{\cal
Q}=\Gamma^{I}\big{(}i\nabla_{I}+\sum_{p}m_{E}^{(p)}K_{I}^{p}\big{)}\,.$ (4.19)
Taking square of this supercharge, one finds
$\\{{\cal Q},{\cal Q}\\}={\cal H}-{\cal Z}\,,$ (4.20)
where the central charge (to be distinguished from $Z$ of the field theory) is
defined via Lie-derivatives
${\cal Z}=-i\sum_{p}m_{E}^{(p)}{\cal L}_{K^{p}}\,,$ (4.21)
with respect to the Killing vectors, whose action is part of the global
$U(1)^{N}$ flavor rotations acting the kinks.
Since the BPS state must saturate the bound ${\cal H}-{\cal Z}=0$, the search
for BPS states in any given kink sector boils down to finding zero modes of
${\cal Q}$ on the moduli space. This task is in principle very complicated.
However, one can reduce counting problem to that of two-body problems, at
least for the index of such quantum mechanics.
With $m_{E}\neq 0$, the operator ${\cal H}-{\cal Z}$ has a massgap which
separate the continuum from the ground state. Such operators are called
Fredholm operators, for which usual index theorem applies; one simply choose
to scale up the values of $a_{p}$’s, thus increasing the mass gap
indefinitely, while keeping the index unaffected. This localizes the index
computation to the fixed points of the vector fields $K^{a}$’s. Once this
happens, the counting problem becomes that of harmonic oscillators and
factorizes into minimal units with two bosonic and two fermionic coordinates
[19]. The latter is a two-kink problem, so it suffices to count BPS bound
states in a two-kink problem in order to compute index for arbitrary multi-
kink states.
For flavored 20-kink state problem, we have seen that the supercharge reduces
to
$\displaystyle{\cal Q}=\Gamma^{I}\big{(}i\nabla_{I}+m_{E}K_{I}\big{)}\,,$
(4.22)
when $m_{20}=2m$ and $m_{10}=m+im_{E}$ with real $m$ and real $m_{E}$. The
Hamiltonian is nonnegative and has the general form
${\cal
H}=\frac{1}{2}(g_{rel})_{\mu\nu}\left(\pi^{\mu}\pi^{\nu}+m_{E}^{2}K^{\mu}K^{\nu}\right)+\cdots\,,$
(4.23)
where the ellipsis denotes terms involving fermions and $\pi^{\mu}$’s are the
canonical conjugate momenta of the moduli coordinates $y$’s.
With $\zeta_{rel}=e^{\rho+i\varphi}$, the metric for ${\cal M}_{2}$ is
$g_{\text{rel}}=f(\rho)^{2}\left(d\rho^{2}+d\varphi^{2}\right)\,,$ (4.24)
where
$f(\rho)^{2}\equiv\frac{2r}{m}e^{-4\rho}F(e^{4\rho})\,.$ (4.25)
In the relevant orthonormal frame,
$\displaystyle e^{\hat{\rho}}=f(\rho)d\rho\ ,\ \
e^{\hat{\varphi}}=f(\rho)d\varphi\ ,\ \ \omega^{\hat{\varphi}}_{\
\hat{\rho}}=\frac{\partial_{\rho}f(\rho)}{f(\rho)}d\varphi\,,$ (4.26)
the supercharge reduces to
$\displaystyle{\cal
Q}=\Gamma^{\hat{\rho}}\frac{1}{f(\rho)}\bigg{[}\partial_{\rho}+\frac{1}{2}\frac{\partial_{\rho}f(\rho)}{f(\rho)}+i\Gamma^{\hat{\rho}\hat{\varphi}}\Big{(}q-m_{E}f(\rho)^{2}\Big{)}\bigg{]}\
,$ (4.27)
in the charge $q$ sector, that is, when $-i\partial/\partial\varphi\to q$. A
supersymmetric state in this sector has the central charge $qm_{E}$, which
must be saturated by the nonnegative Hamiltonian. Thus, a BPS bound state is
possible only if $qm_{E}\geq 0$.
Denoting two chiral components of $\Psi$ under
$i\Gamma^{\hat{\rho}\hat{\varphi}}$ by $u_{\pm}$, the zero-mode solves
$\displaystyle\partial_{\rho}\big{[}\sqrt{f(\rho)}u_{\pm}\big{]}\pm\Big{(}q-m_{E}f(\rho)^{2}\Big{)}\big{[}\sqrt{f(\rho)}u_{\pm}\big{]}=0\
,$ (4.28)
from which one can obtain one and only one normalizable solution
$\displaystyle
u_{-}=\frac{u_{0}}{\sqrt{f(\rho)}}e^{iq\varphi}\text{exp}\bigg{[}\int^{\rho}_{\rho_{0}}d\rho^{\prime}\Big{(}q-m_{E}f(\rho)^{2}\Big{)}\bigg{]}\,,$
(4.29)
whenever
$\displaystyle 0\leq
q<q_{\text{cr}}=m_{E}f(\infty)^{2}=2r\frac{|m_{\text{E}}|}{m}$ (4.30)
The upper bound comes from the asymptotic normalizability while the lower
bound is required by normalizability at origin ($\rho\to-\infty$),
$\displaystyle u_{-}\simeq
u_{0}\left(\frac{8m}{r\pi}\right)^{\frac{1}{4}}e^{(q-1/2)\rho+iq\varphi}\exp{\bigg{[}-\frac{r|m_{\text{E}}|\pi}{16m}e^{2\rho}\bigg{]}}\
.$ (4.31)
Although $q=0$ wavefunction is mildly singular at origin, it is still
normalizable.#4#4#4Note that the upper bound on the electric charge is
precisely the critical charge obtained from the classical construction of
flavored composite dyons in the section 3 $\displaystyle
q_{cr}=Q_{1}=-(Q_{0}+Q_{2})=r\big{(}\tan\theta+\tan\tilde{\theta}\big{)}\simeq
2r\frac{|m_{\text{E}}|}{m}\ .$ (4.32)
In summary, we found exactly one flavored bound state of the 10-kink and
21-kink for each integral relative charge $q$ from 0 up to
$q_{cr}=2r{|m_{\text{E}}|}/{m}$ and for arbitrary half-integral (odd $q$) or
integral (even $q$) $q_{cm}$. Each of such bound states complete into a BPS
multiplet, thanks to the Goldstino mode. These flavored kinks become unstable
against decay to a pair of simple flavored kinks (10- and 21) when the mass
parameters are changed such that the critical relative charge $q_{cr}$ becomes
smaller or equal to $q$.
Index computation for more general flavored multi-kink states follows
immediately. As argued above, the problem factorizes into several two-body
problems. We consider general flavored $L0$-kink, viewed as bound state of
10-kink, 21-kink, 32-kink, etc. For the $p$-th pair, there is one “relative”
flavor charge $q^{(p)}$. When this charge obeys the conditions,
$0\leq|q^{(p)}|<q^{(p)}_{cr}(m_{i0})\;\;\;\hbox{and}\;\;\;0<m^{(p)}_{E}q^{(p)}\,,$
(4.33)
the above two-body result tells us that the index is unit. The total index for
this $L$-body problem is a product of all such two-body indices, so we learn
finally that
$\Omega=(-1)^{f}\,,$ (4.34)
where $f$ is the $R$-charge of the soliton, provided that (4.33) is satisfied
for all $p=1,2,\dots,L-1$. Otherwise
$\Omega=0\,,$ (4.35)
which we will take as an evidence that the corresponding BPS does not exist.
### 4.3 wall-crossing
After lengthy computations, we finally arrive at wall-crossing issues at large
mass limit of this massive $D=2$ QED. Since $q^{(p)}_{cr}\sim
rm_{E}^{(p)}/m_{L0}$, there is a wall of marginal stability for these flavored
kink at $rm_{E}^{(p)}/m_{L0}\sim q^{(p)}$, details of which would follow once
we compute the metric and the potential on the moduli space. This is a tedious
but straightforward exercise. For us, it suffices to know that these walls of
marginal stability are determined by $r$ and $q$’s, and they extend to the
asymptotic region of large $r$. Across any such a wall, the flavored multi-
kink states break into a pair of smaller flavored multi-kink states, such as
$L0$-kink interpolating between 0 vacuum and $L$ vacuum breaking up into a
flavored $K0$-kink and a flavored $LK$-kink. The latter two objects exist on
both side of this particular wall, so the jump in the spectrum is only for the
bound state, and we have the simple jumping formula
$|\Delta\Omega|=1\,.$ (4.36)
As we saw in section 2, the marginal stability wall is, as always, defined by
the phase alignment of the two central charges of the flavored $K0$-kink and
the flavored $LK$-kink.
In fact, this simple wall-crossing formula is a special case of general wall-
crossing where we are considering bound states of two BPS particles with unit
degeneracy. For this, let us review a result from [21]. They defined a twisted
partition function of $D=2$ field theories as
${\cal F}(\beta;m^{i})=\lim_{l\to\infty}\frac{i\beta}{l}{\rm
tr}(-1)^{R}Re^{-\beta H}\,,$ (4.37)
where $l$ is the regulated size of the spatial line. Alternatively this may be
thought of as expectation value of $R$ when the theory is defined on
$S^{1}\times{\mathbb{R}}^{1}$ with Euclidean signature and periodic boundary
condition on $S^{1}$. A single-particle BPS state, $Z$, contributes
${\cal
F}_{Z}=i\beta(-1)^{f}\int\frac{dp}{\pi}e^{-\beta\sqrt{p^{2}+|Z|^{2}}}=\frac{i(-1)^{f}}{\pi}\int{d\mu}\;\beta|Z|\cosh\mu\;e^{-\beta|Z|\cosh\mu}\,,$
(4.38)
with the rapidity $\mu=\sinh^{-1}(p/|Z|)$. Note that, as we vary the
parameters of the theory, wall-crossing will occur somewhere and this
contribution from single particle BPS states will have to be disappear in a
discontinuous manner.
On the other hand, $\hat{\Omega}$ also receives contributions from many
particle sectors. In particular, with the decomposition of the central charge
as, $Z=Z_{1}+Z_{2}$, the two-particle contribution is of some interests.
Following Cecotti et.al., we also finds that, when the pair of BPS states
$Z_{1,2}$ backscatter,#5#5#5Even in $D=2$ what one means by forward-scattering
and backward-scattering can be somewhat ambiguous when particles can change
species. However, we are mostly interested in situations when two particles in
question are clearly distinct, with different masses for example, so that the
particles are unambiguously labeled. In this context, backscattering means the
sign flip of the relative rapidity before and after. there is a contribution
from the two-particle sector of the type
$\displaystyle{\cal F}_{Z_{1}+Z_{2}}=$ $\displaystyle
d_{{1}}d_{{2}}\frac{i(-1)^{f_{1}+f_{2}}}{4\pi^{2}}\int\int{d\mu_{1}}d\mu_{2}\;\beta\left(|Z_{1}|\cosh\mu_{1}+|Z_{2}|\cosh\mu_{2}\right)e^{-\beta(|Z_{1}|\cosh\mu_{1}+|Z_{1}|\cosh\mu_{1})}$
$\displaystyle\hskip
113.81102pt\times\frac{\partial}{\partial\mu_{1}}\log\left(\sinh(\mu_{2}-\mu_{1}+i\epsilon)/\sinh(\mu_{1}-\mu_{2}+i\epsilon)\right)\,,$
(4.39)
where $2\epsilon={\rm Im}\log(Z_{2}/Z_{1})$ and $d_{{1,2}}$ are the number of
such BPS supermultiplets of central charge $Z_{1,2}$.
Recall that the wall of marginal stability would be at $\epsilon=0$ where the
two central charges line up in the complex plane. Because of the logarithm,
the two-particle expression ${\cal F}_{Z_{1}+Z_{2}}$ also has a discontinuous
imaginary part, and in fact
$\lim_{\epsilon\to 0^{\pm}}{\cal F}_{Z_{1}+Z_{2}}=\pm
d_{{1}}d_{{2}}\frac{{\cal F}_{Z}}{2}\,,$ (4.40)
so that
$\lim_{\epsilon\to 0^{+}}{\cal F}_{Z_{1}+Z_{2}}-\lim_{\epsilon\to 0^{-}}{\cal
F}_{Z_{1}+Z_{2}}=d_{{1}}d_{{2}}{\cal F}_{Z}\,.$ (4.41)
Although individual contributions are discontinuous, the twisted partition
function $\hat{\Omega}$ itself can be continuous provided that $Z$ state
exists as a one-particle BPS state only on the $\epsilon<0$ side. The
continuity of the twisted partition function seems reasonable, and this would
then imply a rather general wall-crossing behavior. Assuming such a continuity
of ${\cal F}$, and since $\Omega(Z_{1,2})=(-1)^{f_{1,2}}d_{{1,2}}$, we the
find the general wall-crossing formula across $Z\rightarrow Z_{1}+Z_{2}$ walls
of marginal stability,
$\Delta\Omega(Z)=\pm\Omega(Z_{1})\Omega(Z_{2})\,.$ (4.42)
For flavored domain walls in the massive ${\mathbf{C}P}^{N}$ theory, we found
$|\Delta\Omega(Z)|=1$, which is easily explained by this wall-crossing
formula, since elementary excitations and simple kinks all have unit index,
$|\Omega|=1$. Building more complicated flavored kinks out of them can only
generate flavored kinks with $|\Omega|=1$ because the wall-crossing formula
(4.42) is so simple.
Wall-crossing in $D=2$ was originally studied by Cecotti and Vafa for purely
topological kinks [22]. For this case, the central charges simplifies as
differences of “canonical coordinates” which in our case are simply the masses
$m^{i}_{D}\simeq\tau m^{i}$, and ${\cal F}$ can be explicitly solved using the
$tt^{*}$ equations [23]. Introduction of flavor charges to the kink should
modify the latter approach somewhat, if not drastically, which will appear
elsewhere.
## 5 $D=4$ ${\cal N}=2$ $SU(N+1)$ with flavors
This two-dimensional QED shows certain features reminiscent of the Seiberg-
Witten theory of four dimensions. This was first noted by Hanany and Hori [26]
who found that the renormalization of the FI parameters $\tau=-ir+\theta/2\pi$
and the asymptotic form of the four-dimensional $\tau_{SW}$ have a close
resemblance. This was taken up later more seriously by Dorey [24] who argued
that the spectrum of this theory is related to that of $SU(N+1)$ Seiberg-
Witten theory with $N+1$ flavors of masses $m^{i}$. The correspondence was
supposed to be precise at the root of the baryonic branch where the vacuum
expectation values of the Seiberg-Witten scalars match with the quark masses.
This conjecture was further extended by Dorey, Hollowood, and Tong [25].
The most compelling reason for this conjecture comes from the exact central
charge (2.34) of the BPS states, obtained from effective superpotential ${\cal
W}(\Sigma)$ after integrating over all chiral multiplets of (2.24) in the
parameter region $e\ll\Lambda_{\sigma}$. In [26], it has been pointed out that
the periods $m_{D}^{i}-m_{D}^{j}$ (2.36) are in perfect matching with those of
the Seiberg-Witten curve at baryonic root of the corresponding $D=4$ ${\cal
N}=2$ $SU(N+1)$ gauge theory with massive $N+1$ quarks.
This latter observation, strictly speaking, tells us only that the set of
central charges in the two theories may coincides, not necessarily the actual
particle content. Nor does not say much about degeneracies of general BPS
states on the two sides. Yet, one may go a bit further and hope that at least
hypermultiplets of Seiberg-Witten theory may match against $D=2$ spectra,
since these can be potentially massless somewhere in the moduli space (or
parameter space for $D=2$) and can be associated with singular structure of
the latter. This is precisely the conjecture of Dorey and his collaborators.
Now that we found a very rich spectrum of flavored kinks, counted their
degeneracy, and found the wall-crossing behavior, let us come back to this
conjecture and see how it lives up to its promise. In generic Seiberg-Witten
theory of rank large than one, typical BPS dyons are not in the
hypermultiplet. Rather they come with large angular momentum which is already
evident in the classical soliton solutions. As we will see below, under the
proposed correspondence between $D=2$ QED and the Seiberg-Witten theory, a
typical flavored kink we found would be mapped to such dyons with high angular
momenta. Let us explore to what extent and in what sense there might be
an“equivalence” of BPS spectra of the two theories.
Recall the central charge of Seiberg-Witten theory,
$Z_{SW}=\vec{a}_{D}\cdot\vec{Q}_{m}+\vec{a}\cdot\vec{Q}_{e}+\sum_{f}m^{f}S_{f}\,.$
(5.1)
In the asymptotic region, we have $\vec{a}_{D}=\tau_{4D}\vec{a}$. For
$SU(N+1)$ theory with $N+1$ fundamental hypermultiplets, we have a special
point where $a^{i}=m^{f=i}$, where the central charge simplifies to
$Z_{SW}=\tau_{4D}\vec{m}\cdot\vec{Q}_{m}+\vec{m}\cdot\vec{Q}_{e}^{adj}+\sum(m^{i}-m^{j})\tilde{Q}_{ij}\,.$
(5.2)
$Q_{e}^{adj}$ denotes electric charges in the adjoint root lattice and the
combined contribution from the matter multiplet
$\tilde{Q}=S+Q_{e}^{matter}$ (5.3)
effectively lives in a $SU(N+1)$ root lattice, which explains why we wrote the
last term in Eq. (5.2) as mass differences. For “unit” magnetic charges, we
have the following mapping from $D=4$ theories,
$\displaystyle Q_{m}$ $\displaystyle\rightarrow$ $\displaystyle T\,,$
$\displaystyle Q_{e}^{adj}+\tilde{Q}$ $\displaystyle\rightarrow$
$\displaystyle Q\,,$ $\displaystyle\tau_{4D}$ $\displaystyle\rightarrow$
$\displaystyle\tau=\frac{\theta}{2\pi}-ir\,,$
$\displaystyle\big{(}\vec{a},\vec{a}_{D}\big{)}$ $\displaystyle\rightarrow$
$\displaystyle\big{(}\vec{m},\vec{m}_{D}\big{)}\,,$ (5.4)
to $D=2$. Note that $Q$’s we found are always in the root lattice which is
achieved on the left hand side by mixing of $SU(N+1)$ color weights and
$SU(N+1)$ favor weights at this special point in the Seiberg-Witten moduli
space This map forms the basis of the conjectured equivalence of BPS spectra
on the two sides. Writing the root system of $SU(N+1)$ as collection of
$e_{i}-e_{j}$ with $0\leq j<i\leq N$, and mapping the $D=2$ central charge to
this, we see that the $ki$-kink corresponds to a magnetic root of
$e_{k}-e_{i}$ whereas $jl$ flavor charge maps to either a
$(e_{l}-e_{j})$-vector meson, or an $e_{j}$ colored quark of $l$-th flavor (or
vice versa).
Finally, the relevant index for $D=4$ ${\cal N}=2$ theory is the second
helicity trace.
$\Omega_{SW}=-2\,{\rm tr}(-1)^{F}J_{3}^{2}\,.$ (5.5)
which counts various BPS multiplets with some weights. Actual values are
$\Omega_{SW}([s]_{spin}\otimes[{\rm
half\;Hypermultiplet}])=(-1)^{2s}(2s+1)\,,$ (5.6)
where the first factor denotes the angular momentum multiplet under the
$SO(3)$ little group, denoted by its spin. For example, a charged vector gives
$-2$.
### 5.1 BPS dyons in pure $SU(N+1)$ and wall-crossing
What are known in literature about such a large-rank Seiberg-Witten theory
come from weak coupling analysis, that is, in the limit of large vacuum
expectation
values.[SeeRef.~\cite[cite]{[\@@bibref{}{Weinberg:2006q}{}{}]}foracomprehensivereview.]
In this regime, the low energy dynamics of monopoles are easily set up and
reliable for general ${\cal N}=2$ theories. In particular, dyons in pure
$SU(N+1)$ theory whose magnetic charge is a (dual) root, as opposed to
arbitrary linear combinations thereof, are completely classified and counted
by Stern and Yi [19]. Let us summarize their result first.
As in $D=2$, an ordering is possible when the adjoint vacuum expectation
values $a^{i}=m^{i}$ almost line up in the complex plane. By overall $U(1)$
rotation, we can take them to be almost real, such that
${\rm Re}\,m^{0}<{\rm Re}\,m^{1}<\cdots{\rm Re}\,m^{N}\,,$ (5.7)
as we did in the previous sections for $D=2$ theory. Without loss of
generality, take dyons of magnetic charge $e_{L}-e_{0}$. With the above
ordering of vev’s, electric charges of dyons are restricted as
$-\left(\frac{k+\sum
n^{(p)}}{2}\right)e_{0}+n^{(1)}e_{1}+n^{(2)}e_{2}+\cdots+n^{(L-1)}e_{L-1}+\left(\frac{k-\sum
n^{(p)}}{2}\right)e_{L}\,,$ (5.8)
with integers $k$ and $n^{(p)}$’s correlated such that the coefficients of
$e_{L,0}$ are also integral. For a BPS dyon of such a charge to exist, the
charges must obey the inequalities
$n^{(1)}\times{\rm Im}\,m^{1}>0,\quad n^{(2)}\times{\rm
Im}\,m^{2}>0,\quad\dots,\quad n^{(L-1)}\times{\rm Im}\,m^{L-1}>0\,,$ (5.9)
and also that the individual electric charge does not exceed the critical
value, which goes as
$|n^{(p)}|<\frac{8\pi^{2}}{e^{3}}\sum_{q}\mu^{-1}_{pq}{\rm Im}\,m^{q}\,,$
(5.10)
where the matrix $\mu$ is a reduced mass matrix defined in terms of ${\rm
Re}\,m^{q}$’s. See Ref. [19, 4]
When these conditions are satisfied, the degeneracy is known [19].
Furthermore, the angular momentum content is also not difficult to find, and
the end result is that the dyon is in the following multiplet,
$\left(\otimes_{p}\left[\frac{|n^{(p)}|-1}{2}\right]\right)\otimes[{\rm
half\,Hypermultiplet}]\,.$ (5.11)
Note that the dyon appears not as a single supermultiplet but rather as a sum
of many supermultiplets with spins up to $(\sum|n^{(p)}|-L+1)/2$. The index
$\Omega_{2}$ of such a dyon is
$\Omega_{SW}=(-1)^{\sum n^{(p)}-L+1}\prod_{p}|n_{(p)}|\,.$ (5.12)
In fact, the computation of BPS bound states for kinks of previous section is
modeled after the computation here. This result was later reproduced by Denef
from more stringy viewpoint [20].
Recently a startling proposal by Kontsevich and Soibelman (KS) [16] was given
for all wall-crossing behavior of $D=4$ ${\cal N}=2$ theories, which seems to
fit all known examples of wall-crossings of these theories. For our purpose,
we will not really need the full power of KS proposal but a corollary for the
so-called semi-primitive cases. One considers BPS bound states of the form
$\gamma(s)=\gamma_{1}+s\gamma_{2}$, where $\gamma$’s denote electromagnetic
charges of the states and we assume that $\gamma_{1,2}$ are primitive, namely
they are not integer multiple of other charge vector. Denoting
$\Omega_{t,s}\equiv\Omega_{SW}(t\gamma_{1}+s\gamma_{2})$, we have the wall-
crossing formula for $\Omega_{1,s}$ as a consequence of KS formula;
$\Omega_{1,0}+\sum_{s\geq
1}\Delta\Omega_{1,s}y^{s}=\Omega_{1,0}\prod_{s^{\prime}\geq
1}\left(1-(-1)^{s^{\prime}\langle\gamma_{1},\gamma_{2}\rangle}y^{s^{\prime}}\right)^{\pm
s^{\prime}\langle\gamma_{1},\gamma_{2}\rangle\Omega_{0,s^{\prime}}}\,.$ (5.13)
The Schwinger product of the charges $\langle\gamma_{1},\gamma_{2}\rangle$
enters the exponents everywhere. When only $\Omega_{t,0}$ and $\Omega_{0,s}$
are nonzero on one side of the wall, this would determine
$\Omega_{1,s}=\Delta\Omega_{1,s}$ completely on the other side of the wall.
This was first suggested by Denef and Moore [41] as a phenomenological
formula. It can also be derived from the KS formula, which shows how to fix
the sign in the last exponent in terms of the sign of the relative phase of
the two central charges $Z_{1}$ and $Z_{2}$ on the side of the wall. We left
the sign ambiguous since we will presently fit this formula to the known
spectrum where the correct sign appears quite obviously.
A further simplification results if we take $\Omega_{0,s}=0$ for all but
$s=1$. As far as we know, in all $D=4$ ${\cal N}=2$ field theories, no non-
primitive charge state has ever been found as one particle states.#6#6#6This
is one notable difference from the supergravity countings, despite many other
similarities. We do not know of an explicit proof of this statement, although
there were examples where this absence was shown in some cases. Then we have,
$\Omega_{1,0}+\sum_{s\geq
1}\Omega_{1,s}y^{s}=\Omega_{1,0}\left(1-(-1)^{\langle\gamma_{1},\gamma_{2}\rangle}y\right)^{\pm\langle\gamma_{1},\gamma_{2}\rangle\Omega_{1,0}}\,.$
(5.14)
Let us see how this fits with the known spectrum of dyons we discussed above.
Take for example the simplest $L=2$. We will write the charge vectors as
$\gamma_{1}=(e_{2}-e_{0};e_{1}-e_{0})$ and $\gamma_{2}=(0;e_{1}-e_{0})$ so
that
$\gamma(s)=(e_{2}-e_{0};(s+1)e_{1}-(s+1)e_{0})\,.$ (5.15)
In terms of dyons whose degeneracy we saw earlier, this corresponds to $L=2$,
$n^{(1)}=k=s+1$. One may be tempted to take $\gamma_{1}=(e_{2}-e_{0};0)$ but
this state is absent in this corner of moduli space and cannot be used as
$\gamma_{1}$.
From the knowledge of $\Omega_{1,0}=1$ and $\Omega_{0,1}=-2$ (because it is a
vector multiplet), we find
$\sum_{n\geq
0}y^{s}\Omega((e_{2}-e_{0};(s+1)e_{1}-(s+1)e_{0}))=\left(1+y\right)^{\pm
2}\,.$ (5.16)
With the negative sign in the exponent (which is something that can be checked
independently), we find
$\Omega_{SW}((e_{2}-e_{0};n^{(1)}e_{1}-n^{(1)}e_{0}))=(-1)^{n^{(1)}-1}n^{(1)}\,,$
(5.17)
after putting $n^{(1)}=s+1$ in the expression. It is clear that this procedure
can be repeated for more complicated dyons with $L>3$ by taking
$\gamma_{2}=(e_{p}-e_{0})$ for all $p=1,\dots,L-1$, which results in
$\Omega_{SW}((e_{L}-e_{0};\sum_{p=1}^{L-1}n^{(p)}e_{p}-\sum_{p=1}^{L-1}n^{(p)}e_{0}))=(-1)^{\sum(n^{(p)}-1)}\prod_{p}n^{(p)}\,,$
(5.18)
in precise accordance with the general index formulae computed in the low
energy dynamics approach. Now that we have some confidence in how wall-
crossing formula reproduce known spectra, let us move on to the flavored
cases.
### 5.2 flavored dyons from wall-crossing formula
The actual dyons whose spectra was proposed to be equivalent to that of $D=2$
theory are those that appear in $SU(N+1)$ Seiberg-Witten theory with $N+1$
fundamental hypermultiplets with masses $m_{i}$’s. Furthermore, the comparison
can be made only at the root of the baryonic branch. Recall that well inside
the baryonic branch, where electric charges are screened, the vector mesons
and massive hypermultiplets together form a long multiplet. Let us denote them
as
$W_{ij},\;\;q_{i}^{(j)},\;\;\tilde{q}_{j}^{(i)},$ (5.19)
where $q$, $\tilde{q}$ are the two chiral multiplets of the hypermultiplets
and are, respectively, in the representations $(N+1,\overline{N+1})$ and
$(\overline{N+1},{N+1})$ under $SU(N+1)_{gauge}\times SU(N+1)_{flavor}$. Given
the map (5), the correspondence between the flavored kinks and $D=4$ dyons are
easy to see.
Let us first consider the simplest nontrivial case with $L=2$. The kinks of
topological and flavor charge#7#7#7Although general flavored kink in this
simple example would be more like
$(T,Q)=(e_{2}-e_{0};k^{\prime}(e_{2}-e_{0})+n(e_{1}-e_{0}))$ for any integer
$k^{\prime}$, we set $k^{\prime}=0$ because it affects neither the marginal
stability nor degeneracy, at least in the leading order in $1/r$. The same
goes for $L0$-kink cases we later consider.
$(T,Q)=(e_{2}-e_{0};n(e_{1}-e_{0}))\,,$ (5.20)
can be mapped to a monopole of charge $(e_{2}-e_{0})$, which we denote by
$M_{20}$, bound with $n$ electrically charged particles which can be either
$W_{10}$ or $\tilde{q}_{0}^{(1)}$. The other quark, $\tilde{q}_{1}^{(0)}$
cannot bind to this monopole since it does not have the right dynamical
charge. Thus we find the following map,
$(T,Q)=(e_{2}-e_{0};n(e_{1}-e_{0}))\leftarrow
M_{20}+nW_{10}\;\;or\;\;M_{20}+(n-1)W_{10}+\tilde{q}_{0}^{(1)}\,.$ (5.21)
The quark cannot bind more than once due to the Pauli exclusion principle,
although this can also be deduced from the wall-crossing formula. See below.
In figuring out degeneracies of these dyons, one crucial information missing
is with what minimal electric charge the dyon actually exist as a
hypermultiplet. In this asymptotic corner and in the pure $SU(N+1)$ case, we
saw that $M_{20}+W_{10}$ is the first such hypermultiplet. With flavors
present, this need not be true anymore. In fact the original conjecture on
equivalence of $D=2$ and $D=4$ spectra relied heavily on the fact that the two
theories share the same spectral curve, suggesting that at least
hypermultiplet content of $D=4$ theory should be faithfully reflected in $D=2$
theories. This leads us to guess that the first hypermultiplet is the purely
magnetic bound state, $M_{20}$, namely a magnetic monopole of charge
$e_{2}-e_{0}$. Our objective here is to reproduce the rest of BPS spectra from
this single assumption.
We may naively repeat the analysis of the pure case. From the wall-crossing
formula, we deduce that
$\displaystyle\sum y^{s}\Omega_{SW}(M_{20}+s\tilde{q}_{0}^{(1)})=1+y\,,$
(5.22)
which, as promised, shows that quarks can bind to a monopole at most once.
Using the wall-crossing formula one more time, we find
$\displaystyle\Omega_{SW}(M_{20}+nW_{10})=(-1)^{n}(n+1)\,,$
$\displaystyle\Omega_{SW}(M_{20}+(n-1)W_{10}+\tilde{q}_{0}^{(1)})=(-1)^{n-1}n\,.$
Note that individual spectra of these dyons are rather nontrivial and come
with high angular momentum content. However, tt is intriguing that the sum of
these two indices is rather simple
$\Omega_{SW}(M_{20}+nW_{10})+\Omega_{SW}(M_{20}+(n-1)W_{10}+\tilde{q}_{0}^{(1)})=(-1)^{n}\,,$
(5.23)
and actually coincides with the $D=2$ counting of flavored kinks, up to a
sign.
More generally, for dyons with magnetic charge $e_{L}-e_{0}$, the relevant
indices are
$\displaystyle\Omega_{SW}(M_{L0}+\sum_{p=1}^{L-1}l^{(p)}W_{p0}+\sum_{p^{\prime}}\tilde{q}_{0}^{(p^{\prime})})=(-1)^{\sum
l^{(p)}}\prod(l^{(p)}+1)\,,$ (5.24)
where $\\{p^{\prime}\\}$ is a subset of $\\{1,2,\dots,L-1\\}$. The map to
$D=2$ flavored kink follows the same rule as before; These dyons are mapped to
flavored $L0$-kinks with $p0$-flavor charges $q^{(p)}$ being equal to either
$n^{(p)}=l^{(p)}$ (when $p\neq p^{\prime}$) or
$n^{(p^{\prime})}=l^{(p^{\prime})}+1$. Summing over the indices for fixed
$q^{(p)}=n^{(p)}$’s, we find
$\displaystyle\sum_{\\{p^{\prime}\\}}\left(\prod_{p=1,p\neq
p^{\prime}}^{L-1}(-1)^{n^{(p)}}(n^{(p)}+1)\prod_{p^{\prime}}(-1)^{n^{(p^{\prime})}-1}(n^{(p^{\prime})})\right)\,,$
(5.25)
which is the same as
$\displaystyle(-1)^{\sum
n^{(p)}}\prod_{p=1}^{L-1}((n^{(p)}+1)-n^{(p)})=(-1)^{\sum n^{(p)}}\,.$ (5.26)
We thus find that under the proposed map (5), $D=2$ indices equal precisely to
the sum of $D=4$ indices of all corresponding dyons, possibly up to a sign.
Note that this cancellation among $D=4$ indices, and the resulting match
against $D=2$ index, is possible only upon very fine-tuned relationships among
these dyons with different quark contents.
## 6 Conclusion
In this paper, we reviewed $D=2$ ${\cal N}=(2,2)$ QED with twisted masses,
with emphasis on BPS spectra in the large mass limit. With $N+1$ chiral matter
fields, one finds BPS kink solutions endowed with $U(1)^{N}$ flavor charges,
whose stability criteria mimics those of $D=4$ ${\cal N}=2$ dyons. In the
classical limit, this also coincides with that of open string web, or
equivalently 1/4 BPS dyons of ${\cal N}=4$ Yang-Mills theory, giving us a
pictorial way to determine the marginal stability walls. We quantized these
solitons to obtain degeneracies, which turned out to be unit for all such
solitons. This result is consistent with general wall-crossing behavior
expected in $D=2$ ${\cal N}=(2,2)$ theories, namely,
$\Delta\Omega(Z_{1}+Z_{2})=\pm\Omega(Z_{1})\Omega(Z_{2})\,.$
Wall-crossing of $D=2$ topological kinks has been studied in depth where
$tt^{*}$ equation makes a prominent appearance. It would be very interesting
to explore further how this could be refined to situations with conserved
charges (such as flavor charges) other than topological charges.
We also compared the spectrum to the conjectured $D=4$ counterpart, i.e., that
of the $SU(N+1)$ Seiberg-Witten theories with $N+1$ massive fundamental
hypermultiplets, at the root of the baryonic branch. Due to the special nature
of this point in the moduli space, where the gauge symmetry and the flavor
symmetry are locked, one type of flavored kink is mapped to several different
kind of dyons with different quark contents. The degeneracies of the latter,
as counted by the second helicity trace, can be complicated and large unlike
those of the kinks. However, this difference is remedied miraculously once we
sum over the indices of all the corresponding dyons with different quark
content, which gives at the end,
$|\Omega|=1=|\sum_{\rm dyons}\Omega_{SW}|\,,$ (6.1)
for each flavored kink that exists on the left hand side and for all the
corresponding dyons on the right hand side.
One cannot really say that spectra of the two theories are equivalent, since
various dyons that are mapped to one type of flavored kink will generally
carry mutually different electric and flavor charges. Note also that in this
map only a subset of $D=4$ BPS dyons participate. A topological charge of a
kink is always mapped to a dual root of the gauge group; since general dyons
may carry more general (magnetic) weight that lie in the dual root lattice,
there must be dyons that do not fit in this correspondence. Given such obvious
differences, the agreement (6.1) is all the more remarkable.
The question of whether and how wall-crossing behaviors and indices of $D=2$
theories and those of $D=4$ theories might be related deserves further study.
$D=4$ wall-crossing received much attention lately, as we noted already, and
some of mathematical tools there have uncanny resemblance to those of $tt^{*}$
equations. Whether such a mathematical resemblance has anything to do with the
present example is unclear, but it still begs for a clarification. In
particular, the partial agreement (6.1) of $D=2$ and $D=4$ indices, despite
vastly different BPS spectra with their different-looking individual indices,
needs to be understood better. In a recent study [42], Gaiotto pointed out a
relationship between surface operators in $D=4$ ${\cal N}=2$ gauge theories
and $D=2$ sigma model whose UV theory is ${\cal N}=(2,2)$ QED with massive
chiral matters. It would be interesting to see what are the implications in
the present context.
Acknowledgement
We thank Kentaro Hori, Yoon Pyo Hong, Seok Kim, Ki-Myeong Lee, Sangmin Lee,
and Jaemo Park for valuable discussions. P.Y. thanks Yukawa Institute of
Theoretical Physics and organizers of the workshop,“Branes, Strings, and Black
Holes” for hospitality. P.Y. is also grateful to the Center for Theoretical
Physics, Seoul National University, where part of this manuscript was written.
P.Y. was supported in part by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MEST) (No. 2005-0049409).
Appendix
## Appendix A Miscellany
#### notations and conventions
One convenient way to describe two-dimensional supersymmetric theories is to
use the four-dimensional superspace formalism of Wess and Bagger followed by a
suitable dimensional reduction: let us compactify the four-dimensional
theories along $x^{1},x^{2}$ directions so that chiral and anti-chiral spinors
$\psi_{\alpha},\bar{\psi}_{\dot{\alpha}}$ reduce to two-dimensional complex
spinors
$\displaystyle\big{(}\psi_{1},\psi_{2}\big{)}\equiv\big{(}\psi_{+},\psi_{-}\big{)},\qquad\big{(}\bar{\psi}_{\dot{1}},\bar{\psi}_{\dot{2}}\big{)}\equiv\big{(}\bar{\psi}_{-},\bar{\psi}_{+}\big{)}\
.$ (A.1)
Here $\pm$ denote the charges under $U(1)_{\text{A}}$ R-symmetry, arising from
the spatial rotation in the compactified dimensions.
In addition to usual superfields with four supercharges such as vector and
chiral superfields, it is well-known that two-dimensional theories allow a so-
called twisted chiral superfield. The twisted chiral superfield $\hat{\Phi}$
is defined as
$\displaystyle\bar{D}_{+}\hat{\Phi}=D_{+}\hat{\Phi}=0\ .$ (A.2)
Defining twisted fermionic coordinates
$\hat{\theta}_{\alpha}=(\theta_{+},-\bar{\theta}_{+})$, the twisted chiral
superfield has the following component field expansion
$\displaystyle\hat{\Phi}=\hat{\phi}+\sqrt{2}\hat{\theta}\hat{\psi}+\hat{\theta}\hat{\theta}\hat{F}\
.$ (A.3)
As a comment, the chiral/twisted chiral-multiplets are indeed in a mirror
pair.
One peculiar example of such twisted chiral superfields is of the form
$\displaystyle\Sigma=D_{+}\bar{D}_{+}V\ ,$ (A.4)
where $V$ denote the vector multiplet. The component field expansions of the
above superfield $\Sigma$ read
$\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\big{(}A_{1}-iA_{2}\big{)}+2i\bar{\theta}_{+}\lambda_{+}+2i\theta_{+}\bar{\lambda}_{+}+2\theta_{+}\bar{\theta}_{+}\big{(}D+iF_{03}\big{)}+\cdots$
(A.5) $\displaystyle=$
$\displaystyle\hat{\phi}+\sqrt{2}\hat{\theta}\hat{\psi}+\hat{\theta}\hat{\theta}\hat{F}$
with
$\displaystyle\hat{\phi}=A_{1}-iA_{2},\qquad\hat{\psi}_{\alpha}=-\sqrt{2}i\big{(}\lambda_{+},\bar{\lambda}_{+}\big{)},\qquad\hat{F}=D+iF_{03}\
.$
Using $\Sigma$, the Fayet-Iliopoulos term and topological $\theta$-term can be
combined as
$\displaystyle{\cal L}_{\text{FI}}+{\cal L}_{\theta}=-\text{Im}\Big{[}\tau\int
d^{2}\hat{\theta}\ \Sigma\Big{]}=rD-\frac{\theta}{2\pi}F_{03}\ ,$ (A.6)
where $\tau=-ir+\frac{\theta}{2\pi}$.
#### covariant derivative
Using the inhomogeneous parameterization $z^{m}$ of $\mathbb{CP}^{N}$, the
GLSM scalar fields can be expressed up to overall $U(1)$ phase as
$\displaystyle\phi^{0}=\sqrt{\frac{r}{1+\bar{z}_{m}z^{m}}}\
,\qquad\phi^{n}=\sqrt{\frac{r}{1+\bar{z}_{m}z^{m}}}z^{n}\ .$ (A.7)
The $U(1)$ gauge field $A_{\mu}$ (2.18) now in turn becomes
$\displaystyle
A_{\mu}=\frac{\bar{z}_{m}\partial_{\mu}z^{m}-\partial_{\mu}\bar{z}_{m}z^{m}}{2i\big{(}1+\bar{z}_{m}z^{m}\big{)}}.$
(A.8)
The various covariant derivatives are then given by
$\displaystyle D_{\mu}\phi^{0}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{r}{1+\bar{z}_{m}z^{m}}}\
\frac{\bar{z}^{m}\partial_{\mu}z_{m}}{1+\bar{z}^{m}z_{m}}\ ,$ $\displaystyle
D_{\mu}\phi^{n}$ $\displaystyle=$
$\displaystyle+\sqrt{\frac{r}{1+\bar{z}_{m}z^{m}}}\
\Big{[}\partial_{\mu}z^{n}-\frac{z^{n}\big{(}\bar{z}^{m}\partial_{\mu}z_{m}\big{)}}{1+\bar{z}^{m}z_{m}}\Big{]}\
.$ (A.9)
Inserting the above results back into the BPS equation (2.41), one can obtain
(3.2).
#### energy for composite kinks
For the composite kink solution, it needs much elaboration to massage the
energy functional to sum of complete squares and boundary terms. Since two
mass parameters $m_{10}$ and $m_{20}$ are now parallel, let us set them to be
purely real without loss of generality. From the general expression of energy
functional (2.29), one can obtain
$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle\int d{\bf x}^{3}\
\frac{r}{\big{(}1+|z^{1}|^{2}+|z^{2}|^{2}\big{)}^{2}}\left[\frac{(1+|z^{1}|^{2}+|z^{2}|^{2})\big{|}\bar{z}_{2}\partial_{3}z_{1}-\bar{z}_{1}\partial_{3}z_{2}\big{|}^{2}}{|z^{1}|^{2}+|z^{2}|^{2}}\right.$
(A.10) $\displaystyle\hskip
99.58464pt+\frac{\big{|}\bar{z}_{1}\partial_{3}z^{1}+\bar{z}_{2}\partial_{3}z^{2}\big{|}^{2}}{|z^{1}|^{2}+|z^{2}|^{2}}+m_{10}^{2}|z^{1}|^{2}+m_{20}^{2}|z^{2}|^{2}+m_{12}^{2}|z^{1}|^{2}|z^{2}|^{2}\Bigg{]}$
$\displaystyle=$ $\displaystyle\int d{\bf x}^{3}\
\frac{r}{\big{(}1+|z^{1}|^{2}+|z^{2}|^{2}\big{)}^{2}}\Bigg{[}\frac{\big{|}\bar{z}_{1}(\partial_{3}z^{1}-m_{10}z^{1})+\bar{z}_{2}(\partial_{3}z^{2}-m_{20}z^{2})\big{|}^{2}}{|z^{1}|^{2}+|z^{2}|^{2}}$
$\displaystyle\hskip
99.58464pt+\frac{\big{|}\bar{z}_{2}(\partial_{3}z^{1}-m_{10}z^{1})-\bar{z}_{1}(\partial_{3}z^{2}-m_{20}z^{2})\big{|}^{2}}{|z^{1}|^{2}+|z^{2}|^{2}}$
$\displaystyle\hskip
99.58464pt+\big{(}m_{10}+m_{12}|z^{2}|^{2}\big{)}\partial_{3}|z^{1}|^{2}+\big{(}m_{20}-m_{12}|z^{1}|^{2}\big{)}\partial_{3}|z^{2}|^{2}\Bigg{]}$
$\displaystyle\geq$
$\displaystyle\left.\frac{r}{1+|z^{1}|^{2}+|z^{2}|^{2}}\Big{(}m_{0}+m_{1}|z^{1}|^{2}+m_{2}|z^{2}|^{2}\Big{)}\right|^{{\bf
x}^{3}=+\infty}_{{\bf x}^{3}=-\infty}=rm_{20}\ .$
It implies that the composite kink saturating the bound has the same mass as
the simple $(20)$-kink solution.
## Appendix B Low energy dynamics of kinks
### B.1 fermion zero mode counting with aligned masses
We begin by clarifying the number of fermionic zero modes in the simple kink
background. Under the $(20)$-kink background, one can naturally define inner
products of $\chi^{1,2}$ as
$\displaystyle\langle\tilde{\chi}^{1}|\chi^{1}\rangle$ $\displaystyle=$
$\displaystyle\int d{\bf x}^{3}\ \frac{1}{1+e^{2|m_{20}|{\bf
x}^{3}}}\tilde{\chi}^{1\dagger}\chi^{1}\ ,$
$\displaystyle\langle\tilde{\chi}^{2}|\chi^{2}\rangle$ $\displaystyle=$
$\displaystyle\int d{\bf x}^{3}\ \frac{1}{\big{(}1+e^{2|m_{20}|{\bf
x}^{3}}\big{)}^{2}}\tilde{\chi}^{2\dagger}\chi^{2}\ ,$ (B.1)
from which the adjoints of ${\cal D}^{1,2}$ becomes
$\displaystyle\langle{\cal
D}^{(1,2)^{\dagger}}\tilde{\chi}^{1,2}|\chi^{1,2}\rangle=\langle\tilde{\chi}^{1,2}|{\cal
D}^{(1,2)}\chi^{1,2}\rangle\ .$ (B.2)
It will be shown that the redefined fermion fields $\eta^{1,2}$
$\displaystyle\eta^{1}=\frac{1}{\sqrt{1+e^{2|m_{20}|{\bf x}^{3}}}}\chi^{1}\
,\qquad\eta^{2}=\frac{1}{1+e^{2|m_{20}|{\bf x}^{3}}}\chi^{2}\ ,$ (B.3)
are convenient to study their zero-modes in manifest normalizability. Then,
one can rewrite the fermion quadratic pieces in the sigma-model Lagrangian as
$\displaystyle\langle\chi^{1,2}|{\cal D}^{(1,2)}\chi^{1,2}\rangle=\int d{\bf
x}^{3}\ \eta^{\dagger}_{1,2}D^{(1,2)}\eta^{1,2}\ .$ (B.4)
One finds that the equations of motions for $\eta^{1,2}$ can be simplified as
$\displaystyle\omega\eta^{1}$ $\displaystyle\equiv$ $\displaystyle
D^{(1)}\eta^{1}=\Bigg{[}i\tau^{3}\partial_{3}-\hat{\tau}_{m_{10}}+\hat{\tau}_{m_{20}}\bigg{(}\frac{|z^{2}|^{2}}{1+|z^{2}|^{2}}\bigg{)}\Bigg{]}\eta^{1}$
$\displaystyle\omega\eta^{2}$ $\displaystyle\equiv$ $\displaystyle
D^{(2)}\eta^{2}=\Bigg{[}i\tau^{3}\partial_{3}-\hat{\tau}_{m_{20}}\bigg{(}1-\frac{2|z^{2}|^{2}}{1+|z^{2}|^{2}}\bigg{)}\Bigg{]}\eta^{2}\
.$ (B.5)
Inserting the explicit configuration of the kink solution, the above
differential operators can be reduced to
$\displaystyle D^{(1)}$ $\displaystyle=$ $\displaystyle
i\tau^{3}\partial_{3}-\hat{\tau}_{m_{10}}+\hat{\tau}_{m_{20}}f({\bf
x}^{3})\big{)}\ ,$ $\displaystyle D^{(2)}$ $\displaystyle=$ $\displaystyle
i\tau^{3}\partial_{3}-\hat{\tau}_{m_{20}}\big{(}1-2f({\bf x}^{3})\big{)}$
(B.6)
with
$\displaystyle f({\bf x}^{3})=\frac{e^{2|m_{20}|{\bf
x}^{3}}}{1+e^{2|m_{20}|{\bf x}^{3}}}\ ,\qquad\partial_{3}f=2|m_{20}|f(1-f)\geq
0\ .$ (B.7)
Figure B.1: The profiles of the effective potentials (a) $V^{(1)}_{\pm}({\bf
x}^{3})$ and (b) $V^{(2)}({\bf x}^{3})$ in the case of $|m_{20}|>|m_{10}|$.
Assuming the alignment of phases of $m_{10}$ and $m_{20}$, it is then easy to
show that, for $\eta^{1}$,
$\displaystyle D^{(1)\dagger}D^{(1)}=D^{(1)}D^{(1)\dagger}=$
$\displaystyle\bigg{[}-\partial_{3}^{2}+|m_{10}|^{2}-2|m_{10}||m_{20}|f+|m_{20}|^{2}f^{2}\bigg{]}{\bf
1}_{4}+i\partial_{3}f\tau^{3}\hat{\tau}_{m_{20}}$ $\displaystyle\equiv$
$\displaystyle-\partial_{3}^{2}+V^{(1)}_{\pm}({\bf x}^{3})\ ,$ (B.8)
where the effective potentials are given by
$\displaystyle V^{(1)}_{\pm}=\big{(}|m_{10}|-|m_{20}|f\big{)}^{2}\pm
2|m_{20}|^{2}f(1-f)\ ,\qquad i\tau^{3}\hat{\tau}_{m_{20}}\doteq\pm|m_{20}|\ .$
(B.9)
By definition, $V^{(1)}_{+}\geq V^{(1)}_{-}$ always. The profile of the
effective potentials $V^{(1)}_{\pm}({\bf x}^{3})$ is depicted in figure B.1
(a), where you can see their extremum and asymptotic values are given by
$\displaystyle V^{(1)}_{+\text{ min}}$ $\displaystyle=$
$\displaystyle\big{(}|m_{20}|^{2}-|m_{10}|\big{)}^{2}+|m_{10}|^{2}\ ,\ \
\left\\{\begin{array}[]{l}V^{(1)}_{+}({\bf x}^{3}\to-\infty)=|m_{10}|^{2}\\\
V^{(1)}_{+}({\bf
x}^{3}\to+\infty)=\big{(}|m_{20}|-|m_{10}|\big{)}^{2}\end{array}\right.$
(B.12) $\displaystyle V^{(1)}_{-\text{ max}}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}|m_{20}|\big{(}|m_{20}|-|m_{10}|\big{)}-\frac{1}{3}|m_{20}|^{2}\
,\ \left\\{\begin{array}[]{l}V^{(1)}_{+}({\bf
x}^{3}\to-\infty)=|m_{10}|^{2}\\\ V^{(1)}_{+}({\bf
x}^{3}\to+\infty)=\big{(}|m_{20}|-|m_{10}|\big{)}^{2}\end{array}\right.$
(B.15)
from which one can show that $D^{(1)}D^{(1)\dagger}$, $D^{(1)\dagger}D^{(1)}$
with $i\tau^{3}\hat{\tau}_{m_{20}}=+|m_{20}|$ becomes manifestly positive
definite. It implies that there is no normalizable zero-modes for the above
chirality. For another chirality $i\tau^{3}\hat{\tau}_{m_{20}}=-|m_{20}|$, one
can have a normalizable zero-mode $\eta^{(1)}_{0}$
$\displaystyle\eta^{1}_{0}=\frac{e^{|m_{10}|{\bf
x}^{3}}}{\sqrt{1+e^{2|m_{20}|{\bf x}^{3}}}}\epsilon_{0}\ ,\
i\tau^{3}\hat{\tau}_{m_{20}}\epsilon_{0}=-|m_{20}|\epsilon_{0}\ \ \Rightarrow\
\ \chi_{0}^{1}=e^{|m_{10}|{\bf x}^{3}}\ ,$ (B.16)
provided that $|m_{20}|\geq|m_{10}|$.
Let us now in turn consider the Dirac operator for $\eta^{2}$. One can again
easily show that
$\displaystyle D^{(2)\dagger}D^{(2)}=D^{(2)}D^{(2)\dagger}=$
$\displaystyle\bigg{[}-\partial_{3}^{2}+|m_{20}|^{2}\big{(}1-2f\big{)}^{2}\bigg{]}{\bf
1}_{4}+\bigg{[}2|m_{20}|f(1-f)\bigg{]}i\tau^{3}\hat{\tau}_{m_{20}}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lcc}-\partial_{3}^{2}+|m_{20}|^{2}&\text{for}&i\tau^{3}\hat{\tau}_{m_{20}}\doteq+|m_{20}|\\\
-\partial_{3}^{2}+|m_{20}|^{2}\big{(}1-8f(1-f)\big{)}&\text{for}&i\tau^{3}\hat{\tau}_{m_{20}}\doteq-|m_{20}|\end{array}\right.$
(B.19) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lc}-\partial_{3}^{2}+|m_{20}|^{2}\geq
0&\\\ -\partial_{3}^{2}+V^{(2)}({\bf x}^{3})&\end{array}\right.\ ,$ (B.22)
which implies that there is no nomarlizable zero-modes for the former
chirality $i\tau^{3}\hat{\tau}_{m_{20}}=|m_{20}|$. On the other hand, the
effective potential $V^{(2)}({\bf x}^{3})$, depicted in figure B.1 (b), has
its minimum and asymptotic values like
$\displaystyle V^{(2)}_{\text{min}}=-|m_{20}|^{2}\ ,\ \ V^{(2)}\ \to\
|m_{20}|^{2}\text{ as }{\bf x}^{3}\to\pm\infty\ ,$ (B.23)
from which one can expect a normalizable zero-mode $\eta^{2}_{0}$ of chirality
$i\tau^{3}\hat{\tau}_{m_{20}}=-|m_{20}|$ whose the explicit expression becomes
$\displaystyle\eta^{2}_{0}=\frac{1}{\text{cosh}\Big{[}|m_{20}|{\bf
x}^{3}\Big{]}}\epsilon_{0}\ \ \Rightarrow\ \ \chi^{2}_{0}=e^{|m_{20}|{\bf
x}^{3}}\epsilon_{0}\ .$ (B.24)
### B.2 the two-kink moduli space metric
As discussed in literatures, a general kink can decompose into several
fundamental kinks. Each of fundamental kink has two obvious collective
coordinates, position and phase. It implies that the moduli space of kinks is
therefore toric Kähler manifold.
For computational simplicity and concreteness, let us consider the present
model with $m_{20}=2m_{10}\equiv 2m$. From (4.3), the metric components can
read
$\displaystyle g_{1\bar{1}}$ $\displaystyle=$ $\displaystyle
4\frac{|\zeta^{2}|^{2}}{|\zeta^{1}|^{6}}\Bigg{[}\frac{r}{4m}F\big{(}|\zeta^{1}|^{4}/|\zeta^{2}|^{2}\big{)}\Bigg{]}$
$\displaystyle g_{2\bar{2}}$ $\displaystyle=$
$\displaystyle\frac{r}{4m}\frac{1}{|\zeta^{2}|^{2}}+\frac{|\zeta^{1}|^{2}}{|\zeta^{1}|^{6}}\Bigg{[}\frac{r}{4m}F\big{(}|\zeta^{1}|^{4}/|\zeta^{2}|^{2}\big{)}\Bigg{]}$
$\displaystyle g_{1\bar{2}}$ $\displaystyle=$
$\displaystyle-2\frac{\bar{\zeta}_{1}\zeta^{2}}{|\zeta^{1}|^{6}}\Bigg{[}\frac{r}{4m}F\big{(}|\zeta^{1}|^{4}/|\zeta^{2}|^{2}\big{)}\Bigg{]}\
,$ (B.25)
where $F(x)$ is defined in Eqs. (4.4,4.5). Bosonic kinetic terms of
interacting multi-kinks therefore take the following form
$\displaystyle L^{\text{kin}}_{\text{boson}}=L_{\text{com}}+L_{\text{rel}}\ ,$
where
$\displaystyle
L_{\text{com}}=\frac{r}{4m}\Big{|}d\text{log}\zeta^{2}\Big{|}^{2}\ ,\qquad
L_{\text{rel}}=\frac{r}{4m}F\big{(}|\zeta^{1}|^{4}/|\zeta^{2}|^{2}\big{)}\Bigg{|}d\frac{\zeta^{2}}{{\zeta^{1}}^{2}}\Bigg{|}^{2}\
.$ (B.26)
In the limit of $\frac{|\zeta^{2}|}{|\zeta^{1}|^{2}}\to\infty$,
$L_{\text{rel}}$ is asymptotic to
$\displaystyle
L_{\text{rel}}\simeq\frac{r}{4m}\cdot\frac{\pi}{4}\bigg{|}d\frac{\zeta^{1}}{\sqrt{\zeta^{2}}}\bigg{|}^{2}\
.$ (B.27)
Note that the moduli space metric of interacting two-kinks (, or multi kinks
in four-dimensional ${\cal N}=2$ SQED) has been explored by David Tong [38],
although our result appears slightly different from his.
### B.3 supersymmetric low energy dynamics with potential
For completeness, we present in this section a short review on supersymmetric
nonlinear sigma-model quantum mechanics with potential. Let us begin by the
Lagrangian which takes the following form
$\displaystyle{\cal
L}_{\text{kin}}=\frac{1}{2}g_{IJ}\bigg{[}\partial_{0}\Phi^{I}\partial_{0}\Phi^{J}+i\Psi^{I}D_{0}\Psi^{J}\bigg{]}\
,$ (B.28)
where the covariant derivatives are
$\displaystyle
D_{0}\Psi^{I}=\partial_{0}\Psi^{I}+\partial_{0}\Phi^{K}\Gamma^{I}_{JK}\Psi^{K}\
.$ (B.29)
and the fermions are real. Since the kink solitons possess equal number of
bosonic and fermionic collective coordinate, this quantum mechanics is
appropriate for the
The above Lagrangian has a real supersymmetry whose Nöther charge is given by
$\displaystyle{\cal Q}=i\sqrt{2}g_{IJ}\Psi^{I}\partial_{0}\Phi^{I}\ .$ (B.30)
Once we quantize the system. the real fermion fields $\Psi^{I}$ cab be
represented as gamma matrices $\Gamma^{I}$
$\displaystyle\big{\\{}\Psi^{I},\Psi^{J}\big{\\}}=\delta^{IJ}\ \to\
\Psi^{I}\doteq\frac{1}{\sqrt{2}}\Gamma^{I}\ .$ (B.31)
It implies that the supercharge can be represented on the Hilbert space as the
spinorial Dirac operator
$\displaystyle{\cal Q}\doteq
i\Gamma^{I}\nabla_{I}=i\Gamma^{I}\Big{(}\partial_{I}+\frac{1}{4}{\omega_{I}}_{AB}\Gamma^{AB}\Big{)}\
.$ (B.32)
When the geometry has a restricted holonomy, the supersymmetry is enhanced. In
particular, for a Kähler space such as our multi-kink moduli space, the
supersymmetry is enhanced to ${\cal N}=2$.
One may introduce to the above model a supersymmetry-preserving deformation of
the form
$\displaystyle{\cal
L}_{\text{def}}=-\frac{1}{2}\Big{[}g_{IJ}G^{I}G^{J}+i\nabla_{I}G_{J}\Psi^{I}\Psi^{J}\Big{]}\
.$ (B.33)
One can show that the total Lagrangian ${\cal L}={\cal L}_{\text{kin}}+{\cal
L}_{\text{def}}$ is invariant under a supersymmetry whose Nöther charge is
deformed as
$\displaystyle{\cal
Q}=\sqrt{2}\Psi^{I}\Big{[}ig_{IJ}\dot{\Phi}^{J}+G_{I}\Big{]}\ .$ (B.34)
After canonical quantization, demanding the Jacobi identity for the deformed
supercharge tells us that $G^{I}$ in fact turns out to be a Killing vector
field
$\displaystyle\big{[}{\cal Q},\big{\\{}{\cal Q},{\cal Q}\big{\\}}\big{]}=0\
\to\ \nabla_{I}G_{J}+\nabla_{J}G_{I}=0\ .$ (B.35)
When the manifold is Kähler with the complex structure $J$, ${\cal N}=2$
supersymmetry remain consistent with introduction of $G$ provided that $G$ is
not only Killing but also holomorphic,
${\cal L}_{G}J=0\,.$ (B.36)
One can split $\big{\\{}{\cal Q},{\cal Q}\big{\\}}$ into two conserved
quantities as
$\displaystyle\big{\\{}{\cal Q},{\cal Q}\big{\\}}=4\big{(}{\cal H}-{\cal
Z}\big{)}\ ,$ (B.37)
where ${\cal H}$ and ${\cal Z}$ denote Hamiltonian and central charge
$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{IJ}\Big{[}\partial_{0}\Phi^{I}\partial_{0}\Phi^{J}+G^{I}G^{J}\Big{]}+\frac{i}{2}\nabla_{I}G_{J}\Psi^{I}\Psi^{J}\
,$ $\displaystyle{\cal Z}$ $\displaystyle=$ $\displaystyle
G_{I}\partial_{0}\Phi^{I}-\frac{i}{2}\nabla_{I}G_{J}\Psi^{I}\Psi^{J}\ .$
(B.38)
Note here that the positive energy BPS states of real supersymmetry then
preserve all the supercharges of the moduli space dynamics. As a final
comment, the deformed supercharge now in turn can be represented as
$\displaystyle{\cal Q}\doteq\Gamma^{I}\big{(}i\nabla_{I}+G_{I}\big{)}$ (B.39)
since we may view the wavefunctions as sections of the spinor bundle over the
moduli space.
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* [42] D. Gaiotto, “Surface Operators in N=2 4d Gauge Theories,” arXiv:0911.1316 [hep-th].
|
arxiv-papers
| 2009-11-25T01:28:04 |
2024-09-04T02:49:06.669418
|
{
"license": "Public Domain",
"authors": "Sungjay Lee, Piljin Yi",
"submitter": "Sungjay Lee",
"url": "https://arxiv.org/abs/0911.4726"
}
|
0911.4758
|
# Magetic and Superconducting Properties of Single Crystals of Fe1+δTe1-xSex
System
Jinhu Yang yangjinhu@kuchem.kyoto-u.ac.jp Department of chemistry, Graduate
School of Science, Kyoto University, 606 - 8502, Japan Mami Matsui
Department of chemistry, Graduate School of Science, Kyoto University, 606 -
8502, Japan Masatomo Kawa Department of chemistry, Graduate School of
Science, Kyoto University, 606 - 8502, Japan Hiroto Ohta Department of
chemistry, Graduate School of Science, Kyoto University, 606 - 8502, Japan
Chishiro Michioka Department of chemistry, Graduate School of Science, Kyoto
University, 606 - 8502, Japan Chiheng Dong Department of physics, Graduate
School of Science, Zhejiang University, Hangzhou 310027,China Hangdong Wang
Department of physics, Graduate School of Science, Zhejiang University,
Hangzhou 310027,China Huiqiu Yuan Department of physics, Graduate School of
Science, Zhejiang University, Hangzhou 310027,China Minghu Fang Department
of chemistry, Graduate School of Science, Kyoto University, 606 - 8502, Japan
Department of physics, Graduate School of Science, Zhejiang University,
Hangzhou 310027,China Kazuyoshi Yoshimura kyhv@kuchem.kyoto-u.ac.jp
Department of chemistry, Graduate School of Science, Kyoto University, 606 -
8502, Japan
###### Abstract
The spin-fluctuation effect in the Se-substituted system Fe1+δTe1-xSex ($x$ =
0, 0.05, 0.12, 0.20, 0.28, 0.33, 0.45, 0.48 and 1.00; $0<\delta<0.12$) has
been studied by the measurements of the X-ray diffraction, the magnetic
susceptibility under high magnetic fields and the electrical resistivity under
magnetic fields up to 14 T. The samples with $x$ = 0.05, 0.12, 0.20, 0.28,
0.33, 0.45 and 0.48 show superconducting transition temperatures in the ranger
of 10 K$\sim$14 K. We obtained their intrinsic susceptibilities by the Honda-
Owen method. A nearly linear-in-$T$ behavior in magnetic susceptibility of
superconducting samples was observed, indicating the antiferromagnetic spin
fluctuations have a strong link with the superconductivity in this series. The
upper critical field $\mu_{0}H_{c2}^{orb}$ for $T\to$ 0 was estimated to
exceed the Pauli paramagnetic limit. The Kadowaki-Woods and Wilson ratios
indicate that electrons are strongly correlated in this system. Furthermore,
the superconducting coherence length and the electron mean free path were also
discussed. These superconducting parameters indicate that the
superconductivity in the Fe1+δTe1-xSex system is unconventional.
###### pacs:
74.70.-b,75.50.Bb,74.62.Dh
††preprint: APS/123-QED
## I INTRODUCTION
Shortly after the discovery of the iron-based oxypnictide superconductor
LaFeAsO1-xFx with $T_{c}$ of 26 K, Hosono another family of the iron-based
chalcogenide superconductor $\alpha$-FeSe with $T_{c}$ of about 8.5 K was
reported by Wu’s group. Wu Later on, Fang’s group enhanced the $T_{c}$ up to
14 K by substituting Se for Te in the FeTe1-xSex system. Fang Interestingly,
although both FeTe and FeS are not superconductors under ambient pressure, the
superconductivity can be induced by Se doping in FeTe1-xSex as well as S
doping in the FeTe1-xSx system. Fang ; STe ; Mizuguchi While there is no sign
of superconductivity in pure FeTe under high pressure in contrast to that the
superconducting transition temperature was enhanced to 37 K, just below the
McMillan limit in FeSe under high pressure. 37K Superconducting
$\alpha$-Fe1.01Se belongs to the tetragonal symmetry system at room
temperature and undergoes a structural transition to an orthorhombic phase at
90 K, while the non-superconducting Fe1.03Se does not. McQueen FeSe has a
simpler structure by stacking only the conducting Fe2Se2 layers in contrast to
the Fe-As based superconductors having both the conducting Fe2As2 layers and
the blocking R2O2 layers (R = rare earth elements). FeTe has the same
structure as FeSe but with a rather complex magnetic structure. For example,
the stoichiometric sample FeTe has an commensurate antiferromagetic (AF)
ordering at low temperatures after suffering a structural phase transition,
while samples with excess iron Fe1+δTe has an incommensurate AF ordering. Bao
According to the result of density functional calculations, the spin density
wave (SDW) is more stable in FeTe than that in FeSe. Subedi Therefore, the
doped sample FeTe is expected to have higher $T_{c}$. This was indeed observed
in S or Te substituted systems for Se in FeSe. SSe The iso-valent
substitution does not directly introduce extra carriers but may change the
topology of the Fermi surface. Fang Recently, a spin fluctuation spectrum and
a spin gap behavior were observed by neutron scattering. Bao ; nuclear In
fact, the NMR results indicated that the AF spin fluctuations were enhanced
greatly toward $T_{c}$, indicating the importance of AF spin fluctuations for
the superconducting mechanism in FeSe. Cava
As for the sample preparations, the iron-based superconductors, as previously
reported, however, usually contain a very small amount of magnetic impurities,
e.g., Fe7Se8 and Fe3O4. Wu ; Fang ; Li ; Mizuguchi ; McQueen ; Kazumasa ;
Williams ; Taen Therefore, in many cases the Verwey-phase like transition
happens around 120 K due to the existence of the magnetic impurity Fe3O4 which
causes a peak in the magnetic susceptibility. Fe3O4 To elucidate the
superconducting mechanism and its relation with the AF spin fluctuations, it
is vitally important to obtain the intrinsic susceptibility. In this study, we
successfully synthesized the single crystals of Fe1+δTe1-xSex and measured
their magnetic susceptibilities under high magnetic fields and obtained the
intrinsic susceptibilities by using the Honda-Owen plot. Honda We have found
that the magnetic susceptibility of Fe1+δTe1-xSex ($0.12\leqslant x\leqslant
1.00$) decreases with decreasing temperature from 300 K to 20 K, similar to
the results in high temperature cuprate superconductor (La1-xSrx)2CuO4, or Fe-
As based superconductors. Takagi ; Klingeler In addition, we conducted
electrical resistivity measurements of two single crystal samples with very
close composition $x\sim$ 0.3, both of which show the superconductivity at
$T_{c}$ $\sim$ 14 K under magnetic fields up to 14 T in order to estimate the
upper critical field $\mu_{0}H_{c2}^{orb}$ and the coherence length $\xi$. As
a result, the upper critical field is found to be much larger than the Pauli
limit, and the initial slope near $T_{c}$ is comparable with those of Fe-As
based superconductors. Kohama ; Terashima Although Fe(Te-Se) is a layered
superconductor, both the upper critical field and the initial slope near
$T_{c}$ show weak anisotropies. In order to investigate the electron
correlation strength, we have estimated Kadowaki-woods and Wilson ratios which
indicate a strongly correlated electrons picture. The superconducting
coherence length and the electron mean free path are also discussed, leading
to the fact that Fe1+δTe1-xSex is a clean superconductor.
## II EXPERIMENTAL
The high-quality single crystals of Fe1+δTe1-xSex ( $x$ = 0, 0.05, 0.12, 0.20,
0.28, 0.33, 0.45, 0.48 and 1.00; $0<\delta<0.12$) were prepared from Fe
powders (4N purity), Te powders (4N purity) and Se powders (5N purity).
Stoichiometric quantities of about 3g-mixtures were loaded into a small quartz
tube. This small tube was then sealed into second evacuated quartz tube, and
placed in a furnace at room temperature. The temperature was slowly ramped up
to 920∘C over for 36 hours and then held at that temperature for another 12
hours in order to obtain sample homogeneities. Then, the temperature was
reduced to 400∘C over for140 hours. On the other hand, the polycrystalline of
Fe1+δSe was synthesized by previously reported solid state reaction method. Wu
The obtained single crystal samples were ground into powders for measuring
powder X-ray diffraction (XRD) with Cu $K_{\alpha}$ radiation. The detailed
structural parameters were analyzed by Rietveld refinements. The compositions
of the single crystals were analyzed using SEM (JED-2300, JEOL) equipped with
an Energy Dispersive X-Ray (EDX) spectrometer. The DC magnetic measurements
were performed by using a Superconducting Quantum Interference Device (SQUID,
Quantum Design Magnetometer). For the observations of the superconducting
transitions, both the zero-field cooling (ZFC) and field cooling (FC )
measurements were performed under the magnetic field of 20 Oe. The temperature
dependence of resistivity was measured using a standard dc four-probe method
under dc magnetic fields up to 14 T with a Physics Property Measurements
System (PPMS, Quantum Design Magnetometer). The current direction was parallel
to the a- axis of the single crystal sample.
## III RESULTS AND DISCUSSION
### III.1 XRD and EDX Spectroscopy
The obtained single crystal has the layered planes held together by Van der
Waals force only, and thus the crystal can easily be cleaved. X-ray
diffraction pattern of a typical single crystal Fe1.12Te0.72Se0.28 measured
with the scattering vector perpendicular to the cleaved surface was shown in
Fig. 1 (a) and the image of the single crystal is in the inset. Only (00$l$)
reflections appear, indicating that the c-axis is perpendicular to the cleaved
surface. In order to get more structural information from the XRD pattern, the
singe crystal were ground into powders for powder XRD measurements. Figure 1
(b) shows XRD patterns of the selected samples of Fe1+δTe1-xSex( $x$ = 0,
0.05, 0.12, 0.20 and 0.33). The real compositions were analyzed by EDX
measurements as listed in Table I. The EDX spectroscopy results show that
there is a slight excess amount of Fe existing in each sample, and
furthermore, the Se content has a smaller ratio than the nominal one. All the
peaks are well indexed based on a tetragonal cell with the space group of
$P$4/$nmm$, except for a small amount of the magnetic impurity phase of
Fe7Se8-type, indicating that the samples are almost in a single phase. The
impurity phase exits in the surface or in the inter-layers of the single
crystal. It should be pointed out that there is a very small amount of Fe3O4
impurity in each sample, which cannot however be probed in the XRD patterns
but causes a peak in temperature dependence of magnetic susceptibility at
about 120 K due to the Verwey phase transition. Fe3O4 With increasing Se
doping level, the a-axis decreases slightly, while the c-axis shrinks
remarkably, which makes the (001) and (200) diffraction peaks shift to higher
angles monotonously, shown as enlarged views in the inset. The cell volume is
consequently decreased by substituting Te for Se from 92 Å3 to 78 Å3,
indicating that the samples are in a solid solution in which Se enters the
lattice as Fe1+δTe1-xSex successfully.
Figure 1: X-ray diffraction patterns of Fe1+δTe1-xSex. (a) A typical single
crystal XRD pattern for sample Fe1.12Te0.72Se0.28 as well as the single
crystal image in the inset. (b) Powder XRD patterns by using samples of ground
single crystal; the peaks marked by * are Fe7Se8 impurity phase. The enlarged
view of the (001) and (200) peaks and the lattice constants as functions of Se
content $x$ were shown in the inset.
Figure 2: (a) Isothermal magnetization ($M$) with magnetic field ($H$) in the
temperature range 60 K $\leqslant T\leqslant$ 200 K with the step of 10 K,
$M_{s}$ is saturation moment of the impurities. (b) Temperature dependence of
the intrinsic susceptibility for Fe1+δTe1-xSex ($x$ = 0, 0.05, 0.12, 0.20,
0.28 and 0.33) with $H$//c. The intrinsic susceptibility of Fe1.12Te was also
measured under $H$//a. (c) The Fe(I) site contribution to the magnetic
susceptibility: $\chi_{Fe(I)}$ as a function of temperature for
superconducting samples with $x$ = 0.12, 0.20, 0.28, 0.33, 0.45 and 0.48 with
$H$//c as well as the sample with $x$ = 0.28 with $H$//a. The anisotropy
susceptibility in this series is very weak: the susceptibility in sample of
$x$ = 0.28 showed only weak dependence on the magnetic direction of $H$//c or
$H$//a.
### III.2 Magnetic Susceptibility
Since the presence of a small amount of ferromagnetic impurities which have a
profound effect on the low-field magnetization as shown in Fig. 2 (a) for
sample Fe1+δTe. A linear-in-$H$ term in high magnetic fields magnetization
appears in each curve for various temperatures; if we extrapolate the data
from high magnetic fields to $H$ = 0, all the extrapolation ends into almost
the same point, $M_{s}$, indicating the saturation magnitude moment of the
impurity. According to the Honda-Owen plot, by extrapolating the measured
susceptibility $M/H$ = $\chi$ \+ $C_{s}$$M_{s}/$$H$ for $1/H$ $\to$ 0, where
$M/H$ is the measured susceptibility, $\chi$ the intrinsic susceptibility,
$C_{s}$ the presumed ferromagnetic impurity content and $M_{s}$ its saturation
magnetization. The influence of ferromagnetic impurities must be avoided in
order to obtain the intrinsic susceptibility, $\chi$. Therefore, we use the
Honda-Owen method to obtain the $\chi$ of these samples as $\chi(T)$ = $\Delta
M\over\Delta H$. The magnetizations were measured between 5 K and 300 K
separately under magnetic fields of 3 and 4 T, or 4 and 5 T, above which the
magnetizations of the magnetic impurity were supposed to be saturated. Figure
2 (b) shows the temperature dependence of the intrinsic magnetic
susceptibility for samples with $x$ = 0, 0.05, 0.12, 0.20, 0.28, 0.33 and
1.00. The external field is applied parallel along the c-axis. For undoped
sample Fe1.12Te, the magnetic susceptibility $\chi(T)$ increases with
decreasing temperature, and decreases sharply near 69 K, due to the AF phase
transition accompanied by the structural phase transition, then becomes almost
the constant with decreasing temperature, in agreement with the previous
reports. Wang The susceptibility does not show any anisotropy since it has
almost no distinct difference in the paramagnetic phase in cases of $H$//c and
$H$//a, and even at low temperatures $\chi^{H//c}$(5 K)/$\chi^{H//a}$(5 K)
$\sim$ 1.45 shows a weak anisotropy. For the superconducting sample, the
intrinsic susceptibility of the sample with $x$ = 0.28 under $H$//c and $H$//a
also shows a very weak anisotropy as displayed in Fig. 2 (c). On Se-doping,
the AF transition shifts to lower temperature and the peak is broadened, then
is hardly observed in the range $x>$ 0.12, where the superconducting phase
transition occurs at 10 K $\sim$ 14 K. The upturn of $\chi(T)$ at low
temperatures, indicating a Curie-Weiss like behavior, which is naturally
ascribed to a local moment effect. Here, we noticed that the Fe’s possibly
occupy two different sites in Fe1+δTe1-xSex, i. e., Fe(I) occupies (0, 0, 0)
site and has 1.6 $\sim$1.8 $\mu_{B}$ with itinerant characters and Fe(II)
occupies (0.5, 0, z) with a localized moment of 2.5$\mu_{B}$. Bao ; Chiba ;
Zhang The localized moment has a strong competition with superconductivity,
making the superconductivity very sensitive to the excess amount of Fe in the
FeSe compound. Cava Supposed by the band theory, the excess Fe occurs as Fe+,
donating one electron to the Fe(I) layer. Experimentally, there is always
excess iron in Fe(II) site and the number is lager in FeTe than that in FeSe,
Sales which may be the reason for that FeSe has such a high $T_{c}$ under
high pressure while FeTe just changes to a metallic state at low temperatures
in the same situation. Okada In contrast to the doped sample, the end parent
compound Fe1+δSe shows a very weak $T$-linear behavior in $\chi(T)$, in
agreement with the 77Se-nuclear magnetic resonance(NMR) measurement, Kotegawa
confirming a good reliability of Honda-Owen method. In NMR measurement, the
Fe(I) site contributes completely to the Knight shift, strongly indicating
that the Fe(I) site plays the key role to understand the superconducting
mechanism in this system. Kotegawa Furthermore, density functional
calculations show that the electronic states near the Fermi level are mostly
of Fe $3d$ characters from the Fe(I) site and with a smaller contribution from
the excess Fe(II) site. Zhang Herein, we ascribe the temperature dependence
of the magnetic susceptibility in Fe1+δTe1-xSex primarily originates from the
Fe(I) and Fe(II) sites. However, the magnetic susceptibility of Fe(II) will be
dominant since it has a larger local moment than that of Fe(I) site,
especially at low temperatures. We suggest that the upturn in $\chi(T)$ at low
temperatures comes from the excess Fe(II) site. In order to separate the
contributions from the two different Fe sites to the magnetic susceptibility,
We fitted the magnetic susceptibility data with the Curie-Weiss law at low
temperatures for sample with $x$ = 0.12, 0.20, 0.28 and 0.33 in the
temperature range of 20 K $\leqslant$ T $\leqslant$ 50 K as
$\chi(T)=\chi_{0}+\frac{C}{(T-\theta)},$ (1)
where the $T$-independent term $\chi_{0}$ contains the Pauli paramagnetic
susceptibility from itinerant-electron bands, the Van Vleck-orbital
susceptibility and the Larmor diamagnetic susceptibility from ionic cores, $C$
stands for the Curie constant, and $\theta$ the Weiss temperature. Here, the
Curie-Weiss term may be due to the Fe(II) site contribution. Therefore, the
magnetic susceptibility of the Fe(I) can be roughly estimated as
$\chi_{Fe(I)}$ = $\chi(T)$ \- $C_{II}\over(T-\theta_{II})$, as shown in Fig. 2
(c), where $C_{II}$ is Curie constant due to the Fe(II) site, $\theta_{II}$
the Weiss temperature due to the Fe(II) site. We also fitted the data with Eq.
(1) by using different temperature range of 100 K $\leqslant$ T $\leqslant$
300 K for samples with $x$ = 0 and 0.05 (the nominal composition is $x$ = 0
and 0.10). The fitting results are listed in Table I in detail.
Table 1: Fitted parameters using Eq. (1) for Fe1+δTe1-xSex system as well as the real compositions checked by EDXS. $C$ and $\theta$ are obtained from the wider temperature fitting. The units of $C$, $\theta$, $\mu_{eff}$, CII and $\theta_{II}$ are emu K/mol, K, $\mu_{B}$, emu K/mol and K, respectively. sample(nominal) | Fe | Te | Se | $C$ | $\theta$ | $\mu_{eff}$ | C(II) | $\theta_{II}$
---|---|---|---|---|---|---|---|---
0 | 1.12 | 1 | 0 | 2.24 | -319 | 4.2 | - | -
0.10 | 1.00 | 0.95 | 0.05 | 1.6 | -260 | 3.7 | - | -
0.20 | 1.01 | 0.88 | 0.12 | - | - | - | 0.10 | -52
0.30 | 1.07 | 0.80 | 0.20 | - | - | - | 0.02 | -5
0.40(I) | 1.12 | 0.72 | 0.28 | - | - | | 0.12 | -24
0.40(II) | 1.04 | 0.67 | 0.33 | - | - | - | 0.05 | -23
After subtracting the Fe(II) contribution from the susceptibility, it is clear
that $\chi_{Fe(I)}$ decreases gradually from 300 K down to 20 K, as shown in
Fig. 2 (c), qualitatively consistent with our NMR results. Our It is
important to note that there are other systems which also show the linear-
in-$T$ behavior: for example, the geometric frustrated system Na0.5CoO2, Maw
the high temperature cuprate superconductor La2-xSrxCuO2, Takagi the simply
metal Cr as well as its alloys Fawcett and even the new discovered Fe-As
based superconductors. Klingeler All the above systems share a common
feature: having antiferromagnetic spin fluctuations on their backgrounds. Very
recently, Han and his colleagues studied the electronic structure and magnetic
interaction in Fe1+δTe. They found that the small amount of excess Fe played
an important role in determining the magnetic structure and drove the Fermi
surface nesting from ($\pi$, $\pi$) to ($\pi$, 0). Han With increasing Se
doping, the ratio of Fe(II) was depressed in Fe1+δTe1-xSex system. Sales
Thus, the upturn at low temperatures will disappear in Se rich sample. It did
so in the samples with $x$ = 0.45 and 0.48 as shown in Fig. 2 (c) in which we
did not subtract the Fe(II) site contribution to the magnetic susceptibility
but only show the raw data. In LaFeAsO1-xFx, the linear-in-$T$ behavior is
considered to be a strong AF spin fluctuations with multi-orbital character.
Klingeler Korshunov argued that it was universal for systems with the strong
($\pi$, $\pi$) SDW fluctuation. Korshunov In fact, we observed the
$\mathbf{q}\neq 0$ modes of antiferromagnetic spin fluctuations were strongly
enhanced toward $T_{c}$ in the normal state. Our Overall the linear-in-$T$
behavior of $\chi(T)$ observed in our single crystal samples strongly supports
the above model, suggesting the importance of the ($\pi$, $\pi$ ) AF spin
fluctuations originated from the Fe(I) site in superconducting mechanism of
this system.
### III.3 Superconducting State and Upper Critical Field
$\mu_{0}H_{c2}^{orb}$
The superconducting transition temperature was found to be $\sim$10 K for the
sample with $x>$ 0.05 as shown in Figs. 3 (a), (b), (c), (d) and (e). Because
the excess Fe in the Fe(II) site has a localized moment, Chiba ; Zhang ; Liu
where there is the more excess amount of Fe(II), the less superconducting
volume fraction is observed, compared with Figs. 3 (d) and (e), where these
two samples have very close composition. With increasing the Se content $x$,
the superconducting volume fraction was enhanced greatly. The susceptibility
measured in the FC process shows no negative sign but a small positive value
in the superconducting state, indicating an intrinsic pinning effect in this
layered structure compound. Iye
Figure 3: Temperature dependence of susceptibility in superconducting samples
under magnetic field $H$ = 20 Oe applied along the c-axis $H$//c, for ZFC and
FC processes. (a) Fe1.00Te0.95Se0.05. (b) Fe1.01Te0.88Se0.12, (c)
Fe1.07Te0.80Se0.20. (d) Fe1.04Te0.67Se0.33. (e) Fe1.12Te0.72Se0.28.
In order to estimate the superconducting parameters, we selected two samples
with close composition Fe1.12Te0.72Se0.28 (simplified as R1) and
Fe1.04Te0.67Se0.33 (simplified as R2) for resistivity measurements. Figures 4
(a), (b), (c) and (d) show the suppression of the superconducting transition
in the electrical resistivity for $H$//c and $H$//a up to 14 T for the samples
of R1 and R2, respectively. With increasing the magnetic field, the
superconducting transitions shifted to lower temperatures, and became
broadened. The upper critical field $\mu_{0}H_{c2}^{orb}$ determined from the
onset $T_{c}$ were plotted in Figs. 5 (a) and (b) for the samples R1 and R2,
respectively. Here, the onset $T_{c}$ was defined as the resistivity falls to
90% of the $\rho_{0}$ value in the normal state just above $T_{c}$. The
initial slopes $\partial\mu_{0}H_{c2}/\partial T$ near $T_{c}$ are -6 T/K and
-3.9 T/K for the R1 sample with $H$//a and $H$//c, respectively, leading to an
estimation of the upper critical field extrapolated to zero-temperature as
$\mu_{0}H_{c2}^{orb}(0)$ = 57 T and 37 T, by using the Werthamer-Helfand-
Hohenberg (WHH) model as
$\mu_{0}H_{C2}^{orb}(0)=-0.693T_{c}(\frac{\partial\mu_{0}H_{C2}^{orb}}{\partial
T})_{T=T_{c}}.$ (2)
In contrast to the sample R1, the sample R2 shows larger initial slopes of
-8.7 T/K and -4.2 T/K as well as the upper critical fields of 85 T and 40 T
under $H$//a and $H$//c, respectively. The upper critical field in this system
is comparable with the cases of the Fe-As based superconductors
LaFeAsO0.93F0.03 Kohama and KFe2As2. Terashima In addition, the upper
critical field is much lager than the Pauli limit field
$\mu_{0}H_{\mathrm{P}}$ =1.84 $T_{c}$ $\sim$ 25 T. The anisotropy coefficients
$\Gamma$ determined from $\Gamma$ = $H_{c2}^{\perp}$/$H_{c2}^{\parallel}$ are
1.54 and 2.1 for the samples R1 and R2, respectively. In fact, the $\Gamma$ is
rather isotropic at low temperatures, indicating the three dimensional nature
of the Fermi-surface topology.Yuan These results strongly suggest an
unconventional superconducting mechanism in this compound. The estimated
parameters are listed in Table II.
Figure 4: Temperature dependence of resistivity ($\rho$) under magnetic field
($H$) up to 14 T (0, 2, 4, 6, 8, 10, 12 and 14 T) for samples of R1 and R2:
(a) The sample R1 for $H$ // c. (b) The sample R1 for $H$ // a. (c) The sample
R2 for $H$ // c. (d) The sample R2 for $H$ // a.
Figure 5: Temperature dependence of the upper critical fields of (a) the
sample R1 and (b) the sample R2. The dashed line is the estimation by the WHH
theory.
The initial slope of $\mu_{0}H_{c2}^{orb}$ near $T_{c}$ is also weakly
anisotropic. The similar behavior was also observed in the 122-type compounds,
and was thought to be two-band superconductivity. Baily The Sample R2 with
the less Fe shows a larger initial slope and a higher upper critical field in
compared with the sample R1, indicating the existence of Fe(II) affects the
electronic band structure and consequently the superconductivity greatly. The
initial slope near $T_{c}$ is proportional to the square of the electron
effective mass $m^{*2}$, in agreement with a large $\gamma\sim 39$ mJ/mol K2.
Sales In strongly correlated electron systems, the Kadowaki-Woods ratio
$A/\gamma^{2}$ is expected to be a constant $\sim$ 1.0 $\times$
10-5$\mu\Omega$ cm (mJ/mol-K)2, where $A$ is the quadratic term of the
resistivity and $\gamma$ is the linear term coefficient of the specific heat,
so called the electronic specific heat coefficient. We obtained $A\sim 0.03$
$\mu\Omega$ cm /K2 by fitting the data with $\rho=\rho_{0}+AT^{2}$ in the
temperature range of 16 K $\leqslant$ T $\leqslant$ 20 K for the sample R2,
resulting in $A/\gamma^{2}\sim$ 2$\times$ 10-5$\mu\Omega$ cm/(mJ/mol K)2,
which is a little bit larger than the value of heavy fermion compound
UBe${}_{13}.$UBe13 The Wilson ratio
$R_{w}=\pi^{2}k_{B}^{2}\chi_{spin}/3\mu_{B}^{2}\gamma$ is estimated as 5.7 for
the sample R2 with $\chi_{spin}$ of $2\times 10^{-3}$emu/mol from our data,
well exceeding the unity for a free electron system. These results strongly
suggest that the electron in superconducting Fe1+δTe1-xSex is strongly
correlated, being in good agreement with our recent NMR investigation on the
same single crystal of Fe1.04Te0.67Se0.33 which strongly indicates the
unconventional d-wave superconductivity with spin singlet pairing-symmetry.
Our
It is also very important to know whether the Fe1+δTe1-xSex is a clean
superconductor or not. To solve this issue, we need to know the mean free path
$\ell$ and the Pippard coherence length $\xi_{0}$. On the basis of the
Bardeen-Cooper-Schrieffer (BCS) theory and the Drude model, $\ell$ = $\hbar$
(3$\pi^{2}$)1/3/e2$\rho_{0}$n2/3 and $\xi_{0}$ = $\hbar$VF/$\pi$$\Delta$,
where $n$ is the carrier concentration, $\rho_{0}$ the residual resistivity,
VF the Fermi velocity and $\Delta$ the superconducting gap. Giving the
superconducting gap 2$\Delta$/kB$T_{c}$ = 3.52 in the BCS theory, the
$\xi_{0}$ can be written as
$\hbar^{2}$(3$\pi^{2}$n)1/3/1.76$\pi$$m$$k_{B}$$T_{c}$, where $m$ is the free
electron rest mass. Very recently, the angle-resolved photoemission
spectroscopy (ARPES) measurement on Fe1.03Se0.3Te0.7 showed the Fermi velocity
$\sim$ 0.4 eV$\mathrm{\AA}$ for both the hole and the electron bands and the
superconducting gap $\Delta$ $\sim$ 4 meV. Nakayama Therefore the $\xi_{0}$
is estimated as 33.5$\mathrm{\AA}$ and the carrier concentration estimated as
$\sim$ 6.8$\times$1023/m3. Since the composition is very close among the
samples of Fe1.12Te0.72Se0.28, Fe1.04Te0.67Se0.33 and Fe1.03Te0.70Te0.30 (in
ref Nakayama ), it is reasonable to consider that the carrier concentration
does not change very much among these three samples. The residual resistivity
was estimated as $\rho_{0}=0.70\times 10^{-5}\Omega m$, $0.65\times
10^{-5}\Omega m$ for the samples R1 and R2, respectively. Therefore, $\xi_{0}$
is estimated as $\sim$31 $\mathrm{\AA}$ and $\ell$ $\sim$ 2336 $\mathrm{\AA}$
for the sample R1 and $\xi_{0}$ $\sim$ 32 $\mathrm{\AA}$, $\ell$$\sim$ 2516
$\mathrm{\AA}$ for the case of the sample R2. The estimated parameters are
listed in Table II. Therefore, the ratio of $\ell$/$\xi_{0}$ $\sim$ 80 is well
exceeding the unity so that the Fe1+δTe1-xSex is thought to be a clean
superconductor and the estimated superconducting parameters are considered to
be intrinsic. For example, in a clean superconductor, $\xi_{GL}$ $\sim$ 0.74
$\xi_{0}$/(1-$T/T_{c}$)1/2. We estimated $\xi_{GL}$$\sim$ 24 $\mathrm{\AA}$
for sample R2 at T = 0 K along the c direction, in good agreement with the
value of 29 $\mathrm{\AA}$ derived from the upper critical field, where
$\xi_{GL}$ is expressed as $\xi_{GL}^{2}$ = $\phi_{0}$/2$\pi$$H_{c2}^{orb}$ (
$\phi_{0}$ = 2 $\times$ 10-7 Oe cm2). However, it should be pointed out that
those estimations based on the one band theory which may not valid for the
multi-band compound, and much information will be needed for further
discussion on superconductivity in this system.
Table 2: Estimated superconducting parameters for the samples R1 and R2. Tc = 13.7 and 14.1 K for the samples R1 and R2, respectively. sample | $\partial\mu_{0}H^{orb}_{c2}$/$\partial$T | $\mu_{0}$$H^{orb}_{c2}$ | $\mu_{0}H_{\mathrm{P}}$(0) | $\xi$0 | $\xi$GL | $\ell$ | $\ell$/$\xi_{0}$
---|---|---|---|---|---|---|---
($H$//a,c) | ($T/K$) | (T) | (T) | (Å) | (Å) | (Å) |
R1(a) | -6.0 | 57 | 25 | 32 | 24 | 2336 | 73
R1(c) | -3.9 | 37 | 25 | 32 | 30 | |
R2(a) | -8.7 | 85 | 26 | 31 | 20 | 2516 | 79
R2(c) | -4.2 | 40 | 29 | 32 | 29 | |
## IV CONCLUSIONS
In summary, we successfully synthesized the single crystal of Fe1+δTe1-xSex
($x$ = 0, 0.05, 0.12, 0.20, 0.28, 0.33, 0.45, 0.48 and 1.00; $0<\delta<0.12$)
and measured their magnetic susceptibilities. The intrinsic magnetic
susceptibility was obtained though Honda-Owen method for the first time. The
nearly linear-in-$T$ behavior in susceptibility was observed in
superconducting samples with $x$ = 0.12, 0.20, 0.28, 0.33 0.45, 0.48 and 1.00,
indicating a close relationship between the AF spin fluctuations around
($\pi$, $\pi$) and the superconductivity. The excess Fe has a localized moment
which affects the superconducting state greatly. The intrinsic susceptibility
shows a weakly anisotropic behavior. Also, the initial slope near $T_{c}$ and
the upper critical field estimated by measuring the resistivity under high
magnetic fields show a weakly anisotropic behavior. The estimated coherence
length $\xi_{GL}$, the Pippard coherence length $\xi_{0}$ and the mean free
path $\ell$ by the BCS theory and the Drude model support a clean
superconductor scenario. The estimations of Kadowaki-Woods and Wilson ratios
indicate Fe1+δTe1-xSex belongs to a strongly electron-correlated system.
Consequently, the superconductivity in Fe1+δTe1-xSex is considered to be of
unconventional and in the strongly correlated one with the very high value of
$\mu_{0}H_{c2}^{orb}$, which are also supported by our recent NMR study. Our
## Acknowledgement
This research was supported by Grant-in-Aid for the Global COE Program
“International Center for Integrated Research and Advanced Education in
Materials Science” and for Scientific Research on Priority Area “Invention of
anomalous quantum materials”(16076210) from the Ministry of Education,
Culture, Sports, Science and Technology of Japan, and also by Grants-in-Aid
for Scientific Research (19350030) from the Japan Society for Promotion of
Science.
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|
arxiv-papers
| 2009-11-25T04:30:42 |
2024-09-04T02:49:06.681805
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinhu Yang, Mami Matsui, Masatomo Kawa, Hiroto Ohta, Chishiro\n Michioka, Chiheng Dong, Hangdong Wang, Huiqiu Yuan, Minghu Fang, and\n Kazuyoshi Yoshimura",
"submitter": "Jinhu Yang",
"url": "https://arxiv.org/abs/0911.4758"
}
|
0911.4762
|
# Perfect Entanglement Transport in Quantum Spin Chain Systems
Sujit Sarkar 1\. PoornaPrajna Institute of Scientific Research, 4
Sadashivanagar, Bangalore 5600 80, India
###### Abstract
We propose a mechanism for perfect entanglement transport in anti-
ferromagnetic (AFM) quantum spin chain systems with modulated exchange
coupling along the xy plane and in the z direction. We use the principle of
adiabatic quantum pumping process for entanglement transfer in the spin chain
systems. In our proposed mechanism, perfect entanglement transfer can be
achieved over an arbitraly long distance. We explain analytically and
physically why the entanglement hops in alternate sites. We solve this problem
by using the Berry phase analysis and Abelian bosonization methods. We find
the condition for blocking of entanglement transport even in the perfect
pumping condition. We also explain physically why entanglement transfer in AFM
chain out performs the ferromagnetic chain. Our analytical solution
interconnects quantum many body physics and quantum information science.
1\. Introduction: Quantum communication between distant co-ordinates in a
quantum network is an important requirement for quantum computation and
information. One can construct the quantum network in different ways. Optical
systems typically employed in quantum communication and cryptography
application to transfer the state between two distinct co-ordinates directly
via photons kie ; ski . Quantum computing applications work with trapped atoms
to transfer information between distant sites , photons in cavity QED zeili ;
raus ; sack ; bayer ; plas . However we would like to study the entanglement
transfer through the quantum spin chain systems bose ; chris ; osbo ; bayat ;
venuti ; eckert1 ; eckert2 ; srini ; hartmann ; amico . The equivalence of
state transferring and teleporation of information transmission has already
been studied in the literature horo ; abol . The potentiality of the spin
chain system, antiferromagnetic(AFM) and ferromagnetic(FM), as a network of
quantum state and entanglement transport has already been studied by many
groups as referred in the literature. The experimental evidence of nanoscale
spin chain and their properties have discussed in Ref. hein . Our approach in
this study is different from the existing studies in the literature bose ;
chris ; osbo ; bayat ; venuti ; eckert1 ; eckert2 ; srini ; hartmann ; amico .
The literature of quantum entanglement study is quite vast in quantum
computation science. Here we mention very briefly the important works that
have already existed in state and entanglement transport in the literature:
The authors of Ref. eckert1 have shown explicitly that the quality of state
and entanglement transfer through all phases of spin-$1$ chain have been
possible. Some AFM phases are more efficient than the FM phase. The authors of
Ref. eckert2 have shown explicitly that dimerized AFM states of spin-$1$
chains are also able to transfer through an adiabatic modulation of exchange
couplings. The authors of Ref. venuti , have shown explicitly that the quantum
information can be efficiently transferred between weakly coupled end spins of
an AFM chain because of an effective coupling between the end spins. The
authors of Ref. srini ; hartmann have studied the quantum state and
entanglement transfer and the authors of Ref. amico have studied the
entanglement dynamics, considering initial states deviating from the final
states. The authors of Ref. abol ; bose have studied the entanglement
transfer in a uniformly coupled spin-$1/2$ AFM/FM spin chain. They have
claimed a curious result that for the AFM spin chain, the entanglement hops to
skip alternate sites. They have also found that the entanglement transfer in
the AFM chain outperforms the FM chain. We explain in our work that these
theoretical predictions are natural. Here we mention very briefly the basic
mechanism of entanglement transfer through the spin chain system based on the
conventional wisdom in the literature bose ; chris ; osbo ; bayat ; venuti ;
eckert1 ; eckert2 ; srini ; hartmann ; amico and at the same time illustrate
the difference with our approach.
It is well known that entanglement is the manifestation of quantum
correlations between two systems when they are inseparable state. We consider
the spin singlet state as an example of an entangled state.
${|{\psi}^{-}>}_{0,0^{\prime}}=\frac{1}{\sqrt{2}}[{|0>}_{0^{\prime}}{|1>}_{0}~{}-~{}{|1>}_{0^{\prime}}{|0>}_{0}]$
(1)
Typically, the sender holds one member of the state of the pair of qubits
while puting the other member at the near end of the AFM spin chain of length
N. The spin chain is in the ground state. When the spin $0$ starts to interact
with the first spin of the chain then the Hamiltonian includes this additional
interaction term ( ${I}_{0^{\prime}}{\otimes}J{{\sigma}_{0}}.{{\sigma}_{1}}$
), where ${{\sigma}_{0}}$ and ${{\sigma}_{1}}$ are the Pauli spin operators
for the $0$ and $1$ sites respectively and $J$ is the exchange coupling). The
initial state being
$|{\psi}{(0)}>={|{\psi}^{-}>}_{0,0^{\prime}}\otimes|{{\psi}_{g}}>$ (2)
Where $|{\psi}_{g}>$ is the ground state wave function of the AFM Hamiltonian
and $|{\psi}(0)>$ is the ground state wave function of the total Hamiltonian.
This initial state starts to evolve and from that one computes the density
matrix and concurrence to measure the entanglement and purity of states. But
our approach is different. Our main motivation is to interconnect the quantum
many body physics and quantum information science. It is common practice in
quantum many body physics to create a particle at any point in the system and
study the dynamics of that particle to understand the physical behaviour of
the system. Therefore, we consider one of the spin ($\uparrow$ or
$\downarrow$) of the singlet interacts with the spin chain and this spin
itself transports through the chain medium due to the adiabatic variation of
exchange couplings of the Hamiltonian, and reaches the other end of the chain.
Our spin chains are the AFM spin chain with the modulated exchange couplings.
But we consider the monogamous nature of the shared entanglement between the
two spins $0$ and $0^{\prime}$. Before we proceed further we would like to
state the basic aspects of adiabatic pumping process: an adiabatic parametric
quantum pump is a device that generates a dc current by a cyclic variation of
system parameters, the variation being slow enough that the system remains
close to the ground state throughout the pumping cycle thou1 ; thou2 . It is
well known that when a quantum mechanical system evolves, it acquires a time
dependent dynamical phase and time independent geometrical phase berry . The
geometrical phase depends on the geometry of the path in the parameter space.
In the adiabatic entanglement pumping process, the locking potential well
carries a spin of the singlet pairs. As the locking potential well slides
through the adiabatic variation of system parameters, it induces a current
($I$) in the system. In this study we calculate the current of this spin
transport, which transports a spin from one end of the chain to the other and
as a result of which entanglement is transported (because the spin
$0^{\prime}$ and $0$ are singlet and monogonus in nature) from one side to the
other. In our study this entanglement transport is the perfect because the the
adiabatic pumping physics based on Berry phase analysis is topologically
protected against the external perturbations thou1 ; thou2 ; shin .
Here we consider two different Hamiltonian, $H_{1}$ and $H_{2}$ with modulated
exchange coupling in $xy$ and $z$ directions respectively, Hamiltonians of the
systems are the following
$\displaystyle{H_{1}}$ $\displaystyle=$
$\displaystyle-\sum_{n}J(1-(-1)^{n}{{\delta}_{1}}(t))({{S}_{+}}^{n}{{S}_{-}}^{n+1}+{{S}_{+}}^{n+1}{{S}_{-}}^{n})$
(3) $\displaystyle+\sum_{n}{\Delta}{{S}_{z}}^{n}{{S}_{z}}^{n+1}$
This model Hamiltonian has some experimental relevance shin . The other model
Hamiltonian is
$\displaystyle{H_{2}}$ $\displaystyle=$
$\displaystyle-\sum_{n}J({{S}_{x}}^{n}{{S}_{x}}^{n+1}+{{S}_{y}}^{n}{{S}_{y}}^{n+1})$
(4)
$\displaystyle+\sum_{n}{\Delta}{{S}_{z}}^{n}{{S}_{z}}^{n+1}-\frac{1}{2}\sum_{n}{B_{0}}(1-(-1)^{n}{{\delta}_{2}}(t)){{S}_{z}}^{n}$
Here we consider that the fluctuations is periodic over two lattice sites. We
see that this model have essential ingredients to capture the adiabatic
entanglement pumping. One can express spin chain systems to a spinless fermion
systems through the application of Jordan-Wigner transformation. In Jordan-
Wigner transformation the relation between the spin and the electron creation
and annihilation operators are $S_{n}^{z}=\psi_{n}^{\dagger}\psi_{n}-1/2~{}$,
$S_{n}^{-}=\psi_{n}~{}\exp[i\pi\sum_{j=-\infty}^{n-1}n_{j}]~{}$,
$S_{n}^{+}=\psi_{n}^{\dagger}~{}\exp[-i\pi\sum_{j=-\infty}^{n-1}n_{j}]~{}$,
gia2 , where $n_{j}=\psi_{j}^{\dagger}\psi_{j}$ is the fermion number at site
$j$. Spin operators in terms of bosonic field are the following.
$\displaystyle S_{n}^{x}~{}$ $\displaystyle=$
$\displaystyle~{}[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\cos({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{y}~{}$ $\displaystyle=$
$\displaystyle~{}-[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\sin({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{z}~{}$ $\displaystyle=$
$\displaystyle~{}{\sqrt{\frac{\pi}{K}}}~{}\partial_{x}\phi~{}+~{}(-1)^{n}c_{1}\cos(2{\sqrt{\pi
K}}\phi)~{},$ (5)
${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))}$
(6)
$r$ denotes the chirality of the fermionic fields, right (1) or left movers
(-1). The operators $U_{r}$ are operators that commute with the bosonic field.
$U_{r}$ of different species commute and $U_{r}$ of the same species
anticommute. $\phi$ field corresponds to the quantum fluctuations (bosonic) of
spin and $\theta$ is the dual field of $\phi$. They are related by this
relation ${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$.
Using the standard machinery of continuum field theory gia2 , we finally get
the bosonized Hamiltonians as $H_{0}$ is the gapless Tomonoga-Luttinger liquid
part of the Hamiltonian.
After the application of continuum field-theory the Hamiltonian become, in
terms of bosonic fields.
$\displaystyle{H_{1}}$ $\displaystyle=$
$\displaystyle{H_{0}}+\frac{{E_{J_{0}}}{{\delta}_{1}}(t)}{2{{\pi}^{2}}{\alpha}^{2}}\int
dx:cos[2\sqrt{K}{\phi}(x)]:$ (7)
$\displaystyle+\frac{\Delta}{2{{\pi}^{2}}{\alpha}^{2}}\int
dx:cos[4\sqrt{K}{\phi}(x)]:$ $\displaystyle{H_{2}}$ $\displaystyle=$
$\displaystyle{H_{0}}+\frac{{B_{0}}{{\delta}_{2}}(t)}{2{\pi}{\alpha}}\int
dx:cos[2\sqrt{K}{\phi}(x)]:$ (8)
$\displaystyle+\frac{\Delta}{2{{\pi}^{2}}{\alpha}^{2}}\int
dx:cos[4\sqrt{K}{\phi}(x)]:-\frac{B_{0}}{2}\int dx{{\partial}_{x}}{\phi}(x)$
Here, we would like to explain the basic aspects of quantum entanglement
pumping in terms of spin pumping physics of our model Hamiltonians: An
adiabatic sliding motion of one dimensional potential, in gapped Fermi surface
(insulating state), pumps an integer numbers of particle per cycle. In our
case the transport of Jordan-Wigner fermions (spinless fermions) is nothing
but the transport of spin from one end of the chain to the other end because
the number operator of spinless fermions is related to the z-component of spin
density cal . We see that non-zero ${{\delta}_{1}}(t)$ and ${{\delta}_{2}}(t)$
introduce the gap at around the Fermi point and the system is in the
insulating state (Peierls insulator). In this phase spinless fermions form the
bonding orbital between the neighboring sites, which yields a valance band in
the momentum space. It is well known that the physical behavior of the system
is identical at these two Fermi points. We would like to analyse these double
degeneracy point, following the seminal paper of Berry berry : in our model
Hamiltonian there are two adiabatic parameters ${{\delta}_{1}}(t)$ and
${{\delta}_{2}}(t)$. The Hamiltonian starts to evolve under the variation of
these two adiabatic parameters, when the Hamiltonian returns to its original
form after a time $T$, the total geometric phase acquired by the system is
${{\gamma}_{n}}(T)~{}=~{}\frac{i}{2\pi}\int_{C}<{{\psi}_{n}}|{{\nabla}_{R}}|{\psi}_{n}>~{}dR$,
a line integral around a closed loop in two dimensional parameter space. Using
Stokes theorem, one can write
${{\gamma}_{n}}(T)~{}=~{}\frac{i}{2\pi}\int{{\nabla}_{R}}\times<{{\psi}_{n}}|{{\nabla}_{R}}|{\psi}_{n}>~{}dS$.
The flux $\Phi$ through a closed surface C is, $\Phi=\int B.dS$. Therefore one
can think of the Berry phase as flux of a magnetic field. Now we express,
${B_{n}}(K1)={{\nabla}_{K1}}\times{A_{n}}(K1)$, and
${A_{n}}(K1)=\frac{i}{2\pi}<n(K1)|{{\nabla}_{K1}}|n(K1)>$, where
$K1=(k,{\delta}_{1}(t),{\delta}_{2}(t))$. Here $B_{n}$ and $A_{n}$ are the
fictitious magnetic field (flux) and vector potential of the nth Bloch band
respectively. The degenerate points behave as a magnetic monopole in the
generalized momentum space ($K_{1}$) berry , whose magnetic unit can be shown
to be $1$, analytically shin ; berry
$\int_{S1}~{}dS\cdot B_{\pm}~{}=~{}\pm 1$ (9)
positive and negative signs of the above equations are respectively for the
conduction and valance band meet at the degeneracy points. $S_{1}$ represent
an arbitrary closed surface which enclose the degeneracy point. In the
adiabatic process the parameter ${{\delta}_{1}}(t)$ or ${{\delta}_{2}}(t)$ are
changed along a loop ($\Gamma$) enclosing the origin (minima of the system).
We define the expression for spin current ($I$) from the analysis of Berry
phase. It is well known in the literature of adiabatic quantum pumping physics
that two independent parameters are needed to achieve the adiabatic quantum
pumping in a system sharma . Here one may consider these two parameters as the
real and imaginary part of the fourier transform of a modulated coupling
induce potential. When the shape of the potential will change in time, then it
amounts to changing the phase and amplitude in time. The role of adiabatic
parameters are not explicit in our study. Our formalism is different from
others. We define the expression for spin current ($I$) from the analysis of
Berry phase. Then according to the original idea of quantum adiabatic particle
transport thou1 ; thou2 ; shin ; avron , the total number of spinless fermions
($I$) which are transported from one side of this system to the other is equal
to the total flux of the valance band, which penetrates the 2D closed sphere
($S_{2}$) spanned by the $\Gamma$ and Brillioun zone shin .
$I=\int_{S_{2}}dS\cdot B_{+1}~{}=1$ (10)
$B_{+1}$ is directly related with the Berry phase (${{\gamma}_{n}}(T)$) which
is acquired by the system during the adiabatic variation of the exchange
couplings the time period of the adiabatic process. This quantization is
topologically protected against the other perturbation as long as the gap
along the loop remains finite shin ; avron . Therefore the adiabatic
entanglement pumping is constant over the arbitrarily long distance of the
system. This result is in contrast with the existed results in the literature
[8,19]. They have found that the entanglement decay exponentially after a
certain distance.
Now we explain the quantum entanglement transfer for $H_{1}$. The second term
of the Hamiltonian for NN exchange interaction has originated from the $x$ and
$y$ component of exchange interaction. This term implies that infinitesimal
variation of coupling in lattice sites, is sufficient to produce a gap around
the Fermi points. So when ${1/2}<K<1$, only these time dependent exchange
couplings are relevant and lock the phase operator at
${\phi}=0+\frac{n\pi}{\sqrt{K}}$. Now the locking potential slides
adiabatically. The speed of the sliding potential is low enough such that the
system stays in the same valley, i.e., there is no scope to jump onto the
other valley. The system will acquire $2\pi$ phase during one complete cycle
of adiabatic process. This expection is easily verified when we notice the
physical meaning of the phase operator ($\phi$ (x)). Since the spatial
derivative of the phase operator corresponds to the z-component of spin
density, this phase operator is nothing but the minus of the spatial
polarization of the z-component of spin, i.e.,
$P_{s^{z}}~{}=-\frac{1}{N}\sum_{j=1}^{N}j{S_{j}}^{z}$. Shindou has shown
explicitly the equivalence between these two considerations shin . During the
adiabatic process $<{\phi}_{t}>$ changes monotonically and acquires \- $2\pi$
phase. In this process ${P_{s}}^{z}$ increases by 1 per cycle. We define it
analytically as
${\delta}{P_{s}}^{z}=\int_{\Gamma}d{P_{s}}^{z}=-\frac{1}{2\pi}\int
dx{{\partial}_{x}}<{\phi}(x)>=1$ (11)
This physics always hold as far as the system is locked by the sliding
potential and ${\Delta}<1$ shin . The change of the spatial polarization by
unity during a complete evaluation of adiabatic cycle implies that the
transport of entanglement across the system. This is because the spatial
derivative of the phase operator is the Cooper pair density in our system. The
entanglement transport of this scenario can be generalized up to the value of
$\Delta$ for which $K$ is greater than 1/2 . In this limit, z-component of the
exchange interaction has no effect on the entanglement pumping of our system.
But when $K<1/2$ , then the interaction due to $\Delta$ becomes relevant and
creates a gap in the excitation spectrum. This potential profile is static.
Therefore there is no scope to slide the potential and to get a adiabatic
pumping across the system. The authors of Ref. abol ; bose have also found
that when ${\Delta}>1$ for $XXZ$ AFM spin chain, the fidelity of AFM spin
chain also decreases ,i.e., the entanglement transport decreases in this
limit.
Similarly for the Hamiltonian $H_{2}$, the second term of the Hamiltonian
produce the gap and the pumping process is the same as that of $H_{1}$.
Therefore we conclude that the modulations in the in plane exchange coupling
and also for the modulations in the z-directions yield the same adiabatic
entanglement pumping.
In this pumping process the most favourable states of the system are the
antiferromagnetic configuration $|010101....>$ and $|101010,,,,>$ ($0$ stands
for up spin and $1$ stands for down spin). One may start from any
antiferromagnetic states and transfer the spin of every site to the right by
two sites to achieve the pumping. Therefore our test spin which we introduce
at the one end of the spin, it hops to the right by two sites in every step.
Thus when we study the entanglement transport between the spin $0^{\prime}$
and $0$, then it is natural that entanglement also is transported through
every alternate sites. The authors of Ref. abol ; bose have observed a very
peculiar behaviour of entanglement transfer for AFM: the nonanalytical
behaviour as a function of time. It is zero for most of the time and it
suddenly grows up and forms a peak at a regular interval of time. But in our
study the entanglement current is constant and it is almost perfect
entanglement pumping. In their case the spin chain has the spin rotational
symmetry. When one member of an entangled pair of qubits is transmitted
through such a channel , then the two qubits states evolve to a Werner state
benn . But our spin chain systems there is no spin rotational invariant
symmetry and the transport mechanism is also different. The physical scenario
of our study is completely different from the existing physical picture. The
quantized entanglement transport of this scenario can be generalized up to the
value of $\Delta$ for which $K$ is greater than 1/2. In this limit, the
z-component of the exchange interaction has no effect on the entanglement
pumping physics of Hamiltonian. . In this limit, z-component of the exchange
interaction has no effect on the entanglement pumping of our system.
Here, we would like to explain the difference of entanglement transport
between the FM and AFM spin chain, it has mentioned in the literature but the
complete physical explanation is not upto the mark bose ; chris ; osbo ; bayat
; venuti ; eckert1 ; eckert2 ; srini ; hartmann ; amico ; abol . As we know
that entanglement is a quantum mechanical property, Schrodinger singled out
many decades ago as ”the characteristic of quantum mechanics sch and that has
been studied extensively in connection with Bell’s inequality bell . FM ground
state state there is no difference between the classical and quantum
mechanical ground state and the low lying excitations are spin-1 magnons. The
AFM ground state has a complex structure specified by the Bethe-ansatz
solution. There are no similarities between classical and quantum mechanical
ground state and first excited state of the AFM chain and as a result of the
quantum mechanical property of the system the entanglement manifests
prominently in the AFM spin chain. This is the only clear reason why AFM
outperforms the FM spin chain.
Here we discuss possible sources of imperfections in the entanglement pumping
process. The non-adiabatic contributions leave the system in an unknown
superposition of states after the full cycle. Also the appearance of Landau-
Zener transition in the pumping system should be negligible so that the system
is in the ground state. This condition limits the pumping rate of entanglement
by the mathematical relation $\frac{h}{\tau}<<J$. However even then the
entanglement pumping is not perfect due to the non vanishing
$\frac{J}{\Delta}$. Our effort also should take the elimination of
entanglement pumping in the wrong directions. The residual exchange coupling
may lead to a different spin state. An entangled spin transported through a
correct exchange coupling modulation with probability $P$ and through the
residual exchange coupling with the probability $Q=1-P$. Therefore the pumping
error in each site is $\frac{P}{Q}$. Our system consists of $N$ sites.
Therefore the probability of correct entanglement transport is $\sim{P^{N/2}}$
and wrong entanglement transport is $\sim{Q^{N/2}}$. The total pumping error,
$({\frac{Q}{P})}^{N/2}$, decreases with the number of sites in nanoscale spin
chain. Therefore for the spin chain system entanglement transport is better
for larger length compare to the smaller length with same exchange couplings.
Conclusions: we have presented the theoretical explanation of adiabatic
entanglement pumping for our model Hamiltonians. We have found the perfect
entanglement transport condition which cure the existed results in the
literature. We have explained few physical findings of entanglement transport
which were curious before this study.
The author would like to thank, The Center for Condensed Matter Theory of IISc
for extended facility. Finally the author would like to thank Prof. R.
Srikanth, Dr. T. Tulsi and Prof. Indrani Bose.
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|
arxiv-papers
| 2009-11-25T04:52:43 |
2024-09-04T02:49:06.687712
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sujit Sarkar",
"submitter": "Sujit Sarkar",
"url": "https://arxiv.org/abs/0911.4762"
}
|
0911.4798
|
1
# Enrichment by supernovae in globular clusters with multiple populations
Jae-Woo Lee1 Young-Woon Kang1 Jina Lee1 & Young-Wook Lee2
###### Abstract
The most massive globular cluster in the Milky Way, $\omega$ Centauri, is
thought to be the remaining core of a disrupted dwarf galaxy[1, 2], as
expected within the model of hierarchical merging[3, 4]. It contains several
stellar populations having different heavy elemental abundances supplied by
supernovae[5] — a process known as metal enrichment. Although M22 appears to
be similar to $\omega$ Cen[6], other peculiar globular clusters do not[7, 8].
Therefore $\omega$ Cen and M22 are viewed as exceptional, and the presence of
chemical inhomogeneities in other clusters is seen as ‘pollution’ from the
intermediate-mass asymptotic-giant-branch stars expected in normal globular
clusters[9]. Here we report Ca abundances for seven globular clusters and
compare them to $\omega$ Cen. Calcium and other heavy elements can only be
supplied through numerous supernovae explosions of massive stars in these
stellar systems[10], but the gravitational potentials of the present-day
clusters cannot preserve most of the ejecta from such explosions[11]. We
conclude that these globular clusters, like $\omega$ Cen, are most probably
the relics of more massive primeval dwarf galaxies that merged and disrupted
to form the proto-Galaxy.
Department of Astronomy and Space Science, ARCSEC, Sejong University, Seoul
143-747, Korea
Center for Space Astrophysics, Yonsei University, Seoul 120-749, Korea
The Sejong/ARCSEC Ca uvby survey program was initiated in 2006 to investigate
the homogenous metallicity scale for globular clusters and to obtain the
complete metallicity distribution function of the Galactic bulge using the
$hk$ index [= $(Ca-b)-(b-y)$][12]. The Ca filter in the $hk$ index measures
ionized calcium H and K lines, which have been frequently used to calibrate
metallicity scale for globular clusters[13, 14]. The utility of the $hk$ index
is that it is known to be about three times more sensitive to metallicity than
the $m_{1}$ index is for stars more metal-poor than the Sun and it has half
the sensitivity of the $m_{1}$ index to interstellar reddening[12]. During the
last three years, we have used more than 85 nights of CTIO 1.0-m telescope
time for this project. The telescope was equipped with an STA 4k $\times$ 4k
CCD camera, providing a plate scale of 0.289 arcsec/pixel and a field of view
of 20 $\times$ 20 arcmin. All of our targets accompanied with standards were
observed under the photometric weather conditions and most of targets were
repeatedly visited between separate runs. The photometry of our targets and
standards were analyzed using DAOPHOT II, ALLSTAR, and ALLFRAME[15, 16].
In the course of metallicity calibration of red giant branch (RGB) stars in
GCs, we found that many GCs show split in the RGB in their $hk$ versus $V$
color-magnitude diagrams (Figs 1 and 2). The prime examples are M22 and
NGC1851. In particular, the double RGB sequence in M22 is very intriguing. The
differential reddening effect and the contamination from the off cluster
populations cannot explain the double RGB sequences in M22 (see Supplementary
Information). It has been debated for decades whether this cluster is
chemically inhomogeneous or not, but the recent high resolution spectroscopic
study of 17 RGB stars in the cluster suggests that it contains chemically
inhomogeneous subpopulations[6]. The bimodality in the $m_{1}$ index of M22
RGB stars was also known, but it has been argued that it is most likely due to
the bimodal CN abundances, where CN absorption strengths strongly affect the
$m_{1}$ index, not due to the bimodal distribution of heavy elements in the
cluster[17, 18, 19]. The star-to-star light elemental abundance (C, N, O, Na,
Mg and Al) variations have been known for decades and they are now generally
believed to be resulted from chemical pollutions by intermediate-mass
asymptotic giant branch stars[9] or fast rotating massive stars[20]. However,
it should be emphasized that our $hk$ measurements for RGB stars in M22,
NGC1851 and other GCs show discrete or bimodal distributions in calcium
abundance, which cannot be supplied by intermediate-mass asymptotic giant
branch stars or fast rotating massive stars.
As shown in Fig 3, the difference in calcium, silicon, titanium and iron
abundances between the calcium weak (Ca-w hereafter) group with smaller $hk$
index and the calcium strong (Ca-s hereafter) group with larger $hk$ index in
M22 and NGC1851 suggests that they are indeed chemically distinct[24, 22, 21,
23]. (It is not shown in the figure but europium also has a bimodal abundance
distribution in M22, in the sense that the Ca-s group has a higher europium
abundance.) As for the origin of chemical inhomogeneity in globular clusters,
at least four viable chemical enrichment mechanisms have been proposed up to
date. They are, in the order of time required to emerge; (i) fast rotating
massive stars, (ii) Type II supernovae, (iii) intermediate-mass asymptotic
giant branch stars, and (iv) Type Ia supernovae. If the current understanding
of supernovae explosions is correct, only Type Ia and II supernovae can supply
the heavy elements such as calcium and iron[10]. To explain the discrete
calcium abundances seen in M22 and NGC1851, however, the contribution from
Type Ia supernovae can be ruled out for two reasons. First, the longer
timescale ($\geq$ 1 – 2 Gyr) before the onset of Type Ia supernova explosions,
which would produce detectable age spread between two populations; and second,
the enhanced $\alpha$-elemental abundances, indicative of absence of
contributions from Type Ia supernovae[10]. Qualitatively, the differences in
elemental abundances between the two stellar populations in M22 and NGC1851
can be naturally explained by invoking chemical enrichment by Type II
supernovae, where $\alpha$-elements (silicon, calcium, and titanium) and
$r$-process element (europium) are dominantly produced. However, our results
do not necessarily imply that Type II supernovae are solely responsible for
the chemical enrichment in M22 and NGC1851, since all four above-mentioned
mechanisms may be involved. We emphasize that the crux of our results is the
undeniable evidence for Type II supernovae contribution to chemical enrichment
of some globular clusters, in sharp contrast to the widely accepted idea of
chemical pollution only by intermediate-mass asymptotic giant branch or fast
rotating massive stars, with which the chemical enhancement of the $\alpha$\-
and $r$-process elements in the second generation of the stars cannot be
easily explained.
More than half of 37 globular clusters in our sample shows discrete or broad
distributions of the $hk$ index in their RGB sequences. In Fig 2, We show
color-magnitude diagrams for some of exemplary globular clusters in the order
of $hk$ widths of RGB sequences at $V_{HB}$, the $V$ magnitude level at the
horizontal branch: $\omega$ Cen, M22, NGC1851, NGC2808, M4, M5, NGC6752 and
NGC6397 (see also Supplementary Table 3 and Figs 6 – 13). NGC2808 is known to
have multiple main-sequences but no multiple RGB sequences have been reported
to date. Our new results show that NGC2808 shows at least two discrete RGB
sequences with a large spread in calcium abundance. Similarly, M5 has very
broad $hk$ index in the RGB sequence and NGC6752 shows discrete RGB sequences.
It is interesting to note that all the globular clusters with signs of
multiple stellar populations have relatively extended horizontal branch, while
the globular clusters with normal horizontal branch (e.g. NGC6397 in Fig 2 and
Supplementary Fig 13) show no spread or split in RGB. This is consistent with
the suggestion that the extended horizontal branch is a signal of the presence
of multiple stellar populations in globular clusters[25].
The overwhelming problem of the chemical enrichment by Type II supernovae in
globular clusters is that their ejecta are considered to be too energetic to
be retained by less massive systems like typical Galactic globular clusters
($\leq$ a few times 105 $M_{\odot}$)[11]. Our results therefore suggest that
M22, NGC1851 and other globular clusters with RGB split were much more massive
in the past, unless the current understanding of supernovae explosions is in
great error. Perhaps, these globular clusters were once nuclei of dwarf-
galaxy-like fragments and then accreted and dissolved in the Milky Way, as is
widely accepted for $\omega$ Cen[1, 2, 26]. Recent calculations suggest that a
massive ($\geq$ a few times $10^{6}$ $M_{\odot}$) star cluster embedded in a
proto-dwarf galaxy could accrete gas from its host dwarf galaxy which may
cause the formation of the second generation stars, producing multiple stellar
populations[27]. Note that this scenario is also suggesting that the globular
clusters with multiple stellar populations would be the remaining cores of the
proto-galactic building blocks. This idea is supported by the recent
investigations of the extended horizontal branch globular clusters (i.e.
globular clusters with signatures of multiple stellar populations), which has
shown that extended horizontal branch globular clusters are clearly distinct
from the normal globular clusters in orbital kinematics and mass[25].
Extensive photometric surveys for fainter stars in these globular clusters, as
well as spectroscopic surveys for stars in double RGB sequences, would
undoubtedly help to shed more light into the discovery reported here.
## References
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is linked to the online version of the paper at www.nature.com/nature.
J.-W. L. thanks A. Walker for providing the CTIO Ca filter transmission curve,
D. Yong for NGC1851 spectroscopic data before publication and A. Yushchenko
for discussions on spectrum synthesis. Support for this work was provided by
the National Research Foundation of Korea to the Astrophysical Research Center
for the Structure and Evolution of the Cosmos (ARCSEC). This work was based on
observations made with the CTIO 1.0-m telescope, which is operated by the
SMARTS consortium.
J.-W. L. performed observations, data analysis, interpretation, model
simulations and writing of the manuscript. Y.-W. K. participated in
observation planning, J. L. performed a part of observations and data
analysis. Y.-W. L. performed interpretation and writing of the manuscript. All
authors discussed the results and commented on the manuscript.
Reprints and permissions information is available at www.nature.com/reprints.
The authors declare that they have no competing financial interests.
Correspondence should be addressed to J.-W. L. (jaewoolee@sejong.ac.kr) or
Y.-W. L. (ywlee2@yonsei.ac.kr).
Figure 1: Color-magnitude diagrams for M22. a, $V$ versus $b-y$; b, $V$ versus
$hk$. In b, note the distinct and discrete double RGB sequences in M22. This
cannot be due to differential reddening effect across the cluster or the
contamination from the off cluster field but, is due to the difference in
calcium abundance, which was synthesized in supernovae, between the two RGB
sequences. The number ratio between the Ca-w group with smaller $hk$ index and
the Ca-s group with larger $hk$ index is about 70:30. Black arrows in each
panel denote reddening vectors. Figure 2: Color-magnitude diagrams for
$\omega$ Cen, M22, NGC1851, NGC2808, M4, M5, NGC6752 and NGC6397. Note that,
while the distributions of the RGB sequences in the $b-y$ color are relatively
narrow, those in the $hk$ index are either discrete or broad. This is evidence
for the multiple stellar populations with distinct calcium abundances. Among
these globular clusters, NGC6397 appears to be the only normal globular
cluster with simple population (i.e. coeval and monometallic). Figure 3:
Differences in chemical compositions between double RGB sequences in M22 and
NGC1851. a, b, Black ‘plus’ signs denote stars in M22 with proper motion
membership probabilities $P$ $\geq$ 90%; blue filled diamonds and red filled
circles denote RGB stars studied with high-resolution spectroscopy in the Ca-w
and the Ca-s groups, respectively[23, 24]. The green solid line denotes the
fiducial sequence of RGB stars and $\Delta hk$ denotes the difference in the
$hk$ index against the fiducial sequence. The double RGB sequences persist in
proper motion member stars. c – f, Comparisons of elemental abundances between
the Ca-w and the Ca-s groups in M22. Solid lines denote the mean values, and
dashed lines denote standard deviations of each group. The Ca-s group has
higher $\alpha$-elements (Si, Ca, and Ti) and iron abundances, which must be
supplied by numerous Type II supernova explosions. g, h, Black ‘plus’ signs
denote stars in NGC1851; blue filled diamonds and red filled circles denote
RGB stars studied with high-resolution spectroscopy in the Ca-w and the Ca-s
groups, respectively[21, 22]. i – l, As c – f but for NGC1851.
Supplementary Information
## 1 The CTIO Ca Filter System
### 1.1 The Filter Transmission
The Ca filter system was designed to include Ca II H and K lines at $\lambda$
3968 and 3933 Å, respectively, with a full-width half maximum (FWHM) of
approximately 90 Å. The CN band absorption strengths at $\approx$ $\lambda$
3885 Å are often very strong in stellar spectra and the lower limit of the Ca
filter is set to avoid contamination by the CN band12. The Ca filter used at
Cerro Tololo Inter-American Observatory (CTIO) has a similar FWHM,
approximately 90 Å, but its passband is shifted approximately 15 Å to the
longer wavelength (Alistair Walker, private communication) compared to that in
Anthony-Twarog _et al._ 12 In Supplementary Figure 1, we show the transmission
functions of the both Ca filters. In the figure, we also show synthetic
spectra for the CN normal and the CN strong RGB stars in typical intermediate
metallicity globular clusters (GCs) as an illustration. The effect of the CN
band on the $hk$ index is negligible as will be discussed below.
### 1.2 The Central Wavelength Drift of the Ca Filter
The CTIO Ca filter transmission function shown in Supplementary Figure 1 is
that measured with a collimated beam. It is known that the passband of the
narrow band interference filter depends on the angle of the incidence beam
following,
$\lambda=\lambda_{0}\left(1-\frac{\sin^{2}\beta}{n^{*2}}\right)^{1/2},$ (1)
where $\lambda_{0}$ is the wavelength of peak transmittance at normal
incidence, $\beta$ is the angle of incidence of the collimated beam on the
filter and $n^{*}$ is the effective refractive index of the filter[28].
Therefore, when the Ca filter is used with a fast telescope, the filter
passband can be significantly different from that shown in Supplementary
Figure 1. The CTIO 1-m telescope used for our survey is a slow telescope with
$f$/10.5 and the effect resulted from the angular dependency of a converging
beam is expected to be very small. Assuming $\beta$ $\approx$ 1/21 radians for
the converging beam at the CTIO 1-m telescope and $n^{*}$ $\approx$ 1.4 for
the CTIO Ca filter, the peak wavelength of the Ca filter will be shifted by
2.3 Å to the shorter wavelength. Given the much larger FWHM of the CTIO Ca
filter, this may contribute small effect. We investigate contributions to the
$hk$ index resulted from the shifted Ca passband using synthetic spectra for
typical intermediate metallicity RGB stars in our GCs. Our calculations
integrating over the filter transmission curve show that this effect
contributes no larger than 0.011 mag to our $hk$ measurements, in the sense
that the shifted Ca passband to the shorter wavelength gives slightly larger
$hk$ values. We emphasize that, since our results are based on a single
instrument setup (the same telescope, filters and the CCD camera) during the
observations of our science targets and the photometric standards, this effect
is expected to be cancelled out during our photometric calibrations. Also
importantly, our main results presented here rely on the split or the spread
in the $hk$ index of RGB stars of an individual GC. Therefore, the shifted Ca
passband affects similar degree to the $hk$ index of the RGB stars in a GC and
does not contribute to the apparent RGB split or the spread in the $hk$ index
of an individual GC.
### 1.3 Effect of radial motions and internal velocity dispersions of GCs
The mean radial motion of 139 GCs in our Milky Way Galaxy[29] is $|v_{r}|$ =
110 km/s, equivalent to the wavelength shift by $|\Delta\lambda|$ $\approx$
1.4 Å at $\lambda$ 3950Å. We calculate the contribution due to the mean radial
motion of GCs to the $hk$ index using the shifted CTIO Ca passband and
synthetic spectra for typical intermediate metallicity GC RGB stars. We find
that the net effect is negligibly small, $|\Delta hk|$ $<$ 0.006 mag. Among
our eight GCs, NGC1851 has the largest radial velocity, $v_{r}$ = 321 km/s,
equivalent to the wavelength shift by $\Delta\lambda$ = 4.2 Å to the longer
wavelength at $\lambda$ 3950Å. We calculate the contribution due to the radial
motion of NGC1851 using the shifted CTIO Ca passband and the red-shifted
synthetic spectra with a fixed CN abundance for the cluster. We obtain $\Delta
hk$ $\approx$ 0.015 mag, in the sense that the red-shift is resulted in a
slightly larger $hk$ index. As we discussed above, the difference in the $hk$
index due to the high radial motion of NGC1851 does not affect our results
presented here, since the $hk$ indices of RGB stars in NGC1851 will be
affected by similar degree and the mean radial motion of the cluster does not
produce an apparent split or a spread in the $hk$ index. Perhaps, this effect
may become important in the inter-cluster comparisons, which is beyond the
scope of our study.
What concerns us most about the high radial velocities of some GCs, in
particular for red-shift, is the potential contamination by the strong CN band
absorption features at $\lambda$ 3885 Å as shown in Supplementary Figure 1.
For example, NGC1851 has a bimodal CN distribution and some RGB stars show
very strong CN band absorption strengths[30]. Due to its high radial velocity
away from us (i.e. red-shifted), the CN band absorption features in the CN-
strong RGB stars may affect the $hk$ index and, subsequently, may produce an
apparent RGB split of the cluster as shown in Figure 2 or Supplementary Figure
8. We calculate the CN band contributions using the shifted CTIO Ca passband
and the red-shifted synthetic spectra for the CN-normal and the CN-strong RGB
stars (see discussion below). Our calculations suggest that the net effect is
negligibly small, $\Delta hk$ $\leq$ 0.003 mag, and the high radial velocity
of NGC1851 combined with a bimodal CN distributions does not produce the RGB
split in the $hk$ index.
We also investigate the effect of the internal velocity dispersion of an
individual GC. Assuming $\sigma_{LOS}$ = 15 km/s, equivalent to
$\Delta\lambda$ $\leq$ 0.2 Å at $\lambda$ 3950Å, we obtain $\Delta hk$ $\leq$
0.001 mag following the same method described above, and the effect from the
internal velocity dispersions of GCs does not affect our results.
### 1.4 Summary of uncertainties on the $hk$ index
Supplementary Table 1 summarizes the uncertainties in our $hk$ index
measurements relevant to the CTIO Ca passband. (The variations in the $hk$
index due to differences in elemental abundances of GC RGB stars will be
discussed below.) As discussed above, the effects due to the shifted CTIO Ca
passband and the radial motions of GCs do not affect our results, since both
effects contribute similar degree to the $hk$ indices among RGB stars in a GC.
(i.e. They only affect the zero point of the $hk$ index and they do not affect
the $\Delta hk$ distributions). In addition to our photometric measurement
errors which will be discussed below, the effects due to the internal velocity
dispersions of GCs and the differential interstellar Ca II absorption (see
discussion below) can affect our $hk$ index measurements. However, their
contributions to our $hk$ measurements are no larger than 0.022 mag and they
do not affect our main conclusion presented here. Therefore, our results
strongly suggest that the split or the spread in the $hk$ index of RGB stars
in GCs are related to the variations in elemental, in particular calcium,
abundances among RGB stars in a GC, which will be discussed below.
## 2 The Double RGB Sequences of M22
### 2.1 Differential Reddening Effect on the Double RGB Sequences in M22
The continuous interstellar extinction by the interstellar dust and the
discrete interstellar line extinction by the interstellar Ca II atoms may
affect our main results. We considered both effects and will discuss that the
RGB split of M22 in the $hk$ index is indeed due to the difference in calcium
abundances between two stellar populations in M22 and other explanations are
highly unlikely. Also both effects tend to produce spreads in RGB sequences
rather than the distinct and discrete RGB sequences of GCs reported here.
The differential continuous reddening across the cluster can thicken the
apparent RGB sequence of GCs in broad-band optical photometry[31]. In contrast
to other color indices being used in broad-band photometry, the $hk$ index is
known to be insensitive to interstellar reddening12, $E(hk)/E(b-y)$ = $-$0.16
and $E(hk)/E(B-V)$ = $-$0.12. The difference in the $hk$ index between the two
RGB sequences in M22 is about 0.2 mag at the magnitude level of the horizontal
branch. If this $hk$ split is only due to differential reddening effect, we
would expect even larger separation of the two RGB sequences in the $b-y$
color and the $V$ magnitude. The reddening correction value in the $b-y$
color, $E(b-y)$, for the Ca-s group is about $-$1.25 mag, equivalent to
$E(B-V)$ = $-$1.69 mag assuming $E(b-y)/E(B-V)$ = 0.74, making the RGB stars
in the Ca-s group too hot to be RGB stars (see Supplementary Figure 2 – d). At
the same time, the extinction correction value in the $V$ magnitude is $-$5.24
mag, assuming $A_{V}$ = 3.1$\times E(B-V)$, for the Ca-s RGB stars. Applying
this large extinction correction makes the RGB stars in the Ca-s group too
bright to be members of M22 (see Supplementary Figure 2 – e & f). We emphasize
that both the Ca-w and the Ca-s groups are proper motion members of the
cluster as shown in Figure 3. Note also that the reddening vector (see Figure
1 or Supplementary Figure 4) is almost parallel to the slopes of HB and RGB in
the $hk$ versus $V$ CMD, and thus the differential reddening can not produce
the RGB split. Therefore, continuous differential reddening effect can be
completely ruled out to explain the observed bimodal RGB sequences in M22.
Similarly, the interstellar reddening toward NGC1851 is very small, $E(B-V)$ =
0.02 mag[29], but the $hk$ split in RGB stars of the cluster is as large as
0.2 mag (see Supplementary Figure 8), which can not be explained by
differential continuous reddening effect22.
The previous study for the GCs showed that the equivalent width of the
interstellar Ca II K absorption line strength can be as large as several times
100 mÅ[32]. The interstellar Ca II atom is thought to be heavily depleted on
to dust in denser clouds[33], which may cause small-scale differential
discrete reddening effect across M22 and other GCs studied here (see also
Andrew _et al._[34] for the small-scale variations of interstellar Na I D
lines111Note that the number of interstellar Na I atoms appears to be about a
factor of ten larger than that of interstellar Ca II atoms[33]. toward the
less extincted globular cluster M92). We generate synthetic spectrum to
surrogate interstellar Ca II H & K absorption lines. We adopt a gaussian line
profile with a FWHM of 1 Å, equivalent to $\Delta v_{r}$ $\approx$ 76 km/s,
and we assign equivalent widths of 350 mÅ and 650 mÅ for the interstellar Ca
II H & K lines, respectively. Our synthetic spectrum is shown in Supplementary
Figure 1 – (c). Assuming they are linear part of the curve of growth[35], the
column density of the interstellar Ca II can be estimated as
$N({\rm Ca~{}II})=1.13\times 10^{20}\frac{EW}{\lambda^{2}f},$ (2)
where $EW$ and $\lambda$ are the equivalent width and wavelength in Å and $f$
is the oscillator strength. Using the oscillator strengths of 0.681 and 0.341
for Ca II H & K, respectively, the column density for interstellar Ca II is
$\log N$(Ca II) $\approx$ 12.8 cm-2, equivalent to $\Delta E(B-V)$ $\approx$
0.32 mag[33]. If this large amount of small-scale interstellar Ca II variation
exists among our GCs studied here, how much will it affect our $hk$ index
measurements? We calculate the $Ca$ magnitudes with and without the
interstellar Ca II variations using the shifted CTIO Ca transmission function.
The difference in the $Ca$ magnitude (i.e. in the $hk$ index since the
interstellar Ca II H & K lines do not affect $b$ or $y$ passbands) is only
0.010 mag and, therefore, the differential discrete reddening effect due to
the variations in the interstellar Ca II abundances can be completely ruled
out to explain the GC RGB splits in the $hk$ index.
### 2.2 The Spatial Distributions
In Supplementary Figure 3, we show the spatial distributions of Ca-w and Ca-s
RGB stars in M22. As can be seen in the figure, each population does not show
any spatially patched features, supporting our results that differential
reddening is not responsible for the RGB split in M22.
### 2.3 Contamination from the Milky Way’s Bulge Population
M22 is located in the direction of the Milky Way’s bulge and the contamination
from the bulge population may affect our results. However, this is very
unlikely, since the proper motion member RGB stars show discrete double RGB
sequences as shown in Figure 3. In addition, the bulge RGB stars are located
farther from the Sun, more metal-rich and suffering from heavier interstellar
reddening than those in M22 are. In Supplementary Figure 4, we compare M22
CMDs with those of two bulge fields (NGC6528 and OGLEII - 12). As can be seen
in the figure, the RGB stars in the bulge are fainter and redder than those in
M22 are and the contamination from the Milky Way’s bulge population does not
affect our results.
### 2.4 Effects of Metal Contents and Helium Abundances on the M22 RGB
As shown in Figure 3, the stars in the Ca-s group are about 0.2 dex more
metal-rich than those in the Ca-w group. It is suspected that this large
metallicity spread may produce any detectable discrepancy in stellar
evolutionary sequences, in particular for RGB sequence, between two stellar
populations based on broad-band photometry. To explore metallicity effect on
the RGB sequence, we compare $BV$ CMD by Monaco _et al._[31] with the latest
$Y^{2}$ isochrones (Version 3, Yi _et al._ in preparation). In Supplementary
Figure 5, we show model isochrones for [Fe/H] = $-$1.6 and $-$1.4 with the
helium abundance of $Y$ = 0.23 and the age of 11 Gyr, using the reddening
value and the distance modulus for the cluster from Harris[29]. Although the
split in the RGB sequences of two model isochrones is noticeable, the
discrepancy in the RGB sequence does not appear to cause a serious problem to
explain the $BV$ CMD by Monaco _et al._[31] Note that the $V$ magnitude
difference in the sub-giant branch between two model isochrones can be as
large as 0.2 mag, apparently consistent with recent HST/ACS observations of
the cluster[36].
As inferred from the extended HB (EHB) morphology of M22, the second
generation of the stars is expected to have enhanced helium abundance by
$\Delta Y\approx$ 0.05[37]. Since the new version of $Y^{2}$ isochrones
provides models with enhanced helium abundances, we investigate the effect of
helium abundance on the evolutionary sequence. As illustrated in Supplementary
Figure 5 – (c), the discrepancy between two stellar populations alleviates due
to the opposite effect of metal contents and helium abundances on the RGB
temperature. Since the second generation of the stars in M22 shows signs of
the chemical enrichment by Type II supernovae and intermediate-mass asymptotic
giant branch (AGB) stars, the second generation of stars may be slightly
younger than the first generation. Assuming the age difference of 1 Gyr
between two stellar populations, two model isochrones are in excellent
agreement except for bright RGB sequence, where the observed number of stars
is small.
It is intriguing to note that the number ratio between the Ca-w and the Ca-s
RGB stars (70:30) found here is very similar to those found between (1) the
two stellar groups with different [Fe/H] and [$s$-process/Fe] ratios6, (2) the
brighter SGB and the fainter SGB stars[36], and (3) the two groups of HB stars
with the bluer HB being less populated. The population synthesis models (Han
_et al._ 2009, in preparation) suggest that this can be naturally reproduced
by the enhanced metal and helium abundances in the second generation of stars.
## 3 The _hk_ and Metallicity Distributions of GCs
### 3.1 Observations
In Supplementary Table 2, we show the journal of observations for eight GCs.
They were observed under the photometric weather conditions and, for most
cases, the median seeing was about 1.5 – 1.6 arcsec during our observations.
Note that the RGB stars in NGC2808 are roughly 21 times fainter than those in
NGC6397 in the Ca passband for a fixed magnitude, while our total integration
time for NGC2808 is only about three times longer than that for NGC6397 in the
Ca passband. Statistically, the lack of total integration time for NGC2808
will be resulted in a $\approx$ 2.6 times larger $Ca$ measurement error than
that expected for NGC6397 at a given $Ca$ magnitude, particularly for fainter
stars. Although our survey relied on a rather small telescope through a rather
narrow filter at a rather blue wavelength222Fortunately, the CCD camera used
in our survey has rather high quantum efficiency (QE) at shorter wavelength
with QE $\approx$ 0.686 at $\lambda$ 3800Å and $\approx$ 0.770 at $\lambda$
4000Å. (see http://www.astronomy.ohio-state.edu/Y4KCam/OSU4K/index.html#DQE).,
our investigations of the multiple stellar populations of GCs are focused on
bright RGB stars, where the numbers of photon in the Ca passband are enough so
that the measurement errors, including propagation errors during the
photometric calibrations, are less than 0.020 mag (see Supplementary Table 3).
### 3.2 Color Distributions of Bright RGB Stars
Here, we investigate the $b-y$ color and the $hk$ index distributions of RGB
stars brighter than $V-V_{\rm HB}$ = 1.0 mag. We derive lower order ($\approx$
4 – 5) polynomial fits, which are forced to pass through the peak $b-y$ colors
or the peak $hk$ indices of given magnitude bins, as fiducial sequences for
eight GCs and then we calculate differences in the $b-y$ color, $\Delta(b-y)$,
or the $hk$ index, $\Delta hk$, with respect to fiducial sequences of each GC.
We show our results in Supplementary Figures 6 – 13. In the figures, the blue
horizontal bars denote the mean measurement errors including propagation
errors during the photometric calibrations with a 2$\sigma_{*}$ range ($\pm$
1$\sigma_{*}$) for individual stars at given magnitude bins.
In Supplementary Table 3, we show comparisons of the observed FWHMs of RGB
stars and the measurement errors at the magnitude level of horizontal branch
stars, $V_{HB}+0.5$ $\geq$ $V$ $\geq$ $V_{HB}-0.5$, in each GC. To calculate
the FWHMs of RGB stars in GCs, we used the following relation,
${\rm FWHM(RGB)}\approx 2.3548\times\sigma_{\Delta},$ (3)
where $\sigma_{\Delta}$ is the standard deviation of RGB stars in the
$\Delta(b-y)$ or the $\Delta hk$ distributions in a fixed magnitude bin. Note
that the FWHM of RGB stars in the $hk$ index is slightly larger than the $hk$
separation between the double populations such as in M22, NGC1851, NGC2808, M4
and NGC6752. The mean measurement errors, $\sigma_{*}(b-y)$ or
$\sigma_{*}(hk)$, given in the table include propagation errors during the
photometric calibrations and they are those for individual stars. Therefore,
the mean measurement errors for an individual population in GCs,
$\sigma_{p}(b-y)$ or $\sigma_{p}(hk)$, will be given by
$\approx\sigma_{*}(b-y)$/$\sqrt{n_{p}}$ or $\sigma_{*}(hk)$/$\sqrt{n_{p}}$,
where $n_{p}$ ($\geq$ 20) is the number of stars in each population. Note
that, while the FWHMs of most GCs have much larger values [$\geq$
8$\sigma_{*}(hk)$] than the measurement errors for individual stars in the
$hk$ index, the FWHM of NGC6397 is comparable in size to the measurement error
in the $hk$ index, consistent with the idea that NGC6397 is the only normal GC
with a simple stellar population (i.e. coeval and monometallic) in our sample.
Also note that NGC6397 shows similar degree of the full RGB widths in the
$\Delta(b-y)$ and the $\Delta hk$ distributions (see Supplementary Table 3 and
Supplementary Figure 13).
The last two columns of Supplementary Table 3, $E(b-y)_{1/2}$ and
$E(hk)_{1/2}$, are for the contributions due to the continuous differential
reddening effect assuming a 50% variation in the total interstellar reddening
across each GC. The observed FWHMs of GCs in the $(b-y)$ color and in the $hk$
index can not be explained simultaneously, similar to what shown in
Supplementray Figure 2. Therefore, the continuous differential reddening
effect can be ruled out to explain the differences between the observed
FWHMs(RGB) and the measurement errors in GCs.
### 3.3 $\Delta hk$ as a probe of multiple stellar populations in GCs
For $\omega$ Cen, M22 and NGC1851 (see Figure 3 and Supplementary Figure 15),
when the two subpopulations are defined by our $hk$ index (or our $\Delta hk$
distribution), we can also see the clear division in spectroscopic elemental
abundances. Similarly, we will discuss that the split or the spread in the
$\Delta hk$ distributions of RGB stars in other clusters can provide a
powerful method to probe the multiple stellar populations in GCs.
The calcium abundance is the major factor that determines the $hk$ index or
the $\Delta hk$ distribution of RGB stars in a GC (see discussion below) and
Type II supernovae are responsible for the calcium enrichment in a GC. As
discussed, however, our results do not imply that Type II supernovae are
solely responsible for the chemical enrichment in GCs. In an attempt to
explain the observed large star-to-star lighter elemental abundance variations
(in particular O and Na) in GCs, chemical pollution by intermediate mass AGB
stars9 or fast rotating massive (FRM) stars20 has been widely accepted. It
should be reminded that, however, neither AGB nor FRM scenarios can explain
the chemical enrichment of the $\alpha$\- and $r$-process elements in the
second generation of the stars. It is most likely that all three
aforementioned mechanisms (and perhaps including Type Ia supernovae) are
required to explain elemental abundance patterns found in GCs. In addition to
the chemical enrichment by Type II supernovae, which is the main results
presented here, if the second generation of the stars in some of our GCs have
experienced the chemical pollution by intermediate-mass AGB or FRM stars, the
lighter elemental abundances, such as oxygen and sodium, between the two
generatrions of stars must have been different. Furthermore, the variations in
[O/Fe] and [Na/Fe] can be as large as 1 dex in some GCs7 and the differences
in the oxygen and sodium abundances are easily detectable compared to those in
the heavy elements, such as calcium and iron.
During the last few years, tremendous amount of effort has been directed at
spectroscopic study of RGB stars in GCs, in particular, using the multi-object
spectrograph mounted at VLT. Among our eight GCs, NGC2808[38], M4[39] and
NGC6752[40] have been studied using moderately high resolution spectra for
more than 100 RGB stars. In Supplementary Figure 14, we show comparisons of
$\Delta hk$ versus O, Na and Fe distributions of the clusters. In panel (a),
we show the plot of $V-V_{\rm HB}$ versus $\Delta hk$ for NGC2808 RGB stars.
In the figure, the plus signs denote the RGB stars with known [O/Fe] and
[Na/Fe] ratios[38]. From the $\Delta hk$ distribution of RGB stars shown in
panel (b), we define the boundary at $\Delta hk$ = $-$0.05 mag (the vertical
dashed line) assuming that NGC2808 has two major stellar populations as shown
in panel (b) or Supplementary Figure 9. Similar to the procedure employed in
M22 and NGC1851 (see Figure 3), we define the Ca-w group with smaller $hk$
index and the Ca-s group with larger $hk$ index and they are denoted by the
blue and the red plus signs, respectively, in panel (a). In panels (c), (d)
and (e), we show the [O/Fe], [Na/Fe] and [Fe/H] distributions for each group,
where the shaded histograms outlined with blue color are for the Ca-w group
and the blank histrograms outlined with red color are for the Ca-s group. The
Ca-w group has a higher mean oxygen and a lower mean sodium abundances, while
the Ca-s group has a lower mean oxygen and a higher mean sodium abundances,
indicative of the presence of the proton-capture process at high temperature
between the two formation epochs presumably via intermediate-mass AGB or FRM
stars, where oxygen is depleted by the CNO cycle while sodium is enriched from
the 22Ne + 1H $\rightarrow$ 23Na reaction. Our results strongly suggest that
they are truly different stellar populations and not related to, for example,
our photometric measurement errors and differential reddening effect: the Ca-w
group is the first generation of stars while the Ca-s group is the second
generation of stars enriched by Type II supernovae (e.g. calcium) and
intermediate-mass AGB or FRM stars (e.g. sodium). Although the difference in
the [Fe/H] distributions between the two groups does not appear to be as
compelling as those in the [O/Fe] and the [Na/Fe] distributions, the Ca-w
group has a slightly lower mean metallicity than the Ca-s group does. We
performed non-parametric Kolmogorov-Smirnov (K-S) tests to see if the [Fe/H]
distributions of the two populations in NGC2808 are drawn from the same parent
population. Our calculation shows that the probability of being drawn from
identical stellar populations is 5.5% for NGC2808, suggesting that they have
different parent populations.
The same results can be found in M4 and NGC6752. From the comparisons of the
[O/Fe] and the [Na/Fe] distributions between the Ca-w and the Ca-s groups, it
can be seen that the Ca-w groups are the first generations of stars while the
Ca-s groups are the second generations of stars in the clusters. We also
performed K-S tests for the [Fe/H] distribution of M4, we obtained that the
probability of being drawn from identical parent populations is 5.5% for M4,
indicating that each subpopulation in M4 has different parent populations.
### 3.4 Recalibration of [Fe/H]hk Based on RGB Stars in $\omega$ Cen and
Metallicity Distributions of Eight GCs
In our previous study for NGC1851, we showed that the $hk$ index traces the
calcium abundance and, furthermore, it can provide a very powerful method to
distinguish multiple stellar populations in GCs22. However, it can be seen
that the full range of $\Delta hk$ increases with the luminosity of RGB stars
(i.e. different temperature or surface gravity), in particular, in $\omega$
Cen and M22. Due to the temperature dependency on the $hk$ index versus
metallicity relation, the $\Delta hk$ distributions cannot be directly
translated into the absolute metallicity scale. Therefore, we calculate the
photometric metallicity, [Fe/H]hk, of individual RGB stars in eight GCs using
the [Fe/H] relations on the $hk_{0}$ versus $(b-y)_{0}$ plane12,22.
Recently, Johnson _et al._ 5 studied elemental abundances, including calcium,
of large sample of RGB stars in $\omega$ Cen using moderately high resolution
spectra (R $\approx$ 18,000). Since $\omega$ Cen contains multiple stellar
populations with very broad metallicity range, $\Delta$[Fe/H] $\approx$ 1.5
dex, comparisons of our results of RGB stars in $\omega$ Cen with those of
Johnson _et al._ may provide an wonderful opportunity to assess our
photometric metallicity scale using the $hk$ index, [Fe/H]hk. In Supplementary
Figure 15, we show elemental abundances of 40 RGB stars in $\omega$ Cen
studied by Johnson _et al._ as a function of $\Delta hk$. As shown in the
figure, [Ca/H] and [Fe/H] appear to be well correlated with $\Delta hk$,
indicating that $\Delta hk$ can truly be treated as the relative calcium
abundance or metallicity indicators for RGB stars with similar luminosities in
a GC. We also show plots of [Fe/H]spec versus [Fe/H]hk and [Ca/H]spec versus
[Fe/H]hk for 32 RGB stars with sufficiently high signal-to-noise ratios
($\geq$ 100). We derive linear fits to each relation and we find
$\rm{[Fe/H]}_{\rm
spec}=0.533\rm{[Fe/H]}_{hk}-0.775~{}~{}~{}~{}~{}~{}(\sigma=0.087\rm{dex}),$
(4)
and
$\rm{[Ca/H]}_{\rm
spec}=0.587\rm{[Fe/H]}_{hk}-0.403~{}~{}~{}~{}~{}~{}(\sigma=0.106\rm{dex}).$
(5)
We recalibrate our photometric metallicity using the equation (4),
[Fe/H]hk,corr, and we derive metallicity distribution functions (MDFs) for
eight GCs. During our calculations of MDFs, we use RGB stars with $-$2.0
$\leq$ $V$ $-$ $V_{\rm HB}$ $\leq$ $-$0.5 mag in order to minimize
contamination from off-cluster field and red-clump populations. We show our
results in Supplementary Figure 15. As expected from the $\Delta hk$
distributions, the signs of multiple stellar populations persist in our MDFs
for most GCs.
Finally, cautions are advisable on our MDFs of GCs. Our metallicity scale is
not on the traditional Zinn & West14 scale, therefore our MDFs for GCs can be
different from those from other photometric or spectroscopic studies. RGB
stars in $\omega$ Cen of Johnson _et al._ have different individual elemental
abundances, which were not taken into consideration in our [Fe/H]spec versus
[Fe/H]hk or [Ca/H]spec versus [Fe/H]hk relations. Furthermore, each GCs may
have slightly different elemental abundance ratios and our calibrated
photometric metallicities for GCs would be affected. However, it should be
emphasized that the crux of our results is the split or the spread in the $hk$
index in the RGB stars of individual GCs, which is insensitive to other
elemental abundances except calcium as will be discussed below.
## 4 The Influence of Elemental Abundances on the _hk_ Index
The realistic modeling of the resonance Ca II H & K lines requires proper
understanding of stellar atmospheres, including chromospheres, and
hydrodynamic non-local thermodynamic equilibrium treatments, which have posed
difficult problems for decades[41]. Here, we demonstrate that the calcium
abundance is the major factor that determines the $hk$ index of RGB stars
using 1-dimensional plane-parallel stellar atmospheres[42].
### 4.1 Calcium
Using the model atmosphere for the RGB star at the magnitude level of the
horizontal branch with $T_{\rm eff}$ = 4750 K, $\log g$ = 2.0 (in cgs unit),
$v_{\rm turb}$ = 2.0 km/s, [Fe/H] = $-$1.6, we calculate synthetic spectra for
[Ca/Fe] = 0.25, 0.30, 0.35, 0.40, 0.45 and we show some of our synthetic
spectra in Supplementary Figure 17. We convolve the filter transmission
functions with synthetic spectra and we obtain the calcium abundance
sensitivity on the $hk$ index, $\partial(hk)$/$\partial$[Ca/H] $\approx$ 0.422
mag/dex. Note that this result is based on the fixed model parameters, such as
$T_{\rm eff}$, $\log g$, $v_{\rm turb}$, and [Fe/H], except calcium abundance.
To interpret observed $\Delta hk$ between the two stellar populations in a GC
in terms of different calcium abundances, proper atmospheric parameters should
be taken into consideration. For example, the two stellar groups with
different [Fe/H] and [$s$-process/Fe] ratios in M22 by Marino _et al._ 6 have
slightly different elemental abundances and temperatures. The stars in the
metal-poor group by Marino _et al._ have $\langle$[Fe/H]$\rangle$ = $-$1.82,
$\langle$[Ca/Fe]$\rangle$ = +0.25 and those in the metal-rich group have
$\langle$[Fe/H]$\rangle$ = $-$1.68, $\langle$[Ca/Fe]$\rangle$ = +0.35. The
mean temperature of the stars in the metal-poor group is $\sim$ 100 $\pm$ 42 K
hotter than those in the metal-rich group. Using these atmospheric parameters,
we obtain $\Delta hk$ = 0.121 $\pm$ 0.072 mag, which is apparently consistent
with the double peaks in the $\Delta hk$ distribution of M22 within the error
as shown in Supplementary Figure 7.
### 4.2 Helium
As shown in Supplementary Figure 5, helium is very important in stellar
structure and evolution. Very unfortunately, however, there is no direct
method to measure helium abundances of stars in GCs. As we discussed, all the
GCs with signs of multiple stellar populations have relatively EHB, for
example, the second generation of the stars in M22 is expected to have
enhanced helium abundance by $\Delta Y$ $\approx$ 0.05 inferred from its EHB
morphology. Using the model atmospheres with enhanced helium abundance ($Y$
$\approx$ 0.35, equivalent to $\Delta Y$ $\approx$ 0.10) by
Castelli333http://wwwuser.oat.ts.astro.it/castelli/grids.html, we obtain
$\Delta hk$ $\approx$ $-$0.002 mag, in the sense that the $hk$ index decreases
as helium abundance increases, and thus the effect of enhanced helium
abundance on the $hk$ index is negligible. Given the cool temperatures of RGB
stars in GCs, the helium enhancement by $\Delta Y$ = 0.05 – 0.10 does not
appear to be important.
### 4.3 CNO
It is well-known fact that many GCs show large star-to-star elemental
abundance variations. In particular, almost all GCs show variations in the CNO
abundances resulted from the internal evolutionary mixing accompanied with the
CNO-cycle or the primordial pollution by intermediate-mass AGB stars to the
second generation of the stars[43, 44]. In spite of their high abundances, the
CNO abundances are hard to measure in the optical wavelength mainly due to the
lack of atomic transitions. On the other hand, in the form of molecules, the
CNO can affect the $hk$ index, in particular the CN band at $\lambda$ 3885 Å
as shown in Supplementary Figure 1. Both carbon and nitrogen contribute in the
formation of CN molecules. The typical RGB stars in GCs show an
anticorrelation between the CN band and the CH band strengths and a
correlation between the CN band and the NH band strengths, indicating that the
nitrogen controls the CN band strength[45]. Our results suggest that the
variations in the CNO abundances do not affect the $hk$ index significantly.
We obtain $\Delta hk$ $\approx$ $-$0.007, +0.002, $-$0.004 for
$\Delta$[C,N,O/Fe] = +1.0 dex, respectively, and their influence on the $hk$
index appears to be negligible.
### 4.4 Aluminium
It is also well-known fact that many GCs show large star-to-star aluminium
variations by more than $\Delta$[Al/Fe] $\approx$ 1.0 dex, presumably resulted
from the proton-capture process at high temperature or the primordial
pollution by intermediate-mass AGB stars to the second generation of the
stars[46, 44]. The resonance lines of Al I at $\lambda$ 3944.01 and 3961.52 Å
are often very strong (see Supplementary Figure 17) and it may affect our
conclusions that the $hk$ index traces calcium abundances of RGB stars in GCs.
We obtain $\Delta hk$ $\approx$ 0.013 mag for $\Delta$[Al/Fe] = +1.0 dex. The
effect of the variations in aluminium abundances on the $hk$ index is
insignificant compared to our observations, by more than a factor of ten. The
insignificant influence of aluminium on the Ca II H & K lines was also
confirmed by others for the globular cluster NGC675217.
### 4.5 $\alpha$-elements
The $\alpha$-elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) are quite abundant
and they are major donors to the H- opacity in RGB stars. However, their
influence, except for Ca, on the $hk$ index appears to be small. We obtained
$\Delta hk$ $\approx$ $-$0.008 mag for +0.3 dex variation in $\alpha$-elements
excluding calcium.
### 4.6 _s_ -process elements
RGB stars in GCs show large star-to-star $s$-process elemental abundance
variations presumably resulted from the primordial pollution by intermediate-
mass AGB stars to the second generation of the stars9,[44]. For example,
globular clusters M22 and NGC1851 show bimodal $s$-process elemental abundance
distributions with $\Delta$[$s$-process/Fe] $\geq$ 0.5 dex. We obtained
$\Delta hk$ $\approx$ +0.008 mag for $\Delta$[$s$-process/Fe] = +0.5 dex, and
thus their influence on the $hk$ index is small.
## References
* [28] Clarke, D., McLean, I. S. & Wyllie, T. H. A. Stellar Line Profiles by Tilt-scanned Narrow Band Interference Filters. _Astron. Astrophys. J._ 43, 215–221 (1975).
* [29] Harris, W. E. Catalog of Parameters for Globular Clusters in the Milky Way. _Astron. J._ 112, 1487–1488 (1996).
* [30] Hesser, J. E. _et al._ Strong CN stars in the globular cluster NGC 1851. _Astron. J._ 87, 1470–1477 (1982).
* [31] Monaco, L. _et al._ Wide-field photometry of Galactic globular cluster M22. _Mon. Not. R. Astron. Soc._ 349, 1278–1290 (2004).
* [32] Beers, T. C. Estimation of the equivalent width of the interstellar Ca II $K$ absorption line. _Astron. J._ 99, 323–329 (1990).
* [33] Hunter, I. _et al._ Early-type stars observed in the ESO UVES Paranal Observatory Project – I. Interstellar Na I UV, Ti II and Ca II K observations. _Mon. Not. R. Astron. Soc_ 367, 1478–1514 (2006).
* [34] Andrews, S. M., Meyer, D. M. & Lauroesch, J. T. Small-scale interstellar Na I structure toward M92. _Astron. J._ 99, 323–329 (1990).
* [35] Smoker, J. V. _et al._ Ca II $K$ interstellar observations towards early-type disc and halo stars, abundances and distances of intermediate- and high-velocity clouds.. _Mon. Not. R. Astron. Soc_ 367, 1478–1514 (2006).
* [36] Piotto, G. Observations of multiple populations in star clusters. _ArXiv Astrophysics e-prints_ (2009). arXiv:astro-ph/0902.14226v1.
* [37] D’Antona, F., _et al._ Helium variation due to self-pollution among Globular Cluster stars. Consequences on the horizontal branch morphology. _Astron. Astrophys. J._ 395, 69–75 (2002).
* [38] Carretta, E. _et al._ Na-O anticorrelation and HB. I. The Na-O anticorrelation in NGC 2808 _Astron. Astrophys. J._ 450, 523–533 (2006).
* [39] Marino, A. F. _et al._ Spectroscopic and photometric evidence of two stellar populations in the Galactic globular cluster NGC 6121 (M 4) _Astron. Astrophys. J._ 490, 625–640 (2008).
* [40] Carretta, E. _et al._ Na-O anticorrelation and horizontal branches. II. The Na-O anticorrelation in the globular cluster NGC 6752 _Astron. Astrophys. J._ 464, 927–937 (2007).
* [41] Linsky, J. L. & Avrett, E. H. The Solar $H$ and $K$ Lines. _Pub. Astron. Soc. Pacif._ 82, 169–248 (1970).
* [42] Castelli, F. & Kurucz, R. L. New Grids of ATLAS9 Model Atmosphere. _ArXiv Astrophysics e-prints_ (2004). arXiv:astro-ph/0405087.
* [43] Kraft, R. P. Abundance Differences Among Globular-Cluster Giants: Primordial Versus Evolutionary Scenarios. _Pub. Astron. Soc. Pacific._ 113, 553–565 (1994).
* [44] Yong, D. _et al._ A Large C+N+O Abundance Spread in Giant Stars of the Globular Cluster NGC 1851. _Astrophys. J. Lett._ 695, L62–L66 (2009).
* [45] Briley, M. M. & Smith, G. H. NH-, CH-, and CN-band strengths in M5 and M13 bright red giants. _Pub. Astron. Soc. Pacific._ 105, 1260–1268 (1993).
* [46] Kraft, R. P. _et al._ Proton Capture Chains in Globular Cluster Stars. II. Oxygen, Sodium, Magnesium, and Aluminum Abundances in M13 Giants Brighter than the Horizontal Branch. _Astron. J._ 113, 279–295 (1997).
1.5
| Uncertainty on the $hk$ index | Note
---|---|---
Photometry | $\leq$ 0.020 mag | random
Shifted CTIO $Ca$ passband | $<$ 0.011 mag | systematic
Mean radial motion of GCs | $<$ 0.006 mag | systematic
Internal velocity dispersion | $<$ 0.001 mag | random
Interstellar Ca II absorption | $<$ 0.010 mag | random
total | $\leq$ 0.024 mag |
total (random) | $\leq$ 0.022 mag |
Supplementary Table 1: Summary of uncertainties relevant to the $hk$ index.
ID | $V_{HB}$ | $E(B-V)$ | Exposure Time (sec) | | Obs. Pos. | Date
---|---|---|---|---|---|---
| | | $Ca$ | $u$ | $v$ | $b$ | $y$ | | RA | DEC | (MM/YY)
$\omega$ Cen | 14.53 | 0.12 | 12,860 | …… | 4,890 | 2,130 | 1,300 | | 13:26:44 | $-$47:26:28 | 05/07, 02/08
M22 | 14.15 | 0.34 | 8,100 | 2,400 | 1,200 | 2,530 | 1,500 | | 18:36:29 | $-$23:55:34 | 07/08, 08/08
NGC1851 | 16.09 | 0.02 | 19,100 | 12,300 | 7,400 | 7,100 | 3,810 | | 5:14:14 | $-$40:01:49 | 02/08, 08/08
NGC2808 | 16.22 | 0.22 | 10,820 | …… | 3,600 | 4,960 | 3,080 | | 9:11:57 | $-$64:49:24 | 05/07
M4 | 13.45 | 0.36 | 8,400 | 5,400 | 5,570 | 2,920 | 2,060 | | 16:23:33 | $-$26:30:47 | 05/07, 08/08
M5 | 15.07 | 0.03 | 9,660 | 3,900 | 2,100 | 4,010 | 2,390 | | 15:18:29 | 2:04:03 | 05/07, 08/08
NGC6752 | 13.70 | 0.04 | 7,500 | 2,400 | 1,800 | 2,400 | 1,200 | | 19:10:57 | $-$60:00:20 | 07/08, 08/08
NGC6397 | 12.87 | 0.18 | 3,560 | 3,560 | 2,140 | 1,355 | 930 | | 17:40:52 | $-$53:36:06 | 08/06, 09/07
Supplementary Table 2: Journal of observations for eight GCs. Only one field
has been observed for a particular GC and the coordinates are given in columns
(9) and (10).
ID | FWHM(RGB) | | Measurement errors | | Differential Reddening
---|---|---|---|---|---
| $(b-y)$ | $hk$ | | $\sigma_{*}(b-y)$ | $\sigma_{*}(hk)$ | | $E(b-y)_{1/2}$ | $E(hk)_{1/2}$
$\omega$ Cen | 0.079 | 0.534 | | 0.012 | 0.020 | | 0.089 | 0.014
M22 | 0.050 | 0.216 | | 0.004 | 0.008 | | 0.252 | 0.041
NGC1851 | 0.035 | 0.182 | | 0.006 | 0.013 | | 0.015 | 0.002
NGC2808 | 0.042 | 0.159 | | 0.008 | 0.019 | | 0.163 | 0.026
M4 | 0.037 | 0.119 | | 0.005 | 0.010 | | 0.266 | 0.043
M5 | 0.025 | 0.105 | | 0.006 | 0.013 | | 0.022 | 0.004
NGC6752 | 0.022 | 0.090 | | 0.004 | 0.007 | | 0.030 | 0.005
NGC6397 | 0.024 | 0.034 | | 0.007 | 0.012 | | 0.133 | 0.022
Supplementary Table 3: Comparisons of the observed FWHMs of RGB stars and the
measurement errors at the magnitude level of horizontal branch stars,
$V_{HB}+0.5$ $\geq$ $V$ $\geq$ $V_{HB}-0.5$. The measurement errors are those
for individual stars and, therefore, measurement errors for individual
subpopulation in GCs will be given by $\approx\sigma_{*}(b-y)$/$\sqrt{n_{p}}$
or $\sigma_{*}(hk)$/$\sqrt{n_{p}}$, where $n_{p}$ ($\geq$ 20) is the number of
stars in each subpopulation in GCs. Note that, while the FWHMs of most GCs
have much larger values [$\geq$ 8$\sigma_{*}(hk)$] than the measurement errors
for individual stars in the $hk$ index, the FWHM of NGC6397 is comparable in
size to the measurement error in the $hk$ index, consistent with the idea that
NGC6397 is the only normal GC in our sample (see Supplementary Figures 6 –13).
The last two columns, $E(b-y)_{1/2}$ and $E(hk)_{1/2}$, denote contributions
due to the differential reddening effect assuming a 50% variation in the total
interstellar reddening of each GC, with which observed FWHMs of GCs in the
$(b-y)$ color and in the $hk$ index can not be explained simultaneously.
1
Supplementary Figure 1: (a) A comparison of $Ca$ filter transmission functions
between that in Anthony-Twarog _et al._ 12 (the black line) and that for the
CTIO 1-m telescope (the blue line). Both filters have similar FWHMs,
approximately 90 Å, but the passband for the CTIO 1-m telescope is shifted
approximately 15 Å to the longer wavelength. (b) Synthetic spectra for the CN
normal (the blue line) and the CN strong (the red line) RGB stars. The CN band
at $\lambda$ 3885 Å lies on the lower tail of the $Ca$ filter but the
contamination from the CN band is insignificant. (c) Synthetic spectra for the
interstellar Ca II $H$ & $K$ lines. We adopt equivalent widths of 350 mÅ and
650 mÅ for the interstellar Ca II $H$ & $K$ lines, respectively, with a
gaussian line profile with a FWHM of 1 Å (equivalent to $\Delta v_{r}$
$\approx$ 76 km/s). In the inset of the figure, the red line denotes the
velocity profile for the interstellar Ca II $H$ line and the blue line for the
interstellar Ca II $K$ line. This large amount of discrete interstellar
absorption contributes only 0.010 mag to our results.
Supplementary Figure 2: (a & b) Blue crosses and red crosses denote RGB stars
in the Ca-w and the Ca-s groups, respectively, with proper motion membership
probabilities $P$ $\geq$ 90%. (c) RGB stars in the Ca-s group are shifted by
$\Delta hk$ = $-$0.20 mag to match with those in the Ca-w group, assuming the
RGB split in M22 is due to differential reddening. The reddening correction
value of $\Delta hk$ = $-$0.20 mag for the Ca-s group is equivalent to
$E(B-V)$ = $-$1.69. (d) The RGB stars in the Ca-s group are shifted by
$\Delta(b-y)$ = $-$1.25 mag, assuming $E(b-y)/E(B-V)$ = 0.74, and two RGB
sequences do not agree, in the sense that RGB stars in the Ca-s group is too
hot to be in the RGB phase. (e & f) After applying reddening correction in $V$
(= $-$5.24 mag), assuming $A_{V}$ = 3.1$\times E(B-V)$. The RGB stars in the
Ca-s group become too bright to be members of M22, inconsistent with the
proper motion study of the cluster.
Supplementary Figure 3: Spatial distribution of the Ca-w (blue dots) and the
Ca-s (red dots) groups in M22. Note the absence of spatially patched features,
indicating that differential reddening is not responsible for the RGB split in
M22.
Supplementary Figure 4: Color-magnitude diagrams for M22 and two bulge fields
(NGC6528 and OGLEII-12). The black dots represent M22, the red dots and the
blue dots denote NGC6528 and OGLEII-12, respectively. The stars in the Milky
Way bulge are fainter and redder than those in M22 are and they do not affect
the double RGB sequences in M22. Black arrows indicate reddening vectors,
assuming $E(B-V)$ = 0.34 for M22[29].
Supplementary Figure 5: (a) The color-magnitude diagram by Monaco _et al._[31]
(b) Model isochrones for [Fe/H] = $-$1.6 (a blue line), and $-$1.4 (a red
line), $Y$ = 0.23, and 11 Gyr. (c) Model isochrones for [Fe/H] = $-$1.6,
$Y$=0.23, 11 Gyr (a blue line) and [Fe/H] = $-$1.4, $Y$=0.28, 11 Gyr (a red
line). (d) Model isochrones for [Fe/H] = $-$1.6, $Y$=0.23, 11 Gyr (a blue
line) and [Fe/H] = $-$1.4, $Y$=0.28, 10 Gyr (a red line).
Supplementary Figure 6: The $(b-y)$ and $hk$ distributions of RGB stars in
$\omega$ Cen. The red lines denote fiducial sequences of the cluster, which
are forced to pass through the peak $(b-y)$ colors or $hk$ indices of given
magnitude bins. The $\Delta(b-y)$ and the $\Delta hk$ are differences in the
$(b-y)$ color and the $hk$ index, respectively, of each RGB stars from the
fiducial sequences. The blue horizontal bars indicate measurement errors with
a 2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. From the $hk$ distribution, at least five distinct
populations, whose $hk$ splits are much larger than the measurement errors,
can be found in $\omega$ Cen.
Supplementary Figure 7: The $(b-y)$ and $hk$ distributions of RGB stars in
M22. The blue horizontal bars indicate measurement errors with a 2$\sigma_{*}$
range ($\pm$ 1$\sigma_{*}$) of individual stars at given magnitude bins. Two
distinct and discrete populations can be found in M22. At the magnitude of HB,
the $hk$ split between two populations is larger than 25$\times\sigma_{*}(hk)$
or 250$\times\sigma_{p}(hk)$, where $\sigma_{*}(hk)$ and $\sigma_{p}(hk)$
denote measurement errors for individual stars and populations in the $hk$
index, respectively.
Supplementary Figure 8: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC1851. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. Two discrete populations can be found in NGC1851. At the
magnitude of HB, the $hk$ split between two populations is larger than
11$\times\sigma_{*}(hk)$ or 55$\times\sigma_{p}(hk)$.
Supplementary Figure 9: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC2808. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. At least two discrete populations can be found in NGC2808. At
the magnitude of HB, the $hk$ split between two major populations is larger
than 5$\times\sigma_{*}(hk)$ or 50$\times\sigma_{p}(hk)$.
Supplementary Figure 10: The $(b-y)$ and $hk$ distributions of RGB stars in
M4. The blue horizontal bars indicate measurement errors with a 2$\sigma_{*}$
range ($\pm$ 1$\sigma_{*}$) of individual stars at given magnitude bins. Two
discrete populations can be found in M4. At the magnitude of HB, the $hk$
split between two populations is larger than 10$\times\sigma_{*}(hk)$ or
45$\times\sigma_{p}(hk)$.
Supplementary Figure 11: The $(b-y)$ and $hk$ distributions of RGB stars in
M5. The blue horizontal bars indicate measurement errors with a 2$\sigma_{*}$
range ($\pm$ 1$\sigma_{*}$) of individual stars at given magnitude bins. The
RGB sequence of the cluster shows a large spread in the $hk$ index, indicative
of heterogeneous calcium abundances. At the magnitude of HB, the FWHM of RGB
stars in M5 is larger than 8$\times\sigma_{*}(hk)$.
Supplementary Figure 12: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC6752. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. Two discrete populations can be found in NGC6752. At the
magnitude of HB, the $hk$ split between two populations is larger than
10$\times\sigma_{*}(hk)$ or 70$\times\sigma_{p}(hk)$.
Supplementary Figure 13: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC6397. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. It is the only normal GC in Figure 2. Note the similar degree
of the RGB widths in $\Delta(b-y)$ and $\Delta hk$ and the similar degree of
RGB widths as the measurement errors.
Supplementary Figure 14: (a) A plot of $V-V_{\rm HB}$ versus $\Delta hk$ for
RGB stars in NGC2808. The blue and red plus signs denote the Ca-w and the Ca-s
RGB stars. The dashed line denotes the boundary between the two groups at
$\Delta hk$ = $-$0.05 mag. (b) The $\Delta hk$ distribution of NGC2808 RGB
stars. (c) The [O/Fe] distributions of the two RGB populations in NGC2808. The
shaded histogram outlined with blue color is for the Ca-w group and the blank
histrogram outlined with red color is for the Ca-s group. (d) The [Na/Fe]
distributions. (e) The [Fe/H] distributions. (f–j) Same as (a–e), but for M4
RGB stars with the boundary at $\Delta hk$ = $-$0.08 mag. (k–o) Same as (a–e),
but for NGC6752 RGB stars with the boundary at $\Delta hk$ = $-$0.04 mag.
Supplementary Figure 15: (a) A plot of $V-V_{HB}$ versus $\Delta hk$ for 40
RGB stars in $\omega$ Cen of Johnson et al.5 (b – e) Elemental abundances of
40 RGB stars in $\omega$ Cen as functions of $\Delta hk$. (f – g) Comparisons
of our photometric metallicity, [Fe/H]hk, with spectroscopic metallicity,
[Fe/H]spec, and calcium abundance, [Ca/H]spec. The linear fits to the data are
shown with red lines.
Supplementary Figure 16: Metallicity distribution functions for eight GCs
derived from the $hk$ index. In the figure, [Fe/H]hk,corr is our recalibrated
photometric metallicity using the equation (4). Note that we only use bright
RGB stars in order to minimize contamination from off-cluster field and red-
clump populations. For most GCs, signs of multiple stellar populations persist
in our MDFs.
Supplementary Figure 17: Comparisons of synthetic spectra for [Fe/H] = $-$1.6,
Teff = 4750 K, $\log$ g = 2.0. (Upper panel) The red line denotes synthetic
spectrum for [Ca/Fe] = 0.25 dex and the blue line denotes that for [Ca/Fe] =
0.45 dex. We adopt the fixed aluminium abundance of [Al/Fe] = 0.50 dex. (Lower
panel) The red line denotes the synthetic spectrum for [Al/Fe] = 0.00 dex and
the blue line denotes that for [Al/Fe] = 1.00 dex. We adopt the fixed calcium
abundance of [Ca/Fe] = 0.30 dex. The effect of aluminium contamination on the
$hk$ index appears to be negligible.
|
arxiv-papers
| 2009-11-25T10:08:38 |
2024-09-04T02:49:06.694146
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jae-Woo Lee, Young-Woon Kang, Jina Lee, Young-Wook Lee",
"submitter": "Jae-Woo Lee",
"url": "https://arxiv.org/abs/0911.4798"
}
|
0911.4929
|
# The Klein-Gordon Equation for the Coulomb Potential in Non-commutative Space
Amin Rezaei Akbarieh and Hossein Motavalli
Faculty of Physics, University of Tabriz, Tabriz 56554, Iran.
Motavalli@Tabrizu.ac.ir
## 1 Abstract
In this paper the stationary Klein-Gordon equation is considered for the
Coulomb potential in non-commutative space. The energy shift due to non-
commutativeity is obtained via the perturbation theory. Furthermore, we show
that the degeneracy of the initial spectral line is broken in transition from
commutative space to non-commutative space.
Keywords: Klein-Gordon equation; Coulomb potential; Non-commutative
PACS Nos.: 03.65.-w; 03.65.Ge; 03.65.Ta.
## 2 Introduction
Recently, there has been an increased interest in the study of the non-
commutative field theory [1-2]. The most important motivation for studying
these theories, comes mainly from the works that establish a connection
between non-commutative geometry and string theory [3]. The investigation of
these theories gives us the opportunity to understand interesting phenomena,
such as non-locality and IR/UV mixing [4], new physics at very short distances
[1-2], and possible implications of Lorentz violation [5-6]. Among these
theories, the quantum mechanics is one of the simplest theories [7-8]. It is
well-known that solutions of the relativistic wave equation play an essential
role in the relativistic quantum mechanics for some physical potentials of
interest [9-13]. Recently, there has been an increasing interest in finding
exact solutions of the Klein-Gordon (KG) equation [14-18]. In the past few
years, exact solutions and energy eigenvalues of this equation have been
presented for Scarf [19], Rosen-Morse type [20], Hulthen [21], Wood-saxon [22,
23], Posch-Teller [24], five-parameter exponential [25, 26], generalized
symmetrical double-well [27], ring-shape harmonic oscillator [28], and pseudo
harmonic oscillator [29] potentials, etc. In the above cited papers the scalar
and vector potentials are almost taken to be equal in the relativistic
framework. However, there is almost no explicit expression for the energy
eigenvalues. Within the framework of non-commutativity, situation is more
complicated and most models cannot be solved exactly. Accordingly, most of the
available results are based upon perturbation theory [30-31]. This implies
that a simple physical system in the commutative space may be changed into a
complex theory within non-commutative framework.
Inclusion of non-commutativity into the quantum field theory can be achieved
in two different ways: via Moyal product on the space of ordinary functions,
or redefining the field theory on a coordinate operator space which is
intrinsically non-commutative [32-33]. The equivalence between the two
approaches has been described in references [34-35]. In the usual method, we
introduce non-commutativity by means of non-commutative coordinates of
position and momentum $(x,p)$ satisfying the following commutation relations
$\displaystyle[x_{i}\;,\;x_{j}]=i{\theta}_{ij}\;\;,\;\;[x_{i}\;,\;p_{j}]=i{\delta}_{ij}\;\;,\;\;[p_{i}\;,\;p_{j}]=0,\;\;i,j=1,2,3$
(1)
where ${\theta}_{ij}={\epsilon}_{ij}\theta$, in which ${\epsilon}_{ij}$ is
Levichevita symbol and $\theta$ is a parameter that measures the non-
commutativity of coordinates. In the non-commutative space the ordinary
product is replaced by Moyal product
$\displaystyle f(x)\star
g(x)=exp\\{\frac{i}{2}{\theta}^{jk}{\partial_{j}}^{(1)}{\partial_{k}}^{(2)}\\}f(x_{1})g(x_{2})|_{x_{1}=x_{2}=x}$
where $f(x)$ and $g(x)$ are two arbitrary differentiable functions.
## 3 The Non-commutative Klein-Gordon Equation
In this section we consider the three dimensional Klein-Gordon equation for a
long-range $1/r$ interaction in the non-commutative space. For time
independent potentials, the KG equation for a particle of rest mass $M$ can be
written as ($\hbar$=c=1)
$\displaystyle\\{{\nabla}^{2}+[V(r)-E]^{2}-[S(r)+M]^{2}\\}\psi(r)=0$ (2)
in commutative space, where $E$ is the relativistic energy, $V(r)$ and $S(r)$
denote vector and scalar potentials, respectively. Recently, interest for
considering of this equation with equal scalar and vector potentials has been
increased [19-20]. Under assumption $V(r)=S(r)$, Eq. (2) takes the form
$\displaystyle\\{{\nabla}^{2}+(E^{2}-M^{2})-2(E+M)V(r)\\}\psi(r)=0.$ (3)
By using the common separation of variables in the spherical polar coordinate
$\psi(r)=Y(\theta,\phi)R(r)/r$, the radial part of this equation reads
$\displaystyle\\{\frac{d^{2}}{dr^{2}}-[E_{eff}+V_{eff}(r)]\\}R(r)=0$ (4)
where
$\displaystyle
V_{eff}(r)=2(M+E)V(r)+\ell(\ell+1)/r^{2},\;\;\;\;\;E_{eff}=(M^{2}-E^{2}).$ (5)
Now to consider this equation in the non-commutative space, let us introduce
the non-commuting coordinates in terms of the commuting coordinates and their
momenta
$\displaystyle\left\\{\begin{array}[]{ll}\hat{x}_{i}=x_{i}+\frac{1}{2}\theta_{ij}p_{j},\\\
\hat{p}_{i}=p_{i}.\end{array}\right.$ (8)
Under these transformations a radial form potential takes the form
$\displaystyle V(\hat{r})$ $\displaystyle=$ $\displaystyle
V(|\vec{r}-\frac{\vec{p}}{2}|)$ (9) $\displaystyle=$ $\displaystyle
V(\sqrt{(x_{i}-\frac{1}{2}\theta_{ij}p_{j})(x_{i}-\frac{1}{2}\theta_{ij}p_{j})}\;\;)$
$\displaystyle=$ $\displaystyle
V(r)+\frac{1}{2}(\vec{\theta}\times\vec{p})\cdot\vec{\nabla}V(r)+O(\theta^{2})$
$\displaystyle=$ $\displaystyle
V(r)-\frac{\vec{\theta}\cdot\vec{L}}{2r}\frac{\partial V}{\partial
r}+O(\theta^{2})$ $\displaystyle\simeq$ $\displaystyle
V(r)-\frac{\vec{\theta}\cdot\vec{L}}{2r}\frac{\partial V}{\partial r}$
up to the first order of $\theta$, where$\;\;\;r=\sqrt{x_{i}x_{i}}$ and
$\vec{L}=\vec{r}\times\vec{p}$ is the angular momentum operator.
By replacement of the ordinary product with Moyal, Eq.(6) takes the following
form
$\displaystyle\\{\frac{d^{2}}{dr^{2}}-[E_{eff}+V_{eff}(r)]\\}\star
R_{n\ell}(r)=0$
in the non-commutative space, or equivalently
$\displaystyle\\{\frac{d^{2}}{dr^{2}}-[E_{eff}+V_{eff}(|\vec{r}-\frac{1}{2}{\vec{p}}|)]\\}R_{n\ell}(r)=0.$
(10)
Comparing Eq. (6) with Eq. (8) indicates that under the Moyal product the only
modification in the radial part of the KG equation appears in the effective
potential term. By substituting Coulomb potential $V(r)=-\frac{Ze^{2}}{r}$
into relation (5) and using effective potential (7) the last equation can be
rewritten as
$\displaystyle\\{\frac{d^{2}}{dr^{2}}-\frac{\ell(\ell+1)}{r^{2}}+2(E+M)\frac{Ze^{2}}{r}-E_{eff}-\frac{(\vec{\theta}\cdot\vec{L})}{2r}[\frac{2\ell(\ell+1)}{r^{3}}-2(E+M)\frac{Ze^{2}}{r^{2}}]\\}R(r)=0.$
(11)
By introducing dimensionless new variable $\rho=2r\sqrt{E_{eff}}$, Eq. (9) is
transformed into the following form
$\displaystyle\\{\frac{d^{2}}{d\rho^{2}}-\frac{\ell(\ell+1)}{\rho^{2}}+\frac{\varsigma}{\rho}-\frac{1}{4}-(\vec{\theta}\cdot\vec{L})[\frac{4\ell(\ell+1)E_{eff}}{\rho^{4}}-2(1+E/M)\sqrt{E_{eff}}\frac{Z\alpha}{\rho^{3}}]\\}R(\rho)=0$
(12)
where $\;\varsigma=\frac{Z\alpha}{M}\sqrt{1+\frac{2E}{M-E}}$.
## 4 The Solution
The last equation has not yet been solved exactly in the presence of the last
two terms, whereas in their absence, its exact solution is available [36]. To
obtain the solution, we choose $\theta=0$, and get
$\displaystyle\\{\frac{d^{2}}{d\rho^{2}}-\frac{\ell(\ell+1)}{\rho^{2}}+\frac{\varsigma}{\rho}-\frac{1}{4}\\}R^{(0)}(\rho)=0.$
(13)
This is a second order differential equation and can be easily solved via
Nikiforov-Uvarov (NU) mathematical method. In this method a second order
linear differential equation is reduced to a generalized equation of hyper-
geometric type whose exact solutions are expressed in terms of special
orthogonal functions [37], as well as corresponding eigenvalues are obtained.
To apply this method for Eq. (11), we compare this equation with the
generalized hyper-geometric type equation
$\displaystyle\\{\frac{d^{2}}{d\rho^{2}}+\frac{\tilde{\tau}(\rho)}{\sigma(\rho)}\frac{d}{d\rho}+\frac{\tilde{\sigma}(\rho)}{\sigma^{2}(\rho)}\\}R^{(0)}(\rho)=0$
(14)
and get
$\displaystyle\tilde{\tau}(\rho)=0,\;\;\;\sigma(\rho)=2\rho,\;\;\;\tilde{\sigma}(\rho)=-4\ell(\ell+1)-\rho^{2}+4\varsigma\rho.$
(15)
Using these functions it is straightforward to show that the exact solution of
Eq. (11) is [19]
$\displaystyle
R^{(0)}(\rho)=N\rho^{\ell+1}\frac{(n-\ell-1)!}{(n+\ell)!}(2\ell+1)!L^{2\ell+1}_{n-\ell-1}(\rho)e^{-\frac{\rho}{2}},\;\;\;\;\;\;n=0,\;1,\;2,...$
(16)
where $L^{2\ell+1}_{n-\ell-1}(\rho)$ denotes the generalized Laguerre
polynomials and $N$ is normalization constant
$\displaystyle
N=\sqrt{\frac{(n+\ell)!}{2|E^{(0)}|n(n-\ell-1)!}}\frac{1}{(2\ell+1)!}$ (17)
in which $E^{(0)}$ is the energy eigenvalues and is given by
$\displaystyle
E^{(0)}=\\{\frac{(Z\alpha)^{2}-(n-\ell)^{2}M^{2}}{(Z\alpha)^{2}+(n-\ell)^{2}M^{2}}\\}M,\;\;\;\;\;\;n=0,\;1,\;2,....$
(18)
Now, to obtain the modifacation of energy levels as a result of the last two
terms in Eq. (10) due to the non-commutativity, we use perturbation theory.
For simplicity, first of all we take ${\theta}_{3}={\theta}$ and assume that
the other ${\theta}$-components are zero (by rotation or redefinition of
coordinates), such that $\vec{\theta}\cdot\vec{L}=L_{z}{\theta}$. In addition,
we use
$\displaystyle<nlm|L_{z}|nlm^{\prime}>=m{\delta}_{mm^{\prime}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;-l\leq
m\leq l$
and also the fact that in the first order perturbation theory the expectation
value of $\rho^{-3}$ and $\rho^{-4}$ with respect to the exact solution of Eq.
(11), are given by [38]
$\displaystyle<n|{\rho^{-3}}|n>$ $\displaystyle=$
$\displaystyle\int\\{R^{(0)}(\rho)\\}^{2}{\rho}^{-1}d\rho=\frac{1}{2|E^{0}|}\frac{1}{\ell(2\ell+1)(2\ell+2)}$
$\displaystyle<n|{\rho^{-4}}|n>$ $\displaystyle=$
$\displaystyle\int\\{R^{(0)}(\rho)\\}^{2}{\rho}^{-2}d\rho=\frac{1}{n|E^{0}|}\frac{\Gamma(2\ell-1)}{\Gamma(2\ell+4)}[3n^{2}-\ell(\ell+1)].$
Putting these results together, one gets
$\displaystyle\Delta
E_{NC}=\frac{m\theta}{4(2\ell+1)|E^{0}|}\\{\frac{(3n^{2}-\ell(\ell+1))}{n(2\ell-1)(2\ell+3)}-\frac{2(n-\ell)^{2}Z\alpha}{\ell(\ell+1)[(n-\ell)^{2}+(Z\alpha/M)^{2}]}\\},\;\;\;\;\;n=0,1,2,....$
This is energy shift due to the additional last two terms of Eq. (10). The
appearance of the magnetic quantum number $m$ in this expression explicitly
indicates the splitting of states with the same orbital angular momentum into
the corresponding components. In fact each level $\ell$ splits into $2\ell+1$
sublevels and subsequently breaks the degeneracy of the initial spectral line.
The lifting of degeneracy is due to the emergence of a magnetic field
associated with the non-commutative space in transition from commutative space
into non-commutative space. This behavior is similar to the Zeeman effect. In
addition, it is worth noting that the correction terms containing
$\vec{\theta}\cdot\vec{L}$ are very similar to that of the spin orbit
coupling, in which the non-commutative parameter $\vec{\theta}$ plays the role
of the spin.
## 5 Conclusion
In this paper, we have investigated the Klein-Gordon equation for the Coulomb
potential in the non-commutative space. The energy shift, due to the non-
commutativity, is obtained via first order perturbation theory. It is
explicitly shown that the degeneracy of the initial spectral line is broken in
transition from commutative space into non-commutative space by splitting
states into the corresponding components. This behavior is similar to the
Zeeman effect in which a magnetic field is applied to the system. In this
space the non-commutative parameter $\vec{\theta}$ plays the role of the spin.
## References
* [1] R. J. Szabo, Phys. Rep. 378, (2003) 207.
* [2] M. R. Douglas and N.A. Nekrasov, Rev. Mod. Phys. 73, (2002) 977.
* [3] N. Seiberg and E. Witten, J. High Energy Phys. 09, (1999) 032.
* [4] S. Minwalla, M. Van Raamsdonk and N. Seiberg, J. High Energy Phys. 02, (2000) 020.
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* [6] C. E. Carlson, C. D. Carone and R. F. Lebed, Phys.Lett. B 518, (2001) 201; Phys. Lett. B 549, (2002) 337.
* [7] G. Dunne, R. Jackiw, C. Trugenberger, Phys. Rev. D 41, (1990) 661.
* [8] Kang Li, Jianhua Wang, Chiyi Chen, Mod. Phys. Lett. A 20, (2005) 2165.
* [9] B. A. Volodin A. M. Khapayev, Computational Math. And Math. Phys. 31, (1991) 69.
* [10] S.M. Ikhdair, Chinese Phys. 3, (2008) 291.
* [11] R.G. Abdel-Rahman, Chinese Phys. 5, (2008) 495.
* [12] H. Hong-Sheng, C. Jiang, Y. Kong-Qing, Chinese Phys. 14, (2005) 1926.
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* [14] F. Dominguez-Adame, Phys. Lett. A 136, (1989) 175.
* [15] Z. Min-Cang, W. Zhen-Bang, Chinese Phys. 16, (2007) 1863.
* [16] Z. Qiang, Y. Ping, G. Lun-Xun, Chinese Phys. 15, (2006) 35.
* [17] Z. Min-Cang W. Zhen-Bang, Chinese Phys. Lett. 22, (2005) 2994.
* [18] F. Yasuk, A. Durmus, I. Boztosun, J. Math. Phys. 47, (2006) 082302.
* [19] H. Motavali, Mod. Phys. Lett. A 24, (2009) 1227.
* [20] A. Rezaei Akbarieh and H. Motavali, Mod. Phys. Lett. A 23, (2008) 3009\.
* [21] G. Chen, Mod. Phys. Lett. A 19, (2004) 2009.
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* [23] G. Chen, Accta Phys. Sinica 53, (2004) 680.
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* [26] Y.F. Diao, L.Z. Yi, C.S. Jia, Phys. Lett. A 332, (2004) 157.
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* [30] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett. 86, (2001) 2716.
* [31] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Eur. Phys. J. C 36, (2004) 251.
* [32] M. R. Douglas, N. A. Nekrasov Non-commutative Field Theory; hep-th/0106048
* [33] M. Chaichian, A. Demichev, P. Pre snajder Nucl. Phys. B 567, (2000) 360.
* [34] L. Alvarez-Gaume, S. R. Wadia Phys. Lett. B 501, (2001) 319.
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* [36] I. I. Gol’dman and D. V. Krivchenkov, Problems in quantum mechanics (Pergamon, London, 1961).
* [37] A. F. Nikiforov, V. B. Uvarov, Special Functions of Mathematical Physics (Birkhauser, Basel, 1988).
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|
arxiv-papers
| 2009-11-25T17:57:13 |
2024-09-04T02:49:06.705418
|
{
"license": "Public Domain",
"authors": "Amin Rezaei Akbarieh and Hossein Motavalli",
"submitter": "Hossein Motavalli",
"url": "https://arxiv.org/abs/0911.4929"
}
|
0911.5099
|
# The coming of age of X-ray polarimetry
Francesco Lazzarotto a Sergio Fabiani a Enrico Costa a Fabio Muleri a Paolo
Soffitta a Sergio Di Cosimo a Giuseppe Di Persio a Alda Rubini a Ronaldo
Bellazzini b Alessandro Brez b Gloria Spandre b Vincenzo Cotroneo c Alberto
Moretti c Giovanni Pareschi c Giampiero Tagliaferri c
###### Abstract
The INFN and INAF Italian research institutes developed a space-borne X-Ray
polarimetry experiment based on a X-Ray telescope, focussing the radiation on
a Gas Pixel Detector (GPD). The instrument obtains the polarization angle of
the absorbed photons from the direction of emission of the photoelectrons as
visualized in the GPD. Here we will show how we compute the angular resolution
of such an instrument.
## Chapter 0 Angular Resolution of a Photoelectric Polarimeter in the Focus
of an Optical System
a IASF - INAF Roma, b INFN Pisa, c INAF Brera, email:
francesco.lazzarotto@iasf-roma.inaf.it web: http://bigfoot.iasf-
roma.inaf.it/$\sim$agile/Polar/SPSdoc/index.html
### 1 Introduction
The X-ray telescopes are based on the grazing angle principle. The radiation
is reflected with small incidence angles on the surfaces of hyperboloid and
paraboloid mirrors and then is focused. The GPD is a gas detector which is
able to image the photoelectron tracks. The polarization is measured using the
dependence of the photoelectric cross section from the photon polarization
direction [1, Costa 2001] [3, Bellazzini 2007]. The photoelectron is emitted
with more probability in the direction of the electric field of the photon.
The track created by the photoelectron path, is drifted and amplified by the
Gas Electron Multiplier (GEM) and collected on a fine sub-divided pixel
detector. Using different mixtures of gas it is possible to properly select
the energy band of the instrument in the range of about $1-30$ keV. This GPD
has the capability to preserve the imaging while reaching a good sensitivity
in polarization as well as in spectroscopic and timing measurements.
Characteristic | Value | Unit
---|---|---
optics energy band | 0.1-10 | keV
GPD energy band | 1-30 | keV
GPD Area | $15\cdot 15$ | $mm^{2}$
GPD height | 10 | mm
GPD transistors | $16.5\cdot 10^{6}$ | n.
GPD pixels | $105600$ | n.
GPD pixel matrix | $300\cdot 352$ | n.
Table 1: GPD characteristics
### 2 Resolution Calculation and related Simulation Software
We studied a system composed by an X-Ray telescope and the GPD. We considered
only the on-axis radiation. In this case an ideally perfect optical system can
focus the radiation exactly in a single point on the detector, assuming that
it has:
* •
Perfect quality reflective surfaces;
* •
Perfectly coaxial alignment of the mirrors;
* •
A detector with negligible thickness.
Figure 1: Photon path in the GPD Figure 2: Distribution of the absorbtion
points in the gas cell of the GPD causing the gas blurring.
---
Figure 3: Simulation software class tree
For the GPD the thickness of the gas cell is not negligible: 1 cm. To express
the real behavior of radiation intensity distribution, the Point Spread
Function (PSF) is obtained taking into account:
* •
Blurring introduced by imperfections of the optics [2, Citterio, 1993];
* •
Blurring due to the approximations of the photons tracks reconstruction
algorithm;
* •
Blurring due to the radiation absorption in the gas.
We developed a simulation software based on montecarlo techniques to study the
angular resolution of the instrument (see fig. 3 [8, Fabiani, 2008]). At this
level the intrinsic resolution of the detector is neglected, the simulation
program takes as input:
* •
The surface density of the incident radiation ($n.\ of\ photons\cdot
cm^{-2}$);
* •
The geometry and the effective area of the optical system;
* •
The geometrical and physical characteristics of the gas detector.
The program calculates the absorption point of the photons in the gas cell
taking into account the effects of the optical aberrations and gas blurring.
In output it produces graphics and statistics on the photon detection
positions around the focal plane.
Figure 4: Qualitative representation of the resolution results on the image of
the Crab PWN.
---
### 3 Conclusion
We report in the Table 2 the angular resolution results expressed as the HPW
[Half Power Width] of the radiation intensity on the detector plane for a
simulation with a gas mixture composed by 70% of DME and 30% of He. In fig. 4
the related error circles show that small missions as POLARIX and HXMT can be
used to achieve the first results for the angular resolved X-Ray polarimetry.
For instance it will be possible to measure the polarization of the main
regions of extended sources such as the Pulsar Wind Nebulae. Whereas advanced
missions as IXO will be able to investigate the thinner properties of such
sources or to reach the resolution needed to resolve the knots in AGN jets.
Characteristic | POLARIX | HXMT | IXO |
---|---|---|---|---
energy | 3 keV | 3 keV | 1.5 keV |
HPW gas + optics | 19.3 arcsec | 34.7 arcsec | 6.6 arcsec |
HPW only optics | 14.7 arcsec | 23.2 arcsec | 5.0 arcsec |
HPW only gas | 10.0 arcsec | 19.5 arcsec | 3.0 arcsec |
Table 2: Angular resolution, showing the different blurring contributions
99
## References
* [1] Costa et al, ”An efficient photoelectric X-ray polarimeter for the study of black holes and neutron stars”, Nature 411, 662-665, 2001
* [2] Citterio O. et al, ”X-Ray optics for the JET-X experiment aboard the SPECTRUM-X Satellite.”, SPIE 1993 Vol. 2279
* [3] Bellazzini R. et al, ”A sealed Gas Pixel Detector for X-ray astronomy”, NIMPA 579 (853) 2007
* [4] Muleri F. et al, ”The Gas Pixel Detector as an X-ray photoelectric polarimeter with a large field of view” SPIE 2008, vol. 7011-88
* [5] Soffitta et al, ”X-ray polarimetry on-board HXMT”, SPIE 2008, vol. 7011-85
* [6] Costa et al, ”POLARIX: a small mission of x-ray polarimetry”, SPIE 2006, Vol. 6266
* [7] Costa et al, ”XPOL: a photoelectric polarimeter onboard XEUS”, SPIE 2008, vol. 7011-15
* [8] Fabiani et al, ”The Study of PWNe with a photoelectric polarimeter”, PoS(CRAB2008)027, 2008
|
arxiv-papers
| 2009-11-26T15:43:57 |
2024-09-04T02:49:06.711742
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco Lazzarotto, Sergio Fabiani, Enrico Costa, Fabio Muleri,\n Paolo Soffitta, Sergio Di Cosimo, Giuseppe Di Persio, Alda Rubini, Ronaldo\n Bellazzini, Alessandro Brez, Gloria Spandre, Vincenzo Cotroneo, Alberto\n Moretti, Giovanni Pareschi, Giampiero Tagliaferri",
"submitter": "Francesco Lazzarotto PhD",
"url": "https://arxiv.org/abs/0911.5099"
}
|
0911.5113
|
# SuperAGILE data processing services
Lazzarotto F., Costa E., Del Monte E., Donnarumma I., Evangelista Y., Feroci
M., Lapshov I., Pacciani L., Soffitta P. Trifoglio M., Bulgarelli A.,
Gianotti F
###### Abstract
The SuperAGILE (SA) instrument is a X-ray detector for Astrophysics
measurements, part of the Italian AGILE satellite for X-Ray and Gamma-Ray
Astronomy launched at 23/04/2007 from India. SuperAGILE is now studying the
sky in the 18 - 60 KeV energy band. It is detecting sources with advanced
imaging and timing detection capabilities and good spectral detection
capabilities. Several astrophysical sources has been detected and localized,
including Crab, Vela and GX 301-2. The instrument has the skill to resolve
correctly sources in a field of view of [-40, +40] degrees interval, with the
angular resolution of 6 arcmin, and a spectral analysis with the resolution of
8 keV. Transient events are regularly detected by SA with the aid of its
temporal resolution (2 microseconds) and using signal coincidence on different
portions of the instrument, with confirmation from other observatories. The SA
data processing scientific software performing at the AGILE Ground Segment is
divided in modules, grouped in a processing pipeline named SASOA. The
processing steps can be summarized in data reduction, photonlist building,
sources extraction and sources analysis. The software services allow orbital
data processing (near real-time), daily data set integration, Temporal Data
Set (TDS) processing and TDS processing with source target optimization
(TDS_SRC). Automatic data processing monitoring and interactive data analysis
is possible from an internet connected workstation, with the use of SA data
processing Web services. Many solutions were implemented in order to achieve
fault tolerance. Archive management and data storage are performed with the
help of relational database instruments.
National Institute for Astrophysics (INAF) IASF Rome, Italy
National Institute for Astrophysics (INAF) IASF Bologna, Italy
## 1\. The SuperAGILE processing pipeline
SASOA software Data Processing Stages
|
arxiv-papers
| 2009-11-26T16:26:49 |
2024-09-04T02:49:06.715381
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Lazzarotto, E. Costa, E. Del Monte, I. Donnarumma, Y. Evangelista,\n M. Feroci, I. Lapshov, L. Pacciani, P. Soffitta",
"submitter": "Francesco Lazzarotto PhD",
"url": "https://arxiv.org/abs/0911.5113"
}
|
0911.5174
|
# Unified $(r,s)$-relative entropy††thanks: This project is supported by
Natural Science Foundation of China (10771191 and 10471124) and Natural
Science Foundation of Zhejiang Province of China (Y6090105).
Wang Jiamei, Wu Junde Department of Mathematics, Zhejiang University, Hangzhou
310027, P. R. China. E-mail: wjd@zju.edu.cn
###### Abstract
In this paper, we introduce and study unified $(r,s)$-relative entropy and
quantum unified $(r,s)$-relative entropy, in particular, our main results of
quantum unified $(r,s)$-relative entropy are established on the infinite
dimensional separable complex Hilbert spaces.
Key Words. Hilbert space, unified $(r,s)$-relative entropy, state.
1\. Introduction
In 1991, Rathie and Taneja introduced the unified $(r,s)$-entropy which
generalized many classical entropies ([1]), that is, let
$A=(a_{1},a_{2},\cdots,a_{n})$ be a discrete probability distribution
satisfies that $0<a_{i}\leq 1$ and $\sum_{i=1}^{n}a_{i}=1$. If we denote
$p(r)=\sum_{i=1}^{n}a_{i}^{r}$, then for any $r>0$ and any real number $s$,
the unified $(r,s)-$entropy is defined by
$\displaystyle E_{r}^{s}(A)=\left\\{\begin{array}[]{ll}H_{r}^{s}(A),&{\rm if\
}r\neq 1,s\neq 0,\\\ H_{r}(A),&{\rm if\ }r\neq 1,s=0,\\\ H^{r}(A),&{\rm if\
}r\neq 1,s=1,\\\ _{\frac{1}{r}}H(A),&{\rm if\ }r\neq 1,s=1/r,\\\ H(A),&{\rm
if\ }r=1,\end{array}\right.$
where
$H_{r}^{s}(A)=[(1-r)s]^{-1}[p(r)^{s}-1],$ $H_{r}(A)=(1-r)^{-1}\ln p(r),$
$H^{r}(A)=(1-r)^{-1}(p(r)-1),$ ${}_{r}H(A)=(r-1)^{-1}[p(\frac{1}{r})^{r}-1],$
$H(A)=-\sum_{i=1}^{n}a_{i}\ln{a_{i}}$
are the $(r,s)$-entropy, R$\acute{e}$nyi entropy of order $r$, the Tsallis
entropy, the entropy of type $r$ and the well-known Shannon entropy,
respectively.
In 2006, Hu and Ye introduced the quantum version of the unified
$(r,s)$-entropy ([2]), that is, let $H$ be a complex Hilbert space and $\rho$
a state (see [3]) on $H$. If we denote $P(r)=tr(\rho^{r})$, then for any $r>0$
and any real number $s$, the quantum unified $(r,s)-$entropy is defined by
$\displaystyle
E_{r}^{s}(\rho)=\left\\{\begin{array}[]{ll}S_{r}^{s}(\rho),&{\rm if\ }r\neq
1,s\neq 0,\\\ S_{r}(\rho),&{\rm if\ }r\neq 1,s=0,\\\ S^{r}(\rho),&{\rm if\
}r\neq 1,s=1,\\\ _{\frac{1}{r}}S(\rho),&{\rm if\ }r\neq 1,s=1/r,\\\
S(\rho),&{\rm if\ }r=1,\end{array}\right.$
where
$S_{r}^{s}(\rho)=[(1-r)s]^{-1}\left[P(r)^{s}-1\right],$
$S_{r}(\rho)=(1-r)^{-1}\ln P(r),$ $S^{r}(\rho)=(1-r)^{-1}\left(P(r)-1\right),$
${}_{r}S(\rho)=(r-1)^{-1}\left[P(\frac{1}{r})^{r}-1\right],$
$S(\rho)=-tr(\rho\ln\rho)$
are the quantum $(r,s)$-entropy, the quantum R$\acute{e}$nyi entropy of order
$r$, the quantum Tsallis entropy, the quantum entropy of type $r$ and the
well-known Von Neumann entropy, respectively.
On the other hand, although the R$\acute{e}$nyi relative entropy of order $r$
([4]), the Tsallis relative entropy of degree $r$ (([5]), the relative entropy
([3]), even the quantum R$\acute{e}$nyi relative entropy ([4]) and quantum
Tsallis relative entropy of degree $r$ ([5-6]) were studied, respectively,
nevertheless, until now, we do not find the works of unified $(r,s)$-relative
entropy and quantum unified $(r,s)$-relative entropy. In this paper, we fill
this gap.
2\. The unified $(r,s)$-relative entropy
Let $A=(a_{1},a_{2},\cdots,a_{n})$, $B=(b_{1},b_{2},\cdots,b_{n})$ be two
discrete probability distributions satisfying $0<a_{i},b_{i}<1$ and
$\sum\limits_{i=1}^{n}a_{i}=\sum\limits_{i=1}^{n}b_{i}=1$. Then for any $r>0$
and any real number $s$, the unified $(r,s)-$relative entropy is defined by
$\displaystyle
E_{r}^{s}(A\|B)=\left\\{\begin{array}[]{ll}H_{r}^{s}(A\|B),&{\rm if\ }r\neq
1,s\neq 0,\\\ H_{r}(A\|B),&{\rm if\ }r\neq 1,s=0,\\\ H^{r}(A\|B),&{\rm if\
}r\neq 1,s=1,\\\ _{\frac{1}{r}}H(A||B),&{\rm if\ }r\neq 1,s=\frac{1}{r},\\\
H(A\|B),&{\rm if\ }r=1,\end{array}\right.$
where
$\displaystyle
H_{r}^{s}(A\|B)=-[(1-r)s]^{-1}\left[\left(\sum_{i=1}^{n}a_{i}\frac{a_{i}^{r-1}}{b_{i}^{r-1}}\right)^{s}-1\right],r>0,r\neq
1,s\neq 0,$ $\displaystyle
H_{r}(A\|B)=-(1-r)^{-1}\ln\left(\sum_{i=1}^{n}a_{i}\frac{a_{i}^{r-1}}{b_{i}^{r-1}}\right),r>0,r\neq
1,$ $\displaystyle
H^{r}(A\|B)=-(1-r)^{-1}\left(\sum_{i=1}^{n}a_{i}\frac{a_{i}^{r-1}}{b_{i}^{r-1}}-1\right),r>0,r\neq
1,$
${}_{r}H(A\|B)=-(r-1)^{-1}\left[\left(\sum_{i=1}^{n}a_{i}\frac{a_{i}^{\frac{1}{r}-1}}{b_{i}^{\frac{1}{r}-1}}\right)^{r}-1\right],r>0,r\neq
1,$ $\displaystyle H(A\|B)=\sum_{i=1}^{n}a_{i}\ln\frac{a_{i}}{b_{i}}$
are the $(r,s)$-relative entropy, the R$\acute{e}$nyi relative entropy of
order $r$, the Tsallis relative entropy of degree $r$, the relative entropy of
type $r$ and the relative entropy, respectively ([3-5]).
Now, we discuss some elementary properties of the unified $(r,s)$-relative
entropy. First, we point out an important unified $(r,s)$-directed divergence
${\cal F}_{r}^{s}(A\|B)$ which was studied in [7], note that when $r\neq 1$,
$E^{\frac{s-1}{r-1}}_{r}(A\|B)={\cal F}_{r}^{s}(A\|B)$, so by using Theorem 1
in [7], we can prove the nonnegativity, nonadditivity and convexity of
$E_{r}^{s}(A\|B)$ directly:
(i) Let
$\Delta_{n}=\\{A=(a_{1},a_{2},\cdots,a_{n}):a_{i}>0,\sum\limits_{i=1}^{n}a_{i}=1\\}$.
If $A,B\in\Delta_{n}$, then $E_{r}^{s}(A\|B)\geq 0$, and the equality holds
iff $A=B$.
(ii) Let
$\Delta_{m}=\\{B=(b_{1},b_{2},\cdots,b_{m}):b_{i}>0,\sum\limits_{i=1}^{m}b_{i}=1\\}$.
If $A_{1},A_{2}\in\Delta_{n}$, $B_{1},B_{2}\in\Delta_{m}$, and denote
$A*B=(a_{1}b_{1},\cdots,a_{1}b_{m},a_{2}b_{1},\cdots,a_{2}b_{m},\cdots,a_{n}b_{m})$,
then
$E_{r}^{s}(A_{1}*B_{1}\|A_{2}*B_{2})=E_{r}^{s}(A_{1}\|A_{2})+E_{r}^{s}(B_{1}\|B_{2})+(r-1)sE_{r}^{s}(A_{1}\|A_{2})E_{r}^{s}(B_{1}\|B_{2}).$
(iii) If $r=1$ or $r>1,s\geq 1$ or $0<r<1,s\leq 1$, then $E_{r}^{s}(A\|B)$ is
a convex function of $(A,B)$.
Next, we prove the following:
Theorem 2.1. If $r=1$ or $0<r<1,s\geq 0$, then
$E_{r}^{s}(A\|B)\leq H(A\|B)\leq E_{2-r}^{s}(A\|B).$
Proof. That $r=1$ is clear. Let $0<r<1,s=0$. By the convexity of the function
$f(x)=\frac{1}{1-r}\ln x$, we get that
$\displaystyle H_{r}(A\|B)$ $\displaystyle=$
$\displaystyle-(1-r)^{-1}\ln\left(\sum_{i}a_{i}\frac{{a_{i}}^{r-1}}{{b_{i}}^{r-1}}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{i}a_{i}\left[-(1-r)^{-1}\ln\frac{{a_{i}}^{r-1}}{{b_{i}}^{r-1}}\right]$
$\displaystyle=$ $\displaystyle\sum_{i}a_{i}\ln\frac{a_{i}}{b_{i}}=H(A\|B).$
By a similar way, we get that $H(A\|B)\leq H_{2-r}(A\|B)$. Thus, we have
$\displaystyle H_{r}(A\|B)\leq H(A\|B)\leq H_{2-r}(A\|B).$ (4)
If $0<r<1$ and $s>0,$ let
$x_{0}=\sum\limits_{i=1}^{n}a_{i}\frac{b_{i}^{1-r}}{a_{i}^{1-r}}.$ Then
$\sum\limits_{i=1}^{n}a_{i}\frac{b_{i}^{1-r}}{a_{i}^{1-r}}\leq[\sum\limits_{i=1}^{n}(a_{i}\frac{b_{i}}{a_{i}})]^{1-r}=1$.
Note that when $0<x\leq 1$ and $s>0$, we have $\ln x\leq\frac{x^{s}-1}{s}$, so
for any $0<r<1,s>0$, we have $-\frac{x_{0}^{s}-1}{(1-r)s}\leq-\frac{1}{1-r}\ln
x_{0}$, thus,
$-\frac{{\left(\sum\limits_{i=1}^{n}a_{i}\frac{b_{i}^{1-r}}{a_{i}^{1-r}}\right)}^{s}-1}{(1-r)s}\leq-\frac{1}{1-r}\ln\left(\sum\limits_{i=1}^{n}a_{i}\frac{b_{i}^{1-r}}{a_{i}^{1-r}}\right),$
that is,
$\displaystyle H_{r}^{s}(A\|B)\leq H_{r}(A\|B).$ (5)
It follows from (1) and (2) that $H_{r}^{s}(A\|B)\leq H(A\|B)$. By a similar
way, we can prove $H(A\|B)\leq H_{2-r}^{s}(A\|B).$ Thus, we proved the
theorem.
3\. The quantum unified $(r,s)$-relative entropy
Let $H$ be a separable complex Hilbert space and $\rho,\sigma$ be two states
on $H$. Then for any $0\leq r\leq 1$ and any real number $s$, the quantum
unified $(r,s)-$relative entropy is defined by
$\displaystyle
E_{r}^{s}(\rho\|\sigma)=\left\\{\begin{array}[]{ll}H_{r}^{s}(\rho\|\sigma),&{\rm
if\ }0\leq r<1,s\neq 0,\\\ H_{r}(\rho\|\sigma),&{\rm if\ }0\leq r<1,s=0,\\\
H^{r}(\rho\|\sigma),&{\rm if\ }0\leq r<1,s=1,\\\
_{\frac{1}{r}}H(\rho\|\sigma),&{\rm if\ }0<r<1,s=\frac{1}{r},\\\
H(\rho\|\sigma),&{\rm if\ }r=1,\end{array}\right.$
where
$\displaystyle
H_{r}^{s}(\rho\|\sigma)=-[(1-r)s]^{-1}[\left(tr(\rho^{r}\sigma^{1-r})\right)^{s}-1],$
$\displaystyle
H_{r}(\rho\|\sigma)=-(1-r)^{-1}\ln\left(tr(\rho^{r}\sigma^{1-r})\right),$
$\displaystyle H^{r}(\rho\|\sigma)=-(1-r)^{-1}[tr(\rho^{r}\sigma^{1-r})-1],$
${}_{r}H(\rho\|\sigma)=-(r-1)^{-1}[\left(tr({\rho^{\frac{1}{r}}\sigma^{1-\frac{1}{r}}})\right)^{r}-1],$
$\displaystyle H(\rho\|\sigma)=tr(\rho\ln\rho)-tr(\rho\ln\sigma)$
are the quantum $(r,s)$-relative entropy, the quantum R$\acute{e}$nyi relative
entropy of order $r$, the quantum Tsallis relative entropy, the quantum
relative entropy of type $r$ and the quantum relative entropy ([3-6]),
respectively.
We point out that if the state $\sigma$ is invertible, then the definition of
quantum unified $(r,s)-$relative entropy can be extended to $r>1$. Moreover,
we have the following important equalities:
$\displaystyle H^{1}_{r}(\rho\|\sigma)$ $\displaystyle=$ $\displaystyle
H^{r}(\rho\|\sigma),$ (7) $\displaystyle H^{r}_{\frac{1}{r}}(\rho\|\sigma)$
$\displaystyle=$ ${}_{r}H(\rho\|\sigma),$ (8)
(3) and (4) showed that the quantum Tsallis relative entropy and quantum
relative entropy of type $r$ are the particular cases of the quantum
$(r,s)$-relative entropy.
In order to study the properties of quantum unified $(r,s)$-relative entropy,
we need the following lemma.
Lemma 3.1. Let $H$ be a separable complex Hilbert spaces, $A$ and $B$ two
positive trace class operators on $H$. Then for any $\lambda,\mu>0$, we have
$R(\lambda A+\mu B)=R(A+B)$, where $R(A)$ is the range of $A$.
Proof. In fact, if $0\leq A\leq B$, that is, $0\leq
A^{\frac{1}{2}}A^{\frac{1}{2}}\leq B^{\frac{1}{2}}B^{\frac{1}{2}}$, then it
follows from Theorem 1 in [8] that $R({A}^{\frac{1}{2}})\subseteq
R({B}^{\frac{1}{2}})$. Note that $R({A}^{\frac{1}{2}})=R({A})$,
$R({B}^{\frac{1}{2}})=R(B)$, so $R(A)\subseteq R(B)$. Thus, we have
$R(A+B)\subseteq R(A+(1+\alpha)B)$ for any $\alpha>0$. On the other hand, take
$n$ such that $0\leq\frac{A+(1+\alpha)B}{n}\leq A+B$, then
$R(A+(1+\alpha)B)=R(\frac{A+(1+\alpha)B}{n})\subseteq R(A+B).$ Thus,
$R(A+B)=R(A+(1+\alpha)B)$, i.e., $R(A+B)=R(A+\beta B)$ for any $\beta>1.$
Replace $B$ with $\frac{1}{\beta}B$, we have $R(A+\frac{1}{\beta}B)=R(A+B)$
for any $\beta>1$. Hence, $R(A+\mu B)=R(A+B)$ for any $\mu>0$. Furthermore,
$R(\lambda A+\mu B)=R(A+\mu B)=R(A+B)$ for any $\lambda>0$ and $\mu>0$.
Theorem 3.1. Let $H$, $H_{1}$ and $H_{2}$ be separable complex Hilbert spaces.
(I) If $\rho$ and $\sigma$ are two states on $H$, then
$E_{r}^{s}(\rho\|\sigma)\geq 0$. Furthermore, when $0<r\leq 1$,
$E_{r}^{s}(\rho\|\sigma)=0$ iff $\rho=\sigma$; when $r=0$,
$E_{r}^{s}(\rho\|\sigma)=0$ iff $Ker(\rho)\subseteq Ker(\sigma)$.
(II) If $\rho_{j}$ and $\sigma_{j}$ are states on $H$, $\lambda_{i}>0$,
$j=1,2,\cdots,n$, and $\sum_{j=1}^{n}\lambda_{j}=1$, then when $r=1$ or $0\leq
r<1$ and $s\leq 1$, we have
$E_{r}^{s}(\sum_{j}\lambda_{j}\rho_{j}\|\sum_{j}\lambda_{j}\sigma_{j})\leq\sum_{j}\lambda_{j}E_{r}^{s}(\rho_{j}\|\sigma_{j}).$
(III) If $\rho$ and $\sigma$ are two states, $U$ is a unitary operator on $H$,
then
$E_{r}^{s}(U\rho U^{*}\|U\sigma U^{*})=E_{r}^{s}(\rho\|\sigma).$
(IV) If $\rho_{1}$ and $\sigma_{1}$ are two states on $H_{1}$, $\rho_{2}$ and
$\sigma_{2}$ are two states on $H_{2}$, then
$E_{r}^{s}(\rho_{1}\otimes\rho_{2}\|\sigma_{1}\otimes\sigma_{2})=E_{r}^{s}(\rho_{1}\|\sigma_{1})+E_{r}^{s}(\rho_{2}\|\sigma_{2})+(r-1)sE_{r}^{s}(\rho_{1}\|\sigma_{1})E_{r}^{s}(\rho_{2}\|\sigma_{2}).$
Proof. For $r=1$, the conclusion had been proved (see [3], [9-12]). Note that
(3) and (4), we only need to prove the cases of
$E^{s}_{r}(\rho\|\sigma)=H^{s}_{r}(\rho\|\sigma)$ and
$E^{s}_{r}(\rho\|\sigma)=H_{r}(\rho\|\sigma)$.
(I) Note that when $0\leq r<1$ and $s\neq 0$, $h(x)=\frac{1-x^{s}}{(1-r)s}$
and $g(x)=\frac{\ln x}{r-1}$ are monotone decreasing, so it is sufficient to
prove that $0\leq tr(\rho^{r}\sigma^{1-r})\leq 1$.
Let $\rho=0P_{0}+\sum\limits_{i}\lambda_{i}P_{i}$ and
$\sigma=0Q_{0}+\sum\limits_{j}\mu_{j}Q_{j}$ be the spectral decompositions of
states $\rho$ and $\sigma$, where $i,j\in{\bf N}=\\{1,2,\cdots\\}$, $P_{i}$
and $Q_{j}$ are the one dimension projection operators, $P_{0}$ and $Q_{0}$
are the projections on the kernel spaces of $\rho$ and $\sigma$, respectively,
and $\lambda_{i}>0,\mu_{j}>0$. Then $P_{i}P_{0}=0$, $Q_{j}Q_{0}=0$,
$P_{i}P_{j}=Q_{i}Q_{j}=0$ if $i\neq j$,
$P_{0}+\sum\limits_{i}P_{i}=Q_{0}+\sum\limits_{j}Q_{j}=I$ and
$\sum\limits_{i}\lambda_{i}=\sum\limits_{j}\mu_{j}=1$. So, we have
$tr(P_{0}Q_{j})+\sum\limits_{i}tr(P_{i}Q_{j})=tr(Q_{j})=1$ and
$tr(Q_{0}P_{i})+\sum\limits_{j}tr(P_{i}Q_{j})=tr(P_{i})=1$. Thus, when $0\leq
r<1$,
$\displaystyle tr(\rho^{r}\sigma^{1-r})$ $\displaystyle=$
$\displaystyle\sum_{i}\sum_{j}\lambda_{i}^{r}\mu_{j}^{1-r}tr(P_{i}Q_{j})$
$\displaystyle=$
$\displaystyle\sum_{i}\sum_{j}\lambda_{i}^{r}\mu_{j}^{1-r}tr(Q_{j}P_{i}Q_{j})$
$\displaystyle\geq$ $\displaystyle 0.$
When $r=0,$
$\displaystyle tr(\rho^{0}\sigma^{1-0})$ $\displaystyle=$
$\displaystyle\sum_{ij}\mu_{j}tr(P_{i}Q_{j})$ $\displaystyle=$
$\displaystyle\sum_{j}\mu_{j}tr(\sum_{i}P_{i}Q_{j})$ $\displaystyle\leq$
$\displaystyle\sum_{j}\mu_{j}=1,$
and with equality iff for any $j,\sum\limits_{i}tr(P_{i}Q_{j})=1$ iff for any
$j,tr(P_{0}Q_{j})=0$ iff $P_{0}\leq Q_{0}$ iff $Ker(\rho)\subseteq
Ker(\sigma)$.
When $0<r<1$, note that $\sum_{i}tr(P_{i}Q_{j})+tr(P_{0}Q_{j})=1,$ by the
concavity of $f(x)=x^{r}$, we have
$\displaystyle tr(\rho^{r}\sigma^{1-r})$ $\displaystyle=$
$\displaystyle\sum_{i}\sum_{j}\lambda_{i}^{r}\mu_{j}^{1-r}tr(P_{i}Q_{j})$ (9)
$\displaystyle=$
$\displaystyle\sum_{j}\mu_{j}[\sum_{i}(\frac{\lambda_{i}}{\mu_{j}})^{r}tr(P_{i}Q_{j})+(\frac{0}{\mu_{j}})^{r}tr(P_{0}Q_{j})]$
(10) $\displaystyle\leq$
$\displaystyle\sum_{j}\mu_{j}[\sum_{i}\frac{\lambda_{i}}{\mu_{j}}tr(P_{i}Q_{j})+\frac{0}{\mu_{j}}tr(P_{0}Q_{j})]^{r}$
(11) $\displaystyle\leq$
$\displaystyle(\sum_{j}\mu_{j}{\sum_{i}\frac{\lambda_{i}}{\mu_{j}}tr(P_{i}Q_{j})})^{r}$
(12) $\displaystyle=$
$\displaystyle(\sum_{i}\lambda_{i}\sum_{j}tr(P_{i}Q_{j}))^{r}$ (13)
$\displaystyle\leq$ $\displaystyle 1.$ (14)
Thus, we proved that when $0\leq r<1$, $0\leq tr(\rho^{r}\sigma^{1-r})\leq 1$,
and when $r=0$, $tr(\rho^{0}\sigma^{1-0})=1$ iff $Ker(\rho)\subseteq
Ker(\sigma)$. Note that when $0<r<1$, $E_{r}^{s}(\rho\|\sigma)=0$ iff
$tr(\rho^{r}\sigma^{1-r})=1$, so, we only need to prove that if $0<r<1$ and
$tr(\rho^{r}\sigma^{1-r})=1$, then $\rho=\sigma$.
First, if $tr(\rho^{r}\sigma^{1-r})=1$, it follows from (9) and (10) that for
each $i\in\bf N$, $\sum\limits_{j}tr(P_{i}Q_{j})=1$, so
$tr(P_{i}Q_{0})=0=tr(P_{i}Q_{0}P_{i})$, it is easily to know that
$P_{i}Q_{0}P_{i}=0$, so for each $i\in\bf N$, $P_{i}Q_{0}=0$, thus we have
$Q_{0}\leq P_{0}$. Moreover, if $tr(\rho^{r}\sigma^{1-r})=1$, then (7) takes
equality, we get that (i) or (ii) as follows:
(i) For each given $j$, there exists a $i_{j}\in\bf N$ such that
$tr(P_{i_{j}}Q_{j})=1$ and $tr(P_{i}Q_{j})=0$ for all $i\neq i_{j}$.
(ii) For each $j$, we have
$\frac{\lambda_{1}}{\mu_{j}}=\frac{\lambda_{2}}{\mu_{j}}=\cdots$ and
$\sum\limits_{i}tr(P_{i}Q_{j})=1$.
If (i) is satisfied, then for each $j$, we have $P_{0}Q_{j}=0$, so $P_{0}\leq
Q_{0}$, combining this and $Q_{0}\leq P_{0}$ proved before, we get
$Q_{0}=P_{0}$. Moreover, note that $P_{i_{j}}$ and $Q_{j}$ are both one
dimensional projections and $tr(P_{i_{j}}Q_{j})=1$, so, it is easy to know
that $P_{i_{j}}=Q_{j}$. It also follows from $tr(\rho^{r}\sigma^{1-r})=1$ that
$\frac{\lambda_{i_{j}}}{\mu_{j}}=1$, thus, we can prove that $\rho=\sigma$.
If (ii) is satisfied, then for each $j$, we have
$\frac{\lambda_{1}}{\mu_{j}}=\frac{\lambda_{2}}{\mu_{j}}=\cdots$ and
$\sum_{i}tr(P_{i}Q_{j})=1$, so we have $\lambda_{1}=\lambda_{2}=\cdots$ and
for each $j$, $tr(P_{0}Q_{j})=0$, so we can prove that $P_{0}\leq Q_{0}$,
thus, $P_{0}=Q_{0}$. Moreover, it follows from $\frac{\lambda_{i}}{\mu_{j}}$
is a constant, $\sum_{i}tr(P_{i}Q_{j})=1$ and (5)-(10) that
$\mu_{1}=\mu_{2}=\cdots=\lambda_{1}=\lambda_{2}=\cdots$, thus, we have
$\rho=\sigma$, (I) is proved.
(II) Let $\rho$ and $\sigma$ be two states on $H$ and
$f(\rho,\sigma)=tr({\rho}^{r}{\sigma}^{1-r})$. If $0<r<1$, then it follows
from [13, Corollary 1.1] that $f(\rho,\sigma)=tr({\rho}^{r}{\sigma}^{1-r})$ is
a joint concave functional with respect to the states $\rho$ and $\sigma$,
that is, for any states $\rho_{1}$, $\rho_{2}$, $\sigma_{1}$ and $\sigma_{2}$,
when $0<\lambda<1$, we have
$\displaystyle
f(\lambda\rho_{1}+(1-\lambda)\rho_{2},\lambda\sigma_{1}+(1-\lambda)\sigma_{2})\geq\lambda
f(\rho_{1},\sigma_{1})+(1-\lambda)f(\rho_{2},\sigma_{2}).$ (15)
If $r=0$, let $P_{1}$, $P_{2}$ and $P$ be the projection operators on
$R(\rho_{1})$, $R(\rho_{2})$ and $R(\lambda\rho_{1}+(1-\lambda)\rho_{2})$,
respectively, then $\rho_{1}^{0}=P_{1}$, $\rho_{2}^{0}=P_{2}$,
$(\lambda\rho_{1}+(1-\lambda)\rho_{2})^{0}=P$. It follows from Lemma 3.1 that
$P\geq P_{1}$ and $P\geq P_{2}.$ Therefore, we have
$\displaystyle
tr((\lambda\rho_{1}+(1-\lambda)\rho_{2})^{0}(\lambda\sigma_{1}+(1-\lambda)\sigma_{2})^{1})$
$\displaystyle=$ $\displaystyle
tr(P(\lambda\sigma_{1}+(1-\lambda)\sigma_{2}))$ $\displaystyle=$
$\displaystyle\lambda tr(P\sigma_{1})+(1-\lambda)tr(P\sigma_{2})$
$\displaystyle\geq$ $\displaystyle\lambda
tr(P_{1}\sigma_{1})+(1-\lambda)tr(P_{2}\sigma_{2})$ $\displaystyle=$
$\displaystyle\lambda tr(P_{1}\sigma_{1})+(1-\lambda)tr(P_{2}\sigma_{2})$
$\displaystyle=$ $\displaystyle\lambda
tr((\rho_{1})^{0}\sigma_{1})+(1-\lambda)tr((\rho_{2})^{0}\sigma_{2}).$
This shows that the inequality (11) also holds when $r=0$.
If $0\leq r<1,s=0$, by the monotone decreasing property and convexity of the
function $g(x)=\frac{\ln x}{r-1}$, we have
$\displaystyle
H_{r}(\lambda\rho_{1}+(1-\lambda)\rho_{2}\|\lambda\sigma_{1}+(1-\lambda)\sigma_{2})$
$\displaystyle=$
$\displaystyle\frac{1}{r-1}\ln(f(\lambda\rho_{1}+(1-\lambda)\rho_{2},\lambda\sigma_{1}+(1-\lambda)\sigma_{2}))$
$\displaystyle\leq$ $\displaystyle\frac{1}{r-1}\ln(\lambda
f(\rho_{1},\sigma_{1})+(1-\lambda)f(\rho_{2},\sigma_{2}))$ $\displaystyle\leq$
$\displaystyle\lambda\frac{1}{r-1}\ln(f(\rho_{1},\sigma_{1})+(1-\lambda)\frac{1}{r-1}\ln(f(\rho_{2},\sigma_{2}))$
$\displaystyle=$ $\displaystyle\lambda
H_{r}(\rho_{1}\|\sigma_{1})+(1-\lambda)H_{r}(\rho_{2}\|\sigma_{2}).$
If $0\leq r<1,s\neq 0$ and $s\leq 1$, then $h(x)=\frac{1-x^{s}}{(1-r)s}$ is
also a monotone decreasing convex function, so
$\displaystyle
H_{r}^{s}(\lambda\rho_{1}+(1-\lambda)\rho_{2}\|\lambda\sigma_{1}+(1-\lambda)\sigma_{2})$
$\displaystyle=$
$\displaystyle[(1-r)s]^{-1}[1-f^{s}(\lambda\rho_{1}+(1-\lambda)\rho_{2},\lambda\sigma_{1}+(1-\lambda)\sigma_{2})]$
$\displaystyle\leq$ $\displaystyle[(1-r)s]^{-1}[1-(\lambda
f^{s}(\rho_{1},\sigma_{1})+(1-\lambda)f^{s}(\rho_{2},\sigma_{2}))]$
$\displaystyle=$ $\displaystyle\lambda
H_{r}^{s}(\rho_{1}\|\sigma_{1})+(1-\lambda)H_{r}^{s}(\rho_{2}\|\sigma_{2}).$
Thus, (II) is proved. (III) and (IV) can be proved easily, we omit them.
In order to study the other properties of quantum unified $(r,s)$-relative
entropy, we need the following:
Let $H_{1}$ and $H_{2}$ be two separable complex Hilbert spaces and
$H_{1}\otimes H_{2}$ their tensor product. The set of all trace class
operators on $H_{1}\otimes H_{2}$ is denoted by $T(H_{1}\otimes H_{2})$, the
set of all trace class positive operators on $H_{1}\otimes H_{2}$ is denoted
by $T_{+}(H_{1}\otimes H_{2})$. If $A\in T_{+}(H_{1}\otimes H_{2})$, by the
following form, we can define a trace class positive operator $A_{1}$ on
$H_{1}$:
$(x,A_{1}y)=\sum_{i}(x\otimes e_{i},A(y\otimes e_{i})),$
where $x,y\in H_{1}$, $\\{e_{i}\\}$ is an orthonormal basis of $H_{2}$. We
call $A_{1}$ to be the partial trace of $A$ on $H_{1}$ and denoted by
$A_{1}=tr_{2}A$. Similarly, we can define the partial trace $A_{2}$ of $A$ on
$H_{2}$. Note that when $A$ is a state, $A_{1}$ and $A_{2}$ are also states.
It follows from Theorem 3.1(III), Theorem 3.1(IV) and the methods in the proof
of [3, Theorem 11.17], we have
Lemma 3.2. Let $H_{1},H_{2}$ be two finite dimensional complex Hilbert space.
If $r=1$ or $0\leq r<1$ and $s\leq 1$, then for any states $\rho$ and $\sigma$
on $H_{1}\otimes H_{2}$,
$E_{r}^{s}(\rho_{1}||\sigma_{1})\leq E_{r}^{s}(\rho||\sigma),$
where $\rho_{1}$ and $\sigma_{1}$ are the partial traces of $\rho$ and
$\sigma$ on $H_{1}$, respectively.
Lemma 3.3. Let $H$ be a finite dimensional complex Hilbert space, $\Phi$ a
trace-preserving completely positive map of $T(H)$ into itself. If $r=1$ or
$0\leq r<1$ and $s\leq 1$, then for any states $\rho$ and $\sigma$ on $H$,
$E_{r}^{s}(\Phi(\rho)||\Phi(\sigma))\leq E_{r}^{s}(\rho||\sigma).$
Proof. Taking a finite dimensional complex Hilbert space $H_{0}$ such that the
dimension of $H_{0}$ is bigger than 1. Then it follows from ([9,11-12]) that
there are a unitary operator $U$ on $H\otimes H_{0}$ and a projection operator
$P$ on $H_{0}$ such that for any state $\rho$ on $H$, we have
$\Phi(\rho)=tr_{2}(U(\rho\otimes P)U^{*}),$
thus, it follows from Lemma 3.2 that
$E_{r}^{s}(\Phi(\rho)||\Phi(\sigma))\leq E_{r}^{s}(U(\rho\otimes
P)U^{*}||U(\sigma\otimes P)U^{*})=E_{r}^{s}(\rho\otimes P||\sigma\otimes
P)=E_{r}^{s}(\rho||\sigma).$
Lemma 3.4 ([11]). Let $H$ be a separable complex Hilbert space, $\Phi$ a
trace-preserving completely positive map of $T(H)$ into itself, and
$\\{P_{n}\\}$ a family of finite-dimensional projections such that $P_{m}\leq
P_{n}$ for $m\leq n$ and $P_{n}\rightarrow I$ strongly when
$n\rightarrow\infty$. Then there is a family $\\{\Phi_{n}\\}$ of completely
positive maps such that $\\{\Phi_{n}\\}$ is trace-preserving on $P_{n}(H)$ and
$\Phi_{n}(A)\rightarrow\Phi(A)$ uniformly for each $A\in T_{+}(H)$.
Theorem 3.2. Let $H$ be a separable complex Hilbert space and $\Phi$ a trace-
preserving completely positive map of $T(H)$ into itself. If $r=1$ or $0\leq
r<1$ and $s\leq 1$, then for any state $\rho$ and $\sigma$, we have
$\displaystyle E_{r}^{s}(\Phi(\rho)\|\Phi(\sigma))\leq
E_{r}^{s}(\rho\|\sigma).$
Proof. Let $P_{n}$ and $\Phi_{n}$ satisfy the conditions of Lemma 3.4. Then
$\Phi_{n}(\rho)\rightarrow\Phi(\rho)$ uniformly for each state $\rho$. Since
function $x^{r}y^{1-r}$ is continuous, we have
$(\Phi_{n}(\rho))^{r}(\Phi_{n}(\sigma))^{1-r}\rightarrow(\Phi(\rho))^{r}(\Phi(\sigma))^{1-r}$
uniformly, hence $tr((\Phi_{n}(\rho))^{r}(\Phi_{n}(\sigma))^{1-r})\rightarrow
tr((\Phi(\rho))^{r}(\Phi(\sigma))^{1-r}).$ This shows that
$\displaystyle
E_{r}^{s}(\Phi(\rho)\|\Phi(\sigma))=\lim_{n\rightarrow\infty}E_{r}^{s}(\Phi_{n}(\rho)\|\Phi_{n}(\sigma)).$
Let $\rho_{n}=\frac{P_{n}\rho P_{n}}{tr(\rho
P_{n})},\sigma_{n}=\frac{P_{n}\sigma P_{n}}{tr(\sigma P_{n})}$. By the proof
of Lemma 4 in [10], $P_{n}\rho P_{n}\rightarrow\rho$ and $P_{n}\sigma
P_{n}\rightarrow\sigma$ uniformly. Hence $tr(P_{n}\rho P_{n})\rightarrow
tr(\rho)=1$ and $tr(P_{n}\sigma P_{n})\rightarrow tr(\sigma)=1.$ Therefore
$\displaystyle\lim_{n\rightarrow\infty}(\Phi_{n}(\rho))^{r}(\Phi_{n}(\sigma))^{1-r}$
$\displaystyle=$ $\displaystyle\lim_{n\rightarrow\infty}(\Phi_{n}(P_{n}\rho
P_{n}))^{r}(\Phi_{n}(P_{n}\sigma P_{n}))^{1-r}$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{(\Phi_{n}(P_{n}\rho
P_{n}))^{r}(\Phi_{n}(P_{n}\sigma P_{n}))^{1-r}}{(tr(P_{n}\rho
P_{n}))^{r}(tr(P_{n}\sigma P_{n}))^{1-r}}$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}(\Phi_{n}(\rho_{n}))^{r}(\Phi_{n}(\sigma_{n}))^{1-r}.$
Hence we get that
$\lim\limits_{n\rightarrow\infty}E_{r}^{s}(\Phi_{n}(\rho)\|\Phi_{n}(\sigma))=\lim\limits_{n\rightarrow\infty}E_{r}^{s}(\Phi_{n}(\rho_{n})\|\Phi_{n}(\sigma_{n})).$
By Lemma 3.2, $E_{r}^{s}(\Phi_{n}(\rho_{n})\|\Phi_{n}(\sigma_{n}))\leq
E_{r}^{s}(\rho_{n}\|\sigma_{n})$. Again
$\rho_{n}\rightarrow\rho,\sigma_{n}\rightarrow\sigma$ uniformly, we get that
$\lim\limits_{n\rightarrow\infty}E_{r}^{s}(\rho_{n}\|\sigma_{n})=E_{r}^{s}(\rho\|\sigma).$
Therefore $E_{r}^{s}(\Phi(\rho)\|\Phi(\sigma))\leq E_{r}^{s}(\rho\|\sigma).$
That completes the proof.
Theorem 3.3 (Monotonicity). Let $H_{1},H_{2}$ be separable complex Hilbert
space, $H=H_{1}\otimes H_{2}$. If $r=1$ or $0\leq r<1$ and $s\leq 1$, then for
any state $\rho$ and $\sigma$ on $H$,
$E_{r}^{s}(\rho_{1}\|\sigma_{1})\leq E_{r}^{s}(\rho\|\sigma),$
where $\rho_{1}$ and $\sigma_{1}$ are the partial traces of $\rho$ and
$\sigma$ on $H_{1}$, respectively.
Proof. Since $H_{2}$ is a separable complex Hilbert space, so there is a
sequence of $\\{P_{n}\\}$ of finite-dimensional projection operators on
$H_{2}$ such that $P_{m}\leq P_{n}$ for $m\leq n$ and $P_{n}\rightarrow I$
strongly when $n\rightarrow\infty$. Let
$H_{2}^{n}=P_{n}(H_{2}),H^{n}=H_{1}\otimes H_{2}^{n}$. It follows from the
proof of Lemma 4 in [10] again that
$\rho_{n}=\frac{(I\otimes P_{n})\rho(I\otimes P_{n})}{tr(\rho(I\otimes
P_{n}))}\rightarrow\rho,$ $\sigma_{n}=\frac{(I\otimes P_{n})\sigma(I\otimes
P_{n})}{tr(\sigma(I\otimes P_{n}))}\rightarrow\sigma,$
$\rho_{1n}=tr_{2}\rho_{n}\rightarrow\rho_{1},$
$\sigma_{1n}=tr_{2}\sigma_{n}\rightarrow\sigma_{1}$
uniformly. Hence $E_{r}^{s}(\rho_{1n}\|\sigma_{1n})\rightarrow
E_{r}^{s}(\rho_{1}\|\sigma_{1}),$ and
$E_{r}^{s}(\rho_{n}\|\sigma_{n})\rightarrow E_{r}^{s}(\rho\|\sigma).$
Define $\Phi:B(H^{n})\rightarrow B(H_{1})\otimes\\{\lambda I_{2}^{n}\\}$ by
$\Phi(\rho)=(tr_{2}\rho)\otimes C_{2n},$ where $I_{2}^{n}$ is the identity
operator on $H_{2}^{n}$) and $C_{2n}=(dimH_{2}^{n})^{-1}I^{n}_{2}.$ Then
$E_{r}^{s}(\Phi(\rho_{n})\|\Phi(\sigma_{n}))=E_{r}^{s}(\rho_{1n}\otimes
C_{2n}\|\sigma_{1n}\otimes C_{2n})=E_{r}^{s}(\rho_{1n}\|\sigma_{1n}).$
It is obvious that $\Phi$ is a trace-preserving completely positive map from
$B(H^{n})$ into itself. By Theorem 3.2,
$E_{r}^{s}(\Phi(\rho_{n})\|\Phi(\sigma_{n}))\leq
E_{r}^{s}(\rho_{n}\|\sigma_{n})$, so
$E_{r}^{s}(\rho_{1n}\|\sigma_{1n})\leq E_{r}^{s}(\rho_{n}\|\sigma_{n}).$
Note that $E_{r}^{s}(\rho_{1n}\|\sigma_{1n})\rightarrow
E_{r}^{s}(\rho_{1}\|\sigma_{1}),$ $E_{r}^{s}(\rho_{n}\|\sigma_{n})\rightarrow
E_{r}^{s}(\rho\|\sigma),$ thus we have
$E_{r}^{s}(\rho_{1}\|\sigma_{1})\leq E_{r}^{s}(\rho\|\sigma)$
and the theorem is proved.
Theorem 3.4. Let $H$ be a separable complex Hilbert space, $\rho$ and $\sigma$
two states on $H$ and $\sigma$ invertible. Then for $r=1$ or $0\leq r<1,s\geq
0,$ we have
$\displaystyle E_{r}^{s}(\rho\|\sigma)\leq H(\rho\|\sigma)\leq
E_{2-r}^{s}(\rho\|\sigma).$ (16)
Proof. That $r=1$ is clear. If $0\leq r<1,s=0$, we need to prove that
$\displaystyle H_{r}(\rho\|\sigma)\leq H(\rho\|\sigma)\leq
H_{2-r}(\rho\|\sigma).$ (17)
Let $\rho=0P_{0}+\sum\limits_{i}\lambda_{i}P_{i}$ and
$\sigma=\sum\limits_{j}\mu_{j}Q_{j}$ be the spectral decompositions of $\rho$
and $\sigma$, where $P_{i}$ and $Q_{j}$ be the one dimension projection
operators, $P_{0}$ be the projection operator on the kernel space of $\rho$,
and $P_{i}P_{0}=0$, $\lambda_{i}>0,\mu_{j}>0$ when $i,j\in\bf N$, and
$P_{i}P_{j}=Q_{i}Q_{j}=0$ if $i\neq j$,
$\sum\limits_{i}\lambda_{i}=\sum\limits_{j}\mu_{j}=1$,
$P_{0}+\sum\limits_{i}P_{i}=\sum\limits_{j}Q_{j}=I$. Then
$\displaystyle H_{2-r}(\rho\|\sigma)$ $\displaystyle=$
$\displaystyle-\frac{\ln tr(\rho^{2-r}\sigma^{r-1})}{r-1}$ $\displaystyle=$
$\displaystyle-\frac{1}{r-1}\ln\sum_{ij}\lambda_{i}^{2-r}\mu_{j}^{r-1}tr(P_{i}Q_{j})$
$\displaystyle=$
$\displaystyle-\frac{1}{r-1}\ln\sum_{ij}\lambda_{i}tr(P_{i}Q_{j})(\frac{\lambda_{i}}{\mu_{j}})^{1-r}$
Let $g(x)=-\frac{1}{r-1}\ln x,\ \alpha_{ij}=\lambda_{i}tr(P_{i}Q_{j})$ and
$x_{ij}=(\frac{\lambda_{i}}{\mu_{j}})^{1-r}.$ Then
$\sum\limits_{ij}\alpha_{ij}=\sum\limits_{ij}\lambda_{i}tr(P_{i}Q_{j})=\sum\limits_{j}tr(\rho
Q_{j})=tr(\rho)=1.$ By the concavity of the function $g(x)=-\frac{1}{r-1}\ln
x,$ we have
$\displaystyle H_{2-r}(\rho\|\sigma)$ $\displaystyle=$ $\displaystyle
g(\sum_{ij}\alpha_{ij}x_{ij})$ $\displaystyle\geq$
$\displaystyle\sum_{ij}\alpha_{ij}g(x_{ij})$ $\displaystyle=$
$\displaystyle\sum_{ij}\lambda_{i}tr(P_{i}Q_{j})\left(-\frac{1}{r-1}\ln(\frac{\lambda_{i}}{\mu_{j}})^{1-r}\right)$
$\displaystyle=$
$\displaystyle\sum_{ij}\lambda_{i}tr(P_{i}Q_{j})(\ln\lambda_{i}-\ln\mu_{j})$
$\displaystyle=$ $\displaystyle H(\rho\|\sigma).$
The left-hand side inequality of (13) is proven by a similar way.
If $0\leq r<1,s>0$, we need to prove that
$\displaystyle H_{r}^{s}(\rho\|\sigma)\leq H(\rho\|\sigma)\leq
H_{2-r}^{s}(\rho\|\sigma).$ (18)
Let $tr(\rho^{r}\sigma^{1-r})=x_{0}.$ Since
$\ln x\leq\frac{x^{s}-1}{s},$
for any $x>0,s>0,$ so, for any $0\leq r<1,s>0$, we have
$-\frac{x_{0}^{s}-1}{[(r-1)s]}\geq-(r-1)^{-1}\ln x_{0}.$
That is,
$H_{2-r}^{s}(\rho\|\sigma)\geq H_{2-r}(\rho\|\sigma).$
Combining this with (13), we have $H_{2-r}^{s}(\rho\|\sigma)\geq
H(\rho\|\sigma)$. Similarly, the left-hand side inequality of (14) can be
proven.
Note that when $0\leq r<1,s=1$, the inequalities (12) degenerate into
convexity inequalities for estimating free energy and relative entropy given
by Ruskai and Stillinger in [14].
Theorem 3.5. Let $\rho$ and $\sigma$ be two states on the separable complex
Hilbert space ${\mathcal{H}}$. We have
(1) If $\rho$ is an invertible state, then $E_{0}^{s}(\rho\|\sigma)=0$.
(2) If $s\geq 0$, then $E_{r}^{s}(\rho\|\sigma)$ is monotone increasing with
respect to $r\in[0,1]$; if $s<0$, then $E_{r}^{s}(\rho\|\sigma)$ is monotone
decreasing with respect to $r\in[0,1]$.
(3) For each $0\leq r\leq 1$, $E_{r}^{s}(\rho\|\sigma)$ is monotone decreasing
with respect to $s$.
(4) For each $0\leq r\leq 1$, $E_{r}^{s}(\rho\|\sigma)$ is a convex function
of $s$.
Proof. (1) If $\rho$ is invertible, we have $\rho^{0}=I$, so
$E_{0}^{s}(\rho\|\sigma)=0$.
(2) It follows from Theorem 3.4 that $E_{r}^{s}(\rho\|\sigma)\leq
H(\rho\|\sigma)=E_{1}^{s}(\rho\|\sigma)$, so it is sufficient to prove the
conclusion for $0\leq r<1$ and any $s$.
Let $\rho=0P_{0}+\sum\limits_{i}\lambda_{i}P_{i}$ and
$\sigma=0Q_{0}+\sum\limits_{j}\mu_{j}Q_{j}$ be the spectral decompositions of
$\rho$ and $\sigma$, where $P_{i}$ and $Q_{j}$ are the one dimension
projection operators, $P_{0}$ and $Q_{0}$ are the projection operators on the
zero spaces of $\rho$ and $\sigma$ respectively, and for all $i,j\in\bf N$,
$\lambda_{i}>0,\mu_{j}>0$. Then
$\sum\limits_{i}\lambda_{i}=\sum\limits_{j}\mu_{j}=1$,
$P_{0}+\sum\limits_{i}P_{i}=Q_{0}+\sum\limits_{j}Q_{j}=I$.
Let $f(x)=x\ln x$, $\alpha_{ij}=\lambda_{i}tr(P_{i}Q_{j})$,
$x_{ij}=(\frac{\mu_{j}}{\lambda_{i}})^{1-r}$. Then
$\sum\limits_{i}[\sum\limits_{j}\alpha_{ij}+\lambda_{i}tr(P_{i}Q_{0})]=\sum\limits_{i}\lambda_{i}tr(P_{i}(\sum\limits_{i}Q_{j}+Q_{0}))=tr(\rho)=1.$
Because $f(x)$ is a convex function, we have
$\sum\limits_{ij}\alpha_{ij}f(x_{ij})\geq
f(\sum\limits_{ij}\alpha_{ij}x_{ij}).$ Therefore,
$\sum\limits_{ij}\lambda_{i}tr(P_{i}Q_{j})(\frac{\mu_{j}}{\lambda_{i}})^{1-r}\ln(\frac{\mu_{j}}{\lambda_{i}})^{1-r}\geq\sum\limits_{ij}\lambda_{i}tr(P_{i}Q_{j})(\frac{\mu_{j}}{\lambda_{i}})^{1-r}\ln\sum\limits_{ij}\lambda_{i}tr(P_{i}Q_{j})(\frac{\mu_{j}}{\lambda_{i}})^{1-r},$
that is,
$\displaystyle-(1-r)tr(\rho^{r}(\ln\rho-\ln\sigma)\sigma^{1-r})\geq
tr(\rho^{r}\sigma^{1-r})\ln tr(\rho^{r}\sigma^{1-r}).$ (19)
(i) If $s=0$, then
$E_{r}^{0}(\rho\|\sigma)=H_{r}(\rho\|\sigma)=-(1-r)^{-1}\ln(tr(\rho^{r}\sigma^{1-r}))$.
Note that $tr(\rho^{r}\sigma^{1-r})=0$ iff for any $i,j\in{\bf
N},P_{i}Q_{j}=0.$ Hence, if for some $0\leq r_{0}<1$ such that
$tr(\rho^{r_{0}}\sigma^{1-{r_{0}}})=0$, then it is easily to see that for any
$0\leq r<1$, $tr(\rho^{r}\sigma^{1-{r}})=0$, so for any $0\leq r<1$,
$E_{r}^{0}(\rho\|\sigma)=H_{r}(\rho\|\sigma)=+\infty$, thus, the conclusion is
also true in this case. If for each $0\leq r<1$, $tr(\rho^{r}\sigma^{1-r})>0$,
then
$\displaystyle\frac{dH_{r}(\rho\|\sigma)}{dr}$ $\displaystyle=$
$\displaystyle\frac{H_{r}(\rho\|\sigma)}{1-r}-\frac{tr(\rho^{r}(\ln\rho-\ln\sigma)\sigma^{1-r})}{(1-r)tr(\rho^{r}\sigma^{1-r})}$
$\displaystyle=$ $\displaystyle\frac{-\ln
tr(\rho^{r}\sigma^{1-r})}{(1-r)^{2}}-\frac{tr(\rho^{r}(\ln\rho-\ln\sigma)\sigma^{1-r})}{(1-r)tr(\rho^{r}\sigma^{1-r})}.$
By (15), we know that
$-tr(\rho^{r}(\ln\rho-\ln\sigma)\sigma^{1-r})\geq\frac{1}{1-r}{tr(\rho^{r}\sigma^{1-r})\ln
tr(\rho^{r}\sigma^{1-r})},$ so
$\frac{dH_{r}(\rho\|\sigma)}{dr}\geq\frac{-\ln
tr(\rho^{r}\sigma^{1-r})}{(1-r)^{2}}+\frac{\ln
tr(\rho^{r}\sigma^{1-r})}{(1-r)^{2}}=0.$
This conclusion is proved when $s=0$.
(ii) If $s\neq 0$, and for some $0\leq r_{0}<1$,
$tr(\rho^{r_{0}}\sigma^{1-{r_{0}}})=0$, then for any $0\leq r<1$,
$tr(\rho^{r}\sigma^{1-{r}})=0$, so for any $0\leq r<1$,
$E_{r}^{s}(\rho\|\sigma)=H_{r}^{s}(\rho\|\sigma)=\frac{1}{(1-r)s}$, thus, the
conclusion is true in this case. If for each $0\leq r<1$,
$tr(\rho^{r}\sigma^{1-r})>0$, then
$E_{r}^{s}(\rho\|\sigma)=H_{r}^{s}(\rho\|\sigma)=-[(1-r)s]^{-1}[(tr(\rho^{r}\sigma^{1-r}))^{s}-1]$,
so by (15) again that
$\displaystyle\frac{dH_{r}^{s}(\rho\|\sigma)}{dr}$ $\displaystyle=$
$\displaystyle\frac{H_{r}^{s}(\rho\|\sigma)}{1-r}-\frac{(tr(\rho^{r}\sigma^{1-r}))^{s-1}tr\rho^{r}(\ln\rho-\ln\sigma)\sigma^{1-r}}{1-r}$
$\displaystyle=$
$\displaystyle\frac{1}{(1-r)^{2}}\left[\frac{1-(tr(\rho^{r}\sigma^{1-r}))^{s}}{s}-(1-r)(tr(\rho^{r}\sigma^{1-r}))^{s-1}tr(\rho^{r}(\ln\rho-\ln\sigma)\sigma^{1-r})\right]$
$\displaystyle\geq$
$\displaystyle\frac{1}{(1-r)^{2}}\left[\frac{1-(tr(\rho^{r}\sigma^{1-r}))^{s}}{s}+\frac{s(tr(\rho^{r}\sigma^{1-r})\ln(tr\rho^{r}\sigma^{1-r}))(tr(\rho^{r}\sigma^{1-r}))^{s-1}}{s}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{(1-r)^{2}}\left[\frac{1-(tr(\rho^{r}\sigma^{1-r}))^{s}+s(tr(\rho^{r}\sigma^{1-r}))^{s}\ln
tr(\rho^{r}\sigma^{1-r})}{s}\right].$
Let $tr(\rho^{r}\sigma^{1-r})=x,\ f(x)=\frac{sx^{s}\ln x-x^{s}+1}{s}.$ Then
$0<x\leq 1,\ f^{\prime}(x)=sx^{s-1}\ln x.$ Note that If $s>0$, then
$f^{\prime}(x)\leq 0,\ f(x)\geq f(1)=0.$ Thus,
$\frac{dH^{s}_{r}(\rho\|\sigma)}{dr}\geq 0.$ Similarly,
$\frac{dH^{s}_{r}(\rho\|\sigma)}{dr}\leq 0$ if $s<0$.
The conclusion is proved finally.
(3) If $r=1$, then $E_{r}^{s}(\rho\|\sigma)=H(\rho\|\sigma)$ is a constant, so
the conclusion is true in this case.
If for some $0\leq r_{0}<1$, $tr(\rho^{r_{0}}\sigma^{1-{r_{0}}})=0$, then for
any $0\leq r<1$, $tr(\rho^{r}\sigma^{1-{r}})=0$, thus,
$E_{r}^{0}(\rho\|\sigma)=+\infty>\frac{1}{(1-r)s}=E_{r}^{s}(\rho\|\sigma)$ for
any $s$.
If for each $0\leq r<1$, $tr(\rho^{r}\sigma^{1-r})>0$, by the inequality
$\ln x\leq\frac{x^{s}-1}{s}$
for any $x>0,s>0$, we can prove that for any $s>0$ and $0\leq r<1$,
$E_{r}^{s}(\rho\|\sigma)\leq E_{r}^{0}(\rho\|\sigma)$. Similarly, we have
$E_{r}^{s}(\rho\|\sigma)\geq E_{r}^{0}(\rho\|\sigma)$ for any $s<0$ and $0\leq
r<1$.
Thus, in order to prove the conclusion, it is sufficient to show that
$\frac{dE^{s}_{r}(\rho\|\sigma)}{ds}\leq 0$
if $tr(\rho^{r}\sigma^{1-{r}})>0$ for any $s\neq 0,0\leq r<1.$ Note that
$\displaystyle\frac{dE^{s}_{r}(\rho\|\sigma)}{ds}=\frac{[-(tr(\rho^{r}\sigma^{1-r}))^{s}\ln
tr(\rho^{r}\sigma^{1-r})]s-[1-(tr(\rho^{r}\sigma^{1-r}))^{s}]}{(1-r)s^{2}}.$
(20)
Let $tr(\rho^{r}\sigma^{1-r})=x$, $f(x)=\frac{x^{s}-1-sx^{s}\ln
x}{(1-r)s^{2}}.$ Then $0<x\leq 1$,
$\frac{dE^{s}_{r}(\rho\|\sigma)}{ds}=\frac{{x}^{s}-1-s{x}^{s}\ln{x}}{(1-r)s^{2}}$,
$f^{{}^{\prime}}(x)=\frac{-x^{s-1}\ln x}{1-r}\geq 0,$ so $f(x)\leq f(1)=0.$
Thus we get $\frac{dE^{s}_{r}(\rho\|\sigma)}{ds}\leq 0.$
(4) When $r=1$, the conclusion is clear.
When $s=0$, $E_{r}^{s}(\rho\|\sigma)=H_{r}(\rho\|\sigma)$ is a constant
function of $s$. Hence, the conclusion is clear.
When $s\neq 0$, if for some $0\leq r_{0}<1$,
$tr(\rho^{r_{0}}\sigma^{1-{r_{0}}})=0$, then for any $0\leq r<1$,
$tr(\rho^{r}\sigma^{1-{r}})=0$, thus,
$E_{r}^{s}(\rho\|\sigma)=\frac{1}{(1-r)s}$ is a convex function of $s$, hence,
the conclusion is true in this case.
When $s\neq 0$, if $tr(\rho^{r}\sigma^{1-r})>0$ for each $0\leq r<1$. Let
$tr(\rho^{r}\sigma^{1-r})=x$. Then $0<x\leq 1$. Moreover,
$\frac{d^{2}E^{s}_{r}(\rho\|\sigma)}{ds^{2}}=\frac{-s^{2}{x}^{s}\ln^{2}{x}+2s{x}^{s}\ln{x}+2(1-{x}^{s})}{(1-r)s^{3}}.$
Let $g(x)=-s^{2}x^{s}\ln^{2}x+2sx^{s}\ln x+2(1-x^{s}).$ Then
$g^{{}^{\prime}}(x)=-s^{3}x^{s-1}\ln^{2}x.$ Thus, $g^{{}^{\prime}}(x)\leq 0$
if $s>0$, $g^{{}^{\prime}}(x)\geq 0$ if $s<0$. Correspondingly, $g(s)\geq
g(1)=0$ if $s>0$, $g(s)\leq g(1)=0$ if $s<0$. Hence
$\frac{d^{2}E^{s}_{r}(\rho\|\sigma)}{ds^{2}}\geq 0$ for $s\neq 0$. The
conclusion is proved finally.
References
[1]. P. N. Rathie, Unified $(r,s)$-entropy and its bivariate measures, Inf.
Sci. 54, 23-39, (1991)
[2]. X. H. Hu and Z. X. Ye, Generalized quantum entropy, J. Math. Phys. 47,
023502-1-023502-7, (2006)
[3]. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum
Information. Cambrige University Press, Cambrige, (2000)
[4]. M. Ohya and D. Petz, Quantum Entropy and its Use. Springer-Verlag,
Berlin, (1991)
[5]. S. Furuichi, K. Yanagi and K. Kuriyama, Fundamental properties of Tsallis
relative entropy, J. Math. Phys. 45, 4868-4877, (2004)
[6]. S. Furuichi, A note on a parametrically extended entanglement-measure due
to Tsallis relative entropy, INFORMATION. 9, 837-844, (2006)
[7]. I. J. Taneja, L. Pardo, D. Morales and M. L. Men$\acute{e}$ndez, On
generalized information and divergence measures and their applications: a
brief review, Q$\ddot{U}$ESTII$\acute{O}$, 13, 47-73, (1989)
[8]. R. G. Douglas, On majorization and range inclusion of operators in
Hilbert space, Proc. Amer. Math. Soc. 17, 413-416, (1966)
[9]. A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221-260,
(1978)
[10]. G. Lindblad, Expectations and Entropy Inequalities for Finite Quantum
Systems, Commun. math. Phys. 39, 111-119, (1974)
[11]. G. Lindblad, Completely Positive Maps and Entropy Inequalities, Commun.
math. Phys. 40, 147-151, (1975)
[12]. M. B. Ruskai, Inequalities for quantum entropy: A review with conditions
for equality, J. Math. Phys. 43, 4358-4375, (2002)
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Conjecture, Advan. Math. 11, 267-288, (1973)
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|
arxiv-papers
| 2009-11-26T23:33:04 |
2024-09-04T02:49:06.719629
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Wang Jiamei and Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0911.5174"
}
|
0911.5189
|
# Curvature Dependent Diffusion Flow on Surface with Thickness
Naohisa Ogawa 111ogawanao@hit.ac.jp Hokkaido Institute of Technology, Sapporo
006-8585 Japan
###### Abstract
Particle diffusion in a two dimensional curved surface embedded in $R_{3}$ is
considered. In addition to the usual diffusion flow, we find a new flow with
an explicit curvature dependence. New diffusion equation is obtained in
$\epsilon$ (thickness of surface) expansion. As an example, the surface of
elliptic cylinder is considered, and curvature dependent diffusion coefficient
is calculated.
###### pacs:
87.10.-e, 02.40.Hw, 02.40.Ma, 82.40.Ck
## I Motivation
The particle motion on a given curved surface is old but interesting problem
in wide range of physics. Especially the diffusion process of particles on
such a manifold is still an open problem, and related to various kinds of
phenomena.
For example the motion of protein on cell membrane has great importance in
biophysics. There are several research papers discussing on this problem. Some
of them are treating this problem by using usual diffusion equation with
curved coordinate, and discuss the curvature (Gauss curvature) dependence of
its solution diffusion_equation . Other of them use the Langevin equation on
curved surface and calculating the curvature dependence of diffusion
coefficient langevin_equation .
The quantum mechanics of particle motion on such a curved manifold is also
considered by many authors. This problem is usually explained by the
Schroedinger equation with Laplace-Beltrami operator. However, when we treat
the curved surface as embedded one in 3 dimensional Euclidean space, situation
is changed and then we have a quantum potential term related to the curvature
additional to the kinetic operator da Costa ,ogawa_fujii ,fujii .
Another example is in larger scale physics in which our consideration is
devoted. Patterns of animal skins are well expressed by the reaction diffusion
equation Turing . But the patterns are different for each parts even in one
individual. For example, Char fish, the side part has white spot pattern, but
the back part has labyrinth pattern. (For these two patterns, see for example
Shoji_Iwasa .) One of the reasons might come from the curvature difference
between side part and back part. If the diffusion is influenced by the
curvature, this difference of patterns might be explained. Furthermore, the
cross section of fish has form of ellipsoid and the surface can be
approximated as the one of elliptic cylinder. In two dimensional space, we
have only two kinds of curvature, one is Gauss curvature and other is mean
curvature. Both are constructed from second fundamental tensor by taking
determinant or trace. Gauss curvature can also be constructed only by metric
tensor and its derivatives, but this is not the case for the mean curvature.
The elliptic cylinder, in which we have much interest, has zero Gauss
curvature and non-zero mean curvature. Therefore to explain the pattern change
of Char fish, solution of the diffusion equation should depend on mean
curvature. This is impossible if we start from usual diffusion equation
because it depends only on metric but not on second fundamental tensor.
Therefore we need some new diffusion equation, which bring not only Gauss
curvature but also mean curvature. In this article, we discuss how to
construct such curvature dependent diffusion equation.
## II Coordinate and Metric
The simple extension of diffusion equation in Euclidean space to Riemannian
space can be done by changing Laplacian with Cartesian coordinate to the one
with curved coordinate, i.e. Laplace Beltrami operator. This coordinate change
is not enough for our purpose, however. The way of construction of new
diffusion equation in this paper is the followings. We re-identify the two
dimensional diffusion as the limiting process from three dimensional
diffusion. We place the curved surface $\Sigma$ in three dimensional Euclidean
space $R_{3}$, and we put two similar copies of $\Sigma$, called
$\tilde{\Sigma}$ and $\Sigma^{\prime}$ at a small distance of $\epsilon/2$.
Our particles can only move between these two surfaces, and later we take a
limit $\epsilon\to 0$. We look for the form of diffusion equation in this
limit. The coordinates we use hereafter is the followings. (See fig.1)
$\vec{X}$ is the Cartesian coordinate in $R_{3}$. $\vec{x}$ is the Cartesian
coordinate which specifies only the points on $\Sigma$. $q^{i}$ is the curved
coordinate on $\Sigma$. (Small Latin indices $i,j,k,\cdots$ runs from 1 to 2.)
$q^{0}$ is the coordinate in $R_{3}$ normal to $\Sigma$. Further by using the
normal unit vector $\vec{n}(q^{1},q^{2})$ on $\Sigma$ at point
$(q^{1},q^{2})$, we can identify any points between two surfaces
$\Sigma^{\prime}$ and $\tilde{\Sigma}$ by the following thin-layer
approximation fujii .
$\vec{X}(q^{0},q^{1},q^{2})=\vec{x}(q^{1},q^{2})+q^{0}\vec{n}(q^{1},q^{2}),$
(1)
where $-\epsilon/2\leq q^{0}\leq\epsilon/2$.
Figure 1: Embedding and Coordinate
From this relation we can obtain the curvilinear coordinate system between two
surfaces ($\subset R_{3}$) by the coordinate $q^{\mu}=(q^{0},q^{1},q^{2})$,
and metric $G_{\mu\nu}$. (Hereafter Greek indices $\mu,\nu,\cdots$ runs from 0
to 2.)
$G_{\mu\nu}=\frac{\partial\vec{X}}{\partial
q^{\mu}}\cdot\frac{\partial\vec{X}}{\partial q^{\nu}}.$ (2)
Each part of $G_{\mu\nu}$ is the following.
$G_{ij}=g_{ij}+q^{0}(\frac{\partial\vec{x}}{\partial
q^{i}}\cdot\frac{\partial\vec{n}}{\partial
q^{j}}+\frac{\partial\vec{x}}{\partial
q^{j}}\cdot\frac{\partial\vec{n}}{\partial
q^{i}})+(q^{0})^{2}\frac{\partial\vec{n}}{\partial
q^{i}}\cdot\frac{\partial\vec{n}}{\partial q^{j}},$ (3)
where
$g_{ij}=\frac{\partial\vec{x}}{\partial
q^{i}}\cdot\frac{\partial\vec{x}}{\partial q^{j}}$ (4)
is the metric on $\Sigma$. Hereafter indices $i,j,k\cdots$ are lowered or
rised by $g_{ij}$ and its inverse $g^{ij}$. We also obtain
$G_{0i}=G_{i0}=0,~{}~{}G_{00}=1.$ (5)
We can proceed the calculation by using the new variables. We first define the
tangential vector to $\Sigma$ by
$\vec{B}_{k}=\frac{\partial\vec{x}}{\partial q^{k}}.$ (6)
Note that $\vec{n}\cdot\vec{B}_{k}=0$. Then we obtain two relations.
Gauss equation:
$\frac{\partial\vec{B}_{i}}{\partial
q^{j}}=-\kappa_{ij}\vec{n}+\Gamma^{k}_{ij}\vec{B}_{k},$ (7)
Weingarten equation:
$\frac{\partial\vec{n}}{\partial q^{j}}=\kappa_{j}^{m}\vec{B}_{m},$ (8)
where
$\Gamma^{k}_{ij}\equiv\frac{1}{2}g^{km}(\partial_{i}g_{mj}+\partial_{j}g_{im}-\partial_{m}g_{ij}).$
$\kappa_{ij}$ is called Euler-Schauten tensor, or second fundamental tensor
defined as
$\kappa_{ij}=\frac{\partial\vec{n}}{\partial q^{i}}\cdot\vec{B}_{j}.$ (9)
The second fundamental tensor $\kappa_{ij}$ is the projection of
$\partial\vec{n}$ into the surface. Furthermore, the mean curvature is given
by
$\kappa=g^{ij}\kappa_{ij},$ (10)
and Ricci scalar curvature $R$ is obtained by
$R/2=\det(g^{ik}\kappa_{kj})=\det(\kappa^{i}_{j})=\frac{1}{2}(\kappa^{2}-\kappa_{ij}\kappa^{ij}).$
(11)
Then we have the formula for metric of curvilinear coordinate in a
neighborhood of $\Sigma$.
$G_{ij}=g_{ij}+2q^{0}\kappa_{ij}+(q^{0})^{2}\kappa_{im}\kappa^{m}_{j}.$ (12)
Under the inversion $q^{0}\to-q^{0}$, we have $\kappa_{ij}\to-\kappa_{ij}$ as
well as $\vec{n}\to-\vec{n}$ from
$\vec{n}=\partial_{0}\vec{X}/\mid\partial_{0}\vec{X}\mid$. Therefore $G_{ij}$
is invariant under $q^{0}\to-q^{0}$.
Now we have the total metric tensor such as,
$G_{\mu\nu}=\left(\begin{array}[]{cc}1&~{}~{}~{}0{}{}{}\\\
0&G_{ij}\end{array}\right).$ (13)
## III Embedding of Diffusion field
Let us denote 3 dimensional diffusion field as $\phi^{(3)}$, and Laplacian as
$\Delta^{(3)}$. Then we have the equation with normalization condition
$\displaystyle\frac{\partial\phi^{(3)}}{\partial t}=D\Delta^{(3)}\phi^{(3)},$
(14) $\displaystyle 1$ $\displaystyle=$
$\displaystyle\int\phi^{(3)}(q^{0},q^{1},q^{2})\sqrt{G}~{}d^{3}q,$ (15)
where $D$ is the diffusion constant, and $G=\det(G_{\mu\nu})=\det(G_{ij})$.
Our aim is to construct the effective two dimensional diffusion equation from
3D equation above.
$\displaystyle\frac{\partial\phi^{(2)}}{\partial
t}=D\Delta^{(eff)}\phi^{(2)},$ (16) $\displaystyle 1$ $\displaystyle=$
$\displaystyle\int\phi^{(2)}(q^{1},q^{2})\sqrt{g}~{}d^{2}q,$ (17)
where $\phi^{(2)}$ is the two dimensional diffusion field, $g=\det(g_{ij})$,
and $\Delta^{(eff)}$ is unknown effective 2D diffusion operator which might
not be equal to simple 2D Laplace Beltrami operator.
From two normalization conditions, we obtain
$\displaystyle 1$ $\displaystyle=$
$\displaystyle\int\phi^{(3)}(q^{0},q^{1},q^{2})\sqrt{G}~{}d^{3}q,$
$\displaystyle=$
$\displaystyle\int[\int_{-\epsilon/2}^{\epsilon/2}dq^{0}(\phi^{(3)}\sqrt{G/g})]~{}\sqrt{g}~{}d^{2}q,$
$\displaystyle=$ $\displaystyle\int\phi^{(2)}(q^{1},q^{2})\sqrt{g}~{}d^{2}q.$
Therefore we obtain the relation,
$\phi^{(2)}(q^{1},q^{2})=\int_{-\epsilon/2}^{\epsilon/2}\tilde{\phi}^{(3)}dq^{0},$
(18)
where
$\tilde{\phi}^{(3)}\equiv\phi^{(3)}\sqrt{G/g}.$ (19)
We multiply $\sqrt{G/g}$ to equation (14) and integrate by $q^{0}$, then we
obtain
$\frac{\partial\phi^{(2)}}{\partial
t}=D\int_{-\epsilon/2}^{\epsilon/2}\tilde{\Delta}^{(3)}\tilde{\phi}^{(3)}dq^{0},$
(20)
where
$\tilde{\Delta}^{(3)}\equiv\sqrt{G/g}~{}~{}\Delta^{(3)}\sqrt{g/G}.$ (21)
Next we analyze new operator $\tilde{\Delta}^{(3)}$. From the form of Laplace
Beltrami operator
$\Delta^{(3)}=G^{-1/2}\frac{\partial}{\partial
q^{\mu}}G^{1/2}G^{\mu\nu}\frac{\partial}{\partial q^{\nu}},$
we have
$\displaystyle\tilde{\Delta}^{(3)}$ $\displaystyle=$ $\displaystyle
g^{-1/2}\frac{\partial}{\partial
q^{\mu}}G^{1/2}G^{\mu\nu}\frac{\partial}{\partial q^{\nu}}(g/G)^{1/2}$ (22)
$\displaystyle=$ $\displaystyle\tilde{\Delta}^{(2)}+\tilde{\Delta}^{(1)},$
where
$\tilde{\Delta}^{(2)}\equiv g^{-1/2}\frac{\partial}{\partial
q^{i}}G^{1/2}G^{ij}\frac{\partial}{\partial q^{j}}(g/G)^{1/2},$ (23)
and
$\tilde{\Delta}^{(1)}\equiv\frac{\partial}{\partial
q^{0}}G^{1/2}\frac{\partial}{\partial q^{0}}G^{-1/2}.$ (24)
Then our diffusion equation has form
$\frac{\partial\phi^{(2)}}{\partial
t}=D\int_{-\epsilon/2}^{\epsilon/2}\tilde{\Delta}^{(2)}\tilde{\phi}^{(3)}dq^{0}.$
(25)
The contribution from $\tilde{\Delta}^{(1)}$ vanishes because
$\displaystyle\int_{-\epsilon/2}^{\epsilon/2}\tilde{\Delta}^{(1)}\tilde{\phi}^{(3)}dq^{0}$
$\displaystyle=$ $\displaystyle
g^{-1/2}\int_{-\epsilon/2}^{\epsilon/2}\frac{\partial}{\partial
q^{0}}(G)^{1/2}\frac{\partial}{\partial q^{0}}\phi^{(3)}~{}dq^{0}$ (26)
$\displaystyle=$ $\displaystyle
g^{-1/2}[~{}(G)^{1/2}\frac{\partial\phi^{(3)}}{\partial
q^{0}}]\mid_{-\epsilon/2}^{\epsilon/2}=0.$
The last equality is the requirement that diffusion flow does not pass through
the surface: $\Sigma^{\prime}$ and $\tilde{\Sigma}$.
Now we calculate r.h.s of (25) up to ${\cal O}(\epsilon^{2})$. Since we have
$\tilde{\phi}^{(3)}={\cal O}(\epsilon^{-1}),$ (27)
from (18), we need to expand $\tilde{\Delta}^{(2)}$ up to ${\cal
O}(\epsilon^{2})$. The following relations are useful
$\displaystyle G_{ij}$ $\displaystyle=$ $\displaystyle
g_{ij}+2q^{0}\kappa_{ij}+(q^{0})^{2}\kappa_{im}\kappa^{m}_{j},$ (28)
$\displaystyle G^{ij}$ $\displaystyle=$ $\displaystyle
g^{ij}-2q^{0}\kappa^{ij}+3(q^{0})^{2}\kappa^{i}_{m}\kappa^{mj}+{\cal
O}(\epsilon^{3}),$ (29) $\displaystyle G_{~{}~{}}$ $\displaystyle=$
$\displaystyle g~{}\\{1+2q^{0}\kappa+(q^{0})^{2}(\kappa^{2}+R)+{\cal
O}(\epsilon^{3})\\},$ (30) $\displaystyle G^{1/2}$ $\displaystyle=$
$\displaystyle g^{1/2}\\{1+q^{0}\kappa+\frac{1}{2}(q^{0})^{2}R+{\cal
O}(\epsilon^{3})\\},$ (31)
where $R=\kappa^{2}-\kappa_{ij}\kappa^{ij}$ is used.
Then the operator $\tilde{\Delta}^{(2)}$ can be expanded as follows
$\tilde{\Delta}^{(2)}=\Delta^{(2)}+q^{0}\hat{A}+(q^{0})^{2}\hat{B}+{\cal
O}(\epsilon^{3}),$ (32)
where,
$\hat{A}=-g^{-1/2}\frac{\partial}{\partial
q^{i}}g^{1/2}(2\kappa^{ij}\frac{\partial}{\partial
q^{j}}+g^{ij}\frac{\partial\kappa}{\partial q^{j}}),$ (33)
$\displaystyle\hat{B}$ $\displaystyle=$ $\displaystyle
g^{-1/2}\frac{\partial}{\partial
q^{i}}g^{1/2}(3\kappa^{im}\kappa_{m}^{j}\frac{\partial}{\partial q^{j}}$ (34)
$\displaystyle+$
$\displaystyle\frac{1}{2}g^{ij}\frac{\partial(\kappa^{2}-R)}{\partial
q^{j}}+2\kappa^{ij}\frac{\partial\kappa}{\partial q^{j}}).$
Then our two dimensional effective diffusion equation up to ${\cal
O}(\epsilon)$ is,
$\displaystyle\frac{\partial\phi^{(2)}}{\partial t}$ $\displaystyle=$
$\displaystyle D\Delta^{(2)}\phi^{(2)}$ (35) $\displaystyle+$ $\displaystyle
D\hat{A}\int_{-\epsilon/2}^{\epsilon/2}q^{0}\tilde{\phi}^{(3)}dq^{0}$
$\displaystyle+$ $\displaystyle
D\hat{B}\int_{-\epsilon/2}^{\epsilon/2}(q^{0})^{2}\tilde{\phi}^{(3)}dq^{0}+{\cal
O}(\epsilon^{3}).$
To proceed the $q^{0}$ integration, we suppose there is no diffusion flow in
normal direction in layer , that is,
$0=\frac{\partial\phi^{(3)}}{\partial q^{0}}=g^{1/2}\frac{\partial
G^{-1/2}\tilde{\phi}^{(3)}}{\partial q^{0}}.$ (36)
Solution is,
$\displaystyle\tilde{\phi}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{1}{N}(G/g)^{1/2}\phi^{(2)}(q^{1},q^{2}),$ (37)
$\displaystyle N$ $\displaystyle\equiv$
$\displaystyle\int_{-\epsilon/2}^{\epsilon/2}(G/g)^{1/2}dq^{0}.$ (38)
Each integration can be explicitly performed, and we obtain
$\displaystyle N~{}~{}~{}~{}~{}$ $\displaystyle=$
$\displaystyle\epsilon+\frac{R}{24}\epsilon^{3}+{\cal O}(\epsilon^{5}),$ (39)
$\displaystyle<q^{0}>~{}$ $\displaystyle=$
$\displaystyle\frac{\kappa\epsilon^{2}}{12}+{\cal O}(\epsilon^{4}),$ (40)
$\displaystyle<(q^{0})^{2}>$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}}{12}+{\cal O}(\epsilon^{4}),$ (41)
where we have used the definition
$<f(q^{0})>\equiv\frac{1}{N}\int_{-\epsilon/2}^{\epsilon/2}f(q^{0})(G/g)^{1/2}dq^{0}.$
(42)
We obtain the final form of equation up to ${\cal O}(\epsilon^{2})$ as
$\displaystyle\frac{\partial\phi^{(2)}}{\partial t}$ $\displaystyle=$
$\displaystyle
D\Delta^{(2)}\phi^{(2)}+\tilde{D}(\hat{A}\kappa+\hat{B})\phi^{(2)}$ (43)
$\displaystyle=$ $\displaystyle
D\Delta^{(2)}\phi^{(2)}+\tilde{D}g^{-1/2}\frac{\partial}{\partial
q^{i}}~{}g^{1/2}$ $\displaystyle\times$
$\displaystyle\\{(3\kappa^{im}\kappa_{m}^{j}-2\kappa\kappa^{ij})\frac{\partial}{\partial
q^{j}}-\frac{1}{2}g^{ij}\frac{\partial R}{\partial
q^{j}}\\}\phi^{(2)},~{}~{}~{}$
where $\tilde{D}=\frac{\epsilon^{2}}{12}D$.
We give two comments here. First, ${\cal O}(\epsilon)$ term disappears. Since
$\epsilon$ has the dimension of length, it always appears with curvature
$\kappa$. Therefore the 1st order term, if it exists, it contains 1st order of
curvature $\kappa$. But this curvature depends on unphysical choice of normal
unit vector $\vec{n}$, and so it does not appear.
Second, additional potential term disappears. In quantum mechanics, the
similar embedding techniques leads to the appearance of additional potential
term written by curvature. But in our classical case we have no such terms.
Because in diffusion equation, potential term breaks probability conservation
law, i.e.
$\partial\phi/\partial t=(D\Delta+V(x))\phi,$ $\to~{}\frac{d}{dt}\int
d^{3}x~{}\phi=\int d^{3}xV(x)\phi\neq 0.$
The normal diffusion flow can be written in general coordinate,
$J_{N}^{i}=-Dg^{ij}\frac{\partial\phi^{(2)}}{\partial q^{j}}.$ (44)
The anomalous diffusion flow is,
$J_{A}^{i}=-\tilde{D}\\{(3\kappa^{im}\kappa_{m}^{j}-2\kappa\kappa^{ij})\frac{\partial\phi^{(2)}}{\partial
q^{j}}-\frac{1}{2}g^{ij}\frac{\partial R}{\partial q^{j}}\phi^{(2)}\\}.$ (45)
The Diffusion equation is written as
$\displaystyle-\frac{\partial\phi^{(2)}}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla_{i}(J_{N}^{i}+J_{A}^{i}),$ (46) $\displaystyle=$
$\displaystyle g^{-1/2}\frac{\partial}{\partial
q^{j}}~{}g^{1/2}(J_{N}^{i}+J_{A}^{i}),$
where $\nabla_{i}$ is the covariant derivative. By using a suitable boundary
condition, we can prove
$\frac{d}{dt}\int\phi^{(2)}g^{1/2}d^{2}q=0.$
## IV Properties of curvature dependent flow
The anomalous flow equals to zero for the flat surface. The last term in
equation (45) shows that curvature gradient generate the flow without particle
density gradient. From the signature of this term, this flow goes from the
smaller Ricci scalar point to the larger Ricci scalar point. Ricci scalar can
take the negative, zero, and positive values. (Ricci scalar $R$ is related to
Gauss curvature by $R/2=\det[\kappa^{i}_{j}]$.) Let us work with the
coordinate which satisfies
$g_{ij}=\delta_{ij},~{}~{}\kappa^{i}_{j}=~{}\mbox{diag}[1/r_{1},~{}1/r_{2}],$
at point $P$, where $r_{i}$ is the curvature radius along the $q^{i}$
coordinate and it takes positive or negative value for convex or concave. (The
metric can be diagonalized by choosing the two coordinates as to satisfy
orthogonality, and it can be normalized by using the re-parametrization. The
second fundamental tensor is diagonalized by rotation of coordinate.)
Then at point $P$, we have $R=\frac{2}{r_{1}r_{2}}$ and,
* •
$R>0$ if the surface is convex or concave.
* •
$R=0$ if the surface is essentially flat.
* •
$R<0$ if the surface is hyperbolic.
Therefore the flow goes from hyperbolic or flat points to convex or concave
points with positive larger Ricci scalar value.
Next we consider the first term in (45). We have positive or negative value
for
$f^{ij}\equiv 3\kappa^{im}\kappa_{m}^{j}-2\kappa\kappa^{ij},$
depending on the value of curvature. In our coordinate, we can immediately
write it in the simple form
$f^{ij}=\delta^{ij}(\frac{1}{(r_{i})^{2}}-\frac{2}{r_{1}r_{2}}).$ (47)
When the surface is hyperbolic ($R<0$),
$f^{11}=\frac{1}{(r_{1})^{2}}+\frac{2}{\mid
r_{1}r_{2}\mid}>0,~{}~{}f^{22}=\frac{1}{(r_{2})^{2}}+\frac{2}{\mid
r_{1}r_{2}\mid}>0,$
usual diffusion occurs (See fig. 2).
Figure 2: Wave packet on hyperbolic surface diffuses in two directions.
When the surface is convex or concave ($R>0$),
$f^{11}=\frac{1}{(r_{1})^{2}}-\frac{2}{\mid r_{1}r_{2}\mid}\\\ =\frac{\mid
r_{2}\mid-2\mid r_{1}\mid}{\mid r_{1}\mid^{2}\mid r_{2}\mid},$
$f^{22}=\frac{1}{(r_{2})^{2}}-\frac{2}{\mid r_{1}r_{2}\mid}\\\ =\frac{\mid
r_{1}\mid-2\mid r_{2}\mid}{\mid r_{2}\mid^{2}\mid r_{1}\mid}.$
In this case, we have three possibilities. One possibility is that both are
negative, if
$1/2<\mid\frac{r_{2}}{r_{1}}\mid<2.$
Then we have no diffusion but concentration occurs (See fig. 3).
Figure 3: Wave packet on convex surface concentrates.
The second possibility is that one of two is positive and the other is
negative, if
$\mid\frac{r_{2}}{r_{1}}\mid<1/2,~{}~{}\mbox{or}~{}\mid\frac{r_{2}}{r_{1}}\mid>2.$
Then we have diffusion in one direction, but concentration in another
direction (See fig. 4).
The third possibility for $R>0$ is,
$\mid\frac{r_{2}}{r_{1}}\mid=1/2,~{}~{}\mbox{or}~{}\mid\frac{r_{2}}{r_{1}}\mid=2.$
This is critical point, where the flow stops for larger curvature direction
and flow concentrates for smaller curvature direction.
Figure 4: Wave packet on convex surface with one direction curvature is over
two times higher than another. Packet diffuses in higher curvature direction
and concentrates in smaller curvature direction.
When the Ricci scalar is zero ($R=0$), for example $r_{2}=\infty$, $f^{22}=0$
and $f^{11}>0$ follows. The diffusion occurs only in $q^{1}$ direction but not
in another direction. (See fig. 5).
Figure 5: Wave packet on elliptic cylinder. Packet diffuses only in curved
direction but not in another direction.
In this way, this anomalous diffusion flow has much varieties depending on the
curvature.
## V One Example: Elliptic Cylinder
Let us consider one simple example where diffusion coefficient depends on
curvature. The surface of elliptic cylinder is the case of $R=0$ just as
figure 5, but the surface has non zero mean curvature.
Ellipsoid is given by the equation
$(\frac{x}{a})^{2}+(\frac{y}{b})^{2}=1.$ (48)
Figure 6: Elliptic cylinder
Then any points on cylinder are specified by curved coordinate $\theta$ and
$z$;
$x=a\cos\theta,~{}~{}y=b\sin\theta.$ (49)
Another choice of coordinate instead of $\theta$ is,
$du=\sqrt{dx^{2}+dy^{2}}=f(\theta)~{}d\theta.$ (50)
where $f(\theta)$ is defined as
$f(\theta)\equiv\sqrt{a^{2}\sin^{2}\theta+b^{2}\cos^{2}\theta}.$ (51)
The length of $u$ is given by
$\displaystyle u(\phi)$ $\displaystyle=$
$\displaystyle\int_{0}^{\phi}f(\theta)d\theta$ (52) $\displaystyle=$
$\displaystyle b\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}\theta}d\theta\equiv
bE(k,\phi),$
with $k=\sqrt{1-(a/b)^{2}},~{}a\leq b.$ $E(k,\phi)$ is the Elliptic integral
of the second kind. (See appendix)
The total length of $u$ is given by
$U\equiv 4bE(k,\pi/2).$
We use the normalized value for $u$ hereafter such that,
$\tilde{u}=u/U,~{}~{}0\leq\tilde{u}\leq 1.$ (53)
The normal unit vector $\vec{n}$ is given as
$\displaystyle\vec{n}$ $\displaystyle=$
$\displaystyle(\frac{x}{a^{2}\sqrt{(x^{2}/a^{4})+(y^{2}/b^{4})}},\frac{y}{b^{2}\sqrt{(x^{2}/a^{4})+(y^{2}/b^{4})}})$
(54) $\displaystyle=$
$\displaystyle\frac{1}{f(\theta)}(b\cos\theta,a\sin\theta).$
$\frac{\partial\vec{n}}{\partial\theta}=\frac{1}{f}(-b\sin\theta,a\cos\theta)-\frac{\partial_{\theta}f}{f}\vec{n}.$
(55)
$\vec{B}_{\theta}=\frac{\partial\vec{x}}{\partial\theta}=(-a\sin\theta,b\cos\theta).$
(56)
Then we obtain the second fundamental tensor.
$\kappa_{\theta\theta}=\vec{B}_{\theta}\cdot\frac{\partial\vec{n}}{\partial\theta}=\frac{ab}{f}.$
(57)
Then we collect all the necessary quantities as follows
$\displaystyle g_{\theta\theta}=f^{2},~{}~{}g_{zz}=1,~{}~{}g_{\theta z}=0,$
$\displaystyle\kappa_{\theta\theta}=\frac{ab}{f},~{}~{}\kappa_{zz}=\kappa_{\theta
z}=0,~{}~{}\kappa=\frac{ab}{f^{3}}.$ (58)
Then we obtain the total diffusion equation expressed by the parameters
$\theta$ and $z$.
$\frac{\partial\phi^{(2)}}{\partial
t}=(\frac{1}{f}\frac{\partial}{\partial\theta})D_{\theta}(\frac{1}{f}\frac{\partial}{\partial\theta})\phi^{(2)}+D\frac{\partial^{2}}{\partial
z^{2}}\phi^{(2)},$ (59)
where the effective diffusion coefficient depends on mean curvature.
$D_{\theta}=D(1+\frac{\epsilon^{2}\kappa^{2}}{12})=D(1+\varepsilon^{2}(b\kappa)^{2}),$
where $\varepsilon\equiv\epsilon/(2\sqrt{3}b)$.
Under this equation, we obtain the following particle number conservation law.
$\frac{d}{dt}\int
dz\int_{0}^{2\pi}d\theta~{}f(\theta)~{}\phi^{(2)}(\theta,z)~{}=0$
with suitable Neumann boundary condition.
By using the variable $\tilde{u}$ instead of $\theta$ we have simple
dimensionless equation,
$\frac{\partial\phi^{(2)}}{\partial\tau}=\frac{\partial}{\partial\tilde{u}}(1+V)\frac{\partial}{\partial\tilde{u}}\phi^{(2)}+\frac{\partial^{2}}{\partial\eta^{2}}\phi^{(2)}$
(60)
where
$\tau=tD/U^{2},~{}\eta=z/U,~{}V=\varepsilon^{2}(b\kappa)^{2},~{}U=4bE(k,\pi/2)$.
By using the approximation of elliptic function given in appendix, curvature
dependent potential $V$ can be written as function of $\tilde{u}$. The
simulation can be done as usual diffusion equation. For $0\leq\tilde{u}\leq 1$
and $0\leq\eta\leq 4$ using periodic boundary condition, since the length of
$\eta$ is larger than one of $\tilde{u}$, the diffusion in $u$ direction
occurs fastly and then the diffusion in $\eta$ direction follows slowly just
like $\phi\sim a+\sum_{k}b(k)e^{-k^{2}\tau}\cos(k\eta)$. The $u$-directional
diffusion can not occur uniformly, because at $\tilde{u}=0.25,$ and $0.75$ the
diffusive coefficient is higher than other points. Therefore the slope of
diffusion field is small especially at these two points during the diffusion
process. (fig. 7)
Figure 7: Snap shot of diffusion process starting from the wave packet
$\phi=\sin^{10}(\pi\tilde{u})$ as the initial condition. At two points (0.25
and 0.75), diffusion occurs quickly and its slope is smaller than other.
Figure 8: Mean curvature as a function of $\tilde{u}$ when $b/a=2$
## VI Conclusion
We have discussed on the diffusion equation on curved surface embedded in
$R_{3}$. We obtained the new diffusion equation up to ${\cal O}(\epsilon^{2})$
which includes anomalous diffusive flow additional to the usual one. This
anomalous flow depends on the second fundamental tensor, and it has not only
diffusion but also concentration properties depending on the curvature of its
manifold.
At the point with negative Ricci scalar $R<0$, surface is hyperbolic, and
diffusion in both direction occurs. (fig.2)
When Ricci scalar is positive $R>0$, we have three possibilities. $r_{i}$
appearing below is curvature radius in each direction ($i=1,2$).
* •
Concentration in both direction (fig.3), when
$1/2<\mid\frac{r_{2}}{r_{1}}\mid<2.$
* •
Concentration in smaller curvature direction, and diffusion in higher
curvature direction (fig.4), when
$\mid\frac{r_{2}}{r_{1}}\mid<1/2,~{}~{}\mbox{or}~{}\mid\frac{r_{2}}{r_{1}}\mid>2.$
* •
Concentration in smaller curvature direction, and no flow in higher curvature
direction, when
$\mid\frac{r_{2}}{r_{1}}\mid=1/2,~{}~{}\mbox{or}~{}\mid\frac{r_{2}}{r_{1}}\mid=2.$
When Ricci scalar is zero $R=0$, surface is essentially flat, but we have
finite curvature radius in one direction. Then we have diffusion only in this
direction (fig.5).
In the case of surface of elliptic cylinder, we gave a concrete form of
equation and we showed the curvature dependent diffusion coefficient.
$D_{u}=D(1+\frac{\epsilon^{2}\kappa^{2}}{12}),~{}~{}~{}D_{z}=D,$
where $\kappa$ is the mean curvature. In this case curvature dependence is
simply included into diffusion coefficient. However this is not true in
general case, where situation is much more complicated, and this can be seen
from the form of anomalous flow.
The application to the pattern formation by reaction diffusion using this
obtained equation is not yet finished. This will be done in further
publication.
## VII Appendix
We approximate the elliptic integral of the second kind.
$E(k,\phi)\equiv\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}\theta}~{}d\theta$ (61)
with
$k=\sqrt{1-(a/b)^{2}}.$
Under the expansion in powers of $k^{2}$, we obtain the power series of
Elliptic integral of the second kind.
$E(k,\phi)=\phi-\sum_{n=1}^{\infty}\frac{k^{2n}(2n-3)!!}{n!~{}2^{n}}\int_{0}^{\phi}\sin^{2n}\theta
d\theta.$ (62)
Since the integration part can be expanded by $\phi$ and $\sin 2n\phi$, we
have
$E(k,\phi)=a_{0}\phi+\sum_{n=1}^{\infty}a_{n}\sin 2n\phi.$ (63)
with the relation
$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle\frac{2}{\pi}E(k,\pi/2),$
(64) $\displaystyle a_{n}$ $\displaystyle=$
$\displaystyle(-1)^{n}\frac{2E(k,\pi/2)}{n\pi}$ (65)
$\displaystyle+\frac{4}{\pi}\int_{0}^{\pi/2}\sin
2n\phi~{}E(k,\phi)d\phi.~{}~{}(n\geq 1)$
For the real Char fishes, $b/a$ takes values $1.5\sim 2.5$. Then the value of
$k$ takes $0.75\sim 0.92$. When $b/a=2$, each values of $a_{n}$ is given
numerically
$a_{0}=0.771,~{}~{}a_{1}=0.123,~{}~{}a_{2}=-0.00506,~{}~{}a_{3}=0.000558.$
Now we have for $u$ given in (52),
$u/b=E(k,\phi)=a_{0}\phi+a_{1}\sin 2\phi+a_{2}\sin 4\phi+\cdots.$ (66)
And we rewrite it by using normalized $u$,
$\phi=2\pi\tilde{u}-\frac{a_{1}}{a_{0}}\sin 2\phi-\frac{a_{2}}{a_{0}}\sin
4\phi-\cdots,$ (67)
where $\tilde{u}=u/(4bE(k,\pi/2)).$
The iteration method up to order $(a_{1}/a_{0})^{1}$ gives
$\phi=2\pi\tilde{u}-\frac{a_{1}}{a_{0}}\sin(4\pi\tilde{u}).$ (68)
Then we take the derivative by $u$ in both hand sides.
$\frac{1}{f}=\frac{1}{ba_{0}}(1-\frac{2a_{1}}{a_{0}}\cos(4\pi\tilde{u})),$
(69)
where the following relation is used.
$\frac{du}{d\phi}=f(\phi)\equiv\sqrt{a^{2}\sin^{2}\phi+b^{2}\cos^{2}\phi}.$
Then the mean curvature given by (58) is obtained as function of $u$.
$b\kappa=\frac{ab^{2}}{f^{3}}=\frac{1}{\beta(a_{0})^{3}}(1-\frac{2a_{1}}{a_{0}}\cos(4\pi\tilde{u}))^{3},$
(70)
where $\beta=b/a$. This function is shown in figure 8.
## References
* (1) J. Faraudo, J. Chem. Phys, 116 (2002) 5831-5841; J. Balakrishnan, arXiv:physics/0308089, 25 Aug 2003.
* (2) A. Naji and F. Brown, J. Chem. Phys. 126 (2007) 235103; E. Reister and U. Seifert, arXive:cond-mat/0503568, 23 Mar 2005.
* (3) R. C. T. da Costa, Phys. Rev. 23 (1981) 1982; J. Tolar, 1988 Lecture Notes in Physics 313, ed. H. D. Doever, J. D. Henning and T. D. Raev, (Springer-Verlag, Berlin, Heidelberg) 268.
* (4) N. Ogawa, K. Fujii, and K. P. Kobushkin, Prog. Theor. Phys. 83 (1990) 894; N. Ogawa, K. Fujii, N. M. Chepilko, and K. P. Kobushkin, Prog. Theor. Phys. 85 (1991) 1189; N. Ogawa, Prog. Theor. Phys. 87 (1992) 513.
* (5) K. Fujii and N. Ogawa, Prog. Theor. Phys. 89 (1993) 575.
* (6) A. M. Turing, Phil. Trans. R. Soc. London B 237 (1952) 37-72; H. Meinhardt, Models of Biological Pattern Formation., Academic Press, London (1982); J. D. Murray, Mathematical Biology. 2nd ed. Springer, New York (1989).
* (7) H. Shoji, Y. Iwasa and S. Kondo, J. Theor. Biol. 224 (2003) 339-350; H. Shoji and Y. Iwasa, J. Theor. Biol. 237 (2005) 104-116.
|
arxiv-papers
| 2009-11-27T02:17:27 |
2024-09-04T02:49:06.725509
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Naohisa Ogawa",
"submitter": "Naohisa Ogawa",
"url": "https://arxiv.org/abs/0911.5189"
}
|
0911.5262
|
# Quantifying Resource Use in Computations
R.J.J.H. van Son111 ACLC/University of Amsterdam, Spuistraat 210-212, 1012 VT
Amsterdam, The Netherlands, R.J.J.H.vanSon@gmail.com. Licensed under the
Creative Commons Attribution license
###### Abstract
It is currently not possible to quantify the resources needed to perform a
computation. As a consequence, it is not possible to reliably evaluate the
hardware resources needed for the application of algorithms or the running of
programs. This is apparent in both computer science, for instance, in
cryptanalysis, and in neuroscience, for instance, comparative neuro-anatomy. A
System versus Environment game formalism is proposed based on Computability
Logic that allows to define a computational work function that describes the
theoretical and physical resources needed to perform any purely algorithmic
computation. Within this formalism, the cost of a computation is defined as
the sum of information storage over the steps of the computation. The size of
the computational device, eg, the action table of a Universal Turing Machine,
the number of transistors in silicon, or the number and complexity of synapses
in a neural net, is explicitly included in the computational cost. The
proposed cost function leads in a natural way to known computational trade-
offs and can be used to estimate the computational capacity of real silicon
hardware and neural nets. The theory is applied to a historical case of 56 bit
DES key recovery, as an example of application to cryptanalysis. Furthermore,
the relative computational capacities of human brain neurons and the C.
elegans nervous system are estimated as an example of application to neural
nets.
keywords: computation, compatibility logic, neural nets, cryptanalysis
## 1 Introduction
In June 1998, a high ranking USA official, Robert S. Litt, testified before a
Senate judicial subcommittee that …decrypting one single message that had been
encrypted with a 56-bit [Data Encryption Standard] key took 14,000 Pentium-
level computers over four months; obviously these kinds of resources are not
available to the FBI. Later the same year, a 56 bit DES key was recovered in
56 hours at a cost of less than $250,000 using 1536 custom chips [Ele98].
The DES example points to the lack of a computational work function as a
fundamental problem in the theory of algorithms and computation. At the time,
questions were raised about the security of 56 bit DES. In this debate, there
was no way to estimate the resources needed to find a 56 bit key based on the
available technology. So the above predictions could neither be supported nor
defeated in a quantitative way, except by going to the expenses of actually
cracking the keys.
A decade later, there still is no theoretical model for the abstract
computational needs, or costs, of running an algorithm, nor a way to evaluate
the computational capacity of customized hardware. This problem crops up more
generally in game theory, eg, when defining costs in computational Nash
equilibria [Hal08, HP08], and in computational complexity theory when modeling
time and space bounded automata [DKV08, FH02]. In a practical sense, those who
want to perform extensive computations have few tools to evaluate the
computational power that current technology could (theoretically) provide.
At the other end of the spectrum of computational devices, neuro-informatics
studies how neural networks and the brain compute [Gor03]. There is an acute
interest in understanding how nervous systems compute behavioral responses to
environmental challenges [Gor03, OHS09, Leh09, Eisnt]. Brain imaging and
activity recording techniques, eg, fMRI, MER, and ERPs, can show subsets of
neurons computing specific mental functions in real time. The local and long
range connections between neurons can be mapped in detail [OHS09, Leh09,
Eisnt]. The underlying questions are what is computed where, and how? One
obvious intermediate question is what can actually be computed by a certain
subset of neurons in a certain animal in a given time? This is again a
question on resource use in computations, but now based on neurons instead of
silicon gates.
In principle, it should be possible to compare the computational capacities of
the nerve systems of different animals like it is possible to compare their
metabolic rates. A human brain has on the order of $10^{11}$ neurons, whereas
the nematode Caenorhabditis elegans has only 302 neurons in total (adult
hermaphrodite, eg, [CP97]). But how can the computational work these different
neurons perform be compared? This is a question that is currently impossible
to formulate in a quantitatively meaningful manner.
The remainder of this paper is structured as follows. In section 2, a model is
proposed for quantifying the resource use, or cost function, for performing a
computation on theoretical and real devices (see also the Appendix). This
model is applied to examples from cryptanalysis and neural physiology in
section 3. The results are discussed in section 4.
## 2 A computational work function
Any universal computational work function should have a few general features.
It should describe the resource needs of a computation in terms of costs. It
should be abstract enough to be applicable to both theoretical and real
devices. It must be able to add and remove resources during a computation. The
cost must increase strictly monotonically and must be additive in serial and
parallel computations. And finally, it must be possible to emulate any
computational device efficiently, where “efficient” is formalized here as a
linear cost dependency. An efficient emulator allows comparisons between
different devices by comparing the sizes of emulator programs, independent of
the emulated devices.
First, in section 2.1 a game model of computation will be formulated that
identifies resources and deliminates what is part of the computation for which
the costs must be calculated and what is not. Section 2.2 proposes a cost
function which has the desired features. The proposed cost function defines a
least-cost implementation for any computation for which an algorithm is known,
which is explained in section 2.3. The cost function is then used to model the
computational resources of silicon hardware (section 2.4) and neural networks
(section 2.5).
### 2.1 The computability logic game model
Real computations need some material structure to carry and process the
information, time and energy to allow state changes and to remove state
information while increasing the entropy of the environment [LT07, Llo02,
Llo05, Llo00]. So it is important to check whether the physics of computation
does set limits on the resources needed in terms of the time, energy, and
temperatures that are required and entropy that is generated. Of these
factors, the minimum amount of energy $E$ to drive a bit sized state change in
time $\Delta t$ is $E\geq h/\Delta t$ and the minimum dissipation needed to
erase a bit is of the order $\Delta E\geq kT\ln(2)$, with $h$ Planck’s
constant, $k$ the Boltzmann constant, and $T$ the absolute temperature. These
values are important on a molecular scale, or in quantum computers, but not in
current computers [Llo00]. So this study will ignore these physical
constraints.
A theoretic framework that describes the qualitative use of resources in
computations well is computability logic [Jap05, Jap06]. In computability
logic, computability is defined in terms of games. The “computer”, or System,
plays against the Environment and “wins” if it can complete the requested
computation successfully using the available resources. This game model of
computability explicitly defines what the responsibilities of the Environment
are and how it interfaces with the System. It also accounts for the resources
that are used by the System to perform the computation and how the system
communicates the results. Therefore, it is very well suited to delimit and
define the costs of computations.
Ignoring purely physical constraint, e.g., absolute time, temperature, and
energy, in the cost function allows the use of a purely algorithmic game model
from computability logic [Jap05, Jap06]. On this game model, a computational
work function can be defined analogous to the cryptanalysis work function of
Shannon [Sha49].
The current study will restrict itself to such a purely algorithmic and
deterministic games where the speed of the moves is not relevant and the
environment has unlimited capacities to execute moves [Jap05, Jap06]. In the
framework of computability logic, the System doing the computation is further
simplified by describing it as a collection of Universal Turing Machines
[Tur36], UTMs, each with a Finite State Machine, FSM, doing the processing and
three or more tapes: one or more work tapes, a valuation tape, and a run tape.
The work tape(s) correspond(s) to the working memory of a computer and
contains the program and all related data in use. The run tape corresponds to
an input/output medium that stores the moves written to it by the System and
the Environment. The System can not move backwards on the run tape. That is,
the System must use its own memory and cannot use the (free) run tape to store
the in- and output history. The valuation tape contains the game specific
parameters supplied by the Environment and used by the program. A more general
interpretation of the valuation tape is that it contains any public
information outside the control of the System.
The System can recruit as many computational devices, UTMs, as it wants by
specifying them on the run tape. Every daughter device of the System can
itself play against the Environment on its personal run tape and receives a
personal valuation tape. Both the personal run and valuation tapes of each
daughter device will be copies of the original System tapes. The communication
between the UTMs that make up the System is modelled by simply letting their
work tapes overlap, but other solutions are possible. Any UTM request should
consist of a full description of the finite state machine, initial state,
contents of the work tape, position of the heads, and the overlap between work
tapes.
The computational model is completely interactive, so there are no general
rules limiting what can be written to the run tape. To make the resource use
explicit, it will be assumed that all moves are written as either fixed size
or self delimited strings. Scanning the run tape for moves of the Environment
is a computational cost that must be born by the System. To minimize that
cost, the moments at which the Environment can write to a run tape are
restricted. The Environment will only write to a run tape in response to a
move of the device that “plays” on that run tape. Any daughter device of the
System will go to sleep after it has written a move, and it wakes up only
after the Environment has responded. The computational costs are defined on
the work tape(s) and the processing units (UTMs), but not on the valuation and
run tapes.
### 2.2 A simple cost function
A very simple work, or cost, function for a single UTM that has all the above
features is
$C=\sum_{\lambda=1}^{\Lambda}I_{UTM}(\lambda)$ (1)
Where $C$ is the cost of a computation, $\Lambda$ is the number of steps
needed to complete the computation, and $I_{UTM}(\lambda)$ is the information
in bits, stored in that UTM at step $\lambda$ ($\lambda\leq\Lambda$). In a
situation with parallel UTMs, the cost is calculated for each UTM separately
using the step cycles of that UTM. Shared memory is attributed to the UTM that
makes the most steps.
The cost function in equation 1 replaces memory or time limited computations
with a limitation in $time\cdot memory$ (c.f., [DKV08, FH02]). $I_{UTM}$ is an
information measure that is linear in its components and always $I_{UTM}>0$
for any computation in progress. Therefore, $C$ in equation 1 is strictly
monotonically increasing over “time” for any computation. The cost of a
computation under equation 1 is linear in time and computational resources. So
the cost of doing computations in parallel on different computational devices
or in series on a single device is simply the sum of the costs of doing the
individual computations in isolation (provided the Environment takes care of
initialization of the System between computations). So equation 1 has indeed
the compositional features requested above.
The information $I_{UTM}(\lambda)$ is the information needed to specify a UTM
in the current state. That is, the information needed to specify at step
$\lambda$ the
* •
action table
* •
the current state
* •
the position of the heads
* •
the current contents of the working tape
The working tape of a UTM is potentially of infinite size. But at any moment
of time, only a finite part of it is in actual use. For the cost calculations,
it is assumed that only part of the working tape is actually “in use” and the
contribution of each work tape cell is proportional to $\sim\log_{2}(N)$
(where $N$ is the total number of possible symbols). Memory locations are
considered “in use”, and part of the cost equation if they have been written
to during initialization or during operation of the UTM.
This can be compared to the System “leasing” new stretches of tape as needed.
It is here assumed new memory automatically enters equation 1 when an empty
cell is written to. Some means for ending the “lease”, i.e., “freeing up” tape
is allowed. This could simply be a special request on the run tape with an
indicator of the working tape cells to be freed (eg, $X$ cells from the
current head position). After such a request, the specified part of the work
tape is not part of the cost equation anymore. The valuation and run tape are
not factored in, as these are considered part of the Environment.
In a game context, the output moves of a UTM are only valid in a certain
context where, in some sense, the output symbols get a meaning. To be able to
compare the costs of a computation using different UTMs, they must all adhere
to the same language on the output. A rigorous definition of the cost of
running a program can most easily be given on a single computational device.
An efficient emulater bridges the gap between different computational devices
For every finite set of UTMs, it is straightforward to define a UTM that can
efficiently emulate them all (see Appendix A).
If the cost of doing a computation on the original UTM in $\Lambda$ steps was
$C$, then the cost of doing that computation on the emulator, $C^{\prime}$
will be:
$C^{\prime}\leq 4\cdot\left(C+\Lambda(\alpha+\epsilon)\right)+\beta$ (2)
The constants $\alpha$ and $\beta$ are specific for the emulator whereas
$\epsilon$ is the “rounding error” of representing the original symbols and
states in the symbols of the emulator. All three constants can be determined
from the emulator program and structure. Examples of efficient emulators for
UTMs and neural nets are given in Appendix A.
The cost function in equation 1 incorporates several trade-off relations. Most
notably, a trade-off between processor complexity and length of computation in
steps. A more complex computing device that processes more bits per step can
reduce the cost of a computation if the memory use is large and vice versa. A
specific case consists of a more complex device that can reduce the number of
steps in a computation without increasing the amount of memory used. In such a
case, the most efficient set up would be to select a processor with a size
that is comparable to the average size of the memory used, $I_{device}\sim
I_{\text{eff}}$ (see Appendix B).
The cost function of equations 1 emphasizes a drawback of standard UTMs. No
practical computer will enumerate all memory positions to access a specific
memory site, as a standard UTM does, as this is not cost effective. Therefore,
it will be assumed here that the UTM can extract a relative address from the
action table that will let the head skip a number of cells on tape in a single
clock cycle (i.e., processing step). This ability is related to the indirect
addressing of register machines (e.g., Random Access Stored Program, RASP, or
RAM machines).
Instead of adding a head skip with every entry in the action table, one or
more accumulator/index registers could be added with some special states to
manipulate them. However, the UTM with skip uses relative addressing, i.e.,
move head $i$ cells forward or backward, with a limited maximal skip.
Furthermore, the relative position of the head over the tape is not explicitly
stored (as a symbol) and is not accessible to the System. It might depend on
the computation and UTM formulation whether the cost of the added complexity
of the registers would be offset by the benefits.
The maximal number of cells that can be skipped in a single step affects the
size of the action table, and the number of states and symbols, so this
ability does not come for free. Going back from a UTM with $N$ symbols which
allows for $D$ skipped cells to an equivalent UTM with only single cell moves,
requires adding “move” states which remember the original state and read
symbol, and move one step. The addition of these move states increases the
total number of states needed by a factor $O(N\cdot D)$ and computation time
by a factor $O(D)$. So the cost of a computation without skipped cells grows
by a factor $O(N\cdot D^{2})$ compared to a UTM with upto $D$ skipped cells
(ignoring logarithmic terms).
To make the cost of performing a computation on a UTM complete, the cost of
operating the read/write head of the UTM should be taken into account. The
structure of the head follows directly from the action table. So the head of a
UTM does not have to be specified separately. However, the head is the actual
processing element and as such constructing and operating one adds costs to a
computation. A model of the computational cost of a UTM head is presented in
Appendix C. The head is specified by the action table, and, for larger
systems, the cost of operating the head is generally smaller than the costs
associated with the action table. Therefore, the contribution of the head to
the cost of computations is ignored in the current study.
A quantitative example of complete cost calculations is presented in Appendix
D for a minimal Tit-for-Tat game.
### 2.3 Least-cost implementation
A least-cost implementation can be defined in the same way as the algorithmic
or Kolmogorov complexity [Cha69, LV97]. If a program $P$ is known that can
perform a computation on a UTM in finite time, then the least-cost program can
be found in a finite time too. The procedure is very simple and based on the
fact that a program with a size larger than $C$ cannot run for a single step
using less than $C$ resources. Run the original program $P$ and determine it’s
cost $C$. Now run all programs $p_{i}$ with sizes smaller than $C$ (a finite
number of programs) and stop them if they have consumed $C$ in resources. All
of these programs will stop executing either because they halt on their own,
or because they overrun the cost limit. The program which needs the least
resources to complete the computation is by definition the least-cost program.
To be able to compare different computational devices, eg, UTMs, all devices
are required to generate their output in the same alphabet. This fits in the
game formalism which requires the game participants to communicate in a shared
language of “moves”. In the current context, a least-cost combination of
$\\{P,UTM\\}$ can be defined within the set of UTMs that can be emulated by a
specific emulator. The cost $C$ to be minimized is that of equation 2. In this
setting, a program on tape and a “program” inside the processing unit become
interchangeable.
The same procedure used between programs on a single UTM can now be repeated
over all UTMs. Any UTM with a FSM size larger than $C$ cannot run even a
single step within fixed cost bounds of $C$. Determine the set of all UTMs
with a size of their FSM $S\leq C$. This is a finite set and can be emulated
efficiently on a single device (see Appendix A). Run each of them with all
programs $p_{i}$ with a size $I_{p_{i}}+S<C$ until they halt or have consumed
$C$ in resources. Again, all these programs will stop. Select the pair
$\\{P,UTM\\}$ which consumed the least resources as the least-cost option.
### 2.4 Relations with real hardware
The cost function of equation 1 is set in terms of stored information times
number of steps the information is used. For non-storage hardware, this
translates to the information put into the device, in terms of components and
connections, and the operating frequency, ie, time per step. That is, the
hardware of the computational device is treated as a “program”. The
connections between the active elements, eg, transistors, are “programmable”
to the degree they can be freely chosen during design.
Although it might be difficult to model a modern complex CPU in terms of
component UTMs, it is possible to estimate the computational resources they
generate by looking at the transistor counts. As the cost function only looks
at memory “use”, the CPU complexity can be reduced to the information needed
to describe the CPU state. That is, the variable state of the transistors and
the fixed structure of the connections, ie,
$I_{CPU}\approx\log_{2}(\\#states)+\log_{2}(\\#possible\ connections)$. It
will be assumed, rather arbitrarily, that transistors are mainly connected
locally (small world topology) and each transistor could on average have been
connected in a hundred different ways ($\sim 7$ bits). Also, a transistor has
a 1 bit state, on or off, and the size of the “state machine” is ignored as it
can be covered by the state+connections. Under these assumptions these numbers
are $\log_{2}(\\#states)=O(\\#transistors)$ and $\log_{2}(\\#possible\
connections)=O(\\#transistors)\cdot 7$. Taken together, each transistor is
guessed to contribute around $\sim 8$ bits to $I_{CPU}$. This is, of course,
just a very crude, ball-park estimate. It is straightforward to estimate the
size of real computer systems from these principles.
As an example, the computational capacity of an off-the-shelf 2007 desktop
system is estimated. An AMD 64 X2 CPU core is made up of around 50 million
transistors, corresponding to $\sim 50\cdot 10^{6}$ byte of memory running at
3 GHz (2007, source Wikipedia). So the resources produced by two such cores on
a CPU could be estimated at $\sim 3\cdot 10^{17}$ byte/second. 2 GB high speed
dynamic RAM running at 400 MHz produces around $8\cdot 10^{17}$ byte/s. It is
rather difficult to quantify magnetic disks, as it is not immediately clear
what clock-speed would be most appropriate. A terabyte disk system would need
a $10^{5}$ Hz clock speed to get in the same order of magnitude as the other
subsystems, so it will be ignored for the moment. The on-chip caches are small
in comparison ($\sim 10^{15}$ byte/s) and will be ignored here too. All
together, a modern system with dual-core CPU and 2 GB RAM will run at around
$\sim 10^{18}$ byte/s, ie, at around 1 exabyte/s.
These data for general purpose CPUs can be compared to other types of devices.
Recently, GPUs (Graphical Processing Units), originally designed to render
graphics in personal computers and game consoles, are becoming popular in high
performance computing [Str09, Val09]. A GPU can have half a billion
transistors and runs at a half GHz with many parallel on-chip modules (data
from 2007). For instance, the NVIDIA GeForce 8800 GT chip set contains 750 M
transistors and runs at 0.6 GHz (source, Wikipedia, fall 2007). The crude
metrics used here puts such a GPU at delivering $4.5\cdot 10^{17}$ byte/s
without memory. This is close to half what a AMD 64 could deliver, but
optimized for its task.
According to these measures, the original IBM PC with an Intel 8088 CPU (5
MHz, 29,000 transistors) and 0.1 MB memory would come in at about $\sim
10^{12}$ byte/s. Given the growth of computing power, decibels would seem to
be a more convenient measure of resource size for a single computer in
byte/second, eg, $10\cdot\log_{10}(I_{device}/10^{12})$, using the scale of
the original IBM PC as a reference. A dual-core AMD 64 system with 2GB RAM
would then count as $\sim 60$ dB. Of course, equation 1 cannot be expected to
reflect cost differences in real monetary terms. $60$ dB over 23 years
(1984-2007) corresponds to an increase of roughly $2.6$ dB/year.
### 2.5 Relations with neurons
The same models as described above can in principle also be used to estimate
the capacity of neurons in the brain. However, in neurons it is not yet clear
what anatomical scale, and therefore, temporal scale, would be relevant to
computation: the cell, the synapse, or even the neurotransmitter receptor. In
addition, the current knowledge of neural computational functions and their
relation to the neuro-physiology is fragmentary at best. Therefore, the
estimates described below are only intended as illustrations of how the
computational capacity of real neural nets might be modelled.
Assume the synapse is the relevant active element [yAFB03, RGnt] (“… a neuron
is defined by synaptic connections” [RGnt]). Synapses are the contact points
between neurons and it is generally believed that they mediate most of the
computational and learning activity of the nervous system. The neuroanatomy of
the human brain is far from settled [OHS09, Eisnt, Leh09] and it is difficult
to put numbers on the populations of neurons and synapses with any precision.
For this example, only general estimates will be used as can be found in
textbooks. And the estimates will be limited to connections using chemical
synapses. Each neuron receives input from up to $10^{4}$ synapses (eg,
[MH07]). There are approximately $10^{11}$ neurons in the human brain (e.g.,
[Leh09]). So there are around $10^{15}$ synapses in a human brain. In general,
a synapse will originate from a local, nearby, neuron. Take this local set to
contain around $10^{6}$ neurons, which corresponds to connection distances of
around 2 millimeters. The relative position of a synapse on the neural body is
important for its function. For simplicity, the spatial structure of the
neuron is reduced to the relative position of the synapses. Both the pre-
synaptic and the post-synaptic part of the synapse can be in several (many)
states describing it’s sensitivity to incoming action potentials and it’s
ability to (de)polarize the post-synaptic membrane. As a last factor, the
runtime delay of incoming action potentials will differ between different axon
end points of the originating neuron. These differences have to be modeled
too.
The above description treats the synapse as a static, passive, device and the
estimates are in line with [DA06]. But biological neurons are dynamic, active,
devices. This aspect of synaptic function is important to computations
[PTK01]. This means that the computational capacity should include the
complexity of the synaptic “device”. At the moment, it is completely unclear
how the size in bytes of the complexity of the synapse should be estimated
from physiological data.
## 3 Applications
### 3.1 Understanding the DES cracker example
The above theory might in future help support an informed discussion about the
potential capabilities of modern computer hardware. That way, it might become
less necessary to implement costly demonstrations just to show that a certain
prediction is wrong, like the one presented by the FBI analysts. Looking at
the DES cracker example from the Introduction, it is possible to estimate the
computational resources available to the FBI and others at the time [Ele98].
The protagonists in the example used two well known approaches to estimate the
costs of performing a computation. The public FBI approach was to take off-
the-shelf systems, and estimate the run time and number of systems needed to
perform the computation. The EFF approach was to design special purpose
hardware and determine empirically what the requirements are in terms of
number of systems and run time. The current study tries to base estimates on a
combination of these approaches. This is done by trying to estimate what
performance could be achieved if the most complex or powerful hardware
available could be redesigned and optimized for the desired computation. That
is, first estimate what, according to equation 1, the maximum computational
costs are that can be handled by existing hardware in a given time on any
computation (the maximal performance). Next estimate what the minimum cost is
to perform the desired computation on optimized hardware. Then compare these
two under the assumption that the existing hardware could be redesigned to be
as good as the optimized hardware.
A 1998 Pentium II processor would have contained around $7.5\cdot 10^{6}$
transistors and ran at 400 MHz. This would account for approximately $3.0\cdot
10^{15}$ byte/s. A high end system in 1998 would have up to 256 Mbyte of 100
MHz main memory, which equates to $2.6\cdot 10^{16}$ byte/s. This brings the
whole system up to around $3\cdot 10^{16}$ byte/s. 14,000 Pentium computers
running for 4 months deliver $4.4\cdot 10^{27}$ byte (steps).
The Electronic Frontier Foundation, EFF, succeeded in designing a search unit
in silicon that could check a 56 bit DES key in 16 clock cycles [Ele98]. The
EFF were able to fit 24 such search units onto a single chip with around
10,000 transistors and use the units in parallel to check all possible keys.
Many such chips can be used in parallel. Using the earlier ball-park estimate
of a contribution to $I_{CPU}$ of 8 bit per transistor, the computational
effort for a single encryption can therefore be estimated as $16\cdot
10^{4}/24\approx 6.7\cdot 10^{3}$ byte (ignoring memory).
As one of the design goals of DES was easy implementation, this low figure
should not be a surprise. If a general office computer of 1998 would have been
a very efficient DES encryptor for its complexity, it would have been able to
test $4.5\cdot 10^{11}$ keys a second (again, ignoring memory). A single such
computer should find a key in less than 30 hours.
To evaluate the DES cracker, the housekeeping, communication and other
functions are deliberately ignored. Attention is focused on the key search.
The DES cracker chip could run with a clock speed of 40 Mhz. In total, 1536
chips were used each with around 0.5 Mbyte of memory. Together, this is
$10^{4}\cdot 4\cdot 10^{7}\cdot 1536$ or around $6.1\cdot 10^{14}$ byte/s for
the chips and $1536\cdot 0.5$ Mbyte memory on 40 MHz or only $3.1\cdot
10^{10}$ byte/s for the memory. Together, these specialized chips produce less
as a computational resource than a single Pentium computer of the time, or
less than the workstation used to coordinate the search. Running for 56 hours,
the DES cracker chips delivered $1.2\cdot 10^{20}$ byte (steps). From this it
can be concluded that the DES cracker set-up was seven orders of magnitude
more efficient in DES encryption than a conventional computer of the day.
Which is not really remarkable given the simplicity of the DES encryption
algorithm.
Obviously, general office computers are all but efficient DES encryptors.
Basically, the EFF used the fact that silicon is equivalent to a program: it
is relatively easy to “program” a new chip to do exactly what is needed. If
the FBI analysts [Ele98], or their critics, had been able to factor in the
simplicity of the DES algorithm and the complexity of hardware of the time,
they would have been better able to predict the vulnerability of the DES
encryption.
### 3.2 Comparing human and C. elegans neurons
To describe each human brain synapse, an estimated 20 bits are needed to
address the originating neuron out of a potential local population of 1
million. Some 10-13 bits might be needed to indicate the synapse’s relative
position on the neural body. These 10-13 bits incorporate some of the spatial
organization of the neural body. 8 bits each are allocated for the pre- and
post-synaptic states, which might be a conservative estimate, given the
complexity of synapses [RGnt]. The timing differences between synapses
originating in the same neuron could be described in, eg, 4 bits. So a
conservative estimate of the information needed to uniquely describe each
synapse would be around 50 bit, or in the order of 6 byte. In total, on the
order of $6\cdot 10^{15}$ byte (six petabyte) are needed to describe the state
of all the synapses in a human brain. This is in accordance with the $\sim
10^{15}$ bit of [DA06] (Note that the estimate in [WLW03] is unphysical as it
exceeds the Beckenstein bound for a brains sized object [Llo00])
The number of neurons is four orders of magnitude less than the number of
synapses, so their contributions to the number of states are ignored. Action
potentials have a maximum rate of approximately 500 Hz. So it would be prudent
to estimate the step timing of synapses in the same range. That would mean
that a conservative estimation of the human brain indicates that it calculates
at a rate of $3\cdot 10^{18}$ byte/s.
The above estimations are based on a static synapse model. In reality,
synapses are dynamic entities that adapt to stimulation [PTK01]. It is
estimated here that two bytes are needed to describe the state of the synapse.
To simplify matters, it is assumed that 10 bits of these are needed to
describe dynamic state parameters. To get at least an order of magnitude
estimate, the action table size of a UTM with the same number of states,
$2^{10}$, is used as a proxy measure. From this it follows that the complexity
of the synapse is of the order of $10^{3}$ bytes (on the order of $\sim$10
bits per state). This increases the estimated capacity of the human brain to
something in the order of $10^{21}$ byte/s.
Compare the human central nervous system to the neural system of C. elegans
[CP97]. An adult hermaphrodite contains 302 neurons and around 7000 synapses.
Each neuron has on average around 25 incoming synapses. That is, the
originating neuron can be described in 8 bit and the position of the incoming
synapse on the neural body in around 4 bits. Timing differences in incoming
synapses can probably be ignored (0 bit). It is unclear how the pre- and
postsynaptic state information relates between nematodes and mammals, but here
it is arbitrarily assumed that nematodes will need less bits, just to put a
number on it, 10 instead of 16 bits. In total, around 22 bit would be needed
to completely describe the state and position in each synapse in a nematode,
or less than 3 bytes. This is assuming only static synapses. Again, the
contribution of the neurons is partly included in the post-synaptic state, and
partly ignored.
As nematodes are not homeiotherm, the switching speed of the synapses will be
lower than in mammals. For a ten degrees difference in body temperature
($37^{\circ}$ versus $25^{\circ}$ C), at least a halving of the metabolic
rate, and switching speed, is expected. Using only the values for static
synapses, the nervous system of a complete hermaphrodite adult C. elegans
would then have a computational capacity of $7000\cdot 3\cdot 0.25\cdot
10^{3}\sim 5\cdot 10^{6}$ byte per second. A single human neuron would have
$10^{4}$ synapses each needing around 6 bytes to describe statically, working
at $0.5\cdot 10^{3}$ Hz for a total of $3\cdot 10^{7}$ byte per second. So,
according to these crude, ballpark, estimations, a single human brain neuron
processes, or computes, around six times as much information than the complete
neural system of a C. elegans adult.
It is informative to look at what makes individual human neutrons perform at a
higher level than the complete neural system of C. elegance. The important
factors are 1) number of synapses, 2) population of possible originating
neurons, 3) spatial interactions between synapses on a neuron, 4) metabolic
speed.
1) The number of synapses ending on a single human neuron and in the complete
C. elegance body are comparable (7,000 versus 10,000). As it is assumed that
the synapses are the computational entities, this fact alone predicts
comparable performance.
2) Each synapse in C. elegance can originate in some 300 other neurons. This
corresponds to some 8 bit to describe the possible information processing
wirings. Each human synapse can originate, potentially, from $10^{11}$ other
neurons. Here it is assumed that connections in the human brain are in general
local (a small world network) and the real, or effective, number of
originating neurons in the human brain is much more limited. But any realistic
number for the human brain will be way larger than the 300 in C. elegance. In
our example this is simply limited to a million originating neurons, i.e., 20
bits. But even with only a 20,000 possible originating neurons this would
still be double the contribution of a C. elegance synapse.
3) With $\sim 10,000$ synapses contacting each human neuron compared to the 25
synapses contacting each C. elegance neuron, the options for spatial
interactions between synapses increases. In our simple model this increases
the computational power of human neurons from approximately 4 to 13 bits.
4) Last, there is an expected metabolic speed doubling, from 25∘C to 37∘C,
which would double computational performance.
## 4 Discussion and conclusions
Almost from the start of the computer era, questions about the time and memory
needed to complete a computation were raised [FH02]. A lot of theoretical
progress has been made towards these questions in the fields of game theory,
logic, and computational complexity. The current study tries to bring these
developments a step closer to the practical developments in other fields, eg,
cryptanalysis, neuro-imaging, and neuro-informatics. A pressing need in these
latter fields is an evaluation of the computational resources of an actual
processor, eg, the electronic hardware or neuronal wet-ware, and to link these
to the theoretical powers of Turing Complete theoretical devices, eg, the UTM.
This inclusion of the processing hardware in the accounting of the resources
is a challenge which requires a way to valuate memory and processing hardware
in a uniform currency that can be integrated with, or in, time.
Based on a few natural requirements, a simple formula for a computational work
function for quantified resource use emerges with the features of Memory times
Steps, ie, a dimension of bytes (equation 1). In more intuitive physical
terms, the computational resources are counted as an integration of
information (entropy) over a normalized interaction time. This count includes
the information frozen into the computational device itself, eg, the UTM
action table, the silicon of the CPU, or the neurons and synapses in a nervous
system.
This definition of the cost of a computation directly leads to the concept of
a least-cost implementation, both for a single computational device and
between devices. Such a least-cost implementation can always be found within a
finite time given a single example program that can perform the computation.
As such, the least-cost is a universal invariant of the computation.
In the end, computing is done using some physical substrate. This substrate,
eg, silicon chips or neural tissue, will need to have some, non-random,
structure to be able to run a program, eg, transistors, synapses, and most of
all connections. The information stored in this structure, as far as it is
relevant to computations, is the $I_{device}$ needed to calculate the
computational costs of equation 1.
Reducing silicon CPU complexity to concrete hardware design features like
transistor count and connectivity, and memory capacity, it is possible to
roughly guess both the capacity of real computer hardware and the hardware
needs of (simple) algorithms. A more refined model, that takes into account
what level of complexity can be achieved by custom hardware, can be used to
estimate the real costs of implementing and executing abstract mathematical
algorithms. This would allow, for instance, security analysts to be better
prepared to the increasing power of computer hardware than they currently are.
The same models can also be used to estimate the capacity of neurons in the
human or animal nervous system. These estimations are currently rather
speculative. But it can be easily shown on elementary neuro-anatomical and
neuro-physiological arguments that each individual human brain neuron should
outperform the complete nervous system of C. elegans by almost an order of
magnitude.
It is even possible to compare the computational capacity of neural nets and
silicon. However, this does not lead to a lot of insight immediately. Neurons
and silicon are on different ends of the computational spectrum. The
computational capacity of silicon is dominated by it’s clock speed. On the
other hand, neurons are slow, but the capacity of neural nets is dominated by
their connectivity. This formalizes the well known fact that the computational
strengths of human brains and silicon computers lie in completely different
problem areas. Simulating the one in the other has always proved to be
extremely inefficient.
## 5 Funding
Netherlands Organization for Scientific Research (276-75-002)
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## Appendix A Efficient emulators
The Efficient Emulator requirement can be defined as follows: Given a finite
set of computational devices of a certain complexity, eg, UTMs with up to $t$
tapes and action table sizes of unto $S$ bits, that can perform computation
$A$ in $\Lambda$ steps with cost $C$, there exists a computing device which
can emulate any of these devices performing $A$ at a cost $C^{\prime}$ such
that (by definition)
$C^{\prime}\leq\gamma\cdot\left(C+O(\Lambda)\right)+O(1)\;\;\text{with}\
\gamma>0$ (A-1)
For a UTM emulator, $O(\Lambda)$ can be interpreted as
$\Lambda\cdot(\alpha+\epsilon)$ and $O(1)$ as $\beta$. For a UTM, $\alpha$,
$\beta$, and $\gamma$ are fixed emulator cost factors for all emulated devices
and computations and $\epsilon$ represents the “rounding error” in
representing the original states and symbols on the emulated device, eg, UTMa,
in the symbols of the emulator, eg, UTMb. The size of the rounding error
$\epsilon$ can be estimated from the encoding of the emulated device, eg, the
action table.
Efficient emulation according to equation A-1 is possible using equation 1 as
a cost function at least for some Turing Complete devices. Which means that
any algorithm that can be computed efficiently by one device, eg, a UTM, can
also be computed efficiently by other devices. It is a weak condition as it
does not ensure that there will always be an efficient emulator of a specific
type.
If the lowest cost, $C^{\prime}$, of a certain computation on any efficient
emulator is known, it can be shown that the cheapest program on any emulated
device, eg, UTMa, that can perform the same computation in $\Lambda$ steps has
a cost $C$ of at least
$C+\Lambda\cdot(\alpha+\epsilon)\geq(C^{\prime}-\beta)/\gamma$ (A-2)
Where $C$ itself depends on $\Lambda$ (see equation 1).
Below, two examples of efficient emulators are given. One emulates all single
tape UTMs up to a given number of symbols and states. The other emulates
simplified neural nets unto a maximal number of nodes and connections.
### A.1 An efficient emulator of UTMs
For any UTMa with $\leq t$ tapes, it is possible to design a $t+1$ tape UTMb
that can emulate it efficiently. Here this is proven for $t=1$, but other
cases and types of devices follow directly from this case. The dual tape UTMb
uses one tape, $T_{a}$ with head $H_{a}$, to store the tape of UTMa. The other
tape, $T_{b}$ with head $H_{b}$, contains the action table of UTMa, organized
as a table of
{New symbol $T_{a}$, Move $H_{a}$, Move $H_{b}$} addressed by row addresses
{State UTMa, Symbol $\sigma_{a}$ on $T_{a}$}. The action table of UTMb has a
simple structure, and will not be described here.
At the start of an emulated read-write-move cycle of UTMa, the position of the
head, $H_{b}$, over the $T_{b}$ tape indicates the current state of UTMa. UTMb
performs the following steps:
1. 1.
Move $H_{b}$ to correct row on $T_{b}$ in stored action table of UTMa
* •
Read current symbol $\sigma_{a}$ from tape $T_{a}$
* •
Move $H_{b}$ by $\sigma_{a}$ rows
2. 2.
Read and write new symbol
* •
Read new symbol $\sigma^{\prime}_{a}$ from $T_{b}$
* •
Write $\sigma^{\prime}_{a}$ to tape $T_{a}$
* •
Move $H_{b}$ to next field in row
3. 3.
Move $H_{a}$
* •
Read $H_{a}$ movement $D_{a}$ from $T_{b}$
* •
Move $H_{a}$ by $D_{a}$
* •
Move $H_{b}$ to next field in row
4. 4.
Move $H_{b}$ to new state of UTMa
* •
Read $H_{b}$ movement $D_{b}$ from $T_{b}$
* •
Move $H_{b}$ by $D_{b}$ to the row position that indicates the new state of
UTMa
The $Halt$ state of UTMa will move UTMb to a program tape area that will halt
UTMb.
It is obvious that UTMb can only efficiently emulate single tape UTMs for
which it can handle all symbols, states, and head movements inside it’s own
tape symbols. This sets an upper size limit to the UTMs it can emulate. But
within these size limits, this emulator clearly works according to equation
A-1. UTMb can emulate every single read-write-move cycle of any UTMa in four
of its own read-write-move cycles ($\gamma=4$). The action table of UTMa can
be stored in $N_{a}\cdot M_{a}$ rows of 3 symbols of UTMb, which takes more
space than $log_{2}(N_{a}M_{a}D_{a})$ bits by $\epsilon=O(1)$. The action
table, state and other work tape contents of UTMb are fixed, contributing
$\gamma\cdot\alpha(=O(1))$ per emulated read-write-move cycle of UTMa for a
total of $\gamma\cdot\Lambda\alpha$ ($=O(\Lambda)$). Starting and halting
costs are also of order $\beta=O(1)$.
### A.2 An efficient emulator for neural nets
An efficient emulator for a simplified neural network can be build from
parallel processors. Such an emulator will be generated as above for a UTM. In
a general game model, a UTM is recruited for each neural node and then a UTM
for each synapse or connection between neural nodes. There are a maximum of
$N_{max}$ nodes available each with at most $k$ incoming connections
(synapses). In total, a maximum of $k\cdot N_{max}$ synapses will be
available. Unused nodes and synapses are unconnected and have empty worktapes,
but they do contribute to the computational cost of the emulator. All UTMs
have dual work tapes, $T_{\alpha}$ stores the node and synapse states and
$T_{\beta}$ a state transformation table. The $T_{\alpha}$ work tapes of the
synapse UTMs will overlap with a shared state field in the $T_{\alpha}$ work
tape of the originating neural node UTM and a ”personal” activity field in the
target neural node UTM. It is assumed that an unlimited number of UTMs can
read concurrently from the same field of a shared work tape. However, only one
UTM at a time can write to a shared field.
Each synapse UTM has a table that tells how a current synapse state $\sigma$
changes to a new state $\sigma^{\prime}$ under influence of the state, $\eta$,
of the originating node ($0$ or $1$, for firing a spike or not). The table
contains a row for every possible synapse state. Each row contains fields:
{Activation, Head movement for $\eta=0$, Head movement for $\eta=1$}
The relation between the synapse state (row number) and the Activation is the
weight of the synapse. “Learning” could result in changing the activation
entries (not implemented here).
Start at the work tape position of the head $H_{\alpha}$ over $T_{\alpha}$
that contains the state of the originating node and head $H_{\beta}$ over the
start of the table row on $T_{\beta}$ that contains the current state of the
synapse. Then first update all the synapse UTMs in parallel.
1. 1.
Synapse UTMs read the state of the originating node UTMs
* •
Read node state $\eta$, which is either $0$ or $1$ (fire spike)
* •
Move $H_{\alpha}$ to next field containing the current activation
* •
Move $H_{\beta}$ to the row field corresponding to the node state $\eta$
2. 2.
Read new synapse states $\sigma^{\prime}$ from $T_{\beta}$
* •
Read $\sigma^{\prime}$ from $T_{\beta}$ as a relative head movement
$D_{\beta}$
* •
Move $H_{\beta}$ by $D_{\beta}$
3. 3.
Read and write activation of corresponding synapse states
* •
Read Activation from $T_{\beta}$
* •
Write Activation to $T_{\alpha}$
* •
Move $H_{\alpha}$ to previous field
Then update the neural node UTMs. The head, $H_{\alpha}$ starts at the first
Activation field (of $k$ fields) on $T_{\alpha}$. $T_{\beta}$ contains a table
to relate the new state to the activation level. The table is organized in
rows with {New State, New Activation}. The new activation level which follows
the current, is stored as a movement of $H_{\beta}$. The position of
$H_{\beta}$ indicates the current activation of the node.
1. 4.
Sum activation fields ($k$ steps), end over node state field
* •
Read activation $\rho$ from $T_{\alpha}$
* •
Move $H_{\beta}$ by $\rho$ rows
* •
Move $H_{\alpha}$ to next field
2. 5.
Read new state and update node state
* •
Read new state $\eta^{\prime}$ from $T_{\beta}$
* •
Write new state $\eta^{\prime}$ to $T_{\alpha}$
* •
Move $H_{\beta}$ to next field on $T_{\beta}$
3. 6.
Update node activation state
* •
Read new activation state from $T_{\beta}$
* •
Move $H_{\alpha}$ back to first activation field (ie, by $k$ fields)
* •
Move $H_{\beta}$ to new activations state $\eta$
(eg, start of the current row for state $0$, and to the start row of the table
after state $1$, spike generation)
With the exception of step 4, all steps take a single cycle of the UTMs. In
total, a single cycle on the original neuron can be emulated in $\gamma=k+5$
steps of all the UTMs in parallel.
Step 4 is extremely inefficient because it needs $k$ steps to sum the
activations, and every neural network solves the problem by using a fast
integrator. Such an integrator will sum the synapse activations in a short
time. This integration can be done by a fast or parallel accumulator.
Assume that all activation symbols are two’s complement bit numbers (to allow
for inhibitting synapses) that indicate the size of the activation. The
accumulator would contain a register with $A$ bits representing the current
activation and an adder with $A$ full bit adders. A one bit full adder has a
truth table of $8\cdot 2=16$ bits, 3 bits for inputs and carry-in to indicate
the row and 2 bits for output and carry-out. The first and last bit adders
need only half as much, 8 bits, because they lack a carry-in or carry-out.
If the truth tables are used as the complexity of the adders, the $A$ bit
accumulator would need $3\cdot A-1$ bit registers (accumulator, input, and
$A-1$ carry bits) and $16\cdot(A-1)$ bit truth tables, or, $19\cdot A-17$
bits. In this calculation the two half bit adders are combined.
So for a 32 bit activation size, the accumulator would need around 591 bits. A
parallel integrator can be simulated by a fast accumulator which sums the $k$
activation fields in a single clock step. That is, step 4 is performed in a
single step by the accumulator which sums all the activations and prints out a
selection of bits from the accumulator (not necessarily all $A$ bits). The
cost of such a fast accumulator would be $k\cdot(A\cdot 19-17)=k\cdot I_{A}$
per clock step. Note that this is approximately the same cost as would be
needed for $k$ parallel accumulators working in a single step. Then step 5 is
changed to read the activation and generate the spike (1) or not (0).
The original cost, $C$, of a computation of a neural network with $N$ nodes
and $k$ synapses per node, from $N$ originating nodes, over $\Lambda$ steps is
$C=\Lambda N\left(I_{node}+k(I_{syn}+\log_{2}(N))\right)$ (A-3)
Where $I_{node}$ and $I_{syn}$ are the total information size of the neuron
nodes and synapses, respectively. Here, the complexities are estimated as the
sizes of the action tables of equivalent UTMs, as there is currently no
sensible estimate based on physiological data. With a fast accumulator, the
simulation of a node splits the complexity into an accumulator part ($k\cdot
I_{A}$) and “the rest” ($I_{B}$), ie, $I_{node}=I_{B}+k\cdot I_{A}$.
To calculate the cost of the emulation, using the fast accumulator, the sizes
of the emulator UTMs and accumulator without tapes are $\alpha_{B}$,
$\alpha_{A}$, and $\alpha_{syn}$ for emulating the node body, accumulator, and
synapse, respectively. The corresponding rounding errors for emulating the
real neural states and symbols in the emulator are $\epsilon_{B}$,
$\epsilon_{A}$, and $\epsilon_{syn}$. For $N$ nodes with each $k$ synapses and
$\log_{2}(N)$ bits to designate the originating node, the emulator cost
becomes:
$\displaystyle C^{\prime}$ $\displaystyle\leq$ $\displaystyle 6\Lambda
N(I_{B}+\epsilon_{B}+k(I_{syn}+\log_{2}(N)+I_{A}+\epsilon_{syn}+\epsilon_{A}))+\beta$
$\displaystyle+\ 6\Lambda
N_{max}(\alpha_{B}+k_{max}(\alpha_{A}+\alpha_{syn}))$ $\displaystyle\leq$
$\displaystyle 6C+6\Lambda
N(\epsilon_{B}+k(\epsilon_{A}+\epsilon_{syn}))+6\Lambda
N_{max}(\alpha_{B}+k_{max}(\alpha_{A}+\alpha_{syn}))+\beta$
The cost in equation A.2 is indeed linear in $C$, and $\Lambda$ according to
equation A-1, for fixed maximum $N_{max}$ and $k_{max}$.
## Appendix B Complexity versus time trade-off
Using a more complex Finite-State-Machine (FSM) often reduces the time and
cost needed to complete a lengthy computation. On the other hand, moving a
short computation to a smaller device can reduce costs too. The boundaries of
such trade-offs follow from the cost function. As an example, consider a UTMα
with $M$ states, $N$ symbols and $D$ possible head movements. UTMα has a FSM
size $S=MN(m+n+d)+m$, where $m,n,d$ are the bit sizes needed to store,
respectively, states, symbols, and head movements. Assume there is an
efficient, low-cost, program $P$ for UTMα that computes $A$ in $\Lambda$ steps
effectively using $I_{\text{eff}}$ bits on tape (where $\Lambda
I_{\text{eff}}\equiv\sum^{\Lambda}_{1}I(\lambda)$) with cost
$C\simeq\Lambda(S+I_{\text{eff}}+\log_{2}(I_{\text{eff}}))$. At each step,
UTMα can process $b=m+n+d$ bits. Assume that $A$ depends on the total number
of bits processed, $\Lambda\cdot b$.
Construct a new UTMβ that can process $b^{\prime}$ bits per step, or
$b^{\prime}=\delta
b=\delta_{1}m+\delta_{2}n+\delta_{3}d\;\;\text{with}\;\;\delta>0$ (B-1)
and take $\\{S,I_{\text{eff}}\\}\gg\\{b,\log_{2}(I_{\text{eff}})\\}$. Very
simple examples of such operations would be to combine program steps for
parallel execution to increase $b$, or to split program steps into smaller
components to decrease $b$.
The new UTMβ is chosen such as to reduce the cost of computation $A$. UTMβ has
a FSM size, $S^{\prime}$, of
$S^{\prime}=M^{\delta_{1}}N^{\delta_{2}}\delta(m+n+d)+\delta_{1}m\approx\delta
M^{\delta_{1}-1}N^{\delta_{2}-1}S$ (B-2)
Simplify the new FSM size to $S^{\prime}=\delta\Gamma_{\delta}S$ where
$\Gamma_{\delta}\equiv M^{\delta_{1}-1}N^{\delta_{2}-1}$ can be roughly
approximated as an exponential function of $\delta$, $\Gamma_{\delta}\sim
2^{(m+n)(\delta-1)}$. A new, efficient, program $P^{\prime}$ on UTMβ can
calculate $A$ by processing $\Lambda^{\prime}\cdot b^{\prime}\geq\Lambda\cdot
b$ bits or in $\frac{1}{\delta}\Lambda\leq\Lambda^{\prime}\leq\Lambda$ steps.
As the total number of bits processed remain the same, it is assumed that
$I_{\text{eff}}^{\prime}\geq I_{\text{eff}}$.
The cost, $C^{\prime}$, of computing $A$ using $P^{\prime}$ on UTMβ can be
estimated as
$C^{\prime}\geq\Lambda^{\prime}(S^{\prime}+I_{\text{eff}}^{\prime}+log_{2}(I_{\text{eff}}^{\prime}))$.
After ignoring small components $\delta_{1}m$ and $log_{2}(I_{\text{eff}})$,
the new cost becomes
$\displaystyle C^{\prime}$ $\displaystyle\geq$
$\displaystyle\dfrac{1}{\delta}\cdot\dfrac{\delta\cdot\Gamma_{\delta}S+I_{\text{eff}}}{S+I_{\text{eff}}}\cdot
C$ (B-3)
For large $I_{\text{eff}}\gg\delta\Gamma_{\delta}S$, the new cost becomes
$C^{\prime}\geq\frac{C}{\delta}$ which is a decrease if $\delta>1$. For small
$I_{\text{eff}}\ll\delta\Gamma_{\delta}S$, the new cost becomes
$C^{\prime}\geq\Gamma_{\delta}C$ which is a decrease if $\delta<1$. Note that
in the limits of $I_{\text{eff}}\rightarrow\infty$ and
$I_{\text{eff}}\rightarrow 0$ the costs can be made very small indeed by,
respectively, increasing or decreasing $S$.
The optimal size of the FSM can be estimated by calculating the minimum of
equation B-3. Express the effective memory size in terms of the FSM size,
$I_{\text{eff}}=\omega S$ and assume that
$\delta\approx\delta_{1}\approx\delta_{2}\approx\delta_{3}$. Differentiate
with respect to $\delta$. The minimum cost is reached if:
$\omega=\delta^{2}\cdot 2^{(m+n)(\delta-1)}\cdot\dfrac{(m+n)}{\ln(2)}$ (B-4)
The optimal size of a FSM is reached if $\delta=1$, which means that the
minimal cost is reached if $S=\frac{\ln(2)}{(m+n)}\cdot I_{\text{eff}}$.
The above boundaries on the cost are for the ideal cases, where both the
memory use, $I^{\prime}_{\text{eff}}$, as the number of steps,
$\Lambda^{\prime}$, are minimal. In general, a cost reduction to
$1/\delta_{\text{eff}}$ can be found for large, $I_{\text{eff}}$, if
$\dfrac{\Lambda^{\prime}I^{\prime}_{\text{eff}}}{\Lambda
I_{\text{eff}}}\equiv\dfrac{1}{\delta_{\text{eff}}}<1$ (B-5)
These results suggest that the optimum results are found for choices of $b$
for which $S\approx I_{\text{eff}}/(m+n)$. This implies that $MN\sim
O(I_{\text{eff}})$ for $d\sim O(m,n)$.
The above modelling refers to computations that are processor bound, ie, the
computations depend on the number of bits processed. For such a computation,
the most efficient implementation should try to reduce the number of
computational steps by equalizing the complexity (“size”) of the central
processor and the amount of memory used.
## Appendix C The cost of operating the UTM head
In a UTM, the head is the “processing element”. The head reads and writes
symbols, and steps forward and backward. It can also be seen as responsible
for changing the state of a UTM. The structure of the head is fully determined
by the actions table, ie, number of states, symbols, and possible head
movements. So it does not have to be specified in the definition of a UTM.
However, the complexity of the moving head adds to the real costs of operating
a UTM.
The complexity of the UTM head can be estimated, in symbolic terms, from the
number of symbols and head movements. For $N$ symbols, at least
$n\geq\log_{2}(N)$ bits are needed for each of the read and the write
functions. Head movements over the tape and state changes in the action table
will be implemented as counters that keep track of the relative movements over
the tape and the action table and signals when zero is reached (count down).
For each bit in a counter, 2 bits are needed for the register and carry-in, 2
bits for the output and carry-out and $4\cdot 2=8$ bits for the truth table.
In a counter, only the carry-in bits are counted as the carry-out bits are the
same bits. In total, 11 bits are needed per counter bit. The last counter bit
does not need a carry-out bit and only needs a 4 bit truth table. So a counter
of width $w$, needs $11w-4$ bit of ”content” for the bare counter. A compare-
to-zero can be implemented as a logical OR over the $w$ bits of the counter
that is triggered by the result 0/false. This can be implemented by an OR of
each bit with the result of the higher order bits. For each bit, except the
highest order bit, two inputs and one output and a 4 bit (OR) truth table are
needed, where all but the last output are shared with the next input.
Together, 6 bits per counter bit for a total of $6(w-1)+1$ bits for a counter
of width $w$. So a counter plus zero comparator with $w$ bits needs
$I_{counter}=17w-9$ bits of logic storage. Note that the information needed to
describe the connections is ignored here for simplicity.
With $M$ states, the state counter into the rows of the action table needs a
width of $m\geq\log_{2}(M)$ bits and a total content of $17m-9$ bit. To
address the columns in the action table with $N$ symbols, the counter width is
$n\geq\log_{2}(N)$ with a total content of $17n-9$ bit. For a maximal range of
$D$ steps, the tape counter will need $d\geq\log_{2}(D)$ bits width and a
total content of $17d-9$ bits. With one bit dedicated to the direction of
movement, the latter might be reduced by 18 bit. For the purpose of
generality, the full $17d-9$ bits will be used here. Operating a state or tape
counter running $M$, $N$, and $D$ steps would cost, respectively, $M(17m-9)$,
$N(17n-9)$, and $D(17d-9)$ bit (steps) of our work function.
In total, $\sim 2n+17(m+n+d)-27$ bits are needed to specify the state of the
head during operation for a cost of:
$\dfrac{C_{head}}{\Lambda}=2n+M(17m-9)+N(17n-9)+D(17d-9)$ (C-1)
Where $C_{head}$ is the cost of running the head in bits and $\Lambda$ is the
length of the computation in clock steps. Equation C-1 only represents the
minimum cost in symbolic (bit) terms.
Reducing the UTM head to one that does not skip tape cells, $D=2$ (i.e.,
$\\{-1,1\\}$), increases the number of steps needed to complete the
computation by a factor of $O(D/2)$ and increases the number of states needed,
and the size of the action table, by a factor of $O(ND/2)$ to store state and
symbol information while stepping to the desired tape cell. So the cost of
running the computation increases by a factor of $O(ND^{2}/4)$, both when
accounting for the action table size and when accounting for the state counter
cost (ignoring logarithmic terms). The tape counter will run at approximate
the same cost as the decrease in the number of counts compensates for the
increased duration of the computation, ignoring logarithmic factors. The cost
of the symbol read and write heads and the symbol counter will increase by a
factor $O(D/2)$ due to the longer compute times.
The cost of using a UTM can be divided into the size of the tape and action
table, and the cost of deploying the head. For a fully 8 bit UTM, $m=n=d=8$,
the head will account for just over 6% of the non-tape cost (107 kb versus
1.57 Mb for the action table), down to under 0.03% for a fully 16 bit UTM
($m=n=d=16$).
From the definitions it can be derived that, for large $N$ and $M$, the cost
of running the head becomes small compared to the cost of the action table if
$17(N+M+D)\ll NM$. This is satisfied if $D\leq\max(M,N)$ and $\min(M,N)\gg
51$. Both conditions are not unreasonable for practical systems doing long
computations. See Appendix B for trade-offs between $M$, $N$,
$I_{\text{eff}}$, and $\Lambda$.
The information to prescribe the UTM head can be extracted from the action
table and does not have to be specified independently. Moreover, for UTMs
which are not minimalist, the contribution of the head to the costs of the
computation will be relatively small. To simplify this study, the
contributions of the head to the costs of computations will, therefore, be
ignored in this paper.
## Appendix D A quantitative cost example: Tit-for-Tat
To illustrate the cost computations in the game model, it will be applied to
the Iterated Prisoner’s Dilemma game with Tit-for-Tat as the strategy [AH81].
The game is played on a single Run tape, where the moves of the System and
Environment are written in pairs of cells.
There are three symbols: $C$ for cooperate, $D$ for defect and $H$ for halt.
The environment starts a turn by writing a string of two cells, one with a
random symbol $C$ or $D$, and one with the Environment’s move, either $C$ or
$D$. Then the Environment wakes up the System which is always positioned on
the first cell of the string where it is supposed to write a move. The System
completes the turn in a two step cycle. Note that the System has no private
tape and cannot move backward over the Run tape.
First, the system reads the content of the cell it is positioned over and
writes its current move, $C$ or $D$, into the cell. If the symbol read was
$H$, the System halts and the game is over, else the System moves the head to
the next cell. In the next step, the System reads the symbol in the underlying
cell, moves to the next (empty) cell and goes to sleep (if that cell is
empty). Then the Environment generates the next turn.
The Tit-for-Tat strategy is implemented in a simplified Turing Machine with
five states: Cooperate ($c$), Defect ($d$), Read ($r$), and Halt ($h$). There
are three symbols, $C$ (cooperate), $D$ (defect), and $H$ (halt). The System
cycles through the turns as follows:
1. 1.
cycle
* •
Wake up by Environment
* •
Read content of Run tape
* •
Depending on the current state write:
* $C$
if state is $c$
* $D$
if state is $d$
* •
Move to next cell
* •
If read symbol was:
* $H$
switch to $h$ $\rightarrow\ halt$
* $C$
switch to $r$
* $D$
switch to $r$
2. 2.
cycle
* •
Read content of Run tape
* •
Do not write (or write back the read symbol)
* •
Move to next cell
* •
If read symbol was:
* $H$
switch to $h$ $\rightarrow\ halt$
* $C$
switch to $c$ $\rightarrow\ sleep$
* $D$
switch to $d$ $\rightarrow\ sleep$
The game starts by writing the specification of the Tit-for-Tat program on the
valuation tape with $c$ as the initial state. The Environment loads the
program, writes its first move and positions the System over the first cell on
the Run tape in state $c$, and starts the System. The game ends when the
Environment writes an $H$ symbol which halts the System.
The System goes to “sleep” when it reaches an uninitialized tape cell. If the
Environment writes all its moves in one go, the System will not go to sleep
and play until it reaches an $H$ symbol. Else it will sleep until the cell
under its head is initialized.
The action table of the Tit-for-Tat player is presented in table 1.
Table 1: Action table for Tit-for-Tat player. The System is not actually allowed to overwrite the move of the Environment. For completeness, the system is set to rewrite the Environment’s move in state $r$. | Symbol read
---|---
State | $C$ | | | $D$ | | | H | |
c | $C$ | 1 | $r$ | $C$ | 1 | $r$ | $C$ | 0 | $h$
d | $D$ | 1 | $r$ | $D$ | 1 | $r$ | $D$ | 0 | $h$
r | $C$ | 1 | $c$ | $D$ | 1 | $d$ | C | 0 | $h$
| symbol write - move - new state
This simplified implementation is very small, 4 states and 3 symbols. After
entering the $h$ state the System halts and the game is over. The System
cannot write the $H$ symbol. There is a rule that the Environment is not
allowed to write an $H$ symbol in its second, move, cell because it would lead
to an incomplete game. If the Environment does make this illegal move, it
loses. The fact that the System writes a $C$ symbol in the same cell afterward
(lower right hand side cell of table 1), effectively breaking the rule that is
not allowed to change the move of the Environment, does not change this
outcome. In compatibility logic, the player who makes the first illegal move
loses.
The total action table could fit in 36 bit ($3\cdot 4$ bits per row and 3
rows) and the current state in 2 bits. One turn would take two clock cycles.
The cost for the System of running the game would be 76 bits per turn ($2\cdot
38$) as sleep time is not counted. It is easy to see how the complexity of the
System’s game playing strategy can be increased by including one or more
private work tapes and more states. However, such a more complex strategy
would increase the costs of the computation, potentially by a very large
amount.
If the cost of running the head is included, with $N=3$ symbols, $M=4$ states,
and $D=2$ movement options, the complexity of the run tape head would be
$n_{w}=1$ bit for writing (2 symbols) and $n_{r}\sim 2$ bit for reading (3
symbols), $17\cdot 1-9=8$ bit for the head movement counter, and $17\cdot
2-9=25$ bit each for the state change and symbol counter. The cost of running
the head would be $2\cdot 219=438$ bit per move (per step:
$n_{w}+n_{r}+M(17m-9)+N(17n-9)+D(17d-9)=1+2+4\cdot 25+4\cdot 25+2\cdot 8=219$
bit). So running the head would be the major cost of running this Tit-for-Tat
Machine.
Some aspects of computability logic have been used implicitly in this example.
Most notably the fact that any player who breaks the rules loses. So if any of
the players would rewrite any of the moves, it would lose. The Environment can
read any cell, and therefore, has to write its moves first in every turn or
else it could cheat. The System cannot move back over the Run tape, so it has
to write down its own move before it can read the Environment’s move or else
forgo this turn. It is a free design choice to go for a game structure that
prevents this type of cheating instead of a rule to bind the players. Both
approaches would work. It is the Environment who determines whether the System
has completed the computation and, therefor, “wins”. This means the System is
not required to keep track of the score, which would be costly. In this Tit-
for-Tat game, the condition for winning could be anything from not breaking
the rules to actually getting the most points.
|
arxiv-papers
| 2009-11-27T13:22:55 |
2024-09-04T02:49:06.732611
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R.J.J.H. van Son",
"submitter": "Rob Van Son",
"url": "https://arxiv.org/abs/0911.5262"
}
|
0911.5348
|
# Characteristics and Estimates of Double Parton Scattering at the Large
Hadron Collider
Edmond L. Berger berger@anl.gov High Energy Physics Division, Argonne
National Laboratory, Argonne, IL 60439 C. B. Jackson cb.jackson@mac.com High
Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439 Gabe
Shaughnessy g-shaughnessy@northwestern.edu High Energy Physics Division,
Argonne National Laboratory, Argonne, IL 60439 Department of Physics &
Astronomy, Northwestern University, Evanston, IL 60208
###### Abstract
We evaluate the kinematic distributions in phase space of 4-parton final-state
subprocesses produced by double parton scattering, and we contrast these with
the final-state distributions that originate from conventional single parton
scattering. Our goal is to establish the distinct topologies of events that
arise from these two sources and to provide a methodology for experimental
determination of the relative magnitude of the double parton and single parton
contributions at Large Hadron Collider energies. We examine two cases in
detail, the $b~{}\bar{b}~{}\rm{jet~{}jet}$ and the 4 jet final states. After
full parton-level simulations, we identify a few variables that separate the
two contributions remarkably well, and we suggest their use experimentally for
an empirical measurement of the relative cross section. We show that the
double parton contribution falls off significantly more rapidly with the
transverse momentum $p_{T}^{j1}$ of the leading jet, but, up to issues of the
relative normalization, may be dominant at modest values of $p_{T}^{j1}$ .
††preprint: ANL-HEP-PR-09-109, NU-HEP-TH/09-14
## I Introduction
Double parton scattering (DPS) means that two short-distance subprocesses
occur in a given hadronic interaction, with two initial partons being active
from each of the incident protons in a collision at the Large Hadron Collider
(LHC). The concept is shown for illustrative purposes in Fig. 1, and it may be
contrasted with conventional single parton scattering (SPS) in which one
short-distance subprocess occurs, with one parton active from each initial
hadron. Since the probability of single parton scattering is itself small, it
is often expected that the chances are considerably suppressed for two or more
short-distance interactions in a given collision. However, expectations such
as these bear quantitative re-examination at the LHC where the high overall
center-of-mass energy provides access to very small values of the fractional
momentum $x$ carried by partons, a region in which parton densities grow
rapidly. A large contribution from double parton scattering could result in a
larger than otherwise anticipated rate for multi-jet production and produce
significant backgrounds in searches for signals of new phenomena. The high
energy of the LHC also provides an increased dynamic range of available phase
space for detailed investigations of DPS.
Figure 1: Sketch of a double-parton process in which the active partons are
$i$ and $k$ from one proton and $j$ and $l$ from the second proton. The two
hard scattering subprocess are $A(i~{}j\rightarrow a~{}b)$ and
$B(k~{}l\rightarrow c~{}d)$.
Investigations of double parton scattering have a long history theoretically
Goebel:1979mi ; Paver:1982yp ; Humpert:1983pw ; Mekhfi:1983az ; Humpert:1984ay
; Ametller:1985tp ; Halzen:1986ue ; Mangano:1988sq ; Godbole:1989ti ;
Drees:1996rw ; Eboli:1997sv ; Yuan:1997tr ; Calucci:1997uw ; DelFabbro:1999tf
; Kulesza:1999zh ; Korotkikh:2004bz ; Cattaruzza:2005nu ; Hussein:2006xr ;
Maina:2009sj ; Domdey:2009bg ; d'Enterria:2009hd ; Gaunt:2009re , and there is
evidence for their presence in collider data from the CERN Intersecting
Storage Rings Akesson:1986iv and Fermilab Tevatron Abe:1997xk ; D0:2009 . A
significantly greater role for double-parton processes may be expected at the
LHC where higher luminosities are anticipated along with the higher collision
energies. Of substantial importance is to know empirically how large the
double parton contribution may be and its dependence on relevant kinematic
variables.
Our aim is to calculate characteristic final states at LHC energies in which
it may be straightforward to discern a double parton signal. We show in this
paper that double parton scattering produces an enhancement of events in
regions of phase space in which the “background” from single parton scattering
is relatively small. If such enhancements are observed experimentally, with
the kinematic dependence we predict, then we will have a direct empirical
means to measure the size of the double parton contribution. In addition to
its role in general LHC phenomenology, this measurement will have an impact on
the development of partonic models of hadrons, since the effective cross
section for double parton scattering measures the size in impact parameter
space of the incident hadron’s partonic hard core.
From the perspective of sensible rates and experimental tagging, a good
process to examine should be the 4 parton final state in which there are $2$
hadronic jets plus a $b$ quark and a $\bar{b}$ antiquark, viz.
$b~{}\bar{b}~{}j_{1}~{}j_{2}$. If the final state arises from double parton
scattering, then it is plausible that one subprocess produces the
$b~{}\bar{b}$ system and another subprocess produces the two jets. There are,
of course, many single parton scattering (2 to 4 parton) subprocesses that can
result in the $b~{}\bar{b}~{}j_{1}~{}j_{2}$ final state, and we look for
kinematic distributions that show notable separations of the two
contributions. As we show, the correlations in the final state are predicted
to be quite different between the double parton and the single parton
subprocesses. For example, the plane in which the $b~{}\bar{b}$ pair resides
is uncorrelated with the $j_{1}~{}j_{2}$ plane in double parton scattering,
but not in the single parton case.
The state-of-the-art of calculations of single parton scattering is well
developed whereas the phenomenology of double parton scattering is as yet much
less advanced. In the remainder of this Introduction, we first describe the
approach we adopt for the calculation of double parton scattering,
specializing to the proton-proton situation of the LHC. Then we outline the
paper and summarize our main results. Our calculations are done at leading-
order in perturbative QCD, adequate for the points we are trying to make.
Making the usual factorization assumption, we express the single-parton hard-
scattering differential cross section for $p~{}p\rightarrow a~{}b~{}X$ as
$\displaystyle d\sigma^{SPS}=\sum_{i,j}\int
f^{i}_{p}(x_{1},\mu)f^{j}_{p}(x_{1}^{\prime},\mu)d\hat{\sigma}_{(ij\rightarrow
ab)}(x_{1},x_{1}^{\prime},\mu)dx_{1}dx_{1}^{\prime}.$ (1)
Indices $i$ and $j$ run over the different parton species in each of the
incident protons. The parton-level subprocess cross sections
$d\hat{\sigma}_{(ij\rightarrow ab)}(x_{1},x_{1}^{\prime},\mu)$ are functions
of the fractional partonic longitudinal momenta $x_{1}$ and $x_{1}^{\prime}$
from each of the incident hadrons and of the partonic
factorization/renormalization scale $\mu$. The parton distribution functions
$f^{i}_{p}(x_{1},\mu)$ express the probability that parton $i$ is found with
fractional longitudinal momentum $x_{1}$ at scale $\mu$ in the proton; they
are integrated over the intrinsic transverse momentum (equivalently, impact
parameter) carried by the parton in the parent hadron.
A formal theoretical treatment of double parton scattering would begin with a
discussion of the hadronic matrix element of four field operators and an
explicit operator definition of two-parton correlation functions. This
procedure would lead to a decomposition of the hadronic matrix element into
non-perturbative two-parton distribution functions and the corresponding hard
partonic cross sections for $\hat{\sigma}(ijkl\rightarrow abcd)$. An operator
definition of two-parton correlation functions may be found in Ref.
Mueller:1985wy where the two-parton correlation function is reduced to a
product of single parton distributions. An explicit operator definition of
two-parton distributions with different values of the two fractional momenta
$x_{1}$ and $x_{2}$ is presented in Ref. Guo:1997it , along with a model for
the two-parton distributions in terms of normal parton distributions. In this
paper, we follow a phenomenological approach along lines similar to Refs.
Goebel:1979mi ; Paver:1982yp ; Humpert:1983pw ; Mekhfi:1983az ; Humpert:1984ay
; Ametller:1985tp ; Halzen:1986ue ; Mangano:1988sq ; Godbole:1989ti ;
Drees:1996rw ; Eboli:1997sv ; Yuan:1997tr ; Calucci:1997uw ; DelFabbro:1999tf
; Kulesza:1999zh ; Korotkikh:2004bz ; Cattaruzza:2005nu ; Hussein:2006xr ;
Maina:2009sj ; Domdey:2009bg ; d'Enterria:2009hd ; Gaunt:2009re .
In a double parton process, partons $i$ and $k$ are both active in a given
incident proton. We require the joint probability that parton $k$ carries
fractional momentum $x_{2}$, given that parton $i$ carries fractional momentum
$x_{1}$. In general, this joint probability
$H^{i,k}(x_{1},x_{2},\mu_{A},\mu_{B})$ should also depend on the intrinsic
transverse momenta $k_{T,i}$ and $k_{T,k}$ of the two partons (or,
equivalently, their impact parameters). The hard scales $\mu_{A}$ and
$\mu_{B}$ are characteristic of the two hard subprocesses in which partons $i$
and $k$ participate. In the sections below, we discuss the choice we make of
the hard-scale and do not explore in this paper theoretical uncertainties
associated with higher-order perturbative contributions.
In contrast to single parton distributions functions $f^{i}_{p}(x_{1},\mu)$
for which global analyses have produced detailed information, very little is
known phenomenologically about the magnitude and functional dependences of
joint probabilities $H^{i,k}(x_{1},x_{2},\mu_{A},\mu_{B})$. A common
assumption made in estimates of double parton rates is to ignore possibly
strong correlations in longitudinal momentum and to use the approximation
$\displaystyle
H^{i,k}_{p}(x_{1},x_{2},\mu_{A},\mu_{B})=f^{i}_{p}(x_{1},\mu_{A})f^{k}_{p}(x_{2},\mu_{B}).$
(2)
For reasons of energy-momentum conservation, if not dynamics, the simple
factorized form of Eq. (2) cannot be true for all values of the fractional
momenta $x$. The values of $x_{2}$ available to the second interaction are
always limited by the values of $x_{1}$ in the initial interaction since
$x_{1}+x_{2}\leq 1$. The approximation certainly fails even at the kinematic
level if both partons carry a substantial fraction of the momentum of the
parent hadron. However, it may be adequate for applications in which the
values of $x_{1}$ and $x_{2}$ are small. We remark that the momentum integral
$\displaystyle\sum_{i,k}\int
x_{1}x_{2}H^{i,k}_{p}(x_{1},x_{2},\mu_{A},\mu_{B})dx_{1}dx_{2}=1,$ (3)
as long as we can run the upper limits of the $x_{1}$ and $x_{2}$ integrations
to $1$, independently. The large phase space at the LHC may make it possible
to explore dynamic correlations that break Eq. (2).
In Fig. 2, for the region of phase space of interest to us, we show the
contributions to the $b\bar{b}jj$ cross section as a function of $x$ from both
DPS and SPS, after minimal acceptance cuts are imposed (Sec. II). The center-
of-mass energy is $\sqrt{s}=10$ TeV. It is evident that the majority of DPS
events are associated with low $x$ values, in essence never exceeding $0.2$.
The momentum carried off by the beam remnant is $(1-x_{1}-x_{2})$ in DPS and
$(1-x)$ in SPS. The results in Fig. 2 show that this remnant momentum is not
too different in DPS and SPS. Thus, the use of Eq. (2) in calculations of
event rates at the LHC appears adequate as a good first approximation. While
available Tevatron data on double parton scattering Abe:1997xk ; D0:2009 are
insensitive to possible correlations in $x$, the greater dynamic range at the
LHC may make it possible to observe them. 111As emphasized in Refs.
Korotkikh:2004bz ; Gaunt:2009re , even if the approximation in Eq. (2) holds
at one hard scale, evolution of the parton densities with $\mu$ will induce
violations at larger scales.
Figure 2: Values of the parton longitudinal momentum fractions $x$ in the DPS
and SPS events. Most DPS events have low $x$ values. The events used for this
plot include the requirements $n_{\rm jet}=4$, $n_{\rm btag}=2$, and the
threshold cuts discussed in Sec. II.
Assuming next that the two subprocesses $A(i~{}j\rightarrow a~{}b)$ and
$B(k~{}l\rightarrow c~{}d)$ are dynamically uncorrelated, we express the
double parton scattering differential cross section as:
$\displaystyle d\sigma^{DPS}=\dfrac{m}{2\sigma_{\rm eff}}\sum_{i,j,k,l}\int
H^{ik}_{p}(x_{1},x_{2},\mu_{A},\mu_{B})H_{p}^{jl}(x_{1}^{\prime},x_{2}^{\prime},\mu_{A},\mu_{B})$
(4) $\displaystyle\times
d\hat{\sigma}^{A}_{ij}(x_{1},x_{1}^{\prime},\mu_{A})d\hat{\sigma}^{B}_{kl}(x_{2},x_{2}^{\prime},\mu_{B})dx_{1}dx_{2}dx_{1}^{\prime}dx_{2}^{\prime}.$
The symmetry factor $m$ is $1$ if the two hard-scattering subprocesses are
identical and is $2$ otherwise. In the denominator, there is a factor
$\sigma_{\rm eff}$ with the dimensions of a cross section. Given that one
hard-scatter has taken place, $\sigma_{\rm eff}$ measures the size of the
partonic core in which the flux of accompanying short-distance partons is
confined. It should be at most proportional to the transverse size of a
proton. For the first process of interest in this paper, $pp\rightarrow
b\bar{b}j_{1}j_{2}$, Eq. (4) reduces to
$\displaystyle d\sigma^{DPS}(pp\rightarrow
b\bar{b}j_{1}j_{2}X)=\dfrac{d\sigma^{SPS}(pp\rightarrow
b\bar{b}X)d\sigma^{SPS}(pp\rightarrow j_{1}j_{2}X)}{\sigma_{\rm eff}}.$ (5)
Tevatron collider data Abe:1997xk ; D0:2009 yield values in the range
$\sigma_{\rm eff}\sim 12$ mb. We use this value for the estimates we make, but
we emphasize that the goal should be to make an empirical determination of its
value at LHC energies.
In Sec. II, we present our calculation of the double parton and the single
parton contributions to $p~{}p\rightarrow b~{}\bar{b}~{}j_{1}~{}j_{2}~{}X$. We
identify variables that discriminate the two contributions quite well. In Sec.
III, we treat the double parton and the single parton contributions to $4$ jet
production, again finding that good separation is possible despite the
combinatorial uncertainty in the pairing of jets. We show in both cases that
the double parton contribution falls off significantly more rapidly with
$p_{T}^{j1}$, the transverse momentum of the leading jet. For the value of
$\sigma_{\rm eff}\sim 12$ mb and the cuts that we use, we find that, in the
region in which it is most identifiable, double parton scattering is dominant
for $p_{T}^{j1}<30$ GeV in $b~{}\bar{b}~{}j_{1}~{}j_{2}$ at LHC energies, and
$p_{T}^{j1}<50$ GeV in $4$ jet production. Our conclusions are found in Sec.
IV.
## II Heavy quark pair and jet pair production in QCD.
In this section, we describe the calculation of the DPS and SPS event rates
for $b\bar{b}jj$ production at the LHC. For our purposes, light jets (denoted
by $j$) are assumed to originate only from gluons or one of the four lighter
quarks ($u,d,s$ or $c$) and, as stated above, we perform all calculations for
the LHC with a center-of-mass energy of $\sqrt{s}=10$ TeV. Event rates are
quoted for 10 pb-1 of data.
### II.1 Outline of the method
The prediction for the DPS event rate is based on the assumption that the two
partonic interactions which produce the $b\bar{b}$ and $jj$ systems occur
independently (as expressed in Eq. (4)). At leading order, the only
contribution is:
$(ij\rightarrow b\bar{b})\otimes(kl\rightarrow jj)$ (6)
where the symbol $\otimes$ denotes the combination of one event each from the
$b\bar{b}$ and the $jj$ final states. In an attempt to model some of the
effects expected from initial- and final-state radiation, we also account for
the possibility of an additional jet which is undetected because it is either
too soft or outside of the accepted rapidity range. Thus, we include several
other contributions to the DPS event:
$\displaystyle b\bar{b}(j)\otimes
jj\,\,\,,\,\,\,b\bar{b}j\otimes(j)j\,\,\,,\,\,\,b\bar{b}j\otimes j(j)$ (7)
$\displaystyle b\bar{b}\otimes(j)jj\,\,\,,\,\,\,b\bar{b}\otimes
j(j)j\,\,\,,\,\,\,b\bar{b}\otimes jj(j)\,,$ (8)
where the parentheses surrounding a jet indicate that it is undetected. We
compute processes such as $jj(j)$ and $b\bar{b}(j)$ at LO as 3 parton final-
state processes.
The 2 to 3 parton amplitudes for $b\bar{b}(j)$ [and $jj(j)$] diverge as the
undetected jet $(j)$ becomes soft or collinear to one of the other final state
partons or to an initial parton. The divergences are removed in a full next-
to-leading order (NLO) treatment, in which real emission and virtual (loop)
contributions are incorporated, and the finite $b\bar{b}$, $b\bar{b}(j)$, and
$b\bar{b}j$ contributions are present with proper relative normalization. In
the LO parton level simulations done in this paper, we employ a cut at the
generator level to remove the divergences. All the final state objects in the
processes listed above are required to have transverse momentum $p_{T}\geq 20$
GeV. In this fashion, we model some aspects of the expected momentum imbalance
between the $b$ and $\bar{b}$ arising from the 2 to 3 process $ij\rightarrow
b\bar{b}j$, but we cannot claim to include the relative normalization between
the $b\bar{b}$ and $b\bar{b}j$ contributions that would result from a full NLO
treatment. We leave a complete NLO analysis for future work.
The SPS cross section is computed according to Eq. (1). It receives
contributions at lowest order from the 2 parton to 4 jet final state process:
$ij\rightarrow b\bar{b}jj\,,$ (9)
and, in the case where a jet is undetected, from the 5-jet final states
(computed at LO):
$b\bar{b}(j)jj\,\,\,,\,\,\,b\bar{b}j(j)j\,\,\,,\,\,\,b\bar{b}jj(j)\,.$ (10)
We also investigate the possibility of $jjjj$ and $jjjj(j)$ final state
contributions to the SPS cross section where two of the jets “fake” $b$ jets.
We find that the effects from these final states are subdominant compared to
the processes listed in Eqs. (9) and (10).
In our numerical analysis, we use the leading-order CTEQ6L1 parton
distribution functions (PDFs) Pumplin:2002vw to compute both DPS and SPS
cross sections, and we evaluate all cross sections using one-loop evolution of
$\alpha_{s}(\mu)$. For the renormalization and factorization scales, we choose
the dynamic scale:
$\mu^{2}=\sum_{i}p_{T,i}^{2}+m_{i}^{2}\,,$ (11)
where $p_{T,i}$ is the transverse momentum of the $i^{th}$ jet and $m_{i}=0$
($m_{i}=4.7$ GeV) for light (bottom) jets. In the case of roughly equal values
of the transverse momenta $p_{T,i}$, Eq. (11) yields $\mu\sim 2p_{T}$ in SPS
and $\mu\sim\sqrt{2}p_{T}$ in DPS. At LO there is no obviously “right” hard
scale, and the choice in Eq. (11) seems as good as any other.
The DPS events are generated as two separate sets of events with
Madgraph/Madevent Maltoni:2002qb and then combined as described above. For
example, at leading order, we generate events separately for $pp\to b\bar{b}X$
and $pp\to jjX$, and these events are then combined as indicated in Eq. (6).
To increase the speed of the simulations, the SPS events are generated with
Alpgen Mangano:2002ea since the SPS processes of interest are hard-coded in
Alpgen, which contains more compact expressions for the squared-matrix-
elements than Madgraph.
The events accepted after generation are required to have 4 jets $n_{\rm
jet}=4$ with 2 of these tagged as $b$’s $n_{\rm btag}=2$. At the generator
level, all the final state objects in the processes listed in Eq. (6) through
Eq. (10) must have transverse momentum $p_{T}\geq 20$ GeV, as mentioned above.
Furthermore, at the analysis level, all events (DPS and SPS) are required to
pass the following acceptance cuts:
$\displaystyle p_{T,j}$ $\displaystyle\geq$ $\displaystyle
25\,\,\,\mbox{GeV},\,\,\,|\eta_{j}|\leq 2.5$ (12) $\displaystyle p_{T,b}$
$\displaystyle\geq$ $\displaystyle 25\,\,\,\mbox{GeV},\,\,\,|\eta_{b}|\leq
2.5$ (13) $\displaystyle\Delta R_{jj}$ $\displaystyle\geq$ $\displaystyle
0.4,\,\,\,\Delta R_{bb}\geq 0.4$ (14)
where $\eta_{i}$ is the jet’s pseudorapidity, and $\Delta R_{ij}$ is the
separation in the azimuthal angle ($\phi$) - pseudorapidity plane between jets
$i$ and $j$:
$\Delta R_{ij}=\sqrt{(\eta_{i}-\eta_{j})^{2}+(\phi_{i}-\phi_{j})^{2}}\,.$ (15)
We model detector resolution effects by smearing the final state energy
according to:
${\delta E\over E}={a\over\sqrt{E/\rm{GeV}}}\oplus b,$ (16)
where we take $a=50\%$ and $b=3\%$ for jets. To account for $b$ jet tagging
efficiencies, we assume a $b$-tagging rate of 60% for $b$-quarks with
$p_{T}>20\text{ GeV}$ and $|\eta_{b}|<2.0$. We also apply a mistagging rate
for charm-quarks as:
$\epsilon_{c\to b}=10\%\quad\quad\text{ for }p_{T}(c)>50\text{ GeV}\\\ $ (17)
while the mistagging rate for a light quark is:
$\displaystyle\epsilon_{u,d,s,g\to b}$ $\displaystyle=0.67\%\quad\quad\text{
for }$ $\displaystyle p_{T}(j)<100\text{ GeV}$ (18)
$\displaystyle\epsilon_{u,d,s,g\to b}$ $\displaystyle=2\%\quad\quad\quad\text{
for }$ $\displaystyle p_{T}(j)>250\text{ GeV}.$ (19)
Over the range $100\text{ GeV}<p_{T}(j)<250\text{ GeV}$, we linearly
interpolate the fake rates given above Baer:2007ya .
### II.2 Properties of SPS and DPS in $b~{}\bar{b}~{}j~{}j$
Having detailed the calculation of the $b\bar{b}jj$ event rates from DPS and
SPS, we now discuss some of the distinguishing characteristics of the two
contributions. First, however, it is important to check that our simulations
of DPS events are not introducing an artificial correlation between the
$b\bar{b}$ and $jj$ final states. We do this by inspecting the angle $\Phi$
between the plane defined by the $b\bar{b}$ system and the plane defined by
the $jj$ system. If the two scattering processes $ij\rightarrow b\bar{b}$ and
$kl\rightarrow jj$ which produce the DPS final state are truly independent,
one would expect to see a flat distribution in the angle $\Phi$. By contrast,
many diagrams, including some with non-trivial spin correlations, contribute
to the 2 parton to 4 parton final state in SPS, and naively one would expect
some correlation between the two planes. To avoid possible effects from
boosting to the lab frame, we define the two planes in the partonic center-of-
mass frame.
We specify the planes by using the three-momenta of the outgoing jets. Then,
the angle between the two planes defined by the $jj$ and $b\bar{b}$ systems
is:
$\cos\Phi=\hat{n}_{3}(j_{1},j_{2})\cdot\hat{n}_{3}(b_{1},b_{2}),$ (20)
where $\hat{n}_{3}(x,y)$ is the unit three-vector normal to the plane defined
by the $x-y$ system.
The normal is undefined when $j_{1}$ and $j_{2}$ are back-to-back or $b_{1}$
and $b_{2}$ are back-to-back, as occurs in a large fraction of the DPS events.
Therefore, when $\cos\phi_{(x,y)}<-0.9$, we use a different procedure. We use
the three-momentum of one of the incoming partons along with the three-
momentum of one of the outgoing $b$ quarks to define the $b\bar{b}$ plane. Let
$q_{b}$ be the three-momentum of an incoming parton, and $p_{b}$ be the three-
momentum of the final-state $b$ (or $\bar{b}$) quark. We then define
$\phi_{p_{b},q_{b}}$ to be the azimuthal angle of the three-vector normal to
the $q_{b}-p_{b}$ plane. Note that we use $\phi$ here since the normal to any
three-vector and the beam-line will be transverse to the beam-line (not the
case in the SPS process). In this way, the jet which is not used to define the
plane is guaranteed to lie in the plane. The plane for the $jj$ system is
defined in an analogous manner. Finally, the angle between the planes is then:
$\Phi=|\phi_{p_{j},q_{j}}-\phi_{p_{b},q_{b}}|\,.$ (21)
In Fig. 3, we display the number of events as a function of the angle between
the two planes. There is an evident correlation between the two planes in SPS,
while the distribution is flat in DPS, indicative that the two planes are
uncorrelated.
Figure 3: Event rate as a function of the angle between the two planes defined
by the $b\bar{b}$ and $jj$ systems. In SPS events, there is a correlation
among the planes which is absent for DPS events.
Another interesting difference between DPS and SPS is the behavior of event
rates as a function of transverse momentum. As an example of this, in Fig. 4,
we show the transverse momentum distribution for the leading jet (either a $b$
or light $j$) for both DPS and SPS. Several characteristics are evident.
First, SPS produces a relatively hard spectrum, and for the value of
$\sigma_{\rm eff}$ and the cuts that we use, we see that SPS tends to dominate
over the full range of transverse momentum considered. On the other hand, DPS
produces a much softer spectrum which (up to issues of normalization in the
form of $\sigma_{\rm eff}$) can dominate at small values of transverse
momentum. The cross-over between the two contributions to the total event rate
is $\sim 30$ GeV for the acceptance cuts considered here. A smaller (larger)
value of $\sigma_{\rm eff}$ would move the cross-over to a larger (smaller)
value of the transverse momentum $p_{T}^{j1}$ of the leading jet.
Figure 4: The transverse momentum $p_{T}$ distribution of the leading jet in
$jjb\bar{b}$ after minimal cuts.
### II.3 Distinguishing variables
We turn next to the search for variables that may allow for a clear separation
of the DPS and SPS contributions. Since the topology of the DPS events
includes two $2\to 2$ hard scattering events, the two pairs of jet objects are
roughly back-to-back. We expect the azimuthal angle between the pairs of jets
corresponding to each hard scattering event to be strongly peaked near
$\Delta\phi_{jj}\sim\Delta\phi_{bb}\sim\pi$. Real radiation of an additional
jet, where the extra jet is missed because it fails the threshold or
acceptance cuts, allows smaller values of $\Delta\phi_{jj}$. The relevant
distribution is shown for light jets (non $b$-tagged) in Fig. 5a. There is a
clear peak near $\Delta\phi_{jj}=\pi$ for DPS events, while the events are
more broadly distributed in SPS events. The secondary peak near small
$\Delta\phi_{jj}$ arises from gluon splitting which typically produces nearly
collinear jets. The suppression at still lower $\Delta\phi_{jj}$ comes from
the isolation cut $\Delta R_{jj}>0.4$.
Figure 5: (a) The difference $\Delta\phi$ in the azimuthal angles of light jet
pairs for DPS and both SPS+DPS events. The dijet pairs are back-to-back in DPS
events. (b) The variable $S_{\phi}$ for DPS and SPS+DPS events provides a
stronger separation of the underlying DPS events from the total sample when
compared to $\Delta\phi$ for any pair.
The separation of DPS events from SPS events becomes more pronounced if
information is used from both the $b\bar{b}$ and $jj$ systems. As an example,
we consider the distribution built from a combination of the azimuthal angle
separations of both $jj$ and $b\bar{b}$ pairs, using a variable adopted from
Ref. D0:2009 :
$S_{\phi}={1\over\sqrt{2}}\sqrt{\Delta\phi(b_{1},b_{2})^{2}+\Delta\phi(j_{1},j_{2})^{2}}.$
(22)
In Fig. 5b, we present a distribution in $S_{\phi}$ for both DPS and SPS+DPS
events. Again, as in the case of the $\Delta\phi$ distribution, we see that
the SPS events are broadly distributed across the allowed range of $S_{\phi}$.
However, the combined information from both the $b\bar{b}$ and $jj$ systems
shows that the DPS events produce a sharp and substantial peak near
$S_{\phi}\simeq\pi$ which is well-separated from the total sample.
The narrow peaks near $\Delta\phi_{jj}=\pi$ in Fig. 5a and near $S_{\phi}=1$
in Fig. 5b will be smeared somewhat once soft QCD radiation and other higher-
order terms are included in the calculation.
Another possibility for discerning DPS is the use of the total transverse
momentum of both the $b\bar{b}$ and $jj$ systems. At lowest order for a $2\to
2$ process, the vector sum of the transverse momenta of the final state pair
vanishes. In reality, radiation and momentum mismeasurement smear the expected
peak near zero. Nevertheless, we still expect DPS events to show a
distribution in the transverse momenta of the jet pairs that is reasonably
well-balanced. To encapsulate this expectation for both light jet pairs and
$b$-tagged pairs, we use the variable D0:2009 :
$S_{p_{T}}^{\prime}={1\over\sqrt{2}}\sqrt{\left({|p_{T}(b_{1},b_{2})|\over|p_{T}(b_{1})|+|p_{T}(b_{2})|}\right)^{2}+\left({|p_{T}(j_{1},j_{2})|\over|p_{T}(j_{1})|+|p_{T}(j_{2})|}\right)^{2}}.$
(23)
Here $p_{T}(b_{1},b_{2})$ is the vector sum of the transverse momenta of the
two final state $b$ jets, and $p_{T}(j_{1},j_{2})$ is the vector sum of the
transverse momenta of the two (non $b$) jets.
The distribution in $S_{p_{T}}^{\prime}$ is shown in Fig. 6. As expected, we
observe that the DPS events are peaked near $S_{p_{T}}^{\prime}\sim 0$ and are
well-separated from the total sample. The SPS events, on the other hand, tend
to be far from a back-to-back configuration and, in fact, are peaked near
$S_{p_{T}}^{\prime}\sim 1$. This behavior of the SPS events is presumably
related to the fact that a large number of the $b\bar{b}$ or $jj$ pairs arise
from gluon splitting which yields a large $p_{T}$ imbalance and, thus, larger
values of $S_{p_{T}}^{\prime}$.
Figure 6: Distribution of events in $S_{p_{T}}^{\prime}$ for the DPS and SPS
samples. Due to the back-to-back nature of the $2\to 2$ events in DPS
scattering, the transverse momenta of the jet pair and of the $b$-tagged jet
pair are small, resulting in a small value of $S_{p_{T}}^{\prime}$. In (a) we
show the $S_{p_{T}}^{\prime}$ distribution for our standard cuts, and in (b)
we increase the cut on the transverse momentum of the leading jet,
$p_{T}^{j1}>40$ GeV. The fraction of DPS events in the whole sample decreases
with increasing $p_{T}^{j1}$.
In this subsection, we find that extraction of the DPS “signal” for
$b\bar{b}jj$ production from the SPS “background” can be enhanced by combining
information from both $b\bar{b}$ and $jj$ systems. Our simulations suggest
that the variable $S_{p_{T}}^{\prime}$ may be a more effective discriminator
than $S_{\phi}$. However, given the leading order nature of our calculations
and the absence of smearing associated with initial state soft radiation, this
picture may change and a variable such as $S_{\phi}$ (or some other variable)
may become a clearer signal of DPS at the LHC. Realistically, it would be
valuable to study both distributions once LHC data are available in order to
determine which is more instructive. In the following, we use the clear
separation shown in Fig. 6 in our exploration of the distinct properties of
DPS and SPS events.
### II.4 Two-dimensional distributions
The evidence in Fig. 5 and Fig. 6 for distinct regions of DPS dominance
prompts the search for greater discrimination in a plane represented by a two
dimensional distribution of one variable against another. We examined scatter
plots involving the inter-plane angle $\Phi$, the jet-jet azimuthal angle
difference $\Delta\phi_{jj}$, $S_{\phi}$, and $S^{\prime}_{p_{T}}$. Strong
kinematic correlations are evident in the plot of $S_{\phi}$ vs.
$S^{\prime}_{p_{T}}$ at the level of our leading order calculation, and we
observe no additional separation of DPS and SPS beyond that evident in Figs. 5
and 6. Likewise, there are strong correlations between $\Delta\phi_{jj}$ and
$S_{\phi}$.
One scatter plot with interesting features is displayed in Fig. 7. The DPS
events are seen to be clustered near $S^{\prime}_{p_{T}}=0$ and are uniformly
distributed in $\Phi$. The SPS events peak toward $S^{\prime}_{p_{T}}=1$ and
show a roughly $\sin\Phi$ character. While already evident in Figs. 3 and 6,
these two features are more apparent in the scatter plot Fig. 7. Moreover, the
scatter plot shows a valley of relatively low density between
$S^{\prime}_{p_{T}}\sim 0.1$ and $\sim 0.4$. In an experimental one-
dimensional $\Phi$ distribution such as Fig. 3, one would see the sum of the
DPS and SPS contributions. If structure is seen in data similar to that shown
in the scatter plot Fig. 7, one could make a cut at $S^{\prime}_{p_{T}}<0.1$
or $0.2$ and verify whether the experimental distribution in $\Phi$ is flat as
expected for DPS events.
Figure 7: Two-dimensional distribution of events in the variables $\Phi$ and
$S_{p_{T}}^{\prime}$ for the DPS and SPS samples.
In Fig. 4, we show that DPS produces a softer transverse momentum distribution
for the leading jet (either a $b$ or light $j$). In data one would see only
the sum of the DPS and SPS components in a plot like Fig. 4. A scatter plot of
$S_{p_{T}}^{\prime}$ vs. the transverse momentum of the leading jet motivates
an empirical separation of the two components. In Figs. 6(a) and 6(b) we
compare the $S_{p_{T}}^{\prime}$ distributions for two different selections on
the transverse momentum $p_{T}^{j1}$ of the leading jet in the $b\bar{b}jj$
sample. This comparison of the distributions confirms that events in the DPS
region, defined empirically by the region $S_{p_{T}}^{\prime}<0.1$ or $0.2$,
fall off more steeply with $p_{T}^{j1}$ than the rest of the sample. It will
be important and interesting to see whether the selection
$S_{p_{T}}^{\prime}<0.1$ or $0.2$ in LHC data also produces events that show a
more rapid decrease with $p_{T}^{j1}$.
The leading-jet transverse momentum distributions are shown in Figs. 8(a) and
8(b) for two different cuts on $S_{p_{T}}^{\prime}$. In both cases, we see
that the SPS sample has a broader distribution in $p_{T}^{j1}$ and that the
DPS sample dominates for small enough values of $p_{T}^{j1}$. For our chosen
value of $\sigma_{\rm eff}\sim 12$ mb, and for cuts we employ, the crossover
points are roughly $80$ GeV for $S_{p_{T}}^{\prime}<0.2$ and $40$ GeV for
$S_{p_{T}}^{\prime}<0.4$.
Figure 8: The distribution in the transverse momentum of the leading jet
$p_{T}^{j1}$ for (a) $S_{p_{T}}^{\prime}<0.2$ and (b)
$S_{p_{T}}^{\prime}<0.4$. As the signal region becomes more dominated by SPS
events (i.e. moving from (a) to (b)), the resulting distribution becomes
harder and shifts the SPS-DPS cross-over from $\sim 80$ GeV to $\sim 40$ GeV.
## III Four Jet Production
In addition to $b\bar{b}jj$, we can also ask how important DPS can be for a
generic $4j$ final-state, where none of the jets are $b$-tagged. In this
section, we describe our calculation of the double parton scattering and the
single parton scattering contributions to the production of a $4j$ final
state, for which the cross section is larger. Our exposition can be brief
since we repeat the procedure described in some detail in Sec. II.
### III.1 Outline of the method
The DPS process for $4j$ production is topologically equivalent to
$b\bar{b}jj$. However, in the $4j$ system, we lose the $b$-tagging ability
that reduces the combinatorial background in $b\bar{b}jj$, and the prospects
for isolating and measuring DPS over the SPS background may appear less
promising. Fortunately, in going from the $b\bar{b}$ subprocess to the $jj$
subprocess, a much larger DPS rate is possible due to the much larger cross
section for $jj$ production. As we show below, we find that the DPS signature
can be extracted in this $4j$ mode as well.
The DPS cross section for $4j$ production receives contributions from the
following sub-processes at the lowest order:
$jj\otimes jj\,\,\,,\,\,\,b\bar{b}\otimes jj,$ (24)
where both $b$-quarks fail the $b$-tag. We do not include the $b\bar{b}\otimes
b\bar{b}$ process due to its relatively small rate ($\sim 0.14$ nb). This rate
is further reduced by requiring no $b$-tags, yielding roughly 40 events in the
10 pb-1 of luminosity assumed here.
Following Sec. II, we account for the possibility of an additional jet which
is undetected because it is too soft or outside of the accepted rapidity
range. Thus, we include several other contributions to the DPS cross section:
$\displaystyle jjj\otimes(j)j\,\,\,,\,\,\,jj(j)\otimes jj\,,$ (25)
$\displaystyle b\bar{b}j\otimes j(j)\,\,\,,\,\,\,b\bar{b}(j)\otimes jj,$ (26)
$\displaystyle b\bar{b}\otimes j(j)j\,\,\,,\,\,\,b(\bar{b})\otimes
jjj\,\,\,,\,\,\,(b)\bar{b}\otimes jjj\,.$ (27)
where the parentheses surrounding a jet signify that it is not detected.
The SPS cross section receives contributions at lowest order from the final
state:
$jjjj\,\,\,,b\bar{b}jj\,,$ (28)
where both $b$-quarks fail the $b$-tag, and, in the case where a jet is not
detected, from the final states:
$b\bar{b}(j)jj\,\,\,,\,\,\,(b)\bar{b}jjj\,\,\,,\,\,\,b(\bar{b})jjj\,\,\,,\,\,\,(j)jjjj\,.$
(29)
We refer to Sec. II for the specification of acceptance cuts and detector
resolution, and for our treatment of the potential divergences present in the
amplitudes for the processes in Eqs. (24)-(29).
### III.2 Results
Similar to the $b\bar{b}jj$ process, the leading jet in the $4j$ DPS sample is
typically softer than in the SPS channels (see Fig. 9). In this case, again
using $\sigma_{\rm eff}=12$ mb, we find that the cross-over between DPS and
SPS dominance occurs near $p_{T}\simeq 50$ GeV, higher than in the
$b\bar{b}jj$ case shown in Fig. 4.
Figure 9: As in Fig. 4, but for $4j$ events. Similar to the $b\bar{b}jj$
sample, the SPS sample exhibits a harder $p_{T}$ spectrum.
Improvement in the separation between DPS and SPS in the $4j$ case can be
achieved with an analogous version of the $S_{p_{T}}^{\prime}$ variable
introduced in Eq. (23):
$S_{p_{T}}^{\prime}={1\over\sqrt{2}}\sqrt{\left({|p_{T}(j_{a},j_{b})|\over|p_{T}(j_{a})|+|p_{T}(j_{b})|}\right)^{2}+\left({|p_{T}(j_{c},j_{d})|\over|p_{T}(j_{c})|+|p_{T}(j_{d})|}\right)^{2}}.$
(30)
Here $p_{T}(j_{a},j_{b})$ is the vector sum of the transverse momenta of two
final state jets, $a$ and $b$, chosen among the four. The remaining $c$ and
$d$ jets are then fixed. This choice is unique if a separation of the two hard
interactions is possible. In the $b\bar{b}jj$ system, the separation into the
$b\bar{b}$ and $jj$ subsystems via $b$-tagging removed most of the degeneracy
(some degeneracy still remained via tagging efficiencies or light jet
mistagging). In the $4j$ system, the degeneracy can at first glance be
problematic as there are 3 possible pairings of the four jets.
Figure 10: The democratic $S_{p_{T}}^{\prime}$ distribution for $4j$ events
shows much more combinatorial background than in the $b\bar{b}jj$ events. Even
after accepting two mis-matched jet pairs, we see that the DPS and SPS samples
can still be separated well.
One might be tempted to take the pairing of jets which minimizes the value of
$S_{p_{T}}^{\prime}$. Unfortunately, this choice places a bias on the
distribution that makes it potentially problematic to trust the
discrimination. Instead, to construct $S_{p_{T}}^{\prime}$ we take all three
combinations of pairings, which includes one “correct” pairing and two
incorrect pairings in the DPS process. This “democratic” $S_{p_{T}}^{\prime}$
distribution is shown in Fig. 10 and is re-weighted by 1/3 for proper
normalization. As in the $b\bar{b}jj$ case, we see that the DPS distribution
peaks near $S_{p_{T}}^{\prime}\sim 0$, indicative that two back-to-back hard
interactions are present. In addition to this expected feature, we also see a
continuum that extends above $S_{p_{T}}^{\prime}\sim 0.1$, associated with the
wrong combination taken in the democratic approach. In Fig. 10 we see that DPS
produces a secondary peak at $S_{p_{T}}^{\prime}\sim 1$, not present in the
$b\bar{b}jj$ case in Fig. 6. It appears to arise from the wrong pairings of
jets associated with the combinatorial background. In these instances, the
wrong combination of two jets that are close together in $\Delta R$, meaning
that their momenta are aligned, can maximize the value of
$S_{p_{T}}^{\prime}$. Overall, we see that the DPS peak near
$S_{p_{T}}^{\prime}=0$ provides a good means to separate DPS events from SPS
events.
Figure 11: As in Fig. 8, but for $4j$ events with (a) democratic
$S_{p_{T}}^{\prime}<0.2$ and (b) democratic $S_{p_{T}}^{\prime}<0.4$. As in
$b\bar{b}jj$ events, as one increases the cut on $S_{p_{T}}^{\prime}$, the SPS
fraction increases and the total distribution is harder.
As in the $b\bar{b}jj$ case, we inspect the distribution in the $p_{T}$ of the
leading jet after cuts on the $S_{p_{T}}^{\prime}$ variable. Since there are
three jet pairings per event, we now require that at least one of the three
pairings has $S_{p_{T}}^{\prime}$ in the given window. Due to this softer
constraint, the hardening of the $p_{T}$ spectrum of the leading jet is less
dramatic than in the $b\bar{b}jj$ case (e.g. compare Figs. 8 and 11). The
crossover of the SPS and DPS contributions occurs near $80$ GeV for
$S_{p_{T}}^{\prime}<0.2$ and near $50$ GeV for $S_{p_{T}}^{\prime}<0.4$
## IV Discussion and conclusions
Our goal is to develop a method to search for a double parton scattering
contribution in the $b~{}\bar{b}~{}j~{}j$ and 4 jet final states at LHC
energies and to measure the magnitude of its contribution relative to the
single parton contribution to the same final states. Based on our parton level
simulations, we find that variables such as $S_{p_{T}}^{\prime}$ and
$S_{\phi}$ that take into account information from the entire final state,
thereby including both of the hard subprocesses in DPS, are more effective at
discrimination than variables such as $\Delta\phi_{jj}$ that reflect only a
subset of the final-state. The enhancement at low values of
$S_{p_{T}}^{\prime}$ shown in Figs. 6, 7 and 10 provides a good signature for
the presence of double parton scattering. We urge experimenters to search for
such a concentration of events in data at the LHC. Having found this
enhancement, we then suggest that the magnitude of this peak be examined as a
function of the transverse momentum $p_{T}^{j1}$ of the leading jet in the
event sample. The double parton scattering contribution in the peak region
should fall off more rapidly with $p_{T}^{j1}$ than the rest of the sample.
The distribution of events in the region of small values of
$S_{p_{T}}^{\prime}$ should also be examined as a function of the inter-plane
angle $\Phi$ to see whether the flat behavior is seen, as expected for two
independent production processes. Once these characteristics of double parton
scattering are established, the data can be used to determine the effective
normalization $\sigma_{\rm eff}$, defined and discussed in the Introduction.
It will be interesting to see whether the values extracted for $\sigma_{\rm
eff}$ are about the same in the $b~{}\bar{b}~{}j~{}j$ and 4 jet final states
and how they compare with values measured at the Fermilab Tevatron.
Once double parton scattering is established in data, and $\sigma_{\rm eff}$
is determined, in a relatively clean process such as $b\bar{b}jj$, double
parton contributions to a wide range of other processes can be computed with
more certainty about their expected rates at LHC energies. To be sure, given
the approximations described in the Introduction, some variation in the values
of $\sigma_{\rm eff}$ might be expected and appropriate for different
processes and in different kinematic regions. The connection of $\sigma_{\rm
eff}$ with the effective size of the hard-scattering core of the proton may
mean that $\sigma_{\rm eff}$ will have different values for $gg$, $qq$, and
$q\bar{q}$ scattering.
There are several avenues for future work. Of great importance is the proper
inclusion of next-to-leading order contributions NLO . They are needed to make
more robust predictions of the relative normalization of the DPS and SPS
contributions, of the shape of the $p_{T}$ distribution of the leading jet,
and for proper softening of the sharp peaks seen near $S_{p_{T}}^{\prime}=1$
in Figs. 6 and 10, and near $S_{\phi}=\pi$ in Fig. 5b.
It will also be important to develop joint probabilities
$H^{i,k}(x_{1},x_{2},\mu_{A},\mu_{B})$ that are more sophisticated
theoretically than the first approximation represented by Eq. (2) in which
parton-parton correlations are absent. A valuable development in this
direction are the studies presented in Refs. Korotkikh:2004bz ; Gaunt:2009re .
Double parton contributions are potentially relevant for a wide range of
standard model processes, many already considered in the literature
Goebel:1979mi ; Paver:1982yp ; Humpert:1983pw ; Mekhfi:1983az ; Humpert:1984ay
; Ametller:1985tp ; Halzen:1986ue ; Mangano:1988sq ; Godbole:1989ti ;
Drees:1996rw ; Eboli:1997sv ; Yuan:1997tr ; Calucci:1997uw ; DelFabbro:1999tf
; Kulesza:1999zh ; Cattaruzza:2005nu ; Hussein:2006xr ; Maina:2009sj ;
Domdey:2009bg ; d'Enterria:2009hd ; Akesson:1986iv ; Abe:1997xk ; D0:2009 ,
and they may also feed pertinent standard model backgrounds to new physics
processes Sullivan:2008ki . They could be an issue in studies of Higgs boson
production in weak-boson-fusion since the “forward” jets could come from a
second hard subprocess.
## V Acknowledgments
We benefited greatly from discussions with Dr. Thomas LeCompte and from
communications with Dr. John Campbell during the early development of this
project. We also thank Tom, John, and Professor Jianwei Qiu for valuable
comments and suggestions on an earlier draft of this paper. Research in the
High Energy Physics Division at Argonne is supported by the U. S. Department
of Energy under Contract No. DE-AC02-06CH11357. The research of GS at
Northwestern is supported by the U. S. Department of Energy under Contract No.
DE-FG02-91ER40684.
## References
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|
arxiv-papers
| 2009-11-27T21:07:12 |
2024-09-04T02:49:06.744234
|
{
"license": "Public Domain",
"authors": "Edmond L. Berger, C. B. Jackson (Argonne), Gabe Shaughnessy (Argonne\n and Northwestern)",
"submitter": "Edmond Berger",
"url": "https://arxiv.org/abs/0911.5348"
}
|
0911.5419
|
ON CREATING MASS/MATTER BY EXTRA DIMENSIONS
IN THE EINSTEIN-GAUSS-BONNET GRAVITY
A.N.Petrov
Inter-University Center for Astronomy and Astrophysics,
Post Bag 4, Ganeshkhind Pune 411 007, INDIA
and
Relativistic Astrophysics group, Sternberg Astronomical institute,
Universitetskii pr., 13, Moscow, 119992, RUSSIA
E-mail: anpetrov@rol.ru
PACS numbers: 04.50+h; 11.30.-j
###### Abstract
Kaluza-Klein (KK) black hole solutions in the Einstein-Gauss-Bonnet (EGB)
gravity in $D$ dimensions obtained in the current series of the works by
Maeda, Dadhich and Molina are examined. Interpreting their solutions, the
authors claim that the mass/matter is created by the extra dimensions. To
support this claim, one needs to show that such objects have classically
defined masses. We calculate the mass and mass flux for 3D KK black holes in
6D EGB gravity whose properties are sufficiently physically interesting.
Superpotentials for arbitrary types of perturbations on arbitrary curved
backgrounds, recently obtained by the author, are used, and acceptable mass
and mass flux are obtained. A possibility of considering the KK created matter
as dark matter in the Universe is discussed.
## 1 Introduction
We study new exact solutions in the Einstein-Gauss-Bonnet (EGB) gravity in $D$
dimensions, which are $d$-dimensional Kaluza-Klein (KK) black holes (BHs) with
$(D-d)$-dimensional submanifold, presented recently in [1] \- [4] by Maeda,
Dadhich and Molina. The authors treat them as a classical example of creating
matter by curvature. The idea of such a kind is not new. Thus, to make
inflation possible, a pioneer proposal was advanced by Starobinsky [5] that a
high-energy density state was achieved by curved space corrections. Many other
problems of modern cosmology may be solved in the framework of
multidimensional gravity using high-order curvature invariants of KK type
spacetimes, see, e.g., [6] and references there in.
To support the claim on creating ‘matter without matter’, it is necessary to
calculate the mass and the mass flux by classical methods. It is the main goal
of the present paper. Here, we concentrate on 3D BHs in 6D EGB gravity [4].
These toy objects are rich enough in physical properties, e.g., they can have
a radiative regime. For calculations we use the conservation laws developed by
us in [7] \- [9], where in the framework of EGB gravity, superpotentials
(antisymmetric tensor densities) for arbitrary types of perturbations on
arbitrary curved backgrounds have been constructed. Three important types of
superpotentials [9] are used, those based on (i) Nœther’s canonical theorem,
(ii) Belinfante’s symmetrization rule and (iii) a field-theoretical
derivation.
The paper is organized as follows. In section 2, we outline the solutions
obtained in [1] \- [4] and describe necessary properties of the 3D objects in
6D EGB gravity. In particular, in a natural way, we define a spacetime where a
BH is placed. It can be considered as a possible background against which
perturbations are studied. In section 3, in the preliminaries, the main
notions and properties of the applied formalism are presented. Then we study
the objects themselves: (a) as vacuum 6D solutions; (b) as 3D KK solutions
with a ‘matter’ created by extra dimensions. Calculating the mass and the mass
flux we support the second viewpoint. In section 4, we discuss (a) an
ambiguity in the canonical approach related to a divergence in the Lagrangian;
(b) a possibility of applying the KK BH solutions in cosmology. The Appendix
presents explicit general expressions for all three types of superpotentials
in EGB gravity.
## 2 Kaluza-Klein 3D black holes
We consider the action of the EGB gravity in the form:
$S=-\frac{1}{2\kappa_{D}}\int d^{D}x{\hat{\cal
L}}_{EGB}=-\frac{1}{2\kappa_{D}}\int
d^{D}x\sqrt{-g}\left[R-2\Lambda_{0}+\alpha\underbrace{\left(R^{2}_{\mu\nu\rho\sigma}-4R^{2}_{\mu\nu}+R^{2}\right)}_{\mbox{$L_{GB}$}}\right]\,$
(2.1)
where $\alpha>0$. Here and below, curvature tensor
$R^{\mu}{}_{\nu\rho\sigma}$, Ricci tensor $R_{\mu\nu}$ and scalar curvature
$R$ are related to the dynamic metric $g_{\mu\nu}$; a ‘hat’ means densities of
the +1, e.g., $\hat{g}^{\mu\nu}=\sqrt{-g}g^{\mu\nu}$;
$({,\alpha})\equiv\partial_{\alpha}$ means ordinary derivatives; the
subscripts ‘E’ and ‘GB’ are related to the Einstein and the Gauss-Bonnet parts
in (2.1).
The main assumption in [1] \- [4] is that the spacetime is locally
homeomorphic to ${\cal M}^{d}\times{\cal K}^{D-d}$ with the metric
$g_{\mu\nu}={\rm diag}(g_{AB},r^{2}_{0}\gamma_{ab})$,
$A,B=0,\cdots,d-1;~{}a,b=d,\cdots,D-1$. Thus, $g_{AB}$ is an arbitrary
Lorentzian metric on ${\cal M}^{d}$, $\gamma_{ab}$ is the unit metric on the
$(D-d)$-dimensional space of constant curvature ${\cal K}^{D-d}$ with
$k=0,\,\pm 1$. Factor $r_{0}$ is a small scale of extra dimensions
compactified by appropriate identifications. The gravitational equations
corresponding to the EGB gravity action (2.1) have the form:
${\cal G}^{\mu}{}_{\nu}\equiv G^{\mu}{}_{\nu}+\alpha
H^{\mu}{}_{\nu}+\delta^{\mu}{}_{\nu}\Lambda_{0}=0\,,$ (2.2)
where the Einstein tensor $G^{\mu}{}_{\nu}$ and $\delta^{\mu}{}_{\nu}$
correspond to the Einstein part and $H^{\mu}{}_{\nu}$ corresponds to the GB
part in (2.1). After all assumptions their decomposition is as follows:
$\displaystyle{\cal G}^{A}{}_{B}$ $\displaystyle\equiv$
$\displaystyle\left[1+\frac{2k\alpha}{r_{0}^{2}}(D-d)(D-d-1)\right]{}_{(d)}{G}^{A}{}_{B}+\alpha\,{}_{(d)}\\!{H}^{A}{}_{B}$
(2.3) $\displaystyle+$
$\displaystyle\left[\Lambda_{0}-\frac{k}{2r^{2}_{0}}(D-d)(D-d-1)\left(1+\frac{k\alpha}{r_{0}^{2}}(D-d-2)(D-d-3)\right)\right]{\delta}^{A}{}_{B}=0\,;$
$\displaystyle{\cal G}^{a}{}_{b}$ $\displaystyle\equiv$
$\displaystyle{\delta}^{a}{}_{b}\left\\{-\frac{{}_{(d)}\\!{R}}{2}+\Lambda_{0}-\frac{k}{2r^{2}_{0}}(D-d-1)(D-d-2)-\alpha\left[\frac{k}{r^{2}_{0}}(D-d-1)(D-d-2)\\!\times\right.\right.$
(2.4)
$\displaystyle\times\\!\\!\left.\left.\left({}_{(d)}\\!{R}+\frac{k}{2r^{2}_{0}}(D-d-3)(D-d-4)\right)+\frac{{}_{(d)}\\!{L}_{GB}}{2}\right]\right\\}=0$
where the subscript ‘(d)’ means that a quantity is constructed with the use of
$g_{AB}$ only. As a result, one can see that (2.3) is a tensorial equation on
${\cal M}^{d}$, whereas (2.4) is a constraint for it. However to obtain more
interesting solutions one has to consider a special case that the quantity
${\cal G}^{A}{}_{B}$ disappears identically. This is possible for $d\leq 4$
only because then ${}_{(d)}\\!H_{\mu\nu}\equiv 0$. Next, constants are chosen
so as to suppress the coefficients in (2.3), which is possible if $D\geq d+2$,
$k=-1$ and $\Lambda_{0}<0$. Taking into account all the above, there remains a
single governing equation, the scalar equation (2.4) on ${\cal M}^{d}$.
Here, we consider the solutions for $D=6$ and $d=3$ presented in [4]. A
suitable set of constraints for the constants is
$r^{2}_{0}=12\alpha=-3/\Lambda_{0}$. Then, the left hand side of (2.3)
disappears identically. Keeping in mind that ${}_{(3)}\\!{L}_{GB}\equiv 0$,
one simplifies (2.4) to obtain
${}_{(d)}\\!{R}=2\Lambda_{0}\,,$ (2.5)
to which the static solution $g_{AB}(r)$ has been found:
$ds^{2}=-fdt^{2}+f^{-1}dr^{2}+r^{2}d\phi\,,\qquad f\equiv
r^{2}/l^{2}+q/r-\mu\,.$ (2.6)
Here, $\mu$ and $q$ are integration constants, and
$l^{2}\equiv-3/\Lambda_{0}$. The Einstein tensor components for the solution
(2.6) are
$G^{0}_{0}=G^{1}_{1}=1/l^{2}-q/2r^{3},~{}~{}G^{2}_{2}=1/l^{2}+q/r^{3}\,.$
(2.7)
As a space of a constant curvature, $(D-d=3)$-sector is completely presented
by its scalar curvature:
${}_{(D-d)}\\!{R}=6k/r_{0}^{2}=2\Lambda_{0}=-1/2\alpha\,.$ (2.8)
For comparison we consider the BTZ BH [10]. Its metric is presented in the
form
$ds^{2}=-fdt^{2}+f^{-1}dr^{2}+r^{2}d\phi\,,\qquad
f\equiv-r^{2}\Lambda_{0}-\mu\,,$ (2.9)
which is a solution to the 3D pure Einstein equations. The horizon radius
$r_{+}$ of the BH is defined as $r^{2}_{+}=-\mu/\Lambda_{0}$, thus $r_{+}$
(and consequently a BH itself) disappears for vanishing $\mu$. Therefore the
integration constant $\mu$ can be called the mass parameter. For
$\mu\rightarrow 0$, the so-called real vacuum related to the BH (in another
word, a spacetime where a BH is placed) is defined by (2.9) with
$\overline{f}=-r^{2}\Lambda_{0}$. However, such a spacetime is not maximally
symmetric, unlike AdS one. The latter with $\overline{f}=-r^{2}\Lambda_{0}+1$
is approached when $\mu=-1$. A difference between a real vacuum and a
maximally symmetric vacuum is usual in BH solutions of modified metric
theories (see, e.g., [11, 12]); the BTZ BH is the simplest illustration.
The solution (2.6) is more complicated than (2.9), although one has clear
analogies with the BTZ case. Considering BH solutions for simulating dark
matter (see a discussion in section 4) we are more interested in the cases
with a horizon. In (2.6), the equation for the event horizon is
$l^{2}q+r_{+}(r_{+}^{2}-l^{2}\mu)=0$. It is again natural to choose a mass
parameter $\tilde{\mu}$ in such a way that the BH horizon disappears under
vanishing $\tilde{\mu}$. This gives $\tilde{\mu}=\mu-q/r_{+}$ and
$r_{+}^{2}=l^{2}\tilde{\mu}$ (compare with the BTZ case), and consequently
$\tilde{\mu}>0$. Then a real vacuum is defined by (2.6) with
$\overline{f}\equiv r^{2}/l^{2}+q/r-q/r_{+}$, it is again not maximally
symmetric. The maximally symmetric AdS vacuum is defined by (2.6) with
$\overline{f}\equiv r^{2}/l^{2}+1$. For the latter, parameter $q$ is
considered entirely as a perturbation together with $\mu+1$. For
$\tilde{\mu}\leq 0$ a horizon does not exist, this takes place, when $\mu>0$
with $q>2l\left(\mu/3\right)^{3/2}$ or $\mu\leq 0$ with $q\geq 0$.
The scalar equation (2.5) is also satisfied by the radiative Vaidya metric
$g_{AB}(v,r)$:
$ds^{2}=-fdv^{2}+2dvdr+r^{2}d\phi\,,\qquad f\equiv
r^{2}/l^{2}+q(v)/r-\mu(v)\,$ (2.10)
where $\mu(v)$ and $q(v)$ now depend on the retarded/advanced time $v$.
Keeping in mind a possibility to form KK black holes [1] \- [4], advanced time
is more interesting. Then (2.10) can be connected with the solution of the
form (2.6) by the transformation $dt=dv-dr/f(v,r)$. After that, for every
constant $v_{0}$, one can define its own horizon (if it exists) and a
corresponding real vacuum analogously to the static case. The Einstein tensor
components corresponding to (2.10) are
$G^{0}_{0}=G^{1}_{1}=1/l^{2}-q/2r^{3},~{}~{}G^{1}_{0}=(\dot{\mu}r-\dot{q})/2r^{2},~{}~{}G^{2}_{2}=1/l^{2}+q/r^{3}\,,$
(2.11)
where dot means $\partial/\partial v$. The scalar curvature of
$(D-d=3)$-sector is expressed again by (2.8).
Considering (2.6) and (2.10) as solutions to the Einstein 3D equations on
${\cal M}^{3}$ (or, the same, EGB equations because in (2.2) one has
${}_{(3)}\\!H_{\mu\nu}\equiv 0$), one concludes that they are not vacuum
equations with a redefined cosmological constant
$\Lambda=\Lambda_{0}/3=-1/l^{2}$. Indeed, both (2.7) and (2.11) show that a
‘matter’ source ${\cal T}_{AB}$ with zero trace ${\cal T}^{A}{}_{A}=0$ should
exist, and the Einstein equations corresponding to (2.5) could be rewritten as
${}_{(3)}\\!{R}_{AB}-{\textstyle{1\over
2}}g_{AB}{}_{(3)}\\!{R}+g_{AB}\Lambda=\kappa_{3}{\cal T}_{AB}\,.$ (2.12)
A natural treating in [1] \- [4] is that ${\cal T}_{AB}$ is created by the
compact extra dimensions.
## 3 The mass and the mass flux for 3D black holes
### 3.1 Preliminaries
Our calculation is based on differential conservation laws for perturbations
in a given background spacetime in the form:
$\hat{\cal I}^{\alpha}(\xi)=\partial_{\beta}\hat{\cal I}^{\alpha\beta}(\xi)\,$
(3.1)
where $\xi^{\alpha}$ is a displacement vector, $\hat{\cal I}^{\alpha}$ is a
vector density (carrent) and $\hat{\cal I}^{\alpha\beta}$ is an antisymmetric
tensor density (superpotential). Thus, $\partial_{\alpha\beta}\hat{\cal
I}^{\alpha\beta}\equiv 0$ and $\partial_{\alpha}\hat{\cal I}^{\alpha}=0$. The
current contains energy-momentum of both matter and metric perturbations,
whereas the superpotential depends on metric perturbations only. Integrating
$\partial_{\alpha}\hat{\cal I}^{\alpha}=0$ and using the Gauss theorem one
obtains the integral conserved charges in a generalized form:
${\cal P}(\xi)=\int_{\Sigma}d^{D-1}x\,\hat{\cal
I}^{0}(\xi)=\oint_{\partial\Sigma}dS_{i}\,\hat{\cal I}^{0i}(\xi)\,$ (3.2)
where $\Sigma$ is a $(D-1)$-dimensional hypersurface $x^{0}=\rm const$,
$\partial\Sigma$ is its $(D-2)$-dimensional boundary, the zero indices
correspond to time or lightlike coordinates, and small Latin indices
correspond to space coordinates. Since we consider spherically symmetric
systems, we need $01$-components of the superpotentials in (3.2) only.
The formalism describes exact (not infinitesimal) perturbations in general.
This is achieved if one one solution (dynamical) is considered as a perturbed
system with respect to another (background) solution of the same theory. Thus
conserved quantities are defined with respect to a fixed (thought as known)
spacetime, e.g., a mass of a perturbed system on a given background. A
background can be both vacuum and non-vacuum, and usually is to be chosen to
correspond with problems under consideration. The task of the present paper is
calculating a global mass of the KK BHs presented above. It is more important
the mass defined with respect to a spacetime, in which BH is placed because
then with vanishing BH, one obtains a zero mass. Therefore, first of all a
real vacuum described in previous section is chosen as a natural background.
Although such backgrounds are curved and nonsymmetric, the technique used is
powerful. Besides, as interesting and important backgrounds we consider the
AdS space. For such kinds of backgrounds, perturbations are not infinitesimal
in general. However, we need in appropriate asymptotic of superpotentials in
(3.2) only. As one can see below, the fall-off integrands in (3.2) both at
spatial and at null infinity turns out to be sufficiently strong to allow
surface integrals to converge and to give reasonable results.
In the previous section, the bar meant a quantity related to a spacetime where
a BH is ‘placed’; here and below, without contradictions the bar means a
quantity related to a background spacetime as a structure of the formalism. As
a natural choice, for the above described static and radiative solutions we
use the background metric in the same forms (2.6) and (2.10), respectively,
where $\overline{f}=\overline{f}(r)$ can be arbitrary in general but should be
static. For calculating the global mass $M$ we use the timelike Killing vector
$\xi^{\alpha}=(-1,{\bf 0})\,.$ (3.3)
It has this unique form for the above two generalized types of background
metrics: the zero component in (3.3) can be both timelike and lightlike; ${\bf
0}$ includes 5 or 2 space dimensions in a 6D or 3D derivation, respectively.
The metrics of the real vacuum and AdS space just belong to the aforementioned
two types of background metrics and consequently also have a timelike Killing
vector of the unique form (3.3). Then, since (3.3) is used every time, we will
not recall this frequently.
### 3.2 The BTZ solution
As an example, we calculate the mass of the BTZ BH [10] with the metric (2.9).
We take the Einstein parts of each of the superpotentials (A.1), (A.5) and
(A.7), and, keeping in mind a 3D consideration, calculate their
$01$-components
$\displaystyle{}_{E}{\hat{\cal I}}^{01}_{C}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-\overline{g}_{3}}}{2\kappa_{3}r}(f-\overline{f})\left[\frac{r\overline{f}^{\prime}}{2\overline{f}f}\left(f-\overline{f}\right)-1\right]\,,$
(3.4) $\displaystyle{}_{E}{\hat{\cal I}}^{01}_{B}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-\overline{g}_{3}}}{2\kappa_{3}r}(f-\overline{f})\left[\frac{r\overline{f}^{\prime}}{2\overline{f}f}\left(3f+\overline{f}\right)-\frac{r{f}^{\prime}}{f^{2}}\left(f+\overline{f}\right)-1\right]\,,$
(3.5) $\displaystyle{}_{E}{\hat{\cal I}}^{01}_{S}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{-\overline{g}_{3}}}{2\kappa_{3}r}(f-\overline{f})\frac{\overline{f}}{f}\,,$
(3.6)
where the prime means $\partial/\partial r$. Taking into account a background
with $\overline{f}=-r^{2}\Lambda_{0}$, for which $f-\overline{f}=-\mu$, and
substituting (3.4) - (3.6) into (3.2), we obtain, as $r\rightarrow\infty$, the
unique result
$M=\oint_{r\rightarrow\infty}{}_{E}\hat{\cal
I}^{01}d\phi=\frac{\pi\mu}{\kappa_{3}}\,,$ (3.7)
which is quite acceptable for the global mass of the BTZ BH (see, e.g., [13]).
The canonical superpotential has already been checked for calculating (3.7) in
[14], for the other superpotentials the result (3.7) could be considered as a
nice test. Using the AdS background with $\overline{f}=-r^{2}\Lambda_{0}+1$
one obtains $M=\pi(\mu+1)/\kappa_{3}$.
### 3.3 The static KK solution
Now let us turn to (2.6); since it is the solution of the EGB theory one
should try to calculate the mass with using the full formulae (A.1), (A.5) and
(A.7) for this theory. The full background metric is to be chosen as
$\overline{g}_{\mu\nu}=\overline{g}_{AB}\times r^{2}_{0}\gamma_{ab}$. Many
formulae below take place for arbitrary $\overline{f}$ in (2.6), although in
specific calculations we choose $\overline{f}\equiv r^{2}/l^{2}+q/r-q/r_{+}$.
Let us turn to the $(D-2)$-dimensional surface integral (3.2). Really, the
distant surface is considered in $(d=3)$-dimensional spacetime only, whereas
the integral over the $(D-d=3)$-dimensional compact space could be interpreted
as a constant, which ‘normalizes’ the 6D Einstein constant $\kappa_{6}$ to the
3D one $\kappa_{3}$. Indeed, one has for the global mass constructed by (3.2):
$M=\oint_{\partial\Sigma}dx^{D-2}\,\sqrt{-\overline{g}_{D}}\,{\cal
I}^{01}_{D}=\oint_{r\rightarrow\infty}d\phi\sqrt{-\overline{g}_{d}}\,{\cal
I}^{01}_{D}\,\oint_{r_{0}}dx^{D-d}\sqrt{-\overline{g}_{D-d}}=V_{r_{0}}\oint_{r\rightarrow\infty}d\phi\sqrt{-\overline{g}_{d}}\,{\cal
I}^{01}_{D}.$ (3.8)
Thus, since ${\cal I}^{01}_{D}\sim 1/\kappa_{6}$ one could set
$\kappa_{3}=\kappa_{6}/V_{r_{0}}$. At first we follow this prescription.
With our assumptions, we find out that the Einstein parts of the
$01$-components of the superpotentials (A.1), (A.5) and (A.7) for the solution
(2.6) are described only by the $d$-sector. Therefore, to calculate the
Einstein parts, it is sufficient to use Eqs. (3.4) - (3.6), but only with
$\sqrt{-\overline{g}_{3}}/\kappa_{3}$ replaced by
$\sqrt{-\overline{g}_{D}}/\kappa_{6}$ . For all cases, in the natural
background, the Einstein part in (3.8) gives a result corresponding to (3.7):
$M_{E}=\pi\tilde{\mu}V_{r_{0}}/\kappa_{6}\,.$ (3.9)
We now construct the GB $01$-components of the superpotentials (A.1), (A.5)
and (A.7) for the solution (2.6). They consist of two parts. The first one is
pure $(d=3)$-dimensional:
$\displaystyle{}_{(d)}{\hat{\cal I}}^{01}_{C}$ $\displaystyle\equiv$
$\displaystyle 0\,,$ (3.10) $\displaystyle{}_{(d)}{\hat{\cal I}}^{01}_{B}$
$\displaystyle=$
$\displaystyle\frac{\alpha\sqrt{-\overline{g}_{D}}}{\kappa_{6}r^{2}}\frac{\overline{f}}{f}(f-\overline{f})(rf^{\prime\prime}-f^{\prime})\,,$
(3.11) $\displaystyle{}_{(d)}{\hat{\cal I}}^{01}_{S}$ $\displaystyle\equiv$
$\displaystyle 0\,$ (3.12)
(for brevity we suppress the subscript ‘GB’). For $\overline{f}\equiv
r^{2}/l^{2}+q/r-q/r_{+}$, the behavior of (3.11) as $r\rightarrow\infty$ is
$\sim 1/r^{3}$, thus each of the variants (3.10) - (3.12) gives a zero
contribution into the integral (3.8). The other part of the GB $01$-components
is determined by the intersecting terms of the $(d=3)$-sector and the scalar
curvature of the $(D-d=3)$-sector (2.8):
$\displaystyle{}_{(D-d)}{\hat{\cal I}}^{01}_{C}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-\overline{g}_{D}}}{4\kappa_{6}}\left[(f-\overline{f})^{\prime}-\frac{\overline{f}^{\prime}}{f\overline{f}}(f-\overline{f})^{2}+\frac{2(f-\overline{f})}{r}\right]\,,$
(3.13) $\displaystyle{}_{(D-d)}{\hat{\cal I}}^{01}_{B}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-\overline{g}_{D}}}{2\kappa_{6}}\left[\frac{(f-\overline{f})^{2}}{2f\overline{f}}(f+\overline{f})^{\prime}+\frac{f^{2}-\overline{f}^{2}}{f}\left(\frac{f^{\prime}}{f}-\frac{\overline{f}^{\prime}}{\overline{f}}\right)+\frac{f-\overline{f}}{r}\right]\,,$
(3.14) $\displaystyle{}_{(D-d)}{\hat{\cal I}}^{01}_{S}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-\overline{g}_{D}}}{2\kappa_{6}r}(f-\overline{f})\frac{\overline{f}}{f}\,,$
(3.15)
where the subscript ‘(D-d)’ means that a quantity is without pure ‘(d)’-terms.
We remark that both (3.10) and (3.13) are unique for each of (A.3) and (A.4).
The asymptotic of each of (3.13) - (3.15) at spatial infinity in the natural
background is $\sim-\tilde{\mu}$, and their substitution into (3.8) gives the
unique result:
$M_{GB}=-\pi\tilde{\mu}V_{r_{0}}/\kappa_{6}\,.$ (3.16)
Thus, keeping in mind (3.9) one can see that the global mass defined in the
natural background by the total integral (3.8) is zero in all the three
approaches. The same result is valid if the AdS background with
$\overline{f}=r^{2}/l^{2}+1$ is chosen.111The zero result has been recently
obtained for a similar situation by other methods as well by R.G. Cai, L.M.
Cao, and N. Ohta, “Black holes without mass and entropy in Lovelocj gravity”,
Phys. Rev. D, 81, 024018; (Preprint arXiv:0911.0245 [hep-th]).
At least, this result could be anticipated for the field-theoretical approach.
Indeed, the superpotential (A.7) can be connected directly with the linearized
equations [7]. Contracting the latter with $\xi^{\alpha}$ in (3.3), one
selects the $d$-sector only. However, under the present assumptions, the
tensor in (2.3) is equal to zero identically, therefore its linearization is
equal to zero identically as well. This conclusion is supported by combining
the expressions (3.6), with the replacement
$\sqrt{-\overline{g}_{3}}/\kappa_{3}\rightarrow\sqrt{-\overline{g}_{D}}/\kappa_{6}$,
(3.12) and (3.15), which leads to zero identically. At the same time, the
canonical and Belinfante corrected approaches give a zero result only
asymptotically.
Of course, the zero result cannot be acceptable. Analyzing (2.6), one can find
out that, considering this system from the point of view of the Newtonian-like
limit in 3 dimensions (see, e.g., [13]), this system must have a total mass.
Thus one should conclude that a vacuum 6D interpretation (2.2) with (3.8) is
not successful. By this argument, one should consider the 3D Einstein
interpretation (2.12) with a created ‘matter’. Calculating the global
conserved quantity basing on (3.2), we can use only the surface integral,
whereas a source (maybe not determined explicitly, as in (2.12)) is included
into the current in the volume integral. Thus, considering the solution (2.6),
we can be restricted to only the Einstein parts of each of the superpotentials
(A.1), (A.5) and (A.7) related to the non-vacuum equations (2.12). As a full
background metric, one must again consider $\overline{g}_{AB}$ in (2.6)
(without $r^{2}_{0}\gamma_{ab}$); we choose
$\overline{f}=r^{2}/l^{2}+q/r-q/r_{+}$ again and use the Killing vector (3.3).
Then, since the parameter $q$ describes a ‘created matter’ in (2.12), such a
background is not vacuum in 3 dimensions now. Nevertheless, the meaning of the
notion ‘real vacuum’ is not changed, although it could be called wider as a
‘real background’ now. Also, the applied formalism remains powerful in non-
vacuum backgrounds, and the structure of the superpotentials remains the same.
Then again we use (3.4) - (3.6) and obtain the acceptable result of the type
(3.7):
$M=\pi\tilde{\mu}/\kappa_{3}\,.$ (3.17)
If AdS space with $\overline{f}=r^{2}/l^{2}+1$ is chosen as a background, the
mass of the system is $M=\pi(\mu+1)/\kappa_{3}$. Note that in both cases the
parameter $q$ makes no contribution.
### 3.4 The radiative Vaidya KK solution
For the radiative solution (2.10) we have carried out calculations similar to
those in Subsection 3.3. Though, in this case the lightlike $v$-coordinate is
used instead of the time $t$-coordinate. We again calculate $01$-components
for the superpotentials, however, now $\Sigma$ in (3.2) is defined as
$x^{0}=v={\rm constant}$, and the mass calculation is related to null
infinity. In Eqs. (3.18) - (3.23) below, an arbitrary
$\overline{f}=\overline{f}(r)$ is considered. However, now there is no sense
to connect a background (which must be static) with a horizon (which is
changed in time). Therefore, in specific calculations we consider the AdS
background with $\overline{f}=r^{2}/l^{2}+1$ only.
We first derive out the Einstein parts of all superpotentials:
${}_{E}{\hat{\cal I}}^{01}_{C}={}_{E}{\hat{\cal I}}^{01}_{B}={}_{E}{\hat{\cal
I}}^{01}_{S}=-\frac{\sqrt{-\overline{g}_{D}}}{2\kappa_{6}r}(f-\overline{f})\,$
(3.18)
where $f=f(v,r)$, which looks surprisingly simple, see, e.g., (3.4) - (3.6).
The GB $01$-components of the superpotentials (A.1), (A.5) and (A.7) for the
solution (2.10) consist of two parts again. The pure $(d=3)$-dimensional part
is
$\displaystyle\left.{}_{(d)}{\hat{\cal I}}^{01}_{C}\right|_{(A.3)}$
$\displaystyle=$
$\displaystyle\frac{\alpha\sqrt{-\overline{g}_{D}}}{\kappa_{6}r^{2}}(f-\overline{f})(f^{\prime}-rf^{\prime\prime})\,,$
(3.19) $\displaystyle\left.{}_{(d)}{\hat{\cal I}}^{01}_{C}\right|_{(A.4)}$
$\displaystyle\equiv$ $\displaystyle 0\,,$ (3.20)
$\displaystyle{}_{(d)}{\hat{\cal I}}^{01}_{B}$ $\displaystyle=$
$\displaystyle\frac{\alpha\sqrt{-\overline{g}_{D}}}{\kappa_{6}r^{2}}\left[(f-\overline{f})(rf^{\prime\prime}-f^{\prime})+2\left(r\overline{f}^{\prime}+\overline{f}+2r\frac{\partial}{\partial
v}\right)\left(r(f-\overline{f})^{\prime}\right)^{\prime}\right],$ (3.21)
$\displaystyle{}_{(d)}{\hat{\cal I}}^{01}_{S}$ $\displaystyle\equiv$
$\displaystyle 0\,.$ (3.22)
For the AdS background one has as $r\rightarrow\infty$: for (3.19) $\sim
1/r^{3}$ and for (3.21) $\sim 1/r^{2}$, thus all (3.19) - (3.22) again give a
zero contribution into the integral (3.8). As in the static case, the other
part of the GB $01$-components is determined by the intersection terms of the
$(d=3)$-sector and the scalar curvature of the $(D-d=3)$-sector (2.8):
${}_{(D-d)}{\hat{\cal I}}^{01}_{C}={}_{(D-d)}{\hat{\cal
I}}^{01}_{B}={}_{(D-d)}{\hat{\cal
I}}^{01}_{S}=\frac{\sqrt{-\overline{g}_{D}}}{2\kappa_{6}r}(f-\overline{f})\,.$
(3.23)
One can see that these components precisely compensate the components (3.18).
Thus, as in the previous subsection, the global mass defined in 6 dimensions
is zero. Then one should follow the interpretation of the static case and
reject the vacuum 6D derivation (2.2) with (3.8) as unacceptable one.
We again consider Eq. (2.12) as a governing one. Restricting ourselves to the
$d$-sector only and repeating the steps of Subsection 3.3, we obtain in the
AdS background $M=\pi(\mu(v)+1)/\kappa_{3}$. This is in a correspondence with
the static case. The mass flux for the radiating metric (2.10) is obtained
simply by differentiating with respect to $v$:
$\dot{M}=\pi\dot{\mu}(v)/\kappa_{3}$. Comparing with the known BMS flux
derivation [15], this looks acceptable.
Concluding the section we assert that since the KK BH objects have classically
defined global mass and flux, they bring ‘matter’ created by extra dimensions
and a special structure of the objects themselves. If we set $q=0$ and
$q(v)=0$, then, at least in the static case, ${\cal T}_{AB}=0$ in (2.2).
However, this does not influence on our assertion because in all the cases $q$
and $q(v)$ do not contribute into the global mass. Thus, mass/matter is
created in a more wide sense than creating ${\cal T}_{AB}$ in (2.2).
## 4 Concluding remarks
We will first discuss a well-known ambiguity in the canonical approach related
to a choice of a divergence in the Lagrangian. We consider this problem in [9]
and do not make a definite choice between [14] (or (A.3)) and [16] (or (A.4)).
Indeed, both choices give an acceptable mass for the Schwarzschild-AdS BH
tested in [9]. Here, the study of KK objects also does not give an answer
because in all cases we have a unique result. However, in [16] arguments in
favor (A.4) are given. In multitimendional GR, the Katz and Livshits
superpotential [16] turns out uniquely the KBL superpotential [17]; in EGB
gravity, their superpotential naturally transfers into the KBL superpotential
for $D=4$. This is in a correspondence with the Olea arguments [18] where GB
terms in the Lagrangian regularize conserved quantities even if $D\leq 4$.
Lastly, the choice (A.4) looks more preferable because (a) it is more
‘symmetric’ than (A.3), (b) the canonical superpotential with (3.20) gives a
zero global integral in 6 dimensions identically, as in the field-theoretical
approach.
Now we turn to cosmological problems. As well known, the properties of dark
energy and of dark matter are very weakly constrained by the cosmological
observable data, therefore their derivation remains very uncertain. Thus a
search for acceptable models describing the cosmic ingredients is very
important, it is carried out very intensively, and even dramatically, see,
e.g., the recent papers, reviews [19] \- [27] and references there in.
As an example, in the recent paper [28], recalling the ’t Hooft ideas of 1985,
so-called ‘quantum black holes’ are discussed as elementary particles playing
the role of the dark matter particles. The latter are assumed as weak
interacting matter particles (WIMPs), which can have desirable TeV energies
(see the aforementioned reviews). ‘Quantum black holes’ can be presented just
like WIMPs, they can be stable and do not radiate in the Hawking-Bekenstein
regime, unlike usual black holes.
Our main results show that the solutions (2.6) and (2.10) have a classically
defined mass and mass flux. This just presents a possibility for the KK BHs to
be presented in the regime of ‘quantum black holes’. Thus, the topic of the
present paper, as we think, could be related to the dark matter problems.
Concerning this, we remark the following. First, since the parameter $q$ can
describe additional (to gravity) interactions, its presence can suppress the
WIMP idea. Then one needs to set $q=0$, which is permissible, as has been
remarked above. Second, it is desirable to have a positive mass for WIMP
objects. We support this because, if a BH exists, one has $\tilde{\mu}>0$ that
leads to $M>0$. Third, basing on the radiating regime, in [1] \- [4] a
scenario of forming KK BHs in EGB gravity was suggested. One could try to
develop this scenario for various epochs. Keeping in mind all that, in future
studies we plan an examination of more realistic models presented in [1] \-
[4]: they are 4D KK objects in 6 and more dimensions of EGB gravity.
### Acknowledgments
The author thanks very much Naresh Dadhich, Alexey Starobinsky, Joseph Katz,
Nathalie Deruelle and Rong-Gen Cai for fruitful discussions and useful
comments and recommendations. Also, the author expresses his gratitude to
professors and administration of IUCAA, where the work was mainly elaborated
and finalized, for nice hospitality. The work is supported by the grant No.
09-02-01315-a of the Russian Foundation for Basic Research.
## Appendix A Superpotentials in the EGB gravity
In this Appendix, we represent an explicit form of the three types of
superpotentials for perturbations in the EGB gravity [9]. The background
quantities: Christoffel symbols $\overline{\Gamma}^{\sigma}_{\tau\rho}$,
covariant derivatives $\overline{D}_{\alpha}$, the Riemannian tensor
$\overline{R}^{\sigma}{}_{\tau\rho\pi}$ and its contractions are constructed
on the basis of a background $D$-dimensional spacetime metric
$\overline{g}_{\mu\nu}$. It is a known (fixed) solution of EGB gravity; the
bar means that a quantity is a background one. One can find a detail
derivation in [9]. We first present the superpotential in the canonical
prescription:
$\displaystyle\hat{\cal I}^{\alpha\beta}_{C}$ $\displaystyle=$
$\displaystyle{}_{E}\hat{\cal I}^{\alpha\beta}_{C}+{}_{GB}\hat{\cal
I}^{\alpha\beta}_{C}={\kappa}^{-1}\left({\hat{g}^{\rho[\alpha}\overline{D}_{\rho}\xi^{\beta]}}+\hat{g}^{\rho[\alpha}\Delta^{\beta]}_{\rho\sigma}\xi^{\sigma}-\overline{D^{[\alpha}\hat{\xi}^{\beta]}}+\xi^{[\alpha}{}_{E}\hat{d}^{\beta]}\right)$
(A.1) $\displaystyle+$
$\displaystyle{}_{GB}{\hat{\imath}^{\alpha\beta}_{C}}-{}_{GB}\overline{\hat{\imath}^{\alpha\beta}_{C}}+\kappa^{-1}\xi^{[\alpha}{}_{GB}\hat{d}^{\beta]}\,$
where
$\Delta^{\alpha}_{\mu\nu}\equiv\Gamma^{\alpha}_{\mu\nu}-\overline{\Gamma}^{\alpha}_{\mu\nu}={\textstyle{1\over
2}}g^{\alpha\rho}\left(\overline{D}_{\mu}g_{\rho\nu}+\overline{D}_{\nu}g_{\rho\mu}-\overline{D}_{\rho}g_{\mu\nu}\right)$
and222The expression (A.2) is differed from the correspondent one in [9],
where the mistake has been found. Nevertheless, the main results and
conclusions in [9] are not changed; see Corrigendum: Class. Quantum Grav. 27
(2010) 069801 (2pp); Preprint arXiv:0905.3622 [gr-qc] .
$\displaystyle{}_{GB}\hat{\imath}^{\alpha\beta}_{C}=$ $\displaystyle-$
$\displaystyle\frac{2\alpha\sqrt{-g}}{\kappa}\left\\{\Delta^{\rho}_{\lambda\sigma}R_{\rho}{}^{\lambda\alpha\beta}+4\Delta^{\rho}_{\lambda\sigma}g^{\lambda[\alpha}R^{\beta]}_{\rho}+\Delta^{[\alpha}_{\rho\sigma}g^{\beta]\rho}R\right\\}\xi^{\sigma}$
(A.2) $\displaystyle-$
$\displaystyle\frac{2\alpha\sqrt{-g}}{\kappa}\left\\{R_{\sigma}{}^{\lambda\alpha\beta}+4g^{\lambda[\alpha}R^{\beta]}_{\sigma}+\delta_{\sigma}^{[\alpha}g^{\beta]\lambda}R\right\\}\overline{D}_{\lambda}\xi^{\sigma}\,.$
The vector density
$\hat{d}^{\lambda}={}_{E}\hat{d}^{\lambda}+{}_{GB}\hat{d}^{\lambda}$ could be
defined as in [14] or following the prescription of [16]:
$\displaystyle\hat{d}^{\lambda}_{1}$ $\displaystyle=$
$\displaystyle{2\sqrt{-g}}\Delta^{[\alpha}_{\alpha\beta}g^{\lambda]\beta}+4\alpha\sqrt{-g}\left(R_{\sigma}{}^{\alpha\beta\lambda}-4R^{[\alpha}_{\sigma}g^{\lambda]\beta}+\delta^{[\alpha}_{\sigma}g^{\lambda]\beta}R\right)\Delta^{\sigma}_{\alpha\beta}\,,$
(A.3) $\displaystyle\hat{d}^{\lambda}_{2}$ $\displaystyle=$
$\displaystyle{2\sqrt{-g}}\Delta^{[\alpha}_{\alpha\beta}g^{\lambda]\beta}+4\alpha\sqrt{-g}\left(R_{\sigma}{}^{\alpha\beta\lambda}-2R^{[\alpha}_{\sigma}g^{\lambda]\beta}-2\delta^{[\alpha}_{\sigma}R^{\lambda]\beta}+\delta^{[\alpha}_{\sigma}g^{\lambda]\beta}R\right)\Delta^{\sigma}_{\alpha\beta}\,.$
(A.4)
The Einstein part in (A.1) is precisely the KBL superpotential [14, 17], which
in 4D general relativity (GR) for the Minkowski background in the Cartesian
coordinates and with the translation Killing vectors
$\xi^{\alpha}=\delta^{\alpha}_{(\beta)}$ is just the well-known Freud
superpotential [29].
The Belinfante corrected superpotential in EGB gravity is
$\hat{\cal I}^{\alpha\beta}_{B}={}_{E}\hat{\cal
I}^{\alpha\beta}_{B}+{}_{GB}\hat{\cal
I}^{\alpha\beta}_{B}={\kappa}^{-1}\left(\xi^{[\alpha}\overline{D}_{\lambda}\hat{l}^{\beta]\lambda}-\overline{D}^{[\alpha}\hat{l}^{\beta]}_{\sigma}\xi^{\sigma}+\hat{l}^{\lambda[\alpha}\overline{D}_{\lambda}\xi^{\beta]}\right)+{}_{GB}{\hat{\imath}^{\alpha\beta}_{B}}-{}_{GB}\overline{\hat{\imath}^{\alpha\beta}_{B}}$
(A.5)
where
$\hat{l}^{\alpha\beta}=\hat{g}^{\alpha\beta}-\overline{\hat{g}}^{\alpha\beta}$
and
$\displaystyle{}_{GB}\hat{\imath}^{\alpha\beta}_{B}$ $\displaystyle=$
$\displaystyle{\alpha\over\kappa}\overline{D}_{\lambda}\left\\{\hat{R}_{\sigma}{}^{\lambda\alpha\beta}+4g^{\lambda[\alpha}\hat{R}^{\beta]}_{\sigma}+\left[2\hat{R}_{\tau}{}^{\rho\lambda[\alpha}-2\hat{R}^{\rho\lambda}{}_{\tau}{}^{[\alpha}-8\hat{R}^{\lambda}_{\tau}g^{\rho[\alpha}\right.\right.$
(A.6) $\displaystyle+$
$\displaystyle\left.\left.4\hat{R}^{\rho}_{\tau}g^{\lambda[\alpha}+4g^{\rho\lambda}\hat{R}^{[\alpha}_{\tau}+2\hat{R}\left(\delta^{\lambda}_{\tau}g^{\rho[\alpha}-\delta^{\rho}_{\tau}g^{\lambda[\alpha}\right)\right]\overline{g}^{\beta]\tau}\overline{g}_{\rho\sigma}\right\\}\xi^{\sigma}$
$\displaystyle-$
$\displaystyle{2\alpha\over\kappa}\left\\{{\hat{R}}_{\sigma}{}^{\lambda\alpha\beta}+4{g^{\lambda[\alpha}\hat{R}^{\beta]}_{\sigma}}+\delta_{\sigma}^{[\alpha}g^{\beta]\lambda}\hat{R}\right\\}\overline{D}_{\lambda}\xi^{\sigma}\,.$
The Einstein part, ${}_{E}\hat{\cal I}^{\alpha\beta}_{B}$, being constructed
in arbitrary $D$ dimensions, has precisely the form of the Belinfante
corrected superpotential in 4D GR [30]. In the Minkowski background in the
Cartesian coordinates and with the translation Killing vectors
${}_{E}\hat{\cal I}^{\alpha\beta}_{B}$, it transforms to the well-known
Papapetrou superpotential [31].
Lastly, the superpotential in the field-theoretical derivation in EGB gravity
is
$\displaystyle\hat{\cal I}_{S}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle{}_{E}\hat{\cal I}_{S}^{\alpha\beta}+{}_{GB}\hat{\cal
I}_{S}^{\alpha\beta}={\kappa}^{-1}\left(\xi_{\nu}\overline{D}^{[\alpha}\hat{h}^{\beta]\nu}-\xi^{[\alpha}\overline{D}_{\nu}\hat{h}^{\beta]\nu}+\xi^{[\alpha}\overline{D}^{\beta]}\hat{h}-\hat{h}^{\nu[\alpha}\overline{D}_{\nu}\xi^{\beta]}+{\textstyle{1\over
2}}\hat{h}\overline{D}^{[\alpha}\xi^{\beta]}\right)$ (A.7) $\displaystyle+$
$\displaystyle{{4\over
3}}\left(2\xi_{\sigma}\overline{D}_{\lambda}\hat{N}_{{GB}}^{\sigma[\alpha|\beta]\lambda}-\hat{N}_{{GB}}^{\sigma[\alpha|\beta]\lambda}\overline{D}_{\lambda}\xi_{\sigma}\right)\,.$
where
$\hat{h}_{\alpha\beta}={\sqrt{-\overline{g}}}(g_{\alpha\beta}-\overline{g}_{\alpha\beta})$
and
$\displaystyle\hat{N}^{\rho[\lambda|\mu]\nu}_{GB}=$ $\displaystyle-$
$\displaystyle\frac{3\alpha\sqrt{-\overline{g}}}{4\kappa}\left\\{h^{\sigma}_{\sigma}\left[\overline{g}^{\nu[\lambda}\overline{g}^{\mu]\rho}\overline{R}+2\overline{g}{}^{\rho[\lambda}\overline{R}{}^{\mu]\nu}-2\overline{g}{}^{\nu[\lambda}\overline{R}{}^{\mu]\rho}-\overline{R}^{\rho\nu\lambda\mu}\right]+\left(\overline{g}^{\rho[\lambda}h^{\mu]\nu}-\overline{g}^{\nu[\lambda}h^{\mu]\rho}\right)\overline{R}\right.$
$\displaystyle+$ $\displaystyle
2\left(h{}^{\nu[\lambda}\overline{R}{}^{\mu]\rho}-h{}^{\rho[\lambda}\overline{R}{}^{\mu]\nu}\right)+2\left(h{}^{\sigma[\lambda}\overline{g}^{\mu]\rho}\overline{R}{}^{\nu}_{\sigma}-h{}^{\sigma[\lambda}\overline{g}^{\mu]\nu}\overline{R}{}^{\rho}_{\sigma}\right)+2\left(h{}^{\sigma\rho}\overline{g}^{\nu[\lambda}\overline{R}{}^{\mu]}_{\sigma}-h{}^{\sigma\nu}\overline{g}^{\rho[\lambda}\overline{R}{}^{\mu]}_{\sigma}\right)$
$\displaystyle-$
$\displaystyle\left.2\overline{g}^{\nu[\lambda}\overline{g}^{\mu]\rho}h^{\sigma}_{\tau}\overline{R}_{\sigma}^{\tau}+4\left(\overline{R}{}_{\sigma}{}^{[\lambda\mu][\rho}h{}^{\nu]\sigma}+\overline{R}{}_{\sigma}{}^{[\rho\nu][\lambda}h{}^{\mu]\sigma}\right)+2h_{\sigma\tau}\left(\overline{R}{}^{\sigma\nu\tau[\lambda}\overline{g}^{\mu]\rho}-\overline{R}{}^{\sigma\rho\tau[\lambda}\overline{g}^{\mu]\nu}\right)\right\\}\,.$
One obtains from (A.7) the Deser-Tekin superpotential [32] if one chooses the
AdS background. Again, doing simplifications in 4 dimensions as above, one
obtains the Papapetrou superpotential [31] (note, see [8], that in 4D GR the
Belinfante and field-theoretical approaches give the same result). Under
weaker restrictions, say, to AdS/dS backgrounds in 4D GR, the superpotential
(A.7) goes to the Abbott-Deser expression [33].
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|
arxiv-papers
| 2009-11-28T19:27:39 |
2024-09-04T02:49:06.752788
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.N.Petrov",
"submitter": "Alexander Petrov Nikolaevich",
"url": "https://arxiv.org/abs/0911.5419"
}
|
0911.5512
|
# On some fundamental problems of the theory of gravitation
L. V. Verozub, Kharkov National University
###### Abstract
Cosmological observations indicate that the Einstein equation may not be
entirely correct to describe gravity. However, numerous modifications of these
equations usually do not affect foundations of the theory. In this paper two
important issue that lead to a substantial revision of the theory are
considered :
1\. The significance of relativity of space-time geometry with respect to
measuring instruments for theory of gravitation.
2\. The gauge transformations of the field variables in correct theory of
gravitation.
## 1 Relativity of Space-Time
Einstein’s theory of gravity is a realization of the idea of the relativity of
the properties of space-time with respect to the distribution of matter.
However, before the advent of Einstein’s theory, Henri Poincaré showed that
the properties of space and time are also relative to the properties of the
used measuring instruments. Of course now it can be said also about the
properties of space-time too. However, these convincing arguments have never
been implemented in physical theory.
We can make a step towards the realization of this idea, if we will pay
attention that the properties of measuring instruments are one of the
characteristics of the reference frame used. We can, therefore, expect that we
deal with the manifestation of a fundamental property of physical reality —
with space-time relativity with respect to the reference frame used.
The following simple example shows that this rather unexpected statement makes
sense. Consider two reference frames, and two observers which proceed from the
notion of the relativity of space-time in the sens of Berkley-Leibnitz-Mach-
Poincaré (BLMP). Let the reference body of the first, inertial frame (IRF), is
associated with the surface of a non-rotating planet, and the reference body
of the second frame formed by a set of material points, falling freely under
the influence of the planet gravity. (It can be named by proper reference
frame of the given force field (PRF)).
The observer, located in the first, inertial, frame of reference, of course
will examine the fall of test bodies as happens under the action of a force
field $\mathcal{F}$ in the Minkowski space-time, the source of which is the
planet. He sees no need to explain the motion of test bodies with curvature.
However, the observer, located in the second reference frame, does not detect
this force field. Instead, he observes rapprochement of points of the
reference body of his frame which for him are points of his physical space.
If he is denied the opportunity to see the planets and stars, it seems
impossible for him to find another explanation of this fact, which is
different from the generally accepted explanation — of an evidence of space-
time curvature.
Thus, if an observer in a IRF can consider space-time as flat, then the
observer in the PRF of the force field $\mathcal{F}$, who proceeds from
relativity of space and time in the BLMP meaning, is forced to consider it as
an non-Euclidean.
Some quantitative results on the metric of space-time in PRFs were obtained
earlier by the author [4]. Namely, we postulate that space-time $E$ in
inertial frames is the Minkowski one, according to the spacial relativity.
From our point of view, space-time geometry and properties of the reference
frame do not have meaning by themselves. Therefore, this postulate means that
only a complex “Minkowski space-time $E$ \+ inertial reference system” makes a
physical sense. Starting from this postulate and based on the relativity of
space-time, it is possible to find the line element of space-time $V$ in a PRF
of any given in the $E$ force field.
Consider a PRF, the reference body of which formed by material points with
masses $m$ moving under the action of the force field $\mathcal{F}$. If we
proceed from relativity of space and time in the BLMP sense, then the line
element of space-time in PRFs can be expected to have the following form [4]
$ds=-(mc)^{-1}\,dS(x,dx).,$ (1)
where $dS=\mathcal{L}(x,\dot{x})dt$, and $\mathcal{L}(x,\dot{x})$ is a
Lagrange function describing in Minkowski space-time the motion of the
identical point masses $m$.
## 2 Examples
1\. Suppose that in the Minkowski space-time gravitation can be described as a
tensor field $\psi_{\alpha\beta}(x)$ in $E$, and the Lagrangian, describing
the motion of a test particle with the mass $m$ in $E$ is given by the form
$\mathcal{L}=-mc[g_{\alpha\beta}(\psi)\;\dot{x}^{\alpha}\;\dot{x}^{\beta}]^{1/2},$
(2)
where $\dot{x}^{\alpha}=dx^{\alpha}/dt$ and $g_{\alpha\beta}$ is a symmetric
tensor whose components are functions of $\psi_{\alpha\beta}$ [2].
If particles move under influence of the force field $\psi_{\alpha\beta}(x)$,
then according to (1) the space-time line element in PFRs of this field takes
the form
$ds^{2}=g_{\alpha\beta}(\psi)\;dx^{\alpha}\;dx^{\beta}$ (3)
Consequently, the space-time in such PRFs is Riemannian $V$ with curvature
other than zero. The tensor $g_{\alpha\beta}(\psi)$ is a space-time metric
tensor in the PRFs.
Viewed by an observer located in the IRF, the motion of the particles, forming
the reference body of the PRF, is affected by the force field
$\psi_{\alpha\beta}$. Let $x^{i}(t,\chi)$ be a set of the particles paths,
depending on the parameter $\chi$. Then, for the observer located in the IRF
the relative motion of a pair of particles from the set is described in non-
relativistic limit by the differential equations [3]
$\frac{\partial^{2}n^{i}}{\partial t^{2}}+\frac{\partial^{2}U}{\partial
x^{i}\partial x^{k}}n^{k}=0,$ (4)
where $n^{k}=\partial x^{k}/\partial{\chi}$ and $U$ is the gravitational
potential.
However, the observer in a PRF of this field will not feel the existence of
the field.The presence of the field $\psi_{\alpha\beta}$ will be displayed for
him differently — as space-time curvature which manifests itself as a
deviation of the world lines of nearby points of the reference body.
For a quantitative description of this fact it is natural for him to use the
Riemannian normal coordinates. 111This and the above consideration does not
depend on the used coordinate system, it can be performed by a covariant
method. In these coordinates spatial components of the deviation equations of
geodesic lines are
$\frac{\partial n^{i}}{\partial t^{2}}+R_{0k0}^{i}n^{k}=0,$ (5)
where $R_{0k0}^{i}$ are the components of the Riemann tensor. In the Newtonian
limit these equations coincide with (4).
Thus, in two frames of reference being used we have two different descriptions
of particles motion — as moving under the action of a force field in the
Mankowski space-time, and as moving along the geodesic line in a Riemann
space-time with the curvature other than zero.
2\. Another, rather unexpected example, give the recent results on the motion
of small elements of a perfect isentropic fluid [4].
Instead of the traditional continuum assumption, the behavior of the fluid
flow can be considered as the motion of a finite mumber of particles uder the
influence of interparticles forces which mimic effects of pressure, viscosity,
etc. [5]. Owing to replacement of integration by summation over a number of
particles, continual derivatives become simply time derivatives along the
particles trajectories. The velocity of the fluid at a given point is the
velocity of the particle at this point. The continuity equation is always
fulfilled and can consequently be omitted. Owing to such discratization the
motion of particles is governed by means of solutions of ordinary differential
equations of classical or relativistic dynamics.
In [4] it was shown that the following Lagrangian describes the motion of
small elements of a perfect isentropic fluid in adiabatic processes is given
by
$L=-mc\left(G_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}\right)^{1/2}d\lambda.$
(6)
In this equation
$G_{\alpha\beta}=\varkappa^{2}\eta_{\alpha\beta}$,$\eta_{\alpha\beta}$ is the
metric tensor of the space-time $E$,
$\varkappa=\frac{w}{nmc^{2}}=1+\frac{\varepsilon}{\rho c^{2}}+\frac{P}{\rho
c^{2}},$ (7)
$\varepsilon$ is the fluid density energy, $m$ is the mass of the fluid
particles, $c$ is speed of light, and $\rho=mn$, $n$ is the particles number
density, $P$ is the pressure i the fluid, $\lambda$ is a parameter along
4-pathes of particles.
In an inertial reference drame (i.e. in Minkowski space-time $E$) we can set
the parameter $\lambda=\sigma$ which yields the following Lagrange equations:
$\frac{d}{d\sigma}\left(\varkappa
u_{\alpha}\right)-\frac{\partial\varkappa}{\partial x^{\alpha}}=0$ (8)
where $u_{\alpha}=\eta_{\alpha\beta}u^{\beta}$, and
$u^{\alpha}=dx^{\alpha}/d\sigma$. For adiabatic processes [6]
$\frac{\partial}{\partial
x^{\alpha}}\left(\frac{w}{n}\right)=\frac{1}{n}\frac{\partial P}{\partial
x^{\alpha}},$ (9)
and we arrive at the equations of the motion of the set of the particles in
the form
$w\frac{du_{\alpha}}{d\sigma}+u_{\alpha}u^{\beta}\frac{\partial P}{\partial
x^{\beta}}-\frac{\partial P}{\partial x^{\alpha}}=0.$ (10)
where $du_{\alpha}/d\sigma=\left(\partial u_{\alpha}/\partial
x^{\epsilon}\right)u^{\epsilon}.$ It is the general accepted relativistic
equations of the motion of fluid [6].
In a comoving reference frame the space-time the line element is of the form
$ds^{2}=G_{\alpha\beta}dx^{\alpha}dx^{\beta}.$ (11)
In this case the element of the proper time is $ds$. After the seting
$\lambda=s$, the Lagrangian equation of the motion takes the standard form of
a congruence of geodesic lines :
$\frac{du^{\alpha}}{ds}+\Gamma_{\beta\gamma}^{\alpha}u^{\beta}u^{\gamma}=0,$
(12)
where $du_{\alpha}/ds=\left(\partial u_{\alpha}/\partial
x^{\epsilon}\right)u^{\epsilon}$, $u^{\alpha}=du^{\alpha}/ds$, and
$\Gamma_{\beta\gamma}^{\alpha}=\frac{1}{2}G^{\alpha\epsilon}\left(\frac{\partial
G_{\epsilon\beta}}{\partial x^{\gamma}}+\frac{\partial
G_{\epsilon\gamma}}{\partial x^{\beta}}-\frac{\partial
G_{\beta\gamma}}{\partial x^{\epsilon}}\right).$ (13)
In the Cartesian coordinates
$\Gamma_{\alpha\beta}^{\gamma}=\frac{1}{\varkappa}\left(\frac{\partial\varkappa}{\partial
x^{\gamma}}\delta_{\beta}^{\alpha}+\frac{\partial\varkappa}{\partial
x^{\beta}}\delta_{\gamma}^{\alpha}-\eta^{\alpha\epsilon}\frac{\partial\varkappa}{\partial
x^{\epsilon}}\eta_{\beta\gamma}\right),$ (14)
so that
$\Gamma_{00}^{1}=-\frac{1}{\rho c^{2}}\frac{\partial P}{\partial x^{1}}$ (15)
In the spherical coordinates the scalar curvarure $R$ is given by
$R=\frac{6}{\varkappa^{3}r^{2}}(r^{2}\varkappa^{\prime})^{\prime},$ (16)
where the prime denotes a derivative with respect to $r$.
Therefore, the motion of small elements of the fluid in a comoving reference
frame can be viewed as the motion in a Riemannian space-time with a nonzero
curvature.
Of course, (1) refers to any classical field $\mathcal{F}$. For instance,
space-time in PRFs of an electromagnetic field is Finslerian. However, since
$ds$, in this case, depends on the mass and charge of the particles forming
the reference body, this fact is not of great significance.
Thus any force field can be considered based on the aggregate ”IRF + Minkowski
space”, and based on the aggregate ”PRF + non-Euclidean space-time with metric
(1)” From this point of view of geometrization of gravity is the second
possibility, which was discovered by Einstein’s intuition.
It is important to realize that the relativity of space-time geometry to the
frame of reference is the same important and fundamental property of physical
relativity as relativity to act of measurement, the physical realization of
which is quantum mechanics. Full implementation of these ideas can have far-
reaching implications for fundametal physics.
## 3 Gravity equations and gauge-invariance
In the theory of gravitation, the equations of motion of test particles play a
fundamental role. Notion of ”gravitational field” emerged as something
necessary to correctly describe the motion of bodies. The values that appear
in the equations of motion, become the main characteristic of the field. The
field equations have emerged as a tool for finding these values for a given
distribution of masses.
All this is very similar to classical electrodynamics. It is very important in
this case that the equations of motion of test charges are invariant under
gauge transformations of 4-potentials. For this reason, all 4-potentials,
obtained from a given by a gauge transformation, describe the same field. That
is why the field equations of classical electrodynamics are invariant under
gauge transformations.
Einstein’s equations of the motions of test particles in gravitational field
are also invariant with respect to some class of transormations of the field
variables in any given coordinate system — with respect to geodesic
transformations of Christoffel symbols (or metric tensor). [7] Such
transformations for the Christoffel symbols are of the form
$\overline{\Gamma}_{\beta\gamma}^{\alpha}(x)=\Gamma_{\beta\gamma}^{\alpha}(x)+\delta_{\beta}^{\alpha}\
\phi_{\gamma}(x)+\delta_{\gamma}^{\alpha}\phi_{\beta}(x),$ (17)
where $\phi_{\alpha}(x)$ is a continuously differentiable vector field. (The
transformations for the metric tensor are solutions of some complicate partial
differential equations).
Consequently, all Christoffel symbols obtained from a given by geodesic
transformations, describe the same gravitational field. The equations for
determining the gravitational field must be invariant under such
transformations, and the physical meaning can only have values which are
invariant under geodesic transformations.
However, Einstein’s gravitational equations are not consistent completely with
the requirements which imposes on them the main hypothesis of the motion of
test particles along geodesics, because they are not geodesically invariant
[8].
Therefore, we can assume that in a fully correct theory of gravity, based on
the hypothesis of the motion of test particles along geodesics, geodesic
transformations should play the role of gauge transformations, and coordinate
transformation should play the same role as in electrodynamics.
Einstein equations are in good agreement with observations in weak and
moderately strong fields. Therefore, if there are more correct equation of
gravitation, then deriving from them physical results should differ observably
from Einstein’s equations only in strong fields.
Simplest vacuum equation of this kind were first proposed (from a different
point of view) in [9], and discussed in greater detail in [4], their physical
implications discussed in [10] \- [12], and the equations in the presence of
matter - in [13]. They are some geodesic-invariant modification of Einstein’s
equations.
From a theoretical point of view, the most satisfactory are the vacuum
equations.
They predict some fundamentally new physical consequences which can be tested
experimentally.
Under geodesic transformations the Ricci tensor $R_{\alpha\beta}$ of space-
time $V$ in PRFs of gravitational field transforms as follows:
$\overline{R}_{\alpha\beta}=R_{\alpha\beta}+(n-1)\psi_{\alpha\beta},$ (18)
where
$\psi_{\alpha\beta}=\psi_{\alpha;\beta}-\psi_{\alpha}\psi_{\beta,}$ (19)
and a semicolon denotes a covariant differetiation in $V$. Therefore, the
simpest generalization of the Einstein equations is of the form
$R_{\alpha\beta}+(n-1)\Gamma_{\alpha\beta}=0,$ (20)
where $\Gamma_{\alpha\beta}$ is a tesor transformed under geodesic
transformations as follows
$\overline{\Gamma}_{\alpha\beta}=\Gamma_{\alpha\beta}-\psi_{\alpha\beta}.$
(21)
Due to the fact that our space-time is a bimetric, there exists a vector field
$Q_{\alpha}=\Gamma_{\alpha}-\overset{\circ}{\Gamma}_{\alpha}$ (22)
where $\Gamma_{\alpha}=\Gamma_{\alpha\beta}^{\beta}$ ,
$\overset{\circ}{\Gamma}_{\alpha}=\overset{\circ}{\Gamma}_{\alpha\beta}^{\beta}$
, $\Gamma_{\alpha\beta}^{\gamma}$ and
$\overset{\circ}{\Gamma}_{\alpha\beta}^{\gamma}$ are the Christoffel symbols
in $V$ and $E$, respectively.
Under geodesic transformations in $V$ the quantities $\Gamma_{\alpha}$ are
transformed as follows:
$\overline{\Gamma}_{\alpha}=\Gamma_{\alpha}+(n+1)\,\psi_{\alpha}$ (23)
For this reason, a tensor object
$A_{\alpha\beta}=Q_{\alpha;\beta}-Q_{\alpha}Q_{\beta},$ (24)
where $Q_{\alpha;\beta}$ is a covariant derivative of $Q_{\alpha}$ in $V$, has
the same transformation properties under geodesic transformations as must have
the above vector field $\Gamma_{\alpha\beta}$.
The line element of space-time in PRFs was obtained from the Lagrangian motion
of test particles in the Minkowski space-time $E$. If we want to find the
equation of gravity in space-time $E$, you must realize that in this space,
the Christoffel symbols $\Gamma_{\alpha\beta}^{\gamma}$ can be regarded as
components of the tensor
$\Gamma_{\alpha\beta}^{\gamma}-\overset{\circ}{\Gamma}_{\alpha\beta}^{\gamma}$
in the Cartesian coordinate system, i.e. as components of
$\Gamma_{\alpha\beta}^{\gamma}$, where the ordinary derivatives replaced by
covariant in the metric of space-time $E$. (Just as in bimetric Rosen’s theory
[14]).
Given this, we arrive at the conclusion that the equation
$R_{\alpha\beta}-A_{\alpha\beta}=0$ (25)
is the simplest geodesic invariant modification of the vacuum Einstein
equations, considered from the point of view of flat space-time.
These equations can be written in another form. The simplest geodesic-
invariant object in $V$ is a Thomas symbols:
$\Pi_{\alpha\beta}^{\gamma}=\Gamma_{\alpha\beta}^{\gamma}-\frac{1}{n+1}\left(\delta_{\alpha}^{\eta}\Gamma_{\beta}+\delta_{\beta}^{\eta}\Gamma_{\alpha}\right).$
(26)
It is not a tensor. However, from point of view of flat space-time $E$, they
can be considered as components of the tensor
$B_{\alpha\beta}^{\gamma}=\Pi_{\alpha\beta}^{\gamma}-\overset{\circ}{\Pi}_{\alpha\beta}^{\gamma}$,
where $\overset{\circ}{\Pi}_{\alpha\beta}^{\gamma}$ is the Thomas simbols in
$E$. In another words, $B_{\alpha\beta}^{\gamma}$ can be considered as the
Thomas symbols where derivatives replaced by the covariant ones with respect
to the metric $\eta_{\alpha\beta}$. This geodesic-invariant tensor can be
named by strength tensor of gravitational field.
The above gravitation equation can be written by tensor
$B_{\alpha\beta}^{\gamma}$ as follows:
$\bigtriangledown_{\gamma}B_{\alpha\beta}^{\gamma}-B_{\alpha\delta}^{\gamma}B_{\beta\gamma}^{\delta}=0.$
(27)
where $\bigtriangledown$ denotes a covariant derivative in $E$.
The physical consequences following from these equations do not contradict any
observational data, however, lead to some unexpected results, which allow to
us to test the theory. The first result is that they predict the existence of
supermassive compact objects without event horizon which are an alternative to
supermassive black holes in the centers of galaxies. The second result is that
they provide a simple and natural explanation for the fact of an acceleration
of the universe as of a consequence of the gravity properties.
## 4 Remaks on the equations inside matter
We can not claim that the particles inside the material medium move along
geodesics. The exception is the case of dust matter and perfect fluid.
Consequently, it is unclear whether the field equations inside the matter to
be a generalization of the geodesic equations of Einstein. However, such
equations have been proposed in the work [13]. Comparison of the results
obtained from them with observations of the binary pulsar PSR 1913+16 shows
good agreement with observations. Despite this, doubts as to their correctness
are still remain. The problem is that the writing of generalization of the
equations in the matter requires significantly narrow the class of admissible
geodesic transformations of the metric tensor of space-time $V$. It is not
clear whether such space-time is Riemannian. It is possible, geodesic
invariance is violated in a material medium. For this reason, we do not
consider these equations here in more detail, assuming that this is still a
subject for further research.
## References
* [1] L. Verozub, Ann. Phys. Berlin, 17, 28, (2008).
* [2] W. Thirring, Ann. Phys., 16, 96 (1961).
* [3] Ch. Misner, K. Thorne, J. Wheeler, Gravitation, (San Francisco, Feeman and Comp.) (1973).
* [4] L. Verozub, Int. Journ. Mod. Phys. D, 17, 337 (2008).
* [5] J. Monaghan and D. Rice, Month. Not. Royal Astron. Soc. , 328, 381 (2001).
* [6] L. Landau, and E. Lifshitz, Fluid Mechanics, (Pergamon, Oxford) (1987)
* [7] L. Eisenhart, Riemannian geometry, (Princeton, Univ. Press) (1950).
* [8] A. Petrov, Einstein Spaces , (New-York-London, Pergamon Press. (1969).
* [9] L. Verozub, L. Phys. Lett. A, 156, 404 (1991).
* [10] L. Verozub, Astr. Nachr., 317, 107 (1996).
* [11] L. Verozub, & A. Kochetov, Astr. Nachr., 322, 143 (2001).
* [12] L. Verozub, Astr. Nachr., 327, 355 (2006).
* [13] L. Verozub & A. Kochetov, Grav and Cosmol., 6, 246 (2000).
* [14] H. Treder, Gravitationstheorie und Äquivalenzprinzip, (Berlin, Akademie-Verlag-Berlin) (1971).
|
arxiv-papers
| 2009-11-29T20:11:19 |
2024-09-04T02:49:06.760441
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leonid V. Verozub",
"submitter": "Leonid V. Verozub",
"url": "https://arxiv.org/abs/0911.5512"
}
|
0911.5604
|
# Some considerations on the nonabelian tensor square of crystallographic
groups
Ahmad Erfanian Mathematics Department and
Centre of Excellence in Analysis on Algebraic Structures
Ferdowsi University of Mashhad
P.O.Box 1159, 91775, Mashhad, Iran erfanian@math.um.ac.ir , Francesco G.
Russo Laboratorio di Dinamica e Geotecnica - Strega
Universitá del Molise
via Duca degli Abruzzi, Termoli (CB) francescog.russo@yahoo.com and Nor
Haniza Sarmin Department of Mathematics, Faculty of Science
Universiti Teknologi Malaysia
81310 UTM Johor Bahru, Johor, Malaysia nhs@utm.my
###### Abstract.
The nonabelian tensor square $G\otimes G$ of a polycyclic group $G$ is a
polycyclic group and its structure arouses interest in many contexts. The same
assertion is still true for wider classes of solvable groups. This motivated
us to work on two levels in the present paper: on a hand, we investigate the
growth of the Hirsch length of $G\otimes G$ by looking at that of $G$, on
another hand, we study the nonabelian tensor product of pro–$p$–groups of
finite coclass, which are a remarkable class of solvable groups without
center, and then we do considerations on their Hirsch length. Among other
results, restrictions on the Schur multiplier will be discussed.
###### Key words and phrases:
Hirsch length, Schur multiplier, crystallographic groups, pro–$p$–groups of
finite coclass, Bieberbach groups.
Mathematics Subject Classification 2010: 20J99, 20F18.
## 1\. Introduction
The $nonabelian$ $tensor$ $square$ $G\otimes G$ of a group $G$ is the group
generated by the symbols $x\otimes y$ and subject to the relations
(1.1) $xy\otimes z=(^{x}y\otimes\hskip 1.13791pt^{x}z)(x\otimes z)\ \ \
\textrm{and}\ \ \ x\otimes zt=(x\otimes z)\ (^{z}x\otimes\hskip
1.13791pt^{z}t)$
for all $x,y,z,t\in G,$ where $G$ acts on itself via conjugation
${}^{x}y=xyx^{-1}$. In particular, if $G$ is abelian and acts trivially on
itself, we have the usual abelian tensor product $G\otimes_{\mathbb{Z}}G$. The
group $G\wedge G=G\otimes G/\nabla(G)$ is called $nonabelian$ $exterior$
$square$ of $G$, where $\nabla(G)=\langle x\otimes x\ |\ x\in G\rangle$. From
[5] the maps $\kappa:x\otimes y\in G\otimes G\longmapsto[x,y]\in[G,G]$ and
$\kappa^{\prime}:x\wedge y\in G\wedge G\longmapsto[x,y]\in[G,G]$ are
epimorphisms and the topological meaning of $\ker\kappa=J_{2}(G)$ is described
in [6]. Still from [5] the following diagram is commutative with exact rows
and central extensions as columns:
(1.2) $\setcounter{MaxMatrixCols}{11}\begin{CD}11\\\ @V{}V{}V@V{}V{}V\\\
H_{3}(G)@>{}>{}>\Gamma(G^{ab})@>{\psi}>{}>J_{2}(G)@>{}>{}>H_{2}(G)@>{}>{}>1\\\
\Big{\|}\Big{\|}@V{}V{}V@V{}V{}V\\\
H_{3}(G)@>{}>{}>\Gamma(G^{ab})@>{\psi}>{}>G\otimes G@>{}>{}>G\wedge
G@>{}>{}>1\\\ @V{\kappa}V{}V@V{\kappa^{\prime}}V{}V\\\ [G,G]@=[G,G]\\\
@V{}V{}V@V{}V{}V\\\ 11\\\ \end{CD}$
where $H_{2}(G)=\ker\kappa^{\prime}$ is the second integral homology group of
$G$, $H_{3}(G)$ is the third integral homology group of $G$ and $\Gamma$ is
the Whitehead’s quadratic functor in [5, Section 2]. We note that $H_{2}(G)$
is exactly the Schur multiplier $M(G)$ of $G$.
After the initial work [7], many authors investigated the structure of
$G\otimes G$ by looking at that of $G$ and in the last times there is a
significant production which is devoted to the classes $\mathfrak{P}$ of all
polycyclic groups, $\mathfrak{F}$ of all finite groups and $\mathfrak{S}$ of
all solvable groups (see [3, 4, 8, 16, 21]). In a solvable group $G$ we recall
that the number of infinite cyclic factors $h(G)$ is an invariant, called
Hirsch length, or torsion–free rank, of $G$ (see [14, pp.14, 15, 16, 85]). If
$G\in\mathfrak{P}$, we have $h(G)=0$ if and only if
$G\in\mathfrak{P}\cap\mathfrak{F}$. Now, if $G$ is abelian, then $G\otimes G$
is abelian by [21, Theorem 3.1]; if $G\in\mathfrak{P}$, then $G\otimes
G\in\mathfrak{P}$ (see [4, 8, 16]) and, so far as we know, the structure of
$G\otimes G$ is widely described in terms of the upper central series of $G$.
For instance, [12] classifies $G\otimes G$, when $G$ is a 2–generator 2–group,
and so, $G$ is a particular type of polycyclic group with nontrivial center.
[19] describes $G\otimes G$, where $G$ is an infinite nonabelian 2–generator
nilpotent group of class 2, and so, $G$ is still a polycyclic group with
nontrivial center. There are several contributions on this line of research
but it is hard to find information on $G\otimes G$ when $G$ is a polycyclic
group with trivial center: we found the initial idea in [2] and a recent
interest in [3, 8, 9, 15].
The aim of the present work is to detect the structure of $G\otimes G$, when
$G$ is a polycyclic group with trivial center, or more generally an infinite
solvable group with trivial center, starting from bounds on $h(G\otimes G)$
and $h(G)$. The absence of literature on such a line of investigation has
motivated us to write the present paper. On another hand, R. F. Morse has
kindly pointed out (see [17]) that the same question was posed by C. Rover at
the Conference on Computational Group Theory and Cohomology at the Harlaxton
College (Harlaxton Lincolnshire, UK) in 2008. We end this introduction, noting
that the terminology and the notations of the present paper are standard and
can be found in [5, 6, 7, 11].
## 2\. The growth of the Hirsch length in the nonabelian tensor square
The following (unpublished) lemma was communicated to us by D. Ramras and
describes some classical situations, which we may encounter, when we deal with
the presentations of polycyclic groups. Further details can be found in [18].
###### Lemma 2.1.
Let $l,p,k,m,n_{1},n_{2},\ldots,n_{m}$ be integers. Consider an extension of
groups $1\rightarrow A\rightarrowtail\Gamma\twoheadrightarrow Q\rightarrow 1$
in which $A$ is a finitely generated abelian group and $Q$ is finite. If
(2.1) $Q=\langle q_{1},\ldots,q_{l}\ |\
r_{1}(q_{1},\ldots,q_{l})=\ldots=r_{p}(q_{1},\ldots,q_{l})=1\rangle$
for some words $r_{1},\ldots,r_{p}$ in the free group on $l$ letters and
(2.2) $A=\langle a_{1},\ldots,a_{k+m}\ |\ [a_{i},a_{j}]=1\ (1\leq i\leq j\leq
k+m),\ a^{n_{1}}_{1}=\ldots=a^{n_{m}}_{m}=1\ (1\leq n_{1}\leq\ldots\leq
n_{m})\rangle,$
then for some words $w_{i}$ and $u_{ij}$ (not uniquely determined) in the free
group on $k+m$ letters,
(2.3)
$\Gamma=\langle\alpha_{1},\ldots,\alpha_{k+m},\gamma_{1},\ldots,\gamma_{l}\ |\
r_{1}(\gamma_{1},\ldots,\gamma_{l})=w_{1}(\alpha_{1},\ldots,\alpha_{k+m}),\ldots,$
(2.4)
$r_{p}(\gamma_{1},\ldots,\gamma_{l})=w_{p}(\alpha_{1},\ldots,\alpha_{k+m}),[\alpha_{i},\alpha_{j}]=1,\alpha^{n_{j}}_{j}=1,\gamma_{i}\alpha_{j}\gamma_{i}^{-1}=u_{ij}(\alpha_{1},\ldots,\alpha_{k+m}),$
(2.5) $\ (1\leq i\leq j\leq k+m)\rangle.$
###### Proof.
To begin, we must specify the words $u_{ij}$ and $w_{i}$. Choose elements
$\widetilde{q}_{i}\in\Gamma$ lying over $q_{i}\in Q$. Since $A$ is normal in
$\Gamma$, we know that $\widetilde{q}_{i}a_{j}\widetilde{q}^{-1}_{i}\in A$,
and hence $\widetilde{q}_{i}a_{j}\widetilde{q}^{-1}_{i}=u_{ij}(a_{1},\ldots
a_{k+m})$ for some word $u_{ij}$. Next, since $r_{i}(q_{1},\ldots,q_{l})=1$ in
$Q$, we know that $r_{i}(\widetilde{q}_{1},\ldots,\ldots{q}_{l})\in A$, and
hence $r_{i}(q_{1},\ldots,q_{l})=w_{i}(a_{1},\ldots,a_{k+m})$ for some word
$w_{i}$. Now, let $\widetilde{\Gamma}$ denote the group presented by
(2.3)–(2.5), and let $\widetilde{A}$ denote the subgroup generated by
$\alpha_{1},\ldots,\alpha_{k+m}$. Let
$\Phi:\widetilde{\Gamma}\rightarrow\Gamma$ be the homomorphism defined by
$\Phi(\alpha_{i})=a_{i}$ and $\Phi(\gamma_{i})=\widetilde{q}_{i}$. Then $\Phi$
is surjective, and its restriction to $\widetilde{A}$ is a surjection onto
$A\leq\Gamma$. The third set of relations in (2.3)–(2.5) ensures that
$\widetilde{A}$ is normal in $\widetilde{\Gamma}$, and we define
$\widetilde{Q}=\widetilde{\Gamma}/\widetilde{A}$. The map $\Phi$ induces a
surjection $\widetilde{\Phi}:\widetilde{Q}\twoheadrightarrow Q$, and we have a
commutative diagram
(2.6) $\begin{CD}\
1@>{}>{}>\widetilde{A}@>{}>{}>\widetilde{\Gamma}@>{}>{}>\widetilde{Q}@>{}>{}>1\\\
@V{}V{}V@V{\Phi}V{}V@V{\widetilde{\Phi}}V{}V\\\ \ 1@>{}>{}>A@>{}>{}>\Gamma
@>{}>{}>Q@>{}>{}>1\\\ \end{CD}$
The map $\widetilde{\Gamma}\rightarrow\widetilde{Q}$ induces a surjection from
the free group on the generators $\gamma_{i}$ onto $\widetilde{Q}$, and this
surjection factors through the quotient group
$\langle\gamma_{1},\ldots\gamma_{l}\ |\
r_{i}(\gamma_{1},\ldots,\gamma_{l})=1\rangle\simeq Q$. Hence we have a
surjection $Q\twoheadrightarrow\widetilde{Q}$, meaning that $\widetilde{Q}$ is
a finite group of order at most $|Q|$. The existence of the surjection
$\widetilde{\Phi}:\widetilde{Q}\twoheadrightarrow Q$ now shows that both of
these surjections must in fact be isomorphisms. Next, we show that the map
$\widetilde{A}\rightarrow A$ is injective. Each element
$\alpha\in\widetilde{A}$ has the form
$\alpha^{p_{1}}_{1}\alpha^{p_{2}}_{2}\ldots\alpha^{p_{k+m}}_{k+m}$ for some
integers $p_{i}$. Our presentation for $A$ shows that, if $\Phi(\alpha)=0$,
then $p_{i}$ is a multiple of $n_{i}$ for $1\leq i\leq m$, and $p_{i}=0$ for
$i>m$. But such elements are already trivial in $\widetilde{\Gamma}$, so
$\phi$ is injective when restricted to $\widetilde{A}$. We have now shown that
the two outer maps in (2.6) are isomorphisms, and the 5-lemma shows that
$\Phi$ is an isomorphism as well. ∎
Lemma 2.1 can be specialized in various ways. For instance, assume that the
cyclic group $C_{n}=\langle t\ |\ t^{n}=1\rangle$ of order $n>1$ is equal to
$Q$; the free abelian group
$\mathbb{Z}^{n-1}=\underbrace{\mathbb{Z}\times\ldots\times\mathbb{Z}}_{(n-1)-\textrm{times}}=\langle
a_{1},\ldots,a_{n-1}\ |\ [a_{i},a_{j}]=1;1\leq i,j\leq n-1\rangle$ of rank
$n-1$ is equal to $A$; $C_{n}$ acts on $\mathbb{Z}^{n-1}$ via the following
homomorphism
(2.7) $\xi:t\in
C_{n}\mapsto\xi(t)=\left(\begin{array}[]{ccccccc}0&1&0\ldots&0&0\\\
0&0&1\ldots&0&0\\\ \ldots&\ldots&\ldots&\ldots&\ldots\\\ 0&0&0\ldots&0&1\\\
-1&-1&-1\ldots&-1&-1\\\ \end{array}\right)\in GL_{n-1}(\mathbb{Z}).$
We have the crystallographic group $G_{n}=C_{n}\ltimes\mathbb{Z}^{n-1}$ $of$
$holonomy$ $n$ and several information on it can be found in [8, §6.3], or [1,
Proposition 3.3]. Looking at its construction, $G_{n}\in\mathfrak{P}$,
$h(G_{n})=n-1$, $Z(G_{n})=\\{1\\}$ and $G_{n}$ is metabelian (in particular,
$[G_{n},G_{n}]$ is abelian). On another hand, we may get a presentation for
$G_{n}$, taking a generating set for $C_{n}$, another for $\mathbb{Z}^{n-1}$
and considering the action (2.7). We have as follows.
###### Corollary 2.2.
$G_{n}=\langle a_{1},\ldots,a_{n-1},t\ |\ t^{n}=1,t^{-1}a_{i}t=a_{i+1}\ (1\leq
i\leq n-2),t^{-1}a_{n-1}t=a_{1}^{-1}\ldots a_{n-1}^{-1},[a_{i},a_{j}]=1\
(1\leq j<i\leq n-1)\rangle.$
We can be more accurate in the description of $[G_{n},G_{n}]$ and of the
abelianization $G^{ab}_{n}=G_{n}/[G_{n},G_{n}]$. In fact $[G_{n},G_{n}]$ is
generated by the commutators of generators of $G_{n}$ and their inverses.
Since all the $a_{i}$ commute and $t$ has finite order, one has only to
consider commutators of the form $[t,a_{i}]$ and thus
(2.8) $[G_{n},G_{n}]=\langle a^{-1}_{i}a_{i+1},a^{-1}_{1}\ldots
a^{-1}_{n-2}a^{-2}_{n-1}\ |$ (2.9) $\
[a^{-1}_{i}a_{i+1},a^{-1}_{j}a_{j+1}]=[a^{-1}_{i}a_{i+1},a^{-1}_{1}\ldots
a^{-1}_{n-2}a^{-2}_{n-1}]=1\ (1\leq i,j\leq n-2)\rangle.$
We note that $[G_{n},G_{n}]$ is free abelian of rank $n-1$. On another hand,
when we factor $G_{n}$ through $[G_{n},G_{n}]$, we have that $t$ is an
independent generator and $a_{1}=a_{2}=\ldots=a_{n-1}$. So
$a_{n-1}=a_{n-1}^{-(n-1)}$ which implies $a^{n}_{n-1}=1$ and $a_{n-1}$ is a
second independent generator. We conclude that $G^{ab}_{n}=C_{n}\times C_{n}$.
On another hand, if $G_{n}$ has $n=p^{s}$ ($p$ prime and $s\geq 1$) and we
replace $\mathbb{Z}^{p-1}$ with $\mathbb{Z}^{d_{s}}_{p}$, where
$\mathbb{Z}_{p}$ denotes the $group$ $of$ $p$–$adic$ $integers$ and
$d_{s}=p^{s-1}(p-1)$, then we have the $pro$–$p$–$group$
$K_{s}=C_{p^{s}}\ltimes\mathbb{Z}^{d_{s}}_{p}$ $of$ $finite$ $coclass$ $with$
$central$ $exponent$ $s$, studied in [9, 13]. This time we cannot apply Lemma
2.1, but computational arguments are still true. We recall a result in this
direction, to convenience of the reader.
###### Lemma 2.3 (See [9], Theorem 7).
For an integer $i$ let $e_{i}=1$, if $p^{s-1}$ divides $i-1$, and $e_{i}=0$,
otherwise. Then $K_{s}=\langle a_{1},\ldots,a_{d_{s}},t\ |\ t^{p^{s}}=1,\
t^{-1}a_{1}t=a^{-1}_{d_{s}},\ t^{-1}a_{i}t=a_{i-1}a^{-e_{i}}_{d_{s}}\ (1<i\leq
d_{s}),[a_{i},a_{j}]=1\ (1\leq j<i\leq d_{s})\rangle$. Furthermore,
$M(K_{s})\simeq\mathbb{Z}^{\frac{d_{s}}{2}}_{p}$, unless $p=2$ and $s=1$ in
which case $M(K_{s})=1$.
We may use the above arguments in order to note that $K_{s}$ is a metabelian
group with $h(K_{s})=d_{s}$, $[K_{s},K_{s}]\simeq\mathbb{Z}_{p}^{d_{s}}$,
$K^{ab}_{s}=C_{p^{s}}\times C_{p^{s}}$ and $Z(K_{s})=\\{1\\}$. However,
$K_{s}\not\in\mathfrak{P}$, but $K_{s}\in\mathfrak{S}$.
B. Eick and W. Nickel [8] have studied the nonabelian tensor square of
$G_{n}$, when $n=p$. For $p=2$ we have the infinite dihedral group
$G_{2}=D_{\infty}=\langle a,x\ |\
a^{x}=a^{-1},x^{2}=1\rangle=C_{2}\ltimes\mathbb{Z}$. Quoting [8, Figure at
p.943], the following list holds:
(2.10) $h(G_{2}\otimes G_{2})=h(G_{2})=1,h(G_{3}\otimes
G_{3})-h(G_{3})=3-2=1,$ (2.11) $h(G_{5}\otimes
G_{5})-h(G_{5})=6-4=2,h(G_{7}\otimes G_{7})-h(G_{7})=9-6=3,\ldots.$
With the help of GAP [20] one can see that the same list is true when $s=1$,
$p=2,3,5,7$ and we deal with
$K_{2}=C_{2}\ltimes\mathbb{Z}_{2},K_{3}=C_{3}\ltimes\mathbb{Z}^{2}_{3},K_{5}=C_{5}\ltimes\mathbb{Z}^{4}_{5},K_{7}=C_{7}\ltimes\mathbb{Z}^{6}_{7}$.
Then it would be interesting to detect the properties of the following
function from the set of the integers onto the set of the integers
(2.12) $f:h(S)\in\\{h(S)\ |\ S\in\mathfrak{S}\\}\mapsto f(h(S))=h(S\otimes
S)-h(S).$
###### Remark 2.4.
I. Nakaoka and M. Visscher show that $S\otimes S\in\mathfrak{S}$, whenever
$S\in\mathfrak{S}$ (see [4, 16, 21]) and so $f$ is well–posed. On another
hand, G. Ellis [10] and P. Moravec [16] show that $F\otimes
F\in\mathfrak{F}\cap\mathfrak{P}$, whenever
$F\in\mathfrak{F}\cap\mathfrak{P}$. Then $0=h(C_{2})\mapsto f(0)=0$, or more
generally, $0=h(F)\mapsto f(0)=0$ for all $F\in\mathfrak{F}\cap\mathfrak{P}$,
but also $1=h(G_{2})\mapsto f(1)=0$. Hence $f$ is not injective. In fact
$N(f)=\\{h(S)\ |\ f(h(S))=0\\}=\\{h(S\otimes S)=h(S)\ |\ S\in\mathfrak{S}\\}$.
Finally, one can note that $f$ is neither additive nor multiplicative.
The next property of the Hirsch length is well–known.
###### Lemma 2.5 (See [14], §1.3).
If $A,B\in\mathfrak{S}$ and $\varphi:A\rightarrow B$ is a homomorphism of
groups, then $h(A)=h(\varphi(A))+h(\ker\varphi)$. In particular, the Hirsch
length is additive on the extensions.
We have immediately the next consequence.
###### Corollary 2.6.
$f(h(S))\leq h(J_{2}(S))$ for all $S\in\mathfrak{S}$.
###### Proof.
(1.2) shows that $S\otimes S\in\mathfrak{S}$ is a central extension of
$J_{2}(S)$ by $[S,S]$. From Lemma 2.5, $h(S\otimes S)=h(J_{2}(S))+h([S,S])$.
On another hand, $[S,S]\leq S$ implies $h([S,S])\leq h(S)$ and so $h(S\otimes
S)\leq h(J_{2}(S))+h(S)$ from which the result follows. ∎
We recall the following information on the structure of $J_{2}(G)$,
$\nabla(G)$ and $G\otimes G$.
###### Proposition 2.7 (See [3], Corollary 1.4).
Let $G$ be a group such that $G^{ab}$ is abelian finitely generated with no
elements of square order. Then $J_{2}(G)=\Gamma(G^{ab})\times M(G)$.
###### Proposition 2.8 (See [3], Theorem 1.3 (iii)).
Let $G$ be a group such that either $G^{ab}$ has no elements of square order
or $G^{\prime}$ has a complement in $G$. Then $\nabla(G)\simeq\nabla(G^{ab})$
and $G\otimes G\simeq\nabla(G)\times(G\wedge G)$.
The linear growth of (2.12) is described by the next result.
###### Proposition 2.9.
In Lemma 2.3 let $s=1$, $p\not=2$ and $K_{p}=C_{p}\ltimes\mathbb{Z}^{p-1}_{p}$
be the corresponding pro–$p$–group. Then $f(h(K_{p}))=\frac{1}{2}(p-1)$. In
particular, $f(h(K_{p}))=h(J_{2}(K_{p}))=h(M(K_{p}))$ has a linear growth.
###### Proof.
We claim that (1.2) is equivalent to the following diagram
(2.13) $\begin{CD}11\\\ @V{}V{}V@V{}V{}V\\\
H_{3}(K_{p})@>{}>{}>C^{2}_{p}\times C_{p^{2}}@>{\psi}>{}>C^{2}_{p}\times
C_{p^{2}}\times\mathbb{Z}_{p}^{\frac{p-1}{2}}@>{}>{}>\mathbb{Z}_{p}^{\frac{p-1}{2}}@>{}>{}>1\\\
\Big{\|}\Big{\|}@V{}V{}V@V{}V{}V\\\ H_{3}(K_{p})@>{}>{}>C^{2}_{p}\times
C_{p^{2}}@>{\psi}>{}>C^{2}_{p}\times
C_{p^{2}}\times\mathbb{Z}^{p-1}_{p}\times\mathbb{Z}^{\frac{p-1}{2}}_{p}@>{}>{}>\mathbb{Z}^{p-1}_{p}\times\mathbb{Z}^{\frac{p-1}{2}}_{p}@>{}>{}>1.\\\
@V{\kappa}V{}V@V{\kappa^{\prime}}V{}V\\\
\mathbb{Z}_{p}^{p-1}@=\mathbb{Z}_{p}^{p-1}\\\ @V{}V{}V@V{}V{}V\\\ 11\\\
\end{CD}$
From [5, §2, (13), p.181],
(2.14) $\Gamma(K_{p}^{ab})=\Gamma(C_{p}\times C_{p})=C_{p}\times
C_{p}\times(C_{p}\otimes_{\mathbb{Z}}C_{p})=C_{p}\times C_{p}\times
C_{p^{2}}.$
Note that $C_{p}\otimes_{\mathbb{Z}}C_{p}=C_{p^{2}}$ is an elementary fact on
the usual abelian tensor product. Still by [5, §2],
(2.15) $\psi(\Gamma(C_{p}\times C_{p}))=\nabla(K_{p})=C_{p}\times C_{p}\times
C_{p^{2}}.$
From Lemma 2.3, $M(K_{p})=\mathbb{Z}_{p}^{\frac{p-1}{2}}$. We do not have
elements of square order in $K_{p}^{ab}=C_{p}\times C_{p}$ and Proposition 2.7
yields $J_{2}(K_{p})\simeq\Gamma(K_{p}^{ab})\times M(K_{p})\simeq C_{p}\times
C_{p}\times C_{p^{2}}\times\mathbb{Z}_{p}^{\frac{p-1}{2}}$.
The commutativity of (1.2) shows that $K_{p}\wedge K_{p}$ is a central
extension of $M(K_{p})=\ker\kappa^{\prime}$ by $[K_{p},K_{p}]$, which are both
normal abelian subgroups of $K_{p}\wedge K_{p}$, then $K_{p}\wedge
K_{p}=\langle
M(K_{p}),[K_{p},K_{p}]\rangle=M(K_{p})[K_{p},K_{p}]=\langle\mathbb{Z}_{p}^{p-1},\mathbb{Z}_{p}^{\frac{p-1}{2}}\rangle=\mathbb{Z}_{p}^{p-1}\mathbb{Z}_{p}^{\frac{p-1}{2}}.$
On another hand,
(2.16) $[M(K_{p}),M(K_{p})]=[[K_{p},K_{p}],[K_{p},K_{p}]]=1$
implies
(2.17) $[K_{p}\wedge K_{p},K_{p}\wedge
K_{p}]=[M(K_{p})[K_{p},K_{p}],M(K_{p})[K_{p},K_{p}]]=[M(K_{p}),M(K_{p})][M(K_{p}),[K_{p},K_{p}]]$
(2.18)
$[[K_{p},K_{p}],[K_{p},K_{p}]][M(K_{p}),[K_{p},K_{p}]]=[M(K_{p}),[K_{p},K_{p}]].$
Since
$M(K_{p})=\mathbb{Z}_{p}^{\frac{p-1}{2}}\leq\mathbb{Z}_{p}^{p-1}=[K_{p},K_{p}]$,
we deduce $C_{K_{p}\wedge K_{p}}([K_{p},K_{p}])\leq C_{K_{p}\wedge
K_{p}}(M(K_{p}))$ and then
(2.19) $[K_{p},K_{p}]\leq C_{K_{p}\wedge K_{p}}([K_{p},K_{p}])\leq
C_{K_{p}\wedge K_{p}}(M(K_{p})),$
which implies $[M(K_{p}),[K_{p},K_{p}]]=1$. We conclude that $K_{p}\wedge
K_{p}$ is abelian and then the central extension is actually a direct product
of the form $K_{p}\wedge
K_{p}=\mathbb{Z}^{p-1}_{p}\times\mathbb{Z}^{\frac{p-1}{2}}_{p}$. From
Proposition 2.8,
(2.20) $K_{p}\otimes K_{p}=C_{p}\times C_{p}\times
C_{p^{2}}\times\mathbb{Z}^{p-1}_{p}\times\mathbb{Z}^{\frac{p-1}{2}}_{p}.$
Then
(2.21)
$h(M(K_{p}))=h(J_{2}(K_{p})/\nabla(K_{p}))=h(J_{2}(K_{p}))=\frac{p-1}{2}.$
We conclude from (2.13) and Lemma 2.5 that
(2.22) $h(K_{p}\otimes K_{p})=h(\kappa(K_{p}\otimes
K_{p}))+h(J_{2}(K_{p}))=(p-1)+h(M(K_{p}))=\frac{3}{2}(p-1).$
Therefore $f(h(K_{p}))=h(J_{2}(K_{p}))=\frac{1}{2}(p-1)$. ∎
The methods in the above proof continue to be valid when $s>1$. Therefore we
draw the following result, which has independent interest and, in view of [13,
Theorem 7.4.12, Corollary 7.4.13], describes the nonabelian tensor square of
all pro–$p$–groups of finite coclass with trivial center.
###### Theorem 2.10.
If $s>1$ and $p$ is an odd prime, then $K_{s}\otimes K_{s}=C^{2}_{p^{s}}\times
C_{p^{2s}}\times\mathbb{Z}^{\frac{3}{2}d_{s}}_{p}$.In particular,
$f(h(K_{s}))=h(J_{2}(K_{s}))=h(M(K_{s}))$ has a linear growth.
###### Proof.
Mutatis mutandis, we may argue as in the proof of Proposition 2.9. ∎
The computational data show that $M(G_{p})=\mathbb{Z}^{\frac{p-1}{2}}$. In
alternative, an argument as in [9, Proof of Theorem 7] can be applied, that
is, we may express the Schur’s Formula for $M(G_{p})$, beginning from the
presentation in Corollary 2.6. Equivalently, we may work via duality, since
the cohomology of $G_{p}$ is known by [1]. This justifies the assumption of
the next result.
###### Corollary 2.11.
Assume $M(G_{p})=\mathbb{Z}^{\frac{p-1}{2}}$ for all primes $p\not=2$. Then
$f(h(G_{p}))=\frac{1}{2}(p-1)$. In particular,
$f(h(G_{p}))=h(J_{2}(G_{p}))=h(M(G_{p}))$ has a linear growth.
###### Proof.
We may argue as in the proof of Proposition 2.9, replacing $K_{p}$ with
$G_{p}$. ∎
The above results prove that there are crystallographic groups of holonomy
$p\not=2$ which achieve the bound in Corollary 2.6. The same is true for the
pro–$p$–group $K_{p}$ with $p\not=2$. Note that Proposition 2.9 describes
rigorously the structure of $K_{p}\otimes K_{p}$ with respect to that of
$K_{p}$ in terms of their torsion–free factors. The same is true for $G_{p}$
by Corollary 2.11. The fact that (2.12) has a linear growth can be translated
in terms of restrictions on the Schur multiplier as follows.
###### Corollary 2.12.
If $f(h(S))=c\ h(S)\ $ for some integer $c\geq 0$ and $S\in\mathfrak{S}$, then
$h(M(S))\leq h(S)^{2}+(c+1)h(S)$. The equality holds, whenever
$S\in\mathfrak{F}$.
###### Proof.
We have $f(h(S))=h(S\otimes
S)-h(S)=\Big{(}h(J_{2}(S))+h([S,S])\Big{)}-h(S)=h(M(S))-h(\nabla(S))+h([S,S])-h(S)$.
Now we may always write $s\otimes s=(s\otimes 1)(1\otimes s)$ in a unique way
and then the map $\iota:s\otimes s\in\nabla(S)\mapsto\iota(s\otimes
s)=\iota((s\otimes 1)(1\otimes s))=\iota(s\otimes 1)\iota(1\otimes s)=(s,s)\in
S\times S$ is a monomorphism. Therefore $h(\nabla(S))\leq h(S)^{2}$ and so
$h(M(S))=f(h(S))+h(\nabla(S))-h([S,S])+h(S)\leq f(h(S))+h(\nabla(S))+h(S)\leq
c\ h(S)+h(S)^{2}+h(S)$ from which the result follows. ∎
Unfortunately, (2.12) has not a linear growth for all groups in $\mathfrak{S}$
and we cannot predict its form in general. Already in $\mathfrak{P}$ there are
examples in this sense (see [8, Figure at p.943]). However, a nice
circumstance is described below.
###### Corollary 2.13.
There exists a metabelian group $G$ with trivial center for which
$f(h(G))=h(M(G))=\frac{1}{2}p^{s-1}(p-1)$, where $s>1$ and $p$ is an odd
prime.
###### Proof.
Consider $G=K_{s}$. By Lemma 2.3, $h(M(K_{s}))=\frac{1}{2}p^{s-1}(p-1)$. From
Theorem 2.10, $f(h(K_{s}))=h(K_{s}\otimes
K_{s})-h(K_{s})=\big{(}p^{s-1}(p-1)+\frac{1}{2}p^{s-1}(p-1)\big{)}-p^{s-1}(p-1)=\frac{1}{2}p^{s-1}(p-1)=h(M(K_{s})).$
∎
We end the section with an explicit description for (2.12), modifying a
classic argument of N. Rocco, which can be found in [3, Theorem 1] (see also
[3, Observation]).
###### Theorem 2.14.
Let $G$ be a group in $\mathfrak{P}$ such that
$G^{ab}=\displaystyle\prod_{i=1}^{n}C_{p^{e_{i}}}$, for integers $1\leq
e_{i}\leq e_{j}$ such that $1\leq i<j\leq n$, $p$ odd prime and
$d=\sum_{i=1}^{n}(n-i)e_{i}$.
* (a)
If $G$ is finite, then $|G\otimes G|=p^{d}|G||M(G)|$.
* (b)
If $G$ is infinite, then $f(h(G))=h(M(G))$.
###### Proof.
(a). Assume $G$ is finite. Since $G^{ab}$ is finitely generated and has no
elements of order two, all the hypotheses of [3, Theorem 1] are satisfied and
so $G\otimes G\simeq\nabla(G)\times G\wedge G$. From this and (1.2) we deduce
(2.23) $|G\otimes
G|=\frac{|\nabla(G)|}{|G^{ab}|}|G||M(G)|=\frac{|\Gamma(G^{ab})|}{|G^{ab}|}|G||M(G)|=|\prod_{i=1}^{n}(C_{p^{e_{i}}})^{n-i}||G||M(G)|=p^{d}|G||M(G)|$
where $d=\sum_{i=1}^{n}(n-i)e_{i}.$
(b). Assume $G$ is infinite. From Proposition 2.7 and Lemma 2.5 we conclude
that $h(J_{2}(G))=h(\Gamma(G^{ab}))+h(M(G))=h(M(G))$, where the last equality
is due to the fact that $\Gamma(G^{ab})$ is periodic. Proceeding as in (2.22),
(2.24) $h(G\otimes G)=h(\kappa(G\otimes
G))+h([G,G])=h(J_{2}(G))+h([G,G])=h(M(G))+h([G,G]).$
Subtracting $h(G)$, we find
(2.25) $f(h(G))=h(G\otimes
G)-h(G)=h(M(G))-(h(G)-h([G,G]))=h(M(G))-h(G^{ab})=h(M(G)),$
since $G^{ab}$ is periodic. ∎
###### Remark 2.15.
It is not used the hypothesis $G\in\mathfrak{P}$ in Theorem 2.14 (a) and so
this part of the result is true for an arbitrary finite group.
## 3\. Some evidences
The present section is devoted to evaluate (2.12) for other classes of groups
for which it is known their nonabelian tensor product. A $Bieberbach$ $group$
$B$ is an extension of a free abelian group $L$ (called $lattice$ $group$) of
finite rank by a group $P$ (called $holonomy$ $group$). Following the notation
of Lemma 2.1, we are fixing $A=L$, $B=\Gamma$ and $Q=P$, imposing a precise
choice for these groups. The $dimension$ of $B$ is the rank of $L$. It is easy
to see that $G_{p}$, studied in the previous section, is of this form, once
$L=\mathbb{Z}^{n-1}$ and $P=C_{n}$. It is known that
(3.1) $B_{1}(2)=\langle a,x,y\ |\
a^{2}=y,axa^{-1}=x^{-1},[a,y]=[x,y]=1\rangle$
is a $Bieberbach$ $group$ $of$ $dimension$ $2$ $with$ $point$ $group$ $C_{2}$
and that the groups
(3.2) $B_{1}(n)=B_{1}(2)\times\mathbb{Z}^{n-2}\,\,\,\,\textrm{for}\,\,\,n\geq
2$
are $Bieberbach$ $groups$ $of$ $dimension$ $n$ $with$ $point$ $group$ $C_{2}$.
More details can be found in [15]. The next two results check (2.12) on
$B_{1}(2)$ and $B_{1}(n)$.
###### Corollary 3.1.
In $B_{1}(2)$ we have that $f$ is constant to 0.
###### Proof.
From [15, Theorem 4.1] we have
(3.3) $B_{1}(2)\otimes B_{1}(2)=C_{2}\times C_{4}\times\mathbb{Z}^{2}.$
Still from [15] we know that $M(B_{1}(2))$ is trivial. Now
$f(h(B_{1}(2)))=h(B_{1}(2)\otimes B_{1}(2))-h(B_{1}(2))=2-2=0$ and the result
follows. ∎
###### Corollary 3.2.
In $B_{1}(n)$ we have that $f(h(B_{1}(n)))=n^{2}-3n+4$ for all $n>2$.
###### Proof.
From [15, Corollary 4.1] we have
(3.4) $B_{1}(n)\otimes B_{1}(n)=C^{2n-3}_{2}\times
C_{4}\times\mathbb{Z}^{(n-1)^{2}+1}.$
Still from [15] we know that $M(B_{1}(2))=n-2$ and so it is nontrivial. Now
$f(h(B_{1}(n)))=h(B_{1}(n)\otimes
B_{1}(n))-h(B_{1}(n))=((n-1)^{2}+1)-(n-2)=n^{2}-2n+2-n+2=n^{2}-3n+4$ and the
result follows. ∎
In a certain sense Theorem 2.14 (b) forces the growth of (2.12) to be equal to
that of the Schur multiplier, when the abelianization of the group is the
direct product of finite cyclic groups. Is this condition really necessary?
Unfortunately, the answer is positive and $B_{1}(n)$ for $n>2$ shows it.
###### Corollary 3.3.
For all $n>2$, $f(h(B_{1}(n)))$ has not a linear growth but $h(M(B_{1}(n)))$
has a linear growth.
###### Proof.
$f(h(B_{1}(n)))=n^{2}-3n+4$ and $h(M(B_{1}(n)))=n-2$. ∎
Recent progresses in [3, 4] show that the nonabelian tensor product of
Bieberbach groups has a similar structure with respect to that of the free
solvable groups of finite rank and free nilpotent groups of finite rank.
Therefore we have the following results.
###### Corollary 3.4.
Let $F$ be the free group of finite rank $r\geq 1$ and $G=F/F^{(d)}$ be the
free solvable group of derived length $d\geq 1$ and rank $r$. If $F^{\prime}$
is periodic, then $f(h(G))\leq\frac{1}{2}r(r-1)$. In particular, if $h(G)=r$,
then the equality holds and $f(h(G))=\frac{1}{2}r(r-1)$.
###### Proof.
We may apply [3, Corollary 2.4] and so $G\otimes
G=\mathbb{Z}^{\frac{1}{2}r(r+1)}\times F^{\prime}/[F,F^{(d)}]$. Lemma 2.5
implies $h(G\otimes G)=\frac{1}{2}r(r+1)$. Of course $h(G)\leq r$. Then
$f(h(G))\leq\frac{1}{2}r(r+1)-r=\frac{1}{2}r(r-1)$, as claimed. ∎
###### Corollary 3.5.
Let $G$ be the free nilpotent group of rank $r\geq 1$ and class $c\geq 1$. If
$G^{\prime}$ is periodic, then $f(h(G))\leq\frac{1}{2}r(r-1)$. In particular,
if $h(G)=r$, then the equality holds and $f(h(G))=\frac{1}{2}r(r+1)$.
###### Proof.
Note that nilpotent groups are solvable and so it is meaningful to consider
$f(h(G))$. Applying [3, Corollary 2.3], $G\otimes
G=\mathbb{Z}^{\frac{1}{2}r(r+1)}\times G^{\prime}$ and the remainder is
similar to the previous corollary. ∎
However, Lemma 2.1 imposes the following question, which we leave open in its
generality.
###### Open Question 3.6.
What is the growth of $h(\Gamma\otimes\Gamma)$ with respect to $h(\Gamma)$,
where $\Gamma$ is an arbitrary extension of two abelian groups $A$ and $Q$ as
in Lemma 2.1?
## References
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* [2] J.R. Beuerle, L.–C. Kappe, Infinite metacyclic groups and their non-abelian tensor squares, Proc. Edinb. Math. Soc. 43 (2000) 651 -662.
* [3] R. Blyth, F. Fumagalli and M. Morigi, Some structural results on the non-abelian tensor square of groups, Preprint, Cornell University, arXiv:0810.4620, 2008.
* [4] R. Blyth and R. Morse, Computing the nonabelian tensor squares of polycyclic groups, J. Algebra 321 (2009), 2139 -2148.
* [5] R. Brown, D.L. Johnson and E. F. Robertson, Some computations of non-abelian tensor products of groups, J. Algebra 111 (1987), 177–202.
* [6] R. Brown and J.–L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311–335.
* [7] K. Dennis, In search of new homology functors having a close relationship to K-theory, Preprint, Cornell University, 1976.
* [8] B. Eick and W. Nickel, Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group, J. Algebra 320 (2008), 927 -944.
* [9] B. Eick, Schur multiplicators of infinite pro-$p$-groups with finite coclass, Israel J. Math. 166 (2008), 147 -156.
* [10] G. Ellis, The nonabelian tensor product of finite groups is finite, J. Algebra 111 (1987), 203–205.
* [11] G. Ellis, Tensor products and $q$-crossed modules, J. London Math. Soc. 2 (1995), 241–258.
* [12] L.–C. Kappe, N.H. Sarmin and M.P. Visscher, Two-generator two-groups of class two and their nonabelian tensor squares, Glasgow Math. J. 41 (1999), 417- 430.
* [13] C. R. Leedham–Green and S. McKay, The Structure of Groups of Prime Power Order, Oxford University Press, Oxford, 2002.
* [14] J.C. Lennox and D.J.S. Robinson, The Theory of Infinite Soluble Groups, Oxford Univerisity Press, Oxford, 2004.
* [15] R. Masri, The nonabelian tensor squares of certain Bieberbach groups with cyclic point group of order 2, Phd thesis, Universiti Teknologi Malaysia, 2009.
* [16] P. Moravec, The nonabelian tensor product of polycyclic groups is polycyclic, J. Group Theory 10 (2007), 795–798.
* [17] R. F. Morse, Private communication, 2009.
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* [19] N.H. Sarmin, Infinite two generator groups of class two and their non-abelian tensor squares, Int. J. Math. Math. Sci. 32 (10) (2002), 615–625.
* [20] The GAP Group, GAP—Groups, Algorithms and Programming, version 4.4, available at http://ww.gap-system.org, 2005\.
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### Acknowledgment
The first and the second author are grateful to the Department of Mathematics
and the Ibnu Sina Institute of the Universiti Teknologi Malaysia for the
hospitality in the summer of 2009, when the initial part of this manuscript
was written. We also thank Prof. B. Eick, who suggested [8, 9], Prof. R. F.
Morse and Dr. P. Moravec, who communicated some inaccuracies in the original
version of the present paper. Finally, we appreciated some email contributions
of R. Brown, A. Caranti, D. Feirtenschlager, R. Hartung, M. Horn and D. Ramras
in 2010.
|
arxiv-papers
| 2009-11-30T10:15:53 |
2024-09-04T02:49:06.766361
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Erfanian (Ferdowsi University of Mashhad, Mashhad, Iran),\n Francesco G. Russo (Universita' degli Studi di Palermo, Palermo, Italy) and\n Nor Haniza Sarmin (Universiti Teknologi Malaysia, Johor Bahru, Malaysia)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/0911.5604"
}
|
0912.0076
|
Automation of PRL's
Astronomical Optical Polarimeter with a GNU/Linux based distributed control system
SHASHIKIRAN GANESH1,
U. C. JOSHI, K. S. BALIYAN, S. N. MATHUR, P. S. PATWAL and R. R. SHAH
Physical Research Laboratory
Astronomy & Astrophysics,
Ahmedabad, INDIA 380 009
1Email: shashi@prl.res.in
Astronomy & Astrophysics Division
Physical Research Laboratory
Ahmedabad, INDIA 380 009
This document was created by the authors using with a style file based on the IEEE style file
and fancyhdr, graphicx packages
Original version submitted September 2008, accepted March 2009
<http://www.prl.res.in/ shashi/>
<http://www.prl.res.in/ library/>
PRL Technical Report # PRL-TN-2008-93
This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
Short contents 1
PRL's Optical Polarimeter has been used on various telescopes in India since its development in-house in the mid 1980s. To make the instrument more efficient and effective we have designed the acquisition and control system and written the software to run on the GNU/Linux Operating System. CCD cameras have been used, in place of eyepieces, which allow to observe fainter sources with smaller apertures. The use of smaller apertures provides dramatic gains in the signal-to-noise ratio. The polarimeter is now fully automated resulting in increased efficiency. With the advantage of networking being built-in at the operating system level in GNU/Linux, this instrument can now be controlled from anywhere on the PRL local area network which means that the observer can be stationed in Ahmedabad / Thaltej as well or via ssh anywhere on the internet. The current report provides an overview of the system as implemented.
§ INTRODUCTION
The Optical Polarimeter (see Fig. <ref>) has been in use as one of the backend instruments (Deshpande et al., 1985, Joshi et al., 1987) at the 1.2m telescope operated by the Astronomy & Astrophysics Division of the Physical Research Laboratory (PRL) at Gurushikhar near Mt Abu.
This instrument enables the study of polarisation at optical wavelengths of a wide variety of astronomical subjects ranging from comets and stars to blazars. To minimize the error due to the background sky light, one should use the smallest apertures possible (say 6 to 10 arc sec), however in the case of visual centring of the source this is not always possible due to the very small contrast between the typical sources of interest (such as blazars / quasars) and the sky. In the absence of on-axis guiding, it was not possible to make long integrations to improve the signal-to-noise ratio.
[Optical polarimeter on 1.2m Telescope (1997)]
The optical polarimeter mounted on the Cassegrain focus of the 1.2m Mt Abu telescope (circa 1997).
In order to improve the efficiency of the instrument and to address the above shortcomings we have completely overhauled and rebuilt the acquisition system and added new subsystems. To reduce human interaction and thus human error we have used CCD cameras, in place of eyepieces, which enable to look at the location of star vis-a-vis the edge of the aperture on a monitor. The instrument has been fully automated using GNU/Linux with an RTAI (Real Time Application Interface) enabled kernel running on a PC/104 based embedded CPU board[PC/104 format is a compact ($96 \times 90 $mm$^{2}$ size) board with the ISA bus' 104 pins arranged in four rows in a condensed format. With a pin and socket connection, the PC/104 systems are self-stackable and are extremely rugged when compared to normal ISA bus based motherboards]. Onyx PC/104 counter/timer and digital I/O boards were utilized to record the counts coming from photo-multipliers in photon counting mode. One PC/104 board has been developed in house for rotating a half-wave plate to generate fast modulation of incoming light beam.
Other mechanical operations (such as changing of optical filters, apertures etc) have been achieved through the use of stepper motors driven by Atmel microcontrollers. Another PC/104 CPU board controls a USB interfaced CCD camera to provide a view of the observing aperture and the field being observed by the instrument. An additional independent embedded computer controls the auto-guider CCD. With the use of GNU/Linux and in-house developed control software (both kernel device driver module as well as user space) the instrument can be operated from anywhere on the local area network (LAN). Since the automation, the instrument has been used extensively from the fully enclosed telescope control room adjacent to the telescope floor(dome) and also remotely from PRL campus at Ahmedabad. In principle, the operator/observer can be stationed anywhere on the PRL computer network including Ahmedabad / Thaltej.
In this report we provide a brief overview of the techniques employed to upgrade the instrument. While the concepts discussed here have been implemented for an astronomical instrument, they are general enough to be applicable to other experimental sciences wherever remote control is desired. Further reports in this series will elaborate on the various aspects in much greater depth. This is done with a view to make the reports as modular and self-contained as possible so that information of interest is easy to locate for developing other instruments in future.
§ PRINCIPLE OF OPERATION
Optical schematic of the polarimeter. The narrow and broadband filters (typically 80Åand 1000Åbandwidth respectively) are the only components which need to be changed based on science requirement. The half-wave plates are good from 3500 to over 10000Å. The neutral density filter reduces the incoming starlight by 2.5 magnitudes.
The principle of operation of the instrument is described by Frecker & Serkowski (1976). The basic idea is to measure the optical polarisation by the use of an analyser, with photo-multiplier tubes (PMTs) recording the counts in photon counting mode. The light path through the instrument is shown in the schematic layout in Fig. <ref>. In order to minimise the influence of varying atmospheric conditions, fast modulation of the incoming light beam is used. For this purpose a half-wave plate is rotated by a stepper motor at 5 or 10 rotations per second with 96 steps per rotation resulting in sampling time of 1 or 2msec per step. The modulated beam is then split into ordinary and extra-ordinary polarised components by a Wollaston prism and the respective counts registered by two independent photon counting photo-multiplier tubes. Due to the modulation, the recorded counts exhibit a sine wave pattern in 24 steps. Since this includes contribution from the sky, an equally large, vacant area of the sky adjacent to the source is observed. These counts are subtracted from the counts recorded for the object of interest. A function of the form
\begin{equation}
I_{j}~=~{{1}\over{2}}{\Big\{I_{0}~\pm~Q~\cos 4\theta_{j}~\pm~U~\sin 4\theta_{j}\Big\}}
\end{equation}
is fitted to the counts $I_{j}$ recorded at different positions of the half-wave plate (angle $\theta_{j}$) and the Stokes parameters describing linearly polarised light $I_{0}$, $Q$ and $U$ are obtained. From these the degree, $p$, and position angle, $\Theta$, of polarisation are readily obtained using the simple formulae:
\begin{equation}
\end{equation}
§ HARDWARE
A distributed embedded control system (see block diagram in Fig. <ref>) has been developed as described in the following sections. The eyepieces have been replaced by CCD cameras from Starlight-Xpress which have proved to be extremely efficient in detection of the source and the edge of the aperture being used. With 16 bit data contrast levels even very faint sources can be observed with ease now and they can be accurately centred even in the smallest of apertures. For changing the apertures we have implemented a stepper motor driven rack and pinion coupled mechanical extension to the existing aperture slide. A similar mechanism has been implemented for changing the optical filters. A third motor unit allows to pull in and out the mirror which directs light to the CCD or to the photo-multiplier tubes.
All of the power supplies, support electronics, computer boards are contained completely in an Embedded Control System box. This makes the instrument a very efficient self-contained unit. The entire instrument is mounted as a single unit on the Cassegrain focus of the telescope and only three cables need to be connected to the instrument : A.C. mains power supply input, ethernet connectivity cable and finally a cable to interface the telescope guiding with the instrument.
[Schematic of control and acquisition system]
Schematic of control and acquisition system of the polarimeter. Dashed lines indicate ethernet connectivity between subsystems. TCC is Telescope Control Computer. The observer's computer is shown as a Linux laptop; this can be located anywhere on the local area network (LAN).
§.§ PC-104 control system
[The inner view of the embedded distributed control system showing the PC/104 stack]
The inner view of the embedded control system showing the PC/104 stack. The counter interface board has been removed from it's usual position across the ONYX boards to show the PC/104 stack clearly.
The embedded control system mounted as a part of the instrument consists of two embedded PC/104 CPU boards. PC/104 specification is a compact ($90 \times 96$ mm$^2$) size bus based system. The Prometheus PC/104 CPU board (manufactured by M/s Diamond Systems) controls the data acquisition process and distributes jobs to the other subsystems. It is a Zfx86 CPU (equivalent to a 100MHz 486-DX2) board with 32MB RAM, 10/100 Mbps ethernet, 4 serial and one IDE port apart from other peripherals. A 32MB solid-state IDE flash disk is connected to the IDE interface. This 32MB disk is sufficient for the entire operating system and control software (as discussed in the next section). This is a self-stackable rugged system. The ruggedness comes from the 104 pins of the PC/104 bus which are arranged in four rows on one side of the board. The CPU board has both male and female bus connectors and other peripheral cards can be stacked on top of / below or on both sides of the CPU board. The boards are supported on each corner by threaded PC/104 stand-off supports or spacers. Thus the entire stack is electrically as well as mechanically ruggedly supported. One needs special board separator or extractor tools to separate the PC/104 boards without damaging the bus pins. In the stack that we have implemented (see Fig. <ref>) there are two CPU boards in the same physical stack but with the bus connections being independent. The Prometheus CPU board is connected to several PC/104 peripheral boards - a VGA display board (used only for debugging purpose), a 5 port 100 Mbps ethernet switch, two ONYX digital I/O and counter/timer boards and an in house developed 8-phase stepper driver board (see next section). The other PC/104 CPU board in this stack controls the CCD Camera connected via USB interface. Since the data storage devices are flash based (i.e. semiconductor based), the reliability is orders of magnitude better than the earlier hard disk based systems. This advantage arises from the lack of moving components in flash based storage devices.
§.§.§ Stepper motor driver board
An 8 bit PC/104 card (schematic shown in Fig. <ref>) was developed in-house for driving the legacy 8-phase stepper motor for the rotating half-wave plate. This motor + gearbox coupling to the rotating half-wave plate has been working very smoothly for a very long time with occasional requirement to replace (once in 8-10 years) the precision carbon bearings of the rotating half-wave plate. This card was designed and built as a double layer PCB and uses an 8254 timer chip. A 74LS164 8 bit-serial-to-parallel shift register is used for providing the 8 phase timing waveforms to the stepper motor via a ULN2803 driver IC. The 8254 timer chip provides the clock pulses and is also a source of hardware interrupts to the Prometheus CPU every 2 msec corresponding to the duration between each step of the stepper motor / half-wave plate. For sensing a reference point in the rotation of the half wave plate the following arrangement has been made. The gear wheel coupled to the stepper motor has a tiny hole near the edge of the wheel (rest of it being a solid block). Fixed on the top of the gearbox is a light emitting diode (LED) and on the bottom side across where the hole passes in front of the LED is a light dependent resistor (LDR). The output of the LDR is suitably amplified and shaped as a pulse and this digital pulse is monitored by a digital I/O bit of one of the Onyx interface boards discussed below.
§.§.§ Onyx counter boards
Two Onyx PC/104 counter boards were obtained commercially from the same vendor as the Prometheus board. The Onyx counter and digital I/O board provides 16 bit counter timer functionality with the use of an 8254 timer chip. For each of the two PMTs, we used two counters of the 8254 on one counter board. Using suitable gating inputs derived from the interrupt pulse, described above, each of the two counters per PMT input is alternately enabled and disabled for counting in binary down counting mode. When one counter is being read and reset, the photon counts are being recorded by the other counter. We also tried the 9513 timer chip (which has 5 16bit counter/timers on a single chip) but this was not as successful at recording the PMT output pulses as the 8254 chip. The 8254 (with 3 16-bit counter/timers) is able to record pulses as narrow as a few nano sec, while the 9513 chip requires that the pulse width be much larger (typically a few 100 nano sec).
§.§ CCD cameras
Two CCD cameras have been used in place of the eyepieces shown in Fig. <ref>.
§.§.§ Starlight Xpress SXV-H9
An SXV-H9 CCD camera from Starlight-Xpress is used to view the source and accurately centre it in the aperture being used. This camera is a $1392 \times 1040$ pixel 16-bit thermoelectrically cooled CCD device with exceptionally compact driver electronics. This is mounted in the location where the aperture eyepiece was previously located. An other PC/104 CPU board (PCM-5330) sourced from M/s Aaeon Technology is used for controlling the CCD camera. This CPU board is based around an STPC Atlas System-on-Chip (SoC : x86 equivalent) running at 133MHz. It has 64MB on board RAM and 10/100 Mbps Ethernet, 4 serial and 2 USB ports along with an IDE and compact flash interface. A 128MB solid-state compact flash disk is used for the operating system and control software with this board. It is also connected to the main telescope controls via a 4 bit channel corresponding to North/South/East/West movements from the guider interface of the CCD camera.
§.§.§ Starlight Xpress SXVF-M25C
The SXVF-M25C CCD Camera from Starlight-Xpress is a one-shot-colour CCD camera with a large field of view. It has $3024 \times 2016 $ pixels in a Bayer matrix. This CCD shares the same USB interfacing techniques as the SXV-H9 and is also connected with the telescope movements via the guider interface. With it's large field of view it is used for the field acquisition and source identification. This CCD is mounted in place of the field acquisition eyepiece (see Fig <ref>). If need be, both CCDs can be interchanged.
§.§ Micro-controller subsystems
An AVR micro-controller PCB board has been developed in house. We have designed and built this board around an Atmel AVR (ATMEGA8 or ATMEGA88) as the micro-controller and with a MAX 232 serial interfacing chip. The PCB supports in circuit programming via a 5 pin connector (programming port). This programming port can be connected to a parallel or USB port of a host PC with the appropriate cables. This board has been used in all the modules discussed in the subsections below. The basic PCB remains the same and minor changes are hand made by using the general purpose pin outs made available on the PCB. The variations are mainly in the firmware for each application. A separate technical note is in preparation which will present the hardware / software / firmware details of the AVR board based stepper motor controller.
§.§.§ Stepper motor with discrete position encoding
Three stepper motors are used in the movement of the various components i.e. the filter selection slide, the aperture selection slide and thirdly a sliding mirror to divert light from the rest of the instrument. These are controlled by three identical stepper motor control cards and interface to the Prometheus PC-104 board via serial interfaces. These stepper motor control cards are based on an Atmel Atmega 8 AVR microcontroller and were developed in house (schematic of the stepper motor control card is shown in Fig. <ref>).
§.§.§ Monitoring temperature of cold chamber holding the PMTs
One of the serial ports of the Prometheus motherboard interfaces to an Atmega 8 microcontroller which monitors other system parameters such as the voltage levels and temperatures (both ambient as well as temperature inside the cold box holding the PMTs). The temperature monitoring is done with a DS18S20 one-wire sensor interfaced to one of the I/O pins of the microcontroller. ADCs on the microcontroller are used for monitoring the various voltage levels (supply, control etc.) required for operating the PMTs. This AVR board uses a copy of the same PCB as the stepper motor controller discussed in the previous section (without the ULN2003 driver IC being mounted).
§.§.§ LCD driver
Yet another serial port on the Prometheus board is used to display status information on a character matrix LCD mounted on the embedded system. This is again coupled via another AVR board which takes serial input and provides suitable glue logic to display it on the LCD.
§.§ Power supplies
Two power supplies are used for powering the different subsystems in the embedded control system. One compact 55 Watt SMPS with 5 and 12V output powers most of the electronics including the two PC/104 CPUs and the stepper motors. Another SMPS provides 5V supply for powering the 8-phase stepper motor and 12V as input supply for the CCD cameras. Compact high voltage power supply modules (total weight few hundred gm) from Electron Tubes have been used in place of the original bulky (several Kg) power supply. These h.v. power supply modules require 24V input and their output can be monitored. The PMTs are housed in a cold box where the temperatures are held at about 30 degree below ambient temperature. The linear power supply being used for the thermoelectric cooling unit has been replaced with a high current SMPS power supply which weighs a fraction of the original supply and is also much reduced in terms of volume. All these supplies which were earlier housed in individual chassis and mounted separately on the telescope or kept on the observing floor table are now made an integral and permanent part of the instrument and need not be disconnected for storage between observing runs.
§ SOFTWARE
§.§ Operating system
[Software block diagrams]
Block diagram of the kernel and user space software on the Prometheus and Aaeon PC-104 linux systems. The different kernel and user modules are discussed in the text.
We make use of the GNU/Linux Operating System for the control of the instrument. This is a unix like operating system available for a large variety of microprocessors. It is easily scalable from 32 bit AVR microprocessors to high end clusters (super-computers). We have been using this OS for the analysis of astronomical data from most of our instruments (CCD and NICMOS images etc.) and all analysis and developmental software are freely available for this OS under the GNU General Public License (GPL) or other similar open source licenses. One of the biggest advantages of this operating system environment (compared to the single tasking MS-DOS) is that it is fully multi-tasking and network interfacing is fully built-in at the very basic level (Kernel) of the operating system. Highly advanced graphical user interfaces are available and high level libraries (both general computation as well as scientific application related) are easily available with full documentation. Virus related problems seen in other operating systems such as MS-DOS and MS-Windows are not present in GNU/Linux. The scalability of the OS is such that a minimal system with networking support can be fit in to less than 2MB of disk space.
The instrument is controlled by a dedicated control PC over LAN. This control PC is usually kept in the control room at the observatory and serves as the operator console hosting the X-Window graphical interface. It is a Pentium III running at 800MHz with 512MB RAM with standard Redhat 7.3 Linux distribution along with all required developmental tools/software. The PC is connected via Ethernet cable to one of the ports of the 5-port Ethernet switch of the embedded system. This control PC exports it's home partition as a network file system(NFS). However, the instrument is completely independent of this PC and can be controlled from any system with a network reachable X-Window display. In the event of the home partition not being available via NFS one can save the data on USB flash drives or other devices connected directly to the PC/104 stack.
The Prometheus board runs GNU/Linux with real time extensions (RTAI : Real Time Application Interface) added to a standard Linux kernel (version 2.4.19) from <www.kernel.org>. The file system on the 32MB flash disk is based on white-dwarf Linux. The base operating system requires only 16MB. Additional space is taken up by the GTK graphical interface libraries and the application software. The data recorded by the system is saved on the NFS (network file system) partition mounted as /home on the embedded Prometheus CPU board.
In Fig. <ref> we show the block diagrams of the software implementation on the two PC104 systems (Prometheus for the main polarimeter system and Aaeon for the CCD sub-systems). As shown, both user and kernel space codes have been developed for this instrument and are described in the following sub-sections.
§.§ Kernel space drivers
§.§.§ Stepper driver board and Onyx counter boards
The control software consists of both kernel space as well as user space code. Kernel level codes (marked as rtopal in Fig.<ref> initialise all the 3 PC-104 interface boards (2 Onyx boards and the 8-phase stepper driver board). The integration or exposure starts after the starting position has been sensed by monitoring the status of one of the digital I/O bits connected to the LDR via a pulse shaping circuitry. Thereafter the exposure continues until the specified time interval has been completed. The job of reading out the counts in synchronisation with the interrupts received at each step of the half wave plate is carried out in real time kernel module code written in C. In order to remove the effects due to the jitter in the interrupt response and the finite time it takes to record the counts from the 8254 counter, we use two counters per PMT as mentioned earlier. During the first 2msec one of the counters is enabled and is down counting. At the end of the 2msec the first counter is disabled by a suitable gating level and it is read out by the host processor and then reset. At the same time an inverted gate is supplied to the second counter which starts counting down until the end of 2msec and so on. The same process is followed for the second PMT+counter board combination at the same time. All other functions are disabled during the time the system is recording the counts from the celestial sources (which can be of typically few seconds to few minutes in duration). By using hardware gating we have precisely equal intervals for each readout independent of any interrupt jitter that is always present in a multi-tasking OS (although that in itself is also minimised with the RTAI extensions). The device driver software code is available from the authors.
§.§.§ CCD USB device driver
The USB device driver for the two CCD cameras is derived from the code originally written by David Schmenk. The SXV-H9C CCD camera is directly supported by ccd_kernel version 1.8. In the case of the SXVF-M25C camera, the ccd_kernel driver had to be modified to include appropriate device parameters and also to adapt the code to the special read-out mode of the CCD chip.
§.§ User space GUIs
The instrument is controlled by Graphical User Interface (GUI) software which run on the PC/104 sub-systems with the display being provided by the local X-Window terminal of the observer. This could typically be the observer's laptop or any desktop on the local-area-network. Two graphical interfaces are launched. One, called OPAL, is for controlling the basic instrument and acquiring and displaying the data. The second one is GCCD for the control of the CCD cameras.
§.§.§ GUI : OPAL controls
Graphical interface software OPAL designed using the GLADE software runs in user space. It is written in C, and uses only the GTK graphical libraries. This complies with tight memory and execution time constraints. The OPAL software also interfaces with the telescope control computer (TCC) over LAN, using network sockets, to record the telescope parameters (time, direction etc.) at the time of observation.
The aperture and filter selection is menu-driven. The callbacks from the respective menu option send command codes to the respective microcontroller boards via independent serial ports. The status feedback from each microcontroller is displayed in the window and also recorded with the computed output of each observation. Several C code and header files implement the details of the user interface and callbacks etc. The compilation is via the standard GNU `make' mechanism. A tar file containing the entire source code is available on request from the authors.
100% polarisation as observed with a Glan prism for star $69~\nu~ Cnc$
The individual observation records are saved incrementally to a text format file along with the telescope parameters. The file name is derived from the date of observation and is opened in append mode so that all observations of a given night are contained in one file. For each observation we also save the individual counts recorded at the 24 folded positions of the half-wave plate in a separate file. Fig. <ref> shows a plot of test observation for 100% polarisation. This is a plot saved in postscript format by the OPAL GUI. The test for 100% polarisation is done by introducing a Glan prism in the light path and observing a bright star. Typical measurements range from 97.5 to 99.5%. Compliance with observation of 100% polarisation demonstrates the overall linearity of the system (from very low counts to a few million counts) in the polarisation measurements.
§.§.§ GUI : CCD Controls
The Aaeon PC/104 CPU board has more resources in terms of memory and operating system base space so we have installed a very stripped down version of Redhat 7.3 Linux distribution on the 128MB compact flash disk of this board. The Starlight-Xpress SXV-H9 CCD is operated by a free software called GCCD written by David Schmenk. The software uses the gnome library files and so has a little larger memory requirement than the OPAL GUI. GCCD is available on the website listed in the resources below.
As already mentioned it is also possible to make small movements of the telescope to accurately centre the source in the aperture while monitoring the CCD view. Thus the instrument is fully integrated with the telescope control system.
§.§ AVR firmware
All the five Atmega 8 microcontrollers used in this instrument were programmed using a version of C (GNU-compiler collection - gcc) for the AVR again on a GNU/Linux PC with the appropriate developmental tools (compilers / libraries). Useful programming tips and tools (including a bootable live-cd with compilers and other software tools for AVR programming are available on the PHOENIX project (Physics with Home-made Equipment & Innovative Experiments) website of the Inter-University Accelerator Centre (IUAC) and also on the Tuxgraphics.org websites. A separate technical note is in preparation and details the software(firmware) and hardware aspects of the use of the AVR Atmega PCB board. The PCB is general purpose enough to be usable as a microcontroller experimental and developmental board.
§ SUMMARY
We have designed and built the electronics and control system of the Optical Polarimeter. It is controlled by software running on a GNU/Linux/RTAI platform and can be controlled from anywhere on the local area network. Hardware and software including firmware for the micro-controllers were developed completely in-house. Use of CCD cameras in place of the conventional eyepieces allows to observe very faint sources systematically and efficiently. Precise centring of the source in the observing aperture is now possible routinely. This has also allowed to use much smaller apertures (6 to 10 arc sec) than what was being used earlier (15 to 20 arc sec) for the observations. With the smaller apertures, sky (background) contribution reduces and thus the noise due to the background also reduces. This provides significant gain in the signal-to-noise (S/N) ratio and also enables to observe much fainter sources than was possible earlier. Human error has been nearly completely taken out of the picture as far as the observational aspects are concerned.
§ ACKNOWLEDGEMENTS
The 8-phase stepper motor driver board was built as part of an M.Sc. project
carried out by students (Nirmit Dudhia and Prashant Raghuvanshi of Gujarat
University) while some of the AVR microcontroller codes were implemented by
Gagan Mallik, student of Nirma University as part of his B.E. project under our
supervision and guidance. We acknowledge useful discussions with N.M. Vadher,
A. B. Shah and C. R. Shah. We are thankful to our colleagues in the Astronomy
and Astrophysics Division and the staff members at Mt Abu Observatory for their
help and support. We also acknowledge the help provided by PRL workshop. This report was prepared with with a style file modified from the IEEE format. This
work is supported by the Dept. of Space, Govt. of India.
§ REFERENCES
* Deshpande, M. R., Joshi, U. C., Kulshrestha, A., Banshidhar, Vadher, N.
M., 1985, An astronomical polarimeter, Bulletin of the Astronomical Society of
India, v. 13, pp. 157-161.
* Frecker, J. E., Serkowski, K., 1976, Linear polarimeter with rapid modulation, achromatic in the 0.3-1.1-micron range, Applied Optics, v. 15. pp. 605-606.
* Joshi, U. C., Deshpande, M. R., Sen, A. K., Kulshrestha, A., 1987, Polarisation
investigations in four peculiar supergiants with high IR excess, Astronomy &
Astrophysics, v. 181. pp. 31-33.
§ WWW RESOURCES
* Free Software Foundation website : <http://www.fsf.org/>
* RTAI : <http://www.rtai.org/>
* Linux kernel website : <http://www.kernel.org/>
* White Dwarf Linux web pages : <http://www.blast.com/index.php?id=66>
* GLADE user interface design software : <http://glade.gnome.org>
* David Schmenk's GCCD software website : <http://schmenk.is-a-geek.com/>
* <http://www.iuac.res.in/ elab/phoenix/>
* <http://tuxgraphics.org/electronics>
* The source codes accompanying this report and further figures are available on the first author's webpages at <http://www.prl.res.in/ shashi/inst.html>
[Schematic of 8-phase stepper driver board]
Schematic of the 8-phase stepper driver PC-104 board
Schematic of the AVR micro-controller stepper driver board
|
arxiv-papers
| 2009-12-01T05:41:19 |
2024-09-04T02:49:06.778891
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Ganesh, U. C. Joshi, K. S. Baliyan, S. N. Mathur, P. S. Patwal, R.\n R. Shah",
"submitter": "Shashikiran Ganesh",
"url": "https://arxiv.org/abs/0912.0076"
}
|
0912.0270
|
# Single-Agent On-line Path Planning in Continuous, Unpredictable and Highly
Dynamic Environments
Nicolás Arturo Barriga Richards
| Universidad Técnica Federico Santa María
---|---
Departamento de Informática |
Valparaíso – Chile |
SINGLE-AGENT ON-LINE PATH PLANNING IN CONTINUOUS, UNPREDICTABLE AND HIGHLY
DYNAMIC ENVIRONMENTS
Tesis presentada como requerimiento parcia
para optar al grado académico de
MAGÍSTER EN CIENCIAS DE LA INGENIERíA INFORMáTICA y al título profesional de
INGENIERO CIVIL EN INFORMÁTICA por Nicolás Arturo Barriga Richards
Comisión Evaluadora:
Dr. Mauricio Solar (Guía, UTFSM) Dr. Horst H. von Brand (UTFSM) Dr. John
Atkinson (UdeC)
NOV 2009
> Departamento de Informática Valparaíso – Chile
TITULO DE LA TESIS: SINGLE-AGENT ON-LINE PATH PLANNING IN CONTINUOUS,
UNPREDICTABLE AND HIGHLY DYNAMIC ENVIRONMENTS
AUTOR: NICOLÁS ARTURO BARRIGA RICHARDS
Tesis presentada como requerimiento parcial para optar al grado académico de
Magíster en Ciencias de la Ingeniería Informática y al título profesional de
Ingeniero Civil en Informática de la Universidad Técnica Federico Santa María.
Dr. Mauricio Solar
Profesor Guía
Dr. Horst H. von Brand
Profesor Correferente
Dr. John Atkinson
Profesor Externo
Nov 2009.
Valparaíso, Chile.
Real stupidity beats artificial intelligence every time
Terry Pratchett
## Index of Contents
toc
###### List of Tables
1. 4.1 Dynamic Environment Results, map 1.
2. 4.2 Dynamic Environment Results, map 2.
3. 4.3 Partially Known Environment Results, map 1.
4. 4.4 Partially Known Environment Results, map 2.
5. 4.5 Unknown Environment Results
###### List of Figures
1. 2.1 RRT during execution
2. 2.2 The roles of the genetic operators
3. 3.1 A Multi-stage Strategy for Dynamic Path Planning
4. 3.2 The arc operator
5. 3.3 The mutation operator
6. 4.1 The dynamic environment, map 1
7. 4.2 The dynamic environment, map 2
8. 4.3 The partially known environment, map 1
9. 4.4 The partially known environment, map 2
10. 4.5 The unknown environment
11. 4.6 Dynamic environment time
12. 4.7 Dynamic environment success rate
## Chapter 1 Introduction
The _dynamic path-planning_ problem consists in finding a suitable plan for
each new configuration of the environment by recomputing a collision-free path
using the new information available at each time step [HA92]. This kind of
problem has to be solved for example by a robot trying to navigate through an
area crowded with people, such as a shopping mall or supermarket. The problem
has been widely addressed in its several flavors, such as cellular
decomposition of the configuration space [Ste95], partial environmental
knowledge [Ste94], high-dimensional configuration spaces [KSLO96] or planning
with non-holonomic constraints [LKJ99]. However, even simpler variations of
this problem are complex enough that they can not be solved with deterministic
techniques, and therefore are worthy of study.
This thesis is focused on algorithms for finding and traversing a collision-
free path in two dimensional space, for a holonomic robot111A holonomic robot
is a robot in which the controllable degrees of freedom is equal to the total
degrees of freedom., without kinodynamic restrictions222Kinodynamic planning
is a problem in which velocity and acceleration bounds must be satisfied, in a
highly dynamic environment, but for comparison purposes three different
scenarios will be tested:
* •
Several unpredictably moving obstacles or adversaries.
* •
Partially known environment, where some obstacles become visible when the
robot approaches each one of them.
* •
Totally unknown environment, where every obstacle is initially invisible to
the planner, and only becomes visible when the robot approaches it.
Besides the obstacles in the second and third scenario we assume that we have
perfect information of the environment at all times.
We will focus on continuous space algorithms and will not consider algorithms
that use a discretized representation of the configuration space333the space
of possible positions that a physical system may attain, such as D* [Ste95],
because for high dimensional problems the configuration space becomes
intractable in terms of both memory and computation time, and there is the
extra difficulty of calculating the discretization size, trading off accuracy
versus computational cost. Only single agent algorithms will be considered
here. On-line as well as off-line algorithms will be studied. An on-line
algorithm is one that is permanently adjusting its solution as the environment
changes, while an off-line algorithm computes a solution only once (however,
it can be executed many times).
The offline Rapidly-Exploring Random Tree (RRT) is efficient at finding
solutions, but the results are far from optimal, and must be post-processed
for shortening, smoothing or other qualities that might be desirable in each
particular problem. Furthermore, replanning RRTs are costly in terms of
computation time, as are evolutionary and cell-decomposition approaches.
Therefore, the novelty of this work is the mixture of the feasibility benefits
of the RRTs, the repairing capabilities of local search, and the computational
inexpensiveness of greedy algorithms, into our lightweight multi-stage
algorithm. Our working hypothesis will be that a multi-stage algorithm, using
different techniques for initial planning and navigation, outperforms current
probabilistic sampling techniques in highly dynamic environments
### 1.1 Problem Formulation
At each time-step, the problem could be defined as an optimization problem
with satisfiability constraints. Therefore, given a path our objective is to
minimize an evaluation function (i.e., distance, time, or path-points), with
the $C_{\text{free}}$ constraint. Formally, let the path
$\rho=p_{1}p_{2}\ldots p_{n}$ be a sequence of points, where
$p_{i}\in\mathbb{R}^{n}$ is a $n$-dimensional point
($p_{1}=q_{\text{init}},p_{n}=q_{\text{goal}}$), $O_{t}\in\mathcal{O}$ the set
of obstacles positions at time $t$, and
$\operatorname{eval}\colon\mathbb{R}^{n}\times\mathcal{O}\mapsto\mathbb{R}$ an
evaluation function of the path depending on the object positions. Our ideal
objective is to obtain the optimum $\rho^{*}$ path that minimizes our
$\operatorname{eval}$ function within a feasibility restriction in the form
$\displaystyle\rho^{*}=\underset{\rho}{\operatorname{argmin}}[\operatorname{eval}(\rho,O_{t})]\textrm{
with }\operatorname{feas}(\rho,O_{t})=C_{\text{free}}$ (1.1)
where $\operatorname{feas}(\cdot,\cdot)$ is a _feasibility_ function that
equals $C_{\text{free}}$ if the path $\rho$ is collision free for the
obstacles $O_{t}$. For simplicity, we use very naive
$\operatorname{eval}(\cdot,\cdot)$ and $\operatorname{feas}(\cdot,\cdot)$
functions, but our approach could be extended easily to more complex
evaluation and feasibility functions. The $\operatorname{feas}(\rho,O_{t})$
function used assumes that the robot is a point object in space, and therefore
if no segments $\overrightarrow{p_{i}p_{i+1}}$ of the path collide with any
object $o_{j}\in O_{t}$, we say that the path is in $C_{\text{free}}$. The
$\operatorname{eval}(\rho,O_{t})$ function is the length of the path, i.e.,
the sum of the distances between consecutive points. This could be easily
changed to any other metric such as the time it would take to traverse this
path, accounting for smoothness, clearness or several other optimization
criteria.
### 1.2 Document Structure
In the following sections we present several path planning methods that can be
applied to the problem described above. In section 2.1 we review the basic
offline, single-query RRT, a probabilistic method that builds a tree along the
free configuration space until it reaches the goal state. Afterwards, we
introduce the most popular replanning variants of RRT: Execution Extended RRT
(ERRT) in section 2.3, Dynamic RRT (DRRT) in section 2.4 and Multipartite RRT
(MP-RRT) in section 2.5. The Evolutionary Planner/Navigator (EP/N), along with
some variants, is presented in section 2.8. Then, in section 3.1 we present a
mixed approach, using a RRT to find an initial solution and the EP/N to
navigate, and finally, in section 3.2 we present our new hybrid multi-stage
algorithm, that uses RRT for initial planning and informed local search for
navigation, plus a simple greedy heuristic for optimization. Experimental
results and comparisons that show that this combination of simple techniques
provides better responses to highly dynamic environments than the standard RRT
extensions are presented in section 4.3. The conclusions and further work are
discussed in section 5.
## Chapter 2 State of the Art
In this chapter we present several path planning methods that can be applied
to the problem described above. First we will introduce variations of the
Rapidly-Exploring Random Tree (RRT), a probabilistic method that builds a tree
along the free configuration space until it reaches the goal state. This
family of planners is fast at finding solutions, but the solutions are far
from optimal, and must be post-processed for shortening, smoothing or other
qualities that might be desirable in each particular problem. Furthermore,
replanning RRTs are costly in terms of computation time. We then introduce an
evolutionary planner with somewhat opposite qualities: It is slow in finding
feasible solutions in difficult maps, but efficient at replanning when a
feasible solution has already been found. It can also optimize the solution
according to any given fitness function without the need for a post-processing
step.
### 2.1 Rapidly-Exploring Random Tree
One of the most successful probabilistic methods for offline path planning
currently in use is the Rapidly-Exploring Random Tree (RRT), a single-query
planner for static environments, first introduced in [Lav98]. RRTs works
towards finding a continuous path from a state $q_{\text{init}}$ to a state
$q_{\text{goal}}$ in the free configuration space $C_{\text{free}}$ by
building a tree rooted at $q_{\text{init}}$. A new state $q_{\text{rand}}$ is
uniformly sampled at random from the configuration space $C$. Then the nearest
node, $q_{\text{near}}$, in the tree is located, and if $q_{\text{rand}}$ and
the shortest path from $q_{\text{rand}}$ to $q_{\text{near}}$ are in
$C_{\text{free}}$, then $q_{\text{rand}}$ is added to the tree (algorithm 1).
The tree growth is stopped when a node is found near $q_{\text{goal}}$. To
speed up convergence, the search is usually biased to $q_{\text{goal}}$ with a
small probability.
In [KL00], two new features are added to RRT. First, the EXTEND function
(algorithm 2) is introduced, which instead of trying to add $q_{\text{rand}}$
directly to the tree, makes a motion towards $q_{\text{rand}}$ and tests for
collisions.
Algorithm 1 $\operatorname{BuildRRT}(q_{\text{init}},q_{\text{goal}})$
1: $T\leftarrow\text{empty tree}$
2: $T.\operatorname{init}(q_{\text{init}})$
3: while $\operatorname{Distance}(T,q_{\text{goal}})>\text{threshold}$ do
4: $q_{\text{rand}}\leftarrow\operatorname{RandomConfig}()$
5: $\operatorname{Extend}(T,q_{\text{rand}})$
6: return $T$
Algorithm 2 $\operatorname{Extend}(T,q)$
1: $q_{\text{near}}\leftarrow\operatorname{NearestNeighbor}(q,T)$
2: if $\operatorname{NewConfig}(q,q_{\text{near}},q_{\text{new}})$ then
3: $T.\operatorname{add\\_vertex}(q_{\text{new}})$
4: $T.\operatorname{add\\_edge}(q_{\text{near}},q_{\text{new}})$
5: if $q_{\text{new}}=q$ then
6: return Reached
7: else
8: return Advanced
9: return Trapped
Then a greedier approach is introduced (the CONNECT function, shown in
algorithms 3 and 4), which repeats EXTEND until an obstacle is reached. This
ensures that most of the time we will be adding states to the tree, instead of
just rejecting new random states. The second extension is the use of two
trees, rooted at $q_{\text{init}}$ and $q_{\text{goal}}$, which are grown
towards each other (see figure 2.1). This significantly decreases the time
needed to find a path.
Figure 2.1: RRT during execution Algorithm 3
$\operatorname{RRTConnectPlanner}(q_{\text{init}},q_{\text{goal}})$
1: $T_{a}\leftarrow\text{tree rooted at $q_{\text{init}}$}$
2: $T_{b}\leftarrow\text{tree rooted at $q_{\text{goal}}$}$
3: $T_{a}.\operatorname{init}(q_{\text{init}})$
4: $T_{b}.\operatorname{init}(q_{\text{goal}})$
5: for $k=1$ to $K$ do
6: $q_{\text{rand}}\leftarrow\operatorname{RandomConfig}()$
7: if not ($\operatorname{Extend}(T_{a},q_{\text{rand}})=\text{Trapped}$) then
8: if $\operatorname{Connect}(T_{b},q_{\text{new}})=\text{Reached}$ then
9: return $\operatorname{Path}(T_{a},T_{b})$
10: $\operatorname{Swap}(T_{a},T_{b})$
11: return Failure
Algorithm 4 $\operatorname{Connect}(T,q)$
1: repeat
2: $S\leftarrow\operatorname{Extend}(T,q)$
3: until $(S\neq\text{Advanced})$
### 2.2 Retraction-Based RRT Planner
The Retraction-based RRT Planner presented in [ZM08] aims at improving the
performance of the standard offline RRT in static environments with narrow
passages. The basic idea of the $\operatorname{Optimize}(q_{r},q_{n})$
function in algorithm 5 is to iteratively retract a randomly generated
configuration that is in $C_{\text{obs}}$ to the closest boundary point in
$C_{\text{free}}$. So, instead of using the standard extension that tries to
extend in a straight line from $q_{\text{near}}$ to $q_{\text{rand}}$, it
extends from $q_{\text{near}}$ to the closest point in $C_{\text{free}}$ to
$q_{\text{rand}}$. This gives more samples in narrow passages. This technique
could easily be applied to on-line RRT planners.
Algorithm 5 Retraction-based RRT Extension
1: $q_{r}\leftarrow\text{a random configuration in $C_{\text{space}}$}$
2: $q_{n}\leftarrow\text{the nearest neighbor of $q_{r}$ in $T$}$
3: if $\operatorname{CollisionFree}(q_{n},q_{r})$ then
4: $T.\operatorname{addVertex}(q_{r})$
5: $T.\operatorname{addEdge}(q_{n},q_{r})$
6: else
7: $S\leftarrow\operatorname{Optimize}(q_{r},q_{n})$
8: for all $q_{i}\in S$ do
9: Standard RRT Extension$(T,q_{i})$
10: return $T$
### 2.3 Execution Extended RRT
The Execution Extended RRT presented in [BV02] introduces two extensions to
RRT to build an on-line planner, the waypoint cache and adaptive cost penalty
search, which improve re-planning efficiency and the quality of generated
paths. ERRT uses a kd-tree (see section 2.7) to speed nearest neighbor look-
up, and does not use bidirectional search. The waypoint cache is implemented
by keeping a constant size array of states, and whenever a plan is found, all
the states in the plan are placed in the cache with random replacement. Then,
when the tree is no longer valid, a new tree must be grown, and there are
three possibilities for choosing a new target state, as shown in algorithm 6,
which is used instead of $\operatorname{RandomConfig}()$ in previous
algorithms. With probability P[goal], the goal is chosen as the target; with
probability P[waypoint], a random waypoint is chosen, and with the remaining
probability a uniform state is chosen as before. In [BV02] the values used are
P[goal]$=0.1$ and P[waypoint]$=0.6$.
Another extension is adaptive cost penalty search, where the planner
adaptively modified a parameter to help it find shorter paths. A value of 1
for beta will always extend from the root node, while a value of 0 is
equivalent to the original algorithm. However, the paper [BV02] lacks
implementation details and experimental results on this extension.
Algorithm 6 $\operatorname{ChooseTarget}(q,{\text{goal}})$
1: $p\leftarrow\operatorname{UniformRandom}(0.0,1.0)$
2: $i\leftarrow\operatorname{UniformRandom}(0.0,\text{NumWayPoints})$
3: if $0<p<\text{GoalProb}$ then
4: return $q_{\text{goal}}$
5: else if $\text{GoalProb}<p<\text{GoalProb}+\text{WayPointProb}$ then
6: return $\text{WayPointCache}[i]$
7: else if $\text{GoalProb}+\text{WayPointProb}<p<1$ then
8: return $\text{RandomConfig}()$
### 2.4 Dynamic RRT
The Dynamic Rapidly-Exploring Random Tree described in [FKS06] is a
probabilistic analog to the widely used D* family of algorithms. It works by
growing a tree from $q_{\text{goal}}$ to $q_{\text{init}}$, as shown in
algorithm 7. This has the advantage that the root of the tree does not have to
be moved during the lifetime of the planning and execution. In some problem
classes the robot has limited range sensors, thus moving or newly appearing
obstacles will be near the robot, not near the goal. In general this strategy
attempts to trim smaller branches that are farther away from the root. When
new information concerning the configuration space is received, the algorithm
removes the newly-invalid branches of the tree (algorithms 9 and 10), and
grows the remaining tree, focusing, with a certain probability (empirically
tuned to $0.4$ in [FKS06]) to a vicinity of the recently trimmed branches, by
using the waypoint cache of the ERRT (algorithm 6). In experiments presented
in [FKS06] DRRT vastly outperforms ERRT.
Algorithm 7 $\operatorname{DRRT}()$
1: $q_{\text{robot}}\leftarrow\text{the current robot position}$
2: $T\leftarrow\operatorname{BuildRRT}(q_{\text{goal}},q_{\text{robot}})$
3: while $q_{\text{robot}}\neq q_{\text{goal}}$ do
4: $q_{\text{next}}\leftarrow\operatorname{Parent}(q_{\text{robot}})$
5: Move from $q_{\text{robot}}$ to $q_{\text{next}}$
6: for all obstacles that changed $O$ do
7: $\operatorname{InvalidateNodes}(O)$
8: if Solution path contains an invalid node then
9: $\operatorname{ReGrowRRT}()$
Algorithm 8 $\operatorname{ReGrowRRT}()$
1: $\operatorname{TrimRRT}()$
2: $\operatorname{GrowRRT}()$
Algorithm 9 $\operatorname{TrimRRT}()$
1: $S\leftarrow\emptyset,i\leftarrow 1$
2: while $i<T.\operatorname{size}()$ do
3: $q_{i}\leftarrow T.\operatorname{node}(i)$
4: $q_{p}\leftarrow\operatorname{Parent}(q_{i})$
5: if $q_{p}.\text{flag}=\text{INVALID}$ then
6: $q_{i}.\text{flag}\leftarrow\text{INVALID}$
7: if $q_{i}.\text{flag}\neq\text{INVALID}$ then
8: $S\leftarrow S\bigcup\\{q_{i}\\}$
9: $i\leftarrow i+1$
10: $T\leftarrow\operatorname{CreateTreeFromNodes}(S)$
Algorithm 10 $\operatorname{InvalidateNodes}(obstacle)$
1: $E\leftarrow\operatorname{FindAffectedEdges}(\text{obstacle})$
2: for all $e\in E$ do
3: $q_{e}\leftarrow\operatorname{ChildEndpointNode}(e)$
4: $q_{e}.\text{flag}\leftarrow\text{INVALID}$
### 2.5 Multipartite RRT
Multipartite RRT presented in [ZKB07] is another RRT variant which supports
planning in unknown or dynamic environments. MP-RRT maintains a forest $F$ of
disconnected sub-trees which lie in $C_{\text{free}}$, but which are not
connected to the root node $q_{\text{root}}$ of $T$, the main tree. At the
start of a given planning iteration, any nodes of $T$ and $F$ which are no
longer valid are deleted, and any disconnected sub-trees which are created as
a result are placed into $F$ (as seen in algorithms 11 and 12). With given
probabilities, the algorithm tries to connect $T$ to a new random state, to
the goal state, or to the root of a tree in $F$ (algorithm 13). In [ZKB07], a
simple greedy smoothing heuristic is used, that tries to shorten paths by
skipping intermediate nodes. The MP-RRT is compared to an iterated RRT, ERRT
and DRRT, in 2D, 3D and 4D problems, with and without smoothing. For most of
the experiments, MP-RRT modestly outperforms the other algorithms, but in the
4D case with smoothing, the performance gap in favor of MP-RRT is much larger.
The authors explained this fact due to MP-RRT being able to construct much
more robust plans in the face of dynamic obstacle motion. Another algorithm
that utilizes the concept of forests is Reconfigurable Random Forests (RRF)
presented in [LS02], but without the success of MP-RRT.
Algorithm 11 $\operatorname{MPRRTSearch}(q_{\text{init}})$
1: $T\leftarrow\text{the previous search tree, if any}$
2: $F\leftarrow\text{the previous forest of disconnected sub-trees}$
3: $q_{\text{init}}\leftarrow\text{the initial state}$
4: if $T=\emptyset$ then
5: $q_{\text{root}}\leftarrow q_{\text{init}}$
6: $\operatorname{Insert}(q_{\text{root}},T)$
7: else
8: $\operatorname{PruneAndPrepend}(T,F,q_{\text{init}})$
9: if $\operatorname{TreeHasGoal}(T)$ then
10: return true
11: while search time/space remaining do
12: $q_{\text{new}}\leftarrow\operatorname{SelectSample}(F)$
13:
$q_{\text{near}}\leftarrow\operatorname{NearestNeighbor}(q_{\text{new},T})$
14: if $q_{\text{new}}\in F$ then
15:
$b_{\text{connect}}\leftarrow\operatorname{Connect}(q_{\text{near}},q_{\text{new}})$
16: if $b_{\text{connect}}$ and $\operatorname{TreeHasGoal}(T)$ then
17: return true
18: else
19:
$b_{\text{extend}}\leftarrow\operatorname{Extend}(q_{\text{near}},q_{\text{new}})$
20: if $b_{\text{extend}}$ and $\operatorname{IsGoal}(q_{\text{new}})$ then
21: return true
22: return false
Algorithm 12 $\operatorname{PruneAndPrepend}(T,F,q_{\text{init}})$
1: for all $q\in T,F$ do
2: if not $\operatorname{NodeValid}(q)$ then
3: $\operatorname{KillNode}(q)$
4: else if not $\operatorname{ActionValid}(q)$ then
5: $\operatorname{SplitEdge}(q)$
6: if not $T=\emptyset$ and $q_{\text{root}}\neq q_{\text{init}}$ then
7: if not $\operatorname{ReRoot}(T,q_{\text{init}})$ then
8: $F\leftarrow F\bigcup T$
9: $T.\operatorname{init}(q_{\text{init}})$
Algorithm 13 $\operatorname{SelectSample}(F)$
1: $p\leftarrow\operatorname{Random}(0,1)$
2: if $p<p_{\text{goal}}$ then
3: $q_{\text{new}}\leftarrow q_{\text{goal}}$
4: else if $p<(p_{\text{goal}}+p_{\text{forest}})$ and not
$\operatorname{Empty}(F)$ then
5: $q_{\text{new}}\leftarrow q\in\operatorname{SubTreeRoots}(F)$
6: else
7: $q_{\text{new}}\leftarrow\operatorname{RandomState}()$
8: return $q_{\text{new}}$
### 2.6 Rapidly Exploring Evolutionary Tree
The Rapidly Exploring Evolutionary Tree, introduced in [MWS07] uses a
bidirectional RRT and a kd-tree (see section 2.7) for efficient nearest
neighbor search. The modifications to the $\operatorname{Extend}()$ function
are shown in algorithm 14. The re-balancing of a kd-tree is costly, and in
this paper a simple threshold on the number of nodes added before re-balancing
was used. The authors suggest using the method described in [AL02] and used in
[BV02] to improve the search speed. The novelty in this algorithm comes from
the introduction of an evolutionary algorithm [BFM97] that builds a population
of biases for the RRTs. The genotype of the evolutionary algorithm consists of
a single robot configuration for each tree. This configuration is sampled
instead of the uniform distribution. To balance exploration and exploitation,
the evolutionary algorithm was designed with 50% elitism. The fitness function
is related to the number of left and right branches traversed during the
insertion of a new node in the kd-tree. The goal is to introduce a bias to the
RRT algorithm which shows preference to nodes created away from the center of
the tree. The authors suggest combining RET with DRRT or MP-RRT.
Algorithm 14 $\operatorname{ExtendToTarget}(T)$
1: static $p$: population, $inc\leftarrow 1$
2: $p^{\prime}$: temporary population
3: if $\text{inc}>\operatorname{length}(p)$ then
4: $\operatorname{SortByFitness}(p)$
5: $p^{\prime}\leftarrow\text{null}$
6: for all $i\in p$ do
7: if i is in upper 50% then
8: $\operatorname{AddIndividual}(i,p^{\prime})$
9: else
10: $i\leftarrow\operatorname{RandomState}()$
11: $\operatorname{AddIndividual}(i,p^{\prime})$
12: $p\leftarrow p^{\prime}$
13: $\text{inc}\leftarrow 1$
14: $q_{r}\leftarrow p(\text{inc})$
15: $q_{\text{near}}\leftarrow\operatorname{Nearest}(T,q_{r})$
16: $q_{\text{new}}\leftarrow\operatorname{Extend}(T,q_{\text{near}})$
17: if $q_{\text{new}}\neq\emptyset$ then
18: $\operatorname{AddNode}(T,q_{\text{new}})$
19:
$\operatorname{AssignFitness}(p(\text{inc}),\operatorname{fitness}(q_{\text{new}})$
20: else
21: $\operatorname{AssignFitness}(p(\text{inc}),0)$
22: return $q_{\text{new}}$
### 2.7 Multidimensional Binary Search Trees
The kd-tree, first introduced in [Ben75], is a binary tree in which every node
is a k-dimensional point. Every non-leaf node generates a splitting hyperplane
that divides the space into two subspaces. In the RRT algorithm, the number of
points grows incrementally, unbalancing the tree, thus slowing nearest-
neighbor queries. Re-balancing a kd-tree is costly, so in [AL02] the authors
present another approach: A vector of trees is constructed, where for $n$
points there is a tree that contains $2^{i}$ points for each $"1"$ in the
$i^{th}$ place of the binary representation of $n$. As bits are cleared in the
representation due to increasing $n$, the trees are deleted, and the points
are included in a tree that corresponds to the higher-order bit which is
changed to $"1"$. This general scheme incurs in logarithmic-time overhead,
regardless of dimension. Experiments show a substantial performance increase
compared to a naive brute-force approach.
### 2.8 Evolutionary Planner/Navigator
An evolutionary algorithm [BFM97] is a generic population-based meta-heuristic
optimization algorithm. It is inspired in biological evolution, using methods
such as individual selection, reproduction and mutation. The population is
composed of candidate solutions and they are evaluated according to a fitness
function.
The Evolutionary Planner/Navigator presented in [XMZ96], [XMZT97], and [TX97]
is an evolutionary algorithm for path finding in dynamic environments. A high
level description is shown in algorithm 15. A difference with RRT is that it
can optimize the path according to any fitness function defined (length,
smoothness, etc), without the need for a post-processing step. Experimental
tests have shown it has good performance for sparse maps, but no so much for
difficult maps with narrow passages or too crowded with obstacles. However,
when a feasible path is found, it is very efficient at optimizing it and
adapting to the dynamic obstacles.
Algorithm 15 EP/N
1: $P(t)$: population at generation $t$
2: $t\leftarrow 0$
3: $\operatorname{Initialize}(P(t))$
4: $\operatorname{Evaluate}(P(t))$
5: while (not termination-condition) do
6: $t\leftarrow t+1$
7: Select operator $o_{j}$ with probability $p_{j}$
8: Select parent(s) from $P(t)$
9: Produce offspring applying $o_{j}$ to selected parent(s)
10: Evaluate offspring
11: Replace worst individual in $P(t)$ by new offspring
12: Select best individual $p$ from $P(t)$
13: if $\operatorname{Feasible}(p)$ then
14: Move along path $p$
15: Update all individuals in $P(t)$ with current position
16: if changes in environment then
17: Update map
18: $\operatorname{Evaluate}(P(t))$
19: $t\leftarrow t+1$
Every individual in the population is a sequence of nodes, representing nodes
in a path consisting of straight-line segments. Each node consists of an
$(x,y)$ pair and a state variable $b$ with information about the feasibility
of the point and the path segment connecting it to the next point. Individuals
have variable length.
Since a path $p$ can be either feasible or unfeasible, two evaluation
functions are used. For feasible paths (equation 2.1), the goal is to minimize
distance traveled, maintain a smooth trajectory and satisfy a clearance
requirement (the robot should not approach the obstacles too closely). For
unfeasible paths, we use equation 2.2, taken from [Xia97], where $\mu$ is the
number of intersections of a whole path with obstacles and $\eta$ is the
average number of intersections per unfeasible segment.
$\operatorname{eval}_{f}(p)=w_{d}\cdot\operatorname{dist}(p)+w_{s}\cdot\operatorname{smooth}(p)+w_{c}\cdot\operatorname{clear}(p)$
(2.1)
$\operatorname{eval}_{u}(p)=\mu+\eta$ (2.2)
Figure 2.2: The roles of the genetic operators
EP/N uses eight different operators, as shown in figure 2.2 (description taken
from [XMZ96]):
Crossover: Recombines two (parent) paths into two new paths. The parent paths
are divided randomly into two parts respectively and recombined: The first
part of the first path with the second part of the second path, and the first
part of the second path with the second part of the first path. Note that
there can be different numbers of nodes in the two parent paths.
Mutate_1: Used to fine tune node coordinates in a feasible path for shape
adjustment. This operator randomly adjusts node coordinates within some local
clearance of the path so that the path remains feasible afterwards.
Mutate_2: Used for large random changes of node coordinates in a path, which
can be either feasible or unfeasible.
Insert-Delete: Operates on an unfeasible path by inserting randomly generated
new nodes into unfeasible path segments and deleting unfeasible nodes (i.e.,
path nodes that are inside obstacles).
Delete: Deletes nodes from a path, which can be either feasible or unfeasible.
If the path is unfeasible, the deletion is done randomly. Otherwise, the
operator decides whether a node should definitely be deleted based on some
heuristic knowledge, and if a node is not definitely deletable, its deletion
will be random.
Swap: Swaps the coordinates of randomly selected adjacent nodes in a path,
which can be either feasible or unfeasible.
Smooth: Smoothens turns of a feasible path by “cutting corners,” i.e., for a
selected node, the operator inserts two new nodes on the two path segments
connected to that node respectively and deletes that selected node. The nodes
with sharper turns are more likely to be selected.
Repair: Repairs a randomly selected unfeasible segment in a path by “pulling”
the segment around its intersecting obstacle.
The probabilities of using each operator is set randomly at the beginning, and
then are updated according to the success ratio of each operator, so more
successful operators are used more often, and automatically chosen according
to the instance of the problem, eliminating the difficult problem of hand
tuning the probabilities.
In [TX97], the authors include a memory buffer for each individual to store
good paths from its ancestors, which gave a small performance gain.
In [EAA04], the authors propose strategies for improving the stability and
controlling population diversity for a simplified version of the EP/N. An
improvement proposed by the authors in [XMZT97] is using heuristics for the
initial population, instead of random initialization. We will consider this
improvement in section 3.1.
Other evolutionary algorithms have also been proposed for similar problems, in
[NG04] a binary genetic algorithm is used for an offline planner, and [NVTK03]
presents an algorithm to generate curved trajectories in 3D space for an
unmanned aerial vehicle.
EP/N has been adapted to an 8-connected grid model in [AR08] (with previous
work in [AR05] and [Alf05]). The authors study two different crossover
operators and four asexual operators. Experimental results for this new
algorithm (EvP) in static unknown environments show that it is faster than
EP/N.
## Chapter 3 Proposed Techniques
### 3.1 Combining RRT and EP/N
As mentioned in section 2, RRT variants produce suboptimal solutions, which
must later be post-processed for shortening, smoothing or other desired
characteristics. On the other hand, EP/N, presented in section 2.8, can
optimize a solution according to any given fitness function. However, this
algorithm is slower at finding a first feasible solution. In this section we
propose a combined approach, that uses RRT to find an initial solution to be
used as starting point for EP/N, taking advantage of the strong points of both
algorithms.
#### 3.1.1 The Combined Strategy
##### Initial Solution
EP/N as presented in section 2.8 can not find feasible paths in a reasonable
amount of time in any but very sparse maps. For this reason, RRT will be used
to generate a first initial solution, ignoring the effects produced by dynamic
objects. This solution will be in the initial population of the evolutionary
algorithm, along with random solutions.
##### Feasibility and Optimization
EP/N is the responsible of regaining feasibility when it is lost due to a
moving obstacle or a new obstacle found in a partially known or totally
unknown environment. If a feasible solution can not be found in a given amount
of time, the algorithm is restarted, keeping its old population, but adding a
new individual generated by RRT.
#### 3.1.2 Algorithm Implementation
Algorithm 16 $\operatorname{Main}()$
1: $q_{\text{robot}}\leftarrow\text{is the current robot position}$
2: $q_{\text{goal}}\leftarrow\text{is the goal position}$
3: while $q_{\text{robot}}\neq q_{\text{goal}}$ do
4: $\operatorname{updateWorld}(\text{time})$
5: $\operatorname{processRRTEPN}(\text{time})$
The combined RRT-EP/N algorithm proposed here works by alternating environment
updates and path planning, as can be seen in algorithm 16. The first stage of
the path planning (see algorithm 17) is to find an initial path using a RRT
technique, ignoring any cuts that might happen during environment updates.
Thus, the RRT ensures that the path found does not collide with static
obstacles, but might collide with dynamic obstacles in the future. When a
first path is found, the navigation is done by using the standard EP/N as
shown in algorithm 15.
Algorithm 17 $\operatorname{processRRTEPN}(time)$
1: $q_{\text{robot}}\leftarrow\text{the current robot position}$
2: $q_{\text{start}}\leftarrow\text{the starting position}$
3: $q_{\text{goal}}\leftarrow\text{the goal position}$
4: $T_{\text{init}}\leftarrow\text{the tree rooted at the robot position}$
5: $T_{\text{goal}}\leftarrow\text{the tree rooted at the goal position}$
6: $\text{path}\leftarrow\text{the path extracted from the merged RRTs}$
7: $q_{\text{robot}}\leftarrow q_{\text{start}}$
8: $T_{\text{init}}.\operatorname{init}(q_{\text{robot}})$
9: $T_{\text{goal}}.\operatorname{init}(q_{\text{goal}})$
10: while time elapsed $<$ time do
11: if First path not found then
12: $\operatorname{RRT}(T_{\text{init}},T_{\text{goal}})$
13: else
14: $\operatorname{EP/N}()$
### 3.2 A Simple Multi-stage Probabilistic Algorithm
In highly dynamic environments, with many (or a few but fast) relatively small
moving obstacles, regrowing trees are pruned too fast, cutting away important
parts of the trees before they can be replaced. This dramatically reduces the
performance of the algorithms, making them unsuitable for these classes of
problems. We believe that better performance could be obtained by slightly
modifying a RRT solution using simple obstacle-avoidance operations on the new
colliding points of the path by informed local search. The path could be
greedily optimized if the path has reached the feasibility condition.
#### 3.2.1 A Multi-stage Probabilistic Strategy
If solving equation 1.1 is not a simple task in static environments, solving
dynamic versions turns out to be even more difficult. In dynamic path planning
we cannot wait until reaching the optimal solution because we must deliver a
“good enough” plan within some time restriction. Thus, a heuristic approach
must be developed to tackle the on-line nature of the problem. The heuristic
algorithms presented in sections 2.3, 2.4 and 2.5 extend a method developed
for static environments, which produces poor response to highly dynamic
environments and unwanted complexity of the algorithms.
We propose a multi-stage combination of simple heuristic and probabilistic
techniques to solve each part of the problem: Feasibility, initial solution
and optimization.
Figure 3.1: A Multi-stage Strategy for Dynamic Path Planning. This figure
describes the life-cycle of the multi-stage algorithm presented here. The RRT,
informed local search, and greedy heuristic are combined to produce a cheap
solution to the dynamic path planning problem.
##### Feasibility
The key point in this problem is the hard constraint in equation 1.1 which
must be met before even thinking about optimizing. The problem is that in
highly dynamic environments a path turns rapidly from feasible to unfeasible —
and the other way around — even if our path does not change. We propose a
simple _informed local search_ to obtain paths in $C_{\text{free}}$. The idea
is to randomly search for a $C_{\text{free}}$ path by modifying the nearest
colliding segment of the path. As we include in the search some knowledge of
the problem, the _informed_ term is coined to distinguish it from blind local
search. The details of the operators used for the modification of the path are
described in section 3.2.2. If a feasible solution can not be found in a given
amount of time, the algorithm is restarted, with a new starting point
generated by a RRT variant.
##### Initial Solution
The problem with local search algorithms is that they repair a solution that
it is assumed to be near the feasibility condition. Trying to produce feasible
paths from scratch with local search (or even with evolutionary algorithms
[XMZT97]) is not a good idea due the randomness of the initial solution.
Therefore, we propose feeding the informed local search with a _standard RRT_
solution at the start of the planning, as can be seen in figure 3.1.
##### Optimization
Without an optimization criterion, the path could grow infinitely large in
time or size. Therefore, the $\operatorname{eval}(\cdot,\cdot)$ function must
be minimized when a (temporary) feasible path is obtained. A simple greedy
technique is used here: We test each point in the solution to check if it can
be removed maintaining feasibility; if so, we remove it and check the
following point, continuing until reaching the last one.
#### 3.2.2 Algorithm Implementation
Algorithm 18 $\operatorname{Main}()$
1: $q_{\text{robot}}\leftarrow\text{the current robot position}$
2: $q_{\text{goal}}\leftarrow\text{the goal position}$
3: while $q_{\text{robot}}\neq q_{\text{goal}}$ do
4: $\operatorname{updateWorld}(\text{time})$
5: $\operatorname{processMultiStage}(\text{time})$
The multi-stage algorithm proposed in this thesis works by alternating
environment updates and path planning, as can be seen in algorithm 18. The
first stage of the path planning (see algorithm 19) is to find an initial path
using a RRT technique, ignoring any cuts that might happen during environment
updates. Thus, RRT ensures that the path found does not collide with static
obstacles, but might collide with dynamic obstacles in the future. When a
first path is found, the navigation is done by alternating a simple informed
local search and a simple greedy heuristic as shown in figure 3.1.
Algorithm 19 $\operatorname{processMultiStage}(\text{time})$
1: $q_{\text{robot}}\leftarrow$ is the current robot position
2: $q_{\text{start}}\leftarrow$ is the starting position
3: $q_{\text{goal}}\leftarrow$ is the goal position
4: $T_{\text{init}}\leftarrow$ is the tree rooted at the robot position
5: $T_{\text{goal}}\leftarrow$ is the tree rooted at the goal position
6: $\text{path}\leftarrow$ is the path extracted from the merged RRTs
7: $q_{\text{robot}}\leftarrow q_{\text{start}}$
8: $T_{\text{init}}.\operatorname{init}(q_{\text{robot}})$
9: $T_{\text{goal}}.\operatorname{init}(q_{\text{goal}})$
10: while time elapsed $<$ time do
11: if First path not found then
12: $\operatorname{RRT}(T_{\text{init}},T_{\text{goal}})$
13: else
14: if path is not collision free then
15: firstCol $\leftarrow$ collision point closest to robot
16: $\operatorname{arc}(\text{path},\text{firstCol})$
17: $\operatorname{mut}(\text{path},\text{firstCol})$
18: $\operatorname{postProcess}(\text{path})$
Figure 3.2: The arc operator. This operator draws an offset value $\Delta$
over a fixed interval called vicinity. Then, one of the two axes is selected
to perform the arc and two new consecutive points are added to the path.
$n_{1}$ is placed at a $\pm\Delta$ of the point $b$ and $n_{2}$ at $\pm\Delta$
of point $c$, both of them over the same selected axis. The axis, sign and
value of $\Delta$ are chosen randomly from an uniform distribution. Figure
3.3: The mutation operator. This operator draws two offset values $\Delta_{x}$
and $\Delta_{y}$ over a vicinity region. Then the same point $b$ is moved in
both axes from $b=[b_{x},b_{y}]$ to
$b^{\prime}=[b_{x}\pm\Delta_{x},b_{y}\pm\Delta_{y}]$, where the sign and
offset values are chosen randomly from an uniform distribution.
The second stage is the informed local search, which is a two step function
composed by the _arc_ and _mutate_ operators (algorithms 20 and 21). The first
one tries to build a square arc around an obstacle, by inserting two new
points between two points in the path that form a segment colliding with an
obstacle, as shown in figure 3.2. The second step in the function is a
mutation operator that moves a point close to an obstacle to a random point in
the vicinity, as explained graphically in figure 3.3. The mutation operator is
inspired by the ones used in the Adaptive Evolutionary Planner/Navigator
(EP/N) presented in [XMZT97], while the arc operator is derived from the arc
operator in the Evolutionary Algorithm presented in [AR05].
Algorithm 20 $\operatorname{arc}(\text{path},\text{firstCol})$
1: $\text{vicinity}\leftarrow\text{some vicinity size}$
2:
$\text{randDev}\leftarrow\operatorname{random}(-\text{vicinity},\text{vicinity})$
3: $\text{point1}\leftarrow\text{path}[\text{firstCol}]$
4: $\text{point2}\leftarrow\text{path}[\text{firstCol}+1]$
5: if $\operatorname{random}()\%2$ then
6:
$\text{newPoint1}\leftarrow(\text{point1}[X]+\text{randDev},\text{point1}[Y])$
7:
$\text{newPoint2}\leftarrow(\text{point2}[X]+\text{randDev},\text{point2}[Y])$
8: else
9:
$\text{newPoint1}\leftarrow(\text{point1}[X],\text{point1}[Y]+\text{randDev})$
10:
$\text{newPoint2}\leftarrow(\text{point2}[X],\text{point2}[Y]+\text{randDev})$
11: if path segments point1-newPoint1-newPoint2-point2 are collision free then
12: Add new points between point1 and point2
13: else
14: Drop new points
Algorithm 21 $\operatorname{mut}(\text{path},\text{firstCol})$
1: vicinity $\leftarrow$ some vicinity size
2: path[firstCol][X] $+=$ random$(-\text{vicinity},\text{vicinity})$
3: path[firstCol][Y] $+=$ random$(-\text{vicinity},\text{vicinity})$
4: if path segments before and after path[firstCol] are collision free then
5: Accept new point
6: else
7: Reject new point
The third and last stage is the greedy optimization heuristic, which can be
seen as a post-processing for path shortening, that eliminates intermediate
nodes if doing so does not create collisions, as is described in the algorithm
22.
Algorithm 22 postProcess$(path)$
1: $i\leftarrow 0$
2: while $i<\operatorname{path.size}()-2$ do
3: if segment $\operatorname{path}[i]\text{\ to\
}\operatorname{path}[i+2]\text{\ is collision free}$ then
4: Delete path[i+1]
5: else
6: $i\leftarrow i+1$
## Chapter 4 Experimental Setup and Results
### 4.1 Experimental Setup
Although the algorithms developed in this thesis are aimed at dynamic
environments, for the sake of completeness they will also be compared in
partially known environments and in totally unknown environments, where some
or all of the obstacles become visible to the planner as the robot approaches
each one of them, simulating a robot with limited sensor range.
#### 4.1.1 Dynamic Environment
The first environment for our experiments consists on two maps with 30 moving
obstacles the same size of the robot, with a random speed between 10% and 55%
the speed of the robot. Good performance in this environment is the main focus
of this thesis. This _dynamic environments_ are illustrated in figures 4.1 and
4.2.
Figure 4.1: The dynamic environment, map 1. The _green_ square is our robot,
currently at the start position. The _blue_ squares are the moving obstacles.
The _blue_ cross is the goal. Figure 4.2: The dynamic environment, map 2. The
_green_ square is our robot, currently at the start position. The _blue_
squares are the moving obstacles. The _blue_ cross is the goal.
#### 4.1.2 Partially Known Environment
The second environment uses the same maps, but with a few obstacles, three to
four times the size of the robot, that become visible when the robot
approaches each one of them. This is the kind of environment that most dynamic
RRT variants were designed for. The _partially known environments_ are
illustrated in figure 4.3 and 4.4.
Figure 4.3: The partially known environment, map 1. The _green_ square is our
robot, currently at the start position. The _yellow_ squares are the suddenly
appearing obstacles. The _blue_ cross is the goal. Figure 4.4: The partially
known environment, map 2. The _green_ square is our robot, currently at the
start position. The _yellow_ squares are the suddenly appearing obstacles. The
_blue_ cross is the goal.
#### 4.1.3 Unknown Environment
For completeness sake, we will compare the different technique in a third
environment, were we use one of the maps presented before, but all the
obstacles will initially be unknown to the planners, and will become visible
as the robot approaches them, forcing several re-plans. This _unknown
environment_ is illustrated in figure 4.5.
Figure 4.5: The unknown environment. The _green_ square is our robot,
currently at the start position. The _blue_ cross is the goal. None of the
obstacles is visible initially to the planners
### 4.2 Implementation Details
The algorithms where implemented in C++ using the MoPa framework111MoPa
homepage: https://csrg.inf.utfsm.cl/twiki4/bin/view/CSRG/MoPa partly developed
by the author. This framework features exact collision detection, three
different map formats (including .pbm images from any graphic editor),
dynamic, unknown and partially known environments and support for easily
adding new planners. One of the biggest downsides is that it only supports
rectangular objects, so several objects must be used to represent other
geometrical shapes, as in figure 4.4, composed of 1588 rectangular objects.
There are several variations that can be found in the literature when
implementing RRT. For all our RRT variants, the following are the details on
where we departed from the basics:
1. 1.
We always use two trees rooted at $q_{init}$ and $q_{goal}$.
2. 2.
Our EXTEND function, if the point cannot be added without collisions to a
tree, adds the mid point between the nearest tree node and the nearest
collision point to it.
3. 3.
In each iteration, we try to add the new randomly generated point to both
trees, and if successful in both, the trees are merged, as proposed in [KL00].
4. 4.
We believe that there might be significant performance differences between
allowing or not allowing the robot to advance towards the node nearest to the
goal when the trees are disconnected, as proposed in [ZKB07].
In point 4 above, the problem is that the robot would become stuck if it
enters a small concave zone of the environment (like a room in a building)
while there are moving obstacles inside that zone, but otherwise it can lead
to better performance. Therefore we present results for both kinds of
behavior: DRRT-adv and MP-RRT-adv move even when the trees are disconnected,
while DRRT-noadv and MP-RRT-noadv only move when the trees are connected.
In MP-RRT, the forest was handled by simply replacing the oldest tree in it if
the forest had reached the maximum allowed size.
Concerning the parameter selection, the probability for selecting a point in
the vicinity of a point in the waypoint cache in DRRT was set to 0.4 as
suggested in [FKS06]. The probability for trying to reuse a subtree in MP-RRT
was set to 0.1 as suggested in [ZKB07]. Also, the forest size was set to 25
and the minimum size of a tree to be saved in the forest was set to 5 nodes.
For the combined RRT-EP/N, it was considered the planner was stuck after two
seconds without a feasible solution in the population, at which point a new
solution from a RRT variant is inserted into the population. For the simple
multi-stage probabilistic algorithm, the restart is made after one second of
encountering the same obstacle along the planned path. This second approach,
which seems better, cannot be applied to the RRT-EP/N, because there is no
single path to check for collisions, but instead a population of paths. The
restart times where manually tuned.
### 4.3 Results
The three algorithms were run a hundred times in each environment and map
combination. The cutoff time was five minutes for all tests, after which the
robot was considered not to have reached the goal. Results are presented
concerning:
* •
Success rate (S.R.): The percentage of times the robot arrived at the goal,
before reaching the five minutes cutoff time. This does not account for
collisions or time the robot was stopped waiting for a plan.
* •
Number of nearest neighbor lookups performed by each algorithm (N.N.): One of
the possible bottlenecks for tree-based algorithms
* •
Number of collision checks performed (C.C.), which in our specific
implementation takes a significant percentage of the running time
* •
Time it took the robot to reach the goal, $\pm$ the standard deviation.
#### 4.3.1 Dynamic Environment Results
The results in tables 4.1 and 4.2 show that the multi-stage algorithm takes
considerably less time than the DRRT and MP-RRT to reach the goal, with far
less collision checks. The combined RRT-EP/N is a close second. It was
expected that nearest neighbor lookups would be much lower in both combined
algorithms than in the RRT variants, because they are only performed in the
initial phase and restarts, not during navigation. The combined algorithms
produce more consistent results within a map, as shown by their smaller
standard deviations, but also across different maps. An interesting fact is
that in map 1 DRRT is slightly faster than MP-RRT, and in map 2 MP-RRT is
faster than DRRT. However the differences are too small to draw any
conclusions. Figures 4.6 and 4.7 show the times and success rates of the
different algorithms, when changing the number of dynamic obstacles in map 1.
The simple multi-stage algorithm and the mixed RRT-EP/N clearly show the best
performance, while the DRRT-adv and MP-RRT-adv significantly reduce their
success rate when confronted to more than 30 moving obstacles.
Table 4.1: Dynamic Environment Results, map 1. Algorithm | S.R.[%] | C.C. | N.N. | Time[s]
---|---|---|---|---
Multi-stage | 99 | 23502 | 1122 | 6.62$\ \pm\ $ | 0.7
RRT-EP/N | 100 | 58870 | 1971 | 10.34$\ \pm\ $ | 14.15
DRRT-noadv | 100 | 91644 | 4609 | 20.57$\ \pm\ $ | 20.91
DRRT-adv | 98 | 107225 | 5961 | 23.72$\ \pm\ $ | 34.33
MP-RRT-noadv | 100 | 97228 | 4563 | 22.18$\ \pm\ $ | 14.71
MP-RRT-adv | 94 | 118799 | 6223 | 26.86$\ \pm\ $ | 41.78
Table 4.2: Dynamic Environment Results, map 2. Algorithm | S.R.[%] | C.C. | N.N. | Time[s]
---|---|---|---|---
Multi-stage | 100 | 10318 | 563 | 8.05$\ \pm\ $ | 1.47
RRT-EP/N | 100 | 21785 | 1849 | 12.69$\ \pm\ $ | 5.75
DRRT-noadv | 99 | 134091 | 4134 | 69.32$\ \pm\ $ | 49.47
DRRT-adv | 100 | 34051 | 2090 | 18.94$\ \pm\ $ | 17.64
MP-RRT-noadv | 100 | 122964 | 4811 | 67.26$\ \pm\ $ | 42.45
MP-RRT-adv | 100 | 25837 | 2138 | 16.34$\ \pm\ $ | 13.92
Figure 4.6: Times for different number of moving obstacles in map 1. Figure
4.7: Success rate for different number of moving obstacles in map 1.
#### 4.3.2 Partially Known Environment Results
Taking both maps into consideration, the results in tables 4.3 and 4.4 show
that both combined algorithms are faster and more consistent than the RRT
variants, with the simple multi-stage algorithm being faster in both. These
results were unexpected, as the combined algorithms were designed for dynamic
environments. It is worth to notice though, that in map 1 DRRT-adv is a close
second, but in map 2 it is a close last, so its lack of reliability does not
make it a good choice in this scenario. In this environment, as in the dynamic
environment, in map 1 DRRT is faster than MP-RRT, while the opposite happens
in map 2.
Table 4.3: Partially Known Environment Results, map 1. Algorithm | S.R.[%] | C.C. | N.N. | Time[s]
---|---|---|---|---
Multi-stage | 100 | 12204 | 1225 | 7.96$\ \pm\ $ | 2.93
RRT-EP/N | 99 | 99076 | 1425 | 9.95$\ \pm\ $ | 2.03
DRRT-noadv | 100 | 37618 | 1212 | 11.66$\ \pm\ $ | 15.39
DRRT-adv | 99 | 12131 | 967 | 8.26$\ \pm\ $ | 2.5
MP-RRT-noadv | 99 | 49156 | 1336 | 13.82$\ \pm\ $ | 17.96
MP-RRT-adv | 97 | 26565 | 1117 | 11.12$\ \pm\ $ | 14.55
Table 4.4: Partially Known Environment Results, map 2. Algorithm | S.R.[%] | C.C. | N.N. | Time[s]
---|---|---|---|---
Multi-stage | 100 | 12388 | 1613 | 17.66$\ \pm\ $ | 4.91
RRT-EP/N | 100 | 42845 | 1632 | 22.01$\ \pm\ $ | 6.65
DRRT-noadv | 99 | 54159 | 1281 | 32.67$\ \pm\ $ | 15.25
DRRT-adv | 100 | 53180 | 1612 | 32.54$\ \pm\ $ | 19.81
MP-RRT-noadv | 100 | 48289 | 1607 | 30.64$\ \pm\ $ | 13.97
MP-RRT-adv | 100 | 38901 | 1704 | 25.71$\ \pm\ $ | 12.56
#### 4.3.3 Unknown Environment Results
Results in table 4.5 present the combined RRT-EP/N clearly as the faster
algorithm in unknown environments, with the multi-stage algorithm in second
place. In contrast to dynamic and partially known environments in this same
map, MP-RRT is faster than DRRT.
Table 4.5: Unknown Environment Results Algorithm | S.R.[%] | C.C. | N.N. | Time[s]
---|---|---|---|---
Multi-stage | 100 | 114987 | 2960 | 13.97$\ \pm\ $ | 3.94
RRT-EP/N | 100 | 260688 | 2213 | 10.69$\ \pm\ $ | 2.08
DRRT-noadv | 98 | 89743 | 1943 | 18.38$\ \pm\ $ | 22.01
DRRT-adv | 100 | 104601 | 2161 | 19.64$\ \pm\ $ | 34.87
MP-RRT-noadv | 99 | 129785 | 1906 | 21.82$\ \pm\ $ | 27.23
MP-RRT-adv | 100 | 52426 | 1760 | 16.05$\ \pm\ $ | 10.87
## Chapter 5 Conclusions and Future Work
The new multi-stage algorithm proposed here has good performance in very
dynamic environments. It behaves particularly well when several small
obstacles are moving around at random. This is explained by the fact that if
the obstacles are constantly moving, they will sometimes move out of the way
by themselves, which our algorithm takes advantage of, while RRT based ones do
not, they just drop branches of the tree that could prove useful again just a
few moments later. The combined RRT-EP/N, although having more operators, and
automatic adjustment of the operator probabilities according to their
effectiveness, is still better than the RRT variants, but about 55% slower
than the simple multi-stage algorithm. This is explained by the number of
collision checks performed, more than twice than the multi-stage algorithm,
because collision checks must be performed for the entire population, not just
a single path.
In the partially known environment, even though the difference in collision
checks is even greater than in dynamic environments, the RRT-EP/N performance
is about 25% worse than the multi-stage algorithm. Overall, the RRT variants
are closer to the performance of both combined algorithms.
In the totally unknown environment, the combined RRT-EP/N is about 30% faster
than the simple multi-stage algorithm, and both outperform the RRT variants,
with much smaller times and standard deviations.
All things considered, the simple multi-stage algorithm is the best choice in
most situations, with faster and more predictable planning times, a higher
success rate, fewer collision checks performed and, above all, a much simpler
implementation than all the other algorithms compared.
This thesis shows that a multi-stage approach, using different techniques for
initial plannning and navigation, outperforms current probabilistic sampling
techniques in dynamic, partially known and unknown environments.
Part of the results presented in this thesis are published in [BALS09].
### 5.1 Future Work
We propose several areas of improvement for the work presented in this thesis.
#### 5.1.1 Algorithms
The most promising area of improvement seems to be to experiment with
different on-line planners such as a version of the EvP ([AR05] and [AR08])
modified to work in continuous configuration space or a potential field
navigator. Also, the local search presented here could benefit from the use of
more sophisticated operators and the parameters for the RRT variants (such as
forest size for MP-RRT), and the EP/N (such as population size) could benefit
from being tuned specifically for this implementation, and not simply reusing
the parameters found in previous work.
Another area of research that could be tackled is extending this algorithm to
higher dimensional problems, as RRT variants are known to work well in higher
dimensions.
Finally, as RRT variants are suitable for kinodynamic planning, we only need
to adapt the on-line stage of the algorithm to have a new multi-stage planner
for problems with kinodynamic constraints.
#### 5.1.2 Framework
The MoPa framework could benefit from the integration of a third party logic
layer, with support for arbitrary geometrical shapes, a spatial scene graph
and hierarchical maps. Some candidates would be OgreODE [Ogr], Spring RTS
[Spr] and ORTS [ORT].
Other possible improvements are adding support for other map formats,
including discrimination of static and moving obstacles, limited sensor range
simulation and integration with external hardware such as the Lego NXT [Leg],
to run experiments in a more realistic scenario.
## Bibliography
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* [AR05] T. Alfaro and M. Riff. An on-the-fly evolutionary algorithm for robot motion planning. Lecture Notes in Computer Science, 3637:119, 2005.
* [AR08] T. Alfaro and M. Riff. An evolutionary navigator for autonomous agents on unknown large-scale environments. Intelligent Automation and Soft Computing, 14(1):105, 2008.
* [BALS09] N.A. Barriga, M. Araya-Lopez, and M. Solar. Combining a probabilistic sampling technique and simple heuristics to solve the dynamic path planning problem. In Proceedings XXVIII International Conference of the Chilean Computing Science Society (SCCC), 2009.
* [Ben75] J.L. Bentley. Multidimensional binary search trees used for associative searching. Communications of the ACM, 18(9):517, 1975.
* [BFM97] T. Bäck, DB Fogel, and Z. Michalewicz. Handbook of Evolutionary Computation. Taylor & Francis, 1997.
* [BV02] J. Bruce and M. Veloso. Real-time randomized path planning for robot navigation. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, volume 3, pages 2383–2388 vol.3, 2002.
* [EAA04] A. Elshamli, HA Abdullah, and S. Areibi. Genetic algorithm for dynamic path planning. In Proceedings of the Canadian Conference on Electrical and Computer Engineering, volume 2, 2004.
* [FKS06] D. Ferguson, N. Kalra, and A. Stentz. Replanning with RRTs. In Proceedings of the IEEE International Conference on Robotics and Automation, pages 1243–1248, 15-19, 2006.
* [HA92] Yong K. Hwang and Narendra Ahuja. Gross motion planning — a survey. ACM Computing Surveys, 24(3):219–291, 1992.
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* [KSLO96] L.E. Kavraki, P. Svestka, J.-C. Latombe, and M.H. Overmars. Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Transactions on Robotics and Automation, 12(4):566–580, August 1996.
* [Lav98] S.M. Lavalle. Rapidly-Exploring Random Trees: A new tool for path planning. Technical report, Computer Science Department, Iowa State University, 1998\.
* [Leg] Lego Mindstorms. http://mindstorms.lego.com/.
* [LKJ99] S.M. LaValle and J.J. Kuffner Jr. Randomized kinodynamic planning. In Proceedings of the IEEE International Conference on Robotics and Automation, volume 1, 1999.
* [LS02] Tsai-Yen Li and Yang-Chuan Shie. An incremental learning approach to motion planning with roadmap management. In Proceedings of the IEEE International Conference on Robotics and Automation, volume 4, pages 3411–3416 vol.4, 2002.
* [MWS07] S.R. Martin, S.E. Wright, and J.W. Sheppard. Offline and online evolutionary bi-directional RRT algorithms for efficient re-planning in dynamic environments. In Proceedings of the IEEE International Conference on Automation Science and Engineering, pages 1131–1136, September 2007.
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* [Ogr] OgreODE. http://www.ogre3d.org/wiki/index.php/OgreODE.
* [ORT] ORTS – A free software RTS game engine. http://www.cs.ualberta.ca/~mburo/orts/.
* [Spr] The Spring Project. http://springrts.com/.
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* [Ste95] A. Stentz. The focussed D* algorithm for real-time replanning. In International Joint Conference on Artificial Intelligence, volume 14, pages 1652–1659. LAWRENCE ERLBAUM ASSOCIATES LTD, 1995.
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* [XMZT97] J. Xiao, Z. Michalewicz, L. Zhang, and K. Trojanowski. Adaptive Evolutionary Planner/Navigator for mobile robots. Proceedings of the IEEE Transactions on Evolutionary Computation, 1(1):18–28, April 1997.
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|
arxiv-papers
| 2009-12-01T21:13:36 |
2024-09-04T02:49:06.788886
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nicolas A. Barriga",
"submitter": "Nicolas A. Barriga",
"url": "https://arxiv.org/abs/0912.0270"
}
|
0912.0352
|
# Enhanced spin injection efficiency in a four-terminal double quantum dot
system
Ling Qin,1 Hai-Feng Lü,2 and Yong Guo1,a) 1Department of Physics and Key
Laboratory of Atomic and Molecular NanoSciences, Ministry of Education,
Tsinghua University, Beijing 100084, People’s Republic of China
2Department of Applied Physics, University of Electronic Science and
Technology of China, Chengdu 610054, People’s Republic of China
###### Abstract
Within the scheme of quantum rate equations, we investigate the spin-resolved
transport through a double quantum dot system with four ferromagnetic
terminals. It is found that the injection efficiency of spin-polarized
electrons can be significantly improved compared with single dot case. When
the magnetization in one of four ferromagnetic terminals is antiparallel with
the other three, the polarization rate of the current through one dot can be
greatly enhanced, accompanied by the drastic decrease of the current
polarization rate through the other one. The mechanism is the exchange
interaction between electrons in the two quantum dots, which can be a
promising candidate for the improvement of the spin injection efficiency.
###### pacs:
73.23.-b, 73.63.Kv, 75.30.Et
## I introduction
How to improve the injection efficiency of spin-polarized electrons from a
ferromagnetic (FM) contact into a semiconductor microstructure has puzzled the
researchers in the field of spintronics for many years.Zut04 Due to the
mismatch of conductivity between FM metal and semiconductor, spin polarization
is almost lost at the interface,Sch00 and spin injection efficiency is very
low.Ham99 ; Mon98 ; Fil00 ; Zhu01 To now, various ideas have been proposed to
solve this problem. RashbaRas00 suggested that tunnel contacts can
dramatically increase spin injection efficiency, which was supported by
subsequent theoretical works.Fer01 ; Smi01 ; Joh03 ; Tak03 Jiang et al.Jia05
demonstrated that the spin injection efficiency could be improved dramatically
by inserting a MgO tunnel barrier between the ferromagnetic contact and the
semiconductor. Optical injection of spin-polarized carriers across a
mismatched heterostructure is an effective method. By using circular polarized
excitation and detection, it has been demonstrated that the injected spin-
polarized carriers are quite robust and maintain their polarization memory
even after passing through a dense array of misfit dislocations.Fie99 ; Ohn99
; Han02 ; Gha01 However, it is still desirable to establish electrical,
rather than optical, methods to achieve effective spin injection.
In strongly-correlated electron systems, spin dipole-dipole interactions
between electrons play important roles, which determine the systems’
magnetism, specific heat, and other ground-state properties. In the weak
coupling and strong Coulomb repulsion regime, the Heisenberg-type exchange
interaction $J\textbf{S}_{1}\cdot\textbf{S}_{2}$ can be derived through
perturbation analysis (e.g., Schrieffer-Wolf transformation). For electronic
transport in mesoscopic systems, electronic spin correlation drastically
affects the conductance and the current correlation.Bus00 ; Don02 ; Kau06 ;
Chu07 ; Chu08 ; Fra07 ; Tol07 ; Koe07 For instance, the double quantum dot
(QD) system enables the realization of the two-impurity Kondo problem, in
which a competition between Kondo correlation and antiferromagnetic impurity-
spin correlation leads to a quantum critical phenomenon.Lop02 For the case of
spin-polarized transport, the polarized spin in one dot behaves like an
effective magnetic field and affects the spin transport in another dot through
indirect spin-spin interaction between two dots.Lu08 Therefore, it is
expected that exchange interaction can induce efficient spin injection in QD
systems.
In this work we propose an electrical and internal scheme to improve the spin
injection efficiency based on a double quantum dot system, where each dot is
connected with two FM electrodes. Two different configurations are examined,
one is the magnetizations of four FM electrodes are parallel with each other,
and the other is one of them has antiparallel magnetization with other three
ones. We find that in the latter case, due to the exchange interaction between
electrons in the double dot, the spin-polarization rate of the current through
one dot is greatly enhanced, while the spin-polarization rate through the
other one is drastically suppressed. As for the case of two parallel and two
antiparallel, spin-down electrons can hardly occupy the two dots, while the
spin-up ones dominate in both of the two dots during transport processes, thus
the exchange interaction cannot greatly enhance the current polarization.
## II model and formula
The structure is depicted in Fig. 1. Dot $i$ ($i=$1,2) is connected to FM
leads $i$L and $i$R. The magnetizations of leads 1L, 2L, and 2R are parallel,
while that of lead 1R can be parallel or antiparallel with the other three. We
model this system with the Hamiltonian $H=H_{lead}+H_{dot}+H_{T}$. The FM
leads are described by the Hamiltonian $H_{lead}=\sum\limits_{i\alpha
k\sigma}\varepsilon_{i\alpha k\sigma}a_{i\alpha k\sigma}^{\dagger}a_{i\alpha
k\sigma}$, where $a_{i\alpha k\sigma}^{\dagger}$ ($a_{i\alpha k\sigma}$) is
the creation (annihilation) operator for electrons with wave vector $k$ in
lead $i\alpha$, $\alpha=$L,R. The isolated double dot are described by
$H_{dot}=\sum\limits_{i\sigma}\varepsilon_{i}d_{i\sigma}^{\dagger}d_{i\sigma}+\sum\limits_{i}U_{i}n_{i\uparrow}n_{i\downarrow}+J\textbf{S}_{1}\cdot\textbf{S}_{2}$.
Here $d^{\dagger}_{i\sigma}$ ($d_{i\sigma}$) is the creation (annihilation)
operator for electrons with spin $\sigma$ in dot $i$,
$n_{i\sigma}=d^{\dagger}_{i\sigma}d_{i\sigma}$ is the occupation operator, and
$U_{i}$ stands for the intradot Coulomb repulsion. The last term denotes the
Heisenberg exchange coupling with the exchange coupling parameter $J$ and the
spin operator
$\textbf{S}_{i}=(\hbar/2)\sum\limits_{\sigma\sigma^{\prime}}d^{\dagger}_{i\sigma}$$\sigma$${}_{\sigma\sigma^{\prime}}d_{i\sigma^{\prime}}$.
For simplicity, we neglect the direct interdot tunneling and interdot Coulomb
repulsion.Lop02 ; Fra07 ; Lu08 The tunneling Hamiltonian between dots and
leads is $H_{T}=\sum\limits_{i\alpha k\sigma}(V_{i\alpha
k\sigma}a^{\dagger}_{i\alpha k\sigma}d_{i\sigma}+\textrm{H.c.})$. In the
following, we assume the coupling coefficient $V_{i\alpha k\sigma}$ to be
independent of $k$ and $U_{1},U_{2}\rightarrow\infty$, thus the double
occupation of each dot is forbidden.
Since the exchange interaction is considered, it is natural to describe the
double dot system by triplet and singlet states, which are defined as
$|T_{\uparrow}\rangle=|\uparrow\rangle_{1}|\uparrow\rangle_{2}$,
$|T_{\downarrow}\rangle=|\downarrow\rangle_{1}|\downarrow\rangle_{2}$,
$|T_{0}\rangle=(1/\sqrt{2})(|\uparrow\rangle_{1}|\downarrow\rangle_{2}+|\downarrow\rangle_{1}|\uparrow\rangle_{2})$
(triplet states), and
$|S\rangle=(1/\sqrt{2})(|\uparrow\rangle_{1}|\downarrow\rangle_{2}-|\downarrow\rangle_{1}|\uparrow\rangle_{2})$
(singlet state). Following the procedure in previous works,Don04 ; Qin08 we
use nine slave-boson operators to represent these Dirac brackets:
$e^{\dagger}=|0\rangle_{1}|0\rangle_{2}$,
$f^{\dagger}_{1\sigma}=|\sigma\rangle_{1}|0\rangle_{2}$,
$f^{\dagger}_{2\sigma}=|0\rangle_{1}|\sigma\rangle_{2}$,
$d^{\dagger}_{T_{\sigma}}=|T_{\sigma}\rangle$,
$d^{\dagger}_{T_{0}}=|T_{0}\rangle$, and $d^{\dagger}_{S}=|S\rangle$. Thus,
$d_{i\sigma}=e^{\dagger}f_{i\sigma}+\sigma
f^{\dagger}_{\bar{i}\sigma}d_{T_{\sigma}}+(1/\sqrt{2})\sigma
f^{\dagger}_{\bar{i}\bar{\sigma}}[d_{T_{0}}+(-1)^{i}\bar{\sigma}d_{s}]$ and
$H_{dot}=\sum\limits_{i\sigma}\varepsilon_{i}f_{i\sigma}^{\dagger}f_{i\sigma}+(\varepsilon_{1}+\varepsilon_{2}+J/4)\sum\limits_{\gamma=\uparrow,\downarrow,0}d^{\dagger}_{T_{\gamma}}d_{T_{\gamma}}+(\varepsilon_{1}+\varepsilon_{2}-3J/4)d^{\dagger}_{S}d_{S}$
with $\bar{1}(\bar{2})=2(1)$ and
$\bar{\uparrow}(\bar{\downarrow})=\downarrow(\uparrow)$.
Using equation of motion, one can derive the dynamical equations of elements
of the density matrix.Don04 Their statistical expectations involve the time-
diagonal parts of the less Green’s functions, which can be calculated with the
help of the Langreth analytic continuation rules and the Fourier
transformation. Submitting the uncoupled dot’s Green’s function into the
equations, the mater equations describe the electronic transport can be
derived as
$\displaystyle\dot{\hat{\rho}}_{0}$ $\displaystyle=$
$\displaystyle\sum_{i\alpha\sigma}\Gamma^{\sigma}_{i\alpha}\displaystyle\big{\\{}[1-f_{i\alpha}(\varepsilon_{i})]\rho_{i\sigma}-f_{i\alpha}(\varepsilon_{i})\rho_{0}\displaystyle\big{\\}},$
$\displaystyle\dot{\hat{\rho}}_{i\sigma}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\displaystyle\bigg{\\{}\Gamma^{\sigma}_{i\alpha}f_{i\alpha}(\varepsilon_{i})\rho_{0}-\displaystyle\big{\\{}\Gamma^{\sigma}_{i\alpha}[1-f_{i\alpha}(\varepsilon_{i})]+\Gamma^{\sigma}_{\bar{i}\alpha}f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}+J/4)$
$\displaystyle+\frac{1}{2}\Gamma^{\bar{\sigma}}_{\bar{i}\alpha}[f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}+J/4)+f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}-3J/4)]\displaystyle\big{\\}}\rho_{i\sigma}+\Gamma^{\sigma}_{\bar{i}\alpha}[1-f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}+J/4)]\rho_{T_{\sigma}}$
$\displaystyle+\frac{1}{2}\Gamma^{\bar{\sigma}}_{\bar{i}\alpha}[1-f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}+J/4)]\rho_{T_{0}}+\frac{1}{2}\Gamma^{\bar{\sigma}}_{\bar{i}\alpha}[1-f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}-3J/4)]\rho_{S}$
$\displaystyle+(-1)^{i}\frac{\bar{\sigma}}{2}\Gamma^{\bar{\sigma}}_{\bar{i}\alpha}[1-\frac{1}{2}f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}+J/4)-\frac{1}{2}f_{\bar{i}\alpha}(\varepsilon_{\bar{i}}-3J/4)](\rho_{S,T_{0}}+\rho_{T_{0},S})\displaystyle\bigg{\\}},$
$\displaystyle\dot{\hat{\rho}}_{T_{\sigma}}$ $\displaystyle=$
$\displaystyle\sum_{i\alpha}\Gamma^{\sigma}_{i\alpha}\displaystyle\big{\\{}f_{i\alpha}(\varepsilon_{i}+J/4)\rho_{\bar{i}\sigma}-[1-f_{i\alpha}(\varepsilon_{i}+J/4)]\rho_{T_{\sigma}}\displaystyle\big{\\}},$
$\displaystyle\dot{\hat{\rho}}_{T_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i\alpha\sigma}\Gamma^{\sigma}_{i\alpha}\displaystyle\big{\\{}f_{i\alpha}(\varepsilon_{i}+J/4)\rho_{\bar{i}\bar{\sigma}}-[1-f_{i\alpha}(\varepsilon_{i}+J/4)]\rho_{T_{0}}$
$\displaystyle+\frac{1}{4}(-1)^{i}\sigma[1-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}+J/4)-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}-3J/4)](\rho_{S,T_{0}}+\rho_{T_{0},S})\displaystyle\big{\\}},$
$\displaystyle\dot{\hat{\rho}}_{S}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i\alpha\sigma}\Gamma^{\sigma}_{i\alpha}\displaystyle\big{\\{}f_{i\alpha}(\varepsilon_{i}-3J/4)\rho_{\bar{i}\bar{\sigma}}-[1-f_{i\alpha}(\varepsilon_{i}-3J/4)]\rho_{S}$
$\displaystyle+\frac{1}{4}(-1)^{i}\sigma[1-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}+J/4)-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}-3J/4)](\rho_{S,T_{0}}+\rho_{T_{0},S})\displaystyle\big{\\}},$
$\displaystyle\dot{\hat{\rho}}_{T_{0},S}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{i\alpha\sigma}(-1)^{i}\sigma\Gamma^{\sigma}_{i\alpha}\displaystyle\big{\\{}[1-f_{i\alpha}(\varepsilon_{i}+J/4)]\rho_{T_{0}}+[1-f_{i\alpha}(\varepsilon_{i}-3J/4)]\rho_{S}$
(1)
$\displaystyle-[f_{i\alpha}(\varepsilon_{i}+J/4)+f_{i\alpha}(\varepsilon_{i}-3J/4)]\rho_{\bar{i}\bar{\sigma}}\displaystyle\big{\\}}$
$\displaystyle+\displaystyle\big{\\{}iJ-\frac{1}{2}\sum_{i\alpha\sigma}\Gamma^{\sigma}_{i\alpha}[1-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}+J/4)-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}-3J/4)]\displaystyle\big{\\}}\rho_{T_{0},S},$
where the elements of the density matrix are defined as
$\hat{\rho}_{0}=e^{\dagger}e$,
$\hat{\rho}_{i\sigma}=f^{\dagger}_{i\sigma}f_{i\sigma}$,
$\hat{\rho}_{T_{\gamma}}=d^{\dagger}_{T_{\gamma}}d_{T_{\gamma}}$, and
$\hat{\rho}_{S}=d^{\dagger}_{S}d_{S}$. These elements represent the
probability that both dots are empty, one electron with spin $\sigma$ occupies
dot $i$, and two electrons form the triplet states and the singlet state,
respectively. They satisfy the completeness relation
$\rho_{0}+\sum\limits_{\sigma}(\rho_{1\sigma}+\rho_{2\sigma}+\rho_{T_{\sigma}})+\rho_{T_{0}}+\rho_{S}=1$.
$\rho_{S,T_{0}}$ is induced by the exchange interaction.
$f_{i\alpha}(\omega)=[1+e^{(\omega-\mu_{i\alpha})/k_{B}T}]^{-1}$ is the Fermi
distribution function of lead $i\alpha$, and
$\Gamma^{\sigma}_{i\alpha}=\sum\limits_{k}2\pi|V_{i\alpha
k\sigma}|^{2}\delta(\omega-\varepsilon_{i\alpha k\sigma})$ is the coupling
strength between lead $i\alpha$ and dot $i$. In the stationary situation, the
elements of the density matrix can be derived, and the spin component of
current in lead $i\alpha$ can be obtained as
$\displaystyle I^{\sigma}_{i\alpha}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\Gamma^{\sigma}_{i\alpha}\displaystyle\big{\\{}f_{i\alpha}(\varepsilon_{i})\rho_{0}-[1-f_{i\alpha}(\varepsilon_{i})]\rho_{i\sigma}+f_{i\alpha}(\varepsilon_{i}+J/4)\rho_{\bar{i}\sigma}+\frac{1}{2}[f_{i\alpha}(\varepsilon_{i}+J/4)$
(2)
$\displaystyle+f_{i\alpha}(\varepsilon_{i}-3J/4)]\rho_{\bar{i}\bar{\sigma}}-[1-f_{i\alpha}(\varepsilon_{i}+J/4)]\rho_{T_{\sigma}}-\frac{1}{2}[1-f_{i\alpha}(\varepsilon_{i}+J/4)]\rho_{T_{0}}$
$\displaystyle-\frac{1}{2}[1-f_{i\alpha}(\varepsilon_{i}-3J/4)]\rho_{S}+(-1)^{i}\frac{\sigma}{2}[1-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}+J/4)$
$\displaystyle-\frac{1}{2}f_{i\alpha}(\varepsilon_{i}-3J/4)](\rho_{S,T_{0}}+\rho_{T_{0},S})\displaystyle\big{\\}}.$
When $J\rightarrow 0$, these quantum rate equations reduce to the equations
describing two separate dots.Sou08 ; Bul99 For a single dot, interplay
between Coulomb interaction and spin accumulation in the dot can result in a
bias-dependent current polarization, which can be suppressed in the P
alignment and enhanced in the AP case.Sou08 Furthermore, the spin flip
process make the occupations of spin-up and spin-down electrons in the dots
tend to be equal, which can weaken the enhancement of current spin-
polarization rate.
## III numerical results and discussions
For numerical calculations, we choose meV to be the energy unit and set
$k_{B}T=0.002$. The polarization rates of all leads are assumed to be $P=0.4$,
and the coupling strength is $\Gamma^{\sigma}_{i\alpha}=(1+\sigma P)\Gamma$,
except for lead $1$R it becomes $(1\pm\sigma P)\Gamma$, where $+$ for the
parallel (P) configuration and $-$ for the antiparallel (AP) one. $\Gamma$ and
$J$ are set to be 0.01 and 0.2, respectively,Hat08 ; Tol07 ; Lu08 and the
current are normalized to $e\Gamma/h$. The exchange coupling $J$ between two
dots is the key interaction to improve the spin injection efficiency. Its
strength sensitively depend on the e-e Coulomb interaction, interdot coupling,
Bychkov-Rashba spin-orbit interaction, and magnetic field. $J$ can reach
several hundreds eV and can be tuned to ferromagnetic ($J<0$) type in the
presence of magnetic field.Fqu09 Typical value of the dot-lead coupling
strength $\Gamma$ is order of 1$\mu$eV, therefore, $J/\Gamma\gg 1$, which
makes sure that the quantum rate equations are valid in every bias region.
For clarity, first we show relevant results for single QD system connected to
two FM leads.Sou08 The spin components of the current are
$I^{\sigma}=(e/h)(\Gamma^{\sigma}_{L}\Gamma^{\uparrow}_{R}\Gamma^{\downarrow}_{R})/(\Gamma^{\uparrow}_{L}\Gamma^{\downarrow}_{R}+\Gamma^{\downarrow}_{L}\Gamma^{\uparrow}_{R}+\Gamma^{\uparrow}_{R}\Gamma^{\downarrow}_{R})$.
Thus, the spin-polarization rate is
$\eta=(I^{\uparrow}-I^{\downarrow})/(I^{\uparrow}+I^{\downarrow})=P_{L}=P$,
regardless of whether the system is in P or AP configuration.Sou08 ; Sou07
However, for the four-terminal structure, when the exchange interaction is
absent, $n_{1\sigma}=n_{2\sigma}=1/3$ for the P configuration, while
$n_{1\uparrow}>n_{1\downarrow}$ and $n_{2\uparrow}=n_{2\downarrow}$ for the AP
one. Since the exchange interaction is sensitive to the spin-dependent
occupation numbers in the two dots, we expect that in the P configuration the
exchange interaction has little influence on the current polarization, while
in the AP one it can affect the transport properties greatly. Further, we
apply a large bias between leads 1L and 1R to make sure that $\varepsilon_{1}$
is deeply in the bias window.
Fig. 2(a) shows variations of $I^{\sigma}_{2}$ and $n_{2\sigma}$ with the bias
voltage in the P configuration. In the following, $I^{\sigma}_{2}$ is denoted
by $I^{\sigma}$, for convenience. As expected, both $I^{\uparrow}$ and
$I^{\downarrow}$ increase monotonously with the bias, and three steps occur
when $\mu_{2L}$ crosses $\varepsilon_{2}-3J/4$, $\varepsilon_{2}$, and
$\varepsilon_{2}+J/4$, respectively. They correspond to the situations that
electrons tunnel through dot 2 via the singlet state, the energy level
$\varepsilon_{2}$, and the triplet states. Here we mark the bias regions
$\varepsilon_{2}-3J/4<V/2<\varepsilon_{2}$,
$\varepsilon_{2}<V/2<\varepsilon_{2}+J/4$, and $V/2>\varepsilon_{2}+J/4$ as I,
II, and III, respectively. In each region, $I^{\uparrow}>I^{\downarrow}$.
However, in region I, $n_{2\downarrow}>n_{2\uparrow}$, which is different from
the case of isolated single dot, where $n_{\uparrow}=n_{\downarrow}$ and
$\eta=P=0.4$. Since $n_{2\downarrow}>n_{2\uparrow}$, $\eta_{2}$ is suppressed
from 0.4, accompanied by the increase of $\eta_{1}$. When the bias rises
beyond region I, both $\eta_{1}$ and $\eta_{2}$ return to 0.4. So in the P
configuration we can not enhance $\eta_{2}$ from its original value in single
dot case.
In the AP configuration, $\eta_{2}$ can be strongly modified from the single
dot case by the exchange interaction (see Fig. 3). Figs. 3(a) indicates both
$I^{\uparrow}$ and $I^{\downarrow}$ increase monotonously with the bias, which
is similar to that in the P configuration. However, from region I to region
III, the discrepancy between $I^{\uparrow}$ and $I^{\downarrow}$ keeps
increasing, resulting in the enhancement of $\eta_{2}$ in Fig. 3(b). In region
III, $\eta_{2}$ approaches 0.7, which is much larger than its original value
0.4 in single dot system. At the same time, $\eta_{1}$ keeps decreasing when
bias increases from region I to region III, and finally becomes smaller than
0.1. It is concluded that in the AP configuration one can greatly enhance the
current polarization rate through one dot, accompanied by decrease of the
current polarization rate through another dot. Such phenomenon looks as if the
current polarization rate is “transferred” from one circuit to the other.
The enhancement of the current polarization rate can be understood with the
aid of the expression of the current. Due to the absence of intradot spin
flips, both the amplitude and spin polarization of the total current through
dot $2$ are conserved, i.e., $I_{2L}^{\sigma}=I_{2R}^{\sigma}$. For
simplicity, the current $I_{2R}^{\sigma}$ is chosen in the calculation because
it has an uniform expression in all three regions:
$I^{\sigma}=(e/h)\Gamma^{\sigma}_{2R}[\rho_{2\sigma}+\rho_{T_{\sigma}}+(1/2)\rho_{T_{0}}+(1/2)\rho_{S}]$.
The first term denotes the process that one electron tunnels through dot 2 via
the energy level $\varepsilon_{2}$, and the second to fourth terms denote the
processes that one electron with spin $\sigma$ transports through dot 2 via
the triplet states and the singlet state. Because in $T_{0}$ and $S$ states,
electrons with spin $\sigma$ or $\bar{\sigma}$ have the same probability to
occupy dot 2, both the third and the fourth terms have a factor $1/2$. From
Fig. 3(b), in region I we can see $\eta_{2}$ is slightly larger than $P=0.4$.
In this region, only the energy level $\varepsilon_{2}-3J/4$ enters the bias
window, and electrons can only form the singlet state, which makes $\rho_{S}$
much larger than other elements [see Figs. 3(c) and 3(d)]. Thus, the forth
term dominates in expression of the current, and we have
$I^{\sigma}=(e/2h)\Gamma^{\sigma}_{2R}(\rho_{S}+\rho_{T_{0}})$,
$\eta_{2}=({I^{\uparrow}-I^{\downarrow}})/({I^{\uparrow}+I^{\downarrow}})=P=0.4$.
When the effects of $\rho_{2\sigma}$ and $\rho_{T_{\sigma}}$ are considered,
the value of $\eta_{2}$ is slightly modified. From Eq. (1) we can obtain
$\rho_{2\sigma}\approx\Gamma^{\bar{\sigma}}_{1R}\rho_{S}/[2(\Gamma^{\sigma}_{1L}+\Gamma^{\bar{\sigma}}_{1L}+\Gamma^{\sigma}_{2L}+\Gamma^{\sigma}_{2R})]$.
Here we denote $(1+\sigma P)\Gamma=\Gamma^{\sigma}$, then
$\Gamma^{\sigma}_{i\alpha}=\Gamma^{\sigma}$, except for
$\Gamma^{\sigma}_{1R}=\Gamma^{\bar{\sigma}}$. Thus,
$\rho_{2\uparrow}\approx\rho_{S}/[2(3+\Gamma^{\downarrow}/\Gamma^{\uparrow})]>\rho_{2\downarrow}\approx\rho_{S}/[2(3+\Gamma^{\uparrow}/\Gamma^{\downarrow})]$,
and $\eta_{2}$ is enhanced from 0.4, as shown in Fig. 3(b). In region I,
$\rho_{S}$ is much larger than other elements, which means that during most of
the time electrons in the double dot form the singlet state. So
$\rho_{2\sigma}$ is mainly contributed by the process that an electron in dot
1 tunnels to lead 1R and breaks the singlet state. Noticing that in such a
configuration, $\Gamma^{\downarrow}_{1R}>\Gamma^{\uparrow}_{1R}$, electron
with spin $\downarrow$ can tunnel to lead 1R more easily, and left an electron
with spin $\uparrow$ in dot 2, which makes
$\rho_{2\uparrow}>\rho_{2\downarrow}$.
When the bias locates in region II, the direct tunneling channel at
$\varepsilon_{2}$ opens. We can see the enhancement of $\rho_{2\uparrow}$
($\rho_{T_{\uparrow}}$) is larger than $\rho_{2\downarrow}$
($\rho_{T_{\downarrow}}$), which results in further increase of $\eta_{2}$.
Here
$\rho_{2\sigma}=[\Gamma^{\sigma}_{2L}\rho_{0}+\Gamma^{\sigma}_{1R}\rho_{T_{\sigma}}+(1/2)\Gamma^{\bar{\sigma}}_{1R}(\rho_{T_{0}}+\rho_{S})]/(\Gamma^{\sigma}_{1L}+\Gamma^{\bar{\sigma}}_{1L}+\Gamma^{\sigma}_{2R})$.
It is obvious that the increase of $\rho_{2\sigma}$ is mainly owing to the
term $\Gamma^{\sigma}_{2L}\rho_{0}$ in the numerator, which is absent in
region I. Following the same procedure, this term reads
$\Gamma^{\sigma}_{2L}\rho_{0}/(\Gamma^{\sigma}_{1L}+\Gamma^{\bar{\sigma}}_{1L}+\Gamma^{\sigma}_{2R})=\rho_{0}/(2+\Gamma^{\bar{\sigma}}/\Gamma^{\sigma})$,
so the increase of $\rho_{2\uparrow}$ is larger than that of
$\rho_{2\downarrow}$, and $\eta_{2}$ is enhanced from its value in region I.
When the bias enters region III, $\rho_{2\sigma}$ and $\rho_{T_{\sigma}}$ keep
increasing, and the enhancement of $\rho_{T_{\uparrow}}$ is much more than
other elements. This is because now the channel at $\varepsilon_{2}+J/4$
opens, and if dot 1 is occupied, electrons in lead 2L can directly tunnel into
dot 2 and form the triplet state $T_{\sigma}$. Since lead 1R is in
antiparallel with lead 1L, in most of the time, dot 1 is occupied by one
electron with spin $\uparrow$. As a consequence, electrons with spin
$\uparrow$ in lead 2L is more available to tunnel into dot 2 and form the
triplet state $T_{\uparrow}$, which makes
$\rho_{T_{\uparrow}}\gg\rho_{T_{\downarrow}}$. This can also be seen in the
formula
$\rho_{T_{\sigma}}=(\Gamma^{\sigma}_{2L}\rho_{1\sigma}+\Gamma^{\sigma}_{1L}\rho_{2\sigma})/(\Gamma^{\sigma}_{1R}+\Gamma^{\sigma}_{2R})$,
where the first term in the numerator makes $\rho_{T_{\uparrow}}$ increase
intensively in region III. Thus, $\eta_{2}$ is greatly enhanced in region III.
In the case of $J/\Gamma\gg 1$, the analytical expressions in region I, II,
and III are $\eta_{2}\sim 191P/(165-34P^{2})$, $120P/(84+5P^{2})$, and
$51P/(27+6P^{2})$, respectively. For $P=0.4$, $\eta_{2}\sim$ 0.454, 0.542, and
0.673, which is consistent with our numerical results. As expected, when
$P\rightarrow 1$, $\eta_{2}\rightarrow 1$ in all regions. If we tune the bias
into region III, the injection efficiency can be enhanced to almost twice of
its original value. In the inset of Fig. 3(a), we present the variations of
$\eta_{2}$ with $P$ for different situations. It can be seen that when $P$ is
small, $\eta_{2}$ is greatly enhanced by the exchange interaction.
## IV conclusions
In summary, we propose a scheme based on a four-terminal double quantum dot
system to improve the spin injection efficiency greatly. We find that in the
antiparallel configuration, the spin-polarization rate through one quantum dot
can be dramatically enhanced, while the polarization rate through the other
one is suppressed. The operating mechanism is the exchange interaction between
the two quantum dots.
This project was supported by the NSFC (No. 10774083 and No.10974109) and by
the 973 Program (No. 2006CB605105).
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* (1) Corresponding author: guoy66@tsinghua.edu.cn.
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Figure 1: (color online) The system with two quantum dots coupled to four
external FM leads. The magnetizations of three leads are parallel with each
other, while the magnetization of lead 1R can be parallel (P) or antiparallel
(AP) with the other three. Figure 2: (color online) The spin component of the
current in dot 2 (a) and the spin-polarization rate (b) versus bias in the P
configuration. The inset in (a) shows the variations of the occupation numbers
in dot 2. Figure 3: (color online) The transport properties in the AP
configuration. (a) The spin component of the current versus bias. Ihe inset
shows the variations of the spin-polarization rate with $P$ in different
situations. The solid line corresponds to the single dot case, and the dashed,
dotted, and dash-dotted lines correspond to the situations that the bias
locates in region I, II, and III, respectively. (b) The spin-polarization
versus bias. (c) and (d) The corresponding elements of the density matrix
versus the bias.
|
arxiv-papers
| 2009-12-02T08:04:33 |
2024-09-04T02:49:06.798605
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ling Qin, Hai-Feng Lu, and Yong Guo",
"submitter": "Yong Guo",
"url": "https://arxiv.org/abs/0912.0352"
}
|
0912.0423
|
# Transient response under ultrafast interband excitation of an intrinsic
graphene
P.N. Romanets F.T. Vasko ftvasko@yahoo.com Institute of Semiconductor
Physics, NAS of Ukraine, Pr. Nauky 41, Kiev, 03028, Ukraine
###### Abstract
The transient evolution of carriers in an intrinsic graphene under ultrafast
excitation, which is caused by the collisionless interband transitions, is
studied theoretically. The energy relaxation due to the quasielastic acoustic
phonon scattering and the interband generation-recombination transitions due
to thermal radiation are analyzed. The distributions of carriers are obtained
for the limiting cases when carrier-carrier scattering is negligible and when
the intercarrier scattering imposes the quasiequilibrium distribution. The
transient optical response (differential reflectivity and transmissivity) on a
probe radiation and transient photoconductivity (response on a weak dc field)
appears to be strongly dependent on the relaxation and recombination dynamics
of carriers.
###### pacs:
72.80.Vp, 78.67.Wj, 81.05.ue
## I Introduction
The transient response of photoexcited carriers under ultrafast interband
pumping has been studied during the last decades in bulk semiconductors and
heterostructures (see Ref. 1 for review). The unusual transport of carriers in
graphene is caused by a neutrinolike energy spectrum in gapless semiconductor,
which is described by the Weyl-Wallace model 2 , and a substantial
modification of scattering processes. Recently, the properties of graphene
after ultrafast interband excitation attract special attention. The
experimental results in relaxation dynamics of photoexcited electrons and
holes were published in 3 ; 4 ; 5 and 6 for epitaxial and exfoliated
graphene, respectively. The relaxation of nonequilibrium optical phonons,
which are emitted by carriers after photoexcitation, is studied in Ref. 7. The
theoretical consideration of the carrier relaxation and generation-
recombination processes caused by optical phonons is performed in 8 ; 9 . The
quasielastic energy relaxation of carriers due to acoustic phonons was
considered in 10 ; 11 for low energy carriers (at low temperatures or under
mid-IR excitation). In particular, an interplay between energy relaxation and
generation-recombination processes determines the relaxation dynamics of
photoexcited carrier distribution. 10 To the best of our knowledge, both this
interplay and the relaxation dynamics at low temperatures are not considered
so far. Thus, the investigation of the transient response of carriers under
these conditions is timely now.
In this paper, we consider the transient response of an intrinsic graphene in
case of ultrafast interband excitation in passive region, where the carrier
energies are smaller than the optical phonon energy. Such a regime can be
realized under the pumping in the mid-infrared (IR) spectral region or at low
temperatures, when the peak of photoexcited carriers formed after the process
of optical phonon emission, remains a narrow one. Describing the
photoexcitation process, we restrict ourselves by the collisionless regime,
when a pulse duration, $\tau_{p}$, is shorter than the momentum relaxation
time. Considering the low-temperature transient dynamics of photoexcited
carriers, one takes into account the intraband quasielastic energy relaxation
due to acoustic phonons and generation-recombination interband transitions due
to thermal radiation. The carrier-carrier scattering is described within two
limiting regimes: ($i$) when the Coulomb interaction is unessential, and
($ii$) when intercarrier scattering imposes the quasiequilibrium distribution
of carriers. With the obtained transient distribution of carriers, we analyze
a time-dependent response on the probe field, i.e. we consider the transient
reflection and transmission in THz and mid-IR spectral regions. The transient
photoconductivity is also analyzed below, because the energy relaxation
corresponds to a nanosecond scale (the radiative recombination remains
essential up to microsecond).
Since the electron-hole energy spectrum and scattering processes are symmetric
in an intrinsic graphene, the phenomena under consideration are described by
the same distribution functions for electrons and holes, $f_{pt}$. Such
distribution is governed by the general kinetic equation 12 :
$\frac{\partial f_{pt}}{\partial
t}=\sum_{k}J_{k}\left\\{f_{t}|p\right\\}+G\\{f|pt\\},$ (1)
where the collision integrals $J_{k}\left\\{f_{t}|p\right\\}$ describe the
relaxation of carriers caused by the carrier-carrier scattering ($k=cc$), the
acoustic phonons ($k=ac$), and the thermal radiation ($k=r$), respectively.
The photogeneration rate, $G\\{f|pt\\}$, describes the interband excitation of
electron-hole pairs by the mid-IR ultrafast pulse. Below Eq. (1) is solved
with the initial condition $f_{pt\to-\infty}=f_{p}^{(eq)}$, where
$f_{p}^{(eq)}$ is the equilibrium distribution. The transient response on a
probe radiation is described by the dynamic conductivity due to interband
transitions. The transient response on a weak dc field (photoconductivity) is
considered with the use of the phenomenological model of momentum scattering
suggested in 13 .
The analysis carried out below is organized as follows. The photoexcitation
process under the inerband pumping is described in Sec. II. The transient
evolution distributions are given in Sec. III for the cases ($i$) and ($ii$).
Section IV presents a set of results of transient reflectivity and
transmittivity, and also the transient photoconductivity. The discussion of
the assumptions used and concluding remarks are given in the last section.
Appendix contains the microscopical evaluation of the interband
photogeneration rate under ultrafast interband excitation.
## II Ultrafast excitation
In the framework of the Weyl-Wallace model (spin- and valley-degenerate linear
energy spectrum of carriers which is determined by the characteristic velocity
$v_{W}$), the interband photoexcitation is caused by the in-plane electric
field, $w_{t}{\bf E}\exp(-i\Omega t)+c.c.$ where $\bf E$ is the field
strength, $\omega$ is the frequency, and $w_{t}$ is the envelope form-factor.
Eq. (1) is transformed to the collisionless form on the initial intervals,
when scattering mechanisms are not essential: $\partial f_{pt}/\partial
t=G\left\\{{f|pt}\right\\}$. Using the boundary condition of Eq. (1), one can
rewrite this equation in the integral form
$f_{pt}=f_{p}^{(eq)}+\int_{-\infty}^{t}dt^{\prime}G\\{f|pt^{\prime}\\}$. The
photogeneration rate here is evaluated in Appendix as follows:
$\displaystyle
G\left\\{f|pt\right\\}=\left(\frac{eEv_{W}}{\hbar\Omega}\right)^{2}w_{t}\int\limits_{-\infty}^{0}d\tau
w_{t+\tau}$
$\displaystyle\times\cos\left[\left(\frac{2v_{W}p}{\hbar}-\Omega\right)\tau\right]\left(1-2f_{pt+\tau}\right),$
(2)
where the Pauli blocking factor $(1-2f_{pt+\tau})$ is responsible for the
coherent Rabi oscillations of the excited carriers.
Introducing the dimensionless intensity,
$I_{ex}=(eE\tau_{p}v_{W}/\hbar\Omega)^{2}$, we consider below the linear
regime of excitation which takes place if $I_{ex}\ll 1$ and $f_{pt}\ll 1$, so
that the Pauli factor can be neglected (if $\hbar\Omega$ comparable to the
equilibrium temperature $T$ one has to use the equilibrium Pauli factor in Eq.
(2)). Using the Gaussian form-factor
$w_{t}=\sqrt[4]{2/\pi}\exp\left[-\left(t/\tau_{p}\right)^{2}\right]$ with the
pulse duration $\tau_{p}$, 14 one obtaines the photoexcited distribution in
the form
$f_{pt}^{(ex)}\approx
I_{ex}\int\limits_{-\infty}^{t}{dt^{\prime}}w_{t^{\prime}}\int\limits_{-\infty}^{0}{d\tau
w_{t^{\prime}+\tau}\cos\left({\frac{{2v_{W}\delta p}}{\hbar}\tau}\right)}.$
(3)
Here $\delta p=p-p_{\Omega}$ is centred in the characteristic momentum
$p_{\Omega}=\hbar\Omega/2v_{W}$. The evolution of photoexcited distribution,
$f_{pt}^{(ex)}/I_{ex}$, is shown in Fig. 1. The distribution is dependent on
$t/\tau_{p}$ and $\delta p/\Delta p$, where $\Delta p=\hbar/2v_{W}\tau_{p}$
determines the width of distribution which is proportional to $\tau_{p}^{-1}$.
For $t\gg\tau_{p}$, the integrations in Eq. (3) can be exactly performed and
we obtain steady-state distribution after the photoexcitation pulse,
$f_{p}^{(ex)}=f_{pt\to\infty}^{(ex)}$, as the Gaussian peak of width
$\propto\Delta p$:
$f_{p}^{(ex)}=\sqrt{\frac{\pi}{2}}I_{ex}e^{-(\delta p/\sqrt{2}\Delta p)^{2}}.$
(4)
Thus, at $t\geq 2\tau_{p}$ (see Fig. 1) one can omit the photogeneration rate
in Eq. (1) using instead the initial condition:
$f_{pt=0}=f_{p}^{(eq)}+f_{p}^{(ex)},$ (5)
which is given as a sum of the equilibrium and photoexcited contributions. The
condition (5) can be used directly in case of weak intercarrier scattering. In
case of optical excitation, with a subsequent emission of cascade of $2{\cal
N}$ optical phonons of energy $\hbar\omega_{0}$, the photoexcited distribution
can be written in the form (5) where $\delta p$ is centred in
$p_{\overline{\omega}}=(\hbar\Omega-2{\cal N}\hbar\omega_{0})/2v_{W}$ and
$\Delta p$ is included an additional broadening during the cascade emission.
Figure 1: Temporal evolution of photoexcited distribution $f_{pt}^{(ex)}$
normalized to $I_{ex}$ versus dimensionless momentum and time, $\delta
p/2\Delta p$ and $t/\tau_{p}$.
Under an effective intercarrier scattering, one needs to calculate the initial
temperature and concentration of carriers. The photoexcited concentration and
energy of carriers, which are described by the peak of distribution (4) are
given by
$\left|\begin{array}[]{*{20}c}\Delta n_{ex}\\\ \Delta
E_{ex}\end{array}\right|=\frac{4}{L^{2}}\sum\limits_{\bf
p}\left|\begin{array}[]{*{20}c}1\\\
{v_{W}p}\end{array}\right|f_{p}^{(ex)}\simeq\frac{I_{ex}(\overline{\omega}/v_{W})^{2}}{2\overline{\omega}\tau_{p}}\left|\begin{array}[]{*{20}c}1\\\
\hbar\overline{\omega}/2\end{array}\right|,$ (6)
where $L^{2}$ is the normalization area. One obtains $\Delta
E_{ex}/n_{ex}=\hbar\overline{\omega}/2$, for the Gaussian shape of pulse, i.e.
the averaged energy per generated particle is equal to the excitation energy.
In case of optical excitation, with $\cal N$ optical phonons emitted, the
energy per photoexcited particle, $\Delta E_{ex}/\Delta n_{ex}$, agrees
closely with $\hbar\omega-{\cal N}\hbar\omega_{0}$ (see above).
Figure 2: Initial maximum distribution (a) and effective temperature (b),
$f_{ex}$ and $T_{ex}$, versus pumping ($\Delta n_{ex}/n_{T}\propto I_{ex}$ for
$\hbar\overline{\omega}=60$ meV and 120 meV (solid and dashed curves,
respectively).
If $\tau_{p}\ll\tau_{cc}\ll\tau_{ac,r}$ , where $\tau_{cc}$, $\tau_{ac}$, and
$\tau_{r}$ correspond to the intercarrier scattering, the energy relaxation,
and the generation-recombination processes, respectively [the Coulomb-
controlled case ($ii$)], the dominanting carrier-carrier scattering imposes
the quasi-equilibrium distribution
$f_{pt}=\left[\exp\left(\frac{v_{W}p-\mu_{t}}{T_{t}}\right)+1\right]^{-1}$ (7)
with the effective temperature $T_{t}$ and the quasichemical potential
$\mu_{t}$. If $\tau_{cc}\ll t\ll\tau_{ac,r}$, the initial values
$T_{ex}=T_{t\to 0}$ and $f_{ex}=f_{p=0t\to 0}$ are determined from the
concentration and energy conservation requirements:
$\frac{2}{\pi}\left(\frac{T_{ex}}{\hbar
v_{W}}\right)^{2}\int\limits_{0}^{\infty}dxxf_{x}\left|{\begin{array}[]{*{20}c}1\\\
{T_{ex}x}\\\ \end{array}}\right|=\left|{\begin{array}[]{*{20}c}{n_{T}+\Delta
n_{ex}}\\\ {E_{T}+\Delta E_{ex}}\\\ \end{array}}\right|,$ (8)
where the function $f_{x}$ is introduced according to $f_{x}\equiv
f_{ex}/[e^{x}(1-f_{ex})+f_{ex}]$. Using $\Delta n_{ex}$ and $\Delta E_{ex}$
given by Eq. (6) and solving the transcendental system (8) one obtains the
initial values $f_{ex}$ and $T_{ex}$. The calculations here and below are
performed for the nitrogen temperature, $T=$77 K, the excitation energies
$2v_{W}p_{\overline{\omega}}=$120 meV (CO2 laser) and 60 meV (as an example of
interband excitation with subsequent optical phonon emission), and the
broadening energy $\hbar/\tau_{p}\simeq$6.6 meV, which corresponds to the
pulse duration $\simeq$0.1 ps. In Fig. 2 we plot $f_{ex}$ and $T_{ex}$ versus
the pumping level which is proportional to $\Delta n_{ex}/n_{T}$. Fast
increase of $T_{ex}$ and fast decrease of $f_{ex}$ take place for $\Delta
n_{ex}/n_{T}<1$, while a linear increase of these values are realized if
$\Delta n_{ex}/n_{T}>1$.
## III Energy relaxation and recombination
In this section we analyze the transient evolution of $f_{pt}$ caused by the
energy relaxation and recombination processes. We consider the cases ($i$) and
($ii$), when the initial condition is given by Eq. (5) and written through
$f_{ex}$ and $T_{ex}$ plotted in Fig. 2.
### III.1 Weak intercarrier scattering
If the carrier-carrier scattering is ineffective [case ($i$)], the
distribution $f_{pt}$ is governed by the kinetic equation (1) without the
$cc$-contribution
$\displaystyle\frac{\partial f_{pt}}{\partial
t}=\frac{\nu_{p}^{\ss(qe)}}{p^{2}}\frac{d}{dp}\left\\{p^{4}\left[\frac{df_{pt}}{dp}+\frac{f_{pt}(1-f_{pt})}{p_{T}}\right]\right\\}$
$\displaystyle+\nu_{p}^{(r)}[N_{2p/p_{T}}(1-2f_{pt})-f_{pt}^{2}]$ (9)
and with the initial condition (5) used instead of generation rate. Here we
substituted the explicit expressions of the collision integrals for the
quasielastic acoustic scattering approximation (written in the Fokker-Planck
form) and for the generation-recombination processes, see discussion in 10 .
The Planck distribution $N_{2p/p_{T}}$ is written through $p_{T}=T/v_{W}$
while the energy relaxation rate $\nu_{p}^{(qe)}=v_{qe}p/\hbar$ and the rate
of radiative transitions $\nu_{p}^{(r)}=v_{r}p/\hbar$ are written through the
characteristic velocities $v_{qe}\propto T$ and $v_{r}$ 15 .
The boundary conditions are imposed by both the condition $f_{p\to\infty
t}=0$, which is transformed into the requirement
$p^{4}\left(\frac{\partial f_{pt}}{\partial
p}+\frac{f_{pt}}{p_{T}}\right)_{p\to\infty}<{\rm const},$ (10)
and Eq. (9) at $p=0$ which is transformed into the initial condition
$f_{p=0t}=1/2+f_{p=0}^{(ex)}\exp[-(v_{r}/v_{W})Tt/\hbar]$. According to Eq.
(4) one obtains
$f_{p=0}^{(ex)}=\sqrt{\pi/2}I_{ex}\exp[-(\Omega\tau_{p})^{2}/2]\ll 1$ and one
can neglect the second contribution in this initial condition, so that
$f_{p=0t}=1/2$. Numerical solution of the Cauchy problem given by Eqs. (5),
(9), and (10) is obtained below by the use of the iteration procedure. 16
Figure 3: Distribution $f_{pt}$ versus carrier energy $pv_{W}$ for different
delay times (marked) and excitation conditions: (a)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.26, (b)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.05, and (c)
$\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.21.
In Fig.3 we demonstrate the evolution of the distribution $f_{pt}$ at 77 K for
the cases when carriers are excited around the energies 60 meV and 30 meV. The
delay times are marked in panels a-c and the pumping levels are determined
through the initial peak value, given by $\sqrt{\pi/2}I_{ex}$, see Eq. (4)).
Under mid-IR pumping with pulse duration $\tau_{p}=$0.1 ps and the spot sizes
$\sim$0.5 mm the above-used pumping levels correspond to the pulse energies
$\sim$85 pJ and $\sim$17 pJ for Figs. 3a and 3b, respectively, see 17 for
experimental details. Under optical pumping ($\hbar\Omega\sim$1.6 eV) and
subsequent emission of phonon cascade, the pumping level in Fig. 3c
corresponds to the pulse energy $\sim$12 nJ (duration and size are the same as
above). One can see that the transient evolution of distribution occurs in two
stages: energy relaxation and recombination. During the first stage (about
$t\lesssim$50 ns, which is dependent on position and maximum value
$f_{p}^{(ex)}$; compare with Figs. 3a-c) the initial peak is tranformed into
the quasiequilibrium high-energy tail (with the equilibrium temperature caused
by the energy relaxation) which is connected to the low-energy equilibrium
distrbution. During the next stage (up to 1 $\mu$s) the high-energy tail
shifts to the lower energies and transforms into the equilibrium distribution
due to effective radiative recombination in low-energy region.
### III.2 Coulomb-controlled case
In the carrier-carrier scattering case ($ii$), one has to describe the
transient evolution of the effective temperature $T_{t}$ and the maximum
distribution $f_{t}=f_{p=0t}$, that replaces the chemical potential. Since the
intercarrier scattering change neither the concentration,
$n_{t}=(4/L^{2})\sum_{\bf p}f_{pt}$, nor the energy of carriers,
$E_{t}=(4/L^{2})\sum_{\bf p}v_{W}pf_{pt}$, the balance equations for $n_{t}$
and $E_{t}$ take forms: 18
$\frac{d}{{dt}}\left|{\begin{array}[]{*{20}c}{n_{t}}\\\ {E_{t}}\\\
\end{array}}\right|=\frac{4}{{L^{2}}}\sum\limits_{\bf
p}{\left|{\begin{array}[]{*{20}c}{J_{r}\\{f_{t}|p\\}}\\\
{v_{W}p\left[{J_{ac}\\{f_{t}|p\\}+J_{r}\\{f_{t}|p\\}}\right]}\\\
\end{array}}\right|}.$ (11)
Further, we transform the balance equations, expressing the left-hand side of
(11) through $T_{t}$ and $f_{t}$ as follows:
$\displaystyle\frac{d}{dt}\left(T_{t}^{2}A_{t}^{(1)}\right)=R_{t}^{(1)},$ (12)
$\displaystyle\frac{d}{dt}\left(T_{t}^{3}A_{t}^{(2)}\right)=R_{t}^{(2)}+Q_{t}.$
Here the coefficients $A_{t}^{(1,2)}$ are written as
$A_{t}^{(q)}=\int_{0}^{\infty}dxx^{q}f_{xt}$, where the quasiequilibrium
distribution is given by $f_{xt}=f_{t}/\left[e^{x}(1-f_{t})+f_{t}\right]$, so
that $A_{t}^{(q)}/T_{t}^{l}$ are only depend on $f_{t}$. After substitution of
the collision integrals $J_{r}$ 10 ; 18 and integration, the generation-
recombination contributions to Eq. (12) are obtained in the form
$R_{t}^{(q)}=\frac{2v_{r}T_{t}^{q+2}}{v_{W}\hbar}\int\limits_{0}^{\infty}dxx^{q+2}f_{xt}^{2}\left[\frac{e^{2x}(1-f_{t})^{2}}{(e^{x2T_{t}/T}-1)f_{t}^{2}}-1\right].$
(13)
Similarly, the energy relaxation contribution is written by the use of
$J_{ac}$ as follows
$Q_{t}=\frac{T-T_{t}}{T}\frac{v_{qe}T_{t}^{4}}{v_{W}\hbar}\int\limits_{0}^{\infty}dxx^{4}e^{x}f_{xt}^{2}\frac{1-f_{t}}{f_{t}}.$
(14)
The initial conditions for the system (11) are written as $T_{t=0}=T_{ex}$ and
$f_{t=0}=f_{ex}$.
Figure 4: Temporal evolution of effective temperature, $T_{t}$ (a), and
maximum distribution, $f_{t}$ (b), for different excitation conditions: (1)
$\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.21, (2)
$\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.1, (3)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.052, and (4)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.026. Figure 5: Energy per
carrier (a) and concentration (b) versus time. Solid and dotted curves
correspond to the cases ($i$) and ($ii$) , respectively; excitation conditions
(1) - (4) are the same as in Fig. 4.
Numerical solution of the nonlinear system (11) is performed using the
iteration procedure. In Fig. 4 we plot $T_{t}/T$ and $f_{t}$ versus time.
Temperature relaxes to the equilibrium one during the energy relaxation times
($\lesssim$ 100 ns) while $f_{t}$, which is determined by the chemical
potential $\mu_{t}$, relaxes to 1/2 over 1 $\mu$s (the recombination time
scale), in analogy with the case ($i$). Notice, that after the fast energy
relaxation, one obtains $f_{t}>$1/2 [dotted line in Fig. (4b)], i.e. the low-
energy electron-hole pairs appear to be unstable. 19
Fig. 5 shows the plot of temporal evolutions of the energy per particle and
concentration, $E_{t}/n_{t}$ and $n_{t}$ [see the definitions before Eq.
(11)], for the cases ($i$) and ($ii$). The relaxation processes to the
equilibrium (at nitrogen temperature,
$E_{t\to\infty}/n_{t\to\infty}\simeq$14.5 meV and $n_{t\to\infty}\simeq
5.3\cdot 10^{9}$ cm-2) occur during the same scales as in Figs. 3 and 4. The
temporal dependencies of $n_{t}$ obtained for both cases are in good agreement
(the carreir-carrier scattering does not change concentration) while $E_{t}$
demonstrates a different evolution for cases ($i$) and ($ii$) at $t<$50 ns.
This is because of drift and decrease of photoexcited peak during the energy
relaxation time, see Fig. 3.
## IV Transient response
Here we turn to consideration of the response of photoexcited carriers on a
probe radiation (reflection and transmission in the THz and mid-IR spectral
regions) and on a weak dc electric field (photoconductivity). The transient
electrodynamics of graphene is described using the time-dependent dynamic
conductivity, $\sigma_{\omega t}$, which is caused by the collisionless
interband transitions, see Appendix B. The transient photoconductivity is
calculated by the use of the phenomenological model of momentum relaxation
suggested in 13 .
### IV.1 Reflection and transmission
To calculate the transient reflectance and transmittance of the graphene sheet
placed at $z=0$ on the in-plane electric field ${\bf E}_{zt}\exp(-i\omega t)$
propagated along $0Z$, we apply the wave equation, see 20 and references
therein. The induced current density, $\sigma_{\omega t}E_{z=0}$, is located
around $z=0$ and direction of in-plane field ${\bf E}_{zt}$ is not essential
due to the in-plane isotropy of the problem. Separating the incident
radiation, $E_{in}e^{ik_{\omega}z}$, with the wave vector
$k_{\omega}=\omega/c$, we write the field distribution outside of the graphene
sheet in the form:
$E_{zt}=\left\\{\begin{array}[]{*{20}c}E_{in}e^{ik_{\omega}z}+E_{t}^{(t)}e^{-ik_{\omega}z},&{z<-0}\\\
E_{t}^{(t)}e^{i\overline{k}_{\omega}z},&{z>+0}\end{array}\right.,$ (15)
where $\overline{k}_{\omega}=\sqrt{\epsilon}\omega/c$ is the wave vector in
the substrate with the dielectric permittivity $\epsilon$. The transmitted and
reflected electric fields, $E_{t}^{(t)}$ and $E_{t}^{(r)}$, are determined
from the boundary conditions at $z\to 0$ as follows:
$\frac{E_{t}^{(t)}}{E_{in}}=\frac{2}{1+A_{\omega
t}},~{}~{}~{}~{}\frac{E_{t}^{(t)}}{E_{in}}=\frac{1-A_{\omega t}}{1+A_{\omega
t}}.$ (16)
Here we introduce the dimensionless factor $A_{\omega
t}=\sqrt{\epsilon}+(4\pi/c)\sigma_{\omega t}$. The reflection and transmission
coefficients, $R_{\omega t}=|E_{t}^{(r)}|^{2}/E_{in}^{2}$ and $T_{\omega
t}=|E_{t}^{(t)}|^{2}/E_{in}^{2}$, are written through $A_{\omega t}$ according
to
$R_{\omega t}=\left|\frac{1-A_{\omega t}}{1+A_{\omega
t}}\right|^{2},~{}~{}~{}~{}~{}T_{\omega
t}=\frac{4\sqrt{\epsilon}}{\left|1+A_{\omega t}\right|^{2}}.$ (17)
Using $\sigma_{\omega t}$ determined by Eqs. (B3) and (B4), we consider below
the differential changes in reflectivity and transmissivity, $(\Delta
R/R)_{\omega t}=(R_{\omega t}-R_{\omega}^{(eq)})/R_{\omega}^{(eq)}$ and
$(\Delta T/T)_{\omega t}=(T_{\omega t}-T_{\omega}^{(eq)})/T_{\omega}^{(eq)}$,
which are written through the equilibrium reflection and transmission
coefficients, $R_{\omega}^{(eq)}$ and $T_{\omega}^{(eq)}$.
Figure 6: (a) Spectral dependencies of differential reflectivity, $(\Delta
R/R)_{\omega t}$, for different delays (marked) at the excitation conditions:
(a) $\hbar\overline{\omega}=$120 meV and $I_{ex}=0.052$ in the case ($i$), (b)
$\hbar\overline{\omega}=$120 meV and $I_{ex}=$0.052 in the case ($ii$), and
(c) $\hbar\overline{\omega}=$60 meV and $I_{ex}=$0.21 in the case ($ii$).
The evolution of the differential reflectivity for the cases ($i$) and ($ii$)
are shown in Figs. 6a and 6b, 6c, respectively. If the Coulomb scattering is
not effective [case ($i$)], the distribution of carriers relaxes during the
energy relaxation time scale (around 10 ns, cf. with Fig. 3), when a quenching
of photoexcited peak takes place (if $\hbar\omega$ is comparable with the peak
energy). In case ($ii$) any peculiarities of the spectrsal dependencies at
stort times are absent because the initial distribution is transformed into
the quasiequilibrium one during times $\sim\tau_{cc}\to 0$. The further
evolution of $(\Delta R/R)_{\omega t}$ is limited by the generation-
recombination process and extended up to microseconds. In the THz spectral
region ($\hbar\omega\geq$10 meV is considered here because we neglect the
intraband relaxation), the differential reflectivity increases and changes a
sign. In the high-energy region, $(\Delta R/R)_{\omega t}$ decreases
monotonically with $\omega$ and $t$ and does not exceed $\sim 10^{-4}$ for the
near-IR spectral region. Beside of this, the response is approximately
proportional to the pumping intensity, $I_{ex}$, and $(\Delta R/R)_{\omega t}$
increases with increasing of the photoexcitation energy,
$\hbar\overline{\omega}$ (cf. Figs. 6b and 6c).
Figure 7: (a) Differential transmissivity, $(\Delta T/T)_{\omega t}$, versus
$\hbar\omega$ and $t$ for cases ($i$) and ($ii$) [panels (a) and (b),
respectively] at the same excitation conditions: $\hbar\overline{\omega}=$120
meV, $I_{ex}=$0.052.
In Fig. 7 we plot the differential transmissivity for the cases ($i$) and
($ii$) under the same excitation conditions. Once again, in the high-frequency
region the differential transmissivity decreases slowly (during a microsecond
time scale) and $(\Delta T/T)_{\omega t}$ does not exceed $\sim 10^{-4}$ for
the near-IR spectral region. In the THz spectral region, $(\Delta T/T)_{\omega
t}$ increses and changes the sing in the same manner as $(\Delta R/R)_{\omega
t}$ (cf. Figs. 6 and 7). The dependencies on the excitation parameters
($I_{ex}$ and $\hbar\overline{\omega}$) are also similar to the reflectivity.
Additionally, in case ($i$) a fast (at $t<$10 ns) quenching of the
photoexcited peak contribution in the spectral region $\sim\hbar\Omega$ takes
place.
### IV.2 Photoconductivity
Finally, we consider the transient photoconductivity, i.e. the response of the
photoexcited carriers to the weak dc electric field. Since the momentum
relaxation is governed by elastic scattering mechanisms, 13 one can use the
following expression for the dc conductivity $\sigma_{t}$:
$\sigma_{t}=\sigma_{0}\left[2f_{p=0t}-\frac{l_{c}}{\hbar}\int_{0}^{\infty}dpf_{pt}\frac{\Psi^{\prime}(pl_{c}/\hbar)}{\Psi(pl_{c}/\hbar)^{2}}\right].$
(18)
Here $l_{c}$ is the correlation length characterizing the disorder scattering
and the function $\Psi(z)=e^{-z^{2}}I_{1}(z^{2})/z^{2}$ is written through the
first order Bessel function of imaginary argument, $I_{1}(z)$. The normalized
conductivity, $\sigma_{0}$, is introduced for the case of short-range
scattering, when $l_{c}=0$. The distribution $f_{p=0t}$ is shown in Fig. 4b
for the case ($ii$) while $2f_{p=0t}=1$ for the case ($i$). If $l_{c}=0$, one
obtains $\sigma_{t}/\sigma_{0}=1$, i.e. there is no transient
photoconductivity for the case ($i$); for the case ($ii$) one obtains
$\sigma_{t}/\sigma_{0}=2f_{t}$ and the transient photoconductivity is clear
from Fig. 4b.
Figure 8: Temporal evolution of conductivity for excitation conditions (1)-(4)
which are the same as in Fig. 4 for the correlation length $l_{c}=$30 nm and
10 nm. Solid and dashed curves are correspondent to the cases ($ii$) and
($i$).
If $l_{c}\neq 0$, the transient evolution of conductivity is shown in Fig. 8.
For the definiteness, it was assumed that $l_{c}$=10 and 30 nm and variations
of $\sigma_{t}$ are increased with $l_{c}$ essentially due to contribution of
high-energy carriers. Similar to Sec. IVA, one can separate two stages of
evolution: the fast decrease of $\sigma_{t}$ due to energy relaxation (up to
$\sim 30\div 50$ ns for the conditions considered) and the slow quenching of
$\sigma_{t}$ due to carrier recombination. If $t>1~{}\mu$s, the conductivity
approaches to the equilibrium values: $\sigma_{t\to\infty}/\sigma_{0}=$1.445
if $l_{c}$=30 nm and $\sigma_{t\to\infty}/\sigma_{0}=$1.035 if $l_{c}$=10 nm.
Since the transient conductivity can be measured for the subnanosecond time
scale 21 , such a scheme can be used for verification both energy relaxation
and recombination mechanisms.
## V Concluding remarks
To summarize, we have considered both the interband ultrafast photoexcitation
and the relaxation dynamics of the carriers in an intrinsic graphene. In
contrast to the measurements 3 ; 4 ; 5 ; 6 and calculations 8 ; 9 performed,
where the evolution corresponds to the subpicosecond time scales due to the
opticlal phonon contribution, here we consider the slow relaxation of the low-
energy carriers. The distribution of carriers at $T=$77 K is obtained for the
limiting cases with negligible or dominating intercarrier scattering when the
energy relaxation and generation-recombination processes are caused by the
quasielastic acoustic phonon scattering and thermal radiation, respectively.
The initial distribution is obtained in the framework of the linear, with
respect to pumping, approximation for the collisionless regime of the
interband transitions. The transient optical response on the probe radiation
(transmission and reflection) as well as on the weak dc field (transient
photoconductivity) appears to be strongly dependent on the relaxation and
recombination dynamics of carriers.
Next, we discuss the assumptions made. The main restrictions of the results
presented are the consideration of the low-energy carriers, when the
interaction with optical phonons is unessential, and the single generation-
recombination mechanism (due to thermal radiation) is taken into account.
These conditions are realized at low temperatures under the mid-IR ultrafast
excitation 17 of the clean sample (e.g. suspended graphene 22 ). Such an
approach can be used for the case of optical interband excitation, when the
low-energy initial distrbution, with a phenomenological broadening, is formed
after the cascade process of optical phonon emission. The consideration is
restricted by the radiative recombination (the Auger processes are forbidden
due to the symmetry of electron-hole states 23 ), with the characteristic time
scales up to microseconds. Any visible contribution of other generation-
recombination mechanism (e.g., because of disorder-induced interband
transitions with acoustic phonons, or under intercarrier scattering) leads to
fast decrease of photoresponse. Such a regime requires an additional
investigation but the quasielastic energy relaxation stage is described by the
presented results.
The rest of assumptions are rather standard. The consideration in Sec. III is
limited by the simple cases ($i$) and ($ii$), with and without the
intercarrier scattering. The main peculiarities of the response under
consideration are similar for both cases but the complete description of the
nonequilibrium carriers had been performed neither under optical excitation,
nor under high dc field, see 18 ; 24 and Refs. therein. The description of
the momentum relaxation in Sec. IV is based on the phenomenological model of
Ref. 13. The utilization of the quasielastic energy scattering and the
collisionless interband photoexcitation appear to be rather natural. The
listed assumptions do not change either the character of the response or the
numerical estimates.
In closing, the peculiarities of the transient optical response (transmission
and reflection) as well as of the transient photoconductivity appear to be
useful tool in order to verify the relaxation and generation-recombination
mechanisms of carriers. Thus, in addition to the recently obtained
experimental results 3 ; 4 ; 5 ; 6 ; 7 measurements under mid-IR excitation
and at low-temperature will be useful for characterization of graphene.
## Appendix A Generation rate
Below we describe the interband carrier excitation under ultrafast mid-IR
pumping ${\bf E}_{t}\exp(-i\Omega t)$ for the collisionless case, when
$\tau_{p}$ is shorter than relaxation times. The photogeneration rate into the
$\alpha$-state is based on the general expression (see 1 and Sec. 54 in Ref.
12)
$\displaystyle G_{\alpha
t}=2Re\left(\frac{e}{\hbar\Omega}\right)^{2}\int\limits_{-\infty}^{0}d\tau
e^{\lambda\tau-i\Omega\tau}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (19)
$\displaystyle\times\left\langle\alpha\left|\left[e^{i\hat{h}\tau/\hbar}\left[\left({\bf
E}_{t+\tau}\cdot{\bf\hat{v}}\right),\hat{\rho}_{t+\tau}\right]e^{-i\hat{h}\tau/\hbar},\left({\bf
E}_{t}\cdot{\bf\hat{v}}\right)^{+}\right]\right|\alpha\right\rangle,$
where $\hat{\rho}_{t}$ is the density matrix, $\hat{\bf v}$ is the velocity
operator, and $\lambda\to+0$. Since the collisionless regime of
photoexcitation, we calculate (A1) with the use of the free states $|l{\bf
p}\rangle$ and the energy $\varepsilon_{lp}$ where $l=\pm 1$ stands for $c$\-
or $v$-bands and $\bf p$ is the 2D momentum. Neglecting the nondiagonal
components of the density matrix $\hat{\rho}_{t}$ and using the distribution
functions $f_{l{\bf p}t}$, one obtains the generation rate
$\displaystyle G\\{f|1{\bf
p}t\\}=\left(\frac{e}{\hbar\Omega}\right)^{2}\int\limits_{-\infty}^{0}d\tau
e^{\lambda\tau-i\Omega\tau}e^{i(\varepsilon_{1p}-\varepsilon_{-1p})\tau/\hbar}$
$\displaystyle\times\langle 1{\bf p}|({\bf
E}_{t+\tau}\cdot{\bf\hat{v}})|-1{\bf p}\rangle\langle-1{\bf p}|({\bf
E}_{t}\cdot{\bf\hat{v}})^{+}|1{\bf p}\rangle~{}~{}~{}~{}~{}$ (20)
$\displaystyle\times\left(f_{-1{\bf p}t+\tau}-f_{1{\bf
p}t+\tau}\right)+c.c.~{},$
moreover $G\\{f|-1{\bf p}t\\}=-G\\{f|1{\bf p}t\\}$ according to the particle
concervation law. Next, we separate the envelope form-factor $w_{t}$ using
${\bf E}_{t}={\bf E}w_{t}$ and take into account the in-plane isotropy of the
problem, when one arrives to the averaged matrix element
$\overline{\left|\left\langle+1{\bf p}\left|({\bf
E}\cdot{\bf\hat{v}})\right|-1{\bf p}\right\rangle\right|^{2}}=(Ev_{W})^{2}/2$.
As a result, we obtain the in-plane isotropic generation rate $G\\{f|pt\\}=\pm
G\\{f|\pm lpt\\}$ in the following form:
$\displaystyle
G\\{f|pt\\}=\left(\frac{eEv_{W}}{\hbar\Omega}\right)^{2}\frac{w_{t}}{2}\int\limits_{-\infty}^{0}d\tau
w_{t+\tau}e^{\lambda\tau-i\Omega\tau}$ $\displaystyle\times
e^{i(2v_{W}p)\tau/\hbar}\left(f_{-1{\bf p}t+\tau}-f_{1{\bf
p}t+\tau}\right)+c.c.~{}.$ (21)
Finally, using the electron-hole representation and replacing the filling
factor here by $(1-2f_{pt})$, we arrive to Eq. (2).
## Appendix B Dynamic conductivity
The response of graphene on the in-plane probe field ${\bf E}\exp(-i\omega t)$
is described by the dynamic conductivity 20 ; 25
$\displaystyle\sigma_{\omega t}\approx i\frac{2(ev_{W})^{2}}{\omega
L^{2}}\sum\limits_{\bf p}(1-2f_{pt})~{}~{}~{}~{}$ (22)
$\displaystyle\times\left(\frac{1}{\hbar\omega+2v_{W}p+i\lambda}-\frac{1}{\hbar\omega-2v_{W}p+i\lambda}\right)$
with $\lambda\to+0$. The parametric time dependency of $\sigma_{\omega t}$ is
valid if the time scales under consideration exceed $\omega^{-1}$. It is
convenient to separate the time-independent contribution,
$\overline{\sigma}_{\omega}$, described the undoped graphene in the absence of
photoexcitation, when $f_{pt}$ vanishes. Using the energy conservation law one
obtains ${\rm Re}\overline{\sigma}_{\omega}=e^{2}/4\hbar$. In the framework of
the Weyl-Wallace model, the ${\rm Im}$-contribution into
$\overline{\sigma}_{\omega}$ appears to be divergent at $p\to\infty$. It is
convenient to approximate ${\rm Im}\overline{\sigma}_{\omega}$ as a sum of
$\propto\omega^{-1}$ and $\propto\omega$ terms, which correspond to the
contributions of the virtual interband transitions and the ion background,
correspondingly. As a result, we obtain:
${\rm
Im}\overline{\sigma}_{\omega}\approx\frac{e^{2}}{\hbar}\left(\frac{\varepsilon_{m}}{\hbar\omega}-\frac{\hbar\omega}{\varepsilon_{i}}\right),$
(23)
where the characteristic energies, $\varepsilon_{m}\simeq$0.1 eV, and
$\varepsilon_{i}\simeq$ 6.8 eV are correspondent to the recent measurements of
the graphene optical spectrum. 26
Next, substituting the time-dependent distribution $f_{pt}$ obtained in Sec.
III into the dynamic conductivity (B1) one transforms the real and imagional
parts of $\sigma_{\omega t}$ as follows
$\displaystyle{\rm Re}\sigma_{\omega
t}=\frac{e^{2}}{4\hbar}\left[1-2F\left(p_{\omega},t\right)\right],~{}~{}~{}~{}~{}~{}~{}$
(24) $\displaystyle{\rm Im}\sigma_{\omega t}={\rm
Im}\overline{\sigma}_{\omega}-\frac{e^{2}}{\pi\hbar}{\cal
P}\int\limits_{0}^{\infty}\frac{dyy^{2}}{1-y^{2}}F(p_{\omega}y,t).$
Here $\cal P$ means the principal value of integral. We also introduced the
function $F(p,t)=f_{pt}$ for the case ($i$) and
$F\left(p_{\omega}y,t\right)=\frac{f_{t}}{\exp[(\hbar\omega/T_{t})y](1-f_{t})+f_{t}}$
(25)
for the case ($ii$), when $\sigma_{\omega t}$ is determined both the effective
temperature and the carrier concentration, $T_{t}$ and $f_{t}$.
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* (14) The form-factor $w_{t}$ is normalized according to the condition $\int_{-\infty}^{\infty}dtw_{t}^{2}=\tau_{p}$. The shape of normalized form-factor has little effect on the transient photoexcitation under consideration because the ultrafast response is determined fundamentally by the pulse duration, $\tau_{p}$.
* (15) Here we use the characteristic velocities $v_{ac}\simeq$2.5$\times 10^{5}$ cm/s (for the nitrogen temperature) and $v_{r}\simeq$41.6 cm/s (for graphene sheet placed between SiO2 substrate and cover layer), see 10 .
* (16) D. Potter, Computational Physics (J. Wiley, London, 1973).
* (17) T. Elsaesser and M. Woerner, Physics Reports 321, 253 (1999).
* (18) O. G. Balev, F. T. Vasko and V. Ryzhii, Phys. Rev. B 79, 165432 (2009).
* (19) According to Eq.(B3), the negative interband absorption ($\propto{\rm Re}\sigma_{\omega t}$) takes place, if $f_{pt}>1/2$, see 10 and Refs. therein. This low-energy instability is suppressed due to an effective intraband (Drude) absorption.
* (20) L. A. Falkovsky, Phys. Usp. 51, 887 (2008); T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B78, 085432 (2008); M. V. Strikha and F. T. Vasko, submitted.
* (21) T. Yao, K. Inagaki, and S. Maekawa, in Proceedings of the 11th International Conference on the Physics of Semiconductors (Polish Scientific Publishers, Warszawa, 1972), Vol. 1, p. 417.
* (22) G. Li, A. Luican, and E. Y. Andrei, Phys. Rev. Lett. 102, 176804 (2009); P. Neugebauer, M. Orlita, C. Faugeras, A. L. Barra, and M. Potemski, Phys. Rev. Lett. 103, 136403 (2009).
* (23) M. S. Foster and I. L. Aleiner, Phys. Rev. B 79, 085415 (2009).
* (24) A. Akturka and N. Goldsman, J. Appl. Phys. 103, 053702 (2008); R. S. Shishir and D. K. Ferry, J. Phys.: Condens. Matter, 21, 344201 (2009).
* (25) R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, Science 320, 1308 (2008); T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B78, 085432 (2008); K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
* (26) M. Bruna and S. Bonilla, Appl. Phys. Lett. 94, 031901 (2009).
|
arxiv-papers
| 2009-12-02T13:27:53 |
2024-09-04T02:49:06.805289
|
{
"license": "Public Domain",
"authors": "P.N. Romanets and F.T. Vasko",
"submitter": "Fedir Vasko T",
"url": "https://arxiv.org/abs/0912.0423"
}
|
0912.0500
|
# Gamma-ray and Cosmic-ray Tests of Lorentz Invariance Violation and Quantum
Gravity Models and Their Implications
Floyd W. Stecker
###### Abstract
The topic of Lorentz invariance violation (LIV) is a fundamental question in
physics that has taken on particular interest in theoretical explorations of
quantum gravity scenarios. I discuss various $\gamma$-ray observations that
give limits on predicted potential effects of Lorentz invariance violation.
Among these are spectral data from ground based observations of the multi-TeV
$\gamma$-rays from nearby AGN, INTEGRAL detections of polarized soft
$\gamma$-rays from the vicinity of the Crab pulsar, Fermi Gamma Ray Space
Telescope studies of photon propagation timing from $\gamma$-ray bursts, and
Auger data on the spectrum of ultrahigh energy cosmic rays. These results can
be used to seriously constrain or rule out some models involving Planck scale
physics. Possible implications of these limits for quantum gravity and Planck
scale physics will be discussed.
###### Keywords:
quantum gravity
###### :
04.60Bc
## 1 Introduction
It has been the major goal of particle physics to discover a theoretical
framework for unifying gravity with the other three known forces, viz.,
electromagnetism, and the weak and strong nuclear forces. Such a theory must
be compatible with quantum theory at very small scales corrsponding to very
high energies. Even the possibly less ambitious goal of reconciling general
relativity with quantum theory has been elusive and may require new concepts
to accomplish.
There has been a particular interest in the possibility that a quantum gravity
theories will lead to Lorentz invariance violation (LIV) at the Planck scale,
$\lambda_{Pl}=\sqrt{G\hbar/c^{3}}\sim 1.6\times 10^{-35}$ m. This scale
corresponds to a mass (energy) scale of $M_{Pl}=\hbar/(\lambda_{Pl}c)\sim
1.2\times 10^{19}$ GeV/c2. It is at the Planck scale where quantum effects are
expected to play a key role in determining the effective nature of space-time
that emerges as general relativity in the classical continuum limit. The idea
that Lorentz invariance (LI) may indeed be only approximate has been explored
within the context of a wide variety of suggested Planck-scale physics
scenarios. These include the concepts of deformed relativity, loop quantum
gravity, non-commutative geometry, spin foam models, and some string theory (M
theory) models. Such theoretical explorations and their possible consequences,
such as observable modifications in the energy-momentum dispersion relations
for free particles and photons, have been discussed under the general heading
of “Planck scale phenomenology”. There is an extensive literature on this
subject. (See ma05 for a review; some recent references are Refs. el08 –
he09 . For a non-technical treatment of the present basic approaches to a
quantum gravity theory, see Ref. smolin ). One should keep in mind that in a
context that is separate from quantum gravity considerations, it is important
to test LI for its own sake co98 ; cg99 . LIV gratia LIV. The significance of
such an approach is evident when one considers the unexpected discoveries of
the violation of $P$ and $CP$ symmetries. In fact, it has been shown that a
violation of $CPT$ would imply LIV gr02
We will consider here some of the consequent searches for such effects using
high energy astrophysics observations, particularly observations of high
energy cosmic $\gamma$-rays and ultrahigh energy cosmic rays.
## 2 LIV Perturbations
We know that Lorentz invariance has been well validated in particle physics;
indeed, it plays an essential role in designing machines such as the new LHC
(Large Hadron Collider). Thus, any LIV extant at accelerator energies (“low
energies”) must be extremely small. This consideration is reflected by adding
small Lorentz-violating terms in the free particle Lagrangian. Such terms can
be postulated to be independent of quantum gravity theory, e.g., Refs. co98 ;
cg99 . Alternatively, it can be assumed that the terms are small because they
are suppressed by one or more powers of $p/M_{Pl}$ (with the usual convention
that $c=1$.) In the latter case, in the context of effective field theory
(EFT), such terms are assumed to approximate the effects of quantum gravity at
“low energies” when $p\ll M_{Pl}$.
One result of such assumptions is a modification of the dispersion relation
that relates the energy and momentum of a free particle or photon. This, in
turn, can lead to a maximmum attainable velocity (MAV) of a particle different
from $c$ or a variation of the velocity of a photon in vacuo with photon
energy. Both effects are clear violations of relativity theory. Such
modifications of kinematics can result in changes in threshold energies for
particle interactions, suppression of particle interactions and decays, or
allowance of particle interactions and decays that are kinematically forbidden
by Lorentz invariance cg99 .
A simple formulation for breaking LI by a small first order perturbation in
the electromagnetic Lagrangian which leads to a renormalizable treatment has
been given by Coleman and Glashow cg99 . The small perturbative noninvariant
terms are both rotationally and translationally invariant in a preferred
reference frame which one can assume to be the frame in which the cosmic
background radiation is isotropic. These terms are also taken to be invariant
under $SU(3)\otimes SU(2)\otimes U(1)$ gauge transformations in the standard
model.
Using the formalism of Ref. cg99 , we denote the MAV of a particle of type $i$
by $c_{i}$, a quantity which is not necessarily equal to $c\equiv 1$, the low
energy in vacua velocity of light. We further define the difference
$c_{i}-c_{j}\equiv\delta_{ij}$. These definitions can be generalized and can
be used to discuss the physics implications of cosmic-ray and cosmic
$\gamma$-ray observations sg01--st09.
## 3 Electroweak Interactions
In general then, $c_{e}\neq c_{\gamma}$. The physical consequences of such a
violation of LI depend on the sign of the difference between these two MAVs.
Defining
$c_{e}\equiv c_{\gamma}(1+\delta)~{},~{}~{}~{}~{}0<|\delta|\ll 1\;,$ (1)
one can consider the two cases of positive and negative values of $\delta$
separately cg99 ; sg01 .
Case I: If $c_{e}<c_{\gamma}$ ($\delta<0$), the decay of a photon into an
electron-positron pair is kinematically allowed for photons with energies
exceeding
$E_{\rm max}=m_{e}\,\sqrt{2/|\delta|}\;.$ (2)
The decay would take place rapidly, so that photons with energies exceeding
$E_{\rm max}$ could not be observed either in the laboratory or as cosmic
rays. From the fact that photons have been observed with energies
$E_{\gamma}\geq$ 50 TeV from the Crab nebula, one deduces for this case that
$E_{\rm max}\geq 50\;$TeV, or that -$\delta<2\times 10^{-16}$.
Case II: For this possibility, where $c_{e}>c_{\gamma}$ ($\delta>0$),
electrons become superluminal if their energies exceed $E_{\rm max}/2$.
Electrons traveling faster than light will emit light at all frequencies by a
process of ‘vacuum Čerenkov radiation.’ This process occurs rapidly, so that
superluminal electron energies quickly approach $E_{\rm max}/2$. However,
because electrons have been seen in the cosmic radiation with energies up to
$\sim\,$2 TeV, it follows that $E_{\rm max}\geq 2$ TeV, which leads to an
upper limit on $\delta$ for this case of $3\times 10^{-14}$. Note that this
limit is two orders of magnitude weaker than the limit obtained for Case I.
However, this limit can be considerably improved by considering constraints
obtained from studying the $\gamma$-ray spectra of active galaxies sg01 .
### 3.1 Constraints on LIV from AGN Spectra
A constraint on $\delta$ for $\delta>0$ follows from a change in the threshold
energy for the pair production process $\gamma+\gamma\rightarrow e^{+}+e^{-}$.
This follows from the fact that the square of the four-momentum is changed to
give the threshold condition
$2\epsilon
E_{\gamma}(1-cos\theta)~{}-~{}2E_{\gamma}^{2}\delta~{}\geq~{}4m_{e}^{2},$ (3)
where $\epsilon$ is the energy of the low energy photon and $\theta$ is the
angle between the two photons. The second term on the left-hand-side comes
from the fact that $c_{\gamma}=\partial E_{\gamma}/\partial p_{\gamma}$. It
follows that the condition for a significant increase in the energy threshold
for pair production is $E_{\gamma}\delta/2$ $\geq$ $m_{e}^{2}/E_{\gamma}$, or
equivalently, $\delta\geq{2m_{e}^{2}/E_{\gamma}^{2}}$. The observed
$\gamma$-ray spectrum of the active galaxies Mkn 501 and Mkn 421 while flaring
ah01 exhibited the high energy absorption expected from $\gamma$-ray
annihilation by extragalactic pair-production interactions with extragalactic
infrared photons ds02 ; ko03 . This led Stecker and Glashow sg01 to point out
that the Mkn 501 spectrum presents evidence for pair-production with no
indication of LIV up to a photon energy of $\sim\,$20 TeV and to thereby place
a quantitative constraint on LIV given by
$\delta<2m_{e}^{2}/E_{\gamma}^{2}\simeq 10^{-15}$.
## 4 Gamma-ray Constraints on Quantum Gravity and Extra Dimension Models
As previously mentioned, LIV has been proposed to be a consequence of quantum
gravity physics at the Planck scale ga95 ; al02 . In models involving large
extra dimensions, the energy scale at which gravity becomes strong can occur
at a quantum gravity scale, $M_{QG}<<M_{Pl}$, even approaching a TeV ar98 . In
the most commonly considered case, the usual relativistic dispersion relations
between energy and momentum of the photon and the electron are modified al02 ;
ac98 by a term of order $p^{3}/M_{QG}$.
Generalizing the LIV parameter $\delta$ from equation (1) to an energy
dependent form, we find
$\delta~{}\equiv~{}{\partial E_{e}\over{\partial p_{e}}}~{}-~{}{\partial
E_{\gamma}\over{\partial
p_{\gamma}}}~{}\simeq~{}{E_{\gamma}\over{M_{QG}}}~{}-~{}{m_{e}^{2}\over{2E_{e}^{2}}}~{}-~{}{E_{e}\over{M_{QG}}}.$
(4)
It follows that the threshold condition for pair production given by equation
(3) implies that $M_{QG}~{}\geq~{}E_{\gamma}^{3}/8m_{e}^{2}.$ Since pair
production occurs for energies of at least 20 TeV, we find a constraint on the
quantum gravity scale st03 $M_{QG}\geq 0.3M_{Pl}$. This constraint
contradicts the predictions of some proposed quantum gravity models involving
large extra dimensions and smaller effective Planck masses. In a variant model
of Ref. el04 , the photon dispersion relation is changed, but not that of the
electrons. In this case, we find the even stronger constraint $M_{QG}\geq
0.6M_{Pl}$.
## 5 Energy Dependent Photon Delays from GRBs and Tests of Lorentz Invariance
Violation
One possible manifestation of Lorentz invariance violation, from Planck scale
physics produced by quantum gravity effects, is a change in the energy-
momentum dispersion relation of a free particle or a photon. If this results
from the linear Planck-supressed term as in equation (4) above, this results
in a photon velocity retardation that is of first order in $E_{\gamma}/M_{QG}$
ac98 ; el00 . In a $\Lambda CDM$ cosmology, where present observational data
indicate that $\Omega_{\Lambda}\simeq 0.7$ and $\Omega_{m}\simeq 0.3$, the
resulting difference in the propagation times of two photons having an energy
difference $\Delta E_{\gamma}$ from a $\gamma$-ray burst (GRB) at a redshift
$z$ will be
$\Delta t_{LIV}=H_{0}^{-1}{{\Delta E_{\gamma}}\over
M_{QG}}{\int_{0}^{z}}{{dz^{\prime}(1+z^{\prime})}\over{\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+z^{\prime})^{3}}}}$
(5)
for a photon dispersion of the form $c_{\gamma}=c(1-E_{\gamma}/M_{QG}$), with
$c$ being the usual low energy velocity of light ja08 . In other words,
$\delta$, as defined earlier, is given by $-E_{\gamma}/M_{QG}$.
The Fermi Gamma-ray Space Telescope, (see Figure 1), with its $\gamma$-ray
Burst Monitors (GBM) covers an energy range from 8 keV to 40 MeV and its Large
Area Telescope (LAT) covers an energy range from 20 MeV to $>300$ GeV. 111See
paper the of Silvia Rainò, these proceedings. It can observe and study both
GRBs and flares from active galactic nuclei over a large range of both energy
and distance. This was the case with the GRB 090510, a short burst at a
cosmological distance corresponding to a redshift of 0.9 that produced photons
with energies extending from the X-ray range to a $\gamma$-ray of energy
$\sim$ 31 GeV. This burst was therefore a perfect subject for the application
of equation (5). Fermi observations of GRB090510 have yielded the best
constraint on any first order retardation of photon velocity with energy
$\Delta t\propto(E/M_{QG})$. This result would require a value of
$M_{QG}\mathrel{\raise 2.15277pt\hbox{$>$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}1.2M_{Pl}$ Fermi2009 222See also the paper of Francesco
de Palma, these proceedings. In large extra dimension scenarios, one can have
effective Planck masses smaller than $1.22\times 10^{19}$ GeV, whereas in most
QG scenarios, one expects that the minimum size of space-time quanta to be
$\lambda_{Pl}$. This implies a value for $M_{QG}\mathrel{\raise
1.29167pt\hbox{$<$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}M_{Pl}$ in all
cases.
In particular, we note the string theory inspired model of Ref. el08 . This
model invisions space-time as a gas of D-particles in a higher dimensional
bulk where the observable universe is a D3 brane. The photon is represented as
an open string that interacts with the D-particles, resulting a retardation
$\propto E_{\gamma}/M_{QG}$. The new Fermi data appear to rule out this model
as well as other models that predict such a retardation.
The dispersion effect will be smaller if the dispersion relation has a
quadratic dependence on $E_{\gamma}/M_{QG}$ as suggested by effective field
theory considerations my03 ; ja04 . This will obviate the limits on $M_{QG}$
given above. These considerations also lead to the prediction of vacuum
birefringence (see next section).
Figure 1: Schematic of the Fermi satellite, launched in June of 2008\. The LAT
is located at the top (yellow area) and the GBM array is located directly
below.
## 6 Looking for Birefringence Effects from Quantum Gravity
A possible model for quantizing space-time which has been actively
investigated is loop quantum gravity (see the review given in Ref. pe04 and
references therein.) A signature of this model is that the quantum nature of
space-time can produce a vacuum birefringence effect. (See also the EFT
treatment in Ref. my03 .) This is because electromagnetic waves of opposite
circular polarizations will propagate with different velocities, which leads
to a rotation of linear polarization direction through the angle
$\theta(t)=\left[\omega_{+}(k)-\omega_{-}(k)\right]t/2=\xi k^{2}t/2M_{Pl}$ (6)
for a plane wave with wave-vector $k$ ga99 . Again, for simple Planck-
suppressed LIV, we would expect that $\xi\simeq 1$.
Some astrophysical sources emit highly polarized radiation. It can be seen
from equation (6) that the rotation angle is reduced by the large value of the
Planck mass. However, the small rotations given by equation (6) can add up
over astronomical or cosmological distances to erase the polarization of the
source emission. Therefore, if polarization is seen in a distant source, it
puts constraints on the parameter $\xi$. Observations of polarized radiation
from distant sources can therefore be used to place an upper bound on $\xi$.
Equation (6) indicates that the higher the wave number $|k|$, the stronger the
rotation effect will be. Thus, the depolarizing effect of space-time induced
birefringence will be most pronounced in the $\gamma$-ray energy range. It can
also be seen that the this effect grows linearly with propoagation time.
The difference in rotation angles for wave-vectors $k_{1}$ and $k_{2}$ is
$\Delta\theta=\xi(k_{2}^{2}-k_{1}^{2})d/2M_{Pl},$ (7)
replacing the time $t$ by the distance from the source to the detector,
denoted by $d$.
The best secure bound on this effect, $|\xi|\mathrel{\raise
1.29167pt\hbox{$<$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}10^{-9}$, was
obtained using the observed 10% polarized soft $\gamma$-ray emission from the
region of the Crab Nebula ma08 .
Clearly, the best tests of birefringence would be to measure the polarization
of $\gamma$-rays from GRBs. We note that linear polarization in X-ray flares
from GRBs has been predicted fa05 . Most $\gamma$-ray bursts have redshifts in
the range 1-2 corresponding to distances of greater than a Gpc. Should
polarzation be detected from a burst at distance $d$, this would place a limit
on $|\xi|$ of
$|\xi|\mathrel{\raise 1.29167pt\hbox{$<$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}5\times 10^{-15}/d_{0.5}$ (8)
where $d_{0.5}$ is the distance to the burst in units of 0.5 Gpc ja04 .
Detectors that are dedicated to polarization measurements in the X-ray and
$\gamma$-ray energy range and which can be flown in space to study the
polarization from distant astronomical sources are now being designed mi05 ;
pr05 .
## 7 LIV and the Ultrahigh Energy Cosmic Ray Spectrum
### 7.1 The “GZK Effect”
Shortly after the discovery of the 3K cosmogenic background radiation (CBR),
Greisen gr66 and Zatsepin and Kuz’min za66 predicted that pion-producing
interactions of such cosmic ray protons with the CBR should produce a spectral
cutoff at $E\sim$ 50 EeV. The flux of ultrahigh energy cosmic rays (UHECR) is
expected to be attenuated by such photomeson producing interactions. This
effect is generally known as the “GZK effect”. Owing to this effect, protons
with energies above $\sim$100 EeV should be attenuated from distances beyond
$\sim 100$ Mpc because they interact with the CBR photons with a resonant
photoproduction of pions st68 .
### 7.2 Modification of the GZK Effect Owing to LIV
Let us consider the photomeson production process leading to the GZK effect.
Near threshold, where single pion production dominates,
$p+\gamma\rightarrow p+\pi.$ (9)
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
$4\omega E=m(2M+m)$ (10)
for the proton, and
$4\omega E_{\pi}={{m^{2}(2M+m)}\over{M+m}}$ (11)
in terms of the pion energy, where M is the rest mass of the proton and m is
the rest mass of the pion st68 .
If LI is broken so that $c_{\pi}~{}>~{}c_{p}$, the threshold energy for
photomeson is altered.333This requirement precludes the ‘quasi-vacuum Čerenkov
radiation’ of pions, via the rapid, strong interaction, pion emission process,
$p\rightarrow N+\pi$. This process would be allowed by LIV in the case where
$\delta_{\pi p}$ is negative, producing a sharp cutoff in the UHECR proton
spectrum. (For more details, see Refs. cg99 ; st09 ; alt07 .
Because of the small LIV perturbation term, the square of the four-momentum is
shifted from its LI form so that the threshold condition in terms of the pion
energy becomes
$4\omega E_{\pi}={{m^{2}(2M+m)}\over{M+m}}+2\delta_{\pi p}E_{\pi}^{2}$ (12)
where $\delta_{\pi p}\equiv c_{\pi}~{}-~{}c_{p}$, again in units where the low
energy velocity of light is unity.
Equation (12) is a quadratic equation with real roots only under the condition
$\delta_{\pi
p}\leq{{2\omega^{2}(M+m)}\over{m^{2}(2M+m)}}\simeq\omega^{2}/m^{2}.$ (13)
Defining $\omega_{0}\equiv kT_{CBR}=2.35\times 10^{-4}$ eV with
$T_{CBR}=2.725\pm 0.02$ K, equation (13) can be rewritten
$\delta_{\pi p}\leq 3.23\times 10^{-24}(\omega/\omega_{0})^{2}.$ (14)
### 7.3 Kinematics
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 (14). This condition, together with equation (12), implies that while
photomeson interactions leading to GZK suppression can occur for “lower
energy” UHE protons interacting with higher energy CBR photons on the Wien
tail of the spectrum, other interactions involving higher energy protons and
photons with 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. We now consider a more detailed quantitative
treatment of this possibility, viz., GZK coexisting with LIV.
The kinematical relations governing photomeson interactions are changed in the
presence of even a small violation of Lorentz invariance. The modified
kinematical relations containing LIV have a strong effect on the amount of
energy transfered from a incoming proton to the pion produced in the
subsequent interaction, i.e., the inelasticity st09 ; al03 ; ss08 .
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 integrated over in order to obtain the total
inelasticity. For real-root solutions for interactions involving higher energy
protons, the range of kinematically allowed angles becomes 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.
Figure 2 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 ss08 . 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 $\sim 200$ EeV. Above this energy, the inelasticity
precipitously drops as the LIV term in the pion rest energy approaches
$m_{\pi}$.
Figure 2: 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 ss08
.
With this modified inelasticity, the proton energy loss rate by photomeson
production is given by
${{1}\over{E}}{{dE}\over{dt}}=-{{\omega_{0}c}\over{2\pi^{2}\gamma^{2}}\hbar^{3}c^{3}}\int\limits_{\eta}^{\infty}d\epsilon~{}\epsilon~{}\sigma(\epsilon)K(\epsilon)\ln[1-e^{-\epsilon/2\gamma\omega_{0}}]$
(15)
where we now use $\epsilon$ to designate the energy of the photon in the cms,
$\eta$ is the photon threshold energy for the interaction in the cms,
$K(\epsilon)$ denotes the inelasticity, 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 $\ell=cE/r(E)$, where
the energy loss rate $r(E)\equiv(dE/dt)$. This attenuation length is plotted
in Figure 3 for various values of $\delta_{\pi p}$ along with the unmodified
pair production attenuation length from pair production interactions,
$p+\gamma_{CBR}\rightarrow e^{+}+e^{-}$.
Figure 3: 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) ss08 .
## 8 UHECR Spectra with LIV and Comparison with Present Observations
The effect of by a very small amount of LIV on the UHECR spectrum was
analytically calculated in Ref. ss08 in order to determine the resulting
spectral modifications. It can be demonstrated that there is little difference
between the results of using an analytic calculation vs. a Monte Carlo
calculation (e.g., see Ref. ta09 ). In order to take account of the probable
redshift evolution of UHECR production in astronomical sources, they took
account of the following considerations:
(i) 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
$\displaystyle r_{\gamma p}(E,z)=(1+z)^{3}r[(1+z)E].$ (16)
(ii) They assumed that the average UHECR volume emissivity is of the energy
and redshift dependent form given by $q(E_{i},z)=K(z)E_{i}^{-\Gamma}$ where
$E_{i}$ is the initial energy of the proton at the source and $\Gamma=2.55$.
For the source evolution, we assume $K(z)\propto(1+z)^{3.6}$ with $z\leq 2.5$
so that $K(z)$ is roughly proportional to the empirically determined
$z$-dependence of the star formation rate. $K(z=0)$ and $\Gamma$ are
normalized fit the data below the GZK energy.
Using these assumptions, one can calculate the effect of LIV on the UHECR
spectrum. The results are actually insensitive to the assumed redshift
dependence because evolution does not affect the shape of the UHECR spectrum
near the GZK cutoff energy be88 ; st05 . At higher energies where the
attenuation length may again become large owing to an LIV effect, the effect
of evolution turns out to be less than 10%. The curves calculated in Ref. st09
assuming various values of $\delta_{\pi p}$, are shown in Figure 4 along with
the latest Auger data from Ref. sch09 . They show that even a very small
amount of LIV that is consistent with both a GZK effect and with the present
UHECR data can lead to a “recovery” of the UHECR spectrum at higher energies.
Figure 4: Comparison of the latest Auger data with calculated spectra for
various values of $\delta_{\pi p}$, taking $\delta_{p}=0$ (see text). 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) st09 .
### 8.1 Allowed Range for the LIV Parameter $\delta_{\pi p}$
Stecker and Scully st09 have updated compared the theoretically predicted
UHECR spectra with various amounts of LIV to the latest Auger data from the
procedings of the 2009 International Cosmic Ray Conference sch09 , data . This
update is shown in Figure 4. The amount of presently observed GZK suppression
in the UHECR data is consistent with the possible existence of a small amount
of LIV. The value of $\delta_{\pi p}$ that results in the smallest $\chi^{2}$
for the modeled UHECR spectral fit using the observational data from Auger
sch09 above the GZK energy. The best fit LIV parameter found was in the range
given by $\delta_{\pi p}$ = $3.0^{+1.5}_{-3.0}\times 10^{-23}$, corresponding
to an upper limit on $\delta_{\pi p}$ of $4.5\times 10^{-23}$. 444The HiRes
data ab08 do not reach a high enough energy to further restrict LIV. 555We
note that the overall fit of the data to the theoretically expected spectrum
is somewhat imperfect, even below the GZK energy and even for the case of no
LIV. It appears that the Auger spectrum seems to steepen even below the GZK
energy. As a conjecture, one can assume that the derived energy may be too low
by about 25%, within the uncertainty of both systematic-plus statistical error
given for the energy determination. This gives better agreement between the
theoretical curves and the shifted data st09 . The constraint on LIV would be
only slightly reduced if this shift is assumed.
### 8.2 Implications for Quantum Gravity Models
An effective field theory approximation for possible LIV effects induced by
Planck-scale suppressed quantum gravity for $E\ll M_{Pl}$ was considered in
Ref. ma09 . These authors explored the case where a perturbation to the
energy-momentum dispersion relation for free particles would be produced by a
CPT-even dimension six operator suppressed by a term proportional to
$M_{Pl}^{-2}$. The resulting dispersion relation for a particle of type $a$ is
$E_{a}^{2}=p_{a}^{2}+m_{a}^{2}+\eta_{a}\left({{p^{4}}\over{M_{Pl}^{2}}}\right)$
(17)
In order to explore the implications of our constraints for quantum gravity,
one can take the perturbative terms in the dispersion relations for both
protons and pions, to be given by the dimension six dispersion terms in
equation (17) above. Making this identification, the LIV constraint of
$\delta_{\pi p}<4.5\times 10^{-23}$ in the fiducial energy range around
$E_{f}=100$ EeV indirectly implies a powerful limit on the representation of
quantum gravity effects in an effective field theory formalism with Planck
suppressed dimension six operators. Equating the perturbative terms in both
the proton and pion dispersion relations, one obtains the relation st09
$2\delta_{\pi
p}\simeq(\eta_{\pi}-25\eta_{p})\left({{0.2E_{f}}\over{M_{Pl}}}\right)^{2},$
(18)
where the pion fiducial energy is taken to to be $\sim 0.2E_{f}$, as at the
$\Delta$ resonance that dominates photopion production and the GZK effect st68
. Equation (18), together with the constraint $\delta_{\pi p}<4.5\times
10^{-23}$, indicates that any LIV from dimension six operators is suppressed
by a factor of at least ${\cal{O}}(10^{-6}M_{Pl}^{-2})$, except in the
unlikely case that $\eta_{\pi}-25\eta_{p}\simeq 0$. These results are in
agreement with those obtained independently by Maccione et al. from the Monte
Carlo runs ma09 . It can thus be concluded that an effective field theory
representation of quantum gravity with dimension six operators that suppresses
LIV by only a factor of $M_{Pl}^{2}$ i.e. $\eta_{p},\eta_{\pi}\sim 1$, is
effectively ruled out by the UHECR observations.
## 9 Beyond Constraints: Seeking LIV
As we have seen (see Figure 4), even a very small amount of LIV that is
consistent with both a GZK effect and with the present UHECR data can lead to
a “recovery” of the primary UHECR spectrum at higher energies. This is the
clearest and the most sensitive evidence of an LIV signature. The “recovery”
effect has also been deduced in Refs. ma09 and bi09 666In Ref. bi09 , a
recovery effect is also claimed for high proton energies in the case when
$\delta_{\pi p}<0$. However, we have noted that the ‘quasi-vacuum Čerenkov
radiation’ of pions by protons in this case will cut off the proton spectrum
and no “recovery” effect will occur.. In order to find it (if it exists) three
conditions must exist: (i) sensitive enough detectors need to be built, (ii) a
primary UHECR spectrum that extends to high enough energies ($\sim$ 1000 EeV)
must exist, and (iii) one much be able to distinguish the LIV signature from
other possible effects.
### 9.1 Obtaining UHECR Data at Higher Energies
We now turn to examining the various techniques that can be used in the future
in order to look for a signal of LIV using UHECR observations. As can be seen
from the preceding discussion, observations of higher energy UHECRs with much
better statistics than presently obtained are needed in order to search for
the effects of miniscule Lorentz invariance violation on the UHECR spectrum.
#### 9.1.1 Auger North
Such an increased number of events may be obtained using much larger ground-
based detector arrays. 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).
#### 9.1.2 Space Based Detectors
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. We look forward to these
developments that may have important implications for fundamental high energy
physics.
Two potential spaced-based missions have been proposed to extend our knowledge
of UHECRs to higher energies. One is JEM-EUSO (the Extreme Universe Space
Observatory) EUSO , a one-satellite telescope mission proposed to be placed on
the Japanese Experiment Module (JEM) on the International Space Station. The
other is OWL (Orbiting Wide-angle Light Collectors) OWL , a two satellite
mission for stereo viewing, proposed for a future free-flyer mission. Such
orbiting space-based telescopes with UV sensitive cameras will have wide
fields-of-view (FOVs) in order to observe and use large volumes of the Earth’s
atmosphere as a detecting medium. They will thus trace the atmospheric
fluorescence trails of numbers of giant air showers produced by ultrahigh
energy cosmic rays and neutrinos. Their large FOVs will allow the detection of
the rare giant air showers with energies higher than those presently observed
by ground-based detectors such as Auger. Such missions will thus potentially
open up a new window on physics at the highest possible observed energies.
## 10 Conclusions
The Fermi timing results for GRB090510 rule out and string-inspired D-brane
model predictions as well as other quantum gravity predictions of a
retardation of photon velocity that is simply proportional to $E/M_{QG}$
because they would require $M_{QG}>M_{Pl}$. More indirect results from
$\gamma$-ray birefringence limits, the non-decay of 50 TeV $\gamma$-rays from
the Crab Nebula, and the TeV spectra of nearby AGNs also place severe limits
on violations of special relativity (LIV). Limits on Lorentz invariance
violation from observations of ultrahigh energy cosmic-rays provide severe
constraints for other quantum gravity models, appearing to rule out
retardation that is simply proportional to $(E/M_{QG})^{2}$. Various effective
field theory frameworks lead to such energy dependences.
New theoretical models of Planck scale physics and quantum gravity need to
meet all of the present observational constraints. One scenario that may be
considered is that gravity, i.e. $G$, becomes weaker at high energies. We know
that the strong, weak and electromagnetic interactions all have energy
dependences, given by the running of the coupling constants. If $G$ decreases,
then the effective $\lambda_{Pl}=\sqrt{G\hbar/c^{3}}$ would decrease and the
effective $M_{Pl}=\hbar/(\lambda_{Pl}c)$ would increase. In that case, the
space-time quantum scale would be less than the usual definition of
$\lambda_{Pl}$. Such speculation is presently cogitare ex arcis, but might be
plausible if a transition to a phase where the various forces are unified
occurs at very high energies st80 .
At the time of the present writing, high energy astrophysics observations have
led to strong constraints on LIV. Currently, we have no positive evidence for
LIV. This fact, in itself, should help guide theoretical research on quantum
gravity, already ruling out some models. Will this lead to a new null result
comparable to Michelson-Morley? Will a totally new concept be needed to
describe physics at the Planck scale? If all of the known forces are unified
at the Planck scale, this would not be surprising. One thing is clear: a
consideration of all empirical data will be necessary in order to finally
arrive at a true theory of physics at the Planck scale.
## References
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|
arxiv-papers
| 2009-12-02T19:36:42 |
2024-09-04T02:49:06.813509
|
{
"license": "Public Domain",
"authors": "Floyd W. Stecker (NASA/GSFC)",
"submitter": "Floyd Stecker",
"url": "https://arxiv.org/abs/0912.0500"
}
|
0912.0598
|
# C-axis critical current of a PrFeAsO0.7 single crystal
H. Kashiwaya,1 K. Shirai,1,2 T. Matsumoto,1 H. Shibata,1
H. Kambara,1 M. Ishikado,3,4 H. Eisaki,1,4 A. Iyo,1,4
S. Shamoto,4 I. Kurosawa,2 and S. Kashiwaya1
###### Abstract
The $c$-axis transport properties of a high-pressure synthesized PrFeAsO0.7
single crystal are studied using s-shaped junctions. Resistivity anisotropy of
about 120 detected at 50 K shows the presence of strong anisotropy in the
electronic states. The obtained critical current density for the $c$-axis of
2.9$\times$105 A/cm2 is two orders of magnitude larger than that in
Bi2Sr1.6La0.4CuO6+δ. The appearance of a hysteresis in the current-voltage
curve below $T_{c}$ is the manifestation of the intrinsic Josephson effect
similar to that in cuprate superconductors. The suppression of the critical
current-normal resistance ($I_{c}R_{n}$) product is explained by an inspecular
transport in s±-wave pair potential.
1National Institute of Advanced Industrial Science and Technology (AIST),
Ibaraki, 305-8568, Japan
2Japan Women’s University, Tokyo 112-8681, Japan
3Japan Atomic Energy Agency, Ibaraki 319-1195, Japan
4JST, Transformative Research-Project on Iron Pnictides (TRIP), Tokyo
102-0075, Japan
The discovery of a family of iron-pnictide superconductors has renewed our
interests in unconventional superconductors.[1] The stack of superconducting
FeAs sheets sandwiched between blocking layers characterizes the crystal
structures of the iron-pnictides. The similarity of the crystal structures to
those of the cuprate superconductors suggests the realization of strong
anisotropic electronic states. In contrast, previously reported experimental
data on (Ba,K)Fe2As2 ($T_{c}$ $\sim$ 28 K) have suggested nearly isotropic
features in the temperature range between 10-27 K based on $H_{c2}$
measurements. [2] Since anisotropy has been the bottleneck in several possible
applications, such as power supply cables, the exact evaluation of the
$c$-axis transports on iron-pnictides is an important issue.
Here, we present the $c$-axis transport properties and the anisotropy of an
oxygen deficient PrFeAsO0.7 single crystal evaluated using s-shaped junctions
fabricated by a focused ion beam (FIB) process. PrFeAsO0.7 is one of the
LnFeAsO (Ln=lanthanide, so-called $`$1111$`$) compounds having relatively
higher anisotropy among the iron-pnictides.[3, 4, 5] We also applied the same
measurements to Bi2Sr1.6La0.4CuO6+δ (Bi2201) single crystals as a reference.
Both compounds are single layer systems with similar $Tc$’s, which makes it
easier to clarify the differences between iron-pnictides and cuprates. The
main differences between the two are the pair potential symmetry and the band
structure as depicted in Table I. In the case of cuprates, the $c$-axis
transport below $T_{c}$ is dominated by the interlayer Josephson effect, so-
called intrinsic Josephson effect, that has been identified by a large
hysteresis in the current-voltage ($I$-$V$) curve.[6] The applications of the
intrinsic Josephson junction (IJJ) include a terahertz radiation source and a
qubit. Therefore, one aspect of our investigation is whether a similar
Josephson effect can be observed in the iron-pnictides.
The single crystals of oxygen deficient PrFeAsO0.7 and Bi2201 were prepared by
a high-pressure synthesis method using belt-type anvil apparatus and by a
floating zone method, respectively. The crystals were fixed on SrTiO3
substrates after they were cut into pellets with a size of 10-100$\mu$m. Then
the center parts of the crystals were necked down to 2-3 $\mu$m from the top
using a FIB. The $ab$-plane resistivity $\rho_{ab}(T)$ was measured in this
configuration. The necked devices were processed further by a FIB radiated
from the horizontal direction to form two slits. The slits were designed to
have an overlap along the $c$-axis so that the current direction was
restricted to the $c$-axis in the necked region. Typical scanning ion
microscopy images of s-shaped junctions are shown in Fig. 1. The junction
sizes of 1-2$\mu$m were small enough to be regarded as short junctions. The
present device configuration has widely been used for the IJJ in recent
experiments.[7] Details of the crystal growth condition and the device
fabrication process have been described elsewhere.[3, 4, 5, 8] It should be
noted that one of the essential advantages of the present device is that the
influence of surface or interface degradation can be completely eliminated,
because the present junction does not rely on the hetero-structure. In
addition, junction size of a few micrometers is small enough to exclude the
unanticipated presence of grain boundaries inside the junction. Thus the
present method permits the unambiguous detection of the intrinsic crystal
nature.
Figure 1 shows the temperature dependences of resistivity for the $c$-axis
$\rho_{c}(T)$ and the resistivity anisotropy $\gamma_{\rho}(T)$ determined by
the ratio of $\rho_{c}(T)$ to $\rho_{ab}(T)$. The resistance was measured with
an ac current modulation of about 10$\mu$A. In the case of Bi2201,
$\rho_{c}(T)$ below 140K is insulating whereas that above 140K is metallic. A
similar feature has been detected widely in various cuprates. In contrast,
$\rho_{c}(T)$ for PrFeAsO0.7 is insulating for the entire temperature range.
The variation of resistivity of less than 10$\%$ across the temperature range
from 50K to 300K is far smaller than that of Bi2201. For both compounds,
values of $\gamma_{\rho}(T)$ in Fig.1 show a monotonic increment with lowering
temperature. The $\gamma_{\rho}(T)$ of about 120 at 50K is far larger than
that detected in (Ba,K)Fe2As2,[2] compatible with those of the 1111
compounds,[9, 10, 11, 12] and far smaller than those of Bi-based cuprates.[6]
This fact implies that the block layer in PrFeAsO0.7 works as an insulating
barrier although the barrier height is relatively low as compared to Bi-based
cuprates.
Figure 2 shows the temperature dependences of the critical current
$I_{c}$($T$) obtained below $T_{c}$. The detected Josephson currents in both
compounds monotonically increase with lowering temperature. The temperature
dependences mostly follow Ambegaokar-Baratoff (AB) formula shown as solid
lines.[13] For more detailed comparison, we need fittings by taking account of
the probability distribution of the switching current.[14, 15] The critical
current density for the $c$-axis direction $J_{c}$($T$) of 2.9$\times$105
A/cm2 in PrFeAsO0.7 is two orders of magnitude larger than that of Bi2201.
Assuming that $J_{c}$($T$) for the $ab$-plane is given by the product of the
$c$-axis $J_{c}$($T$) and $\sqrt{\gamma_{\rho}}$, $J_{c}$($T$) of several
MA/cm2 could be attainable at 4.2K. This value is comparable to that obtained
in Ba(Fe1-xCox)2As2 thin films.[16]
Figure 1: (color-online) Temperature dependences of $\rho_{c}$ and
$\gamma_{\rho}$ for (a) PrFeAsO0.7 and (b) Bi2Sr1.6La0.4CuO6+δ. The scanning
ion microscopy images of the s-shaped junctions used for the measurement are
also shown.
$I$-$V$ curves in the inset of Fig. 2 show the appearance of Josephson
switching and the hysteresis for both Bi2201 and PrFeAsO0.7. Josephson
switching means the discontinuous transition from the zero-voltage state to
the finite voltage quasiparticle branch as the bias current increases. We can
evaluate damping of the junction from the switching dynamics. The Q values
estimated from the ratio of the switching and the retrapping current[17] for
Bi2201 and PrFeAsO0.7 are 50 and 2 at 4.2K. The low Q value in PrFeAsO0.7 can
be attributed to the low barrier height of the block layer and is consistent
with the weakly insulating c-axis transport shown in Fig. 1. An important
question is whether the Josephson effect arises from the interlayer tunneling
between adjacent FeAs layers similar to the intrinsic Josephson effect in
cuprates.[6] We believe this interpretation is true for PrFeAsO0.7 based on
the reasons described below. Firstly, the normal resistance ($R_{n}$) of the
Josephson junction deduced from the gradient of the quasiparticle branch in
the $I$-$V$ curve is about 10m$\Omega$. In contrast, the transport measurement
just above $T_{c}$ indicates that the resistance per one layer is about
30m$\Omega$ assuming that the s-shaped junction contains 1600 FeAs layers.
Since these two values are comparable, the origin of resistance in the
Josephson junction is reasonably ascribed to the interlayer transfer. This
fact supports the intrinsic Josephson effect picture. Secondly, we observed
the appearance of the multiple branch structure by increasing the bias
current. The structure reflects the stacking of the Josephson junction in the
$c$-axis direction, which is one of the manifestations of the intrinsic
Josephson effect.[6]
Figure 2: (color-online) Temperature dependences of $I_{c}$ for (a)
PrFeAsO0.7 (b) Bi2Sr1.6La0.4CuO6+δ. Solid lines represent $I_{c}$ based on the
AB formula. The insets show the typical $I$-$V$ curves obtained at 4.2K.
Table I summarizes the data obtained in the present measurements. One
important difference between Bi2201 and PrFeAsO0.7 is the $I_{c}R_{n}$
product. In the case of Bi2201, the gap amplitude of 10-18mV has been obtained
by scanning tunneling spectroscopy on the low temperature cleaved
surfaces.[18] This value corresponds not to the quasiparticle gap (40-100mV)
but to the kink inside the quasiparticle gap.[19] The $I_{c}R_{n}$ product of
6mV estimated from the $I$-$V$ curve is comparable to the gap amplitude. In
contrast, the $I_{c}R_{n}$ product of 0.125mV in PrFeAsO0.7 is two orders of
magnitude smaller than the gap amplitude of 13.3mV detected by Andreev
spectroscopy.[20] Following conventional theories of Josephson junctions that
assume a simple barrier structure, such as no localized states inside the
barrier, $I_{c}R_{n}$ at the zero point corresponds approximately to the gap
amplitude both at the tunneling limit junction[13] and in the weak links.[21]
Therefore, such a small $I_{c}R_{n}$ cannot be attributed to the low barrier
height of the block layer. Another possibility is the suppression of the
superconductivity near the junction interface. Actually, the detection of the
small $I_{c}R_{n}$ has been reported for the hetero-junctions of iron-
pnictides[22]. However, since the present result does not rely on the
artificial interface or the cleaved surfaces, we can exclude this possibility.
Table 1: Summary of experimental data for PrFeAsO0.7 and Bi2201. The values in the upper columns have been obtained in the present experiment, and those in the lower columns are cited from references. $J_{c}$, $Q$ and $I_{c}R_{n}$ are the values at 4.2K and $\gamma_{\rho}$ just above $T_{c}$. | PrFeAsO0.7 | Bi2201
---|---|---
Tc[K] | 35 | 33
Anisotropy $\gamma_{\rho}$ | 120 | $\sim$10000
$dR/dT$ | Insulating | Insulating(T$<$140K)
C-axis $J_{c}$ [A/cm2] | 2.9$\times$105 | 1000-2000
Q | 2 | 50
$I_{c}R_{n}$[mV] | 0.125 | 6
Gap amplitude [mV] | 13.3[20] | 10-18(9K)[18]
Pair potential symmetry | $s_{\pm}$-wave | $d$-wave
Band structure | multi-band | single band
A plausible origin is the effect of the internal phase of s±-wave
symmetry.[23, 24] For an intuitive explanation, we assume a simplified
superconductor having two isotropic pair potentials with a phase difference of
$\pi$, $\Delta_{1}$ and -$\Delta_{2}$. In such case, $I_{c}R_{n}$ is roughly
expressed by $I_{c}R_{n}\propto\
\alpha\Delta_{1}+\beta\Delta_{2}-2\gamma\sqrt{\Delta_{1}\Delta_{2}}$, where
$\alpha$ and $\beta$ are parameters representing the Fermi surface information
and the barrier height, and $\gamma$ is a parameter corresponding to the
interband hopping due to the inspecularity ($\alpha$, $\beta$, $\gamma\geq
0$). It is important to note that the minus in the equation comes from the
phase difference of the two pair potentials. For a system having complete
translational symmetry for the $ab$-plane, the momentum in the plane is
conserved through interlayer hopping. $I_{c}R_{n}$ is approximately
proportional to the amplitude of the pair potential integrated over the Fermi
surface because $\gamma$ is zero even if the pair potential has
anisotropy.[25] While in a real material, since the inspecular components
inevitably exist, the deviation of $\gamma$ from zero reduces $I_{c}$. The
influence of such an effect is estimated to be small for the cuprates although
it does exist.[26] The present experimental result with PrFeAsO0.7 implies
that such an effect is far larger than that in cuprates, which results in the
serious suppression of $I_{c}R_{n}$. Y. Ota $et$ $al$. have discussed a
similar effect at grain boundaries of an s±-wave superconductor.[27] Since the
present mechanism must be sensitive to the nature of the block layers, a
systematic measurement for various iron-pnictides will reveal this effect more
clearly.
We would like to thank Y. Yoshida, S. Kawabata and Y. Tanaka for fruitful
discussion. This work was financially supported by Grant-in-Aid for Scientific
Research (No.21710100, 20540392, 70393725) from JSPS, Japan and by Mitsubishi
foundation.
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|
arxiv-papers
| 2009-12-03T08:47:31 |
2024-09-04T02:49:06.821885
|
{
"license": "Public Domain",
"authors": "H. Kashiwaya, K. Shirai, T. Matsumoto, H. Shibata, H. Kambara, M.\n Ishikado, H. Eisaki, Y. Iyo, S. Shamoto, I. Kurosawa and S. Kashiwaya",
"submitter": "Satoshi Kashiwaya",
"url": "https://arxiv.org/abs/0912.0598"
}
|
0912.0635
|
Astronomy Letters, 2010 Vol. 36, No. 1, pp. 27-43
Analysis of Peculiarities of the Stellar Velocity Field
in the Solar Neighborhood
V.V. Bobylev1, A.T. Bajkova1, and A. A. Mylläri2
1Pulkovo Astronomical Observatory, Russian Academy of Sciences, St-Petersburg
2Turku University, Turku, Finland
Abstract–Based on a new version of the Hipparcos catalogue and an updated
Geneva-Copenhagen survey of F and G dwarfs, we analyze the space velocity
field of $\approx$17000 single stars in the solar neighborhood. The main known
clumps, streams, and branches (Pleiades, Hyades, Sirius, Coma Berenices,
Hercules, Wolf 630-$\alpha$Ceti, and Arcturus) have been identified using
various approaches. The evolution of the space velocity field for F and G
dwarfs has been traced as a function of the stellar age. We have managed to
confirm the existence of the recently discovered KFR08 stream. We have found
19 Hipparcos stars, candidates for membership in the KFR08 stream, and
obtained an isochrone age estimate for the stream, 13 Gyr. The mean stellar
ages of the Wolf 630-$\alpha$Ceti and Hercules streams are shown to be
comparable, 4–6 Gyr. No significant differences in the metallicities of stars
belonging to these streams have been found. This is an argument for the
hypothesis that these streams owe their origin to a common mechanism.
DOI: 10.1134/S1063773710010044
## INTRODUCTION
Studying the stellar velocity field in the solar neighborhood is of great
importance in understanding the kinematics and evolution of various structural
components in the Galaxy. At present, it is well known that the stellar space
velocity distribution has a complex small-scale structure. This may be
attributable to various dynamical factors (the influence of a spiral density
wave, the Galactic bar, etc.).
The stellar velocity field in the solar neighborhood was analyzed by Chereul
et al. (1998),Dehnen (1998), Asiain et al. (1999), Skuljan et al. (1999), and
Torra et al. (2000) using Hipparcos (ESA 1997) data. The space velocities of K
and M giants were studied by Famaey et al. (2005) using data from the
Hipparcos and Tycho-2 (Hog et al., 2000) catalogues in combination with the
radial velocities measured by the CORAVEL spectrovelocimeter. Based on data
from the first version of the Geneva-Copenhagen survey (Nordström et al.,
2004), Bobylev and Bajkova (2007a) analyzed the space velocities of F and G
dwarfs as a function of the stellar age. Antoja et al. (2008) studied an
extensive sample of stars of various spectral types, from O to M, using the
stellar ages and space velocities.
The theory of stellar streams has long been used to explain the nature of the
observed inhomogeneity of the stellar velocity field. Therefore, the names to
the peaks were given by association with open star clusters (OSCs), such as
the Pleiades (with an age of 70–125 Myr; Soderblom et al. 1993), the Sirius-
Ursa Majoris cluster (500 Myr; King et al. 2003), or the Hyades (650 Myr;
Castellani et al. 2001).
The theory of stellar streams suggests a common origin of the stars in a
specific stream (Eggen 1996). The clumpy structure of the observed velocity
field in the solar neighborhood is explained by a superposition of stars
belonging to different streams.
As numerical simulations of the dynamical evolution of such OSCs as the
Hyades, the Pleiades, and Coma Berenices show (Chumak et al. 2005; Chumak and
Rastorguev 2006a, 2006b), stellar tails elongated along the Galactic orbit of
the cluster appear during their evolution. However, in a time $\approx$2 Gyr,
the OSC remnants existing in the form of tails must completely disperse and
mix with the stellar background (Küpper et al. 2008).
The theory of stellar streams runs into great difficulties in explaining the
existence of peaks or clumps in velocity space containing old (older than 2–4
Gyr) stars. Analysis of the stellar metallicities performed by Taylor (2000)
for nine old streams (Hercules, Wolf 630, 61 Cyg, Arcturus, HR 1614, and
others) composed according to Eggen s lists showed such a large spread in
metallicity that a common origin of the stars in each of the streams is out of
the question. With regard to HR 1614, there is still the opinion based on the
chemical homogeneity of the stars that this is an OSC remnant with an age of
about 2 Gyr (De Silva et al. 2007).
In recent years, nonaxisymmetric models of the Galaxy (a spiral structure, a
bar, a triaxial halo) have been invoked to account for peculiarities in the
distribution of stellar velocities in the solar neighborhood. For example, the
Galactic spiral structure gives rise to clumpiness in the observed velocity
field (De Simone et al. 2004; Quillen and Minchev 2005). The bar at the
Galactic center (Dehnen 1999, 2000; Fux 2001; Chakrabarty 2007) leads to a
bimodal distribution of the observed $UV$ velocities.
At present, clumps of a completely different nature to which the Sirius,
Hercules, and Arcturus streams belong are distinguished.
In the opinion of Klement et al. (2008), the Sirius stream contains not only
stars formed simultaneously and evolving as an OSC but also a sizeable
fraction of field stars that fell into this region through the impact of a
spiral density wave.
Numerical simulations have shown that the existence of the Hercules stream
$(V\approx-50$ km s${}^{-1})$ can be explained by the fact that its stars have
resonant orbits induced by the Galactic bar (Dehnen 1999, 2000; Fux 2001). In
this case, the Sun must be located near the outer Lindblad resonance. A
detailed analysis performed by Bensby et al. (2007) using high-resolution
spectra of nearby F and G dwarfs showed this stream to contain stars of
various ages, metallicities, and elemental abundances. Bensby et al. (2007)
concluded that the influence of a bar-type dynamical factor is the most
acceptable explanation for the existence of the Hercules stream.
Several authors (Navarro et al. 2004; Helmi et al. 2006; Arifyanto and Fuchs
2006) concluded that the Arcturus stream $(V\approx-100$ km s${}^{-1})$ is the
old ($\approx$15 Gyr) debris of a dwarf galaxy captured by the Galaxy and
disrupted by its tidal effect. Data on the kinematics and metallicities of the
stars being analyzed served as arguments for this conclusion.
Analysis of the RAVE DR1 experimental data (Steinmetz et al. 2006) revealed a
hitherto unknown stream (Klement et al. 2008) with an age of $\approx 13$ Gyr
in the region of “rapidly flying” stars $(V\approx-160$ km s${}^{-1})$ whose
origin has not yet been established.
The goal of this paper is to analyze peculiarities of the stellar velocity
field in the solar neighborhood based on a new version of the Hipparcos
catalogue, the OSACA and PCRV catalogs of radial velocities, and an updated
Geneva-Copenhagen survey of F and G dwarfs, which provide the currently most
accurate data on the individual distances, space velocities, and ages of
stars.
## THE COORDINATE SYSTEM
In this paper, we use a rectangular Galactic coordinate system with the axes
directed away from the observer toward the Galactic center $(l=0^{\circ},$
$b=0^{\circ},$ the $X$ axis), along the Galactic rotation $(l=90^{\circ},$
$b=0^{\circ},$ the $Y$ axis), and toward the North Galactic Pole
$(b=90^{\circ},$ the $Z$ axis). The corresponding space velocity components of
the object $U,V,$ and $W$ are also directed along the $X,Y,$ and $Z$ axes.
## THE DATA
We use stars from the Hipparcos catalog (ESA 1997). We took the proper-motion
components and parallaxes from an updated version of the Hipparcos catalog
(van Leeuwen 2007), the stellar radial velocities from the OSACA compilation
catalog of radial velocities (Bobylev et al. 2006) and the Pulkovo Compilation
of Radial Velocities (Gontcharov 2006); improved age estimates and metallicity
indices [Fe/H] for F and G dwarfs were taken from an updated Geneva-Copenhagen
survey (Holmberg et al. 2007, 2008).
As a result, we have data of various quality on 34359 stars of various
spectral types. Among them, 16737 stars are single ones with the most reliable
distance estimates, i.e., $e_{\pi}/\pi<0.1$ for them. We chose the constraint
on the parallax errors from the considerations of selecting a sufficiently
large number of stars at the minimal effect of Lutz and Kelker (1973). These
stars constitute our main working sample that we designate as “all” (Figs. 1,
2, 4, 5). The stellar UV-velocity distribution for this sample is presented in
Fig. 1a.
For the selected stars, we, nevertheless, made a statistical estimate of the
$U$ and $V$ velocity biases caused by the measurement errors of the stellar
parallaxes. For this purpose, we used the method of Monte Carlo simulations.
We generated 1000 random realizations of parallax errors for each star that
satisfied a normal law. Figures 1b and 1c present the derived histograms
separately for the U and V velocities, respectively. The number of stars whose
velocity bias lies in a certain bin along the horizontal axis is indicated
along the vertical axis. As we see from the histograms, the statistical U and
V velocity biases caused by the parallax errors are generally insignificant;
for 70% of the stars, they lie in the interval $[-0.05,0.05]$ km s-1. The
maximum bias (given the asymmetry of the derived distributions) does not
exceed 0.5 km s-1. This value is approximately a factor of 2–3 lower than the
statistical uncertainty caused by the measurement errors of the proper motions
and radial velocities (Skuljan et al. 1999).
The stellar velocities were corrected for the differential rotation of the
Galaxy. The Galactic differential rotation effect is known to manifest itself
in its influence on the $U$ velocity via the gradient $dU/dY=-\Omega_{0},$
then $\Delta U=(dU/dY)Y=-\Omega_{0}Y,$ where $\Omega_{0}=B-A\approx-30$ km s-1
kpc-1. This means that for a typical error in the stellar space velocities of
$\varepsilon=1$ km s-1, this effect may be disregarded only for the stars
within $d<\varepsilon/\Omega_{0}=33$ pc. Since the stars used also have
greater distances, the differential rotation of the Galaxy should be taken
into account.
The Galactic rotation parameters (the Oort constants A and B) have been
repeatedly determined by various authors (Zabolotskikh et al. 2002; Olling and
Dehnen 2003; Bobylev 2004); they are known with an error $\sigma\approx 1$ km
s-1 kpc-1. This means that for a typical error in the stellar space velocities
of $\varepsilon=1$ km s-1, the influence of an uncertainty in determining
$\Omega_{0}$ is significant for the stars located at distances
$d>\varepsilon/3\sigma=333$ pc. Fortunately, the number of such distant stars
in our “all” sample is small (only two or three dozen OB stars), and their
influence may be neglected. In this paper, we use the Oort constants
$A=13.7\pm 0.6$ km s-1 kpc-1 and $B=-12.9\pm 0.4$ km s-1 kpc-1 that were
determined by Bobylev (2004) from an analysis of the independent estimates
obtained by various authors.
## THE METHODS
### The Adaptive Kernel Method
We use an adaptive kernel method to obtain an estimate of the velocity
distribution $f(U,V)$ similar to that of the probability density distribution
from the initial velocity distribution presented in Fig. 1. In contrast to the
approach of Skuljan et al. (1999), we use a two-dimensional, radially
symmetric Gaussian kernel function expressed as
$K(r,\sigma)=\frac{1}{2\pi\sigma^{2}}\exp\Biggl{(}-{\frac{r^{2}}{2\sigma^{2}}}\Biggr{)},$
(1)
where $r^{2}=x^{2}+y^{2}$ and $\sigma$ is a positive bandwidth parameter; in
this case, the relation $\int K(r)dr=1$ needed to estimate the probability
density holds. Obviously, the larger the parameter $\sigma$, the larger the
bandwidth and the lower the amplitude.
The basic idea of the adaptive kernel method is that at each point of the map,
the operation of convolution with a band of the width specified by the
parameter $\sigma$ that varies in accordance with the data density near this
point is performed. Thus, in zones with an enhanced density, the smoothing is
done by a comparatively narrow band; the bandwidth increases with decreasing
data density.
We will use the following definition of the adaptive kernel estimator at an
arbitrary point $\xi=(U,V)$ (Silverman 1986; Skuljan et al. 1999) adapted to a
Gaussian kernel function:
$\hat{f}(\xi)=\frac{1}{n}\sum_{i=1}^{n}K\left(|\xi-\xi_{i}|,{h\lambda_{i}}\right),$
where $\xi_{i}=(U_{i},V_{i}),\lambda_{i}$ is the local dimensionless bandwidth
parameter at point $\xi_{i},h$ is a general smoothing parameter, $n$ is the
number of data points $\xi_{i}=(U_{i},V_{i}).$ The parameter $\lambda_{i}$ at
each point of the two-dimensional $UV$ plane is defined as
$\lambda_{i}=\sqrt{\frac{g}{\hat{f}(\xi_{i})}},$ (2)
where $g$ is the geometric mean of $\hat{f}(\xi_{i})$:
$\ln g=\frac{1}{n}\sum_{i=1}^{n}\ln\hat{f}(\xi_{i}).$ (3)
Obviously, to determine $\lambda_{i}$ from Eqs. (2)–(3), we must know the
distribution $\hat{f}(\xi)$ which, in turn, can be determined if all
$\lambda_{i}$ are known. Therefore, the problem of finding the sought-for
distribution is solved iteratively. As the first approximation, we use the
distribution obtained by smoothing the initial $UV$ map with a band of an
arbitrary fixed width. The optimal value of the parameter $h$ can be found
from the condition for the rms deviation of the estimator $\hat{f}(\xi)$ from
the true distribution $f(\xi)$ being at a minimum. In contrast to Skuljan et
al. (1999), to determine $\lambda_{i}$ at each iteration, we used the values
of the function $\hat{f}(\xi)$ determined not at the specified points
$\xi_{i}$ but at all points of an equidistant grid on which the smoothed $UV$
distribution is sought. As our comparison showed, both smoothing methods yield
approximately the same results, but, at the same time, our approach requires
much less computation. The value of $h$ for all maps was taken to be 5.0. To
obtain each map, we made 20 iterations.
The sampling interval of the two-dimensional maps was chosen from a typical
uncertainty in the $U$ and $V$ velocities (Skuljan et al. 1999). In our case,
it is 2 km s-1, since the velocity errors for most of the stars in the solar
neighborhood (about 80%) do not exceed $\pm$1 km s-1. The sampling interval of
the maps in our analysis of the velocity distributions for age separated
samples was taken to be $d=2$ km s-1. In analyzing the “all” sample of stars,
we chose $d=1$ km s-1 from a large number of stars as an optimal one from the
standpoint of providing the necessary detail of the derived smoothed
distribution. To obtain distributions similar to the probability density
distribution, the smoothed two-dimensional velocity distributions must be
scaled by the factor $n\times s,$ where $s=d\times d$ km2 s${}^{2}.$ The map
size was $256\times 256$ pixels at the square bin size $s=2\times 2=4$ km2 s2
in the first case and $512\times 512$ pixels at $s=1\times 1=1$ km2 s2 in the
second case.
### Wavelet Analysis
To identify statistically significant signals of the main inhomogeneities in
the distributions of $UV$ velocities, we also use the wavelet transform
technique. This is known as a powerful tool for filtering spatially localized
signals (Chui 1997; Vityazev 2001).
The wavelet transform of a two-dimensional distribution $f(U,V)$ consists in
its decomposition into analyzing wavelets $\psi(U/a,V/a),$ where $a$ is the
scale parameter that allows a wavelet of a particular scale to be selected
from the entire family of wavelets characterized by the same shape $\psi$. The
wavelet transform $w(\xi,\eta)$ is defined as a correlation function, so that
we have one real value of the following integral at any given point
$(\xi,\eta)$ in the $UV$ plane:
$w(\xi,\eta)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(U,V)\psi\Biggl{(}\frac{(U-\xi)}{a},\frac{(V-\eta)}{a}\Biggr{)}dUdV,$
which is called the wavelet coefficient at $(\xi,\eta)$. Obviously, in our
case of finite discrete maps, their number is finite and equal to the number
of square bins on the map.
As the analyzing wavelet, we use a standard wavelet called a Mexican hat
(MHAT). A two-dimensional MHAT wavelet is given by
$\psi(r/a)=\Biggl{(}2-\frac{r^{2}}{a^{2}}\Biggr{)}e^{-r^{2}/2a^{2}},$ (4)
where $r^{2}=U^{2}+V^{2}.$ Wavelet (4) is obtained by doubly differentiating
the Gaussian function. The parameter $a$ that specifies the spatial scale
(width) of the wavelet $\psi$ is analogous to the parameter $\sigma$ in Eq.
(1). The main property of the wavelet $\psi$ is that its integral over $U$ and
$V$ is equal to zero, which allows any inhomogeneities to be detected in the
investigated distribution. If the distribution being analyzed is
inhomogeneous, then all coefficients of the wavelet transform will be zero.
For our wavelet analysis of various samples in the planes of $UV,VW,UW$
velocities and in the $(V,\sqrt{U^{2}+2V^{2}})$ plane, we chose the scale
parameter a to be 8.37 km s-1. The value of this parameter allowed us to
reliably identify the most significant structural features of the velocity
distribution that are the subject of our investigation. Note that for our
analysis of the velocities in the $(V,\sqrt{U^{2}+2V^{2}})$ plane, the map
size was $1024\times 1024$ pixels, with the square bin size being $s=1\times
1=1$ km2 s${}^{2}.$
## RESULTS
Figure 2a presents the $UV$-velocity distribution for the selected 16737
single stars (the “all” sample) obtained by the adaptive kernel method applied
to the initial velocity distribution shown in Fig. 1. The contour lines are
drawn with a uniform step equal to 2% of the distribution peak.
The classical Pleiades, $(U,V)=(-14,-23)$ kms${}^{-1},$ Hyades,
$(U,V)=(-43,-20)$ km s${}^{-1},$ Sirius, $(U,V)=(-8,2)$ km s${}^{-1},$ and
Coma Berenices, $(U,V)=(-11,-8)$ km s${}^{-1},$ streams as well as the
Hercules, $(U,V)=(-31,-49)$ km s${}^{-1},$ stream are clearly distinguished in
Fig. 2a. In addition, there is a blurred clump elongated along the U axis in a
wide region $(U,V)\approx(37,-22)$ km s-1. In the opinion of Antoja et al.
(2008), the Wolf 630 peak $(U,V)=(25,-33)$ km s-1 (Eggen 1996) and the
nameless peak $(U,V)=(50,-25)$ km s-1 (Dehnen 1998) are associated with this
new clump. Francis and Anderson (2008) designated this clump as the
$\alpha$Ceti stream; the UV coordinates of the star $\alpha$Ceti,
$(U,V)=(25,-23)$ km s-1, are also far from the characteristic clump center, as
for Wolf 630. As a compromise, we suggest calling this structure the Wolf
630-$\alpha$Ceti stream or branch.
Figure 2b presents the sections of map 2a perpendicular to the $(U,V)$ plane
that pass through the main peaks and that make $+16^{\circ}$ with the $U$ axis
if measured clockwise (this axis is designated in the figure as $U);$ the
distribution density in units of $7\times 10^{-4}$ is along the vertical axis.
The orientation of the sections coincides with the direction of the “branches”
detected on the smoothed maps (see also Skuljan et al. 1999; Antoja et al.
2008).
As we see from Fig. 2, the Hyades peak dominates in amplitude, although the
Pleiades peak is integrally more powerful, as can be seen from the wavelet
distribution for the “all” sample shown in Fig. 4.
Figure 3 present the $UV$-velocity distributions for eight samples (t1–t8) of
F and G dwarfs as a function of the stellar age, which allow the evolution of
the main peaks and clumps to be traced. We used a total of 6079 single stars
with distance and age errors $e_{\pi}/\pi<0.2$ and $e_{\pi}/\pi<0.3,$
respectively. The mean ages $\tau$ of samples t1–t8 are 1.2, 1.7, 2.2, 2.7,
3.4, 4.9, 7.2, and 11.2 Gyr, respectively. The numbers of stars in samples
t1–t8 are 509, 1105, 1184, 823, 803, 558, 586, and 511, respectively. The step
of the contour lines in Fig. 3 is 6.7% of the peak value.
As we see from Fig. 3, the ratio of the amplitudes of the main peaks changes
with age. For example, for the samples of comparatively young stars
(t1,t2,t3), the Hyades peak is dominant; the Pleiades peak is gradually
enhanced with stellar age (t4,t5) and is already dominant for sample t6. The
Hyades and Pleiades peaks form an elongated structure in the shape of a
“branch” whose orientation remains unchanged. Such structures in the
$UV$-velocity distribution for a large number of Hipparcos stars were first
described by Skuljan et al. (1999).
Numerical simulations of the disk heating by stochastic spiral waves performed
by De Simone et al. (2004) showed that the stratification of the UV
distribution into “branches” and peaks could be explained by irregularities in
the Galactic potential rather than by irregularities in the star formation
rate. As was shown by Fux (2001), the presence of a bar at the Galactic center
gives rise to branches. It is currently believed that the formation of the
Hercules branch is related precisely to the influence of a bar.
Figure 4 presents the wavelet maps of $UV,UW,$ and $VW$ velocities for the
“all” sample. The contour lines are given on a logarithmic scale: 1, 2, 4, 8,
16, 32, 64, 90, and 99% of the peak value. Note that only the positive
contours that describe the clump regions are shown on the maps. Since the
negative values of the wavelet distributions describe the regions of a sparse
distribution of stars, they are of no interest to us and are not shown in the
figures. Such clumps as HR 1614 $(U,V)=(15,-60)$ km s-1 and no. 13
$(U,V)=(50,0)$ km s-1 are marked in Fig. 4 according to the list by Dehnen
(1998). In addition, clumps no. 8 $(U,V)=(-40,-50)$ km s-1, no. 9
$(U,V)=(-25,-50)$ km s-1, and no. 12 $(U,V)=(-70,-50)$ km s-1 fall into the
Hercules stream, while clump no. 14 $(U,V)=(50,-25)$ km s-1 falls into the
Wolf630-$\alpha$Ceti stream. As a result, out of the 14 clumps marked in
Dehnen (1998), we cannot confirm the presence of isolated clump no. 11
$(U,V)=(-70,-10)$ km s-1 in the region of “high velocity” stars. According to
Navarro et al. (2004), the Arcturus stream is located in the fairly narrow
interval $-150$ km s${}^{-1}<V<-100$ km s${}^{{}_{1}}$ and in the considerably
wider interval $-150$ km s${}^{-1}<U<150$ km s-1; thus, the region marked in
Fig. 4 fits into these limits.
Figure 4 indicates features W1 and W2 for the Wolf 630-$\alpha$Ceti branch and
features H1 and H2 for the Hercules branch. According to these data, we
selected the stars belonging to these features and calculated their mean ages
and metallicities, which are given in Table 1. For the selection of stars, we
used our probabilistic approach described in detail in Bobylev and Bajkova
(2007b). Note that our samples were comparable in the number of stars — 525
and 625 stars are contained in the Wolf 630-$\alpha$Ceti and Hercules
branches, respectively. To calculate the means and dispersions listed in Table
1, we used only the stars with available age and metallicity estimates, in
fact, these are F and G dwarfs; the constraints $e_{\pi}/\pi<0.1$ and
$e_{\pi}/\pi<0.3$ were used.
The last columns in Table 1 give parameters of the $1\sigma$ ellipses: the
semimajor and semiminor axes $a_{i}$ and $b_{i}$ as well as the angle
$\beta_{i}$ between the vertical and semimajor axes (measured from the
vertical axis clockwise). The selection of stars with these parameters was
made within the boundaries of the $3\sigma$ ellipses.
Note that no significant concentrations of stars are observed in the $W-U$ and
$W-V$ planes outside the central “ellipse”.
Next, we apply a technique proposed by Arifyanto and Fuchs (2006) that
consists in identifying velocity field inhomogeneities in the plane of
$V,\sqrt{U^{2}+2V^{2}}$ coordinates. It allows low-power streams to be
reliably identified in the range of high space velocities.
Figure 5 shows the wavelet distributions for the “all” sample in the
$(V,\sqrt{U^{2}+2V^{2}})$ plane. The contour lines are given on a logarithmic
scale: 0.05, 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, . . ., 50, and 99% of the peak
value. In Figs. 5–8, we give the stellar velocities relative to the local
standard of rest (LSR) whose coordinates are $(U,V,W)=(10.0,5.2,7.2)$ km s-1
(Dehnen and Binney 1998); the cited coordinates of the clumps are also given
relative to the LSR. Figure 5 marks the AF06 stream with coordinates
$(-80,130)$ km s-1 (Arifyanto and Fuchs 2006), the Arcturus stream with
coordinates $(-125,185)$ km s-1 (Arifyanto and Fuchs 2006), and the KFR08
stream with coordinates $(-160,225)$ km s-1 (Klement et al. 2008). On this
diagram, the Wolf 630-$\alpha$Ceti stream mergers with the Hyades-Pleiades
branch.
Figure 6 presents the wavelet maps in $V,\sqrt{U^{2}+2V^{2}}$ coordinates for
samples of F and G dwarfs as a function of the stellar age; the set of levels
is similar to that in Fig. 5. As we see from the figure, a prominent clump of
KFR08 stream stars is observed for sample t8, which includes the oldest stars
considered. The central point in the KFR08 region marked on the plot (t8) has
the eighth level; all of the remaining clumps at $\sqrt{U^{2}+2V^{2}}>250$ km
s-1 have one level fewer and, hence, their significance is considerably lower.
Still, it is interesting to note that there is a clump close to the KFR08
region in Fig. 6 for sample t4. However, the significance of the levels in
this case is negligible, corresponding to the presence of only one or two
stars. A special search showed that one star from sample t4, HIP 77946, for
which [Fe/H]$=-0.83$ and $\tau=2.5$ Gyr (Holmberg et al. 2007), falls into the
neighborhood of KFR08 with a radius of 30 km s-1.
Table 2 gives parameters of the stars that are probable members of the KFR08
stream.We selected the candidates for membership in this stream based on the
distribution of the expanded “all” sample with $e_{\pi}/\pi<0.15$ in the plane
of $(V,\sqrt{U^{2}+2V^{2}})$ coordinates. As a result, 19 stars were selected
from the neighborhood of the clump center with coordinates $(-159,227)$ km s-1
and a neighborhood radius of 30 km s-1.
To determine the probability that each of the selected stars belonged to the
KFR08 and Arcturus streams, we performed Monte Carlo simulations of the
distribution of stars in the plane of $V,\sqrt{U^{2}+2V^{2}}$ coordinates by
taking into account the random errors in the stellar space velocities. We
generated 3000 random realizations for each star. In our simulations of the
KFR08 and Arcturus streams, we took the following parameters of their
distribution in the $V,\sqrt{U^{2}+2V^{2}}$ plane obtained by analyzing Fig.
5: (1) the coordinates of the centers are $(-159,227)$ km s-1 for KFR08 and
$(-124,178)$ km s-1 for Arcturus; (2) the velocity dispersion is 5 km s-1 for
both streams. The results of our simulations are reflected in Fig. 7 and in
the last column of Table 2, which gives the probability that a star belongs to
the KFR08 stream, p. Obviously, the probability that a star belongs to the
Arcturus stream is $1-p.$ As we see from Table 2, eleven stars constituting
the core of the KFR08 stream have probabilities $p\geq 0.99$ and only two
stars have $p=0.65.$ The positions of these two stars (HIP 74033 and HIP
58357) are marked in Fig. 7. As we see from the figure, their random errors
are such that they have almost equal chances of being attributed to both the
KFR08 and Arcturus streams. Therefore, it is not surprising that the star HIP
74033 in Arifyanto and Fuchs (2006) was attributed to the Arcturus stream.
Since we have failed to find information about the metallicities of several
stars from this sample in the literature, we calculated the metallicity
indices based on Strömgren uvby? photometry from the compilation by Hauck and
Mermilliod (1998) using the calibration by Schuster and Nissen (1989).
The distribution of $U,V,W$ velocities for KFR08 stream members is shown in
Fig. 8. As can be seen from this figure, the stars are located in a narrow
range of velocities V and in wider ranges of $U$ and $W$ than are typical of
the Arcturus stream stars (Navarro et al. 2004).
Figure 9 presents a color-absolute magnitude diagram for KFR08 stream members
with the Yonsei-Yale (Yi et al. 2003) 11-, 13-, and 15-Gyr isochrones for
$Z=0.007$ (Fe/H$=-0.43).$ We can see that the stream stars fall nicely on the
13-Gyr isochrone; the deviations are most pronounced only for two stars, HIP
87101 and HIP 93269. Our isochrone age estimate for the stream is in good
agreement with the available age estimates for individual stars (Table 2).
## DISCUSSION
(1) Using currently available data, we have been able to confirm the presence
of main known clumps, streams, and branches in the stellar velocity field in
the solar neighborhood and to trace the evolution of the velocity field for F
and G dwarfs as a function of the stellar age. Note that there is a very wide
range of stellar ages in each of the classical Pleiades, Hyades, Sirius, Coma
Berenices, and Hercules streams (Fig. 3). This is in good agreement with the
results of a detailed analysis of the metallicity distribution and age
estimates for stars performed recently by Antoja et al. (2008) and Francis and
Anderson (2008).
(2) The Wolf 630-$\alpha$Ceti and Hercules streams are interesting in that
they both could be produced by a common mechanism related to the influence of
a bar at the Galactic center (Dehnen 1999, 2000; Fux 2001; Chakrabarty 2007).
As can be seen from Fig. 3, both streams begin to manifest themselves at a
mean age of the sample stars $>2$ Gyr. They are most pronounced at a mean
stellar age of $\approx$7 Gyr (sample t7). Using improved stellar age
estimates from an updated version of the Geneva-Copenhagen survey (Holmberg et
al. 2007, 2008) led to a noticeable shift of the mean stellar age for the
Hercules branch in the direction of its decrease. For example, in Bobylev and
Bajkova (2007a), where the age estimates from the first version of the catalog
(Nordström et al. 2004) were used, a similar development of the Hercules
branch was achieved at a mean age of the sample stars $\approx$8.9 Gyr.
According to the data by Taylor (2000), the mean stellar metallicity is
[Fe/H]$=-0.11\pm 0.02\pm 0.15$ dex (the error of the mean and dispersion) for
the Wolf 630 stream ($\approx$40 stars selected according to Eggen s lists)
and [Fe/H]$=-0.12\pm 0.04\pm 0.18$ dex (the error of the mean and dispersion)
for the Hercules stream ($\approx$10 stars).
An extensive analysis of the distribution of stars in age and metallicity in
various streams performed recently by Antoja et al. (2008) showed that the
highest (compared to other branches) stellar metallicity dispersion is
characteristic of the Hercules branch. The mean and dispersion are
[Fe/H]$=-0.15\pm 0.27$ dex.
This structure was shown to be distinguished increasingly clearly in the form
of a branch starting from an age of 2 Gyr. Our results are generally in good
agreement with those of Antoja et al. (2008).
The mean stellar metallicity and age for features H1 and H2 of the Hercules
stream as well as W1 and W2 of the Wolf 630-$\alpha$Ceti branch (Table 1) are
consistent with the hypothesis of a dynamical nature of the streams related to
the influence of the Galactic bar. This is seen most clearly for features H1
and H2. Thus, for example, feature H1, which is closer to the local standard
of rest, is youngest. Since young field stars fall into the samples under
consideration, the mean ages of the streams are underestimated, especially for
features W1 and W2.
Note that the existence of the HR 1614 clump cannot be explained only by the
presence of a OSC remnant with an age of $\sim 2$ Gyr (De Silva et al. 2007),
since this clump is traceable in the $UV$ distributions for samples of
considerably older stars. Thus, for example, it is clearly seen on the $UV$
map for stars with an age of $\approx 7$ Gyr (t7, Figs. 3 and 4), suggesting
that the HR 1614 clump is an outgrowth of the Hercules branch and can be
dynamical in nature.
(3) The KFR08 stream was discovered by Klement et al. (2008) from their
analysis of the data on faint (compared to Hipparcos) stars of the RAVE
experiment. These authors identified 15 stream candidates. Since the distances
of the stars in the analyzed sample were estimated from photometric data, they
are less reliable than the trigonometric distances of Hipparcos stars. At the
same time, Klement et al. (2008) analyzed 13440 stars from the first version
of the Geneva-Copenhagen survey (Nordström et al. 2004) and showed that the
presence of about 30 stars (among the Hipparcos stars) in the KFR08 clump
might be expected in the $V,\sqrt{U^{2}+2V^{2}}$ plane. However, no specific
stars were selected.
The number of candidates for membership in the KFR08 stream we found is in
satisfactory agreement with the expected estimates. The results of our search
based on more accurate data are of great interest in establishing the nature
of the KFR08 stream. In contrast to the samples by Klement et al. (2008), our
“all” sample contains not only dwarfs but also giants.
As a result, we can see the main-sequence turnoff on the color-absolute
magnitude diagram for KFR08 stream members (Fig. 9), which increases the
reliability of the stream age estimate $(\approx 13$ Gyr).
According to the available data (Table 2), the metallicity indices for an
overwhelming majority of stars lie within a fairly narrow range,
$-1<$[Fe/H]$<-0.3.$ A similar homogeneity is also observed for the stars of
the Arcturus stream (Navarro et al. 2004). This is one of the arguments for a
common nature of these two streams. Obviously,much greater statistics is
required to make the final decision.
Note that Minchev et al. (2009) suggested an alternative hypothesis about the
nature of the AF06, Arcturus, and KFR08 streams. It is based on the assumption
that the disk has not yet relaxed and it is “shaken” after the disruption of
the dwarf galaxy captured by our Galaxy; therefore, waves are observed in the
plane of $UV$ velocities.
## CONCLUSIONS
Based on the most recent data, we studied the space velocity field of
$\approx$17000 stars in the solar neighborhood.We used data from a new version
of the Hipparcos catalogue (van Leeuwen 2007), stellar radial velocities from
the OSACA (Bobylev et al. 2006) and PCRV (Gontcharov 2006) catalogs reduced to
a common system, and improved estimates of the ages and metallicity indices
for F and G dwarfs from an updated Geneva-Copenhagen survey (Holmberg et al.
2007, 2008).
We identified all of the main clumps, streams, and branches known to date
using various approaches. Among the stars with a relatively low velocity
dispersion, these are the Pleiades, Hyades, and Coma Berenices streams or
branches. Among the stars with an intermediate velocity dispersion, these are
the Hercules and Wolf 630-$\alpha$Ceti branches. Among the stars with a high
velocity dispersion, these are the Arcturus and AF06 streams (Arifyanto and
Fuchs 2006) and the KFR08 stream (Klement et al. 2008).
Our attention was focused on the most poorly studied structures, the Wolf
630-$\alpha$Ceti and Hercules branches, and on the KFR08 stream discovered
quite recently.
The present view of the nature of the Wolf 630-$\alpha$Ceti and Hercules
streams is that they could be produced by the same mechanism related to the
influence of a bar at the Galactic center. Indeed, these structures begin to
manifest themselves as independent branches at a mean age of the sample stars
$>2$ Gyr, which is in conflict with the hypothesis of their origin based on
the theory of stellar streams (Eggen s hypothesis). Our estimates showed that
the mean stellar ages of these structures are quite comparable and are 4–6
Gyr. We revealed now significant differences in the metallicities of the stars
belonging to these streams.
We found 19 Hipparcos stars belonging to the new KFR08 stream and obtained an
isochrone age estimate for the stream, 13 Gyr. The homogeneity of the
kinematics, chemical composition, and age of the sample stars is consistent
with the hypothesis that the stream is a relic remnant of the galaxy captured
and disrupted by the tidal effect of our own Galaxy. Data from the GAIA
experiment will undoubtedly play a major role for a further study of this
structure.
ACKNOWLEDGMENTS
We are grateful to the referees for helpful remarks that contributed to a
improvement of the paper. The SIMBAD search database was very helpful in the
work. This study was supported by the Russian Foundation for Basic Research
(project no. 08–02–00400) and in part by the “Origin and Evolution of Stars
and Galaxies” Program of the Presidium of the Russian Academy of Sciences and
the Program for State Support of Leading Scientific Schools of Russia
(NSh–6110.2008.2).
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Table 1: Characteristics of the Wolf 630-$\alpha$Ceti branch (features W1 and
W2) and the Hercules stream (features H1 and H2)
Obj. | $N_{\star}$ | [Fe/H], | Age, | $U,$ | $V,$ | $W,$ | $a_{i},$ | $b_{i},$ | $\beta_{i}$
---|---|---|---|---|---|---|---|---|---
| | dex | Gyr | km s-1 | km s-1 | km s-1 | km s-1 | km s-1 | deg.
W1 | 88 | $-0.06~{}(0.20)$ | $3.9~{}(2.7)$ | $23$ | $-28$ | $-5$ | $7.4$ | $5.6$ | $148^{\circ}$
W2 | 95 | $-0.13~{}(0.19)$ | $3.6~{}(2.3)$ | $41$ | $-26$ | $-8$ | $8.9$ | $6.3$ | $120^{\circ}$
H1 | 136 | $-0.09~{}(0.17)$ | $4.6~{}(3.2)$ | $-33~{}$ | $-51$ | $-8$ | $14.2$ | $5.4$ | $103^{\circ}$
H2 | 71 | $-0.16~{}(0.27)$ | $5.7~{}(3.4)$ | $-77~{}$ | $-49$ | $-7$ | $21.2$ | $7.9$ | $80^{\circ}$
Note. $N$ is the number of stars with available age and metallicity estimates,
the velocities $U,V,$ and $W$ are given relative to the Sun (see Fig. 4), the
corresponding dispersions are given for the mean metallicity indices and mean
ages of the sample stars.
Table 2: Parameters of the Hipparcos stars that are probable members of the
KFR08 stream
HIP | | [Fe/H] | Ref | Age | $U\pm e_{U}$ | $V\pm e_{V}$ | $W\pm e_{W}$ | $p$
---|---|---|---|---|---|---|---|---
5336 | | $-0.84$ | (1) | | $-32\pm 1~{}$ | $-153\pm 1~{}$ | $-28\pm 1~{}$ | 1.00
15495 | | $-0.36$ | (2) | | $58\pm 4~{}$ | $-174\pm 8~{}$ | $-3\pm 3~{}$ | 1.00
18235 | | $-0.71$ | (3) | 11 | $-16\pm 3~{}$ | $-161\pm 4~{}$ | $-19\pm 2~{}$ | 1.00
19143 | | $-0.49$ | (2) | | $-140\pm 3~{}$ | $-143\pm 11$ | $-42\pm 2~{}$ | 0.98
54469 | * | $-0.72$ | (4) | 11 | $91\pm 5~{}$ | $-159\pm 16$ | $-64\pm 16$ | 1.00
55988 | | | | | $50\pm 4~{}$ | $-154\pm 6~{}$ | $-25\pm 4~{}$ | 0.99
58357 | * | $-0.71$ | (1) | | $-123\pm 16$ | $-134\pm 23$ | $45\pm 1~{}$ | 0.65
58708 | | $-0.30$ | | | $-14\pm 3~{}$ | $-160\pm 4~{}$ | $15\pm 1~{}$ | 0.99
58843 | | $-0.80$ | | | $122\pm 9~{}$ | $-138\pm 14$ | $-58\pm 7~{}$ | 0.81
59785 | | $-0.37$ | | | $-117\pm 9~{}$ | $-136\pm 6~{}$ | $-109\pm 7~{}$ | 0.92
60747 | * | $-0.77$ | (6) | | $110\pm 7~{}$ | $-146\pm 14$ | $91\pm 7~{}$ | 0.91
64920 | | $-0.42$ | | | $66\pm 5~{}$ | $-159\pm 5~{}$ | $43\pm 5~{}$ | 0.99
74033 | | $-0.75$ | (4) | 13 | $-113\pm 10$ | $-132\pm 10$ | $42\pm 7~{}$ | 0.65
81170 | * | $-1.26$ | (5) | | $-77\pm 2~{}$ | $-157\pm 9~{}$ | $-123\pm 3~{}$ | 0.99
87101 | | $-1.31$ | (6) | | $-76\pm 5~{}$ | $-159\pm 18$ | $-3\pm 2~{}$ | 0.91
93269 | | | | | $70\pm 3~{}$ | $-140\pm 1~{}$ | $-4\pm 3~{}$ | 0.99
93623 | | $-0.60$ | (2) | | $130\pm 5~{}$ | $-149\pm 16$ | $-20\pm 1~{}$ | 0.96
96185 | | $-0.60$ | (4) | 12 | $-56\pm 1~{}$ | $-156\pm 1~{}$ | $66\pm 1~{}$ | 1.00
117702 | | $-0.43$ | (7) | | $12\pm 7~{}$ | $-159\pm 7~{}$ | $124\pm 5~{}$ | 0.99
Note. The age is in Gyr, the velocities $U,V,$ and $W$ are in km s-1 and are
given relative to the LSR (Dehnen and Binney 1998); the asterisk $*$ marks the
candidates with $e_{\pi}/\pi<0.15;$ the stellar metallicities and age
estimates were taken from the following papers: 1, Soubiran et al. (2008); 2,
Ibukiyama, and Arimoto (2002); 3, Bensby et al. (2005); 4, Holmberg et al.
(2007); 5, Borkova and Marsakov (2005); 6, Schuster et al. (2006); 7, Jenkins
et al. (2008).
Fig. 1. (a) $UV$ velocity distribution for the “all” sample of 16737 single
stars with reliable distance estimates $(e_{\pi}/\pi<0.1);$ the velocities are
given relative to the Sun. Distributions of the (b) $U$ and (c) $V$ velocity
biases caused by the measurement errors of the stellar parallaxes.
Fig. 2. Density of the $UV$-velocity distribution corresponding to Fig. 1
obtained by the adaptive kernel method; the velocities are given relative to
the Sun (a); the sections of map (a) perpendicular to the $(U,V)$ plane that
pass through the main peaks and that make $+16^{\circ}$ with the $U$ axis if
measured clockwise (this axis is designated as $U);$ the distribution density
in units of $7\times 10^{-4}$ is along the vertical axis, the numbers denote
the sections passing through the Sirius (1), Coma Berenices (2), Pleiades-
Hyades (3), and Hercules (4) branches)(b).
Fig. 3. Densities of the $UV$-velocity distribution for samples of F and G
dwarfs as a function of the stellar age; the velocities are given relative to
the Sun.
Fig. 4. Wavelet maps of $UV,WU,$ and $WV$ velocities for a sample of 16737
stars; the velocities are given relative to the Sun. See also the text.
Fig. 5. Wavelet maps in the system of $(V,\sqrt{U^{2}+2V^{2}})$ coordinates
for a sample of 16737 stars; the velocities are given relative to the LSR.
Fig. 6. Wavelet maps in the system of $(V,\sqrt{U^{2}+2V^{2}})$ coordinates
for samples of F and G dwarfs as a function of the stellar age; the velocities
are given relative to the LSR.
Fig. 7. Positions of KFR08 stream members in the $(V,\sqrt{U^{2}+2V^{2}})$
plane, the velocities are given relative to the LSR, three contours
corresponding to probabilities of 0.683, 0.954, and 0.997
$(1\sigma,2\sigma,3\sigma)$ are given for the KFR08 and Arcturus streams.
Fig. 8. Velocity distribution for KFR08 stream members, the velocities are
given relative to the LSR.
Fig. 9. Color-absolute magnitude diagram for KFR08 stream members.
|
arxiv-papers
| 2009-12-03T13:42:28 |
2024-09-04T02:49:06.828981
|
{
"license": "Public Domain",
"authors": "V. V. Bobylev, A. T. Bajkova, and A. A. Myllari",
"submitter": "Anisa Bajkova",
"url": "https://arxiv.org/abs/0912.0635"
}
|
0912.0667
|
# Minimal non-nilpotent groups which are supersolvable
Francesco G. Russo Mathematics Department
University of Naples Federico II
via Cinthia, 80126, Naples, Italy francesco.russo@dma.unina.it
###### Abstract.
The structure of a group which is not nilpotent but all of whose proper
subgroups are nilpotent has interested the researches of several authors both
in the finite case and in the infinite case. The present paper generalizes
some classic descriptions of M. Newman, H. Smith and J. Wiegold in the context
of supersolvable groups.
###### Key words and phrases:
Minimal non-nilpotent groups, Schmidt groups, critical groups, groups with
many nilpotent subgroups
###### 1991 Mathematics Subject Classification:
Primary 20E34, 20E45; Secondary 20D10
## 1\. Introduction
Let $\mathfrak{N}$ be the class of all nilpotent groups. A group $G$ is said
to be a $minimal$ $non$-$nilpotent$ $group$, or $\mathfrak{N}$-$critical$
$group$, or $Schmidt$ $group$, or $MNN$-$group$, if it doesn’t belong to
$\mathfrak{N}$ but all of whose proper subgroups belong to $\mathfrak{N}$. We
will use the last terminology in the present paper. It is evident already from
these 4 ways to call the same mathematical object that there is a wide
literature on the topic. Many authors are still interested in studying
$MNN$-groups, because they play an important role from the point of view of
the general theory. The first example of finite $MNN$-group is probably the
symmetric group $S_{3}$ of order 6. We know that $S_{3}$ can be written as the
semidirect product of a cyclic group $C_{3}$ of order 3 by a cyclic group
$C_{2}$ of order $2$, which acts by inversion on $C_{3}$. Already for $S_{3}$
the condition of being an $MNN$-group determines its structure, in fact, we
have a semidirect product and this allows us to have a deep knowledge of the
whole group. At this point the following question becomes natural.
What is the influence of being an $MNN$-group on the group structure?
In the finite case a first answer is due to a famous contribution of O. Yu.
Schmidt and more details can be found in [8]. His methods and techniques
showed that the question can be seen from a different prospective, involving
the theory of classes of groups and conditions which are weaker of being
nilpotent. A recent contribution in this direction has been given by
J.C.Beidleman and H.Heineken in [1, Theorem 2], where they generalize the
description of O. Yu. Schmidt to the context of saturated formation of finite
groups.
On another hand, classic descriptions of $MNN$-groups in the infinite case
were given by M. Newman, H. Smith and J. Wiegold in [4, 9, 10]. Among these
groups, there are Tarski groups [5] so it is a common use the imposition of
suitable finiteness conditions in order to treat separately the Tarski groups.
Now we illustrate the new idea of the present paper. Consider the following
subset of the subgroup lattice $\mathcal{L}(G)$ of $G$
(1.1) $\mathcal{M}(G)=\\{H\leq G:H\not\in\mathfrak{N}\\}.$
$\mathcal{M}(G)=\\{G\\}$ if and only if $|\mathcal{M}(G)|=1$, that is, $G$ is
the unique non-nilpotent subgroup, that is, $G$ is an $MNN$-group. It turns
out that we may extend significatively the classifications in [4, 8, 9],
dealing with (1.1) when $|\mathcal{M}(G)|=m\geq 1$. For the case $m=2$ we can
be more precise and details are illustrated in Section 2, preparing the main
results which are in Section 3. For higher values of $m$ we have not found
deep restrictions on the group structure and, to the best of our knowledge, it
is an open problem.
## 2\. The Case $m=2$
The motivation of studying (1.1) is clear once we note that $|\mathcal{M}(G)|$
gives a measure of how many $MNN$-subgroups are contained in $G$, and so , of
how $G$ is far from the usual classifications in [4, 8, 9]. Of course,
$|\mathcal{M}(G)|=2$ if and only if $G\not\in\mathfrak{N}$ and we have just 1
$MNN$-subgroup $K$ of $G$. Going on, the situation is a little bit more
complicated. Already the case $|\mathcal{M}(G)|=3$ needs of more attention.
###### Lemma 2.1.
$|\mathcal{M}(G)|=2$ if and only if $G\not\in\mathfrak{N}$ and $G$ contains a
maximal normal subgroup $K$ which is an $MNN$-group.
###### Proof.
Since $|\mathcal{M}(G)|=2$, we have $\mathcal{M}(G)=\\{G,K\\}$, where $K<G$.
So $K$ is an $MNN$-group. If there is a subgroup $H$ of $G$ such that $K<H<G$,
then $H\in\mathfrak{N}$ and so $K\in\mathfrak{N}$. This contradiction implies
that $K$ is a maximal subgroup of $G$. Now for each $x\in G$, $K^{x}\leq G$.
But $K^{x}\simeq K\not\in\mathfrak{N}$, so $K^{x}=K$. Then $K$ is normal in
$G$. ∎
###### Lemma 2.2.
Assume $|\mathcal{M}(G)|=2$ and $K$ as in Lemma 2.1. Then $G/K$ is of prime
order and $G^{\prime}\leq K$.
###### Proof.
Since $G/K$ has only two subgroups, $G/K$ is of prime order. Since $G/K$ is
abelian, $G^{\prime}\leq K$. ∎
###### Remark 2.3.
Assume $|\mathcal{M}(G)|=2$. Then $K$ in Lemma 2.1 is a characteristic
subgroup of $G$.
###### Proof.
Let $\alpha\in$Aut$(G)$. Then $\alpha(K)\simeq K\not\in\mathfrak{N}$, so
$\alpha(K)=K$. ∎
In order to proceed we recall the Hall’s Criterion of nilpotence in [6,
5.2.10].
###### Theorem 2.4 (P.Hall, see [6]).
Let $N$ be a normal subgroup of a group $G$. If $N\in\mathfrak{N}$ and
$G/N^{\prime}\in\mathfrak{N}$, then $G\in\mathfrak{N}$.
###### Remark 2.5.
Assume $|\mathcal{M}(G)|=2$ and $G^{\prime}<K$ with $K$ as in Lemma 2.1. Then
$G$ is solvable with a non-trivial non-nilpotent homomorphic image.
###### Proof.
Since $G^{\prime}<K$ and $K$ is an $MNN$-group, $G^{\prime}\in\mathfrak{N}$
and so $G$ is solvable. Theorem 2.4 implies
$G/G^{\prime\prime}\not\in\mathfrak{N}$, which is the requested homomorphic
image. ∎
###### Remark 2.6.
Let $K$ be as in Remark 2.5. If $\mathcal{M}(G)=\\{G,K\\}$, then
$\mathcal{M}(G/G^{\prime\prime})=\\{G/G^{\prime\prime},K/G^{\prime\prime}\\}$.
###### Proof.
Remark 2.5 shows that $G/G^{\prime\prime}\not\in\mathfrak{N}$. Each subgroup
of $G/G^{\prime\prime}$ is of the form $H/G^{\prime\prime}$, where
$G^{\prime\prime}\leq H\leq G$. Then $H/G^{\prime\prime}\in\mathfrak{N}$,
whenever $H\not=K$ and $H\not=G$. Therefore,
$K/G^{\prime\prime}\not\in\mathfrak{N}$ by Theorem 2.4. ∎
A group $G$ is $locally$ $graded$ if every nontrivial finitely generated
subgroup of $G$ have a finite image. The next result recalls [10, Theorem 2].
###### Theorem 2.7 (H. Smith, see [10]).
Let $G$ be a locally graded group and suppose that, for some positive integer
$b(G)$, every non-nilpotent subgroup of $G$ is subnormal of defect $\leq b(G)$
in $G$. Then $G$ is solvable.
Now Remark 2.5 can be reformulated in the following way.
###### Proposition 2.8.
Assume $|\mathcal{M}(G)|=2$. If $G$ is locally graded, then $G$ is solvable.
###### Proof.
Let $K$ be as in Lemma 2.1. All non-nilpotent subgroups of $G$ are subnormal.
Then $G$ is solvable by Theorem 2.7. ∎
###### Lemma 2.9.
Assume $|\mathcal{M}(G)|=2$ and $K$ as in Remark 2.5. If $M\not=K$ is a
maximal normal subgroup of $G$, then $[K,M]\not=1$.
###### Proof.
Assume $[K,M]=1$. Then $M\leq C_{G}(K)$. If $M=C_{G}(K)$, then $M\cap K=Z(K)$
and so $MK/M\simeq K/(M\cap K)\simeq K/Z(K)$ is cyclic. This gives $K$
abelian. If $G=C_{G}(K)$, then $K\leq Z(G)$ and again $K$ is abelian. Both
cases contradict $K\not\in\mathfrak{N}$. The result follows. ∎
###### Proposition 2.10.
Assume $|\mathcal{M}(G)|=2$ and $K$ as in Remark 2.5. If $M$ is a maximal
subgroup of $G$ whose elements have coprime order with those of $K$, then $K$
is the unique maximal subgroup of $G$.
###### Proof.
$K$ is periodic by the classification of M.Newman and J.Wiegold in [4]. Then
$G$ is periodic and so $M$. $M\cap K=\\{1\\}$ from the relation $(|\langle
m\rangle|,|\langle k\rangle|)=1$ for each $m\in M$ and $k\in K$. Then,
$[M,K]\leq M\cap K=\\{1\\}$. Lemma 2.9 implies that $M=K$ and the result
follows. ∎
Recall that $\pi(G)$ denotes the set of prime divisors of the order of the
elements of $G$.
###### Corollary 2.11.
Assume $|\mathcal{M}(G)|=2$ and $K$ non-finitely generated. If $K$ has maximal
subgroups, then $G$ is a Chernikov group of derived length at most 3 with
$|\pi(G)|\leq 2$.
###### Proof.
By the classification of M.Newman and J.Wiegold in [4], $K$ is a metabelian
Chernikov $p$-group for some prime $p$ (see [4, p.242, lines +5 and +6]). From
Lemma 2.2 the result follows easily. ∎
###### Lemma 2.12.
Assume $|\mathcal{M}(G)|=2$ and $K$ non-finitely generated. Then $G$ is
locally nilpotent. In particular, each maximal subgroup of $G$ is normal and
of prime index.
###### Proof.
Every finitely generated subgroup $H$ of $G$, such that $H\not=G$ and
$H\not=K$, is nilpotent. Then $G$ is locally nilpotent. The remaining part of
the result follows easily. ∎
###### Corollary 2.13.
Assume $|\mathcal{M}(G)|=2$ and $K$ non-finitely generated. Then $G$ is
solvable.
###### Proof.
This follows from Lemma 2.12 and Proposition 2.8. ∎
A concrete situation is described as follows.
###### Example 2.14.
Write $A=C_{2^{\infty}}$ for the quasicyclic $2$-group, $B=\langle x\rangle$
and $C=\langle y\rangle$, where $x$ and $y$ have order 2. Consider $K=A\rtimes
B$, which is the well-known locally dihedral 2-group [6, p.344], and
$G=K\times C$. By construction, $\mathcal{L}(K)-\mathcal{L}(A)=\\{K,B,\langle
H,B\rangle\\}$, where $\\{1\\}\not=H<A$. Of course, $B\in\mathfrak{N}$. On the
other hand, $\langle H,B\rangle\leq Z_{i}(K)$ for some $i\geq 1$, since $K$ is
$\omega$-hypercentral. Then $\langle H,B\rangle\in\mathfrak{N}$. We conclude
that $K$ is an $MNN$-group. Now, the presence of $K$ implies that $G$ is not
an $MNN$-group. By construction, $\mathcal{L}(G)-\mathcal{L}(K)=\\{G,C,\langle
L,C\rangle\\}$, where $\\{1\\}\not=L<K$. Noting that $\langle
L,C\rangle=L\times C$, we have $L\times C\in\mathfrak{N}$. Then
$\mathcal{M}(G)=\\{G,K\\}$. Note that $K$ is the unique maximal subgroup of
$G$. Note also that $A$ is the unique maximal subgroup of $K$. We have all it
is needed in order to state that $G$ satisfies Proposition 2.8 and Corollaries
2.11, 2.13.
Example 2.14 shows that we may get groups as in Proposition 2.8 and
Corollaries 2.11, 2.13, adding a finite cyclic group to a given $MNN$-group.
Then, choosing a suitable order for the cyclic group, we may give examples for
Proposition 2.10.
## 3\. Main Theorems
In order to proceed with the proof of the main theorem of the present section,
we recall [4, Lemma 3.2] and [4, Theorem 2.12], respectively.
###### Lemma 3.1 (M.Newman–J.Wiegold, see [4]).
Let $G$ be a finitely generated non-nilpotent group all of whose proper
subgroups are locally nilpotent and $\gamma_{\infty}(G)$ be the last term of
the lower central series of $G$. If $G/\gamma_{\infty}(G)$ is nontrivial, then
$G$ is finite.
###### Theorem 3.2 (M.Newman–J.Wiegold, see [4]).
If $G$ is a group in which every pair of proper normal subgroups generates a
proper subgroup, then $G/G^{\prime}$ is a locally cyclic $p$-group for some
prime $p$ and $G^{\prime}=\gamma_{\infty}(G)$.
We should recall also some notations from [11]. Let $n\geq 1$, $i$ and $j$ be
two distinct integers in $\\{1,2,\dots,n\\}$, $p_{i},p_{j}$ primes,
$d_{i},d_{j}\geq 1$, $\pi(d_{i})$ be the set of prime divisors of $d_{i}$ and
$q_{i}\in\pi(d_{i})$. An $F_{\\{p_{i},d_{i}\\}}$-$group$ is a Frobenius group
whose kernel is an elementary abelian group of order $p_{i}^{m_{i}}$ with
cyclic complement of order $d_{i}$, where $m_{i}$ is the exponent of $p_{i}$
modulo $q_{i}$. The next result quotes [11, Theorem 1].
###### Theorem 3.3.
In a non-nilpotent finite group $G$, all $MNN$-subgroups are subnormal if and
only if
(3.1) $G/Z_{\infty}(G)=G_{1}\times G_{2}\times\ldots\times G_{n},$
where $G_{i}$ is an $F_{\\{p_{i},d_{i}\\}}$-group, and $(d_{i},d_{j})=1$ for
any $i\not=j$ with $i,j\in\\{1,2,\ldots,n\\}$.
Our main result is the following and describes (1.1) in a special case.
###### Theorem 3.4.
Assume $K$ as in Remark 2.5. If $K$ is finitely generated, then $G$ is a
finite supersolvable group. Furthermore,
(3.2) $G/Z_{\infty}(G)=G_{1}\times G_{2}\times\ldots\times G_{n},$
where $G_{i}$ is an $F_{\\{p_{i},d_{i}\\}}$-group and $(d_{i},d_{j})=1$ for
any $i\not=j$ with $i,j\in\\{1,2,\ldots,n\\}$.
###### Proof.
An application of Lemma 3.1 to $K$ implies that $K$ is finite. Then $G$ is
finite by Lemma 2.2. More precisely, $G=K\langle x\rangle$, where $|\langle
x\rangle|=|G/K|=q$ for some prime $q$. By Theorem 3.2 we may deduce that
$|K/K^{\prime}|$ is a cyclic group of order $p^{r}$ for some prime $p$ and
some $r\geq 1$. Then $K=K^{\prime}\langle y\rangle$, where $|\langle
y\rangle|=p^{r}$, and so $G=\langle K^{\prime},x,y\rangle=K^{\prime}\langle
x,y\rangle$, where $K^{\prime}$ is nilpotent finitely generated of class $c$.
We know from [6, 5.2.18] that a finitely generated nilpotent group has a
central series whose factors are cyclic with prime or infinite orders and so
$K^{\prime}=S$ is supersolvable and we have the following series
$\\{1\\}=Z_{0}(S)\triangleleft Z_{1}(S)\triangleleft\ldots\triangleleft
Z_{c}(S)=S\triangleleft K\triangleleft G,$ where
$Z_{1}(S)/Z_{0}(S),\ldots,Z_{c}(S)/Z_{c-1}(S)$ are cyclic groups of prime
order. We have just seen that $K/S$ is a cyclic group. $G/K$ is cyclic by
Lemma 2.2. Note that each term of this series is normal in $G$. Therefore $G$
is supersolvable.
Independently, the only fact that $G$ is finite allows us to apply [11,
Theorem 1] and so $G/Z_{\infty}(G)$ is the direct product of Frobenius groups
as claimed. ∎
It is interesting the following consequence of Theorem 3.4.
###### Corollary 3.5.
If $G$ is a finite solvable group with $|\mathcal{M}(G)|=2$, then it is
supersolvable.
###### Remark 3.6.
Theorem 3.4 relates $G/Z_{\infty}(G)$ with $|\mathcal{M}(G)|$. Recall that
nilpotent finitely generated groups are supersolvable (see [6]). Then we are
saying in Theorem 3.4 that small values of $|\mathcal{M}(G)|$ imply that $G$
is a (finite) supersolvable group which is not nilpotent. Furthermore we are
describing, thanks to $G/Z_{\infty}(G)$, how much is big the difference from
being supersolvable and not being nilpotent.
The remainder of this section illustrates another aspect of (1.1).
We recall from [6, §13.3] that
(3.3) $\omega(G)={\underset{SsnG}{\bigcap}N_{G}(S)}$
is the $Wielandt$ $subgroup$ of a group $G$. $\omega(G)$ is always a
$T$-$group$, that is, a group in which the normality is a transitive relation.
Solvable $T$-groups were classified by D.J.Robinson in 1964 (see [2]) and more
generally the groups in which all the subgroups are subnormal were classified
by W. Möhres in [3] (see also [2, §12.2]). These are related to $MNN$-groups
by [9, Theorem 3.1], which is quoted below.
###### Theorem 3.7 (H.Smith, see [9]).
Let $G$ be a soluble $MNN$-group and suppose that $G$ has no maximal
subgroups. Then:
* (i)
$G$ is a countable $p$-group for some prime $p$ and $G/G^{\prime}\simeq
C_{p^{\infty}}$;
* (ii)
every subgroup of $G$ is subnormal;
* (iii)
every hypercentral image of $G$ is abelian and
$G^{\prime}=\gamma_{\infty}(G)$;
* (iv)
every radicable subgroup of $G$ is central;
* (v)
$HG^{\prime}=G$ implies $H=G$ for every subgroup $H$ of $G$ and
$C_{G}(G^{\prime})$ is abelian. In particular, $G$ has no proper subgroups of
finite index;
* (vi)
$G^{\prime}$ is not the normal closure in $G$ of a finite subgroup;
* (vii)
$Z(G)=Z_{\infty}(G)$.
We have all it is necessary in order to prove the second main result of this
section.
###### Theorem 3.8.
Assume $|\mathcal{M}(G)|=2$ and $K$ non-finitely generated as in Lemma 2.1. If
$K$ has no maximal subgroups, then $K/\omega(K)$ is non-trivial, non-abelian,
of infinite exponent and at least of countable abelian rank.
###### Proof.
From Corollary 2.13, $G$ is solvable. Assume $\omega(K)=K$. $K$ is a $T$-group
and Theorem 3.7 (ii) every subgroup of $K$ is subnormal. Both these conditions
imply that $K$ is a Dedekind group, then $K\in\mathfrak{N}$. This
contradiction shows that $K/\omega(K)$ is non-trivial.
Assume $[K/\omega(K),K/\omega(K)]=1$. Then
(3.4) $[K,K]\leq\omega(K)={\underset{SsnK}{\bigcap}N_{K}(S)}={\underset{S\leq
K}{\bigcap}N_{K}(S)}=norm(K)\leq Z_{2}(K),$
where the last inequality is due to a famous result of E. Schenkman [7].
Therefore $K\in\mathfrak{N}$, which is a contradiction. This implies that
$K/\omega(K)$ cannot be abelian.
The fact that $K/\omega(K)$ is of infinite exponent follows by the
classification of W.Möhres and precisely by [3, Theorem].
Note that $K/\omega(K)$ has no maximal subgroups. Then $K/\omega(K)$ has no
proper subgroups of finite index. On another hand , we know from [2, 5.3.6]
that a solvable group with finite abelian rank and no proper subgroups of
finite index must be nilpotent. This implies that $K/\omega(K)$ cannot be of
finite abelian rank, and so, at least of countable abelian rank. ∎
Unfortunately, we cannot think to Example 2.14 in case of Theorem 3.8, since
in Example 2.14 there are maximal subgroups. However, a satisfactory
description is offered by the following result.
###### Corollary 3.9.
Assume $|\mathcal{M}(G)|=2$ and $K$ non-finitely generated as in Lemma 2.1. If
$K$ has no maximal subgroups, then $K$ has the series
(3.5) $\\{1\\}\triangleleft\omega(G)=K^{(d)}\triangleleft
K^{(d-1)}\triangleleft\ldots\triangleleft K^{\prime}\triangleleft
K\triangleleft G,$
where $\omega(K)=\gamma_{3}(\omega(K))\rtimes L$, $L$ is the subgroup
generated by the involutions of $\omega(K)$, $K/K^{\prime}\simeq
C_{p^{\infty}}$ for some prime $p$, there exists some $i\in\\{1,\ldots,d\\}$
such that $K^{(i+1)}/K^{i}$ is the direct product of infinitely many copies of
$C_{p^{\infty}}$, $G/K$ is of prime order.
###### Proof.
$G$ is solvable by Corollary 2.13. $K$ is a solvable $MNN$-group with no
maximal subgroups and it must be a periodic $p$-group, by Theorem 3.7. The
fact that $\omega(K)$ is a semidirect product of $L$ and
$\gamma_{3}(\omega(K))$ follows from the classification of periodic solvable
$T$-groups and can be found for instance in [6, Exercises 13.4, n.10, p.394].
Now the rest of the result follows from the combination of Lemma 2.1, Theorem
3.8, [6, Exercises 13.4, n.10, p.394] and Theorem 3.7. ∎
###### Acknowledgement.
The author is grateful to the Monastero di S.Pasquale a Chiaja of Naples for
hospitality in the period in which the present paper was written.
## References
* [1] J.C. Beidleman and H. Heineken, Minimal non-$\mathcal{F}$-groups, Ricerche Mat. 58 (2009), 33–41
* [2] J.C. Lennox and D.J. Robinson, The theory of infinite soluble groups, Clarendon Press, Oxford, 2004\.
* [3] W. Möhres, Torsionsgruppen, deren Untergruppen alle subnormal sind, Geom. Dedicata 31 (1989), 237–244.
* [4] M. Newman and J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15 (1964), 241–250.
* [5] A. Yu. Ol’shanskii, Infinite groups with cyclic subgroups, Soviet Math. Dokl. 20 (1979), 343–346.
* [6] D.J. Robinson, A Course in the Theory of Groups, Springer, Berlin, 1981.
* [7] E. Schenkman, On the norm of a group, Illinois J. Math. 4 (1960), 150 -152.
* [8] L. A. Shemetkov, O. Yu. Schmidt and finite groups, Ukr. Math. J. 23 (1971), 482 -486.
* [9] H. Smith, Groups with few non-nilpotent subgroups, Glasgow Math. J. 39 (1997), 141 -151.
* [10] H. Smith, Groups with all non-nilpotent subgroups subnormal, In: Topics in Infinite Groups, Quad. di Mat. vol.8, Caserta (Italy), Eds.: F. de Giovanni and M. Newell, Second University of Naples, 2000, Caserta, pp. 311 -326.
* [11] V.A. Vedernikov, Finite groups with subnormal Schmidt subgroups, Algebra Logic (6) 46 (2007), 363 -372.
|
arxiv-papers
| 2009-12-03T14:23:24 |
2024-09-04T02:49:06.836933
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco G. Russo (Universita' degli Studi di Palermo, Palermo,\n Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/0912.0667"
}
|
0912.0763
|
# Atomic coherent state in Schwinger bosonic realization for optical Raman
coherent effect
Hong-yi Fan${}^{1},$Xue-xiang Xu1,2, and Li-yun Hu2
1Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030,
China;
2College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang, 330022, China. Corresponding author. E-mail address:
hlyun2008@126.com.
###### Abstract
For optical Raman coherent effect we introduce the atomic coherent state (or
the angular momentum coherent state with various angular momemtum values) in
Schwinger bosonic realization, they are the eigenvectors of the Hamiltonian
describing the Raman effect. Similar to the fact that the photon coherent
state describes laser light, the atomic coherent state is related to Raman
process.
## 1 Introduction
Atomic coherent states (or the angular momentum coherent state with various
angular momemtum values) are sometimes referred to in the literature as spin
coherent states or Bloch states [1, 2, 3, 4, 5, 6]. They have been
successfully applied to many branches of physics [7, 8, 9, 10]. For example,
Arecchi et al. applied atomic coherent states to describe interactions between
radiation field and an assembly of two-level atoms [4]. Narducci, Bowden,
Bluemel, Garrazana and Tuft [7] used atomic coherent state to study multitime
correlation function for systems with observables satisfying an angular
momentum algebra, which suggested a convenient classical-quantum
correspondence rule for angular momentum degrees of freedom. Takahashi and
Shibata [9] transformed some equation of motion for density matrix of a damped
spin system into that of a quasi-distribution. Gerry and Benmoussa [10] have
studied the generation of spin squeezing by the repeated action of the angular
momentum Dicke lowering operator on an atomic coherent state. In this work we
shall introduce the atomic coherent state in Schwinger bosonic realization to
study Raman coherent effect in the context of quantum optics.
It is known that the Raman coherent effect, a monochromatic light wave
incident on a Raman active medium gives rise to a parametric coupling between
an optical vibrational mode and the mode of the radiation field, the so-called
Stocks mode. (In the case of Brillouin scattering, there is a similar
coupling, where the vibrations are at acoustical, rather than optical
frequencies.) The simplest Hamiltonian model for describing Raman coherent
effect is
$H=\omega_{1}a^{\dagger}a+\omega_{2}b^{\dagger}b-i\lambda\left(a^{\dagger}b-ab^{\dagger}\right),$
(1)
which is a two coupled oscillator model. In this work we shall show that the
atomic coherent state (some assembly of angular momentum states, so named
angular momentum coherent state) expressed in terms of Schwinger bosonic
realization of angular momentum [11] has its obvious physical background,
i.e., a set of energy eigenstates of two coupled bosonic oscillators with the
Hamiltonian can be classified as the atomic coherent state
$\left|\tau\right\rangle_{j}$ according to the angular momentum value of $j$,
where $\tau$ is determined by the dynamic parameters
$\omega_{1},\omega_{2},\lambda$. Thus the Raman coherent effect is closely
related to atomic coherent state theory, while the laser is described by the
coherent state theoretically.
## 2 Brief review of the atomic coherent state (ACS) in Schwinger bosonic
realization
The atomic coherent state with angular momentum value $j$ is defined as [4, 5,
6, 7]
$\left|\tau\right\rangle=\exp(\mu
J_{+}-\mu^{\ast}J_{-})\left|j,-j\right\rangle=(1+\left|\tau\right|^{2})^{-j}e^{\tau
J_{+}}\left|j,-j\right\rangle,$ (2)
where $J_{+}$ is the raising operator of the angular momentum state
$\left|j,m\right\rangle$, $\left|j,-j\right\rangle$ is the lowest weight state
annihilated by $J_{-}$, and
$\mu=\frac{\theta}{2}\text{e}^{-\text{i}\varphi},\text{
}\tau=\text{e}^{-\text{i}\varphi}\tan(\frac{\theta}{2}).$ (3)
In the $j$-subspace the completeness relation for $\left|\tau\right\rangle$ is
$\int\frac{\text{d}\Omega}{4\pi}\left|\tau\right\rangle\left\langle\tau\right|=\sum_{m=-j}^{j}\left|j,m\right\rangle\left\langle
j,m\right|=1_{j},$ (4)
where d$\Omega=\sin\theta$d$\theta$d$\varphi$, and
$\left\langle\tau^{\prime}\right.\left|\tau\right\rangle=\frac{(1+\tau^{\prime}\tau^{\ast})^{2j}}{(1+\left|\tau\right|^{2})^{j}(1+\left|\tau^{\prime}\right|^{2})^{j}}.$
(5)
Using $[J_{+},J_{-}]=2J_{z},$ $[J_{\pm},J_{z}]=\mp J_{\pm}$, one can show that
$\left|\tau\right\rangle$ obeys the following eigenvector equations,
$\displaystyle(J_{-}+\tau^{2}J_{+})\left|\tau\right\rangle$
$\displaystyle=2j\tau\left|\tau\right\rangle,$ $\displaystyle(J_{-}+\tau
J_{z})\left|\tau\right\rangle$ $\displaystyle=j\tau\left|\tau\right\rangle,$
(6) $\displaystyle(\tau J_{+}-J_{z})\left|\tau\right\rangle$
$\displaystyle=j\left|\tau\right\rangle.$
Employing the Schwinger Bose operator realization of angular momentum
$J_{+}=a^{\dagger}b,\text{ }J_{-}=ab^{\dagger},\text{
}J_{z}=\frac{1}{2}\left(a^{\dagger}a-b^{\dagger}b\right),$ (7)
where $[a,a^{\dagger}]=1,$ $[b,b^{\dagger}]=1$ and $\left|j,m\right\rangle\
$is realized as
$\displaystyle\left|j,m\right\rangle$ $\displaystyle=\frac{a^{\dagger
j+m}b^{\dagger j-m}}{\sqrt{(j+m)!(j-m)!}}\left|00\right\rangle$
$\displaystyle=\left|j+m\right\rangle\otimes\left|j-m\right\rangle,\text{ \ \
}(m=-j,\cdots,j),$ (8)
note that the last ket is written in two-mode Fock space, then
$\left|j,-j\right\rangle=\left|0\right\rangle\otimes\left|2j\right\rangle,$
and the atomic coherent state $\left|\tau\right\rangle$ is expressed as
$\displaystyle\left|\tau\right\rangle$ $\displaystyle=e^{\mu
J_{+}-\mu^{\ast}J_{-}}\left|0\right\rangle\otimes\left|2j\right\rangle$
$\displaystyle=\frac{1}{\sqrt{(2j)!}}[b^{\dagger}\cos(\frac{\theta}{2})+a^{\dagger}\text{e}^{-\text{{i}}\varphi}\sin(\frac{\theta}{2})]^{2j}\left|00\right\rangle$
$\displaystyle=\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{j}}\sum_{l=0}^{2j}\sqrt{\frac{(2j)!}{l!(2j-l)!}}\tau^{2j-l}\left|2j-l\right\rangle\otimes\left|l\right\rangle$
(9)
Especially when $j=0$, $\left|\tau\right\rangle=\left|00\right\rangle$ is just
the two-mode vacuum state in Fock space. Using the normal ordering form of the
two-mode vacuum projector $\left|00\right\rangle\left\langle
00\right|=:e^{-a^{\dagger}a-b^{\dagger}b}:$, we can use the technique of
integration within an ordered product of operators [12, 13] to prove in
$j$-subspace,
$\displaystyle\int\frac{\text{d}\Omega}{4\pi}\left|\tau\right\rangle\left\langle\tau\right|$
$\displaystyle=\frac{1}{\left(2j\right)!}\int_{0}^{\pi}d\theta\sin\theta\int_{0}^{2\pi}d\phi:\left(b^{\dagger}\cos\frac{\theta}{2}+a^{\dagger}e^{-i\phi}\sin\frac{\theta}{2}\right)^{2j}$
$\displaystyle\times\left.\left(b\cos\frac{\theta}{2}+ae^{i\phi}\sin\frac{\theta}{2}\right)^{2j}\exp\left(-a^{\dagger}a-b^{\dagger}b\right):\right.$
$\displaystyle=:\frac{\left(a^{\dagger}a+b^{\dagger}b\right)^{2j}}{\left(2j+1\right)!}e^{-a^{\dagger}a-b^{\dagger}b}:,$
(10)
the completeness relation of $\left|\tau\right\rangle$ in the whole two-mode
Fock space can be obtained after summing over $j$:
$\displaystyle\sum_{2j=0}^{\infty}(2j+1)\int\frac{\text{d}\Omega}{4\pi}\left|\tau\right\rangle\left\langle\tau\right|$
$\displaystyle=\sum_{2j=0}^{\infty}:\frac{\left(a^{\dagger}a+b^{\dagger}b\right)^{2j}}{\left(2j\right)!}e^{-a^{\dagger}a-b^{\dagger}b}:=1,$
(11)
which means that atomic coherent states in Schwinger bosonic realization with
all values of $j$ forms a complete set.
## 3 Atomic coherent state as energy eigenstates of H
Now we inquire whether the atomic coherent state with a definite angular
momentum value $j$ is the solution of the stationary Schrodinger equation
$H\left|\tau\right\rangle=E\left|\tau\right\rangle.$ (12)
In order to solve Eq.(12) we directly use Eq.(9) and the relation
$a^{\dagger}\left|n\right\rangle=\sqrt{n+1}\left|n+1\right\rangle,\text{
}a\left|n\right\rangle=\sqrt{n}\left|n-1\right\rangle,$ (13)
to calculate
$\displaystyle H\left|\tau\right\rangle$
$\displaystyle=\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{j}}\sum_{l=0}^{2j}\sqrt{\frac{(2j)!}{l!(2j-l)!}}\left[\omega_{1}\left(2j-l\right)+\omega_{2}l\right]\tau^{2j-l}\left|2j-l\right\rangle\otimes\left|l\right\rangle$
$\displaystyle-i\lambda\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{j}}\sum_{l=1}^{2j}\sqrt{\frac{(2j)!}{\left(l-1\right)!(2j-l+1)!}}\left(2j-l+1\right)\tau^{2j-l}\left|2j-l+1\right\rangle\otimes\left|l-1\right\rangle$
$\displaystyle+i\lambda\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{j}}\sum_{l=0}^{2j-1}\sqrt{\frac{(2j)!}{\left(l+1\right)!(2j-l-1)!}}\tau^{2j-l}\left(l+1\right)\left|2j-l-1\right\rangle\otimes\left|l+1\right\rangle.$
(14)
Let $l\mp 1\rightarrow l$ in the second and third term of the r.h.s. of
Eq.(14), respectively, we have
$\displaystyle H\left|\tau\right\rangle$
$\displaystyle=\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{j}}\sum_{l=0}^{2j}\sqrt{\frac{(2j)!}{l!(2j-l)!}}\tau^{2j-l}\left\\{\left[\omega_{1}\left(2j-l\right)+\omega_{2}l\right]-i\lambda\left(2j-l\right)\frac{1}{\tau}+i\lambda\tau
l\right\\}\left|2j-l\right\rangle\otimes\left|l\right\rangle$
$\displaystyle=\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{j}}\sum_{l=0}^{2j}\sqrt{\frac{(2j)!}{l!(2j-l)!}}\tau^{2j-l}\left\\{2\left(\omega_{1}-i\frac{\lambda}{\tau}\right)j+\left[\left(\omega_{2}-\omega_{1}\right)+i\lambda\left(\tau+\frac{1}{\tau}\right)\right]l\right\\}\left|2j-l\right\rangle\otimes\left|l\right\rangle$
$\displaystyle=2\left(\omega_{1}-i\frac{\lambda}{\tau}\right)j\left|\tau\right\rangle+\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{j}}\sum_{l=0}^{2j}\sqrt{\frac{(2j)!}{l!(2j-l)!}}\tau^{2j-l}\left[\left(\omega_{2}-\omega_{1}\right)+i\lambda\left(\tau+\frac{1}{\tau}\right)\right]l\left|2j-l\right\rangle\otimes\left|l\right\rangle.$
(15)
We see when the following condition is satisfied,
$i\lambda\tau^{2}+\tau\left(\omega_{2}-\omega_{1}\right)+i\lambda=0\Rightarrow\tau_{\pm}=\frac{\left(\omega_{1}-\omega_{2}\right)\pm\sqrt{\left(\omega_{1}-\omega_{2}\right)^{2}+4\lambda^{2}}}{2i\lambda}.$
(16)
then $\left|\tau_{\pm}\right\rangle,$ expressed by Eq.(9), is the eigenstate
of $H$ with eigenvalue
$\displaystyle E$
$\displaystyle=2\left(\omega_{1}-i\frac{\lambda}{\tau}\right)j$
$\displaystyle=j\left[\left(\omega_{1}+\omega_{2}\right)\pm\sqrt{\left(\omega_{1}-\omega_{2}\right)^{2}+4\lambda^{2}}\right]$
(17)
Hence $H$’s eigenvectors are classifiable according to the angular momentum
value $j$. Especially, when $\omega_{1}=\omega_{2}=\omega$, from Eqs.(16)-(17)
we know $\tau_{\pm}=\mp i,$ $E_{\pm}=2j\left(\omega\pm\lambda\right).$
## 4 Some fundamental atomic coherent states as H’s eigenstates
We now investigate some fundamental atomic coherent states as $H$’s
eigenstates. In the case of $j=1/2,$ from Eq.(9) we know the eigenstate of $H$
is
$\displaystyle\left|\tau_{\pm}\right\rangle_{j=1/2}$
$\displaystyle=\frac{1}{\left(1+\left|\tau_{\pm}\right|^{2}\right)^{1/2}}\left(\tau_{\pm}\left|1\right\rangle\otimes\left|0\right\rangle+\left|0\right\rangle\otimes\left|1\right\rangle\right)$
$\displaystyle\overset{\omega_{1}=\omega_{2}}{\rightarrow}\left|i_{\pm}\right\rangle_{j=1/2}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mp
i\left|1\right\rangle\otimes\left|0\right\rangle+\left|0\right\rangle\otimes\left|1\right\rangle\right).$
(18)
Indeed, one can check
$H\left|i_{+}\right\rangle_{j=1/2}=\frac{\omega+\lambda}{\sqrt{2}}\left(-i\left|1,0\right\rangle+\left|0,1\right\rangle\right).$
In the case of $j=1,$
$\displaystyle\left|\tau_{\pm}\right\rangle_{j=1}$
$\displaystyle=\frac{1}{1+\left|\tau_{\pm}\right|^{2}}\sum_{l=0}^{2}\sqrt{\frac{(2)!}{l!(2-l)!}}\tau_{\pm}^{2-l}\left|2-l\right\rangle\otimes\left|l\right\rangle$
$\displaystyle=\frac{1}{1+\left|\tau_{\pm}\right|^{2}}\left(\tau_{\pm}^{2}\left|2\right\rangle\otimes\left|0\right\rangle+\sqrt{2}\tau_{\pm}\left|1\right\rangle\otimes\left|1\right\rangle+\left|0\right\rangle\otimes\left|2\right\rangle\right)$
$\displaystyle\overset{\omega_{1}=\omega_{2}}{\rightarrow}\left|i_{\pm}\right\rangle_{1}$
$\displaystyle=\frac{1}{2}\left(-\left|2\right\rangle\otimes\left|0\right\rangle\mp
i\sqrt{2}\left|1\right\rangle\otimes\left|1\right\rangle+\left|0\right\rangle\otimes\left|2\right\rangle\right).$
(19)
In the case of $j=3/2,$
$\displaystyle\left|\tau_{\pm}\right\rangle_{j=3/2}$
$\displaystyle=\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{3/2}}\left(\tau_{\pm}^{3}\left|3\right\rangle\otimes\left|0\right\rangle+\sqrt{3}\tau_{\pm}^{2}\left|2\right\rangle\otimes\left|1\right\rangle+\sqrt{3}\tau_{\pm}\left|1\right\rangle\otimes\left|2\right\rangle+\left|0\right\rangle\otimes\left|3\right\rangle\right)$
$\displaystyle\overset{\omega_{1}=\omega_{2}}{\rightarrow}\left|i_{\pm}\right\rangle_{j=3/2}$
$\displaystyle=\frac{1}{2^{3/2}}\left(\pm
i\left|3\right\rangle\otimes\left|0\right\rangle-\sqrt{3}\left|2\right\rangle\otimes\left|1\right\rangle\mp
i\sqrt{3}\left|1\right\rangle\otimes\left|2\right\rangle+\left|0\right\rangle\otimes\left|3\right\rangle\right).$
(20)
In the case of $j=2$
$\displaystyle\left|\tau_{\pm}\right\rangle_{j=2}$
$\displaystyle=\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{2}}\sum_{l=0}^{4}\sqrt{\frac{4!}{l!(4-l)!}}\tau_{\pm}^{4-l}\left|4-l\right\rangle\otimes\left|l\right\rangle$
$\displaystyle=\frac{1}{\left(1+\left|\tau\right|^{2}\right)^{2}}\left(\tau_{\pm}^{4}\left|4\right\rangle\otimes\left|0\right\rangle+2\tau_{\pm}^{3}\left|3\right\rangle\otimes\left|1\right\rangle+\sqrt{6}\tau_{\pm}^{2}\left|2\right\rangle\otimes\left|2\right\rangle+2\tau_{\pm}\left|1\right\rangle\otimes\left|3\right\rangle+\left|0\right\rangle\otimes\left|4\right\rangle\right)$
$\displaystyle\overset{\omega_{1}=\omega_{2}}{\rightarrow}\left|i_{\pm}\right\rangle_{j=2}$
$\displaystyle=\frac{1}{4}\left(\left|4\right\rangle\otimes\left|0\right\rangle\pm
2i\left|3\right\rangle\otimes\left|1\right\rangle-\sqrt{6}\left|2\right\rangle\otimes\left|2\right\rangle\mp
2i\left|1\right\rangle\otimes\left|3\right\rangle+\left|0\right\rangle\otimes\left|4\right\rangle\right).$
(21)
Thus we know how the eigenstate of $H$ is composed of the Fock states.
## 5 Partition function and the Internal energy for H
Knowing that $H$ is diagonal in the basis of atomic coherent state
$\left|\tau_{\pm}\right\rangle$, we can directly calculate its partition
function by virtue of its energy level.
$\displaystyle Z_{+}\left(\beta\right)$
$\displaystyle=\mathtt{Tr}_{+}\left(e^{-\beta
H}\right)=\sum_{2j=0}^{\infty}\left.{}_{j}\left\langle\tau_{+}\right|e^{-\beta
H}\left|\tau_{+}\right\rangle_{j}\right.$
$\displaystyle=\sum_{2j=0}^{\infty}e^{-\beta
A2j}=\frac{1}{e^{\eta}-1}|_{\eta=-\beta A}$ $\displaystyle=\frac{1}{e^{-\beta
A}-1},$ (22)
and
$\displaystyle Z_{-}\left(\beta\right)$
$\displaystyle=\mathtt{Tr}_{-}\left(e^{-\beta
H}\right)=\sum_{2j=0}^{\infty}\left.{}_{j}\left\langle\tau_{-}\right|e^{-\beta
H}\left|\tau_{-}\right\rangle_{j}\right.$ $\displaystyle=\frac{1}{e^{-\beta
B}-1}$ (23)
where
$\displaystyle A$
$\displaystyle=\frac{\left(\omega_{1}+\omega_{2}\right)+\sqrt{\left(\omega_{1}-\omega_{2}\right)^{2}+4\lambda^{2}}}{2},$
$\displaystyle B$
$\displaystyle=\frac{\left(\omega_{1}+\omega_{2}\right)-\sqrt{\left(\omega_{1}-\omega_{2}\right)^{2}+4\lambda^{2}}}{2}.$
(24)
satisfying
$H\left|\tau_{+}\right\rangle=2Aj\left|\tau_{+}\right\rangle,H\left|\tau_{-}\right\rangle=2Bj\left|\tau_{-}\right\rangle.$
Thus the total partition function is
$Z\left(\beta\right)=Z_{+}\left(\beta\right)Z_{-}\left(\beta\right)=\left(\frac{1}{e^{-\beta
A}-1}\right)\left(\frac{1}{e^{-\beta B}-1}\right),$ (25)
and the internal energy of system is
$\displaystyle\left\langle H\right\rangle_{e}$
$\displaystyle=-\frac{\partial}{\partial\beta}\ln Z\left(\beta\right)$
$\displaystyle=-\frac{\partial}{\partial\beta}\left[\ln\left(\frac{1}{e^{-\beta
A}-1}\right)+\ln\left(\frac{1}{e^{-\beta B}-1}\right)\right]$
$\displaystyle=\frac{A}{e^{A\beta}-1}+\frac{B}{e^{\beta B}-1}.$ (26)
In summary, similar to the fact that the photon coherent state describes laser
light, the atomic coherent state is useful to classify the energy eigenstates
of the Hamiltonian describing the Raman effect.This may be useful to further
study stimulated Raman scattering since the scattered light behaves as laser
light.
ACKNOWLEDGEMENT: We sincerely thank the referees for their constructive
suggestion. Work supported by the National Natural Science Foundation of China
under grants: 10775097 and 10874174, and the Research Foundation of the
Education Department of Jiangxi Province of China.
## References
* [1] Perelomov A., _Generalized Coherent States and Their Applications_ (Springer, Berlin) 1986.
* [2] Agarwal G. S., Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions. Phys. Rev. A 24 (1981) 2889
* [3] Wang X., Sanders B. C., Pan, S., Entangled coherent states for systems with SU(2) and SU(1,1) symmetries. J. Phys. A: Math. Gen. 33 (2000) 7451
* [4] Arecchi F. T., Courtens E., Gilmore R. and Thomas H., Atomic coherent states in quantum optics. Phys. Rev. A 6 (1972) 2211.
* [5] Radcliffe J. M., Some properties of coherent spin states. J. Phys. A 4 (1971) 313.
* [6] Zhang W. M., Feng D. H. and Gilmore R., Coherent states: Theory and some applications. Rev. Mod. Phys. 62 (1990) 867.
* [7] Narducci L. M., Bowden C. M., Bluemel V., Garrazana G. P., and Tuft R. A., Multitime-correlation functions and the atomic coherent-state representation. Phys. Rev. A 11 (1975) 973.
* [8] Hepp K., Lieb E. H., Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field. Phys. Rev. A 8 (1973) 2517.
* [9] Takahashi Y., Shibata F., Spin Coherent State Representation in Non-Equilibrium Statistical Mechanics. J. Phys. Soc. Jap. 38 (1975) 656.
* [10] Gerry C. C. and Benmoussa A., Spin squeezing via ladder operations on an atomic coherent state. Phys. Rev. A 77 (2008) 062341.
* [11] Schwinger J., _Quantum Theory of Angular momentum_ (Academic Press, New York) 1965.
* [12] FAN H.-Y., Europhys. Lett. 17 (1992) 285; 19 (1992) 443; 23 (1993) 1
* [13] HU L.-Y. and FAN H.-Y., New n-mode squeezing operator and squeezed states with standard squeezing. Europhys. Lett. 85 (2009) 60001.
|
arxiv-papers
| 2009-12-04T00:59:58 |
2024-09-04T02:49:06.843204
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hong-Yi Fan, Xue-xiang Xu and Li-yun Hu",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0912.0763"
}
|
0912.0768
|
# The emission positions of kHz QPOs and Kerr spacetime influence
ZHANG Chengmin1 Corresponding author. zhangcm@bao.ac.cn WEI Yingchun1 YIN
Hongxing1 ZHAO Yongheng1 LEI YaJuan2 SONG Liming2 ZHANG Fan2 YAN Yan3 1\.
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China 2\. Institute of High Energy Physics, Chinese Academy of
Sciences, 19B Yuquan Road, Beijing 100049, China 3.Urumqi Observatory,
National Astronomical Observatories, CAS, Urumqi 830011, China
###### Abstract
Based the Alfven wave oscillation model (AWOM) and relativistic precession
model (RPM) for twin kHz QPOs, we estimate the emission positions of most
detected kHz QPOs to be at $r=18\pm 3km$ $(R/15km)$ except Cir $X-1$ at $r\sim
30\pm 5km(R/15km)$. For the proposed Keplerian frequency as an upper limit to
kHz QPO, the spin effects in Kerr Spacetime are discussed, which have about a
5% (2%) modification for that of the Schwarzchild case for the spin frequency
of 1000 (400) Hz.The application to the four typical QPO sources, Cir $X-1$,
Sco $X-1$, SAX J1808.4-3658 and XTE 1807-294, is mentioned.
###### keywords:
kHz QPO, neutron star, low-mass X-ray binaries
††journal: Phys Mech Astron
,
## 1 Introduction
In thirty more low-mass X-ray binaries (LMXBs), the kiloHertz quasi-periodic
oscillations (kHz QPOs) have been found , where $2/3$ of them show the twin
peak kHz QPOs[1], upper and lower frequencies, in the ranges of $\sim 100$ Hz
- 1300 Hz for the sources with the different spectrum states, e.g. Atoll and
$Z$[2]. The separations of twin kHz QPOs are not constant[1, 3, 4, 5, 6, 7],
which are inconsistent with the beat model[8, 9]. The low frequency QPOs have
also been found, which follow the tight correlations with the kHz QPOs [1, 4].
Some kHz QPO models have been proposed,most of which are ascribed to the
accretion flow[10], and the Alfven wave mode oscillation[11, 12]. To account
for the varied kHz QPO separation, the relativistic precession model (RPM) is
proposed by Stella and Vietri[15], which ascribes the upper frequency to the
Keplerian frequency of orbiting material in an accretion disk and the lower
frequency to the periastron precession of the same matter.
However, for the detected twin kHz QPOs of neutron star (NS) in a LMXB, their
average ratio value is also $3:2$, but varies with the accretion, which may
indicate some distinctions between BHC and NS[1]. In this short letter, we
will investigate the orbital positions of kHz QPO emissions, based on the
Alfven Wave Oscillation Model (AWOM) [13, 14] and RPM[15]. The Kerr spacetime
modification is discussed by considering the spin influence on the Keplerian
frequency.
## 2 AWOM/RPM for kHz QPOs
AWOM ascribes an upper frequency to the Keplerian frequency of orbiting matter
at radius r, and a lower frequency to the Alfven wave oscillation frequency at
the same radius, as described in the following) [13, 14],
$\nu_{2}=\nu_{k}=1850AX^{\frac{3}{2}}$ (1)
with the parameter $X=R/r$ (ratio between star radius R and disk radius $r$)
and $A=(m/{R_{6}^{\ 3}})^{1/2}$ with $R_{6}=R/10^{6}(cm)$ and $m$ the mass $M$
in the units of solar masses. The ratio of twin kHz QPO frequencies can be
obtained as,
${\nu}/{\nu}=(1+(1-x)^{{1/2}})^{\frac{1}{2}}/X^{\frac{4}{5}}$ (2)
which only depends on the position parameter $X=R/r$, and has nothing to do
with the other parameters. The twin kHz QPO separation is obtained as,
$\nu_{2}-\nu_{1}=\nu_{2}[1-(1-(1-x)^{\frac{1}{2}})^{\frac{1}{2}}]^{\ast}X^{\frac{3}{4}}$
(3)
Figure 1: Upper kHz QPO frequency vs. the position function
$(X=R/r,Y=3Rs/r,Rs=2GM/C2)$. The upper (down) solid curve represents AWOM with
the mass density parameters $A=0.7$ $(A=0.45)$, where the maximum frequency is
1850A (Hz); The upper (down) dashed curve represents RPM with the mass
parameters $m=2$ $(m=3)$ solar masses, where the maximum frequency is 2200/m
(Hz). Figure 2: Twin kHz QPO separation vs. the position function . Curve 1
and 2 represent AWOM with mass density parameters $A=0.7$ and $A=0.45$
respectively. Curve 3 (4) represents RPM with mass parameter $m=2$ $(m=3)$,
respectively. Figure 3: Twin kHz QPO ratio vs. the position function (Same
meaning as shown in FIG.1). The ratio 1.5 (1) is the averaged (minimum limit)
value of the detected twin kHz QPOs.
In FIG.1, the upper kHz QPO frequency is plotted against the position
parameter $X=R/r$ ($Y=3Rs/r$, $Rs$ is the Schwarzschild radius) for AWOM
(RPM). For the detected twin kHz QPOs, the mass density parameter A is found
to be about 0.7 (e.g. Sco $X-1$) [13, 14]. In most cases (except Cir $X-1$),
the position parameter $X=R/r$ is lies in the range from 0.7 to 0.92, or
radius from $r=1.1R$ to $r=1.4R$. This implies that the emission positions of
most kHz QPOs are close to the surface of the NS $X=1$ for AWOM (for RPM the
emission positions are close to $3Rs$), which means that the maximum kHz QPO
frequency occurs at the surface (ISCO of star $r=R$ for AWOM (or $r=3Rs$ for
RPM). In FIG.2, the twin kHz QPO separation vs. position parameter is plotted,
where the maximum separation 375 (200) Hz is achieved for $A=0.7$ (0.45) at
$X=0.7$ for AWOM. The kHz QPO data of two accretion powered millisecond X-ray
pulsars (AMXPs), Sax J 1808.4-3658 and XTE 1807-294, approximately hint at the
condition of $A=0.45$, which presents relatively low kHz QPO separations. For
RPM, the maximum kHz QPO separations are 360 Hz (210 Hz) for the different
choices of mass parameter $m=2(3)$ solar masses, which occurs at $Y=0.76$. For
the two AMXPs, Sax J 1808.4-3658 and XTE 1807-294, RPM has to assume their
star masses are close to the NS mass upper limit, 3 solar masses, if
consistent theoretical curves with the detected data can be fitted. FIG.3 is
the diagram of twin kHz QPO ratios vs. position parameter. It can be noticed
that the averaged ratio 1.5 of the detected kHz QPOs corresponds to the
position $X=0.83$ for AWOM ($Y=0.89$ for RPM). The ratios of all sources but
Cir $X-1$ lie in the regimes between $ratio=1$ and $ratio=2$. The kHz QPO data
of Cir $X-1$ implies that its kHz QPO emitting positions are far away from the
star, i.e. $0.4<X<0.6$ or $2.5R>r>1.6R$, centered at about $2R$.
## 3 Kerr spacetime effect on the kHz QPO
If the influence of Kerr spacetime on the Keplerian frequency is taken into
account, then the orbital frequency of a spinning point mass $M$ with angular
momentum $J$ is expressed as below[1]
$\nu_{2}=\nu\nu_{k}\xi;\quad\nu_{k}=(GM/4\Pi r^{3})^{\frac{1}{2}}$ (4)
with the Kerr modification parameter
$\xi=1+jR_{g}^{\ \frac{2}{3}};\quad R_{g}=R_{s}/2$ (5)
$j=Jc/GM^{2};\quad J=2\pi I\nu_{s}$ (6)
where $I$ is the moment of inertia, with the maximum value for the homogeneous
sphere $I=(2/5)MR^{2}$. In the Schwarzschild geometry, $j=0$, Eq.4 recovers
the conventional Keplerian frequency; $0<j<1$ represents a prograde orbit. To
put the NS mass $(m=M/M_{\odot}$, radius and spin fre-quency parameters, we
have the following simplified expressions,
$j=4\Pi\nu_{2}R^{2}/R_{g}C=(0.22/m)R_{6}^{\ 2}(\nu_{s}/400HZ)$ (7)
$\xi=1+(0.0013m)R_{6}(\nu_{s}/400hz)$ (8)
If we set the conventional values $M=1.4M_{\odot}$, $R=15km$ and $s=400Hz$,
then the Kerr modification parameter has about a 2% contribution to the
Keplerian frequency, which cannot have too much influence on the kHz QPO model
based on the Keplerian frequency. For the maximum spin fre-quency $1122Hz$,
the Kerr modification contributes about 5% to the Schwarzschild spacetime, so
this influence should be considered when we estimate the NS parameters.
## 4 Discussions and results
The kHz QPO emission positions are analyzed by the models (AWOM and RPM),
which shows that most kHz QPOs (e.g. Sco $X-1$) come from the regimes of
several kilometers away from the stellar surface. This may correspond to the
condition of a spinning up NS, since the detected NS spin frequencies are
averaging $400Hz$[16], less than the upper frequencies. In RPM, the star mass
can be derived by the detected twin kHz QPOs, then it usually gives a value of
2 solar masses, higher than the typical NS mass of 1.4 solar masses. One
reason for RPM’s prediction of high NS mass may be originating from its
assumption of the vacuum circumstance around the star in introducing the
perihelion precession term[15], but the accretion disk does not satisfy this
clean condition. A value of about 3 solar masses for SAX J1808.4-3658 [17]
(e.g. XTE 1807-294) is obtained, which seems to suggest that RPM should be
modified. AWOM cannot predict a stellar mass by QPO but rather an averaged
mass density $(A\sim M^{1/2}/R^{3/2)}$, by which one can evaluate the equation
of state (EOS) of the star. For the presently known kHz QPO frequencies, AWOM
cannot give the prediction of quark matter[18] inside the star unless the QPO
frequency over 1500 Hz is detected. In addition, the Kerr spacetime influence
is investigated, and a 5% modification factor in Keplerian frequency exists
for a high spin frequency of 1000 Hz, which will increase the estimation of
the mass density parameter. Though the spectral properties of Cir $X-1$ are
typical of those of Z sources[19, 20], its detected 11 pairs of kHz QPOs are
generally low frequencies, 230 Hz to 500 Hz for the upper QPO and 56 Hz to 225
Hz for the lower QPO, increasing with accretion rate, which is contrary to
those of the other LMXBs. The peak separation lies at 175-340 Hz, similar to
those of other LMXBs [6]. Since the kHz QPO emitting positions of Cir $X-1$
are estimated to be beyond the orbit of 25 kilometers, we guess that its
rotating frequency is low, e.g. a hundred Hz.
This work is supported by National Basic Research Program of China-973 Program
2009CB824800; National Natural Science Foundation of China (10773017).
## References
* [1] van der klis M. Rapid X-ray variability, in Lewin W H G, van der Klis M, eds, Compact Stellar X-ray Sources. Cambridge Univ. Press, Cambridge, 39-112.
* [2] Hasinger G, & van der Klis M. Two patterns of correlated X-ray timing and spectral behavior in low-mass X-ray binaries. Astron Astrophys, 1989, 225: 79 96.
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|
arxiv-papers
| 2009-12-04T03:56:20 |
2024-09-04T02:49:06.848227
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.M. Zhang, Y.C. Wei, H.X. Yin, Y.H. Zhao, Y.J. Lei, L.M. Song, F.\n Zhang, Y.Yan",
"submitter": "Chengmin Zhang",
"url": "https://arxiv.org/abs/0912.0768"
}
|
0912.0812
|
The $n$-tangle of odd $n$ qubits111The paper was supported by NSFC (Grant No.
10875061) and Tsinghua National Laboratory for Information Science and
Technology.
Dafa Li222email address:dli@math.tsinghua.edu.cn
Dept of mathematical sciences
Tsinghua University, Beijing 100084 CHINA
###### Abstract
Coffman, Kundu and Wootters presented the 3-tangle of three qubits in [Phys.
Rev. A 61, 052306 (2000)]. Wong and Christensen extended the 3-tangle to even
number of qubits, known as $n$-tangle [Phys. Rev. A 63, 044301 (2001)]. In
this paper, we propose a generalization of the 3-tangle to any odd $n$-qubit
pure states and call it the $n$-tangle of odd $n$ qubits. We show that the
$n$-tangle of odd $n$ qubits is invariant under permutations of the qubits,
and is an entanglement monotone. The $n$-tangle of odd $n$ qubits can be
considered as a natural entanglement measure of any odd $n$-qubit pure states.
Keywords: 3-tangle, $n$-tangle of odd $n$ qubits, concurrence, residual
entanglement
PACS numbers: 03.67.Mn, 03.65.Ud
## 1 Introduction
Quantum entanglement is a key quantum mechanical resource in quantum
computation and information, such as quantum cryptography, quantum dense
coding and quantum teleportation [1]. Entanglement measure, which
characterizes the degree of entanglement contained in a quantum state, has
been a subject under intensive research.
The entanglement of bipartite systems is well understood. The concurrence [2]
is a good entanglement measure for two-qubit states and is an entanglement
monotone, i.e., it is non-increasing under local quantum operations and
classical communication (LOCC). Generalizations of the concurrence to higher
dimensions can be found, for example, in [3, 4]. The residual entanglement, or
the 3-tangle has been constructed in terms of the concurrences as a widely
accepted entanglement measure to quantify the entanglement in three-qubit pure
states [5]. The 3-tangle is permutationally invariant, is an entanglement
monotone, and is a SLOCC (stochastic local operations and classical
communication) polynomial of degree 4. Furthermore, the 3-tangle is bounded
between 0 and 1, and it assumes value 1 for the GHZ state and vanishes for the
W state [5, 6]. Several other measures have been constructed specifically for
the entanglement of the three-qubit pure states [7, 8, 9]. The partial tangle,
reported in [7], represents the residual two-qubit entanglement of a three-
qubit pure state and reduces to the two-qubit concurrence for the W state. The
$\sigma$-measure [8] and $\pi$-tangle [9] have been introduced as entanglement
monotones for genuine three-qubit entanglement. Whereas the 3-tangle vanishes
for the W state, both $\sigma$-measure and $\pi$-tangle take non-zero values
for the W state as well as the GHZ state. Many other entanglement measures for
quantifying the entanglement of multipartite pure states have been proposed
[10, 11, 12, 13, 14] (see also the review [1] and references therein).
Hyperdeterminant, as a generalization of the concurrence and the 3-tangle, has
been shown to be an entanglement monotone and describes the genuine
multipartite entanglement [10]. The $n$-tangle is a straightforward extension
of 3-tangle to even number of qubits [11]. As has been previously noted, the
$n$-tangle is the square of generalization of the concurrence, is invariant
under permutations, and is an entanglement monotone. Like the 3-tangle, the
$n$-tangle is equal to 1 for the GHZ state and vanishes for the W state [11].
However the $n$-tangle is not residual entanglement for four or more qubits
[15]. It has been found that the 4-tangle for four-qubit states can be
interpreted as a type of residual entanglement similar to the interpretation
of 3-tangle for three-qubit states as the residual tangle [16]. An alternative
4-tangle has recently been obtained by using negativity fonts and the 4-tangle
is a genuine entanglement measure of four-qubit pure states [12]. In [13], the
residual entanglement of odd $n$ qubits has been proposed as an entanglement
measure for odd $n$-qubit pure states and shown to be an entanglement monotone
[14]. The odd $n$-tangle (although called odd $n$-tangle, it is not defined in
the same way as has been done for the $n$-tangle by directly extending the
definition of 3-tangle to even $n$ qubits) has been defined by taking the
average of the residual entanglement with respect to qubit $i$, which is
obtained from the residual entanglement of odd $n$ qubits under transposition
on qubits 1 and $i$ [14]. It has been shown that the odd $n$-tangle is
permutationally invariant, $SL$-invariant and $LU$-invariant, and is an
entanglement monotone [14].
In this paper, we give an alternative formulation of the 3-tangle. We extend
the formulation in a straightforward way to any odd $n$-qubit pure states and
define the $n$-tangle with respect to qubit $i$. By taking the average of the
$n$-tangle with respect to qubit $i$, we define the $n$-tangle of odd $n$
qubits, which is invariant under permutations of the qubits. The extended
formulation is then reduced by using simple mathematics. It turns out that the
$n$-tangle with respect to qubit $i$ and the $n$-tangle of odd $n$ qubits are
equal to the residual entanglement with respect to qubit $i$ and the odd
$n$-tangle respectively, and consequently the former inherit the properties of
the latter, like the monotonicity, invariance under $SL$ and $LU$ operations
as well as the property of satisfying SLOCC equation. Moreover, the $n$-tangle
with respect to qubit $i$ is a SLOCC polynomial of degree 4. Like the
3-tangle, the $n$-tangle of odd $n$ qubits takes value 1 for the GHZ state and
vanishes for the W state. Finally we extend the $n$-tangle of odd $n$ qubits
to mixed states via the convex roof construction. This work will extend our
understanding of multipartite entanglement.
The rest of the paper is organized as follows. In Section 2 we briefly review
the definitions and the formulations of the concurrence, the 3-tangle and the
$n$-tangle. We then give an alternative formulation of the 3-tangle and extend
it to odd $n$ qubits. We also introduce the definitions of the $n$-tangle with
respect to qubit $i$ and the $n$-tangle of odd $n$ qubits. In Section 3, we
study the $n$-tangle with respect to qubit $i$ and the $n$-tangle of odd $n$
qubits in more detail and we discuss their properties. Finally, we draw our
conclusion in Section 4.
## 2 The $n$-tangle of odd $n$ qubits
### 2.1 Preliminaries
The concurrence for two-qubit pure states is defined as
$C(\psi)=\left|\langle\psi\right|\tilde{\psi}\rangle|^{2}$ [2], where
$|\tilde{\psi}\rangle$ denotes the resulting state after applying the operator
$\sigma_{y}\otimes\sigma_{y}$ to the complex conjugate of $|\psi\rangle$ [2],
i.e. $|\tilde{\psi}\rangle=\sigma_{y}\otimes\sigma_{y}|\psi^{\ast}\rangle$.
Here the asterisk indicates complex conjugatation in the standard basis. For
three-qubit pure states, the 3-tangle $\tau_{ABC}$ (or $\tau_{123}$) can be
calculated by means of concurrences and is given by
$\tau_{ABC}=C_{A(BC)}^{2}-C_{AB}^{2}-C_{AC}^{2}$ [5], where $C_{AB}$ and
$C_{AC}$ are the concurrences of the corresponding two-qubit subsytems
$\rho_{AB}$ and $\rho_{AC}$, respectively, and $C_{A(BC)}^{2}=4\det\rho_{A}$.
Here $\rho_{AB}$, $\rho_{AC}$ and $\rho_{A}$ are the reduced density matrices.
Let $|\psi\rangle=\sum_{i=0}^{7}a_{i}|i\rangle$, where
$\sum_{i=0}^{7}|a_{i}|=1$. An expression of the 3-tangle in terms of the
coefficients for the state $|\psi\rangle$ is given by [5]
$\tau_{123}=4\bigl{|}d_{1}-2d_{2}+4d_{3}\bigr{|},$ (2.1)
where
$\displaystyle d_{1}$
$\displaystyle=a_{0}^{2}a_{7}^{2}+a_{1}^{2}a_{6}^{2}+a_{2}^{2}a_{5}^{2}+a_{3}^{2}a_{4}^{2},$
(2.2) $\displaystyle d_{2}$
$\displaystyle=a_{0}a_{7}a_{3}a_{4}+a_{0}a_{7}a_{2}a_{5}+a_{0}a_{7}a_{1}a_{6}+a_{3}a_{4}a_{2}a_{5}+a_{3}a_{4}a_{1}a_{6}+a_{2}a_{5}a_{1}a_{6},$
(2.3) $\displaystyle d_{3}$
$\displaystyle=a_{0}a_{6}a_{5}a_{3}+a_{7}a_{1}a_{2}a_{4}.$ (2.4)
A more standard form of the 3-tangle is given as follows [5]:
$\tau_{123}=2\Bigl{|}\sum
a_{\alpha_{1}\alpha_{2}\alpha_{3}}a_{\beta_{1}\beta_{2}\beta_{3}}a_{\gamma_{1}\gamma_{2}\gamma_{3}}a_{\delta_{1}\delta_{2}\delta_{3}}\times\epsilon_{\alpha_{1}\beta_{1}}\epsilon_{\alpha_{2}\beta_{2}}\epsilon_{\gamma_{1}\delta_{1}}\epsilon_{\gamma_{2}\delta_{2}}\epsilon_{\alpha_{3}\gamma_{3}}\epsilon_{\beta_{3}\delta_{3}}\Bigr{|},$
(2.5)
where the sum is over all the indices, $\alpha_{l}$, $\beta_{l}$,
$\gamma_{l}$, and $\delta_{l}$ $\in\\{0,1\\}$,
$\epsilon_{00}=\epsilon_{11}=0$, and $\epsilon_{01}=-\epsilon_{10}=1$. The
above formulation of the 3-tangle is invariant under permutations of the
qubits.
Let $|\psi\rangle$ be any state of $n$ qubits and
$|\psi\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$, where
$\sum_{i=0}^{2^{n}-1}|a_{i}|=1$. The $n$-tangle is defined for the state
$|\psi\rangle$ as follows [11]:
$\displaystyle\tau_{12\cdots n}$ $\displaystyle=2\Bigl{|}\sum
a_{\alpha_{1}\cdots\alpha_{n}}a_{\beta_{1}\cdots\beta_{n}}a_{\gamma_{1}\cdots\gamma_{n}}a_{\delta_{1}\cdots\delta_{n}}$
$\displaystyle\quad\times\epsilon_{\alpha_{1}\beta_{1}}\epsilon_{\alpha_{2}\beta_{2}}\cdots\epsilon_{\alpha_{n-1}\beta_{n-1}}\epsilon_{\gamma_{1}\delta_{1}}\epsilon_{\gamma_{2}\delta_{2}}\cdots\epsilon_{\gamma_{n-1}\delta_{n-1}}\epsilon_{\alpha_{n}\gamma_{n}}\epsilon_{\beta_{n}\delta_{n}}\Bigr{|},$
(2.6)
for all even $n$ and $n=3$. However the above formula is not invariant under
permutations of qubits for odd $n>3$, and therefore, the $n$-tangle remains
undefined for odd $n>3$ [11].
### 2.2 Alternative formulation of the 3-tangle
Here we let
$\displaystyle\tau_{123}^{(1)}$ $\displaystyle=2\Bigl{|}\sum
a_{\alpha_{1}\alpha_{2}\alpha_{3}}a_{\beta_{1}\beta_{2}\beta_{3}}a_{\gamma_{1}\gamma_{2}\gamma_{3}}a_{\delta_{1}\delta_{2}\delta_{3}}\times\epsilon_{\alpha_{2}\beta_{2}}\epsilon_{\alpha_{3}\beta_{3}}\epsilon_{\gamma_{2}\delta_{2}}\epsilon_{\gamma_{3}\delta_{3}}\epsilon_{\alpha_{1}\gamma_{1}}\epsilon_{\beta_{1}\delta_{1}}\Bigr{|},$
(2.7) $\displaystyle\tau_{123}^{(2)}$ $\displaystyle=2\Bigl{|}\sum
a_{\alpha_{1}\alpha_{2}\alpha_{3}}a_{\beta_{1}\beta_{2}\beta_{3}}a_{\gamma_{1}\gamma_{2}\gamma_{3}}a_{\delta_{1}\delta_{2}\delta_{3}}\times\epsilon_{\alpha_{1}\beta_{1}}\epsilon_{\alpha_{3}\beta_{3}}\epsilon_{\gamma_{1}\delta_{1}}\epsilon_{\gamma_{3}\delta_{3}}\epsilon_{\alpha_{2}\gamma_{2}}\epsilon_{\beta_{2}\delta_{2}}\Bigr{|},$
(2.8) $\displaystyle\tau_{123}^{(3)}$ $\displaystyle=2\Bigl{|}\sum
a_{\alpha_{1}\alpha_{2}\alpha_{3}}a_{\beta_{1}\beta_{2}\beta_{3}}a_{\gamma_{1}\gamma_{2}\gamma_{3}}a_{\delta_{1}\delta_{2}\delta_{3}}\times\epsilon_{\alpha_{1}\beta_{1}}\epsilon_{\alpha_{2}\beta_{2}}\epsilon_{\gamma_{1}\delta_{1}}\epsilon_{\gamma_{2}\delta_{2}}\epsilon_{\alpha_{3}\gamma_{3}}\epsilon_{\beta_{3}\delta_{3}}\Bigr{|}.$
(2.9)
Inspection of Eqs. (2.5) and (2.9) reveals that $\tau_{123}=\tau_{123}^{(3)}$.
Indeed, a direct calculation gives
$\tau_{123}^{(1)}=\tau_{123}^{(2)}=\tau_{123}^{(3)}$. Now, let us look at the
formulas from a different perspective. We note that $\tau_{123}^{(2)}$ can be
obtained from $\tau_{123}^{(1)}$ by taking the transposition $(1,2)$ on qubits
$1$ and $2$. Analogously, $\tau_{123}^{(3)}$ can be obtained from
$\tau_{123}^{(1)}$ by taking the transposition $(1,3)$ on qubits $1$ and $3$.
It turns out that we can also obtain
$\tau_{123}^{(1)}=\tau_{123}^{(2)}=\tau_{123}^{(3)}$ by using the fact that
the 3-tangle $\tau_{123}$ is invariant under permutations of the three qubits
[5]. We may thus rewrite the 3-tangle as follows:
$\tau_{123}=(\tau_{123}^{(1)}+\tau_{123}^{(2)}+\tau_{123}^{(3)})/3.$ (2.10)
### 2.3 The $n$-tangle with respect to qubit $i$ and the $n$-tangle of odd
$n$ qubits
We extend Eqs. (2.7)-(2.9) to any odd $n$ qubits. Let
$\displaystyle\tau_{12\cdots n}^{(i)}$ $\displaystyle=2\bigl{|}W_{12\cdots
n}^{(i)}\bigr{|},$ (2.11) $\displaystyle W_{12\cdots n}^{(i)}$
$\displaystyle=\sum
a_{\alpha_{1}\cdots\alpha_{n}}a_{\beta_{1}\cdots\beta_{n}}a_{\gamma_{1}\cdots\gamma_{n}}a_{\delta_{1}\cdots\delta_{n}}\times\epsilon_{\alpha_{i}\gamma_{i}}\epsilon_{\beta_{i}\delta_{i}}$
$\displaystyle\quad\times\epsilon_{\alpha_{1}\beta_{1}}\cdots\epsilon_{\alpha_{i-1}\beta_{i-1}}\epsilon_{\alpha_{i+1}\beta_{i+1}}\cdots\epsilon_{\alpha_{n}\beta_{n}}$
$\displaystyle\quad\times\epsilon_{\gamma_{1}\delta_{1}}\cdots\epsilon_{\gamma_{i-1}\delta_{i-1}}\epsilon_{\gamma_{i+1}\delta_{i+1}}\cdots\epsilon_{\gamma_{n}\delta_{n}},$
(2.12)
where the sum is over all the indices and $i=1$, $\cdots$, $n$. One can verify
that $\tau_{12\cdots n}^{(i)}$ with $n\geq 5$ is invariant under any
permutation of all but qubit $i$. So, we call $\tau_{12\cdots n}^{(i)}$ the
$n$-tangle with respect to qubit $i$. One can show that $\tau_{12\cdots
n}^{(1)}$ turns into $\tau_{12\cdots n}^{(i)}$ under the transposition $(1,i)$
on qubits 1 and $i$, $i=2,3,\cdots,n$.
In analogy to Eq. (2.10), we define the $n$-tangle of odd $n$ qubits as
follows:
$\tau_{12\cdots n}=\frac{1}{n}\sum_{i=1}^{n}\tau_{12\cdots n}^{(i)}.$ (2.13)
It is not hard to see that $\tau_{12\cdots n}$ is invariant under all the
permutations of the qubits, and the values of $\tau_{12\cdots n}^{(i)}$ and
$\tau_{12\cdots n}$ are bounded between $0$ and $1$. Note also that when
$n=3$, $\tau_{12\cdots n}^{(i)}$ and $\tau_{12\cdots n}$ become $\tau_{123}$.
### 2.4 Reduction of the formulation
We observe that it takes $3\cdot 2^{4n}$ multiplications to compute
$\tau_{12\cdots n}^{(i)}$ by Eqs. (2.11) and (2.12). Next we reduce the
formulation of $\tau_{12\cdots n}^{(1)}$. From Eq. (2.12), we have
$\displaystyle W_{12\cdots n}^{(1)}$ $\displaystyle=\sum
a_{\alpha_{1}\cdots\alpha_{n}}a_{\beta_{1}\cdots\beta_{n}}a_{\gamma_{1}\cdots\gamma_{n}}a_{\delta_{1}\cdots\delta_{n}}\times\epsilon_{\alpha_{1}\gamma_{1}}\epsilon_{\beta_{1}\delta_{1}}$
$\displaystyle\quad\times\epsilon_{\alpha_{2}\beta_{2}}\cdots\epsilon_{\alpha_{n}\beta_{n}}\epsilon_{\gamma_{2}\delta_{2}}\cdots\epsilon_{\gamma_{n}\delta_{n}}.$
(2.14)
After some calculations, we obtain (we refer the reader to Appendix A for
details)
$\displaystyle W_{12\cdots n}^{(1)}$ $\displaystyle=2(PQ-T^{2})\text{,}$
(2.15) $\displaystyle\tau_{12\cdots n}^{(1)}$
$\displaystyle=4\bigl{|}T^{2}-PQ\bigr{|},$ (2.16)
where
$\displaystyle T$
$\displaystyle=\sum_{i=0}^{2^{n-1}-1}(-1)^{N(i)}a_{i}a_{2^{n}-i-1},$ (2.17)
$\displaystyle P$
$\displaystyle=2\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2i}a_{2^{n-1}-2i-1},$
(2.18) $\displaystyle Q$
$\displaystyle=2\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}+2i}a_{2^{n}-2i-1}.$
(2.19)
Here $N(l)$ is the number of 1s in the $n$-bit binary representation
$l_{n-1}...l_{1}l_{0}$ of $l$. We further note that it takes $(2^{n}+3)$
multiplications to compute $\tau_{12\cdots n}^{(1)}$ using Eqs. (2.16)-(2.19).
A plain calculation yields that $\tau_{12\cdots n}^{(1)}=1$ for the $n$-qubit
state $GHZ$ and $\tau_{12\cdots n}^{(1)}=0$ for the $n$-qubit state $W$.
## 3 The $n$-tangle of odd $n$ qubits is an entanglement monotone
Let $|\psi^{\prime}\rangle$ be also any state of $n$ qubits and
$|\psi^{\prime}\rangle=\sum_{i=0}^{2^{n}-1}b_{i}|i\rangle$, where
$\sum_{i=0}^{2^{n}-1}|b_{i}|^{2}=1$. Two states $|\psi\rangle$ and
$|\psi^{\prime}\rangle$ are SLOCC entanglement equivalent if and only if there
exist invertible local operators $\mathcal{\alpha},\mathcal{\beta},\cdots$
such that [6]
$|\psi^{\prime}\rangle=\underbrace{\mathcal{\alpha}\otimes\mathcal{\beta}\otimes\cdots}_{n}|\psi\rangle.$
(3.1)
The residual entanglement of odd $n$ qubits for the state $|\psi\rangle$ is
defined as follows [13]:
$\tau(\psi)=4\bigl{|}(\overline{\mathcal{I}}(a,n))^{2}-4\mathcal{I}^{\ast}(a,n-1)\mathcal{I}_{+2^{n-1}}^{\ast}(a,n-1)\bigr{|},$
(3.2)
where (see [13, 14])
$\displaystyle\overline{\mathcal{I}}(a,n)$
$\displaystyle=\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}\Bigl{[}\bigl{(}a_{2i}a_{(2^{n}-1)-2i}-a_{2i+1}a_{(2^{n}-2)-2i}\bigr{)}$
$\displaystyle\quad-\bigl{(}a_{(2^{n-1}-2)-2i}a_{(2^{n-1}+1)+2i}-a_{(2^{n-1}-1)-2i}a_{2^{n-1}+2i}\bigr{)}\Bigr{]},$
(3.3)
and (see [13, 14])
$\displaystyle\mathcal{I}^{\ast}(a,n-1)$
$\displaystyle=\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}\bigl{(}a_{2i}a_{(2^{n-1}-1)-2i}-a_{2i+1}a_{(2^{n-1}-2)-2i}\bigr{)},$
(3.4) $\displaystyle\mathcal{I}_{+2^{n-1}}^{\ast}(a,n-1)$
$\displaystyle=\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}\bigl{(}a_{2^{n-1}+2i}a_{(2^{n}-1)-2i}-a_{2^{n-1}+1+2i}a_{(2^{n}-2)-2i}\bigr{)}.$
(3.5)
It has been also proven that if states $|\psi\rangle$ and
$|\psi^{\prime}\rangle$ are SLOCC equivalent, then the following SLOCC
equation holds [13]:
$\tau(\psi^{\prime})=\tau(\psi)\underbrace{\bigl{|}\det(\alpha)\det(\beta)\det(\gamma)\cdots\bigr{|}^{2}}_{n}.$
(3.6)
We now argue that $\tau_{12\cdots n}^{(1)}=\tau(\psi)$. This can be seen as
follows. A simple calculation shows that $\overline{\mathcal{I}}(a,n)=T$ (see
(i) in Appendix A). Inspection of Eqs. (2.18) and (A21) (the reduced form of
Eq. (3.4)) reveals that $\mathcal{I}^{\ast}(a,n-1)=P/2$. Furthermore,
inspection of Eqs. (2.19) and (A24) (the reduced form of Eq. (3.5)) reveals
that $\mathcal{I}_{+2^{n-1}}^{\ast}(a,n-1)=Q/2$. Substituting these results
into Eq. (3.2) yields
$\tau(\psi)=4\bigl{|}T^{2}-PQ\bigr{|}.$ (3.7)
Therefore,
$\tau_{12\cdots n}^{(1)}=\tau(\psi).$ (3.8)
Next we recall that the residual entanglement with respect to qubit $i$ is
defined as (see [14]) $\tau^{(i)}(\psi)$, which is obtained from $\tau(\psi)$
under the transposition $(1,i)$ on qubits 1 and $i$. The odd $n$-tangle is
defined by taking the average of the residual entanglement with respect to
qubit $i$ [14]:
$R(\psi)=\frac{1}{n}\sum_{i=1}^{n}\tau^{(i)}(\psi).$ (3.9)
Note that $R(\psi)$ is considered as an entanglement measure for odd $n$
qubits [14].
It follows immediately from Eq. (3.8) and the definitions of $\tau_{12\cdots
n}^{(i)}$ and $\tau^{(i)}(\psi)$ that
$\tau_{12\cdots n}^{(i)}=\tau^{(i)}(\psi),\quad i=1,2,\cdots,n.$ (3.10)
Further, Eq. (2.13), together with Eqs. (3.9) and (3.10), yields
$\tau_{12\cdots n}=R(\psi).$ (3.11)
A direct consequence of Eqs. (3.10) and (3.11) is that the $n$-tangle with
respect to qubit $i$ and the $n$-tangle of odd $n$ qubits inherit the
properties of the residual entanglement with respect to qubit $i$ and the odd
$n$-tangle. We highlight that the $n$-tangle with respect to qubit $i$ and the
$n$-tangle of odd $n$ qubits are $SL$-invariant and $LU$-invariant, and are
entanglement monotones (see [14] for details).
Clearly, both $\tau_{12\cdots n}^{(i)}$ and $\tau_{12\cdots n}$ satisfy Eq.
(3.6). The $n$-tangle with respect to qubit $i$ is called a SLOCC polynomial
of degree 4 of odd $n$ qubits. It should be noted that there are no polynomial
invariants of degree 2 for odd $n$ qubits [18]. In view of the SLOCC equation
(3.6), it is easy to see that if one of $\tau_{12\cdots
n}^{(i)}(\psi^{\prime})$ (resp. $\tau_{12\cdots n}(\psi^{\prime})$) and
$\tau_{12\cdots n}^{(i)}(\psi)$ (resp. $\tau_{12\cdots n}(\psi)$) vanishes
while the other does not, then $|\psi\rangle$ and $|\psi^{\prime}\rangle$
belong to different SLOCC classes. This reveals that the $n$-tangle with
respect to qubit $i$ and the $n$-tangle of odd $n$ qubits can be used for
SLOCC classification.
We exemplify the results for the GHZ state and the W state. In our previous
work [19] it has been shown that $\tau(GHZ)=1$ and $\tau(W)=0$ for any
$n$-qubit GHZ and W states. The above analysis directly gives rise to the
conclusion that the $n$-tangle of odd $n$ qubits $\tau_{12\cdots n}$ is equal
to 1 for the GHZ state and 0 for the W state.
Finally, we extend the $n$-tangle of odd $n$ qubits to mixed states via the
convex roof construction (see, e.g., the review [1]):
$\tau_{12\cdots n}(\rho)=\min\sum_{i}p_{i}\tau_{12\cdots n}(\psi_{i}),$ (3.12)
where $p_{i}\geq 0$ and $\sum_{i}p_{i}=1$, and the minimum is taken over all
possible decompositions of $\rho$ into pure states, i.e.
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$,
## 4 Conclusion
In summary, we have proposed the $n$-tangle of odd $n$ qubits, which is a
generalization of the standard form of the 3-tangle to any odd $n$-qubit pure
states. We have argued that the $n$-tangle of odd $n$ qubits is invariant
under permutations of the qubits, is an entanglement monotone. The $n$-tangle
of odd $n$ qubits takes value 1 for the GHZ state and vanishes for the W
state. The $n$-tangle of odd $n$ qubits is considered as a natural
entanglement measure of any odd $n$-qubit pure states. Finally, we have
extended the $n$-tangle of odd $n$ qubits to mixed states via the convex roof
construction. Our results will provide more insight into the nature of
multipartite entanglement.
As is well known, two SLOCC inequivalent classes of three-qubit pure states,
namely the GHZ class and the W class, can be distinguished via the 3-tangle
[6, 17]. Polynomial invariants of degree 2 have been recently exploited for
SLOCC classification of four-qubit pure states [20, 21] and of the symmetric
Dicke states with $l$ excitations of $n$ qubits [19]. More recently, four
polynomial invariants of degree $2^{n/2}$ of any even $n$ qubits have been
presented and several different genuine entangled states inequivalent to the
GHZ, the W, or the symmetric Dicke states with $l$ excitations under SLOCC
have been obtained by using the polynomials [22]. Further attempts have been
made to build connections between polynomial (algebraic) invariants and SLOCC
classification [23, 24]. We expect the $n$-tangle of odd $n$ qubits proposed
in this paper can be used for SLOCC classification of any odd $n$ qubits.
## Appendix A
We first give proofs of Eqs. (2.15)-(2.16).
Let $\bar{\alpha}_{i}$ be the complement of $\alpha_{i}$. That is,
$\bar{\alpha}_{i}=0$ when $\alpha_{i}=1$. Otherwise, $\bar{\alpha}_{i}=1$. In
view of that $\epsilon_{00}=\epsilon_{11}=0$ and
$\epsilon_{01}=-\epsilon_{10}=1$, to compute $W_{12\cdots n}^{(1)}$ in Eq.
(2.14), we only need to consider $\beta_{i}=\bar{\alpha}_{i}$,
$\delta_{i}=\bar{\gamma}_{i}$, $i=2,\cdots,n$,
$\gamma_{1}={\bar{\alpha}}_{1}$, and $\delta_{1}=\bar{\beta}_{1}$. Thus, Eq.
(2.14) becomes
$W_{12\cdots n}^{(1)}=\sum
a_{\alpha_{1}\alpha_{2}\cdots\alpha_{n}}a_{\beta_{1}{\bar{\alpha}}_{2}\cdots\bar{\alpha}_{n}}a_{\bar{\alpha}_{1}\gamma_{2}\cdots\gamma_{n}}a_{\bar{\beta}_{1}\bar{\gamma}_{2}\cdots\bar{\gamma}_{n}}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}}\epsilon_{\gamma_{2}\bar{\gamma}_{2}}\cdots\epsilon_{\gamma_{n}\bar{\gamma}_{n}}\epsilon_{\alpha_{1}\bar{\alpha}_{1}}\epsilon_{\beta_{1}\bar{\beta}_{1}}.$
(A1)
We distinguish two cases.
Case 1. $\beta_{1}=\alpha_{1}$.
In this case,
$\epsilon_{\alpha_{1}\bar{\alpha}_{1}}\epsilon_{\beta_{1}\bar{\beta}_{1}}=1$.
Thus, from Eq. (A1), we have
$W_{12\cdots n}^{(1)}=\sum
a_{\alpha_{1}\alpha_{2}\cdots\alpha_{n}}a_{\alpha_{1}\bar{\alpha}_{2}\cdots\bar{\alpha}_{n}}a_{\bar{\alpha}_{1}\gamma_{2}\cdots\gamma_{n}}a_{\bar{\alpha}_{1}\bar{\gamma}_{2}\cdots\bar{\gamma}_{n}}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}}\epsilon_{\gamma_{2}\bar{\gamma}_{2}}\cdots\epsilon_{\gamma_{n}\bar{\gamma}_{n}}.$
(A2)
Letting
$\displaystyle P$
$\displaystyle=\sum_{\alpha_{2}\cdots\alpha_{n}}a_{0\alpha_{2}\cdots\alpha_{n}}a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n}}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}},$
(A3) $\displaystyle Q$
$\displaystyle=\sum_{\alpha_{2}\cdots\alpha_{n}}a_{1\alpha_{2}\cdots\alpha_{n}}a_{1\bar{\alpha}_{2}\cdots\bar{\alpha}_{n}}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}},$
(A4)
yields
$W_{12\cdots n}^{(1)}=2PQ.$ (A5)
Case 2. $\beta_{1}=\bar{\alpha}_{1}$.
In this case,
$\epsilon_{\alpha_{1}\bar{\alpha}_{1}}\epsilon_{\beta_{1}\bar{\beta}_{1}}=-1$.
Thus, from Eq. (A1), we have
$W_{12\cdots n}^{(1)}=-\sum
a_{\alpha_{1}\alpha_{2}\cdots\alpha_{n}}a_{\bar{\alpha}_{1}\bar{\alpha}_{2}\cdots\bar{\alpha}_{n}}a_{\bar{\alpha}_{1}\gamma_{2}\cdots\gamma_{n}}a_{\alpha_{1}\bar{\gamma}_{2}\cdots\bar{\gamma}_{n}}\mathcal{\times}\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}}\epsilon_{\gamma_{2}\bar{\gamma}_{2}}\cdots\epsilon_{\gamma_{n}\bar{\gamma}_{n}}.$
(A6)
Let
$\displaystyle T$ $\displaystyle=\sum
a_{0\alpha_{2}\cdots\alpha_{n}}a_{1\bar{\alpha}_{2}\cdots\bar{\alpha}_{n}}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}},$
(A7) $\displaystyle S$ $\displaystyle=\sum
a_{1\alpha_{2}\cdots\alpha_{n}}a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n}}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}}.$
(A8)
From that $\epsilon_{01}=-\epsilon_{10}=1$,
$\epsilon_{\alpha_{i}\bar{\alpha}_{i}}=-\epsilon_{\bar{\alpha}_{i}\alpha_{i}}$,
and therefore
$S=\sum
a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n}}a_{1\alpha_{2}\cdots\alpha_{n}}\times\epsilon_{\bar{\alpha}_{2}\alpha_{2}}\cdots\epsilon_{\bar{\alpha}_{n}\alpha_{n}}=T.$
(A9)
Hence
$W_{12\cdots n}^{(1)}=-2T^{2}.$ (A10)
Eq. (A10), together with Eq. (A5), yields
$W_{12\cdots n}^{(1)}=2(PQ-T^{2})\text{.}$ (A11)
Inserting Eq. (A11) into Eq. (2.11) leads to
$\tau_{12\cdots n}^{(1)}=4\bigl{|}T^{2}-PQ\bigr{|}.$ (A12)
Next, let $\alpha_{2}\cdots\alpha_{n}\ $be the binary representation of $i$.
Noting that
$(-1)^{N(i)}=\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n}\bar{\alpha}_{n}}$,
we may rewrite $T$ as
$T=\sum_{i=0}^{2^{n-1}-1}(-1)^{N(i)}a_{i}a_{2^{n}-i-1}.$ (A13)
(i). Proof of $T=\overline{\mathcal{I}}(a,n)$
Expanding Eq. (A7), we obtain
$\displaystyle T$ $\displaystyle=\sum
a_{0\alpha_{2}\cdots\alpha_{n-1}0}a_{1\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle\quad-\sum
a_{0\alpha_{2}\cdots\alpha_{n-1}1}a_{1\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}0}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle=\sum
a_{00\alpha_{3}\cdots\alpha_{n-1}0}a_{11\bar{\alpha}_{3}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{3}\bar{\alpha}_{3}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle\quad-\sum
a_{01\alpha_{3}\cdots\alpha_{n-1}0}a_{10\bar{\alpha}_{3}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{3}\bar{\alpha}_{3}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle\quad-\sum
a_{00\alpha_{3}\cdots\alpha_{n-1}1}a_{11\bar{\alpha}_{3}\cdots\bar{\alpha}_{n-1}0}\times\epsilon_{\alpha_{3}\bar{\alpha}_{3}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle\quad+\sum
a_{01\alpha_{3}\cdots\alpha_{n-1}1}a_{10\bar{\alpha}_{3}\cdots\bar{\alpha}_{n-1}0}\times\epsilon_{\alpha_{3}\bar{\alpha}_{3}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle=\overline{\mathcal{I}}(a,n),$ (A14)
where the third equality follows by letting $\alpha_{3}\cdots\alpha_{n-1}$ be
the binary number of $i$ and noting that$\
(-1)^{N(i)}=(-1)^{N(\alpha_{3}\cdots\alpha_{n-1})}=\epsilon_{\alpha_{3}\bar{\alpha}_{3}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$.
(ii). Reduction of $P$
Expanding Eq. (A3), we obtain
$\displaystyle P$ $\displaystyle=\sum
a_{0\alpha_{2}\cdots\alpha_{n-1}0}a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle\quad-\sum
a_{0\alpha_{2}\cdots\alpha_{n-1}1}a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}0}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle=2\sum
a_{0\alpha_{2}\cdots\alpha_{n-1}0}a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle=2\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2i}a_{2^{n-1}-2i-1},$
(A15)
where the second equality follows from
$\displaystyle\sum
a_{0\alpha_{2}\cdots\alpha_{n-1}1}a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}0}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle=-\sum
a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}0}a_{0\alpha_{2}\cdots\alpha_{n-1}1}\times\epsilon_{\bar{\alpha}_{2}\alpha_{2}}\cdots\epsilon_{\bar{\alpha}_{n-1}\alpha_{n-1}}$
(A16) $\displaystyle=-\sum
a_{0\alpha_{2}\cdots\alpha_{n-1}0}a_{0\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}},$
(A17)
and the third equality follows by letting $\alpha_{2}\cdots\alpha_{n-1}$ be
the binary number of $i$ and noting that
$(-1)^{N(i)}=\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$.
(iii). Reduction of $Q$
Eq. (A4) gives, by analogy with Eq. (A15),
$\displaystyle Q$ $\displaystyle=\sum
a_{1\alpha_{2}\cdots\alpha_{n-1}0}a_{1\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle\quad-\sum
a_{1\alpha_{2}\cdots\alpha_{n-1}1}a_{1\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}0}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle=2\sum
a_{1\alpha_{2}\cdots\alpha_{n-1}0}a_{1\bar{\alpha}_{2}\cdots\bar{\alpha}_{n-1}1}\times\epsilon_{\alpha_{2}\bar{\alpha}_{2}}\cdots\epsilon_{\alpha_{n-1}\bar{\alpha}_{n-1}}$
$\displaystyle=2\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}+2i}a_{2^{n}-2i-1}.$
(A18)
(iv). Reduction of $\mathcal{I}^{\ast}(a,n-1)$
By Eq. (3.4), we have
$\mathcal{I}^{\ast}(a,n-1)=\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}a_{2i}a_{(2^{n-1}-1)-2i}-\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}a_{2i+1}a_{(2^{n-1}-2)-2i}.$
(A19)
Let $k=2^{n-2}-1-i$. Then $N(k)+N(i)=n-2$, and hence
$(-1)^{N(i)}=-(-1)^{N(k)}$, and
$\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}a_{2i+1}a_{(2^{n-1}-2)-2i}=-\sum_{k=2^{n-3}}^{2^{n-2}-1}(-1)^{N(k)}a_{2k}a_{(2^{n-1}-1)-2k}.$
(A20)
This leads to
$\mathcal{I}^{\ast}(a,n-1)=\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2i}a_{(2^{n-1}-1)-2i}.$
(A21)
(v). Reduction of $\mathcal{I}_{+2^{n-1}}^{\ast}(a,n-1)$
By Eq. (3.5), we have
$\mathcal{I}_{+2^{n-1}}^{\ast}(a,n-1)=\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}a_{2^{n-1}+2i}a_{(2^{n}-1)-2i}-\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}a_{2^{n-1}+1+2i}a_{(2^{n}-2)-2i}.$
(A22)
Letting $k=2^{n-2}-1-i$, we have
$\sum_{i=0}^{2^{n-3}-1}(-1)^{N(i)}a_{2^{n-1}+1+2i}a_{(2^{n}-2)-2i}=-\sum_{k=2^{n-3}}^{2^{n-2}-1}(-1)^{N(k)}a_{2^{n-1}+2k}a_{(2^{n}-1)-2k}.$
(A23)
This leads to
$\mathcal{I}_{+2^{n-1}}^{\ast}(a,n-1)=\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}+2i}a_{(2^{n}-1)-2i}.$
(A24)
## References
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* [13] D. Li, X. Li, H. Huang, and X. Li, Phys. Rev. A 76, 032304 (2007) [arXiv:quant-ph/0704.2087].
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|
arxiv-papers
| 2009-12-04T10:35:55 |
2024-09-04T02:49:06.852738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Li",
"submitter": "Dafa Li",
"url": "https://arxiv.org/abs/0912.0812"
}
|
0912.0851
|
# Electrooptics of graphene: field-modulated reflection and birefringence
M.V. Strikha F.T. Vasko ftvasko@yahoo.com Institute of Semiconductor
Physics, NAS of Ukraine, Pr. Nauky 41, Kyiv, 03028, Ukraine
###### Abstract
The elecrooptical response of graphene due to heating and drift of carriers is
studied theoretically. Real and imaginary parts of the dynamic conductivity
tensor are calculated for the case of effective momentum relaxation, when
anisotropic contributions are small. We use the quasiequilibrium distribution
of electrons and holes, characterized by the effective temperature of carriers
and by concentrations, which are controlled by gate voltage and in-plane
electric field. The geometry of normal propagation of probe radiation is
considered, spectral and field dependences of the reflection coefficient and
the relative absorption are analyzed. The ellipticity degree of the reflected
and transmitted radiation due to small birefringence of graphene sheet with
current have also been determined.
###### pacs:
78.20.Jq, 78.67.Wj, 81.05.ue
## I Introduction
Study of electrooptical response both of bulk semiconductors and of
heterostructures (see Refs. in 1 and 2 , respectively) is a convenient method
to characterize these materials. Such a response is used to modulate both the
intensity of radiation and its polarization. As it was demonstrated more than
30 years ago 3 ; 4 (see also Sect. 17 in 5 ), the main contribution to the
elecrooptical response of narrow gap semiconductors is caused by the
modulation of the interband transitions, both virtual and real one, under
heating and drift of nonequilibrium carriers. The electrooptical properties of
two-dimensional carriers in heterostructures have also been studied 6 . Since
graphene is a gapless semiconductor with linear energy spectrum 7 , the direct
interband transitions in graphene are allowed with the characteristic
interband velocity $v_{W}=10^{8}$ cm/s, which corresponds the Weyl-Wallace
model 8 , degenerated over spin and valleys. Therefore, optical properties of
graphene should be modulated essentially by temperature and carriers
concentration 9 and these dependences were studied recently. 10 The applied
electric field not only changes carriers temperature and concentration, but
also causes the anisotropy of distribution due to carriers drift 11 ; 12 .
Therefore, the birefringence effect can be essential for radiation propagating
across a graphene sheet with current. To the best of our knowledge, no
measurement of electrooptical response was performed until recently, and a
theoretical study of these phenomena is timely now.
The results obtained below are based on the tensor of dynamic conductivity,
determined by interband transitions of non-equilibrium carriers. This tensor
is determined by Kubo formula in collisionless approximation 4 ; 5 with the
use of weakly anisotropic distributions of electrons and holes calculated in
11 ; 13 . The case of normal propagation of the incident ($in$), reflected
($r$), and transmitted ($t$) waves of probe radiation (see Fig. 1) is studied,
and the reflection and transition coefficients, controlled by carriers heating
conditions, are obtained. It is demonstrated, that the heating level
dependence on applied field, temperature of phonons, and sheet charge,
controlled by gate voltage $V_{g}$, can be verified from electrooptical
measurements. Moreover, graphene is to be considered due to carriers drift as
an uniaxial plane, and the elliptically polarized $r$\- and $t$-waves appear
under linear polarization of $in$-radiation, if $\theta\neq 0$ or $\pi/2$, see
Fig.1. Due to an effective relaxation of carriers momenta the distribution
anisotropy and the induced birefringence are small, but a high sensitivity of
polarization measurements enables one to determine drift characteristics of
nonequilibrium carriers using a field-induced Kerr effect.
Figure 1: Schematic view on incident ($in$), reflected ($r$), and transmitted
($t$) radiation for the case of normal propagation through the graphene sheet
with applied electric field $\bf E$. Angle $\theta$ defines the polarization
direction of $in$-wave while $r$-, and $t$-contributions are elliptically
polarized.
The consideration below is organised in the following way. In Sec.II we
present both the basic equations for the complex tensor of dynamic
conductivity, and the electrodynamics equations for the uniaxial graphene
sheet on a substrate. Numerical results, describing the electromodulation
spectra and Kerr effect, are discussed in Sec.III. The concluding remarks and
the list of assumptions are presented in the last Section. In Appendix, the
dynamic conductivity of the undoped graphene is considered.
## II Basic equations
The description of graphene response on the probe in-plane electric field
${\bf E}_{\omega}\exp(-i\omega t)$ is based on the consideration of the high-
frequency dynamic conductivity and on the examination of the electrodynamics
problem for propagation of such a field through graphene sheet. When
performing these calculations, we take into account a modification of
interband transitions due to carriers heating, and an anisotropy of response
due to carriers drift.
### II.1 Anisotropic dynamic conductivity
Contribution of the interband transitions of non-equilibrioum carriers with
the distribution function $f_{l{\bf p}}$, into the response at frequency
$\omega$ is described by the dynamic conductivity tensor
$\sigma_{\alpha\beta}(\omega)$ given by Kubo formula:
$\displaystyle\sigma_{\alpha\beta}(\omega)=i\frac{4(ev_{W})^{2}}{\omega
L^{2}}\sum_{ll^{\prime}{\bf p}}\left(f_{l{\bf p}}-f_{l^{\prime}{\bf
p}}\right)$ (1) $\displaystyle\times\frac{\left\langle l{\bf
p}\left|\hat{\sigma}_{\alpha}\right|l^{\prime}{\bf p}\right\rangle\left\langle
l^{\prime}{\bf p}\left|\hat{\sigma}_{\beta}\right|l{\bf
p}\right\rangle}{\varepsilon_{lp}-\varepsilon_{l^{\prime}p}+\hbar\omega+i\lambda}.$
Here $\left|l{\bf p}\right\rangle$ and $\varepsilon_{lp}$ are the state and
energy of $l$th band ($c$\- or $v$-) with the $2D$ momentum $\bf p$, and
$\lambda\rightarrow+0$. We also use the velocity operator
$v_{W}\hat{\mbox{\boldmath$\sigma$}}$ and the normalizing area $L^{2}$. The
expression (1) is written in a collisionless approximation
$\omega\overline{\tau}>>1$ ($\overline{\tau}$ is the relaxation time), when
intraband transitions are inefficient. In this case the density matrix,
averaged over scattering, should be used in Kubo formula, and
$\sigma_{\alpha\beta}(\omega)$ appears to be written through the stationary
distribution function $f_{l{\bf p}}$. 5 Due to effective momentum relaxation
the anisotropy of non-equilibrium electrons and holes distributions is weak
and the expansion
$f_{l{\bf p}}=f_{lp}+\Delta f_{lp}^{(1)}\cos\varphi+\Delta f_{lp}^{(2)}\cos
2\varphi+\ldots$ (2)
should be substituted into Eq. (1). Here $\varphi$ angle determines
orientation of $\bf p$ and $\Delta f_{lp}^{(r)}\propto E^{r}$, where ${\bf
E}\|OX$ is a dc field applied. The linear in $E$ contribution can be omitted
from $\sigma_{\alpha\beta}(\omega)$ in the case, when the small spatial
dispersion, responsible for the radiation drag by current (see Ref. 14), is
neglected. Thus, with an accuracy of the contributions of $\propto E^{2}$
order, tensor (1) is determined by the transition matrix elements:
$\displaystyle\overline{\left|\left\langle 1{\bf
p}\left|\hat{\sigma}_{x,y}\right|-1{\bf p}\right\rangle\right|^{2}}=1/2,$
$\displaystyle\overline{\cos 2\varphi\left|{\left\langle{1{\bf
p}\left|\hat{\sigma}_{x}\right|-1{\bf p}}\right\rangle}\right|^{2}}=-1/4,$ (3)
$\displaystyle\overline{\cos 2\varphi\left|{\left\langle{1{\bf
p}\left|\hat{\sigma}_{y}\right|-1{\bf p}}\right\rangle}\right|^{2}}=1/4,$
where overline means the averaging over angle.
Since the non-diagonal components of tensor (1) vanish, the $XX$\- and
$YY$-components of dynamic conductivity:
$\sigma_{xx}(\omega)=\sigma_{\omega}-\frac{\Delta\sigma_{\omega}}{2},~{}~{}~{}~{}~{}\sigma_{yy}(\omega)=\sigma_{\omega}+\frac{\Delta\sigma_{\omega}}{2}$
(4)
describe the response of graphene sheet with the field-induced uniaxial
anisotropy. Further, we substitute Eqs. (2) and (3) into (1), and we use the
electron-hole representation, when $f_{c{\bf p}}\rightarrow f_{e{\bf p}}$, and
$f_{v{\bf p}}\rightarrow 1-f_{h,-{\bf p}}$, see 11 . It is convenient to
separate the contributions of the undoped graphene and of the free carriers
(electrons and holes) into the isotropic part of conductivity,
$\overline{\sigma}_{\omega}$ and $\sigma_{\omega}^{(c)}$, so that
$\sigma_{\omega}=\overline{\sigma}_{\omega}+\sigma_{\omega}^{(c)}$. The first
contribution is discussed in the Appendix. Separating the real and imaginary
parts of $\overline{\sigma}_{\omega}$, and using the energy conservation law,
one obtains the frequency-independent result ${\rm
Re}\overline{\sigma}=e^{2}/4\hbar$. The imaginary contribution into
$\overline{\sigma}_{\omega}$ is given by the phenomenological expression
(A.2), which is written through the fitting parameters corresponding to the
recent measurements. 15
The electron-hole contributions to the isotropic part,
$\sigma_{\omega}^{(c)}$, and the anisotropic addition,
$\Delta\sigma_{\omega}$, are written as follows:
$\displaystyle{\rm Re}\left|\begin{array}[]{*{20}c}\sigma_{\omega}^{(c)}\\\
\Delta\sigma_{\omega}\end{array}\right|=-\frac{2\pi(ev_{W})^{2}}{\omega
L^{2}}\sum\limits_{\bf p}\delta(\hbar\omega-2v_{W}p)$ (7)
$\displaystyle\times\left|\begin{array}[]{*{20}c}f_{ep}+f_{hp}\\\ \Delta
f_{ep}^{(2)}+\Delta f_{hp}^{(2)}\end{array}\right|,~{}~{}~{}~{}~{}$ (10)
$\displaystyle{\rm Im}\left|\begin{array}[]{*{20}c}\sigma_{\omega}^{(c)}\\\
\Delta\sigma_{\omega}\end{array}\right|=-\frac{2(ev_{W})^{2}}{\omega
L^{2}}\sum\limits_{\bf p}\frac{\cal P}{\hbar\omega-2v_{W}p}$ (13)
$\displaystyle\times\frac{4v_{W}p}{\hbar\omega+2v_{W}p}\left|\begin{array}[]{*{20}c}f_{ep}+f_{hp}\\\
\Delta f_{ep}^{(2)}+\Delta f_{hp}^{(2)}\end{array}\right|.$ (16)
The real parts of conductivity given by Eqs. (5) are expressed directly
through isotropic distribution (2) at the characteristic momentum for
interband transitions, $p_{\omega}\equiv\hbar\omega/2v_{W}$, according to:
${\rm Re}\left|\begin{array}[]{*{20}c}\sigma_{\omega}^{(c)}\\\
\Delta\sigma_{\omega}\\\
\end{array}\right|=-\frac{e^{2}}{4\hbar}\left|\begin{array}[]{*{20}c}f_{ep_{\omega}}+f_{hp_{\omega}}\\\
\Delta f_{ep_{\omega}}^{(2)}+\Delta f_{hp_{\omega}}^{(2)}\\\
\end{array}\right|.$ (17)
The imaginary parts of $\sigma_{\omega}^{(c)}$, and $\Delta\sigma_{\omega}$,
given by Eq. (6) are transformed into:
$\displaystyle{\rm Im}\left|\begin{array}[]{*{20}c}\sigma_{\omega}^{(c)}\\\
\Delta\sigma_{\omega}\end{array}\right|=-\frac{e^{2}}{2\pi\hbar
p_{\omega}}\int\limits_{0}^{\infty}\frac{dpp^{2}}{p_{\omega}+p}\frac{\cal
P}{p_{\omega}-p}$ (20)
$\displaystyle\times\left|\begin{array}[]{*{20}c}f_{ep}+f_{hp}\\\ \Delta
f_{ep}^{(2)}+\Delta f_{hp}^{(2)}\end{array}\right|$ (23)
and the principal value integrals here should be calculated numerically.
Below, we restrict ourselves to the case of quasielastic distribution of non-
equilibrium electrons and holes ($k=e,h$) with effective temperature $T_{c}$
and chemical potential $\mu_{k}$:
$f_{kp}=\\{\exp[(v_{W}p-\mu_{k})/T_{c}]+1\\}^{-1}.$ (24)
The dependences of distribution (9) on electric field $E$, temperature $T$,
and gate voltage $V_{g}$ are presented in 11 ; 13 . For the anisotropic
addition $\Delta f_{k{\bf p}}^{(2)}$ in the case of momentum relaxation
through elastic scattering we use:
$\Delta
f_{kp}^{(2)}=-\frac{(eE)^{2}p}{2\nu_{p}^{(2)}}\frac{d}{dp}\left[\frac{1}{p\nu_{p}^{(1)}}\left(-\frac{df_{kp}}{dp}\right)\right].$
(25)
For the case of short-range scattering on static defects the relaxation rates
$\nu_{p}^{(1,2)}$ are proportional to the density of states, so that
$\nu_{p}^{(1)}=v_{d}p/\hbar+\nu_{0}$, and
$\nu_{p}^{(2)}=2\nu_{p}^{(1)}+\nu_{0}$. Here $v_{d}$ is a characteristic
velocity, that determines an efficiency of momentum scattering, 16 and
$\nu_{0}$ is a residual rate, which describes the scattering process for low-
energy carriers.
### II.2 Electrodynamics
For normal propagation of probe radiation, the Fourier component of the field,
${\bf E}_{\omega z}$, is governed by the wave equation:
$\frac{d^{2}{\bf E}_{\omega
z}}{dz^{2}}+\epsilon_{z}\left(\frac{\omega}{c}\right)^{2}{\bf E}_{\omega
z}+i\frac{4\pi\omega}{c^{2}}{\bf j}_{\omega z}=0,$ (26)
where $\epsilon_{z}$ is dielectric permittivity. In this paper we examine the
case of graphene on the thick SiO2 substrate, when $\epsilon_{z<0}=1$, and
$\epsilon_{z>0}=\epsilon$. The induced current density in (11) is localized
around $z=0$ plane, so that ${\bf j}_{\omega z}\approx\hat{\sigma}_{\omega
z}{\bf E}_{\omega z=0}$, while $\int\limits_{-0}^{+0}dz\hat{\sigma}_{\omega
z}=\hat{\sigma}_{\omega}$ is determined through the dynamic conductivity
tensor, being examined above. Outside the graphene sheet the solution of (11)
can be written as:
${\bf E}_{\omega z}=\left\\{{\begin{array}[]{*{20}c}{{\bf
E}^{(in)}e^{ik_{\omega}z}+{\bf E}^{(r)}e^{-ik_{\omega}z},}&{z<0}\\\ {{\bf
E}^{(t)}e^{i\overline{k}_{\omega}z}}&{z>0}\\\ \end{array}}\right..$ (27)
Here the wave vectors $k_{\omega}=\omega/c$ and
$\overline{k}_{\omega}=\sqrt{\epsilon}\omega/c$, and the amplitudes of
incident, reflected, and transmitted waves (${\bf E}^{(in)}$, ${\bf E}^{(r)}$,
and ${\bf E}^{(t)}$ respectively) were introduced. This amplitudes are
governed by the boundary condition,
$\left.\frac{d{\bf E}_{\omega
z}}{dz}\right|_{-0}^{+0}+ik_{\omega}\frac{4\pi}{c}\hat{\sigma}_{\omega}{\bf
E}_{\omega z=0}=0,$ (28)
which is obtained after integration of Eq.(11) over $z$ through the graphene
layer $(-0<z<+0)$. The second boundary condition is the requirement of
continuity: ${\bf E}_{\omega z=-0}={\bf E}_{\omega z=+0}$.
Taking into account the diagonality of $\hat{\sigma}_{\omega}$ tensor, we get
the solutions from the boundary conditions as follows:
$E_{\alpha}^{(t)}=\frac{2}{1+A_{\alpha}(\omega)}E_{\alpha}^{(in)},~{}~{}~{}~{}E_{\alpha}^{(r)}=\frac{1-A_{\alpha}(\omega)}{1+A_{\alpha}(\omega)}E_{\alpha}^{(in)},$
(29)
where factor
$A_{\alpha}(\omega)=\sqrt{\epsilon}+4\pi\sigma_{\alpha\alpha}(\omega)/c$ was
introduced. After substitution of Eqs. (12) and (14) into the general
expression for Poynting vector ${\bf S}=c^{2}{\rm Re}\left[{\bf E}\times{\rm
rot}{\bf E}^{*}\right]/8\pi$, we obtain the incident, reflected, and
transmitted fluxes $S_{in}$, $S_{r}$, and $S_{t}$ respectively, which are
parallel to $OZ$. After multiplication of Eq. (13) by ${\bf E}_{t}^{*}$, we
get the relation between these fluxes as follows:
$S_{in}=S_{r}+S_{t}+\frac{1}{2}{\rm Re}\left({\bf
E}_{t}^{*}\cdot\hat{\sigma}_{\omega}\cdot{\bf E}_{t}\right),$ (30)
where the last term describes absorption.
The polarization characteristics of $r$-, and $t$-waves are determined by
solutions (14). It is convenient to present them in complex form
$E_{\alpha}={\cal E}_{\alpha}\exp(i\psi_{\alpha})$, where ${\cal E}_{\alpha}$,
and $\psi_{\alpha}$ give the amplitude and phase of $\alpha$-component of the
field respectively. At $\theta=0$, or at $\theta=\pi/2$, when the response is
described by $\sigma_{xx}$, or by $\sigma_{yy}$, the linearly modulated $r$-,
and $t$-waves occur. For other $\theta$, the reflected and transmitted waves
are elliptically polarized. The ellipticity degree $\varepsilon(\omega)$ is
determined by the phases difference between $X$\- and $Y$-components of the
field, $\Delta\psi=\psi_{x}-\psi_{y}$, see the general expressions in Ref. 17.
Under weak anisotropy, with the accuracy up to first order in
$\Delta\sigma_{\omega}$, we get $\varepsilon(\omega)=\Delta\psi/2$.
Figure 2: Spectral dependences of $\sigma_{\omega}^{(c)}$ (a) and
$\Delta\sigma_{\omega}$ (b) for intrinsic graphene with $f_{p=0}=$0.5, 0.3,
and 0.1. Solid and dashed curves correspond real and imaginary parts of
conductivity, respectively.
## III Results
Now we examine the spectral and polarization characteristics of the
electrooptical response. We study the reflection, transmission, and relative
absorption coefficients, determined as $R_{\omega\theta}=S_{r}/S_{in}$,
$T_{\omega\theta}=S_{t}/S_{in}$, and $\xi_{\omega}={\rm Re}\left({\bf
E}_{t}^{*}\cdot\hat{\sigma}_{\omega}\cdot{\bf E}_{t}\right)/2S_{in}$,
respectively, 18 as well as the ellipticity degree, $\varepsilon(\omega)$,
for the case of weak anisotropy. The final expressions for the coefficients
under consideration are obtained with the use of complex conductivities
$\sigma_{\omega}^{(c)}$, and $\Delta\sigma_{\omega}$, given by Eqs.(7), and
(8), and they depend both on $\hbar\omega/T_{c}$, and on carriers
concentration. In Fig.2. we plot these dependences and one can see that the
response modify essentially with temperature and concentration. The smallness
of anisotropic additions is determined by dimensionless factor
$F=\left(\frac{eE\hbar v_{W}^{2}}{2T_{c}^{2}v_{d}}\right)^{2},$ (31)
which arises from $\propto E^{2}$ contribution to the distribution function
(10). Note also, that ${\rm Im}\Delta\sigma_{\omega}$ depends weakly on the
cutting parameter $(\hbar\nu_{0}v_{W})/(T_{c}v_{d})$, taken below as 0.1.
### III.1 Reflection and absorption
For the examination of $R_{\omega\theta}$, and $\xi_{\omega\theta}$ it is
convenient to separate the isotropic and $\theta$-dependent contributions, so
that
$R_{\omega\theta}=R_{\omega}+\Delta R_{\omega}\cos
2\theta,~{}~{}~{}~{}\xi_{\omega\theta}=\xi_{\omega}+\Delta\xi_{\omega}\cos
2\theta,$ (32)
where the small (of the order of $\Delta\sigma_{\omega}/\sigma_{\omega}$)
anisotropic additions, proportional to $\cos 2\theta$, have been separated,
see Fig. 1. The coefficients in Eq. (17) are written below through
$\sigma_{\omega}$, $\Delta\sigma_{\omega}$ and the factor
$A_{\omega}=\sqrt{\epsilon}+4\pi\sigma_{\omega}/c$. For the isotropic parts of
reflection, and relative absorption coefficiens we get 9 :
$R_{\omega}\simeq\left|\frac{1-A_{\omega}}{1+A_{\omega}}\right|^{2},~{}~{}~{}~{}\xi_{\omega}\simeq\frac{16\pi}{\sqrt{\epsilon}c}\frac{{\rm
Re}\sigma_{\omega}}{|1+A_{\omega}|^{2}},$ (33)
so that these characteristics depend on $T$, $E$, and $V_{g}$.
Figure 3: Spectral dependencies of relative absorption (a), reflection (b) and
differential reflectivity (c) for intrinsic graphene at 77 K and at different
electric fields, $E$ (marked). Solid and dashed curves in (b) are plotted at
$E=$0 and 30 V/cm, respectively, for $\varepsilon_{m}=$60 meV (1), 80 meV (2),
and 100 meV (3).
Spectral dependences of the relative absorption, $\xi_{\omega}/\xi_{max}$, the
reflection, $R_{\omega}$, and the differential reflectivity $(\delta
R/R)_{\omega}\equiv(R_{\omega}-R_{\omega}^{(eq)})/R_{\omega}^{(eq)}$, are
plotted in Fig.3 for intrinsic graphene at $T=$ 77 K and different electric
fields (the data for $T_{c}$ and carriers concentration were used from Ref.
11). Here $\xi_{max}$ is the maximum value of relative absorption for high
frequences, when the free carriers contribution is unessential. One can see,
that due to the increase of average energy of carriers with the increase of
$E$ the absorption increases at high frequencies and decreases for the low
ones. The relative change of $\xi_{\omega}$ is reasonably large, and for
$\hbar\omega\sim T_{c}$ it can be measured directly. At the same time, the
reflection coefficient depends on field in more weak way, see Figs. 3b, c, and
$(\delta R/R)_{\omega}$ can be 10-2 in THz spectral region; in near-IR
spectral region it decreases down to value $\leq 10^{-3}$. Note, that for
$\hbar\omega\leq$0.1 eV $R_{\omega}$ increases essentially (at high
frequencies $R_{\omega}\sim$0.075) due to the contribution of the first
summand of Eq. (A2). Fig. 3b presents the dependence of $R_{\omega}$ on
phenomenological parameter $\varepsilon_{m}$; $\xi_{\omega}$ depends weakly on
this parameter.
Figure 4: The same as in Fig. 3 for doped graphene at room temperature and
different $V_{g}$ (marked). Solid and dashed curves correspond $E=$0 and 30
V/cm, respectively.
The dependences of $\xi_{\omega}$, $R_{\omega}$, and $(\delta R/R)_{\omega}$
on doping level are presented in Fig.4. The data for the room temperature are
presented for $V_{g}=$3 V and 10 V, which correspond the difference between
electron and hole concentrations $1.65\times 10^{11}$ cm-2 and $5.5\times
10^{11}$ cm-2, respectively. Similarly to field dependences at $T=$77 K (see
Fig.3), with the increase of $V_{g}$ (the doping level $\propto V_{g}$) the
response moves towards the high energy region. The dependences on the level of
heating (the applied field $E$), and on carriers concentration (the gate
voltage $V_{g}$) correspond the measurements of spectra for different
temperatures and $V_{g}$, see 10 . For the range of parameters under
examination the field modulation of $\xi_{\omega}$ is of 20$\div$50 % order up
to mid-IR spectral region. These modifications should be observed rather
easily. The carriers contribution into reflection increases as well: at
$\hbar\omega>$0.1 eV the decrease of $R_{\omega}$ occurs, which almost does
not depend on $\varepsilon_{m}$; in this case the shape of the differential
reflectivity is similar to low temperature case, with the shift into the high
energy region.
Figure 5: Spectral dependences of anisotropic contributions to relative
absorption, $\Delta\xi_{\omega}/F$ (a), and to reflectivity, $\Delta
R_{\omega}/(R_{\omega}F)$, (b) for intrinsic graphene at $T=$300 K and at
different electric fields, $E$.
Later we shall examine the anisotropic contributions in Eq.(17), which are
proportional to $\Delta\sigma_{\omega}$. Such a contribution into reflection
coefficient is given by:
$\Delta
R_{\omega}=\frac{R_{\omega}}{c}\left(\frac{4\pi\Delta\sigma_{\omega}}{1-A_{\omega}^{2}}+c.c.\right)$
(34)
and the addition to relative absorption takes form:
$\displaystyle\Delta\xi_{\omega}=\frac{16\pi}{\sqrt{\epsilon}c}\frac{{\rm
Re}\sigma_{\omega}}{|1+A_{\omega}|^{2}}\left(\frac{2\pi\Delta\sigma_{\omega}/c}{1+A_{\omega}}+c.c.\right)$
$\displaystyle-\frac{16\pi}{\sqrt{\epsilon}c}\frac{{\rm
Re}\Delta\sigma_{\omega}}{|1+A_{\omega}|^{2}}.$ (35)
Spectral dependences for anisotropic contributions to the relative absorption
and reflectivity, $\Delta\xi_{\omega}/F$ and $\Delta
R_{\omega}/(R_{\omega}F)$, are shown in Figs. 5 (a) and (b). One can see, that
in the range of fields under examination the parameter $F$ given by Eq. (16)
does not exceed 0.05, so that $\Delta\xi_{\omega}$ and $\Delta
R_{\omega}/R_{\omega}$ are of $10^{-4}$ order for the mid-IR spectral region
($\sim 0.1\div$0.2 eV) and the response increases up to $\sim 10^{-2}$ in THz
spectral region. The anisotropy of such order of value can be analyzed with
the use of the modulation methods.
### III.2 Kerr effect
Besides the cases of parallel or transverse orientation of the probe radiation
polarization with respect to the drift direction (i. e. at $\theta\neq
0,~{}\pi/2$), the reflected and transmitted fields are elliptically polarized.
The maximal Kerr effect occurs if the $i$-wave is polarized along
$\theta=\pi/4$, and below we consider this case only. In the approximation of
the weakly anisotropic distribution (2) the ellipse orientation does not
differ essentially from $\theta\simeq\pi/4$, and the ellipticity degree can be
written as 17 :
$\varepsilon(\omega)=\sin\beta{\rm Re}\Phi(\omega)-\cos\beta{\rm
Im}\Phi(\omega).$ (36)
Here the $\beta$ angle, and the complex function $\Phi(\omega)$ can be
expressed through the difference of the phases of $r$-, and $t$-waves (see the
end of Sec.II). The smallness of the ellipticity is determined by the relation
$\Phi(\omega)\propto F$, while the $\beta$ angle is not small.
For the reflected wave the $\Phi(\omega)$ function is given by the expression
$\Phi_{r}(\omega)=\frac{4\pi\Delta\sigma_{\omega}}{c\left(1+A_{\omega}\right)^{2}}\left|\frac{1+A_{\omega}}{1-A_{\omega}}\right|,$
(37)
while the $\beta$ angle is introduced through the relation:
$\tan\beta_{r}=-\frac{2{\rm Im}A_{\omega}}{1-|A_{\omega}|^{2}}.$ (38)
Similarly, for the transmitted wave, (20) is expressed through the function:
$\Phi_{t}(\omega)=\frac{2\pi\Delta\sigma_{\omega}}{c(1+A_{\omega})^{2}}|1+A_{\omega}|$
(39)
and the $\beta$ angle is given by the expression:
$\tan\beta_{t}=-\frac{{\rm Im}A_{\omega}}{1+{\rm Re}A_{\omega}}.$ (40)
Substitution of these expressions into Eq. (21) gives the ellipticity degrees
for $r$\- and $t$-waves, $\varepsilon_{r}(\omega)$ and
$\varepsilon_{t}(\omega)$.
Figure 6: Spectral dependences of ellipticity degrees of $r$\- and $t$-waves
[panels (a) and (b), respectively] for intrinsic graphene at $T=$300 K and
different electric fields, $E$.
Spectral and field dependences of $\varepsilon_{r}(\omega)$ and
$\varepsilon_{t}(\omega)$ are shown in Figs. 6a and 6b for intrinsic graphene
at $T=$300 K. In mid-IR spectral region $\varepsilon_{r,t}(\omega)$ decreases
with $\omega$ and for $\hbar\omega\sim 0.1\div$0.2 eV the value of ellipticity
degree does not exceed $\sim 10^{-4}$ at $F\sim$0.05. In THz spectral region
$\varepsilon_{r,t}(\omega)$ increases up to $\sim 5\cdot 10^{-3}$, wherein the
direction of rotation for the reflected wave changes at $\hbar\omega\sim$25
meV. Such value of ellipticity degree can be detected by modulation methods
only. However, in stronger fields, when the distribution function is strongly
anisotropic, 12 the ellipticity degree can increase essentially.
## IV Conclusions
Summarizing the consideration performed, we have examined the graphene
electooptical response due to the interband electron transitions under the
carriers heating and drift. It was found, that an essential modulation of the
reflection, and the relative absorption take place starting from the field
strength $\sim$30 V/c at liquid nitrogen and room temperatures (with the
increase of field, the modulation should increase essentially). Due to
current-induced birefringence of graphene sheet the weak ellipticity of the
reflected and transmitted radiation arise.
Next, we list and discuss the assumptions used in our calculations. First, the
dynamic conductivity tensor (1) is written in collisionless approximation. For
the case of short-range scattering, when $\omega>>v_{d}p_{\omega}/\hbar$, one
arrives to the condition $v_{d}/2v_{W}\ll 1$ and the collisionless
approximation is not valid for a strongly disordered material. Also, the
interband response of a pure graphene is described with the use of the
phenomenological expression (A.2) and a low-frequency restriction for this
approximation is not clear.
Second, the quasiequilibrium distribution of carriers (9) was used for the
numerical estimation of electrooptical response. This means the assumption of
an effective intercarrier scattering. The complete description of the carriers
heating under such conditions had not been performed yet 11 ; 19 . However,
the approximation (9) gives a good estimation for the response magnitude, and
the peculiarities of spectral dependences enable us to determine the
contributions of the different relaxation mechanisms. Similarly, the use of
short-range scattering model in the drift-induced contribution (10) gives the
estimation for optical anisotropy magnitude, and the spectral dependences
peculiarities contain information about the momentum relaxation mechanism
(despite the short-range scattering can be treated as a dominant one within
the phenomenological description of momentum relaxation, 16 the microscopic
mechanism have not been verified until now 20 ).
Third, we have examined the heating of carriers with low energies only, (the
results for $E\leq$30 V/cm have been presented), when the essential
electrooptical response occurs in THz spectral region only. With the increase
of field (up to tens kV/cm, see 12 ) the electrooptic effect increases and
shifts into near-IR spectral region. The theoretical approach developed here
can be applied for this region as well, however the calculation of the
distribution of hot carriers for this case have not been performed yet.
Forth, the case of graphene on a thick substrate have been examined. The
consideration of the interference effects for graphene, placed on substrate of
limited thickness, needs more complicated calculations (however, an accuracy
of measurements can increase for near-IR spectral region [21]), and is beyond
the frame of this paper. And the last, we have limited ourselves to the
examination of the geometry of normal propagation of radiation only. The study
of the response dependence on the angle of radiation falling gives additional
experimental data, however it is more complicated and needs special treatment.
Finally, the results obtained demonstrate, that the electrooptical response
due to heating and drift of carriers is large enough, and it can be measured.
Because of strong dependence of the response on the applied field,
temperature, and gate voltage, these measurements can give an information on
relaxation and recombination mechanisms. In addition, the electrooptical
response of graphene can be applied for modulation of intensity and
polarization of radiation in THz and mid-IR spectral regions.
*
## Appendix A Response of undoped graphene
The dynamic conductivity for the case of undoped graphene is described by Eqs.
(1) and (3) after replacement of $f_{v{\bf p}}$ by 1 and of $f_{c{\bf p}}$ by
0. As a result we get the expression:
$\overline{\sigma}_{\omega}=\frac{2(ev_{W})^{2}}{\omega L^{2}}\sum\limits_{\bf
p}\left[\pi\delta(\hbar\omega-2v_{W}p)+\frac{i\cal
P}{\hbar\omega-2v_{W}p}\right],$ (41)
where the real and imaginary parts of conductivity have been separated. The
direct integration with the use of the energy conservation law gives the
frequency-independent real part of (A1): ${\rm
Re}\overline{\sigma}=e^{2}/4\hbar$. The imaginary contribution into
$\overline{\sigma}_{\omega}$ appears to be divergent at $p\to\infty$; moreover
${\rm Im}\overline{\sigma}_{\omega}\propto v_{W}p_{m}/\hbar\omega$, where
$p_{m}$ is a cut-off momentum. 22 On the contrary to the case of bulk
material, 9 this cut-off appears to be too rough for the description of the
response in graphene. It is convenient to approximate
$Im\overline{\sigma}_{\omega}$, by separating the terms $\propto\omega^{-1}$,
and $\propto\omega$, which correspond the contributions of the virtual
interband transitions, and of ions, correspondingly. As a result, we get:
${\rm
Im}\overline{\sigma}_{\omega}\approx\frac{e^{2}}{\hbar}\left(\frac{\varepsilon_{m}}{\hbar\omega}-\frac{\hbar\omega}{\varepsilon_{i}}\right),$
(42)
where the characteristic energies, $\varepsilon_{m}$, and $\varepsilon_{i}$,
have been introduced. The comparison of the response, described by
$\overline{\sigma}_{\omega}$, with the recent measurements of the graphene
optical spectra, yields: $\varepsilon_{m}\sim$0.08 eV, and
$\varepsilon_{i}\sim$ 6.75 eV. 15
## References
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|
arxiv-papers
| 2009-12-04T13:20:39 |
2024-09-04T02:49:06.858886
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.V. Strikha and F.T. Vasko",
"submitter": "Fedir Vasko T",
"url": "https://arxiv.org/abs/0912.0851"
}
|
0912.0953
|
# Can Thermal Nonequilibrium Explain Coronal Loops?
James A. Klimchuk, Judy T. Karpen, and Spiro K. Antiochos NASA Goddard Space
Flight Center, Greenbelt, MD 20771
###### Abstract
Any successful model of coronal loops must explain a number of observed
properties. For warm ($\sim 1$ MK) loops, these include: 1\. excess density,
2. flat temperature profile, 3. super-hydrostatic scale height, 4.
unstructured intensity profile, and 5. 1000–5000 s lifetime. We examine
whether thermal nonequilibrium can reproduce the observations by performing
hydrodynamic simulations based on steady coronal heating that decreases
exponentially with height. We consider both monolithic and multi-stranded
loops. The simulations successfully reproduce certain aspects of the
observations, including the excess density, but each of them fails in at least
one critical way. Monolithic models have far too much intensity structure,
while multi-strand models are either too structured or too long-lived. Our
results appear to rule out the widespread existence of heating that is both
highly concentrated low in the corona and steady or quasi-steady (slowly
varying or impulsive with a rapid cadence). Active regions would have a very
different appearance if the dominant heating mechanism had these properties.
Thermal nonequilibrium may nonetheless play an important role in prominences
and catastrophic cooling events (e.g., coronal rain) that occupy a small
fraction of the coronal volume. However, apparent inconsistencies between the
models and observations of cooling events have yet to be understood.
hydrodynamics — Sun: activity — Sun: corona — Sun: UV radiation — Sun: X-rays,
gamma rays
††slugcomment: Submitted to the Astrophysical Journal
## 1 Introduction
It is well known that much of the plasma in the Sun’s corona is confined in
distinct loop structures. The arching shape of these loops is defined by the
magnetic field, but their thermodynamic properties are determined by the yet-
to-be-established mechanism of coronal heating. Our understanding of coronal
loops and coronal heating has advanced considerably in recent years, but a
number of important questions remain. We report here on an investigation into
whether ordinary coronal loops can be explained by a phenomenon known as
thermal nonequilibrium. Thermal nonequilibrium occurs whenever steady or
quasi-steady heating is highly concentrated at low coronal heights in both
legs of a loop. It is believed to play an important role in prominences (e.g.,
Antiochos & Klimchuk, 1991; Karpen, Antiochos, & Klimchuk, 2006), and it seems
reasonable to consider that its occurrence is more widespread. Note that
quasi-steady heating is here taken to mean heating that changes slowly
compared to a cooling time or that is impulsive but repeats rapidly compared
to a cooling time.
Early observations of coronal loops were made primarily in soft X-rays and
suggested that the loops are in states of static equilibrium (e.g., Rosner,
Tucker, & Vaiana, 1978). This implies that the heating is steady. Soft X-ray
emission is mostly produced by hot ($>2$ MK) plasma, but more recent
observations made in the extreme ultraviolet (EUV) have revealed a much
different picture at lower temperatures. Most warm ($\sim 1$ MK) loops are
clearly inconsistent with static equilibrium. We are referring explicitly to
those warm loops that appear as complete structures within the interiors of
active regions. We do not consider partial loops, sometimes called “fan”
loops, that are often seen at the perimeters of active regions.
A number of discrepancies with static equilibrium have been identified.
Perhaps the most significant concerns the density. Static equilibrium theory
predicts a well defined relationship among the density, temperature, and
length of a loop. Warm loops are observed to have a much higher density than
is expected given the observed temperature and length (Aschwanden et al.,
1999; Aschwanden, Schrijver, & Alexander, 2001; Winebarger, Warren, & Mariska,
2003; Klimchuk, 2006). The density excess is typically a factor of about ten,
but factors ranging from near unity to more than a thousand have been
measured.
A second discrepancy between observations and theory concerns the variation of
temperature along the loop. Observations from broad-band and narrow-band
imagers such as the Transition Region and Coronal Explorer (TRACE) can be used
to measure temperature with a method known as the filter ratio technique. The
ratio of intensities obtained in two filters, or band-passes, is related to a
temperature under the assumption that the emitting plasma is isothermal. When
measured in this way, warm loops tend to have a temperature profile that is
much flatter than expected for static equilibrium (Lenz et al. 1999;
Aschwanden et al. 1999; Aschwanden, Schrijver, & Alexander 2001; although see
Reale & Peres 2000).
A third inconsistency is that the density of warm loops decreases with height
much more slowly than expected for a gravitationally stratified plasma at the
measured temperature. The scale height is too large by up to a factor of two
(Aschwanden, Schrijver, & Alexander, 2001). As a consequence, loops have a
more uniform brightness than static equilibrium would predict.
There are two additional properties of observed loops that prove extremely
important for constraining the models. Both are consistent with static
equilibrium. Most loops do not have small-scale intensity structure. With
occasional exception, there are no localized bright spots or abrupt
transitions in brightness. This is true for both warm loops (López Fuentes,
Démoulin, & Klimchuk, 2008) and hot loops (Kano & Tsuneta, 1996; Klimchuk,
2000).
Finally, there is the loop lifetime. Warm loops are typically visible for
1000–5000 s (Winebarger, Warren, & Seaton, 2003; Winebarger & Warren, 2005;
Ugarte-Urra, Winebarger, & Warren, 2006; Ugarte-Urra, Warren, & Brooks, 2009),
though some can live considerably longer. Hot loops have a much larger range
of lifetimes, with many persisting for multiple hours (López Fuentes,
Klimchuk, & Mandrini, 2007). In all cases the loop lives longer than the
cooling time expected from the measured temperature, density, and loop length.
Explaining all five of these observed properties is very challenging. One
model that does so successfully postulates that loops are bundles of
unresolved strands that are heated impulsively by storms of nanoflares; see
Klimchuk (2006, 2009) for a discussion of the basic idea and references to key
papers. In this picture, each strand is heated once and allowed to cool, but
many different strands are energized over a finite time window, which is the
storm duration. Impulsive heating is very appealing both because it is able to
explain the observations and because all current theories of heating
mechanisms predict that the heating is short lived on individual magnetic
strands (Klimchuk, 2006). This includes wave heating. A critical aspect of the
nanoflare storm idea is that strands do not get reheated. The plasma must be
allowed to cool from high temperatures to less than 1 MK in order to explain
over-dense warm loops. If nanoflares recur in a given strand with a delay that
is significantly shorter than a cooling time, then the conditions are similar
to steady heating.
In the work presented here, we assume that the heating is truly steady or that
the cadence of impulsive heating is sufficiently rapid that a steady
approximation is valid. We further assume that the heating is highly
concentrated near both footpoints of the loop. Such conditions are known to
produce a state of thermal nonequilibrium (Antiochos & Klimchuk, 1991;
Antiochos et al., 1999; Karpen et al., 2001, 2003, 2005; Müller, Hansteen, &
Peter, 2003; Müller, Peter, & Hansteen, 2004; Karpen, Antiochos, & Klimchuk,
2006; Mok et al., 2008). As the name implies, no equilibrium exists. The loop
is inherently dynamic and undergoes periodic convulsions as it searches for a
nonexistent equilibrium. Cold, dense condensations form, slide down the loop
leg, and later reform in a cycle that repeats with periods of several tens of
minutes to a few hours. It has been firmly established that rapidly repeating,
low-altitude nanoflares also produce a state of thermal nonequilibrium (Testa,
Peres, & Reale, 2005; Karpen & Antiochos, 2008; Susino et al., 2010; Antolin &
Shibata, 2009).
Our objective here is to determine whether thermal nonequilibrium can
reproduce the observations described above, in particular the EUV observations
of warm loops: 1. excess density, 2. flat temperature profile, 3. super-
hydrostatic scale height, 4. unstructured intensity profile, and 5. 1000–5000
s lifetime. Our approach is to perform numerical loop simulations by solving
the 1D time-dependent hydrodynamic equations. From these, we generate
synthetic data representing observations made in the 171 and 195 channels of
TRACE and the Al.1 and AlMg channels of the Soft X-ray Telescope (SXT) on
Yohkoh. We then measure temperature and density using precisely the same
filter ratio technique that is applied to real data.
Our study treats two fundamentally different types of loops. We first consider
monolithic structures in which the plasma is uniform over the loop cross
section. We then consider bundles of very thin unresolved strands, similar to
what is envisioned in the nanoflare storm picture described above. Just as the
nanoflares are assumed to occur at different times, we assume that the
condensation cycles of thermal nonequilibrium are out of phase in the
different strands. We describe the details of the numerical model in the next
section and present and discuss the simulation results in Sections 3 and 4.
## 2 Numerical Model
Because the solar corona is highly ionized and because the magnetic field
dominates the plasma within active regions (i.e., the plasma $\beta$ is
small), we can treat coronal loop strands as one-dimensional structures. We
therefore perform our numerical simulations with the Adaptively Refined
Godunov Solver (ARGOS) hydro code (Antiochos et al., 1999), which solves the
1D hydrodynamic equations of mass, momentum, and energy conservation:
$\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial s}(\rho\upsilon)=0,$
(1)
$\frac{\partial}{\partial t}(\rho\upsilon)+\frac{\partial}{\partial
s}(\rho{\upsilon}^{2})=\rho g_{\|}-\frac{\partial P}{\partial s},$ (2)
$\frac{\partial E}{\partial t}+\frac{\partial}{\partial
s}[(E+P)\upsilon]=\rho\upsilon g_{\|}+\frac{\partial}{\partial
s}(\kappa_{0}T^{5/2}\frac{\partial T}{\partial s})-n^{2}\Lambda(T)+Q,$ (3)
where
$E=\frac{1}{2}\rho{\upsilon}^{2}+\frac{P}{\gamma-1}.$ (4)
Here, $s$ is the spatial coordinate along the loop; $n$ is the electron number
density; $\rho=1.67\times{10}^{-24}\times n$ is the mass density assuming a
fully ionized hydrogen plasma; $T$ is the temperature; $P=2nkT$ is the total
pressure; $\upsilon$ is the bulk velocity; $\kappa_{0}={10}^{-6}$ is the
coefficient of thermal conduction for Spitzer conductivity; $\gamma=5/3$ is
the ratio of the specific heats; $g_{\|}$ is the component of gravity parallel
to the loop axis; $Q$ is the volumetric heating rate; and $\Lambda(T)$ is the
optically thin radiative loss function as given in Klimchuk, Patsourakos, &
Cargill (2008) with the exception that there is a $T^{3}$ dependence below 0.1
MK to account approximately for optical depth effects.
ARGOS uses adaptive mesh refinement to dynamically modify the numerical grid
in response to changes in the density gradients. This is critically important
for simulations of this type. The cold condensations which form and move along
the loop are bounded by thin transition regions similar to the classical
transition regions at the footpoints of loops. Only by subdividing and merging
grid cells is it possible to resolve these structures with a grid of
reasonable size. Our simulations have approximately 3500 total cells while the
condensations are present. The smallest cell length is 6 km. This is
approximately one-third the temperature scale length in the lower transition
region where $T=0.1T_{max}$.
The loop is assumed to be a vertical semi-circle with a footpoint-to-apex half
length of $L=75$ Mm. This half length is characteristic of warm loops and is
the value we have used for some of our nanoflare studies (e.g., Klimchuk,
Patsourakos, & Cargill, 2008). The cross sectional area is constant,
consistent with observations of both EUV and soft X-ray loops (Klimchuk, 2000;
López Fuentes, Klimchuk, & Démoulin, 2006). Attached to each end of the
coronal semi-circle is a 50 Mm chromosphere/photosphere that is maintained at
a nearly constant temperature of $3\\!\times\\!10^{4}$ K by a radiative loss
function that decreases precipitously to zero between $3\\!\times\\!10^{4}$
and $2.95\\!\times\\!10^{4}$ K. This loss function applies to the entire loop,
including the cold condensations that form in the coronal portion. Although
the radiative properties of the chromosphere and condensations are treated in
a highly simplified manner (there is no detailed radiative transfer), the
interaction with the rest of the loop is modeled rigorously. In particular,
the exchange of mass by the important processes of evaporation and
condensation is fully included. Radiative transfer effects are important for
explosive evaporation that is driven by energetic particle beams penetrating
deep into the chromosphere during flares, but the gentle evaporation in our
simulations is due to a heat flux that is mostly dissipated in the transition
region. Only a small fraction of the heat flux reaches the chromosphere.
We begin each simulation by allowing the loop to relax to a static equilibrium
with a spatially uniform background heating, $Q_{b}$. The choice
$Q_{b}=6\\!\times\\!10^{-4}$ erg cm-3 s-1 produces an apex temperature of 3.0
MK. Over the next 1000 s, we slowly turn on a localized heating that decreases
exponentially with height above the chromosphere at both ends of the loop and
is spatially uniform below:
$Q_{l}(s\geq s_{0})=Q_{0}\exp[-(s-s_{0})/\lambda]$ (5)
on the left side, where $s_{0}=50$ Mm is the top of the chromosphere. The
right side is a mirror image with the exception of amplitude (see below). Both
the background and localized heating are held constant thereafter. The scale
length of the exponential decrease is $\lambda=5$ Mm, which is 1/15 of the
loop half length. Its maximum amplitude at the left footpoint is
$Q_{0}=8.0\\!\times\\!10^{-2}$ erg cm-3 s-1. We impose an asymmetry by making
the amplitude at the right footpoint only 50, 75, or 90% as large. The
localized heating provides nearly an order of magnitude more total energy
(spatially integrated over the loop) than does the uniform background heating,
and therefore it dominates.
Our volumetric heating function (Eq. 5) has two broad but crucial constraints:
it must be spatially localized above the chromosphere with a characteristic
scale smaller than 10% of the loop length, and quasi-steady in comparison to
the ambient radiative cooling time. Earlier studies of thermal nonequilibrium,
as well as the physical explanation of thermal nonequilibrium (see below), all
indicate that the basic phenomenon is otherwise independent of the details of
the heating. Therefore, any physical heating mechanism that satisfies these
constraints is capable of producing thermal nonequilibrium. Identifying which
of the many candidates for coronal heating meet these criteria is an important
long-term objective, but it is beyond the scope of this paper. Our sole aim is
to investigate whether thermal nonequilibrium can explain ordinary coronal
loops, for which purpose our heating function is appropriate.
## 3 Results
### 3.1 Monolithic Loops
The first loop we consider is monolithic and has a 75% heating asymmetry. It
exhibits quasi-periodic behavior with condensations forming roughly every 6000
s. Figure 1 shows the temperature profile at four different times during the
sixth condensation cycle, long after any memory of the initial static
conditions has disappeared. The evolution is typical of thermal nonequilibrium
and has been well documented in our other work. After the condensation from
the fifth cycle falls to the chromosphere ($t=0$ s), the loop rapidly heats
and attempts to establish an equilibrium. A peak temperature of 4.4 MK is
reached at $t=650$ s. This is followed by a relatively long period of slow
cooling. The solid curve in Figure 1 shows the temperature profile at $t=2950$
s, well into the cooling phase.
The reason for the slow cooling and eventual formation of a condensation can
be understood as follows. We begin by noticing that the maximum temperature
$T_{max}$ occurs close to the left footpoint, at a height comparable to the
heating scale length $\lambda$. Let us hypothetically divide the loop into two
unequal parts: a short section to the left of $T_{max}$ and a much longer
section to the right. Imagine that the short section is one-half of a small
symmetric loop. If this hypothetical loop were in static equilibrium, it would
satisfy the scaling law
$n=1.32\\!\times\\!10^{6}\,\frac{T_{max}^{2}}{\ell},$ (6)
where $\ell$ is the half length, approximately equal to $\lambda$. Equation
(6) follows from the well-known scaling law
$T_{max}=1.4\\!\times\\!10^{3}\,(P\ell)^{1/3}$ (Rosner, Tucker, & Vaiana,
1978) upon substituting for $P$ using the ideal gas law. The downward heat
flux from $T_{max}$ is very large due to the steep temperature gradient.
Correspondingly large density is required in order for radiation from the
transition region to balance the heat flux. If the actual density is smaller
than the equilibrium value given in equation (6), the radiation will be too
weak, and chromospheric evaporation will occur, as it does in our simulation.
Now consider the other section of the original loop, to the right of
$T_{max}$. Imagine that it is half of a different hypothetical loop. It has
the same maximum temperature as the short loop, but because it is much longer,
its equilibrium density is much smaller according to equation (6). Of course
the long and short “loops” are really attached. Evaporation in the short
section drives up the density in the long section to values that exceed the
local equilibrium conditions. Radiation is enhanced at the elevated densities,
so the plasma cools.
The above argument based on the static equilibrium theory shows why static
conditions are not possible with highly localized footpoint heating, but the
actual energy balance in the evolving loop is more involved due to the
presence of flows. The evaporating material carries an enthalpy flux that
plays a very important role. It provides nearly enough energy to power the
enhanced coronal radiation. This is the reason why the evolution is so slow
during most of the cooling phase. In fact, if only the left leg were subjected
to localized heating, the loop would establish a dynamic equilibrium with a
steady end-to-end flow and no cooling (Patsourakos, Klimchuk, & MacNeice,
2004). Our loop has localized heating on both sides, which drives evaporative
upflows from both ends. Because material continually accumulates in the
corona, the plasma must cool, and no steady state is possible.
We see from Figure 1 that the cooling is not symmetric. Because evaporation is
stronger on the left side than on the right, the flows converge to the right
of the loop midpoint. Cooling is fastest at this location, and a dip develops
in the temperature profile. The dip grows at an accelerating pace until a cold
condensation is ultimately formed at $t=4850$ s (dashed curve). The final
collapse resembles a thermal instability; only 350 s are required for the
temperature to drop from 2.0 to 0.03 MK. Once formed, the condensation is
pushed to the right by a small pressure imbalance. It hits the chromosphere
approximately 1300 s after first appearing, and a new condensation cycle
begins.
#### 3.1.1 Excess Density Factor
The model loop is characterized by over-dense conditions during most of its
evolution. We wish to make a quantitative comparison with observations, and
because many studies of observed loops involve spatial and temporal averages,
we define an excess density factor $\Psi$ as follows:
$\Psi=\frac{\bar{n}}{\bar{n}_{eq}},$ (7)
where
$\bar{n}_{eq}=1.32\\!\times\\!10^{6}\,\frac{\bar{T}^{2}}{L}$ (8)
and $\bar{n}$ and $\bar{T}$ are the density and temperature averaged over the
upper 50% of the loop and over one or more condensation cycles. Equation (8)
comes from the Rosner, Tucker, & Vaiana (1978) scaling law, analogous to
equation (6). Averaging over the 11 cycles of our simulation gives
$\Psi=4.09$. Note that these are simple averages using densities and
temperatures taken directly from the simulation output. The excess density
factor obtained in this way is different from what we would get from observed
intensities, which provide nonlinear averages of density and temperature.
Later we will perform a more rigorous analysis that takes this into account.
Because many observed loops are shorter or longer than our model loop, it is
important to examine how $\Psi$ depends on loop length. We therefore consider
two additional models that are half and twice as long as the original:
$L=37.5$ and 150 Mm. The heating scale length $\lambda$ and 75% asymmetry are
the same as before, but we modify the heating amplitudes $Q_{b}$ and $Q_{0}$
so that the peak temperatures of the initial equilibrium and of the
condensation cycles are similar in all three cases. The resulting excess
density factors are 2.90 and 6.62 for the short and long loops, respectively.
These three cases suggest the relationship $\Psi\propto L^{1/2}$. We can
understand the square-root dependence by considering the period of slow
cooling that dominates the evolution. As discussed above, a strong downward
heat flux drives an upward enthalpy flux in the lower legs:
$\kappa_{0}\frac{T_{max}^{7/2}}{\lambda}\approx\frac{5}{2}Pv.$ (9)
The enthalpy is nearly enough to power the radiative losses from the rest of
the loop:
$\frac{5}{2}Pv\approx n^{2}\Lambda(T_{max})L.$ (10)
Combining, we get
$n\approx\left[\frac{\kappa_{0}T_{max}^{7/2}}{\Lambda(T_{max})}\frac{1}{\lambda
L}\right]^{1/2}$ (11)
for the actual loop density.
In static equilibrium, the energy loss rates from radiation and thermal
conduction are approximately equal in the corona (Vesecky, Antiochos, &
Underwood, 1979):
$n_{eq}^{2}\Lambda(T_{max})\approx\kappa_{0}\frac{T_{max}^{7/2}}{L^{2}}.$ (12)
This gives
$n_{eq}\approx\left[\kappa_{0}\frac{T_{max}^{7/2}}{\Lambda(T_{max})}\frac{1}{L^{2}}\right]^{1/2},$
(13)
for the equilibrium density corresponding to $T_{max}$ and $L$.111Comparing
equations 13 and 8, we see that the Rosner, Tucker, & Vaiana scaling law uses
$\Lambda(T)\propto T^{-1/2}$. The excess density factor is therefore
$\Psi=\frac{n}{n_{eq}}\approx\left(\frac{L}{\lambda}\right)^{1/2}.$ (14)
Note that it depends not on the loop length alone, but on the ratio of the
loop length to heating scale length. In principle, we could reproduce model
loops with any $L$ and $\Psi$ simply by adjusting the value of $\lambda$. It
seems, therefore, that the observed excess densities of warm loops can be
readily explained with thermal nonequilibrium.
#### 3.1.2 Intensity
A successful loop model must also explain the intensity properties of observed
loops, both temporal and spatial. We therefore generate light curves and
intensity profiles for simulated TRACE observations of the models made in the
171 channel. We assume that the loops are viewed from the side, so the
intensity at any point along the loop axis is given by $I=n^{2}G(T)$. Here,
$G(T)$ is the instrument response function, which for the 171 channel is
reasonably sharply peaked near 1 MK. We have ignored the loop diameter and a
possible filling factor because we are concerned only with relative
intensities, and both the diameter and filling factor are assumed to be
constant along the loop and unchanging in time.
The solid curve in Figure 2 is the light curve for the sixth condensation
cycle of the original $L=75$ Mm loop. This is the same condensation cycle
shown in Figure 1. We have averaged the intensity over the upper 80% of the
loop to exclude the “moss” emission from the transition regions at the
footpoints. The dashed and dotted curves show the corresponding evolution of
the spatially averaged temperature and density. It is readily apparent how
evaporation slowly fills the loop with plasma.
Before the condensation forms, the coronal plasma is too hot to be easily
detected in the 171 channel, and the light curve is extremely faint. It
brightens dramatically when the condensing plasma cools rapidly through 1 MK
(sharp spike at $t=4700$ s). This contrasts with the much more gradual
brightening exhibited by most observed loops (Winebarger, Warren, & Seaton,
2003). The light curve remains bright after the condensation has fully formed
because transition regions are present on either side of the cold mass. After
about 1000 s the light curve suddenly dims as the condensation moves out of
the 80% averaging window. The spatially-averaged density drops at the same
time since the condensation contains most of the loop’s mass. Bright emission
is actually present in the loop for another 300 s as the condensation
traverses the remaining 20% of the leg before hitting the chromosphere. The
total lifetime in 171 emission is therefore approximately 1300 s. This is at
the extreme low end of the range of observed lifetimes.
The abrupt appearance and disappearance of the 171 emission disagrees with
observations, which show a more gradually evolving light curve. The spatial
distribution of the emission presents an even bigger problem. Figure 3 shows
profiles of intensity (solid) and temperature (dashed) at $t=5000$ s, after
the condensation has formed. The emission is highly concentrated in transition
region layers at the loop footpoints (“moss”) and on either side of the
condensation. This contrasts sharply with observed loops, which tend to have a
far more uniform appearance. Falling bright knots are sometimes observed, but
these are only detected at much cooler temperatures ($\leq 0.1$ MK)
(Schrijver, 2001; De Groof et al., 2004; O’Shea, Banerjee, & Doyle, 2007). We
return to the subject of these knots later in the Discussion section.
To determine whether the extreme nonuniformity in the intensity distribution
is affected by the degree of heating asymmetry, we perform two additional
simulations using the same heating amplitude and scale length as before, but
with asymmetries of 50% and 90%. The results are qualitatively similar to the
75% case. The intensity profiles are highly structured and in gross
disagreement with observations.
The primary reason for the nonuniform intensity is that most of the loop is
too hot to be easily detected in the 171 channel (i.e., significantly hotter
than 1 MK). To obtain temperatures more suitable to 171, we perform three new
simulations with the heating amplitude reduced by an order of magnitude. All
other parameters are as before. The model with 75% asymmetry reaches a maximum
temperature of 1.8 MK and has an excess density factor $\Psi=4.69$. The
results for the 50% and 90% cases are similar.
Figure 4 shows the 171 light curve and the temperature and density evolution
for the second condensation cycle of the 75% case. The cycle lasts
approximately 11,000 s, nearly twice as long as the strong heating
counterpart. The light curve has three rather distinct phases—faint, bright,
and intermediate—which is not consistent with the slowly varying intensities
of most observed warm loops. The bright and intermediate phases together last
about 7000 s, which is longer than most observed loop lifetimes.
An interesting aspect of this simulation is that two condensations are present
at the same time, as was seen in earlier studies (Müller, Peter, & Hansteen,
2004; Karpen et al., 2005). Figures 5 and 6 show intensity and temperature
profiles at $t=5000$ and 7000 s, before and after the condensations form. More
of the loop is visible than in the strong heating models, but the intensity
still is far more structured than is observed. In particular, the region
between the condensations is extremely faint. We can understand this behavior
as follows. When the condensations form, the central region between them is
cut off from the evaporative upflows and associated enthalpy flux that powers
the radiative losses. The plasma cools and drains onto the condensations. The
condensations behave like chromospheres, and a quasi-static loop equilibrium
is established between them. Because the heating rate is so small, the
equilibrium state has a low temperature and very low density, so the 171
emission is minimal. The precise value of the temperature and density depend
on the magnitude of the uniform background heating, which dominates in this
part of the loop. Note that the two condensations remain separate at all times
and do not merge, as is sometimes seen in other simulations (e.g., Karpen et
al., 2005).
Thermal nonequilibrium clearly cannot explain observed loops if the loops are
monolithic structures, at least not with steady, exponential heating of the
type we have considered.
### 3.2 Multi-Strand Loops
#### 3.2.1 Excess Density Factor
Because our monolithic models fail, we now consider loops that are bundles of
many unresolved strands. To start, we assume that all of the strands in a
given loop are identical except for the phasing of the condensation cycles,
which we take to be random. We can then approximate an instantaneous snapshot
of the composite loop by simply time averaging one simulation over one or more
cycles.
A wide variety of temperatures coexist within the cross section of such a
multi-stranded loop. The single temperature that is measured by an instrument
like TRACE or SXT/Yohkoh is a weighted average, where the weighting depends on
both the temperature response function, $G(T)$, and the differential emission
measure distribution, $DEM(T)$. To simulate realistic measurements from our
models, we first compute intensity profiles for the individual strands (i.e.,
for each time in the simulation), and then we average them together to obtain
a single intensity profile for the loop bundle. We do this separately for the
171 and 195 channels of TRACE and the Al.1 and AlMg channels of SXT. We next
infer temperature and emission measure, $EM$, at each position along the loop
using both the 171/195 and Al.1/AlMg ratios. From the emission measure, we
compute density according to
$n=\left(\frac{EM}{df}\right)^{1/2},$ (15)
where $d$ is the loop diameter and $f$ is the filling factor. The diameter
plays no role, since $EM$ is derived from the loop intensity, which scales
with the assumed diameter. We take a filling factor of unity, precisely as
done for real data, which means that the density given by equation (15) is a
lower limit. Finally, we average $T$ and $n$ along the upper 50% of the loop
222The reader may wonder why we use 50% averages here and 80% averages for the
light curves. 50% was used in the observational studies of loop density to
which we will compare our models. For the light curves, we are only concerned
with excluding the bright moss emission at the footpoints. and use equations
(7) and (8) to obtain the excess density factors that would be measured by
TRACE and SXT, designated $\Psi_{TRACE}$ and $\Psi_{SXT}$. We follow this
procedure separately for all 6 of the $L=75$ Mm models (2 heating amplitudes
and 3 heating asymmetries).
It seems unlikely that all of the strands in a given loop bundle would be
identical except for their phases. Therefore, we also build a composite loop
with strong heating and a composite loop with weak heating by averaging
together the models with 50, 75, and 90% asymmetry. The averages include both
the original models, which have greater heating in the left leg, and their
mirror images, which have greater heating in the right leg. The two composite
loops so obtained have a mixture of strands of different types, which is
perhaps more realistic. We simulate temperature and density measurements of
these loops using the same procedure described above, first averaging the
intensities and then applying the filter ratio technique.
Results for the “homogeneous” multi-strand models and the composite multi-
strand models are presented in Table 1. The first column gives the loop half
length, which is the same for all except the last two cases. The second column
gives the amplitude of the localized heating together with an indication of
whether it is strong (produces a peak temperature near 4.4 MK) or weak
(produces a peak temperature near 1.8 MK). The third column gives the heating
asymmetry. The fourth column gives the number of condensation cycles used in
the temporal averages. The fifth column gives the average period of the
cycles, which are only quasi-regular in most cases. The sixth and seventh
columns give the temperatures that would be measured with TRACE and SXT filter
ratios. The last three columns give the excess density factors obtained
directly from the temperatures and densities of the models, equation (7), and
from the simulated TRACE and SXT measurements. The values differ because TRACE
preferentially detects the warm plasma and SXT preferentially detects the hot
plasma.
Figure 7 shows the excess density factors of real loops plotted against
temperature. The factors were determined precisely as described above, i.e.,
using equation (15). The loops near 1 MK (pluses) were observed by TRACE and
analyzed originally by Aschwanden, Nightingale, & Alexander (2000). The hotter
loops (crosses) were observed by SXT and analyzed originally by Porter &
Klimchuk (1995) (also Klimchuk & Porter, 1995). These are the same loops
presented in Figure 4 of Winebarger, Warren, & Mariska (2003) and Figure 6 of
Klimchuk (2006). Also shown are the excess density factors of the model loops
as determined from simulated TRACE observations (diamonds) and simulated SXT
observations (squares).
There is good agreement between the models and observations for the excess
density factors obtained from TRACE. Values range between about 3 and 11 for
the models and between about 1 and 12 for the observations (note that
logarithms are plotted in the figure). The temperatures measured by TRACE are
also in good agreement, but this is expected because the 171 and 195 filters
have a narrow temperature response and are only sensitive to plasma close to 1
MK.
The agreement between the models and observations is much worse for the SXT
measurements. Excess density factors from the models are tightly clustered
between 1 (no excess) and 3, whereas those from the observations range all the
way from 0.02 (highly under dense) to 16\. The agreement is better if we
restrict ourselves to the temperature range of the models ($1.1<T_{SXT}<3.4$
MK), in which case the observed excess density factors are all $>0.4$
(slightly under dense). However, the observed factors have a strong tendency
to decrease with temperature, while the model factors have a weak tendency to
increase. We conclude that the models are consistent with at most a subset of
observed SXT loops. According to equation (14), thermal nonequilibrium can
never produce the under-dense conditions observed at high temperatures because
$\lambda>L$ gives rise to static equilibrium (in fact, static equilibrium
occurs whenever $\lambda>L/5$ approximately). It is worth pointing out that
the nanoflare storm model is capable of explaining both over-dense warm loops
and under-dense hot loops (Klimchuk, 2006).
The quantity $\Psi$ defined in equations (7) and (8) very likely
underestimates the true degree to which TRACE loops are over-dense (cf.
Winebarger, Warren, & Mariska, 2003). Equation (8) is an idealized scaling law
based on: (1) an approximate and somewhat outdated form for the radiative loss
function; (2) the assumption of no gravitational stratification; and (3) the
assumption of spatially uniform heating. The coefficient of the scaling law
should be treated with particular caution. Furthermore, the “actual” density
determined from equation (15) assumes a filling factor $f=1$ and is therefore
a lower limit, but the equilibrium density determined from equation (8) does
not depend on $f$. Despite these limitations, $\Psi$ is a useful tool for
evaluating the agreement between models and observations.
#### 3.2.2 Intensity and Temperature
We rejected the monolithic models on the basis of their 171 intensity
properties, and we now examine whether the multi-strand models fare any
better. We limit our discussion to the composite models because we believe
they are more realistic and, more importantly, because they agree better with
the observations.
The biggest failing of the monolithic models is their highly structured
intensity profile. The problem is especially severe for models with strong
heating, which have localized bright emission immediately flanking the cold
condensation. Multi-strand models have a much more uniform appearance because
they include many unresolved condensations that are spread out along the loop.
Condensations tend to form in the upper two-thirds of the loop, at a location
that depends on the level of heating asymmetry and on $\lambda$. The weaker
the asymmetry, the closer to the apex they form, with perfectly symmetric
heating producing a condensation right at the apex. Once formed, the
condensations move downward toward the footpoints. If all phases of the cycles
are represented in the strands, the entire loop will be filled in with bright
emission, including the lower legs, consistent with observations. It is
critical, however, that some of the strands have nearly symmetric heating so
that a dark gap is not present at the top.
Figure 8 shows the intensity profile for the composite model with strong
heating. Except for the bright spikes at the footpoints (note the logarithmic
scale), the emission is reasonably uniform. Intensity variations along the
loop are less than a factor of 2 and would be smaller still if the bundle
included a larger variety of heating asymmetries.
Figure 9 shows the 171 intensity profile for the composite model with weak
heating (linear scale). The profile is very smooth, due largely to the fact
that the individual strands are reasonably uniform up to the time when the
condensations form. The profile is nonetheless inconsistent with observations
because the intensity decreases too rapidly with height. The scale height in
the model corresponds to a hydrostatic loop at 1 MK, whereas observed scale
heights are super-hydrostatic by up to a factor of 2.
Figure 10 shows three temperature profiles for the strong heating composite
model. The solid curve is the average of the actual temperatures in the
strands; the dashed curve is the temperature that would be measured by TRACE;
and the dotted curve is the temperature that would be measured by SXT. The
temperature profiles are very flat, in excellent agreement with TRACE
observations and not inconsistent with SXT observations (Kano & Tsuneta,
1996). The composite model with weak heating also has a flat TRACE temperature
profile. Its SXT profile is not relevant, since the loop would be extremely
faint in soft X-rays.
The multi-strand models presented here were obtained by temporally averaging
over two or more condensation cycles. As such, they represent very long-lived
loops, inconsistent with observations. We could instead average over a portion
of a cycle to obtain a shorter lived loop, but then the intensity and
temperature profiles would be less uniform. Averaging over a portion of the
cycle corresponds to condensations forming at roughly the same time in the
different strands. If they form at the same time, they move together as a
group. The lower legs of such a loop would be dark in the early stages of
evolution, and the apex would be dark in the later stages, neither of which
agrees with observations. Whether it is possible to build a loop that is both
sufficiently short lived and sufficiently uniform to match observed loops is a
question that we examine in more detail below.
## 4 Discussion and Conclusions
We have modeled monolithic and multi-strand loops undergoing thermal
nonequilibrium with the hope of reproducing the salient features of observed
loops, especially those seen in warm ($\sim 1$ MK) emissions by instruments
like TRACE. A fundamental property of these warm loops is their excess density
compared to static equilibrium. We find that many of our models can
successfully explain the observed densities. Some can also explain the
unstructured intensities and flat temperature profiles that are typically
observed. However, none of the models is able to successfully reproduce all of
the observed properties. The monolithic models fail dramatically in that they
have far too much intensity structure. This is not a problem for the multi-
strand models, but these models, as presented, are far too long-lived. The
competing requirements of uniform intensity and short-to-modest lifetime
(1000–5000 s) are extremely difficult to satisfy. It may be possible to
construct a model that satisfies both, but the conditions are too contrived to
be a plausible explanation for real loops, as we now show.
It is instructive to briefly discuss the nanoflare storm concept (Klimchuk,
2009), because it shares several common features with the thermal
nonequilibrium scenario we are now considering. In the nanoflare case, each
loop is envisioned to be a bundle of strands that are heated impulsively at
different times (but only once). At any given moment, the many strands are in
different stages of cooling and therefore only some of them are detectable in
the 171 channel. If the duration of the nanoflare storm (time delay between
the first and last nanoflare) is long compared to the lifetime of each strand
(duration of visibility), then the lifetime of the entire loop bundle will be
determined primarily by the duration of the storm. If, on the other hand, the
duration of the storm is short compared to the lifetime of each strand, then
the lifetime of the bundle will be determined primarily by the lifetime of the
strands. It is straightforward to see that the loop lifetime is approximately
equal to the sum of the storm duration and the strand lifetime.
We can apply these same ideas to a bundle of strands undergoing thermal
nonequilibrium. In place of the nanoflare storm duration, we have the time
delay $\Delta t$ between the formation of the first and last condensations.
Just as there is only one nanoflare per strand, there can be only one
condensation (or condensation pair) per strand, because the period of the
cycles is considerably longer than observed loop lifetimes. Let $\tau$
represent the time that each strand is visible in the 171 channel. To
reproduce the observed loop lifetimes, $\Delta t$ must satisfy $\Delta
t+\tau\approx$ 1000–5000 s. Model strands with weak heating have $\tau>5000$
and can be immediately ruled out. Model strands with strong heating have
$\tau\approx$ 1000–2000 s (2000 s for the case with 90% heating asymmetry).
Observed loop lifetimes can perhaps be reproduced if $\Delta t\approx$ 0–4000
s.
The condition on $\Delta t$ is necessary but not sufficient. Loops will have
uniform intensity only if the strands are sufficiently out of phase. There is
a problem when $\Delta t$ is small because then all of the strands are roughly
in phase. The condensations form together in the upper part of the loop and
move together down the leg. The requirement of uniform brightness places a
lower limit on $\Delta t$ that is approximately the time it takes a single
condensation to traverse the entire half length of the loop. Only then will
one condensation appear near the apex at the same time that another is about
to disappear into the chromosphere. In the simulation with 90% heating
asymmetry, the condensation takes approximately 2000 s to traverse this
distance. This is also how long the strand is visible in 171. To have a
uniform loop bundle made from these strands implies a loop lifetime of at
least $\Delta t+\tau\approx 2000+2000=4000$ s. A majority of observed warm
loops are shorter lived than this. We conclude that they cannot be explained
by thermal nonequilibrium.
Even the longer lived loops are problematic. To produce a condensation, the
heating in each strand must be steady or quasi-steady for at least one cycle,
which lasts approximately 2 hours. If the heating is steady for 2 hours, then
it seems reasonable to expect that it might remain steady for 4 hours, or even
longer. This would allow additional cycles to occur and the loop to reappear
multiple times. We can rule this out, however. As shown in Table 1, strands
with different heating parameters have different cycle periods. The period
also varies from one cycle to the next for a given set of parameters (i.e.,
the cycles are only quasi-periodic). Therefore, even if the phasing of the
strands were correct for the first appearance of the loop—itself a
challenge—it would not be correct for the second and subsequent appearances.
To reproduce the observations, the heating must turn on, remain steady for one
full cycle, and then turn off before any new condensations can form. This
seems highly implausible. We conclude that thermal nonequilibrium is not a
reasonable explanation for any observed warm coronal loops, even those that
are relatively long-lived. Thermal nonequilibrium is also incapable of
explaining hot loops, since it cannot produce the under-dense conditions that
are characteristic of these loops.
An important implication of our results is that the dominant heating mechanism
in active regions cannot be both highly concentrated low in the corona and
steady or quasi-steady (slowly varying or impulsive with a rapid cadence).
Active regions would look much different if this were the case. Loops
resembling our models—and therefore unlike those observed—would be common.
This claim must be qualified with some caveats. It is acceptable for the
heating to decrease with height as long as the scale length is greater than
about 20% of the loop half length. Only shorter scale lengths produce thermal
nonequilibrium. Even these short lengths might be allowed if only one leg of
the loop is heated, because then a steady flow equilibrium can be established.
It is unclear, however, whether these steady equilibria can reproduce the
excess densities, intensity scale heights, and temperature profiles that are
observed (Patsourakos, Klimchuk, & MacNeice, 2004; Winebarger et al., 2002).
Finally, we cannot exclude the possibility that thermal nonequilibrium is
occurring in the diffuse corona between loops. The properties of this part of
the corona are not well understand, and evidence of thermal nonequilibrium
might not be obvious if there is a multitude of unresolved strands with random
phasing of the cycles. One consequence of many unresolved condensations would
be the absorption of the EUV radiation from below. Evidence of absorption from
unresolved cold material in the corona has been reported (e.g., Schmahl &
Orrall, 1979; Klimchuk & Gary, 1995), but whether the quantities are
consistent with widespread thermal nonequilibrium has not yet been
investigated.
We close by emphasizing that thermal nonequilibrium is likely to play an
important role in the solar atmosphere under more limited circumstances. It is
almost certainly responsible for prominences (Antiochos & Klimchuk, 1991;
Karpen, Antiochos, & Klimchuk, 2006), and it may also explain “catastrophic
cooling events,” including coronal rain (Schrijver, 2001; De Groof et al.,
2004; O’Shea, Banerjee, & Doyle, 2007). During these events, a cold “blob”
condenses out of the hot corona at the top of a loop. It appears sequentially
in the 195, 171, 1600, and 1216 channels of TRACE, which have maximum
sensitivity at temperatures of 1.5, 1.0, 0.1, and 0.02 MK, respectively. The
blob is visible only in the two coolest channels as it falls down the leg at
speeds of 20–100 km s-1. Though fascinating, these events are relatively
uncommon. The number of blobs observed at any one time is much less than the
number of warm loops (C. Schrijver, 2009, private communication).
Müller, Peter, & Hansteen (2004) and Antolin & Shibata (2009) have suggested
that these blobs are formed by thermal nonequilibrium. Our monolithic models
with strong heating have many similarities to the observations, including the
downward velocities, but key differences are not yet explained. The observed
time delay between the blob’s appearance in the 171 and 1600 channels is more
than twice what our models predict. More significantly, our models predict 171
and 195 emission from the transition regions that flank the blob as it falls,
but such emission is apparently not seen. It is clear that more work is needed
before we fully understand the origin of catastrophic cooling events.
This work was supported by the NASA Living With a Star program. A portion of
it was performed while the authors were on the staff of the Naval Research
Laboratory. We gratefully acknowledge useful conversations with Roberto
Lionello, Jon Linker, Yung Mok, Karel Schrijver, and Daniele Spadaro.
Table 1: Model Parameters $L$ | $Q_{0}$ | Asymmetry | Cycles | Period | $T_{TRACE}$ | $T_{SXT}$ | $\Psi$ | $\Psi_{TRACE}$ | $\Psi_{SXT}$
---|---|---|---|---|---|---|---|---|---
[Mm] | [erg cm-3 s-1] | [$\%$] | | [s] | [MK] | [MK] | | |
75 | $8.0\\!\times\\!10^{-2}(strong)$ | 90 | 9 | 7330 | 1.30 | 3.35 | 5.07 | 9.69 | 3.00
75 | $8.0\\!\times\\!10^{-2}(strong)$ | 75 | 11 | 6370 | 1.31 | 3.16 | 4.09 | 6.79 | 3.15
75 | $8.0\\!\times\\!10^{-2}(strong)$ | 50 | 2 | 6800 | 1.36 | 3.15 | 3.56 | 4.74 | 3.02
75 | $8.0\\!\times\\!10^{-2}(strong)$ | Composite | 2-11 | | 1.24 | 3.24 | 4.26 | 8.88 | 3.01
75 | $8.0\\!\times\\!10^{-3}(weak)$ | 90 | 2 | 10,280 | 1.20 | 1.65 | 2.73 | 2.42 | 1.25
75 | $8.0\\!\times\\!10^{-3}(weak)$ | 75 | 2 | 10,280 | 1.19 | 1.47 | 4.69 | 2.53 | 1.52
75 | $8.0\\!\times\\!10^{-3}(weak)$ | 50 | 3 | 6370 | 1.06 | 1.11 | 5.18 | 3.66 | 2.99
75 | $8.0\\!\times\\!10^{-3}(weak)$ | Composite | 2-3 | | 1.14 | 1.51 | 4.05 | 2.85 | 1.45
150 | $2.0\\!\times\\!10^{-2}(strong)$ | 75 | 3 | 7700 | 1.22 | 3.37 | 6.62 | 11.34 | 3.10
37.5 | $3.2\\!\times\\!10^{-1}(strong)$ | 75 | 5 | 6110 | 1.29 | 3.28 | 2.90 | 4.98 | 2.35
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Figure 1: Temperature versus position at four times during the sixth
condensation cycle of the loop with strong heating and 75% asymmetry: $t$ =
2950 (solid), 4500 (dotted), 4850 (dashed), and 5750 (dot-dashed) seconds
after the previous condensation hits the chromosphere.
Figure 2: TRACE 171 intensity (solid), temperature (dashed), and density
(dotted) versus time for the sixth condensation cycle of the loop with strong
heating and 75% asymmetry. Values are averaged over the upper 80% of the loop
and are normalized to their respective maxima.
Figure 3: TRACE 171 intensity (solid) and temperature (dashed) versus position
at $t=5000$ s in the sixth condensation cycle of the loop with strong heating
and 75% asymmetry.
Figure 4: TRACE 171 intensity (solid), temperature (dashed), and density
(dotted) versus time for the second condensation cycle of the loop with weak
heating and 75% asymmetry. Values are averaged over the upper 80% of the loop
and are normalized to their respective maxima.
Figure 5: TRACE 171 intensity (solid) and temperature (dashed) versus position
at $t=5000$ s in the second condensation cycle of the loop with weak heating
and 75% asymmetry.
Figure 6: TRACE 171 intensity (solid) and temperature (dashed) versus position
at $t=7000$ s in the second condensation cycle of the loop with weak heating
and 75% asymmetry.
Figure 7: Excess density factor versus temperature for real loops observed by
TRACE (pluses) and SXT (crosses) and for model loops with simulated
observations by TRACE (diamonds) and SXT (squares).
Figure 8: Logarithm of the TRACE 171 intensity versus position for the
composite loop with strong heating.
Figure 9: TRACE 171 intensity versus position for the composite loop with weak
heating.
Figure 10: Temperature versus position for the composite loop with strong
heating: average of the actual temperatures (solid), simulated TRACE
temperature (dashed), and simulated SXT temperature (dotted).
|
arxiv-papers
| 2009-12-04T21:43:12 |
2024-09-04T02:49:06.866249
|
{
"license": "Public Domain",
"authors": "J. A. Klimchuk, J. T. Karpen, S. K. Antiochos",
"submitter": "James Klimchuk",
"url": "https://arxiv.org/abs/0912.0953"
}
|
0912.1088
|
††thanks: Electronic address: xjzhou@pku.edu.cn
# High Order Momentum Modes by Resonant Superradiant Scattering
Xiaoji Zhou School of Electronics Engineering $\&$ Computer Science, Peking
University, Beijing 100871, China Jiageng Fu School of Electronics
Engineering $\&$ Computer Science, Peking University, Beijing 100871, China
Xuzong Chen School of Electronics Engineering $\&$ Computer Science, Peking
University, Beijing 100871, China
###### Abstract
The spatial and time evolutions of superradiant scattering are studied
theoretically for a weak pump beam with different frequency components
traveling along the long axis of an elongated Bose-Einstein condensate.
Resulting from the analysis for mode competition between the different
resonant channels and the local depletion of the spatial distribution in the
superradiant Rayleigh scattering, a new method of getting a large number of
high-order forward modes by resonant frequency components of the pump beam is
provided, which is beneficial to a lager momentum transfer in atom
manipulation for the atom interferometry and atomic optics.
###### pacs:
03.75.Kk, 42.50.Gy, 32.80.Lg
## I Introduction
Atom interferometry is a valuable tool for studying scientific and technical
fields such as precision measurements and quantum information, and a very
bright source is Bose-Einstein Condensate (BEC) of atomic gases. In which an
important technique is to manipulate the translational motion of atoms and
transfer atoms coherently between two localities in position and momentum
cronin . To obtain the momentum transfer, one useful method is two-photon
Bragg diffraction, where two laser beams impinge upon atoms, whose atoms can
undergo stimulated light-scattering events by absorbing a photon from one of
the beams and emitting into the other. The momentum transfer is determined by
the difference in the wave vectors of the beams, and the frequency difference
defines the corresponding energy transfer Brunello . We will introduce another
method to obtain a large number of high-order momentum modes by the resonant
superradiant scattering from a BEC for a weak pump beam with several frequency
components.
A typical superradiance experiment consists in a far off-resonant laser pulse
traveling along the short axis of a cigar-shaped BEC sample Inouye1999science
, the scattered lights, called end-fire modes, propagate along the long axis
of the condensate, and the recoiled atoms are refereed to as side modes. A
series of experiments Schneble2003scince ; 1999 ; Bar-Gill2007arxiv ;
Courteille2 ; sadler have sparked related interests in phase-coherent
amplification of matter waves Schneble2003scince ; 1999 , quantum information
Bar-Gill2007arxiv , collective scattering instability Courteille2 , and
coherent imaging sadler . Several theoretical descriptions of these
cooperative scattering in BEC with single-frequency pump have also been
presented Moore1999prl ; Pu2003prl ; Zobay2006pra ; guo .
For the long and weak pump beam, we can observe the forward peaks correspond
to Bragg diffraction of atoms Inouye1999science , where the high order
scattering is limited by detuning barriers for the end-fire mode radiation
Zobay1 . On the other hand, a X-shaped recoiling pattern is demonstrated in a
short and strong pulse as Kapitza-Dirac diffraction of atoms
Schneble2003scince , where an atom in the condensate absorbs a photon from the
pump laser, then emits a photon into an end-fire mode, and recoils forwardly.
Meanwhile another atom absorbs a photon from the end-fire modes, emits into
the pump beam and finally recoils backwardly. In this case, there is an energy
mismatch of four times the one-photon recoil kinetic energy $\hbar\omega_{r}$
in backward scattering, which then remains very weak unless a short pumping
pulse with a broad spectrum is used. Hence, two phase-locked incident lasers
with the frequency difference $\Delta\omega$ compensating for the energy
mismatch has been used Bar-Gill2007arxiv ; yang ; Cola , which is named
resonant superradiance, where a large number of backward recoiling atoms can
be produced.
Followed that, it is attractive to extent this idea to achieve a high momentum
transfer by overcoming the detuning barriers, by a weak and long pump beams
with the resonant frequency. It requires to analysis the competition between
the different transition channels and the spatial distribution of different
modes. Because the above traditional superradiant-scattering configuration
involves many atomic side modes coupled together, to simplify it, we chose
another configuration where a pump beam travels along the long axis of the
BEC. This scheme is widely studied in photon echo Piovella , decoherence
2005Italy , spatial distribution effects li and self-organized formation of
dynamic gratings Hilliard . Since the pulse length is far longer than the
initial spontaneous process Zobay2006pra , we choose the semi-classical theory
which can well describe the experimental results Zobay2006pra ; Bar-
Gill2007arxiv ; yang .
In this paper, we first introduce the semi-classical theory for the
superradiance scattering with a several-frequency pump in the weak coupling.
Then the spatial and time evolutions of scattered modes are analyzed for one-
frequency pump beam. Followed that, in the case of two-frequency pump, we find
the backward first order scattering mode is suppressed at the resonant
condition $\Delta\omega=8\omega_{r}$ and the forward second order mode is
enhanced, resulting from the combination of mode competition effects and
spatial distribution of the modes. The case of the three-frequency pump beams
for a lager number of the forward third order scattering modes, and the higher
modes for more resonant frequencies are studied, which supplies a new method
to get a large number of atoms in higher order forward modes. Finally, some
discussion and conclusion are given.
Figure 1: (Color online) Our experimental scheme. A cigar-shape BEC is
illuminated by a far off-resonant laser pulse along its long axis
$\mathbf{\hat{z}}$. Collective Rayleigh scattering induces superradiance. Two
end-fire modes, which are also along $\mathbf{\hat{z}}$ axis, form in
superradiance process and the 1st-order recoiled atoms obtain a momentum of
$2\hbar\mathbf{k}$.
## II Model for a multiple-frequency end-pumped beam
We consider the pump laser, with amplitude $\mathcal{E}_{l}(t)$, polarization
$\mathbf{e_{y}}$, wave vector $\mathbf{k_{l}}$, frequencies $\omega_{l}$ and
$\omega_{l}-\Delta\omega_{n}$, propagating along the long axis
$\mathbf{\hat{z}}$ of an elongated BEC,
$\mathbf{E}_{l}=\mathcal{E}_{l}(t)\mathbf{e_{y}}[(1+\Sigma_{n}e^{i\Delta\omega_{n}t})e^{i(k_{l}z-\omega_{l}t)}+c.c.]/2$,
as shown in Fig. 1. When supperradiant Rayleigh scattering happens, end-fire
modes spread along the same axis. The $\mathcal{E}_{+}$ mode has the same
direction as the incident light and mainly interacts with the right part of
the condensate, and the $\mathcal{E}_{-}$ mode overlaps with the left part of
the condensate. The atoms are recoiled to some discrete momentum states with
momentum $2m\hbar\mathbf{k}$, where $m$ is an integer and the wave vector of
end-fire mode light $k$ is approximated as $k_{l}$ for energy conservation.
The total electric field
$\mathbf{E}(\mathbf{r},t)=\mathbf{E}^{(+)}+\mathbf{E}^{(-)}$ is given by Bar-
Gill2007arxiv ; Zobay2006pra ; yang ; li
$\displaystyle\mathbf{E}^{(+)}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle[(1+\sum_{n}e^{i\Delta\omega_{n}t})\mathcal{E}_{l}(t)e^{-\mathrm{i}(\omega_{l}t-k_{l}z)}/2$
(1) $\displaystyle+$ $\displaystyle\mathcal{E}_{-}(z,t)e^{-\mathrm{i}(\omega
t+kz)}]\mathbf{e_{y}}$
where $\omega=ck$, $\mathbf{E}^{(-)}=\mathbf{E}^{(+)*}$, and $\mathcal{E}_{+}$
is ignored because it has the same wave vector as the pump beam but is very
small in comparison to $\mathcal{E}_{l}$. $\Delta\omega_{n}$ satisfies the
condition $\Delta\omega_{n}\ll\omega_{l}$ Bar-Gill2007arxiv and the initial
phases of the different frequency components are assumed to be zero.
Since the BEC is tightly constrained in its short axis ($\mathbf{\hat{x}},\
\mathbf{\hat{y}}$) in the present superradiance setup and the Fresnel number
of the optical field is around 1, one dimensional approximation is usually
used Inouye1999science ; Bar-Gill2007arxiv ; li ; Hilliard . We expand the
wavefunction of the condensate $\psi(\mathbf{r},t)$ in momentum space,
$\psi(\mathbf{r},t)=\sum_{m}{\phi_{m}(z,t)}e^{-i(\omega_{m}t-2mkz)}$, where
$\phi_{m}(z,t)=\psi_{m}(z,t)/\sqrt{A}$, $\omega_{m}=2\hbar m^{2}k^{2}/M$,
$m=0$ corresponds to the residual condensates, $m\neq 0$ denotes the side
modes, and $A$ is the average cross area of the condensate perpendicular to
$\mathbf{\hat{z}}$. Using the Maxwell-Schrödinger equations, we obtain
dynamics equations for $\phi_{m}(z,t)$,
$\displaystyle\mathrm{i}\frac{\partial\phi_{m}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\hbar}{2M}\frac{\partial^{2}\phi_{m}}{\partial
z^{2}}-\frac{2\mathrm{i}m\hbar k}{M}\frac{\partial\phi_{m}}{\partial z}$ (2)
$\displaystyle+$
$\displaystyle\bar{g}\left[\mathcal{E}_{-}^{*}\phi_{m-1}e^{-4\mathrm{i}(1-2m)\omega_{r}t}+\mathcal{E}_{-}\phi_{m+1}e^{-4\mathrm{i}(1+2m)\omega_{r}t}\right],$
where $\omega_{r}=\hbar k_{l}^{2}/2M$ is the recoil frequency, the coupling
between modes is given by
$\bar{g}(t)=g\left(1+\sum_{n}e^{i\Delta\omega_{n}t}\right),$ (3)
with the coupling factor $g=\sqrt{3\pi c^{3}R/(2\omega_{l}^{2}AL)}$, $R$ is
the Rayleigh scattering rate of the pump components, and $L$ is the BEC
length.
The first term on the right-hand-side of Eq.(2) describes the dispersion of
$\phi_{m}$, and the second term gives rise to their translation. The terms in
square brackets describe the atom exchange between $\phi_{m}$ and $\phi_{m+1}$
or $\phi_{m-1}$ through the pump laser and end-fire mode fields. An atom in
mode $m$ may absorb a laser photon and emit it into end-fire mode
$\mathcal{E}_{-}$, and the accompanying recoil drives the atom into $m+1$
mode, hence atoms with mode $m+1$ can emerge in forward scattering. On the
other hand, in the backward scattering, atoms with mode $m$ absorb one
$\mathcal{E}_{-}$ mode photon, deposit it into the laser mode and go into mode
$m-1$. The envelope function of end-fire mode $\mathcal{E}_{-}$ is given by
$\mathcal{E}_{-}=-\mathrm{i}\frac{\omega_{r}\bar{g}}{2c\varepsilon_{0}}\int^{+\infty}_{z}\mathrm{d}z^{\prime}\sum_{m}\phi_{m}(z^{\prime},t)\phi_{m+1}^{*}(z^{\prime},t)e^{\mathrm{i}4(2m+1)\omega_{r}t},$
(4)
indicating that the end-fire mode field $\mathcal{E}_{-}$ is due to the
transition between $m$ and $m+1$ mode and the magnitude of $\mathcal{E}_{-}$
depends on the spatial overlap between the two modes. In addition, there is a
frequency difference of $8\omega_{r}$ between adjacent modes.
## III The spatial and time evolution of scattered modes with a single-
frequency pump beam
For explaining effects of spatial distribution and the depletion mechanism in
the scattering from BEC, we first study the case of a single-frequency pump in
the weak coupling regime. The evolution of spatial distribution of atomic side
modes and optical end-fire mode are depicted in Fig.2, where the original BEC
is assumed to be symmetrical.
Figure 2: (Color online) Spatial distribution of the atomic side modes
$|\psi^{2}|$ and the light end-fire mode $|\varepsilon_{-}|$ in the weak
coupling $(g=1.25\times 10^{6}s^{-}1)$ in case of a single-frequency pump for
different pulse durations: 150us (a); 200us (b); 250us (c); 350us (d). The
condensate mode $m=0$ is the solid line, the forward first-order side mode
$m=1$ is the dash-dotted line, and the end-fire mode is the dashed line.
Superradiance first starts at the leading-edge of the BEC, as shown in Fig.2
(a). The end-fire mode $\mathcal{E}_{-}$ monotonically increases at the
beginning and becomes strong on the side of the end-pump, and it has a large
overlap with the BEC. The atomic side modes and the optical-field modes are
well localized at the condensate edge. Hence, the recoiled atoms mainly come
from this edge of the condensate, and the forward first order mode $m=1$
emerges due to the overlap between the condensate at $m=0$ and the end fire
mode $\mathcal{E}_{-}$. Then at some point the condensate is completely
scattered to mode $m=1$ and the atoms are transferred back to the edge,
leading to a minimum in the condensate density and regrowth at the edge, as
shown in Fig.2 (b), which appears like a Rabi oscillation between the
condensate and first-order side mode. When the overlap between mode $m=0$ and
$\mathcal{E}_{-}$ is significantly large, the minimum point of mode $m=0$ and
the peak of mode $m=1$ move from the leading-edge to the center of the BEC, as
shown in Fig.2 (c). When the regrowth part of mode $m=0$ is comparable to mode
$m=1$, it will be scattered to mode $m=1$ again. Hence mode $m=1$ also has an
edge regrowth. Due to the movement of the first peak and the regrowth from the
edge, mode $m=1$ will have a minimum point too, as shown in Fig.2 (d). This
distribution shows the evolution of side modes in space and the absence of
backward-scattering modes in the weak coupling regime.
The distribution of the first-order side mode closely connects with the end-
fire mode, and $\mathcal{E}_{-}$ is simply the result of the coupling between
$\phi_{0}$ and $\phi_{1}$. When the condensate population at some point z is
completely pumped to the first-order side mode, the population of mode $m=1$
and $\mathcal{E}_{-}$ are at maximum. When the first-order side mode absorbs
end-fire mode photons leading to the regrowth of the condensate, the
populations of mode $m=1$ and $\mathcal{E}_{-}$ will reach minimum. The
asymmetry could be explained by Eq.(2), where $\phi_{0}$ and $\phi_{1}$ are
coupled through $\mathcal{E}_{-}$ which is very small at the tailing-edge of
the condensate.
The evolution of the side modes and the end-fire mode indicates that the
scattering is a localized process. In this end-pumping configuration, the
scattering first starts on the leading edge of the BEC and then moves towards
the tailing edge.
## IV Mode competition for a two-frequency pump beam
To investigate the effect of the two-frequency pump beam in the case of end-
pumping, the different frequency components of the end-fire mode which
indicate the energy change during the scattering are depicted in Fig. 3. The
momentum of side mode $m=n$ is $2n\hbar\textbf{k}$, and its kinetic energy is
$4n^{2}\hbar^{2}\textbf{k}^{2}/2M=4n^{2}\hbar\omega_{r}$. For the pump
component with frequency $\omega_{l}$, atoms from the condensate are pumped to
the side mode $m=1$ and emit end-fire mode photons with frequency
$\omega_{l}-4\omega_{r}$ spontaneously. However, in the backward scattering
process, an atom in the condensate absorbs the end-fire mode
($\omega_{l}-4\omega_{r}$) and emits a photon with frequency $\omega_{l}$ back
into the pump laser. Since energy is not conserved in backward-scattering, the
backward side mode is not populated in weak-pulse regime. Side mode $m=2$ is
also not populated due to the energy barrier. However, if we use the two
components pump laser with frequency difference $8\omega_{r}$, i.e. resonant
frequency difference, the energy mismatch can be compensated by the pump
laser.
Figure 3: (Color online) Light-field components of a two-frequency pump laser.
The broad arrows are the pump laser and narrow ones are the end-fire mode
(scattering optical field). In a spontaneous process, atoms in the condensate
absorb photons from the pump laser with frequencies $\omega_{l}$ and
$\omega_{l}-8\omega_{r}$, are scattered to side mode $m=1$ and emit end-fire
mode photons with frequency $\omega_{l}-4\omega_{r}$ (dashed arrow) and
$\omega_{l}-12\omega_{r}$ (dotted arrow), respectively. Meanwhile, atoms in
the condensate can also absorb end-fire mode photons with frequency
$\omega_{l}-4\omega_{r}$, be scattered back to side mode $m=-1$ and emit
photons with frequency $\omega_{l}-8\omega_{r}$(solid arrow), resonating to
one of the pump laser components. The side mode $m=1$ can absorb pump laser
photons with frequency $\omega_{l}$ and be scattered to mode $m=2$, emitting
photons with frequency $\omega_{l}-12\omega_{r}$ resonating to the existing
end-fire mode.
Although the resonant condition for the backward mode is satisfied, it should
be noticed that two scattering channels exist almost simultaneously. One is
atoms scattered from side mode $m=0$ to $m=-1$ and the other is from $m=1$ to
$m=2$, resulting in mode competition. The transition from mode $m=1$ to $m=2$
requires absorbtion of photons from pump laser, while the backward transition
takes photons from the end-fire mode. Because the intensity of the pump laser
is far greater than that of the end-fire mode, the transition from $m=1$ to
$m=2$ has a bigger probability than the transition from $m=0$ to $m=-1$. Thus
the population of the backward mode $m=-1$ is suppressed even at the resonant
condition, while the forward mode $m=2$ is enhanced.
However, the existence of competition between these two channels may not lead
to the suppression of the backward mode. If these two channels happen in
different spacial parts of the condensate, then both of side mode $m=-1$ and
$m=2$ will be enhanced. The suppression of backward mode $m=-1$ and the
enhancement of mode $m=2$ need that these two scattering channels happen in
the same area. Therefore, the spatial distribution effect should be
considered.
Figure 4: (Color online) Spatial distribution of the side modes $|\psi^{2}|$
and the end-fire mode $|\varepsilon_{-}|$ in the weak coupling regime
$(g=1.25\times 10^{6}s^{-}1)$ with the two-frequency pump for different pulse
durations: 150us (a); 200us (b); 250us (c); 300us (d). Condensate mode $m=0$
is the solid line-1, backward first-order side mode $m=-1$ is the solid
line-2, forward first-order side mode $m=1$ is the dash-dotted line, forward
second-order side mode $m=2$ is the dashed line, and end-fire mode is the
dotted line.
We analyze the spatial effect when second-order forward side mode and backward
side mode are populated at the resonant condition $\Delta\omega=8\omega_{r}$.
The evolution of spatial distribution of side modes and end-fire mode is shown
in Fig.4. Superradiance first starts on the leading edge of the BEC, as shown
in Fig.4(a). Although the backward first-order side mode $m=-1$ is populated
through the overlap between end-fire mode $\mathcal{E}_{-}$ and side mode
$m=0$, it is very small and emerges at the leading-edge of the BEC. Since the
overlap between end-fire mode and side mode $m=1$ is in the same area, the
population of side mode $m=2$ is obvious on this edge, as shown in Fig.4(b).
Side mode $m=2$ grows more rapidly than side mode $m=-1$, which means more
atoms are scattered from side mode $m=1$ to $m=2$ than that from $m=0$ to
$m=-1$.Then the first peaks of side modes $m=1$ and $m=2$ move to the center
of the BEC, as shown in Fig.4(c). Though the movement of the peaks is similar
to that in the case of a single-frequency pump laser, one major difference is
that the regrowth of side mode $m=0$ is very little, hence nearly all the
atoms on this edge are forwardly scattered. Due to the nearly-complete
depletion of the condensate, atoms are mainly transferred between side mode
$m=1$ and $m=2$. The apparent regrowth of side mode $m=1$ on the leading-edge
shown in Fig.4(d) indicates that there are Rabi oscillations between side
modes $m=1$ and $m=2$ in the depleted area of the condensate.
The above phenomenon is different from the case of the pump laser traveling
along the short axis. In the latter case, a correlation between the center
depletion of the BEC and backward mode was reported in Ref. Zobay2006pra .
However, in our case, such correlation does not exist because side mode $m=2$
emerges on the edge of the BEC. As a consequence of the edge depletion of the
BEC, backward side mode $m=-1$ is not populated significantly, because the
end-fire mode mainly distributes in the leading edge where side mode $m=0$
suffers the strongest depletion. Thus only a small number of atoms in the
residual condensate can absorb end-fire mode photons and be scattered
backwardly. In another word, the emergence of side mode $m=2$ suppress
backward-scattering atoms. Therefore the efficiency of getting $m=2$ mode with
this two-frequency pump is strongly enhanced while greatly suppressed for the
backward side mode.
Figure 5: (Color online) Normalized side mode populations versus time: (a) for
a single-frequency pump beam; (b) for a two-frequency resonant pump beam. In
both cases the coupling constant is kept $g=1.55\times 10^{6}s^{-1}$. The side
mode are: m=-1 (solid); m=1 (dotted); m=2 (dashed).
The time evolution of several side modes populations normalized by the total
atom number are depicted by Fig.5. Fig.5 (a) shows that using a single-
frequency pump laser cannot produce backward mode $m=-1$ or forward higher
mode $m=2$ in the weak-pulse regime. Using a resonant two-frequency pump beam
with the same intensity, modes $m=-1$ and $m=2$ increased, as shown in Fig.5
(b), however, the forward mode is greatly enhanced while the backward mode
remains very small.
## V The third order forward modes Enhanced with a three-frequency pump beam
Figure 6: (Color online) The light-field components of a three-frequency pump
laser. The broad arrows are the pump laser and narrow ones are the end-fire
mode.
The second forward side mode $m=2$ is greatly enhanced with a resonant two-
frequency pump beam, however, the populations of higher forward modes such as
$m=3$ are very small as the channel from the second forward mode to the third
forward mode is not resonant with the exiting optical field. To get a large
number mode for $m=3$, Fig.6 depicts the scheme of the three-frequency pump
beam with the frequencies of the pump laser $\omega_{l}$,
$\omega_{l}-8\omega_{r}$ and $\omega_{l}-16\omega_{r}$. The frequency
components $\omega_{l}$, $\omega_{l}-8\omega_{r}$ and
$\omega_{l}-16\omega_{r}$ both have the resonant frequency difference. Hence,
there could be two channels to form the backward side mode $m=-1$ but the
enhancement of the backward scattering is small because of the formation of
higher forward side modes. There are also two channels to form side mode
$m=2$. One thing different from the two-frequency pump beam is that there is
also a channel to form side mode $m=3$ for the reason that atoms in side mode
$m=2$ absorb pump laser photons with frequency $\omega_{l}$, are then
scattered to mode $m=3$ and eventually emit end-fire mode photons with
frequency $\omega_{l}-20\omega_{r}$ which is resonant to an existing end-fire
mode. This means that more atoms in side mode $m=2$ will be pumped to side
mode $m=3$ and less will be transferred back to side mode $m=1$, a competition
between side mode $m=3$ and $m=1$ is set up. As a result, side mode $m=3$ will
be enhanced and $m=1$ will be reduced relatively.
Figure 7: (Color online) Normalized side mode populations versus time with the
coupling constant $g=1.55\times 10^{6}s^{-1}$: (a) for a three-frequency pump
laser: $m=-1$ (dash-dotted), $m=1$ (dotted), $m=2$ (dashed), $m=3$ (solid) ;
(b) for a five-frequency pump laser: $m=1$ (dotted), $m=3$ (dash-dotted),
$m=4$ (dashed), $m=5$ (solid) .
Fig.7(a) is the simulated result of the time evolution of normalized side mode
populations for a three-frequency pump beam. We could see that side mode $m=3$
would be strongly enhanced at long time while side mode $m=1$ reduced.
## VI Momentum transfer in the high order forward modes
From the above discussion we know that using multi-resonant frequencies is a
promising way to get a large number of higher forward modes. When a pump laser
has frequency components
$\omega_{l},\omega_{l}-8\omega_{r},\cdots,\omega_{l}-(n-1)*8\omega_{r}$,
satisfying $(n-1)*8\omega_{r}\ll\omega_{l}$, with the kinetic energy of mode
$m=n$ equal to $4n^{2}\hbar\omega_{r}$, then after the condensate atoms
spontaneously scattered to mode $m=1$, the end-fire mode will have frequency
components
$\omega_{l}-4\omega_{r},\omega_{l}-12\omega_{r},\cdots,\omega_{l}-(2n-1)*4\omega_{r}$.
For resonance concern, mode $m=1$ will absorb photons from the pump components
$\omega_{l},\omega_{l}-8\omega_{r},\cdots,\omega_{l}-(n-2)*8\omega_{r}$ and
emits end-fire mode photons with frequency
$\omega_{l}-12\omega_{r},\cdots,\omega_{l}-(2n-1)*4\omega_{r}$ which are
resonant with existing end-fire mode, so mode $m=2$ is produced. Like mode
$m=1$, modes $m=2,m=3,\cdots,m=n-1$ can absorb pump photons and emit photons
resonant to the existing end-fire mode. For example, mode $m=n-1$ will absorb
photons with frequency $\omega_{l}$ and emits photons with frequency
$\omega_{l}-(2n-1)*4\omega_{r}$. Therefore atoms could finally be transferred
to mode $m=n$. Note that mode $m=n$ cannot emit resonant end-fire mode, so
mode $m=n$ will be enhanced. To show it, Fig.7(b) is the simulated result of
the time evolution of normalized side mode populations for a five-frequency
pump beam. We could see that side mode $m=5$ would be strongly enhanced.
## VII Discussion and Conclusion
In the experiment, to get several resonant frequencies, the laser beam from an
external cavity diode laser can be split into several parts, and their
frequencies are shifted individually by acoustic-optical modulators (AOMs)
which are driven by phase-locked radio frequency signals, as demonstrated in
the case of two resonant frequency Bar-Gill2007arxiv ; yang . Therefore, the
frequency difference between the beams can be controlled precisely.
Furthermore, to avoid the reflection from the glass tube and formation of
Bragg scattering in the experiment, the pump beam can actually deviate a few
degrees from the long axis, as shown in the experiments 2005Italy ; li ;
Hilliard .
Different to the works in the configuration where the pump beam travels along
the short axis of the condensate with the resonant frequency yang , where a
large number of backward scattering is obvious in a two-frequency pump beam,
the backward scattering is suppressed and the forward second-order mode is
obviously enhanced in our case. This is due to mode competition between the
forward second-order mode and the backward mode and local depletion of the
superradiant process.
We have not considered the initial quantum process because its time scale is
very small, shorter than $1\mu s$. In this quantum process there is also mode
competition to form the end-fire modes along the long axis and suppress the
emission on the other direction. This is different concept from what has been
discussed above, in which case mode competition exists in the different
channels satisfying the energy match and spatial condition.
For the pump beam with several resonant frequencies, not only can we obtain
the high order momentum transfer which is important in the momentum
manipulation for atom interferometry, but also the above analysis is useful to
understand the interplay between the matter wave and light in the matter wave
amplification Schneble2003scince ; 1999 , atomic cooperative scattering in the
optical lattice xu , and by the pump with a noisy laser robb ; zhou .
In conclusion, superradiant scattering from BEC is studied with incident light
having different frequency components traveling along the long axis of the BEC
in the weak coupling regime. It provides a method to get high forward modes by
adding different frequency components to the pump beam. This is the result of
both mode competition for the concern of energy and the local depletion of the
spatial distribution. Our results shows that the spatial effects and mode
competition are very important even in the case of resonant superradiance.
We thank Thibault Vogt, Lan Yin for critical reading of the manuscript and
comments. Thank L. You for his helpful discussion. This work is partially
supported by the state Key Development Program for Basic Research of China
(No.2005CB724503, and 2006CB921402,921401), and by NSFC (No.10874008, 10934010
and 60490280).
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|
arxiv-papers
| 2009-12-06T09:56:40 |
2024-09-04T02:49:06.877132
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaoji Zhou, Jiageng Fu, Xuzong Chen",
"submitter": "Xiaoji Zhou",
"url": "https://arxiv.org/abs/0912.1088"
}
|
0912.1200
|
# Extending Karger’s randomized min-cut Algorithm for a Synchronous
Distributed setting
Shine S
K. Murali Krishnan
Dept. of Computer Science and Engineering College of Engineering Trivandrum
Kerala, India shine@cs.cet.ac.in Dept. of Computer Science and Engineering
National Institute of Technology Calicut Kerala, India kmurali@nitc.ac.in
###### Abstract
A min-cut that seperates vertices $s$ and $t$ in a network is an edge set of
minimum weight whose removal will disconnect $s$ and $t$. This problem is the
dual of the well known $s-t$ max-flow problem. Several algorithms for the min-
cut problem are based on max-flow computation although the fastest known min-
cut algorithms are not flow based. The well known Karger’s randomized
algorithm for min-cut is a non-flow based method for solving the (global) min-
cut problem of finding the min $s-t$ cut over all pair of vertices $s,t$ in a
weighted undirected graph. This paper presents an adaptation of Karger’s
algorithm for a synchronous distributed setting where each node is allowed to
perform only local computations. The paper essentially addresses the
technicalities involved in circumventing the limitations imposed by a
distributed setting to the working of Karger’s algorithm. While the
correctness proof follows directly from Karger’s algorithm, the complexity
analysis differs significantly. The algorithm achieves the same probability of
success as the original algorithm with $O(mn^{2})$ message complexity and
$O(n^{2})$ time complexity, where $n$ and $m$ denote the number of vertices
and edges in the graph.
###### category:
Distributed Algorithms Metrics
###### keywords:
complexity measures
###### category:
Graph Theory Miscellaneous
###### keywords:
Max-flow, Min-cut
††terms: Network-flow
## 1 Introduction
The problem of computing the minimum-cut in a weighted graph has been
classically studied in literature as the dual of the well known max-flow
problem for networks [5] and classical solutions to the max-flow problem were
used to solve the min-cut problem. These algorithms could be classified as
those based on augmenting paths [5, 4], improvements to the augmenting path
approach based on blocking flows[3, 12] and those based on pre-flow method
introduced by Goldberg and Tarjan[6]. The best known algorithms for the max-
flow problem are based on the preflow approach[1, 17, 7]. The max-flow problem
also has been recently studied in a distributed setting in [2].
Further investigations revealed that there are more efficient direct solutions
to the min-cut problem (without solving max-flow and taking the dual).
Nagamochi and Ibaraki[13] published the first deterministic global minimum cut
algorithm that is not based on flow, but was rather complicated. Stoer and
Wagner[16] presented a simple deterministic global minimum cut algorithm which
runs in $O(mn+n^{2}\log{n})$.
Karger[8] presented the first randomized global min-cut algorithm which runs
in $O(mn^{2}\log^{3}n)$. The running time of a single trial of the algorithm
is $O(m\log^{2}{n})$. The algorithm has to be repeated $n^{2}\log{n}$ times to
achieve a high success probability of $1-\frac{1}{n}$. Karger and Stein[9]
further improved its running time to $O(n^{2}\log^{3}n)$ for the same
probability.
Recently there has been revived interest in the min-cut problem owing to its
applications to network coding and wireless sensor networks [15, 10, 14].
Sensor networks operate in a distributed setting and motivates a solution to
the problem in a distributed setting.
In this paper, we show how Karger’s algorithm[8] can be adapted to efficiently
solve the min-cut problem in a distributed setting. We assume a very general
model of a graph where each node knows only information about its neigbours.
It is assumed that the storage capacity of a node is bounded linearly in the
size of the number of its neigbours and the computing capacity of a node is
bounded polynomially in the number of its neighbours. The assumption is
reasonable as each node must have storage and processing capacity sufficient
to keep track of communication with its neighbours. The nodes can perform
local computations and can communicate only with its neighbours along the
edges of the graph. Our objective is to find the value of the global min-cut
and communicate the same to all the nodes. Moreover, each node must know which
among the edges incident on it are present in the min-cut computed. While the
correctness proof follows directly from Karger’s algorithm, the complexity
analysis differs significantly. We show that for a graph of $n$ vertices and
$m$ edges, the algorithm computes the global min-cut with probability atleast
$1-\frac{1}{n}$ with $O(mn^{2})$ message complexity and $O(n^{2})$ time
complexity when there is a global clock for synchronization. We note that
although the assumption of a global clock may be impractical in applications
like sensor networks, there are standard techniques for converting synchronous
distributed algorithms to asynchronous algorithms, with some loss in
computational efficiency[11]. We pursue the simpler synchronous setting here
as it allows a less cumbersome presentation of the algorithm and a simple
analysis.
## 2 The Algorithm
### 2.1 A Brief Description
Assume that given a weighted graph $G=(V,E,w)$ where $E\subseteq V\times V$
and $w:E\rightarrow R^{+}\cup\\{0\\}$ is given(We use the terms network and
graph interchangeably). In our algorithm $N_{u}$ represents the neighbourhood
of vertex $u$, $weight_{u}$ represents the present edge weights of $N_{u}$,
that is, for each $v\in N_{u},weight_{u}[v]$ indicates the weight of edge
$(u,v)$. $rank_{u}[v]$ is the rank of edge $(u,v)$, a random number which is
uniformly chosen between 1 and $m^{k}$(for some fixed $k\geq 5$), on each
trial. $maxrank$ represents the maximum value of rank among all the edges.
Initially $maxrank_{u}$ is defined as the maximum rank of the edges connected
to vertex $u$. The algorithm sets $maxrank=Max_{u\in V(G)}maxrank_{u}$. The
$status$ of a vertex may be $ACTIVE$ or $INACTIVE$ (initially $ACTIVE$).
$status_{u}=INACTIVE$ if all neighbouring edge weights of vertex $u$ are 0,
which means that vertex cannot initiate the contraction process. We call an
edge active if at least one of its end points is active.
The algorithm proceeds by simulating edge contractions as in[8], by collecting
vertices joined together by contraction into vertex groups. Edges within a
group are inactive as they cannot be further contracted. At each step, an
active edge of maximum rank is chosen for contraction. Since edge ranks are
assigned uniformly at random, each active edge has equal probability for
getting contracted. The algorithm continues contractions till only two vertex
groups remain and the set of edges across the two groups is chosen as the
mincut for that trial. The smallest cut found in $n^{2}\log{n}$ trials will be
the mincut with probability $1-\frac{1}{n}$.
The variable $lastmsg_{u}$ stores the last message received at vertex $u$(used
to reduce message flooding) and the boolean variable $stop_{u}$ is set to
$true$ when only two vertex groups are remaining and no more contraction can
be made, and set to $false$ otherwise.
The variable $g_{u}$ represents the present group id of vertex $u$, initially
$g_{u}=u$. Initially there are $n$ groups, one for each vertex. As
contractions progress, the number of groups reduces and we set
$weight_{u}[v]=0$ if $g_{u}=g_{v}$ and $weight_{u}[v]\neq 0$ otherwise. The
following description presents a high level view of the algorithm.
Algorithm 1 distributed-mincut-in-a-nutshell()
assign a $rank$ (between 1 and $m^{k}$) to each non-zero weighted edge.
{Algorithm 4}
At each node $u$ of the network execute the following:
find $maxrank_{u}$ of each vertex $u$ locally. {Algorithm 5}
find the vertex $x$(with largest vertex id) having the maximum value of
$maxrank$. {Algorithms 6, 14}
if there are only two groups then {Algorithms 7, 9, 15, 16, 21}
compute local mincut $mc_{u}$ by summing the non-zero edge weights of vertex
$u$. {Algorithm 10}
compute global mincut by summing up all local mincuts. {Algorithms 11, 22}
broadcast the mincut to all nodes and stop. {Algorithms 12, 23}
else
contract two vertex groups by making the edge weights between them zero and
group ids equal to the value of $maxrank$ (The contraction process is
initiated by the vertex $x$). {Algorithms 7, 8, 17}
repeat the algorithm
end if
### 2.2 Details of the Algorithm
Each node in the network executes Algorithm 2 described below. Here, the
function $initialize()$ initializes the group id of each vertex with its
vertex id. The function $assign$-$rank()$ assigns a $rank$ to each non-zero
weighted edge with in the network, with a random value between 1 and $m^{k}$.
The time complexity for this function is $O(n)$. The function
$find$-$local$-$maxrank()$ computes the maximum rank within its neighbourhood,
with time complexity $O(n)$. The function $find$-$global$-$maxrank()$ computes
the maximum of all the $local$-$maxranks$ within the network, with time
complexity $O(n)$ and message complexity $O(mn)$.
The function $check$-$eligibility$-$and$-$contract()$ checks whether there are
more than two groups within the network and if so, contracts two groups by
making all the edge weights between them zero and their group ids the same.
This can be accomplished with time complexity $O(n)$ and message complexity
$O(m)$. The function $check$-$termination$-$status()$ checks whether there are
only two groups within the network and if so, invokes mincut computation and
halts, otherwise the algorithm is repeated. This can be accomplished with time
complexity $O(n)$ and message complexity $O(m)$. All the above mentioned
functions except $initialize()$ has to be repeated $n-2$ times.
The function $find$-$local$-$mincut()$ computes the sum of edge weights within
its neighbourhood, with time complexity $O(n)$. The function
$find$-$global$-$mincut()$ computes the the sum of all $local$-$mincuts$
within the network, with time complexity $O(n^{2})$ and message complexity
$O(mn)$. Node $u$ messages to node $u+2^{i-1}$ in step $i$, for
$i\in\\{1,...\log{n}\\}$ to ensure that the messages propagate to all nodes in
$O(n^{2})$ time with only $O(mn)$ messages. The function
$broadcast$-$mincut()$ broadcasts the computed mincut value to all the nodes
within the network, which is done with time complexity $O(n)$ and message
complexity $O(m)$. The function $synchronize()$ allows the nodes to wait for
some time so that the same instruction can be executed by each node, in the
next time step. This function waits for $O(n)$ steps.
Algorithm 2 distributed-mincut() //To be executed at each node
initialize()
repeat
assign-rank()
find-local-maxrank()
find-global-maxrank()
synchronize()
check-eligibility-and-contract()
synchronize()
check-termination-status()
synchronize()
until $stop_{u}=true$
find-local-mincut()
find-global-mincut()
synchronize()
broadcast-mincut()
Algorithm 3 initialize()
$g_{u}\leftarrow u$
Algorithm 4 assign-rank()
{Rank of an edge to be assigned by higher numbered end-point}
for each $v\in N_{u}$ do
if $u>v$ then
if $weight_{u}[v]\neq 0$ then
$rank_{u}[v]\leftarrow$ a random number between 1 and $m^{k}$
else
$rank_{u}[v]\leftarrow$ 0
end if
send(SET-RANK, $rank_{u}[v]$) to $v$. {See Algorithm 13 for receipt of
message}
end if
end for
Algorithm 5 find-local-maxrank()
$maxrank_{u}\leftarrow max_{v\in N_{u}}(rank_{u}[v])$
Algorithm 6 find-global-maxrank()
send(FIND-MAX-RANK, $maxrank_{u}$) to each $v\in N_{u}$. {See Algorithm 14 for
receipt of message}
Algorithm 7 check-eligibility-and-contract()
$stop_{u}\leftarrow true$
if $maxrank_{u}=max_{v\in N_{u}}(rank_{u}[v])$ and $u>v$ then
if $\exists w\in N_{u}$ with $weight_{u}[w]\neq 0$ and $v\neq w$ and
$g_{v}\neq g_{w}$ then
$stop_{u}\leftarrow false$
contract()
else
send(IS-ELIGIBLE-CONTRACT, $u$, $g_{u}$, $g_{v}$) to each $x\in N_{u}$. {See
Algorithm 15 for receipt of message}
end if
end if
Algorithm 8 contract()
if $maxrank_{u}=max_{v\in N_{u}}(rank_{u}[v])$ and $u>v$ then
$weight_{u}[v]\leftarrow 0$
check-active()
$g_{u}\leftarrow maxrank_{u}$
send(SET-GROUP-ID, $g_{u}$, $g_{v}$, $maxrank_{u}$) to each $x\in N_{u}$ with
$weight_{u}[x]=0$. {See Algorithm 17 for receipt of message}
end if
Algorithm 9 check-termination-status()
send(STOP, $stop_{u}$) to each $x\in N_{u}$. {See Algorithm 21 for receipt of
message}
Algorithm 10 find-local-mincut()
if $status_{u}=ACTIVE$ then
$mc_{u}\leftarrow\sum_{v\in N_{u}}{weight_{u}[v]}$
else
$mc_{u}\leftarrow 0$
end if
Algorithm 11 find-global-mincut()
for $i\leftarrow 1$ to $\log{n}$ step by 1 do
for $j\leftarrow 2^{i-1}$ to $n-1$ step by $2^{i}$ do
if $u=j$ then
send(LOCAL-MC, $mc_{u}$, $u$, min($u+2^{i-1}$, $n$)) to each $v\in N_{u}$.
{See Algorithm 22 for receipt of message}
end if
synchronize()
end for
end for
Algorithm 12 broadcast-mincut()
if $u=n$ then
$mc_{u}\leftarrow mc_{u}/2$
send(MINCUT, $mc_{u}$, $u$) to each $v\in N_{u}$. {See Algorithm 23 for
receipt of message}
end if
Algorithm 13 upon receipt of (SET-RANK, $num$) msg from $w$
$rank_{u}[w]\leftarrow num$
Algorithm 14 upon receipt of (FIND-MAX-RANK, $m$) msg from $w$
{find maximum rank among all vertices}
if $m>maxrank_{u}$ then
$maxrank_{u}\leftarrow m$
send(FIND-MAX-RANK, $m$) to each $v\in N_{u}$ where $v\neq w$
end if
Algorithm 15 upon receipt of (IS-ELIGIBLE-CONTRACT, $v$, $g^{\prime}$,
$g^{\prime\prime}$) msg from $w$
{checks the eligibility of contraction}
if (IS-ELIGIBLE-CONTRACT, $v$, $g^{\prime}$, $g^{\prime\prime}$) $\neq
lastmsg_{u}$ then
if $\exists y\in N_{u}$ with $weight_{u}[y]\neq 0$ and $g_{y}\neq g^{\prime}$
and $g_{y}\neq g^{\prime\prime}$ then
send(ELIGIBLE-CONTRACT, $v$, $g^{\prime}$) to each $z\in N_{u}$ with
$weight_{u}[z]=0$ or ($weight_{u}[z]\neq 0$ and $g_{z}=g^{\prime}$). {See
Algorithm 16 for receipt of message}
else
send(IS-ELIGIBLE-CONTRACT, $v$, $g^{\prime}$, $g^{\prime\prime}$) to each
$z\in N_{u}$ with $weight_{u}[z]=0$ and $z\neq w$
end if
$lastmsg_{u}\leftarrow$ (IS-ELIGIBLE-CONTRACT, $v$, $g^{\prime}$,
$g^{\prime\prime}$)
end if
Algorithm 16 upon receipt of (ELIGIBLE-CONTRACT, $v$, $g^{\prime}$) msg from
$w$
if (ELIGIBLE-CONTRACT, $v$, $g^{\prime}$) $\neq lastmsg_{u}$ then
if $u=v$ then
$stop_{u}\leftarrow false$
contract()
else
send(ELIGIBLE-CONTRACT, $v$, $g^{\prime}$) to each $z\in N_{u}$, $z\neq w$
with $weight_{u}[z]=0$ or ($weight_{u}[z]\neq 0$ and $g_{z}=g^{\prime}$)
end if
$lastmsg_{u}\leftarrow$ (ELIGIBLE-CONTRACT, $v$, $g^{\prime}$)
end if
Algorithm 17 upon receipt of (SET-GROUP-ID, $g^{\prime}$, $g^{\prime\prime}$,
$newrank$) msg from $w$
{update group id of all vertices in the groups g’ and g” by $maxrank_{u}$ by
sending messages}
if $g_{u}\neq newrank$ then
$weight_{u}[w]\leftarrow 0$
check-active()
$g_{u}\leftarrow newrank$
if $status_{u}=ACTIVE$ then
for all $v\in N_{u}$ with $weight_{u}[v]\neq 0$ do
if $g_{v}=g^{\prime}$ or $g_{v}=g^{\prime\prime}$ or $g_{v}=newrank$ then
$weight_{u}[v]\leftarrow 0$
check-active()
send(SET-WEIGHT) to $v$. {See Algorithm 19 for receipt of message}
end if
end for
end if
send(SET-GROUP-ID, $g^{\prime}$, $g^{\prime\prime}$, $newrank$) to each $x\in
N_{u}$ where $weight_{u}[x]=0$
end if
Algorithm 18 synchronize()
{waits for all nodes to reach the same step of algorithm}
wait for $n$ pulses
Algorithm 19 upon receipt of (SET-WEIGHT) msg from $w$
$weight_{u}[w]\leftarrow 0$
check-active()
Algorithm 20 check-active()
if $\forall v\in N_{u},weight_{u}[v]=0$ then
$status_{u}=INACTIVE$
end if
Algorithm 21 upon receipt of (STOP, $t$) msg from $w$
{broadcast the information on the number of groups in the network}
if (STOP, $t$)$\neq lastmsg_{u}$ then
if $t=false$ then
$stop_{u}\leftarrow false$
send(STOP, $t$) to each $x\in N_{u}$
$lastmsg_{u}\leftarrow$ (STOP, $t$)
end if
end if
Algorithm 22 upon receipt of (LOCAL-MC, $mcut$, $x$, $v$) msg from $w$
{computes mincut partially}
if (LOCAL-MC, $mcut$, $x$, $v$)$\neq lastmsg_{u}$ then
if $u=v$ then
$mc_{u}\leftarrow mc_{u}+mcut$
else
send(LOCAL-MC, $mcut$, $x$, $v$) to each $y\in N_{u}$
end if
$lastmsg_{u}\leftarrow$ (LOCAL-MC, $mcut$, $x$, $v$)
end if
Algorithm 23 upon receipt of (MINCUT, $v$, $mincut$) msg from $w$
{broadcasts the mincut to all nodes}
if (MINCUT, $v$, $mincut$)$\neq lastmsg_{u}$ then
$mc_{u}\leftarrow mincut$
send(MINCUT, $v$, $mincut$) to each $y\in N_{u}$
$lastmsg_{u}\leftarrow$ (MINCUT, $v$, $mincut$)
end if
### 2.3 Correctness
First, we bound the probability of error created by edges getting the same
rank.
###### Lemma 2.3.1
The probability that two edges get the same rank in $n$ trials is $O(n^{-2})$.
###### Proof 2.1.
The rank is a value from the set $\\{1...m^{k}\\}$. The probability that two
edges $m$ and $m^{\prime}$ having the same rank,
$Pr[rank(m)=rank(m^{\prime})]\leq\frac{1}{m^{k}}$
Hence, $Pr[\exists(m,m^{\prime}):rank(m)=rank(m^{\prime})]\leq\\\
\sum_{(m,m^{\prime})\in E\times
E}{Pr[rank(m)=rank(m^{\prime})]}\leq\frac{m^{2}}{m^{k}}=\frac{1}{m^{k-2}}$
Thus, using the union bound, probability that there exists two edges $m$ and
$m^{\prime}$ having the same rank in $n$ iterations is
$\leq\frac{n}{m^{k-2}}\leq\frac{m}{m^{k-2}}=\frac{1}{m^{k-3}}$. Now choose
$k\geq 5$. Then, $Pr[rank(m)=rank(m^{\prime})]\leq\frac{1}{m^{2}}=O(n^{-2})$.
The following Lemma proceeds exactly as in [8].
###### Lemma 2.3.2.
A particular min-cut in G is produced by the contraction algorithm with
probability $\Omega(n^{-2})$.
###### Proof 2.2.
Let $c$ be the value of the mincut in $G$. Each contraction reduces the number
of vertices in the graph by one. Consider the contraction executed when the
graph has $r$ vertices. Since the contracted graph has a min-cut of at least
$c$, it must have minimum degree $c$, and thus atleast $\frac{rc}{2}$ edges.
However, only $c$ of these edges are in min-cut. Thus, a randomly chosen edge
is in the min-cut with probability at most $\frac{2}{r}$. The probability that
we never contract a min-cut edge through all $n-2$ contractions is atleast
$(1-\frac{2}{n})(1-\frac{2}{n-1})(1-\frac{2}{n-2})....(1-\frac{2}{3})=\binom{n}{2}^{-1}=\Omega(n^{-2})$
### 2.4 Complexity Analysis
#### 2.4.1 Message complexity
###### Theorem 2.4.1.
The Karger’s distributed algorithm uses $O(mn^{2})$ messages, in a single
trial.
###### Proof 2.3.
It is not hard to see that the most expensive steps in a trial are those of
determination of $maxrank$ from local maxranks(find-global-maxrank()) and that
of computing the mincut at the end(find-global-mincut()). In find-global-
maxrank(), each node sends its local maxrank value to its neighbours and this
is repeated atmost $n$ times(number of times equal to the diameter of the
graph sufficies). Hence the total number of messages is bounded by
$nO(m+n)=O(mn)$. Thus the message complexity for $n-2$ iterations per trial is
$O(mn^{2})$. Finally, in step $i$ of find-global-mincut(), $\frac{n}{2^{i}}$
nodes send messages to its neighbours. The total number of messages sent at
each step is bounded by $O(m)$. Thus, the total number of messages is
$\Sigma_{i=1}^{\log{n}}\frac{nm}{2^{i}}=O(mn)$. Hence the overall message
complexity is $O(mn^{2})+O(mn)=O(mn^{2})$.
#### 2.4.2 Time complexity
###### Theorem 2.4.2.
The Karger’s distributed algorithm computes mincut in $O(n^{2})$ time, in a
single trial.
###### Proof 2.4.
Before contraction, the algorithm assigns a rank (random number) to each edge
and finds the max-rank among all the vertices in the graph. This requires
atmost $n-1$ steps(strictly, number of steps equal to the diameter of the
graph). For contraction, a message is sent from a vertex within one group to
other group and the message is propagated to all the vertices within the
second group and the neighbouring vertices of that group, which also takes
atmost $n-1$ pulses. Since only one contraction can take place at any time and
there are $n-2$ such contractions, the running time is $O(n^{2})$. To estimate
time for computing the mincut, the function find-global-mincut() runs
$O(\log{n})$ steps and in step $i$, $\frac{n}{2^{i}}$ nodes flood the network.
Thus the time per step is $\frac{n^{2}}{2^{i}}$. Hence the total complexity is
$\Sigma_{i=1}^{\log{n}}\frac{n^{2}}{2^{i}}=O(n^{2})$.
## 3 Conclusion and Future work
A synchronous distributed version of the Karger’s randomized algorithm under
network setting is presented in this paper with a proof of correctness and
complexity analysis. The present algorithm appears not to make use of the full
power of parallelism available. It is interesting to look at how to
efficiently reduce time and message complexity by conducting edge contractions
in parallel.
## References
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* [8] D. R. Karger. “Global min-cuts in RNC, and other ramifications of a simple min-out algorithm”. In Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms, pages 21–30, 25-27 January 1993.
* [9] D. R. Karger and C. Stein. “An Õ$(n^{2})$ algorithm for minimum cuts”. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 757–765, 1993.
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|
arxiv-papers
| 2009-12-07T10:50:50 |
2024-09-04T02:49:06.884965
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Shine, K. Murali Krishnan",
"submitter": "Shine S",
"url": "https://arxiv.org/abs/0912.1200"
}
|
0912.1234
|
# Full tomography from compatible measurements
J. Řeháček Department of Optics, Palacky University, 17. listopadu 50, 772 00
Olomouc, Czech Republic Z. Hradil Department of Optics, Palacky University,
17. listopadu 50, 772 00 Olomouc, Czech Republic Z. Bouchal Department of
Optics, Palacky University, 17. listopadu 50, 772 00 Olomouc, Czech Republic
R. Čelechovský Department of Optics, Palacky University, 17. listopadu 50,
772 00 Olomouc, Czech Republic I. Rigas Departamento de Óptica, Facultad de
Física, Universidad Complutense, 28040 Madrid, Spain L. L. Sánchez-Soto
Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040
Madrid, Spain
###### Abstract
We put forward a reconstruction scheme prompted by the relation between a von
Neumann measurement and the corresponding informationally complete measurement
induced in a relevant reconstruction subspace. This method is specially suited
for the full tomography of complex quantum systems, where the intricacies of
the detection part of the experiment can be greatly reduced provided some
prior information is available. In broader terms this shows the importance of
this often-disregarded prior information in quantum theory. The proposed
technique is illustrated with an experimental tomography of photonic vortices
of moderate dimension.
###### pacs:
03.65.Wj, 03.65.Ta, 42.50.Tx
Introduction. The quantum state is a mathematical object that encodes complete
information about a system Peres : once it is known, the outcomes of any
possible measurement can be predicted. Apart from fundamental reasons,
acquiring the system state is invaluable for verifying and optimizing
experimental setups. For instance, in some protocols of quantum key
distribution, the knowledge of the entangled state distributed between the
parties greatly limits the ability of a third party to eavesdrop on the
communication channel eve .
The reconstruction of the unknown state from a suitable set of measurements is
called quantum tomography lnp . Over the past years, this technique has
evolved from the first theoretical vogel and experimental raymer concepts to
a widely acknowledged and fairly standard method extensively used for both
discrete james ; thew and continuous lvovsky variables.
In this work, we focus on measurement strategies for the tomographic
reconstruction, leaving aside data post-processing issues. In practice, a
sufficient number of independent observations must be included in the set of
measurements so that all physical aspects of the measured system are
addressed. When dealing with complicated systems, such measurements may be
difficult to implement in the laboratory due to various physical and technical
limitations on the available controlled interactions between the system and
the meter.
The goal of this Letter is to present a method of generating a tomographically
complete measurement set from a simple von Neumann measurement that is readily
implemented in the laboratory. Obviously, a von Neumann measurement is not
complete, as all the measured projections are compatible and hence provide
information only about the same aspects. However, as we shall show here,
things are radically different when only a part of the full Hilbert space is
of interest: In this subspace, even a simple von Neumann projection may become
informationally complete. This should not be taken as an approximation, in the
sense that some accuracy is traded for experimental feasibility. First of all,
the energy of any system is always bounded, so one can restrict the attention
to the subspace spanned by low-energy states. Second, due to the finite
resources, all quantum systems are de facto discrete and may be represented by
a relatively small number of parameters. In that case, there is no necessity
of sophisticated measurements that are informationally complete in the
original large Hilbert space: since only a small subset is accessible, even
much simpler observations are able to supply the information needed. This is
the main idea behind the present contribution.
Quantum tomography. Let us consider a density matrix $\varrho$ describing a
$d$-dimensional quantum system. A convenient representation of $\varrho$ can
be obtained with the help of a traceless Hermitian operator basis
$\\{\Gamma_{i}\\}$, satisfying $\mathop{\mathrm{Tr}}\nolimits(\Gamma_{i})=0$
and $\mathop{\mathrm{Tr}}\nolimits(\Gamma_{i}\Gamma_{j})=\delta_{ij}$ sun :
$\varrho=\frac{1}{d}+\sum_{i=1}^{d^{2}-1}a_{i}\Gamma_{i}\,,$ (1)
where $\\{a_{i}\\}$ are real numbers. The set $\\{\Gamma_{i}\\}$ coincides
with the orthogonal generators of SU($d$), which is the associated symmetry
algebra.
In general, the measurements performed on the system are described by positive
operator-valued measures (POVMs), which are a set of operators $\\{\Pi_{j}\\}$
(with $\Pi_{j}\geq 0$ and $\sum_{j}\Pi_{j}=\openone$), such that each POVM
element represents a single output channel of the measuring apparatus. The
probability of detecting the $j$th output is given by a generalized projection
postulate $p_{j}=\mathop{\mathrm{Tr}}\nolimits(\varrho\Pi_{j})$.
By decomposing the POVM elements in the same basis $\\{\Gamma_{i}\\}$, we get
$\Pi_{j}=b_{j}+\sum_{i=1}^{d^{2}-1}c_{ji}\Gamma_{i}\,,$ (2)
where $\\{b_{j}\\}$ are again known real numbers and $\mathbf{C}=\\{c_{ji}\\}$
is a real matrix.
Informational completeness. A set of measurements will be called
informationally complete if any quantum state $\varrho$ is unambiguously
assigned to the corresponding theoretical probabilities $p_{j}$. Since the
projection postulate can be rewritten as
$p_{j}-b_{j}=\sum_{i}c_{ji}a_{i}\,,$ (3)
informational completeness requires the matrix $\mathbf{C}$ to have at least
$d^{2}-1$ linearly independent rows. Numerically, this can be easily verified
by calculating the rank of $\mathbf{C}$, given by the number of nonzero
singular values. These are readily computed from the singular value
decomposition of $\mathbf{C}$. Thus, a set of measurements is informationally
complete provided
$\mathop{\mathrm{rank}}\nolimits\mathbf{C}\geq d^{2}-1\,.$ (4)
For example, a light mode can be treated as a harmonic oscillator. The
eigenstates of the rotated quadrature operators
$Q(\theta)=x\cos\theta+p\sin\theta$ comprise an informationally complete POVM.
Naturally, only a finite set of projections can be done, so that a truncation
of the original infinite-dimensional Hilbert space is necessary grangier ;
polzik . In consequence, consider a von Neumann projection defined in the
infinite-dimensional space $\mathcal{H}$: $\sum_{k=0}^{\infty}|k\rangle\langle
k|=\openone$, where $|k\rangle$ is an orthonormal basis. Experimentally such
measurements do not pose any difficulty: all that has to be done is to
determine the spectrum of a single observable. Nevertheless, this simple von
Neumann measurement is not informationally complete in $\mathcal{H}$, for all
the observations are in this case mutually compatible and consequently no
information about any of the existing complementary observables is available.
Generating informationally complete measurements. As we will now show, this
interpretation no longer holds when only a subspace $\mathcal{S}$ of
$\mathcal{H}$ is considered. Let us specify $\mathcal{S}$ by introducing the
projector $P_{S}=\sum_{s=0}^{S}|s\rangle\langle s|\,,$ where $|s\rangle$ are
eigenstates of $P_{S}$ and $S$ is the dimension. By projecting the original
measurement on $\mathcal{S}$, a POVM is induced in this subspace, namely
$\sum_{k}\Pi_{k}=\sum_{k}P_{S}\,|k\rangle\langle k|\,P_{S}=\openone_{S}\,,$
(5)
whose elements, in general, no longer commute $[\Pi_{k},\Pi_{k^{\prime}}]\neq
0$. Indeed, since the original commuting projections have different overlaps
with the subspace $\mathcal{S}$, their mutual properties (commutators) are not
preserved. In this way, an informationally complete POVM may be generated.
Obviously, this observation has many potential applications beyond tomography,
although, due to strict space limitation only that topic will be discussed.
The protocol we propose consists of the following steps: (i) A reconstruction
subspace $\mathcal{S}$ is selected according to the particular experiment, in
such a way that all the relevant states are included. (ii) An experimentally
feasible von Neumann projection is chosen. (iii) The effective POVM induced in
$\mathcal{S}$, as given by Eq. (5), is calculated and its informational
completeness is checked with the help of condition (4). If the induced POVM is
informationally complete, the task is finished, otherwise the whole procedure
is repeated with different choices of either the von Neumann projection or the
reconstruction subspace or both.
Before we proceed further, let us comment on the differences between our
protocol and the Naimark extension naimark , which is another way of
representing POVMs by projective measurements. This extension works by
enlarging the Hilbert space with an ancilla, so the projective measurement
acts on the product space of the system and ancilla. In our approach, the
possibility of representing a tomographical scheme by a projective measurement
stems from the available prior information. In fact, the unpopulated states or
unused range of variables play the role of ancilla here and, consequently, the
measurement acts on a sum rather than a product space.
Optical vortices. As a relevant example, we use our protocol for the
tomography of optical vortices. As the wave function (or density matrix) in
quantum theory, any transverse distribution of complex amplitude (or coherence
matrix) can be decomposed in a complete basis; the Laguerre-Gauss modes being
a very convenient one
$\mathrm{LG}^{\ell}_{p}(x,y)=\langle x,y|\ell,p\rangle\propto
r^{|\ell|}L_{p}^{|\ell|}(2r^{2})e^{-r^{2}}e^{i\ell\phi}\,,$ (6)
where $r^{2}=x^{2}+y^{2}$ and $\phi=\arctan(y/x)$ are polar coordinates in the
transverse plane and $L_{p}^{\ell}$ is a generalized Laguerre polynomial. It
is well known LG that $\mathrm{LG}^{\ell}_{p}$ beams exhibit helicoidal
wavefronts that induce a vortex structure and carry orbital angular momentum
of $\hbar\ell$ per photon. Suppose a photon has been emitted into a
superposition of modes, and we need to identify the resulting state. In
general, this is an involved task zeilinger ; white ; calvo requiring the use
of complicated optical devices. However, provided that only beams with bounded
vorticities (i.e., values of $|\ell|$) are considered, as it is usually the
case, our protocol can be employed and an informationally complete measurement
can be generated from a very basic one, such as a single transverse intensity
scan that is easy to record. In the language of quantum theory, this intensity
scan is just $I(x,y)\propto\mathrm{Tr}(\varrho|x,y\rangle\langle x,y|),$ where
$x$ and $y$ denote now the coordinates of a given pixel of the position-
sensitive detector. Although detections in any pair of pixels are always
compatible, in a subspace with bounded vorticities noncommuting POVM elements
can be induced.
Figure 1: Incompatibility (computed as the norm of commutator) of the
detections at two spatially separated pixels of a CCD camera in a truncated
Hilbert space $p=0,\ldots,p_{\mathrm{cutoff}}$,
$\ell=-\ell_{\mathrm{cutoff}},\ldots,\ell_{\mathrm{cutoff}}$. Black (white)
color means compatible (strongly incompatible), respectively.
This is illustrated in Fig. 1, which shows the noncommutativity
(incompatibility) corresponding to the positions $(x,y)=(0,0)$, and $(0,1)$
[in the same units of Eq. (6)]. Truncating the Hilbert space at smaller
vorticities typically leads to stronger noncommutativity, although some
nonmonotonicity is also observed as oscillations of gray shades appearing from
the top-right to the bottom-left corner.
Figure 2: Experimental setup of vortex tomography by means of compatible
observations.
Experiment. To demonstrate the potential of the procedure, a full tomography
of an optical vortex field from a single intensity scan has been performed in
a controlled experiment. The experimental scheme is shown in Fig. 2. The beam
generated by a He-Ne laser is spatially filtered by a microscope objective and
a pinhole. After the beam is expanded and collimated by a lens, it impinges on
an amplitude spatial light modulator (CRL Opto, $1024\times 768$ pixels)
displaying a hologram computed as an interference pattern of the required
light and the inclined reference plane wave.
Light behind the hologram consists of three diffraction orders $(-1,0,+1)$,
which can be separated and Fourier filtered by means of the 4$f$ optical
system consisting of the lenses L1 and L2, and an iris diaphragm. The
undesired 0th and -1st orders are removed by an aperture placed at the back
focal plane of the lens L1. This completes the preparation of a given state of
light.
Finally, a collimated beam with the required complex amplitude profile is
obtained at the back focal plane of the second Fourier lens L2, where a
transverse intensity scan $I(x,y)$ is acquired by a CCD camera. In the image
plane, each pixel detection can be approximated by a projection on the
position eigenstates $|x,y\rangle\langle x,y|$. As it has been shown above,
while such detections are compatible in the full infinite-dimensional Hilbert
space, an informationally complete POVM is induced in a subspace of truncated
vorticities.
In our experiment the superposition
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|\ell=1,p=0\rangle+|\ell=2,p=0\rangle)$ (7)
was prepared by letting an amplitude spatial light modulator to display an
interference pattern of the transverse amplitude $\langle x,y|\Psi\rangle$ and
a reference plane wave, as mentioned above. Results for this state are shown
in Fig. 3. The ideal intensity distribution in the detection plane
$I(x,y)\propto|\langle x,y|\Psi\rangle|^{2}$ is shown in the left panel. This
should be compared to the corresponding noisy recorded image shown in the
middle panel. Finally, the right panel shows the best fit obtained with a
maximum-likelihood algorithm prl in the subspace $p=0$ and $\ell=0,\ldots,4$.
The reconstructed $5$-dimensional density matrix is shown in Fig. 4. Notice
that, due to experimental imperfections (such as a discrete structure of the
spatial light modulator, detection noise, etc.), the reconstructed state
slightly differs from the ideal one (typical fidelities in our experiment are
$F\approx 96\%$). In view of the complexity of the system and the simplicity
of the experiment, we consider this to be a very good result.
Figure 3: Experimental tomography of optical vortex fields. From left: ideal
intensity distribution, measured intensity distribution, and the corresponding
best theoretical fit of measured data.
Figure 4: Real (on the left) and imaginary (on the right) elements of the
reconstructed density matrix.
Figure 5: Informational completeness of measurements on vortex beams
generated by a CCD camera with $11\times 11$ pixels. The number of independent
measurements $\Pi_{k}$ generated from those $121$ CCD detections are shown by
circles for different truncations of the Hilbert space $P_{S}$. The number of
independent measurements required for a complete tomography in the same
reconstruction subspace is indicated by crosses. The reconstruction subspaces
are truncated as follows. Upper panel: $p=0$,
$\ell=0,\ldots,\ell_{\mathrm{cutoff}}$; bottom panel: $p=0$,
$\ell=-\ell_{\mathrm{cutoff}},\ldots,\ell_{\mathrm{cutoff}}$.
Given the promising performance of the proposed scheme in this proof-of-
principle experiment, the natural question is whether an experimentally
feasible von Neumann measurement (such as a single intensity scan by a CCD
camera with possibly very fine resolution) would furnish an informationally
complete measurement for any reconstruction subspace. To get some insights
into this problem, we consider two different scenarios related to the
experiment above (see Fig. 5). In the first case, only photons with
nonnegative vorticities are considered: the full tomography from a single
intensity scan is always possible. In the second case, both positive and
negative vorticities are allowed. Here a single intensity scan fails to
provide complete information. It is easy to see why: since the intensity
profiles of the Laguerre-Gauss modes $\mathrm{LG}^{\ell}_{p}$ and
$\mathrm{LG}^{-\ell}_{p}$ are the same, perfect discrimination between states
with positive and negative vorticities is not possible. Interestingly enough,
some information about the negative part of the angular momentum spectrum is
still available (see, e. g., the crosses in the plots for the same truncation
$\ell_{\mathrm{cutoff}}$), as it is also obvious from the fact that the phases
$\exp(i\ell\phi)$ and $\exp(-i\ell\phi)$ in superpositions like
$\mathrm{LG}^{\ell}_{0}+\mathrm{LG}^{1}_{0}$ and
$\mathrm{LG}^{-\ell}_{0}+\mathrm{LG}^{1}_{0}$ can be distinguished via
interference with the other mode. This partial information is however not
sufficient for the full characterization of this part of the reconstruction
subspace. Provided one wants to keep the simple intensity detection, it is
always possible to use a fixed unitary transformation prior to detection to
optimize the scheme. For instance, by increasing angular momentum of the
measured beam by $\ell_{\mathrm{cutoff}}$ (using, e. g., a charged fork-like
hologram) the reconstruction subspace can be moved inside the nonnegative part
of the angular momentum spectrum. This example nicely illustrates the role of
prior information in experimental quantum tomography.
Conclusions. We have shown that simple compatible observations may provide
full information about the measured system when some prior information is
available. This prior information does not only bring about a quantitative
improvement of our knowledge, but may also make feasible a no-go task. Based
on this observation, an efficient protocol was sketched providing the full
characterization of complex systems from simple measurements. This was
demonstrated in an experiment with photonic vortices. In our opinion, this
constitutes an improvement that will have a significant benefit in the number
of different physical architectures where quantum information experiments are
being performed.
This work was supported by the Czech Ministry of Education, Projects
MSM6198959213 and LC06007, the Spanish Research Directorate, Grants
FIS2005-06714 and FIS2008-04356.
## References
* (1) A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993).
* (2) Y.C. Liang, D. Kaszlikowski, B.-G. Englert, L.C. Kwek, and C.H. Oh, Phys. Rev. A, 68, 022324 (2003).
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* (6) D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
* (7) R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Phys. Rev. A 66, 012303 (2002).
* (8) A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
* (9) F. T. Hioe and J.H. Eberly, Phys. Rev. Lett. 47, 838 (1981); G. Kimura, Phys. Lett. A 314, 339 (2004).
* (10) A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 96, 213601 (2006).
* (11) J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Molmer, and E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006).
* (12) M. A. Naimark, Izv. Akad. Nauk SSSR, Ser. Mat 4, 277 (1940).
* (13) L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol, 2003).
* (14) A. Vaziri, G. Weihs, and A. Zeilinger, J. Opt. B 4, S47 (2002).
* (15) N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
* (16) G. F. Calvo, A. Picón, and R. Zambrini, Phys. Rev. Lett. 100, 173902 (2008).
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|
arxiv-papers
| 2009-12-07T13:28:55 |
2024-09-04T02:49:06.890687
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Rehacek, Z. Hradil, Z. Bouchal, R. Celechovsky, I. Rigas, L. L.\n Sanchez-Soto",
"submitter": "Luis L. Sanchez. Soto",
"url": "https://arxiv.org/abs/0912.1234"
}
|
0912.1408
|
# Thermodynamical description of the interacting new agegraphic dark energy
A. Sheykhi 1,2111sheykhi@mail.uk.ac.ir and M.R. Setare 3,2222rezakord@ipm.ir
1Department of Physics, Shahid Bahonar University, Kerman 76175, Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
3 Department of Science, Payame Noor University, Bijar, Iran
###### Abstract
We describe the thermodynamical interpretation of the interaction between new
agegraphic dark energy and dark matter in a non-flat universe. When new
agegraphic dark energy and dark matter evolve separately, each of them remains
in thermodynamic equilibrium. As soon as an interaction between them is taken
into account, their thermodynamical interpretation changes by a stable thermal
fluctuation. We obtain a relation between the interaction term of the dark
components and this thermal fluctuation.
Keywords: dark energy; thermodynamics; entropy.
## I Introduction
The dark energy puzzle is one of the biggest challenges of the modern
cosmology in the past decade. There is an ample evidences on the observational
side that our universe is currently experiencing a phase of accelerated
expansion Rie1 ; Rie2 ; Rie3 ; Rie4 . These observations suggest that nearly
three quarters of our universe consists of a mysterious energy component (dark
energy) which is responsible for this expansion, and the remaining part
consists of pressureless dark matter. Nevertheless, despite the mounting
observational evidences, the nature of such dark energy remains elusive and it
has become a source of much debate except for the fact that it has negative
pressure. Most discussions on dark energy rely on the assumption that it
evolves independently of dark matter. Given the unknown nature of both dark
energy and dark matter there is nothing in principle against their mutual
interaction and it seems very special that these two major components in the
universe are entirely independent. Indeed, this possibility has received a lot
of attention recently (see Ame1 ; Ame2 ; Ame3 ; Ame4 ; Ame5 ; Zim1 ; Zim2 ;
Zim3 ; Seta1 ; Set2 ; Set3 ; Set4 ; Set5 ; wang1 ; wang11 ; Shey0 and
references therein). In particular, it has been shown that the coupling can
alleviate the coincidence problem Pav1 . Furthermore, it was argued that the
appropriate coupling between dark components can influence the perturbation
dynamics and the cosmic microwave background (CMB) spectrum and account for
the observed CMB low $l$ suppression wang2 . It was shown that in a model with
interaction the structure formation has a different fate as compared with the
non-interacting case wang2 . It was also discussed that with strong coupling
between dark energy and dark matter, the matter density perturbation is
stronger during the universe evolution till today, which shows that the
interaction between dark energy and dark matter enhances the clustering of
dark matter perturbation compared to the noninteracting case in the past.
Therefore, the coupling between dark components could be a major issue to be
confronted in studying the physics of dark energy. However, so long as the
nature of these two components remain unknown it will not be possible to
derive the precise form of the interaction from first principles. Therefore,
one has to assume a specific coupling from the outset Ads ; Amen1 ; Amen2 or
determine it from phenomenological requirements Chim1 ; Chim2 .
Thermodynamical description of the interaction (coupling) between holographic
dark energy and dark matter has been studied in wang3 ; SetVag .
Among the various candidates to explain the accelerated expansion, the
agegraphic and new agegraphic dark energy (NADE) models condensate in a class
of quantum gravity may have interesting cosmological consequences. These
models take into account the Heisenberg uncertainty relation of quantum
mechanics together with the gravitational effect in general relativity. The
agegraphic dark energy models assume that the observed dark energy comes from
the spacetime and matter field fluctuations in the universe Cai1 ; Wei2 ; Wei1
. Since in agegraphic dark energy model the age of the universe is chosen as
the length measure, instead of the horizon distance, the causality problem in
the holographic dark energy is avoided. The agegraphic models of dark energy
have been examined and constrained by various astronomical observations age1 ;
age2 ; age3 ; age4 ; age5 ; age6 ; age7 ; sheykhi1 ; sheykhi2 ; sheykhi3 ;
sheykhi4 ; sheykhi5 ; Setare2 ; Setare22 . Although going along a fundamental
theory such as quantum gravity may provide a hopeful way towards understanding
the nature of dark energy, it is hard to believe that the physical foundation
of agegraphic dark energy is convincing enough. Indeed, it is fair to say that
almost all dynamical dark energy models are settled at the phenomenological
level, neither holographic dark energy model nor agegraphic dark energy model
is exception. Though, under such circumstances, the models of holographic and
agegraphic dark energy, to some extent, still have some advantage comparing to
other dynamical dark energy models because at least they originate from some
fundamental principles in quantum gravity.
The main purpose of this Letter is to study thermodynamical interpretation of
the interaction between dark matter and NADE model for a universe enveloped by
the apparent horizon. It was shown that for an accelerating universe the
apparent horizon is a physical boundary from the thermodynamical point of view
Jia . In particular, it was argued that for an accelerating universe inside
the event horizon the generalized second law does not satisfy, while the
accelerating universe enveloped by the apparent horizon satisfies the
generalized second law of thermodynamics Jia ; sheywang1 ; sheywang2 ;
sheywang3 . Therefore, the event horizon in an accelerating universe might not
be a physical boundary from the thermodynamical point of view. This Letter is
outlined as follows. In the next section we consider the thermodynamical
picture of the non-interacting NADE in a non-flat universe. In section III, we
extend the thermodynamical description in the case where there is an
interaction term between the dark components. We also present an expression
for the interaction term in terms of a thermal fluctuation. The last section
is devoted to summary and discussion.
## II Thermodynamical description of the non-interacting NADE
We consider the Friedmann-Robertson-Walker (FRW) universe which is described
by the line element
$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(1)
where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe1 ; spe2 ; spe3 ; spe4 . The first Friedmann
equation takes the form
$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right),$ (2)
where $H=\dot{a}/a$ is the Hubble parameter, $\rho_{m}$ and $\rho_{D}$ are the
energy density of dark matter and dark energy, respectively. We define, as
usual, the fractional energy densities such as
$\displaystyle\Omega_{m}=\frac{\rho_{m}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}}.$ (3)
Thus, the Friedmann equation can be written
$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (4)
Let us first review the origin of the agegraphic dark energy model. Following
the line of quantum fluctuations of spacetime, Karolyhazy et al. Kar1 ; Kar2 ;
Kar3 argued that the distance $t$ in Minkowski spacetime cannot be known to a
better accuracy than $\delta{t}=\beta t_{p}^{2/3}t^{1/3}$ where $\beta$ is a
dimensionless constant of order unity. Based on Karolyhazy relation, Sasakura
discussed that the energy density of metric fluctuations of the Minkowski
spacetime is given by Sas (see also Maz1 ; Maz2 )
$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (5)
where $t_{p}$ is the reduced Planck time and $t$ is a proper time scale. On
these basis, Cai wrote down the energy density of the original agegraphic dark
energy as Cai1
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{T_{A}^{2}},$ (6)
where $T_{A}$ is the age of the universe,
$T_{A}=\int_{0}^{a}{\frac{da}{Ha}},$ (7)
and the numerical factor $3n^{2}$ is introduced to parameterize some
uncertainties, such as the species of quantum fields in the universe, the
effect of curved space-time, and so on. However, to avoid some internal
inconsistencies in the original agegraphic dark energy model, the so-called
“new agegraphic dark energy” was proposed, where the time scale is chosen to
be the conformal time $\eta$ instead of the age of the universe Wei2 . The
NADE contains some new features different from the original agegraphic dark
energy and overcome some unsatisfactory points. For instance, the original
agegraphic dark energy suffers from the difficulty to describe the matter-
dominated epoch while the NADE resolved this issue Wei2 . The energy density
of the NADE can be written
$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (8)
where the conformal time $\eta$ is given by
$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (9)
Consider the FRW universe filled with dark energy and dust (dark matter) which
evolve according to their conservation laws
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D}^{0})=0,$ (10)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (11)
where $w_{D}=p_{D}/\rho_{D}$ is the equation of state parameter of NADE. The
superscript above the equation of state parameter, $w_{D}$, denotes that there
is no interaction between the dark components. The fractional energy density
of the NADE is given by
$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}},$ (12)
where its evolution behavior is governed by sheykhi1
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{n}\sqrt{\Omega_{D}}\right)+\Omega_{k}\right].$
(13)
Here the prime stands for the derivative with respect to $x=\ln{a}$. Taking
the derivative with respect to the cosmic time of Eq. (8) and using Eq. (12)
we get
$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}^{0}}}{na}\rho_{D}.$
(14)
Inserting this relation into Eq. (10) we obtain the equation of state
parameter of the NADE
$\displaystyle 1+w_{D}^{0}=\frac{2}{3na}\sqrt{\Omega_{D}^{0}}.$ (15)
We also 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 $T$ of the energy content inside the apparent
horizon should be in equilibrium with the temperature $T_{h}$ associated with
the apparent horizon, so we have $T=T_{h}$. 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 and the local equilibrium hypothesis will no longer hold. This is
also at variance with the FRW geometry. Thus, when we consider the thermal
equilibrium state of the universe, the temperature of the universe is
associated with the horizon temperature. In this picture the equilibrium
entropy of the NADE is connected with its energy and pressure through the
first law of thermodynamics
$TdS_{D}=dE_{D}+p_{D}dV,$ (16)
where the volume enveloped by the apparent horizon is given by
$V=\frac{4\pi}{3}r_{A}^{3},$ (17)
and $r_{A}$ is the apparent horizon radius. The apparent horizon was argued as
a causal horizon for a dynamical spacetime and is associated with
gravitational entropy and surface gravity Hay1 ; Hay2 ; Bak . For the FRW
universe the apparent horizon radius reads sheyahmad1 ; sheyahmad2
${r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (18)
The total energy of the NADE inside the apparent horizon is
$E_{D}=\rho_{D}V=\frac{4\pi n^{2}m_{p}^{2}r_{A}^{3}}{\eta^{2}}.$ (19)
Taking the differential form of Eq. (19) and using Eq. (12), we find
$dE_{D}=4\pi
m_{p}^{2}({{r}^{0}_{A}})^{2}H_{0}^{2}\Omega_{D}^{0}\left[3d{r}^{0}_{A}-2\frac{{r}^{0}_{A}}{n}H_{0}\sqrt{\Omega_{D}^{0}}d\eta^{0}\right].$
(20)
The associated temperature on the apparent horizon can be written as
$T=\frac{1}{2\pi r_{A}}.$ (21)
Inserting Eqs. (17), (20) and (21) into (16), we obtain
$dS_{D}^{(0)}=8\pi^{2}m_{p}^{2}({{r}^{0}_{A}})^{3}H_{0}^{2}\Omega_{D}^{0}\left[3(1+w_{D}^{0})d{r}^{0}_{A}-2\frac{{r}^{0}_{A}}{n}H_{0}\sqrt{\Omega_{D}^{0}}d\eta^{0}\right],$
(22)
Using Eq. (15) as well as relation $H_{0}d\eta^{0}=dx^{0}/a_{0}$, we find
$dS_{D}^{(0)}=16\pi^{2}m_{p}^{2}({{r}^{0}_{A}})^{3}H_{0}^{2}\Omega_{D}^{0}\frac{\sqrt{\Omega_{D}^{0}}}{na_{0}}\left[d{r}^{0}_{A}-{r}^{0}_{A}dx^{0}\right].$
(23)
Here the superscript/subscript $(0)$ denotes that in this picture our universe
is in a thermodynamical stable equilibrium.
## III Thermodynamical description of the interacting NADE
In this section we study the case where the pressureless dark matter and the
NADE interact with each other. In this case $\rho_{m}$ and $\rho_{D}$ do not
conserve separately; they must rather enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (24)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q.$ (25)
Here $Q$ denotes the interaction term and can be taken as
$Q=3b^{2}H(\rho_{m}+\rho_{D})$ with $b^{2}$ being a coupling constant Pav1 .
Inserting Eq. (14) into (25), we obtain the equation of state parameter of the
interacting NADE
$\displaystyle
1+w_{D}=\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{Q}{9m_{p}^{2}H^{3}\Omega_{D}}.$
(26)
The evolution behavior of the NADE is now given by sheykhi1
$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
(27)
Comparing Eq. (26) with Eq. (15), we see that the presence of the interaction
term $Q$ has provoked a change in the equation of state parameter and
consequently in the dimensionless density parameter of the dark energy
component and thus now there is no subscript above the aforesaid quantities to
denote the absence of interaction. The interacting NADE model in the non-flat
universe as described above is not anymore thermodynamically interpreted as a
state in thermodynamical equilibrium. Indeed, as soon as an interaction
between dark components is taken into account, they cannot remain in their
respective equilibrium states. The effect of interaction between the dark
components is thermodynamically interpreted as a small fluctuation around the
thermal equilibrium. It was shown Das that due to the fluctuation, there is a
leading logarithmic correction $S^{(1)}_{D}=-\frac{1}{2}\ln(CT^{2})$ to the
thermodynamic entropy around equilibrium in all thermodynamical systems.
Therefore, the entropy of the NADE is connected with its energy and pressure
through the first law of thermodynamics
$TdS_{D}=dE_{D}+p_{D}dV,$ (28)
where now the entropy has been assigned an extra logarithmic correction Das
$S_{D}=S^{(0)}_{D}+S^{(1)}_{D},$ (29)
where the leading logarithmic correction is
$S^{(1)}_{D}=-\frac{1}{2}\ln(CT^{2}),$ (30)
and $C$ is the heat capacity defined by
$C=T\frac{\partial S^{(0)}_{D}}{\partial T}.$ (31)
It is a matter of calculation to show that
$C=-16\pi^{2}m_{p}^{2}({{r}^{0}_{A}})^{4}H_{0}^{2}\Omega_{D}^{0}\frac{\sqrt{\Omega_{D}^{0}}}{na_{0}},$
(32)
and therefore
$S^{(1)}_{D}=-\frac{1}{2}\ln\left(-4m_{p}^{2}({{r}^{0}_{A}})^{2}H_{0}^{2}\Omega_{D}^{0}\frac{\sqrt{\Omega_{D}^{0}}}{na_{0}}\right).$
(33)
Substituting the expressions for the volume, energy, and temperature in Eq.
(28) for the interacting case, we get
$dS_{D}=8\pi^{2}m_{p}^{2}{{r}^{3}_{A}}H^{2}\Omega_{D}\left[3(1+w_{D})d{r}_{A}-\frac{2{r}_{A}}{n}H\sqrt{\Omega_{D}}d\eta\right],$
(34)
or in another way
$dS_{D}=8\pi^{2}m_{p}^{2}{{r}^{3}_{A}}H^{2}\Omega_{D}\left[3(1+w_{D})d{r}_{A}-\frac{2{r}_{A}}{na}\sqrt{\Omega_{D}}dx\right],$
(35)
and thus one gets
$\displaystyle 1+w_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}m_{p}^{2}{{r}^{3}_{A}}H^{2}\Omega_{D}}\frac{dS_{D}}{d{r}_{A}}+\frac{2{r}_{A}}{3na}\sqrt{\Omega_{D}}\frac{dx}{d{r}_{A}},$
(36)
$\displaystyle=\frac{1}{24\pi^{2}m_{p}^{2}{{r}^{3}_{A}}H^{2}\Omega_{D}}\left[\frac{dS^{(0)}_{D}}{d{r}_{A}}+\frac{dS^{(1)}_{D}}{d{r}_{A}}\right]+\frac{2{r}_{A}}{3na}\sqrt{\Omega_{D}}\frac{dx}{d{r}_{A}}.$
Employing Eqs. (23), (30)-(33), we can easily find
$\displaystyle\frac{dS^{(0)}_{D}}{d{r}_{A}}$ $\displaystyle=$
$\displaystyle\frac{\partial
S^{(0)}_{D}}{\partial{r}^{0}_{A}}\frac{d{r}^{0}_{A}}{d{r}_{A}}+\frac{\partial
S^{(0)}_{D}}{\partial{x}^{0}}\frac{d{x}^{0}}{d{r}_{A}}=16\pi^{2}m_{p}^{2}{({r_{A}^{0}})^{3}}H_{0}^{2}\frac{({\Omega_{D}^{0}})^{3/2}}{na_{0}}\left(\frac{d{r}^{0}_{A}}{d{r}_{A}}-{r}^{0}_{A}\frac{d{x}^{0}}{d{r}_{A}}\right),$
(37) $\displaystyle\frac{dS^{(1)}_{D}}{d{r}_{A}}$ $\displaystyle=$
$\displaystyle\frac{\partial
S^{(1)}_{D}}{\partial{r}^{0}_{A}}\frac{d{r}^{0}_{A}}{d{r}_{A}}=-\frac{1}{{r}^{0}_{A}}\frac{d{r}^{0}_{A}}{d{r}_{A}}.$
(38)
Finally, by equating expressions (26) and (36) for the equation of state
parameter of the interacting NADE evaluated on cosmological and
thermodynamical sides, respectively, one gets an expression for the
interaction term
$\displaystyle\frac{Q}{9m_{p}^{2}H^{3}}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{\Omega_{D}}}{3na}{\Omega_{D}}\left(1-r_{A}\frac{dx}{dr_{A}}\right)-\frac{1}{24\pi^{2}m_{p}^{2}{{r_{A}}^{3}}H^{2}}\left[\frac{dS^{(0)}_{D}}{dr_{A}}+\frac{dS^{(1)}_{D}}{dr_{A}}\right]$
(39) $\displaystyle=$
$\displaystyle\frac{2\sqrt{\Omega_{D}}}{3na}{\Omega_{D}}\left(1-r_{A}\frac{dx}{dr_{A}}\right)-\frac{2}{3na_{0}}\frac{H_{0}^{2}}{H^{2}}({\Omega_{D}^{0}})^{3/2}\left(\frac{{r}^{0}_{A}}{{r}_{A}}\right)^{3}\left(\frac{d{r}^{0}_{A}}{d{r}_{A}}-{r}^{0}_{A}\frac{d{x}^{0}}{d{r}_{A}}\right)$
$\displaystyle+\frac{1}{24\pi^{2}m_{p}^{2}{{r_{A}}^{3}}{r}^{0}_{A}H^{2}}\frac{d{r}^{0}_{A}}{d{r}_{A}}.$
In this way we provide the relation between the interaction term of the dark
components and the thermal fluctuation.
## IV Summary and discusion
One of the important questions concerns the thermodynamical behavior of the
accelerated expanding universe driven by dark energy. It is interesting to ask
whether thermodynamics in an accelerating universe can reveal some properties
of dark energy. 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. The profound connection between the thermodynamics and
the gravitational field equations has also been observed in the cosmological
situations Cai2 ; Cai3 ; CaiKim ; Wang1 ; Wang2 ; Wang3 ; Cai4 ; sheyahmad3 .
This connection implies that the thermodynamical properties can help
understand the dark energy, which gives strong motivation to study
thermodynamics in the accelerating universe.
Although at this point the interaction between dark energy and dark matter
looks purely phenomenological, but in the absence of a symmetry that forbids
the interaction there is nothing, in principle, against it. Further, the
interacting dark mater dark energy (the latter in the form of a quintessence
scalar field and the former as fermions whose mass depends on the scalar
field) has been investigated at one quantum loop with the result that the
coupling leaves the dark energy potential stable if the former is of
exponential type but it renders it unstable otherwise. Thus, microphysics
seems to allow enough room for the coupling; however, this point is not fully
settled and should be further investigated. Recently evidence was provided by
the Abell Cluster A$586$ in support of the interaction between dark energy and
dark matter Ber1 ; Ber2 .
In this Letter, we provided a thermodynamical description for the NADE model
in a universe with spacial curvature. It was shown that for an accelerating
universe the apparent horizon is a physical boundary from the thermodynamical
point of view. We explored the thermodynamical picture of the interacting NADE
model for a FRW universe enveloped by the apparent horizon. The NADE contains
some new features different from the original agegraphic dark energy and
overcome some unsatisfactory points. For instance, the original agegraphic
dark energy suffers from the difficulty to describe the matter-dominated epoch
while the NADE resolved this issue. We assumed that in the absence of a
coupling, the two dark components remain in separate thermal equilibrium and
that the presence of a small coupling between them can be described as stable
fluctuations around equilibrium. Finally, resorting to the logarithmic
correction to the equilibrium entropy we derived an expression for the
interaction term in terms of a thermal fluctuation.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha.
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|
arxiv-papers
| 2009-12-08T05:59:35 |
2024-09-04T02:49:06.897939
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi, M. R. Setare",
"submitter": "Mohammad Reza Setare",
"url": "https://arxiv.org/abs/0912.1408"
}
|
0912.1432
|
# Quantized Quasi-Two Dimensional Bose-Einstein Condensates with Spatially
Modulated Nonlinearity
Deng-Shan Wang1, Xing-Hua Hu1, Jiangping Hu2 and W. M. Liu1 1Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing $100190$, P.R. China 2Department of Physics, Purdue
University, West Lafayette, Indiana $47907$, U.S.A.
###### Abstract
We investigate the localized nonlinear matter waves of the quasi-two
dimensional Bose-Einstein condensates with spatially modulated nonlinearity in
harmonic potential. It is shown that the whole Bose-Einstein condensates,
similar to the linear harmonic oscillator, can have an arbitrary number of
localized nonlinear matter waves with discrete energies, which are
mathematically exact orthogonal solutions of the Gross-Pitaevskii equation.
Their novel properties are determined by the principle quantum number $n$ and
secondary quantum number $l$: the parity of the matter wave functions and the
corresponding energy levels depend only on $n$, and the numbers of density
packets for each quantum state depend on both $n$ and $l$ which describe the
topological properties of the atom packets. We also give an experimental
protocol to observe these novel phenomena in future experiments.
###### pacs:
03.75.Hh, 05.45.Yv, 67.85.Bc
_Introduction_.—Since the remarkable experimental realization Anderson ;
Hulet1 ; Davis of Bose-Einstein condensations (BEC), there has been an
explosion of the experimental and theoretical activity devoted to the physics
of dilute ultracold bosonic gases. It is known that the properties of BEC
including their shape, collective nonlinear excitations are determined by the
sign and magnitude of the $s$-wave scattering length. A prominent way to
adjust scattering length is to tune an external magnetic field in the vicinity
of a Feshbach resonance Inouye1 . Alternatively, one can use a Feshbach
resonance induced by optical or electric field Theis . Since all quantities of
interest in the BEC crucially depend on scattering length, a tunable
interaction suggests very interesting studies of the many-body behavior of
condensate systems.
In the past years, techniques for adjusting the scattering length globally
have been crucial to many experimental achievements Herbig ; Bartenstein .
More recently, condensates with a spatially modulated nonlinearity by
manipulating scattering length locally have been proposed Rodas-Verde ;
Sakaguchi ; Konotop ; Qian ; Belmonte-Beitia12 . This is experimentally
feasible due to the flexible and precise control of the scattering length with
tunable interactions. The spatial dependence of scattering length can be
implemented by a spatially inhomogeneous external magnetic field in the
vicinity of a Feshbach resonance Xiong .
However, so far, the studies of BEC with spatially modulated nonlinearity are
limited in the quasi-one dimensional cases Rodas-Verde ; Sakaguchi ; Konotop ;
Qian ; Belmonte-Beitia12 . Moreover, in the study of nonlinear problems no one
discusses their quantum properties which are common in linear systems such as
the linear harmonic oscillator. In this Letter, we extend the similarity
transformation Belmonte-Beitia12 to the quasi-two dimensional (quasi-2D) BEC
with spatially modulated nonlinearity in harmonic potential, and find a family
of stable localized nonlinear matter wave solutions. Similar to the linear
harmonic oscillator, we discover that the whole BEC can be quantized which is
unexpected before. Their quantum and topological properties can be simply
described by two quantum numbers. We also formulate an experimental procedure
for the realization of these novel phenomena in 7Li condensate Hulet1 ; Hulet2
. This opens the door to the investigation of new matter waves in the high
dimensional BEC with spatially modulated nonlinearities.
Model and exact localized solutions.—The system considered here is a BEC
confined in a harmonic trap
$V(\textbf{r})=m(\omega_{\perp}^{2}r^{2}+\omega_{z}^{2}z^{2})/2$, where $m$ is
atomic mass, $r^{2}=x^{2}+y^{2}$, and $\omega_{\perp},\omega_{z}$ are the
confinement frequencies in the radial and axial directions, respectively. In
the mean-field theory, the BEC system at low temperature is described by the
Gross-Pitaevskii (GP) equation in three dimensions. If the trap is pancake-
shaped, i.e. $\omega_{z}\gg\omega_{\perp},$ it is reasonable to reduce the GP
equation for the condensate wave function to a quasi-2D equation Kivshar ;
Ueda1 ; Garcia-Ripoll
$i\psi_{t}=-\frac{1}{2}(\psi_{xx}+\psi_{yy})+\frac{1}{2}\omega^{2}(x^{2}+y^{2})\psi+g(x,y)|\psi|^{2}\psi,$
(1)
where $\omega=\omega_{\perp}/\omega_{z}$, the length, time and wave function
$\psi$ are measured in units of
$a_{h}=\sqrt{\hbar/m\omega_{z}},\omega_{z}^{-1},a_{h}^{-1}$ and $g(x,y)=4\pi
a_{s}(x,y)$ represents the strength of interatomic interaction characterized
by the $s$-wave scattering length $a_{s}(x,y)$, which can be spatially
inhomogeneous by magnetically tuning the Feshbach resonances Inouye1 ; Rodas-
Verde ; Sakaguchi ; Konotop ; Qian ; Belmonte-Beitia12 ; Xiong .
Figure 1: (color online). The interaction parameter $g(x,y)$ for two secondary
quantum numbers: $(a)~{}l=0$ and $(b)~{}l=1$ with $\omega=0.02,\nu=0.1$. It is
seen that $g(x,y)$ is a smooth function when $l=0$ and develops singularity
when $l$ gets large.
Now we consider the spatially localized stationary solution
$\psi(x,y,t)=\phi(x,y)e^{-i\mu t}$ of Eq. (1) with $\phi(x,y)$ being a real
function for $\lim_{|x|,|y|\rightarrow\infty}\phi(x,y)=0.$ This maps Eq. (1)
onto a stationary nonlinear Schrödinger equation
$\frac{1}{2}\phi_{xx}+\frac{1}{2}\phi_{yy}-\frac{1}{2}\omega^{2}(x^{2}+y^{2})\phi-g(x,y)\phi^{3}+\mu\phi=0$
Pethick . Here $\mu$ is the real chemical potential. Solving this stationary
equation by similarity transformation Belmonte-Beitia12 , we obtain a families
of exact localized nonlinear wave solutions for Eq. (1) as
$\psi_{n}={\frac{(n+1)K(k)\eta}{\sqrt{\nu}}}\,{\rm cn}(\theta,k)e^{-i\mu
t},n=0,2,4,\cdots$ (2) $\psi_{n}={\frac{(n+1)K(k)\eta}{\sqrt{2\nu}}}\,{\rm
sd}(\theta,k)e^{-i\mu t},n=1,3,5,\cdots$ (3)
where $k=\sqrt{2}/{2}$ is the modulus of elliptic function, $\nu$ is a
positive real constant,
$K(k)=\int_{0}^{\frac{\pi}{2}}[1-k^{2}\sin^{2}\varsigma]d\varsigma$ is
elliptic integral of the first kind, ${\rm sd}={\rm sn}/{\rm dn}$ with ${\rm
sn},{\rm cn}$ and ${\rm dn}$ being Jacobi elliptic functions, $\theta,\eta$
and $g$ are determined by
$\theta=(n+1)K(k){\rm erf}[\sqrt{2\omega}\left(x+y\right)/2],$
$\eta={e^{\omega\,xy}}{\rm
KummerU}[-\mu/(2\omega),1/2,\omega\,\left(x-y\right)^{2}/2],$ (4)
$g(x,y)=-2\omega\,\nu/(\pi\eta^{2}){e^{-\omega\,\left(x+y\right)^{2}}},$
here ${\rm erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-\tau^{2}}d\tau$ is
error function, and ${\rm KummerU}(a,c,s)$ Abramowitz is Kummer function of
the second kind which is a solution of ordinary differential equation
$s\Lambda^{{}^{\prime\prime}}(s)+(c-s)\Lambda^{{}^{\prime}}(s)-a\Lambda(s)=0.$
It is easy to see that when $|x|,|y|\rightarrow\infty$ we have
$\psi_{n}\rightarrow 0$ for solutions $\psi_{n}$ in Eqs. (2)-(3) with Eq. (4),
thus they are localized bound state solutions.
In the above construction, it is observed that the number of zero points of
function $\eta$ in Eq. (4) is equal to that of function ${\rm
KummerU}[-\mu/(2\omega),1/2,\omega\,\left(x-y\right)^{2}/2],$ which strongly
depends on $\omega$ and the ratio $\mu/\omega.$ We assume the number of zero
points in $\eta$ along line $y=-x$ is $l.$ In the following, we will see that
integer $n$ is associated with the energy levels of the atoms and integers
$n,l$ determine the topological properties of atom packets, so $n$ and $l$ are
named the principal quantum number and secondary quantum number in quantum
mechanics. In addition, the three free parameters $\omega$, $\mu$ and $\nu$
are positive, so the dimensionless interaction function $g(x,y)$ is negative,
which indicates an attractive interaction between atoms. There are known
atomic gases with attractive interactions realized by modulating magnetic
Inouye1 technique, for examples, the 85Rb Cornish and 7Li atoms Hulet1 ;
Hulet2 .
Figure 2: (color online). The density distributions of the quasi-2D BEC with
spatially modulated nonlinearities in harmonic potential, for different
principle quantum numbers $n$ in Eqs. (2)-(3) with Eq. (4), where the
secondary quantum number $l=0,$ the parameters $\omega,\nu$ are $0.02$ and
$0.1,$ respectively. The unit of space length $x,y$ is $1.69~{}\mu m.$ Figs.
2(a)-2(c) show the density profiles of the even parity wave function (2) for
$n=0,2$ and $4,$ respectively. Figs. 2(d)-2(f) demonstrate the density
profiles of the odd parity wave function (3) for $n=1,3$ and $5,$
respectively.
To translate our results into units relevant to the experiments Hulet1 ;
Hulet2 , we take the 7Li condensate containing $10^{3}\sim 10^{5}$ atoms in a
pancake-shaped trap with radial frequency $\omega_{\perp}=2\pi\times 10$ Hz
and axial frequency $\omega_{z}=2\pi\times 500$ Hz Rychtarik . In this case,
the ratio of trap frequency $\omega$ in Eq. (1) is $0.02$ which is determined
by $\omega_{\perp}/\omega_{z}.$ The unit of length is $1.69~{}\mu m$, the unit
of time is $0.32~{}ms$ and the unit of chemical potential is $nK$. The
spatially inhomogeneous interaction parameter $g(x,y)$ is independent of
principal quantum number $n$ but is strongly related to the secondary quantum
number $l$. In the Fig. 1, we show that for $\omega=0.02,\nu=0.1,$ function
$g(x,y)$ is smooth in space when $l=0$ and develops singularity when the $l$
gets large.
Quantized quasi-$2$D BEC.—In order to investigate the quantum and topological
properties of the localized nonlinear matter waves in quasi-2D BEC described
by Eqs. (2)-(3) with Eq. (4), we plot their density distributions by
manipulating the principal quantum number $n$ or secondary quantum number $l$.
Figure 3: (color online). The density distributions of the quasi-2D BEC in
harmonic potential for different secondary quantum number $l$. Figs. 3(a)-3(d)
show the density profiles of the even parity wave function (2) for principle
quantum number $n=0,$ and Figs. 3(e)-3(h) show the density profiles of the odd
parity wave function (3) for $n=1,$ corresponding to $l=0,1,2,3$. The other
parameters are the same as that of Fig. 2.
Firstly, when the secondary quantum number $l$ is fixed, we can modulate the
principal quantum number $n$ to analyze the novel matter waves in quasi-2D
BEC. Fig. 2 shows the density profiles in quasi-2D BEC with spatially
modulated nonlinearities in harmonic potential for $l=0$. It is easy to see
that the matter wave functions in Eq. (2) satisfy
$\psi_{n}(-x,-y)=\psi_{n}(x,y),$ so they are even parity and are invariant
under space inversion. Figs. 2(a)-2(c) demonstrate the density profiles of the
even parity wave functions (2) with Eq. (4) for $n=0,2,4,$ which correspond to
a low energy state and two highly excited states. The numbers of atoms
$N_{n}=\int\int dxdy|\psi_{n}(x,y,t)|^{2}$ for the three states are
$N_{0}=3.76\times 10^{3},N_{2}=6.84\times 10^{4},N_{4}=2.633\times 10^{5}$,
respectively. The matter wave functions in Eq. (3) satisfy
$\psi_{n}(-x,-y)=-\psi_{n}(x,y),$ which denotes that they are odd parity.
Figs. 2(d)-2(f) demonstrate the density profiles of the odd parity wave
functions (3) with Eq. (4) for $n=1,3,5,$ which correspond to three highly
excited states. The numbers of atoms for the three states are
$N_{1}=4.016\times 10^{4},N_{3}=2.493\times 10^{5},N_{5}=7.28\times 10^{5},$
respectively. It is observed that when the secondary quantum number $l=0$, the
number of nodes along line $y=x$ for each quantum state is equal to the
corresponding principal quantum number $n$, i.e. the $n$th level quantum state
has $n$ nodes along $y=x$. And the number of density packets increases one by
one along line $y=x$ when the $n$ increases. This is similar to the quantum
properties in the linear harmonic oscillator.
Secondly, when the principal quantum number $n$ is fixed, we can tune the
secondary quantum number $l$ to observe the novel quantum phenomenon in
quasi-2D BEC. In Fig. 3 we demonstrate the density distributions of quasi-2D
BEC in harmonic potential for different secondary quantum number. Figs.
3(a)-3(d) show the density profiles of the even parity wave function (2) with
Eq. (4) for $n=0,$ and $l=0,1,2$ and $3,$ respectively. It is seen that the
number of nodes for the density packets along line $y=-x$ is equal to the
corresponding secondary quantum number $l$ which describes the topological
patterns of the atom packets, and the number of density packets increases one
by one when $l$ increases. Figs. 3(e)-3(h) show the density profiles of the
odd parity wave function (3) with Eq. (4) for $n=1$ and $l=0,1,2,3.$ We see
that the number of density packets increases pair by pair when $l$ increases.
The number of density packets for each quantum state is equal to
$(n+1)\times(l+1),$ and all the density packets are symmetrical with respect
to lines $y=\pm x,$ as shown in Figs. 2-3.
Normalization energy vs chemical potential. Next we calculate the
normalization energy of each quantum states numerically. The total energy of
the quasi-2D BEC is $E(\psi)=\int\int
dxdy[|\nabla\psi|^{2}+\frac{1}{2}\omega^{2}(x^{2}+y^{2})|\psi|^{2}+\frac{1}{2}g(x,y)|\psi|^{4}]$.
So the normalized energy is given by $E(\psi)/N=\mu-\frac{1}{2N}\int\int
dxdyg(x,y)|\psi|^{4}$ with $N=\int\int dxdy|\psi|^{2}$. Fig. 4 shows the
relations of the normalization energy $E(\psi)/N$ with chemistry potential for
different principle quantum numbers $n$. It is observed that for the fixed
$n$, the normalization energy is approximatively linear increase with respect
to chemistry potential, i.e., $d(E(\psi)/N)/d\mu>0.$ Fig. 4(a) demonstrates
that the normalization energy for the even parity wave function (2) increases
when the principal quantum number $n$ increases. So does the odd parity wave
function (3), as shown in Fig. 4(b). It is shown that the energy levels of the
atoms are only associated with the principle quantum number $n$. These are
similar to energy level distribution of the energy eigenvalue problem for the
linear harmonic oscillator described by linear Schrödinger equation.
Figure 4: (color online). The normalization energy $E(\psi)/N$ vs chemical
potential $\mu$, with $N=\int\int dxdy|\psi|^{2}$. (a) even parity wave
function (2) with principal quantum numbers $n=0,2,4$ and (b) odd parity wave
function (3) with $n=1,3,5$. Here the parameters $\omega=0.02$ and $\nu=0.1.$
Stability analysis.—Stability of exact solutions with respect to perturbation
is very important, because only stable localized nonlinear matter waves are
promising for experimental observations and physical applications. To study
the stability of our exact solutions (2)-(3) with Eq. (4), we consider a
perturbed solution $\psi(x,y,t)=[\phi_{n}(x,y)+\Psi(x,y,t)]e^{-i\mu t}$ of Eq.
(1). Here $\phi_{n}(x,y)$ are the exact solutions of the stationary nonlinear
Schrödinger equation
$\frac{1}{2}\phi_{xx}+\frac{1}{2}\phi_{yy}-\frac{1}{2}\omega^{2}(x^{2}+y^{2})\phi-g(x,y)\phi^{3}+\mu\phi=0$.
$\Psi(x,y,t)\ll 1$ is a small perturbation to the exact solutions and
$\Psi(x,y,t)=[R(x,y)+I(x,y)]e^{i\lambda t}$ is decomposed into its real and
imaginary parts Bronski . Substituting this perturbed solution to the quasi-2D
GP equation (1) and neglecting the higher-order terms in $(R,I)$, we obtain a
standard eigenvalue problem $L_{+}R=\lambda I,~{}L_{-}I=\lambda R,$ where
$\lambda$ is eigenvalue, $R,I$ are eigenfunctions with
$L_{+}=-\frac{1}{2}(\partial_{x}^{2}+\partial_{y}^{2})+3g(x,y)\phi_{n}(x,y)^{2}+\frac{1}{2}\omega^{2}(x^{2}+y^{2})-\mu$
and
$L_{-}=-\frac{1}{2}(\partial_{x}^{2}+\partial_{y}^{2})+g(x,y)\phi_{n}(x,y)^{2}+\frac{1}{2}\omega^{2}(x^{2}+y^{2})-\mu.$
Numerical experiments show that when $\omega=0.02$ and $\mu,\nu$ are arbitrary
non-negative constants, only for principle quantum number $n=0,1,2,3,4,5$ are
the eigenvalues $\lambda$ of this eigenvalue problem real. This suggests that
for $\omega=0.02$ the exact localized nonlinear matter wave solution (2) is
linear stability only for $n=0,2,4$ and solution (3) is linear stability only
for $n=1,3,5,$ see Fig. 5. It is seen that when the frequencies of pancake-
shaped trap is fixed, the stability of the exact solutions (2)-(3) with Eq.
(4) rests only on the principle quantum number $n.$
Experimental protocol.—We now provide an experimental protocol for creating
the quasi-2D localized nonlinear matter waves. To do so, we take the
attractive 7Li condensate Hulet1 ; Hulet2 , containing about $10^{3}\sim
10^{5}$ atoms, confined in a pancake-shaped trap with radial frequency
$\omega_{\perp}=2\pi\times 10$ Hz and axial frequency $\omega_{z}=2\pi\times
500$ Hz Rychtarik . This trap can be determined by combination of
spectroscopic observations, direct magnetic field measurement, and the
observed spatial cylindrical symmetry of the trapped atom cloud Rychtarik .
The next step is to realize the spatial variation of the scattering length.
Near the Feshbach resonance Inouye1 ; Xiong ; Ueda1 ; Khaykovich , the
scattering length $a_{s}(B)$ varies dispersively as a function of magnetic
field $B,$ i.e. $a_{s}(B)=a[1+\Delta/(B_{0}-B)],$ with $a$ being the
asymptotic value of the scattering length far from the resonance, $B_{0}$
being the resonant value of the magnetic field, and $\Delta$ being the width
of the resonance. For the magnetic field in $z$ direction with gradient
$\alpha$ along $x$-$y$ direction, we have $\vec{B}=[B_{0}+\alpha
B_{1}(x,y)]\vec{e_{z}}$. In this case, the scattering length is dependent on
$x$ and $y$. In real experiments, the spatially dependent magnetic field may
be generated by a microfabricated ferromagnetic structure integrated on an
atom chip Vengalattore1 ; Vengalattore2 , such that interaction in Fig. 1 can
be realized. In order to observe the density distributions in Figs. 2-3
clearly in experiment, the 7Li atoms should be evaporatively cooled to low
temperatures, say in the range of 10 to 100 $nK$. After the interaction
parameter in Fig. 1(a) is realized by modulating magnetic field properly, the
density distributions in Fig. 2 can be observed for different numbers of atoms
by evaporative cooling, for example, the numbers of atoms in Fig. 2(a)-2(c)
are $3.76\times 10^{3},6.84\times 10^{4},2.633\times 10^{5}$, respectively.
The density distributions in Fig. 3 can also be observed by changing the
scattering lengths through magnetic field for various atom numbers.
Figure 5: (color online). Eigenvalue for different principal quantum number
$n$ with parameters $\omega=0.02,\mu=0.001$ and $\nu=0.1.$ It is shown that
only for $n=0,1,2,3,4,5$ are the localized nonlinear matter wave solutions
(2)-(3) with Eq. (4) linear stability.
Conclusion.—In summary, we have discovered a new family of stable exact
localized nonlinear matter wave solutions of the quasi-2D BEC with spatially
modulated nonlinearities in harmonic potential. Similar to the linear harmonic
oscillator, we introduce two classes of quantum numbers: the principle quantum
number $n$ and secondary quantum number $l$. The matter wave functions have
even parity for the even principle quantum number and odd parity for the odd
one, the energy levels of the atoms are only associated with the principle
quantum number, and the number of density packets for each quantum state is
equal to $(n+1)\times(l+1)$. We also provide an experimental scheme to observe
these novel phenomena in future experiments. Our results are of particular
significance to matter wave management in high dimensional BEC.
This work was supported by NSFC under Grants No. 10874235, No. 10934010, No.
60978019 and by NKBRSFC under Grants No. 2006CB921400, No. 2009CB930704 and
No. 2010CB922904.
## References
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|
arxiv-papers
| 2009-12-08T08:38:48 |
2024-09-04T02:49:06.904241
|
{
"license": "Public Domain",
"authors": "Deng-Shan Wang, Xing-Hua Hu, Jiangping Hu, W. M. Liu",
"submitter": "Deng-Shan Wang DSW",
"url": "https://arxiv.org/abs/0912.1432"
}
|
0912.1606
|
# Temperature Dependence of the Diffusive Conductivity for Bilayer Graphene
Shaffique Adam and M. D. Stiles Center for Nanoscale Science and Technology,
National Institute of Standards and Technology, Gaithersburg, Maryland
20899-6202, USA
###### Abstract
Assuming diffusive carrier transport, and employing an effective medium
theory, we calculate the temperature dependence of bilayer graphene
conductivity due to Fermi surface broadening as a function of carrier density.
We find that the temperature dependence of the conductivity depends strongly
on the amount of disorder. In the regime relevant to most experiments, the
conductivity is a function of $T/T^{*}$, where $T^{*}$ is the characteristic
temperature set by disorder. We demonstrate that experimental data taken from
various groups collapse onto a theoretically predicted scaling function.
###### pacs:
72.80.Vp,73.23.-b,72.80.Ng
## I Introduction
Monolayer and bilayer graphene are distinct electronic materials. Monolayer
graphene is a sheet of carbon in a honeycomb lattice that is one atom thick,
while bilayer graphene comprises two such sheets, with the first lattice
$0.3~{}\rm{nm}$ above the second. Since the first transport measurements
Novoselov et al. (2005); Zhang et al. (2005) in 2005, we have come a long way
in understanding the basic transport mechanisms of carriers in these new
carbon allotropes. (For recent reviews, see Refs. Castro Neto et al., 2009;
Das Sarma et al., 2010a).
A unique feature of both monolayer and bilayer graphene is that the density of
carriers can be tuned continuously by an external gate from electron-like
carriers at positive doping to holes at negative doping. The behavior at the
crossover depends strongly on the amount of disorder. In the absence of any
disorder and at zero temperature, there are no free carriers at precisely zero
doping. However, ballistic transport through evanescent modes should give rise
to a universal minimum quantum limited conductivity $\sigma_{\rm min}$ in both
monolayer Katsnelson (2006); Tworzydło et al. (2006) and bilayer
graphene.Snyman and Beenakker (2007); Cserti (2007); Trushin et al. (2010) The
“ballistic regime” should hold so long as the disorder-limited mean-free path
is larger than the distance between the contacts. Miao et al. (2007); Danneau
et al. (2008) At finite temperature, the thermal smearing of the Fermi surface
gives a density $n(T)\sim T^{2}$ for monolayer graphene. For ballistic
transport in these monolayers, the conductivity $\sigma\sim\sqrt{|n|}$ for
large $n$, so $\sigma(T)\sim T$.Bolotin et al. (2008); Du et al. (2008) In the
absence of disorder, $\sigma(T)$ interpolates from the universal $\sigma_{\rm
min}$ to the linear in $T$ regime following a function that depends only on
$T/T_{\rm F}$; ($T_{\rm F}$ is the Fermi temperature). Müller et al. (2009)
Most experiments, however, are in the dirty or diffusive limit, which is
characterized by a conductivity that is linear in density (i.e.
$\sigma=ne\mu_{c}$, with a mobility $\mu_{c}$ that is independent of both
temperature and carrier density Morozov et al. (2008); Zhu et al. (2009)), and
the existence of a minimum conductivity plateau Adam and Das Sarma (2008) in
$\sigma(n)$, with $\sigma_{\rm min}=n_{\rm rms}e\mu_{c}/\sqrt{3}$. $n_{\rm
rms}$ is the root-mean-square fluctuation in carrier density induced by the
disorder. In bilayer graphene, to our knowledge, all experiments are in the
diffusive limit.
The purpose of the current work is to calculate the temperature dependence of
the minimum conductivity plateau in bilayer graphene. The temperature
dependent conductivity of diffusive graphene monolayers is understood to
depend largely on phonons, Chen et al. (2008) but monolayer and bilayer
graphene are distinct electronic materials and phonons are not expected to be
important for bilayer graphene transport at the experimentally relevant
temperatures. kn: (a)
## II Theoretical Model
An important difference between monolayer and bilayer graphene is the band
structure near the Dirac point. Monolayer graphene has the conical band
structure and a density of states that vanishes linearly at the Dirac point.
Bilayer graphene has a constant density of states close to the Dirac point
from a hyperbolic dispersion. The tight-binding description for bilayer
graphene McCann and Fal’ko (2006); Nilsson et al. (2006) results in a
hyperbolic band dispersion
$E_{\rm F}(n)=v_{\rm F}^{2}m\left[\sqrt{1+n/n_{0}}-1\right],$ (1)
that is completely specified by two parameters, $v_{\rm F}\approx 1.1~{}\times
10^{8}~{}{\rm cm/s}$ and $n_{0}=v_{\rm F}^{2}m^{2}/(\hbar^{2}\pi)\approx
2.3~{}\times 10^{12}~{}{\rm cm}^{-2}$ (where $h=2\pi\hbar$ is Planck’s
constant). For very small carrier density $n\ll n_{0}$, one can approximate
bilayer graphene as having a parabolic dispersion, although most experiments
typically approach carrier densities as large as $5~{}\times 10^{12}$. The
density of states for bilayer graphene is
$D(E)=\frac{2m}{\pi\hbar^{2}}\left[1+\frac{|E|}{v_{\rm F}^{2}m}\right],$ (2)
where the parabolic approximation keeps only the first term.
Understanding the temperature dependence of the conductivity minimum is
complicated for two reasons. First, there is activation of both electron and
hole carriers at finite temperature. Second, the disorder induces regions of
inhomogeneous carrier density (i.e. puddles of electrons and holes). Moreover,
tuning the carrier density with a gate changes the ratio between electron-
puddles and hole-puddles, until at very high density there is only a single
type of carrier. The temperature dependence of the conductivity for bilayer
graphene was studied in Ref. Nilsson et al., 2006 using a coherent potential
approximation. While this approach better captures the impurity scattering and
electronic screening properties of graphene, it does not account for the
puddle physics which is our main focus. Reference Zhu et al., 2009 modeled the
temperature dependence of the Dirac point conductivity by assuming that the
graphene samples comprised just two big “puddles” each with the same number of
carriers. In the appropriate limits, our results agree with these previous
works. Below we will provide a semi-analytic expression for the graphene
conductivity by averaging over the random distribution of puddles with
different carrier densities. This result is valid throughout the crossover
from the Dirac point (where fluctuations in carrier density dominate) to high
density (where these fluctuations are irrelevant), both with and without the
thermal activation of carriers.
Figure 1: (Color online) Bilayer graphene mobility as a function of back-gate
voltage $V_{g}$, normalized by the mobility at $V_{g}=40~{}{\rm V}$. Solid
lines use bilayer graphene’s hyperbolic dispersion relation, while dashed
lines are the parabolic approximation valid only for low carrier density.
Upper panel – long-ranged Coulomb impurities. From bottom to top: over-
screened (parabolic), RPA (parabolic), Thomas-Fermi (parabolic), over-screened
(hyperbolic), Thomas-Fermi (hyperbolic). Lower panel: short-range (i.e.
“delta-correlated” or “white noise”) impurities. From bottom to top: RPA
(parabolic), Thomas-Fermi (parabolic), Thomas-Fermi (hyperbolic), unscreened
(hyperbolic), unscreened (parabolic). See Ref. Das Sarma et al., 2010a for
definitions of the different approximations.
Given a microscopic model for the disorder, one can compute both $\mu_{c}$ and
$n_{\rm rms}$. Shown in Fig. 1 are results for bilayer graphene mobility
assuming both short-range and Coulomb disorder with different approximations
for the screening, and for both parabolic and hyperbolic dispersion relations.
As seen from the figure, generically, Coulomb impurities show a super-linear
dependence on carrier density while short-range scattereres are sub-linear.
Similar to monolayer graphene,Adam et al. (2007); Jang et al. (2008);
Ponomarenko et al. (2009) increasing the dielectric constant tends to decrease
(increase) the scattering of electrons off long (short) range impurities,
except in the over-screened and unscreened limits. All experiments to date
find the mobility to be linear in gate voltage, so it is unclear what the
dominant scattering mechanism in bilayer graphene is (see also discussion in
Ref. Xiao et al., 2010). Further experiments along the lines of Refs. Jang et
al., 2008; Ponomarenko et al., 2009 are needed.
In what follows we take $\mu_{c}$ and $n_{\rm rms}$ to be parameters of the
theory that can be determined directly from experiments: $\mu_{c}$ can be
obtained from low temperature transport measurements and $n_{\rm rms}$ from
local probe measurements.Deshpande et al. (2009); Martin et al. (2008); Zhang
et al. (2009); Miller et al. (2009) Lacking such microscopic measurements for
the samples we compare with, we treat $n_{\rm rms}$ as a fitting parameter,
while taking $\mu_{c}$ from experiment. As a consequence of this
parameterization, the results reported here do not depend on the microscopic
details of the impurity potential, provided this parameterization reasonably
characterizes the properties of the impurity potential. Until more information
about the important scattering centers is determined from experiment, all
microscopic models will require a similar number of parameters such as the
concentration of impurities $n_{\rm imp}$ and their typical distance $d$ from
the graphene sheet. Further, the results will disagree with experiment unless
the choices give a constant mobility.
A key assumption in this work is the applicability of Effective Medium Theory
(EMT), which describes the bulk conductivity $\sigma_{\rm EMT}$ of an
inhomogeneous medium by the integral equation Rossi et al. (2009)
$\int dnP[n]\frac{\sigma(n)-\sigma_{\rm EMT}}{\sigma(n)+\sigma_{\rm EMT}}=0.$
(3)
$P[n]$ is the probability distribution of the carrier density in the
inhomogeneous medium – positive (negative) $n$ corresponds to (electrons)
holes, and $\sigma(n)$ is the local conductivity of a small patch with a
homogeneous carrier density $n$. Ignoring the denominator, Eq. 3 gives
$\sigma_{\rm EMT}$ equal to the average conductivity. The denominator weights
the integral to cancel the build-up of any internal electric fields. The EMT
description has been shown to work well whenever the transport is
semiclassical and quantum corrections and any additional resistance caused by
the $p-n$ interfaces between the electron and hole puddles can be
ignored.Rossi et al. (2009); Adam et al. (2009a); Fogler (2009) It is assumed
that the band structure is not altered by the disorder, which is to be
expected for the experimentally relevant disorder concentrations.Pershoguba et
al. (2009) Since we are concerned with diffusive transport in the dirty limit,
we expect that the EMT results hold for bilayer graphene.
## III Results
Figure 2: (Color online) Conductivity vs. gate voltage for clean and dirty
graphene bilayers calculated from Eq. 3. Solid curves use the hyperbolic
dispersion relation while dashed lines (only distinguishable at high
temperature) show the parabolic approximation. Choice of parameters were based
on experiments of Ref. Morozov et al., 2008 (clean) and Ref. Fuhrer, 2009
(dirty). Left panel: $\mu_{c}=6,750~{}{\rm cm}^{2}/{\rm Vs}$, $n_{\rm
rms}=4\times 10^{11}~{}{\rm cm}^{-2}$ and (from bottom to top) T = 20 K, 100
K, 180 K and 260 K. Right panel: $\mu_{c}=1,100~{}{\rm cm}^{2}/{\rm Vs}$,
$n_{\rm rms}=1.25\times 10^{12}~{}{\rm cm}^{-2}$ and (from bottom to top) T =
12 K, 105 K, 171 K and 290 K.
Figure 3: (Color online) Minimum conductivity as a function of temperature
for linear dispersion (upper curve) and parabolic dispersion (lower curve)
graphene. Dashed lines show the high temperature asymptotes $\sigma_{\rm
min}\rightarrow\pi e\mu_{c}T^{2}/(3\hbar^{2}v_{\rm F}^{2})$ for linear and
$\sigma_{\rm min}\rightarrow me\mu_{c}4\ln 2T/(\pi\hbar^{2})$ parabolic cases.
Solid (red) line shows the hyperbolic result for $n_{\rm rms}=10^{12}{\rm
cm}^{-2}$. Also shown is that the hyperbolic result extrapolates from the
parabolic theory at large $\alpha=m^{2}v_{\rm F}^{2}/(\hbar^{2}\pi n_{\rm
rms})$ becoming similar to the linear dispersion for small $\alpha$. (Red
squares show results for $\alpha=100$ and red circles are for $\alpha=0.01$;
here we ignore the contribution from higher bands).
To solve Eq. 3 we make the additional assumption that the distribution
function $P[n,n_{g}]$ is Gaussian centered at $n_{g}$, (i.e. the field effect
carrier density induced by the back gate that is proportional to $V_{g}$),
with width $n_{\rm rms}$. (This assumption is justified both
theoreticallyMorgan (1965); Stern (1974); Galitski et al. (2007); Adam et al.
(2009b) and empiricallyDeshpande et al. (2009)). Our results are shown in Fig.
2, where as discussed earlier, the temperature dependence comes from the
smearing of the Fermi surface.
At first glance, it is not obvious that the results for clean bilayer graphene
(left panel of Fig. 2) and dirty bilayer graphene (right panel) are closely
related. However, if we consider scaling the conductivity as
${\tilde{\sigma}_{\rm EMT}}=\sigma_{\rm EMT}/(n_{\rm rms}e\mu_{c})$, scaling
temperature as $t=T/T^{*}$, where we define $k_{\rm B}T^{*}=E_{\rm F}(n=n_{\rm
rms})$, and scaling carrier density as $z=n/n_{\rm rms}$, we find that for
both the linear band dispersion $(n\gg n_{0})$ and the parabolic band
dispersion $(n\ll n_{0})$, the scaled functions ${\tilde{\sigma}_{\rm
EMT}}(z,t)$ each follow a universal curve. This is illustrated in Fig. 3 where
we show the temperature dependence of the minimum conductivity. The results
for the hyperbolic dispersion (which is the correct approximation at
experimentally relevant carrier densities), depends on an additional parameter
$\alpha=n_{0}/n_{\rm rms})$.kn: (b)
The scaling function for the hyperbolic dispersion extrapolates from the
parabolic theory at large $\alpha$ becoming similar to the linear result for
small $\alpha$. For the experimentally relevant regime $\alpha\approx 1$ the
hyperbolic result depends only weakly on $\alpha$ and is indistinguishable
from the parabolic result for $T\lesssim 0.5~{}T^{*}$.
Figure 4: (Color online) Same results as in Fig. 3 showing comparison with
experimental data from several groups. Inset shows the unscaled experimental
data, while the main panel shows that the data collapses onto the theoretical
curve with one scaling parameter ($n_{rms}$), where for each of these samples,
we also use the value of mobility reported by the authors and obtained from a
separate low temperature measurement. Green triangles show suspended bilayer
data from Ref. Feldman et al., 2009 using $\mu_{c}=1.4~{}\mbox{\rm
m}^{2}/\mbox{Vs}$ and $T^{*}=36~{}{\rm K}$. Orange squares (Ref. Fuhrer, 2009)
and diamonds (Ref. Zhu et al., 2009) are bilayers on a SiO2 substrate with
$\mu_{c}=0.11~{}\mbox{\rm m}^{2}/\mbox{Vs}$, $T^{*}=530~{}{\rm K}$ and
$\mu_{c}=0.045~{}\mbox{\rm m}^{2}/\mbox{Vs}$ and $T^{*}=290~{}{\rm K}$. Cyan
circles show the four data points of Ref. Morozov et al., 2008, with
$\mu_{c}=0.675~{}\mbox{\rm m}^{2}/\mbox{Vs}$, $T^{*}=80~{}{\rm K}$, which are
off-scale in the main panel.
This analysis suggests that $\sigma_{\rm min}(T)/(e\mu_{c})$, which can be
taken directly from experiment, is not a function of $\mu_{c}$, but only
$n_{\rm rms}$. We take results from a set of experiments in very different
regimes (see the inset of Fig. 4) and choose $n_{\rm rms}$ to fix the value of
$\sigma_{\rm min}(T)/(n_{\rm rms}e\mu_{c})$ at $T=0$. Then using $k_{\rm
B}T^{*}(n_{\rm rms})=E_{\rm F}(n_{\rm rms})$ to scale the temperature, all of
the results lie on top of the theoretical curve computed using the hyperbolic
dispersion, see Fig. 4. The theoretical curve with which they agree is
distinct from similar curves calculated for a linear dispersion and for the
purely parabolic dispersion at high $T/T^{*}$. We note that the scaling
function is more complicated than a line. The calculation reproduces not only
the initial slope as a function of temperature, but the crossover to higher
temperature behavior. For the parabolic dispersion, which agrees at low
temperatures, the conductivity extrapolates from $\sigma_{\rm
min}(T\rightarrow 0)/(n_{\rm rms}e\mu_{c})\approx 3^{-1/2}$ at low temperature
to $\sigma_{\rm min}(t\gg 1)/(n_{\rm rms}e\mu_{c})\approx(2\ln 2)t$ at high
temperature, with a crossover temperature scale of $T\approx T^{*}/2$. In the
future, it should be possible to further test this agreement by measuring
$n_{\rm rms}$ experimentally. Deshpande et al. (2009); Martin et al. (2008);
Zhang et al. (2009); Miller et al. (2009)
Figure 5: (Color online) Bilayer layer graphene conductivity as a function of
temperature and carrier density for $T/T^{*}=0,0.5,1,1.5$, and $2$. Inset
shows a close-up of the zero temperature minimum conductivity (which is the
same for both monolayer and bilayer graphene). The dashed horizontal line
shows the result for $\sigma_{\rm min}$, while the other dashed line is the
high-density transport regime. The solid line (Eq. 4) captures the full
crossover from the regime where the conductivity is dominated by the disorder
induced carrier density fluctuations, to the semiclassical Boltzmann transport
regime.
One feature of Fig. 2 and Fig. 4 is that for most of the experimentally
relevant regime, the temperature dependence of the conductivity calculated
using the parabolic approximation provides an adequate solution. This limit
has been treated in contemporaneous work Das Sarma et al. (2010b); Lv and Wan
(2010) treating this problem with different approximations and reaching
similar conclusions. To better understand the emergence of a universal scaling
form, we consider the conductivity for a parabolic band dispersion. Using the
scaled variables defined above, we can manipulate Eq. (3) into the
dimensionless form
$\displaystyle\int_{0}^{\infty}dz\exp\left[-z^{2}/2\right]\cosh\left[z_{g}z\right]\frac{H[z,t]-{\bar{\sigma}}[z_{g},t]}{H[z,t]+{\bar{\sigma}}[z_{g},t]}=0,$
(4)
where $z_{g}=n_{g}/n_{\rm rms}$ and we have written the local conductivity as
$\sigma(n,T)=n_{\rm rms}e\mu_{c}H(z,t)$. Below we calculate the dimensionless
function $H(z,t)$ assuming thermally activated carrier transport with constant
$n_{\rm rms}$ and $\mu_{c}$ and explicitly show that it depends only on scaled
variables $z=n/n_{\rm rms}$ and $t=T/T^{*}$. With the analytical results for
$H(z,t)$ discussed below, this implicit equation can be solved either
perturbatively or by numerical integration to give $\sigma_{\rm EMT}$. The
results of this calculation are shown in Fig. 5.
To proceed, we calculate the function $H(z,t)$. For thermal activation of
carriers, the chemical potential $\mu$ is determined by solving for
$n_{g}=n_{e}-n_{h}$,Hwang and Das Sarma (2009) where
$\displaystyle n_{e}(T)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dE~{}D(E)f(E,\mu,k_{\rm B}T),$ $\displaystyle
n_{h}(T)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{0}dE~{}D(E)\left[1-f(E,\mu,k_{\rm B}T)\right],$
(5)
where $f(E,\mu,k_{\rm B}T)$ is the Fermi-Dirac function and $k_{\rm B}$ is the
Boltzmann constant. For $T=0$, only majority carriers are present, while for
$T\rightarrow\infty$, activated carriers of both types are present in equal
number. Within the parabolic approximation, we find $n_{e(h)}=n_{g}(T/T_{\rm
F})\ln\left[1+\exp(\mp\mu/k_{\rm B}T)\right]$ and $\mu=E_{\rm F}$. Using
$\sigma(n,T)=(n_{e}+n_{h})e\mu_{c}$, we obtain
$H(z,t)=z+2t\ln\left[1+e^{-z/t}\right].$ (6)
This demonstrates that Eq. 4 depends only on the scaled variables,
guaranteeing that ${\tilde{\sigma}}_{\rm EMT}$ is a function only of $T/T^{*}$
and $n_{g}/n_{\rm rms}$ as shown in Fig. 5.
A similar analysis can be done for the hyperbolic dispersion. We find
$\displaystyle H(z,t,\alpha)=$
$\displaystyle\frac{z}{\xi+2}\left[4tg\ln[1+e^{-y/tg}]+2y\frac{}{}\right.$ (7)
$\displaystyle\mbox{}\left.+\frac{(tg\pi)^{2}\xi}{3}+\xi y^{2}\right],$
where $g(z,\alpha)=T^{*}/T_{\rm F}$, $\xi(z,\alpha)=-1+\sqrt{1+z/\alpha}$, and
the scaled chemical potential $y=\mu/E_{\rm F}$ is given by
$\displaystyle y=\frac{1}{2}\left[2+\xi-2\xi(tg)^{2}({\rm
Li}_{2}(-e^{-y/tg})-{\rm Li}_{2}(-e^{+y/tg}))\right],$ (8)
where ${\rm Li}_{2}(z)=\int_{z}^{0}dt~{}t^{-1}\ln(1-t)$ is the dilogarithm
function. Only for $\alpha\gg 1$ and $\alpha\ll 1$ does $H(z,t,\alpha)$ become
independent of $\alpha=n_{0}/n_{\rm rms}$ giving the universal scaling forms
for linear and parabolic dispersions, respectively.
## IV Conclusion
In summary, we have developed an effective medium theory that captures the
gate voltage and temperature dependence of the conductivity for bilayer
graphene. The theory depends on two parameters: $n_{\rm rms}$ that sets the
scale of the disorder, and $\mu_{c}$ the carrier mobility. These could be
computed a priori by assuming a microscopic model for the disorder potential
and its coupling to the carriers in graphene. Alternatively, one could use an
empirical approach where one uses experimental data at $T=0$ to determine the
parameters and use the theory to predict the temperature dependence.
Our main finding is that experimental data taken from various groups collapse
onto our calculated scaling function where the disorder sets the scale of the
temperature dependence of the conductivity. This further suggests that even
some suspended bilayer samples are still the the diffusive (rather than
ballistic) transport regime.
###### Acknowledgements.
We thank M. Fuhrer and K. Bolotin for suggesting this problem and for useful
discussions. SA also acknowledges a National Research Council (NRC)
postdoctoral fellowship.
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|
arxiv-papers
| 2009-12-09T18:56:44 |
2024-09-04T02:49:06.912128
|
{
"license": "Public Domain",
"authors": "Shaffique Adam and M. D. Stiles",
"submitter": "Shaffique Adam",
"url": "https://arxiv.org/abs/0912.1606"
}
|
0912.1718
|
# Study of Decay Modes $B\to K_{0}^{*}(1430)\phi$
C. S Kim,1 111Email: cskim@yonsei.ac.kr Ying Li,1,2,4 222Email:
liying@ytu.edu.cn Wei Wang3 333Email: wei.wang@ba.infn.it
1.Department of Physics, Yonsei University, Seoul 120-479, Korea
2\. Department of Physics, Yantai University, Yantai 264-005, China
3.Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari 70126, Italy
4\. Kavli Institute for Theoretical Physics China (KITPC), Beijing,100-080,
China
Within the framework of perturbative QCD approach based on $\mathbf{k_{T}}$
factorization, we investigate the charmless decay mode $B\to
K_{0}^{*}(1430)\phi$. Under two different scenarios (S1 and S2) for the
description of scalar meson $K_{0}^{*}(1430)$, we explore the branching
fractions and related $CP$ asymmetries. Besides the dominant contributions
from the factorizable emission diagrams, penguin operators in the annihilation
diagrams could also provide considerable contributions. The central values of
our predictions are larger than those from the QCD factorization in both
scenarios. Compared with the experimental measurements of the BaBar
collaboration, the result of neutral channel in the S1 agrees with
experimental data, while the result of the charged one is a bit smaller than
the data. In the S2 scenario, although the central value for the branching
fractions of both channels are much larger than the data, the predictions
could agree with the data due to the large uncertainties to the branching
fractions from the hadronic input parameters. The $CP$ asymmetry in the
charged channel is small and not sensitive to CKM angle $\gamma$. With the
accurate data in near future from the various $B$ factories, these predictions
will be under stringent tests.
PACS numbers:12.38.Bx, 11.10.Hi, 12.38.Qk, 13.25.Hw
## 1 Introduction
The $b\to ss\bar{s}$ transition, inducing many non-leptonic charmless $B$
meson decay processes such as $B\to K_{S}\phi$, $B\to
K_{S}\eta(\eta^{\prime})$ and $B\to K^{*}\phi$, has attracted much interest
because it serves as an ideal platform to probe the possible new physics (NP)
beyond the standard model (SM). However, the kind of transition involving a
scalar meson have more ambiguities due to intriguing but mysterious underlying
nature of scalar mesons. In the spectroscopy study, there are two different
scenarios to describe the scalar mesons. The scenario-1 (S1) is the naive
2-quark model: the nonet mesons below 1 GeV are treated as the lowest lying
states, and the ones near 1.5 GeV are the first orbitally excited state. In
the scenario-2 (S2), the nonet mesons near 1.5 GeV are viewed as the lowest
lying states, while the mesons below 1 GeV may be viewed as exotic states
beyond the quark model such as four-quark bound states. Under these two
pictures, many $B\to SP$ modes, such as $B\to f_{0}K$, induced by $b\to
ss\bar{s}$ transition have been calculated in both QCD factorization (QCDF)
approach [1, 2] and perturbative QCD (PQCD) approach [3, 4, 5, 6]. Within
proper regions for the input parameters, many theoretical results could agree
with the experimental data.
In this work, we will study the $B\to K^{*}_{0}(1430)\phi$ decays in the
perturbative QCD approach [7]. On the experimental side, the branching ratios
of $B\to K_{0}^{*}(1430)\phi$ have been measured with good precision [8, 9]:
$\displaystyle{\cal B}(\overline{B}^{0}\to\overline{K}_{0}^{*0}(1430)\phi)$
$\displaystyle=$ $\displaystyle(4.6\pm 0.7\pm 0.6)\times 10^{-6}~{},$ (1)
$\displaystyle{\cal B}(B^{\pm}\to K_{0}^{*\pm}(1430)\phi)$ $\displaystyle=$
$\displaystyle(7.0\pm 1.3\pm 0.9)\times 10^{-6}~{},$ (2)
where the result for the neutral channel has been updated as [10]
$\displaystyle{\cal B}(\overline{B}^{0}\to\overline{K}_{0}^{*0}(1430)\phi)$
$\displaystyle=$ $\displaystyle(3.9\pm 0.5\pm 0.6)\times 10^{-6}~{}.$ (3)
Compared with the $B\to K\phi$ decay [11]
$\displaystyle{\cal B}(\overline{B}^{0}\to\overline{K}^{0}\phi)$
$\displaystyle=$ $\displaystyle(8.3^{+1.2}_{-1.0})\times 10^{-6}~{},$ (4)
$\displaystyle{\cal B}(B^{\pm}\to K^{\pm}\phi)$ $\displaystyle=$
$\displaystyle(8.30\pm 0.65)\times 10^{-6}~{},$ (5)
we can see that the decay channels with a scalar meson in the final state,
$B\to K_{0}^{*}(1430)\phi$, seem to have a bit smaller branching fractions. In
Refs. [12, 13], the decay $\overline{B}^{0}\to\overline{K}_{0}^{*0}(1430)\phi$
has been studied within the framework of generalized factorization in which
the non-factorizable effects are described by the parameter $N_{c}^{\rm eff}$,
the effective number of colors. The predicted branching ratio (BR) varies from
$10^{-7}$ to $10^{-5}$, depending on the different values for $N_{c}^{\rm
eff}$. Without the information for non-factorizable effects, one cannot make a
precise prediction of the BR. The QCDF calculation of this and other modes has
also been presented in Ref. [14], and the predicted central value of ${\cal
B}(\overline{B}^{0}\to\overline{K}^{*0}_{0}(1430)\phi)$ deviates from the
experimental data, though it can be accommodated within large theoretical
errors. It is necessary to analyze these channels in the PQCD approach with
different treatments for the matrix elements of the four-quark operators,
which is helpful to probe the structure of the scalar meson model-
independently.
The layout of the present paper is as follows: In Sec. 2 we introduce the
input parameters including the decay constants and light-cone distribution
amplitudes. The factorization formulae in the perturbative QCD approach are
given in Sec. 3. Numerical results and discussions are presented in Sec. 4.
Summary of this work is also given in Sec. 4.
## 2 Input Parameters
In the $B$ meson rest frame, the $B$ meson momentum $P_{1}$, the $\phi$ meson
momentum $P_{2}$, the longitudinal polarization vector $\epsilon_{L}$, and the
kaon momentum $P_{3}$ are chosen, in light-cone coordinates, as
$\displaystyle P_{1}=\frac{M_{B}}{\sqrt{2}}(1,1,{\bf 0}_{T})\;,\;\;\;P_{2}$
$\displaystyle=$
$\displaystyle\frac{M_{B}}{\sqrt{2}}(1-r_{K_{0}^{*}}^{2},r_{\phi}^{2},{\bf
0}_{T})\;,\;\;\;P_{3}=\frac{M_{B}}{\sqrt{2}}(r_{K_{0}^{*}}^{2},1-r_{\phi}^{2},{\bf
0}_{T})\;,\;\;\;$ $\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}r_{\phi}}(1-r_{K_{0}^{*}}^{2},-r_{\phi}^{2},{\bf
0}_{T})\;,$ (6)
with the ratio $r_{\phi(K_{0}^{*})}=m_{\phi({K_{0}^{*}})}/M_{B}$, and
$m_{\phi}$, $m_{K_{0}^{*}}$ being the $\phi$ meson mass and $K_{0}^{*}$ meson
mass, respectively. The momentum of the light antiquark in the $B$ meson and
the light quarks in the final mesons are denoted as $k_{1}$, $k_{2}$ and
$k_{3}$ respectively. Using the intrinsic variables (momentum fractions and
the transverse momentum), we can choose
$\displaystyle k_{1}=(0,x_{1}P_{1}^{-},{\bf k}_{1T}),\quad
k_{2}=(x_{2}P_{2}^{+},0,{\bf k}_{2T}),\quad k_{3}=(0,x_{3}P_{3}^{-},{\bf
k}_{3T}).$ (7)
The decay constants of scalar meson are defined by
$\displaystyle\langle
S(p)|\bar{q}_{2}\gamma_{\mu}q_{1}|0\rangle=f_{S}p_{\mu}\;,\;\langle
S|\bar{q}_{2}q_{1}|0\rangle=m_{S}\bar{f}_{S},$ (8)
where the decay constant $f_{S}$ of the vector current and $\bar{f}_{S}$ of
the scalar current are related by equations of motion
$\mu_{s}f_{S}=\bar{f}_{S}$, with
$\mu_{s}=\frac{m_{S}}{m_{2}(\mu)-m_{1}(\mu)}$. The parameter $m_{S}$ is the
mass of the scalar meson, and $m_{1}$, $m_{2}$ are the running current quark
masses. Inputs of the scalar mesons in our calculation, including the decay
constants, running quark masses and the Gegenbauer moments defined in the
following, are quoted from Ref. [2].
For the scalar meson wave function, the twist-2 light-cone distribution
amplitude (LCDA) $\phi_{S}(x)$ and twist-3 LCDAs $\phi_{S}^{s}(x)$ and
$\phi_{S}^{\sigma}$ for the scalar mesons can be combined into a single matrix
element:
$\displaystyle\langle K_{0}^{*+}(p)|\bar{u}_{\beta}(z)s_{\alpha}(0)|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{6}}\int^{1}_{0}dxe^{ixp\cdot
z}\bigg{\\{}p\\!\\!\\!/\phi_{K^{*+}_{0}}(x)+m_{S}\phi^{S}_{K^{*+}_{0}}(x)+\frac{1}{6}m_{S}\sigma_{\mu\nu}p^{\mu}z^{\nu}\phi^{\sigma}_{K^{*+}_{0}}(x)\bigg{\\}}_{\alpha\beta}$
(9) $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{6}}\int^{1}_{0}dxe^{ixp\cdot
z}\bigg{\\{}p\\!\\!\\!/\phi_{K^{*+}_{0}}(x)+m_{S}\phi^{S}_{K^{*+}_{0}}(x)+m_{S}(n\\!\\!\\!/v\\!\\!\\!/-1)\phi^{T}_{K^{*+}_{0}}(x)\bigg{\\}}_{\alpha\beta},$
where $v$ and $n$ are dimensionless vectors on the light cone, and $n$ is
parallel with the moving direction of the scalar meson. The distribution
amplitudes $\phi_{K^{*}_{0}}(x)$, $\phi^{S}_{K^{*}_{0}}(x)$ and
$\phi^{\sigma}_{K^{*}_{0}}(x)$ are normalized as:
$\displaystyle\int^{1}_{0}dx\phi_{K^{*}_{0}}(x)=\frac{f_{K^{*}_{0}}}{2\sqrt{6}},\,\,\,\,\,\,\int^{1}_{0}dx\phi^{S}_{K^{*}_{0}}(x)=\int^{1}_{0}dx\phi^{\sigma}_{K^{*}_{0}}(x)=\frac{\bar{f}_{K^{*}_{0}}}{2\sqrt{6}},$
(10)
and
$\phi^{T}_{K^{*}_{0}}(x)=\frac{1}{6}\frac{d}{dx}\phi^{\sigma}_{K^{*}_{0}}(x)$.
For the $K^{*+}_{0}$ meson, the decay constant $f_{K^{*}_{0}}$ has the
opposite sign with that of the $K^{*-}_{0}$ meson.
Under the conformal spin symmetry, the twist-2 LCDA $\phi_{K^{*}_{0}}(x)$ can
be expanded as:
$\displaystyle\phi_{K^{*}_{0}}(x,\mu)$ $\displaystyle=$
$\displaystyle\frac{\bar{f}_{K^{*}_{0}}(\mu)}{2\sqrt{6}}6x(1-x)\bigg{[}B_{0}(\mu)+\sum\limits^{\infty}_{m=1}B_{m}(\mu)C_{m}^{3/2}(2x-1)\bigg{]}$
(11) $\displaystyle=$
$\displaystyle-\frac{{f}_{K^{*}_{0}}(\mu)}{2\sqrt{6}}6x(1-x)\bigg{[}-1+\mu_{S}\sum\limits^{\infty}_{m=1}B_{m}(\mu)C_{m}^{3/2}(2x-1)\bigg{]},$
where $B_{m}(\mu)$ and $C_{m}^{3/2}(x)$ are the Gegenbauer moments and
Gegenbauer polynomials, respectively. The Gegenbauer moments $B_{1}$, $B_{3}$
of distribution amplitudes for $K^{*}_{0}$ and the decay constants have been
calculated in the QCD sum rules [2] as
$\displaystyle\mbox{S}\,{\rm 1}$ $\displaystyle:$
$\displaystyle\,\,\,\,B_{1}=0.58\pm 0.07,\;\;\;\;\;\;B_{3}=-1.20\pm
0.08,\;\;\;\;\;\;\bar{f}_{K^{*}_{0}}(1\mathrm{GeV})=-(300\pm
30)~{}~{}\mathrm{MeV};$ $\displaystyle\mbox{S}\,{\rm 2}$ $\displaystyle:$
$\displaystyle\,\,\,\,B_{1}=-0.57\pm 0.13,\;\;\;\;B_{3}=-0.42\pm
0.22,\;\;\;\;\;\;\bar{f}_{K^{*}_{0}}(1\mathrm{GeV})=(445\pm
50)~{}~{}\mathrm{MeV}.$ (12)
All the above values are taken at $\mu=1$ GeV.
For the twist-3 LCDAs, they have been promoted in the Ref. [15] with large
uncertainties, so we take the asymptotic form in our numerical calculation for
simplicity:
$\displaystyle\phi^{S}_{S}(x)=\frac{\bar{f}_{S}}{2\sqrt{6}},\,\,\,\,\,\phi^{T}_{S}(x)=\frac{\bar{f}_{S}}{2\sqrt{6}}(1-2x).$
(13)
Up to twist-3 accuracy, the vector meson’s wave functions are collected as
$\displaystyle\langle\phi(P_{2},\epsilon_{L})|\bar{s}_{\beta}(z)s_{\alpha}(0)|0\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{6}}\int_{0}^{1}dxe^{ixP_{2}\cdot
z}\left[m_{\phi}\not\\!\epsilon^{*}_{L}\phi_{\phi}(x)+\not\\!\epsilon^{*}_{L}\not\\!P_{2}\phi_{\phi}^{t}(x)+m_{\phi}\phi_{\phi}^{s}(x)\right]_{\alpha\beta},$
(14)
for longitudinal polarization. The distribution amplitudes can be parametrized
as:
$\displaystyle\phi_{\phi}(x)$ $\displaystyle=$
$\displaystyle\frac{3f_{\phi}}{\sqrt{6}}x(1-x)\left[1+a_{2\phi}^{||}C_{2}^{3/2}(2x-1)\right],\;$
$\displaystyle\phi^{t}_{\phi}(x)$ $\displaystyle=$
$\displaystyle\frac{3f^{T}_{\phi}}{2\sqrt{6}}(2x-1)^{2},$
$\displaystyle\phi^{s}_{\phi}(x)$ $\displaystyle=$
$\displaystyle\frac{3f_{\phi}^{T}}{2\sqrt{6}}(1-2x)~{},$ (15)
with the Gegenbauer coefficient $a_{2\phi}^{||}(1{\rm GeV})=0.18\pm 0.08$
[16].
Since the $B$ meson is a pseudo-scalar heavy meson, the structure
$(\gamma^{\mu}\gamma_{5})$ and $\gamma_{5}$ components remain as leading
contributions. Then, $\Phi_{B}$ is written by
$\Phi_{B}=\frac{i}{\sqrt{6}}\left\\{(\not\\!P_{B}\gamma_{5})\phi_{B}^{A}+\gamma_{5}\phi_{B}^{P}\right\\},$
(16)
where $P_{B}$ is the corresponding meson’s momentum, and $\phi_{B}^{A,P}$ are
Lorentz scalar distribution amplitudes. As heavy quark effective theory leads
to $\phi_{B}^{P}\simeq M_{B}\phi_{B}^{A}$, $B$ meson’s wave function can be
expressed by
$\phi_{B}(x,b)=\frac{i}{\sqrt{6}}\left[(\not\\!P_{B}\gamma_{5})+M_{B}\gamma_{5}\right]\phi_{B}(x,b).$
(17)
For the $B$ meson distribution amplitude, we adopt the model:
$\displaystyle\phi_{B}(x,b)=N_{B}x^{2}(1-x)^{2}\exp\left[-\frac{1}{2}\left(\frac{xM_{B}}{\omega_{B}}\right)^{2}-\frac{\omega_{B}^{2}b^{2}}{2}\right]\;,$
(18)
with the shape parameter $\omega_{B}=0.4$ GeV, which has been tested in many
channels such as $B\to\pi\pi,K\pi$ [7]. The normalization constant
$N_{B}=91.784$ GeV is related to the decay constant $f_{B}=190$ MeV. In the
above model, $\phi_{B}$ has a sharp peak at $x\sim\bar{\Lambda}/M_{B}\sim
0.1$.
## 3 Analytical Formulae
In the PQCD approach, after the integration over $k_{1}^{+}$, $k_{2}^{+}$, and
$k_{3}^{-}$, the decay amplitude for $B\to{K^{*}_{0}}\phi$ decay can be
conceptually written as
$\displaystyle{\cal A}$ $\displaystyle\sim$
$\displaystyle\int\\!\\!dx_{1}dx_{2}dx_{3}b_{1}db_{1}b_{2}db_{2}b_{3}db_{3}$
(19)
$\displaystyle~{}~{}\times\mathrm{Tr}\left[C(t)\Phi_{B}(x_{1},b_{1})\Phi_{\phi}(x_{2},b_{2})\Phi_{K^{*}_{0}}(x_{3},b_{3})H(x_{i},b_{i},t)S_{t}(x_{i})\,e^{-S(t)}\right],$
where $x_{i}$ are momenta fraction of light quarks in each meson.
$\mathrm{Tr}$ denotes the trace over Dirac and color indices, $C(t)$ is the
Wilson coefficient evaluated at scale $t$, and the hard kernel
$H(k_{1},k_{2},k_{3},t)$ is the hard part and can be calculated
perturbatively. And the function $\Phi_{M}$ is the wave function, the function
$S_{t}(x_{i})$ describes the threshold resummation which smears the end-point
singularities on $x_{i}$, and the last term, $e^{-S(t)}$, is the Sudakov form
factor which suppresses the soft dynamics effectively.
Figure 1: The leading order Feynman diagrams for $B^{+}\to K^{*+}_{0}\phi$
decay in PQCD approach
In the standard model, the effective weak Hamiltonian mediating flavor-
changing neutral current transitions of the type $b\to s$ has the form:
$\displaystyle{\cal
H}_{eff}={G_{F}\over\sqrt{2}}\Big{[}\sum\limits_{p=u,c}V_{pb}V^{*}_{ps}\Big{(}C_{1}O_{1}^{p}+C_{2}O_{2}^{p}\Big{)}-V_{tb}V^{*}_{ts}\sum\limits_{i=3}^{10,7\gamma,8g}C_{i}O_{i}\Big{]},$
(20)
where the explicit form of the operator $O_{i}$ and the corresponding Wilson
coefficient $C_{i}$ can be found in Ref. [17]. $V_{p(t)b}$, $V_{p(t)s}$ are
the CKM matrix elements. According to effective Hamiltonian (20), we draw the
lowest order diagrams of this channel in Fig. 1.
We first calculate the usual factorizable emission diagrams (a) and (b). If we
insert the $(V-A)(V-A)$ or $(V-A)(V+A)$ operators in the corresponding
vertexes, the amplitude associated to these currents is given as:
$F_{e}=-8\pi
C_{F}m_{B}^{4}f_{\phi}\int_{0}^{1}dx_{1}dx_{3}\int_{0}^{\infty}b_{1}db_{1}\,b_{2}db_{3}\,\phi_{B}(x_{1},b_{1})\\\
\bigg{\\{}\left[(1+x_{3})\phi_{K^{*}_{0}}(x_{3})+r_{K^{*}_{0}}(1-2x_{3})\left(\phi_{K^{*}_{0}}^{S}(x_{3})+\phi_{K^{*}_{0}}^{T}(x_{3})\right)\right]a(t_{a})E_{e}(t_{a})h_{e}(x_{1},x_{3},b_{1},b_{3})\\\
+2r_{K^{*}_{0}}\phi_{K^{*}_{0}}^{S}({x_{3}})a(t_{b})E_{e}(t_{b})h_{e}(x_{3},x_{1},b_{3},b_{1})\bigg{\\}}.$
(21)
In the above formulae, $C_{F}=4/3$ is the group factor of the $SU(3)_{c}$
gauge group. We will use the same conventions for the functions $h_{e}$ and
$E_{e}(t^{\prime})$ including the Sudakov factor and jet function as those in
Ref. [18]. The $(S-P)(S+P)$ operator does not contribute to this decay as the
emission particle is a vector particle. For the non-factorizable diagrams (c)
and (d), all three meson wave functions are involved. For the $(V-A)(V-A)$
operators, the result can be read as:
$M_{e}^{LL}=\frac{32\pi}{\sqrt{2N_{C}}}C_{F}m_{B}^{4}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}\,b_{2}db_{2}\,\phi_{B}(x_{1},b_{1})\phi_{\phi}(x_{2})\\\
\bigg{\\{}\left[(x_{2}-1)\phi_{K^{*}_{0}}(x_{3})+r_{K^{*}_{0}}x_{3}\bigg{(}\phi_{K^{*}_{0}}^{S}(x_{3})-\phi_{K^{*}_{0}}^{T}(x_{3})\bigg{)}\right]a(t_{c})E^{\prime}_{e}(t_{c})h_{n}(x_{1},1-x_{2},x_{3},b_{1},b_{2})\\\
+\left[(x_{3}+x_{2})\phi_{K^{*}_{0}}(x_{3})-r_{K^{*}_{0}}x_{3}\left(\phi_{K^{*}_{0}}^{S}(x_{3})+\phi_{K^{*}_{0}}^{T}(x_{3})\right)\right]a(t_{d})E^{\prime}_{e}(t_{d})h_{n}(x_{1},x_{2},x_{3},b_{1},b_{2})\bigg{\\}}\;.$
(22)
For $(V-A)(V+A)$ and the $(S-P)(S+P)$ operators, the formulae are listed as:
$M_{e}^{LR}=\frac{32\pi}{\sqrt{2N_{C}}}C_{F}m_{B}^{4}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}\,b_{2}db_{2}\,\phi_{B}(x_{1},b_{1})r_{\phi}\\\
\bigg{\\{}\bigg{[}(1-x_{2})\phi_{K^{*}_{0}}(x_{3})(\phi_{\phi}^{s}(x_{2})+\phi_{\phi}^{t}(x_{2}))+r_{K^{*}_{0}}\bigg{(}\phi_{\phi}^{s}(x_{2})\left[(x_{3}-x_{2}+1)\phi_{K^{*}_{0}}^{S}(x_{3})+(x_{3}+x_{2}-1)\phi_{K^{*}_{0}}^{T}(x_{3})\right]\\\
-\phi_{\phi}^{t}(x_{2})\left[(x_{3}+x_{2}-1)\phi_{K^{*}_{0}}^{S}(x_{3})+(x_{3}-x_{2}+1)\phi_{K^{*}_{0}}^{T}(x_{3})\right]\bigg{)}\bigg{]}a(t_{c})E^{\prime}_{e}(t_{c})h_{n}(x_{1},1-x_{2},x_{3},b_{1},b_{2})\\\
+\bigg{[}x_{2}\phi_{K^{*}_{0}}(x_{3})(\phi_{\phi}^{t}(x_{2})-\phi_{\phi}^{s}(x_{2}))-r_{K^{*}_{0}}\bigg{(}\phi_{\phi}^{s}(x_{2})\left[(x_{3}+x_{2})\phi_{K^{*}_{0}}^{S}(x_{3})+(x_{3}-x_{2})\phi_{K^{*}_{0}}^{T}(x_{3})\right]\\\
+\phi_{\phi}^{t}(x_{2})\left[(x_{3}-x_{2})\phi_{K^{*}_{0}}^{S}(x_{3})+(x_{3}+x_{2})\phi_{K^{*}_{0}}^{T}(x_{3})\right]\bigg{)}\bigg{]}a(t_{d})E^{\prime}_{e}(t_{d})h_{n}(x_{1},x_{2},x_{3},b_{1},b_{2})\bigg{\\}}\;,$
(23) ${\cal
M}_{e}^{SP}=-\frac{32\pi}{\sqrt{2N_{C}}}C_{F}m_{B}^{4}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}\,b_{2}db_{2}\,\phi_{B}(x_{1},b_{1})\phi_{\phi}(x_{2})\\\
\bigg{\\{}\left[(1-x_{2}+x_{3})\phi_{K^{*}_{0}}(x_{3})-r_{K^{*}_{0}}x_{3}\bigg{(}\phi_{K^{*}_{0}}^{S}(x_{3})+\phi_{K^{*}_{0}}^{T}(x_{3})\bigg{)}\right]a(t_{c})E^{\prime}_{e}(t_{c})h_{n}(x_{1},1-x_{2},x_{3},b_{1},b_{2})\\\
+\left[-x_{2}\phi_{K^{*}_{0}}(x_{3})+r_{K^{*}_{0}}x_{3}\left(\phi_{K^{*}_{0}}^{S}(x_{3})-\phi_{K^{*}_{0}}^{T}(x_{3})\right)\right]a(t_{d})E^{\prime}_{e}(t_{d})h_{n}(x_{1},x_{2},x_{3},b_{1},b_{2})\bigg{\\}}\;,$
(24)
Diagrams (e) and (f) are the factorizable annihilation diagrams, and the
$(V-A)(V-A)$ kind of operators’ contributions are
$F^{L}_{a}(a)=-8\pi
C_{F}m_{B}^{4}f_{B}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}db_{2}\,b_{3}db_{3}\
\\\
\times\Bigg{\\{}\Big{[}(x_{3}-1)\phi_{K^{*}_{0}}(x_{3})\phi_{\phi}(x_{2})-2r_{\phi}r_{K^{*}_{0}}\left((x_{3}-2)\phi_{K^{*}_{0}}^{S}(x_{3})-x_{3}\phi_{K^{*}_{0}}^{T}(x_{3})\right)\phi_{\phi}^{s}(x_{2})\Big{]}a(t_{e})E_{a}(t_{e})h_{a}(x_{2},1-x_{3},b_{2},b_{3})\\\
+\Big{[}x_{2}\phi_{K^{*}_{0}}(x_{3})\phi_{\phi}(x_{2})-2r_{\phi}r_{K^{*}_{0}}\phi_{K^{*}_{0}}^{S}(x_{3})\left((x_{2}+1)\phi_{\phi}^{s}(x_{2})+(x_{2}-1)\phi_{\phi}^{t}(x_{2})\right)\Big{]}a(t_{f})E_{a}(t_{f})h_{a}(1-x_{3},x_{2},b_{3},b_{2})\Bigg{\\}},$
(25)
and the result from $(S-P)(S+P)$ currents is:
$F^{SP}_{a}(a)=16\pi
C_{F}m_{B}^{4}f_{B}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}db_{2}\,b_{3}db_{3}\
\\\
\times\Bigg{\\{}\Big{[}2r_{\phi}\phi_{K^{*}_{0}}(x_{3})\phi_{\phi}^{s}(x_{2})+r_{K^{*}_{0}}(x_{3}-1)\left(\phi_{K^{*}_{0}}^{S}(x_{3})+\phi_{K^{*}_{0}}^{T}(x_{3})\right)\phi_{\phi}(x_{2})\Big{]}a(t_{e})E_{a}(t_{e})h_{a}(x_{2},1-x_{3},b_{2},b_{3})\\\
-\Big{[}2r_{K^{*}_{0}}\phi_{K^{*}_{0}}^{S}(x_{3})\phi_{\phi}(x_{2})+r_{\phi}x_{2}\left(\phi_{\phi}^{t}(x_{2})-\phi_{\phi}^{s}(x_{2})\right)\phi_{K^{*}_{0}}(x_{3})\Big{]}a(t_{f})E_{a}(t_{f})h_{a}(1-x_{3},x_{2},b_{3},b_{2})\Bigg{\\}}.$
(26)
The last two diagrams in Fig. 1 are the non-factorizable annihilation
diagrams, whose contributions are
$M^{LL}_{a}(a)=\frac{32\pi}{\sqrt{2N_{C}}}C_{F}m_{B}^{4}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}\,b_{2}db_{2}\,\phi_{B}(x_{1},b_{1})\\\
\bigg{\\{}\bigg{[}x_{2}\phi_{K^{*}_{0}}(x_{3})\phi_{\phi}(x_{2})+r_{\phi}r_{K^{*}_{0}}\phi_{\phi}^{s}(x_{2})\left((x_{3}-x_{2}-3)\phi_{K^{*}_{0}}^{S}(x_{3})+(x_{3}+x_{2}-1)\phi_{K^{*}_{0}}^{T}(x_{3})\right)\\\
-r_{\phi}r_{K^{*}_{0}}\phi_{\phi}^{t}(x_{2})\left((x_{3}+x_{2}-1)\phi_{K^{*}_{0}}^{S}(x_{3})+(x_{3}-x_{2}+1)\phi_{K^{*}_{0}}^{T}(x_{3})\right)\bigg{]}a(t_{g})E^{\prime}_{e}(t_{g})h_{na}(x_{1},x_{3},x_{2},b_{1},b_{2})\\\
+\bigg{[}(x_{3}-1)\phi_{K^{*}_{0}}(x_{3})\phi_{\phi}(x_{2})-r_{\phi}r_{K^{*}_{0}}\phi_{\phi}^{t}(x_{2})\left((x_{3}+x_{2}-1)\phi_{K^{*}_{0}}^{S}(x_{3})+(-x_{3}+x_{2}+1)\phi_{K^{*}_{0}}^{T}(x_{3})\right)\\\
-r_{\phi}r_{K^{*}_{0}}\phi_{\phi}^{s}(x_{2})\left((x_{3}-x_{2}-1)\phi_{K^{*}_{0}}^{S}(x_{3})-(x_{3}+x_{2}-1)\phi_{K^{*}_{0}}^{T}(x_{3})\right)\bigg{]}a(t_{h})E^{\prime}_{e}(t_{h})h^{\prime}_{na}(x_{1},x_{3},x_{2},b_{1},b_{2})\bigg{\\}},$
(27)
$M^{LR}_{a}(a)=\frac{32\pi}{\sqrt{2N_{C}}}C_{F}m_{B}^{4}\int_{0}^{1}dx_{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}db_{1}\,b_{2}db_{2}\,\phi_{B}(x_{1},b_{1})\\\
\bigg{\\{}\bigg{[}(2-x_{2})r_{\phi}\phi_{K^{*}_{0}}(x_{3})(\phi_{\phi}^{s}(x_{2})+\phi_{\phi}^{t}(x_{2}))+(x_{3}+1)r_{K^{*}_{0}}(\phi_{K^{*}_{0}}^{S}(x_{3})-\phi_{K^{*}_{0}}^{T}(x_{3}))\phi_{\phi}(x_{2})\bigg{]}a(t_{g})E^{\prime}_{e}(t_{g})h_{na}(x_{1},x_{3},x_{2},b_{1},b_{2})\\\
+\bigg{[}x_{2}r_{\phi}\phi_{K^{*}_{0}}(x_{3})\left(\phi_{\phi}^{s}(x_{2})+\phi_{\phi}^{t}(x_{2})\right)-(x_{3}-1)r_{K^{*}_{0}}\left(\phi_{K^{*}_{0}}^{S}(x_{3})-\phi_{K^{*}_{0}}^{T}(x_{3})\right)\phi_{\phi}(x_{2})\bigg{]}a(t_{h})E^{\prime}_{e}(t_{h})h^{\prime}_{na}(x_{1},x_{3},x_{2},b_{1},b_{2})\bigg{\\}}.$
(28)
By combining the contributions from different diagrams with corresponding
Wilson coefficients, one obtains the total decay amplitudes as
$\displaystyle{\cal A}(\overline{B}\to\overline{K}_{0}^{*0}(1430)\phi)$
$\displaystyle=$ $\displaystyle
V_{tb}^{*}V_{ts}\Bigg{\\{}F_{e}\left[a_{3}+a_{4}+a_{5}-\frac{1}{2}(a_{7}+a_{9}+a_{10})\right]$
(29)
$\displaystyle+M_{e}^{LL}\left[C_{3}+C_{4}-\frac{1}{2}C_{9}-\frac{1}{2}C_{10}\right]+M_{e}^{LR}\left[C_{5}-\frac{1}{2}C_{7}\right]+M_{e}^{SP}\left[C_{6}-\frac{1}{2}C_{8}\right]$
$\displaystyle+F_{a}^{LL}\left[a_{4}-\frac{1}{2}a_{10}\right]+F_{a}^{SP}\left[a_{6}-\frac{1}{2}a_{8}\right]$
$\displaystyle+M_{a}^{LL}\left[C_{3}-\frac{1}{2}C_{9}\right]+M_{a}^{LR}\left[C_{5}-\frac{1}{2}C_{7}\right]\Bigg{\\}};$
$\displaystyle{\cal A}(B^{+}\to K_{0}^{+*}(1430)\phi)$ $\displaystyle=$
$\displaystyle
V_{tb}^{*}V_{ts}\Bigg{\\{}F_{e}\left[a_{3}+a_{4}+a_{5}-\frac{1}{2}(a_{7}+a_{9}+a_{10})\right]$
(30)
$\displaystyle+M_{e}^{LL}\left[C_{3}+C_{4}-\frac{1}{2}C_{9}-\frac{1}{2}C_{10}\right]+M_{e}^{LR}\left[C_{5}-\frac{1}{2}C_{7}\right]+M_{e}^{SP}\left[C_{6}-\frac{1}{2}C_{8}\right]$
$\displaystyle+F_{a}^{LL}\left[a_{4}+a_{10}\right]+F_{a}^{SP}\left[a_{6}+a_{8}\right]+M_{a}^{LR}\left[C_{5}+C_{7}\right]+M_{a}^{LL}\left[C_{3}+C_{9}\right]\Bigg{\\}}$
$\displaystyle-
V_{ub}^{*}V_{us}\Bigg{\\{}F_{a}^{LL}\left[C_{2}+\frac{1}{3}C_{1}\right]+M_{a}^{LL}C_{1}\Bigg{\\}},$
where $C_{i}$ are the Wilson coefficients for the four-quark operators and
$a_{i}$ is defined as the combination of the Wilson coefficients:
$\displaystyle a_{i}=C_{i}+\frac{C_{i\pm 1}}{N_{c}}$ (31)
for an odd (even) value of $i$.
## 4 Numerical Results
The CKM phase $\gamma$ is defined via
$\displaystyle V_{ub}=|V_{ub}|e^{-i\gamma},$ (32)
and the CKM matrix elements that we used in the calculation are
$|V_{ub}|=3.51\times 10^{-3}$, $|V_{us}|=0.225$, $|V_{cb}|=41.17\times
10^{-3}$ and $|V_{cs}|=0.973$ [19]. Moreover, we employ the unitary angle
$\gamma=70^{\circ}$, the masses $m_{B}=5.28$ GeV and $m_{\phi}=1.02$ GeV. The
longitudinal decay constant of $\phi$ could be extracted through the leptonic
$\phi\to e^{+}e^{-}$ decay [20]
$\displaystyle\Gamma(\phi\to e^{+}e^{-})=\frac{4\pi\alpha_{\rm
em}^{2}e_{s}^{2}f_{\phi}^{2}}{3m_{\phi}},$ (33)
which gives
$\displaystyle f_{\phi}$ $\displaystyle=$ $\displaystyle 215~{}{\rm MeV}.$
(34)
For the transverse decay constant, we use the recent Lattice QCD result [21]
at 2 GeV
$\displaystyle\frac{f_{\phi}^{T}}{f_{\phi}}=0.750\pm 0.008,$ (35)
which corresponds to $f_{\phi}^{T}(1~{}\mathrm{GeV})=(178\pm 2)$ MeV. The
${\bar{B}}_{d}^{0}$ ($B^{-}$) meson lifetime $\tau_{B^{0}}=1.530$ ps
($\tau_{B^{-}}=1.638$ ps) [20].
With the above input parameters, the $B\to K^{*}_{0}$ form factors are given
as
$\displaystyle
F_{1}(q^{2}=0)=-0.42^{+0.04+0.03-0.09}_{-0.04-0.03+0.07},\,\,\,\,\,\,\,\,\,\,\,\,S1;$
$\displaystyle
F_{1}(q^{2}=0)=~{}~{}0.73^{+0.08-0.10+0.15}_{{-0.08}+0.09-0.12},\,\,\,\,\,\,\,\,\,\,\,\,S2;$
(36)
where the first two uncertainties are from decay constants and the
distribution amplitudes of the scalar meson, and the last uncertainty is from
the $\omega_{B}$ in the distribution amplitude of $B$ meson. The decay
constant in S2 is larger than that in S1, and contributions from the two terms
proportional to $B_{1}$ and $B_{3}$ are constructive in S2 but destructive in
S1. Thus the result for the form factor of $B\to K^{*}_{0}$ in S2 is almost
twice larger than that in S1. Compared with the previous study of transition
form factors [22], we can see that the present results for these form factors
are a bit larger due to a weaker suppression for the endpoint region from the
jet function $S_{t}(x)$.
The total decay amplitude for $B^{+}\to K_{0}^{*+}(1430)\phi$ can be written
as:
${\cal
A}=V_{ub}^{*}V_{us}T-V_{tb}^{*}V_{ts}P=V_{ub}^{*}V_{us}T[1+ze^{i(\delta-\gamma)}],$
(37)
where $z=|V_{tb}^{*}V_{ts}/V_{ub}^{*}V_{us}||P/T|$ and $\delta$ is the
relative strong phase between tree diagrams ($T$) and penguin diagrams ($P$).
The decay width is expressed as:
$\Gamma(B^{+}\to K_{0}^{*+}(1430)\phi)=\frac{G_{F}^{2}}{32\pi M_{B}}|{\cal
A}|^{2}=\frac{G_{F}^{2}}{32\pi
M_{B}}|V_{ub}^{*}V_{us}T|^{2}[1+z^{2}+2z\cos(\delta-\gamma)].$ (38)
Similarly, we can get the decay width for $B^{-}\to K_{0}^{*-}(1430)\phi$,
$\Gamma(B^{-}\to K_{0}^{*-}(1430)\phi)=\frac{G_{F}^{2}}{32\pi
M_{B}}|\overline{{\cal A}}|^{2},$ (39)
where
$\overline{{\cal
A}}=V_{ub}V_{us}^{*}T-V_{tb}V_{ts}^{*}P=V_{ub}V_{us}^{*}T[1+ze^{i(\delta+\gamma)}].$
(40)
From Eqs. (38) and (39), we get the averaged decay width:
$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle\frac{G_{F}^{2}}{32\pi
M_{B}}(|{\cal A}|^{2}/2+|\overline{\cal A}|^{2}/2)\hskip 28.45274pt$ (41)
$\displaystyle=$ $\displaystyle\frac{G_{F}^{2}}{32\pi
M_{B}}|V_{ub}^{*}V_{us}T|^{2}[1+z^{2}+2z\cos\gamma\cos\delta].$
Using Eqs. (38) and (39), the direct $CP$ violation parameter is defined as
$A_{CP}^{dir}=\frac{\Gamma(B^{-}\to K_{0}^{*-}(1430)\phi)-\Gamma(B^{+}\to
K_{0}^{*+}(1430)\phi)}{\Gamma(B^{-}\to K_{0}^{*-}(1430)\phi)+\Gamma(B^{+}\to
K_{0}^{*+}(1430)\phi)}=\frac{2z\sin\gamma\sin\delta}{1+2z\cos\gamma\cos\delta+z^{2}}.$
(42)
Since only penguin operators work on the neutral decay mode, there is no
direct $CP$ asymmetry in the decay $B^{0}\to K_{0}^{*0}(1430)\phi$, and its
branching ratio can be calculated straightforwardly.
Using the parameters, we get the branching ratios in scenario 1 (S1):
$\displaystyle{\cal B}(B^{0}\to K_{0}^{*0}(1430)\phi)$ $\displaystyle=$
$\displaystyle 3.7\times 10^{-6},$ $\displaystyle{\cal B}(B^{\pm}\to
K_{0}^{*\pm}(1430)\phi)$ $\displaystyle=$ $\displaystyle 4.3\times 10^{-6},$
(43)
while in scenario 2 (S2), the results are:
$\displaystyle{\cal B}(B^{0}\to K_{0}^{*0}(1430)\phi)$ $\displaystyle=$
$\displaystyle 23.6\times 10^{-6},$ $\displaystyle{\cal B}(B^{\pm}\to
K_{0}^{*\pm}(1430)\phi)$ $\displaystyle=$ $\displaystyle 25.6\times 10^{-6}.$
(44)
From the above equations, we can see that the branching ratios in S2 are about
8 times larger than those in S1. There are three main reasons: (i) the larger
decay constant in S2; (ii) contributions in emission diagrams from the two
terms $B_{1}$ and $B_{3}$ are constructive in S2 but destructive in S1; (iii)
the annihilation diagrams could cancel the contribution from the emission
diagram. This kind of contribution in annihilation diagram is proportional to
$B_{3}$. The larger value for $B_{3}$ in S1 will results in more sizable
cancelation and the branching fractions are correspondingly reduced.
To be more explicit, we present values of the factorizable and non-
factorizable amplitudes from the emission and annihilation topologies in
Table. 1. As expected, the factorizable amplitudes are the largest, however
the annihilation magnitudes are only few times smaller than that of
factorizable emission diagrams. The non-factorizable amplitudes are down by a
power of $\bar{\Lambda}/M_{B}\sim 0.1$ compared to the factorizable ones. The
cancelation between the twist-2 and twist-3 contributions makes them even
smaller. We demonstrate the importance of penguin enhancement in the Table. 1.
It has been known that the RG evolution of the Wilson coefficients
$C_{4,6}(t)$ dramatically increases as $t<m_{b}/2$, while that of $C_{1,2}(t)$
almost remains constant [17].
Table 1: Decay amplitudes for $B\to K_{0}^{*+}(1430)\phi$ ($\times 10^{-2}~{}\mbox{GeV}^{3}$) $B^{+}\to K_{0}^{*+}(1430)\phi$ | | $F_{e}$ | $M_{e}$ | $F_{a}^{T}$ | $F_{a}$ | $M_{a}^{T}$ | $M_{a}$
---|---|---|---|---|---|---|---
$S1$ | | $-13.4$ | $-0.3+i0.0$ | $-1.0-i4.0$ | $8.1+i4.0$ | $-2.8+i3.0$ | $0.2+i0.0$
$S2$ | | $20.4$ | $-0.8+i0.9$ | $0.4+i0.8$ | $-7.1-i12.0$ | $9.3+i2.1$ | $-0.3-i0.2$
$B^{0}\to K_{0}^{*0}(1430)\phi$ | | $F_{e}$ | $M_{e}$ | $F_{a}^{T}$ | $F_{a}$ | $M_{a}^{T}$ | $M_{a}$
$S1$ | | $-13.4$ | $-0.3+i0.0$ | $0$ | $8.3+i4.0$ | $0$ | $0.2-i0.1$
$S2$ | | $20.4$ | $-0.8+i0.9$ | $0$ | $-7.2-i12.2$ | $0$ | $-0.5-i0.3$
In both scenarios, the branching ratio of $B^{+}\to K_{0}^{*+}(1430)\phi$ is a
bit larger than that of $B^{0}\to K_{0}^{*0}(1430)\phi$, and the difference is
from the tree contribution in $B^{+}\to K_{0}^{*+}(1430)\phi$. Since there
exists interference between tree and penguin diagrams in the charged channel,
the direct $CP$ asymmetry appears. So, we get the $CP$ asymmetry of
$B^{\pm}\to K_{0}^{*\pm}(1430)\phi$ in the different scenarios as follows:
$\displaystyle{\cal A}_{dir}(B^{\pm}\to K_{0}^{*\pm}(1430)\phi)$
$\displaystyle=$ $\displaystyle
1.6\%,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S1$ $\displaystyle{\cal
A}_{dir}(B^{\pm}\to K_{0}^{*\pm}(1430)\phi)$ $\displaystyle=$ $\displaystyle
1.9\%.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S2$ (45)
As the neutral channel as concerned, there is no $CP$ asymmetry as only
penguin operators contribute to this channel.
Figure 2: The dependence of the branching ratios($\times 10^{-6}$) for $B\to
K_{0}^{*}(1430)\phi$ on the CKM angle $\gamma$, where the solid (dashed) curve
is for charged (neutral) channel. The left (right) panel is plotted in S1(S2)
scenario. Figure 3: The dependence of the $CP$ asymmetry for $B^{\pm}\to
K_{0}^{*}{\pm}(1430)\phi$ on the CKM angle $\gamma$, where the solid (dashed)
curve is for S1 (S2) scenario
Although we set $\gamma=70^{\circ}$ in the above discussions, it is not
measured accurately. In the following, we choose $\gamma$ as a free parameter
and plot the branching ratios as a function of the angle $\gamma$ in both S1
and S2, as shown in the Fig. 2 and Fig. 3. As seen from the figures, we note
that both the branching ratios and the $CP$ asymmetries in different scenarios
are not sensitive to the phase $\gamma$. In the decay mode $B^{\pm}\to
K_{0}^{*\pm}(1430)\phi$, the tree contribution only appears in the
annihilation diagrams, which are suppressed compared with the emission
diagrams. Moreover, the CKM element $|V_{ub}V_{us}|$ of tree diagrams is
smaller than $|V_{tb}V_{ts}|$ of penguin diagrams. From this point of view, we
can understand why the branching ratios and the $CP$ asymmetries are not
sensitive to the $\gamma$.
In our calculation, the major uncertainties come from our lack of information
about the scalar meson and heavy meson, involving the decay constants and the
distribution amplitudes. The latter can be fitted from the well measured
channels such as $B\to\pi\pi,K\pi$, the scalar one is not well ascertained.
These uncertainties from the scalar meson can give sizable effects on the
branching ratio, but the $CP$ asymmetries are less sensitive to these
parameters. In this work, for instance, the twist-3 distribution amplitudes of
the scalar mesons are taken as the asymptotic form, which may give large
uncertainties. The characters of the scalar mesons need to be studied in
future work. The another uncertainty comes from the sub-leading order
contributions in PQCD approach, which have also been neglected in the
calculation. In Ref. [23], parts of sub-leading order of $B\to\pi\pi,\pi K$
have been calculated, and the results show that corrections can change the
penguin dominated processes, for example, the quark loops and magnetic-penguin
correction decrease the branching ratio of $B\to\pi K$ by about $20\%$. We
expect the similar size of uncertainty in the decays we analyzed , since they
are also dominated by the penguin operators.
Here we give the results with the uncertainties as follows:
$\displaystyle{\cal B}(B^{0}\to K_{0}^{*0}(1430)\phi)$ $\displaystyle=$
$\displaystyle(3.7^{+0.8+0.1+3.7}_{-0.7-0.1-1.7})\times 10^{-6},$
$\displaystyle{\cal B}(B^{-}\to K_{0}^{*-}(1430)\phi)$ $\displaystyle=$
$\displaystyle(4.3^{+0.9+0.1+4.3}_{-0.8-0.1-2.0})\times
10^{-6}\,\,\,\,\,\,\,\,\,\,\,\,S1;$ $\displaystyle{\cal B}(B^{0}\to
K_{0}^{*0}(1430)\phi)$ $\displaystyle=$
$\displaystyle(23.6^{+5.6+0.8+10.9}_{-5.0-0.6-5.8})\times 10^{-6},$
$\displaystyle{\cal B}(B^{-}\to K_{0}^{*-}(1430)\phi)$ $\displaystyle=$
$\displaystyle(25.6^{+6.2+0.9+12.1}_{-5.4-0.8-6.5})\times
10^{-6}\,\,\,\,\,\,\,\,\,\,S2.$ (46)
In the above results, the first uncertainty comes from the decay constants,
and the second one is from the uncertainties of B1 (B3) in the amplitude
distributions of the scalar meson. The last one comes from the uncertainty in
the $B$ meson shape parameter $\omega=(0.40\pm 0.05)$ GeV. This kind of
uncertainties is extremely large. The change of the shape parameter will
mainly affect the emission diagram including the $B\to K^{*}_{0}$ form factor
while the annihilation diagram, especially factorizable diagram, will not be
affected sizably. Remember that the annihilation diagram could cancel part of
contributions from emission diagram and thus the branching fractions are
sizably changed due to the shape parameter.
In the QCD factorization approach, the results are listed as [14]:
$\displaystyle{\cal B}(B^{0}\to K_{0}^{*0}(1430)\phi)$ $\displaystyle=$
$\displaystyle(0.9^{+0.3+0.4+19.3}_{-0.3-0.3-0.5})\times 10^{-6},$
$\displaystyle{\cal B}(B^{-}\to K_{0}^{*-}(1430)\phi)$ $\displaystyle=$
$\displaystyle(1.0^{+0.3+0.4+20.2}_{-0.3-0.3-0.5})\times
10^{-6}\,\,\,\,\,\,\,\,\,\,\,\,S1;$ $\displaystyle{\cal B}(B^{0}\to
K_{0}^{*0}(1430)\phi)$ $\displaystyle=$
$\displaystyle(16.9^{+6.2+1.7+51.8}_{-4.7-1.6-12.0})\times 10^{-6},$
$\displaystyle{\cal B}(B^{-}\to K_{0}^{*-}(1430)\phi)$ $\displaystyle=$
$\displaystyle(17.3^{+6.2+1.7+52.4}_{-4.7-1.7-12.1})\times
10^{-6},\,\,\,\,\,\,\,\,\,S2.$ (47)
Comparing two group of results, we note that our central values are much large
than the results from QCDF in both two scenarios. It is mostly because that
the form factor derived from Eq. (21) is larger than $F_{1}^{B\to
K^{*}_{0}}(q^{2}=0)=0.21~{}(0.26)$ used in QCDF, which is calculated under S1
(S2) scenario in the covariant light-front model [24]. In addition, our
results suffer from contribution from the annihilation diagrams, as
demonstrated in the Table. 1. In fact, the contribution from annihilation can
take the major uncertainties in the QCDF, as shown in the Eq. (4).
In the S1, for the neutral channel, our result is agree with experimental data
well, but the result of the charged one is smaller than the data, though it is
consistent within theoretical uncertainties. In the S2, both results are much
larger than the data. The predictions in both scenarios suffer from very large
uncertainties from the hadronic input parameters. Fortunately, most of these
uncertainties will cancel out when we consider the ratio of branching
fractions. It is convenient to define the ratio
$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{\tau(B^{0})}{\tau(B^{+})}\frac{{\cal B}(B^{\pm}\to\phi
K^{*\pm}_{0})}{{\cal B}(B^{0}\to\phi K^{*0}_{0})},$ (48)
which is predicted as
$\displaystyle R$ $\displaystyle=$ $\displaystyle 1.08\pm
0.01,\,\,\,\,\,\,\,\,\,\,\,\,S1;$ $\displaystyle R$ $\displaystyle=$
$\displaystyle 1.01\pm 0.01.\,\,\,\,\,\,\,\,\,\,\,\,S2;$ (49)
Using the two experimental results, one can easily obtain the experimental
data for this ratio
$\displaystyle R_{\rm exp}=1.68\pm 0.51,$ (50)
where all uncertainties are added in quadrature. For this ratio, the
uncertainties from theoretical predictions are small while the experimental
data has large uncertainties.
As a summary, we have studied the hadronic charmless decay mode $B\to
K_{0}^{*}(1430)\phi$ within the framework of perturbative QCD approach in the
standard model. Under two different scenarios, we explored the branching
ratios and related $CP$ asymmetries. We find that besides the dominant
contributions from the factorization emission diagrams, the penguin operators
in annihilation can change the ratio remarkably. The central value of our
results are larger than those from QCD factorization. Compared with
experimental data from BaBar, in the S1, the result of neutral channel is
agree with experimental data well, but the result of the charged one is a bit
smaller than the data, though it is consistent within theoretical
uncertainties. In the S2, both results are much larger than the data but the
uncertainties are typically large. The ratio of branching fractions is found
to have small uncertainties in the theoretical side.
## Acknowledgments
The work of C.S.K. was supported in part by Basic Science Research Program
through the NRF of Korea funded by MOEST (2009-0088395) and in part by KOSEF
through the Joint Research Program (F01-2009-000-10031-0). The work of Ying Li
was supported by the Brain Korea 21 Project and by the National Science
Foundation under contract Nos.10805037 and 10735080.
## References
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* [6] X. Liu, Z. Q. Zhang and Z. J. Xiao, arXiv:0904.1955 [hep-ph].
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* [8] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 98, 051801 (2007) [arXiv:hep-ex/0610073].
* [9] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 101, 161801 (2008) [arXiv:0806.4419 [hep-ex]].
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* [11] E. Barberio et al. [Heavy Flavor Averaging Group (HFAG)], arXiv:hep-ex/0603003. The updated results can be found at www.slact.stanford.edu/xorg/hfag.
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* [16] P. Ball and G. W. Jones, JHEP 0703, 069 (2007) [arXiv:hep-ph/0702100].
* [17] For a review, see G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
* [18] A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wang, Phys. Rev. D 76, 074018 (2007) [arXiv:hep-ph/0703162].
* [19] J. Charles et al. [CKMfitter Group], Eur. Phys. J. C 41, 1 (2005) [arXiv:hep-ph/0406184]. The updated results can be found at http://ckmfitter.in2p3.fr/.
* [20] C. Amsler et al. (Particle Data Group), Physics Letters B667, 1 (2008)
* [21] C. Allton et al. [RBC-UKQCD Collaboration], Phys. Rev. D 78, 114509 (2008) [arXiv:0804.0473 [hep-lat]].
* [22] R. H. Li, C. D. Lu, W. Wang and X. X. Wang, Phys. Rev. D 79, 014013 (2009) [arXiv:0811.2648 [hep-ph]].
* [23] H. n. Li, S. Mishima and A. I. Sanda, Phys. Rev. D 72, 114005 (2005) [arXiv:hep-ph/0508041]; H. n. Li and S. Mishima, Phys. Rev. D 74, 094020 (2006) [arXiv:hep-ph/0608277].
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|
arxiv-papers
| 2009-12-09T20:25:42 |
2024-09-04T02:49:06.919863
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.s Kim, Ying Li, Wei Wang",
"submitter": "Ying Li",
"url": "https://arxiv.org/abs/0912.1718"
}
|
0912.1752
|
# Spin squeezing and concurrence
Xiaolei Yin Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University, Hangzhou 310027, China Xiaoqian Wang Zhejiang Institute
of Modern Physics, Department of Physics, Zhejiang University, Hangzhou
310027, China Department of Physics, Changchun University of Science and
Technology, Changchun 130022, China Jian Ma Zhejiang Institute of Modern
Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China
Xiaoguang Wang xgwang@zimp.zju.edu.cn Zhejiang Institute of Modern Physics,
Department of Physics, Zhejiang University, Hangzhou 310027, China
###### Abstract
We study the relations between spin squeezing and concurrence, and find that
they are qualitatively equivalent for an ensemble of spin-1/2 particles with
exchange symmetry and parity, if we adopt the spin squeezing criterion given
by the recent work (G. Tóth et al. Phys. Rev. Lett. 99, 250405 (2007)). This
suggests that the spin squeezing has more intimate relations with pairwise
entanglement other than multipartite entanglement. We exemplify the result by
considering a superposition of two Dicke states.
spin squeezing,concurrence,entanglement
###### pacs:
03.65.Ud,03.67.2a
## I Introduction
As an important resource of quantum information and computation, entanglement
Einstein ; Schrodinger has attracted much attention in recent years
Bennett1993 ; Bennett1992 ; Ekert1991 ; Wang2001 ; WangSolomon ; Pan2001 ;
Vidal2003 ; VidalPalacios ; Leibfried ; Andre . How to measure and detect
entanglement is crucial for both theoretical investigations and potential
practical applications Bennett1996 ; Vedral1997 . The entanglement of a two-
qubit system can be well quantified by the concurrence Wootters1997 ;
Wootters1998 . However, quantification of many-body entanglement is still an
open question in quantum information.
It is well known that there are close relations between entanglement and spin
squeezing SorensenMolmer ; Sorrensen2001 ; Kitagawa1993 ; Wineland1994 ;
Kitagawa2001 ; WangSangders2003 ; Jin2007 ; Jafarpour . There are several
definitions of spin squeezing parameters Sorrensen2001 ; Kitagawa1993 ;
Wineland1994 , which are studied in different papers. The squeezing parameter
$\xi_{R}^{2}$ defined by Wineland et al. is closely related to multipartite
entanglement. It has been proven that Sorrensen2001 , for an ensemble of
spin-1/2 particles, if this squeezing parameter is less than one, the state is
entangled. The advantages of spin squeezing parameters in detecting
entanglement have been shown in both theoretical and experimental aspects.
The squeezing parameter $\xi_{S}^{2}$ defined by Kitagawa and Ueda is relevant
to pairwise entanglement Kitagawa2001 . And for states with exchange symmetry
and parity, a simple quantitative relation between $\xi_{S}^{2}$ and
concurrence was given WangSangders2003 . Furthermore, it has been shown that
the spin squeezing and pairwise entanglement are equivalent for states
generated by the one-axis twisting Hamiltonian WangSangders2003 . However,
even for states with a fixed parity, such as the states generated by one-axis
twisting Hamiltonian with a transverse field, $\xi_{S}^{2}$ is not always
equivalent to concurrence Wang2004 . Inspired by recent works Toth2007 ;
Toth2009 , where a set of generalized spin squeezing inequalities are
developed, one can define another spin squeezing parameter $\xi_{T}^{2}$ from
one of the inequalities Wang2009 . Similar to parameter $\xi_{R}^{2}$, one
advantage of the parameter $\xi_{T}^{2}$ is that we can firmly say that the
state is entangled if $\xi_{T}^{2}<1$. However, if parameter $\xi_{S}^{2}<1$,
we cannot say the state is entangled, although this parameter is closely
related to entanglement.
Reference Kitagawa2001 found that spin squeezing according to parameter
$\xi_{S}^{2}$ is equivalent to the minimal pairwise correlation
$\mathcal{C}_{\vec{n}_{\perp},\vec{n}_{\perp}}$ along the direction
$\vec{n}_{\perp}$ (which is perpendicular to the mean spin direction) for
symmetric states. It was further found Xiaoqian10 that for the symmetric
states, the spin squeezing defined via $\xi_{T}^{2}$ is equivalent to minimal
pairwise correlation $\mathcal{C}_{\vec{n},\vec{n}}$ along an arbitrary
direction $\vec{n}$. For states with a fixed parity, the relations between the
two parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$ are more evident. It will be
seen from Sec. 3 that $\xi_{T}^{2}$ contains the term $\xi_{S}^{2}$, and the
spin squeezing results from just the competition between pairwise correlation
along the direction $\vec{n}_{\perp}$ and that along the mean spin direction.
So, in this sense, the parameter $\xi_{T}^{2}$ is a natural generalization of
$\xi_{S}^{2}$.
We find that for states with exchange symmetry and parity, the spin squeezing
parameter $\xi_{T}^{2}$ is qualitatively equivalent to the concurrence in
characterizing pairwise entanglement. In other words, the spin squeezing
parameter and concurrence emerge and vanish at the same time. This finding is
of significance to entanglement detection in experiments. As we all know,
entanglement detectors such as entropy and concurrence are generally not easy
to measure, especially for physical systems like BEC, for which we cannot
address individual atoms. However, spin squeezing parameters are relatively
easy to measure in experiments, since they only consist of expectations and
variances of collective angular momentum operators. Nevertheless, the
traditional spin squeezing parameter $\xi_{S}^{2}$ is not always equivalent to
concurrence even for states with exchange symmetry and parity. As
$\xi_{T}^{2}$ is qualitatively equivalent to concurrence for an ensemble of
spin-1/2 particles with exchange symmetry and parity, it is better than
$\xi_{S}^{2}$ in detecting pairwise entanglement.
The paper is organized as follows: In Sec. II, we give the definitions of the
two spin squeezing parameters and concurrence. In Sec. III, we give the
concrete forms of the spin squeezing parameters and the concurrence for states
with exchange symmetry and parity. The relations between these two parameters
and concurrence were given in Sec. IV. We exemplify the result by considering
superpositions of Dicke states in Sec. V. Finally, Sec. VI is devoted to
conclusion.
## II Spin squeezing parameters and concurrence
To study spin squeezing, we consider an ensemble of $N$ spin-1/2 particles.
For the sake of describing many-particle systems, we use the total angular
momentum operators
$J_{\alpha}=\frac{1}{2}\sum_{k=1}^{N}\sigma_{k\alpha},~{}~{}~{}\left(\alpha=x,y,z\right),$
(1)
where $\sigma_{k\alpha}$ are the Pauli matrices for the $k$-th spin. Now, we
give the definitions of the two spin squeezing parameters. The first one is
defined as Kitagawa1993 ,
$\xi_{S}^{2}=\frac{4\min(\Delta J_{\vec{n}_{\perp}})^{2}}{N},$ (2)
where $\vec{n}_{\perp}$ is perpendicular to the mean spin direction
$\vec{n}=\frac{\langle\vec{J}\rangle}{|\langle\vec{J}\rangle|}$. Since the
system has the exchange symmetry, the total angular momentum is
$j=\frac{N}{2}$. For spin coherent states Kitagawa1993 , $\Delta
J_{\vec{n}_{\perp}}=\frac{j}{2}$, and $\xi_{S}^{2}=1$. In the following, we
consider states with exchange symmetry.
The next spin squeezing parameter is based on the generalized spin squeezing
inequalities given by Tóth et al. Toth2009 , and is defined as Wang2009
$\xi_{T}^{2}=\frac{\lambda_{\min}}{\langle\vec{J}^{2}\rangle-\frac{N}{2}},$
(3)
where $\lambda_{\min}$ is the minimum eigenvalue of
$\Gamma=(N-1)\gamma+G$ (4)
with $G_{kl}=\frac{1}{2}\left\langle J_{k}J_{l}+J_{l}J_{k}\right\rangle$,
$(k,l=x,y,z)$ the correlation matrix, and $\gamma_{kl}=G_{kl}-\left\langle
J_{k}\right\rangle\left\langle J_{l}\right\rangle$ the covariance matrix. For
our states, $\langle\vec{J}^{2}\rangle-\frac{N}{2}=j\left(j+1\right)-j=j^{2}$.
The two-qubit entanglement is quantified by the concurrence, whose definition
is given by Wootters1998
$C=\max\\{\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4},0\\},$ (5)
where $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$ are the
square roots of eigenvalues of $\tilde{\rho}\rho$. Here $\rho$ is the reduced
density matrix of the system, and
$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{\ast}(\sigma_{y}\otimes\sigma_{y}),$
(6)
where $\rho^{\ast}$ is the conjugate of $\rho$. If $C>0$, the system displays
pairwise entanglement.
## III States with parity
To study the relations between spin squeezing parameters and concurrence, we
consider a class of states with even (odd) parity, which means a state in the
$(2j+1)$-dimensional space with only even (odd) excitations of spins. These
kinds of states are widely studied, e.g., the states generated by the one-axis
twisting model Kitagawa1993 . The states with even parity possess important
properties, $\left\langle J_{\alpha}\right\rangle=0$, $\left\langle
J_{\alpha}J_{z}\right\rangle=\left\langle J_{z}J_{\alpha}\right\rangle=0$,
$\alpha=x,y$, which means the mean spin direction is along the $z$-axis, and
the covariances between $J_{z}$ and $J_{\alpha}$ are zero. Thus, equation (4)
becomes
$\Gamma=\left(\begin{array}[]{ccc}N\left\langle
J_{x}^{2}\right\rangle&\frac{N\left\langle\left[J_{x},J_{y}\right]_{+}\right\rangle}{2}&0\\\
\frac{N\left\langle\left[J_{x},J_{y}\right]_{+}\right\rangle}{2}&N\left\langle
J_{y}^{2}\right\rangle&0\\\ 0&0&N(\Delta J_{z})^{2}+\langle
J_{z}\rangle^{2}\end{array}\right),$ (7)
where $\left[A,B\right]_{+}=AB+BA$, and equation (3) reduces to Wang2009
$\xi_{T}^{2}=\min\left\\{\xi_{S}^{2},\varsigma^{2}\right\\},$ (8)
where
$\displaystyle\varsigma^{2}$ $\displaystyle=$
$\displaystyle\frac{4}{N^{2}}\left[N(\Delta J_{z})^{2}+\langle
J_{z}\rangle^{2}\right]$ (9) $\displaystyle=$ $\displaystyle
1+\left(N-1\right)\left(\left\langle\sigma_{1z}\sigma_{2z}\right\rangle-\left\langle\sigma_{1z}\right\rangle^{2}\right)$
$\displaystyle=$ $\displaystyle 1+(N-1)C_{zz},$
with $C_{zz}$ the two-spin correlation function along $z$ direction. The
explicit form of $\xi_{S}^{2}$ could be obtained as WangSangders2003
$\displaystyle\xi_{S}^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{N}\left(\langle J_{x}^{2}+J_{y}^{2}\rangle-|\langle
J_{-}^{2}\rangle|\right)$ (10) $\displaystyle=$ $\displaystyle
1-2\left(N-1\right)$
$\displaystyle\times\left[\left|\langle\sigma_{1-}\sigma_{2-}\rangle\right|-\frac{1}{4}\left(1-\left\langle\sigma_{1z}\sigma_{2z}\right\rangle\right)\right],$
where we have used the following relations
$\displaystyle\left\langle J_{\alpha}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{N}{2}\left\langle\sigma_{1\alpha}\right\rangle,$
$\displaystyle\langle J_{\alpha}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{N}{4}+\frac{N(N-1)}{4}\langle\sigma_{1\alpha}\sigma_{2\alpha}\rangle,$
$\displaystyle\langle J_{-}^{2}\rangle$ $\displaystyle=$ $\displaystyle
N(N-1)\langle\sigma_{1-}\sigma_{2-}\rangle,$ (11)
which connect the local expectations with collective ones.
For such states, the significance of $\xi_{S}^{2}$ and $\xi_{T}^{2}$ and the
relations between them are clear. According to the parameter $\xi_{S}^{2}$, a
state is squeezed when the minimum variance of angular momentum in the
$\vec{n}_{\perp}$-plane is smaller than $\frac{j}{2}$, while according to
$\xi_{T}^{2}$, the variance in the mean spin direction $\vec{n}$ is also
considered. Equation (9) represents the pairwise correlation along the mean
spin direction, and this can be viewed as a complement of $\xi_{S}^{2}$, which
only considers squeezing in the $\vec{n}_{\perp}$-plane. Thus, $\xi_{T}^{2}$
can be regarded as a generalization of $\xi_{S}^{2}$, and when
$\xi_{S}^{2}<\varsigma^{2}$, the parameter $\xi_{T}^{2}$ reduces to
$\xi_{S}^{2}$.
To calculate concurrence, we first need to calculate the two-body reduced
density matrix, which can be written as WangSangders2003
$\rho=\left(\begin{array}[]{cccc}v_{+}&0&0&u^{\ast}\\\ 0&y&y&0\\\ 0&y&y&0\\\
u&0&0&v_{-}\end{array}\right),$ (12)
where
$\displaystyle v_{\pm}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left(1\pm
2\langle\sigma_{1z}\rangle+\langle\sigma_{1z}\sigma_{2z}\rangle\right),$
$\displaystyle u$ $\displaystyle=$
$\displaystyle\langle\sigma_{1-}\sigma_{2-}\rangle,~{}~{}~{}y=\frac{1}{4}\left(1-\left\langle\sigma_{1z}\sigma_{2z}\right\rangle\right),$
(13)
in the basis
$\left\\{\left|00\right\rangle,\left|01\right\rangle,\left|10\right\rangle,\left|11\right\rangle\right\\}$.
Then the concurrence is given by
$C=2\max\left\\{0,~{}|u|-y,~{}y-\sqrt{v_{+}v_{-}}\right\\}.$ (14)
One key observation is that
$y^{2}-v_{+}v_{-}=-\frac{1}{4}C_{zz}.$ (15)
Thus,
$\varsigma^{2}=1-4(N-1)(y+\sqrt{v_{+}v_{-}})(y-\sqrt{v_{+}v_{-}}).$ (16)
From equations (9), (10), and (13), we obtain
$\displaystyle\xi_{S}^{2}$ $\displaystyle=$ $\displaystyle
1-2(N-1)\left(\left|u\right|-y\right),$ $\displaystyle\xi_{T}^{2}$
$\displaystyle=$ $\displaystyle\min\\{1-2(N-1)\left(\left|u\right|-y\right),$
(17) $\displaystyle 1-4(N-1)(y+\sqrt{v_{+}v_{-}})(y-\sqrt{v_{+}v_{-}})\\}.$
Now, one can see that the squeezing parameters are related to the concurrence
shown in equation (14). The relations between $\xi_{S}^{2}$ and $C$ have been
studied WangSangders2003 . In the following, we consider the squeezing
parameter $\xi_{T}^{2}$, and prove that it is qualitatively equivalent to the
concurrence in detecting pairwise entanglement.
## IV Relations between spin squeezing parameters and concurrence
Firstly, we prove that for a state with exchange symmetry and parity, if
concurrence $C>0$, it must be spin squeezed according to the criterion
$\xi_{T}^{2}<1$. From equation (14) we note that when $C>0$, there are two
cases, $C=\left|u\right|-y>0$ or $C=y-\sqrt{v_{+}v_{-}}>0$. However, since the
density matrix $\rho$ is positive, we find
$\sqrt{v_{+}v_{-}}\geq\left|u\right|$, then immediately
$\left(|u|-y\right)\left(y-\sqrt{v_{+}v_{-}}\right)\leq 0,$ (18)
which means $\left|u\right|-y$ and $y-\sqrt{v_{+}v_{-}}$ cannot be positive
simultaneously. Therefore, if $C>0$, we have Vidal2006
$C=\left\\{\begin{array}[]{ll}2\left(\left|u\right|-y\right),{}{}{}&\left|u\right|>y,\\\
2\left(y-\sqrt{v_{+}v_{-}}\right),{}{}{}&y>\sqrt{v_{+}v_{-}}.\end{array}\right.$
(19)
According to equations (8) and (17), we get the following relations
$\xi_{T}^{2}=\left\\{\begin{array}[]{ll}1-\left(N-1\right)C,{}{}{}&\left|u\right|>y,\\\
1-2\left(N-1\right)\left(y+\sqrt{v_{+}v_{-}}\right)C,{}{}{}&y>\sqrt{v_{+}v_{-}},\end{array}\right.$
(20)
since $C>0$, there always be $\xi_{T}^{2}<1$.
Now, we prove that if the state is spin squeezed $\left(\xi_{T}^{2}<1\right)$,
concurrence $C>0$. If $\xi_{T}^{2}<1$, there are two cases,
$\xi_{T}^{2}=\xi_{S}^{2}<1$ or $\xi_{T}^{2}=\varsigma^{2}<1$. As discussed
above, according to equations (17) and (18), $\xi_{S}^{2}<1$ and
$\varsigma^{2}<1$ could not occur simultaneously. Therefore, if
$\xi_{T}^{2}=\xi_{S}^{2}<1$, we have Vidal2004
$C=\frac{1-\xi_{T}^{2}}{N-1},$ (21)
while if $\xi_{T}^{2}=\varsigma^{2}<1$, we have
$C=\frac{1-\xi_{T}^{2}}{2\left(N-1\right)\left(y+\sqrt{v_{+}v_{-}}\right)}.$
(22)
Therefore, if the state is squeezed, concurrence $C>0$.
Table 1: Spin squeezing parameters and concurrence for states with exchange symmetry and parity. | Pairwise entangled ($C>0$) | Unentangled
---|---|---
Concurrence | $C=2(\left|u\right|-y)>0$ | $C=2(y-\sqrt{v_{+}v_{-}})>0$ | $C=0$
$\xi_{S}^{2}$ | $\xi_{S}^{2}=1-(N-1)C<1$ | $\xi_{S}^{2}>1$ | $\xi_{S}^{2}\geq 1$
$\xi_{T}^{2}$ | $\xi_{T}^{2}=1-\left(N-1\right)C<1$ | $\xi_{T}^{2}=1-2(N-1)(y+\sqrt{v_{+}v_{-}})\times{C<1}$ | $\xi_{T}^{2}\geq 1$
The relations between spin squeezing and concurrence is displayed in Table 1,
and we can see that, for a symmetric state, $\xi_{T}^{2}<1$ is qualitatively
equivalent to $C>0$, that means spin squeezing according to $\xi_{T}^{2}$ is
equivalent to pairwise entanglement. Although $\xi_{S}^{2}<1$ indicates $C>0$,
when $C=2(y-\sqrt{v_{+}v_{-}})>0$, we find $\xi_{S}^{2}>1$. Therefore, a spin-
squeezed state ($\xi_{S}^{2}<1$) is pairwise entangled, while a pairwise
entangled state may not be spin-squeezed according to the squeezing parameter
$\xi_{S}^{2}$. Then, we come to the conclusion that for states with exchange
symmetry and parity, the spin squeezing parameter $\xi_{T}^{2}$ is
qualitatively equivalent to the concurrence in characterizing pairwise
entanglement. In the following, we will give some examples and applications of
our result.
## V Examples and Applications
We first consider a superposition of Dicke states with parity, and then
consider states without a fixed parity. The states under consideration are all
based on Dicke states Dicke1954 , and are defined as
$|n\rangle_{N}\equiv|\frac{N}{2},-\frac{N}{2}+n\rangle,~{}~{}~{}n=0,\ldots,N,$
(23)
where $|0\rangle_{N}\equiv|\frac{N}{2},-\frac{N}{2}\rangle$ denotes a state
for which all spins are in the ground states, and $n$ is the excitation number
of spins. Such states are elementary in atomic physics, and may be
conditionally prepared in experiments with quantum non-demolition technique
Molmer1998 ; Mandel ; Lemer2009 .
As we consider the state with even parity, we choose a simple superposition of
Dicke states as
$|\psi_{D}\rangle=\cos\theta|n\rangle_{N}+e^{i\varphi}\sin\theta|n+2\rangle_{N},~{}~{}~{}n=0,\ldots,N-2$
(24)
with the angle $\theta\in[0,\pi)$ and the relative phase $\varphi\in[0,2\pi)$.
We can easily check that, for the superposition state in equation (24), the
mean spin direction is along the $z$-axis. The expressions for the relevant
spin expectation values can be obtained as
$\displaystyle\left\langle J_{z}\right\rangle$ $\displaystyle=$ $\displaystyle
m+2\sin^{2}\theta,$ $\displaystyle\langle J_{z}^{2}\rangle$ $\displaystyle=$
$\displaystyle m^{2}+\left(4m+1\right)\sin^{2}\theta,$ $\displaystyle\langle
J_{+}^{2}\rangle$ $\displaystyle=$ $\displaystyle\langle
J_{-}^{2}\rangle=\frac{1}{2}e^{i\varphi}\sin 2\theta\sqrt{\mu_{n}},$ (25)
where $m=n-\frac{N}{2},$ and
$\mu_{n}=\left(n+1\right)\left(n+2\right)\left(N-n\right)\left(N-n-1\right)$.
By substituting equations (25) to equation (9) and (10), it is easy to get
$\displaystyle\xi_{S}^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{2}{N}\\{\left|\sin\theta\cos\theta\right|\sqrt{\mu_{n}}$ (26)
$\displaystyle-4[m^{2}+4(m+1)\sin^{2}\theta]-N^{2}\\}$
and
$\displaystyle\varsigma^{2}$ $\displaystyle=$
$\displaystyle\frac{4}{N}\left[m^{2}+4\left(m+1\right)\sin^{2}\theta\right]$
(27) $\displaystyle-\frac{4(N-1)}{N^{2}}\left[m+2\sin^{2}\theta\right]^{2}.$
From the results in Wang2002 we can easily get Vidal2006
$\displaystyle u$ $\displaystyle=$ $\displaystyle\frac{e^{i\varphi}\sin
2\theta}{2N(N-1)}\sqrt{\mu_{n}},$ $\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{N}{4(N-1)}-\frac{[m^{2}+4(m+1)\sin^{2}\theta]}{N(N-1)},$
$\displaystyle\sqrt{v_{+}v_{-}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(N^{2}-2N+4\left\langle
J_{z}^{2}\right\rangle)^{2}-16(N-1)^{2}\left\langle
J_{z}\right\rangle^{2}}}{4N(N-1)}.$
Insert equation (LABEL:rho_results) to equation (14), one can get the
expression of concurrence.
Figure 1: Spin squeezing parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$, and
concurrence as functions of $\theta$ for $N=3$ and $n=0.$ Figure 2: Spin
squeezing parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$, and concurrence as
functions of $\theta$ for $N=3$ and $n=0.$ The numerical result give
$\xi_{S}^{2}=7/3$, $\xi_{T}^{2}=1/9$, and $C=2/3$, when $\theta=\pi/2.$
In figure 1, we plot these two spin squeezing parameters and concurrence
versus $\theta$ in one period. We observe that for
$\theta\in(0,\pi/3)\cup(2\pi/3,\pi)$, $\xi_{T}^{2}=\xi_{S}^{2}<1$, therefore
the state is spin squeezed in the $x$-$y$ plane, moreover, as $C>0$, the state
is pairwise entangled. For $\theta\in(\pi/3,2\pi/3)$, it is obviously that the
state is also pairwise entangled, since $C>0,$ while spin squeezing occurs in
the $z$-axis since $\xi_{T}^{2}<1$ while $\xi_{S}^{2}>1$. The results show
clearly that $\xi_{T}^{2}<1$ is equivalent to $C>0$. But if we adopt
$\xi_{S}^{2}<1$ as squeezing parameter, the spin squeezing is not
qualitatively equivalent to concurrence.
The equivalence of $\xi_{T}^{2}<1$ and $C>0$ for states with parity has been
demonstrated above. Here, we discuss states without parity to see the
relations between spin squeezing and entanglement. For simplicity, we choose
$|\psi_{D}\rangle=\cos\theta|n\rangle_{N}+e^{i\varphi}\sin\theta|n+1\rangle_{N},~{}~{}~{}n=0,\ldots,N-1.$
(29)
Specifically, if $\theta=\frac{\pi}{2}$, $n=0$ or $n=N-2$, the above state
degenerates to the W state. Moreover, when $N=3$, equation (29) reduces to
$|\psi_{D}\rangle=\frac{1}{\sqrt{3}}(|110\rangle+|101\rangle+|011\rangle).$
(30)
The two-qubit reduced density matrix becomes
$\rho=\frac{1}{3}\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&1&1&0\\\ 0&1&1&0\\\
0&0&0&1\end{array}\right),$ (31)
and using equation (14) we find $C=\frac{2}{3}$. We can also get the
expectations of spin components, $\left\langle
J_{z}\right\rangle=-\frac{1}{2}$, $\left\langle
J_{x}^{2}\right\rangle=\left\langle J_{y}^{2}\right\rangle=\frac{7}{4}$,
$\left\langle J_{y}^{2}\right\rangle=\frac{1}{4}$, and then we can easily get
the spin squeezing parameters, $\xi_{S}^{2}=\frac{7}{3}$ and
$\xi_{T}^{2}=\frac{1}{9}$. The numerical results for $\xi_{T}^{2}$ is
displayed in figure 2, which coincide with the special result. It is
interesting to see that, although $|\psi_{D}\rangle$ has no parity, the state
is entangled $\left(C>0\right)$ and is spin squeezed according to
$\xi_{T}^{2}$ in the entire interval. However, according to parameter
$\xi_{S}^{2}$ the state is not squeezed in the middle region. Therefore, we
find that $\xi_{T}^{2}$ is more effective than $\xi_{S}^{2}$ in detecting
pairwise entanglement.
## VI Conclusion
In conclusion, we have studied the relations between spin squeezing and
pairwise entanglement. We have considered two types of spin squeezing
parameters $\xi_{S}^{2}$ and $\xi_{T}^{2}$ , and the pairwise entanglement is
characterized by concurrence $C$. We find that, for states with exchange
symmetry and parity, spin squeezing according to $\xi_{T}^{2}$ is
qualitatively equivalent to pairwise entanglement. In detecting pairwise
entanglement, parameter $\xi_{T}^{2}$ is more effective than parameter
$\xi_{S}^{2}$.
It is important to emphasize that, the above conclusion can be extended to the
states without (even or odd) parity. For states with properties $\left\langle
J_{\alpha}\right\rangle=0$, $\left\langle
J_{\alpha}J_{z}\right\rangle=\left\langle J_{z}J_{\alpha}\right\rangle=0$,
$\alpha=x,y$, we can have the same conclusion that spin squeezing and pairwise
entanglement are qualitatively equivalent. The following superposition of
Dicke states are examples:
$\left|\psi_{D^{\prime}}\right\rangle=\cos\theta|n\rangle_{N}+e^{i\varphi}\sin\theta|n+n^{\prime}\rangle_{N},$
$n=0,\ldots,N-n^{\prime}$, for all $n^{\prime}\geq 3$. As we have seen,
parameter $\xi_{S}^{2}$ is a key factor in $\xi_{T}^{2}$ for our states. The
present results imply that the spin squeezing has more intimate relations with
pairwise entanglement.
## Acknowledgements
This work is supported by NSFC with grant No. 10874151 and 10935010; and the
Fundamental Research Funds for the Central Universities.
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|
arxiv-papers
| 2009-12-09T14:11:20 |
2024-09-04T02:49:06.926603
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaolei Yin, Xiaoqian Wang, Jian Ma, and Xiaoguang Wang",
"submitter": "Xiaolei Yin",
"url": "https://arxiv.org/abs/0912.1752"
}
|
0912.1757
|
# Strongly Prime Submodules
A. R. Naghipour A.R. Naghipour, Department of Mathematics, Shahrekord
University, P.O.Box 115, Shahrekord, Iran Naghipour@sci.sku.ac.ir
###### Abstract.
Let $R$ be a commutative ring with identity. For an $R$-module $M$, the notion
of strongly prime submodule of $M$ is defined. It is shown that this notion of
prime submodule inherits most of the essential properties of the usual notion
of prime ideal. In particular, the Generalized Principal Ideal Theorem is
extended to modules.
###### Key words and phrases:
Prime submodule, strongly prime submodule, strongly prime radical.
###### 2000 Mathematics Subject Classification:
13A10, 13C99, 13E05.
The author is supported by Shahrekord University
## 0\. Introduction
Throughout this paper all rings are commutative with identity and all modules
are unitary. Also we consider $R$ to be a ring and $M$ a unitary $R$-module.
For a submodule $N$ of $M$, let $(N:M)$ denote the set of all elements $r$ in
$R$ such that $rM\subseteq N$. Note that $(N:M)$ is an ideal of $R$, in fact,
$(N:M)$ is the annihilator of the $R$-module $M/N$. A proper submodule $N$ of
$M$ is called prime if $rx\in N$, for $r\in R$ and $x\in M$, implies that
either $x\in N$ or $r\in(N:M)$. This notion of prime submodule was first
introduced and systematically studied in Dauns (1978) and recently has
received a good deal of attention from several authors, see for example Man
and Smith (2002), McCasland and Smith (1993), McCasland et al. (1997) and
Moore and Smith (2002).
In this article, we introduce a slightly different notion of prime submodule
and call it strongly prime submodule. First of all, we bring a notation.
Notation. Let $N$ be a submodule of $M$ and let $x\in M$. We denote the ideal
$(N+Rx:M)$ by $I_{x}^{N}$. Therefore, $I_{x}^{N}=\\{r\in R|rM\subseteq
N+Rx\\}$.
Let $P$ be a proper submodule of $M$. We say that $P$ is a strongly prime
submodule if $I_{x}^{P}y\subseteq P$, for $x,y\in M$, implies that either
$x\in P$ or $y\in P$. We call a proper submodule $C$ of $M$ to be a strongly
semiprime submodule if $I_{x}^{C}x\subseteq C$, for $x\in M$, implies that
$x\in C$.
Note that if we consider $R$ as an $R$-module, then strongly prime
(respectively, semiprime) submodules are exactly prime (respectively,
semiprime) ideals of $R$.
Our definition of strongly prime (respectively, semiprime) submodule seems
more natural, comparing to the usual notion of prime (respectively, semiprime)
ideal of a ring. We will show that every strongly semiprime submodule of $M$
is an intersection of strongly prime submodules. Note that this result is not
true for semiprime submodules, see Jenkins and Smith (1992).
This article consists of two sections. In the first section we prove some
preliminary facts about strongly prime submodules, which one could expect. In
Section 2, as an application of our result in Section 1, we state and prove a
module version of the Generalized Principal Ideal Theorem.
## 1\. Strongly Prime Submodules
We begin with the following proposition.
###### Proposition 1.1.
Let $M$ be an $R$-module. Then the following hold.
(1) Any strongly prime submodule of $M$ is prime.
(2) Any maximal submodule of $M$ is strongly prime.
###### Proof.
(1) Suppose on the contrary that $P$ is not a prime submodule. Then there
exist $x\in M\setminus P$ and $r\in R$ such that $rx\in P$ and $rM\nsubseteq
P$. So there exits $y\in M$ such that $ry\not\in P$. We have
$I_{x}^{P}ry=rI_{x}^{P}y\subseteq r(P+Rx)\subseteq P.$
Since $P$ is a strongly prime submodule, we should have $x\in P$ or $ry\in P$,
which is a contradiction.
(2) Let $x,y\in M$ and $I_{x}^{P}y\subseteq P$. If $x\not\in P$, then $P+Rx=M$
and hence $I_{x}^{P}=R$. It follows that $y\in P$, which completes the proof.
∎
Before we continue, let us show that a prime submodule need not be a strongly
prime (or even a strongly semiprime) submodule.
###### Example 1.2.
Let $R$ be a ring and ${\mathfrak{p}}\in{\operatorname{Spec}}(R)$. Then
$({\mathfrak{p}},{\mathfrak{p}})$ is a prime submodule of the $R$-module
$R\times R$. But it is not a strongly prime (or strongly semiprime) submodule
because
$I_{(1,0)}^{({\mathfrak{p}},{\mathfrak{p}})}(1,0)\subseteq{\mathfrak{p}}(1,0)\subseteq({\mathfrak{p}},{\mathfrak{p}})$,
and $(1,0)\not\in({\mathfrak{p}},{\mathfrak{p}})$.
Notation. The set of all strongly prime submodules of $M$ is denoted by
${\operatorname{S-Spec}}_{R}(M)$.
###### Proposition 1.3.
Let $V$ be a vector space over a field $F$. Then
${\operatorname{S-Spec}}_{F}(V)=\\{W|W\;\;{\mbox{is a maximal subspace
of}}\;\;V\\}.$
###### Proof.
By the above proposition, every maximal subspace is strongly prime. For the
converse, suppose to the contrary that $W$ is a strongly prime subspace of $V$
which is not a maximal subspace. Then there exists $x\in V\setminus W$ such
that $Fx+W\neq V$. For any $y\in M$, we have
$I_{x}^{W}y=\\{r\in F|rV\subseteq Fx+W\\}y=\\{0\\}y=\\{0\\}\subseteq W.$
It follows that $y\in W$ and hence $W=V$, which is a contradiction. Thus every
strongly prime subspace is maximal. ∎
Following Dauns (1980), we say that a proper submodule $N$ of an $R$-module
$M$ is semiprime if whenever $r^{2}x\in N$, where $r\in R$ and $x\in M$, then
$rx\in N$. The ring $R$ is called Max-ring if every $R$-module has a maximal
submodule. Max-Rings, which also called $B$-rings, were introduced by Hamsher
(1967) and has been studied by several authors, see for example Camillo
(1975), Faith (1973, 1995), Hirano (1998) and Koifmann (1970).
The following corollary provides characterizations of Max-rings.
###### Corollary 1.4.
Let $R$ be a ring. Then the following are equivalent.
(1) $R$ is Max-ring.
(2) Every $R$-module has a strongly prime submodule.
(3) Every $R$-module has a prime submodule.
(4) Every $R$-module has a semiprime submodule.
###### Proof.
(1)$\Longrightarrow$(2) and (2)$\Longrightarrow$(3) follow easily from
Proposition 1.1.
(3)$\Longrightarrow$(4) is trivial and (4)$\Longrightarrow$(1) follows from
Behboodi et al. (2004, Theorem 3.9). ∎
Next, we observe that strongly prime submodules behave naturally under
localization.
###### Theorem 1.5.
Let $M$ be an $R$-module, and let $U$ be a multiplicatively closed subset of
$R$. Then
${\operatorname{S-Spec}}_{U^{-1}R}U^{-1}M=\\{U^{-1}P|P\in{\operatorname{S-Spec}}_{R}M\,{\mbox{
and}}\,\,U^{-1}P\neq U^{-1}M\\}.$
If, moreover, $M$ is finitely generated, then
${\operatorname{S-Spec}}_{U^{-1}R}U^{-1}M=\\{U^{-1}P|P\in{\operatorname{S-Spec}}_{R}M\,{\mbox{
and}}\,\,(P:M)\cap U=\emptyset\\}.$
###### Proof.
First assume that $P\in{\operatorname{S-Spec}}_{R}M$ and $U^{-1}P\neq
U^{-1}M$. We show that $U^{-1}P\in{\operatorname{S-Spec}}_{U^{-1}R}U^{-1}M$.
Let $I_{x_{1}/u_{1}}^{U^{-1}P}x_{2}/u_{2}\subseteq U^{-1}P$, where
$x_{1}/u_{1},x_{2}/u_{2}\in U^{-1}M$. We claim that
$I_{x_{1}}^{P}x_{2}\subseteq P$. If $r\in I_{x_{1}}^{P}$, then $rM\subseteq
P+Rx_{1}$ and hence
$(r/1)U^{-1}M\subseteq U^{-1}P+U^{-1}R(x_{1}/1)=U^{-1}P+U^{-1}R(x_{1}/u_{1}).$
Therefore $(r/1)(x_{2}/u_{2})\subseteq U^{-1}P$ and so there exist $p\in P$
and $v_{1},v_{2}\in U$ such that $v_{2}(v_{1}rx_{2}-pu_{2})=0$. This implies
that $(v_{1}v_{2})rx_{2}\in P$. On the other hand, it is easy to see that
$U^{-1}P\neq U^{-1}M$ implies $(P:M)\cap U=\emptyset$. So we have $rx_{2}\in
P$. Thus $I_{x_{1}}^{P}x_{2}\subseteq P$. It follows that $x_{1}\in P$ or
$x_{2}\in P$ and hence $(x_{1}/u_{1})\in U^{-1}P$ or $(x_{2}/u_{2})\in
U^{-1}P$, as desired.
Now let $Q\in{\operatorname{S-Spec}}_{U^{-1}R}U^{-1}M$. Set $P=\\{x\in
M|x/1\in Q\\}$. It is easy to see that $Q=U^{-1}P$ and
$P\in{\operatorname{S-Spec}}_{R}M$ and thus we are done.
For the second assertion, it is enough to show that $(P:M)\cap U=\emptyset$
implies that $U^{-1}P\neq U^{-1}M$. Suppose on the contrary that
$U^{-1}P=U^{-1}M$. Since $M$ is finitely generated, we may assume that there
exist elements $x_{1},x_{2},\ldots,x_{n}\in M$ that generate $M$. For each
$1\leq i\leq n$ there exist $u_{i},v_{i}\in U$ and $p_{i}\in P$ such that
$v_{i}(u_{i}x_{i}-p_{i})=0$. If $t=v_{1}\ldots v_{n}u_{1}\ldots u_{n}$, then
$t\in(P:M)\cap U$, which is a contradiction. ∎
The following is an immediate consequence of Theorem 1.5.
###### Corollary 1.6.
Let $M$ be a finitely generated $R$-module and $U$ be a multiplicatively
closed subset of $R$. Then there is a bijective inclusion-preserving mapping
$\displaystyle\\{P\in{\operatorname{S-Spec}}_{R}M|(P:M)\cap U=\emptyset\\}$
$\displaystyle\longrightarrow$
$\displaystyle{\operatorname{S-Spec}}_{U^{-1}R}U^{-1}M$ $\displaystyle P$
$\displaystyle\longmapsto$ $\displaystyle U^{-1}P.$
whose inverse is also inclusion-preserving.
Let $N$ be a proper submodule of $M$. The strongly prime radical of $N$ in
$M$, denoted ${\operatorname{s-rad}}(N)$, is defined to be the intersection of
all strongly prime submodules of $M$ containing $N$. If there is no strongly
prime submodule containing $N$, then we put ${\operatorname{s-rad}}(N)=M$.
We conclude this section with a good justification for the study of strongly
prime submodules. In fact, as it mentioned in the introduction it is not true
that every semiprime submodule of an $R$-module $M$ is an intersection of
prime submodules, see Jenkins and Smith (1992), but our next theorem shows
that as in the ideal case, this is true for strongly semiprime submodules.
###### Theorem 1.7.
Let $C$ be a strongly semiprime submodule of an $R$-module $M$. Then $C$ is an
intersection of some strongly prime submodules of $M$.
###### Proof.
It is enough to show that ${\operatorname{s-rad}}(C)\subseteq C$. Let $x\in
M\setminus C$. We define $T=\\{x_{0},x_{1},\ldots\\}$ inductively as follows:
$x_{0}=x$, $x_{1}\in I_{x_{0}}^{C}x_{0}\setminus C$, $x_{2}\in
I_{x_{1}}^{C}x_{1}\setminus C$,$\ldots$, etc. Set
$\Omega=\\{K\leq M\,|\,C\subseteq K,\,\,K\cap T=\emptyset\\}.$
$\Omega\neq\emptyset$, since $C\in\Omega$. Then by Zorn’s lemma $\Omega$ has a
maximal element, say $P$. We claim that $P$ is a strongly prime submodule of
$M$. Suppose on the contrary that $x,y\in M\setminus P$ and
$I_{x}^{P}y\subseteq P$. Since $x,y\not\in P$, we have $(P+Rx)\cap
T\neq\emptyset$ and $(P+Ry)\cap T\neq\emptyset$. So there exist
$r_{1},r_{2}\in R$ and $p_{1},p_{2}\in P$ and $x_{i},x_{j}\in T$ such that
$p_{1}+r_{1}x=x_{i}$ and $p_{2}+r_{2}y=x_{j}$. We have
$I_{x_{i}}^{C}(x_{j}-p_{2})\subseteq
I_{x_{i}}^{P}(x_{j}-p_{2})=I_{r_{1}x+p_{1}}^{P}(x_{j}-p_{2})\subseteq
I_{x}^{P}(x_{j}-p_{2})=I_{x}^{P}r_{2}y\subseteq I_{x}^{P}y\subseteq P$
If $i\geq j$, then there exists $a\in R$ such that $x_{i}=ax_{j}$ and hence
$I_{x_{i}}^{C}(x_{i}-p_{2})=I_{x_{i}}^{C}(ax_{j}-p_{2})\subseteq
I_{x_{i}}^{C}(x_{j}-p_{2})\subseteq P.$
Since $x_{i+1}\in I_{x_{i}}^{C}x_{i}$, we have $x_{i+1}\in P$, which is a
contradiction. If $i<j$, then there exists $b\in R$ such that $x_{j}=bx_{i}$
and hence
$I_{x_{j}}^{C}(x_{j}-p_{2})=I_{bx_{i}}^{C}(x_{j}-p_{2})\subseteq
I_{x_{i}}^{C}(x_{j}-p_{2})\subseteq P.$
Since $x_{j+1}\in I_{x_{j}}^{C}x_{j}$, we have $x_{j+1}\in P$, which is again
a contradiction. Therefore $P$ is a strongly prime and hence
$x\not\in{\operatorname{s-rad}}(C)$ and the proof is complete. ∎
## 2\. A Generalized Principal Ideal Theorem for Modules
The Generalized Principal Ideal Theorem (GPIT) states that if $R$ is a
Noetherian rings and ${\mathfrak{p}}$ is a minimal prime ideal of an ideal
$(a_{1},\ldots,a_{n})$ generated by $n$ elements of $R$, then
${\mbox{ht}}{\mathfrak{p}}\leq n$. Consequently,
${\mbox{ht}}(a_{1},\ldots,a_{n})\leq n$, where for an ideal $I$ of $R$,
${\mbox{ht}}I$ denotes the height of $I$.
Krull proved this theorem by induction on $n$. The case $n=1$ is then the
hardest part of the proof. Krull called the $n=1$ case the Principal Ideal
Theorem (PIT).
###### Remark 2.1.
The PIT is one of the cornerstones of dimension theory for Noetherian rings,
see Eisenbud (1995, Theorem 10.1). Indeed, Kaplansky (1974, page 104) call it
“the most important single theorem in the theory of Noetherian rings”.
It is natural to ask if the GPIT can be extended to modules. Nishitani (1998),
has proved that the GPIT holds for modules. The aim of this section is to give
an alternative generalization of GPIT to modules. For this purpose we need to
define some notions.
Let $P$ be a strongly prime submodule of $M$. We shall say that $P$ is
strongly minimal prime over a submodule $N$ of $M$, if $N\subseteq P$ and
there does not exist a strongly prime submodule $L$ of $M$ such that
$N\subseteq L\subset P$.
###### Definition 2.2.
(1) Let $P$ be a strongly prime submodule of $M$. The strong height of $P$,
denoted ${0pt}_{R}{P}$, is defined by
${0pt}_{R}P={\sup}\\{n|\exists\;{P}_{0},{P}_{1},\ldots,{P}_{n}\in{\operatorname{S-Spec}}_{R}M\;\;{\mbox{such
that}}\;\;{P}_{0}\subset{P}_{1}\subset\cdots\subset{P}_{n}={P}\\}.$
(2) Let ${N}$ be a proper submodule of an $R$-module $M$. The strong height of
${N}$, denoted ${0pt}_{R}{N}$, is defined by
${0pt}_{R}{N}={\min}\\{{0pt}_{R}{P}|{P}\in{\operatorname{S-Spec}}_{R}{M},\;P\;{\mbox{is
strongly minimal prime over}}\;N\\}.$
###### Theorem 2.3.
Let $R$ be a ring and $M$ be a Noetherian flat $R$-module. Let $N$ be a proper
submodule of $M$ generated by $n$ elements $x_{1},\ldots,x_{n}\in M$. Then
${0pt}_{R}N\leq n.$
###### Proof.
Replacing $R/(0:M)$ by $R$, we can assume that $R$ is a Noetherian ring. Let
${0pt}_{R}N=\ell$. Then there is a submodule $P$ of $M$ such that $P$ is
strongly minimal prime over $N$ and ${0pt}_{R}P=\ell$. Let
${\mathfrak{p}}=(P:M)$ and $U=R\setminus{\mathfrak{p}}$. By Corollary 1.6,
${0pt}_{R}N={0pt}_{U^{-1}R}U^{-1}N$. Thus replacing $U^{-1}R$ by $R$, we can
assume that $R$ is a Noetherian local ring with maximal ideal
${\mathfrak{p}}$. Because $M$ is a flat module over a local ring, it is free
with finite rank, say $m$. Since $M/P$ is an $R/{\mathfrak{p}}$-vector space
and $(0)$ is a strongly prime submodule of $M/P$, by Proposition 1.3, we have
${\dim}_{R/{\mathfrak{p}}}M/P=1$. Hence there exists a basis
$\\{e_{1},e_{2},\ldots,e_{m}\\}$ for $M$ such that
$e_{1},e_{2},\ldots,e_{m-1}\in P$ and $e_{m}\not\in P$. We have
$P=Re_{1}+Re_{2}+\ldots+Re_{m}+{\mathfrak{p}}e_{m}$. There are elements
$a_{1j},a_{2j},\ldots,a_{m-1j}\in R$ and $a_{mj}\in{\mathfrak{p}}$ such that
$x_{j}=a_{1j}e_{1}+a_{2j}e_{2}+\ldots+a_{mj}e_{m}$. Let ${\mathfrak{q}}$ be a
minimal prime ideal over an ideal $(a_{m1},a_{m2},\ldots,a_{mn})$ and $Q$
denotes the submodule $Re_{1}+Re_{2}+\ldots+Re_{m}+{\mathfrak{q}}e_{m}$. Since
$M/Q\cong R/{\mathfrak{q}}$, $Q$ is a strongly prime submodule and hence
$P=Q$, by the minimality of $P$. Hence ${\mathfrak{p}}={\mathfrak{q}}$ holds
and so ${\mathfrak{p}}$ is a minimal prime over an ideal generating by $n$
elements. Since ${0pt}_{R}P=\ell$, we can consider the following chain of
distinct strongly prime submodules of $M$
$P_{0}\subset P_{1}\subset\ldots\subset P_{\ell}=P.$
We claim that the above chain induces a chain
$(P_{0}:M)\subset(P_{1}:M)\subset\ldots\subset(P_{\ell}:M)={\mathfrak{p}}$
of distinct prime ideals of $R$. It is enough to show that
$(P_{0}:M)\subset(P_{1}:M)$. The containment $(P_{0}:M)\subseteq(P_{1}:M)$ is
always true. Suppose that $(P_{0}:M)=(P_{1}:M)$. Then for any $x\in
P_{1}\setminus P_{0}$ and any $y\in M$, we have
$I_{x}^{P_{0}}y\subseteq I_{x}^{P_{1}}y=\\{r\in R|rM\subseteq
P_{1}\\}y=(P_{1}:M)y=(P_{0}:M)y\subseteq P_{0}.$
Since $P_{0}$ is strongly prime and $x\not\in P_{0}$, we have $y\in P_{0}$ and
hence $P_{0}=M$ which is a contradiction. Thus $(P_{0}:M)\subset(P_{1}:M)$.
Now by the GPIT for rings, we have $\ell\leq{\mbox{ht}}_{R}{\mathfrak{p}}\leq
n$. This completes the proof. ∎
## Acknowledgments
I thank Javad Asadollahi for his suggestions and comments. I also thank the
referee for many careful comments, and Sharekord University for the financial
support.
## References
* [1] Behboodi, M., Karamzadeh, O. A. S., Koohy, H. (2004). Modules whose certain submodules are prime, Vietnam J. Math, 32(3):303-317.
* [2] Camillo, V. P. (1975). On some rings whose modules have maximal submodules, Proc. Amer. Math. Soc. 55:97-100.
* [3] Dauns, J. (1978). Prime modules, J. Reine Angew. Math. 298:156-181.
* [4] Dauns, J. (1980). Prime modules and one-sided ideals. Lecture Notes Pure Appl. Math. 55:301-344.
* [5] Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York.
* [6] Faith, C. (1976). Algebra II, Ring Theory, Springer-Verlag, Berlin-New York.
* [7] Faith, C. (1995). Locally perfect commutative rings are those whose modules have maximal submodules, Comm. Algebra 23(13):4885-4886.
* [8] Hamsher, R. (1967). Commutative rings over which every module has a maximal submodule, Proc. Amer. Math. Soc. 18:1133-1137.
* [9] Hirano, Y. (1998). On rings over which each module has a maximal submodule, Comm. Algebra 26(10):3435-3445.
* [10] Jenkins, J., Smith, P. F. (1992). On the prime radical of a module over a commutative ring, Comm. Algebra 20(12):3593-3602.
* [11] Kaplansky, I. (1974). Commutative Rings, Revised Edition, The University of Chicago Press, Chicago.
* [12] Koifman, L. A. (1970). Rings over which every module has a maximal submodule, Math. Zametki 7:359-367.
* [13] Man, S. H., Smith, P. F. (2002). On chains of prime submodules, Israel J. Math. 127:131-155.
* [14] McCasland, R. L., Moore, M. E., Smith, P. F. (1997). On the spectrum of a module over a commutative ring, Comm. Algebra 25(1):79-103.
* [15] McCasland, R. L., Smith, P. F. (1993). Prime submodules of Noetherian modules, Rocky Mountain J. Math. 23(3):1041-1062.
* [16] Moore, M. E., Smith, S. J. (2002). Prime and radical submodules of modules over commutative rings. Comm. Algebra 30(10):5037-5064.
* [17] Nishitani, I. (1998). A generalized principal ideal theorem for modules over a commutative ring, Comm. Algebra 26(6):1999-2005.
|
arxiv-papers
| 2009-12-09T14:24:40 |
2024-09-04T02:49:06.932827
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.R. Naghipour",
"submitter": "Ali Reza Naghipour",
"url": "https://arxiv.org/abs/0912.1757"
}
|
0912.1898
|
# Singlet-triplet transitions in highly correlated nanowire quantum dots
Y. T. Chen Department of Electrophysics, National Chiao Tung University,
Hsinchu 30010, Taiwan, Republic of China C. C. Chao S. Y. Huang C. S. Tang
Department of Mechanical Engineering, National United University, 1, Lienda,
Miaoli 36003, Taiwan, Republic of China S. J. Cheng sjcheng@mail.nctu.edu.tw
###### Abstract
We consider a quantum dot embedded in a three-dimensional nanowire with
tunable aspect ratio $a$. A configuration interaction theory is developed to
calculate the energy spectra of the finite 1D quantum dot systems charged with
two electrons in the presence of magnetic fields $B$ along the wire axis.
Fruitful singlet-triplet transition behaviors are revealed and explained in
terms of the competing exchange interaction, correlation interaction, and spin
Zeeman energy. In the high aspect ratio regime, the singlet-triplet
transitions are shown designable by tuning the parameters $a$ and $B$. The
transitions also manifest the highly correlated nature of long nanowire
quantum dots.
###### keywords:
Nanowire quantum dot , Exchange , Correlation , Singlet-triplet transition
††journal: Physica E
## 1 Introduction
For years, few electron charged quantum dots have attracted extensive
attention due to the controllable electronic and spin properties [1]. However,
only few attempts have so far been made for studies of finite 1D nanowire
quantum dots (NWQDs). More recently, it was shown that the NWQDs formed in the
heterostructures in nanowires can be fabricated as single electron transistors
and successively charged with controlled number of electrons [2]. The
successful experimental works motivate us to explore the possible geometric
effects of NWQDs characterized by their aspect ratios, $a$, on the electronic
and spin properties of two-electron charged NWQDs.
In this work, we focus on the study of the singlet-triplet (ST) transitions in
two-electron charged NWQDs [3], conducted by using a developed configuration
interaction (CI) theory in combination with the exact diagonalization
techniques based on a 3D asymmetric parabolic model. It will be illustrated
that the ST transitions in InAs-based NWQDs driven by an appropriate magnetic
field are associated with the competing effects of large spin-Zeeman energies
as well as the exchange and correlation energies. The correlation-dominated
nature of a long NWQD (i.e. with high aspect ratio) will be identified by the
spin phase diagram with respect to the applied magnetic fields and the tunable
aspect ratio.
## 2 Theoretical Model
### 2.1 single-electron spectrum
We begin with the single electron problem of a NWQD with axial magnetic field
${\bf B}=(0,0,B)$, described by the Hamiltonian [4]
$H_{0}=\frac{1}{2m^{\ast}}({\bf p}+e{\bf A})^{2}+V(x,y,z)+H_{\rm Z}\,,$ (1)
where ${\bf A}=(B/2)(y,-x,0)$ denotes the vector potential and $m^{\ast}$
stands for the effective mass of an electron with charge $-e$. The spin-Zeeman
Hamiltonian $H_{\rm Z}=g^{\ast}\mu_{B}Bs_{z}$ is in terms of the z-component
of electron spin $s_{z}$ and the effective Lande g-factor of electron
$g^{\ast}$ and the Bohr magneton $\mu_{B}$. In addition, the confining
potential
$V(x,y,z)=m^{\ast}\left[\omega_{0}^{2}\left(x^{2}+y^{2}\right)+\omega_{z}^{2}z^{2}\right]/2$
is assumed of the parabolic form with $\omega_{0}$ and $\omega_{z}$
parametrizing, respectively, the transverse and the longitudinal confining
strength.
In this work, we assume a constant $g^{\ast}$ and take the $g^{\ast}=-8.0$ for
InAs [3, 6]. The single electron Hamiltonian (1) leads to the extended Fock-
Darwin single-particle spectrum
$\displaystyle\epsilon_{n,m,q,s_{z}}$ $\displaystyle=$
$\displaystyle\hbar\omega_{+}\left(n+\frac{1}{2}\right)+\hbar\omega_{-}\left(m+\frac{1}{2}\right)$
(2) $\displaystyle+\hbar\omega_{z}\left(q+\frac{1}{2}\right)+E_{\rm Z}$
where $n,m,q=0,1,2\cdots$ denote oscillator quantum numbers,
$s_{z}=+\frac{1}{2}$ ($s_{z}=-\frac{1}{2}$) the projection of electron spin
$\uparrow$ ($\downarrow$) $E_{\rm Z}=g^{\ast}\mu_{B}Bs_{z}$ the spin Zeeman
energy, and $\omega_{\pm}=\omega_{h}\pm\omega_{c}/2$ is defined in terms of
the hybridized frequency
$\omega_{h}\equiv(\omega_{0}^{2}+\omega_{c}^{2}/4)^{1/2}$ and the cyclotron
frequency $\omega_{c}={eB}/{m^{\ast}}$. .
The eigenstate $|n,m,q\rangle$ possesses the orbital angular momentum
$l_{z}=\hbar(n-m)$ and the parity $P=1$ ($P=-1$) with respect to $z-$axis for
the even (odd) $q$ number. The wave function of the lowest orbital is given by
$\psi_{000}({\bf
r})=\exp\left[-\left(\left(x^{2}+y^{2}\right)/{l_{h}^{2}}+{z^{2}}/{l_{z}^{2}}\right)/4\right]/\left(2\pi^{3/4}{l_{h}}\sqrt{l_{z}}\right)$
with the characteristic lengths of the wave function extents
$l_{h}=\sqrt{{\hbar}/{2m^{\ast}\omega_{h}}}$ and
$l_{z}=\sqrt{{\hbar}/{2m^{\ast}\omega_{z}}}$, from which one can generate the
wave functions of any other excited states by successively applying raising
operators [5]. For most synthesized NWQDs, the diameter of the cross section
is of the scales $50$ nm, while the length of wire could be tunable over a
wide range from $10$ nm to $300$ nm [6]. To characterize the geometry of
NWQDs, we define the aspect ratio parameter,
$a\equiv\frac{l_{z}}{l_{0}}=\sqrt{\frac{\omega_{0}}{\omega_{z}}}\sim\frac{L_{z}}{L_{x}}\,.$
(3)
according to the extents of the wave function.
Figure 1: Single-electron energy spectrum as a function of aspect ratio $a$
with no magnetic field. The dot diameter $L_{x}$=$50$ nm and the dot height
$L_{z}$ varies from $40\rm{nm}$ to $150\rm{nm}$. The finite difference results
are shown in the inset for comparison.
Figure 1 shows the single electron energy spectra, as a function of $a$, of
NWQDs at zero magnetic field. To examine the validity of the model, we carry
out a numerical finite difference (FD) simulation for the electronic structure
of InAs/InP heterostructure NWQD, as shown in the inset of Figure 1. The InAs
NWQD is embedded in InP barriers with the diameter $L_{x}=50$ nm and varying
the length from $L_{z}=40$ nm to $L_{z}=150$ nm. The effective mass
$m^{\ast}=0.023m_{0}$ and the barrier offset $V_{b}=0.6$ eV are taken [7]. The
confining strength parameter $\hbar\omega_{0}$ is fit by the ground state
energy from the FD simulation. We set $\hbar\omega_{0}=13.3$ meV for
$L_{x}=50$ nm NWQDs throughout this paper. The numerically calculated energy
spectrum is in good agreement with that given by parabolic model. The
schematic illustration of the engineered single electron energy levels of
NWQDs by tunable $a$ and applied $B$ is shown in Figure 2.
Figure 2: Schematic diagram of the lowest two orbitals occupied by two
relevant electrons for the cases: (a) isotropic ($a=1$) and $B=0$; (b) “rod-
like” ($a>1$) and $B=0$; (c) with nonzero electronic $g$-factor and $B\neq 0$.
$2E_{Z}=g^{\ast}\mu_{B}B$ indicates the spin Zeeman energy splitting.
### 2.2 Interacting NWQD
To investigate the few-electron interaction effects in a NWQD, we express the
few electron Hamiltonian in second quantization as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i,\sigma}\epsilon_{i\sigma}c_{i\sigma}^{{\dagger}}c_{i\sigma}$
(4) $\displaystyle+\frac{1}{2}\sum_{ijkl,\sigma\sigma^{\prime}}\langle
ij|V|kl\rangle
c_{i\sigma}^{{\dagger}}c_{j\sigma^{\prime}}^{{\dagger}}c_{k\sigma^{\prime}}c_{l\sigma}\,,$
where $i,j,k,l$ stand for the composite indices of single electron orbitals
(e.g. $|i\rangle=|n_{i},m_{i},q_{i}\rangle$), $\sigma=\uparrow/\downarrow$
denotes the electron spin with $s_{z}=+\frac{1}{2}/-\frac{1}{2}$, and
$c_{i\sigma}^{{\dagger}}$ ($c_{i\sigma}$) is the electron creation
(annihilation) operators. The first (second) term on the right hand side of
Eq.(4) represents the kinetic energy of electrons (the Coulomb interactions
between electrons) and the Coulomb matrix elements are defined as $\langle
ij|V|kl\rangle\equiv e^{2}\left(4\pi\kappa\right)^{-1}\int\int d{\bf
r_{1}}d{\bf r_{2}}\psi_{i}^{*}({\bf r_{1}})\psi_{j}^{*}({\bf
r_{2}})\left(|{\bf r_{1}}-{\bf r_{2}}|\right)^{-1}\psi_{k}({\bf
r_{2}})\psi_{l}({\bf r_{1}})$, where $\kappa$ is the dielectric constant of
dot material ( $\kappa=15.15\epsilon_{0}$ is taken for InAs throughout this
work). After lengthy derivation, for NWQDs with $a\geq 1$, we obtain the
following formulation of the Coulomb matrix elements:
$\displaystyle\langle
n_{i}m_{i}q_{i};n_{j}m_{j}q_{j}|V|n_{k}m_{k}q_{k};n_{l}m_{l}q_{l}\rangle$
$\displaystyle=$ $\displaystyle(\frac{1}{\pi
l_{h}})\frac{\delta_{R_{L},R_{R}}\cdot\delta_{q_{i}+q_{j}+q_{l}+q_{k},\rm{even}}}{\sqrt{n_{i}!m_{i}!q_{i}!n_{j}!m_{j}!q_{j}!n_{k}!m_{k}!q_{k}!n_{l}!m_{l}!q_{l}!}}$
$\displaystyle\times$
$\displaystyle\sum_{p_{1}=0}^{\min(n_{i},n_{l})}\sum_{p_{2}=0}^{\min(m_{i},m_{l})}\sum_{p_{3}=0}^{\min(q_{i},q_{l})}\sum_{p_{4}=0}^{\min(n_{j},n_{k})}\sum_{p_{5}=0}^{\min(m_{j},m_{k})}\sum_{p_{6}=0}^{\min(q_{j},q_{k})}$
$\displaystyle\times$ $\displaystyle p_{1}!p_{2}!p_{3}!p_{4}!p_{5}!p_{6}!$
$\displaystyle\times$ $\displaystyle{n_{i}\choose p_{1}}{n_{l}\choose
p_{1}}{m_{i}\choose p_{2}}{m_{l}\choose p_{2}}{q_{i}\choose
p_{3}}{q_{l}\choose p_{3}}$ $\displaystyle\times$ $\displaystyle{n_{j}\choose
p_{4}}{n_{k}\choose p_{4}}{m_{j}\choose p_{5}}{m_{k}\choose
p_{5}}{q_{j}\choose p_{6}}{q_{k}\choose p_{6}}$ $\displaystyle\times$
$\displaystyle(-1)^{u+v/2+n_{j}+m_{j}+q_{j}+n_{k}+m_{k}+q_{k}}\times(\frac{1}{2})^{u}\times
x^{u+1/2}$ $\displaystyle\times$
$\displaystyle\frac{\Gamma(\frac{1+2u+v}{2})\Gamma(1+u)\Gamma(\frac{1+v}{2})}{\Gamma(\frac{3+2u+v}{2})}$
$\displaystyle\times$
${}_{2}F_{1}(1+u,\frac{1+2u+v}{2};\frac{3+2u+v}{2};1-x)\,,$
where we have defined $u=m_{i}+m_{j}+n_{l}+n_{k}-(p_{1}+p_{2}+p_{4}+p_{5})$,
$v=(q_{i}+q_{l}+q_{j}+q_{k})-2(p_{3}+p_{6})$,
$R_{L}=(m_{i}+m_{j})-(n_{i}+n_{j})=-(L_{z,i}+L_{z,j})$,
$R_{R}=(m_{l}+m_{k})-(n_{l}+n_{k})=-(L_{z,l}+L_{z,k})$,
$x\equiv\omega_{z}/\omega_{h}$, and ${}_{2}F_{1}$ is the hypergeometric
function. The $\delta$-functions $\delta_{q_{i}+q_{j}+q_{l}+q_{k},\rm{even}}$
and $\delta_{R_{L},R_{R}}$ in the formulation ensure the conservation of the
parity with respect to $z$-axis and the $z$-component of angular momentum of
system $L_{z}$, respectively. The formulation of Eq. (2.2) is reexamined by
computing the Coulomb integral numerically.
### 2.3 Exact diagonalization
The energy spectrum of an interacting two-electron NWQD is calculated using
the standard numerical exact diagonalization technique [8]. The numerically
exact results are obtained by increasing the numbers of chosen single electron
orbital basis and the corresponding two-electron configurations until a
numerical convergence is achieved. In our full configuration interaction (FCI)
calculations, we chose the typical orbital number from $20$ to $26$ and the
number of corresponding configurations is from $190$ to $325$. In order to
highlight the Coulomb correlations, we also carry out the partial CI (PCI)
calculations in which only the lowest energy $N_{e}$ configuration is taken
and compare the PCI results with those obtained from the FCI calculations.
## 3 Results and discussion
Figure 3: Correlated two-electron energy spectrum as a function of magnetic
field in a NWQD with diameter $L_{x}=50$ nm and aspect ratio $a=3$.
Figure 3 presents the FCI result of magneto-energy spectrum of two interacting
electrons in a NWQD with $a=3$. The ST transition of the two-electron ground
state is shown to happen as $B_{\rm ST}\sim 0.9$ T. As the applied magnetic
field is weak, the spin Zeeman splitting is small and the two electrons mostly
doubly fill the lowest S-orbital. With increasing magnetic field increases,
the energy difference between triplet and singlet states of the two electrons
decreases because of increasing spin Zeeman and exchange energies, both of
which energetically favor the triplet states $|T^{+}\rangle$. As the applied
field is higher than $B_{c}\sim 0.9$T, the ground state of two electrons
transit from the singlet state to the triplet one.
Figure 4: (a) Singlet-triplet splitting $\Delta_{{\rm ST}^{i}}$ as a function
of aspect ratio $a$ with $B=1$ T. (b) The spin phase diagram for electrons
making singlet-triplet transition with respect to magnetic field $B$ versus
aspect ratio $a$.
Now we turn to study the spin singlet-triplet splitting as a function of $a$,
defined by $\Delta_{{\rm ST}^{i}}\equiv E_{T^{i}}-E_{S}$, with $i=-$, $0$, and
$+$ corresponding to the $T^{-}$, $T^{0}$, and $T^{+}$ triplet states,
respectively. In Figure 4(a), we show the $\Delta_{{\rm ST}^{i}}$ as a
function of aspect ratio $a$ under a fixed magnetic field $B=1$ T. In the non-
interacting case, $\Delta^{0}_{\rm ST^{i}}$ are shown to decrease
monotonically with increasing aspect ratio $a$. Since only $T^{+}$ energy is
decreased by spin-Zeeman term, the ST transition could only occur between $S$
and $T^{+}$ states. That is, only $\Delta_{{\rm ST}^{+}}$ crosses zero as $a$
is very large, while $\Delta_{{\rm ST}^{0}}$ remains positive always. Thus
below we shall only consider $\Delta_{{\rm ST}^{+}}$ for the discussion of ST
transition. The non-interacting ST splitting can be derived as
$\Delta^{0}_{\rm ST^{+}}=\hbar\omega_{0}/a^{2}-2E_{\rm Z}$, explicitly showing
the quadratic decrease of $\Delta^{0}_{\rm ST^{+}}$ with respect to $a$.
Accordingly, in the non-interacting picture, the critical aspect ratio $a_{\rm
ST}$ where the ST transition occurs is predicted as $a_{\rm
ST}=\sqrt{\hbar\omega_{0}/g^{*}\mu_{B}B}$.
However, the PCI calculation predicts a much smaller value of critical aspect
ratio $a_{\rm ST}=2.5$. In the PCI result, the ST splitting is substantially
reduced by the energy reduction of the T state due to the reduced direct
Coulomb interaction and the negative exchange interaction between the two
electrons in the state. The FCI calculation shows $a_{\rm ST}=2.9$, as
indicated by the dashed vertical line in Figure 4(a). In fact, the difference
in the values of $\Delta_{\rm ST^{+}}$ obtained from the PCI and FCI
calculations increases as $a$ increases. This indicates that the Coulomb
correlation effect tends to increase the ST splitting again and becomes even
more pronounced in long NWQD with high $a$.
Figure 4(b) shows the calculated spin phase diagram of two-electron NWQDs with
respect to the aspect ratio $a$ and applied magnetic field $B$. The spin
singlet and triplet phases, appearing in the low $a$-$B$ and high $a$-$B$
regimes, respectively, are distinguished by the curve of $B_{\rm ST}$ which
show a monotonic decrease with $a$. For noninteracting electrons, the critical
magnetic field can be derived as $B_{\rm ST}=\hbar\omega_{0}/g\mu_{B}a^{2}$,
showing a quadratic decay with $a$.
In comparison with the non-interacting cases, the PCI calculations obtain the
$B_{\rm ST}$ that is significantly reduced and goes to zero for $a>2.9$. In
the one-configuration approximation used in the PCI calculation, the ST
splitting is given by $\Delta_{\rm ST^{+}}\approx\Delta^{0}_{\rm
ST^{+}}+\Delta_{\rm ST}^{\rm dir}-V_{\rm T}^{\rm ex}$, where $\Delta^{0}_{\rm
ST^{+}}$ is the ST splitting in the non-interacting cases, $\Delta_{\rm
ST}^{\rm dir}\equiv V_{\rm T}^{\rm dir}-V_{\rm S}^{\rm dir}<0$ is the direct
energy difference between the triplet and the singlet states, and $V_{\rm
T}^{\rm ex}$ is the exchange energy between electrons in the $T^{+}$ state.
Accordingly, we obtain $B_{\rm ST}=(\hbar\omega_{0}/a^{2}+\Delta_{\rm ST}^{\rm
dir}-V_{\rm T}^{\rm ex})/g^{\ast}\mu_{B}$. In the large aspect ratio regime,
the negative $\Delta_{\rm ST}^{\rm dir}$ and $V_{\rm T}^{\rm ex}$ reduce the
$E_{{\rm T}^{+}}$ and $B_{\rm ST}=0$ results for $a>3$. However, the FCI
calculation predict larger and always positive $B_{ST}$. In fact, as
increasing $a$, the relative strength of electronic Coulomb correlations
increases because of reduced $\hbar\omega_{z}$ and strong configuration
interactions. Such $a$-engineered Coulomb correlations energetically favor the
singlet two-electron states and result in the non-zero $B_{ST}$ in the high
aspect ratio regime.
## 4 Summary
In conclusion, a configuration interaction (CI) theory is developed for
studying the magneto-energy spectra and the singlet-triplet transitions of
two-electron NWQDs with longitudinal magnetic field $B$ and tunable aspect
ratio $a$. For short NWQDs of low aspect ratio $a<3$, the ST transition
behaviors are dominated by the spin Zeeman, Coulomb direct and exchange
energies, and can be well studied by using PCI calculation. However, our FCI
calculations show the increasing importance of Coulomb correlations in long
NWQDs with increasing aspect ratio $a$ over $3$. The FCI calculation present
the spin phase diagram of a two-electron NWQD which are highly dependent on
$a$, and suggests the controllability of singlet or triplet spin states by
changing the aspect ratio of NWQD.
## 5 Acknowledgment
This work was supported in part by the National Science Council of the
Republic of China through Contracts No. NSC95-2112-M-009-033-MY3 and No.
NSC97-2112-M-239-003-MY3.
## References
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* [2] M. T. Bj$\rm\ddot{o}$rk, et al., 4 (2004) 1621.
* [3] C. Fasth, et al., Phys. Rev. Lett. 98 (2007) 266801.
* [4] Y. T. Chen, Few-Electron Theory of Semiconductor Nanocrystal and Nanorod Systems, MSc thesis submmitted to National Chiao Tung University, 2006.
* [5] P. Hawrylak, Solid State Comm. 88 (1993) 475.
* [6] M. T. Bj$\rm\ddot{o}$rk, et al., Phys. Rev. B 72 (2005) 201307.
* [7] M. T. Bj$\rm\ddot{o}$rk, et al., Appl. Phys. Lett. 81 (2002) 4458.
* [8] A. Wensauer, M. Korkusinski, P. Hawrylak, Solid State Comm. 130 (2003) 1155.
|
arxiv-papers
| 2009-12-10T01:49:36 |
2024-09-04T02:49:06.940336
|
{
"license": "Public Domain",
"authors": "Y. T. Chen, C. C. Chao, S. Y. Huang, C. S. Tang, S. J. Cheng",
"submitter": "Yan-Ting Chen",
"url": "https://arxiv.org/abs/0912.1898"
}
|
0912.1919
|
# Engineered spin phase diagram of two interacting electrons
in semiconductor nanowire quantum dots
Yan-Ting Chen Department of Electrophysics, National Chiao Tung University,
Hsinchu 30010, Taiwan, Republic of China Shun-Jen Cheng
sjcheng@mail.nctu.edu.tw Department of Electrophysics, National Chiao Tung
University, Hsinchu 30010, Taiwan, Republic of China Chi-Shung Tang
Department of Mechanical Engineering, National United University, Miaoli
36003, Taiwan, Republic of China
###### Abstract
Spin properties of two interacting electrons in a quantum dot (QD) embedded in
a nanowire with controlled aspect ratio and longitudinal magnetic fields are
investigated by using a configuration interaction (CI) method and exact
diagonalization (ED) techniques. The developed CI theory based on a three-
dimensional (3D) parabolic model provides explicit formulations of the Coulomb
matrix elements and allows for straightforward and efficient numerical
implementation. Our studies reveal fruitful features of spin singlet-triplet
transitions of two electrons confined in a nanowire quantum dot (NWQD), as a
consequence of the competing effects of geometry-controlled kinetic energy
quantization, the various Coulomb interactions, and spin Zeeman energies. The
developed theory is further employed to study the spin phase diagram of two
quantum-confined electrons in the regime of “cross over” dimensionality, from
quasi-two-dimensional (disk-like) QDs to finite one-dimensional (rod-like)
QDs.
## I Introduction
Stimulated by recent success in coherent control of two-electron spin in
laterally coupled quantum dots (QDs), Petta05 the spin states of two
interacting electrons in semiconductor QDs have received increasingly
considerable attention. Accessible and engineerable spin states of few
electrons in QDs thus have become one of the basic features required by the
quantum information applications in which electron spins are utilized as
quantum bit. Loss98 ; Loss09 For two-dimensional (2D) epitaxial QDs, magnetic
field induced spin singlet-triplet (ST) transitions of two-electron ground
states have been studied extensively for years. Kouwenhoven97 ; Kouwenhoven01
; Reimann02 ; Ellenberger06 The underlying physics of the ST transitions is
usually associated with the energetic competition between quantized kinetic
energies, the coulomb interactions, and spin Zeeman energies. Reversely
switching the singlet and triplet spin states of a lateral two-electron QD is
feasible by utilizing electrical control. Kyriakidis02 Moreover, it has been
both theoretically and experimentally shown that more complex oscillating spin
phases can be generated either by reducing the lateral confinement or by
increasing an applied magnetic field. Wagner92 ; Hawrylak93-prl ; Peeters99 ;
Tarucha07
Recently, the local-gate electrical depletion Fasth05 ; Ensslin06 ; Ensslin07
and the bottom-up grown techniques Bjork04 ; Bjork05 have been developed for
the fabrication of few-electron QDs embedded in a nanowire. These experimental
developments open up an opportunity of exploring the cross over mechanisms
from the 2D (disk-like) to the finite 1D (rod-like) QD regimes. Such nanowire
quantum dots (NWQDs) are advantageous for geometrical control over a wide rage
of aspect ratio $a$ (typically from $a\sim 10^{-1}$ to $a\gg 1$). Bjork04 ;
Bjork05 The excellent versatility of shape and dimensionality makes NWQDs a
suitable nanomaterial for scalable quantum electronics. Very recently,
successful fabrication of single electron transistors made of InAs based gate-
defined NWQDs and observations of the singlet-triplet transitions of two
electrons in the QDs have been demonstrated. Fasth07 How the highly tunable
longitudinal confinement of NWQD affects and can be utilized to tailor the
spin properties of few electrons in NWQDs are interesting subjects worth
studying.
The above experimental efforts motivate us to perform a theoretical
investigation of the spin states of two electrons in InAs-based NWQDs Fasth07
by using a developed configuration interaction (CI) theory and exact
diagonalization techniques. Hawrylak03 The developed CI theory is based on
the 3D parabolic model with arbitrary transverse and longitudinal confinement
strengths Nazmitdinov97 ; Lin01 and provides explicit generalized
formulations of the Coulomb matrix, and thus allows for straightforward and
efficient numerical or even semi-analytical implementation widely applicable
for various cylindrically symmetric QDs. Our exact diagonalization studies of
two-electron charged NWQDs with controlled geometric aspect ratios and
longitudinal magnetic fields reveal fruitful features of spin singlet-triplet
transitions, as a consequence of the competing effects of geometry-engineered
kinetic energy quantization, the various Coulomb interactions, and spin Zeeman
energies. The developed theory is further employed to study the spin phase
diagram of two quantum-confined electrons in the regime of “cross over”
dimensionality from quasi-2D (disk-like) QDs to finite 1D (rod-like) QDs.
This article is organized as follows: Section II describes the theoretical
model and the developed configuration interaction theory for few-electron
problems of three-dimensionally confining quantum dots. In Sec. III, we
present and discuss the calculated results of magneto-energy spectrum, the ST
transitions and geometry-engineered spin phase diagrams of two-electron
charged quantum dots embedded in nanowires. Concluding remarks are presented
in Sec. IV.
## II Model
### II.1 Single-particle model
We begin with the problem of a single electron in a NWQD with a uniform
longitudinal magnetic field ${\bf B}=(0,0,B)$, which is described by the
single-electron Hamiltonian,
$H_{0}=\frac{1}{2m^{\ast}}({\bf p}+e{\bf A})^{2}+V_{c}(x,y,z)+H_{\rm Z}.$ (1)
Here the first term indicates the term of kinetic energy with ${{\bf
A}}=(B/2)(-y,x,0)$ being the vector potential in symmetric gauge, $m^{\ast}$
the effective mass of electron and $e$ the charge of an elctron. The second
term is the confining potential of NWQD modeled by
$V_{c}(x,y,z)=\frac{1}{2}m^{\ast}\left[\omega_{0}^{2}\left(x^{2}+y^{2}\right)+\omega_{z}^{2}z^{2}\right]$
(2)
with $\omega_{0}$ and $\omega_{z}$ parametrizing, respectively, the transverse
and the longitudinal confining strength. The last term is the spin Zeeman
energy $H_{\rm Z}=g^{\ast}\mu_{B}Bs_{z}$, in terms of the $z$-component of
electron spin $s_{z}=\pm 1/2$, the effective Lande $g$-factor of electron
$g^{\ast}$ and the Bohr magneton $\mu_{B}$.
Figure 1: (Color online) Single-electron energy spectrum as a function of
aspect ratio $a$ of a NWQD with fixed lateral confinement
$\hbar\omega_{0}=13.3$ meV at zero magnetic field obtained from the 3D
parabolic model. The considered lateral confinement strength
$\hbar\omega_{0}=13.3$ meV corresponds to the cross section diameter ${\bf
L}_{0}\sim 65$ nm for a cylindrical InAs nanowire. The low-lying $s$-,
$p^{\pm}$-, and $p^{0}$-orbitals are relevant to a two-electron problem. The
energy quantization for a short (long) NWQD with $a<1$ ( $a>1$) is
characterized by the energy difference between the lowest and first excited
orbitals $\hbar\omega_{0}$ ($\hbar\omega_{z}$).
The single-particle Hamiltonian (1) leads to the extended Fock-Darwin single-
particle spectrum
$\displaystyle\epsilon_{n,m,q,s_{z}}$ $\displaystyle=$
$\displaystyle\hbar\omega_{+}\left(n+\frac{1}{2}\right)+\hbar\omega_{-}\left(m+\frac{1}{2}\right)$
(3) $\displaystyle+\hbar\omega_{z}\left(q+\frac{1}{2}\right)+E_{\rm Z}$
where $n,m,q=0,1,2\cdots$ denote oscillator quantum numbers, $E_{\rm
Z}=g^{\ast}\mu_{B}Bs_{z}$ is the spin Zeeman energy,
$\omega_{\pm}=\omega_{h}\pm\omega_{c}/2$ are in terms of the hybridized
frequency $\omega_{h}\equiv(\omega_{0}^{2}+\omega_{c}^{2}/4)^{1/2}$ and the
cyclotron frequency $\omega_{c}={eB}/{m^{\ast}}$. The corresponding eigenstate
$|n,m,q\rangle$ possesses the orbital angular momentum projection
${\ell}_{z}=\hbar(n-m)$ and the parity $P=1$ ($P=-1$) with respect to $z$-axis
for an even (odd) $q$ number. The wave function of the lowest orbital is given
by
$\displaystyle\psi_{000}({\bf r})$ $\displaystyle=$
$\displaystyle\left[(2\pi)^{3/4}{l_{h}}\sqrt{l_{z}}\right]^{-1}$ (4)
$\displaystyle\times{\rm{exp}}\left[-\frac{1}{4}\left(\frac{x^{2}+y^{2}}{l_{h}^{2}}+\frac{z^{2}}{l_{z}^{2}}\right)\right]\,,$
with the characteristic lengths of the wave function extents
$l_{h}=\sqrt{{\hbar}/{2m^{\ast}\omega_{h}}}$ and
$l_{z}=\sqrt{{\hbar}/{2m^{\ast}\omega_{z}}}$. The wave functions of other
excited states can be generated by successively applying the following defined
raising operators Hawrylak03
$\displaystyle a^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\frac{x+iy}{2l_{h}}-l_{h}(\partial_{x}+i\partial_{y})\right],$
$\displaystyle b^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\frac{x-iy}{2l_{h}}-l_{h}(\partial_{x}+i\partial_{y})\right],$
(5) $\displaystyle a_{z}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{z}{2l_{z}}-l_{z}\partial_{z}$
onto the ground state $|0,0,0\rangle$, i.e.
$|n,m,q\rangle=\frac{(\hat{a}^{\dagger})^{n}(\hat{b}^{\dagger})^{m}(\hat{a_{z}}^{\dagger})^{q}}{\sqrt{n!m!q!}}|0,0,0\rangle.$
(6)
The diameter of cross section of bottom-up synthesized nanowire is typically
$\sim 50-70$ nm. By contrast, the length of a QD in a nanowire, defined by
imposed electrodes or heterostructure potential barriers, is highly tunable
over a wide range from $10$ to $300$ nm. Bjork05 For characterizing the
geometry of a NWQD, it is convenient to define the parameter of aspect ratio,
$a\equiv\frac{l_{z}}{l_{0}}=\sqrt{\frac{\omega_{0}}{\omega_{z}}}$ (7)
according to the characteristic length of the lowest orbital wave function
based on the 3D parabolic model. A rod-like (disk-like) NWQD is characterized
by the value of aspect ratio $a>1$ ($a<1$), where the longitudinal extent of
the electron wave function is longer (shorter) than the transverse one on the
cross section of the nanowire. Notably, the effective aspect ratio
$a=l_{z}/l_{0}$ defined here is not but very close to the geometric aspect
ratio $a_{\rm geom}$, namely $a\simeq a_{\rm geom}={\bf L}_{z}/{\bf L}_{0}$
with ${\bf L}_{0}$ (${\bf L}_{z}$) being the cross section diameter (length)
of NWQD.
Figure 2: (Color online) Schematic illustration of the electronic structures,
consisting of few relevant low lying orbitals (one $s$\- and three
$p$-orbitals), of long rod-like NWQDs [(a)(c)(e)] and short disk-like NWQDs
[(b)(d)(f)] with or without longitudinal magnetic field $B$ and including or
excluding the spin Zeeman splitting $E_{\rm{Z}}$ ($g^{\ast}=0$ or
$g^{\ast}\neq 0$). (a) $a>1$ and $B=0$; (b) $a<1$ and $B=0$; (c) $a>1$, $B\neq
0$ and $g^{\ast}=0$; (d)$a<1$, $B\neq 0$ and $g^{\ast}=0$; (e)$a>1$, $B\neq 0$
and $g^{\ast}\neq 0$; (f)$a<1$, $B\neq 0$ and $g^{\ast}\neq 0$.
Figure 1 presents the calculated single-electron energy spectrum as a function
of aspect ratio $a$ for a NWQD with fixed lateral confinement
$\hbar\omega_{0}=13.3$ meV at zero magnetic field according to Eq.(3). The
chosen parameter of lateral confinement $\hbar\omega_{0}=13.3$ meV is
determined by fitting the numerically calculated energy separation between the
two lowest single-electron orbitals of a cylindrical InAs/InP NWQD of cross
section diameter ${\bf L}_{0}=65$ nm by 3D finite difference simulation. In
the simulation, the $\rm{Schr\ddot{o}dinger}$ equation for a single electron
confined in a 3D cylindrical potential well is solved by using finite
difference method, with the used parameters: the effective mass
$m^{\ast}=0.023m_{0}$ of electron for InAs and the InAs/InP band edge offset
$V_{b}=0.6$ eV as the barrier height of the confining potential. Bjork05 ;
Bjork02
In a two-electron (2e) problem, the most relevant orbitals are the two lowest
ones because the kinetic energy difference between the two orbitals is the
main energy cost, in competition with the coulomb or spin Zeeman energies, for
a spin triplet state to be the ground state of two-electron. By convention, we
from now on name the lowest single electron state
$|n,m,q\rangle=|0,0,0\rangle$ as $s$-orbital, and the next three $p$-shell
states $|0,0,1\rangle$, $|1,0,0\rangle$, and $|0,1,0\rangle$ as $p^{0}$-,
$p^{+}$-, and $p^{-}$-orbitals, respectively. According to Eq. (3), the energy
of the lowest $s$-orbital is explicitly given by
$\epsilon_{s,s_{z}}=\frac{1}{2}\left(\hbar\omega_{+}+\hbar\omega_{-}+\hbar\omega_{z}\right)+g^{\ast}\mu_{B}Bs_{z},$
(8)
and those of the three p-shell orbitals are respectively given by
$\displaystyle\epsilon_{p^{0},s_{z}}=\epsilon_{s,s_{z}}+\hbar\omega_{z},$
$\displaystyle\epsilon_{p^{+},s_{z}}=\epsilon_{s,s_{z}}+\hbar\omega_{+},$
$\displaystyle\epsilon_{p^{-},s_{z}}=\epsilon_{s,s_{z}}+\hbar\omega_{-}.$ (9)
For $B=0$, we have
$\epsilon_{s,s_{z}}=\hbar\omega_{0}\left(1+1/2a^{2}\right)$,
$\epsilon_{p^{0},s_{z}}=\epsilon_{s,s_{z}}+\hbar\omega_{0}/a^{2}$, and
$\epsilon_{p^{+},s_{z}}=\epsilon_{p^{-},s_{z}}=\epsilon_{s,s_{z}}+\hbar\omega_{0}$
according to Eqs.(7) and (9). Here, the $p^{+}$\- and $p^{-}$-orbitals are
degenerate with the same energy separation from the $s$-orbital,
$\hbar\omega_{\pm}=\hbar\omega_{0}$, while the $p^{0}$-orbital is
energetically higher than $s$-orbital by
$\hbar\omega_{z}=\hbar\omega_{0}/a^{2}$. Obviously, $p^{0}$ ($p^{\pm}$) is the
second lowest orbital for a long (short) NWQD with $a>1$ ($a<1$) at zero
magnetic field, as shown in Fig. 1. For a symmetric NWQD with $a=1$, the
$p^{0}$\- and $p^{\pm}$-orbitals form a 3-fold orbital-degenerate shell.
Figure 2 (a) [(b)] schematically depicts the low-lying orbitals of a long
[short] NWQD with $a>1$ [$a<1$] at zero magnetic field.
Applying a longitudinal magnetic field onto a cylindrical NWQD breaks the
degeneracy of $p^{+}$\- and $p^{-}$-orbitals. The orbital Zeeman effect lowers
(raises) the energy level of the $p^{-}$($p^{+}$)-orbital from
$\hbar\omega_{0}$ to $\hbar\omega_{-}$ ($\hbar\omega_{+}$). Thus, if a long
NWQD is subjected to a sufficiently strong magnetic field, the second lowest
orbital of the dot could be changed from the $p^{0}$ to $p^{-}$. By contrast,
the second lowest orbital of a short NWQD is always the $p^{-}$-orbital.
Therefore, the characteristic energy quantization of the $p^{-}$-orbitals,
$\hbar\omega_{-}=\hbar\left(\omega_{0}^{2}+\omega_{c}^{2}/4\right)^{1/2}-\hbar\omega_{c}/2$,
is often a key parameter for a short NWQD or a moderately long NWQD with
strong magnetic field. Considering wide-band gap materials such as GaAs, the
$g$-factors are usually small and the spin Zeeman effect on the energy shift
of orbital is negligible. Figure 2(c) [(d)] depicts the $B$-dependent
electronic orbitals of a long [short] NWQDs, where vanishing spin Zeeman
splitting is assumed ($g^{\ast}=0$ is set).
For a low energy gap material with larger $g^{\ast}$, like InAs, the spin
Zeeman effect could be significant in the spin ST transition of two-electron
QD. Figure 2(e) [(f)] schematically shows the spin-resolved electronic
orbitals of a long [short] NWQDs with $B\neq 0$ and $g^{\ast}\neq 0$ by the
spin Zeeman splitting $2E_{\rm Z}$. With the spin Zeeman effect, all the spin-
up (spin-down) orbitals are energetically lowered (raised) by $E_{\rm
Z}=g^{\ast}\mu_{B}B/2$ according to Eq. (3). If the applied magnetic field or
the $g$-factor of material is so large that the spin Zeeman splittings exceed
the kinetic energy quantization of QD, both of the two lowest single-electron
states are the spin-up ones and the ground state of the 2e dot is ensured to
be a spin triplet state simply according to spin Pauli exclusion principle.
In this work, the following formulation for the $g$-factor of an InAs-based QD
is adopted Hermann77 ; Bjork05
$g^{\ast}=g\left[1-\frac{P^{2}}{3}\frac{\Delta_{\rm SO}}{E_{g}^{\rm
eff}\left(E_{g}^{\rm eff}+\Delta_{\rm SO}\right)}\right]\,,$ (10)
where $E_{g}^{\rm eff}$ is the effective energy gap of semiconductor QD,
$g=2.0$ is the Lande $g$-factor for free electron, $\Delta_{\rm SO}$ is the
spin-orbit splitting in the valence band, and $P$ is the parameter of
interband transition matrix element. Hermann77
Here, the effective energy gap of a QD can be estimated as $E_{g}^{\rm
eff}=E_{g}^{\rm bulk}+\epsilon_{s,s_{z}}$, where $E_{g}^{\rm bulk}$ is the
bulk energy gap and $\epsilon_{s,s_{z}}$ is the quantization energy of the
lowest electronic orbital of the QD with $B=0$ measured from the conduction
band edge. For InAs-based QDs, we take the following parameter values:
$E_{g}^{\rm bulk}=460$ meV, $\Delta_{\rm SO}=390$ meV, $P^{2}=21.5$ eV.
Bjork05 Accordingly, the value of $g^{\ast}$ for a symmetric NWQD with ${\bf
L}_{0}={\bf L}_{z}=65$ nm is estimated as large as $g^{\ast}\approx-11$.
Bjork05
### II.2 Interacting few-electron model
The interacting Hamiltonian of few electrons in a NWQD can be expressed in the
form of second quantization as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i,\sigma}\epsilon_{i\sigma}c_{i\sigma}^{{\dagger}}c_{i\sigma}$
(11) $\displaystyle+\frac{1}{2}\sum_{ijkl,\sigma\sigma^{\prime}}\langle
ij|V|kl\rangle
c_{i\sigma}^{{\dagger}}c_{j\sigma^{\prime}}^{{\dagger}}c_{k\sigma^{\prime}}c_{l\sigma}\,,$
where $i,j,k,l$ denote the composite indices of single electron orbitals such
as $|i\rangle=|n_{i},m_{i},q_{i}\rangle$, $c_{i\sigma}^{{\dagger}}$
($c_{i\sigma}$) the electron creation (annihilation) operators, and
$\sigma=\pm$ the electron spins $s_{z}=\pm\frac{1}{2}$. The first (second)
term on the right hand side of Eq.(11) represents the kinetic energy of
electrons (the Coulomb interactions between electrons) and the Coulomb matrix
elements are defined as
$\displaystyle\langle ij|V|kl\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{e^{2}}{4\pi\kappa}\int\int d{\bf r_{1}}d{\bf
r_{2}}\psi_{i}^{*}({\bf r_{1}})\psi_{j}^{*}({\bf r_{2}})$ (12)
$\displaystyle\times\frac{1}{|{\bf r_{1}}-{\bf r_{2}}|}\psi_{k}({\bf
r_{2}})\psi_{l}({\bf r_{1}})\,,$
where $\kappa$ is the dielectric constant of dot material. For InAs material,
we take $\kappa=15.15$. After lengthy derivation, one can obtain the
generalized Coulomb matrix elements for the case of $a\geq 1$:
$\displaystyle\langle
n_{i}m_{i}q_{i};n_{j}m_{j}q_{j}|V|n_{k}m_{k}q_{k};n_{l}m_{l}q_{l}\rangle=$
$\displaystyle\left(\frac{1}{\pi
l_{h}}\right)\frac{\delta_{R_{L},R_{R}}\cdot\delta_{q_{i}+q_{j}+q_{l}+q_{k},\rm{even}}}{\sqrt{n_{i}!m_{i}!q_{i}!n_{j}!m_{j}!q_{j}!n_{k}!m_{k}!q_{k}!n_{l}!m_{l}!q_{l}!}}$
$\displaystyle\times\sum_{p_{1}=0}^{\min(n_{i},n_{l})}\sum_{p_{2}=0}^{\min(m_{i},m_{l})}\sum_{p_{3}=0}^{\min(q_{i},q_{l})}\sum_{p_{4}=0}^{\min(n_{j},n_{k})}\sum_{p_{5}=0}^{\min(m_{j},m_{k})}\sum_{p_{6}=0}^{\min(q_{j},q_{k})}\
p_{1}!p_{2}!p_{3}!p_{4}!p_{5}!p_{6}!$ $\displaystyle\times{n_{i}\choose
p_{1}}{n_{l}\choose p_{1}}{m_{i}\choose p_{2}}{m_{l}\choose
p_{2}}{q_{i}\choose p_{3}}{q_{l}\choose p_{3}}{n_{j}\choose
p_{4}}{n_{k}\choose p_{4}}{m_{j}\choose p_{5}}{m_{k}\choose
p_{5}}{q_{j}\choose p_{6}}{q_{k}\choose p_{6}}$
$\displaystyle\times(-1)^{u+v/2+n_{j}+m_{j}+q_{j}+n_{k}+m_{k}+q_{k}}{\left(\frac{1}{2}\right)^{u}}x^{u+1/2}$
$\displaystyle\times\frac{\Gamma(\frac{1+2u+v}{2})\Gamma(1+u)\Gamma(\frac{1+v}{2})}{\Gamma(\frac{3+2u+v}{2})}{{}_{2}F_{1}}\left(1+u,\frac{1+2u+v}{2};\frac{3+2u+v}{2};1-x\right)\,,$
(13)
where we define $u=m_{i}+m_{j}+n_{l}+n_{k}-(p_{1}+p_{2}+p_{4}+p_{5})$,
$v=(q_{i}+q_{l}+q_{j}+q_{k})-2(p_{3}+p_{6})$,
$R_{L}=(m_{i}+m_{j})-(n_{i}+n_{j})=-(\ell_{z,i}+\ell_{z,j})$,
$R_{R}=(m_{l}+m_{k})-(n_{l}+n_{k})=-(\ell_{z,l}+\ell_{z,k})$,
$x\equiv\omega_{z}/\omega_{h}$, and ${}_{2}F_{1}$ is the hypergeometric
function. The $\delta$-functions $\delta_{q_{i}+q_{j}+q_{l}+q_{k},\rm{even}}$
and $\delta_{R_{L},R_{R}}$ in the formulation ensure the conservation of the
parity with respect to $z$-axis and the $z$-component of angular momentum of
system $L_{z}$, respectively. The formulation of Eq. (13) is confirmed by
computing the Coulomb integral numerically.
For short NWQDs with $a<1$, the formulations of the Coulomb matrix elements
are obtained by simply taking Euler’s hypergeometric transformation for the
hypergeometric function in Eq. (13), i.e., replacing
${{}_{2}F_{1}}\left(1+u,\frac{1+2u+v}{2};\frac{3+2u+v}{2};1-x\right)\,$
by
$x^{-\frac{1+2u+v}{2}}{{}_{2}F_{1}}\left(\frac{1+v}{2},\frac{1+2u+v}{2};\frac{3+2u+v}{2};1-\frac{1}{x}\right)\,.$
The generalized formulations for the Coulomb matrix elements based on the 3D
asymmetric parabolic model are probably for the first time derived, which
allows for straightforward implementation of the CI theory and is widely
applicable to arbitrary 3D confining semiconductor nanostructures.
### II.3 Exact diagonalization
Based on the CI theory presented above, we follow the standard numerical exact
diagonalization procedure to calculate the energy spectrum of $N_{e}$
interacting electrons in a NWQD . Hawrylak03 The numerically exact results
are obtained by increasing the numbers of chosen single electron orbital basis
and the corresponding $N_{e}$-electron configurations until a numerical
convergence is achieved. In the full configuration interaction (FCI)
calculation for a 2e problem, we usually take the number of single electron
orbitals typically from $20$ to $26$ and that of the corresponding 2e
configurations from $190$ to $325$ to have a satisfactory numerical
convergence.
## III Numerical results and discussion
### III.1 Magnetic-field induced ST transitions
Figure 3: (Color online) Magneto-energy spectrum of two interacting electrons
in a NWQD with transverse confining strength $\hbar\omega_{0}=13.3$ meV and
aspect ratio $a=3$.
Let us first consider two interacting electrons in a rod-like NWQD with the
aspect ratio $a=3$ and the transverse confining strength
$\hbar\omega_{0}=13.3$ meV using FCI calculation. The low-lying magneto-energy
spectrum of the two-electron NWQD is shown in Fig. 3, which consists of a spin
singlet state branch, labeled by $\rm{S}$, and three triplet state branches
split by the spin Zeeman energy, labeled by $\rm{T_{L0}^{+}}$,
$\rm{T_{L0}^{0}}$ and $\rm{T_{L0}^{-}}$ according to the z-component of total
spin ($S_{z}=+1$, $S_{z}=0$ and $S_{z}=-1$), respectively. Fasth07 ; Hanson07
Since usually only triplet states with $S_{z}=+1$ are involved in ST
transitions, we shall use $\rm T_{\rm L\it{\left|L_{z}\right|}}$ to denote the
triplet states with angular momentum $L_{z}$ through out this article,
skipping the superscript $+$ of $\rm{T_{L\it{\left|L_{z}\right|}}^{+}}$ for
brevity.
The main configurations of the two-electron ground states around the critical
magnetic field are schematically shown in the lower right corner of Fig. 3. In
the weak magnetic field regime $\left(B<B_{\rm{ST_{L0}}}\sim 0.9~{}\rm
T\right)$, the two electrons in the NWQD mainly occupy the lowest $s$-orbital
simply following the Aufbau principle, and form a spin singlet ground state.
With increasing $B$, the triplet state $\rm{{T_{L0}}}$ is more energetically
favorable than the singlet state because of the increasing spin Zeeman energy,
the reduced Coulomb repulsion, and exchange energy between the two spin
polarized electrons. A crossing of the singlet branch and the triplet state
branch $\rm{T_{L0}}$ is observed at the critical magnetic field
$B_{\rm{ST_{L0}}}=0.9$ T. Such magnetic-field induced ST transitions are
attributed to the energetic competition between single particle energy
quantization, the spin Zeeman energy, and the various Coulomb interactions
including the direct, exchange, and correlation interactions as well.
Hawrylak93-prl
Other weak spin-related terms, such as the spin-orbital coupling (SOC) with
1-2 order of magnitude smaller than the kinetic quantization of QD are
neglected in the Hamiltonian of Eq.(11). The SOC mixes the spin of the
$\rm{S}$ and $\rm{{T_{L0}}}$ states and creates an anti-crossing of the S- and
$\rm{{T_{L0}}}$-branches around the $B_{ST}$ with a small energy gap,
typically only $\sim 0.1-0.5$ meV as observed in previous experiments. Fasth07
### III.2 Spin phase diagram
Figure 4: (Color online) Spin phase diagrams of two-electron NWQDs of lateral
confinement $\hbar\omega_{0}=13.3$ meV with respect to tunable magnetic field
$B$ and aspect ratio $a$. The phases are distinguished by the curves of
critical magnetic field $B_{\rm{ST}}$ obtained from non-interacting (black
dotted), PCI (blue dashed), and FCI (red solid) calculations.
Figure 4 shows the calculated spin phase diagrams of the two-electron ground
state of the NWDQs with a fixed cross section diameter (fixed
$\hbar\omega_{0}=13.3$ meV) but various lengths (various $\hbar\omega_{z}$)
with respect to the applied magnetic field $B$ and the aspect ratio $a$. Three
phases ($\rm S$, $\rm{T_{L0}}$, and $\rm{T_{L1}}$) are distinguished by the
curves of critical magnetic field $B_{\rm{ST}}$ in Fig. 4. Correspondingly,
the main configurations of the 2e ground states are depicted inside the
colored regions of the phases. To identify the various underlying mechanisms
in the phase diagrams, including the spin Zeeman effect and the inter-particle
Coulomb interactions, the spin phase diagrams are calculated by using non-
interacting, full CI, and partial CI calculations, respectively.
In the non-interacting calculation, the coulomb interactions are artificially
disabled and the considered ST transitions are induced only by the spin Zeeman
effect. The comparison between the results of non-interacting and FCI
calculations allows us to distinguish effects of the Coulomb interaction and
spin Zeeman coupling on the ST transitions. In particular, to highlight the
Coulomb correlation effect, a partial configuration interaction (PCI)
calculation is also performed for the spin phase diagrams, in which only the
lowest energy configuration is taken as the sole basis and the couplings from
higher energy configurations are excluded.
The essential features of the phase diagrams can be realized based on the non-
interacting picture. For a not very long (small or moderate $a$) NWQD with
weak $B$, the 2e ground state is likely to be the spin singlet state $\rm S$,
simply following Aufbau principle (the yellow region in Fig. 4). Starting from
the singlet phase $\rm S$, the two-electron ground state of a NWQD might be
switched to the spin triplet phases (the pink region $\rm{T_{L0}}$ or the cyan
region $\rm{T_{L1}}$) by increasing either $a$ or $B$ (see the horizontal and
vertical dashed lines with arrows in Fig. 4 for the guidance of eyes).
Following the horizontal dashed line, the longitudinal energy quantization
$\hbar\omega_{z}$ is decreased by the increase of $a$. With the addition of
spin Zeeman term, the spin-up level of $p^{0}$-orbital could become even lower
than the spin-down level of $s$-orbital if the decreasing $\hbar\omega_{z}$ is
so small as that $\hbar\omega_{z}<2|E_{z}|$ (see the difference between the
schematic configurations for the $\rm S$ and $\rm{T_{L0}}$ states). In this
situation, the 2e ground state can transit to the spin triple states
$\rm{T_{L0}}$, simply following spin Pauli exclusion principle. On the other
hand, the transition of a 2e ground state of NWQD from the singlet state $\rm
S$ to the triple one $\rm{T_{L1}}$ is shown also possible by increasing the
strength of applied magnetic field. Following the vertical dashed line,
increasing $B$ reduce the energy separation between the $s$\- and
$p^{-}$-orbital levels, i.e. $\hbar\omega_{-}$. Similar to the case of
$\rm{S}$-$\rm{T_{L0}}$ transition, a $\rm S$-$\rm{T_{L1}}$ transition can
happen as the decreased $\hbar\omega_{-}$ is so small as that
$\hbar\omega_{-}<2|E_{z}|$ . In the non-interacting picture, $B_{\rm ST_{L0}}$
is explicitly given by $B_{\rm ST_{L0}}=\hbar\omega_{0}/g^{\ast}\mu_{B}a^{2}$,
showing a quadratic decay with $a$, while the critical magnetic field $B_{\rm
ST_{L1}}$ for $\rm{S}$-$\rm{T_{L1}}$ transitions is dependent only on
$\hbar\omega_{0}$ and remains nearly constant in the $B-a$ plot.
The Coulomb interactions are shown to reduce the singlet phase area in the
diagrams from the comparison between the non-interacting and CI results. For
example, the segment of vertical solid line at $a=3$ in Fig. 4 indicates that
the critical magnetic field is significantly reduced from $B_{\rm
ST_{L0}}(\rm{Non-interacting})=3.2$ T to $B_{\rm ST_{L0}}(\rm{FCI})=0.9$ T as
the Coulomb interactions are taken into account. This is because the spin
triplet states gain additional negative exchange energies while the singlet
state does not. We also notice that the $B_{\rm ST_{L1}}$ for the
$\rm{S}$-$\rm{T_{L1}}$ transition no longer remains constant but slightly
increases with increasing $a$ because the strength of the coulomb interactions
is reduced by the increase of dot volume.
Basically, the results obtained from the FCI and PCI calculations have similar
features except for those in the regime of high $a$ ($a>3$). While the PCI
calculation shows the vanishing $B_{\rm ST_{L0}}$ for $a\sim 3$, the FCI
calculation yields the always non-zero $B_{\rm ST_{L0}}$. This means that the
Coulomb correlations energetically favor the spin singlet state as ground
state and become more pronounced in long NWQDs.
### III.3 Crossover from disk-like to rod-like QDs
The spin phase diagrams of Fig. 4 suggest that purposely accessing a specific
spin phase of two-electron is feasible through the geometrical control of
NWQDs. For instance, the ground state of a two-electron NWQD can be switched
from the singlet $\rm{S}$ to the triplet state $\rm{T_{L0}}$ by increasing the
aspect ratio $a$ at the fixed $B=5$ T (trace the horizontal dashed line in
Fig. 4).
Figure 5: (Color online) Spin phase diagrams of doubly charged NWQDs with
respect to the lateral and longitudinal confinements, parametrized by
$\hbar\omega_{0}$ and $\hbar\omega_{z}$, respectively, in a fixed magnetic
field $B=5$ T for (a) non-interacting two electrons with $g^{\ast}\neq 0$, (b)
interacting two electrons with $g^{\ast}\neq 0$ and (c) interacting two
electrons with $g^{\ast}=0$.
Figure 5 presents the spin phase diagrams of two-electron NWQDs with respect
to the lateral and longitudinal confinements, parametrized by
$\hbar\omega_{0}$ and $\hbar\omega_{z}$, respectively, in a fixed magnetic
field $B=5$ T for (a) non-interacting two electrons with $g^{\ast}\neq 0$, (b)
interacting two electrons with $g^{\ast}\neq 0$, and (c) interacting two
electrons with $g^{\ast}=0$. In Figs. 6(a) and (b), we present the relevant
two-electron configurations to the spin phase diagrams of Figs. 5(a) and (b)
with the inclusion of spin Zeeman effect $\left(g^{\ast}\neq 0\right)$, while
in Figs. 6(c) and (d) we present the relevant two-electron configurations to
the spin phase diagrams of Fig. 5(c) for $g^{\ast}=0$
Figure 6: (Color online) Relevant two-electron configurations possibly being
the main components in the ground states of NWQDs in an uniform magnetic field
$B=5$ T for (a) $a>1$ and $g^{\ast}\neq 0$, (b) $a<1$ and $g^{\ast}\neq 0$,
(c) $a>1$ and $g^{\ast}=0$, and (d) $a<1$ and $g^{\ast}=0$.
The non-interacting spin phase diagram is first shown in Fig. 5(a) in order to
identify the spin Zeeman effect and also contrast the Coulomb interaction
effects on the interacting spin phase diagrams presented in Fig. 5(b). In the
non-interacting case, the features of the spin phases of Fig. 5(a) are purely
determined by the competition between geometry-dependent quantized electronic
structures of dots and the spin Zeeman splitting, which is nearly a constant
here created by the fixed $B$. Three distinctive spin phases, $\rm{S}$,
$\rm{T_{L0}}$, and $\rm{T_{L1}}$, are marked in different colors in Fig. 5(a).
In the yellow region where both $\hbar\omega_{0}$ ($\hbar\omega_{-}$) and
$\hbar\omega_{z}$ are large, the kinetic quantizations in both longitudinal
and lateral directions are stronger than the spin Zeeman splitting and
$\rm{S}$ remains as a ground state. Reducing the longitudinal confinement,
$\hbar\omega_{z}$, can lead to the $\rm{S}$-$\rm{T_{L0}}$ (from the yellow to
the pink region) transition as $\hbar\omega_{z}\lesssim 2|E_{\rm{Z}}|$.
Similarly, reducing the transverse confinement leads to the
$\rm{S}$-$\rm{T_{L1}}$ transition as $\hbar\omega_{-}\lesssim
2E_{\rm{Z}}$(from the yellow to the light cyan region).
Compared with Fig. 5(a), the interacting spin phase diagram of Fig. 5(b) shows
the following additional features:
(i) Larger areas of both $\rm{T_{L0}}$ and $\rm{T_{L1}}$ phases are observed
because of the additional negative exchange energies and the reduced direct
Coulomb repulsions gained by the triplet states.
(ii) A NWQD with $\hbar\omega_{0}\approx 12$ meV could experience a three-
phase transitions from $\rm{T_{L1}}$ (cyan) to $\rm{S}$ (yellow), and then to
$\rm{T_{L0}}$ (pink) with increasing the length of wire, from
$\hbar\omega_{z}>25$ meV to $\hbar\omega_{z}<5$ meV (see the vertical line
positioned at $\hbar\omega_{0}=12$ meV in Fig. 5(b)).
(iii) In the regime of small $\hbar\omega_{0}$ and large $\hbar\omega_{z}$
(i.e. flat quasi-2D dots with $a\ll 1$), a series of transitions from the spin
single states to various triplet states, $\rm{T_{L1}}$, $\rm{T_{L3}}$,
$\rm{T_{L5}}$, etc. [see Fig. 6(b)] and a staircase increase of total orbital
angular momentum are observed with reducing the lateral confinement
$\hbar\omega_{0}$.
In the weak laterally confining regime, few electrons in the quasi-2D QD in a
high magnetic field successively fill the orbitals with negative
$z$-projection of orbital angular momentum, i.e. the orbitals of lowest Landau
level (LLL), with small kinetic energy separation $\hbar\omega_{-}$. The
inter-particle Coulomb interactions thus become particularly pronounced among
the particles on the nearly degenerate LLL orbitals with alomost quenched
kinetic energies. In order to minimize the coulomb repulsion, the particles on
the quasi-degenerate orbitals tend to spread the occupancy of orbitals as far
as possible, but in competition with the cost of increase of kinetic energy.
As a result, with reducing $\hbar\omega_{0}$ or increasing $B$, the total
angular momentum of two-electron increases, as previously discussed by Wagner
et al. Wagner92 for gated 2D QDs.
Figure 5(c) shows the phase diagram of two interacting electrons calculated by
FCI method but with the vanishing spin Zeeman term, i.e. $g^{\ast}=0$. This
allows us to distinguish the effects of spin Zeeman energy and the coulomb
interactions on the spin phase diagram of Fig. 5(b), and also to study the
spin phases of QD made of a material with small $g^{\ast}$ such as GaAs.
Without spin Zeeman splitting, the significant features of Fig. 5(c) are
completely determined by the many-body effects and geometry-engineered
electronic structures of NWQDs.
In the $a>1$ regime, unlike the result shown in Fig. 5(b), the $\rm{T_{L0}}$
phase disappears and naturally there is no $\rm{S}$-$\rm{T_{L0}}$ transition
observed. This is because the Coulomb correlations that energetically favor
spin-singlet state as mentioned previously, become dominant and compensate the
negative exchange energy gained by the $\rm{T_{L0}}$ states. Reimann02 ;
Hanson07 However, in the small $\hbar\omega_{0}$ regime, an additional
singlet-triplet state oscillation with decreasing $\hbar\omega_{0}$ is
observed. Compared with Figure 5(b), the difference is the emergences of
various singlet states between the triplet phases. This is due to the removal
of spin Zeeman splittings, which energetically favor only the triplet states.
Such a singlet-triplet state oscillation is evidenced as a main feature of a
flat 2D QD with small spin Zeeman effect, as shown both theoretically Wagner92
and experimentally Tarucha07 in the previous studies.
## IV Summary
In conclusion, we present exact diagonalization studies of spin phase
transitions of two electrons confined in nanowire quantum dots with highly
tunable aspect ratio and external magnetic field. A configuration interaction
theory based on a 3D parabolic model for such three dimensionally confining
QDs is developed, which provides generalized explicit formulation of the
Coulomb matrix elements and allows for straightforward implementation of
direct diagonalization. The exact diagonalization study reveals fruitful
features of spin ST transitions with respect to the tunable geometric aspect
ratio and applied magnetic field.
For disk-like QDs, the ST transition behaviors may be dominated by the spin
Zeeman, the direct-Coulomb, and the exchange energies. The pronounced Coulomb
correlations are identified in rod-like QDs with aspect ratio $a>3$, which
energetically favor singlet spin states and yield the always non-zero critical
magnetic fields of ST transitions. The developed theory is further employed to
study spin phase diagram in the dimensional “cross over” regime from the 2D
(disk-like) QDs to finite 1D (rod-like) QDs. In the 2D disk-like QD regime,
various distinctive spin phases are emerged under the conditions of
appropriate lateral confinement strength and magnetic fields. In the rod-like
QD regime, switching the ST transitions is shown feasible by controlling both
lateral and/or longitudinal confinement strength.
## V Acknowledgement
This work was financially supported by the National Science Council in Taiwan
through Contracts No. NSC-98-2112-M-009-011-MY2 (SJC) and No.
NSC97-2112-M-239-003-MY3 (CST). The authors are grateful to the facilities
supported by the National Center of Theoretical Sciences in Hsinchu and the
National Center for High-Performance Computing in Taiwan.
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|
arxiv-papers
| 2009-12-10T06:13:10 |
2024-09-04T02:49:06.945880
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yan-Ting Chen, Shun-Jen Cheng, and Chi-Shung Tang",
"submitter": "Yan-Ting Chen",
"url": "https://arxiv.org/abs/0912.1919"
}
|
0912.1934
|
2010287-298Nancy, France 287
Paul Dütting Monika Henzinger Ingmar Weber
# Sponsored Search, Market Equilibria, and the Hungarian Method
P. Dütting , M. Henzinger and I. Weber Ecole Polytechnique Fédérale de
Lausanne, Switzerland paul.duetting,monika.henzinger,ingmar.weber@epfl.ch
University of Vienna, Austria monika.henzinger@univie.ac.at Yahoo! Research
Barcelona, Spain ingmar@yahoo-inc.com
###### Abstract.
Two-sided matching markets play a prominent role in economic theory. A prime
example of such a market is the sponsored search market where $n$ advertisers
compete for the assignment of one of $k$ sponsored search results, also known
as “slots”, for certain keywords they are interested in. Here, as in other
markets of that kind, market equilibria correspond to stable matchings. In
this paper, we show how to modify Kuhn’s Hungarian Method (Kuhn, 1955) so that
it finds an optimal stable matching between advertisers and advertising slots
in settings with generalized linear utilities, per-bidder-item reserve prices,
and per-bidder-item maximum prices. The only algorithm for this problem
presented so far (Aggarwal et al., 2009) requires the market to be in “general
position”. We do not make this assumption.
###### Key words and phrases:
stable matching, envy-free allocation, general auction mechanism, general
position
###### 1991 Mathematics Subject Classification:
F.2.2 (Nonnumerical Algorithms and Problems)
This work was conducted as part of a EURYI scheme award (see
http://www.esf.org/euryi/).
## 1\. Introduction
Two-sided matching markets play a prominent role in economic theory. A prime
example of such a market is the sponsored search market [14] where $n$
advertisers (or bidders) compete for the assignment of one of $k$ sponsored
search results, also known as “slots”, for certain keywords (or items) they
are interested in. Here, as in other markets of that kind, market equilibria
correspond to stable matchings. A stable matching that is preferred by all
bidders over all other stable matchings is bidder optimal. Mechanisms that
compute bidder optimal matchings typically provide the bidders with the
incentive to reveal their true preferences, i.e., they are truthful.
In the most basic model of a two-sided matching market, known as the stable
marriage problem [9], each bidder has a strict preference ordering over the
items and each item has a strict preference ordering over the bidders. In a
more general model, see e.g. [16], each bidder has a linear utility function
for each item that depends on the price of the item and every item can have a
reserve price, i.e., a price under which the item cannot be sold to any
bidder. In the even stronger model that we study here every bidder-item pair
can have a reserve price, i.e., a price under which the item cannot be sold to
this specific bidder, and a maximum price, i.e., a price above which this
bidder does not want to buy this specific item. We call this model the
sponsored search market. An interesting property of this model is that it
generalizes standard auction formats such as VCG [17, 4, 10] and GSP [7].
While the problem of finding a bidder optimal matching in the first two models
has been largely solved in the 60s, 70s, and 80s [9, 16, 5, 15], the problem
of finding a bidder optimal matching in the sponsored search market has been
addressed only recently [2].
The main finding of [2] is that if the market is in “general position”, then
(a) there is a unique bidder optimal matching and (b) it can be found in
$O(nk^{3})$ steps by a truthful mechanism. For a market to be in “general
position”, however, any two reserve prices and/or maximum prices must be
distinct. In practice, this will rarely be the case and so we typically have
to deal with markets that are not in general position. The authors of [2]
propose to bring such markets into “general position” using random
perturbations and/or symbolic tie-breaking. The problem with this approach,
however, is that there is no guarantee that a bidder optimal solution of the
perturbed market leads to a bidder optimal solution of the original market. In
fact, such a solution may not even exist (see Section 3). Additionally, a
pertubation-based mechanism may not be truthful.
We improve upon the results of [2] as follows: First, in Section 3, we show
how to modify the definition of stability so that a bidder optimal matching is
guaranteed to exist for arbitrary markets. Then, in Section 5, 6, and 7, we
show how to modify Kuhn’s Hungarian Method [13, 8] so that it finds a bidder
optimal matching in time $O(nk^{3}\log(k))$. Afterwards, in Section 8, we show
that with our notion of stability bidder optimality no longer implies
truthfulness, unless further restrictions are imposed on the model. Finally,
in Section 9, we show how to reduce more general linear utility functions to
our setting.111These utilities can be used to model that the click probability
in the pay-per-click model has a bidder-dependent component $c_{i}$ and an
item-dependent component $c_{j}$. See [1, 7] for details.
Independently of us Ashlagi et al. [3] also improved upon the results of [2]
by (a) showing the existence of a unique feasible, envy free, and Pareto
efficient solution for position auctions with budgets and by (b) providing a
truthful mechanism that finds it. The notion of envy-freeness is equivalent to
our notion of stability. Their model, however, is a special case of our model
as it requires a common preference ordering over the items, it does not
incorporate reserve prices, it does not allow the maximum prices to depend on
the bidder and the item, and it requires the maximum prices to be distinct.
Recently, Kempe et al. [12] presented an efficient algorithm that finds the
minimum envy-free prices (if they exist) for a given matching.
To summarize our main contributions are: (1) We show how to modify the
Hungarian Method so that it finds a bidder optimal solution for arbitrary
markets, including markets that are not in “general position”. (2) We show how
different definitions of stability affect the existence of a bidder optimal
solution. (3) We show how to reduce more general linear utility functions to
the setting that we study in this paper with no loss in performance.
## 2\. Problem Statement
We are given a set $I$ of $n$ bidders and a set $J$ of $k$ items. We use
letter $i$ to denote a bidder and letter $j$ to denote an item. For each
bidder $i$ and item $j$ we are given a valuation $v_{i,j}$, a reserve price
$r_{i,j}$, and a maximum price $m_{i,j}.$ We assume that the set of items $J$
contains a dummy item $j_{0}$ for which all bidders have a valuation of zero,
a reserve price of zero, and a maximum price of $\infty.$222Reserve utilities,
or outside options $o_{i}$, can be incorporated by setting $v_{i,j_{0}}=o_{i}$
for all bidders $i.$
We want to compute a matching $\mu\subseteq I\times J$ and per-item prices
$p=(p_{1},\dots,p_{k}).$ We require that every bidder $i$ appears in exactly
one bidder-item pair $(i,j)\in\mu$ and that every non-dummy item $j\neq j_{0}$
appears in at most one such pair. We allow the dummy item $j_{0}$ to appear
more than once. We call bidders (items) that are not matched to any non-dummy
item (bidder) unmatched. We regard the dummy item as unmatched.
We define the utility $u_{i}$ of bidder $i$ to be $u_{i}=0$ if bidder $i$ is
unmatched and $u_{i}=u_{i,j}(p_{j})$ if bidder $i$ is matched to item $j$ at
price $p_{j}.$ We set $u_{i,j}(p_{j})=v_{i,j}-p_{j}$ if $p_{j}<m_{i,j}$ and
$u_{i,j}(p_{j})=-\infty$ if $p_{j}\geq m_{i,j}$. We say that a matching $\mu$
with prices $p$ is feasible if (1) $u_{i}\geq 0$ for all $i$, (2)
$p_{j_{0}}=0$ and $p_{j}\geq 0$ for all $j\neq j_{0}$, and (3) $r_{i,j}\leq
p_{j}<m_{i,j}$ for all $(i,j)\in\mu$. We say that a feasible matching $\mu$
with prices $p$ is stable if $u_{i}\geq u_{i,j}(p_{j})$ for all $(i,j)\in
I\times J.$333Since we have $u_{i}\geq 0$ and $u_{i,j}(p_{j})=-\infty$ if
$p_{j}\geq m_{i,j}$, this definition is equivalent to requiring $u_{i}\geq
v_{i,j}-p_{j}$ for all items $j$ with $p_{j}<m_{i,j}.$ Finally, we say that a
stable matching $\mu$ with prices $p$ is bidder optimal if $u_{i}\geq
u^{\prime}_{i}$ for all $i$ and stable matchings $\mu^{\prime}$ with prices
$p^{\prime}.$
We say that an algorithm is truthful if for every bidder $i$ with utility
functions $u_{i,1}(\cdot),\dots,$ $u_{i,k}(\cdot)$ and any two inputs
$(u^{\prime}_{i,j}(\cdot),r_{i,j},m^{\prime}_{i,j})$ and
$(u^{\prime\prime}_{i,j}(\cdot),r_{i,j},m^{\prime\prime}_{i,j})$ with
$u^{\prime}_{i,j}(\cdot)=u_{i,j}(\cdot)$ for $i$ and all $j$ and
$u^{\prime}_{k,j}(\cdot)=u^{\prime\prime}_{k,j}(\cdot)$ for $k\neq i$ and all
$j$ and matchings $\mu^{\prime}$ with $p^{\prime}$ and $\mu^{\prime\prime}$
with $p^{\prime\prime}$ we have that
$u_{i,j^{\prime}}(p^{\prime}_{j^{\prime}})\geq
u_{i,j^{\prime\prime}}(p^{\prime\prime}_{j^{\prime\prime}})$ where
$(i,j)\in\mu$ and $(i,j^{\prime\prime})\in\mu^{\prime\prime}$. This definition
formalizes the notion that “lying does not pay off” as follows: Even if bidder
$i$ claims that his utility is $u^{\prime\prime}_{i,j}$ instead of $u_{i,j}$
he will not achieve a higher utility with the prices and the matching computed
by the algorithm. Thus, the algorithm “encourages truthfulness”.
## 3\. Motivation
The definition of stability in [2], which we call relaxed stability to
indicate that every stable solution is also relaxed stable (but not vice
versa), requires that for every pair $(i,j)\in I\times J$ either (a)
$u_{i}\geq v_{i,j}-\max(p_{j},r_{i,j})$ or (b) $p_{j}\geq m_{i,j}$. The
disadvantage of relaxed stability is that there can be situations where no
bidder optimal solution exists if the market is not in “general position” (see
[2] for a formal definition). Here are two canonical examples:
1. $\bullet$
Example 1. There are three bidders and two items. The valuations and reserve
prices are as follows: $v_{1,1}=1$, $v_{2,1}=4$, $v_{2,2}=4$, $v_{3,2}=1$,
$r_{1,1}=0$, $r_{2,1}=r_{2,2}=2$, and $r_{3,2}=0$. While
$\mu=\\{(1,1),(2,2)\\}$ with $p=(0,2)$ is “best” for bidder 1,
$\mu=\\{(2,1),(3,2)\\}$ with $p=(2,0)$ is “best” for bidder 3.
2. $\bullet$
Example 2. There are two bidders and one item. The valuations and maximum
prices are as follows: $v_{1,1}=10$, $v_{2,1}=10$, and $m_{1,1}=m_{2,1}=5.$
While $\mu=\\{(1,1)\\}$ with $p_{1}=5$ is “best” for bidder 1,
$\mu=\\{(2,1)\\}$ with $p_{1}=5$ is “best” for bidder 2.
unmatchedmatched$0$$2$$1$$0$$2$$4,2$$4,2$$1,0$$1,0$$2$$4,2$$1,0$$1$$0$$0$$4,2$$1,0$$2$$5$$0$$10,5$$10,5$$5$$0$$5$$10,5$$10,5$$5$
Figure 1. The left two graphs illustrate Example 1. The right two graphs
illustrate Example 2. Bidders are on the left side, items on the right side of
the bipartite graph. The numbers next to the bidder indicate her utility, the
numbers next to the item indicate its price. The labels along the edge show
valuations and reserve prices for the left two graphs and valuations and
maximum prices for the right two graphs. With relaxed stability a bidder
optimal matching does not exist.
In the market of the first example no bidder optimal solution exists as long
as there exists a bidder that has the same utility functions and reserve
prices for two items and two other bidders that are only interested in one of
the items. In the market of the second example no bidder optimal solution
exists as long as both bidders have the same maximum price and a non-zero
utility at the maximum price. Since these cases are quite general, we
conjecture that they occur rather frequently in practice.
With our notion of stability a bidder optimal solution is guaranteed to exist
(e.g. $\mu=\\{(2,1)\\}$ with $p_{1}=p_{2}=2$ in Example 1 and $\mu=\emptyset$
with $p_{1}=5$ in Example 2) for all kinds of markets, including markets that
are not in general position.
## 4\. Preliminaries
We define the _first choice graph_ $G_{p}=(I\cup J,F_{p})$ at prices $p$ as
follows: There is one node per bidder $i$, one node per item $j$, and an edge
from $i$ to $j$ if and only if item $j$ gives bidder $i$ the highest utility
possible, i.e., $u_{i,j}(p_{j})\geq u_{i,j^{\prime}}(p_{j^{\prime}})$ for all
$j^{\prime}.$ For $i\in I$ we define $F_{p}(i)=\\{j:\exists\ (i,j)\in
F_{p}\\}$ and similarly $F_{p}(j)=\\{i:\exists\ (i,j)\in F_{p}\\}$.
Analogously, for $T\subseteq I$ we define $F_{p}(T)=\cup_{i\in T}F_{p}(i)$ and
for $S\subseteq J$ we define $F_{p}(S)=\cup_{j\in S}F_{p}(j)$. Note that (1)
$p_{j}<m_{i,j}$ for all $(i,j)\in F_{p}$ and (2) if the matching $\mu$ with
prices $p$ is stable then $\mu\subseteq F_{p}.$
We define the _feasible first choice graph_ $\tilde{G}_{p}=(I\cup
J,\tilde{F}_{p})$ at prices $p$ as follows: There is one node per bidder $i$,
one node per item $j$, and an edge from $i$ to $j$ if and only if item $j$
gives bidder $i$ the highest utility possible, i.e., $u_{i,j}(p_{j})\geq
u_{i,j^{\prime}}(p_{j^{\prime}})$ for all $j^{\prime}$, and $p_{j}\geq
r_{i,j}.$ Note that $\tilde{F}_{p}\subseteq F_{p}.$ For $i\in I$ we define
$\tilde{F}_{p}(i)=\\{j:\exists\ (i,j)\in\tilde{F}_{p}\\}$ and similarly
$\tilde{F}_{p}(j)=\\{i:\exists\ (i,j)\in\tilde{F}_{p}\\}$. Analogously, for
$T\subseteq I$ we define $\tilde{F}_{p}(T)=\cup_{i\in T}\tilde{F}_{p}(i)$ and
for $S\subseteq J$ we define $\tilde{F}_{p}(S)=\cup_{j\in S}\tilde{F}_{p}(i).$
Note that (1) $r_{i,j}\leq p_{j}<m_{i,j}$ for all $(i,j)\in\tilde{F}_{p}$ and
(2) the matching $\mu$ with prices $p$ is stable if and only if
$\mu\subseteq\tilde{F}_{p}.$ Also note that the edges in
$F_{p}(i)\setminus\tilde{F}_{p}(i)$ are all the edges $(i,j)$ with maximum
$u_{i,j}(p_{j})$ but $p_{j}<r_{i,j}.$
We define an alternating path is a sequence of edges in $\tilde{F}_{p}$ that
alternates between matched and unmatched edges. We require that all but the
last item on the path are non-dummy items. The last item can (but does not
have to) be the dummy item. A tree in the feasible first choice graph
$\tilde{G}_{p}$ is an alternating tree rooted at bidder $i$ if all paths from
its root to a leaf are alternating paths that either end with the dummy item,
an unmatched item, or a bidder whose feasible first choice items are all
contained in the tree. We say that an alternating tree with root $i$ is
maximal if it is the largest such tree. See Figure 2 for an example.
$i_{1}$$i_{2}$$i_{3}$$i_{4}$$i_{5}$$i_{6}$$j_{1}$$j_{2}$$j_{3}$$j_{4}$$j_{5}$$j_{0}$$i_{1}$$j_{2}$$j_{1}$$i_{2}$$i_{3}$$j_{3}$$j_{4}$$j_{5}$$j_{0}$in
$F_{p}\setminus\tilde{F}_{p}$in $\tilde{F}_{p}$in
$\mu\cap\tilde{F}_{p}$$j_{0}$dummy item
Figure 2. The graph on the left is the (feasible) first choice graph. The
bidders $i_{1}$ to $i_{6}$ are on the left. The items $j_{1}$ to $j_{5}$ are
on the right. The dummy item is $j_{0}.$ Edges in $\mu\cap\tilde{F}_{p}$ are
thick. Edges in $\tilde{F}_{p}$ are thin. Edges in
$F_{p}\setminus\tilde{F}_{p}$ are dashed. The graph on the right is a maximal
alternating tree rooted at $i_{1}$.
## 5\. Algorithm
Our algorithm starts with an empty matching and prices all zero. It then
matches one bidder after the other by augmenting the current matching along an
alternating path. If there is no such path, it repeatedly raises the price of
all items in the maximal alternating tree under consideration by the minimum
amount (a) to make some item $j\not\in F_{p}(i)$ desirable for some bidder $i$
in the tree, or (b) to make some item $j\in F_{p}(i)\setminus\tilde{F}_{p}(i)$
feasible for some bidder $i$ in the tree, or (c) to make some item
$j\in\tilde{F}_{p}(i)$ no longer desirable for some bidder $i$ in the tree.
Thus it ensures that eventually an alternating path will exist and the
matching can be augmented. Note that a matched bidder $i$ can become unmatched
if the price of the item $j$ she is matched to reaches $m_{i,j}$. Case (a)
corresponds to $\delta_{\mbox{out}}$, Case (b) corresponds to
$\delta_{\mbox{res}}$, and Case (c) corresponds to $\delta_{\mbox{max}}$ in
the pseudocode below.
> Modified Hungarian Method
> 1 set $p_{j}:=0$ for all $j\in J$,
> $u_{i}:=\max_{j^{\prime}}v_{i,j^{\prime}}$ for all $i\in I$, and
> $\mu:=\emptyset$,
> 2 while $\exists$ unmatched bidder $i$ do
> 3 find a maximal alternating tree rooted at bidder $i$ in $\tilde{G}_{p}$
> 4 let $T$ and $S$ be the set of bidders and items in this tree
> 5 while all items $j\in S$ are matched and $j_{0}\not\in S$ do
> 6 compute
> $\delta:=\min(\delta_{\mbox{out}},\delta_{\mbox{res}},\delta_{\mbox{max}})$
> where
> 7 $\delta_{\mbox{out}}\ :=\min_{i\in T,j\not\in
> F_{p}(i)}(u_{i}+p_{j}-v_{i,j})^{~{}4}$
> 8 $\delta_{\mbox{res}}\ :=\min_{i\in T,j\in
> F_{p}(i)\setminus\tilde{F}_{p}(i)}(r_{i,j}-p_{j})$ 444We need to define
> $\min_{i\in T,j\in\emptyset}(...)=\infty$ as we might have $F_{p}(I)=J$ or
> $F_{p}(i)\setminus\tilde{F}_{p}(i)=\emptyset$.
> 9 $\delta_{\mbox{max}}:=\min_{i\in T,j\in F_{p}(i)}(m_{i,j}-p_{j})$
> 10 update prices, utilities, and matching by setting
> 11 $p_{j}:=p_{j}+\delta$ for all $j\in F_{p}(T)$ \\\ leads to a new graph
> $\tilde{G}_{p}$
> 12 $u_{i}:=\max_{j^{\prime}}(v_{i,j^{\prime}}-p_{j^{\prime}})$ for all $i\in
> I$
> 13 $\mu\ :=\mu\cap\tilde{F}_{p}$ \\\ removes unfeasible edges from $\mu$
> 14 find a maximal alternating tree rooted at bidder $i$ in $\tilde{G}_{p}$
> 15 let $T$ and $S$ be the set of bidders and items in this tree
> 16 end while
> 17 augment $\mu$ along alternating path rooted at $i$ in $\tilde{G}_{p}$
> 18 end while
> 19 output $p$, $u$, and $\mu$
## 6\. Feasibility and Stability
###### Theorem 6.1.
The Modified HM finds a feasible and stable matching. It can be implemented to
run in $O(nk^{3}\log(k))$.
###### Proof 6.2.
The matching $\mu$ constructed by the Modified HM is a subset of the feasible
first choice graph $\tilde{G}_{p}$ at all times. Hence it suffices to show
that after $O(nk^{3}\log(k))$ steps all bidders are matched.
The algorithm consists of two nested loops. We analyze the running time in two
steps: (1) The time spent in the outer loop without the inner loop (ll. 2–4
and 17–18) and (2) the time spent in the inner loop (ll. 5–16). Note that
after each execution of the outer while loop the number of matched bidder
increases by one. A matched bidder $i$ can only become unmatched if the price
of the item $j$ she is matched to reaches $m_{i,j}.$ This can happen only once
for each pair $(i,j)$, which implies that each bidder can become at most $k$
times unmatched. Thus, the outer loop is executed at most $nk$ times. Since
$|S|\leq k$, it follows that $|T|\leq k.$ Thus it is straightforward to
implement the outer while loop in time $O(k^{2}).$
We call an execution of the inner while loop special if (a) right before the
start of the execution the outer while loop was executed, (b) in the previous
iteration of the inner while loop the maximum price of a pair $(i,j)$ was
reached, or (c) the reserve price of a pair $(i,j)$ was reached. As each of
these cases can happen at most $nk$ times, there are at most $3nk$ special
executions of the inner while loop. Non-special executions increase the number
of items in the maximal alternating tree by at least one. Thus there are at
most $k$ non-special executions between any two consecutive special
executions. We present next a data structure that (1) can be built in time
$O(k^{2})$ and (2) allows to implement all non-special executions of the inner
while loop between two consecutive special iterations in time $O(k^{2}\log
k)$. Thus the total time of the algorithm is $O(nk^{3}\log k).$
Data structure:
1. (1)
Keep a list of all bidders in $T$ and a bit vector of length $n$ where bit $i$
is set to 1 if bidder $i$ belongs currently to $T$ and to 0 otherwise. Keep a
list of all items in $S$ and bit vector of length $k$, where bit $j$ is set of
1 if item $j$ belongs currently to $S$ and to 0 otherwise. Finally also keep a
list and a bit vector of length $k$ representing all items in $F_{p}(T).$
2. (2)
Keep a heap $H_{\mbox{out}}$ and a value $\delta_{\mbox{out}}$, such that
$H_{\mbox{out}}$ stores $x_{i}+p_{j}-v_{i,j}$ for all pairs $(i,j)$ with $i\in
T$ and $j\not\in F_{p}(i)$ and $\delta_{\mbox{out}}+x_{i}$ equals $u_{i}$ for
every $i\in T.$ Keep a heap $H_{\mbox{res}}$ and a value
$\delta_{\mbox{res}}$, such that $H_{\mbox{res}}$ stores $r_{i,j}-y_{j}$ for
all pairs $(i,j)$ with $i\in T$ and $j\in F_{p}(i)\setminus\tilde{F}_{p}(i)$
and $\delta_{\mbox{res}}+y_{j}$ equals $p_{j}$ for every $j\in
F_{p}(i)\setminus\tilde{F}_{p}(i)$. Keep a heap $H_{\mbox{max}}$ and a value
$\delta_{\mbox{max}}$, such that $H_{\mbox{max}}$ stores $m_{i,j}-y_{j}$ for
all pairs $(i,j)$ with $i\in T$ and $j\in F_{p}(i)$ and
$\delta_{\mbox{max}}+y_{j}$ equals $p_{j}$ for every $j\in F_{p}(i).$
3. (3)
We also store at each bidder $i$ its current $u_{i}$, at each item $j$ its
current $p_{j}.$ Thus given a pair $(i,j)$ we can decide in constant time
whether $u_{i}=v_{i,j}-p_{j}$, i.e., whether $j\in F_{p}(i).$ Finally we keep
a list of edges in $\mu.$
At the beginning of each special execution of the inner while loop a list of
bidders and items currently in $T$ and $S$ are passed in either from the
preceding execution of the outer while loop (where $T$ and $S$ are constructed
in time $O(k^{2})$) or from the previous execution of the inner while loop.
Recall that $|S|\leq k$ and thus $|T|\leq k.$ Thus we can build the above data
structures from scratch in time $O(k^{2})$ as follows. To initialize the bit
vector for $T$ we use the following approach: At the beginning of the
algorithm the vector is once initialized to 0, taking time $O(n).$ Then at the
beginning of all but the first special execution of the inner while loop the
bit vector is “cleaned” by setting the bit of all elements of $T$ in the
previous iteration to 0 using the list of elements of $T$ of the previous
iteration. Then the list of elements currently in $T$ is used to set the
appropriate bits to 1. This takes time $O(k)$ per special execution. The bit
vector of items in $S$ has only $k$ entries and thus is simply initialized to
0 at the beginning of each special execution. Then the list of elements
currently in $S$ is used to set the appropriate bits to 1. Given the list of
bidders in $T$ we decide in constant time for each pair $(i,j)$ with $i\in T$
into which heap(s) its appropriate values should be inserted. If $j\in
F_{p}(i)$ we also add $j$ to $F_{p}(T)$ if it is not already in this set
update the bit vector and the list. When we have processed all pairs $(i,j)$
with $i\in T$ we build the three heaps in time linear in their size such that
all $\delta$ values are 0. Since $|S|=k$ we know that $|T|=k.$ Thus, the
initialization takes time $O(k^{2}).$
To implement each iteration of the inner while loop we first perform a find-
min operation on all three heaps to determine $\delta$. Then we remove all
heap values that equal $\delta.$ Afterwards we update the price of all items
in $F_{p}(T)$ using the list of $F_{p}(T)$. We also update the utility of all
items in $T$ as follows. If $\delta\not=\delta_{\mbox{max}}$ updating the
utilities is just a simple subtraction per bidder. If
$\delta=\delta_{\mbox{max}}$, i.e., $p_{j}$ becomes $m_{i,j}$ for some pair
$(i,j)$, then updating $u_{i}$ requires computing $v_{i,j}-p_{j}$ for all $j$
and potentially removing the edge $(i,j)$ from $\mu$, which in turn might cut
a branch of the alternating tree. Thus, in this case we completely rebuild the
alternating tree, including $S$, $T$, and $F_{p}(T)$ from scratch. Note
however that this can only happen in a special execution of the inner while
loop. If $\delta\not=\delta_{\mbox{max}}$ the elements removed from the heaps
tell us which new edges are added to $\tilde{F}_{p}(T)$ and which new items to
add to $F_{p}(T)$. The new items in $F_{p}(T)$ gives a set of items from which
we start to augment the alternating tree in breadth first manner. For each new
item $j$, we add to $\tilde{F}_{p}(T)$ the bidder it is matched to as new
bidder to $S$ and to $\tilde{F}_{p}(T)$. For each new bidder $i$ added to
$\tilde{F}_{p}(T)$ we spend time $O(k)$ to determine its adjacent edges in
$F_{p}(i)$ and insert the suitable values for the pairs $(i,j)$ into the three
heaps. This process repeats until no new items and no new bidders are added to
$F_{p}(i)$. During this traversal we also update the bit vectors and lists
representing $T$, $S$, and $F_{p}(T).$ Let $T_{\mbox{new}}$ be the set of
bidders added to $T$ during an execution of the inner while loop and let $r$
be the number of elements removed from the heaps during the execution. Then
the above data structures implement the inner while loop in time $O(r*\log
k+|T_{\mbox{new}}|*k.)$ Now note that during a sequence of non-special
executions of the inner while loop between two consecutive special executions
bidders are never removed from $T$ and each $(i,j)$ pair with $i\in T$ is
added (and thus also removed) at most once from each heap. Thus the total
number of heap removals during all such non-special executions is $3k^{2}$ and
the total number of elements added to $T$ is $k$, giving a total running time
of $O(k^{2}\log k)$ for all such non-special executions. Since there are at
most $3nk$ special executions, the total time for all inner while loops is
$O(nk^{3}\log k).$
## 7\. Bidder Optimality
###### Theorem 7.1.
The Modified HM finds a bidder optimal matching in $O(nk^{3}\log(k))$ steps.
We say that a (possibly empty) set $S\subseteq J$ is _strictly overdemanded_
for prices $p$ wrt $T\subseteq I$ if (i) $\tilde{F}_{p}(T)\subseteq S$ and
(ii) $\forall\ R\subseteq S$ and $R\neq\emptyset:|\tilde{F}_{p}(R)\cap
T|>|R|$. Using Hall’s Theorem [11] one can show that a feasible and stable
matching exists for given prices $p$ if and only if there is no strictly
overdemanded set of items $S$ in $\tilde{F}_{p}.$
The proof strategy is as follows: In Lemma 7.2 we show that a feasible and
stable matching $\mu$ with prices $p$ is bidder optimal if we have that
$p_{j}\leq p^{\prime}_{j}$ for all items $j$ and all feasible and stable
matchings $\mu^{\prime}$ with prices $p^{\prime}.$ Afterwards, in Lemma 7.4,
we establish a lower bound on the price increase of strictly overdemanded
items. Finally, in Lemma 7.6 we argue that whenever the Modified HM updates
the prices it updates the prices according to Lemma 7.4. This completes the
proof.
###### Lemma 7.2.
If the matching $\mu$ with prices $p$ is stable and $p_{j}\leq p^{\prime}_{j}$
for all $j$ and all stable matchings $\mu^{\prime}$ with prices $p^{\prime}$,
then the matching $\mu$ with prices $p$ is bidder optimal.
###### Proof 7.3.
For a contradiction suppose that there exists a feasible and stable matching
$\mu^{\prime}$ with prices $p^{\prime}$ such that $u^{\prime}_{i}>u_{i}$ for
some bidder $i.$ Let $j$ be the item that bidder $i$ is matched to in $\mu$
and let $j^{\prime}$ be the item that bidder $i$ is matched to in
$\mu^{\prime}$. Since $p_{j^{\prime}}\leq p^{\prime}_{j^{\prime}}$ and
$p^{\prime}_{j^{\prime}}<m_{i,j^{\prime}}$ we have that
$u_{i,j^{\prime}}(p_{j^{\prime}})=v_{i,j^{\prime}}-p_{j^{\prime}}$. Since the
matching $\mu$ with prices $p$ is stable we have that
$u_{i}=u_{i,j}(p_{j})=v_{i,j}-p_{j}\geq
u_{i,j^{\prime}}(p_{j^{\prime}})=v_{i,j^{\prime}}-p_{j^{\prime}}.$ It follows
that
$u^{\prime}_{i}=v_{i,j^{\prime}}-p^{\prime}_{j^{\prime}}>u_{i}=v_{i,j}-p_{j}\geq
v_{i,j^{\prime}}-p_{j^{\prime}}$ and, thus,
$p^{\prime}_{j^{\prime}}<p_{j^{\prime}}$. This gives a contradiction.
###### Lemma 7.4.
Given $p=(p_{1},\dots,p_{k})$ let $u_{i}=\max_{j}u_{i,j}(p_{j})$ for all $i.$
Suppose that $S\subseteq J$ is strictly overdemanded for prices $p$ with
respect to $T\subseteq I$ and let
$\delta=\min(\delta_{\mbox{out}},\delta_{\mbox{res}},\delta_{\mbox{max}})$,
where $\delta_{\mbox{out}}=\min_{i\in T,j\not\in
F_{p}(i)}(u_{i}+p_{j}-v_{i,j})$, $\delta_{\mbox{res}}=\min_{i\in T,j\in
F_{p}(i)\setminus\tilde{F}_{p}(i)}(r_{i,j}-p_{j})$, and
$\delta_{\mbox{max}}=\min_{i\in T,j\in F_{p}(i)}(m_{i,j}-p_{j}).$ Then, for
any stable matching $\mu^{\prime}$ with prices $p^{\prime}$ with
$p^{\prime}_{j}\geq p_{j}$ for all $j$, we have that $p^{\prime}_{j}\geq
p_{j}+\delta$ for all $j\in F_{p}(T).$
###### Proof 7.5.
We prove the claim in two steps. In the first step, we show that
$p^{\prime}_{j}\geq p_{j}+\delta$ for all $j\in\tilde{F}_{p}(T)$. In the
second step, we show that $p^{\prime}_{j}\geq p_{j}+\delta$ for all $j\in
F_{p}(T)\setminus\tilde{F}_{p}(T)$.
Step 1. Consider the set of items $A=\\{j\in\tilde{F}_{p}(T)\ |\ \forall
k\in\tilde{F}_{p}(T):p^{\prime}_{j}-p_{j}\leq p^{\prime}_{k}-p_{k}\\}$ and the
set of bidders $B=\tilde{F}_{p}(A)\cap T.$ Assume by contradiction that
$\delta^{\prime}=\min_{j\in\tilde{F}_{p}(T)}(p^{\prime}_{j}-p_{j})<\delta.$ We
show that this implies that $|B|>|A|\geq|\tilde{F}_{p^{\prime}}(B)|$, which
gives a contradiction.
The set of items $S$ is strictly overdemanded for prices $p$ wrt to $T$ and
$A$. Thus, since $A\subseteq S$ and $A\neq\emptyset$,
$|B|=|\tilde{F}_{p}(A)\cap T|>|A|.$ Next we show that
$A\supseteq\tilde{F}_{p^{\prime}}(B)$ and, thus,
$|A|\geq|\tilde{F}_{p^{\prime}}(B)|$. It suffices to show that
$\tilde{F}_{p^{\prime}}(i)\setminus A=\emptyset$ for all bidders $i\in B.$ For
a contradiction suppose that there exists a bidder $i\in B$ and an item
$k\in\tilde{F}_{p^{\prime}}(i)\setminus A$. Recall that we must have (1)
$u_{i,k}(p^{\prime}_{k})\geq 0$, (2) $u_{i,k}(p^{\prime}_{k})\geq
u_{i,k^{\prime}}(p^{\prime}_{k^{\prime}})$ for all $k^{\prime}$, and (3)
$p_{k}\geq r_{i,k}.$ Recall also that (1)–(3) imply that $r_{i,k}\leq
p^{\prime}_{k}<m_{i,k}$ and so
$u_{i,k}(p^{\prime}_{k})=v_{i,k}-p^{\prime}_{k}.$
We know that there exists $j\in A$ such that $j\in\tilde{F}_{p}(i)$. Since
$j\in A$ we have that $p^{\prime}_{j}<p_{j}+\delta\leq m_{i,j}$ and so
$u_{i,j}(p^{\prime}_{j})=v_{i,j}-p^{\prime}_{j}$. Thus, since
$k\in\tilde{F}_{p^{\prime}}(i)$, $v_{i,k}-p^{\prime}_{k}\geq
v_{i,j}-p^{\prime}_{j}$. Finally, since $j\in\tilde{F}_{p}(i)$ and $p_{k}\leq
p^{\prime}_{k}<m_{i,k}$, we have that $u_{i,j}(p_{j})=v_{i,j}-p_{j}\geq
u_{i,k}(p_{k})=v_{i,k}-p_{k}$.
Case 1: $k\in J\setminus F_{p}(B)$. Since $\delta\leq\delta_{\mbox{out}}\leq
u_{i}+p_{k}-v_{i,k}$ and $u_{i}=v_{i,j}-p_{j}$ we have that $\delta\leq
v_{i,j}-p_{j}+p_{k}-v_{i,k}.$ Rearranging this gives $v_{i,k}-p_{k}+\delta\leq
v_{i,j}-p_{j}.$ Since $p^{\prime}_{k}\geq p_{k}$ and
$p_{j}>p^{\prime}_{j}-\delta$ this implies that
$v_{i,k}-p^{\prime}_{k}<v_{i,j}-p^{\prime}_{j}$. Contradiction!
Case 2: $k\in F_{p}(B)\setminus\tilde{F}_{p}(B)$. If $p^{\prime}_{k}-p_{k}\leq
p^{\prime}_{j}-p_{j}=\delta^{\prime}$ then $p^{\prime}_{k}\leq
p_{k}+\delta^{\prime}<p_{k}+\delta.$ Since $\delta\leq\delta_{\mbox{res}}\leq
r_{i,k}-p_{k}$ this implies that $p^{\prime}_{k}<r_{i,k}$. Contradiction!
Otherwise, $p^{\prime}_{k}-p_{k}>p^{\prime}_{j}-p_{j}.$ Since
$v_{i,j}-p_{j}\geq v_{i,k}-p_{k}$ this implies that
$v_{i,j}-p^{\prime}_{j}>v_{i,k}-p^{\prime}_{k}$. Contradiction!
Case 3: $k\in\tilde{F}_{p}(B)\setminus A$. Since $j\in A$ and $k\not\in A$ we
have that $p^{\prime}_{k}-p_{k}>\delta^{\prime}=p^{\prime}_{j}-p_{j}.$ Since
$v_{i,j}-p_{j}\geq v_{i,k}-p_{k}$ this implies that
$v_{i,j}-p^{\prime}_{j}>v_{i,k}-p^{\prime}_{k}$. Contradiction!
Step 2. Consider an arbitrary item $j\in F_{p}(T)\setminus\tilde{F}_{p}(T)$
such that $p^{\prime}_{j}-p_{j}\leq p^{\prime}_{j^{\prime}}-p_{j^{\prime}}$
for all $j^{\prime}\in F_{p}(T)\setminus\tilde{F}_{p}(T)$ and a bidder $i\in
T$ such that $j\in F_{p}(i)$. Assume by contradiction that
$\delta^{\prime}=p^{\prime}_{j}-p_{j}<\delta$. We show that this implies that
$\tilde{F}_{p^{\prime}}(i)=\emptyset$, which gives a contradiction.
First observe that $\delta^{\prime}<\delta\leq\delta_{\mbox{res}}\leq
r_{i,j}-p_{j}$ and, thus, $p^{\prime}_{j}<p_{j}+\delta\leq r_{i,j}$, which
shows that $j\not\in\tilde{F}_{p^{\prime}}(i).$ Next consider an arbitrary
item $k\neq j.$ For a contradiction suppose that
$k\in\tilde{F}_{p^{\prime}}(i)$. It follows that $r_{i,k}\leq
p^{\prime}_{k}<m_{i,k}$ and
$u_{i,k}(p^{\prime}_{k})=v_{i,k}-p^{\prime}_{k}\geq u_{i,j}(p^{\prime}_{j})$.
Since $p^{\prime}_{j}=p_{j}+\delta^{\prime}<p_{j}+\delta\leq m_{i,j}$ we have
that $u_{i,j}(p^{\prime}_{j})=v_{i,j}-p^{\prime}_{j}$ and so
$v_{i,k}-p^{\prime}_{k}\geq v_{i,j}-p^{\prime}_{j}.$ Finally, since $j\in
F_{p}(i)$ and $p_{k}\leq p^{\prime}_{k}<m_{i,k}$, we have that
$u_{i,j}(p_{j})=v_{i,j}-p_{j}\geq u_{i,k}(p_{k})=v_{i,k}-p_{k}$.
Case 1: $k\in J\setminus F_{p}(T)$. Since $\delta\leq\delta_{\mbox{out}}\leq
u_{i}+p_{k}-v_{i,k}$ and $u_{i}=v_{i,j}-p_{j}$ we have that $\delta\leq
v_{i,j}-p_{j}+p_{k}-v_{i,k}.$ Rearranging this gives $v_{i,k}-p_{k}+\delta\leq
v_{i,j}-p_{j}.$ Since $p^{\prime}_{k}\geq p_{k}$ and
$p_{j}>p^{\prime}_{j}-\delta$ this implies that
$v_{i,k}-p^{\prime}_{k}<v_{i,j}-p^{\prime}_{j}$. Contradiction!
Case 2: $k\in F_{p}(T)\setminus\tilde{F}_{p}(T)$. If $p^{\prime}_{k}-p_{k}\leq
p^{\prime}_{j}-p_{j}=\delta^{\prime}$ then $p^{\prime}_{k}\leq
p_{k}+\delta^{\prime}<p_{k}+\delta.$ Since $\delta\leq\delta_{\mbox{res}}\leq
r_{i,k}-p_{k}$ this implies that $p^{\prime}_{k}<r_{i,k}$. Contradiction!
Otherwise, $p^{\prime}_{k}-p_{k}>p^{\prime}_{j}-p_{j}.$ Since
$v_{i,j}-p_{j}\geq v_{i,k}-p_{k}$ this implies that
$v_{i,j}-p^{\prime}_{j}>v_{i,k}-p^{\prime}_{k}$. Contradiction!
Case 3: $k\in\tilde{F}_{p}(T)$. From Step 1 we know that
$p^{\prime}_{k}-p_{k}\geq\delta>\delta^{\prime}=p^{\prime}_{j}-p_{j}.$ Since
$v_{i,j}-p_{j}\geq v_{i,k}-p_{k}$ this implies that
$v_{i,j}-p^{\prime}_{j}>v_{i,k}-p^{\prime}_{k}$. Contradiction!
###### Lemma 7.6.
Let $p$ be the prices computed by the Modified HM. Then for any stable
matching $\mu^{\prime}$ with prices $p^{\prime}$ we have that $p_{j}\leq
p^{\prime}_{j}$ for all $j.$
###### Proof 7.7.
We prove the claim by induction over the price updates. Let $p^{t}$ denote the
prices after the $t$-th price update.
For $t=0$ the claim follows from the fact that $p^{t}=0$ and
$p^{\prime}_{j}\geq 0$ for all items $j$ and all feasible matchings
$\mu^{\prime}$ with prices $p^{\prime}$.
For $t>0$ assume that the claim is true for $t-1.$ Let $S$ be the set of items
and let $T$ be the set of bidders considered by the matching mechanism for the
$t$-th price update. We claim that $S$ is strictly overdemanded for prices
$p^{t-1}$ wrt to $T.$ This is true because: (1) $S$ and $T$ are defined as the
set of items resp. bidders in a maximal alternating tree and, thus, there are
no edges in $\tilde{F}_{p^{t-1}}$ from bidders in $T$ to items in $J\setminus
S$ which shows that $\tilde{F}_{p^{t-1}}(T)\subseteq S.$ (2) For all subsets
$R\subset S$ and $R\neq\emptyset$ the number of “neighbors” in the alternating
tree under consideration is strictly larger than $|R|$ which shows that
$|\tilde{F}_{p^{t-1}}(R)\cap T|>|R|.$ By the induction hypothesis
$p^{\prime}_{j}\geq p^{t-1}_{j}$ for all items $j\in J$ and, thus, Lemma 7.4
shows that $p^{\prime}_{j}\geq p^{t-1}_{j}+\delta$ for all items $j\in
F_{p^{t-1}}(t)$. The Modified HM sets $p^{t}_{j}=p^{t-1}_{j}+\delta$ for all
items $j\in F_{p^{t-1}}(T)$ and $p^{t}_{j}=p^{t-1}_{j}$ for all items
$j\not\in F_{p^{t-1}}(T)$ and so $p^{\prime}_{j}\geq p^{t}_{j}$ for _all_
items $j\in J$.
## 8\. Truthfulness
The following example shows that with our notion of stability bidder
optimality no longer implies truthfulness, even if (i) there are no reserve
prices, i.e., $r_{i,j}=0$ for all $i$ and $j$, (ii) maximum prices depend only
on the item, i.e., for all $i$ there exists a constant $m_{i}$ such that
$m_{i,j}=m_{i}$ for all $j$, and (iii) no two bidders have the same maximum
price, i.e., $m_{i}\neq m_{k}$ for any two bidders $i\neq k.$ More
specifically, it shows that a bidder can improve her utility by lying about
the valuation of a single item. Since the bidder optimal utilities are
uniquely defined, this shows that no mechanism that computes a bidder optimal
matching $\mu$ with prices $p$ can be truthful. Note that if (i) to (iii) hold
and there exists constants $\alpha_{1}\geq\dots\geq\alpha_{k}$ and
$v_{1},\dots,v_{k}$ such that $v_{i,j}=v_{i}\cdot\alpha_{j}$ for all $i$ and
$j$, then Ashlagi et al. [3] show the existence of a truthful mechanism.
matchedunmatched$2$$5,4$$4,3$$2$$2$$2$$4$$3$$0$$1$$0$$6,6$$5,4$$4,3$$6$$4$$9$$6,6$$10,3$
$11,4$
$4,4$$5,6$$5,6$
$0,4$
$4,4$$10,3$
Figure 3. Bidders are on the left and items are on the right. The numbers next
to the bidders indicate their utilities. The numbers next to the items
indicate their prices. The labels along the edges show valuations and maximum
prices. The graph on the left depicts the bidder optimal matching for the
“true” valuations. The graph on the right depicts the bidder optimal matching
for the “falsified” valuations. Specifically, in the matching on the right
bidder 2 misreports her valuation for item 1. This gives her a strictly higher
utility, and shows that lying “pays off”.
## 9\. Generalized Linear Utilities
The following theorem generalizes our results to utilities of the form
$u_{i,j}(p_{j})=v_{i,j}-c_{i}\cdot c_{j}\cdot p_{j}$ for $p_{j}<m_{i,j}$ and
$u_{i,j}(p_{j})=-\infty$ otherwise. This reduction does not work if
$u_{i,j}(p_{j})=v_{i,j}-c_{i,j}\cdot p_{j}$ for $p_{j}<m_{i,j}$ and
$u_{i,j}(p_{j})=-\infty$ otherwise. We prove the existence of a bidder optimal
solution for more general utilities in [6].
###### Theorem 9.1.
The matching $\hat{\mu}$ with prices $\hat{p}$ is bidder optimal for
$\hat{v}=(\hat{v}_{i,j})$, $\hat{r}=(\hat{r}_{i,j})$,
$\hat{m}=(\hat{m}_{i,j})$ and utilities $u_{i,j}(p_{j})=v_{i,j}-c_{i}\cdot
c_{j}\cdot p_{j}$ if $p_{j}<m_{i,j}$ and $u_{i,j}(p_{j})=-\infty$ otherwise if
and only if the matching $\mu$ with prices $p$, where $\mu=\hat{\mu}$ and
$p=(c_{j}\cdot\hat{p}_{j})$, is bidder optimal for $v=(\hat{v}_{i,j}/c_{i})$,
$r=(c_{j}\cdot\hat{r}_{i,j})$, $m=(c_{j}\cdot\hat{m}_{i,j})$ and utilities
$u_{i,j}(p_{j})=v_{i,j}-p_{j}$ if $p_{j}<m_{i,j}$ and $u_{i,j}(p_{j})=-\infty$
otherwise.
###### Proof 9.2.
Since $\hat{p}_{j}<\hat{m}_{i,j}$ if and only if $p<m_{i,j}$ we have that
$\hat{u}_{i,j}(\hat{p}_{j})=c_{i}\cdot u_{i,j}(p_{j}).$ Since $\hat{\mu}=\mu$
this implies that $\hat{u}_{i}=c_{i}\cdot u_{i}$ for all $i.$
Feasibility. Since $c_{i}>0$ for all $i$ we have that $\hat{u}_{i}\geq 0$ for
all $i$ if and only if $u_{i}=\hat{u}_{i}/c_{i}\geq 0$ for all $i.$ Since
$c_{j}>0$ for all $i$ we have that $\hat{p}_{j}\geq 0$ for all $j$ if and only
if $p_{j}=c_{j}\cdot\hat{p}_{j}\geq 0$ for all $j.$ Since $\mu=\hat{\mu}$ and
$r_{i,j}=c_{j}\cdot\hat{r}_{i,j}$, $p_{j}=c_{j}\cdot\hat{p}_{j}$, and
$m_{i,j}=c_{j}\cdot\hat{m}_{i,j}$ for all $i$ and $j$ we have that
$\hat{r}_{i,j}\leq\hat{p}_{j}<\hat{m}_{i,j}$ for all $(i,j)\in\hat{\mu}$ if
and only if $r_{i,j}\leq p_{j}<m_{i,j}$ for all $(i,j)\in\mu$.
Stability. If $\hat{\mu}$ with $\hat{p}$ is stable then $\mu$ with $p$ is
stable because $u_{i}=c_{i}\cdot\hat{u}_{i}\geq
c_{i}\cdot\hat{u}_{i,j}(\hat{p}_{j})=u_{i,j}(p_{j})$ for all $i$ and $j.$ If
$\mu$ with $p$ is stable then $\hat{\mu}$ with $\hat{p}$ is stable because
$\hat{u}_{i}=u_{i}/c_{i}\geq u_{i,j}(p_{j})/c_{i}=\hat{u}_{i,j}(\hat{p}_{j})$
for all $i$ and $j.$
Bidder Optimality. For a contraction suppose that $\hat{\mu}$ with $\hat{p}$
is bidder optimal but $\mu$ with $p$ is not. Then there must be a feasible and
stable matching $\mu^{\prime}$ with $p^{\prime}$ such that
$u^{\prime}_{i}>u_{i}$ for at least one bidder $i.$ By transforming
$\mu^{\prime}$ with $p^{\prime}$ into $\hat{\mu}^{\prime}$ with
$\hat{p}^{\prime}$ we get a feasible and stable matching for which
$\hat{u}^{\prime}_{i}=c_{i}\cdot u^{\prime}_{i}>c_{i}\cdot u_{i}=\hat{u}_{i}.$
Contradiction!
For a contraction suppose that $\mu$ with $p$ is bidder optimal but
$\hat{\mu}$ with $\hat{p}$ is not. Then there must be a feasible and stable
matching $\hat{\mu}^{\prime}$ with $\hat{p}^{\prime}$ such that
$\hat{u}^{\prime}_{i}>\hat{u}_{i}$ for at least one bidder $i.$ By
transforming $\hat{\mu}^{\prime}$ with $\hat{p}^{\prime}$ into $\mu^{\prime}$
with $p^{\prime}$ we get a feasible and stable matching for which
$u^{\prime}_{i}=\hat{u}^{\prime}_{i}/c_{i}>\hat{u}_{i}/c_{i}=u_{i}.$
Contradiction!
## References
* [1] G. Aggarwal, A. Goel, and R. Motwani. Truthful auctions for pricing search keywords. Proceedings of the Conference on Electronic Commerce, pages 1–7, 2006.
* [2] G. Aggarwal, S. Muthukrishnan, D. Pál, and M. Pál. General auction mechanism for search advertising. Proceedings of the World Wide Web Conference, pages 241–250, 2009\.
* [3] I. Ashlagi, M. Braverman, A. Hassidim, R. Lavi, and M. Tennenholtz. Position auctions with budgets: Existence and uniqueness. Working Paper, 2009.
* [4] E. H. Clarke. Multipart pricing of public goods. Public Choice, 11:17–33, 1971.
* [5] G. Demange, D. Gale, and M. Sotomayor. Multi-item auctions. Political Economy, 94(4):863–72, 1986.
* [6] P. Dütting, M. Henzinger, and I. Weber. Bidder optimal assignments for general utilities. Proceedings of the Workshop on Internet and Network Economics, pages 575–582, 2009.
* [7] B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. American Economic Review, 97(1):242–259, 2007.
* [8] A. Frank. On Kuhn’s Hungarian Method. Naval Research Logistics, 51:2–5, 2004.
* [9] D. Gale and L. S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69:9–15, 1962.
* [10] T. Groves. Incentives in teams. Econometrica, 41:617–631, 1973.
* [11] P. Hall. On representatives of subsets. London Mathematical Society, 10:26–30, 1935.
* [12] D. Kempe, A. Mu’alem, and M. Salek. Envy-free allocations for budgeted bidders. Proceedings of the Workshop on Internet and Network Economics, pages 537–544, 2009.
* [13] H. W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics, 2:83–97, 1955.
* [14] S. Lahaie, D. M. Pennock, A. Saberi, and R. V. Vohra. Algorithmic Game Theory, chapter 28, pages 699–716. Cambridge University Press, 2007.
* [15] A. E. Roth and M. Sotomayor. Two-sided matching: A study in game-theoretic modeling and analyis. Cambridge University Press, 1990.
* [16] L. S. Shapley and M. Shubik. The assignment game: The core I. Game Theory, 29:111–130, 1972.
* [17] W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Finance, 16:8–27, 1961.
|
arxiv-papers
| 2009-12-10T08:18:10 |
2024-09-04T02:49:06.953392
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Paul D\\\"utting, Monika Henzinger and Ingmar Weber",
"submitter": "Paul D\\\"utting",
"url": "https://arxiv.org/abs/0912.1934"
}
|
0912.2147
|
# Study of $\pi^{0}$ and $\eta$ decays containing dilepton
Chong-Chung Lih1,2
1Department of Optometry, Shu-Zen College of Medicine and Management,
Kaohsiung Hsien 452,Taiwan
2Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan
###### Abstract
We calculate the momentum dependent form factors of
$M\to\gamma^{*}\gamma^{*}$($M=\pi^{0},\eta$) within the light-front quark
model. Using the form factors, we examine the decays of $M\to l^{+}l^{-}$,
$M\to l^{+}l^{-}\gamma$ and $M\to l^{+}l^{-}l^{+}l^{-}$($l=e$ or $\mu$) and
compare our results with the experimental data and other theoretical
predictions. In particular, for $\pi^{0}\to e^{+}e^{-}$, we find that the
decay branching ratio is $6.68\times 10^{-8}$, which is closed to the recent
measurement of $(7.48\pm 0.29\pm 0.25)\times 10^{-8}$ by E799 of
KTeV/Fermilab.
## I Introduction
The neutral pseudoscalar meson decays of $M\to l^{+}l^{-}$, in particular
$K_{L}\to\mu^{+}\mu^{-}$, have played very important roles to understand the
Standard Model (SM). For the light pseudoscalar mesons of $\pi^{0}$ and
$\eta$, the decays are dominated by the long distance (LD) contributions,
described by the two photon intermediate state at the lowest order of QED.
Since the short distance (SD) contributions in the SM are many orders of
magnitude smaller, they can be neglected. Therefore, these decay modes are
good processes to explore new physics beyend the SM.
The measurement on this process by the KTeV-E799 experiment at Fermilab has
givenex1
$\displaystyle{\cal B}(\pi^{0}\to e^{+}e^{-},\,x_{D}>0.95)=(6.44\pm 0.25\pm
0.22)\times 10^{-8}$ (1)
where $x_{D}\equiv(m_{2e}/m_{\pi})^{2}$ is the Dalitz variable with $m_{2e}$
being the $e^{+}e^{-}$ mass. By extrapolating the Dalitz branching ratio to
the full range of $x_{D}$ with the overall radiative correction, one gets
$\displaystyle{\cal B}^{KTeV}_{\pi^{0}\to e^{+}e^{-}}=(7.48\pm 0.29\pm
0.25)\times 10^{-8}\,.$ (2)
The decay of $\pi^{0}\to e^{+}e^{-}$ has been well studied theoretically over
the years. However, the KTeV result in Eq. (2) disagrees with the some
theoretical predictions about 1.5 $\sim$ 3.3 standard deviations ex2 ; chpt ;
chpt2 ; vmd ; qm ; qed .
At the lowest order of QED, the decay branching ratio of $\pi^{0}\to
e^{+}\,e^{-}$ is found to beim1 ; im2 ; im3 :
$\displaystyle{\cal B}_{\pi^{0}\to e^{+}e^{-}}\equiv{\Gamma(\pi^{0}\to
e^{+}e^{-})\over{\Gamma(\pi^{0}\to
2\gamma)}}=2\beta\bigg{(}{\alpha\,m_{e}\over{\pi
m_{\pi}}}\bigg{)}^{2}\,|\,{\cal A}(m_{\pi}^{2})|^{2},$ (3)
where $\beta\equiv\sqrt{1-4m^{2}_{e}/m^{2}_{\pi}}$ and $|\,{\cal
A}(m_{\pi}^{2})|^{2}$ can be generally decomposed into $|{\rm Im}\,\,{\cal
A}(m_{\pi}^{2})|^{2}+|{\rm Re}\,\,{\cal A}(m_{\pi}^{2})|^{2}$. Here, ${\rm
Im}\,{\cal A}$ denotes the absorptive contribution from the real photon in the
intermediate state, which can be determined in a model-independent formim1 ;
im2 ; im3 ; im4
$\displaystyle|{\rm Im}\,\,{\cal
A}|^{2}={\pi^{2}\over{4\beta^{2}}}\,\Bigg{[}\ln{1-\beta\over{1+\beta}}\Bigg{]}^{2},$
(4)
leading to the unitary bound on the branching ratio as
$\displaystyle{\cal B}_{\pi^{0}\to e^{+}e^{-}}>2\beta\bigg{(}\frac{\alpha
m_{e}}{\pi m_{\pi}}\bigg{)}^{2}|{\rm Im}\,\,{\cal A}|^{2}=4.75\times
10^{-8}\,\,.$ (5)
The real part ${\rm Re}\,{\cal A}$ is given by the dispersive one, which can
be written as the sum of SD and LD contributions,
$\displaystyle{\rm Re}\,\,{\cal A}={\rm Re}\,\,{\cal A}_{SD}+{\rm Re}\,\,{\cal
A}_{LD}\,.$ (6)
In the SM, the SD part is given by one-loop box and penguin diagramssd1 ; sd3
. The LD one involves the form factor related to the $\pi^{0}\gamma\gamma$
vertex. Using the form factor, the LD amplitude one has
$\displaystyle{\cal A}_{LD}={2i\over{\pi^{2}m_{\pi}^{2}}}\int
d^{4}q\,{[P^{2}q^{2}-(P\cdot
q)^{2}]\over{q^{2}\,(P-q)^{2}\,[(q-p_{e})^{2}-m^{2}_{e}]}}\,{F(q^{2},(P-q)^{2})\over{F(0,0)}}\,,$
(7)
where $P$ and $p_{e}$ are the pion and electron monenta, respectively. The
function $F(q^{2},(P-q)^{2})$ is the double form factor of
$\pi^{0}\to\gamma^{*}\gamma^{*}$. This form factor contains the nontrivial
dynamics of the process and has been studied in various modelsqed4e1 ; qed4e2
; vmd4e ; chpt ; vmd ; qm ; qed . In this paper, we calculate the form factor
$F(q^{2},(P-q)^{2})$ within the light-front quark model (LFQM) and use this
form factor to evaluate the decays of $\pi^{0}\to e^{+}e^{-}$ and
$e^{+}e^{-}\gamma$. We will also study $\eta$ decays, which contain a dilepton
or dileptons.
This paper is organized as follows: In Sec. II, we present the relevant
formulas for the matrix elements and form factors for
$M\to\gamma^{*}\gamma^{*}\ (M=\pi^{0},\eta)$. In Sec. III, we show our
numerical results on the form factors and the branching ratios of meson $M$
decays with dilepton. We give our conclusions in Sec. IV.
## II The form factors
To calculate $M\to\gamma^{*}\gamma^{*}(M=\pi^{0},\eta)$ transition from
factors within the LFQM, we have to decompose the mesons into $Q\bar{Q}$ Fock
states. Explicitly, $\pi^{0}$ may be described as
$(u\bar{u}-d\bar{d})/\sqrt{2}$ and the valence state of $\eta$ can be written
asflavor
$\displaystyle|\eta\rangle=\Phi^{8}\cos\theta_{P}|u\bar{u}+d\bar{d}-2s\bar{s}\rangle/\sqrt{6}-\Phi^{1}\sin\theta_{P}|u\bar{u}+d\bar{d}+s\bar{s}\rangle/\sqrt{3}\,,$
(8)
where $\Phi^{1,8}$ are the wave functions of the Fock states and
$\theta_{P}\sim-20^{o}$ is the mixing angle. In the scheme of the $Q\bar{Q}$
state, the amplitude of $M\to\gamma^{*}\gamma^{*}$ with $CP$ conservation is
given by:
$\displaystyle
A(Q\bar{Q}(P)\to\gamma^{*}(q_{1},\epsilon_{1})~{}\gamma^{*}(q_{2},\epsilon_{2}))=ie^{2}F_{Q\bar{Q}}(q^{2}_{1},q^{2}_{2})~{}\varepsilon_{\mu\nu\rho\sigma}~{}\epsilon^{\mu}_{1}~{}\epsilon^{\nu}_{2}~{}q^{\rho}_{1}~{}q^{\sigma}_{2}\,,$
(9)
where $F_{Q\bar{Q}}(q^{2}_{1},q^{2}_{2})$ in Eq. (9) is a symmetric function
under the interchange of $q^{2}_{1}$ and $q^{2}_{2}$. From the quark-meson
diagram depicted in Fig. 1, we get
Figure 1: Loop diagrams that contribute of $\pi^{0}\to\gamma^{*}\gamma^{*}$.
$\displaystyle A(Q\bar{Q}\to\gamma^{*}(q_{1})~{}\gamma^{*}(q_{2}))$
$\displaystyle=$ $\displaystyle
e_{Q}e_{\bar{Q}}N_{c}\int{d^{4}p_{3}\over{(2\pi)^{4}}}\Lambda_{P}\Bigg{\\{}{\rm
Tr}\Bigg{[}\gamma_{5}{i(-\not{\\!p_{3}}+m_{\bar{Q}})\over{p_{3}^{2}-m^{2}_{\bar{Q}}+i\epsilon}}\not{\\!\epsilon_{2}}{i(\not{\\!p_{2}}+m_{Q})\over{p_{2}^{2}-m^{2}_{Q}+i\epsilon}}$
(10)
$\displaystyle\times\not{\\!\epsilon_{1}}{i(\not{\\!p_{1}}+m_{Q})\over{p_{1}^{2}-m^{2}_{Q}+i\epsilon}}\Bigg{]}+(\epsilon_{1}\leftrightarrow\epsilon_{2}\,,\,q_{1}\leftrightarrow
q_{2})\Bigg{\\}}$ $\displaystyle+(\,p_{1(3)}\leftrightarrow
p_{3(1)}\,,\,m_{Q}\leftrightarrow m_{\bar{Q}})\,,$
where $N_{c}$ is the number of colors and $\Lambda_{P}$ is a vertex function
which related to the $Q\bar{Q}$ meson. In the light front (LF) approach, the
LF meson wave function can be expressed by an anti-quark $\bar{Q}$ and a quark
$Q$ with the total momentum $P$ as:
$\displaystyle|M(P,S,S_{z})\,\rangle$ $\displaystyle=$
$\displaystyle\sum_{\lambda_{1}\lambda_{2}}\int[dp_{1}][dp_{2}]2(2\pi)^{3}\delta^{3}(P-p_{1}-p_{2})$
(11)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\Phi_{M}^{SS_{z}}(z,k_{\bot})b_{\bar{Q}}^{+}(p_{1},\lambda_{1})d_{Q}^{+}(p_{2},\lambda_{2})|0\,\rangle\,,$
and
$\displaystyle[d^{3}p]={dp^{+}d^{2}p_{\bot}\over 2(2\pi)^{3}}\,,$ (12)
where $\Phi_{M}^{\lambda_{1}\lambda_{2}}$ is the amplitude of the
corresponding $\bar{Q}(Q)$ and $p_{1(2)}$ is the on-mass shell LF momentum of
the internal quark. In the momentum space, the wave function
$\Phi_{M}^{SS_{z}}$ is given by
$\displaystyle\Phi_{M}^{SS_{z}}(k_{1},k_{2},\lambda_{1},\lambda_{2})=R^{SS_{z}}_{\lambda_{1}\lambda_{2}}(z,k_{\bot})~{}\phi(z,k_{\bot}),$
(13)
where $\phi(z,k_{\bot})$ describes the momentum distribution amplitude of the
constituents in the bound state and $R^{SS_{z}}_{\lambda_{1}\lambda_{2}}$
constructs a spin state $(S,S_{z})$ out of light front helicity eigenstates
$(\lambda_{1}\lambda_{2})$melosh . The LF relative momentum variables
$(z,k_{\bot})$ are defined by
$\displaystyle p^{+}_{1}=zP^{+},\quad p^{+}_{2}=(1-z)P^{+}\,,$ $\displaystyle
p_{1\bot}=zP_{\bot}-k_{\bot},\quad p_{2\bot}=(1-z)P_{\bot}+k_{\bot}\,.$ (14)
The normalization condition of the meson state is given by
$\displaystyle\langle
M(P^{\prime},S^{\prime},S^{\prime}_{z})|M(P,S,S_{z})\rangle=2(2\pi)^{3}P^{+}\delta^{3}(P^{\prime}-P)\delta_{S^{\prime}S}\delta_{S^{\prime}_{z}S_{z}}\,,$
(15)
which leads the momentum distribution amplitude $\phi(z,k_{\bot})$ to
$\displaystyle N_{c}\int{dz\,d^{2}k_{\bot}\over
2(2\pi)^{3}}|\phi(z,k_{\bot})|^{2}=1\,.$ (16)
We note that Eq. (13) can, in fact, be expressed as a covariant formvex1 ;
vex2 ; lf1
$\displaystyle\Phi_{M}^{SS_{z}}(z,k_{\bot})$ $\displaystyle=$
$\displaystyle\left(\frac{p_{1}^{+}p_{2}^{+}}{2[M_{0}^{2}-\left(m_{Q}-m_{\bar{Q}}\right)^{2}]}\right)^{\frac{1}{2}}\overline{u}\left(p_{1},\lambda_{1}\right)\gamma^{5}v\left(p_{2},\lambda_{2}\right)\phi(z,k_{\bot})\,,$
$\displaystyle M_{0}^{2}$ $\displaystyle=$
$\displaystyle{m_{\bar{Q}}^{2}+k_{\bot}^{2}\over
z}+{m_{Q}^{2}+k_{\bot}^{2}\over 1-z}\,.$ (17)
In principle, the momentum distribution amplitude $\phi(z,k_{\bot})$ can be
obtained by solving the light-front QCD bound state equation lf1 . However,
before such first-principle solutions are available, we would have to be
contented with phenomenological amplitudes. One example that has been used is
the Gaussian type wave functionlf2 ; lf3 ; lf4 :
$\displaystyle\phi(z,k_{\bot})=N\sqrt{\frac{1}{N_{c}}\frac{dk_{z}}{dz}}\exp\left(-\frac{\vec{k}^{2}}{2\omega_{M}^{2}}\right)\,,$
(18)
where $N=4(\pi/\omega_{M}^{2})^{\frac{3}{4}}$, $\vec{k}=(k_{\bot},k_{z})$, and
$k_{z}$ defined through
$\displaystyle z={E_{1}+k_{z}\over E_{1}+E_{2}}\,,~{}~{}\ \
1-z={E_{2}-k_{z}\over E_{1}+E_{2}}\,,~{}~{}\ \
E_{i}=\sqrt{m_{i}^{2}+\vec{k}^{2}}\,$ (19)
by
$\displaystyle\ \
k_{z}=\left(z-\frac{1}{2}\right)M_{0}+\frac{m_{\bar{Q}}^{2}-m_{Q}^{2}}{2M_{0}}~{}\,,~{}~{}M_{0}=E_{1}+E_{2}\,.$
(20)
and $dk_{z}/dz=E_{1}E_{2}/z(1-z)M_{0}$. After integrating over $p_{3}^{-}$ in
Eq. (10), we obtain
$\displaystyle A(Q\bar{Q}\to\gamma^{*}(q_{1})~{}\gamma^{*}(q_{2}))$
$\displaystyle=$ $\displaystyle
e_{Q}e_{\bar{Q}}N_{c}\int^{q_{2}^{+}}_{0}dp_{3}^{+}\int{d^{2}p_{3\bot}\over
2(2\pi)^{3}\prod^{3}_{i=1}p^{+}_{i}}\bigg{[}{\Lambda_{P}\over
P^{-}-p^{-}_{1{\rm on}}-p^{-}_{3{\rm on}}}(I|_{p^{-}_{3}=p^{-}_{3{\rm on}}})$
(21) $\displaystyle{1\over q^{-}_{2}-p^{-}_{2{\rm on}}-p^{-}_{3{\rm
on}}}+(\epsilon_{1}\leftrightarrow\epsilon_{2},\,q_{1}\leftrightarrow
q_{2})\bigg{]}+(p_{1(3)}\leftrightarrow p_{3(1)})\,,$
and
$\displaystyle I$ $\displaystyle=$ $\displaystyle{\rm
Tr}[\gamma_{5}(-\not{\\!p_{3}}+m_{\bar{Q}})\not{\\!\epsilon_{2}}(\not{\\!p_{2}}+m_{Q})\not{\\!\epsilon_{1}}(\not{\\!p_{1}}+m_{Q})]\,,~{}~{}~{}~{}~{}~{}p_{ion}^{-}={m_{i}^{2}+p_{i\bot}^{2}\over
p_{i}^{+}}$ (22)
where the subscript $\\{on\\}$ represents the on-shell particles. One can
extracted the vertex function $\Lambda_{P}$ from Eqs. (10), (17) and (21),
given by lf6 ; vex1 ; vex2 :
$\displaystyle\frac{\Lambda_{P}}{{P^{-}-p^{-}_{1{\rm on}}-p^{-}_{3{\rm on}}}}$
$\displaystyle=$
$\displaystyle{\sqrt{p_{1}^{+}p_{3}^{+}}\over\sqrt{2[M_{0}^{2}-\left(m_{Q}-m_{\bar{Q}}\right)^{2}]}}\,\phi(z,k_{\bot})~{}\,,$
(23)
To calculated the trace $I$, we have used the definitions of the LF momentum
variables $(z(x),k_{\bot}(k^{\prime}_{\bot}))$ and taken the frame with the
transverse monentum $(P-q_{2})_{\perp}=0$ for the $Q\bar{Q}$ state($P$) and
photon($q_{2}$) in Fig. 1a. Hence, the relevant quark variables are:
$\displaystyle
p_{1}^{+}=zP^{+},~{}~{}p_{3}^{+}=(1-z)P^{+},~{}~{}p_{1\perp}=zP_{{\perp}}-k_{\perp},~{}~{}p_{3\perp}=(1-z)P_{{\perp}}+k_{\perp}\,.$
$\displaystyle~{}p_{2}^{+}=xq_{2}^{+},~{}p_{3}^{+}=(1-x)q_{2}^{+},~{}p_{2\perp}=xq_{2_{\perp}}-k^{{}^{\prime}}_{\perp},~{}p_{3\perp}=(1-x)q_{2_{\perp}}+k^{{}^{\prime}}_{\perp}\,.$
(24)
At the quark loop, it requires that
$\displaystyle k_{\perp}=(z-x)q_{2_{\perp}}+k^{{}^{\prime}}_{\perp}\,.$ (25)
The trace $I$ in Eq. (22) can be easily carried out. Thus, the form factor
$F(q^{2}_{1},q^{2}_{2})$ in Eq. (9) can be found to be:
$\displaystyle F_{Q\bar{Q}}(q_{1}^{2},q_{2}^{2})$ $\displaystyle=$
$\displaystyle-8\sqrt{N_{c}\over
3}\int\frac{dx\,d^{2}k_{\bot}}{2\left(2\pi\right)^{3}}\Phi\left(z,k_{\bot}^{2}\right){c^{2}_{Q}\over
1-z}\frac{m_{Q}}{x(1-x)q_{2}^{2}-m_{Q}^{2}-k_{\bot}^{2}}+(q_{2}\leftrightarrow
q_{1})\,,$ (26)
where $c_{Q}$ is the quark electric charge factor and
$\displaystyle\Phi(z,k_{\bot}^{2})$ $\displaystyle=$ $\displaystyle
N\sqrt{{\frac{z(1-z)}{2M_{0}^{2}}}}\sqrt{{\frac{dk_{z}}{dz}}}\exp\left(-{\frac{\vec{k}^{2}}{2\omega_{M}^{2}}}\right)\,,$
$\displaystyle\vec{k}$ $\displaystyle=$
$\displaystyle(\vec{k}_{\bot},\vec{k}_{z})\,,~{}~{}z=xr\,,~{}~{}$
$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{q_{2}^{+}}{P^{+}}=\frac{(m_{P}^{2}+q_{2}^{2}-q_{1}^{2})+\sqrt{(m_{P}^{2}+q_{2}^{2}-q_{1}^{2})^{2}-4q_{2}^{2}m_{P}^{2}}}{2m_{P}^{2}}\,\,.$
(27)
If $q_{1}$ and $q_{2}$ are on mass shell where $r=1$, the form factors of
$\pi\to\gamma\gamma$ and $\eta\to\gamma\gamma$ can be written as
$\displaystyle F_{\pi\to\gamma\gamma}(0,0)$ $\displaystyle=$ $\displaystyle
8\sqrt{2}\sqrt{N_{c}\over
3}\int\frac{dx\,d^{2}k_{\bot}}{2\left(2\pi\right)^{3}}{\Phi\left(x,k_{\bot}^{2}\right)\over
1-x}\left\\{\frac{4}{9}\frac{m_{u}}{m_{u}^{2}+k_{\bot}^{2}}-\frac{1}{9}\frac{m_{d}}{m_{d}^{2}+k_{\bot}^{2}}\right\\}\,,$
$\displaystyle F_{\eta\to\gamma\gamma}(0,0)$ $\displaystyle=$ $\displaystyle
16\sqrt{N_{c}\over
3}\int\frac{dx\,d^{2}k_{\bot}}{2\left(2\pi\right)^{3}}\bigg{\\{}{\Phi^{8}\left(x,k_{\bot}^{2}\right)\cos\theta_{P}\over(1-x)\sqrt{6}}\bigg{(}\frac{4}{9}\frac{m_{u}}{m_{u}^{2}+k_{\bot}^{2}}+\frac{1}{9}\frac{m_{d}}{m_{d}^{2}+k_{\bot}^{2}}-\frac{2}{9}\frac{m_{s}}{m_{s}^{2}+k_{\bot}^{2}}\bigg{)}$
(28)
$\displaystyle-{\Phi^{1}\left(x,k_{\bot}^{2}\right)\sin\theta_{P}\over(1-x)\sqrt{3}}\bigg{(}\frac{4}{9}\frac{m_{u}}{m_{u}^{2}+k_{\bot}^{2}}+\frac{1}{9}\frac{m_{d}}{m_{d}^{2}+k_{\bot}^{2}}+\frac{1}{9}\frac{m_{s}}{m_{s}^{2}+k_{\bot}^{2}}\bigg{)}\bigg{\\}}\,.$
## III Numerical Result
To numerically calculate the transition form factors of $\pi^{0}$ and $\eta$
in Eq.(26) and (28), we need to specify the parameters appearing in
$\phi(x,k_{\bot})$. To constrain the quark masses of $m_{u,d,s}$ and the meson
scale parameters of $\omega_{M}$ in Eq. (26), we use the meson decay constants
$f_{M}$ and its branching ratios of $M\to 2\gamma$, given bypdg
$\displaystyle f_{\pi^{0}}$ $\displaystyle=$ $\displaystyle\,132\,{\rm
MeV},~{}~{}f_{\eta}^{8}=\,169\,{\rm MeV}\,,~{}~{}f_{\eta}^{1}=\,145\,{\rm
MeV}\,.$ (29)
and
$\displaystyle Br_{\pi^{0}\to 2\gamma}=\,(98.832\pm
0.034)\%\,,~{}~{}Br_{\eta\to 2\gamma}=\,(39.30\pm 0.2)\%\,\,,$ (30)
respectively. Here, the explicit expression of $f_{M}$ is given byfp
$\displaystyle f_{M}$ $\displaystyle=$
$\displaystyle\,4{\sqrt{N_{c}}\over\sqrt{2}}\int{dx\,d^{2}k_{\perp}\over
2(2\pi)^{3}}\,\phi(x,k_{\perp})\,{m\over\sqrt{m^{2}+k_{\perp}^{2}}}\,.$ (31)
From
$\displaystyle{\cal B}_{M\to 2\gamma}$ $\displaystyle=$
$\displaystyle\frac{(4\pi\alpha)^{2}}{64\pi\Gamma_{P}}m_{P}^{3}|F(0,0)_{P\to
2\gamma}|^{2}\,,$ (32)
we find that $|F(0,0)_{\pi^{0}(\eta)\to 2\gamma}|=0.274(0.272)$ in $GeV^{-1}$.
As an illustration, we extracte $m_{u}=m_{d}=0.24$, $m_{s}=0.38$ and
$\omega_{\pi}=0.33$, $\omega_{\eta 1}=0.42$, $\omega_{\eta 8}=0.58$ in GeV,
which will be used in our following numerical calculations.
### III.1 $\pi^{0}(\eta)\to e^{+}e^{-}\gamma$
We now examine process of $\pi^{0}\to e^{+}e^{-}\gamma$ with the form factor
in Eq.(26). The interaction between the photon and leptons is given by the
conventional QEDqed4e1 ; cqed . One easily obtains the differential decay rate
$\displaystyle{d\,\Gamma(\pi^{0}\to
e^{+}e^{-}\,\gamma)\over{\Gamma(\pi^{0}\to\gamma\gamma)\,dq^{2}_{1}}}=\frac{2\,\alpha}{3\,\pi}\frac{1}{q_{1}^{2}}\,\left(1-\frac{q_{1}^{2}}{m_{\pi}^{2}}\right)^{3}\,\left(1-{4\,m^{2}_{e}\over{q_{1}^{2}}}\right)^{1/2}\left(1+{2\,m^{2}_{e}\over{q_{1}^{2}}}\right)\,|f(t)|^{2}\,,$
(33)
where $f(t)=F_{\pi}(q_{1}^{2},0)/F_{\pi}(0,0)$ and $t=q_{1}^{2}/m_{\pi}^{2}$.
Obviously, the branching ratio of $\pi^{0}\to e^{+}e^{-}\gamma$ in the Eq.(33)
depends on the factor of $1/q_{1}^{2}$. The function of $f(t)$ is an analytic
function in the entire physics region of $4m^{2}_{e}\leq q_{1}^{2}\leq
m_{\pi}^{2}$, related to
$\displaystyle F_{\pi}(q_{1}^{2},0)$ $\displaystyle=$
$\displaystyle-4\sqrt{2}\int\frac{dx\,d^{2}k_{\bot}}{2\left(2\pi\right)^{3}}\Phi\left(z,k_{\bot}^{2}\right){1\over
1-z}$ (34)
$\displaystyle\bigg{\\{}\frac{4}{9}\bigg{[}\frac{m_{u}}{x(1-x)q_{1}^{2}-m_{u}^{2}-k_{\bot}^{2}}+\frac{m_{u}}{m_{u}^{2}+k_{\bot}^{2}}\bigg{]}-\frac{1}{9}(m_{u}\leftrightarrow
m_{d})\bigg{\\}}\,.$
Integrating over $q^{2}_{1}$ in Eq. (33), we obtain the branching ratio
$\displaystyle{\Gamma(\pi^{0}\to
e^{+}e^{-}\gamma)\over{\Gamma(\pi^{0}\to\gamma\gamma)}}=1.18\times 10^{-2}\,,$
(35)
which agrees well with those by QEDqed4e1 ; qed4e2 and vector meson
dominance(VMD) modelvmd4e . Our result is also close the experimental data:
${\cal B}_{\pi^{0}\to e^{+}e^{-}\gamma}^{exp}=(1.198\pm 0.032)\times 10^{-2}$
pdg .
Similarly, the branching ratios of $\eta\to e^{+}e^{-}\gamma$ and
$\eta\to\mu^{+}\mu^{-}\gamma$ which normalized with $\eta$ tatal width are
found to be
$\displaystyle{\cal B}_{\eta\to e^{+}e^{-}\gamma}$ $\displaystyle=$
$\displaystyle{\Gamma(\eta\to
e^{+}e^{-}\gamma)\over{\Gamma_{\eta}}}=6.95\times 10^{-3}\,,$
$\displaystyle{\cal B}_{\eta\to\mu^{+}\mu^{-}\gamma}$ $\displaystyle=$
$\displaystyle{\Gamma(\eta\to\mu^{+}\mu^{-}\gamma)\over{\Gamma_{\eta}}}=2.94\times
10^{-4}\,.$ (36)
Ours result of $\eta\to e^{+}e^{-}\gamma$ is smaller than that in the CLEO
datacleo but larger than the one in Ref.mpp . However, for the mode of
$\eta\to\mu^{+}\mu^{-}\gamma$, our result agrees with Ref.mpp as well as that
by the effective mass theory(EMT)emt . Furthermore, our predictions in the two
decay modes agree well with the experimental data in CELSIUSwasa and the
PDGpdg .
### III.2 $\pi^{0}\to e^{+}e^{-}e^{+}e^{-}$ and
$\eta\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}\ (\ell=e,\mu)$
We examine the double lepton-pair decay of $\pi^{0}\to e^{+}e^{-}e^{+}e^{-}$
with the form factors in Eq. (26). The decay matrix element is calculated by
the conventional QED with the interaction of $\pi^{0}$ and two photons and the
differential decay rate is given by
$\displaystyle{d\,\Gamma(\pi^{0}\to
e^{+}e^{-}e^{+}e^{-})\over{\Gamma(\pi^{0}\to\gamma\gamma)\,dq_{1}^{2}\,dq_{2}^{2}}}={2\over{q_{1}^{2}q_{2}^{2}}}\left({\alpha\over{3\pi}}\right)^{2}\left|{F_{\pi}(q_{1}^{2},q_{2}^{2})\over{F_{\pi}(0,0)}}\right|^{2}\,\lambda^{3/2}\left(1,{q_{1}^{2}\over{m^{2}_{\pi}}},{q_{2}^{2}\over{m^{2}_{\pi}}}\right)\,G_{l}(q_{1}^{2})\,G_{l^{\prime}}(q_{2}^{2}).$
(37)
where
$\displaystyle\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+bc+ca),$ $\displaystyle
G_{l}(q^{2})=\left(1-{4\,m^{2}_{e}\over{q^{2}}}\right)^{1/2}\left(1+{2\,m^{2}_{e}\over{q^{2}}}\right)$
(38)
After the integrations over $q^{2}_{1}$ and $q^{2}_{2}$, we obtain the
branching ratio as follows:
$\displaystyle{\cal B}_{\pi^{0}\to e^{+}e^{-}e^{+}e^{-}}$
$\displaystyle\equiv$ $\displaystyle{\Gamma(\pi^{0}\to
e^{+}e^{-}e^{+}e^{-})\over{\Gamma(\pi^{0}\to\gamma\gamma)}}=3.29\times
10^{-5}\,,$ (39)
which is smaller than that in Ref.qed4e1 , but larger than the one in
Ref.qed4e2 slightly. However, all results are consistent with the
experimental data. We note that even if the form factor is replaced by an on-
shell constant with $F(q_{1}^{2},q_{2}^{2})=F(0,0)$, the branching ratio is
found to be very close to the result in Eq. (39). It might be a good
approximation to neglect the momentum dependence of the form factor for the
decay.
We can also perform the similar calculations for $\eta\to
l^{+}l^{-}l^{+}l^{-}$($l=e$ or $\mu$) and we find
$\displaystyle{\cal B}_{\eta\to e^{+}e^{-}e^{+}e^{-}}$ $\displaystyle=$
$\displaystyle 2.47\times 10^{-5}\,,$ $\displaystyle{\cal B}_{\eta\to
e^{+}e^{-}\mu^{+}\mu^{-}}$ $\displaystyle=$ $\displaystyle 5.83\times
10^{-7}\,,$ $\displaystyle{\cal B}_{\eta\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}}$
$\displaystyle=$ $\displaystyle 1.68\times 10^{-9}\,.$ (40)
Our result on ${\cal B}_{\eta\to e^{+}e^{-}e^{+}e^{-}}$ is in good agreement
with the experimental data ${\cal B}_{\eta\to
e^{+}e^{-}e^{+}e^{-}}^{exp}=(2.7^{+2.1}_{-2.7stat}\pm 0.1_{syst})\times
10^{-5}$wasa and Ref.mpp . For other modes, currently, our theoretical
predictions are many orders of magnitude smaller than the experimental upper
bounds pdg ; wasa .
### III.3 $\pi^{0}(\eta)\to\ell^{+}\ell^{-}$
We first calculate the real part of ${\rm Re}\,\,{\cal A}_{LD}$ in Eq. (7) at
the pion momentum limit of $P^{2}\to 0$. At this limit, the relevant form
factor of Eq. (26), given by a triangular quark loop, would be simplify to
$\displaystyle F(q^{2},q^{2})$ $\displaystyle=$
$\displaystyle-8\sqrt{2}\int\frac{dx\,d^{2}k_{\bot}}{2\left(2\pi\right)^{3}}\Phi\left(z,k_{\bot}^{2}\right){1\over
1-z}$ (41)
$\displaystyle\left\\{\frac{4}{9}\frac{m_{u}}{x(1-x)q^{2}-m_{u}^{2}-k_{\bot}^{2}}-\frac{1}{9}\frac{m_{d}}{x(1-x)q^{2}-m_{d}^{2}-k_{\bot}^{2}}\right\\}\,.$
One could easily find
$\displaystyle{\rm Re}\,\,{\cal A}_{LD}(0)\simeq-20.74\,\,.$ (42)
The numerical result is in agreement with the most vector meson dominance(VMD)
model at $P^{2}\to 0$. This implies the equivalence between the VMD and LFQM
descriptions on the form factors of hadrons with the relevant vector meson
mass of $M_{V}\sim 2m_{u}$ in the VMD. To illustrate ${\rm Re}\,\,{\cal
A}_{LD}(q^{2})$ in the range $-m_{\pi}^{2}\geq q^{2}\geq m_{\pi}^{2}$, we use
the dispersive framework proposed in Ref.qm . The real part may be written by
a once-subtracted dispersion relationex2 ; qm ; cleo2
$\displaystyle{\rm Re}\,\,{\cal A}_{LD}(q^{2})={\rm Re}\,\,{\cal
A}(0)+\frac{q^{2}}{\pi}\,\int_{0}^{\infty}dq^{\prime 2}\frac{{\rm Im}\,\,{\cal
A}(q^{\prime 2})}{(q^{\prime 2}-q^{2})q^{\prime 2}}$ (43)
Extrapolating from $q^{2}=0$ to $m_{\pi}^{2}$, we find ${\rm Re}\,\,{\cal
A}_{LD}(m_{\pi}^{2})=11.18$. Since the SD part of ${\rm Re}\,{\cal A}_{SD}$
can be neglected, we get the branching ratio of the real part in Eq.(1) to be
$1.93\times 10^{-8}$. The total decay branching ratio is about $6.68\times
10^{-8}$. Our prediction is smaller than the experimental value of ${\cal
B}^{\rm{KTeV}}_{\pi^{0}\to e^{+}e^{-}}=(7.48\pm 0.29\pm 0.25)\times 10^{-8}$
measured by KTeV. We note that our result is larger than the values of
$(6.41\pm 0.19)\times 10^{-8}$ and $6\times 10^{-8}$ calculated in Ref.vmd ;
qm with the VMD and quark model(QM), respectively, but closed to $(7\pm
1)\times 10^{-8}$ in the Chiral Perturbation Theory(ChPT)chpt . It is clear
that we provide a method to calculate the form factor of
$\pi^{0}\to\gamma^{*}\gamma^{*}$ and get a result in $\pi^{0}\to e^{+}e^{-}$
within the LFQM.
The $\eta\to l^{+}l^{-}$ decay can be analyzed in a similar technique as
$\pi^{0}\to e^{+}e^{-}$. In the momentum limit $P^{2}\to 0$, we obtained
$\displaystyle{\rm Re}\,\,{\cal A}_{(2e)LD}(0)$ $\displaystyle\simeq$
$\displaystyle-22.43\,\,,$ $\displaystyle{\rm Re}\,\,{\cal A}_{(2\mu)LD}(0)$
$\displaystyle\simeq$ $\displaystyle-6.48\,\,.$ (44)
Form the dispersive integral in Eq.(43) and Eq.(44), one obtains
$\displaystyle{\rm Re}\,\,{\cal A}_{(2e)LD}(m_{\eta}^{2})$
$\displaystyle\simeq$ $\displaystyle 27.11\,\,,$ $\displaystyle{\rm
Re}\,\,{\cal A}_{(2\mu)LD}(m_{\eta}^{2})$ $\displaystyle\simeq$
$\displaystyle-2.81\,\,.$ (45)
The SD contributions to the decays can be still ignored and the total
branching ratios are given by
$\displaystyle{\cal B}_{\eta\to e^{+}e^{-}}$ $\displaystyle=$ $\displaystyle
4.47\times 10^{-9}\,,$ $\displaystyle{\cal B}_{\eta\to\mu^{+}\mu^{-}}$
$\displaystyle=$ $\displaystyle 5.47\times 10^{-6}\,.$ (46)
One notes that the value of ${\cal B}_{\eta\to e^{+}e^{-}}$ is larger than the
CLEO resultex2 . For the mode of $\eta\to\mu^{+}\mu^{-}$, it is consistent
with the CLEOex2 and VMD resultscpt . It also agrees with the PDG data of
$5.8\pm 0.8\times 10^{-5}$.
We summarized the related experimental and theoretical values of the decay
branching ratios of $\pi^{0}\to e^{+}e^{-}\gamma$, $\pi^{0}\to
e^{+}e^{-}e^{+}e^{-}$ and $\pi^{0}\to e^{+}e^{-}$ in Table I and $\eta\to
l^{+}l^{-}\gamma$, $\eta\to l^{+}l^{-}l^{+}l^{-}$ and $\eta\to l^{+}l^{-}$ in
Table II.
Table 1: Summary of the decays of $\pi^{0}$ with lepton pair. Br | Exp. data | This work | Other models
---|---|---|---
$10^{2}~{}{\cal B}_{e^{+}e^{-}\gamma}$ | $1.174\pm 0.035$pdg | $1.18$ | $1.18$qed4e1 qed4e2 vmd4e
$10^{5}~{}{\cal B}_{e^{+}e^{-}e^{+}e^{-}}$ | $3.34\pm 0.16$pdg | $3.29$ | $3.28$qed4e1 , $3.46$qed4e2
$10^{8}~{}{\cal B}_{e^{+}e^{-}}$ | $7.48\pm 0.29\pm 0.25$ex1 ; ex2 | $6.68$ | $7\pm 1$chpt , $8.3\pm 0.4$chpt2 , $6.41\pm 0.19$vmd , $6$qm ,
| $6.46\pm 0.33$pdg | | $<4.7$qed , $6.23\pm 0.09$ex2 ; cleo2
Table 2: Summary of the decays of $\eta$ with lepton pair. Br | Exp. data | This work | Other models
---|---|---|---
$10^{3}~{}{\cal B}_{e^{+}e^{-}\gamma}$ | $7.8\pm 0.5_{stat}\pm 0.7_{syst}$wasa | $6.95$ | $9.4\pm 0.7$cleo ,
| $7.0\pm 0.7$pdg | | $6.31-6.46$mpp , $6.5$emt
$10^{4}~{}{\cal B}_{\mu^{+}\mu^{-}\gamma}$ | $3.1\pm 0.4$pdg | $6.95$ | $2.14-3.01$mpp , $3.0$emt
$10^{5}~{}{\cal B}_{e^{+}e^{+}e^{-}e^{-}}$ | $2.7^{+2.1}_{-2.7stat}\pm 0.1_{syst}$wasa | $2.47$ | $2.49-2.62$mpp
| $<6.9$pdg | |
$10^{7}~{}{\cal B}_{\mu^{+}\mu^{-}e^{+}e^{-}}$ | $<1.6\times 10^{3}$pdg | $5.83$ | $1.57-2.21$mpp
$10^{9}~{}{\cal B}_{\mu^{+}\mu^{-}\mu^{+}\mu^{-}}$ | $<3.6\times 10^{5}$pdg | $1.68$ |
$10^{9}~{}{\cal B}_{e^{+}e^{-}}$ | $<2.7\times 10^{4}$pdg | $4.47$ | $13.7$vmd , $4.60\pm 0.06$ex2 ; cleo2
$10^{6}~{}{\cal B}_{\mu^{+}\mu^{-}}$ | $5.8\pm 0.8$pdg | $5.47$ | $5.8\pm 0.2$chpt , $11.4$vmd
| | | $5.11\pm 0.20$ex2 ; cleo2 , $5.2\pm 1.2$cpt
## IV Conclusions
We have calculated the form factors of
$P\to\gamma^{*}\gamma^{*}$($P=\pi^{0},\eta$) directly within the LFQM. In our
calculations, we have adopted the Gaussian-type wave function and evaluated
the form factors for the momentum dependences in the energy regions from
$q^{2}=0$ to $m_{P}^{2}$. Using the form factors, we have examined $\pi^{0}\to
e^{+}e^{-}\gamma$ and $\pi^{0}\to e^{+}e^{-}e^{+}e^{-}$ and shown that our
results on the decay branching ratios agree well with the experimental data
shown in Table. I. Our predicted values are also close to those in the QED and
VMD modelsqed4e1 ; qed4e2 ; vmd4e . For $\pi^{0}\to e^{+}e^{-}$, we have found
that ${\cal B}_{\pi^{0}\to e^{+}e^{-}}$ is $6.68\times 10^{-8}$, which agrees
with $(7\pm 1)\times 10^{-8}$ in the ChPT chpt but larger than those in
Refs.vmd ; qm ; qed . We have demonstrated that the long-distance dispersive
contribution in this model is possibly small. However, like other theoretical
predictions, our result for $\pi^{0}\to e^{+}e^{-}$ is also slightly smaller
than the experimental data. Clearly, further theoretical studies as well as
more precise experimental data such as those from the KTeV-E799 experiment at
Fermilab on the spectra of the decays with lepton pair are needed. About the
$\eta$ decays, our results are all consistent with the experimental data. In
particular, the branching ratios of $\eta\to 2e2\mu$, $\eta\to 4\mu$ and
$\eta\to 2e$ are expected to be 4$\sim$5 orders of magnitude lower than the
current experimental upper limits.
## V Acknowledgments
This work is supported in part by the National Science Council of R.O.C. under
Contract NSC-97-2112-M-471-002-MY3.
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|
arxiv-papers
| 2009-12-11T03:48:58 |
2024-09-04T02:49:06.963557
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chong-Chung Lih",
"submitter": "Chong Chung Lih",
"url": "https://arxiv.org/abs/0912.2147"
}
|
0912.2272
|
# The Structure and Dynamics of the Upper Chromosphere and Lower Transition
Region as Revealed by the Subarcsecond VAULT Observations
A. Vourlidas1B. Sanchez Andrade-Nuño1,2E. Landi1S. Patsourakos2L. Teriaca3U.
Schühle3C.M. Korendyke1I. Nestoras4 1 Space Science Division, Naval Research
Laboratory, 4555 Overlook Ave, SW, Washington, D.C., USA 11email:
vourlidas@nrl.navy.mil2 George Mason University, 4400 University Dr, Fairfax,
VA, USA 11email: bsanchez@ssd5.nrl.navy.mil3 MPI for Solar System Research,
37191 Katlenburg-Lindau, Germany 11email: teriaca@mps.mpg.de,
schuele@mps.mpg.de4 Max-Plank-Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany
###### Abstract
The Very high Angular resolution ULtraviolet Telescope (VAULT) is a sounding
rocket payload built to study the crucial interface between the solar
chromosphere and the corona by observing the strongest line in the solar
spectrum, the Ly$\alpha$ line at 1216Å. In two flights, VAULT succeeded in
obtaining the first ever sub-arcsecond ($0.5\hbox{${}^{\prime\prime}$}$)
images of this region with high sensitivity and cadence. Detailed analyses of
those observations have contributed significantly to new ideas about the
nature of the transition region. Here, we present a broad overview of the
Ly$\alpha$ atmosphere as revealed by the VAULT observations, and bring
together past results and new analyses from the second VAULT flight to create
a synthesis of our current knowledge of the high-resolution Ly$\alpha$ Sun. We
hope that this work will serve as a good reference for the design of upcoming
Ly$\alpha$ telescopes and observing plans.
###### keywords:
line: Hydrogen Ly alpha —- atomic data —- Sun: corona —- Sun: UV radiation —-
Sun: transition region
## 1 Introduction s:Intro
The structure of the solar atmosphere as a function of temperature has been a
’thorny’ issue of solar physics research for decades. As the density
decreases, the temperature, instead of decreasing, abruptly increases from
$\sim 10^{4}$ K to a million K within a thousand km. It is known since the
first solar imaging space missions that this so-called temperature transition
region (TR) between the chromosphere and the corona, is also where the
morphology of the atmospheric structures changes strongly. At the base of the
atmosphere, the photosphere consists of small scale convective granules
interlaced with occasional smaller intergranular lanes concentrating strong
magnetic flux elements($|B|\leq 1kG$, e.g. 2004Natur.430..326T). The
chromosphere (T$\leq 10^{4}$ K for the discussion here) consists of a very
rugged, inhomogeneous, and very filamentary layer blanketing the photosphere.
Beginning at the chromosphere, the geometry of the individual structures is
increasingly dominated by the local magnetic field. At the lower transition
region (T$\leq 2\times 10^{5}$ K), the structures strongly reflect the
morphology of the underlying supergranular network. As the magnetic pressure
overtakes the gas pressure leading to the low beta corona, the percentage of
emission in filamentary loops steadily increases until the network completely
disappears at temperatures above $10^{6}$ K. It may seem that a
straightforward interplay between heating and morphology takes place in the
transition region but this is not the case.
The traditional picture of the transition region as the interface between the
footpoints of large-scale structures and their coronal tops has been
contradicted by the weakness of its observed emission [Landi and Feldman
(2004)]. While the emission in the upper TR ($T>2\times 10^{5}$K) can be
understood in terms of heat conduction from the corona along magnetic field
lines, the lower TR ($T<2\times 10^{5}$ K) cannot. Instead, this plasma forms
a completely separate component of the solar atmosphere [Feldman (1983),
Feldman (1987)]. This component could consists of small “cool” loops
[Antiochos and Noci (1986), Dowdy, Rabin, and Moore (1986), Feldman, Dammasch,
and Wilhelm (2000), Peter (2001)] that are best seen in the Quiet Sun and that
probably correspond to the upper reaches of the mixed polarity magnetic carpet
[Schrijver et al. (1997)]. 2009ApJ…693.1474F showed that the Differential
Emission Measure (DEM) of the TR has the same shape everywhere (coronal holes,
Quiet Sun, active regions) while coronal DEM of the very same regions are very
different. Why and how are transition region loops different from higher
arching coronal loops? Are they also comprised of unresolved strands? Are they
heated in a fundamentally different way? Recently, Judge (2008) proposed a
radically different view of the transition region emission, suggesting that it
might result from cross-field diffusion of plasma from very fine cool threads
extending into the corona (e.g. spicules), and its subsequent ionization. Cool
threads gradually expand in thickness as the ionizing front expands across the
field lines and emits at TR temperatures, and provide images of the transition
region similar to those observed by the SUMER [Wilhelm et al. (1995)]
spectrometer aboard SOHO.
Hinode observations revealed a dramatically new picture of the solar
chromosphere and demonstrated its potential importance for the dynamics,
energy and mass supply of the transition region and corona. High temporal
($\simeq$5s) and spatial ($\simeq$0.2′′) Hinode/SOT observations have shown
that the chromosphere is much more structured and dynamic than previously
believed. SOT has revealed a chromosphere hosting a wealth of wave and
oscillatory phenomena manifested as longitudinal and transverse motions within
structures at the resolution limit [de Pontieu et al. (2007a), Ofman and Wang
(2008), Okamoto et al. (2007)]. Even a fraction of the inferred wave energy
flux could account for the coronal energy losses if it reached the corona. SOT
also showed that a significant fraction of observed spicules (’type II’),
known for decades to dominate the chromospheric landscape, disappear very
rapidly (De Pointieu et al. 2007b). This was interpreted as a signature of the
plasma heating up to transition region and coronal temperatures; the mass
contained in these disappearing spicules is sufficient to account for the mass
present in the corona.
Capturing the fine spatial scales and rapid temporal evolution of the
chromosphere and transition region plasmas represents a considerable
observational and technical challenge. Nonetheless, recent significant
improvements on instrumentation and image processing has been achieved both
from ground (e.g., 2006A&A…454.1011P, 2008SoPh..251..533R, 2007ASPC..368…65D)
or spaceborne instruments (e.g. 2007PASJ…59S.655D), reaching in all cases
spatial resolution under 1′′ for plasmas at chromospheric regimes. Reaching
these resolution on the TR involves the use of strong UV lines, accessible
only above the Earth’s atmosphere.
The Very high Angular resolution ULtraviolet Telescope (VAULT,
2001SoPh..200…63K), a sounding rocket payload, is the only instrument that has
observed this critically important layer of the solar atmosphere at such high
resolution. VAULT is specifically designed to obtain high spectral purity,
zero dispersion spectroheliograms in the Lyman-$\alpha$ (1216 Å) resonance
line of hydrogen. This emission line emanates from plasmas at 8000 to 30000K
[Gouttebroze, Vial, and Tsiropoula (1986)]. The Ly$\alpha$ radiation directly
maps the dominant energy loss from plasmas at these temperatures which
correspond to the lower TR [Fontenla, Reichmann, and Tandberg-Hanssen (1988)].
This instrument is the latest in a long and distinguished line of solar
optical instruments obtaining observations in the Ly$\alpha$ emission line
[Purcell and Widing (1972), Prinz (1974), Bartoe and Brueckner (1975), Bonnet
et al. (1980)]. The VAULT observations are the highest quality UV observations
of the solar atmosphere ever obtained and are a considerable improvement over
previous instruments. Each rocket flight obtained observations with observable
structures of $<0.5\hbox{${}^{\prime\prime}$}$ spatial scale, exposure times
of 1 second with a 17 second cadence and a 355′′$\times$235′′ instantaneous
field of view (FoV).
The VAULT data and, more recently, the Hinode/SOT observations have
invigorated the debate about the nature of the solar Transition Region. Not
surpsingly, Ly$\alpha$ telescopes are planned for the upcoming Solar Orbiter
mission, and, possibly, the proposed Solar-C mission. It is therefore, an
appropriated time for a review of the VAULT observations. We believe that as a
trailblazer project in the exploration of the upper chromosphere-corona
interface, the VAULT experiences will be a useful reference for the instrument
design and science operations for those missions. We also take this
opportunity to present the final calibration of the data and introduce the
project website where all the data are publicly available. This paper presents
a detailed examination of the Ly$\alpha$ structures near the base of the solar
corona obtained during the second flight of the payload (hereafter, VAULT-II).
We are specifically concerned with those plasmas whose temperatures lie
between 8 000 and 30 000K, ranging roughly from $\sim$2 000 km to $\sim$60 000
km above the photosphere.
The paper is organized as follows. Section s:calib describes the latest
instrument calibration and the observations from the second VAULT flight.
Section s:ly summarizes the importance of the Ly$\alpha$ in the frame of
Coronal and TR models. Section s:inten discusses the sources of Ly$\alpha$
emission as determined in the VAULT images. Sections s:prom, s:qs, s:spicules
focus on, respectively, prominences, Quiet Sun, and spicules. We discuss our
findings and conclude in Sections s:dis-s:con.
Figure 1.: The total solar field of view observed during the second VAULT
flight. The image is a composite of all VAULT-II observations after dark
current subtraction and flatfielding. It covers a $\sim 600\times 450$ arcsecs
area, with $0.12\hbox{${}^{\prime\prime}$}$ pixel size. Solar North is to the
right and Solar East at the top of the image. The image is plotted with
histogram equalization of the intensities.fig:composite
## 2 Data Analysis and Observationss:calib
VAULT has been successfully launched twice (May, 7, 1999 and June, 14, 2002).
Using the experience from the first flight [Korendyke et al. (2001)], the
instrument performance during the second flight was improved by using a higher
transmission Ly$\alpha$ filter (higher throughput) and better filtering of the
power converter output (lower noise/higher quality data). So we concentrate on
the VAULT-II images for the remainder of the paper.
VAULT-II was flown on June 14, 2002 from White Sands Missile Range onboard a
Black Brant sounding rocket. The observations took place around the apogee of
the parabolic trajectory while the rocket was above 100 km. This minimum
altitude was chosen to minimize absorption effects from the geocorona [Prinz
and Brueckner (1977)]. The duration of science operations was $363$ sec and
the rocket peaked at an altitude of 182 miles (294 km). The entire flight,
from launch to recovery, lasted 15 minutes.
Figure 2.: _Left:_ Alignment of VAULT-TRACE Ly$\alpha$ images. We find the
best correlation by optimizing the position, rotation and scale in both x-y
directions. We derive a pixel size of $0.125\hbox{${}^{\prime\prime}$}\times
0.110\hbox{${}^{\prime\prime}$}$. The TRACE Ly$\alpha$ was taken at 18:18:54
UT (B&W figure and white contour), the VAULT was taken at 18:17:30 UT (three
red contours). _Right:_ Estimation of the VAULT resolution using the thin
spicule located at the center of the image. The median normalized cross
section (in arcseconds) of the spicule along its length (plot inset in figure)
is fitted with a gaussian, which leads to a FWHM $\approx
0.49\hbox{${}^{\prime\prime}$}$ as an upper limit. fig:align
fig:resolution
### 2.1 VAULT-II Observations s:obs
VAULT-II obtained 21 images from 18:12:01 to 18:17:47 UT with a cadence of 17
seconds. The integration time was 1 second for all frames except for a 5-sec
image (the 2nd in the series, not shown here). The target was an old active
region complex near the east limb which included NOAA regions 9997-9999, Quiet
Sun, filaments, plage and the limb.
Figure fig:composite is a composite image of all VAULT-II frames. The
composite field of view (FOV) covers nearly 10% of the total visible solar
disc area. To investigate possible center-to-limb variation [Miller, Mercure,
and Rense (1956)] we have calculated the radial median intensity of non-active
region areas (excluding plage region, prominences, flaring regions). We do not
find any significant center-to-limb gradient in agreement with Curdt et al.
(2008).
The VAULT flight was supported by several other instruments. All corresponding
data (see table tab:JOP) are available online or per request.
All VAULT data are publicly available online in FITS format and compatible
with _SolarSoft_ mapping routines. The images are interaligned, the dark level
is subtracted and an _ad-hoc_ synthetic flat-field is also created and
provided with the data, but not applied on the online set. The flat-field is
generated by retrieving the median (in time) pixel value as the solar image
moves during the observations. It therefore accounts for flatfield and
scattered light. Intensities are left in DN.
To improve the visibility of faint, small scale structures, we have applied a
wavelet enhancement technique Stenborg, Vourlidas, and Howard (2008). This
method decomposes the image into frequency components (scales). The frequency
decomposition is achieved by means of the so-called a-trous algorithm. With
this method we can then obtain an edge-enhanced version of the original image
by assigning different weights to the different scales upon reconstruction. We
note that the aforementioned decomposition does not create orthogonal
components, and therefore the reconstruction does not conserve the flux. It is
also possible to retain the low-scale information by adding a model background
image. Both processed data, with and without the model are freely available in
the VAULT website. The level 0.9 VAULT-II data, together with the IDL-
_SolarSoft_ routines, composite full field image, flat-field and wavelets
processed data are available under:
`http://wwwsolar.nrl.navy.mil/rockets/vault/`
Telescope | Channel | co-temporal time serie
---|---|---
SOHO | MDI/EIT/CDS | EIT-304 Å partial FoV
TRACE | 171, WL,1600 | 171 Å, partial FoV
BBSO | H$\alpha$,Ca,BLOS,WL | H$\alpha$, partial FoV
Kitt-Peak | Mgram | Photospheric magnetogram
Table 1.: Joint observing campaign supporting the VAULT-II launchtab:JOP
### 2.2 Spatial resolution s:spatial
The rocket pointing accuracy was $\sim 1$ arcmin with exceptional pointing
stability of 0.25′′peak-to-peak over 10 sec. To obtain the solar coordinates,
rotation relative to solar North, and pixel size for the VAULT-II images we
used TRACE Ly$\alpha$ images taken only minutes apart from the VAULT images.
Figure fig:align shows the alignment results. The resulting VAULT pixel size
is 0.125′′$\times$0.110′′ which is in excellent agreement with the optical
design expectations Korendyke et al. (2001). During the flight, a small
thermal expansion of the spectrograph structure relative to the primary mirror
resulted in an apparent pointing drift which was variable but less than
$\approx$3 pixels/second. If it was uniform during the flight, this drift
would place a conservative lower limit on the instrument resolution of $\sim
0.75\hbox{${}^{\prime\prime}$}$ (0.375′′/pixel). However, we could visually
identify smaller structures (always in absorption on disk) in several images.
To better estimate the actual image resolution we measured the FWHM of the
smallest structure we could find in the images. We used the median cross
section along the 110-pixel length of the thinnest spicule, located at the
center of Figure fig:resolution (right), and fitted it with a Gaussian. The
$FWHM\approx 2.35\cdot\sigma$, where $\sigma$ is the standard deviation of the
fitted gaussian profile. This leads to a VAULT-II resolution
$0.49\hbox{${}^{\prime\prime}$}$. However, the upper limit may be dictated by
opacity effects rather than instrumental ones.
The photometric calibration of the instrument was originally determined from
the observations during its first flight in May 1999\. The calibration factor
from digital units (DNs) to intensity (ergs s-1 cm2 sr-1) was deduced by
comparing the average emission (in DNs) of an area of the Quiet Sun to the
Quiet Sun intensity obtained by Prinz (1974). This was a reasonable assumption
since both observations were made at similar phases of the cycle; the majority
of the VAULT I field of view contained Quiet Sun and the 1974ApJ…187..369P
measurements are well calibrated ($\sim 20\%$).
A comparison to SUMER Ly-$\alpha$ observations was made to improve on the
radiometric accuracy of our measurements. The first issue was the spectral
purity of the signal. The VAULT gratings transmit solar light in the range of
$1140-1290$Å. In this range there are only few relatively bright lines, the
brightest of which is Si iii at 1206.51Å. SUMER Quiet Sun spectra show that
almost all ($\sim 95$%) of the emission in this range comes from the
Ly-$\alpha$ (assuming a rectangular filter). Concerning the spectral purity on
the full range above 120 nm, our calculations show that the signal should be
70% pure.
Figure 3.: Normalized H i Ly$\alpha$ distributions as obtained from SUMER data
(dashed line) and the VAULT data (solid line) rebinned and convolved to match
the SUMER spatial resolution after subtracting a background signal level of 80
DN s-1 (see section s:spatial). fig:sumer
Since the SUMER instrument has a photon counting detector with no dark signal,
there is no background to be removed from the SUMER data. To establish the
comparison with SUMER we assumed that the normalized radiance frequency
distributions over quiet-Sun areas produced with data from the two instruments
should be equal or very similar to each other. To account for the different
spatial resolution we have also computed the radiance frequency distributions
after convolving the VAULT data with a 2-D Gaussian function (of 12
pixel=1.5′′ FWHM, equal to the SUMER spatial resolution) and binning over
$8\times 8$ pixels to yield the SUMER pixel size of $\approx
1^{\prime\prime}$. The comparison revealed that a low level signal of about
$\sim 80$ DN s-1 needs to be removed from the VAULT images to bring them in
accordance to the SUMER measurements (Figure fig:sumer. After a careful
examination of the VAULT-II images, we found a noise pattern of $83\pm 20$ DN
s-1 which is variable from image to image and cannot therefore be removed with
the dark current subtraction. We have traced the source of the noise to
interference from a faulty ground when the payload is switched to battery
power.
The final step is a comparison of the Quiet Sun level in our images with an
average Quiet Sun radiance measured at Earth as we did for the first flight.
For the VAULT Quiet Sun level we used the peak of the histogram of the image
intensities (in DN s-1) minus the 83 DNs of the background signal. The Quiet
Sun level was $217\pm 20$ DN s-1. The SUMER average Ly$\alpha$ radiance on the
Quiet Sun in 2008 was $73\pm 16$ W m-2 sr-1. This is well within uncertainties
with the 1974ApJ…187..369P measurement of $78\pm 16$ W m-2 sr-1. We adopt the
latter value for consistency with our VAULT-I results and because it was
obtained about two years after maximum and may better compare with our 2002
data. In this case, we derive a calibration factor of 1 DN s-1 = $0.359\pm
0.081$ W m2 sr-1.
Figure 4.: VAULT-II, TRACE Ly$\alpha$, and BBSO H$\alpha$ comparison of the
solar limb. All intensities are individually trimmed and scaled to emphasize
the fainter details. _Left:_ VAULT-II after alignment and calibration. The
black contour follows the visible limb, as aligned with the TRACE White light
channel. _Center:_ TRACE Ly$\alpha$. The black grid denotes the photosphere.
The prominence is much fainter than in the VAULT images but it is still
visible. The white contour close to noise level helps to correlate with VAULT.
_Right:_ H$\alpha$ channel from ground-based BBSO. The BBSO FOV is slightly
smaller than the other instruments. The dotted contour denotes the VAULT edge.
Note that the H$\alpha$ spicule heights are up to $\sim
2\hbox{${}^{\prime\prime}$}$ shorter than in Ly$\alpha$. fig:offlimb
Figure 5.: Prominence as seen in almost simultaneous observations with various
instruments. The contours mark the outer envelope of the Ly$\alpha$
prominence. The field of view is the same. _Top left_ : VAULT Ly$\alpha$._Top
right_ : TRACE 171Å. _Bottom right_ : SOHO/MDI photospheric magnetogram.
_Bottom left_ : BBSO H$\alpha$ center.fig:4promin
## 3 The interpretation of the Ly$\alpha$ emission s:ly
The hydrogen Lyman-$\alpha$ line, the strongest line of the solar spectrum, is
a $1s~{}^{2}S_{1/2}$ \- $2p~{}^{2}P_{1/2,3/2}$ doublet resonant line at
1215.67Å. The FWHM of the line core is very broad ($\sim 1$Å ) due to Stark
and Doppler broadening and the high optical thickness. The line center
probably forms in the lower TR ($\sim 40000$ K; 1981ApJS…45..635V) while the
wings form in the chromosphere ($\sim 6000$ K) by partial redistribution of
the core emission. Thus, the Ly$\alpha$ line plays a critical role in the
radiation transport in the chromosphere/TR interface. Below 8000 K, model
calculations show that the line is very close to detailed balance. For
temperatures between approximately 8000 and 30000 K, the dominant energy loss
is through Ly$\alpha$ emission. For temperatures higher than about 30000 K,
Ly$\alpha$ is transparent Gouttebroze (2004). The physics of this line have
been explored in a number of papers Vernazza, Avrett, and Loeser (1981);
Gouttebroze, Vial, and Tsiropoula (1986); Woods et al. (1995); Fontenla,
Avrett, and Loeser (2002); Gouttebroze (2004) , and the average full-Sun line
profile and its variation over the solar cycle has been measured by the SUMER
instrument Lemaire et al. (2004) but most deal with the spectral
characteristics and are of more interest to spectroscopic analysis. On the
contrary, VAULT data consist of the integrated line intensity over a wide
bandpass which includes contributions from other lines such as Si iii, N i, N
v, and C iii. Because of the complexity of the line, model calculations are
the easiest way to interpret imaging observations. Past analysis was based on
plane parallel radiative transfer models using the Ly$\alpha$ contrast (the
ratio of the Ly$\alpha$ emission of a structure relative to the average Quiet
Sun) to derive estimates of pressure and temperature within the observed
structures Bonnet and Tsiropoula (1982); Tsiropoula et al. (1986). Recent
computational and theoretical improvements have enabled the calculation of the
emission from models with more realistic cylindrical geometries Gouttebroze
(2004) and therefore direct comparison with observed Ly$\alpha$ intensities
Gunár et al. (2006); Patsourakos, Gouttebroze, and Vourlidas (2007). However,
calculations from the latter models remain time-consuming and difficult to
apply over the wide range of structures seen in the VAULT images. Since the
scope of our paper is to present a broad overview of the Ly$\alpha$
atmosphere, we return to the plane parallel assumption and adopt the approach
of 1986A&A…167..351T to estimate physical paramaters for the structures in our
images. More careful analyses of specific features will be undertaken in the
future.
The calculations in 1986A&A…154..154G require the calculation of the ratio of
the intensity of a given structure over the average intensity over the solar
disk or “Ly$\alpha$ relative intensity” (LRI). Since we do not have full disk
images in Ly$\alpha$ we cannot compute directly a solar disk average. However,
the disk emission is dominated by the Quiet Sun (Figure 5 in
1974ApJ…187..369P) and we therefore need only to calculate the Quiet Sun
level. Thanks to the large FOV, the VAULT images contain large Quiet Sun
areas. So, we use the median of the lower part of the FoV
$\sim(x\in[-550,-400],\forall y)$ in Figure fig:composite as the ”Quiet Sun”
level. We then calculate the LRI range for several representative features.
The results are shown in Table tbl:meas. The corresponding pressure,
temperature and optical thicknesses derived from 1986A&A…154..154G are also
included. The numbers suggest that most solar structures are optically thick
in Ly$\alpha$ even at temperatures departing significantly from chromospheric
ones ($\geq 10^{4}$ K). Quiet Sun emission seems to arise at the chromosphere
while plage, prominence and offlimb structures have lower TR temperatures and
are presumably located at larger heights. These results are in agreement with
the earlier measurements of 1986A&A…167..351T except of the minimum LRI
values. 1986A&A…167..351T reported values as low as 0.05 but do not observe
LRI below about 0.2 anywhere but at the edges of offlimb loops. The difference
is most likely due to higher sensitivity and spectra purity of the VAULT
instrument which should increase the detected counts of the fainter structures
and minimize the continuum contribution to the Quiet Sun levels relative to
past instruments. The faintest structures (LRI $\sim 0.2$) seen in the VAULT
images are long, thin strands seen in absorption against the network. These
strands are also the smallest resolved structures with the lowest temperatures
(Table tbl:meas). They are very similar to chromospheric filaments but they do
not seem to be associated with any large scale structure. Their origin is
currently a mystery but they could be cooling loops. The best candidates for
optically thin emission are the offlimb loops seen in the northeastern edge of
the VAULT FOV. The observed LRI range of 0.4–0.5 could be consistent with
either chromospheric ($<10^{4}$ K) or TR emission. These loops were not
detected in the BBSO H$\alpha$ images and thus we selected the higher
temperature solutions (T$\sim 3-4\times 10^{4}$K) for them.
Structure | Intensity | Radiance | Opt. Depth | T | Pressure
---|---|---|---|---|---
| [LRI]† | $[10^{12}{ergs\over cm^{2}s\ sr}]$ | $Log$ | [$10^{3}$K] | [dyn/cm2]
Quiet Sun | 0.5 — 5 | 3.3 — 32.5 | 4 — 5 | 8 — 10 | 0.1 — 1
Quiet Sun Prom. | 0.2 — 1.4 | 1.8 — 9.5 | 6 — 3 | 7 — 9 (20)* | 1 (0.1)*
Plage | 5.7 — 12 | 37.5—75.0 | 4 | 10—13 | 1
Plage Prom. | 1—5 | 6.7—32.5 | 3—0 | 8—40 | 0.1—1
Offlimb Prom. | 0.8—1.1 | 5.8—7.8 | 3—0 | 15—80 | 0.1—1
Offlimb Loops | 0.4—0.5 | 2.8—3.8 | 0 | 30—40 | 0.1
Table 2.: Qualitative plasma diagnostics for several types of structures. See
Section s:ly.
* Likely to have reduced optical thickness. High values reflect underlying plage.
$\dagger$ Ly$\alpha$ relative intensity (LRI). LRI=1 represents median of
Quiet Sun region.tbl:meas
Table tbl:meas serves as a concise description of the physical parameters of
Ly$\alpha$ structures and we will refer to it in our subsequent discussions of
individual features starting the contribution of each of these features to the
overall Ly$\alpha$ intensity.
## 4 Sources of the Ly$\alpha$ Intensity s:inten
Figure 6.: Ly$\alpha$ emission histogram (black line). Different colors
represent partial histograms from the labeled subregions. In particular we
note that the ”Quiet Sun” emissions spans one order of magnitude. Intensities
are scaled to the median Quiet Sun level. Plot ordinates scaled to total
number of data pixels. Cover area for each type (integral over the histogram
curve) is: Total (black line): 100%, Quiet Sun: 61%, Plage:13%, Filament: 2%,
Flaring region: 1%, Offlimb: 1%, Rest: 23%. See text on Section s:dis for more
details.histogram
Ly$\alpha$ is a very optically thick line and results in both emission and
absorption depending on the properties of the surrounding plasma. This
interplay is at the region where the plasma starts to be dominated by the
magnetic fields, creating a wide range of intensities. On the other hand, the
strength and variability of the Ly$\alpha$ irrandiance has important effects
on Earth because it affects the chemistry of the mesosphere (e.g., ozon layer)
as well as the climate on longer time scales. Only the central part of the
broad spectral profile of the solar Ly$\alpha$ emission is effective for the
geo-environment. But there is a clear relationship between the central
radiance of the solar Lya line and the total irradiance of the line Emerich et
al. (2005).To understand changes in Ly$\alpha$ irradiance we first need to
identify the contributions of the various solar sources of this emission to
the total Ly$\alpha$ irradiance.
We attempt a first cut at this problem using our spatially resolved,
calibrated images. As we discussed before, we are able to differentiate among
Quiet Sun, Plage, Prominence over Plage, Offlimb and Flaring regions. Figure
histogram shows the corresponding intensity histograms for each domains (color
coded), relative to the overall histogram (black line). The values are
constructed from the pixels inside each region, and considering the median
value for each pixel in time (from Figure fig:composite):
_Quiet Sun (blue line):_ We select a region around the lower right corner in
Figure fig:composite as typical Quiet Sun. Based on this selection, the Quiet
Sun covers 61% of the pixels. We use the median value of the Quiet Sun as a
normalizing factor. Normalized values inside this region, however, span from
0.5 to 5. The Quiet Sun exhibits a wide range of intensities, as it can be
expected by the high optical thickness and strong structuring of the plasma.
The low end of the histogram reaches the edge detection of offlimb
prominences, while the high end reaches the plage levels. Scattered around
this Quiet Sun we find several cases of localized brightenings which may be
related to explosive events, which we discuss in Section s:qs.
_Plage (green line):_ The central part of the VAULT FOV shows a bright plage.
Following a similar method as for the Quiet Sun we find that the plage covers
13% of the pixels, without considering the central overlying filament. Typical
normalized intensities range from 5 to 15\. The only other contribution at
these levels comes from the flaring region at the north edge of the image.
This means that one approximation to the total solar Ly$\alpha$ irradiance can
be obtained using the Quiet Sun level adding a multiplying factor $\sim$7 for
the percentage of the disc corresponding to plages (which could be obtained
from other lines like Ca).
_Filaments over plage (red line):_ Our results show that the plasmas in the
filaments over the plage are sufficiently opaque to reduce the observed
intensity to Quiet Sun values. This particular filament blocks the central 22%
of the plage area.
_Offlimb (purple line):_ The VAULT images contain several examples of limb
structures, including spicules. As discussed later, we find higher heights for
the spicules compared to H$\alpha$. Large overlying loops reaching projected
heights of 60′′ can also be observed. The emission from these structures
indeed shows Quiet Sun levels, down to our detection threshold for the
histogram (0.5). It is likely that these structures are nearly optically thin,
implying temperature $\gtrsim$30,000K. The large heights imply a dynamic state
for these loops and they are probably associated with catastrophic cooling
episodes studied previously with TRACE Schrijver (2001).
## 5 Prominence and Filament Observations s:prom
The images contain a large number of filaments, filamentary structures and a
prominence and it is the first time that the fine scale structure of the
filaments is resolved in this wavelength. Figure fig:threads reveals a highly
organized filament comprised of parallel threads with little, if any, twist.
No obvious twist is evident in any of the other filaments as well. The threads
have a typical width of around 0.5′′or less, and are seen as intensity
enhancement profiles of about 5%. Figure fig:threads also shows a stable and
detached thread with a width reaching the instrument resolution and 30%
absorption over the underlying plage. The filament is further analyzed in
Millard:2009jk where the comparison with the H$\alpha$ observations suggests
that Ly$\alpha$ traces the cool outer plasma while H$\alpha$ originates from
the coolest part of the filament. There is also evidence for uneven absorption
across the filament axis. The northern side shows evidence of Ly$\alpha$
absorption while the southern side shows absorption only in the coronal lines
(171Å ) consistent with the presence of a void or cavity around the filament.
The northern absorption could be understood as a line of sight effect from
low-lying absorbing plasma at the filament flanks.
Figure 7.: Prominence shows reduced absorption threads along its axis below
0.5′′. This supports a non-axisymetric prominence. _Top_ : Single VAULT frame,
with labeled segments for the perpendicular cuts (labeled as ’A’ and ’B’).
Intensities are not scaled. _Bottom_ : Intensity profile perpendicular to the
prominence axis for ’A’ segment and ’B’ detached thread (plot inset).
Intensities are scaled to surrounding median plage value.fig:threads
The last panel in Figure fig:offlimb shows the size discrepancy between
Ly$\alpha$ and H$\alpha$ observations. Only a small knot, $\sim
5\hbox{${}^{\prime\prime}$}$ width, of H$\alpha$ emission is visible whereas
the Ly$\alpha$ prominence extends for almost 50′′.
## 6 Quiet Sun observations s:qs
Figure 8.: Detail of a supergranular cell in the Quiet Sun in Ly$\alpha$. The
horizontal extent is 121′′ and the vertical is 117′′ and the field of view is
centered at around (-500, 250) in Figure fig:composite. These are the first
spatially resolved images of the Ly$\alpha$ emission of a cell interior.fig:qs
The Quiet Sun has been the testing ground for the various theories and
concepts of the structure of the solar atmosphere. It is not surprising then,
that it is also the area where VAULT observations have generated the most
interesting results Patsourakos, Gouttebroze, and Vourlidas (2007); Judge
(2008). Earlier observations showed that Ly$\alpha$ emission is concentrated
along the supergranular lanes in clumps with small loop-like extentions
towards the cell interiors. Faint emission without spatial structures was
detected at the cell centers. VAULT images, especially VAULT-I which covered a
much larger Quiet Sun area, resolved the spatial stucture in the clumps along
the supergranular boundaries (Figure fig:qs). The Quiet Ly$\alpha$ Sun area
shows groupings of filamentary plasma, similar to the H$\alpha$ rosettes, with
a typical diameter of $\sim 23\hbox{${}^{\prime\prime}$}$. These rosettes show
filamentary structure up to resolution limit of the instrument, of about
$0.4\hbox{${}^{\prime\prime}$}$. This grouping in rosettes is stable through
the observations ($\sim 6$ min) but shows the presence of localized
brightening events with a timescale variation $60-120$ sec and sizes of a
couple of arcseconds. The network structures rise above the chromosphere about
7100 km or 10′′ as seen in Figure fig:offlimb. This measured value is
consistent with previously measured values of the height of the transition
region above the limb. Their location at the supergranular cell boundary
uniquely identifies these loops as being the byproduct of convective motion
driving together magnetic fields at the edges of the supergranular cell.
The outer areas consist of short loop-like structures while the centers of the
clumps have a more point-like nature. This morphology is consistent with loops
of progressively higher inclination towards the center of the boundary. The
obvious question is whether the more extended Ly$\alpha$ loops are full loops
or just the lower part of a larger structure, possibly extending to higher
temperatures. 2007ApJ…664.1214P applied an analysis method used for coronal
loops to a detailed Ly$\alpha$ emission model and found that the short loops
at the edges of the boundary channel were consistent with full Ly$\alpha$
loops and therefore could account for the “cool” loops predicted by models of
the transition region Dowdy, Rabin, and Moore (1986). However, the magnetic
footpoints of these loops could not be identified in photospheric magnetograms
due to the lower spatial resolution and reduced sensitivity of the MDI data.
Although these problems should not affect the larger loops, their footpoints
remain ambiguous. To address these problems, 2008ApJ…687.1388J decided to
investigate the magnetic origin of the extended Ly$\alpha$ loops using
magnetic field extrapolations. They found that the longer Ly$\alpha$ loops
originate near the boundary center and are more likely the lower extensions of
large scale loops that connect areas much more distant than the neighboring
cells. The extrapolations showed that the smaller loops at the edge of the
network lanes are indeed small scale loops supporting the interpretations of
2007ApJ…664.1214P.
### 6.1 Cell Interior
Another new observation from VAULT is the imaging of Ly$\alpha$ emission from
the cell interiors for the first time. As can be seen in the example of Figure
fig:qs, the emission extends over the full interior area and is structured in
various spatial scales. The emission is filamentary, optically thick with some
apparent dependence on the local radiation field. The associated time series
(movies available in the online VAULT archive) reveal significant evolution in
these structures, like flows and jets. The material within the filamentary
structures shows an overall motion towards the network boundary similar to the
motions of emerging magnetic field elements in photospheric magnetograms and
white light images. As magnetic field of opposing direction accumulates in the
boundary, it is expected that some cancellation is taking place. Indeed, there
are a few cases where Ly$\alpha$ material appears to jet out from smaller
emission clumps creating a bright point. These events are never seen in the
cell center and could originate from magnetic reconnection closer to the
photosphere. Some examples can be seen along the column at
$-500\hbox{${}^{\prime\prime}$}$ in Figure fig:composite. The limited
resolution of available magnetograms has not allowed us to locate the origins
of these jets.
Figure 9.: Microflaring event in the Quiet Sun detected in Ly$\alpha$ and He
ii. Top panels: EIT He ii images of the event. Middle panel: Comparison of the
energy curves from VAULT to the EIT light curve. The total counts within the
black outline (VAULT) and within the white boxes (EIT) were used to calculate
the curves. Bottom panels: Ly$\alpha$ images of the event at its initiation
(left) and peak (right). fig:blob
### 6.2 Mircoflaring in the Quiet Sun
Although the VAULT time series show continuous motions and brightness
evolution thoughout the full field of view, there are very few strong
enhancements that could qualify as flaring emission. The short duration of the
flight may be a reason for this but we were able to isolate only $2-3$ events.
Figure fig:blob shows an example from a Quiet Sun feature which gives rise to
a plasma jet rising from the cell center. The brightening was detected by
SOHO/EIT which classifies it as a regular bright point. The event lasts for
$\sim 500$ s.
Since we have calibrated images, we could estimate the thermal energy of the
Ly$\alpha$ flaring under some assumptions. We adopted equation (5) in
2002ApJ…568..413B
$E_{th}=3k_{B}T\sqrt{EMV}$ (1)
where the energy $E_{th}$ corresponds to Ly$\alpha$ plasma at temperature $T$
and emission measure, $EM$ integrated over volume $V$. We assumed $T=2\times
10^{4}K$, $V=~{}{\mathrm{d}}A~{}{\mathrm{d}}l$,
$dA=21\hbox{${}^{\prime\prime}$}\times 21\hbox{${}^{\prime\prime}$}$ area, and
$dl=0.5\hbox{${}^{\prime\prime}$}$ equal to the mean free path of a Ly$\alpha$
photon for optically thick emission. For the estimation of $EM$ we adopted the
calculations in 2001ApJ…563..374V but used the updated photometric calibration
reported here. The new $EM$ calibration for VAULT is 1 DN s-1 pix-1 =
$3.74\times 10^{26}$ cm-5. To account for integrating the energies over an
area which may contain both flaring and background (likely optically thin)
emission, we have subtracted the emission from the first, pre-event image from
the plots. The resulting energy levels are very similar to those for coronal
bright points Krucker and Benz (1998) as the EIT observation of plasma at
$T\geq 8\times 10^{4}$ K suggests. Unfortunately, we cannot tell whether there
is any coronal emission from this bright point because it lies outside the
TRACE field of view and EIT was observing solely in He i during the VAULT
flight. We only report counts for the EIT light curves because there is only
one wavelength available and the emission measure cannot be calculated (right
axis in Figure fig:blob). The VAULT and EIT curves are aligned at the pre-
event emission level along the intensity axis to allow a comparison. The main
conclusions from Figure fig:blob are that the Ly$\alpha$ and He ii have a
similar impulsive phase and the He ii emission seems to be the extention of
the cooler Ly$\alpha$ emission. This is also in agreement with the earlier
results showing a delay from cooler to hotter coronal lines and extends the
detection of heating events to a much lower layer of the atmosphere.
Figure 10.: Microflaring event in the Ly$\alpha$ plage. Left: The symbol ’x’
marks the location of the brightening. Right: The energy estimate for this
event.fig:blob2
The energy estimates in Figure fig:blob are in the range of microflares which
seems reasonable for the lower TR. An inspection of the plage area around the
filament shows fainter brightenings that could still be classified as
impulsive based on their light curves. Energy estimates for those brightenings
are around $<5\times 10^{23}$ ergs, lower than a microflare. Figure fig:blob2
shows an example of such a brightening. The energy was estimated over an area
of $\sim 1.8\hbox{${}^{\prime\prime}$}\times 1.8\hbox{${}^{\prime\prime}$}$;
all other assumptions are the same as above. Because these brightness changes
are very close to the overall brightness variability of the plage, it is
difficult to say with certainty that these are flaring events. A more
sophisticated analysis is required but it is beyond the scope of the paper.
## 7 Plage and Spicules in $Ly\alpha$s:spicules
The active plage has been studied in some detail using the first VAULT
observations Vourlidas et al. (2001). The large degree of spatial structuring
and the variability of these structures combined with the complex radiative
character of Ly$\alpha$ emission complicate the detailed analysis of the
plage. The plage has clearly a different morphology than the Quiet Sun. It
lacks extended loop-like structures, but contains many point-like brightenings
reminiscent of the 171 Å moss. Actually the TRACE 171 Å images show moss over
the majority of plage with large scale loops located only in the periphery
(Figure fig:4promin). As expected, the moss underlies hotter loops seen in the
EIT Fe XV 284 Å images but the Ly$\alpha$ brightness is not correlated with
the degree of coronal heating above. A quick inspection of the Ly$\alpha$ and
171 Å images in Figure fig:4promin shows that despite the largely similar
mossy appearance, there are several areas without a detailed correlation
between corona and lower TR as noted before (e.g., region R2 in Figure 2 of
2001ApJ…563..374V). Neutral hydrogen diffusion across field lines as proposed
by 2008ApJ…683L..87J maybe an explanation of the uniform brightness of the
plage in Ly$\alpha$ but better calculations are needed before we can establish
the viability of this mechanism.
### 7.1 Detection of Proper Motions
A significant part of the variability seems quite random. For a given pixel,
the brightness change could be due to the weakening of the emission, the
lateral motion of the bright point or the appearance of dark (likely
absorbing) features. We believe that these changes can be understood as the
buffeting of the Ly$\alpha$ moss by chromospheric H$\alpha$ jets similarly to
the picture proposed by 1999SoPh..190..419D for the 171 Å moss but extending
it to much smaller spatial scales.
On the other hand, we can identify coherent motions in several places. The
most obvious ones can be found at or near the filament footpoints and along
their backbone structure. Blobs of weakly emitting Ly$\alpha$ seem to flow
towards the lower atmosphere. At the same time, apparently upward moving blobs
can be seen also at the filament footpoints as well as along the boundaries of
the small network cell within the plage and basically in most locations where
there is high contrast with the background.
Figure 11.: Detection of proper motions in Ly$\alpha$ plage. The lengths of
the displacement vectors are proportional to the esimated speed. Only pixels
with correlation coefficients $\geq 0.3$ and intensity changes $\geq 3\sigma$
above the background are considered. The units are VAULT pixels
($0.112\hbox{${}^{\prime\prime}$}$/pixel). Left: Filament and neaby plage.
Upward motions can be seen along the western footpoint. Right: Plage detail.
Note the counterstreaming motions along the filament boundary and diverging
(explosive?) motions at certain bright points.fig:motions
In an attempt to quantify these motions we used a local correlation method to
track the blobs in time. To suppress the influence of the background buffeting
motions we calculated the standard deviation, $\sigma$, of the intensity
variability for each pixel at the peak of the emission and then considered
only pixels with $\geq 3\sigma$ as inputs to the cross correlation algorithm.
The large degree of variability and spatial structuring results in many
correlations. So we kept only the pixel with correlation coefficients higher
than 0.3 and estimated their speeds and velocity vectors. We derive speeds in
the range of 5-20 km s-1 which are similar to the speeds of chromospheric
fibrils and spicules (e.g., 2000A&A…360..351W; 2008ApJ…673.1194L). In general,
the cross correlation results showed motions in all directions reinforcing the
visual impressions of the large degree of randomness in the Ly$\alpha$
structures. However, a closer inspection of the displacement vector revealed
several instances of coherent motions. In the example of Figure fig:motions,
we can see upward motions along the western filament footpoints and the
filament boundaries. There was clear evidence of counterstreaming motions
along the filament. Some of those were in the upper range of our estimated
speeds ($\sim 20$ km s-1) and are very close to H$\alpha$ measurements in
filaments Engvold (1998); Lin, Engvold, and Wiik (2003). The nearby plage
showed motions that followed the curvature of the filament (Figure
fig:motions, right panel). They may lie along thin, dark strands that are part
of the filament rather than the plage. Coherent apparently upward motions were
also detected at network boundaries along spicular-like structures. The most
interesting results were at locations of diverging motions as can be seen
towards the upper end of the field (Figure fig:motions). Some were associated
with moderate flare-like brightenings (right panel in Figure fig:motions and
Figure fig:blob2) and may suggest an explosive nature for these intensity
changes. It is possible that some of the TR variability seen in TR lines with
coarser resolution and attributed to stationary brightenings could actually be
an effect of spatial smoothing of the above mentioned flows. In other places,
we found diverging vectors suggesting rotation. In the wavelet-processed
movies, we see unwinding features at those areas. They are very suggestive of
the so-called mini-CMEs detected recently by STEREO and associated with vortex
flows at supergranular boundaries Innes et al. (2009).
A big advantage of VAULT’s large FOV is the observation of similar structures
both on disk and at the limb. An obvious candidate are the spicules.
2009A&A…499..917K measured the dynamics of several Ly$\alpha$ spicules and
found many similarities to the H$\alpha$ dynamic fibrils despite the short
VAULT time series. Based on the TRACE co-alignement we measured Ly$\alpha$
spicules to be $8\hbox{${}^{\prime\prime}$}-12\hbox{${}^{\prime\prime}$}$ in
height, from VAULT spicule edge to the TRACE limb position. When we consider
the co-aligned cotemporal BBSO H$\alpha$ channel, Ly$\alpha$ spicules can be
up to $\sim 2$′′ higher than in the comparatively optically thinner H$\alpha$.
Although scattered light may play a role in the ground-based observations, the
height difference between Ly$\alpha$ and H$\alpha$ appears to be significant.
These results imply that Ly$\alpha$ spicules could be the outer sheaths of the
H$\alpha$ fibrils Koza, Rutten, and Vourlidas (2009). When we take into
account similar results between H$\alpha$ and C iv de Wijn and De Pontieu
(2006) it becomes obvious that chromospheric mass is propelled to the corona
via the fibrils and undergoes heating appearing in successively higher
temperatures (Ly$\alpha$ to C iv, for example). This scenario seems to
corroborate the very recent results of 2009ApJ…701L…1D where it is proposed
that type-II spicules may be the means of chromospheric plasma transport to
coronal levels and temperatures and may play an important role in the coronal
heating problem. Further observations of both fibrils and type-II spicules
are, therefore, highly desirable in Ly$\alpha$ (in addition to chromospheric
and coronal lines) to provide a more robust connection between the evolution
of the chromospheric and coronal structures. For the moment, the above
discussion suggests that spicules/fibrils may provide the mass heated to
coronal temperatures (e.g., 2007PASJ…59S.655D).
Overall, our initial attempt to characterize the variability seen in the VAULT
images seems to provide reasonable results. The most serious problem is the
large amount of variability in all intensity and spatial levels. We plan to
revisit the analysis of proper motions using our newly available wavelet-
processed images which supress the background “noise” and may enhance the
effectiveness of cross-correlation techinques.
## 8 Discussion s:dis
The VAULT data, being taken from a sounding rocket platform, do not permit
long time series investigations of the Ly$\alpha$ atmosphere. However, they do
provide several tantalizing clues about the dynamics and morphology of the
crucial interface of the upper chromosphere/lower TR at least for long-lived
structures and for variability at a time scale of a few minutes.
The improved photometric analysis of the VAULT data combined with better
Ly$\alpha$ models show that, for wideband imaging at least, most of the
emission originates from the lower TR ($\geq 10^{4}$ K) and only the darker
areas contain much chromospheric material (Table tbl:meas). Therefore,
Ly$\alpha$ imaging observations are a great probe for the structure of the
transition region Teriaca et al. (2005). It seems that the Ly$\alpha$ Quiet
Sun is dominated by longer thread-like structures reminiscent of H$\alpha$
fibrils. The VAULT observations have provoked new ideas about the nature of
the TR as the region where neutral hydrogen atoms from these threads diffuse
across magnetic field lines, interact with nearby electrons and subsequently
excite, ionize, and/or radiate to provide the emission we see in TR lines
Judge (2008). These ideas remain to be tested in detail but they demonstrate
the value of sounding rocket observations.
The high spatial resolution of the VAULT data resolves a great deal of
variability, mostly associated with lateral motions, in the plage. We believe
that the majority of this variability can be explained as buffeting of the
Ly$\alpha$ structures by cooler material, such as H$\alpha$ jets. In addition,
the VAULT observation of spicules show that they extend higher and have larger
widths but otherwise similar dynamics Koza, Rutten, and Vourlidas (2009) with
their H$\alpha$ counterparts. These observations verify past SUMER results
Budnik et al. (1998) and provide significant support for an interesting idea
put forth recently by 2009ApJ…702.1016D to explain the large emission measure
discrepancies between coronal and lower TR structures Vourlidas et al. (2001)
as a result of EUV absorption from chromospheric material injected in the
corona. When we consider these observations/ideas together; namely, the long
network loops and neutral cross-field diffusion, the continuous buffeting, and
the Ly$\alpha$ jets as extension of H$\alpha$ dynamic fibrils, we come to the
conclusion that the transition region may be nothing more than the transient,
evaporating part of the chromosphere rather than the stable layer in the
simple 1D models, such as Vernazza, Avrett, and Loeser (1981), long favoured
in our discipline. The VAULT and more recently Hinode/SOT observations are
making us reassess our views on the structure of the lower solar atmosphere.
The large field of view of the instrument led to observations of basically
every solar structure, with the exception of coronal holes, which enabled us
to estimate the contribution of various Ly$\alpha$ sources to the observed
intensity and thereby introducing the first empirical segmentation of
Ly$\alpha$ irradiance to its sources (Sec. s:inten). We found that Quiet Sun
features can have intensities several times the intensity of the average Quiet
Sun and that filaments exhibit both absorption and emission in Ly$\alpha$. The
latter can be as bright as weak bright points. Optically thin structures, up
to 50% fainter than the average Quiet Sun may exist in the center of cell
interiors and as off limb loops. We did find that high temperatures are likely
in off-limb Ly$\alpha$ loops which may explain their large heights ($\sim
60^{\prime\prime}$, corresponding to $\approx 45,000$ km) in the VAULT images.
We also found that active region filament partially absorbs plage emission, by
around 20% to 30%, and this effect may need to be considered carefully in
irradiance studies. These segmentation results may be useful to irradiance
studies until a full disk Ly$\alpha$ imaging becomes available.
The VAULT images provide the first ever unambiguous Ly$\alpha$ imaging of the
fine structure of filaments/prominences and show that both emission and
absorption takes places along the prominence backbone. It is interesting to
note, that the underlying plage is visible through several locations along the
prominence suggesting that Ly$\alpha$ is optically thin and that the
distribution of hydrogen is highly anisotropic through these structures. It is
also clear that the Ly$\alpha$ filament is larger than the H$\alpha$ one
Millard, Vial, and Vourlidas (2009) and is likely to reach a higher altitude.
The high LRI measurements in the filaments (up to 5, Table tbl:meas) are again
consistent with a decreased optical thickness, even to the point of being
optically thin. According to 1986A&A…154..154G and their Figure 5, a very hot
temperature of $\sim 5\times 10^{4}$K is also possible. For this study we have
chosen the cooler more plausible solution of the curve. Nevertheless, with the
lack of other observational constraints, it remains unsolved whether the hot
solution is possible. One approach would be a point-to-point correlation with
other chromospheric-TR lines. This calls for a high-resolution spectrograph
which is not currently available in space.
VAULT images also reveal a wealth of activity in both the plage and the Quiet
Sun regions. In the latter, we see evidence of braiding in the loop structures
outlining the cell boundaries. However, we do not see any direct unambiguous
evidence of reconnection as would be expected from such activity. It may be
that longer time series are needed to evidence such events. Alternatively, a
mixture of cool absorbing structures propagating alongside these loops may
create the appearance of braiding. Those structures may be the same absorbing
structures that create the buffeting motions in the plage. On the other hand,
we see frequent brightenings and even jets in the interior of the cells. This
is the first time that the Ly$\alpha$ emission from these areas has been
imaged and the amount of observed activity was unexpected. The brightenings
seem to be associated with the emergence of magnetic field elements and their
subsequent movement towards the cell boundary. These motions are regularly
seen with sub-arcsecond resolution magnetographs (e.g., SOUP instrument,
1989ApJ…336..475T) but we did not have any available during the flight. The
relation between the emerging flux and the Ly$\alpha$ brigtenings remains to
be confirmed in a future flight but if it is true it suggests that the effects
of even such small magnetic elements reach substantial heights in the solar
atmosphere. We wonder whether some of those jets are the Ly$\alpha$
counterparts of the Type II spicules seen in the SOT observations de Pontieu
et al. (2007a).
Another rather surprising observation is the relative scarcity of microflaring
events. We have been able to identify a handful in the five minutes of
observation. These have been previously identified by 1998A&A…336.1039B. They
suggest a possible link with atmospheric turbulence. In our observations we
have observed them in both active regions and Quiet Sun regions. The largest
of them had a light curve and energy consistent with a microflare and was
detected in He ii as well (Figure fig:blob). Others had energies in the range
of $10^{24}$ ergs. Although the short duration of the observation does not
allow proper statistics for the occurrence of these events, our field of view
covers a substantial part of the solar disk. Therefore, it seems unlikely that
microflares are a common occurrence in this temperature range.
## 9 Conclusions s:con
We conclude our overview of Ly$\alpha$ imaging observations with a set of
“lessons learned” that may be useful in the design of future Ly$\alpha$
instruments or observation campaigns.
* •
Ly$\alpha$ is formed at the critical interface between the upper chormosphere
and the low TR. Thus, imaging is very useful and the well-known difficulties
surrounding the interpretation of Ly$\alpha$ emission are not insurmountable
anymore. We can rely on models to derive reasonable physical parameters for
the observed structures.
* •
We see few Ly$\alpha$ structures close to the instrument resolution limit of
$0.5\hbox{${}^{\prime\prime}$}$. Only absorbing (dark) features and off-limb
structures (in emission) can be identified at that resolution. Most of the on-
disk structures are much larger. This could be due to the high optical
thickness of the line throughout these structures. In any case, this
observation should be considered in the design of future Ly$\alpha$
telescopes. Extreme resolutions may not be useful unless the instruments can
spectrally resolve the line or their science objectives include absorption or
off-limb features.
* •
There is evidence of optically thin emission in many locations besides the
obvious limb structures. Areas around filaments are especially interesting.
This would require observations of the spectral profiles to be confirmed.
* •
There is considerable structure and variability within the cell interiors
which is probably linked to photospheric flux emergence. This is a new area
for Ly$\alpha$ studies and to understand it will require a future telescope
sensitivity equal or better than VAULT.
* •
Even if flaring activity is relatively unimportant, there is variability.
Future instruments should achieve both high signal-to-noise ratio and cadence
to allow the study of both.
* •
Both types of spicules are observable and given the significant temperature
range of Ly$\alpha$ formation, observations in Ly$\alpha$ are excellent
tracers of the injection of material from the chomosphere to the oorona.
For the near future, the advent of Hinode/SOT has created a new and unique
opportunity to address the nature of the transition region by combining VAULT
and SOT observations of Quiet Sun structures and spicules. We plan to seek
funding for refurbishment of the VAULT payload, which was damaged during its
last flight, and for an underflight with SOT with the specific objectives of
addressing the nature of the long Ly$\alpha$ fibrils over the quiet network
and investigate type-II spicule dynamics particularly at coronal holes.
However the dynamics of the type-I spicules and macro-spicules may need longer
time series due to their longer lifetimes Xia et al. (2005).
To summarize, we presented a broad overview of the morphology and dynamics of
the Sun’s Ly$\alpha$ atmosphere; an important but rarely imaged region. These
were the first sub-arcsecond, high sensitivity observations of this line and,
at that time, the highest resolution observations of any solar structure from
space. The VAULT observations showed that Ly$\alpha$ emission arises from
every location and in every solar feature, and generated new ideas about the
nature of the transition region and coronal heating. These results demonstrate
the wideranging value of sounding rocket experiments despite their short
observing windows.
#### Acknowledgements
This work is dedicated to the memories of D. Prinz, G. Bruecker, and D. Lilley
whose efforts have contributed enormously to the success of the NRL sounding
rocket programs. We are grateful to V. Yurchyshyn for providing calibrated
BBSO H$\alpha$ images, and to J. Cook, J. Koza, S. Martin, R. Rutten, J.C.
Vial, for useful discussions and constant encouragement. The achievements
presented in this paper are the product of many years of development work at
the Naval Research Laboratory Solar Physics Branch and the NASA sounding
rocket program. The VAULT instrument borrows heavily from the High Resolution
Telescope and Spectrograph. The NRL rocket team of J. Smith, R. Moye, R.
Hagood, R. Feldman, J. Moser, D. Roberts, T. Spears and R. Waymire did a
superb job in preparing and launching the instrument. We would like to
acknowledge the efforts of the sounding rocket support team that made the
VAULT launches possible. We would like to particularly acknowledge the
following individuals. Tracy Gibb did a superb job managing the launch of the
VAULT payload. Frank Lau managed the development of the Mark 7 digital SPARCS
attitude control system. We would like to acknowledge the SPARCS team for
their superb efforts in the support of our launch. Jesus and Carlos Martinez
developed and operated the Mark 7 SPARCS. Richard Garcia, Shelby Elborn and
Kenneth Starr developed the VAULT telemetry section. The support from the
White Sands Missile Range and Wallops Flight Facility NASROC personnel was of
the highest caliber. The VAULT instrument development work has been supported
by the ONR task area SP033-02-43 and by NASA defense procurement request
S-84002F.
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|
arxiv-papers
| 2009-12-11T16:47:53 |
2024-09-04T02:49:06.972801
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Vourlidas, B. S\\'anchez-Andrade Nu\\~no, E. Landi, S. Patsourakos,\n L. Teriaca, U. Sch\\\"uhle, C.M. Korendyke, I. Nestoras",
"submitter": "Bruno S\\'anchez-Andrade Nu\\~no",
"url": "https://arxiv.org/abs/0912.2272"
}
|
0912.2283
|
# A Dynamic Renormalization Group Study of Active Nematics
Shradha Mishra smishr02@syr.edu Physics Department, Syracuse University,
Syracuse NY 13244 USA R. Aditi Simha aditi@physics.iitm.ac.in Department of
Physics, Indian Institute of Technology Madras, Chennai 600 036, India Sriram
Ramaswamy sriram@physics.iisc.ernet.in Centre for Condensed Matter Theory,
Department of Physics, Indian Institute of Science, Bangalore 560 012 India
###### Abstract
We carry out a systematic construction of the coarse-grained dynamical
equation of motion for the orientational order parameter for a two-dimensional
active nematic, that is a nonequilibrium steady state with uniaxial, apolar
orientational order. Using the dynamical renormalization group, we show that
the leading nonlinearities in this equation are marginally irrelevant. We
discover a special limit of parameters in which the equation of motion for the
angle field of bears a close relation to the $2d$ stochastic Burgers equation.
We find nevertheless that, unlike for the Burgers problem, the nonlinearity is
marginally irrelevant even in this special limit, as a result of of a hidden
fluctuation-dissipation relation. $2d$ active nematics therefore have quasi-
long-range order, just like their equilibrium counterparts
###### pacs:
Active nematics sradititoner ; chateginellimontagne ; vjmenonsr are the
simplest example of spontaneously broken rotation-invariance in a
nonequilibrium system. Analytical studies of their statistical properties have
mainly been confined to a linearized approximation sradititoner , whose
predictions of anomalous density fluctuations have largely been confirmed in
experiments vjmenonsr and numerical simulations chateginellimontagne . Within
the theory of sradititoner the density fluctuations were driven by the
broken-symmetry modes associated with orientational order. In this paper we
ignore density fluctuations and focus on the effect of the broken-symmetry
modes on the strength of orientational order. We ask: can a noisy two-
dimensional system of active particles display long-range nematic order?
Let us see why this question is worth asking. It is well known that at thermal
equilibrium, in two space dimensions, neither XY models nor nematic liquid
crystals can have long-range order. Instead of a true ordered phase, these
systems have a critical low-temperature state in which the fluctuation-
averaged order parameter vanishes in the thermodynamic limit at all nonzero
temperatures, but order-parameter correlations decay as a power of distance MW
; hohenberg ; KT ; frenkels . The simplest generalization of the $2d$ XY model
to a nonequilibrium steady state is the Vicsek model vicsek of flocks in two
dimensions, in which the local velocity of the flock is the XY order-parameter
field. Toner and Tu tonertu showed that the resulting advection of the order-
parameter field by its own fluctuations tonertu stabilizes long-range order
even in two dimensions. Technically, the mechanism amounted to a singular
renormalization of the XY stiffness by nonlinearities of a type not permitted
in the equilibrium XY model. The ordered state of a Vicsek flock can be
thought of as a collection of arrows all pointing on average in the same
direction; this is known as polar order. One can imagine a different ordered
state, in which the axes of the arrows are on average parallel to an
arbitrarily chosen spatial direction, call it $\hat{\bf n}$, but the arrows
point indifferently along $+\hat{\bf n}$ and $-\hat{\bf n}$ or, equivalently,
one could simply lop the heads off the arrows. The resulting state is apolar,
and has purely nematic order. The Vicsek flock moves on average in the
$\hat{\bf n}$ direction, while a nonequilibrium steady state with nematic
order – an active nematic – cannot tell forward from back, and so does not
drift on the average. The nature of order in such active nematics is the
subject of our study. Our main concern is whether the interplay of
nonlinearity and fluctuations stiffens the order-parameter fluctuations in
active nematics as it does tonertu in polar ordered phases, leading to true
long-range order in two dimensions.
Here are our main results. (i) We elucidate the route to the equation of
motion for the nematic orientational order parameter, taking care to
distinguish the constraints introduced purely by rotation invariance, and
hence applicable to both active and equilibrium systems, from those arising
specifically in the thermal-equilibrium limit. (ii) We show that the two
quadratic nonlinearities in the equation of motion have independent
coefficients, unlike in the equilibrium case where they are determined by a
single parameter. In both equilibrium and active nematics power-counting shows
that the nonlinearities are marginal, but such analysis cannot distinguish
marginally relevant from marginally irrelevant. (iii) In a certain limit of
parameter values, our equation of motion can be mapped to the noisy two-
dimensional Burgers FNS and KPZ BurgerKPZ equations, but with a velocity
field ${\bf v}$ satisfying the peculiar condition
$\partial_{x}v_{x}-\partial_{z}v_{z}=0$, which is neither solenoidal nor
irrotational. (iv) The similarity to the Burgers problem ends there: our
dynamical renormalization-group treatment shows that the nonlinearities are
marginally irrelevant in our theory, in the Burgers limit as well as in
general. Active nematics thus have only quasi-long-range order. Although
disappointing if one is looking for novelty in nonequilibrium systems, this
negative result reinforces the findings of a numerical study
chateginellimontagne of an apolar generalization of the Vicsek model.
This paper is organized as follows. In section I we construct the coarse-
grained equations of motion for the nematic order parameter, highlighting the
differences between equilibrium and active systems. In section I.2 we examine
the relation of our equations of motion to the Burgers and KPZ equations, in a
special high-symmetry limit. In section II we outline the dynamic
renormalization group (DRG) treatment with which we extract the long-time,
long-wavelength properties of correlation functions in our system. Further
calculational details are relegated to the Appendix. The paper closes in
section III with a discussion of possible future directions.
## I Equation of motion
We now construct the equations of motion for an active nematic. Since we are
considering a system that can undergo apolar orientational ordering, one of
the slow variables for a coarse-grained description of the dynamics is the
traceless symmetric second-rank tensor nematic order parameter Q degp . The
magnitude of Q is slow upon approach to the ordering transition, and the
fluctuations of its principal axis are the broken-symmetry modes of the
ordered phase. If the system were isolated, mass and momentum would be
conserved within the system and the corresponding densities $\rho$ and ${\bf
J}=\rho{\bf v}$, ${\bf v}$ being the velocity field, would be slow variables
as well temperaturefootnote . However, we will consider a system adsorbed on a
solid surface which acts as a momentum sink, thus turning ${\bf J}$ or ${\bf
v}$ into a fast variable, and allow deposition and evaporation khandkarbarma ,
i.e., birth and death tonerpc , thus rendering $\rho$ fast as well. We will
start from a complete dynamical description, eliminate the fast $\rho$ and
${\bf J}$, and obtain the dynamics of Q alone.
For a system where particles can enter and leave the system in the bulk, the
density obeys
$\frac{\partial\rho}{\partial t}=-\gamma\rho+\beta-\nabla\cdot{\bf
J}+f_{\rho}.$ (1)
The third term on the right of (1) contains the number-conserving motion of
particles on the substrate. The random adsorption and desorption of discrete
particles has two effects. In the mean, conditioned on a given local density
$\rho({\bf r},t)$, it leads to the $\gamma$ and $\beta$ terms. Fluctuations
about this average effect lead to the nonconserving spatiotemporally white
noise $f_{\rho}$. A steady, spatially uniform state has mean density
$\rho_{0}\equiv\beta/\gamma$. Newton’s second law for the momentum density
$m{\bf J}$ reads
$m\frac{\partial{\bf J}}{\partial t}=-\Gamma{\bf v}+{\bf
f}_{R}-\nabla\cdot\sigma$ (2)
The first term on the right hand side of (2) is friction due to the substrate,
with a kinetic coefficient $\Gamma$. The random agitation of the particles as
a result of thermal motions, biochemical stochasticity, or dynamical chaos is
modelled in the simplest possible manner by the spatiotemporally white
Gaussian noise ${\bf f}_{R}$. This noise is nonconserving, i.e., its strength
is nonvanishing at zero wavenumber, since the dynamics is not momentum-
conserving. The last term contains all effects arising from interactions of
the particles with each other, and thus takes the momentum-conserving form of
the divergence of a stress tensor $\sigma$. In principle $\sigma$ contains
stresses coming from the free-energy functional for Q (see below)qviscous
These, however, are readily seen sradititoner to be irrelevant at large
lengthscales compared to the contribution $\sigma^{a}=w_{1}\rho\mbox{Q}$
coming from the active nature of the particles activestressrefs .
The equation of motion for the orientational order parameter Q including
coupling to the velocity field dforster ; pdolmsted is
$\frac{\partial\mbox{Q}}{\partial t}+{\bf v}\cdot\nabla\mbox{Q}=\Gamma
G+(\alpha_{0}{\bf\kappa}+\alpha_{1}{\bf\kappa}\cdot\mbox{Q})_{ST}+\Omega\cdot\mbox{Q}-\mbox{Q}\cdot\Omega$
(3)
where ${\bf\kappa}=[\nabla{\bf v}+(\nabla{\bf v})^{T}]/2$ and
$\Omega=[\nabla{\bf v}-(\nabla{\bf v})^{T}]/2$ are the shear rate and
vorticity tensor respectively, $\Gamma$ is a kinetic coefficient
kincoeffootnote , and the parameters $\alpha_{0}$ and $\alpha_{1}$
characterise the coupling of orientation to flow. The molecular field
$G=-\delta F/\delta\mbox{Q}$ is obtained from an extended Landau-de Gennes
free energy
$\displaystyle F$ $\displaystyle=\int d^{d}x[{a\over
2}\mbox{Tr}\mbox{Q}^{2}+{u\over 4}(\mbox{Tr}\mbox{Q}^{2})^{2}+{K\over
2}(\nabla_{i}Q_{kl})^{2}$
$\displaystyle+\bar{K}Q_{ij}\nabla_{i}Q_{kl}\nabla_{j}Q_{kl}+CQ_{ij}\nabla_{i}\nabla_{j}\rho]+\Phi[\rho]$
(4)
where we have left out terms cubic in Q as these vanish degp in dimension
$d=2$. The density $\rho$ enters $F$ through the functional $\Phi$, the
quadrupolar coupling term with coefficient $C$, and the $\rho$-dependence of
parameters in $f$. On timescales much larger than $1/\gamma$ and $m/\Gamma$,
the density and momentum equations (1) and (2) become constitutive relations
determining $\rho$ and ${\bf J}$ in terms of the slow field Q. Eq. (1 tells us
we can replace $\rho$ everywhere by $\rho_{0}$ to leading order in gradients,
and (2) becomes
${\bf v}\simeq-{w_{1}\rho_{0}\over\Gamma}\nabla\cdot\mbox{Q}$ (5)
apart from noise terms. The molecular field $G$ in (3) contains a term of the
form $\mbox{Q}\nabla\nabla\mbox{Q}$, and one of the form
$\nabla\mbox{Q}\nabla\mbox{Q}$, whose coefficients will be related as both
terms arise as variational derivatives of the single $\bar{K}$ term in $F$
[Eq. (4). Replacing ${\bf v}$ by its expression (5) in Eq. (3) will give rise
to additional terms of that form, controlled by the activity parameter
$w_{1}$. As a result, the $\mbox{Q}\nabla\nabla\mbox{Q}$ and
$\nabla\mbox{Q}\nabla\mbox{Q}$ terms in the effective equation of motion for Q
cannot be combined into the variational derivative of a scalar functional, and
will have two independent coefficients. We will explore below the consequences
of the existence of two independent nonlinear couplings. In space dimension
$d=2$ the order-parameter tensor has the simple form
$\mbox{Q}={S\over 2}\left(\begin{array}[]{cc}\cos 2\theta&\sin 2\theta\\\ \sin
2\theta&-\cos 2\theta\end{array}\right),$ (6)
where the scalar order parameter $S$ measures the magnitude of nematic order
and $\theta$ is the angle from a reference direction. Let us work in the
nematic phase, where we can take $S=$ constant and define $\theta=0$ along
axis of mean macroscopic orientation. Eq. (5) for small $\theta$ becomes
${\bf v}=-\bar{\Gamma}^{-1}(\partial_{z}\theta,\partial_{x}\theta),$ (7)
neither a gradient nor a curl, $\bar{\Gamma}$ being a constant determined by
those in (1) - (6). Substituting ${\bf v}$ in (3) by its expression (7),
writing Q in terms of $\theta$ as in (6), treating $S$ as constant, and
including noise terms, we obtain
$\frac{\partial\theta}{\partial
t}=A_{1}\partial_{x}^{2}\theta+A_{2}\partial_{z}^{2}\theta+\lambda_{1}\partial_{x}\theta\partial_{z}\theta+\lambda_{2}\theta\partial_{x}\partial_{z}\theta+f_{\theta}$
(8)
to order $\theta^{2}$, where the additivemultimplicativenoisefootnote non-
conserving Gaussian white noise$f_{\theta}$ satisfies
$<f_{\theta}({\bf r},t)f_{\theta}({\bf
r^{{}^{\prime}}},t^{{}^{\prime}})>=2D_{0}\delta({\bf r}-{\bf
r^{{}^{\prime}}})\delta(t-t^{{}^{\prime}})$ (9)
with a noise strength $D_{0}$. All the coefficients in (8) and (9) are related
to those in (1) - (3), the corresponding noise strengths, and the scalar order
parameter $S$. As a consequence of rotation invariance, i.e., the fact that
the underlying equation of motion in terms of Q has a frame-independent form,
we find
$2(A_{1}-A_{2})=\lambda_{2}.$ (10)
It is therefore convenient to re-express them as
$A_{1}=A_{0}+\lambda_{2}/4;\qquad A_{2}=A_{0}-\lambda_{2}/4$ (11)
Without the detailed derivation above, it would have been hard to guess the
form of the equations of motion and the constraints on the parameters. Note
that $\lambda_{1}$ and $\lambda_{2}$ are in general independent, as we argued
above. We will comment below on the relation they satisfy in the special case
of an equilibrium nematic. Eqs. (8) and (10) can also be obtained from a
microscopic model of collisional dynamics of apolar particles
collisionalderivation .
### I.1 Equilibrium limit
The energy cost of elastic deformations and, hence, the thermal equilibrium
statistics of configurations, of a two-dimensional nematic are governed by the
Frank free energy frankandors ; degp ; NP
$H=\int{[\frac{K_{1}}{2}(\nabla\cdot{\bf
n})^{2}+\frac{K_{3}}{2}(\nabla\times{\bf n})^{2}]d^{2}r},$ (12)
a functional of the director field ${\bf n}=(\cos\theta,\sin\theta)$, with
splay and bend elastic moduli $K_{1}$ and $K_{3}$. To cubic order in
$\theta({\bf r})$
$\displaystyle H/k_{B}T$ $\displaystyle={A_{3}\over 2}\int{d_{2}\bf
r}[[\partial_{x}\theta({\bf r})]^{2}+(1+\Delta)[\partial_{z}\theta({\bf
r})]^{2}$ $\displaystyle-2\Delta\theta({\bf r})[\partial_{x}\theta({\bf
r})\partial_{z}\theta({\bf r})]]$ (13)
where $A_{3}=K_{3}/k_{B}T$ and $\Delta=\frac{(K_{1}-K_{3})}{K_{3}}$. The
purely relaxational dynamics of the angle field $\theta$, at thermal
equilibrium consistent with (13), reads
$\displaystyle\frac{\partial\theta}{\partial t}$
$\displaystyle=A_{3}\partial_{x}^{2}\theta+(1+\Delta)A_{3}\partial_{z}^{2}\theta+\lambda_{1}\partial_{x}\theta\partial_{z}\theta$
$\displaystyle+\lambda_{2}\theta\partial_{x}\partial_{z}\theta+f_{\theta}$
(14)
where $\langle f_{\theta}({\bf r},t)f_{\theta}({\bf 0},0)\rangle=2\delta({\bf
r})\delta(t)$, and a kinetic coefficient has been absorbed into a time-
rescaling. The nonlinearities in (14) have the same form as in (8), but the
couplings are not independent: $2\lambda_{1}=\lambda_{2}=-2A_{3}$, since both
come from the same anharmonic term in the free energy (13). In addition, the
nonlinearity is connected to the diffusion anisotropy:
$2[A_{3}-(1+\Delta)A_{3}]=\lambda_{2}$ as required by rotation invariance. Eq.
(14) is simply the limit $2\lambda_{1}=\lambda_{2}$ of (8).
A static renormalization-group treatment of the $2d$ equilibrium nematic NP
with Hamiltonian (13) showed that $\Delta$ was marginally irrelevant, and that
the large-scale behaviour of the system was governed by a fixed point with
$\Delta=0$, i.e., a single, finite Frank constant for both splay and bend. The
dynamics of the active nematic does not correspond to downhill motion with
respect to a free-energy functional, and the two nonlinear terms thus have
independent coefficients. Their (marginal) relevance or otherwise must be
established by a dynamic renormalization-group study of the equation of motion
(8), which we present in section II.
### I.2 Burgers equation
The structure of (8) in a certain special limit merits some attention. If we
switch off the $\lambda_{2}$-nonlinearity, equation (8) has a higher symmetry
than in general, viz., under $\theta\rightarrow\theta+\mbox{constt}$ without a
corresponding transformation of the coordinates. In addition, it is invariant
under $x\leftrightarrow z$, which allows us to rescale the equations so that
the diffusion of $\theta$ is isotropic:
$\frac{\partial\theta}{\partial
t}=A{\bf\nabla}^{2}\theta+\lambda\partial_{x}\theta\partial_{z}\theta+f_{\theta}$
(15)
with a spatiotemporally white noise $f_{\theta}$ as in (9). This equation for
$\lambda\neq 0$ cannot correspond to an equilibrium system, because the sole
surviving nonlinear term $\lambda\partial_{x}\theta\partial_{z}\theta$ cannot
be written as $\delta A/\delta\theta({\bf x})$ for any scalar functional
$A[\theta]$noneqmtermfootnote Note the similarity of (15) to the KPZ equation
BurgerKPZ for the height field of a driven interface. Extending the analogy,
it is easy to see that the velocity field ${\bf
v}=(\partial_{z}\theta,\partial_{x}\theta)$ as in (7) obeys the Burgers-like
equation BurgerKPZ ; FNS
$\frac{\partial{\bf v}}{\partial t}=A\nabla^{2}{\bf v}+\lambda({\bf
v}\cdot\nabla){\bf v}+{\bf f}_{{\bf v}}$ (16)
with a conserving noise ${\bf f}_{{\bf
v}}=(\partial_{z}f_{\theta},\partial_{x}f_{\theta})$. The curl-free condition
of a traditional Burgers velocity field is replaced in our case by
$\partial_{x}v_{x}-\partial_{z}v_{z}=0$, which amounts to equal extension
rates along $x$ and $z$. In the $2d$ randomly-forced Burgers-KPZ problem, the
nonlinearity is known FNS ; BurgerKPZ to be marginally relevant, so that the
large-scale long-time behaviour is governed by a strong-coupling fixed point
inaccessible to a perturbative RG. It is natural to ask what happens in the
seemingly similar problem at hand.
#### I.2.1 Galilean invariance
Eqns. (15) and (16) are invariant under the infinitesimal Galilean boost
${\bf x}\to{\bf x}-{\bf u}t$ (17) $\theta\to\theta+\tilde{\bf u}\cdot{\bf x}$
(18)
or equivalently
${\bf v}\to{\bf v}+{\bf u}$ (19)
where
$\tilde{\bf u}=(u_{z},u_{x})$ (20)
inverts the vector components of ${\bf u}$. By analogy to the results of FNS
and BurgerKPZ this invariance implies that the nonlinear-coupling $\lambda$
does not renormalise in this special limit.
## II Renormalization group theory
In this section we outline our one-loop dynamic renormalization group (DRG)
analysis of the large-scale, long-time behaviour of Eq. (8). Our treatment is
general, allowing for two independent coupling strengths $\lambda_{1}$,
$\lambda_{2}$, but we will examine the $\lambda_{2}\to 0$ limit of section I.2
as well. We present only the key steps of the calculation, relegating details
to the Appendices.
The momentum-shell dynamical renormalization group (DRG) SMa ; SMab ; HH ; FNS
consists of two steps. Consider a system with physical fields described by
Fourier modes with wavevector ${\bf q}$ with $0\leq q\equiv|{\bf q}|<\Lambda$,
the ultraviolet (UV) cutoff. First: eliminate modes with $\Lambda e^{-l}\leq
q<\Lambda$, by solving for them in terms of those in $0\leq q<\Lambda e^{-l}$
and the noise, and average over that part of the noise whose wavenumber lies
in $[\Lambda e^{-l},\,\Lambda)$. Second: rescale space, time, and dynamical
variables to restore the cutoff $\Lambda$ and to preserve the form of the
equations of motion to the extent possible. The result is an equation of
motion in which the parameters have changed from their initial values, call
them $\\{K_{0}\\}$, to $l$-dependent values $\\{K(l)\\}$. Now, correlation
functions at small wavenumber can be calculated either from the original
equations of motion or from those obtained after the above two steps. This key
observation leads to a homogeneity relation between correlation functions
$C({\bf q},\omega;\\{K_{0}\\})=e^{fl}C({\bf q}e^{l},\omega
e^{zl};\\{K(l)\\}).$ (21)
that can be used to calculate long-wavelength correlations with particular
ease if the couplings flow to a small fixed-point value $\\{K(\infty)\\}$
under iteration of the above transformation. Let us carry out this process for
our model, Eq. (8).
We insert the decomposition intkomagefootnote $\theta({\bf
r},t)=\int_{q<\Lambda,\omega}\theta({\bf q},\omega)\exp{(i{\bf q}\cdot{\bf
r}-i\omega t)}$ into (8) to obtain the $\theta$ equation in Fourier space:
$\displaystyle\theta({\bf q},\omega)$ $\displaystyle=G_{0}({\bf
q},\omega)f_{\theta}({\bf q},\omega)-G_{0}({\bf q},\omega)$
$\displaystyle\int_{k\Omega}M({\bf k},{\bf q}-{\bf k}){\theta({\bf
k},\Omega)\theta({\bf q}-{\bf k},\omega-\Omega)}$ (22)
where
$G_{0}({\bf q},\omega)=[-i\omega+A_{1}q_{x}^{2}+A_{2}q_{z}^{2}]^{-1}$ (23)
is the bare propagator,
$\displaystyle M({\bf k},{\bf q}-{\bf k})$
$\displaystyle=\frac{\lambda_{1}}{2}[k_{x}(q_{z}-k_{z})+k_{z}(q_{x}-k_{x})]$
$\displaystyle+\frac{\lambda_{2}}{2}[k_{x}k_{z}+(q_{x}-k_{x})(q_{z}-k_{z})]$
(24)
the bare vertex, and the Fourier transform $f_{\theta}({\bf q},\omega)$ of the
Gaussian spatiotemporally white noise in (8) has autocorrelation
$\langle f_{\theta}({\bf q},\omega)f_{\theta}({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})\rangle=2D_{0}(2\pi)^{2+1}\delta({\bf
q}+{\bf q}^{{}^{\prime}})\delta(\omega+\omega^{{}^{\prime}})$ (25)
(a)
(b)
Figure 1: (a) Definition of symbols. (b) Diagram for full non-linear equation
(22) in Fourier space. The left hand side of the pictorial equation is the
full solution to $\theta({\bf q},\omega)=G({\bf q},\omega)f_{\theta}({\bf
q},\omega)$, where $G({\bf q},\omega)$ is the full propagator. The first part
on the right hand side is the zeroth order solution to (22) $\theta({\bf
q},\omega)=G_{0}({\bf q},\omega)f_{\theta}({\bf q},\omega)$ and the second
term is the contribution of the nonlinearity.
Eq. (22) can be represented graphically as in Fig. 1. A perturbative approach
to solving (22) generates corrections that can be expressed in terms of
Feynman graphs of three types – propagator, noise strength and nonlinearities
– given in Fig. 2.
(
(
(
Figure 2: (a) Graph for propagator $G({\bf q},\omega)$. The left hand side
with a double line is the full propagator, the first term on the right hand
side is the zeroth order and the second term is the one-loop correction. (b)
Graph for force density $D({\bf q},\omega)$ defined by (25). The second term
on the right hand side is the one-loop correction. (c) Graph for the three-
point vertex function. The structure with three legs with one incoming and two
outgoing is the vertex $-\frac{1}{(2\pi)^{2+1}}\int{M({\bf k},{\bf q}-{\bf
k})}$. The three graphs are $\Gamma_{a}$, $\Gamma_{b}$ and $\Gamma_{c}$.
a) b) c)
### II.1 Propagator calculation
The effective propagator $G({\bf q},\omega)$ [defined by $\theta({\bf
q},\omega)\equiv G({\bf q},\omega)f_{\theta}({\bf q},\omega)$] is given
perturbatively in Fig. 2(a). The averaging over the noise is performed using
(25). The one-loop correction to the propagator is
$\displaystyle G({\bf q},\omega)$ $\displaystyle=G_{0}({\bf
q},\omega)+4G_{0}^{2}(({\bf q},\omega)\times 2D_{0}$
$\displaystyle\int_{k\Omega}M({\bf k},{\bf q}-{\bf k})M(-{\bf k},{\bf
q})G_{0}({\bf k},\Omega)$ $\displaystyle G_{0}(-{\bf k},-\Omega)G_{0}({\bf
q}-{\bf k},\omega-\Omega)$ (26)
or
$G^{-1}({\bf q},\omega)=G_{0}^{-1}({\bf q},\omega)-\Sigma({\bf q},\omega)$
(27)
with a self-energy
$\displaystyle\Sigma({\bf q},\omega)$ $\displaystyle=4\times
2D_{0}\int_{k\Omega}M({\bf k},{\bf q}-{\bf k})M(-{\bf k},{\bf q})$
$\displaystyle G_{0}({\bf k},\Omega)G_{0}(-{\bf k},-\Omega)G_{0}({\bf q}-{\bf
k},\omega-\Omega)$ (28)
where the combinatorial factor of four represents possible noise contractions
leading to Fig 2 (a). A few steps of calculation of the integrals are
performed in Appendix A. For small wavenumber ${\bf q}$ and for
$\omega\rightarrow 0$, the result of integrating out a shell between $\Lambda
e^{-l}$ and $\Lambda$ in ${\bf q}$ space is the self-energy.
$\displaystyle\Sigma({\bf q},0)$
$\displaystyle=\frac{l}{4\pi}\bigg{[}-\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{8}(A_{1}q_{x}^{2}+A_{2}q_{y}^{2})$
$\displaystyle+\frac{G_{3}(\bar{\lambda}_{1},\bar{\lambda}_{2})A_{1}A_{2}}{(\sqrt{A_{1}}+\sqrt{A_{1}})^{2}}\bigg{]}$
(29)
where
$\displaystyle
G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})=(2\bar{\lambda}_{1}^{2}+\bar{\lambda}_{2}^{2}-3\bar{\lambda}_{1}\bar{\lambda}_{2})$
$\displaystyle
G_{3}(\bar{\lambda}_{1},\bar{\lambda}_{2})=(\bar{\lambda}_{2}^{2}-\bar{\lambda}_{1}\bar{\lambda}_{2})$
(30)
The dimensionless quantities $\bar{\lambda}_{1}$ and $\bar{\lambda}_{2}$ are
defined by
$\bar{\lambda}_{i}\bar{\lambda}_{j}=\frac{\lambda_{i}\lambda_{j}D_{0}}{(A_{1}A_{2})^{3/2}},\qquad
i,j=1,2$ (31)
When we implement the dynamical renormalization group, terms of order $q^{2}$
and of order 1 are generated though the self-energy. Terms of order $q^{2}$
will give corrections to the diffusion constants $(A_{1},A_{2})$. What about
the terms comparisonburgerfootnose of order 1, which also arise in the
analysis of Pelcovits et al. NP ? As in NP , we proceed by first ignoring the
terms of order 1, whose coefficient is proportional to one nonlinear coupling
$\lambda_{2}$, and then, post facto, realise they too are (marginally)
irrelevant because $\lambda_{2}$ itself is found to be marginally irrelevant.
Proceeding in this manner we find
$\displaystyle G^{-1}({\bf q},0)$ $\displaystyle=G_{0}^{-1}({\bf
q},0)-\Sigma({\bf q},0)$
$\displaystyle\sim\tilde{A_{1}}q_{x}^{2}+\tilde{A_{2}}q_{z}^{2}$
$\displaystyle=A_{1}q_{x}^{2}+A_{2}q_{z}^{2}$
$\displaystyle+\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})(A_{1}q_{x}^{2}+A_{2}q_{z}^{2})l}{4\times
8\pi}$ (32)
That is,
$\displaystyle\tilde{A_{1}}=A_{1}[1+\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})l}{4\times
8\pi}];$
$\displaystyle\tilde{A_{2}}=A_{2}[1+\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})l}{4\times
8\pi}].$ (33)
These are the intermediate (one-loop graphical) corrections for anisotropic
diffusion constants.
### II.2 Vertex calculation
From the full equation (22) and (Fig 1), the diagrams contributing to the
vertex correction are shown in (Fig 2(b)). There will be three types of
diagrams, all with multiplicity 4, denoted by $\Gamma_{a}$, $\Gamma_{b}$ and
$\Gamma_{c}$. The details of the calculation are given in Appendix B. The full
vertex is defined as a combination of $\lambda_{1}$ and $\lambda_{2}$ equation
(24). We study how this vertex evolves under the DRG and at the end of the
calculation we can separate terms corresponding to $\lambda_{1}$ and
$\lambda_{2}$. From (Fig 2(b)), expression for
$\displaystyle\Gamma_{a}({\bf q},{\bf k_{1}})$ $\displaystyle=4\times
2D_{0}\int_{k\Omega}M({\bf k},{\bf q}-{\bf k})$ $\displaystyle\times
M(\frac{{\bf q}}{2}+{\bf k}_{1},{\bf k}-\frac{{\bf q}}{2}-{\bf k}_{1})$
$\displaystyle\times M(\frac{{\bf q}}{2}-{\bf k_{1}},-{\bf k}+\frac{{\bf
q}}{2}+{\bf k_{1}})$ $\displaystyle\times\bigg{|}G_{0}({\bf k}-\frac{{\bf
q}}{2}-{\bf k}_{1},\Omega-\frac{\omega}{2}-\Omega_{1})\bigg{|}^{2}$
$\displaystyle\times G_{0}({\bf k},\Omega)G_{0}({\bf q}-{\bf
k},\omega-\Omega)$ (34)
The integral as usual is over $\Lambda e^{-l}<q<\Lambda$. Similarly one can
get expressions for $\Gamma_{b}({\bf q},{\bf k_{1}})$ and $\Gamma_{c}({\bf
q},{\bf k_{1}})$. Hence, adding contributions to all diagrams for the vertex,
$\Gamma_{a}({\bf q},{\bf k_{1}})+\Gamma_{b}({\bf q},{\bf
k_{1}})+\Gamma_{c}({\bf q},{\bf k_{1}})$ we can get the graphical corrections
to the couplings $\lambda_{1}$ and $\lambda_{2}$. After a calculation as in
Appendix B, the graphical corrections to $\lambda_{1}$ and $\lambda_{2}$ are
$\displaystyle\tilde{\lambda}_{1}=\lambda_{1}[1-\frac{F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})l}{2\times
8\pi}]$
$\displaystyle\tilde{\lambda}_{2}=\lambda_{2}[1-\frac{F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})l}{2\times
8\pi}]$ (35)
$F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})$,
$F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})$ defined by,
$\displaystyle
F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})=-2\bar{\lambda}_{1}\bar{\lambda}_{2}+3\bar{\lambda}_{2}^{2}+\bar{\lambda}_{2}^{3}/\bar{\lambda}_{1}$
$\displaystyle
F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})=-4\bar{\lambda}_{2}\bar{\lambda}_{1}+6\bar{\lambda}_{2}^{2}$
(36)
Note from (36) that $F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})=0$, if
$\lambda_{2}$ is zero. This says that there is no graphical correction to
$\lambda_{1}$ if $\lambda_{2}$ is zero. This is a result of the Galilean
invariance in this limit, as pointed out in section I.2.1.
### II.3 Noise strength renormalization
An effective noise strength $\tilde{D}$ can be defined by
$\langle\theta^{*}({\bf q},\omega)\theta({\bf
q},\omega)\rangle=2\tilde{D}G({\bf q},\omega)G(-{\bf q},-\omega).$ (37)
This quantity is calculated perturbatively by the series shown in (Fig 2(c)).
To one-loop order
$\displaystyle 2\tilde{D}$
$\displaystyle=2D_{0}+2(2D_{0})^{2}\int_{k\Omega}M({\bf k},{\bf q}-{\bf
k})M(-{\bf k},{\bf k}-{\bf q})$ $\displaystyle\times\bigg{|}G_{0}({\bf
k},\Omega)\bigg{|}^{2}\bigg{|}G_{0}({\bf q}-{\bf
k},\omega-\Omega)\bigg{|}^{2}$ (38)
The integral in equation (38) is performed in Appendix C. After doing the
integrals, the graphical correction to $D_{0}$ is
$\tilde{D}=D_{0}\bigg{[}1+\frac{(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}l}{2\times
8\pi}\bigg{]}$ (39)
#### II.3.1 The detailed balance limit
From equations (33) and (39), for $\lambda_{2}=0$ ($A_{1}=A_{2}=A$,
$G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})=(2\bar{\lambda}_{1}^{2}+\bar{\lambda}_{2}^{2}-3\bar{\lambda}_{1}\bar{\lambda}_{2})=2\bar{\lambda}_{1}^{2}$
and $(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}=\bar{\lambda}_{1}^{2}$), i.e.,
$A$ and $D$ have the same graphical corrections. This suggests that detailed
balance should obtain in the limit $\lambda_{2}=0$. To discover this detailed
balance let us write the Fokker-Planck equation FP for the probability
distribution functional $P[\theta,t]$ of the $\theta$-field:
$\displaystyle\frac{\partial P}{\partial t}+\sum_{{\bf
q}}\frac{\partial}{\partial\theta_{{\bf
q}}}\bigg{[}D_{0}\frac{\partial}{\partial\theta_{-{\bf q}}}+A{\bf
q}^{2}\theta_{{\bf q}}$
$\displaystyle+\frac{\lambda_{1}}{\sqrt{\Omega}}\sum_{{\bf l},{\bf m}}M({\bf
l},{\bf m})\theta_{{\bf l}}\theta_{{\bf m}}\delta_{{\bf q},{\bf l}+{\bf
m}}\bigg{]}P=0.$ (40)
We guess that a Gaussian probability distribution function
$P_{st}=N\exp\bigg{[}-\frac{1}{2}\sum_{{\bf q}}\frac{\theta_{{\bf
q}}\theta_{-{\bf q}}}{<\theta_{{\bf q}}\theta_{-{\bf q}}>}\bigg{]}$ (41)
is a steady solution to equation (40), $M({\bf l},{\bf
m})=(l_{x}m_{y}+m_{x}l_{y})$, $N$ is a normalization factor and the two-point
function $<\theta_{{\bf q}}\theta_{-{\bf q}}>=(D_{0}/A)q^{-2}$. If this is so,
the last term on the right of equation (40) should vanish if $P_{st}$ from
equation (41) is inserted for $P$. Let us check this:
$\displaystyle\bigg{[}\sum_{q,l,m}\frac{\partial}{\partial\theta_{q}}M({\bf
l},{\bf m})\theta_{{\bf l}}\theta_{{\bf m}}\delta_{{\bf q},{\bf l}+{\bf
m}}\bigg{]}P_{0}$ $\displaystyle=\sum_{q,l,m}M({\bf l},{\bf m})\theta_{{\bf
l}}\theta_{{\bf m}}\delta_{{\bf q},{\bf l}+{\bf m}}\frac{\partial
P_{0}}{\partial\theta_{q}}$ $\displaystyle=-P_{0}\frac{D_{0}}{A}\sum_{{\bf
q},{\bf l},{\bf m}}{\bf q}^{2}M({\bf l},{\bf m})\theta_{{\bf l}}\theta_{{\bf
m}}\theta_{-{\bf q}}\delta_{{\bf q},{\bf l}+{\bf m}}$ (42)
Using the symmetry $-{\bf q}\rightleftharpoons{\bf l}\rightleftharpoons{\bf
m}$ in (42) we get
$\displaystyle\sum_{q,l,m}{\bf q}^{2}M({\bf l},{\bf m})\theta_{{\bf
l}}\theta_{{\bf m}}\theta_{-{\bf q}}\delta_{q,l+m}$
$\displaystyle=\frac{1}{3}\sum_{l,m}[M({\bf l},{\bf m})({\bf l}+{\bf
m})^{2}+{\bf l}^{2}M(-{\bf m},{\bf l}+{\bf m})$ $\displaystyle+{\bf
m}^{2}M(-{\bf l},{\bf l}+{\bf m})]\theta_{{\bf l}}\theta_{{\bf
m}}\theta_{-{\bf l}-{\bf m}}$ (43)
The summation inside the square bracket in (43) is zero. This means that for
$\lambda_{2}=0$ the Gaussian defined in (41), is a steady solution of the FP
equation (40), consistent with the detailed balance noted after equation (39)
in this limit. In particular, we can already conclude that there is no
singular renormalization of the stiffnesses in the Burgers-like limit of the
model, as the equal-time correlators of $\theta$ can be obtained directly from
the Gaussian probability distribution function (41).
### II.4 Full RG Analysis
We now return to the general case $\lambda_{1}$, $\lambda_{2}$ nonzero.
Substituting results from (33), (35) and (39) to (22), gives the intermediate
equation for $\theta^{<}({\bf q},\omega)$ (without rescaling)
$\displaystyle\theta^{<}_{l}({\bf q},\omega)$ $\displaystyle=G_{l}({\bf
q},\omega)(f_{l\theta}({\bf q},\omega)+\Delta f_{\theta}({\bf q},\omega))$
$\displaystyle-G_{l}({\bf q},\omega)\int_{k\Omega}M_{l}({\bf k},{\bf q}-{\bf
k})$ $\displaystyle\times\theta^{<}({\bf k},\Omega)\theta^{<}({\bf q}-{\bf
k},\omega-\Omega),$ (44)
where the propagator at this intermediate stage is
$G_{l}({\bf
q},\omega)=(-i\omega+\tilde{A}_{1}q_{x}^{2}+\tilde{A}_{2}q_{z}^{2})^{-1},$
(45)
with $\tilde{A}_{1}$ and $\tilde{A}_{2}$ given by (33) and $0<|{\bf
q}|<\Lambda e^{-l}$, unlike the original equation, which is defined on the
large range $0<|{\bf q}|<\Lambda$.
Next rescale variables to preserve the form of the original equation:
$\displaystyle q^{{}^{\prime}}=qe^{l};\qquad\omega^{{}^{\prime}}=\omega
e^{\alpha(l)};\qquad$ $\displaystyle\theta^{<}({\bf
q},\omega)=\xi(l)\theta^{{}^{\prime}}({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}}).$ (46)
Thus the new variable ${\bf q}^{\prime}$ is defined on the same interval
$0<|{\bf q}^{\prime}|<\Lambda$ as the wave-vector ${\bf q}$ in the original
equation. In terms of the new variables, the intermediate equation for
$\theta^{\prime}({\bf q}^{\prime},\omega^{\prime})$ is
$\displaystyle\theta^{{}^{\prime}}({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})$ $\displaystyle=G(l)({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})f^{{}^{\prime}}_{\theta}({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})$ $\displaystyle-G(l)({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})\int_{k^{{}^{\prime}}\Omega^{{}^{\prime}}}M(l)({\bf
k}^{{}^{\prime}},{\bf q}^{{}^{\prime}}-{\bf k}^{{}^{\prime}})$
$\displaystyle\times\theta^{{}^{\prime}}({\bf
k}^{{}^{\prime}},\Omega^{{}^{\prime}})\theta^{{}^{\prime}}({\bf
q}^{{}^{\prime}}-{\bf
k}^{{}^{\prime}},\omega^{{}^{\prime}}-\Omega^{{}^{\prime}}),$ (47)
where
$G(l)({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})=[-i\omega+A_{1}(l)q_{x}^{{}^{\prime}2}+A_{2}(l)q_{z}^{{}^{\prime}2}]^{-1}$
(48)
with
$A_{1}(l)=\tilde{A}_{1}e^{\alpha(l)-2l};\qquad
A_{2}(l)=\tilde{A}_{2}e^{\alpha(l)-2l};$ (49) $f^{{}^{\prime}}_{\theta}({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})=f^{<}_{\theta}({\bf
q},\omega)e^{\alpha(l)}\xi^{-1}(l)$ (50) $\displaystyle M(l)({\bf
k}^{{}^{\prime}},{\bf q}^{{}^{\prime}}-{\bf k}^{{}^{\prime}})$
$\displaystyle=\frac{\lambda_{1}(l)}{2}\bigg{[}k_{x}^{{}^{\prime}}(q_{z}^{{}^{\prime}}-k_{z}^{{}^{\prime}})$
$\displaystyle+k_{z}^{{}^{\prime}}(q_{x}^{{}^{\prime}}-k_{x}^{{}^{\prime}})\bigg{]}+\frac{\lambda_{2}(l)}{2}$
$\displaystyle\times\bigg{[}k_{x}^{{}^{\prime}}k_{z}^{{}^{\prime}}+(q_{x}^{{}^{\prime}}-k_{x}^{{}^{\prime}})(q_{z}^{{}^{\prime}}-k_{z}^{{}^{\prime}})\bigg{]}$
(51)
where $\lambda_{1}(l)$ and $\lambda_{2}(l)$ are rescaled nonlinearities given
by
$\lambda_{1}(l)=\tilde{\lambda}_{1}\xi(l)e^{-(d+2)l};\qquad\lambda_{2}(l)=\tilde{\lambda}_{2}\xi(l)e^{-(d+2)l}$
(52)
The correlation function characterising the force
$f^{{}^{\prime}}_{\theta}({\bf q}^{{}^{\prime}},\omega^{{}^{\prime}})$, given
by expression (50), can be constructed using definition (25) and the new set
of variables (46)
$\displaystyle<f_{\theta}^{{}^{\prime}}({\bf
q},\omega)f_{\theta}^{{}^{\prime}}({\bf
q}^{{}^{\prime}},\omega^{{}^{\prime}})>=$ $\displaystyle
2D(l)(2\pi)^{2+1}\delta({\bf q}+{\bf q}^{{}^{\prime}})$
$\displaystyle\delta(\omega+\omega^{{}^{\prime}})$ (53)
with
$D(l)=\tilde{D}e^{(3\alpha(l)+dl)}\xi^{-2}(l)$ (54)
where $d=2$ and all tilde variables correspond to the graphically corrected
quantities in (33), (35) and (39). Substituting for the expressions for all
tilde variables
$\displaystyle
A_{1}(l)=A_{1}[1+\frac{lG_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}]e^{\alpha(l)-2l},$ $\displaystyle
A_{2}(l)=A_{2}[1+\frac{lG_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}]e^{\alpha(l)-2l},$
$\displaystyle\lambda_{1}(l)=\lambda_{1}[1-\frac{lF_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}]e^{-4l}\xi(l),$
$\displaystyle\lambda_{2}(l)=\lambda_{2}[1-\frac{lF_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}]e^{-4l}\xi(l),$ $\displaystyle
D(l)=D[1+\frac{l(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}}{2\times
8\pi}]e^{3\alpha(l)+2l}\xi^{-2}(l).$ (55)
### II.5 Recursion relation
Here we calculate the recursion relation for all five parameters. From (55),
the constraint of rotational invariance $2(A_{1}-A_{2})=\lambda_{2}$ requires
$\xi(l)=\exp(\alpha(l)+2l)\bigg{(}1+\frac{lG_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}+\frac{lF_{2}((\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}\bigg{)}$ (56)
where the functions $G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})$ and
$F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})$ are already defined in (30) and
(36). With this choice of $\xi(l)$, substituting in (55), recursion relations
for all five variables given by,
$\displaystyle\frac{dA_{1}}{dl}=A_{1}[-2+z(l)+\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}],$
$\displaystyle\frac{dA_{2}}{dl}=A_{2}[-2+z(l)+\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}],$
$\displaystyle\frac{d\lambda_{1}}{dl}=\lambda_{1}[-2+z(l)+\frac{F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}+\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times 8\pi}$
$\displaystyle-\frac{F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}],$
$\displaystyle\frac{d\lambda_{2}}{dl}=\lambda_{2}[-2+z(l)+\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}],$
$\displaystyle\frac{dD}{dl}=D[-2+z(l)+\frac{(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}}{2\times
8\pi}-\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{2\times 8\pi}$
$\displaystyle-\frac{F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{2\times
8\pi}].$ (57)
where $z(l)$ is defined by
$\alpha(l)=\int_{0}^{l}z(l^{{}^{\prime}})dl^{{}^{\prime}}$, and the
dimensionless variables $\bar{\lambda}_{1}$ and $\bar{\lambda}_{2}$ were
defined in (31). The functions $G_{2}$, $F_{1}$ and $F_{2}$ are already
defined in (30) and (36). In these recursion relations the function $z(l)$ is
unknown at this point. It will drop out in the recursion relation for the
dimensionless variables, $\bar{\lambda}_{1}$ and $\bar{\lambda}_{2}$, for
which the recursion relations are
$\frac{d\bar{\lambda}_{1}}{dl}=\bar{\lambda}_{1}\bigg{[}\frac{(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}}{4\times
8\pi}-\frac{3}{2}\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}\bigg{]}-\frac{F_{1}^{*}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi},$ (58)
$\frac{d\bar{\lambda}_{2}}{dl}=\bar{\lambda}_{2}\bigg{[}\frac{(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}}{4\times
8\pi}-\frac{3}{2}\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}-\frac{F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{4\times
8\pi}\bigg{]}.$ (59)
Equations (58) and (59) are coupled nonlinear equations for
$\bar{\lambda}_{1}$ and $\bar{\lambda}_{2}$.
In the special, high-symmetry case $\lambda_{2}=0$, from (30),
$G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})=2\lambda_{1}^{2}$ and
$F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})=0$. Then the dimensionless
coupling $\bar{\lambda}_{1}^{2}(l)=\lambda_{1}^{2}(l)D(l)/A^{3/2}(l)$ obeys
$\frac{d\bar{\lambda}_{1}}{dl}=\bar{\lambda}_{1}\bigg{[}-2+2-\frac{\bar{\lambda}_{1}^{2}(l)}{2\times
8\pi}\bigg{]}=-\frac{\bar{\lambda}_{1}^{3}(l)}{2\times 8\pi}$ (60)
which tells us $\bar{\lambda}_{1}$ is marginally irrelevant. By contrast, for
the Burgers equation in 2-d, the nonlinearity is marginally relevant. This is
surprising, given the similarities of the two models in the limit
$\lambda_{2}=0$. A second special case is
$\bar{\lambda}_{2}=2\bar{\lambda}_{1}$, when the problem reduces to an
equilibrium problem, as remarked in section I.1. At this particular choice of
$\bar{\lambda}_{1}$ and $\bar{\lambda}_{2}$,
$G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})=0$,
$(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}=\bar{\lambda}_{1}^{2}=\frac{\bar{\lambda}_{2}^{2}}{4}$,
$F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})=16\lambda_{1}^{3}$ and
$F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})=4\bar{\lambda}_{2}^{2}$.
Substituting these expressions for all functions in (58) and (59), the flow
equations for the equilibrium limit are
$\frac{d\bar{\lambda}_{1}}{dl}=-\frac{15}{4\times
8\pi}\bar{\lambda}_{1}^{3};\qquad\frac{d\bar{\lambda}_{2}}{dl}=-\frac{15}{4\times
4\times 8\pi}\bar{\lambda}_{2}^{3}$ (61)
We can draw the flow-diagram in $(\bar{\lambda}_{1},\bar{\lambda}_{2})$ plane.
(Fig 3) shows that for three special cases, $\lambda_{2}=0$,
$\lambda_{2}=2\lambda_{1}$ and $\lambda_{2}=\lambda_{1}$ flow is towards zero.
For other points also flow is towards zero. This means (0, 0) is the only
fixed point and it is stable. We have checked this numerically as well.
Since the nonlinearities are marginally irrelevant the effective stiffness
$A_{1}$ and $A_{2}$ become equal at large scales, and are nonsingular.
Therefore $<|\theta_{q}|^{2}>\sim q^{-2}$ for small ${\bf q}$, i.e. the
renormalized theory still has only quasi long-ranged order.
## III Conclusion and Discussion
In this paper we have provided a systematic analysis of the large-scale, long-
time behaviour of the stochastic nonlinear partial differential equation for
the angle field of an active nematic on a 2-dimensional substrate. We
constructed the general equation of motion for the order parameter, starting
from a description that included the velocity, density as well. We then
reduced the model to focus on the director or small-angle fluctuations about
an ordered active nematic, and studied the evolution of the parameters therein
under the dynamic renormalization group SMa ; FNS ; BurgerKPZ . The equation
has five parameters, $A_{1}$ and $A_{2}$ which are director diffusivities for
two directions, the nonlinear couplings $\lambda_{1}$ and $\lambda_{2}$ and
$D_{0}$ the noise strength. Two special cases are of interest:
$\lambda_{2}=2\lambda_{1}$, for which the dynamics is that of an equilibrium
two-dimensional nematic where static properties are shown to agree with NP .
The second case is $\lambda_{2}=0$, for which the equation can be mapped to a
Burgers equation, for a velocity field ${\bf v}$ given in (7), with
$\partial_{x}v_{x}-\partial_{z}v_{z}=0$. Despite this resemblance the
dimensionless nonlinear coupling parameter
$\bar{\lambda}^{2}=\frac{\lambda^{2}D_{0}}{A^{3}}$ is found to be marginally
irrelevant, whereas for the Burgers equation in $d=2$ (see FNS ) the
nonlinearity was marginally relevant. Interestingly in this limit the
diffusion constant and noise strength renormalize the same way, implying the
system has a hidden detailed balance, which we exposed via a Fokker-Planck
analysis. The complete one-loop recursion relation for the five parameters
constrained only by rotational-invariance show that the nonlinearities are
always marginally irrelevant.
Figure 3: RG flow diagram in the phase plane of dimensionless nonlinear
couplings $\bar{\lambda}_{1}$ and $\bar{\lambda}_{2}$ defined in (31). The
solid line represents line $\bar{\lambda}_{2}=2\bar{\lambda}_{1}$ (equilibrium
limit), the dot dashed line represents $\bar{\lambda}_{1}=\bar{\lambda}_{2}$
and dashed line represents $\bar{\lambda}_{2}=0$ (limit when equation is
similar to Burgers equation). For these three cases, it is particularly easy
to show analytically that the flow is inward (i.e. nonlinearities are
marginally irrelevant). In fact for all $\bar{\lambda}_{1}$,
$\bar{\lambda}_{2}$ the flow is towards (0, 0).
In Appendix D we present the equation of motion for the angle field starting
from a velocity field which satisfies incompressibility. This provides
another, inequivalent, situation in which the density is fast and can
therefore be suitably eliminated. The procedure leads to a slightly different
equation from (8) or (22) with nonlocality due to transverse projectors. We
have not analysed the properties of the incompressible version. Our results,
despite the neglect of the density, are consistent with the numerical findings
of chateginellimontagne , that active nematic order in $d=2$ is quasi long-
range. A complete treatment of the coupled behaviour of angle and density
correlators in steady state, beyond the linearized analysis of sradititoner ,
as well as a study of the incompressible model, are left for future work.
###### Acknowledgements.
SM thanks the CSIR, India for financial support. SR acknowledges support from
CEFIPRA project 3504-2, and from the DST, India through the Centre for
Condensed Matter Theory and Math-Bio Centre grant SR/S4/MS:419/07
## Appendix A Propagator renormalization
We start from the symmetrised version of (26) (by substituting ${\bf
k}\equiv\frac{{\bf q}}{2}+{\bf k}$ and $\Omega\equiv\frac{\omega}{2}+\Omega$)
$\displaystyle\Sigma({\bf q},\omega)$ $\displaystyle=4\times
2D_{0}\int_{k\Omega}M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf k})$
$\displaystyle\times M(-\frac{{\bf q}}{2}-{\bf k},{\bf q})\times
G_{0}(\frac{{\bf q}}{2}+{\bf k},\frac{\omega}{2}+\Omega)$ $\displaystyle\times
G_{0}(-\frac{{\bf q}}{2}-{\bf k},-\frac{\omega}{2}-\Omega)G_{0}(\frac{{\bf
q}}{2}-{\bf k},\frac{\omega}{2}-\Omega)$ (62)
where $G_{0}({\bf q},\omega)=(-i\omega+A_{1}q_{x}^{2}+A_{2}q_{z}^{2})^{-1}$ is
the unrenormalized propagator. It is easy to evaluate the $\Omega$ integral
first in (62). Separating the $\Omega$-integral
$I_{\Omega}^{P}({\bf k})=\int_{-\infty}^{+\infty}\bigg{|}G_{0}(\frac{{\bf
q}}{2}+{\bf k},\frac{\omega}{2}+\Omega)\bigg{|}^{2}G_{0}(\frac{{\bf
q}}{2}-{\bf k},\frac{\omega}{2}-\Omega)d\Omega$ (63)
After substituting the expressions for the unrenormalized propagator in (63)
$I_{\Omega}({\bf
k})=\int^{+\infty}_{-\infty}\frac{i(\frac{\omega}{2}-\Omega)+a}{[(\frac{\omega}{2}+\Omega)^{2}+b^{2}]\times[(\frac{\omega}{2}-\Omega)^{2}+a^{2}]}d\Omega$
(64)
where
$\displaystyle
a=[A_{1}(\frac{q_{x}}{2}-k_{x})^{2}+A_{1}(\frac{q_{z}}{2}-k_{z})^{2}]$
$\displaystyle
b=[A_{1}(\frac{q_{x}}{2}+k_{x})^{2}+A_{1}(\frac{q_{z}}{2}+k_{z})^{2}]$ (65)
After integrating $I_{\Omega}({\bf k})$ over $\Omega$, for
$\omega\longrightarrow 0$, we see that,
$I_{\Omega}({\bf k})=\frac{\pi}{b(a+b)}.$ (66)
Substituting this $\Omega$ integral in the calculation of the self-energy (62)
$\displaystyle\Sigma({\bf q},\omega)$ $\displaystyle=4\times
2D_{0}\pi\frac{1}{(2\pi)^{2+1}}\int M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf
q}}{2}-{\bf k})$ $\displaystyle M(-\frac{{\bf q}}{2}-{\bf k},{\bf
q})\times\frac{1}{b(a+b)}d{\bf k},$ (67)
where $a$ and $b$ are defined in (65). Since we are interested in long-
wavelength properties, we can do small $q_{x}$ and $q_{y}$ expansions. For
calculating $\Sigma({\bf q},\omega)$, we need to perform the ${\bf k}$
integral. Defining small parameters $x=\frac{q_{x}}{k_{x}}$ and
$z=\frac{q_{z}}{k_{z}}$, and expanding up to lowest order in $x$ and $z$
$\displaystyle\frac{1}{b(a+b)}$
$\displaystyle=\frac{1}{2k_{x}^{4}\alpha^{2}}\bigg{[}1-\frac{x^{2}}{2\alpha}A_{1}-\frac{z^{2}}{2\alpha}A_{2}\tan^{2}\theta$
$\displaystyle-\frac{x}{\alpha}A_{1}-\frac{z}{\alpha}A_{2}\tan^{2}\theta+\frac{x^{2}}{\alpha^{2}}A_{1}^{2}$
$\displaystyle+\frac{z^{2}}{\alpha^{2}}A_{2}^{2}\tan^{4}\theta+\frac{2xz}{\alpha^{2}}A_{1}A_{2}\tan^{2}\theta\bigg{]}$
(68)
where $\theta=\tan^{-1}(\frac{k_{z}}{k_{x}})$ and
$\alpha=(A_{1}+A_{2}\tan^{2}\theta)$. The next step for the calculation of the
integral is the product of two propagators $M\times M$ in (67).
$\displaystyle M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf k})\times
M(-\frac{{\bf q}}{2}-{\bf k},{\bf q})$
$\displaystyle=\frac{k_{x}^{2}k_{z}^{2}}{4}\bigg{[}xzG_{1}(\lambda_{1},\lambda_{2})+(x+z)G_{2}(\lambda_{1},\lambda_{2})$
$\displaystyle+2G_{3}(\lambda_{1},\lambda_{2})\bigg{]}$ (69)
From (68) and (69) integrand of (67) is,
$\displaystyle\frac{M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf
k})\times M(-\frac{{\bf q}}{2}-{\bf k},{\bf q})}{b(a+b)}$
$\displaystyle=\frac{k_{x}^{2}k_{z}^{2}}{4\times
2k_{x}^{4}\alpha^{2}}\bigg{[}xzG_{1}+G_{2}\bigg{(}-\frac{x^{2}}{\alpha}A_{1}$
$\displaystyle-\frac{z^{2}}{\alpha}A_{2}\tan^{2}\theta-\frac{xz}{\alpha}A_{1}-\frac{xz}{\alpha}A_{2}\tan^{2}\theta\bigg{)}+2G_{3}$
$\displaystyle\bigg{(}1-\frac{x^{2}}{2\alpha}A_{1}-\frac{z^{2}}{2\alpha}A_{2}\tan^{2}\theta-\frac{x}{\alpha}A_{1}-\frac{z}{\alpha}A_{2}\tan^{2}\theta$
$\displaystyle+\frac{x^{2}}{\alpha^{2}}A_{1}^{2}+\frac{z^{2}}{\alpha^{2}}A_{2}^{2}\tan^{4}\theta+\frac{2xz}{\alpha^{2}}A_{1}A_{2}\tan^{2}\theta\bigg{)}\bigg{]}$
(70)
On integration (inside the $[\qquad]$) only term of $O(x^{2})$, of $O(z^{2})$
and $O(1)$ survive. Hence terms which will contribute to the integration are
$\displaystyle
G_{2}\bigg{(}-\frac{x^{2}}{\alpha}A_{1}-\frac{z^{2}}{\alpha}A_{2}\tan^{2}\theta\bigg{)}$
$\displaystyle+2G_{3}\bigg{(}1-\frac{x^{2}}{2\alpha}A_{1}-\frac{z^{2}}{2\alpha}A_{2}\tan^{2}\theta+\frac{x^{2}}{\alpha^{2}}A_{1}^{2}$
$\displaystyle+\frac{z^{2}}{\alpha^{2}}A_{2}^{2}\tan^{4}\theta\bigg{)}$ (71)
where $G_{2}=(2\lambda_{2}^{2}+\lambda_{2}^{2}-3\lambda_{1}\lambda_{2})$ and
$G_{3}=(\lambda_{2}^{2}-\lambda_{1}\lambda_{2})$. $k_{x}=k\cos\theta$ and
$k_{z}=k\sin\theta$ and $\alpha=(A_{1}+A_{2}\tan^{2}\theta)$. After performing
the integration for these two types of terms in (71),
$\displaystyle\Sigma({\bf q},\omega\rightarrow 0)$
$\displaystyle=\frac{l}{4\pi}\bigg{[}-\frac{G_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})}{8}(A_{1}q_{x}^{2}+A_{2}q_{y}^{2})$
$\displaystyle+\frac{G_{3}(\bar{\lambda}_{1},\bar{\lambda}_{2})A_{1}A_{2}}{(\sqrt{A_{1}}+\sqrt{A_{1}})^{2}}\bigg{]}$
(72)
This is the expression for the self-energy as given in (29).
## Appendix B Vertex renormalization
Here we calculate the three-point symmetrised vertex function $\Gamma$. There
are three distinct one-loop diagrams $\Gamma_{a}$, $\Gamma_{b}$ and
$\Gamma_{c}$ contributing to the correction to the vertex as shown in (Fig
2(b)). These diagrams all have multiplicity 4. In this Appendix we will go
into the details of the calculation of $\Gamma_{a}$. The calculations for
$\Gamma_{b}$ and $\Gamma_{c}$ are the same as for $\Gamma_{a}$. Small
variables $x$ and $z$ are as defined in Appendix A: for self-energy. We start
from the symmetrised version of (34)
$\displaystyle\Gamma_{a}({\bf q},{\bf k_{1}})$ $\displaystyle=4\times
2D_{0}\int_{k\Omega}M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf k})$
$\displaystyle\times M(\frac{{\bf q}}{2}+{\bf k}_{1},{\bf k}-{\bf k}_{1})$
$\displaystyle\times M(\frac{{\bf q}}{2}-{\bf k_{1}},-{\bf k}+{\bf k_{1}})$
$\displaystyle\times\bigg{|}G_{0}({\bf k}-{\bf
k}_{1},\Omega-\Omega_{1})\bigg{|}^{2}\times$ $\displaystyle G_{0}(\frac{{\bf
q}}{2}+{\bf k},\frac{\omega}{2}+\Omega)\times G_{0}(\frac{{\bf q}}{2}-{\bf
k},\frac{\omega}{2}-\Omega)$ (73)
Separating the $\Omega$ integral part from the full integration in (73)
$\displaystyle I^{V}_{a\Omega}({\bf k})$
$\displaystyle=\int^{+\infty}_{-\infty}\bigg{|}G_{0}({\bf k}-{\bf
k}_{1},\Omega-\Omega_{1})\bigg{|}^{2}\times$ $\displaystyle G_{0}(\frac{{\bf
q}}{2}+{\bf k},\frac{\omega}{2}+\Omega)G_{0}(\frac{{\bf q}}{2}-{\bf
k},\frac{\omega}{2}-\Omega)d\Omega$ (74)
for $\omega\longrightarrow 0$ and $\Omega_{1}\longrightarrow 0$ limit and
writing in terms of real and imaginary parts,
$Re(I^{V}_{a\Omega}({\bf
k}))=\int\frac{ab+\Omega^{2}}{(\Omega^{2}+b^{2})(\Omega^{2}+a^{2})(\Omega^{2}+c^{2})}$
(75)
For $\omega\longrightarrow 0$ and $\Omega_{1}\longrightarrow 0$ limits
$Im(I^{p}_{\Omega}({\bf k}))=0$. where $a$ and $b$ are as defined in (65), and
$c=[A_{1}(k_{x}-k_{x_{1}})^{2}+A_{1}(k_{z}-k_{z_{1}})^{2}]$ (76)
Performing the integral over $\Omega$,
$I^{V}_{a\Omega}({\bf k})=\frac{\pi(2c+a+b)}{c(a+c)(b+c)(a+b)}$ (77)
Similarly for $\Gamma_{b}$ and $\Gamma_{c}$,
$\displaystyle I^{V}_{b\Omega}({\bf k})=\frac{\pi}{a(a+c)(a+b)}$
$\displaystyle I^{V}_{c\Omega}({\bf k})=\frac{\pi}{b(b+c)(a+b)}$ (78)
Substituting this $I^{V}_{a\Omega}({\bf k})$ from (77) in the calculation of
$\Gamma_{a}$,
$\displaystyle\Gamma_{a}({\bf q},{\bf k_{1}})$ $\displaystyle=4\times
2D_{0}\int_{k\Omega}M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf k})$
$\displaystyle\times M(\frac{{\bf q}}{2}+{\bf k}_{1},{\bf k}-{\bf k}_{1})$
$\displaystyle\times M(\frac{{\bf q}}{2}-{\bf k_{1}},-{\bf k}+{\bf k_{1}})$
$\displaystyle\times\frac{\pi(2c+a+b)}{c(a+c)(b+c)(a+b)}$ (79)
We are interested in long wavelength properties. By defining the small
quantities $x=\frac{q_{x}}{k_{x}}$, $z=\frac{q_{z}}{k_{z}}$,
$x_{1}=\frac{k_{x_{1}}}{k_{x}}$ and $z_{1}=\frac{k_{z_{1}}}{k_{z}}$, where
$k_{x}=k\cos\theta$ and $k_{z}=k\sin\theta$, up to lowest order in $x$, $z$,
$x_{1}$ and $z_{1}$,
$\displaystyle I^{V}_{a\Omega}({\bf k})$
$\displaystyle=\frac{\pi(2c+a+b)}{c(a+c)(b+c)(a+b)}$
$\displaystyle=\frac{\pi}{2k_{x}^{6}\alpha^{3}}\bigg{[}1+\frac{3x_{1}}{\alpha}A_{1}+\frac{3z_{1}}{\alpha}A_{2}\tan^{2}\theta$
$\displaystyle+\frac{xz}{2\alpha^{2}}A_{1}A_{2}\tan^{2}\theta+\frac{14x_{1}z_{1}}{\alpha^{2}}A_{1}A_{2}\tan^{2}\theta\bigg{]}$
(80)
The next step for the calculation of the integral is the product of three
propagators $M\times M\times M$
$\displaystyle M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf k})\times
M(\frac{{\bf q}}{2}+{\bf k}_{1},{\bf k}-{\bf k}_{1})$ $\displaystyle\times
M(\frac{{\bf q}}{2}-{\bf k_{1}},-{\bf k}+{\bf k_{1}})$
$\displaystyle=2k_{x}^{3}k_{z}^{3}\bigg{[}\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}^{3}\bigg{(}2(\frac{xz}{4}-x_{1}z_{1})\bigg{)}$
$\displaystyle+\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}^{2}\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}\bigg{(}-\frac{xz}{2}+10x_{1}z_{1}-2x_{1}-2z_{1}\bigg{)}$
$\displaystyle+\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}^{2}\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}\bigg{(}-\frac{xz}{4}-14x_{1}z_{1}+4x_{1}+4z_{1}-1\bigg{)}$
$\displaystyle+\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}^{3}\bigg{(}\frac{3xz}{4}+6x_{1}z_{1}-2x_{1}-2z_{1}+1\bigg{)}\bigg{]}$
(81)
From (80) and (81), the product inside the integral for $\Gamma_{a}({\bf
q},{\bf k_{1}})$ is
$\displaystyle\frac{\pi(2c+a+b)}{c(a+c)(b+c)(a+b)}\times M(\frac{{\bf
q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf k})$ $\displaystyle M(\frac{{\bf
q}}{2}+{\bf k}_{1},{\bf k}-{\bf k}_{1})\times M(\frac{{\bf q}}{2}-{\bf
k_{1}},-{\bf k}+{\bf k_{1}})$ $\displaystyle=\frac{\pi
2k_{x}^{3}k_{z}^{3}}{2k_{x}^{6}\alpha^{3}}\bigg{[}2\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}^{3}\bigg{(}2(\frac{xz}{4}-x_{1}z_{1})\bigg{)}$
$\displaystyle+\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}^{2}\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}\bigg{(}-\frac{xz}{2}+10x_{1}z_{1}-\frac{6x_{1}z_{1}}{\alpha}A_{1}$
$\displaystyle-\frac{6x_{1}z_{1}}{\alpha}A_{2}\tan^{2}\theta\bigg{)}$
$\displaystyle+\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}^{2}\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}\bigg{(}-\frac{xz}{4}-14x_{1}z_{1}+\frac{12x_{1}z_{1}}{\alpha}A_{1}$
$\displaystyle+\frac{12x_{1}z_{1}}{\alpha}A_{2}\tan^{2}\theta-\frac{xz}{2\alpha^{2}}A_{1}A_{2}\tan^{2}\theta$
$\displaystyle-\frac{14x_{1}z_{1}}{\alpha^{2}}A_{1}A_{2}\tan^{2}\theta\bigg{)}$
$\displaystyle+\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}^{3}\bigg{(}\frac{3xz}{4}+6x_{1}z_{1}-\frac{6x_{1}z_{1}}{\alpha}A_{1}$
$\displaystyle-\frac{6x1z1}{\alpha}A_{2}\tan^{2}\theta+\frac{xz}{2\alpha^{2}}A_{1}A_{2}\tan^{2}\theta$
$\displaystyle+\frac{14x_{1}z_{1}}{\alpha^{2}}A_{1}A_{2}\tan^{2}\theta\bigg{)}$
(82)
We display only those terms which give a nonzero contribution after
integrating over ${\bf k}$. Similarly we can obtain expressions for
$\Gamma_{b}$ and $\Gamma_{c}$
The total $\Gamma=\Gamma_{a}+\Gamma_{b}+\Gamma_{c}=\Gamma_{a}+2\Gamma_{b}$.
After doing the integration over ${\bf k}$, the final expression for $\Gamma$,
$\displaystyle\Gamma({\bf q},{\bf k_{1}})$
$\displaystyle=2(k_{x}k_{z})\frac{1}{2}\bigg{[}\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}^{2}\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}\bigg{(}-\frac{xz}{16\pi}$
$\displaystyle+\frac{x_{1}z_{1}}{4\pi}\bigg{)}+\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}^{2}\bigg{(}\frac{\lambda_{1}}{2}\bigg{)}\bigg{(}-\frac{xz}{32\pi}$
$\displaystyle-\frac{7x_{1}z_{1}}{8\pi}\bigg{)}+\bigg{(}\frac{\lambda_{2}}{2}\bigg{)}^{3}\bigg{(}-\frac{7xz}{32\pi}+\frac{x_{1}z_{1}}{8\pi}\bigg{)}\bigg{]}$
(83)
The bare vertex is
$\displaystyle\Gamma_{0}({\bf q},{\bf k_{1}})$
$\displaystyle=2(k_{x}k_{z})\bigg{[}\frac{\lambda_{1}}{2}\bigg{(}\frac{xz}{4}-x_{1}z_{1})\bigg{)}$
$\displaystyle+\frac{\lambda_{2}}{2}\bigg{(}\frac{xz}{4}+x_{1}z_{1})\bigg{)}\bigg{]}$
(84)
Decomposing expression in (84) into parts of the form
$(\frac{xz}{4}-x_{1}z_{1})$ and $(\frac{xz}{4}+x_{1}z_{1})$, we get the
corrections to $\frac{\lambda_{1}}{2}$ and $\frac{\lambda_{2}}{2}$. Hence with
this decomposition (84) can be rewritten as
$\displaystyle\Gamma({\bf q},{\bf k_{1}})$
$\displaystyle=2(k_{x}k_{z})\bigg{(}\frac{xz}{4}-x_{1}z_{1}\bigg{)}\bigg{[}-\frac{\lambda_{1}^{2}\lambda_{2}}{4\times
2\times 8\pi}$ $\displaystyle+\frac{3\lambda_{2}^{2}\lambda_{1}}{2\times
8\times 8\pi}+\frac{\lambda_{2}^{3}}{2\times 8\times 8\pi}\bigg{]}$
$\displaystyle+2(k_{x}k_{z})\bigg{(}\frac{xz}{4}+x_{1}z_{1}\bigg{)}$
$\displaystyle\bigg{[}-\frac{4\lambda_{2}^{2}\lambda_{1}}{2\times 8\times
8\pi}+\frac{6\lambda_{2}^{3}}{2\times 8\times 8\pi}\bigg{]}$ (85)
Comparing with the expression for the original vertex, the corrections to
$\frac{\lambda_{1}}{2}$ and $\frac{\lambda_{2}}{2}$ are
$\displaystyle\tilde{\lambda}_{1}=\lambda_{1}\bigg{[}1-\frac{F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})l}{2\times
8\pi}\bigg{]}$
$\displaystyle\tilde{\lambda}_{2}=\lambda_{2}\bigg{[}1-\frac{F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})l}{2\times
8\pi}\bigg{]}$ (86)
where functions $F_{1}(\bar{\lambda}_{1},\bar{\lambda}_{2})$ and
$F_{2}(\bar{\lambda}_{1},\bar{\lambda}_{2})$ are defined in (36)
## Appendix C Noise strength renormalization
Here we will compute the leading-order correction to the noise strength. The
relevant diagram which will contribute to the integral is shown in (Fig 2(c));
it has multiplicity of 2. Calculating the integral with this symmetrised
vertex,
$\displaystyle\Delta{D}$
$\displaystyle=2\times(2D_{0})^{2}\int_{k\Omega}M(\frac{{\bf q}}{2}+{\bf
k},\frac{{\bf q}}{2}-{\bf k})$ $\displaystyle M(-\frac{{\bf q}}{2}-{\bf
k},{\bf k}-\frac{{\bf q}}{2})\bigg{|}G_{0}(\frac{{\bf q}}{2}+{\bf
k},\frac{\omega}{2}+\Omega)\bigg{|}^{2}\bigg{|}$ $\displaystyle
G_{0}(\frac{\bf q}{2}-{\bf k},\frac{\omega}{2}-\Omega)\bigg{|}^{2}$ (87)
Separating the $\Omega$ integral from the full integration and taking
$\omega\longrightarrow 0$,
$I^{D}_{\Omega}{\bf k}=\frac{\pi}{ab(a+b)}$ (88)
Expanding $\frac{1}{ab(a+b)}$ as in the calculation of the propagator in terms
of small variables $x$ and $z$, the terms which will contribute to lowest
order are of order 1. Hence to lowest order,
$\frac{1}{ab(a+b)}\simeq\frac{1}{2k_{x}^{6}\alpha^{3}}$ (89)
The next step of the calculation of the integral is the product of two
propagators, $M\times M$. To lowest order,
$M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf q}}{2}-{\bf k})M(-\frac{{\bf
q}}{2}-{\bf k},{\bf k}-\frac{{\bf
q}}{2})=k_{x}^{2}k_{z}^{2}(\lambda_{2}-\lambda_{1})^{2}$ (90)
The final expression for the product
$\displaystyle\frac{1}{ab(a+b)}\times M(\frac{{\bf q}}{2}+{\bf k},\frac{{\bf
q}}{2}-{\bf k})M(-\frac{{\bf q}}{2}-{\bf k},{\bf k}-\frac{{\bf q}}{2})$
$\displaystyle=\frac{k_{x}^{2}k_{z}^{2}(\lambda_{2}-\lambda_{1})^{2}}{2k_{x}^{6}\alpha^{3}}$
(91)
After performing the integration over ${\bf k}$ in the integral (87),
$\Delta{D}=\frac{D_{0}^{2}(\lambda_{2}-\lambda_{1})^{2}l}{8\pi(A_{1}A_{2})^{3/2}}$
(92)
This gives
$\tilde{D}=D_{0}\bigg{[}1+\frac{(\bar{\lambda}_{2}-\bar{\lambda}_{1})^{2}l}{2\times
8\pi}\bigg{]}$ (93)
## Appendix D An incompressible active nematic
In this section we give the equation for the angle field $\theta$, obtained
from an incompressible velocity field ${\bf v}$ ($\nabla\cdot{\bf v}=0$). From
(2), imposing $\rho=\mbox{constt}$ and $\nabla\cdot{\bf v}=0$, and defining
the transverse projector $P=({\bf 1}-\hat{\bf q}\hat{\bf q})$, we see that
${\bf v}=-\bar{\Gamma}^{-1}P\cdot(\nabla\cdot{\mbox{Q}})$ (94)
writing Q in terms of $\theta$
${\bf v}=-\bar{\Gamma}^{-1}P\cdot(\partial_{z}\theta,\partial_{x}\theta)$ (95)
Substituting the expression for ${\bf v}$ in (3) to linear order in $\theta$
the equation of motion
$\displaystyle\bar{G}_{0}^{-1}({\bf q},\omega)\theta_{{\bf q},\omega}$
$\displaystyle=f_{\theta}({\bf q},\omega)-\int_{{\bf k},\Omega}\theta_{{\bf
k},\Omega}\theta_{{{\bf q}-{\bf k}},\omega-\Omega}\bigg{[}\bigg{(}\gamma_{1}$
$\displaystyle-\frac{\alpha_{0}}{2}[P_{22}(\hat{\bf k})+P_{22}(\hat{\bf
q}-\hat{\bf k})-P_{11}(\hat{\bf k})$ $\displaystyle-P_{11}(\hat{\bf
q}-\hat{\bf k})]\bigg{)}\times\bigg{(}M({\bf k},{\bf q}-{\bf k})\bigg{)}$
$\displaystyle+\gamma_{2}\bigg{(}[P_{12}(\hat{\bf k})+P_{12}(\hat{\bf
q}-\hat{\bf k})]{\bf k}\cdot({\bf q}-{\bf k})$ $\displaystyle+[P_{11}(\hat{\bf
k})+P_{22}(\hat{\bf q}-\hat{\bf k})]k_{y}(q_{x}-k_{x})$
$\displaystyle+[P_{22}(\hat{\bf k})+P_{11}(\hat{\bf q}-\hat{\bf
k})]k_{x}(q_{y}-k_{y})\bigg{)}\bigg{]}$ (96)
where $f_{\theta}({\bf q},\omega)$ is Gaussian random nonconserving noise with
noise-noise correlation as defined in (25). $\bar{G}_{0}^{-1}({\bf q},\omega)$
is inverse propagator, defined by
$\displaystyle\bar{G}_{0}^{-1}({\bf q},\omega)$
$\displaystyle=\bigg{(}-i\omega+\frac{\alpha_{0}}{2}{\bf q}^{2}$
$\displaystyle+A_{1}P_{11}(\hat{\bf q})q_{z}^{2}-A_{2}P_{22}(\hat{\bf
q})q_{x}^{2}\bigg{)}^{-1}$ (97)
$M({\bf k},{\bf q}-{\bf k})$ as defined in (24), $P_{11}(\hat{\bf q})$,
$P_{22}(\hat{\bf q})$ are diagonal components and $P_{12}(\hat{\bf q})$ is the
off-diagonal component of projection operator. We have not studied further the
properties of this equation.
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* (11) M. Kardar et al., Phys. Rev. Lett. 56, 889 (1986); E. Medina et al., Phys. Rev. A 39, 3053 (1989); M. Kardar et al., Phys. Rev. Lett. 58, 2087 (1987); E. Frey at al., Phys. Rev. E 50, 1024 (1994) ; H. E. Stanley et al. Fractal Concept in Surface growth, Cambridge University Press (1995)
* (12) P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1995).
* (13) For simplicity we consider a strictly isothermal system, so that the energy density or temperature field can be ignored.
* (14) M. Khandkar and M. Barma, Phys. Rev. E 2005 consider a model of needles, i.e., rods of vanishing thickness, in two dimensions, with deposition and evaporation.
* (15) J. Toner, arXiv:0909.1954v1.
* (16) As well as viscous dissipative terms of order $\nabla{\bf v}$ as in a bulk fluid, with a corresponding momentum-conserving noise, that are subdominant to the wavenumber-independent damping $\Gamma$ term and the noise ${\bf f}_{R}$.
* (17) S. Ramaswamy and R. A. Simha, Phys. Rev. Lett. 89, 058101 (2002); Physica A 306, 262-269 (2002); B. Manneville, P. Bassereau, S. Ramaswamy and J. Prost, Phys. Rev. E, 64, 021908 (2001).
* (18) D. Forster, Phys. Rev. Lett. 32 , 1161 (1974); M. Doi , J. Polym. Sci. Polym. Phys. Ed. 19, 229 (1981).
* (19) P. D. Olmsted and P. M. Goldbart , Phys. Rev. A 41, 4578 (1990).
* (20) The alert reader will argue that in an active system the relaxation rate of Q should not be dictated wholly by a conjugate thermodynamic force determined by the free-energy functional $F$. We should allow an additional relaxational term of the form $-\lambda\mbox{Q}$ on the right-hand side of (3). This is true, but such a term can be absorbed into a redefinition of $F$, as far as the equation of motion (3) is concerned. The point is that the same redefinition will not transform the active stress into a form derivable from $F$.
* (21) For simplicity we ignore the dependence of the noise strength on the dynamical variables. This would give rise to multiplicative noise effects that are beyond the scope of this work.
* (22) If we derive the equation for $\theta$ from a collisional model, where each particle moves forward or backward along its length and where two particles which come within a certain radius of each other try to align parallel to each other, we find
$\frac{\partial\theta}{\partial
t}=\lambda_{1}(\theta)\partial_{x}\theta\partial_{z}\theta+A_{1}(\theta)\partial_{x}^{2}\theta+A_{2}(\theta)\partial_{z}^{2}\theta+\lambda_{2}\theta\partial_{x}\partial_{z}\theta$
upto quadratic order in $\theta$ and gradients, and ignoring derivatives of
the density. The coefficients are
$\lambda_{1}(\theta)=\frac{8}{3}S^{2}\cos 2\theta-\frac{1}{3}S(1-\cos
4\theta)\simeq\frac{8}{3}S^{2},$
$A_{1}(\theta)=-\frac{1}{2}S^{2}\cos 2\theta+\frac{1}{6}S-\frac{1}{3}S\cos
4\theta\simeq-\frac{1}{2}S^{2}-\frac{1}{6}S,$
$A_{2}(\theta)=\frac{1}{2}S^{2}\cos 2\theta+\frac{1}{6}S-\frac{1}{3}S\cos
4\theta\simeq\frac{1}{2}S^{2}-\frac{1}{6}S,$
$\lambda_{2}\simeq 2S^{2}.$
where $S$ is the scalar order parameter. Comparing with (10), we see it
satisfy the relation $2(A_{2}-A_{1})=\lambda_{2}$.
* (23) C.W. Oseen. Trans. Faraday Soc. 29 (1933) 883; H. Zocher. ibid. 29 (1933) 945; F.C. Frank. Disc. Faraday Soc. 25 (1958) 19.
* (24) D. R. Nelson and R. Pelcovits, Phys. Rev. B 16 2191 (1977).
* (25) This can be confirmed by checking that the functional curl is nonzero:
* (26) S. K. Ma and G. F. Mazenko, Phys. Rev. B 11, 4077 (1975).
* (27) S. K. Ma, Modern Theory of Critical Phenomena (Benjamin, Reading, Mass., 1976).
* (28) P. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
* (29) In what follows, $\int_{k\Omega}\equiv\int_{k<\Lambda}(d^{2}k/(2\pi)^{2})\int_{-\infty}^{+\infty}d\Omega/2\pi$.
* (30) This term is absent in the Burgers-like limit $\lambda_{2}=0$.
* (31) H. Risken, The Fokker-Planck equation: Methods of Solution and Applications Springer (1989).
|
arxiv-papers
| 2009-12-11T17:15:19 |
2024-09-04T02:49:06.982391
|
{
"license": "Public Domain",
"authors": "Shradha Mishra, R. Aditi Simha, Sriram Ramaswamy",
"submitter": "Shradha Mishra",
"url": "https://arxiv.org/abs/0912.2283"
}
|
0912.2358
|
# Dark Matter Direct Detection with Non-Maxwellian Velocity Structure
Michael Kuhlena,b, Neal Weinerc, Jürg Diemandd, Piero Madaue, Ben Moored, Doug
Potterd, Joachim Stadeld, Marcel Zempf
a School of Natural Science, Institute for Advanced Study, Princeton, NJ 08540
b Theoretical Astrophysics Center, University of California Berkeley,
Berkeley, CA 94720
c Center for Cosmology and Particle Physics, Department of Physics, New York
University, New York, NY 10003
d Institute for Theoretical Physics, University of Zurich, 8057 Zurich,
Switzerland
e Department of Astronomy & Astrophysics, University of California Santa Cruz,
Santa Cruz, CA 95064
f Department of Astronomy, University of Michigan, Ann Arbor, Michigan 48109
E-mail mqk@astro.berkeley.edu,neal.weiner@nyu.edu
###### Abstract:
The velocity distribution function of dark matter particles is expected to
show significant departures from a Maxwell-Boltzmann distribution. This can
have profound effects on the predicted dark matter - nucleon scattering rates
in direct detection experiments, especially for dark matter models in which
the scattering is sensitive to the high velocity tail of the distribution,
such as inelastic dark matter (iDM) or light (few GeV) dark matter (LDM), and
for experiments that require high energy recoil events, such as many
directionally sensitive experiments. Here we determine the velocity
distribution functions from two of the highest resolution numerical
simulations of Galactic dark matter structure (Via Lactea II and GHALO), and
study the effects for these scenarios. For directional detection, we find that
the observed departures from Maxwell-Boltzmann increase the contrast of the
signal and change the typical direction of incoming DM particles. For iDM, the
expected signals at direct detection experiments are changed dramatically: the
annual modulation can be enhanced by more than a factor two, and the relative
rates of DAMA compared to CDMS can change by an order of magnitude, while
those compared to CRESST can change by a factor of two. The spectrum of the
signal can also change dramatically, with many features arising due to
substructure. For LDM the spectral effects are smaller, but changes do arise
that improve the compatibility with existing experiments. We find that the
phase of the modulation can depend upon energy, which would help discriminate
against background should it be found.
dark matter, direct detection, numerical simulations
## 1 Introduction
Direct detection experiments aim to detect the low energy nuclear recoil from
rare scattering events between dark matter (hereafter DM) particles, assumed
to be weakly interacting massive particles (WIMPs), and target nuclei. The
event rate and its energy spectrum depend on the properties of the DM
distribution at Earth’s location, about 8.5 kpc from the Galactic Center [1].
Typical calculations of the scattering rate assume a “standard halo model”
(SHM) consisting of a local DM density of $0.3\pm 0.1$ GeV cm-3 and a Maxwell-
Boltzmann (MB) velocity distribution function with a (three-dimensional)
dispersion of $270\pm 70$ km/s [2] truncated at an escape speed of $\sim 550$
km/s. Recent numerical simulations of the formation of Galactic-scale DM halos
have reached the necessary resolution to directly test these assumptions.
At the same time, absent a positive signal, a set of uniform halo assumptions
allows a simple means to compare different experiments. However, in light of
the recent results from DAMA/LIBRA [3], confirming earlier results from
DAMA/NaI [4], it is important to consider halo model uncertainties when
discussing exclusion limits from experiments with different targets, or energy
ranges. This is particularly important because proposals such as light dark
matter (LDM) [5, 6] and inelastic dark matter (iDM) [7, 8], which aim to
reconcile DAMA with null results from other experiments, sample the high
velocity component of the WIMPs preferentially, and it is especially here that
the Maxwell-Boltzmann distribution is expected to break down.
In the hierarchical structure formation paradigm of standard cold dark matter
cosmology the DM halo of a typical galaxy is built up through the merger of
many individual gravitationally bound progenitor halos, which themselves were
assembled in a hierarchical fashion. High resolution cosmological simulations,
such as Via Lactea [9, 10], GHALO [11], and Aquarius [12], have shown that
this merging process is in general incomplete, with the dense cores of many of
the merging halos surviving as subhalos orbiting within their respective host
halos. During pericenter passages tidal forces can strip off a large fraction
of a subhalo’s material, but the resulting cold tidal streams can readily be
identified as velocity space substructure. The resulting DM halos are not
perfectly phase mixed, and the assumption of a smooth halo is in general not a
good one, neither in configuration space nor in velocity space [13].
At $8.5$ kpc the Sun is located quite close to the Galactic Center, at least
when compared to the overall extent of the Milky Way’s DM halo ($r_{\rm
vir}\sim 200-300$ kpc). This central region is notoriously difficult for
cosmological numerical simulations to resolve, as very high particle numbers
and very short time steps are required to avoid the so-called “over-merging
problem” [14], which has until recently resulted in an artifically smooth
central halo devoid of any substructure. With the advent of
$\mathcal{O}(10^{9})$ particle simulations at the Galactic scale this problem
finally seems to have been overcome, with hundreds of subhalos identified at
$\lesssim 20$ kpc. Nevertheless the local phase space structure is far from
completely resolved, and likely never will be through direct numerical
simulation. Any estimation of the importance of local density or velocity
substructure based on cosmological simulation must thus rely on extrapolations
over many orders of magnitude below its resolution limit.
Local density variations due to the clumpiness of the DM halo are unlikely to
significantly affect the direct detection scattering rate. Based on the
Aquarius Project suite of numerical simulations, Vogelsberger et al. (2008)
[15] report that at more than 99.9% confidence the DM density at the Sun’s
location differs by less than 15% from the average over a constant density
ellipsoidal shell. Extrapolating from their numerical convergence study they
estimate a probability of $10^{-4}$ of the Sun residing in a bound subhalo of
any mass. Analytical work by Kamionkowski & Koushiappas (2008) [16] predicts a
positively skewed density distribution with local densities as low as one
tenth the mean value, but probably not much less than half.
The situation for velocity substructure is less clear. It is well established
that numerically simulated dark matter halos exhibit significant velocity
anisotropy and global departures from a Maxwell-Boltzmann distribution [17,
18, 19, 20]. The implications for direct detection experiments of these global
departures from the standard Maxwellian model have previously been
investigated in the context of a standard WIMP model [21, 22, 23, 24] and for
inelastic dark matter [25, 26], and they were found to result in appreciable
differences (factor of a few) in the total event rates and the annual
modulation signal. Most recently Vogelsberger et al. (2008) reported
significant structure (“wiggles”) in the velocity distribution function
measured in their high-resolution Aquarius simulations, which they attributed
to events in the halo’s mass assembly history. Their analysis concluded that
velocity substructure due to bound subhalos or unbound tidal streams, however,
does not influence the detector signals, since it makes up a highly sub-
dominant mass fraction locally.
The aim of this paper is take a closer look at this velocity space
substructure and to examine its impact on the direct detection signal for
models that are particularly sensitive to the high velocity tail, such as LDM
or iDM. In contrast to Vogelsberger et al. (2008), we find that both global
and local departures from the best-fit Maxwell-Boltzmann distribution can
significantly affect the total event rate, the annual modulation, and the
recoil energy spectrum. Parameter exclusion limits derived using a standard MB
halo model are likely to be overly restrictive.
This paper is organized as follows: in Section 2 we present velocity
distribution functions derived from the high-resolution numerical simulations
Via Lactea II and GHALO. In Section 3 we look at the implications of high
velocity substructure for direct detection experiments with directional
sensitivity. In Section 4 we consider iDM and LDM models and show how the
observed local and global departures from the MB model affect scattering event
rates and recoil spectra at several ongoing direct detection experiments, and
how this modifies parameter exclusion limits. A summary and discussion of our
results can be found in Section 5.
## 2 Results from Numerical Simulations
The nuclear scattering event rate depends on the size of the detector, the
type of target material, the scattering cross section, and the number density
and velocity distribution of the impinging DM particles. We defer calculations
of the expected event rate for various experimental setups and types of DM
models to section 4, and focus in this section on the particle velocity
distributions, which we determine directly from numerical simulations.
### 2.1 The Via Lactea and GHALO simulations
Our analysis is based on two of the currently highest resolution numerical
simulations of Galactic DM structure: Via Lactea II (VL2) [10] and GHALO [11].
Both are cosmological cold DM N-body simulations that follow the hierarchical
growth and evolution of a Milky-Way-scale halo and its substructure from
initial conditions in the linear regime ($z=104$ for VL2, $z=58$ for GHALO)
down to the present epoch. For details about the setup of the simulations we
refer the reader to the above references. The VL2 host halo is resolved with
$\sim 400$ million particles of mass $m_{p}=4,100\,\rm M_{\odot}$ within its
virial radius111$r_{\rm vir}$ is defined as the radius enclosing a density of
$\Delta_{\rm vir}=389$ times the background density. of $r_{\rm vir}=309$ kpc
and has a mass of $M_{\rm halo}=1.7\times 10^{12}\,\rm M_{\odot}$ and peak
circular velocity $V_{\rm max}=201.3$ km/s. The GHALO host is somewhat less
massive, $M_{\rm halo}=1.1\times 10^{12}\,\rm M_{\odot}$ and $V_{\rm
max}=152.7$ km/s, but even more highly resolved, with 1.1 billion particles of
mass $m_{p}=1,000\,\rm M_{\odot}$ within its $r_{\rm vir}=267$ kpc. For
reference we show the circular velocity of the two halos in Fig.1. In order to
facilitate a more direct comparison between the two halos, we have also scaled
GHALO to match VL2’s $V_{\rm max}$ by multiplying the simulation’s length and
velocity units by a factor $f=V_{\rm max}({\rm VL2})/V_{\rm max}({\rm
GHALO})=1.32$, and the mass unit by $f^{3}$. We refer to this model as GHALOs.
The circular velocity of these three halos at 8.5 kpc is 158.1, 121.7, and
148.9 km/s for VL2, GHALO, and GHALOs, respectively.
Figure 1: Circular velocity profiles of the VL2, GHALO, and GHALOs host halos.
### 2.2 Velocity Modulus Distributions
The DM-nucleon scattering event rate is directly proportional to
$g(v_{\rm min})=\int_{v_{\rm min}}^{\infty}\frac{f(v)}{v}dv,$ (1)
where $f(v)$ is the DM velocity distribution function in the Earth’s rest
frame and $v_{\rm min}(E_{R})$ is the minimum velocity that can result in a
scattering with a given nuclear recoil energy $E_{R}$. For a target with
nuclear mass $m_{N}$ and a WIMP/nucleon reduced mass
$\mu=m_{N}m_{\chi}/(m_{N}+m_{\chi})$, $v_{\rm min}(E_{R})$ is given by
$\left(\frac{v_{\rm
min}}{c}\right)^{2}=\frac{1}{2}\frac{m_{N}E_{R}}{\mu^{2}}\left(1+\frac{\mu}{m_{N}E_{R}}\delta\right)^{2}.$
(2)
The $\delta$ refers to the possible mass splitting between the incoming and
outgoing DM particle, which would be 0 for standard and light DM and
$\mathcal{O}$(100 keV) for inelastic DM.
We determine $f(v)$ in the halo rest frame directly from the particle
velocities in our numerical simulations, and in the Earth’s rest frame by
first applying a Galilean velocity boost by $v_{\oplus}(t)$. The Earth’s
velocity with respect to the Galactic center is the sum of the local standard
of rest (LSR) circular velocity around the Galactic center, the Sun’s peculiar
motion with respect to the LSR, and the Earth’s orbital velocity with respect
to the Sun,
$\vec{v}_{\oplus}(t)=\vec{v}_{\rm LSR}+\vec{v}_{\rm pec}+\vec{v}_{\rm
orbit}(t).$ (3)
We follow the prescription given in Chang et al. (2008) [27] and set
$\vec{v}_{\rm LSR}=(0,220,0)$ km/s, $\vec{v}_{\rm pec}=(10.00,5.23,7.17)$ km/s
[28], and $\vec{v}_{\rm orbit}(t)$ as specified in reference [29]. The
velocities are given in the conventional $(U,V,W)$ coordinate system where $U$
refers to motion radially inwards towards the Galactic center, $V$ in the
direction of Galactic rotation, and $W$ vertically upwards out of the plane of
the disk. We associate these three velocity coordinates with the
$(v_{r},v_{\theta},v_{\phi})$ coordinates of the simulation particles.
The Earth’s orbital motion around the Sun results in the well-known annual
modulation of the scattering rate, which the DAMA collaboration claims to have
detected at very high statistical significance [3]. For the SHM the peak of
this modulation occurs around June 2nd, when the Earth’s relative motion with
respect to the Galactic DM halo is maximized.
Figure 2: Velocity distribution functions: the left panels are in the host
halo’s restframe, the right panels in the restframe of the Earth on June 2nd,
the peak of the Earth’s velocity relative to Galactic DM halo. The solid red
line is the distribution for all particles in a 1 kpc wide shell centered at
8.5 kpc, the light and dark green shaded regions denote the 68% scatter around
the median and the minimum and maximum values over the 100 sample spheres, and
the dotted line represents the best-fitting Maxwell-Boltzmann distribution.
We have measured the DM velocity distribution from all particles in a 1 kpc
wide spherical shell (8 kpc $<r<$ 9 kpc), containing 2.1, 5.4, and 3.6 million
particles in VL2, GHALO, and GHALOs, respectively. The large particle numbers
in these measurements result in a very small statistical uncertainty, but fail
to capture any local variations. To address this we have also determined
$f(v)$ from the particles in 100 randomly distributed sample spheres centered
at 8.5 kpc. These sample spheres have radii of 1.5 kpc for VL2 and 1 kpc for
GHALO and GHALOs, and contain a median of 31,281, 21,740, and 14,437 particles
in the three simulations.222Tables of $g(v_{\rm min})$ determined from the
spherical shell and the 100 sample spheres, and tracing the annual modulation
over 12 evenly spaced output times, are available for download at
http://astro.berkeley.edu/$\sim$mqk/dmdd/.
The resulting distributions, both in the halo rest frame and translated into
Earth’s rest frame, are shown in Fig. 2. The shell averaged distribution is
plotted with a solid line, while the light and dark green shaded regions
indicate the 68% scatter around the median and the absolute minimum and
maximum values of the distribution over the 100 sample spheres. For comparison
we have also overplotted the best-fitting Maxwell-Boltzmann (hereafter MB)
distributions, with 1D velocity dispersion of $\sigma_{\rm 1D}=$ 130, 100, and
130 km/s. These clearly underpredict both the low and high velocity tails of
the actual distribution. This is not a new result and has previously been
found in cosmological numerical simulations [17, 18, 19, 15]. Actually there
is no reason to assume that a self-gravitating, dissipationless system would
have a locally Maxwellian velocity distribution, and in fact it has been shown
that self-consistent, stable models of cuspy DM structures require just such
non-Gaussianity [30, 31].
In addition to its overall non-Maxwellian nature, we notice several broad
bumps present in both the shell averaged and, at very similar speeds, in the
sub-sample $f(v)$. Similar features were reported by Vogelsberger et al.
(2008) [15] for the host halos of their completely independent Aquarius
simulations, and thus appear to be robust predictions of hierarchically formed
collisionless objects. Vogelsberger et al. also showed that the broad bumps
are independent of location and persistent in time and hence reflect the
detailed assembly history of the host halo, rather than individual streams or
subhalos. The extrema of the sub-sample distributions, however, exhibit
numerous distinctive narrow spikes at certain velocities, and these are due to
just such discrete structures. Note that although only a small fraction of
sample spheres exhibits such spikes, they are clearly present in some spheres
in all three simulations. The Galilean transform into the Earth’s rest frame
washes out most of the broad bumps, but the spikes remain visible, especially
in the high velocity tails, where they can profoundly affect the scattering
rates for inelastic and light DM models (see Section 4).
In order to assess the dependence of these features on the sample sphere size,
we also considered for the VL2 simulation sphere radii of 1 and 2 kpc,
containing a median of 9,200 and 74,398 particles, respectively. The coarse
features of the distributions persist, but of course the prevalence of the
spikes increases with sample sphere size, as more of the substructure is
probed. It is difficult to assess with our simulations the true likelihood of
significant local velocity substructure, as it depends on the abundance and
physical extent of subhalos and tidal streams many orders of magnitude below
the length scales that we can accurately resolve. Higher resolution numerical
simulations, as well as analytical models [32], perhaps in conjunction with
simulations [33], will be necessary to settle this question.
### 2.3 The effects of neglected baryonic physics
The values of $\sigma_{\rm 1D}$ we report here may appear surprisingly low to
a reader familiar with the standard isothermal MB halo assumption of $\langle
v^{2}\rangle=3\,\sigma_{\rm 1D}^{2}=3/2\,v_{0}^{2}$, where $v_{0}$, the peak
of the MB distribution, i.e. the most probable speed, is assumed to be equal
to the rotation velocity of the Sun around the galaxy, $v_{0}\simeq 220$ km/s.
In this standard model $\sigma_{\rm 1D}$ would be 156 km/s, considerably
higher than our values of 130 km/s and 100 km/s, respectively. In fact, the
local circular velocity $v_{c}$ and velocity dispersion $\sigma$ are only
indirectly related and not necessarily equal. While $v_{c}$ is set by the
local radial gradient of the potential, $\sigma$ depends on the shape of the
potential at exterior radii. For a non-rotating spherical system the relation
between $v_{c}(r)$ and the radial velocity dispersion $\sigma_{r}(r)$ is given
by
$v_{c}^{2}=-\sigma_{r}^{2}\left(\frac{d\ln\rho}{d\ln
r}+\frac{d\ln\sigma_{r}^{2}}{d\ln r}+2\beta\right),$ (4)
where $\frac{d\ln\rho}{d\ln r}\equiv\gamma(r)$ is the logarithmic slope of the
density profile and $\beta(r)\equiv 1-\sigma_{\theta}^{2}/\sigma_{r}^{2}$ is
the velocity anisotropy. In a singular isothermal sphere ($\gamma=-2$,
$\frac{d\ln\sigma_{r}^{2}}{d\ln r}=0$, $\beta=0$) we have $v_{c}=v_{0}$, but
for an NFW profile $v_{c}/v_{0}\approx 0.88$ at $r=r_{s}/2$. In the VL2 and
GHALO host halos the relation is $v_{c}/v_{0}=0.85$ and $0.86$, respectively.
A central baryonic condensation in the form of a Galactic disk and bulge will
deepen the central potential, raise the local circular velocity to $\sim 220$
km/s, and increase the velocity dispersion of DM particles at the Sun’s
location. Since the VL2 and GHALO simulations do not include baryonic physics,
it is not surprising that the values of $v_{c}$ and $v_{0}$ at 8.5 kpc are
lower than appropriate for our Milky Way galaxy. The main focus of our work
here is to investigate the effects of global and local variations from the MB
assumption, and therefore we compare our results to the best-fitting MB model
($v_{0}=184$ km/s for VL2 and GHALOs and 141 km/s for GHALO) instead of the
standard MB halo model ($v_{0}=220$ km/s). In principle we could have scaled
just the velocities up to give $v_{0}=220$ km/s, but this would remove many of
the effects of substructure by pushing it above the escape velocity. Thus, we
use VL2 and GHALOs simulations for the parameter exclusions plots in Section
4, but it should be recognized, that the velocity dispersion is somewhat lower
than in conventional MB halo parameterizations.
### 2.4 Local Escape Speed
The escape speed from the Sun’s location in the Galactic halo is another
factor that can strongly affect scattering rates, especially for inelastic and
light DM models. The RAVE survey’s sample of high-velocity stars constrains
the Galactic escape velocity to lie between 498 and 608 km/s at 90%
confidence, with a median likelihood of 544 km/s [34]. This is in good
agreement with the highest halo rest frame speed of any particle in our 8.5
kpc spherical shells, namely 550 km/s in VL2 and 586 km/s in GHALOs. The lower
$V_{\rm max}$ of the GHALO host is reflected in a significantly lower escape
speed, only 433 km/s. The corresponding maximum speeds in the Earth rest frame
are 735-761, 773-802, and 634-660 km/s, where the range refers to the
modulation introduced by the Earth’s orbit around the Sun.
### 2.5 Radial and Tangential Distributions
Figure 3: Radial and tangential velocity distribution functions. The solid
lines show the shell average, the shaded range the 68% scatter around the
median, and the dotted curve shows the best-fit MB model.
For comparison with past work [25, 24] we also present separately the
distribution functions of the radial $v_{r}$ and tangential
$v_{t}=\sqrt{v_{\theta}^{2}+v_{\phi}^{2}}$ velocity components in Fig. 3. We
fit these distributions with functions like the ones in Fairbairn & Schwetz
(2009) [24], except that we don’t normalize the velocities by the square root
of the gravitational potential at the particles’ location and instead provide
separate fits for each simulation. We also include an estimate of the variance
due to local velocity substructure by separately fitting the distribution in
each of the sample spheres. The fitting functions are
$\displaystyle f(v_{r})$ $\displaystyle=$
$\displaystyle\frac{1}{N_{r}}\exp\left[-\left(\frac{v_{r}^{2}}{\bar{v}_{r}^{2}}\right)^{\alpha_{r}}\right]$
(5) $\displaystyle f(v_{t})$ $\displaystyle=$
$\displaystyle\frac{v_{t}}{N_{t}}\exp\left[-\left(\frac{v_{t}^{2}}{\bar{v}_{t}^{2}}\right)^{\alpha_{t}}\right],$
(6)
and the parameters of these fits are listed in Table 1. The normalizations
$N_{r}$ and $N_{t}$ can readily be obtained numerically for a given set of
parameters by ensuring that the distributions integrate to unity.
| | radial | tangential
---|---|---|---
| | shell | median | $16^{\rm th}$ | $84^{\rm th}$ | shell | median | $16^{\rm th}$ | $84^{\rm th}$
VL2 | $\bar{v}_{r,t}$ [km/s] | 202.4 | 199.9 | 185.5 | 212.7 | 128.9 | 135.1 | 124.2 | 148.9
$\alpha_{r,t}$ | 0.934 | 0.941 | 0.877 | 0.985 | 0.642 | 0.657 | 0.638 | 0.674
GHALO | $\bar{v}_{r,t}$ [km/s] | 167.9 | 163.6 | 156.4 | 173.0 | 103.1 | 114.3 | 93.21 | 137.0
$\alpha_{r,t}$ | 1.12 | 1.11 | 1.02 | 1.20 | 0.685 | 0.719 | 0.666 | 0.819
GHALOs | $\bar{v}_{r,t}$ [km/s] | 217.9 | 213.8 | 202.3 | 226.6 | 138.2 | 162.2 | 125.1 | 183.1
$\alpha_{r,t}$ | 1.11 | 1.11 | 1.01 | 1.18 | 0.687 | 0.759 | 0.664 | 0.842
Table 1: Radial and tangential velocity distribution fit parameters (see Eqs.
5 and 6). The columns labeled “shell” refer to the fit for all particles in
the 1 kpc wide shell centered on 8.5 kpc, whereas the following three columns
give the median as well as the $16^{\rm th}$ and $84^{\rm th}$ percentile of
the distribution of fits over the 100 sample spheres (see text for more
detail).
### 2.6 Modulation Amplitude and Peak Day
The annual modulation of the scattering rate $g(v_{\rm min})$ (Eq. 1) grows
with $v_{\rm min}$, since a reduction in the number of particles able to
scatter makes the summer-to-winter difference relatively more important. At
sufficiently high velocities the modulation amplitude can even reach unity,
when during the winter there simply aren’t any particles with velocities above
$v_{\rm min}$.
The expected modulation amplitude for a given experiment is then determined by
the relation between the measured recoil energy $E_{R}$ and $v_{\rm min}$, see
Eq. 2. There are important qualitative difference between standard DM models
($\delta=0$) and inelastic models ($\delta\sim 100$ keV). Inelastic DM models
require a higher $v_{\rm min}$ for a given $E_{R}$, resulting in a lower total
scattering rate and a more pronounced modulation. Furthermore, while for
standard DM $v_{\rm min}$ grows with $E_{R}$, in inelastic models $v_{\rm
min}$ typically falls with $E_{R}$ (for $\delta>E_{R}m_{N}/\mu$).
Figure 4: The amplitude (left panels) and peak day (right) as a function of
$v_{\rm min}$. The solid red line indicates the shell averaged quantities, and
the dotted line the best-fit MB distribution. The light and green shaded
regions cover the central 68% region around the median and the minimum and
maximum values of the distribution over the 100 sample spheres. The thin black
line shows the behaviour of one example sample sphere.
In the left panels of Fig. 4 we show the $v_{\rm min}$ dependence of the
annual modulation amplitude, $({\rm max}(g(v_{\rm min}))-{\rm min})/({\rm
max}+{\rm min})$ as measured in our simulations. We focus on the high $v_{\rm
min}$ region that is relevant for inelastic and light dark matter models, as
well as directional detection experiments. In VL2 and GHALOs the shell
averaged modulation amplitude (solid red line) rises from about 20% at $v_{\rm
min}=400$ km/s to unity at $\sim 750$ km/s. The GHALO amplitudes are shifted
to lower $v_{\rm min}$, growing from 40% at 400 km/s to 100% already at $\sim
600$ km/s. The strong high velocity tails of $f(v)$ (Fig. 2) result in
somewhat lower modulation amplitudes compared to the best-fit MB distributions
(dotted line). At the very highest velocities $f(v)$ drops below the
Maxwellian distribution in VL2 and GHALO, and this leads to the rise in
amplitudes above the Maxwellian case for $v_{\rm min}>600$ and 550 km/s,
respectively. In GHALOs the distribution more closely follows the MB fit, and
only barely rises above it at $v_{\rm min}>670$ km/s. Interestingly, all three
simulations exhibit a pronounced dip in the modulation amplitude at close to
the highest $v_{\rm min}$. These correspond to bumps in $f(v)$ discussed in
Section 2.2.
As before, the light and dark shaded green regions in Fig. 4 cover the 68%
region around the median and the extrema of the distribution over the 100
sample spheres. Over most of the range of $v_{\rm min}$, the typical
modulation amplitude in a sample sphere differs from the spherical shell
average by less than 10%. The extrema of the distribution, however, can differ
much more significantly, especially at higher velocities. If the Sun happens
to be passing through a fast moving subhalo or tidal stream, the typical
velocity of impinging DM particles can greatly differ from the smooth halo
expectation, leading to an increase (or decrease) in the overall scattering
rate and modulation amplitude. For a conventional massive ($>10$ GeV) DM
particle, scattering events are dominated by the peak of the velocity
distribution function and the effects from streams or subhalos are washed out
and typically negligible [15]. However, whenever scattering events are
dominated by particles on the high velocity tail of the distribution, either
because of a velocity threshold (inelastic DM), a particularly low particle
mass (light DM), or for detectors intrinsically sensitive to only high recoil
energies (e.g. many directionally sensitive detectors), the presence of
velocity substructure can have a profound impact on the event rates.
Another potentially interesting signature of velocity space structure is the
shift in the peak day of the annual modulation, as was explored in [22]. For
an isotropic velocity distribution (e.g. MB) the peak day is independent of
$v_{\rm min}$ and occurs around the beginning of June. Any kind of departure
from isotropy, however, would leave its signature as a change in the phase of
the modulation. In the most extreme case of a very massive DM stream moving in
exactly the same direction as the Sun, the phase of modulation could flip
completely, with the maximum scattering amplitude occurring in the winter.
In the right panel of Fig. 4 we show the $v_{\rm min}$ dependence of the peak
day as measured in our simulations. The spherical shell average is consistent
with a phase shift of zero, except at the very highest velocities where a few
discrete velocity structures dominate the shell average and introduce small
velocity dependent phase shifts. The peak days in the sample spheres, however,
often differ by $\sim 20$ days from the fiducial value. As an example we have
plotted a curve for one of the sample spheres. The peak day determination
appears to be quite noisy here, but since the particle numbers are still
fairly large ($N(>v_{\rm min})=9114,802,105$ for $v_{\rm min}=400,600,670$
km/s in the VL2 sample sphere shown), we believe that the variations in peak
day arise from actual velocity structure rather than discreteness noise. The
68% scatter of the phase shifts over the 100 sample spheres remains remarkable
constant at $\pm 10-20$ days throught most of the $v_{\rm min}$ range. At
higher velocities ($v_{\rm min}>600-700$ km/s) the typical phase shifts grow
to $\sim 50$ days, which could be due to the increasing relative importance of
individual streams, but may also be due to particle discreteness noise. Based
on our measurements we would expect some amount of variation in the peak day
as a function recoil energy. As it is hard to imagine any background
contamination to exhibit such a phase shift, this raises the possibility of
such a measurement confirming a DM origin of the DAMA modulation signal.
We conclude this section by noting that our simulations reveal both global
(shell averaged) and local (sample spheres) departures from the standard halo
model in velocity space. These can have a significant effects on the overall
scattering rate, as well as on the amplitude and peak day of the annual
modulation thereof. In the next section we go on to explore how these effects
translate into predicted detection rates for actual direct detection
experiments, and how they modify parameter exclusions plots based on existing
null-detections.
## 3 Velocity Substructure and Directional Detection
Up to this point we have focused on the integrated features of the halo, i.e.,
$f(v)$ or $g(v_{\rm min})$, but there are many more structures that appear
that are not pronounced in these measures. In particular, the presence of
clumps and streams, while contributing to the bumps and wiggles of these
functions, can be more pronounced when the direction of the particles’
motions, rather than just the amplitude, is considered. These features are
especially important for directional WIMP detectors, such as DRIFT [35],
NEWAGE [36] and DMTPC [37], where the presence of such structures could show
up as a dramatic signal. Although we do not discuss it here, such directional
experiments have been argued to be especially important for testing inelastic
models [38, 39].
Figure 5: $f(v)$ and HealPix skymaps of the fraction of particles above
$v_{\rm min}$ coming from a given direction, for VL2 sample sphere #03 which
contains a fast-moving subhalo. The top row is in the halo rest frame ($v_{\rm
min}=400$ km/s), and the middle translated into the Earth rest frame ($v_{\rm
min}=500$ km/s). For comparison the bottom row shows the Maxwell-Boltzmann
halo case without substructure.
To quantify this, we begin by searching for “hotspots” on the sky. To do this,
we make a map of the sky in HealPix, and consider the flux of WIMPs from each
direction in the sky. We divide the sky into 192 equal regions of 215 square
degrees, and determine $p_{i}$, the fraction of particles above $v_{\rm min}$
with a velocity vector pointing towards bin $i$. We take the same 100 sample
spheres as before, but note that beyond determining sample sphere membership
we do not consider the location of a particle: the assignment to a given sky
pixel is based solely on the direction of its velocity vector. All of the
structure in a given sample sphere is considered local, i.e. able to influence
the signal at an Earth-bound direct detection experiment.
As an example of the kind of effects high velocity substructure can produce,
we show in Fig. 5 the speed distribution $f(v)$ and skymaps for VL2 sample
sphere #03. In the halo rest frame (top row, $v_{\rm min}=400$ km/s) a very
pronounced feature is visible due to the presence of a subhalo moving with
galacto-centric velocity modulus of $\sim 440$ km/s. This feature persists in
the Earth rest frame (center row, $v_{\rm min}=500$ km/s), where the direction
of the subhalo’s motion is “hotter” than the “DM headwind” hotspot in the
direction of Earth’s motion. When translating to the Earth’s rest frame, there
is an additional degree of freedom arising from the unspecified plane of the
Galactic disk. We associate radial motion towards (away from) the Galactic
Center with the center (anti-center) of the map, but the hotspot arising from
Earth’s motion is free to be rotated around this axis. In this example we have
chosen this rotation to maximize the angle between the subhalo hotspot and
Earth’s motion. In the bottom row we show for comparison the Earth rest frame
map for the MB-case without any substructure.
Figure 6: The distribution of HR$(v_{\rm min})$ as a function of $v_{\rm min}$
(left) and of $\psi$ for $v_{\rm min}=0$ and 500 km/s (right) for the VL2
simulation.
This is just one strong example, and we would like to understand what sorts of
hotspots we might expect. For this purpose we define the hotspot ratio
${\rm HR}(v_{\rm min})=\frac{{\rm max}\left\\{p_{i}(v_{\rm
min})\right\\}}{{\rm max}\left\\{p^{\rm MB}_{i}(v_{\rm min})\right\\}},$ (7)
the ratio of the hottest pixel in the sphere sample skymap above $v_{\rm min}$
to the hottest pixel in the corresponding MB-case, and the hotspot angle
$\psi$ between these two pixels. We calculate HR and $\psi$ for all 100 sample
spheres, and in each sphere for a full $2\pi$ rotation (in one degree
increments) of the direction of Earth’s motion. We show in Fig. 6 for VL2 the
distribution of HR as a function of $v_{\rm min}$ and the distribution of
$\psi$ for the case without a velocity threshold and for $v_{\rm min}=500$
km/s. For small velocities the mean of HR is unity and the r.m.s. variation is
only 10%. As $v_{\rm min}$ increases, HR grows: at $v_{\rm min}=600$ km/s, the
mean HR is $1.3\pm 0.35$, and at $v_{\rm min}=700$ km/s it’s $3.1\pm 1.4$. The
downturn at the very highest velocities is caused by running out of particles
in the sample spheres. There are also marked changes in the direction of the
hottest pixel. Even without a velocity threshold ($v_{\rm min}=0$ km/s) in 38%
of all cases the direction of the hottest spot on the sky is more than 10∘
removed from the direction of Earth’s motion (i.e. the MB halo expectation).
At $v_{\rm min}=500$ km/s the bulk of DM particles are more likely to be
coming from a local hotspot with $\psi>10^{\circ}$ than from the direction of
Earth’s motion. In this case there is only an 18% chance of having
$\psi<10^{\circ}$. Not all of this structure is due to individual subhalos;
some of it may arise from tidal streams, and some from the anisotropic
velocity distribution of the host halo particles.
## 4 Implications for DAMA
In light of the DAMA/LIBRA results [3], two scenarios which have garnered a
great deal of attention of late are light dark matter [5, 6], and inelastic
dark matter [7, 8]. These scenarios were proposed some time ago to address the
conflicts between DAMA [4] and CDMS [40] as well as other experiments at the
time.
Recent studies [27, 25, 41, 42] have shown that iDM remains a viable
explanation of the DAMA data, consistent with recent results from CDMS [43],
XENON10 [44], KIMS [45], ZEPLIN II [46], ZEPLIN III [47] and CRESST [48]. Such
models have some tension with CRESST (see the discussion in [27, 49]), which
observed 7 events in the tungsten band, while approximately 13 would be
expected [27]. While lighter masses have no tension with CDMS, higher mass
($\gtrsim 250$ GeV) WIMPs do.
Light dark matter no longer works in its original incarnation [5, 6], but
instead relies upon “channeling,” [50, 51], a difficult to quantify effect
whereby some fraction of nuclear recoils have essentially all of the energy
converted into scintillation light, rather than just a fraction. Even
including channeling, such scenarios seem to have strong tension with the data
[52, 24, 53], both in the spectrum of the modulation, and constraints from the
unmodulated event rate at low (1-2 keVee) energies333keVee stands for
“electron equivalent keV”, a unit of energy used for scintillation light,
which is produced by interactions of the recoiling nucleus with electrons. It
is related to the full nuclear recoil energy (in keVr) through the quenching
factor $q$: $E({\rm scintillation})/{\rm keVee}=q\;E({\rm recoil})/{\rm
keVr}$. $q$ is a material-dependent quantity that must be experimentally
determined.. If one disregards the lowest (2-2.5 keVee) bin from DAMA,
however, the fit improves [52, 54], but there is still tension for the
lightest particles from the presence of modulation above 4 keVee and an
overprediction of an unmodulated signal at lower energies. It is important to
note that the uncertainties in $L_{\rm eff}$, the scintillation efficiency of
liquid Xenon (see [55] for a discussion), are not completely included in these
analyses, and the most recent measurements of $L_{\rm eff}$ [55] suggest that
the low energy analysis threshold may be somewhat higher than the 4.5keVr used
with $L_{\rm eff}=0.19$, weakening the limits. On the other hand, the most
recent analysis from XENON10 [56] using a more advanced rejection of double-
scatter events shows no events at all in the signal region all the way down to
the S2 threshold, strengthening the limits. In light of these uncertainties,
and in view of the importance of the result, we believe it is best to maintain
an open mind about the viability of the light WIMP explanation.
For our purposes, the crucial feature is that these scenarios sample the high
($v\sim 600$ km/s) component of the velocity distribution, and so are
especially sensitive to the departures from a simple Maxwell-Boltzmann
distribution. Consequently, it is important to investigate whether these
changes affect the tensions described above.
### 4.1 Inelastic Dark Matter
The inelastic Dark Matter scenario [7] was proposed to explain the origin of
the DAMA annual modulation signal. The scenario relies upon three basic
elements: first, the presence of an excited state $\chi^{*}$ of the dark
matter $\chi$, with a splitting $\delta=m_{\chi^{*}}-m_{\chi}\sim
m_{\chi}v^{2}$ comparable to the kinetic energy of the WIMP. Second, the
absence of, or at most a suppressed elastic scattering cross section off of
nuclei, i.e., the process $\chi N\rightarrow\chi N$ should be small. Third, an
allowed cross section for inelastic scatterings, i.e., $\chi
N\rightarrow\chi^{*}N$, with a size set roughly by the weak scale.
Although these properties may seem odd at first blush, they are in fact
perfectly natural if the scattering occurs through a gauge interaction [7, 8],
where the splitting is between the two Majorana components of a massive
pseudo-Dirac fermion, or between the real and imaginary components of a
complex scalar. Simple models can be constructed where the mediating
interaction is the Z-boson [7, 8, 57, 41]. Models with new vector interactions
to explain the PAMELA positron excess also naturally provide models of iDM
[58], and often where all scales arise naturally from radiative corrections
[58, 59, 60, 61, 62]. Composite models provide a simple origin for the excited
states as well [63, 64].
The principle change is a kinematical requirement on the scattering,
$\displaystyle\beta_{\rm min}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2m_{N}E_{R}}}\left(\frac{m_{N}E_{R}}{\mu}+\delta\right),$
(8)
where $m_{N}$ is the mass of the target nucleus and $\mu$ is the reduced mass
of the WIMP/target nucleus system. For $\delta\sim\mu\beta^{2}/2$, the
consequences can be significant. This requirement has three principal effects:
first, the kinematical constraint depends on the target nucleus mass, and is
more stringent for lighter nuclei. If we are sampling dominantly the tail of
the velocity distribution, the relative effect between heavy and light targets
(e.g., iodine versus germanium) can be significant. Second, again because the
signal is sampling the tail of the velocity distribution, the modulated
amplitude can be significantly higher than the few percent expected for a
standard WIMP. In fact, in the cases where there are particles kinematically
scattering in the summer, but not the winter, the modulation can reach 100%.
Third, the inelasticity suppresses or eliminates the event rate at lower
energies. Because standard WIMPs have exponentially more events at lower
energies, most experiments have focused on controlling background in this
region and lowering the threshold. In particular, the XENON10 and ZEPLINIII
experiments have few events at low energies, but a significant background at
higher energies. Consequently, their limits for standard WIMPs are quite
strong, but significantly weaker for inelastic WIMPs. The combination of these
three elements allows an explanation of the DAMA result that is consistent
with all other current experiments [27, 25, 41].
#### 4.1.1 Fitting the DAMA signal
DAMA reports an annual modulation in the range of $2-6$ keVee range. This can
be interpreted as arising from either Na or I scattering events. In the former
case (Na), it has been shown that the “light inelastic” region (an
approximately 15 GeV WIMP with $\delta\approx 30{\rm~{}keV}$) can open up
significant parameter space [52, 49]. In the latter case (I), we find the
“heavy inelastic” region (a 100+ GeV WIMP with
$\delta\mathop{}_{\textstyle\sim}^{\textstyle>}100{\rm~{}keV}$) opens
significant parameter space. Since the constraints are stronger on the heavy
case, we focus on this region.
When determining the precise values of parameters that might agree with DAMA,
we must convert from keVee to keVr, which already introduces significant
uncertainties, a point which has been recently discussed by [25]. The
quenching factor for iodine has been found to have a range of different
values, including $q=0.09\pm 0.01$ [65], $q=0.05\pm 0.02$ [66], $q=0.08\pm
0.02$ [67] and $q=0.08\pm 0.01$ [68] 444The value quoted by [68] is generally
$q=0.086\pm 0.007$, but this value averages over a wide range of energies. The
two measured values for the DAMA energy range specifically are $q=0.08\pm
0.01$ and $q=0.08\pm 0.02$.. The first two measurements of the quenching
factor are somewhat indirect, fitting event distributions at low energies. The
last two are more direct. Of the last two, we should note that the first
includes non-linearity in its stated uncertainty, while the second explicitly
fixes to a signal electron energy, and thus this effect is not included. We
adopt $q=0.08\pm 0.02$ as the value for quenching, which corresponds to a
range of $25-75$ keVr for DAMA.
Figure 7: The ratio of DAMA signal in a given simulation to that in a MB
distribution. The solid red line is the spherical shell average, the dashed
line the median of the distribution over the sample spheres.
#### 4.1.2 Constraining the iDM interpretation of DAMA
While iDM allows DAMA to exist consistently with the other experiments, the
reach of other experiments is at or near the predicted DAMA level. In
particular, as we see in Fig. 14, within the context of a MB halo, at high
masses, CDMS excludes the entire range of the parameter space that fits DAMA.
At the same time, the CRESST results (with a tungsten target) seem very tense
with the data over the whole mass range. These two experiments provide the
greatest present tension with the iDM interpretation of DAMA. An immediate
question we can ask is then whether the velocity-space distribution of
particles in the simulations makes the constraints more or less significant
than a MB distribution.
This is not an easy question to answer. Since there can be significant spatial
variation in the velocity distribution, we would like to quantify this without
making full exclusion plots for every data point. The three principle
constraints on the iDM interpretation of DAMA come from a) Xe experiments
(XENON10, ZEPLIN-II and ZEPLIN-III), b) CDMS (Germanium) and c) CRESST
(Tungsten). While varying the halo model can have significant effects on each
experiment, including rates and spectra, the variation of velocity structures
in the different halos affects these limits differently, and it is difficult
to quantify in aggregate what the effects are.
Before proceeding into a detailed analysis of the DAMA signal, we can already
study a preliminary question: how does the cross section needed to explain
DAMA compare between a MB halo and a simulation? We consider the ratio of the
modulation between MB and the simulations in Fig. 7. Overall, we see that the
signal at DAMA is increased in the simulation, as much as a factor of a few.
Of course, increasing the DAMA signal does not change the constraints if the
signal in any of the other experiments is changed, as well. We proceed by next
considering a ratio of ratios (RoR). Since Germanium (CDMS) has a higher
velocity threshold than Iodine (DAMA), while Tungsten (CRESST) has a lower
velocity threshold, a simple test is to look at the variation of the signal at
different experiments as a fraction of the DAMA modulation amplitude. We thus
calculate for each of the simulation samples (spherical shell and 100
spheres), as well as for the best-fitting MB model, the ratios of the CDMS and
CRESST signals, integrated from 10-100 keV, to the total DAMA modulation in
either of the high-$q$ and low-$q$ benchmark regions described above. We then
divide the simulation ratio by the MB ratio: (CDMS/[DAMA])sim/(CDMS/[DAMA])MB
and (CRESST/[DAMA])sim/(CRESST/[DAMA])MB. If these RoR’s are smaller than one,
it suggests that using the simulation’s velocity distribution instead of the
best-fitting MB distribution weakens the limits relative to DAMA, while values
larger than one imply stronger limits. The effects on Xenon limits are harder
to quantify, because of the added uncertainties in the conversion from keVee
to keVr at those experiments, and where, precisely, the backgrounds lie. Thus,
we consider first only CDMS and CRESST limits.
Figure 8: Scatter plots of the ratios of ratios (RoR, see text for details)
for the CRESST and CDMS experiments. RoR’s less than one indicate that using
the simulation’s velocity distribution instead of the best-fitting MB model
weakens the CRESST or CDMS limits relative to the DAMA (high-$q$) signal.
Figure 9: Left: The differential recoil energy distribution (in
dru$=$events/kg/day/keV) at DAMA for selected spheres from VL2, GHALO and
GHALOs, normalized to unity in the DAMA signal range 2-6 keVee Right: The
distribution at CRESST for the same spheres, normalized to unity from 0-100
keV. The models are $m_{\chi}=100{\rm~{}GeV}$ and $\delta=130\,({\rm
top}),150\,({\rm middle}),170\,({\rm bottom}){\rm~{}keV}$. Figure 10: As in
Fig. 9, but with $m_{\chi}=700{\rm~{}GeV}$.
In Fig. 8 we show scatter plots of the CRESST RoR against the CDMS RoR. Large
filled symbols indicate the spherical shell sample and small symbols are used
for the sample spheres. Note that in some cases the CDMS RoR is zero,
indicating that the simulation velocity distribution resulted in no CDMS
signal at all. A few things are immediately obvious from this plot. First, the
limits from CDMS can vary wildly between simulations, and even between
different spheres within a single simulation. This simply represents how
dramatically the velocity distribution can change at the highest velocities.
Second, we see that the CRESST rate is much more weakly affected, with
typically suppressions of (0.6-1), (0.7-1.1), and (0.7-1.3) for Via Lactea II,
GHALO and GHALOs respectively. Thus, from the perspective of Poisson limits,
these results would suggest that those derived from Maxwellian halos are
possibly excessively aggressive by almost a factor of two. However, many
limits are placed using one of Yellin’s techniques [69], where not only the
overall rate, but also the distribution of signal versus background is
important. Here we find that the halo uncertainties can be at their largest.
We show in Fig. 9 the spectra at DAMA and CRESST for a 100 GeV WIMP with
$\delta=130,150,170{\rm~{}keV}$. We employ energy smearing at DAMA by assuming
that the smearing reported for the one of 25 targets of DAMA/LIBRA [70] is
characteristic of all of them. We assume a smearing of 1 keV at CRESST.
One can see that the peak positions and properties can change by quite a large
amount. At DAMA, the effect of this is principally to shift the peak. Such an
effect is largely degenerate with the quenching uncertainty, which can
reasonably range from $q=0.06$ to $q=0.09$. (For a lower quenching value, for
instance, the peak will shift to lower energy in keVee.) The effects would be
similar at XENON10, where the energy smearing will eliminate most interesting
structures, and the shift in the location of the peak will be comparable to
the uncertainties induced from $L_{\rm eff}$. In contrast, the spectrum for
CRESST can vary dramatically, with the peak location moving from below
$30{\rm~{}keV}$ to above $60{\rm~{}keV}$. The implication of this is that
techniques such as optimum interval, maximum gap and ${\rm p_{max}}$ are
likely all overly aggressive in that a peak in the spectrum might arise at a
specific location in the real Milky Way halo, but that would not be reproduced
at the appropriate position, for instance in a Maxwellian halo. In general,
any technique that relies on knowing precisely the predicted spectrum is
unreliable when considering these variations.
Figure 11: Allowed parameter space for DAMA at 90% (purple) and 99% (blue)
with $m_{\chi}=70{\rm~{}GeV}$. For comparison the 90% and 99% regions for
DAMA/LIBRA rates only (as opposed to DAMA/NaI + DAMA/LIBRA) are shown in red
and green lines. Constraints are CDMS (solid), ZEPLIN-III (long dashed, thin),
ZEPLIN-II (long-dashed, thick), CRESST (medium-dashed) and XENON10 (short-
dashed). The regions are MB with $v_{0}=220{\rm~{}km/s}$ and $v_{\rm
esc}=550{\rm~{}km/s}$ (top left), VL2 (top right), and a sample sphere from
VL2 (bottom, left) and GHALOs (bottom, right). Figure 12: As in 11, but with
$m_{\chi}=150{\rm~{}GeV}$. Figure 13: As in 11, but with
$m_{\chi}=300{\rm~{}GeV}$. Figure 14: As in 11, but with
$m_{\chi}=700{\rm~{}GeV}$.
Although we are still limited by our sample of simulations, we can study the
effects on various limits by looking at the allowed parameter space and limits
in a variety of halos. To do so, we largely follow [27] in calculating the
allowed parameter space and limits, with a few important differences. As
pointed out by [25], the uncertainty in quenching factor for iodine can be
extremely important. We thus consider the range $q=0.06$ to $q=0.09$ and take
the 90% allowed region to be the union of 90% allowed regions, and similarly
for the 99% region. We smear the signal as described above.
For CDMS we employ the maximum gap method with the data set specified in [27].
For limits arising from XENON10, we use the data presented in the recent
reanalysis of [56], and calculate the maximum gap limit (as in ZEPLIN-II and
ZEPLIN-III) for both [71] and [55] values of $L_{\rm eff}$, taking at every
point the more conservative of the two. For limits, we employ the maximum gap
method. Since the data are smeared and there are great uncertainties in
$L_{\rm eff}$, the detailed spectral information arising from structure is
lost.
For ZEPLIN-II, we take the data described in [46], but (conservatively) take
the energy values to be shifted by a factor of $0.24/0.19$ for a maximum gap
analysis, which employs the value of $L_{\rm eff}$ from [71] at higher
energies.555We could parameterize $L_{\rm eff}$ as a function of energy, and
use different shifts at different energies, but the limits are essentially the
same, being dominated by the events at the highest energy.
For ZEPLIN-III, we utilize the data within the published acceptance box up to
15 keVee for the maximum gap analysis. We additionally employ data outside the
blind acceptance box up to 30 keVee, the maximum to which efficiencies were
provided in [47]. We use the delineated $1\sigma$ region for the data in this
high energy range since the $3\sigma$ region is not specified. To convert from
keVee to keVr, we use
$E_{r}=(0.142E_{\rm ee}+0.005)\,\exp(-0.305\,E_{\rm ee}^{0.564})$ (9)
below 10 keVee [25, 42], and a constant ratio of 0.55 above that. (We use a
larger value than [42] at high energies to account for the energy dependent
value of $L_{\rm eff}$ suggested by [71], which tends to weaken limits
slightly.) Note that we do not employ a background subtraction using the
science data as in [72], nor do we do so for ZEPLIN-II as the same basic
technique seems to have led to a significant overprediction of background for
ZEPLIN-III.
For all Xe based experiments, we take a fixed smearing of $\sqrt{30}$ keVr,
which is typical of the smearing of the experiments in the range of the peak
signal.
For the CRESST experiment, we consider the data from the commissioning run
[48], as well as the additional events at $\sim$ 22 keV, 33 keV and 88 keV
presented for Run 31 by [73], and take a smearing of 1 keV. As described
above, Yellin-style techniques are ironically inappropriate for experiments
with high energy resolution. Because of the significant variation in the
possible spectrum when varying halo models with the high energy resolution, we
use a binned Poisson analysis, rather than a maximum-gap technique. We divide
up the signal region into a low region ($E_{R}<35{\rm~{}keV}$), a middle
region ($35{\rm~{}keV}<E_{R}<50{\rm~{}keV}$) and a high region
$(50{\rm~{}keV}<E_{R}<100{\rm~{}keV})$. In this way, we do not overweight the
events near the zero of the form factor (in the mid region), but still include
some broad spectral information. We require that the rate be lower than 95%
Poisson upper limits for each bin, which is a 95% limit when the signal is
exclusively in the low bin and $\sim$ 90% when there is signal in both the low
and high bins, and $\sim$ 85% in the (rarer) instances when there is signal in
all three bins. Although stronger limits can be set using for instance,
optimum interval, because of the uncertainties in the halo distribution, this
seems inappropriate. In other words, if some of the events at CRESST are real,
then the distribution of events may well be telling us the properties of the
halo.
We show the results of these limits in Figs. 11-14. Even when accounting for
smearing, and using a binned Poisson limit instead of optimum interval, it is
remarkable how significantly the allowed parameter space can change from
simulation to simulation, and between spheres in the same simulation. We see
that CRESST remains the most constraining, as found previously, but we see
important differences. First, agreement is generally improved in these
spheres. This is due to the presence of structure at high velocities which
increases the modulated fraction at DAMA. As a consequence, small regions of
allowed parameter space exist at high mass, the size of which depends
significantly on the particular halo. Larger regions exist at smaller mass
($\sim 150{\rm~{}GeV}$). Moreover, the allowed region of parameter space can
shift dramatically in $\sigma$ and $\delta$, with pockets appearing at large
$\delta$ from high velocity particles. Should iDM be relevant for nature, the
detailed nature of the halo is clearly significant in determining the
properties of the direct detection signals.
Figure 15: Spectra for a sample of spheres best fit to the DAMA data (shown)
within the different simulations for LDM models with $m_{\chi}=3{\rm~{}GeV}$
(top, left), $7{\rm~{}GeV}$ (top, right), and $13{\rm~{}GeV}$ (bottom). Figure
16: The allowed parameter space for LDM models. Top, left: MB with
$v_{0}=220{\rm~{}km/s}$ and $v_{\rm esc}=550{\rm~{}km/s}$. Top, right: VL2.
Middle, left: GHALO. Middle, right GHALOs. Bottom two sample spheres taken
from the simulations.
### 4.2 Light dark matter
Light ($\sim$ GeV) dark matter [5] has been proposed as a means to reconcile
DAMA with the other existing experiments [6, 74, 51]. The current iteration of
LDM involves “channelling” [75, 50] whereby for some small fraction of the
time the entirety of the nuclear recoil energy is converted to scintillation
light. Since the observed energy is lower, it allows lighter WIMPs to scatter
at the DAMA experiment. Such light WIMPs may be incapable of depositing energy
above the threshold at XENON10 or CDMS-Si, for instance, allowing DAMA to
evade these bounds.
Such an explanation is not without difficulty, however. These light WIMPs have
a rapidly falling event spectrum, due to the decreasing probability of
channeling at higher energies, and the increasingly suppressed number of
particles with higher kinetic energy. As a consequence the spectrum of LDM
seems generally to fall too rapidly at light masses [52, 24, 53]. At higher
masses, the spectrum is acceptable, but is excluded by other experiments. We
should note again here that this is under the assumption that the errors in
the data are statistical only. Should there be, e.g., significant changes in
the efficiency, it is possible that LDM could give a better fit. As noted
earlier, the uncertainties in $L_{\rm eff}$ should weaken these limits, while
the most recent analysis from XENON10 (with no events to the S2 threshold)
would strengthen the limits. Thus, while the models do not seem to describe
the data well, there are adequate uncertainties that this scenario remains an
interesting possibility as an explanation of the DAMA signal.
Since these models are also extremely sensitive to the presence of high
velocity particles, one might wonder whether, just as in the iDM case, the
properties of halos at high velocities might modify the spectrum and the
relative signal strength at other experiments.
We begin again by addressing the question of the effects on the spectrum. We
show these for three different masses in Fig. 15. One can see that, while
there are noticeable changes in the spectrum, its overall shape remains
basically unchanged. The difference between LDM and iDM here is simple: in
iDM, the “threshold” (i.e., minimum velocity) scattering is actually at large
energy, and to go to lower energies one requires higher velocities. Competing
against this are the significant form factor effects. The competition between
these two sensitive quantities leads to dramatic changes and features. For
LDM, the threshold scattering is at zero, and all effects (channeling, form
factor, velocity distribution) contribute to the falling (unmodulated)
spectrum. Thus, it can fall somewhat more or less rapidly, depending on the
number of particles at high velocity, but without very significant changes.
To determine the spectrum, we follow the analysis of [52]. The most proper
analysis would be to include the updated measurement of $L_{\rm eff}$ (which
will weaken the limits) and the reanalysis to remove multi-hit events from
[56] (which strengthens the limits). However, incorporating the S2 threshold,
resulting in contribution to an energy-dependent acceptance at low energies is
complicated. Lacking detailed information on this, we will restrict ourselves
to the previous XENON10 limits. Since our point is the relative change from
halo model to halo model, this is reasonable, so long as we recognize that the
limits should be taken with a grain of salt.
We show in Fig. 16 the allowed parameter space for the LDM scenario for the
canonical MB model, for the (averaged) simulations VL2, GHALO and GHALO-
scaled, as well as two sample spheres from the halos. We see that the small
spectral effects translate into small effects in the allowed parameter space.
There can be some improvement, however, and although the exclusion curves
technically cover the allowed regions, uncertainties in the low-energy
scintillation of Xe, for instance, and the ambiguity of employing statistical
errors without a covariance matrix suggest that a small region may yet be
allowed near 10 GeV. We should again emphasize that there is no sense in which
the sampling we have done is statistically complete, and our failure to find
an allowed region does not preclude the possibility that one exists, or might
arise in another simulation. A proper reanalysis using the data from [56] is
warranted.
## 5 Discussion
Direct detection of dark matter intimately ties up questions of astrophysics
and particle physics. In recent years, the range of particle physics models
has exploded, many with new interactions and properties. Some of these, in
particular light dark matter and inelastic dark matter, are particularly
sensitive to the high velocity tail of the WIMP velocity distribution. The
same is true, even for standard WIMPs, for many directionally sensitive
detection experiments (e.g. DRIFT, NEWAGE, DMTPC), which rely on high recoil
energies in order to measure the direction of the scattering DM particle. It
is especially at these high velocities that deviations from the standard
isothermal Maxwell-Boltzmann assumption are expected to be most significant.
Velocity substructure, arising from nearby subhalos and tidal streams, as well
as from the host halo’s velocity anisotropy, can have profound effects on the
expected event rates, the recoil energy spectrum, and the typical direction of
scattering DM particles.
We have studied the effects of these deviations by employing data from N-body
simulations directly, rather than with analytic fits that miss interesting
structures. Our analysis is based on two of the highest resolution numerical
simulations of Galactic dark matter structure, Via Lactea II and GHALO. We
confirm previously reported global departures from the Maxwell-Boltzmann
distribution, with a noticeable excess of particles on the high velocity tail.
In many of our local samples we find additional discrete features due to
subhalos and tidal streams.
These global and local departures from the standard Maxwellian model have a
number of interesting consequences:
* •
For directionally sensitive detection, we have found that the fraction of
particles coming from the “hottest” direction in the sky increases with the
velocity threshold. More importantly, the direction from which most DM
particles are coming can be affected significantly. At high velocity
thresholds ($\gtrsim 500$ km/s in the Earth’s frame) the typical direction is
more often than not shifted by $>10^{\circ}$ away from the direction of
Earth’s motion, and up to $80^{\circ}$ in extreme cases.
* •
For inelastic DM the effects are dramatic. The relative strength of CDMS
limits versus DAMA can change by an order of magnitude, while the relative
strength of CRESST limits can change by almost a factor of two. For CRESST
most of this effect is due to an enhanced modulation, which can be a factor of
two larger than for a Maxwellian halo. Such effects would persist at Xenon
detectors as well, changing their sensitivity relative to DAMA by
$\mathcal{O}(1)$.
* •
For light DM, we have seen that small changes in the spectrum are possible,
although not so much as to allow very light ($\sim$ 1 GeV) WIMPs to fit the
existing data, but only somewhat heavier
($\mathop{}_{\textstyle\sim}^{\textstyle>}$ 10 GeV).
* •
In general, dramatic variations in the spectrum can appear for iDM models,
complicating attempts to employ Yellin-type analyses, which rely upon a
detailed knowledge of the predicted spectrum of the model. Instead we advocate
a binned Poisson analysis, which is less sensitive to assumption about the
shape of the signal spectrum.
* •
A further important effect that we have found is the possible variation of the
phase of modulation as a function of energy. This has been noted before in the
context of streams [76]. Since most background modulations would be expected
to have the same phase as a function of energy, such a change could be a
strong piece of evidence that a modulated signal was arising from DM.
It is important to keep in mind that our simulations are nowhere close to
resolving the phase-space structure at the relevant sub-parsec scales, and we
rely on extrapolation from a coarser sampling (1-1.5 kpc radius spheres). If
velocity substructure does not persist on smaller scales, then our analysis
may overestimate the likelihood of these effects. On the other hand, we are
probably underestimating the significance of the effects, since at the moment
we are diluting the substructure signal by the host halo particles, while it
would completely dominate the background host halo, should the Earth be
passing through one of these substructures.
We have seen here how, should DM be conclusively discovered with direct
detection experiments, it may begin to help unlock questions about the
formation of the galaxy, precisely because of these dramatic sensitivities.
With the fantastic improvements in direct detection on the horizon it is more
important than ever to increase our awareness of the significant impact that
the detailed structure of our Galaxy’s dark matter halo may have.
## Acknowledgments
The authors thank S.Chang and A.Pierce for discussions and their collaboration
in the development of the DM direct detection analysis tools. MK gratefully
acknowledges support from William L. Loughlin at the Institute for Advanced
Study, and from the Theoretical Astrophysics Center at UC Berkeley. NW is
supported by DOE OJI grant # DE-FG02-06ER41417 and NSF CAREER grant
PHY-0449818.
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|
arxiv-papers
| 2009-12-12T09:53:32 |
2024-09-04T02:49:06.992893
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Kuhlen, N. Weiner, J. Diemand, P. Madau, B. Moore, D. Potter, J.\n Stadel, M. Zemp",
"submitter": "Michael Kuhlen",
"url": "https://arxiv.org/abs/0912.2358"
}
|
0912.2407
|
# Gaussian Covariance faithful Markov Trees
Dhafer Malouche
Ecole Supérieure de la Statistique
et de l’Analyse de l’Information, Tunisia.
dhafer.malouche@essai.rnu.tn
and
Bala Rajaratnam
Standford University, USA.
brajarat@stanford.edu
###### Abstract
A covariance graph is an undirected graph associated with a multivariate
probability distribution of a given random vector where each vertex represents
each of the different components of the random vector and where the absence of
an edge between any pair of variables implies marginal independence between
these two variables. Covariance graph models have recently received much
attention in the literature and constitute a sub-family of graphical models.
Though they are conceptually simple to understand, they are considerably more
difficult to analyze. Under some suitable assumption on the probability
distribution, covariance graph models can also be used to represent more
complex conditional independence relationships between subsets of variables.
When the covariance graph captures or reflects all the conditional
independence statements present in the probability distribution the latter is
said to be faithful to its covariance graph - though no such prior guarantee
exists. Despite the increasingly widespread use of these two types of
graphical models, to date no deep probabilistic analysis of this class of
models, in terms of the faithfulness assumption, is available. Such an
analysis is crucial in understanding the ability of the graph, a discrete
object, to fully capture the salient features of the probability distribution
it aims to describe. In this paper we demonstrate that multivariate Gaussian
distributions that have trees as covariance graphs are necessarily faithful.
The method of proof is original as it uses an entirely new approach and in the
process yields a technique that is novel to the field of graphical models.
## 1 Introduction
Markov random fields or graphical models are widely used to represent
conditional independences in a given multivariate probability distribution
(see Kunsch et al., (1995), Ji & Seymour, (1996), Spitzer, (1975), Kindermann
& Snell, (1980), Lauritzen, (1996) to name just a few). Many different types
of Markov Random fields or graphical models have been studied in the
literature. For example, directed acyclic graphs or DAGs are commonly referred
to as “Bayesian networks” (see Pearl, (1988)). When the graph is undirected
and when such graphs are constructed using marginal independence relationships
between pairs of random variables in a given random vector these graphical
models are called “covariance graph” models (see Cox & Wermuth, (1993), Cox &
Wermuth, (1996), Kauermann, (1996), Malouche & Rajaratnam, (2009) and Khare &
Rajaratnam, (2009)). Covariance graph models are commonly represented by
graphs with exclusively bi-directed or dashed edges (see Kauermann, (1996)).
This representation is used in order to distinguish them from the traditional
and widely used concentration graph models. Concentration graphs encode
conditional independence between pairs of variables given the remaining ones.
Formally, if we consider a random vector $\mathbf{X}=(X_{v},v\in V)^{\prime}$
with a probability distribution $P$ where $V$ is a finite set representing the
random variables in $\mathbf{X}$. The concentration graph associated with $P$
is an undirected graph $G=(V,E)$ where
* •
$V$ is the set of vertices.
* •
Each vertex represents one variable in $\mathbf{X}$.
* •
$E$ is the set of edges (between the verices in $V$) constructed using the
pairwise rule : for pair $(u,v)\in V\times V$, $u\not=v$
$(u,v)\not\in
E\;\iff\;X_{u}\,\bot\bot\,X_{v}\mid\mathbf{X}_{V\setminus\\{u,v\\}}$ (1)
where $\mathbf{X}_{V\setminus\\{u,v\\}}:=(X_{w},\,w\not=u\mbox{ and
}w\not=v)^{\prime}$.
Note that $(u,v)\not\in E$ means that the vertices $u$ and $v$ are not
adjacent in $G$.
An undirected graph $G_{0}=(V,E_{0})$ is called the covariance graph
associated with the probability distribution $P$ if the set of edges $E_{0}$
is constructed as follows
$(u,v)\not\in E\;\iff\;X_{u}\,\bot\bot\,X_{v}$ (2)
The subscript zero is invoked for covariance graphs (i.e., $G_{0}$ vs $G$) as
the definition of covariance graphs does not involve conditional
independences.
Both concentration and covariance graphs are not only used to encode pairwise
relationships between pairs of variables in the random vector $\mathbf{X}$,
but as we will see below, these graphs can be used to encode conditional
independences that exist between subsets of variables of $\mathbf{X}$. First
we introduce some definitions:
The multivariate distribution $P$ is said to satisfy the “intersection
property” if for any subsets $A$, $B$ $C$ and $D$ of $V$ which are pairwise
disjoint,
$\left\\{\begin{array}[]{lcl}\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mid\mathbf{X}_{C\cup
D}&&\\\ \mbox{and }&\mbox{ then }&\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B\cup
C}\mid\mathbf{X}_{D}\\\
\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{C}\mid\mathbf{X}_{B\cup D}&&\\\
\end{array}\right.$ (3)
We will call the intersection property (see Lauritzen, (1996)) in (3) above
the concentration intersection property in this paper in order to
differentiate it from another property that is satisfied by $P$ when studying
covariance graph models.
Let $P$ satisfy the concentration intersection property. Then for any triplet
$(A,B,S)$ of subsets of $V$ pairwise disjoint, if $S$ separates111We say that
$S$ separates $A$ and $B$ if any path connecting $A$ and $B$ in $G$ intersects
$S$, i.e., $A\bot_{G}B\mid S$, and is not to be confused with stochastic
independence which is denoted by $\,\bot\bot\,$ as compared to $\bot_{G}$. $A$
and $B$ in the concentration graph $G$ associated with $P$ then the random
vector $\mathbf{X}_{A}=(X_{v},\,v\in A)^{\prime}$ is independent of
$\mathbf{X}_{B}=(X_{v},\,v\in B)^{\prime}$ given $\mathbf{X}_{S}=(X_{v},\,v\in
S)^{\prime}$. This latter property is called concentration global Markov
property and is formally defined as,
$A\bot_{G}B\mid
S\;\Rightarrow\;\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mid\mathbf{X}_{S}.$
(4)
Kauermann, (1996) and Banerjee & Richardson, (2003) show that if $P$ satisfies
the following property : for any triplet $(A,B,S)$ of subsets of $V$ pairwise
disjoint,
$\mbox{ if }\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mbox{ and
}\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{C}\;\mbox{ then
}\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B\cup C},$ (5)
then for any triplet $(A,B,S)$ of subsets of $V$ pairwise disjoint, if
$V\setminus(A\cup B\cup S)$ separates $A$ and $B$ in the covariance graph
$G_{0}$ associated with $P$ then
$\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mid\mathbf{X}_{S}$. This latter
property is called the covariance global Markov property and can be written
formally as follows
$A\bot_{G_{0}}B\mid V\setminus(A\cup B\cup
S)\;\Rightarrow\;\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mid\mathbf{X}_{S}.$
(6)
In parallel to the concentration graph case, property (5) will be called the
covariance intersection property.
Even if $P$ satisfies both intersection properties, the covariance and
concentration graphs may not be able to capture or reflect all the conditional
independences present in the distribution, i.e., there may exist one or more
conditional independences present in the probability distribution that does
not correspond to any separation statement in either $G$ or $G_{0}$.
Equivalently, a lack of a separation statement in the graph does not
necessarily imply conditional independences. On the contrary case when no
other conditional independence exist in $P$ except the ones encoded by the
graph, we classify $P$ as a faithful probability distribution to its graphical
model. More precisely we say that $P$ is concentration faithful to its
concentration graph if for any triplet $(A,B,S)$ of subsets of $V$ pairwise
disjoint, the following statement holds :
$S\mbox{ separates }A\mbox{ and
}B\;\iff\;\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mid\mathbf{X}_{S}.$ (7)
Similarly, $P$ is said to be covariance faithful to its covariance graph
$G_{0}$ if for any triplet $(A,B,S)$ of subsets of $V$ pairwise disjoint, the
following statement holds :
$V\setminus(A\cup B\cup S)\mbox{ separates }A\mbox{ and
}B\;\iff\;\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mid\mathbf{X}_{S}.$ (8)
A natural question of both theoretical and applied interest in probability
theory is to understand the implications of the faithfulness assumption. This
assumption is fundamental since it yields a bijection between the probability
distribution $P$ and the graph $G$ in terms of the independences that are
present in the distribution. In this paper we show that when $P$ is a
multivariate Gaussian distribution whose covariance graph are trees are
necessarily covariance faithful, i.e., these probability distributions satisfy
property (8), i.e., the associated covariance graph $G$ is fully able to
capture all the conditional independences present in the multivariate
distribution $P$. This result can be considered as a dual of a previous
probabilistic result proved by Becker et al., (2005) for concentration graphs
that demonstrates that Gaussian distributions having concentration trees,
i.e., the concentration graph is a tree are necessarily concentration faithful
to its concentration graph (implying property (7) is satisfied). This result
was proved by showing that Gaussian distributions satisfy an additional
intersection property. The approach in the proof of the main result of this
paper is vastly different from the one used for concentration graphs by Becker
et al., (2005).
The outline of this paper is follows. Section 2 presents graph theory
preliminaries. Section 3 gives a brief overview of covariance and
concentration graphs associated with multivariate Gaussian distributions.
Furthermore, an easier way to encode conditional independence using covariance
graphs is given in Section 3. The prove of the main result of this paper is
given in Section 4. Section 5 concludes by summarizing the results in the
paper and the implications thereof.
## 2 Graph theory preliminaries
This section introduces notation and terminology that is required in
subsequent sections. An undirected graph $G=(V,E)$ consists of two sets $V$
and $E$, with $V$ representing the set of vertices, and $E\subseteq(V\times
V)\setminus\\{(u,u),\,u\in V\\}$ the set of edges satisfying :
$\forall\;(u,v)\in E\,\iff\,(v,u)\in E$
For $u,\,v\in V$, we write $u\sim_{G}v$ when $(u,v)\in E$ and we say that $u$
and $v$ are adjacent in $G$.
###### Definition 1
A path connecting two distinct vertices $u$ and $v$ in $G$ is a sequence of
distinct vertices $\left(u_{0},u_{1},\ldots,u_{n})\right)$ where $u_{0}=u$ and
$u_{n}=v$ where for every $i=0,\ldots,n-1$, $u_{i}\sim_{G}u_{i+1}.$
Such a path will be denoted $p=p(u,v,G)$ and we say that $p(u,v,G)$ connects
$u$ and $v$ or alternatively $u$ and $v$ are connected by $p(u,v,G)$. Its
length, denoted by $|p(u,v,G)|$, is defined as the number of edges connecting
the vertices of $p$. So, in this case $|p(u,v,G)|=n$. We also denote by
$\mathcal{P}(u,v,G)$ the set of paths between $u$ and $v$.
Trees are a particular class of graphs that are studied in this paper. This
class of graphs are formally defined below.
###### Definition 2
Let $G=(V,E)$ be an undirected graph. The graph $G$ is called a tree if any
pair of vertices $(u,v)$ in $G$ are connected by exactly one path, i.e.,
$|\mathcal{P}(u,v,G)|=1\;\;\forall\;u,v\in V$.
A subgraph of $G$ induced by a subset $U\subseteq V$ is denoted by
$G_{U}=(U,E_{U})$, $U\subseteq V$ and $E_{U}=E\cap(U\times U)$.
###### Definition 3
A connected component of a graph $G$ is the largest subgraph $G_{U}=(U,E_{U})$
of $G$ such that each pair of vertices can be connected by at least one path
in $G_{U}$.
We now state a Lemma needed in the proof of the main result of this paper.
###### Lemma 1
Let $G=(V,E)$ be an undirected graph. If $G$ is a tree, any subgraph of $G$
induced by a subset of $V$ is a union of connected components, each of which
are trees (or what we shall refer to as a “union of tree connected
components”).
Proof. Consider $U\subset V$, the induced graph $G_{U}$ and a pair of vertices
$(u,v)\in U\times U$. Let us assume to the contrary that $u$ and $v$ are
connected by two distinct paths $p_{1}$ and $p_{2}$ in $G_{U}$ (i.e., $G_{U}$
is not a tree). As the set of edges $E_{U}$ of the graph $G_{U}$ is included
in the set of edges $E$ of $G$, i.e., $E_{U}=E\cap(U\times U)\subseteq E$,
then $p_{1}$ and $p_{2}$ are also paths in $G$. Hence $u$ and $v$ are vertices
in $G$ which are connected by two distinct paths, i.e., $p_{1}$ and $p_{2}$.
This of course yields a contradiction with the fact that $G$ is a tree. Thus
any pair of vertices in $G_{U}$ are connected by at most one path and, hence
$G_{U}$ is a union of connected components, each of which are trees (or a
“union of tree connected components”).
###### Definition 4
For a connected graph, a separator is a subset $S$ of $V$ such that there
exists a pair of non-adjacent vertices $u$ and $v$ such that $u,$ $v\not\in S$
and
$\forall p\in\mathcal{P}(u,v,G),\;\;p\cap S\not=\emptyset$ (9)
If $S$ is a separator then it is easily verified that every
$S^{\prime}\supseteq S$ such that $S^{\prime}\subseteq V\setminus\\{u,v\\}$ is
also a separator. We are thus lead to the notion of a minimal separator.
###### Definition 5
The separator $S$ is defined to be a minimal separator between two non-
adjacent vertices $u$ and $v$ if for any $w\in S$, the subsets
$S\setminus\\{w\\}$ is not a separator of $u$ and $v$.
Note that in the case where $G$ contains more than two connected components
and if $u$ and $v$ belong to different connected components the empty set is
the only possible separator of $u$ and $v$. Finally, let $A$, $B$ and $S$ be
pairwise disjoint subsets of $V$. We say that $S$ separates $A$ and $B$ if for
any pair of vertices $(u,v)\in A\times B$, any path connecting $u$ and $v$
intersects $S$. In the case where $A$ and $B$ belong to different connected
components of $G$ the subset $S$ can be empty because the set of paths between
any pair of vertices $(u,v)\in A\times B$ is empty.
## 3 Gaussian Concentration and Covariance Graphs
In this section we present a brief overview of concentration and covariance
graphs in the case when the probability distribution $P$ is multivariate
Gaussian. Such graphical models are commonly referred to as Gaussian
covariance or Gaussian concentration graph models.
### 3.1 Gaussian concentration graph models
Consider a probability space with triplet $(\Omega,{\cal F},\mathbb{P})$ and
let $\mathbf{X}\,:\,\Omega\rightarrow\mathbb{R}^{|V|}$ be a random vector
where $\mathbf{X}=(X_{v},\,v\in V)^{\prime}$ and $P$ represents the induced
measure of $\mathbb{P}$ by $\mathbf{X}$. If $\mathbf{X}$ follows a Gaussian
distribution then it has the following density function with respect to
Lebesgue measure :
$f(\mathbf{x})=\frac{1}{(2\pi)^{|V|/2}|\Sigma|^{1/2}}\,\exp\left(-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})^{\prime}\Sigma^{-1}(\mathbf{x}-\mathbf{\mu})^{\prime}\right),$
(10)
where $\mathbf{x}=(x_{u},\,u\in V)^{\prime}\in{\rm IR}^{|V|}$,
$\mathbf{\mu}\in{\rm IR}^{|V|}$ is the mean vector and
$\Sigma=(\sigma_{uv})\in\mathcal{P}^{+}$ is the covariance matrix with
$\mathcal{P}^{+}$ denoting the cone of symmetric positive definite matrices.
Without loss of generality we will assume that $\mathbf{\mu}=\mathbf{0}$. As
any Gaussian distribution with $\mathbf{\mu}=\mathbf{0}$ is completely
determined by its covariance matrix $\Sigma$, this set of multivariate
Gaussian distributions can therefore be identified by the set of symmetric
positive definite matrices. Gaussian distributions can also be parameterized
by the inverse of the covariance matrix $\Sigma$ denoted by
$K=\Sigma^{-1}=(k_{uv})$. The matrix $K$ is called the precision or
concentration matrix. It is well known (see Lauritzen, (1996)) that for any
pair of variables $(X_{u},X_{v})$, where $u\not=v$
$X_{u}\,\bot\bot\,X_{v}\mid\mathbf{X}_{V\setminus\\{u,v\\}}\;\iff\;k_{uv}=0.$
Hence the concentration graph $G=(V,E)$ can be constructed simply using the
precision matrix $K$ and the following rule
$(u,v)\not\in E\;\iff\;k_{uv}=0.$
Furthermore it can be easily deduced from a classical result in Hammersly &
Clifford, (1971), that is reproved in Lauritzen, (1996), that any multivariate
random vector with a positive density necessarily satisfies the concentration
intersection property (3). Hence for Gaussian concentration graph models the
pairwise Markov property in (1) is equivalent to the concentration global
Markov property in (4).
### 3.2 Gaussian covariance graph models
As seen earlier in (2) covariance graphs are constructed using pairwise
marginal independence relationships. It is also well known that for
multivariate Gaussian distributions :
$X_{u}\,\bot\bot\,X_{v}\;\iff\,\sigma_{uv}=0.$
Hence in the Gaussian case the covariance graph $G_{0}=(V,E_{0})$ can be
constructed using the following rule :
$(u,v)\not\in E_{0}\;\iff\;\sigma_{uv}=0.$
It is also easily seen that Gaussian distributions satisfy the covariance
intersection property defined in (5). Hence Gaussian covariance graphs can
also encode conditional independences according to the following rule : for
any triplet $(A,B,S)$ of subsets of $V$ pairwise disjoint, if
$V\setminus(A\cup B\cup S)$ separates $A$ and $B$ in the covariance graph
$G_{0}$ then $\mathbf{X}_{A}\,\bot\bot\,\mathbf{X}_{B}\mid\mathbf{X}_{S}$. We
now show (see proposition 2 below) that there is a simple way to read
conditional independence statements from the covariance graph. This result
holds true for any probability distribution that satisfy the covariance
intersection property given in (5).
###### Proposition 2
Let $\textbf{X}_{V}=(X_{v},\,v\in V)^{\prime}$ be a random vector with
probability distribution $P$ satisfying the covariance intersection property
in (5) and let $G_{0}=(V,E_{0})$ be the covariance graph associated with $P$.
Then the following statements are equivalent,
* i.
for any pairwise disjoint subsets $A$, $B$ and $S$ of $V$ : if
$V\setminus(A\cup B\cup S)$ separates $A$ and $B$ in $G_{0}$ then
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{S}$
* ii.
for any pairwise disjoint subsets $A$, $B$ and $S$ of $V$ : if $S$ separates
$A$ and $B$ in $G_{0}$ then
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(A\cup
B\cup S)}$
Proof. Let us first assume that (i) is satisfied and let us prove (ii).
Let $A$, $B$ and $S$ be three pairwise disjoint subsets of $V$ such that $S$
separates $A$ and $B$ in $G_{0}$. Note that we can write $S$ as follows:
$S=V\setminus(V\setminus(A\cup B\cup S))\cup A\cup B)$
Since $(V\setminus(A\cup B\cup S)\cup A\cup B=V\setminus S$ and
$V\setminus(V\setminus S)=S$.
By hypothesis $S$ separates $A$ and $B$ in $G_{0}$. Let
$S^{\prime}=V\setminus(A\cup B\cup S)$ and since $S=V\setminus(S^{\prime}\cup
A\cup B)$ we can apply property (i) to the triplet $(A,B,S^{\prime})$. Hence
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{S^{\prime}}$. Hence
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(S\cup
A\cup B)}$ since $S^{\prime}:=V\setminus(S\cup A\cup B)$. We have therefore
proved that if $S$ separates $A$ and $B$ in $G_{0}$, then
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(S\cup
A\cup B)}$.
Assume now that property (ii) is satisfied and let $A$, $B$ and $S$ be three
pairwise disjoint subsets of $V$ such that $V\setminus(S\cup A\cup B)$
separates $A$ and $B$ in $G_{0}$. Let us denote by
$S^{\prime}=V\setminus(S\cup A\cup B)$ which is a subset separating $A$ and
$B$ in $G_{0}$. Since (ii) is satisfied, we deduce that
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(A\cup
B\cup S^{\prime})}$. However
$V\setminus(A\cup B\cup S^{\prime})=V\setminus((V\setminus(A\cup B\cup S))\cup
A\cup B)=S$
Hence we conclude that $V\setminus(A\cup B\cup S)$ separates $A$ and $B$ in
$G_{0}$ implies that
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{S}$. Thus property
(i) is satisfied.
Proposition 2 can be used to formulate an equivalent definition of the
covariance faithfulness property.
###### Definition 6
Let $\textbf{X}_{V}=(X_{v},\,v\in V)^{\prime}$ be a random vector with
probability distribution $P$ satisfying the covariance intersection property
in (5) and let $G_{0}=(V,E_{0})$ be the covariance graph associated with $P$.
We say that $P$ is covariance faithful to $G_{0}$ if for any pairwise disjoint
subsets $A$, $B$ and $S$ of $V$ the following condition is satisfied
$S\mbox{ separates }A\mbox{ and
}B\;\iff\;\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(A\cup
B\cup S)}$
The above reformulation of the covariance faithfulness property is an
important ingredient in the proofs in the next section.
## 4 Gaussian Covariance faithful trees
We now proceed to study the faithfulness assumption in the context of
multivariate Gaussian distributions and when the associated covariance graphs
are trees.
The main result of this paper, presented in Theorem 3, proves that
multivariate Gaussian probability distributions having tree covariance graphs
are necessarily faithful to their covariance graphs. The analogous result for
concentration graphs was demonstrated by Becker et al., (2005) where the
authors proved that Gaussian distributions having tree concentration graphs
are necessarily faithful to these graphs. We now formally state Theorem 3. The
proof follows shortly after a series of lemmas/theorem(s) and an illustrative
example.
###### Theorem 3
Let $\textbf{X}_{V}=(X_{v},\,v\in V)^{\prime}$ be a random vector with
Gaussian distribution $P=\mathcal{N}_{|V|}(\mu,\Sigma^{-1})$. Let
$G_{0}=(V,E_{0})$ be the covariance graph associated with $P$. If $G_{0}$ is a
tree or more generally a union of connected components each of which are trees
(or a union of “tree connected components”), then $P$ is $g_{0}-$faithful to
$G_{0}$.
The proof of Theorem 3 requires among others a result proved by Jones & West,
(2005). This result gives a method that can be used to compute the covariance
matrix $\Sigma$ from the precision matrix $K$ using the paths in the
concentration graph $G$. The result can also be easily extended to show that
the precision matrix $K$ can be computed from the covariance matrix $\Sigma$
using the paths in the covariance graph $G_{0}$. We now state the result by
Jones & West, (2005).
###### Theorem 4
Jones & West, (2005).
Let $\textbf{X}_{V}=(X_{v},\,v\in V)^{\prime}$ be a random vector with
Gaussian distribution $P=\mathcal{N}_{|V|}(\mu,\Sigma)$ where $\Sigma$ and
$K=\Sigma^{-1}$ are positive definite matrices. Let $G=(V,E)$ and
$G_{0}=(V,E_{0})$ denote respectively the concentration and covariance graph
associated with the probability distribution of $\textbf{X}_{V}$.
For all $(u,v)$ in $V\times V$
$k_{uv}=\displaystyle\sum_{p\in\mathcal{P}(u,v,G_{0})}(-1)^{|p|+1}|\sigma|_{p}\,\frac{|\Sigma\setminus
p|}{|\Sigma|}$
and
$\sigma_{uv}=\displaystyle\sum_{p\in\mathcal{P}(u,v,G)}(-1)^{|p|+1}|k|_{p}\frac{|K\setminus
p|}{|K|}$
where, if $p=(u_{0},\ldots,u_{n}),$
$|\sigma|_{p}=\sigma_{u_{0}u_{1}}\sigma_{u_{1}u_{2}}\ldots\sigma_{u_{n-1}u_{n}},\;\;|k|_{p}=k_{u_{0}u_{1}}k_{u_{1}u_{2}}\ldots
k_{u_{n-1}u_{n}},$
$K\setminus p=\left(k_{uv},\,(u,v)\in(V\setminus p)\times(V\setminus
p)\right)$ and $\Sigma\setminus p=\left(\sigma_{uv},\,(u,v)\in(V\setminus
p)\times(V\setminus p)\right)$ denote respectively $K$ and $\Sigma$ with rows
and columns corresponding to variables in path $p$ omitted. The determinant of
a zero-dimensional matrix is defined to be $1$.
The proof of our main theorem (Theorem 3) also requires the results proved in
the lemma below.
###### Lemma 5
Let $\textbf{X}_{V}=(X_{v},\,v\in V)^{\prime}$ be a random vector with
Gaussian distribution $P=\mathcal{N}_{|V|}(\mu,K=\Sigma^{-1})$. Let
$G_{0}=(V,E_{0})$ and $G=(V,E)$ denote respectively the covariance and
concentration graphs associated with $P$, then
* i.
$G$ and $G_{0}$ have the same connected components
* ii.
If a given connected component in $G_{0}$ is a tree then the corresponding
connected component in $G$ is complete and vice-versa.
Proof.
* Proof of (i).
The fact that $G_{0}$ and $G$ have the same connected components can be
deduced from the matrix structure of the covariance and the precision matrix.
The connected components of $G_{0}$ correspond to block diagonal matrices in
$\Sigma$. Since $K=\Sigma^{-1}$, then by properties of inverting partitioned
matrices, $K$ also has the same block diagonal matrices as $\Sigma$ in terms
of the variables that constitute these matrices. These blocks corresponds to
distinct components in $G$ and $G_{0}$. Hence both matrices have the same
connected components.
* Proof of (ii).
Let us assume now that the covariance graph $G_{0}$ is a tree, hence it is a
connected graph with only one connected component. We shall prove that the
concentration graph $G$ is complete by using Theorem 4 by Jones & West, (2005)
and computing any coefficient $k_{uv}$ ($u\not=v$). Since $G_{0}$ is a tree,
there exists exactly one path between between any two vertices $u$ and $v$. We
shall denote this path as $p=(u_{0}=u,\ldots,u_{n}=v)$. Then by Theorem 4
$k_{uv}=(-1)^{n+1}\sigma_{u_{0}u_{1}}\ldots\sigma_{u_{n-1}u_{n}}\displaystyle\frac{\left|\Sigma\setminus
p\right|}{\left|\Sigma\right|}$ (11)
First note that the determinant of the matrices in (11) are all positive since
principal minors of positive definite matrices are positive. Second since we
are considering a path in $G_{0}$, $\sigma_{u_{i-1}u_{i}}\not=0$,
$\forall\;i=1,\ldots,n$. Using these two facts we deduce from (11) that
$k_{uv}\not=0$ for all $(u,v)\in E$. Hence $u$ and $v$ are adjacent in $G$ for
all $(u,v)\in E$. The concentration graph $G$ is therefore complete. The proof
that when $G$ is assumed to be a tree implying that $G_{0}$ is complete
follows similarly.
Remark. We further note that Theorem 4 is also directly useful in deducing the
completeness of the concentration graph by using the covariance graph in other
settings. As a concrete example consider the case when $G_{0}$ is a cycle with
an even number of edges s.t. $|V|=2k$ for some odd integer $k$, and assume
that all the coefficients in the covariance matrix $\Sigma$ of
$\textbf{X}_{V}$ are positive. Hence a given pair of vertices $(u,v)$ in
$G_{0}$ are connected by two paths which are both of odd length. Let us denote
these paths as $p_{1}$ and $p_{2}$. Using Theorem 4, it is easily deduced that
$k_{uv}=\sigma_{|p_{1}|}\frac{|\Sigma\setminus
p_{1}|}{|\Sigma|}+\sigma_{|p_{2}|}\frac{|\Sigma\setminus p_{2}|}{|\Sigma|}$
Here $|\sigma_{p_{1}}|$ and $|\sigma_{p_{1}}|$ are different from zero as they
are both equal to a product of positive coefficients. Hence $k_{uv}\not=0$.
The same argument can also be used in the case when $p_{1}$ and $p_{2}$ both
have even length (i.e., $|V|=2k$ for some even integer $k$) to deduce that
$k_{uv}\not=0$. Hence $u$ and $v$ are adjacent in the concentration graph $G$;
thus $G$ is necessarily complete.
We now give an example illustrating the main result in this paper (Theorem 3).
###### Example 1
Consider a Gaussian random vector $\textbf{X}=(X_{1},\ldots,X_{8})^{\prime}$
with covariance matrix $\Sigma$ and its associated covariance graph as given
in Figure 1.
$1$$2$$3$$4$$5$$6$$7$$8$ Figure 1: An $8-$vertex covariance tree $G_{0}$.
Consider the sets $A=\\{1,2\\}$, $B=\\{5\\}$ and $S=\\{4,6\\}$. Note that $S$
does not separate $A$ and $B$ in $G_{0}$ as any path from $A$ and $B$ does not
intersect $S$. In this case we cannot use the covariance global Markov
property to claim that $\textbf{X}_{A}$ is not independent of $\textbf{X}_{B}$
given $\textbf{X}_{V\setminus(A\cup B\cup S)}$. This is because the covariance
global Markov property allows us to read conditional independences present in
a distribution if a separation is present in the graph. It is not an “if and
only if” property in the sense that the lack of a separation in the graph does
not necessarily imply the lack of the corresponding conditional independence.
We shall show however that in this example that $\textbf{X}_{A}$ is indeed not
independent of $\textbf{X}_{B}$ given $\textbf{X}_{V\setminus(A\cup B\cup
S)}$. In other words we shall show that the graph has the ability to capture
this conditional dependence present in the probability distribution $P$.
Let us now examine the relationship between $X_{2}$ and $X_{5}$ given
$\textbf{X}_{\\{3,7,8\\}}$. Note that in this example $V\setminus(A\cup B\cup
S)=\\{3,8,7\\}$, $2\in A$ and $5\in B$. Note that the covariance graph
associated with the probability distribution of the random vector
$(X_{2},X_{5},\textbf{X}_{\\{3,8,7\\}})^{\prime}$ is the subgraph represented
in Figure 2 and can be obtained directly as a subgraph of $G_{0}$ induced by
the subset $\\{2,5,3,7,8\\}$.
$2$$3$$5$$7$$8$ Figure 2: the covariance graph $(G_{0})_{\\{2,5,3,8,7\\}}$
Since $2$ and $5$ are connected by exactly one path in
$(G_{0})_{\\{2,5,3,7,8\\}}$, that is $p=(2,3,5)$, then the coefficient
$k_{25\mid 387}$, i.e., the coefficient between $2$ and $5$ in inverse of the
covariance matrix of $(X_{2},X_{5},\textbf{X}_{\\{3,8,7\\}})^{\prime}$, can be
computed using Theorem 4 as follows
$k_{25\mid
387}=(-1)^{2+1}\sigma_{23}\,\sigma_{35}\displaystyle\frac{|\Sigma(\\{8,7\\})|}{|\Sigma(\\{2,5,3,8,7\\})|}$
(12)
where $\Sigma(\\{7,8\\})$ and $\Sigma(\\{2,5,3,8,7\\})$ are respectively the
covariance matrices of the Gaussian random vectors $(X_{7},X_{8})^{\prime}$
and $(X_{2},X_{5},\textbf{X}_{\\{3,8,7\\}})^{\prime}$. Hence $k_{25\mid
387}\not=0$ since the right hand side of the equation in (12) is different
from zero. Hence
$X_{2}\,\not\\!\\!\\!\bot\bot\,X_{5}\mid\textbf{X}_{\\{3,8,7\\}}$.
Now recall that for any Gaussian random vector vector
$\textbf{X}_{V}=(X_{u},\,u\in V)^{\prime}$ ,
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{C}\mbox{ if and only
if }\;\forall\,(u,v)\in A\times
B,\;\;X_{u}\,\bot\bot\,X_{v}\mid\textbf{X}_{C}$ (13)
where $A$, $B$ and $C$ are pairwise disjoint subsets of $V$. The
contrapositive of (13) yields
$X_{2}\,\not\\!\\!\\!\bot\bot\,X_{5}\mid\textbf{X}_{\\{3,7,8\\}}\;\Rightarrow\;\textbf{X}_{\\{1,2\\}}\,\not\\!\\!\\!\bot\bot\,X_{5}\mid\textbf{X}_{\\{3,7,8\\}}.$
Hence we conclude that since $\\{3,7,8\\}$ does not separate $\\{1,2\\}$ and
$\\{5\\}$ therefore $\textbf{X}_{\\{1,2\\}}$ is not independent of $X_{5}$
given $\textbf{X}_{\\{3,7,8\\}}$, i.e.,
$\\{1,2\\}\not\perp_{G_{0}}\\{5\\}\mid\\{3,7,8\\}\Rightarrow\textbf{X}_{\\{1,2\\}}\,\not\\!\\!\\!\bot\bot\,X_{5}\mid\textbf{X}_{\\{3,7,8\\}}$
.
We now proceed to the proof of Theorem 3. Proof. of Theorem 3. Without loss of
generality we assume that $G_{0}$ is a connected tree. Let us assume to the
contrary that $P$ is not covariance faithful to $G_{0}$, then there exists a
triplet $(A,B,S)$ of pairwise disjoint subsets of $V$, such that
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(A\cup
B\cup S)}$, but $S$ does not separate $A$ and $B$ in $G_{0}$, i.e.,
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(A\cup
B\cup S)}\;\mbox{ and }\;A\not\perp_{G_{0}}B\mid S$
As $S$ does not separate $A$ and $B$ and since $G_{0}$ is a connected tree,
then there exists a pair of vertices $(u,v)\in A\times B$ such that the single
path $p$ connecting $u$ and $v$ in $G_{0}$ does not intersect $S$, i.e.,
$S\cap p=\emptyset$. Hence $p\subseteq V\setminus S=(A\cup
B)\cup(V\setminus(A\cup B\cup S))$. Thus two cases are possible with regards
to where the path $p$ can lie : either $p\subseteq A\cup B$ or
$p\cap(V\setminus(A\cup B\cup S))\not=\emptyset$. Let us examine both cases
separately.
* •
Case 1 : $p\subseteq A\cup B$
In this case the entire path between $u$ and $v$ lies in $A\cup B$ and hence
we can find a pair of vertices222As an illustration of this point consider the
graph presented in Figure 1. Let $A=\\{1,2\\}$, $B=\\{3,5\\}$ and
$S=\\{4,6\\}$. We note that the path $p=(1,2,3,5)$ lies entirely in $A\cup B$
and hence we can find two vertices, namely, $2\in A$ and $3\in B$, belonging
to path $p$ that are adjacent in $G_{0}$. $(u^{\prime},v^{\prime})$ belonging
to $p$ and $(u^{\prime},v^{\prime})\in A\times B$ such that
$u^{\prime}\sim_{G_{0}}v^{\prime}$.
Recall that since $G_{0}$ is a tree, any induced graph of $G_{0}$ by a subset
of $V$ is a union of tree connected components (see Lemma 1). Hence the
subgraph $(G_{0})_{W}$ of $G_{0}$ induced by
$W=\\{u^{\prime},v^{\prime}\\}\cup V\setminus(A\cup B\cup S)$ is a union of
tree connected components. As $u^{\prime}$ and $v^{\prime}$ are adjacent in
$G_{0}$, they are also adjacent in $(G_{0})_{W}$ and belong to the same
connected component333In our example in Figure 1 with $W=\\{2,3,8,7\\}$,
$(G_{0})_{W}$ consists a union of two connected components with its respective
vertices being $\\{2,3\\}$ and $\\{8,7\\}$. of $(G_{0})_{W}$. Hence the only
path between $u^{\prime}$ and $v^{\prime}$ is precisely the edge
$(u^{\prime},v^{\prime})$. Using theorem 4 to compute the coefficient
$k_{u^{\prime}v^{\prime}\mid V\setminus(A\cup B\cup S)}$, i.e.,
$(u^{\prime},v^{\prime})th$ coefficient in the inverse of the covariance
matrix of the random vector $\textbf{X}_{W}=(X_{w},\,w\in
W)^{\prime}=(X_{u^{\prime}},X_{v^{\prime}},\textbf{X}_{V\setminus(A\cup B\cup
S)})^{\prime}$, we obtain,
$k_{u^{\prime}v^{\prime}\mid V\setminus(A\cup B\cup
S)}=(-1)^{1+1}\sigma_{u^{\prime}v^{\prime}}\,\displaystyle\frac{\left|\Sigma(W\setminus\\{u^{\prime},v^{\prime}\\})\right|}{|\Sigma(W)|},$
(14)
where $\Sigma(W)$ denotes the covariance matrix of $\textbf{X}_{W}$, and
$\Sigma(W\setminus\\{u^{\prime},v^{\prime}\\})$ denotes the matrix $\Sigma(W)$
with the rows and the columns corresponding to variables $X_{u^{\prime}}$ and
$X_{v^{\prime}}$ omitted. We can therefore deduce from (14) that
$k_{u^{\prime}v^{\prime}\mid V\setminus(A\cup B\cup S)}\not=0$. Recall that at
the start of the proof we assumed to the contrary that
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(A\cup
B\cup S)}$. Now since $P$ is Gaussian, for pairwise disjoint subsets
$A,B,V\setminus(A\cup B\cup C)$ then
$\textbf{X}_{A}\,\bot\bot\,\textbf{X}_{B}\mid\textbf{X}_{V\setminus(A\cup
B\cup C)}\Leftrightarrow\forall\,(u,v)\in A\times
B,\;X_{u}\,\bot\bot\,X_{v}\mid\textbf{X}_{V\setminus(A\cup B\cup C)}$ (15)
Note however that we have established that
$X_{u^{\prime}}\,\not\\!\\!\\!\bot\bot\,X_{v^{\prime}}\mid\textbf{X}_{V\setminus(A\cup
B\cup S)}$ since $k_{u^{\prime}v^{\prime}\mid V\setminus(A\cup B\cup S)}\neq
0$. Hence we obtain a contradiction to (15) since $u^{\prime}\in A$ and
$v^{\prime}\in B$.
* •
Case 2 : $p\cap(V\setminus(A\cup B\cup S))\not=\emptyset$ & $V\setminus(A\cup
B\cup S)$ is not empty.
Now if $V\setminus(A\cup B\cup S)$ is empty then $p$ has to lie entirely in
$A\cup B$. This is because by assumption $p$ does not intersect $S$. The case
when $p$ lies in $A\cup B$ is covered in Case 1 and hence it is assumed that
$V\setminus(A\cup B\cup S)\not=\emptyset.$ 444As an illustration of this point
consider once more the graph presented in Figure 1. Consider $A=\\{1,2\\}$,
$B=\\{7,8\\}$ and $S=\\{4,6\\}$. Here $V\setminus(A\cup B\cup S)=\\{3,5\\}$
and the path $p=(1,2,3,5,7,8)$ connecting $A$ and $B$ intersects
$V\setminus(A\cup B\cup S)$.
In this case there exists a pair of vertices $(u^{\prime},v^{\prime})\in
A\times B$ with $u^{\prime},v^{\prime}\in p$, such that the vertices
$u^{\prime}$ and $v^{\prime}$ are connected by exactly one path
$p^{\prime}\subseteq p$ in the induced graph $(G_{0})_{W}$ of $G_{0}$ by
$W=\\{u^{\prime},v^{\prime}\\}\cup V\setminus(A\cup B\cup S)$ (see Lemma 1)
555In our example in figure 1 with $A=\\{1,2\\}$, $B=\\{7,8\\}$ and
$S=\\{4,6\\}$ , the vertices $u^{\prime}$ and $v^{\prime}$ will correspond to
vertices $2$ and $7$ respectively, and $p^{\prime}=(2,3,5,7)$, which is a path
entirely contained in $V\setminus(A\cup B\cup
S)\cup\\{u^{\prime},v^{\prime}\\}$..
Let us now use Theorem 4 to compute the coefficient
$k_{u^{\prime}v^{\prime}\mid V\setminus(A\cup B\cup S)}$, i.e., the
$(u^{\prime},v^{\prime})-$coefficient in the inverse of the covariance matrix
of the random vector $\textbf{X}_{W}=(X_{w},\,w\in
W)^{\prime}=(X_{u^{\prime}},X_{v^{\prime}},\textbf{X}_{V\setminus(A\cup B\cup
S)})^{\prime}$. We obtain that
$k_{u^{\prime}v^{\prime}\mid V\setminus(A\cup B\cup
S)}=(-1)^{|p^{\prime}|+1}|\sigma_{p^{\prime}}|\,\displaystyle\frac{\left|\Sigma(W\setminus
p^{\prime})\right|}{|\Sigma(W)|},$ (16)
where $\Sigma(W)$ denotes the covariance matrix of $\textbf{X}_{W}$ and
$\Sigma(W\setminus p^{\prime})$ denotes $\Sigma(W)$ with the rows and the
columns corresponding to variables in path $p^{\prime}$ omitted. One can
therefore easily deduce from (16) that $k_{u^{\prime}v^{\prime}\mid
V\setminus(A\cup B\cup S)}\not=0$. Thus $X_{u^{\prime}}$ is not independent of
$X_{v^{\prime}}$ given $\textbf{X}_{V\setminus(A\cup B\cup S)}$. Hence once
more we obtain a contradiction to (15) since $u^{\prime}\in A$ and
$v^{\prime}\in B$.
Remark. The dual result of the theorem above for the case of concentration
trees was proved by Becker et al., (2005). We note however that the argument
used in the proof of Theorem 3 cannot also be used to prove faithfulness of
Gaussian distributions that have trees as concentration graphs. The reason for
this is as follows. In our proof we employed the fact that the sub-graph
$(G_{0})_{\\{u,v\\}\cup S}$ of $G_{0}$ induced by a subset ${\\{u,v\\}\cup
S}\subseteq V$ is also the covariance graph associated with the Gaussian sub-
random vector of $\textbf{X}_{V}$ as denoted by $\textbf{X}_{\\{u,v\\}\cup
S}=(X_{w},\,w\in\\{u,v\\}\cup S)^{\prime}$. Hence it was possible to compute
the coefficient $k_{uv\mid S}$ which quantifies the conditional (in)dependence
between $u$ and $v$ given $S$, in terms of the paths in
$(G_{0})_{\\{u,v\\}\cup S}$ and the coefficients of the covariance matrix of
$\textbf{X}_{\\{u,v\\}\cup S}=(X_{w},\,u\in\\{u,v\\}\cup S)^{\prime}$. On the
contrary, in the case of concentration graphs the sub-graph $G_{\\{u,v\\}\cup
S}$ of the concentration graph $G$ induced by $\\{u,v\\}\cup S$ is not in
general the concentration graph of the random vector
$\textbf{X}_{\\{u,v\\}\cup S}=(X_{w},\,u\in\\{u,v\\}\cup S)^{\prime}$. Hence
our approach is not directly applicable in the concentration graph setting.
## 5 Conclusion
Faithfulness of a probability distribution to a graph is a crucial assumption
that is often made in the probabilistic treatment of graphical models. This
assumption describes the ability of a graph to reflect or encode the
multivariate dependencies that are present in a joint probability
distribution. Much of the methodology in this area often do not undertake a
detailed analysis of the faithfulness assumption, as such an endeavor requires
a more careful and rigorous probabilistic study of the joint distribution at
hand. In this note we looked at the class of multivariate Gaussian
distributions that are Markov with respect to covariance graphs and prove that
Gaussian distributions which have trees as their covariance graphs are
necessarily faithful. The method of proof that is employed in this paper is
novel in the sense that it is self contained and yields a completely new
approach to demonstrating faithfulness - as compared to the methods that are
traditionally used in the literature. Moreover, it is also vastly different in
nature from the proof of the analogous result for concentration graph models.
Hence the approach used in this paper promises to have further implications
and give other insights. Future research in this area will explore if the
techniques used in this paper can be modified to prove or disprove
faithfulness for other classes of graphs.
## Acknowledgments
The authors gratefully acknowledge the faculty at Stanford University for
their feedback and tremendous enthusiasm for this work.
## References
* Banerjee & Richardson, (2003) Banerjee, M., & Richardson, T. 2003. On a Dualization of Graphical Gaussian Models: A Correction Note. Scand. J. Statist., Vol 30, 817–820.
* Becker et al., (2005) Becker, Ann, Geiger, Dan, & Meek, Christopher. 2005. Perfect Tree-like Markovian Distributions. Probability and Mathematical Statistics, 25(2), 231–239.
* Cox & Wermuth, (1996) Cox, D. R., & Wermuth, N. 1996. Multivariate Depencies : Models, Analysis and Interpretations. Chapman and Hall.
* Cox & Wermuth, (1993) Cox, D.R., & Wermuth, M. 1993. Linear dependencies represented by chain graphs (with Discussion). Statist. Sci., 8, 204–218, 247–277.
* Hammersly & Clifford, (1971) Hammersly, J. M., & Clifford, P. E. 1971. Markov fields on finite graphs and lattices. Unpublished manuscript.
* Ji & Seymour, (1996) Ji, C., & Seymour, L. 1996. A consistent model selection procedure for Markov random fields based on penalized pseudolikelihood. The Annals of Applied Probability, 6(2), 423–443.
* Jones & West, (2005) Jones, B., & West, M. 2005. Covariance decomposition in undirected Gaussian graphical models. Biometrika, 92, 770–786.
* Kauermann, (1996) Kauermann, G. 1996. On a dualization of graphical Gaussian models. Scand. J. Statist., 23, 105–116.
* Khare & Rajaratnam, (2009) Khare, K., & Rajaratnam, B. 2009. Wishart distributions for decomposable covariance graph models. under review in the Annals of Statistics.
* Kindermann & Snell, (1980) Kindermann, R., & Snell, J. L. 1980. Markov Random Fields and Their Applications. American Mathematical Society, Providence, Rhode Island.
* Kunsch et al., (1995) Kunsch, H., Gemanand, S., & Kehagias, A. 1995. Hidden Markov Random Fields. The Annals of Applied Probability, 5(3), 577–602.
* Lauritzen, (1996) Lauritzen, S. L. 1996. Graphical Models. New York : Oxford University Press.
* Malouche & Rajaratnam, (2009) Malouche, D., & Rajaratnam, B. 2009. Analysis of the faithfulness assumption in Graphical Models. Technical Report, Department of Statistics, Stanford University.
* Pearl, (1988) Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems. Tech. rept. Morgan Kaufman.
* Spitzer, (1975) Spitzer, C. 1975. Markov random fields on an infinite tree. The Annals of Probability, 3, 387–398.
|
arxiv-papers
| 2009-12-14T05:30:53 |
2024-09-04T02:49:07.004027
|
{
"license": "Public Domain",
"authors": "Dhafer Malouche and Bala Rajaratnam",
"submitter": "Dhafer Malouche DM",
"url": "https://arxiv.org/abs/0912.2407"
}
|
0912.2427
|
# SU(5) Grand Unified Model and Dark Matter
Shi-Hao Chen Institute of Theoretical Physics, Northeast Normal University,
Changchun 130024, P.R.China; shchen@nenu.edu.cn
([; date; date; date; date)
###### Abstract
A dark matter model which is called w-matter or mirror dark matter is
concretely constructed based on (f-SU(5))X(w-SU(5)) symmetry. There is no
Higgs field and all masses originate from interactions in the present model.
W-matter is dark matter relatively to f-matter and vice versa. In high-energy
processes or when temperature is very high, visible matter and dark matter can
transform from one into another. In such process energy seems to be non-
conservational, because dark matter cannot be detected. In low-energy
processes or when temperature is low, there is only gravitation interaction of
dark matter for visible matter.
Dark matter
††preprint:
year number number identifier Date text]date
LABEL:FirstPage1 LABEL:LastPage#17
###### Contents
1. I Introduction
2. II Lagrangian of the $SU_{f}(5)\times SU_{w}(5)$ model
3. III Symmetry spontaneously breaking and temperature effects
4. IV The physical significance of the present model
5. V Conclusion
## I Introduction
What is the origin of mass? A possible answer is spontaneous symmetry-
breaking. Higgs fields can cause spontaneous symmetry-breaking. But it is
difficult to understand $\left(-\mu^{2}\right)$ in Higgs potentials. Hence
dynamical breaking is considered[1]. It is not realized to construct a
realistic grand unified model based the dynamical breaking according to the
conventional theory.
There are many sorts of grand unified models. There are some difficulties such
as proton decay in the simple $SU(5)$ model. There are not the proton decay
and quark confinement problems in a $SU(5)$ model with hadrons as
nontopological solitons${}^{[2]}.$ This model is not contradictory to given
experiments and astronomical observations up to now. Hence a $SU(5)$ model is
still possible.
Many astronomical observations show that there is dark matter. Many dark
matter models were presented. A necessary inference of a quantum field theory
without divergence is just that there must be dark matter ($w-matter$) which
and visible matter are symmetric and there is no interaction except the
gravitation between both[3]. The energy density $\rho_{0}$ is zero without
normal ordering of operators and all loop corrections are finite in the
quantum field theory. The sort of dark matter ($w-matter$) is called mirror
matter which is discussed in detail in Refs[4].
A dark matter model which is called $w-matter$ or mirror dark matter is
concretely constructed based on $SU_{f}(5)\times SU_{w}(5)$ symmetry in the
present paper. There is no Higgs field and all masses originate from
interactions in the present model. $W-matter$ is dark matter relatively to
$f-matter$ and vice versa. In high-energy processes or when temperature is
very high, visible matter and dark matter can transform from one into another.
In such process energy seems to be non-conservational, because dark matter
cannot be detected. In low-energy processes or when temperature is low, there
is only gravitation interaction of dark matter for visible matter.
In section 2, Lagrangian of the $SU_{f}(5)\times SU_{w}(5)$ model is
constructed; In section 3, symmetry spontaneously breaking is discussed; In
section 4, the physical significance of the present model is given; Section 5
is the conclusion.
## II Lagrangian of the $SU_{f}(5)\times SU_{w}(5)$ model
###### Conjecture 1
There are two sorts of matter which are called $fire-matter$ ($f-matter$) and
$water-matter$ ($w-matter$), respectively. Both are symmetric and have
$SU_{f}(5)\times SU_{w}(5)$ symmetry. There is no other interaction except the
gravitation between both and the coupling $\left(5\right)$ of f-scalar fields
and w-scalar fields.
The conjecture, in fact, is a necessary inference of a quantum field theory
without divergence in which all loop-corrections are finite and the energy
density $\rho_{0}$ of the vacuum state must be zero without normal ordering of
operators[3]. It is obvious that the conjecture is consistent with a sort of
dark matter model which is called $w-matter^{\left[3\right]}$ or mirror dark
matter[4].
Based the conjecture, the Lagrangian density of the $SU_{f}(5)\times
SU_{w}(5)$ model can be taken as
$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{f}\left(\chi_{f},\Psi_{f},G_{f},\Phi_{f},H_{f}\right)+\mathcal{L}_{w}\left(\chi_{w},\Psi_{w},G_{w},\Phi_{w},H_{w}\right)+\mathcal{L}_{\Omega}+V,$
(1) $\displaystyle V$ $\displaystyle=V_{f}+V_{w}+V_{\Omega}+V_{I},$
$V_{f}=\frac{1}{4}a\left(Tr\Phi_{f}^{2}\right)^{2}+\frac{1}{2}bTr\left(\Phi_{f}^{4}\right)+\frac{1}{4}\xi\left(H_{f}^{+}H_{f}\right)^{2}+\frac{1}{2}\varsigma
H_{f}^{+}H_{f}Tr\Phi_{f}^{2}-\frac{1}{2}\varkappa H_{f}^{+}\Phi_{f}^{2}H_{f},$
(2)
$V_{w}=\frac{1}{4}a\left(Tr\Phi_{w}^{2}\right)^{2}+\frac{1}{2}bTr\left(\Phi_{w}^{4}\right)+\frac{1}{4}\xi\left(H_{w}^{+}H_{w}\right)^{2}+\frac{1}{2}\varsigma
H_{w}^{+}H_{w}Tr\Phi_{w}^{2}-\frac{1}{2}\varkappa
H_{w}^{+}\Phi_{w}^{2}H_{w},,$ (3)
$V_{\Omega}=\frac{1}{4}\lambda\Omega^{4},\text{ \ \
}\mathcal{L}_{\Omega}=\frac{1}{2}\partial_{\mu}\Omega\partial^{\mu}\Omega,$
(4)
$V_{I}=-\frac{1}{15}w\Omega^{2}\left(Tr\Phi_{f}^{2}+Tr\Phi_{w}^{2}\right)-\frac{2A}{225}Tr\Phi_{f}^{2}Tr\Phi_{w}^{2},$
(5)
where $\chi$ and $\Psi$ denote fermion fields, and $G$ the $SU(5)$ gauge
fields. $\Omega,$ $\Phi$ and $H$ are the $\underline{1},$ $\underline{24}$ and
$\underline{5}$ representations, respectively. It should be pointed out that
all the scalar fields are not Higgs fields because they are all massless
before symmetry breaking.
Similarly to the conventional $SU(5)$ model, the possible fermion states for
the first generation are
$\Psi_{fL}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}0\text{ \ \ \ \
}u_{f3}^{c}\text{\ }-u_{f2}^{c}\text{ }-u_{f1}\text{ }-d_{f1}\\\
-u_{f3}^{c}\text{ \ \ }0\text{ \ \ \ }u_{f1}^{c}\text{\ }-u_{f2}\text{\
}-d_{f2}\\\ u_{f2}^{c}\text{ }-u_{f2}^{c}\text{ \ \ }0\text{ \ }-u_{f3}\text{\
}-d_{f3}\\\ u_{f1}\text{ \ \ \ }u_{f2}\text{ \ \ }u_{f3}\text{ \ \ \ }0\text{
}-e_{f}^{+}\\\ d_{f1}\text{ \ \ }d_{f2}\text{ \ \ }d_{f3}\text{ \ \ \
}e_{f}^{+}\text{ \ \ }0\end{array}\right)_{L},\text{ \
}\Psi_{fR}=\left(\begin{array}[c]{c}d_{f1}\\\ d_{f1}\\\ d_{f1}\\\ e_{f}^{+}\\\
-\nu_{fe}^{c}\end{array}\right)_{R}$ (6)
$\Psi_{wR}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}0\text{ \ \ \ \ \ \
}u_{w3}^{c}\text{\ \ }-u_{w2}^{c}\text{ }-u_{w1}\text{ }-d_{w1}\\\
-u_{w3}^{c}\text{ \ \ \ }0\text{ \ \ \ \ }u_{w1}^{c}\text{\ \ }-u_{w2}\text{\
}-d_{w2}\\\ u_{w2}^{c}\text{ \ }-u_{w2}^{c}\text{ \ \ }0\text{ \ \
}-u_{w3}\text{\ }-d_{w3}\\\ u_{w1}\text{ \ \ \ }u_{w2}\text{ \ \ }u_{w3}\text{
\ \ \ }0\text{ }-e_{w}^{+}\\\ d_{w1}\text{ \ \ }d_{w2}\text{ \ \ }d_{w3}\text{
\ \ \ }e_{w}^{+}\text{ \ \ }0\end{array}\right)_{R},\text{ \
}\Psi_{wL}=\left(\begin{array}[c]{c}d_{w1}\\\ d_{w1}\\\ d_{w1}\\\ e_{w}^{+}\\\
-\nu_{we}^{c}\end{array}\right)_{L}$ (7)
The other possible model is an $SU(5)$ grand unified model with hadrons as
nontopological solitons${}^{[2]}.$ The conclusions of the present paper are
independent of a concrete model.
## III Symmetry spontaneously breaking and temperature effects
For simplicity, we do not consider the couplings $\Omega$ and $\Phi$ with
$\chi$ for a time. Ignoring the contributions of the scalar fields and the
fermion fields to one loop correction and only considering the contribution of
the gauge fields to one-loop correction, when $\overline{\varphi}_{s}\ll kT$,
$k$ is the Boltzmann constant, similarly to Ref. $[1],$ the finite-temperature
effective potential approximate to 1-loop in flat space can be obtained
$\displaystyle V$
$\displaystyle=\frac{\lambda}{8}T^{2}\Omega^{2}+\frac{1}{4}\lambda\Omega^{4}-\frac{A}{2}\varphi_{f}^{2}\varphi_{w}^{2}-\frac{1}{2}w\Omega^{2}\left(\varphi_{f}^{2}+\varphi_{w}^{2}\right)$
$\displaystyle+\frac{D}{4!}\varphi_{f}^{4}+B\varphi_{f}^{4}\left(\ln\frac{\varphi_{f}^{2}}{\sigma^{2}}-\frac{1}{2}\right)+CT^{2}\varphi_{f}^{2}$
$\displaystyle+\frac{D}{4!}\varphi_{w}^{4}+B\varphi_{w}^{4}\left(\ln\frac{\varphi_{w}^{2}}{\sigma^{2}}-\frac{1}{2}\right)+CT^{2}\varphi_{w}^{2},$
(8)
where
$\Phi_{s}=Diagonal\left(1,1,1,-\frac{3}{2},-\frac{3}{2}\right)\overline{\varphi}_{s},$
(9) $B\equiv\frac{5625}{1024\pi^{2}}g^{4},\text{ \
}\frac{\left(225a+105b\right)}{16}\equiv\frac{D}{4!}+\frac{11}{3}B,\text{ \
}C\equiv\frac{75}{16}\left(kg\right)^{2},$
$\sigma$ is regarded as a constant, and the terms independent of $\Omega$ and
$\Phi$ are neglected.
According to the mirror dark matter model, the temperature of mirror matter is
strikingly lower than that of visible matter. But this is not necessary when a
cosmological model is considered. We will discuss the problem in another
paper. The temperature $T_{f}$ of $f-matter$ may be different from $T_{w}$ of
$w-matter$ in the present model as well, but for simplicity we take
$T_{f}=T_{w}.$
The conditions by which $V$ takes its extreme values are
$\displaystyle\left[\lambda\overline{\Omega}^{2}-w\left(\overline{\varphi}_{f}^{2}+\overline{\varphi}_{w}^{2}\right)+\frac{\lambda}{4}T^{2}\right]\overline{\Omega}$
$\displaystyle=0,$ (10a)
$\displaystyle-w\overline{\Omega}^{2}-A\overline{\varphi}_{w}^{2}+\frac{D}{6}\overline{\varphi}_{f}^{2}+4B\overline{\varphi}_{f}^{2}\ln\frac{\overline{\varphi}_{f}^{2}}{\sigma^{2}}+2CT^{2}$
$\displaystyle=0,$ (10b)
$\displaystyle-w\overline{\Omega}^{2}-A\overline{\varphi}_{f}^{2}+\frac{D}{6}\overline{\varphi}_{w}^{2}+4B\overline{\varphi}_{w}^{2}\ln\frac{\overline{\varphi}_{w}^{2}}{\sigma^{2}}+2CT^{2}$
$\displaystyle=0.$ (10c)
When $T\sim 0$,
$\displaystyle\overline{\varphi}_{f}^{2}$
$\displaystyle=\overline{\varphi}_{w}^{2}\equiv\sigma_{0}^{2}=\sigma^{2}\exp
M,\text{ \ \
}M\equiv\frac{1}{4B}\left(A+\frac{2w^{2}}{\lambda}-\frac{D}{6}\right),$
$\displaystyle\overline{\Omega}_{0}^{2}$
$\displaystyle=\upsilon_{0}^{2}=\frac{2w}{\lambda}\sigma^{2}\exp M,$ (11a)
$\displaystyle V$ $\displaystyle=V_{\min}=-B\sigma^{4}\exp 2M.$ (11b)
$\sigma^{2}\left(T\right)$ and $\upsilon^{2}\left(T\right)$ will decrease and
$V_{\min}$ will increase as temperature rises. There must be the critical
temperature $T_{cr}$ so that when $T>T_{cr},$ the least value of $V$ is
$V\left(\overline{\varphi}_{f}=\overline{\varphi}_{w}=\overline{\Omega}=0\right)=0.$
$T_{cr}$ is rough estimated to be
$T_{cr}=\frac{8B}{w+8C}\sigma^{2}\exp\left(M-\frac{1}{2}\right).$ (12)
$\Omega$ is not absolutely necessary for the symmetry breaking of the present
model, but it is necessary for some a cosmological model${}^{[5]}.$
After spontaneous symmetry-breaking, the reserved symmetry is
$\left[SU_{f}(3)\times SU_{f}(2)\times
U_{f}(1)\right]\times\left[SU_{w}(3)\times SU_{w}(2)\times U_{w}(1)\right].$
The breaking is a sort of dynamical breaking. In other words, the interactions
of the scalar fields with the gauge fields make the massless scalar fields
become ‘Higgs fields’, and finally cause the spontaneous symmetry-breaking. As
a consequence, the $f-particles$ ($w-particles$) can get their masses
determined by the reserved symmetry $SU(3)\times SU(2)\times U(1)$ as the
conventional $SU(5)$ $GUT$ theory in which there are Higgs fields.
## IV The physical significance of the present model
1\. The model implies that all masses originate from interactions.
2\. $W-matter$ is dark matter for $f-matter$ in low energy process, vice
versa. This is because the masses of the scalar particles to be very large in
low temperature so that the transformation of the $f-$ and the $w-scalar$
particles from one into another and their effects may be ignored and there is
no interaction except the coupling $\left(5\right)$ and the gravitation
between $f-matter$ and $w-matter$. This sort of dark matter is called mirror
dark matter in Refs.[4].
3\. In high-energy processes or when temperature is very high, visible matter
and dark matter can transform from one into another. In such process energy
seems to be non-conservational, because dark matter cannot be detected. The
following reaction originating from $\left(1\right)$ and $\left(5\right)$ is
an example in which visible matter transforms into dark matter.
$p+\overline{p}\longrightarrow\varphi_{fA}\longrightarrow\varphi_{fB}+\varphi_{wC}+\varphi_{wD}.$
(13)
In the reaction $\varphi_{wC}$ and $\varphi_{wD}$ and the $w-particles$ coming
from the decay of $\varphi_{wC}$ and $\varphi_{wD}$ cannot be detected.
## V Conclusion
A dark matter model which is called $w-matter$ or mirror dark matter is
concretely constructed based on $SU_{f}(5)\times SU_{w}(5)$ symmetry. There is
no Higgs field and all masses originate from interactions in the present
model. $W-matter$ is dark matter relatively to $f-matter$ and vice versa. In
high-energy processes or when temperature is very high, visible matter and
dark matter can transform from one into another. In such process energy seems
to be non-conservational, because dark matter cannot be detected. In low-
energy processes or when temperature is low, there is only gravitation
interaction of dark matter for visible matter.
Acknowledgement
I am very grateful to professor Zhao Zhan-yue and professor Wu Zhao-yan for
their helpful discussions and best support.
## References
* (1) S. Coleman and E. Weinberg, Phys. Rev. D 6, (1972) 1888; R. H. Brandenberger, Rev. of Mod. Phys. 57, (1985) 1.
* (2) S-H. Chen, High Energy Phys. and Nuc. Phys., 18, (1994) 317, 18, (1994) 409.
* (3) S-H. Chen, 2002a, ‘Quantum Field Theory Without Divergence A’, hep-th/0203220; S-H, Chen, 2002b ‘Significance of Negative Energy State in Quantum Field Theory A’ hep-th/0203230; S-H, Chen, 2005a, ‘Quantum Field Theory :New Research’, O. Kovras Editor, Nova Science Publishers, Inc. p103-170; S-H, Chen, 2001, ‘A Possible Candidate for Dark Matter’, hep-th/0103234; S-H, Chen, 2005b, ‘Progress in Dark Matter Research’ Editor: J. Val Blain, pp.65-72. Nova Science Publishers, Inc.
* (4) Z. Berezhiani, D Comelli and F. L. Villante, Phys. Lett. B, 503, (2001) 362; A Y. Ignatiev and R. R. Volkas, Phys. Rev. D 68, (2003). 023518; P. Ciarcelluti, astro-ph/0409630; astro-ph/0409633.
* (5) S-H, Chen, 2006, ‘A Possible Universal Model without Singularity and its Explanation for Evolution of the Universe’, hep-th/0611283; 2009, ‘Discussion of a Possible Universal Model without Singularity’, arXiv. 0908.1495.
|
arxiv-papers
| 2009-12-12T14:54:58 |
2024-09-04T02:49:07.010909
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shi-Hao Chen",
"submitter": "Shihao Chen",
"url": "https://arxiv.org/abs/0912.2427"
}
|
0912.2473
|
The Second Main Theorem Concerning Small Algebroid Functions.∗
Daochun Sun
( School of Mathematics, South China Normal University, Guangzhou 510631,
China)
Zongsheng Gao
( LMIB and Department of Mathematics, Beijing University of Aeronautics and
Astronautics, Beijing 100083, China)
Huifang Liu
( School of Mathematics, South China Normal University, Guangzhou 510631,
China)
Abstract. In this paper, we firstly give the definition of meromorphic
function element and algebroid mapping. We also construct the algebroid
function family in which the arithmetic, differential operations is closed. On
basis of these works, we firstly proved the Second Main Theorem concerning
small algebroid functions for $v$-valued algebroid functions.
Keywords. algebroid function, algebroid mapping, corresponding addition, the
Second Main Theorem.
MSC(2000). 32C20, 30D45.
000
∗This work is supported by the National Nature Science Foundation of
China(No.10771011, 10871076).
## 1\. Introduction
In 1925, R. Nevanlinna obtained the Second Main Theorem for meromorphic
functions and posed the problem whether the the Second Main Theorem can be
extended to small functions (See [1].). Dealing with the problem, Q. T. Chuang
proved the Second Main Theorem still holds for small entire functions (See
[2], [3].). Until 1986, the problem was solved by N. Steinmetz (See [4].). In
2000, M. Ru proved the Second Main Theorem concerning small meromorphic
functions for algebroid functions (See [5].).
It is natural to consider the problem whether the Second Main Theorem for
algebroid functions is still true when we replace the small meromorphic
functions by small algebroid functions. Before considering the problem, we
must define the arithmetic, differential operations over algebroid functions.
Hence we give the definition of meromorphic function element, algebroid
mapping and construct the algebroid function family $H_{W}$. In $H_{W}$ the
arithmetic, differential operations is closed. On basis of these works, by
using the method of Reference [6], we proved the Second Main Theorem
concerning small algebroid functions.
Suppose that $A_{v}(z),\cdots,A_{0}(z)$ are analytic functions without common
zeros in the complex plane $C$. Then the binary complex equation
$\Psi(z,W)=A_{v}(z)W^{v}+A_{v-1}(z)W^{v-1}+\cdots+A_{0}(z)=0$
defines a $v$-valued algebroid function $W(z)$ in the complex plane $C$. The
above equation can be transformed to the standard equation
$\Psi^{*}(z,W)=W^{v}+A^{*}_{v-1}(z)W^{v-1}+\cdots+A^{*}_{0}(z)=0,$
where $A^{*}_{t}(z):=\frac{A_{t}(z)}{A_{v}(z)}~{}~{}(t=0,1,2,\cdots,v-1)$ are
meromorphic functions in the complex plane $C$. Note that for a $v$-valued
algebroid function $W(z)$, its standard equation is unique.
If $\Psi(z,W)$ is irreducible, then the corresponding $W(z)$ is called a
$v$-valued irreducible algebroid function. For an irreducible algebroid
function $W(z)$, the points in the complex plane can be divided to two
classes. One is a set $T_{W}$ of regular points of $W(z)$, the other is a set
$S_{W}=C-T_{W}$ of critical points of $W(z)$. The set $S_{W}$ is an isolated
set (See [7], [8].).
In this paper, $\Psi(z,W)$ needn’t be irreducible in the usual case. A
$v$-valued algebroid function $W(z)$ may decompose to $n(\geq 1)$ number of
$v_{n}$-valued irreducible algebroid functions(containing the case $W$ is a
complex constant) and $v=\sum^{v}_{j=1}v_{j}$.
For a $v$-valued reducible algebroid function $W(z)$, its corresponding binary
complex equation $\Psi(z,W)=0$ can be decomposed to the product of $q(\leq v)$
non-meromorphic coprime factors, namely
$\Psi(z,W)=\Psi_{1}(z,W)\Psi_{2}(z,W)\cdots\Psi_{q}(z,W)=0.$
Let $S_{j}$ denote the set of critical points of the irreducible complex
equation $\Psi_{t}(z,W)=0$. We define the set of critical points of reducible
algebroid function $W(z)$ by $S_{W}:=\cup^{q}_{j=1}S_{j}$ (Since
$\\{S_{j}\\}(j=1,\cdots,q)$ are all isolated sets, $S_{W}$ is also an isolated
set.), the set of regular points of reducible algebroid function $W(z)$ by
$T_{W}:=C-S_{W}$.
###### Remark 1.1.
If $q=1$, then $W(z)$ is an irreducible algebroid function.
###### Remark 1.2.
If $(q(z),b)$ is a polar element or a multivalent algebraic function element,
then $b\in S_{W}$.
###### Remark 1.3.
For every $a\in T_{W}$, there exist and only exist $v$ number of regular
function elements $\\{(w_{t}(z),a)\\}^{v}_{t=1}$. In this paper, we usually
denote $W(z)=\\{w_{j}(z)\\}^{v}_{j=1}$. If there exists $1\leq t<j\leq v$ such
that $w_{t}(z)\equiv w_{j}(z)$, then the complex equation $\psi(z,W)=0$ must
have non-meromorphic function multiple factor.
In this paper, we use the standard notations of the value distribution for
algebroid functions (See [7].).
## 2\. Some basic properties of algebroid functions
###### Definition 2.1.
Let $W(z)$ and $M(z)$ be two algebroid functions defined by
$\Psi(z,W)=A_{v}(z)W^{v}+A_{v-1}(z)W^{v-1}+\cdots+A_{0}(z)=A_{v}(z)\prod^{v}_{j=1}(W-w_{j}(z))=0,~{}~{}A_{v}(z)\not\equiv
0$ $None$
and
$\Phi(z,M)=B_{s}(z)M^{s}+B_{s-1}(z)M^{s-1}+\cdots+B_{0}(z)=B_{s}(z)\prod^{s}_{t=1}(M-m_{t}(z))=0,~{}~{}B_{s}(z)\not\equiv
0,$ $None$
respectively, $W(z)$ and $M(z)$ are called identical, write $W(z)\equiv M(z)$,
provided that $v=s$ and the corresponding coefficients are proportional,
namely
$E(z):=\frac{A_{v}(z)}{B_{v}(z)}=\frac{A_{v-1}(z)}{B_{v-1}(z)}=\cdots=\frac{A_{0}(z)}{B_{0}(z)}.$
Since the coefficients of the equations (2.1) and (2.2) haven’t common zeros,
$E(z)$ is a nonzero constant or an analytic function without zeros.
###### Theorem 2.1.
Suppose that $W(z)=\\{w_{j}(z)\\}^{v}_{j=1}$ and
$M(z)=\\{m_{t}(z)\\}^{s}_{t=1}$ are two irreducible algebroid functions
defined by (2.1) and (2.2), respectively. The following conditions are
equivalent:
(1) $W(z)\equiv M(z)$.
(2) There exist some regular function elements $(w_{j}(z),b)$ of $W(z)$ and
$(m_{j}(z),b)$ of $M(z)$ such that $(w_{j}(z),b)=(m_{j}(z),b)$.
(3) The eliminant $R(\Psi,\Phi)\equiv 0$.
###### Proof.
(1)$\Rightarrow$(3):
$R(\Psi,\Phi)=A^{s}_{v}(z)\prod^{v}_{j=1}\Phi(z,w_{j}(z))=E(z)A^{s}_{v}(z)\prod^{v}_{j=1}\Psi(z,w_{j}(z))\equiv
0.$
By the property of the eliminant, the first equal sign holds (See [9].). Then
by Definition 2.1, we get the second equal sign. Since
$(w_{j}(z),z)(j=1,\cdots,v)$are regular function elements belong to (2.1),
$\Psi(z,w_{j}(z))\equiv 0$ in some neighborhood of $z$. Combining the
identical principle of analytic functions, we get the third equal sign.
(3)$\Rightarrow$(2):Since
$R(\Psi,\Phi)=A^{s}_{v}(z)\prod^{v}_{j=1}\Phi(z,w_{j}(z))=A^{s}_{v}(z)B^{v}_{s}(z)\prod^{v}_{j=1}\prod^{s}_{t=1}(w_{j}(z)-m_{t}(z))\equiv
0.$
there at least exists some term $w_{j}(z)-m_{t}(z)\equiv 0$. Hence there exist
some regular function element $(w_{j}(z),a)$ of $W(z)$ and $(m_{j}(z),a)$ of
$M(z)$ such that $(w_{j}(z),a)=(m_{j}(z),a)$.
(2)$\Rightarrow$(1): Since the irreducible algebroid function is a connected
Riemann surface, the two identical regular function elements can be continued
analytically to their Riemann surface respectively, such that the
corresponding regular function elements are all identical. Hence $v=s$. Then
combining the Viete theorem, we get
$\frac{A_{t}(z)}{A_{v}(z)}=\frac{B_{t}(z)}{B_{s}(z)}=\sum(-1)^{v-t}w_{n_{1}}(z)w_{n_{2}}(z)\cdots
w_{n_{v-t}}(z)(t=0,1,2,\cdots,v-1),$
where $w_{n_{1}}(z),w_{n_{2}}(z),\cdots,w_{n_{v-t}}(z)$ denote any given $v-t$
distinct elements among $w_{1}(z),\cdots,w_{v}(z)$. From this we can obtain
(1). ∎
Note that by Theorem 2.1, an irreducible algebroid function $W(z)$ can not
contain two same regular function elements.
###### Theorem 2.2.
Suppose that $W(z)=\\{(w_{j}(z),B(a,r_{a}))\\}^{v}_{j=1}$ is a $v$-valued
algebroid function defined by (2.1). If it contains two same regular function
elements, then there exist two same $m$-valued ($2m\leq v$) algebroid
functions decomposed from $W(z)$. Hence $W(z)$ is reducible.
###### Proof.
Suppose that $(w_{j}(z),B(a,r_{a}))\equiv(w_{t}(z),B(a,r_{a}))$. Then
$R(\Psi,\Psi_{W})=(-1)^{\frac{v(v-1)}{2}}A^{2v-1}_{v}(z)\prod_{1\leq j<t\leq
v}(w_{j}(z)-w_{t}(z))^{2}\equiv 0.$
By Theorem 2.4 in reference [7], $\Psi(z,W)$ must have the non-meromorphic
function multiple factor. Hence there exist two same $m$-valued ($2m\leq v$)
algebroid function decomposed from $W(z)$. So $W(z)$ is reducible. ∎
###### Definition 2.2.
Meromorphic function element is defined by $(q(z),B(a,r))$, where $q(z)$ is
analytic in the disc $B_{0}(a,r):=\\{0<|z-a|<r\\}$ and $a$ is not a essential
point. So $q(z)$ can be expressed by Laurent series
$q(z)=\sum^{\infty}_{n=t}a_{n}(z-a)^{n}~{}(a_{t}\neq 0)$. We also denote it by
$(q(z),a)$. If the above $t<0$, then we call $(q(z),a)$ is a truth meromorphic
function element. Especially if $q(z)\equiv c$ ($c$ denotes a constant.).
Two meromorphic function elements $(q(z),a)$ and $(p(z),b)$ are called
identical provided that $a=b$ and there exists $r>0$ such that $q(z)\equiv
p(z)$ in the disc $B_{0}(a,r)$.
If $\Psi(z,q(z))=0$ holds for any $z\in B_{0}(a,r)$, then $(q(z),a)$ is called
a meromorphic function element of algebroid function $W(z)$ or $\Psi(z,W)=0$.
###### Remark 2.1.
The regular function element is also the meromorphic function element.
###### Definition 2.3.
The regular function element $(p(z),B(b,R_{b}))$ is called the direct
continuation of meromorphic function element $(q(z),B(a,R_{a}))$ provided that
$b\in B(a,R_{a})$ and in the domain $B(a,R_{a})\cap B(b,R_{b})$ we have
$p(z)\equiv q(z)$.
For any $\epsilon\in(0,R_{a})$, the set of meromorphic function element
$(q(z),B(a,R_{a}))$ and all direct continuation of meromorphic function
element $(q(z),B(a,R_{a}))$ in the disc $B_{0}(a,\epsilon)$ is called a
neighborhood of $(q(z),B(a,R_{a}))$. We denote it by $V_{\epsilon}(q(z),a)$.
###### Remark 2.2.
For any given point in $B_{0}(a,R_{a})$, the direct continuation is
uniqueness.
###### Remark 2.3.
The direct continuation of meromorphic function element must be regular
function element. Hence the truth meromorphic function element is isolated.
###### Definition 2.4.
Let $W(z)=\\{(w_{a,j}(z),a)\\}$ be a $v$-valued algebroid function. $h$ is
called an algebroid mapping of $W(z)$ if $h$ satisfies the following
conditions.
(i)Uniqueness: For any regular function element $(w_{a,j}(z),a)$, its image
element $h\circ(w_{a,j}(z),a)=(h\circ w_{a,j}(z),a)$ is meromorphic function
element and unique.
(ii)Continuation: For any image element $(h\circ w_{a,j}(z),a)$, there exists
$\epsilon=\epsilon(h\circ w_{a,j}(z),a)>0$ such that for any regular function
element $(w_{b}(z),b)\in V_{\epsilon}(w_{a,j},a)$, we have $(h\circ
w_{b}(z),b)\in V_{\epsilon}(h\circ w_{a,j},a)$.
(iii)Weak boundary: If $a\in S_{W}$, then $h$ is weak bounded at the
neighborhood of $a$. Namely there exist integer $p>0$, real numbers $r>0$ and
$M>0$, such that for any $b\in B_{0}(a,r):=\\{z;0<|z-a|<r\\}\subset T_{W}$ and
any $t=1,2,\cdots,v$, the corresponding image element $(h\circ w_{b,t}(z),b)$
are all the regular function elements and satisfies $|(b-a)^{p}h\circ
w_{b,t}(b)|<M$.
###### Theorem 2.3.
Let $h$ be an algebroid mapping of $v$-valued algebroid function
$W(z)=\\{(w_{a,j}(z),a)\\}$. Then
(1)$h\circ W(z):=\\{(h\circ w_{a,j}(z),a)\\}$ is a $v$-valued algebroid
function.
(2)If $W(z)$ is irreducible, then $h\circ W(z)$ is irreducible if and only if
$h$ is injective. Namely $h\circ(w(z),a)\neq h\circ(m(z),b)$) when
$(w(z),a)\neq(m(z),b)$, where $(w(z),a)$ and $(m(z),b)$ are regular function
elements.
###### Proof.
For any $z_{0}\in T_{W}$, if there exists some truth meromorphic function
element among the corresponding meromorphic image elements $\\{(h\circ
w_{z_{0},j}(z),z_{0})\\}^{v}_{j=1}$, then $z_{0}$ is called a pole of $h$. We
denote by $P_{h}$ the set of poles of $h$. By the continuation of $h$, we know
that $P_{h}$ is an isolated set.
(1)Firstly we define the analytic functions $\\{H^{*}_{t}(z)\\}^{v-1}_{t=0}$
in $T_{W}-P_{h}$. For any $z_{0}\in T_{W}-P_{h}$, the corresponding image
elements $\\{(h\circ w_{z_{0},j}(z),z_{0})\\}^{v}_{j=1}$ are all regular
function elements. Set
$H^{*}_{t}(z_{0})=\sum(-1)^{v-t}[h\circ w_{z_{0},j_{1}}(z_{0})]\cdot[h\circ
w_{z_{0},j_{2}}(z_{0})]\cdot...\cdot[h\circ w_{z_{0},j_{v-t}}(z_{0})],\hskip
8.5359ptt=0,1,2,...,v-1.$
By the continuation of $h$, there exists $\epsilon$, such that for any $y\in
B(z_{0},\epsilon)$, the corresponding image elements $\\{(h\circ
w_{y,j}(z),y)\\}$ are the direct continuation of $\\{(h\circ
w_{z_{0},j}(z),z_{0})\\}$ respectively. Namely we have $h\circ
w_{y,j}(z)\equiv h\circ w_{z_{0},j}(z)$ in the neighborhood of $y$. So we have
$H^{*}_{t}(y)=\sum(-1)^{v-t}[h\circ w_{y,j_{1}}(y)]\cdot[h\circ
w_{y,j_{2}}(y)]\cdot...\cdot[h\circ w_{y,j_{v-t}}(y)]$ $=\sum(-1)^{v-t}[h\circ
w_{z_{0},j_{1}}(y)]\cdot[h\circ w_{z_{0},j_{2}}(y)]\cdot...\cdot[h\circ
w_{z_{0},j_{v-t}}(y)].$
Hence in $B(z_{0},\epsilon)$, for any $t=0,1,...,v-1$ we have
$H^{*}_{t}(z)\equiv\sum(-1)^{v-t}[h\circ w_{z_{0},j_{1}}(z)]\cdot[h\circ
w_{z_{0},j_{2}}(z)]\cdot...\cdot[h\circ w_{z_{0},j_{v-t}}(z)].$
So $\\{H^{*}_{t}(z)\\}$ is analytic in $B(z_{0},\epsilon)$. By Viete theorem,
they define the following complex equation
$\Phi^{*}(z,W)=W^{v}+H^{*}_{v-1}(z)W^{v-1}+...+H^{*}_{0}(z)=\prod^{v}_{j=1}[W-h\circ
w_{z_{0},j}(z)]=0$
and $\Phi^{*}(z,h\circ w_{z_{0},j}(z))=0$ in $B(z_{0},\epsilon)$. Since
$z_{0}$ is arbitrary, $\\{H^{*}_{t}(z)\\}^{v-1}_{t=0}$ are analytic in
$T_{W}-P_{h}$.
When $z_{0}\in S_{W}\cup P_{h}$, since $h$ is weak bounded, $z_{0}$ is the
isolated singular point and is not the essential isolated singular point of
$\\{H^{*}_{t}(z)\\}$. This shows that $\\{H^{*}_{t}(z)\\}^{v-1}_{t=0}$ are
meromorphic in the complex plane and the corresponding complex equation
$\Phi^{*}(z,W)=0$ defines the algebroid function $h\circ W(z)$.
(2)Suppose that $h$ is injective. For any two regular image elements $(h\circ
w_{a,j}(z),a)\neq(h\circ w_{b,t}(z),b)$, they define uniquely two distinct
regular primary image elements $(h\circ w_{a,j}(z),a)\neq(h\circ
w_{b,t}(z),b)$. Take a path $\gamma\subset T_{W}\cap T_{h\circ W}$ such that
two primary image elements can be continued analytically each other along
$\gamma$. By the continuation of $h$, we know that $(h\circ w_{a,j}(z),a)$ and
$(h\circ w_{b,t}(z),b)$ can be connected by $\gamma$. Hence $h\circ W(z)$ is
irreducible.
Conversely suppose that there exist two different regular function elements
$(w_{a,j}(z),a)\neq(w_{a,t}(z),a)$($j\neq t$) such that the corresponding
image elements $(h\circ w_{j}(z),a)=(h\circ w_{t}(z),a)$). Then by Theorem
2.2, $h\circ W(z)$ is reducible. ∎
###### Definition 2.5.
Suppose that $W(z)=\\{(w_{j}(z),a)\\}$ is a $v$-valued algebroid function
defined by the following complex equation
$\Psi(z,w)=A_{v}(z)W^{v}+A_{v-1}(z)W^{v-1}+...+A_{1}(z)W+A_{0}(z)$
$=A_{v}(z)(W-w_{1}(z))(W-w_{2}(z))...(W-w_{v}(z))=0,$
and $f(z)$ is meromorphic in the complex plane $C$.
1) Define $h_{-W}\circ(w_{j}(z),a):=(-w_{j}(z),a)$. By Viete theorem, the
complex equation with respect to $h_{-W}\circ W(z)$ is
$\Psi_{-W}(z,w):=A_{v}(z)(W-(-w_{1}(z)))(W-(-w_{2}(z)))...(W-(-w_{v}(z)))$
$=A_{v}(z)W^{v}-A_{v-1}(z)W^{v-1}+...+(-1)^{v}A_{0}(z)=0.$
The $v$-valued algebroid function $h_{-W}\circ W(z)$ is called the negative
element of $W(z)$. We denote it by $-W(z)$, denote the algebroid mapping
$h_{-W}$ by $-h$.
2) Define $h_{1/W}\circ(w_{j}(z),a):=(\frac{1}{w_{j}(z)},a)$.By Viete theorem,
the complex equation with respect to $h_{1/W}\circ W(z)$ is
$\Psi_{1/W}(z,w):=A_{v}(z)(W-\frac{1}{w_{1}(z)})(W-\frac{1}{w_{2}(z)})...(W-\frac{1}{w_{v}(z)})$
$=A_{0}(z)W^{v}-A_{1}(z)W^{v-1}+...+A_{v}(z)=0.$
The $v$-valued algebroid function $h_{1/W}\circ W(z)$ is called the inverse
element of $W(z)$. We denote it by $\frac{1}{W(z)}$, denote the algebroid
mapping $h_{1/W}$ by $\frac{1}{h}$.
###### Remark 2.4.
Especially, $W(z)\equiv 0$ is also the algebroid function. Its inverse element
is defined as $\frac{1}{W(z)}\equiv\infty$ and $\frac{1}{W(z)}$ is also the
algebroid function.
3) Define $h_{f}\circ(w_{j},a)=(f(z),a)$. It is easy to prove that $h_{f}$
satisfies Definition 2.4. So $h_{f}$ is an algebroid mapping. By Theorem 2.3,
The $v$-valued algebroid function $h_{f}\circ W(z)=\\{f(z)\\}$ are $v$ same
meromorphic functions $f(z)$. Especially, if $f(z)\equiv c\in{\overline{C}}$,
then the algebroid function $h_{c}\circ W(z)=\\{c\\}$ degenerates into $v$
same finite or infinite complex constants.
4) Define $h_{W^{\prime}}\circ(w_{j}(z),a)=(w^{\prime}_{j}(z),a)$. It is easy
to prove that $h_{W^{\prime}}$ satisfies the conditions 1, 2 of Definition
2.4. If $z_{0}\in S_{W}$, then in $B_{0}(z_{0},r):=\\{0<|z-z_{0}|<r\\}$ we
have
$q_{t}(z):=\sum^{\infty}_{n=u_{t}}a_{n,t}(z-a_{0})^{n/\lambda_{t}},~{}t=1,2,...,m,$
where $\lambda_{t}$ is a positive integer, $u_{t}$ is an integer and
$\sum^{m}_{t=1}\lambda_{t}=v$. It is easy to see that
$h_{W^{\prime}}\circ
q_{t}(z)=\sum^{\infty}_{n=u_{t}}\frac{na_{n,t}}{\lambda_{t}}(z-a_{0})^{\frac{n-\lambda_{t}}{\lambda_{t}}},~{}t=1,2,...,m$
is weak bounded. By Theorem 2.3, $h_{W^{\prime}}\circ W(z)$ defines a
$v$-valued algebroid function. We call it the derivative of $W(z)$. We denote
it by $h_{W^{\prime}}\circ W(z)=W^{\prime}(z)$. The complex equation with
respect to $W^{\prime}(z)$ is
$\Psi^{\prime}(z,w):=B_{v}(z)(W^{\prime}-w^{\prime}_{1}(z))(W^{\prime}-w^{\prime}_{2}(z))...(W^{\prime}-w^{\prime}_{v}(z))$
$:=B_{v}(z)(W^{\prime})^{v}+B_{v-1}(z)(W^{\prime})^{v-1}+...+B_{1}(z)W^{\prime}+B_{0}(z)=0.$
###### Definition 2.6.
Let $W(z)=\\{(w_{j}(z),a)\\}^{v}_{j=1}$ be a $v$-valued algebroid function.
The set of all algebroid mappings of $W(z)$ is denoted by $Y_{W}$. The set
$H_{W}:=\\{h\circ W(z);h\in Y_{W}\\}$
is called the algebroid function class of $W(z)$.
Set
$X_{W}:=\\{f\in H_{W};T(r,f)=o[T(r,W)]~{}(r\rightarrow\infty,~{}r\not\in
E_{f})\\},$
where $E_{f}$ is a real number set of finite linear measure depending on $f$.
$X_{W}$ is called the small algebroid function set of $W(z)$. The element in
$X_{W}$ is called the small algebroid function of $W(z)$.
Note that the set $X_{W}$ contains all the finite or infinite complex
constants, all the small meromorphic functions and all the small algebroid
functions.
###### Definition 2.7.
Let the set of all algebroid mappings of $W(z)$ be $Y_{W}$ and
$H_{W}:=\\{h\circ W(z);h\in Y_{W}\\}$. For any $h_{1},h_{2}\in Y_{W}$, define
1)Addition: $(h_{1}+h_{2})\circ W(z)=h_{1}\circ W(z)+h_{2}\circ W(z)$.
2)Subtraction: $(h_{1}-h_{2})\circ W(z)=h_{1}\circ W(z)-h_{2}\circ W(z)$.
3)Multiplication: $(h_{1}\cdot h_{2})\circ W(z)=(h_{1}\circ
W(z))\cdot(h_{2}\circ W(z))$.
4)Division: $(\frac{h_{1}}{h_{2}})\circ W(z)=h_{1}\circ
W(z)\cdot\frac{1}{h_{2}}\circ W(z)$.
It is easy to prove that they satisfy Definition 2.4. Hence they are all
algebroid mappings. So $H_{W}$ is a linear space and is closed with respect to
Multiplication and Division.
Suppose that $\\{a_{j}(z)\\}$,$\\{b_{i}(z)\\}$ are two group of analytic
functions defined in the complex plane $C$, without no common zeros. The
function
$q[z,w]:=\frac{a_{n}(z)w^{n}+a_{n-1}(z)w^{n-1}+...+a_{0}(z)}{b_{m}(z)w^{m}+b_{m-1}(z)w^{m-1}+...+b_{0}(z)}$
is called rational complex function with meromorphic coefficients. The set of
all rational complex functions with meromorphic coefficients is denoted by
$Q[z,w]$. By the above definition, Definitions 2.5 and 2.6, for any $q[z,w]\in
Q[z,w]$, $q\circ\\{(w_{j}(z),a)\\}=\\{(q[z,w_{j}(z)],a)\\}\in H_{W}$ is the
algebroid function. So $q[z,w]\in Y_{W}$.
Especially, when $Q(z)$ is a single valued rational function defined in the
complex plane, $Q\circ W(z):=\\{Q\circ w_{j}(z),a\\}$ is the $v$-valued
algebroid function. If $W(z)$ is irreducible and $Q$ is linear, then $Q\circ
W(z)$ is irreducible. If $q[z,w]=w\in Q[z,w]$, then
$q\circ\\{(w_{j}(z),a)\\}=\\{(w_{j}(z),a)\\}=W(z)$ is an identical mapping.
###### Theorem 2.4.
Suppose that $h$ is an algebroid mapping of $v$-valued irreducible algebroid
function $W(z)=\\{(w_{j}(z),a)\\}$. If $h\circ W(z)$ is reducible, then it can
split to $n(\geq 1)$ number of $m$-valued irreducible algebroid functions and
$v=mn$.
###### Proof.
By Theorem 2.3, we know that $h$ isn’t injective.Namely there exist two
regular function element $(w_{1}(z),a)\neq(w_{2}(z),a)$, such that the image
elements $(h\circ w_{1}(z),a)=(h\circ w_{2}(z),a)$. By Theorem 2.2, $h\circ
W(z)=\\{(h\circ w_{j}(z),a)\\}$ can split at least two equal $m$-valued
($2m\leq v$) algebroid functions
$h\circ W_{1}(z)=\\{(h\circ w_{1}(z),a)\\}=h\circ W_{2}(z)=\\{(h\circ
w_{2}(z),a)\\}.$
If $2m<v$, then there exist the regular function elements
$(h\circ w_{3}(z),a)\in h\circ W(z)-h\circ W_{1}(z)-h\circ W_{2}(z)$
and $(h\circ w_{4}(z),a)\in h\circ W_{1}(z)=\\{(h\circ w_{1}(z),a)\\}$ such
that $(h\circ w_{3}(z),a)=(h\circ w_{4}(z),a)$(Otherwise, since the primary
images $(w_{3}(z),a)$ and $(w_{4}(z),a)$ are connected, $(h\circ w_{3}(z),a)$
and $(h\circ w_{4}(z),a)$ are also connected, which contradicts the fact that
$W_{1}(z)$ is an alhgebroid function.). Hence by Theorem 2.1 from $(h\circ
w_{3}(z),a)$ we can continue a $m$-valued algebroid function $h\circ W_{3}(z)$
such that it equals to $h\circ W_{1}(z)$. This work doesn’t stop until we get
$n$ same $m$-valued algebroid functions with $nm=v$. ∎
###### Corollary 2.1.
Suppose that $h$ is an algebroid mapping of $v$-valued irreducible algebroid
function $W(z)=\\{(w_{j}(z),a)\\}$. If $v$ is prime, then $h\circ W(z)$ is
irreducible or $v$ same meromorphic functions.
Dealing with the addition of two $v$-valued algebroid functions, we get the
following result.
###### Theorem 2.5.
Let $W(z)=\\{(w_{t}(z),a)\\}$ and $M(z)=\\{(m_{t}(z),a)\\}\in H_{W}$ be two
$v$-valued algebroid functions. Then
$T(r,W+M)\leq T(r,W)+T(r,M)+\log 2.$ $T(r,W\cdot M)\leq T(r,W)+T(r,M).$
###### Proof.
Suppose that $W(z)$ and $M(z)$ are decomposed to $v$ simple-valued branch
$\\{W_{t}(z)\\}$ and $\\{M_{t}(z)\\}$ in the cutting complex plane. Then
$m(r,W+M)=\frac{1}{v}\sum_{1\leq t\leq v}m(r,W_{t}(z)+M_{t}(z))$
$=\frac{1}{v}\sum_{1\leq t\leq
v}\frac{1}{2\pi}\int^{2\pi}_{0}\log^{+}|W_{t}(re^{i\theta})+M_{t}(re^{i\theta})|d\theta$
$\leq\frac{1}{v}(v\log
2+\sum^{v}_{t=1}\frac{1}{2\pi}\int^{2\pi}_{0}\log^{+}|W_{t}(re^{i\theta})|d\theta+\sum^{v}_{t=1}\frac{1}{2\pi}\int^{2\pi}_{0}\log^{+}|M_{t}(re^{i\theta})|d\theta)$
$=m(r,W(z))+m(r,M(z))+\log 2.$
$N(r,W+M)=\frac{1}{v}\int^{r}_{0}\frac{n(t,W+M)-n(0,W+M)}{t}dt+\frac{n(0,W+M)}{v}\ln
r$ $\leq\frac{1}{v}\int^{r}_{0}\frac{n(t,W)-n(0,W)}{t}dt+\frac{n(0,W)}{v}\ln
r+\frac{1}{v}\int^{r}_{0}\frac{n(t,M)-n(0,M)}{t}dt+\frac{n(0,M)}{v}\ln r$
$=N(r,W)+N(r,M).$ $m(r,W\cdot M)=\frac{1}{v}\sum_{1\leq t\leq
v}\frac{1}{2\pi}\int^{2\pi}_{0}\log^{+}|W_{t}(re^{i\theta})M_{t}(re^{i\theta})|d\theta$
$\leq\frac{1}{v}(\sum^{v}_{t=1}\frac{1}{2\pi}\int^{2\pi}_{0}\log^{+}|W_{t}(re^{i\theta})|d\theta+\sum^{v}_{t=1}\frac{1}{2\pi}\int^{2\pi}_{0}\log^{+}|M_{t}(re^{i\theta})|d\theta)$
$=m(r,W(z))+m(r,M(z)).$ $N(r,W\cdot M)=\frac{1}{v}\int^{r}_{0}\frac{n(t,W\cdot
M)-n(0,W\cdot M)}{t}dt+\frac{n(0,W\cdot M)}{v}\ln r$
$\leq\frac{1}{v}\int^{r}_{0}\frac{n(t,W)-n(0,W)}{t}dt+\frac{n(0,W)}{v}\ln
r+\frac{1}{v}\int^{r}_{0}\frac{n(t,M)-n(0,M)}{t}dt)+\frac{n(0,M)}{v}\ln r$
$=N(r,W)+N(r,M).$
Hence we get the conclusions of Theorem 2.5. ∎
## 3\. Nevanlinna’s second main theorem concerning small algebroid functions
Since in $H_{W}$, elements in $X_{W}$ can make addition, subtraction,
multiplication, division and differential, we have conditions to investigate
the theorem concerning small algebroid functions. Referring to the method in
[2, 6], we firstly obtain the Second Main Theorem concerning small algebroid
functions.
###### Lemma 3.1.
Suppose that $W(z)=\\{(w_{t}(z),a)\\}$ is a $v$-valued nonconstant algebroid
functin in $\\{|z|<R\\}$, and $\\{a_{j}(z)\\}^{p}_{j=0}\subset X_{W}$ are $q$
distinct small algebroid function with respect to $W(z)$. Then for any
$r\in(0,R)$, we have
$|m(r,\sum^{q}_{j=1}\frac{1}{W(z)-a_{j}(z)})-\sum^{q}_{j=1}m(r,\frac{1}{W(z)-a_{j}(z)})|=S(r,W),$
where
$S(r,W)=O(\log(rT(r,f))),~{}(r\rightarrow\infty,~{}r\not\in E),$
$E$ is a positive real number set of finite linear measure.
###### Proof.
By using the tree $Y$ through all branch points of $W(z)$, we cut $W(z)$ into
$v$ singule-valued branch $\\{W_{t}(z)\\}^{v}_{t=1}$. Accordingly, we cut
every $a_{j}(z)$ into $v$ singule-valued branch $\\{a_{j,t}(z)\\}^{v}_{t=1}$.
For any $t=1,2,...,v$, set
$F_{t}(z):=\sum^{q}_{j=1}\frac{1}{W_{t}(z)-a_{j,t}(z)}$ $None$
and
$m(r,F_{t})\leq\sum^{q}_{j=1}m(r,\frac{1}{W_{t}(z)-a_{j,t}(z)})+\log q.$
$None$
In order to obtain the lower bound of $m(r,F_{t})$, for any $z$, set
$\delta_{t}(z):=\min_{1\leq j<u\leq q}\\{|a_{j,t}(z)-a_{u,t}(z)|\\}\geq 0.$
Note that $\delta_{t}(z)$ is the function of $z$, by the uniqueness theorem,
its zeros must be isolated. Take arbitrary $z\in\\{z;\delta_{t}(z)\neq 0\\}$.
Case 1. If for any $j\in\\{1,2,...,q\\}$, we have
$|W_{t}(z)-a_{j,t}(z)|\geq\frac{\delta_{t}(z)}{2q},$
then
$\sum^{q}_{j=1}\log^{+}\frac{1}{|W_{t}(z)-a_{j,t}(z)|}\leq
q\log^{+}\frac{2q}{\delta_{t}(z)}.$ $None$
Case 2. If there exists some $u\in\\{1,2,...,q\\}$ such that
$|W_{t}(z)-a_{u,t}(z)|\leq\frac{\delta_{t}(z)}{2q}.$ $None$
Then when $j\neq u$, we have
$|W_{t}(z)-a_{j,t}(z)|\geq|a_{u,t}(z)-a_{j,t}(z)|-|W_{t}(z)-a_{u,t}(z)|\geq\delta_{t}(z)-\frac{\delta_{t}(z)}{2q}=\frac{2q-1}{2q}\delta_{t}(z).$
Hence by (3.4) we get
$\frac{1}{|W_{t}(z)-a_{j,t}(z)|}\leq\frac{1}{2q-1}\frac{2q}{\delta_{t}(z)}$
$None$ $<\frac{1}{2q-1}\frac{1}{|W_{t}(z)-a_{u,t}(z)|}.$ $None$
By (3.1) and (3.6) we get
$|F_{t}(z)|\geq\frac{1}{|W_{t}(z)-a_{u,t}(z)|}-\sum_{j\neq
u}\frac{1}{|W_{t}(z)-a_{j,t}(z)|}$
$\geq\frac{1}{|W_{t}(z)-a_{u,t}(z)|}-\frac{q-1}{2q-1}\frac{1}{|W_{t}(z)-a_{u,t}(z)|}>\frac{1}{2|W_{t}(z)-a_{u,t}(z)|}.$
Then by (3.5) we get
$\log^{+}|F_{t}(z)|>\log^{+}\frac{1}{|W_{t}(z)-a_{u,t}(z)|}-\log 2$
$=\sum^{q}_{j=1}\log^{+}\frac{1}{|W_{t}(z)-a_{j,t}(z)|}-\sum_{j\neq
u}\log^{+}\frac{1}{|W_{t}(z)-a_{j,t}(z)|}-\log 2$
$\geq\sum^{q}_{j=1}\log^{+}\frac{1}{|W_{t}(z)-a_{j,t}(z)|}-\sum_{j\neq
u}\log^{+}\frac{2q}{(2q-1)\delta_{t}(z)}-\log 2$
$>\sum^{q}_{j=1}\log^{+}\frac{1}{|W_{t}(z)-a_{j,t}(z)|}-q\log^{+}\frac{2q}{\delta_{t}(z)}-\log
2.$
Combining (3.3), in two cases we have
$\log^{+}|F_{t}(z)|>\sum^{q}_{j=1}\log^{+}\frac{1}{|W_{t}(z)-a_{j,t}(z)|}-q\log^{+}\frac{2q}{\delta_{t}(z)}-\log
2.$ $None$
By definition, for any $z\in\\{z;\delta_{t}(z)\neq 0\\}$, there exists
$j(z)\neq u(z)$ such that $\delta_{t}(z)=a_{j(z),t}(z)-a_{u(z),t}(z)$. Hence
we get
$\frac{1}{\delta_{t}(z)}=\frac{1}{|a_{j(z),t}(z)-a_{u(z),t}(z)|}\leq\sum_{1\leq
j<u\leq q}\frac{1}{|a_{j,t}(z)-a_{u,t}(z)|}.$
So
$\frac{1}{2\pi}\int^{2\pi}_{0}\ln^{+}\frac{d\theta}{\delta_{t}(re^{i\theta})}\leq\sum_{1\leq
j<u\leq
q}\frac{1}{2\pi}\int^{2\pi}_{0}\ln^{+}\frac{d\theta}{|a_{j,t}(re^{i\theta})-a_{u,t}(re^{i\theta})|}+O(1)$
$=\sum m(r,a_{j,t}-a_{u,t})+O(1)\leq\sum T(r,a_{j,t}-a_{u,t})+O(1)\leq$
$=\sum[T(r,a_{j,t})+T(r,a_{u,t})]+O(1)=S(r,W).$ $None$
Write $z=re^{i\theta}$, integrating (3.7) and combining (3.8), we get
$m(r,F_{t})>\sum^{q}_{j=1}m(r,\frac{1}{|W_{t}(z)-a_{j,t(z)}|})+S(r,W).$
Then by (3.2), we get
$|m(r,F_{t})-\sum^{q}_{j=1}m(r,\frac{1}{|W_{t}(z)-a_{j,t(z)}|})|<S(r,W).$
So
$|m(r,F_{t}(z))-\sum^{q}_{j=1}m(r,\frac{1}{W(z)-a_{j,t(z)}})|$
$=|\frac{1}{v}\sum^{v}_{t=1}m(r,F_{t})-\sum^{q}_{j=1}[\frac{1}{v}\sum^{v}_{t=1}m(r,\frac{1}{|W_{t}(z)-a_{j,t(z)}|})]|$
$\leq\frac{1}{v}\sum^{v}_{t=1}|m(r,F_{t})-\sum^{q}_{j=1}m(r,\frac{1}{|W_{t}(z)-a_{j,t(z)}|})|<S(r,W).$
∎
###### Lemma 3.2.
Suppose that $W(z)$ is a $v$-valued nonconstant algebroid functin in
$\\{|z|<R\\}$ and $n$ is a positive integer. Then $\frac{W^{(n)}}{W}$ is the
differential polynomial of $\frac{W^{\prime}}{W}$.
###### Proof.
When $n=1$, the conclusion holds cleary.
Suppose that for $n=t$ we have
$\frac{W^{(t)}}{W}=P(\frac{W^{\prime}}{W}),$
where $P(\frac{W^{\prime}}{W})$ is the differential polynomial of
$\frac{W^{\prime}}{W}$. Since
$(\frac{W^{(t)}}{W})^{\prime}=\frac{W^{(t+1)}}{W}-\frac{W^{(t)}}{W}\cdot\frac{W^{\prime}}{W},$
$\frac{W^{(t+1)}}{W}=(\frac{W^{(t)}}{W})^{\prime}+\frac{W^{(t)}}{W}\cdot\frac{W^{\prime}}{W}$
$=[P(\frac{W^{\prime}}{W})]^{\prime}+P(\frac{W^{\prime}}{W})\cdot\frac{W^{\prime}}{W}$
is the differential polynomial of $\frac{W^{\prime}}{W}$. ∎
###### Lemma 3.3.
Let $f_{1},f_{2},...,f_{k},g\in H_{W}$. Then
$W(f_{1},f_{2},...,f_{k}):=\left|\begin{array}[]{llll}f_{1}&f_{2}&\cdots&f_{k}\\\
f^{\prime}_{1}&f^{\prime}_{2}&\cdots&f^{\prime}_{k}\\\ \cdots&\cdots&&\\\
f^{(k-1)}_{1}&f^{(k-1)}_{2}&\cdots&f^{(k-1)}_{k}\\\
\end{array}\right|=g^{k}W(\frac{f_{1}}{g},\frac{f_{2}}{g},...,\frac{f_{k}}{g}).$
###### Proof.
(1) When $k=2$, we have
$g^{2}W(\frac{f_{1}}{g},\frac{f_{2}}{g})=g^{2}\left|\begin{array}[]{ll}\frac{f_{1}}{g}&\frac{f_{2}}{g}\\\
\\\ (\frac{f_{1}}{g})^{\prime}&(\frac{f_{2}}{g})^{\prime}\\\
\end{array}\right|=g^{2}\left|\begin{array}[]{ll}\frac{f_{1}}{g}&\frac{f_{2}}{g}\\\
\\\
\frac{f^{\prime}_{1}g-f_{1}g^{\prime}}{g^{2}}&\frac{f^{\prime}_{2}g-f_{2}g^{\prime}}{g^{2}}\\\
\end{array}\right|$
$=g^{2}[\frac{f_{1}f^{\prime}_{2}g-f_{1}f_{2}g^{\prime}}{g^{3}}-\frac{f_{2}f^{\prime}_{1}g-f_{2}f_{1}g^{\prime}}{g^{3}}]=f_{1}f^{\prime}_{2}-f_{2}f^{\prime}_{1}=W(f_{1},f_{2}).$
(2) Suppose that for positive integer $k$, we have
$g^{k}W(\frac{f_{1}}{g},\frac{f_{2}}{g},...,\frac{f_{k}}{g})=W(f_{1},f_{2},...,f_{k}).$
Then for $k+1$, we have
$g^{k+1}W(\frac{f_{1}}{g},\frac{f_{2}}{g},...,\frac{f_{k}}{g},\frac{f_{k+1}}{g})=g^{k+1}\left|\begin{array}[]{lllll}\frac{f_{1}}{g}&\frac{f_{2}}{g}&\cdots&\frac{f_{k}}{g}&\frac{f_{k+1}}{g}\\\
(\frac{f_{1}}{g})^{\prime}&(\frac{f_{2}}{g})^{\prime}&\cdots&(\frac{f_{k}}{g})^{\prime}&(\frac{f_{k+1}}{g})^{\prime}\\\
\cdots&\cdots&&\\\
(\frac{f_{1}}{g})^{(k-1)}&(\frac{f_{2}}{g})^{(k-1)}&\cdots&(\frac{f_{k}}{g})^{(k-1)}&(\frac{f_{k+1}}{g})^{(k-1)}\\\
(\frac{f_{1}}{g})^{(k)}&(\frac{f_{2}}{g})^{(k)}&\cdots&(\frac{f_{k}}{g})^{(k)}&(\frac{f_{k+1}}{g})^{(k)}\\\
\end{array}\right|$
$=g^{k+1}\sum^{k+1}_{n=1}(-1)^{k+1-n}(\frac{f_{n}}{g})^{(k)}W(\frac{f_{1}}{g},...,\frac{f_{n-1}}{g},\frac{f_{n+1}}{g},...,\frac{f_{k+1}}{g})$
$=g\sum^{k+1}_{n=1}(-1)^{k+1-n}(\frac{f_{n}}{g})^{(k)}W(f_{1},...,f_{n-1},f_{n+1},...,f_{k+1})$
$=g\sum^{k+1}_{n=1}(-1)^{k+1-n}[\sum^{k}_{j=0}C^{k}_{j}f^{(j)}_{n}(\frac{1}{g})^{(k-j)}]W(f_{1},...,f_{n-1},f_{n+1},...,f_{k+1})$
$=g\sum^{k}_{j=0}C^{k}_{j}(\frac{1}{g})^{(k-j)}[\sum^{k+1}_{n=1}(-1)^{k+1-n}f^{(j)}_{n}W(f_{1},...,f_{n-1},f_{n+1},...,f_{k+1})]$
$=g\sum^{k}_{j=0}C^{k}_{j}(\frac{1}{g})^{(k-j)}\left|\begin{array}[]{lllll}f_{1}&f_{2}&\cdots&f_{k}&f_{k+1}\\\
(f_{1})^{\prime}&(f_{2})^{\prime}&\cdots&(f_{k})^{\prime}&(f_{k+1})^{\prime}\\\
\cdots&\cdots&&\\\
(f_{1})^{(k-1)}&(f_{2})^{(k-1)}&\cdots&(f_{k})^{(k-1)}&(f_{k+1})^{(k-1)}\\\
(f_{1})^{(j)}&(f_{2})^{(j)}&\cdots&(f_{k})^{(j)}&(f_{k+1})^{(j)}\\\
\end{array}\right|$
$=gC^{k}_{k}(\frac{1}{g})^{(k-k)}\left|\begin{array}[]{lllll}f_{1}&f_{2}&\cdots&f_{k}&f_{k+1}\\\
(f_{1})^{\prime}&(f_{2})^{\prime}&\cdots&(f_{k})^{\prime}&(f_{k+1})^{\prime}\\\
\cdots&\cdots&&\\\
(f_{1})^{(k-1)}&(f_{2})^{(k-1)}&\cdots&(f_{k})^{(k-1)}&(f_{k+1})^{(k-1)}\\\
(f_{1})^{(k)}&(f_{2})^{(k)}&\cdots&(f_{k})^{(k)}&(f_{k+1})^{(k)}\\\
\end{array}\right|$ $=W(f_{1},...,f_{n-1},f_{n},f_{n+1},...,f_{k+1}).$
So the conclusion of Lemma 3.3 holds. ∎
###### Lemma 3.4.
Suppose that $A_{q}=\\{a_{j}:=a_{j}(z)\\}^{q}_{j=1}\subset X_{W}$ are $q\geq
1$ distinct small algebroid fuctions. Let $L(s,A_{q})$ denote the vector space
spanned by finitely many $a^{p_{1}}_{1}a^{p_{2}}_{2}...a^{p_{q}}_{q}$, where
integer $p_{j}\geq 0$($j=1,2,...,q$) and $\sum^{q}_{j=1}p_{j}=s(\geq 1)$. Let
$\dim L(s,A_{q})$ denote the dimension of the vector space $L(s,A_{q})$. Then
for any $\epsilon>0$, there exists $s\geq 1$ such that
$\frac{\dim L(s+1,A_{q})}{\dim L(s,A_{q})}<1+\epsilon.$
###### Proof.
Let $G(s,A_{q})$ denote the set of the form
$a^{p_{1}}_{1}a^{p_{2}}_{2}...a^{p_{q}}_{q}$, and let $\\#(s,A_{q})$ denote
the number of distinct element of $G(s,A_{q})$.
Using mathematical induction, we firstly prove that for any $q>0,s>0$, we have
$\\#(s+1,A_{q})=C^{s+1}_{q+s}.$ $None$
When $q=1$, for any integer $s\geq 1$, $\\#(s+1,A_{1})=1=C^{s+1}_{1+s}$. (3.9)
holds.
When $q=2$, for any integer $s\geq 1$, $\\#(s+1,A_{2})=s+2=C^{s+1}_{2+s}$.
(3.9) holds.
Suppose that for $q=k$ and any integer $s\geq 1$, we have
$\\#(s+1,A_{k})=C^{s+1}_{k+s}$. Then for $q=k+1$, we have
$\\#(s+1,A_{k+1})=\\#(s+1,A_{k})+\\#(s,A_{k})\cdot\\#(1,A_{1})+\\#(s-1,A_{k})\cdot\\#(2,A_{1})$
$+...+\\#(1,A_{k})\cdot\\#(s,A_{1})+\\#(s+1,A_{1})$
$=\\#(s+1,A_{k})+\\#(s,A_{k})+\\#(s-1,A_{k})+...+\\#(2,A_{k})+\\#(1,A_{k})+1$
$=C^{s+1}_{k+s}+C^{s}_{k+s-1}+C^{s-1}_{k+s-2}+...+C^{2}_{k+1}+C^{1}_{k}+1=1+\sum^{s}_{j=0}C^{j+1}_{k+j}.$
Since $C^{j+1}_{k+j+1}=C^{j+1}_{k+j}+C^{j}_{k+j}$,
$C^{j+1}_{k+j}=C^{j+1}_{k+j+1}-C^{j}_{k+j}$. Substituting it into the above
equality, we get
$\\#(s+1,A_{k+1})=1+\sum^{s}_{j=0}(C^{j+1}_{k+j+1}-C^{j}_{k+j}).$
$=1+(C^{s+1}_{k+s+1}-C^{s}_{k+s})+(C^{s}_{k+s}-C^{s-1}_{k+s-1})+(C^{s-1}_{k+s-1}-C^{s-2}_{k+s-2})+(C^{s-2}_{k+s-2}-C^{s-3}_{k+s-3})+...$
$+(C^{4}_{k+4}-C^{3}_{k+3})+(C^{3}_{k+3}-C^{2}_{k+2})+(C^{2}_{k+2}-C^{1}_{k+1})+(C^{1}_{k+1}-C^{0}_{k})$
$=1+C^{s+1}_{k+s+1}-C^{0}_{k}=C^{s+1}_{k+s+1}.$
Then we prove that for any $q>0,s>0$, we have
$C^{s+1}_{q+s}\leq q(q+1)s^{q}.$ $None$
When $q=1$, for any integer $s\geq 1$, $C^{s+1}_{1+s}=1\leq 2s$. (3.10) holds.
When $q=2$, for any integer $s\geq 1$, $C^{s+1}_{2+s}=s+2\leq 6s^{2}$. (3.10)
holds.
Suppose that for $q=k$ and any integer $s\geq 1$, we have $C^{s+1}_{k+s}\leq
k(k+1)s^{k}$. Then for $q=k+1$, we get
$C^{s+1}_{k+s+1}=C^{s+1}_{k+s}\frac{k+s+1}{k}\leq k(k+1)s^{k}\frac{k+s+1}{k}$
$=(k+1)s^{k}(k+s+1)=(k+1)(k+2)s^{k+1}\frac{k+s+1}{ks+2s}\leq(k+1)(k+2)s^{k+1}.$
This shows that (3.10) holds. Combining (3.9), for any $q>0,s>0$ we have
$\dim L(s+1,A_{q})\leq\\#(s+1,A_{q})=C^{s+1}_{q+s}\leq q(q+1)s^{q}.$ $None$
Finally if Lemma 3.4 doesn’t hold, then for any integer $s\geq 1$, we have
$\dim L(s+1,A_{q})\geq(1+\epsilon)\dim L(s,A_{q}).$
Hence
$\dim L(s+1,A_{q})\geq(1+\epsilon)\dim
L(s,A_{q})\geq...\geq(1+\epsilon)^{s}\dim L(1,A_{q})\geq(1+\epsilon)^{s}.$
Combining (3.11), we get
$(1+\epsilon)^{s}\leq q(q+1)s^{q}.$ $None$
But
$\lim_{s\rightarrow\infty}\frac{(1+\epsilon)^{s}}{s^{q}}=\infty.$
This contradicts (3.12). ∎
###### Theorem 3.5.
(Nevanlinna’s Second Main Theorem)
Suppose that $W(z)=\\{(w_{j}(z),a)\\}$ is a $v$-valued nonconstant algebroid
function in the complex plane $C$. $\\{a_{j}\\}^{q}_{j=1}\subset X_{W}$ are
$q\geq 2$ distinct small algebroid functions of $W(z)$. Then for any
$\epsilon\in(0,1)$ and $r>0$, we have
$m(r,W)+\sum^{q}_{j=1}m(r,\frac{1}{W(z)-a_{j}})=(2+\epsilon)T(r,W)+2N_{x}(r,W)+S(r,W).$
$None$
Its equivalent form is
$(q-1-\epsilon)T(r,W)\leq
N(r,W)+\sum^{q}_{j=1}N(r,\frac{1}{W-a_{j}})+2N_{x}(r,W)+S(r,W)$ $None$
or
$(q-4v+3-\epsilon)T(r,W)\leq
N(r,W)+\sum^{q}_{j=1}N(r,\frac{1}{W-a_{j}})+S(r,W).$ $None$
###### Proof.
Let $A_{q}=\\{a_{1},a_{2},...,a_{q}\\}$ and $L(s,A_{q})$ denote the vector
space spanned by finitely many $a^{n_{1}}_{1}a^{n_{2}}_{2}...a^{n_{q}}_{q}$,
where $n_{j}\geq 0$($j=1,2,...,q$) and $\sum^{q}_{j=1}n_{j}=s$. For given $s$,
set $\dim L(s,A_{q})=n$. Let $b_{1},b_{2},...,b_{n}$ denote a basis of
$L(s,A_{q})$. Set $\dim L(s+1,A_{q})=k$. Let $B_{1},B_{2},...,B_{k}$ denote a
basis of $L(s+1,A_{q})$. By Lemma 3.4, for any $\epsilon>0$, there exists some
$s$ such that
$1\leq\frac{k}{n}<1+\epsilon.$ $None$
Let
$P(W):=W(B_{1},B_{2},...,B_{k},Wb_{1},Wb_{2},...,Wb_{n}).$
Since $B_{1},B_{2},...,B_{k},Wb_{1},Wb_{2},...,Wb_{n}$ are linearly
independent, $P(W)\not\equiv 0$. By the definition of the Wronskian
determinant, we get
$P(W)=\sum C_{p}(z)\prod^{n+k-1}_{j=0}(W^{(j)})^{p_{j}}=W^{n}\sum
C_{p}(z)\prod^{n+k-1}_{j=0}(\frac{W^{(j)}}{W})^{p_{j}}.$ $None$
Since $m(r,W^{\prime}/W)=S(r,W)$, we get
$m(r,P(W)\leq nm(r,W)+S(r,W).$ $None$
By Lemma 3.3, we get
$W(B_{1},...,B_{k},Wb_{1},...,Wb_{n})=P(W)=W^{n+k}W(\frac{B_{1}}{W},...,\frac{B_{k}}{W},b_{1},...,b_{n}).$
(i) Suppose that $(q(z),z_{0})$ is a meromorphic fuction element or
multivalent algebraic function element of $W(z)$. If $z_{0}$ is a $\tau$-fold
pole of $q(z)$, by the right of the above equality, it can be see that outside
the poles of the small algebroid functions $\\{B_{i}\\}$,$\\{b_{j}\\}$, the
order of pole of $P(W)$ at $(q(z),z_{0})$ is $(n+k)\tau$.
If $z_{0}$ is a zero of $q(z)$, by the left of the above equality, it can be
see that outside the poles of the small algebroid functions
$\\{B_{i}\\}$,$\\{b_{j}\\}$, $(q(z),z_{0})$ isn’t the pole of $P(W)$.
(ii) For any $1\leq t\leq k$, set
$W_{t}(B_{1},...,B_{k},Wb_{1},...,Wb_{n}):=W(B_{1},...,B_{t-1},B_{t+1},...,B_{k},Wb_{1},...,Wb_{n}).$
When $k<t\leq n+k$, set
$W_{t}(B_{1},...,B_{k},Wb_{1},...,Wb_{n}):=W(B_{1},...,B_{k},Wb_{1},...,Wb_{t-1},Wb_{t+1},...,Wb_{n}).$
Suppose that $(q(z),z_{0})$ is any $\lambda$-sheeted algebraic function
element of $W(z)$ and $z_{0}$ isn’t the pole of $q(z)$. Then $z_{0}$ is at
most the pole of $q^{\prime}(z)$ with the order $\lambda-1$. By Lemma 3.3 we
get
$P(W)=\sum^{k}_{t=1}[(-1)^{t+1}B_{t}\cdot
W_{t}(B^{\prime}_{1},...,B^{\prime}_{k},(Wb_{1})^{\prime},...,(Wb_{n})^{\prime})]$
$+\sum^{k+n}_{t=k+1}[(-1)^{t+1}Wb_{t}\cdot
W_{t}(B^{\prime}_{1},...,B^{\prime}_{k},(Wb_{1})^{\prime},...,(Wb_{n})^{\prime})]$
$=\sum^{k}_{t=1}[(-1)^{t+1}B_{t}\cdot(W^{\prime}b_{t}+Wb^{\prime}_{t})^{n+k-1}W_{t}(\frac{B^{\prime}_{1}}{(Wb_{t})^{\prime}},...,\frac{B^{\prime}_{k}}{(Wb_{t})^{\prime}},\frac{(Wb_{1})^{\prime}}{(Wb_{t})^{\prime}},...,\frac{(Wb_{k})^{\prime}}{(Wb_{t})^{\prime}})$
$+\sum^{k+n}_{t=k+1}[(-1)^{t+1}Wb_{t}\cdot(W^{\prime}b_{t}+Wb^{\prime}_{t})^{n+k-1}W_{t}(\frac{B^{\prime}_{1}}{(Wb_{t})^{\prime}},...,\frac{B^{\prime}_{k}}{(Wb_{t})^{\prime}},\frac{(Wb_{1})^{\prime}}{(Wb_{t})^{\prime}},...,\frac{(Wb_{k})^{\prime}}{(Wb_{t})^{\prime}}).$
Hence outside the poles of the small algebroid functions
$\\{B_{i}\\}$,$\\{b_{j}\\}$, the order of pole of $P(W)$ at $(q(z),z_{0})$ is
at most $(\lambda-1)(n+k-1)$.
Combining (i) and (ii), we get
$N(r,P(W))\leq(n+k)N(r,W)+(n+k-1)N_{x}(r,W)+S(r,W).$
By (3.18) we get
$T(r,P(W))\leq nT(r,W)+kN(r,W)+(n+k-1)N_{x}(r,W)+S(r,W).$ $None$
Suppose that $a$ is a linear combination of $\\{a_{j}\\}$, then
$P(W-a)=W(B_{1},B_{2},...,B_{k},Wb_{1}-ab_{1},Wb_{2}-ab_{2},...,Wb_{n}-ab_{n})$
$=W(B_{1},B_{2},...,B_{k},Wb_{1},Wb_{2},...,Wb_{n})\pm\sum
W(B_{1},B_{2},...,B_{k},...),$
where the element ”…” behind $B_{k}$ in $\sum W(B_{1},B_{2},...,B_{k},...)$
consists of $ab_{j}$. But $ab_{j}$ and $B_{1},B_{2},...,B_{k}$ are linearly
dependent, so we get $\sum W(B_{1},B_{2},...,B_{k},...)=0$. Hence we get
$P(W-a)=P(W).$ $None$
By (3.17) and Lemma 3.2, we get
$P(W)=W^{n}\cdot Q(\frac{W^{\prime}}{W}),$ $None$
where $Q(\frac{W^{\prime}}{W})$ is the differential polynomial of
$\frac{W^{\prime}}{W}$. Set
$u_{j}:=W-a_{j},\hskip 8.5359ptQ_{j}:=Q(\frac{u^{\prime}_{j}}{u_{j}}),\hskip
8.5359ptj=1,2,...,q.$
By (3.20) and (3.21) we get $P(W)=P(u_{j})=u^{n}_{j}Q_{j}$, namely
$\frac{1}{(W-a_{j})^{n}}=\frac{Q_{j}}{P(W)}$. Hence we get
$\frac{1}{|W-a_{j}|}=\frac{|Q_{j}|^{\frac{1}{n}}}{|P(W)|^{\frac{1}{n}}}.$
$None$
Set
$F(z):=\sum^{q}_{j=1}\frac{1}{W(z)-a_{j}}.$
By Lemma 3.1, we get
$m(r,F)=m(r,\sum^{q}_{j=1}\frac{1}{W(z)-a_{j}})=\sum^{q}_{j=1}m(r,\frac{1}{W(z)-a_{j}})+O_{1}(1).$
$None$
By (3.22)we get
$|F(z)|\leq\sum^{q}_{j=1}\frac{1}{|W(z)-a_{j}|}\leq\frac{1}{|P(W)|^{\frac{1}{n}}}\sum^{q}_{j=1}|Q_{j}|^{\frac{1}{n}}.$
Then by (3.19) and (3.16), we get
$m(r,F)\leq\frac{1}{n}m(r,\frac{1}{P(W)})+\frac{1}{n}\sum^{q}_{j=1}m(r,Q_{j})+O(1)$
$\leq\frac{1}{n}T(r,P(W))-\frac{1}{n}N(r,\frac{1}{P(W)})+S(r,W)$ $\leq
T(r,W)+\frac{k}{n}N(r,W)+\frac{n+k-1}{n}N_{x}(r,W)-\frac{1}{n}N(r,\frac{1}{P(W)})+S(r,W)$
$<T(r,W)+\frac{k}{n}N(r,W)+2N_{x}(r,W)-\frac{1}{n}N(r,\frac{1}{P(W)})+S(r,W).$
$None$
By (3.16),(3.23) and (3.24), we get
$m(r,W)+\sum^{q}_{j=1}m(r,\frac{1}{W(z)-a_{j}})\leq\frac{k}{n}m(r,W)+m(r,F)$
$\leq(1+\frac{k}{n})T(r,W)+2N_{x}(r,W)+S(r,W)$
$<(2+\epsilon)T(r,W)+2N_{x}(r,W)+S(r,W).$
Hence we get (3.13).
Note that
$m(r,\frac{1}{W(z)-a_{j}})\leq T(r,W-a_{j})-N(r,\frac{1}{W-a_{j}})+O(1)$
$=T(r,W)-N(r,\frac{1}{W-a_{j}})+S(r,W).$ $None$
Substituting (3.25) into (3.13), we get (3.14). ∎
## References
* [1] R. Nevanlinna, _Zur theorie der meromophen funktionen_ , Acta Math., 1925, 45: 1-9.
* [2] Q. T. Chuang, _Une generalisation d’une inegalite de Nevanlinna_ , Sci. Sinica, 1964, 13: 887-895.
* [3] L. Yang, _Value distribution theory and its new research_ , Beijing, Science Press, 1982(in Chinese).
* [4] N. Steinmetz, _Eine Verallgemeimerung des zweiten Navanlinnaschen Hauptsatzes_. J. Reine Angew. Math., 1986, 386: 134-141.
* [5] M. Ru, _Algebroid functions,Wirsing’s theorem and their relations_ , Math.Z., 2000, 233(1): 137-148.
* [6] C. C. Yang, H. X. Yi, _Uniqueness theory of meromorphic functions_ , Kluwer Academic Publishers, 2003.
* [7] Y. Z. He, X. Z. Xiao, _Algebroidal Function and Ordinary Differential Equations_ , Science Press, Beijing, 1988 (in Chinese).
* [8] Y. N. Lü, X. L. Zhang, _Riemann surface_ , Science Press, Beijing, 1997 (in Chinese).
* [9] A. I. Kostrikin, _Introduction to algebra_ , Higher Education Press, 2007\.
* [10] D. C. Sun, Z. S. Gao, _On the theorems of algebroid functions_ , Acta Math. Sin., Chinese Series, 2006, 49: 1027-1032.
|
arxiv-papers
| 2009-12-13T03:06:11 |
2024-09-04T02:49:07.018287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daochun Sun, Zongsheng Gao, Huifang Liu",
"submitter": "Sun DaoChun",
"url": "https://arxiv.org/abs/0912.2473"
}
|
0912.2598
|
# Mechanisms of proton-proton inelastic cross-section growth in multi-
peripheral model within the framework of perturbation theory. Part 3
I.V. Sharf Odessa National Polytechnic University, Shevchenko av. 1, Odessa,
65044, Ukraine. G.O. Sokhrannyi Odessa National Polytechnic University,
Shevchenko av. 1, Odessa, 65044, Ukraine. A.V. Tykhonov Odessa National
Polytechnic University, Shevchenko av. 1, Odessa, 65044, Ukraine. Department
of Experimental Particle Physics, Jozef Stefan Institute,Jamova 39, SI-1000
Ljubljana, Slovenia. K.V. Yatkin Odessa National Polytechnic University,
Shevchenko av. 1, Odessa, 65044, Ukraine. N.A. Podolyan Odessa National
Polytechnic University, Shevchenko av. 1, Odessa, 65044, Ukraine. M.A.
Deliyergiyev Odessa National Polytechnic University, Shevchenko av. 1,
Odessa, 65044, Ukraine. Department of Experimental Particle Physics, Jozef
Stefan Institute,Jamova 39, SI-1000 Ljubljana, Slovenia. V.D. Rusov
siiis@te.net.ua Odessa National Polytechnic University, Shevchenko av. 1,
Odessa, 65044, Ukraine. Department of Mathematics, Bielefeld University,
Universitatsstrasse 25, 33615 Bielefeld, Germany.
###### Abstract
We develop a new method for taking into account the interference contributions
to proton-proton inelastic cross-section within the framework of the simplest
multi-peripheral model based on the self-interacting scalar ${\phi^{3}}$ field
theory, using Laplace‘s method for calculation of each interference
contribution.
We do not know any works that adopted the interference contributions for
inelastic processes. This is due to the generally adopted assumption that the
main contribution to the integrals expressing the cross section makes multi-
Regge domains with its characteristic strong ordering of secondary particles
by rapidity. However, in this work, w e find what kind of space domains makes
a major contribution to the integral and these space domains are not multi-
Regge. We demonstrated that because these interference contributions are
significant, so they cannot be limited by a small part of them. With the help
of the approximate replacement the sum of a huge number of these contributions
by the integral were calculated partial cross sections for such numbers of
secondary particles for which direct calculation would be impossible.
The offered model qualitative agrees with experimental dependence of total
scattering cross-section on energy $\sqrt{s}$ with a characteristic minimum in
the range $\sqrt{s}\approx 10$ GeV. However, quantitative agreement was not
achieved; we assume that due to the fact that we have examined the simplest
diagrams of $\phi^{3}$ theory.
inelastic scattering cross-section, total scattering cross-section, Laplace
method, virtuality, multi-peripheral model, Regge theory
††preprint: AIP/123-QED
## I Introduction
This paper is the sequel to [Sharf and Rusov, 2006; Sharf, Rusov _et al._ ,
2007], where to calculate proton-proton scattering partial cross-sections
within the framework of multi-peripheral model the Laplace method was applied.
The inelastic scattering amplitude with production of a specified multiplicity
of secondary particles, in framework of the multi-peripheral model can be
represented as a sum of diagrams demonstrated on Fig.1.
Figure 1: Diagram representation of an inelastic scattering amplitude when
the $n$ secondary particles are formed. Here $P_{1}$ and $P_{2}$ are the four-
momenta of primary particles before scattering; $P_{3}$ and $P_{4}$ are the
four-momenta of primary particles after scattering;
${p_{{i_{1}}}},{p_{{i_{2}}}},\cdots,{p_{{i_{n}}}}$ are the four-momenta of
secondary particles. Symbol
$\sum\limits_{\hat{P}({i_{1}},\;{i_{2}},...,\,{i_{n}})}{}$ denote a sum over
all permutations of indices ${i_{1}}=1,{i_{2}}=2,...,{i_{n}}=n$.
To calculate the partial cross-section ${\sigma_{n}}$ is necessary to evaluate
an integral of the squared modulus of a sum of contributions shown in Fig.1.
After simple transformations [Sharf, Rusov _et al._ , 2007], the expression
for the partial cross-section can be represented as a sum of “cut” diagrams in
Fig.2. We call summands entering into the sum Fig.2 the interference
contributions. Approximate calculation of their sum is the purpose of this
paper.
At present time the inelastic scattering processes are considered without the
interference contributions [Kuraev, Lipatov, and Fadin, 1976; Bartels,
Lipatov, and Vera, 2009]. This due to the generally adopted assumption that
the main contribution to the integrals expressing an inelastic processes makes
multi-Regge domains [Kuraev, Lipatov, and Fadin, 1976; Bartels, Lipatov, and
Vera, 2009; Kozlov, Reznichenko, and Fadin, 2007; Danilov and Lipatov, 2006]
with its characteristic strong ordering of secondary particles by rapidity.
This means that the rapidity of neighboring particles on the “comb” should be
different from each other by a large value. Thus the amplitude of the right-
hand and left-hand parts of the diagram on Fig.2 for different orders of
connecting lines would be significantly different from zero to almost non-
overlapping regions of phase space and integral of their product would be a
small quantity.
However, as it was shown in [Sharf and Rusov, 2006] near the threshold of the
$n$ particles production at the maximum point of the scattering amplitude
Fig.1 difference between neighboring particle‘s of rapidities is close to zero
and at higher energies increases logarithmically with energy $\sqrt{s}$
growth. This difference has factor $1/(n+1)$, so for high numbers of secondary
particles it increases slowly with energy. Moreover, even if each of
interference terms is insignificant, all of them are positive and a huge
amount $n!$ of them not only makes it impossible to discard them, but also
leads to the conclusion that the contribution of a “ladder” diagram Fig.2,
which is usually only taken into account, is negligibly small compared with
the sum of the remaining interference terms. This was shown in [Sharf, Rusov
_et al._ , 2007]. For the relatively small number of secondary particles
($n\leq 8$) we are able to calculate all the interference contributions in the
direct way without any approximations.
Further in this paper we will demonstrate method for approximate calculation
of the sum of the interference contributions for large numbers of secondary
particles, when direct numerical calculation is not feasible.
## II Method description
Figure 2: Representation of the partial cross-section as a sum of “cut”
diagrams. The order of joining of lines with four-momenta $p_{k}$ from the
left-hand side of the cut is as following: the line with $p_{1}$ is joined to
the first vertex, the lines with $p_{2}$ is joined to the second vertex, etc.
The order of joining of lines from the right side of cut corresponds to one of
the $n!$ possible permutations of the set of numbers $1,2,\ldots,n$. Where
$\hat{P}_{j}(k),k=1,2,\ldots,n$ denote the number into which a number $k$ goes
due to permutation $\hat{P}_{j}$. An integration is performed over the four-
momenta $p_{k}$ for all “cut lines” taking into account the energy-momentum
conservation law and mass shell condition for each of $p_{k}$.
Using the Laplace‘s method we have found [Sharf and Rusov, 2006; Sharf, Rusov
_et al._ , 2007] the mechanism of partial cross-section growth, which was not
taken into account in the previously known variants of multi-peripheral model.
This mechanism may be responsible for the experimentally observed increase of
hadron-hadron total cross-section. However, in this approach based on the
Laplace‘s method, it was found out that the calculation of partial cross-
sections in the multi-peripheral model can be limited just to contributions
from the “cut ladder diagram”. Because for any number of the secondary
particles $n$ there is the wide range of energies $\sqrt{s}$, where such
contribution is negligibly small compared to the sum of $n!$ positive
interference contributions. At the same time, as we will demonstrated further,
the allowance for the interference contributions results in the appearance of
multipliers in expression for the partial cross-section, which are decrease
with the energy $\sqrt{s}$ rise (see below Eq.7). Thereupon the question
arises: “Will the sum of partial cross-sections increase with energy rise if
we take interference summands into account?”
As shown in [Sharf and Rusov, 2006], each term in sum shown in Fig.1 with
accuracy up to the fixed factor is a function with real and positive values,
which has a constrained maximum if its arguments satisfy the mass-shell
conditions and energy-momentum conservation law. Therefore, in the c.m.s. of
initial particles function corresponding to the left-hand part of cut diagram
in Fig.2 can be rewritten in the neighborhood of maximum point in the form
[Sharf and Rusov, 2006; Sharf, Rusov _et al._ , 2007]
$\displaystyle
A\left({\hat{X}}\right)=A\left({{{\hat{X}}^{\left(0\right)}}}\right)$
$\times\exp\left({-\frac{1}{2}{{\left({\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}^{T}}\hat{D}\left({\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}\right)$
(1)
where $\hat{X}$ is the column composed of $3n+2$ independent variables, on
which the scattering amplitude depends after consideration of mass-shell
conditions and energy-momentum conservation law; the first $n$ components of
column are the rapidities of secondary particles; the next $n$ components are
the $x$ components of transversal momenta of secondary particles (it is
supposed that the reference system is chosen so that $Z$-axis is directed in
the line of the three-dimensional momentum $P_{1}$ of initial particle in
Fig.1), the $y$ \- components of secondary particle transversal momenta and
the two last variables are the antisymmetric combinations of particle
transversal momenta $P_{3}$ and $P_{4}$, i.e.,
${X_{3n+1}}=\frac{1}{2}\left({{P_{3\bot x}}-{P_{4\bot x}}}\right)$
${X_{3n+2}}=\frac{1}{2}\left({{P_{3\bot y}}-{P_{4\bot y}}}\right)$ We denote
the column of the values of variables in a maximum point through
${\hat{X}^{\left(0\right)}}$ and a matrix with the elements
${D_{ab}}=-{\left.{\frac{{{\partial^{2}}}}{{\partial{X_{a}}\partial{X_{b}}}}\left({\ln\left({A\left({\hat{X}}\right)}\right)}\right)}\right|_{\hat{X}={{\hat{X}}^{\left(0\right)}}}}$
(2)
where
$a=1,2,\cdots,3n+2,b=1,2,\ldots,3n+2$ (3)
are the coefficients of the Taylor series expansion of amplitude logarithm in
the neighborhood of maximum point. As it was shown in [Sharf and Rusov, 2006],
if we do our computations in the c.m.s.of initial particles, the maximum is
reached when transversal momenta is zero and secondary particle rapidities are
close to numbers that formed an arithmetic progression.
If we denote the difference of this progression through $\Delta
y\left({n,\sqrt{s}}\right)$ and the value of particle‘s rapidity to which the
line attached to the $k$-th vertex of diagram in Fig.1 corresponds, through
$\Delta
y\left({n,\sqrt{s}}\right)=y_{k}^{\left(0\right)}-y_{k+1}^{\left(0\right)},\quad
k=1,2,\cdots,n-1$ (4)
we get [Sharf and Rusov, 2006]:
$y_{k}^{\left(0\right)}=\left({\frac{{n+1}}{2}-k}\right)\Delta
y\left({n,\sqrt{s}}\right),\quad k=1,2,\cdots,n$ (5)
The form of the function $\Delta y\left({n,\sqrt{s}}\right)$ has been
discussed in [Sharf and Rusov, 2006]. For further consideration, it is
important that it is a slowly increasing function on $s$ and decreasing
function on the number $n$ of the secondary particles and vanishes when $s$ is
equal to the threshold of $n$ particle production. Thus, the column
${\hat{X}^{\left(0\right)}}$ contains only the first nonzero $n$ rapidity
components, which are defined by Eq.5.
The following expression corresponds to the right-hand part of cut diagram in
Fig.2:
${{\hat{P}}_{j}}\left({A\left({\hat{X}}\right)}\right)=A\left({{{\hat{X}}^{\left(0\right)}}}\right)$
$\displaystyle\times\mbox{$\exp\left({-\frac{1}{2}{{\left({{{\hat{P}}_{j}}\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}^{T}}\hat{D}\left({{{\hat{P}}_{j}}\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}\right)$
}$ (6)
The interference contribution corresponding to whole “cut” diagram, which
correlates with the $j$-th summand in Fig.2, is proportional to an integral of
the product of functions Eq.1 and Eq.6 over all variables. Denoting an
interference summand corresponding to the permutation ${\hat{P}_{j}}$ through
${\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)$ and calculating its
Gaussian integral (at the same time, other multipliers besides the squared
modulus of scattering amplitude in an integrand are approximately replaced by
their values at the maximum point [Sharf, Rusov _et al._ , 2007]), we get
$\displaystyle{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)=\frac{{{{\left({A\left({{{\hat{X}}^{\left(0\right)}}}\right)}\right)}^{2}}v\left({\sqrt{s}}\right)}}{{\sqrt{\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}}$
$\displaystyle\times\exp\left({-\frac{1}{2}\left({{{\left({\Delta\hat{X}_{j}^{\left(0\right)}}\right)}^{T}}{{\hat{D}}^{\left(j\right)}}\Delta\hat{X}_{j}^{\left(0\right)}}\right)}\right)$
(7)
where we use the following notations:
$\displaystyle\Delta\hat{X}_{j}^{\left(0\right)}={\hat{X}^{0}}-{\hat{P}_{j}}^{-1}\left({{{\hat{X}}^{\left(0\right)}}}\right)$
(8a)
$\displaystyle{\hat{D}^{\left(j\right)}}={\left({{{\hat{D}}^{-1}}+{{\hat{P}}_{j}}^{T}{{\hat{D}}^{-1}}{{\hat{P}}_{j}}}\right)^{-1}}$
(8b)
$v\left({\sqrt{s}}\right)=\frac{1}{2}\frac{1}{{\sqrt{s}\sqrt{s/4-{M^{2}}}\left({E_{P}/2}\right)\sqrt{{{\left({E_{P}/2}\right)}^{2}}-{M^{2}}}}}$
(8c) $\displaystyle{E_{P}}=\sqrt{s}-\sum\limits_{k=1}^{n}{{\mathop{\rm
ch}\nolimits}\left({y_{k}^{\left(0\right)}}\right)}$ (8d)
$M$ is the mass of initial particle, which is made dimensionless by the mass
of secondary particle (it is supposed that the energy $\sqrt{s}$ is also made
dimensionless by the mass of the secondary particle).
Note, that here and in the following sections we will use the “prime” sign in
ours notation to indicate that we use a dimensionless quantity that
characterized the dependence of the cross-sections on energy, but not their
absolute values.
The value of amplitude at the maximum point
$A\left({{{\hat{X}}^{\left(0\right)}}}\right)$ increases with the $\sqrt{s}$
growth due to mechanism of virtuality reduction [Sharf and Rusov, 2006].
However, the distance $\Delta\hat{X}_{j}^{\left(0\right)}$ between maximum
points of “cut” diagram also increases with the $\sqrt{s}$ growth. Therefore,
the exponential factor entering in Eq.7 can decrease with energy growth. This
makes considered above question. How competition of these two multipliers will
result on the dependence of the sum of partial cross-sections on $\sqrt{s}$?
Thus, each interference contribution can be computed numerically. However due
to the huge number of contributions and large number of secondary particles
$n$ the direct numerical calculation of the sum of interference terms in Fig.2
is impossible.
Figure 3: The interference contributions dependence on ${z_{l}}$ at
$\sqrt{s}=1000$ GeV: (a) $n=8$, (b) $n=9$. Here and in subsequent figures the
interference contributions divided by the common multiplier
$\exp\left({-\sum\limits_{a=1}^{3n+2}{\sum\limits_{b=1}^{3n+2}{X_{a}^{\left(0\right)}{D_{ab}}X_{b}^{\left(0\right)}}}}\right)$
are indicated on the $Y$-axis. Obviously, that to the one value of $z_{l}$
correspond a lot of different contributions, as well as that the average
values of the logarithms of these contributions are placed approximately on
a straight line (see below Eq.26 and Fig.4).
We can avoid this difficulty in the following way. The maximum in the right
part of cut diagram in Fig.2 is attained at
$\hat{X}=\hat{P}_{j}^{-1}\left(\hat{X}^{(0)}\right)$. In other words, a
maximum of function, which is associated with the right-hand part of cut
diagram, can be obtained from a maximum of function, which maps with the left-
hand part of cut diagram, by the rearrangement of arguments. Then the value of
each interference contribution is determined by the distance between points of
maximum in the right-hand and left-hand part of cut diagram as well as by the
relative position of these maximum points, since in different directions
contributions to scattering amplitude fall off with distance from point of
maximum, in general, with different rate, and also by the relative position of
proper directions of the matrices $\hat{D}$ and
$\hat{P}_{j}^{T}\hat{D}\hat{P}_{j}$. In other words, multiplying Gaussian
functions corresponding to the right-hand and to the left-hand part of
interference diagrams in Fig.2 each time we will obtain as a result Gaussian
function, which has the proper value at the maximum point (which we call the
“height” of the maximum) and the proper multidimensional volume cutout by
resulting Gaussian function from an integration domain (which we call the
“width” of the maximum).
Figure 4: Comparing the values of
$\ln\left({\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle}\right)$
obtained by a direct numerical calculation with consideration of all
interference contributions (circles) and by approximation Eq.25 (straight
line) at $n=8$, $\sqrt{s}=10$ GeV (a); $n=9$, $\sqrt{s}=10$ GeV (b); $n=8$,
$\sqrt{s}=100$ GeV (c); $n=9$, $\sqrt{s}=100$ GeV (d).
Figure 5: The values of $\left\langle{w\left({{z_{l}}}\right)}\right\rangle$
obtained by direct calculation values Eq.25 for all interference contributions
for $n=8$ and $n=9$ at $\sqrt{s}=10$ GeV 5,5 accordingly; for the same number
$n$, but at $\sqrt{s}=100$ GeV 5,5 and the ratio
$\left\langle{w({z_{l}}}\right\rangle/\left\langle{w({z_{l}}_{0}}\right\rangle$
for $n=8$ and $n=9$ at $\sqrt{s}=10$ GeV 5,5; $\sqrt{s}=100$ GeV 5,5.
We assume that summands in Fig.2 are arranged in ascending order of the
distance between the maximum points in the right-hand part and left-hand part
of cut diagram (we denote this distance through $r$) so that ”cut” diagram
with the initial attachment of lines to the right-hand part of diagram
corresponds to $j=1$. In other words, the line of secondary particle with the
four-momentum $p_{i}$ is attached to the $i$-th top in the right-hand part of
cut diagram in Fig.2. As follows from Eq.7, the interference contributions
exponentially decrease with the $r^{2}$ growth. However, in spite of this the
interference contributions do not become negligible due to their huge number,
which, as discussed below, are increases very rapidly with $r^{2}$ growth. The
value of $r^{2}$ is proportional to the square of magnitude $\Delta
y(n,\sqrt{s})$, which, as was noted above, is zero on the threshold of $n$
particle production and slowly increases with distance from this threshold.
Therefore, for each number $n$ there is the fairly wide range of energies
close to the threshold, in which the sharpness of decrease of the interference
contributions with the $r^{2}$ increase is small in the sense that it is less
important factor than the increase in their number. At such energies, which we
call “low”, the partial cross-section $\sigma^{\prime}_{n}$ is determined by
the sum of huge number of small interference contributions. When the magnitude
$\Delta y(n,\sqrt{s})$ is increased with the further growth of energy
$\sqrt{s}$, the decrease rate of interference contributions increases, while
the growth rate of their number with the $r^{2}$ increase does not change with
energy. At such energies, which we call “high”, the main contribution to the
partial cross-section is made by the relatively small number of interference
terms corresponding to the small $r^{2}$, which can be calculated by Eq.7.
If we compose the $n$-dimensional vector (we denote it through
$\vec{y}^{(0)}$) from the particle rapidities Eq.5, which constrainedly
maximizes the function associated with the diagram with the initial
arrangement of momenta in Fig.2, vectors maximizing the functions with another
momentum arrangement will differ from the initial vector only by the
permutation of components, i.e., these vectors have the same length. Consider
two such $n$-dimensional vectors, one of which corresponds to the initial
arrangement, and another - to some permutation, then in the $n$-dimensional
space it is possible to “pull on” a two-dimensional plane on them (as a set of
their various linear combinations), where two-dimensional geometry takes
place. Therefore, the distance $r$ will be determined by cosine of an angle
between the considered equal on length $n$-dimensional rapidity vectors in the
two-dimensional plane, “pulled” on them. An angle corresponding to the
$\hat{P}_{j}$ permutation we designate through $\theta_{j}$,
$0\leq\theta_{j}\leq\pi$.
Thus, each of the terms in the sum Fig.2 can be uniquely matched to its angle
$\theta_{j}$. At the same time the variable $z=cos(\theta)$ is more handy for
consideration than an angle $\theta_{j}$. Using Eq.5, can be shown that the
variable $z$ can take discrete set of values:
$\displaystyle{z_{l}}=1-\frac{{12}}{{\left({n-1}\right)n\left({n+1}\right)}}l\quad$
(9) $\displaystyle
l=0,1,\cdots,\frac{{\left({n-1}\right)n\left({n+1}\right)}}{6}$ (10)
Note that although the relation Eq.5 for the rapidities of secondary particles
is satisfied with high accuracy at the maximum point, it is still approximate.
This means that those contributions, to which matched one and the same value
of variable $z$ in Eq.5, in fact, matched a slightly different from each other
values of $z$.
Consequently, to such contributions correspond a similar but unequal to each
other distances between maximum points in a “cut” diagram. In addition, this
distance, as was discussed above, is not a unique factor affecting to the
value of interference contribution. Therefore, if each interference
contribution is associated with the value of variable $z$ by the approximation
Eq.5, it appears, that the different values of interference contributions
correspond to the one and the same value of $z_{l}$ (see Fig.3).
Thus, while each contribution is associated to some value of variable $z$ in
the approximation Eq.5, the value of contribution is not the unique function
of $z$. However, the sum expressing the partial cross-section
$\sigma^{\prime}_{n}$ can be written in the following way
$\displaystyle{\sigma^{\prime}_{n}}=\sum\limits_{l=0}^{\frac{{\left({n-1}\right)n\left({n+1}\right)}}{6}}{\Delta{N_{l}}\left({\frac{{\sum\limits_{{z_{j}}={z_{l}}}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}}}{{\Delta{N_{l}}}}}\right)}$
(11)
where $\Delta{N_{l}}$ the number of summands to which the value
${z_{j}}={z_{l}}$ is corresponds in the approximation Eq.5. The average value
of all interference contributions in Eq.11 is already the unique function of
$z_{l}$. Therefore, we introduce notation
$\displaystyle\frac{{\sum\limits_{{z_{j}}={z_{l}}}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}}}{{\Delta{N_{l}}}}=\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle$
(12)
where $\left\langle\sigma^{\prime}_{n}(z_{l})\right\rangle$ is some function,
whose form at “low” energies can be determined from the following
considerations.
For any multiplicity $n$ when the values of parameter $l$ in Eq.10 are small
and when number of corresponding interference contributions is relatively
small, we can directly calculate these elements and their sum. Denote the
maximum value $l$, for which all interference contributions are calculated
through $l_{0}$. In particular, in this paper we managed to calculate the
interference contributions up to $l_{0}=6$. Partial cross-section can be
written as
$\displaystyle{\sigma^{\prime}_{n}}=\sigma_{n}^{\prime(h)}+\sigma_{n}^{\prime(l)}=$
$=\sum\limits_{\scriptstyle{z_{j}}={z_{l}},\hfill\atop\scriptstyle
l=0,1,\cdots{l_{0}}\hfill}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}+\sum\limits_{l={l_{0}}+1}^{\frac{{\left({n-1}\right)n\left({n+1}\right)}}{6}}{\Delta{N_{l}}\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle}$
(13)
where $\sigma_{n}^{\prime(h)}$ is the sum of contributions sufficient at
“high” energies, and $\sigma_{n}^{\prime(l)}$ is the sum of contributions
sufficient at “low” energies. Thus, the difficulties in the calculations of
the huge number of interference contributions mainly relates to the range of
“low” energies and can be reduced to the approximate calculation of
$\left\langle\sigma^{\prime}_{n}(z_{l})\right\rangle$ and $\Delta N_{l}$.
## III The approximate calculation of
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle$.
As follows from Eq.7, the exponential factor exerts the most significant
effect on the dependence of
$\left\langle\sigma^{\prime}_{n}(z_{l})\right\rangle$ on $z_{l}$. Note that
the expression
$\left(\Delta\hat{X}_{j}^{(0)}\right)^{T}\hat{D}^{(j)}\Delta\hat{X}_{j}^{(0)}$
entering into the exponent in Eq.7 depends only on those matrix
$\hat{D}^{(j)}$ components, which are at the intersection of the first $n$
rows and first $n$ columns, since all column $\Delta\hat{X}_{j}^{(0)}$
components starting with $n+1$ are zero, because they are the particle
momentum transverse components at the maximum point. If we denote the matrix
composed of elements located at the intersection of the first $n$ rows and
first $n$ columns of the matrix $\hat{D}^{(j)}$ through $\hat{D}_{y}^{(j)}$
and a matrix, which is obtained from the matrix $\hat{D}$ in analogy, through
$\hat{D}_{y}$, we have
$\displaystyle{\hat{D}^{\left(j\right)}}:\hat{D}_{y}^{\left(j\right)}={\left({\hat{D}_{y}^{-1}+{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}}\right)^{-1}}$
(14)
The matrices $\hat{D}_{y}^{-1}$ and
${\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}$ have one and the same
eigenvalues, but they correspond to different eigenvectors. We denote the
normalized to unit eigenvector corresponding to the minimal eigenvalue of
matrix $\hat{D}_{y}^{-1}+{\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}$
through ${\hat{u}_{\min}}$ and the eigenvalue itself - through
${\lambda_{\min}}$. This implies
${\lambda_{\min}}=\hat{u}_{\min}^{T}\hat{D}_{y}^{-1}{{\hat{u}}_{\min}}+\hat{u}_{\min}^{T}{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}{{\hat{u}}_{\min}}$
(15)
Since the minimum eigenvalue of matrix $\hat{D}_{y}^{-1}$ is equal to the
minimum values of quadratic form ${\hat{u}^{T}}\hat{D}_{y}^{-1}\hat{u}$ for
the unit vectors $\hat{u}$, the magnitude
$\hat{u}_{\min}^{T}\hat{D}_{y}^{-1}{\hat{u}_{\min}}$ is not less than the
minimum eigenvalue of matrix $\hat{D}_{y}^{-1}$. By analogy the magnitude
$\hat{u}_{\min}^{T}{\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}{\hat{u}_{\min}}$
is not less than the minimum eigenvalue of matrix
${\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}$, which coincides with the
minimal eigenvalue of matrix $\hat{D}_{y}^{-1}$ and is reciprocal of the
maximum eigenvalue of matrix ${\hat{D}_{y}}$ denoted through $d_{y}^{\max}$.
Thus, ${\lambda_{\min}}\geq\frac{2}{{d_{y}^{\max}}}$. From this it follows
that, the maximum eigenvalue of matrix
${\left({\hat{D}_{y}^{-1}+{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}}\right)^{-1}}$
does not exceed $d_{y}^{\max}/2$. By analogy we obtain that the minimum
eigenvalue of matrix
${\left({\hat{D}_{y}^{-1}+{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}}\right)^{-1}}$
is no smaller than $d_{y}^{\min}/2$, where $d_{y}^{\min}$ is the minimum
eigenvalue of matrix ${\hat{D}_{y}}$. Thus, an interval enclosing the
eigenvalues of matrix $\hat{D}_{y}^{\left(j\right)}$ is, at least, twice
smaller than an interval enclosing the eigenvalues of matrix ${\hat{D}_{y}}$.
We can demonstrate that at approximation of an equal denominators [Sharf and
Rusov, 2006] the value of $d_{y}^{\max}$ can be estimated in the following way
$\displaystyle d_{y}^{\max}\approx\frac{2}{{4{{{\mathop{\rm
sh}\nolimits}}^{2}}\left({\frac{{\Delta y\left({n,s}\right)}}{2}}\right)+1}}$
(16)
i.e., an interval enclosing the eigenvalues of matrix
$\hat{D}_{y}^{\left(j\right)}$ at any energies and number of particles is less
than unity, whereas at the considerable values of $\Delta
y\left({n,s}\right)$, i.e. at a distance from the threshold, this interval is
much less than unity.
Therefore, if we reduce matrix $\hat{D}_{y}^{\left(j\right)}$ to diagonal
form, it will be close to a matrix multiple of unit matrix. If we represent
this matrix in the form
$\displaystyle\hat{D}_{y}^{\left(j\right)}=\frac{1}{n}Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)\hat{E}+\Delta\hat{D}_{y}^{\left(j\right)}$
(17)
where $\hat{E}$ is unit matrix, the eigenvalues of the traceless matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$ will be small. Then
$\frac{1}{2}{\left({\Delta\hat{X}_{j}^{\left(0\right)}}\right)^{T}}{{\hat{D}}^{\left(j\right)}}\Delta\hat{X}_{j}^{\left(0\right)}=\frac{1}{n}Sp\left({D_{y}^{\left(j\right)}}\right){\left|{{{\vec{y}}^{\left(0\right)}}}\right|^{2}}\left({1-\cos\left({{\theta_{j}}}\right)}\right)+\frac{1}{2}\sum\limits_{k=1}^{n}{\Delta
d_{y,k}^{\left(j\right)}}{\left({{V_{kn}}\left({y_{n}^{\left(0\right)}-{{\hat{P}}_{j}}^{-1}\left({y_{n}^{\left(0\right)}}\right)}\right)}\right)^{2}}$
(18)
where $\Delta d_{y,k}^{\left(j\right)}$ are the eigenvalues of matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$, ${V_{kn}}$ is the transformation matrix
to the basis composed from the eigenvectors of matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$ (the summation over reheated indices is
supposed). The second term in this sum is small in comparison with the first
one due to the smallness of eigenvalues $\Delta d_{y,k}^{\left(j\right)}$ as
well as due to their different signs (since the trace of matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$ is zero, the different terms over $k$
partially compensate each other). Therefore, we can adopt the following
approximation:
$\frac{1}{2}{\left({\Delta\hat{X}_{j}^{\left(0\right)}}\right)^{T}}{\hat{D}^{\left(j\right)}}\Delta\hat{X}_{j}^{\left(0\right)}\approx$
$\approx\frac{1}{n}Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)\times{\left|{{{\vec{y}}^{\left(0\right)}}}\right|^{2}}\left({1-\cos\left({{\theta_{j}}}\right)}\right)$
(19)
To approximately calculate the trace of matrix $\hat{D}_{y}^{\left(j\right)}$
we select the spherically symmetric part of matrix ${\hat{D}_{y}}$
representing it in the form
$\displaystyle{\hat{D}_{y}}=\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)\hat{E}+\Delta{\hat{D}_{y}}$
(20)
The results of numeral calculation of the eigenvalues of matrix
${\hat{D}_{y}}$ (which are denoted through
$d_{k}^{\left(y\right)},k=1,2,\cdots,n$) are shown in Table.1. It is obvious
that most eigenvalues are close between themselves with the exception of a few
eigenvalues, which are substantially smaller. Therefore, these smallest
eigenvalues have the highest absolute value of deviations from mean eigenvalue
$\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)$. Since all the eigenvalues of
matrix ${\hat{D}_{y}}$ are positive, the deviation of eigenvalues from average
value is less than this average in absolute value (see Table.1). Note that the
matrix $\hat{D}_{y}^{\left(j\right)}$ can be represented in the following
form:
$\hat{D}_{y}^{\left(j\right)}=\frac{1}{{2n}}Sp\left({{{\hat{D}}_{y}}}\right)\left({\hat{E}+\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}}\right){\left({\hat{E}+\frac{{\Delta{{\hat{D}}_{y}}+{{\hat{P}}_{j}}^{T}\Delta{{\hat{D}}_{y}}{{\hat{P}}_{j}}}}{{\frac{2}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}}\right)^{-1}}\left({\hat{E}+\frac{{{{\hat{P}}_{j}}^{T}\Delta{{\hat{D}}_{y}}{{\hat{P}}_{j}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}}\right)$
(21)
By analogy we can conclude that the minimum eigenvalue of matrix
$\displaystyle\Delta{\hat{D}_{y}}+{\hat{P}_{j}}^{T}\Delta{\hat{D}_{y}}{\hat{P}_{j}}$
(22)
(which is maximum in absolute value, see Table.1) is greater than the doubled
minimum eigenvalue of matrix $\Delta{\hat{D}_{y}}$. This means that all the
eigenvalues of matrix
$\frac{{\Delta{{\hat{D}}_{y}}+{{\hat{P}}_{j}}^{T}\Delta{{\hat{D}}_{y}}{{\hat{P}}_{j}}}}{{\frac{2}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
are less than unity in absolute value. It applies equally to the eigenvalues
of matrices
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
and
${\hat{P}_{j}}^{T}\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}{\hat{P}_{j}}$.
Therefore, we can represent the matrix $\hat{D}_{y}^{\left(j\right)}$ as the
expansion in powers of
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$.
Since matrix
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
is traceless by definition, then a nonzero contribution to
$Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)$ in addition to the term of
“zero” order $\frac{1}{{2n}}Sp\left({{{\hat{D}}_{y}}}\right)\hat{E}$ can give
terms starting with the second-order. As it follows from Table.1, the maximum
in absolute value eigenvalue of matrix
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
increases with the energy growth. Therefore, we can expect that at “low”
energies higher-order terms will make negligibly small contributions. In such
an approximation we have:
$\displaystyle
Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)\approx\frac{1}{2}Sp\left({{{\hat{D}}_{y}}}\right)$
(23)
Let Eq.7 is taken in place of Eq.12 in approximation Eq.23, then we have
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle={\left({A\left({{{\hat{X}}^{\left(0\right)}}}\right)}\right)^{2}}v\left({\sqrt{s}}\right)$
$\times\exp\left({-\frac{{{{\left|{{{\vec{y}}^{\left(0\right)}}}\right|}^{2}}Sp\left({{{\hat{D}}_{y}}}\right)}}{{2n}}\left({1-{z_{l}}}\right)}\right)$
$\times\frac{1}{{\Delta{N_{l}}}}\sum\limits_{{z_{j}}={z_{l}}}{\frac{1}{{\sqrt{\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}}}$
(24)
Let us introduce the following notation
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle=\frac{1}{{\Delta{N_{l}}}}\sum\limits_{{z_{j}}={z_{l}}}{\frac{1}{{\sqrt{\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}}}$
(25)
If we assume that multiplier
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ is weakly dependent on
${z_{l}}$, we obtain
$\displaystyle\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle=\left\langle{{\sigma^{\prime}_{n}}\left({{z_{{l_{0}}}}}\right)}\right\rangle\exp\left({\frac{{{{\left|{{{\vec{y}}^{\left(0\right)}}}\right|}^{2}}Sp\left({{{\hat{D}}_{y}}}\right)}}{{2n}}\left({{z_{l}}-{z_{{l_{0}}}}}\right)}\right)$
(26)
where ${z_{l}}$ is the minimum value of ${z_{l}}$ for which can be numerically
calculated all interference contributions. Therefore, the magnitude
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{{l_{0}}}}}\right)}\right\rangle$
can be directly calculated numerically. The results of numerical calculation
of
$\ln\left({\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle}\right)$
over all interference contributions in comparison with the results obtained by
Eq.25 are demonstrated on Fig.4, it follows that such an approximation is
acceptable at “low” energies.
Results shown in Fig.4 confirm also our assumption that
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ weakly depends on
${z_{l}}$. To analyze this dependence we turn to Fig.5. It is obvious, that
the magnitude $\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ takes small
values at “low” energies. This means that
$\displaystyle\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)$
(27)
takes large values at the same energies. Indeed, as it follows from the
expression for the matrix $\hat{D}$, Eq.27 tends to infinity on the threshold
of $n$ particle production, and this means that at threshold the phase space
of physical area of the inelastic process with $n$ particles production takes
place is equal to zero.
Because of symmetry with respect to direction inversion in a plane of
transversal momenta the mixed second derivatives with respect to rapidities
and transversal momentum components are zeros. As a consequence, the
determinant Eq.27 is equal to the product of the three determinants, first of
which is composed from second derivatives with respect to rapidities, the
second is composed from the second derivatives with respect to the transversal
momentum $x$-components and the third one is composed from derivatives with
respect to the transversal momentum $y$-components. All the three factors tend
to infinity at the threshold energy. As it follows from a numerical
calculation, a matrix determinant composed from the second derivatives with
respect to rapidities reduced quite rapidly with energy growth. Matrix
determinants composed from the second derivatives with respect to transversal
momentum components also reduced, but in a wide energy range, they remain
quite large. Therefore, the value of Eq.27 is great at all $j$. Since the
function
${1\mathord{\left/{\vphantom{1{\sqrt{x}}}}\right.\kern-1.2pt}{\sqrt{x}}}$
varies slightly at the great values of argument, the function
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ weakly depends on
${z_{l}}$.
To estimate roughly the function
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ we can replace it by the
Taylor expansion taking into account just linear contributions. The expansion
coefficients are found by the calculating of
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ for ${z_{l}}$ close to
$1$ and $(-1)$. In these cases the values of
$\displaystyle{1/\sqrt{\det\left({1/2\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}$
(28)
were obtained directly for all proper interference contributions, and after
that we obtain the values of
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ by averaging using Eq.25.
Figure 6: A sphere ${S_{2}}$ and figure ${F_{4!}}$ (is shown by points). Basis
in the four-dimensional space is chosen so that the one of vectors coincides
with the vector
${\vec{e}_{4}}=\left({\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}}\right)$,
and the three basis vectors of three-dimensional subspace, into which depicted
sphere is embedded, are perpendicular to ${\vec{e}_{4}}$.
Figure 7: The partition of sphere ${S_{2}}$ by shortest arcs joining the
points of figure ${F_{4!}}$ into the two “hexagonal” and one “tetragonal”
regions 7; (b) areas, which is located on the borders of 4 or 6 points
belonging to figure ${F_{4!}}$ can be divided between those points into
figures of equal area; (c) whole sphere $S_{2}$ is divided into figures of
equal area, each of which contains the one point of figure ${F_{4!}}$ one of
these shapes are painted in white.
The values in Fig.5 have been obtained by the direct calculation of
$\frac{1}{{\Delta{N_{l}}}}\sum\limits_{{z_{j}}={z_{l}}}{1/\sqrt{\det\left({1/2\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}$
(29)
with consideration of all interference contributions at different $\sqrt{s}$.
So, we have the following expression instead of Eq.26
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle=\left\langle{{\sigma^{\prime}_{n}}\left({{z_{{l_{0}}}}}\right)}\right\rangle\left({{w_{0}}+{w_{1}}\left({1-{z_{l}}}\right)}\right)\times\exp\left({\frac{{{{\left|{{{\vec{y}}^{\left(0\right)}}}\right|}^{2}}Sp\left({{{\hat{D}}_{y}}}\right)}}{{2n}}\left({{z_{l}}-{z_{{l_{0}}}}}\right)}\right)$
(30)
where the coefficients ${w_{0}}$ and ${w_{1}}$ are found by above mentioned
method.
## IV Approximate calculation of the $\Delta{N_{l}}$ values
Let us turn to the new variables
$\displaystyle Y_{k}^{\left(0\right)}=\frac{{y_{k}^{\left(0\right)}}}{{\Delta
y\left({n,\sqrt{s}}\right)\sqrt{\frac{{\left({n+1}\right)n\left({n-1}\right)}}{{12}}}}}$
(31)
where $y_{k}^{\left(0\right)}$ are determined by Eq.5,
$Y_{k}^{\left(0\right)},k=1,2,\cdots,n$ are considered as the components of
vector ${\vec{Y}^{\left(0\right)}}$, which, as it follows from Eq.31 is of
unit length.
Thus, the angle ${\theta_{j}}$ between the vector
${\vec{y}^{\left(0\right)}}=\left({y_{1}^{\left(0\right)},y_{2}^{\left(0\right)},\cdots,y_{n}^{\left(0\right)}}\right)$
and vector $\hat{P}_{j}^{-1}\left({{{\vec{y}}^{\left(0\right)}}}\right)$
obtained by the permutation of corresponding components is the same as the
angle between the vector
${\vec{Y}^{\left(0\right)}}=\left({Y_{1}^{\left(0\right)},Y_{2}^{\left(0\right)},\cdots,Y_{n}^{\left(0\right)}}\right)$
and vector $\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$.
Moreover, as it follows from Eq.5
$\displaystyle
y_{1}^{\left(0\right)}=-y_{n}^{\left(0\right)},y_{2}^{\left(0\right)}=-y_{n-1}^{\left(0\right)},\cdots,y_{k}^{\left(0\right)}=-y_{n-k+1}^{\left(0\right)};$
$\displaystyle k=1,2,\cdots,n$ (32)
It follows that all vectors $\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$ are
orthogonal to vector
$\displaystyle\vec{e}_{n}=\left(\underbrace{1/\sqrt{n},1/\sqrt{n},\ldots,1/\sqrt{n}}_{n\quad\scriptsize{components}}\right)$
(33)
Figure 8: Diagrams, which correspond $(n-1)$ vectors
$\hat{P}_{l}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ closest to
vector ${\vec{Y}^{\left(0\right)}}$.
Figure 9: Comparison of the interference contribution distribution by the
variable $z=\cos\left(\Theta\right)$ (histogram) and the plot of function
$\rho\left(z\right)=\frac{{dN\left({z,z+dz}\right)}}{{dz}}$ (solid line) at
$n=8$, $\Delta z=0.1$ (a); $n=9$, $\Delta z=0.1$ (b); $n=9$, ${\Delta z}=0.05$
(c). Here $\Delta N$ is the number of interference contributions corresponding
to value of $z$ in the proper interval of $\Delta z$ width.
Therefore, considering vectors $\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$ as
the elements of $n$-dimensional euclidean space, which we denote through
$E_{n}$, then the ends of all vectors
$\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$ are lie on the unit sphere
embedded into the $(n-1)$-dimensional subspace of $E_{n}$. We denote this
sphere through $S_{n-2}$ and shape formed by the set of points in which the
ends of vectors $\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$
($j=1,2,\ldots,n!$) come, denote through $F_{n!}$. In particular, when $n=4$
the sphere $S_{2}$ and figure $F_{4!}$ graphically look like in Fig.6.
Figure 10: Comparison of the values of right-hand side and left-hand side of
approximate equality Eq.42 at $n=8$ (a, b) and $n=9$ (c, d). Circles are the
values of $\Delta{N_{l}}$ calculated with consideration of for all
interference contributions; crosses are the values of function
$\rho\left({{z_{l-1}}}\right)\Delta z$ from Eq.40.
We examine some geometrical properties of figure ${F_{n!}}$ at arbitrary $n$.
If we apply the permutation transformation component to all vectors in the
$n$-dimensional space, where the vectors ${\vec{Y}^{\left(0\right)}}$ are
primordially defined, the examined $(n-1)$-dimensional subspace as well as a
sphere ${S_{n-2}}$ and figure ${F_{n!}}$ go into themselves. As it follows
from the group properties of permutation group, the each point of figure
${F_{n!}}$ can be obtained from any other point by some transformation
$\hat{P}_{j}^{-1}$. This means that the configuration of the points of figure
${F_{n!}}$ relative to each of these points must be identical, that can be
clearly seen in Fig.7.
As it follows from Eq.32, besides the end of each vector
$\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ a figure
${F_{n!}}$ contains also the end of vector
$\left({-\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)$,
i.e., a figure ${F_{n!}}$ has a center of symmetry, which coincides with the
center of sphere ${S_{n-2}}$. In this case, if we using point of ${F_{n!}}$
form path from the point
$\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ to the point
$\left({-\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)$,
then it will be simultaneously formed a centro-symmetrical path, that leads
from
$\left({-\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)$
to $\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ of figure
${F_{n!}}$.
Joining these paths we will obtain the closed path, which “girdles” the sphere
${S_{n-2}}$. If we assume that there is such a “girdling” path, inside of
which are concentrated all points of figure ${F_{n!}}$, we would find that the
figure ${F_{n!}}$ has a “boundary” and “internal” points, that would
contradict the fact that spacing of all points relative to each point of the
${F_{n!}}$ should be the same. In other words, the points of figure ${F_{n!}}$
must “crawl away” all over the sphere ${S_{n-2}}$ and can not be concentrated
on some area of the sphere.
If we consider a vector ${\vec{Y}^{\left(0\right)}}$, then closest to it are
the vectors corresponding to permutations $\hat{P}_{l}^{-1},l=1,2,\cdots,n-1$
defined by the following relation
${\left({\hat{P}_{l}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)_{k}}=\left\\{\begin{array}[]{l}Y_{k}^{\left(0\right)},{\rm{if}}\quad
k<l,\\\ Y_{l+1}^{\left(0\right)},{\rm{if}}\quad k=l,\\\
Y_{l}^{\left(0\right)},{\rm{if}}\quad k=l+1,\\\
Y_{k}^{\left(0\right)},{\rm{if}}\quad k>l+1.\\\ \end{array}\right.$ (38)
The type of “cut” diagrams corresponding to such permutations is shown on
Fig.8.
At the same time, all the components of vector
$\hat{P}_{l}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)-{\vec{Y}^{\left(0\right)}}$
, except the $l$-th and $l+1$, are zero, whereas these two components take on
the least values in modulus
$\sqrt{\frac{{12}}{{\left({n+1}\right)n\left({n-1}\right)}}}$ and
$\left({-\sqrt{\frac{{12}}{{\left({n+1}\right)n\left({n-1}\right)}}}}\right)$,
respectively.
Thus, we can conclude that the each point of figure ${F_{n!}}$ has $(n-1)$
nearest neighboring points, which lying at distance of from it:
$\displaystyle{r_{n}}=\sqrt{\frac{{24}}{{\left({n+1}\right)n\left({n-1}\right)}}}$
(39)
Connecting the each point of figure ${F_{n!}}$ with its (n-1) nearest
neighbors points by shortest arc thereby we divide the sphere ${S_{n-2}}$ into
closed regions as is shown in Fig.7. Indeed, let us choose the some point
${A_{0}}$ of figure ${F_{n!}}$, and will move from it to the nearest point
$A_{1}$ along a shortest arc, then we move from the point $A_{1}$ to the
nearest point $A_{2}$ etc. At the same time, motion in a backward direction is
prohibited. Thus, there are $(n-1)$ paths going out from each point, and
$(n-2)$ paths are allowed at each step. But since figure ${F_{n!}}$ has the
finite number of points at some step we will surely come back to the point
${A_{0}}$.
Moreover, since shortest arcs joining two nearest points are subtended by
equal chords ${r_{n}}$ in length (see Eq.39), this arcs are of the same
length. Let us consider any two neighboring points ${A_{i}}$ and ${A_{i+1}}$
of figure ${F_{n!}}$. Under any transformation $\hat{P}_{j}^{-1}$ the shortest
arc, which joins the points ${A_{i}}$ and ${A_{i+1}}$, and an arc joining the
points $\hat{P}_{j}^{-1}\left({{A_{i}}}\right)$ and
$\hat{P}_{j}^{-1}\left({{A_{i+1}}}\right)$ are of the same length. This means
that the boundaries of closed regions formed by shortest arcs, which join
neighboring points, replaced into one another under any transformation
$\hat{P}_{j}^{-1}$. It follows that, if we examine closed areas which include
any point of figure ${F_{n!}}$, then the adjacent areas to all points of this
figure will have the same “area”.
Figure 11: Comparison of the values of
$\sum\limits_{{z_{j}}={z_{l}}}{{\sigma_{n}}\left({{{\hat{P}}_{j}}}\right)}$
obtained with consideration of all interference contributions (circles) and
the approximate values of
$\left\langle{{\sigma_{n}}\left({{z_{l}}}\right)}\right\rangle\rho\left({{z_{l-1}}}\right)\Delta
z$ (crosses) for 11 \- for $n=8$ at $\sqrt{s}=10$ GeV, 11 \- for $n=8$ at
$\sqrt{s}=100$ GeV, 11 \- for $n=9$ at $\sqrt{s}=10$ GeV, 11 \- for $n=9$ at
$\sqrt{s}=100$ GeV.
Figure 12: The partial cross-section dependence on energy $\sqrt{s}$
calculated over all interference contributions (solid line) and by Eq.13 with
the application of approximations Eqs.30, 40, 42 (dashed line): 12 \-
$\sigma^{\prime}_{8}(\sqrt{s})$; 12 \- $\sigma^{\prime}_{9}(\sqrt{s})$; 12 \-
$\sigma^{\prime}_{10}(\sqrt{s})$; 12 \- $\sigma^{\prime}_{11}(\sqrt{s})$; 12
\- $\sigma^{\prime}_{11}(\sqrt{s})$. This approximation is acceptable at least
in the range of parameters in which they are can be verified.
Figure 13: Theoretical dependences of the $\sigma^{\prime I}(\sqrt{s})$ 13
and $\sigma^{\prime\Sigma}(\sqrt{s})$ 13 obtained for the energy range
$\sqrt{s}=1\div 100$ Gev at $L=5.51$. First minimum for the total cross-
section can be obtained only when we take into account contributions from the
high multiplicities. Experimental data for the inelastic 13 and for the total
13 pp scattering cross-section [Nakamura and Group, 2010; Aad _et al._ ,
2011] presented for qualitative comparison with the prediction from our model.
Note: data-points for the inelastic cross-section, obtained from the
definition $\sigma_{inel}=\sigma_{total}-\sigma_{elastic}$.
There is one more requirement, to which the areas obtained by partition of the
sphere ${S_{n-2}}$ must satisfy: they must not overlap, i.e., these regions do
not have common internal points. Indeed, otherwise, at least any two of the
examined arcs would intersect in some internal point of these arcs. As it
follows from Eq.39, when $n$ is large the value of ${r_{n}}$ is small. This
means that when we join the each point of figure ${F_{n!}}$ with its nearest
neighbors by the shortest arcs of sphere ${S_{n-2}}$, these arcs practically
coincide with chords, which tights them.
If we assume, that any two chords ${A_{{i_{1}}}}{A_{{i_{1}}+1}}$ and
${A_{{i_{2}}}}{A_{{i_{2}}+1}}$ intersect in an internal point, then it is
possible “to pull” on them a two-dimensional plane. Then we get a flat
rectangle ${A_{{i_{1}}}}{A_{{i_{2}}}}{A_{{i_{1}}+1}}{A_{{i_{2}}+1}}$, which
has at least one angle no smaller than $90^{\circ}$. This means that square of
diagonal lying opposite it is not less than sum of squares of the parties that
make up the corner. Denoting the lengths of these sides through $a$ and $b$,
we have ${a^{2}}+{b^{2}}\leq r_{n}^{2}$. In this case, either $a$ or $b$ would
not exceed ${r_{n}}/\sqrt{2}$, i.e., the figure ${F_{n!}}$ contains points,
which are at distance less then ${r_{n}}$ but that cannot happen due to
minimality of this distance.
Thus, we can conclude that at an arbitrary $n$ a sphere ${S_{n-2}}$ can be
divided into the parts of equal area, each of which contains only one point of
figure ${F_{n!}}$, as it shown in Fig.7-7.
Let us introduce a multidimensional spherical coordinate system so that the
end of vector ${\vec{Y}^{\left(0\right)}}$ is the “north pole” of sphere
${S_{n-2}}$. Then the number of points of figure ${F_{n!}}$, to which the
values of variable $z=\cos\left(\Theta\right)$ in the interval $[z,z+dz]$
correspond, is equal
$\displaystyle dN\left({z,dz}\right)=\rho\left(z\right)dz$ (40)
where
$\displaystyle\rho\left(z\right)=\frac{{n!}}{{\sqrt{\pi}}}\frac{{\Gamma\left({\frac{{n-1}}{2}}\right)}}{{\Gamma\left({\frac{{n-2}}{2}}\right)}}{\left({1-{z^{2}}}\right)^{\frac{{n-4}}{2}}}$
(41)
$\Gamma$ is the Euler gamma function.
To verify the validity of Eq.40 we can calculate all interference
contributions and corresponding values of $z$ at $n=8$ and $n=9$ (since for
the larger number of particles this can not be realized).
The distributions of interference contribution from the variable
$z=\cos\left(\Theta\right)$ and the graphs of function
$\rho\left(z\right)=\frac{{dN\left({z,z+dz}\right)}}{{dz}}$ from Eq.40 are
shown in Fig.9. Obtained results of numerical calculation of interference
contributions and by Eq.40 are in a good agreement.
Moreover, as it follows from Fig.9 and from Fig.9 this fitness is improved
with increasing number of particles $n$, i.e., Eq.40 is suitable for large
$n$, when the direct numerical calculation of all interference contributions
is impossible.
Taking Eq,40 and Eq.10 into account we obtain the following the approximate
equality
$\displaystyle\Delta{N_{l}}\approx\rho\left({{z_{l-1}}}\right)\Delta z$ (42)
where
$\displaystyle\Delta z=\frac{{12}}{{\left({n-1}\right)n\left({n+1}\right)}}$
(43)
Verification results of Eq.42 at $n=8$ and $n=9$ are presented in Fig.10.
Another verification of considered above equations is presented in Fig.11,
where the values of
$\sum\limits_{{z_{j}}={z_{l}}}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}$
and approximating magnitudes
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle\rho\left({{z_{l-1}}}\right)\Delta
z$ (here
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle$ is
calculated by Eq.30) are compared.
From results demonstrated on Fig.4 and Fig.10-12, we can conclude that the at
least for those numbers of particles for which it can be directly tested Eq.13
with Eq.30, Eqs.40 \- 42 yields an acceptable approximation. As is obvious
from Fig.4, than closer energy to the threshold of $n$ particle production,
the better approximation Eq.30. Therefore, if we choose the range of low
energies, for example, up to 100 GeV, because in this range total cross-
section growth is observed, it is expected that the considered approximations
will be acceptable for the large numbers of particles than those for which
they were tested. In addition, as it follows from Fig.10-10, the accuracy of
approximation Eq.42, as expected, increases with the growth of $n$.
Thus, within the framework of examined approximations is possible to calculate
the interference contributions at sufficiently large $n$, and we can consider
the dependence of total inelastic cross-section on energy $\sqrt{s}$ in the
simplest case of multi-peripheral model taking into account all significant
interference contributions.
## V The model of dependence of hadron inelastic scattering total cross-
section on energy $\sqrt{s}$
Let us consider the magnitude
$\displaystyle{\sigma^{\prime\Sigma}}\left({\sqrt{s}}\right)=\sum\limits_{n=1}^{{n_{\max}}}{{L^{n}}{\sigma^{\prime}_{n}}\left({\sqrt{s}}\right)}$
(44)
which within the framework of the discussed above model is an analogue of
total inelastic scattering cross-section. Here ${n_{\max}}$ is the maximum
number of secondary particles allowed by energy-momentum conservation law and
$L$ is the dimensionless coupling constant, which we considered as a fitting
parameter (see Eq.32 [Sharf, Rusov _et al._ , 2007]). Since the calculation
of ${\sigma^{\prime}_{n}}$ up to $n={n_{\max}}$ takes a long time, so in
practice we restrict the upper bound of summation by those values of $n$,
beyond which the neglected contributions known to be smaller than the
experimental error of cross-section measurements.
The constant $L$ can be fitted so that the dependence
${\sigma^{\prime}_{\Sigma}}\left({\sqrt{s}}\right)$ looks like the behavior of
total hadron-hadron scattering cross-section with a minimum about
$\sqrt{s}=10$ GeV. The result of such a fitting is shown in Fig.13 (in that
calculations we take proton mass as mass of primary particles and pion mass as
mass of secondary particles).
Quantitative comparison with experimental data requires the consideration of
more realistic model than the self-interacting scalar ${\phi^{3}}$ field
model.
## VI Conclusions
From obtained result, one might conclude that the considered in [Sharf and
Rusov, 2006] mechanism of virtuality reduction at the constrained maximum
point of multi-peripheral scattering amplitude may be responsible for proton-
proton total cross-section growth when all the considerable interference
contributions are taken into account.
Just the revelation of mechanism of cross-section growth we consider as the
main result of earlier papers [Sharf and Rusov, 2006; Sharf, Rusov _et al._ ,
2007] and present work, since this mechanism is intrinsic not only to the
diagrams of the “comb” type, but also to different modifications of considered
model.
Application the Laplace method allow to calculate another types of diagrams
corresponding to various scenarios of hadron-hadron inelastic scattering and
compare it with experimental data.
REFERENCES
## References
* Sharf and Rusov (2006) I. Sharf and V. Rusov, “Mechanism of hadron inelastic scattering cross-section growth in the multiperipheral model within the framework of perturbation theory. part 1,” (2006), arXiv:0605110 [hep-ph] .
* Sharf, Rusov _et al._ (2007) I. Sharf, V. Rusov, _et al._ , “Mechanism of hadron inelastic scattering cross-section growth in the multiperipheral model within the framework of perturbation theory. part 2,” (2007), arXiv:0711.3690 [hep-ph] .
* Kuraev, Lipatov, and Fadin (1976) E. Kuraev, L. Lipatov, and V. Fadin, “Multi reggeon processes in the yang-mills theory,” Sov. Phys. JETP. 44, 443–450 (1976).
* Bartels, Lipatov, and Vera (2009) J. Bartels, L. N. Lipatov, and A. S. Vera, “Bfkl pomeron, reggeized gluons, and bern-dixon-smirnov amplitudes,” Phys. Rev. D 80, 045002 (2009), arXiv:0802.2065 [hep-th] .
* Kozlov, Reznichenko, and Fadin (2007) M. G. Kozlov, A. V. Reznichenko, and V. S. Fadin, “Quantum chromodynamics at high energies,” Vestnik NSU 2, 3–31 (2007).
* Danilov and Lipatov (2006) G. S. Danilov and L. N. Lipatov, “BFKL pomeron in string models,” Nucl. Phys. B754, 187–232 (2006), arXiv:hep-ph/0603073 .
* Nakamura and Group (2010) K. Nakamura and P. D. Group, “Review of particle physics,” Journal of Physics G: Nuclear and Particle Physics 37, 075021 (2010).
* Aad _et al._ (2011) G. Aad _et al._ (ATLAS Collaboration), “Measurement of the Inelastic Proton-Proton Cross-Section at $\sqrt{s}=7$ TeV with the ATLAS Detector,” (2011), * Temporary entry *, arXiv:1104.0326 [hep-ex] .
Table 1: Results of numerical calculations of the eigenvalues of matrix
${\hat{D}_{y}}$. $n=20$
---
$\sqrt{s}=10$ GeV | $\sqrt{s}=300$ GeV | $\sqrt{s}=10$ TeV
$d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$
1.317 | -0.417 | 0.181 | -0.864 | 0.064 | -0.928
3.078 | 0.352 | 0.551 | -0.586 | 0.227 | -0.746
3.006 | 0.321 | 0.878 | -0.34 | 0.421 | -0.527
1.883 | -0.173 | 1.099 | -0.174 | 0.604 | -0.321
2.53 | 0.111 | 1.238 | 0.342 | 0.745 | -0.163
2.527 | 0.11 | 1.785 | 0.342 | 0.849 | -0.047
2.401 | 0.055 | 1.785 | -0.07 | 1.26 | 0.415
2.399 | 0.054 | 1.324 | -0.005 | 1.26 | 0.415
2.061 | -0.094 | 1.38 | 0.037 | 0.92 | 0.033
2.312 | 0.016 | 1.416 | 0.064 | 0.967 | 0.087
2.311 | 0.015 | 1.441 | 0.083 | 1.001 | 0.124
2.124 | -0.067 | 1.573 | 0.183 | 1.022 | 0.147
2.248 | -0.012 | 1.573 | 0.183 | 1.037 | 0.164
2.247 | -0.013 | 1.458 | 0.096 | 1.046 | 0.175
2.152 | -0.055 | 1.47 | 0.105 | 1.053 | 0.183
2.161 | -0.05 | 1.478 | 0.111 | 1.06 | 0.19
2.203 | -0.032 | 1.485 | 0.116 | 1.065 | 0.196
2.203 | -0.032 | 1.483 | 0.115 | 1.057 | 0.188
2.174 | -0.045 | 1.504 | 0.131 | 1.075 | 0.207
$n=10$
$\sqrt{s}=10$ GeV | $\sqrt{s}=300$ GeV | $\sqrt{s}=10$ TeV
$d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$
0.955 | -0.457 | 0.147 | -0.809 | 0.037 | -0.901
2.124 | 0.207 | 0.435 | -0.433 | 0.13 | -0.65
2.121 | 0.205 | 0.665 | -0.133 | 0.242 | -0.351
1.529 | -0.131 | 0.794 | 0.036 | 0.34 | -0.087
1.707 | -0.03 | 0.855 | 0.115 | 0.412 | 0.107
1.77 | 0.006 | 0.882 | 0.15 | 0.46 | 0.236
1.805 | 0.026 | 0.893 | 0.164 | 0.503 | 0.352
1.891 | 0.075 | 1.052 | 0.372 | 0.489 | 0.313
|
arxiv-papers
| 2009-12-14T09:27:15 |
2024-09-04T02:49:07.026894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. V. Sharf, G. O. Sokhrannyi, A. V. Tykhonov, K. V. Yatkin, N. A.\n Podolyan, M. A. Deliyergiyev, V. D. Rusov",
"submitter": "Vladimir Smolyar",
"url": "https://arxiv.org/abs/0912.2598"
}
|
0912.2625
|
2010537-548Nancy, France 537
Dietrich Kuske
# Is Ramsey’s theorem $\omega$-automatic?
Dietrich Kuske Centre national de la recherche scientifique (CNRS) and
Laboratoire Bordelais de Recherche en Informatique (LaBRI), Bordeaux, France
###### Abstract.
We study the existence of infinite cliques in $\omega$-automatic
(hyper-)graphs. It turns out that the situation is much nicer than in general
uncountable graphs, but not as nice as for automatic graphs.
More specifically, we show that every uncountable $\omega$-automatic graph
contains an uncountable co-context-free clique or anticlique, but not
necessarily a context-free (let alone regular) clique or anticlique. We also
show that uncountable $\omega$-automatic ternary hypergraphs need not have
uncountable cliques or anticliques at all.
###### Key words and phrases:
Logic in computer science, Automata, Ramsey theory
###### 1991 Mathematics Subject Classification:
F.4.1
These results were obtained when the author was affiliated with the
Universität Leipzig.
## Introduction
Every infinite graph has an infinite clique or an infinite anticlique – this
is the paradigmatic formulation of Ramsey’s theorem [Ram30]. But this theorem
is highly non-constructive since there are recursive infinite graphs whose
infinite cliques and anticliques are all non-recursive (not even in
$\Sigma^{0}_{2}$, [Joc72], cf. [Gas98, Thm. 4.6]). Recall that a graph is
recursive if both its set of nodes and its set of edges can be decided by a
Turing machine. Replacing these Turing machines by finite automata, one
obtains the more restrictive notion of an _automatic graph_ : the set of nodes
is a regular set and whether a pair of nodes forms an edge can be decided by a
synchronous two-tape automaton (this concept is known since the beginning of
automata theory, a systematic study started with [KN95, BG04], see [Rub08] for
a recent overview). In this context, the situation is much more favourable:
every infinite automatic graph contains an infinite regular clique or an
infinite regular anticlique (cf. [Rub08]).
Soon after Ramsey’s paper from 1930, authors got interested in a quantitative
analysis. For finite graphs, one can ask for the minimal number of nodes that
guarantee the existence of a clique or anticlique of some prescribed size.
This also makes sense in the infinite: how many nodes are necessary and
sufficient to obtain a clique or anticlique of size ${\aleph_{0}}$ (Ramsey’s
theorem tells us: ${\aleph_{0}}$) or $\aleph_{1}$ (here one needs more than
${2^{\aleph_{0}}}$ nodes [Sie33, ER56]).
Since automatic graphs contain at most ${\aleph_{0}}$ nodes, we need a more
general notion for a recursion-theoretic analysis of this situation. For this,
we use Blumensath & Grädel’s [BG04] $\omega$-automatic graphs: the names of
nodes form a regular $\omega$-language and the edge relation (on names) as
well as the relation “these two names denote the same node” can be decided by
a synchronous 2-tape Büchi-automaton. In this paper, we answer the question
whether these $\omega$-automatic graphs are more like automatic graphs (i.e.,
large cliques or anticliques with nice properties exist) or like general
graphs (large cliques need not exist).
Our answer to this question is a clear “somewhere in between”: We show that
every $\omega$-automatic graph of size ${2^{\aleph_{0}}}$ contains a clique or
anticlique of size ${2^{\aleph_{0}}}$ (Theorem 3.1) – this is in contrast to
the case of arbitrary graphs where such a subgraph need not exist [Sie33]. But
in general, there is no regular clique or anticlique (Theorem 3.23) – this is
in contrast with the case of automatic graphs where we always find a large
regular clique or anticlique. Finally, we also provide an $\omega$-automatic
“ternary hypergraph” of size ${2^{\aleph_{0}}}$ without any clique or
anticlique of size $\aleph_{1}$, let alone ${2^{\aleph_{0}}}$ (Theorem 3.19).
For Theorem 3.1, we re-use the proof from [BKR08] that was originally
constructed to deal with infinity quantifiers in $\omega$-automatic
structures. The proof of Theorem 3.23 makes use of the “ultimately equal”
relation. This relation was also crucial in the separation of injectively from
general $\omega$-automatic structures [HKMN08] as well as in the handling of
infinity quantifiers in [KL08] and [BKR08]. In the ternary hypergraph from
Theorem 3.19, a 3-set $\\{x,y,z\\}$ of infinite words with
$x<_{\mathrm{lex}}y<_{\mathrm{lex}}z$ forms an undirected hyperedge iff the
longest common prefix of $x$ and $y$ is shorter than the longest common prefix
of $y$ and $z$.
¿From Theorem 3.1 (i.e., the existence of large cliques or anticliques in
$\omega$-automatic graphs), we derive that any $\omega$-automatic partial
order of size ${2^{\aleph_{0}}}$ contains an antichain of size
${2^{\aleph_{0}}}$ or a copy of the real line.
## 1\. Preliminaries
### 1.1. Ramsey-theory
For a set $V$ and a natural number $k\geq 1$, let $[V]^{k}$ denote the set of
$k$-element subsets of $V$. A _$(k,\ell)$ -partition_ is a pair
$G=(V,E_{1},\dots,E_{\ell})$ where $V$ is a set and $(E_{1},\dots,E_{\ell})$
is a partition of $[V]^{k}$ into (possibly empty) sets. For $1\leq i\leq\ell$,
a set $W\subseteq V$ is _$E_{i}$ -homogeneous_ if $[W]^{k}\subseteq E_{i}$; it
is _homogeneous_ if it is $E_{i}$-homogeneous for some $1\leq i\leq\ell$. The
case $k=\ell=2$ is special: any $(2,2)$-partition $G=(V,E_{1},E_{2})$ can be
considered as an (undirected loop-free) graph $(V,E_{1})$. Homogeneous sets in
$G$ are then complete or discrete induced subgraphs of $(V,E_{1})$.
Ramsey theory is concerned with the following question: Does every
$(k,\ell)$-partition $G=(V,E_{1},\dots,E_{\ell})$ with $|V|=\kappa$ have a
homogeneous set of size $\lambda$ (where $\kappa$ and $\lambda$ are cardinal
numbers and $k,\ell\geq 2$ are natural numbers). If this is the case, one
writes
$\kappa\to(\lambda)^{k}_{\ell}$
(a notation due to Erdős and Rado [ER56]). This allows to formulate Ramsey’s
theorem concisely:
###### Theorem 1.1 (Ramsey [Ram30]).
If $k,\ell\geq 2$, then ${\aleph_{0}}\to({\aleph_{0}})^{k}_{\ell}$.
In particular, every graph with ${\aleph_{0}}$ nodes contains a complete or
discrete induced subgraph of the same size. If one wants to find homogeneous
sets of size $\aleph_{1}$, the base set has to be much larger:
###### Theorem 1.2 (Sierpiński [Sie33]).
If $k,\ell\geq 2$, then ${2^{\aleph_{0}}}\not\to(\aleph_{1})^{k}_{\ell}$ and
therefore in particular
${2^{\aleph_{0}}}\not\to({2^{\aleph_{0}}})^{k}_{\ell}$.
Erdős and Rado [ER56] proved that partitions of size properly larger than
${2^{\aleph_{0}}}$ have homogeneous sets of size $\aleph_{1}$. For more
details on infinite Ramsey theory, see [Jec02, Chapter 9].
### 1.2. $\omega$-languages
Let $\Gamma$ be a finite alphabet. With $\Gamma^{*}$ we denote the set of all
finite words over the alphabet $\Gamma$. The set of all nonempty finite words
is $\Gamma^{+}$. An _$\omega$ -word_ over $\Gamma$ is an infinite
$\omega$-sequence $x=a_{0}a_{1}a_{2}\cdots$ with $a_{i}\in\Gamma$, we set
$x[i,j)=a_{i}a_{i+1}\dots a_{j-1}$ for natural numbers $i\leq j$. In the same
spirit, $x[i,\omega)$ denotes the $\omega$-word $a_{i}a_{i+1}\dots$. The set
of all $\omega$-words over $\Gamma$ is denoted by $\Gamma^{\omega}$ and
$\Gamma^{\infty}=\Gamma^{*}\cup\Gamma^{\omega}$. For a set
$V\subseteq\Gamma^{+}$ of finite words let
$V^{\omega}\subseteq\Gamma^{\omega}$ be the set of all $\omega$-words of the
form $v_{0}v_{1}v_{2}\cdots$ with $v_{i}\in V$. Two infinite words
$x,y\in\Gamma^{\omega}$ are _ultimately equal_ , briefly $x\sim_{e}y$, if
there exists $i\in{\mathbb{N}}$ with $x[i,\omega)=y[i,\omega)$. By
$\leq_{\mathrm{lex}}$, we denote the lexicographic order on the set
$\Sigma^{\omega}$ (with some, implicitly assumed linear order on the letters
from $\Sigma$) and $\leq_{\mathrm{pref}}$ the prefix order on
$\Sigma^{\infty}$.
For $\Sigma=\\{0,1\\}$, the support ${\mathrm{supp}}(x)\subseteq{\mathbb{N}}$
is the set of positions of the letter $1$ in the word $x\in\Sigma^{\omega}$.
A (nondeterministic) _Büchi-automaton_ $M$ is a tuple
$M=(Q,\Gamma,\delta,\iota,F)$ where $Q$ is a finite set of states, $\iota\in
Q$ is the initial state, $F\subseteq Q$ is the set of final states, and
$\delta\subseteq Q\times\Gamma\times Q$ is the transition relation. If
$\Gamma=\Sigma^{n}$ for some alphabet $\Sigma$, then we speak of an _$n$
-dimensional Büchi-automaton over $\Sigma$_. A _run_ of $M$ on an
$\omega$-word $x=a_{0}a_{1}a_{2}\cdots$ is an $\omega$-word
$r=p_{0}p_{1}p_{2}\cdots$ over the set of states $Q$ such that
$(p_{i},a_{i},p_{i+1})\in\delta$ for all $i\geq 0$. The run $r$ is
_successful_ if $p_{0}=\iota$ and there exists a final state from $F$ that
occurs infinitely often in $r$. The $\omega$-language
$L(M)\subseteq\Gamma^{\omega}$ defined by $M$ is the set of all $\omega$-words
that admit a successful run. An $\omega$-language $L\subseteq\Gamma^{\omega}$
is _regular_ if there exists a Büchi-automaton $M$ with $L(M)=L$.
Alternatively, regular $\omega$-languages can be represented algebraically. To
this end, one defines _$\omega$ -semigroups_ to be two-sorted algebras
$S=(S_{+},S_{\omega};\cdot,*,\pi)$ where $\cdot:S_{+}\times S_{+}\to S_{+}$
and $*:S_{+}\times S_{\omega}\to S_{\omega}$ are binary operations and
$\pi:(S_{+})^{\omega}\to S_{\omega}$ is an $\omega$-ary operation such that
the following hold:
* •
$(S_{+},\cdot)$ is a semigroup,
* •
$s*(t*u)=(s\cdot t)*u$,
* •
$s_{0}\cdot\pi((s_{i})_{i\geq 1})=\pi((s_{i})_{i\geq 0})$,
* •
$\pi((s^{1}_{i}\cdot s^{2}_{i}\cdots s^{k_{i}}_{i})_{i\geq
0})=\pi((t_{j})_{j\geq 0})$ whenever
$(t_{j})_{j\geq
0}=(s^{1}_{0},s^{2}_{0},\dots,s^{k_{0}}_{0},s^{1}_{1},\dots,s^{k_{1}}_{1},\dots)\
.$
The $\omega$-semigroup $S$ is _finite_ if both, $S_{+}$ and $S_{\omega}$ are
finite. The free $\omega$-semigroup generated by $\Gamma$ is
$\Gamma^{\infty}=(\Gamma^{+},\Gamma^{\omega};\cdot,*,\pi)$
where $u\cdot v$ and $u*x$ are the natural operations of prefixing a word by
the finite word $u$, and $\pi((u_{i})_{i\geq 0})$ is the omega-word
$u_{0}u_{1}u_{2}\dots$. A homomorphism $h:\Gamma^{\infty}\to S$ of
$\omega$-semigroups maps finite words to elements of $S_{+}$ and
$\omega$-words to elements of $S_{\omega}$ and commutes with the operations
$\cdot$, $*$, and $\pi$. The algebraic characterisation of regular
$\omega$-languages then reads as follows.
###### Proposition 1.3.
An $\omega$-language $L\subseteq\Gamma^{\omega}$ is regular if and only if
there exists a finite $\omega$-semigroup $S$, a set $T\subseteq S_{\omega}$,
and a homomorphism $\eta:\Gamma^{\infty}\to S$ such that $L=\eta^{-1}(T)$.
Hence, every Büchi-automaton is “equivalent” to a homomorphism into some
finite $\omega$-semigroup together with a distinguished set $T$ (and vice
versa).
For $\omega$-words $x_{i}=a_{i}^{0}a_{i}^{1}a_{i}^{2}\dots\in\Gamma^{\omega}$,
the _convolution_ $x_{1}\otimes x_{2}\otimes\cdots\otimes
x_{n}\in(\Gamma^{n})^{\omega}$ is defined by
$(x_{1},\dots,x_{n})^{\otimes}=(a_{1}^{0},\ldots,x_{n}^{0})\,(a_{1}^{1},\ldots,a_{n}^{1})\,(a_{1}^{2},\ldots,a_{n}^{2})\cdots\
.$
An $n$-ary relation $R\subseteq(\Gamma^{\omega})^{n}$ is called _$\omega$
-automatic_ if the $\omega$-language
$\\{(x_{1},\dots,x_{n})^{\otimes}\mid(x_{1},\ldots,x_{n})\in R\\}$ is regular.
To describe the complexity of $\omega$-languages, we will use language-
theoretic terms. Let $\mathrm{LANG}$ denote the class of all languages (i.e.,
sets of finite words over some finite set of symbols) and
$\omega\mathrm{LANG}$ the class of all $\omega$-languages. By $\mathrm{REG}$
and $\omega\mathrm{REG}$, we denote the regular languages and
$\omega$-languages, resp. An $\omega$-language is _context-free_ if it can be
accepted by a pushdown-automaton with Büchi-acceptance (on states), it is _co-
context-free_ if its complement is context-free. We denote by
$\omega\mathrm{CF}$ the set of context-free $\omega$-languages and by
$\mathrm{co}\text{-}\omega\mathrm{CF}$ their complements. An $\omega$-language
belongs to $\mathrm{LANG}^{*}$ if it is of the form $\bigcup_{1\leq i\leq
n}U_{i}V_{i}^{\omega}$ with $U_{i},V_{i}\in\mathrm{LANG}$. Then
$\omega\mathrm{REG}\subseteq\mathrm{LANG}^{*}$ and
$\omega\mathrm{CF}\subseteq\mathrm{LANG}^{*}$ where the sets $U_{i}$ and
$V_{i}$ are regular and context-free, resp [Sta97]. In between these two
classes, we define the class $\omega\mathrm{erCF}$ of _eventually regular
context-free_ $\omega$-languages that comprises all sets of the form
$\bigcup_{1\leq i\leq n}U_{i}V_{i}^{\omega}$ with $U_{i}\in\mathrm{LANG}$
context-free and $V_{i}\in\mathrm{LANG}$ regular. Alternatively, eventually
regular context-free $\omega$-languages are the finite unions of
$\omega$-languages of the form $C\cdot L$ where $C$ is a context free-language
and $L$ a regular $\omega$-language. Let
$\mathrm{co}\text{-}\omega\mathrm{erCF}$ denote the set of complements of
eventually regular context-free $\omega$-languages.
A final, rather peculiar class of $\omega$-languages is $\Lambda$: it is the
class of $\omega$-languages $L$ such that $(\mathbb{R},\leq)$ embeds into
$(L,\leq_{\mathrm{lex}})$ (the name derives from the notation $\lambda$ for
the order type of $(\mathbb{R},\leq)$).
### 1.3. $\omega$-automatic $(k,\ell)$-partitions
An _$\omega$ -automatic presentation of a $(k,\ell)$-partition
$(V,E_{1},\dots,E_{\ell})$_ is a pair $(L,h)$ consisting of a regular
$\omega$-language $L$ and a surjection $h:L\to V$ such that
$\\{(x_{1},x_{2},\dots,x_{k})\in
L^{k}\mid\\{h(x_{1}),h(x_{2}),\dots,h(x_{k})\\}\in E_{i}\\}$ for $1\leq i\leq
k$ and $R_{\approx}=\\{(x_{1},x_{2})\in L^{2}\mid h(x_{1})=h(x_{2})\\}$ are
$\omega$-automatic. An $\omega$-automatic presentation is _injective_ if $h$
is a bijection. A $(k,\ell)$-partition is _(injectively) $\omega$-automatic_
if it has an (injective) $\omega$-automatic presentation. From [BKR08], it
follows that an uncountable $\omega$-automatic $(k,\ell)$-partition has
${2^{\aleph_{0}}}$ elements.
This paper is concerned with the question whether every (injective)
$\omega$-automatic presentation $(L,h)$ of a $(k,\ell)$-partition admits a
“simple” set $H\subseteq L$ such that $h(H)$ has $\lambda$ elements and is
homogeneous. More precisely, let $\mathcal{C}$ be a class of
$\omega$-languages, $k,\ell\geq 2$ natural numbers, and $\kappa$ and $\lambda$
cardinal numbers. Then we write
$(\kappa,\omega\mathsf{A})\to(\lambda,\mathcal{C})^{k}_{\ell}$
if the following partition property holds: for every $\omega$-automatic
presentation $(L,h)$ of a $(k,\ell)$-partition $G$ of size $\kappa$, there
exists $H\subseteq L$ in $\mathcal{C}$ such that $h(H)$ is homogeneous in $G$
and of size $\lambda$.
$(\kappa,\omega\mathsf{iA})\to(\lambda,\mathcal{C})^{k}_{\ell}$
is to be understood similarly where we only consider injective
$\omega$-automatic presentations.
###### Remark 1.4.
Let $G=(V,E_{1},\dots,E_{\ell})$ be some $(k,\ell)$-partition with
$\omega$-automatic presentation $(L,h)$. Then the partition property above
requires that there is a “large” homogeneous set $X\subseteq V$ and an
$\omega$-language $H\in\mathcal{C}$ such that $h(H)=X$, in particular, every
element of $X$ has at least one representative in $H$. Alternatively, one
could require that $h^{-1}(X)\subseteq L$ is an $\omega$-language from
$\mathcal{C}$. In this paper, we only encounter classes $\mathcal{C}$ of
$\omega$-languages such that the following closure property holds: if
$H\in\mathcal{C}$ and $R$ is an $\omega$-automatic relation, then also
$R(H)=\\{y\mid\exists x\in H:(x,y)\in R\\}\in\mathcal{C}$. Since
$h^{-1}h(H)=R_{\approx}(H)$, all our results also hold for this alternative
requirement $h^{-1}(X)\in\mathcal{C}$.
This paper shows
1. (0)
if $k,\ell\geq 2$, then
$({\aleph_{0}},\omega\mathsf{A})\to({\aleph_{0}},\omega\mathrm{REG})^{k}_{\ell}$,
but
$({2^{\aleph_{0}}},\omega\mathsf{A})\not\to({\aleph_{0}},\omega\mathrm{REG})^{k}_{\ell}$,
see Theorem 2.1.
2. (1)
if $\ell\geq 2$, then
$({2^{\aleph_{0}}},\omega\mathsf{A})\to({2^{\aleph_{0}}},\mathrm{co}\text{-}\omega\mathrm{erCF})^{2}_{\ell}$,
see Theorem 3.1.
3. (2)
if $k\geq 3$, $\ell\geq 2$, and $\lambda>{\aleph_{0}}$, then
$({2^{\aleph_{0}}},\omega\mathsf{iA})\not\to(\lambda,\omega\mathrm{LANG})^{k}_{\ell}$,
see Theorem 3.19.
4. (3)
if $k,\ell\geq 2$ and $\lambda>{\aleph_{0}}$, then
$({2^{\aleph_{0}}},\omega\mathsf{iA})\not\to(\lambda,\omega\mathrm{CF})^{k}_{\ell}$,
see Theorem 3.23.
Here, the first part of (0) is a strengthening of Ramsey’s theorem since the
infinite homogeneous set is regular. The second part might look surprising
since larger $(k,\ell)$-partitions should have larger homogeneous sets – but
not necessarily regular ones! In contrast to Sierpiński’s result, (1) shows
that $\omega$-automatic $(2,\ell)$-partitions have a larger degree of
homogeneity than arbitrary $(2,\ell)$-partitions. Even more, the complexity of
the homogeneous set can be bound in language-theoretic terms (there is always
a homogeneous set that is the complement of an eventually regular context-free
$\omega$-language). Statement (2) is an analogue of Sierpiński’s Theorem 1.2
showing that (injective) $\omega$-automatic $(k,\ell)$-partitions are as in-
homogeneous as arbitrary $(k,\ell)$-partitions provided $k\geq 3$. The
complexity bound from (1) is shown to be optimal by (3) proving that one
cannot always find context-free homogeneous sets. Hence, despite the existence
of large homogeneous sets for $k=2$, for some $\omega$-automatic
presentations, they are bound to have a certain (low) level of complexity that
is higher than the regular $\omega$-languages.
## 2\. Countably infinite homogeneous sets
Let $k,\ell\geq 2$ be arbitrary. Then, from Ramsey’s theorem, we obtain
immediately
$({\aleph_{0}},\omega\mathsf{A})\to({\aleph_{0}},\omega\mathrm{LANG})^{k}_{\ell}$
and
$({2^{\aleph_{0}}},\omega\mathsf{A})\to({\aleph_{0}},\omega\mathrm{LANG})^{k}_{\ell}$,
i.e., all infinite $\omega$-automatic $(k,\ell)$-partitions have homogeneous
sets of size ${\aleph_{0}}$. In this section, we ask whether such homogeneous
sets can always be chosen regular:
###### Theorem 2.1.
Let $k,\ell\geq 2$. Then
1. (a)
$({\aleph_{0}},\omega\mathsf{A})\to({\aleph_{0}},\omega\mathrm{REG})^{k}_{\ell}$.
2. (b)
$({2^{\aleph_{0}}},\omega\mathsf{iA})\to({\aleph_{0}},\omega\mathrm{REG})^{k}_{\ell}$.
3. (c)
$({2^{\aleph_{0}}},\omega\mathsf{A})\not\to({\aleph_{0}},\mathrm{LANG}^{*})^{k}_{\ell}$,
and therefore in particular
$({2^{\aleph_{0}}},\omega\mathsf{A})\not\to({\aleph_{0}},\omega\mathrm{CF})^{k}_{\ell}$
and
$({2^{\aleph_{0}}},\omega\mathsf{A})\not\to({\aleph_{0}},\omega\mathrm{REG})^{k}_{\ell}$.
###### Proof 2.2.
Let $(L,h)$ be an $\omega$-automatic presentation of some $(k,\ell)$-partition
$G=(V,E_{1},\dots,E_{\ell})$ with $|V|={\aleph_{0}}$. By [BKR08], there exists
$L^{\prime}\subseteq L$ regular such that $(L^{\prime},h)$ is an injective
$\omega$-automatic presentation of $G$. From a Büchi-automaton for
$L^{\prime}$, one can compute a finite automaton accepting some language $K$
such that $(K,h^{\prime})$ is an injective automatic presentation of $G$
[Blu99]. Hence, by [Rub08], there exists a regular set $H^{\prime}\subseteq K$
such that $h^{\prime}(H^{\prime})$ is homogeneous in $G$ and countably
infinite. From this set, one obtains a regular $\omega$-language $H\subseteq
L^{\prime}\subseteq L$ with $h(H)=h^{\prime}(H^{\prime})$, i.e., $h(H)$ is a
homogeneous set of size ${\aleph_{0}}$. This proves (a).
To prove (b), let $(L,h)$ be an injective $\omega$-automatic presentation of
some $(k,\ell)$-partition $G=(V,E_{1},\dots,E_{\ell})$ of size
${2^{\aleph_{0}}}$. Then there exists a regular $\omega$-language
$L^{\prime}\subseteq L$ with $|L^{\prime}|={\aleph_{0}}$. Consider the sub-
partition $G^{\prime}=(h(L^{\prime}),E_{1}^{\prime},\dots,E_{\ell}^{\prime})$
with $E_{i}^{\prime}=E_{i}\cap[h(L^{\prime})]^{k}$. This $(k,\ell)$-partition
has as $\omega$-automatic presentation the pair $(L^{\prime},h)$. Then, by
(a), there exists $L^{\prime\prime}\subseteq L^{\prime}$ regular and infinite
such that $h(L^{\prime\prime})$ is homogeneous in $G^{\prime}$ and therefore
in $G$. Since $h$ is injective, this implies
$|h(L^{\prime})|=|L^{\prime}|={\aleph_{0}}$.
Finally, we show (c) by a counterexample. Let $L=\\{0,1\\}^{\omega}$,
$V=L/\mathord{\sim_{e}}$, and $h:L\to V$ the canonical mapping. Furthermore,
set $E_{1}=[L]^{k}$. Then $G=(V,E_{1},\emptyset,\dots,\emptyset)$ is a
$(k,\ell)$-partition with $\omega$-automatic presentation $(L,h)$.
Now let $H=\bigcup_{1\leq i\leq n}U_{i}V_{i}^{\omega}\subseteq L$ for some
non-empty languages $U_{i},V_{i}\subseteq\\{0,1\\}^{+}$ such that $h(H)$ is
homogeneous and infinite.
If $|V_{i}^{\omega}|=1$, then $U_{i}V_{i}^{\omega}/\mathord{\sim_{e}}$ is
finite. Since $h(H)$ is infinite, there exists $1\leq i\leq n$ with
$|V_{i}^{\omega}|>1$ implying the existence of words $v,w\in V_{i}^{+}$ such
that $|v|=|w|$ and $v\neq w$. For $u\in U_{i}$, the set
$u\\{v,w\\}^{\omega}\subseteq H$ has ${2^{\aleph_{0}}}$ equivalence classes
wrt. $\sim_{e}$. Hence $|h(H)|={2^{\aleph_{0}}}$.
## 3\. Uncountable homogeneous sets
### 3.1. A Ramsey theorem for $\omega$-automatic $(2,\ell)$-partitions
The main result of this section is the following theorem that follows
immediately from Prop. 3.11 and Lemma 3.7.
###### Theorem 3.1.
For all $\ell\geq 2$, we have
$({2^{\aleph_{0}}},\omega\mathsf{A})\to({2^{\aleph_{0}}},\mathrm{co}\text{-}\omega\mathrm{erCF}\cap{\text{\boldmath$\Lambda$}})^{2}_{\ell}$.
#### 3.1.1. The proof
The proof of this theorem will construct a language from
$\mathrm{co}\text{-}\omega\mathrm{erCF}$ that describes a homogeneous set.
This language is closely related to the following language
$N=1\\{0,1\\}^{\omega}\cap\bigcap_{n\geq 0}\\{0,1\\}^{n}(0\\{0,1\\}^{n}00\cup
10^{n}\\{01,10\\})\\{0,1\\}^{\omega}\ ,$
i.e., an $\omega$-word $x$ belongs to $N$ iff it starts with $1$ and, for
every $n\geq 0$, we have $x[n,2n+3)\in 0\\{0,1\\}^{*}00\cup 10^{*}01\cup
10^{*}10$. We first list some useful properties of this language $N$:
###### Lemma 3.2.
The $\omega$-language $N$ is contained in $(1^{+}0^{+})^{\omega}$, belongs to
$\mathrm{co}\text{-}\omega\mathrm{erCF}\cap\text{\boldmath$\Lambda$}$, and
${\mathrm{supp}}(x)\cap{\mathrm{supp}}(y)$ is finite for any $x,y\in N$
distinct.
###### Proof 3.3.
Let $b_{i}\in\\{0,1\\}$ for all $i\geq 0$ and suppose the word
$x=b_{0}b_{1}\dots$ belongs to $N$. Then $b_{0}=1$, hence the word $x$
contains at least one occurrence of $1$. Note that, whenever $b_{n}=1$, then
$\\{b_{2n+1},b_{2n+2}\\}=\\{0,1\\}$, hence $x$ contains infinitely many
occurrences of $1$ and therefore infinitely many occurrences of $0$, i.e.,
$N\subseteq(1^{+}0^{+})^{\omega}$.
Note that the complement of $N$ equals
$\displaystyle\ 0\\{0,1\\}^{\omega}\cup\bigcup_{n\geq
0}\Big{(}\\{0,1\\}^{n}(0\\{0,1\\}^{n}\\{01,10,11\\}\cup
1\\{0,1\\}^{n}\\{00,11\\})\\{0,1\\}^{\omega}\Big{)}$ $\displaystyle=$
$\displaystyle\left[0\cup\bigcup_{n\geq
0}\\{0,1\\}^{n}(0\\{0,1\\}^{n}\\{01,10,11\\}\cup
1\\{0,1\\}^{n}\\{00,11\\})\right]\\{0,1\\}^{\omega}\ .$
Since the expression in square brackets denotes a context-free language,
$\\{0,1\\}^{\omega}\setminus N$ is an eventually regular context-free
$\omega$-language.
Note that a word $10^{n_{0}}10^{n_{1}}10^{n_{2}}\dots$ belongs to $N$ iff, for
all $k\geq 0$, we have $0\leq n_{k}-|10^{n_{0}}10^{n_{1}}\dots
10^{n_{k-1}}|\leq 1$. Hence, when building a word from $N$, we have two
choices for any $n_{k}$, say $n_{k}^{0}$ and $n_{k}^{1}$ with
$n_{k}^{0}<n_{k}^{1}$. But then $a_{0}a_{1}a_{2}\dots\mapsto
10^{n_{0}^{a_{0}}}10^{n_{1}^{a_{1}}}10^{n_{2}^{a_{2}}}\dots$ defines an order
embedding
$(\\{0,1\\}^{\omega},\leq_{\mathrm{lex}})\hookrightarrow(N,\leq_{\mathrm{lex}})$.
Since
$(\mathbb{R},\leq)\hookrightarrow(\\{0,1\\}^{\omega},\leq_{\mathrm{lex}})$, we
get $N\in\text{\boldmath$\Lambda$}$.
Now let $x,y\in N$ with ${\mathrm{supp}}(x)\cap{\mathrm{supp}}(y)$ infinite.
Then there are arbitrarily long finite words $u$ and $v$ of equal length such
that $u1$ and $v1$ are prefixes of $x$ and $y$, resp. Since $u1$ is a prefix
of $x\in N$, it is of the form $u1=u^{\prime}10^{|u^{\prime}|}1$ (if $|u|$ is
even) or $u1=u^{\prime}10^{|u^{\prime}|}01$ (if $|u|$ is odd) and analogously
for $v$. Inductively, one obtains $u^{\prime}=v^{\prime}$ and therefore $u=v$.
Since $u$ and $v$ are arbitrarily long, we showed $x=y$.
###### Lemma 3.4.
Let $\sim$ and $\approx$ be two equivalence relations on some set $L$ such
that any equivalence class $[x]_{\sim}$ of $\sim$ is countable and $\approx$
has ${2^{\aleph_{0}}}$ equivalence classes. Then there are elements
$(x_{\alpha})_{\alpha<{2^{\aleph_{0}}}}$ of $L$ such that
$[x_{\alpha}]_{\sim_{e}}\cap[x_{\beta}]_{\approx}=\emptyset$ for all
$\alpha<\beta$.
###### Proof 3.5.
We construct the sequence $(x_{\alpha})_{\alpha<{2^{\aleph_{0}}}}$ by ordinal
induction. So assume we have elements $(x_{\alpha})_{\alpha<\kappa}$ for some
ordinal $\kappa<{2^{\aleph_{0}}}$ with
$[x_{\alpha}]_{\sim}\cap[x_{\beta}]_{\approx}=\emptyset$ for all
$\alpha<\beta<\kappa$.
Suppose
$\bigcup_{\alpha<\kappa}[x_{\alpha}]_{\sim}\cap[x]_{\approx}\neq\emptyset$ for
all $x\in L$. For $x,y\in L$ with $x\not\approx y$, we have
$(\bigcup_{\alpha<\kappa}[x_{\alpha}]_{\sim}\cap[x]_{\approx})\cap(\bigcup_{\alpha<\kappa}[x_{\alpha}]_{\sim}\cap[y]_{\approx})\subseteq[x]_{\approx}\cap[y]_{\approx}=\emptyset$.
Since $\bigcup_{\alpha<\kappa}[x_{\alpha}]_{\sim}$ has
$\kappa\cdot{\aleph_{0}}\leq\max(\kappa,\aleph_{0})<{2^{\aleph_{0}}}$
elements, we obtain $|L|<{2^{\aleph_{0}}}$, contradicting
$|L|\geq|L/{\approx}|={2^{\aleph_{0}}}$. Hence there exists an element
$x_{\kappa}\in L$ with
$[x_{\alpha}]_{\sim}\cap[x_{\kappa}]_{\approx}=\emptyset$ for all
$\alpha<\kappa$.
###### Definition 3.6.
Let $u$, $v$, and $w$ be nonempty words with $|v|=|w|$ and $v\neq w$. Define
an $\omega$-semigroup homomorphism $h:\\{0,1\\}^{\infty}\to\Sigma^{\infty}$ by
$h(0)=v$ and $h(1)=w$ and set
$H_{u,v,w}=u\cdot h(N)$
where $N$ is the set from Lemma 3.2.
###### Lemma 3.7.
Let $u$, $v$, and $w$ be as in the previous definition. Then
$H_{u,v,w}\in\mathrm{co}\text{-}\omega\mathrm{erCF}\cap\text{\boldmath$\Lambda$}$.
###### Proof 3.8.
Assume $v<_{\mathrm{lex}}w$. Then the mapping
$\chi:\\{0,1\\}^{\omega}\to\Sigma^{\omega}:x\mapsto uh(x)$ (where $h$ is the
homomorphism from the above definition) embeds $(N,\leq_{\mathrm{lex}})$ (and
hence $(\mathbb{R},\leq)$) into $(H_{u,v,w},\leq_{\mathrm{lex}})$. If
$w<_{\mathrm{lex}}v$, then
$(\mathbb{R},\leq)\cong(\mathbb{R},\geq)\hookrightarrow(N,\geq_{\mathrm{lex}})\hookrightarrow(H_{\alpha,\beta,\gamma},\leq_{\mathrm{lex}})$.
This proves that $H_{u,v,w}$ belongs to $\Lambda$.
Since $v\neq w$, the mapping $\chi$ is injective. Hence
$\Sigma^{\omega}\setminus
H_{\alpha,\beta,\gamma}=\Sigma^{\omega}\setminus\chi(N)=\Sigma^{\omega}\setminus\chi(\\{0,1\\}^{\omega})\cup\chi(\\{0,1\\}^{\omega}\setminus
N)\ .$
Since $\chi$ can be realized by a generalized sequential machine with Büchi-
acceptance, $\chi(\\{0,1\\}^{\omega})$ is regular and
$\chi(\\{0,1\\}^{\omega}\setminus N)$ (as the image of an eventually regular
context-free $\omega$-language) is eventually regular context-free. Hence
$\Sigma^{\omega}\setminus H_{u,v,w}$ is eventually regular context-free.
###### Proposition 3.9.
Let $G=(L,E_{0},E_{1},\dots,E_{\ell})$ be some $(2,1+\ell)$-partition with
injective $\omega$-automatic presentation $(L,\mathrm{id})$ such that
$\\{(x,y)\mid\\{x,y\\}\in E_{0}\\}\cup\\{(x,x)\mid x\in L\\}$ is an
equivalence relation on $L$ (denoted $\approx$) with ${2^{\aleph_{0}}}$
equivalence classes. Then there exist nonempty words $u$, $v$, and $w$ with
$v$ and $w$ distinct, but of the same length, such that $H_{u,v,w}$ is
$i$-homogeneous for some $1\leq i\leq\ell$.
###### Proof 3.10.
There are finite $\omega$-semigroups $S$ and $T$ and homomorphisms
$\gamma:\Sigma^{\infty}\to S$ and $\delta:(\Sigma\times\Sigma)^{\infty}\to T$
such that
1. (a)
$x\in L$, $y\in\Sigma^{\omega}$, and $\gamma(x)=\gamma(y)$ imply $y\in L$ and
2. (b)
$x,x^{\prime},y,y^{\prime}\in L$, $\\{h(x),h(x^{\prime})\\}\in E_{i}$, and
$\delta(x,x^{\prime})=\delta(y,y^{\prime})$ imply $\\{h(y),h(y^{\prime})\\}\in
E_{i}$ (for all $0\leq i\leq\ell$).
By Lemma 3.4, there are words $(x_{\alpha})_{\alpha<{2^{\aleph_{0}}}}$ in $L$
such that $[x_{\alpha}]_{\sim_{e}}\cap[x_{\beta}]_{\approx}=\emptyset$ for all
$\alpha<\beta$.
In the following, we only need the words $x_{0},x_{1},\dots,x_{C}$ with
$C=|S|\cdot|T|$. Then [BKR08, Sections 3.1-3.3]111The authors of [BKR08]
require $[x_{i}]_{\sim_{e}}\cap[x_{j}]_{\approx}=\emptyset$ for all $0\leq
i,j\leq C$ distinct, but they use it only for $i<j$. Hence we can apply their
result here. first constructs two $\omega$-words $y_{1}$ and $y_{2}$ and an
infinite sequence $1\leq g_{1}<g_{2}<\dots$ of natural numbers such that in
particular $y_{1}[g_{1},g_{2})<_{\mathrm{lex}}y_{2}[g_{1},g_{2})$. Set
$u=y_{2}[0,g_{1})$, $v=y_{1}[g_{1},g_{2})$, and $w=y_{2}[g_{1},g_{2})$. In the
following, let $h:\\{0,1\\}^{\infty}\to\Sigma^{\infty}$ be the homomorphism
from Def. 3.6 and set $\chi(x)=uh(x)$ for $x\in\\{0,1\\}^{*}$. As in [BKR08],
one can then show that all the words from $H_{u,v,w}$ belong to the
$\omega$-language $L$. In the following, set
$x_{\circ\bullet}=\chi((01)^{\omega})$ and
$x_{\bullet\circ}=\chi((10)^{\omega})$. Then obvious alterations in the proofs
by Bárány et al. show:
1. (1)
[BKR08, Lemma 3.4]222The authors of [BKR08] only require one of the two
differences to be infinite, but the proof uses that they both are infinite. If
$x,y\in\\{0,1\\}^{\omega}$ with
${\mathrm{supp}}(x)\setminus{\mathrm{supp}}(y)$ and
${\mathrm{supp}}(y)\setminus{\mathrm{supp}}(x)$ infinite, then
$\\{\delta(\chi(x),\chi(y)),\delta(\chi(y),\chi(x))\\}=\\{\delta(x_{\bullet\circ},x_{\circ\bullet}),\delta(x_{\circ\bullet},x_{\bullet\circ})\\}\
.$
2. (2)
[BKR08, Lemma 3.5] $x_{\bullet\circ}\not\approx x_{\circ\bullet}$.
There exists $0\leq i\leq\ell$ with
$\\{x_{\bullet\circ},x_{\circ\bullet}\\}\in E_{i}$. Then (2) implies $i>0$.
Let $x,y\in N$ be distinct. Then ${\mathrm{supp}}(x)\cap{\mathrm{supp}}(y)$ is
finite by Lemma 3.2. Since, on the other hand, ${\mathrm{supp}}(x)$ and
${\mathrm{supp}}(y)$ are both infinite, the two differences
${\mathrm{supp}}(x)\setminus{\mathrm{supp}}(y)$ and
${\mathrm{supp}}(y)\setminus{\mathrm{supp}}(x)$ are infinite. Hence we obtain
$\delta(\chi(x),\chi(y))\in\\{\delta(x_{\bullet\circ},x_{\circ\bullet}),\delta(x_{\circ\bullet},x_{\bullet\circ})\\}$
from (1). Hence (b) implies $\\{\chi(x),\chi(y)\\}\in E_{i}$, i.e.,
$H_{u,v,w}$ is $E_{i}$-homogeneous.
Since
$H_{u,v,w}\in\mathrm{co}\text{-}\omega\mathrm{erCF}\cap\text{\boldmath$\Lambda$}$
by Lemma 3.7, the result follows.
###### Proposition 3.11.
Let $G=(V,E_{1}^{\prime},\dots,E_{\ell}^{\prime})$ be some
$(2,\ell)$-partition with automatic presentation $(L,h)$. Then there exist
$u,v,w\in\Sigma^{+}$ with $v$ and $w$ distinct of equal length such that
$h(H_{u,v,w})$ is homogeneous and of size ${2^{\aleph_{0}}}$.
###### Proof 3.12.
To apply Prop. 3.9, consider the following $(2,1+\ell)$-partition
$G=(L,E_{0},\dots,E_{\ell})$:
* •
The underlying set is the $\omega$-language $L$,
* •
$E_{0}$ comprises all sets $\\{x,y\\}$ with $h(x)=h(y)$ and $x\neq y$, and
* •
$E_{i}$ (for $1\leq i\leq\ell$) comprises all sets $\\{x,y\\}$ with
$\\{h(x),h(y)\\}\in E_{i}^{\prime}$.
Then $(L,\mathrm{id})$ is an injective $\omega$-automatic presentation of the
$(2,1+\ell)$-partition $G$. By Prop. 3.9, there exists $1\leq i\leq\ell$ and
words $u$, $v$ and $w$ such that $H_{u,v,w}$ is $i$-homogeneous in $G$. Since
$(E_{0},\dots,E_{\ell})$ is a partition of $[L]^{2}$, we have $\\{x,y\\}\notin
E_{0}$ (and therefore $h(x)\neq h(y)$) for all $x,y\in H_{u,v,w}$ distinct.
Hence $h$ is injective on $H_{u,v,w}$. Furthermore $[H_{u,v,w}]^{2}\subseteq
E_{i}$ implies $[h(H_{u,v,w})]^{2}\subseteq E_{i}^{\prime}$. Hence
$h(H_{u,v,w})$ is an $i$-homogeneous set in $G^{\prime}$ of size
${2^{\aleph_{0}}}$.
This finishes the proof of Theorem 3.1.
#### 3.1.2. Effectiveness
Note that the proof above is non-constructive at several points: Lemma 3.4 is
not constructive and the proof proper uses Ramsey’s theorem [BKR08, page 390]
and makes a Ramseyan factorisation coarser [BKR08, begin of section 3.2]. We
now show that nevertheless the words $u$, $v$, and $w$ can be computed. By
Prop. 3.11, it suffices to decide for a given triple $(u,v,w)$ whether
$h(H_{u,v,w})$ is $i$-homogeneous for some fixed $1\leq i\leq\ell$.
To be more precise, let $(V,E_{1},\dots,E_{\ell})$ be some
$(2,\ell)$-partition with $\omega$-automatic presentation $(L,h)$.
Furthermore, let $u,v,w\in\Sigma^{+}$ with $v\neq w$ of the same length and
write $H$ for $H_{u,v,w}$. We have to decide whether $H\subseteq L$ and
$H\otimes H\subseteq L_{i}\cup L_{=}$. Note that $H\subseteq L$ iff
$L\cap\Sigma^{\omega}\setminus H=\emptyset$. But $\Sigma^{\omega}\setminus H$
is context-free, so the intersection is context-free. Hence the emptiness of
the intersection can be decided.
Towards a decision of the second requirement, note that
$\displaystyle(\Sigma\times\Sigma)^{\omega}\setminus(H\otimes H)$
$\displaystyle=(\Sigma^{\omega}\setminus
H\otimes\Sigma^{\omega})\cup(\Sigma^{\omega}\cup\Sigma^{\omega}\setminus H)$
is the union of two context-free $\omega$-languages and therefore context-free
itself. Since $L_{i}\cup L_{=}$ is regular, the intersection $(L_{i}\cup
L_{=})\cap(\Sigma\times\Sigma)^{\omega}\setminus(H\otimes H)$ is context-free
implying that its emptiness is decidable. But this emptiness is equivalent to
$H\otimes H\subseteq L_{1}\cup L_{=}$.
#### 3.1.3. $\omega$-automatic partial orders
¿From Theorem 3.1, we now derive a necessary condition for a partial order of
size ${2^{\aleph_{0}}}$ to be $\omega$-automatic. A partial order
$(V,\sqsubseteq)$ is $\omega$-automatic iff there exists a regular
$\omega$-language $L$ and a surjection $h:L\to V$ such that the relations
$R_{=}=\\{(x,y)\in L^{2}\mid h(x)=h(y)\\}$ and $R_{\sqsubseteq}=\\{(x,y)\in
L^{2}\mid h(x)\sqsubseteq h(y)\\}$ are $\omega$-automatic.
###### Corollary 3.13 ([BKR08]333As pointed out by two referees, the
paragraph before Sect. 4.1 in [BKR08] already hints at this result, although
in a rather implicit way.).
If $(V,\sqsubseteq)$ is an $\omega$-automatic partial order with
$|V|\geq\aleph_{1}$, then $(\mathbb{R},\leq)$ or an antichain of size
${2^{\aleph_{0}}}$ embeds into $(V,\sqsubseteq)$.
###### Proof 3.14.
Let $(V,\sqsubseteq)$ be a partial order, $L\subseteq\Sigma^{\omega}$ a
regular $\omega$-language and $h:L\to V$ a surjection such that $R_{=}$ and
$R_{\sqsubseteq}$ are $\omega$-automatic. Define an injective
$\omega$-automatic $(2,4)$-partition $G=(L,E_{0},E_{1},E_{2},E_{3})$:
* •
$E_{0}$ comprises all pairs $\\{x,y\\}\in[L]^{2}$ with $h(x)=h(y)$,
* •
$E_{1}$ comprises all pairs $\\{x,y\\}\in[L]^{2}$ with $h(x)\sqsubset h(y)$
and $x<_{\mathrm{lex}}y$,
* •
$E_{2}$ comprises all pairs $\\{x,y\\}\in[L]^{2}$ with $h(x)\sqsupset h(y)$
and $x<_{\mathrm{lex}}y$, and
* •
$E_{3}=[L]^{2}\setminus(E_{0}\cup E_{1}\cup E_{2})$ comprises all pairs
$\\{x,y\\}\in[L]^{2}$ such that $h(x)$ and $h(y)$ are incomparable.
From $|L|\geq|V|>{\aleph_{0}}$, we obtain $|L|={2^{\aleph_{0}}}$. Hence, by
Prop. 3.9, there exists $H\subseteq L$ $1$-, $2$\- or $3$-homogeneous with
$(\mathbb{R},\leq)\hookrightarrow(H,\leq_{\mathrm{lex}})$. Since
$[H]^{2}\subseteq E_{1}\cup E_{2}\cup E_{3}$ and since $G$ is a partition of
$L$, the mapping $h$ acts injectively on $H$. If $[H]^{2}\subseteq E_{1}$ (the
case $[H]^{2}\subseteq E_{2}$ is symmetrical) then
$(\mathbb{R},\leq)\hookrightarrow(H,\leq_{\mathrm{lex}})\cong(h(H),\sqsubseteq)$.
If $[H]^{2}\subseteq E_{3}$, then $h(H)$ is an antichain of size
${2^{\aleph_{0}}}$.
A linear order $(L,\sqsubseteq)$ is _scattered_ if $(\mathbb{Q},\leq)$ cannot
be embedded into $(L,\sqsubseteq)$. Automatic partial orders are defined
similarly to $\omega$-automatic partial orders with the help of finite
automata instead of Büchi-automata.
###### Corollary 3.15 ([BKR08]33footnotemark: 3).
Any scattered $\omega$-automatic linear order $(V,\sqsubseteq)$ is countable.
Hence,
* •
a scattered linear order is $\omega$-automatic if and only if it is automatic,
and
* •
an ordinal $\alpha$ is $\omega$-automatic if and only if
$\alpha<\omega^{\omega}$.
###### Proof 3.16.
If $(V,\sqsubseteq)$ is not countable, then it embeds $(\mathbb{R},\leq)$ by
the previous corollary and therefore in particular $(\mathbb{Q},\leq)$. The
remaining two claims follow immediately from [BKR08] (“countable
$\omega$-automatic structures are automatic”) and [Del04] (“an ordinal is
automatic iff it is properly smaller than $\omega^{\omega}$”), resp.
Contrast Theorem 3.1 with Theorem 1.2: any uncountable $\omega$-automatic
$(k,\ell)$-partition contains an uncountable homogeneous set of size
${2^{\aleph_{0}}}$. But we were able to prove this for $k=2$, only. One would
also wish the homogeneous set to be regular and not just from
$\mathrm{co}\text{-}\omega\mathrm{erCF}$. We now prove that these two
shortcomings are unavoidable: Theorem 3.1 does not hold for $k=3$ nor is there
always an $\omega$-regular homogeneous set. These negative results hold even
for injective presentations.
### 3.2. A Sierpiński theorem for $\omega$-automatic $(k,\ell)$-partitions
with $k\geq 3$
We first concentrate on the question whether some form of Theorem 3.1 holds
for $k\geq 3$. The following lemma gives the central counterexample for $k=3$
and $\ell=2$, the below theorem then derives the general result.
###### Lemma 3.17.
$({2^{\aleph_{0}}},\omega\mathsf{iA})\not\to(\aleph_{1},\omega\mathrm{LANG})^{3}_{2}$.
###### Proof 3.18.
Let $\Sigma=\\{0,1\\}$, $V=L=\\{0,1\\}^{\omega}$. Furthermore, for $H\subseteq
L$, we write $\bigwedge H\in\Sigma^{\infty}$ for the longest common prefix of
all $\omega$-words in $H$, $\bigwedge\\{x,y\\}$ is also written $x\wedge y$.
Then let $E_{1}$ consist of all 3-sets $\\{x,y,z\\}\in[L]^{3}$ with
$x<_{\mathrm{lex}}y<_{\mathrm{lex}}z$ and $x\wedge y<_{\mathrm{pref}}y\wedge
z$; $E_{2}$ is the complement of $E_{1}$. This finishes the construction of
the $(3,2)$-partition $(V,E_{1},E_{2})$ of size ${2^{\aleph_{0}}}$ with
injective $\omega$-automatic presentation $(L,\mathrm{id})$.
Note that $1^{*}0^{\omega}$ is a countable $E_{1}$-homogeneous set and that
$0^{*}1^{\omega}$ is a countable $E_{2}$-homogeneous set. But there is no
uncountable homogeneous set: First suppose $H\subseteq L$ is infinite and
$x\wedge y<_{\mathrm{pref}}y\wedge z$ for all
$x<_{\mathrm{lex}}y<_{\mathrm{lex}}z$ from $H$. Let $u\in\Sigma^{*}$ such that
$H\cap u0\Sigma^{\omega}$ and $H\cap u1\Sigma^{\omega}$ are both nonempty and
let $x,y\in H\cap u0\Sigma^{\omega}$ with $x\leq_{\mathrm{lex}}y$ and $z\in
H\cap u1\Sigma^{\omega}$. Then $x\wedge y>_{\mathrm{pref}}u=y\wedge z$ and
therefore $x=y$ (for otherwise, we would have
$x<_{\mathrm{lex}}y<_{\mathrm{lex}}z$ in $H$ with $x\wedge
y>_{\mathrm{pref}}y\wedge z$). Hence we showed $|H\cap u0\Sigma^{\omega}|=1$.
Let $u_{0}=\bigwedge H$ and $H_{1}=H\cap u_{0}1\Sigma^{\omega}$. Since $H\cap
u_{0}0\Sigma^{\omega}$ is finite, the set $H_{1}$ is infinite. We proceed by
induction: $u_{n}=\bigwedge H_{n}$ and $H_{n+1}=H_{n}\cap
u_{n}1\Sigma^{\omega}$ satisfying $|H_{n}\cap u_{n}0\Sigma^{\omega}|=1$. Then
$u_{0}<_{\mathrm{pref}}u_{0}1\leq_{\mathrm{pref}}u_{1}<_{\mathrm{pref}}u_{1}1\leq_{\mathrm{pref}}u_{2}\cdots$
with
$H=\bigcup_{n\geq 0}(H\cap u_{n}0\Sigma^{\omega})\cup\bigcap_{n\geq 0}(H\cap
u_{n}1\Sigma^{\omega})\ .$
Then any of the sets $H\cap u_{n}0\Sigma^{\omega}=H_{n}\cap
u_{n}0\Sigma^{\omega}$ and $\bigcap(H\cap u_{n}1\Sigma^{\omega})$ is a
singleton, proving that $H$ is countable. Thus, there cannot be an uncountable
$E_{1}$-homogeneous set.
So let $H\subseteq L$ be infinite with $x\wedge y\geq_{\mathrm{pref}}y\wedge
z$ for all $x<_{\mathrm{lex}}y<_{\mathrm{lex}}z$. Since we have only two
letters, we get $x\wedge y>_{\mathrm{pref}}y\wedge z$ for all
$x<_{\mathrm{lex}}y<_{\mathrm{lex}}z$ which allows to argue symmetrically to
the above. Thus, indeed, there is no uncountable homogeneous set in $L$.
###### Theorem 3.19.
For all $k\geq 3$, $\ell\geq 2$, and $\lambda>{\aleph_{0}}$, we have
$({2^{\aleph_{0}}},\omega\mathsf{iA})\not\to(\lambda,\omega\mathrm{LANG})^{k}_{\ell}$.
###### Proof 3.20.
Let $G$ be the $(3,2)$-partition from Lemma 3.17 that does not have
homogeneous sets of size $\lambda$ and let $(L,\mathrm{id})$ be an injective
$\omega$-automatic presentation of $G=(V,E_{1},E_{2})$ (in particular, $V=L$).
For a set $X\in[L]^{k}$, let $X_{1}<_{\mathrm{lex}}X_{2}<_{\mathrm{lex}}X_{3}$
be the three lexicographically least elements of $X$. Then set
$G^{\prime}=(V,E_{1}^{\prime},E_{2}^{\prime},\dots,E_{\ell}^{\prime})$ with
$\displaystyle E_{1}^{\prime}$
$\displaystyle=\\{X\in[V]^{k}\mid\\{X_{1},X_{2},X_{3}\\}\in E_{1}\\},$
$\displaystyle E_{2}^{\prime}$
$\displaystyle=\\{X\in[V]^{k}\mid\\{X_{1},X_{2},X_{3}\\}\in E_{2}\\},\text{
and }$ $\displaystyle E_{i}^{\prime}$ $\displaystyle=\emptyset\text{ for
}3\leq i\leq\ell\ .$
Then $(L,\mathrm{id})$ is an injective $\omega$-automatic presentation of
$G^{\prime}$. Now suppose $H^{\prime}\subseteq L$ is homogeneous in
$G^{\prime}$ and of size $\lambda$. Then there exists $H\subseteq H^{\prime}$
of size $\lambda$ such that for any words
$x_{1}<_{\mathrm{lex}}x_{2}<_{\mathrm{lex}}x_{3}$ from $H$, there exists
$X\subseteq H^{\prime}$ with $X_{i}=x_{i}$ for $1\leq i\leq 3$ (if necessary,
throw away some lexicographically largest elements of $H^{\prime}$). Hence $H$
is homogeneous in $G$, contradicting Lemma 3.17.
### 3.3. Complexity of homogeneous sets in $\omega$-automatic
$(2,\ell)$-partitions
Having shown that $k=2$ is a central assumption in Theorem 3.1, we now turn to
the question whether homogeneous sets of lower complexity can be found.
##### Construction
Let $V=L$ denote the regular $\omega$-language $(1^{+}0^{+})^{\omega}$.
Furthermore, $E_{1}\subseteq[L]^{2}$ comprises all 2-sets $\\{x,y\\}\subseteq
L$ such that ${\mathrm{supp}}(x)\cap{\mathrm{supp}}(y)$ is finite or
$x\sim_{e}y$. The set $E_{2}$ is the complement of $E_{1}$ in $[L]^{2}$. This
completes the construction of the $(2,2)$-partition $G=(L,E_{1},E_{2})$. Note
that $(L,\mathrm{id}_{L})$ is an injective $\omega$-automatic presentation of
$G$.
By Theorem 3.1, $G$ has an $E_{1}$\- or an $E_{2}$-homogeneous set of size
${2^{\aleph_{0}}}$. We convince ourselves that $G$ has large homogeneous sets
of both types. By Lemma 3.2, there is an $\omega$-language
$N\subseteq(1^{+}0^{+})^{\omega}$ of size ${2^{\aleph_{0}}}$ such that the
supports of any two words from $N$ have finite intersection. Hence
$[N]^{2}\subseteq E_{1}$ and $N$ has size ${2^{\aleph_{0}}}$. But there is
also an $E_{2}$-homogeneous set $L_{2}$ of size ${2^{\aleph_{0}}}$: Note that
the words from $N$ are mutually non-$\sim_{e}$-equivalent and let $L_{2}$
denote the set of all words $1a_{1}1a_{2}1a_{3}\dots$ for
$a_{1}a_{2}a_{3}\dots\in N$. Then for any $x,y\in L_{2}$ distinct, we have
$2{\mathbb{N}}\subseteq{\mathrm{supp}}(x)\cap{\mathrm{supp}}(y)$ and
$x\not\sim_{e}y$, i.e., $\\{x,y\\}\in E_{2}$.
###### Lemma 3.21.
Let $H\in\mathrm{LANG}^{*}$ have size $\lambda>{\aleph_{0}}$. Then $H$ is not
homogeneous in $G$.
###### Proof 3.22.
By definition of $\mathrm{LANG}^{*}$, there are languages
$U_{i},V_{i}\in\mathrm{LANG}$ with $H=\bigcup_{1\leq i\leq
n}U_{i}V_{i}^{\omega}$.
Since $H$ is infinite, there are $1\leq i\leq n$ and $x,y\in
U_{i}V_{i}^{\omega}$ distinct with $x\sim_{e}y$ and therefore $\\{x,y\\}\in
E_{1}$.
Since $|H|>{\aleph_{0}}$, there is $1\leq i\leq n$ with
$|U_{i}V_{i}^{\omega}|>{\aleph_{0}}$; we set $U=U_{i}$ and $V=V_{i}$. From
$|U|\leq{\aleph_{0}}$, we obtain $|V^{\omega}|>{\aleph_{0}}$. Hence there are
$v_{1},v_{2}\in V^{+}$ distinct with $|v_{1}|=|v_{2}|$. Since
$uv_{1}^{\omega}\in H$ and each element of $H$ contains infinitely many
occurrences of $1$, the word $v_{1}$ belongs to $\\{0,1\\}^{*}10^{*}$. Let
$u\in U$ be arbitrary (such a word exists since $UV^{\omega}\neq\emptyset$)
and consider the $\omega$-words $x^{\prime}=u(v_{1}v_{2})^{\omega}$ and
$y^{\prime}=u(v_{1}v_{1})^{\omega}$ from $UV^{\omega}\subseteq H$. Then
$x^{\prime}\not\sim_{e}y^{\prime}$ since $v_{1}\neq v_{2}$ and
$|v_{1}|=|v_{2}|$. At the same time,
${\mathrm{supp}}(x^{\prime})\cap{\mathrm{supp}}(y^{\prime})$ is infinite since
$v_{1}$ contains an occurrence of $1$. Hence $\\{x^{\prime},y^{\prime}\\}\in
E_{2}$.
Thus, we found $\omega$-words $x,y,x^{\prime},y^{\prime}\in H$ with
$\\{x,y\\}\in E_{1}$ and $\\{x^{\prime},y^{\prime}\\}\notin E_{1}$ proving
that $H$ is not homogeneous.
Thus, we found a $(2,2)$-partition $G=(V,E_{1},E_{2})$ with ${2^{\aleph_{0}}}$
elements and an injective $\omega$-automatic presentation $(L,h)$ such that
1. (1)
$G$ has sets $L_{1}$ and $L_{2}$ in $\mathrm{co}\text{-}\omega\mathrm{erCF}$
of size ${2^{\aleph_{0}}}$ with $[L_{i}]^{2}\subseteq E_{i}$ for $1\leq i\leq
2$.
2. (2)
There is no $\omega$-language $H\in\mathrm{LANG}^{*}$ with $H\subseteq L$ such
that $h(H)$ is homogeneous of size ${2^{\aleph_{0}}}$.
Since all context-free $\omega$-languages belong to $\mathrm{LANG}^{*}$, the
following theorem follows the same way that Lemma 3.17 implied Theorem 3.19.
###### Theorem 3.23.
For all $k,\ell\geq 2$ and $\lambda>{\aleph_{0}}$, we have
$({2^{\aleph_{0}}},\omega\mathsf{iA})\not\to(\lambda,\omega\mathrm{CF})^{k}_{\ell}$
and
$({2^{\aleph_{0}}},\omega\mathsf{iA})\not\to(\lambda,\omega\mathrm{REG})^{k}_{\ell}$.
This result can be understood as another Sierpiński theorem for
$\omega$-automatic $(k,\ell)$-partitions. This time, it holds for all $k\geq
2$ (not only for $k\geq 3$ as Theorem 3.19). The price to be paid for this is
the restriction of homogeneous sets to “simple” ones. In particular the non-
existence f regular homogeneous sets provides a Sierpiński theorem in the
spirit of automatic structures.
## Open questions
Our positive result Theorem 3.1 guarantees the existence of some clique or
anticlique of size ${2^{\aleph_{0}}}$ (and such a clique or anticlique can
even be constructed). But the following situation is conceivable: the
$\omega$-automatic graph contains large cliques without containing large
cliques that can be described by a language from
$\mathrm{co}\text{-}\omega\mathrm{erCF}$. In particular, it is not clear
whether the existence of a large clique is decidable.
A related question concerns Ramsey quantifiers. Rubin [Rub08] has shown that
the set of nodes of an automatic graph whose neighbors contain an infinite
anticlique is regular (his result is much more general, but this formulation
suffices for our purpose). It is not clear whether this also holds for
$\omega$-automatic graphs. A positive answer to this second question (assuming
that it is effective) would entail an affirmative answer to the decidability
question above.
## References
* [BG04] A. Blumensath and E. Grädel. Finite presentations of infinite structures: Automata and interpretations. Theory of Computing Systems, 37(6):641–674, 2004.
* [BKR08] V. Bárány, Ł. Kaiser, and S. Rubin. Cardinality and counting quantifiers on omega-automatic structures. In STACS’08, pages 385–396. IFIB Schloss Dagstuhl, 2008.
* [Blu99] A. Blumensath. Automatic structures. Technical report, RWTH Aachen, 1999.
* [Del04] Ch. Delhommé. Automaticité des ordinaux et des graphes homogènes. C. R. Acad. Sci. Paris, Ser. I, 339:5–10, 2004.
* [ER56] P. Erdős and R. Rado. A partition calculus in set theory. Bull. AMS, 62:427–489, 1956.
* [Gas98] W. Gasarch. A survey of recursive combinatorics. In Handbook of recursive mathematics vol. 2, volume 139 of Stud. Logic Found. Math., pages 1040–1176. North-Holland, Amsterdam, 1998\.
* [HKMN08] G. Hjorth, B. Khoussainov, A. Montalbán, and A. Nies. From automatic structures to borel structures. In LICS’08, pages 431–441. IEEE Computer Society Press, 2008.
* [Jec02] Th. Jech. Set Theory. Springer Monographs in Mathematics. Springer, 3rd edition, 2002.
* [Joc72] C.G. Jockusch. Ramsey’s theorem and recursion theory. Journal of Symbolic Logic, 37:268–280, 1972.
* [KL08] D. Kuske and M. Lohrey. First-order and counting theories of $\omega$-automatic structures. Journal of Symbolic Logic, 73:129–150, 2008.
* [KN95] B. Khoussainov and A. Nerode. Automatic presentations of structures. In Logic and Computational Complexity, Lecture Notes in Comp. Science vol. 960, pages 367–392. Springer, 1995.
* [Ram30] F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264–286, 1930.
* [Rub08] S. Rubin. Automata presenting structures: A survey of the finite string case. Bulletin of Symbolic Logic, 14:169–209, 2008.
* [Sie33] W. Sierpiński. Sur un problème de la thèorie des relations. Ann. Scuola Norm. Sup. Pisa, 2(2):285–287, 1933.
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|
arxiv-papers
| 2009-12-14T12:46:45 |
2024-09-04T02:49:07.035782
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dietrich Kuske",
"submitter": "Dietrich Kuske",
"url": "https://arxiv.org/abs/0912.2625"
}
|
0912.2768
|
# Anomalous low temperature ambipolar diffusion and Einstein relation
A. L. Efros efros@physics.utah.edu Department of Physics, University of Utah,
Salt Lake City UT, 84112 USA
###### Abstract
Regular Einstein relation, connecting the coefficient of ambipolar diffusion
and the Dember field with mobilities, is generalized for the case of
interacting electron-hole plasma. The calculations are presented for a non-
degenerate plasma injected by light in semiconductors of silicon and germanium
type. The Debye-Huckel correlation and the Wigner-Seitz exchange terms are
considered. The corrections to the mobilities of carriers due to difference
between average and acting electric fields within the electron-hole plasma is
taken into account. The deviation of the generalized relation from the regular
Einstein relation is pronounced at low temperatures and can explain anomaly of
the coefficient of ambipolar diffusion, recently discovered experimentally.
###### pacs:
71.27.+a,71.35.Ee, 78.20.Jq, 78.56.-a
## .1 Introduction
The Einstein relation for electronsLandau and Lifshitz (1984) $nUd\mu/dn=eD$
connects mobility $U$ with diffusion coefficient $D$ by the thermodynamic
function $d\mu/dn$ and electron charge $e$. Here $\mu$ is chemical potential,
$n$ is electron density. For the case of the Boltzmann gas of non-interacting
electrons, where $d\mu/dn=kT/n$, one gets a regular form of the Einstein
relation, presented by Einstein Einstein (1905) and von Smoluchowsky von
Smoluchowsky (1906) for the Brownian particles. Here $kT$ is the temperature
in energy units. However, the Einstein relation can be used in a general case
of interacting particlesA.L.Efros (2008).
The focus of this paper is the ambipolar diffusivity under condition that
interaction between electrons and holes is substantial. I have been initiated
by the paper of Hui ZhaoZhao (2008) who has measured the coefficient of the
ambipolar diffusivity (CAD) by optical method in silicon-on-insulator (SOI)
structure with 750 nm silicon layer. The doping density is $10^{15}{\rm
cm^{-3}}$ while the density $n$ of electron-hole pairs excited by light is
between $(0.5-3)\times 10^{17}{\rm cm^{-3}}$. The temperature is in the range
$90K-400K$.
The regular Einstein relation for the CAD in non-degenerate and non-
interacting electron-hole plasma has a form
$D_{a}=\frac{2kT}{e}\frac{U_{p}U_{n}}{U_{p}+U_{n}},$ (1)
where $U_{n},U_{p}$ are mobilities of electrons and holes respectively. The
CAD observed in Ref.Zhao (2008) approximately follows Eq.(1) at $400K>T>300K$.
At lower temperatures the CAD goes down and deviates from the Eq.(1)
approximately 7 times at $T=90K$. The mobilities were taken from experiments
with a pure bulk silicon. Author explained this anomalous behavior as a result
of uncontrolled defects that are absent in a bulk silicon.
I propose here alternative and more universal intrinsic explanation of the low
temperature anomaly. This explanation is based upon a novel form of Einstein’s
fundamental relation for the CAD, that takes into account interaction between
excited carriers.
Under conditions of the experiment the electron-hole plasma can be considered
as a non-degenerate. On the other hand, the classical correlation energy per
particle $\sim e^{2}n^{1/3}/\kappa$ is of the order of 200K. Here $\kappa$ is
dielectric constant. Thus, in the region of the observed anomaly the
interaction energy becomes of the order of temperature.
The paper is organized as follows. First the novel form of the fundamental
Einstein relation connecting the CAD and the Dember field with mobilities of
electrons and holes is derived. The relation contains the derivatives of the
Helmholtz energy density (HED) of the interacting plasma with respect to
particle densities. The HED is calculated taken into account correlation and
exchange between particles. Then the corrections to the mobilities due to
deviation of acting electric field from the applied field are considered.
Finally the theoretical results are compared with the experimental data.
## .2 Einstein relation for ambipolar diffusivity. Thermodynamics approach
Silicon is an example of semiconductor with a long recombination time of
interband excitation. Assume that this system is in the thermodynamic
equilibrium with respect to all relevant parameters except the total numbers
of electrons and holes. To derive the Einstein relation I use here the same
method as in Ref(A.L.Efros (2008)). The difference is that excited electrons
and holes have two independent electrochemical potentials $\Phi^{n}$ and
$\Phi^{p}$ respectively. The Helmholtz energy has a form
$\displaystyle F$ $\displaystyle=$ $\displaystyle\int f(n,p)d^{3}r+\int
e(p({\bf r})-n({\bf r}))\psi d^{3}r$ (2) $\displaystyle-$
$\displaystyle\Phi^{p}\int p({\bf r})d^{3}r-\Phi^{n}\int n({\bf r})d^{3}r.$
Here $n$ and $p$ are electron and hole densities. Function $f(n,p)$ is the HED
of the almost neutral and microscopically homogenous electron-hole plasma, the
function $\psi({\bf r})$ is a potential of a static electric field.
Using conditions $\delta F/\delta p=0$ and $\delta F/\delta n=0$ one gets
$\Phi^{p}=\frac{\partial f(n,p)}{\partial p}+e\psi$ (3)
and
$\Phi^{n}=\frac{\partial f(n,p)}{\partial n}-e\psi.$ (4)
It follows from the general principles of statistical physics that that in the
state of equilibrium both $\Phi^{p},\Phi^{n}$ should be constant along the
system. Exploring Einstein’s idea that electric field is equivalent to a
certain density gradient one can write the fluxes of holes and electrons as
${\bf q}_{p}=-\frac{\sigma_{p}}{e^{2}}\nabla\Phi^{p},{\bf
q}_{n}=-\frac{\sigma_{n}}{e^{2}}\nabla\Phi^{n},$ (5)
where $\sigma_{p}$ and $\sigma_{n}$ are conductivities of holes and electrons
respectively. Then the fluxes are
${\bf q}_{p}=\frac{\sigma_{p}}{e}{\bf
E}-\frac{\sigma_{p}}{e^{2}}\left(\frac{\partial^{2}f}{\partial p^{2}}\nabla
p+\frac{\partial^{2}f}{\partial p\partial n}\nabla n\right)$ (6)
and
${\bf q}_{n}=-\frac{\sigma_{n}}{e}{\bf
E}-\frac{\sigma_{n}}{e^{2}}\left(\frac{\partial^{2}f}{\partial n^{2}}\nabla
n+\frac{\partial^{2}f}{\partial p\partial n}\nabla p\right).$ (7)
Note that separation of the conductivities of electrons and holes are possible
only if their mutual scattering is small. We assume here that the mobilities
of the carriers are controlled by the lattice scattering. But even in this
case the above expressions predict a drag effect due to the interaction terms
in the HED. The flux of holes is proportional to gradient of electron density
and to gradient of hole density. The same is true for the flux of electrons.
For example, if electric field is zero and gradient of electron density is
zero, there is an electron flux proportional to a gradient of hole density.
The ambipolar diffusion is measured under condition that electrical circuit is
open. Then the total electric current ${\bf j}$ is zero and ${\bf q}_{p}={\bf
q}_{n}$. Taking into account the continuity equation
$e\partial(p-n)/\partial t+{\rm div}{\bf j}=0.$ (8)
one finds that since electron and holes are excited by light in equal amounts
the system is neutral everywhere. Then $n({\bf r})=p({\bf r})$ and $\nabla
n=\nabla p$. A small separation of charges at the boundaries of the sample
appears due to a difference of diffusion coefficients of electrons and holes.
This difference is compensated by an electric field called the Dember field.
Nevertheless, the electron-hole plasma in the bulk of the sample is neutral.
The expressions for flaxes can be written in a form
${\bf q_{p}}=\frac{\sigma_{p}}{e}{\bf E}-D_{p}\nabla p$ (9)
and
${\bf q_{n}}=-\frac{\sigma_{n}}{e}{\bf E}-D_{n}\nabla n.$ (10)
It follows from Eqs.(6,7,9,10) that diffusion coefficients of electrons and
holes $D_{n}$ and $D_{p}$ are connected with corresponding mobilities
$U_{n},U_{p}$ by the relations
$D_{n}=\left(\frac{\partial^{2}f}{\partial
n^{2}}+\frac{\partial^{2}f}{\partial n\partial p}\right)\frac{nU_{n}}{e}$ (11)
and
$D_{p}=\left(\frac{\partial^{2}f}{\partial
p^{2}}+\frac{\partial^{2}f}{\partial n\partial p}\right)\frac{nU_{p}}{e}.$
(12)
Eqs.(11,12) give a novel form of the Einstein relation that is valid in the
case of interacting plasma.
The mixed partial derivatives in these equations describe the drag effect.
Using Eqs.(9,10) and condition ${\bf q}_{p}={\bf q}_{n}$ one gets expression
for the Dember field in terms of $D_{n}$, $D_{p}$
${\bf E}_{D}=\frac{e(D_{p}-D_{n})}{\sigma_{p}+\sigma_{n}}\nabla n.$ (13)
This regular form becomes more complicated if $D_{n}$ and $D_{p}$ are
expressed through mobilities $U_{n}$ and $U_{p}$ using Eqs.(11, 12). Then
$\displaystyle{\bf E}_{D}$ $\displaystyle=$
$\displaystyle\left[e(U_{n}+U_{p})\right]^{-1}\left(\frac{\partial^{2}f}{\partial
n^{2}}U_{n}-\frac{\partial^{2}f}{\partial p^{2}}U_{p}\right.$ (14)
$\displaystyle+$ $\displaystyle\left.\frac{\partial^{2}f}{\partial n\partial
p}(U_{n}-U_{p})\right)\nabla n.$
Due to the interaction of carriers the Dember field is not necessarily
proportional to the difference of mobilities $U_{n}-U_{p}$.
Substituting Eq.(15) into Eqs.(9,10) one gets
${\bf q_{n}=q_{p}}=-D_{a}\nabla n,$ (15)
where the CAD
$D_{a}=\frac{D_{n}U_{p}+D_{p}U_{n}}{U_{p}+U_{n}}.$ (16)
Using Eqs.(11,12) we get the generalized Einstein relation for the CAD
$D_{a}=\frac{2kT}{e}\frac{U_{p}U_{n}}{U_{p}+U_{n}}Q(n,T),$ (17)
where
$Q(n,T)=\frac{n}{2kT}\left(\frac{\partial^{2}f}{\partial
n^{2}}+\frac{\partial^{2}f}{\partial p^{2}}+2\frac{\partial^{2}f}{\partial
n\partial p}\right)$ (18)
is a ratio of the coefficients of ambipolar diffusivity calculated with and
without interaction (cp Eq.(17) with Eq.(1)).
It is important to put $n=p$ after calculation of the second partial
derivatives in Eqs.(11, 12,14,18).
## .3 HED of semiconductors with band structure of Si and Ge
Analytical calculations of the HED of interacting carriers are possible in the
framework of perturbation theory only. One should keep in mind, however, that
the region of applicability of these calculations does not cover all
temperature range of the experiment.
The HED can be written in a form $f=f_{id}+f_{i}$, where the first term
describes the ideal gas, while the second one takes into account interaction.
The non-interacting carriers are independent and
$f_{id}=f^{p}_{id}(p)+f^{n}_{id}(n)$. Since only the second derivatives of the
HED are necessary, one can write $f_{id}(p)=pkT(1+\ln p)$ and
$f_{id}(n)=nkT(1+\ln n)$.
The largest interaction term for the non-degenerate plasma describes
correlation effect. It was calculated by Debye and HuckelDebye and Huckel
(1923) in a form
$f_{c}(n+p)=-(2e^{3}/3\kappa^{3/2})\sqrt{\pi/{kT}}(n+p)^{3/2}.$ (19)
This term is independent of the spectra of electrons and holes. In our
approximation this is the only term that has non-zero $\partial^{2}f/(\partial
n\partial p)$ and contributes to the drag effect (See Eqs.(6,7)).
I also take into account the exchange interaction between electrons in each
ellipsoid, between heavy holes and between light holes. This interaction term
has a higher power of $T$ in the denominator of the HED than the correlation
term. The thermodynamic potential density $\Omega(\mu,T)$ for this interaction
can be written in a form of the Wigner-Seitz integral (SeeL.D.Landau and
E.M.Lifshitz (1980))
$\Omega_{ex}=-\frac{4\pi
e^{2}}{\hbar^{4}\kappa}\int\int\frac{n_{p1}n_{p2}d^{3}p_{1}d^{3}p_{2}}{(\vec{p_{1}}-\vec{p_{2}})^{2}(2\pi)^{6}},$
(20)
where $n_{p}$ is the Fermi function that has the Boltzmann form in this case.
To find the HED one should express chemical potentials through the density of
carriers.
For a conduction band consisting of $g$ equivalent ellipsoids of rotation one
gets
$f_{ex}(n)=-\frac{n^{2}e^{2}\hbar^{2}I(a)}{4\kappa\sqrt{\pi}gm_{\perp}kT},$
(21)
where masses $m_{\perp}$ and $m_{\parallel}$, are perpendicular and parallel
to the rotation axis of an ellipsiod, $a=m_{\parallel}/m_{\perp}$. At $a>1$
$I(a)=\frac{2\sqrt{\pi}\arctan(\sqrt{a-1}}{\sqrt{a-1}}.$ (22)
For a parabolic valence band with light hole $m_{l}$ and heavy hole $m_{h}$
$f_{ex}(p)=-\frac{\pi e^{2}\hbar^{2}p^{2}(m_{h}^{2}+m_{l}^{2})}{2\kappa
kT(m_{h}^{3/2}+m_{l}^{3/2})^{2}}.$ (23)
My final result for corrections to a regular Einstein relation reads
$\displaystyle Q(n,T)$ $\displaystyle=$ $\displaystyle 1-\frac{e^{3}\sqrt{\pi
n}}{\sqrt{2}\kappa^{3/2}(kT)^{3/2}}-\frac{e^{2}\hbar^{2}nI(a)}{4\sqrt{\pi}\kappa
m_{\perp}g(kT)^{2}}$ (24) $\displaystyle-$ $\displaystyle\frac{\pi
e^{2}\hbar^{2}n(m_{h}^{2}+m_{l}^{2})}{2\kappa(kT)^{2}(m_{h}^{3/2}+m_{h}^{3/2})^{2}}.$
The Dember field in terms of mobilities has a form
$\displaystyle E_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{(eU_{n}+eU_{p})}\left(\left(\frac{kT}{n}+\frac{e^{3}\sqrt{\pi}}{\kappa^{3/2}\sqrt{2kTn}}\right)(U_{p}-U_{n})\right.$
(25) $\displaystyle-$ $\displaystyle\left.\frac{\pi
e^{2}\hbar^{2}(m_{h}^{2}+m_{l}^{2})U_{p}}{\kappa
kT(m_{h}^{3/2}+m_{l}^{3/2})^{2}}+\frac{e^{2}\hbar^{2}I(a)U_{n}}{2\kappa\sqrt{\pi}gm_{\perp}kT}\right)\nabla
n.$
## .4 Mobility of carriers in a plasma
It is assumed above that mobilities of the carriers are controlled by the
lattice scattering. But even in this case there are important corrections to
these mobilities due to the interaction of electrons and holes. Each carrier
is surrounded by a screening atmosphere of the opposite sign. This atmosphere
is polarized by an applied electric field. The field of this polarization is
opposite to the applied field so that the effective field acting on the
carrier is less than applied field. It can be interpreted as a decrease of the
mobility. The theory of this effect was created by Debye, Huckel, and Onsager
(See Ref.E.M.Lifshitz and L.P.Pitaevskii (1981)). The resulting changes of the
mobilities are
$S(n,T)=\frac{\Delta U_{n}}{U_{n}}=\frac{\Delta
U_{p}}{U_{p}}=-\frac{\sqrt{2\pi}e^{3}n^{1/2}}{3(1+\sqrt{0.5})\kappa^{3/2}(kT)^{3/2}}.$
(26)
Coming back to the Einstein relation Eq.(17) one should note that if the
mobilities $U_{n},U_{p}$ are measured in the presence of plasma, the above
corrections are irrelevant because the experimental values $U_{n},U_{p}$
contain them. However, if the mobilities are known from experiments without
light excitation, as in the case of Ref.Zhao (2008), the Einstein relation
Eq.(17) takes a form
$D_{a}=\frac{2T}{e}\frac{U_{p}U_{n}}{U_{p}+U_{n}}P(n,T),$ (27)
where $P(n,T)=Q(n,T)+S(n,T)$. One can see that this change increases the
numerical factor in the second term of $Q$, originated from correlation, by
1.39.
## .5 Discussion and Conclusion
Figure 1: (Color online)Function $P(n,T)$ for Si at three different values of
$n_{0}$ defined as $n=n_{0}\times 2.3\times 10^{17}\rm{cm}^{-3}$; $n_{0}=2$
for lower curve (blue), $n_{0}=1$ for two intermediate curves, and $n_{0}=0.5$
for upper curve(green). The upper intermediate curve does not take into
account exchange interaction.
Figure 2: (Color online)Temperature dependence of $D_{a}$ obtained under
assumption that $U_{p}\sim T^{-2.2}$ at $n=2.3\times 10^{17}\rm{cm}^{-3}$. The
absolute values of the hole mobility is chosen such that maximum value of
$D_{a}=20{\rm cm^{2}/sec}$. Maximum occurs at $T\approx 230K$, which is very
close to the experimental result.
Now I discuss the low temperature anomaly of the CAD in Si. Function $P(n,T)$
is shown in Fig.1 in a proper temperature range. The comparison of the first
approximation (correlation) with the second one (exchange) shows that the
perturbation theory looks reasonable at $T\geq 150$K and at $n_{0}=1$.
To estimate CAD as a function of $T$ one should know mobilities $U_{n}$ and
$U_{p}$. The experimental and theoretical data of Ref.Jacoboni et al. (1976)
show that in silicon $U_{p}/U_{n}\approx 0.25$, at $T\approx 200$K. Then
$U_{p}U_{n}/(U_{p}+U_{n})\approx U_{p}$. The hole mobility in the pure silicon
is due to the phonon scattering and it depends on temperature as $T^{-2.2}$ in
all temperature range considered. The deviation from a usual law $T^{-1.5}$ is
due to the warping of the top of the valence band. The mobility of doped
silicon with the hole density $2\times 10^{17}cm^{-3}$ has $T^{-1.5}$
dependenceJacoboni et al. (1976) at $T>180K$ that may be interpreted as a
phonon scattering but with the warping smeared by the doping. A pure silicon
is considered here, and $T^{-2.2}$ mobility dependence is used.
The expression $D_{a}=P(T)U_{p}T$ with $U_{p}=RT^{-2.2}$ is used to get
T-dependence of $D_{a}$ that follows from the above theory. To make comparison
with experimental result easier the factor $R$ is chosen such that
$D_{a}=20{\rm cm^{2}/sec}$ in the maximum, similar to the experimental data of
RefZhao (2008). The theoretical result is shown in Fig. 2.
Since the HED is calculated using perturbation theory, the discussion of the
low temperature behavior might be doubtful, and the most important argument is
position of the maximum of CAD. Clearly the way factor $R$ is chosen has no
effect on the maximum position. Theoretical position of maximum is 230K, which
is close to the experimental position that has some uncertainty because of the
large error bars. I think this similarity is a strong argument in favor of the
proposed explanation.
In the range $T>150K$ the the theoretical curve is similar to experimental
points of Ref(Zhao (2008)). At lower $T$ theoretical values become negative.
This definitely means collapse of the perturbation theory, because negative
CAD leads to an absolute instability of a neutral plasmaA.L.Efros (2008). It
is intriguing that just near this temperature the character of experimental
data changes: CAD becomes independent of both $T$ and $n$. This might be a
manifestation of a new phase. The speculations about this phase are outside
the scope of this paper. I would only mention that amount of excitons at these
temperatures, as given by the Saha equation, is negligible but the Saha
equation is not reliable for the case of non-ideal plasma. It is known also
that the phase of exciton gas-liquid coexistence corresponds to lower
temperature at these densitiesKittel (2005).
Finally, I proposed an explanation of the low temperature anomaly of the CAD
based upon the Einstein relation for interacting carriers. The applicability
of the theory is limited at low enough temperatures because the calculation of
the HED is perturbational. Using fundamental thermodynamics relations
Eqs.(15,17,18) one could restore unknown thermodynamic functions of the non-
ideal plasma at low temperatures by measuring mobilities, CAD, and the Dember
field.
I am grateful to M.I. Dyakonov for an important critical comment and to Hui
Zhao for a valuable discussion.
## References
* Landau and Lifshitz (1984) L. D. Landau and E. M. Lifshitz, _Electrodynamics of Continuous Media_ (Butterworth-Heinenann, 1984), chapter III.
* Einstein (1905) A. Einstein, Annalen der Physik 17, 549 (1905).
* von Smoluchowsky (1906) M. von Smoluchowsky, Annalen der Physik 21, 756 (1906).
* A.L.Efros (2008) A.L.Efros, Phys. Rev. B 78, 155130 (2008).
* Zhao (2008) H. Zhao, Appl. Phys. Lett. 92, 112104 (2008).
* Debye and Huckel (1923) P. Debye and E. Huckel, Physik Z. 24, 185 (1923).
* L.D.Landau and E.M.Lifshitz (1980) L.D.Landau and E.M.Lifshitz, _Statistical Physics_ (Butterworth-Heinenann Ltd, 1980), 3rd Edition Part1.
* E.M.Lifshitz and L.P.Pitaevskii (1981) E.M.Lifshitz and L.P.Pitaevskii, _Physical Kinetics_ (Butterworth-Heinenann Ltd, 1981), chapter 2.
* Jacoboni et al. (1976) C. Jacoboni, C.Canali, G. Ottaviani, and A. A. Quaranta, Solid State Electronics 20, 77 (1976).
* Kittel (2005) C. Kittel, _Introduction to Solid State Physics_ (John Willey&Sons, Inc., 2005), eight Edition, p.443.
|
arxiv-papers
| 2009-12-14T22:25:59 |
2024-09-04T02:49:07.045245
|
{
"license": "Public Domain",
"authors": "A. L. Efros (University of Utah, USA)",
"submitter": "Alexei Efros",
"url": "https://arxiv.org/abs/0912.2768"
}
|
0912.2815
|
2010609-620Nancy, France 609
David Peleg
Liam Roditty
# Relaxed spanners for directed disk graphs
D. Peleg Department of Computer Science and Applied Mathematics,
The Weizmann Institute of Science, Rehovot 76100, Israel
david.peleg@weizmann.ac.il and L. Roditty Department of Computer Science,
Bar-Ilan University,
Ramat-Gan 52900, Israel liamr@macs.biu.ac.il
###### Abstract.
Let $(V,\delta)$ be a finite metric space, where $V$ is a set of $n$ points
and $\delta$ is a distance function defined for these points. Assume that
$(V,\delta)$ has a constant doubling dimension $d$ and assume that each point
$p\in V$ has a disk of radius $r(p)$ around it. The disk graph that
corresponds to $V$ and $r(\cdot)$ is a _directed_ graph $I(V,E,r)$, whose
vertices are the points of $V$ and whose edge set includes a directed edge
from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In [8] we presented an algorithm
for constructing a $(1+\epsilon)$-spanner of size $O(n/\epsilon^{d}\log M)$,
where $M$ is the maximal radius $r(p)$. The current paper presents two
results. The first shows that the spanner of [8] is essentially optimal, i.e.,
for metrics of constant doubling dimension it is not possible to guarantee a
spanner whose size is independent of $M$. The second result shows that by
slightly relaxing the requirements and allowing a small perturbation of the
radius assignment, considerably better spanners can be constructed. In
particular, we show that if it is allowed to use edges of the disk graph
$I(V,E,r_{1+\epsilon})$, where $r_{1+\epsilon}(p)=(1+\epsilon)\cdot r(p)$ for
every $p\in V$, then it is possible to get a $(1+\epsilon)$-spanner of size
$O(n/\epsilon^{d})$ for $I(V,E,r)$. Our algorithm is simple and can be
implemented efficiently.
###### Key words and phrases:
Spanners, Directed graphs
###### 1991 Mathematics Subject Classification:
F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, F.2.0 General
Thanks: Supported in part by grants from the Minerva Foundation and the Israel
Ministry of Science
## Introduction
This paper concerns efficient constructions of spanners for disk graphs, an
important family of directed graphs. A spanner is essentially a skeleton of
the graph, namely, a sparse spanning subgraph that faithfully represents
distances. Formally, a subgraph $H$ of a graph $G$ is a $t$-spanner of $G$ if
$\delta_{H}(u,v)\leq t\cdot\delta_{G}(u,v)$ for every two nodes $u$ and $v$,
where $\delta_{G^{\prime}}(u,v)$ denotes the distance between $u$ and $v$ in
$G^{\prime}$. We refer to $t$ as the stretch factor of the spanner. Graph
spanners have received considerable attention over the last two decades, and
were used implicitly or explicitly as key ingredients of various distributed
applications. It is known how to efficiently construct a $(2k-1)$-spanner of
size $O(n^{1+1/k})$ for every weighted undirected graph, and this size-stretch
tradeoff is conjectured to be tight. Baswana and Sen [BaSe07] presented a
linear time randomized algorithm for computing such a spanner. In directed
graphs, however, the situation is different. No such general size-stretch
tradeoff can exist, as indicated by considering the example of a directed
bipartite graph $G$ in which all the edges are directed from one side to the
other; clearly, the only spanner of $G$ is $G$ itself, as any spanner for $G$
must contain every edge.
The main difference between undirected and directed graphs is that in
undirected graphs the distances are symmetric, that is, a path of a certain
length from $u$ to $v$ can be used also from $v$ to $u$. In directed graphs,
however, the existence of a path from $u$ to $v$ does not imply anything on
the distance in the opposite direction from $v$ to $u$. Hence, in order to
obtain a spanner for a directed graph one must impose some restriction either
on the graph or on its distances. In order to bypass the problem of asymmetric
distances of directed graphs, Cowen and Wagner [5] introduced the notion of
roundtrip distances in which the distance between $u$ and $v$ is composed of
the shortest path from $u$ to $v$ plus the shortest path from $v$ to $u$. It
is easy to see that under this definition distances are symmetric also in
directed graphs. It is shown by Cowen and Wagner [5] and later by Roditty,
Thorup and Zwick [6] that methods of path approximations from undirected
graphs can work using more ideas also in directed graphs when roundtrip
distances are considered. Bollobás, Coppersmith and Elkin [BoCoEl05]
introduced the notion of distance preservers and showed that they exist also
in directed graphs.
In [8] we presented a spanner construction for directed graphs without
symmetric distances. The restriction that we imposed on the graph was that it
must be a disk graph. More formally, let $(V,\delta)$ be a finite metric space
of constant doubling dimension $d$, where $V$ is a set of $n$ points and
$\delta$ is a distance function defined for these points. A metric is said to
be of constant doubling dimension if a ball with radius $r$ can be covered by
at most a constant number of balls of radius $r/2$. Every point $p\in V$ is
assigned with a radius $r(p)$. The disk graph that corresponds to $V$ and
$r(\cdot)$ is a directed graph $I(V,E,r)$, whose vertices are the points of
$V$ and whose edge set includes a directed edge from $p$ to $q$ if $q$ is
inside the disk of $p$, that is, $\delta(p,q)\leq r(p)$. In [8] we presented
an algorithm for constructing a $(1+\epsilon)$-spanner with size
$O(n/\epsilon^{d}\log M)$, where $M$ is the maximal radius. In the case that
we remove the radius restriction the resulted graph is the complete undirected
graph where the weight of every edge is the distance between its endpoint. In
such a case it is possible to create $(1+\epsilon)$-spanners of size
$O(n/\epsilon^{d})$, see [4], [2] and [9] for more details. Moreover, when the
radii are all the same and the graph is the unit disk graph then it is also
possible to create $(1+\epsilon)$-spanners of size $O(n/\epsilon^{d})$, see
[3], [8].
As a result of that, a natural question is whether a spanner size of
$O(n/\epsilon^{d}\log M)$ in the case of directed disk graph is indeed the
best possible or maybe it is possible to get a spanner of size
$O(n/\epsilon^{d})$ as in the cases of the complete graph and the unit disk
graph. For the case of the Euclidean metric space, the answer turns out to be
positive; a simple modification of the Yao graph construction [11] to fit the
directed case yields a directed spanner of size $O(n/\epsilon^{d})$. However,
the question remains for more general metric spaces, and in particular for the
important family of metric spaces of bounded doubling dimension.
In this paper we provide an answer for this question. We show that our
construction from [8] is essentially optimal by providing a metric space with
a constant doubling dimension and a radius assignment whose corresponding disk
graph has $\Omega(n^{2})$ edges and none of its edges can be removed. (This
does not contradict our spanner construction from [8] as the maximal radius in
that case is $\Theta(2^{n})$ and hence $\log M=n$.)
This (essentially negative) optimality result motivates our main interest in
the current paper, which focuses on attempts to slightly relax the assumptions
of the model, in order to obtain sparser spanner constructions. Indeed, it
turns out that such sparser spanner constructions are feasible under a
suitably relaxed model. Specifically, we demonstrate the fact that if a small
perturbation of the radius assignment is allowed, then a
$(1+\epsilon)$-spanner of size $O(n/\epsilon^{d})$ is attainable. More
formally, we show that if we are allowed to use edges of the disk graph
$I(V,E,r_{1+\epsilon})$, where $r_{1+\epsilon}(p)=(1+\epsilon)\cdot r(p)$ for
every $p\in V$, then it is possible to get a $(1+\epsilon)$-spanner of size
$O(n/\epsilon^{d})$ for the original disk graph $I(V,E,r)$. This approach is
similar in its nature to the notation of emulators introduced by Dor, Halperin
and Zwick [1]. An emulator of a graph may use any edge that does not exist in
the graph in order to approximate its distances. It was used in the context of
spanners with an additive stretch.
The main application of disk graph spanners is for topology control in the
wireless ad hoc network model. In this model the power required for
transmitting from $p$ to $q$ is commonly taken to be $\delta(p,q)^{\alpha}$,
where $\delta(p,q)$ denotes the distance between $p$ and $q$ and $\alpha$ is a
constant typically assumed to be between $2$ and $4$. Most of the ad hoc
network literature makes the assumption that the transmission range of all
nodes is identical, and consequently represents the network by a _unit disk
graph_ (UDG), namely, a graph in which two nodes $p,q$ are adjacent if their
distance satisfies $\delta(p,q)\leq 1$. A unit disk graph can have as many as
$O(n^{2})$ edges.
There is an extensive body of literature on spanners of unit disk graphs. Gao
et al. [3], Wang and Yang-Li [10] and Yang-Li et al. [7] considered the
restricted Delaunay graph, whose worst-case stretch is constant (larger than
$1+\epsilon$). In [8] we showed that any $(1+\epsilon)$-geometric spanner can
be turned into a $(1+\epsilon)$-UDG spanner.
Disk graphs are a natural generalization of unit disk graphs, that provide an
intermediate model between the complete graph and the unit disk graph. Our
size efficient spanner construction for disk graphs whose radii are allowed to
be slightly larger falls exactly into the model of networks in which the
stations can change their transmission power. In particular our constriction
implies that if any station increases its transmission power by a small
fraction then a considerably improved topology can be built for the network.
Our result has both practical and theoretical implications. From a practical
point of view it shows that, in certain scenarios, extending the transmission
radii even by a small factor can significantly improve the overall quality of
the network topology. The result is also very intriguing from a theoretical
standpoint, as to the best of our knowledge, our relaxed spanner is the first
example of a spanner construction for directed graphs that enjoys the same
properties as the best constructions for undirected graphs. (As mentioned
above, it is easy to see that for general directed graphs, it is not possible
to have an algorithm that given any directed graph produces a sparse spanner
for it.) In that sense, our result can be viewed as a significant step towards
gaining a better understanding for some of the fundamental differences between
directed and undirected graphs. Our result also opens several new research
directions in the relaxed model of disk graphs. The most obvious research
questions that arise are whether it is possible to obtain other objects that
are known to exist in undirected graphs, such as compact routing schemes and
distance oracles, for disk graphs as well.
The rest of this paper is organized as follows. In the next section we present
a metric space of constant doubling dimension in which no edge can be removed
from its corresponding disk graph. Section 2 first describes a simple variant
of our construction from [8], and then uses it together with new ideas in
order to obtain our new relaxed construction. Finally, in Section 3 we present
some concluding remarks and open problems.
## 1\. Optimality of the spanner construction
In this section we build a disk graph $G$ with $2n$ vertices and
$\Omega(n^{2})$ edges that is non-sparsifiable, namely, whose only spanner is
$G$ itself. In this graph $M=\Omega(2^{n})$ hence our spanner construction
from [8] has a size of $\Omega(n^{2})$ and is essentially optimal.
Given a set of points, we present a distance function such that for a given
assignment of radii for the points any spanner of the resulting disk graph
must have $\Omega(n^{2})$ edges. We then prove that the underlying metric
space has a constant doubling dimension.
We partition the points into two types, $Y=\\{y_{1},\ldots,y_{n}\\}$ and
$X=\\{x_{1},\ldots,x_{n}\\}$. We now define the distance function
$\delta(\cdot,\cdot)$ and the radii assignment $r(\cdot)$. The main idea is to
create a bipartite graph $G(X,Y,E)$ in which every point of $Y$ is connected
by a directed edge to all the points of $X$.
The distance between any two points $x_{i}$ and $x_{j}$ is at least
$1+\epsilon$ for some small $0<\epsilon<1$ and the radius assignment of every
point $x_{i}$ is exactly $1$. Thus, there are no edges between the points of
$X$.
We now define the distances between the points of $Y$ and the points of $X$.
We start with the point $y_{1}$. Let $\delta(y_{1},x_{i})=n$ for every
$x_{i}\in X$ and let $r(y_{1})=n$. Place the points of $X$ on the boundary of
a ball of radius $n$ centered at $y_{1}$ such that the distance between any
two consecutive points $x_{i}$ and $x_{i+1}$ is exactly $1+\epsilon$. This is
depicted in Figure 1(a).
\begin{picture}(0.0,0.0)\end{picture}
$y_{1}$$x_{n}$$x_{1}$$x_{2}$$x_{i}$$y_{n}$$X$$y_{3}$$y_{2}$$y_{1}$
Figure 1. (a) First step in constructing the non-sparsifiable disk graph $G$.
(b) The non-sparsifiable disk graph $G$.
Turning to the point $y_{2}$, let $\delta(y_{2},x_{i})=2n$ for every $x_{i}\in
X$, $\delta(y_{2},y_{1})=2n+\epsilon$, and $r(y_{2})=2n$. Hence there is an
edge from $y_{2}$ to all the points of $X$, but no edge connects $y_{2}$ and
$y_{1}$.
We now turn to define the general case. Consider $y_{i}\in Y$. Let
$r(y_{i})=2^{i-1}n$ and $\delta(y_{i},x_{j})=2^{i-1}n$ for every $x_{j}\in X$.
Let $\delta(y_{i},y_{i-1})=2^{i-1}n+\epsilon$, and in general, for every
$0<j<i$ we have
$\delta(y_{i},y_{j})~{}=~{}\sum_{k=j}^{i-1}\delta(y_{k+1},y_{k})~{},$ (1)
implying that
$\delta(y_{i},y_{j})~{}<~{}2^{i}n.$ (2)
It is easy to verify that $y_{i}$ has outgoing edges to the points of $X$ (and
to them only) and it does not have any incoming edges. See Figure 1(b).
The resulting disk graph $G$ has $2n$ vertices and $\Omega(n^{2})$ edges.
Clearly, removing any edge from $G$ will increase the distance between its
head and its tail to infinity, and thus the only spanner of $G$ is $G$ itself.
It is left to show that the metric space defined above for $G$ has a constant
doubling dimension. Given a metric space $(V,\delta)$, its doubling dimension
is defined to be the minimal value $d$ such that every ball $B$ of radius $r$
in the metric space can be covered by $2^{d}$ balls of radius $r/2$. In the
next Theorem we prove that for the metric space described above, $d$ is
constant.
###### Theorem 1.1.
The metric space $(X\cup Y,\delta)$ defined for $G$ has a constant doubling
dimension.
###### Proof 1.2.
Let $B$ be a ball with an arbitrary radius $r$. We show that it is possible to
cover all the points of $X\cup Y$ within $B$ using a constant number of balls
whose radius is $r/2$. The proof is divided into two cases.
Case a: There is some $y_{j}\in Y$ within the ball $B$. (If there is more than
one such point, then let $y_{j}$ be the point whose index is maximal.) Let
$B^{\prime}$ be a ball of radius $R=2r$ centered at $y_{j}$. Clearly $B\subset
B^{\prime}$, so $B^{\prime}$ contains all the points of $B$. In what follows
we show that all the points of $X\cup Y$ within $B^{\prime}$ can be covered by
a constant number of balls of radius $r/2$. Let $y_{i}$ be the point within
$B^{\prime}$ whose index is maximal. We have to consider two possible
scenarios. The first is that $y_{j}=y_{i}$. This implies that $y_{j+1}\notin
B^{\prime}$, hence $R<\delta(y_{j+1},y_{j})=2^{j}n+\epsilon$. We now show that
it is possible to cover $B^{\prime}$ by a constant number of balls of radius
$R/4$. If $R<2^{j-1}n$, then only $y_{j}$ is within $B^{\prime}$ and it is
covered by a ball of radius $R/4$ centered at itself. If $2^{j-1}n\leq
R<2^{j-1}n+\epsilon$, then $B^{\prime}$ contains all the points of $X$ and
$y_{j}$. From packing arguments it follows that it is possible to cover all
the points of $X$ by a constant number of balls of radius $n/4$, hence also by
a constant number of balls of radius $R\geq n$. The point $y_{j}$ itself is
covered by a ball centered at it. Finally, if $2^{j-1}n+\epsilon\leq
R<2^{j}n+\epsilon$, then $R/4$ is at least $2^{j-3}n+\epsilon/4$. A ball
centered at $y_{j-3}$ of radius $R/4$ covers every $y_{k}$ within
$B^{\prime}$, where $1\leq k\leq j-3$, as $\delta(y_{j-3},y_{k})\leq
2^{j-3}n$. Hence, we cover $Y\cap B^{\prime}$ by balls of radius $R/4$ whose
centers are $y_{j}$, $y_{j-1}$, $y_{j-2}$ and $y_{j-3}$. We cover $X\cap
B^{\prime}$ as before. This completes the first scenario, where $y_{i}=y_{j}$.
Assume now that $y_{i}\neq y_{j}$. This implies that $\delta(y_{i},y_{j})\leq
R$ and that $R<\delta(y_{i+1},y_{j})$, where the first inequality follows from
the fact that $y_{i}\in B^{\prime}$ and the second inequality follows from the
fact that $y_{i}$ is the point with maximal index inside $B^{\prime}$, hence,
$y_{i+1}\notin B^{\prime}$. As $\delta(y_{i},y_{i-1})\leq\delta(y_{i},y_{j})$,
we get that $2^{i-1}n+\epsilon\leq R$. Also, by (2),
$\delta(y_{i+1},y_{j})<2^{i+1}n$. We conclude that $2^{i-1}n\leq R<2^{i+1}n$
and that $R/4\geq 2^{i-3}n$. A ball centered at $y_{i-3}$ of radius $R/4$
covers every $y_{k}$ within $B^{\prime}$, where $k\leq i-3$, as
$\delta(y_{i-3},y_{k})\leq 2^{i-3}n$. Hence, we can cover $B^{\prime}\cap Y$
by balls of radius $R/4$ whose centers are $y_{i}$, $y_{i-1}$, $y_{i-2}$ and
$y_{i-3}$. We cover $X\cap B^{\prime}$ as before. This completes the first
case.
Case b: The ball $B$ does not contain any point from $Y$. The points of $X$
are spread as appears in Figure 1(a), thus by standard packing arguments, any
ball that contains only points from $X$ is covered by a constant number of
balls of half the radius.
## 2\. Improved spanner in the relaxed disk graph model
The (negative) optimality result from the previous section motivates us to
look for a slightly relaxed definition of disk graphs in which it will still
be possible to create a spanner of size $O(n/\epsilon^{d})$.
Let $(V,\delta)$ be a metric space of constant doubling dimension $d$ with a
radius assignment $r(\cdot)$ for its points and let $I=(V,E,r)$ be its
corresponding disk graph. Assume that we multiply the radius assignment of
every point by a factor of $1+\epsilon$, for some $\epsilon>0$, and let
$I^{\prime}=(V,E^{\prime},r_{1+\epsilon})$ be the corresponding disk graph. It
is easy to see that $E\subseteq E^{\prime}$. In this section we show that it
is possible to create a $(1+\epsilon)$-spanner of size $O(n/\epsilon^{d})$ if
we are allowed to use edges of $I^{\prime}$. As a first step we present a
simple variant of our $(1+\epsilon)$-spanner construction of size
$O(n/\epsilon^{d}\log M)$ from [8]. This variation is needed in order to
obtain the efficient construction in the relaxed model which is presented
right afterwards.
### 2.1. Spanners for general disk graphs
Let $(V,\delta)$ be a metric space of constant doubling dimension and assume
that any point $p\in V$ is the center of a ball of radius $r(p)$, where $r(p)$
is taken from the range $[1,M]$. In this section we describe a simple variant
of our construction from [8], which computes a $(1+\epsilon)$-spanner with
$O(n/\epsilon^{d}\log M)$ edges for a given disk graph. We then use this
variant, together with new ideas, in order to obtain (in the next section) our
main result, namely, a spanner with only $O(n/\epsilon^{d})$ edges.
The spanner construction algorithm receives as input a directed graph
$I(V,E,r)$ and an arbitrarily small (constant) approximation factor
$\epsilon>0$, and constructs a set of spanner edges $E_{\mbox{\tiny
SP}}^{\mbox{\tiny DIR}}$, returning the spanner subgraph $H^{\mbox{\tiny
DIR}}(V,E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}})$. The construction of the
spanner is based on a hierarchical partition of the points of $V$ that takes
into account the different radius of each point. The construction operates as
follows. Let $\alpha$ and $\beta$ be two small constants depending on
$\epsilon$, to be fixed later on. Assume that the ball radii are scaled so
that the smallest edge in the disk graph is of weight $1$. Let $i$ be an
integer from the range $[0,\lfloor\log_{1+\alpha}M\rfloor]$ and let
$M_{i}=M/(1+\alpha)^{i}$. The edges of $I(V,E,r)$ are partitioned into classes
by length, letting $E(M_{i+1},M_{i})=\\{(x,y)\mid M_{i+1}\leq\delta(x,y)\leq
M_{i}\\}$. Let $\ell(x,y)$ be the level of the edge $(x,y)$, that is,
$\ell(x,y)=i$ such that $(x,y)\in E(M_{i+1},M_{i})$. Let $p$ be a point whose
ball is of radius $r(p)\in[M_{i+1},M_{i}]$. It follows that level $i$ is the
first level in which $p$ can have outgoing edges. We denote this level by
$\ell(p)$.
For every $i\in[0,\lfloor\log_{1+\alpha}M\rfloor]$, starting from $i=0$, the
edges of the class $E(M_{i+1},M_{i})$ are considered by the algorithm in a
non-decreasing order. (Assume that in each class the edges are sorted by their
weight.) In each stage of the construction we maintain a set of pivots
$P_{i}$. Let $x\in V$ and let $\mbox{\sf NN}(x,P_{i})$ be the nearest neighbor
of $x$ among the points of $P_{i}$. For a pivot $p\in P_{i}$, define
$\Gamma_{i}(p)=\\{x\mid x\in V,\mbox{\sf
NN}(x,P_{i})=p,r(x)\geq\delta(x,p)\\}$, namely, all the points that have a
directed edge to $p$ and $p$ is their nearest neighbor from $P_{i}$. We refer
to $\Gamma_{i}(p)$ as the close neighborhood of $p$.
The algorithm is given in Figure 2.1. Let $(x,y)$ be an edge considered by the
algorithm in the $i$th iteration. The algorithm first checks whether $x$ or
$y$ or both should be added to the pivots set $P_{i}$. The main change with
respect to [8] is that if $y$ is assigned with a large enough radius it might
become a pivot when the edge $(x,y)$ is examined. When considering the edge
$(x,y)$, the algorithm acts according to the following rule: If the distance
from $x$ to its nearest neighbor in $P_{i}$ is greater than $\beta M_{i+1}$
then $x$ is added to $P_{i}$. If the distance from $y$ to its nearest neighbor
in $P_{i}$ is greater than $\beta M_{i+1}$ and the radius of $y$ is at least
$M_{i+1}$ then $y$ is added to $P_{i}$. To decide whether the edge $(x,y)$ is
added to the spanner, the following two cases are considered. The first case
is when $r(y)\geq M_{i+1}$. In this case, if there is no edge from the close
neighborhood of $x$ to the close neighborhood of $y$ then $(x,y)$ is added to
the spanner. The second case is when $r(y)<M_{i+1}$. In this case, if there is
no edge from the close neighborhood of $x$ to $y$ then $(x,y)$ is added to the
spanner. When $i$ reaches $\lfloor\log_{1+\alpha}M\rfloor$, the algorithm
handles all the edges that belong to
$E(M_{\lfloor\log_{1+\alpha}M\rfloor+1},M_{\lfloor\log_{1+\alpha}M\rfloor})$.
This includes also edges whose weight is $1$, the minimal possible weight. The
algorithm returns the directed graph $H^{\mbox{\tiny DIR}}(V,E_{\mbox{\tiny
SP}}^{\mbox{\tiny DIR}})$.
In what follows we prove that for suitably chosen $\alpha$ and $\beta$,
$H^{\mbox{\tiny DIR}}(V,E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}})$ is a
$(1+\epsilon)$-spanner with $O(n/\epsilon^{d}\log M)$ edges of the directed
graph $I(V,E,r)$.
Algorithm disk-spanner $E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}}\leftarrow\phi$
$P_{0}\leftarrow\phi$ for $i\leftarrow 0$ to $\lfloor\log_{1+\alpha}M\rfloor$
for each $(x,y)\in E(M_{i+1},M_{i})$ do if $\delta(\mbox{\sf
NN}(x,P_{i}),x)>\beta M_{i+1}$ then $P_{i}\leftarrow P_{i}\cup\\{x\\}$ if
$\delta(\mbox{\sf NN}(y,P_{i}),y)>\beta M_{i+1}\wedge r(y)\geq M_{i+1}$ then
$P_{i}\leftarrow P_{i}\cup\\{y\\}$ if $r(y)\geq M_{i+1}$ if
$\nexists(x^{\prime},y^{\prime})\in E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}}$
s.t. $x^{\prime}\in\Gamma_{i}(\mbox{\sf NN}(x,P_{i}))\wedge$
$y^{\prime}\in\Gamma_{i}(\mbox{\sf NN}(y,P_{i}))$ then $E_{\mbox{\tiny
SP}}^{\mbox{\tiny DIR}}\leftarrow E_{\mbox{\tiny SP}}^{\mbox{\tiny
DIR}}\cup\\{(x,y)\\}$ if $r(y)<M_{i+1}$ if $\nexists(x^{\prime},y)\in
E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}}$ s.t.
$x^{\prime}\in\Gamma_{i}(\mbox{\sf NN}(x,P_{i}))$ then $E_{\mbox{\tiny
SP}}^{\mbox{\tiny DIR}}\leftarrow E_{\mbox{\tiny SP}}^{\mbox{\tiny
DIR}}\cup\\{(x,y)\\}$ $P_{i+1}\leftarrow P_{i}$ return $H^{\mbox{\tiny
DIR}}(V,E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}})$
Figure 2. A high level implementation of the spanner construction algorithm
for _general_ disk graphs
###### Lemma 2.1 (Stretch).
Let $\epsilon>0$, set $\alpha=\beta<\epsilon/6$ and let $H=H^{\mbox{\tiny
DIR}}(V,E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}})$ be the graph returned by
Algorithm $\mbox{\bf disk-spanner}(I(V,E,r),\epsilon)$. If $(x,y)\in E$ then
$\delta_{H}(x,y)\leq(1+\epsilon)\delta(x,y)$.
###### Proof 2.2.
Recall that the radii are scaled so that the shortest edge is of weight $1$.
We prove that every directed edge of an arbitrary node $x\in V$ is
approximated with $1+\epsilon$ stretch. Let
$i\in[0,\lfloor\log_{1+\alpha}M\rfloor$]. The proof is by induction on $i$.
For a given node $x$, the base of the induction is the maximal value of $i$ in
which $x$ has an edge in $E(M_{i+1},M_{i})$. Let $j$ be this value for $x$,
that is, the set $E(M_{j+1},M_{j})$ contains the shortest edge that touches
$x$. Every other node is at distance at least $M_{j+1}$ away from $x$, hence
$x$ is a pivot at this stage and every edge that touches $x$ from the set
$E(M_{j+1},M_{j})$ is added to $E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}}$.
Let $(x,y)\in E(M_{i+1},M_{i})$ for some $i<j$ and let $p=\mbox{\sf
NN}(x,P_{i})$. Assume that $r(y)\geq M_{i+1}$ and let $q=\mbox{\sf
NN}(y,P_{i})$. It follows from definition that $\delta(x,p)\leq\beta M_{i+1}$
and $\delta(y,q)\leq\beta M_{i+1}$.
If the edge $(x,y)$ is not in the spanner, then there must be an edge
$(\hat{x},\hat{y})\in E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}}$, where
$\hat{x}\in\Gamma_{i}(p)$ and $\hat{y}\in\Gamma_{i}(q)$. The crucial
observation is that the radius of $x$ and $\hat{y}$ is at least $M_{i+1}$. By
the choice of $\beta$, it follows that $2\beta M_{i+1}<M_{i+1}$ and
$(x,\hat{x}),(\hat{y},y)\in E$. Thus, there is a (directed) path from $x$ to
$y$ of the form $\langle x,\hat{x},\hat{y},y\rangle$ whose length is $4\beta
M_{i+1}+M_{i}$. However, only its middle edge, $(\hat{x},\hat{y})$, is in
$E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}}$. The length of this edge is bounded
by the length of the edge $(x,y)$ since the algorithm picked the minimal edge
that connects between the neighborhoods. This implies that the length of
$(\hat{x},\hat{y})$ is at most $M_{i}$.
By the inductive hypothesis, the edges $(x,\hat{x})$ and $(\hat{y},y)$ whose
weight is at most $2\beta M_{i+1}$ are approximated with $1+\epsilon$ stretch.
Thus, there is a path in the spanner from $x$ to $y$ whose length is at most
$(1+\epsilon)\delta(x,\hat{x})+M_{i}+(1+\epsilon)\delta(\hat{y},y),$ and this
can be bounded by
$(1+\epsilon)4\beta
M_{i+1}+M_{i}~{}=~{}((1+\epsilon)4\beta+(1+\alpha))M_{i+1}.$
As the edge $(x,y)\in E(M_{i+1},M_{i})$ it follows that $\delta(x,y)\geq
M_{i+1}$. It remains to prove that $1+4\epsilon\beta+4\beta+\alpha\leq
1+\epsilon$, which follows directly from the choice of $\alpha$ and $\beta$.
If $r(y)<M_{i+1}$ then there must be an edge $(\hat{x},y)\in E_{\mbox{\tiny
SP}}^{\mbox{\tiny DIR}}$, where $\hat{x}\in\Gamma_{i}(p)$. Following similar
arguments to those used above it can be shown that there is a path in the
spanner from $x$ to $y$ of length at most $(1+\epsilon)2\beta M_{i+1}+M_{i}$
and hence bounded by $(1+\epsilon)M_{i+1}$.
#### The size of the spanner.
We now prove that the size of the spanner $H^{\mbox{\tiny
DIR}}(V,E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}})$ is $O(n/\epsilon^{d}\log M)$.
As a first step, we state the following well-known lemma, cf. [2].
###### Lemma 2.3.
[Packing Lemma] If all points in a set $U\in\mathbb{R}^{d}$ are at least $r$
apart from each other, then there are at most $(2R/r+1)^{d}$ points in $U$
within any ball $X$ of radius $R$.
The next lemma establishes a bound on the number of incoming spanner edges
that a point may be assigned on stage $i\in[0,\lfloor\log_{1+\alpha}M\rfloor]$
of the algorithm.
###### Lemma 2.4.
Let $i\in[0,\lfloor\log_{1+\alpha}M\rfloor]$ and let $y\in V$. The total
number of incoming edges of $y$ that were added to the spanner on stage $i$ is
$O(\epsilon^{-d})$.
###### Proof 2.5.
Let $(x,y)$ be a spanner edge and let $\mbox{\sf NN}(x,P_{i})=p$. We associate
$(x,y)$ to $p$. From the spanner construction algorithm it follows that this
is the only incoming edge of $y$ whose source is in $\Gamma_{i}(p)$. Thus,
this is the only incoming edge of $y$ which is associated to $p$. Now consider
all the incoming edges of $y$ on stage $i$. The source of each of these edges
is associated to a unique pivot within distance of at most $M_{i}+2\beta
M_{i+1}$ away from $y$ and any two pivots are $\beta M_{i+1}$ apart from each
other. Using Lemma 2.3, we get that the number of edges entering $y$ is
$(\frac{M_{i}+2\beta M_{i+1}}{\beta
M_{i+1}}+1)^{d}=((1+\alpha)/\beta+3)^{d}=O(\epsilon^{-d})$.
It follows from the above lemma that the total number of edges that were added
to $E_{\mbox{\tiny SP}}^{\mbox{\tiny DIR}}$ in the main loop is
$O(n/\epsilon^{d}\log M)$. The total cost of the construction algorithm is
$O(m\log n)$. For more details on the construction time see [8].
### 2.2. Spanner for relaxed disk graphs
Let $(V,\delta)$ be a metric space of constant doubling dimension $d$ with a
radius assignment $r(\cdot)$ for its points and let $I=(V,E,r)$ be its
corresponding disk graph. Assume that we multiply the radius assignment of
every point by a factor of $1+\epsilon$, for some $\epsilon>0$, and let
$I^{\prime}=(V,E^{\prime},r_{1+\epsilon})$ be the corresponding disk graph. In
this section we show that it is possible to create a $(1+\epsilon)$-spanner of
$I$ of size $O(n/\epsilon^{d})$ if we are allowed to use edges of
$I^{\prime}$.
Our construction consists of two stages: a building stage and a pruning stage.
The building stage creates two spanners, $H$ and $H^{\prime}$, using the
algorithm of Section 2.1, where $H$ is the spanner of $I$ and $H^{\prime}$ is
the spanner of $I^{\prime}$. In the pruning stage we prune the union of these
two spanners. Throughout the pruning stage we use the radius assignment of
each point before the increase. Let $q\in V$ and let $\ell(q)$ be the first
level in which $q$ can have outgoing edges, that is,
$r(q)\in[M_{\ell(q)+1},M_{\ell(q)}]$ (recall that as the levels get larger the
edges get shorter). In the pruning stage we only prune incoming edges of $q$
whose level is below $\ell(q)$. In other words, we do not touch the incoming
edges of $q$ that are shorter than the radius of $q$. The pruning is done as
follow. Let $\gamma=\log_{1+\alpha}{1/\beta}+1$. We keep in the spanner the
incoming edges of $q$ that come from the first $4\gamma$ different levels
below $\ell(q)$.
Let $\hat{H}$ be the resulting spanner and let $\hat{E}$ be the remaining set
of edges after the pruning step. In the remainder of this section we show that
the size of $\hat{H}$ is $O(n/\epsilon^{d})$ and its stretch with respect to
the distances in $I(V,E,r)$ is $1+\epsilon$. We start by showing that the size
of $\hat{H}$ is $O(n/\epsilon^{d})$. Notice that the first part of the proof
below is possible only due to the change we have done in the previous section
to our spanner construction from [8]. Roughly speaking, given an edge
$(p,q)\in E$ that is shorter than $r(q)$ we use pivot selection also on $q$’s
side (and not only on $p$’s) to sparisify the graph. This allows us to deal
separately with edges of $q$ of length larger than $r(q)$ and those of length
smaller than $r(q)$.
###### Lemma 2.6.
$|\hat{E}|=O(n/\epsilon^{d})$.
###### Proof 2.7.
Let $(p,q)$ be a spanner edge that survived the pruning step. There are two
possible cases to consider.
The first case is that $\ell(p,q)>\ell(q)$. Let $i=\ell(p,q)$ and let
$x=\mbox{\sf NN}(p,P_{i})$ and $y=\mbox{\sf NN}(q,P_{i})$. By packing
considerations similar to Lemma 2.4 it follows that the total number of edges
at level $i$ that connects between two pivots as the edge $(p,q)$ that are
associated with $x$ (and with $y$) is $O(1/\epsilon^{d})$. The distance
between $x$ and $y$ is at most $2\beta M_{i+1}+M_{i}$, therefore at level
$i-2\gamma$ either $x$ or $y$ are no longer pivots.
Let $x\in P_{j}$ and $x\not\in P_{j-1}$, that is, $P_{j}$ is the first pivot
set that contains $x$. Then we charge $x$ with every (incoming and outgoing)
edge of this type from levels $[j,j+2\gamma]$ that is incident to $x$. Now
given such an edge $(p,q)$ whose level is $i$, either $x$ or $y$ are not
pivots in level $i-2\gamma$, which means that either $x$ or $y$ has been
charged for this edge, since one of them first becomes a pivot between levels
$i-2\gamma$ and $i$.
The second case is that $\ell(p,q)\leq\ell(q)$. In this case, it must be that
level $\ell(p,q)$ is among the $4\gamma$ first different levels below
$\ell(q)$ from which an incoming edge is allowed to enter $q$. Subsequently,
we associate the edge $(p,q)$ with $q$, as the total number of such edges that
$q$ can have is $O(\gamma/\epsilon^{d})$.
We now turn to prove that the stretch of the spanner $\hat{H}$ with respect to
the disk graph $I$ is $1+\epsilon$.
###### Lemma 2.8.
Let $(p,q)$ be an edge of the spanner $H$ that was pruned. We show that there
is a path in $\hat{H}$ whose length is at most $(1+\epsilon)\delta(p,q)$.
###### Proof 2.9.
The proof is by induction on the lengths of the pruned edges. For the
induction base let $(p,q)$ be the shortest edge that was pruned. For every
$x\in V$, let $s(x)$ be the head of an edge whose level is the $\gamma$-th
level below $\ell(x)$ from which $x$ has an incoming edge. Let $q_{1},\ldots
q_{i},\ldots$ be a sequence of points, where $q_{1}=q$ and $q_{i}=s(q_{i-1})$.
As $q_{i+1}=s(q_{i})$, it follows that
$\ell(q_{i+1},q_{i})\leq\ell(q_{i})-\gamma$. Combining this with the fact that
$\ell(q_{i})\leq\ell(q_{i},q_{i-1})$ we get that
$\ell(q_{i+1},q_{i})\leq\ell(q_{i},q_{i-1})-\gamma$. Therefore,
$\delta(q_{i},q_{i-1})\leq\beta\delta(q_{i+1},q_{i})$.
The analysis distinguishes between two cases.
Case a: There is a point $q_{t}$ such that $\delta(q_{t},q)>\beta\delta(p,q)$.
This situation is depicted in Figure 3. (If there is more than one point that
satisfies this requirement, take the one whose index is minimal.)
Claim: $\delta(q_{t},q_{t-1})\geq\frac{\beta}{2}\delta(p,q)$.
###### Proof 2.10.
For the sake of contradiction, assume that
$\delta(q_{t},q_{t-1})<\frac{\beta}{2}\delta(p,q)$. This implies that
$2\delta(q_{t},q_{t-1})~{}<~{}\beta\delta(p,q)~{}<~{}\delta(q_{t},q)~{}\leq~{}\sum_{i=2}^{t}\delta(q_{i},q_{i-1})~{},$
(3)
where the last inequality follows from the triangle inequality as the distance
between $q$ and $q_{t}$ is at most $\sum_{i=2}^{t}\delta(q_{i-1},q_{i})$. For
every $2\leq i\leq t-1$ we have
$\delta(q_{i},q_{i-1})\leq\beta\delta(q_{i+1},q_{i})$, which implies that
$\delta(q_{i},q_{i-1})\leq\beta^{t-i}\delta(q_{t},q_{t-1})$. Combined with
(3), we get
$\delta(q_{t},q_{t-1})~{}<~{}\sum_{i=2}^{t-1}\delta(q_{i},q_{i-1})~{}\leq~{}\delta(q_{t},q_{t-1})\sum_{i=2}^{t-1}\beta^{t-i}~{}.$
If $\beta<1/2$ we have $\sum_{i=2}^{t-1}\beta^{t-i}<1$ and this yields a
contradiction.
We now focus our attention on the point $q_{t-1}$. The minimality of $q_{t}$
implies that $\delta(q,q_{t-1})\leq\beta\delta(p,q)$. By combining it with the
triangle inequality we get that
$\delta(p,q_{t-1})\leq\delta(p,q)+\beta\delta(p,q)$. Therefore, in the graph
$I^{\prime}$ there must be an edge from $p$ to $q_{t-1}$.
\begin{picture}(0.0,0.0)\end{picture}
$p$$q$$\beta\delta(p,q)$$>\beta/2\delta(p,q)$$q_{t-1}$$q_{t}$
Figure 3. The case in which $q_{t}$ exists
Let $i=\ell(p,q_{t-1})$. There are two possible scenarios for the spanner
$H^{\prime}$. The first scenario is when $r^{\prime}(q_{t-1})<M_{i+1}$. In
this case, there is an edge in $H^{\prime}$ from some
$x\in\Gamma_{i}(\mbox{\sf NN}(p,i))$ to $q_{t-1}$, whose length is at most
$\delta(p,q)+\beta\delta(p,q)$.
There are $4\gamma$ different levels below $\ell(q_{t-1})$ from which edges
that belong to the spanners $H$ and $H^{\prime}$ are not being pruned and
survived to the spanner $\hat{H}$. We know that the edge $(q_{t},q_{t-1})$ is
such an edge from the $\gamma$-th non-empty level below $\ell(q_{t-1})$. We
also know that $\delta(q_{t},q_{t-1})>\frac{\beta}{2}\delta(p,q)$. Therefore,
as the length of the edge $(x,q_{t-1})$ is at most
$\delta(p,q)+\beta\delta(p,q)$ it is within the $4\gamma$ non-empty levels
below $\ell(q_{t-1})$ and it is not pruned. We can now build a path from $p$
to $q$ by concatenating three segments as follows: A path from $p$ to $x$, the
edge $(x,q_{t-1})$ and a path from $q_{t-1}$ to $q$. The point $x$ is at most
$2\beta\delta(p,q)+2\beta^{2}\delta(p,q)$ away from $p$ and for the right
choice of $\beta$ it is less than $\delta(p,q)/(1+\epsilon)$, hence the weight
of every edge on the path that approximates the distance between $x$ and $p$
in $H\cup H^{\prime}$ is less than $\delta(p,q)$, the shortest pruned edge,
and the entire path survived the punning stage. Similarly, the point $q_{t-1}$
is at most $\beta\delta(p,q)$ away from $q$ and again for the right choice of
$\beta$ every edge on the path that approximates the distance between
$q_{t-1}$ and $q$ survived the punning stage. Thus, we get that there is a
path whose length is at most
$(1+\epsilon)(3\beta\delta(p,q)+2\beta^{2}\delta(p,q))+\delta(p,q)+\beta\delta(p,q)~{},$
which is less than $(1+\epsilon)\delta(p,q)$ for $\beta<\epsilon/11$.
The second scenario is when $r^{\prime}(q_{t-1})\geq M_{i+1}$. In this case,
there is an edge in $H^{\prime}$ from some $x\in\Gamma_{i}(\mbox{\sf
NN}(p,i))$ to some $y\in\Gamma_{i}(\mbox{\sf NN}(q_{t-1},i))$ whose length is
at most $\delta(p,q)+\beta\delta(p,q)$, which is not being pruned. We can
build a path from $p$ to $q$ by concatenating three segments as follows: A
path from $p$ to $x$, the edge $(x,y)$ and a path from $y$ to $q$. As before,
for the right choice of $\beta$ the paths from $p$ to $x$ and from $y$ to $q$
are composed from edges that are shorter from $\delta(p,q)$, the length of the
shortest pruned edge, hence, from the minimality $\delta(p,q)$ every edge on
these paths survived the punning stage. We get that there is a path whose
length is at most
$(1+\epsilon)(4\beta\delta(p,q)+5\beta^{2}\delta(p,q))+\delta(p,q)+\beta\delta(p,q)~{},$
which is less than $(1+\epsilon)\delta(p,q)$ for $\beta<\epsilon/19$. This
completes the proof for case a.
Case b: There is no point $q_{t}$ such that
$\delta(q_{t},q)>\beta\delta(p,q)$. In this case, let $q_{t-1}$ be the last
point in the sequence of points $q_{1},\ldots q_{i},\ldots$, where
$q_{i}=s(q_{i-1})$ and $q_{1}=q$. Similarly to before, there are two possible
scenarios for the spanner $H^{\prime}$. Let $i=\ell(p,q_{t-1})$. The first
scenario is when $r^{\prime}(q_{t-1})<M_{i+1}$. In this case, there is an edge
in $H^{\prime}$ from some $x\in\Gamma_{i}(\mbox{\sf NN}(p,i))$ to $q_{t-1}$
whose length is at most $\delta(p,q)+\beta\delta(p,q)$. This edge could not be
pruned, since if it was pruned then $q_{t-1}$ could not have been the last
point in the sequence. Hence we can construct a path from $p$ to $q$ exactly
as we have done in the first scenario of case a, described above. The second
scenario is when $r^{\prime}(q_{t-1})\geq M_{i+1}$. In this case, we can
construct a path from $p$ to $q$ exactly as we have done in the second
scenario of case a, described above.
This completes the proof of the induction base. The proof of the general
inductive step is almost identical. The only difference is that when a path is
constructed from $p$ to $q$, its portions from $p$ to $x$ and from $q_{t-1}$
to $q$ in the first scenario and from $p$ to $x$ and from $y$ to $q$ in the
second scenario exist in $\hat{H}$ by the induction hypothesis and not by the
minimality of $\delta(p,q)$.
We end this section by stating its main Theorem. The proof of this Theorem
stems from Lemma 2.6 and Lemma 2.8.
###### Theorem 2.11.
Let $(V,\delta)$ be a metric space of constant doubling dimension with a
radius assignment $r(\cdot)$ for its points and let $I=(V,E,r)$ be its
corresponding disk graph. Let $I^{\prime}=(V,E^{\prime},r_{1+\epsilon})$ be
the corresponding disk graph in the relaxed model. It is possible to create a
$(1+\epsilon)$-spanner of size $O(n/\epsilon^{d})$ for $I$ using edges of
$I^{\prime}$.
## 3\. Concluding remarks and open problems
This paper presents a spanner construction for disk graphs in a slightly
relaxed model that is as good as spanners for complete graphs and unit disk
graphs. This result opens many other research directions for disk graphs. We
list here two questions that we find particularly intriguing: Is it possible
to design an efficient compact routing scheme for disk graphs? And is it
possible to build an efficient distance oracle for disk graphs?
## References
* [1] U. Zwick D. Dor, S. Halperin. All-pairs almost shortest paths. SIAM J. Comput., 29(5):1740–1759, 2000.
* [2] J. Gao, L. Guibas, and A. Nguyen. Deformable spanners and applications. In Proc. 20th ACM Symp. on Computational Geometry, pages 179–199, 2004.
* [3] Jie Gao, Leonidas J. Guibas, John Hershberger, Li Zhang, and An Zhu. Geometric spanners for routing in mobile networks. IEEE J. on Selected Areas in Communications, 23(1):174–185, 2005\.
* [4] Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Computing, 35:1148–1184, 2006.
* [5] C. Wagner L. Cowen. Compact roundtrip routing for digraphs. In SODA ’99: Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, pages 885–886, Philadelphia, PA, USA, 1999. Society for Industrial and Applied Mathematics.
* [6] U. Zwick L. Roditty, M. Thorup. Roundtrip spanners and roundtrip routing in directed graphs. ACM Trans. Algorithms, 4(3):1–17, 2008.
* [7] Xiang-Yang Li, Gruia Calinescu, Peng-Jun Wan, and Yu Wang. Localized delaunay triangulation with application in ad hoc wireless networks. IEEE Trans. on Parallel and Distributed Systems, 14(10):1035–1047, 2003.
* [8] D. Peleg and L. Roditty. Localized spanner construction for ad hoc networks with variable transmission range. In Proc. 7th Int. Conf. on Ad-Hoc Networks and Wireless (AdHoc-NOW), pages 622–633, 2008.
* [9] L. Roditty. Fully dynamic geometric spanners. symposium on computational geometry. pages 373–380, 2007.
* [10] Yu Wang and Xiang-Yang Li. Efficient delaunay-based localized routing for wireless sensor networks. Int. J. of Communication Systems, 20(7):767–789, 2006.
* [11] Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput., 11(4):721–736, 1982.
|
arxiv-papers
| 2009-12-15T07:31:51 |
2024-09-04T02:49:07.051346
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David Peleg, Liam Roditty",
"submitter": "Liam Roditty",
"url": "https://arxiv.org/abs/0912.2815"
}
|
0912.2966
|
# Azimuthally Symmetric Theory of Gravitation (I)
On the Perihelion Precession of Planetary Orbits
G. G. Nyambuya
Email: gadzirai@gmail.com
(Received: 5 Sept. 2009 / Accepted with Moderate Revision: 9 Dec. 2009 )
###### Abstract
From a purely none-general relativistic standpoint, we solve the empty space
Poisson equation ($\nabla^{2}\Phi=0$) for an azimuthally symmetric setting,
i.e., for a spinning gravitational system like the Sun. We seek the general
solution of the form $\Phi=\Phi(r,\theta)$. This general solution is
constrained such that in the zeroth order approximation it reduces to Newton’s
well known inverse square law of gravitation. For this general solution, it is
seen that it has implications on the orbits of test bodies in the
gravitational field of this spinning body. We show that to second order
approximation, this azimuthally symmetric gravitational field is capable of
explaining at least two things (1) the observed perihelion shift of solar
planets (2) that the mean Earth-Sun distance must be increasing – this
resonates with the observations of two independent groups of astronomers
(Krasinsky & Brumberg 2004; Standish 2005) who have measured that the mean
Earth-Sun distance must be increasing at a rate of about $7.0\pm 0.2\,m/cy$
(Standish 2005) to $15.0\pm 0.3\,m/cy$ (Krasinsky & Brumberg 2004). In-
principle, we are able to explain this result as a consequence of loss of
orbital angular momentum – this loss of orbital angular momentum is a direct
prediction of the theory. Further, we show that the theory is able to explain
at a satisfactory level the observed secular increase Earth Year ($1.70\pm
0.05\,ms/yr$; Miura et al. 2009). Furthermore, we show that the theory makes a
significant and testable prediction to the effect that the period of the solar
spin must be decreasing at a rate of at least $8.00\pm 2.00\,s/cy$.
###### keywords:
astronomical unit, azimuthal symmetry, orbit, perihelion shift, solar spin
††volume: 0000††pagerange: Azimuthally Symmetric Theory of Gravitation (I) On
the Perihelion Precession of Planetary Orbits–References††pubyear: 2009
## 1 Introduction
From as way back as the $1850s$, it has been known that the orbit of the
planet Mercury exhibits a peculiar motion of its perihelion, specifically, the
perihelion of Mercury advances by ${43.1}\pm{0.5}$ $\rm{arcsec/century}$. When
Newton’s theory of gravitation is applied to try and explain this (by making
use of the oblateness of the planets because when the Sun’s gravitational
force acts on the oblate-planets, the oblateness causes torque [on the
planets] and this torque is thought to give rise to the anomalous motion of
the planets); it was found first by Leverrier in ${1859}$ see e.g. Kenyon
(1990) that it predicted a precession of ${532}$ $\rm{arcsec/century}$ which
is larger than the observed (Kenyon, 1990). With the failure of Newton’s
theory to explain this, it was proposed that a small undetected planet was the
cause. Careful scrutiny of the terrestrial heavens by telescopes and spaces
probes reveals no such object – the meaning of which is that the cause may
very well be a hitherto unknown gravitational phenomena – Einstein was to
demonstrate that this was the case, that there existed a hitherto unknown
gravitational phenomena that is the cause of this peculiar motion.
With the herald of Einstein’s General Theory of Relativity (GTR) in ${1915}$,
Einstein immediately applied his GTR to this problem; much to his elation
which caused him heart palpitations – he obtained the unprecedented value of
${43.0}\,\rm{arcsec/century}$ and this was (and is still) hailed as one of the
greatest triumphs for the GTR and this lead to its quick acceptance. Venus,
the Earth, and other planets show such peculiar motion of their perihelion.
Observations reveal a shift of ${8.40}\pm{4.80}$ and
${5.00}\pm{1.00}\,\rm{arcsec/century}$ respectively (see e.g. Kenyon
${1994}$). Einstein’s theory is able to explain the perihelion shift of the
other planets well, so much that it is now a well accepted paradigm that the
perihelion shift of planetary orbits is a general relativistic phenomena.
Einstein’s GTR explains the perihelion shift of planetary orbits as a result
of the curvature of spacetime around the Sun. It does not take into account
the spin of the Sun and at the same time it assumes all the planets lay on the
same plane. The assumption that the planets lay on the same plane is in the
GTR solution only taken as a first order approximation – in reality, planets
do not lay on the same plane. In this reading we set forth what we believe is
a new paradigm; we have coined this paradigm the Azimuthally Symmetric Theory
of Gravitation (ASTG) and this is derived from Poisson’s well accepted
equation for empty space – namely $\nabla^{2}\Phi=0$. Poisson’s Law is a
differential form of Newton’s Law of Gravitation. We explain the perihelion
shift of the orbits of planets as a consequence of the spin of the Sun – i.e.
solar spin. It is well known that the Sun does exhibit some spin angular
momentum – specifically, it [the Sun] undergoes differential rotation. On the
average, it spins on its spin axis about once in every $\sim 25.38$ days (see
e.g. Miura et al. 2009). Its spin axis makes an angle of about
$83\hbox{${}^{\circ}$$$}$ with the ecliptic plane. It is important that we
state clearly here that by no means have we discovered a new theory or a set
of new equations; we have merely applied Poisson’s well known azimuthally
symmetric solution to gravity for a spinning gravitating body.
Further, with regard to Einstein’s GTR – vis; in its solution to the problem
of the perihelion shift of planetary orbits, it [the GTR] assumes the
traditional Newtonian gravitational potential, namely:
$\Phi(r)=-G\mathcal{M}/r$, where $G=6.667\times 10^{-11}kg^{-1}ms^{-2}$ is
Newton’s universal constant of gravitation, $\mathcal{M}$ is the mass of the
central gravitating body and $r$ is the radial distance from this gravitating
body. Einstein’s GTR which is embodied in Einstein’s law of gravitation,
namely:
$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu}+\Lambda g_{\mu\nu},$ (1)
is designed such that in the low energy limit and low spacetime curvature such
as in the Solar System, this equation reduces directly to Poison’s equation.
In Einstein’s law above, $R_{\mu\nu}$ is the Ricci tensor, $R$ the Ricci
scalar, $g_{\mu\nu}$ the metric of spacetime, $\Lambda$ is Einstein’s
controversial cosmological constant which at best can be taken to be zero
unless one is making computations of a cosmological nature where darkenergy is
involved, and $\kappa=8\pi G/c^{4}$ where $c=2.99792458\times 10^{8}ms^{-1}$
is the speed of light in vacuum; and Poisson’s equation is given by:
$\nabla^{2}\Phi=4\pi G\rho,$ (2)
where $\rho$ is the density of matter and the operator $\vec{\nabla}^{2}$
written for spherical coordinate system (see figure 1 for the coordinate
setup) is given by:
$\vec{\nabla}^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial
r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}.$
(3)
Figure 1: This figure shows a generic spherical coordinate system, with the
radial coordinate denoted by $r$, the zenith (the angle from the North Pole;
the colatitude) denoted by $\theta$, and the azimuth (the angle in the
equatorial plane; the longitude) by $\varphi$.
As already been said, our solution or paradigm, hails directly from Poisson’s
equation, which in itself is a first order approximate solution to Einstein’s
GTR, albeit with the important difference that we have taken into account
solar spin. This fact that our paradigm explains reasonably well – within the
confines of its error margins; the precession of planetary orbits as a
consequence of solar spin and at the sametime the GTR explains this same
phenomena well as a consequence of the curvature of spacetime raises the
question “Is the precession of the perihelion of solar orbits a result of (1)
solar spin or (2) is it a result of the curvature of spacetime?” If anything,
this is the question that this reading seems to raise. An answer to it, will
only come once the meaning of the ASTG is fully understood.
In the above we say the ASTG “explains reasonably well – within the confines
of its error margins” – what immediately comes to mind is that can a theory
have error margins or is it not experiments that have error margins? As will
be seen, certain undetermined constants ($\lambda_{\ell}$) in the theory
emerge and at present, one has to infer these from observations and it is here
that the error margins of the ASTG come into play.
Further, we show, that in-principle, the ASTG does explain (1) the increase in
the mean Earth-Sun distance, (2) the increase in the mean Earth-Moon distance
etc, and these emerge as a consequence of the fact that from the ASTG, the
orbital angular momentum is not a conserved quantity as is the case in
Newtonian’s gravitational theory and Einstein’s GTR. That the orbital angular
momentum is not a conserved quantity may lead one to think that the ASTG
violets the Law of Conservation of angular momentum – no, this is not the
case. The lost angular momentum is transferred to the spin of the orbiting
body and as well as the Sun.
## 2 Theory
For empty space: $\nabla^{2}\Phi=0$; and for a spherically symmetric setting
we have $\Phi=\Phi(r)$ and this leads directly to Newtonian gravitation. For a
scenario or setting that exhibits azimuthal symmetry such as a spinning
gravitating body as the Sun we must have: $\Phi=\Phi(r,\theta)$, we thus shall
solve the Poisson equation: $\nabla^{2}\Phi(r,\theta)=0$. The Poisson equation
for this setting is readily soluble and its solution can readily be found in
most of the good textbooks of electrodynamics and quantum mechanics for
example – it is instructive that we present this solution here.
We shall solve Poisson’s equation for empty space ($\nabla^{2}\Phi=0$)
exactly; by means of separation of variables, i.e. we shall set:
$\Phi(r,\theta)=\Phi(r)\Phi(\theta)$. Inserting this into the Poisson equation
we will have after some basic algebraic operations:
$\frac{1}{\Phi(r)}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial\Phi(r)}{\partial
r}\right)+\frac{1}{\Phi(\theta)}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Phi(\theta)}{\partial\theta}\right)={0}.$
(4)
The radial and the angular portions of this equation must equal some constant
since they are independent of each other. Following tradition, we must set:
$\frac{1}{\Phi(r)}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial\Phi(r)}{\partial r}\right)=\ell(\ell+1),$ (5)
and the solution to this is:
$\Phi_{\ell}(r)=A_{\ell}r^{\ell}+\frac{B_{\ell}}{r^{\ell+1}},$ (6)
where $A_{\ell}$ and $B_{\ell}$ are constants and $\ell={0,1,2,3},...$ . If we
set the boundary conditions; $\Phi_{\ell}(r=\infty)={0}$, then $A_{\ell}={0}$
for all $\ell$. Now, just as Einstein demanded of his GTR to reduce to the
well known Poisson equation in the low energy regime of minute curvature, we
must demand that $\Phi(r)$, in its zeroth order approximation – where
$\ell={0}$ and the terms $\ell\geq{1}$ are so small that they can be
neglected; the theory must reduce to Newton’s inverse square law; for this to
be so, we must have:
$B_{\ell}=-\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}}{c^{2}}\right)^{\ell+1},$
(7)
where $\lambda_{\ell}$ is an infinite set of dimensionless parameters such
that $\lambda_{0}=1$ and the rest of the parameters $\lambda_{\ell}$ for
$\ell>1$ will take values different from unity and these constants will have –
for now, until such a time that we are able to deduce them directly from
theory; to be determined from the experience of observations. In the
discussion section, we shall hint at our current thinking on the nature of
these constants. With this given, it means we will have:
$\Phi_{\ell}(r)=-\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}}{rc^{2}}\right)^{\ell+1}.$
(8)
Now, moving onto the angular part, we will have:
$\frac{\sin\theta}{\Phi(\theta)}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Phi(\theta)}{\partial\theta}\right)+\left[\ell(\ell+{1})\right]\sin^{2}\theta={0},$
(9)
and a solution to this is a little complicated; it is given by the spherical
harmonic function:
$\Phi(\theta)=P_{\ell}(\cos\theta),$ (10)
of degree $\ell$ and $P_{l}(\cos\theta)$ is associated Legendre polynomial. As
already said, the derivation of $\Phi(r,\theta)$ just presented can be found
in most good standard textbooks of quantum mechanics and classical
electrodynamics. Since equation (9) is a second order differential equation,
one would naturally expect there to exist two independent solutions for every
$\ell$. It so happens that the other solutions give infinity at
$\theta=({0},\pi)$, which is physically meaningless (see e.g. Grifitts
$2008$). Now, putting all the things together, the most general solution is
given:
$\Phi(r,\theta)=-\sum^{\infty}_{\ell=0}\left[\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}}{rc^{2}}\right)^{\ell+1}P_{l}(\cos\theta)\right],$
(11)
which is a linear combination of all the solutions for $\ell$. In the case of
ordinary bodies such as the Sun, the higher orders terms [i.e. $\ell>{1}$: of
the term $(G\mathcal{M}/rc^{2})^{\ell+1}$], will be small and in these cases,
the gravitational field will tend to Newton’s gravitational theory. Equation
(11) is the embodiment of the ASTG, and from this, we shall show that one is
able to explain the precession of the perihelion of planetary orbits.
In this equation [i.e., (11)], nowhere does the value of the Sun’s spin
($\mathcal{T}_{\tiny\odot}\simeq 25.38$ days) enter into our equation. This
may lead one to asking “So where has this been taken into account?”. To answer
this, it is important to note that if the potential is a function of $r$ only
i.e., $\Phi=\Phi(r)$, then, it technically is a function of $r$ and $\theta$
as well (with the $\theta$-dependence being trivial). What this means is that
spherical symmetry implies an azimuthal symmetry around any arbitrarily chosen
axis. If a specific axis is singled out, e.g., by the spin of a body about the
spin axis, then, the spherical symmetry of the static body is broken, and only
an azimuthal symmetry remains and this azimuthal symmetry is only about the
plane cutting the body into hemispheres such that this plane is normal to the
spin axis. For any other plane cutting the body into hemispheres, the two
hemispheres are asymmetric. From this we see that the azimuthally symmetric
solution is consequence of the breaking of the spherical symmetry by the
introduction of a spin axis, hence thus one is automatically lead to consider
the solutions for which $\Phi=\Phi(r,\theta)$. In this way, the spin has been
taken into account.
Figure 2: The elliptical planetary orbits have the Sun at one focus. As the
planets describe their orbits, their major axes slowly rotate about the Sun in
the process shifting the line from the Sun to the perihelion through an angle
$\Delta\varphi$ during each orbit. This shift is referred to as the precession
of the perihelion.
### 2.1 Equations of Motion
We shall derive here the equations of motion for the azimuthally symmetric
gravitational field, $\Phi(r,\theta)$. We know that the force per unit mass
[or the acceleration i.e., $\vec{\textbf{g}}=-\nabla\Phi(r,\theta)$] is given
by
$\vec{\textbf{a}}=(\ddot{r}-r\dot{\varphi}^{2})\hat{\textbf{r}}+(r\ddot{\varphi}+2\dot{r}\dot{\varphi})\hat{\mbox{\boldmath$\theta$}}$
(see any good textbook on Classical Mechanics) where a single dot represents
the time derivative $d/dt$ and likewise a double dot presents the second time
derivative $d^{2}/dt^{2}$. Comparison of
$\vec{\textbf{a}}=(\ddot{r}-r\dot{\varphi}^{2})\hat{\textbf{r}}+(r\ddot{\varphi}+2\dot{r}\dot{\varphi})\hat{\mbox{\boldmath$\theta$}}$
with ($\vec{\textbf{g}}$); i.e.: $\vec{\textbf{a}}\equiv\vec{\textbf{g}}$,
leads to the equations:
$\frac{d^{2}r}{dt^{2}}-r\left(\frac{d\varphi}{dt}\right)^{2}=-\frac{d\Phi}{dr},$
(12)
for the $\hat{\bf{r}}$-component and for the
$\hat{\mbox{\boldmath$\theta$}}$-component we will have:
$r\frac{d^{2}\varphi}{dt^{2}}+{2}\frac{dr}{dt}\frac{d\varphi}{dt}=-\frac{1}{r}\frac{d\Phi}{d\theta}.$
(13)
Now, taking equation (13) and dividing throughout by $r\dot{\varphi}$ and
remembering that the specific angular momentum $J=r^{2}\dot{\varphi}$, we will
have:
$\frac{1}{\dot{\varphi}}\frac{d\dot{\varphi}}{dt}+\frac{2}{r}\frac{dr}{dt}=-\frac{1}{J}\frac{d\Phi}{d\varphi}\Longrightarrow\frac{1}{J}\frac{dJ}{dt}=-\frac{1}{J}\frac{d\Phi}{d\theta},$
(14)
hence thus:
$\frac{dJ}{dt}=-\frac{d\Phi}{d\theta}.$ (15)
The specific orbital angular momentum is the orbital angular momentum per unit
mass and unless otherwise specified, we shall refer to it as angular momentum.
Digressing a little: what the above equation (15) means is that the orbital
angular momentum of a planet around the Sun is not a conserved quantity. If it
is not conserved, then the sum of the orbital and spin angular momentum must
be a conserved quantity (if this angular momentum is not say transfered to the
Sun or other solar bodies), the meaning of which is that at the different
$r$-positions, the spin of a planet about its own axis must vary. This could
mean the length of the day must vary depending on the radial position away
from the Sun. We shall come to this later, all we simple want to do is to
underline this, as it points to the possibility of a secular change in the
mean length of the day.
Now moving on, if we make the transformation $u=1/r$, then for $\dot{r}$ and
$\ddot{r}$ we will have:
$\frac{dr}{dt}=-J\frac{du}{d\varphi}\,\,\textrm{and}\,\,\frac{d^{2}r}{dt^{2}}=-\frac{dJ}{dt}\frac{du}{d\varphi}-J^{2}u^{2}\frac{d^{2}u}{d\varphi^{2}},$
(16)
respectively. Inserting these into (12) and then dividing the resultant
equation by $-u^{2}J$ and remembering (15) and also that $dr=-du/u^{2}$, one
is lead to:
$\frac{d^{2}u}{d\varphi^{2}}-\left(\frac{1}{J^{2}u^{2}}\frac{d\Phi(u,\theta)}{d\varphi}\right)\frac{du}{d\varphi}+u=\frac{1}{J^{2}}\frac{d\Phi(u,\theta)}{du}.$
(17)
The solutions that we shall consider are those for which $\theta$ is a time
constant, i.e. $r=r(\varphi)$ and for the convenience we shall write $\theta$
with subscript $p$, i.e., $\theta_{p}$. This is just to remind us that it
($\theta$) is not a variable in the equations of motion as this is a constant
for a particular planet $p$, hence:
$\frac{d^{2}u}{d\varphi^{2}}-\left(\frac{1}{J^{2}u^{2}}\frac{d\Phi(u,\theta_{p})}{d\theta_{p}}\right)\frac{du}{d\varphi}+u=\frac{1}{J^{2}}\frac{d\Phi(u,\theta_{p})}{du},$
(18)
and:
$\frac{dJ}{dt}=-\frac{d\Phi(u,\theta_{p})}{d\theta_{p}}.$ (19)
This ends our derivation of the equations of motion for the field
$\Phi(r,\theta)$. Before we proceed to our main task of showing how equations
(18 and 19) explain the precession of planetary orbits, let us – for
instructive purposes, first lay down Einstein’s solution.
## 3 Einstein’s Solution
When Einstein applied his newly discovered GTR to the problem of the
precession of the perihelion of the planet mercury he obtained that the
trajectory of solar planets must be described by the equation:
$\frac{d^{2}u}{d\varphi^{2}}+u-\frac{G\mathcal{M}}{J^{2}}=\frac{{3}G\mathcal{M}u^{2}}{c^{2}},$
(20)
where again $u=1/r$. To obtain a solution to this equation, we note that the
left hand side is the usual Newtonian equation for the orbit of planets, i.e.:
$\frac{d^{2}u}{d\varphi^{2}}+u-\frac{G\mathcal{M}}{J^{2}}=0,$ (21)
and the solution to this equation is: $u=({1}+\epsilon\cos\varphi)/l$ where
$\epsilon$ is the eccentricity of the orbit and
$l=({1}-\epsilon^{2})\mathcal{R}$ where $\mathcal{R}$ is half the size of the
major axis of the ellipse. Written in different form, this solution is:
$r=\left(\frac{1+\epsilon}{1+\epsilon\cos\theta}\right)\mathcal{R}_{min}.$
(22)
where $\mathcal{R}_{min}$ is the planet’s distance of closest approach to the
Sun [see figure (2) for an illustration]. This solution is a good approximate
solution to (20) because the orbit of Mercury is nearly Newtonian.
Consequently, we can rewrite the small term on the right hand side of (20) as:
${3}G\mathcal{M}({1}+\epsilon\cos\varphi)^{2}/l^{2}c^{2}$; and in so doing, we
make an entirely negligible error. With this substitution (20) becomes:
$\frac{d^{2}u}{d\varphi^{2}}+u-\frac{G\mathcal{M}}{J^{2}}=\frac{{3}G\mathcal{M}}{l^{2}c^{2}}\left({1}+{2}\epsilon\cos\varphi+\epsilon^{2}\cos^{2}\varphi\right).$
(23)
and the solution to this equation is:
$u=\frac{{1}+\epsilon\cos\varphi}{l}+\frac{{3}G\mathcal{M}}{l^{2}c^{2}}\left[{1}+\frac{\epsilon^{2}}{{2}}+\frac{\epsilon^{2}\cos{2}\varphi}{{{6}}}+\epsilon\varphi\sin\varphi\right],$
(24)
Of the additional terms, the first i.e. ($1+\epsilon^{2}/2$) is a constant and
the second oscillates through two cycles on each orbit; both these terms are
immeasurably small. However, the last term increases steadily in amplitude
with $\varphi$, and hence with time, whilst oscillating through one cycle per
orbit; clearly this term is responsible for the precession of the perihelion.
Dropping all unimportant terms we will have:
$u=\frac{{1}+\epsilon\cos\varphi+\epsilon\eta\varphi\sin\varphi}{l},$ (25)
where $\eta={3}G\mathcal{M}/lc^{2}$ is extremely small. Thus all this leads us
to:
$u=\frac{1+\epsilon\cos\left(\beta_{E}\varphi\right)}{l},$ (26)
where: $\beta_{E}=\left({1}-\eta\right)$. At the perihelion we will have:
$\beta_{E}\varphi={2}n\pi$ and this implies:
$\varphi={2}n\pi\beta_{E}^{-1}\simeq{2}n\pi+{6}n\pi G\mathcal{M}/lc^{2}$.
Essentially, this means that the perihelion advances by $\Delta\varphi={6}\pi
G\mathcal{M}/lc^{2}$ per revolution and the resultant equation for the orbit
is:
$r=\frac{l}{1+\epsilon\cos\left(\varphi+\Delta\varphi\right)},$ (27)
hence thus the rate of precession of the perihelion is given by:
$\left<\frac{\Delta\varphi}{\tau}\right>_{E}=\frac{{6}\pi G\mathcal{M}}{\tau
c^{2}({1}-\epsilon^{2})\mathcal{R}}.$ (28)
This is Einstein’s formula derived in $1916$ soon after he discovered the GTR.
He [Einstein] concluded in the reading containing this formula:
> “Calculation gives for the planet Mercury a rotation of the orbit of
> $43\arcsec$ per century, corresponding exactly to the astronomical
> observation (Leverrier); for the astronomers have discovered in the motion
> of the perihelion of this planet, after allowing for disturbances by the
> other planets, an inexplicable remainder of this magnitude. ”
$\Phi(u,\theta)=-G\mathcal{M}u\left[1+\lambda_{1}\left(\frac{G\mathcal{M}u}{c^{2}}\right)\cos\theta+\lambda_{2}\left(\frac{G\mathcal{M}u}{c^{2}}\right)^{2}\left(\frac{{3}\cos^{2}\theta-{1}}{{2}}\right)\right].$
(29)
$\frac{d^{2}u}{d\varphi^{2}}+\left(\frac{\dot{J}}{J^{2}u^{2}}\right)\frac{du}{d\varphi}+u=-G\mathcal{M}u^{2}\left[{1}+\lambda_{1}\left(\frac{{2}G\mathcal{M}u\cos\theta}{c^{2}}\right)+\lambda_{2}\left(\frac{{3}G\mathcal{M}u}{c^{2}}\right)^{2}\left(\frac{{3}\cos^{2}\theta-{1}}{{2}}\right)\right],$
(30)
## 4 Solution from the ASTG
For the present, we shall take the second order approximation of the potential
$\Phi(r,\theta)$ in-order to make our calculation for the precession of the
perihelion of planetary orbits and this potential has been written down in
(29). As has already been said; we shall consider only those solutions for
which $\theta$ is a time constant, i.e. $r=r(\varphi)$ and for the convenience
that we do not think of $\theta$ as a variable we have set
$\theta:=\theta_{p}$. The solutions $r=r(\varphi)$ are those solutions for
which the orbit of a planet stays put in the same $\theta$-plane. Now from the
potential (29) we shall have:
$\frac{dJ}{dt}=-\left(\frac{G\mathcal{M}u}{c}\right)^{2}\left[\lambda_{1}\sin\theta_{p}+\lambda_{2}\left(\frac{{3}G\mathcal{M}\sin{2}\theta_{p}}{{2}rc^{2}}\right)\right].$
(31)
Now making the transformation $r={1}/u$, the first term on the left hand side
of equation (30) transforms to:
$\frac{d^{2}u}{d\varphi^{2}}+u-\frac{G\mathcal{M}}{J^{2}}=\beta_{1}u+\beta_{2}u^{2},$
(32)
where:
$\beta_{1}=\left(\frac{G\mathcal{M}}{J}\right)^{2}\left(\frac{{2}\lambda_{1}\cos\theta_{p}}{c^{2}}\right),$
(33)
and:
$\beta_{2}l=\lambda_{2}\left(\frac{{3}G\mathcal{M}}{c^{4}}\right)\left(\frac{G\mathcal{M}}{J}\right)^{2}\left(\frac{{3}\cos^{2}\theta_{p}-{1}}{\textit{2}}\right)$
(34)
The left hand side of this equation (i.e. 32) is what one gets from pure
Newtonian theory and the term on the right is the new term due to the first
order term in the corrected Newtonian potential and likewise the second term
on the right is a new term due to the second order term in the corrected
Newtonian potential.
Now, taking the term $\beta_{1}u$ in equation (32) to the right hand side, we
will have:
$\frac{d^{2}u}{d\varphi^{2}}+({1}-\beta_{1})u-\frac{G\mathcal{M}}{J^{2}}=\beta_{2}lu^{2}.$
(35)
We know that the solution of the right hand side of the above equation when
set to zero, i.e.:
$\frac{d^{2}u}{d\varphi^{2}}+({1}-\beta_{1})u-\frac{G\mathcal{M}}{J^{2}}=0,$
(36)
is given by:
$r=\frac{l}{{1}+\epsilon\cos(\eta_{1}\varphi)},$ (37)
where:
$\eta_{1}=\sqrt{{1}-\beta_{1}}=\sqrt{{1}-\left(\frac{G\mathcal{M}}{J}\right)^{2}\left(\frac{{2}\lambda_{1}\cos\theta_{p}}{c^{2}}\right)}.$
(38)
To obtain a solution to (35) to first order approximation, we note that the
left hand side has solution (37) and that for nearly Newtonian orbits this
solution $u=({1}+\epsilon\cos\varphi)/l$, is a good approximation to (35) for
nearly Newtonian orbits such as Mercury for example. Consequently, we can
rewrite the small term on the right hand side of (35) as:
${3}G\mathcal{M}({1}+\epsilon\cos\varphi)^{2}/l^{2}$; and make an entirely
negligible error (see e.g. Kenyon 1990). With this substitution, equation (35)
becomes:
$\frac{d^{2}u}{d\varphi^{2}}+\eta_{1}^{2}u-\frac{G\mathcal{M}}{J^{2}}=\frac{\beta_{2}}{l}\left({1}+{2}\epsilon\cos\varphi+\epsilon^{2}\cos^{2}\varphi\right),$
(39)
and the solution to this equation is:
$u=\frac{{1}+\epsilon\cos\eta_{1}\varphi}{l}+\frac{\beta_{2}}{l}\left[\left({1}+\frac{\epsilon^{2}}{\textit{2}}\right)+\frac{\epsilon^{2}\cos{2}\varphi}{6}+\epsilon\varphi\sin\varphi\right].$
(40)
As before, i.e., as in the steps leading to Einstein’s solution; of the
additional terms, the first is a constant and the second oscillates through
two cycles on each orbit; both these terms are immeasurably small. However,
the last term increases steadily in amplitude with $\varphi$, and hence with
time, whilst oscillating through one cycle per orbit; clearly this term is
responsible for the precession of the perihelion. Now, dropping all the
unimportant terms one is lead to:
$u=\frac{{1}+\epsilon\cos\eta_{1}\varphi+\epsilon\eta_{2}\varphi\sin\eta_{1}\varphi}{l},$
(41)
where for the convenience we have set $\eta_{2}=\beta_{2}$ and this quantity
is extremely small, in which case $\cos\eta_{2}\varphi\simeq 1$ and
$\sin\eta_{2}\varphi\simeq\eta_{2}\varphi$ and using these approximations (in
the cosine addition formula
$\cos\eta_{1}\varphi+\eta_{2}\varphi\sin\eta_{1}\varphi\simeq\cos\eta_{2}\varphi\cos\eta_{1}\varphi+\sin\eta_{2}\varphi\sin\eta_{1}\varphi=\cos\left[(\eta_{1}+\eta_{2})\varphi\right]$),
we will have:
$u=\frac{{1}+\epsilon\cos\left[(\eta_{1}+\eta_{2})\varphi\right]}{l}.$ (42)
Now, at the perihelion we are going to have:
$(\eta_{1}+\eta_{2})\varphi=2n\pi$ where $n={1},{2},{3},\dots$ and this
implies $\varphi={2}\pi n\left(\eta_{1}+\eta_{2}\right)^{-1}=2\pi
n[\sqrt{1-\beta_{1}}+\beta_{2}]^{-1}\simeq 2\pi
n[1-(\beta_{1}-2\beta_{2})/2]^{-1}=2\pi n[1+(\beta_{1}/2-\beta_{2})+...]$
hence: $\varphi\simeq{2}\pi n+n\lambda_{1}h_{1}+n\lambda_{2}h_{2}$, where:
$h_{1}=\left(\frac{{6}\pi
G\mathcal{M}}{lc^{2}}\right)\left(\frac{\cos\theta_{p}}{3}\right),$ (43)
and:
$h_{2}=-\left(\frac{{3}\cos^{2}\theta_{p}-{1}}{{12}\pi}\right)\left(\frac{{6}\pi
G\mathcal{M}}{lc^{2}}\right)^{2}.$ (44)
This shows that per every revolution, the perihelion advances by:
$\frac{\Delta\varphi}{\tau}=\left(\frac{{6}\pi
G\mathcal{M}}{lc^{2}}\right)\left(\frac{\lambda_{1}\cos\theta_{p}}{3}\right)-\lambda_{2}\left(\frac{{3}\cos^{2}\theta_{p}-{1}}{{12}\pi}\right)\left(\frac{{6}\pi
G\mathcal{M}}{lc^{2}}\right)^{2},$
and this can be written more neatly and conveniently as:
$\left<\frac{\Delta\varphi}{\tau}\right>_{O}=\left<\frac{\Delta\varphi}{\tau}\right>_{E}\left[\frac{\cos\theta_{p}}{{3}}\lambda_{1}-\left<\frac{\Delta\varphi}{\tau}\right>_{E}\frac{{3}\cos^{2}\theta_{p}-{1}}{{12}\pi\tau^{-1}}\lambda_{2}\right].$
(45)
This formula – which is a second order approximation; tells us of the
perihelion shift of the planets. In the next section we will use this to
deduce an estimate of the values of $\lambda_{1}$ and $\lambda_{2}$ for the
Solar System and thereafter proceed to calculate the predicted values of the
perihelion shift. As a way of showing that these are solar values, let us
denote ($\lambda_{1}$ and $\lambda_{2}$) as ($\lambda_{1}^{\tiny\odot}$ and
$\lambda_{2}^{\tiny\odot}$) respectively.
## 5 An Estimate for $\lambda_{1}^{\tiny\odot}$ and
$\lambda_{2}^{\tiny\odot}$
If $\mathscr{P}_{p}$ is the precession per century of the perihelion of planet
$p$, i.e.:
$\mathscr{P}_{p}=\left<\frac{\Delta\varphi}{\tau}\right>_{O},$ (46)
then equation (45) can be written as:
$\mathscr{P}_{p}=\mathscr{A}_{p}\lambda^{\tiny\odot}_{1}+\mathscr{B}_{p}\lambda^{\tiny\odot}_{2},$
(47)
where:
$\mathscr{A}_{p}=\left<\frac{\Delta\varphi}{\tau}\right>_{E}\left(\frac{\cos\theta_{p}}{{3}}\right),$
(48)
and:
$\mathscr{B}_{p}=-\left(\frac{{3}\cos^{2}\theta_{p}-{1}}{{12}\pi\tau^{-1}}\right)\left(\left<\frac{\Delta\varphi}{\tau}\right>_{E}\right)^{2}.$
(49)
Given a set of the observed values for the size ($l_{p}$), the period of
revolution $\tau_{p}$, the tilt ($\theta_{p}$) and the known precessional
values of the perihelion of planets ($\mathscr{P}^{obs}_{p}$); these values
are listed in columns ${2}$, ${3}$, ${4}$ and ${8}$ of table (I) respectively;
we can solve for $\lambda^{\tiny\odot}_{1}$ and $\lambda^{\tiny\odot}_{2}$
since $\mathscr{P}_{p}$, $\mathscr{A}_{p}$ and $\mathscr{B}_{p}$ will all be
known, thus one simple has to solve equation (47) for any pair of planets as a
simultaneous equation.
The values of $\mathscr{A}_{p}$ and $\mathscr{B}_{p}$ for all the solar
planets are listed in columns ${6}$ and ${7}$ of table (I) respectively. It is
important that we state that the values of the Inclination listed in column
$4$ of table (I) are the inclination of the planetary orbits relative to the
ecliptic plane and in-order to compute the inclination of these orbits
relative to the solar equator we have to add ${7}\hbox{${}^{\circ}$$$}$ to
this because the ecliptic plane and the solar equator are subtended at this
angle. The solar equator is here defined as the plane cutting the Sun into
hemispheres and this plane is normal to the spin axis of the Sun.
Perihelion Precession of Solar Planetary Orbits According to the ASGT
| Precession ($1\arcsec/100{yrs}$)
---|---
| ———————————————————————
Planet | ${}^{(b)}l_{p}$ | ${}^{(b)}\tau_{p}$ | (b)Incl. | (b) $\epsilon$ | $\mathscr{A}_{p}$ | $\mathscr{B}_{p}$ | $\mathscr{P}_{p}^{obs}$ | $\mathscr{P}^{E}_{p}$ | $\mathscr{P}_{p}$
| $(\textrm{AU})$ | $({yrs})$ | (∘) | | | | | |
Mercury | ${0.39}$ | ${0.24}$ | ${7.0}$ | ${0.206}$ | ${3.50}\times{10}^{0}$ | ${1.72}\times{10}^{2}$ | $43.1000\pm 0.5000^{(c)}$ | $43.50000$ | $42.80000\pm 0.10000$
Venus | ${0.72}$ | ${0.62}$ | ${3.4}$ | ${0.007}$ | ${5.19}\times{10}^{-1}$ | ${2.88}\times{10}^{1}$ | $8.0000\pm 5.0000^{(c)}$ | $\,\,\,8.62000$ | $12.00000\pm 3.00000$
Earth | ${1.00}$ | ${1.00}$ | ${0.0}$ | ${0.017}$ | ${1.57}\times{10}^{-1}$ | ${3.80}\times{10}^{-1}$ | $5.0000\pm 1.0000^{(c)}$ | $\,\,\,3.87000$ | $\,\,\,4.00000\pm 1.00000$
Mars | ${1.52}$ | ${1.88}$ | ${1.9}$ | ${0.093}$ | ${7.02}\times{10}^{-2}$ | ${2.43}\times{10}^{-2}$ | $1.3624\pm 0.0005^{(e)}$ | $\,\,\,1.36000$ | $\,\,\,1.70000\pm 0.50000$
Jupiter | ${5.20}$ | ${11.86}$ | ${1.3}$ | ${0.048}$ | ${3.02}\times{10}^{-3}$ | ${1.00}\times{10}^{-5}$ | $0.0700\pm 0.0040^{(e)}$ | $\,\,\,0.06280$ | $\,\,\,0.07000\pm 0.02000$
Saturn | ${9.54}$ | ${29.46}$ | ${2.5}$ | ${0.056}$ | ${7.59}\times{10}^{-4}$ | ${1.72}\times{10}^{-7}$ | $0.0140\pm 0.0020^{(e)}$ | $\,\,\,0.01380$ | $\,\,\,0.01900\pm 0.00050$
Uranus | ${19.2}$ | ${84.10}$ | ${0.8}$ | ${0.046}$ | ${1.09}\times{10}^{-4}$ | ${9.76}\times{10}^{-5}$ | $--^{(f)}$ | $\,\,\,0.00240$ | $\,\,\,0.00250\pm 0.00070$
Neptune | ${30.1}$ | ${164.80}$ | ${1.8}$ | ${0.009}$ | ${3.98}\times{10}^{-5}$ | ${9.13}\times{10}^{-11}$ | $--^{(f)}$ | $\,\,\,0.00078$ | $\,\,\,0.00270\pm 0.00070$
Pluto(a) | ${39.4}$ | ${247.70}$ | ${17.2}$ | ${0.250}$ | ${5.77}\times{10}^{-5}$ | ${9.48}\times{10}^{-12}$ | $--^{(f)}$ | $\,\,\,0.00042$ | $\,\,\,0.00140\pm 0.00040$
Notes: ${}^{\textbf{(a)}}$ At the 2006 annual meeting of the International
Astronomical Union, it was democratically decided that the solar test body
Pluto is not a planet but a dwarf planet. For our purpose, its inclusion here
as a planet is not affected by this decision for as long as this test body
orbits the Sun like other planets. ${}^{\textbf{(b)}}$ The values of
$l_{p},\tau_{p},$ Inc. and Ecc. are adapted from Sagan (1974).
${}^{\textbf{(c)}}$ Adapted from Kenyon (1990). ${}^{\textbf{(d)}}$ Adapted
from Pitjeva (2005). ${}^{\textbf{(e)}}$ Obtained by adding the extra
precession determined by Pitjeva (2005) and found in Iorio (2008b) to the
standard Einsteinian perihelion precession. ${}^{\textbf{(f)}}$ Because of
their long orbital duration covering at least $2$ human lifetimes, no data is
currently available covering one full orbital revolution for Neptune and Pluto
hence there is not yet any observational values for the precession of their
perihelia. The data for Uranus is unreliable (see e.g. Iorio 2008b).
Table I: Above, column 1 gives the name of the planet $p$, column 2 gives
$l_{p}$ which is the observed value for the orbital size of planet $p$, column
3 gives $\tau_{p}$ which is the period of revolution of the planet for one
full orbit, column 4 is the tilt $\theta_{p}$ in degrees of the planet’s orbit
orbit relative to the ecliptic plane, column 5 gives the eccentricity of the
orbit of the planet, while columns 6 and column 7 give the computed values
$\mathscr{A}_{p}$ and $\mathscr{B}_{p}$ and column 8,9 and 10 give (1) the
observed, (2) the GTR and (3) the ASTG precessional values of the planet.
Now, having calculated the values of $\lambda^{\tiny\odot}_{1}$ and
$\lambda^{\tiny\odot}_{2}$, we will have to use these values
($\lambda^{\tiny\odot}_{1}$ and $\lambda^{\tiny\odot}_{2}$) to check what are
the predictions for the precession of the perihelion of the other seven
planets. If the predictions of our theory are in agreement with the observed
precession of the perihelion of these seven planets, then our theory is
correct and if the predictions are otherwise then, our theory cannot be
correct – it must be wrong!
For the present, we have calculated $\lambda^{\tiny\odot}_{1}$ and
$\lambda^{\tiny\odot}_{2}$ for the different planet pairs were we have all the
information to do so and these values are displayed in table (II). The final
adopted values are:
$\centering\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lambda_{1}^{\tiny\odot}={24.0\pm
7.0}\,\,\textrm{and}\,\,\lambda_{2}^{\tiny\odot}={-0.200\pm
0.100}.\@add@centering$ (50)
and these values are the mean and the standard deviation – there is a $27\%$
error in $\lambda_{1}^{\tiny\odot}$ and about twice ($50\%$) that error margin
in $\lambda_{2}^{\tiny\odot}$. From the values given in (50), the predicted
values of the precession of the perihelion of the other seven planets i.e.
Earth, Mars, …, Pluto; where computed and are listed in column 10 of table
(I). The equivalent predictions of these values from Einstein’s theory are
listed in column 9 of the same table. Inspection of the predictions of our
theory reveals that our predicted values are – as Einstein’s predictions; in
good agreement with observations. We believe that this does not mean the
theory is correct but merely that it contains an element of truth in it. It
means we have a reason to believe in it and as well a reason to peruse it
further from the present exploration to its furthest reaches if this were at
all possible!
The reader should take note that in our derivation, we have assumed as a first
order approximation the Newtonian result namely that the angular momentum is a
time constant. From the preceding section, clearly this is not the case. We
have only assumed this as starting point of our exploration. It is hoped that
taking into account the fact arising from the ASTG that orbital angular
momentum is not a conserved quantity should lead to improved results that
hopefully come closer to the observed values.
Estimation of the Values $\lambda_{1}^{\tiny\odot}$ and
$\lambda_{2}^{\tiny\odot}$
Planet Plair | $\lambda_{1}$ | $\lambda_{2}$
---|---|---
Mercury-Venus | $15.8$ | $-0.0716$
Mercury-Earth | $32.8$ | $-0.4174$
Mercury-Mars | $20.0$ | $-0.1574$
Mercury-Jupiter | $26.1$ | $-0.2895$
Mercury-Saturn | $27.6$ | $-0.3112$
Mean | :$\,24.0$ | $-0.200$
Standard Deviation | : $\,\,7.0$ | $-0.100$
Percentage Error | : $27\%$ | $50\%$
Table II: Column 1 in the first part of the table gives the name of the pair
of the planets which have been used to obtain the pair of $\lambda$-values
listed in columns 2 and 3. In the second part of the table we compute the
Mean, Standard Deviation and Percentage Error of the $\lambda$-values
## 6 None Conserved Orbital Angular Momentum and its Implications
Through equation (31) which clearly states that the orbital angular momentum
of a planet must change with time; three immediate consequences of this are
(1) a change in the mean Sun-Planet distance (2) a changing length of a
planet’s day and (3) a secular change in solar spin. In the subsequent
subsection, we shall go through these implied phenomena.
### 6.1 Increase in Mean Sun-Planet Distance
One of the most accurately determined physical parameters in astronomy is the
mean Earth-Sun distance which is about the size of the Astronomical Unit
($AU$) where $1\,AU=149597870696.1\pm 0.1m$ (Pitjeva, 2005) and this is known
to an accuracy of $10\,cm$ (Pitjeva, 2005). The Astronomical Unit according to
the International Astronomical Union (Resolution No. 10 1976111see
http://www.iau.org/static/resolutions/IAU1976 French.pdf) is defined as the
radius of an unperturbed circular orbit that a massless body would revolve
about the Sun in $2\pi/k$ days where $k=01720209895AU^{3/2}day^{-1}$ is Gauss’
constant. This definition is such that there is an equivalence between the AU
and the mass of the Sun $\mathcal{M}_{\tiny\odot}$ which is given by
$G\mathcal{M}_{\tiny\odot}=k^{2}A^{3}$. So, if $\mathcal{M}_{\tiny\odot}$ is
fixed, it is technically incorrect to speak of a change AU.
Before it was noticed that the mean Earth-Sun distance was changing it made
perfect sense to refer to the mean Earth-Sun distance as the Astronomical
Unit. Now, (1) because units must not change, and (2) because of this fact
that the mean Earth-Sun distance is changing; then, until such a time that the
Astronomical Unit is correctly defined so that it is a true constant as
physical unit must be, it makes sense only to talk of the mean Earth-Sun
distance instead of the Astronomical Unit.
That the mean Earth-Sun distance is changing, this has been measured by
Krasinsky & Brumberg (2004) and Standish (2005). Krasinsky & Brumberg (2004)
finds $15.0\pm 4.0\,m/cy$ which in SI units is $(4.75\pm 1.27)\times
10^{-9}m/s$ and Standish (2005) finds $7.00\pm 0.20\,m/cy$ which in SI units
is $(2.22\pm 0.06)\times 10^{-9}\,m/s$ where $1cy=100\,yr$.
To this rather surprising result, i.e., the apparent secular change in the
mean Earth-Sun distance, Iorio (2008a) states that the secular increase in the
mean Earth-Sun distance can not be explained within the realm of classical
physics. Contrary to this, we believe and hold that the ASTG can in-principle
explain this result. The ASTG is well within the provinces of classical
physics hence thus this result is explainable from within the domains and
confines of classical physics. In his reading (Iorio, 2008a) argues that the
Dvali-Gabadadze-Porrati braneworld scenario – a none-classical theory, which
is a multi-dimensional model of gravity aimed to the explanation of the
observed cosmic acceleration without darkenergy, predicts, among other things,
a perihelion secular shift, due to Lue-Starkman Effect of $5\times
10^{-4}arcsec/cy$ for all the planets of the Solar System. It yields a
variation of about $6m/cy$ for the increase in mean Earth-Sun distance; this
is compatible with the observed time rate of change of the mean Earth-Sun
distance hence giving the Dvali-Gabadadze-Porrati braneworld theory some
breath.
Iorio (2008a) goes on to say that the recently measured corrections to the
secular motions of the perihelia of the inner planets of the Solar System are
in agreement with the predicted value of the Lue-Starkman effect for Mercury,
Mars and, at a slightly worse level, the Earth. We shall show that in-
principle, the ASTG can explain this result as a consequence of the none-
conservation of orbital angular momentum of planets in this azimuthally
symmetric gravitational setting. The none-conversation of the orbital angular
momentum leads directly to a time variation in the eccentricity of planetary
orbits. This makes the secular change a purely classical result.
Now, given the definition of the eccentricity of an orbit:
$\epsilon^{2}=1-\left(\frac{\mathcal{R}_{min}}{\mathcal{R}_{max}}\right)^{2}$
(51)
where $\mathcal{R}_{min}$ and $\mathcal{R}_{max}$ are the spatial extent of
the minor and major axis respectively; and then, differentiating this with
respect to time, one is lead to:
$\epsilon\frac{d\epsilon}{dt}=-\frac{\mathcal{R}_{min}}{\mathcal{R}_{max}^{2}}\left(\frac{d\mathcal{R}_{min}}{dt}-\frac{\mathcal{R}_{min}}{\mathcal{R}_{max}}\frac{d\mathcal{R}_{max}}{dt}\right).$
(52)
There is no reason to assume that the rate of change of the minor and major
axis be the same, thus we must set:
$\frac{d\mathcal{R}_{max}}{dt}=\left(\gamma+1\right)\left(\frac{d\mathcal{R}_{min}}{dt}\right),$
(53)
and from this it follows that:
$\epsilon\frac{d\epsilon}{dt}=-\left(\frac{\mathcal{R}_{min}}{\mathcal{R}_{max}^{2}}\right)\left(1-\left(\gamma+1\right)\frac{\mathcal{R}_{min}}{\mathcal{R}_{max}}\right)\left(\frac{d\mathcal{R}_{min}}{dt}\right).$
(54)
and multiplying by $\mathcal{R}_{min}$ both sides, and thereafter substituting
$\mathcal{R}_{min}/\mathcal{R}_{max}$ on the right hand side we will have:
$\epsilon\mathcal{R}_{min}\frac{d\epsilon}{dt}=-(1-\epsilon^{2})\left(1-\left(\gamma+1\right)\sqrt{(1-\epsilon^{2})}\right)\frac{d\mathcal{R}_{min}}{dt},$
(55)
therefore:
$\frac{d\mathcal{R}_{min}}{dt}=-\frac{\mathcal{R}_{min}}{(1-\epsilon^{2})\left(1-\left(\gamma+1\right)\sqrt{1-\epsilon^{2}}\right)}\left(\epsilon\frac{d\epsilon}{dt}\right).$
(56)
Now, on the average, the time change of the minor axis must to a large extend
be a good measure of the time change of the average distance
$\left<\mathcal{R}\right>$ between the planet and the Sun, hence thus:
$\frac{d\left<\mathcal{R}\right>}{dt}=\frac{\left<\mathcal{R}\right>}{(1-\epsilon^{2})\left(\left(\gamma+1\right)\sqrt{1-\epsilon^{2}}-1\right)}\left(\epsilon\frac{d\epsilon}{dt}\right).$
(57)
In the realm of Newtonian gravitation where spherical symmetry is assumed thus
producing equations only dependent on the radial distance $r$, the
eccentricity is an absolute time constant, i.e. $d\epsilon/dt\equiv 0$, and
this directly leads to $d\left<\mathcal{R}\right>/dt\equiv 0$, hence when one
finds that the mean Earth-Sun distance is increasing, it comes more as a
surprise. If we consider azimuthally symmetry in Poisson’s equation as has
been done here, the result emerges naturally because the eccentricity is
expected to increase with the passage of time – this we shall demonstrate very
soon.
In §(4), against the clear message from the ASTG, we assumed that the orbital
angular momentum of a planet is a conserved quantity. It turns out that taking
this into account leads us to two type of orbits (1) spiral orbits (2) the
normal elliptical orbits with the important difference that the eccentricity
of these orbits varies with time and it is this variation of eccentricity that
we believe the secular increase of the mean Earth-Sun distance is rooted.
Doing the right thing and taking into account the predicted change in the
angular momentum, then equation (35) will be:
$\frac{d^{2}u}{d\varphi^{2}}+\left(\frac{1}{J^{2}u^{2}}\frac{dJ}{dt}\right)\frac{du}{d\varphi}+\eta_{1}^{2}u-\frac{G\mathcal{M}}{J^{2}}=\beta_{2}lu^{2},$
(58)
and taking the change of angular momentum to first order approximation from
equation (31), one will have:
$\frac{dJ}{dt}=-\left[\lambda_{1}\left(\frac{G\mathcal{M}}{c}\right)^{2}\sin\theta_{p}\right]u^{2}=-2\alpha
u^{2},$ (59)
where $\alpha$ is clearly defined from this equation i.e.:
$\alpha=\frac{1}{2}\left[\lambda_{1}\left(\frac{G\mathcal{M}}{c}\right)^{2}\sin\theta_{p}\right],$
(60)
it therefore follows that:
$\frac{d^{2}u}{d\varphi^{2}}-\frac{2\alpha}{J^{2}}\frac{du}{d\varphi}+\eta_{1}^{2}u-\frac{G\mathcal{M}}{J^{2}}=\beta_{2}lu^{2}.$
(61)
and writing $k=\alpha/J^{2}$ which is:
$k=\frac{\lambda_{1}}{2}\left(\frac{G\mathcal{M}}{cJ}\right)^{2}\sin\theta_{p}=\frac{\lambda_{1}}{2}\left(\frac{G\mathcal{M}}{lc^{2}}\right)\sin\theta_{p},$
(62)
where the Newtonian approximation $J^{2}=G\mathcal{M}l$ has been used and
$K=G\mathcal{M}/J^{2}$, the above becomes:
$\frac{d^{2}u}{d\varphi^{2}}-k\frac{du}{d\varphi}+\eta_{1}^{2}u-K=\beta_{2}lu^{2}.$
(63)
If the orbital angular momentum varies constantly with time, then
$J=\dot{J}t+J_{0}$ where $J_{0}$ is the angular momentum at time $t=0$ and
$\dot{J}$ is a time constant, then $k=k(t)$ and $K=K(t)$, meaning $k(t)$ and
$K(t)$ will dependent not on the coordinates $r,\theta,\varphi$ but only on
time, hence in solving the above equation we can treat these as constants
since they do not dependent on $r,\theta,\varphi$. We believe the assumption
that $\dot{J}=constant$ is justified because if that was not the case, there
could be an accelerated increase in the orbital angular momentum and this
could have been noticed by now. In this assumption that $\dot{J}=constant$, we
must have $\dot{J}$ being so small that it is not easily noticeable as it
appears to be the case since we have had to relay on delicate observations to
deduce the secular increase of the mean Earth-Sun distance.
Now, to obtain a solution to this equation (i.e. 63), we need first to get a
solution to:
$\frac{d^{2}u}{d\varphi^{2}}-2k\frac{du}{d\varphi}+\eta_{1}^{2}u-\frac{G\mathcal{M}}{J^{2}}=0,$
(64)
and to obtain a solution to this, we need first to solve:
$\frac{d^{2}u}{d\varphi^{2}}-2k\frac{du}{d\varphi}+\eta_{1}^{2}u=0,$ (65)
and to its solution we add $G\mathcal{M}/J^{2}$. The axillary differential
equation to this differential equation is: $X^{2}-2kX+\eta_{1}^{2}=0$ and its
(i.e. equation 65) solutions are:
$X=k\pm\sqrt{k^{2}-\eta_{1}^{2}}=k\pm i\eta_{3},$ (66)
where $\eta_{3}=\sqrt{\eta_{1}^{2}-k^{2}}$. If $(\eta_{3})^{2}<0$ the solution
is:
$u=Ae^{\left(k+\eta_{3}\right)\varphi}+Be^{\left(k-\eta_{3}\right)\varphi}$
where $A$ and $B$ are constants, thus adding $G\mathcal{M}/J^{2}$ we have:
$u=Ae^{\left(k+\eta_{3}\right)\varphi}+Be^{\left(k-\eta_{3}\right)\varphi}+\frac{G\mathcal{M}}{J^{2}},$
(67)
and if $(\eta_{3})^{2}=0$ the solution is: $u=(A\varphi+B)e^{k\varphi}$ thus
adding $G\mathcal{M}/J^{2}$ we have:
$u=(A\varphi+B)e^{k\varphi}+\frac{G\mathcal{M}}{J^{2}}.$ (68)
The solutions (67) and (68) are clearly spiral orbits. These solutions are
obvious very interesting but because our focus is not on them, but on the
solutions giving elliptical orbits in which the eccentricity varies, we shall
not be looking into these spiral orbit solutions any further than we have
already done.
Now, in the event that $(\eta_{3})^{2}>0$ the solution to equation (64) is:
$u=\frac{1+\epsilon e^{k\varphi}\cos(\eta_{3}\varphi)}{l},$ (69)
Now, using the same strategy as that used in §(3) and (4) to solving equations
(20) and (35) respectively, one finds that the resultant orbit equation will
be:
$r=\frac{l}{1+\epsilon e^{k\varphi}\cos[(\eta_{2}+\eta_{3})\varphi]},$ (70)
and as before, at the perihelion we will have $(\eta_{2}+\eta_{3})\varphi=2\pi
n$ and this implies $\varphi=2\pi n(\eta_{2}+\eta_{3})^{-1}\simeq 2\pi
n[\beta_{2}+\sqrt{\eta_{1}^{2}-k^{2}}]^{-1}=2\pi
n[\beta_{2}+\sqrt{1-\beta_{1}-k^{2}}]^{-1}\simeq 2\pi
n[1+(2\beta_{2}-\beta_{1})/2-k^{2}/2]^{-1}$ and taking only first order terms
we will have: $\varphi\simeq 2\pi n[1+(\beta_{1}-2\beta_{2})/2+k^{2}/2]$ and
this shows that the perihelion will precess by an amount
$\Delta\varphi=2\pi[(\beta_{1}/2-\beta_{2})+k^{2}/2]$, and in comparison with
$\Delta\varphi\simeq 2\pi[\beta_{1}/2-\beta_{2}]$ obtained without taking into
account the changing angular momentum, there is an additional precession of
$(\Delta\varphi)_{\textbf{+}}\simeq~{}\pi k^{2}$. The value of $k^{2}$ for the
Solar System is so small that in practice, one can neglect it, thus, we have
not missed out much in our calculation in which we have assumed a constant
orbital angular momentum. While this result is important our main thrust is to
deduce the variation of the eccentricity of elliptical orbits (we shall shelf
any deliberations on this result for a further reading).
In equation (70), the term $\epsilon e^{k\varphi}$ in the denominator is the
eccentricity, let us write this as $\epsilon_{*}=\epsilon e^{k\varphi}$, and
from this we see that the eccentricity varies with time – i.e.; as the orbital
angular momentum changes with the passage of time, so does the eccentricity.
Now plucking this into equation (57) we can determine the variation of the
mean Earth-Sun distance if we have knowledge of $\gamma$, unfortunately we do
not have this. However, if we are to reproduce the observed variation of the
Earth-Sun distance, one finds that if they were to set $\gamma_{E}=1.48\times
10^{-4}$, which practically means that the orbit grows evenly at every point,
one is able to explain the secular increase of the mean Earth-Sun distance.
It should be said that, if the ASTG is to stand on its own – i.e., independent
of observations, then it must be able to explain the result
$\gamma_{E}=1.48\times 10^{-4}$ from within its own provinces. It is for this
reason that we say, in-principle, the ASTG is able to explain the secular
increase in the mean Earth-Sun distance and only until such a time when one is
able to derive say the value $\gamma_{E}=1.48\times 10^{-4}$ from within the
theory itself, will we be able to say the ASTG explains the secular increase
in the Earth-Sun distance.
Other than the secular increase in the mean Earth-Sun distance, there is also
the increase in the mean Earth-Moon distance. This has been measured by
Williams & Boggs (2009) to be $\sim 3.50\times 10^{-3}\,m/cy$ and in SI units
this is $1.11\times 10^{-12}\,m/s$. This observation provides a test for the
ASGT, but unfortunately, we do not have the value of $\lambda_{1}$ so as to
check what the ASTG says about this. We believe one cannot use the same
$\lambda$’s values obtained for the Sun because these values must be specific
to the gravitating body and may very well be connected to the spin or the
gravitating body in question. We are working on these ideas to improve the
ASTG and at present we can only say it is prudent to assume that the
$\lambda$-values are specific to the body in question hence one has to
calculate them from observational data. For the Earth, this increase in the
Earth-Moon distance is but the only observations we have in-order for us to
deduce $\lambda_{1}^{\tiny{\earth}}$ hence the ASTG is unable to make any
predictions on this as it stands in the present. We hope in the future one
will be able to deduce a general form of the $\lambda_{\ell}$-values, thus
placing the ASTG on a level where it is able to make predictions that are
independent from observations.
Important to note from $\epsilon_{*}=\epsilon e^{k\varphi}$ is that, as
$\varphi\longmapsto-\infty$, the eccentricity will decrease and the reserve is
that the eccentricity will increase as $\varphi\longmapsto+\infty$ decreases.
An increasing eccentricity leads to a secular decrease in the Planet-Sun
distance and a decreasing eccentricity leads to a secular increase in the
Planet-Sun distance. This means the sense in which the planet orbits the Sun
is important! Because we believe from Krasinsky & Brumberg (2004) and
Standish (2005), that there is a secular increase in the Earth-Sun distance,
this means the current direction of rotation of the Earth around the Sun must
be such that $\varphi\longmapsto-\infty$. This must be true for other planets
rotating in the same sense as the Earth; and to any (object in the Solar
System) that rotates in the direction opposite to this, this body will
experience a secular decrease in its distance from the Sun.
### 6.2 Secular Increase in the Orbital Period of Planets
Given that through the passage of time – what is suppose to be a sacrosanct
parameter – the mean Earth-Sun distance; is changing, and that the time change
of the specific orbital angular momentum is given
$\dot{J}=2r\dot{r}\dot{\theta}+r^{2}\ddot{\theta}$, then, if as in the case of
Newtonian gravitation the specific orbital angular momentum of a planet is a
conserved quantity, i.e. $\dot{J}=2r\dot{r}\dot{\theta}+r^{2}\ddot{\theta}=0$,
then accompanying this result of a changing mean Earth-Sun distance must be an
increase in the length of a planet’s duration for one complete orbit since
$\ddot{\theta}/\dot{\theta}=-2\dot{r}/r$. Given that
$\dot{\theta}=2\pi/\mathcal{T}_{Y}$ where $\mathcal{T}_{Y}$ is the orbital
period of a planet, the equation $\ddot{\theta}/\dot{\theta}=-2\dot{r}/r$
becomes: $\dot{\mathcal{T}}_{Y}/\mathcal{T}_{Y}=2\dot{r}/r$. Plucking in the
relevant values for the Earth, one is lead to
$\dot{\mathcal{T}}^{\tiny{\earth}}_{Y}=2.97\,ms/cy$. Since
$\mathcal{T}_{Y}^{\tiny{\earth}}=365.25\mathcal{T}_{D}$ where
$\mathcal{T}_{D}^{\tiny{\earth}}$ is the period of an Earth day, it follows
that:
$\dot{\mathcal{T}}_{Y}^{\tiny{\earth}}=365.25\dot{\mathcal{T}}_{D}^{\tiny{\earth}}$,
further, follows that we must have: $\mathcal{T}_{D}^{\tiny{\earth}}=8.13\,\mu
s/cy$ – this value is at odds with physical reality; for records held for over
$2700\,yrs$ indicate that the Earth day changes by an amount
$\dot{\mathcal{T}}^{\tiny{\earth}}_{D}=+1.70\pm 0.05\,ms/cy$ (see e.g. Miura
et al. 2009), which is about $200$ times that expected if the orbital angular
momentum where a conserved quantity as in Newtonian gravitation – clearly,
this suggests that the orbital angular momentum may not be conserved.
If say the conserved quantity where the total angular momentum of a planet,
i.e. the sum total of the spin angular momentum ($S$) and the orbital angular
momentum, then $\dot{S}=-\dot{J}$ and if the radius of the planet is not
changing with time, then
$\dot{\mathcal{T}}_{D}=-2\pi\mathcal{R}^{2}\dot{J}\mathcal{T}_{D}^{2}$. For
the Earth, one finds that $\dot{\mathcal{T}}_{D}^{\tiny{\earth}}=-5.18\,s/cy$
which is $\sim 3000$ times the observed value – this can not be, sure
something must be wrong. We shall explain this observational value
$\dot{\mathcal{T}}^{\tiny{\earth}}_{D}=1.70\pm 0.05\,ms/cy$ from the ASTG.
From the ASTG, we have:
$\left(\frac{\dot{J}}{J}\right)_{\tiny{\earth}}^{theory}=-(6.00\pm 2.00)\times
10^{-15}s^{-1},$ (71)
and we know that:
$\left(\frac{\dot{J}}{J}\right)_{p}=2\left(\frac{\dot{\mathcal{R}}}{\mathcal{R}}\right)_{p}-\left(\frac{\dot{\mathcal{T}_{Y}}}{\mathcal{T}_{Y}}\right)_{p}$
(72)
hence plucking in the observed values and remembering not to forget that for
the Earth
$\dot{\mathcal{T}}_{Y}^{\tiny{\earth}}=365.25\dot{\mathcal{T}}_{D}^{\tiny{\earth}}$,
then we will have:
$\left(\frac{\dot{J}}{J}\right)_{\tiny{\earth}}^{obs}=-(2.28\pm 0.07)\times
10^{-15}s^{-1}.$ (73)
This value – vis, the order of magnitude, is on a satisfactory level in good
agreement with observations. We take this as further indication that the ASTG
contains in it, a grail of the truth.
### 6.3 Secular Increase in Solar Spin
We know that angular momentum must be conserved but according to (31), it is
not conserved. This lost orbital angular momentum must go somewhere – it
cannot just disappear into the thin interstices of spacetime or into the
wilderness of spacetime thereof. Let $\mathcal{L}_{tot}$ be the sum total
angular momentum of the Solar System, were we consider that the Solar System
is composed of the planets. If the sum total of the angular momentum of a
planet and its system of satellite is $J_{p}^{tot}$, then
$\mathcal{L}_{tot}=\mathcal{M}_{\tiny\odot}S_{\tiny\odot}+\sum_{i}\mathcal{M}_{i}J_{i}^{tot}$.
We would expect that the total angular momentum of the Solar System be
conversed, that is $d\mathcal{L}_{tot}/dt=0$. From this we must have:
$\frac{\dot{S}_{\tiny\odot}}{S_{\tiny\odot}}=-\frac{\dot{\mathcal{M}}_{\tiny\odot}}{\mathcal{M}_{\tiny\odot}}-\frac{1}{S_{\tiny\odot}}\sum_{i}\left[\frac{\mathcal{M}_{i}}{\mathcal{M}_{\tiny\odot}}\left(\frac{dJ_{i}^{tot}}{dt}\right)\right],$
(74)
and $dJ^{tot}_{p}/dt=dJ_{p}/dt$ hence thus:
$\frac{\dot{\mathcal{T}}_{\tiny\odot}}{\mathcal{T}_{\tiny\odot}}=\frac{2\dot{\mathcal{R}}_{\tiny\odot}}{\mathcal{R}_{\tiny\odot}}+\frac{\dot{\mathcal{M}}_{\tiny\odot}}{\mathcal{M}_{\tiny\odot}}+\frac{\mathcal{T}_{\tiny\odot}}{2\pi\mathcal{R}_{\tiny\odot}^{2}}\sum_{i}\frac{\mathcal{M}_{i}}{\mathcal{M}_{\tiny\odot}}\frac{dJ_{i}}{dt},$
(75)
and this means the orbital period of the Sun must be changing. If we assume
that the Sun’s radius has remained constant through the passage of time, i.e.
$\dot{\mathcal{R}}_{\tiny\odot}=0$ (which is certainly not true), then what we
obtain from the above is a minimum value for the secular change in the Sun’s
spin. The reason for invoking this assumption is because there currently is no
information on the secular change of the Sun’s radius (see e.g. Miura et al.
2009), hence we make this assumption so that we can proceed with our
calculation. As already said, what we get is not the exact secular change in
the Sun’s spin but a lower limit to this.
The second term in equation (75), i.e.
$\dot{\mathcal{M}}_{\tiny\odot}/\mathcal{M}_{\tiny\odot}$; represents the
effect of solar mass loss, which can be evaluated in the following way. The
Sun has a luminosity of at least $3.939\times 10^{26}\,W$, or $4.382\times
10^{9}\,kg/s$; this includes electromagnetic radiation and the contribution
from neutrinos (Noerdlinger 2008). The particle mass loss rate due to the
solar wind is $\sim 1.374\times 10^{9}\,kg/s$ (see e.g. Noerdlinger 2008).
From this information, it follows that
$\dot{\mathcal{M}}_{\tiny\odot}/\mathcal{M}_{\tiny\odot}\simeq 9.10\times
10^{-12}cy^{-1}$.
Now, the last term in equation (75) can be evaluated from the ASTG since
$\dot{J}$ is known – so doing, one finds that it is equal to $\sim-(4.00\pm
1.00)\times 10^{-6}cy^{-1}$; this implies
$\dot{\mathcal{T}}_{\tiny\odot}=8.00\pm 2.00\,s/cy$. This result is a
significant $10^{6}$ times larger than the term emerging from the solar mass
loss so much that we can neglect this altogether and consider only the last
term in equation (75) hence $\dot{\mathcal{T}}_{\tiny\odot}=8.00\pm
2.00\,s/cy$. This value is significantly larger compared to that calculated by
Miura et al. (2009) where these authors find a value of $21.0\,ms/cy$.
Currently, no serious measurements on the secular change in the period of the
solar spin has been made. It should be possible to undertake this effort and
with respect to the ASTG, and the result of Miura et al. (2009), this
experiment would act an arbiter.
Furthermore, the authors Miura et al. (2009) propose that the Sun and the
Earth are literally pushing each other away (leading to the increase in the
Earth-Sun distance) due to their tidal interaction and they believe that this
same process is what’s gradually driving the moon’s orbit outward: they say
“Tides raised by the moon in our oceans are gradually transferring Earth’s
rotational energy to lunar motion. As a consequence, each year the moon’s
orbit expands by about $4\,cm$ and Earth’s rotation slows by about $30\mu s$”.
Further Miura et al. (2009) assumes that our planet’s mass is raising a tiny
but sustained tidal bulge in the Sun. They calculate that, thanks to Earth,
the Sun’s rotation rate is slowing by $30\mu s/cy$. Thus according to their
explanation, the distance between the Earth and Sun is growing because the Sun
is losing its angular momentum – the ASTG gives a different explanation
altogether and this is in our opinion, very interesting.
## 7 Discussion and Conclusions
We have considered Poisson’s equation for empty space and solved this for an
azimuthally symmetric setting – we have coined the term Azimuthally Symmetric
Theory of Gravitation (ASTG) for the emergent theory thereof. From the
emergent solution, we have shown that the ASTG is capable of explaining
certain observed (and yet to be observed) anomalies:
1. (1).
Precession of the perihelion of planets.
2. (2).
Secular increase in the Earth-Sun distance.
3. (3).
Secular increase in the Earth Year.
4. (4).
Secular decrease in solar spin.
5. (5).
Spiral orbits must exist.
One of the draw-backs of the ASTG as it currently stands is that it is heavily
dependent on observations; for the values of $\lambda_{\ell}$ need (have) to
be determined from observations. Without knowledge of the
$\lambda_{\ell}^{\prime}s$, one is unable to produce the hard numbers required
to make any quantifications. Clearly, a theory incapable of making any
numerical quantifications is useless. This must be averted. We shall make use
of the solar values of the $\lambda_{\ell}^{\prime}s$ in shading some light
into our current thinking on this, i.e. finding a general form for the
constants $\lambda_{\ell}$; In the subsequent paragraphs, we shall make what
we believe is a reasonable suggestion and give our current envisage-ment on
the general form for these constants.
(1) First things first, if the constants $\lambda_{\ell}$ where all
independent of each other, then, the theory would clearly be horribly
complicated. If we take as guide the Principle of Occam’s Razor which in most
if not all cases, leads to the simplest theory, then, these constants must be
dependent on each other somehow so as to reduce the labyrinth of complications
thereof. The simplest imaginable such dependence is
$\lambda_{\ell}=F(\ell)\lambda_{1}$; in this way, the entire system of
constants $\lambda_{\ell}$ is dependent on just the one constant
$\lambda_{1}$. This idea that the system of constants be dependent on just one
constant is drawn from the theory of polynomial functions where for a
polynomial function $F(x)=\sum_{n=0}^{\infty}c_{n}x^{n}$, one can have “well
behaved” polynomial functions for which the constants $c_{n}$ have a general
form, i.e., were they dependent on $n$; e.g.,
$e^{x}=\sum_{n=0}^{\infty}x^{n}/n!$. We envisage the function $\Phi(r,\theta)$
to be a “well behaved” function. By “well behaved” we simple mean its system
of constants, $\lambda_{\ell}$, is critically dependent on $\ell$ just as the
constants $c_{n}$ depend on $n$.
(2) Second, we could like that on a practical level, only the second order
approximation of the theory must suffice, this means the terms $\ell>3$ must
be practically negligible. We have already shown herein that the second order
approximation of the ASTG is able to explain a sizable amount of anomalous
observations. With the ASTG written in its second order approximation and as
will be shown in the second reading (i.e., a follow-up reading that we hope
will be published in the present journal), one is able without much
difficulties, to explain from this second order approximation, the emergence
of molecular bipolar outflows in star forming systems, as a gravitational
phenomena. If the other terms beyond the second order approximation become
practically significant, one will have difficulties to explain outflows. So in
a way, we are not going to pretend but clearly state that, we want – albeit
with a priori and posteriori justification; to fine tune the theory so that it
is able to explain the emergence of bipolar. This is the strongest reason we
want the terms for which $\ell>3$ to be so small such that in practice one can
neglect them entirely.
(3) Third and most important, the only data point we have of these constants
is the determined values for the Sun, i.e.,
$\lambda_{\ell}^{\tiny\odot}=24.0\pm 7$ and $\lambda_{2}=-0.2\pm 0.1$. If
logic is to hold – as it must; then, our suggestion,
$\lambda_{\ell}=F(\ell)\lambda_{1}$; must be able to explain this. We find
that the following proposal:
$\lambda_{\ell}=\left(\frac{(-1)^{\ell+1}}{\left(\ell^{\ell}\right)!\left(\ell^{\ell}\right)}\right)\lambda_{1},$
(76)
meets (1), (2) and (3). We shall assume this result until such a time evidence
to the contrary is brought forth. Checking on (3) we see that within the error
margins
$\lambda_{2}^{\tiny\odot}\simeq[(-1)^{2+1}/\left((2^{2})!(2^{2}\right)]\lambda_{1}^{\tiny\odot}$.
Further checking on (2); from (76) we will have $\lambda_{4}=3.40\times
10^{-30}\lambda_{1}$ which is practically small; the meaning of which is that
all terms for which $\ell>3$ can in practice be neglected entirely.
If the above proposal proves itself to be correct, then, the resultant theory
will have just one undetermined parameter $\lambda_{1}$. We are not going to
try and deduce what this parameter depends on but simple hint at our current
thinking. We believe this parameter must depend on the angular frequency of
the spin of the gravitating body in question. If we can find the correct
dependence, then, the ASTG will stand on it own thus positioning itself on the
podium to make testable predictions. We have left the task to make this
deduction an exercise for the follow-up reading.
The fact the we have deduced the crucial parameters $\lambda_{1}^{\tiny\odot}$
& $\lambda_{2}^{\tiny\odot}$ from experience, means we have in the current
reading done some reserve engineering. Normally, a theory must give these
values and make clear predictions, just as when Einstein wrote down his
equations and found that his theory predicted a factor $2$ difference when
compared to Newton’s theory when it come to the bending of light by the Sun
and when applied to the Sun-Mercury system, it accounted very well for the
then unexplained $43.0\arcsec$ per century for the precession of the
perihelion of the orbit of this planet; it just came out right. There were no
free parameters that needed fitting as is the case of the ASTG. As argued
above, once a general form for the $\lambda_{\ell}$ is found, this setback of
the ASTG will be solved. Because we were able to obtain the values
$\lambda_{1}^{\tiny\odot}$ & $\lambda_{2}^{\tiny\odot}$ which lead acceptable
values for the perihelion precession, means that the values $\mathscr{A}$ &
$\mathscr{B}$ are not random but systematic. If the theory was all wrong,
then, only luck would make the obtained values for $\mathscr{A}$ &
$\mathscr{B}$ give values of $\lambda_{1}^{\tiny\odot}$ &
$\lambda_{2}^{\tiny\odot}$ such that equation (47) give in general, acceptable
values for the precession of the perihelion of the planets.
With regard to the values obtained for the precession of the perihelion of
solar planets, it can be said that, the values obtained from the ASTG as shown
in column 10 (table I) when weighed against the observational values listed in
column 10 (of the same table) are acceptable. Given that we have taken into
account the fact the orbits of these planets are not found laying in the same
plane, this can hardly be a coincident or an accident since changing their
inclination by just $1\hbox{${}^{\circ}$$$}$ will alter the predicted values
of the precession of their perihelion.
Iorio (2008a) states that the secular increase in the mean Earth-Sun distance
cannot be explained within the realm of classical physics. Contrary to this,
we believe and hold that herein – we have shown from within the provinces of
classical physics that this result is explainable from within the domains and
confines of classical physics. Before the present, the reason why perhaps this
observation appeared beyond the reach of classical physics is because
classical physics has not really considered gravitation as an azimuthally
symmetric phenomenon as has been done in present reading. This strongly
suggests that the ideas presented herein need to be explored further for they
contain a debris of the truth.
One of the interesting outcome that was not explored in this reading for fear
of digression is that the ASTG has a provision for spiral orbits (equation 67
and 68). These orbits occur when $(n_{3})^{2}\leq 0$. This condition implies
the existence of a region ($r\leq\mathcal{R}_{crit}$) in which spiral orbits
will occur. Evaluating the inequality $(n_{3})^{2}\leq 0$, leads to:
$\mathcal{R}_{crit}=(2\lambda_{1}G\mathcal{M}/c^{2})\cos^{2}(\theta/2)$, and
from this, it is easy for one to deduce that spiral orbits are unlikely in the
Solar System since these will have to occur inside the Sun because
$\mathcal{R}_{crit}\leq\mathcal{R}_{\odot}$.
At this point as we approach the end of this reading, we feel strongly that we
must address the question; “Does the spin along the azimuthal axis of a
gravitating body induce an azimuthal symmetry into the gravitational field for
this spinning body?” To answer this, we must ask the question; “Will a
contracting none-spinning cloud of gas experience any bulge alone its
equator?” First, we know that the equatorial bulge will occur on a plane
perpendicular to the spin axis. Since a none-spinning gas cloud is going to
have to spin axis, there is going to be no spin axis about which the
equatorial bulge will occur. If the material in the cloud is randomly and
uniformly distributed, the cloud will exhibit a spherically symmetric
distribution of mass and its gravitational field is expected to be spherically
symmetric. A spherically spherically symmetric gravitational field is one that
only has a radial dependence, i.e. $\varphi=\varphi(r)$.
Now, if the gas cloud is spinning, the centrifugal forces will cause there to
exist a disk and the material distribution will have an azimuthal symmetry,
i.e. $\rho=\rho(r,\theta)$. Should not this azimuthal symmetric distribution
of matter induce an azimuthal gravitational field? From Poison’s equation (2),
$\rho=\rho(r,\theta)$ implies $\Phi=\Phi(r,\theta)$; should not this, i.e.
$\Phi=\Phi(r,\theta)$, hold as-well for a body spinning gravitating body in a
vacuum? From this, clearly, a spinning gravitating body ought to exhibit an
azimuthal symmetry. We have argued in the last paragraph of §(2) that the spin
of a gravitating body breaks the existing spherical symmetry of the non-
spinning body and the above argument is just adding more to this. It is from
this that the subtitle and running head finds its justification.
If the ASTG turns out to be correct – as we believe it will; then, we have an
important question to ask; “What is the speed of light doing in a theory of
gravitation because from (7) we see that the constants $G$ and $c$ are
intimately tied-up together? This is a similar if not a congruent question
that has been asked by Martin & Anderson (2009) in their expository work on
Earth Flyby Anomalies (AFA). The empirical formula deduced to quantify EFA
contains in it the speed of light, $c$, so in their exposition of the
phenomena of AFA, Martin & Anderson (2009) have asked the perdurable question
“What is the speed of light doing there?”. EFA are thought to be a
gravitational phenomena, so, what does the speed of light have to do with
gravitation – really? If there is an intimate relationship between the speed
of light and gravitation, then, one will be forgiven to think this suggests a
link between gravitation and the theory of light – electromagnetism. The speed
of light, $c$, appears to be dire to the ASTG presented herein. Why not
another value but the speed of light, $c$? We shall leave these matters
hanging in-limbo.
In relation to the question above, i.e., “What is the speed of light doing in
a theory of gravitation”, one notes that Newtonian gravitation – which
requires instantaneous interaction as a postulate; does not imply the
dependence of the gravitational potential on the azimuthal angle for a
spinning body as is the case in the ASTG, because at any instant $t$, the
gravitating body appears spherically symmetric. Here we have the speed of
light $c$ coming in because of the azimuthal symmetry. Does this speed of
light $c$ link (or not) the propagation of the gravitational phenomena to the
speed of light? In the present, we can only pause this as a question, for we
still have to do further work on these ideas.
In closing, allow me to say that we find it hard to call what has been
presented herein as “A New Theory of Gravitation”. When one tells you they
have come up with a new theory of gravitation, what immediately comes to mind
is that they have discovered a new principle upon which gravitation can
further be understood from the present understanding. The ASTG is not founded
on any new physical principle but on the well known vintage equation of
Poisson. What we have done is simply taken the azimuthally symmetric equations
of this equation and applied them to gravitation. Based on this understanding,
it is difficult to call it a new theory. Yes, the azimuthally symmetric
equations of Poisson have brought new and exciting physics – perhaps only
because of this, the title of this reading finds its qualification.
Acknowledgments: I am grateful to Mkoma George – Baba va Panashe, and his wife
– Mai va Panashe, for their kind hospitality they offered while working on
this reading and to Mr. Isak D. Davids & Ms. M. Christina Eddington for proof
reading the grammar and spelling. Further, I am grateful to the anonymous
reviewers – for their invaluable criticism that has helped in the refinement
of the arguments presented. Last and certainly not least, I am very grateful
to my Professor, D. Johan van der Walt and Professor Pienaar Kobus, for the
strength and courage that they have given me.
## References
* Miura et al. (2009) Miura T., Arakida H., Kasai H. & Shuichi Kuramata, 2009, Secular Increase of the Astronomical Unit: A possible Explanation in terms of the Total Angular Momentum Conservation Law, accepted for publication in Publication of the Astronomical Society of Japan: arXiv:0905.3008.
* Grifitts (2008) Grifitts D. J., 2008, Introduction to Electrodynamics: $3^{th}$ Edition: ISBN 0-13-919960-8, Pearson Benjamin Cummings, pp.137-139.
* Kenyon (1990) Kenyon I. R., 1990, General Relativity, Oxford Univ. Press: ISBN 0-19-851995-8, pp.87-93.
* Krasinsky & Brumberg (2004) Krasinsky G. A. & Brumberg V. A., 2004, Secular Increase of Astronomical Unit from Analysis of the Major Planets Motions, and its Interpretation, Celest. Mech. & Dyn. Astron., Vol. 90, pp.267-288.
* Iorio (2008a) Iorio L., 2008a, Secular increase of the Astronomical Unit and Perihelion Precessions as Tests of the Dvali-Gabadadze-Porrati Multi-dimensional Braneworld Scenario: arXiv:gr-qc/0508047v2.
* Iorio (2008b) Iorio L., 2008b, Solar System Tests of Some Models of Modified Gravity Proposed to Explain Galactic Rotation Curves without Dark Matter, Scholarly Research Exchange, Vol. 2008, Article ID 968393, doi:10.3814/2008/968393.
* Lue (2003) Lue A. & Starkmann G., 2003, Gravitational Leakage into Extra Dimensions Probing Dark Energy Using Local Gravity, Phys. Rev. D, Vol. 67, p.064002.
* Noerdlinger (2008) Noerdlinger P. D., 2008, Solar Mass Loss - the Astronomical Unit and the Scale of the Solar System: arXiv:0801.3807.
* Martin & Anderson (2009) Martin M. & Anderson J. D., 2009, Earth Flyby Anomalies, preprint: arXiv:0910.1321.
* Pitjeva (2005) Pitjeva E. V., 2005, High-Precision Ephemerids of Planets-EPM and Determinations of Some Astronomical Constants, Sol. Sys. Res., Vol. 39, pp.176-186.
* Sagan (1974) Sagan C., 1974, The Solar System, Scientific American: ISBN 0-7167-0550-8, p.6.
* Standish (2005) Standish E. M., 2005, The Astronomical Unit Now, in Transits of Venus: New Views of the Solar System and Galaxy, Proceedings IAU Colloquium, Ed. Kurtz, D. W., No. 196, (Cambridge University Press, Cambridge), pp.163-179.
* Williams & Boggs (2009) Williams, J. G. & Boggs D. H., 2009, in Proceedings of 16th International Workshop on Laser Ranging, Ed. S. Schillak, (Space Research Centre, Polish Academy of Sciences).
|
arxiv-papers
| 2009-12-15T17:42:26 |
2024-09-04T02:49:07.061373
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. G. Nyambuya",
"submitter": "Golden Gadzirayi Nyambuya Mr.",
"url": "https://arxiv.org/abs/0912.2966"
}
|
0912.3429
|
2010597-608Nancy, France 597
Angelo Montanari Gabriele Puppis Pietro Sala Guido Sciavicco
# Decidability of the interval temporal
logic $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ over the natural numbers
A. Montanari Department of Mathematics and Computer Science, Udine
University, Italy angelo.montanari—pietro.sala@dimi.uniud.it , G. Puppis
Computing Laboratory, Oxford University, England
Gabriele.Puppis@comlab.ox.ac.uk , P. Sala and G. Sciavicco Department of
Information, Engineering and Communications, Murcia University, Spain
guido@um.es
###### Abstract.
In this paper, we focus our attention on the interval temporal logic of the
Allen’s relations “meets”, “begins”, and “begun by”
($A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ for short), interpreted over
natural numbers. We first introduce the logic and we show that it is
expressive enough to model distinctive interval properties, such as
accomplishment conditions, to capture basic modalities of point-based temporal
logic, such as the until operator, and to encode relevant metric constraints.
Then, we prove that the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ over natural numbers is decidable by
providing a small model theorem based on an original contraction method.
Finally, we prove the EXPSPACE-completeness of the problem.
###### Key words and phrases:
interval temporal logics, compass structures, decidability, complexity
###### 1991 Mathematics Subject Classification:
F.3: logics and meaning of programs; F.4: mathematical logic and formal
languages
The work has been partially supported by the GNCS project: “Logics, automata,
and games for the formal verification of complex systems”. Guido Sciavicco has
also been supported by the Spain/South Africa Integrated Action N. HS2008-0006
on: “Metric interval temporal logics: Theory and Applications”.
## 1\. Introduction
Interval temporal logics are modal logics that allow one to represent and to
reason about time intervals. It is well known that, on a linear ordering, one
among thirteen different binary relations may hold between any pair of
intervals, namely, “ends”, “during”, “begins”, “overlaps”, “meets”, “before”,
together with their inverses, and the relation “equals” (the so-called Allen’s
relations [1])111We do not consider here the case of ternary relations.
Amongst the multitude of ternary relations among intervals there is one of
particular importance, which corresponds to the binary operation of
concatenation of meeting intervals. The logic of such a ternary interval
relation has been investigated by Venema in [20]. A systematic analysis of its
fragments has been recently given by Hodkinson et al. [13].. Allen’s relations
give rise to respective unary modal operators, thus defining the modal logic
of time intervals HS introduced by Halpern and Shoham in [12]. Some of these
modal operators are actually definable in terms of others; in particular, if
singleton intervals are included in the structure, it suffices to choose as
basic the modalities corresponding to the relations “begins” $B$ and “ends”
$E$, and their transposes $\,\overline{\\!B\\!}\,$, $\,\overline{\\!E\\!}\,$.
HS turns out to be highly undecidable under very weak assumptions on the class
of interval structures over which its formulas are interpreted [12]. In
particular, undecidability holds for any class of interval structures over
linear orderings that contains at least one linear ordering with an infinite
ascending or descending chain, thus including the natural time flows
$\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$. Undecidability of
HS over finite structures directly follows from results in [15]. In [14],
Lodaya sharpens the undecidability of HS showing that the two modalities $B,E$
suffice for undecidability over dense linear orderings (in fact, the result
applies to the class of all linear orderings [11]). Even though HS is very
natural and the meaning of its operators is quite intuitive, for a long time
such sweeping undecidability results have discouraged the search for practical
applications and further investigations in the field. A renewed interest in
interval temporal logics has been recently stimulated by the identification of
some decidable fragments of HS, whose decidability does not depend on
simplifying semantic assumptions such as locality and homogeneity [11]. This
is the case with the fragments $B\,\overline{\\!B\\!}\,$,
$E\,\overline{\\!E\\!}\,$ (logics of the “begins/begun by” and “ends/ended by”
relations) [11], $A$, $A\,\overline{\\!A\\!}\,$ (logics of temporal
neighborhood, whose modalities capture the “meets/met by” relations [10]), and
$D$, $D\,\overline{\\!D\\!}\,$ (logics of the subinterval/superinterval
relations) [3, 16].
In this paper, we focus our attention on the product logic
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$, obtained from the join of
$B\,\overline{\\!B\\!}\,$ and $A$ (the case of
$\,\overline{\\!A\\!}\,\mspace{-0.3mu}E\,\overline{\\!E\\!}\,$ is fully
symmetric), interpreted over the linear order $\mathbb{N}$ of the natural
numbers (or a finite prefix of it). The decidability of
$B\,\overline{\\!B\\!}\,$ can be proved by translating it into the point-based
propositional temporal logic of linear time with temporal modalities $F$
(sometime in the future) and $P$ (sometime in the past), which has the finite
(pseudo-)model property and is decidable, e.g., [9]. In general, such a
reduction to point-based temporal logics does not work: formulas of interval
temporal logics are evaluated over pairs of points and translate into binary
relations. For instance, this is the case with $A$. Unlike the case of
$B\,\overline{\\!B\\!}\,$, when dealing with $A$ one cannot abstract way from
the left endpoint of intervals, as contradictory formulas may hold over
intervals with the same right endpoint and a different left endpoint. The
decidability of $A\,\overline{\\!A\\!}\,$, and thus that of its fragment $A$,
over various classes of linear orderings has been proved by Bresolin et al. by
reducing its satisfiability problem to that of the two-variable fragment of
first-order logic over the same classes of structures [4], whose decidability
has been proved by Otto in [18]. Optimal tableau methods for $A$ with respect
to various classes of interval structures can be found in [6, 7]. A decidable
metric extension of $A$ over the natural numbers has been proposed in [8]. A
number of undecidable extensions of $A$, and $A\,\overline{\\!A\\!}\,$, have
been given in [2, 5].
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ retains the simplicity of its
constituents $B\,\overline{\\!B\\!}\,$ and $A$, but it improves a lot on their
expressive power (as we shall show, such an increase in expressiveness is
achieved at the cost of an increase in complexity). First, it allows one to
express assertions that may be true at certain intervals, but at no
subinterval of them, such as the conditions of accomplishment. Moreover, it
makes it possible to easily encode the until operator of point-based temporal
logic (this is possible neither with $B\,\overline{\\!B\\!}\,$ nor with $A$).
Finally, meaningful metric constraints about the length of intervals can be
expressed in $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$, that is, one can
constrain an interval to be at least (resp., at most, exactly) $k$ points
long. We prove the decidability of $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$
interpreted over $\mathbb{N}$ by providing a small model theorem based on an
original contraction method. To prove it, we take advantage of a natural
(equivalent) interpretation of $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$
formulas over grid-like structures based on a bijection between the set of
intervals over $\mathbb{N}$ and (a suitable subset of) the set of points of
the $\mathbb{N}\times\mathbb{N}$ grid. In addition, we prove that the
satisfiability problem for $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ is
EXPSPACE-complete (that for $A$ is NEXPTIME-complete). In the proof of
hardness, we use a reduction from the exponential-corridor tiling problem.
The paper is organized as follows. In Section 2 we introduce
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$. In Section 3, we prove the
decidability of its satisfiability problem. We first describe the application
of the contraction method to finite models and then we generalize it to
infinite ones. In Section 4 we deal with computational complexity issues.
Conclusions provide an assessment of the work and outline future research
directions. Missing proofs can be found in [17].
## 2\. The interval temporal logic $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$
In this section, we briefly introduce syntax and semantics of the logic
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$, which features three modal
operators ${\langle A\rangle}$, ${\langle B\rangle}$, and
${\langle\,\overline{\\!B\\!}\,\rangle}$ corresponding to the three Allen’s
relations $A$ (“meets”), $B$ (“begins”), and $\,\overline{\\!B\\!}\,$ (“begun
by”), respectively. We show that $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ is
expressive enough to capture the notion of accomplishment, to define the
standard until operator of point-based temporal logics, and to encode metric
conditions. Then, we introduce the basic notions of atom, type, and
dependency. We conclude the section by providing an alternative interpretation
of $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ over labeled grid-like
structures.
### 2.1. Syntax and semantics
Given a set $\mathcal{P}\mathit{rop}$ of propositional variables, formulas of
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ are built up from
$\mathcal{P}\mathit{rop}$ using the boolean connectives $\neg$ and $\;\vee\;$
and the unary modal operators ${\langle A\rangle}$, ${\langle B\rangle}$,
${\langle\,\overline{\\!B\\!}\,\rangle}$. As usual, we shall take advantage of
shorthands like
$\varphi_{1}\;\wedge\;\varphi_{2}=\neg(\neg\varphi_{1}\;\vee\;\neg\varphi_{2})$,
$[A]\varphi=\neg{\langle A\rangle}\neg\varphi$, $[B]\varphi=\neg{\langle
B\rangle}\neg\varphi$, $\top=p\vee\neg p$, and $\bot=p\wedge\neg p$, with
$p\in\mathcal{P}\mathit{rop}$. Hereafter, we denote by ${\lvert\varphi\rvert}$
the size of $\varphi$.
We interpret formulas of $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ in interval
temporal structures over natural numbers endowed with the relations “meets”,
“begins”, and “begun by”. Precisely, we identify any given ordinal
$N\leq\omega$ with the prefix of length $N$ of the linear order of the natural
numbers and we accordingly define $\mathbb{I}_{N}$ as the set of all non-
singleton closed intervals $[x,y]$, with $x,y\in N$ and $x<y$. For any pair of
intervals $[x,y],[x^{\prime},y^{\prime}]\in\mathbb{I}_{N}$, the Allen’s
relations “meets” $A$, “begins” $B$, and “begun by” $\,\overline{\\!B\\!}\,$
are defined as follows (note that $\,\overline{\\!B\\!}\,$ is the inverse
relation of $B$):
* •
“meets” relation: $[x,y]\;A\;[x^{\prime},y^{\prime}]$ iff $y=x^{\prime}$;
* •
“begins” relation: $[x,y]\;B\;[x^{\prime},y^{\prime}]$ iff $x=x^{\prime}$ and
$y^{\prime}<y$;
* •
“begun by” relation: $[x,y]\;\,\overline{\\!B\\!}\,\;[x^{\prime},y^{\prime}]$
iff $x=x^{\prime}$ and $y<y^{\prime}$.
Given an _interval structure_
$\mathcal{S}=(\mathbb{I}_{N},A,B,\,\overline{\\!B\\!}\,,\sigma)$, where
$\sigma:\mathbb{I}_{N}\;\rightarrow\;\mathscr{P}(\mathcal{P}\mathit{rop})$ is
a labeling function that maps intervals in $\mathbb{I}_{N}$ to sets of
propositional variables, and an initial interval $I$, we define the semantics
of an $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ formula as follows:
* •
$\mathcal{S},I\vDash a$ iff $a\in\sigma(I)$, for any
$a\in\mathcal{P}\mathit{rop}$;
* •
$\mathcal{S},I\vDash\neg\varphi$ iff $\mathcal{S},I\not\vDash\varphi$;
* •
$\mathcal{S},I\vDash\varphi_{1}\;\vee\;\varphi_{2}$ iff
$\mathcal{S},I\vDash\varphi_{1}$ or $\mathcal{S},I\vDash\varphi_{2}$;
* •
for every relation $R\in\\{A,B,\,\overline{\\!B\\!}\,\\}$,
$\mathcal{S},I\vDash{\langle R\rangle}\varphi$ iff there is an interval
$J\in\mathbb{I}_{N}$ such that $I\;R\;J$ and $\mathcal{S},J\vDash\varphi$.
Given an interval structure $\mathcal{S}$ and a formula $\varphi$, we say that
$\mathcal{S}$ _satisfies_ $\varphi$ if there is an interval $I$ in
$\mathcal{S}$ such that $\mathcal{S},I\vDash\varphi$. We say that $\varphi$ is
_satisfiable_ if there exists an interval structure that satisfies it. We
define the _satisfiability problem_ for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ as the problem of establishing
whether a given $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$-formula $\varphi$ is
satisfiable.
We conclude the section with some examples that account for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ expressive power. The first one
shows how to encode in $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ conditions of
accomplishment (think of formula $\varphi$ as the assertion: “Mr. Jones flew
from Venice to Nancy”): ${\langle
A\rangle}\bigl{(}\varphi\;\wedge\;[B](\neg\varphi\;\wedge\;[A]\neg\varphi)\;\wedge\;[\,\overline{\\!B\\!}\,]\neg\varphi\bigr{)}$.
Formulas of point-based temporal logics of the form $\psi\;U\;\varphi$, using
the standard until operator, can be encoded in
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ (where atomic intervals are two-
point intervals) as follows: ${\langle
A\rangle}\bigl{(}[B]\bot\\!\;\wedge\;\\!\varphi\bigr{)}\;\vee\;{\langle
A\rangle}\bigl{(}{\langle
A\rangle}([B]\bot\\!\;\wedge\;\\!\varphi)\;\wedge\;[B]({\langle
A\rangle}([B]\bot\\!\;\wedge\;\\!\psi))\bigr{)}.$ Finally, metric conditions
like: “$\varphi$ holds over a right neighbor interval of length greater than
$k$ (resp., less than $k$, equal to $k$)” can be captured by the following
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ formula: ${\langle
A\rangle}\bigl{(}\varphi\;\wedge\;{\langle B\rangle}^{k}\top\bigr{)}$ (resp.,
${\langle A\rangle}\bigl{(}\varphi\;\wedge\;[B]^{k-1}\bot\bigr{)}$, ${\langle
A\rangle}\bigl{(}\varphi\;\wedge\;[B]^{k}\bot\;\wedge\;{\langle
B\rangle}^{k-1}\top\bigr{)}$)222It is not difficult to show that
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ subsumes the metric extension of $A$
given in [8]. A simple game-theoretic argument shows that the former is in
fact strictly more expressive than the latter..
### 2.2. Atoms, types, and dependencies
Let $\mathcal{S}=(\mathbb{I}_{N},A,B,\,\overline{\\!B\\!}\,,\sigma)$ be an
interval structure and $\varphi$ be a formula of
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$. In the sequel, we shall compare
intervals in $\mathcal{S}$ with respect to the set of subformulas of $\varphi$
they satisfy. To do that, we introduce the key notions of $\varphi$-atom,
$\varphi$-type, $\varphi$-cluster, and $\varphi$-shading.
First of all, we define the _closure_ $\mathcal{C}\mathit{l}(\varphi)$ of
$\varphi$ as the set of all subformulas of $\varphi$ and of their negations
(we identify $\neg\neg\alpha$ with $\alpha$, $\neg{\langle A\rangle}\alpha$
with $[A]\neg\alpha$, etc.). For technical reasons, we also introduce the
_extended closure_ $\mathcal{C}\mathit{l}^{+}(\varphi)$, which is defined as
the set of all formulas in $\mathcal{C}\mathit{l}(\varphi)$ plus all formulas
of the forms ${\langle R\rangle}\alpha$ and $\neg{\langle R\rangle}\alpha$,
with $R\in\\{A,B,\,\overline{\\!B\\!}\,\\}$ and
$\alpha\in\mathcal{C}\mathit{l}(\varphi)$.
A _$\varphi$ -atom_ is any non-empty set
$F\subseteq\mathcal{C}\mathit{l}^{+}(\varphi)$ such that (i) for every
$\alpha\in\mathcal{C}\mathit{l}^{+}(\varphi)$, we have $\alpha\in F$ iff
$\neg\alpha\not\in F$ and (ii) for every
$\gamma=\alpha\;\vee\;\beta\in\mathcal{C}\mathit{l}^{+}(\varphi)$, we have
$\gamma\in F$ iff $\alpha\in F$ or $\beta\in F$ (intuitively, a _$\varphi$
-atom_ is a maximal locally consistent set of formulas chosen from
$\mathcal{C}\mathit{l}^{+}(\varphi)$). Note that the cardinalities of both
sets $\mathcal{C}\mathit{l}(\varphi)$ and $\mathcal{C}\mathit{l}^{+}(\varphi)$
are linear in the number ${\lvert\varphi\rvert}$ of subformulas of $\varphi$,
while the number of $\varphi$-atoms is at most exponential in
${\lvert\varphi\rvert}$ (precisely, we have
${\lvert\mathcal{C}\mathit{l}(\varphi)\rvert}=2{\lvert\varphi\rvert}$,
${\lvert\mathcal{C}\mathit{l}^{+}(\varphi)\rvert}=14{\lvert\varphi\rvert}$,
and there are at most $2^{7{\lvert\varphi\rvert}}$ distinct atoms).
We also associate with each interval $I\in\mathcal{S}$ the set of all formulas
$\alpha\in\mathcal{C}\mathit{l}^{+}(\varphi)$ such that
$\mathcal{S},I\vDash\alpha$. Such a set is called _$\varphi$ -type_ of $I$ and
it is denoted by $\mathcal{T}\mathit{ype}_{\mathcal{S}}(I)$. We have that
every $\varphi$-type is a $\varphi$-atom, but not vice versa. Hereafter, we
shall omit the argument $\varphi$, thus calling a $\varphi$-atom (resp., a
$\varphi$-type) simply an atom (resp., a type).
Given an atom $F$, we denote by $\mathcal{O}\mathit{bs}(F)$ the set of all
_observables_ of $F$, namely, the formulas
$\alpha\in\mathcal{C}\mathit{l}(\varphi)$ such that $\alpha\in F$. Similarly,
given an atom $F$ and a relation $R\in\\{A,B,\,\overline{\\!B\\!}\,\\}$, we
denote by $\mathcal{R}\mathit{eq}_{R}(F)$ the set of all _$R$ -requests_ of
$F$, namely, the formulas $\alpha\in\mathcal{C}\mathit{l}(\varphi)$ such that
${\langle R\rangle}\alpha\in F$. Taking advantage of the above sets, we can
define the following two relations between atoms $F$ and $G$:
$\begin{array}[]{rcl}F\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$}}\,G&\;\quad\text{iff}&\mathcal{R}\mathit{eq}_{A}(F)\;=\;\mathcal{O}\mathit{bs}(G)\,\cup\,\mathcal{R}\mathit{eq}_{B}(G)\,\cup\,\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}(G);\vspace{2mm}\\\
F\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,G&\;\quad\text{iff}&\begin{cases}\mathcal{O}\mathit{bs}(F)\,\cup\,\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}(F)\;\subseteq\;\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}(G)\;\subseteq\;\mathcal{O}\mathit{bs}(F)\,\cup\,\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}(F)\,\cup\,\mathcal{R}\mathit{eq}_{B}(F),\vspace{1mm}\\\
\mathcal{O}\mathit{bs}(G)\,\cup\,\mathcal{R}\mathit{eq}_{B}(G)\;\subseteq\;\mathcal{R}\mathit{eq}_{B}(F)\;\subseteq\;\mathcal{O}\mathit{bs}(G)\,\cup\,\mathcal{R}\mathit{eq}_{B}(G)\,\cup\,\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}(G).\end{cases}\end{array}$
Note that the relation
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$
is transitive, while
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$
is not. Moreover, both
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$
and
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$
satisfy a _view-to-type dependency_ , namely, for every pair of intervals
$I,J$ in $\mathcal{S}$, we have that
$\begin{array}[]{rcl}I\;A\;J&\;\quad\text{implies}&\mathcal{T}\mathit{ype}_{\mathcal{S}}(I)\,\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$}}\,{}\,\mathcal{T}\mathit{ype}_{\mathcal{S}}(J)\vspace{1mm}\\\
I\;B\;J&\;\quad\text{implies}&\mathcal{T}\mathit{ype}_{\mathcal{S}}(I)\,\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,{}\,\mathcal{T}\mathit{ype}_{\mathcal{S}}(J).\end{array}$
Relations
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$
and
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$
will come into play in the definition of consistency conditions (see
Definition 2.1).
### 2.3. Compass structures
Figure 1. Correspondence between intervals and points of a discrete grid.
The logic $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ can be equivalently
interpreted over grid-like structures (the so-called compass structures [20])
by exploiting the existence of a natural bijection between the intervals
$I=[x,y]$ and the points $p=(x,y)$ of an $N\times N$ grid such that $x<y$. As
an example, Figure 1 depicts four intervals $I_{0},...,I_{3}$ such that
$I_{0}\;A\;I_{1}$, $I_{0}\;B\;I_{2}$, and
$I_{0}\;\,\overline{\\!B\\!}\,\;I_{3}$, together with the corresponding points
$p_{0},...,p_{3}$ of a discrete grid (note that the three Allen’s relations
$A,B,\,\overline{\\!B\\!}\,$ between intervals are mapped to corresponding
spatial relations between points; for the sake of readability, we name the
latter ones as the former ones).
###### Definition 2.1.
Given an $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ formula $\varphi$, a
(consistent and fulfilling) _compass_ ($\varphi$-)_structure_ of length
$N\leq\omega$ is a pair $\mathcal{G}=(\mathbb{P}_{N},\mathcal{L})$, where
$\mathbb{P}_{N}$ is the set of points $p=(x,y)$, with $0\leq x<y<N$, and
$\mathcal{L}$ is function that maps any point $p\in\mathbb{P}_{N}$ to a
($\varphi$-)atom $\mathcal{L}(p)$ in such a way that
* •
for every pair of points $p,q\in\mathbb{P}_{N}$ and every relation
$R\in\\{A,B\\}$, if $p\;R\;q$ holds, then
$\mathcal{L}(p)\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{R\,}$}}}{\longrightarrow}$}}\,\mathcal{L}(q)$
follows (consistency);
* •
for every point $p\in\mathbb{P}_{N}$, every relation
$R\in\\{A,B,\,\overline{\\!B\\!}\,\\}$, and every formula
$\alpha\in\mathcal{R}\mathit{eq}_{R}\bigl{(}\mathcal{L}(p)\bigr{)}$, there is
a point $q\in\mathbb{P}_{N}$ such that $p\;R\;q$ and
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(q)\bigr{)}$ (fulfillment).
We say that a compass ($\varphi$-)structure
$\mathcal{G}=(\mathbb{P}_{N},\mathcal{L})$ _features_ a formula $\alpha$ if
there is a point $p\in\mathbb{P}_{N}$ such that $\alpha\in\mathcal{L}(p)$. The
following proposition implies that the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ is reducible to the problem of
deciding, for any given formula $\varphi$, whether there exists a
$\varphi$-compass structure that features $\varphi$.
###### Proposition 2.2.
An $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$-formula $\varphi$ is satisfied by
some interval structure if and only if it is featured by some
($\varphi$-)compass structure.
## 3\. Deciding the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$
In this section, we prove that the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ is decidable by providing a “small-
model theorem” for the satisfiable formulas of the logic. For the sake of
simplicity, we first show that the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ interpreted over finite interval
structures is decidable and then we generalize such a result to all (finite or
infinite) interval structures.
As a preliminary step, we introduce the key notion of shading. Let
$\mathcal{G}=(\mathbb{P}_{N},\mathcal{L})$ be a compass structure of length
$N\leq\omega$ and let $0\leq y<N$. The _shading of the row $y$ of
$\mathcal{G}$_ is the set
$\mathcal{S}\mathit{hading}_{\mathcal{G}}(y)=\bigl{\\{}\mathcal{L}(x,y)\,:\,0\leq
x<y\bigr{\\}}$, namely, the set of the atoms of all points in $\mathbb{P}_{N}$
whose vertical coordinate has value $y$ (basically, we interpret different
atoms as different colors). Clearly, for every pair of atoms $F$ and
$F^{\prime}$ in $\mathcal{S}\mathit{hading}_{\mathcal{G}}(y)$, we have
$\mathcal{R}\mathit{eq}_{A}(F)=\mathcal{R}\mathit{eq}_{A}(F^{\prime})$.
### 3.1. A small-model theorem for finite structures
Let $\varphi$ be an $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ formula. Let us
assume that $\varphi$ is featured by a finite compass structure
$\mathcal{G}=(\mathbb{P}_{N},\mathcal{L})$, with $N<\omega$. In fact, without
loss of generality, we can assume that $\varphi$ belongs to the atom
associated with a point $p=(0,y)$ of $\mathcal{G}$, with $0<y<N$. We prove
that we can restrict our attention to compass structures
$\mathcal{G}=(\mathbb{P}_{N},\mathcal{L})$, where $N$ is bounded by a double
exponential in ${\lvert\varphi\rvert}$. We start with the following lemma that
proves a simple, but crucial, property of the relations
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$
and
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$
(the proof can be found in [17]).
###### Lemma 3.1.
If
$F\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$}}\,H$
and
$G\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,H$
hold for some atoms $F,G,H$, then
$F\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$}}\,G$
holds as well.
The next lemma shows that, under suitable conditions, a given compass
structure $\mathcal{G}$ may be reduced in length, preserving the existence of
atoms featuring $\varphi$.
###### Lemma 3.2.
Let $\mathcal{G}$ be a compass structure featuring $\varphi$. If there exist
two rows $0<y_{0}<y_{1}<N$ in $\mathcal{G}$ such that
$\mathcal{S}\mathit{hading}_{\mathcal{G}}(y_{0})\subseteq\mathcal{S}\mathit{hading}_{\mathcal{G}}(y_{1})$,
then there exists a compass structure $\mathcal{G}^{\prime}$ of length
$N^{\prime}<N$ that features $\varphi$.
Figure 2. Contraction $\mathcal{G}^{\prime}$ of a compass structure
$\mathcal{G}$.
###### Proof 3.3.
Suppose that $0<y_{0}<y_{1}<N$ are two rows of $\mathcal{G}$ such that
$\mathcal{S}\mathit{hading}_{\mathcal{G}}(y_{0})\subseteq\mathcal{S}\mathit{hading}_{\mathcal{G}}(y_{1})$.
Then, there is a function
$f:\\{0,...,y_{0}-1\\}\;\rightarrow\;\\{0,...,y_{1}-1\\}$ such that, for every
$0\leq x<y_{0}$, $\mathcal{L}(x,y_{0})=\mathcal{L}(f(x),y_{1})$. Let
$k=y_{1}-y_{0}$, $N^{\prime}=N-k$ ($<N$), and $\mathbb{P}_{N^{\prime}}$ be the
portion of the grid that consists of all points $p=(x,y)$, with $0\leq
x<y<N^{\prime}$. We extend $f$ to a function that maps points in
$\mathbb{P}_{N^{\prime}}$ to points in $\mathbb{P}_{N}$ as follows:
* •
if $p=(x,y)$, with $0\leq x<y<y_{0}$, then we simply let $f(p)=p$;
* •
if $p=(x,y)$, with $0\leq x<y_{0}\leq y$, then we let $f(p)=(f(x),y+k)$;
* •
if $p=(x,y)$, with $y_{0}\leq x<y$, then we let $f(p)=(x+k,y+k)$.
We denote by $\mathcal{L}^{\prime}$ the labeling of $\mathbb{P}_{N^{\prime}}$
such that, for every point $p\in\mathbb{P}_{N^{\prime}}$,
$\mathcal{L}^{\prime}(p)=\mathcal{L}(f(p))$ and we denote by
$\mathcal{G}^{\prime}$ the resulting structure
$(\mathbb{P}_{N^{\prime}},\mathcal{L}^{\prime})$ (see Figure 2). We have to
prove that $\mathcal{G}^{\prime}$ is a consistent and fulfilling compass
structure that features $\varphi$ (see Definition 2.1). First, we show that
$\mathcal{G}^{\prime}$ satisfies the consistency conditions for the relations
$B$ and $A$; then we show that $\mathcal{G}^{\prime}$ satisfies the
fulfillment conditions for the $\,\overline{\\!B\\!}\,$-, $B$-, and
$A$-requests; finally, we show that $\mathcal{G}^{\prime}$ features $\varphi$.
Consistency with relation $B$. Consider two points $p=(x,y)$ and
$p^{\prime}=(x^{\prime},y^{\prime})$ in $\mathcal{G}^{\prime}$ such that
$p\;B\;p^{\prime}$, i.e., $0\leq x=x^{\prime}<y^{\prime}<y<N^{\prime}$. We
prove that
$\mathcal{L}^{\prime}(p)\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,\mathcal{L}^{\prime}(p^{\prime})$
by distinguishing among the following three cases (note that exactly one of
such cases holds):
1. (1)
$y<y_{0}$ and $y^{\prime}<y_{0}$,
2. (2)
$y\geq y_{0}$ and $y^{\prime}\geq y_{0}$,
3. (3)
$y\geq y_{0}$ and $y^{\prime}<y_{0}$.
If $y<y_{0}$ and $y^{\prime}<y_{0}$, then, by construction, we have $f(p)=p$
and $f(p^{\prime})=p^{\prime}$. Since $\mathcal{G}$ is a (consistent) compass
structure, we immediately obtain
$\mathcal{L}^{\prime}(p)=\mathcal{L}(p)\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,\mathcal{L}(p^{\prime})=\mathcal{L}^{\prime}(p^{\prime})$.
If $y\geq y_{0}$ and $y\geq y_{0}$, then, by construction, we have either
$f(p)=(f(x),y+k)$ or $f(p)=(x+k,y+k)$, depending on whether $x<y_{0}$ or
$x\geq y_{0}$. Similarly, we have either
$f(p^{\prime})=(f(x^{\prime}),y^{\prime}+k)=(f(x),y^{\prime}+k)$ or
$f(p^{\prime})=(x^{\prime}+k,y^{\prime}+k)=(x+k,y^{\prime}+k)$. This implies
$f(p)\;B\;f(p^{\prime})$ and thus, since $\mathcal{G}$ is a (consistent)
compass structure, we have $\mathcal{L}^{\prime}(p)=\mathcal{L}(f(p))$
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$
$\mathcal{L}(f(p^{\prime}))=\mathcal{L}^{\prime}(p^{\prime})$.
If $y\geq y_{0}$ and $y^{\prime}<y_{0}$, then, since $x<y^{\prime}<y_{0}$, we
have by construction $f(p)=(f(x),y+k)$ and $f(p^{\prime})=p^{\prime}$.
Moreover, if we consider the point $p^{\prime\prime}=(x,y_{0})$ in
$\mathcal{G}^{\prime}$, we easily see that (i)
$f(p^{\prime\prime})=(f(x),y_{1})$, (ii) $f(p)\;B\;f(p^{\prime\prime})$
(whence
$\mathcal{L}(f(p))\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,\mathcal{L}(f(p^{\prime\prime}))$),
(iii) $\mathcal{L}(f(p^{\prime\prime}))=\mathcal{L}(p^{\prime\prime})$, and
(iv) $p^{\prime\prime}\;B\;p^{\prime}$ (whence
$\mathcal{L}(p^{\prime\prime})\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,\mathcal{L}(p^{\prime})$).
It thus follows that
$\mathcal{L}^{\prime}(p)=\mathcal{L}(f(p))\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,\mathcal{L}(f(p^{\prime\prime}))$
$=\mathcal{L}(p^{\prime\prime})$
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$
$\mathcal{L}(p^{\prime})=\mathcal{L}(f(p^{\prime}))=\mathcal{L}^{\prime}(p^{\prime})$.
Finally, by exploiting the transitivity of the relation
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$
, we obtain
$\mathcal{L}^{\prime}(p)\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,\mathcal{L}^{\prime}(p^{\prime})$.
Consistency with relation $A$. Consider two points $p=(x,y)$ and
$p^{\prime}=(x^{\prime},y^{\prime})$ such that $p\;A\;p^{\prime}$, i.e.,
$0\leq x<y=x^{\prime}<y^{\prime}<N^{\prime}$. We define
$p^{\prime\prime}=(y,y+1)$ in such a way that $p\;A\;p^{\prime\prime}$ and
$p^{\prime}\;B\;p^{\prime\prime}$ and we distinguish between the following two
cases:
1. (1)
$y\geq y_{0}$,
2. (2)
$y<y_{0}$.
If $y\geq y_{0}$, then, by construction, we have
$f(p)\;A\;f(p^{\prime\prime})$. Since $\mathcal{G}$ is a (consistent) compass
structure, it follows that $\mathcal{L}^{\prime}(p)=\mathcal{L}(f(p))$
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$
$\mathcal{L}(f(p^{\prime\prime}))=\mathcal{L}^{\prime}(p^{\prime\prime})$.
If $y<y_{0}$, then, by construction, we have
$\mathcal{L}(p^{\prime\prime})=\mathcal{L}(f(p^{\prime\prime}))$. Again, since
$\mathcal{G}$ is a (consistent) compass structure, it follows that
$\mathcal{L}^{\prime}(p)=\mathcal{L}(f(p))=\mathcal{L}(p)$
$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$
$\mathcal{L}(p^{\prime\prime})=\mathcal{L}(f(p^{\prime\prime}))=\mathcal{L}^{\prime}(p^{\prime\prime})$.
In both cases we have
$\mathcal{L}^{\prime}(p)\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$}}\,\mathcal{L}^{\prime}(p^{\prime\prime})$.
Now, we recall that $p^{\prime}\;B\;p^{\prime\prime}$ and that, by previous
arguments, $\mathcal{G}^{\prime}$ is consistent with the relation $B$. We thus
have
$\mathcal{L}^{\prime}(p^{\prime})\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{B\,}$}}}{\longrightarrow}$}}\,\mathcal{L}^{\prime}(p^{\prime\prime})$.
Finally, by applying Lemma 3.1, we obtain
$\mathcal{L}^{\prime}(p)\,\text{\raisebox{-0.86108pt}{$\overset{\text{\raisebox{-0.43057pt}[0.0pt][-0.43057pt]{${}_{A\,}$}}}{\longrightarrow}$}}\,\mathcal{L}^{\prime}(p^{\prime})$.
Fulfillment of $B$-requests. Consider a point $p=(x,y)$ in
$\mathcal{G}^{\prime}$ and some $B$-request
$\alpha\in\mathcal{R}\mathit{eq}_{B}\bigl{(}\mathcal{L}^{\prime}(p)\bigr{)}$
associated with it. Since, by construction,
$\alpha\in\mathcal{R}\mathit{eq}_{B}\bigl{(}\mathcal{L}(f(p))\bigr{)}$ and
$\mathcal{G}$ is a (fulfilling) compass structure, we know that $\mathcal{G}$
contains a point $q^{\prime}=(x^{\prime},y^{\prime})$ such that
$f(p)\;B\;q^{\prime}$ and
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(q^{\prime})\bigr{)}$. We
prove that $\mathcal{G}^{\prime}$ contains a point $p^{\prime}$ such that
$p\;B\;p^{\prime}$ and
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}^{\prime}(p^{\prime})\bigr{)}$
by distinguishing among the following three cases (note that exactly one of
such cases holds):
1. (1)
$y<y_{0}$
2. (2)
$y^{\prime}\geq y_{1}$,
3. (3)
$y\geq y_{0}$ and $y^{\prime}<y_{1}$.
If $y<y_{0}$, then, by construction, we have $p=f(p)$ and
$q^{\prime}=f(q^{\prime})$. Therefore, we simply define
$p^{\prime}=q^{\prime}$ in such a way that $p=f(p)\;B\;q^{\prime}=p^{\prime}$
and
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}^{\prime}(p^{\prime})\bigr{)}$
($=\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(f(p^{\prime}))\bigr{)}=\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(q^{\prime})\bigr{)}$).
If $y^{\prime}\geq y_{1}$, then, by construction, we have either
$f(p)=(f(x),y+k)$ or $f(p)=(x+k,y+k)$, depending on whether $x<y_{0}$ or
$x\geq y_{0}$. We define $p^{\prime}=(x,y^{\prime}-k)$ in such a way that
$p\;B\;p^{\prime}$. Moreover, we observe that either
$f(p^{\prime})=(f(x),y^{\prime})$ or $f(p^{\prime})=(x+k,y^{\prime})$,
depending on whether $x<y_{0}$ or $x\geq y_{0}$, and in both cases
$f(p^{\prime})=q^{\prime}$ follows. This shows that
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}^{\prime}(p^{\prime})\bigr{)}$
($=\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(f(p^{\prime})\bigr{)}=\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(q^{\prime})\bigr{)}$).
If $y\geq y_{0}$ and $y^{\prime}<y_{1}$, then we define
$\,\overline{\\!p\\!}\,=(x,y_{0})$ and
$\,\overline{\\!q\\!}\,=(x^{\prime},y_{1})$ and we observe that
$f(p)\;B\;\,\overline{\\!q\\!}\,$, $\,\overline{\\!q\\!}\,\;B\;q^{\prime}$,
and $f(\,\overline{\\!p\\!}\,)=\,\overline{\\!q\\!}\,$. From
$f(p)\;B\;\,\overline{\\!q\\!}\,$ and $\,\overline{\\!q\\!}\,\;B\;q^{\prime}$,
it follows that
$\alpha\in\mathcal{R}\mathit{eq}_{B}\bigl{(}\mathcal{L}(\,\overline{\\!q\\!}\,)\bigr{)}$
and hence
$\alpha\in\mathcal{R}\mathit{eq}_{B}\bigl{(}\mathcal{L}(\,\overline{\\!p\\!}\,)\bigr{)}$.
Since $\mathcal{G}$ is a (fulfilling) compass structure, we know that there is
a point $p^{\prime}$ such that $\,\overline{\\!p\\!}\,\;B\;p^{\prime}$ and
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(\,\overline{\\!p\\!}\,^{\prime})\bigr{)}$.
Moreover, since $\,\overline{\\!p\\!}\,\;B\;p^{\prime}$, we have
$f(p^{\prime})=p^{\prime}$, from which we obtain $p\;B\;p^{\prime}$ and
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}(p^{\prime})\bigr{)}$.
Fulfillment of $\,\overline{\\!B\\!}\,$-requests. The proof that
$\mathcal{G}^{\prime}$ fulfills all $\,\overline{\\!B\\!}\,$-requests of its
atoms is symmetric with respect to the previous one.
Fulfillment of $A$-requests. Consider a point $p=(x,y)$ in
$\mathcal{G}^{\prime}$ and some $A$-request
$\alpha\in\mathcal{R}\mathit{eq}_{A}\bigl{(}\mathcal{L}^{\prime}(p)\bigr{)}$
associated with $p$ in $\mathcal{G}^{\prime}$. Since, by previous arguments,
$\mathcal{G}^{\prime}$ fulfills all $\,\overline{\\!B\\!}\,$-requests of its
atoms, it is sufficient to prove that either
$\alpha\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}^{\prime}(p^{\prime})\bigr{)}$
or
$\alpha\in\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}\bigl{(}\mathcal{L}^{\prime}(p^{\prime})\bigr{)}$,
where $p^{\prime}=(y,y+1)$. This can be easily proved by distinguishing among
the three cases $y<y_{0}-1$, $y=y_{0}-1$, and $y\geq y_{0}$.
Featured formulas. Recall that, by previous assumptions, $\mathcal{G}$
contains a point $p=(0,y)$, with $0<y<N$, such that
$\varphi\in\mathcal{L}(p)$. If $y\leq y_{0}$, then, by construction, we have
$\varphi\in\mathcal{L}^{\prime}(p)$ ($=\mathcal{L}(f(p))=\mathcal{L}(p)$).
Otherwise, if $y>y_{0}$, we define $q=(0,y_{0})$ and we observe that
$q\;\,\overline{\\!B\\!}\,\;p$. Since $\mathcal{G}$ is a (consistent) compass
structure and
${\langle\,\overline{\\!B\\!}\,\rangle}\varphi\in\mathcal{C}\mathit{l}^{+}(\varphi)$,
we have that
$\varphi\in\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}\bigl{(}\mathcal{L}(q)\bigr{)}$.
Moreover, by construction, we have $\mathcal{L}^{\prime}(q)=\mathcal{L}(f(q))$
and hence
$\varphi\in\mathcal{R}\mathit{eq}_{\,\overline{\\!B\\!}\,}\bigl{(}\mathcal{L}^{\prime}(q)\bigr{)}$.
Finally, since $\mathcal{G}^{\prime}$ is a (fulfilling) compass structure, we
know that there is a point $p^{\prime}$ in $\mathcal{G}^{\prime}$ such that
$f(q)\;\,\overline{\\!B\\!}\,\;p^{\prime}$ and
$\varphi\in\mathcal{O}\mathit{bs}\bigl{(}\mathcal{L}^{\prime}(p^{\prime})\bigr{)}$.
On the grounds of the above result, we can provide a suitable upper bound for
the length of a minimal finite interval structure that satisfies $\varphi$, if
there exists any. This yields a straightforward, but inefficient, 2EXPSPACE
algorithm that decides whether a given
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$-formula $\varphi$ is satisfiable
over finite interval structures.
###### Theorem 3.4.
An $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$-formula $\varphi$ is satisfied by
some finite interval structure iff it is featured by some compass structure of
length $N\leq 2^{2^{7{\lvert\varphi\rvert}}}$ (i.e., double exponential in
${\lvert\varphi\rvert}$).
###### Proof 3.5.
One direction is trivial. We prove the other one (“only if” part). Suppose
that $\varphi$ is satisfied by a finite interval structure $\mathcal{S}$. By
Proposition 2.2, there is a compass structure $\mathcal{G}$ that features
$\varphi$ and has finite length $N<\omega$. Without loss of generality, we can
assume that $N$ is minimal among all finite compass structures that feature
$\varphi$. We recall from Section 2.2 that $\mathcal{G}$ contains at most
$2^{7{\lvert\varphi\rvert}}$ distinct atoms. This implies that there exist at
most $2^{2^{7{\lvert\varphi\rvert}}}$ different shadings of the form
$\mathcal{S}\mathit{hading}_{\mathcal{G}}(y)$, with $0\leq y<N$. Finally, by
applying Lemma 3.2, we obtain $N\leq 2^{2^{7{\lvert\varphi\rvert}}}$
(otherwise, there would exist two rows $0<y_{0}<y_{1}<N$ such that
$\mathcal{S}\mathit{hading}_{\mathcal{G}}(y_{0})=\mathcal{S}\mathit{hading}_{\mathcal{G}}(y_{1})$,
which is against the hypothesis of minimality of $N$).
### 3.2. A small-model theorem for infinite structures
In general, compass structures that feature $\varphi$ may be infinite. Here,
we prove that, without loss of generality, we can restrict our attention to
sufficiently “regular” infinite compass structures, which can be represented
in double exponential space with respect to ${\lvert\varphi\rvert}$. To do
that, we introduce the notion of periodic compass structure.
###### Definition 3.6.
An infinite compass structure $\mathcal{G}=(\mathbb{P}_{\omega},\mathcal{L})$
is _periodic_ , with _threshold_ $\widetilde{y}_{0}$, _period_
$\widetilde{y}$, and _binding_
$\widetilde{g}:\\{0,...,\widetilde{y}_{0}+\widetilde{y}-1\\}\;\rightarrow\;\\{0,...,\widetilde{y}_{0}-1\\}$,
if the following conditions are satisfied:
* •
for every $\widetilde{y}_{0}+\widetilde{y}\leq x<y$, we have
$\mathcal{L}(x,y)=\mathcal{L}(x-\widetilde{y},y-\widetilde{y})$,
* •
for every $0\leq x<\widetilde{y}_{0}+\widetilde{y}\leq y$, we have
$\mathcal{L}(x,y)=\mathcal{L}(\widetilde{g}(x),y-\widetilde{y})$.
Figure 3 gives an example of a periodic compass structure (the arrows
represent some relationships between points induced by the binding function
$\widetilde{g}$). Note that any periodic compass structure
$\mathcal{G}=(\mathbb{P}_{\omega},\mathcal{L})$ can be finitely represented by
specifying (i) its threshold $\widetilde{y}_{0}$, (ii) its period
$\widetilde{y}$, (iii) its binding $\widetilde{g}$, and (iv) the labeling
$\mathcal{L}$ restricted to the portion
$\mathbb{P}_{\widetilde{y}_{0}+\widetilde{y}-1}$ of the domain.
Figure 3. A periodic compass structure with threshold $\widetilde{y}_{0}$,
period $\widetilde{y}$, and binding $\widetilde{g}$.
The following theorem leads immediately to a 2EXPSPACE algorithm that decides
whether a given $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$-formula $\varphi$ is
satisfiable over infinite interval structures (the proof is provided in [17]).
###### Theorem 3.7.
An $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$-formula $\varphi$ is satisfied by
an infinite interval structure iff it is featured by a periodic compass
structure with threshold $\widetilde{y}_{0}<2^{2^{7{\lvert\varphi\rvert}}}$
and period $\widetilde{y}<2{\lvert\varphi\rvert}\cdot
2^{2^{7{\lvert\varphi\rvert}}}\cdot 2^{2^{7{\lvert\varphi\rvert}}}$.
## 4\. Tight complexity bounds to the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$
In this section, we show that the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ interpreted over (either finite or
infinite) interval temporal structures is EXPSPACE-complete.
The EXPSPACE-hardness of the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ follows from a reduction from the
_exponential-corridor tiling problem_ , which is known to be EXPSPACE-complete
[19]. Formally, an instance of the exponential-corridor tiling problem is a
tuple $\mathcal{T}=(T,t_{\bot},t_{\top},H,$ $V,n)$ consisting of a finite set
$T$ of tiles, a bottom tile $t_{\bot}\in T$, a top tile $t_{\top}\in T$, two
binary relations $H,V$ over $T$ (specifying the horizontal and vertical
constraints), and a positive natural number $n$ (represented in unary
notation). The problem consists in deciding whether there exists a tiling
$f:\mathbb{N}\times\\{0,...,2^{n}-1\\}\;\rightarrow\;T$ of the infinite
discrete corridor of height $2^{n}$, that associates the tile $t_{\bot}$
(resp., $t_{\top}$) with the bottom (resp., top) row of the corridor and that
respects the horizontal and vertical constraints $H$ and $V$, namely,
1. i)
for every $x\in\mathbb{N}$, we have $f(x,0)=t_{\bot}$,
2. ii)
for every $x\in\mathbb{N}$, we have $f(x,2^{n}-1)=t_{\top}$,
3. iii)
for every $x\in\mathbb{N}$ and every $0\leq y<2^{n}$, we have
$f(x,y)\;H\;f(x+1,y)$,
4. iv)
for every $x\in\mathbb{N}$ and every $0\leq y<2^{n}-1$, we have
$f(x,y)\;V\;f(x,y+1)$.
The proof of the following lemma, which reduces the exponential-corridor
tiling problem to the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$, can be found in [17]. Intuitively,
such a reduction exploits (i) the correspondence between the points $p=(x,y)$
inside the infinite corridor $\mathbb{N}\times\\{0,...,2^{n}-1\\}$ and the
intervals of the form $I_{p}=[y+2^{n}x,y+2^{n}x+1]$, (ii) ${\lvert T\rvert}$
propositional variables which represent the tiling function $f$, (iii) $n$
additional propositional variables which represent (the binary expansion of)
the $y$-coordinate of each row of the corridor, and (iv) the modal operators
${\langle A\rangle}$ and ${\langle B\rangle}$ by means of which one can
enforce the local constrains over the tiling function $f$ (as a matter of
fact, this shows that the satisfiability problem for the $A\mspace{-0.3mu}B$
fragment is already hard for EXPSPACE).
###### Lemma 4.1.
There is a polynomial-time reduction from the exponential-corridor tiling
problem to the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$.
As for the EXPSPACE-completeness, we claim that the existence of a compass
structure $\mathcal{G}$ that features a given formula $\varphi$ can be decided
by verifying suitable local (and stronger) consistency conditions over all
pairs of contiguous rows. In fact, in order to check that these local
conditions hold between two contiguous rows $y$ and $y+1$, it is sufficient to
store into memory a bounded amount of information, namely, (i) a counter $y$
that ranges over
$\bigl{\\{}1,...,2^{2^{7{\lvert\varphi\rvert}}}+{\lvert\varphi\rvert}\cdot
2^{2^{7{\lvert\varphi\rvert}}}\bigr{\\}}$, (ii) the two guessed shadings $S$
and $S^{\prime}$ associated with the rows $y$ and $y+1$, and (iii) a function
$g:S\;\rightarrow\;S^{\prime}$ that captures the horizontal alignment relation
between points with an associated atom from $S$ and points with an associated
atom from $S^{\prime}$. This shows that the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ can be decided in exponential space,
as claimed by the following lemma. Further details about the decision
procedure, including soundness and completeness proofs, can be found in [17].
###### Lemma 4.2.
There is an EXPSPACE non-deterministic procedure that decides whether a given
formula of $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ is satisfiable or not.
Summing up, we obtain the following tight complexity result.
###### Theorem 4.3.
The satisfiability problem for $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$
interpreted over (prefixes of) natural numbers is EXPSPACE-complete.
## 5\. Conclusions
In this paper, we proved that the satisfiability problem for
$A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ interpreted over (prefixes of) the
natural numbers is EXPSPACE-complete. We restricted our attention to these
domains because it is a common commitment in computer science. Moreover, this
gave us the possibility of expressing meaningful metric constraints in a
fairly natural way. Nevertheless, we believe it possible to extend our results
to the class of all linear orderings as well as to relevant subclasses of it.
Another restriction that can be relaxed is the one about singleton intervals:
all results in the paper can be easily generalized to include singleton
intervals in the underlying structure $\mathbb{I}_{N}$. The most exciting
challenge is to establish whether the modality $\,\overline{\\!A\\!}\,$ can be
added to $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ preserving decidability
(and complexity). It is easy to show that there is not a straightforward way
to lift the proof for $A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,$ to
${A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,\,\overline{\\!A\\!}\,}$ (notice that
${\langle A\rangle}$, ${\langle B\rangle}$, and
${\langle\,\overline{\\!B\\!}\,\rangle}$ are all future modalities, while
${\langle\,\overline{\\!A\\!}\,\rangle}$ is a past one).
## References
* [1] J.F. Allen. Maintaining knowledge about temporal intervals. Communications of the Association for Computing Machinery, 26(11):832–843, 1983.
* [2] D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G. Sciavicco. Decidable and undecidable fragments of Halpern and Shoham’s interval temporal logic: towards a complete classification. In Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR), volume 5330 of Lecture Notes in Computer Science, pages 590–604. Springer, 2008.
* [3] D. Bresolin, V. Goranko, A. Montanari, and P. Sala. Tableau-based decision procedures for the logics of subinterval structures over dense orderings. Journal of Logic and Computation, doi:10.1093/logcom/exn063, 2008\.
* [4] D. Bresolin, V. Goranko, A. Montanari, and G. Sciavicco. On decidability and expressiveness of propositional interval neighborhood logics. In Proceedings of the International Symposium on Logical Foundations of Computer Science (LFCS), volume 4514 of Lecture Notes in Computer Science, pages 84–99. Springer, 2007.
* [5] D. Bresolin, V. Goranko, A. Montanari, and G. Sciavicco. Propositional interval neighborhood logics: expressiveness, decidability, and undecidable extensions. Annals of Pure and Applied Logic, 161(3):289–304, 2009.
* [6] D. Bresolin, A. Montanari, P. Sala, and G. Sciavicco. Optimal tableaux for right propositional neighborhood logic over linear orders. In Proceedings of the 11th European Conference on Logics in Artificial Intelligence (JELIA), volume 5293 of Lecture Notes in Artificial Intelligence, pages 62–75. Springer, 2008.
* [7] D. Bresolin, A. Montanari, and G. Sciavicco. An optimal decision procedure for Right Propositional Neighborhood Logic. Journal of Automated Reasoning, 38(1-3):173–199, 2007.
* [8] D. Bresolin, V. Goranko A. Montanari, and G. Sciavicco. Right propositional neighborhood logic over natural numbers with integer constraints for interval lengths. In Proceedings of the 7th IEEE International Conference on Software Engineering and Formal Methods (SEFM), pages 240–249. IEEE Comp. Society Press, 2009.
* [9] D. Gabbay, I. Hodkinson, and M. Reynolds. Temporal Logic: mathematical foundations and computational aspects. Oxford University Press, 1994.
* [10] V. Goranko, A. Montanari, and G. Sciavicco. Propositional interval neighborhood temporal logics. Journal of Universal Computer Science, 9(9):1137–1167, 2003.
* [11] V. Goranko, A. Montanari, and G. Sciavicco. A road map of interval temporal logics and duration calculi. Applied Non-classical Logics, 14(1-2):9–54, 2004.
* [12] J.Y. Halpern and Y. Shoham. A propositional modal logic of time intervals. Journal of the Association for Computing Machinery, 38:279–292, 1991.
* [13] I. Hodkinson, A. Montanari, and G. Sciavicco. Non-finite axiomatizability and undecidability of interval temporal logics with C, D, and T. In Proceedings of the 17th Annual Conference of the EACSL, volume 5213 of Lecture Notes in Computer Science, pages 308–322. Springer, 2008.
* [14] K. Lodaya. Sharpening the undecidability of interval temporal logic. In Proceedings of the 6th Asian Computing Science Conference on Advances in Computing Science, volume 1961 of Lecture Notes in Computer Science, pages 290–298. Springer, 2000.
* [15] C. Lutz and F. Wolter. Modal logics of topological relations. Logical Methods in Computer Science, 2(2), 2006.
* [16] A. Montanari, G. Puppis, and P. Sala. A decidable spatial logic with cone-shaped cardinal directions. In Proceedings of the 18th Annual Conference of the EACSL, volume 5771 of Lecture Notes in Computer Science, pages 394–408. Springer, 2009.
* [17] A. Montanari, G. Puppis, P. Sala, and G. Sciavicco. Decidability of the interval temporal logic ${A\mspace{-0.3mu}B\,\overline{\\!B\\!}\,}$ over the natural numbers. Research Report UDMI/2009/07, Department of Mathematics and Computer Science, University of Udine, Udine, Italy, 2009, http://users.dimi.uniud.it/$\sim$angelo.montanari/rr200907.pdf.
* [18] M. Otto. Two variable first-order logic over ordered domains. Journal of Symbolic Logic, 66(2):685–702, 2001.
* [19] P. Van Emde Boas. The convenience of tilings. In Complexity, Logic and Recursion Theory, volume 187 of Lecture Notes in Pure and Applied Mathematics, pages 331–363. Marcel Dekker Inc., 1997.
* [20] Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1(4):453–476, 1991.
|
arxiv-papers
| 2009-12-17T15:22:45 |
2024-09-04T02:49:07.080159
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Montanari, G. Puppis, P. Sala, G. Sciavicco",
"submitter": "Pietro Sala Mr.",
"url": "https://arxiv.org/abs/0912.3429"
}
|
0912.3509
|
# Path integral representation of the quantum evolution in dynamical systems
with a symmetry for the non-zero momentum level reduction
S.N.Storchak
###### Abstract
For the case of reduction onto the non-zero momentum level, in the problem of
the path integral quantization of a scalar particle motion on a smooth compact
Riemannian manifold with the given free isometric action of the compact
semisimle Lie group, the path integral representation of the matrix Green’s
function, which describes the quantum evolution of the reduced motion, has
been obtained. The integral relation between the path integrals representing
the fundamental solutions of the parabolic differential equation defined on
the total space of the principal fiber bundle and the linear parabolic system
of the differential equations on the space of the sections of the associated
covector bundle has been derived.
## 1 Introduction
There is a number of a remarkable properties in dynamical systems with a
symmetry. The main property of these systems manifests itself in a
relationship between an original system and another system (a reduced one)
obtained from the original system after “removing” the group degrees of
freedom.
One of the system of this class is the dynamical system which describes a
motion of a scalar particle on a smooth compact finite-dimensional Riemannian
manifold with a given free isometric smooth action of a semisimple compact Lie
group. In fact, the original motion of the particle takes place on the total
space of a principal fiber bundle, and the reduced motion — on the orbit space
of this bundle.
This system bears close resemblance to the gauge field models, where the
reduced evolution is given on the orbit space of a gauge group action. That is
why a great deal of attention has been devoted to the quantization of the
finite-dimensional system related to the particle motion on a manifold with a
group action [1, 2, 3, 4].
In gauge theories, the motion on the orbit space is described in terms of the
gauge fields that are restricted to a gauge surface. Moreover, a description
of this motion is only possible by means of dependent variables.
Such a description is used in a heuristic method of the path integral
quantization of the gauge fields proposed by Faddeev and Popov [5]. However,
at present, it is not even quite clear how to define correctly the path
integral measure on the space of the gauge fields. Therefore, in order to
establish the final validity of the method it would be desirable to carry out
its additional investigation from the standpoint of a general approach
developed in the integration theory. There is a hope that it gives us an
answer on yet unsolved questions of the Faddeev–Popov method.
As a first step in that direction, it was studied the path integral reduction
in the aforementioned finite-dimensional dynamical system [6]. We have used
the methods of the stochastic process theory for definition of a path integral
measure and in order to study the path integral transformation under the
reduction. That is, we dealt with diffusion on a manifold with a given group
action and with the path integral representation of the solution of the
backward Kolmogorov equation.
Path integral reduction is based on the separation of the variables or, in
other words, on the factorization of the original path integral measure into
the ‘group’ measure and the measure that is given on the orbit space. In our
papers, it was fulfilled with the help of the nonlinear filtering stochasic
differential equation. Note that a similar approach to the measure
factorization was developed in [7]. Also, the questions related to the
factorization have been studied in [8].
As a result of the reduction, the integral relation between the wave functions
of the corresponding ‘quantum’ evolutions (the reduced and original
diffusions) was obtained.
It was found that the Hamilton operator of the reduced dynamical system (the
differential generator of the stochastic process) has an extra potential term.
This term comes from from the reduction Jacobian.
In [9], the path integral reduction has been considered in the case when the
reduced motion is described in terms of the dependent variables.
As in gauge theories, we have suggested that the principal bundle is a trivial
one. Then, in the principal fiber bundle, there is a global cross-section. The
cross-section may be determined with the choice of the special gauge surface.
The evolution on this gauge surface serves for description of the
corresponding reduced evolution on the orbit space.
In this paper we will study the case of the non-zero momentum level reduction
in the path integral for the discussed finite-dimensional dynamical system.
The path integral, which describes the evolution of the reduced motion on the
orbit space, will be represent the fundamental solution of the linear
parabolic system of the differential equations.
## 2 Definitions
In our papers [6], we have considered the diffusion of a scalar particle on a
smooth compact Riemannian manifold $\cal P$. The backward Kolmogorov equation
for the original diffusion was as follows
$\left\\{\begin{array}[]{l}\displaystyle\left(\frac{\partial}{\partial
t_{a}}+\frac{1}{2}\mu^{2}\kappa\triangle_{\cal
P}(p_{a})+\frac{1}{\mu^{2}\kappa
m}V(p_{a})\right){\psi}_{t_{b}}(p_{a},t_{a})=0\\\
{\psi}_{t_{b}}(p_{b},t_{b})=\phi_{0}(p_{b}),\qquad\qquad\qquad\qquad\qquad(t_{b}>t_{a}),\end{array}\right.$
(1)
where $\mu^{2}=\frac{\hbar}{m}$ , $\kappa$ is a real positive parameter,
$\triangle_{\cal P}(p_{a})$ is a Laplace–Beltrami operator on $\cal P$, and
$V(p)$ is a group–invariant potential term. In a chart with the coordinate
functions $Q^{A}={\varphi}^{A}(p)$, $p\in{\cal P}$, the Laplace – Beltrami
operator is written as
$\triangle_{\cal P}(Q)=G^{-1/2}(Q)\frac{\partial}{\partial
Q^{A}}G^{AB}(Q)G^{1/2}(Q)\frac{\partial}{\partial Q^{B}},$
with $G=det(G_{AB})$, $G_{AB}(Q)=G(\frac{\partial}{\partial
Q^{A}},\frac{\partial}{\partial Q^{B}})$.
In accordance with the theory developed by Daletskii and Belopolskaya [10],
the solution of (1) is given by the global semigroup which is a limit (under
the refinement of the subdivision of the time interval) of a superposition of
the local semigroups
$\psi_{t_{b}}(p_{a},t_{a})=U(t_{b},t_{a})\phi_{0}(p_{a})={\lim}_{q}{\tilde{U}}_{\eta}(t_{a},t_{1})\cdot\ldots\cdot{\tilde{U}}_{\eta}(t_{n-1},t_{b})\phi_{0}(p_{a}).$
(2)
Each local semigroup is determined by the path integrals with the integration
measures defined by the local representatives $\eta^{A}(t)$ of the global
stochastic process $\eta(t)$. The local stochastic process $\eta^{A}(t)$ are
given by the solutions of the following stochastic differential equations:
$d\eta^{A}(t)=\frac{1}{2}\mu^{2}\kappa G^{-1/2}\frac{\partial}{\partial
Q^{B}}(G^{1/2}G^{AB})dt+\mu\sqrt{\kappa}{\mathfrak{X}}_{\bar{M}}^{A}(\eta(t))dw^{\bar{M}}(t),$
(3)
where the matrix ${\mathfrak{X}}_{\bar{M}}^{A}$ is defined by the local
equality $\sum^{n_{P}}_{\bar{{\scriptscriptstyle
K}}\scriptscriptstyle=1}{\mathfrak{X}}_{\bar{K}}^{A}{\mathfrak{X}}_{\bar{K}}^{B}=G^{AB}$.
(We denote the Euclidean indices by over–barred indices.)
Therefore, the behavior of the global semigroup (2) is completely defined by
these stochastic differential equations. The global semigroup can be written
symbolically as follows
$\displaystyle{\psi}_{t_{b}}(p_{a},t_{a})$ $\displaystyle=$ $\displaystyle{\rm
E}\Bigl{[}\phi_{0}(\eta(t_{b}))\exp\\{\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}V(\eta(u))du\\}\Bigr{]}$ (4) $\displaystyle=$
$\displaystyle\int_{\Omega_{-}}d\mu^{\eta}(\omega)\phi_{0}(\eta(t_{b}))\exp\\{\ldots\\},$
where ${\eta}(t)$ is a global stochastic process on a manifold $\cal P$.
$\Omega_{-}=\\{\omega(t):\omega(t_{a})=0,\eta(t)=p_{a}+\omega(t)\\}$ is the
path space on this manifold. The path integral measure ${\mu}^{\eta}$ is
defined by the probability distribution of a stochastic process ${\eta}(t)$.
### 2.1 Geometry of the problem
Since in our case, there is a free isometric smooth action of a semisimple
compact Lie group $\cal G$ on the original manifold $\cal P$, this manifold
can be viewed as a total space of the principal fiber bundle $\pi:\cal
P\to{\cal P}/{\cal G}=\cal M$.
At the first step of the reduction procedure, we have transformed the original
coordinates $Q^{A}$ given on a local chart of the manifold $\cal P$ for new
coordinates $(Q^{\ast}{}^{A},a^{\alpha})$ ($A=1,\ldots,N_{\cal P},N_{\cal
P}=\dim{\cal P};{\alpha}=1,\ldots,N_{\cal G},N_{\cal G}=\dim{\cal G}$) related
to the fiber bundle. In order to meet a requirement of a one-to-one mapping
between $Q^{A}$ and $(Q^{\ast}{}^{A},a^{\alpha})$, we are forced to introduce
the additional constraints, ${\chi}^{\alpha}(Q^{\ast})=0$.
These constraints define the local submanifolds in the manifold $\cal P$. On
the assumption that these local submanifolds (local sections) can be ‘glued’
into the global manifold $\Sigma$, we come to a trivial principal fiber bundle
$P({\cal M},\cal G)$.
We note that this bundle is locally isomorphic to the trivial bundle
$\Sigma\times{\cal G}\to{\Sigma}$. It allows us to use the coordinates
$Q^{\ast}{}^{A}$ for description of the evolution on the manifold $\cal M$.
If we replace the coordinate basis $(\frac{\partial}{\partial Q^{A}})$ for a
new coordinate basis $(\frac{\partial}{\partial
Q^{\ast}{}^{A}},\frac{\partial}{\partial a^{\alpha}})$, we get the following
representation for the original metric ${\tilde{G}}_{\cal A\cal
B}(Q^{\ast},a)$ of the manifold $\cal P$:
$\left(\begin{array}[]{cc}G_{CD}(Q^{\ast})(P_{\perp})^{C}_{A}(P_{\perp})^{D}_{B}&G_{CD}(Q^{\ast})(P_{\perp})^{D}_{A}K^{C}_{\mu}\bar{u}^{\mu}_{\alpha}(a)\\\
G_{CD}(Q^{\ast})(P_{\perp})^{C}_{A}K^{D}_{\nu}\bar{u}^{\nu}_{\beta}(a)&{\gamma}_{\mu\nu}(Q^{\ast})\bar{u}_{\alpha}^{\mu}(a)\bar{u}_{\beta}^{\nu}(a)\end{array}\right).$
(5)
To obtain this expression we have used the right action of the group $\cal G$
on a manifold $\cal P$. It was given by functions $F^{A}(Q,a)$, performing an
action, and their derivatives: $F^{C}_{B}(Q,a)\equiv\frac{\partial
F^{C}}{\partial Q^{B}}(Q,a)$. For example, $G_{CD}(Q^{\ast})\equiv
G_{CD}(F(Q^{\ast},e))$ is defined as
$G_{CD}(Q^{\ast})=F^{M}_{C}(Q^{\ast},a)F^{N}_{D}(Q^{\ast},a)G_{MN}(F(Q^{\ast},a)),$
($e$ is an identity element of the group $\cal G$). In (5), the Killing vector
fields $K_{\mu}$ for the Riemannian metric $G_{AB}(Q)$ are also taken on the
submanifold $\Sigma\equiv\\{{\chi}^{\alpha}=0\\}$, i.e. the components
$K^{A}_{\mu}$ depend on $Q^{\ast}$. By ${\gamma}_{\mu\nu}$, defined as
${\gamma}_{\mu\nu}=K^{A}_{\mu}G_{AB}K^{B}_{\nu}$, we denote the metric given
on the orbit of the group action.
The operator $P_{\perp}(Q^{\ast})$, which projects the vectors onto the
tangent space to the gauge surface $\Sigma$, has the following form:
$(P_{\perp})^{A}_{B}=\delta^{A}_{B}-{\chi}^{\alpha}_{B}(\chi\chi^{\top})^{-1}{}^{\beta}_{\alpha}(\chi^{\top})^{A}_{\beta},$
$(\chi^{\top})^{A}_{\beta}$ is a transposed matrix to the matrix
$\chi^{\nu}_{B}\equiv\frac{\partial\chi^{\nu}}{\partial Q^{B}}$,
$(\chi^{\top})^{A}_{\mu}=G^{AB}{\gamma}_{\mu\nu}\chi^{\nu}_{B}.$
The pseudoinverse matrix ${\tilde{G}}^{\cal A\cal B}(Q^{\ast},a)$ to the
matrix (5) is determined by the equality
$\displaystyle\displaystyle{\tilde{G}}^{\cal A\cal B}{\tilde{G}}_{\cal B\cal
C}=\left(\begin{array}[]{cc}(P_{\perp})^{A}_{C}&0\\\
0&{\delta}^{\alpha}_{\beta}\end{array}\right).$
It follows that ${\tilde{G}}^{\cal A\cal B}$ is equal to
$\displaystyle\left(\begin{array}[]{cc}G^{EF}N^{C}_{E}N^{D}_{F}&G^{SD}N^{C}_{S}{\chi}^{\mu}_{D}(\Phi^{-1})^{\nu}_{\mu}{\bar{v}}^{\sigma}_{\nu}\\\
G^{CB}{\chi}^{\gamma}_{C}(\Phi^{-1})^{\beta}_{\gamma}N^{D}_{B}{\bar{v}}^{\alpha}_{\beta}&G^{CB}{\chi}^{\gamma}_{C}(\Phi^{-1})^{\beta}_{\gamma}{\chi}^{\mu}_{B}(\Phi^{-1})^{\nu}_{\mu}{\bar{v}}^{\alpha}_{\beta}{\bar{v}}^{\sigma}_{\nu}\end{array}\right).$
(7)
The matrix $(\Phi^{-1}){}^{\beta}_{\mu}$ is inverse to the Faddeev – Popov
matrix $\Phi$, which is given by
$(\Phi){}^{\beta}_{\mu}(Q)=K^{A}_{\mu}(Q)\frac{\partial{\chi}^{\beta}(Q)}{\partial
Q^{A}}.$
In (7),
$N^{A}_{C}\equiv{\delta}^{A}_{C}-K^{A}_{\alpha}(\Phi^{-1}){}^{\alpha}_{\mu}{\chi}^{\mu}_{C}$
is a projection operator with the following properties:
$N^{A}_{B}N^{B}_{C}=N^{A}_{C},\,\,\,\,\,N^{A}_{B}K^{B}_{\mu}=0,\,\,\,\,\,(P_{\perp})^{\tilde{A}}_{B}N^{C}_{\tilde{A}}=(P_{\perp})^{C}_{B},\,\,\,\,\,\,\,N^{\tilde{A}}_{B}(P_{\perp})^{C}_{\tilde{A}}=N^{C}_{B}.$
The matrix ${\bar{v}}^{\alpha}_{\beta}(a)$ is an inverse matrix to matrix
${\bar{u}}^{\alpha}_{\beta}(a)$. The $\det{\bar{u}}^{\alpha}_{\beta}(a)$ is a
density of a right invariant measure given on the group $\cal G$.
The determinant of the matrix (5) is equal to
$\displaystyle(\det{\tilde{G}}_{\cal A\cal B})=\det
G_{AB}(Q^{\ast})\det{\gamma}_{\alpha\beta}(Q^{\ast})(\det{\chi}{\chi}^{\top})^{-1}(Q^{\ast})(\det{\bar{u}}^{\mu}_{\nu}(a))^{2}$
$\displaystyle\,\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times(\det{\Phi}^{\alpha}_{\beta}(Q^{\ast}))^{2}\det(P_{\perp})^{C}_{B}(Q^{\ast})$
$\displaystyle\,\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\det\Bigl{(}(P_{\perp})^{D}_{A}\;G^{\rm
H}_{DC}\,(P_{\perp})^{C}_{B}\Bigr{)}\det{\gamma}_{\alpha\beta}\,\,(\det{\bar{u}}^{\mu}_{\nu})^{2},$
where the “horizontal metric” $G^{\rm H}$ is defined by the relation $G^{\rm
H}_{DC}={\Pi}^{\tilde{D}}_{D}\,{\Pi}^{\tilde{C}}_{C}\,G_{{\tilde{D}}{\tilde{C}}}$,
in which
${\Pi}^{A}_{B}={\delta}^{A}_{B}-K^{A}_{\mu}{\gamma}^{\mu\nu}K^{D}_{\nu}G_{DB}$
is the projection operator. (From the definition of ${\Pi}^{A}_{B}$ it follows
that ${\Pi}^{A}_{L}N^{L}_{C}={\Pi}^{A}_{C}$ and
${\Pi}^{L}_{B}N^{A}_{L}=N^{A}_{B}$.)
Note also that $\det{\tilde{G}}_{\cal A\cal B}$ does not vanish only on the
surface $\Sigma$. On this surface $\det(P_{\perp})^{C}_{B}$ is equal to unity.
### 2.2 The semigroup on $\Sigma$ and its path integral representation
Transition to the bundle coordinates on $\cal P$ leads to the replacement of
the local stochastic process ${\eta}^{A}_{t}$ for the process
${\zeta}^{A}_{t}=({Q_{t}^{\ast}}^{A},a^{\alpha}_{t})$.111This phase space
transformation of the stochastic processes does not change the path integral
measures in the evolution semigroups. Instead of the stochastic differential
equation for the process ${\eta}^{A}_{t}$ we get the system of equations for
the processes ${Q_{t}^{\ast}}^{A}$ and $a^{\alpha}_{t}$:
$dQ_{t}^{*}{}^{\small
A}={\mu}^{2}\kappa\biggl{(}-\frac{1}{2}G^{EM}N^{C}_{E}N^{B}_{M}\,{}^{\rm
H}{\Gamma}^{A}_{CB}+j^{\small A}+j^{\small
A}\biggr{)}dt+\mu\sqrt{\kappa}N^{A}_{C}\tilde{\mathfrak{X}}^{C}_{\bar{M}}dw^{\bar{M}}_{t},$
(8) $\displaystyle
da_{t}^{\alpha}=-\frac{1}{2}{\mu}^{2}\kappa\biggl{[}G^{RS}\tilde{\Gamma}^{B}_{RS}(Q^{*}){\Lambda}^{\beta}_{B}{\bar{v}}^{\alpha}_{\beta}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\beta}_{B}K^{B}_{\sigma
P}{\bar{v}}^{\alpha}_{\beta}-G^{CA}N^{M}_{C}{\Lambda}^{\beta}_{AM}{\bar{v}}^{\alpha}_{\beta}$
$\displaystyle-G^{MB}{\Lambda}^{\epsilon}_{M}{\Lambda}^{\beta}_{B}{\bar{v}}^{\nu}_{\epsilon}\frac{\partial}{\partial
a^{\nu}}\bigl{(}{\bar{v}}^{\alpha}_{\beta}\bigr{)}\biggr{]}dt+\mu\sqrt{\kappa}{\bar{v}}^{\alpha}_{\beta}{\Lambda}^{\beta}_{B}\tilde{\mathfrak{X}}^{B}_{\bar{M}}dw_{t}^{\bar{M}}.$
(9)
In these equations, ${\bar{v}}\equiv{\bar{v}}(a)$, and the other coefficients
depend on $Q^{*}$.
In equation (8), ${}^{\rm H}{\Gamma}^{B}_{CD}$ are the Christoffel symbols
defined by the equality
$G^{\rm H}_{AB}\,{}^{\rm H}{\Gamma}^{B}_{CD}=\frac{1}{2}\left(G^{\rm
H}_{AC,D}+G^{\rm H}_{AD,C}-G^{\rm H}_{CD,A}\right),$ (10)
in which by the derivatives we mean the following: $G^{\rm
H}_{AC,D}\equiv\left.{{\partial G^{\rm H}_{AC}(Q)}\over{\partial
Q^{D}}}\right|_{Q=Q^{*}}$. Also, by $j$ we have denoted the mean curvature
vector of the orbit space, and by $j^{A}(Q^{*})$ — the projection of the mean
curvature vector of the orbit onto the submanifold $\Sigma$. This vector can
be defined as
$\displaystyle j^{A}(Q^{*})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,G^{EU}N^{A}_{E}N^{D}_{U}\left[{\gamma}^{\alpha\beta}G_{CD}({\tilde{\nabla}}_{K_{\alpha}}K_{\beta})^{C}\right](Q^{\ast})$
(11) $\displaystyle=$
$\displaystyle-\frac{1}{2}\,N^{A}_{C}\left[{\gamma}^{\alpha\beta}({\tilde{\nabla}}_{K_{\alpha}}K_{\beta})^{C}\right](Q^{\ast}),$
where
$({\tilde{\nabla}}_{K_{\alpha}}K_{\beta})^{C}(Q^{\ast})=K^{A}_{\alpha}(Q^{\ast})\left.\frac{\partial}{\partial
Q^{A}}K^{C}_{\beta}(Q)\right|_{Q=Q^{\ast}}+K^{A}_{\alpha}(Q^{\ast})K^{B}_{\beta}(Q^{\ast}){\tilde{\Gamma}}^{C}_{AB}(Q^{\ast})$
with
${\tilde{\Gamma}}^{C}_{AB}(Q^{\ast})=\frac{1}{2}\
G^{CE}(Q^{\ast})\Bigl{(}\frac{\partial}{\partial{Q^{\ast}}^{A}}G_{EB}(Q^{\ast})+\frac{\partial}{\partial{Q^{\ast}}^{B}}G_{EA}(Q^{\ast})-\frac{\partial}{\partial{Q^{\ast}}^{E}}G_{AB}(Q^{\ast})\Bigr{)}.$
Note also that in equation (9),
${\Lambda}^{\alpha}_{B}=({\Phi}^{-1})^{\alpha}_{\mu}{\chi}^{\mu}_{B}$,
${\Lambda}^{\beta}_{AM}=\frac{\partial}{\partial
Q^{*}{}^{M}}\bigl{(}{\Lambda}^{\beta}_{A}\bigr{)}$, and $K^{B}_{\sigma
P}=\frac{\partial}{\partial Q^{*}{}^{P}}(K^{B}_{\sigma})$.
The superposition of the local semigroup ${\tilde{U}}_{\zeta}$, together with
a subsequent limiting procedure, gives the global semigroup determined on the
submanifold $\Sigma$.
Our next transformation in the path integral reduction procedure, performed in
[9], was related to the factorization of the path integral measure generated
by the process $\zeta_{t}$. First of all, it was done in the path integrals
for the local evolution semigroups. In each semigroup, we have separated the
local evolution, given on the orbit of the group action, from the evolution on
the orbit space. Then, we extended the factorization onto the global semigroup
by taking an appropriate limit in the superposition of new-obtained local
semigroups.
In case of the reduction onto non-zero momentum level, that is when
$\lambda\neq 0$, it have led us to the integral relation between the path
integrals for the Green’s functions defined on the global manifolds $\Sigma$
and $\cal P$:
$G^{\lambda}_{pq}(Q^{*}_{b},t_{b};Q^{*}_{a},t_{a})=\displaystyle\int_{\cal
G}G_{\cal
P}(p_{b}\theta,t_{b};p_{a},t_{a})D_{qp}^{\lambda}(\theta)d\mu(\theta),\;\;\;(Q^{*}=\pi_{\Sigma}(p)).$
(12)
Here $D^{\lambda}_{pq}(a)$ are the matrix elements of an irreducible
representation $T^{\lambda}$ of a group $\cal G$:
$\sum_{q}D_{pq}^{\lambda}(a)D_{qn}^{\lambda}(b)=D_{pn}^{\lambda}(ab)$.
The Green’s function ${G}_{\cal P}(Q_{b},t_{b};Q_{a},t_{a})$ is defined222We
have assumed that equation (1) has a fundamental solution. by semigroup (4):
$\psi(Q_{a},t_{a})=\int{G}_{\cal
P}(Q_{b},t_{b};Q_{a},t_{a})\,\varphi_{0}(Q_{b})\,dv_{\cal P}(Q_{b})$
($dv_{\cal P}(Q)=\sqrt{G(Q)}\,dQ^{1}\cdot\dots\cdot dQ^{N_{\cal P}})$.
The probability representation of the kernel ${G}_{\cal
P}(Q_{b},t_{b};Q_{a},t_{a})$ of the semigroup (4) (the path integral for
${G}_{\cal P}$) may be obtained from the path integral (4) by choosing
$\varphi_{0}(Q)=G^{-1/2}(Q)\,\delta(Q-Q^{\prime})$ as an initial function.
The Green’s function $G^{\lambda}_{pq}$ is presented by the following path
integral
$\displaystyle G^{\lambda}_{pq}(Q^{*}_{b},t_{b};Q^{*}_{a},t_{a})=$
$\displaystyle{\tilde{\rm
E}}_{{\xi_{\Sigma}(t_{a})=Q^{*}_{a}}\atop{\xi_{\Sigma}(t_{b})=Q^{*}_{b}}}\left[(\overleftarrow{\exp})_{mn}^{\lambda}(\xi_{\Sigma}(t),t_{b},t_{a})\exp\left\\{\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}\tilde{V}(\xi_{\Sigma}(u))du\right\\}\right]$
$\displaystyle=\int\limits_{{\xi_{\Sigma}(t_{a})=Q^{*}_{a}\atop{\xi_{\Sigma}(t_{b})=Q^{*}_{b}}}}d{\mu}^{{\xi}_{\Sigma}}\exp\left\\{\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}\tilde{V}(\xi_{\Sigma}(u))du\right\\}$
$\displaystyle\times\overleftarrow{\exp}\int_{t_{a}}^{t_{b}}\Bigl{\\{}\frac{1}{2}{\mu}^{2}\kappa\Bigl{[}{\gamma}^{\sigma\nu}(\xi_{\Sigma}(u))(J_{\sigma})_{pr}^{\lambda}(J_{\nu})_{rq}^{\lambda}$
$\displaystyle-\bigl{(}G^{RS}\tilde{\Gamma}^{B}_{RS}{\Lambda}^{\beta}_{B}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\beta}_{B}K^{B}_{\sigma
P}-G^{CA}N^{M}_{C}{\Lambda}^{\beta}_{AM}\bigr{)}\,\,(J_{\beta})_{pq}^{\lambda}\Bigr{]}du$
$\displaystyle+\mu\sqrt{\kappa}{\Lambda}^{\beta}_{C}(J_{\beta})_{pq}^{\lambda}{\Pi}^{C}_{K}\tilde{\mathfrak{X}}^{K}_{\bar{M}}dw^{\bar{M}}(u)\Bigr{\\}}.$
(13)
The measure in this path integral is generated by the global stochastic
process ${\xi}_{{\Sigma}}(t)$ given on the submanifold $\Sigma$. This process
is described locally by equations (8).
In equation (13), $\overleftarrow{\exp}(...)_{pq}^{\lambda}$ is a
multiplicative stochastic integral. This integral is a limit of the sequence
of time–ordered multipliers that have been obtained as a result of breaking of
a time interval $[s,t]$, $[s=t_{0}\leq t_{1}\ldots\leq t_{n}=t]$. The time
order of these multipliers is indicated by the arrow directed to the
multipliers given at greater times. We note that, by definition, a
multiplicative stochastic integral represents the solution of the linear
matrix stochastic differential equation.
On the right-hand side of (13), by
$\left.(J_{\mu})_{pq}^{\lambda}\equiv(\frac{\partial
D_{pq}^{\lambda}(a)}{\partial a^{\mu}})\right|_{a=e}$ we denoted the
infinitesimal generators of the representation $D^{\lambda}(a)$:
$\bar{L}_{\mu}D_{pq}^{\lambda}(a)=\sum_{q^{\prime}}(J_{\mu})_{pq^{\prime}}^{\lambda}D_{q^{\prime}q}^{\lambda}(a)$
($\bar{L}_{\mu}={\bar{v}}^{\alpha}_{\beta}(a)\frac{\partial}{\partial
a^{\mu}}$ is a right-invariant vector field).
The differential generator (the Hamiltonian operator) of the matrix semigroup
with the kernel (13) is
$\displaystyle\frac{1}{2}\mu^{2}\kappa\left\\{\left[G^{CD}N^{A}_{C}N^{B}_{D}\frac{{\partial}^{2}}{\partial
Q^{*}{}^{A}\partial
Q^{*}{}^{B}}-G^{CD}N^{E}_{C}N^{M}_{D}\,{}^{H}{\Gamma}^{A}_{EM}\frac{\partial}{\partial
Q^{*}{}^{A}}\right.\right.$
$\displaystyle+\left.2\left(j^{A}+j^{A}\right)\frac{\partial}{\partial
Q^{*}{}^{A}}+\frac{2{\tilde{V}}}{(\mu^{2}\kappa)^{2}m}\right](I^{\lambda})_{pq}+2N^{A}_{C}G^{CP}{\Lambda}^{\alpha}_{P}(J_{\alpha})_{pq}^{\lambda}\frac{\partial}{\partial
Q^{*}{}^{A}}$
$\displaystyle-\left(G^{RS}{\tilde{\Gamma}}^{B}_{RS}{\Lambda}^{\alpha}_{B}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\alpha}_{B}K^{B}_{\sigma
P}-G^{CA}N^{M}_{C}{\Lambda}^{\alpha}_{AM})\right)(J_{\alpha})_{pq}^{\lambda}$
$\displaystyle+\biggl{.}G^{SB}{\Lambda}^{\alpha}_{B}{\Lambda}^{\sigma}_{S}(J_{\alpha})_{pq^{\prime}}^{\lambda}(J_{\sigma})_{q^{\prime}q}^{\lambda}\biggr{\\}},$
(14)
where $(I^{\lambda})_{pq}$ is a unity matrix.
The operator acts in the space of the sections
${\Gamma}(\Sigma,V^{*}_{\lambda})$ of the associated covector bundle with the
scalar product333Another form of this scalar product is as follows
$(\psi_{n},\psi_{m})=\\!\\!\int\langle\psi_{n},\psi_{m}{\rangle}_{V^{\ast}_{\lambda}}\det{\Phi}^{\alpha}_{\beta}\prod_{\alpha=1}^{N_{\cal
G}}\delta({\chi}^{\alpha}(Q^{*})){\det}^{1/2}G_{AB}\,dQ^{*}{}^{1}\wedge\dots\wedge
dQ^{*}{}^{N_{\cal P}}.$
$\displaystyle(\psi_{n},\psi_{m})$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\langle\psi_{n},\psi_{m}{\rangle}_{V^{\ast}_{\lambda}}\,{\det}^{1/2}\bigl{(}(P_{\perp})^{D}_{A}\;G^{\rm
H}_{DC}\,(P_{\perp})^{C}_{B}\bigr{)}\,{\det}^{1/2}{\gamma}_{\alpha\beta}$ (15)
$\displaystyle\times\,dQ^{*1}\wedge\ldots\wedge dQ^{*N_{\cal P}}.$
${\Gamma}(\Sigma,V^{*}_{\lambda})$ is isomorphic to the space of the
equivariant functions on $\cal P$. The isomorphism between the functions
${\tilde{\psi}}_{n}(p)$, such that
${\tilde{\psi}}_{n}(pg)=D_{mn}^{\lambda}(g){\tilde{\psi}}_{m}(p),$
is given by the following equality:
$\;\;{\tilde{\psi}}_{n}(F(Q^{*},e))={\psi}_{n}(Q^{*})$.
## 3 Girsanov transformation
In the case of the reduction onto the zero momentum level, our goal is to
obtain the description of true evolution on the orbit space $\cal M$ in terms
of the evolution given on an additional gauge surface $\Sigma$. By true
evolution we mean such a diffusion on $\cal M$ which has the Laplace—Beltrami
operator as a differential generator.
A required correspondence between the diffusion on $\cal M$ and the diffusion
on $\Sigma$ can be achieved only in that case when the stochastic process
$\tilde{\xi}_{\Sigma}$ related to the diffusion on $\Sigma$ is described by
the stochastic differential equations, which look as equations (8), but
without the “$j$-term” in the drift:
$dQ^{*}_{t}{}^{\small
A}={\mu}^{2}\kappa\biggl{(}-\frac{1}{2}G^{EM}N^{C}_{E}N^{B}_{M}\,{}^{H}{\Gamma}^{A}_{CB}+j^{\small
A}\biggr{)}dt+\mu\sqrt{\kappa}N^{A}_{C}\tilde{\mathfrak{X}}^{C}_{\bar{M}}dw_{t}^{\bar{M}}.$
(16)
Note that in case of the reduction onto the zero-momentum level, the
differential generator of the process $\tilde{\xi}_{\Sigma}$ could be
transformed into the Laplace—Beltrami operator (a differential generator of
the process on $\cal M$), if we succeded in finding the independent variables
that parametrize $\Sigma$.
In the same way, in order to come to the correct description of the reduced
diffusion on $\cal M$ for the reduction onto the non-zero momentum level, we
should properly transform the semigroup, given by the kernel (13).
In the path integral (13), such a transformation, in which we perform the
transition to the process $\tilde{\xi}_{\Sigma}$ with the local stochastic
differential equations (16) from the process $\xi_{\Sigma}$ defined by the
equation (8), is known as the Girsanov transformation. In spite of the fact
that in the equations (13) and (16), the diffusion coefficients are
degenerated, the Girsanov transformation formula can be nevertheless derived
by making use of the Itô’s differentiation formula for the composite function.
It is necessary only to take into account the predefined ambiguities, which
exist in the problem.
When we deal with the system of the linear parabolic differential equations,
as in our case, the multiplicative stochastic integral should be also involved
in the Girsanov transformation. Assuming a new form of this integral for the
process $\tilde{\xi}_{\Sigma}$, we compare the differential generators for the
processes $\xi_{\Sigma}$ and $\tilde{\xi}_{\Sigma}$. The existence and
uniqueness solution theorem for the the system of the differential equations
allows us to determine a new multiplicative stochastic integral for the
process $\tilde{\xi}_{\Sigma}$.
After lengthy calculation which we omit for brevity and because of its
resemblance to the calculation performed in [9, 11] for $\lambda=0$ case, we
come to the following expression for the multiplicative stochastic integral:
$\displaystyle\overleftarrow{\exp}(...)^{\lambda}_{pq}(\tilde{\xi}_{\Sigma}(t))=\overleftarrow{\exp}\int_{t_{a}}^{t}\Bigl{\\{}\frac{1}{2}{\mu}^{2}\kappa\Bigl{[}{\gamma}^{\sigma\nu}\,(J_{\sigma})_{pr}^{\lambda}(J_{\nu})_{rq}^{\lambda}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\,G^{\rm
H}_{LK}(P_{\bot})^{L}_{A}(P_{\bot})^{K}_{E}j^{A}j^{E}{I}^{\lambda}_{pq}-2{\Pi}^{C}_{L}j^{L}{\Lambda}^{\alpha}_{C}(J_{\alpha})_{pq}^{\lambda}$
$\displaystyle\;\;\;\;\;-\Bigl{(}G^{RS}\tilde{\Gamma}^{B}_{RS}{\Lambda}^{\beta}_{B}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\beta}_{B}K^{B}_{\sigma
P}-G^{CA}N^{M}_{C}{\Lambda}^{\beta}_{A,M}\Bigr{)}(J_{\beta})_{pq}^{\lambda}\Bigr{]}du$
$\displaystyle\;\;\;\;\;\;\;\;+\mu\sqrt{\kappa}\Bigl{[}G^{\rm
H}_{KL}\,({P}_{\bot})^{L}_{A}\,j^{A}\,{I}^{\lambda}_{pq}+{\Pi}^{C}_{K}\,{\Lambda}^{\beta}_{C}\,(J_{\beta})_{pq}^{\lambda}\,\Bigr{]}\tilde{\mathfrak{X}}^{K}_{\bar{M}}dw^{\bar{M}}_{u}\Bigr{\\}}.$
(17)
We note that terms that are proportional to ${I}^{\lambda}_{pq}$ can be factor
out of the multiplicative stochastic integral. Hence the right-hand side of
(17) can be presented as a product of two factors:
$\displaystyle\overleftarrow{\exp}(...)^{\lambda}_{pq}(\tilde{\xi}_{\Sigma}(t))=\exp\int^{t}_{t_{a}}\left[-\frac{1}{2}{\mu}^{2}\kappa\left((P_{\bot})^{L}_{A}G^{H}_{LK}(P_{\bot})^{K}_{E}\right)j^{A}j^{E}du\right.$
$\displaystyle\left.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\mu\sqrt{\kappa}G^{H}_{LK}(P_{\bot})^{L}_{A}j^{A}\tilde{\mathfrak{X}}^{K}_{\bar{M}}dw^{\bar{M}}_{u}\right]{I}^{\lambda}_{pq^{\prime}}$
$\displaystyle\times\overleftarrow{\exp}\int_{t_{a}}^{t}\Bigl{\\{}\frac{1}{2}{\mu}^{2}\kappa\Bigl{[}{\gamma}^{\sigma\nu}\,(J_{\sigma})_{q^{\prime}r}^{\lambda}(J_{\nu})_{rq}^{\lambda}-2{\Pi}^{C}_{L}j^{L}{\Lambda}^{\alpha}_{C}(J_{\alpha})_{q^{\prime}q}^{\lambda}$
$\displaystyle\;\;\;\;\;-\Bigl{(}G^{RS}\tilde{\Gamma}^{B}_{RS}{\Lambda}^{\beta}_{B}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\beta}_{B}K^{B}_{\sigma
P}-G^{CA}N^{M}_{C}{\Lambda}^{\beta}_{A,M}\Bigr{)}(J_{\beta})_{q^{\prime}q}^{\lambda}\Bigr{]}du$
$\displaystyle\;\;\;\;\;\;\;\;+\mu\sqrt{\kappa}\,{\Pi}^{C}_{K}{\Lambda}^{\beta}_{C}\,(J_{\beta})_{q^{\prime}q}^{\lambda}\,\tilde{\mathfrak{X}}^{K}_{\bar{M}}dw^{\bar{M}}_{u}\Bigr{\\}}.$
(18)
The first factor of (18) coinsides with the path integral reduction Jacobian
for the $\lambda=0$ case. It was obtained in [11] in the following way.
We first rewrote the exponential of the Jacobian for getting rid of the
stochastic integral: the stochastic integral was replaced by an ordinary
integral taken with respect to the time variable. It was made with the help of
the Itô’s identity. Then it was obtained the geometrical representation of the
Jacobian:
$\displaystyle\Bigl{(}\frac{{\gamma}(Q^{\ast}(t_{b}))}{{\gamma}(Q^{\ast}(t_{a}))}\Bigr{)}^{\frac{1}{4}}{\exp}\Bigl{\\{}-\frac{1}{8}{\mu}^{2}{\kappa}\int\limits_{t_{a}}^{t_{b}}{\tilde{J}}dt\Bigr{\\}},$
(19)
where the integrand $\tilde{J}$ is equal to
${\tilde{J}}=R_{\mathcal{P}}-{}^{\rm
H}R-R_{\mathcal{G}}-\frac{1}{4}{\mathcal{F}}^{2}-||j||^{2}.$ (20)
In this expression, $R_{\mathcal{P}}$ is a scalar curvature of the original
manifold $\mathcal{P}$. ${}^{\rm H}R$ is a scalar curvature of the manifold
with the degenerated metric $G^{\rm H}_{AB}$. More exactly,
${}^{\rm H}R\equiv
G^{A^{\prime}C^{\prime}}N^{S}_{A^{\prime}}N^{C}_{C^{\prime}}N^{E}_{M}\,{}^{\rm
H}R_{SEC}^{\;\;\;\;\;\;\;\;M},$
where $N^{S}_{A}N^{E}_{M}\,{}^{\rm H}R^{\;\;\;\;\;\;\;M}_{SEC}$ is equal to
$N^{S}_{A}N^{E}_{M}\left(\frac{\partial}{\partial Q^{\ast}{}^{S}}{}^{\rm
H}{\Gamma}^{M}_{CE}-\frac{\partial}{\partial Q^{\ast}{}^{E}}{}^{\rm
H}{\Gamma}^{M}_{CS}+{}^{\rm H}{\Gamma}^{K}_{CE}\,{}^{\rm
H}{\Gamma}^{M}_{KS}-{}^{\rm H}{\Gamma}^{P}_{CS}\,{}^{\rm
H}{\Gamma}^{M}_{PE}\right).$
$R_{\mathrm{\mathcal{G}}}$ is the scalar curvature of the orbit:
$R_{\mathrm{\mathcal{G}}}\equiv\frac{1}{2}{\gamma}^{\mu\nu}c^{\sigma}_{\mu\alpha}c^{\alpha}_{\nu\sigma}+\frac{1}{4}{\gamma}_{\mu\sigma}{\gamma}^{\alpha\beta}{\gamma}^{\epsilon\nu}c^{\mu}_{\epsilon\alpha}c^{\sigma}_{\nu\beta}.$
By ${\mathcal{F}}^{2}$ we denote the following expression:
${\mathcal{F}}^{2}\equiv\bigl{(}G^{ES}N^{F}_{S}N^{B}_{E}\bigr{)}\,\bigl{(}G^{MQ}N^{P}_{M}N^{A}_{Q}\bigr{)}\,{\gamma}_{\mu\nu}\,{\mathcal{F}}^{\mu}_{PF}{\mathcal{F}}^{\nu}_{AB},$
in which the curvature ${\mathcal{F}}^{\alpha}_{EP}$ of the connection
${\mathscr{A}}^{\nu}_{P}={\gamma}^{\nu\mu}K^{R}_{\mu}\,G_{RP}$ 444In the case
of the reduction, this connection is naturally defined on the principal fiber
bundle. is given by
${\mathcal{F}}^{\alpha}_{EP}=\displaystyle\frac{\partial}{\partial
Q^{\ast}{}^{E}}\,{\mathscr{A}}^{\alpha}_{P}-\frac{\partial}{\partial{Q^{\ast}}^{P}}\,{\mathscr{A}}^{\alpha}_{E}+c^{\alpha}_{\nu\sigma}\,{\mathscr{A}}^{\nu}_{E}\,{\mathscr{A}}^{\sigma}_{P}.$
The last term of (20) , the “square” of the fundamental form of the orbit, is
$||j||^{2}=G^{\rm
H}_{AB}\,{\gamma}^{\alpha\mu}\,{\gamma}^{\beta\nu}\,j^{A}_{\alpha\beta}\,j^{B}_{\mu\nu}\,,$
where
$j^{B}_{\alpha\beta}(Q^{\ast})=-\frac{1}{2}G^{PS}N^{B}_{P}N^{E}_{S}\,\bigl{(}{\mathcal{D}}_{E}{\gamma}_{\alpha\beta}\bigr{)}(Q^{\ast})$
with
${\mathscr{D}}_{E}{\gamma}_{\alpha\beta}=\Bigl{(}\frac{\partial}{\partial
Q^{\ast}{}^{E}}{\gamma}_{\alpha\beta}-c^{\sigma}_{\mu\alpha}{\mathscr{A}}^{\mu}_{E}{\gamma}_{\sigma\beta}-c^{\sigma}_{\mu\beta}{\mathscr{A}}^{\mu}_{E}{\gamma}_{\sigma\alpha}\,\Bigr{)}.$
To obtain $j^{B}_{\alpha\beta}(Q^{\ast})$ we have projected the second
fundamental form $j^{C}_{\alpha\beta}(Q)$ of the orbit onto the direction
which is parallel with the orbit space. In other words, we calculated the
following expression:
${\tilde{G}}^{AB}{\tilde{G}}\left({\Pi}^{C}_{D}(Q)\bigl{(}{\nabla}_{K_{\alpha}}K_{\beta}\bigr{)}^{D}\frac{\partial}{\partial
Q^{C}},\frac{\partial}{\partial Q^{\ast}{}^{A}}\right)$, where $\tilde{G}$ was
the metric of the manifold $\cal P$.
Therefore, the Girsanov transformation allows us to rewrite the integral
relation (12) as follows
$\bigl{(}{\gamma}(Q^{*}_{b})\,{\gamma}(Q^{*}_{a})\bigr{)}^{-1/4}\,{\tilde{G}}^{\lambda}_{pq}(Q^{*}_{b},t_{b};Q^{*}_{a},t_{a})=\displaystyle\int_{\cal
G}G_{\cal
P}(p_{b}\theta,t_{b};p_{a},t_{a})D_{qp}^{\lambda}(\theta)d\mu(\theta),$
where the Green’s function ${\tilde{G}}^{\lambda}_{pq}$ is given by the
following path integral
$\displaystyle{\tilde{G}}^{\lambda}_{pq}(Q^{*}_{b},t_{b};Q^{*}_{a},t_{a})=\int\limits_{{{\tilde{\tilde{\xi}}}_{\Sigma}(t_{a})=Q^{*}_{a}\atop{{\tilde{\xi}}_{\Sigma}(t_{b})=Q^{*}_{b}}}}d{\mu}^{{{\tilde{\xi}}}_{\Sigma}}\exp\biggl{\\{}\int_{t_{a}}^{t_{b}}\Bigl{(}\frac{\tilde{V}({\tilde{\xi}}_{\Sigma}(u))}{\mu^{2}\kappa
m}-\frac{1}{8}\mu^{2}\kappa{\tilde{J}}\Bigr{)}du\biggr{\\}}$
$\displaystyle\times\overleftarrow{\exp}\int_{t_{a}}^{t_{b}}\Bigl{\\{}\frac{1}{2}{\mu}^{2}\kappa\Bigl{[}{\gamma}^{\sigma\nu}\,(J_{\sigma})_{pr}^{\lambda}(J_{\nu})_{rq}^{\lambda}-2{\Pi}^{C}_{L}j^{L}{\Lambda}^{\alpha}_{C}(J_{\alpha})_{pq}^{\lambda}$
$\displaystyle\;\;\;\;\;-\Bigl{(}G^{RS}\tilde{\Gamma}^{B}_{RS}{\Lambda}^{\beta}_{B}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\beta}_{B}K^{B}_{\sigma
P}-G^{CA}N^{M}_{C}{\Lambda}^{\beta}_{A,M}\Bigr{)}(J_{\beta})_{pq}^{\lambda}\Bigr{]}du$
$\displaystyle\;\;\;\;\;\;\;\;+\mu\sqrt{\kappa}\,{\Pi}^{C}_{K}{\Lambda}^{\beta}_{C}\,(J_{\beta})_{pq}^{\lambda}\,\tilde{\mathfrak{X}}^{K}_{\bar{M}}dw^{\bar{M}}_{u}\Bigr{\\}}.$
(21)
${\tilde{G}}^{\lambda}_{pq}$ is the kernel of the evolution semigroup which
describes the true reduced evolution on the orbit space $\cal M$. This
semigroup acts in the space of sections ${\Gamma}(\Sigma,V^{*}_{\lambda})$ of
the associated covector bundle $P\times_{\mathcal{G}}V^{*}_{\lambda}$ with the
following scalar product:
$\displaystyle(\psi_{n},\psi_{m})$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\langle\psi_{n},\psi_{m}{\rangle}_{V^{\ast}_{\lambda}}\,{\det}^{1/2}\bigl{(}(P_{\perp})^{D}_{A}\;G^{\rm
H}_{DC}\,(P_{\perp})^{C}_{B}\bigr{)}\,$ (22)
$\displaystyle\times\,dQ^{*1}\wedge\ldots\wedge dQ^{*N_{\cal P}}.$
The differential generator of the matrix semigroup with the kernel
${\tilde{G}}^{\lambda}_{pq}$ is
$\displaystyle\frac{1}{2}\mu^{2}\kappa\left\\{\left[G^{CD}N^{A}_{C}N^{B}_{D}\frac{{\partial}^{2}}{\partial
Q^{*}{}^{A}\partial
Q^{*}{}^{B}}-G^{CD}N^{E}_{C}N^{M}_{D}\,{}^{H}{\Gamma}^{A}_{EM}\frac{\partial}{\partial
Q^{*}{}^{A}}\right.\right.$
$\displaystyle+\left.2j^{A}\frac{\partial}{\partial
Q^{*}{}^{A}}+\frac{2{\tilde{V}}}{(\mu^{2}\kappa)^{2}m}-\frac{1}{4}{\tilde{J}}\right](I^{\lambda})_{pq}+2N^{A}_{C}G^{CP}{\Lambda}^{\alpha}_{P}(J_{\alpha})_{pq}^{\lambda}\frac{\partial}{\partial
Q^{*}{}^{A}}$
$\displaystyle-\left(G^{RS}{\tilde{\Gamma}}^{B}_{RS}{\Lambda}^{\alpha}_{B}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\alpha}_{B}K^{B}_{\sigma
P}-G^{CA}N^{M}_{C}{\Lambda}^{\alpha}_{AM}\right)(J_{\alpha})_{pq}^{\lambda}$
$\displaystyle+{\Lambda}^{\alpha}_{C}{\gamma}^{\mu\nu}[{\triangledown}_{K_{\mu}}K_{\nu}]^{C}(J_{\alpha})_{pq}^{\lambda}+\biggl{.}G^{SB}{\Lambda}^{\alpha}_{B}{\Lambda}^{\sigma}_{S}(J_{\alpha})_{pq^{\prime}}^{\lambda}(J_{\sigma})_{q^{\prime}q}^{\lambda}\biggr{\\}}.$
(23)
The first term of the last line in (23) comes from
$(-2{\Pi}^{C}_{L}j^{L}{\Lambda}^{\alpha}_{C}$) - term of the multiplicative
stochastic integral given in (21). Its derivation is based on the following
relations:
${\Pi}^{R}_{C}{\Lambda}^{\alpha}_{R}={\Lambda}^{\alpha}_{C}-{\mathscr{A}}^{\alpha}_{C},\quad{\mathscr{A}}^{\alpha}_{C}\,{\gamma}^{\mu\nu}[{\triangledown}_{K_{\mu}}K_{\nu}]^{C}=0.$
In the next section we will obtain another representation for the
multiplicative stochastic integral. For this purpose, it is sufficient to
consider the transformation of the differential operator (23), since there
exists a quite definite relationship between the integrand of the path
integral and the corresponding differential generator.
## 4 The horizontal Laplacian
It is well-known that the horizontal Laplacian ${\triangle}^{{\mathcal{E}}}$
$\displaystyle\left({\triangle}^{{\mathcal{E}}}\right)^{\lambda}_{pq}$
$\displaystyle=$ $\displaystyle{\sum}_{\bar{k}=1}^{n_{\cal
M}}\left({\nabla}^{\mathcal{E}}_{X^{i}_{\bar{k}}{\rm
e_{i}}}{\nabla}^{\mathcal{E}}_{X^{j}_{\bar{k}}{\rm
e_{j}}}-{\nabla}^{\mathcal{E}}_{{\nabla}^{\mathcal{M}}_{X^{i}_{\bar{k}}{\rm
e_{i}}}{X^{j}_{\bar{k}}{\rm e_{j}}}}\right)^{\\!\\!\lambda}_{\\!\\!pq}$
$\displaystyle=$ $\displaystyle{\triangle}_{\cal M}\,{\rm
I}^{\lambda}_{pq}+2h^{ij}({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{ipq}\,{\partial}_{j}$
$\displaystyle-\,h^{ij}\left[{\partial}_{i}({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{jpq}-({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{ip{q}^{{}^{\prime}}}({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{j{q}^{{}^{\prime}}q}+({\rm{\Gamma}^{\cal
M}})^{m}_{ij}\,({\rm{\Gamma}}^{\mathcal{E}})^{\lambda}_{mpq}\right],$
determined on the space of the sections of the associated vector bundle
$\mathcal{E}=P\times_{\mathcal{G}}V_{\lambda}$, is an invariant operator which
can be considered as a generalization of the Lapalace–Beltrami operator given
on the base manifold $\mathcal{M}$. It would be naturally to expect that in
the case of description of the evolution by means of dependent variables,
there is also a corresponding operators which may be refer to as the
horizontal Laplacian.
For the covector bundle, such an operator may be given by the following
expression:
$\displaystyle\\!\\!\\!\\!\\!\left({\triangle}^{{\mathcal{E}^{*}}}\right)^{\lambda}_{pq}={\sum}_{\bar{\scriptscriptstyle
M}=1}^{n_{\cal P}}\left({\nabla}^{\mathcal{E}^{*}}_{Y^{A}_{\bar{M}}{\rm
e_{A}}}{\nabla}^{\mathcal{E}^{*}}_{Y^{B}_{\bar{M}}{\rm
e_{B}}}-{\nabla}^{\mathcal{E}^{*}}_{{\nabla}^{\mathcal{M}}_{Y^{A}_{\bar{M}}{\rm
e_{A}}}{Y^{B}_{\bar{M}}{\rm e_{B}}}}\right)^{\\!\\!\lambda}_{\\!\\!pq}$
$\displaystyle\\!\\!\\!\\!\\!\\!={\triangle}_{\cal M}\,{\rm
I}^{\lambda}_{pq}-2\,G^{LM}N^{E}_{L}N^{C}_{M}\,({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{Epq}\,{\partial}_{Q^{*C}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!-G^{LM}N^{E}_{L}N^{B}_{M}\left[{\partial}_{Q^{*}{}^{E}}(N^{C}_{B}({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{Cpq})-({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{E\,p{q}^{{}^{\prime}}}({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{B{q}^{{}^{\prime}}q}-{}^{\rm
H}{\rm{\Gamma}}^{C}_{EB}N^{D}_{C}({\rm{\Gamma}}^{\mathcal{E}})^{\lambda}_{Dpq}\right],$
in which ${Y}^{A}_{\bar{M}}=N^{A}_{P}{\mathfrak{X}}^{P}_{\bar{M}}$ is defined
by the equality $\sum^{n_{P}}_{\bar{{\scriptscriptstyle
M}}\scriptscriptstyle=1}Y_{\bar{M}}^{A}Y_{\bar{M}}^{B}=G^{PR}N^{A}_{P}N^{B}_{R}$
and where
$({\rm{\Gamma}}^{\mathcal{E}})^{\lambda}_{Bpq}={\mathscr{A}}^{\alpha}_{B}\,(J_{\alpha})^{\lambda}_{pq}$.
The covariant derivative ${\nabla}^{{\pi}^{*}}$ is defined as
${\nabla}^{{\pi}^{*}}_{{\rm e}_{B}}u_{p}=N^{D}_{B}\left({\rm
I}^{\lambda}_{pq}\frac{\partial}{\partial
Q^{*D}}-{\mathscr{A}}^{\alpha}_{D}(J_{\alpha})^{\lambda}_{pq}\right)u_{q},$
and
${\nabla}^{\scriptscriptstyle{\mathcal{M}}}_{\rm e_{A}}\,{\rm
e_{B}}={}^{H}{\Gamma}^{C}_{AB}\,{\rm e_{C}}.$
The horizontal Laplacian ${\triangle}^{{\mathcal{E}^{*}}}$ can be also written
as follows
$\displaystyle
G^{LM}N^{E}_{L}N^{C}_{M}\left\\{\left(\frac{{\partial}^{2}}{{\partial
Q^{*E}}{\partial Q^{*C}}}+\frac{\partial}{\partial
Q^{*C}}(N^{B}_{E})\frac{\partial}{\partial Q^{*B}}-\,{}^{\scriptstyle{\rm
H}}{\Gamma}^{B}_{EC}N^{D}_{B}\frac{\partial}{\partial Q^{*D}}\right){\rm
I}^{\lambda}_{pq}\right.$
$\displaystyle\left(-2{\mathscr{A}}^{\alpha}_{E}\frac{\partial}{\partial
Q^{*C}}-{\mathscr{A}}^{\alpha}_{B}\frac{\partial}{\partial
Q^{*E}}(N^{B}_{C})-\frac{\partial}{\partial
Q^{*E}}({\mathscr{A}}^{\alpha}_{C})+\,{}^{\scriptstyle{\rm
H}}{\Gamma}^{B}_{EC}N^{D}_{B}{\mathscr{A}}^{\alpha}_{D}\right)(J_{\alpha})^{\lambda}_{pq}$
$\displaystyle\left.+({\mathscr{A}}^{\beta}_{E}J_{\beta})^{\lambda}_{pq^{{}^{\prime}}}({\mathscr{A}}^{\alpha}_{C}J_{\alpha})^{\lambda}_{q^{{}^{\prime}}q}\right\\}.$
(24)
It turns out, that operator (24) is intrinsically related to the the operator
(23).
First note that diagonal parts of these operator (without taking into account
the potential terms $\tilde{V}$ and $\tilde{J}$ in (23)) are equal. It may be
checked with the help of the following identity
$\displaystyle-\frac{1}{2}N^{A}_{A^{\prime}}\,{}^{\scriptstyle{\rm
H}}{\Gamma}^{A^{\prime}}_{CD}\,N^{C}_{C^{\prime}}N^{D}_{D^{\prime}}G^{C^{\prime}D^{\prime}}+\frac{1}{2}N^{A}_{LM}\,N^{L}_{L^{\prime}}N^{M}_{M^{\prime}}\,G^{L^{\prime}M^{\prime}}$
$\displaystyle=-\frac{1}{2}G^{EM}N^{C}_{E}N^{B}_{M}\,{}^{\scriptstyle{\rm
H}}{\Gamma}^{A}_{CB}+j^{A}.$
The off-diagonal matrix elements of the operators (23) and (24), that include
the generator $(J_{\alpha})^{\lambda}_{pq}$, are also equal. In order to show
this, in the operator (23), one should rewrite such terms in the following way
$-\left({}^{\perp}G^{RS}{\tilde{\Gamma}}^{P}_{RS}{\Lambda}^{\alpha}_{P}+G^{RP}{\Lambda}^{\sigma}_{R}{\Lambda}^{\alpha}_{B}K^{B}_{\sigma
P}-G^{CA}N^{M}_{C}{\Lambda}^{\alpha}_{AM}-{\gamma}^{\mu\sigma}{\Lambda}^{\alpha}_{P}K^{A}_{\mu}K^{P}_{\sigma
A}\right),$ (25)
where ${}^{\perp}G^{RS}=G^{RS}-K^{R}_{\mu}{\gamma}^{\mu\nu}K^{S}_{\nu}$, and
the analagous terms of the operator (24) as follows
$-G^{PQ}N^{E}_{P}N^{B}_{Q}N^{C}_{BE}{\mathscr{A}}^{\alpha}_{C}-G^{PQ}N^{E}_{P}N^{C}_{Q}\frac{\partial}{\partial
Q^{*E}}({\mathscr{A}}^{\alpha}_{C})+G^{PQ}N^{A}_{P}N^{C}_{Q}\,{}^{\rm
H}{\Gamma}^{B}_{AC}N^{D}_{B}{\mathscr{A}}^{\alpha}_{D}.$ (26)
Replacing the term, which involve the derivative of
${\mathscr{A}}^{\alpha}_{C}$, with the expression
$\displaystyle G^{LM}N^{E}_{L}N^{B}_{M}\frac{\partial}{\partial
Q^{*}{}^{E}}\,({\mathscr{A}}^{\alpha}_{B})=N^{E}_{R}\,{\Gamma}^{R}_{ES}{\gamma}^{\alpha\sigma}K^{S}_{\sigma}+G^{PB}N^{E}_{P}{\gamma}^{\alpha\sigma}K^{S}_{\sigma}\,{\Gamma}^{R}_{EB}G_{RS}$
$\displaystyle\qquad\qquad+N^{E}_{P}{\gamma}^{\alpha\sigma}K^{P}_{\sigma
E}+G^{LM}N^{E}_{L}{\Lambda}^{\sigma}_{M}{\gamma}^{\alpha\mu}K^{C}_{\mu}G_{CD}K^{D}_{\sigma
E}$
and making use of the identity
$\displaystyle
N^{A}_{\tilde{A}}\,\,{}^{H}{\Gamma}^{\tilde{A}}_{CD}\,N^{C}_{\tilde{C}}N^{D}_{\tilde{D}}\,G^{\tilde{C}\tilde{D}}=$
$\displaystyle\;\;N^{A}_{LM}N^{L}_{\tilde{L}}N^{M}_{\tilde{M}}G^{\tilde{L}\tilde{M}}-G^{CT}N^{U}_{C}N^{A}_{TU}+{}^{\bot}G^{CR}{\Lambda}^{\beta}_{C}N^{A}_{T}K^{T}_{\beta
R}+{}^{\bot}G^{LM}{\tilde{\Gamma}}^{D}_{LM}N^{A}_{D},$
one can arrive at the equality of the transformed expressions. It will be
noted that in the expression obtained as a result of the transformation of
(26), besides of the necessary terms, that are equal to the corresponding
terms coming from (25), there are redundent terms. But, it can be verified
that these terms are mutually cancelled. It follows from the calculation in
which one should takes into account the Killing identity, the equality
${\gamma}^{\beta\nu}({\triangle}_{K_{\nu}}K_{\beta})^{P}{\mathscr{A}}^{\alpha}_{P}=0,$
which is obtained from the identity
$-{\gamma}^{\beta\nu}({\triangle}_{K_{\nu}}K_{\beta})^{T}=\frac{1}{2}G^{PT}N^{E}_{P}\;\Bigl{(}{\gamma}^{\mu\nu}\frac{\partial}{\partial
Q^{*E}}{\gamma}_{\mu\nu}\Bigr{)},$
and the condition $c^{\alpha}_{\beta\alpha}=0$, which is valid for the
structure constants of the semisimple Lie groups.
Except for the potential terms, the only distinction between (23) and (24)
consists of the terms that involve the product of two group generators. But,
since
$G^{LM}N^{E}_{L}N^{P}_{M}\,{\mathscr{A}}^{\mu}_{E}{\mathscr{A}}^{\nu}_{P}=G^{EP}{\Lambda}^{\mu}_{E}{\Lambda}^{\nu}_{P}-{\gamma}^{\mu\nu},$
we can present the operator (23) as
$\frac{1}{2}{\mu}^{2}{\kappa}\left[\bigl{(}{\triangle}^{{\mathcal{E}^{*}}}\bigr{)}_{pq}^{\lambda}+{\gamma}^{\mu\nu}(J_{\mu})_{pq^{\prime}}^{\lambda}(J_{\nu})_{q^{\prime}q}^{\lambda}\right]+\left(\frac{1}{\mu^{2}\kappa
m}{\tilde{V}}-\frac{1}{8}\mu^{2}\kappa{\tilde{J}}\right)(I^{\lambda})_{pq}\,.$
## 5 The path integral for the matrix Green’s function
${\tilde{G}}^{\lambda}_{pq}$
Now we can rewrite the multiplicative stochastic integral in the path integral
(21). We already know that $(J_{\alpha})_{pq}^{\lambda}$–terms of the drift in
the integrand of the multiplicative stochastic integral are equal to the
corresponding terms (26) of the operator (24). These terms can be rewritten as
follows
$\displaystyle-G^{PQ}N^{A}_{P}N^{B}_{Q}N^{E}_{A}N^{C}_{B,E}{\mathscr{A}}^{\alpha}_{C}-G^{PQ}N^{A}_{P}N^{B}_{Q}N^{C}_{B}N^{E}_{A}\frac{\partial}{\partial
Q^{*E}}({\mathscr{A}}^{\alpha}_{C})$
$\displaystyle\;\;\;\;+G^{PQ}N^{A}_{P}N^{C}_{Q}\,{}^{\rm
H}{\Gamma}^{B}_{AC}N^{D}_{B}{\mathscr{A}}^{\alpha}_{D},$
and also as
$-G^{PQ}N^{E}_{P}N^{B}_{Q}\,{\nabla}^{\rm H}_{{\rm
e}_{E}}(N^{C}_{B}{\mathscr{A}}^{\alpha}_{C}).$
The coefficient ${\Pi}^{C}_{K}{\Lambda}^{\beta}_{C}$ of the diffusion term of
the integrand may be written in the form
${\Pi}^{C}_{K}{\Lambda}^{\beta}_{C}={\Lambda}^{\beta}_{K}-{\mathscr{A}}^{\beta}_{K}.$
Thus, we obtain the following path integral representation of the matrix
Green’s function ${\tilde{G}}^{\lambda}_{pq}$:
$\displaystyle{\tilde{G}}^{\lambda}_{pq}(Q^{*}_{b},t_{b};Q^{*}_{a},t_{a})=\int\limits_{{{\tilde{\tilde{\xi}}}_{\Sigma}(t_{a})=Q^{*}_{a}\atop{{\tilde{\xi}}_{\Sigma}(t_{b})=Q^{*}_{b}}}}d{\mu}^{{{\tilde{\xi}}}_{\Sigma}}\exp\biggl{\\{}\int_{t_{a}}^{t_{b}}\Bigl{(}\frac{\tilde{V}}{\mu^{2}\kappa
m}-\frac{1}{8}\mu^{2}\kappa{\tilde{J}}\Bigr{)}du\biggr{\\}}$
$\displaystyle\times\overleftarrow{\exp}\int_{t_{a}}^{t_{b}}\Bigl{\\{}\frac{1}{2}{\mu}^{2}\kappa\Bigl{[}{\gamma}^{\sigma\nu}\,(J_{\sigma})_{pr}^{\lambda}(J_{\nu})_{rq}^{\lambda}-G^{PQ}N^{E}_{P}N^{B}_{Q}\,{\nabla}^{\rm
H}_{{\rm
e}_{E}}(N^{C}_{B}{\mathscr{A}}^{\alpha}_{C})(J_{\alpha})_{pq}^{\lambda}\Bigr{]}du$
$\displaystyle\;\;\;\;\;\;\;\;-\mu\sqrt{\kappa}\,N^{B}_{K}{\mathscr{A}}^{\alpha}_{B}\,(J_{\alpha})_{pq}^{\lambda}\,\tilde{\mathfrak{X}}^{K}_{\bar{M}}dw^{\bar{M}}_{u}\Bigr{\\}}.$
(27)
In $(Q^{*}_{b},t_{b})$-variables this Green’s function satisfies the forward
Kolmogorov equation with the operator
${\hat{H}}_{\kappa}=\frac{\hbar\kappa}{2m}\left[({\triangle}^{\mathcal{E}})^{\lambda}_{pq}+{\gamma}^{\mu\nu}(J_{\mu})_{pq^{\prime}}^{\lambda}(J_{\nu})_{q^{\prime}q}^{\lambda}\right]-\frac{\hbar\kappa}{8m}[\tilde{J}\,]I^{\lambda}_{pq}+\frac{\tilde{V}}{\hbar\kappa}I^{\lambda}_{pq},$
where the horizontal Laplacian $({\triangle}^{{\mathcal{E}}})^{\lambda}_{pq}$
is
$\displaystyle\\!\\!\\!\\!\\!\left({\triangle}^{{\mathcal{E}}}\right)^{\lambda}_{pq}={\sum}_{\bar{\scriptscriptstyle
M}=1}^{n_{\cal P}}\left({\nabla}^{\mathcal{E}}_{Y^{A}_{\bar{M}}{\rm
e_{A}}}{\nabla}^{\mathcal{E}}_{Y^{B}_{\bar{M}}{\rm
e_{B}}}-{\nabla}^{\mathcal{E}^{*}}_{{\nabla}^{\mathcal{M}}_{Y^{A}_{\bar{M}}{\rm
e_{A}}}{Y^{B}_{\bar{M}}{\rm e_{B}}}}\right)^{\\!\\!\lambda}_{\\!\\!pq}$
$\displaystyle\\!\\!\\!\\!\\!\\!={\triangle}_{\cal M}\,{\rm
I}^{\lambda}_{pq}+2\,G^{LM}N^{E}_{L}N^{C}_{M}\,({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{Epq}\,{\partial}_{Q^{*C}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!-G^{LM}N^{E}_{L}N^{B}_{M}\left[{\partial}_{Q^{*}{}^{E}}(N^{C}_{B}({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{Cpq})-({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{E\,p{q}^{{}^{\prime}}}({\rm{\Gamma}^{\mathcal{E}}})^{\lambda}_{B{q}^{{}^{\prime}}q}+{}^{\rm
H}{\rm{\Gamma}}^{C}_{EB}N^{D}_{C}({\rm{\Gamma}}^{\mathcal{E}})^{\lambda}_{Dpq}\right].$
The Laplace operator ${\triangle}_{\mathcal{M}}$ is
${\triangle}_{\mathcal{M}}=G^{CD}N^{A}_{C}N^{B}_{D}\frac{{\partial}^{2}}{\partial
Q^{*}{}^{A}\partial
Q^{*}{}^{B}}-G^{CD}N^{E}_{C}N^{M}_{D}\,{}^{H}{\Gamma}^{A}_{EM}\frac{\partial}{\partial
Q^{*}{}^{A}}+2j^{A}\frac{\partial}{\partial Q^{*}{}^{A}}.$
At $\kappa=i$ the forward Kolmogorov equation becomes the Schrödinger equation
with the Hamilton operator
$\hat{H}_{\mathcal{E}}=-\frac{\hbar}{\kappa}{\hat{H}}_{\kappa}|_{\kappa=i}$.
The operator $\hat{H}_{\mathcal{E}}$ acts in the Hilbert space of the sections
of the associated vector bundle
${\mathcal{E}}=P\times_{\mathcal{G}}V_{\lambda}$. The scalar product in this
space has the same volume measure as in (22).
## 6 Conclusion
In this paper, we have considered the transformation of the path integral
obtained as a result of the reduction of the finite-dimensional dynamical
system with a symmetry. We have dealt with the reduction, which in the
constrained dynamical system theory is called the reduction onto the non-zero
momentum level.
Because of exploiting the dependent variables for the description of the local
reduced motion, we were forced to consider only the trivial principal fiber
bundles. Thereby, our consideration is a global one only for the trivial
principal bundle. For the nontrivial principal fiber bundle, that may be
related to the dynamical system with a symmetry, the dependent variable
description of the evolution is valid in a some local domain.
Although for the nontrivial principal fiber bundles, there is a method [12]
which allows us to extend the local evolution to a global one, but in general
this problem remains unsolved, especially for the reason of a possible
existence of the non-trivial topology of the orbit space.
In conclusion, we note that besides of the application of the obtained path
integral representation (and the integral relation) in the quantization of the
finite-dimensional dynamical systems with a symmetry, this representation may
be useful for a quantum description (in the Schrödinger’s approach) of the
excited modes in the gauge fields models.
## References
* [1] Landsman N P and Linden N 1991 Nucl. Phys. B365 121;
Tanimura S and Tsutsui I 1995 Mod. Phys. Lett. A34 2607;
McMullan D and Tsutsui I 1995 Ann. Phys. 237 269\.
* [2] Kunstatter G 1992 Class. Quant.Grav. 9 1466-86.
* [3] Falck N K and Hirshfeld A C 1982 Ann. Phys. 144 34;
Gavedzki K 1982 Phys.Rev. D26 3593\.
* [4] Lott J 1984 Comm. Math. Phys. 95 289\.
* [5] L. D. Faddeev, Teor. i Mat. Fyz. 1 (1969) 3 (in Russian);
L. D. Faddeev, V. N. Popov, Phys. Lett. 25B (1967) 30.
* [6] S. N. Storchak, J. Phys. A: Math. Gen. 34 (2001) 9329,
IHEP Preprint 96-110, Protvino, 1996;
S. N. Storchak. Bogolubov transformation in path integral on manifold with a
group action. IHEP Preprint 98-1, Protvino, 1998;
S. N. Storchak, Physics of Atomic Nuclei 64 n.12 (2001) 2199
* [7] K. D. Elworthy, Y. Le Jan, Xue-Mei Li The Geometry of Filtering (Preliminary Version) (2008), arXiv:0810.2253
* [8] M. Arnaudon, S. Paycha, Stochastic and Stochastic Reports 53 (1995) 81.
* [9] S. N. Storchak, J. Phys. A: Math. Gen. 37 (2004) 7019,
IHEP Preprint 2000-54, Protvino, 2000; arXiv:math-ph/0311038
* [10] Ya. I. Belopolskaya, Yu. L. Daletskii, Russ. Math. Surveys 37 109 (1982); Usp. Mat. Nauk 37 n.3 (1982) 95 (in Russian);
Yu. L. Daletskii, Usp. Mat. Nauk 38 n.3 (1983) 87 (in Russian);
Ya. I. Belopolskaya and Yu. L. Daletskii, Stochastic equations and
differential geometry (Kluwer, Dordrecht, 1990), Mathematics and Its
Applications, Soviet Series, 30.
* [11] S. N. Storchak, J. of Geometry and Physics 59 (2009) 1155.
* [12] H. Hüffel, G. Kelnhofer, Ann. of Phys. 266 (1998) 417;
Ann. of Phys. 270 (1998) 231.
|
arxiv-papers
| 2009-12-17T20:28:14 |
2024-09-04T02:49:07.088937
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. N. Storchak",
"submitter": "Sergey Storchak",
"url": "https://arxiv.org/abs/0912.3509"
}
|
0912.3596
|
# Vertical Structure of Neutrino-Dominated Accretion Disk and Applications to
Gamma-Ray Bursts
Tong Liu11affiliation: Department of Astronomy, Nanjing University, Nanjing,
Jiangsu 210093, China; tongliu@nju.edu.cn , Wei-Min Gu22affiliation:
Department of Physics and Institute of Theoretical Physics and Astrophysics,
Xiamen University, Xiamen, Fujian 361005, China; guwm@xmu.edu.cn , Zi-Gao
Dai11affiliation: Department of Astronomy, Nanjing University, Nanjing,
Jiangsu 210093, China; tongliu@nju.edu.cn , and Ju-Fu Lu22affiliation:
Department of Physics and Institute of Theoretical Physics and Astrophysics,
Xiamen University, Xiamen, Fujian 361005, China; guwm@xmu.edu.cn
###### Abstract
We revisit the vertical structure of neutrino-dominated accretion flows in
spherical coordinates. We stress that the flow should be geometrically thick
when advection becomes dominant. In our calculation, the luminosity of
neutrino annihilation is enhanced by one or two orders of magnitude. The empty
funnel along the rotation axis can naturally explain the neutrino annihilable
ejection.
accretion, accretion disks - black hole physics - gamma rays: bursts
## 1 Introduction
Gamma-Ray Bursts (GRBs) are short-lived bursts of gamma-ray photons occurring
at cosmological distances. GRBs are usually sorted of two classes (Kouveliotou
et al. 1993): short-hard GRBs ($T_{90}<2\rm s$) and long-soft GRBs
($T_{90}>2\rm s$). The likely progenitors are the merger of two neutron stars
or a neutron star and a black hole (Eichler et al. 1989; Paczyński 1991;
Narayan et al. 1992) and collapsar (Woosley 1993; Paczyński 1998),
respectively. The popular model of the central engine, namely neutrino
dominated accretion flows (NDAFs), involves a hyperaccreting black hole with
mass accretion rates in the range of $0.01\sim 10M_{\odot}{\rm s}^{-1}$. Such
a model has been widely investigated in the past decade (see, e.g., Popham et
al. 1999; Narayan et al. 2001; Kohri & Mineshige 2002; Di Matteo et al. 2002;
Rosswog et al. 2003; Kohri et al. 2005; Lee et al. 2005; Gu et al. 2006; Chen
& Beloborodov 2007; Liu et al. 2007; Kawanaka & Mineshige 2007; Janiuk et al.
2007; Liu et al. 2008). The model can provide a good understanding of both the
energetics of GRBs and the processes of making the relativistic and baryon-
poor fireballs by neutrino annihilation or magnetohydrodynamic processes (see,
e.g., Popham et al. [1999] and Di Matteo et al. [2002] for references).
In cylindrical coordinates ($R$, $z$, $\varphi$), Gu & Lu (2007) discussed the
potential importance of taking the explicit form of the gravitational
potential for calculating slim disk (Abramowicz et al. 1988) solutions, and
pointed out that the Hōshi form of the potential (Hōshi 1977),
$\displaystyle\psi(r,z)\simeq\psi(r,0)+\frac{1}{2}\Omega_{\rm K}^{2}z^{2}\ ,$
(1)
is valid only for geometrically thin disks with $H/R\lesssim 0.2$. Thus the
well-known relationship $c_{s}/\Omega_{\rm K}H=$ constant does not hold for
slim disks with $H/R\lesssim 1$, where $c_{s}$ is the sound speed, and
$\Omega_{\rm K}$ is the Keplerian angular velocity. Moreover, with the
explicit form of the gravitational potential, Liu et al. (2008) found that
NDAFs have both a maximal and a minimal possible mass accretion rate at their
each radius, and presented a unified description of all the three known
classes of optically thick accretion disks around black holes, namely Shakura-
Sunyaev disks (Shakura & Sunyaev 1973), slim disks, and NDAFs. These works
are, however, based on the following simple vertical hydrostatic equilibrium:
$\displaystyle\frac{1}{\rho}\frac{\partial p}{\partial
z}+\frac{\partial\psi}{\partial z}=0\ ,$ (2)
instead of the general form (Abramowicz et al. 1997):
$\displaystyle\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\ ,$ (3)
where $\rho$ is the mass density, $p$ is the pressure, $v_{R}$ is the
cylindrical radial velocity, and $v_{z}$ is the vertical velocity. Since
$v_{z}$ is not negligible for geometrically thick or slim disks, the solutions
in Gu & Lu (2007) and Liu et al. (2008) are still not self-consistent.
Recently, Gu et al. (2009) revisited the vertical structure in spherical
coordinates and showed that advection-dominated accretion disks should be
geometrically thick rather than being slim. However, the detailed radiative
cooling was not considered in that work, and therefore no thermal equilibrium
solution was established.
The purpose of this paper is to investigate the vertical structure of NDAFs
with detailed neutrino radiation. In section 2, with the self-similar
assumption in the radial direction, we numerically solve the differential
equations of NDAFs in the vertical direction. In section 3, we present the
vertical distribution of physical quantities and show the geometrical
thickness and the energy advection of the disk. In section 4, we estimate the
luminosity of neutrino annihilation and discuss some applications to GRBs.
Conclusions are made in section 5.
## 2 Equation
We consider a steady state axisymmetric accretion flow in spherical
coordinates ($r$, $\theta$, $\phi$), i.e., $\partial/\partial
t=\partial/\partial\phi=0$. We adopt the Newtonian potential $\psi=-GM/r$
since it is convenient for self-similar assumption, where $M$ is the mass of
the central black hole. The basic equations of continuity and momentum are the
following (see, e.g., Xue & Wang 2005; Gu et al. 2009):
$\displaystyle\ \frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\rho
v_{r})+\frac{1}{r^{2}{\rm sin}\theta}\frac{\partial}{\partial\theta}({\rm
sin}\theta\rho v_{\theta})=0,$ (4) $\displaystyle\ v_{r}\frac{\partial
v_{r}}{\partial r}+\frac{v_{\theta}}{r}(\frac{\partial
v_{r}}{\partial\theta}-v_{\theta})-\frac{{v_{\phi}}^{2}}{r}=-\frac{GM}{r^{2}}-\frac{1}{\rho}\frac{\partial
p}{\partial r},$ (5) $\displaystyle\ v_{r}\frac{\partial v_{\theta}}{\partial
r}+\frac{v_{\theta}}{r}(\frac{\partial
v_{\theta}}{\partial\theta}+v_{r})-\frac{{v_{\phi}}^{2}}{r}\cot\theta=-\frac{1}{\rho
r}\frac{\partial p}{\partial\theta},$ (6) $\displaystyle\ v_{r}\frac{\partial
v_{\phi}}{\partial r}+\frac{v_{\theta}}{r}\frac{\partial
v_{\phi}}{\partial\theta}+\frac{v_{\phi}}{r}(v_{r}+v_{\theta}\cot\theta)=\frac{1}{\rho
r^{3}}\frac{\partial}{\partial r}(r^{3}T_{r\phi}),$ (7)
where $v_{r}$, $v_{\theta}$, and $v_{\phi}$ are the three components of the
velocity. Here, we only consider the $r\phi$-component of the viscous stress
tensor, $T_{r\phi}=\rho\nu r\partial(v_{\phi}/r)/\partial r$. The kinematic
coefficient of viscosity takes the form: $\nu=\alpha c_{s}^{2}/\Omega_{\rm K}$
(e.g., Narayan & Yi 1995), where the sound speed $c_{s}$ is defined as
$c_{s}^{2}=p/\rho$, the Keplerian angular velocity is $\Omega_{\rm
K}=(GM/r^{3})^{1/2}$, and $\alpha$ is a constant viscosity parameter.
To avoid directly solve the above partial differential equations, some radial
simplification is required since our main interest is the vertical
distribution. Based on the radial self-similar assumption, Begelman & Meier
(1982) studied the vertical structure of geometrically thick, optically thick,
supercritical accretion disks. Under the same self-similar assumption, Narayan
& Yi (1995) investigated the vertical structure of optically thin advection-
dominated accretion flows (ADAFs). In fact, since the well-known self-similar
solutions of ADAFs (Narayan & Yi 1994), such type of solutions has been widely
investigated for different classes of accretion, such as slim disks (Wang &
Zhou 1999), convection-dominated accretion flows (Narayan et al. 2000), NDAFs
(Narayan et al. 2001), and accretion flows with ordered magentic field and
outflows (Bu et al. 2009). Even though the detailed radiation was considered
in some works (e.g., Di Matteo et al. 2002; Chen & Beloborodov 2007), and
therefore the solutions cannot be regarded as self-similar solutions, the
self-similar assumption was still adopted such that the original differential
energy equation can be simplified as an algebraic one. Furthermore, for
optically thick flows, Ohsuga et al. (2005) showed that their simulations are
close to the self-similar solutions of the slim disk model (e.g., the density
profile in their Fig. 11 ). In our opinion, the radial simplification is
necessary for the study of vertical structure and it is a good choice to take
the well-known self-similar assumption.
Similar to Narayan & Yi (1995), we adopt the following radial self-similar
assumption:
$\displaystyle\ \rho(r,\theta)\propto r^{-3/2},$ (8) $\displaystyle\
c_{s}(r,\theta),v_{r}(r,\theta),v_{\phi}(r,\theta)\propto r^{-1/2},$ (9)
$\displaystyle\ v_{\theta}(r,\theta)=0.$ (10)
With the above assumption, equations (5-7) can be simplified as follows:
$\displaystyle\
\frac{1}{2}{v_{r}}^{2}+\frac{5}{2}{c_{s}}^{2}+{v_{\phi}}^{2}-r^{2}{\Omega_{\rm
K}}^{2}=0,$ (11) $\displaystyle\
\frac{1}{\rho}\frac{dp}{d\theta}={v_{\phi}}^{2}\cot\theta,$ (12)
$\displaystyle\ v_{r}=-\frac{3}{2}\frac{\alpha{c_{s}}^{2}}{r\Omega_{\rm K}}.$
(13)
Integrating equation (4) over angle we obtain the mass accretion rate,
$\displaystyle\ \dot{M}=-4\pi r^{2}\int_{\theta_{0}}^{\frac{\pi}{2}}\rho
v_{r}\sin\theta d\theta,$ (14)
where $\theta_{0}$ is the polar angle of the surface.
The equation of state is
$\displaystyle\ p=p_{\rm gas}+p_{\rm rad}+p_{\rm e}+p_{\nu},$ (15)
where $p_{\rm gas}$, $p_{\rm rad}$, $p_{\rm e}$, and $p_{\nu}$ are the gas
pressure from nucleons, the radiation pressure of photons, the degeneracy
pressure of electrons, and the radiation pressure of neutrinos, respectively.
Detailed expressions of the pressure components were given in Liu et al.
(2007). We assume a polytropic relation in the vertical direction,
$p=K\rho^{4/3}$ , where $K$ is a constant.
The energy equation is written as
$\displaystyle\ Q_{\rm vis}=Q_{\rm adv}+Q_{\nu},$ (16)
where $Q_{\rm vis}$, $Q_{\rm adv}$, and $Q_{\nu}$ are the viscous heating rate
per unit area, the advective cooling rate per unit area, and the cooling rate
per unit area due to the neutrino radiation, respectively. Here we ignore the
cooling of photodisintegration of $\alpha$-particles and other heavier nuclei.
The viscous heating rate per unit volume $q_{\rm vis}=\nu\rho
r^{2}[\partial(v_{\phi}/r)/\partial r]^{2}$ and the advective cooling rate per
unit volume $q_{\rm adv}=\rho v_{r}(\partial e/\partial
r-p/\rho^{2}\partial\rho/\partial r)$ ($e$ is the internal energy per unit
volume) are expressed in the self-similar formalism as
$\displaystyle\ q_{\rm vis}=\frac{9}{4}\frac{\alpha
pv_{\phi}^{2}}{r^{2}\Omega_{\rm K}},$ (17) $\displaystyle\ q_{\rm
adv}=-\frac{3}{2}\frac{(p-p_{\rm e})v_{r}}{r}.$ (18)
where the entropy of degenerate particles is negligible. Thus the vertical
integration of $Q_{\rm vis}$ and $Q_{\rm adv}$ are the following:
$\displaystyle\ Q_{\rm vis}=2\int_{\theta_{0}}^{\frac{\pi}{2}}q_{\rm
vis}r\sin{\theta}d\theta\ ,$ (19) $\displaystyle\ Q_{\rm
adv}=2\int_{\theta_{0}}^{\frac{\pi}{2}}q_{\rm adv}r\sin{\theta}d\theta.$ (20)
The cooling due to the neutrino radiation $Q_{\nu}$ can be written as
$\displaystyle\
Q_{\nu}=2\int_{\theta_{0}}^{\frac{\pi}{2}}q_{\nu}r\sin{\theta}d\theta\ ,$ (21)
where $q_{\nu}$ is the sum of Urca processes, electron-positron pair
annihilation, nucleon-nucleon bremsstrahlung, and Plasmon decay (see, e.g.,
Liu et al. 2007). We therefore can obtain the luminosity of neutrino radiation
$L_{\nu}$ by integrating $Q_{\nu}$.
In our system, we have six physical quantities varying with $\theta$, i.e.,
$v_{r}$, $v_{\phi}$, $c_{s}$, $\rho$, $p$, and $T$. The six equations for
solving these quantities are equations (11-13), (15), the polytropic relation,
and the definition of $c_{s}$ ($c_{s}^{2}=p/\rho$). Three boundary conditions
are required to solve the system since there is one differential equation, and
the boundary $\theta_{0}$ and the constant parameter $K$ in the polytropic
relation are unknown. Now we have already two boundary conditions, i.e.,
equations (14) and (16), thus one more boundary condition is required for
solving the system, which is set to be $c_{s}=0$ (accordingly $\rho=0$ and
$p=0$, e.g., Kato et al. 2008, p. 244) at the surface of the disk, i.e.,
$\theta=\theta_{0}$. The numerical method is as follows. For given $\alpha$,
$M$, $\dot{M}$, $r$, and a test $\theta_{0}$, from the above six equations and
two boundary conditions (except the energy equation, Eq. [16]), we can
numerically obtain the vertical distribution of the above six quantities. With
equations (19-21), we then check whether equation (16) is satisfied for the
test $\theta_{0}$. By varying $\theta_{0}$ we can find the exact value of
$\theta_{0}$ for which equation (16) is matched, and therefore we obtain the
exact vertical distribution of all the variables. In our calculations we take
$\alpha=0.1$ and $M=3M_{\odot}$.
## 3 Numerical Results
Figure 1 shows the variations of the density $\rho$, temperature $T$, electron
fraction $Y_{\rm e}$, and radial velocity $v_{r}$ with the polar angle
$\theta$ for $\dot{M}=1M_{\odot}\rm s^{-1}$. Here $Y_{\rm e}$ is defined as
$Y_{\rm e}\equiv n_{\rm p}/(n_{\rm p}+n_{\rm n})$, where $n_{\rm p}$ and
$n_{\rm n}$ are the total number density of protons and of neutrons,
respectively (e.g., Beloborodov 2003; Liu et al. 2007). The solid, dashed, and
dotted lines represent the solutions at $r/r_{g}=10,40$, and $100$,
respectively. The profiles of $\rho$ and $v_{r}$ are similar to that of the
optically thin advection-dominated accretion flows (Narayan & Yi 1995), i.e.,
$\rho$ and $v_{r}$ (the absolute value) decrease from the equatorial plane to
the surface. On the contrary, electron fraction $Y_{\rm e}$ increases from the
equatorial plane to the surface and approaches $0.5$ near the surface, which
means that the matter is non-degenerate. The vertical distribution of $v_{r}$,
as shown in Fig. 1($d$), indicates a multilayer flow with the matter close to
the equatorial plane being accreted much faster than that near the surface.
Figure $2(a)$ shows the variation of the half-opening angle of the disk
$(\pi/2-\theta_{0})$ with radius $r/r_{g}$, where $r_{g}=2GM/c^{2}$ is the
Schwarzschild radius. The solid, dashed, and dotted lines represent the
solutions with $\dot{M}/M_{\odot}\rm s^{-1}=0.1,1$, and 10, respectively. It
is seen that, in the inner region of the disk, the half-opening angle
increases as increasing accretion rates. For $\dot{M}=10M_{\odot}\rm s^{-1}$,
the inner disk is extremely thick with the half-opening angle is $\sim 1.4$
radian, which implies that there exists a narrow empty funnel $\sim
20^{\circ}$ along the rotation axis. Figure $2(b)$ shows the variation of the
energy advection factor $f_{\rm adv}$ ($\equiv Q_{\rm adv}/Q_{\rm vis}$) with
$r/r_{g}$. It is seen that advection becomes important in the inner disk for
$\dot{M}\gtrsim 1M_{\odot}\rm s^{-1}$. Comparing Figs. $2(a)$ and $2(b)$, we
find that the curves of the half-opening angle and the advection factor are
similar, which indicates that the geometrical thickness is relevant to the
advection. For $f_{\rm adv}=0.5$, it is seen from Fig. 2 that the half-opening
angle is around $1.3$ radian. We therefore stress that NDAFs should be
significantly thick when advection becomes dominant, which is in agreement
with Narayan & Yi (1995) since their solutions imply that the flows are
extremely thick with the half-opening angle approaching $\pi/2$.
## 4 Applications to GRBs
In our calculations, the inner disk will be quite thick for large mass
accretion rates, $\dot{M}\gtrsim 1M_{\odot}\rm s^{-1}$. Thus the volume above
the disk shrinks and the radiated neutrino density increases. Accordingly, the
neutrino annihilation efficiency also increases. We have obtained the neutrino
luminosity $L_{\nu}$ (before annihilation), thus the luminosity of neutrino
annihilation $L_{\nu\bar{\nu}}$ can be roughly evaluated by the assumption:
$\eta\propto V_{\rm ann}^{-1}$ (see, e.g., Mochkovitch et al 1993), where
$\eta\equiv L_{\nu\bar{\nu}}/L_{\nu}$ is the annihilation efficiency, and
$V_{\rm ann}$ is the volume above the disk. For a given outer boundary $r_{\rm
out}$, we calculate $V_{\rm ann}$ by integrating the region of
$\theta<\theta_{0}$ and $r<r_{\rm out}$. The variations $L_{\nu}$ and
$L_{\nu\bar{\nu}}$ with $\dot{M}$ are shown in figure 3. The solid lines
correspond to the present solutions whereas the dashed lines correspond to
those in Liu et al. (2007). As shown in Fig. 3, for the same $\dot{M}$,
$L_{\nu}$ is comparable, whereas $L_{\nu\bar{\nu}}$ in the present results is
significantly larger than that in Liu et al. (2007) by one or two orders of
magnitude. Moreover, we find that for $\dot{M}=5M_{\odot}\rm s^{-1}$,
$L_{\nu\bar{\nu}}$ is very close to $L_{\nu}$, which means that the density of
radiated neutrino is so large that the annihilation efficiency is close to 1.
Thus we can expect that $L_{\nu\bar{\nu}}$ is roughly equal to $L_{\nu}$ for
$\dot{M}\gtrsim 5M_{\odot}\rm s^{-1}$.
Many previous works have calculated the annihilation luminosity and claimed
that the NDAF mode can provide enough energy for GRBs. However, GRBs are
generally believed to be a jet with a small opening angle $\theta_{\rm jet}$.
The problem is that, the annihilation could not be limited into such a small
angle even though the region well above the inner disk have larger luminosity
than other place. We argue that our model is preferably to explain the
ejection-like radiation, because the disk is adequately thick and there exists
a narrow empty funnel along the rotation axis, which can naturally explain the
neutrino annihilable ejection.
## 5 Conclusions
In this paper we revisit the vertical structure of NDAFs in spherical
coordinates. The major points we wish to stress are as follows:
1. 1.
We show the vertical structure of NDAFs and stress that the flow should be
significantly thick when advection becomes dominant.
2. 2.
The luminosity of neutrino annihilation is enhanced by one or two orders of
magnitude.
3. 3.
The narrow empty funnel ($\sim 20^{\circ}$) along the rotation axis can
naturally explain the neutrino annihilable ejection.
We thank Katsuaki Asano, H.-Thomas Janka and Yi-Zhong Fan for beneficial
discussion and comments. This work was supported by the National Basic
Research Program (973 Program) of China under Grant 2009CB824800 (JFL and
WMG), the National Natural Science Foundation of China under grants 10778711
(WMG), 10833002 (JFL and WMG), 10873009 (ZGD), and the China Postdoctoral
Science Foundation funded project 20080441038 (TL).
## References
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Figure 1: Variations of the density $\rho$, temperature $T$, electron
fraction $Y_{\rm e}$, and radial velocity $v_{r}$ with the polar angle
$\theta$, for which the given parameters are $\dot{M}/M_{\odot}\rm s^{-1}=1$
and $r/r_{g}=10$ (solid lines), $40$ (dashed lines), $100$ (dotted lines).
Figure 2: Variations of the half-opening angle of the disk
$(\pi/2-\theta_{0})$ and the advection factor $f_{\rm adv}$ with radius
$r/r_{g}$, for which the given parameter is $\dot{M}/M_{\odot}\rm s^{-1}=0.1$
(solid line), $1$ (dashed line), $10$ (dotted line).
Figure 3: Neutrino luminosity $L_{\nu}$ (thick lines) and annihilation
luminosity $L_{\nu\bar{\nu}}$ (thin lines) for varying mass accretion rates
$\dot{M}$. The solid lines correspond to the present solutions, whereas the
dashed lines correspond to the solutions of Liu et al. (2007).
|
arxiv-papers
| 2009-12-18T07:26:20 |
2024-09-04T02:49:07.097961
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tong Liu, Wei-Min Gu, Zi-Gao Dai, and Ju-Fu Lu",
"submitter": "Liu Tong",
"url": "https://arxiv.org/abs/0912.3596"
}
|
0912.3696
|
# Searches for high energy solar flares with Fermi-LAT
G. Iafrate INAF - Astronomical Observatory of Trieste, Italy and INFN
Trieste, Italy F. Longo INFN Trieste, Italy and Dipartimento di Fisica,
Trieste, Italy on behalf of the Fermi Large Area Telescope Collaboration
###### Abstract
The Fermi Large Area Telescope (LAT) has been surveying the sky in gamma rays
from 30 MeV to more than 300 GeV since August 2008\. Fermi is the only mission
able to detect high energy $(>\text{few hundreds MeV})$ emission from the Sun
during the new solar cycle 24: the Solar System Science Group of the Fermi
team is continuously monitoring high energy emission from the Sun searching
for flare events. Preliminary upper limits $(>100\text{ MeV})$ have been
derived for all solar flares detected so far by other missions and experiments
(RHESSI, Fermi GBM, GOES). Upper limit for flaring Sun emission (integrated
over one year of data) has also been derived. Here we present the analysis
techniques as well as the details of this search and the preliminary results
obtained so far.
Figure 1: Solar flares detected by RHESSI and analysed in this search,
superimposed on a count map of LAT data ($E>200\textnormal{ MeV}$) for a
easier localization in the sky. There is no evidence of correlation between
flare positions and excesses of the LAT events.
## I Introduction
_Fermi_ was successfully launched from Cape Canaveral on 2008 June 11. It is
currently in an almost circular orbit around the Earth at an altitude of 565
km having an inclination of 25.6∘ and an orbital period of 96 minutes. After
an initial period of engineering data taking and on-orbit calibrationFermi ,
the observatory was put into a sky-survey mode in August 2008. The observatory
has two instruments onboard, the Large Area Telescope (LAT)LAT , a pair-
conversion gamma-ray detector and tracker (energy range 30 MeV - $>300$ GeV)
and a Gamma-ray Burst Monitor (GBM), dedicated to the detection of gamma-ray
bursts (energy range 8 keV - 40 MeV). The instruments on _Fermi_ provide
coverage over the energy range measurements from few keV to several hundreds
of GeV.
Here we report Fermi LAT limits on emission $>100\text{ MeV}$ for the few
flares detected by other missions over the past year. Solar flares are the
most energetic phenomena that occur within our Solar System. A flare is
characterized by the impulsive release of a huge amount of energy, previously
stored in the magnetic fields of active regions. During a flare plasma of the
solar corona and chromosphere is accelerated and electromagnetic radiation
covering the entire spectrum is emitted. The production of $\gamma$-rays
involves flare-accelerated charged-particle (electrons, protons and heavier
nuclei) interactions with the ambient solar atmosphere. Electrons accelerated
by the flare, or from the decay of $\pi^{\pm}$ secondaries produced by nuclear
interactions, yield X and $\gamma$-ray bremsstrahlung radiation with a
spectrum that extends to the energies of the primary particles. Proton and
heavy ion interactions also produce $\gamma$-rays through $\pi^{0}$ decay,
resulting in a spectrum that has a maximum at 68 MeVshare .
The frequency of solar flares follows the 11 year solar activity cycle. Most
intense flares occur during the maximum, but intense flares can occur also in
the rising and decreasing phases of the cycle. The new solar activity cycle 24
has started at the beginning of year 2008, the maximum is predicted in year
2012. _Fermi_ has been launched during the minimum of the solar cycle, so the
frequency and the intensity of solar flares will increase throughout most of
the mission. If the goal of a 10-year mission life is achieved, _Fermi_ will
operate for nearly the entire duration of solar cycle 24. During this time,
_Fermi_ will be the only high-energy observatory ($>\text{few hundreds MeV}$)
to complement several solar missions at lower energies: RHESSI, GOES, SoHO,
Coronas.
## II Previous observations
The 2005 January 20 solar flare produced one of the most intense, fastest
rising and hardest solar energetic particle events ever observed in space or
on the ground. $\gamma$-ray measurements of the flareshare06 grechnev
revealed what appear to be two separate components of particle acceleration at
the Sun: i) an impulsive release lasting $\sim 10$ min with a power-law index
of $\sim 3$ observed in a compact region on the Sun and, ii) an associated
release of much higher energy particles having a spectral index $\leq 2.3$
interacting at the Sun for about two hours. Pion-decay $\gamma$-rays appear to
dominate the latter component. Such long-duration high-energy events have been
observed before, most notably on 1991 June 11 when the EGRET instrument on
CGRO observed $>50$ MeV emission for over 8 hourskanbach . It is possible that
these high-energy components are directly related to the particle events
observed in space and on Earth.
_Fermi_ will improve our understanding of the mechanisms of the $\gamma$-ray
emission by solar flares thanks to its large effective area, sensitivity and
high spatial and temporal resolution.
## III Monitor of solar cycle 24
The solar cycle 24 has started at the beginning of 2008, but actually we are
in an extended period of minimal solar activity. We are seeing an interesting
diminished level of activity. There are some discussion ongoing if sunspots
and flares ever return and how unusual is this behaviornugget . A closer look
at the daily values of three indices: F10.7 (10 cm radio flux from the Sun),
the total solar irradiance TSI, and the classical sunspot number give only a
little appearance of a up-turn. In the modern era there is no precedent for
such a protracted activity minimum, but there are historical records from a
century ago of a similar pattern (transition between cycles 13 and 14, 107
years ago).
Activity is expected to pick up in the next months. In the meanwhile is a good
opportunity to use the excellent data available from many satellites to
improve LAT analysis of solar flares and practise in flare monitoring and
analysis, to be ready when the first intense flare of cycle 24 will arrive.
## IV Data selection
Since August 2008 flares detected by RHESSI and GOES have been continuously
monitored, analysing LAT data for flare events potentially detectable by the
LAT and computing upper limits on the solar high energy emission. Solar flares
have been searched in LAT data from Augusr 2008 to the end of August 2009. LAT
data have been analysed in the time intervals of flares detected by GBM,
RHESSI and GOES. A zenith cut of $105^{\circ}$ has been applied to eliminate
photons from the Earth’s albedo. For this analysis the “Diffuse” classLAT
selection has been adopted, corresponding to the events with the highest
photon classification probability, using the IRFs (Instrumental Response
Functions) version P6_V3.
## V Analysis method
The list of flares detected by RHESSIrhessi and the _Solar Monitor_ web
sitesolarmonitor were monitored constantly at a daily basis. Flares seen by
RHESSI and GOES with more than $10^{5}$ counts (detected by RHESSI) have been
selected. For each of these flares start and end time of the event in _Fermi_
MET (Mission Elapsed Time), the position of the Sun during the flare and the
angle of the Sun direction with the LAT boresight have been computed.
The excess of events in the LAT data has been searched for flares within the
LAT field of view (angle with the LAT boresight $<80^{\circ}$). Although the
Sun is a moving source in the sky, covering about $1^{\circ}$ per day, in this
analysis the Sun has been considered as a fixed source, due to the short
duration of the flare events ($<1$ h). As analysis method a likelihood fitting
technique has been used, performed with a model that includes the Sun as a
point source and fixed galactic and extragalactic diffuse emission.
Moreover, the upper limit of high energy solar emission integrated on more
than one year of flares has been computed. LAT data of flares detected by
RHESSI have been collected (data selected one hour before and five hours later
with respect RHESSI flares, because of the long duration of high energy
emission). The position of the Sun has been computed using a JPL library
interfaceJPL and then data have been centered on the istantaneous solar
position. Successively these data have been merged and the analysis has been
performed, using the standard likelihood technique provided by the LAT
ScienceTools package (v9r15). Since the Sun is a moving source in the sky, the
problem is to compute the correct galactic and extragalactic diffuse
background emission as the Sun moves through the sky. In order to evaluate the
diffuse background in proper way the fake source method has been used. The
fake Sun follows the real Sun along the same path (i.e. the ecliptic) but at
an angular distance of $30^{\circ}$. The fake Sun is therefore exposed to the
same celestial sources as the true Sun and the events observed in the frame
centered on the fake Sun make a good description of the diffuse background.
The model for the likelihood analysis is composed by two fixed components
(quiet Sun and fake Sun) obtained in previous analysisquietSun moriond AIP
and the flaring Sun free component.
## VI Results
At 20:14:42.77 UT on 02 November 2008, _Fermi_ -GBM triggered and located a
very soft and bright eventgcn . The event location was RA = 217.6 deg, Dec =
-15.7 deg ($\pm 1.1$ deg), in excellent agreement with the Sun location. The
time of the event coincides with the solar activity reported in GOES solar
reports (event 9790: onset at 20:12 UT, max at 20:15 UT, end at 20:17, B5.7
flare). This is the first GBM detection of a solar flare. _Fermi_ -GBM
triggered on a solar flare a second time at 19:37:46.39 UT on 28 October
2009gcn2 . LAT data have been selected in the energy range 100 MeV - 300 GeV,
according to the solar activity detected by GOES and RHESSI: no high energy
emission has been detected by the LAT for both events.
From August 2008 to August 2009 RHESSI has detected 200 flares with $>10^{5}$
counts. The highest energy band in which most of these flare have been
observed by RHESSI is 3-6 keV. Few flares ($<20$) have been observed in the
energy band 6-12 or 12-25 keV. Flares outside the LAT field of view and the
ones that occurred while the LAT was transiting in the SAA have been
discarded. As a result LAT data of 80 flares have been analysed and the upper
limit on the high energy ($>100\textnormal{ MeV}$) emission has been computed
for each of these flares. No significant emission has been detected.
The preliminary upper limit on the emission of the flaring Sun integrated over
one year of flares in LAT data is $5.67\cdot 10^{-7}\text{photons
cm}^{-2}\text{ s}^{-1}$. This value is derived from a cumulative analysis of
all the 80 flares with a time of six hour around each trigger time (one hour
before and five hours later), taking into account the quiet sun
componentquietSun2 . A more detailed analysis is in preparation.
## VII Conclusions
Solar flare events have been searched in the first year of LAT data (August
2008 - August 2009). Up until now there is no evidence of high energy emission
from solar flares detected by the LAT, while the quiet Sun emission has been
detectedquietSun moriond AIP . However, the Sun is at the minimum of its
activity cycle and no intense flare has occurred. The solar activity is
expected to rise in the next months, reaching the maximum in 2012.
We will continue to monitor the active regions of the Sun and to improve our
analysis techniques, waiting for an intense flare detectable by the LAT.
###### Acknowledgements.
The _Fermi_ LAT Collaboration acknowledges support from a number of agencies
and institutes for both development and the operation of the LAT as well as
scientific data analysis. These include NASA and DOE in the United States,
CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA
in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council
and the National Space Board in Sweden. Additional support from INAF in Italy
for science analysis during the operations phase is also gratefully
acknowledged. Work supported by Department of Energy contract DE-
AC03-76SF00515.
## References
* (1) A. Abdo _et al._ , 2009, _Astroparticle Physics_ , Volume 32, Issue 3-4, p. 193-219.
* (2) W. B. Atwood _et al._ , 2009, _The Astrophysical Journal_ , 697 1071.
* (3) G. Share, R. Murphy, 2007, _GLAST FIRST SYMPOSIUM_ , AIP Conference Proceedings, 921.
* (4) G. Share _et al._ , 2006, _BAAS_ , 38, 255.
* (5) V.V. Grechnev, 2008, _Sol. Phys._ , 252, 149.
* (6) G. Kanbach _et al._ , 1993, _A &AS_, 97, 349.
* (7) L. Svalgaard, 2009, _RHESSI Science Nugget 99_.
* (8) _Rhessi flare list_ , http://hesperia.gsfc.nasa.gov/hessidata/dbase/hessi_flare_list.txt.
* (9) _Solar Monitor_ , http://www.solarmonitor.org/.
* (10) http://iau-comm4.jpl.nasa.gov/access2ephs.html.
* (11) E. Orlando _Fermi-LAT Observation of quiescent solar emission_ , proceedings of $31^{\text{st}}$ ICRC.
* (12) M. Brigida, 2009, 44th Rencontres de Moriond Proceedings.
* (13) N. Giglietto, 2009, AIP Conference Proceedings, 1112 238.
* (14) C. Kouveliotou, _GCN Circular 8477_.
* (15) P.N. Bath, _GCN Circular 10105_.
* (16) E. Orlando _Fermi-LAT Observation of Quiescent Solar Emission_ , these proceedings.
|
arxiv-papers
| 2009-12-18T15:04:03 |
2024-09-04T02:49:07.104897
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Iafrate (1), F. Longo (2) (for the FERMI Large Area Telescope\n Collaboration) ((1) INAF - Astronomical Observatory of Trieste, Italy, (2)\n INFN Trieste, Italy and Dipartimento di Fisica, Trieste, Italy)",
"submitter": "Nicola Giglietto",
"url": "https://arxiv.org/abs/0912.3696"
}
|
0912.3802
|
2010335-346Nancy, France 335
László Egri
Andrei Krokhin
Benoit Larose
Pascal Tesson
# The complexity of the list homomorphism problem for graphs
L. Egri School of Computer Science, McGill University, Montréal, Canada
laszlo.egri@mail.mcgill.ca , A. Krokhin School of Engineering and Computing
Sciences, Durham University, Durham, UK andrei.krokhin@durham.ac.uk , B.
Larose Department of Mathematics and Statistics, Concordia University,
Montréal, Canada larose@mathstat.concordia.ca and P. Tesson Department of
Computer Science, Laval University, Quebec City, Canada
pascal.tesson@ift.ulaval.ca
###### Abstract.
We completely classify the computational complexity of the list
$\mathbf{H}$-colouring problem for graphs (with possible loops) in
combinatorial and algebraic terms: for every graph $\mathbf{H}$ the problem is
either NP-complete, NL-complete, L-complete or is first-order definable;
descriptive complexity equivalents are given as well via Datalog and its
fragments. Our algebraic characterisations match important conjectures in the
study of constraint satisfaction problems.
###### Key words and phrases:
graph homomorphism, constraint satisfaction problem, complexity, universal
algebra, Datalog
## 1\. Introduction
Homomorphisms of graphs, i.e. edge-preserving mappings, generalise graph
colourings, and can model a wide variety of combinatorial problems dealing
with mappings and assignments [Hell04:book]. Because of the richness of the
homomorphism framework, many computational aspects of graph homomorphisms have
recently become the focus of much attention. In the list
$\mathbf{H}$-colouring problem (for a fixed graph $\mathbf{H}$), one is given
a graph $\mathbf{G}$ and a list $L_{v}$ of vertices of $\mathbf{H}$ for each
vertex $v$ in $\mathbf{G}$, and the goal is to determine whether there is a
homomorphism $h$ from $\mathbf{G}$ to $\mathbf{H}$ such that $h(v)\in L_{v}$
for all $v$. The complexity of such problems has been studied by combinatorial
methods, e.g., in [Feder99:list, Feder03:bi-arc]. In this paper, we study the
complexity of the list homomorphism problem for graphs in the wider context of
classifying the complexity of constraint satisfaction problems (CSP), see
[Barto09:digraphs, Feder98:monotone, Hell08:survey]. It is well known that the
CSP can be viewed as the problem of deciding whether there exists a
homomorphism from a relational structure to another, thus naturally extending
the graph homomorphism problem.
One line of CSP research studies the non-uniform CSP, in which the target (or
template) structure ${\bf T}$ is fixed and the question is whether there
exists a homomorphism from an input structure to $\bf T$. Over the last years,
much work has been done on classifying the complexity of this problem, denoted
$\operatorname{Hom}(\bf T)$ or $\operatorname{CSP}({\bf T})$, with respect to
the fixed target structure, see surveys [Bulatov08:duality,
Bulatov07:UAsurvey, Cohen06:handbook, Hell08:survey]. Classification here is
understood with respect to both computational complexity (i.e. membership in a
given complexity class such as P, NL, or L, modulo standard assumptions) and
descriptive complexity (i.e. definability of the class of all positive, or all
negative, instances in a given logic).
The best-known classification results in this direction concern the
distinction between polynomial-time solvable and NP-complete CSPs. For
example, a classical result of Hell and Nešetřil (see [Hell04:book,
Hell08:survey]) shows that, for a graph ${\bf H}$, $\operatorname{Hom}({\bf
H})$ (aka ${\bf H}$-colouring) is tractable if ${\bf H}$ is bipartite or
admits a loop, and is NP-complete otherwise, while Schaefer’s dichotomy
[Schaefer78:complexity] proves that any Boolean CSP is either in P or NP-
complete. Recent work [Allender09:refining] established a more precise
classification in the Boolean case: if ${\bf T}$ is a structure on $\\{0,1\\}$
then $\operatorname{CSP}({\bf T})$ is either NP-complete, P-complete, NL-
complete, $\oplus$L-complete, L-complete or in AC0.
Much of the work concerning the descriptive complexity of CSPs is centred
around the database-inspired logic programming language Datalog and its
fragments (see [Bulatov08:duality, Dalmau05:linear, Egri07:symmetric,
Feder98:monotone, Kolaitis08:logical]). Feder and Vardi initially showed
[Feder98:monotone] that a number of important tractable cases of
$\operatorname{CSP}(\bf T)$ correspond to structures for which
$\neg\operatorname{CSP}(\bf T)$ (the complement of $\operatorname{CSP}(\bf
T)$) is definable in Datalog. Similar ties were uncovered more recently
between the two fragments of Datalog known as linear and symmetric Datalog and
structures ${\bf T}$ for which $\operatorname{CSP}({\bf T})$ belongs to NL and
L, respectively [Dalmau05:linear, Egri07:symmetric].
Algebra, logic and combinatorics provide three angles of attack which have
fueled progress in this classification effort [Bulatov08:duality,
Bulatov07:UAsurvey, Cohen06:handbook, Hell04:book, Hell08:survey,
Kolaitis08:logical]. The algebraic approach (see [Bulatov07:UAsurvey,
Cohen06:handbook]) links the complexity of $\operatorname{CSP}({\bf T})$ to
the set of functions that preserve the relations in ${\bf T}$. In this
framework, one associates to each ${\bf T}$ an algebra $\mathbb{A}_{\bf T}$
and exploits the fact that the properties of $\mathbb{A}_{\bf T}$ completely
determine the complexity of $\operatorname{CSP}({\bf T})$. This angle of
attack was crucial in establishing key results in the field (see, for example,
[Barto09:bounded, Bulatov03:conservative, Bulatov07:UAsurvey]).
Tame Congruence Theory, a deep universal-algebraic framework first developed
by Hobby and McKenzie in the mid 80’s [Hobby88:structure], classifies the
local behaviour of finite algebras into five types (unary, affine, Boolean,
lattice and semilattice.) It was recently shown (see [Bulatov08:duality,
Bulatov07:UAsurvey, Larose09:universal]) that there is a strong connection
between the computational and descriptive complexity of
$\operatorname{CSP}({\bf T})$ and the set of types that appear in
$\mathbb{A}_{\bf T}$ and its subalgebras. There are strong conditions
involving types which are sufficient for NL-hardness, P-hardness and NP-
hardness of $\operatorname{CSP}({\bf T})$ as well as for inexpressibility of
$\neg\operatorname{CSP}({\bf T})$ in Datalog, linear Datalog and symmetric
Datalog. These sufficient conditions are also suspected (and in some cases
proved) to be necessary, under natural complexity-theoretic assumptions. For
example, (a) the presence of unary type is known to imply NP-completeness,
while its absence is conjectured to imply tractability (see
[Bulatov07:UAsurvey]); (b) the absence of unary and affine types was recently
proved to be equivalent to definability in Datalog [Barto09:bounded]; (c) the
absence of unary, affine, and semilattice types is proved necessary, and
suspected to be sufficient, for membership in NL and definability in linear
Datalog [Larose09:universal]; (d) the absence of all types but Boolean is
proved necessary, and suspected to be sufficient, for membership in L and
definability in symmetric Datalog [Larose09:universal]. The strength of
evidence varies from case to case and, in particular, the conjectured
algebraic conditions concerning CSPs in NL and L (and, as mentioned above,
linear and symmetric Datalog) still rest on relatively limited evidence
[Bulatov08:duality, Dalmau05:linear, Dalmau08:symDatalog, Dalmau08:majority,
Larose09:universal].
The aim of the present paper is to show that these algebraic conditions are
indeed sufficient and necessary in the special case of list
$\mathbf{H}$-colouring for undirected graphs (with possible loops), and to
characterise, in this special case, the dividing lines in graph-theoretic
terms (both via forbidden subgraphs and through an inductive definition). One
can view the list $\mathbf{H}$-colouring problem as a CSP where the template
is the structure $\mathbf{H}^{L}$ consisting of the binary (edge) relation of
$\mathbf{H}$ and all unary relations on $H$ (i.e. every subset of $H$).
Tractable list homomorphism problems for general structures were characterised
in [Bulatov03:conservative] in algebraic terms. The tractable cases for graphs
were described in [Feder03:bi-arc] in both combinatorial and (more specific)
algebraic terms; the latter implies, when combined with a recent result
[Dalmau08:majority], that in these cases
$\neg\operatorname{CSP}({\mathbf{H}^{L}})$ definable in linear Datalog and
therefore $\operatorname{CSP}(\mathbf{H}^{L})$ is in fact in NL. We complete
the picture by refining this classification and showing that
$\operatorname{CSP}({\mathbf{H}^{L}})$ is either NP-complete, or NL-complete,
or L-complete or in AC0 (and in fact first-order definable). We also remark
that the problem of recognising into which case the problem
$\operatorname{CSP}({\mathbf{H}^{L}})$ falls can be solved in polynomial time.
As we mentioned above, the distinction between NP-complete cases and those in
NL follows from earlier work [Feder03:bi-arc], and the situation is similar
with distinction between L-hard cases and those leading to membership in AC0
[Larose07:FOlong, Larose09:universal]. Therefore, the main body of technical
work in the paper concerns the distinction between NL-hardness and membership
in L. We give two equivalent characterisations of the class of graphs
$\mathbf{H}$ such that $\operatorname{CSP}({\mathbf{H}^{L}})$ is in L. One
characterisation is via forbidden subgraphs (for example, the reflexive graphs
in this class are exactly the $(P_{4},C_{4})$-free graphs, while the
irreflexive ones are exactly the bipartite $(P_{6},C_{6})$-free graphs), while
the other is via an inductive definition. The first characterisation is used
to show that graphs outside of this class give rise to NL-hard problems; we do
this by providing constructions witnessing the presence of a non-Boolean type
in the algebras associated with the graphs. The second characterisation is
used to prove positive results. We first provide operations in the associated
algebra which satisfy certain identities; this allows us to show that the
necessary condition on types is also sufficient in our case. We also use the
inductive definition to demonstrate that the class of negative instances of
the corresponding CSP is definable in symmetric Datalog, which implies
membership of the CSP in L.
## 2\. Preliminaries
### 2.1. Graphs and relational structures
In the following we denote the underlying universe of a structure
$\mathbf{S}$, $\mathbf{T}$, … by its roman equivalent $S$, $T$, etc. A
signature is a (finite) set of relation symbols with associated arities. Let
$\mathbf{T}$ be a structure of signature $\tau$; for each relation symbol
$R\in\tau$ we denote the corresponding relation of $\mathbf{T}$ by
$R(\mathbf{T})$. Let $\mathbf{S}$ be a structure of the same signature. A
homomorphism from $\mathbf{S}$ to $\mathbf{T}$ is a map $f$ from $S$ to $T$
such that $f(R(\mathbf{S}))\subseteq R(\mathbf{T})$ for each $R\in\tau$. In
this case we write $f:\mathbf{S}\rightarrow\mathbf{T}$. A structure
$\mathbf{T}$ is called a core if every homomorphism from $\mathbf{T}$ to
itself is a permutation on $T$. We denote by $\operatorname{CSP}(\mathbf{T})$
the class of all $\tau$-structures $\mathbf{S}$ that admit a homomorphism to
$\mathbf{T}$, and by $\neg\operatorname{CSP}(\mathbf{T})$ the complement of
this class.
The direct $n$-th power of a $\tau$-structure $\mathbf{T}$, denoted
$\mathbf{T}^{n}$, is defined to have universe $T^{n}$ and, for any (say
$m$-ary) $R\in\tau$, $({\bf a}_{1},\ldots,{\bf a}_{m})\in R(\mathbf{T}^{n})$
if and only if $({\bf a}_{1}[i],\ldots,{\bf a}_{m}[i])\in R(\mathbf{T})$ for
each $1\leq i\leq n$. For a subset $I\subseteq T$, the substructure induced by
$I$ on $\mathbf{T}$ is the structure $\mathbf{I}$ with universe $I$ and such
that $R(\mathbf{I})=R(\mathbf{T})\cap I^{m}$ for every $m$-ary $R\in\tau$.
For the purposes of this paper, a graph is a relational structure
${\mathbf{H}}=\langle H;\theta\rangle$ where $\theta$ is a symmetric binary
relation on $H$. The graph $\mathbf{H}$ is reflexive (irreflexive) if
$(x,x)\in\theta$ ($(x,x)\not\in\theta$) for all $x\in H$. Given a graph
$\mathbf{H}$, let $S_{1},\dots,S_{k}$ denote all subsets of $H$; let
$\mathbf{H}^{L}$ be the relational structure obtained from $\mathbf{H}$ by
adding all the $S_{i}$ as unary relations; more precisely, let $\tau$ be the
signature that consists of one binary relational symbol $\theta$ and unary
symbols $R_{i}$, $i=1,\dots,k$. The $\tau$-structure $\mathbf{H}^{L}$ has
universe $H$, $\theta(\mathbf{H}^{L})$ is the edge relation of $\mathbf{H}$,
and $R_{i}(\mathbf{H}^{L})=S_{i}$ for all $i=1,\dots,k$. It is easy to see
that $\mathbf{H}^{L}$ is a core. We call $\operatorname{CSP}(\mathbf{H}^{L})$
the list homomorphism problem for $\mathbf{H}$. Note that if $\mathbf{G}$ is
an instance of this problem then $\theta(\mathbf{G})$ can be considered as a
digraph, but the directions of the arcs are unimportant because $\mathbf{H}$
is undirected. Also, if an element $v\in G$ is in $R_{i}(\mathbf{G})$ then
this is equivalent to $v$ having $S_{i}$ as its list, so $\mathbf{G}$ can be
thought of as a digraph with $\mathbf{H}$-lists.
In [Feder03:bi-arc], a dichotomy result was proved, identifying bi-arc graphs
as those whose list homomorphism problem is tractable, and others as giving
rise to NP-complete problems. Let $C$ be a circle with two specified points
$p$ and $q$. A bi-arc is a pair of arcs $(N,S)$ such that $N$ contains $p$ but
not $q$ and $S$ contains $q$ but not $p$. A graph $\mathbf{H}$ is a bi-arc
graph if there is a family of bi-arcs $\\{(N_{x},S_{x}):x\in H\\}$ such that,
for every $x,y\in H$, the following hold: (i) if $x$ and $y$ are adjacent,
then neither $N_{x}$ intersects $S_{y}$ nor $N_{y}$ intersects $S_{x}$, and
(ii) if $x$ is not adjacent to $y$ then both $N_{x}$ intersects $S_{y}$ and
$N_{y}$ intersects $S_{x}$.
### 2.2. Algebra
An $n$-ary operation on a set $A$ is a map $f:A^{n}\rightarrow A$, a
projection is an operation of the form $e_{n}^{i}(x_{1},\ldots,x_{n})=x_{i}$
for some $1\leq i\leq n$. Given an $h$-ary relation $\theta$ and an $n$-ary
operation $f$ on the same set $A$, we say that $f$ preserves $\theta$ or that
$\theta$ is invariant under $f$ if the following holds: given any matrix $M$
of size $h\times n$ whose columns are in $\theta$, applying $f$ to the rows of
$M$ will produce an $h$-tuple in $\theta$.
A polymorphism of a structure $\mathbf{T}$ is an operation $f$ that preserves
each relation in $\mathbf{T}$; in this case we also say that $\mathbf{T}$
admits $f$. In other words, an $n$-ary polymorphism of $\mathbf{T}$ is simply
a homomorphism from $\mathbf{T}^{n}$ to $\mathbf{T}$. With any structure
$\mathbf{T}$, one associates an algebra $\mathbb{A}_{\mathbf{T}}$ whose
universe is $T$ and whose operations are all polymorphisms of $\mathbf{T}$.
Given a graph $\mathbf{H}$, we let $\mathbb{H}$ denote the algebra associated
with $\mathbf{H}^{L}$. An operation on a set is called conservative if it
preserves all subsets of the set (as unary relations). So, the operations of
$\mathbb{H}$ are the conservative polymorphisms of $\mathbf{H}$. Polymorphisms
can provide a convenient language when defining classes of graphs. For
example, it was shown in [Brewster08:nuf] that a graph is a bi-arc graph if
and only if it admits a conservative majority operation where a majority
operation is a ternary operation $m$ satisfying the identities
$m(x,x,y)=m(x,y,x)=m(y,x,x)=x$.
In order to state some of our results, we will need the notions of a variety
and a term operation. Let $I$ be a signature, i.e. a set of operation symbols
$f$ each of a fixed arity (we use the term “signature” for both structures and
algebras, this will cause no confusion). An algebra of signature $I$ is a pair
$\mathbb{A}=\langle A;F\rangle$ where $A$ is a non-empty set (the universe of
$\mathbb{A}$) and $F=\\{f^{\mathbb{A}}:f\in I\\}$ is the set of basic
operations (for each $f\in I$, $f^{\mathbb{A}}$ is an operation on $A$ of the
corresponding arity). The term operations of $\mathbb{A}$ are the operations
built from the operations in $F$ and projections by using composition. An
algebra all of whose (basic or term) operations are conservative is called a
conservative algebra. A class of similar algebras (i.e. algebras with the same
signature) which is closed under formation of homomorphic images, subalgebras
and direct products is called a variety. The variety generated by an algebra
$\mathbb{A}$ is denoted by $\mathcal{V}(\mathbb{A})$, and is the smallest
variety containing $\mathbb{A}$, i.e. the class of all homomorphic images of
subalgebras of powers of $\mathbb{A}$.
Tame Congruence Theory, as developed in [Hobby88:structure], is a powerful
tool for the analysis of finite algebras. Every finite algebra has a typeset,
which describes (in a certain specified sense) the local behaviour of the
algebra. It contains one or more of the following 5 types: (1) the unary type,
(2) the affine type, (3) the Boolean type, (4) the lattice type and (5) the
semilattice type. The numbering of the types is fixed, and they are often
referred to by their numbers. Simple algebras, i.e. algebras without non-
trivial proper homomorphic images, admit a unique type; the prototypical
examples are: any 2-element algebra whose basic operations are all unary has
type 1. A finite vector space has type 2. The 2-element Boolean algebra has
type 3. The 2-element lattice is the 2-element algebra with two binary
operations $\langle\\{0,1\\};\vee,\wedge\rangle$: it has type 4\. The
2-element semilattices are the 2-element algebras with a single binary
operation $\langle\\{0,1\\};\wedge\rangle$ and $\langle\\{0,1\\};\vee\rangle$:
they have type 5. The typeset of a variety $\mathcal{V}$, denoted
$typ(\mathcal{V})$, is simply the union of typesets of the algebras in it. We
will be mostly interested in type-omitting conditions for varieties of the
form $\mathcal{V}(\mathbb{A}_{\mathbf{T}})$, and Corollary 3.2 of
[Valeriote09:intersection] says that in this case it is enough to consider the
typesets of $\mathbb{A}_{\mathbf{T}}$ and its subalgebras.
On the intuitive level, if $\mathbf{T}$ is a core structure then the typeset
$typ(\mathcal{V}(\mathbb{A}_{\mathbf{T}}))$ contains crucial information about
the kind of relations that $\mathbf{T}$ can or cannot simulate, thus implying
lower/upper bounds on the complexity of $\operatorname{CSP}(\mathbf{T})$. For
our purposes here, it will not be necessary to delve further into the
technical aspects of types and typesets. We only note that there is a very
tight connection between the kind of equations that are satisfied by the
algebras in a variety and the types that are admitted or omitted by a variety,
i.e. those types that do or do not appear in the typesets of algebras in the
variety [Hobby88:structure].
In this paper, we use ternary operations $f_{1},\dots,f_{n}$ satisfying the
following identities:
$\displaystyle x$ $\displaystyle=$ $\displaystyle f_{1}(x,y,y)$ (1)
$\displaystyle f_{i}(x,x,y)$ $\displaystyle=$ $\displaystyle
f_{i+1}(x,y,y)\mbox{ for all $i=1,\ldots n-1$}$ (2) $\displaystyle
f_{n}(x,x,y)$ $\displaystyle=$ $\displaystyle y.$ (3)
The following lemma contains some type-omitting results that we use in this
paper.
###### Lemma 2.1.
[Hobby88:structure] A finite algebra $\mathbb{A}$ has term operations
$f_{1},\ldots,f_{n}$, for some $n\geq 1$, satisfying identities (1)–(3) if and
only if the variety $\mathcal{V}(\mathbb{A})$ omits types 1, 4 and 5.
If a finite algebra $\mathbb{A}$ has a majority term operation then
$\mathcal{V}(\mathbb{A})$ omits types 1, 2 and 5.
We remark in passing that operations satisfying identities (1)–(3) are also
known to characterise a certain algebraic (congruence) condition called
$(n+1)$-permutability [Hobby88:structure].
### 2.3. Datalog
Datalog is a query and rule language for deductive databases (see
[Kolaitis08:logical]). A Datalog program $\mathcal{D}$ over a (relational)
signature $\tau$ is a finite set of rules of the form $h\leftarrow
b_{1}\wedge\ldots\wedge b_{m}$ where $h$ and each $b_{i}$ are atomic formulas
$R_{j}(v_{1},...,v_{k})$. We say that $h$ is the head of the rule and that
$b_{1}\wedge\ldots\wedge b_{m}$ is its body. Relational predicates $R_{j}$
which appear in the head of some rule of $\mathcal{D}$ are called intensional
database predicates (IDBs) and are not part of the signature $\tau$. All other
relational predicates are called extensional database predicates (EDBs) and
are in $\tau$. So, a Datalog program is a recursive specification of IDBs
(from EDBs).
A rule of $\mathcal{D}$ is linear if its body contains at most one IDB and is
non-recursive if its body contains only EDBs. A linear but recursive rule is
of the form $I_{1}(\bar{x})\leftarrow I_{2}(\bar{y})\wedge
E_{1}(\bar{z}_{1})\wedge\ldots\wedge E_{k}(\bar{z}_{k})$ where $I_{1},I_{2}$
are IDBs and the $E_{i}$ are EDBs (note that the variables occurring in
$\bar{x},\bar{y},\bar{z}_{i}$ are not necessarily distinct). Each such rule
has a symmetric $I_{2}(\bar{y})\leftarrow I_{1}(\bar{x})\wedge
E_{1}(\bar{z}_{1})\wedge\ldots\wedge E_{k}(\bar{z}_{k}).$ A Datalog program is
non-recursive if all its rules are non-recursive, linear if all its rules are
linear and symmetric if it is linear and if the symmetric of each recursive
rule of $\mathcal{D}$ is also a rule of $\mathcal{D}$.
A Datalog program $\mathcal{D}$ takes a $\tau$-structure $\bf A$ as input and
returns a structure $\mathcal{D}$$(\bf A)$ over the signature
$\tau^{\prime}=\tau\cup\\{I:I$ is an IDB in $\mathcal{D}$$\\}$. The relations
corresponding to $\tau$ are the same as in $\bf A$, while the new relations
are recursively computed by $\mathcal{D}$ , with semantics naturally obtained
via least fixed-point of monotone operators. We also want to view a Datalog
program as being able to accept or reject an input $\tau$-structure and this
is achieved by choosing one of the IDBs of $\mathcal{D}$ as the goal
predicate: the $\tau$-structure $\bf A$ is accepted by $\mathcal{D}$ if the
goal predicate is non-empty in $\mathcal{D}(\bf A)$. Thus every Datalog
program with a goal predicate defines a class of structures - those that are
accepted by the program.
When using Datalog to study $\operatorname{CSP}(\mathbf{T})$, one usually
speaks of the definability of $\neg\operatorname{CSP}(\mathbf{T})$ in Datalog
(i.e. by a Datalog program) or its fragments (because any class definable in
Datalog must be closed under extension). Examples of CSPs definable in Datalog
and its fragments can be found, e.g., in [Bulatov08:duality,
Egri07:symmetric]. As we mentioned before, any problem
$\operatorname{CSP}(\mathbf{T})$ is tractable if its complement is definable
in Datalog, and all such structures were recently identified in
[Barto09:bounded]. Definability of $\neg\operatorname{CSP}(\mathbf{T})$ in
linear (symmetric) Datalog implies that $\operatorname{CSP}(\mathbf{T})$
belongs to NL and L, respectively [Dalmau05:linear, Egri07:symmetric]. As we
discussed in Section 1, there is a connection between definability of CSPs in
Datalog (and its fragments) and the presence/absence of types in the
corresponding algebra (or variety).
Note that it follows from Lemma 2.1 and from the results in
[Larose09:universal, Larose06:bounded] that if, for a core structure
$\mathbf{T}$, $\neg\operatorname{CSP}(\mathbf{T})$ is definable in symmetric
Datalog then $\mathbf{T}$ must admit, for some $n$, operations satisfying
identities (1)–(3). Moreover, with the result of [Barto09:bounded], a
conjecture from [Larose09:universal] can be restated as follows: for a core
structure $\mathbf{T}$, if $\neg\operatorname{CSP}(\mathbf{T})$ is definable
in Datalog and, for some $n$, $\mathbf{T}$ admits operations satisfying
(1)–(3), then $\neg\operatorname{CSP}(\mathbf{T})$ is definable in symmetric
Datalog. This conjecture is proved in [Dalmau08:symDatalog] for $n=1$.
## 3\. A class of graphs
In this section, we give combinatorial characterisations of a class of graphs
whose list homomorphism problem will turn out to belong to L.
Let $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ be bipartite irreflexive graphs,
with colour classes $B_{1}$, $T_{1}$ and $B_{2}$ and $T_{2}$ respectively,
with $T_{1}$ and $B_{2}$ non-empty. We define the special sum
$\mathbf{H}_{1}\odot\mathbf{H}_{2}$ (which depends on the choice of the
$B_{i}$ and $T_{i}$) as follows: it is the graph obtained from the disjoint
union of $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ by adding all possible edges
between the vertices in $T_{1}$ and $B_{2}$. Notice that we can often
decompose a bipartite graph in several ways, and even choose $B_{1}$ or
$T_{2}$ to be empty. We say that an irreflexive graph $\mathbf{H}$ is a
special sum or expressed as a special sum if there exist two bipartite graphs
and a choice of colour classes on each such that $\mathbf{H}$ is isomorphic to
the special sum of these two graphs.
Let $\mathcal{K}$ denote the smallest class of irreflexive graphs containing
the one-element graph and closed under (i) special sum and (ii) disjoint
union. We call the graphs in $\mathcal{K}$ basic irreflexive.
The following result gives a characterisation of basic irreflexive graphs in
terms of forbidden subgraphs:
###### Lemma 3.1.
Let $\mathbf{H}$ be an irreflexive graph. Then the following conditions are
equivalent:
1. (1)
$\mathbf{H}$ is basic irreflexive;
2. (2)
$\mathbf{H}$ is bipartite, contains no induced 6-cycle, nor any induced path
of length 5.
We shall now describe our main family of graphs, first by forbidden induced
subgraphs, and then in an inductive manner.
Define the class $\mathcal{L}$ of graphs as follows: a graph $\mathbf{H}$
belongs to $\mathcal{L}$ if it contains none of the following as an induced
subgraph:
1. (1)
the reflexive path of length 3 and the reflexive 4-cycle;
2. (2)
the irreflexive cycles of length 3, 5 and 6, and the irreflexive path of
length 5;
3. (3)
${\bf B1}$, ${\bf B2}$, ${\bf B3}$, ${\bf B4}$, ${\bf B5}$ and ${\bf B6}$ (see
Figure 1.)
$\mathbf{B1}$$\mathbf{B2}$$\mathbf{B3}$$\mathbf{B4}$$\mathbf{B5}$$\mathbf{B6}$$c$$b$$a$$c$$b$$a$$d$$c$$b$$a$$e$$d$$c$$b$$a$$a^{\prime}$$b^{\prime}$$c^{\prime}$$a$$b$$c$$a^{\prime}$$b^{\prime}$$c^{\prime}$$a$$b$$c$
Figure 1. The forbidden mixed graphs.
We will now characterise the class $\mathcal{L}$ in an inductive manner.
A connected graph $\mathbf{H}$ is basic if either (i) $\mathbf{H}$ is a single
loop, or (ii) $\mathbf{H}$ is a basic irreflexive graph, or (iii) $\mathbf{H}$
is obtained from a basic irreflexive graph $\mathbf{H}_{1}$ with colour
classes $B$ and $T$ by adding every edge (including loops) of the form
$\\{t,t^{\prime}\\}$ where $t,t^{\prime}\in T$.
Given two vertex-disjoint graphs $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$, the
adjunction of $\mathbf{H}_{1}$ to $\mathbf{H}_{2}$ is the graph
$\mathbf{H}_{1}\oslash\mathbf{H}_{2}$ obtained by taking the disjoint union of
the two graphs, and adding every edge of the form $\\{x,y\\}$ where $x$ is a
loop in $\mathbf{H}_{1}$ and $y$ is a vertex of $\mathbf{H}_{2}$.
###### Lemma 3.2.
Let $\mathcal{L}_{R}$ denote the class of reflexive graphs in $\mathcal{L}$.
Then $\mathcal{L}_{R}$ is the smallest class $\mathcal{D}$ of reflexive graphs
such that:
1. (1)
$\mathcal{D}$ contains the one-element graph;
2. (2)
$\mathcal{D}$ is closed under disjoint union;
3. (3)
if $\mathbf{H}_{1}$ is a single loop and $\mathbf{H}_{2}\in\mathcal{D}$ then
$\mathbf{H}_{1}\oslash\mathbf{H}_{2}\in\mathcal{D}$.
Lemma 3.2 states that the reflexive graphs avoiding the path of length 3 and
the 4-cycle are precisely those constructed from the one-element loop using
disjoint union and adjunction of a universal vertex. These graphs can also be
described by the following property: every connected induced subgraph of size
at most 4 has a universal vertex. These graphs have been studied previously as
those with NLCT width 1, which were proved to be exactly the trivially perfect
graphs [Gurski06:co-graphs]. Our result provides an alternative proof of the
equivalence of these conditions.
###### Theorem 3.3.
The class $\mathcal{L}$ is the smallest class $\mathcal{C}$ of graphs such
that:
1. (1)
$\mathcal{C}$ contains the basic graphs;
2. (2)
$\mathcal{C}$ is closed under disjoint union;
3. (3)
if $\mathbf{H}_{1}$ is a basic graph and $\mathbf{H}_{2}\in\mathcal{C}$ then
$\mathbf{H}_{1}\oslash\mathbf{H}_{2}\in\mathcal{C}$.
###### Proof 3.4.
We start by showing that every basic graph is in $\mathcal{L}$, i.e. that a
basic graph does not contain any of the forbidden graphs. If $\mathbf{H}$ is a
single loop or a basic irreflexive graph, then this is immediate. Otherwise
$\mathbf{H}$ is obtained from a basic irreflexive graph $\mathbf{H}_{1}$ with
colour classes $B$ and $T$ by adding every edge of the form $(t_{1},t_{2})$
where $t_{i}\in T$. In particular, the loops form a clique and no edge
connects two non-loops; it is clear in that case that $\mathbf{H}$ contains
none of ${\bf B1}$, ${\bf B2}$, ${\bf B3}$, ${\bf B4}$. On the other hand if
$\mathbf{H}$ contains ${\bf B5}$ or ${\bf B6}$, then $\mathbf{H}_{1}$ contains
the path of length 5 or the 6-cycle, contradicting the fact that
$\mathbf{H}_{1}$ is basic.
Next we show that $\mathcal{L}$ is closed under disjoint union and adjunction
of basic graphs. It is obvious that the disjoint union of graphs that avoid
the forbidden graphs will also avoid these. So suppose that an adjunction
$\mathbf{H}_{1}\oslash\mathbf{H}_{2}$, where $\mathbf{H}_{1}$ is a basic
graph, contains an induced forbidden graph $\mathbf{B}$ whose vertices are
neither all in $H_{1}$ nor $H_{2}$; without loss of generality $H_{1}$
contains at least one loop, its loops form a clique and none of its edges
connects two non-loops. It is then easy to verify that $\mathbf{B}$ contains
both loops and non-loops. Because the other cases are similar, we prove only
that $\mathbf{B}$ is not ${\bf B3}$: since vertex $d$ is not adjacent to $a$
it must be in $\mathbf{H}_{2}$, and similarly for $c$. Since $b$ is not
adjacent to $d$ it must also be in $\mathbf{H}_{2}$; since non-loops of
$\mathbf{H}_{1}$ are not adjacent to elements of $\mathbf{H}_{2}$ it follows
that $a$ is in $\mathbf{H}_{2}$ also, a contradiction.
Now we must show that every graph in $\mathcal{L}$ can be obtained from the
basic graphs by disjoint union and adjunction of basic graphs. Suppose this is
not the case. If $\mathbf{H}$ is a counterexample of minimum size, then
obviously it is connected, and it contains at least one loop for otherwise it
is a basic irreflexive graph. By Lemma 3.2, $\mathbf{H}$ also contains at
least one non-loop.
For $a\in H$ let $N(a)$ denote its set of neighbours. Let
$\mathbf{R}(\mathbf{H})$ denote the subgraph of $\mathbf{H}$ induced by its
set $R(H)$ of loops, and let $\mathbf{J}(\mathbf{H})$ denote the subgraph
induced by $J(H)$, the set of non-loops of $\mathbf{H}$. Since $\mathbf{H}$ is
connected and neither ${\bf B1}$ nor ${\bf B2}$ is an induced subgraph of
$\mathbf{H}$, the graph $\mathbf{R}(\mathbf{H})$ is also connected, and
furthermore every vertex in $J(H)$ is adjacent to some vertex in $R(H)$. By
Lemma 3.2, we know that $\mathbf{R}(\mathbf{H})$ contains at least one
universal vertex: let $U$ denote the (non-empty) set of universal vertices of
$\mathbf{R}(\mathbf{H})$. Let $J$ denote the set of all $a\in J(H)$ such that
$N(a)\cap R(H)\subseteq U$. Let us show that $J\neq\emptyset$. For every $u\in
U$, there is $w\in J(H)$ not adjacent to $u$ because otherwise $\mathbf{H}$ is
obtained by adjoining $u$ to the rest of $\mathbf{H}$, a contradiction with
the choice of $\mathbf{H}$. If this $w$ has a neighbour $r\in R(H)\setminus U$
then there is some $s\in R(H)\setminus U$ not adjacent to $r$, and the graph
induced by $\\{w,u,s,r\\}$ contains ${\bf B2}$ or ${\bf B3}$, a contradiction.
Hence, $w\in J$. Let $\mathbf{S}$ denote the subgraph of $\mathbf{H}$ induced
by $U\cup J$. The graph $\mathbf{S}$ is connected. We claim that the following
properties also hold:
1. (1)
if $a$ and $b$ are adjacent non-loops, then $N(a)\cap U=N(b)\cap U$;
2. (2)
if $a$ is in a connected component of the subgraph of $\mathbf{S}$ induced by
$J$ with more than one vertex, then for any other $b\in J$, one of $N(a)\cap
U,N(b)\cap U$ contains the other.
The first statement holds because ${\bf B1}$ is forbidden, and the second
follows from the first because ${\bf B4}$ is also forbidden. Let
$J_{1},\dots,J_{k}$ denote the different connected components of $J$ in
$\mathbf{S}$. By (1) we may let $N(J_{i})$ denote the set of common neighbours
of members of $J_{i}$ in $U$. By (2), we can re-order the $J_{i}$’s so that
for some $1\leq m\leq k$ we have $N(J_{i})\subseteq N(J_{j})$ for all $i\leq
m$ and all $j>m$, and, in addition, we have $m=1$ or $|J_{i}|=1$ for all
$1\leq i\leq m$. Let $\mathbf{B}$ denote the subgraph of $\mathbf{S}$ induced
by $B=\bigcup_{i=1}^{m}{(J_{i}\cup N(J_{i}))}$, and let $\mathbf{C}$ be the
subgraph of $\mathbf{H}$ induced by $H\setminus B$. We claim that
$\mathbf{H}=\mathbf{B}\oslash\mathbf{C}$. For this, it suffices to show that
every element in $\bigcup_{i=1}^{m}N(J_{i})$ is adjacent to every non-loop
$c\in C$. By construction this holds if $c\in J\cap C$. Now suppose this does
not hold: then some $x\in J(H)\setminus J$ is not adjacent to some $y\in
N(J_{i})$ for some $i\leq m$. Since $x\not\in J$ we may find some $z\in
R(H)\setminus U$ adjacent to $x$; it is of course also adjacent to $y$. Since
$z\not\in U$ there exists some $z^{\prime}\in R(H)\setminus U$ that is not
adjacent to $z$, but it is of course adjacent to $y$. If $x$ is adjacent to
$z^{\prime}$, then $\\{x,z,z^{\prime}\\}$ induces a subgraph isomorphic to
${\bf B2}$, a contradiction. Otherwise, $\\{x,z,y,z^{\prime}\\}$ induces a
subgraph isomorphic to ${\bf B3}$, also a contradiction.
If every $J_{i}$ with $i\leq m$ contains a single element, notice that
$\mathbf{B}$ is a basic graph: indeed, removing all edges between its loops
yields a bipartite irreflexive graph which contains neither the path of length
5 nor the 6-cycle, since $\mathbf{B}$ contains neither ${\bf B5}$ nor ${\bf
B6}$. Since this contradicts our hypothesis on $\mathbf{H}$, we conclude that
$m=1$. But this means that $N(J_{1})$ is a set of universal vertices in
$\mathbf{H}$. Let $u$ be such a vertex and let $D$ denote its complement in
$\mathbf{H}$: clearly $\mathbf{H}$ is obtained as the adjunction of the single
loop $u$ to $D$, contradicting our hypothesis. This concludes the proof. ∎
## 4\. Classification results
Recall the standard numbering of types: (1) unary, (2) affine , (3) Boolean,
(4) lattice and (5) semilattice. We will need the following auxiliary result
(which is well known). Note that the assumptions of this lemma effectively say
that $\operatorname{CSP}(\mathbf{T})$ can simulate the graph $k$-colouring
problem (with $k=|U|$) or the directed $st$-connectivity problem.
###### Lemma 4.1.
Let $\mathbf{S},\mathbf{T}$ be structures, let $s_{1},s_{2}\in S$, and let
$R=\\{(f(s_{1}),f(s_{2}))\mid f:\mathbf{S}\rightarrow\mathbf{T}\\}$.
1. (1)
If $R=\\{(x,y)\in U^{2}\mid x\neq y\\}$ for some subset $U\subseteq T$ with
$|U|\geq 3$ then $\mathcal{V}(\mathbb{A}_{\mathbf{T}})$ admits type 1.
2. (2)
If $R=\\{(t,t),(t,t^{\prime}),(t^{\prime},t^{\prime})\\}$ for some distinct
$t,t^{\prime}\in T$ then $\mathcal{V}(\mathbb{A}_{\mathbf{T}})$ admits at
least one of the types 1, 4, 5.
* Proof [sketch]: The assumption of this lemma implies that $\mathbb{A}_{\mathbf{T}}$ has a subalgebra (induced by $U$ and $\\{t,t^{\prime}\\}$, respectively) such that all operations of the subalgebra preserve the relation $R$. It is well-known (see, e.g., [Hell04:book]) that all operations preserving the disequality relation on $U$ are essentially unary, while it is easy to check that the order relation on a 2-element set cannot admit operations satisfying identities (1)–(3), so one can use Lemma 2.1. $\blacksquare$
The following lemma connects the characterisation of bi-arc graphs given in
[Brewster08:nuf] with a type-omitting condition.
###### Lemma 4.2.
Let $\mathbf{H}$ be a graph. Then the following conditions are equivalent:
1. (1)
the variety ${\mathcal{V}}(\mathbb{H})$ omits type 1;
2. (2)
the graph $\mathbf{H}$ admits a conservative majority operation;
3. (3)
the graph $\mathbf{H}$ is a bi-arc graph.
The results summarised in the following theorem are known (or easily follow
from known results, with a little help from Lemma 4.2).
###### Theorem 4.3.
Let $\mathbf{H}$ be a graph.
* •
If $typ({\mathcal{V}}(\mathbb{H}))$ admits type 1, then
$\neg\operatorname{CSP}(\mathbf{H}^{L})$ is not expressible in Datalog and
$\operatorname{CSP}(\mathbf{H}^{L})$ is $\mathrm{NP}$-complete (under first-
order reductions);
* •
if $typ({\mathcal{V}}(\mathbb{H}))$ omits type 1 but admits type 4 then
$\neg\operatorname{CSP}(\mathbf{H}^{L})$ is not expressible in symmetric
Datalog but is expressible in linear Datalog, and
$\operatorname{CSP}(\mathbf{H}^{L})$ is $\mathrm{NL}$-complete (under first-
order reductions.)
|
arxiv-papers
| 2009-12-18T21:14:52 |
2024-09-04T02:49:07.113658
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Laszlo Egri, Andrei Krokhin, Benoit Larose, Pascal Tesson",
"submitter": "Laszlo Egri Mr.",
"url": "https://arxiv.org/abs/0912.3802"
}
|
0912.3826
|
# Systematic reduction of sign errors in many-body calculations of atoms and
molecules
Michal Bajdich Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA Murilo L. Tiago Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Randolph Q. Hood Lawrence Livermore National Laboratory, Livermore, CA 94550,
USA Paul R. C. Kent Center for Nanophase Materials Sciences, Oak Ridge
National Laboratory, Oak Ridge, TN 37831, USA Fernando A. Reboredo Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA
###### Abstract
The self-healing diffusion Monte Carlo algorithm (SHDMC) [Phys. Rev. B 79,
195117 (2009), ibid. 80, 125110 (2009)] is shown to be an accurate and robust
method for calculating the ground-state of atoms and molecules. By direct
comparison with accurate configuration interaction results for the oxygen atom
we show that SHDMC converges systematically towards the ground-state wave
function. We present results for the challenging N2 molecule, where the
binding energies obtained via both energy minimization and SHDMC are near
chemical accuracy (1 kcal/mol). Moreover, we demonstrate that SHDMC is robust
enough to find the nodal surface for systems at least as large as C20 starting
from random coefficients. SHDMC is a linear-scaling method, in the degrees of
freedom of the nodes, that systematically reduces the fermion sign problem.
###### pacs:
02.70.Ss,02.70.Tt
Since electrons are fermions, their many-body wave functions must change sign
when the coordinates of any pair are interchanged. In contrast, the sign of a
bosonic wave functions is unchanged for any coordinate interchange. Due to
this misleadingly small difference, the ground-state energy of bosons can be
determined by quantum Monte Carlo (QMC) methods HLRbook ; mfoulkesrmp2001
with an accuracy limited only by computing time, while QMC calculations of
fermions are either exponentially difficult, or are stabilized by imposing a
systematic error, a direct consequence of our lack of knowledge of the
fermionic nodal surface. Therefore, one of the most important problems in
many-body electronic structure theory is to accurately find representations of
the fermion nodes ceperley91 ; mtroyerprl2005 , the locations where the
fermionic wave function changes sign, the so-called “fermion sign problem”.
The sign problem limits (i) the number of physical systems where ab initio QMC
can be applied and (ii) our ability to improve approximations of density
functional theory (DFT) using QMC results ceperley80 . More importantly, it
limits our overall understanding of the effects of interactions in fermionic
systems. Therefore, a method to circumvent the sign problem with reduced
computational cost could transform Condensed Matter Theory, Quantum Chemistry
and Nuclear Physics among other fields.
Arguably the most accurate technique for calculating the ground-state of a
many-body system with more than $20$ fermions is diffusion Monte Carlo (DMC).
The standard DMC algorithm ceperley80 finds the lowest energy of all wave
functions that share the nodal surface $S_{T}({\bf R})$ imposed by a trial
wave function $\Psi_{T}({\bf R})$. This is the fixed-node approximation where
the resultant energy $E_{DMC}$ is a rigorous upper bound of the exact ground-
state energy anderson79 ; reynolds82 . The exact ground-state energy is
obtained only when $\Psi_{T}({\bf R})$ has the same nodal surface as the exact
ground-state wave function.
If the exact nodes are not provided, the implicit fixed-node ground-state wave
function $\Psi_{FN}({\bf R})$ will exhibit discontinuities in its gradient
reynolds82 ; keystone (i.e. kinks) on some parts of $S_{T}({\bf R})$. We
recently proved keystone that by locally smoothing these discontinuities in
$\Psi_{FN}({\bf R})$, a new trial wave function can be obtained with improved
nodes. This proof enables an algorithm that systematically moves the nodal
surface $S_{T}({\bf R})$ towards the one of an eigen-state. If the form of
trial wave function is sufficiently flexible, and given sufficient statistics,
this process leads to an exact eigen-state wave function keystone ;
rockandroll . We named the method self-healing DMC (SHDMC), since the trial
wave function is self-corrected in DMC and can recover even from a poor
starting point.
In this Letter, we report the first applications of SHDMC to real atoms and
molecules (O, N2, C20). SHDMC energies are within error bars of DMC
calculations using the current state of the art approach umrigar05 ; umrigar07
. Tests of SHDMC for C20 demonstrate that our method can be applied at the
nanoscale. Its cost scales linearly with the number of independent degrees of
freedom of the nodes with an accuracy limited only by the achievable
statistics and choice of representation of the nodes.
Brief review of SHDMC — SHDMC is fundamentally different from optimization
methods used in variational Monte Carlo (VMC): HLRbook ; mfoulkesrmp2001 (i)
the wave function is directly optimized based on a property of the nodal
surface and not on the local energy or its variance, and (ii) the nodes are
optimized at the DMC level (as opposed to a VMC based algorithm).
Using a short-time many-body propagator, SHDMC samples the coefficients of an
improved wave function removing the artificial derivative discontinuities of
$\Psi_{FN}({\bf R})$ arising from the inexact nodes. Repeated application of
this method results in the best nodal surface for a given basis. For wave
functions expanded in a complete basis it can be shown that the final accuracy
is limited only by the statistics keystone ; rockandroll .
In SHDMC (see Refs. keystone, ; rockandroll, for details), the weighted
walker distribution is ceperley80
$\displaystyle f({\bf R},\tau^{\prime}+\tau)$ $\displaystyle=\Psi_{T}^{*}({\bf
R},\tau^{\prime})\left[e^{-\tau(\hat{{\mathcal{H}}}_{FN}-E_{T})}\Psi_{T}({\bf
R},\tau^{\prime})\right]$ (1)
$\displaystyle=\lim_{N_{c}\rightarrow\infty}\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}W_{i}^{j}(k)\delta\left({\bf
R-R}_{i}^{j}\right),$
where
$\displaystyle\Psi_{T}({\bf R},\tau^{\prime})=e^{J({\bf
R})}\sum_{n}^{\sim}\lambda_{n}(\tau^{\prime})\Phi_{n}({\bf R})$ (2)
is a trial function where $\sum_{n}^{\sim}$ represents a truncated sum,
$\\{\Phi_{n}({\bf R})\\}$ forms a complete orthonormal basis of the
antisymmetric Hilbert space and $e^{J({\bf R})}$ is a symmetric Jastrow
factor. In Eq. (1), $\hat{{\mathcal{H}}}_{FN}$ is the fixed-node Hamiltonian
[$\hat{{\mathcal{H}}}_{FN}$ is the many-body Hamiltonian with an infinite
potential at the nodes of $\Psi_{T}({\bf R},\tau^{\prime})$] and $E_{T}$ is an
energy reference. Next, ${\bf R}_{i}^{j}$ corresponds to the position of the
walker $i$ at step $j$ of $N_{c}$ equilibrated configurations. The weights
$W_{i}^{j}(k)$ are given by
$\displaystyle
W_{i}^{j}(k)\\!=\\!e^{-\left[E_{i}^{j}(k)-E_{T}\right]\tau}\text{with
}E_{i}^{j}(k)\\!=\\!\frac{1}{k}\\!\sum_{\ell=0}^{k-1}\\!E_{L}({\bf
R}_{i}^{j-\ell}),$ (3)
where $E_{T}$ in Eq. (3) is periodically adjusted so that
$\sum_{i}W_{i}^{j}(k)\approx N_{c}$ and $\tau$ is $k\delta\tau$ (with $k$
being a number of steps and $\delta\tau$ a standard DMC time step).
From Eq. (1), one can formally obtain
$\displaystyle\tilde{\Psi}_{T}({\bf R},\tau^{\prime}+\tau)=f({\bf
R},\tau^{\prime}+\tau)/\Psi_{T}^{*}({\bf R},\tau^{\prime}).$ (4)
We now define the local smoothing function to be
$\displaystyle\tilde{\delta}\left({\bf
R^{\prime},R}\right)=\sum_{n}^{\sim}e^{J({\bf R^{\prime}})}\Phi_{n}({\bf
R^{\prime}})\Phi_{n}^{*}({\bf R})e^{-J({\bf R})}.$ (5)
Applying Eq. (5) to both sides of Eq. (4), using Eq. (1), and integrating over
${\bf R}$ we obtain
$\displaystyle\Psi_{T}({\bf R},\tau^{\prime}+\tau)=e^{J({\bf
R})}\sum_{n}^{\sim}\lambda_{n}(\tau^{\prime}+\tau)\Phi_{n}({\bf R}),$ (6)
with
$\displaystyle\lambda_{n}(\tau^{\prime}\\!+\\!\tau)=\\!\\!\lim_{N_{c}\rightarrow\infty}\frac{1}{\mathcal{N}}\sum_{i}^{N_{c}}W_{i}^{j}(k)e^{-2J({\bf
R}_{i}^{j})}\frac{\Phi_{n}^{*}({\bf R}_{i}^{j})}{\Phi_{T}^{*}({\bf
R}_{i}^{j},\tau^{\prime})}$ (7)
where $\mathcal{N}=\sum_{i=1}^{N_{c}}e^{-2J({\bf R}_{i}^{j})}$ normalizes the
Jastrow factor. These new $\lambda_{n}(\tau^{\prime}+\tau)$ [Eq. (7)] are used
to construct a new trial wave function [Eq. (2)] recursively within DMC
(therefore the name self-healing DMC). The weights in Eq. (3) can be evaluated
within (i) a branching algorithm keystone for
$\tau^{\prime}\rightarrow\infty$ or (ii) a fixed population scheme for small
$\tau^{\prime}$ rockandroll ; umrigar_private . The former method is more
robust, but the latter improves final convergence. Equation (7) can be related
to the maximum-overlap method used for bosonic wave functions reatto82 .
Since SHDMC is targeted for large systems we report validations using
pseudopotentials.
Validation of SHDMC with configuration interaction (CI) calculations for the O
atom — In short, CI is the diagonalization of the many-body Hamiltonian in a
truncated basis of Slater determinants. We chose to study the 3P ground-state
of the O atom because it has at least two valence electrons in both spin
channels burkatzki07 . The single-particle orbitals were expanded in VTZ and
V5Z Gaussian basis sets burkatzki07 using the GAMESS games09 code. To
facilitate a direct comparison between SHDMC and CI, no Jastrow factor was
employed.
Figure 1 shows a direct comparison of the first 250 converged coefficients
$\lambda_{n}$ obtained using SHDMC with those from the largest CI calculation
(see Table 1). The initial SHDMC trial wave function was the Hartree–Fock (HF)
solution, and the final SHDMC coefficients resulted from sampling the 1481
most significant excitations in the CI. We used $\delta\tau=0.01\,a.u.$,
$\tau=0.5\,a.u.$, and $16$ iterations of trial wave function projection (
$\approx 6\times 10^{7}$ sampled configurations).
Table 1: Total energies (and correlation % in {}) for the ground-state of O obtained with CI, coupled-cluster (CCSD(T) ccsdt ) and SHDMC (no Jastrow). Other symbols defined in the text. | VTZ | V5Z
---|---|---
Method | $N_{b}$ | E [Ha]{[%]} | $N_{b}$ | E [Ha]{[%]}
CI111full-CI in VTZ and CISDTQ in V5Z. | 775182 | -15.88258{89.0} | 1762377 | -15.89557{95.7}
CCSD(T)222from Ref. burkatzki07 . | - | -15.88204{88.8} | - | -15.90166{98.8}
SHDMC | 539 | -15.9003(2){98.1(1)} | 1481 | -15.9040(4){100.0(2)}
Figure 1 shows the excellent agreement between the coefficients $\lambda_{n}$
obtained independently by SHDMC and CI. A perfect agreement is guaranteed only
in the limit of a complete basis and $N_{c}\rightarrow\infty$. The small
differences in Fig. 1 are due to the truncation of the expansion and the
stochastic error in $\lambda_{n}$. The inset shows the residual projection as
a function of the total number $N_{b}$ of CSFs included in the expansion,
normalized either using the entire CI expansion (circles) or using a
$\Psi_{\rm CI}$ that included only the $\lambda_{n}$ sampled in SHDMC
(squares). The residual projection is much smaller for the truncated norm than
the full norm illustrating that most of the error in $\Psi_{\rm SHDMC}$ is
from truncation and not limited statistics. Similar results were obtained for
the C atom (not shown).
Figure 1: Comparison of the values of the coefficients $\lambda_{n}$
corresponding to the first 250 excitations of a converged SHDMC trial wave
function (large black circles) with a large CISDTQ wave function (small red
circles) for the oxygen atom. The first coefficient of the expansion, 0.9769,
is not shown. Inset: Residual projection ($R_{P}=1-|\langle\Psi_{\rm
SHDMC}|\Psi_{\rm CI}\rangle/\langle\Psi_{\rm CI}|\Psi_{\rm CI}\rangle|$) as a
function of the number of CSFs included: circles $R_{P}$ obtained with the
full CISDTQ norm, squares $R_{P}$ obtained with the truncated CISDTQ norm.
Validation with Energy Minimization for N2 — We also compared the VMC and DMC
energies of wave functions optimized with energy minimization in VMC (EMVMC)
umrigar05 ; umrigar07 and SHDMC using the QWALK wagner09 code. EMVMC can be
briefly described as a generalized CI with an additional Jastrow factor
(sampling the Hamiltonian stochastically and solving a generalized eigenvalue
problem). Several bases were obtained from series of complete active space
(CAS) and restricted active space (RAS) ras multiconfiguration self-
consistent field (MCSCF) calculations [distributing 10 electrons into $m$
active orbitals: CAS(10,$m$)]. We retained the $N_{b}$ basis functions with
coefficients of absolute value larger than a given cutoff. Subsequently, for
each basis, we performed energy minimization of the Jastrow and the
coefficients of trial wave function using a mixture of 95% of energy and 5% of
variance. We also sampled these $N_{b}$ coefficients in SHDMC recursively
starting from HF solution. For a clear comparison we used the same Jastrow in
EMVMC and SHDMC.
We performed these calculations for the ground-state (${}^{1}\Sigma^{+}_{g}$)
of N2 at the experimental geometry Ruscic . Figure 2 shows the resulting VMC
and DMC energies obtained for wave functions optimized independently with
EMVMC and SHDMC methods for the largest RAS(10,43) (2629447 CSFs yielding
E=-19.921717) Slater-Jastrow wave function (See also Table 2). In EMVMC , as
previously observed for C2 and Si2 umrigar07 , we found a systematic reduction
in the fixed-node errors, even when starting from the smallest CAS wave
function (see Table 2). When we compare with SHDMC optimized wave functions we
find an excellent agreement in both VMC and DMC energies. Therefore, SHDMC
improves the nodes systematically starting from the HF ground-state.
Figure 2: Total energies obtained for N2 with VMC and DMC methods for wave
functions optimized via EMVMC umrigar05 (squares) and SHDMC (circles) as a
function of the square of the norm of the CI coefficients retained in the
basis [$\sum_{n}(\lambda^{\rm MCSCF}_{n})^{2}$]. The lines are parabolic
extrapolations to 1. The dot-dashed line represents the scalar relativistic
core-corrected estimate of the exact energy (see Table 2). The shaded area is
the region of chemical accuracy.
Since retaining all the determinants in the wave function would be costly, we
performed calculations with different $N_{b}$ to extrapolate (quadratically)
the final energies as $\sum_{n}(\lambda^{\rm MCSCF}_{n})^{2}\to 1$ (see Fig.
2). The extrapolated DMC energies reached chemical accuracy (see also Table
2).
Table 2: Comparison of total and binding DMC energies of N2 for wave
functions optimized with EMVMC and SHDMC for increasingly larger basis (see
text). All SHDMC calculations started from the single HF determinant. Binding
energies were obtained using an atomic energyc of -9.80213(5) Ha, a core-
correlations correction of 1.4 mHa Bytautas , and a zero point energy of 5.4
mHa Ruscic .
| Total energy [Ha] | Binding energy [eV]
---|---|---
Wave function | EMVMC | SHDMC | EMVMC | SHDMC
1 determinant | -19.9362(5) | 9.07(1)
CAS(10,14) | -19.9536(6) | -19.9536(6) | 9.54(2) | 9.54(2)
RAS(10,35) | -19.9639(4) | -19.9627(4) | 9.83(1) | 9.79(1)
RAS(10,43) | -19.9654(4) | -19.9647(4) | 9.87(1) | 9.85(1)
Estimated exact | -19.9668(2)111Based on the scalar relativistic core-corrected estimate from Ref. Bytautas . | -9.900(1)222Using the experimental value from Ref. Ruscic .
33footnotetext: Based on a large multi-determinant DMC calculation.
Proof of principle in larger systems — Figures 1 and 2 show that SHDMC
produces reliable and accurate results for small systems starting form the HF
nodes. It is also important to demonstrate that SHDMC is a robust approach
that can find the correct nodal surface topology of much larger systems even
when starting from random nodal surfaces.
Figure 3 shows proof of principle results obtained for a C20 fullerene. These
calculations used the branching SHDMC algorithm[keystone, ] implemented by us
in CASINO casino . Two electrons were removed from the system to obtain a non-
interacting DFT ground-state wave function invariant under any transformation
belonging to the icosahedral group ($I_{h}$) symmetry. The orbitals were
obtained directly with the real space code PARSEC parsec and classified
according to their irreducible representations for $I_{h}$ and its subgroup
$D_{2h}$. For this calculation 694 excitations (determinants) were sampled. No
CI prefiltering of determinants is required; we only use the selection rules
of both $I_{h}$ and $D_{2h}$ symmetries.
The C${}_{20}^{+2}$ system has a large DFT gap (5.53 eV) which is often
associated with a dominant role of the non-interacting solution in the many-
body wave function. The $\lambda_{0}$ coefficient is expected to dominate the
final optimized trial wave function. All initial coefficients $\lambda_{n}$ of
$\Psi_{T}({\bf R})$ were set to random values, but for $\lambda_{0}$ which was
set to zero. New $\lambda_{n}$ values were sampled with $\sim 5094$ walkers
every $100$ DMC steps. We found that when the quality of the wave function is
poor, it is better (i) to update $\lambda_{n}$ frequently (after only $4$
samplings), and (ii) to use the T-moves approximation casula06 which limits
persistent configurations. As the quality of the wave-function improved, we
gradually increased the accumulation time (up to 96 samplings) and removed the
T-moves approximation (which, in practice, hinders the final SHDMC
convergence). Figure 3 shows that SHDMC can correct nodal errors as large as
$0.5$ Ha. The calculation was stopped when we obtained an energy of
$-112.487(2)$ Ha compared with the single determinant energy of $-112.473(1)$
Ha. We have confidence that SHDMC can be applied to cases where the nodal
structure of the ground-state is completely unknown since it is successful and
converges to the expected result starting from random.
The SHDMC recursive runs required $220$ hrs on $1024$ processors (Cray XT4).
This can be reduced to $\sim 100$ hrs starting from the ground state
determinant. Comparable EMVMC calculations with the same basis were
unsuccessful, presumably due to the statistical errors in the Hessian and
overlap matrices. The energy was not improved with EMVMC ($-112.488(3)$ Ha)
even selecting a basis with the largest $104$ coefficients of the $694$
sampled in SHDMC. The estimated running time for EMVMC with CASINO 2.5 using
$N_{b}=694$ and just $400$ configurations fn:configurations on $1024$
processors is already $\sim 100$ hrs, suggesting that for C${}^{+}_{20}$ SHDMC
is faster than EMVMC. However, both methods can be improved for large $N_{b}$
(e.g. as in Ref. nukala09, ), by removing redundant IO etc.
Figure 3: Proof of principle of SHDMC for larger systems. Initial evolution of
the average local energy for a SHDMC run with branchingkeystone generated for
C${}_{20}^{+2}$, with random initial coefficients (see text). Inset:
calculated icosahedral cluster C${}_{20}^{+2}$.
Summary — We have shown that the SHDMC wave function converges to the ground-
state of our best CI calculations and is systematically improved as the number
of coefficients sampled increases and the statistics are improved. SHDMC
presents equivalent accuracy to the EMVMC approach umrigar05 ; umrigar07
starting from random coefficients. SHDMC is numerically robust and can be
automated.
The number of independent degrees of freedom of the nodes increases
exponentially with the number of electrons. rockandroll Since EMVMC is based
on VMC, the prefactor for its computational cost is much smaller than SHDMC.
However, the number of quantities sampled in EMVMC is quadratic with respect
to the number of degrees of freedom. In addition, EMVMC requires inverting a
noisy matrix. These requirements cause EMVMC to scale at least quadratically.
In contrast, SHDMC only requires one to sample a number of quantities linear
in the number of optimized degrees of freedom. Therefore, a crossover between
the methods is expected for systems of sufficient size or complexity. Tests on
the large C${}_{20}^{+2}$ fullerene system demonstrate that SHDMC is robust
and that the nodes are systematically improved even starting from a random
coefficients in the trial wave function. This shows that SHDMC can be used to
find the nodes of unknown complex systems of unprecedented size.
We thank D. Ceperley, R. M. Martin and C. J. Umrigar for critically reading
the manuscript and useful comments. This research used computer resources
supported by the U.S. DOE Office of Science under contract DE-AC02-05CH11231
(NERSC) and DE-AC05-00OR22725 (NCCS). Research sponsored by U.S. DOE BES
Division of Materials Sciences & Engineering (FAR, MLT) and ORNL LDRD program
(MB). The Center for Nanophase Materials Sciences research was sponsored by
the U. S. DOE Division of Scientific User Facilities (PRCK). Research at LLNL
was performed under U.S. DOE contract DE-AC52-07NA27344 (RQH).
## References
* (1) B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry (World Scientific, Singapore-New Jersey-London-Hong Kong, 1994).
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* (8) F. A. Reboredo, R. Q. Hood, and P. R. C. Kent, Phys. Rev. B 79, 195117 (2009).
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* (11) C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, Phys. Rev. Lett. 98, 110201 (2007).
* (12) C. Umrigar (priv. commun.).
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* (17) A symmetry-adapted linear combination of Slater determinants.
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* (19) We included excitations up to quadruple level.
* (20) X. Tang, Yu Hou, C. Y. Ng, and B. Ruscic, J. Chem. Phys., 123, 074330 (2005).
* (21) L. Bytautas and K. Ruedenberg, J. Chem. Phys., 122, 154110 (2005).
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* (23) L. Kronik et al. Phys. Status Solidi B 243, 1063 (2006).
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* (25) These configurations are not enough for $N_{b}=100$.
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|
arxiv-papers
| 2009-12-18T22:48:40 |
2024-09-04T02:49:07.120213
|
{
"license": "Public Domain",
"authors": "Michal Bajdich, Murilo L. Tiago, Randolph Q. Hood, Paul R. C. Kent,\n and Fernando A. Reboredo",
"submitter": "Fernando Reboredo",
"url": "https://arxiv.org/abs/0912.3826"
}
|
0912.3894
|
# The systolic constant of orientable Bieberbach 3-manifolds
Chady El Mir
###### Abstract
The _systole_ of a compact non simply connected Riemannian manifold is the
smallest length of a non-contractible closed curve ; the _systolic ratio_ is
the quotient $(\mathrm{systole})^{n}/\mathrm{volume}$. Its supremum, over the
set of all Riemannian metrics, is known to be finite for a large class of
manifolds, including the $K(\pi,1)$.
We study the optimal systolic ratio of compact, $3$-dimensional orientable
Bieberbach manifolds which are not tori, and prove that it cannot be realized
by a flat metric.
Key words and phrases. Systole; systolic ratio; singular Riemannian metric;
Bieberbach manifold.
Laboratoire de Math matiques et Physique Th orique
CNRS, UMR 6083
Université François Rabelais
UFR Sciences et Techniques
Parc de Grandmont
37200 Tours, France
## 1 Introduction
### 1.1 Motivations and main result
The systole of a compact non simply connected Riemannian manifold $(M^{n},g)$
is the shortest length of a non contractible closed curve, we denote it by
$\mathrm{Sys}(g)$. To get an homogeneous Riemannian invariant, we introduce
the _systolic ratio_ $\frac{\mathrm{Sys}(g)^{n}}{\mathrm{Vol}(g)}$. It is
important to note that this invariant is well defined even if $g$ is only
continuous, i.e. a continuous section of the fiber bundle $S^{2}T^{\ast}M$ of
symmetric forms.
An isosystolic inequality on a manifold $M$ is a inequality of the form
$\frac{\mathrm{Sys}(g)^{n}}{\mathrm{Vol}(g)}\leq C$
that holds for any Riemannian metric $g$ on $M$. The smallest such constant
$C$ is called the _systolic constant_.
A systolic geodesic will be for us a closed curve, not homotopically trivial,
whose length is equal to the systole.
In 1949, in an unpublished work, C. Loewner proved the following result. For
any metric $h$ on the torus $\mathbb{T}^{2}$ we have
$\frac{sys^{2}(\mathbb{T}^{2},h)}{Area(\mathbb{T}^{2},h)}\leq 2/\sqrt{3}$
Furthermore, the equality is achieved if and only if $(\mathbb{T}^{2},h)$ is
isometric to a flat hexagonal torus.
In 1952, P.M. Pu proved an isosystolic inequality for the real projective
plane (c.f. [Pu52]). The extremal metric has constant curvature, too. In the
same paper, he proved a variant of the isosystolic inequality for the Möbius
bands with boundary, valid for each conformal class of any metric.
There exists a third case, solved by C. Bavard in [Bav86], where the upper
bound of the systolic ratio is known, and realized, this is the case of the
Klein bottle. This time the extremal metric (for the isosystolic inequality)
is singular, more precisely piecewise $C^{1}$ (see [Bav86]). Furthermore, it
has curvature $+1$ where it is smooth.
In higher dimension, there exists non simply connected manifolds that do not
satisfy any isosystolic inequality. The simplest example is $S^{2}\times
S^{1}$, or more generally the product of a simply connected manifolds by a non
simply connected one. Making the volume of the simply connected factor tend to
zero insures the explosion of the systolic ratio..
However a fundamental result of M. Gromov (cf. [Gro83]), ensures that
_essential manifolds_ satisfy an isosystolic inequality. A compact manifold
$M$ is _essential_ if there exists a continuous map from $M$ in a $K(\pi,1)$
($\pi=\pi_{1}(M)$) which sends the fundamental class to a non trivial one. The
essential manifolds include notably aspherical manifolds and the real
projectives. Furthermore, I. Babenko proved in [Bab92] a reciprocal of the
theorem of M. Gromov : "In the orientable case, essential manifolds are the
only manifolds that satify isosystolic inequalities".
However, in dimensions $\geq 3$, hardly anything is known about metrics that
realize the systolic constant (extremal metrics). It is not known for example,
in the apparently simple cases of tori and real projective spaces, whether the
metrics of constant curvature are extremal.
In the present work, we are interested in _Bieberbach manifolds_ , i.e.
compact manifolds that carry a flat Riemannian metric. These manifolds are
$K(\pi,1)$, and then the theorem of Gromov can be applied. Our result is the
following
_Let $M$ be a Bieberbach orientable manifold of dimension $3$ that is not a
torus. Then there exists on $M$ a Riemannian metric $g$ such that, for any
flat metric $h$,_
$\frac{(\mathrm{sys}(h))^{3}}{\mathrm{vol}(h)}<\frac{(\mathrm{sys}(g))^{3}}{\mathrm{vol}(g)}$
We recently proved the same result for non-orientable $3$-dimensional
Bieberbach manifolds (see [El-La08]). The main idea consisted in the fact that
we can get any such manifold by suspending a flat Klein bottle. Putting then
the spherical metric of Bavard on these Klein bottles give metrics (locally
isometric to $S^{2}\times\mathbb{R}$) whose systolic ratio is better than the
flat ones.
_Acknowledgements:_ The author is grateful to Jacques Lafontaine and Benoit
Michel for useful discussions and remarks especially on _lemma 5_. He also
thanks Christophe Bavard, Romain Gicquaud and St phane Sabourau for their
interest in this work.
### 1.2 Idea of the proof
The proof relies on the fact that these manifolds contain an isolated systolic
geodesic on one hand, and a lot of surfaces that are flat Klein bottles and
flat Möbius bands (except for $C_{3}$ but this case can be treated similarly)
on the other. To see this we use a theorem of classification of flat manifolds
of dimension $3$, this theorem is a result of the theorem of Bieberbach for
crystallographic groups (see [Wol74] and [Cha86]).
In section 3 we define in the setting of Riemannian polyhedrons the Riemannian
singular spaces. The extremal Klein bottle (for the isosytolic inequality)
fits into this setting. We introduce then Riemannian singular metrics (with
the preceding meaning) on the orientable Bieberbach manifolds of dimension $3$
of type $C_{2}$,$C_{3}$,$C_{4}$ and $C_{6}$. (We follow the notations of W.
Thurston, see [Thu97])
For these metrics the Klein bottles and Möbius bands become singular surfaces
of curvature $+1$ outside the singularity. It is useful to note that the group
of isometries of these metrics is the same as for the flat ones. The case of
the manifold $C_{2,2}$ is treated thanks to the following (probably folk)
result communicated to us by I.Babenko.
_if $g$ is an extremal Riemannian metric (eventually singular) on $M$, the
systolic geodesics do cover $M$_ (see [Cal96]).
This property is satisfied by flat tori, and real projectives endowed with
their metric of constant curvature. On Bieberbach manifolds of dimension $3$,
the metrics that optimize the systolic ratio _among flat metrics_ also satisfy
this property, except for the manifold $C_{2,2}$ (cf. [Wol74] p.117-118, and
the suggestive figure of [Thu97], p.236). This property gives the result for
$C_{2,2}$.
The metrics that we construct also satisfy this property, and so there is no
obvious obstacle that prevents them from being extremal (especially the one on
$C_{2}$, see section 7).
## 2 Flat manifolds
### 2.1 Classification of flat manifolds
Compact flat manifolds are quotients $\mathbb{R}^{n}/\Gamma$, where $\Gamma$
is a discrete cocompact subgroup of affine isometries of $\mathbb{R}^{n}$
acting freely. By the theorem of Bieberbach $\Gamma$ is an extension of a
finite group $G$ by a lattice $\Lambda$ of $\mathbb{R}^{n}$. This lattice is
the subgroup of the elements of $\Gamma$ that are translations, we obtain then
the following exact sequence:
$0\longrightarrow\Lambda\longrightarrow\Gamma\longrightarrow G\longrightarrow
1$
Actually, if $M$ is a flat manifold, $M$ is the quotient of the flat torus
$\mathbb{R}^{n}/\Lambda$ by an isometry group isomorphic to $G$. Two compact
and flat manifolds $\mathbb{R}^{n}/\Gamma$ and
$\mathbb{R}^{n}/\Gamma^{\prime}$ are homeomorphic if and only if the groups
$\Gamma$ and $\Gamma^{\prime}$ are isomorphic. Such groups are then conjugate
by an affine isomorphism of $\mathbb{R}^{n}$: two compact and flat
homeomorphic manifolds are affinely diffeomorphic.
### 2.2 3-dimensional orientable flat manifolds
The classification of flat manifolds of dimension $3$ results of a direct
method of classification of discrete, cocompact subgroups of
$\mathrm{Isom}(\mathbb{R}^{3})$ operating freely. This classification is due
to W. Hantzsche and H. Wendt (1935), and exposed in the book [Wol74] of J.A.
Wolf. There exists ten compact and flat manifolds of dimension $3$ up to
diffeomorphism, six are orientable and four are not.
In the orientable case, these types are caracterized by the holonomy group
$G$, this reason motivates our notation. A rotation of angle $\alpha$ around
an axis $a$ will be denoted by $r_{a,\alpha}$.
i) $G=\\{1\\}$: type $C_{1}$. This is the torus, it is the quotient of
$\mathbb{R}^{3}$ by an arbitrary lattice of $\mathbb{R}^{3}$.
ii) $G=\mathbb{Z}_{2}$: type $C_{2}$. Given a basis $(a_{1},a_{2},a_{3})$ of
$\mathbb{R}^{3}$ with $a_{3}\perp(a_{1},a_{2})$, let $\Gamma$ be the subgroup
of isometries of $\mathbb{R}^{3}$ generated by $t_{a_{3}/2}\circ
r_{a_{3},\pi}$ and the translations $t_{a_{1}}$ and $t_{a_{2}}$. The quotient
$\mathbb{R}^{3}/\Gamma$ is a manifold of type $C_{2}$. Note that the lattice
$\Lambda$ generated by $t_{a_{1}}$, $t_{a_{2}}$ et $t_{a_{3}}$ is of index $2$
in $\Gamma$.
This manifold is the quotient of the torus $\mathbb{R}^{3}/\Lambda$ by the
cyclic group of order $2$ generated by (the image of) $t_{a_{3}/2}\circ
r_{a_{3},\pi}$.
iii) $G=\mathbb{Z}_{4}$: type $C_{4}$. Given an orthogonal basis
$(a_{1},a_{2},a_{3})$ of $\mathbb{R}^{3}$ with $|a_{1}|=|a_{2}|$, let $\Gamma$
be the subgroup of isometries of $\mathbb{R}^{3}$ generated by
$t_{a_{3}/4}\circ r_{a_{3},\pi/2}$ and the translations $t_{a_{1}}$ et
$t_{a_{2}}$. The quotient $\mathbb{R}^{3}/\Gamma$ is a manifold of type
$C_{4}$. Note that the lattice $\Lambda$ generated by $t_{a_{1}}$,$t_{a_{2}}$
and $t_{a_{3}}$ is of index $4$ in $\Gamma$.
This manifold is the quotient of the torus $\mathbb{R}^{3}/\Lambda$ by the
cyclic group of order $4$ generated by (the image of) $t_{a_{3}/4}\circ
r_{a_{3},\pi/2}$. It is also the quotient of $C_{2}$ (the basis
$(a_{1},a_{2},a_{3})$ should be chosen orthogonal with $|a_{1}|=|a_{2}|$), by
the subgroup generated by $t_{a_{3}/4}\circ r_{a_{3},\pi/2}$. This remark will
play an important role in the improvement of the systolic ratio of $C_{4}$.
iv) $G=\mathbb{Z}_{6}$: type $C_{6}$. Given a basis $(a_{1},a_{2},a_{3})$ of
$\mathbb{R}^{3}$ with $a_{3}\perp(a_{1},a_{2})$, $|a_{1}|=|a_{2}|$ and
$(a_{1},a_{2})=\pi/3$, let $\Gamma$ be the subgroup of isometries of
$\mathbb{R}^{3}$ generated by $t_{a_{3}/6}\circ r_{a_{3},\pi/3}$ and the
translations $t_{a_{1}}$ et $t_{a_{2}}$. The quotient $\mathbb{R}^{3}/\Gamma$
is a manifold of type $C_{6}$. This time, the lattice $\Lambda$ generated by
$t_{a_{1}}$,$t_{a_{2}}$ and $t_{a_{3}}$ is of index $6$ in $\Gamma$.
$C_{6}$ is the quotient of the torus $\mathbb{R}^{3}/\Lambda$ by the cyclic
group of order $6$ generated by (the image of) $t_{a_{3}/6}\circ
r_{a_{3},\pi/3}$. It is also the quotient of $C_{2}$ by the subgroup generated
by $t_{a_{3}/6}\circ r_{a_{1},\pi/3}$.
v) $G=\mathbb{Z}_{3}$: type $C_{3}$. Given a basis $(a_{1},a_{2},a_{3})$ of
$\mathbb{R}^{3}$ with $a_{3}\perp(a_{1},a_{2})$, $|a_{1}|=|a_{2}|$ and
$(a_{1},a_{2})=2\pi/3$, let $\Gamma$ be the subgroup of isometries of
$\mathbb{R}^{3}$ generated by $t_{a_{3}/3}\circ r_{a_{3},2\pi/3}$ and the
translations $t_{a_{1}}$ et $t_{a_{2}}$. The quotient $\mathbb{R}^{3}/\Gamma$
is a manifold of type $C_{3}$. The lattice $\Lambda$ generated by
$t_{a_{1}}$,$t_{a_{2}}$ and $t_{a_{3}}$ is of index $3$ in $\Gamma$. This
manifold is the quotient of the torus $\mathbb{R}^{3}/\Lambda$ but it is not a
quotient of $C_{2}$.
vi) $G=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$: type $C_{2,2}$. Given an
orthogonal basis $(a_{1},a_{2},a_{3})$ of $\mathbb{R}^{3}$, let $\Gamma$ be
the subgroup of isometries of $\mathbb{R}^{3}$ generated by $t_{a_{1}/2}\circ
r_{a_{1},\pi}$, $t_{(\frac{a_{1}+a_{2}}{2})}\circ r_{a_{2},\pi}$ and
$t_{(\frac{a_{1}+a_{2}+a_{3}}{2})}\circ r_{a_{3},\pi}$. The quotient
$\mathbb{R}^{3}/\Gamma$ is the manifold $C_{2,2}$. This time, the holonomy
group $G$ is not cyclic, it is equal to $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.
###### Remark 1.
In the case of the manifolds $C_{2}$, $C_{4}$ and $C_{6}$, every plane that
contains $a_{3}$ gives, in the quotient, a flat Klein bottle (if the plane
contains points of the lattice other than those of the axis $a_{3}$) or a flat
Möbius band without boundary (otherwise).
## 3 Singular metrics on Bieberbach manifolds
All the singular metrics we will use give rise to length spaces, i.e. spaces
with a notion of length for simple closed curves. In this section we will
define these metrics in a general case, the Riemannian polyhedrons. For
further details on this notion see [Bab02].
A polyhedron is a topological space endowed with a triangulation, i.e. divided
into simplexes glued together by their faces. We denote by $\sigma$ an
arbitrary simplexe of a polyhedron $P$.
###### Definition 1.
A Riemannian metric on a polyhedron $P$ is a family of Riemannian metrics
$\\{g_{\sigma}\\}_{\sigma\in I}$, where $I$ is in bijection with the set of
simplexes of $P$. These metrics should satisfy the following conditions:
1. 1.
Every $g_{\sigma}$ is a smooth metric in the interior of the simplex $\sigma$.
2. 2.
The metrics $g_{\sigma}$ coïncide on the faces; i.e. for any pair of simplexes
$\sigma_{1}$, $\sigma_{2}$, we have the equality
$g_{\sigma_{1}}|_{\sigma_{1}\bigcap\sigma_{2}}=g_{\sigma_{2}}|_{\sigma_{1}\bigcap\sigma_{2}}$
Such a Riemannian structure on the polyhedron allows us to calculate the
length of any piecewise smooth curve in $P$, this way, the polyhedron $(P,g)$
turns out to be a length space. If $\gamma$ is a piecewise smooth path from an
interval $I$ to $P$, then the length of $\gamma$ is defined as for the
$C^{\infty}$ metrics:
$l(\gamma)=\int_{I}{\big{(}g(\gamma^{\prime}(t),\gamma^{\prime}(t))\big{)}^{1/2}dt}.$
Furthermore $(P,g)$ gains a structure of metric space (and especially a
structure of length space) the same way as for smooth manifolds.
$d_{g}(x,y)=\inf_{\gamma}l(\gamma)$
where $\gamma$ runs the set of piecewise smooth paths from $x$ to $y$.
It is useful to note that the Riemannian measure too, is defined exactly as in
the smooth case. Of course, the volume of the singular part will be zero.
The geodesics of a Riemannian polyhedron are the geodesics of the associated
length structure (see [Bu-Iv01]). In the interior of a simplex
$(\sigma,g_{\sigma})$, the first variation formula shows that such a geodesic
is a geodesic of $g_{\sigma}$ in the Riemannian sense.
### 3.1 The Klein-Bavard bottle
The flat Klein bottle are the manifolds $\mathbf{R}^{2}/\Gamma$, where
$\Gamma$ is the subgroup of isometries of $\mathbb{R}^{2}$ generated by the
glide reflection $(x,y)\mapsto(x+\frac{a}{2},-y)$ and the translation
$(x,y)\mapsto(x,y+b)$. We know by Bavard ([Bav86]) that any flat Klein bottle
is not be extremal for the isosystolic inequality (see also [Gro83]), the
unique extremal one is singular and has curvature $+1$ outside the
singularities:
We start with the round sphere, and we locate the points by their latitude
$\phi$ and their longitude $\theta$. For $\phi_{o}\in]0,\pi/2[$, let
$\Sigma_{\phi_{o}}$ be the domain defined by $|\phi|\leq\phi_{o}$. In
$\Sigma_{\phi_{o}}$, the round metric is given by $d\phi^{2}+\cos^{2}\phi
d\theta^{2}$. Note that the universal cover of $\Sigma_{\phi_{o}}$ is the
strip $\mathbb{R}\times[-\phi_{o},\phi_{o}]$ with the same metric
$(d\phi^{2}+\cos^{2}\phi d\theta^{2})$. Here we introduce in $\mathbb{R}^{2}$
the singular Riemannian metric
$d\phi^{2}+f^{2}(\phi)d\theta^{2},$ (1)
where $f$ is the $2\phi_{0}$ periodic function which agrees with $cos\phi$ in
the interval $[-\phi_{o},\phi_{o}]$.
###### Example 1.
The metric on the Klein bottle that gives the best systolic ratio
($\frac{\pi}{2\sqrt{2}}$) is obtained for $\phi_{o}=\frac{\pi}{4}$ by taking
the quotient of the plane endowed with the metric 1 by the action of the group
generated by
$(\theta,\phi)\mapsto(\theta+\pi,-\phi)\quad\hbox{et}\quad(\theta,\phi)\mapsto(\theta,\phi+4\phi_{0}).$
For more details on the Klein-Bavard bottle (that we denote $(\mathbf{K},b)$)
see [El-La08] and [Bav88].
###### Remark 2.
It may seem more natural to take the quotient of the plane (endowed with the
metric 1) by the group generated by
$(\theta,\phi)\mapsto(\theta+\pi,-\phi)\quad\hbox{and}\quad(\theta,\phi)\mapsto(\theta,\phi+2\phi_{0})$
the surface we obtain is indeed a Klein bottle but it does not give the best
systolic ratio: the geodesics closed by the correspondence
$(\theta,\phi)\mapsto(\theta,\phi+2\phi_{0})$ have a length equal to $\pi/2$
whereas the ones closed by the correspondence
$(\theta,\phi)\mapsto(\theta,\phi+2\phi_{0})$ have length $\pi$. It is then
possible to reduce the volume without shortening the systole by reducing the
metric in the direction of the long closed curves.
### 3.2 Singular metrics on orientable Bieberbach manifolds
Starting with an arbitrary lattice of $\Delta$ of $\mathbb{R}^{2}$, we
introduce the associated Dirichlet-Vorono paving. It is a paving of the plane
by hexagons (or rectangles if the lattice $\Delta$ is rectangle) $A_{p}$
centered at the points $p$ of the lattice. Then a lattice of $\mathbb{R}^{3}$
of the form $\Delta\times c\mathbb{Z}$, where $c\in\mathbb{R}$, allows us to
pave $\mathbb{R}^{3}$ naturally with hexagonal or rectangular prisms that we
denote by $D_{p}$.
Now we introduce on $\mathbb{R}^{3}$ the Riemannian singular metric
$h=dx^{2}+dy^{2}+\psi(m)dz^{2}$, where we set, for $m(x,y,z)\in D_{p}$,
$\psi(m)=\cos^{2}\mathrm{dist}\big{(}(x,y),p\big{)}$, with
$\mathrm{dist}\big{(}(x,y),p\big{)}<\pi/2$. If $m$ is in two domains $D_{p}$
and $D_{p^{\prime}}$ then $p$ and $p^{\prime}$ are at the same distance from
$m$ : the map $\psi$ is well defined. It is continuous, but it is not $C^{1}$.
The connected component of the identity in the group of isometries of
$(\mathbb{R}^{3},h)$ consists of the vertical translations
$(x,y,z)\mapsto(x,y,z+c^{\prime})$. The translations by the vectors of
$\Delta$ are also isometries. It is important to note that the metric $h$ can
also be written in the form
$dx^{2}+dy^{2}+\cos^{2}\big{(}d((x,y),\Delta)\big{)}$, where $d((x,y),\Delta)$
is the distance from the point $(x,y)$ to the lattice $\Delta$.
The quotient of $(\mathbb{R}^{3},h)$ by the group $\Delta\times c\mathbb{Z}$
(where $c>0$) is a singular torus of dimension $3$. We denote by $(T,h)$ this
special torus. The sections of $(T,h)$ by the planes $z=constant$ are
$2-$dimensional totally geodesic flat tori. All these flat tori are isometric
to $\mathbb{R}^{2}/\Delta$. Note that the map from $(T,h)$ into the torus
$\mathbb{R}^{2}/\Delta$, which consists in projecting onto the torus $z=0$, is
a Riemannian submersion.
With a good choice of the lattice $\Delta$, the transformations
$t_{a_{3}/n}\circ r_{a_{3},2\pi/n}$ $(n=2,3,4,6)$, described in the
classification of the flat orientable manifolds, become isometries of $(T,h)$
(The lattice $\Delta$ should be square to get $C_{4}$ and hexagonal to get
$C_{3}$ et $C_{6}$). This way we get a family of singular Riemannian metrics
on the manifolds of type $C_{2}$, $C_{3}$, $C_{4}$ and $C_{6}$.
###### Remark 3.
Actually this construction works if we take the quotient of
$(\mathbb{R}^{3},h)$ by the lattice $n\Delta\times c\mathbb{Z}$. Starting with
this torus, we can re-obtain all the manifolds $C_{i}$ $(i=2,3,4,6)$ exactly
the same way as for the torus $(T,h)$. We will see in the next sections that
taking $n=2$ is more useful to get "good systolic ratios" on these manifolds.
## 4 Two singular tori and their systole
We take the quotient of the Riemannian singular space $(\mathbb{R}^{3},h)$
seen in section 3.2 by the lattice $2\Delta\times 2\pi\mathbb{Z}$. We get a
$3$-dimensional torus $(T,g)$ whose singular area is connected. It consists of
the boundary of four hexagonal (or rectangular) prisms constituting a
fundamental domain for the action of $2\Delta\times c\mathbb{Z}$. If $u$ and
$v$ are two vectors generating the lattice $\Delta$ then the sections of
$(T,g)$ by planes containing a point of the lattice $\Delta$ and parallel to
$u$ (or $v$) are tori of dimension $2$ and of curvature $+1$ outside their
singular area. Taking the good choice of $u$ and $v$ these tori are the
orientable covering of the Klein-Bavard bottle $(\mathbf{K},b)$ introduced in
[Bav86]. In general, the sections of $(T,g)$ by planes containing the axis of
a domain $D_{p}$ are surfaces of curvature $+1$ as long as we stay in the
interior of $D_{p}$. We denote these surfaces by $S_{p}$.
###### Remark 4.
To preserve the systole and reduce the volume of the manifolds of type $C_{i}$
it is crucial to take the quotient of $(\mathbb{R}^{3},h)$ by the lattice
$2\Delta\times 2\pi\mathbb{Z}$ and not by $\Delta\times 2\pi\mathbb{Z}$. This
prevents shortening closed curves at the level of the surfaces $S_{p}$.
First suppose that the lattice $\Delta$ is square and generated by two vectors
of norm $2a>0$. This lattice is generated by three translations
$t_{1}:(x,y,z)\longrightarrow(x+4a,y,z)$,
$t_{2}:(x,y,z)\longrightarrow(x,y+4a,z)$ and
$t_{3}:(x,y,z)\longrightarrow(x,y,z+2\pi)$. We denote by $(T,g_{c})$ the
quotient torus. Its singular area consists of four connected surfaces $x=a$,
$x=3a$, $y=a$ and $y=3a$.
Note that the symmetries with respect to the surfaces $x=pa$ and $y=qa$ where
$p,q\in\mathbb{Z}$, are isometries of $(T,g_{c})$.
###### Lemma 1.
The systole of $(T,g_{c})$ is equal to $\inf\\{4a,2\pi\cos(a\sqrt{2})\\}$.
###### Proof.
Let $\gamma$ be a curve in $(\mathbb{R}^{3},h)$, from $m(x_{0},y_{0},z_{0})$
to $t_{1}(m)$, then
$l(\gamma)\geq\int{\sqrt{x^{\prime 2}+y^{\prime 2}+\psi(x,y)z^{\prime
2}}dt}\geq\int{x^{\prime}dt}\geq 4a$
Now, if $\gamma$ is a curve from $m(x_{0},y_{0},z_{0})$ to $t_{2}(m)$ we find
exactly the same way as before that $l(\gamma)\geq 4a$, just compare the
length of $\gamma$ to its projection on $\\{y=y_{0},z=z_{0}\\}$. Finally, for
a curve $\gamma$ from $m$ to $t_{3}(m)$, we have
$l(\gamma)=\int{\sqrt{x^{\prime 2}+y^{\prime 2}+\psi(x,y)z^{\prime
2}}dt}\geq\int{\inf(\psi)z^{\prime}dt}=2\pi\cos a\sqrt{2}$
the equality is obtained for the points of the edges of the square prism
$D_{p}$. Using exactly the same technique we can prove that the distance
between a point $m$ and its image by the composition of several translations
is greater or equal to $\inf\\{4a,2\pi\cos a\sqrt{2}\\}$.
∎
Suppose now that the lattice $\Delta$ is hexagonal and generated by two
vectors of norm $2a>0$. The lattice $2\Delta\times 2\pi\mathbb{Z}$ is
generated by the translations $T_{1}:(x,y,z)\longrightarrow(x+4a,y,z)$,
$T_{2}:(x,y,z)\longrightarrow(x+2a,y+2a\sqrt{3},z)$ and
$T_{3}:(x,y,z)\longrightarrow(x,y,z+2\pi)$. The manifold we get is a singular
torus that we denote by $(T,g_{hex})$. Its singular area consists of the edges
of the hexagonal prisms $D_{p}$ that pave $\mathbb{R}^{3}$.
###### Remark 5.
The symetries with respect to the surfaces $x=pa$,
$y+\frac{x}{\sqrt{3}}=\frac{2pa}{\sqrt{3}}$ and
$y-\frac{x}{\sqrt{3}}=\frac{2pa}{\sqrt{3}}$, are isometries of
$(T^{3},g_{hex})$.
###### Lemma 2.
The systole of $(T,g_{hex})$ is equal to $\inf\\{4a,2\pi\cos(2a/\sqrt{3})\\}$.
###### Proof.
For any curve $\gamma$ from $m$ to $T_{1}(m)$ we have
$l(\gamma)\geq\int{\sqrt{x^{\prime 2}+y^{\prime 2}+\psi(x,y)z^{\prime
2}}dt}\geq\int{x^{\prime}dt}\geq 4a$
the same inequality holds for any curve from $m$ to $T_{2}(m)$ since the
situation is invariant by the rotation $r_{a_{3},\pi/3}$ of angle $\pi/3$
around the axis $z$.
Finally, for any curve $\gamma$ from $m$ to $T_{3}(m)$, we have
$l(\gamma)=\int{\sqrt{x^{\prime 2}+y^{\prime 2}+\psi(x,y)z^{\prime
2}}dt}\geq\int{\inf(\psi)z^{\prime}dt}=2\pi\cos(2a\sqrt{3})$
the equality is achieved for the points of the edges of the hexagonal prisms
$D_{p}$. The distance between a point $m$ and its image by the composition of
several translations is greater or equal to
$\inf\\{4a,2\pi\cos(2a\sqrt{3})\\}$.
∎
## 5 The systolic ratio of $C_{2}$
### 5.1 The systolic ratio in the case of flat metrics
The volume is equal to $\frac{1}{2}\det(a_{2},a_{1})|a_{3}|$ and the systole
is equal to $\inf\\{|a_{3}|/2,s\\}$, where $s$ is the systole of the flat
torus of dimension $2$ defined by the lattice of basis $a_{1},a_{2}$. We
normalize such that $\frac{1}{2}|a_{3}|=1$, then the systolic ratio is equal
to
$\frac{s^{3}}{\det(a_{1},a_{2})}\quad\hbox{if $s\leq 1$
and}\quad\frac{1}{\det(a_{1},a_{2})}\quad\hbox{if $s\geq 1$,}$
Since we have $\frac{s^{2}}{\det(a_{1},a_{2})}\leq\frac{2}{\sqrt{3}}$ (lattice
of dimension $2$), the systolic ratio is less or equal to $2/\sqrt{3}$.
### 5.2 Klein bottles and Möbius bands in $C_{2}$
We have already seen that the planes containing $a_{3}$ give rise to flat
Klein bottles or flat Möbius bands without boundary. If the plane contains a
point of $\Lambda$ (c.f. 2.2) other than those of the axis $a_{3}$, the
intersection is a Klein bottle, otherwise it is a Möbius band.
To improve the systolic ratio $2/\sqrt{3}$, we should reduce the volume
without touching the systole. This can be done by taking benifit of the non
orientable surfaces embedded in the flat manifold $C_{2}$ and "put" the
spherical metric of Bavard and Pu on them.
### 5.3 A singular metric on $C_{2}$ better than the flat ones
We start with the singular torus $(T,g_{hex})$ seen in section 4.
The transformation $\sigma:(x,y,z)\longrightarrow(-x,-y,z+\pi)$ is an isometry
of the metric $g$. To get a manifold homeomorphic to $C_{2}$ we must take the
quotient of $(T,g_{hex})$ by the subgroup generated by $\sigma$. We denote
this manifold by $(C_{2},g_{hex})$.
In the torus $(T,g_{hex})$, the transformation $\sigma$ keeps $4$ geodesics
globally invariant, these are the vertical axes containing the $4$ centers of
the prisms that constitute a fundamental domain of $C_{2}$ (this is the set
$\\{x=2pa,y=2qa,(p,q)\in\mathbb{Z}^{2}\\}$). Actually, this is a property of
the fundamental group of $C_{2}$ that does not depend on the metric and holds
for any metric on $C_{2}$. The existence of these geodesics is a bit
disturbing since they can shorten some closed curves in the manifold
$(C_{2},g_{hex})$.
We go back now to the metric $g$, it can be written locally (in the domain D)
in cylindrical coordinates (with respect to $x$ and $y$) in the form
$g=dr^{2}+r^{2}d\theta^{2}+\cos^{2}rdz^{2}$ ($r$ is the distance to the
vertical axes going through the center $p$ of the prism $D_{p}$, and $\theta$
is the angle with respect to the axis $"x"$). In the following, we will even
consider the first form or the other depending what we need.
###### Remark 6.
In restriction to a prism $D_{p}$, a surface of equation $\theta=\theta_{0}$
is totally geodesic. To see that, just remark that the length of any curve
$\gamma$ in $D_{p}$ joining two points of $\theta=\theta_{0}$ is always
greater than its projection on this surface. This is simply due to the
expression of the metric in the "cylindrical" coordinates:
$l(\gamma)=\int{\sqrt{{r^{\prime}}^{2}+r^{2}{\theta^{\prime}}^{2}+\cos^{2}r{z^{\prime}}^{2}}dt}\geq\int{\sqrt{{r^{\prime}}^{2}+\cos^{2}r{z^{\prime}}^{2}}}$
The surfaces $\theta=constant$ are not singular as long as we stay in the
interior of a domain $D_{p}$, they are locally isometric to $S^{2}$ and their
geodesics are also geodesics of $(C_{2},g_{hex})$.
###### Lemma 3.
Let $\gamma$ be a curve of the universal Riemannian covering of $(T,g_{hex})$
and $\gamma^{\prime}$ its minimal projection on a hexagonal prism $D_{p}$,
then we have $l(\gamma)\geq l(\gamma^{\prime})$.
###### Proof.
The _minimal projection_ of a point $m$ is here the point of $D_{p}$ at a
minimal distance (Euclidian) of $m$. It is unique since $D_{p}$ is convex.
If the minimal orthogonal projection is completely inside the singularity
$x=pa$, i.e. if $\gamma^{\prime}$ is in such a hypersurface, then
$l(\gamma)=\int{\sqrt{x^{\prime 2}+y^{\prime 2}+\psi(x,y)z^{\prime
2}}dt}\geq\int{\sqrt{y^{\prime 2}+\psi(x,y)z^{\prime
2}}dt}=l(\gamma^{\prime})$
but the situation is invariant by a rotation of angle $\pi/3$ around $p$; this
shows that if $\gamma$ is projected on the surfaces
$y+\frac{x}{\sqrt{3}}=\frac{2pa}{\sqrt{3}}$ or
$y-\frac{x}{\sqrt{3}}=\frac{2pa}{\sqrt{3}}$ of the singularity, we have
$l(\gamma)\geq l(\gamma^{\prime})$. Finally, the result is true for any curve
projected on anywhere on the singularity.
∎
###### Remark 7.
In fact the previous lemma holds even if we take the minimal projection on a
hexagonal prism _inside_ $D_{p}$ and parallel to it. A prism is parallel to
$D_{p}$ if every plane consisting its boundary is parallel to a plane of the
boundary of $D_{p}$. This remark will be used in the improvement of the
systolic ratio of the manifold $C_{3}$.
###### Lemma 4.
For any point $m(r_{0},\theta_{0},z_{0})$ in $(T,g_{hex})$ we have
$d_{(T,g_{hex})}(m,\sigma(m))\geq\pi.$
The equality is achieved for a geodesic of the surface $\theta=\theta_{0}$.
###### Proof.
Let $m(r_{0},\theta_{0},z_{0})$ be a point in $D_{p}$, and $\gamma$ a curve in
$(\mathbb{R}^{3},h)$ from $m$ to $\sigma(m)$. If $\gamma$ stays in $D_{p}$,
then by Remark 6 we have $l(\gamma)\geq l(pr(\gamma))$ where $pr(\gamma)$ is
the projection of $\gamma$ on the surface $\theta=\theta_{0}$. But
$l(pr(\gamma))\geq\pi$ since the metric on this surface is spherical
($dr^{2}+cos^{2}rd\theta^{2}$). Now if $\gamma$ gets out of the prism $D_{p}$,
let $\gamma^{\prime}$ be the curve obtained by taking the projection (minimal)
of the part of $\gamma$ outside $D_{p}$ on the boundary $\partial D_{p}$, and
by leaving the part inside $D_{p}$ unchanged. Then $\gamma^{\prime}$ is a
curve of $D_{p}$ from $m$ to $\sigma(m)$. Its length is greater or equal to
$\pi$ (using the same argument of projection on the surface
$\theta=\theta_{0}$). We conclude that $l(\gamma)\geq l(\gamma^{\prime})$.
Then we have to calculate in $(\mathbb{R}^{3},h)$ (a lower bound of) the
distance to (a lift of) $\sigma(m)$ of the images of $m$ by translations. We
denote by $\sigma_{0}$ any lift of $\sigma$ in $(\mathbb{R}^{3},h)$. If we
translate $m$ by $T_{3}$, the situation will be equivalent to the one above
since $T_{3}(m)$ and $\sigma_{0}(m)$ are conjugate by $\sigma_{0}^{-1}$. Now a
curve $\gamma$ in $(\mathbb{R}^{3},h)$ from $\sigma_{0}(m)$ to $T_{1}(m)$
should go through at least $3$ domains $D_{p}$. Among these let $D^{\prime}$
be the domain that neither contains $\sigma_{0}(m)$ nor $T_{1}(m)$.
* •
If $\gamma$ stays in these three domains, let $\gamma^{\prime}$ be the curve
obtained by taking symmetrics of the parts of $\gamma$ outside $D^{\prime}$
with respect to the singular "plane" of $\partial D^{\prime}$ beside the curve
(see fig.3.1). The curve $\gamma^{\prime}$ is in $D^{\prime}$, it joins two
conjugate points by the transformation $\sigma_{0}$, then $l(\gamma)\geq
l(\gamma^{\prime})\geq\pi$ (above argument).
* •
If $\gamma$ gets out of these domains, let $\gamma^{\prime}$ be the curve
obtained by projecting the part of $\gamma$ outside $D^{\prime}$ on its
boundary $\partial D^{\prime}$. We get a continuous curve in $D^{\prime}$
joining two conjugate points by $\sigma_{0}$, we conclude that $l(\gamma)\geq
l(\gamma^{\prime})\geq\pi$.
Finally, note that the distance to $\sigma_{0}(m)$ of the composition of
several translations of $m$ is too large by arguments similar to those above.
∎
Figure 1: A curve joining $m$ to $m^{\prime}=T_{1}(\sigma(m))$ will go through
$3$ domains $D_{p}$. For the parts of this curve outside $D^{\prime}$ we take
their symetrics with respect to the boundary $D^{\prime}$.
###### Remark 8.
In fact the two preceding lemmas are also true for the torus $(T,g_{c})$ and
can be proven exactly the same way.
###### Theorem 1.
If the real number $a$ is equal to $\pi/4$ then
$\frac{Sys^{3}(C_{2},g_{hex})}{Vol(C_{2},g_{hex})}>\frac{2}{\sqrt{3}}$
###### Proof.
The volume of $(C_{2},g_{hex})$ is equal to
$\int_{0}^{\pi}\iint_{D}\cos\sqrt{x^{2}+y^{2}}dydxdz$
where $D$ is a regular hexagon of shortest distance between its opposite edges
equal to $2a$.
The systole is equal to
$\inf\big{\\{}Sys(T,g_{hex}),\inf\\{dist_{(T,g_{hex})}(m,\sigma(m))\\}\big{\\}}$
By Lemmas 2 and 4, it is equal to $\inf\\{4a,2\pi\cos(a\sqrt{2}),\pi\\}$.
Then, for $a=\pi/4$, we have $Sys(C_{2},g_{hex})=\pi$. Using the software
"Maple" we find an approximation of the volume (2,80) up to $1/100$, then a
simple calculation gives the systolic ratio
$Sys^{3}(C_{2},g_{hex})/Vol(C_{2},g_{hex})\simeq 1,38$.
∎
### 5.4 The manifold $C_{2}$ as a quotient of the torus $(T,g_{c})$
To get a manifold homeomorphic to $C_{2}$, we can take an arbitrary $\Delta$,
then consider the quotient by the same transformations as before. To increase
the most the systolic ratio, $\Delta$ should have the smallest volume
possible, i.e. it should be hexagonal. It is nevertheless interessting to get
this manifold as a quotient of the torus $(T,g_{c})$, i.e. when $\Delta$ is
the "special" square lattice. We denote by $(C_{2},g_{c})$ the quotient of
$(T,g_{c})$ by the subgroup generated by $\sigma$.
When $a=\pi/4$, the intersection of $(T,g_{c})$ with one of the planes $x=0$
or $y=0$ is the covering torus of the Klein-Bavard bottle (c.f. [Bav86], see
also [El-La08]). More generally, the intersection with planes containing the
axis $z$ is a singular surface (a cylinder or a torus) of curvature $+1$ where
it is smooth. It turns out to be true that with a good choice of the parameter
$a$ the manifold $(C_{2},g_{c})$ admits a systolic ratio greater than
$\sqrt{3}/2$, and the calculation is based, as in the case of $(T,g_{hex})$,
on the fact that the distance in $(T,g_{c})$ between a point and its image by
$\sigma$ is greater than $\pi$ (c.f. 4).
###### Proposition 1.
If the real number $a$ satisfies the equation $2a-\pi\cos a\sqrt{2}=0$, then
the systolic ratio $\frac{Sys^{3}(C_{2},g_{c})}{Vol(C_{2},g_{c})}$ is greater
than $2/\sqrt{3}$. It is approximately equal to $1,18$.
## 6 The systolic ratio of $C_{4}$, $C_{6}$, $C_{3}$ and $C_{2,2}$
### 6.1 Type $C_{4}$
In the flat case, we saw that $C_{4}$ is the quotient of $C_{2}$ (the basis
$(a_{1},a_{2},a_{3})$ should be orthogonal with $|a_{1}|=|a_{2}|$) by the
subgroup generated by $t_{a_{3}/4}\circ r_{a_{3},\pi/2}$. It turns out that
this property is true for the metric $g_{c}$, more precisely the
transformation $t_{a_{3}/4}\circ r_{a_{3},\pi/2}$ is indeed an isometry of
$g_{c}$ and the quotient of $(C_{2},g_{c})$ by this isometry gives a manifold
of type $C_{4}$.
The volume of a flat manifold of type $C_{4}$ is equal to
$|a_{1}||a_{2}||a_{3}|/4$, and the systole is equal to
$\inf\\{|a_{1}|,|a_{2}|,|a_{3}|/4\\}$. The systolic ratio is smaller than $1$.
Now, the quotient of $(T,g_{c})$ by the subgroup $\Gamma$ of isometries of $g$
generated by $\tau:(x,y,z)\longrightarrow(-y,x,z+\pi/2)$ gives a manifold
homeomorphic to $C_{4}$, we denote it by $(C_{4},g_{c})$. Actually $C_{4}$ can
be seen in two different ways starting from the isometry $t_{a_{3}/4}\circ
r_{a_{3},\pi/2}$ (which is the same as $\tau$) of $\mathbb{R}^{3}$. This
isometry gives when we go to the quotient a fixed points free isometry of
order $4$ (resp of order $2$) of $(T,g_{c})$ (resp of $(C_{2},g_{c}))$.
The transformations $\tau$ and $\tau^{-1}$ are of order $4$ in $(T,g_{c})$ and
keep $2$ geodesics globally invariant.
The transformation $\tau^{2}$ is of order $2$ in $(T,g_{c})$ and keep, in
addition to the geodesics fixed by the transformation $\tau$, $2$ others
globally invariant. They are the vertical geodesics going through the points
of the lattice $\Delta$ (see fig.2).
Figure 2: The transformations $\tau$ et $\tau^{-1}$ keep fixed the vertical
axes going through the points $O_{1}$ et $O_{4}$. The transformation
$\tau^{2}$ keep fixed these same axes, as well as the vertical ones going
through the points $O_{3}$ and $O_{4}$.
###### Theorem 2.
If $a=\pi/8$, the systole of $(C_{4},g_{c})$ is equal to $\pi/2$ and the
systolic ratio $\frac{Sys^{3}(C_{4},g_{c})}{Vol(C_{4},g_{c})}$ is greater than
$1$.
###### Proof.
The systole of $(T,g_{c})$ is equal to $\inf\\{4a,2\pi\cos(a\sqrt{2})\\}$. By
proposition 8 we know that $d(m,\tau^{2}(m))\geq\pi$ ($\tau^{2}=\sigma$), the
proof is reduced to find a "good" lower bound of $\tau$. Using the triangular
inequality in $(T,g_{c})$, we have
$d(m,\tau^{2}(m))\leq d(m,\tau(m))+d(\tau(m),\tau^{2}(m))$
but $d(p,\tau(p))=d(\tau(p),\tau^{2}(p))$ since $\tau$ is an isometry of
$(T,g_{c})$. Then $d(m,\tau(m))\geq\pi/2$, and the equality is achieved for
the points $m$ of the rotation axis. Note that using the same method, we get a
good lower bound of $\tau^{3}=\tau^{-1}$.
Finally for $a=\pi/8$ the systole of $(C_{4},g_{c})$ is equal to $\pi/2$. The
volume is equal to
$4\int_{0}^{\frac{\pi}{2}}\int_{-\frac{\pi}{8}}^{\frac{\pi}{8}}\int_{-\frac{\pi}{8}}^{\frac{\pi}{8}}\cos\sqrt{x^{2}+y^{2}}dxdydz$
Using Maple, we find the systolic ratio of our manifold, it is approximately
equal to $1,05>1$.
∎
### 6.2 Type $C_{6}$
In the flat case, the volume is equal to $\frac{1}{6}det(a_{1},a_{2})|a_{3}|$
and the systole is equal to $\inf\\{|a_{3}|/6,s\\}$, where $s$ is the systole
of the flat 2-dimensional torus defined by the lattice of basis $a_{1},a_{2}$.
Considering the usual normalisation $\frac{1}{6}|a_{3}|=1$, the systolic ratio
is equal to
$\frac{s^{3}}{\det(a_{1},a_{2})}\quad\hbox{if $s\leq 1$
and}\quad\frac{1}{\det(a_{1},a_{2})}\quad\hbox{if $s\geq 1$,}$
It is smaller than $2/\sqrt{3}$.
Now to improve this systolic ratio, we will start this time with the hexagonal
torus $(T,g_{hex})$ defined in 4, since the lattice $\Delta$ should be
hexagonal. To get the manifold $C_{6}$, we take the quotient of $(T,g_{hex})$
by the subgroup generated by the isometry $\phi$ which sends a point $(p,z)$
to the point $(r_{\pi/3}(p),z+\pi/3)$, the result is the manifold
$(C_{6},g_{hex})$.
The manifold $C_{6}$ too can be seen in two different ways starting with the
isometry $t_{a_{3}/4}\circ r_{a_{3},\pi/2}$. This last one gives, when we go
to the quotient manifold, a fixed point free isometry of order $6$ (resp of
order $3$) of the torus $(T,g_{hex})$ (resp of $(C_{2},g_{hex})$).
The transformations $\phi$ and $\phi^{-1}$ are of order $6$ in $(T,g_{hex})$
and keep only one geodesic globally invariant.
The transformations $\phi^{2}$ and $\phi^{4}$ are of order $3$ in
$(T,g_{hex})$ and keep, in addition to the one of $\phi$, $2$ vertical
geodesics globally invariant.
The transformation $\phi^{3}$ is of order $2$ and keeps, in addition to the
one kept invariant by $\phi$, $3$ vertical geodesics globally invariant (see
fig.3).
Figure 3: The transformations $\phi$ and $\phi^{-1}$ only keep fixed the
vertical axis going through the point $O_{1}$. The transformations $\phi^{2}$
and $\phi^{-2}$ keep fixed, in addition to the axis going through $O_{1}$, the
vertical axes going through the points $A$ et $B$. The transformation
$\phi^{3}$ keeps fixed, in addition to these three axes, the vertical ones
going through the points $O_{i}$, ($i=2,3,4$).
###### Theorem 3.
If the real number $a=\pi/12$, the systolic ratio
$\frac{Sys(C_{6},g_{hex})^{3}}{Vol(C_{6},g_{hex})}$ is greater than
$2/\sqrt{3}$.
###### Proof.
If $a=\pi/12$ we know, by lemma 2, that the systole of $(T,g_{hex})$ is equal
to $\pi/3$. We also know, by Lemma 4, that the distance in $(T,g_{hex})$
between a point $m$ and its image by $\phi^{3}$ is greater or equal to $\pi$.
Using the triangular inequality in $(T,g_{hex})$ we get
$d(m,\phi^{3}(m))\leq
d(m,\phi(m))+d(\phi(m),\phi^{2}(m))+d(\phi^{2}(m),\phi^{3}(m))$
and then $d(m,\phi(m))\geq\pi/3$ ($\phi^{3}=\sigma$). Moreover, the distance
in $(T,g_{hex})$ between a point $m$ of coordinate $(x,y,z)$ and a point
$m^{\prime}$ of coordinate $(x^{\prime},y^{\prime},z+2\pi/3)$ is greater than
$\pi/3$. Indeed, if $\gamma$ is a curve from $m$ to $m^{\prime}$ we have
$l(\gamma)=\int{\sqrt{x^{\prime 2}+y^{\prime 2}+\psi(x,y)z^{\prime
2}}dt}\geq\int{\sqrt{\psi(x,y)z^{\prime
2}}}\geq\cos\frac{2a}{\sqrt{3}}2\pi/3\geq\pi/3$
The curves from $T_{3}(m)$ to $m$ are too much long, and then for any $m$ we
have
$\mathrm{dist}_{(T,g_{hex})}\big{(}m,\phi^{2}(m)\big{)}\geq\pi/3$
Finally, we conclude thar $Sys(C_{6},g_{hex})=\pi/3$. The volume is equal to
$\int_{0}^{\frac{\pi}{3}}\iint_{D}\cos\sqrt{x^{2}+y^{2}}dydxdz$
With an approximation on "Maple" we calculate the systolic ratio
$\frac{Sys^{3}(C_{6},g_{hex})}{Vol(C_{6},g_{hex})}\simeq 1,18$.
∎
### 6.3 Type $C_{2,2}$
It is the easiest case since the systolic geodesics of the best metric among
the flat ones are isolated.
In the flat case, the systole is equal to $\inf\\{a_{1}/2,a_{2}/2,a_{3}/2\\}$.
The volume is equal to $\frac{|a_{1}||a_{2}||a_{3}|}{4}$. The systolic ratio
is smaller than $1/2$, the equality is achieved if and only if
$|a_{1}|=|a_{2}|=|a_{3}|$. In that case the systolic geodesics are isolated
and so they do not cover the manifold $C_{2,2}$.
The criterion seen in the introduction allows us to conclude that the flat
metric on $C_{2,2}$ are not the best for the isosystolic inequality.
### 6.4 Type $C_{3}$
In the flat case, the volume is equal to $\frac{1}{3}\det(a_{1},a_{2})|a_{3}|$
but also to $\frac{\sqrt{3}}{6}|a_{1}||a_{3}|$, and the systole is equal to
$\inf\\{|a_{3}|/3,|a_{1}|\\}$, since the lattice generated by $a_{1}$ and
$a_{2}$ is hexagonal. We conclude easily that the systolic ratio is less or
equal to $2/\sqrt{3}$. The equality is achieved for
$|a_{3}|=3|a_{2}|=3|a_{1}|$.
To improve this systolic ratio, we start with the hexagonal torus
$(T,g_{hex})$ defined in section 4, since the lattice $\Delta$ should be
hexagonal. To get the manifold $C_{3}$, we take the quotient of $(T,g_{hex})$
by the subgroup generated by the isometry $\varphi$ which sends a point
$(p,z)$ to the point $(r_{2\pi/3}(p),z+2\pi/3)$, the result is the manifold
$(C_{3},g_{hex})$.
Since the manifold $C_{3}$ is not a quotient of $C_{2}$, it does not contain
surfaces that are Klein bottles or Möbius bands. Then, our previous methods of
getting a lower bound for the systole cannot be applied. Though, a special and
more general argument is necessary.
Let $\varphi_{c}$ be the isometry of $(T,g_{hex})$ which sends $(p,z)$ to the
point $(r_{2\pi/3}(p),z+c)$. The quotient of $(T,g_{hex})$ by the subgroup
generated by $\varphi_{c}$ is clearly a manifold homeomorphic to $C_{3}$, we
denote it by $(C_{3},g_{hex}^{c})$. Let $\gamma$ be the vertical geodesic in a
domain $D_{p}$ in $(C_{3},g_{hex}^{c})$ going through the point $p$, it has
length equal to $c$. Now let $H$ be a piecewise smooth variation of $\gamma$
through geodesics joining a point $m$ to $\varphi_{c}(m)$, we impose that
these curves stay in $D_{p}$ and do not touch the singularity.
###### Lemma 5.
The second variation of $H$ at the curve $\gamma$ is strictly positive if
$0<c<2\pi/3$.
###### Proof.
Let $O$ be a small tubular neighborhood of $\gamma$ and let $\Omega$ be the
set of geodesics in $D_{p}$ from $m_{t}\in O$ to $\varphi_{c}(m_{t})$ that do
not touch the singularity (one parameter family since the situation is
invariant under rotation around $\gamma$). Then
$H:]-\epsilon,\epsilon[\longrightarrow\Omega$
$t\longrightarrow\gamma_{t}:[0,1]\rightarrow(C_{3},g_{hex}^{c})$
is such that $H(o)=\gamma$. Let $T=\frac{\partial\gamma_{t}}{\partial s}$
(velocity vector of $\gamma_{t}$), and $V=\frac{\partial\gamma}{\partial
t}|_{\gamma_{t}}$ (the Jacobi field along $\gamma$), and set
$L=\int_{0}^{c}{|T|ds}$. We have then
$\frac{\partial L}{\partial
t}=\frac{1}{L(t)}\int_{0}^{c}{g_{hex}^{c}(V,\nabla_{V}T)ds}$
since $\nabla_{T}V-\nabla_{V}T=[V,T]=0$ we get
$L\frac{\partial L}{\partial t}=[g_{hex}^{c}(V,T)]_{0}^{c}\qquad\text{(1st
variation formula)}$
Now
$\frac{\partial}{\partial t}(L\frac{\partial L}{\partial t})=(\frac{\partial
L}{\partial t})^{2}+L\frac{\partial^{2}L}{\partial t^{2}}$
$=\int_{0}^{c}{(|\nabla_{T}V|^{2}+g_{hex}^{c}(T,\nabla_{V}\nabla_{T}V))ds}$
$=\int_{0}^{c}{|\nabla_{T}V|^{2}}+\int_{0}^{c}{g_{hex}^{c}(T,\nabla_{T}\nabla_{V}V)}+\int_{0}^{c}{g_{hex}^{c}(R(V,T)V,T)}$
where $R$ is the curvature tensor of $g_{hex}^{c}$. Now since the curvature in
the direction of the plane $(T,V)$ is equal to $1$ we get
$L\frac{\partial^{2}L}{\partial
t^{2}}=\int_{0}^{c}{|\nabla_{T}V|^{2}}-L^{2}\int_{0}^{c}{|V|^{2}}$
$=\int_{0}^{c}{(\nabla_{T}g_{hex}^{c}(V,\nabla_{T}V)-g_{hex}^{c}(V,R(T,V)T))}-L^{2}\int_{0}^{c}{|V|^{2}}=g_{hex}^{c}(V,\nabla_{T}V)|_{0}^{c}$
(see [Che75] p. 20 for more details on the second variation formula).
Now $V$ is a Jacobi Field orthogonal to $\gamma$ and so can be written in the
form $V=f_{1}E_{1}+f_{2}E_{2}$, where $(E_{1},E_{2})$ is an orthonormal basis
of the (horizontal) plane and parallel along $\gamma$. We can suppose that
$V(0)=E_{1}$ and $V(c)=E_{1}\cos{(2\pi/3)}+E_{2}\sin{(2\pi/3)}$. Now solving
the Jacobi Field equation $V^{\prime\prime}+V=0$ we get
$f_{1}(s)=\cos{(s)}+\frac{\cos{(2\pi/3)-\cos{(c)}}}{\sin{(c)}}\sin{(s)}$ and
$f_{2}(s)=\frac{\sin{(2\pi/3)}}{\sin{(c)}}\sin{(s)}$.
Finally
$g_{hex}^{c}(V,\nabla_{T}V)|_{0}^{c}=f_{2}(c)f_{2}^{\prime}(c)+f_{1}(c)f_{1}^{\prime}(c)-f_{1}^{\prime}(0)$
$=\sin^{2}(2\pi/3)(\cos(c)-\cos(2\pi/3))+\cos(2\pi/3)(\cos^{2}(c)-\cos^{2}(2\pi/3))$
∎
###### Remark 9.
This lemma shows that there exists a neighbourhood $U$ of the geodesic
$\gamma$ in which $\gamma$ is of minimum length among the geodesics joining
any point $m$ to $\varphi_{c}(m)$.
###### Theorem 4.
If $c=2\pi/3$ and $a=\pi/6$, the systolic ratio
$\frac{Sys(C_{3},g_{hex})^{3}}{Vol(C_{3},g_{hex})}$ is greater than
$2/\sqrt{3}$.
###### Proof.
We consider in the neighborhood $U$ a hexagon $H$ "parallel" (c.f. remark 7)
to the boundary $\partial D_{p}$. Let $\delta$ be a curve in
$(\mathbb{R}^{3},g_{hex})$ from a point $m$ in $D_{p}$ to $\varphi_{c}(m)$,
the minimal projection of $\delta$ on the boundary $\partial H$ gives a curve
$\delta^{\prime}$ in $U$ joining two conjugate points by the transformation
$\varphi_{c}$, then we have by lemma 3 and remark 7
$l(\delta)\geq l(\delta^{\prime})\geq c$
The same arguments as the ones used in section 5 show that
$d_{(T,g_{hex})}(m,\varphi_{c}(m))\geq c$.
Now passing to the limit when $c\rightarrow 2\pi/3$ we get
$d_{(T,g_{hex})}(m,\varphi(m))\geq 2\pi/3$
This allows us to calculate the systole of $(C_{3},g_{hex})$, when $a=\pi/6$.
It is equal to $2\pi/3$ (of course we use Lemma 2 too). The volume is equal to
$\int_{0}^{\frac{2\pi}{3}}\iint_{D}\cos\sqrt{x^{2}+y^{2}}dydxdz$
As before we calculate this integral using Maple, and we get an approximation
of $\frac{Sys(C_{3},g_{hex})^{3}}{Vol(C_{3},g_{hex})}\simeq 1.24$.
∎
###### Remark 10.
The previous proof is also valid for the manifolds $(C_{6},g_{hex})$ and
$(C_{4},g_{c})$, and allows us to find the good lower bound of their systoles.
But the method used in section 6 is a lot more simple (we just used the
triangular inequality), this is due to the existence of Klein bottles and
Möbius bands in these manifolds.
## 7 Comparison between $(C_{2},g_{hex})$ and flat hexagonal $3$-dimensional
torus
Among flat tori of dimension $3$, the hexagonal one is the best for the
isosystolic inequality. It is the quotient of $\mathbb{R}^{3}$ by the lattice
that has a basis $(a_{1},a_{2},a_{3})$ such that $(a_{i},a_{j})=\pi/3$ for
$i\neq j$. It is known that this torus, that we denote by $T^{3}_{hex}$, is a
very good candidate to realize the systolic constant of tori of dimension $3$,
it satisfies the following properties:
* •
At any point in $T^{3}_{hex}$ there exists exactly $6$ systolic geodesics
going through the point.
* •
The systolic geodesics of any systolic class of $T^{3}_{hex}$ cover the torus.
A systolic class is an element of the fundamental group that contains at least
one systolic geodesic.
Our singular metric $(C_{2},g_{hex})$ verifies the second property and a
stronger one than the first: At any point outside the singularity of
$(C_{2},g_{hex})$, there exists infinitely many systolic geodesics going
through the point.
For the points on the singularity, there are $5$ systolic geodesics going
through any of these points: $3$ in the horizontal flat 2-torus and $2$ in the
surface $\theta=constant$. The number of systolic geodesics going through the
points of the singularity is less than the case of $T^{3}_{hex}$, but this
does not cause any trouble since the singularity has zero measure.
We think that as for the $3$-dimensional hexagonal torus the manifold
$(C_{2},g_{hex})$ is a very good candidate to realize the systolic constant
because it has an abondance of systolic geodesics that can be seen by the fact
that it satisfies the properties mentioned above.
When speaking about the metrics $(C_{3},g_{hex})$,$(C_{6},g_{hex})$ and
$(C_{4},g_{c})$, they still satisfy the property of being covered by systolic
geodesics mentioned in the introduction. But we cannot say if they satisfy
something stronger as for the manifold $(C_{2},g_{hex})$ since we do not have
much information about the length of vertical geodesics.
The following table allows to do the comparison between the biggest systolic
ratio of the flat metrics ($\tau$(flat)) and the biggest ones of the singular
metrics that we have constructed in this paper ($\tau$(singular)) on the
orientable Bieberbach 3-manifolds of type $C_{2},C_{3},C_{4}$ and $C_{6}$.
type | $\tau$(flat) | approximate value | $\tau$(singular)
---|---|---|---
$C_{2}$ | $\frac{2}{\sqrt{3}}$ | $\thickapprox 1,154$ | $\thickapprox 1,38$
$C_{3}$ | $\frac{2}{\sqrt{3}}$ | $\thickapprox 1,154$ | $\thickapprox 1,24$
$C_{4}$ | $1$ | $1$ | $\thickapprox 1,05$
$C_{6}$ | $\frac{2}{\sqrt{3}}$ | $\thickapprox 1,154$ | $\thickapprox 1,18$
## References
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* [Bab02] Babenko, I.; Souplesse intersystolique forte des variétés fermées et des polyèdres, Ann. Inst. Fourier 52 no. 4, 1259-1284 (2002).
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* [bav93] Bavard, C.; L’aire systolique conforme des groupes cristallographiques du plan, Ann. Inst. Fourier (Grenoble) 43, 815-842 (1993).
* [bav93K] Bavard, C.; Une remarque sur la g om trie systolique de la bouteille de Klein, Arch. Math. (Basel) 87 (2006), No 1, 72-74 (1993).
* [Ber70] Berger, M.; Quelques problèmes de géométrie riemannienne ou deux variations sur les espaces compacts symétriques de rang $1$, L’Ens.Math. (2) 16 73–96 (1970).
* [Ber93] Berger, M.; Systoles et applications selon Gromov, Séminaire N. Bourbaki, exposé 771, Astérisque 216, 279–310 (1993).
* [Ber72P] Berger, M.; Du c t de chez Pu, Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 1–44.
* [Ber72L] Berger, M.; A l’ombre de Lowner, Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 241–260.
* [Ber03] Berger, M.; A Panoramic View of Riemannian Geometry, Springer-Verlag, Berlin, (2003).
* [Bla61F] Blatter, C.; Über extremallängen auf geschlossenen Flächen, Comment. Math. Helvetici 35 (1961), 153–168.
* [Bla61M] Blatter, C.; Zur Riemannschen Geometrie im Grossen auf dem Möbiusband, Compositio Math. 15 (1961), 88–107.
* [Bu-Iv01] Burago, D.; Burago, Y.D.; Ivanov, S.; A course in metric geometry, Graduate studies in Mathematics (33), Amer. Math. Soc., Providence, R.I. (2001).
* [Cal96] Calabi E.; Extremal isosystolic metrics for compact surfaces, Actes de la table ronde de géométrie différentielle, Semin. Congr 1, Soc.Math.France 146–166 (1996).
* [Cro03] Croke, C.; Katz, M.; Universal volume bounds in Riemannian manifolds, Surv. Differ. Geom. VIII (Boston, MA, 2002), 109-137, Surv. Differ. Geom. VIII, Int. Press, Somerville, MA, (2003).
* [Cha86] Charlap, L.S.; Bieberbach Groups and Flat Manifolds, Springer Universitext, Berlin (1986).
* [Che75] Cheeger, J; Ebin, D; Comparison Theorems in Riemannian Geometry, North-Holland Publishing Co., Amsterdam (1975).
* [El-La08] Elmir, C.; Lafontaine,J.; Sur la géométrie systolique des variétés de Bieberbach, Geom. Dedicata. 136, 95–110 (2008)
* [GHL04] Gallot, S.; Hulin, D.; Lafontaine, J.; Riemannian Geometry, 3rd edition, Springer, Berlin Heidelberg (2004).
* [Gro83] Gromov, M.; Filling Riemannian manifolds, J. Diff. Geom. 18, 1–147(1983)
* [Gro] Gromov M.; Systoles and intersystolic inequalities, in : Besse, A.L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Société Mathématique de France, Séminaires et Congrès no. 1, p. 291–362.
* [Hir76] Hirsch M.; Differential Topology, Springer Verlag Universitext, Berlin Heidelberg (1976).
* [Kat07] Katz, M.G, Systolic Geometry and Topology, Math. Surveys and Monographs 137, Amer. Math. Soc., Providence, R.I. (2007).
* [Pu52] Pu, P.M.; Some inequalities in certain non-orientable riemannian manifolds. Pacific J.Math.2, 55–71(1952).
* [Sak88] Sakai, T.; A proof of the isosystolic inequality for the Klein bottle, Proc. Amer. Math. Soc. 104, 589–590 (1988).
* [Thu97] Thurston, W.P.; Three-Dimensional Geometry and Topology, edited by S. Levy, Princeton University Press, Princeton (1997).
* [Wol74] Wolf, J.A.; Spaces of constant curvature, Publish or Perish, Boston (1974).
|
arxiv-papers
| 2009-12-19T13:22:55 |
2024-09-04T02:49:07.127118
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chady Elmir and Jacques Lafontaine",
"submitter": "Chady Elmir",
"url": "https://arxiv.org/abs/0912.3894"
}
|
0912.4044
|
11institutetext: Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425
Jülich, Germany. m.chraibi@fz-juelich.de, a.seyfried@fz-juelich.de
22institutetext: Institute for Theoretical Physics, Universität zu Köln,
D-50937 Köln, Germany. as@thp.uni-koeln.de 33institutetext: Hamburg University
of Technology, 21071 Hamburg, Germany. mackens@tuhh.de
# Quantitative Verification of a Force-based Model for Pedestrian Dynamics
Mohcine Chraibi 11 Armin Seyfried 11 Andreas Schadschneider 22 Wolfgang
Mackens 33
###### Abstract
This paper introduces a spatially continuous force-based model for simulating
pedestrian dynamics. The main intention of this work is the quantitative
description of pedestrian movement through bottlenecks and in corridors.
Measurements of flow and density at bottlenecks will be presented and compared
with empirical data. Furthermore the fundamental diagram for the movement in a
corridor is reproduced. The results of the proposed model show a good
agreement with empirical data.
## 1 Introduction
One application of pedestrian dynamics is the enhancement of the safety of
people in complex buildings and in big mass events e.g., sporting events,
religious pilgrimages, etc. where there is a risk of disaster. Thanks to
computer simulations, it is possible to forecast the emergency egress and
optimise the evacuation of large crowds. Another aspect of pedestrian dynamics
is the comfort of passengers in pedestrian facilities e.g., airports, railway
stations, shopping malls, etc. Those facilities have to be designed in a way
to ensure minimal travel times and maximal capacities. For these applications,
robust and quantitatively validated models are necessary.
A wide spectrum of models have been designed to simulate pedestrian dynamics.
Generally those models can be classified into macroscopic and microscopic
models. In macroscopic models the system is described by mean values of
characteristics of pedestrian streams e.g., density and flow, whereas
microscopic models consider the movement of individual persons separately.
Microscopic models can be subdivided into several classes e.g., rule-based and
force-based models. For a detailed discussion we refer to Schadschneider2009a
. In this work we focus on spatially continuous force-based models.
Force-based models take Newton’s second law of dynamics as a guiding
principle. Thus, the movement of each pedestrian is defined by:
$\overrightarrow{F_{i}}=\sum_{j\neq i}^{\tilde{N}}\overrightarrow{F_{ij}^{\rm
rep}}+\sum_{B}\overrightarrow{F_{iB}^{\rm rep}}+\overrightarrow{F_{i}^{\rm
drv}}=m_{i}\overrightarrow{a_{i}},$ (1)
where $\overrightarrow{F_{ij}^{\rm rep}}$ denotes the repulsive force from
pedestrian $j$ acting on pedestrian $i$, $\overrightarrow{F_{iB}^{\rm rep}}$
is the repulsive force emerging from borders and $\overrightarrow{F_{i}^{\rm
drv}}$ is a driving force. $m_{i}$ is a constant with dimensions of mass and
$\tilde{N}$ the number of neighbouring pedestrians. Repulsive forces model the
collision-avoidance performed by pedestrians. Whereas the driving force models
the intention of a pedestrian to move to some destination. The set of
equations (1) for all pedestrians results in a high-dimensional system of
second order ordinary differential equations. The time evolution of the
positions and velocities of all pedestrians is obtained by numerical
integration.
Most force-based models describe the movement of pedestrians qualitatively
well. Collective phenomena like lane formations Helbing1995 ; Helbing2004 ;
Yu2005 , oscillations at bottlenecks Helbing1995 ; Helbing2004 , the “faster-
is-slower” effect Lakoba2005 ; Parisi2007 , clogging at exit doors Helbing2004
; Yu2005 etc. are reproduced. These achievements indicate that these models
are promising candidates. However, a qualitative description is not sufficient
if reliable statements about critical processes, e.g., emergency egress, are
requested. Moreover, implementations of models do not rely on one sole
approach. Especially in high density situations simple numerical treatment has
to be supplemented by additional techniques to obtain reasonable results.
Examples are restrictions on state variables and sometimes even totally
different procedures replacing the above equations of motion (1) to avoid
partial and total overlapping among pedestrians Lakoba2005 ; Yu2005 or
negative and high velocities Helbing1995 .
We address the possibility of describing reasonably and in a quantitative
manner the movement of pedestrians, with a modelling approach as simple as
possible. For a systematic verification of our model we measure the
fundamental diagram, the flow through bottlenecks and the density inside and
in front of the entrance of a bottleneck. In the next section, we propose such
a model which is solely based on the equation of motion (1). Furthermore the
model incorporates free parameters which allow calibration to fit quantitative
data.
## 2 Definition of the model
Our model is based on the Centrifugal Force Model (CFM) Yu2005 . The CFM takes
into account the distance between pedestrians as well as their relative
velocities. Pedestrians are modelled as circles with constant diameter. Their
movement is a direct result of superposition of repulsive and driving forces
acting on the centre of each pedestrian. Repulsive forces acting on pedestrian
$i$ from other pedestrians in their neighbourhood and eventually from walls,
stairs, etc. to prevent collisions and overlapping (Fig. 1). The driving
force, however, adds a positive term to the resulting force, to enable
movement of pedestrian $i$ in a certain direction with a given desired speed
$\parallel\overrightarrow{V_{i}^{0}}\parallel$. The mathematical expression of
the driving force as introduced initially in Helbing1995 is used:
$\overrightarrow{F_{i}^{\rm
drv}}=m_{i}\frac{\overrightarrow{V_{i}^{0}}-\overrightarrow{V_{i}}}{{\tau}},$
(2)
with a time constant ${\tau}$.
Figure 1: The direction of the repulsive force pedestrian $j$ acting on
pedestrian $i$.
The definition of the repulsive force in the CFM expresses several principles.
First, the force between two pedestrians decreases with increasing distance.
In the CFM it is inversely proportional to their distance. Given the position
of two pedestrians $i$ and $j$, the direction vector between their centers is
defined as:
$\overrightarrow{R_{ij}}=\overrightarrow{R_{j}}-\overrightarrow{R_{i}},\;\;\;\;\overrightarrow{e_{ij}}=\frac{\overrightarrow{R_{ij}}}{\parallel\overrightarrow{R_{ij}}\parallel}\,.$
(3)
Furthermore, the repulsive force takes into account the relative velocity
between pedestrian $i$ and pedestrian $j$. The following special definition
provides that slower pedestrians are not affected by the presence of faster
pedestrians in front of them:
$V_{ij}=\frac{1}{2}[(\overrightarrow{V_{i}}-\overrightarrow{V_{j}})\cdot\overrightarrow{e_{ij}}+|(\overrightarrow{V_{i}}-\overrightarrow{V_{j}})\cdot\overrightarrow{e_{ij}}|].$
(4)
As in general pedestrians react only to obstacles and pedestrians that are
within their perception, the reaction field of the repulsive force is reduced
to the angle of vision of each pedestrian ($180^{\circ}$), by introducing the
coefficient:
$K_{ij}=\frac{1}{2}\frac{\overrightarrow{V_{i}}\cdot\overrightarrow{e_{ij}}+\mid\overrightarrow{V_{i}}\cdot\overrightarrow{e_{ij}}\mid}{\parallel\overrightarrow{V_{i}}\parallel}.$
(5)
With the definitions in Eqs. (3), (4) and (5), the repulsive force between two
pedestrians is formulated as:
$\overrightarrow{F_{ij}^{\rm
rep}}=-m_{i}K_{ij}\frac{V_{ij}^{2}}{\parallel\overrightarrow{R_{ij}}\parallel}\overrightarrow{e_{ij}}\,.$
(6)
In Chraibi2009a it was shown that the introduction of a “collision detection
technique” (CDT), see Yu2005 for the definition, is necessary to mitigate
overlapping among pedestrians.
In the following, we will discuss why volume exclusion is not guaranteed by
Eq. (6) and meanwhile introduce our modifications of the repulsive force. Due
to the quotient in Eq. (6) when the distance is small, low relative velocities
lead to an unacceptably small force. Consequently, partial or total
overlapping are not prevented. Introducing the intended speed in the numerator
of the repulsive force eliminates this side-effect. Furthermore, the modified
repulsive force and driving force (2) compensate at low velocities, which
damps oscillations.
Since faster pedestrians require more space than slower pedestrians, due to
increasing step sizes Seyfried2006 , the diameter of pedestrian $i$ depends
linearly on its velocity:
$D_{i}=d_{a}+d_{b}\parallel\overrightarrow{V_{i}}\parallel,$ (7)
with free parameters $d_{a}$ and $d_{b}$. We define the distance between
pedestrian $i$ and pedestrian $j$ as:
${\rm
dist}_{ij}=\parallel\overrightarrow{R_{ij}}\parallel-\frac{1}{2}(D_{i}(\parallel\overrightarrow{V_{i}}\parallel)+D_{j}(\parallel\overrightarrow{V_{j}}\parallel)).$
(8)
By taking these aspects into account, the definition of the modified repulsive
force reads
$\overrightarrow{F_{ij}^{\rm
rep}}=-m_{i}K_{ij}\frac{(\nu\parallel\overrightarrow{V_{i}^{0}}\parallel+V_{ij})^{2}}{{\rm
dist}_{ij}}\overrightarrow{e_{ij}},$ (9)
where $\nu$ is a parameter which adjusts the strength of the force. Due to
these changes we can do without the extra CDT which dominates the dynamics in
Yu2005 in case of formation of dense crowds.
The repulsive force between two pedestrians $i$ and $j$ is infinite at contact
and decreasing with increasing distance between $i$ and $j$. Since the
repulsive force as defined in Eq. (9) does not vanish, the summation over all
other pedestrians leads to a complexity of $O(N^{2})$. To deal with this
problem and to consider a limited range of pedestrian interaction only the
influence of neighbouring pedestrians is taken into account. Two pedestrians
are said to be neighbours if their distance is within a certain cut-off radius
$R_{c}=2.5\;\mbox{m}$. To guarantee robust numerical integration a two-sided
Hermite-interpolation of the repulsive force is implemented (see Fig. 2).
Figure 2: Left: The interpolation of the repulsive force between pedestrians
$i$ and $j$. Right: Direction of pedestrians in corridors and bottlenecks.
The interpolation guarantees that for each pair $i$, $j$ with a distance in
the interval $[{R^{\prime}}_{c},R_{c}]$ the norm of the repulsive force
between them decreases smoothly to zero. ${R^{\prime}}_{c}$ is set to
$R_{c}-0.1\;\mbox{m}$. For distances in the interval $[S_{\rm max},R_{\rm
eps}]$ the interpolation avoids an increase of the force to infinity, to reach
a maximum value of $F_{\rm max}=1000\;\mbox{N}$. $R_{\rm eps}$ is set to
$0.1\;\mbox{m}$ and $S_{\rm max}$ to $-5\;\mbox{m}$.
The desired direction of a pedestrian is set to be parallel to the walls of
the corridor. In the bottleneck case it is set towards the centre of the
entrance to the bottleneck if the pedestrian is outside the range of the
bottleneck. That is if he can not “see” the exit of the bottleneck. Otherwise,
the desired direction is chosen parallel to the length of the bottleneck (Fig.
2).
## 3 Simulation results
The initial value problem (1) was solved using an Euler scheme with fixed-step
size $\Delta t=0.01\;$s. The desired speeds of pedestrians are Gaussian
distributed with mean $\mu=1.34\;\mbox{m/s}$ and standard deviation
$\sigma=0.26\;\mbox{m/s}$. The constant ${\tau}$ in Eq. (2) is set to
$0.5\;\,\mbox{s}$. For simplicity, the mass, $m_{i}$ is set to unity. Several
parameter values were tested. The free parameters in Eqs. (9) and (7) are set
to $\nu=0.2$, $d_{a}=0.3\mbox{ m}\;\,\mbox{and}\;\;d_{b}=0.2\mbox{ s}$. With
this parameter set the results of the simulations are in good agreement with
empirical data.
To verify the ability of the model to reproduce the fundamental diagram,
measurements in corridors of different widths were performed. The length of
the corridor is $20\;\mbox{m}$ and its width is $2\;\mbox{m}$.
Figure 3: Left: The fundamental diagram in comparison with empirical data. For
other values of the corridor’s width ($1\;\mbox{m}\;\mbox{and}\;4\;\mbox{m}$),
the simulation results are also in good agreement with the empirical data.
Right: Flow measurement with the modified CFM in comparison with empirical
data.
The shape of the reproduced velocity-density relation is in good agreement
with the empirical data Mori1987 ; Helbing2007 ; Oeding1963 ; Hankin1958 , see
Fig. 3).
Furthermore, the flow of $60$ pedestrians through the bottleneck as described
in Seyfried2009b was simulated. The width of the bottleneck was changed from
$0.8\;$m to $1.2\;$m in steps of $0.1\;$m (Fig. 3).
A third validation comes from measurements of density inside the bottleneck as
well as in front of the entrance to the bottleneck. The density in front of
the entrance to the bottleneck is presented in Fig. 4(a). The results are in
good agreement with the experimental data in Seyfried2009b . Additionally, the
measured density values inside the bottleneck are in accordance with the
published empirical results in Rupprecht2007 , see Fig. 4(b). One remarks that
the density in front of the bottleneck is much higher than the density in the
bottleneck. This difference reflects typical dynamics at bottlenecks, which is
reproduced by our model.
(a) Density in front of the entrance to the bottleneck
(b) Density inside the bottleneck
Figure 4: Density measurements: The simulation results (blue lines) are in
good agreement with the empirical data presented in Seyfried2009 and
Rupprecht2007 . The difference between the density in front and inside the
bottleneck as well as the amplitude of the fluctuations are given correctly.
The width of the bottleneck is $1.2\;\mbox{m}$. Also for other values of the
width a good agreement between simulation results and empirical data is found.
## 4 Conclusions
We have proposed modifications of a spatially continuous force-based model
Yu2005 to describe quantitatively the movement of pedestrians in 2D-space.
Besides being a remedy for numerical instabilities in CFM the modifications
simplify the approach of Yu et al. Yu2005 since we can dispense with their
extra “collision detection technique” without deteriorating performance. The
implementation of the model is straightforward and does not use any
restrictions on the velocity. Simulation results show good agreement with
empirical data. Nevertheless, the model contains free parameters that have to
be tuned adequately to adapt the model to a given scenario. Further
improvement of the model could be made by including, for example, a density
dependent repulsive force.
## Acknowledgement
The authors are grateful to the Deutsche Forschungsgemeinschaft (DFG) for
funding this project under Grant-Nr.: SE 1789/1-1.
## References
* (1) Schadschneider A, Klingsch W, Klüpfel H, Kretz T, Rogsch C, Seyfried A (2009) Evacuation Dynamics: Empirical Results, Modeling and Applications. In: Meyers R A (ed.) Encyclopedia of Complexity and System Science. p. 3142-3176. Springer, Berlin Heidelberg
* (2) Helbing D, Molnár P (1995) Phys. Rev. E 51:4282–4286
* (3) Helbing D (2004) Computational Materials Science 30:180–187
* (4) Yu W J, Chen L Y, Dong R, Dai S Q (2005) Phys. Rev. E 72(2):026112
* (5) Lakoba T I, Kaup D J, Finkelstein N M (2005) Simulation 81:339–352
* (6) Parisi D R, Dorso C O (2007) Physica A 385(1):343–355
* (7) Chraibi M, Seyfried A, Schadschneider A, Mackens W (2009) Quantitative Description of Pedestrian Dynamics with a Force-based Model. In: IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology. p 583-586, vol 3
* (8) Seyfried A, Steffen B, Lippert T (2006) Physica A 368:232–238
* (9) Mori M, Tsukaguchi H (1987) Transp. Res. 21A(3):223–234
* (10) Helbing D, Johansson A, Al-Abideen H Z (2007) Phys. Rev. E 75:046109
* (11) Oeding D (1963) Verkehrsbelastung und Dimensionierung von Gehwegen und anderen Anlagen des Fußgängerverkehrs. Forschungsbericht 22, Technische Hochschule Braunschweig
* (12) Hankin B D, Wright R A (1958) Operational Research Quarterly 9:81–88
* (13) Seyfried A, Steffen B, Winkens T, Rupprecht A, Boltes M, Klingsch W (2009) Empirical data for pedestrian flow through bottlenecks. In Appert-Rolland C, Chevoir F, Gondret P, Lassarre S, Lebacque J P, Schreckenberg M (eds) Traffic and Granular Flow ’07. p. 189-199. Springer, Berlin Heidelberg
* (14) Rupprecht T, Seyfried A, Klingsch W, Boltes M (2007) Bottleneck capacity estimation for pedestrian traffic. In Proceedings of the Interflam 2007. p. 1423-1430. Intersience
* (15) Seyfried A, Rupprecht T, Passon O, Steffen B, Klingsch W, Boltes M (2009) Transportation Science 43:395–406
|
arxiv-papers
| 2009-12-20T19:22:00 |
2024-09-04T02:49:07.136757
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mohcine Chraibi, Armin Seyfried, Andreas Schadschneider, and Wolfgang\n Mackens",
"submitter": "Mohcine Chraibi",
"url": "https://arxiv.org/abs/0912.4044"
}
|
0912.4069
|
# Production of new superheavy Z=108-114 nuclei with 238U, 244Pu and 248,250Cm
targets
Zhao-Qing Feng fengzhq@impcas.ac.cn Gen-Ming Jin Jun-Qing Li Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic
of China
###### Abstract
Within the framework of the dinuclear system (DNS) model, production cross
sections of new superheavy nuclei with charged numbers Z=108-114 are analyzed
systematically. Possible combinations based on the actinide nuclides 238U,
244Pu and 248,250Cm with the optimal excitation energies and evaporation
channels are pointed out to synthesize new isotopes which lie between the
nuclides produced in the cold fusion and the 48Ca induced fusion reactions
experimentally, which are feasible to be constructed experimentally. It is
found that the production cross sections of superheavy nuclei decrease
drastically with the charged numbers of compound nuclei. Larger mass
asymmetries of the entrance channels enhance the cross sections in 2n-5n
channels.
PACS number(s)
25.70.Jj, 24.10.-i, 25.60.Pj
The synthesis of superheavy nuclei (SHN) is motivated with respect to search
the ”island of stability” which is predicted theoretically, and has obtained
much experimental research with fusion-evaporation reactions Ho00 ; Og07 .
Neutron-deficient SHN with charged numbers Z=107-112 were synthesized in cold
fusion reactions with the 208Pb and 209Bi targets for the first time and
investigated at GSI (Darmstadt, Germany) with the heavy-ion accelerator UNILAC
and the SHIP separator Ho00 ; Mu99 . Recently, experiments on the synthesis of
element 113 in the 70Zn+209Bi reaction have been performed successfully at
RIKEN (Tokyo, Japan) Mo04 . More neutron-rich SHN with Z=113-116, 118 were
assigned at FLNR in Dubna (Russia) with the double magic nucleus 48Ca
bombarding the actinide nuclei Og07 ; Og04 ; Og06 . New heavy isotopes 259Db
and 265Bh were also synthesized at HIRFL in Lanzhou (China) Ga01 . New SHN
between the isotopes of the cold fusion and the 48Ca induced reactions are of
importance not only for investigating the structure of SHN such as influence
of shell effect on stability of SHN etc, and also as a stepstone for further
synthesizing and identifying heavier superheavy nuclei.
Here we use a dinuclear system (DNS) model Fe06 ; Fe07 , in which the nucleon
transfer is coupled to the relative motion by solving a set of microscopically
derived master equations by distinguishing protons and neutrons, and a barrier
distribution in the capture and fusion process of two colliding nuclei is
introduced in the model. In the DNS model, the evaporation residue cross
section is expressed as a sum over partial waves with angular momentum $J$ at
centre-of-mass energy $E_{c.m.}$ Fe07 ; Fe09 ,
$\displaystyle\sigma_{ER}(E_{c.m.})=$ $\displaystyle\frac{\pi\hbar^{2}}{2\mu
E_{c.m.}}\sum_{J=0}^{J_{max}}(2J+1)T(E_{c.m.},J)$ (1) $\displaystyle\times
P_{CN}(E_{c.m.},J)W_{sur}(E_{c.m.},J).$
Here, $T(E_{c.m.},J)$ is the transmission probability of the two colliding
nuclei overcoming the Coulomb barrier in the entrance channel to form the DNS.
The $P_{CN}$ is the probability that the system will evolve from a touching
configuration to the formation of compound nucleus in competition with the
quasi-fission of the DNS and the fission of heavy fragment. The last term is
the survival probability of the formed compound nucleus, which can be
estimated with the statistical evaporation model by considering the
competition between neutron evaporation and fission Fe06 .
Within the concept of the DNS, the fusion probability was also calculated by
using the multidimensional Kramers-type expression to get the fusion and
quasifission rate by Adamian _et al._ Ad97 . In order to describe the fusion
dynamics as a diffusion process along proton and neutron degrees of freedom,
the fusion probability is obtained by solving a set of master equations
numerically in the potential energy surface of the DNS. The time evolution of
the distribution probability function $P(Z_{1},N_{1},E_{1},t)$ for fragment 1
with proton number $Z_{1}$ and neutron number $N_{1}$ and with excitation
energy $E_{1}$ is described by the following master equations,
$\displaystyle\frac{dP(Z_{1},N_{1},E_{1},t)}{dt}=$
$\displaystyle\sum_{Z_{1}^{\prime}}W_{Z_{1},N_{1};Z_{1}^{\prime},N_{1}}(t)\left[d_{Z_{1},N_{1}}P(Z_{1}^{\prime},N_{1},E_{1}^{\prime},t)-d_{Z_{1}^{\prime},N_{1}}P(Z_{1},N_{1},E_{1},t)\right]+\sum_{N_{1}^{\prime}}W_{Z_{1},N_{1};Z_{1},N_{1}^{\prime}}(t)$
(2)
$\displaystyle\left[d_{Z_{1},N_{1}}P(Z_{1},N_{1}^{\prime},E_{1}^{\prime},t)-d_{Z_{1},N_{1}^{\prime}}P(Z_{1},N_{1},E_{1},t)\right]-\left[\Lambda_{qf}(\Theta(t))+\Lambda_{fis}(\Theta(t))\right]P(Z_{1},N_{1},E_{1},t).$
Here $W_{Z_{1},N_{1};Z_{1}^{\prime},N_{1}}$
($W_{Z_{1},N_{1};Z_{1},N_{1}^{\prime}}$) is the mean transition probability
from the channel $(Z_{1},N_{1},E_{1})$ to
$(Z_{1}^{\prime},N_{1},E_{1}^{\prime})$ (or $(Z_{1},N_{1},E_{1})$ to
$(Z_{1},N_{1}^{\prime},E_{1}^{\prime})$), and $d_{Z_{1},N_{1}}$ denotes the
microscopic dimension corresponding to the macroscopic state
$(Z_{1},N_{1},E_{1})$. The sum is taken over all possible proton and neutron
numbers that fragment $Z_{1}^{\prime},N_{1}^{\prime}$ may take, but only one
nucleon transfer is considered in the model with the relation
$Z_{1}^{\prime}=Z_{1}\pm 1$ and $N_{1}^{\prime}=N_{1}\pm 1$. The excitation
energy $E_{1}$ is determined by the dissipation energy from the relative
motion and the potential energy surface of the DNS. The motion of nucleons in
the interacting potential is governed by the single-particle Hamiltonian Fe06
; Fe07 . The evolution of the DNS along the variable R leads to the quasi-
fission of the DNS. The quasi-fission rate $\Lambda_{qf}$ and the fission rate
$\Lambda_{fis}$ of heavy fragment are estimated with the one-dimensional
Kramers formula Fe07 ; Fe09 .
In the relaxation process of the relative motion, the DNS will be excited by
the dissipation of the relative kinetic energy. The local excitation energy is
determined by the excitation energy of the composite system and the potential
energy surface of the DNS. The potential energy surface (PES) of the DNS is
given by
$\displaystyle U(\\{\alpha\\})=$ $\displaystyle
B(Z_{1},N_{1})+B(Z_{2},N_{2})-\left[B(Z,N)+V^{CN}_{rot}(J)\right]$ (3)
$\displaystyle+V(\\{\alpha\\})$
with $Z_{1}+Z_{2}=Z$ and $N_{1}+N_{2}=N$. Here the symbol $\\{\alpha\\}$
denotes the sign of the quantities
$Z_{1},N_{1},Z_{2},N_{2};J,\textbf{R};\beta_{1},\beta_{2},\theta_{1},\theta_{2}$.
The $B(Z_{i},N_{i})(i=1,2)$ and $B(Z,N)$ are the negative binding energies of
the fragment $(Z_{i},N_{i})$ and the compound nucleus $(Z,N)$, respectively,
which are calculated from the liquid drop model, in which the shell and the
pairing corrections are included reasonably. The $V^{CN}_{rot}$ is the
rotation energy of the compound nucleus. The $\beta_{i}$ represent the
quadrupole deformations of the two fragments. The $\theta_{i}$ denote the
angles between the collision orientations and the symmetry axes of deformed
nuclei. The interaction potential between fragment $(Z_{1},N_{1})$ and
$(Z_{2},N_{2})$ includes the nuclear, Coulomb and centrifugal parts, the
details are given in Ref. Fe07 . In the calculation, the distance R between
the centers of the two fragments is chosen to be the value which gives the
minimum of the interaction potential, in which the DNS is considered to be
formed. So the PES depends on the proton and neutron numbers of the fragments.
The formation probability of the compound nucleus at the Coulomb barrier $B$
and for the angular momentum $J$ is given by Fe06 ; Fe07
$P_{CN}(E_{c.m.},J,B)=\sum_{Z_{1}=1}^{Z_{BG}}\sum_{N_{1}=1}^{N_{BG}}P(Z_{1},N_{1},E_{1},\tau_{int}).$
(4)
The interaction time $\tau_{int}$ in the dissipation process of two colliding
partners is dependent on the incident energy $E_{c.m.}$ and the quantities $J$
and $B$. We obtain the fusion probability as
$P_{CN}(E_{c.m.},J)=\int f(B)P_{CN}(E_{c.m.},J,B)dB,$ (5)
where the barrier distribution function is taken as an asymmetric Gaussian
form.
Neutron-deficient SHN with charged numbers Z=107-113 were synthesized
successfully in the cold fusion reactions. The evaporation residues was
observed by the consecutive $\alpha$ decay until to take place the spontaneous
fission of the known nuclides, in which the fusion dynamics and the structure
properties of the compound nucleus have a strong influence in the production
of SHN. Recently more neutron-rich and heavier SHN with charged numbers
Z=113-116, 118 were produced in the fusion-evaporation reactions of 48Ca
bombarding actinide targets. Superheavy residues were also identified by the
consecutive $\alpha$ decay, unfortunately to spontaneous fission of unknown
nuclides. Neutron-rich projectile-target combinations are necessary to be
chosen so that superheavy residues approach the ”island of stability” with the
doubly magic shell closure beyond 208Pb at the position of protons Z=114-126
and neutrons N=184. New SHN between the isotopes of the cold fusion and the
48Ca induced reactions are of importance for the structure studies themselves
and also as daughter nuclides for identifying heavier SHN in the future.
Figure 1: Evaporation residue excitation functions in the production of
isotopes of superheavy element Mt in the reactions 27Al+248,250Cm, 31P+244Pu
and 37Cl+238U. Figure 2: Calculated production cross sections for the
reactions 30Si+248,250Cm, 36S+244Pu and 40Ar+238U to produce superheavy
element Ds.
The excitation energy of compound nucleus is obtained by
$E^{\ast}_{CN}=E_{c.m.}+Q$, where $E_{c.m.}$ is the incident energy in the
center-of-mass system. The $Q$ value is given by $Q=\Delta M_{P}+\Delta
M_{T}-\Delta M_{C}$, and the corresponding mass defects are taken from Ref.
Mo95 for projectile, target and compound nucleus, respectively. Usually,
neutron-rich projectiles are used to synthesize SHN experimentally, such as
64Ni and 70Zn in the cold fusion reactions, which can enhance the survival
probability $W_{sur}$ in Eq.(1) of the excited compound nucleus owing to
smaller neutron separation energy. Within the framework of the DNS model, we
calculated the evaporation residue cross sections of superheavy element Mt
based on the actinide targets 248,250Cm, 244Pu and 238U with the neutron-rich
projectiles 27Al, 31P and 37Cl as shown in Fig. 1. One can see that the 3n
channel in the reactions 27Al+248Cm and 37Cl+238U, and the 4n and 5n channels
in the system 27Al+250Cm have the larger cross sections in the production of
SHN 272Mt and 273Mt. Superheavy element Ds(Z=110) was successfully synthesized
in the cold fusion reactions Ho00 ; Ho95 . The production of the SHN depends
on the isotopic combinations of projectiles and targets. Calculations were
performed for the reactions 30Si+248,250Cm, 36S+244Pu and 40Ar+238U to produce
superheavy element Ds as shown in Fig. 2. Combination with 248Cm has the
larger cross section in the 4n channel than the isotope 250Cm due to the
larger value of survival probability. The 4n channels in the systems
30Si+248,250Cm and 40Ar+238U and the 3n channel in the reaction 30Si+248Cm are
feasible in the synthesis of new SHN 274-276Ds. These combinations can be
chosen in experimental preparation with the present facilities.
Figure 3: The same as in Fig. 1, but for the reactions 31P+248,250Cm,
37Cl+244Pu and 41K+238U to produce superheavy element Rg. Figure 4:
Comparison of the calculated production cross sections in the reactions
36S+248,250Cm and 40Ar+244Pu to synthesize superheavy element Z=112.
In the DNS model, the isotopic trends are mainly determined by both the fusion
and survival probabilities. When the neutron number of the projectile is
increasing, the DNS gets more symmetrical and the fusion probability decreases
if the DNS does not consist of more stable nuclei due to a higher inner fusion
barrier. A smaller neutron separation energy and a larger shell correction
lead to a larger survival probability. The compound nucleus with closed
neutron shells has larger shell correction energy and neutron separation
energy. The cross section decreases rapidly in the production of the isotopes
of Rg(Z=111). Optical channels are the 4n evaporation for the systems
31P+248,250Cm, 37Cl+244Pu and 41K+238U to produce 275,277Rg as shown in Fig.
3. Superheavy element Z=112 is more difficult to be produced in the selected
systems. Shown in Fig. 5 gives that the possible way is the 4n channel in the
reaction 36S+250Cm, but the cross section is still smaller than the system
48Ca+238U Fe09 although the larger mass asymmetry.
Table 1: Comparisons of calculated maximal evaporation residue cross sections and optimal excitation energies (in bracket) in 2n-5n channels. | Reactions | $\sigma_{ER}^{2n}$(pb) ($E^{\ast}_{CN}$) | $\sigma_{ER}^{3n}$(pb) ($E^{\ast}_{CN}$) | $\sigma_{ER}^{4n}$(pb) ($E^{\ast}_{CN}$) | $\sigma_{ER}^{5n}$(pb) ($E^{\ast}_{CN}$)
---|---|---|---|---|---
| 26Mg+248Cm | 2.50 (40 MeV) | 26.2 (40 MeV) | 719.1 (42 MeV) | 1.23 (51 MeV)
| 26Mg+250Cm | 1.11 (41 MeV) | 10.57 (41 MeV) | 185.2 (42 MeV) | 108.5 (45 MeV)
| 30Si+244Pu | 0.46 (42 MeV) | 5.09 (43 MeV) | 185.1 (44 MeV) | 0.72 (51 MeV)
| 36S+238U | 0.21 (37 MeV) | 1.96 (38 MeV) | 42.97 (42 MeV) | 0.11 (52 MeV)
| 27Al+248Cm | 0.31 (44 MeV) | 27.83 (44 MeV) | 3.59 (47 MeV) | 1.34 (51 MeV)
| 27Al+250Cm | 0.12 (46 MeV) | 1.64 (46 MeV) | 24.31 (46 MeV) | 97.44 (49 MeV)
| 31P+244Pu | 4.71$\times 10^{-2}$ (47 MeV) | 4.25 (47 MeV) | 0.87 (50 MeV) | 0.52 (53 MeV)
| 37Cl+238U | 0.16 (38 MeV) | 13.31 (38 MeV) | 0.67 (44 MeV) | 0.17 (50 MeV)
| 30Si+248Cm | 0.34 (39 MeV) | 1.72 (39 MeV) | 65.32 (43 MeV) | 1.22$\times 10^{-2}$ (56 MeV)
| 30Si+250Cm | 0.11 (41 MeV) | 0.42 (42 MeV) | 3.54 (44 MeV) | 0.93 (48 MeV)
| 36S+244Pu | 3.87$\times 10^{-2}$ (36 MeV) | 0.101 (38 MeV) | 0.61 (41 MeV) | 0.12 (48 MeV)
| 40Ar+238U | 0.26 (32 MeV) | 0.55 (36 MeV) | 2.10 (42 MeV) | 0.45 (53 MeV)
| 31P+248Cm | 4.11$\times 10^{-2}$ (43 MeV) | 0.31 (44 MeV) | 1.85 (45 MeV) | 0.69 (50 MeV)
| 31P+250Cm | 9.91$\times 10^{-3}$ (47 MeV) | 5.49$\times 10^{-2}$ (48 MeV) | 0.41 (51 MeV) | 0.25 (52 MeV)
| 37Cl+244Pu | 4.01$\times 10^{-2}$ (36 MeV) | 0.11 (38 MeV) | 0.33 (42 MeV) | 7.15$\times 10^{-2}$ (49 MeV)
| 41K+238U | 1.96$\times 10^{-2}$ (32 MeV) | 8.67$\times 10^{-2}$ (35 MeV) | 0.21 (41 MeV) | 3.77$\times 10^{-2}$ (49 MeV)
| 36S+248Cm | 4.31$\times 10^{-2}$ (33 MeV) | 7.64$\times 10^{-2}$ (36 MeV) | 7.02$\times 10^{-2}$ (43 MeV) | 2.55$\times 10^{-3}$ (54 MeV)
| 36S+250Cm | 1.76$\times 10^{-2}$ (36 MeV) | 8.25$\times 10^{-2}$ (37 MeV) | 0.24 (42 MeV) | 1.47$\times 10^{-2}$ (51 MeV)
| 40Ar+244Pu | 5.79$\times 10^{-3}$ (31 MeV) | 9.48$\times 10^{-3}$ (36 MeV) | 9.84$\times 10^{-3}$ (44 MeV) | 5.02$\times 10^{-4}$ (56 MeV)
| 37Cl+248Cm | 5.81$\times 10^{-2}$ (31 MeV) | 0.26 (35 MeV) | 0.195 (42 MeV) | 1.05$\times 10^{-2}$ (52 MeV)
| 37Cl+250Cm | 2.08$\times 10^{-2}$ (35 MeV) | 0.21 (36 MeV) | 0.594 (41 MeV) | 7.06$\times 10^{-2}$ (49 MeV)
| 41K+244Pu | 1.11$\times 10^{-2}$ (29 MeV) | 4.22$\times 10^{-2}$ (34 MeV) | 3.31$\times 10^{-2}$ (42 MeV) | 2.3$\times 10^{-3}$ (53 MeV)
| 45Sc+238U | 1.72$\times 10^{-2}$ (27 MeV) | 1.99$\times 10^{-2}$ (35 MeV) | 2.32$\times 10^{-3}$ (45 MeV) | 1.92$\times 10^{-4}$ (57 MeV)
| 40Ar+248Cm | 6.98$\times 10^{-3}$ (26 MeV) | 2.21$\times 10^{-2}$ (33 MeV) | 3.5$\times 10^{-2}$ (41 MeV) | 2.12$\times 10^{-3}$ (51 MeV)
| 40Ar+250Cm | 2.77$\times 10^{-3}$ (29 MeV) | 2.11$\times 10^{-2}$ (33 MeV) | 7.96$\times 10^{-2}$ (40 MeV) | 9.69$\times 10^{-3}$ (48 MeV)
| 48Ti+238U | 4.51$\times 10^{-2}$ (24 MeV) | 1.37$\times 10^{-2}$ (32 MeV) | 5.81$\times 10^{-3}$ (42 MeV) | 1.71$\times 10^{-4}$ (54 MeV)
| 50Ti+238U | 5.11$\times 10^{-2}$ (23 MeV) | 2.18$\times 10^{-2}$ (31 MeV) | 1.07$\times 10^{-2}$ (40 MeV) | 4.11$\times 10^{-4}$ (50 MeV)
The productions of superheavy element Z=113 were successfully performed in the
cold fusion reaction 70Zn+209Bi Mo04 and also in the hot fusion 48Ca+237Np
Og07b with the cross section less than 1 pb. We calculated the evaporation
residue excitation functions for the reactions 37Cl+248,250Cm, 41K+244Pu and
45Sc+238U. The results show that the 3n and 4n channels in the systems
37Cl+248,250Cm have larger cross sections and are possible to synthesize new
isotopes 281-284113 in experimentally. Superheavy element Z=114 is difficulty
to be produced from our calculations for the selected systems because of the
smaller cross sections with less than 0.1 pb for all systems. We list the
maximal production cross sections and the corresponding excitation energies in
the brackets calculated by using the DNS model in Table 1. These selected
systems and evaporation channels are feasible to produce new isotopes between
the cold fusion and the 48Ca induced fusion reactions.
In summary, we systematically investigated the production of superheavy
residues in the fusion-evaporation reactions using the DNS model, in which the
nucleon transfer leading to the formation of superheavy compound nucleus is
described by a set of microscopically derived master equations distinguishing
the proton and neutron transfer that are coupled to the dissipation of
relative motion energy and angular momentum. The production of new isotopes
between the gap of the cold fusion and the 48Ca induced fusion reactions are
discussed for selected systems. Optimal evaporation channels and excitation
energies corresponding to the maximal cross sections are stated and discussed
systematically.
###### Acknowledgements.
We would like to thank Prof. Werner Scheid for carefully reading the
manuscript. This work was supported by the National Natural Science Foundation
of China under Grant Nos. 10805061 and 10775061, the special foundation of the
president fund, the west doctoral project of Chinese Academy of Sciences, and
major state basic research development program under Grant No. 2007CB815000.
## References
* (1) S. Hofmann and G. Münzenberg, Rev. Mod. Phys. 72, 733 (2000); S. Hofmann, Rep. Prog. Phys. 61, 639 (1998).
* (2) Yu. Ts. Oganessian, J. Phys. G 34, R165 (2007).
* (3) G. Münzenberg, J. Phys. G 25, 717 (1999).
* (4) K. Morita, K. Morimoto, D. Kaji _et al._ , J. Phys. Soc. Jpn. 73, 2593 (2004).
* (5) Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov _et al._ , Phys. Rev. C 69, 021601(R) (2004).
* (6) Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov _et al._ , Phys. Rev. C 74, 044602 (2006).
* (7) Z. G. Gan, Z. Qin, H. M. Fan _et al._ , Eur. Phys. J. A10, 21 (2001); Z. G. Gan, J. S. Guo, X. L. Wu _et al._ , Eur. Phys. J. A20, 385 (2004).
* (8) Z. Q. Feng, G. M. Jin, F. Fu and J. Q. Li, Nucl. Phys. A771, 50 (2006).
* (9) Z. Q. Feng, G. M. Jin, J. Q. Li and W. Scheid, Phys. Rev. C 76, 044606 (2007).
* (10) Z. Q. Feng, G. M. Jin, J. Q. Li and W. Scheid, Nucl. Phys. A816, 33 (2009).
* (11) G. G. Adamian, N. V. Antonenko, W. Scheid _et al._ , Nucl. Phys. A627, 361 (1997); Nucl. Phys. A633, 409 (1998).
* (12) P. Möller _et al._ , At. Data Nucl. Data Tables 59, 185 (1995).
* (13) S. Hofmann, V. Ninov, F.P. Heßberger _et al._ , Z. Phys. A 350, 277 (1995); Z. Phys. A 350, 281 (1995).
* (14) Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov _et al._ , Phys. Rev. C 76, 011601(R) (2007).
|
arxiv-papers
| 2009-12-21T02:19:30 |
2024-09-04T02:49:07.141904
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhao-Qing Feng, Gen-Ming Jin, Jun-Qing Li",
"submitter": "Zhaoqing Feng",
"url": "https://arxiv.org/abs/0912.4069"
}
|
0912.4110
|
2010501-512Nancy, France 501
Alexander Kartzow
# Collapsible Pushdown Graphs of Level $\mathbf{2}$ are Tree-Automatic
A. Kartzow TU Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, 64289
Darmstadt, Germany
###### Abstract.
We show that graphs generated by collapsible pushdown systems of level $2$ are
tree-automatic. Even when we allow $\varepsilon$-contractions and add a
reachability predicate (with regular constraints) for pairs of configurations,
the structures remain tree-automatic. Hence, their $\mathrm{FO}$ theories are
decidable, even when expanded by a reachability predicate. As a corollary, we
obtain the tree-automaticity of the second level of the Caucal-hierarchy.
###### Key words and phrases:
tree-automatic structures, collapsible pushdown graphs, collapsible pushdown
systems, first-order decidability, reachability
###### 1991 Mathematics Subject Classification:
F.4.1[Theory of Computation]:Mathematical Logic
## 1\. Introduction
Higher-order pushdown systems were first introduced by Maslov [10, 11] as
accepting devices for word languages. Later, Knapik et al. [8] studied them as
generators for trees. They obtained an equi-expressivity result for higher-
order pushdown systems and for higher-order recursion schemes that satisfy the
constraint of _safety_ , which is a rather unnatural syntactic condition.
Recently, Hague et al. [6] introduced collapsible pushdown systems as
extensions of higher-order pushdown systems and proved that these have exactly
the same power as higher-order recursion schemes as methods for generating
trees.
Both – higher-order and collapsible pushdown systems – also form interesting
devices for generating graphs. Carayol and Wöhrle [3] showed that the graphs
generated by higher-order pushdown systems111The graph generated by a higher-
order pushdown system is the $\varepsilon$-closure of its reachable
configurations. of level $l$ coincide with the graphs in the $l$-th level of
the Caucal-hierarchy, a class of graphs introduced by Caucal [4]. Every level
of this hierarchy is obtained from the preceding level by applying graph
unfoldings and $\mathrm{MSO}$ interpretations. Both operations preserve the
decidability of the $\mathrm{MSO}$ theory whence the Caucal-hierarchy forms a
rather large class of graphs with decidable $\mathrm{MSO}$ theories. If we use
collapsible pushdown systems as generators for graphs we obtain a different
situation. Hague et al. showed that even the second level of the hierarchy
contains a graph with undecidable $\mathrm{MSO}$ theory. But they showed the
decidability of the modal $\mu$-calculus theories of all graphs in the
hierarchy. This turns graphs generated by collapsible pushdown systems into an
interesting class from a model theoretic point of view. There are few natural
classes that share these properties. In fact, the author only knows one
further example, viz. nested pushdown trees. Alur et al.[1] introduced these
graphs for $\mu$-calculus model checking purposes. We proved in [7] that
nested pushdown trees also have decidable first-order theories. We gave an
effective model checking algorithm using pumping techniques, but we also
proved that nested pushdown trees are tree-automatic structures. Tree-
automatic structures were introduced by Blumensath [2]. These structures enjoy
decidable first-order theories due to the good closure properties of finite
automata on trees.
In this paper, we are going to extend our previous result to the second level
of the collapsible pushdown hierarchy. All graphs of the second level are
tree-automatic. This subsumes our previous result as nested pushdown trees are
first-order interpretable in collapsible pushdown graphs of level two.
Furthermore, we show that collapsible pushdown graphs of level $2$ are still
tree-automatic when expanded by a reachability predicate, i.e., by the binary
relation which contains all pairs of configurations such that there is a path
from the first to the second configuration. Thus, first-order logic extended
by reachability predicates is decidable on level $2$ collapsible pushdown
graphs.
In the next section, we introduce the necessary notions concerning tree-
automaticity and in Section 3 we define collapsible pushdown graphs. We
explain the translation of configurations into trees in Section 4. Section 5
is a sketch of the proof that this translation yields tree-automatic
representations of collapsible pushdown graphs, even when enriched with
certain regular reachability predicates. The last section contains some
concluding remarks about questions arising from our result.
## 2\. Preliminaries
We write $\mathrm{MSO}$ for monadic second order logic and $\mathrm{FO}$ for
first-order logic. For words $w_{1},w_{2}\in\Sigma^{*}$, we write $w_{1}\sqcap
w_{2}$ for the greatest common prefix of $w_{1}$ and $w_{2}$. A _$\Sigma$
-labelled tree_ is a function $T:D\rightarrow\Sigma$ for a finite
$D\subseteq\\{0,1\\}^{*}$ which is closed under prefixes.
For $d\in D$ we denote by ${T}_{d}$ the _subtree rooted at $d$_.
Sometimes it is useful to define trees inductively by describing their left
and right subtrees. For this purpose we fix the following notation. Let
$\hat{T}_{0}$ and $\hat{T}_{1}$ be $\Sigma$-labelled trees and
$\sigma\in\Sigma$. Then we write
$T\mathrel{\mathop{:}}=\sigma({\hat{T}_{0}},{\hat{T}_{1}})$ for the
$\Sigma$-labelled tree $T$ with the following three properties
$\displaystyle 1.\ T(\varepsilon)=\sigma,$ $\displaystyle 2.\
{T}_{0}=\hat{T}_{0}\text{, and }$ $\displaystyle 3.\
{T}_{1}=\hat{T}_{1}\enspace.$
In the rest of this section, we briefly present the notion of a tree-automatic
structure as introduced by Blumensath [2].
The _convolution_ of two $\Sigma$-labelled trees $T$ and $T^{\prime}$ is given
by a function
$\displaystyle T\otimes
T^{\prime}:\text{dom}(T)\cup\text{dom}(T^{\prime})\rightarrow(\Sigma\cup\\{\Box\\})^{2}$
where $\Box$ is a new symbol for padding and
$\displaystyle(T\otimes
T^{\prime})(d)\mathrel{\mathop{:}}=\begin{cases}(T(d),T^{\prime}(d))&\text{ if
}d\in\text{dom}(T)\cap\text{dom}(T^{\prime})\\\ (T(d),\Box)&\text{ if
}d\in\text{dom}(T)\setminus\text{dom}(T^{\prime})\\\
(\Box,T^{\prime}(d))&\text{ if
}d\in\text{dom}(T^{\prime})\setminus\text{dom}(T)\end{cases}$
By “tree-automata” we mean a nondeterministic finite automaton that labels a
finite tree top-down.
###### Definition 2.1.
A structure $\mathfrak{B}=(B,E_{1},E_{2},\ldots,E_{n})$ with domain $B$ and
binary relations $E_{i}$ is _tree-automatic_ if there are tree-automata
$A_{B},A_{E_{1}},A_{E_{2}},\ldots,A_{E_{n}}$ and a bijection $f:L\rightarrow
B$ for $L$ the language accepted by $A_{B}$ such that the following hold. For
$T,T^{\prime}\in L$, the automaton $A_{E_{i}}$ accepts $T\otimes T^{\prime}$
if and only if $\big{(}f(T),f(T^{\prime})\big{)}\in E_{i}$.
Tree-automatic structures form a nice class because automata theoretic
techniques may be used to decide first-order formulas on these structures:
###### Lemma 2.2 ([2]).
If $B$ is tree-automatic, then its first-order theory is decidable.
We will use the classical result that regular sets of trees are $\mathrm{MSO}$
definable.
###### Theorem 2.3 ([12], [5]).
For a set $\mathbb{T}$ of finite $\Sigma$-labelled trees, there is a tree
automaton recognising $\mathbb{T}$ if and only if $\mathbb{T}$ is
$\mathrm{MSO}$ definable.
## 3\. Definition of Collapsible Pushdown Graphs (CPG)
In this section we define our notation of collapsible pushdown systems. For a
more comprehensive introduction, we refer the reader to [6].
### 3.1. Collapsible Pushdown Stacks
First, we provide some terminology concerning stacks of (collapsible) higher-
order pushdown systems. We write $\Sigma^{*2}$ for $(\Sigma^{*})^{*}$ and
$\Sigma^{+2}$ for $(\Sigma^{+})^{+}$. We call an $s\in\Sigma^{*2}$ a $2$-word.
Let us fix a $2$-word $s\in\Sigma^{*2}$ which consists of an ordered list
$w_{1},w_{2},\ldots,w_{m}\in\Sigma^{*}$. We separate the words of this list by
colons writing $s=w_{1}:w_{2}:\ldots:w_{m}$. By $\lvert s\rvert$ we denote the
number of words $s$ consists of, i.e., $\lvert s\rvert=m$.
For another word
$s^{\prime}=w_{1}^{\prime}:w_{2}^{\prime}:\ldots:w^{\prime}_{n}\in\Sigma^{*2}$,
we write $s:s^{\prime}$ for the concatenation
$w_{1}:w_{2}:\ldots:w_{m}:w_{1}^{\prime}:w_{2}^{\prime}:\ldots:w_{n}^{\prime}$.
If $w\in\Sigma^{*}$, we write $[w]$ for the $2$-word that consists of a list
of one word which is $w$.
A level $2$ collapsible pushdown stack is a special element of
$(\Sigma\times\\{1,2\\}\times\mathbb{N})^{+2}$ that is generated by certain
stack operations from an initial stack which we introduce in the following
definitions. The natural numbers following the stack symbol represent the so-
called _collapse pointer_ : every element in a collapsible pushdown stack has
a pointer to some substack and applying the collapse operation returns the
substack to which the topmost symbol of the stack points. Here, the first
number denotes the _collapse level_. If it is $1$ the collapse pointer always
points to the symbol below the topmost symbol and the collapse operations just
removes the topmost symbol. The more interesting case is when the collapse
level of the topmost symbol of the stack $s$ is $2$. Then the stack obtained
by the collapse contains the first $n$ words of $s$ where $n$ is the second
number in the topmost element of $s$.
The initial level $1$ stack is $\bot_{1}\mathrel{\mathop{:}}=(\bot,1,0)$ and
the initial level $2$ stack is $\bot_{2}\mathrel{\mathop{:}}=[\bot_{1}]$.
For $k\in\\{1,2\\}$ and for a $2$-word
$s=w_{1}:w_{2}:\ldots:w_{n}\in(\Sigma\times\\{1,2\\}\times\mathbb{N})^{+2}$
such that $w_{n}=a_{1}a_{2}\ldots a_{m}$ with
$a_{i}\in\Sigma\times\\{1,2\\}\times\mathbb{N}$ for all $1\leq i\leq m$:
* •
we define the _topmost $(k-1)$-word of $s$_ as
$\mathrm{top}_{k}(s)\mathrel{\mathop{:}}=\begin{cases}w_{n}&\text{if }k=2\\\
a_{m}&\text{if }k=1\end{cases}$
* •
for
$\mathrm{top}_{1}(s)=(\sigma,i,j)\in\Sigma\times\\{1,2\\}\times\mathbb{N}$, we
define the _topmost symbol_ $\mathrm{Sym}(s)\mathrel{\mathop{:}}=\sigma$, the
_collapse-level of the topmost element_
$\mathrm{CLvl}(s)\mathrel{\mathop{:}}=i$, and the _collapse-link of the
topmost element_ $\mathrm{CLnk}(s)\mathrel{\mathop{:}}=j$.
For $s$, $w_{n}$ and $k$ as before, $\sigma\in\Sigma\setminus\\{\bot\\}$, and
$w_{n}^{\prime}:=a_{1}\ldots a_{m-1}$, we define the stack operations
$\displaystyle{\mathrm{pop}_{k}}(s)\mathrel{\mathop{:}}=$
$\displaystyle\begin{cases}w_{1}:w_{2}:\ldots:w_{n-1}&\text{if }k=2,n\geq 2\\\
w_{1}:w_{2}:\ldots:w_{n-1}:w_{n}^{\prime}&\text{if }k=1,m\geq 2\\\
\text{undefined}&\text{otherwise}\end{cases}$
$\displaystyle{\mathrm{clone}_{2}}(s)\mathrel{\mathop{:}}=$ $\displaystyle\
w_{1}:w_{2}:\ldots:w_{n-1}:w_{n}:w_{n}$
$\displaystyle\mathrm{push}_{\sigma,k}(s)\mathrel{\mathop{:}}=$
$\displaystyle\begin{cases}w_{1}:w_{2}:\ldots:w_{n}(\sigma,2,n-1)&\text{ if
k=2}\\\ w_{1}:w_{2}:\ldots:w_{n}(\sigma,1,m)&\text{ if k=1}\end{cases}$
$\displaystyle\mathrm{collapse}{}(s)\mathrel{\mathop{:}}=$
$\displaystyle\begin{cases}w_{1}:w_{2}:\ldots:w_{r}&\text{if
}\mathrm{CLvl}(s)=2,\mathrm{CLnk}(s)=r>0\\\ {\mathrm{pop}_{1}}(s)&\text{if
}\mathrm{CLvl}(s)=1\\\ \text{undefined}&\text{otherwise}\end{cases}$
The _set of level $2$-operations_ is
$\mathrm{OP}\mathrel{\mathop{:}}=\left\\{\mathrm{push}_{\sigma,1},\mathrm{push}_{\sigma,2},{\mathrm{clone}_{2}},{\mathrm{pop}_{1}},{\mathrm{pop}_{2}},\mathrm{collapse}{}\right\\}$.
The _set of level $2$ stacks_, $\mathrm{Stck}(\Sigma)$, is the smallest set
that contains $\bot_{2}$ and is closed under all operations from
$\mathrm{OP}$.
Note that $\mathrm{collapse}$\- and ${\mathrm{pop}_{k}}$-operations are only
allowed if the resulting stack is in $(\Sigma^{+})^{+}$. This avoids the
special treatment of empty words or stacks. Furthermore, a $\mathrm{collapse}$
on level $2$ summarises a non-empty sequence of
${\mathrm{pop}_{2}}$-operations. For example, starting from $\bot_{2}$, we can
apply a ${\mathrm{clone}_{2}}$, a $\mathrm{push}_{\sigma,2}$, a
${\mathrm{clone}_{2}}$, and finally a $\mathrm{collapse}$. This sequence first
creates a level $2$ stack that contains $3$ words and then performs the
collapse and ends in the initial stack again. This example shows that
${\mathrm{clone}_{2}}$-operations are responsible for the fact that collapse-
operations on level $2$ may remove more than one word from the stack.
For $s,s^{\prime}\in\mathrm{Stck}(\Sigma)$, we call $s^{\prime}$ a substack of
$s$ if there are $n_{1},n_{2}\in\mathbb{N}$ such that
$s^{\prime}={\mathrm{pop}_{1}}^{n_{1}}({\mathrm{pop}_{2}}^{n_{2}}(s))$. We
write $s^{\prime}\leq s$ if $s^{\prime}$ is a substack of $s$.
### 3.2. Collapsible Pushdown Systems and Collapsible Pushdown Graphs
Now we introduce collapsible pushdown systems and graphs (of level $2$) which
are analogues of pushdown systems and pushdown graphs using collapsible
pushdown stacks instead of ordinary stacks.
###### Definition 3.1.
A _collapsible pushdown system_ of level $2$ ($\mathrm{CPS}$) is a tuple
$S=(\Sigma,Q,\Delta,q_{0})$ where $\Sigma$ is a finite stack alphabet with
$\bot\in\Sigma$, $Q$ a finite set of states, $q_{0}\in Q$ the initial state,
and $\Delta\subseteq Q\times\Sigma\times Q\times\mathrm{OP}$ the transition
relation.
For $q\in Q$ and $s\in\mathrm{Stck}(\Sigma)$ the pair $(q,s)$ is called a
_configuration_. We define labelled transitions on pairs of configurations by
setting $(q_{1},s)\mathrel{{\vdash^{(q_{2},op)}}}(q_{2},t)$ if there is a
$(q_{1},\sigma,q_{2},op)\in\Delta$ such that $\mathrm{Sym}(s)=\sigma$ and
$op(s)=t$. The union of the labelled transition relations is denoted as
$\mathrel{{\vdash}}\mathrel{\mathop{:}}=\bigcup_{l\in
Q\times\mathrm{OP}}\mathrel{{\vdash^{l}}}$. We set $C(S)$ to be the set of all
configurations that are reachable from $(q_{0},\bot_{2})$ via
$\mathrel{{\vdash}}$-paths. We call $C(S)$ the set of _reachable_ or _valid_
configurations. The _collapsible pushdown graph ( $\mathrm{CPG}$) generated by
$S$_ is
$\displaystyle\mathrm{CPG}(S)\mathrel{\mathop{:}}=\left(C(S),(C(S)^{2}\cap\mathrel{{\vdash^{\ell}}})_{\ell\in
Q\times\mathrm{OP}}\right)$
###### Example 3.2.
The following example of a collapsible pushdown graph of level $2$ is taken
from [6]. Let
$Q\mathrel{\mathop{:}}=\\{0,1,2\\},\Sigma\mathrel{\mathop{:}}=\\{\bot,a\\}$,
and $\Delta$ given by $(0,*,1,{\mathrm{clone}_{2}})$,
$(1,*,0,\mathrm{push}_{a,2})$, $(1,*,2,\mathrm{push}_{a,2})$,
$(2,a,2,{\mathrm{pop}_{1}})$, and $(2,a,0,\mathrm{collapse})$, where $*$
denotes any letter in $\Sigma$. In our picture (see Figure 1), the labels are
abbreviated as follows:
$\mathrm{cl}\mathrel{\mathop{:}}=(1,{\mathrm{clone}_{2}})$,
$a\mathrel{\mathop{:}}=(0,\mathrm{push}_{a,2})$,
$a^{\prime}\mathrel{\mathop{:}}=(2,\mathrm{push}_{a,2})$,
$p\mathrel{\mathop{:}}=(2,{\mathrm{pop}_{1}})$, and
$\mathrm{co}\mathrel{\mathop{:}}=(0,\mathrm{collapse})$.
$\textstyle{0,\bot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{cl}}$$\textstyle{1,\bot:\bot\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{a^{\prime}}$$\textstyle{0,\bot:\bot
a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{cl}}$$\textstyle{1,\bot:\bot
a:\bot
a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{a^{\prime}}$$\textstyle{0,\bot:\bot
a:\bot
aa\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{cl}}$$\textstyle{1,\bot:\bot
a:\bot aa:\bot
aa\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\scriptstyle{a}$$\textstyle{\ldots}$$\textstyle{2,\bot:\bot
a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\mathrm{co}}$$\textstyle{2,\bot:\bot
a:\bot
aa\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\mathrm{co}}$$\textstyle{2,\bot:\bot
a:\bot aa:\bot
aaa\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\mathrm{co}}$$\textstyle{\ldots}$$\textstyle{2,\bot:\bot}$$\textstyle{2,\bot:\bot
a:\bot
a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\mathrm{co}}$$\textstyle{2,\bot:\bot
a:\bot aa:\bot
aa\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\mathrm{co}}$$\textstyle{\ldots}$$\textstyle{2,\bot:\bot
a:\bot}$$\textstyle{2,\bot:\bot a:\bot aa:\bot
a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\mathrm{co}}$$\textstyle{\ldots}$$\textstyle{2,\bot:\bot
a:\bot aa:\bot}$$\textstyle{\ldots}$
Figure 1. Example of a collapsible pushdown graph
###### Remark 3.3.
Hague et al. [6] showed that modal $\mu$-calculus model checking on level $n$
$\mathrm{CPG}$ is $n$-EXPTIME complete. Note that there is an $\mathrm{MSO}$
interpretation which turns the graph of the previous example into a grid-like
structure. Hence its $\mathrm{MSO}$ theory is undecidable.
The next definition introduces runs of collapsible pushdown systems.
###### Definition 3.4.
Let $S$ be a $\mathrm{CPS}$. A run $r$ of $S$ of length $n$ is a function
$r:\\{0,1,2,\ldots,n\\}\rightarrow
Q\times(\Sigma\times\\{1,2\\}\times\mathbb{N})^{*2}\text{ such that
}r(0)\mathrel{{\vdash}}r(1)\mathrel{{\vdash}}\cdots\mathrel{{\vdash}}r(n).$
We write $\mathrm{ln}(r)\mathrel{\mathop{:}}=n$ and call $r$ a run from $r(0)$
to $r(n)$. We say $r$ visits a stack $s$ at $i$ if $r(i)=(q,s)$.
For runs $r,r^{\prime}$ of length $n$ and $m$, respectively, with
$r(n)=r^{\prime}(0)$, we define the composition $r\circ r^{\prime}$ of $r$ and
$r^{\prime}$ in the obvious manner.
###### Remark 3.5.
Note that we do not require runs to start in the initial configuration.
## 4\. Encoding of Collapsible Pushdown Graphs in Trees
In this section we prove that $\mathrm{CPG}$ are tree-automatic. For this
purpose we have to encode stacks in trees. The idea is to divide a stack into
_blocks_ and to encode different blocks in different subtrees. The crucial
observation is that every stack is a list of words that share the same first
letter. A block is a maximal list of words in the stack that share the same
two first letters222see Figure 2 for an example of blocks and Definition 4.1
for their formal definition. If we remove the first letter of every word of
such a block, the resulting $2$-word decomposes again as a list of blocks.
Thus, we can inductively carry on to decompose parts of a stack into blocks
and code every block in a different subtree. The roots of these subtrees are
labelled with the first letter of the corresponding block. This results in a
tree in which every initial left-closed path represents one word of the stack.
By left-closed, we mean that the last element of the path has no left
successor.
It turns out that – via this encoding – each stack operation corresponds to a
simple $\mathrm{MSO}$-definable tree-operation. The main difficulty is to
provide a tree-automaton that checks whether there is a run to the
configuration represented by some tree. This problem is addressed in Section
5.
As already mentioned, the encoding works by dividing stacks into blocks. The
following definition makes our notion of blocks precise. For $w\in\Sigma^{*}$
and $s=w_{1}:w_{2}:\ldots:w_{n}\in\Sigma^{*2}$, we write
$s^{\prime}\mathrel{\mathop{:}}=w\mathrel{\backslash}s$ for
$s^{\prime}=[ww_{1}]:[ww_{2}]:\ldots:[ww_{n}]$.
$\textstyle{f}$$\textstyle{e}$$\textstyle{g}$$\textstyle{i}$$\textstyle{b}$$\textstyle{d}$$\textstyle{d}$$\textstyle{d}$$\textstyle{h}$$\textstyle{j}$$\textstyle{l}$$\textstyle{a}$$\textstyle{c}$$\textstyle{c}$$\textstyle{c}$$\textstyle{c}$$\textstyle{c}$$\textstyle{c}$$\textstyle{k}$$\textstyle{\bot}$$\textstyle{\bot}$$\textstyle{\bot}$$\textstyle{\bot}$$\textstyle{\bot}$$\textstyle{\bot}$$\textstyle{\bot}$$\textstyle{\bot}$
Figure 2. Example of blocks in a stack. These form a $c$-blockline.
###### Definition 4.1 ($\sigma$-block(line)).
For $\sigma\in\Sigma$, we call $b\in\Sigma^{*2}$ a _$\sigma$ -block_ if
$b=[\sigma]$ or $b=\sigma\tau\mathrel{\backslash}s^{\prime}$ for some
$\tau\in\Sigma$ and $s^{\prime}\in\Sigma^{*2}$. See Figure 2 for examples of
blocks. If $b_{1},b_{2},\ldots,b_{n}$ are $\sigma$-blocks, then we call
$b_{1}:b_{2}:\ldots:b_{n}$ a _$\sigma$ -blockline_.
Note that every stack in $\mathrm{Stck}(\Sigma)$ forms a
$(\bot,1,0)$-blockline. Furthermore, every blockline $l$ decomposes uniquely
as $l=b_{1}:b_{2}:\ldots:b_{n}$ of maximal blocks $b_{i}$ in $l$. Another
crucial observation is that a $\sigma$-block $b\in\Sigma^{*2}\setminus\Sigma$
decomposes as $b=\sigma\mathrel{\backslash}l$ for some blockline $l$ and we
say $l$ is the induced blockline of $b$. For $b\in\Sigma$ the induced
blockline of $[b]$ is just the empty $2$-word.
Now we encode a $(\sigma,n,m)$-blockline $l$ in a tree by labelling the root
with $(\sigma,n)$, by encoding the blockline induced by the first block of $l$
in the left subtree, and by encoding the rest of the blockline in the right
subtree. In order to avoid repetitions, we do not repeat the symbol
$(\sigma,n)$ in the right subtree, but replace it by the default letter
$\varepsilon$.
###### Definition 4.2.
Let
$s=w_{1}:w_{2}:\ldots:w_{n}\in(\Sigma\times\\{1,2\\}\times\mathbb{N})^{+2}$ be
a $(\sigma,l,k)$-blockline. Let $w_{i}^{\prime}$ be words such that
$s=(\sigma,l,k)\mathrel{\backslash}[w_{1}^{\prime}:w_{2}^{\prime}:\ldots:w_{n}^{\prime}]$
and set
$s^{\prime}\mathrel{\mathop{:}}=w_{1}^{\prime}:w_{2}^{\prime}:\ldots:w_{n}^{\prime}$.
As an abbreviation we write
${}_{h}s_{i}\mathrel{\mathop{:}}=w_{h}:w_{h+1}:\ldots:w_{i}$. Furthermore, let
$w_{1}:w_{2}:\ldots:w_{j}$ be a maximal block of $s$. Note that $j>1$ implies
$w_{j^{\prime}}=(\sigma,l,k)(\sigma^{\prime},l^{\prime},k^{\prime})w_{j^{\prime}}^{\prime\prime}$
for all $j^{\prime}\leq j$, some fixed
$(\sigma^{\prime},l^{\prime},k^{\prime})\in\Sigma\times\\{1,2\\}\times\mathbb{N}$,
and appropriate $w_{j^{\prime}}^{\prime\prime}\in\Sigma^{*}$. For
$\rho\in\big{(}\Sigma\times\\{1,2\\}\big{)}\cup\\{\varepsilon\\}$, we define
recursively the
$\big{(}\Sigma\times\\{1,2\\}\big{)}\cup\\{\varepsilon\\}$-labelled tree
$\mathrm{Enc}(s,\rho)$ via
$\displaystyle\mathrm{Enc}(s,\rho)\mathrel{\mathop{:}}=\begin{cases}\rho&\text{if
}\lvert w_{1}\rvert=1,n=1\\\
\rho(\emptyset,{\mathrm{Enc}(_{2}s_{n},\varepsilon)})&\text{if }\lvert
w_{1}\rvert=1,n>1\\\
\rho({\mathrm{Enc}(_{1}s_{n}^{\prime},(\sigma^{\prime},l^{\prime}))},\emptyset)&\text{if
}j=n,\lvert w_{1}\rvert>1\\\
\rho({\mathrm{Enc}(_{1}s_{j}^{\prime},(\sigma^{\prime},l^{\prime}))},{\mathrm{Enc}(_{j+1}s_{n},\varepsilon)})&\text{otherwise.}\end{cases}$
$\mathrm{Enc}(s)\mathrel{\mathop{:}}=\mathrm{Enc}(s,(\bot,1))$ is called the
(tree-)encoding of the stack $s\in\mathrm{Stck}(\Sigma)$.
Figure 3 shows a configuration and its encoding.
$\textstyle{(c,2,1)}$$\textstyle{(e,1,3)}$$\textstyle{(b,2,0)}$$\textstyle{(b,2,0)}$$\textstyle{(c,1,2)}$$\textstyle{(d,2,3)}$$\textstyle{(a,2,0)}$$\textstyle{(a,2,0)}$$\textstyle{(a,2,2)}$$\textstyle{(a,2,2)}$$\textstyle{(a,2,2)}$$\textstyle{(\bot,1,0)}$$\textstyle{(\bot,1,0)}$$\textstyle{(\bot,1,0)}$$\textstyle{(\bot,1,0)}$$\textstyle{(\bot,1,0)}$
$\textstyle{c,2}$$\textstyle{e,1}$$\textstyle{b,2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varepsilon\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c,1}$$\textstyle{d,2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a,2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a,2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varepsilon\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varepsilon}$$\textstyle{\bot,1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varepsilon\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 3. A stack $s$ and its encoding $\mathrm{Enc}(s)$: right arrows lead to
$1$-successors (right successors), upward arrows lead to $0$-successors (left
successors).
###### Remark 4.3.
In this encoding, the first block of a $(\sigma,l,k)$-blockline is encoded in
a subtree whose root $d$ is labelled $(\sigma,l)$. We can restore $k$ from the
position of $d$ in the tree $\mathrm{Enc}(s)$ as follows. If $l=1$ then
$k=\lvert d\rvert_{0}$, i.e., the number of occurrences of $0$ in $d$. This is
due to the fact that level $1$ links always point to the preceding letter and
that we always introduce a left-successor tree in order to encode letters that
are higher in the stack.
The case $l=2$ needs some closer inspection. Assume that some $d\in
T:=\mathrm{Enc}(s)$ is labelled $(\sigma,2)$. Then it encodes a letter
$(\sigma,2,k)$ and this is not a cloned element. Thus, $k$ equals the numbers
of words to the left of this letter $(\sigma,2,k)$. We claim that
$k=\left\lvert\big{\\{}e\in
T\cap\\{0,1\\}^{*}1:e\leq_{lex}d\big{\\}}\right\rvert$. The existence of a
pair $e,e1\in T$ corresponds to the fact that there is some blockline
consisting of blocks $b_{1}:b_{2}:\ldots:b_{n}$ with $n\geq 2$ such that
$b_{1}$ is encoded in ${T}_{e}\setminus{T}_{e1}$ and $b_{2}:\ldots:b_{n}$ is
encoded in ${T}_{e1}$. By induction, one easily sees that for each such pair
$e,e1\in T$ all the letters that are in words left of the letter encoded by
$e1$ are encoded in lexicographically smaller elements. Furthermore, the size
of $((0^{*})1)^{*}\cap T$ corresponds to the number of words in $s$ since the
introduction of a $1$-successor corresponds to the separation of the first
block of some blockline from the other blocks. Each of these separation can
also be seen as the separation of the last word of the first block from the
first word of the second block of this blockline. Note that we separate two
words that are next to each other in exactly one blockline. Putting these
facts together our claim is proved.
Another view on this correspondence is the bijection $f:\\{1,2,\ldots,\lvert
s\rvert\\}\rightarrow R$ where $R:=((0^{*})1)^{*}\cap\text{dom}(T)$ and $i$ is
mapped to the $i$-th element of $R$ in lexicographic order. $f(i)$ is exactly
the position where the $(i-1)$-st word is separated from the $i$-th one for
all $i\geq 2$. In order to state the properties of $f$, we need some more
notation. We write $\pi$ for the canonical projection
$\pi:(\Sigma\times\\{1,2\\}\times\mathbb{N})^{*}\rightarrow(\Sigma\times\\{1,2\\})^{*}$
and $w_{i}$ for the $i$-th word of $s$. Furthermore, let $w_{i}^{\prime}$ be a
word such that, $w_{i}=(w_{i}\sqcap w_{i-1})\circ w_{i}^{\prime}$ (here we set
$w_{0}:=\varepsilon$). Then the word along the path333By the word along a path
from one node to another we mean the word consisting of the non
$\varepsilon$-labels along this path. from the root to $f(i)$ is exactly
$\pi(w_{i}\sqcap w_{i-1})$ for all $2\leq i\leq\lvert s\rvert$ and the path
from $f(j)$ to $f(j)\circ 0^{m}$ for maximal $m\in\mathbb{N}$ is
$\pi(w_{j}^{\prime})$ for all $1\leq j\leq\lvert s\rvert$.
In order to encode a configuration $c:=(q,s)$, we add $q$ as a new root of the
tree and attach the encoding of $s$ as the left subtree, i.e.,
$\mathrm{Enc}(c)\mathrel{\mathop{:}}=q({\mathrm{Enc}(s)},\emptyset)$.
The image of this encoding function contains only trees of a very specific
type. We call this class $\mathbb{T}_{\mathrm{Enc}}$. In the next definition
we state the characterising properties of $\mathbb{T}_{\mathrm{Enc}}$. This
class is $\mathrm{MSO}$ definable, whence automata-recognisable.
###### Definition 4.4.
Let $\mathbb{T}_{\mathrm{Enc}}$ be the class of all trees $T$ that satisfy the
following conditions.
1. (1)
The root of $T$ is labelled by some element of $Q$ ($T(\varepsilon)\in Q$).
2. (2)
Every element of the form $\\{0,1\\}^{*}0$ is labelled by some
$(\sigma,l)\in\Sigma\times\\{1,2\\}$; especially, $T(0)=(\bot,1)$ and there
are no other occurrences of $(\bot,1)$ or $(\bot,2)$.
3. (3)
Every element of the form $\\{0,1\\}^{*}1$ is labelled by $\varepsilon$.
4. (4)
$1\notin\text{dom}(T)$, $0\in\text{dom}(T)$.
5. (5)
For all $t\in T$, if $T(t0)=(\sigma,1)$ then $T(t10)\neq(\sigma,1)$.
###### Remark 4.5.
Note that (5) holds as $T(t0)=T(t10)=(\sigma,1)$ would imply that the subtree
rooted at $t$ encodes a blockline $l$ such that the first block of $l$ induces
a $(\sigma,1,n)$-blockline and the second one induces a
$(\sigma,1,m)$-blockline. But as level $1$ links always point to the preceding
letter, $n$ and $m$ are equal to the length of the prefix of $l$ in the stack
plus $1$, i.e., if $T$ encodes a stack $s$ then
$s=s_{1}:[w\mathrel{\backslash}l]:s_{2}$ and $n=m=\lvert w\rvert+1$. This
would contradict the maximality of the blocks in the encoding.
###### Remark 4.6.
$\mathrm{Enc}:Q\times\mathrm{Stck}(\Sigma)\rightarrow\mathbb{T}_{\mathrm{Enc}}$
is a bijection and we denote its inverse by $\mathrm{Dec}$.
Our encoding turns the transitions of a $\mathrm{CPG}$ into regular tree-
operations. The tree-operations corresponding to ${\mathrm{pop}_{2}}$ and
$\mathrm{collapse}$ can be seen in Figures 4 and 5. For the
${\mathrm{pop}_{2}}$, note that if $v_{1}$ is the $0$-successor of $v_{0}$
then $v_{0}$ and $v_{1}$ encode symbols in the same word of the encoded stack.
As a ${\mathrm{pop}_{2}}$ removes the rightmost word, we have to remove all
the nodes encoding information about this word. As the rightmost leaf
corresponds to the topmost symbol of the stack, we have to remove this leaf
and all its $0$-ancestors.
For the $\mathrm{collapse}$ (on level $2$), we note that each $\varepsilon$
represents a cloned element. The $\mathrm{collapse}$ induced by such an
element produces the same stack as a ${\mathrm{pop}_{2}}$ of its original
version. The original symbol of the rightmost leaf is its first ancestor not
labelled by $\varepsilon$.
Note that the operations corresponding to ${\mathrm{pop}_{2}}$ and
$\mathrm{collapse}$ are clearly $\mathrm{MSO}$ definable. All other
transitions in $\mathrm{CPG}$ correspond to $\mathrm{MSO}$ definable tree-
operations, too. Due to space restrictions we skip the details.
$(\sigma,l)$$\varepsilon$$\ldots$$\varepsilon$$\varepsilon$$(\tau,k)$$\vdots$$(\sigma^{\prime},l^{\prime})$$(\sigma,l)$$\varepsilon$$\ldots$$\varepsilon$
Figure 4. ${\mathrm{pop}_{2}}$-operation
$\vdots$$(\sigma,2)$$\varepsilon$$\ldots$$\varepsilon$$\varepsilon$ Figure 5.
$\mathrm{collapse}$-operation of level $2$.
###### Lemma 4.7.
Let $C$ be the set of encodings of configurations of a $\mathrm{CPS}$ $S$.
Then there are automata $A_{(q,\mathrm{op})}$ for all $q\in Q$ and all
$\mathrm{op}\in\mathrm{OP}$ such that for all $c_{1},c_{2}\in C$
$A_{(q,\mathrm{op})}\text{ accepts
}\mathrm{Enc}(c_{1})\otimes\mathrm{Enc}(c_{2})\text{\quad
iff\quad}c_{1}\mathrel{{\vdash^{(q,\mathrm{op})}}}c_{2}\enspace.$
## 5\. Recognising Reachable Configurations
We show that $\mathrm{Enc}$ maps the reachable configurations of a given
$\mathrm{CPS}$ to a regular set. For this purpose we introduce milestones of a
stack $s$. It turns out that these are exactly those substacks of $s$ that
every run to $s$ has to visit. Furthermore, the milestones of $s$ are
represented by the nodes of $\mathrm{Enc}(s)$: with every
$d\in\mathrm{Enc}(s)$, we can associate a subtree of $s$ which encodes a
milestone. Furthermore, the substack relation on the milestones corresponds
exactly to the lexicographical order $\leq_{lex}$ of the elements of
$\mathrm{Enc}(s)$. For every $d\in\mathrm{Enc}(s)$ we can guess the state in
which the corresponding milestone is visited for the last time by some run to
$s$ and we can check the correctness of this guess using $\mathrm{MSO}$ or,
equivalently, tree-automata.
We prove that we can check the correctness of such a guess by introducing a
special type of run, called _loop_ , which is basically a run that starts and
ends with the same stack. A run from one milestone to the next will mainly
consist of loops combined with a finite number of stack operations.
### 5.1. Milestones
###### Definition 5.1 (Milestone).
A substack $s^{\prime}$ of $s=w_{1}:w_{2}:\ldots:w_{n}$ is a _milestone_ if
$s^{\prime}=w_{1}:w_{2}:\ldots:w_{i}:w^{\prime}$ such that $0\leq i<n$ and
$w_{i}\sqcap w_{i+1}\leq w^{\prime}\leq w_{i+1}$. We denote by
$\mathrm{MS}(s)$ the set of milestones of $s$.
Note that the substack relation $\leq$ linearly orders $\mathrm{MS}(s)$.
###### Lemma 5.2.
If $s,t,m$ are stacks with $m\in\mathrm{MS}(t)$ but $m\not\leq s$, then every
run from $s$ to $t$ visits $m$. Thus, for every run $r$ from the initial
configuration to $s$, the function
$\displaystyle f:\mathrm{MS}(s)\rightarrow\text{dom}(r),$ $\displaystyle
s^{\prime}\mapsto\max\\{i\in\text{dom}(r):r(i)=(q,s^{\prime})\text{ for some
}q\in Q\\}$
is an order embedding with respect to substack relation on the milestones and
the natural order of $\text{dom}(r)$.
In order to state the close correspondence between milestones of a stack $s$
and the elements of $\mathrm{Enc}(s)$, we need the following definition.
###### Definition 5.3.
Let $T\in\mathbb{T}_{\mathrm{Enc}}$ be a tree and $d\in
T\setminus\\{\varepsilon\\}$. Then the _left and downward closed tree induced
by $d$_ is $LT({d,T})\mathrel{\mathop{:}}=T{\restriction}_{D}$ where
$D\mathrel{\mathop{:}}=\\{d^{\prime}\in
T:d^{\prime}\leq_{lex}d\\}\setminus\\{\varepsilon\\}$. Then we denote by
$\mathrm{LStck}(d,T)\mathrel{\mathop{:}}=\mathrm{Dec}(LT({d,T}))$ the _left
stack induced by $d$_.
###### Remark 5.4.
$\mathrm{LStck}(d,s)$ is a substack of $s$ for all
$d\in\text{dom}(\mathrm{Enc}(s))$. This observation follows from Remark 4.3
combined with the fact that the left stack is induced by a lexicographically
downward closed subset. In fact, $\mathrm{LStck}(d,s)$ is a milestone of $s$.
###### Lemma 5.5.
The map given by $g:d\mapsto\mathrm{LStck}(d,\mathrm{Enc}(s))$ is an order
isomorphism between
$\left(\text{dom}(\mathrm{Enc}(q,s))\setminus\\{\varepsilon\\},\leq_{lex}\right)$
and $\left(\mathrm{MS}(s),\leq\right)$.
Lemmas 5.5 and 5.2 imply that every run $r$ decomposes as $r=r_{1}\circ
r_{2}\circ\ldots\circ r_{n}$ where $r_{i}$ is a run from the $i$-th milestone
of $r(\mathrm{ln}(r))$ to the $(i+1)$-st milestone.
In order to describe the structure of the $r_{i}$, we have to introduce the
notion of a loop. Informally speaking, a loop is a run $r$ that starts and
ends with the same stack $s$ and which does not look too much into $s$.
###### Definition 5.6.
Let $r$ be a run of length $n$ with $r(i)=(q_{i},s_{i})$ for all $0\leq i\leq
n$.
* •
$r$ is called a _simple high loop_ if $s_{0}=s_{n}$ and if $s_{0}<s_{i}$ for
all $0<i<n$.
* •
$r$ is called a _simple low loop_ of $s$ if $s_{0}=s_{n}=s$, between $0$ and
$n$ the stack $s$ is never visited, $s_{1}={\mathrm{pop}_{1}}(s)$,
$\mathrm{CLvl}(s)=1$, $\lvert s_{i}\rvert\geq\lvert s\rvert$ for all $0\leq
i\leq n$, and $r{\restriction_{[2,n-1]}}$ is the composition of simple low
loops and simple high loops of ${\mathrm{pop}_{1}}(s)$.
* •
$r$ is called _loop_ if it is a finite composition of low loops and high
loops.
###### Lemma 5.7.
Let $s$ be some stack, $m_{1},m_{2}$ milestones of $s$, and $r$ a run from
$m_{1}$ to $m_{2}$ that never visits any other milestone of $s$. Then either
$r=l_{1}\circ p\circ l_{2}$ or $r=l_{0}\circ c\circ l_{1}\circ p_{1}\circ
l_{2}\circ p_{2}\circ l_{3}\circ\ldots\circ p_{n}\circ l_{n+1}$ where each
$l_{i}$ is a loop, and all $p_{i},p$, and $c$ are runs of length $1$, $p$
performs one $\mathrm{push}_{\sigma,k}$, $c$ performs one
${\mathrm{clone}_{2}}$, and the $p_{i}$ perform one ${\mathrm{pop}_{1}}$ each.
This lemma motivates why we only define low loops for stacks $s$ with
$\mathrm{CLvl}(s)=1$. Whenever the topmost symbol of a milestone $m$ is not a
cloned element, then ${\mathrm{pop}_{1}}(m)$ is another milestone. Hence, the
$l_{i}$ can only contain low loops if they start at a stack with cloned
topmost symbol. But any stack $s$ with cloned topmost symbol and
$\mathrm{CLvl}(s)=2$ cannot be restored from ${\mathrm{pop}_{1}}(s)$ without
passing ${\mathrm{pop}_{2}}(s)$ since a $\mathrm{push}_{\sigma,2}$-operation
would create the wrong link-level.
From Lemma 5.7 we can derive that deciding whether there is a run from one
milestone to the next is possible if we know the pairs of initial and final
states of loops of certain stacks $s$. Hence we are interested in the sets
$\mathrm{Loops}(s)\subseteq Q\times Q$ with
$(q_{1},q_{2})\in\mathrm{Loops}(s)$ if and only if there is a loop from
$(q_{1},s)$ to $(q_{2},s)$. The crucial observation is that
$\mathrm{Loops}(s)$ may be calculated by a finite automaton reading
$\mathrm{top}_{2}(s)$.
###### Lemma 5.8.
For every $\mathrm{CPS}$ there exists a finite automaton $A$ that
calculates444 We consider the final state reached by $A$ on input $w$ as the
value it calculates for $w$. on input $w\in(\Sigma\times\\{1,2\\})^{*}$ the
set $\mathrm{Loops}(s)$ for all stacks $s$ such that
$w=\pi(\mathrm{top}_{2}(s))$. Here,
$\pi:(\Sigma\times\\{1,2\\}\times\mathbb{N})^{*}\rightarrow(\Sigma\times\\{1,2\\})^{*}$
is the projection onto the symbols and collapse-levels.
### 5.2. Detection of Reachable Configurations
We have already seen that every run to a valid configuration $(q,s)$ passes
all the milestones of $s$. Now, we use the last state in which a run $r$ to
$(q,s)$ visits each milestone as a certificate for the reachability of
$(q,s)$. To be precise, _a certificate for the reachability of $(q,s)$_ is a
map
$f:\text{dom}\big{(}\mathrm{Enc}(q,s)\big{)}\setminus\\{\varepsilon\\}\rightarrow
Q$ such that there is some run $r$ from $\bot_{2}$ to $(q,s)$ and $f(d)=q$ if
and only if $r(i)=\big{(}q,\mathrm{LStck}(d)\big{)}$ for $i$ the maximal
position in $r$ where $\mathrm{LStck}(d)$ is visited.
###### Lemma 5.9.
For every $\mathrm{CPG}$ $G$, there is a tree-automaton that checks for each
map
$\displaystyle f:\text{dom}(\mathrm{Enc}(q,s))\setminus\\{\varepsilon\\}$
$\displaystyle\rightarrow Q$
whether $f$ is a certificate of the reachability of $(q,s)$, i.e., whether $f$
is induced by some run $r$ from the initial configuration to $(q,s)$.
The proof of the lemma uses Lemma 5.8 and the fact that the path from the root
to some $d\in\mathrm{Enc}(s)$ encodes the topmost word of
$\mathrm{LStck}(d,\mathrm{Enc}(s))$. Hence, a tree automaton reading
$\mathrm{Enc}(s)$ is able to calculate for each position $d\in\mathrm{Enc}(s)$
the pairs of initial and final states of loops of $\mathrm{LStck}(d)$. As
every run decomposes as a sequence of loops separated by a single operation,
knowing $\mathrm{Loops}(s^{\prime})$ for each $s^{\prime}\leq s$ enables the
automaton to check the correctness of a candidate for a certificate of
reachability.
As a tree-automaton may non-deterministically guess a certificate of the
reachability of a configuration, the encodings of reachable configurations
form a regular set.
### 5.3. Extension to Regular Reachability
By now, we have already established the tree-automaticity of each
$\mathrm{CPG}$ $G$ since we have seen that our encoding yields a regular image
of the vertices of $G$ and the transition relations are turned into regular
relations of the tree encoding. Using similar techniques, we can improve this
result:
###### Theorem 5.10.
If $G$ is the $\varepsilon$-closure of some $\mathrm{CPG}$ $G^{\prime}$ then
$(G,\mathrm{Reach})$ is tree-automatic where $\mathrm{Reach}$ is the binary
predicate that is true on a pair $(c_{1},c_{2})$ of configurations if there is
a path from $c_{1}$ to $c_{2}$ in $G$.
###### Remark 5.11.
Each graph in the second level of the Caucal-hierarchy can be obtained as the
$\varepsilon$-contraction of some level $2$ $\mathrm{CPG}$ (see [3]) whence
all these graphs are tree-automatic.
For a $\mathrm{CPS}$ $S$ let $R\subseteq\Delta^{*}$ be a regular language over
the transitions of $S$. As collapsible pushdown graphs are closed under
products with finite automata even the reachability predicate
$\mathrm{Reach}_{R}$ with restriction to $R$ is tree-automatic. Here,
$\mathrm{Reach}_{R}xy$ holds if there is a path from $x$ to $y$ in
$\mathrm{CPG}(S)$ that uses a sequence of transitions in $R$. If $A$ is the
automaton recognising $R$, we obtain that
$\mathrm{Reach}_{R}(q,s)(q^{\prime},s^{\prime})$ holds in $\mathrm{CPG}(S)$
iff
$\mathrm{Reach}\big{(}(q,q_{i}),s\big{)}\big{(}(q^{\prime},q_{f}),s^{\prime}\big{)}$
holds in $\mathrm{CPG}(S\times A)$ where $q_{i}$ is the initial and $q_{f}$
the unique final state of $A$. Using this idea one can define a $\mathrm{CPG}$
$G^{\prime}$ which is basically $\mathrm{CPG}(S\cup(S\times A))$ extended by
transitions from $(q,s)$ to $((q,q_{i}),s)$ and to $((q,q_{f}),s)$.
$\mathrm{CPG}(S)$ as well as $\mathrm{Reach}_{R}$ w.r.t. $\mathrm{CPG}(S)$ are
$\mathrm{FO}[\mathrm{Reach}]$-interpretable in $G^{\prime}$. Hence we obtain:
###### Theorem 5.12.
Given a collapsible pushdown graph of level $2$, its
$\mathrm{FO}[\mathrm{Reach}_{R}]$ theory is decidable for each regular
$R\subseteq\Delta^{*}$.
### 5.4. Computation of concrete tree-automatic representations of CPG
Up to now, we have only seen that there is a tree-automatic representation for
each $\mathrm{CPG}$. For computing a concrete representation, we rely on the
following lemma.
###### Lemma 5.13.
Given some $\mathrm{CPS}$ $S=(\Gamma,Q,\Delta,q_{0})$, some $q\in Q$, and some
stack $s$, it is decidable whether $(q,s)$ is a vertex of $\mathrm{CPG}(S)$.
The proof is based on the idea that a stack is uniquely determined by its top
element and the information which substacks can be reached via
$\mathrm{collapse}$\- and ${\mathrm{pop}_{i}}$-operations. Hence we can
construct an extension $S^{\prime}$ of $S$ and a modal formula $\varphi_{q,s}$
such that there is some element $v\in\mathrm{CPG}(S^{\prime})$ satisfying
$\mathrm{CPG}(S^{\prime}),v\models\varphi_{q,s}$ iff
$(q,s)\in\mathrm{CPG}(S)$. $S^{\prime}$ basically contains new states for
every substack of $s$ and connects the different states via the appropriate
${\mathrm{pop}_{i}}$-operations which are only applied if the topmost symbol
of the stack agrees with the symbol we would expect when starting the
${\mathrm{pop}_{i}}$-sequence in configuration $(q,s)$.
From this lemma we can derive the computability of the automata in Lemma 5.8.
Having obtained these automata, the construction of a tree-automatic
representation of some $\mathrm{CPG}$ is directly derived from the proofs
yielding the following theorem.
###### Theorem 5.14.
There is an algorithm that, given a level $2$ $\mathrm{CPG}$ $G$ and regular
sets $R_{1},\ldots,R_{n}\subseteq\Delta^{*}$, computes a tree-automatic
representation of $(G,\mathrm{Reach}_{R_{1}},\ldots,\mathrm{Reach}_{R_{n}})$.
## 6\. Conclusion
We have seen that level $2$ collapsible pushdown graphs are tree-automatic.
This result holds also if we apply $\varepsilon$-contractions and if we add
regular reachability predicates. This implies that the second level of the
Caucal-hierarchy is tree-automatic. But our result can only be seen as a
starting point for further investigations of the $\mathrm{CPG}$ hierarchy: are
level $3$ collapsible pushdown graphs tree-automatic? We know an example of a
level $5$ $\mathrm{CPG}$ which is not tree-automatic. But even when tree-
automaticity of all $\mathrm{CPG}$ cannot be expected, the question remains
whether all $\mathrm{CPG}$ have decidable $\mathrm{FO}$ theories. In order to
solve this problem one has to come up with new techniques.
A rather general question concerning our result aims at our knowledge about
tree-automatic structures. Recent developments in the string case [9] show the
decidability of rather large extensions of first-order logic for automatic
structures. It would be interesting to clarify the status of the analogous
claims for tree-automatic structures. Positive answers concerning the
decidability of extensions of first-order logic on tree-automatic structures
would give us the corresponding decidability results for collapsible pushdown
graphs of level $2$.
## References
* [1] R. Alur, S. Chaudhuri, and P. Madhusudan. Languages of nested trees. In Proc. 18th International Conference on Computer-Aided Verification, volume 4144 of LNCS, pages 329–342. Springer, 2006.
* [2] A. Blumensath. Automatic structures. Diploma thesis, RWTH Aachen, 1999.
* [3] A. Carayol and S. Wöhrle. The Caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata. In Proceedings of the 23rd Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2003, volume 2914 of LNCS, pages 112–123. Springer, 2003.
* [4] D. Caucal. On infinite terms having a decidable monadic theory. In MFCS’02, pages 165–176, 2002.
* [5] J. Doner. Tree acceptors and some of their applications. J. Comput. Syst. Sci., 4(5):406–451, 1970.
* [6] M. Hague, A. S. Murawski, C-H. L. Ong, and O. Serre. Collapsible pushdown automata and recursion schemes. In LICS ’08: Proceedings of the 2008 23rd Annual IEEE Symposium on Logic in Computer Science, pages 452–461, 2008.
* [7] A. Kartzow. FO model checking on nested pushdown trees. In MFCS’09, volume 5734 of LNCS, pages 451–463. Springer, 2009.
* [8] T. Knapik, D. Niwinski, and P. Urzyczyn. Higher-order pushdown trees are easy. In FOSSACS’02, volume 2303 of LNCS, pages 205–222. Springer, 2002.
* [9] D. Kuske. Theories of automatic structures and their complexity. In CAI’09, Third International Conference on Algebraic Informatics, volume 5725 of LNCS, pages 81–98. Springer, 2009.
* [10] A. N. Maslov. The hierarchy of indexed languages of an arbitrary level. Sov. Math., Dokl., 15:1170–1174, 1974.
* [11] A. N. Maslov. Multilevel stack automata. Problems of Information Transmission, 12:38–43, 1976.
* [12] J. W. Thatcher and J. B. Wright. Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical Systems Theory, 2(1):57–81, 1968.
|
arxiv-papers
| 2009-12-21T09:09:50 |
2024-09-04T02:49:07.148166
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexander Kartzow",
"submitter": "Alexander Kartzow",
"url": "https://arxiv.org/abs/0912.4110"
}
|
0912.4117
|
2010405-416Nancy, France 405
Stefan Göller
Markus Lohrey
# Branching-time model checking
of one-counter processes
S. Göller Universität Bremen, Fachbereich Mathematik und Informatik
goeller@informatik.uni-bremen.de and M. Lohrey Universität Leipzig, Institut
für Informatik lohrey@informatik.uni-leipzig.de
###### Abstract.
One-counter processes (OCPs) are pushdown processes which operate only on a
unary stack alphabet. We study the computational complexity of model checking
computation tree logic (${\mathsf{CTL}}$) over OCPs. A ${\mathsf{PSPACE}}$
upper bound is inherited from the modal $\mu$-calculus for this problem.
First, we analyze the periodic behaviour of ${\mathsf{CTL}}$ over OCPs and
derive a model checking algorithm whose running time is exponential only in
the number of control locations and a syntactic notion of the formula that we
call leftward until depth. Thus, model checking fixed OCPs against
${\mathsf{CTL}}$ formulas with a fixed leftward until depth is in
${\mathsf{P}}$. This generalizes a result of the first author, Mayr, and To
for the expression complexity of ${\mathsf{CTL}}$’s fragment ${\mathsf{EF}}$.
Second, we prove that already over some fixed OCP, ${\mathsf{CTL}}$ model
checking is ${\mathsf{PSPACE}}$-hard. Third, we show that there already exists
a fixed ${\mathsf{CTL}}$ formula for which model checking of OCPs is
${\mathsf{PSPACE}}$-hard. For the latter, we employ two results from
complexity theory: (i) Converting a natural number in Chinese remainder
presentation into binary presentation is in logspace-uniform
${\mathsf{NC}}^{1}$ and (ii) ${\mathsf{PSPACE}}$ is
${\mathsf{AC}}^{0}$-serializable. We demonstrate that our approach can be used
to answer further open questions.
###### Key words and phrases:
model checking, computation tree logic, complexity theory
###### 1991 Mathematics Subject Classification:
F.4.1; F.1.3
The second author would like to acknowledge the support by DFG research
project GELO
## 1\. Introduction
Pushdown automata (PDAs) (or recursive state machines) are a natural model for
sequential programs with recursive procedure calls, and their verification
problems have been studied extensively. The complexity of model checking
problems for PDAs is quite well understood: The reachability problem for PDAs
can be solved in polynomial time [4, 10]. Model checking modal $\mu$-calculus
over PDAs was shown to be ${\mathsf{EXPTIME}}$-complete in [29], and the
global version of the model checking problem has been considered in [7, 21,
22]. The ${\mathsf{EXPTIME}}$ lower bound for model checking PDAs also holds
for the simpler logic ${\mathsf{CTL}}$ and its fragment $\mathsf{EG}$ [28],
even for a fixed formula (data complexity) [5] or a fixed PDA (expression
complexity). On the other hand, model checking PDAs against the logic
${\mathsf{EF}}$ (another natural fragment of ${\mathsf{CTL}}$) is
${\mathsf{PSPACE}}$-complete [28], and again the lower bound still holds if
either the formula or the PDA is fixed [4]. Model checking problems for
various fragments and extensions of PDL (Propositional Dynamic Logic) over
PDAs were studied in [12].
One-counter processes (OCPs) are Minsky counter machines with just one
counter. They can also be seen as a special case of PDAs with just one stack
symbol, plus a non-removable bottom symbol which indicates an empty stack (and
thus allows to test the counter for zero) and hence constitute a natural and
fundamental computational model. In recent years, model checking problems for
OCPs received increasing attention [13, 15, 23, 25]. Clearly, all upper
complexity bounds carry over from PDAs. The question, whether these upper
bounds can be matched by lower bounds was just recently solved for several
important logics: Model checking modal $\mu$-calculus over OCPs is
${\mathsf{PSPACE}}$-complete. The ${\mathsf{PSPACE}}$ upper bound was shown in
[23], and a matching lower bound can easily be shown by a reduction from
emptiness of alternating unary finite automata, which was shown to be
${\mathsf{PSPACE}}$-complete in [18, 19]. This lower bound even holds if
either the OCP or the formula is fixed. The situation becomes different for
the fragment ${\mathsf{EF}}$. In [13], it was shown that model checking
${\mathsf{EF}}$ over OCPs is in the complexity class
$\mathsf{P}^{\mathsf{NP}}$ (the class of all problems that can be solved on a
deterministic polynomial time machine with access to an oracle from
$\mathsf{NP}$). Moreover, if the input formula is represented succinctly as a
directed acyclic graph, then model checking ${\mathsf{EF}}$ over OCPs is also
hard for $\mathsf{P}^{\mathsf{NP}}$. For the standard (and less succinct) tree
representation for formulas, only hardness for the class
$\mathsf{P}^{\mathsf{NP}[\log]}$ (the class of all problems that can be solved
on a deterministic polynomial time machine which is allowed to make
$O(\log(n))$ many queries to an oracle from $\mathsf{NP}$) was shown in [13].
In fact, there already exists a fixed ${\mathsf{EF}}$ formula such that model
checking this formula over a given OCP is hard for
$\mathsf{P}^{\mathsf{NP}[\log]}$, i.e., the data complexity is
$\mathsf{P}^{\mathsf{NP}[\log]}$-hard.
In this paper we consider the model checking problem for ${\mathsf{CTL}}$ over
OCPs. By the known upper bound for the modal $\mu$-calculus [23] this problem
belongs to ${\mathsf{PSPACE}}$. First, we analyze the combinatorics of
${\mathsf{CTL}}$ model checking over OCPs. More precisely, we analyze the
periodic behaviour of the set of natural numbers that satisfy a given
${\mathsf{CTL}}$ formula in a given control location of the OCP (Thm. 4.1). By
making use of Thm. 4.1, we can derive a model checking algorithm whose running
time is exponential only in the number of control locations and a syntactic
measure on ${\mathsf{CTL}}$ formulas that we call leftward until depth (Thm.
4.3). As a corollary, we obtain that model checking a fixed OCP against
${\mathsf{CTL}}$ formulas of fixed leftward until depth lies in
${\mathsf{P}}$. This generalizes a recent result from [13], where it was shown
that the expression complexity of ${\mathsf{EF}}$ over OCPs lies in
${\mathsf{P}}$. Next, we focus on lower bounds. We show that model checking
${\mathsf{CTL}}$ over OCPs is ${\mathsf{PSPACE}}$-complete, even if we fix
either the OCP (Thm. 5.3) or the ${\mathsf{CTL}}$ formula (Thm. 7.3). The
proof of Thm. 5.3 uses a reduction from QBF. We have to construct a fixed OCP
for which we can construct for a given unary encoded number $i$
${\mathsf{CTL}}$ formulas that express, when interpreted over our fixed OCP,
whether the current counter value is divisible by $2^{i}$ and whether the
$i^{\text{th}}$ bit in the binary representation of the current counter value
is $1$, respectively. For the proof of Thm. 7.3 (${\mathsf{PSPACE}}$-hardness
of data complexity for ${\mathsf{CTL}}$) we use two techniques from complexity
theory, which to our knowledge have not been applied in the context of
verification so far: (i) the existence of small depth circuits for converting
a number from Chinese remainder representation to binary representation and
(ii) the fact that ${\mathsf{PSPACE}}$-computations are serializable in a
certain sense (see Sec. 6 for details). One of the main obstructions in
getting lower bounds for OCPs is the fact that OCPs are well suited for
testing divisibility properties of the counter value and hence can deal with
numbers in Chinese remainder representation, but it is not clear how to deal
with numbers in binary representation. Small depth circuits for converting a
number from Chinese remainder representation to binary representation are the
key in order to overcome this obstruction.
We are confident that our new lower bound techniques described above can be
used for proving further lower bounds for OCPs. We present two other
applications of our techniques in Sec. 8: (i) We show that model checking
${\mathsf{EF}}$ over OCPs is complete for $\mathsf{P}^{\mathsf{NP}}$ even if
the input formula is represented by a tree (Thm. 8.1) and thereby solve an
open problem from [13]. (ii) We improve a lower bound on a decision problem
for one-counter Markov decision processes from [6] (Thm. 8.2). The following
table summarizes the picture on the complexity of model checking for PDAs and
OCPs. Our new results are marked with (*).
Logic | PDA | OCP
---|---|---
modal $\mu$-calculus | ${\mathsf{EXPTIME}}$-complete | ${\mathsf{PSPACE}}$-complete
modal $\mu$-calculus, fixed formula | ${\mathsf{EXPTIME}}$-complete | ${\mathsf{PSPACE}}$-complete
modal $\mu$-calculus, fixed system | ${\mathsf{EXPTIME}}$-complete | ${\mathsf{PSPACE}}$-complete
${\mathsf{CTL}}$, fixed formula | ${\mathsf{EXPTIME}}$-complete | ${\mathsf{PSPACE}}$-complete (*)
${\mathsf{CTL}}$, fixed system | ${\mathsf{EXPTIME}}$-complete | ${\mathsf{PSPACE}}$-complete (*)
${\mathsf{CTL}}$, fixed system, fixed leftward until depth | ${\mathsf{EXPTIME}}$-complete | in ${\mathsf{P}}$ (*)
${\mathsf{EF}}$ | ${\mathsf{PSPACE}}$-complete | $\mathsf{P}^{\mathsf{NP}}$-complete (*)
${\mathsf{EF}}$, fixed formula | ${\mathsf{PSPACE}}$-complete | $\mathsf{P}^{\mathsf{NP}[\log]}$-hard, in $\mathsf{P}^{\mathsf{NP}}$
${\mathsf{EF}}$, fixed system | ${\mathsf{PSPACE}}$-complete | in ${\mathsf{P}}$
Missing proofs due to space restrictions can be found in the full version of
this paper [14].
## 2\. Preliminaries
We denote the naturals by $\mathbb{N}=\\{0,1,2,\ldots\\}$. For
$i,j\in\mathbb{N}$ let $[i,j]=\\{k\in\mathbb{N}\mid i\leq k\leq j\\}$ and
$[j]=[1,j]$. In particular $[0]=\emptyset$. For $n\in\mathbb{N}$ and $i\geq
1$, let $\text{bit}_{i}(n)$ denote the $i^{\text{th}}$ least significant bit
of the binary representation of $n$, i.e., $n=\sum_{i\geq
1}2^{i-1}\cdot\text{bit}_{i}(n)$. For every finite and non-empty subset
$M\subseteq\mathbb{N}\setminus\\{0\\}$, define $\text{LCM}(M)$ to be the least
common multiple of all numbers in $M$. It is known that
$2^{k}\leq\text{LCM}([k])\leq 4^{k}$ for all $k\geq 9$ [20]. As usual, for a
possibly infinite alphabet $A$, $A^{*}$ (resp. $A^{\omega}$) denotes the set
of all finite (resp. infinite) words over $A$. Let $A^{\infty}=A^{*}\cup
A^{\omega}$ and $A^{+}=A^{*}\setminus\\{\varepsilon\\}$, where $\varepsilon$
is the empty word. The length of a finite word $w$ is denoted by $|w|$. For a
word $w=a_{1}a_{2}\cdots a_{n}\in A^{*}$ (resp. $w=a_{1}a_{2}\cdots\in
A^{\omega}$) with $a_{i}\in A$ and $i\in[n]$ (resp. $i\geq 1$), we denote by
$w_{i}$ the $i^{\text{th}}$ letter $a_{i}$. A nondeterministic finite
automaton (NFA) is a tuple $A=(S,\Sigma,\delta,s_{0},S_{f})$, where $S$ is a
finite set of states, $\Sigma$ is a finite alphabet, $\delta\subseteq
S\times\Sigma\times S$ is the transition relation, $s_{0}\in S$ is the initial
state, and $S_{f}\subseteq S$ is a set of final states. We assume some basic
knowledge in complexity theory, see e.g. [1] for more details.
## 3\. One-counter processes and computation tree logic
Fix a countable set $\mathcal{P}$ of propositions. A transition system is a
triple $T=(S,\\{S_{p}\mid p\in\mathcal{P}\\},\rightarrow)$, where $S$ is the
set of states, $\to\,\subseteq S\times S$ is the set of transitions and
$S_{p}\subseteq S$ for all $p\in\mathcal{P}$ with $S_{p}=\emptyset$ for all
but finitely many $p\in\mathcal{P}$. We write $s_{1}\rightarrow s_{2}$ instead
of $(s_{1},s_{2})\in\,\rightarrow$. The set of all finite (resp. infinite)
paths in $T$ is ${\mathrm{path}}_{+}(T)=\\{\pi\in S^{+}\mid\forall
i\in[|\pi|-1]:\pi_{i}\to\pi_{i+1}\\}$ (resp.
${\mathrm{path}}_{\omega}(T)=\\{\pi\in S^{\omega}\mid\forall i\geq
1:\pi_{i}\to\pi_{i+1}\\}$). For a subset $U\subseteq S$ of states, a (finite
or infinite) path $\pi$ is called a $U$-path if $\pi\in U^{\infty}$.
A one-counter process (OCP) is a tuple $\mathbb{O}=(Q,\\{Q_{p}\mid
p\in\mathcal{P}\\},\delta_{0},\delta_{>0})$, where $Q$ is a finite set of
control locations, $Q_{p}\subseteq Q$ for all $p\in\mathcal{P}$ with
$Q_{p}=\emptyset$ for all but finitely many $p\in\mathcal{P}$,
$\delta_{0}\subseteq Q\times\\{0,1\\}\times Q$ is a set of zero transitions,
and $\delta_{>0}\subseteq Q\times\\{-1,0,1\\}\times Q$ is a set of positive
transitions. The size of the OCP $\mathbb{O}$ is
$|\mathbb{O}|=|Q|+\sum_{p\in\mathcal{P}}|Q_{p}|+|\delta_{0}|+|\delta_{>0}|$.
The transition system defined by $\mathbb{O}$ is
$T(\mathbb{O})=(Q\times\mathbb{N},\\{Q_{p}\times\mathbb{N}\mid
p\in\mathcal{P}\\},\rightarrow)$, where $(q,n)\rightarrow(q^{\prime},n+k)$ if
and only if either $n=0$ and $(q,k,q^{\prime})\in\delta_{0}$, or $n>0$ and
$(q,k,q^{\prime})\in\delta_{>0}$. A one-counter net (OCN) is an OCP, where
$\delta_{0}\subseteq\delta_{>0}$. For
$(q,k,q^{\prime})\in\delta_{0}\cup\delta_{>0}$ we usually write
$q\xrightarrow{k}q^{\prime}$.
More details on the temporal logic ${\mathsf{CTL}}$ can be found for instance
in [2]. Formulas $\varphi$ of ${\mathsf{CTL}}$ are defined by the following
grammar, where $p\in\mathcal{P}$:
$\varphi\quad::=\quad p\ \mid\ \neg\varphi\ \mid\ \varphi\wedge\varphi\ \mid\
\exists\mathsf{X}\varphi\ \mid\ \exists\varphi{\mathsf{U}}\varphi\ \mid\
\exists\varphi\mathsf{WU}\varphi.$
Given a transition system $T=(S,\\{S_{p}\mid p\in\mathcal{P}\\},\rightarrow)$
and a ${\mathsf{CTL}}$ formula $\varphi$, we define the semantics
$[\\![\varphi]\\!]_{T}\subseteq S$ by induction on the structure of $\varphi$
as follows: $[\\![p]\\!]_{T}=S_{p}\text{ for each }p\in\mathcal{P}$,
$[\\![\neg\varphi]\\!]_{T}=S\setminus[\\![\varphi]\\!]_{T}$,
$[\\![\varphi_{1}\wedge\varphi_{2}]\\!]_{T}=[\\![\varphi_{1}]\\!]_{T}\cap[\\![\varphi_{2}]\\!]_{T}$,
$[\\![\exists\mathsf{X}\varphi]\\!]_{T}=\\{s\in S\mid\exists
s^{\prime}\in[\\![\varphi]\\!]_{T}:s\rightarrow s^{\prime}\\}$,
$[\\![\exists\varphi_{1}{\mathsf{U}}\varphi_{2}]\\!]_{T}=\\{s\in
S\mid\exists\pi\in{\mathrm{path}}_{+}(T):\pi_{1}=s,\pi_{|\pi|}\in[\\![\varphi_{2}]\\!]_{T},\forall
i\in[|\pi|-1]:\pi_{i}\in[\\![\varphi_{1}]\\!]_{T}\\}$,
$[\\![\exists\varphi_{1}\mathsf{WU}\varphi_{2}]\\!]_{T}=[\\![\exists\varphi_{1}{\mathsf{U}}\varphi_{2}]\\!]_{T}\cup\\{s\in
S\mid\exists\pi\in{\mathrm{path}}_{\omega}(T):\pi_{1}=s,\forall i\geq
1:\pi_{i}\in[\\![\varphi_{1}]\\!]_{T}\\}$. We also write $(T,s)\models\varphi$
(or briefly $s\models\varphi$ if $T$ is clear from the context) for
$s\in[\\![\varphi]\\!]_{T}$. We introduce the usual abbreviations
$\varphi_{1}\vee\varphi_{2}=\neg(\neg\varphi_{1}\wedge\neg\varphi_{2})$,
$\forall\mathsf{X}\varphi=\neg\exists\mathsf{X}\neg\varphi$,
$\exists\mathsf{F}\varphi=\exists(p\vee\neg p){\mathsf{U}}\varphi$, and
$\exists\mathsf{G}\varphi=\exists\varphi\mathsf{WU}(p\wedge\neg p)$ for some
$p\in\mathcal{P}$. Formulas of the ${\mathsf{CTL}}$-fragment ${\mathsf{EF}}$
are given by the following grammar, where $p\in\mathcal{P}$: $\varphi::=p\
\mid\neg\varphi\ \mid\ \varphi\wedge\varphi\ \mid\ \exists\mathsf{X}\varphi\
\mid\exists\mathsf{F}\varphi$. The size of ${\mathsf{CTL}}$ formulas is
defined as follows: $|p|=1$,
$|\neg\varphi|=|\exists\mathsf{X}\varphi|=|\varphi|+1$,
$|\varphi_{1}\wedge\varphi_{2}|=|\varphi_{1}|+|\varphi_{2}|+1$,
$|\exists\varphi_{1}{\mathsf{U}}\varphi_{2}|=|\exists\varphi_{1}{\mathsf{W}}{\mathsf{U}}\varphi_{2}|=|\varphi_{1}|+|\varphi_{2}|+1$.
## 4\. CTL on OCPs: Periodic behaviour and upper bounds
The goal of this section is to prove a periodicity property of
${\mathsf{CTL}}$ over OCPs, which implies an upper bound for ${\mathsf{CTL}}$
on OCPs, see Thm. 4.3. As a corollary, we state that for a fixed OCP,
${\mathsf{CTL}}$ model checking restricted to formulas of fixed leftward until
depth (see the definition below) can be done in polynomial time. We define the
leftward until depth $\mathrm{lud}$ of ${\mathsf{CTL}}$ formulas inductively
as follows: $\mathrm{lud}(p)=0$ for $p\in\mathcal{P}$,
$\mathrm{lud}(\neg\varphi)=\mathrm{lud}(\exists\mathsf{X}\varphi)=\mathrm{lud}(\varphi)$,
$\mathrm{lud}(\varphi_{1}\wedge\varphi_{2})=\max\\{\mathrm{lud}(\varphi_{1}),\mathrm{lud}(\varphi_{2})\\}$,
$\mathrm{lud}(\exists\varphi_{1}{\mathsf{U}}\varphi_{2})=\mathrm{lud}(\exists\varphi_{1}{\mathsf{W}}{\mathsf{U}}\varphi_{2})=\max\\{\mathrm{lud}(\varphi_{1})+1,\mathrm{lud}(\varphi_{2})\\}$.
A similar definition of until depth can be found in [24], but there the until
depth of $\exists\varphi_{1}{\mathsf{U}}\varphi_{2}$ is 1 plus the maximum of
the until depths of $\varphi_{1}$ and $\varphi_{2}$. Note that
$\mathrm{lud}(\varphi)\leq 1$ for every ${\mathsf{EF}}$ formula $\varphi$.
Let us fix an OCP $\mathbb{O}=(Q,\\{Q_{p}\mid
p\in\mathcal{P}\\},\delta_{0},\delta_{>0})$ for the rest of this section. Let
$|Q|=k$ and define $K=\text{LCM}([k])$ and
$K_{\varphi}=K^{\mathrm{lud}(\varphi)}$ for each ${\mathsf{CTL}}$ formula
$\varphi$.
###### Theorem 4.1.
For all ${\mathsf{CTL}}$ formulas $\varphi$, all $q\in Q$ and all
$n,n^{\prime}>2\cdot|\varphi|\cdot k^{2}\cdot K_{\varphi}$ with $n\equiv
n^{\prime}\text{ mod }K_{\varphi}$:
$\displaystyle{}(q,n)\in[\\![\varphi]\\!]_{T(\mathbb{O})}\quad\Longleftrightarrow\quad(q,n^{\prime})\in[\\![\varphi]\\!]_{T(\mathbb{O})}.$
(1)
###### Proof 4.2 (Proof sketch).
We prove the theorem by induction on the structure of $\varphi$. We only treat
the difficult case $\varphi=\exists\psi_{1}{\mathsf{U}}\psi_{2}$ here. Let
$T=\max\\{2\cdot|\psi_{i}|\cdot k^{2}\cdot K_{\psi_{i}}\mid i\in\\{1,2\\}\\}$.
Let us prove equivalence (1). Note that $K_{\varphi}=\text{LCM}\\{K\cdot
K_{\psi_{1}},K_{\psi_{2}}\\}$ by definition. Let us fix an arbitrary control
location $q\in Q$ and naturals $n,n^{\prime}\in\mathbb{N}$ such that
$2\cdot|\varphi|\cdot k^{2}\cdot K_{\varphi}<n<n^{\prime}$ and $n\equiv
n^{\prime}\text{ mod }K_{\varphi}$. We have to prove that
$(q,n)\in[\\![\varphi]\\!]_{T(\mathbb{O})}$ if and only if
$(q,n^{\prime})\in[\\![\varphi]\\!]_{T(\mathbb{O})}$. For this, let
$d=n^{\prime}-n$, which is a multiple of $K_{\varphi}$. We only treat the
“if”-direction here and recommend the reader to consult [14] for helpful
illustrations. So let us assume that
$(q,n^{\prime})\in[\\![\varphi]\\!]_{T(\mathbb{O})}$. To prove that
$(q,n)\in[\\![\varphi]\\!]_{T(\mathbb{O})}$, we will use the following claim.
Claim: Assume some $[\\![\psi_{1}]\\!]_{T(\mathbb{O})}$-path
$\pi=[(q_{1},n_{1})\to(q_{2},n_{2})\to\cdots\to(q_{l},n_{l})]$ with $n_{i}>T$
for all $i\in[l]$ and $n_{1}-n_{l}\geq k^{2}\cdot K\cdot K_{\psi_{1}}$. Then
there exists a $[\\![\psi_{1}]\\!]_{T(\mathbb{O})}$-path from $(q_{1},n_{1})$
to $(q_{l},n_{l}+K\cdot K_{\psi_{1}})$, whose counter values are all strictly
above $T+K\cdot K_{\psi_{1}}$.
The claim tells us that paths that lose height at least $k^{2}\cdot K\cdot
K_{\psi_{1}}$ and whose states all have counter values strictly above $T$ can
be flattened (without changing the starting state) by height $K\cdot
K_{\psi_{1}}$.
Proof of the claim. For each counter value $h\in\\{n_{i}\mid i\in[l]\\}$ that
appears in $\pi$, let $\mu(h)=\min\\{i\in[l]\mid n_{i}=h\\}$ denote the
minimal position in $\pi$ whose corresponding state has counter value $h$.
Define $\Delta=k\cdot K_{\psi_{1}}$. We will be interested in $k\cdot K$ many
consecutive intervals (of counter values) each of size $\Delta$. Define the
bottom $b=n_{1}-(k\cdot K)\cdot\Delta$. Formally, an interval is a set
$I_{i}=[b+(i-1)\cdot\Delta,b+i\cdot\Delta]$ for some $i\in[k\cdot K]$. Since
each interval has size $\Delta=k\cdot K_{\psi_{1}}$, we can think of each
interval $I_{i}$ to consist of $k$ consecutive sub-intervals of size
$K_{\psi_{1}}$ each. Note that each sub-interval has two extremal elements,
namely its upper and lower boundary. Thus all $k$ sub-intervals have $k+1$
boundaries in total. Hence, by the pigeonhole principle, for each interval
$I_{i}$, there exists some $c_{i}\in[k]$ and two distinct boundaries
$\beta(i,1)>\beta(i,2)$ of distance $c_{i}\cdot K_{\psi_{1}}$ such that the
control location of $\pi$’s earliest state of counter value $\beta(i,1)$
agrees with the control location of $\pi$’s earliest state of counter value
$\beta(i,2)$, i.e., formally $q_{\mu(\beta(i,1))}\ =\ q_{\mu(\beta(i,2))}$.
Observe that flattening the path $\pi$ by gluing together $\pi$’s states at
position $\mu(\beta(i,1))$ and $\mu(\beta(i,2))$ (for this, we add $c_{i}\cdot
K_{\psi_{1}}$ to each counter value at a position $\geq\beta(i,2)$) still
results in a $[\\![\psi_{1}]\\!]_{T(\mathbb{O})}$-path by induction
hypothesis, since we reduced the height of $\pi$ by a multiple of
$K_{\psi_{1}}$. Our overall goal is to flatten $\pi$ by gluing together states
only of certain intervals such that we obtain a path whose height is in total
by precisely $K\cdot K_{\psi_{1}}$ smaller than $\pi$’s. Recall that there are
$k\cdot K$ many intervals. By the pigeonhole principle there is some $c\in[k]$
such that $c_{i}=c$ for at least $K$ many intervals $I_{i}$. By gluing
together $\frac{K}{c}\in\mathbb{N}$ pairs of states of distance $c\cdot
K_{\psi_{1}}$ each, we reduce $\pi$’s height by exactly $\frac{K}{c}\cdot
c\cdot K_{\psi_{1}}=K\cdot K_{\psi_{1}}$. This proves the claim.
Let us finish the proof the “if”-direction. Since by assumption
$(q,n^{\prime})\in[\\![\varphi]\\!]_{T(\mathbb{O})}$, there exists a finite
path $\pi\ =\ (q_{1},n_{1})\rightarrow(q_{2},n_{2})\to\cdots\to(q_{l},n_{l})$,
where $\pi[1,l-1]$ is a $[\\![\psi_{1}]\\!]_{T(\mathbb{O})}$-path,
$(q,n^{\prime})=(q_{1},n_{1})$, and where
$(q_{l},n_{l})\in[\\![\psi_{2}]\\!]_{T(\mathbb{O})}$. To prove
$(q,n)\in[\\![\varphi]\\!]_{T(\mathbb{O})}$, we will assume that $n_{j}>T$ for
each $j\in[l]$. The case when $n_{j}=T$ for some $j\in[l]$ can be proven
similarly. Assume first that the path $\pi[1,l-1]$ contains two states whose
counter difference is at least $k^{2}\cdot K\cdot K_{\psi_{1}}+K_{\varphi}$
which is (strictly) greater than $k^{2}\cdot K\cdot K_{\psi_{1}}$. Since
$K_{\varphi}$ is a multiple of $K\cdot K_{\psi_{1}}$ by definition, we can
apply the above claim $\frac{K_{\varphi}}{K\cdot K_{\psi_{1}}}\in\mathbb{N}$
many times to $\pi[1,l-1]$. This reduces the height by $K_{\varphi}$. We
repeat this flattening process of $\pi[1,l-1]$ by height $K_{\varphi}$ as long
as possible, i.e., until any two states have counter difference smaller than
$k^{2}\cdot K\cdot K_{\psi_{1}}+K_{\varphi}$. Let $\sigma$ denote the
$[\\![\psi_{1}]\\!]_{T(\mathbb{O})}$-path starting in $(q,n^{\prime})$ that we
obtain from $\pi[1,l-1]$ by this process. Thus, $\sigma$ ends in some state,
whose counter value is congruent $n_{l-1}$ modulo $K_{\varphi}$ (since we
flattened $\pi[1,l-1]$ by a multiple of $K_{\varphi}$). Since $K_{\varphi}$ is
in turn a multiple of $K_{\psi_{2}}$, we can build a path $\sigma^{\prime}$
which extends the path $\sigma$ by a single transition to some state that
satisfies $\psi_{2}$ by induction hypothesis. Moreover, by our flattening
process, the counter difference between any two states in $\sigma^{\prime}$ is
at most $k^{2}\cdot K\cdot K_{\psi_{1}}+K_{\varphi}\leq 2\cdot k^{2}\cdot
K_{\varphi}$. Recall that $T=\max\\{2\cdot|\psi_{i}|\cdot k^{2}\cdot
K_{\psi_{i}}\mid i\in\\{1,2\\}\\}$. As
$n\ >\ 2\cdot|\varphi|\cdot k^{2}\cdot K_{\varphi}\ =\
2\cdot(|\varphi|-1+1)\cdot k^{2}\cdot K_{\varphi}\ \geq\ T+2\cdot k^{2}\cdot
K_{\varphi},$
it follows that the path that results from $\sigma^{\prime}$ by subtracting
$d$ from each counter value (this path starts in $(q,n)$) is strictly above
$T$. Moreover, since $d$ is a multiple of $K_{\psi_{1}}$ and $K_{\psi_{2}}$,
this path witnesses $(q,n)\in[\\![\varphi]\\!]_{T(\mathbb{O})}$ by induction
hypothesis.
The following result can be obtained basically by using the standard model
checking algorithm for ${\mathsf{CTL}}$ on finite systems (see e.g. [2]) in
combination with Thm. 4.1.
###### Theorem 4.3.
For a given one-counter process $\mathbb{O}=(Q,\\{Q_{p}\mid
p\in\mathcal{P}\\},\delta_{0},\delta_{>0})$, a ${\mathsf{CTL}}$ formula
$\varphi$, a control location $q\in Q$, and $n\in\mathbb{N}$ given in binary,
one can decide $(q,n)\in[\\![\varphi]\\!]_{T(\mathbb{O})}$ in time
$O(\log(n)+|Q|^{3}\cdot|\varphi|^{2}\cdot
4^{|Q|\cdot\mathrm{lud}(\varphi)}\cdot|\delta_{0}\cup\delta_{>0}|)$.
As a corollary, we can deduce that for every fixed OCP $\mathbb{O}$ and every
fixed $k$ the question if for a given state $s$ and a given CTL formula
$\varphi$ with $\mathrm{lud}(\varphi)\leq k$, we have
$(T(\mathbb{O}),s)\models\varphi$, is in ${\mathsf{P}}$. This generalizes a
result from [13], stating that the expression complexity of ${\mathsf{EF}}$
over OCPs is in $\mathsf{P}$.
## 5\. Expression complexity for CTL is hard for PSPACE
The goal of this section is to prove that model checking ${\mathsf{CTL}}$ is
${\mathsf{PSPACE}}$-hard already over a fixed OCN. We show this via a
reduction from the well-known ${\mathsf{PSPACE}}$-complete problem QBF. Our
lower bound proof is separated into three steps. In step one, we define a
family of ${\mathsf{CTL}}$ formulas $(\varphi_{i})_{i\geq 1}$ such that over
the fixed OCN $\mathbb{O}$ that is depicted in Fig. 1 we can express
(non-)divisibility by $2^{i}$. In step two, we define a family of
${\mathsf{CTL}}$ formulas $(\psi_{i})_{i\geq 1}$ such that over $\mathbb{O}$
we can express if the $i^{\text{th}}$ bit in the binary representation of a
natural is set to $1$. In our final step, we give the reduction from QBF. For
step one, we need the following simple fact which characterizes divisibility
by powers of two (recall that $[n]=\\{1,\ldots,n\\}$, in particular
$[0]=\emptyset$):
$\forall n\geq 0,i\geq 1:\text{ $2^{i}$ divides $n$ }\ \Leftrightarrow\
(2^{i-1}\text{ divides }n\;\wedge\;|\\{n^{\prime}\in[n]\mid 2^{i-1}\text{
divides }n^{\prime}\\}|\text{ is even})$ (2)
$\delta_{>0}:$$\overline{t}$$t$$q_{0}$$q_{2}$$q_{1}$$-1$$q_{3}$$-1$$-1$$-1$$-1$$-1$$f$$g$$0$$0$$0$$0$$0$$0$$-1$$-1$$p_{0}$$p_{1}$$+1$$0$$0$$+1$$\delta_{0}:$$\overline{t}$$t$$q_{0}$$f$$0$$0$$p_{0}$$p_{1}$$+1$$0$
Figure 1. The one-counter net $\mathbb{O}$ for which ${\mathsf{CTL}}$ model
checking is ${\mathsf{PSPACE}}$-hard
The set of propositions of $\mathbb{O}$ in Fig. 1 coincides with its control
locations. Recall that $\mathbb{O}$’s zero transitions are denoted by
$\delta_{0}$ and $\mathbb{O}$’s positive transitions are denoted by
$\delta_{>0}$. Since $\delta_{0}\subseteq\delta_{>0}$, $\mathbb{O}$ is indeed
an OCN. Note that both $t$ and $\overline{t}$ are control locations of
$\mathbb{O}$. Now we define a family of ${\mathsf{CTL}}$ formulas
$(\varphi_{i})_{i\geq 1}$ such that for each $n\in\mathbb{N}$ we have: (i)
$(t,n)\models\varphi_{i}$ if and only if $2^{i}$ divides $n$ and (ii)
$(\overline{t},n)\models\varphi_{i}$ if and only if $2^{i}$ does not divide
$n$. On first sight, it might seem superfluous to let the control location $t$
represent divisibility by powers of two and the control location
$\overline{t}$ to represent non-divisibility by powers of two since
${\mathsf{CTL}}$ allows negation. However the fact that we have only one
family of formulas $(\varphi_{i})_{i\geq 1}$ to express both divisibility and
non-divisibility is a crucial technical subtlety that is necessary in order to
avoid an exponential blowup in formula size. By making use of (2), we
construct the formulas $\varphi_{i}$ inductively. First, let us define the
auxiliary formulas $\text{test}=t\vee\overline{t}$ and
$\varphi_{\diamond}=q_{0}\vee q_{1}\vee q_{2}\vee q_{3}$. Think of
$\varphi_{\diamond}$ to hold in those control locations that altogether are
situated in the “diamond” in Fig. 1. We define
$\displaystyle\varphi_{1}$ $\displaystyle=$
$\displaystyle\text{test}\wedge\exists\mathsf{X}\,\left(f\wedge{\mathsf{EF}}(f\wedge\neg\exists\mathsf{X}g)\right)\text{
and }$ $\displaystyle\varphi_{i}$ $\displaystyle=$ $\displaystyle\text{test}\
\wedge\
\exists\mathsf{X}\,\bigl{(}\exists(\varphi_{\diamond}\wedge\exists\mathsf{X}\varphi_{i-1})\
{\mathsf{U}}\ (q_{0}\wedge\neg\exists\mathsf{X}q_{1})\bigr{)}\text{ for }i>1.$
Since $\varphi_{i-1}$ is only used once in $\varphi_{i}$, we get
$|\varphi_{i}|\in O(i)$. The following lemma states the correctness of the
construction.
###### Lemma 5.1.
Let $n\geq 0$ and $i\geq 1$. Then
* •
$(t,n)\models\varphi_{i}$ if and only if $2^{i}$ divides $n$.
* •
$(\overline{t},n)\models\varphi_{i}$ if and only if $2^{i}$ does not divide
$n$.
Proof sketch. The lemma is proved by induction on $i$. The induction base for
$i=1$ is easy to check. For $i>1$, observe that $\varphi_{i}$ can only be true
either in control location $t$ or $\overline{t}$. Note that the formula right
to the until symbol in $\varphi_{i}$ expresses that we are in $q_{0}$ and that
the current counter value is zero. Also note that the formula left to the
until symbol requires that $\varphi_{\diamond}$ holds, i.e., we are always in
one of the four “diamond control locations”. In other words, we decrement the
counter by moving along the diamond control locations (by possibly looping at
$q_{1}$ and $q_{3}$) and always check if $\exists\mathsf{X}\varphi_{i-1}$
holds, just until we are in $q_{0}$ and the counter value is zero. Since there
are transitions from $q_{1}$ and $q_{3}$ to $\overline{t}$ (but not to $t$),
the induction hypothesis implies that the formula
$\exists\mathsf{X}\varphi_{i-1}$ can be only true in $q_{1}$ and $q_{3}$ as
long as the current counter value is not divisible by $2^{i-1}$. Similarly,
since there are transitions from $q_{0}$ and $q_{2}$ to $t$ (but not to
$\overline{t}$), the induction hypothesis implies that the formula
$\exists\mathsf{X}\varphi_{i-1}$ can be only true in $q_{0}$ and $q_{2}$ if
the current counter value is divisible by $2^{i-1}$. With (2) this implies the
lemma. ∎
For expressing if the $i^{\text{th}}$ bit of a natural is set to $1$, we make
use of the following simple fact:
$\forall n\geq 0,i\geq 1:\text{bit}_{i}(n)=1\ \Longleftrightarrow\
|\\{n^{\prime}\in[n]\mid 2^{i-1}\text{ divides }n^{\prime}\\}|\text{ is odd}$
(3)
Let us now define a family of ${\mathsf{CTL}}$ formulas $(\psi_{i})_{i\geq 1}$
such that for each $n\in\mathbb{N}$ we have $\text{bit}_{i}(n)=1$ if and only
if $(\overline{t},n)\models\psi_{i}$. We set $\psi_{1}=\varphi_{1}$ and
$\psi_{i}=\overline{t}\wedge\exists\mathsf{X}\left((q_{1}\vee q_{2})\ \wedge\
\mu_{i}\right)$, where
$\mu_{i}=\exists(\varphi_{\diamond}\wedge\exists\mathsf{X}\varphi_{i-1})\
{\mathsf{U}}\ (q_{0}\wedge\neg\exists\mathsf{X}q_{1})$ for each $i>1$. Due to
the construction of $\psi_{i}$ and since $|\varphi_{i}|\in O(i)$, we obtain
that $|\psi_{i}|\in O(i)$. The following lemma states the correctness of the
construction.
###### Lemma 5.2.
Let $n\geq 0$ and let $i\geq 1$. Then $(\overline{t},n)\models\psi_{i}$ if and
only if $\text{bit}_{i}(n)=1$.
Let us sketch the final step of the reduction from QBF. For this, let us
assume some quantified Boolean formula
$\alpha=Q_{k}x_{k}\,Q_{k-1}x_{k-1}\cdots
Q_{1}x_{1}:\beta(x_{1},\ldots,x_{k})$, where $\beta$ is a Boolean formula over
variables $\\{x_{1},\ldots,x_{k}\\}$ and $Q_{i}\in\\{\exists,\forall\\}$ is a
quantifier for each $i\in[k]$. Think of each truth assignment
$\vartheta:\\{x_{1},\ldots,x_{k}\\}\rightarrow\\{0,1\\}$ to correspond to the
natural number $n(\vartheta)\in[0,2^{k}-1]$, where
$\text{bit}_{i}(n(\vartheta))=1$ if and only if $\vartheta(x_{i})=1$, for each
$i\in[k]$. Let $\widehat{\beta}$ be the CTL formula that is obtained from
$\beta$ by replacing each occurrence of $x_{i}$ by $\psi_{i}$, which
corresponds to applying Lemma 5.2. It remains to describe how we deal with
quantification. Think of this as to consecutively incrementing the counter
from state $(\overline{t},0)$ as follows. First, setting the variable $x_{k}$
to $1$ will correspond to adding $2^{k-1}$ to the counter and getting to state
$(\overline{t},2^{k-1})$. Setting $x_{k}$ to $0$ on the other hand will
correspond to adding $0$ to the counter and hence remaining in state
$(\overline{t},0)$. Next, setting $x_{k-1}$ to $1$ corresponds to adding to
the current counter value $2^{k-2}$, whereas setting $x_{k-1}$ to $0$
corresponds to adding $0$, as expected. These incrementation steps can be
achieved using the formulas $\varphi_{i}$ from Lemma 5.1. Finally, after
setting variable $x_{1}$ either to $0$ or $1$, we verify if the CTL formula
$\widehat{\beta}$ holds. Formally, let $\bigcirc_{i}=\wedge$ if
$Q_{i}=\exists$ and $\bigcirc_{i}=\,\rightarrow$ if $Q_{i}=\forall$ for each
$i\in[k]$ (recall that $Q_{k},\ldots,Q_{1}$ are the quantifiers of our
quantified Boolean formula $\alpha$). Let
$\theta_{1}=Q_{1}\mathsf{X}\,((p_{0}\vee
p_{1})\bigcirc_{1}\exists\mathsf{X}\,\widehat{\beta})$ and for $i\in[2,k]$:
$\theta_{i}=Q_{i}\mathsf{X}\;\left((p_{0}\vee
p_{1})\bigcirc_{i}\exists\left((p_{0}\vee\exists\mathsf{X}\,(\overline{t}\wedge\varphi_{i-1}))\
{\mathsf{U}}\
(\overline{t}\wedge\neg\varphi_{i-1}\wedge\theta_{i-1}))\biggl{.}\right)\right).$
Then, it can be show that $\alpha$ is valid if and only if
$(\overline{t},0)\in[\\![\theta_{k}]\\!]_{T(\mathbb{O})}$.
###### Theorem 5.3.
${\mathsf{CTL}}$ model checking of the fixed OCN $\mathbb{O}$ from Fig. 1 is
${\mathsf{PSPACE}}$-hard.
Note that the constructed ${\mathsf{CTL}}$ formula has leftward until depth
that depends on the size of $\alpha$. By Thm. 4.3 this cannot be avoided
unless ${\mathsf{P}}={\mathsf{PSPACE}}$. Observe that in order to express
divisibility by powers of two, our ${\mathsf{CTL}}$ formulas
$(\varphi_{i})_{i\geq 0}$ have linearly growing leftward until depth.
## 6\. Tools from complexity theory
For Sec. 7 and 8 we need some concepts from complexity theory. By
${\mathsf{P}}^{{\mathsf{NP}}[\log]}$ we denote the class of all problems that
can be solved on a polynomially time bounded deterministic Turing machines
which can have access to an ${\mathsf{NP}}$-oracle only logarithmically many
times, and by ${\mathsf{P}}^{\mathsf{NP}}$ the corresponding class without the
restriction to logarithmically many queries. Let us briefly recall the
definition of the circuit complexity class ${\mathsf{NC}}^{1}$, more details
can be found in [26]. We consider Boolean circuits $C=C(x_{1},\ldots,x_{n})$
built up from AND- and OR-gates. Each input gate is labeled with a variable
$x_{i}$ or a negated variable $\neg x_{i}$. The output gates are linearly
ordered. Such a circuit computes a function
$f_{C}:\\{0,1\\}^{n}\to\\{0,1\\}^{m}$, where $m$ is the number of output
gates, in the obvious way. The _fan-in of a circuit_ is the maximal number of
incoming wires of a gate in the circuit. The _depth of a circuit_ is the
number of gates along a longest path from an input gate to an output gate. A
_logspace-uniform ${\mathsf{NC}}^{1}$-circuit family_ is a sequence
$(C_{n})_{n\geq 1}$ of Boolean circuits such that for some polynomial $p(n)$
and constant $c$: (i) $C_{n}$ contains at most $p(n)$ many gates, (ii) the
depth of $C_{n}$ is at most $c\cdot\log(n)$, (iii) the fan-in of $C_{n}$ is at
most $2$, (iv) for each $m$ there is at most one circuit in $(C_{n})_{n\geq
1}$ with exactly $m$ input gates, and (v) there exists a logspace transducer
that computes on input $1^{n}$ a representation (e.g. as a node-labeled graph)
of the circuit $C_{n}$. Such a circuit family computes a partial mapping on
$\\{0,1\\}^{*}$ in the obvious way (note that we do not require to have for
every $n\geq 0$ a circuit with exactly $n$ input gates in the family,
therefore the computed mapping is in general only partially defined). In the
literature on circuit complexity one can find more restrictive notions of
uniformity, see e.g. [26], but logspace uniformity suffices for our purposes.
In fact, polynomial time uniformity suffices for proving our lower bounds
w.r.t. polynomial time reductions.
For $m\geq 1$ and $0\leq M\leq 2^{m}-1$ let
${\mathrm{BIN}}_{m}(M)=\text{bit}_{m}(M)\cdots\text{bit}_{1}(M)\in\\{0,1\\}^{m}$
denote the $m$-bit binary representation of $M$. Let $p_{i}$ denote the
$i^{\text{th}}$ prime number. It is well-known that the $i^{\text{th}}$ prime
requires $O(\log(i))$ bits in its binary representation. For a number $0\leq
M<\prod_{i=1}^{m}p_{i}$ we define the Chinese remainder representation
${\mathrm{CRR}}_{m}(M)$ as the Boolean tuple
${\mathrm{CRR}}_{m}(M)=(x_{i,r})_{i\in[m],0\leq r<p_{i}}$ with $x_{i,r}=1$ if
$M\text{ mod }p_{i}=r$ and $x_{i,r}=0$ else. By the following theorem, one can
transform a Chinese remainder representation very efficiently into binary
representation.
###### Theorem 6.1 ([9]).
There is a logspace-uniform ${\mathsf{NC}}^{1}$-circuit family
$(B_{m}((x_{i,r})_{i\in[m],0\leq r<p_{i}}))_{m\geq 1}$ such that for every
$m\geq 1$, $B_{m}$ has $m$ output gates and for every $0\leq
M<\prod_{i=1}^{m}p_{i}$ we have that
$B_{m}({\mathrm{CRR}}_{m}(M))={\mathrm{BIN}}_{m}(M\text{ mod }2^{m})$.
By [17], we could replace logspace-uniform ${\mathsf{NC}}^{1}$-circuits in
Thm. 6.1 even by $\mathsf{DLOGTIME}$-uniform ${\mathsf{TC}}^{0}$-circuits. The
existence of a $\mathsf{P}$-uniform ${\mathsf{NC}}^{1}$-circuit family for
converting from Chinese remainder representation to binary representation was
already shown in [3]. Usually the Chinese remainder representation of $M$ is
the tuple $(r_{i})_{i\in[m]}$, where $r_{i}=M\text{ mod }p_{i}$. Since the
primes $p_{i}$ will be always given in unary notation, there is no essential
difference between this representation and our Chinese remainder
representation. The latter is more suitable for our purpose.
The following definition of ${\mathsf{NC}}^{1}$-serializability is a variant
of the more classical notion of serializability [8, 16], which fits our
purpose better. A language $L$ is ${\mathsf{NC}}^{1}$-serializable if there
exists an NFA $A$ over the alphabet $\\{0,1\\}$, a polynomial $p(n)$, and a
logspace-uniform ${\mathsf{NC}}^{1}$-circuit family $(C_{n})_{n\geq 0}$, where
$C_{n}$ has exactly $n+p(n)$ many inputs and one output, such that for every
$x\in\\{0,1\\}^{n}$ we have $x\in L$ if and only if $C_{n}(x,0^{p(n)})\cdots
C_{n}(x,1^{p(n)})\in L(A)$, where “$\cdots$” refers to the lexicographic order
on $\\{0,1\\}^{p(n)}$. With this definition, it can be shown that all
languages in ${\mathsf{PSPACE}}$ are ${\mathsf{NC}}^{1}$-serializable. A proof
can be found in the appendix of [14]; it is just a slight adaptation of the
proofs from [8, 16].
## 7\. Data complexity for CTL is hard for PSPACE
In this section, we prove that also the data complexity of ${\mathsf{CTL}}$
over OCNs is hard for ${\mathsf{PSPACE}}$ and therefore
${\mathsf{PSPACE}}$-complete by the known upper bounds for the modal
$\mu$-calculus [23]. Let us fix the set of propositions
${\mathcal{P}}=\\{\alpha,\beta,\gamma\\}$ for this section. In the following,
w.l.o.g. we allow in $\delta_{0}$ (resp. in $\delta_{>0}$) transitions of the
kind $(q,k,q^{\prime})$, where $k\in\mathbb{N}$ (resp. $k\in\mathbb{Z}$) is
given in unary representation with the expected intuitive meaning.
###### Proposition 7.1.
For the fixed ${\mathsf{EF}}$ formula
$\varphi=(\alpha\to\exists\mathsf{X}(\beta\wedge{\mathsf{EF}}(\neg\exists\mathsf{X}\gamma)))$
the following problem can be solved with a logspace transducer:
INPUT: A list $p_{1},\ldots,p_{m}$ of the first $m$ consecutive (unary
encoded) prime numbers and a Boolean formula $F=F((x_{i,r})_{i\in[m],0\leq
r<p_{i}})$
OUTPUT: An OCN ${\mathbb{O}}(F)$ with distinguished control locations
${\mathsf{in}}$ and ${\mathsf{out}}$, such that for every number $0\leq
M<\prod_{i=1}^{m}p_{i}$ we have that $F({\mathrm{CRR}}_{m}(M))=1$ if and only
if there exists a $[\\![\varphi]\\!]_{T({\mathbb{O}}(F))}$-path from
$({\mathsf{in}},M)$ to $({\mathsf{out}},M)$ in the transition system
$T({\mathbb{O}}(F))$.
###### Proof 7.2.
W.l.o.g., negations occur in $F$ only in front of variables. Then
additionally, a negated variable $\neg x_{i,r}$ can be replaced by the
disjunction $\bigvee\\{x_{i,k}\mid 0\leq k<p_{i},r\neq k\\}$. This can be done
in logspace, since the primes $p_{i}$ are given in unary. Thus, we can assume
that $F$ does not contain negations.
The idea is to traverse the Boolean formula $F$ with the OCN ${\mathbb{O}}(F)$
in a depth first manner. Each time a variable $x_{i,r}$ is seen, the OCN may
also enter another branch, where it is checked, whether the current counter
value is congruent $r$ modulo $p_{i}$. Let
${\mathbb{O}}(F)=(Q,\\{Q_{\alpha},Q_{\beta},Q_{\gamma}\\},\delta_{0},\delta_{>0})$,
where $Q=\\{{\mathsf{in}}(G),{\mathsf{out}}(G)\mid G\text{ is a subformula of
}F\\}\cup\\{{\mathrm{div}}(p_{1}),\ldots,{\mathrm{div}}(p_{m}),\perp\\}$,
$Q_{\alpha}=\\{{\mathsf{in}}(x_{i,r})\mid i\in[m],0\leq r<p_{i}\\}$,
$Q_{\beta}=\\{{\mathrm{div}}(p_{1}),\ldots,{\mathrm{div}}(p_{m})\\}$, and
$Q_{\gamma}=\\{\perp\\}$. We set ${\mathsf{in}}={\mathsf{in}}(F)$ and
${\mathsf{out}}={\mathsf{out}}(F)$. Let us now define the transition sets
$\delta_{0}$ and $\delta_{>0}$. For every subformula $G_{1}\wedge G_{2}$ or
$G_{1}\vee G_{2}$ of $F$ we add the following transitions to $\delta_{0}$ and
$\delta_{>0}$:
$\displaystyle{\mathsf{in}}(G_{1}\wedge
G_{2})\xrightarrow{0}{\mathsf{in}}(G_{1}),\
{\mathsf{out}}(G_{1})\xrightarrow{0}{\mathsf{in}}(G_{2}),\
{\mathsf{out}}(G_{2})\xrightarrow{0}{\mathsf{out}}(G_{1}\wedge G_{2})$
$\displaystyle{\mathsf{in}}(G_{1}\vee
G_{2})\xrightarrow{0}{\mathsf{in}}(G_{i}),\
{\mathsf{out}}(G_{i})\xrightarrow{0}{\mathsf{out}}(G_{1}\vee G_{2})\text{ for
all }i\in\\{1,2\\}$
For every variable $x_{i,r}$ we add to $\delta_{0}$ and $\delta_{>0}$ the
transition ${\mathsf{in}}(x_{i,r})\xrightarrow{0}{\mathsf{out}}(x_{i,r})$.
Moreover, we add to $\delta_{>0}$ the transitions
${\mathsf{in}}(x_{i,r})\xrightarrow{-r}{\mathrm{div}}(p_{i})$. The transition
${\mathsf{in}}(x_{i,0})\xrightarrow{0}{\mathrm{div}}(p_{i})$ is also added to
$\delta_{0}$. For the control locations ${\mathrm{div}}(p_{i})$ we add to
$\delta_{>0}$ the transitions
${\mathrm{div}}(p_{i})\xrightarrow{-p_{i}}{\mathrm{div}}(p_{i})$ and
${\mathrm{div}}(p_{i})\xrightarrow{-1}\perp$. This concludes the description
of the OCN ${\mathbb{O}}(F)$. Correctness of the construction can be easily
checked by induction on the structure of the formula $F$.
We are now ready to prove ${\mathsf{PSPACE}}$-hardness of the data complexity.
###### Theorem 7.3.
There exists a fixed ${\mathsf{CTL}}$ formula of the form
$\exists\varphi_{1}{\mathsf{U}}\varphi_{2}$, where $\varphi_{1}$ and
$\varphi_{2}$ are ${\mathsf{EF}}$ formulas, for which it is
${\mathsf{PSPACE}}$-complete to decide
$(T({\mathbb{O}}),(q,0))\models\exists\varphi_{1}{\mathsf{U}}\varphi_{2}$ for
a given OCN ${\mathbb{O}}$ and a control location $q$ of ${\mathbb{O}}$.
###### Proof 7.4.
Let us take an arbitrary language $L$ in ${\mathsf{PSPACE}}$. Recall from Sec.
6 that ${\mathsf{PSPACE}}$ is ${\mathsf{NC}}^{1}$-serializable. Thus, there
exists an NFA $A=(S,\\{0,1\\},\delta,s_{0},S_{f})$ over the alphabet
$\\{0,1\\}$, a polynomial $p(n)$, and a logspace-uniform
${\mathsf{NC}}^{1}$-circuit family $(C_{n})_{n\geq 0}$, where $C_{n}$ has
$n+p(n)$ many inputs and one output, such that for every $x\in\\{0,1\\}^{n}$
we have:
$x\in L\ \Longleftrightarrow\ C_{n}(x,0^{p(n)})\cdots C_{n}(x,1^{p(n)})\in
L(A),$ (4)
where “$\cdots$” refers to the lexicographic order on $\\{0,1\\}^{p(n)}$. Fix
an input $x\in\\{0,1\\}^{n}$. Our reduction can be split into the following
five steps:
Step 1. Construct in logspace the circuit $C_{n}$. Fix the the first $n$
inputs of $C_{n}$ to the bits in $x$, and denote the resulting circuit by $C$;
it has only $m=p(n)$ many inputs. Then, (4) can be written as
$x\in L\ \Longleftrightarrow\ \prod_{M=0}^{2^{m}-1}C({\mathrm{BIN}}_{m}(M))\in
L(A).$ (5)
Step 2. Compute the first $m$ consecutive primes $p_{1},\ldots,p_{m}$. This is
possible in logspace, see e.g. [9]. Every $p_{i}$ is bounded polynomially in
$n$. Hence, every $p_{i}$ can be written down in unary notation. Note that
$\prod_{i=1}^{m}p_{i}>2^{m}$ (if $m>1$).
Step 3. Compute in logspace the circuit $B=B_{m}((x_{i,r})_{i\in[m],0\leq
r<p_{i}})$ from Thm. 6.1. Thus, $B$ is a Boolean circuit of fan-in 2 and depth
$O(\log(m))=O(\log(n))$ with $m$ output gates and
$B({\mathrm{CRR}}_{m}(M))={\mathrm{BIN}}_{m}(M\text{ mod }2^{m})$ for every
$0\leq M<\prod_{i=1}^{m}p_{i}$.
Step 4. Now we compose the circuits $B$ and $C$: For every $i\in[m]$, connect
the $i^{\text{th}}$ input of the circuit $C(x_{1},\ldots,x_{m})$ with the
$i^{\text{th}}$ output of the circuit $B$. The result is a circuit with fan-in
2 and depth $O(\log(n))$. In logspace, we can unfold this circuit into a
Boolean formula $F=F((x_{i,r})_{i\in[m],0\leq r<p_{i}})$. The resulting
formula (or tree) has the same depth as the circuit, i.e., depth $O(\log(n))$
and every tree node has at most 2 children. Hence, $F$ has polynomial size.
For every $0\leq M<2^{m}$ we have
$F({\mathrm{CRR}}_{m}(M))=C({\mathrm{BIN}}_{m}(M))$ and equivalence (5) can be
written as
$x\in L\ \Longleftrightarrow\ \prod_{M=0}^{2^{m}-1}F({\mathrm{CRR}}_{m}(M))\in
L(A).$ (6)
Step 5. We now apply our construction from Prop. 7.1 to the formula $F$. More
precisely, let $G$ be the Boolean formula $\bigwedge_{i\in[m]}x_{i,r_{i}}$
where $r_{i}=2^{m}\text{ mod }p_{i}$ for $i\in[m]$ (these remainders can be
computed in logspace). For every $1$-labeled transition $\tau\in\delta$ of the
NFA $A$ let ${\mathbb{O}}(\tau)$ be a copy of the OCN
${\mathbb{O}}(F\wedge\neg G)$. For every $0$-labeled transition
$\tau\in\delta$ let ${\mathbb{O}}(\tau)$ be a copy of the OCN
${\mathbb{O}}(\neg F\wedge\neg G)$. In both cases we write
${\mathbb{O}}(\tau)$ as
$(Q(\tau),\\{Q_{\alpha}(\tau),Q_{\beta}(\tau),Q_{\gamma}(\tau)\\},\delta_{0}(\tau),\delta_{>0}(\tau))$.
Denote with ${\mathsf{in}}(\tau)$ (resp. ${\mathsf{out}}(\tau)$) the control
location of this copy that corresponds to ${\mathsf{in}}$ (resp.
${\mathsf{out}}$) in ${\mathbb{O}}(F)$. Hence, for every $b$-labeled
transition $\tau\in\delta$ ($b\in\\{0,1\\}$) and every $0\leq
M<\prod_{i=1}^{m}p_{i}$ there exists a
$[\\![\varphi]\\!]_{T({\mathbb{O}}(\tau))}$-path ($\varphi$ is from Prop. 7.1)
from $({\mathsf{in}}(\tau),M)$ to $({\mathsf{out}}(\tau),M)$ if and only if
$F({\mathrm{CRR}}_{m}(M))=b$ and $M\neq 2^{m}$.
We now define an OCN
${\mathbb{O}}=(Q,\\{Q_{\alpha},Q_{\beta},Q_{\gamma}\\},\delta_{0},\delta_{>0})$
as follows: We take the disjoint union of all the OCNs ${\mathbb{O}}(\tau)$
for $\tau\in\delta$. Moreover, every state $s\in S$ of the NFA $A$ becomes a
control location of ${\mathbb{O}}$, i.e.
$Q=S\cup\bigcup_{\tau\in\delta}Q(\tau)$ and
$Q_{p}=\bigcup_{\tau\in\delta}Q_{p}(\tau)$ for each
$p\in\\{\alpha,\beta,\gamma\\}$. We add to $\delta_{0}$ and $\delta_{>0}$ for
every $\tau=(s,b,t)\in\delta$ the transitions
$s\xrightarrow{0}{\mathsf{in}}(\tau)$ and
${\mathsf{out}}(\tau)\xrightarrow{1}t$. Then, by Prop. 7.1 and (6) we have
$x\in L$ if and only if there exists a
$[\\![\varphi]\\!]_{T({\mathbb{O}})}$-path in $T({\mathbb{O}})$ from
$(s_{0},0)$ to $(s,2^{m})$ for some $s\in S_{f}$. Also note that there is no
$[\\![\varphi]\\!]_{T({\mathbb{O}})}$-path in $T({\mathbb{O}})$ from
$(s_{0},0)$ to some configuration $(s,M)$ with $s\in S$ and $M>2^{m}$. It
remains to add to ${\mathbb{O}}$ some structure that enables ${\mathbb{O}}$ to
check that the counter has reached the value $2^{m}$. For this, use again
Prop. 7.1 to construct the OCN ${\mathbb{O}}(G)$ ($G$ is from above) and add
it disjointly to ${\mathbb{O}}$. Moreover, add to $\delta_{>0}$ and
$\delta_{0}$ the transitions $s\xrightarrow{0}{\mathsf{in}}$ for all $s\in
S_{f}$, where ${\mathsf{in}}$ is the ${\mathsf{in}}$ control location of
${\mathbb{O}}(G)$. Finally, introduce a new proposition $\rho$ and set
$Q_{\rho}=\\{{\mathsf{out}}\\}$, where ${\mathsf{out}}$ is the
${\mathsf{out}}$ control location of ${\mathbb{O}}(G)$. By putting $q=s_{0}$
we obtain: $x\in L$ if and only if
$(T({\mathbb{O}}),(q,0))\models\exists(\varphi\ \mathsf{U}\ \rho)$, where
$\varphi$ is from Prop. 7.1. This concludes the proof of the theorem.
By slightly modifying the proof of Thm. 7.3, one can also prove that the fixed
CTL formula can chosen to be of the form $\exists\mathsf{G}\psi$, where $\psi$
is an ${\mathsf{EF}}$ formula.
## 8\. Two further applications: EF and one-counter Markov decision processes
In this section, we present two further applications of Thm. 6.1 to OCPs.
First, we state that the combined complexity for ${\mathsf{EF}}$ over OCNs is
hard for $\mathsf{P}^{\mathsf{NP}}$. For formulas represented succinctly by
directed acyclic graphs this was already shown in [13]. The point here is that
we use the standard tree representation for formulas.
###### Theorem 8.1.
It is $\mathsf{P}^{\mathsf{NP}}$-hard (and hence
$\mathsf{P}^{\mathsf{NP}}$-complete by [13]) to check
$(T({\mathbb{O}}),(q_{0},0))\models\varphi$ for given OCN ${\mathbb{O}}$,
state $q_{0}$ of ${\mathbb{O}}$, and ${\mathsf{EF}}$ formula $\varphi$.
The proof of Thm. 8.1 is very similar to the proof of Thm. 7.3, but does not
use the concept of serializability. We prove hardness by a reduction from the
question whether the lexicographically maximal satisfying assignment of a
Boolean formula is even when interpreted as a natural number. This problem is
$\mathsf{P}^{\mathsf{NP}}$-hard by [27]. At the moment we cannot prove that
the data complexity of ${\mathsf{EF}}$ over OCPs is hard for
$\mathsf{P}^{\mathsf{NP}}$ (hardness for $\mathsf{P}^{\mathsf{NP}[\log]}$ was
shown in [13]). Analyzing the proof of Thm. 8.1 in [14] shows that the main
obstacle is the fact that converting from Chinese remainder representation
into binary representation is not possible by uniform ${\mathsf{AC}}^{0}$
circuits (polynomial size circuits of constant depth and unbounded fan-in);
this is provably the case.
In the rest of the paper, we sketch a second application of our lower bound
technique based on Thm. 6.1, see [14] for more details. This application
concerns one-counter Markov decision processes. Markov decision processes
(MDPs) extend classical Markov chains by allowing so called nondeterministic
vertices. In these vertices, no probability distribution on the outgoing
transitions is specified. The other vertices are called probabilistic
vertices; in these vertices a probability distribution on the outgoing
transitions is given. The idea is that in an MDP a player Eve plays against
nature (represented by the probabilistic vertices). In each nondeterministic
vertex $v$, Eve chooses a probability distribution on the outgoing transitions
of $v$; this choice may depend on the past of the play (which is a path in the
underlying graph ending in $v$) and is formally represented by a strategy for
Eve. An MDP together with a strategy for Eve defines a Markov chain, whose
state space is the unfolding of the graph underlying the MDP. Here, we
consider infinite MDPs, which are finitely represented by OCPs; this formalism
was introduced in [6] under the name one-counter Markov decision process (OC-
MDP). With a given OC-MDP $\mathcal{A}$ and a set $R$ of control locations of
the OCP underlying $\mathcal{A}$ (a so called reachability constraint), two
sets were associated in [6]: $\text{ValOne}(R)$ is the set of all vertices $s$
of the MDP defined by $\mathcal{A}$ such that for every $\epsilon>0$ there
exists a strategy $\sigma$ for Eve under which the probability of finally
reaching from $s$ a control location in $R$ and at the same time having
counter value $0$ is at least $1-\varepsilon$. $\text{OptValOne}(R)$ is the
set of all vertices $s$ of the MDP defined by $\mathcal{A}$ for which there
exists a specific strategy for Eve under which this probability is $1$. It was
shown in [6] that for a given OC-MDP $\mathcal{A}$, a set of control locations
$R$, and a vertex $s$ of the MDP defined by $\mathcal{A}$, the question if
$s\in\text{OptValOne}(R)$ is ${\mathsf{PSPACE}}$-hard and in
${\mathsf{EXPTIME}}$. The same question for $\text{ValOne}(R)$ instead of
$\text{OptValOne}(R)$ was shown to be hard for each level of the Boolean
hierarchy ${\mathsf{BH}}$, which is a hierarchy of complexity classes between
${\mathsf{NP}}$ and $\mathsf{P}^{\mathsf{NP}[\log]}$. By applying our lower
bound techniques (from Thm. 7.3) we can prove the following.
###### Theorem 8.2.
Membership in $\text{ValOne}(R)$ is ${\mathsf{PSPACE}}$-hard.
As a byproduct of our proof, we also reprove ${\mathsf{PSPACE}}$-hardness for
$\text{OptValOne}(R)$. It is open, whether $\text{ValOne}(R)$ is decidable;
the corresponding problem for MDPs defined by pushdown processes is
undecidable [11].
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|
arxiv-papers
| 2009-12-21T09:45:23 |
2024-09-04T02:49:07.155290
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefan G\\\"oller, Markus Lohrey",
"submitter": "Stefan G\\\"oller",
"url": "https://arxiv.org/abs/0912.4117"
}
|
0912.4175
|
# Shear viscosity in antikaon condensed matter
Rana Nandi1, Sarmistha Banik2 and Debades Bandyopadhyay1 1Theory Division and
Centre for Astroparticle Physics, Saha Institute of Nuclear Physics, 1/AF
Bidhannagar, Kolkata-700064, India 2Variable Energy Cyclotron Centre, 1/AF
Bidhannagar, Kolkata-700064, India
###### Abstract
We investigate the shear viscosity of neutron star matter in the presence of
an antikaon condensate. The electron and muon number densities are reduced due
to the appearance of a $K^{-}$ condensate in neutron star matter, whereas the
proton number density increases. Consequently the shear viscosity due to
scatterings of electrons and muons with themselves and protons is lowered
compared to the case without the condensate. On the other hand, the
contribution of proton-proton collisions to the proton shear viscosity through
electromagnetic and strong interactions, becomes important and comparable to
the neutron shear viscosity.
###### pacs:
97.60.Jd, 26.60.-c, 52.25.Fi, 52.27.Ny
## I Introduction
Shear viscosity plays important roles in neutron star physics. It might damp
the r-mode instability below the temperature $\sim 10^{8}$ K And . The
knowledge of shear viscosity is essential in understanding pulsar glitches and
free precession of neutron stars Glam . The calculation of the neutron shear
viscosity ($\eta_{n}$) for nonsuperfluid matter using free-space nucleon-
nucleon scattering data was first done by Flowers and Itoh Flo1 ; Flo2 .
Cutler and Lindblom Cut fitted the results of Flowers and Itoh Flo2 for the
study of viscous damping of oscillations in neutron stars. Recently the
neutron shear viscosity of pure neutron matter has been investigated in a
self-consistent way Ben .
It was noted that electrons, the lightest charged particles and neutrons, the
most abundant particles in neutron star matter contribute significantly to the
total shear viscosity. Flowers and Itoh found that the neutron viscosity was
larger than the combined viscosity of electrons and muons ($\eta_{e\mu}$) in
non-superfluid matter Flo2 . Further Cutler and Lindblom argued that the
electron viscosity was larger than the neutron viscosity in a superfluid
neutron star Cut . Later Andersson and his collaborators as well as Yakovlev
and his collaborator showed $\eta_{e\mu}>\eta_{n}$ in the presence of proton
superfluidity Glam ; Yak . In the latter calculation, the effects of the
exchange of transverse plasmons in the collisions of charged particles were
included and it lowered the $\eta_{e\mu}$ compared with the case when only
longitudinal plasmons were considered Yak .
So far, all of those calculations of shear viscosity were done in neutron star
matter composed of neutrons, protons, electrons and muons. However, exotic
forms of matter such as hyperon or antikaon condensed matter might appear in
the interior of neutron stars. Negatively charged hyperons or a $K^{-}$
condensate could affect the electron shear viscosity appreciably.
Here we focus on the role of $K^{-}$ meson condensates on the shear viscosity.
No calculation of shear viscosity involving antikaon condensation has been
carried out so far. This motivates us to investigate the shear viscosity in
the presence of an antikaon condensate. The $K^{-}$ condensate appears at 2-3
times the normal nuclear matter density. With the onset of the condensate,
$K^{-}$ mesons replace electrons and muons in the core. As a result, $K^{-}$
mesons along with protons maintain the charge neutrality. It was noted that
the proton fraction became comparable to the neutron fraction in a neutron
star including the $K^{-}$ condensate at higher densities Brow ; Pal ; Banik .
The appearance of the $K^{-}$ condensate would not only influence the electron
and muon shear viscosities but it will also give rise to a new contribution
called the proton shear viscosity.
This paper is organised in the following way. In Sec. II, we describe the
calculation of shear viscosity in neutron stars involving the $K^{-}$
condensate. Results are discussed in Sec. III. A summary is given in Sec. IV.
## II Formalism
Here we are interested in calculating the shear viscosity of neutron star
matter in the presence of an antikaon condensate. We consider neutron star
matter undergoing a first order phase transition from charge neutral and beta-
equilibrated nuclear matter to a $K^{-}$ condensed phase. The nuclear phase is
composed of neutrons, protons, electrons and muons whereas the antikaon
condensed phase is made up of neutrons and protons embedded in the Bose-
Einstein condensate of $K^{-}$ mesons along with electrons and muons.
Antikaons form a s-wave ($\bf p=0$) condensation in this case. Therefore,
$K^{-}$ mesons in the condensate do not take part in momentum transfer during
collisions with other particles. However, the condensate influences the proton
fraction and equation of state (EOS) which, in turn, might have important
consequences for the shear viscosity. The starting point for the calculation
of the shear viscosity is a set of coupled Boltzmann transport equations Flo2
; Yak for the ith particle species (i= n, p, e, $\mu$) with velocity $v_{i}$
and distribution function $F_{i}$,
${\vec{v}_{i}}\cdot{\vec{\bigtriangledown}}F_{i}={\sum}_{j=n,p,e,\mu}I_{ij}.~{}$
(1)
The transport equations are coupled through collision integrals given by,
$I_{ij}={\frac{V^{3}}{(2\pi\hbar)^{9}(1+\delta_{ij})}}{\sum_{{s_{i^{\prime}}},{s_{j}},{s_{j^{\prime}}}}}\int
d{\bf p}_{j}d{\bf p}_{i^{\prime}}d{\bf p}_{j^{\prime}}W_{ij}{\cal F}~{},$ (2)
where
${\cal
F}=[{F_{i^{\prime}}F_{j^{\prime}}(1-F_{i})(1-F_{j})-F_{i}F_{j}(1-F_{i^{\prime}})(1-F_{j^{\prime}})}]~{}.$
(3)
Here ${\bf p}_{i}$, ${\bf p}_{j}$ are momenta of incident particles and ${\bf
p}_{i^{\prime}}$, ${\bf p}_{j^{\prime}}$ are those of final states. The
Kronecker delta in Eq. (2) is inserted to avoid double counting for identical
particles. Spins are denoted by $s$ and $W_{ij}$ is the differential
transition rate. The nonequilibrium distribution function for the i-th species
$F_{i}$ is given by
$F_{i}=f_{i}-\phi_{i}\frac{\partial f_{i}}{\partial\epsilon_{i}}~{},$ (4)
where the equilibrium Fermi-Dirac distribution function
$f(\epsilon_{i})~{}=~{}\frac{1}{1~{}+~{}e^{\frac{\epsilon_{i}~{}-~{}\mu_{i}}{kT}}}$
and the departure from the equilibrium is given by $\phi$. We adopt the
following ansatz for $\phi_{i}$ Yak ; Rup
$\phi_{i}=-\tau_{i}(v_{i}p_{j}-\frac{1}{3}v_{i}p_{i}\delta_{ij})(\bigtriangledown_{i}{\cal
V}_{j}+\bigtriangledown_{j}{\cal
V}_{i}-\frac{2}{3}\delta_{ij}{\vec{\bigtriangledown}}\cdot{\vec{\cal V}})~{},$
(5)
where $\tau_{i}$ is the effective relaxation time for the ith species and
${\cal V}$ is the flow velocity. The transport equations are linearised and
multiplied by
$(2\pi\hbar)^{-3}(v_{i}p_{j}-\frac{1}{3}v_{i}p_{i}\delta_{ij})d{\bf p}_{i}$.
Summing over spin $s_{i}$ and integrating over $d{\bf p}_{i}$ we obtain a set
of relations between effective relaxation times and collision frequencies Yak
$\sum_{j=n,p,e,\mu}(\nu_{ij}\tau_{i}+\nu^{\prime}_{ij}\tau_{j})=1~{},$ (6)
and the effective collision frequencies are
$\displaystyle\nu_{ij}=\frac{3\pi^{2}\hbar^{3}}{{2p_{F_{i}}^{5}{kT}m_{i}^{*}}(1+\delta_{ij})}{\sum_{{s_{i}},{s_{i^{\prime}}},{s_{j}},{s_{j^{\prime}}}}}\int\frac{d{\bf
p}_{i}d{\bf p}_{j}d{\bf p}_{i^{\prime}}d{\bf
p}_{j^{\prime}}}{(2\pi\hbar)^{12}}W_{ij}[f_{i}f_{j}(1-f_{i^{\prime}})(1-f_{j^{\prime}})]$
$\displaystyle\times[\frac{2}{3}p_{i}^{4}+\frac{1}{3}p_{i}^{2}p_{i^{\prime}}^{2}-({\bf
p}_{i}\cdot{\bf p}_{i^{\prime}})^{2}]~{},$ (7)
$\displaystyle\nu^{\prime}_{ij}=\frac{3\pi^{2}\hbar^{3}}{{2p_{F_{i}}^{5}{kT}m_{j}^{*}}(1+\delta_{ij})}{\sum_{{s_{i}},{s_{i^{\prime}}},{s_{j}},{s_{j^{\prime}}}}}\int\frac{d{\bf
p}_{i}d{\bf p}_{j}d{\bf p}_{i^{\prime}}d{\bf
p}_{j^{\prime}}}{(2\pi\hbar)^{12}}W_{ij}[f_{i}f_{j}(1-f_{i^{\prime}})(1-f_{j^{\prime}})]$
$\displaystyle\times[\frac{1}{3}p_{i}^{2}p_{j^{\prime}}^{2}-\frac{1}{3}p_{i}^{2}p_{j}^{2}+({\bf
p}_{i}\cdot{\bf p}_{j})^{2}-({\bf p}_{i}\cdot{\bf p}_{j^{\prime}})^{2}]~{}.$
(8)
The differential transition rate is given by
${\sum_{{s_{i}},{s_{i^{\prime}}},{s_{j}},{s_{j^{\prime}}}}}W_{ij}={4(2\pi)^{4}\hbar^{2}}\delta(\epsilon_{i}+\epsilon_{j}-\epsilon_{i^{\prime}}-\epsilon_{j^{\prime}})\delta({\bf
p}_{i}+{\bf p}_{j}-{\bf p}_{i^{\prime}}-{\bf p}_{j^{\prime}}){\cal
Q}_{ij}~{},$ (9)
where ${\cal Q}_{ij}=<|{\cal M}_{ij}|^{2}>$ is the squared matrix element
summed over final spins and averaged over initial spins Yak ; Yak2 ; Bai .
We obtain effective relaxation times for different particle species solving a
matrix equation that follows from Eq.(6). The matrix equation has the
following form:
$\left(\begin{array}[]{llll}\nu_{e}&\nu^{\prime}_{e\mu}&\nu^{\prime}_{ep}&0\\\
\nu^{\prime}_{{\mu}e}&\nu_{{\mu}}&\nu^{\prime}_{{\mu}p}&0\\\
\nu^{\prime}_{pe}&\nu^{\prime}_{p{\mu}}&\nu_{p}&\nu^{\prime}_{pn}\\\
0&0&\nu^{\prime}_{np}&\nu_{n}\end{array}\right)\left(\begin{array}[]{c}\tau_{e}\\\
\tau_{\mu}\\\ \tau_{p}\\\ \tau_{n}\end{array}\right)=1$ (10)
where,
$\displaystyle\nu_{e}$ $\displaystyle=$
$\displaystyle\nu_{ee}+\nu^{\prime}_{ee}+\nu_{e\mu}+\nu_{ep}~{},$ (11)
$\displaystyle\nu_{\mu}$ $\displaystyle=$
$\displaystyle\nu_{\mu\mu}+\nu^{\prime}_{\mu\mu}+\nu_{{\mu}e}+\nu_{{\mu}p}~{},$
(12) $\displaystyle\nu_{p}$ $\displaystyle=$
$\displaystyle\nu_{pp}+\nu^{\prime}_{pp}+\nu_{pn}+\nu_{pe}+\nu_{p{\mu}}~{},$
(13) $\displaystyle\nu_{n}$ $\displaystyle=$
$\displaystyle\nu_{nn}+\nu^{\prime}_{nn}+\nu_{np}~{}.$ (14)
It is to be noted here that the proton-proton interaction is made up of
contributions from electromagnetic and strong interactions. As there is no
interference of the electromagnetic and strong interaction terms, the
differential transition rate for the proton-proton scattering is the sum of
electromagnetic and strong contributions. This was discussed earlier in
Ref.Flo2 . Therefore, we can write the strong and electromagnetic parts of the
effective collision frequencies of proton-proton scattering as
$\displaystyle\nu_{pp}$ $\displaystyle=$
$\displaystyle\nu_{pp}^{s}+\nu_{pp}^{em}~{},$
$\displaystyle\nu{{}^{\prime}}_{pp}$ $\displaystyle=$
$\displaystyle\nu{{}^{\prime}}_{pp}^{s}+\nu{{}^{\prime}}_{pp}^{em}~{}.$ (15)
Here the superscripts ’$em$’ and ’$s$’ denote the electromagnetic and strong
interactions. Solutions of Eq. (10) are given below
$\displaystyle\tau_{e}=\frac{(\nu_{p}\nu_{n}-\nu^{\prime}_{pn}\nu^{\prime}_{np})(\nu_{\mu}-\nu^{\prime}_{e{\mu}})+(\nu^{\prime}_{pn}-\nu_{n})(\nu_{\mu}\nu^{\prime}_{ep}-\nu^{\prime}_{e\mu}\nu^{\prime}_{{\mu}p})+\nu_{n}\nu^{\prime}_{p\mu}(\nu^{\prime}_{ep}-\nu^{\prime}_{{\mu}p})}{detA}~{},$
$\displaystyle\tau_{p}=\frac{(\nu_{n}-\nu^{\prime}_{pn})(\nu_{e}\nu_{\mu}-\nu^{\prime}_{e\mu}\nu^{\prime}_{{\mu}e})+\nu^{\prime}_{p\mu}\nu_{n}(\nu^{\prime}_{e{\mu}}-\nu_{e})+\nu^{\prime}_{pe}\nu_{n}(\nu^{\prime}_{e{\mu}}-\nu_{{\mu}})}{detA}~{},$
$\displaystyle\tau_{n}=\frac{(\nu_{p}-\nu^{\prime}_{np})(\nu_{e}\nu_{\mu}-\nu^{\prime}_{e\mu}\nu^{\prime}_{{\mu}e})+(\nu^{\prime}_{np}-\nu^{\prime}_{{\mu}p})(\nu^{\prime}_{p{\mu}}\nu_{e}-\nu^{\prime}_{e\mu}\nu^{\prime}_{p{\mu}})+(\nu^{\prime}_{ep}-\nu^{\prime}_{np})(\nu^{\prime}_{{\mu}e}\nu^{\prime}_{p\mu}-\nu^{\prime}_{pe}\nu_{{\mu}})}{detA}~{}.$
(16)
where $A$ is the $4\times 4$ matrix of Eq. (10) and
$detA=[\nu_{e}\nu_{\mu}(\nu_{p}\nu_{n}-\nu^{\prime}_{pn}\nu^{\prime}_{np})-\nu_{e}\nu^{\prime}_{{\mu}p}\nu^{\prime}_{p\mu}\nu_{n}-\nu^{\prime}_{e\mu}\nu^{\prime}_{{\mu}e}(\nu_{p}\nu_{n}-\nu^{\prime}_{pn}\nu^{\prime}_{np})-\nu^{\prime}_{e\mu}\nu^{\prime}_{{\mu}p}\nu^{\prime}_{pe}\nu_{n}+\nu^{\prime}_{ep}\nu^{\prime}_{{\mu}e}\nu^{\prime}_{p\mu}\nu_{n}-\nu^{\prime}_{ep}\nu_{\mu}\nu^{\prime}_{pe}\nu_{n}$].
We obtain $\tau_{\mu}$ from $\tau_{e}$ replacing $e$ by $\mu$. In the next
paragraphs, we discuss the determination of matrix element squared for
electromagnetic and strong interactions.
First we focus on the electromagnetic scattering of charged particles. Here we
adopt the plasma screening of the interaction due to the exchange of
longitudinal and transverse plasmons as described in Refs.Yak ; Yak2 ; Hei .
The matrix element for the collision of identical charged particles is given
by $M_{12}=M^{(1)}_{12}+M^{(2)}_{12}$, where the first term implies the
scattering channel $12\rightarrow 1^{\prime}2^{\prime}$ and the second term
corresponds to that of $12\rightarrow 2^{\prime}1^{\prime}$. The scattering of
charged particles in neutron star interiors involves small momentum and energy
transfers. Consequently both channels contribute equally because the
interference term is small in this case. The matrix element for nonidentical
particles is given by Yak ; Yak2 ; Hei
$M_{12\rightarrow 1^{\prime}2^{\prime}}=\frac{4\pi
e^{2}}{c^{2}}\left(\frac{J_{0}^{11^{\prime}}J_{0}^{22^{\prime}}}{q^{2}+\Pi_{l}^{2}}-\frac{{\bf
J}_{t}^{11^{\prime}}\cdot{\bf
J}_{t}^{22^{\prime}}}{q^{2}-\omega^{2}/c^{2}+\Pi_{t}^{2}}\right)~{},$ (17)
where $\bf q$ and $\omega$ are momentum and energy transfers in the neutron
star interior. Further four-current $(J_{0},{\bf J})$ and longitudinal and
transverse polarization functions $(\Pi_{l},\Pi_{t})$ have the same
expressions as defined in Ref.Yak . It is to be noted that ${\bf J}_{t}$ is
the transverse component of $\bf J$ with respect to $\bf q$ and the
longitudinal component is related to the timelike component $J_{0}$ by the
conservation of current Hei . Polarization functions $\Pi_{l}$ and $\Pi_{t}$
are associated with the plasma screening of charged particles’ interactions
through the exchange of longitudinal and transverse plasmons, respectively.
After evaluating the matrix element squared and doing the angular and energy
integrations, the effective collision frequencies are calculated following the
prescription of Ref.Yak ; Yak2 . The collision frequencies of Eqs. (7) and (8)
for charged particles become
$\displaystyle\nu_{ij}$ $\displaystyle=$
$\displaystyle\nu_{ij}^{||}+\nu_{ij}^{\perp}~{},$
$\displaystyle\nu^{\prime}_{ij}$ $\displaystyle=$
$\displaystyle{\nu^{\prime}}_{ij}^{||\perp}~{},$ (18)
where $\nu_{ij}^{||}$ and $\nu_{ij}^{\perp}$ correspond to the charged
particle interaction due to the exchange of longitudinal and transverse
plasmons and $\nu_{ij}^{||\perp}$ is the result of the interference of both
interactions. For small momentum and energy transfers, different components of
the collision frequency are given by Yak ; Yak2
$\displaystyle\nu_{ij}^{\perp}$ $\displaystyle=$
$\displaystyle\frac{e^{4}\alpha}{\hbar^{4}c^{3}}\frac{p_{F_{j}}^{2}}{p_{F_{i}}m_{i}^{*}c}\left({\frac{\hbar
c}{q_{t}^{2}}}\right)^{1/3}(kT)^{5/3},$ $\displaystyle\nu_{ij}^{||}$
$\displaystyle=$
$\displaystyle\frac{e^{4}\pi^{2}m_{i}^{*}m_{j}^{*2}}{\hbar^{4}p_{F_{i}}^{3}q_{l}}(kT)^{2}~{},$
$\displaystyle\nu{{}^{\prime}}_{ij}^{||\perp}$ $\displaystyle=$
$\displaystyle\frac{2e^{4}\pi^{2}m_{i}^{*}p_{F_{j}}^{2}}{\hbar^{4}c^{2}p_{F_{i}}^{3}q_{l}}(kT)^{2}~{},$
(19)
where $i,j=e,\mu,p$ and longitudinal and transverse wave numbers are given by
$\displaystyle q_{l}^{2}$ $\displaystyle=$
$\displaystyle\frac{4e^{2}}{\hbar^{3}c\pi}\sum_{j=e,\mu,p}{cm_{j}^{*}p_{F_{j}}}~{},$
$\displaystyle q_{t}^{2}$ $\displaystyle=$
$\displaystyle\frac{4e^{2}}{\hbar^{3}c\pi}\sum_{j=e,\mu,p}p_{F_{j}}^{2}~{}.$
(20)
The value of $\alpha=2({\frac{4}{\pi}})^{1/3}\Gamma(8/3)\zeta(5/3)\sim 6.93$
where $\Gamma(x)$ and $\zeta(x)$ are gamma and Riemann zeta functions,
respectively. The shear viscosities of electrons and muons are given by Yak
$\eta_{i(=e,\mu)}=\frac{n_{i}p_{F_{i}}^{2}\tau_{i}}{5m_{i}^{*}}~{}.$ (21)
Here effective masses ($m_{i}^{*}$) of electrons and muons are equal to their
corresponding chemical potentials because of relativistic effects. It was
noted that the shear viscosity was reduced due to the inclusion of plasma
screening by the exchange of transverse plasmons Yak ; Yak2 . It is worth
mentioning here that we extend the calculation of the collision frequencies
for electrons and muons in Refs.Yak ; Yak2 to that of protons due to
electromagnetic interaction. Before the appearance of the condensate in our
calculation, protons may be treated as passive scatterers as was earlier done
by Ref.Yak . However, after the onset of the antikaon condensation, electrons
and muons are replaced by $K^{-}$ mesons and proton fraction increases rapidly
in the system Pal ; Banik . In this situation protons can not be treated as
passive scatterers.
Next we focus on the calculation of collision frequencies of neutron-neutron,
proton-proton and neutron-proton scatterings due to the strong interaction.
The knowledge of nucleon-nucleon scattering cross sections are exploited in
this calculation. This was first done by Ref.Flo2 . Later recent developments
in the calculation of nucleon-nucleon scattering cross sections in the Dirac-
Brueckner approach were considered for this purpose Yak2 ; Bai . Here we adopt
the same prescription of Ref.Bai for the calculation of collision frequencies
due to nucleon-nucleon scatterings. The collision frequency for the scattering
of identical particles under strong interaction is given by
$\nu_{ii}+\nu{{}^{\prime}}_{ii}=\frac{16m_{i}^{*3}(kT)^{2}}{3m_{n}^{2}{\hbar}^{3}}S_{ii}~{},$
(22)
$S_{ii}=\frac{m_{n}^{2}}{16{\hbar}^{4}{\pi}^{2}}\int_{0}^{1}dx^{\prime}\int_{0}^{\sqrt{(1-x^{\prime
2})}}dx\frac{12{x^{2}}{x^{\prime 2}}}{\sqrt{1-x^{2}-x^{\prime
2}}}{\cal{Q}}_{ii}~{},$ (23)
where $i=n,p$ and $m_{n}$ is the bare nucleon mass and ${\cal Q}_{ii}$ is the
matrix element squared which appears in Eq. (9). Similarly we can write the
collision frequency for nonidentical particles as
$\displaystyle\nu_{ij}=\frac{32m_{i}^{*}m_{j}^{*2}(kT)^{2}}{3m_{n}^{2}{\hbar}^{3}}S_{ij}~{},$
$\displaystyle\nu{{}^{\prime}}_{ij}=\frac{32m_{i}^{*2}m_{j}^{*}(kT)^{2}}{3m_{n}^{2}{\hbar}^{3}}S^{\prime}_{ij}~{},$
(24)
and
$\displaystyle
S_{ij}=\frac{m_{n}^{2}}{16{\hbar}^{4}{\pi}^{2}}\int_{0.5-x_{0}}^{0.5+x_{0}}dx^{\prime}\int_{0}^{f}dx\frac{6(x^{2}-x^{4})}{\sqrt{(f^{2}-x^{2})}}{\cal{Q}}_{ij}~{},$
$\displaystyle
S^{\prime}_{ij}=\frac{m_{n}^{2}}{16{\hbar}^{4}{\pi}^{2}}\int_{0.5-x_{0}}^{0.5+x_{0}}dx^{\prime}\int_{0}^{f}dx\frac{[6{x^{4}}+12{x^{2}}{x^{\prime
2}}-(3+12x_{0}^{2})x^{2}]}{\sqrt{(f^{2}-x^{2})}}{\cal{Q}}_{ij}~{}.$ (25)
We define $x_{0}=\frac{p_{F_{j}}}{2p_{F_{i}}}$,
$x=\frac{{\hbar}q}{2p_{F_{i}}}$,
$x^{\prime}=\frac{{\hbar}q^{\prime}}{2p_{F_{i}}}$,
$f=\frac{\sqrt{x_{0}^{2}-(0.25+x_{0}^{2}-x^{\prime 2})^{2}}}{x^{\prime}}$,
where momentum transfers ${\bf q}={\bf p}_{j^{\prime}}-{\bf p}_{j}$ and ${\bf
q}^{\prime}={\bf p}_{j^{\prime}}-{\bf p}_{i}$. We find that the calculation of
$S_{ij}$, $S_{ii}$ and $S^{\prime}_{ij}$ requires the knowledge of ${\cal
Q}_{ii}$ and ${\cal Q}_{ij}$. The matrix elements squared may be extracted
from nucleon-nucleon differential cross sections. A detailed discussion on the
calculation of matrix elements squared from the in-vacuum nucleon-nucleon
differential scattering cross sections computed using Dirac-Brueckner approach
Mach1 ; Mach2 can be found in Ref.Yak2 ; Bai . We follow this procedure in
this calculation. It is to be noted here that $S_{pp}$, $S_{pn}$ and
$S^{\prime}_{ij}$ are the new results of this calculation. As soon as we know
the collision frequencies of nucleon-nucleon scatterings due to the strong
interaction, we can immediately calculate effective relaxation times of
neutrons and protons from Eq. (16). This leads to the calculation of the
neutron and proton shear viscosities as
$\displaystyle\eta_{n}=\frac{n_{n}p_{F_{n}}^{2}\tau_{n}}{5m_{n}^{*}}~{},$
$\displaystyle\eta_{p}=\frac{n_{p}p_{F_{p}}^{2}\tau_{p}}{5m_{p}^{*}}~{}.$ (26)
Finally the total shear viscosity is given by
$\eta_{total}=\eta_{n}+\eta_{p}+\eta_{e}+\eta_{\mu}~{}.$ (27)
The EOS enters into the calculation of the shear viscosity as an input. We
construct the EOS within the framework of the relativistic field theoretical
model walecka ; serot . Here we consider a first order phase transition from
nuclear matter to $K^{-}$ condensed matter. We adopt the Maxwell construction
for the first order phase transition. The constituents of matter are neutrons,
protons, electrons and muons in both phases and also (anti)kaons in the
$K^{-}$ condensed phase. Both phases maintain charge neutrality and $\beta$
equilibrium conditions. Baryons and (anti)kaons are interacting with each
other and among themselves by the exchange of $\sigma$, $\omega$ and $\rho$
mesons Pal ; Banik . The baryon-baryon interaction is given by the Lagrangian
density glendenning ; schaffnerprc
$\displaystyle{\cal L}_{B}$ $\displaystyle=$
$\displaystyle\sum_{B=n,p}\bar{\psi}_{B}\left(i\gamma_{\mu}{\partial}^{\mu}-m_{B}+g_{\sigma
B}\sigma-g_{\omega B}\gamma_{\mu}\omega^{\mu}-g_{\rho
B}\gamma_{\mu}{\mbox{\boldmath
t}}_{B}\cdot{\mbox{\boldmath$\rho$}}^{\mu}\right)\psi_{B}$ (28)
$\displaystyle+\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2}\right)-U(\sigma)$
$\displaystyle-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}{\mbox{\boldmath$\rho$}}_{\mu\nu}\cdot{\mbox{\boldmath$\rho$}}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}{\mbox{\boldmath$\rho$}}_{\mu}\cdot{\mbox{\boldmath$\rho$}}^{\mu}~{}.$
The scalar self-interaction schaffnerprc ; glendenning ; boguta is
$U(\sigma)~{}=~{}\frac{1}{3}~{}g_{1}~{}m_{N}~{}(g_{\sigma
N}\sigma)^{3}~{}+~{}\frac{1}{4}~{}g_{2}~{}(g_{\sigma N}\sigma)^{4}~{},$ (29)
The effective nucleon mass is given by $m_{B}^{*}=m_{B}-g_{\sigma B}\sigma$,
where $m_{B}$ is the vacuum baryon mass. The Lagrangian density for
(anti)kaons in the minimal coupling is given by Pal ; Banik ; Gle99
${\cal L}_{K}=D^{*}_{\mu}{\bar{K}}D^{\mu}K-m_{K}^{*2}{\bar{K}}K~{},$ (30)
where the covariant derivative is $D_{\mu}=\partial_{\mu}+ig_{\omega
K}{\omega_{\mu}}+ig_{\rho K}{\mbox{\boldmath
t}}_{K}\cdot{\mbox{\boldmath$\rho$}}_{\mu}$ and the effective mass of
(anti)kaons is $m_{K}^{*}=m_{K}-g_{\sigma K}\sigma$. The in-medium energies of
$K^{\pm}$ mesons are given by
$\omega_{K^{\pm}}=\sqrt{(p^{2}+m_{K}^{*2})}\pm\left(g_{\omega
K}\omega_{0}+\frac{1}{2}g_{\rho K}\rho_{03}\right)~{}.$ (31)
The condensation sets in when the chemical potential of $K^{-}$ mesons
($\mu_{K^{-}}=\omega_{K^{-}}$) is equal to the electron chemical potential
i.e. $\mu_{e}=\mu_{K^{-}}$.
Using the mean field approximation walecka ; serot and solving equations of
motion self-consistently, we calculate the effective nucleon mass and Fermi
momenta of particles at different baryon densities.
## III Results and Discussions
The knowledge of meson-nucleon and meson-kaon coupling constants are needed
for this calculation. The nucleon-meson coupling constants determined by
reproducing the nuclear matter saturation properties such as binding energy
$E/B=-16.3$ MeV, baryon density $n_{0}=0.153$ fm-3, the asymmetry energy
coefficient $a_{\rm asy}=32.5$ MeV, incompressibility $K=300$ MeV ,and
effective nucleon mass $m^{*}_{N}/m_{N}=0.70$, are taken from Ref.Gle91 . Next
we determine the kaon-meson coupling constants using the quark model and
isospin counting rule. The vector coupling constants are given by
$g_{\omega K}=\frac{1}{3}g_{\omega N}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}g_{\rho K}=g_{\rho N}~{}.$ (32)
The scalar coupling constant is obtained from the real part of $K^{-}$ optical
potential depth at normal nuclear matter density
$U_{\bar{K}}\left(n_{0}\right)=-g_{\sigma K}\sigma-g_{\omega K}\omega_{0}~{}.$
(33)
It is known that antikaons experience an attractive potential and kaons have a
repulsive interaction in nuclear matter Fri94 ; Fri99 ; Koc ; Waa ; Li ; Pal2
. On the one hand, the analysis of $K^{-}$ atomic data indicated that the real
part of the antikaon optical potential could be as large as
$U_{\bar{K}}=-180\pm 20$ MeV at normal nuclear matter density Fri94 ; Fri99 .
On the other hand, chirally motivated coupled channel models with a self-
consistency requirement predicted shallow potential depths of $-40$-$60$ MeV
Ram ; Koch . Recently, the double pole structure of $\Lambda(1405)$ was
investigated in connection with the antikaon-nucleon interaction Mag ; Hyo .
Further, the highly attractive potential depth of several hundred MeV was
obtained in the calculation of deeply bound antikaon-nuclear states Yam ; Akai
. An alternative explanation to the deeply bound antikaon-nuclear states was
given by others Toki . This shows that the value of antikaon optical potential
depth is still a debatable issue. Motivated by the findings of the analysis of
$K^{-}$ atomic data, we perform this calculation for an antikaon optical
potential depth $U_{\bar{K}}=-160$ MeV at normal nuclear matter density. We
obtain kaon-scalar meson coupling constant $g_{\sigma K}=2.9937$ corresponding
to $U_{\bar{K}}(n_{0})=-160$ MeV.
The composition of neutron star matter including the $K^{-}$ condensate as a
function of normalised baryon density is shown in Fig. 1. The $K^{-}$
condensation sets in at 2.43$n_{0}$. Before the onset of the condensation, all
particle fractions increase with baryon density. In this case, the charge
neutrality is maintained by protons, electrons and muons. As soon as the
antikaon condensate is formed, the density of $K^{-}$ mesons in the condensate
rapidly increases and $K^{-}$ mesons replace leptons in the system. The proton
density eventually becomes equal to the $K^{-}$ density. The proton density in
the presence of the condensate increases significantly and may be higher than
the neutron density at higher baryon densities Pal2 . This increase in the
proton fraction in the presence of the $K^{-}$ condensate might result in an
enhancement in the proton shear viscosity and appreciable reduction in the
electron and muon viscosities compared with the case without the condensate.
We discuss this in details in the following paragraphs.
Next we focus on the calculation of $\nu_{ii}$, $\nu_{ij}$ and
$\nu^{\prime}_{ij}$. For the scatterings via the electromagnetic interaction,
we calculate those quantities using Eqs. (18) and (19). On the other hand,
$\nu$s corresponding to collisions through the strong interaction are
estimated using Eqs. (22)-(25). In an earlier calculation, the authors
considered only $S_{nn}$ and $S_{np}$ Yak for the calculation of the neutron
shear viscosity in nucleons-only neutron star matter because protons were
treated as passive scatterers. It follows from the discussion in the preceding
paragraph that protons can no longer be treated as passive scatterers because
of the large proton fraction in the presence of the $K^{-}$ condensate.
Consequently the contributions of $S_{pp}$ and $S_{pn}$ have to be taken into
account in the calculation of the proton and neutron shear viscosities. The
expressions of $S_{nn}$, $S_{pp}$, $S_{np}$ and $S_{pn}$ given by Eqs. (23)
and (25) involve matrix elements squared. We note that there is an one to one
correspondence between the differential cross section and the matrix element
squared Bai . We exploit the in-vacuum nucleon-nucleon cross sections of Li
and Machleidt Mach1 ; Mach2 calculated using Bonn interaction in the Dirac-
Brueckner approach for the calculation of matrix elements squared. We fit the
neutron-proton as well as proton-proton differential cross sections and use
them in Eqs. (23) and (25) to calculate $S_{nn}$, $S_{pp}$, $S_{np}$ and
$S_{pn}$ which are functions of neutron ($p_{F_{n}}$) and proton ($p_{F_{p}}$)
Fermi momenta. The values of $p_{F_{n}}$ ranges from 1.3 to 2.03 $fm^{-1}$
whereas that of $p_{F_{p}}$ spans the interval 0.35 to 1.73 $fm^{-1}$. This
corresponds to the density range $\sim$0.5 to $\sim$ 3.0$n_{0}$. We fit the
results of our calculation. Figures 2 and 3 display the variation of $S_{nn}$,
$S_{pp}$, $S_{np}$ and $S_{pn}$ with baryon density. The value of $S_{nn}$ is
greater than that of $S_{np}$ in the absence of the condensate as evident from
Fig. 2. Our results agree well with those of Ref.Yak . However, $S_{np}$ rises
rapidly with baryon density after the onset of the $K^{-}$ condensation and
becomes higher than $S_{nn}$. It is noted that the effect of the condensate on
$S_{nn}$ is not significant. Figure 3 shows that $S_{pp}$ drops sharply with
increasing baryon density and crosses the curve of $S_{pn}$ in the absence of
the condensate. However $S_{pp}$ and $S_{pn}$ are not influenced by the
antikaon condensate. A comparison of Fig. 2 and Fig. 3 reveals that $S_{pp}$
is almost one order of magnitude larger than $S_{nn}$ at lower baryon
densities. This may be attributed to the smaller proton Fermi momentum. We
also compute $S^{\prime}_{np}$ and $S^{\prime}_{pn}$ (not shown here) and
these quantities have negative values. Further we find that the magnitude of
$S^{\prime}_{pn}$ is higher than that of $S^{\prime}_{np}$. It is to be noted
here that $S^{\prime}_{ij}$ is related to ${\nu}^{\prime}_{ij}$ by Eq. (24).
This is again connected to Eq. (6). Therefore, the values of $S^{\prime}_{ij}$
and ${\nu}^{\prime}_{ij}$ can be made positive by putting a negative sign
between two terms in Eq. (6).
As soon as we know $\nu$s, we can calculate effective relaxation times using
Eq. (16) and shear viscosities using Eqs. (21), (26) and (27). First, we
discuss the total shear viscosity in nuclear matter without a $K^{-}$
condensate. This is shown as a function of baryon density at a temperature
$10^{8}$K in Fig. 4. Here our results indicated by the solid line are compared
with the calculation of the total shear viscosity using the EOS of Akmal,
Pandharipande and Ravenhall (APR) APR denoted by the dotted line and also
with the results of Flowers and Itoh Flo1 ; Flo2 . For the APR case, we
exploit the parametrization of the EOS by Heiselberg and Hjorth-Jensen HHJ .
Further we take density independent nucleon effective masses
$m_{n}^{*}=m_{p}^{*}=0.8m_{n}$ for the calculation with the APR EOS which was
earlier discussed by Shternin and Yakovlev Yak2 . On the other hand, the
results of Flowers and Itoh were parametrized by Cutler and Lindblom (CL) Cut
and it is shown by the dashed line in Fig. 4. It is evident from Fig. 4 that
the total shear viscosity in our calculation is significantly higher than
other cases. This may be attributed to the fact that our EOS is a fully
relativistic one.
We exhibit shear viscosities in the presence of an antikaon condensate as a
function of baryon density in Fig. 5. This calculation is performed at a
temperature $10^{8}$K. In the absence of the $K^{-}$ condensate, the
contribution of the electron shear viscosity to the total shear viscosity is
the highest. The electron, muon and neutron shear viscosities exceed the
proton shear viscosity by several orders of magnitude. Further we note that
the lepton viscosities are greater than the neutron viscosity. On the other
hand, we find interesting results in the presence of the antikaon condensate.
The electron and muon shear viscosities decrease very fast after the onset of
$K^{-}$ condensation whereas the proton shear viscosity rises in this case.
There is almost no change in the neutron shear viscosity. It is interesting to
note that the proton shear viscosity in the presence of the condensate
approaches the value of the neutron shear viscosity as baryon density
increases. The total shear viscosity decreases in the $K^{-}$ condensed matter
due to the sharp drop in the lepton shear viscosities. Here the variation of
shear viscosities with baryon density is shown up to 3$n_{0}$. The neutron and
proton shear viscosities in neutron star matter with the $K^{-}$ condensate
might dominate over the electron and muon shear viscosities beyond baryon
density 3$n_{0}$. Consequently, the total shear viscosity would again
increase.
The temperature dependence of the total shear viscosity is shown in Fig. 6. In
an earlier calculation, electron and muon shear viscosities were determined by
collisions only due to the exchange of transverse plasmons because this was
the dominant contribution Yak2 . Under this approximation, the electron and
muon shear viscosities had a temperature dependence of $T^{-5/3}$, whereas,
the neutron shear viscosity was proportional to $T^{-2}$. The temperature
dependence of the electron and muon shear viscosities deviated from the
standard temperature dependence of the shear viscosity of neutron Fermi
liquid. However, in this calculation we have not made any such approximation.
We have considered all the components of effective collision frequency which
have different temperature dependence as given by Eq. (19). This gives rise to
a complicated temperature dependence in the calculation of shear viscosity.
The total shear viscosity is plotted for $T=10^{7}$, $10^{8}$, and $10^{9}$ K
in Fig. 6. It is noted that the shear viscosity increases as temperature
decreases.
The shear viscosity plays an important role in damping the r-mode instability
in old and accreting neutron stars Nay ; Chat1 ; Chat2 ; Chat3 ; Chat4 . The
suppression of the instability is achieved by the competition of various time
scales associated with gravitational radiation ($\tau_{GR}$), hyperon bulk
viscosity ($\tau_{B}$), modified Urca bulk viscosity ($\tau_{U}$), and shear
viscosity ($\tau_{SV}$). At high temperatures the bulk viscosity damp the
r-mode instability. As neutron stars cool down, the bulk viscosity might not
be the dominant damping mechanism. The shear viscosity becomes significant in
the temperature regime $\leq 10^{8}$K and might suppress the r-mode
instability effectively.
In this calculation, we consider only the antikaon optical potential depth
$U_{\bar{K}}=-160$ MeV. However, this calculation could be performed for other
values of antikaon optical potential depths. As the magnitude of the $K^{-}$
potential depth decreases, the threshold of the antikaon condensation is
shifted to higher densities Pal . On the other hand, hyperons may also appear
in neutron star matter around 2-3$n_{0}$. Negatively charged hyperons might
delay the onset of $K^{-}$ condensation schaffnerprc ; Ell ; Kno . However, it
was noted in an earlier calculation that $\Sigma^{-}$ hyperons were excluded
from the system because of repulsive $\Sigma$-nuclear matter interaction and
$\Xi^{-}$ hyperons might appear at very high baryon density Banik . However,
the appearance of $\Lambda$ hyperons could compete with the threshold of
$K^{-}$ condensation. If $\Lambda$ hyperons appear before $K^{-}$
condensation, the threshold of $K^{-}$ condensation is shifted to higher
baryon density because of softening in the equation of state due to $\Lambda$
hyperons. But the qualitative results of the shear viscosity discussed above
remain the same.
## IV Summary and Conclusions
We have investigated the shear viscosity in the presence of a $K^{-}$
condensate. With the onset of $K^{-}$ condensation, electrons and muons are
replaced by $K^{-}$ mesons rapidly. The proton fraction also increases and
eventually becomes equal to the neutron fraction in the $K^{-}$ condensed
neutron star matter. This has important consequences for the electron, muon
and proton shear viscosities. We have found that the electron and muon shear
viscosities drop steeply after the formation of the $K^{-}$ condensate in
neutron stars. On the other hand, the proton shear viscosity whose
contribution to the total shear viscosity was negligible in earlier
calculations Flo2 ; Yak , now becomes significant in the presence of the
$K^{-}$ condensate. The proton shear viscosity would exceed the neutron as
well as lepton shear viscosities beyond 3$n_{0}$. The total viscosity would be
dominated by the proton and neutron shear viscosities in this case. This
calculation may be extended to neutron stars with strong magnetic fields.
It is worth mentioning here that we adopt the Maxwell construction for the
first order phase transition in this calculation. Such a construction is
justified if the surface tension between two phases is quite large Alf .
Moreover the value of the surface tension between the nuclear and antikaon
condensed phases or between the hadron and quark phases is not known
correctly. Therefore, this problem could also be studied using the Gibbs
construction NKG .
Besides the role of shear viscosity in damping the r-mode instability as well
as in pulsar glitches and free precession of neutron stars, it has an
important contribution in the nucleation rate of bubbles in first order phase
transitions. It was shown earlier that the shear viscosity might control the
initial growth rate of a bubble Kap ; Bom . This needs further study in
connection with antikaon condensation in neutron stars.
## V Acknowledgments
We thank R. Machleidt for providing us with the tables of neutron-proton and
proton-proton differential scattering cross sections. RN and DB thank the
Alexander von Humboldt Foundation for the support under the Research Group
Linkage programme. We also acknowledge the warm hospitality at the Frankfurt
Institute for Advanced Studies where a part of this work was completed.
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Fig. 1. Number densities of different particle species as a function of
normalised baryon density.
Fig. 2. $S_{nn}$ and $S_{np}$ are plotted as a function of normalised baryon
density.
Fig. 3. $S_{pp}$ and $S_{pn}$ are plotted as a function of normalised baryon
density.
Fig. 4. Total shear viscosities in nuclear matter without an antikaon
condensate corresponding to this work (solid line), the parameterization of
Cutler and Lindblom (dashed line) and the EOS of Akmal, Pandharipande and
Ravenhall are shown as a function of normalised baryon density at a
temperature $T=10^{8}$ K.
Fig. 5. The total shear viscosity as well as shear viscosities corresponding
to different particle species are shown as a function of normalised baryon
density at a temperature $T=10^{8}$ K with (solid line) and without (dashed
line) a $K^{-}$ condensate.
Fig. 6. The total shear viscosity as a function of normalised baryon density
at different temperatures with (solid line) and without (dashed line) a
$K^{-}$ condensate.
|
arxiv-papers
| 2009-12-21T14:22:01 |
2024-09-04T02:49:07.164445
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rana Nandi, Sarmistha Banik, and Debades Bandyopadhyay",
"submitter": "Debades Bandyopadhyay",
"url": "https://arxiv.org/abs/0912.4175"
}
|
0912.4184
|
11institutetext: State Key Laboratory of Novel Software Technology
Dept. of Computer Sci. and Tech. Nanjing University
Nanjing, Jiangsu, P.R.China 210093
zhaojh@nju.edu.cn
# Scope Logic: Extending Hoare Logic for Pointer Program Verification
††thanks: This paper is supported by the Chinese HighTech project, grant no.
2009AA01Z148; and the HGJ project, grant no. 2009ZX01036-001-001
ZHAO Jianhua LI Xuandong
###### Abstract
This paper presents an extension to Hoare logic for pointer program
verification. First, the Logic for Partial Function (LPF) used by VDM is
extended to specify memory access using pointers and memory layout of
composite types. Then, the concepts of data-retrieve functions ( DRF ) and
memory-scope functions (MSF) are introduced in this paper. People can define
DRFs to retrieve abstract values from interconnected concrete data objects.
The definition of the corresponding MSF of a DRF can be derived syntactically
from the definition of the DRF. This MSF computes the set of memory units
accessed when the DRF retrieves an abstract value. This memory unit set is
called the memory scope of the abstract value. Finally, the proof rule of
assignment statements in Hoare’s logic is modified to deal with pointers. The
basic idea is that a virtual value keeps unmodified as long as no memory unit
in its scope is over-written. Another proof rule is added for memory
allocation statements. The consequence rule and the rules for control-flow
statements are slightly modified. They are essentially same as their original
version in Hoare logic.
An example is presented to show the efficacy of this logic. We also give some
heuristics on how to verify pointer programs.
## 1 Introduction
To reasoning the correctness of programs, C.A.R. Hoare presented an axiomatic
system for specifying and verifying programs[1][2]. However, this logic can
not deal with pointer programs because of pointer alias, i.e. many pointers
may refer to the same location. A few extensions to Hoare logic have been made
to deal with pointers or shared mutable data structures [3][4][5]. Among them,
separation logic [6] is one of the most successful extensions. That logic uses
a memory model which consists of two parts: the stack and the heap. Pointers
can only refer to data objects in the heap. Separation logic extends the
predicate calculus with the separation operator, which can separate the heap
into different disjoint parts. Then the Hoare logic is extended with a set of
proof rules for heap lookup, heap mutation and variable assignment. Though a
few programs have been used to demonstrate the potential of local reasoning
for scalability[7], verifying programs using separation logic is still very
difficult.
This paper presents an extension to Hoare logic for verification of pointer
programs. This logic uses an extension of the Logic for Partial Functions
(LPF) [8] to describe pre- and post-conditions of code fragments. Three type
constructors are introduced to construct composite types and pointer types
used in programs. Several kinds of function symbols associated with these
types, together with a set of proof rules, are introduced to model and specify
the memory layout/access in pointer programs.
In this logic, people can define recursive functions to retrieve abstract
values from interconnected concrete data objects. These functions are called
data-retrieve functions (DRFs). DRFs are recursively defined based on basic
function symbols and the memory access/layout function symbols. For each DRF
$f$, there is a memory-scope function $\mathfrak{M}(f)$ of which the
definition can be constructed syntactically from the definition of $f$. If an
application of $f$ results in an abstract value, then an application of
$\mathfrak{M}(f)$ to same arguments results in the set of memory units
accessed during the application of $f$. During program executions, the
application of $f$ to same arguments results in same abstract value as long as
no memory unit in this set is modified.
In this logic, program specifications are of the form $\mathbb{P}\vdash
p\\{c\\}r$, where $\mathbb{P}$ is a set of LPF formulae (usually a set of
function definitions), $c$ is a fragment of code, and $q,r$ are the pre-
condition and post-condition respectively. Such a specification means that if
all the formulae in $\mathbb{P}$ hold for arbitrary program states, and $c$
starts its execution from a program state satisfying $q$; then the state must
satisfy $r$ when $c$ stops.
This paper is organized as follows. An extension to LPF is presented in
Section 2. To model memory access and layout in pointer programs, several
kinds of new function symbols and constants are introduced into LPF. A set of
proof rules are introduced to specify these function symbols and constants. In
Section 3, the concept ‘memory scope forms’ of terms and ‘memory scope
functions’ (MSFs) are introduced. A proof rule is introduced to specify how
definitions of MSFs can be constructed. A property about memory scope forms is
also given in this section. The syntax of a small program language is given in
Section 4. The semantic of this program language is also briefly described in
this section. The syntax and meaning of program specifications are given in
Section 5. The extension to Hoare logic is presented in Section 6. The proof
rule for assignment statements is modified to dealing with the pointer alias
problem. Another proof rule is introduced for memory allocation statements.
Section 7 presents a formal verification of the running example in this paper.
Section 8 gives some heuristics on program verifications using our logic.
Section 9 concludes this paper.
In Appendix 0.A, we verify another program which inserts a new node to a
binary search tree. In Appendix 0.B, we use a simplified version of the
Schorre-Waite algorithm to show that our logic can help people think about
program verification in different abstract levels.
### 1.1 Preliminary of the logic for partial functions
The logic for partial functions (LPF) used in Vienna Development Method (VDM)
can reason about undefinedness, (abstract) types, and recursive partial
function definitions. The syntax of LPF terms and formulae is briefly
described here. A term of LPF can be one of the following forms:
1. 1.
a variable symbol;
2. 2.
$f(e_{1},\dots,e_{n})$ if $f$ is a function symbol, $arity(f)=n$ and
$e_{1},\dots,e_{n}$ are terms,
3. 3.
$p\,?\,e_{1}:e_{2}$, where $p$ is a formula.
A formula of LPF can be one of the following forms:
1. 1.
a boolean-typed term,
2. 2.
$\circledast$; ($\circledast$ denotes the neither-true-nor-false value. It is
originally represented by the symbol $\ast$ in LPF papers, but $\ast$ is used
to denote the memory access function in this paper.)
3. 3.
$P(e_{1},\dots,e_{n})$, if $P$ is a predicate symbol and $arity(P)=n$, and
$e_{1},\dots,e_{n}$ are terms. In this paper, we view a predicate symbol as a
boolean-typed function symbol.
4. 4.
$e_{1}=e_{2}$, where $e_{1},e_{2}$ are terms,
5. 5.
$e:t$, where $e$ is a term and $t$ is a type symbol.
6. 6.
$\Delta A$, $\neg A$, $A_{1}\land A_{2}$ are formulae if $A,A_{1},A_{2}$ are
formulae.
7. 7.
$\forall x:t\cdot A$, where $x$ is a variable, $t$ is a type symbol, and $A$
is a formula.
8. 8.
$f(x_{1}:T_{1},\dots,x_{n}:T_{n})\triangleq e$, where $e$ is a term, and all
the free variables in $t$ are in the set $\\{x_{1},\dots,x_{n}\\}$.
For the proof rules, semantics and other detail information of LPF, we refer
readers to [8].
The LPF formulae used in our logic have a constraint: the logical connectives,
$\circledast$, $\Delta$ and quantifiers can not occur in a term. Specifically,
in a conditional form $p?e_{1}:e_{2}$, $p$ contains no logical connective and
quantifier. However, we can use some operators like cand, cor, $\dots$, in
terms. These operators can be defined using conditional forms. This constraint
makes it possible to define the memory scope form of terms.
## 2 The extension of the logic for partial functions
In this paper, LPF is extended to deal with issues about memory access/layout,
composite and pointer program types, data-retrieve functions and memory-scope
functions. Now we first extend LPF with program types and associated function
symbols.
### 2.1 Program types and associated function symbols
In LPF, a type can be either a basic type such as $\mathbf{integer}$ and
$\mathbf{boolean}$, or a type constructed using type constructors such as
SetOf, SeqOf and Map. However, the abstract types constructed using these type
constructors can not be used directly in imperative programs. To deal with
types appeared in programs, we introduce three new type constructors into LPF:
pointer (P), array (ARR), and record (REC). We call the types that can appear
in programs as _P-types_.
1. 1.
$\mathbf{integer}$ and $\mathbf{boolean}$ are P-types;
2. 2.
Let $t,t_{1},\dots,t_{k}$ be P-types, $n_{1},n_{2},\dots,n_{k}$ are $k$
different names, $c$ is an positive integer constant. $\textbf{P}(t)$,
$\textbf{ARR}(t,c)$, and
$\textbf{REC}((n_{1},t_{1})\times\dots\times(n_{k},t_{k}))$ are also P-types.
We allow a record type $t$ has one or more fields with type $\textbf{P}(t)$
such that we can deal with recursive data types in our program language. We
use Ptr as the super type of all pointer types $\textbf{P}(t)$, where $t$ is a
P-type. The abstract type constructors
$\textbf{Map},\textbf{SetOf},\textbf{SeqOf}$ can not be applied to composite
program types. However, these type constructors can be applied to pointer
types to form new abstract types. That is, we can get an abstract
$\textbf{SetOf}(\textbf{P}(t))$ for some P-type $t$, but can not get an
abstract type
$\textbf{SetOf}(\textbf{Rec}((n_{1},t_{1})\times\dots\times(n_{k},t_{k}))$.
The following constant and function symbols associated with P-types are
introduced.
1. 1.
A program can declare a finite set of program variables with P-types. For each
program variable $v$ declared with P-type $t$, $\&v$ is a constant with type
$\textbf{P}(t)$.
2. 2.
For each pointer type $t$, there is a $t$-typed constant $\textbf{nil}_{t}$.
The type subscript $t$ can be omitted if there is no ambiguity caused.
3. 3.
A partial function
$\ast:\textbf{Ptr}\rightarrow\textbf{Ptr}\cup\textbf{integer}\cup\textbf{boolean}$.
We write an application of $\ast$ to $e$ as $\ast e$. For a non-nil pointer
$r$ with type $\textbf{P}(t)$, where $t$ is integer, boolean or a pointer
type, $\ast r$ is a $t$-typed value. An application of this function symbol
models a memory unit access.
4. 4.
For each array type $t=\textbf{ARR}(t^{\prime},c)$, there is a partial
function
$\&[]_{t}:\textbf{P}(t)\times\textbf{integer}\rightarrow\textbf{P}(t^{\prime})$.
We write an application of such function as $\&e[i]_{t}$ instead of
$\&[]_{t}(e,i)$. The type subscripts can be omitted if there is no ambiguity
caused. These function symbols model the memory layout of array types.
Intuitively speaking, if $e$ is a non-nil reference to a $t$-typed data
object, $\&e[i]$ is the reference to the $i$th element. $\&e[i]$ is defined if
and only if $e\neq\textbf{nil}$ and $0\leq i<c$.
5. 5.
For each record type
$t=\textbf{REC}((n_{1},t_{1})\times\dots\times(n_{k},t_{k}))$ and a name
$n_{i}$ $(1\leq i\leq k)$, we have a partial function
$\&\\!\\!\rightarrow_{t}\\!\\!n_{i}:\textbf{P}(t)\rightarrow\textbf{P}(t_{i})$.
It is only undefined on the constant $\textbf{nil}_{t}$. We write an
application of this function symbol to $e$ as $\&e\rightarrow_{t}\\!n_{i}$.
The type subscript $t$ can be omitted if there is no ambiguity caused. These
functions model memory layout of record types. Intuitively speaking, if $e$ is
a non-nil reference to a record-typed data object,
$\&e\rightarrow_{t}\\!n_{i}$ is the reference to the field $n_{i}$.
The above function (and constant) symbols can be used in both programs and
specifications. For conciseness, we use the following abbreviations.
1. 1.
Let $v$ be a program variable declared with type integer, boolean or a pointer
type, $v$ is an abbreviation for $\ast(\&v)$.
2. 2.
For a program variable $v$ declared with an array type $\textbf{ARR}(t,c)$,
and $t$ is integer, boolean, or a pointer type, we use $v[e]$ as an
abbreviation for $\ast(\&(\&v)[e])$.
3. 3.
If $e$ is of type $\textbf{P}(t)$, $t$ is a record type of which $n$ is a
field name, and the field type is integer, boolean or a pointer type, we can
use $e\rightarrow n$ as an abbreviation for $\ast(\&e\rightarrow n)$.
4. 4.
Let $v$ be a program variable declared with a record type of which $n$ is a
field name, the field type is integer, boolean or a pointer type, we can use
$v.n$ as an abbreviation for $\ast(\&(\&v)\rightarrow n)$.
### 2.2 The proof rules about memory access and layout
In this subsection, we present some proof rules to specify memory unit access
and memory layout of composite types. We define an auxiliary function
$\texttt{Block}:\textbf{Ptr}\rightarrow\textbf{SetOf}(\textbf{Ptr})$ to denote
the set of memory units in a memory block. The definition of Block is as
follows.
$\texttt{Block}(r)=\emptyset$ if $r=\textbf{nil}$. Otherwise
$\texttt{Block}(r)=$
$\left\\{\begin{array}[]{rcl}\\{r\\}&&\mbox{if $\ast r$ is of type
$\textbf{integer}$, $\textbf{boolean}$ or $\textbf{Ptr}$}\\\
\bigcup_{\mbox{\tiny$n$: field name of $t$}}\texttt{Block}(\&r\rightarrow
n)&&\mbox{if $r:\textbf{P}(t)$ and $t$ is a record type}\\\
\bigcup_{i=0}^{c-1}\texttt{Block}(\&r[i])&&\mbox{if
$r:\textbf{P}(\textbf{ARR}(t^{\prime},c))$ for some
$t^{\prime}$}\end{array}\right.$
Intuitively speaking, $\texttt{Block}(r)$ is the set of memory units in the
memory block referred by $r$.
The rule MEM-ACC says that if $r$ denotes a non-nil pointer referring to a
memory unit storing basic type values or pointer values, $\ast r$ denotes a
basic type value or a pointer value respectively.
$\framebox{\ \ \ MEM-ACC\ \ \ }\frac{\ \ \ \ \ r:\textbf{P}(t)\ \ \
r\neq\textbf{nil}\ }{\ast r:t}\mbox{\small\ \ $t$ is $\textbf{integer}$,
$\textbf{boolean}$, or $\textbf{P}(t^{\prime})$ for some $t^{\prime}$}$
The rule MEM-BLK specifies how memory blocks are allocated. Given two
arbitrary different memory blocks, they are either disjoint with each other,
or one is contained by the other.
$\framebox{MEM-BLK}\frac{p:\textbf{Ptr}\ \ \ q:\textbf{Ptr}\ \ \ p\neq
q}{\begin{array}[]{c}\texttt{Block}(p)\cap\texttt{Block}(q)=\emptyset\lor\\\
\texttt{Block}(p)\subset\texttt{Block}(q)\lor\texttt{Block}(q)\subset\texttt{Block}(p)\end{array}}$
The following two rules specify how the memory blocks are allocated for
declared program variables. The rule PVAR-1 says that for each program
variable, a memory block with corresponding type is allocated. Furthermore,
this block is not a sub-block of any other memory block. The rule PVAR-2 says
that each program variable is allocated a separate memory block.
$\framebox{PVAR-1}\frac{\ \ \ \
}{\begin{array}[]{c}\&v:\textbf{P}(t)\land\&v\neq\textbf{nil}\land\\\ \forall
x:\textbf{Ptr}\cdot\texttt{Block}(\&v)\not\subset\texttt{Block}(x)\end{array}}\mbox{\small\
$v$ is a program declared with type $t$.}$
$\framebox{PVAR-2}\frac{\ \ \ }{\ \ \ \&v_{1}\neq\&v_{2}\ \ \ \ }\mbox{\small\
\ $v_{1},v_{2}$ are two different program variables}$
The following two rules specify the memory layout for record-typed memory
blocks. The rule RECORD-1 says that a record-typed memory block is allocated
as a whole, i.e. when a record-typed memory block is allocated, all the memory
blocks for its fields are also allocated. The rule RECORD-2 says that the
memory blocks allocated for the fields are disjoint with each other.
$\framebox{RECORD-1}\frac{\ \ \
r:\textbf{P}(\textbf{REC}(\dots\times(n,t)\times\dots))\ \ \ \
r\neq\textbf{nil}}{\ \ \ \ (\&r\rightarrow
n:\textbf{P}(t))\land(\&r\rightarrow n\neq\textbf{nil})}$
$\framebox{RECORD-2}\frac{\ \ \
r:\textbf{P}(\textbf{REC}(\dots\times(n_{1},t_{1})\times\dots\times(n_{2},t_{2})\times\dots))\
\ \ \ r\neq\textbf{nil}}{\ \ \ \ \ \ \ \texttt{Block}(\&r\rightarrow
n_{1})\cap\texttt{Block}(\&r\rightarrow n_{2})=\emptyset\ \ \ }$
The following two rules specify the memory layout for array-typed memory
blocks. The rule ARR-1 says that an array-typed memory block is allocated as a
whole, i.e. when an array-typed memory block is allocated, all of the memory
blocks for its elements are allocated. The rule ARR-2 says that the memory
blocks allocated for different elements are disjoint with each other.
$\framebox{ARR-1}\frac{\ \ \ r:\textbf{P}(\textbf{ARR}(t,c))\ \ \
r\neq\textbf{nil}\ \ \ 0\leq i<c\ \ \
}{(\&r[i]:\textbf{P}(t))\land(\&r[i]\neq\textbf{nil})}$
$\framebox{{ARR}-2}\frac{\ \ \ r:\textbf{P}(\textbf{ARR}(t,c))\ \ \
r\neq\textbf{nil}\ \ \ 0\leq i<c\ \ \ 0\leq j<c\ \ \ i\neq
j}{\texttt{Block}(\&r[i])\cap\texttt{Block}(\&r[j])=\emptyset}$
### 2.3 The interpretation of P-types and new function symbols
Please be noticed that the types of the constant symbols
($\&v,\textbf{nil}_{t}$ ) introduced in this section are integer, boolean, or
pointer types. The argument types and result types of the function symbols
introduced in this section are also integer, boolean, and pointer values. So
the terms in our logic do not denote array or record P-type values. Thus
structures for our logic does not have to interpret record and array types.
For each P-type $t$, $(\textbf{P}(t))^{A}$ is a countable infinite set in the
universal domain $\mathcal{U}^{A}$ satisfying that
$(\textbf{nil}_{\textbf{P}(t)})^{A}\in(\textbf{P}(t))^{A}$. Furthermore, it is
required that for different P-types $t_{1}$ and $t_{2}$,
$(\textbf{P}(t_{1}))^{A}$ and $(\textbf{P}(t_{2}))^{A}$ are disjoint.
$\textbf{Ptr}^{A}$ is the union of all such sets.
The function symbols $\&\\!\rightarrow\\!n$ and $\&[\,]$ model the memory
layout of records and arrays respectively. As we do not go into details about
memory layout of composite types, we just requires that all the proof rules in
the previous subsection are satisfied by the interpretation of these function
symbols.
The function symbol $\ast$ models program states. Its interpretation must
satisfy that $\ast^{A}(x)\in t^{A}$ if $x\in(\textbf{P}(t))^{A}$ and
$x\neq(\textbf{nil}_{\textbf{P}(t)})^{A}$, where $t$ is integer, boolean, or
$\textbf{P}(t^{\prime})$ for some $t^{\prime}$; $\ast^{A}(x)=\bot$ otherwise.
## 3 Memory scope functions
In LPF, a formula $f(x_{1},\dots,x_{n})\triangleq e$ defines a function
denoted by $f$. People can define data-retrieve functions using such formulae.
In definitions for DRFs, we require that for any conditional sub-term
$e_{0}?e_{1}:e_{2}$ of $e$, none of the function symbols occurred in $e_{0}$
is defined (directly or indirectly) based on $f$. So for each DRF definition
$f(x_{1},\dots,x_{n})\triangleq e$, $e$ is continuous in $f$, thus we can use
the proof rule Func-Ind in [8] to prove properties about DRFs.
Given an LPF term $e$, the memory scope form of $e$, denoted as
$\mathfrak{M}(e)$, is defined as follow.
1. 1.
If $e$ is a variable, $\mathfrak{M}(e)$ is $\emptyset$.
2. 2.
If $e$ is of the form $f(e_{1},\dots,e_{n})$, $\mathfrak{M}(e)$ is
$\mathfrak{M}(e_{1})\cup\dots\cup\mathfrak{M}(e_{n})\cup\mathfrak{M}(f)(e_{1},\dots,e_{n})$,
where $\mathfrak{M}(f)$ represents the MSF symbol of $f$, which is defined as
follow
* •
If $f$ is a function symbol associated with basic types or abstract types (for
example, $+,-,\times,/,>,<,\in,\subseteq\dots$), $\mathfrak{M}(f)$ is defined
as the constant $\emptyset$.
* •
If $f$ is $\&\rightarrow n$, $\&[\,]$, $\&v$ for some program variable,
$\textbf{nil}_{t}$ for some type $t$, $\mathfrak{M}(f)$ is defined as the
constant $\emptyset$.
* •
If $f$ is the memory access function $\ast$ introduced in sub-section 2.1,
$\mathfrak{M}(\ast)$ is defined as $\mathfrak{M}(\ast)(x)\triangleq x$.
* •
For any other function symbols, $\mathfrak{M}(f)$ represents a new function
symbol denoting the memory scope function of $f$.
3. 3.
If $e$ is of the form $e_{0}?e_{1}:e_{2}$, $\mathfrak{M}(e)$ is
$\mathfrak{M}(e_{0})\cup(e_{0}?\mathfrak{M}(e_{1}):\mathfrak{M}(e_{2}))$.
Given a DRF $f$ defined as $f(x_{1},\dots,x_{n})\triangleq e$, the memory
scope function $\mathfrak{M}(f)$ of $f$ is defined as
$\mathfrak{M}(f)(x_{1},\dots,x_{n})=\mathfrak{M}(e)$
Formally, it is expressed using the following proof rule.
$\framebox{SCOPE-FUNC}\frac{\ \ \ \ \ \ f(x_{1},\dots,x_{n})\triangleq e}{\ \
\ \ \ \ \ \mathfrak{M}(f)(x_{1},\dots,x_{n})\triangleq\mathfrak{M}(e)}\ \ $
Please be noticed that for any sub-term $e_{0}?e_{1}:e_{2}$ of
$\mathfrak{M}(e)$, no function symbol is recursively defined based on
$\mathfrak{M}(f)$.
###### Definition 1
We say a structure $A$ with signature $\Sigma$ _conforms_ to a set of function
definitions $\mathbb{P}$ iff for each definition
$f(x_{1},\dots,x_{n})\triangleq e$ in $\mathbb{P}$,
$[\\![f(x_{1},\dots,x_{n})\triangleq e]\\!]^{A}_{\alpha}$ is $T$, here
$\alpha$ is the assignment of $A$.
The structure $A$ conforms to $\mathbb{P}$ means that $A$ interprets the
defined function symbols according to their definition in $\mathbb{P}$.
In our logic, the function symbol $\ast$ is used to model program states. The
DRFs used to retrieve abstract values are defined on $\ast$. One of the basic
ideas of our logic is that the abstract values retrieved by these functions
keep unchanged if no memory unit in their memory scopes is over-written during
a program execution. We have the following lemma and theorem about MSFs and
memory scope forms of terms.
###### Lemma 1
Let $\mathbb{P}$ be a set of recursive function definitions. Let $A$ and
$A^{\prime}$ be two structures. They both conform to $\mathbb{P}$ and are
identical except that they may have different interpretations for $\ast$ and
for the function symbols defined in $\mathbb{P}$. Let $e$ be a term satisfying
that all function symbols in $e$ are either defined in $\mathbb{P}$, or
associated with basic types, abstract types or P-types. We have that
$[\\![e]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}$
if $[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}\neq\bot$ and
$\ast^{A}(x)=\ast^{A^{\prime}}(x)$ for all
$x\in[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}$.
###### Proof
By induction, we first prove that the conclusion holds when $e$ contains no
function symbol defined in $\mathbb{P}$.
BASE: The conclusion holds if $e$ is a variable or a constant symbol.
INDUCTION: Assuming the conclusion holds for all terms shorter than $e$. We
prove that $[\\![e]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}$
if $[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}\neq\bot$ and
$\ast^{A}(x)=\ast^{A^{\prime}}(x)$ for all
$x\in[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}$.
* •
If $e$ is of the form $f(e_{1},\dots,e_{n})$, here $f$ is a function symbol
other than $\ast$, and $f$ is not defined in $\mathbb{P}$. $\mathfrak{M}(e)$
is $\mathfrak{M}(e_{1})\cup\dots\cup\mathfrak{M}(e_{n})$ because
$\mathfrak{M}(f)$ is $\emptyset$. So
$[\\![\mathfrak{M}(e_{i})]\\!]^{A}_{\alpha}\neq\bot$ and
$\ast^{A}(x)=\ast^{A^{\prime}}(x)$ for all
$x\in[\\![\mathfrak{M}(e_{i})]\\!]^{A}_{\alpha}$ for $i=1,\dots,n$. According
to the inductive assumption,
$[\\![e_{i}]\\!]^{A}_{\alpha}=[\\![e_{i}]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(e_{i})]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e_{i})]\\!]^{A^{\prime}}_{\alpha}$.
It follows that $[\\![e]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A^{\prime}}_{\alpha}$
and
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}$
because $f^{A}=f^{A^{\prime}}$.
* •
If $e$ is of the form $\ast e_{1}$. $\mathfrak{M}(e)$ is defined as
$\\{e_{1}\\}\cup\mathfrak{M}(e_{1})$. So
$[\\![\mathfrak{M}(e_{1})]\\!]^{A}_{\alpha}\neq\bot$ and
$\ast^{A}(x)=\ast^{A^{\prime}}(x)$ for all
$x\in[\\![\mathfrak{M}(e_{1})]\\!]^{A}_{\alpha}$. From the inductive
assumption, we have that
$[\\![e_{1}]\\!]^{A}_{\alpha}=[\\![e_{1}]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(e_{1})]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e_{1})]\\!]^{A^{\prime}}_{\alpha}$.
Because
$[\\![e_{1}]\\!]^{A}_{\alpha}\in[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}$ if
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}$ is not $\bot$, we have
$[\\![e]\\!]^{A}_{\alpha}=\ast^{A}([\\![e_{1}]\\!]^{A}_{\alpha})=\ast^{A^{\prime}}([\\![e_{1}]\\!]^{A^{\prime}}_{\alpha})=[\\![e]\\!]^{A^{\prime}}_{\alpha}$
and
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}$.
* •
If $e$ is of the form $e_{0}?e_{1}:e_{2}$. $\mathfrak{M}(e)$ is
$\mathfrak{M}(e_{0})\cup(e_{0}?\mathfrak{M}(e_{1}):\mathfrak{M}(e_{2}))$. From
the inductive assumption, we have
$[\\![e_{0}]\\!]^{A}_{\alpha}=[\\![e_{0}]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(e_{0})]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e_{0})]\\!]^{A^{\prime}}_{\alpha}$.
So $[\\![e_{0}]\\!]^{A^{\prime}}_{\alpha}=T$ iff
$[\\![e_{0}]\\!]^{A}_{\alpha}=T$. When
$[\\![e_{0}]\\!]^{A^{\prime}}_{\alpha}=[\\![e_{0}]\\!]^{A}_{\alpha}=T$, we
have
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e_{0})]\\!]^{A}_{\alpha}\cup[\\![\mathfrak{M}(e_{1})]\\!]^{A}_{\alpha}$,
$[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}=[\\![\mathfrak{M}(e_{0})]\\!]^{A^{\prime}}_{\alpha}\cup[\\![\mathfrak{M}(e_{1})]\\!]^{A^{\prime}}_{\alpha}$,
$[\\![e]\\!]^{A}_{\alpha}=[\\![e_{1}]\\!]^{A}_{\alpha}$ and
$[\\![e]\\!]^{A^{\prime}}_{\alpha}=[\\![e_{1}]\\!]^{A^{\prime}}_{\alpha}$.
From the inductive assumption, we have that
$[\\![e]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}$.
We can also prove that
$[\\![e]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}$
when $[\\![e_{0}]\\!]^{A}_{\alpha}$ is $F$ or $N$.
Second, we prove that the conclusion holds if no function symbol defined in
$\mathbb{P}$ is (directly or indirectly) recursively defined on itself. We
give a rank to each term and each function symbol. The rank of a term $e$ is
the highest rank of the function symbols occur in $e$. The ranks of function
symbols associated with basic types and abstract types are $0$. The function
symbols $\ast$, $\&\rightarrow n$, $\&[]$ also have rank $0$. The rank of a
function symbol $f$ defined as $f(x_{1},\dots,x_{n})\triangleq e_{r}$ in
$\mathbb{P}$ is the rank of $e_{r}$ plus $1$. As no function symbol is
recursively defined, each function symbol and term has a rank. Now, the
conclusion is proved by an induction on the ranks and the lengthes of terms.
BASE: According to the conclusion of the first step, this conclusion holds for
$0$-rank terms with any length.
INDUCTION: Let $e$ be a $k$-rank term. If the conclusion holds for all terms
either with a rank less than $k$, and all $k$-rank terms shorter than $e$.
* •
If $e$ is of the form $f(e_{1},\dots,e_{n})$ and $f$ is a function symbol with
a rank non-greater than $k$, and defined as $f(x_{1},\dots,x_{n})\triangleq
e_{r}$. Then the rank of $e_{r}$ is less than or equal to $k-1$. As all the
function-definition formulae in $\mathbb{P}$ are interpreted to $T$, according
to the semantic model of function definitions of LPF, both
$[\\![f(e_{1},\dots,e_{n})]\\!]^{A}_{\alpha}$ and
$[\\![f(e_{1},\dots,e_{n})]\\!]^{A^{\prime}}_{\alpha}$ are $\bot$ if some of
$[\\![e_{i}]\\!]^{A}_{\alpha}$ is $\bot$. Otherwise,
$[\\![f(e_{1},\dots,e_{n})]\\!]^{A}_{\alpha}$ and
$[\\![f(e_{1},\dots,e_{n})]\\!]^{A^{\prime}}_{\alpha}$ are
$[\\![e_{r}]\\!]^{A}_{\alpha^{\prime}}$ and
$[\\![e_{r}]\\!]^{A^{\prime}}_{\alpha^{\prime}}$ respectively, where
$\alpha^{\prime}=\alpha(x_{1}\rightarrow[\\![e_{1}]\\!]^{A}_{\alpha})\dots(x_{n}\rightarrow[\\![e_{n}]\\!]^{A}_{\alpha})$,
i.e. $\alpha^{\prime}$ is same as $\alpha$ except that $\alpha^{\prime}$ maps
$x_{i}$ to $[\\![e_{i}]\\!]^{A}_{\alpha}$;
$[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A}_{\alpha}$ and
$[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A^{\prime}}_{\alpha}$ are
$[\\![\mathfrak{M}(e_{r})]\\!]^{A}_{\alpha^{\prime}}$ and
$[\\![\mathfrak{M}(e_{r})]\\!]^{A^{\prime}}_{\alpha^{\prime}}$ respectively.
Because
$[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A}_{\alpha}\subseteq[\\![e]\\!]^{A}_{\alpha}$,
we have $\ast^{A}(x)=\ast^{A}(x)$ for all
$x\in[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e_{r})]\\!]^{A}_{\alpha^{\prime}}$.
As the rank of $e_{r}$ is less than or equal to $k-1$, from the inductive
assumption, we have
$[\\![e_{r}]\\!]^{A}_{\alpha^{\prime}}=[\\![e_{r}]\\!]^{A^{\prime}}_{\alpha^{\prime}}$
and
$[\\![\mathfrak{M}(e_{r})]\\!]^{A}_{\alpha^{\prime}}=[\\![\mathfrak{M}(e_{r})]\\!]^{A^{\prime}}_{\alpha^{\prime}}$,
i.e. $[\\![e]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A^{\prime}}_{\alpha}$ and
$[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A^{\prime}}_{\alpha}$.
So
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e)]\\!]^{A^{\prime}}_{\alpha}$
because
$[\\![\mathfrak{M}(e_{i})]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(e_{i})]\\!]^{A^{\prime}}_{\alpha}$
and
$[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A}_{\alpha}=[\\![\mathfrak{M}(f)(e_{1},\dots,e_{n})]\\!]^{A^{\prime}}_{\alpha}$.
* •
If $e$ is a conditional form with a rank $k$, the proof is similar to those of
the first step.
Now we are about to prove the general case. For each function symbol $f$
recursively defined in $\mathbb{P}$, we introduce infinite number of function
symbols $f_{0},f_{1},\dots$. For the definition of $f$, i.e.
$f(x_{1},\dots,x_{n})\triangleq e$, we introduce a set of definitions
$f_{i}(x_{1},\dots,x_{n})\triangleq e_{i}$ for $i=1,2,\dots$, where $e_{i}$ is
derived by replacing each function symbol $g$ recursively defined in
$\mathbb{P}$ by $g_{i-1}$ (Here $g$ can be $f$) in $e$. We also introduce a
function definition $f_{0}(x_{1},\dots,x_{n})=\circledast$ for each $f_{0}$.
Notice that $f_{i}$s are not recursively defined. Furthermore,
$\mathfrak{M}(e_{i})$ is same as the term derived by replacing $g$ and
$\mathfrak{M}(g)$ respectively by $g_{i-1}$ and $\mathfrak{M}(g_{i-1})$ in
$\mathfrak{M}(e)$. Because it is required that for each definition
$f(x_{1},\dots,x_{n})=e$ in $\mathbb{P}$, $e$ is continuous in $f$, we have
that $f_{i}$ is less defined than $f_{i+1}$, i.e.
$f_{i+1}(x_{1},\dots,x_{n})=f_{i}(x_{1},\dots,x_{n})$ if
$f_{i}(x_{1},\dots,x_{n})$ is defined for any $x_{1},\dots,x_{n}$. Because
$\mathfrak{M}(e)$ is continuous in $\mathfrak{M}(f)$, we have that
$\mathfrak{M}(f_{i})^{A}$ is less defined than $\mathfrak{M}(f_{i+1})$. So
$f^{A}$ is the least upper-bound of the function sequence
$f_{0}^{A},f_{1}^{A},\dots$, and $\mathfrak{M}(f)^{A}$ is the least upper-
bound of the function sequence
$\mathfrak{M}(f_{0})^{A},\mathfrak{M}(f_{1})^{A},\dots$. Let $e$ be a term
containing recursively defined function symbols. If $[\\![e]\\!]^{A}_{\alpha}$
is not $\bot$, there must be a large-enough integer $i$ such that
$[\\![e_{i}]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A}_{\alpha}$ and
$[\\![\mathfrak{M}(e_{i})]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A}_{\alpha}$, where
$e_{i}$ is derived by replacing each recursively defined function symbol $g$
by $g_{i-1}$. As $e_{i}$ contains no recursively defined symbols, according to
the second conclusion, we have that this lemma holds in general. QED
$\square$
The following theorem 3.1 gives a sufficient condition under which an LPF
formula $p$ keeps unchanged before/after some memory units are modified. A
term occurs in $p$ is called a top-level one if it is not a sub-term of
another term occurs in $p$.
###### Theorem 3.1
Let $\mathbb{P}$ be a set of recursive function definitions. Let $A$ and
$A^{\prime}$ be two structures. They both conform to $\mathbb{P}$ and are
identical except that they may have different interpretations for $\ast$ and
for the function symbols defined in $\mathbb{P}$. Let $p$ be an LPF formula
satisfying that
* •
all function symbols in $p$ are either defined in $\mathbb{P}$, or associated
with basic types, abstract types, or P-types, and
* •
$p$ has no sub-formula of the form $f(x_{1},\dots,x_{n})\triangleq e_{r}$.
We have that $[\\![p]\\!]^{A}_{\alpha}=[\\![p]\\!]^{A^{\prime}}_{\alpha}$ if
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha^{\prime}}\neq\bot$ and
$\ast^{A}(x)=\ast^{A^{\prime}}(x)$ for all
$x\in[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha^{\prime}}$ for each top-level term
$e$ of $p$, and arbitrary assignment $\alpha^{\prime}$.
###### Proof
This theorem can be proved by an induction on the structure of $p$.
BASE:
* •
If $p$ is of the form $f(e_{1},\dots,e_{n})$, and $f$ is a boolean-typed
function symbol (or a predicate symbol). $p$ itself is the only top-level term
of $p$. From Lemma 1,
$[\\![p]\\!]^{A}_{\alpha}=[\\![p]\\!]^{A^{\prime}}_{\alpha}$.
* •
If $p$ is of the form $e_{1}=e_{2}$. From Lemma 1,
$[\\![e_{i}]\\!]^{A}_{\alpha}=[\\![e_{i}]\\!]^{A^{\prime}}_{\alpha}$ for
$i=1,2$. So $[\\![p]\\!]^{A}_{\alpha}=[\\![p]\\!]^{A^{\prime}}_{\alpha}$.
* •
If $p$ is of the form $e:t$. From Lemma 1,
$[\\![e]\\!]^{A}_{\alpha}=[\\![e]\\!]^{A^{\prime}}_{\alpha}$ and
$t^{A}=t^{A^{\prime}}$. So
$[\\![p]\\!]^{A}_{\alpha}=[\\![p]\\!]^{A^{\prime}}_{\alpha}$.
INDUCTION:
* •
If $p$ is of the form $\forall x:t\cdot p^{\prime}$. A top-level term of $p$
is also a top-level term of $p^{\prime}$. From the inductive assumption, for
an assignment $\alpha(x\rightarrow v)$ for an arbitrary $t$-typed value $v$,
$[\\![p^{\prime}]\\!]^{A}_{\alpha(x\rightarrow
v)}=[\\![p^{\prime}]\\!]^{A^{\prime}}_{\alpha(x\rightarrow v)}$. According to
the interpretation rule for $\forall x:t\cdot p^{\prime}$, we conclude that
$[\\![p]\\!]^{A}_{\alpha}=[\\![p]\\!]^{A^{\prime}}_{\alpha}$.
* •
The conclusion can also be proved when $p$ is of the form $\Delta p^{\prime}$,
$\neg p^{\prime}$, and $p_{1}\land p_{2}$.
QED
$\square$
## 4 Syntax of programs
The small program language used in this paper is strong typed. Each expression
in the programs has a static P-type. An expression $e$ has a static P-type $t$
means that at the runtime, either $e$ denotes a value of type $t$ or $e$ is
non-denoting. The argument types and result types of function symbols appeared
in programs are definitely specified. The static types of expressions can be
decided statically and automatically. It also can be statically checked (by a
compiler, for example) that each function symbol is applied to arguments with
suitable static types. In this paper, it is supposed that all programs under
verification have passed such static type check.
### 4.1 The syntax of program expressions
A program expression is an LPF term with following restrictions.
1. 1.
A program expression contains no free variable. Be noticed that a program
variable $v$ occurs in a term is in fact an abbreviation for $\ast(\&v)$.
2. 2.
Only the following function (predicate) symbols can occur in program
expressions.
1. (a)
Constant symbols for basic types (integer, boolean), $\textbf{nil}_{t}$ for
type $t$, $\&v$ for a program variable $v$;
2. (b)
Function symbols associated with integer and boolean, like
$+,-,*,\div,<,\leq,=,\dots$;
3. (c)
Memory access/layout function symbols $\ast$, $\&\rightarrow n$, $\&[\,]$;
4. (d)
Boolean functions not, cand, cor which are defined using conditional forms as
follows.
1. i.
$\textbf{not}\ x\triangleq x\mbox{?}\textbf{false}:\textbf{true}$
2. ii.
$x\ \textbf{cand}\ y\triangleq\neg x\mbox{?}\textbf{false}:y$
3. iii.
$x\ \textbf{cor}\ y\triangleq x\mbox{?}\textbf{true}:y$.
We define these boolean operators because the semantic of logical connectives
$\land$ and $\lor$ of LPF is different from that of the logical operators
commonly used in program languages.
### 4.2 The syntax of program statements
The syntax of program statements is as follows.
$\begin{array}[]{rcl}st&::=&\texttt{skip}\ \ |\ \ \ast e_{1}:=e_{2}\ \ |\ \
\ast e:=\texttt{alloc}(t)\\\ &&|\ \ st;\ st\ \ |\ \ \textbf{if}\ (e)\ st\
\textbf{else}\ st\\\ &&|\ \ \textbf{while}\ (e)\ st\\\ \end{array}$
This programming language has two kinds of primitive statements: assignment
statements and memory-allocation statements.
* •
An assignment statement $\ast e_{1}:=e_{2}$ first evaluates $e_{1}$ and
$e_{2}$, then assigns the value of $e_{2}$ to the memory unit referred by the
value of $e_{1}$. The values stored in other memory units keep unchanged. It
is required that $\ast e_{1}$ and $e_{2}$ has same static type, which is
limited to be integer, boolean, or a pointer type.
* •
A memory-allocation statement $\ast e:=\texttt{alloc}(t)$ allocates a memory
block of type $t$, and assigns the reference to this memory block to the
memory unit referred by the value of $e$. Furthermore, in the new memory
block, all the memory units storing pointer values are initialized to nil. It
is required that the static type of $\ast e$ is $\textbf{P}(t)$.
The semantics of the composite statements $st;st$, $\textbf{if}\ (e)\ st\
\textbf{else}\ st$, and $\textbf{while}\ (e)\ st$ are same as those commonly
used in real program languages. It is required that in $\textbf{if}\ (e)\ st\
\textbf{else}\ st$ and $\textbf{while}\ (e)\ st$, the static type of $e$ must
be boolean.
###### Example 1
The program depicted in Figure 1 is a running example used in this paper. The
type of the program variables k and d is integer. The type of program
variables root and p is P($T$), where $T$ is
$\textbf{REC}((l,\textbf{P}(T))\times(r,\textbf{P}(T))\times(K,\textbf{integer})\times(D,\textbf{integer}))$.
This program first searches a binary search tree for a node of which the field
$K$ equals k. Then it sets the filed $D$ of this node to d. Please be noticed
that p, root, k, d, $\textsf{p}\rightarrow K$, $\textsf{p}\rightarrow D$,
$\textsf{p}\rightarrow l$, $\textsf{p}\rightarrow r$ are respectively
abbreviations for $\ast(\&\textsf{p})$, $\ast(\&\textsf{root})$,
$\ast(\&\textsf{k})$, $\ast(\&\textsf{d})$, $\ast(\&\textsf{p}\rightarrow K)$,
$\ast(\&\textsf{p}\rightarrow D)$, $\ast(\&\textsf{p}\rightarrow l)$,
$\ast(\&\textsf{p}\rightarrow r)$.
p:=root;
---
while | ($\textsf{p}\rightarrow K\neq\textsf{k}$)
{
| if ($\textsf{k}<\textsf{p}\rightarrow K$ )
$\textsf{p}:=\textsf{p}\rightarrow l$ else $\textsf{p}:=\textsf{p}\rightarrow
r$;
}
$\textsf{p}\rightarrow D:=\textsf{d}$;
Figure 1: The program used as a running example
## 5 Syntax of specifications
A program specification is of the form $\mathbb{P}\vdash q\\{c\\}r$, where $c$
is a program, $\mathbb{P}$ is a set of LPF formulae, $q$ and $r$ are LPF
formulae satisfying the following conditions.
* •
They contain only function symbols defined in $\mathbb{P}$, the function
symbols which can occur in program expressions, and the function symbols
associated with abstract types.
* •
$q$ and $r$ contains no sub-formula of the form
$f(x_{1},\dots,x_{n})\triangleq e$.
The formula set $\mathbb{P}$ is called the premise of this specification.
$\mathbb{P}$ usually contains a set of function definitions. The formulae $q$
and $r$ are respectively called the pre-condition and post-condition.
Intuitively speaking, such a specification means that if all the formulae in
$\mathbb{P}$ hold for arbitrary program states, and the program $c$ starts its
execution on a state satisfying $q$, then the state satisfies $r$ when the
program $c$ stops.
###### Example 2
Let $\mathbb{P}$ be the set of formulae depicted in Figure 2. These formulae
define a set of data retrieve functions. The boolean function InHeap is
defined in sub-section 6.3. $\texttt{InHeap}(x)$ means that $x$ refers to a
memory block disjoint with all memory blocks for program variables. Let
$q=\textsf{isHBST}(\textsf{root})\land\textsf{Map}(\textsf{root})=M\land\textsf{k}\in\textsf{Dom}(\textsf{root})$
$r=\textsf{isHBST}(\textsf{root})\land\textsf{Map}(\textsf{root})=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}$
$Prog$ is the program depicted in Figure 1. The specification
$\mathbb{P}\vdash q\\{Prog\\}r$ says that if the program state satisfies the
following conditions when $\\{Prog\\}$ starts.
1. 1.
The value of root points to the root node of a binary search tree stored in
the heap;
2. 2.
The tree represents a finite map $M$ from integer to integer;
3. 3.
The value stored in k is in the domain of this map,
When $Prog$ stops, root still points to the root node of the binary search
tree, and now the finite map represented by the binary search tree becomes
$M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}$.
$\textsf{NodeSet}(x):\textbf{P}(T)\rightarrow\textbf{SetOf}(\textbf{Ptr})$
---
| $\triangleq$ $(x=\textbf{nil})?$ | $\emptyset:(\\{x\\}\cup\textsf{NodeSet}(x\rightarrow l)\cup\textsf{NodeSet}(x\rightarrow r))$
$\textsf{Map}(x):\textbf{P}(T)\rightarrow\textbf{Map integer to integer}$
| $\triangleq(x=\textbf{nil})?\emptyset:\\{x\rightarrow K\mapsto x\rightarrow
D\\}{\dagger}\textsf{Map}(x\rightarrow l){\dagger}\textsf{Map}(x\rightarrow
r)$
$\textsf{MapP}(x,y):\textbf{P}(T)\times\textbf{P}(T)\rightarrow\textbf{Map
integer to integer}$
| $\triangleq(x=\textbf{nil})?\emptyset:\textsf{MapP}(x\rightarrow
l){\dagger}\textsf{MapP}(x\rightarrow r){\dagger}$
| $((x=y)?\emptyset:\\{x\rightarrow K\mapsto x\rightarrow D\\})$
$\textsf{Dom}(x):\textbf{P}(T)\rightarrow\textbf{SetOf}(\textbf{integer})$
| $\triangleq(x=\textbf{nil})?\emptyset:(\\{x\rightarrow
K\\}\cup\textsf{Dom}(x\rightarrow l)\cup\textsf{Dom}(x\rightarrow r))$
$\textsf{isHBST}(x):\textbf{P}(T)\rightarrow\textbf{boolean}$
|
$\triangleq(x=\textbf{nil})?\textbf{true}:\texttt{InHeap}(x)\land\textsf{isHBST}(x\rightarrow
l)\land\textsf{isHBST}(x\rightarrow r)\land$
| | $(\textsf{Dom}(x\rightarrow l)=\emptyset?\texttt{true}:\texttt{MAX}(\textsf{Dom}(x\rightarrow l))<x\rightarrow K)\land$
| | $(\textsf{Dom}(x\rightarrow r)=\emptyset?\texttt{true}:x\rightarrow K<\texttt{MIN}(\textsf{Dom}(x\rightarrow r)))$
Figure 2: The definitions of a set of data retrieve functions
## 6 Proof rules of program statements
In this section, we present the proof rules for program statements. There are
three rules for primitive statements, one rule for consequences, and three
rules for control flow statements.
### 6.1 The proof rule for skip statement
The skip statement changes nothing, so we have the following proof rule.
$\framebox{SKIP-ST}\frac{\ \ \ \ }{\ \ \ \ \ \ \ \ \ \emptyset\vdash
q\\{\texttt{skip}\\}q\ \ \ \ \ \ \ \ \ \ }$
### 6.2 The proof rule for assignment statements
Let $q$ be an LPF formula and $x$ be the only free variable in $q$. Let $t$ be
the static type of $\ast e_{1}$ and $e_{2}$. The type $t$ must be integer,
boolean, or $\textbf{P}(t^{\prime})$ for some $t^{\prime}$. We have the
following proof rule for assignment statements.
$\framebox{ASSIGN-ST}\frac{\begin{array}[]{l}\mathbb{P},q[e_{2}/x]\vdash
e_{1}\neq\textbf{nil}\land e_{1}\not\in\mathfrak{M}(e_{1})\land e_{2}:t\\\
\mathbb{P},q[e_{2}/x]\vdash e_{1}\not\in\mathfrak{M}(e)[e_{2}/x]\mbox{ for
each top-level term $e$ of $q$}\end{array}}{\mathbb{P}\vdash q[e_{2}/x]\\{\ast
e_{1}:=e_{2}\\}q[\ast e_{1}/x]}$
Here, it is required that all bounded variables in $q$ are different from $x$.
A term $e$ of $q$ is called a top-level one if it is not a sub-term of another
term of $q$. Furthermore, it is required that for each conditional term
$e_{0}?e_{1}:e_{2}$ of $q$, $e_{0}$ is a boolean-typed term, so we can
construct a memory form of each top-level term of $q$.
Now we briefly prove the soundness of this rule. We can use two structure $A$
and $A^{\prime}$ to denote the program states before/after the assignment
statement. $A$ and $A^{\prime}$ are only different in the interpretations of
the function symbol $\ast$ and the symbols defined in $\mathbb{P}$. The
semantic of an assignment $\ast e_{1}=e_{2}$ is as follow. It first evaluates
the value of $e_{1}$ and $e_{2}$, i.e. $[\\![e_{1}]\\!]^{A}_{\alpha}$ and
$[\\![e_{2}]\\!]^{A}_{\alpha}$, then the content of the memory unit referred
by $[\\![e_{1}]\\!]^{A}_{\alpha}$ is set to $[\\![e_{2}]\\!]^{A}_{\alpha}$.
Formally, we say
$\ast^{A^{\prime}}([\\![e_{1}]\\!]^{A}_{\alpha})=[\\![e_{2}]\\!]^{A}_{\alpha}$,
and $\ast^{A}(x)=\ast^{A^{\prime}}(x)$ for all
$x\neq[\\![e_{1}]\\!]^{A}_{\alpha}$. According to Lemma 1, the condition
$e_{1}\not\in\mathfrak{M}(e_{1})$ assures that
$[\\![e_{1}]\\!]^{A}_{\alpha}=[\\![e_{1}]\\!]^{A^{\prime}}_{\alpha}$, so
$[\\![e_{2}]\\!]^{A}_{\alpha}=\ast^{A^{\prime}}([\\![e_{1}]\\!]^{A}_{\alpha})=\ast^{A^{\prime}}([\\![e_{1}]\\!]^{A^{\prime}}_{\alpha})=[\\![\ast
e_{1}]\\!]^{A^{\prime}}_{\alpha}$. The condition $e_{1}\neq\textbf{nil}\land
e_{2}:t$ assures that both $[\\![\ast e_{1}]\\!]^{A}_{\alpha}$ and
$[\\![e_{2}]\\!]^{A}_{\alpha}$ are not $\bot$. Together with these conditions,
the condition $e_{1}\not\in\mathfrak{M}(e)[e_{2}/x]$ assures that for each
top-level term $e$,
$[\\![e_{1}]\\!]^{A}_{\alpha}\not\in[\\![\mathfrak{M}(e)[e_{2}/x]]\\!]^{A}_{\alpha}$,
which equals to
$[\\![\mathfrak{M}(e)]\\!]^{A}_{\alpha(x\rightarrow[\\![e_{2}]\\!]^{A}_{\alpha})}$.
From Lemma 1, we have
$[\\![e]\\!]^{A}_{\alpha(x\rightarrow[\\![e_{2}]\\!]^{A}_{\alpha})}=[\\![e]\\!]^{A^{\prime}}_{\alpha(x\rightarrow[\\![e_{2}]\\!]^{A}_{\alpha})}=[\\![e]\\!]^{A^{\prime}}_{\alpha(x\rightarrow[\\![\ast
e_{1}]\\!]^{A^{\prime}}_{\alpha})}$. So we have
$[\\![e[e_{2}/x]]\\!]^{A}_{\alpha}=[\\![e[\ast
e_{1}/x]]\\!]^{A^{\prime}}_{\alpha}$. As $\alpha$ is arbitrary, according to
Theorem 3.1, $[\\![q[e_{2}/x]]\\!]^{A}_{\alpha}=[\\![q[\ast
e_{1}/x]]\\!]^{A^{\prime}}_{\alpha}$. So we conclude that if $q[e_{2}/x]$
holds before the assignment statement, $q[\ast e_{1}/x]$ holds after.
### 6.3 The proof rule for memory allocation statements
The memory allocation statement $\ast e=\textsf{alloc}(t)$ first evaluates
$e$, then allocates an unused memory block and assigns the reference to this
block to the memory unit referred by $e$. All the memory units storing pointer
values are initialized to nil. This block can not be referred by any pointers
stored somewhere before this allocation. Furthermore, this block is disjoint
with all of the memory blocks allocated for program variables. It is required
that the static type of $\ast e$ must be $\textbf{P}(t)$. Let $p$ be an LPF
formula containing no free variable, we have the following proof rule for
memory allocation statements.
$\framebox{ALLOC-ST}\frac{\begin{array}[]{l}\mathbb{P}\land q\vdash
e\neq\textbf{nil}\land e\not\in\mathfrak{M}(e)\\\ \mathbb{P}\land q\vdash
e\not\in\mathfrak{M}(e^{\prime})\mbox{ for each top-level term $e^{\prime}$ of
$q$}\end{array}}{\mathbb{P}\vdash q\
\\{*e=\textsf{alloc}(t)\\}\left(\begin{array}[]{l}q\land\texttt{InHeap}(\ast
e)\land\texttt{Unique}(e)\\\ \land\texttt{PtrInit}(\ast e)\land(\ast
e\neq\textbf{nil})\end{array}\right)}$
The predicts Unique, InHeap, and PtrInit are defined as follows.
$\texttt{Unique}(x)\triangleq\forall y:\textbf{Ptr}\cdot((y\neq x\land
y\neq\textbf{nil}\land\ast y:\textbf{Ptr})\Rightarrow\texttt{Block}(\ast
y)\cap\texttt{Block}(*x)=\emptyset)$
$\texttt{InHeap}(p)\triangleq\bigwedge_{x\mbox{ is a program
variable.}}(\texttt{Block}(\&v)\cap\texttt{Block}(p)=\emptyset)$
$\texttt{PtrInit}(p)\triangleq\forall
x:\textbf{Ptr}\cdot((x\in\texttt{Block}(p)\land x\neq\textbf{nil}\land\ast
x:\textbf{Ptr})\Rightarrow\ast x=\textbf{nil})$
Intuitively speaking, $\texttt{Unique}(p)$ says that the memory block referred
by the reference stored in $p$ can not be accessed by references stored
elsewhere. $\texttt{InHeap}(p)$ says that the memory block referred by $p$ is
disjoint with all the memory blocks for program variables.
$\texttt{PtrInit}(p)$ says that all memory units with pointer types in the
memory block referred by $p$ store nil pointers.
Similarly to the soundness reasoning for the rule ASSIGN-ST, we can conclude
that $q$ still holds after the allocation statement if it holds before.
Because the allocated memory block is unused, it can not be accessed by any
pointers stored somewhere before this memory allocation. This allocation
statement assigns the reference to this block only to the memory unit referred
by $e$. So $\texttt{Unique}(e)$ holds after this allocation statement. The new
allocated block is disjoint with any blocks for program variables. So
$\texttt{InHeap}(\ast e)$ holds after this allocation statement. The post
condition $\texttt{PtrInit}(\ast e)$ holds because the new block is
initialized as described above. So we conclude that this proof rule is sound.
### 6.4 The consequence rule and the rules for control flow statements
The following proof rules are essentially the same as those presented in [1].
The consequence rule is slightly modified such that the premise of a verified
assertion can be strengthened. The rules for if-statement and while-statement
are modified such that the pre-condition ensures that the condition expression
$e$ is evaluated to either $T$ or $F$.
$\framebox{CONSEQ}\frac{\ \ \ \mathbb{P}\vdash q\\{s\\}r\ \ \ \
\mathbb{Q}\vdash\mathbb{P}\ \ \ \ \ \mathbb{P},q^{\prime}\vdash q\ \ \ \ \
\mathbb{P},r\vdash r^{\prime}\ \ \ }{\mathbb{Q}\vdash
q^{\prime}\\{s\\}r^{\prime}}$
$\framebox{SEQ-ST}\frac{\ \ \ \mathbb{P}\vdash q\\{s_{1}\\}r\ \ \ \ \
\mathbb{P}\vdash r\\{s_{2}\\}r^{\prime}\ \ \ }{\mathbb{P}\vdash
q\\{s_{1};s_{2}\\}r^{\prime}}$
$\framebox{IF-ST}\frac{\ \ \ \mathbb{P},q\vdash e\lor\neg e\ \ \ \ \
\mathbb{P}\vdash(q\land e)\\{s_{1}\\}r\ \ \ \ \mathbb{P}\vdash(q\land\neg
e)\\{s_{2}\\}r\ \ \ }{\mathbb{P}\vdash q\\{\mbox{ {if} }(e)\ s_{1}\mbox{
{else} }s_{2}\ \\}r}$
$\framebox{WHILE-ST}\frac{\ \ \ \ \mathbb{P},q\vdash e\lor\neg e\ \ \ \ \
\mathbb{P}\vdash(q\land e)\\{s\\}q}{\ \ \ \mathbb{P}\vdash q\\{\mbox{ {while}
}(e)\ \ s\ \\}q\land\neg e\ \ \ }$
## 7 Verifying the running example
In this section, we verify the program depicted in Figure 1.
### 7.1 The DRFs, MSFs and their properties.
###### Example 3
Figure 2 shows the data-retrieve functions for specifying and verifying the
program depicted in Figure 1. From the proof rule SCOPE-FUNC in Section 3, we
can derive the definitions of all corresponding MSFs. The definitions of MSFs
depicted in Figure 3 are simplified but equivalent to those derived directly
by the rule SCOPE-FUNC. For conciseness, we write
$\mathfrak{M}(\textsf{NodeSet})$ as $\verb"NS"_{m}$,
$\mathfrak{M}(\textsf{Map})$ as $\verb"MP"_{m}$, $\mathfrak{M}(\textsf{MapP})$
as $\verb"MPP"_{m}$, $\mathfrak{M}(\textsf{Dom})$ as $\verb"DM"_{m}$,
$\mathfrak{M}(\textsf{isHBST})$ as $\verb"HBST"_{m}$. Some properties about
these DRFs and MSFs are depicted in Figure 4. These properties can be proved
in the extended LPF.
$\verb"NS"_{m}(x)$ | $\triangleq$ $(x=\textbf{nil})?$ | $\emptyset:(\\{\&x\rightarrow l,\&x\rightarrow r\\}\cup\verb"NS"_{m}(x\rightarrow l)\cup\verb"NS"_{m}(x\rightarrow r))$
---|---|---
$\verb"MP"_{m}(x)\triangleq(x=\textbf{nil})?\emptyset:$
| $\\{\&x\rightarrow l,\&x\rightarrow r,\&x\rightarrow D,\&x\rightarrow
K\\}\cup\verb"MP"_{m}(x\rightarrow l)\cup\verb"MP"_{m}(x\rightarrow r)$
$\verb"MPP"_{m}(x,y)\triangleq(x=\textbf{nil})?\emptyset:$
| $\\{\&x\rightarrow l,\&x\rightarrow r\\}\cup\verb"MPP"_{m}(x\rightarrow
l)\cup\verb"MPP"_{m}(x\rightarrow r)\cup$
| $((x=y)?\emptyset:\\{\&x\rightarrow K,\&x\rightarrow D\\})$
$\verb"DM"_{m}(x)\triangleq(x=\textbf{nil})?\emptyset:(\\{\&x\rightarrow
K,\&x\rightarrow l,\&x\rightarrow r\\}\cup\verb"DM"_{m}(x\rightarrow
l)\cup\verb"DM"_{m}(x\rightarrow r))$
$\verb"HBST"_{m}(x)\triangleq(x=\textbf{nil})?\emptyset:\\{\&x\rightarrow
l,\&x\rightarrow r\\}\cup\verb"HBST"_{m}(x\rightarrow
l)\cup\verb"HBST"_{m}(x\rightarrow r)\cup$
| $\verb"DM"_{m}(x\rightarrow l)\cup(\textsf{Dom}(x\rightarrow
l)=\emptyset?\emptyset:\\{\&x\rightarrow l\\}\cup\\{\&x\rightarrow
K\\}\cup\verb"DM"_{m}(x\rightarrow l))\cup$
| $\verb"DM"_{m}(x\rightarrow r)\cup(\textsf{Dom}(x\rightarrow
r)=\emptyset?\emptyset:\\{\&x\rightarrow r\\}\cup\\{\&x\rightarrow
K\\}\cup\verb"DM"_{m}(x\rightarrow r)))$
Figure 3: The definitons of MSFs
$\mathbb{P},\textsf{isHBST}(x)\vdash\&\textsf{p}\not\in\verb"HSBT"_{m}({x})\cup\verb"MP"_{m}({x})\cup\verb"DM"_{m}({x})$
(1)
$\mathbb{P},\textsf{isHBST}(x)\vdash\&\textsf{p}\rightarrow
D\not\in\verb"HSBT"_{m}({x})\cup\verb"MPP"_{m}({x},\textsf{p})\cup\verb"DM"_{m}({x})$
(2)
$\mathbb{P},\textsf{isHBST}(x),y\in\textsf{Dom}(x),y<x\rightarrow K\vdash
y\in\textsf{Dom}(x\rightarrow l)$ (3)
$\mathbb{P},\textsf{isHBST}(x),y\in\textsf{Dom}(x),y>x\rightarrow K\vdash
y\in\textsf{Dom}(x\rightarrow r)$ (4)
$\mathbb{P},\textsf{isHBST}(x),\textsf{y}\in\textsf{NodeSet}(x)\vdash\textsf{Map}(x)=\textsf{MapP}(x,y){\dagger}\\{y\rightarrow
K\mapsto y\rightarrow D\\}$ (5)
$\mathbb{P},\textsf{NodeSet}(x):\textbf{SetOf}(\textbf{Ptr})\vdash
x\in\textsf{NodeSet}(x)$ (6)
Figure 4: Some properties about DRFs and MSFs
### 7.2 Verifying the program
In this section, we will prove that if root points to a binary search tree,
and we view this binary tree as a finite map, and k is in the domain of this
map, the program depicted in Figure 1 set the co-value of k to d. In this
section, we use $\mathbb{P}$ to denote the set of the function definitions in
Figure 2. The specification is as follow.
$\mathbb{P}\vdash\textsf{PRE-COND}\ \ \\{\textsl{Prog}\\}\ \
\textsf{isHBST}(\textsf{root})\land\textsf{Map}(\textsf{root})=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}$
Here, PRE-CON is the abbreviation for
$\textsf{isHBST}(\textsf{root})\land\textsf{Map}(\textsf{root})=M\land\textsf{k}\in\textsf{Dom}(\textsf{root})$,
$M$ is a constant with type Map integer to integer. The verification steps are
given below.
From ASSIGN-ST, 1 and $\&\textsf{p}\not\in\\{\&\textsf{root},\&\textsf{k}\\}$:
$\mathbb{P}\vdash\left(\begin{array}[]{l}(\textsf{PRE-COND}\land
x\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(x))[\textsf{root}/x]\\\
\ \ \ \ \ \\{\textsf{p}=\textsf{root};\\}\\\ (\textsf{PRE-COND}\land
x\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(x))[\textsf{p}/x]\end{array}\right)$
(7)
From $\land$-I, 7, CONSEQ, 6:
$\mathbb{P}\vdash\textsf{PRE-COND}\ \\{\textsf{p}=\textsf{root};\\}\
\textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})$
(8)
From ASSIGN-ST, 1, and
$\&\textsf{p}\not\in\\{\&\textsf{root},\&\textsf{k}\\}$:
$\mathbb{P}\vdash\left(\begin{array}[]{l}(\textsf{PRE-COND}\land
x\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(x))[\textsf{p}\rightarrow
l/x]\\\ \ \ \ \ \ \\{\textsf{p}:=\textsf{p}\rightarrow l;\\}\\\ (\textsf{PRE-
COND}\land
x\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(x))[\textsf{p}/x]\end{array}\right)$
(9)
From 3, substitution:
$\begin{array}[]{l}\mathbb{P},\textsf{PRE-
COND},\textsf{p}\in\textsf{NodeSet}(\textsf{root}),\textsf{k}\in\textsf{Dom}(\textsf{p}),\textsf{k}<\textsf{p}\rightarrow
K\vdash\\\ \ \ \ \ \ \textsf{p}\rightarrow
l\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p}\rightarrow
l)\end{array}$ (10)
From 9, 10, and CONSEQ:
$\mathbb{P}\vdash\left(\begin{array}[]{l}\textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\land\textsf{k}<\textsf{p}\rightarrow
K\\\ \ \ \ \ \ \\{\textsf{p}:=\textsf{p}\rightarrow l;\\}\\\ \textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\end{array}\right)$
(11)
Similarly, we can prove:
$\mathbb{P}\vdash\left(\begin{array}[]{l}\textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\land\textsf{k}>\textsf{p}\rightarrow
K\\\ \ \ \ \ \ \\{\textsf{p}:=\textsf{p}\rightarrow r;\\}\\\ \textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\end{array}\right)$
(12)
$\textsf{k}\in\textsf{Dom}(\textsf{p})$ implies $\textsf{p}\neq\textbf{nil}$,
thus $\textsf{k}<\textsf{p}\rightarrow
K\lor\textsf{k}\geq\textsf{p}\rightarrow K$. From IF-ST, 11, 12, and :
$\mathbb{P}\vdash\left(\begin{array}[]{l}\textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\land\textsf{p}\rightarrow
K\neq\textsf{k}\\\ \ \ \ \ \ \\{\mbox{{if} }(\textsf{k}<\textsf{p}\rightarrow
K)\ \textsf{p}:=\textsf{p}\rightarrow l;\ \textbf{else}\
\textsf{p}:=\textsf{p}\rightarrow r;\\}\\\ \textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\end{array}\right)$
(13)
$\textsf{k}\in\textsf{Dom}(\textsf{p})$ implies that
$\textsf{p}\neq\textbf{nil}$, thus $\textsf{p}\rightarrow
K\neq\textsf{k}\lor\textsf{p}\rightarrow K=\textsf{k}$. From WHILE-ST, 13:
$\mathbb{P}\vdash\left(\begin{array}[]{l}\textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\\\
\ \ \ \ \ \\{\mbox{\emph{the while statement}}\\}\\\ \textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\land\textsf{p}\rightarrow
K=\textsf{k}\\\ \end{array}\right)$ (14)
From • ‣ 8 and the properties of finite map:
$\begin{array}[]{l}\mathbb{P},\textsf{isHBST}(x),\textsf{p}\in\textsf{NodeSet}(x),\textsf{k}=\textsf{p}\rightarrow
K\vdash\\\ \ \ \ \ \
\textsf{TMapP}(x,\textsf{p}){\dagger}\\{\textsf{p}\rightarrow K\mapsto
y\\}=\textsf{Map}(x){\dagger}\\{\textsf{k}\mapsto y\\}\end{array}$ (15)
From 15, substitution:
$\begin{array}[]{l}\mathbb{P},\textsf{PRE-
COND},\textsf{p}\in\textsf{NodeSet}(\textsf{root}),\textsf{k}=\textsf{p}\rightarrow
K\vdash\\\
\textsf{isHBST}(\textsf{root})\land\textsf{MapP}(\textsf{root},\textsf{p}){\dagger}\\{\textsf{p}\rightarrow
K\mapsto\textsf{d}\\}=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\})\end{array}$
(16)
From the rule ASSIGN-ST, 2 and $\&\textsf{p}\rightarrow
D\not\in\\{\&\textsf{root},\&\textsf{p},\&\textsf{p}\rightarrow
K,\&\textsf{k},\&\textsf{d}\\}$:
$\mathbb{P}\vdash\left(\begin{array}[]{l}(\textsf{isHBST}(\textsf{root})\land\textsf{MapP}(\textsf{root},\textsf{p}){\dagger}\\{\textsf{p}\rightarrow
K\mapsto x\\}=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\})[\textsf{d}/x]\\\ \
\ \ \ \ \\{\textsf{p}\rightarrow D:=\textsf{d}\\}\\\
(\textsf{isHBST}(\textsf{root})\land\textsf{MapP}(\textsf{root},\textsf{p}){\dagger}\\{\textsf{p}\rightarrow
K\mapsto
x\\}=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\})[\textsf{p}\rightarrow
D/x]\end{array}\right)$ (17)
From the rule CONSEQ, 16, 17
$\mathbb{P}\vdash\left(\begin{array}[]{l}\textsf{PRE-
COND}\land\textsf{p}\in\textsf{NodeSet}(\textsf{root})\land\textsf{k}\in\textsf{Dom}(\textsf{p})\land\textsf{p}\rightarrow
K=\textsf{k}\\\ \ \ \ \ \ \\{\textsf{p}\rightarrow D:=\textsf{d}\\}\\\
\textsf{isHBST}(\textsf{root})\land\textsf{Map}(\textsf{root})=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}\end{array}\right)$
(18)
From the rule SEQ-ST, 8, 14, 18:
$\mathbb{P}\vdash\textsf{PRE-COND}\ \ \\{\textsl{Prog}\\}\ \
\textsf{isHBST}(\textsf{root})\land\textsf{Map}(\textsf{root})=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}$
(19)
## 8 Heuristics: virtual variables and pragmatic meaning of program
statements
Generally speaking, a pointer program may create unbounded number of data
objects during its execution. These data objects usually interconnected
through pointers. They are usually used to represent abstract values which can
be retrieved using recursively defined DRFs. We can view a set of
interconnected data objects as a _virtual variable_ , which holds an abstract
value retrieved using a DRF. Usually, such a data object set maintains a set
of structural properties during the program execution. These properties can
also be expressed using a set of boolean-typed DRFs. As we did in our running
example, a DRF isHBST is used to state that a set of data objects form a
binary search tree, while the DRF Map is used to retrieve a finite map from
this binary search tree.
Usually, assigning new values to such a virtual variable is performed by a
group of program statements. These statements change the values stored in a
few number of the data objects, thus change the abstract value ‘stored’ in the
virtual variable. As to the structural properties, either none of the
statements changes their values, or some statements change their values, but
some other statements restore them afterwards. To reasoning the effect of
these statements on the abstract value, we can define some auxiliary data-
retrieve functions.
* •
These auxiliary DRFs do not accessed the memory units modified by these
statements. So the abstract values retrieved by these auxiliary DRFs keep
unchanged.
* •
The relationship between the abstract value retrieved by the main DRFs, those
retrieved by auxiliary DRFs, and the values stored in the modified memory
units can be proved based on the definitions of the DRFs. For example, the
property
$\mathbb{P},\textsf{isHBST}(x),\textsf{y}\in\textsf{NodeSet}(x)\vdash\textsf{Map}(x)=\textsf{MapP}(x,y){\dagger}\\{y\rightarrow
K\mapsto y\rightarrow D\\}$
shows the relation between the main DRF Map, the auxiliary DRF MapP, and the
values stored in $\&y\rightarrow K$ and $\&y\rightarrow D$.
* •
The values retrieved by auxiliary DRFs keep unchanged. The effect of these
statements on the modified memory units can be relatively easily derived. So,
the effect of these statements on the abstract value retrieved by main DRF can
be reason based on the relations between main DRFs, auxiliary DRFs and the
value stored in modified memory units.
To specify and verify these statements, we should understand and reason these
statements as a whole, as these statements work together to assign a new value
to a virtual variable. Understanding the effects of such statement groups can
help us understand the whole program abstractly. In the appendix 0.B, we
briefly describe such an example. We say the effect of a group of program
statements on a virtual variable as the _pragmatic meaning_ of these
statements. Understanding and verifying the pragmatic meanings of small
statement groups first, then we can verify code with larger size step by step.
## 9 Conclusion and future works
In this paper, we present an extension of Hoare logic for verification of
pointer programs. The pre-conditions and post-conditions are formulae of an
extended version of the LPF logic, which can deal with undefinedness,
recursive function definitions, and types. Program types and function symbols
($\ast$, $\&\\!\\!\rightarrow\\!n$ and $\&[\,]$) associated with these types
are introduced to model memory unit access and memory layout for composite
types. A set of proof rules are introduced to specify these function symbols.
Using these functions, people can deal with high-level program types (record,
array) directly.
People can define recursive functions to retrieve abstract values from
concrete interconnected data objects. We call these functions as data-retrieve
functions (DRFs). Such functions can also be defined to specify the properties
of data structures. For each data-retrieve function $f$, we can derive the
definition of its corresponding memory-scope function (MSF) syntactically.
When an abstract value is retrieved by applying $f$ to a set of arguments,
applying the MSF of $f$ to same arguments results in a set of memory units
accessed during the retrievement. As long as no memory unit in this set is
modified during program executions, applying $f$ to same arguments results in
same abstract value.
We present a new proof rule for assignment statements, and another rule for
memory allocation statements. The proof rule for assignment statements says
that after the assignment, the memory unit referred by the left-hand stores
the value of the right-hand computed before the assignment. It also says that
the abstract values keep unchanged if the memory unit referred by the left-
hand is not in their memory scopes. The proof rule for memory allocation says
that after the allocation, the memory unit referred by the left-hand stores a
reference to a newly allocated memory block.
This logic has the following advantages.
* •
This logic is easy to learn. Most of the knowledge encoded in this logic have
been (explicitly or implicitly) taught in undergraduate CS courses. For
examples, the concept of recursive functions and first order logic are already
taught in undergraduate CS courses. The proof rules about program variables,
$\ast$, $\&\\!\\!\rightarrow\\!n$, $\&[]$ are taught informally in the
undergraduate courses about programming languages and compilers.
* •
This logic supports reuse of proofs. Most of the proved properties of DRFs are
about data structures. They are independent of the code under verification. So
these properties can be reused in verification of other code using same data
structures. It is possible to build a library of pre-defined DRFs, MSFs, and
their properties.
* •
Verification can be performed on different abstract levels. A group of
statements change the abstract value represented by a set of interconnected
data objects, but keep the structural properties of these data objects. People
can first understand the _pragmatic meaning_ of these statements, i.e. the
effect of these statements on the relevant abstract values. Then, they may
view these data objects as a virtual variable, and the statements as an
abstract statement assigning new value to this virtual variable. Thus, people
can reasoning the program at a more abstract level.
* •
Make use of the research results on pointer analysis. Many of the premises
when applying proof rules can be proved automatically by pointer analysis. For
example, for all assignment statements of the form $\ast(\&v)=e$, the premise
that $\&v\neq\textbf{nil}$ can be proved by pointer analyer easily. For
assignment statements of the form $\ast p=e$, the premise
$\textsf{p}\neq\textbf{nil}$ of the proof rule ASSIGN-ST can also be verified
automatically in many cases.
In the future, we will extended our logic to deal with more programming
language concepts: function calls, function pointers, class/object, generics,
$\dots$. At the mean time, we will try to build a library of pre-defined DRFs,
MSFs, and their properties for frequently used data structures.
## References
* [1] C.A.R. Hoare. An axiomatic basis for computer programming. _Communications of the ACM_ , 12(10):576-580 and 583, October 1969
* [2] C.A.R. Hoare. Proof of a program: FIND. _Communications of the ACM_ , 14(1):39-45, January 1971.
* [3] Rodney M. Burstall. Some techniques for proving correctness of programs which alter data structures. In _Machine Intelligence_ 7, pages 23-50. Edinburgh University Press, Edinburgh, Scoland, 1972
* [4] Stephen A. Cook and Derek C. Oppen. An assertion language for data structures. In Conference Record of 2nd ACM Symposium on Priciples of Programming Languages, pages 160-166. New York, 1975
* [5] Joseph M. Morris. A general axiom of assignment; assignment and linked data structures; a proof of the Schorr-Waite algorithm. In _Theoretical Foundations of Programming Methodology_ pages 25-51. D. Reidel, Dordrecht, Holland 1982.
* [6] Jonh C. Reynolds An overview of separation logic. In proceedings of _Verified Software: Theories, Tools, Experiments 2005_ , Zurich, Switzerland, October 10-13, 2005 Revised Selected Papers and Discussions
* [7] Hongseok Yang. An example of local reasoning in BI pointer logic: The Schorr-Waite graph marking algorithm. In Fritz Henglein, John Hughes, Henning Makholm, and Henning Niss, editors, _SPACE 2001: Informal Proceedings of Workshop on Semantics, Program Analysis and Computing Environments for Memory Management,_ pages 41 C68. IT University of Copenhagen, 2001
* [8] C.B. Jones and C.A.Middelburg A typed logic of partial functions reconstructed classically. In _Acta Inform_. 31 5 (1994), pp. 399 C430
* [9] David Gries The Schorr-Waite Graph Marking Algorithm. In Program Construction, International Summer School, pp 58-69, LNCS 69.
## Appendix 0.A Another example: inserting a node to a binary search tree
###### Example 4
The program depicted in Figure 5 add a new tuple $(\textsf{k},\textsf{d})$
into the map represented by a binary search tree. The types of the program
variables k and d are both integer. The type of program variables rt and tmp
are P($T$), where $T$ is
$\textbf{REC}((l,\textbf{P}(T))\times(r,\textbf{P}(T))\times(K,\textbf{integer})\times(D,\textbf{integer}))$.
The type of p is $\textbf{P}(\textbf{P}(T))$.
p:=&rt;
---
while | ($\ast\textsf{p}!=\textbf{nil}$)
{
| if ($\textsf{k}<\textsf{p}\rightarrow K$ )
$\textsf{p}:=\&(\ast\textsf{p})\rightarrow l$ else
$\textsf{p}:=\&(\ast\textsf{p})\rightarrow r$;
}
tmp = alloc($T$);
$\textsf{tmp}\rightarrow K:=\textsf{k}$; $\textsf{tmp}\rightarrow
D:=\textsf{d}$;
$\ast\textsf{p}$=tmp;
Figure 5: Another program
The DRFs depicted in Figure 6 are used in the specification and verification
of the program depicted in Figure 5. If $\ast x$ points to the root node of a
binary search tree, and $y$ is the address of a child-field of a node of this
tree. The DRF $\textsf{PNodeSet}(x,y)$ retrieve the set of the children-
pointer-field addresses (i.e. addresses of the fields $l$ and $r$) of all the
nodes in the binary search tree derived by setting $\ast y$ to nil. The
argument $x$ is also in this set. $\textsf{MapPP}(x,y)$ retrieve the map
represented by this modified binary search tree.
The boolean-typed DRF $\textsf{isHBSTK}(x,y)$ says that if we make $\ast y$
point to a newly allocated node
$\\{\textbf{nil},\textbf{nil},\textsf{k},\textsf{d}\\}$, $\ast x$ is still the
root node of a binary search tree. The DRF $\textsf{DomK}(x,y)$ retrieve the
keys stored in this tree.
The (simplified) definitions of the corresponding MSFs are depicted in Figure
7. We use $\verb"DMK"_{m}$, $\verb"STK"_{m}$, $\verb"MPPP"_{m}$ as
$\mathfrak{M}(\textsf{DomK})$, $\mathfrak{M}(\textsf{isSTK})$, and
$\mathfrak{M}(\textsf{MapPP})$ respectively. Let $\mathbb{P}^{\prime}$ be the
set of function definitions depicted in Figure 2 and Figure 6. Some of the
properties about the DRFs in $\mathbb{P}^{\prime}$ and corresponding MSFs are
depicted in Figure 8. Some of the DRFs and MSFs in Section 7, together with
their properties, are reused in this verification.
$\textsf{DomK}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{SetOf}(\textbf{integer})$
---
$\triangleq$ | $(x=y)\ \ \ \ $ | $\,?\ \\{\textsf{k}\\}:$
| $(\ast x=\textbf{nil})$ | $\,?\ \emptyset:\\{(\ast x)\rightarrow K\\}\cup\textsf{DomK}(\&(\ast x)\rightarrow l,y)\cup\textsf{DomK}(\&(\ast x)\rightarrow r,y)$
$\textsf{isHBSTK}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{boolean}$
$\triangleq$ | $(x=y)$ | $\,?\ \texttt{TRUE}\,:$
| $(\ast x=\textbf{nil})$ | $\,?\ \texttt{TRUE}:$
| | $\texttt{InHeap}(\ast x)\land\textsf{isHBSTK}(\&(\ast x)\rightarrow l,y)\land\textsf{isHBSTK}(\&(\ast x)\rightarrow r,y)\land$
| | $(\textsf{DomK}(\&(\ast x)\rightarrow l)=\emptyset?\texttt{TRUE}:\texttt{MAX}(\textsf{DomK}(\&(\ast x)\rightarrow l))<(\ast x)\rightarrow K)\land$
| | $(\textsf{DomK}(\&(\ast x)\rightarrow r)=\emptyset?\texttt{TRUE}:(\ast x)\rightarrow K<\texttt{MIN}(\textsf{DomK}(\&(\ast x)\rightarrow r)))$
$\textsf{MapPP}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{Map
integer to integer})$
$\triangleq$ | $(x=y)$ | $\,?\ \emptyset\,:$
| $(\ast x=\textbf{nil})$ | $\,?\,\emptyset\,:$
| | $\textsf{MapPP}(\&(\ast x)\rightarrow l,y){\dagger}\textsf{MapPP}(\&(\ast x)\rightarrow r,y){\dagger}\\{(\ast x)\rightarrow K\mapsto(\ast x)\rightarrow D\\}$
$\textsf{PNodeSet}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{SetOf}(\textbf{Ptr})$
$\triangleq\\{x\\}\cup((x=y)?\emptyset:(\ast
x=\textbf{nil})\,?\,\emptyset:(\textsf{PNodeSet}(\&(\ast x)\rightarrow
l,y)\cup\textsf{PNodeSet}(\&(\ast x)\rightarrow r,y))))$
Figure 6: DRFs for specifying and verifying the program in Figure 5
$\verb"DMK"_{m}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{SetOf}(\textbf{Ptr})$
---
$\triangleq$ | $(x=y)\ \ \ \ \ $ | $?\\{\&\textsf{k}\\}$
| $(\\{x\\}\cup(\ast x=\textbf{nil})?\emptyset:\\{x,\&(\ast x)\rightarrow
K\\}\cup\verb"DMK"_{m}(\&(\ast x)\rightarrow l,y)\cup\verb"DMK"_{m}(\&(\ast
x)\rightarrow r,y))$
$\verb"HBSTK"_{m}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{SetOf}(\textbf{Ptr})$
$\triangleq$ | $(x=y)$ | $\,?\ \emptyset\,:$
| $\\{x\\}\cup(\ast x=\textbf{nil})\,?\ \emptyset:\verb"HBSTK"_{m}(\&(\ast
x)\rightarrow l,y)\cup\verb"HBSTK"_{m}(\&(\ast x)\rightarrow r,y)\cup$
| | $\verb"DMK"_{m}(\&(\ast x)\rightarrow l)\cup(\textsf{DomK}(\&(\ast x)\rightarrow l)=\emptyset?\emptyset:\\{\&(\ast x)\rightarrow K\\})\cup$
| | $\verb"DMK"_{m}(\&(\ast x)\rightarrow r)\cup(\textsf{DomK}(\&(\ast x)\rightarrow r)=\emptyset?\emptyset:\\{\&(\ast x)\rightarrow K\\})$
$\verb"MPPP"_{m}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{SetOf}(\textbf{Ptr})$
$\triangleq$ | $(x=y)\,?\emptyset\,:$
| $\\{x\\}\cup(\ast x=\textbf{nil})\,?\,\emptyset\,:$
| | $\verb"MPPP"_{m}(\&(\ast x)\rightarrow l,y)\cup\verb"MPPP"_{m}(\&(\ast x)\rightarrow r,y)\cup\\{\&(\ast x)\rightarrow K,\&(\ast x)\rightarrow D\\}$
$\verb"PNS"_{m}(x,y):\textbf{P}(\textbf{P}(T))\times\textbf{P}(\textbf{P}(T))\rightarrow\textbf{SetOf}(\textbf{Ptr})$
$\triangleq(x=y)?\emptyset:\\{x\\}\cup(\ast
x=\textbf{nil})\,?\,\emptyset:(\verb"PNS"_{m}(\&(\ast x)\rightarrow
l,y)\cup\verb"PNS"_{m}(\&(\ast x)\rightarrow r,y))$
Figure 7: MSFs of the DRFs in Figure 6
$\mathbb{P}^{\prime},\textsf{isHBST}(\ast x),\textsf{isHBSTK}(x,y)\vdash
y\not\in\verb"DMK"_{m}(x,y)\cup\verb"HBSTK"_{m}(x,y)\cup\verb"MPPP"_{m}(x,y)\cup\verb"PNS"_{m}(x,y)$
(20)
$\mathbb{P}^{\prime},\textsf{isHBST}(\ast x),y\in\textsf{PNodeSet}(x,y),\ast
y\neq\textbf{nil}\vdash\ast y\in\textsf{NodeSet}(\ast x)$ (21)
$\begin{array}[]{l}\mathbb{P}^{\prime},\textsf{isHBST}(\ast
x),y\in\textsf{PNodeSet}(x,y)\land\textsf{isHBSTK}(x,y)\land\textsf{k}<(\ast
y)\rightarrow K\vdash\\\ \ \ \ \ \ \ \&(\ast y)\rightarrow
l\in\textsf{PNodeSet}(x,\&(\ast y)\rightarrow
l)\land\textsf{isHBSTK}(x,\&(\ast y)\rightarrow l)\end{array}$ (22)
$\begin{array}[]{l}\mathbb{P}^{\prime},\textsf{isHBST}(\ast
x),y\in\textsf{PNodeSet}(x,y)\land\textsf{isHBSTK}(x,y)\land\textsf{k}>(\ast
y)\rightarrow K\vdash\\\ \ \ \ \ \ \ \&(\ast y)\rightarrow
r\in\textsf{PNodeSet}(x,\&(\ast y)\rightarrow
r)\land\textsf{isHBSTK}(x,\&(\ast y)\rightarrow r)\end{array}$ (23)
$\begin{array}[]{l}\mathbb{P}^{\prime},\textsf{isHBSTK}(x,y),y\in\textsf{PNodeSet}(x,y),\texttt{inHeap}(\ast
y),\\\ (\ast y)\rightarrow K=\textsf{k}\land(\ast y)\rightarrow
l=\textbf{nil}\land(\ast y)\rightarrow
r=\textbf{nil}\end{array}\vdash\textsf{isHBST}(\ast x)$ (24)
$\mathbb{P}^{\prime},\textsf{isHBST}(\ast
x),y\in\textsf{PNodeSet}(x,y)\vdash\textsf{Map}(\ast
x)=\textsf{MapPP}(x,y){\dagger}\textsf{Map}(\ast y)$ (25)
$\mathbb{P}^{\prime},\textsf{isHBST}(\ast
x)\vdash\&\textsf{p}\not\in\verb"DMK"_{m}(x,y)\cup\verb"HBSTK"_{m}(x,y)\cup\verb"MPPP"_{m}(x,y)$
(26)
$\mathbb{P}^{\prime},\textsf{isHBST}(\ast
x)\vdash\&\textsf{tmp}\not\in\verb"DMK"_{m}(x,y)\cup\verb"HBSTK"_{m}(x,y)\cup\verb"MPPP"_{m}(x,y)$
(27)
Figure 8: Some properties about the DRFs and MSFs
We use PRE-COND as the abbreviation for
$\textsf{isHBST}(\textsf{rt})\land\textsf{k}\not\in\textsf{Dom}(\textsf{rt})\land\textsf{Map}(\textsf{rt})=M_{0}$.
The specification of this program is
$\textsf{PRE-COND}\\{\texttt{The
Program}\\}\textsf{isHBST}(\textsf{rt})\land\textsf{Map}(\textsf{rt})=M_{0}{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}$
The sketch of the proof is as follows. The common premise of these assertions
is $\mathbb{P}^{\prime}$, which is omitted for conciseness.
From the rule ASSIGN-ST, 1, 26, and
$\&\textsf{p}\not\in\\{\&\textsf{rt},\&\textsf{k}\\}$, we get following two
assertions:
$\begin{array}[]{l}(\textsf{PRE-
COND}\land\textsf{isHBSTK}(\&\textsf{rt},x)\land
x\in\textsf{PNodeSet}(\&\textsf{rt},x))[\&\textsf{rt}/x]\\\ \ \ \ \ \ \ \ \
\\{\textsf{p}=\&\textsf{rt};\\}\\\ (\textsf{PRE-
COND}\land\textsf{isHBSTK}(\&\textsf{rt},x)\land
x\in\textsf{PNodeSet}(\&\textsf{rt},x))[\textsf{p}/x]\end{array}$ (28)
$\begin{array}[]{l}(\textsf{PRE-
COND}\land(x\in\textsf{PNodeSet}(\&\textsf{rt},x))\land\textsf{isHBSTK}(\&\textsf{rt},x))[\&(\ast\textsf{p})\rightarrow
r/x]\\\ \ \ \ \ \ \ \ \\{\textsf{p}=\&(\ast\textsf{p})\rightarrow r\\}\\\
(\textsf{PRE-
COND}\land(x\in\textsf{PNodeSet}(\&\textsf{rt},x))\land\textsf{isHBSTK}(\&\textsf{rt},x))[\textsf{p}/x]\end{array}$
(29)
From the rule CONSEQUENCE, 29, and 23:
$\begin{array}[]{l}\textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\ast\textsf{p}\neq\textbf{nil}\land\\\
(\textsf{k}>\ast y\rightarrow K)\\\ \ \ \ \ \ \ \
\\{\textsf{p}=\&(\ast\textsf{p})\rightarrow r\\}\\\ \textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\end{array}$
(30)
Similarly to the way we get 30, we have:
$\begin{array}[]{l}\textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\ast\textsf{p}\neq\textbf{nil}\land\\\
(\textsf{k}<\ast y\rightarrow K)\\\ \ \ \ \ \ \ \
\\{\textsf{p}=\&(\ast\textsf{p})\rightarrow l\\}\\\ \textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\end{array}$
(31)
As $\ast\textsf{p}\neq\textbf{nil}$ implies $\textsf{k}<\textsf{p}\rightarrow
K\lor\textsf{k}\geq\textsf{p}\rightarrow K$. From the rule IF-ST, 30 and 31:
$\begin{array}[]{l}\textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\ast\textsf{p}\neq\textbf{nil}\\\
\ \ \ \ \ \ \ \\{\texttt{if }(\textsf{k}<\textsf{p}\rightarrow K)\textsf{
p}:=\&(\ast\textsf{p})\rightarrow l\texttt{ else
}\textsf{p}:=\&(\ast\textsf{p})\rightarrow r;\\}\\\ \textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\end{array}$
(32)
$\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))$ implies
$\textsf{p}\neq\textbf{nil}$, thus
$\ast\textsf{p}=\textbf{nil}\lor\ast\textsf{p}\neq\textbf{nil}$, From the rule
WHILE-ST, 32:
$\begin{array}[]{l}\textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\\\
\ \ \ \ \ \ \ \\{\texttt{the while statement}\\}\\\ \textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\ast\textsf{p}=\textbf{nil}\end{array}$
(33)
From the rule CONSEQUENCE, 25, $\ast\textsf{p}=\textbf{nil}$, and
$\textsf{Map}(\textbf{nil})=\emptyset$, we have
$\begin{array}[]{l}\textsf{PRE-
COND}\land(\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p}))\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\\\
\ \ \ \ \ \ \ \\{\texttt{the while statement}\\}\\\
\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\end{array}$
(34)
From the rule ALLOC-ST and the fact that tmp is not relevant to any terms, we
have:
$\begin{array}[]{l}\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\\\
\ \ \ \ \ \ \ \ \\{\textsf{tmp}=\texttt{alloc}(T);\\}\\\
\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\land\\\
\textsf{tmp}\neq\textbf{nil}\land\texttt{InHeap}(\textsf{tmp})\land\texttt{Unique}(\&\textsf{tmp})\land\texttt{PtrInit}(\textsf{tmp})\end{array}$
(35)
$\begin{array}[]{l}\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\land\\\
\textsf{tmp}\neq\textbf{nil}\land\texttt{InHeap}(\textsf{tmp})\land\texttt{Unique}(\&\textsf{tmp})\land\texttt{PtrInit}(\textsf{tmp})\\\
\ \ \ \ \ \ \ \ \\{\textsf{tmp}\rightarrow
K:=\textsf{k};\textsf{tmp}\rightarrow D:=\textsf{d};\\}\\\
\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\land\\\
\textsf{isHBST}(\textsf{tmp})\land\textsf{Map}(\textsf{tmp})=\\{\textsf{k}\mapsto\textsf{d}\\}\\\
\end{array}$ (36)
From the rule ASSIGN-ST, 20, and $\textsf{p}\not\in\\{\&\textsf{p}\\}$, we
have:
$\begin{array}[]{l}\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\land\\\
\textsf{isHBST}(\textsf{tmp})\land\textsf{Map}(\textsf{tmp})=\\{\textsf{k}\mapsto\textsf{d}\\}\\\
\ \ \ \ \ \ \ \ \\{\ast\textsf{p}:=\textsf{tmp};\\}\\\
\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\land\\\
\textsf{isHBST}(\ast\textsf{p})\land\textsf{Map}(\ast\textsf{p})=\\{\textsf{k}\mapsto\textsf{d}\\}\\\
\end{array}$ (37)
From the rule CONSEQUENCE, 24, and 25, we have:
$\begin{array}[]{l}\textsf{p}\in\textsf{PNodeSet}(\&\textsf{rt},\textsf{p})\land\textsf{isHBSTK}(\&\textsf{rt},\textsf{p})\land\textsf{MapPP}(\&\textsf{rt},\textsf{p})=M_{0}\land\\\
\textsf{isHBST}(\textsf{tmp})\land\textsf{Map}(\textsf{tmp})=\\{\textsf{k}\mapsto\textsf{d}\\}\\\
\ \ \ \ \ \ \ \ \\{\ast\textsf{p}:=\textsf{tmp};\\}\\\
\textsf{isHBST}(\textsf{rt})\land\textsf{Map}(\textsf{rt})=M_{0}{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}\end{array}$
(38)
From the rule SEQ-ST, and 28, 34, 35, 36, 38, we prove the specification.
$\textsf{PRE-COND}\\{\texttt{The
Program}\\}\textsf{isHBST}(\textsf{rt})\land\textsf{Map}(\textsf{rt})=M{\dagger}\\{\textsf{k}\mapsto\textsf{d}\\}$
(39)
## Appendix 0.B Verifying programs abstractly: the simplified Schorr-Waite
algorithm
The Schorr-Waite algorithm marks all nodes of a directed graph that are
reachable form one given node. The program depicted in Figure 9 is rewrite
from a simplified version presented by David Gries[9]. The variables
$\textsf{tmp},\textsf{p},\textsf{q},\textsf{root},\textsf{vroot}$ are declared
with type $\textbf{P}(T)$, and
$T=\textbf{REC}((m,\textbf{integer})\times(l,\textbf{P}(T))\times(r,\textbf{P}(T))$.
In this program, it is simplified that each node has exactly two non-nil
pointers (i.e. the field $l$ and $r$). We use this program to show how to
verify a program in an abstract level. The verification presented here is just
a sketch, many details are omitted.
p=root; q=vroot; /*$\textsf{vroot}\rightarrow l=\textsf{vroot}\rightarrow
r=\textsf{root}$*/
---
while | ($\textsf{p}\neq\textsf{vroot}$) |
{
| $\textsf{p}\rightarrow m=\textsf{p}\rightarrow m+1;$
| if ($\textsf{p}\rightarrow m=3\textbf{ or }(\&\textsf{p}\rightarrow
l)\rightarrow m=0$)
| {
| | $\textsf{tmp}:=\textsf{p};\textsf{p}:=\textsf{p}\rightarrow l;$
| | $\textsf{p}\rightarrow l=\textsf{p}\rightarrow r;\textsf{p}\rightarrow r:=\textsf{q};\textsf{q}=\textsf{tmp};$
| }
| else
| {
| | $\textsf{tmp}:=\textsf{p}\rightarrow l;\textsf{p}\rightarrow l:=\textsf{p}\rightarrow r;$
| | $\textsf{p}\rightarrow r:=\textsf{q};\textsf{q}:=\textsf{tmp}$
| }
}
Figure 9: The simplified Schorr-Waite algorithm
The DRFs used in (partial) specification and verification of this algorithm
are depicted in Figure 10. Intuitively speaking, the DRF
$\textsf{StackPath}(\textsf{p})$ retrieve the path from the virtual root vroot
to the current node p. $\textsf{Pred}(x)$ is used to compute the predecessor
of a node in the path. AcyclicSeq(x) is used to assert that the path retrieved
by $\textsf{StackPath}(\textsf{p})$ is acyclic.
$\textsf{StackPath}(x):\textbf{P}(T)\rightarrow\textbf{SeqOf}(\textbf{P}(T))$
$\triangleq(x=\textsf{vroot})\,?\,[\textsf{vroot}]:[x]^{\frown}\textsf{StackPath}(\textsf{Pred}(x)))$
$\textsf{Pred}(x):\textbf{P}(T)\rightarrow\textbf{P}(T)$
$\triangleq(x\rightarrow m=0)\,?\,\textsf{q}:((x\rightarrow
m=1)\,?\,x\rightarrow r:x\rightarrow l)$
$\textsf{AcyclicSeq}(x):\textbf{SeqOf}(\textbf{P}(T))$
$\triangleq\textbf{head}(x)\not\in\textbf{tail}(x)\land\textsf{AcyclicSeq}(\textbf{tail}(x))$
Figure 10: The functions defined to prove Schorr-Waite algorithm
Let $G$ be the node set of the graph; $L(\textsf{p})$ for original value of
$\textsf{p}\rightarrow l$; $R(\textsf{p})$ for original value of
$\textsf{p}\rightarrow r$; $\textsf{SUCC}(x)\triangleq(x\rightarrow
m=1)\,?\,R(x):L(x)$. From [9], the following invariant of the while statement
holds.
$\begin{array}[]{l}\forall x\in G\cdot(\ \ (x\rightarrow m=0\land x\rightarrow
l=L(x)\land x\rightarrow r=R(x))\lor\\\ \mbox{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}(x\rightarrow m=1\land x\rightarrow l=R(x)\land\textsf{SUCC}(x\rightarrow
r)=x)\lor\\\ \mbox{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(x\rightarrow
m=2\land\textsf{SUCC}(x\rightarrow l)=x\land x\rightarrow r=L(x))\lor\\\
\mbox{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(x\rightarrow m=3\land x\rightarrow
l=L(x)\land x\rightarrow r=R(x))\ \ )\\\ \bigwedge(\\\ \mbox{\ \ \ \ \
}(\textsf{p}\rightarrow m=0\land(L(\textsf{q})=\textsf{p}\lor
R(\textsf{q})=\textsf{p}))\lor\\\ \mbox{\ \ \ \ \ }(\textsf{p}\rightarrow
m=1\land\textsf{q}=L(\textsf{p}))\lor(\textsf{p}\rightarrow
m=2\land\textsf{q}=R(\textsf{p}))\ \ )\\\
\bigwedge\textsf{AcyclicSeq}(\textsf{StackPath}(\textsf{p}))\land\textsf{p}=\texttt{head}(\textsf{StackPath}(\textsf{p}))\end{array}$
(40)
We write this invariant as INV. The following specifications of the body of
the while statement can be proved. In these specifications,
$\stackrel{{\scriptstyle\leftarrow}}{{p}}$ and
$\stackrel{{\scriptstyle\leftarrow}}{{S}}$ are constants used to denote the
original value of p and the path.
$\begin{array}[]{l}\textbf{INV}\land\textsf{p}\rightarrow m=0\land
L(\textsf{p})\rightarrow
m=0\land\textsf{StackPath}(\textsf{p})=\stackrel{{\scriptstyle\leftarrow}}{{S}}\land\textsf{p}=\stackrel{{\scriptstyle\leftarrow}}{{p}}\\\
\mbox{}\ \ \ \ \mbox{\\{The body of the while statement\\}}\\\
\textbf{INV}\land\stackrel{{\scriptstyle\leftarrow}}{{p}}\rightarrow
m=1\land\textsf{StackPath}(\textsf{p})=L(\stackrel{{\scriptstyle\leftarrow}}{{p}})\,^{\frown}\stackrel{{\scriptstyle\leftarrow}}{{S}}\end{array}$
(41)
$\begin{array}[]{l}\textbf{INV}\land\textsf{p}\rightarrow m=1\land
R(\textsf{p})\rightarrow
m=0\land\textsf{StackPath}(\textsf{p})=\stackrel{{\scriptstyle\leftarrow}}{{S}}\land\textsf{p}=\stackrel{{\scriptstyle\leftarrow}}{{p}}\\\
\mbox{}\ \ \ \ \mbox{\\{The body of the while statement\\}}\\\
\textbf{INV}\land\stackrel{{\scriptstyle\leftarrow}}{{p}}\rightarrow
m=2\land\textsf{StackPath}(\textsf{p})=R(\stackrel{{\scriptstyle\leftarrow}}{{p}})\,^{\frown}\stackrel{{\scriptstyle\leftarrow}}{{S}}\\\
\end{array}$ (42)
$\begin{array}[]{l}\textbf{INV}\land\textsf{p}\rightarrow
m=2\land\textsf{StackPath}(\textsf{p})=\textsf{p}^{\
\frown}\stackrel{{\scriptstyle\leftarrow}}{{S}}\\\ \mbox{}\ \ \ \mbox{ \\{The
body of the while statement\\}}\\\
\textbf{INV}\land\textsf{p}=R(\stackrel{{\scriptstyle\leftarrow}}{{p}})\land\stackrel{{\scriptstyle\leftarrow}}{{p}}\rightarrow
m=3\land\textsf{StackPath}(\textsf{p})=\stackrel{{\scriptstyle\leftarrow}}{{S}}\\\
\end{array}$ (43)
If we view $\textsf{StackPath}(\textsf{p})$ as a virtual variable, it can be
seen that the body of the while statement have different pragmatic meanings
when the value of $\textsf{p}\rightarrow m$ equals to $0,1,2$. Based on these
properties, we can view the abstract program depicted in Figure 11 as an
abstract version of the program in Figure 9. From this abstract level, it is
clear that the program in Figure 9 is in fact an efficient and elaborative
implementation of the depth-first-search algorithm. We can continue proving
the algorithm based on this abstract program. Though assignment statements to
abstract variables are not allowed in the code, the abstract program can help
us thinking.
p:=root; S=$\emptyset$; $\textsf{push}(\textsf{vroot},\textsf{S})$;
$\textsf{push}(\textsf{p},\textsf{S})$;
---
while | ($\textsf{p}\neq\textsf{vroot}$) | do {
| $\textsf{p}\rightarrow m=\textsf{p}\rightarrow m+1$;
| if ($\textsf{p}\rightarrow m=1\land L(\textsf{p})\rightarrow m=0$ )
{$\textsf{push}(L(\textsf{p}),\textsf{S});$ }
| else | if($\textsf{p}\rightarrow m=2\land R(\textsf{p})\rightarrow m=0$ ) {$\textsf{push}(R(\textsf{p}),\textsf{S});$ }
| | else | if($\textsf{p}\rightarrow m=3$) {$\textsf{pop}(\textsf{S});$}
| | | else skip
| p = top(S)
| }
Figure 11: The abstract version of the simplified Schorr-Waite algorithm
|
arxiv-papers
| 2009-12-21T15:02:41 |
2024-09-04T02:49:07.172231
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianhua Zhao, Xuandong Li",
"submitter": "Jianhua Zhao",
"url": "https://arxiv.org/abs/0912.4184"
}
|
0912.4226
|
2010359-370Nancy, France 359
Javier Esparza Andreas Gaiser Stefan Kiefer
# Computing Least Fixed Points of
Probabilistic Systems of Polynomials
J. Esparza , A. Gaiser and S. Kiefer Fakultät für Informatik, Technische
Universität München, Germany esparza,gaiser,kiefer@model.in.tum.de
###### Abstract.
We study systems of equations of the form
$X_{1}=f_{1}(X_{1},\ldots,X_{n}),\ldots,X_{n}=f_{n}(X_{1},\ldots,X_{n})$ where
each $f_{i}$ is a polynomial with nonnegative coefficients that add up to $1$.
The least nonnegative solution, say $\mu$, of such equation systems is central
to problems from various areas, like physics, biology, computational
linguistics and probabilistic program verification. We give a simple and
strongly polynomial algorithm to decide whether $\mu=(1,\ldots,1)$ holds.
Furthermore, we present an algorithm that computes reliable sequences of lower
and upper bounds on $\mu$, converging linearly to $\mu$. Our algorithm has
these features despite using inexact arithmetic for efficiency. We report on
experiments that show the performance of our algorithms.
###### Key words and phrases:
computing fixed points, numerical approximation, stochastic models, branching
processes
###### 1991 Mathematics Subject Classification:
F.2.1 Numerical Algorithms and Problems, G.3 Probability and Statistics
## 1\. Introduction
We study how to efficiently compute the least nonnegative solution of an
equation system of the form
$\begin{array}[]{ccc}X_{1}=f_{1}(X_{1},\ldots,X_{n})&\ldots&X_{n}=f_{n}(X_{1},\ldots,X_{n})\;,\end{array}$
where, for every $i\in\\{1,\ldots,n\\}$, $f_{i}$ is a polynomial over
$X_{1},\ldots,X_{n}$ with positive rational coefficients that _add up to
1_.111Later, we allow that the coefficients add up to at most $1$. The
solutions are the fixed points of the function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ with $f=(f_{1},\ldots,f_{n})$. We
call $f$ a probabilistic system of polynomials (short: PSP). E.g., the PSP
$f(X_{1},X_{2})=\left(\,\frac{1}{2}X_{1}X_{2}+\frac{1}{2}\;,\;\frac{1}{4}X_{2}X_{2}+\frac{1}{4}X_{1}+\frac{1}{2}\,\right)$
induces the equation system
$\textstyle X_{1}=\frac{1}{2}X_{1}X_{2}+\frac{1}{2}\qquad
X_{2}=\frac{1}{4}X_{2}X_{2}+\frac{1}{4}X_{1}+\frac{1}{2}\;.$
Obviously, $\overline{1}=(1,\ldots,1)$ is a fixed point of every PSP. By
Kleene’s theorem, every PSP has a least nonnegative fixed point (called just
least fixed point in what follows), given by the limit of the sequence
$\overline{0},f(\overline{0}),f(f(\overline{0})),\ldots$
PSPs are important in different areas of the theory of stochastic processes
and computational models. A fundamental result of the theory of branching
processes, with numerous applications in physics, chemistry and biology (see
e.g. [9, 2]), states that extinction probabilities of species are equal to the
least fixed point of a PSP. The same result has been recently shown for the
probability of termination of certain probabilistic recursive programs [7, 6].
The consistency of stochastic context-free grammars, a problem of interest in
statistical natural language processing, also reduces to checking whether the
least fixed point of a PSP equals $\overline{1}$ (see e.g. [11]).
Given a PSP $f$ with least fixed point $\mu_{f}$, we study how to efficiently
solve the following two problems: (1) decide whether $\mu_{f}=\overline{1}$,
and (2) given a rational number $\epsilon>0$, compute
$\mathbf{lb},\mathbf{ub}\in\mathbb{Q}^{n}$ such that
$\mathbf{lb}\leq\mu_{f}\leq\mathbf{ub}$ and
$\mathbf{ub}-\mathbf{lb}\leq\overline{\epsilon}$ (where
$\mathbf{u}\leq\mathbf{v}$ for vectors $\mathbf{u},\mathbf{v}$ means $\leq$ in
all components). While the motivation for Problem (2) is clear (compute the
probability of extinction with a given accuracy), the motivation for Problem
(1) requires perhaps some explanation. In the case study of Section 4.3 we
consider a family of PSPs, taken from [9], modelling the neutron branching
process in a ball of radioactive material of radius $D$ (the family is
parameterized by $D$). The least fixed point is the probability that a neutron
produced through spontaneous fission does not generate an infinite “progeny”
through successive collisions with atoms of the ball; loosely speaking, this
is the probability that the neutron does not generate a chain reaction and the
ball does not explode. Since the number of atoms in the ball is very large,
spontaneous fission produces many neutrons per second, and so even if the
probability that a given neutron produces a chain reaction is very small, the
ball will explode with large probability in a very short time. It is therefore
important to determine the largest radius $D$ at which the probability of no
chain reaction is still $1$ (usually called the critical radius). An algorithm
for Problem (1) allows to compute the critical radius using binary search. A
similar situation appears in the analysis of parameterized probabilistic
programs. In [7, 6] it is shown that the question whether a probabilistic
program almost surely terminates can be reduced to Problem (1). Using binary
search one can find the “critical” value of the parameter for which the
program may not terminate any more.
Etessami and Yannakakis show in [7] that Problem (1) can be solved in
polynomial time by a reduction to (exact) Linear Programming (LP), which is
not known to be strongly polynomial. Our first result reduces Problem (1) to
solving a system of linear equations, resulting in a strongly polynomial
algorithm for Problem (1). The Maple library offers exact arithmetic solvers
for LP and systems of linear equations, which we use to test the performance
of our new algorithm. In the neutron branching process discussed above we
obtain speed-ups of about one order of magnitude with respect to LP.
The second result of the paper is, to the best of our knowledge, the first
practical algorithm for Problem (2). Lower bounds for $\mu_{f}$ can be
computed using Newton’s method for approximating a root of the function
$f(\overline{X})-\overline{X}$. This has recently been investigated in detail
[7, 10, 5]. However, Newton’s method faces considerable numerical problems.
Experiments show that naive use of exact arithmetic is inefficient, while
floating-point computation leads to false results even for very small systems.
For instance, the PReMo tool [12], which implements Newton’s method with
floating-point arithmetic for efficiency, reports $\mu_{f}\geq\overline{1}$
for a PSP with only 7 variables and small coefficients, although
$\mu_{f}<\overline{1}$ is the case (see Section 3.1).
Our algorithm produces a sequence of guaranteed lower and upper bounds, both
of which converge linearly to $\mu_{f}$. Linear convergence means that,
loosely speaking, the number of accurate bits of the bound is a linear
function of the position of the bound in the sequence. The algorithm is based
on the following idea. Newton’s method is an iterative procedure that, given a
current lower bound $\mathbf{lb}$ on $\mu_{f}$, applies a certain operator
$\mathcal{N}$ to it, yielding a new, more precise lower bound
$\mathcal{N}(\mathbf{lb})$. Instead of computing $\mathcal{N}(\mathbf{lb})$
using exact arithmetic, our algorithm computes two consecutive Newton steps,
i.e., $\mathcal{N}(\mathcal{N}(\mathbf{lb}))$, using inexact arithmetic. Then
it checks if the result satisfies a carefully chosen condition. If so, the
result is taken as the next lower bound. If not, then the precision is
increased, and the computation redone. The condition is eventually satisfied,
assuming the results of computing with increased precision converge to the
exact result. Usually, the repeated inexact computation is much faster than
the exact one. At the same time, a careful (and rather delicate) analysis
shows that the sequence of lower bounds converges linearly to $\mu_{f}$.
Computing upper bounds is harder, and seemingly has not been considered in the
literature before. Similarly to the case of lower bounds, we apply $f$ twice
to $\mathbf{ub}$, i.e., we compute $f(f({\bf ub}))$ with increasing precision
until a condition holds. The sequence so obtained may not even converge to
$\mu_{f}$. So we need to introduce a further operation, after which we can
then prove linear convergence.
We test our algorithm on the neutron branching process. The time needed to
obtain lower and upper bounds on the probability of no explosion with
$\epsilon=0.0001$ lies below the time needed to check, using exact LP, whether
this probability is $1$ or smaller than one. That is, in this case study our
algorithm is faster, and provides more information.
The rest of the paper is structured as follows. We give preliminary
definitions and facts in Section 2. Sections 3 and 4 present our algorithms
for solving Problems (1) and (2), and report on their performance on some case
studies. Section 5 contains our conclusions. The full version of the paper,
including all proofs, can be found in [4].
## 2\. Preliminaries
##### Vectors and matrices.
We use bold letters for designating (column) vectors, e.g.
$\mathbf{v}\in\mathbb{R}^{n}$. We write $\overline{s}$ with $s\in\mathbb{R}$
for the vector $(s,\ldots,s)^{\top}\in\mathbb{R}^{n}$ (where ⊤ indicates
transpose), if the dimension $n$ is clear from the context. The $i$-th
component of $\mathbf{v}\in\mathbb{R}^{n}$ will be denoted by
$\mathbf{v}_{i}$. We write $\mathbf{x}=\mathbf{y}$ (resp.
$\mathbf{x}\leq\mathbf{y}$ resp. $\mathbf{x}\prec\mathbf{y}$) if
$\mathbf{x}_{i}=\mathbf{y}_{i}$ (resp. $\mathbf{x}_{i}\leq\mathbf{y}_{i}$
resp. $\mathbf{x}_{i}<\mathbf{y}_{i}$) holds for all $i\in\\{1,\ldots,n\\}$.
By $\mathbf{x}<\mathbf{y}$ we mean $\mathbf{x}\leq\mathbf{y}$ and
$\mathbf{x}\neq\mathbf{y}$. By $\mathbb{R}^{m\times n}$ we denote the set of
real matrices with $m$ rows and $n$ columns. We write $\mathit{Id}$ for the
identity matrix. For a square matrix $A$, we denote by $\rho(A)$ the _spectral
radius_ of $A$, i.e., the maximum of the absolute values of the eigenvalues. A
matrix is nonnegative if all its entries are nonnegative. A nonnegative matrix
$A\in\mathbb{R}^{n\times n}$ is _irreducible_ if for every
$k,l\in\\{1,\ldots,n\\}$ there exists an $i\in\mathbb{N}$ so that
$(A^{i})_{kl}\not=0$.
##### Probabilistic Systems of Polynomials.
We investigate equation systems of the form
$\begin{array}[]{ccc}X_{1}=f_{1}(X_{1},\ldots,X_{n})&\ldots&X_{n}=f_{n}(X_{1},\ldots,X_{n}),\end{array}$
where the $f_{i}$ are polynomials in the variables $X_{1},\ldots,X_{n}$ with
positive real coefficients, and for every polynomial $f_{i}$ the sum of its
coefficients is _at most_ $1$. The vector $f:=(f_{1},\ldots,f_{n})^{\top}$ is
called a _probabilistic system of polynomials_ (PSP for short) and is
identified with its induced function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. If $X_{1},\ldots,X_{n}$ are the
formal variables of $f$, we define $\overline{X}:=(X_{1},\ldots,X_{n})^{\top}$
and $\text{Var}(f):=\\{X_{1},\ldots,X_{n}\\}$. We assume that $f$ is
represented as a list of polynomials, and each polynomial is a list of its
monomials. If $S\subseteq\\{X_{1},\ldots,X_{n}\\}$, then $f_{S}$ denotes the
result of removing the polynomial $f_{i}(X_{1},\ldots,X_{n})$ from $f$ for
every $x_{i}\notin S$; further, given $\mathbf{x}\in\mathbb{R}^{n}$ and
$B\in\mathbb{R}^{n\times n}$, we denote by $\mathbf{x}_{S}$ and $B_{SS}$ the
vector and the matrix obtained from $\mathbf{x}$ and $B$ by removing the
entries with indices $i$ such that $X_{i}\not\in S$. The coefficients are
represented as fractions of positive integers. The size of $f$ is the size of
that representation. The _degree_ of $f$ is the maximum of the degrees of
$f_{1},\ldots,f_{n}$. PSPs of degree $0$ (resp. $1$ resp. $\mathord{>}1$) are
called constant (resp. _linear_ resp. superlinear). PSPs $f$ where the degree
of each $f_{i}$ is at least $2$ are called purely superlinear. We write
$f^{\prime}$ for the _Jacobian_ of $f$, i.e., the matrix of first partial
derivatives of $f$.
Given a PSP $f$, a variable $X_{i}$ depends directly on a variable $X_{j}$ if
$X_{j}$ “occurs” in $f_{i}$, more formally if $\frac{\partial f_{i}}{\partial
X_{j}}$ is not the constant $0$. A variable $X_{i}$ depends on $X_{j}$ if
$X_{i}$ depends directly on $X_{j}$ or there is a variable $X_{k}$ such that
$X_{i}$ depends directly on $X_{k}$ and $X_{k}$ depends on $X_{j}$. We often
consider the strongly connected components (or SCCs for short) of the
dependence relation. The SCCs of a PSP can be computed in linear time using
e.g. Tarjan’s algorithm. An SCC $S$ of a PSP $f$ is constant resp. linear
resp. superlinear resp. purely superlinear if the PSP $\tilde{f}$ has the
respective property, where $\tilde{f}$ is obtained by restricting $f$ to the
$S$-components and replacing all variables not in $S$ by the constant $1$. A
PSP is an scPSP if it is not constant and consists of only one SCC. Notice
that a PSP $f$ is an scPSP if and only if $f^{\prime}(\overline{1})$ is
irreducible.
A fixed point of a PSP $f$ is a vector $\mathbf{x}\geq\overline{0}$ with
$f(\mathbf{x})=\mathbf{x}$. By Kleene’s theorem, there exists a least fixed
point $\mu_{f}$ of $f$, i.e., $\mu_{f}\leq\mathbf{x}$ holds for every fixed
point $\mathbf{x}$. Moreover, the sequence
$\overline{0},f(\overline{0}),f(f(\overline{0})),\ldots$ converges to
$\mu_{f}$. Vectors $\mathbf{x}$ with $\mathbf{x}\leq f(\mathbf{x})$ (resp.
$\mathbf{x}\geq f(\mathbf{x})$) are called pre-fixed (resp. post-fixed)
points. Notice that the vector $\overline{1}$ is always a post-fixed point of
a PSP $f$, due to our assumption on the coefficients of a PSP. By Knaster-
Tarski’s theorem, $\mu_{f}$ is the least post-fixed point, so we always have
$\overline{0}\leq\mu_{f}\leq\overline{1}$. It is easy to detect and remove all
components $i$ with $(\mu_{f})_{i}=0$ by a simple round-robin method (see e.g.
[5]), which needs linear time in the size of $f$. We therefore assume in the
following that $\mu_{f}\succ\overline{0}$.
## 3\. An algorithm for consistency of PSPs
Recall that for applications like the neutron branching process it is crucial
to know exactly whether $\mu_{f}=\overline{1}$ holds. We say a PSP $f$ is
consistent if $\mu_{f}=\overline{1}$; otherwise it is _inconsistent_.
Similarly, we call a component $i$ consistent if $(\mu_{f})_{i}=1$. We present
a new algorithm for the consistency problem, i.e., the problem to check a PSP
for consistency.
It was proved in [7] that consistency is checkable in polynomial time by
reduction to Linear Programming (LP). We first observe that consistency of
general PSPs can be reduced to consistency of scPSPs by computing the DAG of
SCCs, and checking consistency SCC-wise [7]: Take any bottom SCC $S$, and
check the consistency of $f_{S}$. (Notice that $f_{S}$ is either constant or
an scPSP; if constant, $f_{S}$ is consistent iff $f_{S}=1$, if an scPSP, we
can check its consistency by assumption.) If $f_{S}$ is inconsistent, then so
is $f$, and we are done. If $f_{S}$ is consistent, then we remove every
$f_{i}$ from $f$ such that $x_{i}\in S$, replace all variables of $S$ in the
remaining polynomials by the constant $1$, and iterate (choose a new bottom
SCC, etc.). Note that this algorithm processes each polynomial at most once,
as every variable belongs to exactly one SCC.
It remains to reduce the consistency problem for scPSPs to LP. The first step
is:
###### Proposition 3.1.
[9, 7] An scPSP $f$ is consistent iff $\rho(f^{\prime}(\overline{1}))\leq 1$
(i.e., iff the spectral radius of the Jacobi matrix $f^{\prime}$ evaluated at
the vector $\overline{1}$ is at most $1$).
The second step consists of observing that the matrix
$f^{\prime}(\overline{1})$ of an scPSP $f$ is irreducible and nonnegative. It
is shown in [7] that $\rho(A)\leq 1$ holds for an irreducible and nonnegative
matrix $A$ iff the system of inequalities
$A\mathbf{x}\geq\mathbf{x}+\overline{1}\text{ , }\mathbf{x}\geq\overline{0}$
(1)
is infeasible. However, no strongly polynomial algorithm for LP is known, and
we are not aware that (1) falls within any subclass solvable in strongly
polynomial time [8].
We provide a very simple, strongly polynomial time algorithm to check whether
$\rho(f^{\prime}(\overline{1}))\leq 1$ holds. We need some results from
Perron-Frobenius theory (see e.g. [3]).
###### Lemma 3.2.
Let $A\in\mathbb{R}^{n\times n}$ be nonnegative and irreducible.
* (1)
$\rho(A)$ is a _simple_ eigenvalue of $A$.
* (2)
There exists an eigenvector $\mathbf{v}\succ\overline{0}$ with $\rho(A)$ as
eigenvalue.
* (3)
Every eigenvector $\mathbf{v}\succ\overline{0}$ has $\rho(A)$ as eigenvalue.
* (4)
For all $\alpha,\beta\in\mathbb{R}\setminus\\{0\\}$ and
$\mathbf{v}>\overline{0}$: if $\alpha\mathbf{v}<A\mathbf{v}<\beta\mathbf{v}$,
then $\alpha<\rho(A)<\beta$.
The following lemma is the key to the algorithm:
###### Lemma 3.3.
Let $A\in\mathbb{R}^{n\times n}$ be nonnegative and irreducible.
1. (a)
Assume there is $\mathbf{v}\in\mathbb{R}^{n}\setminus\\{\overline{0}\\}$ such
that $(\mathit{Id}-A)\mathbf{v}=\overline{0}$. Then $\rho(A)\leq 1$ iff
$\mathbf{v}\succ\overline{0}$ or $\mathbf{v}\prec\overline{0}$.
2. (b)
Assume $\mathbf{v}=\overline{0}$ is the only solution of
$(\mathit{Id}-A)\mathbf{v}=\overline{0}$. Then there exists a unique
$\mathbf{x}\in\mathbb{R}^{n}$ such that
$(\mathit{Id}-A)\mathbf{x}=\overline{1}$, and $\rho(A)\leq 1$ iff
$\mathbf{x}\geq\overline{1}$ and $A\mathbf{x}<\mathbf{x}$.
###### Proof 3.4.
1. (a)
From $(\mathit{Id}-A)\mathbf{v}=\overline{0}$ it follows
$A\mathbf{v}=\mathbf{v}$. We see that $\mathbf{v}$ is an eigenvector of $A$
with eigenvalue $1$. So $\rho(A)\geq 1$.
($\Leftarrow$): As both $\mathbf{v}$ and $-\mathbf{v}$ are eigenvectors of $A$
with eigenvalue $1$, we can assume w.l.o.g. that
$\mathbf{v}\succ\overline{0}$. By Lemma 3.2(3), $\rho(A)$ is the eigenvalue of
$\mathbf{v}$, and so $\rho(A)=1$.
($\Rightarrow$): Since $\rho(A)\leq 1$ and $\rho(A)\geq 1$, it follows that
$\rho(A)=1$. By Lemma 3.2(1) and (2), the eigenspace of the eigenvalue $1$ is
one-dimensional and contains a vector $\mathbf{x}\succ\overline{0}$. So
$\mathbf{v}=\alpha\cdot\mathbf{x}$ for some
$\alpha\in\mathbb{R},\alpha\not=0$. If $\alpha>0$, we have
$\mathbf{v}\succ\overline{0}$, otherwise $\mathbf{v}\prec\overline{0}$.
2. (b)
With the assumption and basic facts from linear algebra it follows that
$(Id-A)$ has full rank and therefore $(\mathit{Id}-A)\mathbf{x}=\overline{1}$
has a unique solution $\mathbf{x}$. We still have to prove the second part of
the conjunction:
($\Leftarrow$): Follows directly from Lemma 3.2(4).
($\Rightarrow$): Let $\rho(A)\leq 1$. Assume for a contradiction that
$\rho(A)=1$. Then, by Lemma 3.2(1), the matrix $A$ would have an eigenvector
$\mathbf{v}\neq\overline{0}$ with eigenvalue $1$, so
$(\mathit{Id}-A)\mathbf{v}=\overline{0}$, contradicting the assumption. So we
have, in fact, $\rho(A)<1$. By standard matrix facts (see e.g. [3]), this
implies that $(\mathit{Id}-A)^{-1}=A^{*}=\sum_{i=0}^{\infty}A^{i}$ exists, and
so we have
$\mathbf{x}=(\mathit{Id}-A)^{-1}\overline{1}=A^{*}\overline{1}\geq\overline{1}$.
Furthermore,
$A\mathbf{x}=\sum_{i=1}^{\infty}A^{i}\overline{1}<\sum_{i=0}^{\infty}A^{i}\overline{1}=\mathbf{x}$.
∎
In order to check whether $\rho(A)\leq 1$, we first solve the system
$(\mathit{Id}-A)\mathbf{v}=\overline{0}$ using Gaussian elimination. If we
find a vector $\mathbf{v}\not=\overline{0}$ such that
$(Id-A)\mathbf{v}=\overline{0}$, we apply Lemma 3.3(a). If
$\mathbf{v}=\overline{0}$ is the only solution of
$(Id-A)\mathbf{v}=\overline{0}$, we solve
$(\mathit{Id}-A)\mathbf{v}=\overline{1}$ using Gaussian elimination again, and
apply Lemma 3.3(b). Since Gaussian elimination of a rational $n$-dimensional
linear equation system can be carried out in strongly polynomial time using
$O(n^{3})$ arithmetic operations (see e.g. [8]), we obtain:
###### Proposition 3.5.
Given a nonnegative irreducible matrix $A\in\mathbb{R}^{n\times n}$, one can
decide in strongly polynomial time, using $O(n^{3})$ arithmetic operations,
whether $\rho(A)\leq 1$.
Combining Propositions 3.1 and 3.5 directly yields an algorithm for checking
the consistency of scPSPs. Extending it to multiple SCCs as above, we get:
###### Theorem 3.6.
Let $f(X_{1},\ldots,X_{n})$ be a PSP. There is a strongly polynomial time
algorithm that uses $O(n^{3})$ arithmetic operations and determines the
consistency of $f$.
### 3.1. Case study: A family of “almost consistent” PSPs
In this section, we illustrate some issues faced by algorithms that solve the
consistency problem. Consider the following family $h^{(n)}$ of scPSPs, $n\geq
2$:
$h^{(n)}=\left(\;0.5X_{1}^{2}+0.1X_{n}^{2}+0.4\;,\;0.01X_{1}^{2}+0.5X_{2}+0.49\;,\;\ldots\;,0.01X_{n-1}^{2}+0.5X_{n}+0.49\;\right)^{\top}\;.$
It is not hard to show that $h^{(n)}(\mathbf{p})\prec\mathbf{p}$ holds for
$\mathbf{p}=(1-0.02^{n},\ldots,1-0.02^{2n-1})^{\top}$, so we have
$\mu_{h^{(n)}}\prec\overline{1}$ by Proposition 4.4, i.e., the $h^{(n)}$ are
inconsistent.
The tool PReMo [12] relies on Java’s floating-point arithmetic to compute
approximations of the least fixed point of a PSP. We invoked PReMo for
computing approximants of $\mu_{h^{(n)}}$ for different values of $n$ between
$5$ and $100$. Due to its fixed precision, PReMo’s approximations for
$\mu_{h^{(n)}}$ are $\geq 1$ in all components if $n\geq 7$. This might lead
to the wrong conclusion that $h^{(n)}$ is consistent.
Recall that the consistency problem can be solved by checking the feasibility
of the system (1) with $A=f^{\prime}(\overline{1})$. We checked it with
lp_solve, a well-known LP tool using hardware floating-point arithmetic. The
tool wrongly states that (1) has no solution for $h^{(n)}$-systems with
$n>10$. This is due to the fact that the solutions cannot be represented
adequately using machine number precision.222The mentioned problems of PReMo
and lp_solve are not due to the fact that the coefficients of $h^{(n)}$ cannot
be properly represented using basis 2: The problems persist if one replaces
the coefficients of $h^{(n)}$ by similar numbers exactly representable by
machine numbers. Finally, we also checked feasibility with Maple’s Simplex
package, which uses exact arithmetic, and compared its performance with the
implementation, also in Maple, of our consistency algorithm. Table 1 shows the
results. Our algorithm clearly outperforms the LP approach. For more
experiments see Section 4.3.
| $n=25$ | $n=100$ | $n=200$ | $n=400$ | $n=600$ | $n=1000$
---|---|---|---|---|---|---
Exact LP | $<1$ sec | 2 sec | 8 sec | 67 sec | 208 sec | $>$ 2h
Our algorithm | $<1$ sec | $<1$ sec | 1 sec | 4 sec | 10 sec | 29 sec
Table 1. Consistency checks for $h^{(n)}$-systems: Runtimes of different
approaches.
## 4\. Approximating $\mu_{f}$ with inexact arithmetic
It is shown in [7] that $\mu_{f}$ may not be representable by roots, so one
can only approximate $\mu_{f}$. In this section we present an algorithm that
computes two sequences, $(\mathbf{lb}^{(i)})_{i}$ and
$(\mathbf{ub}^{(i)})_{i}$, such that
$\mathbf{lb}^{(i)}\leq\mu_{f}\leq\mathbf{ub}^{(i)}$ and
$\lim_{i\to\infty}\mathbf{ub}^{(i)}-\mathbf{lb}^{(i)}=\overline{0}$. In words:
$\mathbf{lb}^{(i)}$ and $\mathbf{ub}^{(i)}$ are lower and upper bounds on
$\mu_{f}$, respectively, and the sequences converge to $\mu_{f}$. Moreover,
they converge linearly, meaning that the number of accurate bits of
$\mathbf{lb}^{(i)}$ and $\mathbf{ub}^{(i)}$ are linear functions of $i$. (The
number of accurate bits of a vector $\mathbf{x}$ is defined as the greatest
number $k$ such that $|(\mu_{f}-\mathbf{x})_{j}|/|(\mu_{f})_{j}|\leq 2^{-k}$
holds for all $j\in\\{1,\ldots,n\\}$.) These properties are guaranteed even
though our algorithm uses inexact arithmetic: Our algorithm detects numerical
problems due to rounding errors, recovers from them, and increases the
precision of the arithmetic as needed. Increasing the precision dynamically
is, e.g., supported by the GMP library [1].
Let us make precise what we mean by increasing the precision. Consider an
elementary operation $g$, like multiplication, subtraction, etc., that
operates on two input numbers $x$ and $y$. We can compute $g(x,y)$ with
increasing precision if there is a procedure that on input $x,y$ outputs a
sequence $g^{(1)}(x,y),g^{(2)}(x,y),\ldots$ that converges to $g(x,y)$. Note
that there are no requirements on the convergence speed of this procedure — in
particular, we do not require that there is an $i$ with $g^{(i)}(x,y)=g(x,y)$.
This procedure, which we assume exists, allows to implement floating
assignments of the form
$z\hskip
2.84526pt\rotatebox[x=5.69054pt,y=2.84526pt]{180.0}{$\rightsquigarrow$}\,g(x,y)\textbf{
such that }\phi(z)$
with the following semantics: $z$ is assigned the value $g^{(i)}(x,y)$, where
$i\geq 1$ is the smallest index such that $\phi(g^{(i)}(x,y))$ holds. We say
that the assignment is valid if $\phi(g(x,y))$ holds and $\phi$ involves only
continuous functions and strict inequalities. Our assumption on the arithmetic
guarantees that (the computation underlying) a valid floating assignment
terminates. As “syntactic sugar”, more complex operations (e.g., linear
equation solving) are also allowed in floating assignments, because they can
be decomposed into elementary operations.
We feel that any implementation of arbitrary precision arithmetic should
satisfy our requirement that the computed values converge to the exact result.
For instance, the documentation of the GMP library [1] states: “Each function
is defined to calculate with ‘infinite precision’ followed by a truncation to
the destination precision, but of course the work done is only what’s needed
to determine a result under that definition.”
To approximate the least fixed point of a PSP, we first transform it into a
certain normal form. A purely superlinear PSP $f$ is called perfectly
superlinear if every variable depends directly on itself and every superlinear
SCC is purely superlinear. The following proposition states that any PSP $f$
can be made perfectly superlinear.
###### Proposition 4.1.
Let $f$ be a PSP of size $s$. We can compute in time $O(n\cdot s)$ a perfectly
superlinear PSP $\tilde{f}$ with
$\text{Var}(\tilde{f})=\text{Var}(f)\cup\\{\tilde{X}\\}$ of size $O(n\cdot s)$
such that $\mu_{f}=(\mu_{\tilde{f}})_{\text{Var}(f)}$.
### 4.1. The algorithm
The algorithm receives as input a perfectly superlinear PSP $f$ and an error
bound $\epsilon>0$, and returns vectors $\mathbf{lb},\mathbf{ub}$ such that
$\mathbf{lb}\leq\mu_{f}\leq\mathbf{ub}$ and
$\mathbf{ub}-\mathbf{lb}\leq\overline{\epsilon}$. A first initialization step
requires to compute a vector $\mathbf{x}$ with
$\overline{0}\prec\mathbf{x}\prec f(\mathbf{x})$, i.e., a “strict” pre-fixed
point. This is done in Section 4.1.1. The algorithm itself is described in
Section 4.1.2.
#### 4.1.1. Computing a strict pre-fixed point
Algorithm 1 computes a strict pre-fixed point:
Input: perfectly superlinear PSP $f$
Output: $\mathbf{x}$ with $\overline{0}\prec\mathbf{x}\prec
f(\mathbf{x})\prec\overline{1}$
$\mathbf{x}\leftarrow\overline{0}$;
while _$\overline{0}\not\prec\mathbf{x}$_ do $Z\leftarrow\\{i\mid 1\leq i\leq
n,f_{i}(\mathbf{x})=0\\}$;
$P\leftarrow\\{i\mid 1\leq i\leq n,f_{i}(\mathbf{x})>0\\}$;
$\mathbf{y}_{Z}\leftarrow\overline{0}$;
$\mathbf{y}_{P}\hskip
2.84526pt\rotatebox[x=5.69054pt,y=2.84526pt]{180.0}{$\rightsquigarrow$}\,f_{P}(\mathbf{x})$
such that $\overline{0}\prec\mathbf{y}_{P}\prec
f_{P}(\mathbf{y})\prec\overline{1}$;
$\mathbf{x}\leftarrow\mathbf{y}$;
Algorithm 1 Procedure computeStrictPrefix
###### Proposition 4.2.
Algorithm 1 is correct and terminates after at most $n$ iterations.
The reader may wonder why Algorithm 1 uses a floating assignment
$\mathbf{y}_{P}\hskip
2.84526pt\rotatebox[x=5.69054pt,y=2.84526pt]{180.0}{$\rightsquigarrow$}\,f_{P}(\mathbf{x})$,
given that it must also perform exact comparisons to obtain the sets $Z$ and
$P$ and to decide exactly whether $\mathbf{y}_{P}\prec f_{P}(\mathbf{y})$
holds in the such that clause of the floating assignment. The reason is that,
while we perform such operations exactly, we do not want to use the result of
exact computations as input for other computations, as this easily leads to an
explosion in the required precision. For instance, the size of the exact
result of $f_{P}(\mathbf{y})$ may be larger than the size of $\mathbf{y}$,
while an approximation of smaller size may already satisfy the such that
clause. In order to emphasize this, we never store the result of an exact
numerical computation in a variable.
#### 4.1.2. Computing lower and upper bounds
Algorithm 1 uses Kleene iteration
$\overline{0},f(\overline{0}),f(f(\overline{0})),\ldots$ to compute a strict
pre-fixed point. One could, in principle, use the same scheme to compute lower
bounds of $\mu_{f}$, as this sequence converges to $\mu_{f}$ from below by
Kleene’s theorem. However, convergence of Kleene iteration is generally slow.
It is shown in [7] that for the $1$-dimensional PSP $f$ with
$f(X)=0.5X^{2}+0.5$ we have $\mu_{f}=1$, and the $i$-th Kleene approximant
$\mathbf{\boldsymbol{\kappa}}^{(i)}$ satisfies
$\mathbf{\boldsymbol{\kappa}}^{(i)}\leq 1-\frac{1}{i}$. Hence, Kleene
iteration may converge only logarithmically, i.e., the number of accurate bits
is a logarithmic function of the number of iterations.
In [7] it was suggested to use Newton’s method for faster convergence. In
order to see how Newton’s method can be used, observe that instead of
computing $\mu_{f}$, one can equivalently compute the least nonnegative zero
of $f(\overline{X})-\overline{X}$. Given an approximant $\mathbf{x}$ of
$\mu_{f}$, Newton’s method first computes $g^{(\mathbf{x})}(\overline{X})$,
the first-order linearization of $f$ at the point $\mathbf{x}$:
$g^{(\mathbf{x})}(\overline{X})=f(\mathbf{x})+f^{\prime}(\mathbf{x})(\overline{X}-\mathbf{x})$
The next Newton approximant $\mathbf{y}$ is obtained by solving
$\overline{X}=g^{(\mathbf{x})}(\overline{X})$, i.e.,
$\mathbf{y}=\mathbf{x}+(\mathit{Id}-f^{\prime}(\mathbf{x}))^{-1}(f(\mathbf{x})-\mathbf{x})\;.$
We write
$\mathcal{N}_{f}(\mathbf{x}):=\mathbf{x}+(\mathit{Id}-f^{\prime}(\mathbf{x}))^{-1}(f(\mathbf{x})-\mathbf{x})$,
and usually drop the subscript of $\mathcal{N}_{f}$. If
$\mathbf{\boldsymbol{\nu}}^{(0)}\leq\mu_{f}$ is any pre-fixed point of $f$,
for instance $\mathbf{\boldsymbol{\nu}}^{(0)}=\overline{0}$, we can define a
Newton sequence $(\mathbf{\boldsymbol{\nu}}^{(i)})_{i}$ by setting
$\mathbf{\boldsymbol{\nu}}^{(i+1)}=\mathcal{N}(\mathbf{\boldsymbol{\nu}}^{(i)})$
for $i\geq 0$. It has been shown in [7, 10, 5] that Newton sequences converge
at least linearly to $\mu_{f}$. Moreover, we have
$\overline{0}\leq\mathbf{\boldsymbol{\nu}}^{(i)}\leq
f(\mathbf{\boldsymbol{\nu}}^{(i)})\leq\mu_{f}$ for all $i$.
These facts were shown only for Newton sequences that are computed exactly,
i.e., without rounding errors. Unfortunately, Newton approximants are hard to
compute exactly: Since each iteration requires to solve a linear equation
system whose coefficients depend on the results of the previous iteration, the
size of the Newton approximants easily explodes. Therefore, we wish to use
inexact arithmetic, but without losing the good properties of Newton’s method
(reliable lower bounds, linear convergence).
Algorithm 2 accomplishes these goals, and additionally computes post-fixed
points $\mathbf{ub}$ of $f$, which are upper bounds on $\mu_{f}$.
Input: perfectly superlinear PSP $f$, error bound $\epsilon>0$
Output: vectors $\mathbf{lb},\mathbf{ub}$ such that
$\mathbf{lb}\leq\mu_{f}\leq\mathbf{ub}$ and
$\mathbf{ub}-\mathbf{lb}\leq\overline{\epsilon}$
$\mathbf{lb}\leftarrow\texttt{computeStrictPrefix}(f)$;
$\mathbf{ub}\leftarrow\overline{1}$;
while _$\mathbf{ub}-\mathbf{lb}\not\leq\overline{\epsilon}$_ do
$\mathbf{x}\hskip
2.84526pt\rotatebox[x=5.69054pt,y=2.84526pt]{180.0}{$\rightsquigarrow$}\,\mathcal{N}(\mathcal{N}(\mathbf{lb}))$
such that
$f(\mathbf{lb})+f^{\prime}(\mathbf{lb})(\mathbf{x}-\mathbf{lb})\prec\mathbf{x}\prec
f(\mathbf{x})\prec\overline{1}$;
$\mathbf{lb}\leftarrow\mathbf{x}$;
$Z\leftarrow\\{i\mid 1\leq i\leq n,f_{i}(\mathbf{ub})=1\\}$;
$P\leftarrow\\{i\mid 1\leq i\leq n,f_{i}(\mathbf{ub})<1\\}$;
$\mathbf{y}_{Z}\leftarrow\overline{1}$;
$\mathbf{y}_{P}\hskip
2.84526pt\rotatebox[x=5.69054pt,y=2.84526pt]{180.0}{$\rightsquigarrow$}\,f_{P}(f(\mathbf{ub}))$
such that $f_{P}(\mathbf{y})\prec\mathbf{y}_{P}\prec f_{P}(\mathbf{ub})$;
forall _superlinear SCCs $S$ of $f$ with $\mathbf{y}_{S}=\overline{1}$_ do
$\mathbf{t}\leftarrow\overline{1}-\mathbf{lb}_{S}$;
if _$f_{SS}^{\prime}(\overline{1})\mathbf{t}\succ\mathbf{t}$_ then
$\displaystyle\mathbf{y}_{S}\hskip
2.84526pt\rotatebox[x=5.69054pt,y=2.84526pt]{180.0}{$\rightsquigarrow$}\,\overline{1}-\min\left\\{1,\frac{\min_{i\in
S}(f_{SS}^{\prime}(\overline{1})\mathbf{t}-\mathbf{t})_{i}}{2\cdot\max_{i\in
S}(f_{S}(\overline{2}))_{i}}\right\\}\cdot\mathbf{t}$ such that
$f_{S}(\mathbf{y})\prec\mathbf{y}_{S}\prec\overline{1}$;
$\mathbf{ub}\leftarrow\mathbf{y}$;
Algorithm 2 Procedure calcBounds
Let us describe the algorithm in some detail. The lower bounds are stored in
the variable $\mathbf{lb}$. The first value of $\mathbf{lb}$ is not simply
$\overline{0}$, but is computed by $\texttt{computeStrictPrefix}(f)$, in order
to guarantee the validity of the following floating assignments. We use
Newton’s method for improving the lower bounds because it converges fast (at
least linearly) when performed exactly. In each iteration of the algorithm,
two Newton steps are performed using inexact arithmetic. The intention is that
two inexact Newton steps should improve the lower bound at least as much as
one exact Newton step. While this may sound like a vague hope for small
rounding errors, it can be rigorously proved thanks to the such that clause of
the floating assignment in line 2. The proof involves two steps. The first
step is to prove that $\mathcal{N}(\mathcal{N}(\mathbf{lb}))$ is a (strict)
post-fixed point of the function
$g(\overline{X})=f(\mathbf{lb})+f^{\prime}(\mathbf{lb})(\overline{X}-\mathbf{lb})$,
i.e., $\mathcal{N}(\mathcal{N}(\mathbf{lb}))$ satisfies the first inequality
in the such that clause. For the second step, recall that
$\mathcal{N}(\mathbf{lb})$ is the least fixed point of $g$. By Knaster-
Tarski’s theorem, $\mathcal{N}(\mathbf{lb})$ is actually the least post-fixed
point of $g$. So, our value $\mathbf{x}$, the inexact version of
$\mathcal{N}(\mathcal{N}(\mathbf{lb}))$, satisfies
$\mathbf{x}\geq\mathcal{N}(\mathbf{lb})$, and hence two inexact Newton steps
are in fact at least as “fast” as one exact Newton step. Thus, the
$\mathbf{lb}$ converge linearly to $\mu_{f}$.
13
13
13
13
13
13
13
13
13
13
13
13
13
The upper bounds $\mathbf{ub}$ are post-fixed points, i.e.,
$f(\mathbf{ub})\leq\mathbf{ub}$ is an invariant of the algorithm. The
algorithm computes the sets $Z$ and $P$ so that inexact arithmetic is only
applied to the components $i$ with $f_{i}(\mathbf{ub})<1$. In the
$P$-components, the function $f$ is applied to $\mathbf{ub}$ in order to
improve the upper bound. In fact, $f$ is applied twice in line 2, analogously
to applying $\mathcal{N}$ twice in line 2. Here, the such that clause makes
sure that the progress towards $\mu_{f}$ is at least as fast as the progress
of one exact application of $f$ would be. One can show that this leads to
linear convergence to $\mu_{f}$.
The rest of the algorithm (lines 2-2) deals with the problem that, given a
post-fixed $\mathbf{ub}$, the sequence
$\mathbf{ub},f(\mathbf{ub}),f(f(\mathbf{ub})),\ldots$ does not necessarily
converge to $\mu_{f}$. For instance, if $f(X)=0.75X^{2}+0.25$, then
$\mu_{f}=1/3$, but $1=f(1)=f(f(1))=\cdots$. Therefore, the if-statement of
Algorithm 2 allows to improve the upper bound from $\overline{1}$ to a post-
fixed point less than $\overline{1}$, by exploiting the lower bounds
$\mathbf{lb}$. This is illustrated in Figure 1 for a $2$-dimensional scPSP
$f$.
|
---|---
(a) | (b)
Figure 1. Computation of a post-fixed point less than $\overline{1}$.
The dotted lines indicate the curve of the points $(X_{1},X_{2})$ satisfying
$X_{1}=0.8X_{1}X_{2}+0.2$ and $X_{2}=0.4X_{1}^{2}+0.1X_{2}+0.5$. Notice that
$\mu_{f}\prec\overline{1}=f(\overline{1})$. In Figure 1 (a) the shaded area
consists of those points $\mathbf{lb}$ where
$f^{\prime}(\overline{1})(\overline{1}-\mathbf{lb})\succ\overline{1}-\mathbf{lb}$
holds, i.e., the condition of line 2. One can show that $\mu_{f}$ must lie in
the shaded area, so by continuity, any sequence converging to $\mu_{f}$, in
particular the sequence of lower bounds $\mathbf{lb}$, finally reaches the
shaded area. In Figure 1 (a) this is indicated by the points with the square
shape. Figure 1 (b) shows how to exploit such a point $\mathbf{lb}$ to compute
a post-fixed point $\mathbf{ub}\prec\overline{1}$ (post-fixed points are
shaded in Figure 1 (b)): The post-fixed point $\mathbf{ub}$ (diamond shape) is
obtained by starting at $\overline{1}$ and moving a little bit along the
straight line between $\overline{1}$ and $\mathbf{lb}$, cf. line 2. The
sequence $\mathbf{ub},f(\mathbf{ub}),f(f(\mathbf{ub})),\ldots$ now converges
linearly to $\mu_{f}$.
###### Theorem 4.3.
Algorithm 2 terminates and computes vectors $\mathbf{lb},\mathbf{ub}$ such
that $\mathbf{lb}\leq\mu_{f}\leq\mathbf{ub}$ and
$\mathbf{ub}-\mathbf{lb}\leq\overline{\epsilon}$. Moreover, the sequences of
lower and upper bounds computed by the algorithm both converge linearly to
$\mu_{f}$.
Notice that Theorem 4.3 is about the convergence speed of the approximants,
not about the time needed to compute them. To analyse the computation time,
one would need stronger requirements on how floating assignments are
performed.
The lower and upper bounds computed by Algorithm 2 have a special feature:
they satisfy $\mathbf{lb}\prec f(\mathbf{lb})$ and $\mathbf{ub}\geq
f(\mathbf{ub})$. The following proposition guarantees that such points are in
fact lower and upper bounds.
###### Proposition 4.4.
Let $f$ be a perfectly superlinear PSP. Let
$\overline{0}\leq\mathbf{x}\leq\overline{1}$. If $\mathbf{x}\prec
f(\mathbf{x})$, then $\mathbf{x}\prec\mu_{f}$. If $\mathbf{x}\geq
f(\mathbf{x})$, then $\mathbf{x}\geq\mu_{f}$.
So a user of Algorithm 2 can immediately verify that the computed bounds are
correct. To summarize, Algorithm 2 computes provably and even verifiably
correct lower and upper bounds, although exact computation is restricted to
detecting numerical problems. See Section 4.3 for experiments.
### 4.2. Proving consistency using the inexact algorithm
In Section 3 we presented a simple and efficient algorithm to check the
consistency of a PSP. Algorithm 2 is aimed at approximating $\mu_{f}$, but
note that it can also prove the inconsistency of a PSP: when the algorithm
sets $\mathbf{ub}_{i}<1$, we know $(\mu_{f})_{i}<1$. This raises the question
whether Algorithm 2 can also be used for proving consistency. The answer is
yes, and the procedure is based on the following proposition.
###### Proposition 4.5.
Let $f$ be an scPSP. Let $\mathbf{t}\succ\overline{0}$ be a vector with
$f^{\prime}(\overline{1})\mathbf{t}\leq\mathbf{t}$. Then $f$ is consistent.
Proposition 4.5 can be used to identify consistent components.
Use Algorithm 2 with some (small) $\epsilon$ to compute $\mathbf{ub}$ and
$\mathbf{lb}$. Take any bottom SCC $S$.
* •
If
$f^{\prime}(\overline{1})(\overline{1}-\mathbf{lb}_{S})\leq\overline{1}-\mathbf{lb}_{S}$,
mark all variables in $S$ as consistent and remove the $S$-components from
$f$. In the remaining components, replace all variables in $S$ with $1$.
* •
Otherwise, remove $S$ and all other variables that depend on $S$ from $f$.
Repeat with the new bottom SCC until all SCCs are processed.
There is no guarantee that this method detects all $i$ with $(\mu_{f})_{i}=1$.
### 4.3. Case study: A neutron branching process
One of the main applications of the theory of branching processes is the
modelling of cascade creation of particles in physics. We study a problem
described by Harris in [9]. Consider a ball of fissionable radioactive
material of radius $D$. Spontaneous fission of an atom can liberate a neutron,
whose collision with another atom can produce further neutrons etc. If $D$ is
very small, most neutrons leave the ball without colliding. If $D$ is very
large, then nearly all neutrons eventually collide, and the probability that
the neutron’s progeny never dies is large. A well-known result shows that,
loosely speaking, the population of a process that does not go extinct grows
exponentially over time with large probability. Therefore, the neutron’s
progeny never dying out actually means that after a (very) short time all the
material is fissioned, which amounts to a nuclear explosion. The task is to
compute the largest value of $D$ for which the probability of extinction of a
neutron born at the centre of the ball is still $1$ (if the probability is $1$
at the centre, then it is $1$ everywhere). This is often called the critical
radius. Notice that, since the number of atoms that undergo spontaneous
fission is large (some hundreds per second for the critical radius of
plutonium), if the probability of extinction lies only slightly below 1, there
is already a large probability of a chain reaction. Assume that a neutron born
at distance $\xi$ from the centre leaves the ball without colliding with
probability $l(\xi)$, and collides with an atom at distance $\eta$ from the
centre with probability density $R(\xi,\eta)$. Let further $f(x)=\sum_{i\geq
0}p_{i}x^{i}$, where $p_{i}$ is the probability that a collision generates $i$
neutrons. For a neutron’s progeny to go extinct, the neutron must either leave
the ball without colliding, or collide at some distance $\eta$ from the
centre, but in such a way that the progeny of all generated neutrons goes
extinct. So the extinction probability $Q_{D}(\xi)$ of a neutron born at
distance $\xi$ from the centre is given by [9], p. 86:
$Q_{D}(\xi)=l(\xi)+\int_{0}^{D}R(\xi,\eta)f(Q_{D}(\eta))\;d\eta$
Harris takes $f(x)=0.025+0.830x+0.07x^{2}+0.05x^{3}+0.025x^{4}$, and gives
expressions for both $l(\xi)$ and $R(\xi,\eta)$. By discretizing the interval
$[0,D]$ into $n$ segments and replacing the integral by a finite sum we obtain
a PSP of dimension $n+1$ over the variables $\\{Q_{D}(jD/n)\mid 0\leq j\leq
n\\}$. Notice that $Q_{D}(0)$ is the probability that a neutron born in the
centre does not cause an explosion.
##### Results
For our experiments we used three different discretizations $n=20,50,100$. We
applied our consistency algorithm from Section 3 and Maple’s Simplex to check
inconsistency, i.e., to check whether an explosion occurs. The results are
given in the first 3 rows of Table 2: Again our algorithm dominates the LP
approach, although the polynomials are much denser than in the
$h^{(n)}$-systems.
$D$ | 2 | 3 | 6 | 10
---|---|---|---|---
$n$ | 20 | 50 | 100 | 20 | 50 | 100 | 20 | 50 | 100 | 20 | 50 | 100
inconsistent (yes/no) | n | n | n | y | y | y | y | y | y | y | y | y
Cons. check (Alg. Sec. 3) | $<1$ | $<1$ | 2 | $<1$ | $<1$ | 2 | $<1$ | $<1$ | 2 | $<1$ | $<1$ | 2
Cons. check (exact LP) | $<1$ | 20 | 258 | $<1$ | 22 | 124 | $<1$ | 16 | 168 | $<1$ | 37 | 222
Approx. $Q_{D}$ ($\epsilon=10^{-3}$) | $<1$ | $<1$ | 4 | 2 | 8 | 32 | 1 | 5 | 21 | 1 | 4 | 17
Approx. $Q_{D}$ ($\epsilon=10^{-4}$) | $<1$ | $<1$ | 4 | 2 | 8 | 34 | 2 | 7 | 28 | 1 | 6 | 23
Table 2. Runtime in seconds of various algorithms on different values of $D$
and $n$.
We also implemented Algorithm 2 using Maple for computing lower and upper
bounds on $Q_{D}(0)$ with two different values of the error bound $\epsilon$.
The runtime is given in the last two rows. By setting the _Digits_ variable in
Maple we controlled the precision of Maple’s software floating-point numbers
for the floating assignments. In all cases starting with the standard value of
10, Algorithm 2 increased Digits at most twice by $5$, resulting in a maximal
Digits value of $20$. We mention that Algorithm 2 computed an upper bound
$\prec\overline{1}$, and thus proved inconsistency, after the first few
iterations in all investigated cases, almost as fast as the algorithm from
Section 3.
##### Computing approximations for the critical radius.
After computing $Q_{D}(0)$ for various values of $D$ one can suspect that the
critical radius, i.e., the smallest value of $D$ for which $Q_{D}(0)=1$, lies
somewhere between 2.7 and 3. We combined binary search with the consistency
algorithm from Section 3 to determine the critical radius up to an error of
$0.01$. During the binary search, the algorithm from Section 3 has to analyze
PSPs that come closer and closer to the verge of (in)consistency. For the last
(and most expensive) binary search step that decreases the interval to $0.01$,
our algorithm took $\mathord{<}1$, $1$, $3$, $8$ seconds for
$n=20,50,100,150$, respectively. For $n=150$, we found the critical radius to
be in the interval $[2.981,2.991]$. Harris [9] estimates $2.9$.
## 5\. Conclusions
We have presented a new, simple, and efficient algorithm for checking the
consistency of PSPs, which outperforms the previously existing LP-based
method. We have also described the first algorithm that computes reliable
lower and upper bounds on $\mu_{f}$. The sequence of bounds converges linearly
to $\mu_{f}$. To achieve these properties without sacrificing efficiency, we
use a novel combination of exact and inexact (floating-point) arithmetic.
Experiments on PSPs from concrete branching processes confirm the practicality
of our approach. The results raise the question whether our combination of
exact and inexact arithmetic could be transferred to other computational
problems.
#### Acknowledgments
We thank several anonymous referees for pointing out inaccuracies and helping
us clarify certain aspects of the paper. The second author was supported by
the DFG Graduiertenkolleg 1480 (PUMA). We also thank Andreas Reuss for
proofreading the manuscript.
## References
* [1] GMP library. http://gmplib.org.
* [2] K. B. Athreya and P. E. Ney. Branching Processes. Springer, 1972.
* [3] A. Berman and R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. SIAM, 1994.
* [4] J. Esparza, A. Gaiser, and S. Kiefer. Computing least fixed points of probabilistic systems of polynomials. Technical report, Technische Universität München, Institut für Informatik, 2009.
* [5] J. Esparza, S. Kiefer, and M. Luttenberger. Convergence thresholds of Newton’s method for monotone polynomial equations. In Proceedings of STACS, pages 289–300, 2008.
* [6] J. Esparza, A. Kučera, and R. Mayr. Model checking probabilistic pushdown automata. In LICS 2004, pages 12–21. IEEE Computer Society, 2004.
* [7] K. Etessami and M. Yannakakis. Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations. Journal of the ACM, 56(1):1–66, 2009.
* [8] M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer, 1993.
* [9] T. E. Harris. The theory of branching processes. Springer, Berlin, 1963.
* [10] S. Kiefer, M. Luttenberger, and J. Esparza. On the convergence of Newton’s method for monotone systems of polynomial equations. In Proceedings of STOC, pages 217–226. ACM, 2007.
* [11] C. D. Manning and H. Schuetze. Foundations of Statistical Natural Language Processing. MIT Press, June 1999.
* [12] D. Wojtczak and K. Etessami. PReMo: an analyzer for probabilistic recursive models. In TACAS, volume 4424 of Lecture Notes in Computer Science, pages 66–71. Springer, 2007.
|
arxiv-papers
| 2009-12-21T19:14:12 |
2024-09-04T02:49:07.183719
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Javier Esparza, Andreas Gaiser, Stefan Kiefer",
"submitter": "Andreas Gaiser",
"url": "https://arxiv.org/abs/0912.4226"
}
|
0912.4303
|
# The widest contiguous field of view at Dome C and Mount Graham
Jeff Stoesz Corresponding author address: Jeff Stoesz, INAF - Osservatorio di
Arcetri, Largo Enrico Fermi 5, Firenze, FI 50125, Italy.
E-mail: stoesz@arcetri.astro.it Elena Masciadri Franck Lascaux Susanna
Hagelin
INAF - Osservatorio Astrofisica di Arcetri, Florence, Italy
###### Abstract
The image quality from Ground-Layer Adaptive Optics (GLAO) can be gradually
increased with decreased contiguous field of view. This trade-off is dependent
on the vertical profile of the optical turbulence ($C_{n}^{2}$ profiles). It
is known that the accuracy of the vertical distribution measured by existing
$C_{n}^{2}$ profiling techniques is currently quite uncertain for wide field
performance predictions 4 to 20 arcminutes. With assumed uncertainties in
measurements from Generalized-SCIDAR (GS), SODAR plus MASS we quantify the
impact of this uncertainty on the trade-off between field of view and image
quality for photometry of science targets at the resolution limit. We use a
point spread function (PSF) model defined analytically in the spatial
frequency domain to compute the relevant photometry figure of merit at
infrared wavelengths. Statistics of this PSF analysis on a database of
$C_{n}^{2}$ measurements are presented for Mt. Graham, Arizona and Dome C,
Antarctica. This research is part of the activities of ForOT (3D Forecasting
of Optical Turbulence above astronomical sites).
## 1 Introduction
Characterization of the optical turbulence in the first few kilometres above
the telescope is important for predicting the performance of Ground-Layer
Adaptive Optics (GLAO) telescopes as a function of field of view diameter.
Systems that have been proposed will correct visible or near-infrared science
fields that are typically 4 arcminutes, and potentially up to 20 arcminutes in
diameter and contiguous. There are several measurement techniques being
advanced to provide statistics on the vertical distribution of the structure
function coefficient $C_{n}^{2}(h)$ , and in this paper we explore the impact
of a potential bias from generalized-SCIDAR and MASS measurements. The first
of two sites we will investigate is a typical mid-latitude observatory site,
Mount Graham (32.7 N, 109.87 W, 3200 meters), measured with generalized-
SCIDAR. There are conifer trees at the summit with a height similar to the
SCIDAR telescope’s primary mirror, about 8 meters above the ground. The second
is Dome C (75.1 S, 123.3 E, 3260 meters), an Antarctic site with MASS and
SODAR measurements by Lawrence et al. (2004) and balloon measurements by Agabi
et al. (2006).
The GLAO PSF figure of merit that is of particular importance to wide field
astronomy is radius of 50% encircled energy, computed at several points in the
contiguous field of view and then averaged. It will be symbolized as $EE50$
here. $EE50$ is very closely related to the integration time to achieve some
signal to noise ratio in background-limited point source photometry in the
field (Andersen et al. 2006), a rather common science application for fields
of view 4 to 20 arcminutes in diameter. Roughly,
${\rm integration~{}time}\propto EE50^{2}.$ (1)
We will compute $EE50$ starting with an analytically defined phase Power
Spectral Density (PSD) for anisoplanatism and fitting error using established
theory (Jolissaint et al. 2006; Tokovinin 2004). Table 1 lists the model
parameters selected here. Computation from the analytic PSD is a fast method
to discover the performance gradient of $EE50(\theta)$, where $\theta$ is the
diameter of the field of view.
Table 1: The parameters and implicit assumptions of the GLAO PSF model. phase PSD | von Kármán, $L_{o}=30$ meters
---|---
telescope diameter | $D=8$ meters
Beacons | 4 point sources at range $H=90$ km at zenith
Beacons | evenly distributed on a circle of diameter $\theta$ in the field
image wavelength | $\lambda=1.25\mu m$
image locations | sampling a square field of view with vertices that intersect the circle
Deformable Mirror | cartesian grid of actuators with pitch, $\Delta$
Deformable Mirror | each actuator has a sinc-like influence function
Deformable Mirror | conjugated to height = 0
The exact range of altitudes in the first few kilometres where bias has
greatest impact depends on the basic GLAO system parameters, namely the
diameter of the guide star asterism (also $\theta$) whose signal is averaged
and the effective pitch that is controlled by the ground conjugated deformable
mirror ($\Delta$). The ratio $h_{GZ}=\Delta/\theta$ defines the altitude below
which any contribution to anisoplanatism is negligible. The term gray-zone
(GZ) was coined (Tokovinin 2004) to identify the altitudes above $h_{GZ}$,
where the contribution to anisoplanatism is not negligible (also known as
partially corrected zone).111Looking at the approximate error transfer
function in equation (8) of Tokovinin (2004) one can see why this is the case.
Fig.1 helps illustrate this in terms of performance in the focal plane. The
plot shows the $EE50$ figure of merit as a function of the height of one layer
of turbulence added to a typical, smooth profile. The layer contains half of
the total turbulence strength of the smooth profile. Fig.1 shows that the
largest performance gradient is at altitudes just above $h_{GZ}$. The gradient
vanishes above $h_{D}=D/\theta$, where $D$ is the telescope diameter. In the
following sections we will re-compute $EE50(\theta)$ with estimated bias in
the proportion of turbulence attributed to heights above or below $h_{GZ}$.
Figure 1: The gray-zone begins above $h_{GZ}$.
## 2 Mount Graham and Dome C profile monitoring data
The Mt. Graham G-SCIDAR measurements include 851 in High Vertical Resolution
(HVR) mode and 9911 in regular mode, both have been reduced to discretized
turbulence strength $J_{i}$ at height $h_{i}$. These were computed from the
normalized covariance function of the irradiance fluctuations (see Egner et
al. 2006, 2007) which are proportional to $J_{i}$, which are in turn related
to $C_{n}^{2}(h)$ by
$J_{i}=\int_{{h_{b}}_{i}}^{{h_{b}}_{i+1}}dh~{}C_{n}^{2}(h).$ (2)
The intrinsic vertical resolution of SCIDAR is roughly given by
$\frac{0.78}{\rho}\sqrt{\lambda|h+h_{gs}|}$ (3)
where $\rho$ is the binary separation ($35^{\prime\prime}$), $\lambda$ is the
wavelength of the scintillation signal ($0.5\mu m$), and $h_{gs}$ is the
conjugation height of the generalized SCIDAR analysis plane (about $-3500m$).
The regular mode resolution will represent free-atmosphere, above 1000 meters.
The current HVR data set samples the scale height of the boundary-layer and
provides data up to 1000 meters altitude. In a subsequent section we will
describe how the ground-layer and free-atmosphere are reduced to form a
composite statistical model.
For Dome C we will use 1701 MASS+SODAR profile monitoring measurements at Dome
C by Lawrence et al. (2004) during the Antarctic winter of 2004. These data
sample only two grid points between 30 and 1000 meters and do not sample any
turbulence below 30 meters. However, there exist balloon-borne micro-thermal
measurements (Agabi et al. 2006) that give us an estimate of the scale height
and total strength of the ground-layer, and with this information we model the
statistics of eight grid points from a height of zero to 200 meters. The
turbulence measurements recorded by SODAR in the Lawrence et al. (2004) data
we appropriate to a slab concentrated at 250 meters between the modelled
ground layer and the lowest MASS measurement at 500 meters.
For the Dome C altitudes from zero to 200 meters we define the following
exponential model to
$C_{n}^{2}(h)=Ae^{(-h/h_{A})}.$ (4)
Using Eqn.(2) it follows that
$J_{i}=-Ah_{A}\left(e^{(-{h_{b}}_{i+1}/h_{A})}-e^{(-{h_{b}}_{i}/h_{A})}\right).$
(5)
We will choose the boundaries ${h_{b}}_{i}$ in §4. Using a average, weighted
by $C_{n}^{2}(h)$
$\displaystyle h_{i}=$
$\displaystyle\frac{\int_{{h_{b}}_{i}}^{{h_{b}}_{i+1}}dh~{}C_{n}^{2}(h)~{}h}{\int_{{h_{b}}_{i}}^{{h_{b}}_{i+1}}dh~{}C_{n}^{2}(h)}.$
$\displaystyle=$
$\displaystyle\frac{-Ah_{A}\left[({h_{b}}_{i+1}+h_{A})e^{(-{h_{b}}_{i+1}/h_{A})}-({h_{b}}_{i}+h_{A})e^{(-{h_{b}}_{i}/h_{A})}\right]}{J_{i}}.$
(6)
It has been observed with balloon measurements at Cerro Pachon (Tokovinin and
Travouillon 2006) that the strength of ground-layer is governed primarily by
the scale height. In our model we will make the scale height dictate the
strength exclusively. A lognormal distribution of values of the scale height,
$h_{A}$, while $A=740.\times 10^{-16}$ and is fixed, will give a lognormal
distribution in seeing.
The Mt. Graham (MG) scenario has weaker overall seeing (median 0.74
arcseconds) than Dome C (DC, median 1.2 arcseconds). To illustrate the
differences in the vertical distributions for these two sites we reduce the
data to cumulative histograms of seeing in three slabs, shown in Fig.2. The
Dome C free atmosphere (right panel) and even upper ground-layer slab
(middle)are quite calm. Though the left and middle panels of Fig.2 are not
proof, the scale height of the MG turbulence is resolved by the HV-GS
technique in another analysis (Egner et al. 2006) to be between 100 to 250
meters. The DC scenario clearly has most turbulence concentrated between the
telescope and 30 meters range (left panel Fig.2).
Figure 2: Comparison of the Dome C (DC) and Mount Graham (MG) turbulence
profile data used here.
## 3 Reduction to composite profiles
Since the measurements of the ground-layer and free-atmosphere at these sites
is not simultaneous, we must create composite profiles that would closely
reproduce the PSF statistics as though we had computed them on a full set of
$J_{i}(h_{i})$ data, uninterrupted in $h$ and sampled at the same time. To do
this we sort and combine the profiles of as described in Tokovinin and
Travouillon (2006) using the assumption of uncorrelated ground-layer and free-
atmosphere seeing. We will briefly re-describe the process here in the context
of our data.
The Mt. Graham HVR will provide the ground-layer below 1000 meters and the
regular SCIDAR measurements will provide the free-atmosphere above 1000
meters. Three groups of profiles in the ground-layer are identified using the
sum of $J_{i}$. The first group are those profiles within $5\%$ of the
$25^{th}$ percentile are combined in a simple average for $J_{i}$. We call
them the “good” case. The $50^{th}$ and $75^{th}$ percentile profiles area
combined similarly and called “typical” and “bad”. In each group the grid of
$h_{i}$ is identical and hence remains unchanged by the combining process. The
same process is done for the free-atmosphere. The result is a reduction to
three ground layer profiles and three free-atmosphere profiles, which together
have nine permutations for composite profiles that can reproduce the PSF
statistics as though we had computed them on all of the $J_{i}(h_{i})$ data.
For Dome C we sort and combine the MASS+SODAR profile monitoring measurements
of the free-atmosphere above 200 meters in the same way we described for Mt.
Graham. The ground-layer model does not need to be sorted; the choice of three
scale heights $h_{A}=[14,9,22]$ meters provide the median, first and last
quartile of the integrated ground-layer.
## 4 Resampling the Composite Profiles
In all cases the shape of the composite profiles, whether averaged over time
or defined by a function is smooth and well sampled by the grid of
$J_{i}(h_{i})$ defined so far. Hence, we are permitted to resample the the
$J_{i}(h_{i})$ grid for the GLAO PSF model, which is affected by the density
of points in the gray-zone. We increase the number of grid points in the gray-
zone until the PSF figure of merit has reached an asymptote. This is trivial
for the ground-layer of Dome C, we can define the $h_{b}$ grid and then re-
compute $J_{i}(h_{i})$ with Eqn.(5) and Eqn.(6). For the measurements of Mount
Graham and the free-atmosphere of Dome C we divide several measured
$J_{i}(h_{i})$ grid into more numerous $J_{j}(h_{j})$ using linear
interpolation of the original discretized $C_{n}^{2}(h)$ data.
## 5 Predicted GLAO performance gradient
The reduced composite $C_{n}^{2}(h)$ profiles for each site are input for the
computation of field averaged radius of 50% encircled energy of PSFs at a
wavelength of $1.25\mu m$, outlined in §1, and symbolized $EE50$. The aim is
to asses the impact on GLAO performance by potential biases in the measured
vertical distribution of the turbulence strength. We have selected the
performance $EE50(\theta)$ metric to do this. Fig.3 is a 3x3 multi-panel plot
showing $EE50(\theta)$ at Mt. Graham (red) and Dome C (blue). The thicker
lines are the median values while the thinner ones are the first and last
quartiles of the ordinate.
Figure 3: The field averaged radius of 50% encircled energy on PSFs at
$1.25\mu m$, plotted as a function of the GLAO field of view.
Let us first consider the central column of plots to identify the fundamental
differences between weak and strong free-atmosphere sites. In the upper one we
see the Mt. Graham (red) $EE50$ gracefully increasing with $\theta$, as the
bottom of the gray-zone (§1) reaches into the boundary-layer turbulence 100 to
250 meters thick. For this top middle panel the actuator pitch of the DM was
0.5 meters and the Dome C scenario only very weakly affected by
anisoplanatism, a consequence of an inadequate number of actuators for that
site. In the central panel the pitch is 0.38 meters, which improves correction
at Mt. Graham slightly in all conditions, and greatly improves Dome C for
median or better conditions. The median and first quartile $EE50(\theta)$
curves of Dome C and Mt. Graham have similar shape because the ground-layer
profiles at Mt. Graham have similar exponential shape. The bottom plot shows
the potential gain for Dome C when the wavefront is controlled to a pitch of
0.1 meters. In the central column of plots, the important distinction between
the two sites is that Dome C is always under-actuated with $\Delta=0.5$ and
sometimes near the diffraction-limited $EE50$ with $\Delta=0.1$. Mt. Graham on
the other hand has more high altitude turbulence and is always limited by
anisoplanatism for these $\Delta$.
Next, consider the columns of panels to the left and right of Fig.3 showing
uncertainties pertinent to field of view trade-offs in GLAO telescope
design.As indicated in figure 4 in Tokovinin et al. (2005) both MASS and
SCIDAR measurements are believed to produce faithful total integrals of
turbulence, however, the vertical distribution may be biased. The left column
of plots in Fig.3 were computed from the $J_{i}(h_{i})$ times 0.5 in the
domain $h_{gz}<h_{i}<6km$, the balance was conserved by putting turbulence in
the lowest layer, below $h_{GZ}$. Likewise the the right column of plots is
$J_{i}(h_{i})$ times 1.5 in the domain $h_{gz}<h_{i}<6km$, with the balance
conserved by removing turbulence from the lowest layer. The change from the
central column of plots to the left or the right is the slope of the curves,
germane to designing a field of view trade-off. The performance of a wide
field survey can be expressed using the number of square arcminutes of sky
that can be imaged to some limiting magnitude per unit time. For an
theoretical seeing-limited telescope this is of course proportional to
$\theta^{2}$. For a GLAO telescope with field of view $\theta$ it will be
roughly proportional to $(\theta/EE50(\theta))^{2}$. $EE50(\theta)$ in the
middle row of Fig.3 ($\Delta=0.38$ meters) the slope of the median Mt. Graham
$EE50(\theta)$ in the domain $10<\theta<20$ arcminutes is about $45\%$ less or
more in the left or right panels. It is about $\mp 15\%$ for Dome C. In terms
of $integration~{}time(\theta)\propto EE50(\theta)^{2}$ in the domain
$10<\theta<20$ we find the slope is $\pm 60\%$ for Mt. Graham, $\pm 30\%$ for
Dome C. In other words, at a mid-latitude site similar to Mt. Graham, the
predicted survey coverage of the GLAO telescope could potentially be wrong by
as much as 60%.
## 6 Summary
The GLAO telescope scenario simulated here is a common design for wide field
science demanding a contiguous field. The estimate of 50% uncertainty in the
proportion of turbulence strength between the the corrected-zone and the gray-
zone (in the first 6 $km$) is based on a comparison between MASS and SCIDAR
and here we calculate an uncertainty of 60% in the slope function
$EE50(\theta)$. Dome C is truly a unique site, and more immune to the 50%
uncertainty. However, if the true uncertainly is not simply multiplicative the
uncertainty propagated to $EE50(\theta)$ for Dome C might be similar to that
of Mt. Graham.
_Acknowledgements._
We would like to thank the authors of Lawrence et al. (2004) for providing
their SODAR+MASS data. This work has been funded by the Marie Curie Excellence
Grant (ForOT)-MEXT-CT-2005-023878.
## References
* Agabi et al. (2006) Agabi, A., E. Aristidi, M. Azouit, E. Fossat, F. Martin, T. Sadibekova, J. Vernin, and A. Ziad, 2006: First Whole Atmosphere Nighttime Seeing Measurements at Dome C, Antarctica. PASP, 118, 344–348, doi:10.1086/498728, arXiv:astro-ph/0510418.
* Andersen et al. (2006) Andersen, D. R., J. Stoesz, S. Morris, M. Lloyd-Hart, D. Crampton, T. Butterley, B. Ellerbroek, L. Jolissaint, N. M. Milton, R. Myers, K. Szeto, A. Tokovinin, J.-P. Véran, and R. Wilson, 2006: Performance Modeling of a Wide-Field Ground-Layer Adaptive Optics System. PASP, 118, 1574–1590, doi:10.1086/509266, arXiv:astro-ph/0610097.
* Egner et al. (2007) Egner, S. E., E. Masciadri, and D. McKenna, 2007: Generalized SCIDAR measurements at Mt. Graham. PASP accepted.
* Egner et al. (2006) Egner, S. E., E. Masciadri, D. McKenna, and T. M. Herbst, 2006: Beyond conventional G-SCIDAR: the ground-layer in high vertical resolution. Advances in Adaptive Optics II. Edited by Ellerbroek, Brent L.; Bonaccini Calia, Domenico. Proceedings of the SPIE, Volume 6272, pp. 627256 (2006)., Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, Vol. 6272, doi:10.1117/12.671380.
* Jolissaint et al. (2006) Jolissaint, L., J.-P. Véran, and R. Conan, 2006: Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach. Optical Society of America Journal A, 23, 382–394.
* Lawrence et al. (2004) Lawrence, J. S., M. C. B. Ashley, A. Tokovinin, and T. Travouillon, 2004: Exceptional astronomical seeing conditions above Dome C in Antarctica. , 431, 278–281, doi:10.1038/nature02929.
* Tokovinin (2004) Tokovinin, A., 2004: Seeing Improvement with Ground-Layer Adaptive Optics. PASP, 116, 941–951, doi:10.1086/424805.
* Tokovinin and Travouillon (2006) Tokovinin, A. and T. Travouillon, 2006: Model of optical turbulence profile at Cerro Pachón. MNRAS, 365, 1235–1242, doi:10.1111/j.1365-2966.2005.09813.x.
* Tokovinin et al. (2005) Tokovinin, A., J. Vernin, A. Ziad, and M. Chun, 2005: Optical Turbulence Profiles at Mauna Kea Measured by MASS and SCIDAR. PASP, 117, 395–400, doi:10.1086/428930.
|
arxiv-papers
| 2009-12-22T18:46:19 |
2024-09-04T02:49:07.192667
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Stoesz, E. Masciadri, F. Lascaux and S. Hagelin",
"submitter": "Jeffrey Stoesz",
"url": "https://arxiv.org/abs/0912.4303"
}
|
0912.4338
|
# Baryon Fields with $U_{L}(3)\times U_{R}(3)$ Chiral Symmetry: Axial Currents
of Nucleons and Hyperons
Hua-Xing Chen1 hxchen@rcnp.osaka-u.ac.jp V. Dmitrašinović2 dmitra@vinca.rs
Atsushi Hosaka3 hosaka@rcnp.osaka-u.ac.jp 1Department of Physics and State Key
Laboratory of Nuclear Physics and Technology Peking University, Beijing
100871, China
2 Vinča Institute of Nuclear Sciences, lab 010, P.O.Box 522, 11001 Beograd,
Serbia
3 Research Center for Nuclear Physics, Osaka University, Ibaraki 567–0047,
Japan
###### Abstract
We use the conventional $F$ and $D$ octet and decimet generator matrices to
reformulate chiral properties of local (non-derivative) and one-derivative
non-local fields of baryons consisting of three quarks with flavor $SU(3)$
symmetry that were expressed in $SU(3)$ tensor form in Ref. Chen:2008qv . We
show explicitly the chiral transformations of the $[(6,3)\oplus(3,6)]$ chiral
multiplet in the “$SU(3)$ particle basis”, for the first time to our
knowledge, as well as those of the $(3,\overline{3})\oplus(\overline{3},3)$,
$(8,1)\oplus(1,8)$ multiplets, which have been recorded before in Refs.
Lee:1968 ; Bardeen:1969ra . We derive the vector and axial-vector Noether
currents, and show explicitly that their zeroth (charge-like) components close
the $SU_{L}(3)\times SU_{R}(3)$ chiral algebra. We use these results to study
the effects of mixing of (three-quark) chiral multiplets on the axial current
matrix elements of hyperons and nucleons. We show, in particular, that there
is a strong correlation, indeed a definite relation between the flavor-singlet
(i.e. the zeroth), the isovector (the third) and the eighth flavor component
of the axial current, which is in decent agreement with the measured ones.
baryon, chiral symmetry, axial current, $F$/$D$ values
###### pacs:
14.20.-c, 11.30.Rd, 11.40.Dw
## I Introduction
Axial current “coupling constants” of the baryon flavor octet Okun:1982ap are
well known by now, see Ref. Yamanishi:2007zza 111for history and other
references, see Chapter 6.7 of Okun’s book Okun:1982ap and PDG tables
Amsler:2008zzb . The zeroth (time-like) components of these axial currents are
generators of the $SU_{L}(3)\times SU_{R}(3)$ chiral symmetry that is one of
the fundamental symmetries of QCD. The general flavor $SU_{F}(3)$ symmetric
form of the nucleon axial current contains two free parameters, the so called
$F$ and $D$ couplings, which are empirically determined as $F$=$0.459\pm
0.008$ and $D$=$0.798\pm 0.008$, see Ref. Yamanishi:2007zza . The conventional
models of (linearly realized) chiral $SU_{L}(3)\times SU_{R}(3)$ symmetry,
Refs. Lee:1968 ; Bardeen:1969ra , on the other hand appear to fix these
parameters at either ($F$=0,$D$=1), which case goes by the name of
$[(3,\overline{3})\oplus(\overline{3},3)]$, or at ($F$=1,$D$=0), which case
goes by the name of $[(8,1)\oplus(1,8)]$ representation. Both of these chiral
representations suffer from the shortcoming that $F$+$D$=1$\neq
g_{A}^{(3)}=$1.267 without derivative couplings. But, even with derivative
interactions, one cannot change the value of the vanishing coupling, e.g. of
$F$=0, in $[(3,\overline{3})\oplus(\overline{3},3)]$, or of $D$=0, in
$[(8,1)\oplus(1,8)]$. Rather, one can only renormalize the non-vanishing
coupling to 1.267.
Attempts at a reconciliation of the measured values of axial couplings with
the (broken) $SU_{L}(3)\times SU_{R}(3)$ chiral symmetry go back at least 40
years Hara:1965 ; Lee:1968 ; Bardeen:1969ra ; Harari:1966yq ; Harari:1966jz ;
Gerstein:1966zz ; Weinberg:1969hw , but, none have been successful to our
knowledge thus far. As noted above, perhaps the most troublesome problem are
the $SU(3)$ axial current’s $F$,$D$ values, which problem has repercussions
for the meson-baryon interaction $F$,$D$ values, with far-reaching
consequences for hyper-nuclear physics and even astrophysics. Another, perhaps
equally important and difficult problem is that of the flavor-singlet axial
coupling of the nucleon Bass:2007zzb . This is widely thought of as being
disconnected from the $F$,$D$ problem, but we shall show that the three-quark
interpolating fields cast some perhaps unexpected light on this problem. We
shall attack both of these problems from Weinberg’s Weinberg:1969hw point of
view, viz. chiral representation mixing, extended to the $SU_{L}(3)\times
SU_{R}(3)$ and $U_{L}(1)\times U_{R}(1)$ chiral symmetries, with added input
from three-quark baryon interpolating fields Chen:2008qv that are ordinarily
used in QCD calculations.
The basic idea is simple: a mixture of two baryon fields belonging to
different chiral representations/multiplets has axial couplings that lie
between the extreme values determined by the two chiral multiplets that are
being mixed, and depend on the mixing angle, of course. Weinberg used this
idea to fit the iso-vector axial coupling of the nucleon using the
$[(1/2,0)\oplus(0,1/2)]$ and $[(1,1/2)\oplus(1/2,1)]$ multiplets of the
$SU_{L}(2)\times SU_{R}(2)$ chiral symmetry, but the same idea may be used on
any baryon belonging to the same octet, e.g. for the $\Lambda,\Sigma$ and
$\Xi$ hyperons. In other words, the $F$ and $D$ values of the mixture can be
determined from the $F$ and $D$ values of the $SU_{L}(3)\times SU_{R}(3)$
representations corresponding to the $[(1/2,0)\oplus(0,1/2)]$ and
$[(1,1/2)\oplus(1/2,1)]$ multiplets, viz.
$[(3,\overline{3})\oplus(\overline{3},3)]$ or $[(8,1)\oplus(1,8)]$, and
$[(6,3)\oplus(3,6)]$, respectively. The same principle holds for the
$U_{L}(1)\times U_{R}(1)$ symmetry “multiplets” and the value(s) of the flavor
singlet axial charge.
The $SU_{L}(3)\times SU_{R}(3)$ and $U_{L}(1)\times U_{R}(1)$ chiral
transformation properties of three-quark baryon interpolating fields, that are
commonly used in various QCD (lattice, sum rules) calculations, and that have
recently been determined in Ref. Chen:2008qv will be used here as input into
the chiral mixing formalism, so as to deduce as much phenomenological
information about the axial currents of hyperons and nucleons as possible. As
a result we find three “optimal” scenarios all with identical $F$,$D$ values
(see Sect. IV).
First we recast our previous results Chen:2008qv into the language that is
conventional for axial currents, i.e. in terms of octet $F$ and $D$ couplings.
A large part of the present paper is devoted to this notational conversion
(change of basis) and the subsequent check whether and how the resulting
chiral charges actually satisfy the $SU(3)\times SU(3)$ chiral algebra. That
is a non-trivial task for the $[(3,\overline{3})\oplus(\overline{3},3)]$ and
$(6,3)\oplus(3,6)$ representations, because they involve off-diagonal terms,
and in the latter case one of the diagonal terms in the axial current is
multiplied by a fractional coefficient, that appears to spoil the closure of
the $SU(3)\times SU(3)$ chiral algebra; the off-diagonal terms in the axial
current make crucial contributions that restore the closure. Thus, the afore-
mentioned fractional coefficient is uniquely determined.
We use these results to study the effects of mixing of (three-quark) chiral
multiplets on the axial current matrix elements of hyperons and nucleons. We
show, in particular, that there is a strong correlation between the flavor-
singlet (i.e. the zeroth), the isovector (the third) and the eighth flavor
component of the axial current. There are, in principle, three independent
observables here: the flavor-singlet (i.e. the zeroth), the isovector (the
third) and the eighth flavor component of the axial current of the nucleon. By
fitting just one mixing angle to one of these values, e.g. the (best known)
isovector coupling, we predict the other two. These predictions may differ
widely depending on the field that one assumes to be mixed with the
$(6,3)\oplus(3,6)$ field (which must be present if the isovector axial
coupling has any chance of being fit). If one assumes mixing of three fields
(again, always keeping the $(6,3)\oplus(3,6)$ as one of the three) and fits
the flavor-singlet and the isovector axial couplings, then one finds a unique
prediction for the $F$, $D$ values, which is in decent agreement with the
measured ones, modulo $SU(3)$ symmetry breaking corrections, which may be
important (for a recent fit, see Ref. Yamanishi:2007zza ). The uniqueness of
this result is a consequence of a remarkable relation, $g_{A}^{(0)}=3F-D$ that
holds for all three (five) chiral multiplets involved here, and which leads to
the relation: $g_{A}^{(0)}=\sqrt{3}g_{A}^{(8)}$, see Sect. IV.
Most of the ideas used in this paper, such as that of chiral multiplet mixing,
have been presented in mid- to late 1960’s, Refs. Harari:1966yq ;
Weinberg:1969hw ; Harari:1966jz ; Gerstein:1966zz , with the (obvious)
exception of the use of QCD interpolating fields, which arrived only a decade
afterwards/later, and the (perhaps less obvious) question of baryons’ flavor-
singlet axial current (a.k.a. the $U_{A}(1)$), which was (seriously) raised
yet another decade later.
The present paper consists of five parts: after the present Introduction, in
Sect. II we define the $SU(3)\times SU(3)$ chiral transformations of three-
quark baryon fields, with special emphasis on the $SU(3)$ phase conventions
that ensure standard $SU(2)$ isospin conventions for the isospin sub-
multiplets, and we define the ($SU(3)$ symmetric) vector and axial-vector
Noether currents of three-quark baryon fields. In Sect. III we prove the
closure of the chiral $SU_{L}(3)\times SU_{R}(3)$ algebra. In Sect. IV we
apply chiral mixing formalism to the hyperons’ axial currents and discuss the
results. Finally, in Sect. V we offer a summary and an outlook on future
developments.
## II $SU(3)\times SU(3)$ Chiral Transformations of Three-quark Baryon Fields
and their Noether Currents
We must make sure that our conventions ensure that identical isospin
multiplets in different $SU(3)$ multiplets, such as the octet and the
decuplet, have identical isospin algebras/generators. That is a relatively
simple matter of definition, but was not the case with the octet conventions
used in Ref. Chen:2008qv . Our new definitions of the octet and decuplet
fields avoid these problems.
### II.1 Octet and Decuplet State Definition
The new $\Xi^{-}$ wave function comes with a minus sign: that is precisely the
convention used in Eqs. (18) and (19) in Sect. 18 of Gasiorowicz’s textbook
gas . But then we must also adjust the $8\times 10$ $SU(3)$-spurion matrices
for this modification.
$\displaystyle\Sigma^{\mp}\sim{\pm 1\over\sqrt{2}}(N^{1}\pm
iN^{2})\,,\;\;\;N^{3}\sim\Sigma^{0}\,,\;\;\;N^{8}\sim\Lambda_{8}\,,$ (1)
$\displaystyle\left(\begin{array}[]{c}~{}\Xi^{-}\\\
p\end{array}\right)\sim{\mp 1\over\sqrt{2}}(N^{4}\pm
iN^{5}),\;\;\left(\begin{array}[]{c}~{}\Xi^{0}\\\
n\end{array}\right)\;\sim\,{1\over\sqrt{2}}(N^{6}\pm iN^{7}).$ (6)
$\displaystyle\left(\begin{array}[]{c}p\\\ n\\\ \Sigma^{+}\\\ \Sigma^{0}\\\
\Sigma^{-}\\\ \Xi^{0}\\\ \Xi^{-}\\\ \Lambda_{8}\end{array}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\\
\frac{-1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\
\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0&0&0&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\\
0&0&0&\frac{-1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&0&0&1\end{array}\right)\left(\begin{array}[]{c}N^{1}\\\ N^{2}\\\
N^{3}\\\ N^{4}\\\ N^{5}\\\ N^{6}\\\ N^{7}\\\ N^{8}\end{array}\right)\,,$ (31)
or put them into the $3\times 3$ baryon matrix as follows
$\displaystyle{\mathfrak{N}}=\left(\begin{array}[]{c c
c}{\Sigma^{0}\over\sqrt{2}}+{\Lambda^{8}\over\sqrt{6}}&-\Sigma^{+}&p\\\
\Sigma^{-}&-{\Sigma^{0}\over\sqrt{2}}+{\Lambda^{8}\over\sqrt{6}}&n\\\
-\Xi^{-}&\Xi^{0}&-{2\over\sqrt{6}}\Lambda^{8}\end{array}\right)\,.$ (35)
Note the minus signs in front of $\Xi^{-}$ and $\Sigma^{+}$. We also use a new
normalization of the decuplet fields:
$\displaystyle{\Delta^{1}}\sim-{1\over\sqrt{3}}\Delta^{++}\,,{\Delta^{7}}\sim-{1\over\sqrt{3}}\Delta^{-}\,,{\Delta^{10}}\sim-{1\over\sqrt{3}}\Omega^{-}\,,$
(36)
$\displaystyle\Delta^{2}\sim-\Delta^{+}\,,\Delta^{4}\sim-\Delta^{0}\,,\Delta^{3}\sim-\Sigma^{*+}\,,\Delta^{8}\sim-\Sigma^{*-}\,,\Delta^{6}\sim-\Xi^{*0}\,,\Delta^{9}\sim-\Xi^{*-}\,,$
$\displaystyle\Delta^{5}\sim-\sqrt{2}\Sigma^{*0}$
For the singlet $\Lambda$, we use the normalization:
$\Lambda_{1}=\Lambda_{phy}={2\sqrt{2}\over\sqrt{3}}\Lambda\,.$ (37)
For simplicity, we will just use $\Lambda_{1}$ instead of $\Lambda_{phy}$ in
the following sections.
We define the flavor octet and decuplet matrices/column vectors as
$\displaystyle N$ $\displaystyle=$
$\displaystyle(p,n,\Sigma^{+},\Sigma^{0},\Sigma^{-},\Xi^{0},\Xi^{-},\Lambda_{8})^{T}\,,$
(38) $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle(\Delta^{++},\Delta^{+},\Delta^{0},\Delta^{-},\Sigma^{*+},\Sigma^{*0},\Sigma^{*-},\Xi^{*0},\Xi^{*-},\Omega)^{T}$
(39)
In our previous paper, Ref. Chen:2008qv , we found that the baryon
interpolating fields $N_{+}^{a}=N^{a}_{1}+N^{a}_{2}$ belong to the chiral
representation $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$;
$\Lambda$ and $N_{-}^{a}=N^{a}_{1}-N^{a}_{2}$ belong to the chiral
representation
$(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{\overline{3}},\mathbf{3})$;
$N^{a}_{\mu}$ and $\Delta^{P}_{\mu}$ belong to the chiral representation
$(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$; and
$\Delta^{P}_{\mu\nu}$ belong to the chiral representation
$(\mathbf{10},\mathbf{1})\oplus(\mathbf{1},\mathbf{10})$. Here $N^{a}_{1}$ and
$N^{a}_{2}$ are the two independent kinds of nucleon fields. $N^{a}_{1}$
contains the “scalar diquark” and $N^{a}_{2}$ contains the “pseudoscalar
diquark”. Moreover, we calculated their chiral transformations in Ref.
Chen:2008qv . That form, however, is not conventionally used for the axial
currents. So in the following subsections, we use different conventions,
listed above, and display the chiral transformations in these bases.
### II.2 Chiral Transformations of Three-Quark Interpolating Fields
#### II.2.1 $(8,1)\oplus(1,8)$ Chiral Transformations
This chiral representation contains the flavor octet representation
$\mathbf{8}$. For the octet baryon field $N^{a}$ ($a=1,\cdots,8$), chiral
transformations are given by:
$\displaystyle\delta_{5}^{\vec{b}}N_{+}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}{\bf F}_{(8)}^{a}N_{+}\,,\ $ (40)
The $SU(3)$-spurion matrices ${\bf F}_{(8)}^{a}$ are listed in the Appendix
A.2. This corresponds to the chiral transformations of Ref. Chen:2008qv :
$\displaystyle\delta_{5}^{\vec{b}}(N^{a}_{1}+N^{a}_{2})$ $\displaystyle=$
$\displaystyle\gamma_{5}b^{b}f^{bac}(N^{c}_{1}+N^{c}_{2})\,.$
The coefficients $f^{abc}$ are the standard antisymmetric “structure
constants” of $SU(3)$. For completeness’ sake, we show the following equation
which defines the $f$ and $d$ coefficients
$\displaystyle\lambda^{a}_{AB}\lambda^{b}_{BC}$ $\displaystyle=$
$\displaystyle(\lambda^{a}\lambda^{b})_{AC}={1\over
2}\\{\lambda^{a},\lambda^{b}\\}_{AC}+{1\over 2}[\lambda^{a},\lambda^{b}]_{AC}$
(41) $\displaystyle=$ $\displaystyle{2\over
3}\delta^{ab}\delta_{AC}+(d^{abc}+if^{abc})\lambda^{c}_{AC}\,.$
#### II.2.2 $(3,\overline{3})\oplus(\overline{3},3)$ Chiral Transformations
This chiral representation contains the flavor octet and singlet
representations $\mathbf{\bar{3}}\otimes\mathbf{3}=\mathbf{8}\oplus\mathbf{1}$
$\sim(N^{a},\Lambda)$. These two flavor representations are mixed under chiral
transformations as
$\displaystyle\delta_{5}^{\vec{b}}\Lambda_{1}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\sqrt{2\over 3}{\rm\bf T}^{a}_{1/8}N_{-}$
$\displaystyle\delta_{5}^{\vec{b}}N_{-}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\left({\rm{\bf D}}^{a}N_{-}+\sqrt{2\over 3}{\rm\bf
T}^{a\dagger}_{1/8}\Lambda_{1}\right)\,.\ $ (42)
where ${\rm{\bf D}}^{a}$ are defined in the Appendix A.1. The $SU(3)$-spurion
matrices ${\rm\bf T}^{a}_{1/8}$ have the following properties
$\displaystyle{\rm\bf T}^{a}_{1/8}{\rm\bf T}^{a\dagger}_{1/8}$
$\displaystyle=$ $\displaystyle 8$ $\displaystyle{\rm\bf
T}^{a\dagger}_{1/8}{\rm\bf T}^{a}_{1/8}$ $\displaystyle=$
$\displaystyle{\mathbf{1}}_{8\times 8}\,,$ (43)
and are listed in the Appendix A.4. Here ${\mathbf{1}}_{8\times 8}$ is a unit
matrix of $8\times 8$ dimensions.
#### II.2.3 $(6,3)\oplus(3,6)$ Chiral Transformations
This chiral representation contains flavor octet and decuplet representations
$\mathbf{6}\otimes\mathbf{3}=\mathbf{8}\oplus\mathbf{10}$
$\sim(N^{a},\Delta^{b})$. For their chiral transformations we use the results
from Ref. Chen:2008qv , where they were expressed in terms of coefficients
$g$, $g^{\prime}$, $g^{\prime\prime}$ and $g^{\prime\prime\prime}$ that were
tabulated in Table II. For off-diagonal terms (between octet and decuplet),
there is a (new) factor $1\over 6$, which comes from the different
normalization of octet and decuplet. Here we show the final result:
$\displaystyle\delta_{5}^{\vec{b}}N$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\left({\rm({\bf D}^{a}+{2\over 3}{\bf
F}_{(8)}^{a})}N+\frac{2}{\sqrt{3}}{\rm{\bf T}}^{a}\Delta\right)\,,\ $
$\displaystyle\delta_{5}^{\vec{b}}\Delta$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\left(\frac{2}{\sqrt{3}}{\rm{\bf
T}}^{a\dagger}N+\frac{1}{3}{\rm{\bf F}}_{(10)}^{a}\Delta\right)\,.\ $ (44)
These $SU(3)$-spurion matrices ${\bf T}^{a}$ (sometimes we use ${\bf
T}^{a}_{10/8}$) and ${\bf F}_{(10)}^{a}$ have the following properties
$\displaystyle{\rm{\bf F}}_{(10)}^{a}$ $\displaystyle=$
$\displaystyle-\,i\,f^{abc}{\bf T}^{b\dagger}{\bf T}^{c}\,$ $\displaystyle{\bf
T}^{a}{\bf T}^{a\dagger}$ $\displaystyle=$ $\displaystyle\,\frac{5}{2}\
{\mathbf{1}}_{8\times 8}$ $\displaystyle{\bf T}^{a\dagger}{\bf T}^{a}$
$\displaystyle=$ $\displaystyle\,2\ {\mathbf{1}}_{10\times 10}\,,$ (45)
These transition matrices ${\bf T}^{c}$ and the decuplet generators ${\rm{\bf
F}}_{(10)}^{a}$ are listed in Appendices A.2 and A.3, respectively.
### II.3 Noether Currents of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Symmetry
The chiral $SU_{L}(3)\times SU_{R}(3)$ transformations of the baryon fields
$B_{i}$ define eight components of the baryon isovector axial current ${\bf
J}_{\mu 5}^{a}$, by way of Noether’s theorem:
$\displaystyle-{\bm{b}}\cdot{\bm{J}_{\mu 5}}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\partial{\cal
L}}{\partial\partial^{\mu}B_{i}}\delta_{5}^{\vec{b}}B_{i}.\ $ (46)
Similarly, the flavor $SU(3)$ transformations $\delta^{\vec{a}}B_{i}$ define
the Lorentz-vector Noether (flavor) current
$\displaystyle-{\bm{a}}\cdot{\bm{J}_{\mu}}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\partial{\cal
L}}{\partial\partial^{\mu}B_{i}}\delta^{\vec{a}}B_{i}.\ $ (47)
#### II.3.1 The Axial Current in the $(8,1)\oplus(1,8)$ Multiplet
Eqs. (40), the chiral $SU_{L}(3)\times SU_{R}(3)$ transformation rules of the
$B_{i}=N_{+}^{i}$ baryons in the $(8,1)\oplus(1,8)$ chiral multiplet, define
the eight components of the (hyperon) flavor octet axial current ${\bf J}_{\mu
5}^{a}$, by way of Noether’s theorem, Eq. (46), where $B_{i}$ are the octet
$N^{i}$ baryon fields. The axial current ${\bf J}_{\mu 5}$ is
$\displaystyle{\bf J}_{\mu 5}^{a}$ $\displaystyle=$
$\displaystyle\,\overline{N}\gamma_{\mu}\gamma_{5}{\bf F}_{(8)}^{a}N~{}.\ $
(48)
Here ${\bf F}_{(8)}^{i}$ are the $SU(3)$ octet matrices/generators. The
Lorentz vector Noether (flavor-octet) current in this multiplet reads
$\displaystyle{\bf J}_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\overline{N}\gamma_{\mu}\,{\bf F}_{(8)}^{a}\,N~{},\ $ (49)
which are valid if the interactions do not contain derivatives.
#### II.3.2 The Axial Current in the $(\overline{3},3)\oplus(3,\overline{3})$
Multiplet
Eqs. (42), the chiral $SU_{L}(3)\times SU_{R}(3)$ transformation rules of the
$B_{i}=(N_{-}^{i},\,\Lambda)$ baryons in the
$(\overline{3},3)\oplus(3,\overline{3})$ chiral multiplet, define the eight
components of the (hyperon) flavor octet axial current ${\bf J}_{\mu 5}^{a}$,
by way of Noether’s theorem, Eq. (46), where $B_{i}$ are the flavor octet
$N_{-}^{i}$ and the flavor singlet $\Lambda_{1}$ baryon fields. The axial
current ${\bf J}_{\mu 5}$ is
$\displaystyle{\bf J}_{\mu 5}^{a}$ $\displaystyle=$
$\displaystyle\,\overline{N}\gamma_{\mu}\gamma_{5}\left({\rm{\bf
D}}^{a}N+\sqrt{2\over 3}{\rm\bf T}^{a\dagger}_{1/8}\Lambda_{1}\right)$ (50)
$\displaystyle+$
$\displaystyle\,\overline{\Lambda}_{1}\gamma_{\mu}\gamma_{5}\sqrt{2\over
3}{\rm\bf T}^{a}_{1/8}N~{}.\ $
Here ${\bf D}^{i}$ are the $SU(3)$ octet matrices/generators. The Lorentz
vector Noether (flavor-octet) current in this multiplet reads
$\displaystyle{\bf J}_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\overline{N}\gamma_{\mu}\,{\bf F}_{(8)}^{a}\,N~{}.\ $ (51)
#### II.3.3 Axial Current in the $(3,6)\oplus(6,3)$ Multiplet
The chiral $SU_{L}(3)\times SU_{R}(3)$ transformation rules of the
$B_{i}=(N^{i},\,\Delta^{j})$ baryons, Eqs. (II.2.3), in the $(3,6)\oplus(6,3)$
chiral multiplet, define the eight components of the (hyperon) flavor octet
axial current ${\bf J}_{\mu 5}^{a}$, by way of Noether’s theorem (Eq. (46)),
where $B_{i}$ are the octet $N^{i}$ and the decuplet $\Delta^{j}$ baryon
fields. The axial current ${\bf J}_{\mu 5}$ is
$\displaystyle{\bf J}_{\mu 5}^{a}$ $\displaystyle=$
$\displaystyle\,\overline{N}\gamma_{\mu}\gamma_{5}\left({\rm({\bf
D}^{a}+{2\over 3}{\bf F}_{(8)}^{a})}N+\frac{2}{\sqrt{3}}{\rm{\bf
T}}^{a}\Delta\right)$ (52) $\displaystyle+$
$\displaystyle\,\overline{\Delta}\gamma_{\mu}\gamma_{5}\left(\frac{2}{\sqrt{3}}{\rm{\bf
T}}^{a\dagger}N+\frac{1}{3}{\rm{\bf F}}_{(10)}^{a}\Delta\right)~{}.\ $
Here ${\bf D}^{i}$ and ${\bf F}_{(8)}^{i}$ are the $SU(3)$ octet
matrices/generators ${\bf D}^{a}$ and ${\bf F}_{(8)}^{a}$, respectively, ${\bf
F}_{(10)}^{i}$ are the $SU(3)$ decuplet generators, and ${\bf T}^{i}$ are the
so-called $SU(3)$-spurion matrices. The Lorentz vector Noether (flavor-octet)
current in this multiplet reads
$\displaystyle{\bf J}_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\left(\overline{N}\gamma_{\mu}\,{\bf
F}_{(8)}^{a}\,N\right)+\left(\overline{\Delta}\gamma_{\mu}\,{\bf
F}_{(10)}^{a}\,\Delta\right)~{}.\ $ (53)
## III Closure of the chiral $SU_{L}(3)\times SU_{R}(3)$ algebra
The $SU(3)$ vector charges $Q^{a}=\int d{\bf x}J_{0}^{a}(t,{\bf x})$ defined
by Eq. (47), together with the axial charges $Q_{5}^{a}=\int d{\bf
x}J_{05}^{a}(t,{\bf x})$ defined by Eq. (46) ought to close the chiral algebra
$\displaystyle\left[Q^{a},Q^{b}\right]$ $\displaystyle=$ $\displaystyle
if^{abc}Q^{c}$ (54) $\displaystyle\left[Q_{5}^{a},Q^{b}\right]$
$\displaystyle=$ $\displaystyle if^{abc}Q_{5}^{c}$ (55)
$\displaystyle\left[Q_{5}^{a},Q_{5}^{b}\right]$ $\displaystyle=$
$\displaystyle if^{abc}Q^{c}~{}.\ $ (56)
where $f^{abc}$ are the SU(3) structure constants. Eqs. (54) and (55) usually
hold automatically, as a consequence of the canonical (anti)commutation
relations between Dirac baryon fields $B_{i}$, whereas Eq. (56) is not trivial
for the chiral multiplets that are different from the $[(8,1)\oplus(1,8)]$,
because of the (nominally) fractional axial charges and the presence of the
off-diagonal components. When taking a matrix element of Eq. (56) by baryon
states in a certain chiral representation, the axial charge mixes different
flavor states within the same chiral representation. This is an algebraic
version of the Adler-Weisburger sum rule Weinberg:1969hw . In the following we
shall check and confirm the validity of Eq. (56) in the three multiplets of
SU(3)${}_{L}\times$SU(3)R.
### III.1 Closure of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Algebra in the
$(8,1)\oplus(1,8)$ Multiplet
Due to the absence of fractional coefficients in the $(8,1)\oplus(1,8)$
multiplet’s axial charge $Q_{5}^{a}=\int d{\bf x}J_{05}^{a}(t,{\bf x})$
defined by the current given in Eq. (48), the vector charge $Q^{a}=\int d{\bf
x}J_{0}^{a}(t,{\bf x})$ defined by the current given in Eq. (49) and the axial
charge close the chiral algebra defined by Eqs. (54), (55) and (56). The same
comments holds for the $(10,1)\oplus(1,10)$ chiral multiplet for the same
reasons as in the example shown above.
### III.2 Closure of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Algebra in the
$(3,\overline{3})\oplus(\overline{3},3)$ Multiplet
The vector charge $Q^{a}=\int d{\bf x}J_{0}^{a}(t,{\bf x})$ defined by the
current given in Eq. (51), together with the axial charge $Q_{5}^{a}=\int
d{\bf x}J_{05}^{a}(t,{\bf x})$ defined by the current given in Eq. (50) ought
to close the chiral algebra defined by Eqs. (54), (55) and (56). Eqs. (54) and
(55) hold here, whereas Eq. (56) is the non-trivial one: the diagonal $D$
charge of $N$ ($Q_{5D}^{a}(N)$) axial charge,
$\displaystyle Q_{5D}^{a}(N)$ $\displaystyle=$ $\displaystyle~{}~{}~{}\int
d{\bf x}\,\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf D}^{a}\,N\right)\,,$
(57) $\displaystyle Q_{D}^{a}(N)$ $\displaystyle=$ $\displaystyle~{}~{}~{}\int
d{\bf x}\,\left(\overline{N}\gamma_{0}\,{\bf D}^{a}\,N\right)\,,\ $ (58)
lead to
$\displaystyle\left[Q_{5D}^{a}(N),Q_{5D}^{b}(N)\right]$ $\displaystyle=$
$\displaystyle\int d{\bf x}\left(\overline{N}\gamma_{0}\,\left({\bf D}^{a}{\bf
D}^{b}-{\bf D}^{b}{\bf D}^{a}\right)N\right)\,.\ $ (59)
It turns out that the off-diagonal terms in the axial charge
$\displaystyle Q_{5}^{a}(N,\Lambda)$ $\displaystyle=$ $\displaystyle\int d{\bf
x}\,\Bigg{(}\sqrt{\frac{2}{3}}\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf
T}^{a\dagger}_{1/8}\,\Lambda+\overline{\Lambda}\gamma_{0}\gamma_{5}\,{\bf
T}^{a}_{1/8}\,N\right)\Bigg{)}~{},\ $ (60)
play a crucial role in the closure of the chiral commutator Eq. (56). The
additional terms in the commutator add up to
$\displaystyle\left[Q_{5}^{a}(N,\Delta),Q_{5}^{b}(N,\Delta)\right]$
$\displaystyle=$ $\displaystyle{\frac{2}{3}}\int d{\bf
x}\overline{N}\gamma_{0}\,\left({\bf T}^{a\dagger}_{1/8}{\bf T}^{b}_{1/8}-{\bf
T}^{b\dagger}_{1/8}{\bf T}^{a}_{1/8}\right)\,N\,,\ $ (61)
which provide the “missing” factors due to the following properties of the
off-diagonal isospin operators ${\bf T}^{i}_{1/8}$ and ${\bf D}^{i}$ matrices
$\displaystyle i\,f^{ijk}({\bf F}_{(8)}^{k})$ $\displaystyle=$
$\displaystyle({\bf D}^{i}{\bf D}^{j}-{\bf D}^{j}{\bf D}^{i})+{2\over 3}({\bf
T}^{i\dagger}_{1/8}{\bf T}^{j}_{1/8}-{\bf T}^{j\dagger}_{1/8}{\bf
T}^{i}_{1/8})\,.$ (62)
Therefore, the chiral algebra Eqs. (54), (55) and (56) close.
### III.3 Closure of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Algebra in the
$(3,6)\oplus(6,3)$ Multiplet
The vector charge $Q^{a}=\int d{\bf x}J_{0}^{a}(t,{\bf x})$ defined by the
current in Eq. (53), together with the axial charge $Q_{5}^{a}=\int d{\bf
x}J_{05}^{a}(t,{\bf x})$ defined by the current in Eq. (52) ought to close the
chiral algebra defined by Eqs. (54), (55) and (56). Eqs. (54) and (55) hold
here, whereas Eq. (56) is once again the non-trivial one: the fractions
$\frac{2}{3}$ and $\frac{1}{3}$ in the diagonal $F$ charge of $N$
($Q_{5}^{a}(N)$) and $\Delta$ axial charges, respectively, and the diagonal
$D$ charge of $N$ ($Q_{5}^{a}(N)$):
$\displaystyle Q_{5F}^{a}(N)$ $\displaystyle=$ $\displaystyle\frac{2}{3}\int
d{\bf x}\,\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf
F}_{(8)}^{a}\,N\right)~{}\,,$ (63) $\displaystyle Q_{5F}^{a}(\Delta)$
$\displaystyle=$ $\displaystyle\frac{1}{3}\int d{\bf
x}\,\left(\overline{\Delta}\gamma_{0}\gamma_{5}\,{\bf
F}_{(10)}^{a}\,\Delta\right)~{}\,,$ (64) $\displaystyle Q_{5D}^{a}(N)$
$\displaystyle=$ $\displaystyle~{}~{}~{}\int d{\bf
x}\,\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf D}^{a}\,N\right)~{}\,,$ (65)
lead to
$\displaystyle\left[Q_{5D+F}^{a}(N),Q_{5D+F}^{b}(N)\right]$ $\displaystyle=$
$\displaystyle\int d{\bf x}\Bigg{(}\overline{N}\gamma_{0}\,\Big{(}\big{(}{\bf
D}^{a}+\frac{2}{3}{\bf F}_{(8)}^{a}\big{)}\big{(}{\bf D}^{b}+\frac{2}{3}{\bf
F}_{(8)}^{b}\big{)}$ $\displaystyle-\big{(}{\bf D}^{b}+\frac{2}{3}{\bf
F}_{(8)}^{b}\big{)}\big{(}{\bf D}^{a}+\frac{2}{3}{\bf
F}_{(8)}^{a}\big{)}\Big{)}N\Bigg{)}\,,$
$\displaystyle\left[Q_{5F}^{a}(\Delta),Q_{5F}^{b}(\Delta)\right]$
$\displaystyle=$ $\displaystyle if^{abc}\frac{1}{9}Q^{c}(\Delta)\,,\ $ (67)
lead to ”only” one part of the N and Delta vector charges respectively, on the
right-hand side of Eqs. (III.3) and (67).
Once again, it turns out that the off-diagonal terms in the axial charge
$\displaystyle Q_{5}^{a}(N,\Delta)$ $\displaystyle=$ $\displaystyle\int d{\bf
x}\,\Bigg{(}\frac{2}{\sqrt{3}}\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf
T}^{a}\,\Delta+\overline{\Delta}\gamma_{0}\gamma_{5}\,{\bf
T}^{a\dagger}\,N\right)\Bigg{)}\,,$ (68)
play a crucial role in the closure of the chiral algebra Eq. (56). The
additional terms in the commutator add up to
$\displaystyle\left[Q_{5}^{a}(N,\Delta),Q_{5}^{b}(N,\Delta)\right]$
$\displaystyle=$ $\displaystyle\frac{4}{3}\int d{\bf
x}\left(\overline{N}\gamma_{0}\,\left({\bf T}^{a}{\bf T}^{b\dagger}-{\bf
T}^{b}{\bf T}^{a\dagger}\right)\,N+\overline{\Delta}\gamma_{0}\,\left({\bf
T}^{a\dagger}{\bf T}^{b}-{\bf T}^{b\dagger}{\bf T}^{a}\right)\Delta\right)\,,$
(69)
which provide the “missing” factors due to the following properties of the
off-diagonal flavor operators ${\bf T}^{i}$ and ${\bf D}^{i}$ matrices
$\displaystyle i\,f^{ijk}({\bf F}_{(8)}^{k})$ $\displaystyle=$
$\displaystyle\Big{(}\big{(}{\bf D}^{i}+\frac{2}{3}{\bf
F}_{(8)}^{i}\big{)}\big{(}{\bf D}^{j}+\frac{2}{3}{\bf
F}_{(8)}^{j}\big{)}-\big{(}{\bf D}^{j}+\frac{2}{3}{\bf
F}_{(8)}^{j}\big{)}\big{(}{\bf D}^{i}+\frac{2}{3}{\bf
F}_{(8)}^{i}\big{)}\Big{)}+{4\over 3}({\bf T}^{i}_{10/8}{\bf
T}^{j\dagger}_{10/8}-{\bf T}^{j}_{10/8}{\bf T}^{i\dagger}_{10/8})\,,$
$\displaystyle i\frac{2}{3}\,f^{ijk}{\bf F}_{(10)}^{k}$ $\displaystyle=$
$\displaystyle{\bf T}_{10/8}^{i\dagger}{\bf T}_{10/8}^{j}-{\bf
T}_{10/8}^{j\dagger}{\bf T}_{10/8}^{i}\,.$ (70)
Therefore, the chiral algebra Eqs. (54), (55) and (56) closes in spite, or
perhaps because of the apparent fractional axial charges ($\frac{2}{3}$ and
$\frac{1}{3}$).
## IV Chiral mixing and the axial current
A unique feature of the use of the linear chiral representation is that the
axial coupling is determined by the chiral representations, as given by the
coefficients of the axial transformations. For the nucleon (proton and
neutron), chiral representations of $SU_{L}(2)\times SU_{R}(2)$,
$(\frac{1}{2},0)(\sim(8,1),(3,\bar{3}))$ and $(1,\frac{1}{2})(\sim(6,3))$
provide the nucleon isovector axial coupling $g_{A}^{(3)}=1$ and $5/3$
respectively. Therefore, the mixing of chiral $(\frac{1}{2},0)$ and
$(1,\frac{1}{2})$ nucleons leads to the axial coupling
$\displaystyle 1.267$ $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(1)}~{}\cos^{2}\theta+g_{A~{}(1,\frac{1}{2})}^{(1)}~{}\sin^{2}\theta$
(71) $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(1)}~{}\cos^{2}\theta+\frac{5}{3}~{}\sin^{2}\theta\,,$
Table 1: The Abelian and the non-Abelian axial charges (+ sign indicates “naive”, - sign “mirror” transformation properties) and the non-Abelian chiral multiplets of $J^{P}=\frac{1}{2}$, Lorentz representation $(\frac{1}{2},0)$ nucleon and $\Delta$ fields, see Refs. Nagata:2007di ; Nagata:2008zzc ; Dmitrasinovic:2009vp ; Dmitrasinovic:2009vy . case | field | $g_{A}^{(0)}$ | $g_{A}^{(1)}$ | $F$ | $D$ | $SU_{L}(3)\times SU_{R}(3)$
---|---|---|---|---|---|---
I | $N_{1}-N_{2}$ | $-1$ | $+1$ | $~{}~{}0$ | $+1$ | $(3,\overline{3})\oplus(\overline{3},3)$
II | $N_{1}+N_{2}$ | $+3$ | $+1$ | $+1$ | $~{}~{}0$ | $(8,1)\oplus(1,8)$
III | $N_{1}^{{}^{\prime}}-N_{2}^{{}^{\prime}}$ | $+1$ | $-1$ | $~{}~{}0$ | $-1$ | $(\overline{3},3)\oplus(3,\overline{3})$
IV | $N_{1}^{{}^{\prime}}+N_{2}^{{}^{\prime}}$ | $-3$ | $-1$ | $-1$ | $~{}~{}0$ | $(1,8)\oplus(8,1)$
0 | $\partial_{\mu}(N_{3}^{\mu}+\frac{1}{3}N_{4}^{\mu})$ | $+1$ | $+\frac{5}{3}$ | $+\frac{2}{3}$ | $+1$ | $(6,3)\oplus(3,6)$
Three-quark nucleon interpolating fields in QCD have also well-defined, if
perhaps unexpected $U_{A}(1)$ chiral transformation properties, see Table 1,
that can be used to predict the isoscalar axial coupling $g_{A~{}\rm
mix.}^{(0)}$
$\displaystyle g_{A~{}\rm mix.}^{(0)}$ $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(0)}~{}\cos^{2}\theta+g_{A~{}(1,\frac{1}{2})}^{(0)}~{}\sin^{2}\theta$
(72) $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(0)}~{}\cos^{2}\theta+\sin^{2}\theta,$
together with the mixing angle $\theta$ extracted from Eq. (71). Note,
however, that due to the different (bare) non-Abelian $g_{A}^{(1)}$ and
Abelian $g_{A}^{(0)}$ axial couplings, see Table 1, the mixing formulae Eq.
(72) give substantially different predictions from one case to another, see
Table 2.
Table 2: The values of the baryon isoscalar axial coupling constant predicted from the naive mixing and $g_{A~{}\rm expt.}^{(1)}=1.267$; compare with $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.03\pm 0.05$, $F$=$0.459\pm 0.008$ and $D$=$0.798\pm 0.008$, leading to $F/D=0.571\pm 0.005$, Ref. Yamanishi:2007zza . case | ($g_{A}^{(1)}$,$g_{A}^{(0)}$) | $g_{A~{}\rm mix.}^{(1)}$ | $\theta_{i}$ | $g_{A~{}\rm mix.}^{(0)}$ | $g_{A~{}\rm mix.}^{(0)}$ | $F$ | $F$/$D$
---|---|---|---|---|---|---|---
I | $(+1,-1)$ | $\frac{1}{3}(4-\cos 2\theta)$ | $39.3^{o}$ | $-\cos 2\theta$ | -0.20 | 0.267 | 0.267
II | $(+1,+3)$ | $\frac{1}{3}(4-\cos 2\theta)$ | $39.3^{o}$ | $(2\cos 2\theta+1)$ | 2.20 | 0.866 | 2.16
III | $(-1,+1)$ | $\frac{1}{3}(1-4\cos 2\theta)$ | $67.2^{o}$ | $1$ | 1.00 | 0.567 | 0.81
IV | $(-1,-3)$ | $\frac{1}{3}(1-4\cos 2\theta)$ | $67.2^{o}$ | $-(2\cos 2\theta+1)$ | 0.40 | 0.417 | 0.491
We can see in Table 2 that the two best candidates are cases I and IV, with
$g_{A}^{(0)}=-0.2$ and $g_{A}^{(0)}=0.4$, respectively, the latter being
within the error bars of the measured value $g_{A~{}\rm expt.}^{(0)}=0.33\pm
0.08$, Bass:2007zzb ; Ageev:2007du . Moreover, this scheme predicts the $F$
and $D$ values, as well:
$\displaystyle F$ $\displaystyle=$ $\displaystyle
F_{(\frac{1}{2},0)}~{}\cos^{2}\theta+F_{(1,\frac{1}{2})}^{(1)}~{}\sin^{2}\theta,$
(73) $\displaystyle=$ $\displaystyle
F_{(\frac{1}{2},0)}~{}\cos^{2}\theta+\frac{2}{3}~{}\sin^{2}\theta$
$\displaystyle D$ $\displaystyle=$ $\displaystyle
D_{(\frac{1}{2},0)}~{}\cos^{2}\theta+D_{(1,\frac{1}{2})}~{}\sin^{2}\theta$
(74) $\displaystyle=$ $\displaystyle
D_{(\frac{1}{2},0)}~{}\cos^{2}\theta+\sin^{2}\theta,$
where we have used the $F$ and $D$ values for different chiral multiplets as
listed in Table 1.
Cases I and IV, with $F$/$D$ = 0.267 and 0.491, respectively, ought to be
compared with $F$/$D$ = $0.571\pm 0.005$ 222Note that the Ref.
Yamanishi:2007zza values add up to F+D = $1.312\pm 0.002$, more than
2-$\sigma$ away from the experimental constraint $\neq 1.269\pm 0.002$.. Case
I is, of course, the well-known “Ioffe current”, which reproduces the
nucleon’s properties in QCD lattice and sum rules calculations. The latter is
a “mirror” opposite of the orthogonal complement to the Ioffe current, an
interpolating field that, to our knowledge, has not been used in QCD thus far.
Manifestly, a linear superposition of any three fields (except for the
mixtures of cases II and III, IV above, which yield complex mixing angles)
should give a perfect fit to the central values of the experimental axial
couplings and predict the $F$ and $D$ values. Such a three-field admixture
introduces new free parameters (besides the two already introduced mixing
angles, e.g. $\theta_{1}$ and $\theta_{4}$, we have the relative/mutual mixing
angle $\theta_{14}$, as the two nucleon fields I and IV may also mix). One may
subsume the sum and the difference of the two angles $\theta_{1}$ and
$\theta_{4}$ into the new angle $\theta$, and define
$\varphi\doteq\theta_{14}$ (this relationship depends on the precise
definition of the mixing angles $\theta_{1}$, $\theta_{4}$ and $\theta_{14}$);
thus we find two equations with two unknowns of the general form:
$\displaystyle\frac{5}{3}\,{\sin}^{2}\theta+{\cos}^{2}\theta\,\left(g_{A}^{(1)}{\cos}^{2}\varphi+g_{A}^{(1)\prime}{\sin}^{2}\varphi\right)$
$\displaystyle=1.267$ (75)
$\displaystyle{\sin}^{2}\theta+{\cos}^{2}\theta\,\left(g_{A}^{(0)}{\cos}^{2}\varphi+g_{A}^{(0)\prime}\,{\sin}^{2}\varphi\right)$
$\displaystyle=0.33\pm 0.08$ (76)
The solutions to these equations (the values of the mixing angles
$\theta,\varphi$) provide, at the same time, input for the prediction of $F$
and $D$:
$\displaystyle\cos^{2}\theta\,\left(F\,{\cos}^{2}\varphi+F^{\prime}\,{\sin}^{2}\varphi\right)+\frac{2}{3}~{}\sin^{2}\theta$
$\displaystyle=F$ (77)
$\displaystyle\cos^{2}\theta\,\left(D\,{\cos}^{2}\varphi+D^{\prime}\,{\sin}^{2}\varphi\right)+\sin^{2}\theta$
$\displaystyle=D.$ (78)
The values of the mixing angles ($\theta,\varphi$) obtained from this
straightforward fit to the baryon axial coupling constants are shown in Table
3.
Table 3: The values of the mixing angles obtained from the simple fit to the baryon axial coupling constants and the predicted values of axial $F$ and $D$ couplings. The experimental values are $F$=$0.459\pm 0.008$ and $D$=$0.798\pm 0.008$, leading to $F/D=0.571\pm 0.005$, Ref. Yamanishi:2007zza . case | $g_{A~{}\rm expt.}^{(3)}$ | $g_{A~{}\rm expt.}^{(0)}$ | $\theta$ | $\varphi$ | $F$ | $D$ | $F$/$D$
---|---|---|---|---|---|---|---
I-II | 1.267 | $0.33$ | $39.3^{o}$ | $28.0^{o}\pm 2.3^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
I-III | 1.267 | $0.33$ | $50.7^{o}\pm 1.8^{o}$ | $23.9^{o}\pm 2.9^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
I-IV | 1.267 | $0.33$ | $63.2^{o}\pm 4.0^{o}$ | $54^{o}\pm 23^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
Note that all three admissible scenarios (i.e. choices of pairs of fields
admixed to the (6,3) one that lead to real mixing angles) yield the same
values of $F$ and $D$. This is due to the fact that all three-quark baryon
fields satisfy the following relation $g_{A}^{(0)}=3F-D=\sqrt{3}g_{A}^{(8)}$
Jido09 . The first relation $g_{A}^{(0)}=3F-D$ was not expected, as the
flavor-singlet properties, such as $g_{A}^{(0)}$ are generally expected to be
independent of the flavor-octet ones, such as $F,D$. Yet, it is not unnatural,
either, as it indicates the absence of polarized $s\bar{s}$ pairs in these
SU(3) symmetric, three-quark nucleon interpolators. In order to show that, we
define $g_{A}^{(0)}=\Delta u+\Delta d+\Delta s$ and
$g_{A}^{(8)}={1\over\sqrt{3}}(\Delta u+\Delta d-2\Delta s)$, where $\Delta q$
are the (corresponding flavor) quark contributions to the matrix element of
the nucleon’s axial vector current $\Delta q=\langle
N|\bar{q}\gamma_{\mu}\gamma_{5}q|N\rangle$. We see that $g_{A}^{(0)}\sim
g_{A}^{(8)}$ only if $\Delta s=0$.
Thus, the relation $g_{A}^{(0)}=3F-D$ appears to depend on the choice of
three-quark interpolating fields as a source of admixed mirror fields and may
well change when one considers other interpolating fields, such as the five-
quark (“pentaquark”) ones for example333Note, however, that five- and more
quark, and derivative interpolating fields are not the only ones that can
produce mirror fields, however: so can the one-gluon-three-quark “hybrid
baryon” interpolators, which necessarily have the same chiral properties as
the corresponding three-quark fields.. In that sense a deviation of the
measured values of $g_{A}^{(0)}$ and $g_{A}^{(8)}=\frac{1}{\sqrt{3}}(3F-D)$
from this relation may well be seen as a measure of the contribution of
higher-order configurations’ to the baryon ground state. It seems very
difficult, however, to evaluate $F$ and $D$ for specific higher-order
configurations without going through the procedure outlined in Ref.
Chen:2008qv for the “pentaquark” interpolator chiral multiplets 444If one
were to assign one particular source of mirror fields, for example some
“pentaquark” interpolators, then one could try to determine the contribution
of $s\bar{s}$ pairs to the flavor singlet axial coupling..
Some of the ideas used above have also been used in some of the following
early papers: two-chiral-multiplet mixing was considered long ago by Harari
Harari:1966yq , and by Weinberg Weinberg:1969hw , for example. Moreover,
special cases of three-field/configuration chiral mixing have been considered
by Harari Harari:1966jz , and by Gerstein and Lee Gerstein:1966zz in the
context of the (“collinear”) $U(3)\times U(3)$ current algebra at infinite
momentum. One (obvious) distinction from these early precedents is our use of
QCD interpolating fields, which appeared only in the early 1980’s, and the
(perhaps less obvious) issue of baryons’ flavor-singlet axial current (a.k.a.
the $U_{A}(1)$), that was (seriously) raised yet another decade later. We
emphasize here that our results are based on the $U_{L}(3)\times U_{R}(3)$
chiral algebra of space-integrated time components of currents, without any
assumptions about saturation of this algebra by one-particle states, or its
dependence on any one particular frame of reference. Indeed, our nucleon
interpolating fields transform as the $(\frac{1}{2},0)+(0,\frac{1}{2})$
representation of the Lorentz group, thus making the Noether currents (fully)
Lorentz covariant, so that our results hold in any frame.
## V Summary and Outlook
We have re-organized the results of our previous paper Chen:2008qv into the
(perhaps more) conventional form for the baryon octet using $F$ and $D$
coupling ($SU(3)$ structure constants). This means that, inter alia, we have
explicitly written down (perhaps for the first time) the chiral
transformations of the $(6,3)\oplus(3,6)$ octet and decimet fields in the
$SU(3)$ particle (octet and decimet) basis.
In the process we have independently constructed $SU(3)$ generators of the
decimet and derived a set of $SU(3)$ Clebsch-Gordan coefficients in the
“natural” convention, which means that all isospin $SU(2)$ sub-multiplets of
the octet and the decimet have standard isospin $SU(2)$ generators.
Then we used the above mentioned $SU(3)$ Clebsch-Gordan coefficients to
explicitly check the closure of the $SU_{L}(3)\times SU_{R}(3)$ chiral algebra
in the $SU(3)$ particle basis, which forms an independent check/confirmation
of the calculation.
Next, we investigated the phenomenological consequence for the baryon axial
currents, of the chiral $[(6,3)\oplus(3,6)]$ multiplet mixing with other
three-quark baryon field multiplets, such as the
$[(3,\overline{3})\oplus(\overline{3},3)]$ and $[(8,1)\oplus(1,8)]$. The
results of the three-field (“two-angle”) mixing are interesting: all
permissible combinations fields lead to the same $F$/$D$ prediction, that is
in reasonable agreement with experiment. This identity of results is a
consequence of the relation $g_{A}^{(0)}=3F-D$ between the flavor singlet
axial coupling $g_{A}^{(0)}$ and the (previously unrelated) flavor octet $F$
and $D$ values. That relation is a unique feature of the three-quark
interpolating fields and any potential departures from it may be attributed to
fields with a number of quarks higher than three.
The next step, left for the future, is to investigate $SU_{L}(3)\times
SU_{R}(3)$ chiral invariant interactions and the $SU(3)\times SU(3)\to
SU(2)\times SU(2)$ symmetry breaking/reduction and to the study of the chiral
$SU(2)\times SU(2)$ properties of hyperons. Then one may consider explicit
chiral symmetry breaking corrections to the axial and the vector currents,
which are related to the $SU(3)\times SU(3)$ symmetry breaking meson-nucleon
derivative interactions, not just the explicit $SU(3)$ symmetry breaking ones
that have been considered thus far (see Ref. Yamanishi:2007zza and the
previous subsection, above).
## Acknowledgments
We wish to thank Profs. Daisuke Jido, Akira Ohnishi and Makoto Oka for
valuable conversations regarding the present work. One of us (V.D.) wishes to
thank the RCNP, Osaka University, under whose auspices this work was begun,
and the Yukawa Institute for Theoretical Physics, Kyoto, (molecular workshop
“Algebraic aspects of chiral symmetry for the study of excited baryons”) where
it was finished, for kind hospitality and financial support.
## Appendix A SU(3) Octet, Decimet Generators and 8x10 Transition Matrices
### A.1 Octet “Generator” 8x8 Matrices ${\rm{\bf D}^{a}},~{}{\rm{\bf
F}_{(8)}^{a}}$ in the Particle Basis
$\displaystyle{\rm({\bf D}^{1}+{2\over 3}{\bf F}_{(8)}^{1})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll|l}0&\frac{5}{6}&0&0&0&0&0&0&p\\\
\frac{5}{6}&0&0&0&0&0&0&0&n\\\
0&0&0&\frac{\sqrt{2}}{3}&0&0&0&-\frac{1}{\sqrt{6}}&\Sigma^{+}\\\
0&0&\frac{\sqrt{2}}{3}&0&\frac{\sqrt{2}}{3}&0&0&0&\Sigma^{0}\\\
0&0&0&\frac{\sqrt{2}}{3}&0&0&0&\frac{1}{\sqrt{6}}&\Sigma^{-}\\\
0&0&0&0&0&0&-\frac{1}{6}&0&\Xi^{0}\\\ 0&0&0&0&0&-\frac{1}{6}&0&0&\Xi^{-}\\\
0&0&-\frac{1}{\sqrt{6}}&0&\frac{1}{\sqrt{6}}&0&0&0&\Lambda_{8}\\\ \hline\cr\\\
p&n&\Sigma^{+}&\Sigma^{0}&\Sigma^{-}&\Xi^{0}&\Xi^{-}&\Lambda_{8}\end{array}\right)$
(89) $\displaystyle{\rm{\bf D}^{1}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&\frac{1}{2}&0&0&0&0&0&0\\\
\frac{1}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&\frac{1}{\sqrt{6}}\\\
0&0&0&0&0&0&-\frac{1}{2}&0\\\ 0&0&0&0&0&-\frac{1}{2}&0&0\\\
0&0&-\frac{1}{\sqrt{6}}&0&\frac{1}{\sqrt{6}}&0&0&0\end{array}\right)$ (98)
$\displaystyle{\bf F}_{(8)}^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&\frac{1}{2}&0&0&0&0&0&0\\\
\frac{1}{2}&0&0&0&0&0&0&0\\\ 0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}&0&0&0\\\
0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&\frac{1}{2}&0\\\
0&0&0&0&0&\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&0&0\end{array}\right)$ (107)
$\displaystyle{\rm({\bf D}^{2}+{2\over 3}{\bf F}_{(8)}^{2})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&-\frac{5i}{6}&0&0&0&0&0&0\\\
\frac{5i}{6}&0&0&0&0&0&0&0\\\
0&0&0&-\frac{i\sqrt{2}}{3}&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&\frac{i\sqrt{2}}{3}&0&-\frac{i\sqrt{2}}{3}&0&0&0\\\
0&0&0&\frac{i\sqrt{2}}{3}&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&0&0&0&0&\frac{i}{6}&0\\\ 0&0&0&0&0&-\frac{i}{6}&0&0\\\
0&0&-\frac{i}{\sqrt{6}}&0&-\frac{i}{\sqrt{6}}&0&0&0\end{array}\right)$ (116)
$\displaystyle{\rm{\bf D}^{2}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&-\frac{i}{2}&0&0&0&0&0&0\\\
\frac{i}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&0&0&0&0&\frac{i}{2}&0\\\ 0&0&0&0&0&-\frac{i}{2}&0&0\\\
0&0&-\frac{i}{\sqrt{6}}&0&-\frac{i}{\sqrt{6}}&0&0&0\end{array}\right)$ (125)
$\displaystyle{\bf F}_{(8)}^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&-\frac{i}{2}&0&0&0&0&0&0\\\
\frac{i}{2}&0&0&0&0&0&0&0\\\ 0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&\frac{i}{\sqrt{2}}&0&-\frac{i}{\sqrt{2}}&0&0&0\\\
0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&-\frac{i}{2}&0\\\
0&0&0&0&0&\frac{i}{2}&0&0\\\ 0&0&0&0&0&0&0&0\end{array}\right)$ (134)
$\displaystyle{\rm({\bf D}^{3}+{2\over 3}{\bf F}_{(8)}^{3})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{5}{6}&0&0&0&0&0&0&0\\\
0&-\frac{5}{6}&0&0&0&0&0&0\\\ 0&0&\frac{2}{3}&0&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{1}{\sqrt{3}}\\\ 0&0&0&0&-\frac{2}{3}&0&0&0\\\
0&0&0&0&0&-\frac{1}{6}&0&0\\\ 0&0&0&0&0&0&\frac{1}{6}&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\end{array}\right)$ (143)
$\displaystyle{\rm{\bf D}^{3}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{1}{2}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{1}{\sqrt{3}}\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&-\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&\frac{1}{2}&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\end{array}\right)$ (152) $\displaystyle{\bf
F}_{(8)}^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{1}{2}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&0&0&0&0&0\end{array}\right)$ (161)
$\displaystyle{\rm({\bf D}^{4}+{2\over 3}{\bf F}_{(8)}^{4})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{1}{6\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
0&0&0&0&\frac{1}{6}&0&0&0\\\ 0&0&0&0&0&-\frac{5}{6}&0&0\\\
\frac{1}{6\sqrt{2}}&0&0&0&0&0&-\frac{5}{6\sqrt{2}}&0\\\
0&\frac{1}{6}&0&0&0&0&0&0\\\ 0&0&-\frac{5}{6}&0&0&0&0&0\\\
0&0&0&-\frac{5}{6\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&0&0&-\frac{1}{2\sqrt{6}}&0\end{array}\right)$
(170) $\displaystyle{\rm{\bf D}^{4}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
0&0&0&0&\frac{1}{2}&0&0&0\\\ 0&0&0&0&0&-\frac{1}{2}&0&0\\\
\frac{1}{2\sqrt{2}}&0&0&0&0&0&-\frac{1}{2\sqrt{2}}&0\\\
0&\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{1}{2}&0&0&0&0&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&\frac{1}{2\sqrt{6}}\\\
-\frac{1}{2\sqrt{6}}&0&0&0&0&0&\frac{1}{2\sqrt{6}}&0\end{array}\right)$ (179)
$\displaystyle{\bf F}_{(8)}^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
0&0&0&0&-\frac{1}{2}&0&0&0\\\ 0&0&0&0&0&-\frac{1}{2}&0&0\\\
-\frac{1}{2\sqrt{2}}&0&0&0&0&0&-\frac{1}{2\sqrt{2}}&0\\\
0&-\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{1}{2}&0&0&0&0&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}&0\end{array}\right)$
(188)
$\displaystyle{\rm({\bf D}^{5}+{2\over 3}{\bf F}_{(8)}^{5})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&-\frac{i}{6\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
0&0&0&0&-\frac{i}{6}&0&0&0\\\ 0&0&0&0&0&\frac{5i}{6}&0&0\\\
\frac{i}{6\sqrt{2}}&0&0&0&0&0&\frac{5i}{6\sqrt{2}}&0\\\
0&\frac{i}{6}&0&0&0&0&0&0\\\ 0&0&-\frac{5i}{6}&0&0&0&0&0\\\
0&0&0&-\frac{5i}{6\sqrt{2}}&0&0&0&-\frac{i}{2\sqrt{6}}\\\
-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&0&0&\frac{i}{2\sqrt{6}}&0\end{array}\right)$
(197) $\displaystyle{\rm{\bf D}^{5}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
0&0&0&0&-\frac{i}{2}&0&0&0\\\ 0&0&0&0&0&\frac{i}{2}&0&0\\\
\frac{i}{2\sqrt{2}}&0&0&0&0&0&\frac{i}{2\sqrt{2}}&0\\\
0&\frac{i}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{i}{2}&0&0&0&0&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
-\frac{i}{2\sqrt{6}}&0&0&0&0&0&-\frac{i}{2\sqrt{6}}&0\end{array}\right)$ (206)
$\displaystyle{\bf F}_{(8)}^{5}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{i}{2\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
0&0&0&0&\frac{i}{2}&0&0&0\\\ 0&0&0&0&0&\frac{i}{2}&0&0\\\
-\frac{i}{2\sqrt{2}}&0&0&0&0&0&\frac{i}{2\sqrt{2}}&0\\\
0&-\frac{i}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{i}{2}&0&0&0&0&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&-\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}&0\end{array}\right)$
(215)
$\displaystyle{\rm({\bf D}^{6}+{2\over 3}{\bf F}_{(8)}^{6})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&-\frac{1}{6}&0&0&0&0&0\\\
0&0&0&-\frac{1}{6\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
-\frac{1}{6}&0&0&0&0&0&0&0\\\
0&-\frac{1}{6\sqrt{2}}&0&0&0&-\frac{5}{6\sqrt{2}}&0&0\\\
0&0&0&0&0&0&-\frac{5}{6}&0\\\
0&0&0&-\frac{5}{6\sqrt{2}}&0&0&0&\frac{1}{2\sqrt{6}}\\\
0&0&0&0&-\frac{5}{6}&0&0&0\\\
0&-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&\frac{1}{2\sqrt{6}}&0&0\end{array}\right)$
(224) $\displaystyle{\rm{\bf D}^{6}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&-\frac{1}{2}&0&0&0&0&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
-\frac{1}{2}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
0&0&0&0&-\frac{1}{2}&0&0&0\\\
0&-\frac{1}{2\sqrt{6}}&0&0&0&-\frac{1}{2\sqrt{6}}&0&0\end{array}\right)$ (233)
$\displaystyle{\bf F}_{(8)}^{6}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&\frac{1}{2}&0&0&0&0&0\\\
0&0&0&\frac{1}{2\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
\frac{1}{2}&0&0&0&0&0&0&0\\\
0&\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&\frac{\sqrt{\frac{3}{2}}}{2}\\\
0&0&0&0&-\frac{1}{2}&0&0&0\\\
0&-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&\frac{\sqrt{\frac{3}{2}}}{2}&0&0\end{array}\right)$
(242)
$\displaystyle{\rm({\bf D}^{7}+{2\over 3}{\bf F}_{(8)}^{7})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&\frac{i}{6}&0&0&0&0&0\\\
0&0&0&\frac{i}{6\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
-\frac{i}{6}&0&0&0&0&0&0&0\\\
0&-\frac{i}{6\sqrt{2}}&0&0&0&\frac{5i}{6\sqrt{2}}&0&0\\\
0&0&0&0&0&0&\frac{5i}{6}&0\\\
0&0&0&-\frac{5i}{6\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
0&0&0&0&-\frac{5i}{6}&0&0&0\\\
0&-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&-\frac{i}{2\sqrt{6}}&0&0\end{array}\right)$
(251) $\displaystyle{\rm{\bf D}^{7}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&\frac{i}{2}&0&0&0&0&0\\\
0&0&0&\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
-\frac{i}{2}&0&0&0&0&0&0&0\\\
0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&\frac{i}{2}&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&-\frac{i}{2\sqrt{6}}\\\
0&0&0&0&-\frac{i}{2}&0&0&0\\\
0&-\frac{i}{2\sqrt{6}}&0&0&0&\frac{i}{2\sqrt{6}}&0&0\end{array}\right)$ (260)
$\displaystyle{\bf F}_{(8)}^{7}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&-\frac{i}{2}&0&0&0&0&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
\frac{i}{2}&0&0&0&0&0&0&0\\\
0&\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&\frac{i}{2}&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
0&0&0&0&-\frac{i}{2}&0&0&0\\\
0&-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0\end{array}\right)$
(269)
$\displaystyle{\rm({\bf D}^{8}+{2\over 3}{\bf F}_{(8)}^{2})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{1}{2\sqrt{3}}&0&0&0&0&0&0&0\\\
0&\frac{1}{2\sqrt{3}}&0&0&0&0&0&0\\\ 0&0&\frac{1}{\sqrt{3}}&0&0&0&0&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\\\ 0&0&0&0&\frac{1}{\sqrt{3}}&0&0&0\\\
0&0&0&0&0&-\frac{\sqrt{3}}{2}&0&0\\\ 0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{3}}\end{array}\right)$ (278)
$\displaystyle{\rm{\bf D}^{8}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}-\frac{1}{2\sqrt{3}}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2\sqrt{3}}&0&0&0&0&0&0\\\ 0&0&\frac{1}{\sqrt{3}}&0&0&0&0&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\\\ 0&0&0&0&\frac{1}{\sqrt{3}}&0&0&0\\\
0&0&0&0&0&-\frac{1}{2\sqrt{3}}&0&0\\\ 0&0&0&0&0&0&-\frac{1}{2\sqrt{3}}&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{3}}\end{array}\right)$ (287) $\displaystyle{\bf
F}_{(8)}^{8}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0\\\
0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&-\frac{\sqrt{3}}{2}&0&0\\\
0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0\\\ 0&0&0&0&0&0&0&0\end{array}\right)$ (296)
### A.2 Octet-Decimet 8x10 Transition Matrices $T^{a}$
$\displaystyle
T_{1}=\left(\begin{array}[]{llllllllll|l}-\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{6}}&0&0&0&0&0&0&0&p\\\
0&-\frac{1}{\sqrt{6}}&0&\frac{1}{\sqrt{2}}&0&0&0&0&0&0&n\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0&\Sigma^{+}\\\
0&0&0&0&\frac{1}{2\sqrt{3}}&0&\frac{1}{2\sqrt{3}}&0&0&0&\Sigma^{0}\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0&\Sigma^{-}\\\
0&0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&\Xi^{0}\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&0&\Xi^{-}\\\
0&0&0&0&\frac{1}{2}&0&-\frac{1}{2}&0&0&0&\Lambda_{8}\\\ \hline\cr\\\
\Delta^{++}&\Delta^{+}&\Delta^{0}&\Delta^{-}&\Sigma^{*+}&\Sigma^{*0}&\Sigma^{*-}&\Xi^{*0}&\Xi^{*-}&\Omega\end{array}\right)$
(307)
$\displaystyle
T_{2}=\left(\begin{array}[]{llllllllll}-\frac{i}{\sqrt{2}}&0&-\frac{i}{\sqrt{6}}&0&0&0&0&0&0&0\\\
0&-\frac{i}{\sqrt{6}}&0&-\frac{i}{\sqrt{2}}&0&0&0&0&0&0\\\
0&0&0&0&0&-\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&\frac{i}{2\sqrt{3}}&0&-\frac{i}{2\sqrt{3}}&0&0&0\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&\frac{i}{\sqrt{6}}&0\\\
0&0&0&0&0&0&0&-\frac{i}{\sqrt{6}}&0&0\\\
0&0&0&0&\frac{i}{2}&0&\frac{i}{2}&0&0&0\end{array}\right)$ (316)
$\displaystyle
T_{3}=\left(\begin{array}[]{llllllllll}0&\sqrt{\frac{2}{3}}&0&0&0&0&0&0&0&0\\\
0&0&\sqrt{\frac{2}{3}}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{\sqrt{6}}&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&0&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&0\\\
0&0&0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0\\\
0&0&0&0&0&-\frac{1}{\sqrt{2}}&0&0&0&0\end{array}\right)$ (325)
$\displaystyle
T_{4}=\left(\begin{array}[]{llllllllll}0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0&0&0\\\
-\frac{1}{\sqrt{2}}&0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0&0\\\
0&-\frac{1}{\sqrt{3}}&0&0&0&0&0&0&\frac{1}{2\sqrt{3}}&0\\\
0&0&-\frac{1}{\sqrt{6}}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{\sqrt{6}}&0&0&0&0&-\frac{1}{\sqrt{2}}\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&-\frac{1}{2}&0\end{array}\right)$ (334)
$\displaystyle
T_{5}=\left(\begin{array}[]{llllllllll}0&0&0&0&0&-\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&-\frac{i}{\sqrt{6}}&0&0&0\\\
-\frac{i}{\sqrt{2}}&0&0&0&0&0&0&-\frac{i}{\sqrt{6}}&0&0\\\
0&-\frac{i}{\sqrt{3}}&0&0&0&0&0&0&-\frac{i}{2\sqrt{3}}&0\\\
0&0&-\frac{i}{\sqrt{6}}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{i}{\sqrt{6}}&0&0&0&0&\frac{i}{\sqrt{2}}\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&\frac{i}{2}&0\end{array}\right)$ (343)
$\displaystyle
T_{6}=\left(\begin{array}[]{llllllllll}0&0&0&0&-\frac{1}{\sqrt{6}}&0&0&0&0&0\\\
0&0&0&0&0&-\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&-\frac{1}{\sqrt{6}}&0&0&0&0&0&0&0&0\\\
0&0&-\frac{1}{\sqrt{3}}&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0\\\
0&0&0&-\frac{1}{\sqrt{2}}&0&0&0&0&\frac{1}{\sqrt{6}}&0\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0&0&-\frac{1}{\sqrt{2}}\\\
0&0&0&0&0&0&0&\frac{1}{2}&0&0\end{array}\right)$ (352)
$\displaystyle
T_{7}=\left(\begin{array}[]{llllllllll}0&0&0&0&\frac{i}{\sqrt{6}}&0&0&0&0&0\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&-\frac{i}{\sqrt{6}}&0&0&0&0&0&0&0&0\\\
0&0&-\frac{i}{\sqrt{3}}&0&0&0&0&-\frac{i}{2\sqrt{3}}&0&0\\\
0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0&-\frac{i}{\sqrt{6}}&0\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{i}{\sqrt{6}}&0&0&\frac{i}{\sqrt{2}}\\\
0&0&0&0&0&0&0&-\frac{i}{2}&0&0\end{array}\right)$ (361)
$\displaystyle T_{8}=\left(\begin{array}[]{llllllllll}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{2}}&0&0\\\
0&0&0&0&0&0&0&0&-\frac{1}{\sqrt{2}}&0\\\
0&0&0&0&0&0&0&0&0&0\end{array}\right)$ (370)
### A.3 Decimet Generator 10x10 Matrices ${\bf F}_{(10)}^{a}$
$\displaystyle{\bf F}_{(10)}^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll|l}0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0&\Delta^{++}\\\
\frac{\sqrt{3}}{2}&0&1&0&0&0&0&0&0&0&\Delta^{+}\\\
0&1&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&\Delta^{0}\\\
0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&\Delta^{-}\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0&\Sigma^{*+}\\\
0&0&0&0&\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}&0&0&0&\Sigma^{*0}\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0&\Sigma^{*-}\\\
0&0&0&0&0&0&0&0&\frac{1}{2}&0&\Xi^{*0}\\\
0&0&0&0&0&0&0&\frac{1}{2}&0&0&\Xi^{*-}\\\ 0&0&0&0&0&0&0&0&0&0&\Omega\\\
\hline\cr\\\
\Delta^{++}&\Delta^{+}&\Delta^{0}&\Delta^{-}&\Sigma^{*+}&\Sigma^{*0}&\Sigma^{*-}&\Xi^{*0}&\Xi^{*-}&\Omega\end{array}\right)$
(383)
$\displaystyle{\bf F}_{(10)}^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&-\frac{i\sqrt{3}}{2}&0&0&0&0&0&0&0&0\\\
\frac{i\sqrt{3}}{2}&0&-i&0&0&0&0&0&0&0\\\
0&i&0&-\frac{i\sqrt{3}}{2}&0&0&0&0&0&0\\\
0&0&\frac{i\sqrt{3}}{2}&0&0&0&0&0&0&0\\\
0&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&\frac{i}{\sqrt{2}}&0&-\frac{i}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&0&0&-\frac{i}{2}&0\\\
0&0&0&0&0&0&0&\frac{i}{2}&0&0\\\ 0&0&0&0&0&0&0&0&0&0\end{array}\right)$ (394)
$\displaystyle{\bf F}_{(10)}^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}\frac{3}{2}&0&0&0&0&0&0&0&0&0\\\
0&\frac{1}{2}&0&0&0&0&0&0&0&0\\\ 0&0&-\frac{1}{2}&0&0&0&0&0&0&0\\\
0&0&0&-\frac{3}{2}&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0&0&0\\\
0&0&0&0&0&0&0&\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&0&0&0&0&0&0&0\end{array}\right)$ (405)
$\displaystyle{\bf F}_{(10)}^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&\frac{1}{2}&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ \frac{\sqrt{3}}{2}&0&0&0&0&0&0&1&0&0\\\
0&\frac{1}{\sqrt{2}}&0&0&0&0&0&0&\frac{1}{\sqrt{2}}&0\\\
0&0&\frac{1}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&\frac{\sqrt{3}}{2}\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{\sqrt{3}}{2}&0&0\end{array}\right)$ (416)
$\displaystyle{\bf F}_{(10)}^{5}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&-\frac{i\sqrt{3}}{2}&0&0&0&0&0\\\
0&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&-\frac{i}{2}&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ \frac{i\sqrt{3}}{2}&0&0&0&0&0&0&-i&0&0\\\
0&\frac{i}{\sqrt{2}}&0&0&0&0&0&0&-\frac{i}{\sqrt{2}}&0\\\
0&0&\frac{i}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&i&0&0&0&0&-\frac{i\sqrt{3}}{2}\\\
0&0&0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{i\sqrt{3}}{2}&0&0\end{array}\right)$ (427)
$\displaystyle{\bf F}_{(10)}^{6}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{2}&0&0&0&0&0\\\ 0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{\sqrt{3}}{2}&0&0&0\\\ 0&\frac{1}{2}&0&0&0&0&0&0&0&0\\\
0&0&\frac{1}{\sqrt{2}}&0&0&0&0&\frac{1}{\sqrt{2}}&0&0\\\
0&0&0&\frac{\sqrt{3}}{2}&0&0&0&0&1&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&\frac{\sqrt{3}}{2}\\\
0&0&0&0&0&0&0&0&\frac{\sqrt{3}}{2}&0\end{array}\right)$ (438)
$\displaystyle{\bf F}_{(10)}^{7}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&-\frac{i}{2}&0&0&0&0&0\\\ 0&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&-\frac{i\sqrt{3}}{2}&0&0&0\\\ 0&\frac{i}{2}&0&0&0&0&0&0&0&0\\\
0&0&\frac{i}{\sqrt{2}}&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0\\\
0&0&0&\frac{i\sqrt{3}}{2}&0&0&0&0&-i&0\\\
0&0&0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&i&0&0&-\frac{i\sqrt{3}}{2}\\\
0&0&0&0&0&0&0&0&\frac{i\sqrt{3}}{2}&0\end{array}\right)$ (449)
$\displaystyle{\bf F}_{(10)}^{8}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0&0\\\
0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0\\\
0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0\\\
0&0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0&0\\\
0&0&0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0\\\
0&0&0&0&0&0&0&0&0&-\sqrt{3}\end{array}\right)$ (460)
### A.4 Singlet-Octet 1x8 Transition Matrices $T^{a}_{1/8}$
$\displaystyle{\rm\bf
T}^{1}_{1/8}=\left(\begin{array}[]{llllllll|l}0&0&-{1\over\sqrt{2}}&0&{1\over\sqrt{2}}&0&0&0&\Lambda_{1}\\\
\hline\cr\\\
p&n&\Sigma^{+}&\Sigma^{0}&\Sigma^{-}&\Xi^{0}&\Xi^{-}&\Lambda_{8}\end{array}\right)$
(464)
$\displaystyle{\rm\bf
T}^{2}_{1/8}=\left(\begin{array}[]{llllllll}0&0&-{i\over\sqrt{2}}&0&-{i\over\sqrt{2}}&0&0&0\end{array}\right)$
(466)
$\displaystyle{\rm\bf
T}^{3}_{1/8}=\left(\begin{array}[]{llllllll}0&0&0&1&0&0&0&0\end{array}\right)$
(468)
$\displaystyle{\rm\bf
T}^{4}_{1/8}=\left(\begin{array}[]{llllllll}{1\over\sqrt{2}}&0&0&0&0&0&-{1\over\sqrt{2}}&0\end{array}\right)$
(470)
$\displaystyle{\rm\bf
T}^{5}_{1/8}=\left(\begin{array}[]{llllllll}{i\over\sqrt{2}}&0&0&0&0&0&{i\over\sqrt{2}}&0\end{array}\right)$
(472)
$\displaystyle{\rm\bf
T}^{6}_{1/8}=\left(\begin{array}[]{llllllll}0&{1\over\sqrt{2}}&0&0&0&{1\over\sqrt{2}}&0&0\end{array}\right)$
(474)
$\displaystyle{\rm\bf
T}^{7}_{1/8}=\left(\begin{array}[]{llllllll}0&{i\over\sqrt{2}}&0&0&0&-{i\over\sqrt{2}}&0&0\end{array}\right)$
(476)
$\displaystyle{\rm\bf
T}^{8}_{1/8}=\left(\begin{array}[]{llllllll}0&0&0&0&0&0&0&1\end{array}\right)$
(478)
## References
* (1) L. B. Okun, “Leptons And Quarks”, Amsterdam, Netherlands: North-Holland (1982) 361p.
* (2) T. Yamanishi, Phys. Rev. D 76, 014006 (2007) [arXiv:0705.4340 [hep-ph]].
* (3) C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008).
* (4) W. A. Bardeen and B. W. Lee, Phys. Rev. 177, 2389 (1969).
* (5) B. W. Lee, Phys. Rev. 170, 1359 (1968).
* (6) Y. Hara, Phys. Rev. 139, B134 (1965).
* (7) H. Harari, Phys. Rev. Lett. 16, 964 (1966).
* (8) H. Harari, Phys. Rev. Lett. 17, 56 (1966).
* (9) I. S. Gerstein and B. W. Lee, Phys. Rev. Lett. 16 (1966) 1060.
* (10) S. Weinberg, Phys. Rev. 177, 2604 (1969).
* (11) S. D. Bass, “The Spin structure of the proton,” World Scientific, 2007. (ISBN 978-981-270-946-2 and ISBN 978-981-270-947-9). 212 p.
* (12) H. X. Chen, V. Dmitrašinović, A. Hosaka, K. Nagata and S. L. Zhu, Phys. Rev. D 78, 054021 (2008).
* (13) S. Gasiorowicz, “Elementary Particle Physics”, J. Wiley, New York (1966).
* (14) V. Dmitrašinović, K. Nagata, and A. Hosaka, Mod. Phys. Lett. A 23, 2381 (2008) [arXiv:0705.1896 [hep-ph]].
* (15) K. Nagata, A. Hosaka and V. Dmitrašinović, Eur. Phys. J. C 57 (2008) 557.
* (16) V. Dmitrašinović, A. Hosaka and K. Nagata, Mod. Phys. Lett. A 25, no. 4, 233-242 (2010); arXiv:0912.2372 [hep-ph].
* (17) V. Dmitrašinović, A. Hosaka and K. Nagata, Int. J. Mod. Phys. E 19, 91 (2010); arXiv:0912.2396 [hep-ph].
* (18) E. S. Ageev et al. [Compass Collaboration], Phys. Lett. B 647, 330 (2007) [arXiv:hep-ex/0701014].
* (19) We thank D. Jido for pointing out the relation $g_{A}^{(0)}=3F-D$.
|
arxiv-papers
| 2009-12-22T08:33:22 |
2024-09-04T02:49:07.199872
|
{
"license": "Public Domain",
"authors": "Hua-Xing Chen, V. Dmitrasinovic, Atsushi Hosaka",
"submitter": "Hua-Xing Chen",
"url": "https://arxiv.org/abs/0912.4338"
}
|
0912.4465
|
# A quantum spin approach to histone dynamics
C. Gils Samuel Lunenfeld Research Institute, Mount Sinai Hospital, 600
University Ave, Toronto, ON M5G 1X5, Canada J. L. Wrana Samuel Lunenfeld
Research Institute, Mount Sinai Hospital, 600 University Ave, Toronto, ON M5G
1X5, Canada Department of Molecular Genetics, University of Toronto, 1 Kings
College Circle, Room 4396, Toronto, ON M5S 1A8, Canada W. K. Abou Salem
Department of Mathematics, University of British Columbia, 1984 Mathematics
Road, Vancouver, BC V6T 1Z2, Canada Department of Mathematics and Statistics,
University of Saskatchewan, Saskatoon, SK S7N 5E6, Canada
###### Abstract
Post-translational modifications of histone proteins are an important factor
in epigenetic control that serve to regulate transcription, depending on the
particular modification states of the histone proteins. We study the
stochastic dynamics of histone protein states, taking into account a feedback
mechanism where modified nucleosomes recruit enzymes that diffuse to adjacent
nucleosomes. We map the system onto a quantum spin system whose dynamics is
generated by a non-Hermitian Hamiltonian. Making an ansatz for the solution as
a tensor product state leads to nonlinear partial differential equations that
describe the dynamics of the system. Multiple stable histone states appear in
a parameter regime whose size increases with increasing number of modification
sites. We discuss the role of the spatial dependance, and we consider the
effects of spatially heterogeneous enzymatic activity. Finally, we consider
multistability in a model of several types of correlated post-translational
modifications.
## I Introduction
Nuclear chromosomes in eukaryotic organisms consist of the chromatin, a
complex wrap that is primarily composed of DNA and histone proteins. The
fundamental unit of the chromatin is the nucleosome, each of which contains
two copies of the core histones H2A, H2B, H3 and H4, and approximately $150$
base pairs of DNA. Each of the core histone proteins exhibits multiple amino
acid residues that are subject to post-translational modifications (PTM) by
chemical groups such as phospho-, acetyl-, methyl- or ubiquitin-groups that
can be added and removed in a reversible manner. For example, H4 has a
phosphorylation site, four acetylation sites and six methylation sites.
Depending on the particular modification state of histones, certain regions of
DNA in the chromatin are in an active or repressed state. Regulation of the
PTMs of histones lies at the center of epigenetic control Allis:07 ;
Peterson:04 ; Rando:09 .
A commonly observed epigenetic phenomenon is the existence of alternative
regulatory states. For example, in the fission yeast Schizosaccharomyces pombe
the two mating type cassettes, mat2-P and mat3-M are usually in a silenced
state in which the mating type genes are not expressed. When removing a
portion of the silenced region and inserting a ura4+ reporter gene, the
expression of ura4+ and the mating-type genes becomes bistable, with a state
where ura4+ is repressed and a state where ura4+ is expressed Grewal:96 ;
Thon:96 ; Grewal:02 . The silenced state of ura4+ is associated with a high
concentration of methylation marks on lysine of histone H3 (H3K9), while the
active ura4+ state does not exhibit methylation of H3K9 Hall:02 . Each of the
two epigenetic states is preserved under cell divisions, with transitions
between them occuring only at a very low rate.
Post-translational modifications are regulated by various enzymes. In order to
explain the appearance of multiple stable histone states, a non-local positive
feedback mechanism has been put forward Turner:98 ; Grunstein:98 : A
nucleosome that exhibits a particular modification recruits the enzymes that
catalyze this modification. These enzymes then move to adjacent nucleosomes
and cause the modification to be added there, a mechanism that has indeed been
observed for some histone acetyltransferases, histone decacetylases and
histone methyltransferases Jacobsen:00 ; Owen:00 ; Rusche:01 ; Schotta:02 .
Long-range feedback has been implemented in a stochastic simulation of a
three-state model (unmodified state, acetylated state, methylated state) and
it was shown to lead to robust bistability Dodd:07 . Nearest-neighbour
feedback has been considered in deterministic descriptions of two- and three-
state models Sedighi:03 ; DavidRus:09 . The authors of Ref. Sedighi:03
consider a two-state mean-field Mean_field description that takes into
account cooperativity in binding of enzymes, and they discuss the bifurcation
diagram, including the effects of spatial dependence. In Ref. DavidRus:09 ,
the results of a stochastic simulation are compared to those of a mean-field
description that does not explicitly consider spatial dependence.
Perturbations due to cell divisions were considered, and instability of stable
steady states due to such perturbations were found in the stochastic
simulation, but not the mean-field approach. It is an open question how to
obtain mean-field equations in the continuum starting from a stochastic
description that predict the instabilities due to spatial dependance that are
observed in the microscopic simulations. Among other things, this is one of
the questions that we address in this work.
The considerable number of independently regulated modification sites in the
chromatin has been hypothesized to give rise to a “histone code” Jenuwein:01 :
There are $2^{T}$ possible combinations of modified/unmodified configurations
of $T$ independently regulated PTMs, each of which potentially corresponds to
a distinct “read-out” of information and ultimately a different epigenetic
outcome. Recent efforts in identifying abundances of these histone
modification states (also denoted as histone isoforms) have revealed that only
few of the large number of possible isoforms are actually observed
Phanstiel:08 ; Pesavento:08 . It is also well known that regulation of
different PTMs is correlated. For example, phosphorylation of H3 Ser10
stimulates acetylation of H3 Lys14 Lo:00 , and methylation of H3 Lys4 and
Lys79 requires the ubiquitiniation of H2B Lys123 Sun:02 ; Ng:02 . In this
work, we consider how such correlations in the regulation of PTMs reduce the
information capacity of histone states. In particular, we study a model that
is motivated by an interaction in the H3 N terminus where Ser10
phosphorylation inhibits Lys9 methylation Rea:00 .
We consider a master equation description of the stochastic dynamics of
histone states (section II). The system consists of a large number of
nucleosomes, where each nucleosome exhibits several PTMs that are regulated by
a particular class of enzymes. We take into account the reversible addition
and removal of PTMs due to enzymatic activity, as well as on-site (“local”)
and nearest-neighbour (“non-local”) feedback mechanisms where modified
nucleosomes recruit enzymes that either act locally or diffuse to adjacent
nucleosomes. We use a quantum many-body formulation of the master equation à
la DoiDoi:76 and a tensor product state ansatz to obtain a system of
nonlinear difference equations (section III). We believe that the continuum
limit of these equations is a suitable mean-field description that captures
the role of spatial dependance in the master equation. The reader who is not
interested in the derivation of the nonlinear difference equations/partial
differential equations can go directly to Eqs. (10), Eqs. (13) and Eqs. (20).
We numerically study the system of nonlinear partial differential equations
(section IV). When considering one type of post-translational modification,
and including at least two modification sites, bistable steady states are
obtained without the necessity of explicit cooperativity at the level of the
stochastic description (section IV.1). The two stable steady states correspond
to an unmodified state and a state with a high number of PTMs. We observe that
increasing the number of modification sites increases the size of the
parameter regime where bistable steady states exist. For a large number of
modification sites, bistability is possible even if the coupling strength of
the feedback mechanism is weak compared to the coupling strength of local
processes. We observe that the spatial dependance due to the non-local
feedback mechanism leads to instabilities of steady states under certain
spatial perturbations of the histone state (section IV.2). These instabilities
manifest themselves in traveling wave solutions of the system of nonlinear
partial differential equations. We also consider spatially dependent rate
parameters, which arise from adaptor proteins, such as DNA binding
transcription factors, that recruit histone modifying enzymes to specific
regions of chromatin (section IV.3). We discuss how such spatially dependent
enzyme activity gives rise to spatial heterogeneity in the epigenetic state.
Finally, we introduce a model of two types PTMs that are regulated by
different classes of enzymes and mutually inhibit each other (section IV.4).
Such mechanisms are present in the chromatin, for example, in the case of H3
Ser10 phosphorylation that inhibits H3 Lys9 methylation Rea:00 . We find that
inhibition in one direction is sufficient to reduce the full combinatorial set
of four stable steady states to a set of three stable steady states where the
presence of the two types of PTM is mutually exclusive. We conclude by
discussing open problems and future directions.
## II Stochastic dynamics of histone states
Figure 1: Schematic illustration of an array of $N$ nucleosomes, each of which
contains $S^{A}=5$ PTMs A (blue) and $S^{M}=4$ PTMs M (green) where PTMs of
types A are regulated by a certain set of enzymes and PTMs M are regulated by
another set of enzymes. Filled circles symbolize the presence of a PTM, empty
circles indicate the absence of a PTM. In this example, occupations are
$n_{1}^{A}=5$, $n_{1}^{M}=0$, $n_{2}^{A}=1$, $n_{2}^{M}=2$, etc.
We consider a one-dimensional array of $N$ nucleosomes. Each nucleosome
contains several modification sites of one or several independently regulated
classes of PTMs, as schematically illustrated in Fig 1. A system comprised of
$N$ nucleosomes with $S^{A}$ modification sites of type A (e.g., acetylation)
on each nucleosome is described by a state
$|n_{1}^{A},n_{2}^{A},...,n_{N}^{A}\rangle$ where the number of modified
(e.g., acetylated) sites on nucleosome $i$ is given by
$n_{i}^{A}\in\\{0,1,...,S^{A}\\}$. We denote by
$P(n^{A}_{1},n^{A}_{2},...,n^{A}_{N};t)$ the probability of finding the system
in state $|n_{1}^{A},n_{2}^{A},...,n_{N}^{A}\rangle$ at time $t$. In this and
the following sections, we shall restrict ourselves to a single class of PTMs
(i.e., regulated by a particular set of enzymes); however, in section IV.4 we
shall discuss the case of two types of PTM.
In the description of the stochastic dynamics of the histone state, we
consider on-site (“local”) and nearest-neighbour (“non-local” ) processes:
1. 1.
The addition of a PTM $A$ at nucleosome $i$ with a rate $\lambda^{A}$,
$n_{i}^{A}\stackrel{{\scriptstyle\lambda^{A}}}{{\longrightarrow}}n_{i}^{A}+1,$
caused by enzymatic activity.
2. 2.
The removal of a PTM $A$ at nucleosome $i$ with a rate $\mu^{A}n_{i}^{A}$,
$n_{i}^{A}\xrightarrow{\mu^{A}n_{i}^{A}}n_{i}^{A}-1,$
as a result of enzymatic activity.
3. 3.
The addition of a PTM $A$ at nucleosome $i$ with a rate
$f(n^{A}_{i-1},n_{i}^{A},n^{A}_{i+1})$,
$n^{A}_{i}\xrightarrow{f(n_{i-1}^{A},n_{i}^{A},n_{i+1}^{A})}n^{A}_{i}+1.$
The choice
$f(n_{i-1}^{A},n_{i}^{A},n_{i+1}^{A})=\tilde{\alpha}^{A}n_{i}^{A}+\alpha^{A}(n_{i-1}^{A}+n_{i+1}^{A}-2n_{i}^{A}),$
(1)
corresponds to a feedback mechanism that is both local and non-local. The
first term (coupling parameter $\tilde{\alpha}^{A}$) accounts for local
feedback: the more PTMs are present at nucleosome $i$, the more enzymes that
add PTMs of type A (e.g., acetylases) are present at $i$, and the more likely
is the addition of further PTMs of type A. The second term (coupling parameter
$\alpha^{A}$) corresponds to non-local feedback: the enzymes at nearest-
neighbouring nucleosomes $i-1$ and $i+1$ diffuse to nucleosome $i$ and vice
versa and, as in the case of local feedback, make the addition of additional
PTMs more likely.
4. 4.
The removal of a PTM $A$ at nucleosome $i$ with a rate
$n_{i}^{A}g(n^{A}_{i-1},n_{i}^{A},n^{A}_{i+1})$, i.e.,
$n^{A}_{i}\xrightarrow{n_{i}^{A}g(n_{i-1}^{A},n_{i}^{A},n_{i+1}^{A})}n^{A}_{i}-1.$
The choice
$g(n_{i-1}^{A},n_{i}^{A},n_{i+1}^{A})=\tilde{\beta}^{A}(S^{A}-n_{i}^{A})+\beta^{A}(2n_{i}^{A}-n_{i-1}^{A}-n_{i+1}^{A}),$
(2)
corresponds to a feedback mechanism that is both local and non-local. The
first term (coupling parameter $\tilde{\beta}^{A}$) accounts for local
feedback: The fewer PTMs are present at nucleosome $i$ (i.e., the larger
$S-n_{i}^{A}$), the more enzymes that cause the removal of PTM $A$ (e.g.,
deacetylases) are present at $i$, making the removal of further PTMs more
likely. The second term (coupling parameter $\beta^{A}$) corresponds to non-
local feedback: the enzymes that cause the removal of PTMs A at nearest-
neighbouring nucleosomes $i-1$ and $i+1$ diffuse to nucleosome $i$ and vice
versa and, as in the case of local feedback, make the removal of PTMs at site
$i$ more likely.
The master equation for the above processes is given by
$\displaystyle\frac{dP(n_{1}^{A},...,n_{N}^{A};t)}{dt}=\sum_{i=1}^{N}[\lambda^{A}+f(n^{A}_{i-1},n_{i}^{A},n^{A}_{i+1})][P(n^{A}_{1},...,n^{A}_{i-1},n_{i}^{A}-1,n^{A}_{i+1},...,n^{A}_{N};t)-P(n^{A}_{1},...,n^{A}_{i-1},n^{A}_{i},n^{A}_{i+1},...,n^{A}_{N};t)]$
$\displaystyle+\sum_{i=1}^{N}[\mu^{A}+g(n^{A}_{i-1},n_{i}^{A},n^{A}_{i+1})][(n^{A}_{i}+1)P(n^{A}_{1},...,n^{A}_{i-1},n^{A}_{i}+1,n^{A}_{i+1},...,n^{A}_{N};t)-n^{A}_{i}P(n^{A}_{1},...,n^{A}_{i-1},n^{A}_{i},n^{A}_{i+1},...,n^{A}_{N};t)].$
(3)
## III Derivation of nonlinear difference equations
We shall now introduce a notation of the master equation (3) that is motivated
by quantum physics Doi:76 ; Peliti:85 . Standard quantum physics notation is
used, i.e.,
$|n_{1}^{A},...,n^{A}_{N}\rangle=|n_{1}^{A}\rangle\otimes|n_{2}^{A}\rangle\otimes...\otimes|n^{A}_{N}\rangle$.
We define
$|\Psi(t)\rangle=\sum_{\\{n\\}}P(n^{A}_{1},n^{A}_{2},...,n^{A}_{N};t)|n^{A}_{1},n^{A}_{2},...,n^{A}_{N}\rangle,$
where the sum runs over all possible states. We introduce local raising and
lowering operators creation_annihilation $\mathcal{R}_{i}$ and
$\mathcal{L}_{i}$ that are defined by
$\mathcal{L}^{A}_{i}|n^{A}_{i}\rangle=n^{A}_{i}|n^{A}_{i}-1\rangle,\hskip
28.45274pt\mathcal{R}^{A}_{i}|n^{A}_{i}\rangle=|n^{A}_{i}+1\rangle,\hskip
28.45274pt\mathcal{R}^{A}_{i}|S^{A}_{i}\rangle=0,\hskip
28.45274pt\mathcal{L}^{A}_{i}|0^{A}_{i}\rangle=0,$
Indices $A$ and $i$ of operators signify that the operators are applied to
state $|n_{i}^{A}\rangle$. When representing states $|0^{A}\rangle$,
$|1^{A}\rangle$,…, $|S^{A}\rangle$ by the $S^{A}+1$ unit vectors in $S^{A}+1$
dimensions, the lowering and raising operators can be represented by
$(S^{A}+1)\times(S^{A}+1)$ dimensional matrices,
$\mathcal{L}^{A}_{i}=\left(\begin{array}[]{cccccc}0&1&0&...&...&0\\\
0&0&2&0&...&0\\\ 0&0&0&3&...&0\\\ ...&...&...&...&...&...\\\
0&0&...&...&...&S^{A}\\\ 0&0&...&...&0&0\end{array}\right),\hskip
56.9055pt\mathcal{R}^{A}_{i}=\left(\begin{array}[]{cccccc}0&0&...&...&0&0\\\
1&0&...&...&...&0\\\ 0&1&0&...&...&0\\\ ...&...&...&...&...&...\\\
0&0&...&1&...&...\\\ 0&0&...&...&1&0\end{array}\right)\ .$
The number operator is defined by
$\mathcal{N}^{A}_{i}=\mathcal{R}^{A}_{i}\mathcal{L}^{A}_{i}={\rm
Diag}(0,1,2,...,S^{A})$. In this notation, the master equation becomes
$\frac{\partial|\Psi(t)\rangle}{\partial t}=\mathcal{H}|\Psi(t)\rangle,$ (4)
where
$\mathcal{H}=\mathcal{H}_{1}^{A}\otimes\mathcal{E}_{2}^{A}\otimes...\otimes\mathcal{E}_{N}^{A}+\mathcal{E}_{1}^{A}\otimes\mathcal{H}_{2}^{A}\otimes...\otimes\mathcal{E}_{N}^{A}+...+\mathcal{E}_{1}^{A}\otimes...\otimes\mathcal{H}_{N}^{A}$
(in simplified notation: $\mathcal{H}=\sum_{i=1}^{N}\mathcal{H}^{A}_{i}$),
where $\mathcal{E}_{i}^{A}$ denotes the $S^{A}$-dimensional identity operator,
and
$\displaystyle\mathcal{H}^{A}_{i}$ $\displaystyle=$
$\displaystyle\lambda^{A}(\mathcal{R}^{A}_{i}-\mathcal{I}^{A}_{i})+\mu^{A}(\mathcal{L}^{A}_{i}-\mathcal{N}^{A}_{i})+(\mathcal{R}^{A}_{i}-\mathcal{I}^{A}_{i})[\alpha^{A}(\mathcal{N}^{A}_{i-1}+\mathcal{N}^{A}_{i+1}-2\mathcal{N}^{A}_{i})+\tilde{\alpha}^{A}\mathcal{N}_{i}^{A}]$
(5)
$\displaystyle+(\mathcal{L}^{A}_{i}-\mathcal{N}^{A}_{i})[\beta^{A}(\mathcal{M}^{A}_{i-1}+\mathcal{M}^{A}_{i+1}-2\mathcal{M}^{A}_{i})+\tilde{\beta}^{A}\mathcal{M}_{i}^{A}],$
where $\mathcal{I}^{A}={\rm Diag}(1,1,...,1,0)$, and $\mathcal{M}^{A}={\rm
Diag}(S^{A},S^{A}-1,...,1,0)$. In (5), we substituted the functions (1) and
(2). We note that (4) is an imaginary-time Schrödinger equation. The system
corresponds to a quantum spin chain, though with a non-hermitian Hamitonian.
The master equation (4) is equivalent to a functional variation Eyink:96 ,
$\frac{\delta\Gamma}{\delta\Phi}=0,$ (6)
where
$\Gamma=\int dt\langle\Phi|(\partial_{t}-\mathcal{H})|\Psi\rangle.$
Since the system can be viewed as a quantum spin chain, albeit with a non-
hermitian Hamiltonian $\mathcal{H}$, we make an ansatz for the wave-function
in the Schrödinger picture as a tensor product state,
$|\Psi(t)\rangle=\prod_{i=1}^{N}|\Psi_{i}(t)\rangle,\hskip
56.9055pt\langle\Phi|=\prod_{i=1}^{N}\langle\Phi_{i}|\ .$ (7)
and we write $|\Psi_{i}(t)\rangle$ as a superposition of all possible states
(we shall drop indices $A$ from this point on),
$|\Psi_{i}(t)\rangle=\sum_{n=0}^{S}C_{i,n}(t)|n\rangle=\left(\begin{array}[]{c}C_{i,0}(t)\\\
C_{i,1}(t)\\\ .\\\ .\\\ .\\\ C_{i,S}(t)\end{array}\right),\hskip
28.45274pt\langle\Phi_{i}|=\sum_{n=0}^{S}\langle
n|e^{\phi_{i,n}}=(e^{\phi_{i,0}}\;\;e^{\phi_{i,1}}\;\;...\;\;e^{\phi_{i,S}}),$
(8)
where $\sum_{n=0}^{S}C_{i,n}=1$, and $C_{i,n}$ denotes the probability that
nucleosome $i$ has $n$ modified sites. Since $\sum_{n=0}^{S}C_{i,n}=1$, this
ansatz obeys the probabilistic constraint
$\langle\Phi|\Psi\rangle|_{\phi_{i,n}=0}=1$ (i.e., expectation values
$\langle\Phi|O|\Psi\rangle$ of an observable $O$ are properly normalized).
Using this ansatz, the master equation in the formulation of (6) becomes
$\left(\left\langle\frac{\partial\Phi}{\partial\phi_{i,k}}\right|\left.\frac{\partial\Psi}{\partial
C_{i,n}}\right\rangle\frac{dC_{i,n}}{dt}-\left\langle\frac{\partial\Phi}{\partial\phi_{i,k}}\left|\mathcal{H}\right|\Psi\right\rangle\right)_{\phi_{i,k=0}}=0.$
(9)
Evaluating (9) yields a system of nonlinear difference equations for the
probabilities $C_{i,n}$ that the nucleosome $i$ has $n$ modifications,
$\displaystyle\frac{dC_{i,0}}{dt}$ $\displaystyle=$ $\displaystyle-\lambda
C_{i,0}+\mu C_{i,1}-C_{i,0}(\alpha F^{\nabla}_{i}+\tilde{\alpha}\langle
n_{i}\rangle)+C_{i,1}(\beta G^{\nabla}_{i}+\tilde{\beta}\langle
m_{i}\rangle),$ $\displaystyle\frac{dC_{i,n}}{dt}$
$\displaystyle\stackrel{{\scriptstyle 1\leq n<S}}{{=}}$
$\displaystyle-\lambda(C_{i,n}-C_{i,n-1})-\mu(nC_{i,n}-(n+1)C_{i,n+1})\
-(C_{i,n}-C_{i,n-1})(\alpha F^{\nabla}_{i}+\tilde{\alpha}\langle
n_{i}\rangle)$ $\displaystyle-(nC_{i,n}-(n+1)C_{i,n+1})(\beta
G^{\nabla}_{i}+\tilde{\beta}\langle m_{i}\rangle),$
$\displaystyle\frac{dC_{i,S}}{dt}$ $\displaystyle=$ $\displaystyle\lambda
C_{i,S-1}-S\mu C_{i,S}+C_{i,S-1}(\alpha F^{\nabla}_{i}+\tilde{\alpha}\langle
n_{i}\rangle)-SC_{i,S}(\beta G^{\nabla}_{i}+\tilde{\beta}\langle
m_{i}\rangle),$ (10)
where
$\begin{array}[]{llllll}F^{\nabla}_{i}=\langle n_{i-1}\rangle-2\langle
n_{i}\rangle+\langle n_{i+1}\rangle&{\rm
if}\;\;\;1<i<N,&F^{\nabla}_{1}=-2\langle n_{1}\rangle+\langle
n_{2}\rangle&F^{\nabla}_{N}=\langle n_{N-1}\rangle-2\langle
n_{N}\rangle,\\\\[5.69054pt] G^{\nabla}_{i}=\langle m_{i-1}\rangle-2\langle
m_{i}\rangle+\langle m_{i+1}\rangle&{\rm
if}\;\;\;1<i<N,&G^{\nabla}_{1}=-2\langle m_{1}\rangle+\langle
m_{2}\rangle&G^{\nabla}_{N}=\langle m_{N-1}\rangle-2\langle
m_{N}\rangle,&\end{array}$
and
$\displaystyle\langle n_{i}\rangle$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{S}nC_{i,n}$ (11) $\displaystyle\langle m_{i}\rangle$
$\displaystyle=$ $\displaystyle\sum_{n=0}^{S}(S-n)C_{i,n}=S-\langle
n_{i}\rangle,$ (12)
(open boundary conditions).
Equations (10) are a discretization of a system of nonlinear reaction-
diffusion equations. Let $\ell_{0}$ be the lattice spacing (distance between
nucleosomes). In the mean-field/continuum limit
$\alpha\to\alpha/\ell_{0}^{2}$, $\beta\to\beta/\ell_{0}^{2}$, and $\ell_{0}\to
0$, we obtain the system of nonlinear partial differential equations for
variables $C_{n}(x,t)$, $n=0,1,...,S$,
$\displaystyle\frac{\partial C_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle-\lambda C_{0}+\mu
C_{1}-C_{0}\left(\alpha\sum_{s=0}^{S}\left(s\frac{\partial^{2}C_{s}}{\partial
x^{2}}\right)+\tilde{\alpha}\sum_{s=0}^{S}(sC_{s})\right)+C_{1}\left(\beta\sum_{s=0}^{S}\left((S-s)\frac{\partial^{2}C_{s}}{\partial
x^{2}}\right)+\tilde{\beta}\sum_{s=0}^{S}(S-sC_{s})\right),$
$\displaystyle\frac{\partial C_{n}}{\partial t}$
$\displaystyle\stackrel{{\scriptstyle 1\leq n<S}}{{=}}$
$\displaystyle-\lambda(C_{n}-C_{n-1})-\mu(nC_{n}-(n+1)C_{n+1})-(C_{n}-C_{n-1})\left(\alpha\sum_{s=0}^{S}\left(s\frac{\partial^{2}C_{s}}{\partial
x^{2}}\right)+\tilde{\alpha}\sum_{s=0}^{S}(sC_{s})\right)$ (13)
$\displaystyle-(nC_{n}-(n+1)C_{n+1})\left(\beta\sum_{s=0}^{S}\left((S-s)\frac{\partial^{2}C_{s}}{\partial
x^{2}}\right)+\tilde{\beta}\sum_{s=0}^{S}(S-sC_{s})\right)$
$\displaystyle\frac{\partial C_{S}}{\partial t}$ $\displaystyle=$
$\displaystyle\lambda C_{S-1}-S\mu
C_{S}+C_{S-1}\left(\alpha\sum_{s=0}^{S}\left(s\frac{\partial^{2}C_{s}}{\partial
x^{2}}\right)+\tilde{\alpha}\sum_{s=0}^{S}(sC_{s})\right)-SC_{S}\left(\beta\sum_{s=0}^{S}\left((S-s)\frac{\partial^{2}C_{s}}{\partial
x^{2}}\right)+\tilde{\beta}\sum_{s=0}^{S}(S-sC_{s})\right).$
The diffusion terms are multiplied with the probabilities themselves. We note
that the coefficient in front of the diffusion term is degenerate, and it is
of interest to rigorously show the existence and stability of traveling wave
solutions in reaction-diffusion equations of this type.
## IV Results
In what follows, our analysis is based on numerical analysis of the system
(10) over a finite parameter range. In the following, we set parameters
$\tilde{\alpha}=4\alpha$ and $\tilde{\beta}=4\beta$, and we emphasize that
varying the relative strength of local and non-local feedback does not
qualitatively affect the results of our study. We note that as long as one is
interested in the asymptotics (asymptotically long time) behavior of solutions
of difference equations, what matters as input in the equations is the ratio
(relative strength) of various coupling parameters (e.g., $\beta/\alpha$,
$\lambda/\alpha$, etc.). One can always divide by a non-zero coupling
parameters and rescale time to absorb this parameter in the left-hand-side of
the difference equations.
In section IV A, we will first discuss bistability in the model while
neglecting spatial dependence. We will then incorporate spatial effects in
part B, which we note fundamentally alters the picture. In section IV.3, we
discuss the effects of spatial heterogeneity and in section IV.4, we discuss
multiple correlated PTMs.
### IV.1 Multiple stable steady states in the $S$-state model and the role of
$S$
In this section, we discuss the results of the nonlinear difference equations
(10) when neglecting the spatial dependance, i.e.,
$C_{1,n}=C_{2,n}=...=C_{N,n}=C_{n}$. In this case, a system of coupled
nonlinear ordinary differential equations (ODE) is obtained,
$\displaystyle\frac{dC_{0}}{dt}$ $\displaystyle=$ $\displaystyle-\lambda
C_{0}+\mu C_{1}-4\alpha C_{0}^{2}+4\beta C_{0}C_{1},$
$\displaystyle\frac{dC_{n}}{dt}$ $\displaystyle\stackrel{{\scriptstyle 1\leq
n<S}}{{=}}$
$\displaystyle-\lambda(C_{n}-C_{n-1})-\mu(nC_{n}-(n+1)C_{n+1})-4\alpha
C_{n}(C_{n}-C_{n-1})-4\beta C_{n}(nC_{n}-(n+1)C_{n+1}),$
$\displaystyle\frac{dC_{S}}{dt}$ $\displaystyle=$ $\displaystyle\lambda
C_{S-1}-S\mu C_{S}+4\alpha C_{S-1}C_{S}-4S\beta C_{S}^{2}.$ (14)
Using this simplified ODE description, we evaluate steady states by setting
$dC_{n}/dt=0$, and study their stability by analyzing the Jacobian matrix.
Expressions for the steady state probabilities $C_{n}$ as a function of
parameters $\lambda$, $\mu$, $\alpha$ and $\beta$ can be evaluated
analytically. However, the resulting expressions are cumbersome and
increasingly difficult to obtain for increasing $S$, and therefore
calculations have been done numerically over a finite parameter range.
Figure 2: Bifurcation diagram showing the steady state probabilities $C_{0}$
and $C_{S}$ for $S=3$ modification sites and parameters $\lambda=\mu=1$,
$\beta=3$ as a function of parameter $\alpha$. We denote the stable steady
states where $C_{0}\approx 1$ (few PTMs) and $C_{S}\approx 1$ (large number of
PTMs) by X and Z, respectively, and the unstable steady state by Y. For
$\alpha\in[4.4,7.5]$, steady states X, Y and Z appear, while for small
$\alpha$ only X persists and for large $\alpha$ only Z persists.
Figure 3: Bifurcation diagram for $S=50$ modification sites showing the steady
state probabilities $C_{\rm low}=\sum_{n=0}^{4}C_{n}$ (i.e., low number of
PTMs) and $C_{\rm high}=\sum_{n=46}^{50}C_{n}$ (i.e., high number of PTMs) of
the stable steady states X and Z as a function of $\alpha$. The remaining
parameters are $\lambda=5$, $\mu=1$, $\beta=0.01$. For $\alpha\in[0.36,0.53]$,
bistability persists. Note that $\alpha\ll\lambda$ and $\beta\ll\mu$. Inset:
Width of the bistable regime in units of $\alpha$ as a function of the number
of modification sites $S$ ($\lambda=\mu=1$, $\beta=3$). It can be seen to
increase linearly.
For more than one modification site, i.e., $S\geq 2$, and appropriately chosen
parameters (see below) we find that a parameter regime exists where three
steady states coexist. The multistability is a consequence of the
nonlinearities in Eqs.(14) that are introduced by the feedback terms. Two of
the steady states are stable attractors and one steady state is an unstable
saddle point. We note that no explicit cooperativity is required in order to
obtain bistability if $S$ is chosen larger or equal than two.
The bistability is illustrated in the bifurcation diagram of Fig. 3 where the
steady state probabilities $C_{0}$ and $C_{3}$ are shown as a function of
parameter $\alpha$ (the parameters used are $S=3$, $\mu=\lambda=1$,
$\beta=3$). If the feedback term for enzymes that catalyse the addition of
PTMs is weak compared to the feedback term of enzymes that catalyse the
removal of PTMs, only one steady state appears, as can be seen in Fig. 3 for
$\alpha<4.4$. This steady state, which we denote by X, is characterized by
$C_{0}\approx 1$, i.e., it corresponds to a state where very few PTMs are
present. If the effects of the two terms that add PTMs approximately are
roughly equal to the effects of the two terms that remove PTMs, three steady
states exist ($\alpha\in[4.4,7.5]$ in Fig. 3). In addition to steady state X,
a steady state with $C_{S}\approx 1$ appears. This steady state corresponds to
a state with a high number of PTMs, and we shall denote it by Z. A third
steady state (denoted by Y in Fig. 3) is unstable. Finally, for large enough
$\alpha$, only steady state Z persists, as illustrated in Fig. 3 for
$\alpha>7.5$.
We note that in the previous paragraph we referred to the “strengths” of the
four terms (1.-4. in section II) as they can be read from the expectation
values, e.g., $\langle\Phi|\sum_{i}f(n_{i-1},n_{i},n_{i+1})|\Psi\rangle$. In
contrast, in the following paragraph, we shall refer to the magnitudes of the
coupling parameters (i.e., $\alpha$, $\beta$, $\mu$, $\lambda$) themselves.
The values of the coupling parameters are controlled externally (e.g., the
concentration, catalytic rate and diffusion rate of enzymes), while the
expectation values also depend on system-dependent parameters (i.e., the
number of modification sites $S$).
Bistability is obtained only if both feedback terms are present, i.e., if both
$\alpha$ and $\beta$ are non-zero. If the number of modification sites, $S$,
is small, bistable steady states appear only if the coupling parameters of the
feedback terms are large compared to those of the local terms, i.e., only if
the ratios $\lambda/\alpha$ and $\mu/\beta$ are small enough. However, with
increasing number of modification sites $S$, the size of the parameter regime
where multiple steady states appear increases, as shown in the inset of Fig.
3, and for large enough $S$, bistability can be established even if
$\alpha\ll\lambda$ and $\beta\ll\mu$, as shown in Fig. 3. The existence of a
large number of modification sites $S$ that are regulated by a particular set
of enzymes thus allows for a larger parameter regime of bistability.
### IV.2 Spatial dependance
Figure 4: Time evolution of probabilities $C_{i,3}(t)$. The system ($S=3$
modification sites) is initially (time $t=0$, red curve) in steady state Z
(where $C_{3}\approx 0.9$), except for few nucleosomes in the center that are
strongly perturbed and whose probabilities are in the domain of steady state
X. Parameters are $\lambda=\mu=1$, $\beta=3$, $\alpha=5.6$. Two traveling wave
fronts move towards the boundaries and drive the system into steady state X.
The velocity of the waves is constant.
Figure 5: Time evolution of probabilities $C_{i,3}(t)$. Parameters are as in
Fig. 5, except that $\lambda=1$ in part of the system, and $\lambda=2$ in the
remainder, as indicated. The system is initially in steady state Z except for
few nucleosomes in both regions whose states lie in the domain of X (red
circles). In the left region ($\lambda=1$), two wave fronts move towards the
boundaries, however, once the right front hits the $\lambda=2$ region, it is
stopped. In the $\lambda=2$ region, the perturbation does not cause the system
to approach X. The reason is that for $\lambda=1$, steady state X is the
“stronger attractor”, while for $\lambda=2$, Z is the “stronger attractor”
(terminology see text).
In this section we will explicitly take into account spatial dependence, which
is incorporated in the solutions to equations (10). We numerically integrate
(10) and find that the stable steady states that were discussed in the
previous section may become unstable for certain initial conditions. We
illustrate this in Fig. 5: We set the initial probabilities $C_{i,n}$ of the
nucleosomes to those of steady state $Z$ (the steady state where $C_{i,S}$ is
large), except for very few nucleosomes where we set the initial probabilities
to values close to those corresponding to the second steady state X
Initial_state . It can be seen that the system approaches steady state $X$,
i.e., the spatially restricted perturbation of the histone state causes
instability. This instability manifests itself by traveling wave solutions of
the system of equations (10). It can be seen in Fig. 5 that for a perturbation
away from the boundaries, two traveling wave fronts develop which travel at a
constant velocity towards the boundaries of the system. If the perturbation is
located at one of the boundaries of the system, only one wave front develops.
There exists a set of parameters $S$, $\lambda$, $\mu$, $\alpha$ and $\beta$
where the velocity of the traveling wave(s) is zero. At that point, both
steady states, X and Z, are stable with respect to spatial perturbations. For
the parameters set of Fig. 5, this transition occurs at $\alpha^{*}\approx
5.7$ (bistability occurs for $\alpha\in[4.4,7.5]$). For $\alpha<\alpha^{*}$
and within range of bistability, the steady state X is the “stronger
attractor”: If the initial state is Z and at least one nucleosome is perturbed
such that its state is in the domain of fixed point X, the system approaches
X, as is illustrated in Fig. 5. If the initial state is X, and at least one
nucleosome is perturbed such that its state in the domain of steady state Z,
the system bounces back into steady state X. In contrast, for
$\alpha>\alpha^{*}$, steady state Z is the “stronger attractor”: If the
initial state is X and at least one nucleosome is perturbed such that its
state is in the domain of Z, the system approaches Z. If the initial state is
Z, and at least one nucleosome is perturbed such that its state is in the
domain of attraction of X, the system bounces back into steady state Z.
In conclusion, for parameters $\alpha<\alpha^{*}$, steady state X exhibits a
very high degree of stability as any initial state of the system that gives
rise to traveling wave solutions yields traveling waves that drive the system
into state X. In contrast, for parameters $\alpha>\alpha^{*}$, any traveling
wave solution will drive the system into steady state Z. We note that when the
asymptotic behaviour of equations (10) are considered, the number of
nucleosomes in the system is not relevant. However, a larger number of
nucleosomes does result in a longer duration for the traveling wave to spread
over the entire system, which may be relevant if intermediate time scales are
considered.
Instabilitities due to traveling wave solutions could have significant impact
on the stability and inheritance of chromatin steady states in daughter cells
upon division. During cell division, it is thought that the parental
nucleosomes are randomly distributed among the two daughter cells, with the
second half being newly synthesized Annunziato:05 . The modification state of
these new nucleosomes is crucial to the stability of the epigenetic state in
the presence of non-local feedback terms. This can be seen as follows. The
cell division can be modeled by replacing the states of half of the
nucleosomes (randomly selected) at periodic intervals. Assume that the system
is initially in steady state Z and parameters are set to the values of Fig. 5
where X is the “stronger attractor”. If the states of the newly synthesized
nucleosomes are random (i.e., any state is possible), some of these
nucleosomes might be in states that are in the domain of steady state X right
after cell division. In this case, a traveling wave can form, and drive the
system into steady state X (after one, several or many divisions, depending on
the time-scales involved). We have verified this numerically. However, if the
states of the newly synthesized nucleosomes are correlated with the state of
the nucleosomes in the mother cell such that the states of the new nucleosomes
are in the domain of attraction of the original state, such instabilities
cannot arise. In the presence of non-local effects, a sufficient correlation
between mother and daughter nucleosome states is hence necessary to preserve
the chromatin state. This would relate to the notion of epigenetic memory and
in fact there is a relation between daugher cell state and mother state [an
example was discussed in the second paragraph of the introduction]. However,
how this is conveyed at the molecular level remains a challenging open
question.
We conclude this section with a short discussion of the effects of considering
explicit cooperative behaviour in the feedback terms. Explicit cooperative
action of enzymes on-site, as well as of enzymes on nearest-neighbouring
nucleosomes can be implemented using ansatz $f^{\rm
coop}(n_{i-1},n_{i},n_{i+1})=f(n_{i-1},n_{i},n_{i+1})+\delta
n_{i-1}n_{i}n_{i+1}$ and $g^{\rm
coop}(n_{i-1},n_{i},n_{i+1})=g(n_{i-1},n_{i},n_{i+1})+\gamma(S-n_{i-1})(S-n_{i})(S-n_{i+1})$
Using the approach of sections II, III and IV.1, bistable steady states are
observed, as was the case for the model without explicit cooperative action.
However, bistability is possible even for the case $S=1$. This in agreement
with prior studies of two-state models with explicit cooperativity Sedighi:03
; DavidRus:09 . The difference equations that are obtained using this ansatz,
or their continuum version, admit traveling wave solutions, as in the case of
our model without explicit cooperative behaviour (10) where $S\geq 2$.
### IV.3 Spatially heterogeneous enzymatic activity
In biological systems, nucleosome modifying enzymes are typically recruited to
specific regions of the chromatin by adaptor proteins, such as DNA-binding
transcription factors. As a result, the activity of these enzymes depends on
the region of the chromatin. The increased or decreased activity of enzymes at
certain nucleosomes can be taken into account by including a spatial
dependance in parameters $\lambda$ and $\mu$, i.e., $\lambda_{i}$ and
$\mu_{i}$, where $i$ is the nucleosome number. At each space point, the steady
states are determined by the respective $\lambda_{i}$ and $\mu_{i}$, i.e., the
steady states locally correspond to the steady states with homogenous
activity. Hence the parameter regimes where multiple stable steady states
appear vary in size and position, and steady state probabilities $C_{i,n}$
also depend on the nucleosome number $i$. For example, when choosing $S=3$,
$\mu=1$, $\beta=3$, and $\lambda=1$, bistability exists for
$\alpha\in[4.4,7.5]$ and $\alpha^{*}\approx 5.7$, while for parameters $S=3$,
$\mu=1$, $\beta=3$ and $\lambda=2$, bistability persists for
$\alpha\in[4.3,6.9]$, where $\alpha^{*}\approx 5.5$. As a consequence, for
parameter $\alpha=5.6$, steady state X is the stronger attractor (in the sense
explained in section IV.2) for the former choice of parameters, while steady
state Z is the stronger attractor for the latter choice of parameters. When
perturbing a system that is initially in steady state Z in both
$\lambda$-regions, traveling wave solutions drive the system into steady state
X at the nucleosomes where $\lambda=1$, but not in regions where $\lambda=2$,
and the traveling waves in the region where $\lambda=1$ are stopped once they
hit regions where $\lambda=2$, as shown in Fig. 5. Spatial dependence on the
activity of histone modifying enzymes that is conferred by recruitment to
regulatory regions of chromatin by transcription factors may thus stabilize
the histone state from local and non-local perturbation.
Figure 6: Bifurcation diagram showing steady state probabilities $C_{p,m}$ of
stable steady states for model (20) with two types of modifications, labeled
by P and M, as a function of parameter $\beta^{P\to M}$. Note that only P
inhibits M, but not vice versa, i.e., $\beta^{M\to P}=0$. The remaining
parameters are given by $S^{P}=S^{M}=2$,
$\lambda^{P}=\mu^{P}=\lambda^{M}=\mu^{M}=1$, $\beta^{P}=\beta^{M}=3$,
$\alpha^{P}=\alpha^{M}=4.5$. For very small $\beta^{P\to M}$, four stable
steady states (st.st.) exist, labeled by 00 (low P and low M), P0 (high P, low
M), 0M (low P, high M), and PM (high P and high M). In an intermediate
parameter regime, $\beta^{P\to M}\in[0.4,3.3]$, stable steady states 00, P0
and 0M persist, while for $\beta^{P\to M}>3.3$, only steady states 00 and P0
appear. We note that inhibition in only one direction (as $\beta^{M\to P}=0$)
is sufficient to obtain a parameter regime where steady states have either a
high number of PTMs P or M, or neither, but not both.
### IV.4 Multistability in model of several types of correlated PTMs
Most proteins, such as histones, that are subject to PTM-dependent regulation
are regulated via multiple modifications. In this context, we discuss the
results of including several types of modifications where each type is
associated with different sets of enzymes, and thus different rate parameters
$\lambda$, $\mu$, $\alpha$, $\beta$, $\tilde{\alpha}$ and $\tilde{\beta}$. For
example, one might consider different classes of acetylation (or
phosphorylation, ubiquitination, etc.) sites, each of them associated with a
different enzyme. Alternatively, one might consider PTMs of type P (e.g.,
phosphorylation) and PTMs of type M (e.g., methylation), with different rate
parameters, $\lambda^{P}$ and $\lambda^{M}$, $\alpha^{P}$ and $\alpha^{M}$,
etc. We denote by $C_{i,p,m}$ the probability of finding nucleosome $i$ in the
state with $p$ PTMs of type P and $m$ PTMs of type M. In this model, the
number of stable steady states is four: the number of both M and P
modifications is high (labeled by PM in the following), the number of P
modifications is high and the number of M modifications is low (labeled by
P0), the number of M modifications is high and the number of P modifications
is low (labeled by 0M), and the number of both M and P modifications is low
(labeled by 00). More generally, for $T$ independent classes of modification
sites, where a particular class of sites is associated with a particular set
of coupling parameters, $2^{T}$ stable steady states are obtained. These
steady states correspond to all possible combinations of states of high and
low numbers of PTMs, i.e., all possible binary strings of length $T$.
In practice, however, different types and sites of PTMs are often not
independent from each other. There are examples where the presence of a
certain PTM inhibits the addition of another PTM. An example is the H3 N
teminus where Ser10 phosphorylation inhibits Lys9 methylation Rea:00 . In the
following, we derive difference equations using the formalism introduced in
sections II and III for a model of two types of PTMs, P and M, that mutually
inhibit each other. We consider the processes 1.-4. (section II) separately
for each of the two PTMs and add mutual inhibition (note that $m\equiv n^{M}$,
$p\equiv n^{P}$):
$\displaystyle n_{i}^{P}n_{i}^{M}\stackrel{{\scriptstyle\beta^{P\to
M}n_{i}^{P}n_{i}^{M}}}{{\longrightarrow}}n_{i}^{P}(n_{i}^{M}-1),$ (15)
$\displaystyle n_{i}^{P}n_{i}^{M}\stackrel{{\scriptstyle\beta^{M\to
P}n_{i}^{P}n_{i}^{M}}}{{\longrightarrow}}(n_{i}^{P}-1)n_{i}^{M}.$ (16)
In the case of (15), the presence of PTMs of type P leads to the removal of
PTMs of type M, and in the case of (16), the presence of PTMs of type M leads
to the removal of PTMs of type P.
The wave function is of form (7) with the local wave functions given by
$|\Psi_{i}(t)\rangle=\sum_{p=0}^{S^{P}}\sum_{m=0}^{S^{M}}C_{i,p,m}(t)|p\rangle|m\rangle\hskip
28.45274pt\langle\Phi_{i}|=\sum_{p=0}^{S^{P}}\sum_{m=0}^{S^{M}}\langle
p|\langle m|e^{\phi_{i,p,m}},$ (17)
where the normalization condition
$\sum_{p=0}^{S^{P}}\sum_{m=0}^{S^{M}}C_{i,p,m}=1$ applies. The local operators
$\mathcal{R}_{i}$, $\mathcal{L}_{i}$, $\mathcal{N}_{i}$, $\mathcal{M}_{i}$,
$\mathcal{I}_{i}$ are defined as in section III, and we denote the identity
operator by $\mathcal{E}_{i}^{X}$ (unity matrix of size $S^{X}$). Using this
notation, the “non-hermitian Hamiltonian” of the system is given by
$\mathcal{H}=\sum_{i=1}^{N}\mathcal{H}_{i}$, where
$\displaystyle\mathcal{H}_{i}$ $\displaystyle=$
$\displaystyle[\lambda^{P}+\alpha^{P}(\mathcal{N}^{P}_{i-1}+\mathcal{N}^{P}_{i+1}-2\mathcal{N}^{P}_{i})+\tilde{\alpha}^{P}\mathcal{N}_{i}^{P}](\mathcal{R}^{P}_{i}-\mathcal{I}^{P}_{i})\mathcal{E}_{i}^{M}$
(18)
$\displaystyle+\mathcal{E}_{i}^{P}[\lambda^{M}+\alpha^{M}(\mathcal{N}^{M}_{i-1}+\mathcal{N}^{M}_{i+1}-2\mathcal{N}^{M}_{i})+\tilde{\alpha}^{M}\mathcal{N}_{i}^{M}](\mathcal{R}^{M}_{i}-\mathcal{I}^{M}_{i})$
$\displaystyle+[\mu^{P}+\beta^{P}(\mathcal{M}^{P}_{i-1}+\mathcal{M}^{P}_{i+1}-2\mathcal{M}^{P}_{i})+\tilde{\beta}^{P}\mathcal{M}_{i}^{P}](\mathcal{L}^{P}_{i}-\mathcal{N}^{P}_{i})\mathcal{E}_{i}^{M}$
$\displaystyle+\mathcal{E}_{i}^{P}[\mu^{M}+\beta^{M}(\mathcal{M}^{M}_{i-1}+\mathcal{M}^{M}_{i+1}-2\mathcal{M}^{M}_{i})+\tilde{\beta}^{M}\mathcal{M}_{i}^{M}](\mathcal{L}^{M}_{i}-\mathcal{N}^{M}_{i})$
$\displaystyle+\beta^{P\to
M}(\mathcal{L}^{M}_{i}-\mathcal{N}^{M}_{i})\mathcal{N}_{i}^{P}+\beta^{M\to
P}(\mathcal{L}^{P}_{i}-\mathcal{N}^{P}_{i})\mathcal{N}_{i}^{M}.$
The master equation in quantum variational formulation becomes
$\left(\left\langle\frac{\partial\Phi}{\partial\phi_{i,p^{\prime},m^{\prime}}}\right|\left.\frac{\partial\Psi}{\partial
C_{i,p,m}}\right\rangle\frac{dC_{i,p,m}}{dt}-\left\langle\frac{\partial\Phi}{\partial\phi_{i,p^{\prime},m^{\prime}}}\left|\mathcal{H}\right|\Psi\right\rangle\right)_{\phi_{i,p^{\prime},m^{\prime}=0}}=0.$
(19)
Evaluating (19) yields a system of nonlinear difference equations for the
probabilities $C_{i,p,m}$ that the nucleosome at site $i$ has $p$
modifications of type P and $m$ modifications of type $M$,
$\displaystyle\frac{dC_{i,p,m}}{dt}$ $\displaystyle=$
$\displaystyle-(\lambda^{P}+\alpha^{P}F^{\nabla_{P}}_{i}+\tilde{\alpha}^{P}\langle
n_{i}^{P}\rangle)(C_{i,p,m}-C_{i,p-1,m})-(\lambda^{M}+\alpha^{M}F^{\nabla_{M}}_{i}+\tilde{\alpha}^{M}\langle
n_{i}^{M}\rangle)(C_{i,p,m}-C_{i,p,m-1})$ (20)
$\displaystyle-(\mu^{P}+\beta^{P}G^{\nabla_{P}}_{i}+\tilde{\beta}^{P}\langle
m_{i}^{P}\rangle)(pC_{i,p,m}-(p+1)C_{i,p+1,m})-(\mu^{M}+\beta^{M}G^{\nabla_{M}}_{i}+\tilde{\beta}^{M}\langle
m_{i}^{M}\rangle)(mC_{i,p,m}-(m+1)C_{i,p,m+1})$ $\displaystyle-\beta^{M\to
P}\langle n_{i}^{P}\rangle(pC_{i,p,m}-(p+1)C_{i,p+1,m})-\beta^{P\to M}\langle
n^{M}\rangle(mC_{i,p,m}-(m+1)C_{i,p,m+1}).$
Here $p=0,1,...,S^{P}$, $m=0,1,...,S^{M}$, and
$\begin{array}[]{llllll}F^{\nabla_{X}}_{i}=\langle n^{X}_{i-1}\rangle-2\langle
n^{X}_{i}\rangle+\langle n^{X}_{i+1}\rangle&{\rm
if}\;\;\;1<i<N,&F^{\nabla_{X}}_{1}=-2\langle n^{X}_{1}\rangle+\langle
n^{X}_{2}\rangle&F^{\nabla_{X}}_{N}=\langle n^{X}_{N-1}\rangle-2\langle
n^{X}_{N}\rangle,\\\\[5.69054pt] G^{\nabla_{X}}_{i}=\langle
m^{X}_{i-1}\rangle-2\langle m^{X}_{i}\rangle+\langle m^{X}_{i+1}\rangle&{\rm
if}\;\;\;1<i<N,&G^{\nabla_{X}}_{1}=-2\langle m^{X}_{1}\rangle+\langle
m^{X}_{2}\rangle&G^{\nabla_{X}}_{N}=\langle m^{X}_{N-1}\rangle-2\langle
m^{X}_{N}\rangle,&\end{array}$
where $X\in\\{P,M\\}$, and
$\begin{array}[]{rclrcl}\langle
n^{P}_{i}\rangle&=&\sum_{p=0}^{S^{P}}\sum_{m=0}^{S^{M}}pC_{i,p,m},&\hskip
56.9055pt\langle
m^{P}_{i}\rangle&=&\sum_{p=0}^{S^{P}}\sum_{m=0}^{S^{M}}(S^{P}-p)C_{i,p,m}=S^{P}-\langle
n_{i}^{P}\rangle,\\\\[5.69054pt] \langle
n^{M}_{i}\rangle&=&\sum_{p=0}^{S^{P}}\sum_{m=0}^{S^{M}}mC_{i,p,m},,&\hskip
56.9055pt\langle
m^{M}_{i}\rangle&=&\sum_{p=0}^{S^{P}}\sum_{m=0}^{S^{M}}(S^{M}-m)C_{i,p,m}=S^{M}-\langle
n_{i}^{M}\rangle.\end{array}$
In Eqs.(20), corrections for left-hand-side values of $p=0$, $m=0$, $p=S$, and
$m=S$ have to be taken into account, similarly as in the first and third
equation of (10).
We consider the case of inhibition in only one direction by setting
$\beta^{M\to P}=0$ and varying $\beta^{P\to M}$, as is the case in the example
mentioned above where Ser10 phosphorylation inhibits Lys9 methylation. We
evaluate steady states as explained in section (IV.1). For parameter choices
of $S^{P}=S^{M}=2$, $\lambda^{P}=\mu^{P}=\lambda^{M}=\mu^{M}=1$,
$\beta^{P}=\beta^{M}=3$, $\alpha^{P}=\alpha^{M}=4.5$,
$\tilde{\alpha}^{P}=4\alpha^{P}$, $\tilde{\alpha}^{M}=4\alpha^{M}$,
$\tilde{\beta}^{P}=4\beta^{P}$ and $\tilde{\beta}^{M}=4\beta^{M}$, we observe
that for small $\beta^{P\to M}$, all four stable steady states (as listed
above) exist, as shown in Fig. 6. In an intermediate parameter regime only
three stable steady states persist: 00, P0 and 0M, using the notation
introduced above (Fig. 6). For large enough $\beta^{P\to M}$, only steady
states 00 and P0 remain. This means that inhibitory interactions of two types
of PTMs in only one direction are sufficient to obtain a parameter regime
where steady states have either a high number of PTMs P or M, or neither, but
not both. An analysis of traveling wave solutions of equations (20) similar to
the one in section IV.2 applies in this case.
## V Conclusions and Outlook
The main results of this paper are as follows. We offer a robust method to
obtain nonlinear partial differential equations describing the effective
dynamics of histones. The method proceeds by mapping the system onto a quantum
spin system whose dynamics is generated by a non-hermitian Hamiltonian. A
feedback mechanism due to diffusion of enzymes along nucleosomes gives rise to
multiple stable histone states. We study a number of novel aspects in histone
systems that have not been reported before and are of biological relevance. We
show that explicit cooperativity is not required to obtain multiple stable
steady states as long as the number of PTMs is larger or equal to two, and we
study the effects of varying the number of PTMs that are regulated by a
particular set of enzymes. We also study the effect of spatially heterogeneous
enzymatic on the histone state, and we apply our approach to a system of
several correlated PTMs.
Our approach can easily be generalized to higher spatial dimensions and more
complicated network topologies. Processes other than the ones considered in
this work could be included into the master equation and other biological
systems might be studied. In the context of post-translational histone
modifications, it might be of interest to consider more complex and more
realistic systems. For example, the particular structure of the core histones
might be taken into account i.e., the exact arrangement of the different
modifications on the different core histones. Feedback processes among
different types of post-translational modifications might be considered, as
well as feedback loops that arise due to interactions between the histones and
the DNA in the chromatin. It also remains an open question to study the
existence and stability of traveling wave solutions in the nonlinear reaction-
diffusion equations that arise in our model from a mathematically rigorous
point of view.
Acknowledgement.– We thank an anonymous referee for very helpful comments and
suggestions that improved the presentation of the results in the paper.
## References
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* (2) C. Peterson and M. Laniel. Histones and histone modifications. Current Biology 14, 546 (2004).
* (3) O. J. Rando and H. Y. Chang. Genome-Wide Views of Chromatin Structure. Annu. Rev. Biochem. 78, 245 (2009).
* (4) S.I. Grewal and A.J. Klar. Chromosomal inheritance of epigenetic states in fission yeast during mitosis and meiosis. Cell 86, 95 (1996).
* (5) G. Thon and T. Friis. Epigenetic inheritance of transcriptional silencing and switching competence. Genetics 145, 685 (1996).
* (6) S.I. Grewal and S.C. Elgin. Heterochromatin: new possibilities for the inheritance of structure. Curr. Opin. Genet. Dev. 12, 178 (2002).
* (7) I.M. Hall et.al. Establishment and maintenance of a heterochromatin domain. Science 297, 2232 (2002).
* (8) B.M. Turner. Histone acetylation as an epigenetic determinant of long-term transcriptional competence. Cell Mol. Life Sci. 54, 21 (1998).
* (9) M. Grunstein. Yeast heterochromatin: regulation of its assembly and inheritance of histones. Cell 93, 325 (1998).
* (10) R.H. Jacobsen, A.G. Ladurner, D.S. King, and R. Tijan. Structure and function of a human TAFII250 double bromodomain module. Science 288, 1422 (2000).
* (11) D.J. Owen et. al. The structural basis for recognition of acetylated histone H4 by the bromodomain of histone acetyltransferase gcnp5. EMBO J. 19, 6141 (2000).
* (12) L.N. Rusche and J. Rine, Conversion of a gene-specific repressor to a regional silencer. Genes Dev. 15, 955 (2001).
* (13) G. Schotta et. al. Central role of Drosophila SU(VAR)3-9 in histone H3-K9 methylation and heterochromatic gene silencing. EMBO J. 21, 1121 (2002).
* (14) I.B. Dodd, M.A. Micheelsen, K. Sneppen and G. Thon. Theoretical analysis of epigenetic cell memory by nucleosome modification. Cell 129, 813 (2007).
* (15) M. Sedighi and A.M. Sengupta. Epigenetic chromatin silencing: bistability and front propagation. Physical Biology 4, 246-255 (2003).
* (16) In this work, we denote as mean-field description an effective continuum deterministic description (i.e., ordinary or partial differential equations). We obtain our mean-field Eqs. (11) by rescaling parameters in the difference equations (10) as explained, a procedure generally denoted as mean-field in the applied mathematics community.
* (17) D. David-Rus, S. Mukhopadhyay, J.L. Lebowitz, and A.M. Sengupta. Inheritance of epigenetic chromatin silencing. J. Theor. Biol. 258, 112 (2009).
* (18) T. Jenuwein and C. D. Allis, Translating the histone code. Science 293, 1074 (2001).
* (19) D. Phanstiel, et al. Mass spectrometry identifies and quantifies 74 unique histone H4 isoforms in differentiating human embryonic stem cells. Proc. Natl Acad. Sci. USA 105, 4093 (2008).
* (20) J. J. Pesavento, et. al. Combinatorial modification of human histone H4 quantitated by two- dimensional liquid chromatography coupled with top down mass spectrometry. J. Biol. Chem. 283, 14927 (2008).
* (21) W.S. Lo et.al. Phosphorylation of serine 10 in histone H3 is functionally linked in vitro and in vivo to Gcn5-mediated acetylation at lysine 14. Mol. Cell. 5, 917 (2000).
* (22) Z.W. Sun and C.D. Allis. Ubiquitination of histone H2B regulates H3 methylation and gene silencing in yeast. Nature 418, 104 (2002).
* (23) H.H Ng, R.M. Xu, Y. Zhang, and K. Struhl. Ubiquitination of histone H2B by Rad6 is required for efficient Dot1-mediated methylation of histone H3 lysine79. J. Biol. Chem. 277, 34655 (2002).
* (24) S. Rea et. al. Regulation of chromatin structure by site-specific histone H3 methyltransferases. Nature 406, 593 (2000).
* (25) M. Doi. Second quantization representation for classical many-particle system. J. Phys. A 9, 1465 (1976).
* (26) L. Peliti. Path integral approach to birth-death processes on a lattice. J. Phys. France 46, 1469 (1985).
* (27) We note that the operators $\mathcal{R}$ and $\mathcal{L}$ do not correspond to bosonic creation and annihilation operators (bosonic commutation relations are not obeyed), but are the suitable operators for our model where $S$ takes finite values.
* (28) G.L. Eyink, Action principle in nonequilibrium statistical dynamics. Phys. Rev. E 54, 3419 (1996).
* (29) We choose $C_{i3}(t=0)=0.5\bar{C}_{3}(2-\tanh(i-N/2+5)+\tanh(i-N/2-5))$, where $\bar{C}_{3}\approx 0.9$.
* (30) A.T. Annunziato. Split decision: what happens to nucleosomes during DNA replication? J. Biol. Chem. 280, 12065 (2005).
|
arxiv-papers
| 2009-12-22T17:36:07 |
2024-09-04T02:49:07.210533
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Gils, J.L. Wrana, and W.K. Abou Salem",
"submitter": "Charlotte Gils",
"url": "https://arxiv.org/abs/0912.4465"
}
|
0912.4602
|
2010215-226Nancy, France 215
Bireswar Das
Samir Datta
Prajakta Nimbhorkar
# Log-space Algorithms for Paths and Matchings in $k$-trees
B. Das The Institute of Mathematical Sciences
Chennai, India bireswar, prajakta@imsc.res.in , S. Datta Chennai
Mathematical Institute, Chennai, India sdatta@cmi.ac.in and P. Nimbhorkar
###### Abstract.
Reachability and shortest path problems are NL-complete for general graphs.
They are known to be in L for graphs of tree-width $2$ [14]. However, for
graphs of tree-width larger than $2$, no bound better than NL is known. In
this paper, we improve these bounds for $k$-trees, where $k$ is a constant. In
particular, the main results of our paper are log-space algorithms for
reachability in directed $k$-trees, and for computation of shortest and
longest paths in directed acyclic $k$-trees.
Besides the path problems mentioned above, we consider the problem of deciding
whether a $k$-tree has a perfect macthing (decision version), and if so,
finding a perfect matching (search version), and prove that these problems are
L-complete. These problems are known to be in P and in RNC for general graphs,
and in SPL for planar bipartite graphs [8].
Our results settle the complexity of these problems for the class of
$k$-trees. The results are also applicable for bounded tree-width graphs, when
a tree-decomposition is given as input. The technique central to our
algorithms is a careful implementation of divide-and-conquer approach in log-
space, along with some ideas from [14] and [19].
###### Key words and phrases:
k-trees, reachability, matching, log-space
###### 1991 Mathematics Subject Classification:
Computational Complexity
## 1\. Introduction
Reingold’s striking result [21], showed that undirected reachability is in L,
thus collapsing the class SL to L. On the other hand, directed reachability,
which happens to be NL-complete is another similar sounding problem for which
there is only partial progress to report. A result of Allender and Reinhardt,
[22] hints at a partial collapse of NL by showing that directed reachability
is in the formally smaller class UL, although, _non-uniformly_.
In the absence of better constructive upper bounds it is natural to consider
natural restrictions on graphs which allow us to improve the upper bounds on
reachability and related problems. Typical examples of this approach are
[1],[23], where the complexity of various versions of planar and somewhat non-
planar (in the sense of excluding only a $K_{5}$ or only a $K_{3,3}$ minor)
are considered. In the same spirit, but using different techniques, [14]
considers reachability and related questions in series-parallel graphs and
places all of these in L. They leave open the question of complexity of such
problems in bounded tree-width graphs. Series-parallel graphs have tree-width
two and happen to be planar. But higher tree widths graphs are highly non-
planar. In fact, any $k$-tree for $k>4$ contains both $K_{5}$ and $K_{3,3}$.
We resolve the open questions posed in [14] and show a matching L lower bound
to complete the characterization of reachability problems in $k$-trees. Thus
one of the main results of our paper is the following:
###### Theorem 1.1.
The following problems are L-complete:
1\. Computing reachability between two vertices in directed $k$-trees,
2\. Computing shortest and longest paths in directed acyclic $k$-trees.
In this paper, we also consider the perfect matching problem. The parallel
complexity of perfect matching problems is a long standing open problem where
the best known algorithms use randomness as a resource [20],[15]. Even in the
planar case, the search problem for perfect matchings is known to be in NC for
bipartite graphs only [8].
We prove a complete characterization for the decision and search versions of
the perfect matching problem for $k$-trees. This improves significantly upon
previous best known upper bound of LogCFL for bounded tree-width graphs. Thus
another main result of our paper is:
###### Theorem 1.2.
Deciding whether a $k$-tree has a perfect matching, and if so, finding a
perfect matching is L-complete.
Our primary technique is a careful use of divide-and-conquer to enable the
algorithm to run in L. However, for the distance computation we need to import
a constructive version of tree separation from [19] where it is stated in the
context of Visibly Pushdown Automata (VPAs). We believe that porting this
technique for use in general log-space computation is an important
contribution of this paper.
At this point, we must mention an important caveat. All our log-space results
hold directly only for $k$-trees and not for partial $k$-trees which are also
equivalent to tree-width $k$ graphs. The reason being that a tree
decomposition for partial $k$-trees is apparently more difficult to construct
(best known upper bound is LogCFL[24]) as opposed to $k$-trees (for which it
can be done in L [17]). Having mentioned that it is important to observe that
if we are given the tree decomposition of a partial $k$-tree, we can do the
rest of computation in L.
The rest of the paper is organized as follows: Section 2 gives the necessary
background. Section 3 contains log-space algorithms for reachability in
directed $k$-paths and $k$-trees. Section 4 contains log-space algorithms for
shortest and longest path in directed acyclic $k$-paths and $k$-trees. Section
5 contains log-space algorithms for perfect matching problems in a $k$-tree.
## 2\. Preliminaries
We define $k$-trees and a subclass of $k$-trees known as $k$-paths here, and
also describe a suitable representation for the graphs in these two classes.
This representation is used in our algorithms in the rest of the paper. All
the definitions given here are applicable to both directed as well as
undirected graphs. For directed graphs, the directions of the edges can be
ignored while defining $k$-trees and $k$-paths and while computing their
suitable representations.
The class of graphs known as $k$-trees is defined as (cf. [12] ):
###### Definition 2.1.
The class of _$k$ -trees_ is inductively defined as follows.
* •
A clique with $k$ vertices ($k$-clique for short) is a $k$-tree.
* •
Given a $k$-tree $G^{\prime}$ with $n$ vertices, a $k$-tree $G$ with $n+1$
vertices can be constructed by picking a $k$-clique $X$ (called the
_support_)in $G^{\prime}$ and then joining a new vertex $v$ to each vertex $u$
in $X$. Thus, $V(G)=V(G^{\prime})\cup\\{v\\}$,
$E(G)=E(G^{\prime})\cup\\{\\{u,v\\}\mid u\in X\\}$.
A _partial $k$-tree_ is a subgraph of a $k$-tree. The class of partial
$k$-trees coincides with the class of graphs that have tree-width at most $k$.
$k$-trees are recognizable in log-space [2] but partial $k$-trees are not
known to be recognizable in log-space. In literature, several different
representations of $k$-trees have been considered [10, 2, 17]. We use the
following representation given by Köbler and Kuhnert [17]:
###### Definition 2.2.
Let $G=(V,E)$ be a $k$-tree. The tree representation $T(G)$ of $G$ is defined
by
$V(T(G))=\\{M\subseteq V\mid M\textrm{ is a $k$-clique or a
$(k+1)$-clique}\\},\\\ $ $E(T(G))=\\{\\{M_{1},M_{2}\\}\subseteq V\mid
M_{1}\subsetneq M_{2}\\}$
In [17], it is proved that $T(G)$ is a tree and can be computed in log-space.
In the rest of the paper, we use $G$ in place of $T(G)$. Thus, by a $k$-tree
$G$, we always mean that $G$ is in fact represented as $T(G)$. The term
vertices in $G$ refers to the vertices in the original graph, whereas a node
in $G$ and a clique in $G$ refer to the nodes of $T(G)$. Partial $k$-trees
also have a tree-decomposition similar to that of $k$-trees, which is also not
known to be log-space computable.
$k$-paths is a sub-class of $k$-trees (e.g. see [11]). The recursive
definition of $k$-paths is similar to that of $k$-trees. However, a new vertex
can be added only to a particular clique called the _current clique_. After
addition of a vertex, the current clique may remain the same, or may change by
dropping a vertex and adding the new vertex in the current clique. We consider
the following representation of $k$-paths, which is based on the recursive
definition of $k$-paths, and is known to be computable in log-space [2]:
Given a $k$-path $G=(V,E)$, for $i=1,\cdots,m$, let $X_{i}$ be the current
cliques at the $i$th stage of the recursive construction of the $k$-path. Let
$V_{1}=\cup_{i}X_{i}$ and $V_{2}=V\setminus V_{1}$. We call the vertices in
$V_{2}$ as spikes. The following facts are easy to see:
1\. No two spikes have an edge between them.
2\. Each spike is connected to all the vertices of exactly one of the
$X_{i}$’s.
3\. $X_{i}$ and $X_{i+1}$ share exactly $k-1$ vertices
The representation of $G$ consists of a graph
$G^{\prime}=(V^{\prime},E^{\prime})$ where
$V^{\prime}=\\{X_{1},\ldots,X_{m}\\}\cup V_{2}$ and
$E^{\prime}=\\{(X_{i},X_{i+1})\mid 1\leq i<m\\}\cup\\{(X,v)|X\textrm{ is a
clique in }\in V^{\prime},v\in V_{2}\textrm{ has a neighbour in }X\\}$.
## 3\. Reachability
We give log-space algorithms to compute reachability in $k$-paths and in
$k$-trees. Although the graphs considered in this section are directed, when
we refer to any of the definitions or decompositions in Section 2, we consider
the underlying undirected graph.
### 3.1. Reachability in $k$-paths
Without loss of generality, we can assume that $s$ and $t$ are vertices in
some $k$-cliques $X_{i}$ and $X_{j}$, and not spikes. If $s$ ($t$) is a spike,
then it has at most $k$ out-neighbors (resp. in-neighbors) and we can take one
of the out-neighbors (resp. in-neighbors) as the new source $s^{\prime}$ and
new sink $t^{\prime}$ and check reachability. As there are only $k^{2}$ such
pairs, we can cycle through all of them in log-space. The algorithm is based
on the observation that a simple $s$ to $t$ path $\rho$ can pass through any
clique at most $k$ times. We use a divide- and-conquer approach similar to
that used in Savitch’s algorithm (which shows that directed reachability can
be computed in $DSPACE(\log^{2}{n})$). The main steps involved in the
algorithm are as follows:
1\. Preprocessing step: Make the cliques disjoint by labeling different copies
of each vertex with different labels and introducing appropriate edges.
Compute reachabilities within each clique including its spikes, and _remove
the spikes_. Number the cliques $X_{1},\ldots,X_{m}$ left to right.
2\. Now assume that $s$ and $t$ are in cliques $X_{i}$ and $X_{j}$
respectively. Note that $i=j$ is also possible, but without loss of
generality, we can assume $i<j$. This is because, if $i=j$, we can make
another copy $X_{i}^{\prime}$ of $X_{i}$, join the copies of the same vertex
by bidirectional edges to preserve reachabilities, and choose the copy of $s$
from $X_{i}$ and that of $t$ from $X_{i}^{\prime}$.
3\. Divide the $k$-path into three parts $P_{1},~{}P_{2}$ and $P_{3}$ where
$P_{1}$ consists of cliques $X_{1},\ldots,X_{i}$, $P_{2}$ consists of
$X_{i},\ldots,X_{j}$, and $P_{3}$ consists of $X_{j},\ldots,X_{m}$. Note that
$X_{i}$ ($X_{j}$) appears in both $P_{1}$ and $P_{2}$ ($P_{2}$ and $P_{3}$
respectively). Now compute reachabilities of all pairs of vertices in $X_{i}$
($X_{j}$) when the graph is restricted to $P_{1}$ (respectively $P_{3}$). Then
the reachability of $t$ from $s$ within $P_{2}$ is computed, using the
previously computed reachabilities within $P_{1}$ and $P_{3}$.
Each of these steps can be done by a log-space transducer. The details are
given below.
Preprocessing: Although adjacent k-cliques in a k-path decomposition share
$k-1$ vertices, we perform a preprocessing step, where we give distinct labels
to each copy of a vertex. As all the copies of a vertex form a (connected)
sub-path in the k-path decomposition, we join two copies of a vertex appearing
in two adjacent cliques by bidirectional edges. It can be seen that this
preserves reachabilities. Any copy of $s$ and $t$ can be taken as the new $s$
and $t$. Another preprocessing step involves removing the spikes maintaining
reachabilities between all pairs of vertices, and computing reachabilities
within each k-clique. Both of these preprocessing steps can be done by a log-
space transducer. The proof appears in the full version of the paper.
The Algorithm: We describe an algorithm to compute pairwise reachabilities in
$X_{i}$ and $X_{j}$ in $P_{1}$ and $P_{3}$ respectively, and also $s$-$t$
reachability in $P_{2}$ using these previously computed pairwise
reachabilities. Algorithm 1 describes this reachability routine. The routine
gets as input two vertices $u$ and $v$, and two indices $i$ and $j$. It
determines whether $v$ is reachable from $u$ in the sub-path
$P=(X_{i},\ldots,X_{j})$. This input is given in such a way that $u$ and $v$
always lie in $X_{i}$ or $X_{j}$. Consider the case when both $u$ and $v$ are
in $X_{i}$ (or both in $X_{j}$). Let $l$ be the center of $P$. Then a path
from $u$ to $v$ either lies entirely in the sub-path
$P^{\prime}=(X_{i},\ldots,X_{l})$ or it crosses $X_{l}$ at most $k$ times.
Thus if $X_{l}=\\{v_{1},\ldots,v_{k}\\}$ then for
$\\{v_{i_{1}},\cdots,v_{i_{r}}\\}\subseteq X_{l}$ we need to check
reachabilities between $u$ and say $v_{i_{1}}$ in $P^{\prime}$, then between
$v_{i_{1}}$ and $v_{i_{2}}$ in $P^{\prime\prime}=(X_{l},\ldots,X_{j})$ and so
on, and finally between $v_{i_{r}}$ and $v$ in $P^{\prime}$. It suffices to
check all the $r$-tuples in $X_{l}$, where $0\leq r\leq k$. The case when
$u\in X_{i}$ and $v\in X_{j}$ (and vice versa) is analogous. In Algorithm 1,
we present only one case where $u,v\in X_{i}$. Other three cases are
analogous. Thus at each recursive call, the length of the sub-path under
consideration is halved, and $O(\log{m})$ iterations suffice.
Algorithm 1 Procedure IsReach($u$, $v$, $i$, $j$)
1: Input: Pre-processed k-path decomposition of graph $G$, clique indices
$i,j$, vertex labels $u,v\in X_{i}$. {Other three cases are analogous.}
2: Decide: Whether $v$ is reachable from $u$ in sub-path
$P=(X_{i},\ldots,X_{j})$.
3: if $j-i=1$ then
4: Compute the reachability directly, as the sub-path has only $2k$ vertices.
5: Return the result.
6: end if
7: $l=\frac{j+i}{2}$
8: if $u,v\in X_{i}$ then
9: if IsReach($u$, $v$, $i$, $l$) then
10: Return 1;
11: else
12: for $q=1$ to $k$ do
13: $v_{0}\leftarrow u$, $v_{q+1}\leftarrow v$
14: for all $q$-tuples ($v_{1},\ldots,v_{q}$) of vertices in $X_{l}$ do
15: if $\bigwedge_{\begin{subarray}{c}x=0\\\ x\textrm{
even}\end{subarray}}^{q+1}$ IsReach($v_{x}$,$v_{x+1}$,$i$,$l$) $\land$
$\bigwedge_{\begin{subarray}{c}x=1\\\ x\textrm{ odd}\end{subarray}}^{q+1}$
IsReach($v_{x}$,$v_{x+1}$,$l$,$j$) then
16: Return 1;
17: end if
18: end for
19: end for
20: end if
21: end if
The algorithm can be implemented in log-space. The correctness and complexity
analysis of the algorithm appears in the full version.
### 3.2. Reachability in $k$-trees
Given a directed $k$-tree $G$ in its tree decomposition and two vertices $s$
and $t$ in $G$, we describe a log-space algorithm that checks whether $t$ is
reachable from $s$. This algorithm uses Algorithm 1 as a subroutine and
involves the following steps: The complexity analysis is given in Lemma 3.1.
1\. Preprocessing: Like $k$-paths, assign distinct labels to the copies of
each vertex $u$ in different cliques. Introduce a bidirectional edge between
the copies of $u$ in all the adjacent pairs of cliques. As reachabilities are
maintained during this process, any copy of $s$ and $t$ can be taken as the
new $s$ and $t$. Let $X_{i}$ and $X_{j}$ be the cliques containing $s$ and $t$
respectively.
2\. The Procedure: After this preprocessing, we have a tree $T$ with its
nodes as disjoint $k$-cliques of vertices of $G$, and $s$ and $t$ are
contained in cliques $X_{i}$ and $X_{j}$. Compute the unique undirected path
$\rho$ between $X_{i}$ and $X_{j}$ in $T$ in log-space. Each node on $\rho$
has two of its neighbors on $\rho$, except $X_{i}$ and $X_{j}$, which have one
neighbor each. An $s$ to $t$ path has to cross each clique in $\rho$, and
additionally, it can pass through the subtrees attached to each node $X_{l}$
on $\rho$. Hence for each node $X_{l}$ on $\rho$, we pre-compute the pairwise
reachabilities among the $k$ vertices contained in $X_{l}$ when the $k$-tree
is restricted to the subtree rooted at $X_{l}$. We define the subtree rooted
at $X_{l}$ as the subtree consisting of $X_{l}$ and those nodes which can be
reached from $X_{l}$ without going through any node on $\rho$. Note that once
this is done for each node $X_{l}$ on $\rho$, we are left with $\rho$. As
$\rho$ is a $k$-path, we can use Algorithm 1 in Section 3.1 to compute
reachabilities within $\rho$.
3\. Computing reachabilities within the subtree rooted at $X_{l}$: We do this
inductively. If the subtree rooted at $X_{l}$ contains only one node $X_{l}$,
we have only $k$ vertices, and their pairwise reachabilities within $X_{l}$
can be computed in $O(k\log{k})$ space. We recursively find the reachabilities
within the subtrees rooted at each of the children of $X_{l}$. Let the size of
the subtree rooted at $X_{l}$ be $N$. At most one of the children of $X_{l}$
can have a subtree of size larger than $\frac{N}{2}$. Let $X_{a}$ be such a
child. Recursively compute the pairwise reachabilities for each pair of
vertices in $X_{a}$ within the subtree rooted at $X_{a}$. The reachabilities
are represented as a $k\times k$ boolean matrix referred to as the
reachability matrix $M$ for the vertices in $X_{a}$, when the graph is
confined to the subtree rooted at $X_{a}$. $M$ is then used to compute the
pairwise reachabilities of vertices in $X_{l}$, when the graph is confined to
$X_{l}$ and the subtree rooted at $X_{a}$. This gives a new matrix
$M^{\prime}$ of size $k^{2}$. It is stored on stack while computing the
reachability matrix $M^{\prime\prime}$ for another child $X_{b}$ of $X_{l}$.
The matrix $M^{\prime}$ is updated using $M^{\prime\prime}$, so that it
represents reachabilities between each pair of vertices in $X_{l}$ when the
graph is confined to $X_{l}$ and the subtrees rooted at $X_{a}$ and $X_{b}$.
This process is continued till all the children of $X_{l}$ are processed. The
matrix $M^{\prime}$ at this stage reflects the pairwise reachabilities between
vertices of $X_{l}$, when the graph is confined to the subtree rooted at
$X_{l}$. Note that the storage required while making a recursive call is only
the current reachability matrix $M^{\prime}$. Recall that $M^{\prime}$
contains the pairwise reachabilitities among the vertices in $X_{l}$ in the
subgraph corresponding to $X_{l}$ and the subtrees rooted at those children of
$X_{l}$ which are processed so far. We give the complexity analysis in the
full version.
###### Lemma 3.1.
The procedure described above can be implemented in log-space.
#### Hardness for L:
L-hardness of reachability in $k$-trees follows from L-hardness of the problem
of path ordering (proved to be SL-hard in [9], and is L-hard due to SL=L
result of [21]). We give the details in the full version.
## 4\. Shortest and Longest Paths
We show that the shortest and longest paths in weighted directed acyclic
$k$-trees can be computed in log-space, when the weights are positive and are
given in unary. Throughout this section, the terms $k$-path and $k$-tree
always refer to directed acyclic $k$-paths and $k$-trees respectively, with
integer weights on edges and we here onwards omit the specification weighted
directed acyclic. We use the following (weighted) form of the result from
[18]: The proof is exactly similar to that in [18] and we omit it here.
###### Theorem 4.1 (See[18], Theorem $9$).
Let ${\mathcal{C}}$ be any subclass of weighted directed acyclic graphs closed
under vertex deletions. There is a function $f$, computable in log-space with
oracle access to $\mbox{{\sf Reach}}(\mbox{${\mathcal{C}}$})$, that reduces
$\mbox{{\sf Distance}}(\mbox{${\mathcal{C}}$})$ to $\mbox{{\sf Long-
Path}}(\mbox{${\mathcal{C}}$})$ and $\mbox{{\sf Long-
Path}}(\mbox{${\mathcal{C}}$})$ to $\mbox{{\sf
Distance}}(\mbox{${\mathcal{C}}$})$, where ${\mbox{{\sf
Reach}}(\mbox{${\mathcal{C}}$})}$, $\mbox{{\sf
Distance}}(\mbox{${\mathcal{C}}$})$, and $\mbox{{\sf Long-
Path}}(\mbox{${\mathcal{C}}$})$ are the problems of deciding reachability,
computing distance and longest path respectively for graphs in
${\mathcal{C}}$.
We use this theorem to reduce the shortest path problem in $k$-trees to the
longest path problem, and then compute the longest (that is, maximum weight)
$s$ to $t$ path. The reduction involves changing the weights of the edges such
that the shortest path becomes the longest path and vice versa. This gives a
directed acyclic $k$-tree with positive integer weights on edges given in
unary. The class of $k$-trees is not closed under vertex deletions. However,
once a tree decomposition of a $k$-tree is computed, deleting vertices from
the cliques leaves some cliques of size smaller than $k$, which does not
affect the working of the algorithm.
We show that the maximum weight of an $s$ to $t$ path can be computed in log-
space using a technique which uses ideas from [14]. The algorithm to compute
maximum weight $s$ to $t$ path in $k$-trees uses the algorithm for computing
maximum weight path in $k$-paths as subroutine. Therefore we first describe
the algorithm for $k$-paths in Section 4.1
### 4.1. Maximum Weight Path in Directed Acyclic $k$-paths
Let $G$ be a directed acyclic $k$-path and $s$ and $t$ be two designated
vertices in $G$. The computation of maximum weight of an $s$ to $t$ path is
done in five stages, described below in detail. The main idea is to obtain a
log-depth circuit by a suitable modification of Algorithm 1, and to transform
this circuit to an arithmetic formula over integers, whose value is used to
compute the maximum weight of an $s$ to $t$ path in $G$.
Computing the maximum weight $s$ to $t$ path in $G$ involves the following
steps:
1. (1)
Construct a log-depth formula from Algorithm 1: Modify Algorithm 1 so that it
outputs a circuit $\mathcal{C}$ that has nodes corresponding to the recursive
calls made in Line $15$ and the tuples considered in the for loop in Line
$14$. A node $q$ in $\mathcal{C}$ that corresponds to a recursive call
IsReach($u$, $v$, $i$, $j$) has children $q_{1},\cdots,q_{N}$, which
correspond to the tuples considered in that recursive call (for-loop on Line
$12$ of Algorithm 1). We refer to $q$ as a call-node and $q_{1},\ldots,q_{N}$
as tuple-nodes. A tuple-node $q^{\prime}$ corresponding to a tuple
$(v_{1},\ldots,v_{N})$ has call-nodes $q_{1}^{\prime},\ldots,q_{N}^{\prime}$
as its children, which correspond to the recursive calls made while
considering the tuple $(v_{1},\ldots,v_{N})$ (Line $15$ of Algorithm 1). The
leaves of $\mathcal{C}$ are those recursive calls which satisfy the if
condition on Line $3$ of Algorithm 1, thus they are always call-nodes. As the
depth of the recursion in Algorithm 1 is $O(\log{n})$, the circuit
$\mathcal{C}$ also has $O(\log{n})$ depth. Hence it can be converted to a
formula $\mathcal{F}$ by only a polynomial factor blow-up in its size. The
maximum number of children of a node is $O(k^{k})$ and hence the size of
$\mathcal{F}$ is bounded by $O(k^{k\log{n}})$, which is polynomial in $n$ for
constant $k$.
2. (2)
Prune the boolean formula: The internal call-nodes of $\mathcal{F}$ are
replaced by $\lor$ gates and tuple-nodes are replaced by $\land$ gates. The
leaves of $\mathcal{F}$ are replaced by $0$ or $1$ depending on whether the
corresponding recursive call returned $0$ or $1$ in the if block on Line $3$
of Algorithm 1. It can be seen that a sub-formula of $\mathcal{F}$ rooted at a
call-node evaluates to $1$ if and only if the corresponding recursive call
returns $1$ in Algorithm 1. Similarly, the sub-formula rooted at a tuple-node
evaluates to $1$ if and only if the conjunction corresponding to it (on Line
$15$ of Algorithm 1) evaluates to $1$. Now, we evaluate the sub-formula rooted
at each node of $\mathcal{F}$. Note that a node that evaluates to $0$ does not
contribute to any path from $s$ to $t$, and hence its subtree can be safely
removed.
3. (3)
Transformation into a $\\{+,max\\}$-tree: The new, pruned formula obtained in
Step $2$ is then relabeled: Each $\land$ label is replaced with a $+$ label
and each $\lor$ label with a $max$ label. Each leaf corresponds to calls of
the form $IsReach(u,v,i,i+1)$. It is labeled with the length of the maximum
weight $u$ to $v$ path confined within cliques $i$ and $i+1$, which can be
computed in $O(1)$ space. This weight is strictly positive, since the
$0$-weight leaves are removed in Step $2$. Further, all the weights are in
unary. Thus we now have a $\\{+,max\\}$-tree $T$ with positive, unary weights
on its leaves. It is easy to see that the value of the $\\{+,max\\}$-tree $T$
is the maximum weight of any $s$ to $t$ path in $G$.
4. (4)
Transformation into a $\\{+,\times\\}$-tree: The evaluation problem on the
$\\{+,max\\}$-tree $T$ obtained in Step $3$ is then reduced to the evaluation
problem on a $\\{+,\times\\}$-tree $T^{\prime}$ whose leaves are labeled with
positive integer weights coded in binary. This reduction works in log-space
and is similar to that of [14]. The reduction involves replacing a $+$-node of
$T$ with a $\times$-node, and a $max$-node with a $+$ node. The weight $w$ of
a leaf is replaced with $r^{mw}$, where $r$ is the smallest power of $2$ such
that $r\geq n$, and $m$ is the sum of the weights of all the leaves of $T$
plus one. The correctness of the reduction follows from a similar result in
[14], and we omit the proof here.
5. (5)
Evaluation of the $\\{+,\times\\}$ tree: This can be done in log-space due to
[5, 3, 7, 13]. The value of $T$ is
$v=\lfloor\frac{log_{r}v^{\prime}}{m}\rfloor$.
### 4.2. Maximum Weight Path in Directed Acyclic $k$-trees
Given a directed acyclic $k$-tree (in its tree-decomposition) $G$, two
vertices $s$ and $t$ in $G$, and weights on the edges of $G$, encoded in
unary, we show how to compute the maximum weight of an $s$ to $t$ path in $G$.
Unlike the case of $k$-paths, the reachability algorithm for $k$-trees given
in Section 3.2 can not be used to get a log-depth circuit since the recursion
depth of the algorithm is same as the depth of the $k$-tree. Therefore we need
to find another way of recursively dividing the $k$-tree into smaller and
smaller subtrees, as we did for $k$-paths in Sections 3.1 and 4.1. This is
based on the technique used in the following result of [19]:
###### Lemma 4.2.
(Lemma $6$ of [19], also see [4]) Let $M$ be a visibly pushdown automaton
accepting well-matched strings over an alphabet $\Delta$. Given an input
string $x$, checking whether $x\in L(M)$ can be done in log-space.
Using Lemma 4.2, we can compute a set of recursive separators for a tree
defined below:
###### Definition 4.3.
Given a rooted tree $T$, separators of $T$ are two nodes $a$ and $b$ of $T$
such that
1\. The subtrees rooted at $a$ and $b$ respectively are disjoint,
2\. $T$ is split into subtrees $T_{1}$, $T_{2}$, $T_{3}$ where $T_{1}$
consists of $a$, some (or possibly all) of the children of $a$, and subtrees
rooted at them, $T_{2}$ is defined similarly for $b$, and $T_{3}$ consists of
the rest of the tree along with a copy of $a$ and $b$ each.
3\. Each of $T_{1}$, $T_{2}$, $T_{3}$ consists of at most a $\frac{3}{4}$
fraction of the leaves of $T$.
This process is done recursively for $T_{1}$, $T_{2}$, $T_{3}$, until the
number of leaves in the subtrees is two. Such a subtree is in fact a path. A
set of recursive separators of $T$ consists of the separators of $T$ and of
all the subtrees obtained in the recursive process.
The following lemma gives the procedure to compute a set of recursive
separators of a tree $T$:
###### Lemma 4.4.
Given a tree $T$, the set of recursive separators of $T$ can be computed in
log-space.
###### Proof 4.5.
The algorithm of [19] deals with well-matched strings. An example of a well-
matched string is a balanced parentheses expression, which is a string over
$\\{(,)\\}$. In [19], a log-space algorithm is given for membership testing in
those languages which are subsets of well-matched strings and are accepted by
visibly pushdown automata. We restrict ourselves to balanced parentheses
expressions. To check whether a string on parentheses is in the language, the
algorithm of [19] recursively partitions the string into three disjoint
substrings, such that each of the parts forms a balanced parentheses
expression, and length of each part is at most $\frac{3}{4}$th of the length
of the original string. To use this algorithm, we order the children of each
node of $T$ in a specific way, label the leaves with parentheses $`(^{\prime}$
and $`)^{\prime}$ such that the leaves of the subtree rooted at any internal
node form a string on balanced parentheses. We add dummy leaves if needed. The
steps are as follows:
1\. By adding dummy leaves, ensure that each internal node has an even number
of children which are leaves, and there are at least two such children.
2\. Arrange the children of each node from left to right such that the non-
leaves are consecutive, and they have an equal number of leaves to the left
and to the right.
3\. For each internal node, label the left half of its leaf-children with ‘(’
and the right ones by ‘)’. This ensures that the leaves of the subtree rooted
at each internal node form a balanced parentheses expression. Conversely,
leaves which form a balanced parentheses expression are consecutive leaves in
the subtree rooted at an internal node.
The leaves of $T$ now form a balanced parentheses expression, and we run the
algorithm of [19] on this string. The recursive splitting of the string into
smaller substrings corresponds to the recursive splitting of $T$ at some
internal nodes, which satisfies Definition 4.3. This is ensured by the way the
leaves are labeled. Each balanced parentheses expression corresponds to either
a subtree rooted at an internal node or the subtrees rooted at some of the
children of an internal node.
The subtrees obtained by splitting a tree have at most $\frac{3}{4}$th of the
number of leaves in the tree. Thus at each stage of recursion, the number of
leaves in the subtrees is reduced by a constant fraction. Moreover, the
algorithm of [19] can output all the substrings formed at each stage of
recursion in log-space. As a substring completely specifies a subtree of $T$,
our procedure outputs the set of recursive separators for $T$ in log-space.
Once an algorithm to compute the set of recursive separators for $k$-trees is
known, a reachability routine similar to Algorithm 1 can be designed in a
straight forward way. We give the details in the full version. From the
reachability routine, the computation of maximum weight path follows from the
steps $1$ to $5$ described in Section 4.1.
### 4.3. Distance Computation in Undirected $k$-trees
We give a simple log-space algorithm for computing the shortest path between
two given vertices in an undirected $k$-tree. We use the decomposition of
[16], where a $k$-tree is decomposed into layers. We use the following
properties of the decomposition:
1\. Layer $0$ is a $k$-clique. Each vertex in layer $i>0$ has exactly $k$
neighbors in layers $j<i$. Further, these neighbors of $i$ which are in layers
lower than that of $i$ form a $k$-clique.
2\. No two vertices in the same layer share an edge.
This decomposition is log-space computable [17]. Moreover, given two vertices
$s$ and $t$, it is always possible to find a decomposition in which $t$ lies
in layer $0$. This can also be done in log-space. If both $s$ and $t$ are in
layer $0$, then there is an edge between $s$ and $t$, which is the shortest
path from $s$ to $t$. Therefore assume that $s$ lies in a layer $r>0$. The
following claim leads to a simple algorithm. The proof appears in the full
version.
###### Claim 1.
1\. The shortest $s$ to $t$ path never passes through two vertices $u$ and $v$
such that $layer(u)<layer(v)$. 2\. There is a shortest path from $s$ to $t$
passing through the neighbor of $s$ in the lowest layer.
This claim suggests a simple algorithm which can be implemented in log-space:
Start from $s$ and choose the next vertex from the lowest possible layer, at
each step till we reach layer $0$.
## 5\. Perfect Matching in $k$-trees
#### Hardness for L:
To show that the decision version of perfect matching is hard for L, we show
that the problem of path ordering, can be reduced to the perfect matching
problem for $k$-trees. We give the proof in the full version:
###### Lemma 5.1.
Determining whether a $k$-tree has a perfect matching is L-hard.
#### L upper bounds:
We describe a log-space algorithm to decide whether a $k$-tree has a perfect
matching and, if so, output a perfect matching. The algorithm is inspired by
an $O(n^{3})$ algorithm [6] for computing the matching polynomial in series-
parallel graphs. The idea is to exploit the fact that $k$-trees have a tree
decomposition of bounded width, so that any perfect matching of the entire
$k$-tree induces a partial matching on any subtree which leaves at most
constantly many vertices unmatched. Thus we generalize the problem to that of
determining, for each set, $S$, of constantly many vertices in the root of the
subtree, whether there is a matching of the subtree that leaves exactly the
vertices in $S$ unmatched. Now we “recursively” solve the generalized problem
and for this purpose we need to maintain a bit-vector indexed by the sets $S$
which is still of bounded length. The algorithm composes the bit-vectors of
the children of a node to yield the bit-vector for the node. The bit-vector,
which we refer to as matching vector, is defined as follows:
###### Definition 5.2.
Let $G$ be a $k$-tree with tree-decomposition $T$. $T$ has alternate levels of
$k$-cliques and $k+1$-cliques. Root $T$ arbitrarily at a $k$-clique. Let $s$
be a node in $T$ that shares vertices $\\{u_{1},\ldots,u_{k}\\}$ with its
parent. Further, let $H$ be the subgraph of $G$ corresponding to the subtree
of $T$ rooted at $s$. The matching vector for $s$ is a vector
$\vec{v}_{H}=(v_{H}^{(S_{1})},\ldots,v_{H}^{(S_{2^{k}})})$ of dimension
$2^{k}$, where $S_{1},\ldots,S_{2^{k}}$ are all the distinct subsets of
$\\{u_{1},\ldots,u_{k}\\}$, and $v_{H}^{(S_{i})}=1$ if $H$ has a matching in
which all the vertices of $H$ matched, except those in $S_{i}$,
$v_{H}^{(S_{i})}=0$ if there is no such matching.
It can be seen that $G$ has a perfect matching if and only if
$v_{G}^{(\emptyset)}=1$.We show how to compute $\vec{v}_{G}$ in L, and also
show how to construct a perfect matching in $G$, if one exists. We prove Part
1 of the following theorem. For a proof of part 2, we refer to the full
version.
###### Theorem 5.3.
1\. The problem of deciding whether a $k$-tree has a perfect matching is in L.
2\. Finding a perfect matchings in a $k$-tree is in FL.
###### Proof 5.4.
(of $1$) We compute the matching vector for the root by recursively computing
the matching vectors of each of its children. For a leaf node in the tree-
decomposition, the matching vector can be computed in a brute-force way. At an
internal node $s$, the matching vector is computed from the matching vectors
of its children, which we describe here:
Case $1$: $s$ is a $k$-node Let $s$ has vertices
$V_{s}=\\{u_{1},\ldots,u_{k}\\}$. Recall that a $k$-node shares all its
vertices with all its neighbors. Let the children of $s$ in $T$ be
$s_{1},\ldots,s_{r}$. Let the subgraph corresponding to the subtree rooted at
$s$ be $H$ and those at its children be $H_{1},\ldots,H_{r}$. In order to
determine $v_{H}^{(S)}$, we need to know if there is a matching in $H$ that
leaves exactly the vertices in $S$ unmatched. This holds if and only if the
vertices in $S$ are not matched in any of the $H_{j}$’s, and each vertex in
$V_{s}\setminus S$ is matched in exactly one of the $H_{j}$’s. In other words,
we need to determine if there is a partition $T_{1},T_{2},...,T_{r}$ of
$V_{s}\setminus S$, such that $H_{j}$ has a matching in which precisely
$V_{s}\setminus T_{j}$ is unmatched. That is, $v_{H_{j}}^{(V_{s}\setminus
T_{j})}=1$ for all $1\leq j\leq r$. More formally,
$\displaystyle
v_{H}^{(S)}=\bigvee_{\begin{subarray}{c}T_{1},\ldots,T_{r}\subseteq{V_{s}\setminus
S}:\\\ \forall j\neq j^{\prime}\in[r]T_{j}\cap T_{j^{\prime}}=\emptyset:\\\
\cup_{j\in[r]}{T_{j}}=V_{s}\setminus
S\end{subarray}}\bigwedge_{j\in[r]}{v_{H_{j}}^{(V_{s}\setminus T_{j})}}$
$\displaystyle=$
$\displaystyle\bigvee_{\emptyset=U_{0}\subseteq\ldots\subseteq
U_{r}={V_{s}\setminus
S}}\bigwedge_{j\in[r]}{v_{H_{j}}^{(V_{s}\setminus(U_{j}\setminus U_{j-1}))}}$
(1)
where, the second equality follows by defining $U_{0}=\emptyset$ and
$U_{i}=\cup_{j\in[i]}{T_{j}}$ for $i\in[r]$. The size of the above DNF formula
depends on $r$ which is not a constant hence the straightforward
implementation of the above computation would not be in L. However, consider a
conjunct in the big disjunction in the second line above. The $j^{\mbox{th}}$
factor of this conjunct depends only on $U_{j}$ and $U_{j-1}$, each of which
can be represented by a constant number ($=2^{k}$) of bits. Thus, we can
iteratively extend $U_{j-1}$ in all possible ways to $U_{j}$ and use the bit
indexed by $V_{s}\setminus(U_{j}\setminus U_{j-1})$ in the vector for the
child. How to obtain the vector of the child within a log-space bound is
detailed in the full version.
Case $2$: $s$ is a $k+1$ node The procedure is slightly more complex in this
case. Let $s$ have vertices $\\{u_{1},\ldots,u_{k+1}\\}$. Let the subgraph
corresponding to the subtree rooted at $s$ be $H$. Let $s_{1},\ldots,s_{r}$ be
the children of $s$, with corresponding subgraphs $H_{1},\ldots,H_{r}$. Note
that $s$ may share a different subset of $k$ vertices with each of its
children and with its parent. Let the vertices $s$ shares with its parent be
$\\{u_{1},\ldots,u_{k}\\}$. Then its matching vector is indexed by the subsets
of $\\{u_{1},\ldots,u_{k}\\}$, and moreover, $u_{k+1}$ should always be
matched in $H$. To compute $\vec{v}_{H}$, we first extend the matching vectors
of each of its children and make a $2^{k+1}$ dimensional vector $\vec{w}_{H}$.
The matching vector $\vec{v}_{H_{j}}$ of a child $s_{j}$ of $s$ is extended to
the new vector $\vec{w}_{H_{j}}$ as follows: Let $s_{j}$ contain
$\\{u_{1},\ldots,u_{k}\\}$. We consider an entry $v_{H_{j}}^{(S)}$ of
$\vec{v}_{H_{j}}$. The vector $\vec{w}_{H_{j}}$ has two entries corresponding
to it.
$w_{H_{j}}^{(S\cup\\{u_{k+1}\\})}=v_{H_{j}}^{(S)},\qquad
w_{H_{j}}^{(S)}=\bigvee_{\begin{subarray}{c}p\in[k],u_{p}\notin S,\\\
(u_{k+1},v_{p})\in E\end{subarray}}u_{H_{j}}^{(S\cup\\{u_{p}\\})}$
These new vectors of each of the children can be composed similar to that in
the previous case to get $\vec{w}_{H}$. To get $\vec{v}_{H}$, we remove the
$2^{k}$ entries from $\vec{w}_{H}$ which are indexed on subsets containing
$u_{k+1}$. This vector is passed on to the parent of $s$. The complexity
analysis, and a proof of ($2$) appears in the full version.
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|
arxiv-papers
| 2009-12-23T10:58:08 |
2024-09-04T02:49:07.219937
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bireswar Das, Samir Datta, Prajakta Nimbhorkar",
"submitter": "Prajakta Nimbhorkar",
"url": "https://arxiv.org/abs/0912.4602"
}
|
0912.4630
|
010006 2009 A. Goñi 010006
Trapping of hot electron behavior by trap centers located in buffer layer of a
wurtzite phase GaN MESFET has been simulated using an ensemble Monte Carlo
simulation. The results of the simulation show that the trap centers are
responsible for current collapse in GaN MESFET at low temperatures. These
electrical traps degrade the performance of the device at low temperature. On
the opposite, a light-induced increase in the trap-limited drain current,
results from the photoionization of trapped carriers and their return to the
channel under the influence of the built in electric field associated with the
trapped charge distribution. The simulated device geometries and doping are
matched to the nominal parameters described for the experimental structures as
closely as possible, and the predicted drain current and other electrical
characteristics for the simulated device including trapping center effects
show close agreement with the available experimental data.
# Light effect in photoionization of traps in GaN MESFETs
H. Arabshahi [inst1] A. Binesh[inst2] E-mail: arabshahi@um.ac.ir
(10 November 2009; 26 November 2009)
††volume: 1
99 inst1 Department of Physics, Ferdowsi University of Mashhad, P.O. Box
91775-1436, Mashhad, Iran. inst2 Department of Physics, Payam-e-Nour
University, Fariman, Iran.
## 1 Introduction
GaN has become an attractive material for power transistors [1-3] due to its
wide band gap, high breakdown electric field strength, and high thermal
conductivity. It also has a relatively high electron saturation drift velocity
and low relative permitivity, implying potential for high frequency
performance. However, set against the virtues of the material are the
disadvantages associated with material quality. GaN substrates are not readily
available and the lattice mismatch of GaN to the different substrate materials
commonly used means that layers typically contain between 108 and 1010
threading dislocations per cm2. Further, several types of electron traps occur
in the device layers and have a significant effect on GaN devices.
In the search for greater power and speed performance, the consideration of
different aspects that severely limit the output power of GaN FETs must be
accounted for. It is found that presence of trapping centers in the GaN
material is the most important phenomenon which can effect on current collapse
in output drain current of GaN MESFET. This effect was recently experimentally
investigated in GaN MESFET and was observed that the excess charge associated
with the trapped electrons produces a depletion region in the conducting
channel which results in a severe reduction in drain current [4]. The effect
can be reversed by librating trapped electrons either thermally by emission at
elevated temperatures or optically by photoionization. There have been several
experimental studies of the effect of trapping levels on current collapse in
GaN MESFET. For example, Klein et al. [5-6] measured photoionization
spectroscopy of traps in GaN MESFET transistors and calculated that the
current collapse resulted from charge trapping in the buffer layer. Binari et
al. [7] observed decreases in the drain current of a GaN FET corresponding to
the deep trap centers located at 1.8 and 2.85 eV.
In this work, we report a Monte Carlo simulation which is used to model
electron transport in wurtzite GaN MESFET including a trapping centers effect.
This model is based upon the fact that since optical effect can emit the
trapped electrons that are responsible for current collapse, the incident
light wavelength dependence of this effect should reflect the influence of
trap centers on hot electron transport properties in this device. This article
is organized as follows. Details of the device fabrication and trapping model
which is used in the simulated device are presented in section 2, and the
results from the simulation carried out on the device are interpreted in
section 3.
## 2 Model, device and simulations
An ensemble Monte Carlo simulation has been carried out to simulate the
electron transport properties in GaN MESFET. The method simulates the motion
of charge carriers through the device by following the progress of $\rm
10^{4}$ superparticles. These particles are propagated classically between
collisions according to their velocity, effective mass and the prevailing
field. The selection of the propagation time, scattering mechanism and other
related quantities, is achieved by generating random numbers and using these
numbers to select, for example, a scattering mechanism. Our self-consistent
Monte Carlo simulation was performed using an analytical band structure model
consisting of five non-parabolic ellipsoidal valleys. The scattering
mechanisms considered for the model are acoustic and polar optical phonon,
ionized impurity, piezoelectric and nonequivalent intervalley scattering. The
nonequivalent intervalley scattering is between the $\Gamma_{1}$,
$\Gamma_{3}$, U, M and K points.
The parameters used for the present Monte Carlo simulations for wurtzite GaN
are the same as those used by Arabshahi for MESFET transistors [8-9].
Figure 1: (a) Cross section of wurtzite GaN MESFET structure which we have
chosen in our simulation. Source and drain contacts have low resistance ohmic
contacts, while the gate contact forms a Schottky barrier between the metal
and the semiconductor epilayer, (b) The instantaneous distribution of $\rm
10^{4}$ particles at steady forward bias (drain voltage 50 V, gate voltage
$-1$ V), superimposed on the mesh. Note that in the simulation there are two
types of superparticles. The mobile particles which describe unbound electron
flow through the device and trapping center particles which are fixed at the
center of each electric field cell (in this case in the buffer layer only).
The ellipse represents a trap center which is fixed at the center of an
electric field cell and occupied by some mobile charges.
The device structure illustrated in figure 1.a is used in all simulations. The
overall device length is 3.3 $\mu$m in the $\it x$-direction and the device
has a 0.3 $\mu$m gate length and 0.5 $\mu$m source and drain length. The
source and drain have ohmic contacts and the gate is in Shottky contact in 1
eV to reperesent the contact potential at the Au/Pt. The source and drain
regions are doped to $\rm 5\times 10^{23}$ $\rm m^{-3}$ and the top and down
buffer layers are doped to $\rm 2\times 10^{23}$ $\rm m^{-3}$ and $\rm 1\times
10^{22}$ $\rm m^{-3}$, respectively. The effective source to gate and gate to
drain separation are 0.8 $\mu$m and 1.2 $\mu$m, respectively. The large
dimensions of the device need a long simulation time to ensure convergence of
the simulator. The device is simulated at room temperature and 420 K.
In the interests of simplicity it is assumed that there is just a single trap
with associated energy level $E_{T}$ in all or just part of the device.
Further, it is assumed that only electrons may be captured from the conduction
band by the trap centers, which have a capture cross-section $\sigma_{n}$ and
are neutral when unoccupied, and may only be emitted from an occupied center
to the conduction band. We use the standard model of carrier trapping and
emission [9-10].
For including trapping center effects, the following assumption has been
considered. The superparticles in the ensemble Monte Carlo simulation are
assumed to be of two types. There are mobile particles that represent unbound
electrons throughout the device. However, the particles may also undergo
spontaneous capture by the trap centers distributed in the device. The other
type of superparticles are trapping centers that are fixed at the center of
each mesh cell. As illustrated in figure 1.b, each trap center has the
capacity to trap a finite amount of mobile electronic charge from particles
that are in its vicinity and reside in the lowest conduction band valley. The
vicinity is defined as exactly the area covered by the electric field mesh
cell. The finite capacity of the trapping center in each cell of a specific
region in the device is set by a density parameter in the simulation
programme. The simulation itself is carried out by the following sequence of
events. First, the device is initialized with a specific trap which is
characterized by its density as a function of position, a trap energy level
and a capture cross-section. Then at a specific gate bias the source-drain
voltage is applied.
Some of the mobile charges passing from the source to the drain in each
timestep can be trapped by the centers with a probability which is dependent
on the trap cross-section and particle velocity in the cell occupied at the
relevant time t. The quantity of charge that is captured from a passing mobile
particle is the product of this probability and the charge on it. This charge
is deducted from the charge of the mobile particle and added to the fixed
charge of the trap center. The emission of charge is simulated using the
emission probability. Any charge emitted from a trap center is evenly
distributed to all mobile particles in the same field cell. Such capture and
emission simulations are performed for the entire mesh in the device and
information on the ensemble of particles is recorded in the usual way.
## 3 Results
The application of a high drain-source voltage causes hot electrons to be
injected into the buffer layer where they are trapped by trap centers. The
trapped electrons produce a depletion region in the channel of the GaN MESFET
which tends to pinch off the device and reduce the drain current. This effect
can be reversed by any factor which substantially increases the electron
emission rate from the trapped centers, such as the elevated temperatures
considered previously. Here we consider the effect of exposure to light
[11-13].
There have been several experimental investigations of the influence of light
on the device characteristics. Binari et al. [6] were the first to
experimentally study the current collapse in GaN MESFETs as a function of
temperature and illumination. They showed that the photoionization of trapped
electrons in the high-resistivity GaN layers and the subsequent return of
these electrons to the conduction band could reverse the drain current
collapse. Their measurements were carried out as a function of incident light
wavelength with values in the range of 380 nm to 720 nm, corresponding to
photon energies up to 3.25 eV which is close to the GaN band gap. Their
results show that when the photon energy exceeds the trap energy, the
electrons are quickly emitted and a normal set of drain characteristics is
observed.
To examine the photoionization effect in our simulations, the thermal emission
rate $e^{t}_{n}$ was changed to $e^{t}_{n}+e^{o}_{n}$, where
$e^{o}_{n}\sim\sigma^{o}_{n}\Phi$ is the optical emission rate, with
$\sigma^{o}_{n}$, the optical capture cross-section and $\Phi$ the photon flux
density given by
$\Phi=\frac{I}{h\nu}=\frac{I\lambda}{hc}$ (1)
where I is the light intensity, $\nu$ is the radiation frequency and $\lambda$
is the incident light wavelength.
Figure 2: I-V characteristics of a GaN MESFET under optical and thermal
emission of trapped electrons (solid curve) and thermal emission of trapped
electrons (dashed curve) at two different temperatures. (a) At $T=300$ K with
trap centers at 1.8 eV and illuminated with a photon energy of 2.07 eV. (b) At
$T=420$ K with trap centers at 2.85 eV and illuminated with a photon energy of
3.1 eV.
Our modeling of photoionization effects in GaN MESFETs is based on parameters
used by Binari and Klein [5-7]. The simulations were all carried out for two
different deep trap centers, both with a concentration of $\rm 10^{22}$ $\rm
m^{-3}$, and with photoionization threshold energies at 1.8 and 2.85 eV and
capture cross-sections of $\rm 6\times 10^{-21}$ $\rm m^{2}$ and $\rm
2.8\times 10^{-19}$ $\rm m^{2}$, respectively. A fixed incident light
intensity of 5 $\rm Wm^{-2}$ at photon energies of 2.07 eV and 3.1 eV is used.
The simulations have been performed at a sufficiently high temperature (420
K), for both thermal and optical emission, to be significant as well as at
room temperature.
Figure 2a illustrates the effect on the drain current characteristics of
exposure of the device to light at room temperature. The GaN MESFET has a deep
trap center at 1.8 eV and is illuminated at a photon energy of 2.07 eV. It can
be seen that in the light the I-V curves generally exhibit a larger drain
current, especially at higher drain voltages, reflecting the fact that the
density of trapped electrons is much lower.
Simulations have also been performed at 420 K for a device with deep level
traps at 2.85 eV. The simulation results in figure 2b for illumination of a
photon energy of 3.1 eV are compared with the collapsed I-V curves in the
absence of light. Comparison of figures 2a and 2b shows that the currents are
generally higher at 420 K and that the light has less effect at the highest
temperature.
## 4 Conclusions
The dependence upon light intensity (exposure) of the reversal of current
collapse was simulated in a GaN MESFET for a single tapping center. Traps in
the simulated device produce a serious reduction in the drain current and
consequently the output power of GaN MESFET. The drain current behavior as a
function of illumination with photon energy was also studied. Our results show
that as the temperature and photon energy are increased, the collapsed drain
current curve moves up toward the non-collapsed curve due to more emission of
trapped electrons.
###### Acknowledgements.
The authors wish to thank M. G. Paeezi for the helpful comments and critical
reading of the manuscript.
## References
* [1] B Gil, Group-III Nitride Semiconductor Compounds, Oxford Science Pub. (1998).
* [2] M A Khan, M S Shur, AlGaN/GaN Metal Oxide Semiconductor Heterostructure Field Effect Transistor, Mater. Sci. Eng. B 42, 69 (1997).
* [3] P B Klein, S C Binari, J A Freitas, A E Wickenden, Photoionization spectroscopy of traps in GaN metal-semiconductor field-effect transistors, J. Appl. Phys. 88, 2843 (2000).
* [4] M A Khan, M S Shur, Q C Chen, J N Kuznia, Low frequency noise in GaN metal semiconductor and metal oxide semiconductor field effect transistors, Electron. Lett. 30, 2175 (1994).
* [5] P B Klein, S C Binari, J A Freitas, A E Wickenden, Observation of deep traps responsible for current collapse in GaN metal-semiconductor field-effect transistors, J. Appl. Phys. 88, 2843 (2000).
* [6] P B Klein, J A Freitas, S C Binari, A E Wickenden, AlGaN/GaN heterostructure field-effect transistor model including thermal effects, Appl. Phys. Lett. 75, 4016 (1999).
* [7] S C Binari, W Kruppa, H B Dietrich, G Kelner, A E Wickenden, J A Freitas, Trapping effects and microwave power performance in AlGaN/GaN HEMTs, Solid State Electron. 41, 1549 (1997).
* [8] H Arabshahi, Monte Carlo simulations of electron transport in Wurtzite phase GaN MESFET including trapping effect, Modern Phys. Lett. B 20, 787 (2006).
* [9] H Arabshahi, The frequency response and effect of trap parameters on the characteristic of GaN MESFETs, The Journal of Damghan University of Basic Sciences 1, 45 (2007).
* [10] S Trassaert, B Boudart, C Gaquiere, Investigation of traps induced current collapse in GaN devices, a1404 ORSAY France, 127 (1999).
* [11] A Kastalsky, S Luryi, A C Gossard, W K Chan, Switching in NERFET circuits, IEEE Electron Device Lett. 6, 347 (1985).
* [12] J C Inkson, Deep impurities in semiconductors. II. The optical cross section, J. Phys. C: Solid State Phys. 14, 1093 (1981).
* [13] D V Lang, R A Logan, M Jaros, Monte Carlo evaluations of degeneracy and interface roughness effects, Phys. Rev. B 19, 1015 (1979).
|
arxiv-papers
| 2009-12-23T13:11:44 |
2024-09-04T02:49:07.227550
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Arabshahi, A. Binesh",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/0912.4630"
}
|
0912.4636
|
# Delta shock wave interactions via wave front tracking method
Nebojša Dedović and Marko Nedeljkov
###### Abstract.
In this paper we discuss delta shock interaction problem for a pressureless
gas dynamics system with two different ways of approaching the subject. The
first one is by using shadow wave solution concept. The result of two delta
shock interactions is delta shock with non-constant speed in a general case.
The second one is by perturbing the system with a small pressure term. The
obtained perturbed system is strictly hyperbolic and its Riemann problem is
solvable. We compare a limit of a numerical wave front tracking results as
small pressure term vanishes with the shadow wave solution.
Key words: weighted shadow waves, delta shock waves, wave front tracking,
Riemann problem, interactions
## 1\. Introduction
Consider the one-dimensional Euler gas dynamics system given by
$\begin{array}[]{rcl}\partial_{t}\rho+\partial_{x}(\rho u)&=&0\\\
\displaystyle\partial_{t}(\rho u)+\partial_{x}(\rho
u^{2}+p(\varepsilon,\rho))&=&0,\end{array}$ (1)
where $\rho$ is the density, $m=\rho\,u$ is the momentum,
$p(\varepsilon,\rho)=\varepsilon\,p_{0}(\rho)$ is the scalar pressure,
$\varepsilon<<1$ and $p_{0}(\rho)=\rho^{\gamma}/\gamma\,$. Taking
$\varepsilon\to 0$ in (1), we obtain the pressureless gas dynamics model (PGD
model in the rest of the paper), also called sticky particles model (in [13])
$\begin{array}[]{rcl}\partial_{t}\rho+\partial_{x}(\rho u)&=&0\\\
\displaystyle\partial_{t}(\rho u)+\partial_{x}(\rho
u^{2})&=&0,\;(x,t)\in\mathbb{R}\times\mathbb{R}_{+}\,.\end{array}$ (2)
System (1) can be considered as a perturbation of system (2) which is weakly
hyperbolic with a double eigenvalue $\lambda_{1}=\lambda_{2}=u$. All entropy
pairs $(\eta,q)$ with a semiconvex function $\eta$ are given by $\eta:=\rho
S(u)$, $q:=\rho\,uS(u)$, where $S^{\prime\prime}\geq 0$ (the entropy function
$\eta$ is semi-convex with respect to the variable $(\rho,\rho u)$). The
Riemann problem
$\rho(x,0)=\left\\{\begin{array}[]{cc}\rho_{0},&x<0\,,\\\
\rho_{1},&x>0\,,\end{array}\right.\;,\hskip
28.45274ptu(x,0)=\left\\{\begin{array}[]{cc}u_{0},&x<0\,,\\\
u_{1},&x>0\,,\end{array}\right.$ (3)
has a classical entropy solution consisting of two contact discontinuities
connected with the vacuum state ($\rho=0$) if $u_{0}\leq u_{1}$:
$(\rho(x,t),u(x,t))=\begin{cases}(\rho_{0},u_{0}),&x<u_{0}t,\\\
(0,\psi(x/t)),&u_{0}t<x<u_{1}t,\\\ (\rho_{1},u_{1}),&x>u_{1}t,\end{cases}$
where $\psi(y)=y$. We are now turning to the case $u_{0}>u_{1}$ when there is
no classical solution to the Riemann problem (2, 3).
Throughout this paper, the following constants will be fixed:
$\gamma=1+2\varepsilon,\;0<\varepsilon<\frac{1}{2},\;\kappa=\frac{\sqrt{\varepsilon}}{\sqrt{\gamma}}\;\;{\rm
and}\;\;p=\kappa^{2}\rho^{\gamma}\;.$ (4)
## 2\. Elementary waves of the perturbed system
The eigenvalues of system (1) are
$\begin{array}[]{l}\displaystyle\lambda_{1}=u-\kappa\sqrt{\gamma}\rho^{\frac{\gamma-1}{2}},\\\
\\\
\displaystyle\lambda_{2}=u+\kappa\sqrt{\gamma}\rho^{\frac{\gamma-1}{2}},\end{array}$
(5)
and the corresponding eigenvectors are
$\begin{array}[]{l}\displaystyle
r_{1}=(-1,-u+\kappa\sqrt{\gamma}\rho^{\frac{\gamma-1}{2}})^{T},\\\ \\\
\displaystyle
r_{2}=(1,u+\kappa\sqrt{\gamma}\rho^{\frac{\gamma-1}{2}})^{T}\,.\end{array}$
(6)
We have chosen an orientation such that $\nabla\lambda_{i}\cdot
r_{i}>0,\;i=1,2$, since both fields are genuinely nonlinear. The corresponding
Riemann invariants of system (1) are
$\begin{array}[]{c}s=u+\frac{\kappa\sqrt{\gamma}}{\varepsilon}(\rho^{\varepsilon}-1):\mbox{
1-invariant, and}\\\
r=u-\frac{\kappa\sqrt{\gamma}}{\varepsilon}(\rho^{\varepsilon}-1):\mbox{
2-invariant}\,.\end{array}$ (7)
The rarefaction curves through the point $(\rho_{0},u_{0})$ are given by
$\begin{array}[]{c}u-u_{0}=-\frac{\kappa\sqrt{\gamma}}{\varepsilon}(\rho^{\varepsilon}-\rho^{\varepsilon}_{0})\;,\;\;0\leq\rho\leq\rho_{0}:\mbox{
1-rarefaction curve},\\\
u-u_{0}=\;\;\;\frac{\kappa\sqrt{\gamma}}{\varepsilon}(\rho^{\varepsilon}-\rho^{\varepsilon}_{0})\;,\;\;\rho\geq\rho_{0}:\mbox{
2-rarefaction curve},\end{array}$ (8)
while the shock curves through the point $(\rho_{0},u_{0})$ are given by
$u-u_{0}=-\kappa\sqrt{\frac{\rho^{\gamma}-\rho^{\gamma}_{0}}{\rho_{0}\rho(\rho-\rho_{0})}}\;(\rho-\rho_{0}),\;\;\;\rho>\rho_{0}:\mbox{
1-shock curve},$ (9)
and
$u-u_{0}=\kappa\sqrt{\frac{\rho^{\gamma}-\rho^{\gamma}_{0}}{\rho_{0}\rho(\rho-\rho_{0})}}\;(\rho-\rho_{0}),\;\;\;0<\rho<\rho_{0}:\mbox{
2-shock curve}.$ (10)
With the Riemann invariants, shock curves starting from the point
$(r_{0},s_{0})$ are
$S_{1}:\;\;\;\left\\{\begin{array}[]{l}\displaystyle
r_{0}-r=\kappa\rho_{0}^{\varepsilon}\left(\sqrt{\frac{(\alpha-1)(\alpha^{\gamma}-1)}{\alpha}}+\sqrt{\gamma}\frac{\alpha^{\varepsilon}-1}{\varepsilon}\right),\\\
\displaystyle
s_{0}-s=\kappa\rho_{0}^{\varepsilon}\left(\sqrt{\frac{(\alpha-1)(\alpha^{\gamma}-1)}{\alpha}}-\sqrt{\gamma}\frac{\alpha^{\varepsilon}-1}{\varepsilon}\right),\end{array}\right.$
(11)
where $r_{0}=r(\rho_{0},u_{0})$, $s_{0}=s(\rho_{0},u_{0})$ and
$\alpha=\rho/\rho_{0}\geq 1$, and
$S_{2}:\;\;\;\left\\{\begin{array}[]{l}\displaystyle
s_{0}-s=\kappa\rho_{0}^{\varepsilon}\left(\sqrt{\frac{(1-\alpha)(1-\alpha^{\gamma})}{\alpha}}+\sqrt{\gamma}\frac{1-\alpha^{\varepsilon}}{\varepsilon}\right),\\\
\displaystyle
r_{0}-r=\kappa\rho_{0}^{\varepsilon}\left(\sqrt{\frac{(1-\alpha)(1-\alpha^{\gamma})}{\alpha}}-\sqrt{\gamma}\frac{1-\alpha^{\varepsilon}}{\varepsilon}\right),\end{array}\right.$
(12)
where $r_{0}=r(\rho_{0},u_{0})$, $s_{0}=s(\rho_{0},u_{0})$ and
$0<\alpha=\rho/\rho_{0}\leq 1$. The corresponding rarefaction curves are given
by
$R_{1}:\;\;\;r\geq r_{0},\;s=s_{0},$ (13)
and
$R_{2}:\;\;\;s\geq s_{0},\;r=r_{0}.$ (14)
It is clear that from (11, 12) we have that $r_{0}-r\geq s_{0}-s$ holds for
$S_{1}$ curve and $s_{0}-s\geq r_{0}-r$ holds for $S_{2}$ curve, respectively.
The Riemann problem for system (1) with initial data (3) was solved by Riemann
[12], and the result is summarized in the following theorem (the proof can be
found in Courant-Friedrichs [4] and Smoller [14]).
###### Theorem 2.1.
[1] Consider system (1) with initial data (3). Suppose that
$u_{1}-u_{0}<\frac{\kappa\sqrt{\gamma}}{\varepsilon}(\rho^{\varepsilon}_{1}+\rho^{\varepsilon}_{0})$,
or equivalently $s_{0}-r_{1}>-\frac{2\kappa\sqrt{\gamma}}{\varepsilon}$. Then
there exists a unique solution composed of constant states
$(\rho_{0},u_{0})=(r_{0},s_{0})$, $(\rho_{m},u_{m})=(r_{m},s_{m})$ and
$(\rho_{1},u_{1})=(r_{1},s_{1})$ separated by centered rarefaction or shock
waves satisfying the following estimates:
$\begin{array}[]{l}r(x,t)=r(\rho(x,t),u(x,t))\geq\min\\{r_{0},r_{1}\\},\\\
s(x,t)=s(\rho(x,t),u(x,t))\leq\max\\{s_{0},s_{1}\\}.\end{array}$ (15)
The amplitude of the waves is denoted by
$\begin{array}[]{c}\beta:=r_{m}-r_{0}\;\;:\mbox{ amplitude of an 1-wave},\\\
\chi:=s_{1}-s_{m}\;\;:\mbox{ amplitude of a 2-wave}.\end{array}$ (16)
Here $\beta,\chi\geq 0$ for centered rarefaction waves and $\beta,\chi<0$ for
shock waves; absolute values $|\beta|$, $|\chi|$ are called strengths of
$\beta$ and $\chi$, respectively.
We shall use that notation throughout the rest of the paper.
## 3\. Local Interactions Estimates
Our first task is to obtain a sharp estimate of wave strengths with respect to
$\varepsilon$ as much as possible. In order to do that, we shall present some
assertions from [11] together with modified proofs, since certain changes in
estimates will be useful for our investigation.
###### Theorem 3.1.
[11] The shock curve $S_{1}$ starting at the point $(r_{0},s_{0})$ is given by
$\displaystyle
s_{0}-s=g_{1}(r_{0}-r,\rho_{0})=\int_{0}^{r_{0}-r}\;h_{1}(\alpha)|_{\alpha=\alpha_{1}(\beta/\kappa\rho^{\varepsilon}_{0})}\;d\beta,\;\;r<r_{0},$
(17)
where $0\leq g^{\prime}_{1}(\beta,\rho_{0})<1$ and
$g^{\prime\prime}_{1}(\beta,\rho_{0})\geq 0$111The primes denote
differentiation with respect to the first argument.. The shock curve $S_{2}$
starting at the point $(r_{0},s_{0})$ is
$\displaystyle
r_{0}-r=g_{2}(s_{0}-s,\rho_{0})=\int_{0}^{s_{0}-s}\;h_{2}(\alpha)|_{\alpha=\alpha_{2}(\chi/\kappa\rho^{\varepsilon}_{0})}\;d\chi,\;\;s<s_{0},$
(18)
where $0\leq g^{\prime}_{2}(\chi,\rho_{0})<1$ and
$g^{\prime\prime}_{2}(\chi,\rho_{0})\geq 0$.
Proof. We shall repeat the proof from [11] in order to fix the notation for
the rest of the paper. Relation $s_{0}-s=g_{1}(r_{0}-r,\rho_{0})$ implies
$\frac{\partial(s_{0}-s)}{\partial\alpha}=\frac{\partial
g_{1}(r_{0}-r,\rho_{0})}{\partial(r_{0}-r)}\cdot\frac{\partial(r_{0}-r)}{\partial\alpha},\;{\rm
so}\;\;\frac{\partial(s_{0}-s)/\partial\alpha}{\partial(r_{0}-r)/\partial\alpha}=g^{\prime}_{1}(\beta,\rho_{0})\,.$
(19)
If
$h_{1}(\alpha)=\frac{\partial(s_{0}-s)/\partial\alpha}{\partial(r_{0}-r)/\partial\alpha}\,,$
then one can easily see that
$\displaystyle h_{1}(\alpha)=\left(\frac{Y-1}{Y+1}\right)^{2}\;\;\;\mbox{ with
}\;\;Y=\sqrt{\frac{\gamma\alpha^{\gamma}(\alpha-1)}{\alpha^{\gamma}-1}}\;,\;\;\mbox{
for }\;\;\alpha>1\;.$ (20)
From the first equation in (11), we have
$\frac{\beta}{\kappa\rho^{\varepsilon}_{0}}=\sqrt{\frac{(\alpha-1)(\alpha^{\gamma}-1)}{\alpha}}+\sqrt{\gamma}\;\frac{\alpha^{\varepsilon}-1}{\varepsilon}=:f(\alpha)\;.$
(21)
Therefore
$\begin{array}[]{rl}\displaystyle
f^{\prime}(\alpha)>&\displaystyle\frac{1}{2}\sqrt{\frac{\alpha}{(\alpha-1)(\alpha^{\gamma}-1)}}\cdot\frac{\alpha^{\gamma}-1}{\alpha^{2}}+\sqrt{\gamma}\;\alpha^{\varepsilon-1}>0\end{array}$
since $\alpha^{\gamma}>1$ for $\alpha>1$ and $\gamma>1$. Using the fact that
$f^{\prime}(\alpha)>0$ and (21) the Implicit Function Theorem yields that
there exists $\alpha=\alpha_{1}(\beta/\kappa\rho^{\varepsilon}_{0})$ such that
$g_{1}(r_{0}-r,\rho_{0})=\int_{0}^{r_{0}-r}\;h_{1}(\alpha)|_{\alpha=\alpha_{1}(\beta/\kappa\rho_{0}^{\varepsilon})}\;d\beta\;.$
(22)
Since $g^{\prime}_{1}(\beta,\rho_{0})=h_{1}(\alpha)$, $\displaystyle
g^{\prime\prime}_{1}(\beta,\rho_{0})=h^{\prime}_{1}(\alpha)\cdot\frac{d\alpha}{d\beta}$
and $\displaystyle\frac{d\beta}{d\alpha}=\kappa\rho_{0}^{\varepsilon}\cdot
f^{\prime}(\alpha)>0$ it remains to prove that $0\leq h_{1}(\alpha)<1$ and
$0\leq h^{\prime}_{1}(\alpha)$. From (20) we have
$0\leq
h_{1}(\alpha)=\left(\frac{Y-1}{Y+1}\right)^{2}<\left(\frac{Y+1}{Y+1}\right)^{2}=1\;,$
and
$0\leq h^{\prime}_{1}(\alpha)=4\cdot\frac{Y-1}{(Y+1)^{3}}\cdot Y^{\prime}\;,$
(23)
since $Y\geq 1$ and $Y^{\prime}\geq 0$. The second part of the theorem can be
proved using the same technique. $\Box$
###### Lemma 3.2.
Let $\rho_{0}<\rho_{1}$ and
$\beta/\kappa\rho^{\varepsilon}_{1}<\theta<\beta/\kappa\rho^{\varepsilon}_{0}$.
Then
$\frac{d\alpha}{d\theta}=\frac{1}{f^{\prime}(\alpha)}=\frac{2Y}{\sqrt{\gamma}\alpha^{\frac{\gamma-3}{2}}(1+Y)^{2}}\,.$
(24)
We would need an estimate of the difference of Riemann invariants across two
shock waves which is more precise than the one in [11]. It is provided by the
following theorem.
###### Theorem 3.3.
Let $0<\varepsilon<\frac{1}{2}$, $s_{0}<s_{1}$, and take two $S_{1}$ curves
originating at the points $(r_{0},s_{0})=(\rho_{0},u_{0})$ and
$(r_{0},s_{1})=(\rho_{1},u_{1})$, which are continued to the points $(r,s)$
and $(r,s_{2})$, respectively. Then we have
$0\leq(s_{0}-s)-(s_{1}-s_{2})\leq
C_{*}\,\sqrt{\varepsilon}\,(r_{0}-r)\,(s_{1}-s_{0})\,,$ (25)
where $C_{*}$ is a constant independent of $\varepsilon$, $\rho_{0}$ and
$\rho_{1}$ .
Proof. Let $z^{0}=s_{0}-s$, $z^{1}=s_{1}-s_{2}$ and $w=r_{0}-r$ (look at the
diagram shown in Figure 1).
Figure 1. Two 1-shock wave curves in $r-s$ plane.
By Theorem 3.1 and the Mean Value Theorem we know that for
$\rho_{1}>\rho_{0}$, there exists $\theta$ such that
$\begin{array}[]{rl}z^{0}-z^{1}=&\displaystyle\int_{0}^{w}\frac{dh_{1}(\alpha)}{d\alpha}\Big{|}_{\alpha=\alpha(\theta)}\cdot\alpha^{\prime}(\theta)\left(\frac{\beta}{\kappa\rho_{0}^{\varepsilon}}-\frac{\beta}{\kappa\rho_{1}^{\varepsilon}}\right)\;d\beta\;,\end{array}$
(26)
where
$\theta\in\left(\frac{\beta}{\kappa\rho^{\varepsilon}_{1}},\frac{\beta}{\kappa\rho^{\varepsilon}_{0}}\right)$.
The definitions of $h_{1}$ and $\alpha$ imply
$\frac{dh_{1}(\alpha)}{d\alpha}\geq
0,\;\;\;\frac{d\alpha(\theta)}{d\theta}\geq 0\;\;\;\mbox{ and
}\;\;\;\frac{\beta}{\kappa\rho_{0}^{\varepsilon}}-\frac{\beta}{\kappa\rho_{1}^{\varepsilon}}\geq
0$
so $z^{0}-z^{1}\geq 0$ for $\rho_{1}>\rho_{0}$. We need to estimate the
integrand in (26). By (20), we have
$\begin{array}[]{rl}\displaystyle\frac{dh_{1}(\alpha)}{d\alpha}\cdot\frac{d\alpha(\theta)}{d\theta}\leq&\displaystyle\frac{4(Y-1)(\gamma+1)\alpha^{\frac{1-\gamma}{2}}}{\sqrt{\gamma}\;(Y+1)^{3}}\;.\end{array}$
Thus,
$\begin{array}[]{rl}z^{0}-z^{1}\leq&\displaystyle\frac{4(\gamma+1)}{\kappa\sqrt{\gamma}\;\rho_{0}^{\varepsilon}\rho_{1}^{\varepsilon}}(\rho_{1}^{\varepsilon}-\rho_{0}^{\varepsilon})\int_{0}^{w}\beta\;\alpha^{\frac{1-\gamma}{2}}\Big{|}_{\alpha=\alpha(\theta)}\frac{Y-1}{(Y+1)^{3}}\;d\beta\end{array}$
(27)
and
$\begin{array}[]{rl}z^{0}-z^{1}\leq&\displaystyle\frac{4(\gamma+1)}{\kappa\sqrt{\gamma}\;\rho_{0}^{\varepsilon}\rho_{1}^{\varepsilon}}(\rho_{1}^{\varepsilon}-\rho_{0}^{\varepsilon})\int_{0}^{w}\beta\;\alpha^{-\frac{1+\gamma}{2}}\Big{|}_{\alpha=\alpha(\theta)}\;d\beta\;.\end{array}$
(28)
From Lemma 3.2 we know that $d\alpha/d\theta>0$ for
$\beta/\kappa\rho^{\varepsilon}_{1}<\theta<\beta/\kappa\rho^{\varepsilon}_{0}$.
Hence,
$\displaystyle\alpha\left(\frac{\beta}{\kappa\rho_{1}^{\varepsilon}}\right)\leq\alpha\left(\theta\right)\leq\alpha\left(\frac{\beta}{\kappa\rho_{0}^{\varepsilon}}\right).$
(29)
Moreover,
$\begin{array}[]{rl}\displaystyle\frac{\beta}{\kappa\rho^{\varepsilon}_{1}}=f(\alpha)\leq&\displaystyle
2\sqrt{\frac{(\alpha-1)\;\alpha^{\gamma}}{\alpha-1}}=2\alpha^{\gamma/2}\;,\mbox{
for
}\alpha=\alpha\left(\frac{\beta}{\kappa\rho_{1}^{\varepsilon}}\right)\;.\end{array}$
At this point, we use a majorization of $z^{0}-z^{1}$ different from the one
in [11] in order to obtain bounds for $C_{*}$ independent of $\varepsilon$. By
(29) and the above inequality we obtain
$\begin{array}[]{l}\displaystyle\frac{\beta}{2\kappa\rho^{\varepsilon}_{1}}\leq\left(\alpha\left(\frac{\beta}{\kappa\rho_{1}^{\varepsilon}}\right)\right)^{\gamma/2}\Rightarrow\left(\frac{\beta}{2\kappa\rho^{\varepsilon}_{1}}\right)^{-\frac{\gamma+1}{\gamma}}\geq\left(\alpha\left(\frac{\beta}{\kappa\rho^{\varepsilon}_{1}}\right)\right)^{-\frac{\gamma+1}{2}}\,,\;\;\rm{so}\\\
\\\
\displaystyle\min\left\\{1,\left(\frac{\beta}{2\kappa\rho^{\varepsilon}_{1}}\right)^{-\frac{\gamma+1}{\gamma}}\right\\}\geq\left(\alpha\left(\frac{\beta}{\kappa\rho^{\varepsilon}_{1}}\right)\right)^{-\frac{\gamma+1}{2}}\geq\left(\alpha(\theta)\right)^{-\frac{\gamma+1}{2}}\;,\end{array}$
(30)
for $\alpha\left(\frac{\beta}{\kappa\rho_{1}^{\varepsilon}}\right)\geq 1$ and
$\gamma>1$. Since $d\alpha/d\theta>0$, it follows by (28) that
$\begin{array}[]{rl}z^{0}-z^{1}\leq&\displaystyle\frac{4(\gamma+1)}{\kappa\sqrt{\gamma}\;\rho_{0}^{\varepsilon}\rho_{1}^{\varepsilon}}(\rho_{1}^{\varepsilon}-\rho_{0}^{\varepsilon})\int_{0}^{w}\beta\cdot\min\left\\{1,\left(\frac{\beta}{2\kappa\rho^{\varepsilon}_{1}}\right)^{-\frac{\gamma+1}{\gamma}}\right\\}\;d\beta\\\
&\\\
\leq&\displaystyle\frac{8(\gamma+1)}{\sqrt{\gamma}\;\rho_{0}^{\varepsilon}}\;(\rho_{1}^{\varepsilon}-\rho_{0}^{\varepsilon})\int_{0}^{w}\min\left\\{\frac{\beta}{2\kappa\rho^{\varepsilon}_{1}},\left(\frac{\beta}{2\kappa\rho^{\varepsilon}_{1}}\right)^{-\frac{1}{\gamma}}\right\\}\;d\beta\\\
&\\\
\leq&\displaystyle\frac{8(\gamma+1)}{\sqrt{\gamma}\;\rho_{0}^{\varepsilon}}\;(\rho_{1}^{\varepsilon}-\rho_{0}^{\varepsilon})\;w\,.\end{array}$
(31)
Using
$\rho^{\varepsilon}_{1}-\rho^{\varepsilon}_{0}=\frac{\varepsilon}{2\kappa\sqrt{\gamma}}(s_{1}-s_{0})$
and $\kappa\sqrt{\gamma}=\sqrt{\varepsilon}$ together with (31) we finally get
$\begin{array}[]{rl}z^{0}-z^{1}\leq&\displaystyle\frac{4(\gamma+1)}{\sqrt{\gamma}\;\rho_{0}^{\varepsilon}}\cdot\frac{\varepsilon}{\kappa\sqrt{\gamma}}\;(s_{1}-s_{0})\;w\\\
&\\\
=&\displaystyle\frac{4(\gamma+1)}{\sqrt{\gamma}\;\rho_{0}^{\varepsilon}}\cdot\sqrt{\varepsilon}\;(s_{1}-s_{0})\;w\,.\end{array}$
(32)
Suppose that $\rho$ is the first component of the solution of the Riemann
problem (1, 3) for $u_{0}>u_{1}$. Then Lemma 3.1 from [3] yields that for
small $\varepsilon>0$, there exists $C>0$ independent of $\varepsilon$, such
that $\displaystyle\rho\leq C/\varepsilon$. Now, using (32), if $\rho_{0}\sim
1/\varepsilon$ then $\rho_{0}^{\varepsilon}\sim 1$ as $\varepsilon\to 0$. For
$\varepsilon$ small enough we may write $\rho_{0}^{\varepsilon}\geq C_{1}$.
Thus, there exists a constant $C_{*}$ independent of $\kappa$, $\rho_{0}$,
$\rho_{1}$ and $\varepsilon$ such that
$z^{0}-z^{1}\leq C_{*}\,\sqrt{\varepsilon}\;(s_{1}-s_{0})\;w,$ (33)
holds. This completes the proof of the theorem. $\Box$
The theorem that follows can be proved in the same way.
###### Theorem 3.4.
Let $0<\varepsilon<\frac{1}{2}$, $r_{0}>r_{1}$, and take two $S_{2}$ curves
originating at the points $(r_{0},s_{0})=(\rho_{0},u_{0})$ and
$(r_{1},s_{0})=(\rho_{1},u_{1})$, which are continued to the points $(r,s)$
and $(r_{2},s)$, respectively. Then we have
$0\leq(r_{0}-r)-(r_{1}-r_{2})\leq
C_{**}\,\sqrt{\varepsilon}\,(s_{0}-s)\,(r_{0}-r_{1})\,,$ (34)
where $C_{**}$ is a constant independent of $\varepsilon$, $\rho_{0}$ and
$\rho_{1}$.
(A1) We shall use the following convention: $C_{*}$ denotes the maximum of the
con- stants $C_{*}$ and $C_{**}$ from Theorems 3.3 and 3.4, respectively.
In the following theorem $\beta$ and $\chi$ denote $S_{1}$ and $S_{2}$,
respectively, while $o$ and $\pi$ denote $R_{1}$ and $R_{2}$, respectively.
The prime is reserved for after interaction waves. (For example, the
interaction of $S_{2}$ and $S_{1}$ which produces $S_{1}$ and $S_{2}$ is
denoted by $\chi+\beta\rightarrow\beta^{\prime}+\chi^{\prime}$.)
###### Theorem 3.5.
If $0<\varepsilon<\frac{1}{2}$, then the following estimates are valid for the
corresponding interactions:
1. (1)
$S_{2}$ and $S_{1}$ interaction:
1. (a)
$\chi+\beta\rightarrow\beta^{\prime}+\chi^{\prime}$
$|\beta^{\prime}|\leq|\beta|+C_{*}\sqrt{\varepsilon}\,|\chi||\beta|,\;\;\;\;|\chi^{\prime}|\leq|\chi|+C_{*}\sqrt{\varepsilon}\,|\beta||\chi|$,
or
there exist $\eta,\xi$ such that
2. (b)
$\chi+\beta\rightarrow\beta^{\prime}+\chi^{\prime}$
$0\leq|\beta^{\prime}|=|\beta|-\xi,\;\;\;\;|\chi^{\prime}|\leq|\chi|+C_{*}\sqrt{\varepsilon}\,|\beta||\chi|+\eta$,
where $0\leq\eta\leq g^{\prime}_{1}(|\beta|,\rho_{0})\xi<\xi$, or
3. (c)
$\chi+\beta\rightarrow\beta^{\prime}+\chi^{\prime}$
$0\leq|\chi^{\prime}|=|\chi|-\xi,\;\;\;\;|\beta^{\prime}|\leq|\beta|+C_{*}\sqrt{\varepsilon}\,|\chi||\beta|+\eta$,
where $0\leq\eta\leq g^{\prime}_{1}(|\chi|,\rho_{0})\xi<\xi$ .
2. (2)
$S_{2}$ and $R_{1}$ (or $R_{2}$ and $S_{1}$) interaction:
1. (a)
$\chi+o\rightarrow o^{\prime}+\chi^{\prime}$
$|\chi^{\prime}|=|\chi|,\;\;\;\;|o^{\prime}|\leq|o|+C_{*}\sqrt{\varepsilon}\,|\chi||o|$
.
2. (b)
$\pi+\beta\rightarrow\beta^{\prime}+\pi^{\prime}$
$|\beta^{\prime}|=|\beta|,\;\;\;\;|\pi^{\prime}|\leq|\pi|+C_{*}\sqrt{\varepsilon}\,|\beta||\pi|$
.
3. (3)
$S_{2}$ and $S_{2}$ (or $S_{1}$ and $S_{1}$) interaction:
1. (a)
$\chi_{1}+\chi_{2}\rightarrow o^{\prime}+\chi^{\prime}:$
$|\chi^{\prime}|=|\chi_{1}|+|\chi_{2}|,\;\;\;\;|o^{\prime}|\leq|\chi_{1}|+|\chi_{2}|$
.
2. (b)
$\beta_{1}+\beta_{2}\rightarrow\beta^{\prime}+\pi^{\prime}:$
$|\beta^{\prime}|=|\beta_{1}|+|\beta_{2}|,\;\;\;\;|\pi^{\prime}|\leq|\beta_{1}|+|\beta_{2}|$
.
4. (4)
$S_{2}$ and $R_{2}$ (or $R_{1}$ and $S_{1}$) interaction:
1. (a)
$1^{\circ}$ $\chi+\pi\rightarrow\beta^{\prime}+\chi^{\prime}:$ there exist
1-shock $\beta_{0}$ and 2-shock $\chi_{0}$ such that
$|\chi_{0}|=|\chi|-\xi,\;\;|\beta_{0}|=\eta$ and
$\chi_{0}+\beta_{0}\rightarrow\beta^{\prime}+\chi^{\prime}$,
where $0<\eta\leq g^{\prime}_{2}(|\chi|,\rho_{1})\xi<\xi$ .
$2^{\circ}$ $\chi+\pi\rightarrow\beta^{\prime}+\pi^{\prime}$: there exist
$\eta,\xi$ such that
$|\pi^{\prime}|\leq|\pi|,\;\;\;\;|\beta^{\prime}|=\eta<\xi=|\chi|$,
where $0<\eta\leq g^{\prime}_{2}(|\chi|,\rho_{1})\xi<\xi$ .
2. (b)
$1^{\circ}$ $o+\beta\rightarrow\beta^{\prime}+\chi^{\prime}:$ there exist
1-shock $\beta_{0}$ and 2-shock $\chi_{0}$ such that
$|\beta_{0}|=|\beta|-\xi,\;\;|\chi_{0}|=\eta$ and
$\chi_{0}+\beta_{0}\rightarrow\beta^{\prime}+\chi^{\prime}$,
where $0<\eta\leq g^{\prime}_{2}(|\beta|,\rho_{2})\xi<\xi$ .
$2^{\circ}$ $o+\beta\rightarrow o^{\prime}+\chi^{\prime}$
$|o^{\prime}|\leq|o|,\;\;\;\;|\chi^{\prime}|=\eta<\xi=|\beta|$,
where $0<\eta\leq g^{\prime}_{1}(|\beta|,\rho_{0})\xi<\xi$ .
5. (5)
$R_{2}$ and $S_{2}$ (or $S_{1}$ and $R_{1}$) interaction:
1. (a)
$1^{\circ}$ $\pi+\chi\rightarrow\beta^{\prime}+\chi^{\prime}:$ there exist
$\eta,\xi$ such that
$|\chi^{\prime}|=|\chi|-\xi,\;\;\;\;|\beta^{\prime}|=\eta$,
where $0<\eta\leq g^{\prime}_{1}(|\chi|,\rho_{2})\xi<\xi$ .
$2^{\circ}$ $\pi+\chi\rightarrow\beta^{\prime}+\pi^{\prime}$
$|\pi^{\prime}|\leq|\pi|,\;\;\;\;|\beta^{\prime}|=\eta<\xi=|\chi|$,
where $0<\eta\leq g^{\prime}_{2}(|\chi|,\rho_{0})\xi<\xi$ .
2. (b)
$1^{\circ}$ $\beta+o\rightarrow\beta^{\prime}+\chi^{\prime}:$ there exist
$\eta,\xi$ such that
$|\beta^{\prime}|=|\beta|-\xi,\;\;\;\;|\chi^{\prime}|=\eta$,
where $0<\eta\leq g^{\prime}_{1}(|\beta|,\rho_{1})\xi<\xi$ .
$2^{\circ}$ $\beta+o\rightarrow o^{\prime}+\chi^{\prime}$
$|o^{\prime}|\leq|o|,\;\;\;\;|\chi^{\prime}|=\eta<\xi=|\beta|$,
where $0<\eta\leq g^{\prime}_{1}(|\beta|,\rho_{1})\xi<\xi$ .
6. (6)
$R_{2}$ and $R_{1}$ interaction:
$\pi+o\rightarrow o^{\prime}+\pi^{\prime}$
$|o^{\prime}|=|o|,\;\;\;\;|\pi^{\prime}|=|\pi|$.
Here $C_{*}$ is a positive constant defined as in (A1).
Proof. This theorem can be proved using the same tools as in [11] and
therefore will be omitted. The only differences are: the constant $C_{*}$ is
now independent of $\varepsilon,\beta,\chi,\rho_{0},\rho_{1}$ and $\rho_{2}$,
and we have $\sqrt{\varepsilon}$ instead of $\varepsilon$ in the estimates
$(1)(a)-(1)(c)$, $(2)(a)$ and $(2)(b)$. $\Box$
The main part of the paper is the interaction problem of delta shocks via
pressure perturbation. Thus, one needs to control shock and rarefaction
strengths as $\rho$ goes to infinity as $\varepsilon\to 0$ (more precisely,
when $\rho$ is bounded by ${\rm const}/\varepsilon$). Because of that, we give
their estimates in $r-s$ plane based on Theorem 3.3 and Theorem 3.4. Let
$(\rho_{0},u_{0})=(r_{0},s_{0})$ be connected with $(\rho,u)=(r,s)$ by a
1-rarefaction (or 1-shock) wave, while $(\rho,u)=(r,s)$ be connected with
$(\rho_{1},u_{1})=(r_{1},s_{1})$ by a 2-rarefaction (or 2-shock) wave. Then
the strength of 1-rarefaction wave is
$r-r_{0}=\frac{2}{\sqrt{\varepsilon}}(\rho_{0}^{\varepsilon}-\rho^{\varepsilon}),\;\;\rho<\rho_{0}\,,$
(35)
and the strength of 2-rarefaction wave is
$s_{1}-s=\frac{2}{\sqrt{\varepsilon}}(\rho_{1}^{\varepsilon}-\rho^{\varepsilon}),\;\;\rho<\rho_{1}\,.$
(36)
The strength of 1-shock wave is estimated by
$2\,\rho_{0}^{\varepsilon}\,\sqrt{\varepsilon}\,\ln\frac{\rho}{\rho_{0}}\leq
r_{0}-r\leq
2\,\frac{\sqrt{\varepsilon}}{\sqrt{1+2\varepsilon}}\,\left(\frac{\rho}{\rho_{0}}\right)^{\gamma/2}\cdot\rho_{0}^{\varepsilon},\;\;\rho>\rho_{0}\,,$
(37)
while, the strength of 2-shock wave is estimated by
$2\,\rho_{1}^{\varepsilon}\,\sqrt{\varepsilon}\,\ln\frac{\rho}{\rho_{1}}\leq
s-s_{1}\leq
2\,\frac{\sqrt{\varepsilon}}{\sqrt{1+2\varepsilon}}\,\left(\frac{\rho}{\rho_{1}}\right)^{\gamma/2}\cdot\rho_{1}^{\varepsilon},\;\;\rho>\rho_{1}\,.$
(38)
Let us estimate the upper bound of the 1-shock wave given in (37). For the
function $g_{1}$ from (17) we have $0\leq g^{\prime}_{1}(\beta,\rho_{0})<1$
and $0\leq g^{\prime\prime}_{1}(\beta,\rho_{0})$, so
$\lim\limits_{|\beta|\to+\infty}g^{\prime}_{1}(|\beta|,\rho_{0})\leq 1.$
Let us consider two special cases needed for our investigation. The first
case: $\rho>\rho_{0}$ and $\rho\sim 1/\varepsilon$. We have that there exist
constants $\tilde{C}$, $\bar{\bar{C}}$ and $\bar{C}$ independent of
$\varepsilon$ such that
$\begin{array}[]{l}\displaystyle\left(\frac{\rho}{\rho_{0}}\right)^{\gamma/2}\cdot\rho_{0}^{\varepsilon}=\sqrt{\frac{\rho}{\rho_{0}}}\cdot\rho^{\varepsilon}\leq\sqrt{\frac{\tilde{C}}{\varepsilon}}\cdot\left(\frac{\bar{\bar{C}}}{\varepsilon}\right)^{\varepsilon}\leq\sqrt{\frac{1}{\varepsilon}}\cdot\bar{C},\;\;{\rm
so}\\\ \\\ \displaystyle
2\,\frac{\sqrt{\varepsilon}}{\sqrt{1+2\varepsilon}}\,\left(\frac{\rho}{\rho_{0}}\right)^{\gamma/2}\cdot\rho_{0}^{\varepsilon}\leq
2\,\frac{\sqrt{\varepsilon}}{\sqrt{1+2\varepsilon}}\cdot\frac{\bar{C}}{\sqrt{\varepsilon}}\leq{\rm{const.}}\end{array}$
It follows that there exists a constant $C_{2}$, independent of $\varepsilon$
and $\rho_{0}$, such that
$\sup g^{\prime}_{1}(|\beta|,\rho_{0}):=C_{2}<1\;.$ (39)
Hence,
$\frac{1-g^{\prime}_{1}(|\beta|,\rho_{0})}{g^{\prime}_{1}(|\beta|,\rho_{0})}\geq\frac{1-C_{2}}{C_{2}}=:C_{3}>0\,.$
(40)
The second case: $\rho>\rho_{0}$, $\rho\sim 1/\varepsilon$ and $\rho_{0}\sim
1/\varepsilon$. Then
$\sqrt{\frac{\rho}{\rho_{0}}}\cdot\rho^{\varepsilon}\sim\rm{const}\Rightarrow
2\,\frac{\sqrt{\varepsilon}}{\sqrt{1+2\varepsilon}}\,\sqrt{\frac{\rho}{\rho_{0}}}\,\rho^{\varepsilon}\sim{\mathcal{O}}(\sqrt{\varepsilon})\,.$
Again, $|\beta|\to\infty$ is impossible and (40) holds. In order to estimate
the strength of $S_{2}$, we can use the same arguments to prove
$\sup g^{\prime}_{2}(|\chi|,\rho_{0})=:C_{4}<1,$ (41)
and
$\displaystyle\frac{1-g^{\prime}_{2}(|\chi|,\rho_{0})}{g^{\prime}_{2}(|\chi|,\rho_{0})}\geq\frac{1-C_{4}}{C_{4}}=:C_{5}>0\;.$
(42)
From now on, we shall put
$C_{0}=\min\\{C_{3},C_{5}\\}\,.$ (43)
## 4\. Global interaction estimates
This section contains all the necessary assertions from [1] with several
changes in constants. All changes are similar to those from the previous
section.
###### Definition 4.1.
[1] A Lipschitz curve $J$ defined by $t=T(x)$, $x\in\mathbb{R}$ is called an
I-curve, if $|T^{\prime}(x)|<1/\hat{\lambda}$. We denote $J_{2}>J_{1}$, if
$T_{1}\neq T_{2}$ and $T_{2}(x)\geq T_{1}(x),x\in\mathbb{R}$. Denoting by
$S_{j}(J)$ the set of $j$-shock waves crossing $J$ and
$S(J)=S_{1}(J)+S_{2}(J)$, we define
$L^{-}(J)=\sum_{\alpha\in S(J)}|\alpha|,\;\;\;\;Q(J)=\sum_{\beta\in
S_{1}(J),\chi\in S_{2}(J),\;\;\beta,\chi\mbox{\tiny{
approach}}}|\beta||\chi|\;.$ (44)
Set $F(J)=L^{-}(J)+\tilde{K}\cdot Q(J)$, where
$\tilde{K}:=4C_{*}\sqrt{\varepsilon}$. A space-like line lying between the
initial line and the first interaction point is denoted with $O$.
###### Lemma 4.2.
$Q(O)\leq L^{-}(O)^{2}\;.$ (45)
Proof. The proof follows straightforward from Definition 4.1. $\Box$
###### Lemma 4.3.
Assuming $\;4\,C_{*}\,\sqrt{\varepsilon}\,L^{-}(O)\leq 1$, we have
$F(O)\leq 2L^{-}(O)\;.$ (46)
Proof.
$\begin{array}[]{rl}F(O)=&L^{-}(O)+\tilde{K}\;Q(O)\leq
L^{-}(O)+\tilde{K}\;L^{-}(O)^{2}\mbox{ (by (\ref{q0ocena}))}\\\
=&L^{-}(O)(1+\tilde{K}L^{-}(O))=L^{-}(O)(1+4\;C_{*}\,\sqrt{\varepsilon}\,L^{-}(O))\\\
\leq&L^{-}(O)(1+1)=2L^{-}(O)\;.\end{array}$
$\Box$
As in [2], consider a interval ${\mathcal{J}}\subset\mathbb{R}$ and a map
$a:{\mathcal{J}}\to\mathbb{R}^{n}$. The total variation (TV) of $a$ is then
defined as
$TV(a):=\sup\left\\{\sum_{j=1}^{N}|a(x_{j})-a(x_{j-1})|\right\\},$
where the supremum is taken over all $N\geq 1$ and all $(N+1)$-tuples of
points $x_{j}\in{\mathcal{J}}$ such that $x_{0}<x_{1}<\cdots<x_{N}$. Now, we
give a new estimate for $L^{+}(O)$. Here, $L^{+}(O)$ denotes the sum of the
rarefaction waves strengths which cross the line $O$.
###### Lemma 4.4.
We have
$L^{-}(O)\leq TV(r_{0}(x),s_{0}(x))\;\;\mbox{ and }\;\;L^{+}(O)\leq
TV(r_{0}(x),s_{0}(x))\;.$ (47)
The estimates in previous Lemma can easily be verified. The uniform bounds of
$F(J)$ follows from the following theorem.
###### Theorem 4.5.
If
$\displaystyle\;C_{*}\,\sqrt{\varepsilon}\,F(O)\leq\min\left\\{\frac{1}{2},\frac{C_{0}}{4}\right\\}$,
then $F(J_{2})\leq F(J_{1})$ for $J_{2}>J_{1}$. Particulary, $L^{-}(J)\leq
F(O)$.
Proof. This theorem can be proved in the same way as Lemma 5 from [11] and
hence the proof will be omitted. One has just to substitute a constant $K$
from the original proof with the determined value $\tilde{K}$ here. $\Box$
###### Lemma 4.6.
Assume that $\tilde{K}L^{-}(O)\leq 1$ and that
$\sqrt{\gamma-1}\;TV\,(r_{0}(x),s_{0}(x))\leq\frac{1}{C_{*}}\cdot\min\left\\{\frac{\sqrt{2}}{4},\frac{\sqrt{2}}{8}\,C_{0}\right\\}\,.$
(48)
Then $\displaystyle\tilde{K}F(O)\leq\min\left\\{2,C_{0}\right\\}$.
Proof. Using Lemma 4.3 and 4.4 we have
$\frac{\sqrt{2}}{2}\sqrt{\varepsilon}\,F(O)\leq\sqrt{2}\sqrt{\varepsilon}\,L^{-}(O)\leq\sqrt{\gamma-1}\;TV\,(r_{0}(x),s_{0}(x))\leq\frac{1}{C_{*}}\cdot\min\left\\{\frac{\sqrt{2}}{4},\frac{\sqrt{2}}{8}\,C_{0}\right\\}\,.$
Multiplying it with $8\,C_{*}/\sqrt{2}$, one gets
$4\,C_{*}\sqrt{\varepsilon}\,F(O)=\tilde{K}F(O)\leq\min\left\\{2,C_{0}\right\\}\,.$
which proves the claim. $\Box$
The right hand side of (48) does not depend on $\varepsilon$, and then one can
say that $TV\,(r_{0}(x),s_{0}(x))$ may be arbitrarily large since we can
always choose $\varepsilon$ small enough in order to fulfill (48) with
$\gamma=1+2\varepsilon$. Then we can apply wave front tracking procedure from
[1] for each such $\varepsilon$, and obtain a sequence of step functions
converging to the entropic solution. One only needs to replace $C\varepsilon
F(O)$ and $\frac{1-\delta}{\delta}$ from [1] with
$C_{*}\sqrt{\varepsilon}F(O)$ and $C_{0}$, respectively.
## 5\. Approximate delta shock solutions to pressureless gas dynamics
Our main task is to solve delta shock interaction problem for pressureless gas
dynamics model. Accordingly, we will introduce a solution concept from [9]
(somewhat simplified) and check consistency of theoretical and numerical wave
front tracking results by letting $\varepsilon\to 0$.
### 5.1. Basic notions
In this section we shall use the notions and assertions from [9]. It contains
results for a $3\times 3$ system with energy conservation law added, but all
the results can also be applied to system (2), too. Let us start with the
basic definitions. Vector valued function of the form
$U_{\varepsilon}(x,t)=\begin{cases}U_{0},&x<c(t)-a_{\varepsilon}(t)\\\
U_{1,\varepsilon}(t),&c(t)-a_{\varepsilon}(t)<x<c(t)\\\
U_{2,\varepsilon}(t),&c(t)<x<c(t)+b_{\varepsilon}(t)\\\
U_{1},&x>c(t)+b_{\varepsilon}(t)\end{cases}.$ (49)
is called weighted shadow wave (weighted SDW, for short). Here, $U:=(\rho,u)$.
The functions $a_{\varepsilon}$, $b_{\varepsilon}$ are continuous functions
satisfying $a_{\varepsilon}(0)=x_{1,\varepsilon}$ and
$b_{\varepsilon}(0)=x_{2,\varepsilon}$. The SDW is constant if
$U_{1,\varepsilon}$ and $U_{2,\varepsilon}$ are just constants. If, in
addition, $x_{1,\varepsilon}=x_{2,\varepsilon}=0$, then the wave is called
simple.
The value
$\sigma_{\varepsilon}(t):=a_{\varepsilon}(t)U_{1,\varepsilon}(t)+b_{\varepsilon}(t)U_{2,\varepsilon}(t)$
is called the strength and $c^{\prime}(t)$ is called the speed of the shadow
wave. We assume that $\lim_{\varepsilon\rightarrow
0}\sigma_{\varepsilon}(t)=\sigma(t)\in\mathbb{R}^{n}$ exists for every $t\geq
0$ and
$\lim_{\varepsilon\rightarrow 0}\int
U_{\varepsilon}(x,t)\phi(x,t)\,dx\,dt=\langle
U_{0}+(U_{1}-U_{0})\,\theta(x-c(t))+\sigma(t)\,\delta(x-c(t)),\phi(x,t)\rangle,$
for $t\geq 0$, where $\theta$ is a Heaviside function. The SDW central line is
given by $x=c(t)$, while $x=c(t)-a_{\varepsilon}(t)$ and
$x=c(t)+b_{\varepsilon}(t)$ are called the external SDW lines. The values
$x_{1,\varepsilon}$ and $x_{2,\varepsilon}$ are called the shifts, while
$U_{1,\varepsilon}(t)$ and $U_{2,\varepsilon}(t)$ are called the intermediate
states of a given SDW.
Let $i\in\\{1,2,\dots,n\\}$. We assume
$\|U_{\varepsilon}^{i}\|_{L^{\infty}}={\mathcal{O}}(\varepsilon^{-1}),$ if $f$
and $g$ have at most a linear growth with respect to $i$-th component, or
otherwise $\|U_{\varepsilon}^{i}\|_{L^{\infty}}={o}(\varepsilon^{-1})$. The
components of the first kind are called major ones, while the ones of the
second kind are called minor ones.
A delta shock is a SDW associated with a $\delta$ distribution with all minor
components having finite limits as $\varepsilon\rightarrow 0$.
The following lemma is the base of all calculations involving SDWs.
###### Lemma 5.1.
Let $f,g\in{\mathcal{C}}(\Omega:{\mathbb{R}}^{n})$ and
$U:{\mathbb{R}}_{+}^{2}\rightarrow\Omega\subset{\mathbb{R}}^{n}$ be a
piecewise constant function for every $t\geq 0$. Let us also suppose that $f$
and $g$ satisfy
$\max_{i=1,2}\\{\|f(U_{i,\varepsilon})\|_{L^{\infty}},\|g(U_{i,\varepsilon})\|_{L^{\infty}}\\}={\mathcal{O}}(\varepsilon^{-1}).$
(50)
Then
$\begin{split}\langle\partial_{t}f(U_{\varepsilon}),\phi\rangle\approx&\int_{0}^{\infty}\lim_{\varepsilon\rightarrow
0}{d\over
dt}\Big{(}a_{\varepsilon}(t)f(U_{1,\varepsilon}(t))+b_{\varepsilon}(t)f(U_{2,\varepsilon}(t))\Big{)}\,\phi(c(t),t)\,dt\\\
&-\int_{0}^{\infty}c^{\prime}(t)\Big{(}f(U_{1})-f(U_{0})\Big{)}\,\phi(c(t),t)\,dt\\\
&+\int_{0}^{\infty}\lim_{\varepsilon\rightarrow
0}c^{\prime}(t)\Big{(}a_{\varepsilon}(t)f(U_{1,\varepsilon}(t))+b_{\varepsilon}(t)f(U_{2,\varepsilon}(t))\Big{)}\,\partial_{x}\phi(c(t),t)\,dt\end{split}$
(51)
and
$\begin{split}\langle\partial_{x}g(U_{\varepsilon}),\phi\rangle\approx&\int_{0}^{\infty}\Big{(}g(U_{1})-g(U_{0})\Big{)}\,\phi(c(t),t)\,dt\\\
&-\int_{0}^{\infty}\lim_{\varepsilon\rightarrow
0}\Big{(}(a_{\varepsilon}(t)g(U_{1,\varepsilon}(t))+(b_{\varepsilon}(t)g(U_{2,\varepsilon}(t))\Big{)}\,\partial_{x}\phi(c(t),t)\,dt.\end{split}$
(52)
### 5.2. Entropy conditions
Let $\eta(U)$ be a semi-convex entropy function for (2), with entropy-flux
function $q(U)$. We shall use entropy condition in the following form. A weak
or approximate solution $U_{\varepsilon}=(\rho_{\varepsilon},u_{\varepsilon})$
to system (2) with initial data $U|_{t=0}=U_{0,\varepsilon}$ is admissible
provided that for every $T>0$ we have
$\underline{\lim}_{\varepsilon\rightarrow
0}\int_{\mathbb{R}}\int_{0}^{T}\eta(U_{\varepsilon})\partial_{t}\phi+q(U_{\varepsilon})\partial_{x}\phi\,dt\,dx+\int_{\mathbb{R}}\eta(U_{0,\varepsilon}(x,0))\phi(x,0)\,dx\geq
0,$ (53)
for all non-negative test functions $\phi\in
C_{0}^{\infty}(\mathbb{R}\times(-\infty,T))$.
Using Lemma 5.1 with $f$ substituted by $\eta$ and $g$ by $q$ and the fact
that the delta function is a non-negative distribution, the first condition
for SDW $U_{\varepsilon}$ from (49) to be admissible is given by
$\begin{split}-c^{\prime}(t)(\eta(U_{1})-\eta(U_{0}))+(q(U_{1})-q(U_{0}))&\\\
+\lim_{\varepsilon\rightarrow 0}{d\over
dt}(\eta(U_{1,\varepsilon}(t))a_{\varepsilon}+\eta(U_{2,\varepsilon}(t))b_{\varepsilon})&\leq
0.\end{split}$ (54)
The derivative of delta function changes the sign, so $U_{\varepsilon}$ has to
satisfy
$\begin{split}\lim_{\varepsilon\rightarrow
0}c^{\prime}(t)(\eta(U_{1,\varepsilon}(t))a_{\varepsilon}+\eta(U_{2,\varepsilon}(t))b_{\varepsilon})&\\\
-q(U_{1,\varepsilon}(t))a_{\varepsilon}(t)-q(U_{2,\varepsilon}(t))b_{\varepsilon}(t)&=0\end{split}$
(55)
in addition.
These conditions are much simpler in the case of simple SDW when $U_{0}$,
$U_{1}$, $U_{1,\varepsilon}$ and $U_{2,\varepsilon}$ are constants:
$\overline{\lim}_{\varepsilon\rightarrow
0}-c(\eta(U_{1})-\eta(U_{0}))+a_{\varepsilon}\eta(U_{1,\varepsilon})+b_{\varepsilon}\eta(U_{2,\varepsilon})+q(U_{1})-q(U_{0})\leq
0$ (56)
and
$\lim_{\varepsilon\rightarrow
0}-c(a_{\varepsilon}\eta(U_{1,\varepsilon})+b_{\varepsilon}\eta(U_{2,\varepsilon}))+a_{\varepsilon}q(U_{1,\varepsilon})+b_{\varepsilon}q(U_{2,\varepsilon})=0.$
(57)
In most of the papers with delta or singular shock solution, the authors use
overcompressibility as the admissibility condition. A wave is called the
overcompressive one if all characteristics from both sides of the SDW line run
into a shock curve, i.e.
$\lambda_{i}(U_{0})\geq c^{\prime}(t)\geq\lambda_{i}(U_{1}),\;i=1,\ldots,n,$
where $c$ is a shock speed and $x=\lambda_{i}(U)t$, $i=1,\ldots,n$ are the
characteristics of the system. One will see that these notations coincide with
our model case.
The entropy condition is connected with the problem of uniqueness for a weak
solution of the conservation law system. We give a definition of weak
(distributional) uniqueness and some results about it afterward.
###### Definition 5.2.
An SDW solution is called weakly unique if its distributional image is unique.
More precisely, a speed $c$ of the wave has to be unique as well as the limit
$\lim_{\varepsilon\rightarrow
0}a_{\varepsilon}U_{1,\varepsilon}+b_{\varepsilon}U_{2,\varepsilon}.$
Let $i\in\\{1,\ldots,n\\}$. If a limit
$\displaystyle\lim_{\varepsilon\rightarrow
0}a_{\varepsilon}U_{1,\varepsilon}^{i}+b_{\varepsilon}U_{2,\varepsilon}^{i}$
is unique, then we say that the $i$-th component is unique.
Note that all minor components of $U_{\varepsilon}$ are unique by default.
### 5.3. Entropy solutions to Riemann problem for pressureless gas dynamics
model
The proof for the following theorem in the case of $3\times 3$ PGD model is
given in [9]. Its restriction to a $2\times 2$ system is straightforward and
therefore not discussed here.
###### Theorem 5.3.
Suppose that $u_{0}>u_{1}$. Then there exists a unique shadow wave solution of
the form (49) to the Riemann problem (2, 3) satisfying the entropy inequality
(53) with $\eta$ and $q$ as defined above.
Moreover, the validity of (53) for all semi-convex entropies $\eta$ are
equivalent to the overcompressibility of the shadow wave.
Our aim is to show the structure of a solution in order to be able to compare
it with a numerical approximation described above. For our purposes it is safe
to take $a_{\varepsilon}=b_{\varepsilon}=\varepsilon$ in the sequel. In the
proof of Theorem 5.3 we showed that a SDW solution (49) (with $U=(\rho,u)$) to
(2) and initial data (3), with $u_{0}>u_{1}$, had to satisfy
$\begin{split}c=u_{s}=\lim_{\varepsilon\rightarrow
0}u_{\varepsilon}&\equiv\frac{[\rho
u]-[u]\sqrt{\rho_{0}\rho_{1}}}{[\rho]}\;(u_{s}\text{ does not depend on
}\varepsilon)\\\ \lim_{\varepsilon\rightarrow
0}\varepsilon\rho_{\varepsilon}&=c[\rho]-[\rho
u]=(u_{0}-u_{1})\sqrt{\rho_{0}\rho_{1}},\end{split}$
if $\rho_{0}\neq\rho_{1}$, and $c=u_{s}=(u_{0}+u_{1})/2$, if
$\rho_{0}=\rho_{1}$. That defines a weakly unique SDW solution to the problem.
### 5.4. Two SDWs interaction
The main advantage of using weighted SDWs (intermediate states vary with $t$
in addition) is for solving SDW interaction problem. Then we can proceed with
the main part of the paper by showing numerically that such a solution can be
viewed as a limit of gas dynamics model with a vanishing pressure as
perturbation. Note that verification of delta shock existence has already been
obtained in [3] (see [6] for a somewhat general model).
Suppose that two SDWs interact in a point $(X,T)$. The superscript $1$ is used
for data in the left wave while the superscript $2$ is used for the right one.
The first SDW connects the states $\displaystyle U_{0}=(\rho_{0},u_{0})$ with
$\displaystyle U_{1}=(\rho_{1},u_{1})$, while the second one connects the
states $\displaystyle U_{1}=(\rho_{1},u_{1})$ with $\displaystyle
U_{2}=(\rho_{2},u_{2})$.
Again, the following theorem has been proved in [9] for the extended PGD
system, and the proof can easily be adopted for the present one (2).
###### Theorem 5.4.
The result of two SDW interactions for the pressureless system (2) is a weakly
unique single entropic weighted SDW.
We use the following notation: $[x]_{1}:=x_{1}-x_{0}$, $[x]_{2}:=x_{2}-x_{1}$
and $[x]:=x_{2}-x_{0}$. The weighted SDW solution from the above theorem
satisfies the following: The speed is given by
$c^{\prime}(t)=u_{s}(t):=\lim_{\varepsilon\rightarrow 0}u_{\varepsilon}(t)$,
while $u_{s}(t)$ and $\xi(t):=\lim_{\varepsilon\rightarrow
0}\varepsilon\rho_{\varepsilon}(t)$ satisfies the following ODEs system
$\begin{split}\xi^{\prime}(t)&=u_{s}(t)[\rho]-[\rho u]\\\
(\xi(t)u_{s}(t))^{\prime}&=u_{s}(t)[\rho u]-[\rho u^{2}]\end{split}$ (58)
with the initial data
$\begin{split}\xi(T)=&(\xi^{1}+\xi^{2})T=(-[u]_{1}\sqrt{\rho_{0}\rho_{1}}-[u]_{2}\sqrt{\rho_{1}\rho_{2}})T,\\\
\xi(T)u_{s}(T)=&(c^{1}\xi^{1}+c^{2}\xi^{2})T=\Big{(}-\frac{[\rho
u]_{1}-[u]_{1}\sqrt{\rho_{0}\rho_{1}}}{[\rho]_{1}}\cdot[u]_{1}\sqrt{\rho_{0}\rho_{1}}\\\
&-\frac{[\rho
u]_{2}-[u]_{2}\sqrt{\rho_{1}\rho_{2}}}{[\rho]_{2}}\cdot[u]_{2}\sqrt{\rho_{1}\rho_{2}}\Big{)}T.\end{split}$
(59)
Here are some facts regarding the solution $(\xi(t),u_{s}(t))$, $t\geq T$ to
the above initial data problem (see [9]):
1. (1)
$\xi(t)$, for $t>T$, is an increasing function when exists. The initial data
$\xi(T)>0$ and $\xi(t)$ is always positive function for $t>T$ (when exists),
since $u_{0}>u_{1}>u_{2}$.
2. (2)
From the system (58) we have
$u_{s}^{\prime}(t)=-\frac{1}{\xi(t)}([\rho]u_{s}^{2}(t)-2[\rho
u]u_{s}(t)+[\rho u^{2}]).$
The value $-1/\xi(t)$ is now always negative for $t>T$. The roots of the
right-hand side of the above ODE are denoted as $A_{1}<A_{2}$. Then, for
$[\rho]\neq 0$,
$A_{1,2}=\frac{[\rho u]\pm|u_{0}-u_{2}|\sqrt{\rho_{0}\rho_{2}}}{[\rho]}\,.$
Assume that $[\rho]>0$. If $u_{s}(t)\in(A_{1},A_{2})$, then $u_{s}(t)$
increases, and if $u_{s}(t)\in(-\infty,A_{1})\cup(A_{2},+\infty)$, then
$u_{s}(t)$ decreases. The opposite holds if $[\rho]<0$. There are two possible
cases:
* •
If $\rho_{0}>\rho_{2}$, then $u_{2}\leq A_{1}\leq u_{0}\leq A_{2}$. If
$u_{s}(T)\in(u_{2},A_{1})$, then $u_{s}(t)$ increases for $t>T$ but stays
bellow $A_{1}$. If $u_{s}(T)\in(A_{1},u_{0})$, then $u_{s}(t)$ decreases for
$t>T$ but stays above $A_{1}$.
* •
If $\rho_{2}>\rho_{0}$, then $A_{1}\leq u_{2}\leq A_{2}\leq u_{0}$. Again, if
$u_{s}(T)\in(u_{2},A_{2})$, then $u_{s}(t)$ increases for $t>T$ but stays
bellow $A_{2}$. If $u_{s}(T)\in(A_{2},u_{0})$, then $u_{s}(t)$ decreases for
$t>T$ but stays above $A_{2}$.
This implies $u_{0}\geq u_{s}(t)\geq u_{2}$ (the SDW is overcompressive).
Also, one will see that numerical examples resemble these asymptotic
properties of $u_{s}(t)$ as $t\to\infty$.
## 6\. Numerical results
In this section one can find numerical results which show a consistency of
theoretical (in the sense of SDWs) and numerical results. Consider system (1)
with the initial data
$(\rho,u)\left|{}_{t=0}\right.=\left\\{\begin{array}[]{ll}(\rho_{0},u_{0}),&x<a_{1}\\\
(\rho_{1},u_{1}),&a_{1}<x<a_{2}\\\
(\rho_{2},u_{2}),&x>a_{2}\end{array}\right.$ (60)
where $a_{1}<a_{2}$, $u_{0}>u_{1}>u_{2}$. Then (see [3]), for $\varepsilon$
small enough, there exist
$(\rho_{1,\varepsilon},u_{1,\varepsilon})\in\mathbb{R}_{+}\times\mathbb{R}$
and
$(\rho_{2,\varepsilon},u_{2,\varepsilon})\in\mathbb{R}_{+}\times\mathbb{R}$,
so that:
* •
$(\rho_{0},u_{0})$ is connected with
$(\rho_{1,\varepsilon},u_{1,\varepsilon})$ by an 1-shock, and
$(\rho_{1,\varepsilon},u_{1,\varepsilon})$ is connected with
$(\rho_{1},u_{1})$ by a 2-shock,
* •
$(\rho_{1},u_{1})$ is connected with
$(\rho_{2,\varepsilon},u_{2,\varepsilon})$ by an 1-shock, while
$(\rho_{2,\varepsilon},u_{2,\varepsilon})$ is connected with
$(\rho_{2},u_{2})$ by a 2-shock.
A numerical solution is obtained by wave front tracking algorithm described in
[1]. In order to verify two delta shocks interaction, we shall consider two
cases.
Case A. Suppose that $(\rho_{0},u_{0})$ is connected with $(\rho_{1},u_{1})$
by a single delta shock and $(\rho_{1},u_{1})$ is connected with
$(\rho_{2},u_{2})$ by a single delta shock, too. Assume that
$(\rho_{0},u_{0})$ can be connected with $(\rho_{2},u_{2})$ by a single delta
shock (so-called simple SDW, see [9]). The resulting SDW has a constant speed
as a consequence. That can be done by choosing a special value for $\rho_{2}$
provided that $\rho_{0},u_{0},\rho_{1},u_{1}$ and $u_{2}$ are already given.
Case B. We choose arbitrarily $\rho_{2}$, i.e. the resulting SDW has a
variable speed (a central SDW curve is no longer a line). The numerical
results are given in Tables 2, 3 and 4.
Table 1. Parameter description Parameter | Description
---|---
$\kappa$ | Adiabatic constant defined in (4).
$\rho_{\varepsilon}$ | First component of the intermediate state of the solution for (1, 60).
$u_{\varepsilon}$ | Second component of the intermediate state of the solution for (1, 60).
$c_{1}$ | Speed of the first left shock.
$c_{2}$ | Speed of the last right shock.
$|Eq_{1}|$ | Left hand side of the integral on the first equation in (1).
$|Eq_{2}|$ | Left hand side of the integral on the second equation in (1).
### 6.1. Case A
###### Example 6.1.
Let $a_{1}=0$, $a_{2}=2$, $(\rho_{0},u_{0})=(1,1)$,
$(\rho_{1},u_{1})=(1.2,0.8)$ and $u_{2}=0.7$. Now, for $\rho_{2}=1.14286$,
there exists a single simple SDW as a solution to the interaction problem.
Table 2.
$(\rho_{0},u_{0})=(1,1),(\rho_{1},u_{1})=(1.2,0.8),(\rho_{2},u_{2})=(1.14286,0.7)$
$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|}\gamma&\kappa&\varepsilon&\rho_{\varepsilon}&u_{\varepsilon}&c_{1}&c_{2}&|Eq_{1}|&|Eq_{2}|\\\
\hline\cr 2&0.5&0.5&1.29979&0.80062&0.13553&1.53337&2\cdot 10^{-5}&3\cdot
10^{-5}\\\ \hline\cr 1.2&0.29&0.1&1.68612&0.82806&0.57746&1.09746&1\cdot
10^{-4}&2\cdot 10^{-4}\\\ \hline\cr
1.02&0.099&0.01&4.22718&0.84163&0.79256&0.89411&2\cdot 10^{-3}&3\cdot
10^{-3}\\\ \hline\cr 1.01&0.071&0.005&6.71337&0.84311&0.81565&0.87247&8\cdot
10^{-5}&2\cdot 10^{-3}\\\ \hline\cr
1.006&0.055&0.003&9.95729&0.84379&0.82636&0.86254&1\cdot 10^{-2}&6\cdot
10^{-3}\\\ \hline\cr\end{array}$
After interaction, the speed of the resulting wave is $c_{\delta}=0.844994$.
Two SDWs will interact in a point $(X,T)=(12.365,13.8088)$ with such data.
Now, we are going to explain Figures 2, 7 and 12 which are illustrations of
appropriate numerical results. For each $\varepsilon$ we have two piecewise
linear half-lines. The left one originates from the point $(x,t)=(a_{1},0)$,
while the right one originates from the point $(x,t)=(a_{2},0)$. The $i$-th
linear segment of these half-lines can be written in the form
$x=c_{i,j}\,(t-t_{i})+x_{i}$, $i\geq 1$, $j=1,2$, $x_{i}\leq x\leq x_{i+1}$,
$t_{i}\leq t\leq t_{i+1}$, where $c_{i,1}$ stands for the speed of the first
($S_{1}$) wave on the left hand side in phase plane, while $c_{i,2}$ stands
for the speed of the last ($S_{2}$) wave on the left hand side in phase plane
at each $i$-th segment. Interactions of the waves occur at the points
$(x_{i},t_{i}),\;i\geq 1$. After two delta shock interaction, the resulting
delta shock central line in Figure 2 (dashed line) starts from $(X,T)$ and it
is calculated explicitly from system (2).
### 6.2. Case B
###### Example 6.2.
Let $a_{1}=0$, $a_{2}=2$, $(\rho_{0},u_{0})=(1,1)$,
$(\rho_{1},u_{1})=(1.2,0.8)$ and $(\rho_{2},u_{2})=(1.3,0.7)$. Two SDWs will
interact in a point $(X,T)=(12.2291,13.657)$ with such data. After two delta
shock interaction, the resulting delta shock central lines in Figures 7 and 12
(dashed lines) start from $(X,T)$, too. Here,
$x(t)=\int\limits_{T}^{t}u_{s}(p)\,dp+X,\;t\geq T,$
while $u_{s}(t)$ represents the second component of the solution
$(\xi(t),u_{s}(t))$ of system (58) with initial conditions (59).
Table 3.
$\;\;(\rho_{0},u_{0})=(1,1),(\rho_{1},u_{1})=(1.2,0.8),(\rho_{2},u_{2})=(1.3,0.7)$
$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|}\gamma&\kappa&\varepsilon&\rho_{\varepsilon}&u_{\varepsilon}&c_{1}&c_{2}&|Eq_{1}|&|Eq_{2}|\\\
\hline\cr 2&0.5&0.5&1.38131&0.74968&0.09312&1.54395&3\cdot 10^{-4}&5\cdot
10^{-4}\\\ \hline\cr 1.2&0.29&0.1&1.79287&0.80661&0.56268&1.08778&5\cdot
10^{-4}&6\cdot 10^{-4}\\\ \hline\cr
1.02&0.099&0.01&4.49667&0.83356&0.78596&0.88787&3\cdot 10^{-3}&3\cdot
10^{-3}\\\ \hline\cr 1.01&0.071&0.005&7.14038&0.83646&0.80983&0.86684&1\cdot
10^{-3}&3\cdot 10^{-3}\\\ \hline\cr
1.006&0.055&0.003&10.5884&0.83782&0.82091&0.85711&3\cdot 10^{-2}&2\cdot
10^{-2}\\\ \hline\cr\end{array}$
###### Example 6.3.
Let $a_{1}=0$, $a_{2}=2$, $(\rho_{0},u_{0})=(1,1)$,
$(\rho_{1},u_{1})=(0.8,0.9)$ and $(\rho_{2},u_{2})=(0.9,0.7)$. Two SDWs will
interact in a point $(X,T)=(12.2364,12.8427)$ with such data.
Table 4.
$\;\;(\rho_{0},u_{0})=(1,1),(\rho_{1},u_{1})=(0.8,0.9),(\rho_{2},u_{2})=(0.9,0.7)$
$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|}\gamma&\kappa&\varepsilon&\rho_{\varepsilon}&u_{\varepsilon}&c_{1}&c_{2}&|Eq_{1}|&|Eq_{2}|\\\
\hline\cr 2&0.5&0.5&1.16621&0.88674&0.20529&1.51813&8\cdot 10^{-5}&4\cdot
10^{-5}\\\ \hline\cr 1.2&0.29&0.1&1.50419&0.86711&0.60356&1.11606&2\cdot
10^{-4}&1\cdot 10^{-4}\\\ \hline\cr
1.02&0.099&0.01&3.75748&0.85659&0.80459&0.90592&4\cdot 10^{-3}&2\cdot
10^{-3}\\\ \hline\cr 1.01&0.071&0.005&5.96447&0.85544&0.82632&0.88306&1\cdot
10^{-3}&8\cdot 10^{-4}\\\ \hline\cr
1.006&0.055&0.003&8.66370&0.85341&0.83808&0.86341&2\cdot 10^{-2}&3\cdot
10^{-2}\\\ \hline\cr\end{array}$
## References
* [1] F. Asakura, Wave-front tracking method for the equations of isentropic gas dynamics, Quart. Appl. Math. 63 (2005), no. 1, 20–33.
* [2] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensi-onal Cauchy Problem, Oxford University Press, New York, 2000.
* [3] G.Q. Chen and H. Liu, Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations, SIAM J. Math. Anal. 34 (2003), no. 4, 925–938.
* [4] R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 21. Springer-Verlag, New York-Heidelberg, 1976.
* [5] R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1990.
* [6] D. Mitrović and M. Nedeljkov, Delta shock waves as a limit of shock waves, J. Hyperbolic Differ. Equ. 4 (2007), no. 4, 629–653.
* [7] M. Nedeljkov, Delta and singular delta locus for one dimensional systems of conservation laws, Math. Methods Appl. Sci. 27 (2004), no. 8, 931–955.
* [8] M. Nedeljkov, Singular shock waves in interactions, Quart. Appl. Math. 66 (2008), no. 2, 281–302.
* [9] M. Nedeljkov, Shadow waves – entropies and interactions for delta and singular shocks (2009), to appear in Arch. Ration. Mech. Anal.
* [10] M. Nedeljkov, M. Oberguggengberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008), no. 2, 1143–1157.
* [11] T. Nishida, J.A. Smoller, Solutions in the Large for Some Nonlinear Hyperbolic Conservation Laws, Comm. Pure Appl. Math. 26 (1973), 183–200.
* [12] B. Riemann, Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Gott.Abh.Math.Cl. 8 (1860), 43–65.
* [13] E. Weinan, Y.G. Rykov, Ya.G. Sinai, Generalized variotional principles, global weak solutions and behavior with random initial data for systems of consevation laws arising in adhesion particle dynamics, Comm. Math. Phys. 177 (1996), no. 2, 349–380.
* [14] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.
Nebojša Dedović
Department of Agricultural Engineering, University of Novi Sad
Trg D. Obradovića 8, 21000 Novi Sad, Serbia
dedovicn@uns.ac.rs
Marko Nedeljkov
Department of Mathematics and Informatics, University of Novi Sad
Trg D. Obradovića 4, 21000 Novi Sad, Serbia
markonne@uns.ac.rs
## 7\. Appendix
Figure 2. Phase $x-t$ plane, Case A, Example 6.1.
Figure 3. Speed of delta shock formed after double delta shock interaction,
Case A, Example 6.1.
Figure 4. Speed of the first ($S_{1}$) wave on the left hand side and the last
($S_{2}$) wave on the right hand side for various $\varepsilon$, Case A,
Example 6.1.
Figure 5. Solution $\rho(x,t)$ for various $\varepsilon$ at $t=15000$, Case A,
Example 6.1.
Figure 6. Solution $u(x,t)$ for various $\varepsilon$ at $t=15000$, Case A,
Example 6.1.
Figure 7. Phase $x-t$ plane, Case B, Example 6.2.
Figure 8. Speed of delta shock formed after double delta shock interaction,
Case B, Example 6.2.
Figure 9. Speed of the first ($S_{1}$) wave on the left hand side and the last
($S_{2}$) wave on the right hand side for various $\varepsilon$, Case B,
Example 6.2.
Figure 10. Solution $\rho(x,t)$ for various $\varepsilon$ at $t=15000$, Case
B, Example 6.2.
Figure 11. Solution $u(x,t)$ for various $\varepsilon$ at $t=15000$, Case B,
Example 6.2.
Figure 12. Phase $x-t$ plane, Case B, Example 6.3.
Figure 13. Speed of delta shock formed after double delta shock interaction,
Case B, Example 6.3.
Figure 14. Speed of the first ($S_{1}$) wave on the left hand side and the
last ($S_{2}$) wave on the right hand side for various $\varepsilon$, Case B,
Example 6.3.
Figure 15. Solution $\rho(x,t)$ for various $\varepsilon$ at $t=15000$, Case
B, Example 6.3.
Figure 16. Solution $u(x,t)$ for various $\varepsilon$ at $t=15000$, Case B,
Example 6.3.
|
arxiv-papers
| 2009-12-23T13:23:45 |
2024-09-04T02:49:07.233178
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nebojsa Dedovic and Marko Nedeljkov",
"submitter": "Nebojsa Dedovic M",
"url": "https://arxiv.org/abs/0912.4636"
}
|
0912.4775
|
# First eigenvalue of the $p$-Laplace operator
along the Ricci flow
Jia-Yong Wu Department of Mathematics, East China Normal University, Dong
Chuan Road 500, Shanghai 200241, People’s Republic of China jywu81@yahoo.com
, Er-Min Wang Department of Mathematics, East China Normal University, Dong
Chuan Road 500, Shanghai 200241, People’s Republic of China wagermn@126.com
and Yu Zheng Department of Mathematics, East China Normal University, Dong
Chuan Road 500, Shanghai 200241, People’s Republic of China
zhyu@math.ecnu.edu.cn
(Date: July 1, 2009.)
###### Abstract.
In this paper, we mainly investigate continuity, monotonicity and
differentiability for the first eigenvalue of the $p$-Laplace operator along
the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is
strictly increasing and differentiable almost everywhere along the Ricci flow
under some curvature assumptions. In particular, for an orientable closed
surface, we construct various monotonic quantities and prove that the first
$p$-eigenvalue is differentiable almost everywhere along the Ricci flow
without any curvature assumption, and therefore derive a $p$-eigenvalue
comparison-type theorem when its Euler characteristic is negative.
###### Key words and phrases:
Ricci flow; first eigenvalue; $p$-Laplace operator; continuity; monotonicity;
differentiability.
###### 2000 Mathematics Subject Classification:
Primary 58C40; Secondary 53C44.
This work is partially supported by the NSFC10871069.
## 1\. Introduction
Given a compact Riemannian manifold $(M^{n},g_{0})$ without boundary, the
Ricci flow is the following evolution equation
(1.1) $\frac{\partial}{\partial t}g_{ij}=-2R_{ij}$
with the initial condition $g(x,0)=g_{0}(x)$, where $R_{ij}$ denotes the Ricci
tensor of the metric $g(t)$. The normalized Ricci flow is
(1.2)
$\frac{\partial}{\partial\tilde{t}}\tilde{g}_{ij}=-2\tilde{R}_{ij}+\frac{2}{n}\tilde{r}\tilde{g}_{ij},$
where $\tilde{g}(\tilde{t}):=c(t)g(t)$,
$\tilde{t}(t):=\int^{t}_{0}c(\tau)d\tau$ and
(1.3) $\displaystyle
c(t):=\exp\left(\frac{2}{n}\int^{t}_{0}r(\tau)d\tau\right),\quad\quad\tilde{r}:={\int_{M}\tilde{R}d\tilde{\mu}}\Big{/}{\int_{M}d\tilde{\mu}},$
($d\tilde{\mu}$ and $\tilde{R}$ denote the volume form and the scalar
curvature of the metric $\tilde{g}(\tilde{t})$, respectively.) which preserves
the volume of the initial manifold. Both evolution equations were introduced
by R.S. Hamilton to approach the geometrization conjecture in [11]. Recently,
studying the eigenvalues of geometric operator is a very powerful tool for
understanding of Riemannian manifolds. In [23], G. Perelman introduced the
functional
$\mathcal{F}(g(t),f(t)):=\int_{M}\left(R+|\nabla f|^{2}\right)e^{-f}d\mu$
and showed that this functional is nondecreasing along the Ricci flow coupled
to a backward heat-type equation. More precisely, if $g(t)$ is a solution to
the Ricci flow (1.1) and the coupled $f(x,t)$ satisfies the following
evolution equation:
$\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^{2}-R,$
then we have
$\frac{\partial\mathcal{F}}{\partial
t}=2\int_{M}\left|Ric+\nabla^{2}f\right|^{2}e^{-f}d\mu.$
If we define
$\lambda(g(t)):=\inf\limits_{f\neq 0}\left\\{\mathcal{F}(g(t),f(t)):f\in
C^{\infty}(M),\int_{M}e^{-f}d\mu=1\right\\},$
then $\lambda(g(t))$ is the lowest eigenvalue of the operator $-4\Delta+R$,
and the increasing of the functional $\mathcal{F}(g,f)$ implies the increasing
of $\lambda(g(t))$.
Later in [1], X.-D. Cao studied the eigenvalues $\lambda$ and eigenfunctions
$f$ of the new operator $-\Delta+R/2$ satisfying $\int_{M}f^{2}d\mu=1$ on
closed manifolds with nonnegative curvature operator. In fact he introduced
(1.4) $\lambda(f,t):=\int_{M}\left(-\Delta f+\frac{R}{2}f\right)fd\mu,$
where $f$ is a smooth function satisfying $\int_{M}f^{2}d\mu=1$ and obtained
the following
Theorem A. (X.-D. Cao [1]) _On a closed Riemannian manifold with nonnegative
curvature operator, the eigenvalues of the operator $-\Delta+\frac{R}{2}$ are
nondecreasing under the unnormalized Ricci flow, i.e._
(1.5) $\frac{d}{dt}\lambda(f,t)=2\int_{M}Ric(\nabla f,\nabla
f)+\int_{M}|Ric|^{2}f^{2}d\mu\geq 0.$
In (1.5), when $\frac{d}{dt}\lambda(f,t)$ is evaluated at time $t$, $f$ is the
corresponding eigenfunction of $\lambda(t)$. Hence $\lambda(t)$ is
nondecreasing.
Shortly thereafter J.-F. Li in [17] dropped the curvature assumption and also
obtained the above result for the operator $-\Delta+\frac{R}{2}$. In fact, he
used new entropy functionals to derive a general result.
Theorem B. (J.-F. Li [17]) _On a compact Riemannian manifold $(M,g(t))$, where
$g(t)$ satisfies the unnormalized Ricci flow for $t\in[0,T)$, the lowest
eigenvalue $\lambda_{k}$ of the operator $-4\Delta+kR$ $(k>1)$ is
nondecreasing under the unnormalized Ricci flow. The monotonicity is strict
unless the metric is Ricci-flat._
At around the same time, X.-D. Cao in [2] also considered the general operator
$-\Delta+cR$ $(c\geq 1/4)$, and derived the following exact monotonicity
formula.
Theorem C. (X.-D. Cao [2]) _Let $(M^{n},g(t))$, $t\in[0,T)$, be a solution of
the unnormalized Ricci flow (1.1) on a closed manifold $M^{n}$. Assume that
$\lambda(t)$ is the lowest eigenvalue of $-\Delta+cR$ $(c\geq 1/4)$ and
$f=f(x,t)>0$ satisfies_
$-\Delta f(x,t)+cRf(x,t)=\lambda(t)f(x,t)$
_with $\int_{M}f^{2}d\mu=1$. Then under the unnormalized Ricci flow, we have_
(1.6)
$\frac{d}{dt}\lambda(t)=\frac{1}{2}\int_{M}|Ric+\nabla^{2}\varphi|^{2}e^{-\varphi}d\mu+\frac{4c-1}{2}\int_{M}|Ric|^{2}e^{-\varphi}d\mu\geq
0,$
_where $e^{-\varphi}=f^{2}$._
On the other hand, L. Ma in [20] considered the eigenvalues of the Laplace
operator along the Ricci flow and proved the following result.
Theorem D. (L. Ma [20]) _Let $g=g(t)$ be the evolving metric along the
unnormalized Ricci flow with $g(0)=g_{0}$ being the initial metric in $M$. Let
$D$ be a smooth bounded domain in $(M,g_{0})$. Let $\lambda>0$ be the first
eigenvalue of the Laplace operator of the metric $g(t)$. If there is a
constant such that the scalar curvature $R\geq 2a$ in $D\times\\{t\\}$ and the
Einstein tensor_
$E_{ij}\geq-ag_{ij}\quad\quad\mathrm{in}\quad D\times\\{t\\},$
_then we have $\lambda^{\prime}\geq 0$, that is, $\lambda$ is nondecreasing in
$t$, furthermore, $\lambda^{\prime}(t)>0$ for the scalar curvature $R$ not
being the constant $2a$. The same monotonicity result is also true for other
eigenvalues._
Moreover S.-C. Chang and P. Lu in [4] studied the evolution of Yamabe constant
under the Ricci flow and gave a simple application. Motivated by the above
works, in this paper we will study the first eigenvalue of the $p$-Laplace
operator whose metric satisfying the Ricci flow. For the $p$-Laplace operator,
besides many interesting properties between the eigenvalues of the $p$-Laplace
operator and geometrical invariants were pointed out in fixed metrics (e.g.
[10], [14], [16], [21]), the first author in [28] studied the monotonicity for
the first eigenvalue of the $p$-Laplace operator along the Ricci flow on
closed manifolds.
In this paper, on one hand we will improve those results in [28] and discuss
the differentiability for the first eigenvalue of the $p$-Laplace operator
along the unnormalized Ricci flow. Meanwhile we construct some monotonic
quantities along the unnormalized Ricci flow. On the other hand, we will deal
with the case of the normalized Ricci flow in the same way and give an
interesting application. For the unnormalized Ricci flow, we first have
###### Theorem 1.1.
Let $g(t)$, $t\in[0,T)$, be a solution of the unnormalized Ricci flow (1.1) on
a closed manifold $M^{n}$ and $\lambda_{1,p}(t)$ be the first eigenvalue of
the $p$-Laplace operator $(p>1)$ of $g(t)$. If there exists a nonnegative
constant $\epsilon$ such that
(1.7) $R_{ij}-\tfrac{R}{p}g_{ij}\geq-\epsilon g_{ij}\quad\quad\mathrm{in}\quad
M^{n}\times[0,T)$
and
(1.8) $R\geq p\cdot\epsilon\quad\mathrm{and}\quad R\not\equiv
p\cdot\epsilon\quad\quad\mathrm{in}\quad M^{n}\times\\{0\\},$
then $\lambda_{1,p}(t)$ is strictly increasing and differentiable almost
everywhere along the unnormalized Ricci flow on $[0,T)$.
###### Remark 1.2.
(1). In [28], the first author proved a similar result as in Theorem 1.1,
where he assumed $p\geq 2$, inequality (1.7) and $R>p\cdot\epsilon$ in
$M^{n}\times\\{0\\}$, which are a little stronger than assumptions of Theorem
1.1. The key difference is that the proof approach here is different from that
in [28].
(2). As mentioned Remark 1.2 in [28], the time interval $[0,T)$ of Theorem 1.1
here may be not the maximal time interval of existence of the unnormalized
Ricci flow. In fact if we trace (1.7) and assume that $p<n$, then we have an
upper bound estimate for the scalar curvature $(\epsilon\neq 0)$. But as we
all known, curvature operator must be blow-up as $t\rightarrow T$ $(T<\infty)$
when the curvature operator is positive and $[0,T)$ is the maximal time
interval (see Theorem 14.1 in [11]).
(3). Theorem 1.1 still holds if the conditions (1.7) and (1.8) are replaced by
$R_{ij}-\tfrac{R}{p}g_{ij}>-\epsilon g_{ij}$ in $M^{n}\times[0,T)$ and $R\geq
p\cdot\epsilon$ in $M^{n}\times\\{0\\}$.
(4). For any closed $2$-surface and $3$-manifold, we can relax the above
assumptions (1.7) and (1.8) to the only initial curvature assumptions by the
Hamilton’s maximum principle. We refer the reader to [28] for similar results.
###### Remark 1.3.
Most recently, in [3] X.-D. Cao, S.-B. Hou and J. Ling derived a monotonicity
formula for the first eigenvalue of $-\Delta+aR$ $(0<a\leq 1/2)$ on closed
surfaces with nonnegative scalar curvature under the Ricci flow. Meanwhile
they obtained various monotonicity formulae and estimates for the first
eigenvalue on closed surfaces.
Furthermore, if less curvature assumptions are given, we can construct two
classes of monotonic (increasing and decreasing) quantities about the first
eigenvalue of the $p$-Laplace operator along the unnormalized Ricci flow. We
refer the reader to Section 4 for the more detailed discussions (see Theorems
4.3 and 4.5, and Corollary 4.6).
For the normalized Ricci flow, unfortunately we may not get any monotonicity
for the first eigenvalue of the $p$-Laplace operator in general. However, if
we know the first $p$-eigenvalue differentiability along the unnormalized
Ricci flow, from the relation to the unnormalized Ricci flow, we can give
another way to derive the first $p$-eigenvalue differentiability along the
normalized Ricci flow (see Theorem 5.1 of Section 5).
Besides, the most important result is that we can construct various monotonic
quantities about the first eigenvalue of the $p$-Laplace operator along the
normalized Ricci flow on closed $2$-surfaces without any curvature assumption.
This also leads to the first $p$-eigenvalue differentiability along the
normalized Ricci flow on closed $2$-surfaces without any curvature assumption.
###### Theorem 1.4.
Let $\tilde{g}(\tilde{t})$, $\tilde{t}\in[0,\infty)$, be a solution of the
normalized Ricci flow (1.2) on a closed surface $M^{2}$ and let
$\lambda_{1,p}(\tilde{t})$ be the first eigenvalue of the $p$-Laplace operator
of the metric $\tilde{g}(\tilde{t})$. Then each of the following quantities
1. (1)
$\lambda_{1,p}(\tilde{t})\cdot\left(\frac{\rho_{0}}{\tilde{r}}-\frac{\rho_{0}}{\tilde{r}}e^{\tilde{r}\tilde{t}}+e^{\tilde{r}\tilde{t}}\right)^{p/2}$
$(p\geq 2)$,
$\lambda_{1,p}(\tilde{t}){\cdot}\kern-3.0pt\left(\frac{\rho_{0}}{\tilde{r}}{-}\frac{\rho_{0}}{\tilde{r}}e^{\tilde{r}\tilde{t}}{+}e^{\tilde{r}\tilde{t}}\right){\cdot}\exp\left[\left(1{-}\frac{p}{2}\right)\kern-2.0pt\frac{C}{\tilde{r}}e^{\tilde{r}\tilde{t}}\right]$
$(1<p<2)$, $\mathrm{if}$ $\chi(M^{2})<0$;
2. (2)
$\lambda_{1,p}(\tilde{t})\cdot\left(1+C\tilde{t}\right)^{p/2}$ $(p\geq 2)$,
$\lambda_{1,p}(\tilde{t})\cdot\left(1+C\tilde{t}\right)\cdot
e^{\left(1{-}p/2\right)C\tilde{t}}$ $(1<p<2)$, $\mathrm{if}$ $\chi(M^{2})=0$;
3. (3)
$\ln\lambda_{1,p}(\tilde{t})+\frac{p}{2}\cdot\left(\frac{C}{\tilde{r}}e^{\tilde{r}\tilde{t}}+\tilde{r}\tilde{t}\right)$
$(p\geq 2)$,
$\ln\lambda_{1,p}(\tilde{t})+\left(2-\frac{p}{2}\right)\frac{C}{\tilde{r}}e^{\tilde{r}\tilde{t}}+\tilde{r}\tilde{t}$
$(1<p<2)$, $\mathrm{if}$ $\chi(M^{2})>0$
is increasing and therefore $\lambda_{1,p}(\tilde{t})$ is differentiable
almost everywhere along the normalized Ricci flow on $[0,\infty)$, where
$\chi(M^{2})$ denotes its Euler characteristic, $\rho_{0}:=\inf_{M^{2}}R(0)$
and $C>0$ is a constant depending only on the initial metric.
In the same way, we can also obtain the decreasing quantities on closed
$2$-surfaces.
###### Theorem 1.5.
Under the same assumptions as in Theorem 1.4, then each of the following
quantities
1. (1)
$\ln\lambda_{1,p}(\tilde{t})-\frac{p}{2}\cdot\frac{C}{\tilde{r}}e^{\tilde{r}\tilde{t}}$
$(p\geq 2)$,
$\lambda_{1,p}(\tilde{t}){\cdot}\kern-3.0pt\left(\frac{\rho_{0}}{\tilde{r}}{-}\frac{\rho_{0}}{\tilde{r}}e^{\tilde{r}\tilde{t}}{+}e^{\tilde{r}\tilde{t}}\right)^{\kern-2.0pt(\frac{p}{2}-1)}\kern-6.0pt{\cdot}\exp\kern-2.0pt\left({-}\frac{C}{\tilde{r}}e^{\tilde{r}\tilde{t}}\right)$
$(1<p<2)$, $\mathrm{if}$ $\chi(M^{2})<0$;
2. (2)
$\ln\lambda_{1,p}(\tilde{t})-\frac{p}{2}\cdot C\tilde{t}$ $(p\geq 2)$,
$\lambda_{1,p}(\tilde{t})\cdot\left(1+C\tilde{t}\right)^{(\frac{p}{2}-1)}\cdot
e^{-C\tilde{t}}$ $(1<p<2)$ $\mathrm{if}$ $\chi(M^{2})=0$;
3. (3)
$\ln\lambda_{1,p}(\tilde{t})-\frac{p}{2}\cdot\frac{C}{\tilde{r}}e^{\tilde{r}\tilde{t}}$
$(p\geq 2)$,
$\ln\lambda_{1,p}(\tilde{t}){-}\left(2{-}\frac{p}{2}\right)\frac{C}{\tilde{r}}{\cdot}e^{\tilde{r}\tilde{t}}{-}\left(1{-}\frac{p}{2}\right)\tilde{r}\tilde{t}$
$(1<p<2)$ $\mathrm{if}$ $\chi(M^{2})>0$
is decreasing and therefore $\lambda_{1,p}(\tilde{t})$ is differentiable
almost everywhere along the normalized Ricci flow on $[0,\infty)$, where
$\chi(M^{2})$, $\rho_{0}$ and $C$ are as in Theorem 1.4.
###### Remark 1.6.
We may apply similar techniques above to obtain interesting monotonic
quantities about the first eigenvalue of the $p$-Laplace operator along the
normalized Ricci flow in high-dimensional cases under some curvature
assumptions, but the proof needs more computing. Here we omit this aspect.
Some parts of results for $p=2$ above were proved by L. Ma [20] and J. Ling
[19]. But our method of proof is different from theirs. Their proofs strongly
depend on the differentiability for the eigenvalues and the corresponding
eigenfunctions. But in our setting ($p\geq 2$) it is not clear whether the
eigenvalue or the corresponding eigenfunction is differentiable in advance.
Our method is similar to X.-D. Cao’s trick in [1], which does not depend on
the differentiability for the eigenvalues or the corresponding eigenfunctions.
With the help of Theorem 1.4, our below topic is to extend an earlier J.
Ling’s result for $p=2$ (see [18]). Here we call it $p$-eigenvalue comparison-
type theorem. For the convenience of introducing our result, we shall state a
well-known fact, which was proved by R.S. Hamilton and B. Chow (see also [7],
chapter 5 for details).
Theorem E. (Chow-Hamilton, [5] and [12]) _If $(M^{2},g)$ is a closed surface,
there exists a unique solution $g(t)$ of the normalized Ricci flow (1.2). The
solution exists for all the time. As $t\rightarrow\infty$, the metrics $g(t)$
converge uniformly in any $C^{k}$-norm to a smooth metric
$\bar{g}(=g(\infty))$ of constant curvature._
Let $(M^{2},g)$ be a closed surface. Let $K_{g}$, $\kappa_{g}$,
$\mathrm{Area}_{g}(M^{2})$ denote the Gauss curvature, the minimum of the
Gauss curvature, the area of the surface $M^{2}$, respectively.
$\lambda_{1,p}(g)$ denotes the first eigenvalue of the $p$-Laplace operator
with respect to the metric $g$. Then we prove that
###### Theorem 1.7.
($p$-eigenvalue comparison-type theorem). Suppose that $(M^{2},g)$ is a closed
surface with its Euler characteristic $\chi(M^{2})<0$. The Ricci flow with
initial metric $g$ converges uniformly to a smooth metric $\bar{g}$ of
constant curvature. Then for any $p\geq 2$,
(1.9)
$\frac{\lambda_{1,p}(\bar{g})}{\lambda_{1,p}(g)}\geq\left(\frac{\kappa_{\bar{g}}}{\kappa_{g}}\right)^{p/2}$
and the constant Gauss curvature for metric $\bar{g}$ is
$\kappa_{\bar{g}}=2\pi\chi(M^{2})/\mathrm{Area}_{g}(M^{2})$.
In conclusion, our new contribution of this paper is to obtain the
monotonicity for the first eigenvalue of the $p$-Laplace operator, and
construct many monotonic quantities involving the first eigenvalue of the
$p$-Laplace operator along the Ricci flow under some different curvature
assumptions. By the monotonic property, we can judge the differentiability in
some sense for the first eigenvalue of a nonlinear operator with respect to
evolving metrics. Using the same idea of our arguments, we easily see that
Perelman’s eigenvalue is differentiable almost everywhere111Note that many
literatures have pointed out that the differentiability for Perelman’s
eigenvalue follows from eigenvalue perturbation theory (see also Section 2)..
From Theorem 1.4 above and Corollary 5.4 below, we also see that the first
eigenvalue of the $p$-Laplace operator is differentiable almost everywhere
along the Ricci flow on closed $2$-surfaces without any curvature assumption.
For high-dimensional case, the similar differentiability property still holds
as long as some curvature conditions are satisfied. Of course, the proofs of
these results involve many skilled arguments and computations. Finally, it
should be remarked that it is still an open question whether its corresponding
eigenfunction is differentiable with respect to $t$-variable along the Ricci
flow.
The rest of this paper is organized as follows. In Section 2, we will recall
some notations about $p$-Laplace, and prove that $\lambda_{1,p}(g(t))$ is a
continuous function along the Ricci flow. In Section 3, we will give
Proposition 3.1. Using this proposition, we can finish the proof of Theorem
1.1. In Section 4, we will construct two classes of monotonic quantities about
the first eigenvalue of the $p$-Laplace operator along the unnormalized Ricci
flow. In Section 5, we will discuss the normalized Ricci flow case and mainly
prove Theorems 1.4 and 1.5. In Section 6, we shall prove $p$-eigenvalue
comparison-type theorem, i.e., Theorem 1.7. In Section 7, we will use the same
method to study the first eigenvalue of the $p$-Laplace with respect to
general evolving metrics, especially to the Yamabe flow.
## 2\. Preliminaries
In this section, we will first recall some definitions about the $p$-Laplace
operator and give the definition for the first eigenvalue of the $p$-Laplace
operator under the Ricci flow on a closed manifold. Then we will show that the
first eigenvalue of the $p$-Laplace operator is a continuous function along
the Ricci flow.
Let $M^{n}$ be an $n$-dimensional connected closed Riemannian manifold and
$g(t)$ be a smooth solution of the Ricci flow on the time interval $[0,T)$.
Consider the nonzero first eigenvalue of the $p$-Laplace operator $(p>1)$ at
time $t$ (also called the first $p$-eigenvalue), where $0\leq t<T$, i.e.,
(2.1) $\lambda_{1,p}(t):={\inf\limits_{f\neq
0}}\left\\{\frac{\int_{M}|df|^{p}d\mu}{\int_{M}|f|^{p}d\mu}:f\in
W^{1,p}(M),\quad\int_{M}|f|^{p-2}fd\mu=0\right\\}.$
Obviously, this infimum does not change when $W^{1,p}(M)$ is replaced by
$C^{\infty}(M)$. For the fixed time, this infimum is achieved by a
$C^{1,\alpha}$ ($0<\alpha<1$) eigenfunction $f_{p}$ (see [25] and [26]). The
corresponding eigenfunction $f_{p}$ satisfies the following Euler-Lagrange
equation
(2.2) $\Delta_{p}f_{p}=-\lambda_{1,p}(t)|f_{p}|^{p-2}f_{p},$
where $\Delta_{p}$ $(p>1)$ is the $p$-Laplace operator with respect to $g(t)$,
given by
(2.3)
$\Delta_{p_{g(t)}}f:=\mathrm{div}_{g(t)}\left({|df|_{g(t)}^{p-2}}df\right).$
If $p=2$, the $p$-Laplace operator reduces to the Laplace-Beltrami operator.
The most difference between two operators is that the $p$-Laplace operator is
a nonlinear operator in general, but the Laplace-Beltrami operator is a linear
operator.
Note that it is not clear whether the first eigenvalue of the $p$-Laplace
operator or its corresponding eigenfunction is $C^{1}$-differentiable along
the Ricci flow. When $p=2$, where $\Delta_{p}$ is the Laplace-Beltrami
operator, many papers have pointed out that their differentiability follows
from eigenvalue perturbation theory (for example, see [2], [13], [15] and
[24]). But $p\neq 2$, as far as we are aware, the differentiability for the
first eigenvalue of the $p$-Laplace operator or its corresponding
eigenfunction along the Ricci flow has not been known until now. Even we have
not known whether they are locally Lipschitz. So we can not use the method
used by L. Ma to derive the monotonicity for the first eigenvalue of the
$p$-Laplace operator.
Although we do not know the differentiability for $\lambda_{1,p}(t)$, we will
see that $\lambda_{1,p}(g(t))$ in fact is a continuous function along the
Ricci flow on $[0,T)$. This is a consequence of the following elementary
result.
###### Theorem 2.1.
If $g_{1}$ and $g_{2}$ are two metrics which satisfy
$(1+\varepsilon)^{-1}g_{1}\leq g_{2}\leq(1+\varepsilon)g_{1},$
then for any $p>1$, we have
(2.4)
$(1+\varepsilon)^{-(n+\frac{p}{2})}\leq\frac{\lambda_{1,p}(g_{1})}{\lambda_{1,p}(g_{2})}\leq(1+\varepsilon)^{(n+\frac{p}{2})}.$
In particular, $\lambda_{1,p}(g(t))$ is a continuous function in the
$t$-variable.
To prove this theorem, we first need the following fact. Let $(M^{n},g)$ be an
$n$-dimensional closed Riemannian manifold. For any non-constant function $f$,
consider the following $C^{1}$-function on $s\in(-\infty,\infty)$
$F(s):=\int_{M^{n}}\left|f+s\right|^{p}d\mu_{g},\,\,\,(p>1).$
###### Lemma 2.2.
There exists a unique $s_{0}\in(-\infty,\infty)$ such that
(2.5)
$F(s_{0})=\min\limits_{s\in\mathbb{R}}F(s)\,\,\,\,\,\,\mathrm{if\,\,\,and\,\,\,only\,\,\,if}\,\,\,\,\,\,\int_{M}\left|f+s_{0}\right|^{p-2}(f+s_{0})d\mu_{g}=0.$
###### Proof.
Note that the function $|x|^{p}$ $(p>1)$ is a strictly convex function on
$x\in\mathbb{R}$. Meanwhile we can also check that
$\lim_{|s|\rightarrow{+}\infty}F(s)\rightarrow{+}\infty,\quad\quad
F^{\prime}(s)=p\int_{M}\left|f+s\right|^{p-2}\left(f+s\right)d\mu_{g}.$
Therefore $F(s)$ is a strictly convex function and there exists a unique
$s_{0}\in(-\infty,+\infty)$ such that
(2.6)
$F(s_{0})=\min\limits_{s\in\mathbb{R}}F(s)\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,\,F^{\prime}(s)=p\int_{M}\left|f+s_{0}\right|^{p-2}(f+s_{0})d\mu_{g}=0.$
∎
Now using Lemma 2.2, we give the proof of Theorem 2.1.
###### Proof of Theorem 2.1.
Since the volume form $d\mu$ has degree $n/2$ in $g$, we have
(2.7) $(1+\varepsilon)^{-n/2}d\mu_{g_{1}}\leq
d\mu_{g_{2}}\leq(1+\varepsilon)^{n/2}d\mu_{g_{1}}.$
Taking $f$ be the first eigenfunction of $\Delta_{p}$ with respect to the
metric $g_{1}$, we see that
(2.8)
$\displaystyle\lambda_{1,p}(g_{1})=\frac{\int_{M}|df|_{g_{1}}^{p}d\mu_{g_{1}}}{\int_{M}|f|^{p}d\mu_{g_{1}}}\,\,\,\,\,\,\mathrm{and}\,\,\,\int_{M}|f|^{p-2}fd\mu_{g_{1}}=0.$
Since $\int_{M}|f|^{p-2}fd\mu_{g_{1}}=0$, Lemma 2.2 implies
$\int_{M}|f|^{p}d\mu_{g_{1}}=\min\limits_{s\in\mathbb{R}}\int_{M}\left|f+s\right|^{p}d\mu_{g_{1}}.$
Hence by (2.8), we conclude that
(2.9)
$\displaystyle\lambda_{1,p}(g_{1})=\frac{\int_{M}|df|_{g_{1}}^{p}d\mu_{g_{1}}}{\int_{M}|f|^{p}d\mu_{g_{1}}}\geq\frac{\int_{M}|d(f+s)|_{g_{1}}^{p}d\mu_{g_{1}}}{\int_{M}|f+s|^{p}d\mu_{g_{1}}}.$
Keep in mind that under another metric $g_{2}$, for function
$F(s)=\int_{M}\left|f+s\right|^{p}d\mu_{g_{2}}$, there exists a unique
$s_{0}\in(-\infty,+\infty)$ such that
(2.10)
$F(s_{0})=\min\limits_{s\in\mathbb{R}}F(t)\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,\,F^{\prime}(s)=p\int_{M}\left|f+s_{0}\right|^{p-2}(f+s_{0})d\mu_{g_{2}}=0.$
Using (2.7), from (2.9) we conclude that
(2.11)
$\displaystyle\lambda_{1,p}(g_{1})\geq\frac{\int_{M}|d(f+s)|_{g_{1}}^{p}d\mu_{g_{1}}}{\int_{M}|f+s|^{p}d\mu_{g_{1}}}\geq(1+\varepsilon)^{-(n+\frac{p}{2})}\cdot\frac{\int_{M}|d(f+s)|_{g_{2}}^{p}d\mu_{g_{2}}}{\int_{M}|f+s|^{p}d\mu_{g_{2}}}.$
Letting $s=s_{0}$ in (2.11) yields
(2.12)
$\displaystyle\lambda_{1,p}(g_{1})\geq(1+\varepsilon)^{-(n+\frac{p}{2})}\cdot\frac{\int_{M}|d(f+s_{0})|_{g_{2}}^{p}d\mu_{g_{2}}}{\int_{M}|f+s_{0}|^{p}d\mu_{g_{2}}}\geq(1+\varepsilon)^{-(n+\frac{p}{2})}\cdot\lambda_{1,p}(g_{2}),$
where for the last inequality we used
$\int_{M}\left|f+s_{0}\right|^{p-2}(f+s_{0})d\mu_{g_{2}}=0$ and the definition
for the first $p$-eigenvalue with respect to the metric $g_{2}$.
From the course of this proof, we easily see that (2.12) still holds if we
exchange $g_{1}$ and $g_{2}$. Hence
(2.13)
$(1+\varepsilon)^{-(n+\frac{p}{2})}\leq\frac{\lambda_{1,p}(g_{1})}{\lambda_{1,p}(g_{2})}\leq(1+\varepsilon)^{(n+\frac{p}{2})}.$
This completes the proof of Theorem 2.1. ∎
## 3\. Proof of Theorem 1.1
In this section, we will prove Theorem 1.1 in introduction. In order to
achieve this, we first prove the following proposition. Our proof involves
choosing a proper smooth function, which seems to be a delicate trick.
###### Proposition 3.1.
Let $g(t)$, $t\in[0,T)$, be a solution of the unnormalized Ricci flow (1.1) on
a closed manifold $M^{n}$ and let $\lambda_{1,p}(t)$ be the first eigenvalue
of the $p$-Laplace operator along this flow. For any $t_{1},t_{2}\in[0,T)$ and
$t_{2}\geq t_{1}$, we have
(3.1)
$\lambda_{1,p}(t_{2})\geq\lambda_{1,p}(t_{1})+\int^{t_{2}}_{t_{1}}\mathcal{G}(g(\xi),f(\xi))d\xi,$
where
(3.2) $\mathcal{G}(g(t),f(t)):=p\int_{M}|df|^{p-2}Ric(\nabla f,\nabla
f)d\mu-p\int_{M}\Delta_{p}f\frac{\partial f}{\partial
t}d\mu-\int_{M}|df|^{p}Rd\mu$
and where $f(t)$ is any $C^{\infty}$ function satisfying
$\int_{M}|f|^{p}d\mu=1$ and $\int_{M}|f|^{p-2}fd\mu=0$, such that at time
$t_{2}$, $f(t_{2})$ is the corresponding eigenfunction of
$\lambda_{1,p}(t_{2})$.
###### Proof.
Set
$G(g(t),f(t)):=\int_{M}|df(t)|_{g(t)}^{p}d\mu_{g(t)}.$
We _claim_ that, for any time $t_{2}\in(0,T)$, there exists a $C^{\infty}$
function $f(t)$ satisfying
(3.3)
$\int_{M}|f(t)|^{p}d\mu_{g(t)}=1\quad\quad\mathrm{and}\quad\int_{M}|f(t)|^{p{-}2}f(t)d\mu_{g(t)}=0$
and such that at time $t_{2}$, $f(t_{2})$ is the eigenfunction for
$\lambda_{1,p}(t_{2})$ of $\Delta_{p_{g(t_{2})}}$. To see this, at time
$t_{2}$, we first let $f_{2}=f(t_{2})$ be the eigenfunction for the eigenvalue
$\lambda_{1,p}(t_{2})$ of $\Delta_{p_{g(t_{2})}}$. Then we consider the
following smooth function
(3.4)
$h(t)=f_{2}\left[\frac{\mathrm{det}(g_{ij}(t_{2}))}{\mathrm{det}(g_{ij}(t))}\right]^{\frac{1}{2(p-1)}}$
under the Ricci flow $g_{ij}(t)$. Later we normalize this smooth function
(3.5) $f(t)=\frac{h(t)}{\left({\int_{M}|h(t)|^{p}d\mu}\right)^{1/p}}$
under the Ricci flow $g_{ij}(t)$. From above, we can easily check that $f(t)$
satisfies (3.3).
By the definition for $\lambda_{1,p}(t_{2})$, we have
(3.6) $\lambda_{1,p}(t_{2})=G(g(t_{2}),f(t_{2})).$
Notice that under the unnormalized Ricci flow,
(3.7) $\frac{\partial}{\partial
t}|df|^{p}=p|df|^{p-2}\left(R_{ij}f_{i}f_{j}+f_{i}\frac{\partial
f_{i}}{\partial t}\right),\quad\quad\frac{\partial}{\partial
t}\left(d\mu\right)=-Rd\mu,$
where $f_{i}$ and $R_{ij}$ denote the covariant derivative of $f$ and Ricci
curvature with respect to the Levi-Civita connection of $g(t)$, respectively.
Note that $G(g(t),f(t))$ is a smooth function with respect to $t$-variable. So
(3.8) $\displaystyle\mathcal{G}(g(t),f(t)):$
$\displaystyle=\frac{d}{dt}G(g(t),f(t))$
$\displaystyle=\int_{M}\frac{\partial}{\partial
t}|df|^{p}d\mu-\int_{M}|df|^{p}Rd\mu$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu+p\int_{M}|df|^{p-2}f_{i}\frac{\partial}{\partial
t}(f_{i})d\mu-\int_{M}|df|^{p}Rd\mu$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu-p\int_{M}\nabla_{i}(|df|^{p-2}f_{i})\frac{\partial
f}{\partial t}d\mu-\int_{M}|df|^{p}Rd\mu$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu-p\int_{M}\Delta_{p}f\frac{\partial
f}{\partial t}d\mu-\int_{M}|df|^{p}Rd\mu,$
where we used (3.7). Taking integration on the both sides of (3.8) between
$t_{1}$ and $t_{2}$, we conclude that
(3.9)
$G(g(t_{2}),f(t_{2}))-G(g(t_{1}),f(t_{1}))=\int^{t_{2}}_{t_{1}}\mathcal{G}(g(\xi),f(\xi))d\xi,$
where $t_{1}\in[0,T)$ and $t_{2}\geq t_{1}$. Noticing
$G(g(t_{1}),f(t_{1}))\geq\lambda_{1,p}(t_{1})$ and combining (3.6) with (3.9),
we arrive at
$\lambda_{1,p}(t_{2})\geq\lambda_{1,p}(t_{1})+\int^{t_{2}}_{t_{1}}\mathcal{G}(g(\xi),f(\xi))d\xi,$
where $\mathcal{G}(g(\xi),f(\xi))$ satisfies (3.8). ∎
In the following of this section, we will finish the proof of Theorem 1.1
using Proposition 3.1.
###### Proof of Theorem 1.1.
In fact, we only need to show that $\mathcal{G}(g(t),f(t))>0$ in Proposition
3.1. Notice that at time $t_{2}$, $\lambda_{1,p}(t_{2})$ is the first
eigenvalue and $f(t_{2})$ is the corresponding eigenfunction. Therefore at
time $t_{2}$, we have
(3.10) $\displaystyle\mathcal{G}(g(t_{2}),f(t_{2}))$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu-p\int_{M}\Delta_{p}f\frac{\partial
f}{\partial t}d\mu-\int_{M}|df|^{p}Rd\mu$
$\displaystyle=p\int_{M}|df|^{p{-}2}R_{ij}f_{i}f_{j}d\mu+p\lambda_{1,p}(t_{2})\int_{M}|f|^{p-2}f\frac{\partial
f}{\partial t}d\mu\int_{M}|df|^{p}Rd\mu,$
where we used
$\Delta_{p}f(t_{2})=-\lambda_{1,p}(t_{2})|f(t_{2})|^{p-2}f(t_{2})$.
Under the unnormalized Ricci flow, from the constraint condition
$\frac{d}{dt}\int_{M}\left|f(t)\right|^{p}d\mu_{g(t)}=0,$
we know that
(3.11) $p\int_{M}|f|^{p-2}f\frac{\partial f}{\partial
t}d\mu=\int_{M}|f|^{p}Rd\mu.$
Substituting this into the above formula (3.10) and combining the assumption
of Theorem 1.1: $R_{ij}-\tfrac{R}{p}g_{ij}\geq-\epsilon g_{ij}$ in
$M^{n}\times[0,T)$, we obtain
(3.12) $\displaystyle\mathcal{G}(g(t_{2}),f(t_{2}))$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu+p\lambda_{1,p}(t_{2})\int_{M}|f|^{p-2}f\frac{\partial
f}{\partial t}d\mu-\int_{M}|df|^{p}Rd\mu$
$\displaystyle=\lambda_{1,p}(t_{2})\int_{M}|f|^{p}Rd\mu+\int_{M}|df|^{p-2}(pR_{ij}-Rg_{ij})f_{i}f_{j}d\mu$
$\displaystyle\geq\lambda_{1,p}(t_{2})\int_{M}|f|^{p}Rd\mu-p\cdot\epsilon\int_{M}|df|^{p}d\mu$
$\displaystyle=\lambda_{1,p}(t_{2})\int_{M}|f|^{p}(R-p\cdot\epsilon)d\mu.$
Meanwhile we also have another assumption of Theorem 1.1 on the scalar
curvature
$R\geq p\cdot\epsilon\,\,\,\mathrm{and}\,\,\,R\not\equiv
p\cdot\epsilon\quad\quad\mathrm{in}\quad M^{n}\times\\{0\\}.$
It is well-known that $R\geq p\cdot\epsilon$ is preserved by the unnormalized
Ricci flow. Furthermore by the strong maximum principle (for example, see
Proposition 12.47 of Chapter 12 in [8]), we conclude that
(3.13) $R>p\cdot\epsilon\quad\quad\mathrm{in}\quad M^{n}\times[0,T).$
Plugging this into (3.12) implies $\mathcal{G}(g(t_{2}),f(t_{2}))>0$. Notice
that $f(x,t)$ is a smooth function with respect to $t$-variable. Therefore we
can arrive at $\mathcal{G}(g(\xi),f(\xi))>0$ in any sufficient small
neighborhood of $t_{2}$. Hence
(3.14) $\int^{t_{2}}_{t_{1}}\mathcal{G}(g(\xi),f(\xi))d\xi>0$
for any $t_{1}<t_{2}$ sufficiently close to $t_{2}$. In the end, by
Proposition 3.1, we conclude
$\lambda_{1,p}(t_{2})>\lambda_{1,p}(t_{1})$
for any $t_{1}<t_{2}$ sufficiently close to $t_{2}$. Since $t_{2}\in[0,T)$ is
arbitrary, then the first part of Theorem 1.1 follows.
As for the differentiability for $\lambda_{1,p}(t)$, since $\lambda_{1,p}(t)$
is increasing on the time interval $[0,T)$ under curvature conditions of the
theorem, by the classical Lebesgue’s theorem (for example, see Chapter 4 in
[22]), it is easy to see that $\lambda_{1,p}(t)$ is differentiable almost
everywhere on $[0,T)$. ∎
###### Remark 3.2.
(1). Our proof of the first $p$-eigenvalue monotonicity is not derived from
the differentiability for $\lambda_{1,p}(t)$ or its corresponding
eigenfunction. In fact we do not know whether they are differentiable in
advance. It would be interesting to find out whether the corresponding
eigenfunction of the $p$-Laplace operator is a $C^{1}$-differentiable function
with respect to $t$-variable along the Ricci flow on a closed manifold
$M^{n}$. If it is true, we can use L. Ma’s method to get our result.
(2). If $p=2$, the above theorem is similar to L. Ma’s main result for the
first eigenvalue of the Laplace operator in [20].
(3). Using this method, we can not get any monotonicity for higher order
eigenvalues of the $p$-Laplace operator.
## 4\. Monotonic quantities along unnormalized Ricci flow
Motivated by the works of X.-D. Cao [1] and [2], in this section, we first
introduce a new smooth eigenvalue function (see (4.1) below), and then we give
the following useful Lemma 4.1, resembling Proposition 3.1 of Section 3. Using
this lemma, we can obtain two classes of interesting monotonic quantities
along the unnormalized Ricci flow, that is, Theorem 4.3, Theorem 4.5 and
Corollary 4.6. Then by means of those monotonic quantities, we can prove the
differentiability for the first eigenvalue of the $p$-Laplace operator along
the unnormalized Ricci flow.
Let $M^{n}$ be an $n$-dimensional connected closed Riemannian manifold and
$\tilde{g}(\tilde{t})$ be a smooth solution of the normalized Ricci flow on
the time interval $[0,\infty)$. Now we can define a general smooth eigenvalue
function
(4.1)
$\lambda_{1,p}(\tilde{f},\tilde{t}):=\int_{M}\tilde{\Delta}_{p_{\tilde{g}(\tilde{t})}}\tilde{f}\cdot\tilde{f}d\tilde{\mu}=\int_{M}|d\tilde{f}|^{p}d\tilde{\mu},$
where $\tilde{f}$ is a smooth function and satisfies the following equalities
(4.2)
$\int_{M}|\tilde{f}(\tilde{t})|^{p}d\tilde{\mu}_{\tilde{g}(\tilde{t})}=1\quad\quad\mathrm{and}\quad\int_{M}|\tilde{f}(\tilde{t})|^{p-2}\tilde{f}(\tilde{t})d\tilde{\mu}_{\tilde{g}(\tilde{t})}=0.$
From the proof of Proposition 3.1, we see that the above restriction (4.2) can
be achieved.
Obviously, at time $t_{0}$, if $\tilde{f}$ is the corresponding eigenfunction
of the first eigenvalue $\lambda_{1,p}(t_{0})$, then
$\lambda_{1,p}(\tilde{f},t_{0})=\lambda_{1,p}(t_{0}).$
For the convenient of writing, we shall drop the tilde over all the variables
used above to distinguish between the normalized and unnormalized Ricci flow.
###### Lemma 4.1.
If $\lambda_{1,p}(t)$ is the first eigenvalue of $\Delta_{p_{g(t)}}$, whose
metric satisfying the normalized Ricci flow and $f(t_{0})$ is the
corresponding eigenfunction of $\lambda_{1,p}(t)$ at time $t_{0}$, then we
have
(4.3) $\displaystyle\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}=$
$\displaystyle\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu+p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu$
$\displaystyle-\int_{M}|df|^{p}Rd\mu-\frac{p}{n}r\lambda_{1,p}(f(t_{0}),t_{0}).$
In particular, for any closed $2$-surface, we have
(4.4) $\displaystyle\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}$
$\displaystyle=\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu+\left(\frac{p}{2}-1\right)\int_{M}|df|^{p}Rd\mu$
$\displaystyle\,\,\,\,\,\,-\frac{p}{2}r\lambda_{1,p}(f(t_{0}),t_{0}),$
where $f$ evolves by (4.2) with the initial data $f(t_{0})$.
###### Proof.
The proof is by direct computations. Here we need to use
$\frac{\partial}{\partial
t}|df|^{p}=p|df|^{p-2}\left(R_{ij}f_{i}f_{j}-\frac{r}{n}g_{ij}f_{i}f_{j}+f_{i}\frac{\partial
f_{i}}{\partial t}\right),\,\,\,\,\,\,\frac{\partial}{\partial
t}(d\mu)=(r-R)d\mu.$
Then
(4.5) $\displaystyle\frac{d\lambda_{1,p}(f,t)}{dt}\Big{|}_{t=t_{0}}$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu+p\int_{M}|df|^{p-2}f_{i}\frac{\partial\left(f_{i}\right)}{\partial
t}d\mu$
$\displaystyle\,\,\,\,\,\,-p\int_{M}|df|^{p}\frac{r}{n}d\mu+\int_{M}|df|^{p}(r-R)d\mu$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu-p\int_{M}\nabla_{i}\left(|df|^{p-2}f_{i}\right)\frac{\partial
f}{\partial t}d\mu$
$\displaystyle\,\,\,\,\,\,-\frac{p}{n}r\lambda_{1,p}(f(t_{0}),t_{0})+\int_{M}|df|^{p}(r-R)d\mu$
$\displaystyle=p\int_{M}|df|^{p-2}R_{ij}f_{i}f_{j}d\mu+p\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p-2}f\frac{\partial
f}{\partial t}d\mu$
$\displaystyle\,\,\,\,\,\,-\frac{p}{n}r\lambda_{1,p}(f(t_{0}),t_{0})+\int_{M}|df|^{p}(r-R)d\mu,$
where we used $f$ is the eigenfunction at time $t_{0}$, i.e., equation (2.2)
at time $t_{0}$. Note that by (4.2), we have
(4.6) $p\int_{M}|f|^{p-2}f\frac{\partial f}{\partial
t}d\mu=\int_{M}|f|^{p}(R-r)d\mu.$
Plugging this into (4.5) yields the desired (4.3). For any closed $2$-surface,
we have $R_{ij}=\frac{R}{2}g_{ij}$. Hence (4.4) follows from (4.3). ∎
###### Remark 4.2.
In [28], the first author used a similar method and proved a similar result
for the unnormalized Ricci flow (see Proposition 2.1 in [28]).
In the following we first obtain increasing quantities along the unnormalized
Ricci flow by using Lemma 4.1.
###### Theorem 4.3.
Let $g(t)$ and $\lambda_{1,p}(t)$ $(p>1)$ be the same as in Theorem 1.1. If
$\rho_{0}:=\inf_{M}R(0)>0$ and
(4.7) $R_{ij}-\tfrac{R}{p}g_{ij}(t)>0\quad\quad\mathrm{in}\quad
M^{n}\times[0,T),$
then the following quantity
(4.8) $\lambda_{1,p}(t)\cdot\left(\rho_{0}^{-1}-2at\right)^{\frac{1}{2a}},$
is strictly increasing and therefore $\lambda_{1,p}(t)$ is differentiable
almost everywhere along the unnormalized Ricci flow on $[0,T^{\prime})$, where
$a{:=}\max\\{\frac{1}{n},\frac{n}{p^{2}}\\}$ and
$T^{\prime}{:=}\min\\{\frac{1}{2a\rho_{0}},T\\}$.
###### Proof.
We assume that at time $t_{0}\in[0,T)$, if $g$ is the corresponding
eigenfunction of $\lambda_{1,p}(t_{0})$, then under the unnormalized Ricci
flow, we can construct a smooth function $f$ satisfying
$\int_{M}|f(t)|^{p}d\mu_{g(t)}=1\quad\quad\mathrm{and}\quad\int_{M}|f(t)|^{p{-}2}f(t)d\mu_{g(t)}=0,$
and such that at time $t=t_{0}$, $f=g$ is the eigenfunction of
$\lambda_{1,p}(t_{0})$. Meanwhile we can define a general smooth eigenvalue
function $\lambda_{1,p}(f,t)$ as (4.1) under the unnormalized Ricci flow.
Obviously, we have
$\lambda_{1,p}(f(t_{0}),t_{0})=\lambda_{1,p}(t_{0}).$
According to (4.3) of Lemma 4.1, we have
(4.9)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}=\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu+\int_{M}|df|^{p-2}(pR_{ij}-Rg_{ij})f_{i}f_{j}d\mu,$
where $f$ is a smooth function satisfying the above assumptions. By the
assumption $R_{ij}-\frac{R}{p}g_{ij}>0$ of Theorem 4.3, we get
(4.10)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}>\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu.$
The evolution of the scalar curvature $R$ under the unnormalized Ricci flow
$\frac{\partial}{\partial t}R=\Delta R+2|Ric|^{2}$
and inequality $|Ric|^{2}\geq a{R^{2}}$
($a:=\max\\{\frac{1}{n},\frac{n}{p^{2}}\\}$) imply
(4.11) $\displaystyle\frac{\partial}{\partial t}R\geq\Delta R+2aR^{2}.$
Since the solutions to the corresponding ODE
${d\rho}/{dt}=2a\rho^{2}$
are
$\displaystyle\rho(t)=\frac{1}{{\rho_{0}}^{-1}-2at},\quad t\in[0,T^{\prime}),$
where $\rho_{0}:=\inf_{M}R(0)$ and
$T^{\prime}:=\min\\{(2a\rho_{0})^{-1},T\\}$. Using the maximum principle to
(4.11), we have $R(x,t)\geq\rho(t)$. Therefore (4.10) becomes
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}>\lambda_{1,p}(f(t_{0}),t_{0})\cdot\rho(t_{0}).$
Note that $\lambda_{1,p}(f,t)$ and $\rho(t)$ are both smooth functions with
respect to $t$-variable. Hence we have
(4.12) $\frac{d}{dt}\lambda_{1,p}(f,t)>\lambda_{1,p}(f(t),t)\cdot\rho(t)$
in any sufficiently small neighborhood of $t_{0}$. Now integrating the above
inequality with respect to time $t$ on time interval $[t_{1},t_{0}]$, we get
(4.13) $\displaystyle\ln$
$\displaystyle\lambda_{1,p}(f(t_{0}),t_{0})-\ln\lambda_{1,p}(f(t_{1}),t_{1})$
$\displaystyle>\left(-\frac{1}{2a}\right)\cdot\ln\left(\rho_{0}^{-1}-2at\right)\Big{|}_{t=t_{0}}-\left(-\frac{1}{2a}\right)\cdot\ln\left(\rho_{0}^{-1}-2at\right)\Big{|}_{t=t_{1}}$
for any $t_{1}<t_{0}$ sufficiently close to $t_{0}$. Note that
$\lambda_{1,p}(f(t_{0}),t_{0})=\lambda_{1,p}(t_{0})$ and
$\lambda_{1,p}(f(t_{1}),t_{1})\geq\lambda_{1,p}(t_{1})$. Then (4.13) becomes
$\ln\lambda_{1,p}(t_{0})+\ln\left(\rho_{0}^{-1}-2at_{0}\right)^{\frac{1}{2a}}>\ln\lambda_{1,p}(t_{1})+\ln\left(\rho_{0}^{-1}-2at_{1}\right)^{\frac{1}{2a}}.$
Namely,
$\lambda_{1,p}(t_{0})\cdot\left(\rho_{0}^{-1}-2at_{0}\right)^{\frac{1}{2a}}>\lambda_{1,p}(t_{1})\cdot\left(\rho_{0}^{-1}-2at_{1}\right)^{\frac{1}{2a}}$
for any $t_{1}<t_{0}$ sufficiently close to $t_{0}$. Since $t_{0}$ is
arbitrary, then (4.8) follows.
Now we know that
$\lambda_{1,p}(t)\cdot\left(\rho_{0}^{-1}-2at\right)^{\frac{1}{2a}}$
is increasing along the unnormalized Ricci flow. Moreover,
$\left(\rho_{0}^{-1}-2at\right)^{\frac{1}{2a}}$ is a smooth function. Hence by
the Lebesgue’s theorem, $\lambda_{1,p}(t)$ is differentiable almost everywhere
along the unnormalized Ricci flow on $[0,T^{\prime})$. ∎
###### Remark 4.4.
Since function $\left(\rho_{0}^{-1}-2at\right)^{\frac{1}{2a}}$ is decreasing
in $t$-variable, Theorem 4.3 also implies that $\lambda_{1,p}(t)$ is strictly
increasing along the unnormalized Ricci flow on $[0,T^{\prime})$.
We also have decreasing quantities along the unnormalized Ricci flow.
###### Theorem 4.5.
Let $g(t)$ and $\lambda_{1,p}(t)$ $(p>1)$ be the same as in Theorem 1.1. If
(4.14) $0\leq R_{ij}<\tfrac{R}{p}g_{ij}(t)\quad\quad\mathrm{in}\quad
M^{n}\times[0,T),$
then the following quantity
(4.15)
$\lambda_{1,p}(t)\cdot\left(\sigma_{0}^{-1}-\frac{2n}{p^{2}}t\right)^{\frac{p^{2}}{2n}}$
is strictly decreasing and therefore $\lambda_{1,p}(t)$ is differentiable
almost everywhere along the unnormalized Ricci flow on $[0,T^{\prime})$, where
$\sigma_{0}:=\sup_{M}R(0)$ and
$T^{\prime}:=\min\\{\frac{p^{2}}{2n\sigma_{0}},T\\}$.
###### Proof.
The proof is similar to that of Theorem 4.3 with the difference that we need
to estimate the upper bounds of the right hand side of (4.16). Here we only
briefly sketch the proof. According to (4.3) of Lemma 4.1, we have
(4.16)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}=\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu+\int_{M}|df|^{p-2}(pR_{ij}-Rg_{ij})f_{i}f_{j}d\mu,$
where $f$ is a smooth function satisfying the same assumptions as in the proof
of Theorem 4.3.
Note that $0\leq R_{ij}<\frac{R}{p}g_{ij}$ implies
$|Ric|^{2}<\frac{n}{p^{2}}R^{2}$. So the evolution of the scalar curvature $R$
under the unnormalized Ricci flow
$\frac{\partial}{\partial t}R=\Delta R+2|Ric|^{2}$
implies
(4.17) $\displaystyle\frac{\partial}{\partial t}R\leq\Delta
R+\frac{2n}{p^{2}}R^{2}.$
Applying the maximum principle to (4.17), we have
$0\leq R(x,t)\leq\sigma(t),$
where
$\sigma(t)=\frac{1}{{\sigma_{0}}^{-1}-\tfrac{2n}{p^{2}}t},\quad
t\in[0,T^{\prime}),$
and where $\sigma_{0}:=\sup_{M}R(0)$ and
$T^{\prime}:=\min\\{\frac{p^{2}}{2n\sigma_{0}},T\\}$.
Substituting $0\leq R(x,t)\leq\sigma(t)$ and $0\leq R_{ij}<\frac{R}{p}g_{ij}$
into (4.16) yields
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}<\lambda_{1,p}(f(t_{0}),t_{0})\cdot\sigma(t_{0}).$
Hence
$\frac{d}{dt}\lambda_{1,p}(f,t)<\lambda_{1,p}(f(t),t)\cdot\sigma(t)$
in any sufficiently small neighborhood of $t_{0}$. Integrating this inequality
with respect to time $t$ on time interval $[t_{0},t_{1}]$ yields
$\lambda_{1,p}(t_{1})\cdot\left(\sigma_{0}^{-1}-\frac{2n}{p^{2}}t_{1}\right)^{\frac{p^{2}}{2n}}<\lambda_{1,p}(t_{0})\cdot\left(\sigma_{0}^{-1}-\frac{2n}{p^{2}}t_{0}\right)^{\frac{p^{2}}{2n}}$
for any $t_{1}>t_{0}$ sufficiently close to $t_{0}$, where we used
$\lambda_{1,p}(f,t_{0})=\lambda_{1,p}(t_{0})$ and
$\lambda_{1,p}(f,t_{1})\geq\lambda_{1,p}(t_{1})$. Since $t_{0}$ is arbitrary,
then Theorem 4.5 follows. ∎
For any closed $3$-manifold, we have
###### Corollary 4.6.
Let $g(t)$ and $\lambda_{1,p}(t)$ be the same as in Theorem 1.1., where we
assume $n=3$ and $1<p<3$. If
(4.18) $0\leq R_{ij}(0)<\tfrac{R(0)}{p}g_{ij}(0)\quad\quad\mathrm{in}\quad
M^{3}\times\\{0\\},$
then the conclusion of Theorem 4.5 is also true.
###### Remark 4.7.
Note that if $p=2$, condition (4.18) is the same as positive sectional
curvatures of this closed manifold.
###### Proof.
According to Hamilton’s maximum principle for tensors (see Theorem 9.6 in
[11]), for $1<p<3$, we conclude that $0\leq R_{ij}<\tfrac{R}{p}g_{ij}$ is
preserved under the Ricci flow. Therefore the desired conclusion follows from
Theorem 4.5. ∎
## 5\. First $p$-eigenvalue along normalized Ricci flow
In this section, we will first discuss the differentiability for
$\lambda_{1,p}(\tilde{g}(\tilde{t}))$ under normalized Ricci flow by means of
the differentiability for $\lambda_{1,p}(g(t))$ under unnormalized Ricci flow.
Then for closed $2$-surfaces, we obtain many monotonic quantities about the
first eigenvalue of the $p$-Laplace operator along the normalized Ricci flow
without any curvature assumption, that is, Theorems 1.4 and 1.5 in
introduction.
At first we can apply the differentiability for $\lambda_{1,p}(g(t))$ under
the unnormalized Ricci flow to derive the differentiability for
$\lambda_{1,p}(\tilde{g}(\tilde{t}))$ under the normalized case.
###### Theorem 5.1.
Let $\tilde{g}(\tilde{t})$, $\tilde{t}\in[0,\infty)$, be a solution of the
normalized Ricci flow (1.2) on a closed manifold $M^{n}$ and let
$\lambda_{1,p}(\tilde{t})$ be the first eigenvalue of the $p$-Laplace operator
of the metric $\tilde{g}(\tilde{t})$. If the curvature assumptions of Theorem
1.1 (Theorem 4.3, Theorem 4.5 or Corollary 4.6) are satisfied, then
$\lambda_{1,p}(\tilde{t})$ is differentiable almost everywhere along the
normalized Ricci flow on $[0,\infty)$ in each case.
###### Proof of Theorem 5.1.
Under the normalized Ricci flow $\tilde{g}(\tilde{t}):=c(t)g(t)$, we have
(5.1)
$\displaystyle\lambda_{1,p}(\tilde{g}(\tilde{t}))=\frac{\int_{M}|d\tilde{f}|^{p}_{\tilde{g}(\tilde{t})}d\tilde{\mu}}{\int_{M}|\tilde{f}|^{p}d\tilde{\mu}}=\frac{\int_{M}|d\tilde{f}|^{p}_{\tilde{g}(\tilde{t})}d\mu}{\int_{M}|\tilde{f}|^{p}d\mu}=c(t)^{-p/2}\frac{\int_{M}|d\tilde{f}|^{p}_{g(t)}d\mu}{\int_{M}|\tilde{f}|^{p}d\mu},$
where $\tilde{f}$ is the eigenfunction for the first eigenvalue
$\lambda_{1,p}(\tilde{t})$ with respect to $\tilde{g}(\tilde{t})$, which
implies $\int_{M}|\tilde{f}|^{p-2}\tilde{f}d\tilde{\mu}=0$. Since
$\tilde{g}(\tilde{t}):=c(t)g(t)$, we also have
$\int_{M}|\tilde{f}|^{p-2}\tilde{f}d\mu=0.$
Consider the following quantity
(5.2) $\frac{\int_{M}|d\phi|^{p}_{g(t)}d\mu}{\int_{M}|\phi|^{p}d\mu},$
where $\phi$ is any $C^{1}$ function. Clearly, if $\phi=\tilde{f}$, then (5.2)
achieves its minimum. If it is not true, this contradicts (5.1) by choosing
$c(t)=1$. Therefore (5.1) implies that
$\lambda_{1,p}(\tilde{g}(\tilde{t}))=c(t)^{-p/2}\cdot\lambda_{1,p}(g(t)).$
Note that $\lambda_{1,p}(g(t))$ is differentiable almost everywhere under the
curvature assumptions of Theorem 1.1 (Theorem 4.3, Theorem 4.5 or Corollary
4.6) and $c(t)$ is a smooth function. Hence $\lambda_{1,p}(\tilde{t})$ is
differentiable almost everywhere in each case along the normalized Ricci flow
on $[0,\infty)$. ∎
###### Remark 5.2.
For any $2$-surface, we claim that $\lambda_{1,p}(t)$ is differentiable almost
everywhere along the Ricci flow without any curvature assumption (see Theorems
1.4 and 1.5, and Corollary 5.4).
In the rest of this section, we shall discuss the monotonic quantities about
the first eigenvalue of the $p$-Laplace operator along the normalized Ricci
flow on closed $2$-surfaces. From this, we also see that $\lambda_{1,p}(t)$ is
differentiable almost everywhere along the normalized Ricci flow without any
curvature assumption.
We recall the following curvature estimates along the normalized Ricci flow on
closed surfaces (see Proposition 5.18 in [7]).
###### Proposition 5.3.
For any solution $(M^{2},g(t))$ of the normalized Ricci flow on a closed
surface, there exists a constant $C>0$ depending only on the initial metric
such that:
1. (1)
If $r<0$, then $r-Ce^{rt}\leq R\leq r+Ce^{rt}$.
2. (2)
If $r=0$, then $-\frac{C}{1+Ct}\leq R\leq C$.
3. (3)
If $r>0$, then $-Ce^{rt}\leq R\leq r+Ce^{rt}$.
Now using Proposition 5.3, we shall prove Theorem 1.4. The method of proof is
almost the same as that of Theorem 4.3.
###### Proof of Theorem 1.4.
_Step 1_ : we first prove the case $p\geq 2$. Since $n=2$, by (4.4) of Lemma
4.1, under the normalized Ricci flow, we have
(5.3) $\displaystyle\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}$
$\displaystyle=\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu+\left(\frac{p}{2}-1\right)\int_{M}|df|^{p}Rd\mu$
$\displaystyle\,\,\,\,\,\,-\frac{p}{2}r\lambda_{1,p}(f(t_{0}),t_{0}),$
where $f$ is defined by Lemma 4.1.
Case 1: $\chi(M^{2})<0$.
Note that the evolution of the scalar curvature $R$ on a closed surface under
the normalized Ricci flow is
(5.4) $\frac{\partial}{\partial t}R=\Delta R+R(R-r).$
By the Gauss-Bonnet theorem, $r$ is determined by the Euler characteristic
$\chi(M^{2})$, i.e., $r=4\pi\chi(M^{2})/\mathrm{Area}{(M^{2})}$. Now if
$\chi(M^{2})<0$, applying the maximum principle to equation (5.4), we obtain
sharp lower bounds of the scalar curvature $R$:
(5.5)
$R(x,t)\geq\displaystyle\frac{r}{1-(1-\tfrac{r}{\rho_{0}})e^{rt}},\quad\quad
t\in[0,\infty).$
Note that in this setting, we need more accurate lower bounds than Proposition
5.3. By inequality (5.5), we have
(5.6) $R(x,t)>\frac{r}{1-(1-\tfrac{r}{\rho_{0}})e^{rt}}-\epsilon,\quad\quad
t\in[0,\infty)$
for $\epsilon>0$ sufficiently small. Substituting this into the above formula
(5.3), we obtain
(5.7) $\displaystyle\frac{d\lambda_{1,p}(f,t)}{dt}\Big{|}_{t=t_{0}}$
$\displaystyle>\lambda_{1,p}(f(t_{0}),t_{0})\left[\frac{r}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}-\frac{p}{2}r\right]$
$\displaystyle\,\,\,\,\,\,+\left(\frac{p}{2}-1\right)\frac{r\lambda_{1,p}(f(t_{0}),t_{0})}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}-\frac{p\epsilon}{2}\lambda_{1,p}(f(t_{0}),t_{0})$
$\displaystyle=\frac{p}{2}\lambda_{1,p}(f(t_{0}),t_{0})\left[\frac{r}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}-r-\epsilon\right].$
Since $\lambda_{1,p}(f,t)$ is a smooth function with respect to $t$-variable,
we have
(5.8)
$\frac{d}{dt}\lambda_{1,p}(f,t)>\frac{p}{2}\lambda_{1,p}(f(t),t)\left[\frac{r}{1-(1-\frac{r}{\rho_{0}})e^{rt}}-r-\epsilon\right]$
in any sufficiently small neighborhood of $t_{0}$. Integrating the above
inequality with respect to time $t$ on a sufficiently small time interval
$[t_{1},t_{0}]$, we obtain
(5.9) $\displaystyle\ln$
$\displaystyle\lambda_{1,p}(f(t_{0}),t_{0})-\ln\lambda_{1,p}(f(t_{1}),t_{1})$
$\displaystyle>\frac{p}{2}\left[\ln\frac{\frac{r}{\rho_{0}}e^{rt_{0}}}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}-(r+\epsilon)t_{0}\right]-\frac{p}{2}\left[\ln\frac{\frac{r}{\rho_{0}}e^{rt_{1}}}{1-(1-\frac{r}{\rho_{0}})e^{rt_{1}}}-(r+\epsilon)t_{1}\right]$
for any $t_{1}<t_{0}$ sufficiently close to $t_{0}$ (Note that $t_{1}$ may
equal to $0$). Since $\lambda_{1,p}(f(t_{0}),t_{0})=\lambda_{1,p}(t_{0})$ and
$\lambda_{1,p}(f(t_{1}),t_{1})\geq\lambda_{1,p}(t_{1})$, then we have
(5.10) $\displaystyle\ln$
$\displaystyle\lambda_{1,p}(t_{0})-\ln\lambda_{1,p}(t_{1})$
$\displaystyle>\frac{p}{2}\left[\ln\frac{\frac{r}{\rho_{0}}e^{rt_{0}}}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}-(r+\epsilon)t_{0}\right]-\frac{p}{2}\left[\ln\frac{\frac{r}{\rho_{0}}e^{rt_{1}}}{1-(1-\frac{r}{\rho_{0}})e^{rt_{1}}}-(r+\epsilon)t_{1}\right]$
for any $t_{1}<t_{0}$ sufficiently close to $t_{0}$. Since $t_{0}$ is
arbitrary, we conclude that
(5.11)
$\ln\lambda_{1,p}(t)-\frac{p}{2}\left[\ln\frac{\frac{r}{\rho_{0}}e^{rt}}{1-(1-\frac{r}{\rho_{0}})e^{rt}}-(r+\epsilon)t\right]$
is increasing along the normalized Ricci flow. Taking $\epsilon\rightarrow 0$,
we know that
(5.12)
$\ln\left[\lambda_{1,p}(t)\cdot\left(\frac{\rho_{0}}{r}-\frac{\rho_{0}}{r}e^{rt}+e^{rt}\right)^{p/2}\right]$
is non-decreasing along the normalized Ricci flow. By the Lebesgue’s theorem,
(5.12) is differentiable almost everywhere along the normalized Ricci flow on
$[0,\infty)$. We also note that
$\left[\frac{\rho_{0}}{r}-\frac{\rho_{0}}{r}e^{rt}+e^{rt}\right]^{p/2}$
is a smooth function. Hence $\lambda_{1,p}(t)$ is differentiable almost
everywhere along the normalized Ricci flow.
Case 2: $\chi(M^{2})=0$.
If $\chi(M^{2})=0$, i.e., $r=0$, by Proposition 5.3, we have
(5.13) $R(x,t)\geq-\frac{C}{1+Ct}.$
Substituting this into formula (5.3) and applying similar arguments above (in
case of $\chi(M^{2})\neq 0$), we can obtain the desired results.
Case 3: $\chi(M^{2})>0$.
This proof is similar to the proof of Case 2. we still use Proposition 5.3 and
formula (5.3).
_Step 2_ : we consider the case $1<p<2$. Since the method of proof is similar
to the previous discussions, we only give some key computations.
Case 1: $\chi(M^{2})<0$.
By (5.3) and $R\leq r+Ce^{rt}$ of Proposition 5.3, we have
(5.14) $\displaystyle\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}$
$\displaystyle\geq\lambda_{1,p}(f(t_{0}),t_{0})\left[\frac{r}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}+\left(\frac{p}{2}-1\right)\left(r+Ce^{rt_{0}}\right)-\frac{p}{2}r\right]$
$\displaystyle=\lambda_{1,p}(f(t_{0}),t_{0})\left[\frac{r}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}-r+\left(\frac{p}{2}-1\right)Ce^{rt_{0}}\right]$
where $f$ is defined by Lemma 4.1.
Following similar arguments above, we conclude that (5.14) still holds in any
sufficiently small neighborhood of $t_{0}$. Then integrating this inequality
with respect to time $t$ on a sufficiently small time interval
$[t_{1},t_{0}]$, we obtain
(5.15)
$\displaystyle\ln\lambda_{1,p}(f(t_{0}),t_{0}){-}\ln\lambda_{1,p}(f(t_{1}),t_{1})$
$\displaystyle\geq\left[\ln\frac{\frac{r}{\rho_{0}}}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}{+}\left(\frac{p}{2}-1\right)\frac{C}{r}e^{rt_{0}}\right]$
$\displaystyle\,\,\,\,\,\,-\left[\ln\frac{\frac{r}{\rho_{0}}}{1-(1-\frac{r}{\rho_{0}})e^{rt_{1}}}{+}\left(\frac{p}{2}-1\right)\frac{C}{r}e^{rt_{1}}\right]$
for any $t_{1}<t_{0}$ sufficiently close to $t_{0}$. Note that
$\lambda_{1,p}(f(t_{0}),t_{0})=\lambda_{1,p}(t_{0})$ and
$\lambda_{1,p}(f(t_{1}),t_{1})\geq\lambda_{1,p}(t_{1})$. Hence we have
$\displaystyle\ln$
$\displaystyle\left[\lambda_{1,p}(t_{0})\cdot\left(\frac{\rho_{0}}{r}-\frac{\rho_{0}}{r}e^{rt_{0}}+e^{rt_{0}}\right)\right]+\left(1-\frac{p}{2}\right)\frac{C}{r}e^{rt_{0}}$
$\displaystyle\geq\ln\left[\lambda_{1,p}(t_{1})\cdot\left(\frac{\rho_{0}}{r}-\frac{\rho_{0}}{r}e^{rt_{1}}+e^{rt_{1}}\right)\right]+\left(1-\frac{p}{2}\right)\frac{C}{r}e^{rt_{1}}$
for any $t_{1}<t_{0}$ sufficiently close to $t_{0}$. Since $t_{0}$ is
arbitrary, the result follows.
Case 2: $\chi(M^{2})=0$.
Using $-\frac{C}{1+Ct}\leq R\leq C$ of Proposition 5.3, we have
(5.16) $\displaystyle\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}$
$\displaystyle=\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu+\left(\frac{p}{2}-1\right)\int_{M}|df|^{p}Rd\mu$
$\displaystyle\geq\lambda_{1,p}(f(t_{0}),t_{0})\left[-\frac{C}{1+Ct_{0}}+\left(\frac{p}{2}-1\right)C\right]$
where $f$ is defined by Lemma 4.1. Then using similar arguments above, we can
obtain the desired results.
Case 3: $\chi(M^{2})>0$.
Using $-Ce^{rt}\leq R\leq r+Ce^{rt}$ of Proposition 5.3, we get
(5.17) $\displaystyle\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}$
$\displaystyle=\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu+\left(\frac{p}{2}-1\right)\int_{M}|df|^{p}Rd\mu$
$\displaystyle\,\,\,\,\,\,-\frac{p}{2}r\lambda_{1,p}(f(t_{0}),t_{0})$
$\displaystyle\geq\lambda_{1,p}(f(t_{0}),t_{0})\left[-r+\left(\frac{p}{2}-2\right)Ce^{rt_{0}}\right]$
where $f$ is defined by Lemma 4.1. Then using the standard discussions above,
we can obtain the desired results. ∎
In the following we will finish the proof Theorem 1.5.
###### Proof of Theorem 1.5.
_Step 1_ : we first prove the case $p\geq 2$.
The case $\chi(M^{2})=0$.
By Proposition 5.3, we have $R(x,t)\leq C$. Substituting this into formula
(5.3),
(5.18) $\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t{=}t_{0}}\leq\frac{p}{2}\cdot
C\lambda_{1,p}(f(t_{0}),t_{0}).$
Since $\lambda_{1,p}(f,t)$ is a smooth function with respect to $t$-variable,
we have
(5.19)
$\frac{d}{dt}\lambda_{1,p}(f,t)<\frac{p}{2}\left(C+\epsilon\right)\lambda_{1,p}(f(t),t).$
for $\epsilon>0$ sufficiently small in any sufficiently small neighborhood of
$t_{0}$. Integrating the above inequality with respect to time $t$ on a
sufficiently small time interval $[t_{0},t_{1}]$, we get
(5.20)
$\ln\lambda_{1,p}(f(t_{1}),t_{1})-\ln\lambda_{1,p}(f(t_{0}),t_{0})<\frac{p}{2}\left(C+\epsilon\right)t_{1}-\frac{p}{2}\left(C+\epsilon\right)t_{0}$
for any $t_{1}>t_{0}$ sufficiently close to $t_{0}$. Note that
$\lambda_{1,p}(f(t_{0}),t_{0})=\lambda_{1,p}(t_{0})$ and
$\lambda_{1,p}(f(t_{1}),t_{1})\geq\lambda_{1,p}(t_{1})$. So we have
$\ln\lambda_{1,p}(t_{1})-\frac{p}{2}\left(C+\epsilon\right)t_{1}<\ln\lambda_{1,p}(t_{0})-\frac{p}{2}\left(C+\epsilon\right)t_{0}$
for any $t_{1}>t_{0}$ sufficiently close to $t_{0}$. Since $t_{0}$ is
arbitrary, taking $\epsilon\rightarrow 0$, the result follows in the case of
$\chi=0$.
The case $\chi(M^{2})\neq 0$.
The method of the proof is similar to the case of $\chi(M^{2})\neq 0$. Here we
only give some key inequalities. Using $R\leq r+Ce^{rt}$ of Proposition 5.3
and formula (5.3), we have
(5.21)
$\displaystyle\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\leq\frac{p}{2}Ce^{rt_{0}}\lambda_{1,p}(f(t_{0}),t_{0})$
where $f$ is defined by Lemma 4.1. By similar arguments the results follows.
_Step 2_ : we consider the case $1<p<2$. Similarly, we only give some key
computations.
Case 1: $\chi(M^{2})<0$.
Substituting (5.5) and $R\leq r+Ce^{rt}$ of Proposition 5.3 into formula
(5.3),
(5.22)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\leq\lambda_{1,p}(f(t_{0}),t_{0})\left[\left(\frac{p}{2}-1\right)\cdot\left(\frac{r}{1-(1-\frac{r}{\rho_{0}})e^{rt_{0}}}-r\right)+Ce^{rt_{0}}\right]$
where $f$ is defined by Lemma 4.1. Then using the standard discussion as the
case $\chi(M^{2})=0$, we can obtain the desired results.
Case 2: $\chi(M^{2})=0$.
Substituting $-\frac{C}{1+Ct}\leq R\leq C$ of Proposition 5.3 into formula
(5.3), we have
(5.23)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\leq\lambda_{1,p}(f(t_{0}),t_{0})\left[\left(1-\frac{p}{2}\right)\cdot\frac{C}{1+Ct_{0}}+C\right]$
where $f$ is defined by Lemma 4.1. Using similar discussion above, the result
follows.
Case 3: $\chi(M^{2})>0$.
Using $-Ce^{rt}\leq R\leq r+Ce^{rt}$ of Proposition 5.3, we obtain
(5.24)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\leq\lambda_{1,p}(f(t_{0}),t_{0})\left[\left(1-\frac{p}{2}\right)\cdot
r+\left(2-\frac{p}{2}\right)Ce^{rt_{0}}\right]$
where $f$ is defined by Lemma 4.1. Then the desired results follow by the
above similar discussions. ∎
We should point out that for closed $2$-surfaces, we also have the
differentiability result along the unnormalized Ricci flow without any
curvature assumption.
###### Corollary 5.4.
Let $g(t)$ and $\lambda_{1,p}(t)$ be the same as in Theorem 1.1, where $n=2$.
Then $\lambda_{1,p}(t)$ is differentiable almost everywhere along the
unnormalized Ricci flow.
###### Proof.
For closed 2-surfaces, we know that the first eigenvalue of the $p$-Laplace
operator is differentiable almost everywhere along the normalized Ricci flow.
Hence the conclusion follows from the same argument as in the proof of Theorem
5.1. ∎
## 6\. $p$-eigenvalue comparison-type theorem
In Riemannian geometry, a convenient way of understanding a general Riemannian
manifold is by comparison theorems. And many comparison theorems have been
obtained, such as the Hessian comparison theorem, the Laplace comparison
theorem, the volume comparison theorem, etc..
In this section, we will give another interesting comparison-type theorem on a
closed surface with the Euler characteristic $\chi(M^{2})<0$, which is
motivated by the work of J. Ling [18]. However, our proof may be different
from Ling’s. Because we do not know the eigenvalue or eigenfunction
differentiability under the Ricci flow. Fortunately we can follow similar
arguments above and obtain our desired result.
Let $(M^{2},g)$ be a closed surface. Let $K_{g}$, $\kappa_{g}$,
$\mathrm{Area}_{g}(M^{2})$ denote the Gauss curvature, the minimum of the
Gauss curvature, the area of the surface, respectively. $\lambda_{1,p}(g)$
denotes the first eigenvalue of the $p$-Laplace operator $(p\geq 2)$ with
respect to the metric $g$. We now prove the comparison-type theorem for
$\lambda_{1,p}(g)$ on a closed surface with its Euler characteristic is
negative.
###### Proof of Theorem 1.7.
Let $g(t)$ be the solution of the normalized Ricci flow on a closed surface
(6.1) $\displaystyle\frac{\partial g(t)}{\partial t}=(r-R)g(t)$
with the initial condition $g(0)=g$, where $R$ is the scalar curvature of the
metric $g(t)$ and $r={\int_{M^{2}}Rd\mu}\big{/}{\int_{M^{2}}d\mu}$, which
keeps the area of the surface constant. In fact, from (6.1) we have
$\frac{d}{dt}(d\mu)=(r-R)d\mu$
and
$\frac{d}{dt}\mathrm{Area}_{g(t)}(M^{2})=\frac{d}{dt}\int_{M^{2}}d\mu=\int_{M^{2}}(r-R)d\mu=0.$
Set $A:=\mathrm{Area}_{g(t)}(M^{2})=\mathrm{Area}_{g}(M^{2})$. Obviously,
along the normalized Ricci flow, the area $A$ remains constant independent of
time. By the Gauss-Bonnet theorem, $r$ is determined by the Euler
characteristic $\chi(M^{2})$, i.e., $r=4\pi\chi(M^{2})/A<0$. So we know that
$r$ is a negative constant and the lower bounds of the scalar curvature $R$
are also negative. Meanwhile, according to Theorem E in introduction, the
metric $g(t)$ converges to a smooth metric $\bar{g}(=g(\infty))$ of constant
Gauss curvature $r/2$.
Note that $R/2$ is the Gauss curvature $K$ of the metric $g(t)$. Let
$\rho_{0}<0$ be the minimum of $R(0)$, i.e.,
$R(0)=2K(0)\geq\rho_{0}.$
Since $\chi(M^{2})<0$, by Theorem 1.4, we know that
(6.2)
$\lambda_{1,p}(t)\cdot\left[\frac{\rho_{0}}{r}-\frac{\rho_{0}}{r}e^{rt}+e^{rt}\right]^{p/2}$
is increasing along the normalized Ricci flow on $[0,\infty)$, where
$\rho_{0}=\inf\limits_{M^{2}}R(0)$.
Since that $r<0$ and $p\geq 2$, taking $t\rightarrow\infty$ in (6.2) and
noticing that $\lambda_{1,p}(t)$ is continuous, we conclude that
$\lambda_{1,p}(\infty)\geq\lambda_{1,p}(0)\cdot\left(\frac{r}{\rho_{0}}\right)^{p/2}.$
Note that the metric $\bar{g}(=g(\infty))$ has constant Gauss curvature $r/2$.
So we have $\kappa_{\bar{g}}=r/2$. By the definition for $\rho_{0}$, we also
have $\rho_{0}=2\kappa_{g}$. Therefore we conclude the following inequality
$\frac{\lambda_{1,p}(\bar{g})}{\lambda_{1,p}(g)}\geq\left(\frac{\kappa_{\bar{g}}}{\kappa_{g}}\right)^{p/2}.$
This completes the proof of this theorem. ∎
###### Remark 6.1.
(1). By Theorem 1.4 and Theorem 1.5, using the same method above, if
$\chi(M^{2})<0$ , we can also get some rough estimates
$\frac{\lambda_{1,p}(\bar{g})}{\lambda_{1,p}(g)}\geq\exp\left[\left(1-\frac{p}{2}\right)\frac{C}{r}\right]\cdot\frac{\kappa_{\bar{g}}}{\kappa_{g}}\quad(1<p<2);$
and
$\frac{\lambda_{1,p}(\bar{g})}{\lambda_{1,p}(g)}\leq
e^{-\frac{C}{r}}\cdot\left(\frac{\kappa_{\bar{g}}}{\kappa_{g}}\right)^{\frac{p}{2}-1}\,\,\,(1<p<2),\quad\quad\quad\frac{\lambda_{1,p}(\bar{g})}{\lambda_{1,p}(g)}\leq\exp\left(-\frac{p}{2}\cdot\frac{C}{r}\right)\,\,\,(p\geq
2),$
where $C>0$ is a constant depending only on the metric $g$ and
$r=2\kappa_{\bar{g}}$.
(2). It would be interesting to find out if there exists a similar comparison-
type result for high dimensional closed manifolds. It seems to be difficult to
deal with the high-dimensional case. On the other hand, can one have a similar
result as theorem 1.7 if one removes the condition: $\chi(M^{2})<0$?
(3). Though we do not follow J. Ling’s proof, the idea of proof partly belongs
to his. When $p=2$, our result reduces to J. Ling’s (see [18], Theorem 1.1).
## 7\. First $p$-eigenvalue along general evolving metrics
Following similar arguments in the proof of Theorem 1.1, in this section, we
discuss the monotonicity and differentiability for the first eigenvalue of the
$p$-Laplace with respect to general evolving Riemannian metrics.
Let $(M^{n},g(t))$ be a smooth one-parameter family of compact Riemannian
manifolds without boundary evolving for $t\in[0,T)$ by
(7.1) $\frac{\partial}{\partial t}g_{ij}=-2h_{ij}$
with $g(0)=g_{0}$. Let $H:=\mathrm{tr}\,h=g^{ij}h_{ij}$.
We first have a analog of Proposition 3.1 in Section 3.
###### Proposition 7.1.
Let $g(t)$, $t\in[0,T)$, be a smooth family of complete Riemannian metrics on
a closed manifold $M^{n}$ satisfying (7.1) and let $\lambda_{1,p}(t)$ be the
first eigenvalue of the $p$-Laplace operator $(p>1)$ under the evolving
metrics (7.1). For any $t_{1},t_{2}\in[0,T)$ with $t_{2}\geq t_{1}$, we have
(7.2)
$\lambda_{1,p}(t_{2})\geq\lambda_{1,p}(t_{1})+\int^{t_{2}}_{t_{1}}\mathcal{L}(g(\xi),f(\xi))d\xi,$
where
(7.3) $\mathcal{L}(g(t),f(t)):=p\int_{M}|df|^{p-2}h(\nabla f,\nabla
f)d\mu-p\int_{M}\Delta_{p}f\frac{\partial f}{\partial
t}d\mu-\int_{M}|df|^{p}Hd\mu$
and where $f(t)$ is any $C^{\infty}$ function satisfying the restrictions
$\int_{M}|f(t)|^{p}d\mu_{g(t)}=1$ and $\int_{M}|f(t)|^{p-2}f(t)d\mu_{g(t)}=0$,
such that at time $t_{2}$, $f(t_{2})$ is the corresponding eigenfunction of
$\lambda_{1,p}(t_{2})$.
###### Proof.
The proof is by straightforward computation, which is similar to the proof of
Proposition 3.1. Here we omit those details. ∎
Using this proposition, we have
###### Theorem 7.2.
Let $g(t)$ and $\lambda_{1,p}(t)$ be the same as Proposition 7.1. If there
exists a nonnegative constant $\epsilon$ such that
(7.4) $h_{ij}-\tfrac{H}{p}g_{ij}\geq-\epsilon g_{ij}\quad\quad\mathrm{in}\quad
M\times[0,T)$
and
(7.5) $H>p\cdot\epsilon\quad\quad\mathrm{in}\quad M\times[0,T),$
then $\lambda_{1,p}(t)$ is strictly increasing and therefore differentiable
almost everywhere along the evolving Riemannian metrics (7.1) on $[0,T)$.
###### Proof.
This proof is similar to that of the previous theorems. ∎
###### Remark 7.3.
(1). Assumptions (7.4) and (7.5) may not be valid sometimes for some special
curvature flow. For example, for the normalized Ricci flow, the assumptions
(7.4) and (7.5) are not hold in general.
(2). This theorem may be compared to Theorem 1.1 of this paper. In fact, let
$(M^{n},g(t))$ be a complete solution of the unnormalized Ricci flow on
$[0,T)$. This corresponds to $h_{ij}=R_{ij}$ and $H=R$ in Theorem 7.2.
In the following, a general version of Lemma 4.1 is stated as follows.
###### Lemma 7.4.
If $\lambda_{1,p}(t)$ is the first eigenvalue of $\Delta_{p_{g(t)}}$, whose
metric satisfying equation (7.1) and $f(t_{0})$ is the corresponding
eigenfunction of $\lambda_{1,p}(t)$ at time $t_{0}$, then we have
(7.6)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}=\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Hd\mu+\int_{M}|df|^{p-2}(ph_{ij}{-}Hg_{ij})f_{i}f_{j}d\mu,$
where $f(t)$ is any $C^{\infty}$ function satisfying the restrictions
$\int_{M}|f(t)|^{p}d\mu_{g(t)}=1$ and $\int_{M}|f(t)|^{p-2}f(t)d\mu_{g(t)}=0$,
such that at time $t_{0}$, $f(t_{0})$ is the corresponding eigenfunction of
$\lambda_{1,p}(t_{0})$.
In the same way as before, we can use this lemma to construct some monotonic
quantities about the first eigenvalue of the $p$-Laplace operator along
general evolving Riemannian metrics under some curvature assumptions.
Next we turn to study a particular geometric flow, i.e., Yamabe flow. We will
apply Theorem 7.2 and Lemma 7.4 to the Yamabe flow. When $p=2$, the first
author in [27] obtained some interesting results. The Yamabe flow was still
introduced by R.S. Hamilton, which is defined by
(7.7) $\displaystyle\frac{\partial}{\partial t}g(x,t)$
$\displaystyle=-R(x,t)g(x,t),$ $\displaystyle g(x,0)$ $\displaystyle=g_{0}(x)$
where $R$ denotes the scalar curvature of $g(t)$. The normalized Yamabe flow
is defined by
(7.8) $\displaystyle\frac{\partial}{\partial t}g(x,t)$
$\displaystyle=\left(r(t)-R(x,t)\right)g(x,t),$ $\displaystyle g(x,0)$
$\displaystyle=g_{0}(x)$
where $r(t):=\int_{M}Rd\mu\big{/}\int_{M}d\mu$ is the average scalar curvature
of the metric $g(t)$.
For the unnormalized Yamabe flow, we have the following proposition.
###### Proposition 7.5.
In Proposition 7.1, we replace general evolving metrics by the unnormalized
Yamabe flow (7.7). Then for any $t_{1},t_{2}\in[0,T)$ with $t_{2}\geq t_{1}$,
(7.9)
$\lambda_{1,p}(t_{2})\geq\lambda_{1,p}(t_{1})+\int^{t_{2}}_{t_{1}}\mathcal{L}(g(\xi),f(\xi))d\xi,$
where
(7.10)
$\mathcal{L}(g(t),f(t)):=\frac{p-n}{2}\int_{M}|df|^{p}Rd\mu-p\int_{M}\Delta_{p}f\frac{\partial
f}{\partial t}d\mu.$
###### Proof.
Substituting $h_{ij}=\tfrac{R}{2}g_{ij}$ into Proposition 7.1, the result
follows. ∎
Using this proposition, we have
###### Theorem 7.6.
Let $g(t)$ and $\lambda_{1,p}(t)$ be the same as Proposition 7.5, where we
assume $p\geq n$. If
(7.11) $R\geq 0\quad\mathrm{and}\quad R\not\equiv 0\quad\quad\mathrm{in}\quad
M^{n}\times\\{0\\},$
then $\lambda_{1,p}(t)$ is strictly increasing and therefore differentiable
almost everywhere along the unnormalized Yamabe flow (7.7) on $[0,T)$.
###### Proof of Theorem 7.6.
Using basically the same trick as in proving Theorem 1.1, we shall prove this
result. Under the Yamabe flow (7.7), from the constraint condition
$\frac{d}{dt}\int_{M}\left|f(t)\right|^{p}d\mu_{g(t)}=0,$
we have
(7.12) $p\int_{M}|f|^{p-2}f\frac{\partial f}{\partial
t}d\mu=\frac{n}{2}\int_{M}|f|^{p}Rd\mu.$
Note that at time $t_{2}$, $f(t_{2})$ is the eigenfunction for the first
eigenvalue $\lambda_{1,p}(t_{2})$ of $\Delta_{p_{g(t_{2})}}$. Therefore at
time $t_{2}$, we have
(7.13) $\Delta_{p}f(t_{2})=-\lambda_{1,p}(t_{2})|f(t_{2})|^{p-2}f(t_{2}).$
By Proposition 7.5, at time $t_{2}$, we have
(7.14) $\displaystyle\mathcal{L}(g(t_{2}),f(t_{2}))$
$\displaystyle=\frac{p-n}{2}\int_{M}|df|^{p}Rd\mu-p\int_{M}\Delta_{p}f\frac{\partial
f}{\partial t}d\mu$
$\displaystyle=\frac{p-n}{2}\int_{M}|df|^{p}Rd\mu+p\lambda_{1,p}(t_{2})\int_{M}|f|^{p-2}f\frac{\partial
f}{\partial t}d\mu$
$\displaystyle=\frac{p-n}{2}\int_{M}|df|^{p}Rd\mu+\frac{n}{2}\lambda_{1,p}(t_{2})\int_{M}|f|^{p}Rd\mu,$
where we used (7.13) and (7.12). Notice that the evolution of the scalar
curvature $R$ under the Yamabe flow (7.7) (see [6]) is
(7.15) $\frac{\partial}{\partial t}R=(n-1)\Delta R+R^{2}.$
Applying the strong maximum principle, $R(g(0))\geq 0$ and $R(x_{0},0)>0$ for
some $x_{0}\in M^{n}$ imply that $R(x,t)>0$ for all $(x,t)\in
M^{2}\times(0,T)$. Since $p\geq n$, from (7.14), we then have
$\mathcal{L}(g(t_{2}),f(t_{2}))>0$. Then using the same arguments in proving
Theorem 1.1 yields the desired result. ∎
For the normalized Yamabe flow, we have
###### Lemma 7.7.
If $\lambda_{1,p}(t)$ is the first eigenvalue of $\Delta_{p_{g(t)}}$, whose
metric satisfying normalized Yamabe flow (7.8) and $f(t_{0})$ is the
corresponding eigenfunction of $\lambda_{1,p}(t)$ at time $t_{0}$, then we
have
(7.16)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}=\frac{p-n}{2}\int_{M}|df|^{p}(R-r)d\mu+\frac{n}{2}\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}(R-r)d\mu,$
###### Proof.
Substituting $h_{ij}=\frac{R-r}{2}g_{ij}$ into Lemma 7.4, then the result
follows. ∎
In the end of this section, we will apply Lemma 7.7 to construct some
monotonic quantities along the unnormalized Yamabe flow, generalizing earlier
results for $p=2$ derived by the first author in [27].
###### Theorem 7.8.
Let $g(t)$, $t\in[0,T)$, be a solution of the unnormalized Yamabe flow (7.7)
on a closed manifold $M^{n}$ and let $\lambda_{1,p}(t)$ be the first
eigenvalue of the $p$-Laplace operator of the metric $g(t)$. Assume that the
initial scalar curvature $R(g(0))>0$. Then on one hand, if $1<p<n$,
(7.17)
$\lambda_{1,p}(t)\cdot\left(1-\rho_{0}t\right)^{n/2}\cdot\left(1-\sigma_{0}t\right)^{\frac{p-n}{2}}$
is increasing along the unnormalized Yamabe flow on $[0,T^{\prime\prime})$ and
if $p\geq n$,
(7.18) $\lambda_{1,p}(t)\cdot\left(1-\rho_{0}t\right)^{p/2}$
is increasing along the unnormalized Yamabe flow on $[0,T^{\prime})$. On the
other hand, the following quantities
(7.19)
$\lambda_{1,p}(t)\cdot\left(1-\rho_{0}t\right)^{\frac{p-n}{2}}\cdot\left(1-\sigma_{0}t\right)^{n/2}\quad\quad(1<p<n)$
and
(7.20)
$\lambda_{1,p}(t)\cdot\left(1-\sigma_{0}t\right)^{p/2}\quad\quad\quad\quad\quad\quad(p\geq
n)$
are both decreasing along the unnormalized Yamabe flow on
$[0,T^{\prime\prime})$, where $\rho_{0}:=\inf_{M^{2}}R(0)$,
$\sigma_{0}:=\sup_{M^{2}}R(0)$, $T^{\prime}:=\min\\{\rho_{0}^{-1},T\\}$ and
$T^{\prime\prime}:=\min\\{\sigma_{0}^{-1},T\\}$. Therefore $\lambda_{1,p}(t)$
is differentiable almost everywhere along the unnormalized Yamabe flow.
###### Proof.
Since this proof is similar to the proofs of Theorems 1.4 and 1.5, we only
give some key inequalities. Note that under the unnormalized Yamabe flow,
$\frac{\partial}{\partial t}R=(n-1)\Delta R+R^{2}.$
Applying the maximum principle to this equation, we have lower and upper
bounds of the scalar curvature $R$
(7.21) $R(x,t)\geq\frac{\rho_{0}}{1-\rho_{0}t},\quad
t\in[0,T^{\prime});\quad\quad R(x,t)\leq\frac{\sigma_{0}}{1-\sigma_{0}t},\quad
t\in[0,T^{\prime\prime}).$
where $\rho_{0}:=\inf_{M^{n}}R(0)$, $\sigma_{0}:=\sup_{M^{n}}R(0)$,
$T^{\prime}:=\min\\{\rho_{0}^{-1},T\\}$ and
$T^{\prime\prime}:=\min\\{\sigma_{0}^{-1},T\\}$.
By (7.16) of Lemma 7.7, we also have
(7.22)
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}=\frac{p-n}{2}\int_{M}|df|^{p}Rd\mu+\frac{n}{2}\lambda_{1,p}(f(t_{0}),t_{0})\int_{M}|f|^{p}Rd\mu,$
where $f$ is defined by Lemma 7.7.
On one hand, if $1<p<n$, by (7.21) and (7.22) we conclude
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\geq\lambda_{1,p}(f(t_{0}),t_{0})\left[\frac{p-n}{2}\cdot\frac{\sigma_{0}}{1-\sigma_{0}t_{0}}+\frac{n}{2}\cdot\frac{\rho_{0}}{1-\rho_{0}t_{0}}\right].$
Then following the exactly same arguments as in proving Theorem 1.4, we see
that
$\lambda_{1,p}(t)\cdot\left(1-\rho_{0}t\right)^{n/2}\cdot\left(1-\sigma_{0}t\right)^{\frac{p-n}{2}}$
is increasing along the unnormalized Yamabe flow on $[0,T^{\prime\prime})$.
If $p\geq n$, by (7.21) and (7.22) we have
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\geq\frac{p}{2}\lambda_{1,p}(f(t_{0}),t_{0})\cdot\frac{\rho_{0}}{1-\rho_{0}t_{0}}.$
Then using our standard arguments, we conclude that
$\lambda_{1,p}(t)\cdot\left(1-\rho_{0}t\right)^{p/2}$
is increasing along the unnormalized Yamabe flow on $[0,T^{\prime})$.
On the other hand, we consider the decreasing quantities under the
unnormalized Yamabe flow. If $1<p<n$, by (7.21) and (7.22), we can get
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\leq\lambda_{1,p}(f(t_{0}),t_{0})\left[\frac{p-n}{2}\cdot\frac{\rho_{0}}{1-\rho_{0}t_{0}}+\frac{n}{2}\cdot\frac{\sigma_{0}}{1-\sigma_{0}t_{0}}\right].$
Using the same arguments as in proving Theorem 1.5, then (7.19) follows.
If $p\geq n$, by (7.21) and (7.22), we can obatin
$\frac{d}{dt}\lambda_{1,p}(f,t)\Big{|}_{t=t_{0}}\leq\frac{p}{2}\lambda_{1,p}(f(t_{0}),t_{0})\cdot\frac{\sigma_{0}}{1-\sigma_{0}t_{0}}.$
By the standard arguments of Theorem 1.5, we conclude that
$\lambda_{1,p}(t)\cdot\left(1-\sigma_{0}t\right)^{p/2}$
is decreasing along the unnormalized Yamabe flow on $[0,T^{\prime\prime})$. ∎
## Acknowledgment
The authors would like to thank the referee for helpful comments and
suggestions to improve this paper.
## References
* [1] X.-D. Cao, Eigenvalues of $(-\Delta+\frac{R}{2})$ on manifolds with nonnegative curvature operator, Math. Ann., 337(2): 435-441, 2007\.
* [2] X.-D. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc., 136: 4075-4078, 2008.
* [3] X.-D. Cao, S.-B. Hou and J. Ling, Estimate and monotonicity of the first eigenvalue under Ricci flow, preprint.
* [4] S.-C. Chang and P. Lu, Evolution of Yamabe constant under Ricci flow, Ann. Glob. Anal. Geom., 31(2): 147-153, 2007.
* [5] B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom., 33: 325-334, 1991.
* [6] B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure and Appl. Math., 45: 1003-1014, 1992.
* [7] B. Chow and D. Knopf, The Ricci flow: An introduction, Mathematical Surveys and Monographs, AMS, Providence, RI, 2004.
* [8] B. Chow, S. C. Chu, D. Glickenstein, C. Guentheretc, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, The Ricci flow: techniques and applications. Part II: analytic aspects. Mathematical Surveys and Monographs, 144, AMS, Providence, RI, 2008.
* [9] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow, Lectures in Contemporary Mathematics 3, Science Press and Amer. Math. Soc., 2006.
* [10] J.-F. Grosjean, $p$-Laplace operator and diameter of manifolds, Ann. Glob. Anal. Geom., 28: 257-270, 2005.
* [11] R.S. Hamilton, Three manifolds with positive Ricci curvature, J. Diff. Geom., 17: 255-306, 1982.
* [12] R.S. Hamilton, The Ricci flow on surface, Mathematics and General Relativity, Contemporary Mathematics 71: 237-262, 1988.
* [13] T. Kato, Perturbation theory for linear operator, 2nd, Springer, Berlin, Heidelberg, New York, Tokyo, 1984.
* [14] S. Kawai and N. Nakauchi, The first eigenvalue of the $p$-Laplacian on a compact Riemannian manifold, Nonlin. Anal., 55: 33-46, 2003.
* [15] B. Kleiner and J. Lott, Note on Perelman’s papers, arXiv: math.DG/0605.667v2.
* [16] B. Kotschwar and L. Ni, Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula, Ann. Sci. Ec. Norm. Sup., 42(1): 1-36, 2009.
* [17] J.-F. Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann., 338(4): 927-946, 2007\.
* [18] J. Ling, A comparison theorem and a sharp bound via the Ricci flow, arXiv: math.DG/0710.2574.
* [19] J. Ling, A class of monotonic quantities along the Ricci flow, arXiv: math.DG/0710.4291v2.
* [20] L. Ma, Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Glob. Anal. Geom., 29: 287-292, 2006.
* [21] A.-M. Matei, First eigenvalue for the $p$-Laplace operator, Nonlin. Anal., 39: 1051-1068, 2000.
* [22] A. Mukherjea and K. Pothoven, Real and functional analysis, 2nd, Plenum Press, New York and London, 1984.
* [23] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159.
* [24] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press Harcourt Brace Jovanovich Publishers, New York, 1978.
* [25] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta. Math., 111: 247-302, 1964.
* [26] P. Tolksdorff, Regularity for a more general class of quasilinear ellptic equations, J. Diff. Equa., 51: 126-150, 1984.
* [27] Jia-Yong Wu, The first eigenvalue of the Laplace operator under the Yamabe flow, Chin. Ann. Math. Series A, 30: 631-638, 2009.
* [28] Jia-Yong Wu, First eigenvalue monotonicity for the $p$-Laplace operator under the Ricci flow, Acta Mathematica Sinica, English Series, to appear.
|
arxiv-papers
| 2009-12-24T03:46:16 |
2024-09-04T02:49:07.243014
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jia-Yong Wu, Er-Min Wang, Yu Zheng",
"submitter": "Jia-Yong Wu",
"url": "https://arxiv.org/abs/0912.4775"
}
|
0912.4783
|
# Stability of viscous shock wave for compressible Navier-Stokes equations
with free boundary
Feimin Huang†, Xiaoding Shi††, Yi Wang†
†Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100190,
China
††Department of Mathematics, Graduate School of Science, Beijing University of
Technology and Chemical, Beijing 100029, China
Abstract: A free boundary problem for the one-dimensional compressible Navier-
Stokes equations is investigated. The asymptotic stability of the viscous
shock wave is established under some smallness conditions. The proof is given
by an elementary energy estimate.
## 1 Introduction
We consider the system of viscous and heat conductive fluid in the Eulerian
coordinate
$\left\\{\begin{array}[]{llll}{\displaystyle\rho_{t}+(\rho
u)_{\tilde{x}}=0,}\\\ {\displaystyle(\rho u)_{t}+(\rho
u^{2}+p)_{\tilde{x}}=\mu u_{\tilde{x}\tilde{x}},}\\\
{\displaystyle\left[\rho(e+\frac{u^{2}}{2})\right]_{t}+\left[\rho
u(e+\frac{u^{2}}{2})+pu\right]_{\tilde{x}}=\kappa\theta_{\tilde{x}\tilde{x}}+(\mu
uu_{\tilde{x}})_{\tilde{x}},}\end{array}\right.$ $None$
where $u(\tilde{x},t)$ is the velocity, $\rho(\tilde{x},t)>0$ is the density,
$\theta(\tilde{x},t)$ is the absolute temperature, $p=p(\rho,\theta)$ is the
pressure, $e=e(\rho,\theta)$ is the internal energy, $\mu>0$ is the viscosity
constant, $\kappa>0$ is the coefficient of heat conduction. Here we consider
the perfect gas case, that is
$p=R\theta\rho,\quad e=\frac{R\theta}{\gamma-1}+const.,$ $None$
where $\gamma>1$ is the adiabatic constant and $R>0$ the gas constant.
There has been a large literature on the asymptotic behaviors of the solutions
to the system (1.1). However, most results are concerned with the initial
value problem. We refer to [7]-[10], [12]-[13] and references therein.
Recently the initial boundary value problem (IBVP) attracts an increasing
interest because it has more physically meanings and of course produces new
mathematical difficulty due to the boundary effect. We refer to [4], [11],
[14], [15] for $2\times 2$ case and [5], [6], [17] for $3\times 3$ case.
However, there is few result on the asymptotic stability of the viscous shock
wave to IBVP of the full compressible Navier-stokes equation (1.1) due to
various difficulties. Therefore, the asymptotic stability of the viscous shock
wave to IBVP for (1.1) is our main purpose of the present paper. We shall
consider a free boundary problem of the full compressible Navier-Stokes
equations whose boundary conditions read
$\left\\{\begin{array}[]{llll}{\displaystyle(p-\mu
u_{\tilde{x}})\bigl{|}_{\tilde{x}=\tilde{x}(t)}=p_{0},}\\\
{\displaystyle\theta|_{\tilde{x}=\tilde{x}(t)}=\theta_{-}>0,}\\\
{\displaystyle\frac{d\tilde{x}(t)}{dt}=\tilde{u}(\tilde{x}(t),t),\
\tilde{x}(0)=0,\ t>0,}\end{array}\right.$ $None$
and initial data
$(\rho,u,\theta)\bigl{|}_{t=0}=(\rho_{0},u_{0},\theta_{0})(x)\rightarrow(\rho_{+},u_{+},\theta_{+})\
\mathrm{as}\ \tilde{x}\rightarrow+\infty,$ $None$
where $p_{0}>0,\theta_{-}>0,\rho_{+}>0,\theta_{+}>0,u_{+}$ are prescribed
constants. Here the boundary condition (1.3) means the gas is attached at the
boundary $\tilde{x}=\tilde{x}(t)$ to the atmosphere with pressure $p_{0}$(see
[15]). We of course assume the initial data satisfy the boundary condition as
compatibility condition.
Since the boundary condition (1.3) means the particles always stay on the free
boundary $\tilde{x}=\tilde{x}(t)$, if we use the Lagrangian coordinates, then
the free boundary becomes a fixed boundary. Thus we transform the Eulerian
coordinates $(x,t)$ by
$x=\int_{\tilde{x}(t)}^{\tilde{x}}\rho(y,t)dy,\ t=t,$
and then change the free boundary value problem (1.1)-(1.4) into
$\left\\{\begin{array}[]{ll}\displaystyle v_{t}-u_{x}=0,,&x>0,t>0,\\\
\displaystyle u_{t}+p_{x}=\mu(\frac{u_{x}}{v})_{x},&x>0,t>0,\\\
\displaystyle\bigl{(}e+\frac{u^{2}}{2}\bigr{)}_{t}+(pu)_{x}=(\kappa\frac{\theta_{x}}{v}+\mu\frac{uu_{x}}{v})_{x},&x>0,t>0,\\\\[8.53581pt]
\displaystyle(p-\mu\frac{u_{x}}{v})|_{x=0}=p_{0},\qquad\theta|_{x=0}=\theta_{-},&\\\
\displaystyle(v,u,\theta)(x,0)=(v_{0},u_{0},\theta_{0})(x)\rightarrow(v_{+},u_{+},\theta_{+})&\
\mathrm{as}\ x\rightarrow+\infty,\end{array}\right.$ $None$
where $v=\frac{1}{\rho}$ is the specific volume. Since the domain we consider
here in the Lagrange coordinates is $\\{x>0,t>0\\}$, we only need to consider
the stability of the 3-viscous shock wave.
Before formulating our main result, we briefly recall some results of the
shock wave for the inviscid system of (1.1). That is, we consider the system
(1.5) without viscosity
$\left\\{\begin{array}[]{llll}{\displaystyle v_{t}-u_{x}=0,}\\\ {\displaystyle
u_{t}+p_{x}=0,}\\\
{\displaystyle\bigl{(}e+\frac{u^{2}}{2}\bigr{)}_{t}+(pu)_{x}=0,}\end{array}\right.$
$None$
with the Riemann initial data
$(v_{0},u_{0},\theta_{0})(x)=\left\\{\begin{array}[]{llll}(v_{-},u_{-},\theta_{-}),\
x>0,\\\ (v_{+},u_{+},\theta_{+}),\ x<0.\end{array}\right.$ $None$
It is well known (for example, see [16]) that the Riemann problem (1.6)-(1.7)
admits a 3-shock wave if and only if the two states
$(v_{\pm},u_{\pm},\theta_{\pm})$ satisfy the so-called Rankine-Hugoniot
condition
$\left\\{\begin{array}[]{llll}{\displaystyle-s(v_{+}-v_{-})-(u_{+}-u_{-})=0},\\\
{\displaystyle-s(u_{+}-u_{-})+(p_{+}-p_{-})=0,}\\\
{\displaystyle-s\left[(e_{+}+\frac{u_{+}^{2}}{2})-(e_{-}+\frac{u_{-}^{2}}{2})\right]+(p_{+}u_{+}-p_{-}u_{-})=0},\end{array}\right.$
$None$
and the Lax’s entropy condition
$0<\lambda_{3}^{+}<s<\lambda_{3}^{-},$ $None$
where $p_{\pm}=p(v_{\pm},\theta_{\pm}),e_{\pm}=e(v_{\pm},\theta_{\pm})$ and
$\lambda_{3}=\frac{\sqrt{\gamma R\theta}}{v}$ is the third eigenvalue of the
inviscid system (1.6). And the shock speed $s$ is uniquely determined by
$(v_{\pm},u_{\pm},\theta_{\pm})$ with (1.8). If the right state
$(v_{+},u_{+},\theta_{+})$ is given, it is easy to know that there exists a
3-shock curve $S_{3}(v_{+},u_{+},\theta_{+})$ starting from
$(v_{+},u_{+},\theta_{+})$. For any point $(v,u,\theta)\in
S_{3}(v_{+},u_{+},\theta_{+})$, there exists a unique 3-shock wave connecting
$(v,u,\theta)$ with $(v_{+},u_{+},\theta_{+})$. Our assumptions on the
boundary values are
(A1). Let $(v_{+},u_{+},\theta_{+})$ and $\theta_{-}$ be given, there exist
unique $v_{-},u_{-}$ such that $(v_{-},u_{-},\theta_{-})\in
S_{3}(v_{+},u_{+},\theta_{+})$.
(A2). $p_{0}=\frac{R\theta_{-}}{v_{-}}:=p_{-}.$
Remark1. The assumption (A1) is natural.
Remark2. The condition (A2) means that we only consider the stability of a
single viscous shock wave.
It is known that the system (1.5) admits smooth travelling wave solution with
shock profile $(V,U,\Theta)(x-st)$ under the conditions (1.8) and (1.9) (see
[1]). Such travelling wave has been shown nonlinear stable for the initial
value problem, see [7] and [9]. A natural question is whether the travelling
wave is stable or not for the initial boundary value problem. In this paper,
we give a positive answer for the free boundary problem (1.1)-(1.4) or (1.5).
Our main result is, roughly speaking, as follows. The precise statement is
given in theorem 2.1 below.
Let $(v_{+},u_{+},\theta_{+})$ and $\theta_{-}$ be given and the assumptions
(A1) and (A2) hold, then the 3-viscous shock wave connecting
$(v_{-},u_{-},\theta_{-})$ with $(v_{+},u_{+},\theta_{+})$ is asymptotically
stable.
The plan of this paper is as follows. After stating the notations, in section
2, we give some properties of the viscous shock wave and the main Theorem 2.1.
In Section 3, we reformulate the original problem to a new initial boundary
value problem. The proof of the Theorem 2.1 is given in section 4 by the
elementary energy method. In section 5, we prove the local existence of the
solution by the iteration method.
Notation: Throughout this paper, several positive generic constants which are
independent of $T,\beta$ and $\alpha$ are denoted by $C$ without confusions.
For function spaces, $H^{l}(\mathbb{R}^{+})$ denotes the $l$-th order Sobolev
space with its norm
$\|f\|_{l}=(\sum^{l}_{j=0}\|\partial^{j}_{x}f\|^{2})^{\frac{1}{2}},\quad{\rm
when}~{}\|\cdot\|:=\|\cdot\|_{L^{2}(\mathbb{R}^{+})}.$ $None$
## 2 Preliminaries and Main Result
We first recall some properties of the 3-viscous shock wave. The shock profile
$(V,U,\Theta)(\xi),\xi=x-st$, is determined by
$\left\\{\begin{array}[]{lll}{\displaystyle-sV^{\prime}-U^{\prime}=0,}\\\
{\displaystyle-
sU^{\prime}+P^{\prime}=\mu\left(\frac{U^{\prime}}{V}\right)^{\prime},}\\\
{\displaystyle-s\left(E+\frac{U^{2}}{2}\right)^{\prime}+\left(PU\right)^{\prime}=\left(\kappa\frac{\Theta^{\prime}}{V}+\mu\frac{UU^{\prime}}{V}\right)^{\prime},}\\\
{\displaystyle\left(V,U,\Theta\right)(\pm\infty)=(v_{\pm},u_{\pm},\theta_{\pm}),}\end{array}\right.$
$None$
where $P=R\Theta/V$, $E=R\Theta/(\gamma-1)+const.$,
$(v_{\pm},u_{\pm},\theta_{\pm})$ satisfy R-H condition (1.8) and entropy
condition (1.9) and $s$ is determined by (1.8). Integrating (2.1) on
$(-\infty,\xi)$ gives
$\left\\{\begin{array}[]{lll}{\displaystyle\frac{s\mu
V_{\xi}}{V}=-\left[P+s^{2}(V-\frac{b_{1}}{s^{2}})\right],}\\\
{\displaystyle\frac{\kappa\Theta_{\xi}}{sV}=-\left[E-\frac{s^{2}(V-\frac{b_{1}}{s^{2}})^{2}}{2}+\frac{b_{1}^{2}}{2s^{2}}-b_{2}\right],}\\\
U=-(sV+a),\end{array}\right.$ $None$
where $p_{\pm}=R\theta_{\pm}/v_{\pm}$,
$e_{\pm}=R\theta_{\pm}/(\gamma-1)+const.$, $a=-(sv_{\pm}+u_{\pm})$,
$b_{1}=p_{\pm}+s^{2}v_{\pm}$ and
$b_{2}=e_{\pm}+p_{\pm}v_{\pm}+s^{2}v_{\pm}^{2}/2.$ From [1] and [9], we have
the following proposition:
Proposition 2.1. Assume that the two states $(v_{\pm},u_{\pm},\theta_{\pm})$
satisfy the conditions (1.8) and (1.9), then there exists a unique shock
profile $(V,U,\Theta)(\xi)$, up to a shift, of system (2.1). Moreover, there
are positive constants $c_{1}$ and $c_{2}$ independent of $\gamma>1$ such that
for $\xi\in\mathbb{R}$,
$\left\\{\begin{array}[]{lll}{\displaystyle sV_{\xi}=-U_{\xi}>0,\
s\Theta_{\xi}<0,(|V-v_{\pm}|,|U-u_{\pm}|)\leq c_{1}de^{-c_{2}d|\xi|}}\\\
{\displaystyle|\Theta-\theta_{\pm}|\leq
c_{1}(\gamma-1)de^{-c_{2}d|\xi|},(|V_{\xi}|,|V_{\xi\xi}|,|\Theta_{\xi\xi}|)\leq
c_{1}d^{2}e^{-c_{2}d|\xi|},}\\\ {\displaystyle|\Theta_{\xi}|\leq
c_{1}(\gamma-1)d^{2}e^{-c_{2}d|\xi|},\ |\frac{\Theta_{\xi}}{V_{\xi}}|\leq
c_{1}(\gamma-1),}\\\ {\displaystyle s^{2}=\frac{\gamma
R\theta_{-}(1-d_{1})}{v_{+}v_{-}},d_{1}=\frac{d_{2}}{1+d_{2}},d_{2}=\frac{(\gamma-1)d}{2v_{+}},}\end{array}\right.$
$None$
where $d=v_{+}-v_{-}.$
As pointed out by Liu [7], a generic perturbation of viscous shock wave
produces not only a shift $\alpha$ but also diffusion waves, which decay to
zero with a rate $(1+t)^{-\frac{1}{2}}$, for the Cauchy problem. That is the
solution of the compressible Navier-Stokes equations asymptotically tends to
the translated travelling wave $(V,U,\Theta)(x-st+\alpha)$. The shift $\alpha$
is explicitly determined by the initial value. Similar to the Cauchy problem,
the shift $\alpha$ is also expected for IBVP. For a kind of initial boundary
value problem, in which the velocity is zero on the boundary, Matsumura and
Mei [11] developed a new way to determine the shift $\alpha$. A byproduct of
[11] showed that, unlike the Cauchy problem, there is no diffusion wave for
IBVP due to the boundary effect. This new idea has been used by many authors
to treat the initial boundary value problem of the system (1.5) or other
related systems (see [4], [14], [15]). In the spirit of [11], we calculate the
shift $\alpha$ for the IBVP (1.5).
We consider the situation where the initial data $(v_{0},u_{0},\theta_{0})$
are given in a neighborhood of $(V,U,\Theta)(x-\beta)$ for some large constant
$\beta>0$. That is, we require the viscous shock wave is far from the boundary
initially. Here we can not directly apply the idea of [11] to compute the
shift $\alpha$ since the velocity $u(0,t)$ on the boundary is not given,while
in [11], the velocity is zero on the boundary and the conservation of the mass
$(1.5)_{1}$ is then used to determine the shift $\alpha$. instead of
$(1.5)_{1}$, we use the conservation of momentum $(1.5)_{2}$ to determine the
shift $\alpha$ because $p-\mu\frac{u_{x}}{v}$ is given on the boundary for the
IBVP (1.5). From $(1.5)_{2}$ and $(2.1)_{2}$, we have
$(u-U)_{t}=-[p(v,\theta)-P(V,\Theta)]_{x}+\mu\left(\frac{u_{x}}{v}\right)_{x}-\mu\left(\frac{U_{x}}{V}\right)_{x},$
$None$
where $(V,U)=(V,U)(x-st+\alpha-\beta)$. Integrating (2.4) over $[0,\infty)$
with respect to $x$ and using (2.1) and (A2) yield
$\begin{array}[]{lll}{\displaystyle\frac{d}{dt}\int_{0}^{+\infty}[u(x,t)-U(x-st+\alpha-\beta)]dx}\\\
{\displaystyle=p_{-}-P(V,\Theta)(-st+\alpha-\beta)+\mu\frac{U^{\prime}}{V}(-st+\alpha-\beta)}\\\
{\displaystyle=-s(U(-st+\alpha-\beta)-u_{-}).}\end{array}$ $None$
Integrating (2.5) with respect to $t$, we have
$\begin{array}[]{lll}{\displaystyle\int_{0}^{+\infty}[u(x,t)-U(x-st+\alpha-\beta)]dx}\\\
{\displaystyle=\int_{0}^{+\infty}[u_{0}-U(x+\alpha-\beta)]dx-\int_{0}^{t}s(U(-s\tau+\alpha-\beta)-u_{-})d\tau.}\end{array}$
$None$
We define
$\begin{array}[]{ll}I(\alpha):=&{\displaystyle\int_{0}^{+\infty}[u_{0}-U(x+\alpha-\beta)]dx}\\\
&{\displaystyle-\int_{0}^{+\infty}s\left(U(-st+\alpha-\beta)-u_{-}\right)dt.}\end{array}$
$None$
It follows that
$\begin{array}[]{ll}I^{\prime}(\alpha)=&{\displaystyle-\int_{0}^{+\infty}U^{\prime}(x+\alpha-\beta)dx-s\int_{0}^{\infty}U^{\prime}(-s\tau+\alpha-\beta)]d\tau}\\\
&{\displaystyle=u_{-}-u_{+}.}\end{array}$ $None$
Expectation
$\lim_{t\to\infty}\int_{0}^{+\infty}[u(x,t)-U(x-st+\alpha-\beta)]dx=I(\alpha)=0$
gives
$\alpha=\frac{1}{u_{+}-u_{-}}I(0),$ $None$
and
$\begin{array}[]{lll}{\displaystyle\int_{0}^{+\infty}[u(x,t)-U(x-st+\alpha-\beta)]dx}\\\
{\displaystyle=s\int_{t}^{+\infty}[U(-s\tau+\alpha-\beta)-u_{-}]d\tau\leq
c_{1}e^{-c_{2}d|-st+\alpha-\beta|}\ \mathrm{as}\
t\rightarrow+\infty.}\end{array}$ $None$
Therefore the shift$\alpha$ is uniquely determined by the initial value.
To state our main theorem, we suppose that for some $\beta>0$
$\left(v_{0}(x)-V(x-\beta),u_{0}(x)-U(x-\beta),\theta_{0}(x)-\Theta(x-\beta)\right)\in
H^{1}\cap L^{1}.$ $None$
Let
$\begin{array}[]{lll}{\displaystyle(\widetilde{\Phi}_{0},\widetilde{\Psi}_{0})(x)=-\int_{x}^{+\infty}\left[v_{0}(y)-V(y-\beta),u_{0}(y)-U(y-\beta)\right]dy,}\\\
{\displaystyle\widetilde{W}_{0}(x)=-\int_{x}^{+\infty}\left[(e_{0}+\frac{u_{0}^{2}}{2})(y)-(E+\frac{U^{2}}{2})(y-\beta)\right]dy.}\end{array}$
$None$
Assume that
$(\widetilde{\Phi}_{0},\widetilde{\Psi}_{0},\widetilde{W}_{0})\in L^{2}.$
$None$
Our main result is
Theorem 2.1. Suppose that the assumptions (A1) and (A2) hold. Let
$(V,U,\Theta)(\xi)$ be the travelling wave solution satisfying (2.1). Assume
that $1<\gamma\leq 2$ and (2.11-2.13) hold, then there exists positive
constants $\delta_{0}$ and $\varepsilon_{0}$ such that if
$(\gamma-1)d\leq\delta_{0},$ $None$
and
$\|(\widetilde{\Phi}_{0},\widetilde{\Psi}_{0},\frac{\widetilde{W}_{0}}{\sqrt{\gamma-1}})\|_{2}+e^{-c_{2}d\beta}\leq\varepsilon_{0},$
$None$
then the system (1.5) has a unique global solution $(v,u,\theta)(x,t)$
satisfying
$\begin{array}[]{lll}v(x,t)-V(x-st+\alpha-\beta)\in C([0,\infty),H^{1})\cap
L^{2}(0,\infty;H^{1}),\\\ u(x,t)-U(x-st+\alpha-\beta)\in
C([0,\infty),H^{1})\cap L^{2}(0,\infty;H^{2}),\\\
\theta(x,t)-\Theta(x-st+\alpha-\beta)\in C([0,\infty),H^{1})\cap
L^{2}(0,\infty;H^{2}),\\\ \end{array}$ $None$
and
$\sup_{x\in\mathbb{R}_{+}}\bigl{|}(v,u,\theta)(x,t)-(V,U,\Theta)(x-st+\alpha-\beta)\bigr{|}\longrightarrow
0,\mathrm{\ as\ }t\rightarrow+\infty,$ $None$
where $\alpha=\alpha(\beta)$ is determined by (2.9).
## 3 Reformulation of the Original Problem
Let
$(v,u,\theta)(x,t)=(V,U,\Theta)(x-st+\alpha-\beta)+(\phi,\psi,w)(x,t),$ $None$
then we rewrite the system (1.5) as
$\left\\{\begin{array}[]{llll}{\displaystyle\phi_{t}-\psi_{x}=0,}\\\
{\displaystyle\psi_{t}+R\left(\frac{\Theta+w}{V+\phi}-\frac{\Theta}{V}\right)_{x}=\mu\left[\frac{\psi_{x}}{V+\phi}+\left(\frac{1}{V+\phi}-\frac{1}{V}\right)U_{x}\right]_{x},}\\\\[8.53581pt]
{\displaystyle\left(\frac{R}{\gamma-1}w+\frac{\psi^{2}}{2}+U\psi\right)_{t}+R\left[\frac{\Theta+w}{V+\phi}\psi+(\frac{\Theta+w}{V+\phi}-\frac{\Theta}{V})U\right]_{x}}\\\\[8.53581pt]
{\displaystyle\qquad=\kappa\left[\frac{w_{x}}{V+\phi}+(\frac{1}{V+\phi}-\frac{1}{V})\Theta_{x}\right]_{x}}\\\\[8.53581pt]
{\displaystyle\qquad+\mu\left[\frac{\psi\psi_{x}+U\psi_{x}+U_{x}\psi}{V+\phi}+(\frac{1}{V+\phi}-\frac{1}{V})UU_{x}\right]_{x},}\\\\[8.53581pt]
{\displaystyle
w|_{x=0}=\theta_{-}-\Theta(-st+\alpha-\beta),\quad\left(\frac{R\theta_{-}}{V+\phi}-\mu\frac{U_{x}+\psi_{x}}{V+\phi}\right)\bigr{|}_{x=0}=p_{-},}\\\\[8.53581pt]
{\displaystyle(\phi,\psi,w)|_{t=0}=(\phi,\psi,w)(x,0):=(\phi_{0},\psi_{0},w_{0})(x).}\end{array}\right.$
$None$
We define
$\begin{array}[]{lll}{\displaystyle(\Phi,\Psi)(x,t)=-\int_{x}^{+\infty}\left(\phi,\psi\right)(y,t)dy,}\\\
{\displaystyle
W(x,t)=-\int_{x}^{+\infty}\left(e+\frac{u^{2}}{2}\right)(y,t)-\left(E+\frac{U^{2}}{2}\right)(y-st+\alpha-\beta)dy.}\end{array}$
$None$
Then we have
$(\phi,\psi,w)=\left(\Phi_{x},\Psi_{x},\frac{\gamma-1}{R}[W_{x}-(\frac{1}{2}\Psi_{x}^{2}+U\Psi_{x})]\right).$
$None$
Integrating (3.2) with respect to $x$ yields
$\left\\{\begin{array}[]{lll}{\displaystyle\Phi_{t}-\Psi_{x}=0,}\\\
{\displaystyle\Psi_{t}+R\left(\frac{\Theta+w}{V+\Phi_{x}}-\frac{\Theta}{V}\right)=\frac{\mu\Psi_{xx}}{V+\Phi_{x}}+\left(\frac{\mu}{V+\Phi_{x}}-\frac{\mu}{V}\right)U_{x},}\\\
{\displaystyle
W_{t}+R\left(\frac{\Theta+w}{V+\Phi_{x}}-\frac{\Theta}{V}\right)U+R\frac{\Theta+w}{V+\Phi_{x}}\Psi_{x}}\\\
{\displaystyle\quad=\frac{\kappa
w_{x}}{V+\Phi_{x}}+\left(\frac{\kappa}{V+\Phi_{x}}-\frac{\kappa}{V}\right)\Theta_{x}}\\\
{\displaystyle\quad+\frac{\mu}{V+\Phi_{x}}(\Psi_{x}\Psi_{xx}+U_{x}\Psi_{x}+U\Psi_{xx})+(\frac{\mu}{V+\Phi_{x}}-\frac{\mu}{V})UU_{x}}.\end{array}\right.$
$None$
Introduce a new variable
$\widehat{W}=\frac{\gamma-1}{R}(W-U\Psi),$ $None$
then we write $w$ in the form
$w=\widehat{W}_{x}+\frac{\gamma-1}{R}\left(U_{x}\Psi-\frac{\Psi_{x}^{2}}{2}\right),$
$None$
and transform the system (3.5) into
$\left\\{\begin{array}[]{lll}{\displaystyle\Phi_{t}-\Psi_{x}=0,}\\\
{\displaystyle\Psi_{t}-\frac{b_{1}-s^{2}V}{V}\Phi_{x}+\frac{R}{V}\widehat{W}_{x}-\frac{\mu}{V}\Psi_{xx}+\frac{\gamma-1}{V}U_{x}\Psi=F_{1},}\\\
{\displaystyle\frac{R}{\gamma-1}\widehat{W}_{t}+(b_{1}-s^{2}V)\Psi_{x}-\frac{\kappa}{V}\left(\widehat{W}_{x}+\frac{\gamma-1}{R}U_{x}\Psi\right)_{x}}\\\
{\displaystyle\quad-
sU_{x}\Psi+\frac{\kappa}{V^{2}}\Theta_{x}\Phi_{x}=F_{2},}\end{array}\right.$
$None$
where $F_{1}$ and $F_{2}$ are nonlinear terms with respect to
$(\Phi,\Psi,\widehat{W})$, that is
$\left\\{\begin{array}[]{lll}{\displaystyle
F_{1}=\frac{\gamma-1}{2V}\psi^{2}-\frac{\phi}{V(V+\phi)}\left\\{(b_{1}-s^{2}V)\phi-
Rw+\mu\psi_{x}\right\\}},\\\ {\displaystyle
F_{2}=-\frac{\kappa(\gamma-1)}{RV}\psi\psi_{x}+\frac{\psi}{V+\phi}\left\\{(b_{1}-s^{2}V)\phi-
Rw+\mu\psi_{x}\right\\}}\\\
{\displaystyle\qquad\quad-\frac{\kappa\phi}{V(V+\phi)}\left(w_{x}-\frac{\Theta_{x}\phi}{V}\right).}\end{array}\right.$
$None$
By (3.3)-(3.4), the initial values satisfy
$\begin{array}[]{lll}\Phi(x,0)&{\displaystyle=-\int_{x}^{+\infty}[v_{0}(y)-V(y+\alpha-\beta)]dy}\\\
&={\displaystyle\tilde{\Phi}_{0}(x)+\int_{x}^{+\infty}[V(y+\alpha-\beta)-V(y-\beta)]dy}\\\
&={\displaystyle\tilde{\Phi}_{0}(x)+\int_{0}^{\alpha}[v_{+}-V(x+\varsigma-\beta)]d\varsigma=:\Phi_{0}(x).}\end{array}$
$None$
$\begin{array}[]{lll}\Psi(x,0)&{\displaystyle=-\int_{x}^{+\infty}[u_{0}(y)-U(y+\alpha-\beta)]dy}\\\
&={\displaystyle\widetilde{\Psi}_{0}(x)+\int_{0}^{\alpha}[u_{+}-U(x+\varsigma-\beta)]d\varsigma=:\Psi_{0}(x)}.\end{array}$
$None$
$\begin{array}[]{lll}W(x,0){\displaystyle=-\int_{x}^{+\infty}\left[(\frac{R\theta_{0}}{\gamma-1}+\frac{u_{0}^{2}}{2})(y)-(\frac{R\Theta}{\gamma-1}+\frac{U^{2}}{2})(y+\alpha-\beta)\right]dy}\\\\[8.53581pt]
={\displaystyle\widetilde{W}_{0}(x)+\int_{x}^{+\infty}\left[(\frac{R\Theta}{\gamma-1}+\frac{U^{2}}{2})(y+\alpha-\beta)-(\frac{R\Theta}{\gamma-1}+\frac{U^{2}}{2})(y)\right]dy}\\\\[8.53581pt]
={\displaystyle\widetilde{W}_{0}(x)+\int_{0}^{\alpha}\frac{R}{\gamma-1}[\theta_{+}-\Theta(x+\varsigma-\beta)]+\frac{1}{2}[u_{+}^{2}-U^{2}(x+\varsigma-\beta)]d\varsigma}\\\
=:W_{0}(x).\end{array}$ $None$
$\widehat{W}(x,0)=\frac{\gamma-1}{R}[W_{0}(x)-U(x+\alpha-\beta)\Psi_{0}(x)]=:\widehat{W}_{0}(x).$
$None$
Furthermore, by the same way as in [11], we have
Lemma 3.1. Under the assumptions (2.11)and (2.13),
$(\widetilde{\Phi}_{0},\widetilde{\Psi}_{0},\widetilde{W}_{0})\in H^{2}$ and
the shift
$\alpha\rightarrow 0\quad\mathrm{as}\
\|(\widetilde{\Phi}_{0},\widetilde{\Psi}_{0},\widetilde{W}_{0})\|_{2}\rightarrow
0\ \mathrm{and}\ \beta\rightarrow+\infty.$ $None$
Lemma 3.2. Under the assumptions (2.11) and (2.13), the initial perturbations
$(\Phi_{0},\Psi_{0},\widehat{W}_{0})\in H^{2}$and satisfies
$\|(\Phi_{0},\Psi_{0},\widehat{W}_{0})\|\rightarrow 0\quad\mathrm{as}\
\|(\widetilde{\Phi}_{0},\widetilde{\Psi}_{0},\widetilde{W}_{0})\|\rightarrow
0\ \mathrm{and}\ \beta\rightarrow+\infty.$ $None$
By (3.3) (3.5) and (2.5), the boundary values satisfy
$\begin{array}[]{lll}\displaystyle\Psi(0,t)&=&\displaystyle-\int_{0}^{+\infty}\psi(y,t)dy\\\
\displaystyle&=&\displaystyle-s\int_{t}^{+\infty}\left[U(-s\tau+\alpha-\beta)-u_{-}\right]d\tau:=A(t),\end{array}$
$None$
$\widehat{W}_{x}(0,t)-\frac{\gamma-1}{2R}\Psi_{x}^{2}(0,t)=\omega(0,t)-U_{x}(-st+\alpha-\beta)A(t):=B(t).$
$None$
For any $T>0$, we define the solution space of the problem (3.5), with the
initial values (3.10), (3.11), (3.13) and the boundary values (3.16), (3.17)
by
$X_{m,M}(0,T)=\left\\{\begin{array}[]{l}\displaystyle(\Phi,\Psi,\widehat{W}):\
(\Phi,\Psi,\widehat{W})\in C(0,T;H^{2});\\\ \displaystyle\ \Phi_{x}\in
L^{2}(0,T;H^{1});\ (\Psi_{x};\widehat{W}_{x})\in L^{2}(0,T;H^{2});\\\
\displaystyle\ \sup_{t\in[0,T]}\|(\Phi,\Psi,W)(t)\|_{2}\leq M;\
\inf_{x,t}(V+\Phi_{x})\geq m\end{array}\right\\}$ $None$
where $T,M,m$ are the positive constants.
## 4 Proof of Theorem 2.1
In this section, we give the proof of the Theorem 2.1. Without loss of
generality, we may restrict $\beta>1$ and $|\alpha|<1$. First we state the
local existence result for the IBVP (3.8), (3.10)-(3.13) and (3.16)-(3.17),
whose proof is given in section 5.
Proposition 4.1.(Local Existence) There exists a positive constant $b$ such
that if $\|(\Phi_{0},\Psi_{0},\widehat{W}_{0})\|_{2}\leq M$, and if
$\inf_{x,t}(V+\Phi_{0x})\geq m>0$,then there exists a positive constant
$T_{0}=T_{0}(m,M)$ such that the system (3.8), with the initial values (3.10),
(3.11), (3.13) and the boundary values (3.16), (3.17), has a unique solution
$(\Phi,\Psi,\widehat{W})\in X_{\frac{1}{2}m,bM}(0,T_{0})$.
Denote that
$\begin{array}[]{lll}{\displaystyle
N(T)=\sup_{\tau\in[0,T]}(\|\Phi(\tau)\|_{2}+\|\Psi(\tau)\|_{2}+\|W(\tau)\|_{2}),}\\\
N_{0}=\|\Phi_{0}\|_{2}+\|\Psi_{0}\|_{2}+\|W_{0}\|_{2}.\end{array}$
Proposition 4.2.(A Priori Estimates) Let $(\Phi,\Psi,W)\in
X_{\frac{1}{2}m,b\varepsilon}(0,T)$ be a solution of the problem (3.5) and
$1<\gamma\leq 2$. Then there exist positive constants
$\delta_{1},\varepsilon_{1}$ and $C$, which are independent of $T$, such that
if $(\gamma-1)d\leq\delta_{1}$ and
$N_{0}+\varepsilon+\beta^{-1}\leq\varepsilon_{1}$, then the following estimate
holds for $t\in[0,T]$
$\begin{array}[]{lll}{\displaystyle\|(\Phi,\Psi,\frac{\widehat{W}}{(\gamma-1)^{\frac{1}{2}}})(t)\|^{2}+\|(\phi,\psi,\frac{w}{(\gamma-1)^{\frac{1}{2}}})(t)\|_{1}^{2}+\int_{0}^{t}\|(\psi,w)\|^{2}_{2}+\|\phi\|_{1}^{2}d\tau}\\\
{\displaystyle\leq C\left(N_{0}+e^{-cd\beta}\right)}.\end{array}$ $None$
With the local existence Proposition 4.1 in hand, for the proof of the Theorem
2.1 by the standard continuum process, it is sufficient to prove the a priori
estimate Proposition 4.2. In order to prove the Proposition 4.2, we first give
some Lemmas. The following Lemma is about the boundary estimates.
Lemma 4.3. For $0\leq t\leq T$, the following inequalities hold:
$\begin{array}[]{l}\displaystyle\int_{0}^{t}(\Phi\Psi)\bigl{|}_{x=0}d\tau,\int_{0}^{t}(\Psi\Psi_{x})\bigl{|}_{x=0}d\tau,\
\int_{0}^{t}(\widehat{W}\Psi)\bigl{|}_{x=0}d\tau,\int_{0}^{t}(\psi
w)\bigl{|}_{x=0}d\tau\leq Ce^{-cd\beta},\\\\[8.53581pt]
\displaystyle\int_{0}^{t}(\widehat{W}_{x}\widehat{W})\bigl{|}_{x=0}d\tau\leq
Ce^{-cd\beta}+CN(T)\int_{0}^{t}(\|\Psi_{x}\|^{2}+\|\Psi_{xx}\|^{2})d\tau,\\\\[8.53581pt]
\displaystyle\int_{0}^{t}(\phi\psi)\bigl{|}_{x=0}d\tau,\
\int_{0}^{t}(\psi\psi_{x})\bigl{|}_{x=0}d\tau,\
\int_{0}^{t}(\psi_{x}\psi_{\tau})\bigl{|}_{x=0}d\tau\leq
C(e^{-cd\beta}+\|\phi_{0}\|_{1}),\\\
\displaystyle\int_{0}^{t}(ww_{x})\big{|}_{x=0}d\tau,\
\int_{0}^{t}(w_{x}w_{\tau})\big{|}_{x=0}d\tau\leq(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau+Ce^{-cd\beta}.\end{array}$
Proof. Since $s>0$, and $\beta\gg 1,|\alpha|<1$, we have from (2.3) and (3.16)
that
$|\Psi(0,t)|=|A(t)|\leq Ce^{-cd\beta}e^{-cdt}.$
Thus,
$\int_{0}^{t}(\Phi\Psi)\bigl{|}_{x=0}d\tau\leq Cd^{-1}N(T)e^{-cd\beta}\leq
Ce^{-cd\beta}.$
Similarly we can estimate the term
$\displaystyle\int_{0}^{t}(\Psi\Psi_{x})\bigl{|}_{x=0}d\tau,\
\int_{0}^{t}(\widehat{W}\Psi)\bigl{|}_{x=0}d\tau$.
Also,
$\int_{0}^{t}(\psi w)\big{|}_{x=0}d\tau\leq
N(T)\int_{0}^{t}|\theta_{-}-\Theta(-s\tau+\alpha-\beta)|d\tau\leq
Ce^{-cd\beta}.$
From (3.8),
$\widehat{W}_{x}(0,t)=w(0,t)-\frac{\gamma-1}{R}(U_{x}\Psi(0,t)-\frac{\Psi_{x}^{2}(0,t)}{2}),$
so we have from (2.3) that
$\int_{0}^{t}(\widehat{W}_{x}\widehat{W})\bigl{|}_{x=0}d\tau\leq
Ce^{-cd\beta}+CN(T)\int_{0}^{t}(\|\Psi_{x}\|^{2}+\|\Psi_{xx}\|^{2})d\tau.$
By using the free boundary condition in (1.5), one has
$\frac{R\theta_{-}}{v(0,t)}-\mu\frac{v(0,t)_{t}}{v(0,t)}=\frac{R\theta_{-}}{v_{-}},$
and then
$\begin{array}[]{ll}\displaystyle
v(0,t)-v_{-}&\displaystyle=(v_{0}(0)-v_{-})e^{-\frac{p_{0}}{\mu}t}\\\
\displaystyle&=(V(\alpha-\beta)-v_{-}+\phi_{0}(0))e^{-\frac{p_{0}}{\mu}t}\\\
\displaystyle&\leq
C(e^{-cd\beta}+\|\phi_{0}\|_{1})e^{-\frac{p_{0}}{\mu}t}\end{array}$ $None$
By using (2.3) and (4.2), we obtain
$\begin{array}[]{ll}\displaystyle|\phi(0,t)|&\displaystyle=|v(0,t)-V(-st+\alpha-\beta)|\\\
\displaystyle&\leq|v(0,t)-v_{-}|+|V(-st+\alpha-\beta)-v_{-}|\\\
\displaystyle&\leq
C(e^{-cd\beta}+\|\phi_{0}\|_{1})e^{-\frac{p_{0}}{\mu}t}+Ce^{-cd\beta}e^{-cdt},\end{array}$
$\begin{array}[]{ll}\displaystyle|\psi_{x}(0,t)|&\displaystyle=|\frac{p_{0}}{\mu}(v_{-}-v_{0}(0))e^{-\frac{p_{0}}{\mu}t}-U_{x}(-st+\alpha-\beta)|\\\
\displaystyle&\leq
C(e^{-cd\beta}+\|\phi_{0}\|_{1})e^{-\frac{p_{0}}{\mu}t}+Ce^{-cd\beta}e^{-cdt}.\end{array}$
Then we get at once
$\begin{array}[]{ll}\displaystyle\int_{0}^{t}(\phi\psi)\bigl{|}_{x=0}d\tau,\quad\int_{0}^{t}(\psi\psi_{x})\bigl{|}_{x=0}d\tau&\displaystyle\leq
CN(T)(e^{-cd\beta}+\|\phi_{0}\|_{1})\\\ &\displaystyle\leq
C(e^{-cd\beta}+\|\phi_{0}\|_{1}),\end{array}$
and
$\begin{array}[]{lll}\displaystyle\int_{0}^{t}(\psi_{x}\psi_{\tau})\bigl{|}_{x=0}d\tau&=&\displaystyle\int_{0}^{t}\left(\psi_{x}(0,\tau)\psi(0,\tau)\right)_{\tau}d\tau-\int_{0}^{t}(\psi_{x\tau}\psi)\bigl{|}_{x=0}d\tau\\\
&=&\displaystyle\psi_{x}(0,\tau)\psi(0,\tau)\big{|}_{0}^{t}-\int_{0}^{t}(\psi_{x\tau}\psi)\bigl{|}_{x=0}d\tau\\\
&\leq&\displaystyle CN(T)(e^{-cd\beta}+\|\phi_{0}\|_{1})\leq
C(e^{-cd\beta}+\|\phi_{0}\|_{1}).\end{array}$
Finally,
$\begin{array}[]{ll}&\displaystyle\int_{0}^{t}(ww_{x})\big{|}_{x=0}d\tau\leq
C(\gamma-1)de^{-cd\beta}\int_{0}^{t}e^{-cd\tau}\|w_{x}\|^{\frac{1}{2}}\|w_{xx}\|^{\frac{1}{2}}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau+C(\gamma-1)de^{-cd\beta}\int_{0}^{t}\|w_{x}\|^{\frac{2}{3}}(\tau)e^{-{\frac{2}{3}}cd\tau}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau+C(\gamma-1)de^{-cd\beta}[N(T)]^{\frac{2}{3}}\int_{0}^{t}e^{-{\frac{2}{3}}cd\tau}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau+Ce^{-cd\beta},\end{array}$
and
$\begin{array}[]{ll}&\displaystyle\int_{0}^{t}(w_{x}w_{\tau})\big{|}_{x=0}d\tau\leq
C(\gamma-1)d^{2}e^{-cd\beta}\int_{0}^{t}e^{-cd\tau}\|w_{x}\|^{\frac{1}{2}}\|w_{xx}\|^{\frac{1}{2}}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau+C(\gamma-1)d^{2}e^{-cd\beta}\int_{0}^{t}\|w_{x}\|^{\frac{2}{3}}(\tau)e^{-{\frac{2}{3}}cd\tau}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau+C(\gamma-1)d^{2}e^{-cd\beta}[N(T)]^{\frac{2}{3}}\int_{0}^{t}e^{-{\frac{2}{3}}cd\tau}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau+Ce^{-cd\beta}.\end{array}$
We complete the proof of the lemma 4.3.
Lemma 4.4. For $(\gamma-1)d\leq\delta_{0}$ small enough, then
$\begin{array}[]{lll}{\displaystyle\|(\Phi,\Psi,\frac{\widehat{W}}{(\gamma-1)^{\frac{1}{2}}})(t)\|^{2}+\int_{0}^{t}\|\;|V_{x}|^{\frac{1}{2}}(\Psi,\frac{\widehat{W}}{(\gamma-1)^{\frac{1}{2}}})(\tau)\|^{2}d\tau}\\\
{\displaystyle+\int_{0}^{t}\|(\Psi_{x},\widehat{W}_{x})(\tau)\|^{2}d\tau-C(\gamma-1)d\int_{0}^{t}\|\Phi_{x}(\tau)\|^{2}d\tau}\\\
{\displaystyle\leq
C\left\\{\|(\Phi_{0},\Psi_{0},\frac{\widehat{W}_{0}}{(\gamma-1)^{\frac{1}{2}}})\|^{2}+\int_{0}^{t}\int_{0}^{+\infty}|\Psi|\>|F_{1}|+|\widehat{W}|\>|F_{2}|dxd\tau\right\\}}\\\
{\displaystyle\quad+CN(T)\int_{0}^{t}\|\Psi_{xx}\|^{2}d\tau+Ce^{-cd\beta}.}\end{array}$
$None$
Proof. Let
$k(V)=(b_{1}-s^{2}V)^{-1}.$
Multiplying the first equation of (3.8) by $\Phi$, the second equation of
(3.8) by $k(V)V\Psi$ and the third equation of (3.8) by
$Rk(V)^{2}\widehat{W}$, respectively, summing them up, we have
$\begin{array}[]{lll}{\displaystyle
E_{1}(\Phi,\Psi,\widehat{W})_{t}+E_{2}(\Psi,\Psi_{x})+E_{3}(\widehat{W},\widehat{W}_{x})+G(\Psi,\widehat{W},\Phi_{x},\widehat{W}_{x})}\\\
{\displaystyle+\left\\{\mu k(V)\Psi\Psi_{x}-\Phi\Psi-\frac{R\kappa
k(V)^{2}}{V}(\widehat{W}_{x}+\frac{\gamma-1}{R}U_{x}\Psi)\widehat{W}+Rk(V)\widehat{W}\Psi\right\\}_{x}}\\\
{\displaystyle=k(V)V\Psi F_{1}+Rk(V)^{2}\widehat{W}F_{2},}\end{array}$ $None$
where
$\begin{array}[]{lll}{\displaystyle
E_{1}(\Phi,\Psi,\widehat{W})=\frac{1}{2}\left(\Phi^{2}+k(V)V\Psi^{2}+\frac{R^{2}}{\gamma-1}k(V)^{2}\widehat{W}^{2}\right),}\\\
{\displaystyle
E_{2}(\Psi,\Psi_{x})=\left[\frac{s}{2}(k(V)V)_{x}+(\gamma-1)k(V)U_{x}\right]\Psi^{2}+\mu
k(V)\Psi_{x}^{2}+\mu k(V)_{x}\Psi\Psi_{x},}\\\ {\displaystyle
E_{3}(\widehat{W},\widehat{W}_{x})=\frac{sR^{2}}{\gamma-1}k(V)k(V)_{x}\widehat{W}^{2}+\kappa
R\frac{k(V)^{2}}{V}\widehat{W}_{x}^{2}),}\\\ {\displaystyle
G(\Psi,\widehat{W},\Phi_{x},\widehat{W}_{x})=\kappa
R\left(\frac{k(V)^{2}}{V}\right)_{x}\widehat{W}\left(\widehat{W}_{x}+\frac{\gamma-1}{R}U_{x}\Psi\right)}\\\
{\displaystyle\qquad\qquad\qquad\qquad+\kappa
R\frac{k(V)^{2}}{V^{2}}\Phi_{x}\Theta_{x}\widehat{W}+\kappa(\gamma-1)\frac{k(V)^{2}}{V}U_{x}\Psi\widehat{W}_{x},}\end{array}$
Since
$p_{-}\leq k(V)^{-1}=b_{1}-s^{2}V\leq p_{+},$ $None$
one has
$c\left(\Phi^{2}+\Psi^{2}+\frac{\widehat{W}^{2}}{\gamma-1}\right)\leq
E_{1}\leq C\left(\Phi^{2}+\Psi^{2}+\frac{\widehat{W}^{2}}{\gamma-1}\right),$
$None$ $E_{3}\geq
c\left(|V_{x}|\frac{\widehat{W}^{2}}{\gamma-1}+\widehat{W}_{x}^{2}\right),$
$None$
and for $\forall\alpha_{1}>0$, there $\exists$ a constant $C_{\alpha_{1}}$
such that
$|G|\leq\alpha_{1}\left(|V_{x}|\frac{\widehat{W}^{2}}{\gamma-1}+\widehat{W}_{x}^{2}\right)+C_{\alpha_{1}}(\gamma-1)d\left[|V_{x}|\left(\Psi^{2}+\frac{\widehat{W}^{2}}{\gamma-1}\right)+\Phi_{x}^{2}\right].$
$None$
By using the method in [9], for $\gamma\in(1,2]$ and suitably small
$(\gamma-1)d>0$, one has
$\begin{array}[]{lll}{\displaystyle\inf_{x>0}\frac{\frac{s}{2}(k(V)V)_{x}+(\gamma-1)k(V)U_{x}}{V_{x}}>0,}\\\
{\displaystyle\sup_{x>0}\frac{\mu\left\\{\mu|k(V)_{x}|^{2}-4\left[\frac{s}{2}(k(V)V)_{x}+(\gamma-1)k(V)U_{x}\right]k(V)\right\\}}{V_{x}}<0,}\end{array}$
$None$
and then we get
$E_{2}\geq c(|V_{x}\Psi^{2}|+\Psi_{x}^{2}).$ $None$
Combining with the boundary estimates in Lemma 4.3, (4.3) is obtained.
Lemma 4.5. There is a constant $C$ such that
$\begin{array}[]{lll}{\displaystyle\|\phi(t)\|^{2}+\int_{0}^{t}\|\phi(\tau)\|^{2}d\tau}\\\
{\displaystyle-C\\{\|\Psi(t)\|^{2}+\int_{0}^{t}\|\;|V_{x}|^{\frac{1}{2}}\Psi(\tau)\|^{2}+\|(\psi,\widehat{W}_{x})(\tau)\|^{2}d\tau\\}}\\\
{\displaystyle\leq
C\left\\{\|\Psi_{0}\|^{2}+\|\phi_{0}\|^{2}+\int_{0}^{t}\int_{0}^{+\infty}|\phi|\;|F_{1}|dxd\tau\right\\}.}\end{array}$
$None$
Proof. Multiplying $(3.8)_{1}$ by $V\Psi_{x}-V_{x}\Psi$, $(3.8)_{2}$ by
$-V\Phi_{x}$, then applying $\partial_{x}$ to $(3.8)_{1}$ and multiplying the
resulting equation by $\mu\Phi_{x}$, calculating all their sums, we get
$\begin{array}[]{l}\displaystyle(\frac{\mu\Phi_{x}^{2}}{2}-V\Phi_{x}\Psi)_{t}+(V\Psi\Psi_{x})_{x}+(b_{1}-s^{2}V)\Phi_{x}^{2}\\\
\displaystyle\qquad=V_{x}\Psi\Psi_{x}+V\Psi_{x}^{2}+\left[R\widehat{W}_{x}-s(\gamma-1)V_{x}\Psi-
V_{t}\Psi-VF_{1}\right]\Phi_{x}.\end{array}$ $None$
Integrating (4.12) over $[0,+\infty)\times[0,t]$ with respect to $x,t$ and
using the boundary estimates Lemma 4.2, we obtain Lemma 4.5.
From (3.7), we easily have
$\begin{array}[]{lll}{\displaystyle\int_{0}^{t}\|w(\tau)\|^{2}d\tau-C\left\\{\int_{0}^{t}\|\;|V_{x}|^{\frac{1}{2}}\Psi(\tau)\|^{2}+\|\widehat{W}_{x}(\tau)\|^{2}d\tau\right\\}}\\\
{\displaystyle\qquad\leq C\int_{0}^{t}|\psi^{2}w|dxd\tau.}\end{array}$ $None$
Now we rewrite (3.2) in the form
$\left\\{\begin{array}[]{llll}{\displaystyle\phi_{t}-\psi_{x}=0,}\\\
{\displaystyle\psi_{t}-\frac{b_{1}-s^{2}V}{V}\phi_{x}+\frac{R}{V}w_{x}-(\frac{\mu}{V}\psi_{x})_{x}}{\displaystyle-\\{\frac{b_{1}-s^{2}V}{V}\\}_{x}\phi+(\frac{R}{V})_{x}w}=f_{1}\\\
{\displaystyle\frac{R}{\gamma-1}w_{t}+(b_{1}-s^{2}V)\psi_{x}-(\frac{\kappa}{V}w_{x})_{x}+(\frac{\kappa}{V^{2}}\Theta_{x}\phi)_{x}}\\\
{\displaystyle\quad-\frac{1}{V}\\{(b_{1}-s^{2}V)\phi-
Rw+\mu\psi_{x}\\}U_{x}=f_{2},}\end{array}\right.$ $None$
where $f_{1}$ and $f_{2}$ are nonlinear terms with respect to $(\phi,\psi,w)$
$\begin{array}[]{llll}{\displaystyle
f_{1}=-\\{\frac{\phi}{V(V+\phi)}[(b_{1}-s^{2}V)\phi-
Rw+\mu\psi_{x}]\\}_{x},}\\\ {\displaystyle
f_{2}=\frac{1}{V+\phi}\\{(b_{1}-s^{2}V)\phi-
Rw+\mu\psi_{x}\\}(\psi_{x}-\frac{1}{V}U_{x}\phi)}\\\
{\displaystyle\qquad-\\{\frac{\kappa\phi}{V(V+\phi)}(w_{x}-\frac{1}{V}\Theta_{x}\phi)\\}_{x}.}\end{array}$
The following Lemma is the estimates of $(\phi,\psi,\omega)$ and
$(\phi_{x},\psi_{x},\omega_{x})$.
Lemma 4.6. There is a constant $C$ such that
$\begin{array}[]{llll}{\displaystyle\|(\phi,\psi,\frac{w}{(\gamma-1)^{\frac{1}{2}}})(t)\|^{2}+\int_{0}^{t}\|\partial_{x}(\psi,w)(\tau)\|^{2}d\tau-C\int_{0}^{t}\|(\phi,\psi,w)(\tau)\|^{2}d\tau}\\\
{\displaystyle\qquad-(\gamma-1)d\int_{0}^{t}\|w_{xx}\|^{2}(\tau)d\tau\leq
C\|(\phi_{0},\psi_{0},\frac{w_{0}}{(\gamma-1)^{\frac{1}{2}}})\|^{2}}\\\
{\displaystyle\qquad+C\int_{0}^{t}\int_{0}^{+\infty}(|\psi||f_{1}|+|w||f_{2}|)dxd\tau+C(e^{-cd\beta}+\|\phi_{0}\|_{1}).}\\\
{\displaystyle\|\phi_{x}(t)\|^{2}+\int_{0}^{t}\|\phi_{x}(\tau)\|^{2}d\tau-C\left\\{\|\psi(t)\|^{2}+\int_{0}^{t}\|(\psi,w)(\tau)\|_{1}^{2}d\tau\right\\}}\\\
{\displaystyle\qquad\leq
C\left\\{\|\psi_{0}\|^{2}+\int_{0}^{t}\int_{0}^{+\infty}|\phi_{x}||f_{1}|dxd\tau+(e^{-cd\beta}+\|\phi_{0}\|_{1})\right\\}.}\\\
{\displaystyle\|\partial_{x}(\psi,\frac{w}{(\gamma-1)^{\frac{1}{2}}})(t)\|^{2}+\int_{0}^{t}\|\partial_{xx}(\psi,w)(\tau)\|^{2}d\tau-C\int_{0}^{t}\|(\phi,\psi,w)(\tau)\|_{1}^{2}d\tau}\\\
{\displaystyle\qquad\leq
C\|\partial_{x}(\psi_{0},\frac{w_{0}}{(\gamma-1)^{\frac{1}{2}}})\|^{2}+C(e^{-cd\beta}}+\|\phi_{0}\|_{1})\\\
{\displaystyle\qquad+C\int_{0}^{t}\int_{0}^{+\infty}(|\psi_{xx}||f_{1}|+|w_{xx}||f_{2}|)dxd\tau.}\end{array}$
$None$
Proof. Multiplying $(4.14)_{1}$ by $\phi$, $(4.14)_{2}$ by $Vk(V)\psi$,
$(4.14)_{3}$ by $Rk^{2}(V)w$,and adding them and integrating over $x,t$, we
have
$\begin{array}[]{l}{\displaystyle\|(\phi,\psi,\frac{w}{(\gamma-1)^{\frac{1}{2}}})(t)\|^{2}+\int_{0}^{t}\|\partial_{x}(\psi,w)(\tau)\|^{2}d\tau-C\int_{0}^{t}\|(\phi,\psi,w)(\tau)\|^{2}d\tau}\\\
{\displaystyle+\int_{0}^{t}\left[-\phi\psi+Rk(V)\psi w-\mu
k(V)\psi\psi_{x}+\frac{\kappa R}{V^{2}}\Theta_{x}k^{2}(V)\phi w-\frac{\kappa
Rk^{2}(V)}{V}ww_{x}\right]_{x=0}d\tau}\\\ {\displaystyle\quad\leq
C\|(\phi_{0},\psi_{0},\frac{w_{0}}{(\gamma-1)^{\frac{1}{2}}})\|^{2}+C\int_{0}^{t}\int_{0}^{+\infty}(|\psi||f_{1}|+|w||f_{2}|)dxd\tau.}\end{array}$
Using the boundary estimate in Lemma 4.3, we can get the first inequality in
(4.15).
Now we want to get the estimate of $\|\phi_{x}\|^{2}$ in $(4.15)_{2}$.
Multiplying $(4.14)_{1}$ by $V\psi_{x}-V_{x}\psi$, $(4.14)_{2}$ by
$-V\phi_{x}$, then applying $\partial_{x}$ to $(4.14)_{1}$ and multiplying the
resulting equation by $\mu\phi_{x}$, calculating all their sums, we get
$\begin{array}[]{l}\displaystyle(\frac{\mu\phi_{x}^{2}}{2}-V\phi_{x}\psi)_{t}+(V\psi\psi_{x})_{x}+(b_{1}-s^{2}V)\phi_{x}^{2}=V_{x}\psi\psi_{x}+V\psi_{x}^{2}+\\\\[8.53581pt]
\displaystyle\quad\left[R\omega_{x}+V(\frac{\mu}{V})_{x}\psi_{x}+V(\frac{b_{1}-s^{2}V}{V})_{x}\phi-(\frac{R}{V})_{x}\omega-
V_{t}\psi-Vf_{1}\right]\phi_{x}.\end{array}$ $None$
Thus integrating the equation (4.16) and using the boundary estimate Lemma
4.3,we obtain $(4.15)_{2}$.
Multiplying $(4.14)_{2}$ by $-\psi_{xx}$, $(4.14)_{3}$ by $-w_{xx}$ to get
$\begin{array}[]{ll}\displaystyle(\frac{1}{2}\psi_{x}^{2}+\frac{R}{2(\gamma-1)}w_{x}^{2})_{t}-(\psi_{x}\psi_{t}+w_{x}w_{t})_{x}+\frac{\mu}{V}\psi_{xx}^{2}+\frac{\kappa}{V}w_{xx}^{2}\\\\[8.53581pt]
\displaystyle\quad=-\psi_{xx}\left[\frac{b_{1}-s^{2}V}{V}\phi_{x}-\frac{R}{V}w_{x}+(\frac{\mu}{V})_{x}\psi_{x}+(\frac{b_{1}-s^{2}V}{V})_{x}\phi-(\frac{R}{V})_{x}w+f_{1}\right]\\\\[8.53581pt]
\displaystyle\quad=-w_{xx}\left[-(b_{1}-s^{2}V)\phi_{x}+(\frac{\kappa}{V})_{x}w_{x}+(\frac{\mu}{V^{2}}\Theta_{x}\phi)_{x}\right.\\\\[8.53581pt]
\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\frac{1}{V}\\{(b_{1}-s^{2}V)\phi-
Rw+\mu\psi_{x}\\}U_{x}+f_{2}\right]\end{array}$
Integrating the above equality and using Lemma 4.3, we can get the third
inequality of (4.15). The proof of Lemma 4.6 is complete.
Since
$\begin{array}[]{ll}\displaystyle|F_{1},F_{2}|&\displaystyle=O(1)\left[|(\phi,\psi,w)|^{2}+|(\phi,\psi)||(\psi_{x},w_{x})|\right],\\\\[5.69054pt]
\displaystyle|f_{1},f_{2}|&\displaystyle=O(1)\left[|(\phi,w)|^{2}+|(\phi,w)||(\phi_{x},\psi_{x},w_{x})|\right.\\\\[5.69054pt]
&\displaystyle\quad\left.+|(\phi_{x},\psi_{x})||(\psi_{x},w_{x})|+|\phi||(\psi_{xx},w_{xx})|\right],\end{array}$
$None$
combining (4.17) with the estimates (4.3), (4.11), (4.13), (4.15) and using
the a priori assumption $N(T)\leq b\varepsilon$ sufficiently small, and also
letting $(\gamma-1)d$ small enough, we can get the following estimate
$N^{2}(T)+\int_{0}^{T}\|(\psi,w)\|^{2}_{2}+\|\phi\|_{1}^{2}d\tau\leq\bar{C}(N_{0}+e^{-cd\beta}).$
where the constant $\bar{C}$ is independent of $T$. Thus we get the desired a
priori estimate (4.1) if we choose $N_{0}$ and $e^{-cd\beta}$ small enough.
## 5 The Local Existence
In this section, we prove the local existence result Proposition 4.1 by the
iteration method. First we rewrite the equation (3.8) with the initial values
(3.10)-(3.13) and the boundary values (3.16)-(3.17) as the following
$\left\\{\begin{array}[]{l}\displaystyle\Psi_{t}-\frac{\mu}{V+\Phi_{x}}\Psi_{xx}=g_{1}:=g_{1}(\Psi,\Phi_{x},\Psi_{x},\widehat{W}_{x}),\\\\[8.53581pt]
\displaystyle\Psi(0,t)=A(t),\\\\[5.69054pt]
\displaystyle\Psi(x,0)=\Psi_{0}(x),\end{array}\right.$ $None$
$\left\\{\begin{array}[]{l}\displaystyle\frac{R}{\gamma-1}\widehat{W}_{t}-\frac{\kappa}{V+\Phi_{x}}(\widehat{W}_{x}-\frac{\gamma-1}{2R}\Psi_{x}^{2})_{x}=g_{2}:=g_{2}(\Psi,\Phi_{x},\Psi_{x},\widehat{W}_{x},\Psi_{xx}),\\\\[8.53581pt]
\displaystyle\widehat{W}_{x}(0,t)-\frac{\gamma-1}{2R}\Psi_{x}^{2}(0,t)=B(t),\\\\[5.69054pt]
\displaystyle\widehat{W}(x,0)=\widehat{W}_{0}(x),\end{array}\right.$ $None$
and
$\Phi(x,t)=\Phi_{0}(x)+\int_{0}^{t}\Psi_{x}(x,\tau)d\tau,$ $None$
where $A(t),B(t)$ is given in (3.16), (3.17) respectively, and
$g_{1}=\frac{b_{1}-s^{2}V}{V+\Phi_{x}}\Phi_{x}-\frac{R}{V+\Phi_{x}}(\widehat{W}_{x}+\frac{\gamma-1}{R}U_{x}\Psi-\frac{\gamma-1}{2R}\Psi_{x}^{2}),$
$None$
$\begin{array}[]{ll}g_{2}=&\displaystyle-\frac{b_{1}-s^{2}V}{V+\Phi_{x}}\Phi_{x}\Psi_{x}+\frac{\kappa(\gamma-1)}{R(V+\Phi_{x})}(U_{x}\Psi)_{x}+sU_{x}\Psi-\frac{\kappa\Theta_{x}\Phi_{x}}{V(V+\Phi_{x})}\\\
&\displaystyle+\frac{\mu\Psi_{x}\Psi_{xx}}{V+\Phi_{x}}-\frac{R\Psi_{x}}{V+\Phi_{x}}[\widehat{W}_{x}+\frac{\gamma-1}{R}(U_{x}\Psi-\frac{\Psi_{x}^{2}}{2})].\end{array}$
$None$
To use the iteration method, we approximate the initial values
$(\Phi_{0},\Psi_{0},\widehat{W}_{0})\in H^{2}(0,+\infty)$ by
$(\Phi_{0k},\Psi_{0k},\widehat{W}_{0k})\in H^{5}(0,+\infty)$ which will be
determined later. For fixed $k$, we define the sequence
$\\{(\Phi_{k}^{(n)},\Psi_{k}^{(n)},\widehat{W}_{k}^{(n)})(x,t)\\}_{n=1}^{\infty}$
by
$(\Phi_{k}^{(0)},\Psi_{k}^{(0)},\widehat{W}_{k}^{(0)})(x,t)=(\Phi_{0k},\Psi_{0k},\widehat{W}_{0k})(x),$
$None$
and if $(\Phi_{k}^{(n-1)},\Psi_{k}^{(n-1)},\widehat{W}_{k}^{(n-1)})(x,t)$ is
given, then we define
$(\Phi_{k}^{(n)},\Psi_{k}^{(n)},\widehat{W}_{k}^{(n)})(x,t)$ as the following
$\left\\{\begin{array}[]{l}\displaystyle\Psi_{kt}^{(n)}-\frac{\mu}{V+\Phi_{kx}^{(n-1)}}\Psi_{kxx}^{(n)}=g_{1}^{(n-1)}:=g_{1}(\Psi_{k}^{(n-1)},\Phi_{kx}^{(n-1)},\Psi_{kx}^{(n-1)},\widehat{W}_{kx}^{(n-1)}),\\\\[8.53581pt]
\displaystyle\Psi_{k}^{(n)}(0,t)=A(t),\\\\[5.69054pt]
\displaystyle\Psi_{k}^{(n)}(x,0)=\Psi_{0k}(x),\end{array}\right.$ $None$
$\left\\{\begin{array}[]{l}\displaystyle\frac{R}{\gamma-1}\widehat{W}_{kt}^{(n)}-\frac{\kappa}{V+\Phi_{kx}^{(n-1)}}(\widehat{W}_{kx}^{(n)}-\frac{\gamma-1}{2R}{\Psi_{kx}^{(n)}}^{2})_{x}\\\
\displaystyle\qquad=g_{2}^{(n-1)}:=g_{2}(\Psi_{k}^{(n)},\Phi_{kx}^{(n-1)},\Psi_{kx}^{(n)},\widehat{W}_{kx}^{(n-1)},\Psi_{kxx}^{(n)}),\\\\[8.53581pt]
\displaystyle\widehat{W}_{kx}^{(n)}(0,t)-\frac{\gamma-1}{2R}{\Psi_{kx}^{(n)}}^{2}(0,t)=B(t),\\\\[5.69054pt]
\displaystyle\widehat{W}_{k}^{(n)}(x,0)=\widehat{W}_{0k}(x),\end{array}\right.\qquad\qquad$
$None$
and
$\Phi_{k}^{(n)}(x,t)=\Phi_{0k}(x)+\int_{0}^{t}\Psi_{kx}^{(n)}(x,\tau)d\tau.$
$None$
Now we construct the approximate initial values
$(\Phi_{0k},\Psi_{0k},\widehat{W}_{0k})(x)$. Firstly we choose $\Phi_{0k}\in
H^{5}$ such that $\Phi_{0k}\rightarrow\Phi_{0}$ strongly in $H^{2}$ as
$k\rightarrow\infty.$ Let
$\overline{\Psi}_{0}(x):=\Psi_{0}(x)-A(0)e^{-x^{2}}.$
Note that $A(0)=\Psi_{0}(0).$ Then we have $\overline{\Psi}_{0}(x)\in
H_{0}^{2}$. Now we choose $\overline{\Psi}_{0k}(x)\in H_{0}^{3}\cap H^{5}$
such that $\overline{\Psi}_{0k}\rightarrow\overline{\Psi}_{0}$ strongly in
$H^{2}$ as $k\rightarrow\infty$. We construct
$\Psi_{0k}(x):=\overline{\Psi}_{0k}(x)+A(0)e^{-x^{2}},$ $None$
then we have
$\Psi_{0k}\rightarrow\overline{\Psi}_{0}(x)+A(0)e^{-x^{2}}=\Psi_{0}(x)$
strongly in $H^{2}$ as $k\rightarrow\infty$. Moreover, $\Psi_{0k}(x)$
constructed in (5.10) satisfies the compatibility condition
$\Psi_{0k}(0)=A(0)$ for the approximate equation (5.7). Now we turn to the
compatibility condition for the equation (5.8). Let
$\overline{\widehat{W}}_{0}(x):=\widehat{W}_{0}(x)-B(0)xe^{-x^{2}}-\widehat{W}_{0}(0)e^{-x^{2}}.$
It is obvious that $\overline{\widehat{W}}_{0}(x)\in H^{2}_{0}$. So we can
choose $\overline{\widehat{W}}_{0k}(x)\in H_{0}^{3}\cap H^{5}$ such that
$\overline{\widehat{W}}_{0k}(x)\rightarrow\overline{\widehat{W}}_{0}(x)$
strongly in $H^{2}$ as $k\rightarrow\infty$. Set
$\widehat{W}_{0k}(x):=\overline{\widehat{W}}_{0k}(x)+B(0)xe^{-x^{2}}+\widehat{W}_{0}(0)e^{-x^{2}}.$
$None$
Then we have
$\widehat{W}_{0k}(x)\rightarrow\overline{\widehat{W}}_{0}(x)+B(0)xe^{-x^{2}}+\widehat{W}_{0}(0)e^{-x^{2}}=\widehat{W}_{0}(x)$
strongly in $H^{2}$ as $k\rightarrow\infty$. Note that
$B(0)=\widehat{W}_{0x}(0)-\frac{\gamma-1}{2R}\Psi^{2}_{0x}(0)$. We verify that
the approximated initial values $\Psi_{0k}(x),\widehat{W}_{0k}(x)$ satisfy the
following compatibility condition for the equation (5.8),
$\widehat{W}_{0kx}(0)-\frac{\gamma-1}{2R}\Psi_{0kx}^{2}(0)=\overline{\widehat{W}}_{0kx}(0)+B(0)-\frac{\gamma-1}{2R}\overline{\Psi}_{0kx}^{2}(0)=B(0).$
And it is easy to choose that the above approximation
$(\Phi_{0k}(x),\Psi_{0k}(x),\widehat{W}_{0k}(x))$ satisfies
$\|(\Phi_{0k},\Psi_{0k},\widehat{W}_{0k})\|_{2}\leq\frac{3}{2}M$ and
$\inf_{x}(V+\Phi_{0kx})\geq\frac{2}{3}m$ for any fixed $k$.
If $(\Psi_{k}^{(n-1)},\Psi_{k}^{(n-1)},\widehat{W}_{k}^{(n-1)})\in
X_{\frac{1}{2}m,bM}(0,t_{0})\cap C(0,t_{0};H^{5})$, then $g_{1}^{(n-1)}\in
C(0,t_{0};H^{4}).$ By linear parabolic theory, since $\Psi_{0k}\in H^{5}$,
there exists a unique solution to (5.7) satisfying
$\Psi_{k}^{(n)}\in C(0,t_{0};H^{5})\cap C^{1}(0,t_{0};H^{3})\cap
L^{2}(0,t_{0};H^{6}).$
Substituting $\Psi_{k}^{(n)}$ into $g_{2}^{(n-1)}$, we have that
$g_{2}^{(n-1)}\in C(0,t_{0};H^{3})$. Using linear parabolic theory again, we
obtain
$\widehat{W}_{k}^{(n)}\in C(0,t_{0};H^{5})\cap C^{1}(0,t_{0};H^{3})\cap
L^{2}(0,T;H^{6}).$
From (5.9), we also have
$\Phi_{k}^{(n)}\in C(0,t_{0};H^{5})\cap C^{1}(0,t_{0};H^{3})\cap
L^{2}(0,t_{0};H^{6}).$
The elementary energy estimates to the equation (5.7)-(5.8) yield that
$\|(\Psi_{k}^{(n)},\widehat{W}_{k}^{(n)})\|^{2}_{2}\leq(bM)^{2}.$
if the time interval $t_{0}=t_{0}(m,M)$ is suitably small. We omit the
detailed calculations for brevity.
Now from (5.9), we can compute that
$\|\Phi_{k}^{(n)}\|^{2}_{2}\leq(bM)^{2},$
and
$\inf_{x,t\in[0,t_{0}]}(V+\Phi_{kx}^{(n)})\geq\frac{1}{2}m.$
Therefore we have $(\Psi_{k}^{(n)},\Psi_{k}^{(n)},\widehat{W}_{k}^{(n)})\in
X_{\frac{1}{2}m,bM}(0,t_{0})\cap C(0,t_{0};H^{5})$. Since
$\|(\Psi_{k}^{(0)},\Psi_{k}^{(0)},\widehat{W}_{k}^{(0)})\|_{5}$ is uniformly
bounded for fixed $k$, we can show that $(\Psi_{k}^{(n)},$
$\Psi_{k}^{(n)},\widehat{W}_{k}^{(n)})$ is the Cauchy sequence in
$C(0,t_{0};H^{4})$. Letting $n\rightarrow\infty$ in (5.7)-(5.9), we get a
solution $(\Phi_{k},\Psi_{k},\widehat{W}_{k})(x,t)$ of (5.1)-(5.3) with the
initial values replaced by $(\Phi_{0k},\Psi_{0k},\widehat{W}_{0k})(x)$ in the
time interval $[0,t_{0}]$.
In the same way we can show that $(\Phi_{k},\Psi_{k},\widehat{W}_{k})(x)$ is a
Cauchy sequence in $C(0,T_{0};H^{2})$ (takin $T_{0}$ smaller than $t_{0}$ if
necessary). Now letting $k\rightarrow\infty$, we get the desired unique
solution $(\Phi,\Psi,\widehat{W})(x,t)$ to (5.1)-(5.3) in the time interval
$[0,T_{0}]$.
Acknowledgements: The research of F. M. Huang was supported in part by NSFC
Grant No. 10825102 for distinguished youth scholar, NSFC-NSAF Grant No.
10676037 and 973 project of China, Grant No.2006CB805902. The research of Y.
Wang was supported by the NSFC grant (No. 10801128).
## References
* [1] D.Gilbarg, The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73 (1951), 256-274.
* [2] F.Huang, J. Li and X.Shi, Asymptotic behavior of the solutions to the full compressible Navier-Stokes equations in the half space, to appear in Comm. Math. Sci..
* [3] F.Huang, A.Matsumura, Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation. Comm. Math. Phys. 289 (2009), no. 3, 841–861.
* [4] F.Huang, A.Matsumura and X.Shi, Viscous shock wave and boundary layer solution to an infolw problem for compressible viscous gas, Comm. Math. Phys. 239 (2003), 261-285.
* [5] F.Huang, A.Matsumura and X.Shi, On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J. Math. 41 (2004) 193-210.
* [6] F.Huang, A.Matsumura and X.Shi, A gas solid free boundary problem for a compressible viscous gas, SIAM. J. Math. Anal. 34 (2003), 1331-1355.
* [7] T.Liu, Shock waves for compressible Navier-Stokes equations are stable, Comm. Pure Appl. Math. XXXIX (1986), 565-594.
* [8] T.Liu and Z.Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys. 118 (1988), 451-465.
* [9] S.Kawashima and A.Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), 97-127.
* [10] S.Kawashima, A.Matsumura and K.Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. 62A (1986), 249-252.
* [11] A.Matsumura and M.Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Rat. Mech. Anal. 146 (1999), 1-22.
* [12] A.Matsumura and K.Nishihara, On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 2 (1985), 17-25.
* [13] A.Matsumura and K.Nishihara, Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 3 (1986), 1-13.
* [14] A.Matsumura and K.Nishihara, Large time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys. 222 (2001), 449–474.
* [15] T.Pan, H.Liu and K.Nishihara, Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary, SIAM J. Math. Anal. 34 (2002), 172-291.
* [16] J.Smoller, Shock waves and reaction-diffusion equations, Berlin, Heidelberg, New York, Springer 1982.
* [17] P.Zhu, Existence and asymptotic stability of stationary solution to the full compressible Navier-Stokes equations in the half space, Mathematical analysis in fluid and gas dynamics, RIMS kokyuroku 1247,(2002), 187-207.
|
arxiv-papers
| 2009-12-24T04:08:36 |
2024-09-04T02:49:07.252374
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feimin Huang, Xiaoding Shi, Yi Wang",
"submitter": "Yi Wang",
"url": "https://arxiv.org/abs/0912.4783"
}
|
0912.5009
|
# The MacWilliams Theorem for Four-Dimensional Modulo Metrics
Mehmet Özen, Murat Güzeltepe
Department of Mathematics, Sakarya University, TR54187 Sakarya, Turkey
###### Abstract
In this paper, the MacWilliams theorem is stated for codes over finite field
with four-dimensional modulo metrics.
AMS Classification: 94B05, 94B60
Keywords: MacWilliams theorem, Block codes, Weight enumerator, Quaternion
Mannheim metric.
## 1 Introduction
The MacWilliams theorem is one of the most important theorems in coding
theory. It is well known that two of the most famous results in block code
theory are MacWilliams Identity Theorem end Equivalence Theorem [1, 2]. Given
the weight enumerator of an code, the MacWilliams theorem ensure one to obtain
the weight enumerator of the dual code . The MacWilliams theorem very useful
since weight distribution of high rate codes can be obtained from low rate
codes. A well known version of the MacWilliams theorem for codes with respect
to Hamming weight was presented in [3]. The more general version of this
theorem are less often used in practical applications. The impact of this
theorem for practical as well as theoretical purposes is well known, see for
instance [3, Chs. 11.3, 6.5, and 19.2]. In [4], the MacWilliam theorem proved
for codes over finite fields with two-dimensional modulo metric.
In this study, we utilize the MacWilliam theorem for complete weight
enumerators to obtain the MacWilliams theorem for codes over quaternion
integers (QI). The Hamilton quaternion algebra is defined as follows.
###### Definition 1
Let $\mathcal{R}$ be the field of real numbers. The Hamilton Quaternion
Algebra over $\mathcal{R}$ denoted by $H[\mathcal{R}]$ is the associative
unital algebra given by the following representation:
i)$H[\mathcal{R}]$ is the free $\mathcal{R}$ module over the symbols
$1,i,j,k$, that is, $H[\mathcal{R}]=\\{a_{0}+a_{1}i+$
$a_{2}j+a_{3}k:\;a_{0},a_{1},a_{2},a_{3}\in R\\}$;
ii)1 is the multiplicative unit;
iii) $i^{2}=j^{2}=k^{2}=-1$;
iv) $ij=-ji=k,\;ik=-ki=j,\;jk=-kj=i$ [5].
If $q=a_{0}+a_{1}i+a_{2}j+a_{3}k$ is a quaternion integer, its conjugate
quaternion is $\overline{q}=a_{0}-(a_{1}i+a_{2}j+a_{3}k)$. The norm of $q$ is
$N(q)=q.\overline{q}=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}$, which is
multiplicative, that is, $N(q_{1}q_{2})=N(q_{1})N(q_{2})$. It should be noted
that quaternions are not commutative. The ring of the integers of the
quaternions is
$\;H[\mathcal{Z}]=\left\\{{a_{0}+a_{1}i+a_{2}j+a_{3}k:\;a_{0},a_{1},a_{2},a_{3}\in\mathcal{Z}}\right\\}$.
Let $H[\mathcal{Z}]_{\pi}$ be residue class of $H[\mathcal{Z}]$ modulo $\pi$,
where $\pi$ is prime quaternion integer. The set obtained form the elements of
$H[\mathcal{Z}]_{\pi}$ obtains the elements which by the remainders from right
dividing (or left dividing) the elements of $H[\mathcal{Z}]$ by the element
$\pi$. For example, let $p=3,\pi=1+i+j$ then we get
$H[\mathcal{Z}]_{\pi}=\left\\{{\mp 1,\mp i,\mp j,\mp k}\right\\}$. Also
$H[\mathcal{Z}]_{\pi}$ has $N(\pi)^{2}$ elements [6]. More information which
is related with the arithmetic properties of $H[\mathcal{Z}]$ can be found in
[5, pp. 57-71]. The quaternion Mannheim metric also called Lipschitz metric
was defined in [6, 7]. Let
$\alpha-\beta\equiv\delta=a_{0}+a_{2}i+a_{2}j+a_{3}k\,(\bmod\;\pi)$. Then the
weight of $\delta$ which is denoted by $w_{QM}(\delta)$ is equal
$\left|{a_{0}}\right|+\left|{a_{2}}\right|+\left|{a_{2}}\right|+\left|{a_{3}}\right|$.
The distance between $\alpha$ and $\beta$ was defined as
$d_{QM}(\alpha,\beta)=w_{QM}(\delta)$.
Now we recall some notation and definitions on characters and weight
enumerators needed in this paper. Let $\gamma$ be an element of the Galois
field $GF(p^{m})$. Using the primitive element $\alpha$, $\gamma$ can be
represented as $\gamma=\sum\nolimits_{t=0}^{m-1}{g_{t}\alpha^{t}}$ with
$g_{t}$ from $GF(p)$. The character $\chi_{1}(\gamma)$ is defined using the
primitive complex $p-th$ root $\xi$:
$\chi_{1}(\gamma)=\xi^{g_{0}}$
where
$\xi=\exp({{2\pi\sqrt{-1}}\mathord{\left/{\vphantom{{2\pi\sqrt{-1}}p}}\right.\kern-1.2pt}p})$,
$\pi=3,14...$
The complete weight enumerator classifies the codewords of a linear code
according to the number of times each field element $\omega_{t}$ appears in
the codeword. The composition of a vector
$u=\left({\begin{array}[]{*{20}c}{u_{0},}&{u_{1},}&\cdots&{,u_{n-1}}\\\
\end{array}}\right)$ denoted by $comp(u)$ is given by
$s=\left({\begin{array}[]{*{20}c}{s_{0},}&{s_{1},}&\cdots&{,s_{q-1}}\\\
\end{array}}\right)$, where $s_{t}$ is the number of components $u_{t}$ equal
to $\omega_{t}$. Note that there exist a group homomorphism between
$GF(p^{2})$ and $H[\mathcal{Z}]_{\pi}$ using a rational mapping. For example,
assume that $p=3$ then $\pi=1+i+j$,
$GF(p^{2})=\left\\{{0,1,\alpha,\alpha^{2},...,\alpha^{7}}\right\\}$ and
$H[\mathcal{Z}]_{\pi}=\left\\{{0,1,-1,i,-i,j,-j,k,-k}\right\\}$ where
$\alpha^{2}=\alpha+1,\;\alpha^{8}=1$. We obtain a group homomorphism mapping 0
to 0, 1 to 1, $\alpha$ to $i$, $2\alpha$ to $-i$, $2+2\alpha$ to $j$,
$1+\alpha$ to $-j$, $2\alpha+1$ to $k$, $\alpha+2$ to $-k$.
###### Definition 2
The composition of
$u=\left({\begin{array}[]{*{20}c}{u_{0},}&{u_{1},}&\cdots&{,u_{n-1}}\\\
\end{array}}\right)$, denoted by $comp(u)$, is
$s=\left({\begin{array}[]{*{20}c}{s_{0},}&{s_{1},}&\cdots&{s_{q-1}}\\\
\end{array}}\right)$ where $s_{t}=s_{t}(u)$ is the number of components
$u_{t}$ equal to $\omega_{t}$. Thus it is obtain
$\sum\limits_{t=0}^{q-1}{s_{t}(u)=n}.$
Let $C$ be a linear $[n,k]$ code over $GF(p)$. Then the complete weight
enumerator of $C$
$W_{C}(z_{0},z_{1},\cdots,z_{q-1})=\sum\limits_{c\in
C}{\left({\prod\limits_{t=0}^{q-1}{z_{t}^{s_{t}(u)}}}\right)}$
where $z_{t}$ are indeterminates and the sum extends over all compositions.
The MacWilliams theorem for complete weight enumerators [3, pp.143-144, Thm
10] then states:
###### Theorem 1
The complete weight enumerator of the dual code $C^{\bot}$ can be obtained
from the complete weight enumerator of the code $C$ by replacing each $z_{t}$
by
$\sum\limits_{s=0}^{q-1}{\chi_{1}(\omega_{t}\omega_{s})}z_{s}$
and dividing the result by the cardinality of $C$ which is denoted by
$\left|C\right|$.
## 2 The MacWilliams Theorem for codes over Quaternion Integers
Let $GF(q)$ be a finite field with $q=p^{m}$. The field $GF(q)$ is partitioned
as follows:
$GF(q)=\left\\{0\right\\}\cup G_{1}\cup G_{2}\cup G_{3}\cup G_{4}\cup
G_{5}\cup G_{6}\cup G_{7}\cup G_{8}.$
We set $\omega_{0}=0$. $G_{1}$ contains
${{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}$
elements
$\omega_{t},\,t=1,2,...,{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}$
in a fixed way such that for
$t=1,2,...,{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}$
we have
$\begin{array}[]{l}G_{2}=\omega_{2}G_{1},\;\omega_{2}\notin G_{1},\\\
G_{3}=\omega_{3}G_{1},\;\omega_{3}\notin G_{1}\cup G_{2},\\\
G_{4}=\omega_{4}G_{1},\;\omega_{4}\notin G_{1}\cup G_{2}\cup G_{3},\\\
G_{5}=\omega_{5}G_{1},\;\omega_{5}\notin G_{1}\cup G_{2}\cup G_{3}\cup
G_{4},\\\ G_{6}=\omega_{6}G_{1},\;\omega_{6}\notin G_{1}\cup G_{2}\cup
G_{3}\cup G_{4}\cup G_{5},\\\ G_{7}=\omega_{7}G_{1},\;\omega_{7}\notin
G_{1}\cup G_{2}\cup G_{3}\cup G_{4}\cup G_{5}\cup G_{6},\\\
G_{8}=\omega_{8}G_{1},\;\omega_{8}\notin G_{1}\cup G_{2}\cup G_{3}\cup
G_{4}\cup G_{5}\cup G_{6}\cup G_{7}.\\\ \end{array}.$
The quaternion Mannheim weight of a vector $u$ over $GF(p)$ is defined as
$quaternionic(u)=\left({\begin{array}[]{*{20}c}{g_{0},}&{g_{1},}&\cdots&{,g_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}}\\\
\end{array}}\right).$ Note that the quaternion integer enumerator does not
distinguish between the eight elements $\mp\omega,\mp i\omega,\mp j\omega,\mp
k\omega.$ The complete weight enumerator of the dual code $C^{\bot}$ from the
complete weight enumerator of the code $C$ over $H[\mathcal{Z}]_{\pi}$
obtained as follows:
###### Theorem 2
The quaternion integer (QI) weight enumerator of the dual code $C^{\bot}$ can
be obtained from QI weight enumerator of $C$ by replacing $z_{1}$ by
$z_{0}+\sum\limits_{s=1}^{{{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}}{\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{s})+\chi_{1}(-\omega_{1}\omega_{s})+\chi_{1}(i\omega_{1}\omega_{s})+\chi_{1}(-i\omega_{1}\omega_{s})+\chi_{1}(j\omega_{1}\omega_{s})\\\
+\chi_{1}(-j\omega_{1}\omega_{s})+\chi_{1}(k\omega_{1}\omega_{s})+\chi_{1}(-k\omega_{1}\omega_{s})\\\
\end{array}\right]}z_{s}=z_{0}+$
$[\chi_{1}(\omega_{1}\omega_{1})+\chi_{1}(-\omega_{1}\omega_{1})+\chi_{1}(i\omega_{1}\omega_{1})+\chi_{1}(-i\omega_{1}\omega_{1})+\chi_{1}(j\omega_{1}\omega_{1})+\chi_{1}(-j\omega_{1}\omega_{1})+\chi_{1}(k\omega_{1}\omega_{1})+\chi_{1}(-k\omega_{1}\omega_{1})]z_{1}+\cdots$
$+\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
+\chi_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}(-j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
\end{array}\right]z_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}},$
$z_{2}$ by
$[\chi_{1}(\omega_{1}\omega_{1})+\chi_{1}(-\omega_{1}\omega_{1})+\chi_{1}(i\omega_{1}\omega_{1})+\chi_{1}(-i\omega_{1}\omega_{1})+\chi_{1}(j\omega_{1}\omega_{1})+\chi_{1}(-j\omega_{1}\omega_{1})+\chi_{1}(k\omega_{1}\omega_{1})+\chi_{1}(-k\omega_{1}\omega_{1})]z_{2}+\cdots$
$+\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
+\chi_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}(-j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
\end{array}\right]z_{1}...$
and using the same argument, shifting the coefficients of
$z_{1},z_{2},\cdots,z_{{{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}}$,
$z_{{{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}}$
by
$[\chi_{1}(\omega_{1}\omega_{1})+\chi_{1}(-\omega_{1}\omega_{1})+\chi_{1}(i\omega_{1}\omega_{1})+\chi_{1}(-i\omega_{1}\omega_{1})+\chi_{1}(j\omega_{1}\omega_{1})+\chi_{1}(-j\omega_{1}\omega_{1})+\chi_{1}(k\omega_{1}\omega_{1})+\chi_{1}(-k\omega_{1}\omega_{1})]z_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}+\cdots$
$+\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
+\chi_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}(-j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
\end{array}\right]z_{\left({{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}\right)-1}.$
The proof is immediately obtained from MacWilliams theorem for complete weight
enumerators above.
###### Example 1
Let $p=3,\;\pi=1+i+j+k$. Then
$H[\mathcal{Z}]_{\pi}=\left\\{{0,1,-1,i,-i,j,-j,k,-k}\right\\}$. Let us
consider $[2,1,2]$ \- code $C$ over $GF(9)=H[\mathcal{Z}]_{\pi}$. Thus we get
$GF(9)=H[\mathcal{Z}]_{\pi}=G_{0}\cup G_{1}=\left\\{0\right\\}\cup\left\\{{\mp
1,\mp i,\mp j,\mp k}\right\\},\omega_{0}=0,\;\omega_{1}=1$ Assume that the
code $C$ which is an left ideal of $H[\mathcal{Z}]_{\pi}\times
H[\mathcal{Z}]_{\pi}$ is generate by the matrix $(1,1)$. Then the complete
weight enumerator of $C$ is $w_{QM}(C)=z_{0}^{2}+8z_{1}^{2}$. Applying the QI
MacWilliams theorem means that to replace $z_{1}\to
z_{0}+\left({\xi^{1}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{2}+\xi^{1}+\xi^{1}+\xi^{2}}\right)z_{1}=z_{0}-z_{1}$.
$1+\xi^{1}+\xi^{2}=0$ since there is a group homomorphism between $GF(p^{2})$
and $H[\mathcal{Z}]_{\pi}$, where $\xi=e^{{{2\pi
i}\mathord{\left/{\vphantom{{2\pi i}3}}\right.\kern-1.2pt}3}},\;\pi=3,14...$
Thus the complete weight enumerator of the dual code $C^{\bot}$ is equal
$z_{0}^{2}+8z_{1}^{2}=w_{QM}(C)$.
###### Example 2
Let $p=5,\;\pi=2+i$. Then
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
H[\mathcal{Z}]_{\pi}=\left\\{0\right\\}\cup\left\\{{1,-1,i,-i,j,-j,k,-k}\right\\}\cup(1+j)\left\\{{1,-1,i,-i,j,-j,k,-k}\right\\}\cr
0.0pt{\hfil$\displaystyle\cup(1+k)\left\\{{1,-1,i,-i,j,-j,k,-k}\right\\}.\cr}}}$
Let us consider $[3,1,3]$-code over $GF(25)=H[\mathcal{Z}]_{2+i}$. Thus we
get, $\omega_{0}=0,\;\omega_{1}=1,\;\omega_{2}=1+\alpha\leftrightarrow
1+j,\;\omega_{3}=1+2\alpha\leftrightarrow 1+k$. Assume that the code $C$ which
is an left ideal of $H[\mathcal{Z}]_{\pi}\times H[\mathcal{Z}]_{\pi}$ is
generate by the matrix $\left({\begin{array}[]{*{20}c}1&1&1\\\
\end{array}}\right)$. Then the complete weight enumerator of $C$ is
$w_{QM}(C)=z_{0}^{3}+8z_{1}^{3}+8z_{2}^{3}+8z_{3}^{3}$. Applying the QI
MacWilliams theorem means that to replace
$z_{0}\to z_{0}+8z_{1}+8z_{2}+8z_{3},$ $\halign{\hbox
to\displaywidth{$\hfil\displaystyle#\hfil$}\cr 0.0pt{\hfil$\displaystyle
z_{1}\to
z_{0}+(\xi^{1}+\xi^{4}+\xi^{3}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{0}+\xi^{0})z_{1}+\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{2}\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{3},\cr}}}}$
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle z_{2}\to
z_{0}+(\xi^{1}+\xi^{4}+\xi^{3}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{0}+\xi^{0})z_{2}+\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{3}\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{1},\cr}}}}$
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle z_{3}\to
z_{0}+(\xi^{1}+\xi^{4}+\xi^{3}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{0}+\xi^{0})z_{3}+\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{1}\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{2}.\cr}}}}$
$1+\xi^{1}+\xi^{2}+\xi^{3}+\xi^{4}=0$ since there is a group homomorphism
between $GF(5^{2})$ and $H[\mathcal{Z}]_{2+i}$, where $\xi=e^{{{2\pi
i}\mathord{\left/{\vphantom{{2\pi i}5}}\right.\kern-1.2pt}5}},\;\pi=3,14...$
Thus the complete weight enumerator of the dual code $C^{\bot}$ is equal
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
z_{0}^{3}+24z_{0}z_{1}^{2}+24z_{0}z_{2}^{2}+24z_{0}z_{3}^{2}+24z_{1}^{3}+24z_{2}^{3}+24z_{3}^{3}\cr
0.0pt{\hfil$\displaystyle+48z_{1}^{2}z_{2}+48z_{1}z_{2}^{2}+48z_{2}z_{3}^{2}+48z_{1}^{2}z_{3}+48z_{1}z_{3}^{2}+48z_{2}^{2}z_{3}\cr
0.0pt{\hfil$\displaystyle+192z_{1}z_{2}z_{3}.\cr}}}}$
## 3 Conclusion
In this paper, we proved the MacWilliams for four-dimensional modulo metrics.
In fact, the quaternion Mannheim metric can be seen as a four-dimensional
generalization of the Lee metric. Also the quaternion Mannheim metric can be
seen as a four-dimensional generalization of the Mannheim metric. In other
words, if four-dimensional space is restricted to two-dimensional space then
results in [4] are obtained.
## References
* [1] F. J. MacWilliams, ”Combinatorial Problems of Elementary Abelian Groups,” Ph.D. dissertation, Harvard Univ., Cambridge, MA, 1962.
* [2] F. J. MacWilliams, ”A theorem on the distribution of weights in a systematic code,” Bell Syst. Tech. J., vol. 42, pp. 79-94, 1963.
* [3] F. J. Macwilliams and N. J. Sloane, ”The Theory of Error Correcting Codes”, North Holland Pub. Co., 1977.
* [4] K. Huber, ”The MacWilliams Theorem for Two-Dimensional Modulo Metrics,” AAECC, 41-48, 1997. (submitted, 2009).
* [5] G. Davidoff, P. Sarnak, A. Valette, ”Elementary Number Theory, Group Theory, Ramanujan Graphs”, Cambridge University Pres, 2003.
* [6] C. Martinez et al. ”Perfect Codes from Cayley Graphs over Lipschitz Integers,” IEEE Trans. Inform.Theory, vol. 55, pp. 3552-3562, August, 2009.
* [7] M. zen and M. G zeltepe, ”Codes over Quaternion Integers”, e-print arXiv:0905.4160v1.
|
arxiv-papers
| 2009-12-27T11:16:48 |
2024-09-04T02:49:07.263206
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Murat Guzeltepe, Mehmet Ozen",
"submitter": "Murat Guzeltepe Mr",
"url": "https://arxiv.org/abs/0912.5009"
}
|
0912.5030
|
Quasideterminant solutions of the generalized Heisenberg magnet model
U. Saleem 111Tel No: +92-42-99231243, Fax No: +92-42-35856892
e-mail:usaleem@physics.pu.edu.pk, usman_physics@yahoo.com and M. Hassan
222mhassan@physics.pu.edu.pk
Department of Physics, University of the Punjab,
Quaid-e-Azam Campus, Lahore-54590, Pakistan.
In this paper we present Darboux transformation for the generalized Heisenberg
magnet (GHM) model based on general linear Lie group $GL(n)$ and construct
multi-soliton solutions in terms of quasideterminants. Further we relate the
quasideterminant multi-soliton solutions obtained by the means of Darboux
transformation with those of obtained by dressing method. We also discuss the
model based on the Lie group $SU(n)$ and obtain explicit soliton solutions of
the model based on $SU(2)$.
PACS: 11.10.Nx, 02.30.Ik
Keywords: Integrable systems, Heisenberg model, Darboux transformation,
quasideterminants
## 1 Introduction
During the past decades, there has been an increasing interest in the study of
classical and quantum integrability of Heisenberg ferromagnet (HM) model
[1]-[15]. The Heisenberg ferromagnet (HM) model based on Hermitian symmetric
spaces has been studied in [11]-[14]. The integrability of the HM model based
on $SU(2)$ via inverse scattering method is presented in [2]-[3] and its
$SU(n)$ generalization is studied in [4]. The integrability of the GHM model
based on the general linear Lie group $GL(n)$ via Lax formalism has been
investigated in [1]. In this paper we present the Darboux transformation of
the GHM model based on general linear group $GL(n)$ with Lie algebra
$\verb"gl(n)"$ and calculate multi-soliton solutions in term of
quasideterminants. We also establish the relation between the Darboux
transformation and the well-known dressing method [16]. In the last section,
we discuss the model based $SU(n)$ and calculate an explicit expression of the
single-soliton solution of the HM model based on the Lie group $SU(2)$ using
Darboux transformation.
The Hamiltonian of the GHM model is defined by [1]
${\cal
H}=\frac{1}{2}\mbox{Tr}\left(\left(\partial_{x}U\right)^{T}\left(\partial_{x}U\right)\right),$
(1.1)
with $"T"$ is transpose and $U(x,t)$ is a matrix-valued function which takes
values in the Lie algebra $\verb"gl(n)"$ of the general linear group $GL(n)$.
The corresponding equation of motion can be expressed as
$\partial_{t}U=\\{{\cal H},\partial_{x}U\\}.$ (1.2)
The above equation (1.2) can be written as
$\partial_{t}U=\left[U,\partial^{2}_{x}U\right],$ (1.3)
where $\partial_{x}=\frac{\partial}{\partial x}$ and
$\partial_{t}=\frac{\partial}{\partial t}$. Let us assume that $U(x,t)$ is
diagonizable, i.e.,
$U=g\,T\,g^{-1},$ (1.4)
where $g\in GL(n)$ is matrix function of $(x,t)$ and $T$ is a $n\times n$
constant matrix
$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccccc}c_{1}&0&\cdots&0&0&0&\cdots&0&0\\\
0&c_{1}&\cdots&0&0&0&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&c_{1}&0&0&\cdots&0&0\\\ 0&0&\cdots&0&c_{2}&0&\cdots&0&0\\\
0&0&\cdots&0&0&c_{2}&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&\cdots&0&0&0&\cdots&0&c_{2}\\\ \end{array}\right),$ (1.13)
where $1\leq p\leq n$ and $c_{1},c_{2}\in\mathbb{R}$ (or $\mathbb{C}$). From
equations (1.4) and (1.13), we have
$\left[U,\left[U,\left[U,\chi\right]\right]\right]=c^{2}\left[U,\chi\right],$
(1.14)
for an arbitrary matrix function $\chi$ and $c=c_{1}-c_{2}\neq 0$. Since
$\partial_{x}U\equiv U_{x}=\left[\partial_{x}gg^{-1},U\right],$ (1.15)
implies
$\left[U,\left[U,U_{x}\right]\right]=c^{2}U_{x},$ (1.16)
The equation of motion (1.3) can also be written as the zero-curvature
condition i.e.,
$\left[\partial_{x}-\frac{1}{(1-\lambda)}U,\partial_{t}-\frac{c^{2}}{(1-\lambda)^{2}}U-\frac{1}{(1-\lambda)}\left[U,U_{x}\right]\right]=0.$
(1.17)
The above zero-curvature condition (1.17) is equivalent to the compatibility
condition of the following Lax pair
$\displaystyle\partial_{x}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{(1-\lambda)}U(x,t)\Psi(x,t;\lambda)$ (1.18)
$\displaystyle\partial_{t}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\left(\frac{c^{2}}{(1-\lambda)^{2}}U+\frac{1}{(1-\lambda)}\left[U,U_{x}\right]\right)\Psi(x,t;\lambda)$
(1.19)
where $\lambda$ is a real (or complex) parameter and $\Psi$ is an invertible
$n\times n$ matrix-valued function belonging to $GL(n)$.
In the next section, we define the Darboux transformation on matrix solutions
$\Psi$ of the Lax pair (1.18)-(1.19). To write down the explicit expressions
for matrix solutions of the GHM model, we will use the notion of
quasideterminant introduced by Gelfand and Retakh [17]-[21].
Let $X$ be an $n\times n$ matrix over a ring $R$ (noncommutative, in general).
For any $1\leq i$, $j\leq n$, let $r_{i}$ be the $i$th row and $c_{j}$ be the
$j$th column of $X$. There exist $n^{2}$ quasideterminants denoted by
$|X|_{ij}$ for $i,j=1,\ldots,n$ and are defined by
$|X|_{ij}=\left|\begin{array}[]{cc}X^{ij}&c_{j}^{\,\,i}\\\
r_{i}^{\,\,j}&\framebox(0.0,0.0)[bl]{\framebox{$x_{ij}$}}\end{array}\right|=x_{ij}-r_{i}^{\,\,j}\left(X^{ij}\right)^{-1}c_{j}^{\,\,i},$
(1.20)
where $x_{ij}$ is the $ij$th entry of $X$, $r_{i}^{\,\,j}$ represents the
$i$th row of $X$ without the $j$th entry, $c_{j}^{\,\,i}$ represents the $j$th
column of $X$ without the $i$th entry and $X^{ij}$ is the submatrix of $X$
obtained by removing from $X$ the $i$th row and the $j$th column. The
quasideterminats are also denoted by the following notation. If the ring $R$
is commutative i.e. the entries of the matrix $X$ all commute, then
$|X|_{ij}=(-1)^{i+j}\frac{\mathrm{det}X}{\mathrm{det}X^{ij}}.$ (1.21)
For a detailed account of quasideterminants and their properties see e.g.
[17]-[21]. In this paper, we will consider only quasideterminants that are
expanded about an $n\times n$ matrix over a commutative ring. Let
$\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right),$
be a block decomposition of any $K\times K$ matrix where the matrix $D$ is
$n\times n$ and $A$ is invertible. The ring $R$ in this case is the
(noncommutative) ring of $n\times n$ matrices over another commutative ring.
The quasideterminant of $K\times K$ matrix expanded about the $n\times n$
matrix $D$ is defined by
$\left|\begin{array}[]{cc}A&B\\\
C&\framebox(0.0,0.0)[bl]{\framebox{$D$}}\end{array}\right|=D-CA^{-1}B.$ (1.22)
The quasideterminants have found various applications in the theory of
integrable systems, where the multisoliton solutions of various noncommutative
integrable systems are expressed in terms of quisideterminants (see e.g.
[22]-[30]).
## 2 Darboux transformation
The Darboux transformation is one of the well-known method of obtaining multi-
soliton solutions of many integrable models [31]-[33]. We define the Darboux
transformation on the matrix solutions of the Lax pair (1.18)-(1.19), in terms
of an $n\times n$ matrix $D(x,t,\lambda)$, called the Darboux matrix. For a
general discussion on Darboux matrix approach see e.g. [34]-[39]. The Darboux
matrix relates the two matrix solutions of the Lax pair (1.18)-(1.19), in such
a way that the Lax pair is covariant under the Darboux transformation. The
one-fold Darboux transformation on the matrix solution of the Lax pair
(1.18)-(1.19) is defined by
$\Psi\left[1\right](x,t;\lambda)=D(x,t,{\lambda})\Psi(x,t;\lambda),$ (2.1)
where $D(x,t,{\lambda})$ is the Darboux matrix. For our case, we can make the
following ansatz
$D(x,t,\lambda)=\lambda I-M(x,t),\ \ $ (2.2)
where $M(x,t)$ is an $n\times n$ matrix function and $I$ is an $n\times n$
identity matrix. The new solution $\Psi\left[1\right](x,t;\lambda)$ satisfies
the following Lax pair, i.e.
$\displaystyle\partial_{x}\Psi\left[1\right](x,t;\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{1-\lambda}U\left[1\right]\Psi\left[1\right](x,t;\lambda),$
(2.3) $\displaystyle\partial_{t}\Psi\left[1\right](x,t;\lambda)$
$\displaystyle=$
$\displaystyle\left(\frac{c^{2}}{(1-\lambda)^{2}}U\left[1\right]+\frac{1}{1-\lambda}\left[U\left[1\right],U_{x}\left[1\right]\right]\right)\Psi\left[1\right](x,t;\lambda),$
(2.4)
where $U\left[1\right]$ satisfies the equation of motion (1.3). By operating
$\partial_{x}$ and $\partial_{t}$ on equation (2.1) and equating the
coefficients of different powers of $\lambda$, we get the following
transformation on the matrix field $U$
$\displaystyle U\left[1\right]$ $\displaystyle=$ $\displaystyle U+M_{x},$
(2.5)
and the following conditions which $M$ is required to satisfy
$\displaystyle M_{x}\left(I-M\right)$ $\displaystyle=$
$\displaystyle\left[U,M\right],$ (2.6) $\displaystyle
M_{t}\left(I-M\right)^{2}$ $\displaystyle=$
$\displaystyle\left[c^{2}U+\left[U,U_{x}\right],M\right]+M\left[U,U_{x}\right]M-\left[U,U_{x}\right]M^{2}.$
(2.7)
One can solve equations (2.6)-(2.7) to obtain an explicit expression for the
matrix function $M(x,t)$. An explicit expression for $M(x,t)$ can be found as
follows.
Let us take $n$ distinct real (or complex) constant parameters
${\lambda}_{1},\cdots,{\lambda}_{n}(\neq 1)$. Also take $n$ constant column
vectors $e_{1},e_{2},\cdots,e_{n}$ and construct an invertible non-degenerate
$n\times n$ matrix function $\Theta(x,t)$
$\Theta(x,t)=\left(\Psi({\lambda}_{1})e_{1},\cdots,\Psi({\lambda}_{n})e_{n}\right)=\left(\theta_{1},\cdots,\theta_{n}\right).$
(2.8)
Each column $\theta_{i}=\Psi({\lambda}_{i})e_{i}$ in the matrix $\Theta$ is a
column solution of the Lax pair (1.18)-(1.19) when ${\lambda}={\lambda}_{i}$
and $i=1,2,\ldots,n$ i.e.
$\displaystyle\partial_{x}\theta_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\lambda_{i}}U\theta_{i},$ (2.9)
$\displaystyle\partial_{t}\theta_{i}$ $\displaystyle=$
$\displaystyle\left(\frac{c^{2}}{(1-\lambda_{i})^{2}}U+\frac{1}{1-\lambda_{i}}\left[U,U_{x}\right]\right)\theta_{i}.$
(2.10)
Let us take an $n\times n$ invertible diagonal matrix with entries being
eigenvalues $\lambda_{i}$ corresponding to the eigenvectors $\theta_{i}$
$\Lambda=\text{diag}({\lambda}_{1},\ldots,{\lambda}_{n}).$ (2.11)
The $n\times n$ matrix generalization of the Lax pair (2.9)-(2.10) will be
$\displaystyle\partial_{x}\Theta$ $\displaystyle=$ $\displaystyle
U\Theta\left(I-\Lambda\right)^{-1},$ (2.12)
$\displaystyle\partial_{t}\Theta_{i}$ $\displaystyle=$ $\displaystyle
c^{2}U\Theta\left(I-\Lambda\right)^{-2}+\left[U,U_{x}\right]\Theta\left(I-\Lambda\right)^{-1}.$
(2.13)
The $n\times n$ matrix $\Theta$ is a particular matrix solution of the Lax
pair (2.9)-(2.10) with $\Lambda$ being a matrix of particular eigenvalues. In
terms of particular matrix solution $\Theta$ of the Lax pair (2.9)-(2.10), we
make the following choice of the matrix $M(x,t)$
$M(x,t)=\Theta\Lambda\Theta^{-1}.$ (2.14)
Our next step is to check that equation (2.14) is a solution of equations
(2.6)-(2.7). In order to show this, we first operate $\partial_{x}$ on
equation (2.14) to get
$\displaystyle\partial_{x}M$ $\displaystyle=$
$\displaystyle\partial_{x}(\Theta\Lambda\Theta^{-1}),$ (2.15) $\displaystyle=$
$\displaystyle\left(\partial_{x}\Theta\right)\Lambda\Theta^{-1}+\Theta\Lambda\partial_{x}(\Theta^{-1}),$
$\displaystyle=$ $\displaystyle
U\Theta(I-\Lambda)^{-1}\Lambda\Theta^{-1}-\Theta\Lambda\Theta^{-1}U\Theta(I-\Lambda)^{-1}\Theta^{-1},$
$\displaystyle=$
$\displaystyle-U+\Theta(I-\Lambda)\Theta^{-1}j_{+}\Theta(I-\Lambda)^{-1}\Theta^{-1},$
$\displaystyle=$ $\displaystyle-U+\left(I-M\right)U\left(I-M\right)^{-1},$
which is the equation (2.6). Similarly operate $\partial_{t}$ on (2.14), we
get
$\displaystyle\partial_{t}M$ $\displaystyle=$
$\displaystyle\partial_{t}\left(\Theta\Lambda\Theta^{-1}\right)$ (2.16)
$\displaystyle=$
$\displaystyle\left(\partial_{t}\Theta\right)\Lambda\Theta^{-1}+\Theta\Theta\Lambda\partial_{t}(\Theta^{-1})$
$\displaystyle=$
$\displaystyle\left(c^{2}U\Theta\left(I-\Lambda\right)^{-2}+\left[U,U_{x}\right]\Theta\left(I-\Lambda\right)^{-1}\right)\Lambda\Theta^{-1}-$
$\displaystyle\Theta\Lambda\Theta^{-1}\left(c^{2}U\Theta\left(I-\Lambda\right)^{-2}+\left[U,U_{x}\right]\Theta\left(I-\Lambda\right)^{-1}\right)\Theta^{-1},$
which is equation (2.7). This shows that the choice (2.14) of the matrix $M$
satisfies the equations (2.6)-(2.7). In other words we can say that if the
collection $\left(\Psi,U\right)$ is a solution of the Lax pair (1.18)-(1.19)
and the matrix $M$ is defined by (2.14), then $\left(\Psi[1],U[1]\right)$
defined by (2.1) and (2.5) respectively, is also a solution of the same Lax
pair. Therefore we say that
$\displaystyle\Psi[1]$ $\displaystyle=$ $\displaystyle\left(\lambda
I-\Theta\Lambda\Theta^{-1}\right)\Psi,$ $\displaystyle U[1]$ $\displaystyle=$
$\displaystyle\left(I-\Theta\Lambda\Theta^{-1}\right)U\left(I-\Theta\Lambda\Theta^{-1}\right)^{-1},$
is the required Darboux transformation on the solution $\Psi$ to the Lax pair
(1.18)-(1.19) and $U$ to the equation of motion (1.3) respectively.
## 3 Quasideterminant solutions
We have shown that the matrix $M=\Theta\Lambda\Theta^{-1}$ satisfies the
conditions (2.6)-(2.7). Therefore, the one-fold Darboux transformation (2.1)
can also be written in terms of quasideterments as
$\displaystyle\Psi[1]$ $\displaystyle\equiv$ $\displaystyle
D(x,t;\lambda)\Psi=\left(\lambda
I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)\Psi,$ (3.3) $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cc}\Theta_{1}&\Psi\\\
\Theta_{1}\Lambda_{1}&\framebox(0.0,0.0)[bl]{\framebox{$\lambda\Psi$}}\end{array}\right|.$
The above equation defines the Darboux transformation on the matrix solution
$\Psi$ of the Lax pair (1.18)-(1.19). The corresponding one-fold Darboux
transformation on the matrix field $U$ is
$\displaystyle U[1]$ $\displaystyle=$
$\displaystyle\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)U\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)^{-1},$
(3.8) $\displaystyle=$ $\displaystyle\left|\begin{array}[]{cc}\Theta_{1}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|U\left|\begin{array}[]{cc}\Theta_{1}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|^{-1}.$
We write two-fold Darboux transformation on $\Psi$ as
$\displaystyle\Psi[2]$ $\displaystyle\equiv$ $\displaystyle
D(x,t;\lambda)\Psi[1]=\lambda\Psi[1]-\Theta_{2}[1]\Lambda_{2}\Theta^{-1}_{2}[1]\Psi[1]$
(3.12) $\displaystyle=$ $\displaystyle\lambda\left(\lambda
I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)\Psi-$
$\displaystyle\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)\Lambda_{2}\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)^{-1}\left(\lambda
I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)\Psi,$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Theta_{1}&\Theta_{2}&\Psi\\\
\Theta_{1}\Lambda_{1}&\Theta_{2}\Lambda_{2}&\lambda\Psi\\\
\Theta_{1}\Lambda_{1}^{2}&\Theta_{2}\Lambda_{2}^{2}&\framebox(0.0,0.0)[bl]{\framebox{$\lambda^{2}\Psi$}}\end{array}\right|.$
Similarly the expression for two-fold Darboux transformation on the matrix
field $U$ as
$\displaystyle U[2]$ $\displaystyle=$
$\displaystyle\Theta_{2}[1]\left(I-\Lambda_{2}\right)\Theta^{-1}_{2}[1]U[1]\left(\Theta_{2}[1]\left(I-\Lambda_{2}\right)\Theta^{-1}_{2}[1]\right)^{-1},$
$\displaystyle=$
$\displaystyle\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)\left(I-\Lambda_{2}\right)\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)^{-1}\times$
$\displaystyle\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)U\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)^{-1}\times$
$\displaystyle\left(\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)\left(I-\Lambda_{2}\right)\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)^{-1}\right)^{-1},$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Theta_{1}&\Theta_{2}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|\times
U\times$ (3.20)
$\displaystyle\times\left|\begin{array}[]{ccc}\Theta_{1}&\Theta_{2}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|^{-1}.$
The result can be generalized to obtain $N$-fold Darboux transformation on
matrix solution $\Psi$ as
$\displaystyle\Psi[N]$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccccc}\Theta_{1}&\Theta_{2}&\cdots&\Theta_{N}&\Psi\\\
\Theta_{1}\Lambda_{1}&\Theta_{2}\Lambda_{2}&\cdots&\Theta_{N}\Lambda_{N}&\lambda\Psi\\\
\Theta_{1}\Lambda_{1}^{2}&\Theta_{2}\Lambda_{2}^{2}&\cdots&\Theta_{N}\Lambda_{N}^{2}&\lambda^{2}\Psi\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\Theta_{1}\Lambda_{1}^{N}&\Theta_{2}\Lambda_{2}^{N}&\cdots&\Theta_{N}\Lambda_{N}^{N}&\framebox(0.0,0.0)[bl]{\framebox{$\lambda^{N}\Psi$}}\end{array}\right|.$
(3.26)
Similarly the expression for $U[N]$ is
$\displaystyle U[N]$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccccc}\Theta_{1}&\Theta_{2}&\cdots&\Theta_{N}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{2}&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{N}&\Theta_{2}\left(I-\Lambda_{2}\right)^{N}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{N}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|\times
U\times$ (3.38)
$\displaystyle\times\left|\begin{array}[]{ccccc}\Theta_{1}&\Theta_{2}&\cdots&\Theta_{N}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{2}&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{N}&\Theta_{2}\left(I-\Lambda_{2}\right)^{N}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{N}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|^{-1}.$
We now relate the quasideterminant solutions of GHM with the solutions
obtained by dressing method and the inverse scattering method. For this
purpose, we proceed as follows. From the definition of the matrix $M$, we have
$\displaystyle M\Theta$ $\displaystyle=$ $\displaystyle\Theta\Lambda.$ (3.39)
Let $\theta_{i}$ and $\theta_{j}$ be the column solutions of the Lax pair
(1.18)-(1.19) when $\lambda=\lambda_{i}$ and $\lambda=\lambda_{j}$
respectively i.e.
$\displaystyle M\theta_{i}$ $\displaystyle=$
$\displaystyle\lambda_{i}\theta_{i},\quad i=1,2,\dots,p$ $\displaystyle
M\theta_{j}$ $\displaystyle=$ $\displaystyle\lambda_{j}\theta_{j}.\quad
j=p+1,p+2,\dots,n$ (3.40)
Now we take $\lambda_{i}=\mu$ and $\lambda_{j}=\bar{\mu}$, we may write the
matrix $M$ as
$\displaystyle M$ $\displaystyle=$ $\displaystyle\mu P+\bar{\mu}P^{\perp},$
(3.41)
where $P$ is the hermitian projector i.e. $P^{\dagger}=P$. The projector $P$
satisfies $P^{2}=P$ and $P^{\perp}=1-P$. The projector $P$ is hermitian
projection on a complex space and $P^{\perp}$ as projection on orthogonal
space. Now equation (3.41) can also written as
$\displaystyle M$ $\displaystyle=$
$\displaystyle\left(\mu-\bar{\mu}\right)P+\bar{\mu}I,$ (3.42)
where the hermitian projector can be expressed as
$\displaystyle P$ $\displaystyle=$
$\displaystyle\theta_{i}\left(\theta_{i}^{\dagger},\theta_{i}\right)^{-1}\theta_{i}^{\dagger}.$
(3.43)
The one-fold Darboux transformation (3.3) on the matrix solution $\Psi$ can
also be expressed in terms of projector $P$ as
$\displaystyle\Psi[1]$ $\displaystyle\equiv$ $\displaystyle{\cal
D}(x,t;\lambda)\Psi=\left(I-\frac{\mu-\bar{\mu}}{\lambda-\bar{\mu}}P\right)\Psi,$
(3.44)
where ${\cal D}(x,t;\lambda)$ is the rescaled Darboux-dressing function i.e.
${\cal D}(x,t;\lambda)=\left(\lambda-\mu\right)^{-1}D(x,t;\lambda)$. Similarly
the $N$-fold Darboux transformation (3.26) on the matrix solution $\Psi$ can
also be written as (take $P[1]=P$)
$\displaystyle\Psi[N]$ $\displaystyle=$
$\displaystyle\prod_{k=0}^{N-1}\left(I-\frac{\mu_{N-k}-{\bar{\mu}_{N-k}}}{\lambda-{\bar{\mu}_{N-k}}}P[N-k]\right)\Psi.$
(3.45)
Now we can express the $N$-fold Darboux transformation (3.38) on the matrix
field $U$ can be written as
$\displaystyle U[N]$ $\displaystyle=$
$\displaystyle\prod_{k=0}^{N-1}\left(I-\frac{\mu_{N-k}-{\bar{\mu}_{N-k}}}{1-{\bar{\mu}_{N-k}}}P[N-k]\right)U\prod_{l=1}^{N-1}\left(I-\frac{{\bar{\mu}_{l}}-\mu_{l}}{1-{\bar{\mu}_{l}}}P[l]\right),$
(3.46)
and hermitian projector is defined as
$\displaystyle P[k]$ $\displaystyle=$
$\displaystyle\theta_{i}[k]\left(\theta_{i}^{\dagger}[k],\theta_{i}[k]\right)^{-1}\theta_{i}^{\dagger}[k].$
(3.47)
The expressions (3.45) and (3.46) can also be written as sum of $K$ terms [27]
$\displaystyle\Psi[N]$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\left(I-\frac{1}{\lambda-{\bar{\mu}_{k}}}R_{k}\right)\Psi,$
(3.48)
and
$\displaystyle U[N]$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\left(I-\frac{1}{1-{\bar{\mu}_{k}}}R_{k}\right)U\sum_{l=0}^{N-1}\left(I-\frac{1}{1-{\bar{\mu}_{l}}}R_{l}\right)^{-1},$
(3.49)
where
$\displaystyle R_{k}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{N-1}\left(\mu_{l}-\bar{\mu_{k}}\right)\theta_{i}^{(k)}\left(\theta_{i}^{(k)\dagger},\theta_{i}^{(l)}\right)^{-1}\theta_{i}^{(l)\dagger}.$
(3.50)
## 4 The explicit solutions of the GHM model
In this section we calculate explicit expression of soliton solution. First of
all we will study GHM model based on $SU(n)$. In this case the spin function
$U$ takes values in the Lie algebra $\verb"su(n)"$ so that one can decompose
the spin function into components $U=U^{a}T^{a}$, and
$T^{a},a=1,2,\dots,n^{2}$ are anti-hermitian $n\times n$ matrices with
normalization $\mbox{Tr}\left(T^{a}T^{b}\right)=\frac{1}{2}\delta^{ab}$ and
are the generators of the $SU(n)$ in the fundamental representation satisfying
the algebra
$\left[T^{a},T^{b}\right]=f^{abc}T^{c},$ (4.1)
where $f^{abc}$ are the structure constants of the Lie algebra $\verb"su(n)"$.
For any $X\in\verb"su(n)"$, we write $X=X^{a}T^{a}$ and
$U^{a}=-2\mbox{Tr}(UT^{a})$.
The matrix-field $U$ belongs to the Lie algebra $\verb"su(n)"$ of the Lie
group $SU(n)$ therefore
$\displaystyle U^{\dagger}=-U,\quad\quad\mbox{Tr}(U)=0.$ (4.2)
The equations (2.1)-(2.2) and (2.5) define a Darboux transformation for the
GHM model based on the Lie group $SU(n)$. The new solution of the equation of
motion (1.3) $U[1]$ must be $\verb"su(n)"$ valued i.e.
$\displaystyle U^{\dagger}[1]=-U[1],\quad\quad\mbox{Tr}(U[1])=0,$ (4.3)
therefore, we have the following conditions on the matrix $M$
$\displaystyle M^{\dagger}=-M,\quad\quad\mbox{Tr}(M)=0.$ (4.4)
In other words we want to make specific $M$ to satisfy the (4.4). This can be
achieved if we choose the particular solutions $\theta_{i}$ at
$\lambda=\lambda_{i}$, let us first calculate
$\displaystyle\partial_{x}\left(\theta_{i}^{\dagger}\theta_{j}\right)$
$\displaystyle=$
$\displaystyle\left(\partial_{x}\theta_{i}^{\dagger}\right)\theta_{j}+\theta_{i}^{\dagger}\left(\partial_{x}\theta_{j}\right)$
(4.5) $\displaystyle=$
$\displaystyle\left(1-\bar{\lambda_{i}}\right)^{-1}\theta_{i}^{\dagger}U^{\dagger}\theta_{j}+\left(1-\lambda_{j}\right)^{-1}\theta_{i}^{\dagger}U\theta_{j},$
using equation (4.2) the above equation (4.5) becomes
$\displaystyle\partial_{x}\left(\theta_{i}^{\dagger}\theta_{j}\right)$
$\displaystyle=$ $\displaystyle 0,$ (4.6)
when $\lambda_{i}\neq\lambda_{j}$ (i.e. $\bar{\lambda_{i}}=\lambda_{j}$).
Similarly we can check
$\displaystyle\partial_{t}\left(\theta_{i}^{\dagger}\theta_{j}\right)$
$\displaystyle=$ $\displaystyle 0.$ (4.7)
From the definition of the matrix $M$, we have
$\displaystyle\theta_{i}^{\dagger}\left(M^{\dagger}+M\right)\theta_{j}$
$\displaystyle=$
$\displaystyle\left(\bar{\lambda_{i}}+\lambda_{j}\right)\theta_{i}^{\dagger}\theta_{j},$
(4.8)
when $\lambda_{i}\neq\lambda_{j}$ then the above expression (4.8) implies
$\displaystyle\theta_{i}^{\dagger}\theta_{j}=0.$ (4.9)
The column vectors $\theta_{i}$ are linearly independent and the equation
(4.9) holds everywhere.
For the HM model based on $SU(n)$, the constant matrix (1.13) becomes
$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccccc}2-\frac{2}{n}&0&\cdots&0&0&0&\cdots&0&0\\\
0&-\frac{2}{n}&\cdots&0&0&0&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&-\frac{2}{n}&0&0&\cdots&0&0\\\
0&0&\cdots&0&-\frac{2}{n}&0&\cdots&0&0\\\
0&0&\cdots&0&0&-\frac{2}{n}&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&\cdots&0&0&0&\cdots&0&-\frac{2}{n}\\\ \end{array}\right).$ (4.18)
Then $U^{2}$ becomes
$U^{2}=\frac{4\left(n-1\right)}{n^{2}}I+\frac{2\left(n-2\right)}{n}U.$ (4.19)
These are the constraints given in ref. [4]. For the construction of explicit
soliton solution for the $SU(n)$ HM model, we construct the matrix $M$ by
defining a Hermitian projector $P$. For this case, we take the seed solution
to be
$U_{0}\equiv U=\mbox{i}\left(\begin{array}[]{ccc}a_{1}&&\\\ &\ddots&\\\
&&a_{n}\end{array}\right),$ (4.20)
where $a_{i}$ are real constants and $\sum_{i=1}^{n}a_{i}=0$. The
corresponding solution of the Lax pair is expressed in block diagonal matrix
$\Psi(x,t;\lambda)=\left(\begin{array}[]{cc}W_{p}(\lambda)&O\\\
O&W_{n-p}(\lambda)\end{array}\right),$ (4.21)
where
$W_{p}(\lambda)=\left(\begin{array}[]{ccc}e^{\mbox{i}\omega_{1}(\lambda)}&&\\\
&\ddots&\\\ &&e^{\mbox{i}\omega_{p}(\lambda)}\end{array}\right),$ (4.22)
and
$W_{n-p}(\lambda)=\left(\begin{array}[]{ccc}e^{\mbox{i}\omega_{p+1}(\lambda)}&&\\\
&\ddots&\\\ &&e^{\mbox{i}\omega_{n}(\lambda)}\end{array}\right),$ (4.23)
are $p\times p$ and $(n-p)\times(n-p)$ matrices respectively and
$\omega_{i}(\lambda)=a_{i}\left(\frac{1}{1-\lambda}x+\frac{4}{\left(1-\lambda\right)^{2}}t\right).$
(4.24)
Now define a particular matrix solution $\Theta$ of the Lax pair as
$\Theta=\left(\Psi(\mu)L_{1}\ ,\ \Psi(\bar{\mu})L_{2}\right),$ (4.25)
where $L_{1}$ is an $n\times p$ constant matrix of $p$ column vectors and
$L_{2}$ is the orthogonal complementary $n\times(n-p)$ matrix of $(n-p)$
column vectors. The columns of $L_{1}$ span a $p$-dimensional subspace $U$ of
$C^{n}$, and those of $L_{2}$ span the orthogonal subspace $V$. The projector
$P$ is completely characterized by the two subspaces $U=\text{Im}P$ and
$V=\text{Ker}P$ given by the condition $P^{\bot}U=0$ and $PV=0$. Let us write
$L_{1}=\left(\begin{array}[]{c}A\\\ B\end{array}\right)$ and
$L_{2}=\left(\begin{array}[]{c}C\\\ D\end{array}\right),$ where $A$, $B$, $C$
and $D$ are constant $p\times p$, $(n-p)\times n$, $p\times(n-p)$ and
$(n-p)\times(n-p)$ constant matrices respectively. Given this, the $n\times n$
matric $\Theta$ is given by
$\Theta=\left(\begin{array}[]{cc}W_{p}(\mu)A&W_{p}(\bar{\mu})C\\\
W_{n-p}(\mu)B&W_{n-p}(\bar{\mu})D\end{array}\right).$ (4.26)
We now define the projector $P$ in terms of the matrix
$\Phi=\Psi(\mu)L_{1}=(\theta_{1},\cdots\theta_{p})$ given by
$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\left(\theta_{1},\cdots,\theta_{p}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}W_{p}(\mu)A\\\
W_{n-p}(\mu)B\end{array}\right).$
The projector is thus given by
$P=\left(\begin{array}[]{cc}W_{p}(\mu)A\Delta
A^{{\dagger}}W_{p}^{{\dagger}}(\bar{\mu})&W_{p}(\mu)A\Delta
B^{{\dagger}}W_{n-p}^{{\dagger}}(\bar{\mu})\\\ W_{n-p}(\mu)B\Delta
A^{{\dagger}}W_{p}^{{\dagger}}(\bar{\mu})&W_{n-p}(\mu)B\Delta
B^{{\dagger}}W_{n-p}^{{\dagger}}(\bar{\mu})\end{array}\right),$ (4.28)
where
$\Delta^{-1}=A^{{\dagger}}W_{p}^{{\dagger}}(\bar{\mu})W_{p}(\mu)A+B^{{\dagger}}W_{n-p}^{{\dagger}}(\bar{\mu})W_{n-p}(\mu)A$.
The Darboux matrix $D(\lambda)$ can now be constructed to give explicit
soliton solution of the $SU(n)$ HM model. To elaborate the result more
explicitly, we proceed with the example of $SU(2)$ HM model.
For the $SU(2)$ model, the equations (4.18) and (4.19) become
$\displaystyle T=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\ \end{array}\right).$
(4.31)
Then $U^{2}$ becomes
$U^{2}=I.$ (4.32)
The Lax pair (1.18)-(1.19) for the $SU(2)$ model can be written as
$\displaystyle\partial_{x}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{(1-\lambda)}U(x,t)\Psi(x,t;\lambda),$ (4.33)
$\displaystyle\partial_{t}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\left(\frac{4}{(1-\lambda)^{2}}U+\frac{2}{(1-\lambda)}UU_{x}\right)\Psi(x,t;\lambda).$
(4.34)
If we take trivial solution (as seed solution), single soliton and multi-
soliton solutions can be obtained by Darboux transformation as explained
above.
We take the seed solution to be
$\displaystyle U_{0}\equiv U=\left(\begin{array}[]{cc}\mbox{i}&0\\\
0&-\mbox{i}\end{array}\right).$ (4.37)
The corresponding solution of the linear system (4.33)-(4.34) can be written
as
$\displaystyle\Psi(x,t;\lambda)=\left(\begin{array}[]{cc}e^{\mbox{i}\left(\frac{1}{\left(1-\lambda\right)}x+\frac{4}{\left(1-\lambda\right)^{2}}t\right)}&0\\\
0&e^{-\mbox{i}\left(\frac{1}{\left(1-\lambda\right)}x+\frac{4}{\left(1-\lambda\right)^{2}}t\right)}\end{array}\right).$
(4.40)
Take $\lambda_{1}=\mu$ and $\lambda_{2}=\bar{\mu}$, the constant matrix
$\Lambda$ is given by
$\displaystyle\Lambda=\left(\begin{array}[]{cc}\mu&0\\\
0&\bar{\mu}\end{array}\right),$ (4.43)
and corresponding $2\times 2$ matrix solution $\Theta$ becomes
$\displaystyle\Theta\equiv\left(\theta_{1},\theta_{2}\right)=\left(\begin{array}[]{cc}e^{\mbox{i}\left(\frac{1}{\left(1-\mu\right)}x+\frac{4}{\left(1-\mu\right)^{2}}t\right)}&e^{\mbox{i}\left(\frac{1}{\left(1-\bar{\mu}\right)}x+\frac{4}{\left(1-\bar{\mu}\right)^{2}}t\right)}\\\
-e^{-\mbox{i}\left(\frac{1}{\left(1-\mu\right)}x+\frac{4}{\left(1-\mu\right)^{2}}t\right)}&e^{-\mbox{i}\left(\frac{1}{\left(1-\bar{\mu}\right)}x+\frac{4}{\left(1-\bar{\mu}\right)^{2}}t\right)}\end{array}\right).$
(4.46)
The matrix $M$ is given by
$\displaystyle M$ $\displaystyle=$ $\displaystyle\Theta\Lambda\Theta^{-1},$
(4.49) $\displaystyle=$
$\displaystyle\frac{1}{e^{u}+e^{-u}}\left(\begin{array}[]{ll}\mu
e^{u}+{\bar{\mu}}{e^{-u}}&\left({\bar{\mu}-\mu}\right)e^{{i}v}\\\
\left({\bar{\mu}-\mu}\right)e^{-iv}&{\bar{\mu}}{e^{u}}+{\mu}{e^{-u}}\end{array}\right),$
where the functions $u(x,t)$ and $v(x,t)$ are defined by
$\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\mbox{i}\left(\frac{1}{\left(1-\mu\right)}-\frac{1}{\left(1-\bar{\mu}\right)}\right)x+4\mbox{i}\left(\frac{1}{\left(1-\mu\right)^{2}}-\frac{1}{\left(1-\bar{\mu}\right)^{2}}\right)t,$
$\displaystyle v(x,t)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\left(1-\mu\right)}+\frac{1}{\left(1-\bar{\mu}\right)}\right)x+4\left(\frac{1}{\left(1-\mu\right)^{2}}+\frac{1}{\left(1-\bar{\mu}\right)^{2}}\right)t.$
(4.50)
Let us take the eigenvalue to be $\mu=e^{\mbox{i}\theta}.$ The expression
(4.49) then becomes
$M=\left(\begin{array}[]{cc}\cos\theta+\mbox{i}\sin\theta\tanh
u&-\mbox{i}\left(\sin\theta\text{sech}u\right)e^{\mbox{i}v}\\\
-\mbox{i}\left(\sin\theta\text{sech}u\right)e^{-\mbox{i}v}&\cos\theta-\mbox{i}\sin\theta\tanh
u\end{array}\right),$ (4.51)
and the corresponding Darboux matrix $D\left(\lambda\right)$ in this case is
$D\left(\lambda\right)=\left(\begin{array}[]{cc}\lambda-\cos\theta-\mbox{i}\sin\theta\tanh
u&\mbox{i}\left(\sin\theta\text{sech}u\right)e^{\mbox{i}v}\\\
\mbox{i}\left(\sin\theta\text{sech}u\right)e^{-\mbox{i}v}&\lambda-\cos\theta+\mbox{i}\sin\theta\tanh
u\end{array}\right).$ (4.52)
Comparing the above equation with (3.44), we find the following expression for
the projector
$P=\left(\begin{array}[]{cc}2e^{u}\text{sech}u&-2e^{\mbox{i}v}\text{sech}u\\\
-2e^{-\mbox{i}v}\text{sech}u&2e^{-u}\text{sech}u\end{array}\right).$ (4.53)
Using (3.8) and (4.37), we get
$U[1]=\left(\begin{array}[]{cc}\mbox{i}U_{3}&U_{+}\\\ -U_{-}&-\mbox{i}U_{3}\\\
\end{array}\right),$ (4.54)
where
$\displaystyle U_{3}$ $\displaystyle=$ $\displaystyle
1-(1+\cos\theta)\mbox{sech}^{2}u,$ $\displaystyle U_{+}$ $\displaystyle\equiv$
$\displaystyle\overline{U}_{-}=-\mbox{i}e^{\mbox{i}v}\left[(1+\cos\theta)\mbox{tanh}u+\mbox{i}\sin\theta\right]\mbox{sech}u.$
(4.55)
From equation (4.55), we see that $U^{\dagger}[1]=-U[1]$ and
$\mbox{Tr}(U[1])=0$. Therefore equation (4.55) is an explicit expression of
the single-soliton solution of the HM model based on $SU(2)$ obtained by using
Darboux transformation. Similarly one can calculate explicit expression for
the multi-soliton solution of the model. The expression (4.55) is similar to
the expression of the single soliton given in [2].
## 5 Concluding remarks
In this paper, we have studied GHM model based on general linear Lie group
$GL(n)$ and expressed the multi-soliton solutions in terms of the
quasideterminant using the Darboux transformation defined on the solution of
the Lax pair. We have also established equivalence between the Darboux matrix
approach and the Zakharov-Mikhailov’s dressing method. In last section we have
reduced the GHM model into the HM model based on $SU(n)$ and calculated an
explicit expression for the single-soliton solution. It would be interesting
to study the GHM models based on Hermitian symmetric spaces. We shall address
this problem in a separate work.
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|
arxiv-papers
| 2009-12-26T17:07:46 |
2024-09-04T02:49:07.268833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "U. Saleem and M. Hassan",
"submitter": "Usman Saleem",
"url": "https://arxiv.org/abs/0912.5030"
}
|
0912.5068
|
11institutetext: Institut d’Astrophysique Spatiale, CNRS-Université Paris-Sud
11, 91405 Orsay Cedex, France
22institutetext: National Astronomical Observatory, Chinese Academy of
Sciences, Beijing 100012, China
# Hanle signatures of the coronal magnetic field in the linear polarization of
the hydrogen L$\alpha$ line
M. Derouich Present address: Colorado Research Associates Division, NorthWest
Research Associates, Inc., 3380 Mitchell Ln., Boulder, CO 80301, U.S.A.11 F.
Auchère 11 J. C. Vial 11 and M. Zhang 22
(Received 06 July 2009 / Accepted 21 October 2009 )
###### Abstract
Aims. This paper is dedicated to the assessment of the validity of future
coronal spectro-polarimetric observations and to prepare their interpretation
in terms of the magnetic field vector.
Methods. We assume that the polarization of the hydrogen coronal L$\alpha$
line is due to anisotropic scattering of an incident chromospheric radiation
field. The anisotropy is due to geometrical effects but also to the
inhomogeneities of the chromospheric regions which we model by using
Carrington maps of the L$\alpha$. Because the corona is optically thin, we
fully consider the effects of the integration over the line-of-sight (LOS). As
a modeling case, we include a dipolar magnetic topology perturbed by a non-
dipolar magnetic structure arising from a prominence current sheet in the
corona. The spatial variation of the hydrogen density and the temperature is
taken into account. We determine the incident radiation field developed on the
tensorial basis at each point along the LOS. Then, we calculate the local
emissivity vector to obtain integrated Stokes parameters with and without
coronal magnetic field.
Results. We show that the Hanle effect is an interesting technique for
interpreting the scattering polarization of the L$\alpha$ $\lambda$1216 line
in order to diagnose the coronal magnetic field. The difference between the
calculated polarization and the zero magnetic field polarization gives us an
estimation of the needed polarimetric sensitivity in future polarization
observations. We also obtain useful indications about the optimal
observational strategy.
Conclusions. Quantitative interpretation of the Hanle effect on the scattering
linear polarization of L$\alpha$ line can be a crucial source of information
about the coronal magnetic field at a height over the limb $h$
$<0.7\;R_{\sun}$. Therefore, one needs the development of spatial
instrumentation to observe this line.
###### Key Words.:
Line: polarization – Sun: corona – Sun: UV radiation – Scattering
††offprints: M. Derouich
e-mail: moncef.derouich@ias.u-psud.fr
## 1 Introduction
One of the most powerful tools for the diagnostics of magnetic fields in the
Sun is the interpretation of polarimetric observations (e.g. the monograph by
Landi Degl’Innocenti & Landolfi 2004 and the recent review by Trujillo Bueno
2009). However, these diagnostics are mostly concerned with the fields at the
photospheric and chromospheric levels. The coronal magnetic field presents
more intrinsic difficulties to measure and interpret. This is especially true
for the case of the UV coronal lines. Only rather recently, Raouafi et al.
(2002) performed the first measurement and interpretation of the linear
polarization of a UV line (O vi $\lambda$1032 line) polarized under
anisotropic scattering by the underlying solar radiation field. In addition,
Manso Sainz & Trujillo Bueno (2009) proposed a polarizing mechanism showing
the adequate sensitivity of other coronal UV lines to the direction of the
magnetic field. These successful works suggest that new UV polarimeters with
high sensitivity associated with theoretical and numerical modeling obtained
with a high degree of realism are a fundamental step to be performed in order
to extract information on the coronal plasmas. In this context, the Hanle
effect on the L$\alpha$ polarization constitutes an excellent opportunity
which merits to be exploited.
The scattering polarization of the coronal L$\alpha$ line of neutral hydrogen,
which we are revisiting in this paper, has been computed by Bommier & Sahal-
Bréchot (1982) and by Trujillo Bueno et al. (2005). These authors, however,
neglected the effects of the integration over the line-of-sight (LOS) by
considering a local position of the scattering hydrogen atom. Since the corona
is optically thin, the LOS integration problem has to be solved. Fineschi et
al. (1992) treated the case of the L$\alpha$ line polarization and took into
account the LOS integration. However, Fineschi et al. considered the effect of
a deterministic magnetic field vector having a direction and strength
independent of the position of the scattering volume. They also treated the
case of a random magnetic field.
To improve upon these previous works, we take into account the variation of
the direction and the strength of the magnetic field for each scattering event
along the LOS. The calculation of the polarization generated by scattering
depends strongly on the level of anisotropy of the incident radiation, which
in turn depends strongly on the geometry of the scattering process and the
brightness variation of the chromospheric regions. In order to accurately
compute the degree of the anisotropy at each scattering position, we use
Carrington maps of the chromospheric incident radiation of the L$\alpha$ line
obtained by Auchère (2005). In addition, the coronal density of the scattering
atoms and the local temperature are included according to a quiet coronal
model (Cranmer et al. 1999). We perform a comparison of the L$\alpha$ linear
polarization in the zero-field reference case with the amplitude corresponding
to the polarization in the presence of a magnetic field. In our forward
modeling, we adopt a dipolar magnetic distribution as a first step and then we
add a magnetic field associated to an equatorial current sheet.
The paper is organized as follows. We describe the theoretical background and
formulate the problem in Sect. 2. Section 3 deals with the calculations of the
Hanle effect without integration over the LOS in order to compare with known
results. The generalization of these calculations to integrate over the LOS
and the discussion of the possibility of obtaining a coronal magnetic field
through polarization measurements are presented in Sect. 4. The technique that
could be used to measure the scattering polarization of the L$\alpha$ D2 line
is given in Sect. 5; in particular we show how the linear scattering
polarization could be measured using a L$\alpha$ disk imager and coronagraph
called LYOT (LYman Orbiting Telescope). In Sect. 6 we summarize our
conclusions.
## 2 Formulation of the problem
### 2.1 Hanle effect
The term Hanle effect represents the ways in which the scattering polarization
can be modified by weak magnetic fields. The well-known Zeeman effect and the
Hanle effect are complementary because they respond to magnetic fields in very
different parameter regimes. The Zeeman effect depends on the ratio between
the Zeeman splitting and the Doppler line width. The Hanle effect though
depends on the ratio between the Zeeman splitting and the inverse life time of
the atomic levels involved in the process of the formation of the polarized
line. For the permitted UV lines, the Zeeman effect is of limited interest for
the determination of the magnetic fields in the quiet corona. This is because
the ratio between the Zeeman splitting and the Doppler width is small due to
the weakness of the magnetic field and the high Doppler width in such hot
coronal plasmas. On the contrary, the measurement and physical interpretation
of the scattering polarization of the UV lines are a very efficient diagnostic
tool for determining the coronal magnetic field through its Hanle effect.
### 2.2 Atomic linear polarization
The possibility of the creation of a linear polarization by anisotropic
scattering can be only explained correctly in the framework of the quantum-
mechanical scattering theory. In fact, the intrinsic capacity of a line to be
polarized is intimately linked to subtle quantum behaviors pertaining to the
atomic levels involved in the transition. Let us denote by $m_{J}$ the
projection of the orbital angular momentum $J$ of the hydrogen atom; $m_{J}$
takes the values $-J$, $J+1$,…, $J$. The term “atomic linear polarization” in
a $J$-level consists in (e.g. Cohen-Tannoudji & Kastler 1966, Omont 1977,
Sahal-Bréchot 1977, Blum 1981):
–
an unbalance of the populations of the Zeeman sub-levels having different
absolute values $|m_{J}|$
–
a presence of interferences between these Zeeman sub-levels.
This means that by definition, only levels having $J>1/2$ can be linearly
polarized.
### 2.3 Linear scattering polarization in the L$\alpha$ line
The so-called scattering polarization is simply the observational
manifestation of the atomic polarization. The Hanle effect is nothing but a
perturbation of the atomic polarization by a magnetic field. The Hanle
signatures in the spectrum of the linear polarization are a variation of the
polarization degree and a rotation of the polarization plane. These Hanle
signatures can be used to retrieve information on coronal magnetic fields. The
two components D1 and D2 of the L$\alpha$ connect the hydrogen ground state
${}^{2}S_{J=1/2}$ to the electronic excited states ${}^{2}P_{J=1/2}$ and
${}^{2}P_{J=3/2}$, respectively. The upper level ${}^{2}P_{3/2}$ of the D2
line can be polarized due to the difference of the populations between the
Zeeman sub-levels with $|m_{J}|=1/2$ and $|m_{J}|=3/2$. However, the states
${}^{2}S_{1/2}$ and ${}^{2}P_{1/2}$ cannot be polarized since $|m_{J}|$ is
necessarily 1/2 implying that no difference of population inside these states
can be generated by anisotropic scattering. Consequently, the D1 line is not
linearly polarizable.
It is useful to keep in mind that in the description of the emitting hydrogen
atom, we neglect the contribution of the hyperfine structure (HFS). For
instance if the HFS is not neglected, the level $J=1/2$ of the ground state
${}^{2}S_{1/2}$ is split into hyperfine levels $F=0$ and $F=1$ due to coupling
with the nuclear spin of the hydrogen $I=1/2$. The hyperfine level $F=1$ can
be linearly polarized111In other words, population imbalances and quantum
interferences between the sub-levels having $|m_{F}|=1$ and $|m_{F}|=0$ can be
created due to the scattering of anisotropic light. The same is true for the
hyperfine levels of the upper states ${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$.,
which means that the D1 line can be polarized and the polarization of the D2
line can be affected. As previously suggested by Bommier & Sahal-Bréchot
(1982), we neglect the effect of the HFS in the process of formation of
L$\alpha$ line.
Figure 1: Geometry of the scattering of chromospheric L$\alpha$ photons by
residual coronal neutral hydrogen. The anisotropy of the incident light is due
to geometrical effects but also to the inhomogeneities of the chromospheric
regions.
### 2.4 Expression of the Stokes parameters
The emission of the L$\alpha$ $\lambda$1216 line in the solar corona has been
discovered by Gabriel et al. (1971). They concluded that in most coronal
structures the process responsible for the formation of the L$\alpha$ line is
the photo-excitation by underlying radiation. The creation of population
imbalances and the quantum interferences in the ${}^{2}P_{3/2}$ and thus the
existence of the scattering polarization in the D2 L$\alpha$ line are caused
by the photo-excitation of coronal neutral hydrogen by anisotropic
chromospheric radiation (see Fig 1).
The components of the incident radiation field at a frequency $\nu_{0}$ are
usually denoted by $\bar{J}_{q}^{k}(\nu_{0})$ where $k$ is the tensorial order
and $q$ represents the coherences in the tensorial basis ($-k\leq q\leq k$);
the order $k$ can be equal to 0 (with $q=0$) or 2 (with $q=0$, $\pm$1,
$\pm$2). This radiation field with six components constitutes a generalization
of the unpolarized light field where only the quantity
$\bar{J}_{0}^{0}(\nu_{0})$ is considered. In fact, $\bar{J}_{0}^{0}(\nu_{0})$
is proportional to the intensity of the radiation.
If the incident radiation is no longer anisotropic, the components
$\bar{J}_{q}^{k=2}(\nu_{0})$ become zero, which means that no linear
polarization can be created as a result of scattering processes. Regardless of
the anisotropy of the incident radiation, the radiation component associated
with the circular polarization usually denoted by $\bar{J}_{q}^{k=1}$ is
negligible. This means that no odd order $k$ can be created inside the
scattering hydrogen atom. As a result, the Stokes $V$ of the scattered
radiation is zero.
We denote by $\zeta$ the angle between the direction of the incident light MP
and the local vertical through the scattering center OP. The incident
radiation comes from a chromospheric spherical cap limited by an angle
$\zeta_{\textrm{\scriptsize{max}}}$ corresponding to the tangent to the solar
limb (see Fig. 1). $\chi$ is the azimuth angle around the normal with respect
to an arbitrary reference. Note that $0\leq$ $\zeta$ $\leq$
$\zeta_{\textrm{\scriptsize{max}}}$ and 0 $\leq$ $\chi$ $\leq$ 2 $\pi$. When
the distance from the solar surface increases, the anisotropy of the light
becomes larger and the polarization degree increases. The maximum of
polarization is reached when the radiation is purely directive, i.e. the
spherical cap is seen by the scattering hydrogen atom as a point. It is useful
to notice that if the chromosphere is assumed to be uniform the radiation has
a cylindrical symmetry around its preferred direction, implying that the
coherence components with $q\neq 0$ are zero. In fact, $\bar{J}_{q=\pm
1}^{k=2}(\nu_{0})$ and $\bar{J}_{q=\pm 2}^{k=2}(\nu_{0})$ components quantify
the breaking of the cylindrical symmetry around the axis of quantification
which is here the local vertical.
In the framework of the two level approximation, where only the upper level is
polarized, the statistical equilibrium equations are solved analytically. The
upper level density matrix elements are simply proportional to the incident
radiation elements $\bar{J}_{q}^{k}$. The emissivity vector is then expressed
as a function of the incident radiation field. Consequently, we do not
explicitly calculate the density matrix elements, but instead we determine the
incident radiation tensor at each scattering position along the line of sight.
For an unmagnetized atmosphere, in an arbitrary reference, the emissivity
vector can be written as (e.g. Landi Degl’Innocenti & Landolfi 2004):
$\displaystyle\epsilon_{j}(\Omega)=n_{\textrm{\scriptsize{H}}}\frac{h\nu_{0}B_{J_{l}J_{u}}}{4\pi}\sum_{k,q}W_{k}(J_{l},J_{u})\mathcal{T}_{q}^{k}(j,\Omega)(-1)^{q}\bar{J}_{-q}^{k}(\nu_{0})$
(1)
where $\Omega$ is the solid angle giving the direction of the LOS,
$n_{\textrm{\scriptsize{H}}}$ is the local number density of scattering
hydrogen atoms, $h$ is the Planck constant, and $B_{J_{l},J_{u}}$ is the
Einstein coefficient for absorption. We recall that
$\mathcal{T}_{q}^{k}(j,\Omega)$ is the spherical tensor for polarimetry which
contains the angular distribution of the emitted radiation, and $j$ is the
index of the Stokes parameter ($j$ = 0, 1, 2, and 3 for the Stokes $I,Q,U,$
and $V$, respectively).
In order to determine the magnetic field one has to include its Hanle effect
on the polarization of the L$\alpha$ light, then, for a given magnetic field
vector ${\bf B}$, $\epsilon_{j}(\Omega)$ becomes (e.g. Landi Degl’Innocenti &
Landolfi 2004):
$\displaystyle\epsilon_{j}(\Omega,{\bf
B})=n_{\textrm{\scriptsize{H}}}\frac{h\nu_{0}B_{J_{l}J_{u}}}{4\pi}\times$ (2)
$\sum_{k,q}W_{k}(J_{l},J_{u})\mathcal{T}_{q}^{k}(j,\Omega)(-1)^{q}\bar{J}_{-q}^{k}(\nu_{0})\frac{1}{1+\textrm{i}qH_{u}}$
This expression of $\epsilon_{j}(\Omega,{\bf B})$ is correct only in a
reference system having the quantization $z$-axis in the magnetic field
direction. $H_{u}$ is the so-called reduced magnetic field strength,
associated to the level ${}^{2}P_{3/2}$, given by:
$\displaystyle H_{u}=\frac{0.879\;g_{u}\;\textrm{B}}{A_{J_{u}J_{l}}}$ (3)
where the Einstein coefficient for spontaneous emission $A_{J_{u}J_{l}}$ is
given in [$10^{7}$ s-1], $g_{u}=4/3$ is the Landé factor of the level
${}^{2}P_{3/2}$ and the magnetic field strength B is given in Gauss. $H_{u}=1$
corresponds to the magnetic field strength $B=53$ Gauss around which one may
expect a noticeable change in the scattering polarization of L$\alpha$ with
respect to the unmagnetized reference case. The quantity $W_{k}(J_{l},J_{u})$
was first introduced by Landi Degl’Innocenti (1984) and depends only on the
quantum numbers of the lower and upper levels ($J_{l}$ and $J_{u}$) involved
in the transition. For $k$=2, $W_{2}(J_{l},J_{u})$ can be seen as the
efficiency of creation of the linear polarization in the scattering processes.
That is why the $W_{2}(J_{l}=1/2,J_{u}=1/2)$ =0 for the D1 line, which is not
polarizable, but $W_{2}(J_{l}=1/2,J_{u}=3/2)$=1/2 for the polarizable D2 line.
## 3 Hanle effect without integration over the LOS
We developed a numerical code allowing for the calculation of the theoretical
polarization taking into account the effects of the LOS. In order to validate
the code, we considered typical cases of a horizontal magnetic field having
different azimuth angles $\theta$ (angles between the magnetic field vector
and the LOS). LOS integrations are avoided in order to be able to compare our
results with well known Hanle effect results. We retrieve the Hanle behaviors
typically encountered in the literature, for instance:
–
when the magnetic field is zero or very small or oriented along the symmetry
axis of the radiation field, the polarization is not affected
–
when the field increases until reaching the critical value corresponding to
$H_{u}=1$, the polarization decreases rapidly. Moreover, for a very large
$H_{u}$ (i.e. very large magnetic field strength) we obtain an asymptotical
curve of polarization $p[B\to\infty]$ which depends only on the value of
$\theta$ but not on the magnetic field strength. The asymptotic value of
$p[B\to\infty]$ divided by $p[B=0]$ equals 1/5 when the distribution of the
magnetic field is isotropic222The case of isotropic field distribution is
encountered in the photosphere of the Sun (second solar spectrum) where the
magnetic geometries are unresolved within the spatiotemporal resolution of the
current observational capabilities. and 1/4 when the field has a cylindrical
symmetry (i.e. horizontal magnetic field with random azimuth)
–
no rotation of the plane of the polarization in the case of a highly symmetric
distribution (e.g. isotropic or cylindrical) because the contributions of
opposite magnetic polarities tend to cancel out
–
we find that a meridian magnetic vector (i.e. horizontal with
$\theta=\frac{\pi}{2}$) presents a depolarizing effect without rotation of the
polarization direction.
## 4 Hanle effect integrated over the LOS
The corona being an optically thin medium for the L$\alpha$ line, it is then
necessary to consider the effects of the integration over the LOS. We adopt
the analytical magnetic field model proposed by Fong et al. (2002) and Low et
al. (2003). It is a sum of two terms: a purely dipolar term and a term
corresponding to the magnetic field of a current sheet structure. The model is
axisymmetric and the prominence is treated as a cold plasma sheet forming a
flat ring around the Sun. We take into consideration that the current is in
the equator and that it represents a prominence sheet extending from
$r=R_{\sun}$ to $r=\sqrt{4/5}\;R_{\sun}$. In the analytical expression of the
magnetic field the contribution of the current sheet, relative to the dipolar
background field is controlled by a constant ratio $\gamma$ (see Eq. 12 of Low
et al. 2003).
Figure 2: A purely dipolar magnetic field structure presented in the plane
$(x,z)$ in units of the solar radius $R_{\sun}$. We use a system of orthogonal
Cartesian coordinates $(x,y,z)$ with the origin at the Sun center and the
$z$-axis pointing toward the north solar pole.
### 4.1 Purely dipolar magnetic field: $\gamma$=0
As a first step, we avoid the effect of the term associated to the current
sheet by taking $\gamma$=0. Figure 2 represents the dipolar term of the
magnetic field. This configuration represents a typical coronal magnetic field
of 15 to 20 Gauss close to the base of the corona. The Hanle effect of the
dipolar magnetic field depends on the angle $\phi$ between the axis of
symmetry of the incident light333Rigorously speaking, this is only the axis of
symmetry of the spherical cap where the chromospheric radiation is uniform.
The fact that the incident radiation is inhomogeneous implies that this
symmetry around the preferred axis of radiation is broken. and the axis of
symmetry of the magnetic structure. It also depends on the height above the
solar surface $h$ mainly because the magnetic field strength decreases and the
anisotropy of the incident light increases. The parameters $\phi$ and $h$ are
represented in Fig. 3.
Figure 3: The polarization at each position P of the scattering event depends
on the height over the limb $h$ and the angle $\phi$ between the preferred
axis of the radiation and the symmetry axis of the magnetic structure. The LOS
is perpendicular to the plane of the figure.
In theory, the expression of the emissivity vector is valid regardless of the
location of the scattering atom. However, the integration over the LOS must
take into account the inhomogeneities of the solar conditions like the
variation of the hydrogen density, the temperature, the magnetic field and the
variation of the incident radiation field. The density of neutral hydrogen and
the temperature are assumed to be a function of the radial distance $r$ and
the latitude (see Cranmer et al. 1999 for details). In order to model the
inhomogeneities of the chromospheric intensity, we use the Carrington maps of
the L$\alpha$ chromospheric line built by Auchère (2005). In the optically
thin limit, the integrated Stokes parameters of the scattered radiation reduce
to a volume integration over the LOS:
$\displaystyle\mathcal{E}_{j}(\Omega)$ $\displaystyle=$
$\displaystyle\int_{\textrm{\scriptsize{LOS}}}\epsilon_{j}(\Omega)dl$ (4)
then the polarization degree is
$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\mathcal{E}_{1}^{2}+\mathcal{E}_{2}^{2}}}{\mathcal{E}_{0}}$
(5)
and the rotation of the direction of the polarization $\alpha_{0}$ is given
by:
$\displaystyle tg(2\alpha_{0})$ $\displaystyle=$
$\displaystyle\frac{\mathcal{E}_{2}}{\mathcal{E}_{1}}$ (6)
Figure 4 shows the variation of the linear polarization with the inclination
$\phi$ for two different magnetic structures corresponding to two heights
above the solar surface. Note that the ratio $p[B]/p[B=0]$ obtained for the
height $h$=0.3 R☉ is smaller than the one corresponding to $h$=0.5 R☉ since
the magnetic field decreases with $h$. Furthermore, as shown in Fig. 5, a
notable Hanle rotation of about $10^{o}$ is obtained for $h$=0.3 R☉ and for
$h$=0.5 R☉. Both Hanle signatures on the L$\alpha$ line, i.e. depolarization
and rotation (see Figs. 4 and 5), are clearly sizable for $\phi>$ $40^{o}$.
This important result suggests that in order to measure a dipolar magnetic
field by its Hanle effect one should observe regions rather far away from the
pole.
Figure 4: Linear polarization degree obtained for a dipolar magnetic field
divided by zero-field polarization versus the angle $\phi$. Full lines
represent the polarization at $h$=0.5 R☉ and dashed lines represent the
polarization at $h$=0.3 R☉. Figure 5: Rotation angle obtained after
integration over the LOS versus the inclination $\phi$ at $h$=0.3 R☉ and
$h$=0.5 R☉. Full lines represent the rotation at $h$=0.5 R☉ and dashed lines
represent the rotation at $h$=0.3 R☉. Figure 6: Perturbation of the lines of
the dipolar magnetic field due to an equatorial current sheet. The
calculations of the polarization given in Fig. 7 are obtained for
$\phi=$$80^{o}$. Figure 7: Linear polarization degree versus the height from
the solar surface $h$. We put together the results obtained in the zero-field
case and these obtained for (1) a purely dipolar magnetic field (2) the sum of
a dipolar field and a non-dipolar field associated to a current sheet with
$\gamma=$ 0.25.
### 4.2 Perturbed dipolar field: $\gamma\neq$ 0
To the dipolar part of the magnetic field we now add the contribution
resulting from an equatorial current sheet. We adopt a ratio $\gamma$=0.25
between the current sheet and the dipolar background field. In order to
highlight the Hanle effect of the equatorial current sheet, we calculate the
degree of polarization in a position located at $\phi$=$80^{o}$, and we vary
the height above the limb $h$ (see Fig. 6). Figure 7 shows the difference
between the linear polarization in the zero-magnetic field case and the one in
the presence of the magnetic field, $\Delta p$=$|p_{B=0}-p_{B\neq 0}|$. A
polarimetric sensitivity smaller than $\Delta p$ is needed in order to apply
the Hanle effect as a technique of magnetic field investigations. Our results
show that $\Delta p$ $\sim$ 5 %, i.e. well within the typical measurement
sensitivities of a new generation of instruments such as LYOT (see Sect. 5).
We point out that by using the UV SUMER spectrometer aboard SoHO, Raouafi et
al. (1999) measured the linear polarization of the O vi $\lambda$1032 line
with a polarimetric precision equal to 1.7 %. We note in passing that such an
accuracy is reached although SUMER was not initially designed to measure the
polarization.
## 5 Measurement of the linear polarization degree and its direction
### 5.1 Principle of the measurement
Raouafi et al. (1999) used the rotation of the SUMER spectrometer to measure
the linear polarization of the D2 component of the O vi $\lambda$1032 line.
They extracted the polarization of the D2 line from a ratio of the intensities
of the non-polarizable D1 line and of the D2 line (see the Fig. 3 of Raouafi
et al. 1999). This technique was possible because the wavelengths of the two
components D1 and D2 are sufficiently different to be resolved (1031.93 Å for
the D2 line and 1037.62 Å for the D1). However, the wavelengths of the D1 and
D2 lines of the L$\alpha$ line cannot be resolved since they are very close:
in the vacuum $\lambda$(D1)= 1215.668 Å and $\lambda$(D2)= 1215.674 Å. As a
result, the technique presented by Fig. 3 of Raouafi et al. (1999) cannot be
applied to measure the linear polarization of the D2 line of L$\alpha$.
Using the so-called Poincaré representation444A suitable graphical
representation of polarized light conceived by Henri Poincaré in 1892., one
can demonstrate that the intensity observed when the instrument is placed at
an arbitrary position referred by an angle $\beta$ around the LOS, is
$\displaystyle I(\beta)=\frac{1}{2}(Q\cos 2\beta+U\sin 2\beta+I)$ (7)
In this expression the Stokes $V$ is assumed to be zero. The quantity $I$
denotes the unpolarized part of the intensity of the D1 and D2 L$\alpha$
lines. $I(\beta)$ represents the “real” (polarized and unpolarized) observed
intensity of the two resonance lines. In addition, taking into account that
$\alpha_{0}$ corresponds to the direction of linear polarization (i.e.
privileged direction of the electric field), Eq. (7) becomes
$\displaystyle I(\beta)=\frac{1}{2}(\sqrt{Q^{2}+U^{2}}\cos
2(\beta-\alpha_{0})+I)$ (8)
Note that $\cos 2\alpha_{0}=\frac{Q}{\sqrt{Q^{2}+U^{2}}}$ and $\sin
2\alpha_{0}=\frac{U}{\sqrt{Q^{2}+U^{2}}}$.
On the other hand, generally speaking, the linear polarization is defined as
$\displaystyle p=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$ (9)
where $I_{max}$ and $I_{min}$ are the maximum and minimum intensities. Using
Eqs. (8) and (9), one finds that
$\displaystyle p=\frac{\sqrt{Q^{2}+U^{2}}}{I}$ (10)
and
$\displaystyle\frac{I(\beta)}{I}=\frac{1}{2}(p\cos 2(\beta-\alpha_{0})+1)$
(11)
In Eq. (11) we have three unknowns: $I$, $p$, and $\alpha_{0}$. Then,
theoretically, the linear polarization state is fully obtained through only
three measurements of the $I(\beta)$ which corresponds to three rotations of
the polarizer-spectrometer. One takes for example $\beta=0,\frac{\pi}{4},$ and
$\frac{\pi}{2}$, therefore:
$\displaystyle\tan 2\alpha_{0}$ $\displaystyle=$
$\displaystyle\frac{2I(\frac{\pi}{4})-I}{I(0)-I(\frac{\pi}{2})}$ (12)
$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{I(0)-I(\frac{\pi}{2})}{\cos 2\alpha_{0}\times
I}=\frac{2I(\frac{\pi}{4})-I}{\sin 2\alpha_{0}\times I}$
where the intensity of the unpolarized light is given by:
$\displaystyle I$ $\displaystyle=$ $\displaystyle I(0)+I(\frac{\pi}{2})$ (13)
Obviously, more than three measurements of $I(\beta)$ are welcome in order to
increase the accuracy.
### 5.2 The LYOT project
The LYOT project is a L$\alpha$ coronagraph combined with a L$\alpha$ disk
imager (see Vial et al. 2002, Millard et al. 2006, and Vial et al. 2008). In
addition, it is planned to implement a simple polarizer system. The polarizing
measurements will be performed by rotating the polarizer or the whole
instrument to obtain the intensity of the L$\alpha$ light at different $\beta$
angles (previous section). The choice of the L$\alpha$ line is well justified
by its sensitivity to the coronal magnetic field (as demonstrated in this
paper) and by the fact that in the corona the L$\alpha$ emission is very
intense. In fact, a high signal to noise ratio is needed since in the very low
corona the anisotropy of the light is small, which in turn means that the
polarization degree is small (smaller than 5 %, see Fig. 7). We note that no
coronagraph observing as low as 1.15 R☉ is envisaged beyond 2012 except for
LYOT images which should be obtained with an excellent signal to noise ratio.
## 6 Conclusion
Measurement and interpretation of the scattering polarization of UV coronal
lines provide a largely unexplored diagnostic of the coronal magnetic field.
The greatest difficulty facing the UV coronal spectropolarimetry is that the
polarization measurements integrate radiation along the LOS over structures
with different properties but also that the observations of these lines are
impossible from ground-based telescopes; they can only be observed with the
help of high-sensitivity instruments flown on space missions.
We have performed a forward modeling of the coronal Hanle effect on the
polarization of the L$\alpha$ line generated by anisotropic scattering of
chromospheric light. The main feature of this modeling consists in integrating
the effect of the LOS. We show that the information about the coronal magnetic
field is not lost through LOS integration. To confirm these results, we plan
to work with different families of maps of magnetic fields and to add small
scale magnetic perturbations. For instance, one can think of a set of active
loops whose field determination could be compared with field extrapolations.
One should however keep in mind that (1) a realistic thermodynamic model is
required in order to integrate along the LOS and that (2) our modeling is
limited to the case of optically thin plasmas in the L$\alpha$ line.
Finally, we notice that it is suitable to combine measurements in L$\alpha$
with measurements in polarized lines like the Fe xiv $\lambda$5303 which have
a different sensitivity to the magnetic field. An analysis of combined
measurements should give more information data to constrain the magnetic field
topology and strength (an example of the Hanle effect in a multi-line approach
is given in Landi Degl’Innocenti 1982). It is also of interest to remark that
the ground level ${}^{2}S_{J}$ of L$\alpha$ is non polarizable by radiation
anisotropy, but that this is no longer true in the presence of hyperfine
structure and if the depolarizing effect of the isotropic collisions is
negligible. Because the sensitivity to the Hanle effect depends on the level
life-time, the hyperfine polarization of such a long-lived level is much more
vulnerable to very weak magnetic fields than the short-lived upper levels
${}^{2}P$. Consequently, one could distinguish a very small perturbation of
the magnetic field (smaller than 1 Gauss) which corresponds for instance to a
current sheet with a very small $\gamma$ and a background field of the order
of 10 Gauss or larger. In particular, this could be the key to distinguish
potential magnetic field structures from non potential ones.
## References
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* (12) Low, B. C., Fong, B., Fan, Y., 2003, ApJ, 594, 1060
* (13) Millard, A. A., Lemaire, P., & Vial, J. C., 2006, Proceedings of the SPIE, Vol 6266, 62662G.
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|
arxiv-papers
| 2009-12-27T11:43:49 |
2024-09-04T02:49:07.277426
|
{
"license": "Public Domain",
"authors": "M. Derouich (1), F. Auchere (1), J. C. Vial (1), and M. Zhang (2)",
"submitter": "Moncef Derouich",
"url": "https://arxiv.org/abs/0912.5068"
}
|
0912.5138
|
# Light Scalar Meson $\sigma(600)$ in QCD Sum Rule with Continuum
Hua-Xing Chen1,2 chx@water.pku.edu.cn Atsushi Hosaka2
hosaka@rcnp.osaka-u.ac.jp Hiroshi Toki2 toki@rcnp.osaka-u.ac.jp Shi-Lin Zhu1
zhusl@pku.edu.cn 1Department of Physics and State Key Laboratory of Nuclear
Physics and Technology, Peking University, Beijing 100871, China
2Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka
567–0047, Japan
###### Abstract
The light scalar meson $\sigma(600)$ is known to appear at low excitation
energy with very large width on top of continuum states. We investigate it in
the QCD sum rule as an example of resonance structures appearing above the
corresponding thresholds. We use all the possible local tetraquark currents by
taking linear combinations of five independent local ones. We ought to
consider the $\pi$-$\pi$ continuum contribution in the phenomenological side
of the QCD sum rule in order to obtain a good sum rule signal. We study the
stability of the extracted mass against the Borel mass and the threshold value
and find the $\sigma(600)$ mass at 530 MeV $\pm$ 40 MeV. In addition we find
the extracted mass has an increasing tendency with the Borel mass, which is
interpreted as caused by the width of the resonance.
scalar meson, tetraquark, QCD sum rule
###### pacs:
12.39.Mk, 12.38.Lg, 12.40.Yx
## I Introduction
The light scalar mesons, $\sigma(600)$, $\kappa(800)$, $f_{0}(980)$ and
$a_{0}(980)$, have been intensively discussed for many years Amsler:2004ps ;
Bugg:2004xu ; Klempt:2007cp . However, their nature is still not fully
understood Caprini:2005zr ; Hatsuda:1994pi ; Oller:1997ti ; Sugiyama:2007sg ;
Prelovsek:2010gm . They have the same quantum numbers $J^{PC}=0^{++}$ as the
vacuum, and hence the structure of these states is a very important subject in
order to understand non-perturbative properties of the QCD vacuum such as
spontaneous chiral symmetry breaking. They compose of the flavor $SU(3)$ nonet
with the mass below 1 GeV, and have a mass ordering which is difficult to be
explained by using a $q\bar{q}$ configuration in the conventional quark model
Amsler:2008zzb ; Aitala:2000xu ; Ablikim:2004qna ; Aston:1987ir ;
Akhmetshin:1999di . Therefore, several different pictures have been proposed,
such as tetraquark states and meson-meson bound states, etc. Here we note that
hadrons with complex structures such as tetraquarks may exist in the continuum
above the threshold energy of two hadrons with simple quark structure.
The tetraquark structure of the scalar mesons was proposed long time ago by
Jaffe with an assumption of strong diquark correlations Jaffe:1976ig ;
Jaffe:1976ih . It can naturally explain their mass ordering and decay
properties Alford:2000mm ; Maiani:2004uc ; Weinstein:1990gu . Yet the basic
assumption of diquark correlation is not fully established. In this letter, we
study $\sigma(600)$ as a tetraquark state in the QCD sum rule approach as an
example of resonances in the continuum states above the $\pi$-$\pi$ threshold.
In the QCD sum rule, we calculate matrix elements from the QCD (OPE) and
relate them to observables by using dispersion relations. Under suitable
assumptions, the QCD sum rule has proven to be a very powerful and successful
non-perturbative method in the past decades Shifman:1978bx ; Reinders:1984sr .
Recently, this method has been applied to the study of tetraquarks by many
authors Bracco:2005kt ; Narison:2005wc ; Lee:2006vk ; Chen:2007xr .
In our previous paper Chen:2007xr , we have found that the QCD sum rule
analysis with tetraquark currents implies the masses of scalar mesons in the
region of 600 – 1000 MeV with the ordering
$m_{\sigma}<m_{\kappa}<m_{f_{0},a_{0}}$, while the conventional $\bar{q}q$
current is considerably heavier (larger than 1 GeV). To get this result, first
we find there are five independent local tetraquark currents, and then we use
one of these currents or linear combinations of two currents to perform the
QCD sum rule analysis. But these interpolating currents do not describe the
full space of tetraquark currents. In order to complete our previous study, we
use more general currents by taking linear combinations of all these currents.
It describes the full space of local tetraquark currents which can couple to
$\sigma(600)$. Since $\sigma(600)$ meson is closely related to the $\pi$-$\pi$
continuum and it has a wide decay width, we also consider the contribution of
the $\pi$-$\pi$ continuum as well as the effect of the finite decay width.
This paper is organized as follows. In Sec. II, we establish five independent
local tetraquark currents, and perform a QCD sum rule analysis by using linear
combinations of five single currents. In Sec. III, we perform a numerical
analysis, and we also study the contribution of $\pi$-$\pi$ continuum. In Sec.
IV, we consider the effect of the finite decay width. Sec. V is devoted to
summary.
## II QCD Sum Rule
The local tetraquark currents for $\sigma(600)$ have been worked out in Ref
Chen:2007xr . There are two types of currents: diquark-antidiquark currents
$(qq)(\bar{q}\bar{q})$ and meson-meson currents $(\bar{q}q)(\bar{q}q)$. These
two constructions can be proved to be equivalent, and they can both describe
the full space of local tetraquark currents Chen:2007xr . Therefore we shall
just use the first ones. Since we use their linear combinations to perform the
QCD sum rule analysis, we can not distinguish whether it is a diquark-
antidiquark state or a meson-meson bound state. However, we find that
tetraquark currents with a single term do not lead to a reliable QCD sum rule
result which means that $\sigma(600)$ probably has a complicated structure.
The five independent local currents are given by:
$\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})\,.$
The summation is taken over repeated indices ($\mu$, $\nu,\cdots$ for Dirac,
and $a,b,\cdots$ for color indices). 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 (antidiquarks) are combined into the color representations,
$\mathbf{\bar{3}_{c}}$ and $\mathbf{6_{c}}$ ($\mathbf{3_{c}}$ and
$\mathbf{\bar{6}_{c}}$), respectively.
These five diquark-antidiquark currents $(qq)(\bar{q}\bar{q})$ are
independent. In this work we use general currents by taking linear
combinations of these five currents:
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
t_{1}e^{i\theta_{1}}S^{\sigma}_{3}+t_{2}e^{i\theta_{2}}V^{\sigma}_{3}+t_{3}e^{i\theta_{3}}T^{\sigma}_{6}+t_{4}e^{i\theta_{4}}A^{\sigma}_{6}+t_{5}e^{i\theta_{5}}P^{\sigma}_{3}\,,$
(2)
where $t_{i}$ and $\theta_{i}$ are ten mixing parameters, whose linear
combination describes the full space of local currents which can couple to
$\sigma(600)$. We can not determine them in advance and therefore we choose
them randomly for the study of the QCD sum rule.
By using the current in Eq. (2), we calculate the OPE up to dimension eight.
To simplify our calculation, we neglect several condensates, such as $\langle
g^{3}G^{3}\rangle$, etc., and we do not consider the $\alpha_{s}$ correction,
such as $g^{2}\langle\bar{q}q\rangle^{2}$, etc. The obtained OPE are shown in
the following. We find that most of the crossing terms are not important such
as $\rho_{13}$, and even more some of them disappear: $\rho_{15}=0$, etc. For
the most cases, we find that the OPE terms of Dim=6 and Dim=8 give major
contributions in the OPE series in our region of interest. This is because the
condensates $\langle\bar{q}q\rangle^{2}$ (D=6) and
$\langle\bar{q}q\rangle\langle g\bar{q}\sigma Gq\rangle$ (D=8) are much larger
than others.
Since the OPE series should be convergent to give a reliable QCD sum rule, we
also calculate the OPE of Dim=10 and Dim=12. However, we find that these terms
are not important. Using the parameter set (2) and the the values of the
condensates of the next section as an example, we show the convergence of the
two-point correlation function
$\Pi(M_{B},s_{0})\equiv\int_{0}^{s_{0}}\rho(s)e^{-s/M_{B}^{2}}ds$ in Fig. 1 as
functions of $M_{B}^{2}$. The threshold value is taken to be $s_{0}=1$ GeV2,
and we show its behavior up to certain dimensions. We find that the OPE up to
Dim=0 and Dim=2 are very small; the OPE of Dim=4 gives a minor contribution;
the OPE of Dim=6 and Dim=8 are both important; the OPE of Dim=10 and Dim=12
are both small, and so we shall neglect them in the following analysis.
Figure 1: The convergence of the two-point correlation function
$\Pi(M_{B},s_{0})$. The threshold value is taken to be $s_{0}=1$ GeV2, and we
show its behavior up to certain dimensions, as functions of $M_{B}^{2}$. The
solid line is for $\Pi(M_{B},s_{0})$ up to Dim=8. The short-dashed line around
it is for $\Pi(M_{B},s_{0})$ up to Dim=10, and the long-dashed line around it
is for $\Pi(M_{B},s_{0})$ up to Dim=12.
$\displaystyle\rho(s)$ $\displaystyle=$ $\displaystyle
t_{1}^{2}\rho_{11}(s)+t_{2}^{2}\rho_{22}(s)+t_{3}^{2}\rho_{33}(s)+t_{4}^{2}\rho_{44}(s)+t_{5}^{2}\rho_{55}(s)$
$\displaystyle+2t_{1}t_{2}\cos{(\theta_{1}-\theta_{2})}\rho_{12}(s)+2t_{1}t_{3}\cos{(\theta_{1}-\theta_{3})}\rho_{13}(s)+2t_{1}t_{4}\cos{(\theta_{1}-\theta_{4})}\rho_{14}(s)$
$\displaystyle+2t_{2}t_{3}\cos{(\theta_{2}-\theta_{3})}\rho_{23}(s)+2t_{2}t_{4}\cos{(\theta_{2}-\theta_{4})}\rho_{24}(s)+2t_{2}t_{5}\cos{(\theta_{2}-\theta_{5})}\rho_{25}(s)$
$\displaystyle+2t_{3}t_{4}\cos{(\theta_{3}-\theta_{4})}\rho_{34}(s)+2t_{3}t_{5}\cos{(\theta_{3}-\theta_{5})}\rho_{35}(s)\,,$
where
$\displaystyle\rho_{11}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}+(-\frac{{m_{u}}^{2}}{1536\pi^{6}}+\frac{{m_{u}}m_{d}}{1536\pi^{6}}-\frac{{m_{d}}^{2}}{1536\pi^{6}})s^{3}+(\frac{\langle
g^{2}GG\rangle}{6144\pi^{6}}-\frac{{m_{u}}\langle\bar{q}q\rangle}{192\pi^{4}}-\frac{{m_{d}}\langle\bar{q}q\rangle}{192\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}+\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}-\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{12\pi^{2}})s$
$\displaystyle-\frac{7m_{u}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}+\frac{m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{4\pi^{2}}-\frac{7m_{d}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{768\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{768\pi^{4}}+\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{12\pi^{2}}\,,$ $\displaystyle\rho_{22}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{15360\pi^{6}}+(-\frac{{m_{u}}^{2}}{384\pi^{6}}-\frac{{m_{u}}m_{d}}{768\pi^{6}}-\frac{{m_{d}}^{2}}{384\pi^{6}})s^{3}+(\frac{\langle
g^{2}GG\rangle}{3072\pi^{6}}+\frac{{m_{u}}\langle\bar{q}q\rangle}{24\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{24\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}+\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}-\frac{\langle\bar{q}q\rangle^{2}}{6\pi^{2}})s$
$\displaystyle+\frac{11m_{u}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}+\frac{2m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{11m_{d}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}-\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{6\pi^{2}}\,,$ $\displaystyle\rho_{33}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{1280\pi^{6}}+(-\frac{{m_{u}}^{2}}{32\pi^{6}}-\frac{{m_{d}}^{2}}{32\pi^{6}})s^{3}+(\frac{11\langle
g^{2}GG\rangle}{768\pi^{6}}+\frac{{m_{u}}\langle\bar{q}q\rangle}{4\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{4\pi^{4}})s^{2}$
$\displaystyle+(-\frac{11m_{u}^{2}\langle
g^{2}GG\rangle}{128\pi^{6}}-\frac{11m_{d}^{2}\langle
g^{2}GG\rangle}{128\pi^{6}})s$
$\displaystyle+\frac{5m_{u}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{20m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{5m_{d}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{11m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{96\pi^{4}}+\frac{11m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{96\pi^{4}}\,,$
$\displaystyle\rho_{44}(s)$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}+(-\frac{{m_{u}}^{2}}{192\pi^{6}}+\frac{{m_{u}}m_{d}}{384\pi^{6}}-\frac{{m_{d}}^{2}}{192\pi^{6}})s^{3}+\frac{5\langle
g^{2}GG\rangle}{3072\pi^{6}}s^{2}$ $\displaystyle+(-\frac{5m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{5m_{u}m_{d}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{5m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{16\pi^{4}}-\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{16\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{3\pi^{2}})s$
$\displaystyle-\frac{m_{u}^{2}\langle\bar{q}q\rangle^{2}}{6\pi^{2}}+\frac{8m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{3\pi^{2}}-\frac{m_{d}^{2}\langle\bar{q}q\rangle^{2}}{6\pi^{2}}+\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}+\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}+\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{3\pi^{2}}\,,$ $\displaystyle\rho_{55}(s)$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}+(-\frac{{m_{u}}^{2}}{1536\pi^{6}}-\frac{{m_{u}}m_{d}}{1536\pi^{6}}-\frac{{m_{d}}^{2}}{1536\pi^{6}})s^{3}+(\frac{\langle
g^{2}GG\rangle}{6144\pi^{6}}+\frac{{m_{u}}\langle\bar{q}q\rangle}{64\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{64\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{1024\pi^{6}}+\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}+\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{64\pi^{4}}-\frac{\langle\bar{q}q\rangle^{2}}{12\pi^{2}})s$
$\displaystyle+\frac{17m_{u}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}+\frac{7m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}+\frac{17m_{d}^{2}\langle\bar{q}q\rangle^{2}}{48\pi^{2}}+\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{256\pi^{4}}+\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{256\pi^{4}}-\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{12\pi^{2}}\,,$ $\displaystyle\rho_{12}(s)$
$\displaystyle=$
$\displaystyle(\frac{{m_{u}}^{2}}{3072\pi^{6}}+\frac{{m_{u}}m_{d}}{1536\pi^{6}}+\frac{{m_{d}}^{2}}{3072\pi^{6}})s^{3}+(-\frac{{m_{u}}\langle\bar{q}q\rangle}{48\pi^{4}}-\frac{{m_{d}}\langle\bar{q}q\rangle}{48\pi^{4}})s^{2}$
$\displaystyle+(-\frac{m_{u}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}-\frac{m_{d}\langle g\bar{q}\sigma
Gq\rangle}{32\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{6\pi^{2}})s-\frac{5m_{u}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{2\pi^{2}}-\frac{5m_{d}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}+\frac{\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{6\pi^{2}}\,,$ $\displaystyle\rho_{13}(s)$
$\displaystyle=$ $\displaystyle-\frac{\langle
g^{2}GG\rangle}{1024\pi^{6}}s^{2}+(\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,,$ (13)
$\displaystyle\rho_{14}(s)$ $\displaystyle=$
$\displaystyle(\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}}+\frac{3m_{u}m_{d}\langle
g^{2}GG\rangle}{2048\pi^{6}}+\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,,$ (14)
$\displaystyle\rho_{23}(s)$ $\displaystyle=$
$\displaystyle(-\frac{9m_{u}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}}-\frac{9m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{9m_{d}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}})s+\frac{3m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}+\frac{3m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}\,,$ (15)
$\displaystyle\rho_{24}(s)$ $\displaystyle=$ $\displaystyle\frac{\langle
g^{2}GG\rangle}{1024\pi^{6}}s^{2}+(-\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}-\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}})s+\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}+\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,,$ (16)
$\displaystyle\rho_{25}(s)$ $\displaystyle=$
$\displaystyle(\frac{m_{u}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}}+\frac{m_{u}m_{d}\langle
g^{2}GG\rangle}{2048\pi^{6}}+\frac{m_{d}^{2}\langle
g^{2}GG\rangle}{4096\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{384\pi^{4}}\,,$ (17)
$\displaystyle\rho_{34}(s)$ $\displaystyle=$
$\displaystyle(-\frac{{m_{u}}^{2}}{256\pi^{6}}-\frac{{m_{u}}m_{d}}{128\pi^{6}}-\frac{{m_{d}}^{2}}{256\pi^{6}})s^{3}+(\frac{{m_{u}}\langle\bar{q}q\rangle}{4\pi^{4}}+\frac{{m_{d}}\langle\bar{q}q\rangle}{4\pi^{4}})s^{2}$
$\displaystyle+(-\frac{15m_{u}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}}-\frac{15m_{u}m_{d}\langle
g^{2}GG\rangle}{1024\pi^{6}}-\frac{15m_{d}^{2}\langle
g^{2}GG\rangle}{2048\pi^{6}}+\frac{3m_{u}\langle g\bar{q}\sigma
Gq\rangle}{8\pi^{4}}+\frac{3m_{d}\langle g\bar{q}\sigma
Gq\rangle}{8\pi^{4}}-\frac{2\langle\bar{q}q\rangle^{2}}{\pi^{2}})s$
$\displaystyle+\frac{5m_{u}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{6m_{u}m_{d}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{5m_{d}^{2}\langle\bar{q}q\rangle^{2}}{\pi^{2}}+\frac{5m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}+\frac{5m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{64\pi^{4}}-\frac{2\langle\bar{q}q\rangle\langle
g\bar{q}\sigma Gq\rangle}{\pi^{2}}\,,$ $\displaystyle\rho_{35}(s)$
$\displaystyle=$ $\displaystyle-\frac{\langle
g^{2}GG\rangle}{1024\pi^{6}}s^{2}+(\frac{3m_{u}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}}+\frac{3m_{d}^{2}\langle
g^{2}GG\rangle}{512\pi^{6}})s-\frac{m_{u}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}-\frac{m_{d}\langle
g^{2}GG\rangle\langle\bar{q}q\rangle}{128\pi^{4}}\,.$ (19)
## III Numerical Analysis
To perform the numerical analysis, we use the values for all the condensates
from Refs. Yang:1993bp ; Narison:2002pw ; Gimenez:2005nt ; Jamin:2002ev ;
Ioffe:2002be ; Ovchinnikov:1988gk :
$\displaystyle\langle\bar{q}q\rangle=-(0.240\mbox{ GeV})^{3}\,,$
$\displaystyle\langle\bar{s}s\rangle=-(0.8\pm 0.1)\times(0.240\mbox{
GeV})^{3}\,,$ $\displaystyle\langle g_{s}^{2}GG\rangle=(0.48\pm 0.14)\mbox{
GeV}^{4}\,,$ $\displaystyle m_{u}=5.3\mbox{ MeV}\,,m_{d}=9.4\mbox{ MeV}\,,$
$\displaystyle m_{s}(1\mbox{ GeV})=125\pm 20\mbox{ MeV}\,,$ (20)
$\displaystyle\langle g_{s}\bar{q}\sigma
Gq\rangle=-M_{0}^{2}\times\langle\bar{q}q\rangle\,,$ $\displaystyle
M_{0}^{2}=(0.8\pm 0.2)\mbox{ GeV}^{2}\,.$
As usual we assume the vacuum saturation for higher dimensional operators such
as $\langle 0|\bar{q}q\bar{q}q|0\rangle\sim\langle 0|\bar{q}q|0\rangle\langle
0|\bar{q}q|0\rangle$. There is a minus sign in the definition of the mixed
condensate $\langle g_{s}\bar{q}\sigma Gq\rangle$, which is different with
some other QCD sum rule calculation. This is just because the definition of
coupling constant $g_{s}$ is different Yang:1993bp ; Hwang:1994vp .
Altogether we took randomly chosen 50 sets of $t_{i}$ and $\theta_{i}$. Some
of these sets of numbers lead to negative spectral densities in the low energy
region of interest, which should be, however, positive from their definition.
This is due to several reasons. One reason is that the convergence of OPE may
not be achieved yet for those currents for the tetraquark state. Another
reason is that some currents may not couple to the physical states properly.
Except them, there are fifteen sets which lead to positive spectral densities.
We show these fifteen sets of $t_{i}$ and $\theta_{i}$ in Table 1, and label
them as (01), (02), $\cdots$, (15). They are sorted by the fourth column “Pole
Contribution” (PC):
${\rm
Pole\,\,Contribution}\equiv\frac{\int_{0}^{s_{0}}e^{-s/M_{B}^{2}}\rho(s){\rm
d}s}{\int_{0}^{\infty}e^{-s/M_{B}^{2}}\rho(s){\rm d}s}\,.$ (21)
The pole contribution (PC) is an important quantity to check the validity of
the QCD sum rule analysis. Here, $\rho(s)$ denotes the spectral function. It
depends on the ten mixed parameters as well as $M_{B}$ and $s_{0}$. We note
that $\pi$-$\pi$ continuum which we shall study later is not included in the
pole contribution. By fixing $s_{0}=1$ GeV2, we show the PC values in Table 1
for the fifteen sets. “PC(0.5)”, “PC(0.8)” and “PC(1.2)” denote pole
contribution by setting $M_{B}^{2}=0.5$ GeV2, $0.8$ GeV2 and $1.2$ GeV2,
respectively. We find that the pole contribution decreases very rapidly as the
Borel Mass increases. Since we have discussed the convergence of OPE in the
previous section, and found that the Dim=10 and Dim=12 terms are much smaller
than the Dim=6 and Dim=8 terms, and so it is only the pole contribution which
gives a upper limitation on the Borel Mass. The Borel window is wider for the
former parameter sets (1), (2), $\cdots$, and narrower for the latter ones. It
almost disappears for the set (15), whose mass prediction is also much
different from others. The Borel window should be our working region. However,
since the Borel stability is always very good when $M_{B}^{2}>$ 0.5 GeV2, we
shall keep the idea of Borel window in mind and work in the region
$0.5<M_{B}^{2}<$ 2 GeV2. On the other side, we shall care more about the
threshold value $s_{0}$.
Table 1: Values for parameters $t_{i}$, $\theta_{i}$, the mass range $M_{\sigma}$, the pole contribution (PC) and the continuum amplitude $a(t_{i},\,\theta_{i})$. The meaning of these quantities are given in the text. There are altogether fifteen sets, which are sorted by the fourth column “PC”. “PC(0.5)”, “PC(0.8)” and “PC(1.2)” denote pole contribution by setting $M_{B}^{2}=0.5$ GeV2, $0.8$ GeV2 and $1.2$ GeV2, respectively. No | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $t_{5}$ | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ | $\theta_{4}$ | $\theta_{5}$ | $M_{\sigma}$(MeV) | PC(0.5) | PC(0.8) | PC(1.2) | a (GeV4)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | $0.03$ | $0.03$ | $0.73$ | $0.37$ | $0.24$ | $2.7$ | $3.4$ | $4.7$ | $5.5$ | $3.6$ | $510\sim 580$ | 92% | 52% | 13% | $1.2\times 10^{-7}$
(2) | $0.03$ | $0.92$ | $0.75$ | $0.70$ | $0.03$ | $5.6$ | $0.80$ | $4.1$ | $2.9$ | $2.5$ | $510\sim 590$ | 90% | 46% | 11% | $5.5\times 10^{-7}$
(3) | $0.25$ | $0.79$ | $0.16$ | $0.95$ | $0.22$ | $1.8$ | $1.2$ | $6.1$ | $0.44$ | $1.8$ | $510\sim 600$ | 87% | 44% | 11% | $3.6\times 10^{-7}$
(4) | $0.53$ | $0.26$ | $0.93$ | $0.24$ | $0.76$ | $2.9$ | $0.40$ | $2.0$ | $2.5$ | $3.3$ | $510\sim 610$ | 85% | 41% | 10% | $1.7\times 10^{-6}$
(5) | $0.74$ | $0.54$ | $0.74$ | $0.65$ | $0.67$ | $0.15$ | $3.1$ | $1.4$ | $2.7$ | $6.1$ | $520\sim 640$ | 81% | 36% | 8% | $1.9\times 10^{-6}$
(6) | $0.98$ | $0.50$ | $0.12$ | $0.33$ | $0.03$ | $2.0$ | $4.0$ | $6.3$ | $1.3$ | $1.6$ | $510\sim 590$ | 82% | 32% | 6% | $5.8\times 10^{-8}$
(7) | $0.98$ | $0.42$ | $0.84$ | $0.82$ | $0.72$ | $0.095$ | $1.5$ | $3.7$ | $2.4$ | $3.0$ | $540\sim 700$ | 70% | 26% | 6% | $4.2\times 10^{-6}$
(8) | $0.48$ | $0.68$ | $0.58$ | $0.96$ | $0.04$ | $1.8$ | $2.5$ | $3.0$ | $4.3$ | $3.7$ | $530\sim 690$ | 70% | 25% | 6% | $1.9\times 10^{-6}$
(9) | $0.53$ | $1.0$ | $0.99$ | $0.34$ | $0.86$ | $5.6$ | $4.8$ | $5.3$ | $4.1$ | $0.076$ | $540\sim 700$ | 68% | 24% | 5% | $4.5\times 10^{-6}$
(10) | $0.75$ | $0.96$ | $0.32$ | $0.12$ | $0.11$ | $4.3$ | $2.6$ | $0.93$ | $5.1$ | $2.9$ | $560\sim 760$ | 57% | 17% | 4% | $9.5\times 10^{-7}$
(11) | $0.31$ | $0.81$ | $0.71$ | $0$ | $0.10$ | $4.2$ | $1.8$ | $2.8$ | $5.4$ | $5.1$ | $570\sim 780$ | 55% | 17% | 4% | $3.2\times 10^{-6}$
(12) | $0.47$ | $0.40$ | $0$ | $0.46$ | $0.91$ | $0.18$ | $1.9$ | $1.9$ | $0.091$ | $0.94$ | $540\sim 730$ | 58% | 16% | 3% | $2.0\times 10^{-7}$
(13) | $0.60$ | $0.26$ | $0.44$ | $0.27$ | $0.24$ | $3.3$ | $3.6$ | $0.92$ | $5.9$ | $3.7$ | $620\sim 850$ | 43% | 13% | 3% | $1.7\times 10^{-6}$
(14) | $0.74$ | $0.73$ | $0.73$ | $0.32$ | $0.28$ | $1.3$ | $1.3$ | $4.6$ | $3.3$ | $5.6$ | $620\sim 850$ | 42% | 12% | 3% | $4.3\times 10^{-6}$
(15) | $0.65$ | $0.55$ | $0.92$ | $0.19$ | $0.96$ | $4.9$ | $5.2$ | $4.0$ | $5.5$ | $3.3$ | $730\sim 930$ | 25% | 7% | 2% | $5.4\times 10^{-6}$
By using these fifteen sets of numbers, we perform the QCD sum rule analysis.
There are two parameters, the Borel mass $M_{B}$ and the threshold value
$s_{0}$ in the QCD sum rule analysis. We find that the Borel mass stability is
usually good, but the threshold value stability is not always good. We show
the mass range of $\sigma(600)$, $M_{\sigma}$, in Table 1, where the working
region is taken to be $0.8$ GeV${}^{2}<s_{0}<1.2$ GeV2 and $0.8$
GeV${}^{2}<M_{B}^{2}<2$ GeV2. We find the mass range is small when the pole
contribution (PC) is large.
The parameter sets (01)-(06) lead to relatively good threshold value
stability. Taking the set (02) as an example, we show its spectral density
$\rho(s)$ in Fig 2 as function of $s$. It is positive definite, and has a
small value around $s\sim 1.2$ GeV2. Therefore, the threshold value dependence
is weak around this point, as shown in Fig. 3 for the extracted mass as
functions of both $M_{B}^{2}$ and $s_{0}$. We find all the curves are very
stable in the region $0.5$ GeV${}^{2}<M_{B}^{2}<2$ GeV2 and $0.6$
GeV${}^{2}<s_{0}<1.4$ GeV2. From the set (02) we can extract the mass of
$\sigma(600)$ around $550$ MeV. From other good cases, we find that the mass
of $\sigma(600)$ is around $550$ MeV as well.
Figure 2: The spectral density $\rho(s)$ calculated by the mixed current
$\eta$, as a function of $s$. We show the results of the parameter set (02) as
an example.
Figure 3: The extracted mass of $\sigma(600)$ as a tetraquark state calculated
by the mixed current $\eta$, as functions of the Borel mass $M_{B}$ and the
threshold value $s_{0}$. We show the results of the parameter set (02) as an
example. At the left panel, the solid, short-dashed and long-dashed curves are
obtained by setting $s_{0}=0.8,~{}1$ and $1.2$ GeV2, respectively. At the
right panel, the solid and dashed curves are obtained by setting
$M_{B}^{2}=0.5,~{}1$ and $2$ GeV2, respectively.
The parameter sets (07)-(15) lead to the threshold value stability, which is
not good. Taking the set (13) as an example, we show its spectral density in
Fig. 4 as a function of $s$ (left figure), and the extracted mass in Fig. 5 as
a function of $s_{0}$ (upper three curves). The mass increases with $s_{0}$
and we cannot extract the mass from this result.
Figure 4: The spectral density $\rho(s)$ calculated by the mixed current
$\eta$, as a function of $s$. We show the results of the parameter set (13) as
an example. The left figure shows the full spectral density as given on the
left hand side of Eq. (22), while the right figure is the one with
$\rho_{\pi\pi}(s)$ subtracted.
Figure 5: The extracted mass of $\sigma(600)$ as a tetraquark state calculated
by the mixed current $\eta$, as functions of the threshold value $s_{0}$. We
choose the parameter set (13) as an example. The solid, short-dashed and long-
dashed curves are obtained by setting $M_{B}^{2}=0.5,~{}1$ and $2$ GeV2,
respectively. The upper three curves are obtained without adding the
contribution of the $\pi$-$\pi$ continuum in the spectral density in the
phenomenological side, while the lower three curves are obtained after adding
the contribution of the $\pi$-$\pi$ continuum.
Many effects contribute to the mass dependence on the threshold value, but for
$\sigma(600)$ the $\pi$-$\pi$ continuum contribution is probably the dominant
one. Hence, we add a term $\rho_{\pi\pi}(s)$ in the spectral function in the
phenomenological side to describe the $\pi$-$\pi$ continuum:
$\displaystyle\rho(s)$ $\displaystyle=$ $\displaystyle
f^{2}_{Y}\delta(s-M^{2}_{Y})+\rho_{\pi\pi}(s)+\rho_{cont}\,.$ (22)
where $\rho_{cont}$ is the standard expression of the continuum contribution
except the $\pi$-$\pi$ continuum. To find an expression for
$\rho_{\pi\pi}(s)$, we introduce a coupling
$\displaystyle\lambda_{\pi\pi}$ $\displaystyle\equiv$ $\displaystyle\langle
0|\eta|\pi^{+}\pi^{-}\rangle\,.$ (23)
The correlation function of the $\pi$-$\pi$ continuum is
$\displaystyle\Pi_{\pi\pi}(p^{2})$ $\displaystyle=$ $\displaystyle
i\int{d^{4}q\over(2\pi)^{2}}{i\over(p+q)^{2}-m^{2}_{\pi}+i\epsilon}{i\over
q^{2}-m^{2}_{\pi}+i\epsilon}|\lambda_{\pi\pi}|^{2}\,,$ (24)
and the spectral density of the $\pi$-$\pi$ continuum is just its imaginary
part
$\displaystyle\rho_{\pi\pi}(s)={\rm Im}\Pi_{\pi\pi}(s)={1\over
16\pi^{2}}\sqrt{1-{4m_{\pi}^{2}\over s}}|\lambda_{\pi\pi}|^{2}\,.$ (25)
We may calculate $\lambda_{\pi\pi}$ by using the method of current algebra if
we know the property of the resonance state. However, this is not the topic of
this paper. Moreover, in this paper we use a general local tetraquark current
to test the full space of local tetraquark currents, so we again make some try
and error tests, and find that the following function leads to a reasonable
QCD sum rule result, $\lambda_{\pi\pi}\sim s$. Hence, we take the spectral
density of the $\pi$-$\pi$ continuum as
$\displaystyle\rho_{\pi\pi}(s)=a(t_{i},\theta_{i})s^{2}\sqrt{1-{4m_{\pi}^{2}\over
s}}\,.$ (26)
We add the continuum contribution $\rho_{\pi\pi}(s)$ in the phenomenological
side and perform the QCD sum rule analysis. The values of parameter
$a(t_{i},\theta_{i})$ are listed in Table 1. After adding the continuum
contribution, the threshold value stability becomes much better. Still taking
the set (13) as an example, we show its spectral density in Fig. 4 as a
function of $s$ (right figure), and the extracted mass in Fig. 5 as functions
of $s_{0}$ (lower three curves). We see that now the spectral density has a
small value around $s\sim 1.1$ GeV2, and the stability of the threshold value
is significantly improved.
Hence, we made the same analysis for all the other cases. We found all the
cases are good except one, which is the case (15), where we are not able to
get the desired stability as a function of $s_{0}$. The mass function has a
small stability region and increases rapidly with $s_{0}$. Hence, we consider
this case is between the good case and bad case, and remove it from the
further analysis in this paper. We show several results out of all the good
cases in Fig. 6, which are obtained by using the parameter sets (01), (03),
(06), (09), (12) and (14). We list the used $a(t_{i},\theta_{i})$ in Table 1
for all the cases. All the masses behave very nicely as functions of the Borel
mass and $s_{0}$ as shown in Fig. 6. In our working region $0.8$
GeV${}^{2}<s_{0}<1.2$ GeV2 and $0.8$ GeV${}^{2}<M_{B}^{2}<2$ GeV2, all the
cases lead to a mass within the region $495$ MeV$\sim 570$ MeV. From this mass
range, the mass of $\sigma(600)$ is extracted to be $530$ MeV $\pm$ 40 MeV.
Figure 6: The extracted mass of $\sigma(600)$ as a tetraquark state calculated
by the mixed currents $\eta$, as functions of the threshold value $s_{0}$. We
choose the parameter sets (01), (03), (06), (09), (12) and (14). The results
are shown in sequence. The solid, short-dashed and long-dashed curves are
obtained by setting $M_{B}^{2}=0.5,~{}1$ and $2$ GeV2, respectively.
## IV The Effect of Finite Decay Width
After the $s_{0}$ stability has been improved, we notice now that the mass
increases systematically with the Borel mass as seen in Fig. 6 in all the
cases. We therefore try to consider a possible reason of this systematic
result. The $\sigma(600)$ meson has a large decay width. We parametrize it by
a Gaussian distribution instead of the $\delta$-function for the
$\sigma(600)$.
$\displaystyle\rho^{FDW}(s)={f^{2}_{X}\over\sqrt{2\pi}\sigma_{X}}\exp\big{(}-{(\sqrt{s}-M_{X})^{2}\over
2\sigma_{X}^{2}}\big{)}~{}.$ (27)
The Gaussian width $\sigma_{X}$ is related to the Breit-Wigner decay width
$\Gamma$ by $\sigma_{X}=\Gamma/2.4$. We set $\sigma_{X}=200$ MeV, and
$M_{X}=550$ MeV, and calculate the following “mass”:
$\displaystyle
M^{2}(M_{B},s_{0})={\int_{0}^{s_{0}}e^{-s/M_{B}^{2}}s\exp\big{(}-{(\sqrt{s}-M_{X})^{2}\over
2\sigma_{X}^{2}}\big{)}{ds\over
2\sqrt{s}}\over\int_{0}^{s_{0}}e^{-s/M_{B}^{2}}\exp\big{(}-{(\sqrt{s}-M_{X})^{2}\over
2\sigma_{X}^{2}}\big{)}{ds\over 2\sqrt{s}}}~{}.$ (28)
We find that the obtained mass $M$ is not just 550 MeV, but increases as
$M_{B}^{2}$ increases as shown in Fig. 7. Hence, the extracted mass in the QCD
sum rule analysis ought to depend on the Borel mass. The amount of the change
of the extracted mass in the QCD sum rule analysis is similar to the one found
in this model calculation. Moreover, we find that the finite decay width does
not change the final result significantly, which we have also noticed in our
previous paper Chen:2007xr .
Figure 7: The extracted “mass” considering a finite decay width. The solid,
short-dashed and long-dashed curves are obtained by setting
$M_{B}^{2}=0.5,~{}1$ and $2$ GeV2, respectively.
## V Summary
In summary, we have studied the light scalar meson $\sigma(600)$ in the QCD
sum rule. We have used general local tetraquark currents which are linear
combinations of five independent local ones. This describes the full space of
local tetraquark currents which can couple to $\sigma(600)$ either strongly or
weakly. We find some cases where the stability of the Borel mass and threshold
value is both good, while in some cases the threshold value stability is not
so good. The resonance mass has an increasing trend with $s_{0}$, which
indicates a continuum contribution. Hence, we have introduced a contribution
from the $\pi$-$\pi$ continuum, and obtained a good threshold value stability.
The mass of $\sigma(600)$ is extracted to be $530$ MeV $\pm$ 40MeV. Very
interesting observation is that the mass increases slightly with the Borel
mass. We have made a model calculation by taking the Gaussian width of $200$
MeV centered at $550$ MeV and try to make a sum rule analysis. We see a
similar increase trend as seen in the QCD sum rule analysis.
The continuum contribution exists in the background of the $\sigma(600)$ meson
and it is very important to consider this fact in the QCD sum rule analysis
for exotic states. We have seen clear tendency of the mass increase with the
Borel mass after getting good signal of the threshold dependence. The decay
width of $\sigma(600)$ is related to this increase tendency. We are now trying
to calculate this by using the three-point correlation function within the QCD
sum rule approach. The present analysis is very encouraging to apply the QCD
sum rule including the continuum states for other scalar mesons. Moreover, the
continuum contribution should be important in many other resonances such as
$\Lambda(1405)$ etc, which lies in some continuum background. In the future,
we will use the QCD sum rule analysis with continuum to study various
resonances.
## Acknowledgments
This project is supported by the National Natural Science Foundation of China
under Grants No. 10625521 and No. 10721063, the Ministry of Science and
Technology of China (2009CB825200), the Ministry of Education research Grant:
Kakenhi (18540269), and the Grant for Scientific Research ((C) No. 19540297)
from the Ministry of Education, Culture, Science and Technology, Japan.
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|
arxiv-papers
| 2009-12-28T07:35:40 |
2024-09-04T02:49:07.284953
|
{
"license": "Public Domain",
"authors": "Hua-Xing Chen, Atsushi Hosaka, Hiroshi Toki, and Shi-Lin Zhu",
"submitter": "Hua-Xing Chen",
"url": "https://arxiv.org/abs/0912.5138"
}
|
0912.5191
|
# Calculation of two-loop $\beta$-function for general N=1 supersymmetric
Yang–Mills theory with the higher covariant derivative regularization
A.B.Pimenov, E.S.Shevtsova, K.V.Stepanyantz
###### Abstract
For the general renormalizable N=1 supersymmetric Yang–Mills theory,
regularized by higher covariant derivatives, a two-loop $\beta$-function is
calculated. It is shown that all integrals, needed for obtaining this
function, can be easily calculated, because they are integrals of total
derivatives.
Moscow State University, physical faculty,
department of theoretical physics.
$119992$, Moscow, Russia
## 1 Introduction.
It is well known that most quantum field theory models are divergent in the
ultraviolet region. In order to deal with the divergent expressions, it is
necessary to regularize a theory. Although physical results does not depend on
regularization, a proper choice of the regularization can considerably
simplify calculations or reveal some features of quantum corrections. Most
calculations in the quantum field theory where made with the dimensional
regularization [1]. However, the dimensional regularization is not convenient
for calculations in supersymmetric theories, because it breaks the
supersymmetry. That is why in supersymmetric theories one usually uses its
modification, called the dimensional reduction [2]. There are a lot of
calculation, made in supersymmetric theories with the dimensional reduction,
see e.f. [3]. However, it is well known that the dimensional reduction is not
self-consistent [4]. Ways, allowing to avoid such problems, are discussed in
the literature [5]. Other regularizations are sometimes applied for
calculations in supersymmetric theories. For example, in Ref. [6] two-loop
$\beta$-function of the N=1 supersymmetric Yang–Mills theory was calculated
with the differential renormalization [7].
A self-consistent regularization, which does not break the supersymmetry, is
the higher covariant derivative regularization [8], which was generalized to
the supersymmetric case in Ref. [9] (another variant was proposed in Ref.
[10]). However, using this regularization is rather technically complicated.
The first calculation of quantum corrections for the (non-supersymmetric)
Yang–Mills theory was made in Ref. [11]. Taking into account corrections, made
in subsequent papers [12], the result for the $\beta$-function appeared to be
the same as the well-known result, obtained with the dimensional
regularization [13]. In principle, it is possible to prove that in the one-
loop approximation calculations with the higher covariant derivative
regularization always agree with the results of calculations with the
dimensional regularization [14]. Some calculations in the one-loop and two-
loop approximations were made for various theories [15, 16] with a variant of
the higher covariant derivative regularization, proposed in [17]. The
structure of the corresponding integrals was discussed in Ref. [16].
Application of the higher covariant derivative regularization to calculation
of quantum corrections in the N=1 supersymmetric electrodynamics in two and
three loops [18, 19] reveals an interesting feature of quantum corrections:
all integrals, defining the $\beta$-function appear to be integrals of total
derivatives and can be easily calculated. This makes possible analytical
multiloop calculations with the higher covariant derivative regularization in
supersymmetric theories and allows to explain the origin of the NSVZ
$\beta$-function, which relates the $\beta$-function in $n$-th loop with the
$\beta$-function and the anomalous dimensions in the previous loops. Due to
this, application of this regularization is sometimes very convenient in the
supersymmetric case. The fact that the integrals, appearing with the higher
covariant derivative regularization, in the limit of zero external momentum
become integrals of total derivatives, seems to be a general feature of all
supersymmetric theories. Nevertheless, with the higher derivative
regularization even the two-loop $\beta$-function has not yet been calculated
for a general N=1 supersymmetric Yang–Mills theory. This is made in this
paper. Note that in order to do this calculation, it is necessary to introduce
higher covariant derivative terms not only for the gauge field, but also for
the matter superfields.
The paper is organized as follows:
In Sec. 2 we introduce the notation and recall basic information about the
higher covariant derivative regularization. The $\beta$-function for the
considered theory is calculated in Sec. 3. The result is briefly discussed in
the Conclusion.
## 2 N=1 supersymmetric Yang–Mills theory and the higher covariant derivative
regularization
In this paper we calculate $\beta$-function for a general renormalizable N=1
supersymmetric Yang–Mills theory. In the massless case this theory is
described by the action
$\displaystyle S=\frac{1}{2e^{2}}\mbox{Re}\,\mbox{tr}\int
d^{4}x\,d^{2}\theta\,W_{a}C^{ab}W_{b}+\frac{1}{4}\int
d^{4}x\,d^{4}\theta\,(\phi^{*})^{i}(e^{2V})_{i}{}^{j}\phi_{j}+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\Bigg{(}\frac{1}{6}\int
d^{4}x\,d^{2}\theta\,\lambda^{ijk}\phi_{i}\phi_{j}\phi_{k}+\mbox{h.c.}\Bigg{)},\qquad$
(1)
where $\phi_{i}$ are chiral matter superfields in a representation $R$, which
is in general reducible. $V$ is a real scalar gauge superfield. The superfield
$W_{a}$ is a supersymmetric gauge field stress tensor, which is defined by
$W_{a}=\frac{1}{8}\bar{D}^{2}(e^{-2V}D_{a}e^{2V}).$ (2)
In our notation $D_{a}$ and $\bar{D}_{a}$ are the right and left
supersymmetric covariant derivatives respectively, $V=e\,V^{A}T^{A}$, and the
generators of the fundamental representation are normalized by the condition
$\mbox{tr}(t^{A}t^{B})=\frac{1}{2}\delta^{AB}.$ (3)
Action (2) should be invariant under the gauge transformations
$\phi\to e^{i\Lambda}\phi;\qquad e^{2V}\to
e^{i\Lambda^{+}}e^{2V}e^{-i\Lambda},$ (4)
where $\Lambda$ is an arbitrary chiral superfield. As a consequence, the
coefficient $\lambda^{ijk}$ should satisfy the condition
$(T^{A})_{m}{}^{i}\lambda^{mjk}+(T^{A})_{m}{}^{j}\lambda^{imk}+(T^{A})_{m}{}^{k}\lambda^{ijm}=0.$
(5)
For calculation of quantum corrections it is convenient to use the background
field method. In the supersymmetric case it can be formulated as follows [20]:
Let us make the substitution
$e^{2V}\to e^{2V^{\prime}}\equiv
e^{\mbox{\boldmath${\scriptstyle\Omega}$}^{+}}e^{2V}e^{\mbox{\boldmath${\scriptstyle\Omega}$}},$
(6)
in action (2), where ${\Omega}$ is a background superfield. Then the theory is
invariant under the background gauge transformations
$\phi\to e^{i\Lambda}\phi;\quad V\to e^{iK}Ve^{-iK};\quad
e^{\mbox{\boldmath${\scriptstyle\Omega}$}}\to
e^{iK}e^{\mbox{\boldmath${\scriptstyle\Omega}$}}e^{-i\Lambda};\quad
e^{\mbox{\boldmath${\scriptstyle\Omega}$}^{+}}\to
e^{i\Lambda^{+}}e^{\mbox{\boldmath${\scriptstyle\Omega}$}^{+}}e^{-iK},$ (7)
where $K$ is an arbitrary real superfield, and $\Lambda$ is a background-
chiral superfield. This invariance allows to set
$\mbox{\boldmath$\Omega$}=\mbox{\boldmath$\Omega$}^{+}={\bf V}$.
It is convenient to choose a regularization and gauge fixing so that
invariance (7) is unbroken. First, we fix a gauge by adding
$S_{\mbox{\scriptsize gf}}=-\frac{1}{32e^{2}}\,\mbox{tr}\,\int
d^{4}x\,d^{4}\theta\,\Big{(}V\mbox{\boldmath$D$}^{2}\bar{\mbox{\boldmath$D$}}^{2}V+V\bar{\mbox{\boldmath$D$}}^{2}\mbox{\boldmath$D$}^{2}V\Big{)}$
(8)
to the action. The corresponding Faddeev–Popov and Nielsen–Kallosh ghost
Lagrangians are constructed by the standard way.
For regularization we add the terms
$\displaystyle S_{\Lambda}=\frac{1}{2e^{2}}\mbox{tr}\,\mbox{Re}\int
d^{4}x\,d^{4}\theta\,V\frac{(\mbox{\boldmath$D$}_{\mu}^{2})^{n+1}}{\Lambda^{2n}}V+\frac{1}{8}\int
d^{4}x\,d^{4}\theta\,\Bigg{(}(\phi^{*})^{i}\Big{[}e^{\mbox{\boldmath${\scriptstyle\Omega}$}^{+}}e^{2V}\frac{(\mbox{\boldmath$D$}_{\alpha}^{2})^{m}}{\Lambda^{2m}}e^{\mbox{\boldmath${\scriptstyle\Omega}$}}\Big{]}{}_{i}{}^{j}\phi_{j}+$
$\displaystyle+(\phi^{*})^{i}\Big{[}e^{\mbox{\boldmath${\scriptstyle\Omega}$}^{+}}\frac{(\mbox{\boldmath$D$}_{\alpha}^{2})^{m}}{\Lambda^{2m}}e^{2V}e^{\mbox{\boldmath${\scriptstyle\Omega}$}}\Big{]}{}_{i}{}^{j}\phi_{j}\Bigg{)},$
(9)
where $\mbox{\boldmath$D$}_{\alpha}$ is the background covariant derivative
and we assume that $m<n$.111Other choices of the higher derivative terms are
also possible. (Because the considered theory contains a nontrivial
superpotential, it is also necessary to introduce the higher covariant
derivative term for the matter superfields.)
The regularized theory is evidently invariant under the background gauge
transformations. The regularization, described above, is rather simple, but
breaks the BRST-invariance of the action. That is why it is necessary to use a
special subtraction scheme, which restore the Slavnov–Taylor identities in
each order of the perturbation theory [21]. For the supersymmetric case such a
scheme was constructed in Ref. [22].
It is well-known [23] that the higher covariant derivative term does not
remove divergences in the one-loop approximation. In order to cancel the
remaining one-loop divergences, it is necessary to introduce into the
generating functional the Pauli–Villars determinants
$\prod\limits_{I}\Big{(}\int
D\phi_{I}^{*}D\phi_{I}e^{iS_{I}}\Big{)}^{-c_{I}},$ (10)
where $S_{I}$ is the action for the Pauli–Villars fields,222Note that this
action differs from the one, used in [18], because here the quotient of the
coefficients in the kinetic term and in the mass term does not contain the
factor $Z$. Using terminology of Ref. [24], one can say that here we calculate
the canonical coupling $\alpha_{c}$, while in Ref. [18] we calculated the
holomorphic coupling $\alpha_{h}$. Certainly, after the renormalization the
effective action does not depend on the definitions. However, the definitions
used here are much more convenient.
$\displaystyle S_{I}=\frac{1}{8}\int
d^{4}x\,d^{4}\theta\,\Bigg{(}(\phi_{I}^{*})^{i}\Big{[}e^{\mbox{\boldmath${\scriptstyle\Omega}$}^{+}}e^{2V}\Big{(}1+\frac{(\mbox{\boldmath$D$}_{\alpha}^{2})^{m}}{\Lambda^{2m}}\Big{)}e^{\mbox{\boldmath${\scriptstyle\Omega}$}}\Big{]}{}_{i}{}^{j}(\phi_{I})_{j}+(\phi_{I}^{*})^{i}\Big{[}e^{\mbox{\boldmath${\scriptstyle\Omega}$}^{+}}\Big{(}1+\frac{(\mbox{\boldmath$D$}_{\alpha}^{2})^{m}}{\Lambda^{2m}}\Big{)}\times$
$\displaystyle\times
e^{2V}e^{\mbox{\boldmath${\scriptstyle\Omega}$}}\Big{]}{}_{i}{}^{j}(\phi_{I})_{j}\Bigg{)}+\Big{(}\frac{1}{4}\int
d^{4}x\,d^{2}\theta\,M_{I}^{ij}(\phi_{I})_{i}(\phi_{I})_{j}+\mbox{h.c.}\Big{)}.$
(11)
The masses of the Pauli–Villars fields are proportional to the parameter
$\Lambda$:
$M^{ij}_{I}=a_{I}^{ij}\Lambda.$ (12)
This means that $\Lambda$ is the only dimensionful parameter of the
regularized theory. We assume that the mass term does not break the gauge
invariance. Also we will choose the masses so that
$M_{I}^{ij}(M_{I}^{*})_{jk}=M_{I}^{2}\delta_{k}^{i}.$ (13)
The coefficients $c_{I}$ satisfy the conditions
$\sum\limits_{I}c_{I}=1;\qquad\sum\limits_{I}c_{I}M_{I}^{2}=0.$ (14)
The generating functional for connected Green functions and the effective
action are defined by the standard way.
In this paper we will calculate the $\beta$-function. We use the following
notation. Terms in the effective action, corresponding to the renormalized
two-point Green function of the gauge superfield, are written as
$\Gamma^{(2)}_{V}=-\frac{1}{8\pi}\mbox{tr}\int\frac{d^{4}p}{(2\pi)^{4}}\,d^{4}\theta\,{\bf
V}(-p)\,\partial^{2}\Pi_{1/2}{\bf V}(p)\,d^{-1}(\alpha,\lambda,\mu/p).$ (15)
where $\alpha$ is a renormalized coupling constant. We calculate
$\frac{d}{d\ln\Lambda}\,\Big{(}d^{-1}(\alpha_{0},\lambda_{0},\Lambda/p)-\alpha_{0}^{-1}\Big{)}\Big{|}_{p=0}=-\frac{d\alpha_{0}^{-1}}{d\ln\Lambda}=\frac{\beta(\alpha_{0})}{\alpha_{0}^{2}}.$
(16)
The anomalous dimension is defined similarly. First we consider the two-point
Green function for the matter superfield in the massless limit:
$\Gamma^{(2)}_{\phi}=\frac{1}{4}\int\frac{d^{4}p}{(2\pi)^{4}}\,d^{4}\theta\,(\phi^{*})^{i}(-p,\theta)\,\phi_{j}(p,\theta)\,(ZG)_{i}{}^{j}(\alpha,\lambda,\mu/p),$
(17)
where $Z$ denotes the renormalization constant for the matter superfield. Then
the anomalous dimensions is defined by
$\gamma_{i}{}^{j}\Big{(}\alpha_{0}(\alpha,\lambda,\Lambda/\mu)\Big{)}=-\frac{\partial}{\partial\ln\Lambda}\Big{(}\ln
Z(\alpha,\lambda,\Lambda/\mu)\Big{)}_{i}{}^{j}.$ (18)
## 3 Two-loop $\beta$-function
After calculation of the supergraphs, we have obtained the following result
for the two-loop $\beta$-function:
$\displaystyle\beta_{2}(\alpha)=-\frac{3\alpha^{2}}{2\pi}C_{2}+\alpha^{2}T(R)I_{0}+\alpha^{3}C_{2}^{2}I_{1}+\frac{\alpha^{3}}{r}C(R)_{i}{}^{j}C(R)_{j}{}^{i}I_{2}+\alpha^{3}T(R)C_{2}I_{3}+\quad$
$\displaystyle+\alpha^{2}C(R)_{i}{}^{j}\frac{\lambda_{jkl}^{*}\lambda^{ikl}}{4\pi
r}I_{4},$ (19)
where the following notation is used:
$\displaystyle\mbox{tr}\,(T^{A}T^{B})\equiv
T(R)\,\delta^{AB};\qquad(T^{A})_{i}{}^{k}(T^{A})_{k}{}^{j}\equiv
C(R)_{i}{}^{j};$ $\displaystyle f^{ACD}f^{BCD}\equiv
C_{2}\delta^{AB};\qquad\quad r\equiv\delta_{AA}.$ (20)
(Note that $T(R)=C(R)_{i}{}^{i}/r$.) Here
$I_{i}=I_{i}(0)-\sum\limits_{I}c_{I}I_{i}(M_{I})\quad\mbox{for}\quad I=0,2,3,$
(21)
and the integrals $I_{0}(M)$, $I_{1}$, $I_{2}(M)$, $I_{3}(M)$ and $I_{4}$ are
given by
$\displaystyle
I_{0}(M)=4\pi\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\frac{1}{q^{2}}\frac{d}{dq^{2}}\Bigg{[}\ln\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}+$
$\displaystyle+\frac{M^{2}}{q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}}-\frac{2m\,q^{2m}/\Lambda^{2m}q^{2}(1+q^{2m}/\Lambda^{2m})}{q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}}\Bigg{]};$
(22) $\displaystyle
I_{1}=96\pi^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}k}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\frac{1}{k^{2}}\frac{d}{dk^{2}}\Bigg{[}\frac{1}{q^{2}(q+k)^{2}(1+q^{2n}/\Lambda^{2n})(1+(q+k)^{2n}/\Lambda^{2n})}\times$
$\displaystyle\times\Bigg{(}\frac{n+1}{(1+k^{2n}/\Lambda^{2n})}-\frac{n}{(1+k^{2n}/\Lambda^{2n})^{2}}\Bigg{)}\Bigg{]};$
(23) $\displaystyle
I_{2}(M)=-16\pi^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}k}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\frac{1}{q^{2}}\frac{d}{dq^{2}}\frac{(1+(q+k)^{2m}/\Lambda^{2m})}{\Big{(}(q+k)^{2}(1+(q+k)^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}}\times\vphantom{\Bigg{(}}$
$\displaystyle\times\frac{1}{k^{2}(1+k^{2n}/\Lambda^{2n})}\Bigg{[}\frac{q^{4}(2+(q+k)^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})^{2}(1+q^{2m}/\Lambda^{2m})^{3}}{\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}^{2}}+\vphantom{\Bigg{(}}$
$\displaystyle+mq^{2m}/\Lambda^{2m}\Bigg{(}-\frac{2q^{2}(2+(q+k)^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})(1+q^{2m}/\Lambda^{2m})}{q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}}+\vphantom{\Bigg{(}}$
(24)
$\displaystyle+\frac{q^{2}(2+(q+k)^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})^{2}}{q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}}-\frac{2q^{2}M^{2}(2+(q+k)^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})^{2}}{\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}^{2}}\Bigg{)}\Bigg{]};$
$\displaystyle
I_{3}(M)=4\pi^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}k}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\Bigg{\\{}\frac{\partial}{\partial
q_{\alpha}}\Bigg{[}\frac{k_{\alpha}}{(k+q)^{2}(1+(q+k)^{2n}/\Lambda^{2n})}\times\vphantom{\Bigg{(}}$
$\displaystyle\times\Bigg{(}-\frac{(2+k^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})^{2}(1+k^{2m}/\Lambda^{2m})^{3}(1+q^{2m}/\Lambda^{2m})}{\Big{(}k^{2}(1+k^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}^{2}\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}}-$
$\displaystyle-\frac{m\,k^{2m}/\Lambda^{2m}(2+k^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})^{2}(1+q^{2m}/\Lambda^{2m})}{k^{2}\Big{(}k^{2}(1+k^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}}+\vphantom{\Bigg{(}}$
$\displaystyle+\frac{2m\,k^{2m}/\Lambda^{2m}(2+k^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})(1+k^{2m}/\Lambda^{2m})(1+q^{2m}/\Lambda^{2m})}{k^{2}\Big{(}k^{2}(1+k^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}}+\vphantom{\Bigg{(}}$
$\displaystyle+\frac{2m\,M^{2}k^{2m}/\Lambda^{2m}(2+k^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})^{2}(1+q^{2m}/\Lambda^{2m})}{k^{2}\Big{(}k^{2}(1+k^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}^{2}\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}}\Bigg{)}\Bigg{]}-$
$\displaystyle-\frac{1}{k^{2}}\frac{d}{dk^{2}}\Bigg{[}\frac{2(2+(q+k)^{2m}/\Lambda^{2m}+q^{2m}/\Lambda^{2m})^{2}(1+q^{2m}/\Lambda^{2m})(1+(q+k)^{2m}/\Lambda^{2m})}{\Big{(}q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}\Big{(}(q+k)^{2}(1+(q+k)^{2m}/\Lambda^{2m})^{2}+M^{2}\Big{)}}\times$
$\displaystyle\times\Bigg{(}\frac{1}{(1+k^{2n}/\Lambda^{2n})}+\frac{nk^{2n}/\Lambda^{2n}}{(1+k^{2n}/\Lambda^{2n})^{2}}\Bigg{)}\Bigg{]}\Bigg{\\}};$
$\displaystyle
I_{4}=64\pi^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d^{4}k}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\frac{1}{q^{2}}\frac{d}{dq^{2}}\Bigg{[}\frac{1}{k^{2}(q+k)^{2}(1+k^{2m}/\Lambda^{2m})}\times$
$\displaystyle\times\frac{1}{(1+(q+k)^{2m}/\Lambda^{2m})}\Bigg{(}\frac{1}{(1+q^{2m}/\Lambda^{2m})}+\frac{mq^{2m}/\Lambda^{2m}}{(1+q^{2m}/\Lambda^{2m})^{2}}\Bigg{)}\Bigg{]}.$
(25)
It is easy to see that all these integrals are integrals of total derivatives,
due to the identity
$\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}}\frac{d}{dq^{2}}f(q^{2})=\frac{1}{16\pi^{2}}\Big{(}f(q^{2}=\infty)-f(q^{2}=0)\Big{)},$
(26)
which can be easily proved in the four-dimensional spherical coordinates.
Using this identity we find
$\displaystyle
I_{0}=\frac{1}{4\pi}\frac{d}{d\ln\Lambda}\Big{(}\sum\limits_{I}c_{I}\ln
M_{I}^{2}\Big{)}=\frac{1}{2\pi};$ $\displaystyle
I_{1}=-6\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\Bigg{[}\frac{1}{q^{4}(1+q^{2n}/\Lambda^{2n})^{2}}\Bigg{]}=-\frac{3}{4\pi^{2}};$
$\displaystyle
I_{2}=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\Bigg{[}\frac{(2+k^{2m}/\Lambda^{2m})^{2}}{k^{4}(1+k^{2n}/\Lambda^{2n})(1+k^{2m}/\Lambda^{2m})}\Bigg{]}=\frac{1}{2\pi^{2}};$
$\displaystyle
I_{3}=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\Bigg{[}\frac{2}{q^{4}}-\sum\limits_{I}c_{I}\frac{2(1+q^{2m}/\Lambda^{2m})^{4}}{(q^{2}(1+q^{2m}/\Lambda^{2m})^{2}+M_{I}^{2})^{2}}\Bigg{]}=\frac{1}{4\pi^{2}};$
$\displaystyle
I_{4}=-\int\frac{d^{4}k}{(2\pi)^{4}}\frac{d}{d\ln\Lambda}\Bigg{[}\frac{4}{k^{4}(1+k^{2m}/\Lambda^{2m})^{2}}\Bigg{]}=-\frac{1}{2\pi^{2}}.$
(27)
Note that the Pauli–Villars fields nontrivially contributes only to integrals
$I_{0}$ and $I_{3}$, where they are very important. For example, in the two-
loop integral $I_{3}$ the Pauli–Villars contribution cancels the one-loop
subdivergence, produced by the matter superfields.
Thus, in the two-loop approximation
$\displaystyle\beta(\alpha)=-\frac{\alpha^{2}}{2\pi}\Big{(}3C_{2}-T(R)\Big{)}+\frac{\alpha^{3}}{(2\pi)^{2}}\Big{(}-3C_{2}^{2}+T(R)C_{2}+\frac{2}{r}C(R)_{i}{}^{j}C(R)_{j}{}^{i}\Big{)}-$
$\displaystyle-\frac{\alpha^{2}C(R)_{i}{}^{j}\lambda_{jkl}^{*}\lambda^{ikl}}{8\pi^{3}r}+\ldots$
(28)
Taking into account that the one-loop anomalous dimension is given by
$\gamma_{i}{}^{j}(\alpha)=-\frac{\alpha
C(R)_{i}{}^{j}}{\pi}+\frac{\lambda_{ikl}^{*}\lambda^{jkl}}{4\pi^{2}}+\ldots,$
(29)
we see that our result agrees with the exact NSVZ $\beta$-function [25]
$\beta(\alpha)=-\frac{\alpha^{2}\Big{[}3C_{2}-T(R)+C(R)_{i}{}^{j}\gamma_{j}{}^{i}(\alpha)/r\Big{)}\Big{]}}{2\pi(1-C_{2}\alpha/2\pi)}.$
(30)
Up to notation, this result is in agreement with the results of calculations
made with the dimensional reduction, see e.f. [3].
## 4 Conclusion
In this paper we demonstrate, how the two–loop $\beta$-function in N=1
supersymmetric theories can be calculated with the higher covariant derivative
regularization. The most interesting feature of this calculation is the
factorization of rather complicated integrals into integrals of total
derivatives. Partially this fact can be explained substituting solutions of
Slavnov–Taylor identities into the Schwinger–Dyson equations. However, a
complete proof of this fact has not yet been done. Its origin is also so far
unclear. Possibly, this feature appears due to using of the background field
method [26]. Factorization of integrals, obtained with the higher covariant
derivative regularization, into integrals of total derivatives can allow to do
a simple derivation of the Novikov, Shifman, Vainshtein, and Zakharov
$\beta$-function, which relates $n$-loop contribution to the $\beta$-function
with the $\beta$-function and the anomalous dimension in previous loops. In
this paper we have shown how this can be done at the two-loop level.
Acknowledgements.
This work was partially supported by RFBR grant No 08-01-00281a.
K.V.Stepanyantz is very grateful to Dr. O.J.Rosten for a valuable discussion.
## References
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|
arxiv-papers
| 2009-12-28T16:56:14 |
2024-09-04T02:49:07.292324
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.B.Pimenov, E.S.Shevtsova, K.V.Stepanyantz",
"submitter": "Stepanyantz Konstantin",
"url": "https://arxiv.org/abs/0912.5191"
}
|
0912.5201
|
# Albedo heterogeneity on the surface of (1943) Anteros
Joseph Masiero11affiliation: Institute for Astronomy, University of Hawaii,
2680 Woodlawn Dr, Honolulu, HI 96822 22affiliation: Jet Propulsion Lab,
California Institute of Technology, 4800 Oak Grove Dr., MS 264-767, Pasadena,
CA 91106, Joseph.Masiero@jpl.nasa.gov
###### Abstract
We have investigated the effect of rotation on the polarization of scattered
light for the near-Earth asteroid (1943) Anteros using the Dual Beam Imaging
Polarimeter on the University of Hawaii’s $2.2~{}$m telescope. Anteros is an
L-type asteroid that has not been previously observed polarimetrically. We
find weak but significant variations in the polarization of Anteros as a
function of rotation, indicating albedo changes across the surface.
Specifically, we find that Anteros has a background albedo of $p_{v}=0.18\pm
0.02$ with a dark spot of $p_{v}<0.09$ covering $<2\%$ of the surface.
## 1 Introduction
As the last remnants from an epoch of accretive formation, asteroids provide
us windows into the composition and history of the inner Solar System. Except
for the few largest bodies, asteroids did not heat up enough via decay of
short-lived radionuclides or dissipation of gravitational potential energy to
undergo complete differentiation. As such the minerals observed on their
surfaces capture the elemental and temperature history of the local region of
the protoplanetary disk at the time of their formation. By understanding
asteroid surfaces we can directly probe those early disk conditions.
As the illuminated cross section of an asteroid changes the observed
brightness fluctuates. Given a large enough sample of data a full shape model
of a rotating body can be constructed (Kaasalainen, et al., 2001) even though
it is unresolved. Photometric surveys for asteroid light curves have set
limits on the composition and density of asteroids as a population (Pravec, et
al., 2002) and have estimated the average shape distribution of small Main
Belt asteroids (Masiero, et al., 2009a). All of these results however assume
that the light curve is dominated by the object’s shape and that the entire
surface has a uniform composition and albedo.
It is possible to test for albedo variations using optical imaging
polarimetry, color variations, or even simply photometric variations under the
assumption of a regular shape (Akimov, et al., 1983). In the case of
polarimetry there are strong empirical correlations between the albedo of an
asteroid and both the slope of the polarization-phase curve and the location
of the minimum (negative) polarization (most recently: Cellino, et al., 1999).
The polarization of light scattered off of an atmosphereless body as a
function of phase angle depends on the distance between scattering elements
and their index of refraction (Muinonen, 1989; Muinonen, et al., 2002c). Index
of refraction is an inherent mineralogical property and recent work has shown
that the inter-element scattering distance is likewise determined by the
surface chemistry (Masiero, et al., 2009b). It is not unexpected then that
asteroids of different spectral classification show different polarization-
phase curves (Muinonen, et al., 2002b), or that a differentiated-then-broken
object like (4) Vesta would show polarization variations with rotation.
In almost every way investigated so far the asteroid Vesta stands out as an
interesting and unique object, and this is similarly the case for albedo
variation studies. Although a handful of other asteroids have weak detections
of rotational modulation of their polarization and thus albedos (e.g. Broglia
& Manara, 1992, etc.), Vesta represents one clear case of an object with
polarization changes across its surface caused by changes in composition
(Degewij, et al., 1979; Lupishko, et al., 1988; Broglia & Manara, 1989). This,
along with photometry, spectroscopy and adaptive optic imaging, has lead to
the current interpretation of Vesta as a differentiated body that has
undergone a nearly-catastrophic impact event leaving a giant crater in its
southern hemisphere. The crater reveals a now-solid mantle distinctly
different in color and composition from the original crust material (Cellino,
et al., 1987; Thomas, et al., 1997).
Nakayama, et al. (2000) found rotational modulation of the polarization for
the asteroid (9) Metis with amplitude similar to what is observed for Vesta.
Metis is a $D\sim 180~{}$km asteroid that may have two large spots of
significantly higher albedo ($p_{v}\sim 0.24-0.28$) than the background
material ($p_{v}=0.14$). The authors find that both bright areas are on the
leading (for prograde rotation) or trailing (for retrograde rotation) faces of
the model ellipsoid (Mitchell, et al., 1995). The cause of albedo
heterogeneity across the surface of objects smaller than Vesta is still
undetermined. It is possible that non-disruptive collisions with impactors of
different composition can leave localized deviations from the average
mineralogy. Alternatively, a late formation with a history free of melting may
preserve the varied composition of the protoplanetary disk. However this
theory is complicated by recent work showing that asteroids likely were born
big, and most objects smaller than a few hundred kilometers in diameter should
be collisionally-created fragments (Morbidelli, et al., 2009).
Identifying albedo variations for small asteroids allows us to evaluate the
accuracy of the assumption that flux changes are solely dependent on shape.
This has important implications for results based on this assumption,
especially shape models. Additionally we can also quantify the effect of
collisions between small bodies in determining an asteroid’s local regolith
properties.
All asteroid polarization-phase curves follow the same general trend with
increasing phase angle: zero polarization at zero phase, becoming negative to
some minimum value and then increasing in an approximately linear fashion.
Note that as is standard for Solar system polarimetry the reference direction
for the angle of polarization is aligned with the vector perpendicular to the
plane of scattering such that “positive” and “negative” polarization are
defined as perpendicular and parallel to the scattering plane, respectively.
The results presented here follow this convention. Each polarization-phase
curve displays three distinguishing values used to classify its properties:
the minimum negative polarization ($P_{min}$), the phase angle at which the
polarization returns to zero (the inversion angle, $\alpha_{0}$) and the
linear slope of the polarization-phase relation beyond the inversion angle
($h$). Making use of the albedo-polarization relation from Cellino, et al.
(1999),
$\displaystyle\log p_{v}$ $\displaystyle=$ $\displaystyle(-1.12\pm 0.07)\log
h-(1.78\pm 0.06)$ (1)
(where $p_{v}$ is the geometric V-band albedo) we can use imaging polarimetry
to test for changes in polarization that directly indicate albedo
heterogeneity across an asteroid’s surface.
## 2 Observations and Discussion
Changes in the polarization of the scattered light across the surface of an
asteroid will be small even in the best-case scenarios. To obtain a
significant measurement of the largest of these variations we require an
instrument that can attain better than $0.1\%$ polarization accuracy. Our
study made use of the Dual Beam Imaging Polarimeter (DBIP) located on the
University of Hawaii’s $2.2~{}$m telescope on Mauna Kea, Hawaii (Masiero, et
al., 2007). DBIP uses a double-calcite Savart plate in series with a quarter-
wave and a half-wave retarder to simultaneously measure linear and circular
polarization with accuracy better than $0.1\%$ (Masiero, et al., 2008). DBIP
uses a $g^{\prime}+r^{\prime}$ filter with a bandpass of $400-700~{}$nm. While
asteroid polarization does depend on color (Cellino et al., 2005) changes are
usually small in this wavelength range and typically within measurement
errors.
Observation of our target asteroid were supplemented with polarized and
unpolarized standards to confirm consistency of setup, stability of the
instrument, and accuracy of the measurements. Standards were taken from
Fossati, et al. (2007) as well as the standard list for
Keck/LRISp111http://www2.keck.hawaii.edu/inst/lris/polarimeter/polarimeter.html
which includes the Hubble Space Telescope polarimetric standards (Schmidt, et
al., 1992). These measurements all verified that the errors were within the
range expected from previous calibrations.
As albedo is related to the polarization-phase slope $h$, for a given albedo
variation the respective polarization change will be larger when observed at
higher phase angles (for $\alpha>\alpha_{0}$). Geometric restrictions prevent
Main Belt asteroids (MBAs) from ever reaching phases angles larger than
$\alpha\sim 30^{\circ}$, but near-Earth asteroids (NEAs) pass closer to Earth
and so can reach much larger phase angles. For this reason, NEAs are preferred
targets when looking for albedo variations. At high phases the polarization of
scattered light takes on a linear trend that increases up to the level of
$\sim 5-10\%$ polarized depending on surface mineralogy. These large
polarizations mean that any variation with rotation at high phases can be
easily interpreted as changes in the integrated surface albedo using the
slope-albedo relation (Eq 1) and that the absolute value for the range of the
albedo can be determined.
From July to September of 2009 the NEA (1943) Anteros passed through phase
angles of $16-40^{\circ}$ all while brighter than $V=17~{}$mag presenting a
prime opportunity to measure the polarization, slope and albedo with high
accuracy. The optical/NIR spectrum of Anteros displays a spectral slope
comparable to S-types but with a muted $1~{}\mu$m absorption band resulting in
a classification of L-type (Binzel, et al., 2004). With a measured period of
$P=2.8695\pm 0.0002~{}$hr and a single-peaked photometric light curve with
amplitude $A=0.09~{}$mag (Pravec, et al., 1998), Anteros is an excellent
target to test for rotational variation in polarization and albedo in a few
nights of observing. In particular, a single-peaked low-amplitude light curve
indicates a shape very close to spherical. (Pravec, et al. (1998) found an
amplitude of $0.09~{}$mag across phase angles ranging from
$19^{\circ}<\alpha<32^{\circ}$, resulting in a shape approximation of
$a/b<1.1$).
In Table 1 we present our polarimetric observations of Anteros. Included for
each night are the V magnitude, exposure time, number of 6-exposure
polarimetry measurements acquired ($n_{meas}$), phase angle, ecliptic
longitude, summed linear polarization of all measurements and linear
polarization angle with respect to the vector perpendicular to the scattering
plane ($\theta_{p}$). No significant circular polarization was detected on any
of the nights. The average nightly polarizations are shown in Fig 1 along with
generic model polarization-phase curves for typical S-type asteroids (dotted)
and C-type asteroids (dashed). The model curves were made using the linear-
exponential modeling technique presented by Muinonen, et al. (2002a) and
fitted by-eye to the data shown in Fig 1 of Muinonen, et al. (2002b), to act
as useful approximations. The constants used in this case were, for the C-type
model: $P_{a}=5.5$, $P_{d}=6$, $P_{k}=0.3$ and for the S-type model:
$P_{a}=4.3$, $P_{d}=12$, $P_{k}=0.17$.
The polarization of Anteros is clearly most closely related to an S-type
polarization curve as expected from its spectral features. We measure for
Anteros an inversion angle of $\alpha_{0}=20.3\pm 0.3^{\circ}$ and a slope
beyond the inversion angle of $h=0.122\pm 0.001$. Both $\alpha_{0}$ and $h$
(and their respective errors) were found by conducting a least-squares
minimization fit of a line to the four nights of data. As the data span a
large range of phases and have small individual errors, the resultant error on
$h$ is small. Following Eq 1 we derive a bulk albedo of $p_{v}=0.175\pm
0.002\pm 0.02$ (errors relative and absolute, respectively). Note that the
limiting error on albedo ($\sigma_{abs}=0.02$) derives from the uncertainty on
the constants in Eq 1 and will affect absolute albedo measurements. This does
not apply to relative comparisons between measured albedos, which have an
error of $\sigma_{rel}=0.002$ in the above case. For all observations reported
here, the calibration error on Eq 1 is greater than the noise error by nearly
an order of magnitude, and thus dominates the final error on the measured
albedos. (All errors reported in this paper are $1~{}\sigma$.)
Following (Muinonen, et al., 2002c) polarization can be approximated as
$P\sim\frac{\alpha^{2}}{2n}-\left(\frac{n-1}{n+1}\right)^{2}\frac{(k~{}d~{}\alpha)^{2}}{2~{}[1+(k~{}d~{}\alpha)^{2}]}$
and solving this for the case of zero polarization at the inversion angle we
find
$\displaystyle\alpha_{0}\sim\sqrt{n\left(\frac{n-1}{n+1}\right)^{2}-\frac{1}{(kd)^{2}}}$
(2)
where $k=2\pi/\lambda$, $d$ is the inter-element scattering distance, and $n$
the index of refraction. For a central wavelength of $0.55~{}\mu$m for DBIP
and a typical scattering distance for NEAs of $d\sim 4~{}\mu$m (Masiero, et
al., 2009b) the second term on the right in Eq 2 is negligible, and the index
of refraction can be determined for a given inversion angle. For Anteros we
find an index of refraction of $n\sim 1.74$.
In Fig 2 we show the nightly polarization light curves for Anteros.
Observations have been wrapped to the measured $2.8695~{}$hr photometric
period and all measurements within a $0.1~{}$phase bin have been co-added to
reduce measurement error. The zero point for rotation phase was chosen
arbitrarily on the first night. When all nights of data are wrapped to this
zero-point the error on the period translates to a phase error of $\pm
0.01~{}$rotations between each observing night. Thus, features at specific
phases can be compared across nights.
We find weak variation in the polarization at a $4\sigma$ significance level
for the night of 2009-07-22 with an amplitude of $0.3\%$ when comparing the
data in the rotation phase range of $0.6-0.8$ to the data between rotation
phases $0.8-0.4$. These polarization changes most likely indicate a variation
in the albedo across the surface of Anteros. The amplitude of the polarization
variation scales with the absolute polarization so it is not surprising that
the other observing nights at lower phases show no clear variation, e.g. a
variation with amplitude of $P=0.3\%$ at a phase of $\alpha=40.5^{\circ}$
would be expected to have an amplitude of $P=0.1\%$ at a phase of
$\alpha=28.0^{\circ}$ which is below our threshold for significant detection.
Additionally, changes in the observing geometry between observations could
cause an area that was observable on the night of 2009-07-22 to have a reduced
visibility on subsequent observations, or even be beyond the horizon. However
even in the most extreme case, where the rotation axis is parallel the to
plane of Earth’s orbit, the line-of-sight vector only moves a total of $\sim
18^{\circ}$ with respect to the rotation axis over the dates of the
observations (this is equivalent to the change in ecliptic longitude). Though
this could account for the changes in polarization under specific
circumstances, it is not the most likely scenario.
It has been suggested that the change in polarization alternatively could be
due to an extreme topographical feature that deviates significantly from the
surrounding area. At the high phases at which we observed Anteros first-order
scattering is dominant and so we may simply apply basic scattering properties
to the surface (e.g. the angle of incidence and angle of scattering are equal,
etc). Thus the light we observe necessarily must have been scattered by planes
on the surface normal to the vector that bisects the phase angle. Even on
unusual surfaces, microroughness will provide the appropriate scattering
facets. In the extreme argument, should the plane be a perfectly flat surface
(e.g. a mirror) it will scatter no light to the observer when away from a
perfect alignment. This will decrease the overall flux but not change the
percent of the received flux polarized by the surrounding area. If aligned
perfectly, the plane will still behave polarimetrically as the underlying
material it is made of, polarizing the same fraction of the light as
determined by its albedo. Thus even in case of extreme topography, percent
polarization measurements will only be sensitive to the underlying material
composition.
Changes in the absolute level of polarization as the phase angle changes
nightly precludes wrapping all four nights of observing onto a single rotation
phase. However we can correlate features at similar rotation phases across
nights and from this fit different polarization slopes to different locations
on the asteroid. We use the phase range of $0.6-0.8$ to represent the peak
polarization and the phase range of $0.8-0.4$ (wrapped) to represent the
baseline background polarization (as determined from the 2009-07-22 observing
night). We interpret the phase range of $0.4-0.6$ as a transition region (see
below) and do not include it in either measurement. Using the summed maximum
and baseline values for the first night as location benchmarks we determine
the absolute change in albedo across the surface of Anteros.
We find a background surface albedo of $p_{v}=0.181\pm 0.002\pm 0.02$ with a
single spot of much lower albedo. Note that this value does not vary
significantly from the albedo of $p_{v}=0.13\pm 0.03$ found from radiometric
modeling (Veeder, et al., 1981) or the one published in the compilation by
Chapman, et al. (1994) of $0.17$ (no error given). We measure an upper limit
to the albedo for the dark area of $p_{v}=0.160\pm 0.004\pm 0.02$ however this
feature is unresolved and thus the albedo measurement assumes coverage of the
maximal possible area allowable by the observing geometry ($44\%$ of the total
surface area for 2009-07-22, corresponding to a projected area of
$3.7~{}$km2). It is likely that the dark spot covers a much smaller area with
a much lower reflectance.
Reflected polarization from a mottled surface will mix when unresolved to give
a value between the two extremes. For the case of Anteros on the night of
2009-07-22, taking a background polarization for $\alpha=40.5^{\circ}$ of
$P_{bkg}=2.46\%$ (the mean of the baseline polarization value for the night)
and a peak measured polarization of $2.70\%$ we find that the true percent
polarization of the dark spot ($P_{dark}$) on that night can be described as
$P_{dark}*C+P_{bkg}(1-C)=2.7$
where $C$ is the fraction of the projected illuminated surface that the dark
region covers. This is simply because the polarized light from the dark area
is diluted by the signal from any background material also visible. This
relation can be simplified to:
$\displaystyle P_{dark}=\frac{0.24}{C}+2.46$ (3)
The 2009-07-22 data plotted in Fig 2 show a gradual build up in the
polarization value with a rapid falloff from the peak level to the base level
over slightly more than one tenth of a rotation. This is likely due to the
dark region rising over the horizon as seen from Earth and then passing from
the lit side quickly across the terminator and thus out of illumination. This
scenario would require Anteros to have a prograde rotation state. From the
rapid falloff we can calculate that the maximum size of the dark feature along
the direction of rotation is $<0.7~{}$km assuming it is located on the equator
of the asteroid. If the spot is not on the equator this argument would derive
a smaller value for the size.
Using this size as the diameter of a circular crater this corresponds to a
total projected surface coverage of the dark spot of $C<11\%$ (equivalent to
$<2\%$ of the total unprojected surface area, assuming the rotation pole is
parallel to the plane of the sky), giving a polarization value for this area
of $P_{dark}=4.7\%$. If instead the rotation axis is inclined to the plane of
the sky, the projected area would be smaller. This would increase the required
value of $P_{dark}$ which would cause the calculated value for the albedo (see
below) to decrease. Thus the assumptions made here represent the ’brightest-
case’ scenario for the spot.
As the polarization for the other three nights is consistent at all phases we
can only set an upper limit for the value of $P_{dark}$ on those nights based
on a maximum unresolved change in percent polarization of $0.1\%$. Using the
analouges of Eq 3 for the other nights we find limits of $P_{dark}<1.85\%$ for
the night of 2009-08-11, $P_{dark}<0.88\%$ for 2009-08-26, and
$P_{dark}<-0.14\%$ for 2009-09-10. We show the polarization for the dark
region separated out from the background material in Fig 3 as well as the same
S- and C-type generic models from Fig 1. Using these values we calculate a
lower limit on the slope and thus an upper limit on albedo. The albedo of the
dark spot is limited to $p_{v}<0.09$.
## 3 Conclusions
Using DBIP on the University of Hawaii’s $2.2$ m telescope we have
investigated the NEA (1943) Anteros for surface heterogeneity. We find that
Anteros shows significant polarimetric variation as a function of rotation at
high phase angles, implying an albedo gradient and corresponding surface
composition variations. We determine that Anteros has a base albedo of
$p_{v}=0.181\pm 0.002\pm 0.02$ (errors are relative and absolute,
respectively) consistent with literature values as well as a dark spot of
albedo $p_{v}<0.09$ covering $<2\%$ of its surface.
A single small asteroid showing albedo variations does not invalidate the
assumption that shape dominates the light curves of these bodies. Indeed it is
clear from spacecraft visits that shape does play an important role in
determining the reflected flux. Additionally, albedo variations for most
asteroids appear to be small and localized. However the potential for albedo
variations for small asteroids can not be discounted outright.
It is currently unclear what processes could cause localized albedo changes
across the surfaces of small asteroids. Further polarimetric studies of small
NEAs are thus necessary to constrain the frequency of albedo and composition
changes across the surfaces of these bodies. Once an account the population is
established evolutionary pathways to the creation of these features can be
explored.
## Acknowledgments
J.M. was supported under NASA PAST grant NNG06GI46G. The author would like to
thank Rob Jedicke and Alan Tokunaga for providing comments on the manuscript,
as well as V. Rosenbush and an anonymous referee for helpful reviews that
improved the paper. The author wishes to recognize and acknowledge the very
significant cultural role and reverence that the summit on Mauna Kea has
always had within the indigenous Hawaiian community. I am most fortunate to
have the opportunity to conduct observations from this sacred mountain.
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Table 1: Polarimetry of (1943) Anteros
UT Obs Date | V mag | $t_{exp}$ (sec) | $n_{meas}$ | $\alpha$ | Ecliptic Long | Lin $\%~{}$Pol | $\theta_{p}$
---|---|---|---|---|---|---|---
2009-07-22 | 16.6 | 130 | 21 | $40.5^{\circ}$ | $356.9^{\circ}$ | $2.53\pm 0.02$ | $178.8\pm 0.3^{\circ}$
2009-08-11 | 16.3 | 130 | 26 | $28.0^{\circ}$ | $352.9^{\circ}$ | $0.83\pm 0.02$ | $0.8\pm 0.6^{\circ}$
2009-08-26 | 16.1 | 90 | 27 | $19.1^{\circ}$ | $346.3^{\circ}$ | $-0.14\pm 0.02$ | $86\pm 4^{\circ}$
2009-09-10 | 16.3 | 100 | 30 | $16.4^{\circ}$ | $339.3^{\circ}$ | $-0.41\pm 0.02$ | $94\pm 1^{\circ}$
Figure 1: New observations of the polarization of (1943) Anteros along with
model curves for S-type (dotted) and C-type (dashed), for reference. Note the
errors on the percent polarization are comparable to the size of the points.
Figure 2: Polarization of Anteros as a function of rotation phase.
Observations within $0.1~{}$phase-wide bins have been co-added to reduce
errors. UT date of each observation set is listed. Figure 3: Polarization of
the background and the dark region of (1943) Anteros. The values of the
polarization for the dark region for phase angles below $40.5^{\circ}$ are
upper limits. Note that the error bars on the points are comparable to their
size.
|
arxiv-papers
| 2009-12-28T18:28:03 |
2024-09-04T02:49:07.297585
|
{
"license": "Public Domain",
"authors": "Joseph Masiero (JPL/Caltech)",
"submitter": "Joseph Masiero",
"url": "https://arxiv.org/abs/0912.5201"
}
|
0912.5221
|
# The $D_{s}$ and $D^{+}$ Leptonic Decay Constants from Lattice QCD
Fermilab Lattice and MILC Collaborations A. Bazavova, C. Bernardb, C. DeTarc,
E.D. Freelandb, E. Gamizd,e, Steven Gottliebf,g, U.M. Hellerh, J.E. Hetricki,
A.X. El-Khadrad, A.S. Kronfelde, J. Laihob, L. Levkovac, P.B. Mackenziee, M.B.
Oktayc, M. Di Pierroj, e, R. Sugark, D. Toussainta, and R.S. Van de Waterl
aDepartment of Physics, University of Arizona, Tucson, Arizona, USA
bDepartment of Physics, Washington University, St. Louis, Missouri, USA
cPhysics Department, University of Utah, Salt Lake City, Utah, USA
dPhysics Department, University of Illinois, Urbana, Illinois, USA
eFermi National Accelerator Laboratory, Batavia, Illinois, USA
fDepartment of Physics, Indiana University, Bloomington, Indiana, USA
gNational Center for Supercomputing Applications, University of Illinois,
Urbana, Illinois, USA
hAmerican Physical Society, One Research Road, Box 9000, Ridge, New York, USA
iPhysics Department, University of the Pacific, Stockton, California, USA
jSchool of Computer Sci., Telecom. and Info. Systems, DePaul University,
Chicago, Illinois, USA
kDepartment of Physics, University of California, Santa Barbara, California,
USA
lPhysics Department, Brookhaven National Laboratory, Upton New York, USA
cb@lump.wustl.edusimone@fnal.gov
###### Abstract:
We present the leptonic decay constants $f_{D_{s}}$ and $f_{D^{+}}$ computed
on the MILC collaboration’s $2+1$ flavor asqtad gauge ensembles. We use clover
heavy quarks with the Fermilab interpretation and improved staggered light
quarks. The simultaneous chiral and continuum extrapolation, which determines
both decay constants, includes partially-quenched lattice results at lattice
spacings $a\approx 0.09$, $0.12$ and $0.15$ fm. We have made several recent
improvements in our analysis: a) we include terms in the fit describing
leading order heavy-quark discretization effects, b) we have adopted a more
precise input $r_{1}$ value consistent with our other $D$ and $B$ meson
studies, c) we have retuned the input bare charm masses based upon the new
$r_{1}$. Our preliminary results are $f_{D_{s}}=260\pm 10\;\textrm{MeV}$ and
$f_{D^{+}}=217\pm 10\;\textrm{MeV}$.
## 1 Introduction
We report on progress in the Fermilab Lattice and MILC Collaboration
calculation of the $D$ meson decay constants. This work is a continuation of
the program that predicted the decay constants: $f_{D^{+}}=201(3)(17)$ and
$f_{D_{s}}=249(3)(16)\,\textrm{MeV}$ [1], in good agreement with the CLEO-c
value of $f_{D^{+}}=205.8\pm 8.5\pm 2.5$ [2, 3]. We have since extended this
calculation to two additional ensembles at our finest lattice spacing
$a\approx 0.09\;\textrm{fm}$ and we have replaced a limited set of very coarse
($a\approx 0.18\;\textrm{fm}$) ensembles with higher statistic ensembles at a
somewhat finer spacing $a\approx 0.15\;\textrm{fm}$. In our last update [4] we
reported: $f_{D^{+}}=207(11)$ and $f_{D_{s}}=249(11)$, where $f_{D_{s}}$ is
about 0.6$\sigma$ lower than the recent experimental average [5]. The value of
$f_{D_{s}}$ remains an pressing issue given that experimental average is about
2.1$\sigma$ higher than the most precise lattice result from the HPQCD
collaboration [6]. The apparent tension between experiment and lattice
predictions has motivated suggestions of physics beyond the Standard Model
[7].
Smaller statistical uncertainties and better control of systematic effects are
key to resolving the $f_{D_{s}}$ puzzle. In this report, we have doubled
statistics on the most chiral of the $a\approx 0.09\;\textrm{fm}$ lattices;
otherwise, statistics have not changed. A new generation of calculations, now
underway, aims to increase statistics by a factor of four overall. Our
progress includes: a) a better method of accounting for heavy-quark
discretization effects b) a more precise input value for the scale parameter
$r_{1}$, consistent with our other heavy quark studies and c) more precisely
tuned input charm kappa values.
## 2 Staggered chiral perturbation theory for heavy-light mesons
We use the asqtad improved staggered action for both sea and light valence
quarks. Leading discretization effects split the light pseudoscalar meson
masses,
$M^{2}_{ab,\xi}=(m_{a}+m_{b})\mu+a^{2}\Delta_{\xi}\;,$ (1)
where there are sixteen tastes in representations $\xi=P,A,T,V,I$.
Staggered chiral perturbation theory for heavy-light mesons accounts for such
taste breaking effects [8]. At NLO in the chiral expansion, for $2+1$ flavors,
and at leading order in the heavy quark expansion,
$\phi_{H_{q}}=\Phi_{0}\left[1+\Delta
f_{H}(m_{q},m_{l},m_{h})+p_{H}(m_{q},m_{l},m_{h})+c_{L}\alpha_{V}^{2}a^{2}\right]\;,$
(2)
where $\phi_{H_{q}}=f_{H_{q}}\sqrt{m_{H_{q}}}$ and $f_{H_{q}}$ is the decay
constant of a heavy meson $H_{q}$ consisting of a heavy quark and a light
quark of mass $m_{q}$. The heavier sea quark has mass $m_{h}$ and the two
degenerate light sea quarks have mass $m_{l}$. The $\phi_{H_{q}}$, in general,
are partially quenched: $m_{q}\neq m_{l}$ and $m_{q}\neq m_{h}$. The chiral
logarithm terms, $\Delta f_{H}$, are $a$ dependent as a consequence of the
mass splittings in Eq. (1) as well as from “hairpin” terms proportional to the
low energy constants $a^{2}\delta^{\prime}_{A}$ and
$a^{2}\delta^{\prime}_{V}$. The $a$ dependence of the analytic terms, $p_{H}$,
ensures that $\phi_{H_{q}}$ is unchanged by a change in the chiral scale,
$\Lambda_{\chi}$, of the logarithms. The expression in Eq. (2) is used in our
combined chiral and continuum extrapolations. In practice, we add the NNLO
analytic terms to the fit function in order to extend the fit up to $m_{q}\sim
m_{s}$ and extract $f_{D_{s}}$. Priors for the parameters
$a^{2}\delta^{\prime}_{A}$ and $a^{2}\delta^{\prime}_{V}$ as well as values of
the physical light quark masses are obtained from the MILC analysis of
$f_{\pi}$ and $f_{K}$ [9].
## 3 Discretization effects from clover heavy quarks
We use tadpole-improved clover charm quarks. At leading order, discretization
errors are a combination of $\mathcal{O}(a^{2}\Lambda_{\mathit{HQ}}^{2})$ and
$\mathcal{O}(\alpha a\Lambda_{\mathit{HQ}})$ effects where $\alpha$ is the QCD
coupling and $\Lambda_{\mathit{HQ}}$ is the scale in the heavy quark
expansion. Our past studies have estimated heavy quark discretization effects
using such power counting arguments to bound the error at the smallest lattice
spacing, taking $\Lambda_{\mathit{HQ}}\approx 700\;\textrm{MeV}$. This rather
crude method does not effectively use the data to guide the error estimate.
This study introduces a new procedure: the leading order heavy quark
discretization errors are modeled to leading order as part of the combined
chiral and continuum extrapolation. At tree-level, discretization effects
arise from both the quark action and the (improved) current [10, 11, 12]. We
add five extra terms to Equation 2:
$\Phi_{0}\left[a^{2}\Lambda_{\mathit{HQ}}^{2}\left\\{c_{E}f_{E}(am_{Q})+c_{X}f_{X}(am_{Q})+c_{Y}f_{Y}(am_{Q})\right\\}+\alpha_{V}a\Lambda_{\mathit{HQ}}\left\\{c_{B}f_{B}(am_{Q})+c_{3}f_{3}(am_{Q})\right\\}\right]$
(3)
The coefficients $c_{E}$, $c_{X}$, $c_{Y}$, $c_{B}$ and $c_{3}$ are additional
parameters determined in the fit while the $f_{i}$ are (smooth) functions of
the heavy quark mass, $am_{Q}$, known at tree level. We introduce priors for
the coefficients constraining them to be $\mathcal{O}(1)$ while setting
$\Lambda_{\mathit{HQ}}=700\;\textrm{MeV}$ and $m_{c}\sim 1.2\;\textrm{GeV}$.
Currently the data are too noisy and the shapes of the functions $f_{i}$ are
too similar for the fit to prefer a particular $\Lambda_{\mathit{HQ}}$.
Including the heavy-quark discretization terms increases the decay constants
by a few MeV and increases the error from $\sim 1.8\%$ to $\sim 3.8\%$. The
larger error now includes the residual heavy-quark discretization uncertainty
in addition to residual light-quark discretization effects (encoded in Eq.
(2)) as well as statistical errors. We quote a combined uncertainty from all
the three sources of error.
## 4 Lattice spacing determination from $r_{1}$
Figure 1: Values of scale parameter $r_{1}$ in fermi units. The “HPQCD
$\Upsilon(2S$-$1S)$” value uses the HPQCD collaboration $\Upsilon$ spectrum
results to set the physical value [13]. The “MILC $\Upsilon(2S$-$1S)$” value
derives from essentially the same spectrum analysis [14]. MILC determines
$r_{1}$ more precisely from their calculation of $f_{\pi}$: “MILC $f_{\pi}$
2007” [15] and “MILC $f_{\pi}$ 2009” [9]. In a very recent update, “HPQCD
2009”, several physical quantities, including recent $\Upsilon$ results, are
used as inputs [16].
The distance $r_{1}$ is a property of the QCD potential between heavy quarks.
The ratio $r_{1}/a$, for lattice spacing $a$, has been computed for all of the
MILC gauge ensembles. At intermediate stages of the decay constant analysis
quantities are converted from lattice units to $r_{1}$ units using $r_{1}/a$.
The value of $r_{1}$ must then be input in order to convert results to
physical units. The $r_{1}$ value is also an input to the process of
determining other quantities such as the bare charm quark masses as discussed
in the next section.
Figure 1 depicts several $r_{1}$ determinations. The first two determinations
historically (circa 2004–2005) are labeled “HPQCD $\Upsilon(2S$-$1S)$” [13]
and “MILC $\Upsilon(2S$-$1S)$” [14]. They are both based on the same analysis
of the $\Upsilon$ spectrum by the HPQCD Collaboration using a subset of the
current MILC ensembles. The two determinations differ mainly in the details of
the continuum extrapolation. The MILC Collaboration is also able to infer a
value of $r_{1}$ based on the value of $f_{\pi}$ they find in their analysis
of the light mesons. Recent light-meson analyses include results from finer
lattice spacings than the earlier $\Upsilon$ spectrum study and the resulting
$r_{1}$ values are known to better precision. The figure shows the result of
two recent analyses labeled ‘MILC $f_{\pi}$ 2007” [15] and “MILC $f_{\pi}$
2009” [9]. The (preliminary) 2009 result agrees at the $0.9\sigma$ level with
the MILC $\Upsilon$ value but differs from the HPQCD $\Upsilon$ value at the
$1.8\sigma$ level. As these proceedings were being prepared, HPQCD published a
new value for $r_{1}$ [16], labeled “HPQCD 2009” in the figure, in much better
agreement with MILC’s recent $r_{1}$ values.
In this study, we use the MILC $r_{1}$ determinations from $f_{\pi}$ to set
the physical scale. Our central value for $r_{1}$ (the 2007 value) was also
used in our studies of the semileptonic decays on the same lattices [17, 18].
The range of the 2009 MILC $r_{1}$ determination is used to set a symmetric
uncertainty around the central value. Hence, we take $r_{1}=0.3108\pm 0.0022$.
Our previous decay constant studies used the MILC $\Upsilon$ value:
$r_{1}=0.318\pm 0.007$ as an input which is about one $\sigma$ higher.
## 5 Retuning kappa charm
$r_{1}$ [fm]: | 0.3108 | 0.318
---|---|---
$a$ | $\kappa$ run | $\kappa$ tune | $\delta\phi_{s}$ | % $\phi_{s}$ | $\kappa$ tune | $\delta\phi_{s}$ | % $\phi_{s}$
0.09 | 0.127 | 0.1272 | $-0.0043$ | $-0.56$ | 0.1267 | $+0.0065$ | $+0.84$
0.12 | 0.122 | 0.1222 | $-0.0036$ | $-0.50$ | 0.1215 | $+0.0091$ | $+1.26$
0.15 | 0.122 | 0.1222 | $-0.0031$ | $-0.42$ | 0.1213 | $+0.0108$ | $+1.47$
$\delta f_{D_{s}}$ [MeV] | $-1.8$ | $+1.3$
Table 1: Tuning of $\kappa$ charm at the three lattice spacings for two
choices of $r_{1}$. The shift $\delta\phi_{s}$ is the change in $\phi$ at the
strange quark mass when $\kappa$ changes from the run value to tuned $\kappa$
value. The corresponding change in extrapolated $f_{D_{s}}$ is $\delta
f_{D_{s}}$. In each case, all other extrapolation inputs are fixed to their
appropriate ($r_{1}$ dependent) values.
We determine the value of $\kappa$ for the charm quark by requiring that the
spin-averaged kinetic masses of the lattice pseudoscalar and vector mesons
made from a heavy clover quark and strange asqtad valence quark equal the
spin-averaged $D_{s}$ meson mass. The tuning depends upon $r_{1}$ in the
conversion between lattice and physical masses.
In the past year, we have conducted new kappa-tuning runs with at least four
times the statistics of our older tunings. At each lattice spacing, we
simulated mesons for three values of $\kappa$ around charm and three light-
quark masses around strange allowing us to retune $\kappa$ for a given
$r_{1}$.
Table 1 shows preliminary tunings for $\kappa$ charm based upon the two input
values: $r_{1}=0.3108\;\textrm{fm}$ (present value) and
$r_{1}=0.318\;\textrm{fm}$ (past studies). For each $r_{1}$, the (preliminary)
tuned kappa and the corresponding change
$\delta\phi_{s}=\phi_{s}(\kappa\;\mathit{tune})-\phi_{s}(\kappa\;\mathit{run})$
is listed by lattice spacing. The “run” kappa values are those used in the
decay constant simulations. We adjust each $\phi_{q}$ point by
$\delta\phi_{s}$ prior to the chiral extrapolation to correct for the
mistuning of kappa. The bottom row of the table shows the resulting change in
$f_{D_{s}}$. The opposite signs of the differences show that keeping kappa
tuned partly compensates the change in $r_{1}$. We find that changing $r_{1}$
from $0.318\;\textrm{fm}$ to $0.3108\;\textrm{fm}$ while keeping kappa charm
tuned increases $f_{D_{s}}$ by about $4.2\;\textrm{MeV}$.
## 6 The chiral and continuum extrapolation, results and uncertainty budget
Figure 2: The preliminary $D$ meson chiral extrapolation. The $3\times 4$
matrix of plots (top) show the $\phi$ data and corresponding fit including
$a^{2}$ effects. Reading from left to right and top to bottom, plots
correspond to $(a,m_{l}/m_{h})$ values of $(0.15,0.2)$, $(0.15,0.4)$,
$(0.15,0.6)$, $(0.12,0.14)$, $(0.12,0.2)$, $(0.12,0.4)$, $(0.12,0.6)$,
$(0.12,0.1)$, $(0.09,0.2)$, $(0.09,0.4)$ and $(0.09,0.1)$. The larger plot
(bottom) shows an overlay of the $f_{D_{s}}$ and $f_{D^{+}}$ extrapolations.
The extrapolated curves are the fit (with error bands) taking $a^{2}\to 0$ and
fixing/extrapolating the light quarks to their physical masses. The
extrapolations are not expected to go though any of the points which are
computed at finite $a$. None of the data points having $m_{q}$ near $m_{s}$
seen the upper panel are visible in the $D_{s}$ extrapolation.
We fit $\phi_{D_{q}}$ results from lattice simulations on eleven asqtad MILC
ensembles [14] at the three lattice spacings: $a\approx 0.09$, $0.12$ and
$0.15\;\textrm{fm}$. Our valence quark masses are in the range $0.1m_{s}\leq
m_{q}\lesssim m_{s}$. Since our last report, we have doubled the statistics at
the most chiral of the $a\approx 0.09$ ensembles. The $3\times 4$ panel of
plots at the top in Fig. 2 shows the $\phi_{D_{q}}$ points and the fit where
the fit function includes the lattice-spacing effects described in Sections 2
and 3. The plot at the bottom of Fig. 2 shows the extrapolations in the limit
$a=0$. The upper ($D_{s}$) curve shows $m_{l}\to\hat{m}$ setting
$m_{q}=m_{h}=m_{s}$, while the lower ($D^{+}$) curve shows
$m_{q},m_{l}\to\hat{m}$ setting $m_{h}=m_{s}$. The physical quark mass inputs
are from the MILC light meson analysis and $\hat{m}=(m_{u}+m_{d})/2$. The
points denoted by the red triangles correspond to physical $f_{D_{s}}$ and
$f_{D^{+}}$. Our preliminary results are:
$\begin{array}[]{c@{,\quad}c@{\quad\textrm{and}\quad}c}f_{D^{+}}=217\pm
10\;\textrm{MeV}&f_{D_{s}}=260\pm 10\;\textrm{MeV}&f_{D_{s}}/f_{D^{+}}=1.20\pm
0.02\;.\end{array}$ (4)
source | $\phi_{D_{s}}$ | $\phi_{D^{+}}$ | $R_{{D^{+}/D_{s}}}$
---|---|---|---
statistics and discretization effects | 2.6 | 3.4 | 1.3
chiral extrapolation | 2.0 | 2.5 | 0.8
inputs $r_{1}$, $m_{s}$, $m_{d}$ and $m_{u}$ | 0.7 | 0.7 | 0.3
input $m_{c}$ | 1.0 | 1.2 | 0.2
$Z_{V}^{cc}$ and $Z_{V}^{qq}$ | 1.4 | 1.4 | 0
higher-order $\rho_{A_{4}}$ | 0.3 | 0.3 | 0.2
finite volume | 0.2 | 0.6 | 0.6
total | 3.8 | 4.7 | 1.7
Table 2: Uncertainties as a percentage of $\phi$ and the ratio. The total
combines all of the errors in quadrature.
We have combined the statistical and the systematic uncertainties listed in
Table 2 in quadrature. Our largest uncertainty is the combined uncertainty
from statistical and residual discretization effects. The second largest
uncertainty, chiral extrapolation, is an estimate of chiral expansion effects
not included in the fit function and effects from variation in the
extrapolation procedure. The third largest error is the statistical error in
the nonperturbative calculation of the current renormalizations $Z_{V}^{cc}$
and $Z_{V}^{qq}$. The value of $f_{D_{s}}$ is about eleven MeV (one sigma)
higher than our earlier value. Using nominal kappa values rather than tuned
values at the previous $r_{1}$ value accounts for about 1.3 MeV of the
difference. Changing to the new $r_{1}$ while keeping kappa tuned results in a
4.2 MeV increase. Incorporating heavy quark effects into the fit increases
$f_{D_{s}}$ by about 2 MeV. Higher statistics on the most chiral of the
$a\approx 0.09\;\textrm{fm}$ lattice increases $f_{D_{s}}$ by about 1 MeV.
These changes combine nonlinearly in the fit to yield the net increase.
Figure 3: Comparisons of the Fermilab/MILC values of $f_{D^{+}}$ and
$f_{D_{s}}$ to values from the HPQCD Collaboration [6] and recent experimental
values [3][5].
Figure 3 compares the Fermilab and MILC Collaboration values for the decay
constants with the HPQCD Collaboration [6] values and with the experimental
results. The experimental result for $f_{D^{+}}$ is from CLEO [3] while the
$f_{D_{s}}$ value is the Heavy Flavor Averaging Group average [5] of
determinations by CLEO, BaBar and Belle. The Fermilab / MILC results remain in
agreement with experiment. The total error on the experimental average for
$f_{D_{s}}$ is now smaller that our error providing a challenge for future
lattice determinations. The apparent discrepancy between the HPQCD value of
$f_{D_{s}}$ and the other two $f_{D_{s}}$ values is most striking. The HPQCD
value is lower by about 1.8–2.1$\sigma$. The source of this difference may be
clarified by further lattice simulations.
## 7 Summary and future plans
We have made several improvements in our analysis: a) discretization effects
from both heavy and light quarks are modeled in our extrapolation function, b)
we adopted a more precise $r_{1}$ value which derived from the MILC $f_{\pi}$
analysis rather than the $r_{1}$ value related to early $\Upsilon$ spectrum
results c) we have improved the tuning of kappa charm. These improvements to
the analysis will be more crucial in our next generation of decay constant
study. We will increase statistics by a factor of four and extend the analysis
to the finer lattice spacings $a\approx 0.06$ and $0.045\;\textrm{fm}$ which
will reduce our combined statistical plus discretization error as well as help
reduce uncertainties attributed to chiral extrapolation procedures. In
addition, a new high-statistics computation of the nonperturbative part of the
current renormalization aims for an error below the 0.5% level.
## References
* [1] C. Aubin et al., Charmed meson decay constants in three-flavor lattice QCD, Phys. Rev. Lett. 95 (2005) 122002, [hep-lat/0506030].
* [2] CLEO Collaboration, M. Artuso et al., Improved Measurement of $\mathcal{B}(D^{+}\to\mu^{+}\nu)$ and the Pseudoscalar Decay Constant $f_{D^{+}}$, Phys. Rev. Lett. 95 (2005) 251801, [hep-ex/0508057].
* [3] CLEO Collaboration, B. I. Eisenstein et al., Precision Measurement of $\mathcal{B}(D^{+}\to\mu^{+}\nu)$ and the Pseudoscalar Decay Constant $f_{D^{+}}$, Phys. Rev. D78 (2008) 052003, [0806.2112].
* [4] C. Bernard et al., B and D Meson Decay Constants, PoS LATTICE2008 (2008) 278, [0904.1895].
* [5] H. F. A. G. C. Physics), “$f_{D_{s}}$ world average.” www.slac.stanford.edu/xorg/hfag/charm/PIC09/f_ds/results.html, 2009.
* [6] HPQCD Collaboration, E. Follana, C. T. H. Davies, G. P. Lepage, and J. Shigemitsu, High Precision determination of the $\pi$, $K$, $D$ and $D_{s}$ decay constants from lattice QCD, Phys. Rev. Lett. 100 (2008) 062002, [0706.1726].
* [7] B. A. Dobrescu and A. S. Kronfeld, Accumulating evidence for nonstandard leptonic decays of $D_{s}$ mesons, Phys. Rev. Lett. 100 (2008) 241802, [0803.0512].
* [8] C. Aubin and C. Bernard, Staggered chiral perturbation theory for heavy-light mesons, Phys. Rev. D73 (2006) 014515, [hep-lat/0510088].
* [9] The MILC Collaboration, A. Bazavov et al., Results from the MILC collaboration’s $SU(3)$ chiral perturbation theory analysis, PoS LAT2009 (2009) 079, [0910.3618].
* [10] A. S. Kronfeld, Application of heavy-quark effective theory to lattice QCD. I: Power corrections, Phys. Rev. D62 (2000) 014505, [hep-lat/0002008].
* [11] J. Harada et al., Application of heavy-quark effective theory to lattice QCD. II: Radiative corrections to heavy-light currents, Phys. Rev. D65 (2002) 094513, [hep-lat/0112044].
* [12] M. B. Oktay and A. S. Kronfeld, New lattice action for heavy quarks, Phys. Rev. D78 (2008) 014504, [0803.0523].
* [13] A. Gray et al., The Upsilon spectrum and $m_{b}$ from full lattice QCD, Phys. Rev. D72 (2005) 094507, [hep-lat/0507013].
* [14] C. Aubin et al., Light hadrons with improved staggered quarks: Approaching the continuum limit, Phys. Rev. D70 (2004) 094505, [hep-lat/0402030].
* [15] C. Bernard et al., Status of the MILC light pseudoscalar meson project, PoS LAT2007 (2007) 090, [0710.1118].
* [16] C. T. H. Davies, E. Follana, I. D. Kendall, G. P. Lepage, and C. McNeile, Precise determination of the lattice spacing in full lattice QCD, 0910.1229.
* [17] C. Bernard et al., The $\bar{B}\to D^{*}\ell\bar{\nu}$ form factor at zero recoil from three-flavor lattice QCD: A Model independent determination of $|V_{cb}|$, Phys. Rev. D79 (2009) 014506, [0808.2519].
* [18] J. A. Bailey et al., The $B\to\pi\ell\nu$ semileptonic form factor from three-flavor lattice QCD: A Model-independent determination of $|V_{ub}|$, Phys. Rev. D79 (2009) 054507, [0811.3640].
|
arxiv-papers
| 2009-12-28T21:20:44 |
2024-09-04T02:49:07.303263
|
{
"license": "Public Domain",
"authors": "A. Bazavov, C. Bernard, C. DeTar, E.D. Freeland, E. Gamiz, Steven\n Gottlieb, U.M. Heller, J.E. Hetrick, A.X. El-Khadra, A.S. Kronfeld, J. Laiho,\n L. Levkova, P.B. Mackenzie, M.B. Oktay, M. Di Pierro, J.N. Simone, R. Sugar,\n D. Toussaint, and R.S. Van de Water",
"submitter": "James Simone",
"url": "https://arxiv.org/abs/0912.5221"
}
|
0912.5273
|
# On the Drinfeld-Sokolov Hierarchies of D type
Si-Qi Liu Chao-Zhong Wu Youjin Zhang
Department of Mathematical Sciences, Tsinghua University,
Beijing 100084, P. R. China
liusq@mail.tsinghua.edu.cnwucz05@mails.tsinghua.edu.cnyoujin@mail.tsinghua.edu.cn
###### Abstract
We extend the notion of pseudo-differential operators that are used to
represent the Gelfand-Dickey hierarchies, and obtain a similar representation
for the full Drinfeld-Sokolov hierarchies of $D_{n}$ type. By using such
pseudo-differential operators we introduce the tau functions of these bi-
Hamiltonian hierarchies, and prove that these hierarchies are equivalent to
the integrable hierarchies defined by Date-Jimbo-Kashiware-Miwa and Kac-
Wakimoto from the basic representation of the Kac-Moody algebra $D_{n}^{(1)}$.
Key words: pseudo-differential operator, Drinfeld-Sokolov hierarchy, tau
function, bilinear equation, BKP hierarchy
###### Contents
1. 1 Introduction
2. 2 Pseudo-differential operators
1. 2.1 Definitions
2. 2.2 Properties of pseudo-differential operators
3. 3 An integrable hierarchy represented by pseudo-differential operators
1. 3.1 Construction of the hierarchy
2. 3.2 Bihamiltonian structure and tau structure
4. 4 Drinfeld-Sokolov hierarchies and pseudo-differential operators
1. 4.1 Definition of the Drinfeld-Sokolov hierarchies
2. 4.2 Positive flows of the Drinfeld-Sokolov hierarchy of $D_{n}$ type
3. 4.3 Negative flows of the Drinfeld-Sokolov hierarchy of $D_{n}$ type
5. 5 The two-component BKP hierarchy and its reductions
1. 5.1 The two-component BKP hierarchy
2. 5.2 Reductions of the two-component BKP hierarchy
6. 6 Conclusion
## 1 Introduction
For every affine Lie algebra $\mathfrak{g}$ and a choice of a vertex $c_{m}$
of the extended Dynkin diagram, Drinfeld and Sokolov constructed in [6] a
hierarchy of integrable systems which generalizes the prototypical soliton
equation–the Korteweg-de Vries equation. This construction provides a big
class of integrable hierarchies that are important in different areas of
mathematical physics. In particular, the integrable hierarchies that are
associated to the affine Lie algebras of A-D-E type are shown to be closely
related to 2d topological field theory and Gromov-Witten invariants, see [7,
10, 12, 13, 20, 21, 27, 32] and references therein. In establishing such
relationships the tau functions of the integrable hierarchies play a crucial
role, they correspond to the partition functions of topological field theory
models. The unknown functions of the hierarchy are related to some special two
point correlation functions.
The definition of the tau functions for the Drinfeld-Sokolov hierarchies and
their generalizations [22] was given in [23, 15] by using the dressing
operators of the hierarchies. In terms of the tau functions such integrable
hierarchies and their generalizations are represented as systems of Hirota
bilinear equations, they can also be constructed by using the representation
theoretical approach to solition equations developed by Date, Jimbo,
Kashiwara, Miwa [4, 2] and by Kac, Wakimoto [26, 25]. In this approach the
systems of Hirota bilinear equations are constructed from an integrable
highest weight representation of $\mathfrak{g}$ and its vertex operator
realization, the tau functions that satisfy these equations are elements of
the orbit of the highest weight vector of the representation under the action
of the affine Lie group. Note that tau functions of the Drinfeld-Sokolov
hierarchies are also defined in [11, 31] via certain symmetry (called tau-
symmetry in [10]) of the Hamiltonian densities of the hierarchies represented
in forms of modified KdV type. Here the unknown functions of the Drinfeld-
Sokolov hierarchies in forms of modified KdV type and in that of KdV type are
related by Miura type transformations.
For general Drinfeld-Sokolov hierarchies there are no canonical choices for
their unknown functions, and the definition of the tau functions given in [11,
15, 23] in terms of the dressing operators is in certain sense implicit.
However, in the particular case when the affine Lie algebra is $A_{n}^{(1)}$
the Drinfeld-Sokolov hierarchy coincides with the Gelfand-Dickey hierarchy
[17], the unknown functions can be taken as the coefficients of a differential
operator
$L=D^{n+1}+u^{n}D^{n-1}+\ldots+u^{2}D+u^{1},\quad
D=\frac{\mathrm{d}}{\mathrm{d}x},$
and the integrable hierarchy can be represented in the form
$\frac{\partial L}{\partial t_{k}}=[(L^{\frac{k}{n+1}})_{+},L],\quad
k=\mathbb{Z}_{+}\setminus(n+1)\mathbb{Z}_{+}.$ (1.1)
Here $u^{i}$ are functions of the spatial variable $x$ and the time variables
$t_{1},t_{2},\dots$. This integrable hierarchy has the Hamiltonian structure
$\frac{\partial u^{i}}{\partial t_{k}}=\\{u^{i}(x),H_{k+n+1}\\},$
where the Poisson bracket is defined by
$\\{F,G\\}=\int\mathrm{res}\left(\left[\frac{\delta F}{\delta
L},L\right]\frac{\delta G}{\delta L}\right)\mathrm{d}x$
for local functionals $F$, $G$, and the densities of the Hamiltonians
$H_{k}=\int h_{k}(u,u_{x},\dots)\mathrm{d}x$ can be chosen as
$h_{k}=\frac{n+1}{k}\mathrm{res}\,L^{\frac{k}{n+1}}.$
The advantage of such a choice of the Hamiltonian densities lies in the fact
that they satisfy the tau symmetry condition
$\frac{k}{n+1}\frac{\partial h_{k}}{\partial
t_{l}}=\frac{l}{n+1}\frac{\partial h_{l}}{\partial t_{k}}.$
Due to this property of the densities the tau function of the Gelfand-Dickey
hierarchy can be introduced, as it was done in [4, 10, 14, 31], by the
equations
$\frac{\partial^{2}\log\tau}{\partial x\partial
t_{k}}=\frac{k}{n+1}h_{k},\quad k=\mathbb{Z}_{+}\setminus(n+1)\mathbb{Z}_{+}.$
(1.2)
Note that the Hamiltonians for the general Drinfeld-Sokolov hierarchies are
also given in [6], however the densities given there do not satisfy the tau
symmetry condition. In order to fulfill such a condition these densities
should be modified by adding certain terms which are total $x$-derivatives of
some differential polynomials of the unknown functions.
In the above formalism of the Drinfeld-Sokolov hierarchy associated to the
affine Lie algebra $A_{n}^{(1)}$, the integrable hierarchy and the relation of
its unknown functions with the tau function are relatively explicitly given.
The purpose of the present paper is to give a similar representation for the
Drinfeld-Sokolov hierarchy associated to the affine Lie algebra $D_{n}^{(1)}$
and the vertex $c_{0}$ of the Dynkin diagram. Such a formalism is helpful for
people to have a clear picture of the relation of integrable systems with
Gromov-Witten invariants and topological field models associated to A-D-E
singularities [12, 13, 16, 19, 20, 21, 33]. In fact, Drinfeld and Sokolov
already represented in [6] part of the integrable hierarchy in terms of a
pseudo-differential operator of the form
$L=D^{2n-2}+\sum_{i=1}^{n-1}D^{-1}\left(u^{i}D^{2i-1}+D^{2i-1}u^{i}\right)+D^{-1}\rho
D^{-1}\rho,$ (1.3)
where the functions $u^{1},\dots,u^{n-1},u^{n}=\rho$ serve as the unknown
functions of the hierarchy. The integrable systems of the hierarchy can be
labeled by the elements of a chosen base
$\\{\Lambda^{j}\in\mathfrak{g}^{j},\Gamma^{j}\in\mathfrak{g}^{j(n-1)}\mid j\in
2\mathbb{Z}+1\\}$ (1.4)
of the principal Heisenberg subalgebra of $D_{n}^{(1)}$ (see Sec. 4 for the
definition of these symbols). Denote by $P$ the fractional power
$L^{\frac{1}{2n-2}}$ of $L$ which is a pseudo-differential operator of the
form
$P=D+w_{1}D^{-1}+w_{2}D^{-2}+\dots,$
then the part of the integrable hierarchy that corresponds to the elements
$\Lambda^{j}$ can be represented as [6]
$\frac{\partial L}{\partial t_{k}}=[(P^{k})_{+},L],\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$ (1.5)
The other part that corresponds to the elements $\Gamma^{j}$ can not be
represented in this way by using only the pseudo-differential operators $L,P$.
Inspired by the Lax pair representations of the dispersionless integrable
hierarchy that appear in 2D topological field theory [7, 30], we attempt to
represent the flows corresponding to the elements $\Gamma^{j}$ by the square
root $Q$ of $L$ which takes the form
$Q=D^{-1}\rho+\sum_{k\geq 0}w_{k}D^{k}.$
However, this operator is not a pseudo-differential operator in the usual
sense, because it contains infinitely many terms with positive powers of $D$,
so one cannot compute the square of $Q$. We note that in the dispersionless
case, with $D$ replaced by its symbol $p$, one can define the square of $Q$,
and define the dispersionless hierarchy by using $L,P$ and $Q$.
We are to show in this paper that there exists a new kind of pseudo-
differential operators which are allowed to contain infinitely many terms with
positive power of $D$ such as $Q$, so we can define the square root of the
pseudo-differential operator $L$ in the space of such operators. Then by using
the pseudo-differential operators $L$ and $Q$ we can get the Lax pair
representation of the remaining part of the integrable hierarchy and define
its tau function in a way that one does for the Gelfand-Dickey hierarchy, see
Theorem 4.11. By using this new kind of pseudo-differential operators, we also
find a Lax pair representation of the two-component BKP hierarchy (see [3],
c.f. [29]). We show that the Drinfeld-Sokolov hierarchy of $D_{n}$ type
becomes the $(2n-2,2)$-reduction of the two-component BKP hierarchy [2]. In
this way we also prove that the square root of the tau function satisfies the
Hirota bilinear equations that are constructed in [2, 26] from the principal
vertex operator realization of the basic representation of the affine Lie
algebra $D_{n}^{(1)}$, see (5.2), (5.22) and Theorem 5.2.
In order to obtain the above mentioned results, we first extend, in Section 2,
the usual definition of the ring of pseudo-differential operators. Then in
Section 3 we define a hierarchy of integrable systems and its tau function by
using the pseudo-differential operator $L$ of the form (1.3) and its
fractional powers $P,Q$. In Section 4 we show that the constructed hierarchy
coincides with the Drinfeld-Sokolov hierarchy associated to the affine Lie
algebra $D_{n}^{(1)}$ and the vertex $c_{0}$ of its Dynkin diagram. In Section
5 we give a Lax pair representation of the two-component BKP hierarchy, its
tau function, and its $(2n-2,2)$-reductions. In the final section we give some
concluding remarks.
## 2 Pseudo-differential operators
In this section we generalize the concept of pseudo-differential operators and
list some useful properties of them.
### 2.1 Definitions
Let $\mathcal{A}$ be a commutative ring with unity, and
$D:\mathcal{A}\to\mathcal{A}$ be a derivation. The algebra of pseudo-
differential operators over $\mathcal{A}$ is defined to be
$\mathcal{D}^{-}=\left\\{\sum_{i<\infty}f_{i}D^{i}\mid
f_{i}\in\mathcal{A}\right\\}.$
This is a complete topological ring, whose topological basis is given by the
following filtration
$\cdots\subset\mathcal{D}^{-}_{(d-1)}\subset\mathcal{D}^{-}_{(d)}\subset\mathcal{D}^{-}_{(d+1)}\subset\cdots,\quad\mathcal{D}^{-}_{(d)}=\left\\{\sum_{i\leq
d}f_{i}D^{i}\mid f_{i}\in\mathcal{A}\right\\}.$
The product of two pseudo-differential operators $A=\sum_{i\leq
k}f_{i}D^{i}\in\mathcal{D}^{-}$ and $B=\sum_{j\leq
l}g_{j}D^{j}\in\mathcal{D}^{-}$ is defined by
$A\cdot B=\sum_{i\leq k}\sum_{j\leq l}\sum_{r\geq
0}\binom{i}{r}f_{i}D^{r}(g_{j})D^{i+j-r}\in\mathcal{D}^{-}.$ (2.1)
It is easy to see that for every $s\in\mathbb{Z}$, the coefficient of $D^{s}$
in (2.1) is a finite sum of elements of $\mathcal{A}$, so the above product is
well defined.
In our formalism of the Drinfeld-Sokolov hierarchy of $D_{n}$ type below, one
need not only operators in $\mathcal{D}^{-}$ but also operators in the
following larger abelian group
$\mathcal{D}=\left\\{\sum_{i\in\mathbb{Z}}f_{i}D^{i}\mid
f_{i}\in\mathcal{A}\right\\}.$
However, it is impossible to extend the product (2.1) to $\mathcal{D}$ because
when expanding the product of two elements of $\mathcal{D}$ one meets
summations of infinitely many elements of $\mathcal{A}$, which are not well
defined unless $\mathcal{A}$ possesses certain topology.
Now we assume that on $\mathcal{A}$ there is a gradation
$\mathcal{A}=\prod_{i\geq
0}\mathcal{A}_{i},\quad\mathcal{A}_{i}\cdot\mathcal{A}_{j}\subset\mathcal{A}_{i+j}$
such that $\mathcal{A}$ is topologically complete w.r.t. the induced
decreasing filtration
$\mathcal{A}=\mathcal{A}_{0}\supset\cdots\supset\mathcal{A}_{(d-1)}\supset\mathcal{A}_{(d)}\supset\mathcal{A}_{(d+1)}\supset\cdots,\quad\mathcal{A}_{(d)}=\prod_{i\geq
d}\mathcal{A}_{i}.$
Let $D:\mathcal{A}\to\mathcal{A}$ be a derivation of degree one, i.e.
$D(\mathcal{A}_{i})\subset\mathcal{A}_{i+1}$. An operator
$A\in\mathcal{D}^{-}\subset\mathcal{D}$ is said to be homogeneous if there
exists an integer $k\in\mathbb{Z}$ such that
$A=\sum_{i\leq k}f_{i}D^{i},\quad f_{i}\in\mathcal{A}_{k-i},$
and the integer $k$ is called the degree of $A$. We denote by
$\mathcal{D}_{k}$ the subgroup that consists of all homogeneous pseudo-
differential operators of degree $k$, then the abelian group $\mathcal{D}$ has
the following decomposition
$\mathcal{D}=\prod_{k\in\mathbb{Z}}\mathcal{D}_{k}.$
We introduce the following subgroups of $\mathcal{D}$:
$\mathcal{D}^{+}_{(d)}=\prod_{k\geq
d}\mathcal{D}_{k},\quad\mathcal{D}^{+}=\bigcup_{d\in\mathbb{Z}}\mathcal{D}^{+}_{(d)}.$
It is easy to see that $\mathcal{D}^{+}$ is topologically complete w.r.t. the
filtration
$\cdots\supset\mathcal{D}^{+}_{(d-1)}\supset\mathcal{D}^{+}_{(d)}\supset\mathcal{D}^{+}_{(d+1)}\supset\cdots.$
For any $A\in\mathcal{D}_{k}$ and $B\in\mathcal{D}_{l}$, it is easy to see
that their product defined by (2.1) belongs to $\mathcal{D}_{k+l}$, so we can
extend this product to $\mathcal{D}^{+}$ such that $\mathcal{D}^{+}$ becomes a
ring.
###### Definition 2.1
Elements of $\mathcal{D}^{-}$ (resp. $\mathcal{D}^{+}$) are called pseudo-
differential operators of the first type (resp. the second type) over
$\mathcal{A}$. The intersection of $\mathcal{D}^{-}$ and $\mathcal{D}^{+}$ in
$\mathcal{D}$ is denoted by
$\mathcal{D}^{b}=\mathcal{D}^{-}\cap\mathcal{D}^{+},$
and its elements are called bounded pseudo-differential operators.
Sometimes to indicate the algebra $\mathcal{A}$ and the derivation $D$, we
will use the notations $\mathcal{D}^{\pm}(\mathcal{A},D)$ instead of
$\mathcal{D}^{\pm}$.
The general form of $A\in\mathcal{D}$ reads
$A=\sum_{i\in\mathbb{Z}}\sum_{j\geq 0}a_{i,j}D^{i},\quad
a_{i,j}\in\mathcal{A}_{j}.$ (2.2)
The following lemma is obvious.
###### Lemma 2.2
Suppose $A\in\mathcal{D}$ is given in (2.2), then
* i)
$A\in\mathcal{D}_{k}$ iff the coefficients $a_{i,j}$ are supported on the ray
$\\{(i,j)\mid i+j=k,j\geq 0\\}$;
* ii)
$A\in\mathcal{D}^{+}$ iff there exists $m\in\mathbb{Z}$ such that $a_{i,j}$
are supported on the domain $\\{(i,j)\mid j\geq\max\\{0,m-i\\}\\}$;
* iii)
$A\in\mathcal{D}^{-}$ iff there exists $n\in\mathbb{Z}$ such that $a_{i,j}$
are supported on the domain $\\{(i,j)\mid i\leq n,j\geq 0\\}$.
This lemma has a graphic interpretation as follows:
$\begin{array}[]{ccc}\begin{picture}(16.0,12.0)\put(0.0,0.0){\vector(1,0){14.0}}
\put(7.0,-1.0){\vector(0,1){12.0}} \put(14.0,0.5){$i$} \put(7.5,10.0){$j$}
\put(6.0,-1.3){$0$} \put(10.0,-1.0){$k$}
\matrixput(10,0)(-1,1){7}(0,0){1}{\circle{0.2}}
\put(3.0,7.0){\vector(-1,1){1.5}}
\end{picture}&\begin{picture}(16.0,12.0)\put(0.0,0.0){\vector(1,0){15.0}}
\put(7.0,-1.0){\vector(0,1){12.0}} \put(15.0,0.5){$i$} \put(7.5,10.0){$j$}
\put(6.0,-1.3){$0$} \put(8.0,-1.0){$m$}
\matrixput(9,0)(-1,1){7}(1,0){5}{\circle{0.2}}
\matrixput(13,1)(-1,1){6}(1,0){1}{\circle{0.2}}
\matrixput(12,3)(-1,1){4}(1,0){1}{\circle{0.2}}
\matrixput(11,5)(-1,1){2}(1,0){1}{\circle{0.2}}
\put(2.0,7.0){\vector(-1,1){1.5}} \put(6.0,7.0){\vector(-1,1){1.5}}
\put(12.5,2.0){\vector(1,0){2.0}} \put(11.5,5.0){\vector(1,0){2.0}}
\end{picture}&\begin{picture}(16.0,12.0)\put(0.0,0.0){\vector(1,0){14.0}}
\put(7.0,-1.0){\vector(0,1){12.0}} \put(14.0,0.5){$i$} \put(7.5,10.0){$j$}
\put(6.0,-1.3){$0$} \put(10.0,-1.0){$n$}
\matrixput(10,0)(0,1){7}(-1,0){7}{\circle{0.2}}
\matrixput(3,0)(0,1){6}(-1,0){1}{\circle{0.2}}
\matrixput(2,0)(0,1){4}(-1,0){1}{\circle{0.2}}
\matrixput(1,0)(0,1){2}(-1,0){1}{\circle{0.2}}
\put(10.0,7.0){\vector(0,1){2.0}} \put(6.0,7.0){\vector(0,1){2.0}}
\put(1.5,2.0){\vector(-1,0){2.0}} \put(2.5,5.0){\vector(-1,0){2.0}}
\end{picture}\\\ \\\
\mathrm{(a)}~{}A\in\mathcal{D}_{k}&\mathrm{(b)}~{}A\in\mathcal{D}^{+}&\mathrm{(c)}~{}A\in\mathcal{D}^{-}\end{array}$
From this interpretations it is easy to see the following alternative
expressions of the elements of $A\in\mathcal{D}^{\pm}$.
* i)
If $A\in\mathcal{D}^{+}$, then there exists $m\in\mathbb{Z}$ and
$a_{i,j}\in\mathcal{A}_{j}$ such that $A$ can be written as the following two
forms:
$\displaystyle
A=\sum_{i\in\mathbb{Z}}\left(\sum_{j\geq\max\\{0,m-i\\}}a_{i,j}\right)D^{i},$
(2.3) $\displaystyle A=\sum_{j\geq 0}\left(\sum_{i\geq
m-j}a_{i,j}D^{i}\right).$ (2.4)
* ii)
If $A\in\mathcal{D}^{-}$, then there exists $n\in\mathbb{Z}$ and
$a_{i,j}\in\mathcal{A}_{j}$ such that $A$ can be written as follows:
$\displaystyle A=\sum_{i\leq n}\left(\sum_{j\geq 0}a_{i,j}\right)D^{i},$ (2.5)
$\displaystyle A=\sum_{j\geq 0}\left(\sum_{i\leq n}a_{i,j}D^{i}\right).$ (2.6)
We call the expressions (2.3) and (2.5) the _normal expansion_ of $A$, while
the expressions (2.4) and (2.6) the _dispersion expansion_ of $A$.
Properties of pseudo-differential operators of the first type are well known.
Similar to the operators in $\mathcal{D}^{-}$, we can define the adjoint
operator, the residue, the positive part and the negative part of a pseudo-
differential operator of the second type. Let $A\in\mathcal{D}^{+}$ be given
by (2.2), then
$\displaystyle A^{*}=\sum_{i\in\mathbb{Z}}\sum_{j\geq 0}(-1)^{i}D^{i}\cdot
a_{i,j},\quad\mathrm{res}\,A=\sum_{j\geq 0}a_{-1,j},$ $\displaystyle
A_{+}=\sum_{i\geq 0}\sum_{j\geq 0}a_{i,j}D^{i},\quad
A_{-}=\sum_{i<0}\sum_{j\geq 0}a_{i,j}D^{i}.$
It is easy to see that $A^{*},A_{+},A_{-}\in\mathcal{D}^{+}$ and
$\mathrm{res}\,A\in\mathcal{A}$. In particular, if $A\in\mathcal{D}^{\pm}$,
then $A_{\mp}\in\mathcal{D}^{b}$.
An operator $A\in\mathcal{D}^{\pm}$ is called a _differential operator_ if its
negative part $A_{-}$ vanishes. Note that every differential operator in
$\mathcal{D}^{-}$ is of finite order, while the ones in $\mathcal{D}^{+}$ may
be not. The differential operators in $\mathcal{D}^{\pm}$ form subrings of
$\mathcal{D}^{\pm}$ respectively, and they can act on $\mathcal{A}$ in the
obvious way. Given a differential operator $A\in\mathcal{D}^{\pm}$, we denote
by $A(f)$ the action of $A$ on $f\in\mathcal{A}$.
Let us introduce some other notations to be used latter. Elements of the
quotient space $\mathcal{F}=\mathcal{A}/D(\mathcal{A})$ are called _local
functionals_ , and they are represented in the form
$\int\\!\\!f\mathrm{d}x=f+D(\mathcal{A}),\quad f\in\mathcal{A}.$
Introduce the map
$\langle\,\,\rangle:\ \mathcal{D}\to\mathcal{F},\quad A\mapsto\langle
A\rangle=\int\mathrm{res}A\,\mathrm{d}x.$
We then define the pairing
$\langle A,B\rangle=\langle AB\rangle$ (2.7)
on each of the following four spaces:
$\mathcal{D}^{+}\times\mathcal{D}^{+},\quad\mathcal{D}^{-}\times\mathcal{D}^{-},\quad\mathcal{D}^{b}\times\mathcal{D},\quad\mathcal{D}\times\mathcal{D}^{b}.$
It is easy to see that this pairing is symmetric and is nondegenerate on each
of the above spaces.
### 2.2 Properties of pseudo-differential operators
Now we present some useful properties of pseudo-differential operators.
###### Lemma 2.3
Let $A,B\in\mathcal{D}^{\pm}$. If the commutator $[A^{m},B]=0$ for some
positive integer $m$, then $[A,B]=0$.
Proof The $\mathcal{D}^{-}$ case is well known, we only prove the
$\mathcal{D}^{+}$ case. Suppose $C=[A,B]\neq 0$. We take the dispersion
expansions
$A=\sum_{j\geq a}\sum_{i\geq k_{j}}A_{i,j}D^{i},\quad C=\sum_{j\geq
c}\sum_{i\geq l_{j}}C_{i,j}D^{i},$
such that neither $A_{k_{a},a}$ nor $C_{l_{c},c}$ vanishes, then the
coefficient of $D^{(m-1)k_{a}+l_{c}}$ in
$[A^{m},B]=[A,B]A^{m-1}+A[A,B]A^{m-2}+\cdots+A^{m-1}[A,B]$
reads
$mA_{k_{a},a}^{m-1}C_{l_{c},c}+\cdots,$
where $\cdots$ denote the terms with higher degrees in $\mathcal{A}$. This
contradicts with $[A^{m},B]=0$. The lemma is proved. $\Box$
Let $\rho\in\mathcal{A}$ be an invertible element, we consider the operator
$Q=D^{-1}\rho+Q_{+}\in\mathcal{D}^{+},$ (2.8)
where $Q_{+}$ is a differential operator in $\mathcal{D}^{+}$. Such an
operator $Q$ is invertible, whose inverse reads
$\displaystyle Q^{-1}=$
$\displaystyle\left(D^{-1}\rho(1+\rho^{-1}DQ_{+})\right)^{-1}$
$\displaystyle=$
$\displaystyle\left(1-\rho^{-1}DQ_{+}+\rho^{-1}DQ_{+}\rho^{-1}DQ_{+}-\cdots\right)\rho^{-1}D.$
(2.9)
Note that $Q^{-1}$ is a differential operator in $\mathcal{D}^{+}$.
###### Lemma 2.4
Let $Q\in\mathcal{D}^{+}$ be given in (2.8), then $D$ can be uniquely
expressed as the following form
$D=\sum_{i\geq 1}h_{i}Q^{-i},\,\,h_{i}\in\mathcal{A}.$ (2.10)
Moreover, $m\,h_{m}-\mathrm{res}\,Q^{m}\in D(\mathcal{A})$ for every $m\geq
1$.
Proof The first assertion follows from a simple induction. We are going to
prove the second one by using the following fact
$\mathrm{res}\,Q^{m}=(DQ^{m})_{+}-D(Q^{m})_{+}.$
The first assertion shows that
$(DQ^{m})_{+}=\left(\sum_{i\geq 1}h_{i}Q^{m-i}\right)_{+}=\sum_{i\geq
1}h_{i}\left(Q^{m-i}\right)_{+}.$
We assume $(Q^{m})_{+}=\sum_{i\geq 0}a_{m,i}Q^{-i}$ with
$a_{m,i}\in\mathcal{A}$, then
$\displaystyle D(Q^{m})_{+}=$ $\displaystyle\sum_{i\geq
0}a_{m,i}^{\prime}Q^{-i}+\sum_{i\geq 0}a_{m,i}\sum_{j\geq 1}h_{j}Q^{-i-j}$
$\displaystyle=$ $\displaystyle\sum_{i\geq
0}a_{m,i}^{\prime}Q^{-i}+\sum_{j\geq 1}h_{j}(Q^{m})_{+}Q^{-j},$
where $a_{m,i}^{\prime}=D(a_{m,i})$.
By using the above three formulae, one can obtain
$\sum_{m\geq 1}(\mathrm{res}\,Q^{m})Q^{-m}=\sum_{i\geq
1}\sum_{m=1-i}^{0}h_{i}(Q^{m})_{+}Q^{-m-i}-\sum_{m\geq 1}\sum_{i\geq
0}a_{m,i}^{\prime}Q^{-i-m}.$
Note that $(Q^{m})_{+}=Q^{m}$ when $m\leq 0$, so by comparing the coefficients
of $Q^{-m}$ we have
$m\,h_{m}-\mathrm{res}\,Q^{m}=\sum_{i=0}^{m-1}a_{m-i,i}^{\prime}.$
The lemma is proved. $\Box$
###### Lemma 2.5
Let $A$ be a pseudo-differential operator in $\mathcal{D}^{+}$, and
$\rho\in\mathcal{A}$ be an invertible element. Then there exists a unique
pseudo-differential operator $B\in\mathcal{D}^{+}$ such that $A=\rho
BD+DB\rho$. Furthermore, if $A^{*}=\pm A$, then $B^{*}=\mp B$.
Proof Without loss of generality, we can assume $A$ to be homogeneous, i.e.,
$A=\sum_{i\leq k}a_{i}D^{i}$, $a_{i}\in\mathcal{A}_{k-i}$. Suppose
$B=\sum_{i\leq k-1}b_{i}D^{i}$, then one can determine $b_{k-1},b_{k-2},\dots$
recursively by $A=\rho BD+DB\rho$. So we derive the first part of the lemma.
If $A^{*}=\pm A$, then
$\rho(B^{*}\pm B)D+D(B^{*}\pm B)\rho=0,$
hence $B^{*}\pm B=0$ due to the uniqueness in the first part. The lemma is
proved. $\Box$
## 3 An integrable hierarchy represented by pseudo-differential operators
In this section we are to construct a hierarchy of evolutionary partial
differential equations starting from a pseudo-differential operator $L$. This
hierarchy possesses a bihamiltonian structure which coincides with that of the
Drinfeld-Sokolov hierarchy of $D_{n}$ type, moreover, it possesses a tau
function.
### 3.1 Construction of the hierarchy
Let $M$ be an open ball of dimension $n$ with coordinates
$(u^{1},u^{2},\dots,u^{n})$. We define the algebra $\mathcal{A}$ of
differential polynomials on $M$ to be
$\mathcal{A}=C^{\infty}(M)[[u^{i,s}\mid i=1,\dots,n,\ s=1,2,\dots]].$
There is a gradation on $\mathcal{A}$ defined by
$\deg f=0\mbox{ for }f\in C^{\infty}(M),\quad\deg u^{i,s}=s,$
then it is easy to see that $\mathcal{A}$ is topologically complete. We
introduce a derivation $D$ of degree one over $\mathcal{A}$ as follows
$D:\mathcal{A}\to\mathcal{A},\quad D=\sum_{s\geq
0}\sum_{i=1}^{n}u^{i,s+1}\frac{\partial}{\partial u^{i,s}},$
where $u^{i,0}=u^{i}$. Now let us construct the algebras $\mathcal{D}^{\pm}$
starting from $\mathcal{A}$ and $D$ as we did in the last section.
Let $L$ be the following pseudo-differential operator given in (1.3).
Obviously $L$ belongs to $\mathcal{D}^{b}=\mathcal{D}^{-}\cap\mathcal{D}^{+}$
and satisfies $L^{*}=DLD^{-1}$. Here we re-denote the coordinate $u^{n}$ by
$\rho$, and will use this notation frequently in what follows.
Firstly, we regard $L$ as an element of $\mathcal{D}^{-}$, then by using
properties of the usual pseudo-differential operators we have the following
lemma.
###### Lemma 3.1
There exists a unique pseudo-differential operator $P\in\mathcal{D}^{-}$ of
the form
$P=D+u_{1}D^{-1}+u_{2}D^{-2}+\cdots$ (3.1)
such that $P^{2n-2}=L$. Moreover, the operator $P$ satisfies $[P,L]=0$ and
$P^{*}=-DPD^{-1}.$ (3.2)
In [4], Date, Jimbo, Kashiwara and Miwa proved the following lemma.
###### Lemma 3.2 ([4])
The constraint (3.2) to an operator $P$ of the form (3.1) is equivalent to the
condition that for every $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$ the free term of
$(P^{k})_{+}$ vanishes, i.e. $(P^{k})_{+}(1)=0$.
The above two lemmas imply that the following equations
$\frac{\partial L}{\partial t_{k}}=[(P^{k})_{+},L],\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}$ (3.3)
are well defined, and they give evolutionary partial differential equations of
$u^{1},\dots u^{n}$. In particular, $D=\frac{\mathrm{d}}{\mathrm{d}x}$ with
$x=t_{1}$, and by taking residue of $D\left(\frac{\partial L}{\partial
t_{k}}-[(P^{k})_{+},L]\right)$ one has
$\frac{\partial\rho}{\partial t_{k}}=-(P^{k})_{+}^{*}(\rho).$ (3.4)
The flows in (3.3) first appeared in [6] as part of the Drinfeld-Sokolov
hierarchy of $D_{n}$ type.
Note that the Drinfeld-Sokolov hierarchy of $D_{n}$ type contains $n$ series
of commuting flows, but there are only $n-1$ series of flows given in (3.3),
so in this sense the equations (3.3) do not form a complete integrable
hierarchy. One main result in the present paper is that the $n$th series of
flows of the Drinfeld-Sokolov hierarchy of $D_{n}$ type can be represented by
the square root of $L$ regarded as an element of $\mathcal{D}^{+}$.
###### Lemma 3.3
There exists a unique pseudo-differential operator $Q\in\mathcal{D}^{+}$ of
the following form
$Q=D^{-1}\rho+\sum_{m\geq 0}Q_{m}\,D$ (3.5)
such that $Q^{2}=L$. Here $Q_{m}$ are homogeneous differential operators in
$\mathcal{D}^{b}$ with degree $2\,m$, and satisfy $Q_{m}^{*}=Q_{m}$. Moreover,
the operator $Q$ satisfies
$\displaystyle Q^{*}=-DQD^{-1},$ (3.6) $\displaystyle-$ $\displaystyle
Q^{*}_{+}(\rho)=\frac{1}{2}DL_{+}(1).$ (3.7)
Proof By substituting (1.3) and (3.5) into $DQ^{2}=DL$ and comparing the
homogeneous terms, we can obtain
$\rho Q_{m}D+DQ_{m}\rho=A_{m},\quad m=0,1,2,\dots.$ (3.8)
Here $A_{m}$ are differential operators depending on
$L,Q_{0},Q_{1},\dots,Q_{m-1}$ and satisfy $A_{m}+A_{m}^{*}=0$. Then according
to Lemma 2.5, $Q_{m}$ can be determined by induction, and they satisfy
$Q_{m}^{*}=Q_{m}$.
The symmetry property (3.6) is trivial. To show (3.7), we consider the free
terms on both hand sides of (3.8):
$DQ_{m}(\rho)=\left\\{\begin{array}[]{cl}u^{m+1,2m+1},&m=0,1,\ldots,n-2,\\\
0,&m\geq n-1.\end{array}\right.$
Hence
$-Q^{*}_{+}(\rho)=\sum_{m\geq
0}DQ_{m}(\rho)=\sum_{m=0}^{n-2}u^{m+1,2m+1}=\frac{1}{2}DL_{+}(1).$
The lemma is proved. $\Box$
According to Lemmas 2.3 and 3.3, the following evolutionary equations are well
defined:
$\frac{\partial L}{\partial\hat{t}_{k}}=[-(Q^{k})_{-},L]=[(Q^{k})_{+},L],\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$ (3.9)
In particular, we have
$\frac{\partial\rho}{\partial\hat{t}_{k}}=-(Q^{k})^{*}_{+}(\rho),~{}~{}k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$
(3.10)
Whe $k=1$ we obtain
${\partial\rho}/{\partial\hat{t}_{1}}=\frac{1}{2}DL_{+}(1)$, this flow is
linearly independent with ${\partial\rho}/{\partial t_{2i-1}}$ ($1\leq i\leq
n-1$), so from the bihamiltonian recursion relation (see below) we see that
the equations given in (3.3) are linearly independent with that defined in
(3.9).
###### Theorem 3.4
The flows in (3.3), (3.9) commute with each other.
Proof The commutativity of these flows follows from the following equivalent
representations of (3.3), (3.9):
$\displaystyle\frac{\partial
P}{\partial{t}_{k}}=[(P^{k})_{+},P],\quad\frac{\partial
P}{\partial\hat{t}_{k}}=[-(Q^{k})_{-},P],$ (3.11) $\displaystyle\frac{\partial
Q}{\partial t_{k}}=[(P^{k})_{+},Q],\quad\frac{\partial
Q}{\partial\hat{t}_{k}}=[-(Q^{k})_{-},Q],$ (3.12)
which can be verified as Lemma 2.3. The theorem is proved. $\Box$
The dispersionless limit of the flows $\frac{\partial}{\partial\hat{t}_{k}}$
was first given by Takasaki in [30], but the dispersionful one was not given
there. Following [30], we call the flows (3.3) and (3.9) the _positive_ and
the _negative_ flows respectively. The above theorem shows that the negative
and the positive flows form an integrable hierarchy. We will show that it is
equivalent to the Drinfeld-Sokolov hierarchy of $D_{n}$ type.
### 3.2 Bihamiltonian structure and tau structure
In this subsection we show that the hierarchy (3.3), (3.9) carries a
bihamiltonian structure, and the densities of the Hamiltonians can be chosen
to satisfy the tau symmetry condition. We then define the tau function of the
hierarchy by using this tau symmetry following the approach of [10].
Let $\mathcal{L}=DL$, it has the form
$\mathcal{L}=D^{2n-1}+\sum_{i=1}^{n-1}\left(u^{i}D^{2i-1}+D^{2i-1}u^{i}\right)+\rho
D^{-1}\rho.$ (3.13)
Given a local functional $F=\int f\,\mathrm{d}x\in\mathcal{A}/D(\mathcal{A})$,
we define its variational derivative w.r.t. $\mathcal{L}$ to be an element
$X={\delta F}/{\delta\mathcal{L}}\in\mathcal{D}$ such that
$\delta F=\langle X,\delta\mathcal{L}\rangle,\quad X=X^{*}.$ (3.14)
The existence of such an element can be verified by taking
$X=\frac{1}{2}\sum_{i=0}^{n-1}\left(D^{-2i}\frac{\delta F}{\delta
v^{i}(x)}+\frac{\delta F}{\delta v^{i}(x)}D^{-2i}\right).$ (3.15)
where $v^{0}=\rho^{2}$ and $v^{1},\dots,v^{n-1}$ are determined by
representing the operator $\mathcal{L}$ in the following form
$\mathcal{L}=D^{2n-1}+\sum_{i=1}^{n-1}v^{i}D^{2i-1}+\sum_{i=1}^{n-1}\tilde{v}^{i}D^{2i-2}+\rho
D^{-1}\rho.$
Note that the new coordinates $v^{1},\dots,v^{n-1}$ are related to
$u^{1},\dots,u^{n-1}$ by a Miura-type transformation, and the functions
$\tilde{v}^{i}$ determined by the condition $\mathcal{L}+\mathcal{L}^{*}=0$
are linear functions of the derivatives of $v^{1},\dots,v^{n-1}$.
On the other hand, the variational derivative $X$ defined in (3.14) is
determined up to the addition of a kernel part $Z$ that satisfies
$Z_{+}(\rho)=0,\quad Z_{-}=\sum_{i\leq
n}\left(w_{i}D^{-2i}+D^{-2i}w_{i}\right),~{}~{}w_{i}\in\mathcal{A}.$
The following compatible Poisson brackets are given in Proposition 8.3 of [6]
(see also [9]) for the bihamiltonian structure of the Drinfeld-Sokolov
hierarchy of $D_{n}$ type:
$\displaystyle\\{F,G\\}_{1}(\mathcal{L})$ $\displaystyle=\langle
X,(DY_{+}\mathcal{L})_{-}-(\mathcal{L}Y_{+}D)_{-}+(\mathcal{L}Y_{-}D)_{+}-(DY_{-}\mathcal{L})_{+}\rangle,$
(3.16) $\displaystyle\\{F,G\\}_{2}(\mathcal{L})$ $\displaystyle=\langle
X,(\mathcal{L}Y)_{+}\mathcal{L}-\mathcal{L}(Y\mathcal{L})_{+}\rangle,$ (3.17)
where $F$ and $G$ are two arbitrary local functionals, and
$X=\frac{\delta F}{\delta\mathcal{L}},\quad Y=\frac{\delta
G}{\delta\mathcal{L}}.$
Note that in the above formulae of the Poisson brackets the second component
in the pairing $\langle\,,\,\rangle$ belongs to $\mathcal{D}^{b}$ for any
$Y\in\mathcal{D}$, so from the definition of $\langle\,,\,\rangle$ given in
(2.7) we see that the first component $X$ is not restricted to the space
$\mathcal{D}^{+}$ or $\mathcal{D}^{-}$. One can show by a direct computation
that the definition of these Poisson brackets is independent of the choice of
the kernel parts of $X$ and $Y$, so they are well defined.
###### Theorem 3.5
The hierarchy (3.3), (3.9) has the following bihamiltonian representation:
$\displaystyle\frac{\partial F}{\partial
t_{k}}=\\{F,H_{k+2n-2}\\}_{1}=\\{F,H_{k}\\}_{2},$ (3.18)
$\displaystyle\frac{\partial
F}{\partial\hat{t}_{k}}=\\{F,\hat{H}_{k+2}\\}_{1}=\\{F,\hat{H}_{k}\\}_{2}.$
(3.19)
Here $F\in\mathcal{F}$ is any local functional, and the Hamiltonians are given
by
$H_{k}=\frac{2n-2}{k}\langle P^{k}\rangle,~{}~{}\hat{H}_{k}=\frac{2}{k}\langle
Q^{k}\rangle,\quad k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$ (3.20)
Proof Let us start with the computation of the variational derivatives of the
Hamiltonians $H_{k}$. By using the identity $P^{2n-2}=L$ (see Lemma 3.1) and
the symmetric property of the pairing $\langle\,,\,\rangle$ we have
$\displaystyle\delta H_{k}$ $\displaystyle=(2n-2)\langle P^{k-1},\delta
P\rangle=(2n-2)\langle P^{k-2n+2},P^{2n-3}\delta P\rangle$
$\displaystyle=\langle P^{k-2n+2},\delta L\rangle=\langle
P^{k-2n+2}D^{-1},\delta\mathcal{L}\rangle=\langle
Y_{k},\delta\mathcal{L}\rangle,$ (3.21)
where $Y_{k}=P^{k-2n+2}D^{-1}\in\mathcal{D}$. From (3.2) it follows that
$Y_{k}^{*}=Y_{k}$, so we can take
$\frac{\delta H_{k}}{\delta\mathcal{L}}=Y_{k}=P^{k-2n+2}D^{-1}.$ (3.22)
To show (3.18), we first note due to Lemma 3.2 the validity of
$\displaystyle D(P^{k})_{+}D^{-1}=D(P^{k}D^{-1})_{+}=(DP^{k}D^{-1})_{+},$
$\displaystyle(P^{k}D^{-1}D)_{-}=(P^{k}D^{-1})_{-}D$ (3.23)
for any $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$. So from (3.3) we have
$\displaystyle\frac{\partial\mathcal{L}}{\partial t_{k}}$
$\displaystyle=D(P^{k})_{+}L-DL(P^{k})_{+}=(DP^{k}D^{-1})_{+}\mathcal{L}-\mathcal{L}(P^{k})_{+}$
$\displaystyle=(\mathcal{L}Y_{k})_{+}\mathcal{L}-\mathcal{L}(Y_{k}\mathcal{L})_{+}.$
On the other hand, by using the commutativity between $L$ and $P$ (see Lemma
3.1) we can also represent $\frac{\partial\mathcal{L}}{\partial t_{k}}$ in the
following form:
$\displaystyle\frac{\partial\mathcal{L}}{\partial t_{k}}$
$\displaystyle=D(P^{k})_{+}L-DL(P^{k})_{+}$
$\displaystyle=\left(D(P^{k})_{+}L-DL(P^{k})_{+}\right)_{+}+\left(D(P^{k})_{+}L-DL(P^{k})_{+}\right)_{-}$
$\displaystyle=\left(-D(P^{k})_{-}L+DL(P^{k})_{-}\right)_{+}+\left(D(P^{k})_{+}L-DL(P^{k})_{+}\right)_{-}$
$\displaystyle=\left(\mathcal{L}(Y_{k+2n-2})_{-}D-D(Y_{k+2n-2})_{-}\mathcal{L}\right)_{+}$
$\displaystyle\quad+\left(D(Y_{k+2n-2})_{+}\mathcal{L}-\mathcal{L}(Y_{k+2n-2})_{+}D\right)_{-}$
Now the equivalence of the flows (3.3) with (3.18) follows from the above
identities together with the relation
$\frac{\partial F}{\partial t_{k}}=\left\langle\frac{\delta
F}{\delta\mathcal{L}},\frac{\partial\mathcal{L}}{\partial
t_{k}}\right\rangle.$
By using the property (3.6) of the operator $Q$ we know that for any
$k\in\mathbb{Z^{\mathrm{odd}}_{+}}$ the free term of $Q^{k}$ vanishes, then a
similar argument as above leads to the equivalence of the flows (3.9) with
(3.19). The theorem is proved. $\Box$
By using the formula (3.22) and
$\frac{\delta\hat{H}_{k}}{\delta\mathcal{L}}=Q^{k-2}D^{-1},$
we obtain the following proposition.
###### Proposition 3.6
The local functionals $H_{1},H_{3},\ldots,H_{2n-3}$ and $\hat{H}_{1}$ are
linearly independent Casimirs of the first Poisson bracket $\\{\,,\\}_{1}$.
We now verify that the above defined densities of the Hamiltonians satisfy the
tau symmetry condition, and we can thus define the tau function for the
integrable hierarchy (3.3), (3.9). To this end let us introduce a series of
rescaled time variables
$T^{\alpha,p}=\left\\{\begin{array}[]{cl}\dfrac{(2n-2)\Gamma(p+1+\frac{2\alpha-1}{2n-2})}{\Gamma(\frac{2\alpha-1}{2n-2})}t_{(2n-2)p+2\alpha-1},&\alpha=1,\dots,n-1,\\\
\\\
\dfrac{2\Gamma(p+1+\frac{1}{2})}{\Gamma(\frac{1}{2})}\hat{t}_{2p+1},&\alpha=n\end{array}\right.$
with $p=0,1,2,\dots$. Then the Hamiltonian equations (3.18), (3.19) read
$\frac{\partial F}{\partial
T^{\alpha,p}}=\\{F,H_{\alpha,p}\\}_{1}=\left(p+\frac{1}{2}+\mu_{\alpha}\right)^{-1}\\{F,H_{\alpha,p-1}\\}_{2},$
where the densities of the Hamiltonians $H_{\alpha,p}$ are given by
$h_{\alpha,p-1}=\left\\{\begin{array}[]{cl}\dfrac{\Gamma(\frac{2\alpha-1}{2n-2})}{(2n-2)\,\Gamma(p+1+\frac{2\alpha-1}{2n-2})}\mathrm{res}\,P^{(2n-2)p+2\alpha-1},&\alpha=1,\dots,n-1,\\\
\\\
\dfrac{\Gamma(\frac{1}{2})}{2\,\Gamma(p+1+\frac{1}{2})}\mathrm{res}\,Q^{2p+1},&\alpha=n,\end{array}\right.$
and the constants $\mu_{\alpha}$ are the spectrum of the underlying Frobenius
manifold [7, 9], read
$\mu_{\alpha}=\left\\{\begin{array}[]{cl}\dfrac{2\alpha-n}{2n-2},&\alpha=1,\dots,n-1,\\\
0,&\alpha=n.\end{array}\right.$
Then we have tau symmetry
$\frac{\partial h_{\alpha,p-1}}{\partial T^{\beta,q}}=\frac{\partial
h_{\beta,q-1}}{\partial T^{\alpha,p}},$
and the differential polynomials
$\Omega_{\alpha,p;\beta,q}=\partial_{x}^{-1}\frac{\partial
h_{\alpha,p-1}}{\partial
T^{\beta,q}},\quad\alpha,\beta=1,2,\dots,n;~{}~{}p,q\geq 0.$
have the property
$\Omega_{\alpha,p;\beta,q}=\Omega_{\beta,q;\alpha,p}.$
Hence the chosen $h_{\alpha,p}$ give a tau structure, in the sense of [10], of
the bihamiltonian structure of the integrable hierarchy (3.3), (3.9). This tau
structure defines the tau function $\hat{\tau}$ of the integrable hierarchy by
$\frac{\partial^{2}\log\hat{\tau}}{\partial T^{\alpha,p}\partial
T^{\beta,q}}=\Omega_{\alpha,p;\beta,q}.$ (3.24)
## 4 Drinfeld-Sokolov hierarchies and pseudo-differential operators
In this section we first recall some facts about the Drinfeld-Sokolov
hierarchies associated to untwisted affine Lie algebras, see details in [6].
Then we consider the Drinfeld-Sokolov hierarchy of $D_{n}$ type and identify
it with the hierarchy (3.3), (3.9) constructed in the last section.
### 4.1 Definition of the Drinfeld-Sokolov hierarchies
Let $\mathfrak{g}$ be an untwisted affine Lie algebra, and
$\\{e_{i},f_{i},h_{i}\mid i=0,1,2,\ldots,n\\}$ be a set of Weyl generators of
$\mathfrak{g}$. In Drinfeld and Sokolov’s construction, the central element
$c$ is not used, so we always assume $c=0$. We need to use the following two
gradations on $\mathfrak{g}$ [6, 25]:
* i)
the principal/canonical gradation
$\mathfrak{g}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}^{j},\quad\deg{{e}_{i}}=-\deg{{f}_{i}}=1,\quad
i=0,1,\dots,n;$
* ii)
the homogeneous/standard gradation
$\mathfrak{g}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j},\quad\deg{{e}_{i}}=-\deg{{f}_{i}}=\delta_{i0},\quad
i=0,1,\dots,n.$
We will use notations such as $\mathfrak{g}^{<0}=\sum_{i<0}\mathfrak{g}^{i}$
below.
In [6] Drinfeld and Sokolov assigned a standard gradation to any chosen vertex
$c_{i}$ of the Dynkin diagram of $\mathfrak{g}$ and used the standard
gradation to construct an integrable hierarchy. As mentioned in the beginning
of the present paper, we only consider the case that the vertex is chosen to
be $c_{0}$ which is the special one added to the Dynkin diagram of the
corresponding simple Lie algebra. Integrable hierarchies that associated to
different choices of the vertices are related by Miura type transformations.
Denote by $E$ (resp. $E_{+}$) the set of exponents (resp. positive exponents)
of $\mathfrak{g}$. Let $\mathfrak{s}$ be the Heisenberg subalgebra associated
to the principal gradation, which is defined to be the centralizer of
$\Lambda=\sum_{i=0}^{n}e_{i}$. One can fix a basis
$\lambda_{j}\in\mathfrak{g}^{j}\ (j\in E)$ of $\mathfrak{s}$.
Let $C^{\infty}(\mathbb{R},W)$ be the set of smooth functions from
$\mathbb{R}$ to a linear space $W$. We consider operators of the form
$\mathscr{L}=D+\Lambda+q,\quad q\in
C^{\infty}(\mathbb{R},\mathfrak{g}_{0}\cap\mathfrak{g}^{\leq 0}),$ (4.1)
where $D=\frac{\mathrm{d}}{\mathrm{d}x}$, and $x$ is the coordinate on
$\mathbb{R}$.
###### Proposition 4.1 ([6])
There exists an element $U\in C^{\infty}(\mathbb{R},\mathfrak{g}^{<0})$ such
that the operator $\mathscr{L}_{0}=e^{-\mathrm{ad}_{U}}\mathscr{L}$ has the
form
$\mathscr{L}_{0}=D+\Lambda+H,\quad H\in
C^{\infty}(\mathbb{R},\mathfrak{s}\cap\mathfrak{g}^{<0}),$ (4.2)
and for different choices of $U$, the map $H$ differs by the addition of the
total derivative of a differential polynomial of $q$.
We fix a $U$ as given in the above proposition, and introduce a map
$\varphi:C^{\infty}(\mathbb{R},\mathfrak{g})\to
C^{\infty}(\mathbb{R},\mathfrak{g}),\quad A\mapsto e^{\mathrm{ad}_{U}}A.$
(4.3)
The Drinfeld-Sokolov hierarchy is a hierarchy of partial differential
equations of gauge equivalence classes of $\mathscr{L}$ defined by
$\frac{\partial\mathscr{L}}{\partial
t_{j}}=[\varphi(\lambda_{j})^{+},\mathscr{L}],\quad j\in E_{+}.$ (4.4)
Here $\varphi(\lambda_{j})^{+}$ stands for the projection of
$\varphi(\lambda_{j})$ onto $C^{\infty}(\mathbb{R},\mathfrak{g}^{>0})$, and
the gauge transformations of $\mathscr{L}$ read
$\mathscr{L}\mapsto e^{\mathrm{ad}_{N}}\mathscr{L},\quad N\in
C^{\infty}(\mathbb{R},\mathfrak{g}_{0}\cap\mathfrak{g}^{<0}).$ (4.5)
###### Theorem 4.2 ([6])
The Drinfeld-Sokolov hierarchy carries a bihamiltonian structure, and the
Hamiltonian densities are given by the expansion coefficients of the map $H$
(4.2) in the basis $\\{\lambda_{-j}\mid j\in E_{+}\\}$.
For the classical untwisted affine Lie algebras, Drinfeld and Sokolov proposed
a way to represent their hierarchies via certain scalar pseudo-differential
operators over $\mathcal{A}$, the algebra of gauge invariant differential
polynomials of $q$ in (4.1). They gave such representations for the full
hierarchies of the $A_{n}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$ types by using
pseudo-differential operators of the first type. However, for the
$D_{n}^{(1)}$ case, as pointed out by Drinfeld and Sokolov, the pseudo-
differential operators in $\mathcal{D}^{-}$ are not enough to represent the
full hierarchy. Our purpose of introducing the space $\mathcal{D}^{+}$ in the
present paper is to represent the full Drinfeld-Sokolov hierarchy of $D_{n}$
type in terms of scalar pseudo-differential operators.
The following lemma tells how to construct scalar pseudo-differential
operators from the operator $\mathscr{L}$.
###### Lemma 4.3 ([6])
Let $\mathcal{R}$ be a ring with unity. We consider matrices of the form
$R=\left(\begin{array}[]{cc}\alpha^{t}&a\\\ R_{1}&\beta\\\
\end{array}\right)\in\mathcal{R}^{m\times m},$
in which the block $R_{1}\in\mathcal{R}^{(m-1)\times(m-1)}$ is invertible,
$\alpha$, $\beta$ are $(m-1)$-dimensional column vectors, and the superscript
$t$ means the transpose of matrices. Define
$\Delta(R)=a-\alpha^{t}R_{1}^{-1}\beta$, then the following statements are
true.
* i)
Suppose $x_{1},x_{2},\ldots,x_{m},y$ belong to some $\mathcal{R}$-module such
that
$R\cdot(x_{1},x_{2},\ldots,x_{m})^{t}=(y,0,\ldots,0)^{t},$
then $\Delta(R)\cdot x_{m}=y.$
* ii)
For any upper triangular matrix $\tilde{N}\in\mathcal{R}^{m\times m}$ with
unity on the main diagonal one has
$\Delta(\tilde{N}R\tilde{N}^{-1})=\Delta(R)$.
* iii)
Given an anti-isomorphism $*$ of $\mathcal{R}$, one can define an anti-
isomorphism $T$ of $\mathcal{R}^{m\times m}$ by
$(R^{T})_{ij}=R_{m+1-j,m+1-i}^{*}$. It satisfies
$\Delta(R^{T})=\Delta(R)^{*}$.
### 4.2 Positive flows of the Drinfeld-Sokolov hierarchy of $D_{n}$ type
In this subsection, we recall the approach given in [6] that represents part
of the Drinfeld-Sokolov hierarchy of $D_{n}$ type as the positive flows (3.3)
by using pseudo-diferential operators.
We first recall the matrix realization of the affine Lie algebra
$\mathfrak{g}$ of $D_{n}^{(1)}$ type [25, 6]. Denote by $e_{i,j}$ the
$2n\times 2n$ matrix that takes value $1$ at the $(i,j)$-entry and zero
elsewhere, then one can realize $\mathfrak{g}$ by choosing the Weyl generators
as follows:
$\displaystyle
e_{0}=\frac{\lambda}{2}(e_{1,2n-1}+e_{2,2n}),~{}e_{n}=\frac{1}{2}(e_{n+1,n-1}+e_{n+2,n}),$
(4.6) $\displaystyle e_{i}=e_{i+1,i}+e_{2n+1-i,2n-i}~{}(1\leq i\leq n-1),$
(4.7) $\displaystyle
f_{0}=\frac{2}{\lambda}(e_{2n-1,1}+e_{2n,2}),~{}f_{n}={2}(e_{n-1,n+1}+e_{n,n+2}),$
(4.8) $\displaystyle f_{i}=e_{i,i+1}+e_{2n-i,2n+1-i}~{}(1\leq i\leq n-1),$
(4.9) $\displaystyle h_{i}=[e_{i},f_{i}]~{}(0\leq i\leq n).$ (4.10)
In particular, the associated simple Lie algebra $\mathfrak{g}_{0}$ of $D_{n}$
type is realized as
$\mathfrak{g}_{0}=\Big{\\{}A\in\mathbb{C}^{2n\times 2n}\mid
A=-SA^{T}S^{-1}\Big{\\}},$ (4.11)
where $S$ is the following matrix
$S=\sum_{i=1}^{n}(-1)^{i-1}(e_{i,i}+e_{2n+1-i,2n+1-i}),$
and $A^{T}=(a_{l+1-j,k+1-i})$ for any $k\times l$ matrix $A=(a_{ij})$. Note
that in this realization the algebra $\mathfrak{g}$ is just
$\mathfrak{g}_{0}\otimes\mathbb{C}[\lambda,\lambda^{-1}]$.
The set of exponents of $\mathfrak{g}$ is given by
$E=\\{1,3,5,\ldots,2n-3\\}\cup\\{(n-1)^{\prime}\\}+(2n-2)\mathbb{Z},$
where $(n-1)^{\prime}$ indicates that when $n$ is even the multiplicity of
each exponent congruent to $n-1$ modulo $2n-2$ is $2$. A basis of the
principal Heisenberg subalgebra $\mathfrak{s}$ can be chosen as
$\\{-\Lambda^{k}\in\mathfrak{g}^{k},\Gamma^{k}\in\mathfrak{g}^{k(n-1)}\mid
k\in 2\mathbb{Z}+1\\},$
where $\Lambda=\sum_{i=0}^{n}{e}_{i}$, and
$\displaystyle\Gamma=$
$\displaystyle\kappa\Big{(}e_{n,1}-\frac{1}{2}e_{n+1,1}-\frac{\lambda}{2}e_{n,2n}+\frac{\lambda}{4}e_{n+1,2n}$
$\displaystyle+(-1)^{n}\big{(}e_{2n,n+1}-\frac{1}{2}e_{2n,n}-\frac{\lambda}{2}e_{1,n+1}+\frac{\lambda}{4}e_{1,n}\big{)}\Big{)}$
(4.12)
with $\kappa=1$ when $n$ is even and $\sqrt{-1}$ when $n$ is odd. Here
$\Lambda^{j}$ and $\Gamma^{j}$ are define to be the $j$-th power of $\Lambda$
and $\Gamma$ respectively for $j>0$, while for $j<0$
$\Lambda^{j}=(\lambda^{-1}\Lambda^{2n-3})^{-j},\quad\Gamma^{j}=(\lambda^{-1}\Gamma)^{-j}.$
(4.13)
We now rewrite the Drinfeld-Sokolov hierarchy of $D_{n}$ type (4.4) into the
form
$\frac{\partial\mathscr{L}}{\partial
t_{k}}=[\varphi(-\Lambda^{k})^{+},\mathscr{L}],\quad\frac{\partial\mathscr{L}}{\partial\hat{t}_{k}}=[\varphi(\Gamma^{k})^{+},\mathscr{L}],\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$ (4.14)
We call the flows $\frac{\partial}{\partial t_{k}}$ and
$\frac{\partial}{\partial\hat{t}_{k}}$ the positive and the negative flows of
the Drinfeld-Sokolov hierarchy of $D_{n}$ type respectively. We will show
that these flows coincide with the positive and negative flows (3.3) and (3.9)
defined by the pseudo-differential operator $L$.
It is shown in [6] that in the orbit of gauge transformations of
$\mathscr{L}$, one can find a canonical representative
$\mathscr{L}^{\mathrm{can}}$ of the form
$\mathscr{L}^{\mathrm{can}}=D+\Lambda+q^{\mathrm{can}},$
where $q^{\mathrm{can}}$ reads
$\displaystyle q^{\mathrm{can}}=$
$\displaystyle\sum_{j=1}^{[\frac{n-1}{2}]}\Big{(}q_{j}(e_{1,2j}+e_{2n+1-2j,2n})+q_{n-j}(e_{1,2n+1-2j}+e_{2j,2n})\Big{)}+\hat{q}$
(4.15)
with
$\hat{q}=\left\\{\begin{array}[]{ll}\frac{1}{2}(q_{n/2}+\rho)(e_{1,n}+e_{n+1,2n})+(q_{n/2}-\rho)(e_{1,n+1}+e_{n,2n}),&n\hbox{
even},\\\
-\sqrt{-1}\rho(\frac{1}{2}e_{1,n}-e_{1,n+1}+e_{n,2n}-\frac{1}{2}e_{n+1,2n}),&n\hbox{
odd}.\end{array}\right.$
The coefficients $q_{1},\dots,q_{n-1}$ and $\rho$ are gauge invariant
differential polynomials of $q$ that appears in (4.1). They serve as
coordinates of the orbit space of gauge transformations, and we will use them
as unknown functions of the Drinfeld-Sokolov hierarchy.
Let $\mathcal{A}$ be the algebra of differential polynomials of
$q_{1},\dots,q_{n-1}$ and $\rho$, denote
$\mathcal{A}^{-}=\mathcal{A}((\lambda^{-1}))$, we introduce a free
$\mathcal{A}^{-}$-module
$V=\left(\mathcal{A}^{-}\right)^{2n}=\left\\{\sum_{i<\infty}\alpha_{i}\lambda^{i}\mid\alpha_{i}\in\mathcal{A}^{2n}\right\\}.$
Let us fix a basis $\\{\hat{\psi}_{2n},\psi_{2n-1},\dots,\psi_{1}\\}$ of $V$,
where $\hat{\psi}_{2n}=\frac{\lambda}{2}\psi_{1}+\psi_{2n}$, and $\psi_{i}$ is
the column vector whose $i$-th entry is $1$ and others are zero.
In the notions of Lemma 4.3, we let $\mathcal{R}=\mathcal{D}^{-}$ and denote
by $\mathcal{R}_{+}$ the subalgebra of $\mathcal{R}$ consisting of
differential operators. We define an $\mathcal{R}_{+}$-module structure on $V$
by
$D\cdot\alpha=\mathscr{L}^{\mathrm{can}}\alpha,~{}~{}\alpha\in V.$ (4.16)
Note that
$\left.\mathscr{L}^{\mathrm{can}}\right|_{\lambda=0}\in\mathcal{R}_{+}^{2n\times
2n}$, let
$R=\left(\left.\mathscr{L}^{\mathrm{can}}\right|_{\lambda=0}\right)^{T}=-\mathrm{diag}(D,D,\dots,D)+\Lambda|_{\lambda=0}+\left(q^{\mathrm{can}}\right)^{T},$
then it is straightforward to verify that
$R\cdot(\hat{\psi}_{2n},\psi_{2n-1},\ldots,\psi_{1})^{t}=(-\lambda\psi_{2},0,\ldots,0)^{t}=(-\lambda
D\cdot\psi_{1},0,\ldots,0)^{t}.$ (4.17)
Denote $\mathcal{L}=-\Delta(R)$, where $\Delta$ is the operation defined in
Lemma 4.3, then $\mathcal{L}^{*}=-\mathcal{L}$ by using (4.11) and the third
part of Lemma 4.3. It is easy to see that $\mathcal{L}$ has the form (3.13).
This observation gives a Miura-type transformation between $u^{1},\dots,u^{n}$
and $q_{1},\dots,q_{n-1},\rho$, so the algebra $\mathcal{A}$ defined above
coincides with the one that is given in the last section. Moreover, the second
part of Lemma 4.3 implies that $\mathcal{L}$ is invariant w.r.t. the gauge
transformations (4.5), thus the Drinfeld-Sokolov hierarchy can be represented
by the operator $\mathcal{L}$, or equivalently by $L=D^{-1}\mathcal{L}$.
Note that the operator $\mathcal{L}\notin\mathcal{R}_{+}$, since $V$ is only
an $\mathcal{R}_{+}$-module $\mathcal{L}$ cannot act on $V$, and the first
part of Lemma 4.3 cannot be applied directly. To resolve this problem,
Drinfeld and Sokolov decomposed $V$ into two subspaces such that
$\mathcal{D}^{-}$ can act on one of them, then the first part of Lemma 4.3 can
be applied. In this way, the positive flows of the Drinfeld-Sokolov hierarchy
(4.14) are represented in the form (3.3) as the positive flows given by the
pseudo-differential operataor $L$ of the form (1.3).
In the matrix realization of $\mathfrak{g}$, the elements $\Lambda$ and
$\Gamma$ are $2n\times 2n$ matrices with entries in $\mathbb{C}[\lambda]$, so
they can act on the space $V$. One can verify that the following decomposition
holds true
$V=V_{1}\oplus V_{2},\quad
V_{1}=\mathrm{Im}\,\Lambda=\mathrm{Ker}\,\Gamma,\quad
V_{2}=\mathrm{Ker}\Lambda=\mathrm{Im}\,\Gamma.$
Denote $T=e^{U}$, where $U$ is the matrix appeared in Proposition 4.1 with
$\mathscr{L}=\mathscr{L}^{\mathrm{can}}$, then we also have
$V=V_{1}^{\prime}\oplus V_{2}^{\prime},\quad V_{1}^{\prime}=TV_{1},\quad
V_{2}^{\prime}=TV_{2}.$ (4.18)
Since the operator $\lambda^{-1}\Lambda^{2n-2}$ is the identity operator when
restricted to $V_{1}$, let $\mathscr{P}=\varphi(\lambda^{-1}\Lambda^{2n-2})$
with $\varphi$ being defined in (4.3), then $\mathscr{P}$ is the projection
from $V$ to $V_{1}^{\prime}$. We denote the projection of $\alpha\in V$ in
$V_{1}^{\prime}$ by $\alpha^{\prime}=\mathscr{P}\alpha$, and define the action
$\displaystyle
D^{-1}\cdot\alpha^{\prime}=\left(\mathscr{L}^{\mathrm{can}}\right)^{-1}\alpha^{\prime}=T\big{(}\Lambda-(\Lambda-\mathscr{L}_{0})\big{)}^{-1}T^{-1}\alpha^{\prime}$
$\displaystyle=$ $\displaystyle
T\big{(}\Lambda^{-1}+\Lambda^{-1}(\Lambda-\mathscr{L}_{0})\Lambda^{-1}+(\Lambda^{-1}(\Lambda-\mathscr{L}_{0}))^{2}\Lambda^{-1}+\cdots\big{)}T^{-1}\alpha^{\prime}.$
Here the operator $\mathscr{L}_{0}$ defined in (4.2) now reads
$\mathscr{L}_{0}=e^{-U}\mathscr{L}^{\mathrm{can}}e^{U}=D+\Lambda+\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}f_{k}\Lambda^{-k}+\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}g_{k}\Gamma^{-k}$
(4.19)
with $f_{k},g_{k}\in\mathcal{A}$ and the negative powers $\Lambda$, $\Gamma$
defined in (4.13). Note that
$\mathrm{Im}\,\Lambda^{-1}\subset\mathrm{Im}\,\Lambda$,
$\mathrm{Im}\,\Gamma^{-1}\subset\ker\,\Lambda$, then
$D^{-1}\cdot\alpha^{\prime}\in V_{1}^{\prime}$, so $V_{1}^{\prime}$ becomes an
$\mathcal{R}$-module.
It follows from $[\mathscr{L}_{0},\Lambda]=0$ that
$[\mathscr{P},\mathscr{L}^{\mathrm{can}}]=0$, then by acting $\mathscr{P}$ on
both sides of (4.17) one has
$R\cdot(\hat{\psi}_{2n}^{\prime},\psi_{2n-1}^{\prime},\ldots,\psi_{1}^{\prime})^{t}=(-\lambda
D\cdot\psi_{1}^{\prime},0,\ldots,0)^{t}.$
Now the first part of Lemma 4.3 can be employed to prove the following lemma.
###### Lemma 4.4 ([6])
Let $\mathcal{L}=-\Delta(R)$, $L=D^{-1}\mathcal{L}$, then $L$ takes the form
(1.3). Define $P=L^{\frac{1}{2n-2}}\in\mathcal{D}^{-}$ as in Lemma 3.1, then
for any $i\in\mathbb{Z}$ the following equalities hold true
$\displaystyle\varphi(\Lambda^{i})\psi_{1}^{\prime}=P^{i}\cdot\psi_{1}^{\prime},$
(4.20)
$\displaystyle\big{(}\varphi(\Lambda^{2i+1})^{+}\psi_{1}\big{)}^{\prime}=(P^{2i+1})_{+}\cdot\psi_{1}^{\prime}.$
(4.21)
By using the second equality, one can represent the positive flows
$\frac{\partial}{\partial t_{k}}$ of the Drifeld-Sokolov hierarchy (4.14) in
the form (3.3). We are to explain in the next subsection that the negative
flows of (4.14) can be represented as (3.9).
The first equality of the above lemma gives the following result.
###### Proposition 4.5 ([6])
Let $f_{k}$ be the coefficients that appear in (4.19), then
$f_{k}+\frac{1}{k}\mathrm{res}\,P^{k}\in D(\mathcal{A})$ for all
$k\in\mathbb{Z^{\mathrm{odd}}_{+}}$.
From Theorem 4.2 and (3.20) we know that this proposition related the
densities of the Hamiltonians of the positive flows of the Drinfeld-Sokolov
hierarchy with that of the positive flow (3.3) defined in the last section.
### 4.3 Negative flows of the Drinfeld-Sokolov hierarchy of $D_{n}$ type
In the last subsection, the pseudo-differential operator representation for
the positive flows of the Drinfeld-Sokolov hierarchy of $D_{n}$ type is
obtained by introducing a $\mathcal{D}^{-}$-module structure on the space
$V_{1}^{\prime}$ and using Lemma 4.3 as was done in [6]. In order to obtain a
similar representation for the negative flows, we try to assign a
$\mathcal{D}^{+}$-module structure to $V_{2}^{\prime}$. However, it seems that
there is no such a structure on $V_{2}^{\prime}$, so we first extend the space
$V_{2}^{\prime}$ to a larger one $V_{2}^{\prime\prime}$ which admits a
$\mathcal{D}^{+}$-module structure, then we employ Lemma 4.3 and obtain the
pseudo-differential operator representation for the negative flows of the
Drinfeld-Sokolov hierarchy of $D_{n}$ type.
Recall that $V_{2}$ as an $\mathcal{A}^{-}$-module is spanned by the following
two vectors:
$\hat{\psi}_{1}=\frac{1}{2}\psi_{1}-\frac{1}{\lambda}\psi_{2n},\quad\hat{\psi}_{2}=\Gamma\hat{\psi}_{1}=\kappa\left(\psi_{n}-\frac{1}{2}\psi_{n+1}\right).$
(4.22)
The action of $\Gamma$ restricted to $V_{2}$ satisfies $\Gamma^{2}=\lambda$,
so we introduce $\Gamma^{-1}=\lambda^{-1}\Gamma$, see (4.13). It is easy to
see that every vector $\alpha\in V_{2}$ can be uniquely expressed in the form
$\alpha=\sum_{i\leq m}a_{i}\Gamma^{i}\hat{\psi}_{1},\quad
a_{i}\in\mathcal{A},~{}~{}m\in\mathbb{Z}.$ (4.23)
This observation shows that the space $V_{2}$ is in fact a rank-one free
module of the following algebra
$\mathcal{D}^{-}(\mathcal{A},\Gamma)=\left\\{\sum_{i<\infty}a_{i}\Gamma^{i}\mid
a_{i}\in\mathcal{A}\right\\}.$
This is the algebra of “pseudo-differential operators of the first type” (see
Sec. 2.1) over the algebra $\mathcal{A}$ with the derivation “$D$” being the
following trivial map
$\Gamma:\mathcal{A}\to\mathcal{A},\quad f\mapsto 0,$
which surely gives a derivation of degree one over $\mathcal{A}$.
By regarding another trivial map
$\Gamma^{-1}:\mathcal{A}\to\mathcal{A},\quad f\mapsto 0,$
as a derivation of degree one, one can also define the algebra of “pseudo-
differential operators of the second type” with respect to the algebra
$\mathcal{A}$ and the derivation $\Gamma^{-1}$ as
$\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})=\left\\{\sum_{j\geq 0}\sum_{i\leq
m+j}a_{i,j}\Gamma^{i}\mid
a_{i,j}\in\mathcal{A}_{j},~{}m\in\mathbb{Z}\right\\}.$
We denote by $\hat{V}_{2}$ the rank-one free module of the algebra
$\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})$ with generator $\hat{\psi}_{1}$,
which has a linear topology induced from that of
$\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})$. It is easy to see that the algebra
$\mathcal{D}^{-}(\mathcal{A},\Gamma)$ is a subalgebra of
$\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})$ (see Lemma 2.2), hence $V_{2}$ is a
subspace of $\hat{V}_{2}$.
To define the space $V_{2}^{\prime\prime}$, we need to extend the space $V$ to
certain space $\hat{V}$ that involves $\hat{V}_{2}$ as a subspace. Since the
space $V$ is defined to be $\left(\mathcal{A}^{-}\right)^{2n}$, in which the
algebra $\mathcal{A}^{-}=\mathcal{A}((\lambda^{-1}))$ can also be defined as
$\mathcal{D}^{-}(\mathcal{A},\lambda)$ with $\lambda$ being the trivial
derivation, we similarly extend the space $V$ to
$\hat{V}=\hat{\mathcal{A}}^{2n},\quad\hat{\mathcal{A}}=\mathcal{D}^{+}(\mathcal{A},\lambda^{-1}).$
The space $\hat{V}$ has a linear topology induced from that of
$\hat{\mathcal{A}}$. It is easy to see that the linear transformations
$\Lambda,\Gamma,T=e^{U}:V\to V$ can be extended naturally to $\hat{V}$. Then
the expression
$\alpha=\sum_{j\geq 0}\sum_{i\leq
m+j}a_{i,j}\Gamma^{i}\hat{\psi}_{1}\in\hat{V}_{2}$ (4.24)
is also convergent in $\hat{V}$ according to its topology, hence the space
$\hat{V}_{2}$ is indeed a subspace of $\hat{V}$.
Now let us introduce another subspace of $\hat{V}$:
$V_{2}^{\prime\prime}=T\,\hat{V}_{2}\subset\hat{V},$
then $V_{2}^{\prime}$ is a subspace of $V_{2}^{\prime\prime}$. As in the
previous subsection we define a map
$\mathscr{Q}:V\to
V_{2}^{\prime},\quad\mathscr{Q}=\varphi(\lambda^{-1}\Gamma^{2})$
with $\varphi$ defined in (4.3). Then we have the following commutative
diagram
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{-1}\Gamma^{2}}$$\scriptstyle{T}$$\scriptstyle{\cong}$$\textstyle{V_{2}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{T}$$\scriptstyle{\cong}$$\textstyle{\hat{V}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\cong}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{Q}}$$\textstyle{V_{2}^{\prime}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{V_{2}^{\prime\prime}}$
We also denote the composition of $\mathscr{Q}$ and the inclusion
$V_{2}^{\prime}\hookrightarrow V_{2}^{\prime\prime}$ by $\mathscr{Q}$, and
write $\alpha^{\prime\prime}=\mathscr{Q}\alpha$ for any vector $\alpha\in V$.
###### Lemma 4.6
The space $\hat{V}_{2}$ is a free
$\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})$-module with generator
$T^{-1}\psi_{1}^{\prime\prime}$.
Proof To see that $T^{-1}\psi_{1}^{\prime\prime}$ is another generator
besides $\hat{\psi}_{1}$, we only need to show that these two vectors are
related by the action of a unit of the algebra
$\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})$.
Recall $T=e^{U}$, in which according to the present matrix realization the
element $U$ given in Proposition 4.1 has the form $U_{0}+O(\lambda^{-1})$ with
$U_{0}$ being a strictly upper triangular matrix, and that the vector
$\hat{\psi}_{1}$ defined in (4.22) can be represented as
$\hat{\psi}_{1}=\lambda^{-1}\Gamma^{2}\psi_{1},$
so we have
$\psi_{1}^{\prime\prime}=\mathscr{Q}\psi_{1}=T\lambda^{-1}\Gamma^{2}T^{-1}\psi_{1}=T(\hat{\psi}_{1}+O(\lambda^{-1}))\in
V_{2}^{\prime}.$
By using the general form (4.23) of elements of $V_{2}$ and the identity
$\Gamma^{2j+1}|_{V_{2}}=\lambda^{j}\Gamma$, one can represent
$T^{-1}\psi_{1}^{\prime\prime}\in V_{2}$ in the following form:
$T^{-1}\psi_{1}^{\prime\prime}=\left(1+\sum_{i<0}b_{i}\Gamma^{i}\right)\hat{\psi}_{1},\quad
b_{i}\in\mathcal{A}.$ (4.25)
Obviously the element
$1+\sum_{i<0}b_{i}\Gamma^{i}\in\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})$ is
invertible. The lemma is proved. $\Box$
Aiming at a $\mathcal{D}^{+}$-module structure on the space
$V_{2}^{\prime\prime}$ such that the action of $D$ coincides with (4.16) when
restricted to the subspace $V_{2}^{\prime}$, we need to define the action of
$(\mathscr{L}^{\mathrm{can}})^{i}$ $(i\in\mathbb{Z})$ on the space
$V_{2}^{\prime\prime}$. Note that the operator $\mathscr{L}_{0}:V\to V$ given
in (4.19) can be extended to $\hat{V}$, we denote its restriction on the space
$\hat{V}_{2}$ by $\hat{\mathscr{L}}_{0}$, which reads
$\hat{\mathscr{L}}_{0}=\mathscr{L}_{0}|_{\hat{V}_{2}}=D+\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}g_{k}\Gamma^{-k}.$
Here $g_{1}\in\mathcal{A}$ is invertible as indicated in [6], so the operator
$\hat{\mathscr{L}}_{0}$ is invertible on $\hat{V}_{2}$, and its inverse is
given by
$\displaystyle\hat{\mathscr{L}}_{0}^{-1}=$
$\displaystyle\big{(}g_{1}\Gamma^{-1}(1+g_{1}^{-1}\Gamma\,D+M)\big{)}^{-1}$
$\displaystyle=$
$\displaystyle\big{(}1-(g_{1}^{-1}\Gamma\,D+M)+(g_{1}^{-1}\Gamma\,D+M)^{2}-\cdots\big{)}g_{1}^{-1}\Gamma,$
where $M=g_{1}^{-1}\sum_{j\geq 1}g_{2j+1}\,\Gamma^{-2j}$. One can expand the
right hand side and obtain
$\hat{\mathscr{L}}_{0}^{-1}=\sum_{s\geq 0}\sum_{r\leq
s}A_{rs}\,\Gamma^{r+1},\quad
A_{rs}=\sum_{j=0}^{s}c_{rsj}D^{j},~{}~{}c_{rsj}\in\mathcal{A}_{s-j},$ (4.26)
in which $A_{00}=c_{000}=g_{10}^{-1}$ with $g_{10}$ being the projection of
$g_{1}$ onto $\mathcal{A}_{0}$. Note that $g_{10}/\rho$ is a positive
constant, where $\rho$ appears in the definition (4.15) of
$\mathscr{L}^{\mathrm{can}}$, and we have normalized $\Gamma$ such that this
constant is $1$. Since $A_{rs}$ are differential operators of degree $s$,
i.e., $A_{rs}(\mathcal{A}_{d})\subset\mathcal{A}_{d+s}$, then by using the
expressions (4.26) and (4.24) one can verify that the action of
$\hat{\mathscr{L}}_{0}^{-1}$ on $\hat{V}_{2}$ is well defined. Also note that
the image $\hat{\mathscr{L}}_{0}^{-1}(V_{2})$ is not contained in $V_{2}$
though $\hat{\mathscr{L}}_{0}(V_{2})\subset V_{2}$, which is why we extend
$V_{2}$ to $\hat{V}_{2}$.
To go forward, we need to present another expression for vectors in
$\hat{V}_{2}$.
###### Lemma 4.7
Every vector $\alpha\in\hat{V}_{2}$ can be uniquely expressed in the form
$\alpha=\sum_{j\geq 0}\sum_{i\leq
m+j}b_{i,j}\hat{\mathscr{L}}_{0}^{-i}\,T^{-1}\psi_{1}^{\prime\prime},\quad
b_{i,j}\in\mathcal{A}_{j},~{}~{}m\in\mathbb{Z}.$ (4.27)
Proof According to Lemma 4.6, we suppose $\alpha\in\hat{V}_{2}$ has the form
$\alpha=\sum_{j\geq k}\sum_{i\leq
m+j}a_{i,j}\Gamma^{i}\,T^{-1}\psi_{1}^{\prime\prime}+\cdots,\quad
a_{i,j}\in\mathcal{A}_{j},$
where $\cdots$ stands for the terms of the form (4.27). Let us proceed to
prove the lemma by induction on the lower bound $k$ of the index $j$.
First, we have
$\displaystyle\alpha=$ $\displaystyle\sum_{i\leq
m+k}a_{i,k}\Gamma^{i}\,T^{-1}\psi_{1}^{\prime\prime}+\sum_{j\geq
k+1}\sum_{i\leq m+j}a_{i,j}\Gamma^{i}T^{-1}\psi_{1}^{\prime\prime}+\cdots$
$\displaystyle=$ $\displaystyle
a_{m+k,k}\Gamma^{m+k}\,T^{-1}\psi_{1}^{\prime\prime}+\sum_{i\leq
m-1+k}a_{i,k}\Gamma^{i}\,T^{-1}\psi_{1}^{\prime\prime}$
$\displaystyle\quad+\sum_{j\geq k+1}\sum_{i\leq
m+j}a_{i,j}\Gamma^{i}\,T^{-1}\psi_{1}^{\prime\prime}+\cdots.$ (4.28)
From the expansion (4.26) it follows that
$\hat{\mathscr{L}}_{0}^{-l}=\sum_{s\geq 0}\sum_{r\leq
s}A^{(l)}_{rs}\,\Gamma^{r+l},\quad
A^{(l)}_{rs}\in(\mathcal{D}^{-})_{+},~{}~{}\deg A^{(l)}_{rs}=s,$ (4.29)
where $A^{(l)}_{00}=g_{10}^{-l}$, hence by using (4.25) we have
$\displaystyle\hat{\mathscr{L}}_{0}^{-l}T^{-1}\,\psi_{1}^{\prime\prime}-g^{-l}_{10}\Gamma^{l}\,T^{-1}\psi_{1}^{\prime\prime}$
$\displaystyle=$
$\displaystyle\left(\sum_{r\leq-1}A^{(l)}_{r0}\Gamma^{r+l}+\sum_{s\geq
1}\sum_{r\leq s}A^{(l)}_{rs}\Gamma^{r+l}\right)T^{-1}\psi_{1}^{\prime\prime}$
$\displaystyle=$
$\displaystyle\left(\sum_{r\leq-1}A^{(l)}_{r0}\Gamma^{r+l}+\sum_{s\geq
1}\sum_{r\leq
s}A^{(l)}_{rs}\Gamma^{r+l}\right)\left(1+\sum_{i<0}b_{i}\Gamma^{i}\right)\hat{\psi}_{1}$
$\displaystyle=$
$\displaystyle\left(\sum_{r\leq-1}c_{r,0}\Gamma^{r+l}+\sum_{s\geq
1}\sum_{r\leq s}c_{r,s}\Gamma^{r+l}\right)\hat{\psi}_{1}$ $\displaystyle=$
$\displaystyle\left(\sum_{r\leq-1}\tilde{c}_{r,0}\Gamma^{r+l}+\sum_{s\geq
1}\sum_{r\leq
s}\tilde{c}_{r,s}\Gamma^{r+l}\right)T^{-1}\psi_{1}^{\prime\prime},$ (4.30)
where $c_{r,s},\tilde{c}_{r,s}\in\mathcal{A}_{s}$. The above computation
represents the action of the operator $\hat{\mathscr{L}}_{0}^{-l}$ (4.29) on
certain vector in $\hat{V}_{2}$ by an element in
$\mathcal{D}^{+}(\mathcal{A},\Gamma^{-1})$. By using equation (4.30), we can
eliminate the term $a_{m+k,k}\Gamma^{m+k}\,T^{-1}\psi_{1}^{\prime\prime}$ in
(4.3) and arrive at
$\alpha=\sum_{i\leq
m-1+k}\tilde{a}_{i,k}\Gamma^{i}\,T^{-1}\psi_{1}^{\prime\prime}+\sum_{j\geq
k+1}\sum_{i\leq
m+j}\tilde{a}_{i,j}\Gamma^{i}\,T^{-1}\psi_{1}^{\prime\prime}+\cdots,\quad\tilde{a}_{i,j}\in\mathcal{A}_{j}.$
Then by induction on the upper bound of the index $i$ appearing in the first
summation we have
$\alpha=\sum_{j\geq k+1}\sum_{i\leq
m+j}\tilde{\tilde{a}}_{i,j}\Gamma^{i}\,T^{-1}\psi_{1}^{\prime\prime}+\cdots,$
which shows that the lower bound of the index $j$ has increased by one. The
lemma is proved. $\Box$
Now we are ready to introduce a $\mathcal{D}^{+}$-module structure on the
space $V_{2}^{\prime\prime}$ by defining the action
$D^{i}\cdot\alpha^{\prime\prime}=\varphi(\hat{\mathscr{L}}_{0}^{i})\alpha^{\prime\prime},\quad\alpha^{\prime\prime}\in
V_{2}^{\prime\prime},~{}~{}i\in\mathbb{Z},$ (4.31)
which extends the action (4.16) on $V_{2}^{\prime}$ to an action on
$V_{2}^{\prime\prime}$. Then Lemma 4.7 is equivalent to the following theorem.
###### Theorem 4.8
The $\mathcal{D}^{+}$-module $V_{2}^{\prime\prime}$ is a free module with
generator $\psi_{1}^{\prime\prime}$.
Let us apply Lemma 4.3 to the algebra $\mathcal{R}=\mathcal{D}^{+}$ and the
module $V_{2}^{\prime\prime}$. By acting the projection operator $\mathscr{Q}$
to both sides of (4.17), we have
$R\cdot(\hat{\psi}_{2n}^{\prime\prime},\psi_{2n-1}^{\prime\prime},\ldots,\psi_{1}^{\prime\prime})^{t}=(-\lambda
D\cdot\psi_{1}^{\prime\prime},0,\ldots,0)^{t},$
hence $L\cdot\psi_{1}^{\prime\prime}=\lambda\,\psi_{1}^{\prime\prime}$, where
$L=-D^{-1}\Delta(R)$ as given before. According to Lemma 3.3 we introduce a
pseudo-differential operator $Q\in\mathcal{D}^{+}$ such that $L=Q^{2}$, and
consider the action of $Q^{i}$ on $V_{2}^{\prime\prime}$ for any integer $i$.
###### Lemma 4.9
For any integer $i$ the following equality holds true:
$\varphi({\Gamma}^{i})\psi_{1}^{\prime\prime}=Q^{i}\cdot\psi_{1}^{\prime\prime}.$
(4.32)
Proof We only need to prove the case $i=1$. Since $V_{2}^{\prime\prime}$ is a
free $\mathcal{D}^{+}$-module, there exists an element $A\in\mathcal{D}^{+}$
such that
$\varphi(\Gamma)\psi_{1}^{\prime\prime}=A\cdot\psi_{1}^{\prime\prime}$. Note
that $[\varphi(\Gamma),\mathscr{L}^{\mathrm{can}}]=0$, so the action of
$\varphi(\Gamma)$ on $V_{2}^{\prime\prime}$ commutes with
$D\in\mathcal{D}^{+}$, hence
$A^{2}\cdot\psi_{1}^{\prime\prime}=\varphi(\Gamma^{2})\psi_{1}^{\prime\prime}=\lambda\psi_{1}^{\prime\prime}=L\cdot\psi_{1}^{\prime\prime}.$
By using the freeness of $V_{2}^{\prime\prime}$, we have $A^{2}=L=Q^{2}$. It
follows that $A=\pm Q$.
To show $A=Q$, we only need to compare their leading terms. Equation (4.30)
leads to
$\varphi(\Gamma)\psi_{1}^{\prime\prime}=\varphi(g_{10}\hat{\mathscr{L}}_{0}^{-1}+\cdots)\psi_{1}^{\prime\prime}=(g_{10}D^{-1}+\cdots)\cdot\psi_{1}^{\prime\prime},$
which implies that the leading term of $\mathrm{res}\,A$ is $g_{10}$. On the
other hand $g_{10}$ takes the same sign with $\rho=\mathrm{res}\,Q$, thus
$A=Q$. The lemma is proved. $\Box$
By using Lemmas 2.4 and 4.9, one can prove the following proposition. The
argument is almost the same with the one for Proposition 4.5 in [6], so we
omit the details here.
###### Proposition 4.10
Let $g_{k}$ be the coefficients that appear in (4.19), then
$g_{k}-\frac{1}{k}\mathrm{res}\,Q^{k}\in D(\mathcal{A})$ for all
$k\in\mathbb{Z^{\mathrm{odd}}_{+}}$.
This proposition connects the Hamiltonians of the negative flows of the
Drinfeld-Sokolov hierarchy of $D_{n}$ type to those (3.20) corresponding to
the negative flows (3.9).
Now we arrive at the main result of the present section.
###### Theorem 4.11
The flows (4.14) of the Drinfeld-Sokolov hierarchy of $D_{n}$ type coincide
with the flows of the integrable hierarchy (3.3), (3.9).
Proof It is shown in [6] that the Drinfeld-Sokolov hierarchy of $D_{n}$ type
has a bihamiltonian structure given by the two Poisson brackets (3.16),
(3.17). For the flow (4.4) corresponding to the element $\lambda_{j}$, the
Hamiltonian with respect to the second Poisson bracket is given by
$\mathcal{H}_{j}=\int(H\mid\lambda_{j})\mathrm{d}x,\quad j\in E_{+},$
where $H$ is given in (4.2) and $(\cdot\mid\cdot)$ is the trace form defined
by
$(G\mid H)=\mathrm{res}_{\lambda}\left(\lambda^{-1}\mathrm{tr}(G\,H)\right).$
We choose a basis (1.4) of the Heisenberg subalgebra $\mathfrak{s}$. as
$\lambda_{k}=-\Lambda^{k},\quad\lambda_{k(n-1)^{\prime}}=\Gamma^{k},\quad k\in
2\mathbb{Z}+1.$
Note that
$(\Lambda^{k}\mid\Lambda^{l})=(2n-2)\delta_{k,-l},\quad(\Lambda^{k}\mid\Gamma^{l})=0,\quad(\Gamma^{k}\mid\Gamma^{l})=2\,\delta_{k,-l},$
where $k$, $l$ run over all odd integers, hence by using (4.19) we have
$\mathcal{H}_{k}=-(2n-2)\int\\!\\!f_{k}\,\mathrm{d}x,\quad\mathcal{H}_{k(n-1)^{\prime}}=2\int\\!\\!g_{k}\,\mathrm{d}x,\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$
They are the Hamiltonians for the positive and negative flows of the Drinfeld-
Sokolov hierarchy (4.14) w.r.t. the second Poisson bracket (3.17).
According to Propositions 4.5, 4.10 and Theorem 3.5, these Hamiltonians
satisfy
$\mathcal{H}_{k}=H_{k},\quad\mathcal{H}_{k(n-1)^{\prime}}=\hat{H}_{k},\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}},$
where $H_{k}$, $\hat{H}_{k}$ are the Hamiltonians of the integrable hierarchy
(3.3), (3.9) with respect to the second Poisson bracket (3.17). So the
Drinfeld-Sokolov hierarchy of $D_{n}$ type (4.14) and the integrable hierarchy
(3.3),(3.9) coincide. The theorem is proved. $\Box$
## 5 The two-component BKP hierarchy and its reductions
In this section we represent the two-component BKP hierarchy that is
introduced in [3] via pseudo-differential operators, and show that the
hierarchy (3.3), (3.9) is just a reduction, which was considered in [2], of
the two-component BKP hierarchy.
### 5.1 The two-component BKP hierarchy
Let $\tilde{M}$ be an infinite-dimensional manifold with local coordinates
$(a_{1},a_{3},a_{5},\dots,b_{1},b_{3},b_{5},\dots),$
and $\tilde{\mathcal{A}}$ be the algebra of differential polynomials on
$\tilde{M}$:
$\tilde{\mathcal{A}}=C^{\infty}(\tilde{M})[[a_{i}^{s},b_{i}^{s}\mid
i\in\mathbb{Z^{\mathrm{odd}}_{+}},s\geq 1]].$
As in Section 3, we assign a gradation on $\tilde{\mathcal{A}}$ such that
$\tilde{\mathcal{A}}$ is topologically complete. Define a derivation $D$ by
$D=\sum_{s\geq
0}\sum_{i\in\mathbb{Z^{\mathrm{odd}}_{+}}}\left(a_{i}^{s+1}\frac{\partial}{\partial
a_{i}^{s}}+b_{i}^{s+1}\frac{\partial}{\partial b_{i}^{s}}\right),$
then the algebras
$\tilde{\mathcal{D}}^{\pm}=\mathcal{D}^{\pm}(\tilde{\mathcal{A}},D)$ of
pseudo-differential operators can be constructed as we did in Section 2.1.
Introduce two pseudo-differential operators
$\displaystyle\Phi=$ $\displaystyle 1+\sum_{i\geq
1}a_{i}D^{-i}\in\tilde{\mathcal{D}}^{-},$ (5.1) $\displaystyle\Psi=$
$\displaystyle 1+\sum_{i\geq 1}b_{i}D^{i}\in\tilde{\mathcal{D}}^{+},$ (5.2)
where $a_{2},a_{4},a_{6},\dots,b_{2},b_{4},b_{6},\dots\in\tilde{\mathcal{A}}$
are determined by the following condtions
$\Phi^{*}=D\Phi^{-1}D^{-1},\quad\Psi^{*}=D\Psi^{-1}D^{-1}.$ (5.3)
Now let us define a pair of operators
$P=\Phi D\Phi^{-1}\in\tilde{\mathcal{D}}^{-},\quad Q=\Psi
D^{-1}\Psi^{-1}\in\tilde{\mathcal{D}}^{+}.$
###### Lemma 5.1
The operators $P,Q$ have the following expressions (c.f. (3.1), (3.5)):
$P=D+\sum_{i\geq 1}u_{i}D^{-i},\quad Q=D^{-1}\rho+\sum_{i\geq 1}v_{i}D^{i},$
where $\rho=(\Psi^{-1})^{*}(1)$. They satisfy
$P^{*}=-DPD^{-1},\quad Q^{*}=-DQD^{-1},$ (5.4)
and that for any $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$
$(P^{k})_{+}(1)=0,\quad(Q^{k})_{+}(1)=0.$ (5.5)
Proof The expression of $P$ is obvious. To show that of $Q$, we consider its
negative part:
$\displaystyle Q_{-}=$ $\displaystyle\left(\Psi
D^{-1}\Psi^{-1}\right)_{-}=\left(D^{-1}\Psi^{-1}\right)_{-}=\left(\left(D^{-1}\Psi^{-1}\right)^{*}\right)_{-}^{*}$
$\displaystyle=$
$\displaystyle-\left((\Psi^{-1})^{*}D^{-1}\right)_{-}^{*}=-\left((\Psi^{-1})^{*}(1)D^{-1}\right)^{*}=D^{-1}\rho.$
The symmetry property (5.4) is obvious, which implies (5.5). The lemma is
proved. $\Box$
We define the following evolutionary equations:
$\displaystyle\frac{\partial\Phi}{\partial
t_{k}}=-(P^{k})_{-}\Phi,\quad\frac{\partial\Psi}{\partial
t_{k}}=\bigl{(}(P^{k})_{+}-\delta_{k1}Q^{-1}\bigr{)}\Psi,$ (5.6)
$\displaystyle\frac{\partial\Phi}{\partial\hat{t}_{k}}=-(Q^{k})_{-}\Phi,\quad\frac{\partial\Psi}{\partial\hat{t}_{k}}=(Q^{k})_{+}\Psi,$
(5.7)
where $k\in\mathbb{Z^{\mathrm{odd}}_{+}}$. According to (5.3) and (5.5), it is
easy to see that these flows are well defined, and they yield the Lax
equations of the form (3.11), (3.12). By a straightforward calculation one can
verify the commutativity of these flows, hence they form an integrable
hierarchy indeed. We will show that this hierarchy possesses tau functions,
and that these tau functions satisfy the same bilinear equations of the two-
component BKP hierarchy defined in [3].
First, let us introduce two wave functions
$\displaystyle w=w(\mathbf{t},\hat{\mathbf{t}};z)=\Phi
e^{\xi(\mathbf{t};z)},\quad\hat{w}=\hat{w}(\mathbf{t},\hat{\mathbf{t}};z)=\Psi
e^{xz+\xi(\hat{\mathbf{t}};-z^{-1})},$ (5.8)
where $x=t_{1}$, the function $\xi$ is defined by
$\xi(\mathbf{t};z)=\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}t_{k}z^{k},$ (5.9)
and for any $i\in\mathbb{Z}$ the action of $D^{i}$ on $e^{xz}$ is set to be
$D^{i}e^{xz}=z^{i}e^{xz}$.
It is easy to see that $P\,w=zw,\quad Q\,\hat{w}=z^{-1}\hat{w}$, and that the
flows (5.6), (5.7) are equivalent to the following equations
$\displaystyle\frac{\partial w}{\partial
t_{k}}=(P^{k})_{+}w,\quad\frac{\partial\hat{w}}{\partial
t_{k}}=(P^{k})_{+}\hat{w},$ (5.10) $\displaystyle\frac{\partial
w}{\partial\hat{t}_{k}}=-(Q^{k})_{-}w,\quad\frac{\partial\hat{w}}{\partial\hat{t}_{k}}=-(Q^{k})_{-}\hat{w}.$
(5.11)
Here $(Q^{k})_{-}w$ is understood as
$\left((Q^{k})_{-}\Phi\right)e^{\xi(\mathbf{t};z)}$, and $(Q^{k})_{-}\hat{w}$
is defined similarly. The following theorem can be proved as it was done for
the KP hierarchy given in [4, 5].
###### Theorem 5.2
The hierarchy (5.6), (5.7) is equivalent to the following bilinear equation
$\mathrm{res}_{z}z^{-1}w(\mathbf{t},\hat{\mathbf{t}};z)w(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime};-z)=\mathrm{res}_{z}z^{-1}\hat{w}(\mathbf{t},\hat{\mathbf{t}};z)\hat{w}(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime};-z).$
(5.12)
Here and below the residue of a Laurent series is defined as
$\mathrm{res}_{z}\sum_{i}f_{i}z^{i}=f_{-1}$.
Let $\omega$ be the following $1$-form
$\omega=\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}\left(\mathrm{res}\,P^{k}\,\mathrm{d}t_{k}+\mathrm{res}\,Q^{k}\,\mathrm{d}\hat{t}_{k}\right).$
(5.13)
By using the equations (5.6) and (5.7), one can show that $\omega$ is closed,
so given any solution of the hierarchy (5.6), (5.7) there exists a function
$\tau(\mathbf{t},\hat{\mathbf{t}})$ such that
$\omega=\mathrm{d}\left(2\,\partial_{x}\,\log\tau\right).$ (5.14)
Moreover, one can fix a tau function such that the wave functions can be
written as
$\displaystyle
w(\mathbf{t},\hat{\mathbf{t}};z)=\frac{\tau(\ldots,t_{k}-\frac{2}{kz^{k}},\ldots,\hat{\mathbf{t}})}{\tau(\mathbf{t},\hat{\mathbf{t}})}e^{\xi(\mathbf{t};z)},$
(5.15)
$\displaystyle\hat{w}(\mathbf{t},\hat{\mathbf{t}};z)=\frac{\tau(\mathbf{t},\ldots,\hat{t}_{k}+\frac{2z^{k}}{k},\ldots)}{\tau(\mathbf{t},\hat{\mathbf{t}})}e^{\xi(\hat{\mathbf{t}};-z^{-1})}.$
(5.16)
Introduce a vertex operator $X$ as
$X(\mathbf{t};z)=\exp\left(\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}t_{k}z^{k}\right)\exp\left(-\sum_{k\in\mathbb{Z^{\mathrm{odd}}_{+}}}\frac{2}{kz^{k}}\frac{\partial}{\partial
t_{k}}\right),$
then the bilinear equation (5.12) reads
$\displaystyle\mathrm{res}_{z}z^{-1}X(\mathbf{t};z)\tau(\mathbf{t},\hat{\mathbf{t}})X(\mathbf{t}^{\prime};-z)\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime})$
$\displaystyle=$
$\displaystyle\mathrm{res}_{z}z^{-1}X(\hat{\mathbf{t}};-z^{-1})\tau(\mathbf{t},\hat{\mathbf{t}})X(\hat{\mathbf{t}}^{\prime};z^{-1})\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime}),$
which is equivalent to
$\displaystyle\mathrm{res}_{z}z^{-1}X(\mathbf{t};z)\tau(\mathbf{t},\hat{\mathbf{t}})X(\mathbf{t}^{\prime};-z)\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime})$
$\displaystyle=$
$\displaystyle\mathrm{res}_{z}z^{-1}X(\hat{\mathbf{t}};z)\tau(\mathbf{t},\hat{\mathbf{t}})X(\hat{\mathbf{t}}^{\prime};-z)\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime}).$
(5.17)
Recall that in [3, 24], Date, Jimbo, Kashiwara and Miwa defined the two-
component BKP hierarchy from a two-component neutral free fermions realization
of the basic representation of an infinite-dimensional Lie algebra
$\mathfrak{g}_{\infty}$, which corresponds to the Dynkin diagram of
$D_{\infty}$ type [25]. The tau function of their hierarchy satisfies the
bilinear equations (5.17) and defines two wave functions as (5.15), (5.16), so
the equations (5.6), (5.7) give a representation of the two-component BKP
hierarchy in terms of pseudo-differential operators.
###### Remark 5.3
In [29], Shiota gave a Lax pair representation of the two-component BKP
hierarchy as follows. Let $\phi^{(\nu)}\ (\nu=0,1)$ be the following pseudo-
differential operators of the first type
$\phi^{(\nu)}=1+\sum_{i\geq 1}a^{(\nu)}_{i}D_{\nu}^{-i}$
satisfying
$\left(\phi^{(\nu)}\right)^{*}=D_{\nu}\left(\phi^{(\nu)}\right)^{-1}D_{\nu}^{-1}$,
where $D_{0},D_{1}$ are two commuting derivations. Let
$P^{(\nu)}=\phi^{(\nu)}D_{\nu}\left(\phi^{(\nu)}\right)^{-1},$
then the two-component BKP hierarchy can be defined as
$\frac{\partial\phi^{(\nu)}}{\partial
t^{(\nu)}_{k}}=-\left(P^{(\nu)}\right)^{k}_{-}\phi^{(\nu)},\quad\frac{\partial\phi^{(\nu)}}{\partial
t^{(1-\nu)}_{k}}=\left(P^{(1-\nu)}\right)^{k}_{+}\left(\phi^{(\nu)}\right),\quad
k\in\mathbb{Z^{\mathrm{odd}}_{+}}.$ (5.18)
Here on the right hand side of the second equation it means the action of the
differential operator $\left(P^{(1-\nu)}\right)^{k}_{+}$ on the coefficients
of $\phi^{(\nu)}$. It is easy to see that $D_{\nu}=\frac{\partial}{\partial
t^{(\nu)}_{1}}$. We identify $t^{(0)}_{k}=t_{k}$, $t^{(1)}_{k}=\hat{t}_{k}$
henceforth.
Introduce the wave functions
$w^{(\nu)}(\mathbf{t},\hat{\mathbf{t}};z^{(\nu)})=\phi^{(\nu)}e^{\xi^{(\nu)}},\quad\xi^{(\nu)}=\xi(\mathbf{t}^{(\nu)};z^{(\nu)})$
with $\xi$ given in (5.9). The hierarchy (5.18) was shown [29] equivalent to
the following bilinear equation
$\displaystyle\mathrm{res}_{z^{(0)}}\big{(}z^{(0)}\big{)}^{-1}w^{(0)}(\mathbf{t},\hat{\mathbf{t}};z^{(0)})w^{(0)}(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime};-z^{(0)})$
$\displaystyle=$
$\displaystyle\mathrm{res}_{z^{(1)}}\big{(}z^{(1)}\big{)}^{-1}w^{(1)}(\mathbf{t},\hat{\mathbf{t}};z^{(1)})w^{(1)}(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime};-z^{(1)}).$
(5.19)
By comparing the bilinear equations (5.19) and (5.12), it is easy to see that
Shiota’s wave functions are related to ours by
$w^{(0)}(\mathbf{t},\hat{\mathbf{t}};z)=w(\mathbf{t},\hat{\mathbf{t}};z),\quad
w^{(1)}(\mathbf{t},\hat{\mathbf{t}};z)=\hat{w}(\mathbf{t},\hat{\mathbf{t}};-z^{-1}),$
from which one can obtain the relations between $a^{(0)}_{i},a^{(1)}_{i}$ and
$a_{i},b_{i}$.
### 5.2 Reductions of the two-component BKP hierarchy
Given an integer $n\geq 3$, the condition $P^{2n-2}=Q^{2}$ defines a
differential ideal of $\tilde{\mathcal{A}}$, which is denoted by
$\mathcal{I}$. It is easy to see that this ideal is preserved by the flows
(5.6), (5.7), so we obtain a reduction of the two-component BKP hierarchy.
Let $L=P^{2n-2}=Q^{2}$, then according to Lemma 5.1 the operator $L$ has the
form (1.3). Hence the algebra $\mathcal{A}$ defined in Section 3.1 is
isomorphic to $\tilde{\mathcal{A}}/\mathcal{I}$, and the reduced hierarchy is
an integrable hierarchy over $\mathcal{A}$. It is easy to see that the
derivatives of $L$ with respect to $t_{k}$, $\hat{t}_{k}$ are exactly given by
(3.3), (3.9). Namely the hierarchy (3.3),(3.9) is the reduction of the two-
component BKP hierarchy under the condition $P^{2n-2}=Q^{2}$.
It can be shown that the condition $P^{2n-2}=Q^{2}$ reduces the bilinear
equations (5.12) to the form
$\mathrm{res}_{z}z^{(2n-2)j-1}w(\mathbf{t},\hat{\mathbf{t}};z)w(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime};-z)=\mathrm{res}_{z}z^{-2j-1}\hat{w}(\mathbf{t},\hat{\mathbf{t}};z)\hat{w}(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime};-z)$
(5.20)
with $j\geq 0$, and that conversely the equations (5.20) impose the constraint
$P^{2n-2}=Q^{2}$ to the two-component BKP hierarchy. Hence we establish the
equivalence between the bilinear equations (5.20) and the hierarchy (3.3),
(3.9). The proof is lengthy and technical (c.f. the reduction from the KP
hierarchy to the Gelfand-Dickey hierarchies in [5]), so we omit the details
here. In terms of the tau function, the bilinear equations (5.20) can be
expressed as
$\displaystyle\mathrm{res}_{z}z^{(2n-2)j-1}X(\mathbf{t};z)\tau(\mathbf{t},\hat{\mathbf{t}})X(\mathbf{t}^{\prime};-z)\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime})$
$\displaystyle=$
$\displaystyle\mathrm{res}_{z}z^{2j-1}X(\hat{\mathbf{t}};z)\tau(\mathbf{t},\hat{\mathbf{t}})X(\hat{\mathbf{t}}^{\prime};-z)\tau(\mathbf{t}^{\prime},\hat{\mathbf{t}}^{\prime}),\quad
j\geq 0.$ (5.21)
Note that these bilinear equations are precisely the ones obtained from the
$(2n-2,2)$-reduction of the two-component BKP hierarchy [2, 24].
From the definition (3.24) and (5.14) of the tau functions $\hat{\tau}$ and
$\tau$ it follows that they are related by
$\tau^{2}=\hat{\tau}.$ (5.22)
## 6 Conclusion
We represent the full Drinfeld-Sokolov hierarchy of $D_{n}$ type into Lax
equations of pseudo-differential operators, which is analogous to the Gelfand-
Dickey hierarchies. We also give a Lax pair representation for the two-
component BKP hierarchy, and show that the Drinfeld-Sokolov hierarchy of
$D_{n}$ type is the $(2n-2,2)$-reduction of the two-component BKP hierarchy.
The key step in our approach is to introduce the concept of pseudo-
differential operators of the second type, which are defined over a
topologically complete differential algebra, so that they may contain
infinitely many terms with positive power of the derivation $D$.
Our Lax pair representations of the Drinfeld-Sokolov hierarchy of $D_{n}$ type
and the two-component BKP hierarchy are convenient for further studies. In a
subsequent publication [34], we will show that the two-component BKP hierarchy
carries a bihamiltonian structure, which is expected to correspond to an
infinite-dimensional Frobenius manifold (c.f. [1]).
Note that the bilinear equation (5.17) corresponds to the basic representation
of the affine Lie algebra $D_{\infty}^{\prime}$ in the notion of [24]. It is
shown in [28] that the $(2n-2,2)$-reduction (5.2) corresponds to the basic
representation of the affine Lie algebra $D_{n}^{(1)}$. Then according to [25,
26], the bilinear equation (5.2) is equivalent to the Kac-Wakimoto hierarchy
constructed from the principal vertex operator realization of the basic
representation of the affine Lie algebra $D_{n}^{(1)}$ [26]. By comparing the
boson-fermion correspondences, one can obtain the relation between the time
variables $\mathbf{t},\hat{\mathbf{t}}$ of the Drinfeld-Sokolov hierarchy of
$D_{n}$ type (or the Date-Jimbo-Kashiwara-Miwa hierarchy) and the time
variables $s_{j}\ (j\in E_{+})$ of the the Kac-Wakimoto hierarchy
$t_{k}=\sqrt{2}\,s_{k},\quad\hat{t}_{k}=\sqrt{2n-2}\,s_{k(n-1)^{\prime}}.$
In [21], Givental and Milanov proved that the total descendant potential for
semisimple Frobenius manifolds associated to a simple singularity satisfies a
certain hierarchy of Hirota bilinear/quadratic equations, see also [18, 19,
20]. Such a hierarchy of bilinear equation is shown to be equivalent to the
corresponding Kac-Wakimoto hierarchy constructed from the principal vertex
operator realization of the basic representation of the untwisted affine Lie
algebra [21, 33, 16]. So we arrive at the following result.
###### Theorem 6.1
Up to a rescaling of the flows, the following integrable hierarchies are
equivalent:
* i)
the hierarchy (3.3), (3.9);
* ii)
the Drinfeld-Sokolov hierarchy associated to $D_{n}^{(1)}$ and the $c_{0}$
vertex of its Dynkin diagram;
* iii)
the Date-Jimbo-Kashiwara-Miwa hierarchy constructed from the basic
representation of the affine Lie algebra $D_{n}^{(1)}$;
* iv)
the Kac-Wakimoto hierarchy corresponding to the principal vertex operator
realization of the basic representation of the affine Lie algebra
$D_{n}^{(1)}$;
* v)
the Givental-Milanov hierarchy for the simple singularity of $D_{n}$ type.
###### Remark 6.2
The equivalence between the hierarchies ii) and iv) was also contained in a
general result obtained by Hollowood and Miramontes in [23].
Note that the bihamiltonian structure (3.16), (3.17) is of topological type
[8, 10, 9], its leading term comes from the Frobenius manifold associated to
the Coxeter group of $D_{n}$ type. In [10] a hierarchy of dispersionless
bihamiltonian integrable systems is associated to any semisimple Frobenius
manifold, such an integrable hierarchy is called the Principal Hierarchy. It
is also shown that there is a so called topological deformation of the
Principal Hierarchy which satisfies the condition that its Virasoro symmetries
can be represented by the action of some linear operators, called the Virasoro
operators, on the tau function of the hierarchy. We expect that the Drinfeld-
Sokolov hierarchy associated to $D_{n}^{(1)}$ and the $c_{0}$ vertex of its
Dynkin diagram coincides, after a rescaling of the time variables, with the
topological deformation of the Principal Hierarchy of the Frobenius manifold
that is associated to the Coxeter group of type $D_{n}$. We will investigate
this aspect of the hierarchy in a subsequent publication.
Acknowledgments. The authors thank Boris Dubrovin for his interest in this
work and for his advices, they also thank for the hospitality of SISSA where
part of the work was done. This work is partially supported by the National
Basic Research Program of China (973 Program) No.2007CB814800, the NSFC
No.10631050 and No.10801084.
## References
* [1] Carlet, G.; Dubrovin, B.; Mertens L.P. Infinite-dimensional Frobenius manifolds for 2+1 integrable systems, preprint arXiv: math.ph/0902.1245v1.
* [2] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy. Publ. Res. Inst. Math. Sci. 18 (1982), no.3, 1077–1110.
* [3] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type. Phys. D 4 (1981/82), no.3, 343–365.
* [4] Date, E; Kashiwara, M; Jimbo, M; Miwa, T. Transformation groups for soliton equations. Nonlinear integrable systems—classical theory and quantum theory (Kyoto, 1981), 39–119, World Sci. Publishing, Singapore, 1983.
* [5] Dickey, L.A. Soliton equations and Hamiltonian systems. Second edition. Advanced Series in Mathematical Physics, 26. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
* [6] Drinfeld, V.G.; Sokolov, V.V. Lie algebras and equations of Korteweg-de Vries type. (Russian) Current problems in mathematics, Vol. 24, 81–180, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.
* [7] Dubrovin, B. Geometry of $2$D topological field theories. Integrable systems and quantum groups (Montecatini Terme, 1993), 120–348, Lecture Notes in Math., 1620, Springer, Berlin, 1996.
* [8] Dubrovin, B.; Liu, S.Q.; Zhang Y. On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasi-triviality of bi-Hamiltonian perturbations, Commun. Pure and Appl. Math. 59 (2006), 559-615.
* [9] Dubrovin, B.; Liu, S.Q.; Zhang, Y. Frobenius manifolds and central invariants for the Drinfeld-Sokolov biHamiltonian structures. (English summary) Adv. Math. 219 (2008), no.3, 780–837.
* [10] Dubrovin, B.; Zhang, Y. Normal forms of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, preprint arXiv: math.DG/0108160.
* [11] Enriquez, B.; Frenkel, E. Equivalence of two approaches to integrable hierarchies of KdV type. Comm. Math. Phys. 185 (1997), 211–230.
* [12] Faber, C.; Shadrin, S.; Zvonkine, D. Tautological relations and the r-spin Witten conjecture, preprint arXiv:math.AG/0612510.
* [13] Fan, H.; Jarvis, T.J.; Ruan, Y. The Witten equation, mirror symmetry and quantum singularity theory, preprint arXiv: math.AG/0712.4021v3.
* [14] Feigin, B.; Frenkel, E. Kac-Moody groups and integrability of soliton equations. Invent. Math. 120 (1995), 379–408.
* [15] Ferreira, L.A.; Miramontes, J.L.; S nchez Guillén, J. Tau-functions and dressing transformations for zero-curvature affine integrable equations. J. Math. Phys. 38 (1997), no.2, 882–901.
* [16] Frenkel, E.; Givental, A.; Milanov, T. Soliton equations, vertex operators, and simple singularities, preprint arXiv:math.QA/0909.4032, 2009.
* [17] Gelfand, I.M.; Dikii, L. A. Fractional powers of operators, and Hamiltonian systems. (Russian) Funkcional. Anal. i Prilozen. 10 (1976), no.4, 13–29.
* [18] Givental, A. Semi-simple Frobenius structures at higher genus. International Mathematics Research Notices 2001, no.23 (2001): 1265-1286.
* [19] Givental, A. Gromov-Witten invariants and quantization of quadratic Hamiltonians. Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1 (2001), no.4, 551–568, 645.
* [20] Givental, A. $A_{n-1}$-singularities and $n$KdV hierarchies. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. Mosc. Math. J. 3 (2003), no.2, 475–505, 743.
* [21] Givental, A.; Milanov, T. Simple singularities and integrable hierarchies. The breadth of symplectic and Poisson geometry, 173–201, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005.
* [22] de Groot, M.F.; Hollowood, T.J.; Miramontes, J.L. Generalized Drinfeld-Sokolov hierarchies. Comm. Math. Phys. 145 (1992), no.1, 57–84.
* [23] Hollowood, T.J.; Miramontes, J.L. Tau-functions and generalized integrable hierarchies. Comm. Math. Phys. 157 (1993), no.1, 99–117.
* [24] Jimbo, M.; Miwa, T. Solitons and infinite-dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 19 (1983), no.3, 943–1001.
* [25] Kac, V.G. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990, RI, 1989.
* [26] Kac, V.G.; Wakimoto, M. Exceptional hierarchies of soliton equations. Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987), 191–237, Proc. Sympos. Pure Math., 49, Part 1, Amer. Math. Soc.,
* [27] Kontsevich, M. Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147 (1992), no.1, 1–23.
* [28] ten Kroode, F.; van de Leur, J. Bosonic and fermionic realizations of the affine algebra $\widehat{\rm so}_{2n}$. Comm. Algebra 20 (1992), no.11, 3119–3162.
* [29] Shiota, T. Prym varieties and soliton equations. Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), 407–448, Adv. Ser. Math. Phys., 7, World Sci. Publ., Teaneck, NJ, 1989.
* [30] Takasaki, K. Integrable hierarchy underlying topological Landau-Ginzburg models of $D$-type. Lett. Math. Phys. 29 (1993), no.2, 111–121.
* [31] Wilson, G. The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras. Ergodic Theory Dynamical Systems 1 (1981), no.3, 361–380 (1982).
* [32] Witten, E. Two-dimensional gauge theories revisited. J. Geom. Phys. 9 (1992), no.4, 303–368.
* [33] Wu, C.Z. A Remark on Kac-Wakimoto Hierarchies of D-type. J. Phys. A: Math. Theor. 43 (2010), 035201.
* [34] Wu, C.Z.; Xu, D. Bihamiltonian structure of the two-componet BKP hierarchy, in preparation.
|
arxiv-papers
| 2009-12-30T01:23:28 |
2024-09-04T02:49:07.310195
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Si-Qi Liu, Chao-Zhong Wu, Youjin Zhang",
"submitter": "Chaozhong Wu",
"url": "https://arxiv.org/abs/0912.5273"
}
|
0912.5320
|
# Multiwavelength Opportunities and Challenges in the Era of Public Fermi Data
D. J. Thompson NASA Goddard Space Flight Center, Greenbelt, MD 20771 USA
on behalf of the Fermi Large Area Telescope Collaboration
###### Abstract
The gamma-ray survey of the sky by the Fermi Gamma-ray Space Telescope offers
both opportunities and challenges for multiwavelength and multi-messenger
studies. Gamma-ray bursts, pulsars, binary sources, flaring Active Galactic
Nuclei, and Galactic transient sources are all phenomena that can best be
studied with a wide variety of instruments simultaneously or
contemporaneously. Identification of newly-discovered gamma-ray sources is
largely a multiwavelength effort. From the gamma-ray side, a principal
challenge is the latency from the time of an astrophysical event to the
recognition of this event in the data. Obtaining quick and complete
multiwavelength coverage of gamma-ray sources can be difficult both in terms
of logistics and in terms of generating scientific interest. The Fermi LAT
team continues to welcome cooperative efforts aimed at maximizing the
scientific return from the mission through multiwavelength studies.
## I Opportunities
During its first year, the Fermi Large Area Telescope has excelled in
producing scientific results using multiwavelength approaches. Some examples
include:
* •
PSR J1741-2054 is a radio pulsar found based on gamma-ray timing PSR J1741 . A
bright Fermi LAT point source was the first step. An analysis of the LAT
timing discovered gamma-ray pulsations. A follow-up observation with the Swift
X-Ray Telescope (XRT) found an X-ray source that gave better position
information than could be determined from the LAT image. Using the LAT timing
information and the Swift location allowed archival analysis using Parkes
radio data and a deep search using the Green Bank Telescope that found the
radio pulsar.
* •
PMN J0948+0022 is known as a narrow-line quasar or a radio-loud Narrow-Line
Seyfert 1 galaxy, a somewhat different class than the blazars that are
regularly seen in gamma rays. Contemporaneous observations combining the LAT
data with Swift (X-ray, UV, and optical) and Effelsberg (radio) revealed a
Spectral Energy Distribution that showed this source to be similar to a
blazar, indicating the presence of a relativistic jet NLS1 .
* •
Using the public light curves made available by the LAT team (at the Fermi
Science Support Center Web site http://fermi.gsfc.nasa.gov/ssc/), Bonning et
al. 3C454.3 MW studied simultaneous multiwavelength variability of blazar
3C454.3 using Small and Moderate Aperture Research Telescope System (SMARTS)
telescopes for optical and ultraviolet and X-ray data from the Swift
satellite. They found excellent correlation, with a time lag less than a day,
an important parameter for modeling this blazar in terms of an external
Compton model.
These few examples illustrate some of the ways Fermi results enhance
scientific understanding when combined with observations and analysis from
other wavelengths. Because the position uncertainties for gamma-ray sources
are still large compared to those at many other wavelengths, unidentified
gamma-ray sources are inherently subjects for multiwavelength studies,
depending on timing, spectral, and modeling to determine what objects produce
the gamma rays.
## II Enabling Technologies
Multiwavelength opportunities like those described in the previous section
were not possible even a few years ago. Several developments have facilitated
such multiwavelength efforts:
* •
Communication - The ubiquity of network connectivity has allowed rapid
exchange of data and ideas. Wireless Internet access and portable devices of
all sorts have accelerated the exchange of information. Campaigns that once
had to be organized by telephone and letter can now be arranged in a matter of
minutes or hours.
* •
Facilities - Most parts of the electromagnetic spectrum (and several multi-
messenger fields) are now covered by ground-based and space-based
observatories. Fermi is just one of many facilities that produce prompt and
public results that can be used for multiwavelength study.
* •
Consolidated Information Centers - Resources like ADS, NED, Simbad, ASDC,
HEASARC, and others facilitate rapid discoveries of existing coverage of
sources. Scientists can now almost instantaneously review archival results for
nearly any cataloged object.
## III Challenges
Despite the tools and resources now available for multiwavelength studies, the
Fermi LAT presents some challenges in terms of making the best scientific use
of the gamma-ray data. Three of these issues are described in the sections
below.
### III.1 Challenge 1:Time-Criticality of Response
Gamma-ray bursts (GRB), thanks to the the Gamma-ray bursts Coordinates Network
(GCN) http://gcn.gsfc.nasa.gov/, offer a paradigm for rapid response to
transient astrophysical events. The success of the GCN originates in part from
the intensity of GRB, which can be recognized automatically with high
confidence in satellite detectors. The time for disseminating initial
information about a burst is not generally governed by the actual detection
but rather by the speed of communication. Multiwavelength studies can often
begin within seconds of the initial detection.
The situation for other gamma-ray sources is more complicated, because none of
them approaches the instantaneous brightness of a GRB. Nevertheless, dramatic
changes in flux have been see on time scales of a day or less (e.g. PKS
1502+106, 1502 ). On-board analysis of such sources by the LAT is not
practical, and so the response depends on both communication and analysis. The
Fermi LAT data are stored onboard and transmitted through the Tracking and
Data Relay System to the ground in batches, not continuously. This process
introduces some delay, as does the data reduction to extract the gamma-ray
event candidates and compare the gamma-ray sky with its previous appearance.
Half a day can pass before a flare is discovered.
The LAT team has taken several steps to minimize the latency in reporting
events of multiwavelength interest:
* •
Automation - Much of the data handling process is now automated, and efforts
continue to streamline the procedures. The analysis pipeline now produces
preliminary flux values for over 40 sources of interest routinely, and these
results are posted daily at
http://fermi.gsfc.nasa.gov/ssc/data/access/lat/msl_lc/.
* •
Dedication - The LAT team has a group of scientists called Flare Advocates or
Skywatchers, who examine the automatically produced analysis results as soon
as they appear. By applying scientific expertise at this early stage, the
process optimizes the response to findings of astrophysical interest while
minimizing any reaction to statistical fluctuations of steady sources.
* •
Communication - Flare advocates use three avenues to share results quickly
about activity in the gamma-ray sky. The first is the use of Astronomer’s
Telegrams http://www.astronomerstelegram.org/, over 40 of which have been
issued by the LAT team for quickest reporting of results. The second is a
multiwavelength mailing list, gammamw
https://lists.nasa.gov/mailman/listinfo/gammamw, which is used to contact
scientists directly about gamma-ray multiwavelength news. Anyone interested is
welcome to join this list. The third approach is the Fermi Sky Blog,
http://fermisky.blogspot.com/, which posts weekly summaries of the brightest
sources in the high-energy gamma-ray sky.
### III.2 Challenge 2: Finding Enough Multiwavelength Coverage
Although the LAT is an all-sky, every-day monitor for high energy gamma rays,
most telescopes at other wavelengths have much smaller fields of view and sky
coverage. In addition, many telescopes have sun-angle constraints.
Multiwavelength coverage of an active gamma-ray source is not assured. Two
approaches are being used by the multiwavelength community to enhance the
coverage of the sky:
* •
More all-sky or wide-field monitors are becoming available. The RXTE All-Sky
Monitor in X-rays http://xte.mit.edu/ has recently been complemented with the
Japanese MAXI all-sky X-ray monitor on the International Space Station
http://maxi.riken.jp/top/. In optical, the Palomar Quest program regularly
surveys a large area http://www.astro.caltech.edu/~george/pq/, and the Pan-
STARRS http://pan-starrs.ifa.hawaii.edu/public/ and Skymapper
http://msowww.anu.edu.au/skymapper/ programs will be surveying much of the
northern and southern hemispheres repeatedly. At longer wavelengths, Planck
http://www.rssd.esa.int/index.php?project=Planck, Herschel
http://herschel.esac.esa.int/, and WISE
http://www.nasa.gov/mission_pages/WISE/main/index.html are viewing the sky
with fairly long cadence.
* •
Source monitoring programs have also emerged. In radio, many observers are
cooperating to provide multiple-insrument monitoring of many candidate gamma-
ray targets, particularly blazars. A summary of ongoing activity can be found
at http://pulsar.sternwarte.uni-erlangen.de/radiogamma/. Similarly, optical
programs like the one at the University of Arizona
(http://james.as.arizona.edu/~psmith/Fermi/), SMARTS
(http://www.astro.yale.edu/smarts/glast/), and the GLAST-AGILE Support Program
(GASP, http://www.to.astro.it/blazars/webt/gasp/homepage.html) observe many
gamma-ray sources regularly in the optical.
A useful collection of links to multiwavelength information can be found at
https://confluence.slac.stanford.edu/display/GLAMCOG/. The LAT team greatly
appreciates the ongoing cooperative activities of all these groups and
welcomes other telescope teams who participate in particular campaigns.
### III.3 Challenge 3: Deciding When to Work with the LAT Team
With all the Fermi data public, along with software for analysis, anyone can
undertake multiwavelength studies incorporating gamma-ray results. There may
be times when contacting the LAT team could benefit such analysis, however.
Analysis of the LAT data does involve some important caveats (see
http://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT_caveats.html for more
details):
* •
The diffuse Galactic emission is bright and highly structured. The diffuse
model supplied by the LAT team has recently been updated and is likely to
continue to evolve. Separating weaker sources from the diffuse Galactic
emission is non-trivial. There are regions of the sky where the diffuse model
has deficiencies.
* •
The LAT Instrument Response Functions (IRFs) have significant uncertainties at
energies near 100 MeV and a non-negligible charged particle background at
energies above 10 GeV. Improvements in the IRFs are expected but are not
imminent. Analysis of data below 100 MeV with the current IRFs is not
recommended
Some suggestions about when consulting the LAT team might be beneficial:
* •
If you are searching for a source that is not in the LAT catalog, then it is
probably weak enough that a simple analysis will not be adequate.
* •
If you need a detailed energy spectrum or are looking for particular spectral
features, especially at very low or very high energies, the LAT team has
experience with non-standard analysis.
* •
If you are trying to analyze the Galactic Center region, you are strongly
advised not to go it alone!
* •
If you are interested in the most complete multiwavelength coverage, consider
contacting the LAT team. The LAT team knows many cooperating groups across the
spectrum who may be interested in working with you (even if you do not include
the LAT team).
###### Acknowledgements.
The Fermi LAT Collaboration acknowledges generous ongoing support from a
number of agencies and institutes that have supported both the development and
the operation of the LAT as well as scientific data analysis. These include
the National Aeronautics and Space Administration and the Department of Energy
in the United States, the Commissariat à l’Energie Atomique and the Centre
National de la Recherche Scientifique / Institut National de Physique
Nucléaire et de Physique des Particules in France, the Agenzia Spaziale
Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry
of Education, Culture, Sports, Science and Technology (MEXT), High Energy
Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency
(JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research
Council and the Swedish National Space Board in Sweden. Additional support for
science analysis during the operations phase is gratefully acknowledged from
the Istituto Nazionale di Astrofisica in Italy and the Centre National
d’Études Spatiales in France.
## References
* (1) F. Camilo et al., “Radio Detection of LAT PSRs J1741-2054 and J2032+4127: No Longer Just Gamma-ray Pulsars”, ApJ 705, 1, 2009.
* (2) E. W. Bonning et al., “Correlated Variability in the Blazar 3C 454.3”, ApJ 697, L81, 2009.
* (3) A. A. Abdo et al., “Fermi/Large Area Telescope Discovery of Gamma-Ray Emission from a Relativistic Jet in the Narrow-Line Quasar PMN J0948+0022”, ApJ 699, 976, 2009.
* (4) A. A. Abdo et al., “PKS 1502+106: A New and Distant Gamma-ray Blazar in Outburst Discovered by the Fermi Large Area Telescope”, ApJ in press, 2010.
|
arxiv-papers
| 2009-12-29T15:47:42 |
2024-09-04T02:49:07.321234
|
{
"license": "Public Domain",
"authors": "D. J. Thompson (for the Fermi Large Area Telescope Collaboration)",
"submitter": "David J. Thompson",
"url": "https://arxiv.org/abs/0912.5320"
}
|
0912.5353
|
# Diversity-Multiplexing-Delay Tradeoffs in MIMO Multihop Networks with ARQ
Yao Xie1, Andrea Goldsmith1
Email: yaoxie@stanford.edu, andrea@wsl.stanford.edu 1Department of Electrical
Engineering, Stanford University, Stanford, CA.
###### Abstract
Tradeoff in diversity, multiplexing, and delay in multihop MIMO relay networks
with ARQ is studied, where the random delay is caused by queueing and ARQ
retransmission. This leads to an optimal ARQ allocation problem with per-hop
delay or end-to-end delay constraint. The optimal ARQ allocation has to trade
off between the ARQ error that the receiver fails to decode in the allocated
maximum ARQ rounds and the packet loss due to queueing delay. These two
probability of errors are characterized using the diversity-multiplexing-delay
tradeoff (DMDT) (without queueing) and the tail probability of random delay
derived using large deviation techniques, respectively. Then the optimal ARQ
allocation problem can be formulated as a convex optimization problem. We show
that the optimal ARQ allocation should balance each link performance as well
avoid significant queue delay, which is also demonstrated by numerical
examples.
## I Introduction
11101/04/2010. Submitted to The IEEE International Symposium on Information
Theory 2010.
In a multihop relaying system, each terminal receives the signal only from the
previous terminal in the route and, hence, the relays are used for coverage
extension. Multiple input-multiple output (MIMO) systems can provide increased
data rates by creating multiple parallel channels and increasing diversity by
robustness against channel variations. Another degree of freedom can be
introduced by an automatic repeat request (ARQ) protocol for retransmissions.
With the multihop ARQ protocol, the receiver at each hop feeds back to the
transmitter a one-bit indicator on whether the message can be decoded or not.
In case of a failure the transmitter sends additional parity bits until either
successful reception or message expiration. The ARQ protocol provides improved
reliability but also causes transmission delay of packets. Here we study a
multihop MIMO relay system using the ARQ protocol. Our goal is to characterize
the tradeoff in speed versus reliability for this system.
The rate and reliability tradeoff for the point-to-point MIMO system, captured
by the diversity-multiplexing tradeoff (DMT), was introduced in [1].
Considering delay as the third dimension in this asymptotic analysis with
infinite SNR, the diversity-multiplexing-delay tradeoff (DMDT) analysis for a
point-to-point MIMO system with ARQ is studied in [2], and the DMDT curve is
shown to be the scaled version of the corresponding DMT curve without ARQ. The
DMDT in relay networks has received a lot of attention as well (see, e.g.,
[3].) In our recent work [4], we extended the point-to-point DMDT analysis to
multihop MIMO systems with ARQ and proposed an ARQ protocol that achieves the
optimal DMDT.
The DMDT analysis assumes asymptotically infinite SNR. However, in the more
realistic scenario of finite SNR, retransmission is not a negligible event and
hence the queueing delay has to be brought into the picture (see discussions
in [5]). With finite SNR and queueing delay, the DMDT will be different from
that under the infinite SNR assumption. The DMDT with queueing delay is
studied in [5] and an optimal ARQ adapted to the instantaneous queue state for
the point-to-point MIMO system is presented therein.
In this work, we extend the study [5] of optimal ARQ assuming high but finite
SNR and queueing delay in point-to-point MIMO systems to multihop MIMO
networks. This work is also an extention our previous results in [4] to
incorporate queueing delay. We use the same metric as that used in [5], which
captures the probability of error caused by both ARQ error, and the packet
loss due to queueing delay. The ARQ error is characterized by information
outage probability, which can be found through a diversity-multiplexing-delay
tradeoff analysis [2, 4]. The packet loss is given by the limiting probability
of the event that packet delay exceeds a deadline. Unlike the standard queuing
models for networks (e.g., [6, 7]) where only the number of messages awaiting
transmission is studied, here we also need to study the amount of time a
message has to wait in the queue of each node. Our approach is slightly
different from [5], where the optimal ARQ decision is adapted per packet; we
study the queues after they enter the stable condition, and hence we use the
stationary probability of a packet missing a deadline. An immediate tradeoff
in the choice of ARQ round is: the larger the number of ARQ attempts we used
for a link, the higher the diversity and multiplexing gain we can achieve,
meaning a lower ARQ error. However, this is at a price of more packet missing
deadline. Our goal is to find an optimal ARQ allocation that balances these
two conflicting goals and equalizes performance of each hop to minimizes the
probability of error.
The remainder of this paper is organized as follows. Section II introduces
system models and the ARQ protocol. Section III presents our formulation and
main results. Numerical examples are shown in Section IV. Finally Section V
concludes the paper.
## II Models and Background
### II-A Channel and ARQ Protocol Models
Consider a multihop MIMO network consisting of $N$ nodes: with the source
corresponding to $i=1$, the destination corresponding to $i=N$, and
$i=2,\cdots,N-1$ corresponding to the intermediate relays, as shown in Fig. 1.
Each node is equipped with $M_{i}$ antennas. The packets enter the network
from the source node, and exit from the destination node, forming an open
queue. The network uses a multihop automatic repeat request (ARQ) protocol for
retransmission. With the multihop ARQ protocol, in each hop, the receiver
feeds back to the transmitter a one-bit indicator about whether the message
can be decoded or not. In case of a failure the transmitter retransmits. Each
channel block for the same message is called an ARQ round. We consider the
fixed ARQ allocation, where each link $i$ has a maximum of ARQ rounds $L_{i}$,
$i=1,\cdots N-1$. The packet is discarded once the maximum round has been
reached. The total number of ARQ rounds is limited to $L$:
$\sum_{i=1}^{N-1}L_{i}\leq L$. This fixed ARQ protocol has been studied in our
recent paper [4].
Figure 1: Upper: relay network with direct link from source to destination.
Lower: multihop MIMO relay network without direct link.
Assume the packets are delay sensitive: the end-to-end transmission delay
cannot exceed $k$. One strategy to achieve this goal is to set a deadline
$k_{i}$ for each link $i$ with $\sum_{i=1}^{N-1}k_{i}\leq k$. Once a packet
delays more than $k_{i}$ it is removed from the queue. This per-hop delay
constraint corresponds to the finite buffer at each node. Another strategy is
to allow large per-hop delay while imposing an end-to-end delay constraint.
Other assumptions we have made for the channel models are
* (i)
The channel between the $i$th and ($i+1$)th nodes is given by:
$\displaystyle\boldsymbol{Y}_{i,l}=\sqrt{\frac{SNR}{M_{i}}}\boldsymbol{H}_{i,l}\boldsymbol{X}_{i,l}+\boldsymbol{W}_{i,l},\quad
1\leq l\leq L_{i}.$ (1)
The message is encoded by a space-time encoder into a sequence of $L$ matrices
$\\{\boldsymbol{X}_{i,l}\in\mathcal{C}^{M_{i}\times T},:l=1,\cdots,L\\}$,
where $T$ is the block length, and
$\boldsymbol{Y}_{i,l}\in\mathcal{C}^{M_{i+1}\times T}$, $i=1,\cdots,N-1$, is
the received signal at the $(i+1)$th node, in the $l$th ARQ round. The rate of
the space-time code is $R$. Channels are assumed to be frequency non-
selective, block Rayleigh fading and independent of each other, i.e., the
entries of the channel matrices
$\boldsymbol{H}_{i,l}\in\mathcal{C}^{M_{i+1}\times M_{i}}$ are independent and
identically distributed (i.i.d.) complex Gaussian with zero mean and unit
variance. The additive noise terms $\boldsymbol{W}_{i,l}$ are also i.i.d.
complex Gaussian with zero mean and unit variance. The forward links and ARQ
feedback links only exist between neighboring nodes.
* (ii)
We consider both the full-duplex and half-duplex relays (see, e.g., [4]) where
the relays can or cannot transmit and receive at the same time, respectively,
as shown in Fig. 2. Assume the relays use a decode-and-forward protocol (see,
e.g., [4]).
* (iii)
We assume a short-term power constraint at each node for each block code.
Hence we do not consider power control.
* (iv)
We consider both the long-term static channel, where
$\boldsymbol{H}_{i,l}=\boldsymbol{H}_{i}$ for all $l$, i.e. the channel state
remains constant during all the ARQ rounds, and independent for different $i$.
Our results can be extended to the the short-term static channel using the
DMDT analysis given in [4].
Figure 2: Left: full duplex multihop relay network. Right: half duplex relay
multihop MIMO relay network. Figure 3: The logarithm of the cost function (10)
for the (4, 1, 2) multihop MIMO relay networks. SNR is 20 dB.
Figure 4: Allocations of optimal ARQ: $L_{1}^{*}$, $L_{2}^{*}$,
$L_{1}^{*}+L_{2}^{*}$, $k_{1}^{*}$, in a (4, 1, 2) MIMO relay network. SNR is
20 dB. (The optimal $k_{2}^{*}=k-k_{1}^{*}$.)
### II-B Queueing Network Model
We use an $M/M/1$ queue tandem to model the multihop relay networks. The
packets arrive at the source as a Poisson process with mean interarrival time
$\mu$, (i.e., the time between the arrival of the $n$th packet and $(n-1)$th
packet.) The random service time depends on the channel state and is upper
bounded by the maximum ARQ rounds allocated $L_{i}$. As an approximation we
assume the random service time at Node $i$ for each message is i.i.d. with
exponential distribution and mean $L_{i}$. With this assumption we can treat
each node as an $M/M/1$ queue. This approximation makes the problem tractable
and characterizes the qualitative behavior of MIMO multihop relay network.
Node $i$ has a finite buffer size. The packets enter into the buffer and are
first-come-first-served (FCFS). Assume $\mu\geq L_{i}$ so that the queues are
stable, i.e., the waiting time at a node does not go to infinity as time goes
on. Burke’s theorem (see, e.g., [7]) says that the packets depart from the
source and arrive at each relay as a Poisson process with rate $p_{i}/\mu$,
where $p_{i}$ is the probability that a packet can reach the $i$th node. With
high SNR, the packet reaches the subsequent relays with high probability:
$p_{i}\approx 1$ (the probability of a packet dropping is small because it
uses up the maximum ARQ round.) Hence all nodes have packets arrive as a
Poisson process with mean inter-arrival time $\mu$.
### II-C Throughput
Denote by $b$ the size of the information messages in bits, $B[t]$ the number
of bits removed from transmission buffer at the source at time slot $t$.
Define a renewal event as the event that the transmitted message leaves the
source and eventually is received by the destination node possibly after one
or more ARQ retransmissions. We assume that under full-duplex relays the
transmitter cannot send a new message until the previous message has been
decoded by the relay at which point the relay can begin transmission over the
next hop (Fig. 2a.) Under half duplex relays we assume transmitter cannot send
a new message until the relay to the next hop completes its transmission (Fig.
2b.)
The number of bits $\bar{B}$ transmitted in each renewal event, for full-
duplexing $\bar{B}=(N-1)b$, and for half-duplexing $\bar{B}=(N-1)b/2$ when $N$
is odd, and $\bar{B}=Nb/2$ when $N$ is even. The long-term average throughput
of the ARQ protocol is defined as the transmitted bits per channel use (PCU)
[2], which can be found using renewal theory [8]:
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\liminf_{s\rightarrow\infty}\frac{1}{Ts}\sum_{t=1}^{s}B[t]=\frac{\bar{B}}{E(\tau)}\doteq\frac{\bar{B}}{(N-1)T}$
(5) $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}R,&\hbox{Full
duplex;}\\\ \frac{R}{2},&\hbox{Half duplex, $N$ is odd;}\\\
R\left(\frac{1}{2}+\frac{1}{2N}\right),&\hbox{Half duplex, $N$ is even.}\\\
\end{array}\right.$
where $\tau$ is the average duration from the time a packet arrives at the
source until it reaches the destination node, and $\doteq$ denotes asymptotic
equality. A similar argument as in [2] shows that $E(\tau)\doteq(N-1)T$ for
high SNR.
### II-D Diversity-Multiplexing-Delay Tradeoff
The probability of error $P_{e}$ in the transmission has two sources: from the
ARQ error: the packet is dropped because the receiver fails to decode the
message within the allocated number of ARQ rounds, denoted as
$P_{\mbox{\tiny{ARQ}}}$, and the probability that a message misses its
deadline at any node due to large queueing delay, denoted as
$P_{\mbox{\tiny{Queue}}}$. We will give $P_{e}$ for various ARQ relay
networks. Following the framework of [1], we assume the size of information
messages $b(\rho)$ depends on the operating signal-to-noise ratio (SNR)
$\rho$, and a family of space time codes $\\{\mathcal{C}_{\rho}\\}$ with block
rate $R(\rho)=b(\rho)/T\triangleq r\log\rho$. We use the effective ARQ
multiplexing gain and the ARQ diversity gain [2]
$\displaystyle
r_{e}\triangleq\lim_{\rho\rightarrow\infty}\frac{\eta(\rho)}{\log\rho},\quad
d\triangleq-\lim_{\rho\rightarrow\infty}\frac{\log P_{e}(\rho)}{\log\rho}.$
(6)
We cannot assume infinite SNR because otherwise the queueing delay will be
zero, as pointed out in [5]. However we assume high SNR to use the DMDT
results in our subsequent analysis.
## III Diversity, multiplexing, and delay tradeoff via optimal ARQ round
allocation
### III-A Full-Duplex Relay in Multihop Relay Network
#### III-A1 Per-Hop Delay Constraint
The probability of error depends on the ARQ window length allocation $L_{i}$,
deadline constraint $k_{i}$, multiplexing rate $r$, and SNR $\rho$. For a
given $r$ and $\rho$, we have
$\displaystyle P_{e}(\\{L_{i}\\},\\{k_{i}\\}|\rho,r)=$ $\displaystyle
P_{\mbox{\tiny{ARQ}}}(\rho,\\{L_{i}\\})+\sum_{i=1}^{N-1}P_{\mbox{\tiny{Queue}}}(D_{i}>k_{i}).$
(7)
Here $D_{i}$ denotes the random delay at the $i$th link when the queue is
stationary. This $P_{e}$ expression is similar to that given by Equation (33)
of [5]. Our goal is to allocate per-hop ARQ round $\\{L_{i}\\}$ and delay
constraint $\\{k_{i}\\}$ to minimize the probability of error $P_{e}$.
For the long-term static channel, using the DMDT analysis results [4] we have:
$\displaystyle
P_{\mbox{\tiny{ARQ}}}(\rho,\\{L_{i}\\})=\sum_{i=1}^{N-1}\rho^{-f_{i}\left(\frac{r}{L_{i}}\right)}.$
(8)
Here $f_{i}(r)$ is the diversity-multiplexing tradeoff (DMT) for a point-to-
point MIMO system formed by nodes $i$ and $i+1$. Assuming sufficient long
block lengths, $f_{i}(r)$ is given by Theorem 2 in [1] quoted in the
following:
###### Theorem 1
[1] For sufficiently long block lengths, the diversity-multiplexing tradeoff
(DMT) $f(r)$ for a MIMO system with $M_{t}$ transmit and $M_{r}$ receive
antennas is given by the piece-wise linear function connecting the points
$(r,(M_{t}-r)(M_{r}-r)),$ for $r=0,\cdots,\min(M_{t},M_{r})$.
Denote the amount of time spent in the $i$th node by the $n$th message as
$D_{n}^{i}$. The probability of packet loss
$P_{\mbox{\tiny{Queue}}}(D_{i}>k_{i})$ can be found as the limiting
distribution of $\lim_{n\rightarrow\infty}P(D_{n}^{i}>k_{i})$ (adapted from
Theorem 7.4.1 of [8]):
###### Lemma 2
The limiting distribution of the event that the delay at node $i$ exceeds its
deadline $k_{i}$, for $M/M/1$ queue models, is given by:
$\displaystyle
P_{\mbox{\tiny{Queue}}}(D_{i}>k_{i})=\lim_{n\rightarrow\infty}P(D_{n}^{i}>k_{i})=\frac{L_{i}}{\mu}e^{-k_{i}\left(\frac{1}{L_{i}}-\frac{1}{\mu}\right)}.$
(9)
Here the difference in the service rate and packet arrival rate
$\frac{1}{L_{i}}-\frac{1}{\mu}\geq 0$ and utility factor $\frac{L_{i}}{\mu}$
both indicate how “busy” node $m$ is. Using the above results, (7) can be
written as
$\displaystyle
P_{e}\left(\\{L_{i}\\},\\{k_{i}\\}|\rho,r\right)=\sum_{i=1}^{N-1}\left[\rho^{-f_{i}\left(\frac{r}{L_{i}}\right)}+\frac{L_{i}}{\mu}e^{-k_{i}\left(\frac{1}{L_{i}}-\frac{1}{\mu}\right)}\right].$
(10)
Note that the queueing delay message loss error probability is decreasing in
$L_{i}$, and the ARQ error probability is increasing in $L_{i}$. Hence an
optimal ARQ rounds allocation at each node $L_{i}$ should trade off these two
terms. Also, the optimal ARQ allocation should also equalize the performance
of each link, as the weakest link determines the system performance[4].
Hence the optimal ARQ allocation can be formulated as the following
optimization problem:
$\begin{split}\min_{\\{L_{i}\\},\\{k_{i}\\}\in\mathcal{A}}&P_{e}(\\{L_{i}\\},\\{k_{i}\\}|\rho,r)\end{split}$
(11)
where
$\displaystyle\mathcal{A}=\left\\{\begin{array}[]{l}\sum_{i=1}^{N-1}L_{i}\leq
L,\\\ 1\leq L_{i}\leq\mu,\quad i=1,\cdots,N-1\\\ \sum_{i=1}^{N-1}k_{i}\leq
k.\\\ \end{array}\right\\}$ (15)
The following lemma (proof omitted due to the space limit) shows that the
total transmission distortion function (21) is convex in the interior of
$\mathcal{A}$.
###### Lemma 3
The transmission distortion function (21) is convex jointly in $L_{i}$ and
$k_{i}$ in the convex set
$\displaystyle\left\\{\\{L_{i}\\},\\{k_{i}\\}:k_{i}>\frac{L_{i}}{2(\frac{\mu}{L_{i}}-1)},\quad
i=1,\cdots N-1.\right\\},$
Lemma 3 says that except for the “corners” of $\mathcal{A}$ the cost function
is convex. However these “corners” have higher probability of error: $k_{i}$
and $L_{i}$ take extreme values and hence one link may have a longer queueing
delay then the others. So we only need to search the interior of $\mathcal{A}$
where the cost function is convex.
To gain some insights into where the optimal solution resides in the feasible
domain for the above problem, we present a marginal cost interpretation. Note
that the probability of error can be decomposed as a sum of probability of
error on the $i$th link. The optimal ARQ rounds allocated on this link should
equalize the “marginal cost” of the ARQ error and the packet loss due to
queueing delay. For node $i$, with fixed $k_{i}$, the marginal costs (partial
differentials) of the ARQ error probability, and the packet loss probability
due to queueing delay, with respect to $L_{i}$ are given by
$\displaystyle\frac{\partial\rho^{-f_{i}\left(\frac{r}{L_{i}}\right)}}{\partial
L_{i}}=\frac{r}{L_{i}^{2}}f^{\prime}_{i}\left(\frac{r}{L_{i}}\right)\rho^{-f_{i}\left(\frac{r}{L_{i}}\right)}\ln\rho<0,$
(16)
and
$\displaystyle\frac{\partial P_{\mbox{\tiny{Queue}}}(D_{i}>k_{i})}{\partial
L_{i}}=\frac{1}{\mu}\left(1+\frac{k}{L_{i}}\right)e^{-k_{i}\left(\frac{1}{L_{i}}-\frac{1}{\mu}\right)}>0.$
(17)
Note that $f^{\prime}_{i}<0$. The optimal solution equalizes these two
marginal costs by choosing $L_{i}\in[1,\mu]$. Note that these marginal cost
functions are monotone in $L_{i}$, hence the equalizing $L_{i}^{*}$ exists and
$1<L_{i}^{*}<\mu$ if the following two conditions are true for $L_{i}=1$ and
$L_{i}=\mu$:
$\displaystyle(i):\left.\frac{\partial
P_{\mbox{\tiny{Queue}}}(D_{i}>k_{i})}{\partial
L_{i}}\right|_{L=1}<-\left.\frac{\partial\rho^{-f_{i}\left(\frac{r}{L_{i}}\right)}}{\partial
L_{i}}\right|_{L=1},$ (18) $\displaystyle(ii):\left.\frac{\partial
P_{\mbox{\tiny{Queue}}}(D_{i}>k_{i})}{\partial
L_{i}}\right|_{L=\mu}>-\left.\frac{\partial\rho^{-f_{i}\left(\frac{r}{L_{i}}\right)}}{\partial
L_{i}}\right|_{L=\mu},$ (19)
These conditions involve nonlinear inequalities involving $\mu$, $\rho$, $r$,
$M_{i}$ and $M_{i+1}$, which defines the case when the optimal solution is in
the interior of $\mathcal{A}$. Analyzing these conditions reveals that these
conditions tend to satisfy at lower multiplexing gain $r$, small $M_{i}$ or
$M_{i+1}$, small $k_{i}$, and larger $\mu$ (light traffic). Note that with
high SNR condition (ii) is always true for moderate $k$ values. When $(i)$ and
$(ii)$ are violated, which means one error dominates the other, then the
optimal solution lies at the boundary of $\mathcal{A}$. With the total ARQ
rounds constraint in (11), using the Lagrangian multiplier an argument similar
to above still holds.
#### III-A2 End-to-End Delay constraint
When the buffer per node is large enough a per hop delay constraint is not
needed, and we can instead impose an end-to-end delay constraint. The exact
expression for the tail probability of the end-to-end delay is intractable.
However a large deviation result is available. The following theorem can be
derived using the main theorem in [9]:
###### Theorem 4
For a stationary $M/M/1$ queue tandem (with full-duplex relays):
$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\frac{1}{k}\log
P_{\mbox{\tiny{Queue}}}\left(\sum_{i=1}^{N-1}D_{n}^{i}\geq
k\right)=-\theta^{*},$
where
$\theta^{*}=\min_{i=1}^{N-1}\left\\{\frac{1}{L_{i}}-\frac{1}{\mu}\right\\}$.
This theorem says that the bottleneck of the queueing network is the link with
longest mean service time $L_{i}$. Hence the optimal ARQ round allocation
problem can be formulated as:
$\displaystyle\min_{\\{L_{i}\\}\in\mathcal{B}}$ $\displaystyle
P_{e}(\\{L_{i}\\},\\{k_{i}\\}|\rho,r)$ (20)
where
$\displaystyle P_{e}\left(\\{L_{i}\\},\\{k_{i}\\}|\rho,r\right)$
$\displaystyle=P_{\mbox{\tiny{ARQ}}}(\rho,\\{L_{i}\\})+P_{\mbox{\tiny{Queue}}}\left(\sum_{i=1}^{N-1}D_{n}^{i}\geq
k\right),$
$\displaystyle\doteq\sum_{i=1}^{N-1}\rho^{-f_{i}\left(\frac{r}{L_{i}}\right)}+e^{-\theta^{*}k}.$
(21)
$\displaystyle\mathcal{B}=\left\\{\begin{array}[]{l}\sum_{i=1}^{N-1}L_{i}\leq
L,\\\ 1\leq L_{i}\leq\mu,\quad i=1,\cdots,N-1\end{array}\right\\}$ (24)
For high SNR, this can be shown to be a convex optimization problem. A simple
argument can show that the packet loss probability with the per-hop delay
constraint is larger than that using the more flexible end-to-end constraint.
### III-B Half-duplex Relay in Multihop Network
Half-duplex relay is not a standard queue tandem model. However we can also
derive a large deviation result for the tail probability for the end-to-end
delay of a multihop network with half-duplex relays (proof in the Appendix):
###### Theorem 5
For a stationary $M/M/1$ queue tandem (with half-duplex relays), when the
number of node $N$ is large:
$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\frac{1}{k}\log
P_{\mbox{\tiny{Queue}}}\left(\sum_{i=1}^{N-2}D_{n}^{i}\geq
k\right)=-\theta^{*}.$ (25)
From this theorem we conclude that the optimal ARQ allocation problem with the
end-to-end constraint and half-duplex relays can be formulated the same as
that with full-duplex relays (20).
## IV Numerical Examples
Consider a MIMO relay network consists of a source, a relay, and a destination
node. The relay is full-duplex. The number of antennas on each node is
$(M_{1},M_{2},M_{3})$, $M_{1}=4$, $M_{2}=1$, and $M_{3}=2$, where the relay
has a single antenna. Other parameters are: $\rho=20$dB, $k=30$, $L=8$, and
the multiplexing gain is $r=2$. The base 10 logarithm of the cost function
(10) is shown in Fig. 3. We have optimized the cost function with respect to
$L_{2}$ and $k_{2}$ so we can display it in three dimensions. Note that the
surface is convex in the interior of the feasible region. The optimal
$L_{1}^{*}$, $L_{2}^{*}$, $k_{1}^{*}$ are shown in Fig. 4. Also note that as
$r$ increases to the maximum possible $r=4$, the total number of ARQ rounds
allocated $L_{1}^{*}+L_{2}^{*}$ gradually increases to the upper bound $L=8$
as $k$ increases.
## V Conclusions and Future Work
We have studied the diversity-multiplexing-delay tradeoff in multihop MIMO
networks by considering an optimal ARQ allocation problem to minimize the
probability of error, which consists of the ARQ error and the packet loss due
to queueing delay. Our contribution is two-fold: we combine the DMDT analysis
with queueing network theory, and we use the tail probability of random delay
to find the probability of packet loss due to queueing delay. Numerical
results show that optimal ARQ should equalize the performance of each link and
avoid long service times that cause large queueing delay. Future work will
investigate joint source-channel coding in multihop MIMO relay networks,
extending the results of [5].
Proof of Theorem 5
For node $i$, $i=1\cdots N$, let the random variable $S_{n}^{i}$ denotes the
service time required by the $n$th customer at the $i$th node (the number of
ARQs used for the $n$th packet), and $A_{n}^{i}$ be the inter arrival time of
the $n$th packets (i.e., the time between the arrival of the $n$th and
$(n-1)$th packages to this node). The waiting time of the $n$th packet at the
$i$th node $W_{n}^{i}$ satisfies Lindley’s recursion (see [9]):
$\displaystyle W_{n}^{i}=(W_{n-1}^{i}+S_{n-1}^{i+1}-A_{n}^{i})^{+},\quad 2\leq
i\leq N-2,$ (26)
where $(x)^{+}=\max(x,0)$. The total time a message spent in a node is its
waiting time plus its own service time, hence
$\displaystyle D_{n}^{i}=W_{n}^{i}+S_{n}^{i}.$ (27)
The arrival process to the $(i+1)$th node is the departure process from the
$i$th node, which satisfies the recursion:
$\displaystyle A_{n}^{i}=A_{n}^{i-1}+D_{n}^{i-1}-D_{n-1}^{i},\quad 2\leq i\leq
N-2.$ (28)
with $A_{n}^{i}$ a Poisson process with rate $1/\mu$. Also the waiting time at
the source satisfies:
$\displaystyle W_{n}^{1}=(W_{n-1}^{1}+S_{n-1}^{1}+S_{n-1}^{2}-A_{n}^{1})^{+}.$
(29)
A well-known result is that (see, e.g. [9]), if the arrival and service
processes satisfy the stability condition, then the Lindley’s recursion has
the solution:
$\displaystyle W_{n}^{i}$ $\displaystyle=$ $\displaystyle\max_{j_{i}\leq
n}(\sigma^{i}_{j_{i},n-1}-\tau^{i}_{j_{i}+1,n}),\quad i=2,\cdots N-2,$
$\displaystyle W_{n}^{1}$ $\displaystyle=$ $\displaystyle\max_{j_{1}\leq
j_{2}}(\sigma^{1}_{j_{1},j_{2}-1}+\sigma^{2}_{j_{1},j_{2}-1}-\tau^{1}_{j_{1}+1,j_{2}}).$
(30)
where the partial sum $\tau_{l,p}^{i}=\sum_{k=l}^{p}A_{k}^{i}$ and
$\sigma_{l,p}=\sum_{k=l}^{p}S_{k}^{i}$. Hence
$\displaystyle D_{n}^{i}=\max_{j_{i}\leq
n}(\sigma^{i}_{j_{i},n-1}+S_{n}^{i}-\tau^{i}_{j_{i}+1,n}),\quad i=2,\cdots
N-2.$ (31)
From (28) we have $\tau_{l,p}^{i}=\tau_{l,p}^{i-1}+D_{p}^{i-1}-D_{l-1}^{i-1}$
for $l\leq p+1$, and 0 otherwise. Plug this into (31) we have
$\displaystyle D_{n}^{i}=\max_{j_{i}\leq
n}(\sigma^{i+1}_{j_{i},n-1}+S_{n}^{i}-\tau_{j_{i}+1,n}^{i-1}-D_{n}^{i-1}+D_{j_{i}}^{i-1}).$
(32)
Hence the recursive relation if we move $D_{n}^{i-1}$ to the left-hand-side:
$\displaystyle D_{n}^{i}+D_{n}^{i-1}=\max_{j_{i}\leq
n}(\sigma_{j_{i},n-1}^{i+1}+S_{n}^{i}-\tau_{j_{i}+1,n}^{i-1}+D_{j_{i}}^{i-1}).$
(33)
Now from (31) we have $D_{j_{i}}^{i-1}=\max_{j_{i-1}\leq
j_{i}}(\sigma^{i}_{j_{(i-1)},j_{i}-1}+S_{j_{i}}^{i-1}-\tau_{j_{(m-1)}+1,j_{m}}^{i-1})$.
Plug this in the above (33) we have
$\displaystyle D_{n}^{i}+D_{n}^{i-1}$ $\displaystyle=$
$\displaystyle\max_{j_{(i-1)}\leq j_{i}\leq
n}(\sigma_{j_{i},n-1}^{i+1}+S_{n}^{i}+\sigma^{i}_{j_{(i-1)},j_{i}-1}+S_{j_{i}}^{i-1}-\tau_{j_{(i-1)}+1,n}^{i-1})$
Do this inductively, we have
$\displaystyle\sum_{i=2}^{N-2}D_{n}^{i}=\max_{j_{2}\leq\cdots\leq
j_{N-1}=n}\left[\sum_{m=2}^{N-2}(\sigma^{i+1}_{j_{i},j_{(i+1)}-1}+S_{j_{i+1}}^{i})-\tau^{1}_{j_{2}+1,n}\right].$
If we also add $D_{n}^{1}=W_{n}^{1}+S_{n}^{1}$ to the above equation, after
rearranging terms we have:
$\displaystyle\sum_{i=1}^{N-2}D_{n}^{i}=\sum_{i=2}^{N-2}(\sigma^{i}_{j_{(i-1)},j_{i}-1}+S_{j_{(i+1)}}^{i})-\tau^{1}_{j_{2}+1,n}$
$\displaystyle+S_{j_{2}}^{1}+\sigma_{j_{1},j_{2}-1}^{1}+S_{j_{2}}^{2}+\sigma_{j_{N-2},j_{(N-1)}-1}^{N-1}.$
(34)
Note that $\sigma^{i}_{j_{(i-1)},j_{i}-1}$ is independent of
$S_{j_{(i+1)}}^{i}$. For long queue we can ignored the last four terms caused
by edge effect (the source and end queue of the multihop relay network). By
stationarity of the service process
$\sigma^{i}_{j_{(i-1)},j_{i}-1}+S_{j_{(i+1)}}^{i}$ has the same distribution
as $\sigma^{i}_{0,j_{i}-j_{(i-1)}}$. Then (34) reduces to the case studied in
[9] and we can borrow the large deviation argument therein to derive the
exponent $\theta^{*}$.
## References
* [1] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inform. Theory, vol. 49, pp. 1073–1096, 2003.
* [2] H. El Gamal, G. Caire, and M. O. Damen, “The MIMO ARQ channel: Diversity-multiplexing-delay tradeoff,” IEEE Trans. Inform. Theory, vol. 52, pp. 3601–3621, 2006.
* [3] T. Tabet, S. Dusad, and R. Knopp, “Diversity-multiplexing-delay tradeoff in half-duplex ARQ relay channels,” IEEE Transactions on Information Theory, vol. 53, pp. 3797–3805, October 2007.
* [4] Y. Xie, D. Gunduz, and A. Goldsmith, “Multihop MIMO relay networks with ARQ,” IEEE Globecom 2009 Communication Theory Symposium, Dec. 2009.
* [5] T. Holliday, A. J. Goldsmith, and H. V. Poor, “Joint source and channel coding for MIMO systems: Is it better to be robust or quick?,” IEEE Transactions on Information Theory, vol. 54, no. 4, 2008.
* [6] N. Bisnik and A. Abouzeid, “Queueing network models for delay analysis of multihop wireless Ad Hoc networks,” pp. 773 – 778, Proceedings of the 2006 International Conference on Wireless Communications and Mobile Computing, July 2006.
* [7] G. Bolch, S. Greiner, and H. de Meer, Queueing Networks and Markov Chains : Modeling and Performance Evaluation With Computer Science Applications. Springer Series in Statistics, Wiley-Interscience, 2 ed., Aug. 2006.
* [8] S. M. Ross, Stochastic Processes. John Wiley & Sons, 2 ed., 1995.
* [9] A. J. Ganesh, “Large deviations of the sojourn time for queues in series,” Annals of Operations Research, vol. 79, pp. 3–26, Jan. 1998.
|
arxiv-papers
| 2009-12-29T19:07:31 |
2024-09-04T02:49:07.326619
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yao Xie, Andrea Goldsmith",
"submitter": "Yao Xie",
"url": "https://arxiv.org/abs/0912.5353"
}
|
0912.5430
|
11institutetext: Astrophysikalisches Institut Potsdam, An der Sternwarte 16,
D-14482 Potsdam, Germany
11email: csandin@aip.de, rjacob@aip.de, deschoenberner@aip.de,
msteffen@aip.de, mmroth@aip.de
# The evolution of planetary nebulae
VI. On the chemical composition of the metal-poor PN G135.9+55.9††thanks:
Based in part on observations collected at the Centro Astronómico Hispano
Alemán (CAHA), operated jointly by the Max-Planck-Institut für Astronomie and
the Instituto de Astrofisica de Andalucia (CSIC).
C. Sandin R. Jacob D. Schönberner M. Steffen M. M. Roth
(Received February 5, 2009 / Accepted December 22, 2009)
The actual value of the oxygen abundance of the metal-poor planetary nebula PN
G135.9+55.9 has frequently been debated in the literature. We wanted to
clarify the situation by making an improved abundance determination based on a
study that includes both new accurate observations and new models. We made
observations using the method of integral field spectroscopy with the PMAS
instrument, and also used ultraviolet observations that were measured with
HST-STIS. In our interpretation of the reduced and calibrated spectrum we used
for the first time, recent radiation hydrodynamic models, which were
calculated with several setups of scaled values of mean Galactic disk
planetary nebula metallicities. For evolved planetary nebulae, such as PN
G135.9+55.9, it turns out that departures from thermal equilibrium can be
significant, leading to much lower electron temperatures, hence weaker
emission in collisionally excited lines. Based on our time-dependent
hydrodynamic models and the observed emission line $[{O\textsc{iii}}]\,\lambda
5007$, we found a very low oxygen content of about 1/80 of the mean Galactic
disk value. This result is consistent with emission line measurements in the
ultraviolet wavelength range. The C/O and Ne/O ratios are unusually high and
similar to those of another halo object, BoBn-1.
###### Key Words.:
ISM: planetary nebulae: general – ISM: planetary nebulae: individual (BoBn-1,
PN G135.9+55.9) – Hydrodynamics
††offprints: csandin@aip.de
## 1 Introduction
The stellar-like object SBS 1150+599A from the Second Byurakan Survey (Balayan
1997) has been spectroscopically identified by Tovmassian et al. (2001,
hereafter T01; ) to be an old planetary nebula (PN) of the Galactic halo and
was thereafter renamed PN G135.9+55.9. The same authors perform an abundance
study based on photoionization models and come to the conclusion that this
particular object has the lowest oxygen abundance known so far for planetary
nebulae, viz. below about 1/100 of the solar value.
This object appears to be a challenge for any detailed spectroscopic analysis
since very few emission lines are detectable in the optical wavelength range.
Because of a lack of suitable lines, it is impossible to make a direct plasma
diagnostic, and it is also difficult to constrain photoionization models such
that meaningful abundances will emerge. The analysis is, moreover, hampered
since this object is faint, with a mean $\text{H}\beta$ surface brightness of
only $\sim\\!5\\!\times\\!10^{-16}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}\,\text{arcsec}^{-2}$ (adopting a
diameter of 5″ and a total $\text{H}\beta$ flux of $1.9\\!\times\\!10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$; see Table 1), making it difficult
to accurately measure lines close to the 1% level of $\text{H}\beta$
(corresponding to $\simeq\\!2\\!\times\\!10^{-16}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$).
These observational difficulties have led to a dispute about whether PN
G135.9+55.9 is as extremely metal-deficit as T01 claim. A critical point in
this discussion is the determination of the stellar temperature from the
ionization balance, using the line ratio of $[{Ne\textsc{v}}]\,\lambda 3426$
and $[{Ne\textsc{iii}}]\,\lambda 3869$ in the nebula. The line strength of
$[{Ne\textsc{iii}}]\,\lambda 3869$ is uncertain, but decisive when fixing the
effective temperature ($T_{\text{eff}}$) of the ionizing source. Richer et al.
(2002, hereafter R02) and Jacoby et al. (2002, hereafter J02) find a rather
low $T_{\text{eff}}\\!\simeq\\!100\,000$ K, hence low oxygen abundances of
less than 1/100 solar from the weak $[{O\textsc{iii}}]\,\lambda 5007$ line,
the only oxygen line that is observed in the optical.
Tovmassian et al. (2004, hereafter T04) conclude, by means of optical and
ultraviolet (UV) spectra (FUSE), that the central ionizing source of PN
G135.9+55.9 must be a very hot ($T_{\text{eff}}\\!\approx\\!120\,000$ K) pre-
white dwarf of rather low mass ($\approx\\!0.55$ $\text{M}_{\odot}$), which
resides in a short-period binary system with a more massive companion
($P\\!=\\!3.92$ h, Napiwotzki et al. 2005). Recently Tovmassian et al. 2007,
based on X-ray observations, have estimated that the temperature of this
massive component is very high, $T_{\text{eff}}\\!\simeq\\!170\,000\,$K, while
the less massive optical-UV component is cooler,
$T_{\text{eff}}\\!\simeq\\!58\,000\,$K.
Péquignot & Tsamis (2005, hereafter PT05), moreover, critically examine the
situation and come to the conclusion that all evidence favors a higher
temperature of the central source, viz. $T_{\text{eff}}\\!\simeq\\!130\,000$
K, and that the claimed strength of $[{Ne\textsc{iii}}]\,\lambda 3869$ most
likely is wrong. The oxygen abundance would then be 1/30–1/15 solar and not as
extreme as previously estimated. Jacoby et al. (2006, hereafter J06) also find
a higher stellar temperature, using the much stronger UV emission lines of
highly ionized nitrogen and carbon to constrain $T_{\text{eff}}$. Stasińska et
al. (2005, hereafter S05) include UV lines in their study (likely using the
same HST-STIS data as J06) and conclude that the oxygen abundance is lower
than what PT05 find, 1/130–1/40.
Our aim was to better measure the nebular emission line spectrum, which is why
we performed new observations using the integral-field spectrograph PMAS. With
such a spectrum we could determine a reliable line strength, or an upper
limit, of $[{Ne\textsc{iii}}]\,\lambda 3869$. We begin in Sect. 2 with a
description of our observations and data processing, we then present our
results in Sect. 3. We describe the physical setup of our radiation
hydrodynamic models, how we estimate abundances and compare observations with
our models, in Sect. 4. In Sect. 5 we discuss non-equilibrium effects and
compare our abundances with values of previous studies found in the
literature. We close the paper with our conclusions in Sect. 6.
## 2 Observations and data reduction
Our observations were made with the 3.5 m telescope at Calar Alto using the
lens array (LARR) integral field unit (IFU) of the PMAS instrument (Roth et
al. 2005). The V600 grating was used to cover the spectral interval 3490–5150
Å; at a dispersion of 0.81 Å pixel-1 and a resolving power of ${R=1340}$. The
wavelength range was chosen to cover the Balmer lines $\text{H}\beta$ and
H$\gamma$, as well as [O iii]$\,\lambda\lambda\,4959,\,5007$, He ii$\,\lambda
4686$, and [Ne iii]$\,\lambda 3869$. The LARR IFU, furthermore, holds
$16\\!\times\\!16$ separate fibers, where each fiber represents a spatial
element on the sky. We used the 0$\aas@@fstack{\prime\prime}$5 sampling mode
where every pointing with the IFU covers an area of ${8\arcsec\times
8\arcsec=64\arcsec^{2}}$ on the sky. Any number of spatial elements can be co-
added to create a final spectrum.
Two 2700 s exposures, which were both centered on PN G135.9+55.9, were taken
at an airmass of 1.09–1.12 on 2007 February 13. Two 2700 s exposures were also
taken at an airmass of 1.12–1.19 on 2007 February 14. These two latter
exposures were offset by 5″ E from the first two exposures. One additional
1800 s exposure was taken at an airmass of 1.09 and an offset of 5″ W in the
second night. Weather conditions were less than optimal, and the seeing was
1$\aas@@fstack{\prime\prime}$4–1$\aas@@fstack{\prime\prime}$7\. In addition to
the science exposures continuum and arc lamp flatfields were taken, as well as
spectrophotometric standard-star exposures of G191-B2B. In order to correct
for a varying fiber-to-fiber transmission sky flats were taken at the
beginning of the second night.
Reducing the data we used the tool P3d_online that is a part of the PMAS P3d
pipeline (Becker 2002; Roth et al. 2005). At first the bias level was
subtracted and cosmic-ray hits removed (cf. Sandin et al. 2008). Second, a
trace mask was generated from an internal continuum calibration lamp exposure,
identifying the location of each spectrum on the CCD along the direction of
cross-dispersion. Third, a dispersion mask for wavelength calibration was
created using an arc-exposure. In order to minimize effects due to a
significant flexure in the instrument all continuum and arc lamp exposures
were taken within two hours of the respective science exposure. Fourth, a
correction to fiber-to-fiber sensitivity variations was applied by dividing
with an extracted and normalized sky flat-field exposure. In this process the
data was changed from a CCD-based format to a row-stacked-spectra format. As a
final step flux calibration was performed in iraf using standard-star
exposures. We did not correct for differential atmospheric refraction since
we, due to poor spatial resolution and high seeing, have co-added the flux of
147 spatial elements. These elements fully cover the object with an area of
$36.75\arcsec^{2}$. In order to avoid the periodic line shift of stellar
absorption lines of the central star (CS) we made a second spectrum where we
did not add the nine spatial elements, which were the closest to the CS at
$\lambda_{\text{H}\beta}$ (we also used fewer elements on the outer boundary);
this latter spectrum covers an area of $30.75\arcsec^{2}$.
$\text{H}\beta$ is blended by the helium Pickering line
${He\textsc{ii}}\,\lambda 4859$, and the resulting line intensity is hereby
about 5% too high (cf. J02; PT05). The other Balmer lines are likewise blended
by other helium Pickering lines. Since we did not include the CS in our final
spectrum we have not corrected for underlying stellar absorption as PT05 do.
The discussion of the dereddening performed by R02 remains inconclusive, with
the result of a somewhat uncertain and very low extinction towards PN
G135.9+55.9. Also T04 find a low interstellar extinction towards this object
($E(B\\!-\\!V)\\!=\\!0.04$) that is mainly based on the $N_{\text{H\,I}}$
column density they derive from FUSE spectra. Owing to the large differences
of the line strengths found by different observers (see below), however, any
(rather small) correction due to dereddening appears unimportant at this point
of our discussion.
## 3 Results for PN G135.9+55.9
We made all line fits using the IFU analysis package ifsfit (Sandin, in prep.)
that was initially developed for Sandin et al. (2008). We used a second order
polynomial to fit the continuum and Gaussian curves to fit emission lines.
Because of less favorable weather conditions in the second night we only used
the co-added spectrum of the two exposures of the first night, totaling an
exposure time of 5400 s, see Fig. 1.
The intensities of all object lines we measured are presented in Table 1,
together with the values of R02, J02, and T01. We present both raw values, and
values where we subtracted the contribution of He ii Pickering lines from the
Balmer lines. Note that errors of our intensity measurements decrease for
redder emission lines, reflecting the better signal-to-noise of the spectrum
in the redder region. Since the first night was not photometric our value of
the integrated flux of $\text{H}\beta$ is very likely somewhat uncertain.
Nevertheless, we calculated a spatially integrated flux of $\text{H}\beta$,
using the spectrum where the CS is included, of
$1.92\\!\times\\!10^{-14}\,$$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$ (Table
1), and this value compares well with values given in the literature. R02 find
values ranging from $1.0\,\ldots\,2.6\\!\times\\!10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$, and J02 evaluate a total
$\text{H}\beta$ flux of $1.5\\!\times\\!10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$. While J02 correct their flux
estimate for the region that is not covered by the slit, the varying values of
R02 seem to be a result of the width of the slit they use.
We did not detect $[{Ne\textsc{iii}}]\,\lambda 3869$ (cf. the upper inset of
Fig. 1). Since the signal-to-noise of our spectrum appears to be better than
in previous studies, with five new measured lines in the nebula (see Table 1),
it is questionable if anyone has detected $[{Ne\textsc{iii}}]\,\lambda 3869$.
In order to calculate an upper detection limit of emission lines in our data
we proceeded as follows. For a certain wavelength we assumed that we can
detect a line with the maximum flux of $\sigma$ above the continuum, where
$\sigma$ is the error of the measured flux. We calculated the intensity $I$ of
this limiting line by integrating over a triangle of width 7Å, and
$I\\!=\\!7/2\times\sigma$. At the three wavelengths
$\lambda\\!=\\!3790,\,3860,\,\text{and}\,5010\,$Å we measured
$\sigma\\!=\\!6.17,\,4.70,\,\text{and}\,1.46\\!\times\\!10^{-17}\,$$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}\,\text{\AA}^{-1}$.
Dividing the resulting intensities with the intensity of $\text{H}\beta$ we
got the following limiting line ratios ($\times\\!100$): 1.1
($\lambda\\!=\\!3790\,\AA$), 0.85 ($\lambda\\!=\\!3860\,\AA$), and 0.27
($\lambda\\!=\\!5010\,\AA$). Hence, we consider $0.01\text{H}\beta$ as an
upper limit of the line strength of $[{Ne\textsc{iii}}]\,\lambda 3869$.
Figure 1: The co-added blue spectrum of PN G135.9+55.9 using 123 spatial
elements, including the sky. The upper and lower panels show the wavelength
ranges 3640–4350 Å and 4320–5030 Å, respectively. The insets in the upper and
lower panels show a close-up of the spectrum where H8 and a potential
$[{Ne\textsc{iii}}]\,\lambda 3869$, and $[{O\textsc{iii}}]\,\lambda\lambda
4959,\,5007$ are found, respectively. Gray thick lines in the inset panels
show the line fits. The wavelength ranges of the inset panels are indicated in
the spectrum with horizontal lines. The positions of four emission lines of
telluric origin are also indicated (Hg). The diffuse emission features at,
e.g., $\lambda\\!\simeq\\!4510\,$Å and $\lambda\\!\simeq\\!4980\,$Å are also
of telluric origin. For further details see Sect. 3. Table 1: Flux
measurements of PN G135.9+55.9
Emission line | $\lambda_{0}$ [Å] | PMAS${}_{\text{raw}}$ | PMAS${}_{\text{corr.}}$ | R02 (SPM1) | R02 (CFHT) | J02${}^{\text{a}}$ | T01 | $I(\text{case B})$
---|---|---|---|---|---|---|---|---
$[{O\textsc{iii}}]$ | 5006.84 | 3.36 | (0.33) | 3.48 | (0.35) | 2.7 | (1.3) | 2.87 | (0.85) | 3 | (1) | 3.1 |
$[{O\textsc{iii}}]$ | 4958.92 | 1.27 | (0.29) | 1.33 | (0.30) | | | | | | | |
$\text{H}\beta$ | 4861.32 | 100.00 | (0.49) | 100.00 | (0.49) | 100.0 | (2.1) | 100.0 | (1.7) | 100 | | 100 | 100.0
${He\textsc{ii}}$ | 4685.65 | 78.72 | (0.62) | 82.11 | (0.64) | 76.1 | (2.3) | 78.6 | (1.5) | 77 | (3) | 92 |
${He\textsc{ii}}$ | 4541.59 | 2.96 | (0.39) | 3.09 | (0.40) | | | | | | | |
H$\gamma$ | 4340.45 | 45.49 | (0.60) | 45.40 | (0.63) | 41.0 | (1.6) | 42.05 | (0.98) | 39 | (3) | 56 | 47.6
${He\textsc{ii}}$ | 4199.83 | 2.48 | (0.59) | 2.59 | (0.61) | | | | | | | |
H$\delta$ | 4101.74 | 23.81 | (0.64) | 23.69 | (0.66) | 21.4 | (2.8) | 20.6 | (1.2) | 17 | (2) | 30 | 26.6
H7 | 3970.07 | 9.70 | (0.73) | 10.11 | (0.76) | 10.0 | (2.5) | 6.11 | (0.64) | 4 | (1) | 11 | 16.4
H8 | 3889.06 | 8.63 | (0.89) | 9.00 | (0.93) | | | 2.92 | (0.73) | | | | 10.8
$[{Ne\textsc{iii}}]$ | 3868.80 | $<\\!1$ | | $<\\!1$ | | | | 1.04 | (0.52) | 15 | (7) | |
H9 | 3835.40 | 8.22 | (0.99) | 8.6 | (1.0) | | | | | | | | 7.5
H10 | 3797.91 | 3.4 | (1.2) | 3.3 | (1.2) | | | | | | | | 5.8
$F$($\text{H}\beta$)${}^{\text{b}}$ | | 1.924 | (0.009) | | | 1.82 | (0.03) | 2.55 | (0.03) | 1.47 | 1.19 |
${}^{\text{a}}$ The line ratios are corrected for an extinction value of
$E(B-V)\\!=\\!0.03$ and for a $\sim\\!5\%$ contribution of
${He\textsc{ii}}$-lines to Balmer lines.
${}^{\text{b}}$ This is the total flux in emission measured for
$\text{H}\beta$ in units of $10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$, systematic errors are not
considered in the error estimate.
Note.— Errors are given in parentheses. The emission line names and the rest
wavelength $\lambda_{0}$, and Case B line ratios for the Balmer lines to
$\text{H}\beta$, are given in Cols. 1, 2, and 9. We present our raw values in
Col. 3 and values corrected for the contribution of ${He\textsc{ii}}$
Pickering lines to the Balmer lines in Col. 4. Columns 5–8 give the values
presented by R02 (SPM1), R02 (CFHT), J02, and T01, respectively.
## 4 Data analysis using hydrodynamic models
Consequences for estimates of abundances in the context of photoionization
models and the assumption of thermal equilibrium are poorly known. In a study
based on hydrodynamical models using time-dependent ionization Schönberner et
al. (2009, hereafter Paper VII; also see ) demonstrate that metal-poor nebulae
with low densities are prone to deviations from thermal equilibrium, because
heating by photoionization is no more balanced by _line_ cooling only. In
extreme cases the electron temperature is also controlled by _expansion_
cooling, which results in _lower_ electron temperatures compared to the
(standard) equilibrium case by up to 30%, although ionization is still close
to equilibrium. Differences are insignificant when solar abundances are used
instead (Perinotto et al. 1998).
In this section we combine our observed line strengths and UV-data obtained
with HST-STIS (Jacoby, priv. comm.) with outcome of our time-dependent
hydrodynamic models in order to estimate abundances. The UV-spectrum we used
is publicly available and can be retrieved from the HST archive (see proposal
9466, PI: Garnavich); it is also published by J06 (see Fig. 1 therein).
At first we present our modeling approach in Sect. 4.1, thereafter our model
sample in Sect. 4.2 and a discussion of how we determine our abundances in
Sect. 4.3.
### 4.1 Physical properties of our time-dependent models
Our one-dimensional radiation hydrodynamic (RHD) models of envelopes of PNe
are described in detail in Perinotto et al. (1998, 2004, also see references
therein). We emphasize that our models calculate ionization, recombination,
heating, and cooling time-dependently at every time step. The cooling function
is composed of the contribution of all considered ions, and for every
individual ion up to 12 ionization stages are taken into account. Physical
input parameters to the models include properties of the coupled CS model,
element abundances, and the density and velocity structures of matter in the
envelope.
In calculating the models the wind of the CS was used as input at the inner
boundary of the grid. The full model evolution was thereafter followed across
the Hertzsprung-Russell diagram for about 15 000 years until (partly)
recombination sets in close to the turn-around point (at an age of about 10
000 yr), and into subsequent stages of re-ionization due to advanced
expansion. An important feature of our models is that we, at any time, can
switch off all time-dependent terms and simultaneously fix the density
structure and the radiation field. The models are thereafter evolved until
they settle into equilibrium, we are then able to study differences between
dynamical and static models (also see Paper VII). We refer to these models as
equilibrium models, in comparison to the dynamical models. We calculated
surface brightnesses, emission line profiles and strengths of individual lines
using a supplementary code that is based on a version of Gesicki et al.
(1996).
### 4.2 Properties of our model sample
A definite mass estimate for the ionizing source of PN G135.9+55.9 is still
lacking. T04 identify the nucleus of PN G135.9+55.9 as a close binary, which
hampers their analysis, using non-LTE model atmosphere spectra, to derive the
mass of the ionizing star. They fit the photospheric Balmer lines of selected
optical spectra to obtain the surface gravity $\log g$. Assuming an effective
temperature of $T_{\text{eff}}\\!=\\!120\,000\,$K, and that the companion does
not contribute to the optical emission, a comparison with stellar evolutionary
tracks indicate a mass of $0.88\,M_{\odot}$ that they infer as unrealistically
high.
T04 support the final estimate of 0.55–$0.57\,M_{\odot}$ considering only the
Population II characteristics of the object (not being in the Galactic plane,
having a high radial velocity and being metal-poor), and its relatively high
kinematic age that the authors consider a rough estimate for the post-AGB age.
In order to be an evolved CS, which still is in a pre-white dwarf stage, a
kinematic age of $t_{\text{kin}}\\!=\\!16\,000\,$yr demanded a mass range that
low (see Fig. 9 in T04). The claim of Tovmassian et al. (2007) that the binary
core of PN G135.9+55.9 consists of one component of $0.565\,\text{M}_{\odot}$
and $T_{\text{eff}}\\!\simeq\\!58\,000\,$K, and a second component of
$0.85\,\text{M}_{\odot}$ and $T_{\text{eff}}\\!\simeq\\!170\,000\,$K, agrees
poorly with their previous results. For instance, the hydrogen and helium
lines will hardly be fitted by an object where
$T_{\text{eff}}\\!=\\!58\,000\,$K. The ionization of the nebula is likewise
inconsistent with an ionizing source where $T_{\text{eff}}\\!=\\!170\,000\,$K,
as PT05 show.
PT05 set $T_{\text{eff}}$, the object distance and luminosity, and select
models using a range of masses (0.583–0.600$M_{\odot}$) based also on
arguments using a kinematic age. Conclusions, which are based on kinematic
ages ($R_{\text{xxx}}/V_{\text{yyy}}$), are questionable when they are used as
evolutionary ages. Errors emerge from uncertain distances, the confusion of
matter (Doppler) velocities with structure (shock) velocities, inappropriate
(i.e. unrelated) combinations of $R_{\text{xxx}}$ and $V_{\text{yyy}}$, and
neglecting the expansion history of the object (Schönberner et al. 2005a, also
see Fig. 33 in Paper VII). The subscripts (xxx and yyy) denote that various
combinations of $R$ and $V$ could be used in order to obtain a kinematic age.
For $R$ one could use the outermost radius ($R_{\text{out}}$), the rim radius,
or the radius of any other substructure. For $V$ one could, moreover, use
corresponding differential changes ($\dot{R}_{\text{xxx}}$, although this
quantity is rarely available), or alternatively spectroscopic velocities such
as the half-width half-maximum (HWHM), HW10%M of spatially unresolved
profiles, or any velocity component of a decomposition of a spectrum of a
spatially resolved profile.
Because the properties and evolutionary history of PN G135.9+55.9 are
uncertain we restricted our analysis to one stellar evolutionary track as
input for our RHD models, and used a post-AGB (CS) model of
$0.595\,M_{\odot}$. Using a similar approach as T04 we only considered one
stellar component when evolving the nebula, and assumed that the influence
from the companion is weak. The stellar wind from the CS is calculated
according to Marten & Schönberner (1991). Its dependence on the metallicity
(in this case C, N, and O) is approximately accounted for by correction
factors; we used $\dot{M}\\!\propto\\!Z^{0.69}$ (Vink et al. 2001) and
$v_{\infty}\\!\propto\\!Z^{0.13}$ (Leitherer et al. 1992), whereby
$L_{\text{wind}}\\!=\\!0.5\dot{M}v_{\infty}^{2}\\!\propto\\!Z^{0.95}$. The CS
radiates as a black body. For the AGB wind we, moreover, adopted a power-law
density profile $\rho\\!\propto\\!r^{-\alpha}$ ($\alpha\\!=\\!3$–3.25, cf.
Sect. 4.3) and a constant outflow velocity
$v\\!=\\!10\text{km}\,\text{s}^{-1}$ (cf. Schönberner et al. 2005b, hereafter
Paper II). We used the radial domain
$4.0\times\\!10^{14}\\!\leq\\!r\\!\leq\\!2.8\\!\times\\!10^{18}\,\text{cm}$.
The models are normalized such that $n\\!=\\!10^{5}\,\text{cm}^{-3}$ at
$r\\!=\\!3\times 10^{16}\,$cm. The abundances are based on scaled values of
mean Galactic disk PNe abundances ($Z_{\text{GD}}$; this abundance
distribution is first quoted by Perinotto et al. 1998) for nine elements, see
Table 2; except for carbon and nitrogen $Z_{\text{GD}}$ is close to solar. The
abundance values $\epsilon_{i}$ are specified in (logarithmic) number
fractions relative to hydrogen, i.e.
$\epsilon_{i}\\!=\\!\log\,(n_{i}/n_{\text{H}})\\!+\\!12$. Abundance
distributions, which are similar to $Z_{\text{GD}}$, are used by for example,
Perinotto (1991), Kingsburgh & Barlow (1994), Exter et al. (2004), and Hyung
et al. (2004). We summarize all model parameters and properties in Table 4.
Selecting metallicities we made use of the set of six models of Paper VII that
we refined with four additional models (these additional models are marked
with the prefix ⋆): $3Z_{\text{GD}}$, $Z_{\text{GD}}$, $Z_{\text{GD}}/3$,
$Z_{\text{GD}}/10$, ${}^{\star}Z_{\text{GD}}/15$,
${}^{\star}Z_{\text{GD}}/20$, ${}^{\star}Z_{\text{GD}}/25$,
$Z_{\text{GD}}/30$, ${}^{\star}Z_{\text{GD}}/60$, and $Z_{\text{GD}}/100$.
Table 2: Mean Galactic disk element abundance distribution ($Z_{\text{GD}}$) | H | He | C | N | O | Ne | S | Cl | Ar
---|---|---|---|---|---|---|---|---|---
$Z_{\text{GD}}$ | 12.00 | 11.04 | 8.89 | 8.39 | 8.65 | 8.01 | 7.04 | 5.32 | 6.46
In order to illustrate differences between dynamical and equilibrium models we
present line strength ratios of three collisionally excited oxygen lines in
the UV, optical, and infrared wavelength ranges in Fig. 2. Representing the
model evolution the line ratios are shown as a function of the CS effective
temperature $T_{\text{eff}}$. The figure shows that lines of dynamical models
are weaker than those of equilibrium models for $T_{\text{eff}}\ga
50\,000\,$K. The reason is that dynamical models have a lower electron
temperature (cf. Sect. 3.1.3 and Fig. 16, both in Paper VII). We do not
consider young PNe, where $T_{\text{eff}}\\!<\\!50\,000\,$K; in such objects
the ionization equilibrium is disturbed for a short time, with the consequence
that the ionization is somewhat overestimated in the equilibrium models.
Discrepancies are larger with a lower metallicity, compare the thin lines with
the thick lines in Fig. 2, and shorter wavelengths, compare the dotted lines
with the dashed lines. When we compare the line intensities at
$T_{\text{eff}}\\!\simeq\\!130\,000\,$K we see that, as expected, the
difference is the largest for the UV-line ${O\textsc{iv}}]\,\lambda\lambda
1402\\!+\\!1405$ that is up to 250% stronger in the equilibrium case of the
$Z_{\text{GD}}/100$ sequence. The optical line $[{O\textsc{iii}}]\,\lambda
5007$ is stronger by about 70%. The infrared line is the least affected, it is
up to about 30% stronger in thermal equilibrium.
Because differences in line strengths depend strongly on the input density and
the velocity structure, these values should only be regarded as indicative.
Nevertheless, non-equilibrium effects introduce an ambiguity to the
determination of electron temperatures, hence elemental abundances, which is
neglected when using standard photoionization codes (cf. Sect. 5).
Figure 2: Comparison between emission line strengths of dynamical (dyn) and
equilibrium (eq) nebular models around a $0.595\,\text{M}_{\odot}$ central
star evolving across the H-R diagram. The two shown model sequences use
abundances $Z_{\text{GD}}/10$ (thick lines) and $Z_{\text{GD}}/100$ (thin
lines). We show the dyn/eq line strength ratios
(I${}_{\text{eq}}/$I${}_{\text{dyn}}$) of three collisionally excited oxygen
lines, viz. ${O\textsc{iv}}]\,\lambda\lambda 1402\\!+\\!1405$ (dotted lines),
$[{O\textsc{iii}}]\,\lambda 5007$ (solid lines), and
$[{O\textsc{iv}}]\,\lambda 26\,\mu$m (dashed lines), as a function of
$T_{\text{eff}}$. The evolution is only traced until maximum $T_{\text{eff}}$
is reached in order to avoid confusion. See Fig. 9 for absolute values on the
intensities. For further details see Sect. 4.2.
### 4.3 Estimating abundances for PN G135.9+55.9
A full abundance analysis of PN G135.9+55.9 is beyond the scope of this work,
but as an example we determine abundance estimates that are based on our
hydrodynamical models. Our model grid is not large enough to allow iteration
of all parameter dimensions. The primary goal of this study is instead to find
a model that agrees reasonably with the observational quantities, and then use
this _best-match_ model to elaborate on the influence of non-equilibrium
effects.
We used the following four criteria to find such a best-match model: emission
line strengths should match observed values, the model H$\alpha$ emission line
profile should match the observed line profile (such as Fig. 1 of Richer et
al. 2003), the model H$\alpha$ surface-brightness distribution should match
the observed distribution (such as Fig. 3 of R02), and both distance estimates
from the $\text{H}\beta$ flux of the object and from the corresponding
apparent size, which is obtained from the surface-brightness distribution,
should be in fair agreement. In addition to these properties that apply to the
nebula, the visual magnitude of the central star is also indicative of the
distance. However, since we have only used one single stellar track (with a
stellar mass $M\\!=\\!0.595\,M_{\sun}$) such a visual magnitude is difficult
to match properly. The emission lines and their line strengths that we used in
our study, are shown in Cols. 1–3 of Table 3.
Table 3: Observational vs. modeled line strengths
ID | $\lambda_{0}$ [Å] | observed | model | ratio
---|---|---|---|---
| | | | dyn | eq | eq/dyn
${N\textsc{v}}$ | 1238+1242 | 426${}^{\text{a}}$ | (40) | 330 | 697 | 2.11
${N\textsc{iv}}]$ | 1486 | 87${}^{\text{a}}$ | (30) | 124 | 276 | 2.23
${C\textsc{iv}}$ | 1548+1550 | 660${}^{\text{a}}$ | (50) | 668 | 1319 | 1.97
${C\textsc{iii}}]$ | 1906+1909 | 55${}^{\text{a}}$ | (30) | 31 | 64 | 2.06
${O\textsc{iv}}]$ | 1402+1405 | $<\\!37$${}^{\text{a}}$ | | 6 | 13 | 2.17
$[{O\textsc{iii}}]$ | 5007 | 3.5${}^{\text{b}}$ | (0.4) | 3.3 | 4.4 | 1.33
$[{O\textsc{iv}}]$ | 26 $\mu$m | – | | 26 | 30 | 1.15
$[{Ne\textsc{iv}}]$ | 2422+2425 | 25${}^{\text{a}}$ | (12) | 38 | 62 | 1.63
$[{Ne\textsc{v}}]$ | 3426 | 86${}^{\text{a}}$ | (9) | 78 | 119 | 1.52
$[{Ne\textsc{iii}}]$ | 3869 | $<\\!1$${}^{\text{b}}$ | | 0.4 | 0.5 | 1.25
$[{Ne\textsc{v}}]$ | 14 $\mu$m | – | | 90 | 102 | 1.13
${He\textsc{ii}}$ | 4686 | 82${}^{\text{b}}$ | (1) | 82 | 80 | 0.98
${}^{\text{a}}$ Jacoby (priv. comm.), ${}^{\text{b}}$ this paper
Note.— All values are specified in units of $\text{H}\beta\\!=\\!100$. Columns
1 & 2 give the emission line names and the rest wavelength $\lambda_{0}$.
Measured values of PN G135.9+55.9 are given in Col. 3, with errors in
parentheses. Corresponding values of dynamic (dyn) and equilibrium (eq)
models, and their ratio, are given in Cols. 4–6 (cf. Sect. 4.3).
Since we have not been able to measure $[{Ne\textsc{iii}}]\,\lambda\,3869$ we
cannot determine $T_{\text{eff}}$ using that line. Instead we proceeded as
follows. Using the existing measurements of the two neon lines
$[{Ne\textsc{iv}}]\,\lambda\lambda\,2422\\!+\\!2425$ and
$[{Ne\textsc{v}}]\,\lambda\,3426$ we determined a range of effective
temperatures where these measurements match our model emission line strengths
(see Fig. 10). The result is shown in Fig. 3a. The upper limit of the
temperature range corresponding to the $[{Ne\textsc{iv}}]$-line is set by the
maximum stellar temperature possible, in this case
$\max(T_{\text{eff}})\\!=\\!146\,870\,$K. Note that the extent of the two
regions of the effective temperatures of $[{Ne\textsc{iv}}]$ and
$[{Ne\textsc{v}}]$ that do not overlap, depends on both observations and model
initial conditions. We extrapolated both regions in order to find the nearest
point of intersection ($Z_{\text{Ne}}$, $T_{\text{eff}}$) that we then used as
two of the initial values when iterating our models. From this procedure we
found $T_{\text{eff}}\\!=\\!138\,000\,$K.
For carbon, nitrogen and oxygen we evaluated line strengths at
$T_{\text{eff}}$ for all model sequences (Figs. 7–9); the variation of model
line strengths with stellar effective temperature is moderate at
$T_{\text{eff}}\\!=\\!138\,000\,$K (see below). We show the results in Fig.
3b, together with power law fits in relevant regions. By a comparison with the
observational data we then found a first set of object abundances. For sulfur,
chlorine and argon we simply reduced the respective mean Galactic disk value
using the mean scaling value that we found for carbon, nitrogen, oxygen, and
neon.
Figure 3: For each model sequence panel a) shows the range of $T_{\text{eff}}$
that corresponds to the observed line strengths of
$[{Ne\textsc{iv}}]\,\lambda\lambda 2422,\,2425$ and $[{Ne\textsc{v}}]\,\lambda
3426$, compare the plotted values with the neon line strengths in Fig. 10. The
value of the effective temperature of our first custom model is indicated with
an open circle ($\circ$; $T_{\text{eff}}\\!=\\!138\,000\,$K). In panel b) each
sub-panel shows the line strength of all models (but 3Z) that are all
evaluated at $T_{\text{eff}}\\!=\\!138\,000\,$K, for four different emission
lines. The horizontal dashed lines indicate the observed values. In both
panels error intervals are indicated with gray regions. All axes are
logarithmic. For further details see Sect. 4.3.
In a final step, during the creation of our first custom model, we adjusted
the helium abundance. Figure 11 shows that model line strengths of helium are
nearly independent of metal abundances. The same figure also shows that the
variation of each model sequence with the effective temperature is small for
evolved objects (where $T_{\text{eff}}\\!>\\!10^{5}\,$K). We reduced the
initial helium abundance by one third, as is suggested by the ratio of model-
to-observed line strength. With this set of initial abundances we then
calculated additional custom models with modified abundances in order to
converge on the observed line strengths. In this process we iterated element
abundances, which were the closest to the observed values, accounting for the
size of the error bars of the different lines. Setting the carbon abundance we
first gave ${C\textsc{iv}}\,\lambda\lambda 1548\\!+\\!1550$ a stronger weight
than ${C\textsc{iii}}]\,\lambda\lambda 1906\\!+\\!1909$, since its error is
smaller (also see Fig. 3b).
In the course of our iterative calculations we found that the model
$\text{H}\beta$ flux did not match the apparent size of the object at a common
distance. In order to achieve a less extended model we modified the power law
density distribution of the AGB wind by replacing $\alpha\\!=\\!3$, that is
used in all other models of this study, with $\alpha\\!=\\!3.25$; keeping the
density normalization as before (see Sect. 4.1). By this approach changes to
resulting model emission line strengths should be small. In Paper II we show
that a steeper density gradient results in faster expansion rates of the
leading shock of the shell, i.e. the outer edge of the PN. Therefore our new
choice of $\alpha$ may appear counterintuitive when forming a geometrically
smaller PN. The size of a PN, however, depends on the expansion velocity
integrated over the entire expansion period. Early on a high circumstellar
density at the inner edge of the model domain (due to our choice of a power
law) prevents nebular matter from becoming rapidly ionized. The formation of a
D-type ionization front is thereby delayed as the acceleration starts later
with a steeper density gradient, and the overall PN lifetime thereby becomes
shorter. In the case of $\alpha\\!=\\!3.25$ a higher expansion velocity cannot
compensate for the simultaneous shortening of the expansion period.
Figure 4: This figure shows the evolution of emission line strengths of our
best-match model, as a function of the effective temperature,
$T_{\text{eff}}$. Using the ten observed emission lines of Table 3 we show
model-to-observed line ratios (solid lines). The gray regions mark the
corresponding error bars. Note that only upper limits were observed for
${O\textsc{iv}}]\,\lambda\lambda 1402,\,1405$ and $[{Ne\textsc{iii}}]\,\lambda
3869$. For further details see Sect. 4.3. Table 4: Summary of our best-match
model parameters
Parameter | PN G135.9+55.9 | $Z_{\text{GD}}$
---|---|---
Spectrum | black body |
Stellar mass, $M$ | 0.595 $\text{M}_{\odot}$ |
Model age, $t$ | $8982\,$yr |
Stellar effective temperature, $T_{\text{eff}}$ | 138 049 K |
Stellar luminosity, $L$ | $2994\,\text{L}_{\odot}$ |
Central star wind, $L_{\text{wind}}=0.5\dot{M}v^{2}_{\infty}$ | $\dot{M}\propto Z^{0.69}$, $v_{\infty}\propto Z^{0.13}$ |
AGB wind | $\rho\propto r^{-3.25}$, $v\\!=\\!10\,\text{km}\,\text{s}^{-1}$,
| $n_{r=3\times 10^{16}\,\text{cm}}\\!=\\!1\\!\times\\!10^{5}\,\text{cm}^{-3}$
Abundances, $\epsilon_{i}\\!=\\!\log\,(n_{i}/n_{\text{H}})+12$: | |
He | 10.88 | 11.04
C | 7.90 | 8.89
N | 7.47: | 8.39
O | 6.74 | 8.65
Ne | 6.96 | 8.01
S | [5.94] | 7.04
Cl | [4.22] | 5.32
Ar | [5.36] | 6.46
Distance, $d$ | 18 kpc |
Visual magnitude, $m_{\text{V}}$ | 19.5 mag |
Nebular density, $\langle n_{\text{e}}\rangle$ | 65 $\text{cm}^{-3}$ |
Nebular temperature, $\langle T_{\text{e}}\rangle$ | 21 100 K |
Nebular $\text{H}\beta$-luminosity, $L(\text{H}\beta)$ | 0.193 $\text{L}_{\odot}$ |
Model HWHM velocity, $V_{\text{HWHM}}$ | $41.8\,\text{km}\,\text{s}^{-1}$ |
Note.— The element abundances $\epsilon_{i}$ are used as input in the
calculation of our radiation hydrodynamic models; the values of S, Cl, and Ar
are not fitted, but only scaled. $Z_{\text{GD}}$ denotes the mean abundance
distribution in the Galactic disk (cf. Sect. 4.1).
The abundance distribution of our best-match model, and all relevant model
properties, is given in Table 4, and resulting emission line strengths are
given in Col. 4 of Table 3. We also show model-to-observed line strength
ratios in Fig. 4. This figure illustrates a weak dependence with effective
temperature at values about $T_{\text{eff}}\\!\simeq\\!138\,000\,$K. Since
most lines depend only weakly temperatures about this value, the precise value
of $T_{\text{eff}}$, for say $138\,000\\!\pm\\!5000\,$K, is uncritical to the
ionization structure. We could not achieve a simultaneous agreement for the
two nitrogen lines. The high value of 426 for ${N\textsc{v}}\,\lambda\lambda
1238\\!+\\!1242$ could not be reached with any of our models (Fig. 8), which
is why the nitrogen abundance of our best-match model should be considered
approximate. ${O\textsc{iv}}]\,\lambda\lambda 1402\\!+\\!1405$ can,
furthermore, hardly be identified in the spectrum of J06. We consider the
value of 37 a very conservative upper limit – compare with the value of $87\pm
30$ for ${N\textsc{iv}}]\,\lambda 1486$. A possible blending with
${Si\textsc{iv}}\,\lambda\lambda 1394\\!+\\!1403$ should not be excluded.
Figure 5: This figure shows the basic physical properties of our best fit
model at the stellar parameters $L\\!=\\!2\,985\,\text{L}_{\odot}$ and
$T_{\text{eff}}\\!=\\!138\,152\,$K, at an age of $t\\!=\\!8\,988\,\text{yr}$.
The four panels show: a) the radial structure of of the particle density
(thick line) and the gas velocity (dotted line), b) radial structures of the
electron temperature for the dynamical model (solid line) and the equilibrium
model (dashed line), c) a comparison between the H$\alpha$surface-brightness
distribution of the model at a assumed distance of $18.3\,$kpc and the
observational data of R02 (filled/open squares = semi-major/semi-minor axis,
see text for details) with the actual H$\alpha$-image (dotted line) and an
image that results with a seeing of $1\aas@@fstack{\prime\prime}5$ (solid
line), and d) a comparison between the H$\alpha$ emission line profile of the
model (solid line) and the observation of Richer et al. (2003, see Fig. 1 and
slit 7; open circles $\circ$). The simulated profile was additionally
broadened by a Gaussian with FWHM$=\\!26\,\text{km}\,\text{s}^{-1}$, in order
to be consistent with their observation. The observed left wing is due to
emission of ${He\textsc{ii}}\,\lambda 6560$. For further details see Sect.
4.3.
In Fig. 5 we show structural and kinematic properties of our best-match model;
the CS has evolved to $t\\!=\\!8982\,$yr, $L\\!=\\!2994\,\text{L}_{\odot}$,
and $T_{\text{eff}}\\!=\\!138\,049\,$K (Table 4). The mean electron density is
$\langle n_{\text{e}}\rangle\\!=\\!65\,\mbox{cm}^{-3}$. The density (Fig. 5a)
shows a gradual decline with increasing radius. Neither the density nor the
H$\alpha$ surface-brightness structure (Fig. 5c) show a distinct double shell
morphology. The same panel compares the observed data of R02 (see Fig. 3
therein) with the outcome of our model, using a distance of
$d\\!=\\!18.3\,$kpc. We show the real structure with a central dip that is due
to the hot bubble, and the structure under the seeing conditions of the
observations ($1\aas@@fstack{\prime\prime}5$). A comparison of our observed
$\text{H}\beta$ flux, $F(\text{H}\beta)\\!=\\!1.924\times 10^{-14}$
$\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$, and the
$\text{H}\beta$-luminosity of the model,
$L(\text{H}\beta)\\!=\\!0.193\,\text{L}_{\odot}$, yields a distance of
$d\\!=\\!17.95\,$kpc – that agrees well with the one of the surface-brightness
structure. We adopt $d\\!=\\!18\,$kpc as a final value of the distance. In our
models we use exclusively black bodies for the CSs. Applying the distance
estimate of $d\\!\approx\\!18\,$kpc to the bolometric luminosity of the model,
$L\\!=\\!2\,994\,\text{L}_{\odot}$, yields
$m_{\text{V}}\\!=\\!19.5\,$mag111R02 measure $m_{\text{V}}\\!=\\!17.9\,$mag.
Assuming this value instead our best-match model would shift to a
corresponding distance of only $d\\!=\\!8.6\,$kpc. This discrepancy in our
third distance estimate can be explained by our selected stellar mass (one
single track of 0.595 $\text{M}_{\odot}$), which should also be iterated in
order to achieve a better agreement. Of course, our model cannot reproduce the
actual intensity if the nucleus really is a double degenerate..
Throughout the nebula the matter velocity gradient is positive with increasing
radius (Fig. 5a), reaching a maximum velocity of
$v\\!\simeq\\!65\text{km}\,\text{s}^{-1}$. In the adjacent (radiative) shock
layer at $12.5\\!\la\\!r\\!\la\\!14.5\\!\times\\!10^{17}\,\text{cm}$ the
velocity is about constant. The simulated emission line profile (Fig. 5d)
resembles the long slit Echelle spectral analysis of Richer et al. (2003, see
Fig. 1, slit 7). Our one-dimensional model cannot be fully applied to this
slightly non-spherical PN and its asymmetric line profiles. The observed HWHM
velocity, $V_{\text{HWHM}}\\!=\\!42.5\,\text{km}\,\text{s}^{-1}$, is, however,
well matched by our model with
$V_{\text{HWHM}}\\!=\\!41.8\,\text{km}\,\text{s}^{-1}$.
In Fig. 5b we show the radial electron temperature structure. The temperature
peak behind the outer shock does not contribute to the mean temperature due to
low ion densities in that region. We will in the following subsection discuss
the consequences of non-equilibrium conditions for the electron temperature
and, consequently, for the strengths of collisionally excited lines.
## 5 Discussion
In order to study differences in the outcome of our time-dependent and static
(equilibrium) models we should, ideally, also calculate a corresponding best-
match equilibrium model by the same procedure we used to find the best-match
dynamical model. As such an approach is extremely time-consuming we instead
compare our outcome with literature values, which are all based on standard
photoionization codes (these correspond to our equilibrium models). We present
our abundances anew in Table 5 together with a compilation of literature
values, which are derived using observed emission lines. Errors of individual
estimates are specified where such values are provided. Differences between
estimates and physical assumptions of different sources are large in general.
Since we focus on understanding general trends of values, and not on providing
final abundances, we have not estimated errors of our values. This is also
difficult to do with our models where abundances are input parameters, and not
the outcome.
Table 5: Literature compilation of abundances estimates for PN G135.9+55.9
Ref. | spectral | $T_{\text{eff}}$ | $\langle T_{\text{e}}\rangle$ | He | C | N | O | Ne | C/O | N/O | Ne/O
---|---|---|---|---|---|---|---|---|---|---|---
| domain | $[10^{3}\text{K}]$ | $[10^{3}\text{K}]$ | | | | | | | | | | | | | |
T01 | O | 150 | – | | | | | | 6.3 | (0.5) | | | | | | |
J02 | O | 100 | 17.6 | 10.82 | | | | | 6.93 | | 7.47 | | | | | 3.47 |
R02 | O | 100 | 30 | 10.9 | | | | | 6.15 | (0.35) | 6.35 | | | | | 0.5 | (0.3)
PT05 | O | 130 | 30 | 10.91 | | | | | $7.5\phantom{0}$ | (0.3) | 6.65 | | | | | 0.14 | (0.14)
S05 | O+U | – | – | | 7.51 | (0.15) | 6.87 | (0.19) | 6.85 | (0.25) | 6.6 | 4.7 | (1.1) | 1.05 | (0.15) | 0.65 | (0.35)
J06 | O+U | 130 | 30${}^{\text{a}}$ | 10.87 | 7.58 | | 6.94 | | 7.18 | | 6.66 | 2.5 | | 0.58 | | 0.30 |
this work | O+U | 138 | 21.1 | 10.88 | 7.90 | | 7.47: | 6.74 | | 6.96 | 14 | | 5.4: | | 1.7 |
$Z_{\text{GD}}$ | | | | 11.04 | 8.89 | | 8.39 | | 8.65 | | 8.01 | 1.74 | | 0.55 | | 0.23 |
BoBn-1 | | | | 11.05 | 8.85 | | 8.00 | | 7.83 | | 7.72 | 10.5 | | 1.48 | | 0.78 |
${}^{\text{a}}$ Jacoby (priv. comm.) | | | | | | | |
Note.— The table only includes estimates that are based on observed emission
lines. Columns 1–4 specify the source reference, the wavelength range (O –
optical, and U – UV), the stellar effective temperature ($T_{\text{eff}}$) and
the mean electron temperature ($\langle T_{\text{e}}\rangle$) used in the
study. Columns 5–9 give element abundances using the same units as in Table 2.
In Cols. 10–12 we also give abundance ratios relative to oxygen. Uncertainties
are, were provided, given in parentheses. A colon indicates an uncertain value
of our best-match model. The abundances of our best-match model are given in
the row marked _this work_. In the last two rows we, for comparison, give the
mean abundance distribution of the Galactic disk ($Z_{\text{GD}}$; Table 2)
and the halo-PN BoBn-1 (the values of this object are taken from Howard et al.
1997). For further details see Sect. 5.
Previous studies of PN G135.9+55.9 present improvements to different parts of
the abundance analysis. PT05 provide a thorough model analysis, without making
own observations, where they account for several physical issues, which were
not addressed previously. Notably, they study differences in models assuming
case B vs. non-case B photoionization, they use different sets of collisional
recombination coefficients, and use a stellar atmosphere model of the CS, in
addition to the commonly used black-body model. Due to lack of data they
calibrate their models using only observational data in the visual wavelength
range, and therefore they cannot calculate precise values for the abundances
of carbon and nitrogen. The main conclusion of PT05 is that the oxygen
abundance of previous studies is too low. S05 and J06, moreover, add UV lines,
which are sampled with HST-STIS, to their linelist, and can thereby constrain
the abundances of carbon and nitrogen better than PT05 (S05 also announce IR
observations using SPITZER). S05 also argue that PT05 use too high abundances
for carbon and nitrogen, and therefore have to use a higher oxygen abundance
than is necessary; S05 find a lower value on the oxygen abundance than PT05,
which is in better agreement with estimates of previous studies.
In agreement with all previous studies, except for PT05, we found that the
oxygen abundance of PN G135.9+55.9 is very low. It is difficult to make a
meaningful, more detailed, comparison between our abundances and those of T01,
J02, and R02 since we use different stellar effective temperatures (mainly).
PT05 are the latest authors who base their analysis on only optical emission
lines. The very thorough analysis these authors make is found to be of small
use as there is a strong disagreement between the predicted UV-line
intensities of their best model (that agrees best with model M1, cf. Table 3
in PT05) and observed UV line strengths (Table 3); the difference is (with the
exception of ${C\textsc{iii}}]\,\lambda 1908$) in every case about a factor
two. It is in this context worth mentioning that the electron temperature they
adopt is about 9000 K higher than in our study, resulting in a different
ionization structure (see below). J06 use a similar electron temperature as
PT05 in their photoionization models, which is why their results should be
affected to a similar degree. Furthermore, although S05 include UV-intensities
in their analysis they do not provide any temperatures at all and it is
impossible to make a meaningful comparison with their abundances.
Compared to the mean abundances of the Galactic disk our values for PN
G135.9+55.9 are 1/1.45 (He), 1/9.8 (C), 1/8.3 (N), 1/81 (O), and 1/11 (Ne).
Compared to the total mean metallicity of the Galactic disk our value is 1/13.
Our abundance estimates relative to oxygen are in better agreement with the
values of another halo PN, BoBn-1. In this case our values of C/O, N/O, and
Ne/O are 33–260% higher, although PN G135.9+55.9 is considerably more depleted
of metals.
Figure 6: In panel a) we show the ratio between the emission line strength
values of our best-match model (dyn) and the observed values (cf. Table 3 and
Sect. 4.3). For all, but the two nitrogen lines,
${N\textsc{v}}\,\lambda\lambda 1238\\!+\\!1242$ and ${N\textsc{iv}}]\,\lambda
1486$, the model values and observations agree within error bars. In panel b)
we show emission line strengths of the thermally relaxed best-match model (eq)
relative to the dynamical model (dyn) for all line listed in Table 3. The
vertical dashed lines in both panels indicate limits of different spectral
domains.
In Fig. 6 we illustrate differences between line ratios of our best-match
dynamic model and its corresponding equilibrium model, plotted as a function
of wavelength; we present the same data in Table 3. Fig. 6a shows that the
observed line ratios are satisfactorily matched by the model (Sect. 4.3).
Equilibrium-to-dynamic model line ratios are shown in Fig. 6b for all lines
that we used with our best-match model (also including IR lines; at the
assumed temperature $T_{\text{eff}}\\!\simeq\\!138\,000\,$K, compare with Fig.
2 for the models $Z_{\text{GD}}/10$ and $Z_{\text{GD}}/100$). In this case
differences can be larger than 100% in the EUV, about 50% in the UV, about
20–30% in the optical wavelength range, and about 10–20% in the infrared
wavelength range. A hint of the importance of accurate line ratios is given in
Fig. 3b. If a model line ratio changes by a factor two it is easily seen that
different abundances are required to match the change. Although the non-linear
response to the nebula of the full model is complex, which is why plots such
as Fig. 3b are unsuitable when making quantitative estimates of abundances.
Differences in line ratios between dynamical and equilibrium models occur as a
consequence of a different sensitivity of collisionally excited lines to the
electron temperature. The mean temperature in the nebular region of the two
models are $\langle T_{\text{e}}\rangle_{\text{dyn}}\\!=\\!21\,100\,$K and
$\langle T_{\text{e}}\rangle_{\text{eq}}\\!=\\!25\,100\,$K, compare the two
radial structures in Fig. 5b, the difference is significant. The electron
temperature of our evolved metal-poor models is determined by line cooling and
expansion cooling. It is worth noting that although the oxygen abundance lies
closer to $Z_{\text{GD}}/100$ the mean model abundance is closer to
$Z_{\text{GD}}/10$, and it is this higher abundance that determines the
physical structure of the object (see Fig. 2 and Paper VII, Figs. 15 and 16).
In an observational study using the full wavelength range (EUV–infrared),
where measurement errors are sufficiently small, there should be significant
problems determining abundances of metal-poor objects using models that are
unable to account for dynamical effects.
## 6 Conclusions
PN G135.9+55.9 is an extraordinary object as it is a metal-poor PN with the
lowest oxygen abundance known. In order to clarify contradictory abundance
determinations of PN G135.9+55.9 in the literature we made a new study of this
object using a two-fold approach. At first we re-observed the nebula and could
measure a more accurate spectrum in the visual wavelength range than has been
done so far. Unlike previous observational studies we could only measure an
upper limit of $[{Ne\textsc{iii}}]\,\lambda 3869$ of $0.01\text{H}\beta$,
although we measured five new lines in the nebula. We therefore chose to base
our estimate of the stellar effective temperature using supplementary UV-data
of Jacoby (priv. comm.).
In the second part of our study we used a newly calculated set of our
radiation hydrodynamic models in order to determine abundances and study the
influence of time-dependent effects. In this case such effects are found to be
important, causing lower electron temperatures in the nebula. Resulting line
strengths of dynamical models are lower than in the corresponding equilibrium
models (these models are relaxed after all time-dependent terms are set to
zero). Consequently, different abundances are required to match line strengths
when using either approach. We found that it is only possible to make a self-
consistent abundance determination using a dynamical model that is constrained
using measurements in the entire wavelength range. Our final set of abundances
of the five most abundant elements is: 1/1.45 (He), 1/9.8 (C), 1/8.3 (N), 1/81
(O), and 1/11 (Ne), all with respect to the mean abundance of objects in the
Galactic disk ($Z_{\text{GD}}$). The total metallicity is $Z_{\text{GD}}/13$.
Additionally, using our single $0.595\,M_{\odot}$ evolutionary track we found
an effective temperature of PN G135.9+55.9 of
$T_{\text{eff}}\\!\simeq\\!138\,000\,$K, and a distance of $d\\!=\\!18\,$kpc.
This distance is about the double value assumed by PT05, but agrees well with
the value of T04. The mean electron temperature of our dynamical model is
$\langle T_{\text{e}}\rangle\\!=\\!21\,000\,$K; this is 4000 K lower than the
value of the corresponding equilibrium model.
Although we believe that our approach provides a most significant improvement
when determining abundances of metal-poor objects our modeling can be improved
to provide more accurate values. At first one could consider to iterate more
dimensions of the parameter space, such as e.g. the mass of the central star
and properties of the AGB wind. Three additional suggestions for such
improvements that are all considered by PT05 are: using non-CaseB radiative
transfer, replacing the black-body model of the central star with a model
atmosphere, and using improved collision rates. Observationally an accurate
multi-wavelength study including the infrared wavelength range, such as is
announced by S05, will help to constrain the models further. Last, but not
least, it is important to clarify the parameters and evolutionary history of
the ionizing central star(s) unambiguously.
###### Acknowledgements.
C. S. acknowledges support by DFG grant SCHO 394/26. We thank G. Jacoby both
for providing us with UV data prior to their publication, and for providing us
feedback on a late version of the manuscript.
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* Jacoby et al. (2006) Jacoby, G. H., Garnavich, P. M., Bond, H. E., et al. 2006, in Planetary Nebulae in our Galaxy and Beyond, ed. M. J. Barlow & R. H. Méndez, IAU Symp., 234, 431 (J06)
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* Péquignot & Tsamis (2005) Péquignot, D. & Tsamis, Y. G. 2005, A&A, 430, 187 (PT05)
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* Roth et al. (2005) Roth, M. M., Kelz, A., Fechner, T., et al. 2005, PASP, 117, 620
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* Schönberner et al. (2005c) Schönberner, D., Jacob, R., Steffen, M., & Roth, M. M. 2005c, in Planetary Nebulae as Astronomical Tools, ed. R. Szczerba, G. Stasińska, & S. K. Gorny, AIP Conf. Ser., 804, 269
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## Appendix A Intensity evolution of our RHD models
For each model sequence we mention in Sect. 4.2 we show the intensity
evolution of all emission lines of Table 3 in Figs. 7-11. Solid lines show
dynamic models and dotted lines equilibrium models (for those sequences where
they were calculated). The abscissa is in every case the stellar effective
temperature $T_{\text{eff}}$. Horizontal dashed lines mark observed values for
PN G135.9+55.9, and gray shaded regions mark corresponding error intervals.
Additionally we show the evolution of the two infrared lines,
$[{O\textsc{iv}}]\,\lambda 26\,\mu$m and $[{Ne\textsc{v}}]\,\lambda 14\,\mu$m,
despite a lack of currently existing observational data.
Figure 7: The line strength evolution of different carbon lines as a function
of $T_{\text{eff}}$ for all models. Figure 8: The line strength evolution of
different nitrogen lines as a function of $T_{\text{eff}}$ for all models.
Figure 9: The line strength evolution of different oxygen lines as a function
of $T_{\text{eff}}$ for all models. Figure 10: The line strength evolution of
different neon lines as a function of $T_{\text{eff}}$ for all models. Figure
11: The line strength evolution of ${He\textsc{ii}}\,\lambda\,4686$ as a
function of $T_{\text{eff}}$ for all models.
|
arxiv-papers
| 2009-12-30T13:59:04 |
2024-09-04T02:49:07.334939
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Sandin and R. Jacob and D. Sch\\\"onberner and M. Steffen and M.M.\n Roth",
"submitter": "Christer Sandin",
"url": "https://arxiv.org/abs/0912.5430"
}
|
0912.5468
|
2010371-382Nancy, France 371
Jiří Fiala
Marcin Kamiński
Bernard Lidický
Daniël Paulusma
# The $k$-in-a-path problem for claw-free graphs
J. Fiala Charles University, Faculty of Mathematics and Physics,
DIMATIA and Institute for Theoretical Computer Science (ITI)
Malostranské nám. 2/25, 118 00, Prague, Czech Republic fiala@kam.mff.cuni.cz
bernard@kam.mff.cuni.cz , M. Kamiński Computer Science Department,
Université Libre de Bruxelles,
Boulevard du Triomphe CP212, B-1050 Brussels, Belgium
marcin.kaminski@ulb.ac.be , B. Lidický and D. Paulusma Department of
Computer Science, University of Durham,
Science Laboratories, South Road,
Durham DH1 3LE, England daniel.paulusma@durham.ac.uk
###### Abstract.
Testing whether there is an induced path in a graph spanning $k$ given
vertices is already NP-complete in general graphs when $k=3$. We show how to
solve this problem in polynomial time on claw-free graphs, when $k$ is not
part of the input but an arbitrarily fixed integer.
###### Key words and phrases:
induced path, claw-free graph, polynomial-time algorithm
###### 1991 Mathematics Subject Classification:
G.2.2 Graph algorithms, F.2.2 Computations on discrete structures
Research supported by the Ministry of Education of the Czech Republic as
projects 1M0021620808
and GACR 201/09/0197, by the Royal Society Joint Project Grant JP090172 and by
EPSRC as EP/D053633/1.
## 1\. Introduction
Many interesting graph classes are closed under vertex deletion. Every such
class can be characterized by a set of forbidden induced subgraphs. One of the
best-known examples is the class of perfect graphs. A little over 40 years
after Berge’s conjecture, Chudnovsky et al. [18] proved that a graph is
perfect if and only if it contains neither an odd hole (induced cycle of odd
length) nor an odd antihole (complement of an odd hole). This motivates the
research of detecting induced subgraphs such as paths and cycles, which is the
topic of this paper. To be more precise, we specify some vertices of a graph
called the terminals and study the computational complexity of deciding if a
graph has an induced subgraph of a certain type containing all the terminals.
In particular, we focus on the following problem.
$k$-in-a-Path
Instance: a graph $G$ with $k$ terminals.
Question: does there exist an induced path of $G$ containing the $k$
terminals?
Note that in the problem above, $k$ is a fixed integer. Clearly, the problem
is polynomially solvable for $k=2$. Haas and Hoffmann [11] consider the case
$k=3$. After pointing out that this case is NP-complete as a consequence of a
result by Fellows [9], they prove W$[1]$-completeness (where they take as
parameter the length of an induced path that is a solution for $3$-in-a-Path).
Derhy and Picouleau [6] proved that the case $k=3$ is NP-complete even for
graphs with maximum degree at most three.
A natural question is what will happen if we relax the condition of “being
contained in an induced path” to “being contained in an induced tree”. This
leads to the following problem.
$k$-in-a-Tree
Instance: a graph $G$ with $k$ terminals.
Question: does there exist an induced tree of $G$ containing the $k$
terminals?
As we will see, also this problem has received a lot of attention in the last
two years. It is NP-complete if $k$ is part of the input [6]. However,
Chudnovsky and Seymour [4] have recently given a deep and complicated
polynomial-time algorithm for the case $k=3$.
###### Theorem 1.1 ([4]).
The $3$-in-a-Tree problem is solvable in polynomial time.
The computational complexity of $k$-in-a-Tree for $k=4$ is still open. So far,
only partial results are known, such as a polynomial-time algorithm for $k=4$
when the input is triangle-free by Derhy, Picouleau and Trotignon [7]. This
result and Theorem 1.1 were extended by Trotignon and Wei [20] who showed that
$k$-in-a-Tree is polynomially solvable for graphs of girth at least $k$. The
authors of [7] also show that it is NP-complete to decide if a graph $G$
contains an induced tree $T$ covering four specified vertices such that $T$
has at most one vertex of degree at least three.
In general, $k$-in-a-Path and $k$-in-a-Tree are only equivalent for $k\leq 2$.
However, in this paper, we study claw-free graphs (graphs with no induced
4-vertex star). Claw-free graphs are a rich and well-studied class containing,
e.g., the class of (quasi)-line graphs and the class of complements of
triangle-free graphs; see [8] for a survey. Notice that any induced tree in a
claw-free graph is in fact an induced path.
The $k$-in-a-Path and $k$-in-a-Tree problem are equivalent for the class of
claw-free graphs.
Motivation. The polynomial-time algorithm for 3-in-a-Tree [4] has already
proven to be a powerful tool for several problems. For instance, it is used as
a subroutine in polynomial time algorithms for detecting induced thetas and
pyramids [4] and several other induced subgraphs [16]. The authors of [12] use
it to solve the Parity Path problem in polynomial time for claw-free graphs.
(This problem is to test if a graph contains both an odd and even length
induced paths between two specified vertices. It is NP-complete in general as
shown by Bienstock [1].)
Lévêque et al. [16] use the algorithm of [4] to solve the $2$-Induced Cycle
problem in polynomial time for graphs not containing an induced path or
subdivided claw on some fixed number of vertices. The $k$-Induced Cycle
problem is to test if a graph contains an induced cycle spanning $k$
terminals. In general it is NP-complete already for $k=2$ [1]. For fixed $k$,
an instance of this problem can be reduced to a polynomial number of instances
of the $k$-Induced Disjoint Paths problem, which we define below. Paths
$P_{1},\ldots,P_{k}$ in a graph $G$ are said to be mutually induced if for any
$1\leq i<j\leq k$, $P_{i}$ and $P_{j}$ have neither common vertices (i.e.
$V(P_{i})\cap V(P_{j})=\emptyset$) nor adjacent vertices (i.e. $uv\notin E$
for any $u\in V(P_{i}),v\in V(P_{j})$).
$k$-Induced Disjoint Paths
Instance: a graph $G$ with $k$ pairs of terminals $(s_{i},t_{i})$ for
$i=1,\ldots,k$.
Question: does $G$ contain $k$ mutually induced paths $P_{i}$ such that
$P_{i}$ connects $s_{i}$ and $t_{i}$ for $i=1,\ldots,k$?
This problem is NP-complete for $k=2$ [1]. Kawarabayashi and Kobayashi [14]
showed that, for any fixed $k$, the $k$-Induced Disjoint Paths problem is
solvable in linear time on planar graphs and that consequently $k$-Induced
Disjoint Cycle is solvable in polynomial time on this graph class for any
fixed $k$. In [15], Kawarabayashi and Kobayashi improve the latter result by
presenting a linear time algorithm for this problem, and even extend the
results for both these problems to graphs of bounded genus. As we shall see,
we can also solve $k$-Induced Disjoint Paths and $k$-Induced Cycle in
polynomial time in claw-free graphs. The version of the problem in which any
two paths are vertex-disjoint but may have adjacent vertices is called the
$k$-Disjoint Paths problem. For this problem Robertson and Seymour [17] proved
the following result.
###### Theorem 1.2 ([17]).
For fixed $k$, the $k$-Disjoint Paths problem is solvable in polynomial time.
Our Results and Paper Organization. In Section 2 we define some basic
terminology. Section 3 contains our main result: $k$-in-a-Path is solvable in
polynomial time in claw-free graphs for any fixed integer $k$. This, in fact,
follows from a stronger theorem proved in Section 4; the problem is solvable
in polynomial time even if the terminals are to appear on the path in a fixed
order. A consequence of our result is that the $k$-Induced Disjoint Paths and
$k$-Induced Cycle problems are polynomially solvable in claw-free graphs for
any fixed integer $k$. In Section 4 we present our polynomial-time algorithm
that solves the ordered version of $k$-in-a-Path. The algorithm first performs
“cleaning of the graph”. This is an operation introduced in [12]. After
cleaning the graph is free of odd antiholes of length at least seven. Next we
treat odd holes of length five that are contained in the neighborhood of a
vertex. The resulting graph is quasi-line. Finally, we solve the problem using
a recent characterization of quasi-line graphs by Chudnovsky and Seymour [3]
and related algorithmic results of King and Reed [13]. In Section 5 we mention
relevant open problems.
## 2\. Preliminaries
All graphs in this paper are undirected, finite, and neither have loops nor
multiple edges. Let $G$ be a graph. We refer to the vertex set and edge set of
$G$ by $V=V(G)$ and $E=E(G)$, respectively. The neighborhood of a vertex $u$
in $G$ is denoted by $N_{G}(u)=\\{v\in V\ |\ uv\in E\\}$. The subgraph of $G$
induced by $U\subseteq V$ is denoted $G[U]$. Analogously, the neighborhood of
a set $U\subseteq V$ is $N(U):=\bigcup_{u\in U}N(u)\setminus U$. We say that
two vertex-disjoint subsets of $V$ are adjacent if some of their vertices are
adjacent. The distance $d(u,v)$ between two vertices $u$ and $v$ in $G$ is the
number of edges on a shortest path between them. The edge contraction of an
edge $e=uv$ removes its two end vertices $u,v$ and replaces it by a new vertex
adjacent to all vertices in $N(u)\cup N(v)$ (without introducing loops or
multiple edges).
We denote the path and cycle on $n$ vertices by $P_{n}$ and $C_{n}$,
respectively. Let $P=v_{1}v_{2}\ldots v_{p}$ be a path with a fixed
orientation. The successor $v_{i+1}$ of $v_{i}$ is denoted by $v_{i}^{+}$ and
its predecessor $v_{i-1}$ by $v_{i}^{-}$. The segment $v_{i}v_{i+1}\ldots
v_{j}$ is denoted by $v_{i}\overrightarrow{P}v_{j}$. The converse segment
$v_{j}v_{j-1}\ldots v_{i}$ is denoted by $v_{j}\overleftarrow{P}v_{i}$.
A hole is an induced cycle of length at least 4 and an antihole is the
complement of a hole. We say that a hole is odd if it has an odd number of
edges. An antihole is called odd if it is the complement is an odd hole.
A claw is the graph $(\\{x,a,b,c\\},\\{xa,xb,xc\\})$, where vertex $x$ is
called the center of the claw. A graph is claw-free if it does not contain a
claw as an induced subgraph. A _clique_ is a subgraph isomorphic to a complete
graph. A diamond is a graph obtain from a clique on four vertices after
removing one edge. A vertex $u$ in a graph $G$ is simplicial if $G[N(u)]$ is a
clique.
Let $s$ and $t$ be two specified vertices in a graph $G=(V,E)$. A vertex $v\in
V$ is called irrelevant for vertices $s$ and $t$ if $v$ does not lie on any
induced path from $s$ to $t$. A graph $G$ is clean if none of its vertices is
irrelevant. We say that we clean $G$ for $s$ and $t$ by repeatedly deleting
irrelevant vertices for $s$ and $t$ as long as possible. In general,
determining if a vertex is irrelevant is NP-complete [1]. However, for claw-
free graphs, the authors of [12] could show the following (where they used
Observation 1 and Theorem 2.5 for obtaining the polynomial time bound).
###### Lemma 2.1 ([12]).
Let $s,t$ be two vertices of a claw-free graph $G$. Then $G$ can be cleaned
for $s$ and $t$ in polynomial time. Moreover, the resulting graph does not
contain an odd antihole of length at least seven.
The line graph of a graph $G$ with edges $e_{1},\ldots,e_{p}$ is the graph
$L=L(G)$ with vertices $u_{1},\ldots,u_{p}$ such that there is an edge between
any two vertices $u_{i}$ and $u_{j}$ if and only if $e_{i}$ and $e_{j}$ share
an end vertex in $H$. We note that mutually induced paths in a line graph
$L(G)$ are in one-to-one correspondence with vertex-disjoint paths in $G$.
Combining this observation with Theorem 1.2 leads to the following result.
###### Corollary 2.2.
For fixed $k$, the $k$-Induced Disjoint Paths problem can be solved in
polynomial time in line graphs.
A graph $G=(V,E)$ is called a _quasi-line graph_ if for every vertex $u\in V$
there exist two vertex-disjoint cliques $A$ and $B$ in $G$ such that
$N(u)=V(A)\cup V(B)$ (where $V(A)$ and $V(B)$ might be adjacent). Clearly,
every line graph is quasi-line and every quasi-line graph is claw-free. The
following observation is useful and easy to see by looking at the complements
of neighborhood in a graph.
A claw-free graph $G$ is a quasi-line graph if and only if $G$ does not
contain a vertex with an odd antihole in its neighborhood.
A clique in a graph $G$ is called nontrivial if it contains at least two
vertices. A nontrivial clique $A$ is called homogeneous if every vertex in
$V(G)\backslash V(A)$ is either adjacent to all vertices of $A$ or to none of
them. Notice that it is possible to check in polynomial time if an edge of the
graph is a homogeneous clique. This justifies the following observation.
The problem of detecting a homogeneous clique in a graph is solvable in
polynomial time.
Two disjoint cliques $A$ and $B$ form a _homogeneous pair_ in $G$ if the
following two conditions hold. First, at least one of $A,B$ contains more than
one vertex. Second, every vertex $v\in V(G)\setminus(V(A)\cup V(B))$ is either
adjacent to all vertices of $A$ or to none vertex of $A$ as well as either
adjacent to all of $B$ or to none of $B$. The following result by King and
Reed [13, Section 3] will be useful.
###### Lemma 2.3 ([13]).
The problem of detecting a homogeneous pair of cliques in a graph is solvable
in polynomial time.
Let $V$ be a finite set of points of a real line, and ${\mathcal{I}}$ be a
collection of intervals. Two points are adjacent if and only if they belong to
a common interval $I\in{\mathcal{I}}$. The resulting graph is a _linear
interval graph_. Analogously, if we consider a set of points of a circle and
set of intervals (angles) on the circle we get a _circular interval graph_.
Graphs in both classes are claw-free, in fact linear interval graphs coincide
with proper interval graphs (intersection graph of a set of intervals on a
line, where no interval contains another from the set) and circular interval
graphs coincide with proper circular arc graphs (defined analogously). We need
the following result of Deng, Hell, and Huang [5].
Figure 1. Composition of three linear interval strips (only part of the graph
is displayed).
###### Theorem 2.4 ([5]).
Circular interval graphs and linear interval graphs can be recognized in
linear time. Furthermore, a corresponding representation of such graphs can be
constructed in linear time as well.
A _linear interval strip_ $(S,a,b)$ is a linear interval graph $S$ where $a$
and $b$ are the leftmost and the rightmost points (vertices) of its
representation. Observe that in such a graph the vertices $a$ and $b$ are
simplicial. Let $S_{0}$ be a graph with vertices
$a_{1},b_{1},\dots,a_{n},b_{n}$ that is isomorphic to an arbitrary disjoint
union of complete graphs. Let
$(S_{1}^{\prime},a_{1}^{\prime},b_{1}^{\prime}),\dots,(S_{n}^{\prime},a_{n}^{\prime},b_{n}^{\prime})$
be a collection of linear interval strips. The _composition_ $S_{n}$ is
defined inductively where $S_{i}$ is formed from the disjoint union of
$S_{i-1}$ and $S_{i}^{\prime}$, where:
* $\bullet$
all neighbors of $a_{i}$ are connected to all neighbors of $a_{i}^{\prime}$;
* $\bullet$
all neighbors of $b_{i}$ are connected to all neighbors of $b_{i}^{\prime}$;
* $\bullet$
vertices $a_{i},a_{i}^{\prime},b_{i},b_{i}^{\prime}$ are removed.
See Figure 1 for an example. We are now ready to state the structure of quasi-
line graphs as characterized by Chudnovsky and Seymour [3].
###### Theorem 2.5 ([3]).
A quasi-line graph $G$ with no homogeneous pair of cliques is either a
circular interval graph or a composition of linear interval strips.
Finally, we need another algorithmic result of King and Reed [13]. They
observe that the composition of the final strip in a composition of linear
interval graphs is a so-called nontrivial interval 2-join and that every
nontrivial interval 2-join contains a so-called canonical interval 2-join. In
Lemma 13 of this paper they show how to find in polynomial time a canonical
interval 2-join in a quasi-line graph with no homogeneous pair of cliques and
no simplicial vertex or else to conclude that none exists. Recursively
applying this result leads to the following lemma.
###### Lemma 2.6 ([13]).
Let $G$ be a quasi-line graph with no homogeneous pairs of cliques and no
simplicial vertex that is a composition of linear interval strips. Then the
collection of linear interval strips that define $G$ can be found in
polynomial time.
## 3\. Our Main Result
Here is our main result.
###### Theorem 3.1.
For any fixed $k$, the $k$-in-a-Path problem is solvable in polynomial time in
claw-free graphs.
In order to prove Theorem 3.1 we define the following problem.
Ordered-$k$-in-a-Path
Instance: a graph $G$ with $k$ terminals ordered as $t_{1},\ldots,t_{k}$.
Question: does there exist an induced path of $G$ starting in $t_{1}$ then
passing through $t_{2},\ldots,t_{k-1}$ and ending in $t_{k}$?
We can resolve the original $k$-in-a-Path problem by $k!$ rounds of the more
specific version defined above, where in each round we order the terminals by
a different permutation. Hence, since we assume that $k$ is fixed, it suffices
to prove Theorem 3.2 in order to obtain Theorem 3.1.
###### Theorem 3.2.
For any fixed $k$, the Ordered-$k$-in-a-Paths problem is solvable in
polynomial time in claw-free graphs.
We prove Theorem 3.2 in Section 4 and finish this section with the following
consequence of it.
###### Corollary 3.3.
For any fixed $k$, the $k$-Disjoint Induced Paths and $k$-Induced Cycle
problem are solvable in polynomial time in claw-free graphs.
###### Proof 3.4.
Let $G$ be a claw-free graph that together with terminals $t_{1},\ldots,t_{k}$
is an instance of $k$-Induced Cycle. We fix an order of the terminals, say,
the order is $t_{1},\ldots,t_{k}$. We fix neighbors $a_{i}$ and $b_{i-1}$ of
each terminal $t_{i}$. This way we obtain an instance of $k$-Induced Disjoint
Paths with pairs of terminals $(a_{i},b_{i})$ where $b_{0}=b_{k}$. Clearly,
the total number of instances we have created is polynomial. Hence, we can
solve $k$-Induced Cycle in polynomial time if we can solve $k$-Induced
Disjoint Paths in polynomial time.
Let $G$ be a claw-free graph that together with $k$ pairs of terminals
$(a_{i},b_{i})$ for $i=1,\ldots,k$ is an instance of the $k$-Induced Disjoint
Paths problem. First we add an edge between each pair of non-adjacent
neighbors of every terminal in
$T=\\{a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}\\}$. We denote the resulting
graphs obtained after performing this operation on a terminal by
$G_{1},\ldots,G_{2k}$, and define $G_{0}:=G$. We claim that
$G^{\prime}=G_{2k}$ is claw-free and prove this by induction.
The claim is true for $G_{0}$. Suppose the claim is true for $G_{j}$ for some
$0\leq j\leq 2k-1$. Consider $G_{j+1}$ and suppose, for contradiction, that
$G_{j+1}$ contains an induced subgraph isomorphic to a claw. Let
$K:=\\{x,a,b,c\\}$ be a set of vertices of $G_{j+1}$ inducing a claw with
center $x$. Let $s\in T$ be the vertex of $G_{j}$ that becomes simplicial in
$G_{j+1}$. Then $x\neq s$. Since $G_{j}$ is claw-free, we may without loss of
generality assume that at least two vertices of $K$ must be in
$N_{G_{j+1}}(s)\cup\\{s\\}$. Since $N_{G_{j+1}}(s)\cup\\{s\\}$ is a clique of
$G_{j+1}$ and $\\{a,b,c\\}$ is an independent set of $G_{j+1}$, we may without
loss of generality assume that $K\cap(N_{G_{j+1}}(s)\cup\\{s\\})=\\{x,a\\}$
and $\\{b,c\\}\subseteq V(G_{j+1})\setminus(N_{G_{j+1}}(s)\cup\\{s\\})$. Then
$\\{x,b,c,s\\}$ induces a claw in $G_{j}$ with center $x$, a contradiction.
Hence, $G^{\prime}$ is indeed claw-free.
We note that $G$ with terminals $(a_{1},b_{1}),\ldots,(a_{k},b_{k})$ forms a
Yes-instance of $k$-Induced Disjoint Paths if and only if $G^{\prime}$ with
the same terminal pairs is a Yes-instance of this problem. In the next step we
identify terminal $b_{i}$ with $a_{i+1}$, i.e., for $i=1,\ldots,k-1$ we remove
$b_{i},a_{i+1}$ and replace them by a new vertex $t_{i+1}$ adjacent to all
neighbors of $a_{i+1}$ and to all neighbors of $b_{i}$. We call the resulting
graph $G^{\prime\prime}$ and observe that $G$ is claw-free. We define
$t_{1}:=a_{1}$ and $t_{k+1}:=b_{k}$ and claim that $G^{\prime}$ with terminal
pairs $(a_{1},b_{1}),\ldots,(a_{k},b_{k})$ forms a Yes-instance of the
$k$-Induced Paths problem if and only if $G^{\prime\prime}$ with terminals
$t_{1},\ldots,t_{k+1}$ forms a Yes-instance of the Ordered-$(k+1)$-in-a-Path
problem.
In order to see this, suppose $G^{\prime}$ contains $k$ mutually induced paths
$P_{i}$ such that $P_{i}$ connects $a_{i}$ to $b_{i}$ for $1\leq i\leq k$.
Then
$P=t_{1}\overrightarrow{P_{1}}b_{1}^{-}t_{2}a_{2}^{+}\overrightarrow{P_{2}}b_{2}^{-}\ldots
t_{k}a_{k}^{+}\overrightarrow{P_{k}}t_{k}$
is an induced path passing through the terminals $t_{i}$ in prescribed order.
Now suppose $G^{\prime\prime}$ contains an induced path $P$ passing through
terminals in order $t_{1},\ldots,t_{k+1}$. For $i=1,\ldots,k+1$ we define
paths $P_{i}=a_{i}t_{i}^{+}\overrightarrow{P}t_{i+1}^{-}b_{i}$, which are
mutually induced. We now apply Theorem 3.2. This completes the proof.
## 4\. The Proof of Theorem 3.2
We present a polynomial-time algorithm that solves the Ordered-$k$-in-a-Path
problem on a claw-free graph $G$ with terminals in order $t_{1},\ldots,t_{k}$
for any fixed integer $k$. We call an induced path $P$ from $t_{1}$ to $t_{k}$
that contains the other terminals in order $t_{2},\ldots,t_{k-1}$ a solution
of this problem. Furthermore, an operation in this algorithm on input graph
$G$ with terminals $t_{1},\ldots,t_{k}$ preserves the solution if the
following holds: the resulting graph $G^{\prime}$ with resulting terminals
$t_{1}^{\prime},\ldots,t_{k^{\prime}}^{\prime}$ for some $k^{\prime}\leq k$ is
a Yes-instance of the Ordered-$k^{\prime}$-in-a-Path problem if and only if
$G$ is a Yes-instance of the Ordered-$k$-in-a-Path problem. We call $G$ simple
if the following three conditions hold:
* (i)
$t_{1},t_{k}$ are of degree one in $G$ and all other terminals $t_{i}$
($1<i<k$) are of degree two in $G$, and the two neighbors of such $t_{i}$ are
not adjacent;
* (ii)
the distance between any pair $t_{i},t_{j}$ is at least four;
* (iii)
$G$ is connected.
The Algorithm and Proof of Theorem 3.2
Let $G$ be an input graph with terminals $t_{1},\ldots,t_{k}$.
If $k=2$, we compute a shortest path from $t_{1}$ to $t_{2}$. If $k=3$, we use
Theorem 1.1 together with Observation 1. Suppose $k\geq 4$.
Step 1. Reduce to a set of simple graphs.
We apply Lemma 4.1 and obtain in polynomial time a set ${\mathcal{G}}$ that
consists of a polynomial number of simple graphs of size at most $|V(G)|$ such
that there is a solution for $G$ if and only if there is a solution for one of
the graphs in ${\mathcal{G}}$. We consider each graph in ${\mathcal{G}}$. For
convenience we denote such a graph by $G$ as well.
Step 2. Reduce to a quasi-line graph.
We first clean $G$ for $t_{1}$ and $t_{k}$. If during cleaning we remove a
terminal, then we output No. Otherwise, clearly, we preserve the solution. By
Lemma 2.1, this can be done in polynomial time and ensures that there are no
odd antiholes of length at least seven left. Also, $G$ stays simple. Then we
apply Lemma 4.3, which removes vertices $v$ whose neighborhood contain an odd
hole of length five, as long as we can. Clearly, we can do this in polynomial
time. Note that $G$ stays connected since we do not remove cut-vertices due to
the claw-freeness. By condition (i), we do not remove a terminal either.
Afterwards, we clean $G$ again for $t_{1}$ and $t_{k}$. If we remove a
terminal, we output No. Otherwise, as a result of our operations, $G$ becomes
a simple quasi-line graph due to Observation 2.
Step 3. Reduce to a simple quasi-line graph with no homogeneous clique
We first exhaustively search for homogeneous cliques by running the polynomial
algorithm mentioned in Observation 2 and apply Lemma 4.5 each time we find
such a clique. Clearly, we can perform the latter in polynomial time as well.
After every reduction of such a clique to a single vertex, $G$ stays simple
and quasi-line, and at some moment does not contain any homogeneous clique
anymore, while we preserve the solution.
Step 4. Reduce to a circular interval graph or to a composition of interval
strips.
Let $t_{1}^{\prime},t_{k}^{\prime}$ be the (unique) neighbor of $t_{1}$ and
$t_{k}^{\prime}$, respectively. As long as $G$ contains homogeneous pairs of
cliques $(A,B)$ so that $A$ neither $B$ is equal to
$\\{t_{1},t_{1}^{\prime}\\}$ or $\\{t_{k},t_{k}^{\prime}\\}$, we do as
follows. We first detect such a pair in polynomial time using Lemma 2.3 and
reduce them to a pair of single vertices by applying Lemma 4.7. Also
performing Lemma 4.7 clearly takes only polynomial time. After every
reduction, $G$ stays simple and quasi-line, and we preserve the solution. At
some moment, the only homogeneous pairs of cliques that are possibly left in
$G$ are of the form $(\\{t_{1},t_{1}^{\prime}\\},B)$ and
$(\\{t_{k},t_{k}^{\prime}\\},B)$. As $G$ does not contain a homogeneous clique
(see Step 3), the cliques in such pairs must have adjacent vertex sets. Hence,
there can be at most two of such pairs. We perform Lemma 4.7 and afterwards
make the graph simple again. Although this might result in a number of new
instances, their total number is still polynomial because we perform this
operation at most twice. Hence, we may without loss of generality assume that
$G$ stays simple. By Theorem 2.5, $G$ is either a circular interval graph or a
composition of linear interval strips; we deal with theses two cases
separately after recognizing in polynomial time in which case we are by using
Theorem 2.4.
Step 5a. Solve the problem for a circular interval graph.
Let $G$ be a circular interval graph. Observe that the order of vertices in an
induced path must respect the natural order of points on a circle. Hence,
deleting all points that lie on the circle between $t_{k}$ and $t_{1}$
preserves the solution. So, we may even assume that $G$ is a linear interval
graph. We solve the problem in these graphs in Theorem 4.9.
Step 5b. Solve the problem for a composition of linear interval strips.
Let $G$ be a composition of linear interval strips. Because $G$ is assumed to
be clean for $t_{1},\ldots,t_{k}$, $G$ contains no simplicial vertex. Then we
can find these strips in polynomial time using Lemma 2.6 and use this
information in Lemma 4.11. There we create a line graph $G^{\prime}$ with
$|V(G^{\prime})|\leq|V(G)|$, while preserving the solution. Moreover, this can
be done in polynomial time by the same theorem. Then we use Corollary 2.2 to
prove that the problem is polynomially solvable in line graphs in Theorem
4.12.
Now it remains to state and prove Lemmas 4.1–4.11 and Theorems 4.9– 4.12.
###### Lemma 4.1.
Let $G$ be a graph with terminals ordered $t_{1},\ldots,t_{k}$. Then there
exists a set ${\mathcal{G}}$ of $n^{O(k)}$ simple graphs, each of size at most
$|V(G)|$, such that $G$ has a solution if and only if there exists a graph in
${\mathcal{G}}$ that has a solution. Moreover, ${\mathcal{G}}$ can be
constructed in polynomial time.
###### Proof 4.2.
We branch as follows. First we guess the first six vertices after $t_{1}$ in a
possible solution. Then we guess the last six vertices before $t_{n}$.
Finally, for $2\leq i\leq n-1$, we guess the last six vertices preceding
$t_{i}$ and the first six vertices following $t_{i}$. We check if the subgraph
induced by the terminals and all guessed vertices has maximum degree 2. If not
we discard this guess. Otherwise, for every terminal and for every guessed
vertex that is not an end vertex of a guessed subpath, we remove all its
neighbors that are not guessed vertices. This way we obtain a number of graphs
which we further process one by one.
Let $G^{\prime}$ be such a created subgraph. If $G^{\prime}$ does not contain
all terminals, we discard $G^{\prime}$. If $G^{\prime}$ is disconnected then
we discard $G^{\prime}$ if two terminals are in different components, or else
we continue with the component of $G^{\prime}$ that contains all the
terminals. Suppose there is a guessed subpath in $G^{\prime}$ containing more
than one terminal. If the order is not $t_{i},t_{i+1},\ldots,t_{j}$ for some
$i<j$, we discard $G^{\prime}$. Otherwise, if necessary, we place $t_{i}$ and
$t_{j}$ on this subpath such that they are at distance at least four of each
other and also are of distance at least four of each end vertex of the
subpath. Because the guessed subpaths are sufficiently long, such a placement
is possible. We then remove $t_{i+1},\ldots,t_{j-1}$ from the list of
terminals. After processing all created graphs as above, we obtain the desired
set ${\mathcal{G}}$. Since $k$ is fixed, ${\mathcal{G}}$ can be constructed in
polynomial time.
###### Lemma 4.3.
Let $G$ be a simple claw-free graph. Removing a vertex $u\in V(G)$, the
neighborhood of which contains an induced odd hole of length five, preserves
the solution.
###### Proof 4.4.
Because $G$ is simple, $u$ is not a terminal. We first show the following
claim.
Claim 1. Let $G[\\{v,w,x,y\\}]$ be a diamond in which $vw$ is a non-edge. If
there is a solution $P$ that contains $v,x,w$, then there is another solution
that contains $v,y,w$ (and that does not contain $x$).
In order to see this take the original solution $P$ and notice that by claw-
freeness any neighbor of $y$ on $P$ must be in the (closed) neighborhood of
$v$ or $w$. This way the solution can be rerouted via $y$, without using $x$.
This proves Claim 1.
Now suppose that $u$ is a vertex which has an odd hole $C$ of length five in
its neighborhood. Obviously, $G$ is a Yes-instance if $G-u$ is a Yes-instance.
To prove the reverse implication, suppose $G$ is a Yes-instance. Let $P$ be a
solution. If $u$ does not belong to $P$ then we are done. Hence, we suppose
that $u$ belongs to $P$ and consider three cases depending on $|V(C)\cap
V(P)|$.
Case 1. $|V(C)\cap V(P)|\geq 2$. Then $|V(C)\cap V(P)|=2$, as any vertex on
$P$ will have at most two neighbors. We are done by Claim 1.
Case 2. $|V(C)\cap V(P)|=1$. Let $w\in V(C)$ belong to $P$ and let the other
neighbor of $u$ that belongs to $P$ be $x$. We note that $x$ must be adjacent
to at least one of the neighbors of $w$ in $C$. Then we can apply Claim 1
again.
Case 3. $|V(C)\cap V(P)|=0$. Let the two neighbors of $u$ on $P$ be $x$ and
$y$. To avoid a claw at $u$, every vertex of $C$ must be adjacent to $x$ or
$y$. If there is a vertex in $C$ adjacent to both, we apply Claim 1. Suppose
there is no such vertex and that the vertices of the $C$ are partitioned in
two sets $X$ (vertices of $C$ only adjacent to $x$) and $Y$ (vertices of $C$
only adjacent to $y$). We assume without loss of generality that $|X|=3$, and
hence contains a pair of independent vertices which together with $u$ and $y$
form a claw. This is a contradiction.
###### Lemma 4.5.
Let $G$ be a simple quasi-line graph with a homogeneous clique $A$. Then
contracting $A$ to a single vertex preserves the solution and the resulting
graph is a simple quasi-line graph containing the same terminals as $G$.
###### Proof 4.6.
Each vertex in $A$ lies on a triangle, unless $G$ is isomorphic to $P_{2}$,
which is not possible. Hence, by condition (i), $A$ does not contain a
terminal. We remove all vertices of $A$ except one. The resulting graph will
be a simple quasi-line graph containing the same terminals, and we will
preserve the solution.
###### Lemma 4.7.
Let $G$ be a simple quasi-line graph with terminals ordered
$t_{1},\ldots,t_{k}$ that has no homogeneous clique. Contracting the cliques
$A$ and $B$ in a homogeneous pair to single vertices preserves the solution.
The resulting graph is quasi-line; it is simple unless $A$ or $B$ consists of
two vertices $u,u^{\prime}$ with $u\in\\{t_{1},t_{k}\\}$ and
$d(u^{\prime},t_{i})\leq 3$ for some $t_{i}\neq u$.
###### Proof 4.8.
Because $G$ does not contain a homogeneous clique, $V(A)$ and $V(B)$ must be
adjacent. Then, due to condition (ii), there can be at most one terminal in
$V(A)\cup V(B)$. In all the cases discussed below we will actually not
contract edges but only remove vertices from $A$ and $B$. Hence, the resulting
graph will always be a quasi-line graph.
Suppose $A$ contains $t_{1}$ or $t_{k}$, say $t_{1}$. Suppose $|V(A)|=1$, so
$A$ only contains $t_{1}$. Then the neighbor of $t_{1}$ is in $B$ and
$|V(B)|\geq 2$. We delete all vertices from $B$ except this neighbor, because
they will not be used in any solution. Clearly, the resulting graph is simple
and the solution is preserved. Suppose $|V(A)|\geq 2$. Because $t_{1}$ is of
degree one, $A$ consists of two vertices, namely $t_{1}$ and its neighbor
$t_{1}^{\prime}$. Note that $t_{1}^{\prime}$ does not have a neighbor outside
$A$ and $B$, as $t_{1}$ is of degree one. As $V(A)$ and $V(B)$ are adjacent,
$t_{1}^{\prime}$ has a neighbor $u$ in $B$. We delete $t_{1}$ and replace it
by $t_{1}^{\prime}$ in the set of terminals. We delete all vertices of $B$
except $u$, because of the following reasons. If these vertices are not
adjacent to $t_{1}^{\prime}$, they will never appear in any solution. If they
are adjacent to $t_{1}^{\prime}$, they will not appear in any solution
together with $u$, and as such they can be replaced by $u$. Note that
$t_{1}^{\prime}$ has degree one in the new graph and that this graph is only
simple if $d(t_{1}^{\prime},t_{j})\geq 4$ for all $2\leq j\leq k$. Clearly,
the solution is preserved.
Suppose $A$ contains a terminal $t_{i}$ for some $2\leq i\leq k-1$. Suppose
$A$ only contains $t_{i}$. Because $V(A)$ and $V(B)$ are adjacent, $t_{i}$ is
adjacent to a vertex $u$ in $B$. By condition (i), $u$ is the only vertex in
$B$ adjacent to $t_{i}$. We delete all vertices of $B$ except $u$. Clearly,
the resulting graph is simple and the solution is preserved. Suppose
$|V(A)|\geq 2$. By condition (ii), $A$ contains only one other vertex
$t_{i}^{\prime}$ and $t_{i},t_{i}^{\prime}$ do not have a common neighbor.
Then $A$ must be separated of the rest of the graph by $B$. Furthermore, the
other neighbor of $t_{i}$ must be in $B$. We delete $t_{i}^{\prime}$ and all
vertices of $B$ except the neighbor of $t_{i}$. Clearly, the resulting graph
is simple and the solution is preserved.
Suppose $A$ does not contain a terminal. By symmetry, we may assume that $B$
does not contain a terminal either. Let $a^{\prime}b^{\prime}\in E(G)$ with
$a^{\prime}\in V(A)$ and $b^{\prime}\in V(B)$. Let $G^{\prime}$ be the graph
obtained from $G$ by removing all vertices of $A$ except $a^{\prime}$ and $B$
except $a^{\prime},b^{\prime}$. Note that we have kept all terminals and that
the resulting graph is simple. Any solution $P^{\prime}$ for $G^{\prime}$ is a
solution for $G$.
Now assume we have a solution $P$ for $G$. We claim that $|P\cap A|\leq 1$ and
$|P\cap B|\leq 1$. Suppose otherwise, say $|P\cap A|\geq 2$. Then $|P\cap
A|=2$, as $P$ is a path. Since $t_{1}$ and $t_{k}$ are not in $A$, we find
that $P$ contains a subpath $xuvy$ with $u,v\in A$. Since $x$ is adjacent to
$u\in A$, but also non-adjacent to $v\in A$, we find that $x\in B$.
Analogously we get that $y\in B$. However, then $xy\in E(G)$. This is a
contradiction.
Suppose $|P\cap A|=0$ and $|P\cap B|=0$. Then $P$ is a solution for
$G^{\prime}$ as well. Suppose $|P\cap A|=0$ and $|P\cap B|=1$. Then we may
without loss of generality assume that $b^{\prime}\in V(P)$ and find that $P$
is a solution for $G^{\prime}$ as well. The case $|P\cap A|=1$ and $|P\cap
B|=0$ follows by symmetry. Suppose $|P\cap A|=|P\cap B|=1$, say $P$ intersects
$A$ in $a$ and $B$ in $b$. If $ab\in E(G)$ then we replace $ab$ by
$a^{\prime}b^{\prime}$ and obtain a solution for $G^{\prime}$. Suppose
$ab\notin E(G)$. Because $a$ is not a terminal, $a$ has neighbors $x$ and $y$
on $P$. If $x,y\notin N(b)$ then $\\{a^{\prime},x,y,b^{\prime}\\}$ induces a
claw in $G$ with center $a^{\prime}$. This is not possible. Hence, we may
assume without loss of generality that $y$ is adjacent to $b$. Since $A$ or
$B$ contains at least two vertices, $y$ has degree at least three. Then $y$ is
not a terminal. Thus we can skip $y$ and exchange $ayb$ in $P$ with
$a^{\prime}b^{\prime}$ to get the desired induced path $P^{\prime}$.
###### Theorem 4.9.
The Ordered-$k$-in-a-Path problem can be solved in polynomial time in linear
interval graphs.
###### Proof 4.10.
Let $G$ be a linear interval graph. We may assume without loss of generality
that the terminals form an independent set. We use its linear representation
that we obtain in polynomial time by Lemma 2.6. In what follows the notions of
predecessors (left) and successors (right) are considered for the linear
ordering of the points on the line. Without loss of generality we may assume
that $t_{1}$ is the first point and that $t_{k}$ is the last and that no two
points coincide. By our assumption, $t_{i}$ and $t_{i+1}$ are nonadjacent.
From the set of points belonging to the closed interval $[t_{i},t_{i+1}]$ we
remove all neighbors of $t_{i}$ except the rightmost one and all neighbors of
$t_{i+1}$ except the leftmost. Then the shortest path between $t_{i}$ and
$t_{i+1}$ is induced. In addition, these partial paths combined together
provide a solution unless for some terminal $t_{i}$ its leftmost predecessor
is adjacent to its rightmost successor. Hence, no induced path may have
$t_{i}$ among its inner vertices.
###### Lemma 4.11 (proof postponed to journal version).
Let $G$ be a composition of linear interval strips. It is possible to create
in polynomial time a line graph $G^{\prime}$ with $|V(G^{\prime})|\leq|V(G)|$,
while preserving the solution.
###### Theorem 4.12.
For fixed $k$, Ordered-$k$-in-a-Path is polynomially solvable in line graphs.
###### Proof 4.13.
A version of Ordered-$k$-in-a-Path in which the path is not necessarily
induced can be easily translated into an instance of the $k$-Disjoint Paths
problem and solved in polynomial time due to Theorem 1.2. Noting that mutually
induced paths in a line graph $L(G)$ are in one-to-one correspondence with
vertex-disjoint paths in $G$ enables us to solve the Ordered-$k$-in-a-Path
problem in polynomial time for line graphs.
## 5\. Conclusions and Further Research
We showed that, for any fixed $k$, the problems $k$-in-a-Path, $k$-Disjoint
Induced Paths and $k$-Induced Cycle are polynomially solvable on claw-free
graphs. If $k$ is part of the input these problems are known to be NP-
complete. In the journal version we show this is true, even when the input is
restricted to be claw-free. Perhaps the two most fascinating related open
problems are to determine the complexity of deciding if a graph contains an
odd hole (whereas the problem of finding an even hole is polynomially solvable
[2]) and to determine the computational complexity of deciding if a graph
contains two mutually induced holes (whereas it is known that the case of two
mutually induced odd holes is NP-complete [10]). For claw-free graphs these
two problems are solved. Shrem et al. [19] even obtained a polynomial-time
algorithm for detecting a shortest odd hole in a claw-free graph. In the
journal version we will address the second problem for claw-free graphs.
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* [18] M. Chudnovsky, N. Robertson, P.D. Seymour, and R. Thomas. The strong perfect graph theorem. Annals of Mathematics 164 (2006) 51–229.
* [19] S. Shrem, M. Stern and M.C. Golumbic. Smallest odd holes in claw-free graphs. In Proceedings of WG 2009, LNCS 5911 (2009) 329–340.
* [20] N. Trotignon and L. Wei. The $k$-in-a-tree problem for graphs of girth at least $k$, manuscript.
|
arxiv-papers
| 2009-12-30T18:46:27 |
2024-09-04T02:49:07.343594
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiri Fiala, Marcin Kaminski, Bernard Lidicky and Daniel Paulusma",
"submitter": "Bernard Lidick\\'y",
"url": "https://arxiv.org/abs/0912.5468"
}
|
0912.5533
|
# Oriented Straight Line Segment Algebra: Qualitative Spatial Reasoning
about Oriented Objects
$\mbox{Reinhard Moratz}^{1}$ and $\mbox{Dominik L{\"{u}}cke}^{2}$ and
$\mbox{Till Mossakowski}^{2}$
###### Abstract
Nearly 15 years ago, a set of qualitative spatial relations between oriented
straight line segments (dipoles) was suggested by Schlieder. This work
received substantial interest amongst the qualitative spatial reasoning
community. However, it turned out to be difficult to establish a sound
constraint calculus based on these relations. In this paper, we present the
results of a new investigation into dipole constraint calculi which uses
algebraic methods to derive sound results on the composition of relations and
other properties of dipole calculi. Our results are based on a condensed
semantics of the dipole relations.
In contrast to the points that are normally used, dipoles are extended and
have an intrinsic direction. Both features are important properties of natural
objects. This allows for a straightforward representation of prototypical
reasoning tasks for spatial agents. As an example, we show how to generate
survey knowledge from local observations in a street network. The example
illustrates the fast constraint-based reasoning capabilities of the dipole
calculus. We integrate our results into two reasoning tools which are publicly
available.
${}^{1}\mbox{University of Maine,}$
National Center for Geographic Information and Analysis,
Department of Spatial Information Science and Engineering,
348 Boardman Hall, Orono, 04469 Maine, USA.
moratz@spatial.maine.edu
${}^{2}\mbox{University of Bremen,}$
Collaborative Research Center on Spatial Cognition (SFB/TR 8),
Department of Mathematics and Informatics,
Bibliothekstr. 1, 28359 Bremen, Germany.
till$\;|\,$luecke@informatik.uni-bremen.de
Keywords:
Qualitative Spatial Reasoning, Relation Algebra, Affine Geometry
## 1 Introduction
Qualitative Reasoning about space abstracts from the physical world and
enables computers to make predictions about spatial relations, even when
precise quantitative information is not available [1]. A qualitative
representation provides mechanisms which characterize the essential properties
of objects or configurations. In contrast, a quantitative representation
establishes a measure in relation to a unit of measurement which must be
generally available [2]. The constant and general availability of common
measures is now self-evident. However, one needs only recall the history of
length measurement technologies to see that the more local relative measures,
which are represented qualitatively111Compare for example the qualitative
expression ”one piece of material is longer than another” with the
quantitative expression ”this thing is two meters long”, can be managed by
biological/epigenetic cognitive systems much more easily than absolute
quantitative representations.
Qualitative spatial calculi usually deal with elementary objects (e.g.
positions, directions, regions) and qualitative relations between them (e.g.
”adjacent”, ”to the left of”, ”included in”). This is the reason why
qualitative descriptions are quite natural for people. The two main trends in
Qualitative Spatial Reasoning (QSR) are topological reasoning about regions
[3, 4, 5, 6, 7] and positional (e.g. direction and distance) reasoning about
point configurations [8, 9, 10, 11, 12, 13, 14]. Positions can refer to a
global reference system, e.g. cardinal directions, or just to local reference
systems, e.g. egocentric views. Positional calculi can be related to the
results of Psycholinguistic research [15] in the field of reference systems.
In Psycholinguistics, local reference systems are divided into two modalities:
intrinsic reference systems and extrinsic reference systems. Then, the three
resulting options for giving a linguistic description of the spatial
arrangements of objects are: intrinsic, extrinsic, and absolute (i.e. global)
reference systems [16]222In [16], extrinsic references are called relative
references.. Corresponding QSR calculi can be found in Psycholinguistics for
all three types of reference systems. An intrinsic reference system employs an
oriented physical object as the origin of a reference system (relatum). The
orientation of the physical object then serves as a reference direction for
the reference system. For instance, an intrinsic reference system is used in
the calculus of oriented line segments (see Fig. 1) which is the main topic of
this paper. Another calculus corresponding to intrinsic reference systems is
the $\mathcal{OPRA}$ calculus [17]. In the $\mathcal{OPRA}$ calculus, oriented
points are the basic entities (see Fig. 5).
Extrinsic reference systems are closely related to intrinsic reference
systems. Both reference system options share the feature of focusing on the
local context. The difference is that the extrinsic reference system
superimposes the view direction from an external observer as reference
direction to the relatum of the reference system. A typical example for a QSR
calculus corresponding to an extrinsic reference system is Freksa’s double
cross calculus [18]. In the double cross calculus, two points span a reference
system to localize a third point. The first point then projects a view towards
the second point which generates the reference direction.
Since intrinsic and extrinsic references are closely related in the rest of
the paper, we sometimes refer to QSR calculi which use either intrinsic or
extrinsic reference systems as relative position QSR calculi. Then, the two
terms local reference systems and relative reference systems refer to the same
concept. An interesting special case refers to the representation of a
relative orientation without the concept of distance. These relative
orientations can be viewed as decoupled from anchor points. Then there is no
means for distinguishing between different point locations. The great
advantage is that much more efficient reasoning mechanisms become available.
The work by Isli and Cohn [19] consists of a ternary calculus for reasoning
about such pure orientations. In contrast to relative position calculi, their
algebra has a tractable subset containing the base relations.
Absolute (or global) directions can relate directional knowledge from distant
places to each other. Cardinal directions as an example can be registered with
a compass and compared over a large distance. And for that reason Frank’s
cardinal direction calculus corresponds to such an absolute reference system
[9], [20]. There is a variant of a cardinal direction calculus, which has a
flexible granularity, the Star Calculus [21].
In the previous paragraphs, we discussed the representation of spatial
knowledge. Another important aspect are the reasoning mechanisms which are
employed to make use of the represented initial knowledge to infer indirect
knowledge. In Qualitative Spatial Reasoning two main reasoning modes are used:
Conceptual neighbourhood-based reasoning, and constraint-based reasoning about
(static) spatial configurations. Conceptual neighborhood-based reasoning
describes whether two spatial configurations of objects can be transformed
into each other by small changes [22]. The conceptual neighborhood of a
qualitative spatial relation which holds for a spatial arrangement is the set
of relations into which a relation can be changed with minimal
transformations, e.g. by continuous deformation. Such a transformation can be
a movement of one object in the configuration in a short period of time. At
the discrete level of concepts, the neighborhood corresponds to continuity on
the geometric or physical level of description: continuous processes map onto
identical or neighboring classes of descriptions [23]. Spatial conceptual
neighborhoods are very natural perceptual and cognitive entities and other
neighborhood structures can be derived from spatial neighborhoods, e.g.
temporal neighborhoods. The movement of an agent can then be modeled
qualitatively as a sequence of neighboring spatial relations which hold for
adjacent time intervals333This was the reasoning used in the first
investigation of dipole relations by Schlieder [24]. Based on this qualitative
representation of trajectories, neighborhood-based spatial reasoning can for
example be used as a simple, abstract model of the navigation of a spatial
agent444for an application of neighbourhood based reasoning of spatial agents,
we refer the reader to the simulation model SAILAWAY [25].
In constraint-based reasoning about spatial configurations, typically a
partial initial knowledge of a scene is represented in terms of qualitative
constraints between spatial objects. Implicit knowledge about spatial
relations is then derived by constraint propagation555For an application of
constraint-based reasoning for spatial agents, we refer the reader to the AIBO
robot example in [14]. Previous research has found that the mathematical
notion of a _relation algebra_ and related notions are well-suited for this
kind of reasoning. In many cases, relation algebra-based reasoning only
provides approximate results [26] and the constraint consistency problem for
relative position calculi is NP-hard [27]. Hence we use constraint reasoning
with polynomial time algorithms as an approximation of an intractable problem.
The technical details of constraint reasoning are explained in Section 2.3.
In point-based reasoning, all objects are mapped onto the plane. The centers
of projected objects can be used as point-like representation of the objects.
By contrast, Schlieder’s line segment calculus [24] uses more complex basic
entities. Thus, it is based on extended objects which are represented as
oriented straight line segments (see Fig. 1). These more complex basic
entities capture important features of natural objects:
* •
Natural Objects are extended.
* •
Natural Objects often have an intrinsic direction.
Oriented straight line segments (which were called dipoles by Moratz et al.
[28]) are the simplest geometric objects presenting these features. Dipoles
may be specified by their start and end points.
Figure 1: Orientation between two dipoles
Using dipoles as basic blocks, more complex objects can be constructed (e.g.
polylines, polygons) in a straightforward manner. Therefore, dipoles can be
used as the basic units in numerous applications. To give an example, line
segments are central to edge-based image segmentation and grouping in computer
vision. In addition, GIS systems often have line segments as basic entities
[29]. Polylines are particularly interesting for representing paths in
cognitive robotics [30] and can serve as the geometric basis of a mobile robot
when autonomously mapping its working environment [31].
The next sections of this paper present a detailed and technical description
of dipole calculi. In Section 2 we introduce the relations of the dipole
calculi and revisit the theory of relation algebras and non-associate algebras
underlying qualitative spatial reasoning. Furthermore, we investigate quotient
of calculi on a general level as well as for the dipole calculi. Section 3
provides a condensed semantics for the dipole calculus. A condensed semantics,
as we name it, provides spatial domain knowledge to the calculus in the form
of an abstract symbolic model of a specific fragment of the spatial domain. In
this model, possible configurations of very few of the basic spatial entities
of a calculus are enumerated. In our case, we use orbits in the affine group
$\mathbf{GA}(\mathbb{R}^{2})$. This provides a useful abstraction for
reasoning about qualitatively different configurations in Euclidean space. We
use affine geometry at a rather elementary level and appeal to pictures
instead of complete analytic arguments, whenever it is easy to fill in the
details – however, at key points in the argument, careful analytic treatments
are provided. Further, we calculate the composition tables for the dipole
calculi using the condensed semantics and we investigate properties of the
composition. In Section 4 we answer the question whether the standard
constraint resoning method algebraic closure decides consistency for the
dipole calculi. After the presentation of the technical details of dipole
calculi and some of their properties, a sample application of dipole calculi
using a spatial reasoning toolbox is presented in Section 5. The example uses
the reasoning capabilities of a dipole calculus based on constraint reasoning.
Our paper ends with a summary and conclusion and pointers to future work.
## 2 Representation of Dipole Relations and Relation Algebras
In this section, we first present a set of spatial relations between dipoles,
then variants of this set of spatial relations. The final subsection shows
mathematical structures for constraint reasoning about dipole relations.
### 2.1 Basic Representation of Dipole Relations
The basic entities we use are dipoles, i.e. oriented line segments formed by a
pair of two points, a start point and an end point. Dipoles are denoted by
$A,B,C,\ldots$, start points by ${\bf s}_{A}$ and end points by ${\bf e}_{A}$,
respectively (see Fig. 1). These dipoles are used for representing spatial
objects with an intrinsic orientation. Given a set of dipoles, it is possible
to specify many different relations of different arity, e.g. depending on the
length of dipoles, on the angle between different dipoles, or on the dimension
and nature of the underlying space. When examining different relations, the
goal is to obtain a set of jointly exhaustive and pairwise disjoint atomic or
base relations, such that exactly one relation holds between any two dipoles.
The elements of the powerset of the base relations are called _general_
relations. These are used to express uncertainty about the relative position
of dipoles. If these relations form an algebra which fulfills certain
requirements, it is possible to apply standard constraint-based reasoning
mechanisms that were originally developed for temporal reasoning [32] and that
have also proved valuable for spatial reasoning.
So as to enable efficient reasoning, an attempt should be made to keep the
number of different base relations relatively small. For this reason, we will
restrict ourselves to using two-dimensional continuous space for now, in
particular ${\mathbb{R}}^{2}$, and distinguish the location and orientation of
different dipoles only according to a small set of seven different dipole-
point relations. We distinguish between whether a point lies to the left, to
the right, or at one of five qualitatively different locations on the straight
line that passes through the corresponding dipole 666In his introduction of a
set of qualitative spatial relations between oriented line segments, Schlieder
[24] mainly focused on configurations in which no more than two end or start
points were on the same straight line (e.g. all points were in general
position). However, in many domains, we may wish to represent spatial
arrangements in which more than two start or end points of dipoles are on a
straight line.. The corresponding regions are shown on Fig. 2. A corresponding
set of relations between three points was proposed by Ligozat [33] under the
name flip-flop calculus and later extended to the $\mathcal{LR}$ calculus
[34]777The $\mathcal{LR}$ calculus also features the relations dou and tri for
both reference points or all points being equal, respectively. These cases are
not possible for dipoles, since the start and end points cannot coincide by
definition..
Figure 2: Dipole-point relations (= $\mathcal{LR}$ relations)
Then these dipole-point relations describe cases when the point is: to the
left of the dipole ($\rm l$); to the right of the dipole ($\rm r$); straight
behind the dipole ($\rm b$); at the start point of the dipole ($\rm s$);
inside the dipole ($\rm i$); at the end of the dipole ($\rm e$); or straight
in front of the dipole ($\rm f$). For example, in Fig. 1, ${\bf s}_{B}$ lies
to the left of $A$, expressed as $A\;{\rm l}\;{\bf s}_{B}$. Using these seven
possible relations between a dipole and a point, the relations between two
dipoles may be specified according to the following four relationships:
$A\;{\rm R_{1}}\;{\bf s}_{B}\wedge A\;{\rm R_{2}}\;{\bf e}_{B}\wedge B\;{\rm
R_{3}}\;{\bf s}_{A}\wedge B\;{\rm R_{4}}\;{\bf e}_{A},$
where ${\rm R_{i}}\in\left\\{\rm l,r,b,s,i,e,f\right\\}$ with $1\leq i\leq 4$.
Theoretically, this gives us 2401 relations, out of which 72 relations are
geometrically possible, see Prop. 47 below. They are listed on Fig. 3.
Figure 3: The 72 atomic relations of the $\mathcal{DRA}_{f}$ calculus. In the
dipole calculus, orthogonality is not defined, although the graphical
representation may suggest this.
We introduce an operator that constructs a relation between two dipoles out of
four dipole-point-relations:
###### Definition 1.
The operator $\varrho$ takes the four $\mathcal{LR}$ relations between the
start and end points of two dipoles and constructs a relation between dipoles.
It is defined as the textual concatenation: $\varrho({\rm R_{1}},{\rm
R_{2}},{\rm R_{3}},{\rm R_{4}})={\rm R_{1}R_{2}R_{3}R_{4}}$. By $\tau_{i}$
with $1\leq i\leq 4$, we denote the projections to components of the relations
between dipoles, where the identities $\varrho(\tau_{1}{\rm R},\tau_{2}{\rm
R},\tau_{3}{\rm R},\tau_{4}{\rm R})=R$ and $\tau_{i}\circ\varrho({\rm
R_{1}},{\rm R_{2}},{\rm R_{3}},{\rm R_{4}})=R_{i}$ hold.
The relations that have been introduced above in an informal way can also be
defined in an algebraic way. Every dipole $D$ on the plane ${\mathbb{R}}^{2}$
is an ordered pair of two points ${\bf s}_{D}$ and ${\bf e}_{D}$, each of them
being represented by its Cartesian coordinates $x$ and $y$, with
$x,y\in{\mathbb{R}}$ and ${\bf s}_{D}\not={\bf e}_{D}$.
$D=\left({\bf s}_{D},{\bf e}_{D}\right),\qquad{\bf s}_{D}=\left(({\bf
s}_{D})_{x},({\bf s}_{D})_{y}\right)$
The basic relations are then described by equations with the coordinates as
variables. The set of solutions for a system of equations describes all the
possible coordinates for these points. One first such specification was
presented in Moratz et. al. [28].
### 2.2 Several Versions of Sets of Dipole Base Relations
It is an unrealistic goal to provide a single set of qualitative base
relations which is suitable for all possible contexts. In general, the desired
granularity of a representation framework depends on the specific application
[35]. A coarse granularity only needs a small set of base relations. Finer
granularity can lead to a large number of base relations. If it is desired to
apply a spatial calculus to a problem, it is therefore advantageous when a
choice can be made between several versions of sets of base relations. Then a
calculus may be selected which only has the necessary number of base relations
and thus has less representation complexity but is fine-grained enough to
solve the particular spatial reasoning problem. Focussing on the smallest
number of base relations also fits better with the principle of a vocabulary
of concepts which is compatible with linguistic principles [15, 14]. For this
purpose, several versions of sets of dipole base relations can be constructed
based on the base relation set of $\mathcal{DRA}_{f}$.
In their paper on customizing spatial and temporal calculi, Renz and Schmid
[36] investigated different methods for deriving variants of a given calculus
that have better-suited granularity for certain tasks. In the first variant,
unions of base relations or so-called macro relations were used as base
relations. In the second variant, only a subset of base relations was used as
a new set of base relations. In his pioneering work on dipole relations,
Schlieder [24] introduced a set of base relations in which no more than two
start or end points were on the same straight line. As a result, only a subset
of the $\mathcal{DRA}_{f}$ base relations is used, which corresponds to Renz’
and Schmid’s second variant of methods for deriving new base relation sets for
qualitative calculi. We refer to a calculus based on these base relations as
$\mathcal{DRA}_{\mathit{lr}}$ (where lr stands for left/right). The following
base relations are part of $\mathcal{DRA}_{\mathit{lr}}$: rrrr, rrll, llrr,
llll, rrrl, rrlr, rlrr, rllr, rlll, lrrr, lrrl, lrll, llrl, lllr.
Moratz et al. [28] introduced an extension of $\mathcal{DRA}_{\mathit{lr}}$
which adds relations for representing polygons and polylines. In this
extension, two start or end points can share an identical location. While in
this calculus, three points at different locations cannot belong to the same
straight line. This subset of $\mathcal{DRA}_{f}$ was named
$\mathcal{DRA}_{c}$ ($c$ refers to coarse, $f$ refers to fine). The set of
base relations of $\mathcal{DRA}_{c}$ extends the base relations of
$\mathcal{DRA}_{\mathit{lr}}$ with the following relations: ells, errs, lere,
rele, slsr, srsl, lsel, rser, sese, eses.
Another method for deriving a new set of base relations from an existing set
merges unions of base relations to new base relations. At a symbolic level,
sets of base relations are used to form new base relations. In the context of
$\mathcal{DRA}_{f}$, this is done as shown in Fig. 4 (the meaning of the names
of the new base relations is explained in the following paragraphs).
$\displaystyle{\rm\\{llll,\;lllb,\;lllr,\;lrll,\;lbll\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm LEFTleft}$
$\displaystyle{\rm\\{ffff,\;eses,\;fefe,\;fifi,\;ibib,\;fbii,\;fsei,\;ebis,\;iifb,\;eifs,\;iseb\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm FRONTfront}$
$\displaystyle{\rm\\{bbbb\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
BACKback}$ $\displaystyle{\rm\\{llbr\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm LEFTback}$ $\displaystyle{\rm\\{llfl,\;lril,\;lsel\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm LEFTfront}$
$\displaystyle{\rm\\{llrf,\;llrl,\;llrr,\;lfrr,\;lrrr,\;lere,\;lirl,\;lrri,\;lrrl\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm LEFTright}$
$\displaystyle{\rm\\{rrrr,\;rrrl,\;rrrb,\;rbrr,\;rlrr\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm RIGHTright}$
$\displaystyle{\rm\\{rrll,\;rrlr,\;rrlf,\;rlll,\;rfll,\;rllr,\;rele,\;rlli,\;rilr\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm RIGHTleft}$
$\displaystyle{\rm\\{rrbl\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
RIGHTback}$ $\displaystyle{\rm\\{rrfr,\;rser,\;rlir\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm RIGHTfront}$
$\displaystyle{\rm\\{ffbb,\;efbs,\;ifbi,\;iibf,\;iebe\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm FRONTback}$
$\displaystyle{\rm\\{frrr,\;errs,\;irrl\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm FRONTright}$ $\displaystyle{\rm\\{flll,\;ells,\;illr\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm FRONTleft}$
$\displaystyle{\rm\\{blrr\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
BACKright}$ $\displaystyle{\rm\\{brll\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm BACKleft}$
$\displaystyle{\rm\\{bbff,\;bfii,\;beie,\;bsef,\;biif\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm BACKfront}$
$\displaystyle{\rm\\{slsr\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
SAMEleft}$ $\displaystyle{\rm\\{sese,\;sfsi,\;sisf\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm SAMEfront}$ $\displaystyle{\rm\\{sbsb\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm SAMEback}$
$\displaystyle{\rm\\{srsl\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
SAMEright}$ Figure 4: Mapping from $\mathcal{DRA}_{f}$ to
$\mathcal{DRA}_{\mathit{op}}$ relations
$\mathcal{DRA}_{\mathit{op}}$ is the name of the calculus which has the set of
base relations listed in Fig. 4. In [17], a calculus $\mathcal{OPRA}_{1}$
which is isomorphic888Since we have not introduced operations on QSR calculi
yet, we explain the details of the correspondence between
$\mathcal{DRA}_{\mathit{op}}$ and $\mathcal{OPRA}_{1}$ later in our paper, see
Prop. 21. to $\mathcal{DRA}_{\mathit{op}}$ is defined in a complementary
geometric way. The transition from oriented line segments with well-defined
lengths to line segments with infinitely small lengths is the core idea of
this geometric model. In this conceptualization, the length of objects no
longer has any significance. Thus, only the direction of the objects is
modeled [17]. These objects can then be conceptualized as oriented points. An
o-point, our term for an oriented point, is specified as a pair of a point
with a direction in the 2D-plane. Then the ”op” in the symbol
$\mathcal{DRA}_{\mathit{op}}$ stands for oriented points. A single o-point
induces the sectors depicted in Fig. 5. “Front” and “Back” are linear sectors.
“Left” and “Right” are half-planes. The position of the point itself is
denoted as “Same”. A qualitative spatial relative position relation between
two o-points is represented by the sector in which the second o-point lies in
relation to the first one and by the sector in which the first o-point lies in
relation to the second one. For the general case of two points having
different positions, we use the concatenated string of both sector names as
the relation symbol. Then the configuration shown in Fig. 6 is expressed by
the relation $A\;{\rm RIGHTleft}\;B$. If both points share the same position,
the relation symbol starts with the word “Same” and the second substring
denotes the direction of the second o-point relative to the first one as shown
in Fig. 7.
Figure 5: An oriented point and its qualitative spatial relative directions
Figure 6: Qualitative spatial relation between two oriented points at
different positions. The qualitative spatial relation depicted here is $A$
RIGHTleft $B$. Figure 7: Qualitative spatial relation between two oriented
points located at the same position. The qualitative spatial relation depicted
here is $A$ SAMEright $B$.
Altogether we obtain 20 different atomic relations (four times four general
relations plus four with the oriented points at the same positions). The
relation SAMEfront is the identity relation. $\mathcal{DRA}_{\mathit{op}}$ has
fewer base relations and therefore is more compact than $\mathcal{DRA}_{f}$.
Focussing on a smaller set of base relations in this case also fits better
with the principle of using a vocabulary of concepts which is compatible with
linguistic principles [15, 14]. For this reason, many
$\mathcal{DRA}_{\mathit{op}}$ base relations have simple corresponding
linguistic expressions. For example, the qualitative spatial configuration
represented as $A\;{\rm LEFTfront}\;B$ can be translated into the natural
language expression ”B is left of A and A is in front of B”. A and B in this
example would be oriented objects with an intrinsic front like two cars A and
B in a parking lot. However, in general, the correspondence between QSR
expressions and their linguistic counterparts is only an approximation [15,
14].
The two methods for deriving new sets of base relations which we applied above
reduce the number of base relations. Conversely, other methods extend the
number of base relations. For example, Dylla and Moratz [37] have observed
that $\mathcal{DRA}_{f}$ may not be sufficient for robot navigation tasks,
because the dipole configurations that are pooled in certain base relations
are too diverse. Thus, the representation has been extended with additional
orientation knowledge and a more fine-grained $\mathcal{DRA}_{\mathit{fp}}$
calculus with additional orientation distinctions has been derived. It has
slightly more base relations.
Figure 8: Pairs of dipoles subsumed by the same relation
The large configuration space for the rrrr relation is visualized in Fig. 8.
The other analogous relations which are extremely coarse are llrr, rrll and
llll. In many applications, this unwanted coarseness of four relations can
lead to problems999An investigation by Dylla and Moratz into the first
cognitive robotics applications of dipole relations integrated in situation
calculus [37] showed that the coarseness of $\mathcal{DRA}_{f}$ compared to
$\mathcal{DRA}_{\mathit{fp}}$ would indeed lead to rather meandering paths for
a spatial agent.. Therefore, we introduce an additional qualitative feature by
considering the angle spanned by the two dipoles. This gives us an important
additional distinguishing feature with four distinctive values. These
qualitative distinctions are parallelism (P) or anti-parallelism (A) and
mathematically positive and negative angles between $A$ and $B$, leading to
three refining relations for each of the four above-mentioned relations (Fig.
9).
Figure 9: Refined base relations in $\mathcal{DRA}_{\mathit{fp}}$
We call this algebra $\mathcal{DRA}_{\mathit{fp}}$ as it is an extension of
the fine-grained relation algebra $\mathcal{DRA}_{f}$ with additional
distinguishing features due to “parallelism”. For the other relations, a ’$+$’
or ’$-$’, ’P’ or ’A’ respectively, is already determined by the original base
relation and does not have to be mentioned explicitly. These base relations
then have the same relation symbol as in $\mathcal{DRA}_{f}$.
The introduction of parallelism into dipole calculi not only has benefits in
certain applications. The algebraic features also benefit from this extension
(see Sect. 3.7). For analogous reasons, a derivation of
$\mathcal{DRA}_{\mathit{fp}}$ yields an oriented point calculus which
explicitly contains the feature of parallelism, which is isomorphic to the
$\mathcal{OPRA}_{1}^{*}$ calculus[38]. This calculus
$\mathcal{DRA}_{\mathit{opp}}$ (opp stands for oriented points and
parallelism) has the same base relations as $\mathcal{DRA}_{\mathit{op}}$ with
the exception of the relations ${\rm RIGHTright}$, ${\rm RIGHTleft}$, ${\rm
LEFTleft}$, and ${\rm LEFTright}$. The transformation from
$\mathcal{DRA}_{\mathit{fp}}$ to $\mathcal{DRA}_{\mathit{opp}}$ is shown in
Fig. 10.
$\displaystyle{\rm\\{llllA\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
LEFTleftA}$ $\displaystyle{\rm\\{llll+,\;lllb+,\;lllr+\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm LEFTleft+}$
$\displaystyle{\rm\\{lrll,\;lbll\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm LEFTleft-}$
$\displaystyle{\rm\\{ffff,\;eses,\;fefe,\;fifi,\;ibib,\;fbii,\;fsei,\;ebis,\;iifb,\;eifs,\;iseb\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm FRONTfront}$
$\displaystyle{\rm\\{bbbb\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
BACKback}$ $\displaystyle{\rm\\{llbr\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm LEFTback}$ $\displaystyle{\rm\\{llfl,\;lril,\;lsel\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm LEFTfront}$
$\displaystyle{\rm\\{llrrP\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
LEFTrightP}$ $\displaystyle{\rm\\{llrr+\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm LEFTright+}$
$\displaystyle{\rm\\{llrf,\;llrl,\;llrr-,\;lfrr,\;lrrr,\;lere,\;lirl,\;lrri,\;lrrl\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm LEFTright-}$
$\displaystyle{\rm\\{rrrrA\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
RIGHTrightA}$ $\displaystyle{\rm\\{rrrr+,\;rbrr,\;rlrr\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm RIGHTright+}$
$\displaystyle{\rm\\{rrrr-,\;rrrl,\;rrrb\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm RIGHTright-}$ $\displaystyle{\rm\\{rrllP\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm RIGHTleftP}$
$\displaystyle{\rm\\{rrll+,\;rrlr,\;rrlf,\;rlll,\;rfll,\;rllr,\;rele,\;rlli,\;rilr\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm RIGHTleft+}$
$\displaystyle{\rm\\{rrll-\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
RIGHTleft-}$ $\displaystyle{\rm\\{rrbl\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm RIGHTback}$ $\displaystyle{\rm\\{rrfr,\;rser,\;rlir\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm RIGHTfront}$
$\displaystyle{\rm\\{ffbb,\;efbs,\;ifbi,\;iibf,\;iebe\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm FRONTback}$
$\displaystyle{\rm\\{frrr,\;errs,\;irrl\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm FRONTright}$ $\displaystyle{\rm\\{flll,\;ells,\;illr\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm FRONTleft}$
$\displaystyle{\rm\\{blrr\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
BACKright}$ $\displaystyle{\rm\\{brll\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm BACKleft}$
$\displaystyle{\rm\\{bbff,\;bfii,\;beie,\;bsef,\;biif\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm BACKfront}$
$\displaystyle{\rm\\{slsr\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
SAMEleft}$ $\displaystyle{\rm\\{sese,\;sfsi,\;sisf\\}}$ $\displaystyle\mapsto$
$\displaystyle{\rm SAMEfront}$ $\displaystyle{\rm\\{sbsb\\}}$
$\displaystyle\mapsto$ $\displaystyle{\rm SAMEback}$
$\displaystyle{\rm\\{srsl\\}}$ $\displaystyle\mapsto$ $\displaystyle{\rm
SAMEright}$ Figure 10: Mapping from $\mathcal{DRA}_{\mathit{fp}}$ to
$\mathcal{DRA}_{\mathit{opp}}$ relations
Again, the mathematical properties of the oriented point calculus can be
derived from the corresponding dipole calculus, see Corollary 55.
### 2.3 Relation Algebras for Spatial Reasoning
Standard methods developed for finite domains generally do not apply to
constraint reasoning over infinite domains. The theory of relation algebras
[39, 40] allows for a purely symbolic treatment of constraint satisfaction
problems involving relations over infinite domains. The corresponding
constraint reasoning techniques were originally introduced for temporal
reasoning [32] and later proved to be valuable for spatial reasoning [6, 19].
The central data for a calculus is given by:
* •
a list of (symbolic names for) _base relations_ , which are interpreted as
relations over some domain, having the crucial properties of _pairwise
disjointness_ and _joint exhaustiveness_ (a general relation is then simply a
set of base relations).
* •
a table for the computation of the _converses_ of relations.
* •
a table for the computation of the _compositions_ of relations.
Then, the path consistency algorithm [41] and backtracking techniques [42] are
the tools used to tackle the problem of consistency of constraint networks and
related problems. These algorithms have been implemented in both generic
reasoning tools GQR [43] and SparQ [44]. To integrate a new calculus into
these tools, only a list of base relations and tables for compositions and
converses really need to be provided. Thereby, the qualitative reasoning
facilities of these tools become available for this calculus.101010With more
information about a calculus, both of the tools can provide functionality that
goes beyond simple qualitative reasoning for constraint calculi. Since the
compositions and converses of general relations can be reduced to compositions
and converses of base relations, these tables only need to be given for base
relations. Based on these tables, the tools provide a means to approximate the
consistency of constraint networks, list all their atomic refinements, and
more.
Let $b$ be the name of a base relation, and let $R_{b}$ be its set-theoretic
extension. The converse $(R_{b})^{\smallsmile}=\\{(x,y)|(y,x)\in R_{b}\\}$ is
often itself a base relation and is denoted by $b^{\smallsmile}$111111In
Freksa’s double-cross calculus [2] the converses are not necessarily base-
relations, but for the calculi that we investigate this property holds.. In
the dipole calculus, it is obvious that the converse of a relation can easily
be computed by exchanging the first two and second two letters of the name of
a relation, see Table 1. Also for the dipole calculus
$\mathcal{DRA}_{\mathit{fp}}$ with additional orientation distinctions a
simple rule exchanges ’$+$’ with ’$-$’, and vice versa.’P’ and ’A’ are
invariant with respect to the converse operation. Since base relations
generally are not closed under composition, this operation is approximated by
a _weak composition_ :
$b_{1};b_{2}=\\{b\mid(R_{b_{1}}\circ R_{b_{2}})\cap R_{b}\not=\emptyset\\}$
where $R_{b_{1}}\circ R_{b_{2}}$ is the usual set theoretic composition
$R_{b_{1}}\circ R_{b_{2}}=\\{(x,z)|\exists y\,.\,(x,y)\in R_{b_{1}},(y,z)\in
R_{b_{2}}\\}$
The composition is said to be _strong_ if $R_{b_{1};b_{2}}=R_{b_{1}}\circ
R_{b_{2}}$. Generally, $b_{1};b_{2}$ over-approximates the set-theoretic
composition.121212The $R_{\\_}$ operation naturally extends to sets of (names
of) base relations. Computing the composition table is much harder and will be
the subject of Section 3.
$R$ | rrrr | rrrl | rrlr | rrll | rlrr | rllr | rlll | lrrr
---|---|---|---|---|---|---|---|---
$R^{\smile}$ | rrrr | rlrr | lrrr | llrr | rrrl | lrrl | llrl | rrlr
Table 1: The converse ($\smile$) operation of $\mathcal{DRA}_{f}$ can be
reduced to a simple permutation.
The mathematical background of composition in table-based reasoning is given
by the theory of _relation algebras_ [40, 45]. For many calculi, including the
dipole calculus, a slightly weaker notion is needed, namely that of a _non-
associative algebra_ [46]. These algebras treat spatial relations as abstract
entities that can be combined by certain operations and governed by certain
equations. This allows algorithms and tools to operate at a symbolic level, in
terms of (sets of) base relations instead of their set-theoretic extensions.
###### Definition 2 ([46]).
A _non-associative algebra_ $A$ is a tuple
$A=(A,+,-,\cdot,0,1,;,^{\smallsmile},\Delta)$ such that:
1. 1.
$(A,+,-,\cdot,0,1)$ is a Boolean algebra.
2. 2.
$\Delta$ is a constant, ⌣ a unary and ; a binary operation such that, for any
$a,b,c\in A$:
$\begin{array}[]{lll}(a)\leavevmode\nobreak\
(a^{\smallsmile})^{\smallsmile}=a&(b)\leavevmode\nobreak\
\Delta;a=a;\Delta=a&(c)\leavevmode\nobreak\ a;(b+c)=a;b+a;c\\\
(d)\leavevmode\nobreak\
(a+b)^{\smallsmile}=a^{\smallsmile}+b^{\smallsmile}&(e)\leavevmode\nobreak\
(a-b)^{\smallsmile}=a^{\smallsmile}-b^{\smallsmile}&(f)\leavevmode\nobreak\
(a;b)^{\smallsmile}=b^{\smallsmile};a^{\smallsmile}\\\
\lx@intercol(g)\leavevmode\nobreak\ (a;b)\cdot c^{\smallsmile}=0\mbox{ if and
only if }(b;c)\cdot a^{\smallsmile}=0\hfil\lx@intercol\end{array}$
A non-associative algebra is called a _relation algebra_ , if the composition
; is associative.
The elements of such an algebra will be called (abstract) relations. We are
mainly interested in finite non-associative algebras that are _atomic_ , which
means that there is a set of pairwise disjoint minimal relations, called base
relations, and all relations can be obtained as unions of base relations.
Then, the following fact is well-known and easy to prove:
###### Proposition 3.
An atomic non-associative algebra is uniquely determined by its set of base
relations, together with the converses and compositions of base relations.
(Note that the composition of two base relations is in general not a base
relation.)
###### Example 4.
The powerset of the 72 $\mathcal{DRA}_{f}$ base relations forms a boolean
algebra. The relation sese is the identity relation. The converse and (weak)
composition are defined as above. We denote the resulting non-associative
algebra by $\mathcal{DRA}_{f}$. The algebraic laws follow from general results
about so-called partition schemes, see [46]. Similarly, we obtain a non-
associative algebra $\mathcal{DRA}_{\mathit{fp}}$.
However, we do not obtain a non-associative algebra for $\mathcal{DRA}_{c}$,
because $\mathcal{DRA}_{c}$ does not provide a jointly exhaustive set of base
relations over the Euclidean plane. This leads to the lack of an identity
relation, and more severely, weak composition does not lead to an over-
approximation (nor an under-approximation) of set-theoretic composition,
because e.g. ffbb is missing from the composition of llll with itself. In
particular, we cannot expect the algebraic laws of a non-associative algebra
to be satisfied.
For non-associative algebras, we define lax homomorphisms which allow for both
the embedding of a calculus into another one, and the embedding of a calculus
into its domain.
###### Definition 5 (Lax homomorphism).
Given non-associative algebras $A$ and $B$, a _lax homomorphism_ is a
homomorphism $\mathop{\mathrm{h}}:A\longrightarrow B$ on the underlying
Boolean algebras such that:
* •
$\mathop{\mathrm{h}}(\Delta_{A})\geq\Delta_{B}$
* •
$\mathop{\mathrm{h}}(a^{\smile})=\mathop{\mathrm{h}}(a)^{\smile}$ for all
$a\in A$
* •
$\mathop{\mathrm{h}}(a;b)\geq\mathop{\mathrm{h}}(a);\mathop{\mathrm{h}}(b)$
for all $a,b\in A$
Dually to lax homomorphisms, we can define oplax homomorphisms131313The
terminology is motivated by that for monoidal functors., which enable us to
define projections from one calculus to another.
###### Definition 6 (Oplax homomorphism).
Given non-associative algebras $A$ and $B$, an _oplax homomorphism_ is a
homomorphism $\mathop{\mathrm{h}}:A\longrightarrow B$ on the underlying
Boolean algebras such that:
* •
$\mathop{\mathrm{h}}(\Delta_{A})\leq\Delta_{B}$
* •
$\mathop{\mathrm{h}}(a^{\smile})=\mathop{\mathrm{h}}(a)^{\smile}$ for all
$a\in A$
* •
$\mathop{\mathrm{h}}(a;b)\leq\mathop{\mathrm{h}}(a);\mathop{\mathrm{h}}(b)$
for all $a,b\in A$
A proper homomorphism (sometimes just called a homomorphism) of non-
associative algebras is a homomorphism that is lax and oplax at the same time;
the above inequalities then turn into equations.
An important application of homomorphisms is their use in the definition of
qualitative calculus. Ligozat and Renz [46] define a qualitative calculus in
terms of a so-called _weak representation_ [47]:
###### Definition 7 (Weak representation).
A weak representation is an identity-preserving (i.e.
$\mathop{\mathrm{h}}(\Delta_{A})=\Delta_{B}$) lax homomorphism $\varphi$ from
a (finite atomic) non-associative algebra into the relation algebra of a
domain ${\cal U}$. The latter is given by the canonical relation algebra on
the powerset ${\cal P}({\cal U}\times{\cal U})$, where identity, converse and
composition (as well as the Boolean algebra operations) are given by their
set-theoretic interpretations.
###### Example 8.
Let $\mathbb{D}$ be the set of all dipoles in $\mathbb{R}^{2}$. Then the weak
representation of $\mathcal{DRA}_{f}$ is the lax homomorphism
$\varphi_{\mathit{f}}:\mathcal{DRA}_{f}\to{\cal
P}(\mathbb{D}\times\mathbb{D})$ given by
$\varphi_{\mathit{f}}(R)=\\{R_{b}\,|\,b\in R\\}.$
We obtain a similar weak representation $\varphi_{\mathit{fp}}$ for
$\mathcal{DRA}_{\mathit{fp}}$. The following is straightforward:
###### Proposition 9.
A calculus has a strong composition if and only if its weak representation is
a proper homomorphism.
Proof. Since a weak representation is identity-preserving, being proper means
that $\varphi(R_{1};R_{2})=\varphi(R_{1})\circ\varphi(R_{2})$, which is
nothing but $R_{R_{1};R_{2}}=R_{R_{1}}\circ R_{R_{2}}$, which is exactly the
strength of the composition. ∎
The following is straightforward [47]:
###### Proposition 10.
A weak representation $\varphi$ is injective if and only if
$\varphi(b)\not=\emptyset$ for each base relation $b$.
The second main use of homomorphisms is relating different calculi. For
example, the algebra over Allen’s interval relations [32] can be embedded into
$\mathcal{DRA}_{f}$ ($\mathcal{DRA}_{\mathit{fp}}$) via a homomorphism.
###### Proposition 11.
A homomorphism from Allen’s interval algebra to $\mathcal{DRA}_{f}$
($\mathcal{DRA}_{\mathit{fp}}$) exists and is given by the following mapping
of base relations.
$\begin{array}[]{rclcrcl}\qquad\qquad\qquad=&\mapsto&\textnormal{\rm
sese}&&&&\\\ \textnormal{\rm b}&\mapsto&\textnormal{\rm ffbb}&&\textnormal{\rm
bi}&\mapsto&\textnormal{\rm bbff}\\\ \textnormal{\rm
m}&\mapsto&\textnormal{\rm efbs}&&\textnormal{\rm mi}&\mapsto&\textnormal{\rm
bsef}\\\ \textnormal{\rm o}&\mapsto&\textnormal{\rm ifbi}&&\textnormal{\rm
oi}&\mapsto&\textnormal{\rm biif}\\\ \textnormal{\rm
d}&\mapsto&\textnormal{\rm bfii}&&\textnormal{\rm di}&\mapsto&\textnormal{\rm
iibf}\\\ \textnormal{\rm s}&\mapsto&\textnormal{\rm sfsi}&&\textnormal{\rm
si}&\mapsto&\textnormal{\rm sisf}\\\ \textnormal{\rm
f}&\mapsto&\textnormal{\rm beie}&&\textnormal{\rm fi}&\mapsto&\textnormal{\rm
iebe}\end{array}$
Proof. The identity relation $=$ is clearly mapped to the identity relation
sese. For the composition and converse properties, we just inspect the
composition and converse tables for the two calculi.141414This is a (non-
circular) forward reference to Section 3, where we compute the
$\mathcal{DRA}_{f}$ and $\mathcal{DRA}_{\mathit{fp}}$ composition tables. The
mapping of the base-relation is then lifted directly to a mapping of all
relations, where the map is applied component-wise on the relations. Using the
laws of non-associative algebras, the homomorphism property of these relations
follows from that of the base-relations. ∎
In cases stemming from the embedding of Allen’s Interval Algebra, the dipoles
lie on the same straight lines and have the same direction.
$\mathcal{DRA}_{f}$ and $\mathcal{DRA}_{\mathit{fp}}$ also contain 13
additional relations which correspond to the case with dipoles lying on a line
but facing opposite directions.
As we shall see, it is very useful to extend the notion of homomorphisms to
weak representations:
###### Definition 12.
Given weak representations $\varphi:A\to\mathcal{P}({\cal U}\times{\cal U})$
and $\psi:B\to\mathcal{P}({\cal V}\times{\cal V})$, a _lax (oplax, proper)
homomorphism of weak representations_ $(h,i):\varphi\to\psi$ is given by
* •
a proper homomorphism of non-associative algebras $h:A\to B$, and
* •
a map $i:{\cal U}\to{\cal V}$, such that the diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{h}$$\textstyle{\mathcal{P}({\cal
U}\times{\cal
U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{P}(i\times
i)}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathcal{P}({\cal
V}\times{\cal V})}$
commutes laxly (respectively oplaxly, properly). Here, lax commutation means
that for all $R\in A$, $\psi(h(R))\subseteq\mathcal{P}(i\times
i)(\varphi(R))$, oplax commutation means the same with $\supseteq$, and proper
commutation with $=$. Note that $\mathcal{P}(i\times i)$ is the obvious
extension of $i$ to a function between relation algebras; note that (unless
$i$ is bijective) this is not even a homomorphism of Boolean algebras (it may
fail to preserve top, intersections and complements), although it satisfies
the oplaxness property (and the laxness property if $i$ is
surjective).151515The reader with background in category theory may notice
that the categorically more natural formulation would use the contravariant
powerset functor, which yields homomorphisms of Boolean algebras. However, the
present formulation fits better with the examples.
Note that Ligozat [47] defines a more special notion of morphism between weak
representations; it corresponds to our oplax homomorphism of weak
representations where the component $h$ is the identity.
###### Example 13.
The homomorphism from Prop. 11 can be extended to a proper homomorphism of
weak representations by letting $i$ be the embedding of time intervals to
dipoles on the $x$-axis.
###### Example 14.
Let $h$ map each $\mathcal{DRA}_{\mathit{fp}}$ relation to the corresponding
$\mathcal{DRA}_{f}$ relation:
llll+ $\displaystyle\mapsto$ llll llll- $\displaystyle\mapsto$ llll llllA
$\displaystyle\mapsto$ llll rrrr+ $\displaystyle\mapsto$ rrrr rrrr-
$\displaystyle\mapsto$ rrrr rrrrA $\displaystyle\mapsto$ rrrr llrr+
$\displaystyle\mapsto$ llrr llrr- $\displaystyle\mapsto$ llrr llrrP
$\displaystyle\mapsto$ llrr rrll+ $\displaystyle\mapsto$ rrll rrll-
$\displaystyle\mapsto$ rrll rrllP $\displaystyle\mapsto$ rrll
Then $(h,id):\mathcal{DRA}_{\mathit{fp}}\to\mathcal{DRA}_{f}$ is a surjective
oplax homomorphism of weak representations.
Although this homomorphism of weak representations is surjective, it is not a
quotient in the following sense (and in particular, it does _not_ satisfy
Prop. 20, as will be shown in Sections 3.8 and 3.9).
###### Definition 15.
A homomorphism of non-associative algebras is said to be a _quotient
homomorphism_ 161616Maddux [40] does not have much to say on this subject;
instead, we suggest consulting a textbook on universal algebra, e.g. [48]. if
it is proper and surjective. A (lax, oplax or proper) homomorphism of weak
representations is a quotient homomorphism if it is surjective in both
components.
The easiest way to form a quotient of a weak representation is via an
equivalence relation on the domain:
###### Definition 16.
Given a weak representation $\varphi:A\to\mathcal{P}({\cal U}\times{\cal U})$
and an equivalence relation $\sim$ on ${\cal U}$, we obtain the _quotient
representation_ $\varphi\\!/\\!\\!\sim$ as follows:
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{q_{A}}$$\textstyle{\mathcal{P}({\cal
U}\times{\cal
U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{P}(q\times
q)}$$\textstyle{A\\!/\\!\\!\sim_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi\\!/\\!\\!\sim}$$\textstyle{\mathcal{P}({\cal
U}\\!/\\!\\!\sim\times{\cal U}\\!/\\!\\!\sim)}$
* •
Let $q:{\cal U}\to{\cal U}\\!/\\!\\!\sim$ be the factorization of ${\cal U}$
by $\sim$;
* •
$q$ extends to relations: $\mathcal{P}(q\times q):\mathcal{P}({\cal
U}\times{\cal U})\to\mathcal{P}({\cal U}\\!/\\!\\!\sim\times{\cal
U}\\!/\\!\\!\sim)$;
* •
let $\sim_{A}$ be the congruence relation on $A$ generated by
$\mathcal{P}(q\times q)(\varphi(b_{1}))\cap\mathcal{P}(q\times
q)(\varphi(b_{2}))\not=\emptyset\ \Rightarrow\ b_{1}\sim_{A}b_{2}$
for base relations $b_{1},b_{2}\in A$. $\sim$ is called _regular w.r.t.
$\varphi$_ if $\sim_{A}$ is the kernel of $\mathcal{P}(q\times q)\circ\varphi$
(i.e. the set of all pairs made equal by $\mathcal{P}(q\times
q)\circ\varphi$);
* •
let $q_{A}:A\to A\\!/\\!\\!\sim_{A}$ be the quotient of $A$ by $\sim_{A}$ in
the sense of universal algebra [48], which uses proper homomorphisms; hence,
$q_{A}$ is a proper homomorphism;
* •
finally, the function $\varphi\\!/\\!\\!\sim$ is defined as
$\varphi\\!/\\!\\!\sim(R)=\mathcal{P}(q\times q)(\varphi(q_{A}^{-1}(R))).$
###### Proposition 17.
The function $\varphi\\!/\\!\\!\sim$ defined in Def. 16 is an oplax
homomorphism of non-associative algebras.
Proof. To show this, notice that an equivalent definition works on the base
relations of $A\\!/\\!\\!\sim_{A}$:
$\varphi\\!/\\!\\!\sim(R)=\bigcup_{b\in R}\mathcal{P}(q\times
q)(\varphi(q_{A}^{-1}(b))).$
It is straightforward to show that bottom and joins are preserved; since $q$
is surjective, also top is preserved.
Concerning meets, since general relations in $A\\!/\\!\\!\sim_{A}$ can be
considered to be sets of base relations, it suffices to show that $b_{1}\wedge
b_{2}=0$ implies $\mathcal{P}(q\times
q)(\varphi(q_{A}^{-1}(b_{1})))\cap\mathcal{P}(q\times
q)(\varphi(q_{A}^{-1}(b_{2})))=\emptyset$. Assume to the contrary that
$\mathcal{P}(q\times q)(\varphi(q_{A}^{-1}(b_{1})))\cap\mathcal{P}(q\times
q)(\varphi(q_{A}^{-1}(b_{2})))\not=\emptyset$. Then already
$\mathcal{P}(q\times q)(\varphi(b^{\prime}_{1}))\cap\mathcal{P}(q\times
q)(\varphi(b^{\prime}_{2}))\not=\emptyset$ for base relations
$b^{\prime}_{i}\in q_{A}^{-1}(b_{i})$, $i=1,2$. But then
$b^{\prime}_{1}\sim_{A}b^{\prime}_{2}$, hence
$q_{A}(b^{\prime}_{1})=q_{A}(b^{\prime}_{2})\leq b_{1}\wedge b_{2}$,
contradicting $b_{1}\wedge b_{2}=0$.
Preservation of complement follows from this.
Using properness of the quotient, it is then easily shown that the relation
algebra part of the lax homomorphism property carries over from $\varphi$ to
$\varphi\\!/\\!\\!\sim$: Concerning composition, by surjectivity of $q_{A}$,
we know that any given relations $R_{1},R_{2}\in A\\!/\\!\\!\sim_{A}$ are of
the form $R_{1}=q_{A}(S_{1})$ and $R_{2}=q_{A}(S_{2})$. Hence,
$\varphi\\!/\\!\\!\sim(R_{1};R_{2})=\varphi\\!/\\!\\!\sim(q_{A}(S_{1});q_{A}(S_{2}))=\varphi\\!/\\!\\!\sim(q_{A}(S_{1};S_{2}))=\mathcal{P}(q\times
q)(\varphi(S_{1};S_{2}))\geq\mathcal{P}(q\times
q)(\varphi(S_{1});\varphi(S_{2}))=\mathcal{P}(q\times
q)(\varphi(S_{1}));\mathcal{P}(q\times
q)(\varphi(S_{2}))=\varphi\\!/\\!\\!\sim(q_{A}(S_{1}));\varphi\\!/\\!\\!\sim(q_{A}(S_{2}))=\varphi\\!/\\!\\!\sim(R_{1});\varphi\\!/\\!\\!\sim(R_{2})$.
The inequality of the identity is shown similarly. ∎
###### Proposition 18.
$(q_{A},q):\varphi\to\varphi\\!/\\!\\!\sim$ is an oplax quotient homomorphism
of weak representations. If $\sim$ is regular w.r.t. $\varphi$, then the
quotient homomorphism is proper, and satisfies the following universal
property: if $(q_{B},i):\varphi\to\psi$ is another oplax homomorphism of weak
representations with $\psi$ injective and $\sim\subseteq\mathit{ker}(i)$, then
there is a unique oplax homomorphism of weak representations
$(h,k):\varphi\\!/\\!\\!\sim\to\psi$ with $(q_{B},i)=(h,k)\circ(q_{A},q)$.
Proof. The oplax homomorphism property for $(q_{A},q)$ is
$\mathcal{P}(q\times q)\circ\varphi\subseteq\varphi\\!/\\!\\!\sim\circ q_{A}$,
which by definition of $\varphi\\!/\\!\\!\sim$ amounts to
$\mathcal{P}(q\times q)\circ\varphi\subseteq\mathcal{P}(q\times
q)\circ\varphi\circ q_{A}^{-1}\circ q_{A},$
which follows from surjectivity of $q$. Regularity of $\sim$ is w.r.t.
$\varphi$ means that $\sim_{A}$ is the kernel of $\mathcal{P}(q\times
q)\circ\varphi$, which turns the above inequation into an equality. Concerning
the universal property, let $(q_{B},i):\varphi\to\psi$ with the mentioned
properties be given. Since $\sim\subseteq\mathit{ker}(i)$, there is a unique
function $k:{\cal U}\\!/\\!\\!\sim\to{\cal V}$ with $i=k\circ q$. The
homomorphism $h$ we are looking for is determined uniquely by
$h(q_{A}(b))=q_{B}(b)$; this also ensures the proper homomorphism property.
All that remains to be shown is well-definedness. Suppose that
$b_{1}\sim_{A}b_{2}$. By regularity, $\mathcal{P}(q\times
q)(\varphi(b_{1}))=\mathcal{P}(q\times q)(\varphi(b_{2}))$. Hence also
$\mathcal{P}(i\times i)(\varphi(b_{1}))=\mathcal{P}(i\times
i)(\varphi(b_{2}))$ and $\psi(q_{B}(b_{1}))=\psi(q_{B}(b_{2}))$. By
injectivity of $\psi$, we get $q_{B}(b_{1})=q_{B}(b_{2})$. ∎
###### Example 19.
Given dipoles $d_{1},d_{2}\in\mathbb{D}$, let $d_{1}\sim d_{2}$ denote that
$d_{1}$ and $d_{2}$ have the same start point and point in the same direction.
(This is regular w.r.t. $\varphi_{\mathit{f}}$.) Then
$\mathbb{D}\\!/\\!\\!\sim$ is the domain $\mathbb{OP}$ of oriented points in
$\mathbb{R}^{2}$. Let
$\varphi_{\mathit{op}}:\mathcal{DRA}_{\mathit{op}}\to{\cal
P}(\mathbb{OP}\times\mathbb{OP})$ and
$\varphi_{\mathit{opp}}:\mathcal{DRA}_{\mathit{opp}}\to{\cal
P}(\mathbb{OP}\times\mathbb{OP})$ be the weak representations obtained as
quotients of $\varphi_{\mathit{f}}$ and $\varphi_{\mathit{fp}}$, respectively,
see Fig. 11. At the level of non-associative algebras, the quotient is given
by the tables in Figs. 4 and 10.
This way of constructing $\mathcal{DRA}_{\mathit{op}}$ and
$\mathcal{DRA}_{\mathit{opp}}$ by a quotient gives us their converse and
composition tables for no extra effort; we can obtain them by applying the
respective congruences to the tables for $\mathcal{DRA}_{f}$ and
$\mathcal{DRA}_{\mathit{fp}}$, respectively. Moreover, the next result shows
that we also can use the quotient to transfer an important property of
calculi.
$\textstyle{\mathcal{DRA}_{\mathit{fp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\mathit{fp}}}$$\textstyle{\mathcal{P}(\mathbb{D}\times\mathbb{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{DRA}_{\mathit{opp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\mathit{opp}}}$$\textstyle{\mathcal{P}(\mathbb{OP}\times\mathbb{OP})}$
Figure 11: Homomorphisms of weak representations from
$\mathcal{DRA}_{\mathit{fp}}$ to $\mathcal{DRA}_{\mathit{opp}}$
###### Proposition 20.
Quotient homomorphism of weak representations preserve strength of
composition.
Proof. Let $(h,i):\varphi\to\psi$ with $\varphi:A\to\mathcal{P}({\cal
U}\times{\cal U})$ and $\psi:B\to\mathcal{P}({\cal V}\times{\cal V})$ be a
quotient homomorphism of weak representations. According to Prop. 9, the
strength of the composition is equivalent to $\varphi$ (respectively $\psi$)
being a proper homomorphism. We assume that $\varphi$ is a proper homomorphism
and need to show that $\psi$ is proper as well. We also know that $h$ and
$\mathcal{P}(i\times i)$ are proper. Let $R_{2},S_{2}$ be two abstract
relations in $B$. Because of the surjectivity of $h$, there are abstract
relations $R_{1},S_{1}\in A$ with $h(R_{1})=R_{2}$ and $h(S_{1})=S_{2}$. Now
$\psi(R_{2};S_{2})=\psi(h(R_{1});h(S_{1}))=\psi(h(R_{1};S_{1}))=\mathcal{P}(i\times
i)(\varphi(R_{1};S_{1}))=\mathcal{P}(i\times
i)(\varphi(R_{1}));\mathcal{P}(i\times
i)(\varphi(S_{1}))=\psi(h(R_{1}));\psi(h(S_{1}))=\psi(R_{2});\psi(S_{2})$,
hence $\psi$ is proper. ∎
The application of this Proposition must wait until Section 3, where we
develop the necessary machinery to investigate the strength of the calculi.
The domains of $\mathcal{DRA}_{\mathit{op}}$ and $\mathcal{OPRA}_{1}$
obviously coincide. An inspection of the converse and composition tables (that
of $\mathcal{OPRA}_{1}$ is given in [49]) shows:
###### Proposition 21.
$\mathcal{DRA}_{\mathit{op}}$ is isomorphic to $\mathcal{OPRA}_{1}$.
We can also obtain a similar statement for $\mathcal{DRA}_{\mathit{opp}}$. The
calculus $\mathcal{OPRA}^{*}_{1}$ [38] is a refinement of $\mathcal{OPRA}_{1}$
that is obtained along the same features as $\mathcal{DRA}_{\mathit{fp}}$ is
obtained from $\mathcal{DRA}_{f}$. The method how to compute the composition
table for $\mathcal{OPRA}^{*}_{1}$ is described in [38] and a reference
composition table is provided with the tool SparQ [50].
###### Proposition 22.
$\mathcal{DRA}_{\mathit{opp}}$ is isomorphic to $\mathcal{OPRA}^{*}_{1}$.
In the course of checking the isomorphism properties between
$\mathcal{DRA}_{\mathit{opp}}$ and $\mathcal{OPRA}^{*}_{1}$, we discovered
errors in $197$ entries of the composition table of $\mathcal{OPRA}^{*}_{1}$
as it was shipped with the qualitative reasoner SparQ [50]. This emphasizes
our point how important it is to develop a sound mathematical theory to
compute a composition table and to stay as close as possible with the
implementation to the theory. In the composition table for
$\mathcal{OPRA}^{*}_{1}$ it was claimed that
$\displaystyle\textnormal{\rm SAMEright};\textnormal{\rm RIGHTrightA}$
$\displaystyle\Longrightarrow$ $\displaystyle\\{\textnormal{\rm
LEFTright+},\textnormal{\rm LEFTrightP},\textnormal{\rm LEFTright-},$
$\displaystyle\quad\textnormal{\rm BACKright},\textnormal{\rm RIGHTright+},$
$\displaystyle\quad\textnormal{\rm RIGHTrightA},\textnormal{\rm
RIGHTright-}\\}$
were we use the $\mathcal{DRA}_{\mathit{opp}}$ notation for the
$\mathcal{OPRA}^{*}_{1}$-relations for convenience. So the abstract
composition $\textnormal{\rm SAMEright};\textnormal{\rm RIGHTrightA}$ contains
the base relation LEFTrightP, which however is not supported geometrically.
Consider three oriented points $o_{A}$, $o_{B}$ and $o_{C}$ with
$o_{A}\;\textnormal{\rm SAMEright}\;o_{B}$
Figure 12: $\mathcal{OPRA}^{*}_{1}$ configuration
and $o_{B}\;\textnormal{\rm RIGHTrightA}\;o_{C}$, as depicted in Fig. 12. For
the relation $o_{A}\;\textnormal{\rm LEFTrightP}\;o_{C}$ to hold, the carrier
rays of $o_{A}$ and $o_{C}$ need to be parallel, but because of
$o_{B}\;\textnormal{\rm RIGHTrightA}\;o_{C}$, the carrier rays of $o_{B}$ and
$o_{C}$ and hence also those of $o_{A}$ and $o_{B}$ need to be parallel as
well. Since the start point of $o_{A}$ and $o_{B}$ coincide, this can only be
achieved, if $o_{A}$ and $o_{B}$ are collinear, which is a contradiction to
$o_{A}\;\textnormal{\rm SAMEright}\;o_{B}$.
Altogether, we get the following diagram of calculi (weak representations) and
homomorphisms among them:
| | |
---|---|---|---
$\textstyle{\mathcal{IA}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$proper$\textstyle{\mathcal{DRA}_{\mathit{fp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$oplaxoplax
quotient$\textstyle{\mathcal{DRA}_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$oplax
quotient$\textstyle{\mathcal{IA}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$proper$\textstyle{\mathcal{OPRA}_{1}^{*}}$$\textstyle{\cong}$$\textstyle{\mathcal{DRA}_{\mathit{opp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$oplax$\textstyle{\mathcal{DRA}_{\mathit{op}}}$$\textstyle{\cong}$$\textstyle{\mathcal{OPRA}_{1}}$
### 2.4 Constraint Reasoning
Let us now apply the relation-algebraic method to constraint reasoning. Dipole
constraints are written as $xRy$, where $x,y$ are variables for the dipoles
and $R$ is a $\mathcal{DRA}_{f}$ or $\mathcal{DRA}_{\mathit{fp}}$ relation.
Given a set $\Theta$ of dipole constraints, an important reasoning problem is
to decide whether $\Theta$ is consistent, i.e., whether there is an assignment
of all variables of $\Theta$ with dipoles such that all constraints are
satisfied (a solution). We call this problem DSAT. DSAT is a Constraint
Satisfaction Problem (CSP) [51]. We rely on relation algebraic methods to
check consistency, namely the above mentioned path consistency algorithm. For
non-associative algebras, the abstract composition of relations need not
coincide with the (associative) set-theoretic composition. Hence, in this
case, the standard path-consistency algorithm does not necessarily lead to
path consistent networks, but only to algebraic closure [26]:
###### Definition 23 (Algebraic Closure).
A CSP over binary relations is called _algebraically closed_ if for all
variables $X_{1},X_{2},X_{3}$ and all relations $R_{1},R_{2},R_{3}$ the
constraint relations
$R_{1}(X_{1},X_{2}),\quad R_{2}(X_{2},X_{3}),\quad R_{3}(X_{1},X_{3})$
imply
$R_{3}\leq R_{1};R_{2}.$
In general, algebraic closure is therefore only a one-sided approximation of
consistency: if algebraic closure detects an inconsistency, then we are sure
that the constraint network is inconsistent; however, algebraic closure may
fail to detect some inconsistencies: an algebraically closed network is not
necessarily consistent. For some calculi, like Allen’s interval algebra,
algebraic closure is known to exactly decide consistency, for others it does
not, see [26], where it is also shown that this question is completely
orthogonal to the question as to whether the composition is strong. We will
examine these questions for the dipole calculi in Section 3 below.
Fortunately, it turns out that oplax homomorphisms preserve algebraic closure.
###### Proposition 24.
Given non-associative algebras $A$ and $B$, an oplax homomorphism
$\mathop{\mathrm{h}}:A\longrightarrow B$ preserves algebraic closure. If
$\mathop{\mathrm{h}}$ is injective, it also reflects algebraic closure.
Proof. Since an oplax homomorphism is a homomorphism between Boolean
algebras, it preserves the order. So for any three relations
$R_{1},R_{2},R_{3}$ in the algebraically closed CSP over $A$, with
$R_{3}\leq R_{1};R_{2}$
the preservation of the order implies:
$\mathop{\mathrm{h}}(R_{3})\leq\mathop{\mathrm{h}}(R_{1};R_{2}).$
Applying the oplaxness property yields:
$\mathop{\mathrm{h}}(R_{3})\leq\mathop{\mathrm{h}}(R_{1});\mathop{\mathrm{h}}(R_{2}).$
and hence the image of the CSP under $\mathop{\mathrm{h}}$ is also
algebraically closed. If $\mathop{\mathrm{h}}$ is injective, it reflects
equations and inequations, and the converse implication follows. ∎
###### Definition 25.
Following [26], a _constraint network_ over a non-associative algebra $A$ can
be seen as a function $\nu:A\to{\cal P}(N\times N)$, where $N$ is the set of
nodes (or variables), and $\nu$ maps each abstract relation $R$ to the set of
pairs $(n_{1},n_{2})$ that are decorated with $R$. (Note that $\nu$ is a weak
representation only if the constraint network is algebraically closed.)
Constraint networks can be translated along homomorphisms of non-associative
algebras as follows: Given $h:A\to B$ and $\nu:A\to{\cal P}(N\times N)$,
$h(\nu):B\to{\cal P}(N\times N)$ is the network that decorates $(n_{1},n_{2})$
with $h(R)$ whenever $\nu$ decorates it with $R$
A _solution_ for $\nu$ in a weak representation $\varphi:A\to\mathcal{P}({\cal
U}\times{\cal U})$ is a function $j:N\to{\cal U}$ such that for all $R\in A$,
${\cal P}(j\times j)(\nu(R))\subseteq\varphi(R)$, or ${\cal P}(j\times
j)\circ\nu\subseteq\varphi$ for short.
###### Proposition 26.
Oplax homomorphisms of weak representations preserve solutions for constraint
networks.
Proof. Let weak representations $\varphi:A\to\mathcal{P}({\cal U}\times{\cal
U})$ and $\psi:B\to\mathcal{P}({\cal V}\times{\cal V})$ and an oplax
homomorphism of weak representations $(h,i):\varphi\to\psi$ be given.
A given solution $j:N\to{\cal U}$ for $\nu$ in $\varphi$ is defined by ${\cal
P}(j\times j)\circ\nu\subseteq\varphi$. From this and the oplax commutation
property ${\cal P}(i\times i)\circ\varphi\subseteq\psi\circ h$ we infer ${\cal
P}(i\circ j\times i\circ j)\circ\nu\subseteq\psi\circ h$, which implies that
$i\circ j$ is a solution for $h(\nu)$. ∎
An important question for a calculus (= weak representation) is whether
algebraic closure decides consistency. We will now prove that this property is
preserved under certain homomorphisms.
###### Proposition 27.
Oplax homomorphisms $(h,i)$ of weak representations with $h$ injective
preserve the property that algebraic closure decides consistency to the image
of $h$.
Proof. Let weak representations $\varphi:A\to\mathcal{P}({\cal U}\times{\cal
U})$ and $\psi:B\to\mathcal{P}({\cal V}\times{\cal V})$ and an oplax
homomorphism of weak representations $(h,i):\varphi\to\psi$ be given. Further
assume that for $\varphi$, algebraic closure decides consistency.
Any constraint network in the image of $h$ can be written as $h(\nu):B\to{\cal
P}(N\times N)$. If $h(\nu)$ is algebraically closed, by Prop. 24, this carries
over to $\nu$. Hence, by the assumption, $\nu$ is consistent, i.e. has a
solution. By Prop. 26, $h(\nu)$ is consistent as well. Note that the converse
directly always holds: any consistent network is algebraically closed. ∎
For calculi such as RCC8, interval algebra etc., (maximal) _tractable subsets_
have been determined, i.e. sets of relations for which algebraic closure
decides consistency. We can apply Prop. 27 to the homomorphism from interval
algebra to $\mathcal{DRA}_{f}$ (see Example 13). We obtain that algebraic
closure in $\mathcal{DRA}_{f}$ decides consistency of any constraint network
involving (the image of) a maximal tractable subset of the interval algebra
only.
On the other hand, the consistency problem for the $\mathcal{DRA}_{c}$
calculus in the base relations is already NP-hard, see [27], and hence
algebraic closure does not decide consistency in this case. We will resume the
discussion of consistency versus algebraic closure in Sect. 4.
## 3 A Condensed Semantics for the Dipole Calculus
The $72$ base relations of $\mathcal{DRA}_{f}$, or the 80 base relations of
$\mathcal{DRA}_{\mathit{fp}}$, have so far been derived manually. This is a
potentially erroneous procedure171717For this reason, the manually derived
sets of base relations for the finer-grained dipole calculi described in [24,
28] contained errors., especially if the calculus has many base-relations like
the $\mathcal{DRA}_{f}$ and $\mathcal{DRA}_{\mathit{fp}}$ calculi. Therefore,
it is necessary to use methods which yield more reliable results. To start, we
tried verifying the composition table of $\mathcal{DRA}_{f}$ directly, using
the resulting quadratic inequalities as given in [28]. However, it turned out
that it is unfeasible to base the reasoning on these inequalities, even with
the aid of interactive theorem provers such as Isabelle/HOL [52] and HOL-light
[53] (the latter is dedicated to proving facts about real numbers). This
unfeasibility is probably related to the above-mentioned NP-hardness of the
consistency problem for $\mathcal{DRA}_{f}$ base relations. So, we developed a
qualitative abstraction instead. A key insight is that two configurations are
qualitatively different if they cannot be transformed into each other by maps
that keep that part of the spatial structure invariant that is essential for
the calculus. In our case, these maps are (orientation-preserving) affine
bijections. A set of configurations that can be transformed into each other by
appropriate maps is an _orbit_ of a suitable automorphism group. Here, we use
primarily the affine group $\mathbf{GA}(\mathbb{R}^{2})$ and detail how this
leads to qualitatively different spatial configurations.
### 3.1 Seven qualitatively different configurations
Since the domains of most spatial calculi are infinite (e.g. the Euclidean
plane), it is impossible just to enumerate all possible configurations
relative to the composition operation when deriving a composition table. It is
still possible to enumerate a well-chosen subset of all configurations to
obtain a composition table, but it is difficult to show that this subset leads
to a complete table. We have experimented with the enumeration of all
$\mathcal{DRA}_{f}$ scenarios with six points (which are the start- and end-
points of three dipoles), which are equivalent to the entries of the
composition table, in a _finite_ grid over natural numbers. This method led to
a usable composition table, but its computation took several weeks and it is
unclear if it is complete. The goal remains the efficient and automatic
computation of a composition table. To obtain an efficient method for
computing the table, we introduce the _condensed semantics_ for
$\mathcal{DRA}_{f}$ and $\mathcal{DRA}_{\mathit{fp}}$. For these, we observe
the Euclidean plane with respect to all possible line configurations that are
distinguishable within the $\mathcal{DRA}$ calculi. With condensed semantics,
there is already a level of abstraction from the metrics of the underlying
space. All we can see are lines that are parallel or intersect. For the binary
composition operation of $\mathcal{DRA}$ calculi, we have to consider all
qualitatively different configurations of three lines.
In order to formalize “qualitatively different configurations”, we regard the
$\mathcal{DRA}$ calculus as a first-order structure, with the Euclidean plane
as its domain, together with all the base relations.
###### Proposition 28.
All orientation-preserving affine bijections are $\mathcal{DRA}_{f}$ and
$\mathcal{DRA}_{\mathit{fp}}$ automorphisms.
(In [54], the converse is also shown.)
Proof. It suffices to show that orientation preserving affine bijections
preserve the $\mathcal{LR}$ relations. Now, any orientation-preserving affine
bijection can be composed of translations, rotations, scalings and shears. It
is straightforward to see that these mappings preserve the $\mathcal{LR}$
relations. ∎
Recall that an affine map $f$ from Euclidean space to itself is given by
$f(x,y)=A{{x}\choose{y}}+(b_{x},b_{y})$
$f$ is a bijection iff $det(A)$ is non-zero.
Automorphisms and their compositions form a group which acts on the set of
points (and tuples of points, lines, etc.) by function application. Recall
that, if a group $G$ acts on a set, an _orbit_ consists of the set reachable
from a fixed element by performing the action of all group elements:
$O(x)=\\{f(x)|f\in G\\}$. The importance of this notion is the following:
> Qualitatively different configurations are orbits of the automorphism group.
Here, we start with configurations consisting of three lines, i.e. we consider
the orbits for all sets $\\{l_{1},l_{2},l_{3}\\}$ of (at most) three
lines181818We do not require that $l_{1}$, $l_{2}$ and $l_{3}$ are distinct;
hence, the set $\\{l_{1},l_{2},l_{3}\\}$ may also consist of two elements or
be a singleton. in Euclidean space with respect to the group of _all_ affine
bijections (and not just the orientation preserving ones – orientations will
come in at a later stage). This group is usually called the affine group of
$\mathbb{R}^{2}$ and denoted by $\mathbf{GA}(\mathbb{R}^{2})$.
A line in Euclidean space is given by the set of all points $(x,y)$ for which
$y=mx+b$. Given three lines $y=m_{i}x+b_{i}$ ($i=1,2,3$), we list their orbits
by giving a defining property. In each case, it is fairly obvious that the
defining property is preserved by affine bijections. Moreover, in each case,
we show a _transformation property_ , namely that given two instances of the
defining properties, the first can be transformed into the second by an affine
bijection. Together, this means that the defining property exactly specifies
an orbit. The transformation property often follows from the following basic
facts about affine bijections, see [55]:
1. 1.
An affine bijection is uniquely determined by its action on an affine basis,
the result of which is given by another affine basis. Since an affine basis of
the Euclidean plane is a point triple in general position, given any two point
triples in general position, there is a unique affine bijection mapping the
first point triple to the second.
2. 2.
Affine maps transform lines into lines.
3. 3.
Affine maps preserve parallelism of lines.
That is, it suffices to show that an instance of the defining property is
determined by three points in general position and drawing lines and parallel
lines.
We will consider the intersection of line $i$ with line $j$
($i\not=j\in\\{1,2,3\\}$). This is given by the system of equations:
$\\{y=m_{i}x+b_{i},\leavevmode\nobreak\ y=m_{j}x+b_{j}\\}.$
For $m_{i}\not=m_{j}$, this has a unique solution:
$x=-\frac{b_{i}-b_{j}}{m_{i}-m_{j}},\leavevmode\nobreak\
y=\frac{m_{i}b_{j}-m_{j}b_{i}}{m_{i}-m_{j}}.$
For $m_{i}=m_{j}$, there is either is no solution ($b_{i}\not=b_{j}$; the
lines are parallel), or there are infinitely many solutions ($b_{i}=b_{j}$;
the lines are identical).
We can now distinguish seven cases:
1. 1.
All $m_{i}$ are distinct and the three systems of equations
$\\{y=m_{i}x+b_{i},\leavevmode\nobreak\ y=m_{j}x+b_{j}\\}$
($i\not=j\in\\{1,2,3\\}$) yield three different solutions. Geometrically, this
means that all three lines intersect with three different intersection points.
The transformation property follows from the fact that the three intersection
points determine the configuration.
2. 2.
All $m_{i}$ are distinct and at least two of the three systems of equations
$\\{y=m_{i}x+b_{i},\leavevmode\nobreak\ y=m_{j}x+b_{j}\\}$
($i\not=j\in\\{1,2,3\\}$) have a common solution. Then, obviously, the single
solution is common to all three equation systems. Geometrically, this means
that all three lines intersect at the same point.
Take this point and a second point on one of the lines. By drawing parallels
through this second point, we obtain two more points, one on each of the other
two lines, such that the four points form a parallelogram. The transformation
property now follows from the fact that any two non-degenerate parallelograms
can be transformed into each other by an affine bijection.
3. 3.
$m_{i}=m_{j}\not=m_{k}$ and $b_{i}\not=b_{j}$ for distinct
$i,j,k\in\\{1,2,3\\}$. Geometrically, this means that two lines are parallel,
but not coincident, and the third line intersects them. Such a configuration
is determined by three points: the points of intersection, plus a further
point on one of the parallel lines. Hence, the transformation property
follows.
4. 4.
$m_{i}=m_{j}\not=m_{k}$ and $b_{i}=b_{j}$ for distinct $i,j,k\in\\{1,2,3\\}$.
Geometrically, this means that two lines are equal and a third one intersects
them. Again, such a configuration is determined by three points: the
intersection point plus a further point on each of the (two) different lines.
Hence, the transformation property follows.
5. 5.
All $m_{i}$ are equal, but the $b_{i}$ are distinct. Geometrically, this means
that all three lines are parallel, but not coincident. We cannot show the
transformation property here, which means that this case comprises several
orbits. Actually, we get one orbit for each distance ratio
$\frac{b_{1}-b_{2}}{b_{1}-b_{3}}.$
An affine bijection
$f(x,y)=A{{x}\choose{y}}+(b_{x},b_{y})$
transforms a line $y=mx+b$ to $y=m^{\prime}x+b^{\prime}$, with
$b^{\prime}=c_{1}(m)b+c_{2}(m)$, where $c_{1}$ and $c_{2}$ depend non-linearly
on $m$. However, since $m=m_{1}=m_{2}=m_{3}$, this non-linearity does not
matter. This means that
$\frac{b^{\prime}_{1}-b^{\prime}_{2}}{b^{\prime}_{1}-b^{\prime}_{3}}=\frac{c_{1}(m)b_{1}-c_{1}(m)b_{2}}{c_{1}(m)b_{1}-c_{1}(m)b_{3}}=\frac{b_{1}-b_{2}}{b_{1}-b_{3}},$
i.e. the distance ratio is invariant under affine bijections (which is well-
known in affine geometry). Given a fixed distance ratio, we can show the
transformation property: three points suffice to determine two parallel lines,
and the position of the third parallel line is then determined by the distance
ratio. For a distance ratio $1$, this configuration looks as follows:
Actually, for the qualitative relations between dipoles placed on parallel
lines, their distance ratio does not matter. Hence, we will ignore distance
ratios when computing the composition table below. The fact that we get
infinitely many orbits for this sub-case will be discussed below.
6. 6.
All $m_{i}$ are equal and two of the $b_{i}$ are equal but different from the
third. Geometrically, this means that two lines are coincident, and a third
one is parallel but not coincident. Such a configuration is determined by
three points: two points on the coincident lines and a third point on the
third line. Hence, the transformation property follows.
7. 7.
All $m_{i}$ are equal, and the $b_{i}$ are equal as well. This means that all
three lines are equal. The transformation property is obvious.
Since we have exhaustively distinguished the various possible cases based on
relations between the $m_{i}$ and $b_{i}$ parameters, this describes all
possible orbits of three lines w.r.t. affine bijections. Although we get
infinitely many orbits for case (5), in contexts where the distance ratio
introduced in case (5) does not matter, we will speak of seven qualitatively
different configurations, and it is understood that the infinitely many orbits
for case (5) are conceptually combined into one equivalence class of
configurations.
Figure 13: The $17$ qualitatively different configurations of triples of
oriented lines w.r.t. orientation-preserving affine bijections
Recall that we have considered _sets_ of (up to) three lines. If we consider
_triples_ of lines instead, cases (3) to (6) split up into three sub-cases,
because they feature distinguishable lines. We then get 15 different
configurations, which we name 1, 2, 3a, 3b, 3c, 4a, 4b, 4c, 5a, 5b, 5c, 6a,
6b, 6c and 7. While 5a, 5b and 5c correspond to case (5) above and therefore
are comprised of infinitely many orbits, the remaining configurations are
comprised of a single orbit.
The next split appears at the point when we consider qualitatively different
configurations of triples of unoriented lines with respect to _orientation-
preserving_ affine bijections. An affine map
$f(x,y)=A{{x}\choose{y}}+(b_{x},b_{y})$ is orientation-preserving if $det(A)$
is positive. In the above arguments, we now have to consider oriented affine
bases. Let us call an affine base $(p_{1},p_{2},p_{3})$ positively ($+$)
oriented, if the angle $\angle(\overrightarrow{p_{1}\leavevmode\nobreak\
p_{2}},\overrightarrow{p_{1}\leavevmode\nobreak\ p_{3}})$ is positive,
otherwise, it is negatively ($-$) oriented. Two given affine bases with the
same orientation determine a unique orientation-preserving affine bijection
transforming the first one into the second. Thus, the orientation of the
affine base matters, and hence cases 1 and 2 above are split into two sub-
cases each. For all the other cases, we have the freedom to choose the affine
bases such that their orientations coincide. In the end, we get $17$ different
orbits of triples of oriented lines: 1+, 1-, 2+, 2-, 3a, 3b, 3c, 4a, 4b, 4c,
5a, 5b, 5c, 6a, 6b, 6c and 7. They are shown in Fig. 13
The structure of the orbits already gives us some insight into the nature of
the dipole calculus. The fact that sub-case (1) corresponds to one orbit means
that neither angles nor ratios of angles can be measured in the dipole
calculus. By way of contrast, the presence of infinitely many orbits in sub-
case (5) means that ratios of distances in a specific direction, not
distances, _can_ be measured in the dipole calculus. Indeed, in
$\mathcal{DRA}_{\mathit{fp}}$, it is even possible to replicate a given
distance arbitrarily many times, as indicated in Fig. 14.
Figure 14: Replication of a given distance in $\mathcal{DRA}_{\mathit{fp}}$
That is, $\mathcal{DRA}_{\mathit{fp}}$ can be used to generate a one-
dimensional coordinate system. Note however that, due to the lack of well-
defined angles, a two-dimensional coordinate system cannot be constructed.
Note that Cristani’s 2DSLA calculus [56], which can be used to reason about
sets of lines, is too coarse for our purposes: cases (1) and (2) above cannot
be distinguished in 2DSLA.
### 3.2 Computing the composition table with Condensed Semantics
For the composition of (oriented) dipoles, we use the seventeen different
configurations for triples of (unoriented) lines for the automorphism group of
orientation-preserving affine bijections that have been identified in the
previous section (Fig. 13). A _qualitative composition configuration_ consists
of a qualitative configuration for a triple of lines (the lines will serve as
carrier lines for dipoles), carrying qualitative location information for the
start and end points of three dipoles, as detailed in the sequel. While the
notion of qualitative configuration composition is motivated by geometric
notions, it is purely abstract and symbolic and does not refer explicitly to
geometric objects. This ensures that it can be directly represented in a
finite data structure.
Each of the three (abstract) lines $l^{a}_{A},l^{a}_{B},l^{a}_{C}$ of a
qualitative composition configuration carries two abstract segmentation points
$S_{X}$ and $E_{X}$ ($X\in\\{A,B,C\\}$).
$\mathbf{P}=\left\\{S_{A},S_{B},S_{C},E_{A},E_{B},E_{C}\right\\}$ is the set
of all abstract segmentation points.
In the geometric interpretation of these abstract entities (which will be
defined precisely later on), the segmentation points lead to a segmentation of
the lines. So, we introduce five abstract segments $F$, $E$, $I$, $S$, $B$
(the letters are borrowed from the $\mathcal{LR}$ calculus). The set of
abstract segments is denoted by $\mathcal{S}$. It is ordered in the following
sequence:
$F>E>I>S>B.$
The geometric intuition behind this is shown in Fig. 15.
Figure 15: Segmentation on the line.
Having this segmentation of line configurations, we can introduce qualitative
configurations for _abstract dipoles_ by qualitatively locating their start
and end points based on the above segmentation. In the case that two or more
points fall onto the same segment, information on the relative location of
points within that segment is needed; this is provided by an ordering relation
denoted by $<_{p}$.
By $\mathcal{D}$, we denote the set
$\mathcal{S}\times\mathcal{S}\setminus\\{(S,S),(E,E)\\}$ (the exclusion of
$\\{(S,S),(E,E)\\}$ is motivated by the fact that the start and end points of
a dipole cannot coincide). By $st(dp)$ and $ed(dp)$, we denote the projections
to the first and second components of each tuple, respectively. For
convenience, we call the elements of the co-domains of $st$ and $ed$ abstract
points.
Finally, we need information on the points of intersection of lines. Depending
on orbit, there may be none, one, two or three points of intersection. Hence,
we introduce sets $\hat{\mathcal{S}}(i)$ with
$i\in\left\\{1+,1-,2+,2-,3a,3b,3c,4a,4b,4c,5a,5b,5c,6a,6b,6c,7\right\\}$ which
give names to each abstract point of intersection. These sets are defined as:
$\displaystyle\hat{\mathcal{S}}(1+)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{AB},\hat{s}_{BC},\hat{s}_{AC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(1-)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{AB},\hat{s}_{BC},\hat{s}_{AC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(2+)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{ABC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(2-)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{ABC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(3a)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{AB},\hat{s}_{AC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(3b)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{AC},\hat{s}_{BC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(3c)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{AB},\hat{s}_{BC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(4a)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{ABC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(4b)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{ABC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(4c)$ $\displaystyle:=$
$\displaystyle\left\\{\hat{s}_{ABC}\right\\}$
$\displaystyle\hat{\mathcal{S}}(5a)$ $\displaystyle:=$
$\displaystyle\emptyset$ $\displaystyle\hat{\mathcal{S}}(5b)$
$\displaystyle:=$ $\displaystyle\emptyset$
$\displaystyle\hat{\mathcal{S}}(5c)$ $\displaystyle:=$
$\displaystyle\emptyset$ $\displaystyle\hat{\mathcal{S}}(6a)$
$\displaystyle:=$ $\displaystyle\emptyset$
$\displaystyle\hat{\mathcal{S}}(6b)$ $\displaystyle:=$
$\displaystyle\emptyset$ $\displaystyle\hat{\mathcal{S}}(6c)$
$\displaystyle:=$ $\displaystyle\emptyset$ $\displaystyle\hat{\mathcal{S}}(7)$
$\displaystyle:=$ $\displaystyle\emptyset$
where $\hat{s}_{XY}$ denotes the point of intersection of abstract lines
$l^{a}_{X}$ and $l^{a}_{Y}$ and $\hat{s}_{XYZ}$ denotes the the point of
intersection of the three abstract lines $l^{a}_{X}$, $l^{a}_{Y}$ and
$l^{a}_{Z}$.
In the geometric interpretation, we require segmentation points that coincide
with points of intersection whenever possible. This coincidence is expressed
via an _assignment mapping_ , which is a partial mapping
$a:\mathbf{P}\rightharpoonup\hat{\mathcal{S}}(i)$ subject to the following
properties:
* •
if $a(S_{X})=\hat{s}_{y}$, then $y$ contains $X$;
* •
if $a(E_{X})=\hat{s}_{y}$, then $y$ contains $X$;
* •
if both $a(S_{X})$ and $a(E_{X})$ are defined, then $a(S_{x})\neq a(E_{x})$,
for all $X\in\left\\{A,B,C\right\\}$;
* •
the domain of $a$ has to be maximal.
The first two conditions express that each abstract segmentation point is
mapped to the correspondingly named abstract point of intersection. The third
condition requires that the abstract segmentation points of a line cannot be
mapped to the same abstract point of intersection. The last condition ensures
that abstract segmentation points are mapped to abstract points of
intersection whenever possible.
We now arrive at a formal definition:
###### Definition 29 (Qualitative Composition Configuration).
A _qualitative composition configuration_ (qcc) consists of:
* •
An identifier $i$ from the set
$\left\\{1+,1-,2+,2-,3a,3b,3b,4a,4b,4c,5a,5b,5c,6a,6b,6c,7\right\\}$ denoting
one of the qualitatively different configurations of line triples as
introduced in Section 3.1;
* •
An assignment mapping $a:\mathbf{P}\rightharpoonup\hat{\mathcal{S}}(i)$;
* •
A triple $(dp_{A},dp_{B},dp_{C})$ of elements from $\mathcal{D}$, where we
call each such element an _abstract dipole_ ;
* •
A relation $<_{p}$ on all points, i.e. the start and end points of the
abstract dipoles, which is compatible with $<$.
###### Definition 30 (Abstract direction).
For any abstract dipole $dp$, we say that $dir(dp)=+$ if and only if
$ed(dp)>_{p}st(dp)$, otherwise $dir(dp)=-$.
#### 3.2.1 Geometric Realization
In this section, we claim that each qcc has a realization, first of all, we
need to define what such a realization is.
###### Definition 31 (Order on ray).
Given a ray $l$, for two points $A$ and $B$, we say that $A<_{r}B$, if $B$
lies further in the positive direction than $A$.
We construct a map on each ray that reflects the abstract segments shown in
Fig. 15 to provide a link between a qcc and a compatible line scenario.
###### Definition 32 (Segmentation map).
Given a ray $r$ and two points $\tilde{S}$ and $\tilde{E}$ on it, the
segmentation map
$seg:r\longrightarrow\left\\{\tilde{F},\tilde{E},\tilde{I},\tilde{S},\tilde{B}\right\\}$
is defined as:
$\displaystyle r(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{@{\quad}r@{\quad}l}\textnormal{if
}\tilde{S}<_{r}\tilde{E}&\left\\{\begin{array}[]{rl}\tilde{F}&\textnormal{if
}\tilde{E}<_{r}x\\\ \tilde{E}&\textnormal{if }\tilde{E}=_{r}x\\\
\tilde{I}&\textnormal{if }\tilde{x}<_{r}\tilde{E}\wedge\tilde{S}<_{r}x\\\
\tilde{S}&\textnormal{if }\tilde{S}=_{r}x\\\ \tilde{B}&\textnormal{if
}x<_{r}\tilde{S}\\\ \end{array}\right.\\\ \textnormal{if
}\tilde{E}<_{r}\tilde{S}&\left\\{\begin{array}[]{rl}\tilde{F}&\textnormal{if
}x<_{r}\tilde{E}\\\ \tilde{E}&\textnormal{if }x=_{r}\tilde{E}\\\
\tilde{I}&\textnormal{if }\tilde{E}<_{r}x\wedge x<_{r}\tilde{S}\\\
\tilde{S}&\textnormal{if }x=_{r}\tilde{S}\\\ \tilde{B}&\textnormal{if
}\tilde{S}<_{r}x\\\ \end{array}\right.\end{array}\right.$
for any point on $x$ on $r$.
When it is clear that we are talking about segments on an actual ray, we often
omit the $\tilde{\\_}$.
###### Definition 33 (Geometric Realization).
For any qcc $Q$ a _geometric realization_ $R(Q)$ consists of a triple of
dipoles $(d_{A},d_{B},d_{C})$ in $\mathbb{R}^{2}$, three carrier rays $l_{A}$,
$l_{B}$, $l_{C}$ of the dipoles, and two points $\tilde{S}_{X}$ and
$\tilde{E}_{X}$ on $l_{X}$ for each $X\in\\{A,B,C\\}$, such that:
* •
$(l_{A},l_{B},l_{C})$ (more precisely, the corresponding triple of unoriented
lines) belongs to the configuration denoted by the identifier $i$ of $Q$;
* •
the angle between $l_{a}$ and the other two rays must lie in the interval
$(\pi,2\cdot\pi]$;
* •
for any $x,y\in\tilde{\mathbf{P}}$, if $a(p(x))$ and $a(p(y))$ are both
defined and equal, then $x=y$ (where
$p:\tilde{\mathbf{P}}=\\{\tilde{S}_{A},\tilde{S}_{B},\tilde{S}_{C},\tilde{E}_{A},\tilde{E}_{B},\tilde{E}_{C}\\}\to\mathbf{P}$
be the obvious bijection);
* •
for all $X$, $st(dp_{X})=seg(st(d_{X}))$ and $ed(dp_{X})=seg(ed(d_{X}))$;
* •
for all points $x$ and $y$ on $l_{X}$, if $seg(x)<seg(y)$, then $x<_{r}y$;
* •
if $l_{X}=l_{Y}$, the order $<_{p}$ must be preserved for points $st(d_{X})$,
$ed(d_{X})$, $st(d_{Y})$, $ed(d_{Y})$, in such a way that: if
$st(dp_{X})<_{p}st(dp_{Y})$, then $st(d_{Y})<_{r}st(d_{X})$ and in the same
manner between all other points.
must hold.
###### Proposition 34.
Given three dipoles in $\mathbb{R}^{2}$, there is a qcc $Q$ and a geometric
realization of $R(Q)$ which uses these three dipoles.
Proof. For this proof, we construct a qcc from a scenario of three dipoles in
$\mathbb{R}^{2}$. Given three dipoles $d_{A}$, $d_{B}$, $d_{C}$ in
$\mathbb{R}^{2}$, we determine their carrier rays $l_{A}$, $l_{B}$, $l_{C}$ in
such a way that the angles between $l_{A}$ and $l_{B}$ as well as $l_{A}$ and
$l_{C}$ lie in the interval $(\pi,2\cdot\pi]$. We determine the identifier of
the configuration in which the the scenario lies. We determine the points of
intersection of the rays and identify them with $\hat{s}_{XY}$ in
$\hat{\mathcal{S}}(i)$. For all points $X$ in $\mathcal{P}$, for which $a$ is
undefined, the points $\hat{X}$ are placed in such a way, that
$S_{X}<_{r}E_{X}$ (which is equivalent to $S_{X}<E_{X}$). We identify
$st(dp_{X})$ and $ed(dp_{X})$ according to the segmentation map on these rays.
If two carrier rays coincide, we define the order $<_{p}$ w.r.t. $<_{r}$,
otherwise it is arbitrary. This clearly gives a $qcc$.
An example of this construction is given in Fig. 16.
Figure 16: Construction of qcc
On the left-hand-side of Fig. 16, there is a scenario with three dipoles,
lying somewhere in $\mathbb{R}^{2}$. On the right hand side, rays and points
of intersection are added. Comparison with orbits and placement of lines
determine the identifier $3b$ for this scenario. The map $a$ can be defined as
$\displaystyle a(S_{A})$ $\displaystyle=$
$\displaystyle\hat{\mathcal{S}}_{AC}$ $\displaystyle a(S_{B})$
$\displaystyle=$ $\displaystyle\hat{\mathcal{S}}_{BC}$ $\displaystyle
a(E_{C})$ $\displaystyle=$ $\displaystyle\hat{\mathcal{S}}_{AC}$
$\displaystyle a(S_{B})$ $\displaystyle=$
$\displaystyle\hat{\mathcal{S}}_{BC}$
where the assignment is only free for $E_{A}$ and $E_{B}$. $E_{A}$ and $E_{B}$
are lying at the start point of dipole $d_{A}$ and at the end point of dipole
$d_{B}$. In this way, we get:
$\displaystyle st(dp_{A})=E$ $\displaystyle ed(dp_{A})=F$ $\displaystyle
st(dp_{B})=E$ $\displaystyle ed(dp_{B})=I$ $\displaystyle st(dp_{C})=B$
$\displaystyle ed(dp_{C})=B$
and
$\displaystyle dir(dp_{A})=+$ $\displaystyle dir(dp_{B})=-$ $\displaystyle
dir(dp_{C})=-$
In this case the assignment of $<_{p}$ is arbitrary.
This construction gives us the desired qcc and a realization of it. ∎
### 3.3 Primitive Classifiers
The last and most crucial point is the computation of $\mathcal{DRA}$
relations between three dipoles. We can decompose this task into subtasks,
since each $\mathcal{DRA}_{f}$ relation comprises four $\mathcal{LR}$
relations between a dipole and point; these are obtained from a qualitative
composition configuration using so-called _primitive classifiers_. The _basic
classifiers_ apply the _primitive classifiers_ to the abstract dipoles in each
qualitative composition configuration in an adequate manner. For
$\mathcal{DRA}_{\mathit{fp}}$ relations an extension of the _basic
classifiers_ is used in cases where the qualitative angle between several
dipoles has to be determined. Finally, the resulting data is collected in a
(composition) table.
###### Definition 35 (Primitive Qualitative Composition Configuration).
A _primitive qualitative composition configuration_ (pqcc) is a sub-
configuration of a qualitative composition configuration (see Def. 29)
containing two abstract dipoles (where for the second one, only the start or
end point is used for classification). All other data are the same as in Def.
29.
###### Notation 36.
To simplify the explanation of large classifiers, we shall write:
$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rcl}cond_{1}&\longrightarrow&value_{1}\\\
cond_{2}&\longrightarrow&value_{2}\end{array}\right.$
instead of
$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}value_{1}&\mbox{if }cond_{1}\\\
value_{2}&\mbox{if }cond_{2}.\end{array}\right.$
If it is clear which function we are defining, we even omit the “$f(x)=$”.
Given a primitive qualitative composition configuration $Q$, _primitive
classifiers_ map the qualitative locations of a dipole $dp_{1}$ and a point
$pt$ (which is the start or end point of another dipole $dp_{2}$) to a letter
indicating the $\mathcal{LR}$ relation between the dipole and point. We say
that the dipole has positive $pos$ orientation if $dir(dp)=+$, otherwise the
orientation is negative $neg$.
We need three different types of primitive classifiers for our algorithm.
Given two arbitrary dipoles $dp_{1}$ and $dp_{2}$, we construct a primitive
classifier for a pqcc with intersecting carrier rays in its realization. The
classifier itself only works on $dp_{1}$ and $pt$, where $pt$ is either the
start or end point of $dp_{2}$. A realization of this pqcc is given in Fig. 17
for the reader’s convenience, the actual dipoles are omitted from the figure,
since they can be placed arbitrarily.
Figure 17: Line configuration for primitive Classifier
To realize the dipole, this classifier takes dipole $dp_{1}$ and the start or
end point of $dp_{2}$ called $pt$ as well as information on whether $dp_{1}$
is pointing in the same direction as the ray ($pos$) or against it ($neg$) for
both dipoles. The classifier returns an $\mathcal{LR}$-relation determining
the relation between $dp_{1}$ and $pt$.
In this case, the classifier $cli_{x,y}(dp_{1},pt)$ is given by:
$\displaystyle pos$ $\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}pt>y&\longrightarrow&R\\\
pt=y&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<x\wedge
ed(dp_{1})<x&\longrightarrow&F\\\ st(dp_{1})<x\wedge
ed(dp_{1})=x&\longrightarrow&E\\\ st(dp_{1})<x\wedge
ed(dp_{1})>x&\longrightarrow&I\\\ st(dp_{1})=x\wedge
ed(dp_{1})>x&\longrightarrow&S\\\ st(dp_{1})>x\wedge
ed(dp_{1})>x&\longrightarrow&B\end{array}\right.\\\ pt<y&\longrightarrow&L\\\
\end{array}\right.$ $\displaystyle neg$ $\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}pt<y&\longrightarrow&R\\\
pt=y&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>x\wedge
ed(dp_{1})>x&\longrightarrow&F\\\ st(dp_{1})>x\wedge
ed(dp_{1})=x&\longrightarrow&E\\\ st(dp_{1})>x\wedge
ed(dp_{1})<x&\longrightarrow&I\\\ st(dp_{1})=x\wedge
ed(dp_{1})<x&\longrightarrow&S\\\ st(dp_{1})<x\wedge
ed(dp_{1})<x&\longrightarrow&B\end{array}\right.\\\ pt>y&\longrightarrow&L\\\
\end{array}\right.$
The subscripts on the classifier denote the point of intersection of the two
lines. For the case shown in Fig. 17, we have $x=y=S$. We see that the table
for $neg$ is exactly the complement of $pos$. This primitive classifier
assumes that, in the geometric realization, the second dipole (containing
point $pt$) points to the right w.r.t. dipole $d$. If the second dipole points
to the left in the realization, it is sufficient to apply an operation that
interchanges $L$ with $R$ on this classifier, in order to obtain the correct
results. We will call this operation $com$. This is the only primitive
classifier needed for intersecting lines.
Secondly, we give a primitive classifier $cls(dp_{1},pt)$ for two lines that
coincide, see Fig. 18.
Figure 18: Primitive classifier for same line.
$\displaystyle pos$ $\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}pt=F&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<F\wedge
ed(dp_{1})<F&\longrightarrow&F\\\ st(dp_{1})<F\wedge
ed(dp_{1})=F&\longrightarrow&\left\\{\begin{array}[]{rcl}ed(dp_{1})<_{p}pt&\longrightarrow&F\\\
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\
ed(dp_{1})>_{p}pt&\longrightarrow&I\end{array}\right.\\\ st(dp_{1})=F\wedge
ed(dp_{1})=F&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&F\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&I\\\ st(dp_{1})=_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&S\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&B\\\ \end{array}\right.\\\
\end{array}\right.\\\
pt=E&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<E\wedge
ed(dp_{1})<E&\longrightarrow&F\\\ st(dp_{1})<E\wedge
ed(dp_{1})=E&\longrightarrow&E\\\ st(dp_{1})<E\wedge
ed(dp_{1})>E&\longrightarrow&I\\\ st(dp_{1})=E\wedge
ed(dp_{1})>E&\longrightarrow&S\\\ st(dp_{1})>E\wedge
ed(dp_{1})>E&\longrightarrow&B\\\ \end{array}\right.\\\
pt=I&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<I\wedge
ed(dp_{1})<I&\longrightarrow&F\\\ st(dp_{1})<I\wedge
ed(dp_{1})=I&\longrightarrow&\left\\{\begin{array}[]{rcl}ed(dp_{1})<_{p}pt&\longrightarrow&F\\\
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ ed(dp_{1})>_{p}pt&\longrightarrow&I\\\
\end{array}\right.\\\ st(dp_{1})<I\wedge ed(dp_{1})>I&\longrightarrow&I\\\
st(dp_{1})=I\wedge
ed(dp_{1})=I&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&F\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&I\\\ st(dp_{1})=_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&S\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&B\\\ \end{array}\right.\\\
st(dp_{1})=I\wedge
ed(dp_{1})>I&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<_{p}pt&\longrightarrow&I\\\
st(dp_{1})=_{p}pt&\longrightarrow&S\\\ st(dp_{1})>_{p}pt&\longrightarrow&B\\\
\end{array}\right.\\\ st(dp_{1})>I\wedge
ed(dp_{1})>I&\longrightarrow&B\end{array}\right.\\\
pt=S&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<S\wedge
ed(dp_{1})<S&\longrightarrow&F\\\ st(dp_{1})<S\wedge
ed(dp_{1})=S&\longrightarrow&E\\\ st(dp_{1})<S\wedge
ed(dp_{1})>S&\longrightarrow&I\\\ st(dp_{1})=S\wedge
ed(dp_{1})>S&\longrightarrow&S\\\ st(dp_{1})>S\wedge
ed(dp_{1})>S&\longrightarrow&B\\\ \end{array}\right.\\\
pt=B&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})=B\wedge
ed(dp_{1})=B&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&F\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&I\\\ st(dp_{1})=_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&S\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&B\\\ \end{array}\right.\\\
st(dp_{1})=B\wedge
ed(dp_{1})>B&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<_{p}pt&\longrightarrow&I\\\
st(dp_{1})=_{p}pt&\longrightarrow&S\\\ st(dp_{1})>_{p}pt&\longrightarrow&B\\\
\end{array}\right.\\\ st(dp_{1})>B\wedge
ed(dp_{1})>B&\longrightarrow&B\end{array}\right.\end{array}\right.$
$\displaystyle neg$ $\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}pt=B&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>B\wedge
ed(dp_{1})>B&\longrightarrow&F\\\ st(dp_{1})>B\wedge
ed(dp_{1})=B&\longrightarrow&\left\\{\begin{array}[]{rcl}ed(dp_{1})<_{p}pt&\longrightarrow&I\\\
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\
ed(dp_{1})>_{p}pt&\longrightarrow&F\end{array}\right.\\\ st(dp_{1})=B\wedge
ed(dp_{1})=B&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})<_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&B\\\ st(dp_{1})=_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&S\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&I\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&F\\\ \end{array}\right.\\\
\end{array}\right.\\\
pt=S&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>S\wedge
ed(dp_{1})>S&\longrightarrow&F\\\ st(dp_{1})>S\wedge
ed(dp_{1})=S&\longrightarrow&E\\\ st(dp_{1})>S\wedge
ed(dp_{1})<S&\longrightarrow&I\\\ st(dp_{1})=S\wedge
ed(dp_{1})<S&\longrightarrow&S\\\ st(dp_{1})<S\wedge
ed(dp_{1})<S&\longrightarrow&B\\\ \end{array}\right.\\\
pt=I&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>I\wedge
ed(dp_{1})>I&\longrightarrow&F\\\ st(dp_{1})>I\wedge
ed(dp_{1})=I&\longrightarrow&\left\\{\begin{array}[]{rcl}ed(dp_{1})>_{p}pt&\longrightarrow&F\\\
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ ed(dp_{1})<_{p}pt&\longrightarrow&I\\\
\end{array}\right.\\\ st(dp_{1})>I\wedge ed(dp_{1})<I&\longrightarrow&I\\\
st(dp_{1})=I\wedge
ed(dp_{1})=I&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&F\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&I\\\ st(dp_{1})=_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&S\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&B\\\ \end{array}\right.\\\
st(dp_{1})=I\wedge
ed(dp_{1})<I&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>_{p}pt&\longrightarrow&I\\\
st(dp_{1})=_{p}pt&\longrightarrow&S\\\ st(dp_{1})<_{p}pt&\longrightarrow&B\\\
\end{array}\right.\\\ st(dp_{1})<I\wedge
ed(dp_{1})<I&\longrightarrow&B\end{array}\right.\\\
pt=E&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>E\wedge
ed(dp_{1})>E&\longrightarrow&F\\\ st(dp_{1})>E\wedge
ed(dp_{1})=E&\longrightarrow&E\\\ st(dp_{1})>E\wedge
ed(dp_{1})<E&\longrightarrow&I\\\ st(dp_{1})=E\wedge
ed(dp_{1})<E&\longrightarrow&S\\\ st(dp_{1})<E\wedge
ed(dp_{1})<E&\longrightarrow&B\\\ \end{array}\right.\\\
pt=F&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})=F\wedge
ed(dp_{1})=F&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>_{p}pt\wedge
ed(dp_{1})>_{p}pt&\longrightarrow&F\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})=_{p}pt&\longrightarrow&E\\\ st(dp_{1})>_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&I\\\ st(dp_{1})=_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&S\\\ st(dp_{1})<_{p}pt\wedge
ed(dp_{1})<_{p}pt&\longrightarrow&B\\\ \end{array}\right.\\\
st(dp_{1})=F\wedge
ed(dp_{1})<F&\longrightarrow&\left\\{\begin{array}[]{rcl}st(dp_{1})>_{p}pt&\longrightarrow&I\\\
st(dp_{1})=_{p}pt&\longrightarrow&S\\\ st(dp_{1})<_{p}pt&\longrightarrow&B\\\
\end{array}\right.\\\ st(dp_{1})<F\wedge
ed(dp_{1})<F&\longrightarrow&B\end{array}\right.\end{array}\right.$
This classifier looks a little cumbersome, but we decided to use it in this
way, so that all impossible cases w.r.t. the ordering of the line are
excluded. This gives better error handling capabilities in an implementation
of it, since impossible cases can be detected. A more compressed version is
possible, but it cannot detect impossible cases anymore. All cases that are
not listed in the above classifier are cases where the ordering $>_{p}$ is not
compatible with the segmentation, and so they are impossible. This is the only
classifier for coinciding lines.
The third classifier is for parallel lines, i.e. a configuration like that in
Fig. 19. Let the lower line be the line the dipole lies on. The information
about the line on which the dipole lies is handled by a basic classifier which
uses this primitive classifier and exchanges $L$ and $R$ appropriately.
Figure 19: Primitive classifier for parallel lines.
Fortunately this classifier $clpar(dp_{1},pt)$ is simple:
$\displaystyle pos$ $\displaystyle\longrightarrow$ $\displaystyle R$
$\displaystyle neg$ $\displaystyle\longrightarrow$ $\displaystyle L$
This is the only classifier for parallel lines.
This is a complete list of the basic classifiers that are needed.
### 3.4 Basic Classifiers
Based on the primitive classifiers introduced in Sect. 3.3, we construct the
_basic classifiers_ to determine the $\mathcal{DRA}$ relations in scenarios.
For $\mathcal{DRA}_{f}$, we always need exactly four primitive classifiers to
determine the relation. For $\mathcal{DRA}_{\mathit{fp}}$, in some cases we
need an additional fifth classifier to determine the qualitative angle. We
will first focus on the $\mathcal{DRA}_{f}$ case. Given a qcc, we apply four
basic classifiers three times: namely (1) to the first and second abstract
dipole, (2) to the second and third and (3) to the first and third. Thus, we
obtain an entry in the composition table. Consider a qcc with $i=1+$ and
$a(S_{A})=\hat{s}_{AB}$, $a(S_{B})=\hat{s}_{AB}$ and $a(s_{C})=\hat{S}_{AC}$.
Such a configuration has a realization as in Fig. 20.
Figure 20: Line configuration for Basic Classifier
The dipole $d_{X}$ lies on the ray $l_{X}$ for $X\in\left\\{A,B,C\right\\}$.
We now apply primitive classifiers to this scenario in the way defined in
Section 2.1. Hence, we get the basic classifier for such a configuration:
$\displaystyle R(dp_{A},st_{B})$ $\displaystyle=$ $\displaystyle
cli_{s,s}(dp_{A},st_{B})$ $\displaystyle R(dp_{A},ed_{B})$ $\displaystyle=$
$\displaystyle cli_{s,s}(dp_{A},ed_{B})$ $\displaystyle R(dp_{B},st_{A})$
$\displaystyle=$ $\displaystyle com\circ cli_{s,s}(dp_{B},st_{A})$
$\displaystyle R(dp_{B},ed_{A})$ $\displaystyle=$ $\displaystyle com\circ
cli_{s,s}(dp_{B},ed_{A})$ $\displaystyle R(dp_{B},st_{C})$ $\displaystyle=$
$\displaystyle cli_{e,e}(dp_{B},st_{C})$ $\displaystyle R(dp_{B},ed_{C})$
$\displaystyle=$ $\displaystyle cli_{e,e}(dp_{B},ed_{C})$ $\displaystyle
R(dp_{C},st_{B})$ $\displaystyle=$ $\displaystyle com\circ
cli_{e,e}(dp_{C},st_{B})$ $\displaystyle R(dp_{C},st_{B})$ $\displaystyle=$
$\displaystyle com\circ cli_{e,e}(dp_{C},ed_{B})$ $\displaystyle
R(dp_{A},st_{C})$ $\displaystyle=$ $\displaystyle cli_{e,s}(dp_{A},st_{C})$
$\displaystyle R(dp_{A},ed_{C})$ $\displaystyle=$ $\displaystyle
cli_{e,s}(dp_{A},ed_{C})$ $\displaystyle R(dp_{C},st_{A})$ $\displaystyle=$
$\displaystyle com\circ cli_{s,e}(dp_{C},st_{A})$ $\displaystyle
R(dp_{C},ed_{A})$ $\displaystyle=$ $\displaystyle com\circ
cli_{s,e}(dp_{C},ed_{A})$
and we obtain the relation between $dp_{A}$ and $dp_{B}$:
$\varrho(R(dp_{A},st_{B}),R(dp_{A},ed_{B}),R(dp_{B},st_{A}),R(dp_{B},ed_{A}))$.
The relations between $d_{B}$ and $d_{C}$ as well as between $dp_{A}$ and
$dp_{C}$ are derived analogously. The basic classifiers depend on the
configuration in which the qcc realization lies and on the angle between the
rays in the realization. They are constructed for an angle between the rays in
the interval $(\pi,2\cdot\pi]$. If the angle is in the interval $(0,\pi]$, the
$\mathcal{LR}$ relation between any line on the first ray and a point on the
second just swaps. We capture this by introducing the operation $com$ which is
applied in this case. With it, we can limit the number of necessary primitive
classifiers. The construction of the other basic classifiers is done
analogously.
### 3.5 Extended Basic Classifiers for $\mathcal{DRA}_{\mathit{fp}}$
For $\mathcal{DRA}_{\mathit{fp}}$, basically the same classifiers as described
for $\mathcal{DRA}_{f}$ in Section 3.4 are used. We simply extend them for the
relations rrrr, rrll, llll and llrr to classify the information about
qualitative angles. For this purpose, we have to have a look at the angles
between dipoles in the realization of a given qcc. The qualitative angle
between two dipoles $d_{A}$ and $d_{B}$ is called positive $+$ (negative $-$)
if the angle from the carrier ray of $d_{A}$ called $l_{A}$ to the carrier ray
of $d_{B}$ called $l_{B}$ lies in the interval $(0,\pi)$ ($(\pi,2\cdot\pi)$).
We give an example of this. Consider the configuration of a $\mathcal{DRA}$
scenario in Fig. 21 on the left hand side.
| |
---|---|---
Figure 21: $\mathcal{DRA}$ Scenario
On the right-hand side of Fig. 21, the carrier rays are introduced and we can
see that the angle clearly lies in the interval $(0,\pi)$ and hence the
qualitative angle is positive. The definitions of parallel $P$ and anti-
parallel $A$ are straightforward. The set $a^{-1}(\hat{S}_{xy})$ always
contains exactly two elements, if $\hat{S}_{xy}\in\hat{\mathcal{S}}(i)$. To
continue, we need functions
$proj_{x}:\mathcal{P}(\mathbf{P})\longrightarrow\mathcal{P}(\mathbf{P})$
defined as
$proj_{x}=\left\\{a\mid idx_{x}(a)=x\right\\}$
which form the set of all elements with index ($idx$) x. $\mathcal{P}$ denotes
powerset formation. By the definition of $a$ and the sets
$\hat{\mathcal{S}}(i)$, these sets are always singletons, if $proj_{x}\circ
a^{-1}$ is applied to an intersection point and if $a^{-1}$ contains an
element with index $x$, otherwise the set is empty. We shall write
$a_{x}^{-1}$ for $proj_{x}\circ a^{-1}$.
We observed that the qualitative angles between two dipoles can be classified
very easily once the $\mathcal{DRA}_{f}$ relations between the dipoles $d_{A}$
and $d_{B}$ are known. All we need to do is to find out if the ray $l_{B}$
intersects $l_{A}$ in front of or behind $d_{A}$. In the language of qcc and
abstract dipoles $dp_{A}$ and $dp_{B}$, we can say that, if
$a^{-1}_{A}(\hat{S}_{AB})>ed(dp_{A})$ for $dir(dp_{A})=+$, or if
$a^{-1}_{A}(\hat{S}_{AB})<ed(dp_{A})$, if $dir(dp_{A})=-$, then the abstract
point of intersection lies “in front of $dp_{A}$” and, if
$st(dp_{A})>a^{-1}_{A}(\hat{S}_{AB})$ for $dir(dp_{A})=+$ or
$a^{-1}_{A}(\hat{S}_{AB})>st(dp_{A})$ for $dir(dp_{A})=-$, the abstract point
of intersection lies “behind $dp_{A}$”.
###### Proposition 37.
In a realization $R(Q)$ of a qcc $Q$, the carrier rays of any two dipoles
$d_{1}$ and $d_{2}$ intersect in front of $d_{1}$ if and only if, in $Q$ the
property
$(a^{-1}_{1}(\hat{S}_{12})>ed(dp_{1})\wedge
dir(dp_{1})=+)\vee(a^{-1}_{1}(\hat{S}_{12})<ed(dp_{1})\wedge dir(dp_{1})=-)$
is fulfilled.
Proof. This is immediate by inspection of the property and respective
scenarios. ∎
###### Proposition 38.
In a realization $R(Q)$ of a qcc $Q$, the carrier rays of any two dipoles
$d_{1}$ and $d_{2}$ intersect behind $d_{1}$ if and only if, in $Q$ the
property
$(st(dp_{1})>a^{-1}_{1}(\hat{S}_{12})\wedge
dir(dp_{1})=+)\vee(a^{-1}_{1}{\hat{S}_{12}}>st(dp_{1})\wedge dir(dp_{1})=-)$
is fulfilled.
Proof. This is immediate by inspection of the property and respective
scenarios. ∎
The complete extension for the Basic Classifiers is given as:
rrrr $\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}a^{-1}_{A}(\hat{S}_{AB})>ed(dp_{A})\wedge
dir(dp_{A})=+&\longrightarrow&-\\\ a^{-1}_{A}(\hat{S}_{AB})<ed(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&-\\\ st(dp_{A})>a^{-1}_{A}(\hat{S}_{AB})\wedge
dir(dp_{A})=+&\longrightarrow&+\\\ a^{-1}_{A}(\hat{S}_{AB})>st(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&+\end{array}\right.$ rrll
$\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}a^{-1}_{A}(\hat{S}_{AB})>ed(dp_{A})\wedge
dir(dp_{A})=+&\longrightarrow&+\\\ a^{-1}_{A}(\hat{S}_{AB})<ed(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&+\\\ st(dp_{A})>a^{-1}_{A}(\hat{S}_{AB})\wedge
dir(dp_{A})=+&\longrightarrow&-\\\ a^{-1}_{A}(\hat{S}_{AB})>st(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&-\end{array}\right.$ llll
$\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}a^{-1}_{A}(\hat{S}_{AB})>ed(dp_{A})\wedge
dir(dp_{A})=+&\longrightarrow&+\\\ a^{-1}_{A}(\hat{S}_{AB})<ed(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&+\\\ st(dp_{A})>a^{-1}_{A}(\hat{S}_{AB})\wedge
dir(dp_{A})=+&\longrightarrow&-\\\ a^{-1}_{A}(\hat{S}_{AB})>st(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&-\end{array}\right.$ llrr
$\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{rcl}a^{-1}_{A}(\hat{S}_{AB})>ed(dp_{A})\wedge
dir(dp_{A})=+&\longrightarrow&-\\\ a^{-1}_{A}(\hat{S}_{AB})<ed(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&-\\\ st(dp_{A})>a^{-1}_{A}(\hat{S}_{AB})\wedge
dir(dp_{A})=+&\longrightarrow&+\\\ a^{-1}_{A}(\hat{S}_{AB})>st(dp_{A})\wedge
dir(dp_{A})=-&\longrightarrow&+\end{array}\right.$
Constructing the classifiers for qccs based on configurations with parallel
lines is easy, depending on the $\mathcal{DRA}_{f}$-relations, the dipoles can
either be parallel or anti-parallel in such cases, but never both at the same
time.
###### Lemma 39.
Given two intersecting lines, the $\mathcal{LR}$-relations between a dipole on
a first line and a point on the second line are stable under the movement of
the point along the line, unless it moves through the point of intersection of
the two lines.
Proof. By the definition of $\mathcal{LR}$-relations, the point can be in one
of three different relative positions to the carrier ray of the dipole. The
point can lie on either side of the point of intersection, yielding the
relation $L$ or $R$, or on the point of intersection itself, yielding exactly
one relation on the line. ∎
###### Lemma 40.
Given a dipole and a point lying on its carrier line, the
$\mathcal{LR}$-relations between the dipole and point are stable under the
movement of the point along the line, unless it is moved over the start or end
point of the dipole.
Proof. Inspect the definition of $\mathcal{LR}$-relations on a line. ∎
###### Lemma 41.
For dipoles lying on intersecting rays, the $\mathcal{DRA}$ relations are
stable under the movement of the start and end points of the dipoles along the
rays, as long as the segments for the start and end points and the directions
of the dipoles do not change.
Proof. We observe that the segmentation is a stronger property than the one
used in Lemma 39. For $\mathcal{DRA}_{f}$ relations it suffices to apply Lemma
39 four times. For $\mathcal{DRA}_{\mathit{fp}}$ relations, we also need to
take the intersection property of Prop. 46 into account. ∎
###### Lemma 42.
For dipoles on the same line, the $\mathcal{DRA}$-relations are stable under
the movement of the start and end points of the dipoles along the rays, so
long as the relation $<_{r}$ does not change.
Proof. Apply Lemma 40 four times. ∎
###### Lemma 43.
1. 1.
Transforming a given realization of a qcc along an orientation-preserving
affine transformation preserves the segmentation map.
2. 2.
If two dipoles are on the same line, affine transformations also preserve
$<_{r}$.
Proof. 1) According to Prop. 28, any orientation-preserving affine
transformation preserves the $\mathcal{LR}$ relations.
2) This follows from the preservation of length ratios by affine
transformations, i.e. the length ratios between the start and end points of
the dipoles and points $S$ and $E$ on the ray. ∎
###### Lemma 44.
Given a qcc, any two geometric realizations exhibit the same
$\mathcal{DRA}$-relations among their dipoles.
Proof. Let two geometric realizations $R_{1}$, $R_{2}$ of a qcc $Q$ be given.
Since the line triples of $R_{1}$ and $R_{2}$ belong to the same orbit, there
is an orientation-preserving affine bijection $f$ transforming the line triple
of $R$ into that of $R^{\prime}$. In case of configurations 5a, 5b and 5c, we
assume that all distance ratios are adjusted to 1 in order to reach the same
orbit. Note that this adjustment, although not an affine transformation, does
not affect the relations between dipoles.
Since $f$ maps $R_{1}$’s line triple to $R_{2}$’s line triple, it also maps
the corresponding points of intersection to each other. For orbits $1+$ and
$1-$, all segmentation points are points of intersection. Hence, $f$ does not
change the segments given by $r(x)$ in which the start and end points of the
dipoles lie. For the rest of the argument, apply Lemma 41.
For cases $2+$ and $2-$, we just have a single point of intersection, but the
relative directions of the rays are restricted by the definition of a
realization and so is the location of all segmentation points w.r.t. the
intersection point, as are the locations of the start and end points of the
dipoles w.r.t. the segmentation points. For the rest of the argument, apply
Lemma 41.
In cases $3a$, $3b$ and $3c$, we have two intersection points and two
segmentation points that are not points of intersection but, as before, the
directions of the rays and the locations of all segmentation points are
restricted and hence the locations of the start and end points of the dipoles,
and again, we can apply Lemma 41.
In cases $4a$, $4b$ and $4c$, we have one point of intersection and $3$
segmentation points that are not points of intersection. First, we can argue
to restrict the location and direction. In the end, we can apply Lemma 42 and
Lemma 41.
In cases $5a$, $5b$ and $5c$, we only have segmentation points that are not
points of intersection, but all rays have the same directions and the relative
orientations of segmentation points on the line are restricted. Hence, the
directions of the dipoles do not change during the mapping and the relative
direction between dipoles is all that is necessary to determine their
$\mathcal{DRA}$-relations in the case of parallel dipoles.
The proof of cases $6a$, $6b$ and $6c$ is similar to cases $4$ and $5$, with
the argument based on Lemma 42 for dipoles on the same line, and the arguments
of cases $5$ for parallel lines.
For case $7$, we need to apply Lemma 42.
For additional arguments for $\mathcal{DRA}_{\mathit{fp}}$-relations, please
refer to the proof of Prop. 46.
∎
###### Theorem 45 (Correctness of the Construction).
Given a qcc $Q$ and an arbitrary geometric realization $R(Q)$ of it, the
$\mathcal{DRA}_{f}$ relation in $R(Q)$ is the same as that computed by the
basic classifiers on $Q$.
Proof. According to Lemma 44, we can focus on one geometric realization per
qcc.
For this proof, we need to inspect once more the construction of the basic
classifiers above the primitive classifiers. The actual values of $a$, $dir$
and the start and end points of the abstract dipoles as well as the order
$<_{p}$ are not directly used by basic classifiers191919With the exception of
the extended classifiers, but we will discuss these later. They are passed
through to primitive classifiers. The only information that is directly used
in basic classifiers is the identifier $i$ of the configuration.
We divide this proof in two steps. In the first step, we show that the
primitive classifiers are correct and, in the second step, we do the same for
basic classifiers. We will show a proof for the classifier
$cli_{S,S}(dp_{1},pt)$ and a pqcc with $dir_{dp_{1}}=+$, $dp_{1}=(I,I)$ and
$pt=I$. A realization of this configuration is shown in Fig. 22
Figure 22: A realization
and we can easily see that $d_{1}\textnormal{\rm R}pt$ has to be true. By
observing $cli_{S,S}(dp_{1},pt)$, we see that we are in the case $pos$ and
that $pt>S$ and so the primitive classifier also yields
$dp_{1}\;\textnormal{\rm R}\;pt$ as expected. All other proofs for pqccs are
done in an analogous way by inspection of the relations yielded by the
primitive classifiers and their realizations. With primitive classifiers
working correctly, we need to focus on the basic classifiers. Here, we will
show this for the case $i=1+$, all other cases are handled in an analogous
fashion. First we take any realization of $i=1+$ and add directions to the
lines as described in the section about geometric realizations of qccs. For
example, the one depicted in Fig. 23.
Figure 23: A realization for a qcc
In the next step, this realization is decomposed according to the definition
of $\mathcal{DRA}_{f}$-relations and the basic classifiers shown in Fig. 24.
Figure 24: Decomposition of line configuration
The various parts of the decomposed line configuration need to be matched with
the realization of the primitive classifier, here the realization of Fig. 17.
In our case, the classifier matches directly with the orientations from
$l_{A}$ to $l_{B}$, $l_{B}$ to $l_{C}$ and $l_{A}$ to $l_{C}$. In the other
cases, the angle between the lines may be inverted. Then, we need to swap $R$
and $L$ which is done by the operation $com$. Furthermore, we see that the
lines $l_{C}$ and $l_{B}$ both intersect in segment $E$, whereas $l_{A}$ and
$l_{B}$ intersect both in $S$. The intersection for $l_{A}$ and $l_{C}$ is $E$
for $l_{A}$ and $S$ for $l_{C}$, we need to parameterize the respective
primitive classifiers with that information. But in the end, our arguments
yield exactly the basic classifier shown in Section 3.4. The arguments for the
other $16$ basic classifiers are analogous. ∎
###### Proposition 46.
Given any qcc $Q$ and its geometric realization $R(Q)$, the extended basic
classifiers determine the same $\mathcal{DRA}_{\mathit{fp}}$ relation as in
the realization.
Proof. We assume that the $\mathcal{DRA}_{f}$ relation is determined
correctly. All we need to consider here are the “extended” relations.
We will give the proof for rrrr-, the proof for the other cases is analogous.
Consider two dipoles $d_{A}$ and $d_{B}$ in an rrrr configuration on the rays
$l_{A}$ and $l_{B}$. There are two classes of qualitatively distinguishable
configurations for $(d_{A}\;\textnormal{\rm rrrr}\;d_{B})$:
We can see that $l_{B}$ intersects $l_{A}$ either in front of or behind
$d_{A}$. If the intersection point lies in front of $d_{A}$, we are in a
situation like
where $S$ is the intersection point. We can further see that the angle from
$l_{A}$ to $l_{B}$ lies clearly in the interval $(\pi,2\cdot\pi)$.
Furthermore, $l_{B}$ can be rotated in the whole interval $(\pi,2\cdot\pi)$
without changing the $\mathcal{DRA}_{f}$ relation. Using this, we obtain the
$\mathcal{DRA}_{\mathit{fp}}$-relation rrrr- between $d_{A}$ and $d_{B}$ if
the point of intersection $S$ lies in front of $d_{A}$. For any qcc belonging
to such a scenario, the rest of the proof follows from Prop. 37 and Prop. 38
as well as the inspection of the extended classifiers:
$\displaystyle\hat{S}_{AB}>ed(dp_{A})\wedge dir(dp_{A})=+$
$\displaystyle\longrightarrow$ $\displaystyle-$
$\displaystyle\hat{S}_{AB}<ed(dp_{A})\wedge dir(dp_{A})=-$
$\displaystyle\longrightarrow$ $\displaystyle-$
But these also yield $(dp_{A}\;\textnormal{\rm rrrr-}\;dp_{B})$. By the same
arguments, we show that $(d_{A}\;\textnormal{\rm rrrr+}\;d_{B})$ if the point
of intersection of $l_{A}$ and $l_{B}$ lies behind $d_{A}$. The proof for all
other cases is analogous. ∎
###### Corollary 47.
The 72 relations in Fig. 3 are those out of the 2401 formal combinations of
four $\mathcal{LR}$ letters that are geometrically possible.
Proof. By an exhaustive inspection of the primitive classifiers which occur
in the basic classifiers for all pqccs. For the decomposition, we refer to the
proof of Thm. 45. ∎
###### Theorem 48.
Given a qcc $Q$ and an arbitrary geometric realization $R(Q)$ of it, the
$\mathcal{DRA}_{\mathit{fp}}$ relation in $R(Q)$ is the same as that computed
by the basic classifiers on $Q$.
Proof. Follows from Thm. 45 and Prop. 46. ∎
### 3.6 Implementation of the Classification Procedure
Qualitative composition configurations can be naturally represented as a
finite datatype. The classifiers are implemented as simple programs (mainly
case distinctions) that operate on $qccs$ in the sense of Def. 29. The
classifiers are chosen with respect to the identifier $i$ and the assignment
mapping $a$ of the $qcc$. In our particular implementation, we exploited some
symmetries to limit the number of classifiers that we had to implement.
With the condensed semantics, we are able to compute the composition tables of
the $\mathcal{DRA}$ calculi in an efficient way. In fact we have implemented
the computation of composition tables for both $\mathcal{DRA}_{f}$ and
$\mathcal{DRA}_{\mathit{fp}}$ as Haskell programs, making use of Haskell’s
parallelism extensions. The Haskell implementations of the basic classifiers
for $\mathcal{DRA}_{f}$ and $\mathcal{DRA}_{\mathit{fp}}$ are written in such
a way that they share a library of primitive classifiers. In these programs,
we further generate all qccs in an optimized way, i.e. we only generate the
order $<_{p}$ if it is needed, and classify them with our basic classifiers.
In the end, we compose our results into composition tables. For the case where
three lines are collinear, we simply decided to enumerate all possible
locations of points in a certain interval for reasons of simplicity and this
did not increase the overall runtime too much.
The computation of the composition table for $\mathcal{DRA}_{f}$ takes less
than one minute on a Notebook with an Intel Core 2 T7200 with $1.5$ Gbyte of
RAM, and the computation of the composition table for
$\mathcal{DRA}_{\mathit{fp}}$ takes less than two minutes on the same
computer. This is a great advancement compared to the enumeration of scenarios
on a grid, which took several weeks to compute only an approximation to the
composition table.
### 3.7 Properties of the Composition
We have investigated several properties of the composition tables for
$\mathcal{DRA}_{f}$ and $\mathcal{DRA}_{\mathit{fp}}$. For both tables the
properties
$\displaystyle id^{\smile}$ $\displaystyle=$ $\displaystyle id$
$\displaystyle{\left(R^{\smile}\right)}^{\smile}$ $\displaystyle=$
$\displaystyle R$ $\displaystyle id\circ R$ $\displaystyle=$ $\displaystyle R$
$\displaystyle R\circ id$ $\displaystyle=$ $\displaystyle R$
$\displaystyle{\left(R_{1}\circ R_{2}\right)}^{\smile}$ $\displaystyle=$
$\displaystyle R_{2}^{\smile}\circ R_{1}^{\smile}$ $\displaystyle
R_{1}^{\smile}\in R_{2}\circ R_{3}$ $\displaystyle\iff$ $\displaystyle
R_{3}^{\smile}\in R_{1}\circ R_{2}$
hold with $R$, $R_{1}$, $R_{2}$, $R_{3}$ being any base-relation and $id$ the
identical relation. These properties can be automatically tested by the GQR
and SparQ qualitative reasoners. The other properties for a non-associative
algebra follow trivially. Furthermore, we have tested the associativity of the
composition. For $\mathcal{DRA}_{f}$, we have $373248$ triples of relations to
consider of which $71424$ are not associative. So the composition of $19.14\%$
of all possible triples of relations is not associative202020In the master
thesis of one of our students, a detailed analysis of a specific non-
associative dipole configuration is presented [57], e.g.
$\displaystyle(\textnormal{\rm rrrl};\textnormal{\rm rrrl});\textnormal{\rm
llrl}$ $\displaystyle\neq$ $\displaystyle\textnormal{\rm
rrrl};(\textnormal{\rm rrrl};\textnormal{\rm llrl}).$
For $\mathcal{DRA}_{\mathit{fp}}$ all $512000$ triples of base-relations are
associative w.r.t. composition. With this result, we obtain that
$\mathcal{DRA}_{\mathit{fp}}$ is a relation algebra in a strict sense.
### 3.8 $\mathcal{DRA}_{f}$ composition is weak
The failure of $\mathcal{DRA}_{f}$ to be associative may imply that its
composition is also weak. We will investigate this in this section. First,
recall the definition of strong composition. Furthermore, the composition of
$\mathcal{OPRA}_{1}$ is known to be weak [49], but by Ex. 19 and Prop. 20,
then $\mathcal{DRA}_{f}$ also has a weak composition.
###### Definition 49.
A Qualitative Composition is called _strong_ if, for any arbitrary pair of
objects $A$, $C$ in the domain in relation $Ar_{ac}C$, there is for every
entry in the composition table that contains $Ar_{ac}C$ on the right hand
side, an object $B$ such that $Ar_{ab}B$ and $Br_{bc}C$ reconstruct this
entry.
We will show now that the defining property of strong composition (see Sect.
2.3) is violated for $\mathcal{DRA}_{f}$.
###### Proposition 50.
The composition of $\mathcal{DRA}_{f}$ is weak.
Proof. Consider the $\mathcal{DRA}_{f}$ composition $A\;\textnormal{\rm
BFII}\;B;B\;\textnormal{\rm LLLB}\;C\mapsto A\;\textnormal{\rm LLLL}\;C$. We
show that there are dipoles $A$ and $B$ such that there is no dipole $B$ which
reflects the composition. Consider dipoles $A$ and $B$ as shown in Fig. 25.
Figure 25: $\mathcal{DRA}_{f}$ weak composition
We observe that they are in the $\mathcal{DRA}_{\mathit{fp}}$ relation LLLL-
with the dipole $C$ pointing towards the line dipole $A$ lies on. Because of
$A\;\textnormal{\rm BFII}\;B$, dipole $B$ has to lie on the same line as $A$.
But, since $C$ is a straight line and lines $A$ and $B$ lie in front of $C$,
the endpoint of $B$ cannot lie behind $C$. ∎
As expected, the composition of $\mathcal{DRA}_{f}$ turns out to be weak. Let
us have a closer look at the composition of $\mathcal{DRA}_{\mathit{fp}}$ in
the next section.
### 3.9 Strong Composition
We are now going to prove that $\mathcal{DRA}_{\mathit{fp}}$ has a strong
composition. The following lemma will be crucial; note that it does _not_ hold
for $\mathcal{DRA}_{f}$.
###### Lemma 51.
Let $R$ be a $\mathcal{DRA}_{\mathit{fp}}$ base relation. For
$\mathcal{DRA}_{\mathit{fp}}$ base relations $R$ not involving parallelism or
anti-parallelism, betweenness and equality among $\\{{\bf s}_{A},{\bf
e}_{A},S_{A,B}\\}$212121Please remember that ${\bf s}_{A}=st(dp_{A})$ and
${\bf e}_{A}=ed(dp_{A})$. for given dipoles $A\,R\,B$ are independent of the
choice of $A$ and $B$, hence uniquely determined by $R$ alone.
Proof. Let $R=r_{1}r_{2}r_{3}r_{4}r_{5}$, where $r_{5}\in\\{+,-\\}$ even if
$r_{5}$ this is omitted in the standard notation. Note that the assumption
$r_{5}\in\\{+,-\\}$ implies that $S_{A,B}$ is defined. If
$r_{3}\in\\{b,s,i,e,f\\}$, ${\bf e}_{A}\not={\bf s}_{A}=S_{A,B}$, hence there
is no betweenness. Analogously, ${\bf s}_{A}\not={\bf e}_{A}=S_{A,B}$ if
$r_{4}\in\\{b,s,i,e,f\\}$. The remaining possibilities for $r_{3}r_{4}r_{5}$
are:
1. 1.
ll+, rr-: in these cases, ${\bf e}_{A}$ is between ${\bf s}_{A}$ and
$S_{A,B}$;
2. 2.
ll-, rr+: in these cases, ${\bf s}_{A}$ is between ${\bf e}_{A}$ and
$S_{A,B}$;
3. 3.
rl-, lr+: in these cases, $S_{A,B}$ is between ${\bf s}_{A}$ and ${\bf
e}_{A}$.
Note that cases 1 and 2 cannot be distinguished in $\mathcal{DRA}_{f}$.
∎
###### Corollary 52.
Let $R$ be a $\mathcal{DRA}_{\mathit{fp}}$ base relation not involving
parallelism or anti-parallelism. Let $A\,R\,B$ and
$A^{\prime}\,R\,B^{\prime}$. Then, the map $\\{{\bf s}_{A}\mapsto{\bf
s}_{A^{\prime}};{\bf e}_{A}\mapsto{\bf e}_{A^{\prime}};S_{A,B}\mapsto
S_{A^{\prime},B^{\prime}}\\}$ preserves betweenness and equality.
###### Lemma 53.
Let $R$ be a $\mathcal{DRA}_{\mathit{fp}}$ base relation not involving
parallelism or anti-parallelism. Given dipoles $A\,R\,C$ and
$A^{\prime}\,R\,C^{\prime}$ and points $p_{A}$, $p_{A^{\prime}}$, $p_{C}$ and
$p_{C^{\prime}}$ on the lines carrying $A$, $A^{\prime}$, $C$ and $C^{\prime}$
respectively, if the maps $\\{{\bf s}_{A}\mapsto{\bf s}_{A^{\prime}},{\bf
e}_{A}\mapsto{\bf e}_{A^{\prime}},S_{A,C}\mapsto
S_{A^{\prime},C^{\prime}},p_{A}\mapsto p_{A^{\prime}}\\}$ and $\\{{\rm
s}_{C}\mapsto{\bf s}_{C^{\prime}},{\bf e}_{C}\mapsto{\bf
e}_{C^{\prime}},S_{A,C}\mapsto S_{A^{\prime},C^{\prime}},p_{C}\mapsto
p_{C^{\prime}}\\}$ preserve betweenness and equality, then the angles
$\angle(\overrightarrow{S_{A,C}\leavevmode\nobreak\
p_{A}},\overrightarrow{S_{A,C}\leavevmode\nobreak\ p_{C}})$ and
$\angle(\overrightarrow{S_{A,C}\leavevmode\nobreak\
p_{A}},\overrightarrow{S_{A,C}\leavevmode\nobreak\ p_{C}})$ have the same
sign.
Proof. Since $A\,R\,C$ and $A^{\prime}\,R\,C^{\prime}$, the angles
$\angle(\overrightarrow{{\bf s}_{A}\leavevmode\nobreak\ {\bf
e}_{A}},\overrightarrow{{\rm s}_{C}\leavevmode\nobreak\ {\bf e}_{C}})$ and
$\angle(\overrightarrow{{\bf s}_{A^{\prime}}\leavevmode\nobreak\ {\bf
e}_{A^{\prime}}},\overrightarrow{{\bf s}_{C^{\prime}}\leavevmode\nobreak\ {\bf
e}_{C^{\prime}}})$ have the same sign. By the assumption of the preservation
of betweenness and equality, this carries over to angles
$\angle(\overrightarrow{S_{A,C}\leavevmode\nobreak\
p_{A}},\overrightarrow{S_{A,C}\leavevmode\nobreak\ p_{C}})$ and
$\angle(\overrightarrow{S_{A,C}\leavevmode\nobreak\
p_{A}},\overrightarrow{S_{A,C}\leavevmode\nobreak\ p_{C}})$. ∎
###### Theorem 54.
Composition in $\mathcal{DRA}_{\mathit{fp}}$ is strong.
Proof. Let $r_{ac}\in r_{ab}\circ r_{bc}$ be an entry in the composition
table, with $r_{ac}$, $r_{ab}$ and $r_{bc}$ base relations. Given dipoles $A$
and $C$ with $Ar_{ac}C$, we need to show the existence of a dipole $B$ with
$Ar_{ab}B$ and $Br_{bc}C$.
Since $r_{ac}\in r_{ab}\circ r_{bc}$, we know that there are dipoles
$A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ with $A^{\prime}r_{ab}B^{\prime}$,
$B^{\prime}r_{bc}C^{\prime}$ and $A^{\prime}r_{ac}C^{\prime}$. Given dipoles
$X$ and $Y$, let $S_{X,Y}$ denote the point of intersection of the lines
carrying $X$ and $Y$; it is only defined if $X$ and $Y$ are not parallel.
Consider now the three lines carrying $A^{\prime}$, $B^{\prime}$ and
$C^{\prime}$, respectively. According to the results of Section 3.1, for the
configuration of these three lines, there are fifteen qualitatively different
cases 1, 2, 3a, 3b, 3c, 4a, 4b, 4c, 5a, 5b, 5c, 6a, 6b, 6c and 7:
1. 1.
The three points of intersection $S_{A^{\prime},B^{\prime}}$,
$S_{B^{\prime},C^{\prime}}$ and $S_{A^{\prime},C^{\prime}}$ exist and are
different. Since $Ar_{ac}C$ and $A^{\prime}r_{ac}C^{\prime}$, by Corollary 51,
the point sets $\\{{\bf s}_{A},{\bf e}_{A},S_{A,C}\\}$ and $\\{{\bf
s}_{A^{\prime}},{\bf e}_{A^{\prime}},S_{A^{\prime},C^{\prime}}\\}$ are ordered
in corresponding ways on their lines. Hence, it is possible to choose
$S_{A,B}$ in such a way that the point sets $\\{{\bf s}_{A},{\bf
e}_{A},S_{A,C},S_{A,B}\\}$ and $\\{{\bf s}_{A^{\prime}},{\bf
e}_{A^{\prime}},S_{A^{\prime},C^{\prime}},S_{A^{\prime},B^{\prime}}\\}$ are
ordered in corresponding ways on their lines. In a similar way (interchanging
$A$ and $C$), $S_{B,C}$ can be chosen.
Since both $\\{S_{A,B},S_{A,C},S_{B,C}\\}$ and
$\\{S_{A^{\prime},B^{\prime}},S_{A^{\prime},C^{\prime}},S_{B^{\prime},C^{\prime}}\\}$
are affine bases, there is a unique affine bijection
$h\colon\mathbb{R}^{2}\\!\longrightarrow\\!\mathbb{R}^{2}$ with
$h(S_{A^{\prime},B^{\prime}})=S_{A,B}$, $h(S_{A^{\prime},C^{\prime}})=S_{A,C}$
and $h(S_{B^{\prime},C^{\prime}})=S_{B,C}$. By Lemma 53, $h$ preserves
orientation, and thus by Thm. 28 also the $\mathcal{DRA}_{\mathit{fp}}$
relations. Hence, by choosing $B=h(B^{\prime})$, we get $h(A^{\prime})r_{ab}B$
and $Br_{bc}h(C^{\prime})$. Since the sets $\\{{\bf s}_{A},{\bf
e}_{A},S_{A,C},S_{A,B}\\}$ and $\\{h({\bf s}_{A^{\prime}}),h({\bf
e}_{A^{\prime}}),S_{A,C},S_{A,B}\\}$ are on the same line and have
corresponding qualitative (betweenness) relations, and the same holds for the
sets $\\{{\bf s}_{C},{\bf e}_{C},S_{A,C},S_{B,C}\\}$ and $\\{h({\bf
s}_{C^{\prime}}),h({\bf e}_{C^{\prime}}),S_{A,C},S_{B,C}\\}$, we also have
$Ar_{ab}B$ and $Br_{bc}C$ (even though $h(A^{\prime})=A$ and $h(C^{\prime})=C$
do not necessarily hold).
2. 2.
The three intersection points $S_{A^{\prime},B^{\prime}}$,
$S_{B^{\prime},C^{\prime}}$ and $S_{A^{\prime},C^{\prime}}$ exist and
coincide, i.e.
$S_{A^{\prime},B^{\prime}}=S_{B^{\prime},C^{\prime}}=S_{A^{\prime},C^{\prime}}=:S^{\prime}$.
Let $S=S_{A,C}$. Let $x_{A}$ be ${\bf s}_{A}$ and $x_{A^{\prime}}$ be ${\bf
s}_{A^{\prime}}$ if ${\bf s}_{A}\not=S$ (and therefore ${\bf
s}_{A^{\prime}}\not=S^{\prime}$), otherwise, let $x_{A}$ be ${\bf e}_{A}$ and
$x_{A^{\prime}}$ be ${\bf e}_{A^{\prime}}$. $x_{C}$ and $x_{C^{\prime}}$ are
chosen in a similar way. Since both $\\{S,x_{A},x_{C}\\}$ and
$\\{S^{\prime},x_{A^{\prime}},x_{C^{\prime}}\\}$ are affine bases, there is a
unique affine bijection
$h\colon\mathbb{R}^{2}\\!\longrightarrow\\!\mathbb{R}^{2}$ with
$h(S^{\prime})=S$, $h(x_{A^{\prime}})=x_{A}$ and $h(x_{C^{\prime}})=x_{C}$.
The rest of the argument is similar to case (1).
3. 3.
(Two lines are parallel and intersect with the third one.) In the sequel, we
will just specify how two affine bases are chosen; the rest of the argument
(as well as the choice of points on the unprimed side in such a way that
qualitative relations are preserved) is then similar to the previous cases.
Subcases (3a), (3b): The lines carrying $A$ and $C$ intersect. Choose $x_{A}$
and $x_{A^{\prime}}$ as in case (2), and chose an appropriate point $S_{B,C}$.
Then use the affine bases $\\{x_{A},S_{A,C},S_{B,C}\\}$ and
$\\{x_{A^{\prime}},S_{A^{\prime},C^{\prime}},S_{B^{\prime},C^{\prime}}\\}$.
Subcase (3c): The lines carrying $A$ and $C$ are parallel. Choose appropriate
points $S_{A,B}$ and $S_{B,C}$ and use the affine bases $\\{{\bf
s}_{A},S_{A,B},S_{B,C}\\}$ and $\\{{\bf
s}_{A^{\prime}},S_{A^{\prime},B^{\prime}},S_{B^{\prime},C^{\prime}}\\}$.
4. 4.
(Two lines are identical and intersect with the third one.)
Subcases (4a) and (4b): The lines carrying $A$ and $C$ intersect. Choose
$x_{A}$, $x_{A^{\prime}}$, $x_{C}$ and $x_{C^{\prime}}$ as in case (2) and use
the affine bases $\\{S_{A,C},x_{A},x_{C}\\}$ and
$\\{S_{A^{\prime},C^{\prime}},x_{A^{\prime}},x_{C^{\prime}}\\}$.
Subcase (4c): The lines carrying $A$ and $C$ are identical. This means that
$S_{A^{\prime},B^{\prime}}=S_{A^{\prime},C^{\prime}}=:S^{\prime}$. Choose an
appropriate point $S$ and $x_{A}$, $x_{A^{\prime}}$ as in case (2). Moreover,
in a similar way, choose $x_{B^{\prime}}\not=S^{\prime}$, and then some
corresponding $x_{B}$ being in the same $\mathcal{LR}$-relation to $A$ as
$x_{B^{\prime}}$ has to $A^{\prime}$. Then use the affine bases
$\\{S,x_{A},x_{B}\\}$ and $\\{S,x_{A^{\prime}},x_{B^{\prime}}\\}$.
5. 5.
(All three lines are distinct and parallel.) Subcases (5a), (5b) and (5c) can
all be treated in the same way: Use the affine bases $\\{{\bf s}_{A},{\bf
e}_{A},{\bf s}_{C}\\}$ and $\\{{\bf s}_{A^{\prime}},{\bf e}_{A^{\prime}},{\bf
s}_{C^{\prime}}\\}$. Note that the distance ratio does not matter here.
6. 6.
(Two lines are identical and are parallel to the third one.)
Subcases (6a) and (6b): The lines carrying $A$ and $C$ are parallel. Proceed
as in case (5).
Subcase (6c): The lines carrying $A$ and $C$ are identical. Choose some ${\bf
s}_{B}$ in the same $\mathcal{LR}$-relation to $A$ as ${\bf s}_{B^{\prime}}$
is to $A^{\prime}$. Then use the affine bases $\\{{\bf s}_{A},{\bf e}_{A},{\bf
s}_{B}\\}$ and $\\{{\bf s}_{A^{\prime}},{\bf e}_{A^{\prime}},{\bf
s}_{B^{\prime}}\\}$.
7. 7.
(All three lines are identical.) For this case, the result follows from the
fact that Allen’s interval algebra has strong composition (refer to [26]).
∎
###### Corollary 55.
Composition in $\mathcal{DRA}_{\mathit{opp}}$ is strong as well.
Proof. By Example 19 and Prop. 20. ∎
## 4 Constraint Reasoning with the Dipole Calculus
### 4.1 Consistency
We now consider the question of whether algebraic closure decides consistency.
We call the set of constraints between all dipoles at hand a _constraint
network_. If no constraint between two dipoles is given, we agree that they
are in the universal relation. By _scenario_ , we denote a constraint network
in which all constraints are base-relations222222In this case, a base-relation
between every pair of distinct dipoles has to be provided. We construct
constraint-networks which are geometrically unrealizable but still
algebraically closed. We do this by constructing constraint networks that are
consistent and algebraically closed, and then we will change a relation in
them in such a way that they remain algebraically closed but become
inconsistent. We follow the approach of [58] in using a simple geometric shape
for which scenarios exist, where algebraic closure fails to decide
consistency. In our case, the basic shape is a convex hexagon, similar to a
screw head.
Consider a convex hexagon consisting of the dipoles $A$, $B$, $C$, $D$, $E$
and $F$. Such an object is described as
$(A\;\textnormal{\rm errs}\;B)(B\;\textnormal{\rm errs}\;C)(C\;\textnormal{\rm
errs}\;D)(D\;\textnormal{\rm errs}\;E)(E\;\textnormal{\rm
errs}\;F)(F\;\textnormal{\rm errs}\;A)$
where the components $r$ of the relations ensure convexity, since they enforce
an angle between $0$ and $\pi$ between the respective first and second dipole,
i.e., the endpoint of consecutive dipoles always lies to the right of the
preceding dipole. Such an object is given in Fig. 26
Figure 26: Convex hexagon
To this scenario we add a seventh dipole $G$ with the relations
$(G\;\textnormal{\rm rrll}\;A)(G\;\textnormal{\rm lrll}\;F)(G\;\textnormal{\rm
llrr}\;D)(G\;\textnormal{\rm rlrr}\;C)$
We have the overall constraint network:
$\begin{array}[]{l}(A\;\textnormal{\rm errs}\;B)(B\;\textnormal{\rm
errs}\;C)(C\;\textnormal{\rm errs}\;D)(D\;\textnormal{\rm
errs}\;E)(E\;\textnormal{\rm errs}\;F)(F\;\textnormal{\rm errs}\;A)\\\
(G\;\textnormal{\rm rrll}\;A)(G\;\textnormal{\rm lrll}\;F)(G\;\textnormal{\rm
llrr}\;D)(G\;\textnormal{\rm rlrr}\;C)\end{array}$
Because of the relations $(G\;\textnormal{\rm lrll}\;F)$ and
$(G\;\textnormal{\rm rlrr}\;C)$, line $l_{G}$ intersects line $l_{F}$ as well
as line $l_{C}$. Because of the first two components of the relations, dipoles
$F$ and $C$ are oriented into qualitatively antipodal directions. This network
is consistent and is of course algebraically closed.
To construct an inconsistent network, we change the relation
$(G\;\textnormal{\rm rlrr}\;C)$ to $(G\;\textnormal{\rm rlll}\;C)$ and obtain
the constraint network:
$\begin{array}[]{l}(A\;\textnormal{\rm errs}\;B)(B\;\textnormal{\rm
errs}\;C)(C\;\textnormal{\rm errs}\;D)(D\;\textnormal{\rm
errs}\;E)(E\;\textnormal{\rm errs}\;F)(F\;\textnormal{\rm errs}\;A)\\\
(G\;\textnormal{\rm rrll}\;A)(G\;\textnormal{\rm lrll}\;F)(G\;\textnormal{\rm
llrr}\;D)(G\;\textnormal{\rm rlll}\;C)\end{array}$
The relations $(G\;\textnormal{\rm rlll}\;C)$ and $(G\;\textnormal{\rm
lrll}\;F)$ enforce that $G$ must lie in between $F$ and $C$ as shown in Fig.
27.
Figure 27: Position of $G$
In this case, the all convex hexagons $A$, $B$, $C$, $D$, $E$, $F$ have the
endpoints of consecutive dipoles lying to the left of the preceding one, they
are of the form:
$(A\;\textnormal{\rm ells}\;B)(B\;\textnormal{\rm ells}\;C)(C\;\textnormal{\rm
ells}\;D)(D\;\textnormal{\rm ells}\;E)(E\;\textnormal{\rm
ells}\;F)(F\;\textnormal{\rm ells}\;A)$
which is a contradiction of the required form of hexagon in the scenario. In
fact there is no affine transformation which preserves the relative
orientations between dipoles $A$, $B$, $C$, $D$, $E$, $F$, and maps a hexagon
of Fig. 26 to any that can be constructed along dipoles $C$ and $F$ in Fig. 27
in such a way that the edges $C$ and $F$ of both hexagons coincide. Still
algebraic closure with $\mathcal{DRA}_{f}$ yields the refinement:
$\begin{array}[]{l}(F\;\textnormal{\rm lllr}\;G)(E\,\,(\textnormal{\rm
flll},\textnormal{\rm llll},\textnormal{\rm rfll},\textnormal{\rm
rlll},\textnormal{\rm rrll})\,\,G)(D\;\textnormal{\rm
errs}\;E)(D\;\textnormal{\rm rrll}\;G)\\\ (D\,\,(\textnormal{\rm
rbrr},\textnormal{\rm rllr},\textnormal{\rm rlrr},\textnormal{\rm
rrrr})\,\,F)(C\;\textnormal{\rm llrl}\;G)(C\,\,(\textnormal{\rm
lrrl},\textnormal{\rm rllr})\,\,F)(E\;\textnormal{\rm errs}\;F)\\\
(C\,\,(\textnormal{\rm rllr},\textnormal{\rm rrfr},\textnormal{\rm
rrlr},\textnormal{\rm rrrr})\,\,E)(C\,\,\;\textnormal{\rm
errs}\;\,\,D)(B\,\,(\textnormal{\rm llrr},\textnormal{\rm rrrr})\,\,G)\\\
(B\,\,(\textnormal{\rm blrr},\textnormal{\rm llll},\textnormal{\rm
llrf},\textnormal{\rm llrl},\textnormal{\rm llrr},\textnormal{\rm
rfll},\textnormal{\rm rlll},\textnormal{\rm rlrr, rrbl},\textnormal{\rm
rrll},\textnormal{\rm rrrl},\textnormal{\rm rrrr})\,\,E)\\\
(B\,\,(\textnormal{\rm rbrr},\textnormal{\rm rlrr},\textnormal{\rm
rrfr},\textnormal{\rm rrlr},\textnormal{\rm rrrr})\,\,D)(B\;\textnormal{\rm
errs}\;C)(A\;\textnormal{\rm llrr}\;G)(A\;\textnormal{\rm rser}\;F)\\\
(A\,\,(\textnormal{\rm frrr},\textnormal{\rm lrrr},\textnormal{\rm
rrrb},\textnormal{\rm rrrl},\textnormal{\rm
rrrr)\,\,E)(A\,\,(rllr},\textnormal{\rm rlrr},\textnormal{\rm rrrr})\,\,C)\\\
(A\,\,(\textnormal{\rm lfrr},\textnormal{\rm llbr},\textnormal{\rm
llll},\textnormal{\rm lllr},\textnormal{\rm llrr},\textnormal{\rm
lrll},\textnormal{\rm lrrr},\textnormal{\rm rrlf},\textnormal{\rm
rrll},\textnormal{\rm rrlr},\textnormal{\rm rrrr})\,\,D)\\\
(B\,\,(\textnormal{\rm frrr},\textnormal{\rm lrrl},\textnormal{\rm
lrrr},\textnormal{\rm rrrr})\,\,F)(A\;\textnormal{\rm errs}\;B)\end{array}$
A scenario,
$\begin{array}[]{l}(F\;\textnormal{\rm lllr}\;G)(E\;\textnormal{\rm
flll}\;G)(E\;\textnormal{\rm errs}\;F)(D\;\textnormal{\rm
rrll}\;G)(D\;\textnormal{\rm rrrr}\;F)(D\;\textnormal{\rm errs}\;E)\\\
(C\;\textnormal{\rm llrl}\;G)(C\;\textnormal{\rm rllr}\;F)(C\;\textnormal{\rm
rrlr}\;E)(C\;\textnormal{\rm errs}\;D)(B\;\textnormal{\rm
llrr}\;G)(B\;\textnormal{\rm lrrl}\;F)\\\ (B\;\textnormal{\rm
llrl}\;E)(B\;\textnormal{\rm rlrr}\;D)(B\;\textnormal{\rm
errs}\;C)(A\;\textnormal{\rm llrr}\;G)(A\;\textnormal{\rm
rser}\;F)(A\;\textnormal{\rm rrrr}\;E)\\\ (A\;\textnormal{\rm
lrrr}\;D)(A\;\textnormal{\rm rllr}\;C)(A\;\textnormal{\rm
errs}\;B)\end{array}$
can be derived from this algebraically closed network. It is still deemed
algebraically closed, even though it is not consistent with the same argument
given above. Hence algebraic closure does not decide consistency for
$\mathcal{DRA}_{f}$-scenarios. On the other hand, algebraic closure with
$\mathcal{DRA}_{\mathit{fp}}$ detects all possible extensions of this network
to that calculus as being inconsistent. Extending the consistent case with the
relation $(G\;\textnormal{\rm rlrr}\;C)$ yields three possible extensions for
$\mathcal{DRA}_{\mathit{fp}}$ scenarios, of which all are consistent. In fact,
we get the three following consistent refinements.
$\begin{array}[]{c|c|c|c}DRA_{f}\textnormal{-relation}&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\textnormal{refinement}1\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\textnormal{refinement}2\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\textnormal{refinement}3\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\\\
\hline\cr(G\;\textnormal{\rm rrll}\;A)&(G\;\textnormal{\rm
rrll-}\;A)&(G\;\textnormal{\rm rrll-}\;A)&(G\;\textnormal{\rm rrll-}\;A)\\\
\hline\cr(G\;\textnormal{\rm llrr}\;D)&(G\;\textnormal{\rm
llrr-}\;D)&(G\;\textnormal{\rm llrr+}\;D)&(G\;\textnormal{\rm
llrrP}\;D)\end{array}$
We have found an example that shows that algebraic closure for
$\mathcal{DRA}_{\mathit{fp}}$ finds inconsistencies in constraint networks
where it fails for $\mathcal{DRA}_{f}$. Does algebraic closure for
$\mathcal{DRA}_{\mathit{fp}}$ decide consistency? We can also give a negative
result for this. To construct a counterexample, we begin with a configuration
as in Fig. 28
Figure 28: Construction of the counterexample
We ensure with the constraints
$(A\;\textnormal{\rm errs}\;B)(B\;\textnormal{\rm errs}\;C)(C\;\textnormal{\rm
errs}\;D)(D\;\textnormal{\rm errs}\;E)(E\;\textnormal{\rm
errs}\;F)(F\;\textnormal{\rm errs}\;A)$
that the dipoles $A$, $B$, $C$, $D$, $E$ and $F$ form a convex hexagon.
Furthermore, we ensure that the dipoles $I$, $H$ and $G$ form a continuous
line by
$(I\;\textnormal{\rm efbs}\;H)(H\;\textnormal{\rm efbs}\;G).$
The constraints
$(F\;\textnormal{\rm rrrl}\;I)(C\;\textnormal{\rm rrlr}\;G)$
state that the line has to lie inside the hexagon, since its start point and
end point lie inside. To construct the counterexample, we just claim that the
end point of $H$ lies outside the hexagon by $(A\;\textnormal{\rm rllr}\;H)$,
i.e. the lines $A$ and $H$ intersect, this is a contradiction of the convexity
of the hexagon. This network can be refined to a scenario
$\displaystyle\mathtt{SCEN}$ $\displaystyle:=$
$\displaystyle(H\;\textnormal{\rm efbs}\;G)(I\;\textnormal{\rm
ffbb}\;G)(I\;\textnormal{\rm efbs}\;H)(F\;\textnormal{\rm
rrrl}\;G)(F\;\textnormal{\rm rrrl}\;H)$ $\displaystyle(F\;\textnormal{\rm
rrrl}\;I)(E\;\textnormal{\rm rrrr-}\;G)(E\;\textnormal{\rm
rrrr-}\;H)(E\;\textnormal{\rm rrrr-}\;I)$ $\displaystyle(E\;\textnormal{\rm
errs}\;F)(D\;\textnormal{\rm rrrrA}\;G)(D\;\textnormal{\rm
rrrrA}\;H)(D\;\textnormal{\rm rrrrA}\;I)$ $\displaystyle(D\;\textnormal{\rm
rrrr-}\;F)(D\;\textnormal{\rm errs}\;E)(C\;\textnormal{\rm
rrlr}\;G)(C\;\textnormal{\rm rrlr}\;H)$ $\displaystyle(C\;\textnormal{\rm
rrlr}\;I)(C\;\textnormal{\rm rrlr}\;F)(C\;\textnormal{\rm
rrrr+}\;E)(C\;\textnormal{\rm errs}\;D)$ $\displaystyle(B\;\textnormal{\rm
rrrl}\;G)(B\;\textnormal{\rm rrrl}\;H)(B\;\textnormal{\rm
rrrl}\;I)(B\;\textnormal{\rm lrrl}\;F)$ $\displaystyle(B\;\textnormal{\rm
llrr-}\;E)(B\;\textnormal{\rm rlrr}\;D)(B\;\textnormal{\rm
errs}\;C)(A\;\textnormal{\rm lllr}\;G)$ $\displaystyle(A\;\textnormal{\rm
rllr}\;H)(A\;\textnormal{\rm rrlr}\;I)(F\;\textnormal{\rm
errs}\;A)(A\;\textnormal{\rm rrrrA}\;E)$ $\displaystyle(A\;\textnormal{\rm
lrrr}\;D)(A\;\textnormal{\rm rlrr}\;C)(A\;\textnormal{\rm errs}\;B).$
which is still algebraically closed w.r.t. $\mathcal{DRA}_{\mathit{fp}}$, even
though it is not consistent. We see that algebraic-closure does not decide
consistency even for $\mathcal{DRA}_{\mathit{fp}}$-scenarios.
We have run several tests to get some quantitative information on how much
better the $\mathcal{DRA}_{\mathit{fp}}$ calculus performs with respect to the
$\mathcal{DRA}_{f}$ calculus. We have generated several scenarios of size
$\leq n$ with $n\in\left\\{30,40,50,60,70\right\\}$ randomly to obtain this
information. It turns out that a number of $10^{\frac{n}{10}+1}$ scenarios
yield usable data. In fact, we have generated $\mathcal{DRA}_{\mathit{fp}}$
scenarios and checked them with an algebraic reasoner, then we have projected
them to $\mathcal{DRA}_{f}$ and checked these with the same reasoner. In the
end, we compared the per-scenario results. The results are as follows:
Scenarios | $10000$ | $100000$ | $1000000$ | $10000000$ | $100000000$
---|---|---|---|---|---
Maximum Size | $30$ | $40$ | $50$ | $60$ | $70$
Algebraically Closed | $691$ | $5295$ | $40820$ | $346164$ | $3048063$
A-closed w.r.t. $\mathcal{DRA}_{f}$ only | $11$ | $149$ | $1061$ | $8839$ | $78792$
A-closed w.r.t. $\mathcal{DRA}_{\mathit{fp}}$ only | $0$ | $0$ | $0$ | $0$ | $0$
Roughly $2.5\%$ of the scenarios that are algebraically closed w.r.t. to
$\mathcal{DRA}_{f}$ are not algebraically closed w.r.t.
$\mathcal{DRA}_{\mathit{fp}}$. Still, for the smallest checked maximum
scenario size $30$ the factor is only $1.5\%$.
We also investigate the question if algebraic closure decides consistency for
$\mathcal{DRA}_{\mathit{op}}$ and $\mathcal{DRA}_{\mathit{opp}}$.
Figure 29: $\mathcal{DRA}_{\mathit{opp}}$ scenario
###### Proposition 56.
For $\mathcal{DRA}_{\mathit{opp}}$ algebraic closure does not decide
consistency.
Proof. This proof is inspired by the one that shows that algebraic closure
does not decide consistency for $\mathcal{OPRA}$ (ref. to [49]). Consider a
$\mathcal{DRA}_{\mathit{opp}}$ constraint network in three points $A$, $B$ and
$C$ as shown in Fig. 29. Both $A$ and $B$ point at $C$. These three points are
in the relations:
$A\;\textnormal{\rm LEFTright-}\;B\quad\quad A\;\textnormal{\rm
FRONTleft}\;C\quad\quad B\;\textnormal{\rm FRONTleft}\;C.$
We add a point $D$ to our constraint satisfaction problem with
$C\;\textnormal{\rm RIGHTleftP}\;D$. We claim that $D$ also lies in front of
$A$ and $B$ by introducing the constraints $A\;\textnormal{\rm FRONTleft}\;D$
and $B\;\textnormal{\rm FRONTleft}\;D$. By inspecting the composition table of
$\mathcal{DRA}_{\mathit{opp}}$, we can see that it is consistent. Since by the
constraint $A\;\textnormal{\rm LEFTright-}\;B$ the points $A$ and $B$ are not
collinear, $D$ has to lie on the intersection point of the rays $l_{A}$ and
$l_{B}$, but by $A\;\textnormal{\rm FRONTleft}\;C$ and $B\;\textnormal{\rm
FRONTleft}\;C$, $C$ also has to lie on that intersection point. Hence, $C$ and
$D$ have to have the same position, what is a contradiction to the constraint
$C\;\textnormal{\rm RIGHTleftP}\;D$. Hence this scenario is algebraically
closed, but inconsistent. ∎
###### Proposition 57.
For $\mathcal{DRA}_{\mathit{op}}$ algebraic closure does not decide
consistency.
Proof. This proof is analogous to the one of Prop. 56, with substituting
LEFTright- by LEFTright and RIGHTleftP by RIGHTleft. ∎
## 5 A Sample Application of the Dipole Calculus
Figure 30: A street network and two local observations
In this section, we want to demonstrate with an example how spatial knowledge
expressed in $\mathcal{DRA}_{\mathit{fp}}$ can be used for deductive reasoning
based on constraint propagation (algebraic closure), resulting in the
generation of useful indirect knowledge from partial observations in a spatial
scenario. In our sample application, a spatial agent (a simulated robot,
cognitive simulation of a biological system etc.) explores a spatial scenario.
The agent collects local observations and wants to generate survey knowledge.
Fig. 30 shows our spatial environment. It consists of a street network in
which some streets continue straight after a crossing and some streets run
parallel. These features are typical of real-world street networks. Spatial
reasoning in our example uses constraint propagation (e.g. algebraic closure
computation) to derive indirect constraints between the relative location of
streets which are further apart from local observations between neighboring
streets. The resulting survey knowledge can be used for several tasks
including navigation tasks.
The environment is represented as streets $s_{i}$ and crossings $C_{j}$. The
streets and crossings have unique names (e.g. $s_{1}$, … , $s_{12}$, and
$C_{1}$, …, $C_{9}$ in one concrete example). The local observations are
modeled in the following way, based on specific visibility rules (we want to
simulate prototypical features of visual perception): Both at each crossing
and at each straight street segment we have an observation. At each crossing
the agent observes the neighboring crossings. At the middle of each straight
street segment the agent can observe the direction of the outgoing streets at
the adjacent crossings (but not at their other ends). Two specific examples of
observations are marked in Fig. 30. The observation ”s1 errs s7” is marked
green at crossing C1. The observation ”s8 rrllP s9” is marked red at street
s4.
These observations relate spatially neighboring streets to each other in a
pairwise manner, using $\mathcal{DRA}_{\mathit{fp}}$ base relations. The agent
has no additional knowledge about the specific environment. The spatial world
knowledge of the agent is expressed in the converse and composition tables of
$\mathcal{DRA}_{\mathit{fp}}$ .
The following sequence of partial observations could be the result of a tour
made by the spatial agent, exploring the street network of our example (see
Fig. 30):
Figure 31: All observation and resulting uncertainty
---
Observations at crossings
C1: | (s7 errs s1)
C2: | (s1 efbs s2) (s8 errs s2) (s1 rele s8)
C3: | (s2 rele s9)
C4: | (s10 efbs s7) (s10 errs s3) (s7 srsl s3)
C5: | (s3 efbs s4) (s11 efbs s8) (s11 errs s4) (s3 ells s8)
| (s3 rele s11) (s8 srsl s4)
C6: | (s12 efbs s9) (s4 ells s9) (s4 rele s12)
C7: | (s10 srsl s5)
C8: | (s5 efbs s6) (s5 ells s11) (s11 srsl s6)
C9: | (s6 ells s12)
Observations at streets
s1: | (s7 rrllP s8)
s2: | (s8 rrllP s9)
s3: | (s10 rrllP s11)
s4: | (s11 rrllP s12)
s8: | (s3 llrr- s1)
s9: | (s4 llrr- s2)
s10: | (s3 rrll- s5)
s11: | (s4 rrll- s6)
The result of the algebraic closure computation/constraint propagation is a
refined network with the same solution set (the results are computed with the
publicly available SparQ reasoning tool supplied with our newly computed
$\mathcal{DRA}_{\mathit{fp}}$ composition table [50]). We have listed the
results in the appendix. Three different models are the only remaining
consistent interpretations (see the appendix for a list of all the resulting
data). The three different models agree on all but four relations. The
solution set can be explained with the help of the diagram in Fig. 31. The
input crossing observations are marked with green arrows, the input street
observations are marked with red arrows. The result shows that for all street
pairs which could not be observed directly, the algebraic closure algorithm
deduces a strong constraint/precise information. Typically, the resulting
spatial relation between street pairs comprises just one
$\mathcal{DRA}_{\mathit{fp}}$ base relation. The exception consists of four
relations between streets in which the three models differ (marked with dashed
blue arrows in Fig. 31). For these four relations each model from the solution
set agrees on the same $\mathcal{DRA}_{f}$ base relation for a given relation,
but the three consistent models differ on the finer granularity level of
$\mathcal{DRA}_{\mathit{fp}}$ base relations. Since the refinement of one of
these four underspecified relations on a single interpretation
($\mathcal{DRA}_{\mathit{fp}}$ base relation) as a logical consequence also
assigns a single base relation to the other three relations, only three
interpretations are valid models. The uncertainty/indeterminacy is the result
of the specific street configuration in our example. The streets in a North-
South direction are parallel, but the streets in an East-West direction are
not parallel resulting in fewer constraint composition results. However, the
small solution set of consistent models agrees on most of the relative
position relations between street pairs and the differences between models are
small. In our judgement, this means that the system has generated the relevant
survey knowledge about the whole street network from local observations alone.
## 6 Summary and Conclusion
We have presented different variants of qualitative spatial reasoning calculi
about oriented straight line segments which we call dipoles. We have derived
calculi for oriented points from dipole calculi, which turned out to be
isomorphic to some versions of the $\mathcal{OPRA}$ calculi. These spatial
calculi provide a basis for representing and reasoning about qualitative
position information in intrinsic reference systems.
We have computed the composition table for dipole calculi by a new method
based on the algebraic semantics of the dipole relations. We have used a so-
called condensed semantics which uses the orbits of the affine group
$\mathbf{GA}(\mathbb{R}^{2})$ to provide an abstract symbolic notion of
qualitative composition configuration. This can be used to compute the
composition table in a computer-assisted way. The correctness of this
computation is ensured by letting the computer program directly operate with
qualitative composition configurations.
This has been the first computation of the composition table for
$\mathcal{DRA}_{\mathit{fp}}$. So far, only composition tables for
$\mathcal{DRA}_{c}$ and $\mathcal{DRA}_{f}$ exist, which contain many errors
[59]. We have analysed the algebraic features of the various dipole calculi.
We have proved the result that $\mathcal{DRA}_{\mathit{fp}}$ has strong
composition. This is an interesting result, because in this case an
application-motivated calculus extension has been found to also have a certain
mathematical elegance. Moreover, the strength of composition carries over to
$\mathcal{DRA}_{\mathit{opp}}$, the $\mathcal{OPRA}$ variant introduced in
this paper. This transfer of properties from one calculus to another calculus
is an important new general result on quotients of qualitative calculi. To our
knowledge, also the notion of quotient of a qualitative calculus (defined
using methods from universal algebra) appears for the first time in this
paper.
We have demonstrated a prototypical application of reasoning about qualitative
position information in relative reference systems. In this scenario about
cognitive spatial agents and qualitative map building, coarse locally
perceived street configuration information has to be integrated by constraint
propagation in order to get survey knowledge. The well-known path-consistency
method which is implemented with standard QSR tools can make use of our new
dipole calculus composition table and compute the desired result in polynomial
time. Such concrete but generalizable application scenarios for relative
position calculi are the more important since a recent result by Wolter and
Lee [27] showed that relative position calculi are intractable even in base
relations. For this reason, it is necessary to gain experience in which
application contexts the unavoidable approximate reasoning is effective and
produces relevant inference results. With our street network example, we have
a test case which puts emphasis on deriving implicit knowledge as the output
of qualitative spatial reasoning based on observed data. This is a
prototypical application scenario which in the future can also be applied to
other relative position calculi.
Since the observed data in the case of error-free perception leads to
consistent input constraints, the general consistency problem can be avoided –
we instead rely on logical consequence. Now both problems are intractable and
need to be approximated using algebraic closure; however, in our setting,
approximate losses are less harmful, since we do not risk working with
inconsistent scenarios.
Our future work will address the question of how in general the quality of
approximations for relative position reasoning can also be assessed with
quantitative measures. Another part of our future QSR research will apply our
new condensed semantics method to other calculi.
## Acknowledgment
The authors would like to thank Diedrich Wolter, Jochen Renz, Frank Dylla,
Christian Freksa, Franz Kalhoff, Stefan Wölfl, Lutz Schröder, and Brandon
Bennett for interesting and helpful discussions related to the topic of the
paper. Our work was supported by the DFG Transregional Collaborative Research
Center SFB/TR 8 “Spatial Cognition”.
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## Appendix: Computation for the street network application with the SparQ
tool
In this appendix, we demonstrate how to use the publicly available SparQ QSR
toolbox [50] to compute the algebraic closure by constraint propagation for
the street network example from Section 5. For successful relative position
reasoning, the SparQ tool has to be supplied with our newly computed
$\mathcal{DRA}_{\mathit{fp}}$ composition table [50].
The local street configuration observations by the spatial agent are listed in
Section 5. The direct translation of these logical propositions into a SparQ
spatial reasoning command looks as follows232323For technical details of SparQ
we refer the reader to the SparQ manual [50]:
sparq constraint-reasoning dra-fp path-consistency "( (s7 errs s1) (s1 efbs
s2) (s8 errs s2) (s1 rele s8) (s2 rele s9) (s10 efbs s7) (s10 errs s3) (s7
srsl s3) (s3 efbs s4) (s11 efbs s8) (s11 errs s4) (s3 ells s8) (s3 rele s11)
(s8 srsl s4) (s12 efbs s9) (s4 ells s9) (s4 rele s12) (s10 srsl s5) (s5 efbs
s6) (s5 ells s11) (s11 srsl s6) (s6 ells s12) (s7 rrllP s8) (s8 rrllP s9) (s10
rrllP s11) (s11 rrllP s12) (s3 llrr- s1) (s4 llrr- s2) (s3 rrll- s5) (s4 rrll-
s6) )"
242424SparQ refers to $\mathcal{DRA}_{\mathit{fp}}$ with the symbol dra-80.
SparQ does not accept line breaks which we have inserted here for better
readability. All the data for this sample application including the new
composition table can be obtained from the URL http://www.informatik.uni-
bremen.de/~till/Oslsa.tar.gz (which also provides the composition table and
other data for the GQR reasoning tool https://sfbtr8.informatik.uni-
freiburg.de/R4LogoSpace/Resources/).
The result of this reasoning command is a refined network with the same
solution set derived by the application of the algebraic closure/constraint
propagation algorithm (see Section 2.3).
Modified network.
((S5 (EFBS) S6)(S12 (LSEL) S6)(S12 (LLFL) S5)(S11 (SRSL) S6)(S11 (LSEL)
S5)(S11 (RRLLP) S12) (S4 (RRLL-) S6)(S4 (RRLL-) S5)(S4 (RELE) S12)(S4 (RSER)
S11)(S3 (RRLL-) S6)(S3 (RRLL-) S5) (S3 (RFLL) S12)(S3 (RELE) S11)(S3 (EFBS)
S4)(S10 (RRBL) S6)(S10 (SRSL) S5)(S10 (RRLLP) S12) (S10 (RRLLP) S11)(S10
(RRRB) S4)(S10 (ERRS) S3)(S9 (LBLL) S6)(S9 (LLLL-) S5)(S9 (BSEF) S12) (S9
(LLRRP) S11)(S9 (LSEL) S4)(S9 (LLFL) S3)(S9 (LLRRP) S10)(S8 (BRLL) S6)(S8
(LBLL) S5) (S8 (RRLLP) S12)(S8 (BSEF) S11)(S8 (SRSL) S4)(S8 (LSEL) S3)(S8
(LLRRP) S10)(S8 (RRLLP) S9) (S2 (RRLL+ RRLL- RRLLP) S6)(S2 (RRLL+ RRLL- RRLLP)
S5)(S2 (RRLF) S12)(S2 (RRFR) S11)(S2 (RRLL+) S4) (S2 (RRLL+) S3)(S2 (RRRR+)
S10)(S2 (RELE) S9)(S2 (RSER) S8)(S1 (RRLL+ RRLL- RRLLP) S6) (S1 (RRLL+ RRLL-
RRLLP) S5)(S1 (RRLL+) S12)(S1 (RRLF) S11)(S1 (RRLL+) S4)(S1 (RRLL+) S3) (S1
(RRFR) S10)(S1 (RFLL) S9)(S1 (RELE) S8)(S1 (EFBS) S2)(S7 (RRLL-) S6)(S7 (BRLL)
S5) (S7 (RRLLP) S12)(S7 (RRLLP) S11)(S7 (RRBL) S4)(S7 (SRSL) S3)(S7 (BSEF)
S10)(S7 (RRLLP) S9) (S7 (RRLLP) S8)(S7 (RRRB) S2)(S7 (ERRS) S1))
SparQ can output all path-consistent scenarios (i.e. constraint networks in
base relations) via the command:
sparq constraint-reasoning dra-fp scenario-consistency all "( (s7 errs s1) (s1
efbs s2) (s8 errs s2) (s1 rele s8) (s2 rele s9) (s10 efbs s7) (s10 errs s3)
(s7 srsl s3) (s3 efbs s4) (s11 efbs s8) (s11 errs s4) (s3 ells s8) (s3 rele
s11) (s8 srsl s4) (s12 efbs s9) (s4 ells s9) (s4 rele s12) (s10 srsl s5) (s5
efbs s6) (s5 ells s11) (s11 srsl s6) (s6 ells s12) (s7 rrllP s8) (s8 rrllP s9)
(s10 rrllP s11) (s11 rrllP s12) (s3 llrr- s1) (s4 llrr- s2) (s3 rrll- s5) (s4
rrll- s6) )"
For this CSP, only three slightly different path consistent scenarios exist:
((S5 (EFBS) S6)(S12 (LSEL) S6)(S12 (LLFL) S5)(S11 (SRSL) S6)(S11 (LSEL)
S5)(S11 (RRLLP) S12) (S4 (RRLL-) S6)(S4 (RRLL-) S5)(S4 (RELE) S12)(S4 (RSER)
S11)(S3 (RRLL-) S6)(S3 (RRLL-) S5) (S3 (RFLL) S12)(S3 (RELE) S11)(S3 (EFBS)
S4)(S10 (RRBL) S6)(S10 (SRSL) S5)(S10 (RRLLP) S12) (S10 (RRLLP) S11)(S10
(RRRB) S4)(S10 (ERRS) S3)(S9 (LBLL) S6)(S9 (LLLL-) S5)(S9 (BSEF) S12) (S9
(LLRRP) S11)(S9 (LSEL) S4)(S9 (LLFL) S3)(S9 (LLRRP) S10)(S8 (BRLL) S6)(S8
(LBLL) S5) (S8 (RRLLP) S12)(S8 (BSEF) S11)(S8 (SRSL) S4)(S8 (LSEL) S3)(S8
(LLRRP) S10)(S8 (RRLLP) S9) (S2 (RRLLP) S6)(S2 (RRLLP) S5)(S2 (RRLF) S12)(S2
(RRFR) S11)(S2 (RRLL+) S4)(S2 (RRLL+) S3) (S2 (RRRR+) S10)(S2 (RELE) S9)(S2
(RSER) S8)(S1 (RRLLP) S6)(S1 (RRLLP) S5)(S1 (RRLL+) S12) (S1 (RRLF) S11)(S1
(RRLL+) S4)(S1 (RRLL+) S3)(S1 (RRFR) S10)(S1 (RFLL) S9)(S1 (RELE) S8) (S1
(EFBS) S2)(S7 (RRLL-) S6)(S7 (BRLL) S5)(S7 (RRLLP) S12)(S7 (RRLLP) S11)(S7
(RRBL) S4) (S7 (SRSL) S3)(S7 (BSEF) S10)(S7 (RRLLP) S9)(S7 (RRLLP) S8)(S7
(RRRB) S2)(S7 (ERRS) S1))
((S5 (EFBS) S6)(S12 (LSEL) S6)(S12 (LLFL) S5)(S11 (SRSL) S6)(S11 (LSEL)
S5)(S11 (RRLLP) S12) (S4 (RRLL-) S6)(S4 (RRLL-) S5)(S4 (RELE) S12)(S4 (RSER)
S11)(S3 (RRLL-) S6)(S3 (RRLL-) S5) (S3 (RFLL) S12)(S3 (RELE) S11)(S3 (EFBS)
S4)(S10 (RRBL) S6)(S10 (SRSL) S5)(S10 (RRLLP) S12) (S10 (RRLLP) S11)(S10
(RRRB) S4)(S10 (ERRS) S3)(S9 (LBLL) S6)(S9 (LLLL-) S5)(S9 (BSEF) S12) (S9
(LLRRP) S11)(S9 (LSEL) S4)(S9 (LLFL) S3)(S9 (LLRRP) S10)(S8 (BRLL) S6)(S8
(LBLL) S5) (S8 (RRLLP) S12)(S8 (BSEF) S11)(S8 (SRSL) S4)(S8 (LSEL) S3)(S8
(LLRRP) S10)(S8 (RRLLP) S9) (S2 (RRLL-) S6)(S2 (RRLL-) S5)(S2 (RRLF) S12)(S2
(RRFR) S11)(S2 (RRLL+) S4)(S2 (RRLL+) S3) (S2 (RRRR+) S10)(S2 (RELE) S9)(S2
(RSER) S8)(S1 (RRLL-) S6)(S1 (RRLL-) S5)(S1 (RRLL+) S12) (S1 (RRLF) S11)(S1
(RRLL+) S4)(S1 (RRLL+) S3)(S1 (RRFR) S10)(S1 (RFLL) S9)(S1 (RELE) S8) (S1
(EFBS) S2)(S7 (RRLL-) S6)(S7 (BRLL) S5)(S7 (RRLLP) S12)(S7 (RRLLP) S11)(S7
(RRBL) S4) (S7 (SRSL) S3)(S7 (BSEF) S10)(S7 (RRLLP) S9)(S7 (RRLLP) S8)(S7
(RRRB) S2)(S7 (ERRS) S1))
((S5 (EFBS) S6)(S12 (LSEL) S6)(S12 (LLFL) S5)(S11 (SRSL) S6)(S11 (LSEL)
S5)(S11 (RRLLP) S12) (S4 (RRLL-) S6)(S4 (RRLL-) S5)(S4 (RELE) S12)(S4 (RSER)
S11)(S3 (RRLL-) S6)(S3 (RRLL-) S5) (S3 (RFLL) S12)(S3 (RELE) S11)(S3 (EFBS)
S4)(S10 (RRBL) S6)(S10 (SRSL) S5)(S10 (RRLLP) S12) (S10 (RRLLP) S11)(S10
(RRRB) S4)(S10 (ERRS) S3)(S9 (LBLL) S6)(S9 (LLLL-) S5)(S9 (BSEF) S12) (S9
(LLRRP) S11)(S9 (LSEL) S4)(S9 (LLFL) S3)(S9 (LLRRP) S10)(S8 (BRLL) S6)(S8
(LBLL) S5) (S8 (RRLLP) S12)(S8 (BSEF) S11)(S8 (SRSL) S4)(S8 (LSEL) S3)(S8
(LLRRP) S10)(S8 (RRLLP) S9) (S2 (RRLL+) S6)(S2 (RRLL+) S5)(S2 (RRLF) S12)(S2
(RRFR) S11)(S2 (RRLL+) S4)(S2 (RRLL+) S3) (S2 (RRRR+) S10)(S2 (RELE) S9)(S2
(RSER) S8)(S1 (RRLL+) S6)(S1 (RRLL+) S5)(S1 (RRLL+) S12) (S1 (RRLF) S11)(S1
(RRLL+) S4)(S1 (RRLL+) S3)(S1 (RRFR) S10)(S1 (RFLL) S9)(S1 (RELE) S8) (S1
(EFBS) S2)(S7 (RRLL-) S6)(S7 (BRLL) S5)(S7 (RRLLP) S12)(S7 (RRLLP) S11)(S7
(RRBL) S4) (S7 (SRSL) S3)(S7 (BSEF) S10)(S7 (RRLLP) S9)(S7 (RRLLP) S8)(S7
(RRRB) S2)(S7 (ERRS) S1))
3 scenarios found, no further scenarios exist.
This result can be visualized with a diagram and can be interpreted with
respect to the goals of the reasoning task (see Section 5).
|
arxiv-papers
| 2009-12-30T20:38:12 |
2024-09-04T02:49:07.357933
|
{
"license": "Public Domain",
"authors": "Reinhard Moratz, Dominik L\\\"ucke, Till Mossakowski",
"submitter": "Reinhard Moratz",
"url": "https://arxiv.org/abs/0912.5533"
}
|
1001.0042
|
Seven-Dimensional Gravity with Topological Terms
H. Lü$\,{}^{\dagger\ddagger}$ and Yi Pang$\,{}^{\star}$
$\,{}^{\dagger}$China Economics and Management Academy
Central University of Finance and Economics, Beijing 100081
$\,{}^{\ddagger}$Institute for Advanced Study, Shenzhen University, Nanhai Ave
3688, Shenzhen 518060
$\,{}^{\star}$Key Laboratory of Frontiers in Theoretical Physics
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190
ABSTRACT
We construct new seven-dimensional gravity by adding two topological terms to
the Einstein-Hilbert action. For certain choice of the coupling constants,
these terms may be related to the $R^{4}$ correction to the 3-form field
equation of eleven-dimensional supergravity. We derive the full set of the
equations of motion. We find that the static spherically-symmetric black holes
are unmodified by the topological terms. We obtain squashed AdS7, and also
squashed seven spheres and $Q^{111}$ spaces in Euclidean signature.
## 1 Introduction
There has been considerable interest in topological gauge theories [1] because
of their wide application in physics. The most studied example is the three-
dimensional one. In addition to the Einstein-Hilbert term, the theory has the
Chern-Simons term, given by
$S={\frac{1}{\mu}}\int d^{3}x{\rm
Tr}\,(d\omega\wedge\omega+{\textstyle{\frac{\scriptstyle 2}{\scriptstyle
3}}}\omega\wedge\omega\wedge\omega),$ (1)
where $\omega$ can be either a Yang-Mills gauge potential or the connection
for gravity. Topological Yang-Mills theory can provide a fundamental
interpretation for anyons [2]; it can also generate Lorentz violation
dynamically [3]. Topologically massive gravity [4] becomes dynamical with a
propagating massive particle, with the mass proportional to the coupling
constant $\mu$. Recently, a cosmological constant is added and the
corresponding boundary conformal field theory (CFT) is discussed [5]. The
three-dimensional massive topological gravity is conjectured to be unitary for
certain parameter region even though the theory has higher derivatives in time
[6].
The attention on higher dimensional generalizations is considerably less. The
five dimensional Yang-Mills Chern-Simons term was discussed in [7], but there
is no gravity counterpart due to the fact that the holonomy group $SO(1,4)$
has no invariant rank-3 symmetric tensor. In seven dimensions, Yang-Mills
Chern-Simons terms arise naturally from ${\cal N}=4$ supergravity [8]. As in
the case of three dimensions, we find that such terms in the gravity sector
can be obtained directly from those in the Yang-Mills sector by replacing the
gauge potential $A$ to the connection $\Gamma$. As we shall see later, these
topological terms in seven dimensions may be related to the anomaly
cancelation terms in eleven-dimensional supergravity.
In section 2, we present the two topological terms in seven dimensions, and
discuss their properties. Since they are not manifestly invariant under
general coordinate transformation, we find it is more convenient to lift the
system to eight dimensions in order to derive the equations of motion (EOMs).
We obtain the full set. In section 3, we construct large classes of solutions.
We find that the static spherically-symmetric black holes are unmodified by
the topological terms. This is analogous to three dimensions, where the BTZ
black hole remains to be a solution in topologically massive gravity. In
Euclidean signature, we obtain squashed $S^{7}$ and $Q^{111}$ spaces. In
particular, one of the squashed seven sphere can be Wick rotated to become
squashed AdS7. We conclude in section 4.
## 2 The theory
In seven dimensions, there are two topological terms; they are given by
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\tilde{\mu}\int\Omega^{(7)}_{1}=\tilde{\mu}\int{\rm
Tr}(\Gamma\wedge\Theta-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
3}}}\Gamma^{3})\wedge{\rm Tr}(\Theta^{2})=\tilde{\mu}\int\Omega^{(3)}\wedge
d\Omega^{(3)},$ (2) $\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\tilde{\nu}\int\Omega^{(7)}_{2}=\tilde{\nu}\int{\rm
Tr}(\Theta^{3}\wedge\Gamma-{\textstyle{\frac{\scriptstyle 2}{\scriptstyle
5}}}\Theta^{2}\wedge\Gamma^{3}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
5}}}\Theta\wedge\Gamma^{2}\wedge\Theta\wedge\Gamma+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 5}}}\Theta\wedge\Gamma^{5}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 35}}}\Gamma^{7}),$
with $\Omega^{(3)}={\rm Tr}(d\Gamma\wedge\Gamma+{\textstyle{\frac{\scriptstyle
2}{\scriptstyle 3}}}\Gamma^{3})$. Here, $\Theta$ is the curvature 2-form,
defined as $\Theta\equiv d\Gamma+\Gamma\wedge\Gamma$, and
$\tilde{\mu},\tilde{\nu}$ are two parameters of length dimension 5. (We
rescale the total action by the seven-dimensional Newton constant.) The 3-form
$\Omega^{(3)}$ has the same structure as the Chern-Simons term in $D=3$,
except that now $\Gamma$ depends on seven coordinates. $\Omega_{1}^{(7)}$ and
$\Omega_{2}^{(7)}$ are topological in the same sense as $\Omega^{(3)}$ being
topological in $D=3$. We can lift the system to $D=8$, with the seven-
dimensional spacetime as the boundary. Then, we have
$d\Omega^{(7)}_{1}=Y_{1}^{(8)}\equiv{\rm Tr}(\Theta\wedge\Theta)\wedge{\rm
Tr}(\Theta\wedge\Theta)\,,\qquad d\Omega^{(7)}_{2}=Y_{2}^{(8)}\equiv{\rm
Tr}(\Theta\wedge\Theta\wedge\Theta\wedge\Theta)\,.$ (3)
As we have mentioned earlier, these terms can be derived from the Yang-Mills
Chern-Simons terms in [8] by changing the gauge potential to the
connection.111In [8], the field strength 2-form is defined by $F=dB+gB\wedge
B$, with gauge coupling $g=2$. Then by rescaling the field $B\rightarrow B/g$
and $F\rightarrow F/g$ and setting $g=2$, one can obtain the same expressions
as the ones given here. Note that the Pontryagin term is proportional to
$Y_{1}^{(8)}-2Y_{2}^{(8)}$, corresponding to $\tilde{\nu}=-2\tilde{\mu}$. In
eleven-dimensional supergravity, there is an $R^{4}$ correction to the field
equation, namely $d{*F^{(4)}}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}F^{(4)}\wedge F^{(4)}+X^{(8)}$, where $X^{(8)}$ is given by
$X^{(8)}\propto Y_{1}^{(8)}-4Y_{2}^{(8)}\,.$ (4)
Thus for $\tilde{\nu}=-4\tilde{\mu}$, the topological terms can be obtained
from the $S^{4}$ reduction of supergravity in $D=11$, and the coupling
constant is proportional to the 4-form M5-brane fluxes. For large fluxes, this
topological term dominates the higher-order corrections.
To derive the contribution to the EOMs from the Chern-Simons terms, it is
necessary to perform their variation with respect to the metric. These
topological terms are not manifestly invariant under the general coordinate
transformation, but $Y_{1}^{(8)}$ and $Y_{2}^{(8)}$ are. We find that a
convenient way to derive the variation is to lift the system to eight
dimensions. Let us first consider the variation of $S_{1}$. In terms of
coordinate components, we have
$\int d\Omega^{(7)}_{1}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
16}}}\int
d^{8}x\epsilon^{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\nu_{7}\nu_{8}}R^{\mu_{1}}_{~{}\mu_{2}\nu_{1}\nu_{2}}R_{~{}\mu_{1}\nu_{3}\nu_{4}}^{\mu_{2}}R^{\mu_{3}}_{~{}\mu_{4}\nu_{5}\nu_{6}}R_{~{}\mu_{3}\nu_{7}\nu_{8}}^{\mu_{4}}\,.$
(5)
Here we use Greek letters to denote the eight-dimensional coordinates and
Latin letters to represent the seven-dimensional ones hereafter. We adopt the
convention $\epsilon^{12345678}=1$.
For an infinitesimal variation of the metric $\delta g$, using the Bianchi
identity and the following relation
$\delta
R^{\mu}_{~{}\nu\alpha\beta}=\delta\Gamma^{\mu}_{\nu\beta;\alpha}-\delta\Gamma^{\mu}_{\nu\alpha;\beta},$
(6)
we find that
$\displaystyle\int d\delta\Omega^{(7)}_{1}$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\int
d^{8}x\sqrt{g}\Big{(}\frac{1}{\sqrt{g}}\epsilon^{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\nu_{7}\nu_{8}}R^{\mu_{1}}_{~{}\mu_{2}\nu_{1}\nu_{2}}R_{~{}\mu_{1}\nu_{3}\nu_{4}}^{\mu_{2}}R^{\mu_{3}}_{~{}\mu_{4}\nu_{5}\nu_{6}}\delta\Gamma^{\mu_{4}}_{~{}\mu_{3}\nu_{7}}\Big{)}_{;\nu_{8}}$
(7) $\displaystyle\equiv$ $\displaystyle{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\int d{*J}\,,$ (8)
where “;” denotes a covariant derivative and $*$ is the Hodge dual. For
simplicity, we have introduced a 1-form current $J=J_{\alpha}dx^{\alpha}$. Its
components are given by
$J^{\alpha}=\frac{1}{\sqrt{g}}\epsilon^{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\nu_{7}\alpha}R^{\mu_{1}}_{~{}\mu_{2}\nu_{1}\nu_{2}}R_{~{}\mu_{1}\nu_{3}\nu_{4}}^{\mu_{2}}R^{\mu_{3}}_{~{}\mu_{4}\nu_{5}\nu_{6}}\delta\Gamma^{\mu_{4}}_{~{}\mu_{3}\nu_{7}}.$
(9)
Clearly, we have $d{*J}=-\sqrt{g}J^{\alpha}{}_{;\alpha}d^{8}x$, Thus we obtain
$\delta\Omega^{(7)}_{1}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}{*J},$ (10)
up to a total derivative term. Now restricting the coordinate indices to seven
dimensions only, we have
$\delta S_{1}=4\tilde{\mu}\int{\rm Tr}(\Theta\wedge\Theta)\wedge{\rm
Tr}(\Theta\wedge\delta\Gamma).$ (11)
The variation of $S_{2}$ can be obtained in the same manner, given by
$\displaystyle\delta S_{2}=4\tilde{\nu}\int{\rm
Tr}(\Theta\wedge\Theta\wedge\Theta\wedge\delta\Gamma).$ (12)
Finally, we make use of the variation of the connection
$\delta\Gamma^{i}_{mj}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}g^{in}(\delta g_{nm;j}+\delta g_{nj;m}-\delta g_{ml;n}),$ (13)
and after integrating by parts, we obtain the contributions to EOMs from the
Chern-Simons terms, given by
$\displaystyle C_{1}^{ij}$ $\displaystyle=$ $\displaystyle\frac{\delta
S_{1}}{\sqrt{g}\delta
g_{ij}}=\frac{\mu}{4\sqrt{g}}[\epsilon^{ij_{1}j_{2}j_{3}j_{4}j_{5}j_{6}}(R^{i_{1}}_{~{}i_{2}j_{1}j_{2}}R_{~{}i_{1}j_{3}j_{4}}^{i_{2}}R^{jk}_{~{}~{}j_{5}j_{6}})_{;k}+i\leftrightarrow
j],$ (14) $\displaystyle C_{2}^{ij}$ $\displaystyle=$
$\displaystyle\frac{\delta S_{2}}{\sqrt{g}\delta
g_{ij}}=\frac{\nu}{4\sqrt{g}}[\epsilon^{ij_{1}j_{2}j_{3}j_{4}j_{5}j_{6}}(R^{k}_{~{}i_{1}j_{1}j_{2}}R^{i_{1}}_{~{}i_{2}j_{3}j_{4}}R^{ji_{2}}_{~{}~{}~{}j_{5}j_{6}})_{;k}+i\leftrightarrow
j].$ (15)
For the total action $S$, which is the sum of the Einstein-Hilbert action,
cosmological constant $\Lambda$ and $S_{1}+S_{2}$, the corresponding full set
of EOMs is given by
$R^{ij}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}g^{ij}R+\Lambda
g^{ij}+C_{1}^{ij}+C_{2}^{ij}=0.$ (16)
It should be remarked that under a large gauge transformation
$\Gamma\rightarrow\mathcal{O}\Gamma\mathcal{O}^{-1}-d\mathcal{O}\mathcal{O}^{-1}$,
the action transforms as $S\rightarrow
S+\tilde{\mu}v(\mathcal{O})+\tilde{\nu}w(\mathcal{O})$, where
$v(\mathcal{O})=\int{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
3}}}d\Big{(}{\rm
Tr}(d\mathcal{O}\mathcal{O}^{-1})^{3}\wedge\Omega^{(3)}\Big{)};\qquad
w(\mathcal{O})={\textstyle{\frac{\scriptstyle 1}{\scriptstyle 35}}}\int{\rm
Tr}(d\mathcal{O}\mathcal{O}^{-1})^{7}.$ (17)
The $v$ term is trivial and gives no restriction to the parameter
$\tilde{\mu}$, while the $w$ term should be classified by the seventh homotopy
group of $SO(1,6)$
$\pi_{7}[SO(1,6)]\simeq\pi_{7}[SO(6)]\simeq\mathbb{Z}.$ (18)
The invariance of $e^{{\rm i}S}$ requires that
$64\pi^{4}\tilde{\nu}=2\pi n,~{}~{}~{}~{}n=0,\pm 1,\pm 2\ldots.$ (19)
This result is completely different from that in three dimensions, where the
$SO(1,2)$ is homotoplically trivial and the mass parameter is not quantized.
Moreover, since $\tilde{\nu}$ is quantized, $S_{2}$ will not be renormalized
in the quantum theory. This suggests some intriguing properties in the
corresponding CFT dual.
## 3 Solutions
Spherically-symmetric solutions:
Having obtained the full set of EOMs for topological gravity in seven
dimensions, we are in the position to construct solutions. It is clear that
the maximally-symmetric space(time) is unmodified by the inclusion of the
topological terms. The next simplest case is to consider the spherically-
symmetric ansatz, given by
$ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{G(r)}+r^{2}d\Omega_{5}^{2}\,.$ (20)
We find that for this ansatz, the contributions from the topological terms
$C_{1}^{ij}$ and $C_{2}^{ij}$ vanish identically. This implies that the
previously-known static (AdS) black holes, charged or neutral, are still
solutions when the topological terms are added to the action. This is
analogous to three dimensions, where the BTZ black hole is still a solution in
massive topological gravity. However the thermodynamic quantities such as the
mass and entropy will acquire modifications [9, 10].
As we shall discuss presently, there also exist squashed AdS7 solutions.
$S^{3}$ bundle over $S^{4}$:
We now turn our attention to the Euclidean theory. In three dimensions, there
exists a large class of squashed $S^{3}$ or AdS3 [11]. We expect the same in
seven dimensions. Without loss of generality, we set $\Lambda=30$ so that it
can give rise to a unit round $S^{7}$. We first consider the squashed $S^{7}$
that can be viewed as an $S^{3}$ bundle over $S^{4}$. The metric ansatz is
given by
$ds^{2}=\alpha\sum_{i=1}^{3}(\sigma_{i}-\cos^{2}({\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\theta)\,\tilde{\sigma}_{i})^{2}+\beta\Big{(}d\theta^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
4}}}\sin^{2}\theta\sum_{i=1}^{3}\tilde{\sigma}_{i}^{2}\Big{)}\,.$ (21)
where $\sigma_{i}$ and $\tilde{\sigma}_{i}$ are the $SU(2)$ left-invariant
1-forms, satisfying $d\sigma_{i}={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\epsilon^{ijk}\sigma^{j}\wedge\sigma^{k}$ and
$d\tilde{\sigma}_{i}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\epsilon^{ijk}\tilde{\sigma}^{j}\wedge\tilde{\sigma}^{k}$. The metric is
Einstein provided that either $\alpha=\beta={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}$ or $\alpha={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 5}}}\beta={\textstyle{\frac{\scriptstyle 9}{\scriptstyle
100}}}$. The first case corresponds to the round $S^{7}$ and the second is a
squashed $S^{7}$ that is also Einstein. Now with the contribution from the
topological terms, the EOMs can be reduced to
$2\alpha^{2}+4\alpha\,\beta(7\beta-2)-\beta^{2}=0\,,$ (22)
together with
$\sqrt{\alpha}(\alpha-\beta)^{3}(4(10\alpha+\beta)\tilde{\mu}-(55\alpha+7\beta)\nu)+2\beta^{6}(20\alpha\beta-4\alpha-\beta)=0\,.$
(23)
It is clear from (22) that there exists one and only one positive $\alpha$ for
any positive $\beta$. The squashing parameter $\gamma\equiv\alpha/\beta$ lies
in the range $0<\gamma<2+{\frac{3}{\sqrt{2}}}$. Note that when
$2\tilde{\mu}=3\tilde{\nu}$, the squashed $S^{7}$ that is Eisntein remains
Einstein.
$S^{1}$ bundle over ${{\mathbb{C}}{\mathbb{P}}}^{3}$:
There is another way of squashing an $S^{7}$, which can be viewed as an
$S^{1}$ bundle over ${{\mathbb{C}}{\mathbb{P}}}^{3}$. This example can be
generalized to Minkowskian signature to give rise to squashed AdS7 [12]. The
metric ansatz is given by
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\alpha\,(d\tau+\sin^{2}\theta(d\psi+B))^{2}+\beta\,ds_{{{\mathbb{C}}{\mathbb{P}}}^{3}}^{2}\,,$
(24) $\displaystyle ds_{{{\mathbb{C}}{\mathbb{P}}}^{3}}^{2}$ $\displaystyle=$
$\displaystyle
d\theta^{2}+\sin^{2}\theta\,\cos^{2}\theta(d\psi+B)^{2}+\sin^{2}\theta\Big{(}d\tilde{\theta}^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
4}}}\sin^{2}\tilde{\theta}\,\cos^{2}\tilde{\theta}\,\sigma_{3}^{2}$ (26)
$\displaystyle\qquad\qquad+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}\sin^{2}\tilde{\theta}\,(\sigma_{1}^{2}+\sigma_{2}^{2})\Big{)}\,,$
$\displaystyle B$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\sin^{2}\tilde{\theta}\,\sigma_{3}\,.$ (27)
It is of a round $S^{7}$ when $\alpha=\beta=1$. In general, the EOMs imply
that
$\alpha=\beta(8-7\beta)\,,\qquad
8\tilde{\mu}+\tilde{\nu}+{\frac{\beta^{3}}{10976(\beta-1)^{2}\sqrt{\alpha}}}=0\,.$
(28)
The squashing parameter $\gamma\equiv\alpha/\beta$ lies in the range $(0,8)$.
Squashed $Q^{111}$ spaces:
The $Q^{111}$ space is an Einstein-Sasaki space of $U(1)$ bundle over
$S^{2}\times S^{2}\times S^{2}$. We consider the following ansatz
$ds^{2}=\alpha\Big{(}d\psi+\sum_{i=1}^{3}\cos\theta_{i}\,d\phi_{i}\Big{)}^{2}+\beta\sum_{i=1}^{3}(d\theta_{i}^{2}+\sin\theta_{i}^{2}d\phi_{i}^{2})\,.$
(29)
It is of $Q^{111}$ provided that $\alpha={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\beta=1/16$, and it remains so for $\tilde{\nu}=0$. In
general, we have
$\alpha=4\beta(1-7\beta)\,,\qquad
8(\alpha-\beta)(2\alpha-\beta)\tilde{\mu}+\alpha(2\alpha-3\beta)\tilde{\nu}+{\frac{\beta^{5}(\alpha-8\beta+60\beta^{2})}{4\alpha^{3/2}}}=0\,.$
(30)
Thus the squashing parameter $\gamma\equiv\alpha/\beta$ lies in the range
$(0,4)$. We expect that many of the squashed homogeneous spaces in seven
dimensions are now solutions in this new gravity theory, and we shall not
enumerate them further.
## 4 Conclusions
This work is motivated by studying the classical solutions of Einstein-Chern-
Simons gravity with asymptotic AdS structure. In seven dimensions, there are
two topological Chern-Simons terms, and we obtain the full set of equations of
motion. We find that spherically-symmetric solutions are unmodified by the
inclusion of these topological terms. We also obtain squashed AdS7, and
squashed $S^{7}$ and $Q^{111}$ spaces in Euclidean signature, where the
squashing parameter is related to the coupling constants of the topological
terms. It is intriguing to see that these known squashed homogeneous spaces
which appear to have no connection can now be unified under our new gravity
theory.
As in three dimensions, our topological gravity should play an important role
in exploring the AdS7/CFT6 correspondence. The CFT6 that describes the world-
volume theory of multiple M5-branes is yet to be known, and our solutions
provide many new gravity dual backgrounds. The quantization condition for one
of the coupling constant suggests an unusual property of the CFT6 that is
absent in lower dimensions. Additional future directions include a
classification of all topological gravities in $(4k+3)$ dimensions,
investigating the linearization of $D=7$ topological gravity and obtaining the
propagating degrees of freedom.
## Acknowledgement
We are grateful to Chris Pope for useful discussions. Y.P. is supported in
part by the NSFC grant No.1053060/A050207 and the NSFC group grant
No.10821504.
## References
* [1] S. Deser, R. Jackiw and S. Templeton, “Topologically massive gauge theories,” Annals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988 APNYA,281,409-449.2000)].
* [2] F. Wilczek and A. Zee, “Linking numbers, spin, and statistics of solitons,” Phys. Rev. Lett. 51, 2250 (1983).
* [3] S.M. Carroll, G.B. Field and R. Jackiw, “Limits on a Lorentz and parity violating modification of electrodynamics,” Phys. Rev. D 41, 1231 (1990).
* [4] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional Einstein gravity: dynamics of flat space,” Annals Phys. 152, 220 (1984).
* [5] E. Witten, “Three-dimensional gravity revisited,” arXiv:0706.3359 [hep-th].
* [6] W. Li, W. Song and A. Strominger, “Chiral gravity in three dimensions,” JHEP 0804, 082 (2008) [arXiv:0801.4566 [hep-th]]. E.A. Bergshoeff, O. Hohm and P.K. Townsend, “Massive gravity in three dimensions,” Phys. Rev. Lett. 102, 201301 (2009) arXiv: 0901.1766 [hep-th].
* [7] M. Gunaydin, G. Sierra and P.K. Townsend, “Quantization of the gauge coupling constant in a five-dimensional Yang-Mills/Einstein supergravity theory,” Phys. Rev. Lett. 53, 322 (1984).
* [8] M. Pernici, K. Pilch and P. van Nieuwenhuizen, “Gauged maximally extended supergravity in seven-dimensions,” Phys. Lett. B 143, 103 (1984).
* [9] S. Deser and B. Tekin, “Energy in topologically massive gravity,” Class. Quant. Grav. 20, L259 (2003), gr-qc/0307073.
* [10] Y. Tachikawa, “Black hole entropy in the presence of Chern-Simons terms,” Class. Quant. Grav. 24, 737 (2007), hep-th/0611141.
* [11] D.D.K. Chow, C.N. Pope and E. Sezgin, “Exact solutions of topologically massive gravity,” arXiv:0906.3559 [hep-th].
* [12] P. Hoxha, R.R. Martinez-Acosta and C.N. Pope, “Kaluza-Klein consistency, Killing vectors, and Kaehler spaces,” Class. Quant. Grav. 17, 4207 (2000), hep-th/0005172.
|
arxiv-papers
| 2009-12-30T23:05:06 |
2024-09-04T02:49:07.378413
|
{
"license": "Public Domain",
"authors": "H. Lu, Yi Pang",
"submitter": "Yi Pang",
"url": "https://arxiv.org/abs/1001.0042"
}
|
1001.0066
|
# A matter dominated navigation Universe in accordance with the Type Ia
supernova data
Xin Li1,3 lixin@itp.ac.cn 1Institute of Theoretical Physics, Chinese Academy
of Sciences, 100190 Beijing, China Zhe Chang2,3 zchang@ihep.ac.cn Minghua
Li2,3 limh@ihep.ac.cn 2Institute of High Energy Physics, Chinese Academy of
Sciences, 100049 Beijing, China
3Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
###### Abstract
We investigate a matter dominated navigation cosmological model. The influence
of a possible drift (wind) in the navigation cosmological model makes the
spacetime geometry change from Riemannian to Finslerian. The evolution of the
Finslerian Universe is governed by the same gravitational field equation with
the familiar Friedmann-Robertson-Walker one. However, the change of space
geometry from Riemannian to Finslerian supplies us a new relation between the
luminosity distant and redshift. It is shown that the Hubble diagram based on
this new relation could account for the observations on distant Type Ia
supernovae.
###### pacs:
02.40.-k,98.80.-k
## I Introduction
Einstein’s general relativity enables us to come up with a testable theory of
the Universe. HubbleHubble first found that the galaxies are receding from us
not long after the birth of general relativity. The Hubble’s observation
indicates that our Universe is expanding. Over the past decade, two groups
Riess ; Perlmutter observing supernovae reported that the luminosity distance
can not be explained by a matter dominated Universe. If one accepts the
convincible assumption of homogeneity and isotropy of the Universe which is
approximately true on large scales, then the general relativity tells us that
we now live in a dark energy dominated Universe and the Universe is
accelerated expanding. Dark energy, which has the property of negative
pressure, is different from the classical matter and the found particles. A
great amount of models have been proposed to study the possible candidates of
dark energy and their dynamics Copeland . The most famous and acceptable
candidate is the cosmological constant. However, the magnitude of cosmological
constant $10^{-3}{\rm eV}^{4}$ is much smaller than the energy density of
vacuum in quantum field theory.
The theory of dark energy Padmanabhan dominates the modern cosmology in the
past decade. This situation is similar to the raise of the theory of ether,
which has been considered as direct evidence of the aberration of starlight,
an important astronomical effect known since eighteenth century. Although the
rapid progress in technology makes the astronomical observations more and more
accurate, up to now, there is no direct evidence indicate what the dark energy
it is. Since the theory of dark energy is contrived, which requires fine
tuning and apparently cannot be tested in the laboratory or solar system,
several modified theories of general relativity have been developed as the
alternative source of cosmological acceleration Bludman .
Einstein first used the Riemann geometry to describe the theory of
gravitation. In this Letter, we suppose that the spacetime in large scale may
be described by other geometry instead of Riemann geometry. To preserve
redeeming feature of general relativity, this geometry must take Riemann
geometry as its special case. Fortunately, Paul Finsler proposed a natural
generation of Riemann geometry-Finsler geometry.
Finsler geometry as a more general geometry could provide new sight on modern
physics. It is of great interest for physicists to investigated the violation
of Lorentz symmetry Kostelecky . An interesting case of Lorentz violation,
which was proposed by Cohen and GlashowGlashow , is the model of Very Special
Relativity (VSR) characterized by a reduced symmetry SIM(2). In fact, Gibbons,
Gomis and PopeGibbons showed that the Finslerian line element
$ds=(\eta_{\mu\nu}dx^{\mu}dx^{\nu})^{(1-b)/2}(n_{\rho}dx^{\rho})^{b}$ is
invariant under the transformations of the group DISIM${}_{b}(2)$. In the
framework of Finsler geometry, modified dispersion relation has been
discussedGirelli ; Finsler SR . Also, the model based on Finsler geometry
could explain the recent astronomical observations which Einstein’s gravity
could not. A list includes: the flat rotation curves of spiral galaxies can be
deduced naturally without invoking dark matter Finsler DM ; the anomalous
accelerationAnderson in solar system observed by Pioneer 10 and 11
spacecrafts should correspond to Finsler-Randers space Finsler PA ; the
secular trend in the astronomical unitKrasinsky ; Standish and the anomalous
secular eccentricity variation of the Moon’s orbitWilliams should be
correspond to the effect of the length change of unit circle in Finsler
geometryFinsler AU .
In this Letter, we present a matter dominated navigation model. The influence
of a possible drift (wind) in the navigation cosmolgical model makes the
expanding Universe “accelerated”. It is remarkable that the navigation
cosmological model is described in the framework of Finsler geometry. We find
that the predictions of the navigation cosmological model could account for
the observations of Riess and PerlmutterRiess ; Perlmutter on distant
supernovae.
## II formulism
Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is based on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x$ represents position and $y$ represents velocity. The Finsler metric
is given asBook by Bao
$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (1)
In Finsler manifold, there exists a unique linear connection - the Chern
connectionChern . It is torsion freeness and almost metric-compatibility,
$\Gamma^{\alpha}_{\mu\nu}=\gamma^{\alpha}_{\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{\mu}}{F}\right),$
(2)
where $\gamma^{\alpha}_{\mu\nu}$ is the formal Christoffel symbols of the
second kind with the same form of Riemannian connection, $N^{\mu}_{\nu}$ is
defined as
$N^{\mu}_{\nu}\equiv\gamma^{\mu}_{\nu\alpha}y^{\alpha}-A^{\mu}_{\nu\lambda}\gamma^{\lambda}_{\alpha\beta}y^{\alpha}y^{\beta}$
and $A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})$ is the Cartan tensor (regarded as a measurement of deviation
from the Riemannian Manifold). In terms of Chern connection, the curvature of
Finsler space is given as
$R^{~{}\lambda}_{\kappa~{}\mu\nu}=\frac{\delta\Gamma^{\lambda}_{\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\alpha}_{\kappa\nu}-\Gamma^{\lambda}_{\alpha\nu}\Gamma^{\alpha}_{\kappa\mu},$
(3)
where $\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{\mu}\frac{\partial}{\partial y^{\nu}}$. The notion of Ricci
tensor in Finsler geometry was first introduced by Akbar-ZadehAkbar
$Ric_{\mu\nu}=\frac{\partial^{2}\left(\frac{1}{2}F^{2}R\right)}{\partial
y^{\mu}\partial y^{\nu}},$ (4)
where $R=\frac{y^{\mu}}{F}R^{~{}\kappa}_{\mu~{}\kappa\nu}\frac{y^{\nu}}{F}$.
And the scalar curvature in Finsler geometry is given as
$S=g^{\mu\nu}Ric_{\mu\nu}$.
In standard cosmology, following the cosmological principle, one gets the
Friedmann-Robertson-Walker (FRW) metricWeinberg . In another word, the spatial
part of the Universe is a constant sectional curvature space. Here comes our
major assumption, the gravity in large scale should be described in terms of
Finsler geometry. In light of the cosmological principle, the Finsler
structure of the Universe should be written in such form
$\bar{F}^{2}=dt^{2}-R^{2}(t)F^{2},$ (5)
where $R(t)$ is scale factor with cosmic time $t$, the structure $F$ is a
constant flag curvature space. By making use of the geometrical terms we
mentioned above, one obtains the components of Ricci tensor
$\displaystyle Ric_{00}$ $\displaystyle=$
$\displaystyle-\frac{3\ddot{R}}{R}g_{00},$ (6) $\displaystyle Ric_{ij}$
$\displaystyle=$
$\displaystyle-\left(\frac{\ddot{R}}{R}+\frac{2\dot{R}^{2}}{R^{2}}+\frac{2K}{R^{2}}\right)g_{ij},$
(7)
where the dot denotes a derivative with respect to $t$.
The gravitational field equation in Finsler space should reduce to the
Einstein’s gravitational field equation while the Finsler space reduce to the
Riemannian space. Thus, the symmetrical tensor $G_{\mu\nu}\equiv
Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S$ should involve in the gravitational field
equation in Finsler space. In general, Finsler space is an anisotropic space.
Therefore, it has less Killing vectors than the Riemannian space, and it
breaks the Lorentz symmetryLi . It means the angular momentum in Finsler space
is not a conservative quantity. This fact implies that the energy momentum
tensor is not symmetrical in Finsler space. For these reasons, the
gravitational field equation in Finsler space should be taken in such form
$G_{\mu\nu}+A_{\mu\nu}=8\pi G(T^{s}_{\mu\nu}+T^{a}_{\mu\nu}),$ (8)
where $A_{\mu\nu}$ is an asymmetrical tensor and
$T^{s}_{\mu\nu},T^{a}_{\mu\nu}$ are symmetrical part and asymmetrical part of
energy momentum tensor respectively. This general form is agree with the
result of Asanov, its gravitational field equation contains the asymmetrical
term in Finsler space of Landsberg typeAsanov . Dealing with field equation in
the anisotropic and inhomogeneous cosmology is not a simple staff, and it is
hard to find an exact solution of gravitational field equation while its
energy momentum tensor involves the non diagonal termsMisner . At first
glance, we just deal with the symmetrical part of field equation (8)
$G_{\mu\nu}=8\pi GT_{\mu\nu}.$ (9)
By making use of the equations (6) and (7), the solution of equation (9) is
the same with the Einstein’s field equation deduced by FRW metric. Taking the
energy-momentum tensor $T_{\mu\nu}$ to be the form of perfect fluid, one can
see that the evolution of scale factor $R(t)$ is the same with the Riemannian
case. However, since the luminosity distant is related to the spatial geometry
of the Universe, one may expect that it is different from the Riemannian
luminosity distant.
Unlike Riemann space, a complete classification of the constant flag curvature
spaces remain an unsolved problem. However, by making use of the Zermolo
navigation on Riemannian space, Bao et al. Bao gave a complete classification
of Finsler-Randers space Randers of constant flag curvature. The Zermelo
navigation problem Zermelo aims to find the paths of shortest travel time in
a Riemannian manifold ($M,h$) under the influence of a drift (“wind”)
represented by a vector field $W$. In standard cosmology, our Universe is very
flat now. We may imagine that the Universe is a flat Riemannian manifold with
flat Friedmann metric
$ds^{2}=dt^{2}-R_{h}^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (10)
and it influenced by the “wind” $W$. The relation between the Riemannian
manifold ($M,h$) and the Randers metric $F$ is
$F=\frac{1}{\lambda}\left(\sqrt{\lambda
h_{ij}y^{i}y^{j}+(W_{i}y^{i})^{2}}-W_{i}y^{i}\right),$ (11)
where $W_{i}=h_{ij}W^{j}$ and $\lambda=1-h(W,W)$. One should notice that there
is a map between the Riemannian space which influence by the “wind” and the
Randers-Finsler spaceGibbons1 . It means the effect of “wind” already
accounted in the gravitational field equation in Finsler space. The theoremBao
of the classification supplies an interesting case where the Randers metric
$F$ has constant flag curvature $K$: for $K=-\frac{1}{16}\sigma^{2}<0$ and $h$
is flat. And $\sigma$ satisfies the constraint $\mathcal{L}_{W}h=-\sigma h$,
$\mathcal{L}$ denotes Lie differentiation. We set the vector field to be
radial $W=\epsilon(t)(x_{1}\partial x_{1}+x_{2}\partial x_{2}+x_{3}\partial
x_{3})$. Then the flag curvature of Randers metric is
$K=-\frac{1}{4}\epsilon^{2}$. After doing coordinate change and taking
spherical coordinate, we get the Randers metric as
$F=\sqrt{\frac{dr^{2}}{1+\epsilon^{2}r^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})}-\frac{\epsilon rdr}{1+\epsilon^{2}r^{2}}.$ (12)
After scale change $r\rightarrow 2r/\epsilon$, the metric $F$ changes as
$F=\sqrt{\frac{dr^{2}}{1+4r^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})}-\frac{2rdr}{1+4r^{2}},$ (13)
and the cosmic scale factor changes as $R(t)=\frac{2R_{h}(t)}{\epsilon(t)}$.
Taking the spatial part of the Finsler structure (5) to be of the form (13),
we get metric of the Universe in the framework of Finsler geometry, and the
space curvature of the Universe is $-1$. One can deduce directly from such
metric that the relation between the redshift $z$ and the scale factor $R(t)$
is the same with the Riemannian case
$1+z=\frac{R_{0}}{R(t)}=\frac{1}{a(t)},$ (14)
where the subscript zero denotes the quantities given at the present epoch.
Since the non-radial part of the metric (13) is the same with FRW metric, the
luminosity distant in the navigation cosmological model is given as
$d_{L}=R_{0}r(1+z)$. However, the proper distant is not the case, and it is
given as
$\displaystyle d_{p}$ $\displaystyle=$
$\displaystyle\int_{0}^{r}\left(\frac{1}{\sqrt{1+4r^{2}}}-\frac{2r}{1+4r^{2}}\right)dr$
(15) $\displaystyle=$
$\displaystyle\frac{\sinh^{-1}2r}{2}-\frac{1}{4}\ln(1+4r^{2}).$
The light traveling along the radial direction satisfies the geodesic equation
$\bar{F}^{2}=dt^{2}-R^{2}(t)\left(\frac{1}{\sqrt{1+4r^{2}}}-\frac{2r}{1+4r^{2}}\right)^{2}dr^{2}=0.$
(16)
Then, we obtain
$d_{p}=\frac{1}{R_{0}}\int^{1}_{(1+z)^{-1}}\frac{da}{a\dot{a}}$ (17)
The equations (9) and (17) can be solved. Supposing the Universe is matter
dominated (no more cosmological constant and dynamical dark energy), we obtain
$d_{p}=\log\frac{(\sqrt{1+\Omega_{m}^{(0)}z}-\sqrt{1-\Omega_{m}^{(0)}})(1+\sqrt{1-\Omega_{m}^{(0)}})}{(\sqrt{1+\Omega_{m}^{(0)}z}+\sqrt{1-\Omega_{m}^{(0)}})(1-\sqrt{1-\Omega_{m}^{(0)}})},$
(18)
where $H_{0}$ is the Hubble constant and $\Omega_{m}$ is the density parameter
for matter and satisfies
$1-\Omega_{m}^{(0)}=\frac{1}{H_{0}^{2}R_{0}^{2}}.$ (19)
Substituting the equation (18) into (15), we obtain the relation between
luminosity distant and redshift in the navigation cosmological model
$H_{0}d_{L}=\frac{1+z}{2\sqrt{1-\Omega_{m}^{(0)}}}\frac{|e^{2d_{p}}-1|}{e^{d_{p}}\sqrt{2-e^{2d_{p}}}}.$
(20)
## III numerical result
Figure 1: The luminosity distant $Log_{10}(H_{0}d_{L}/c)$ versus the redshift
$z$ for the navigational cosmological model. The data comes from Riess et
al.Riess and Hubble Space Telescope (HST)Kowalski . And the value of Hubble
constant is set as $H_{0}=67.88km\cdot s^{-1}\cdot Mpc^{-1}$.
Here, we present numerical result for the relation between luminosity distant
and redshift in the framework of the navigation cosmological model. The best
fit curve is shown in Fig.1 with the matter density parameter taken as
$\Omega_{m}^{(0)}=0.92$. And the average value of Chisquare is $1.077588$.
Thus, our prediction could account for experiment data given by Riess et
al.Riess . It should be noticed that the Hubble constant $H_{0}$ we took is
different from the Hubble constant $H_{h0}$ measured in flat space. The
relation for $H$ and $H_{h}$ is
$H=H_{h}-\frac{\dot{\epsilon}}{\epsilon}.$ (21)
By making use of the value of the Hubble constant measured in flat space
$H_{h0}=73\pm 0.3~{}km\cdot s^{-1}\cdot Mpc^{-1}$Spergel , we have
$H_{\epsilon 0}\equiv\frac{\dot{\epsilon}_{0}}{\epsilon_{0}}=5.42km\cdot
s^{-1}\cdot Mpc^{-1}$ (22)
This geometrical parameter $H_{\epsilon 0}$ represents an “accelerated” effect
provided by the vector field $W$.
## IV conclusion
Our Letter initiates an exploration of the possibility that the empirical
success of the observations of Type Ia supernovae can be regarded as the
influence of a navigational wind. The particles move on Riemnnian manifold and
influenced by a vector field (the “wind”) which proportion to the curvature
$-\frac{\epsilon^{2}}{4}$ of the space of the Universe. Its geodesic indeed is
a Finslerian geodesic. Thus, the observations of Riess et al.Riess on distant
supernovae may be explained by the effect of the “wind”. Our numerical result
could account for astronomical observations. At last, we point out that the
age of the Universe is about 9.76Gyr in our model. This is contradicted with
the age ($13.5\pm 2Gyr$) of Globular clusters in the MilkyWayJimenez . In the
navigation cosmological model, we only involve the radial “wind”. The non-
radial “wind” should be taken into account in future work, we expect that the
effect of the non-radial “wind” may supply us a reasonable age of the
Universe.
###### Acknowledgements.
We would like to thank Prof. C. J. Zhu and X. H. Mo for useful discussions.
The work was supported by the NSF of China under Grant No. 10525522 and
10875129\.
## References
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* (14) X. Li and Z. Chang, arXiv:gr-qc/0909.3713.
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* (16) E. M. Standish, Proc. IAU Colloq. 196, 163 (2005).
* (17) J. G. Williams and D. H. Boggs, in Proceedings of 16th International Workshop on Laser Ranging, ed. S. Schillak, (Space Research Centre, Polish Academy of Sciences), 2009.
* (18) X. Li and Z. Chang, arXiv:gr-qc/0911.1890.
* (19) D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathmatics 200, Springer, New York, 2000.
* (20) S. S. Chern, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5, 95 (1948); or Selected Papers, vol. II, 194, Springer 1989.
* (21) H. Akbar-Zadeh, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74, 281 (1988).
* (22) S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York, 1972.
* (23) X. Li, Z. Chang and X. H. Mo, arXiv:hep-th/1001.2667.
* (24) G. S. Asanov, Reports on Mathematical Physics 59, 111 (2007).
* (25) C. W. Misner, K, S, Thorne and J. A. Wheeler, Gravitation, Freeman, San Francisco, 1973.
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* (27) G. Randers, Phys. Rev. 59, 195 (1941).
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|
arxiv-papers
| 2009-12-31T02:23:04 |
2024-09-04T02:49:07.385138
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li, Zhe Chang and Minghua Li",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1001.0066"
}
|
1001.0117
|
2010429-440Nancy, France 429
Xiaoyang Gu
John M. Hitchcock
A. Pavan
# Collapsing and Separating Completeness Notions under Average-Case and Worst-
Case Hypotheses
X. Gu LinkedIn Corporation , J. M. Hitchcock Department of Computer
Science, University of Wyoming and A. Pavan Department of Computer Science,
Iowa State University
###### Abstract.
This paper presents the following results on sets that are complete for ${\rm
NP}$.
1. (i)
If there is a problem in ${\rm NP}$ that requires $2^{n^{\Omega(1)}}$ time at
almost all lengths, then every many-one NP-complete set is complete under
length-increasing reductions that are computed by polynomial-size circuits.
2. (ii)
If there is a problem in co-NP that cannot be solved by polynomial-size
nondeterministic circuits, then every many-one complete set is complete under
length-increasing reductions that are computed by polynomial-size circuits.
3. (iii)
If there exist a one-way permutation that is secure against subexponential-
size circuits and there is a hard tally language in ${\rm NP}\cap{\mbox{\rm
co-NP}}$, then there is a Turing complete language for ${\rm NP}$ that is not
many-one complete.
Our first two results use worst-case hardness hypotheses whereas earlier work
that showed similar results relied on average-case or almost-everywhere
hardness assumptions. The use of average-case and worst-case hypotheses in the
last result is unique as previous results obtaining the same consequence
relied on almost-everywhere hardness results.
###### Key words and phrases:
computational complexity, NP-completeness
Gu’s research was supported in part by NSF grants 0652569 and 0728806.
Hitchcock’s research was supported in part by NSF grants 0515313 and 0652601
and by an NWO travel grant. Part of this research was done while this author
was on sabbatical at CWI
Pavan’s research was supported in part by NSF grants 0830479 and 0916797.
## 1\. Introduction
It is widely believed that many important problems in ${\rm NP}$ such as
satisfiability, clique, and discrete logarithm are exponentially hard to
solve. Existence of such intractable problems has a bright side: research has
shown that we can use this kind of intractability to our advantage to gain a
better understanding of computational complexity, for derandomizing
probabilistic computations, and for designing computationally-secure
cryptographic primitives. For example, if there is a problem in ${\rm EXP}$
(such as any of the aforementioned problems) that has $2^{n^{\Omega(1)}}$-size
worst-case circuit complexity (i.e., that for all sufficiently large $n$, no
subexponential size circuit solves the problem correctly on all instances of
size $n$), then it can be used to construct pseudorandom generators. Using
these pseudorandom generators, ${\rm BPP}$ problems can be solved in
deterministic quasipolynomial time [23]. Similar average-case hardness
assumptions on the discrete logarithm and factoring problems have important
ramifications in cryptography. While these hardness assumptions have been
widely used in cryptography and derandomization, more recently Agrawal [1] and
Agrawal and Watanabe [2] showed that they are also useful for improving our
understanding of ${\rm NP}$-completeness. In this paper, we provide further
applications of such hardness assumptions.
### 1.1. Length-Increasing Reductions
A language is ${\rm NP}$-complete if every language in ${\rm NP}$ is reducible
to it. While there are several ways to define the notion of reduction, the
most common definition uses polynomial-time computable many-one functions.
Many natural problems that arise in practice have been shown to be NP-complete
using polynomial-time computable many-one reductions. However, it has been
observed that all known ${\rm NP}$-completeness results hold when we restrict
the notion of reduction. For example, $\mathrm{SAT}$ is complete under
polynomial-time reductions that are one-to-one and length-increasing. In fact,
all known many-one complete problems for ${\rm NP}$ are complete under this
type of reduction [9]. This raises the following question: are there languages
that are complete under polynomial-time many-one reductions but not complete
under polynomial-time, one-to-one, length-increasing reductions? Berman [8]
showed that every many-one complete set for ${\rm E}$ is complete under one-
to-one, length-increasing reductions. Thus for ${\rm E}$, these two
completeness notions coincide. A weaker result is known for ${\rm NE}$.
Ganesan and Homer [17] showed that all ${\rm NE}$-complete sets are complete
via one-to-one reductions that are exponentially honest.
For NP, until recently there had not been any progress on this question.
Agrawal [1] showed that if one-way permutations exist, then all NP-complete
sets are complete via one-to-one, length-increasing reductions that are
computable by polynomial-size circuits. Hitchcock and Pavan [20] showed that
${\rm NP}$-complete sets are complete under length-increasing P/poly
reductions under the measure hypothesis on ${\rm NP}$ [26]. Recently Buhrman
et al. improved the latter result to show that if the measure hypothesis
holds, then all NP-complete sets are complete via length-increasing, ${\rm
P}/$-computable functions with $\log\log n$ bits of advice [10]. More
recently, Agrawal and Watanabe [2] showed that if there exist regular one-way
functions, then all NP-complete sets are complete via one-one, length-
increasing, P/poly-computable reductions. All the hypotheses used in these
works require the existence of an almost-everywhere hard language or an
average-case hard language in ${\rm NP}$.
In the first part of this paper, we consider hypotheses that only concern the
worst-case hardness of languages in ${\rm NP}$. Our first hypothesis concerns
the deterministic time complexity of languages in ${\rm NP}$. We show that if
there is a language in ${\rm NP}$ for which every correct algorithm spends
more than $2^{n^{\epsilon}}$ time at almost all lengths, then NP-complete
languages are complete via P/poly-computable, length-increasing reductions.
The second hypothesis concerns nondeterministic circuit complexity of
languages in co-NP. We show that if there is a language in co-NP that cannot
be solved by nondeterministic polynomial-size circuits, then all NP-complete
sets are complete via length-increasing P/poly-computable reductions. For more
formal statements of the hypotheses, we refer the reader to Section 3. We
stress that these hypotheses require only worst-case hardness. The worst-case
hardness is of course required at every length, a technical condition that is
necessary in order to build a reduction that works at every length rather than
just infinitely often.
### 1.2. Turing Reductions versus Many-One Reductions
In the second part of the paper we study the completeness notion obtained by
allowing a more general notion of reduction—Turing reduction. Informally, with
Turing reductions an instance of a problem can be solved by asking
polynomially many (adaptive) queries about the instances of the other problem.
A language in ${\rm NP}$ is Turing complete if there is a polynomial-time
Turing reduction to it from every other language in ${\rm NP}$. Though many-
one completeness is the most commonly used completeness notion, Turing
completeness also plays an important role in complexity theory. Several
properties of Turing complete sets are closely tied to the separation of
complexity classes. For example, Turing complete sets for EXP are sparse if
and only if EXP contains polynomial-size circuits. Moreover, to capture our
intuition that a complete problem is easy, then the entire class is easy,
Turing reductions seem to be the “correct” reductions to define completeness.
In fact, the seminal paper of Cook [13] used Turing reductions to define
completeness, though Levin [25] used many-one reductions.
This raises the question of whether there is a Turing complete language for
${\rm NP}$ that is not many-one complete. Ladner, Lynch and Selman [24] posed
this question in 1975, thus making it one of the oldest problems in complexity
theory. This question is completely resolved for exponential time classes such
as ${\rm EXP}$ and ${\rm NEXP}$ [33, 12]. We know that for both these classes
many-one completeness differs from Turing-completeness. However progress on
the ${\rm NP}$ side has been very slow. Lutz and Mayordomo [27] were the first
to provide evidence that Turing completeness differs from many-one
completeness. They showed that if the measure hypothesis holds, then the
completeness notions differ. Since then a few other weaker hypotheses have
been used to achieve the separation of Turing completeness from many-one
completeness [3, 30, 31, 21, 29].
All the hypotheses used in the above works are considered “strong” hypotheses
as they require the existence of an almost everywhere hard language in ${\rm
NP}$. That is, there is a language $L$ in ${\rm NP}$ and every algorithm that
decides $L$ takes exponential-time an all but finitely many strings. A
drawback of these hypotheses is that we do not have any candidate languages in
${\rm NP}$ that are believed to be almost everywhere hard.
It has been open whether we can achieve the separation using more believable
hypotheses that involve average-case hardness or worst-case hardness. None of
the proof techniques used earlier seem to achieve this, as the they crucially
depend on the almost everywhere hardness.
In this paper, for the first time, we achieve the separation between Turing
completeness and many-one completeness using average-case and worst-case
hardness hypotheses. We consider two hypotheses. The first hypothesis states
that there exist $2^{n^{\epsilon}}$-secure one-way permutations and the second
hypothesis states that there is a language in
$\mathrm{NEEE}\cap\mathrm{co}\mathrm{NEEE}$ that can not be solved in triple
exponential time with logarithmic advice, i.e,
$\mathrm{NEEE}\cap\mathrm{co}\mathrm{NEEE}\not\subseteq\mathrm{EEE}/\log$. We
show that if both of these hypothesis are true, then there is a Turing
complete language in ${\rm NP}$ that is not many-one complete.
The first hypothesis is an average-case hardness hypothesis and has been
studied extensively in past. The second hypothesis is a worst-case hardness
hypothesis. At first glance, this hypothesis may look a little esoteric,
however, it is only used to obtain hard tally languages in ${\rm
NP}\cap{\mbox{\rm co-NP}}$ that are sufficiently sparse. Similar hypotheses
involving double and triple exponential-time classes have been used earlier in
the literature [7, 15, 19, 14].
We use length-increasing reductions as a tool to achieve the separation of
Turing completeness from many-one completeness. We first show that if one-way
permutations exist then ${\rm NP}$-complete sets are complete via length-
increasing, quasipolynomial-time computable reductions. We then show that if
the second hypothesis holds, then there is a Turing complete language for
${\rm NP}$ that is not complete via quasi polynomial-time, length-increasing
reductions. Combining these two results we obtain our separation result.
## 2\. Preliminaries
In the paper, we use the binary alphabet $\Sigma=\\{0,1\\}$. Given a language
$A$, $A_{n}$ denotes the characteristic sequence of $A$ at length $n$. We also
view $A_{n}$ as a boolean function from $\Sigma^{n}$ to $\Sigma$. For
languages $A$ and $B$, we say that $A=\mbox{\rm\tiny io}{B}$, if $A_{n}=B_{n}$
for infinitely many $n$. For a complexity class $\mathcal{C}$, we say that
$A\in\mbox{\rm\tiny io}{\mathcal{C}}$ if there is a language
$B\in{\mathcal{C}}$ such that $A=\mbox{\rm\tiny io}{B}$.
For a boolean function $f:\Sigma^{n}\rightarrow\Sigma$, $CC(f)$ is the
smallest number $s$ such that there is circuit of size $s$ that computes $f$.
A function $f$ is quasipolynomial time computable (${\rm QP}$-computable) if
can be computed deterministically in time $O(2^{\log^{O(1)}n})$. We will use
the triple exponential time class $\mathrm{EEE}={\rm
DTIME}(2^{2^{2^{O(n)}}})$, and its nondeterministic counterpart
$\mathrm{NEEE}$.
A language $L$ is in ${\rm NP}/\mathrm{poly}$ if there is a polynomial-size
circuit $C$ and a polynomial $p$ such that for every $x$, $x$ is in $L$ if and
only if there is a $y$ of length $p(|x|)$ such that $C(x,y)=1$.
Our proofs make use a variety of results from approximable sets, instance
compression, derandomization and hardness amplification. We mention the
results that we need.
###### Definition 2.1.
A language $A$ is $t(n)$-time 2-approximable [6] if there is a function $f$
computable in time $t(n)$ such that for all strings $x$ and $y$, $f(x,y)\neq
A(x)A(y)$.
A language $A$ is io-lengthwise t(n)-time 2-approximable if there is a
function $f$ computable in time $t(n)$ such that for infinitely many $n$, for
every pair of $n$-bit strings $x$ and $y$, $f(x,y)\neq A(x)A(y)$.
Amir, Beigel, Gasarch [4] proved that every polynomial-time 2-approximable set
is in ${\rm P}/\mathrm{poly}$. Their proof also implies the following
extension for a superpolynomial function $t(n)$.
###### Theorem 2.2 ([4]).
If $A$ is io-lengthwise $t(n)$-time 2-approximable, then for infinitely many
$n$, $CC(A_{n})\leq t^{2}(n)$.
Given a language $H^{\prime}$ in co-NP, let $H$ be $\\{\langle
x_{1},\cdots,x_{n}\rangle~{}|~{}|x_{1}|=\cdots=|x_{n}|=n,x_{i}\in
H^{\prime}\\}$. Observe that a $n$-tuple consisting of strings of length $n$
can be encoded by a string of length $n^{2}$. From now we view a string of
length $n^{2}$ as an $n$-tuple of strings of length $n$.
###### Theorem 2.3 ([16, 11]).
Let $H$ and $H^{\prime}$ be defined as above. Suppose there is a language $L$,
a polynomial-size circuit family $\\{C_{m}\\}$, and a polynomial $p$ such that
for infinitely many $n$, for every $x\in\Sigma^{n^{2}}$, $x$ is in $H$ if and
only if there is a string $y$ of length $p(n)$ such that $C(x,y)$ is in
$L^{\leq n}$. Then $H^{\prime}$ is in $\mbox{\rm\tiny io}{{\rm NP}}/poly$.
The proof of Theorem 2.3 is similar to the proofs in [16, 11]. The difference
is rather than having a polynomial-time many-one reduction, here we have a
${\rm NP}/\mathrm{poly}$ many-one reduction which works infinitely often. The
nondeterminism and advice in the reduction can be absorbed into the final
${\rm NP}/\mathrm{poly}$ decision algorithm. The ${\rm NP}/\mathrm{poly}$
decision algorithm works infinitely often, corresponding to when the ${\rm
NP}/\mathrm{poly}$ reduction works.
###### Definition 2.4.
A function $f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}^{m}$ is $s$-secure if for
every $\delta<1$, every $t\leq\delta s$, and every circuit
$C:\\{0,1\\}^{n}\rightarrow\\{0,1\\}^{m}$ of size $t$, $\Pr[C(x)=f(x)]\leq
2^{-m}+\delta$. A function $f:\\{0,1\\}^{*}\rightarrow\\{0,1\\}^{*}$ is
$s(n)$-secure if it is $s(n)$-secure at all but finitely many length $n$.
###### Definition 2.5.
An $s(n)$-secure one-way permutation is a polynomial-time computable bijection
$\pi:\\{0,1\\}^{*}\rightarrow\\{0,1\\}^{*}$ such that $|\pi(x)|=|x|$ for all
$x$ and $\pi^{-1}$ is $s(n)$-secure.
Under widely believed average-case hardness assumptions about the hardness of
the RSA cryptosystem or the discrete logarithm problem, there is a secure one-
way permutation [18].
###### Definition 2.6.
A pseudorandom generator (PRG) family is a collection of functions
$G=\\{G_{n}:\\{0,1\\}^{m(n)}\rightarrow\\{0,1\\}^{n}\\}$ such that $G_{n}$ is
uniformly computable in time $2^{O(m(n))}$ and for every circuit of $C$ of
size $n$,
$\left|\Pr_{x\in\\{0,1\\}^{n}}[C(x)=1]-\Pr_{y\in\\{0,1\\}^{m(n)}}[C(G_{n}(y))=1\right|\leq\frac{1}{n}.$
There are many results that show that the existence of hard functions in
exponential time implies PRGs exist. We will use the following.
###### Theorem 2.7 ([28, 23]).
If there is a language $A$ in ${\rm E}$ such that $CC(A_{n})\geq
2^{n^{\epsilon}}$ for all sufficiently large $n$, then there exist a constant
$k$ and a PRG family
$G=\\{G_{n}:\\{0,1\\}^{\log^{k}n}\rightarrow\\{0,1\\}^{n}\\}$.
## 3\. Length-Increasing Reductions
In this section we provide evidence that many-one complete sets for NP are
complete via length-increasing reductions. We use the following hypotheses.
Hypothesis 1. There is a language $L$ in ${\rm NP}$ and a constant
$\epsilon>0$ such that $L$ is not in $\mbox{\rm\tiny io}{{\rm
DTIME}}(2^{n^{\epsilon}})$.
Informally, this means that every algorithm that decides $L$ takes more than
$2^{n^{\epsilon}}$-time on at least one string at every length.
Hypothesis 2. There is a language $L$ in co-NP such that $L$ is not in
$\mbox{\rm\tiny io}{{\rm NP}}/\mathrm{poly}$.
This means that every nondeterministic polynomial size circuit family that
attempts to solve $L$ is wrong on on at least one string at each length.
We will first consider the following variant of Hypothesis 1.
Hypothesis 3. There is a language $L$ in ${\rm NP}$ and a constant
$\epsilon>0$ such that for all but finitely many $n$,
$CC(L_{n})>2^{n^{\epsilon}}$.
We will first show that Hypothesis $3$ holds, then ${\rm NP}$-complete sets
are complete via length-increasing reductions. Then we describe how to modify
the proof to derive the same consequence under Hypothesis 1. We do this
because the proof is much cleaner with Hypothesis $3$. To use Hypothesis $1$
we have to fix encodings of boolean formulas with certain properties.
### 3.1. If ${\rm NP}$ has Subexponentially Hard Languages
###### Theorem 3.1.
If there is a language $L$ in ${\rm NP}$ and an $\epsilon>0$ such that for all
but finitely many $n$, $CC(L_{n})>2^{n^{\epsilon}}$, then all ${\rm
NP}$-complete sets are complete via length-increasing, P/poly reductions.
###### Proof 3.2.
Let $A$ be a ${\rm NP}$-complete set that is decidable in time $2^{n^{k}}$.
Let $L$ be a language in ${\rm NP}$ that requires $2^{n^{\epsilon}}$-size
circuits at every length. Since $\mathrm{SAT}$ is complete via polynomial-
time, length-increasing reductions, it suffices to exhibit a length-
increasing, ${\rm P}/\mathrm{poly}$-reduction from $\mathrm{SAT}$ to $A$.
Let $\delta=\frac{\epsilon}{2k}$. Consider the following intermediate language
$S=\left\\{\langle x,y,z\rangle\;\left|\;|x|=|z|,|y|=|x|^{\delta},\mbox{\tt
MAJ}[L(x),\mathrm{SAT}(y),L(z)]=1\right.\right\\}.$
Clearly $S$ is in ${\rm NP}$. Since $A$ is ${\rm NP}$-complete, there is a
many-one reduction $f$ from $S$ to $A$. We will first show that at every
length $n$ there exist strings on which the reduction $f$ must be honest. Let
$T_{n}=\left\\{\langle
x,z\rangle\in\\{0,1\\}^{n}\times\\{0,1\\}^{n}\;\left|\;L(x)\neq
L(z),~{}\forall y\in\\{0,1\\}^{n^{\delta}}~{}|f(\langle
x,y,z\rangle)|>n^{\delta}\right.\right\\}$
###### Lemma 3.3.
For all but finitely many $n$, $T_{n}\neq\varnothing$.
Assuming that the above lemma holds, we complete the proof of the theorem.
Given a length $m$, let $n=m^{1/\delta}$. Let $\langle x_{n},z_{n}\rangle$ be
the first tuple from $T_{n}$. Consider the following reduction from
$\mathrm{SAT}$ to $A$: Given a string $y$ of length $m$, the reduction outputs
$f(\langle x_{n},y,z_{n}\rangle)$. Given $x_{n}$ and $y_{n}$ as advice, this
reduction can be computed in polynomial time. Since $n$ is polynomial in $m$,
this is a P/poly reduction.
By the definition of $T_{n}$, $L(x_{n})\neq L(z_{n})$. Thus $y\in\mathrm{SAT}$
if and only if $\langle x_{n},y,z_{n}\rangle\in S$, and so $y$ is in
$\mathrm{SAT}$ if and only if $f(\langle x_{n},y,z_{n}\rangle)$ is in $A$.
Again, by the definition of $T_{n}$, for every $y$ of length $m$, the length
of $f(\langle x_{n},y,z_{n}\rangle)$ is bigger than $n^{\delta}=m$. Thus there
is a P/poly-computable, length-increasing reduction from $\mathrm{SAT}$ to
$A$. This, together with the proof of Lemma 3.3 we provide next, complete the
proof of Theorem 3.1.
###### Proof 3.4 (Proof of Lemma 3.3).
Suppose $T_{n}=\varnothing$ for infinitely many $n$. We will show that this
yields a length-wise 2-approximable algorithm for $L$ at infinitely many
lengths. This enables us to contradict the hardness of $L$. Consider the
following algorithm:
1. (1)
Input $x$, $z$ with $|x|=|z|=n$.
2. (2)
Find a $y$ of length $n^{\delta}$ such that $|f(\langle x,y,x\rangle)|\leq
n^{\delta}$.
3. (3)
If no such $y$ is found, Output $10$.
4. (4)
If $y$ is found, then solve the membership of $f(\langle x,y,z\rangle)$ in
$A$. If $f(\langle x,y,z\rangle)\in A$, then output $00$, else output $11$.
We first bound the running time of the algorithm. Step 2 takes
$O(2^{n^{\delta}})$ time. In Step 4, we decide the membership of $f(\langle
x,y,z\rangle)$ in $A$. This step is reached only if the length of $f(\langle
x,y,z\rangle)$ is at most $n^{\delta}$. Thus the time taken to for this step
is $(2^{n^{\delta}})^{k}\leq 2^{n^{\epsilon/2}}$ time. Thus the total time
taken by the algorithm is bounded by $2^{n^{\epsilon}/2}$.
Consider a length $n$ at which $T_{n}=\varnothing$. Let $x$ and $z$ be any
strings at this length. Suppose for every $y$ of length $n^{\delta}$, the
length of $f(\langle x,y,z\rangle)$ is at least $n^{\delta}$. Then it must be
the case that $L(x)=L(z)$, otherwise the tuple $\langle x,z\rangle$ belongs to
$T_{n}$. Thus if the above algorithm fails to find $y$ in Step 2, then
$L(x)L(z)\neq 10$.
Suppose the algorithm succeeds in finding a $y$ in Step 2. If $f(\langle
x,y,z\rangle)\in A$, then at least one of $x$ or $z$ must belong to $L$. Thus
$L(x)L(z)\neq 00$. Similarly, if $f(\langle x,y,z\rangle)\notin A$, then at
least one of $x$ or $z$ does not belong to $L$, and so $L(x)L(z)\neq 11$.
Thus $L$ is 2-approximable at length $n$. If there exist infinitely many
lengths $n$, at which $T_{n}$ is empty, then $L$ is infinitely-often, length-
wise, $2^{n^{\epsilon}/2}$-time approximable. By Theorem 2.2, $L$ has circuits
of size $2^{n^{\epsilon}}$ at infinitely many lengths.
Now we will describe how to modify the proof if we assume that Hypothesis 1
holds. Let $L$ be the hard language guaranteed by the hypothesis. We will work
with 3-$\mathrm{SAT}$. Fix an encoding of 3CNF formulas such that formulas
with same numbers of variables can be encoded as strings of same length.
Moreover, we require that the formulas $\phi(x_{1},\cdots,x_{n})$ and
$\phi(b_{1},\cdots,b_{i},x_{i+1},\cdots,x_{n})$ can be encoded as strings of
same length, where $b_{i}\in\\{0,1\\}$. Fix a reduction $f$ from $L$ to
3-$\mathrm{SAT}$ such that all strings of length $n$ are mapped to formulas
with $n^{r}$ variables, $r\geq 1$. Let
$3\mbox{-}\mathrm{SAT}^{\prime}=3\mbox{-}\mathrm{SAT}\cap\cup_{r}\Sigma^{n^{r}}$.
It follows that that if there is an algorithm that decides
3-$\mathrm{SAT}^{\prime}$ such that for infinitely many $n$ the algorithm runs
in $2^{n^{\epsilon}}$ time on all formulas with $n^{r}$ variables, then $L$ is
in $\mbox{\rm\tiny io}{{\rm DTIME}}(2^{n^{\epsilon}})$.
Now the proof proceeds exactly same as before except that we use
3-$\mathrm{SAT}^{\prime}$ instead of $L$, i.e, our intermediate language will
be
$\\{\langle x,y,z\rangle~{}|~{}\mbox{\tt
MAJ}[3\mbox{-}\mathrm{SAT}^{\prime}(x),\mathrm{SAT}(y),3\mbox{-}\mathrm{SAT}^{\prime}(z)]\\}=1.$
Consider the set $T_{n}$ as before. It follows that if $T_{n}$ is empty at
infinitely many lengths, then for infinitely many $n$,
3-$\mathrm{SAT}^{\prime}$ is 2-approximable on formulas with $n^{r}$
variables. Now we can use the disjunctive self-reducibility of
3-$\mathrm{SAT}^{\prime}$ to show that there is a an algorithm that solves
3-$\mathrm{SAT}^{\prime}$ and for infinitely many $n$, this algorithm runs in
${\rm DTIME}(2^{n^{\epsilon}})$-time on formulas with $n^{r}$ variables. This
contradicts the hardness of $L$. This gives the following theorem.
###### Theorem 3.5.
If there is a language in ${\rm NP}$ that is not in $\mbox{\rm\tiny io}{{\rm
DTIME}}(2^{n^{\epsilon}})$, then all ${\rm NP}$-complete sets are complete via
length-increasing P/poly reductions.
### 3.2. If co-NP is Hard for Nondeterministic Circuits
In this subsection we show that Hypothesis 2 also implies that all NP-complete
sets are complete via length-increasing reductions.
###### Theorem 3.6.
If there is a language $L$ in co-NP that is not in $\mbox{\rm\tiny io}{{\rm
NP}}/poly$, then ${\rm NP}$-complete sets are complete via P/poly-computable,
length-increasing reductions.
###### Proof 3.7.
We find it convenient to work with co-NP rather than ${\rm NP}$. We will show
that all co-NP-complete languages are complete via P/poly, length-increasing
reductions.
Let $H^{\prime}$ be a language in co-NP that is not in $\mbox{\rm\tiny
io}{{\rm NP}}/\mathrm{poly}$. Let $H$ be
$\\{\langle x_{1},\cdots,x_{n}\rangle~{}|~{}\forall 1\leq i\leq n,[x_{i}\in
H^{\prime}\mbox{ and }|x_{i}|=n]\\}.$
Note that every $n$-tuple that may potentially belong to $H$ can be encoded by
a string of length $n^{2}$.
Let $S=0H^{\prime}\cup 1\overline{SAT}$. It is easy to show that $S$ is in co-
NP and $S$ is not in $\mbox{\rm\tiny io}{{\rm NP}}/poly$. Observe that $S$ is
co-NP-complete via length-increasing reductions. Let $A$ be any co-NP-complete
language. It suffices to exhibit a length-increasing reduction from $S$ to
$A$.
Consider the following intermediate language:
$L=\\{\langle x,y,z\rangle~{}|~{}|x|=|z|=|y|^{2},\mbox{\tt MAJ}[x\in H,y\in
S,z\in H]=1\\}.$
Clearly the above language is in co-NP. Let $f$ be a many-one reduction from
$L$ to $\overline{A}$. As before we will first show at every length $n$ that
there exits strings $x$ and $z$ such that for every $y$ in $S$ the length of
$f(\langle x,y,z\rangle)$ is at least $n$.
###### Lemma 3.8.
For all but finitely many $n$, there exist two strings $x_{n}$ and $z_{n}$ of
length $n^{2}$ with $H(x_{n})\neq H(z_{n})$ and for every $y\in S^{n}$,
$|f(\langle x_{n},y,z_{n}\rangle)|>n$.
###### Proof 3.9.
Suppose not. Then there exist infinitely many lengths $n$ at which for every
pair of strings (of length $n^{2}$) $x$ and $z$ with $H(x)\neq H(z)$, there
exist a $y$ of length $n$ such that $|f(x,y,z)|\leq n$.
From this we obtain a ${\rm NP}/\mathrm{poly}$-reduction from $H$ to $A$ such
that for infinitely many $n$, for every $x$ of length $n^{2}$, $|f(x)|\leq n$.
By Theorem 2.3, this implies that $H^{\prime}$ is in $\mbox{\rm\tiny io}{{\rm
NP}}/\mathrm{poly}$. We now describe the reduction. Given $n$ let $z_{n}$ be a
string (of length $n^{2}$) that is not in $H$.
1. (1)
Input $x,|x|=n^{2}$. Advice: $z_{n}$.
2. (2)
Guess a string $y$ of length $n$.
3. (3)
If $|f(\langle x,y,z_{n}\rangle)|>n$, the output $\bot$.
4. (4)
Output $f(\langle x,y,z_{n})$.
Suppose $x\in H$. Since $z_{n}\notin H$, there exists a string $y$ of length
$n$ such that $y\in S$ and $|f(\langle x,y,z_{n}\rangle)|\leq n$. Consider a
path that correctly guesses such a $y$. Since $z_{n}\notin H$, and $y\in S$,
$\langle x,y,z_{n}\rangle\in L$. Thus $f(\langle x,y,z_{n}\rangle)\in A^{\leq
n}$. Thus there exists at least one path on which the reduction outputs a
string from $L\cap\Sigma^{\leq n}$. Now consider the case $x\notin H$. On any
path, the reduction either outputs $\bot$ or outputs $f(\langle
x,y,z_{n}\rangle)$. Since both $z_{n}$ and $x$ are not in $H$, $\langle
x,y,z\rangle\notin L$. Thus $f(\langle x,y,z_{n}\rangle)\notin A$ for any $y$.
Thus there is a ${\rm NP}/\mathrm{poly}$ many-one reduction from $H$ to $L$
such that for infinitely many $n$, the output of the reduction, on strings of
length $n^{2}$, on any path is at most $n$. By Theorem 2.3, this places
$H^{\prime}$ in $\mbox{\rm\tiny io}{{\rm NP}}/\mathrm{poly}$.
Thus for all but finitely many lengths $n$, there exist strings $x_{n}$ and
$z_{n}$ of length $n^{2}$ with $H(x_{n})\neq H(z_{n})$ and for every $y\in
S^{n}$, the length of $f(\langle x_{n},y,z_{n}\rangle)$ is at least $n$.
This suggests the following reduction $h$ from $S$ to $A$. The reduction will
have $x_{n}$ and $z_{n}$ as advice. Given a string $y$ of length $n$, the
reductions outputs $f(\langle x_{n},y,z_{n}\rangle)$. This reduction is
clearly length-increasing and is length-increasing on every string from $S$.
Thus we have the following lemma.
###### Lemma 3.10.
Consider the above reduction $h$ from $S$ to $A$, for all $y\in S$,
$|h(y)|>|y|$.
Now we show how to obtain a length-increasing reduction on all strings. We
make the following crucial observation.
For all but finitely many $n$, there is a string $y_{n}$ of length $n$ such
that $y_{n}\notin S$ and $|f(\langle x_{n},y_{n},z_{n}\rangle)|>n$.
###### Proof 3.11.
Suppose not. This means that for infinitely many $n$, for every $y$ from
$\overline{S}\cap\Sigma^{n}$, the length of $f(\langle x_{n},y,z_{n}\rangle)$
is less than $n$. Now consider the following algorithm that solves $S$. Given
a string $y$ of length $n$, compute $f(\langle x_{n},y,z_{n}\rangle)$. If the
length of $f(\langle x_{n},y,z_{n}\rangle)>n$, then accept $y$ else reject
$y$.
The above algorithm can be implemented in P/poly given $x_{n}$ and $z_{n}$ as
advice. If $y\in S$, then we know that that the length of $f(\langle
x_{n},y,z_{n}\rangle)$ is bigger than $n$, and so the above algorithm accepts.
If $y\notin S$, then by our assumption, the length of $f(\langle
x_{n},y,z_{n}\rangle)$ is at most $n$. In this case the algorithm rejects $y$.
This shows that $S$ is in ioP/poly which in turn implies that $H^{\prime}$ is
in ioP/poly. This is a contradiction.
Now we are ready to describe our length increasing reduction from $S$ to $A$.
At length $n$, this reduction will have $x_{n}$, $y_{n}$ and $z_{n}$ as
advice. Given a string $y$ of length $n$, the reduction outputs $f(\langle
x_{n},y,z_{n}\rangle)$ if the length of $f(\langle x_{n},y,z_{n}\rangle)$ is
more than $n$. Else, the reduction outputs $f(\langle
x_{n},y_{n},z_{n}\rangle)$.
Since $H(x_{n})\neq H(z_{n})$, $y\in S$ if and only if $f(\langle
x_{n},y,z_{n}\rangle)\in A$. Thus the reduction is correct when it outputs
$f(\langle x_{n},y,z_{n}\rangle)$. The reduction outputs $f(\langle
x_{n},y_{n},z_{n}\rangle)$ only when the length of $f(\langle
x_{n},y,z_{n}\rangle)$ is at most $n$. We know that in this case $y\notin S$.
Since $y_{n}\notin S$, $f(\langle x_{n},y_{n},z_{n})\notin A$.
Thus we have a P/poly-computable, length-increasing from $S$ to $A$. Thus all
co-NP-complete languages are complete via P/poly, length-increasing
reductions. This immediately implies that all ${\rm NP}$-complete languages
are complete via P/poly-computable, length-increasing reductions.
## 4\. Separation of Completeness Notions
In this section we consider the question whether the Turing completeness
differs from many-one completeness for ${\rm NP}$ under two plausible
complexity-theoretic hypotheses:
1. (1)
There exists a $2^{n^{\epsilon}}$-secure one-way permutation.
2. (2)
$\mathrm{NEEE}\cap\mathrm{co}\mathrm{NEEE}\not\subseteq\mathrm{EEE}/\log$.
It turns out that the first hypothesis implies that every many-one complete
language for ${\rm NP}$ is complete under a particular kind of length-
increasing reduction, while the second hypothesis provides us with a specific
Turing complete language that is not complete under the same kind of length-
increasing reduction. Therefore, the two hypotheses together separate the
notions of many-one and Turing completeness for ${\rm NP}$ as stated in the
following theorem.
###### Theorem 4.1.
If both of the above hypotheses are true, there is is a language that is
polynomial-time Turing complete for ${\rm NP}$ but not polynomial-time many-
one complete for ${\rm NP}$.
Theorem 4.1 is immediate from Lemma 4.2 and Lemma 4.3 below.
###### Lemma 4.2.
Suppose $2^{n^{\epsilon}}$-secure one-way permutations exist. Then for every
${\rm NP}$-complete language $A$ and every $B\in{\rm NP}$, there is a
quasipolynomial-time computable, polynomial-bounded, length-increasing
reduction reduction $f$ from $B$ to $A$.
A function $f$ is polynomial-bounded if there is a polynomial $p$ such that
the length of $f(x)$ is at most $p(|x|)$ for every $x$.
###### Lemma 4.3.
If $\mathrm{NEEE}\cap\mathrm{co}\mathrm{NEEE}\nsubseteq\mathrm{EEE}/\log$,
then there is a polynomial-time Turing complete set for ${\rm NP}$ that is not
many-one complete via quasipolynomial-time computable, polynomial-bounded,
length-increasing reductions.
The proof of Lemma 4.2 will appear in the full paper. The remainder of this
section is devoted to proving Lemma 4.3. It is well known that any set $A$
over $\Sigma^{*}$ can be encoded as a tally set $T_{A}$ such that $A$ is
worst-case hard if and only if $T_{A}$ is worst-case hard. For our purposes,
we need an average-case version of the this equivalence. Below we describe
particular encoding of languages using tally sets that is helpful for us and
prove the average-case equivalence.
Let $t_{0}=2$, $t_{i+1}=t_{i}^{2}$ for all $i\in\mathbb{N}$. Let
$\mathcal{T}=\left\\{0^{t_{i}}\;\left|\;i\in\mathbb{N}\right.\right\\}$. For
each $l\in\mathbb{N}$, let
$\mathcal{T}_{l}=\left\\{0^{t_{i}}\;\left|\;2^{l}-1\leq i\leq
2^{l+1}-2\right.\right\\}$. Observe that
$\mathcal{T}=\bigcup_{l=0}^{\infty}\mathcal{T}_{l}$. Given a set
$A\subseteq\\{0,1\\}^{*}$, let
$T_{A}=\left\\{\left.0^{2^{2^{r_{x}}}}\right|x\in A\right\\},$ where $r_{x}$
is the rank index of $x$ in the standard enumeration of $\\{0,1\\}^{*}$. It is
easy to verify that for all $l\in\mathbb{N}$ and every $x$,
$\displaystyle x\in A\cap\\{0,1\\}^{l}$ $\displaystyle\iff$ $\displaystyle
0^{t_{r_{x}}}\in T_{A}\cap\mathcal{T}_{l}.$ (1)
###### Lemma 4.4.
Let $A$ and $T_{A}$ be as above. Suppose there is a quasipolynomial time
algorithm $\mathcal{A}$ such that for every $l$, on an $\epsilon$ fraction of
strings from $\mathcal{T}_{l}$, this algorithm correctly decides the
membership in $T_{A}$, and on the rest of the strings the algorithm outputs “I
do not know”. There is a $2^{2^{2^{k(l+1)}}}$-time algorithm
$\mathcal{A}^{\prime}$ for some constant $k$ that takes one bit of advice and
correctly decides the membership in $A$ on $\frac{1}{2}+\frac{\epsilon}{2}$
fraction of the strings at every length $l$.
We know several results that establish worst-case to average-case connections
for classes such as ${\rm EXP}$ and ${\rm PSPACE}$ [34, 5, 22, 23, 32]. The
following lemma establishes a similar connection for triple exponential time
classes, and can be proved using known techniques.
###### Lemma 4.5.
If $\mathrm{NEEE}\cap\mathrm{co}\mathrm{NEEE}\not\subseteq\mathrm{EEE}/\log$,
then there is language $L$ in $\mathrm{NEEE}\cap\mathrm{co}\mathrm{NEEE}$ such
that no $\mathrm{EEE}/\log$ algorithm can decide $L$, at infinitely many
lengths $n$, on more than $\frac{1}{2}+\frac{1}{n}$ fraction of strings from
$\\{0,1\\}^{n}$.
Now we are ready to prove Lemma 4.3.
###### Proof 4.6 (Proof of Lemma 4.3).
By Lemma 4.5, there is a language
$L\in(\mathrm{NEEE}\cap\mathrm{co}\mathrm{NEEE})-\mathrm{EEE}/\log$ such that
no $\mathrm{EEE}/\log$ algorithm can decide $L$ correctly on more than a
$\frac{1}{2}+\frac{1}{n}$ fraction of the inputs for infinitely many lengths
$n$.
Without loss of generality, we can assume that $L\in{\rm
NTIME}(2^{2^{2^{n}}})\cap\mathrm{co}{\rm NTIME}(2^{2^{2^{n}}})$ Let
$T_{L}=\left\\{\left.0^{2^{2^{r_{x}}}}\right|x\in L\right\\}.$
Clearly, $T_{L}\in{\rm NP}\cap\mathrm{co}{\rm NP}$.
Define $\tau:\mathbb{N}\rightarrow\mathbb{N}$ such that
$\tau(n)=\max\left\\{i\;\left|\;t_{i}\leq n\right.\right\\}$. Now we will
define our Turing complete language. Let
$\mathrm{SAT}_{0}=\left\\{0x\;\left|\;0^{t_{\tau(|x|)}}\notin T_{L}\text{ and
}x\in\mathrm{SAT}\right.\right\\},$
$\mathrm{SAT}_{1}=\left\\{1x\;\left|\;0^{t_{\tau(|x|)}}\in T_{L}\text{ and
}x\in\mathrm{SAT}\right.\right\\}.$
Let $A=\mathrm{SAT}_{0}\cup\mathrm{SAT}_{1}$. Since $L$ is in ${\rm
NP}\cap{\mbox{\rm co-NP}}$, $A$ is in ${\rm NP}$. The following is a Turing
reduction from $\mathrm{SAT}$ to $A$: Given a formula $x$, ask queries $0x$
and $1x$, and accept if and only if at least one them is in $A$. Thus $A$ is
polynomial-time $2$-$\mathrm{tt}$ complete for ${\rm NP}$.
Suppose $A$ is complete via length-increasing, polynomial-bounded,
quasipolynomial-time reductions. Then there is such a reduction $f$ from
$\\{0\\}^{*}$ to $A$. There is a constant $d$ such that $f$ is $n^{d}$-bounded
and runs in quasipolynomial time.
The following observation is easy to prove. Let $y\in\\{0,1\\}^{*}$ and
$b\in\\{0,1\\}$ be such that $f(0^{t_{i}})=by$. Then $0^{t_{\tau(|y|)}}\in
T_{L}$ if and only if $b=1$.
Fix a length $l$. We will describe a quasipolynomial-time algorithm that will
decide the membership in $T_{L}$ on at least $\frac{1}{\log d}$ fraction of
strings from $\mathcal{T}_{l}$, and says “I do not know” on other strings. By
the Lemma 4.4, this implies that there is $\mathrm{EEE}/1$ algorithm that
decides $L$ on more than $\frac{1}{2}+\frac{1}{2\log d}$ fraction of strings
from $\\{0,1\\}^{l}$. This contradicts the hardness of $L$ and completes the
proof.
Let $s=2^{l}-1$ and $r=2^{l+1}-2$. Recall that
$\mathcal{T}_{l}=\left\\{0^{t_{i}}\;\left|\;s\leq i\leq r\right.\right\\}$.
Divide $\mathcal{T}_{l}$ in sets $T_{0},T_{2},\cdots T_{r}$ where
$T_{k}=\left\\{0^{t_{i}}\;\left|\;s+k\log d\leq r+(k+1)\log
d\right.\right\\}$. This gives at least $\frac{2^{l}}{\log d}$ sets. Consider
the following algorithm that decides $T_{L}$ on strings from
$\mathcal{T}_{l}$: Let $0^{t_{j}}$ be the input. Say, it lies in the set
$T_{k}$. Compute $f(0^{t_{s+k\log d}})=by$. If $t_{\tau(|y|)}\neq t_{j}$, then
output “I do not know”. Otherwise, accept $0^{t_{j}}$ if and only if $b=1$. By
Observation 4.6 this algorithm never errs. Since $f$ is computable in
quasipolynomial time, this algorithm runs in quasipolynomial time. Finally,
observe that $t_{\tau(|y|)}$ lies between $t_{s+k\log d}$ and $t_{s+(k+1)\log
d}$. Thus for every $k$, $0\leq k\leq r$, there is at least one string from
from $T_{k}$ on which the above algorithm correctly decides $T_{L}$. Thus the
above algorithm correctly decides $T_{L}$ on at least $\frac{1}{\log d}$
fraction of strings from $\mathcal{T}_{l}$, and never errs.
## References
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* [8] L. Berman. Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, 1977.
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|
arxiv-papers
| 2010-01-04T20:55:05 |
2024-09-04T02:49:07.392756
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaoyang Gu, John M. Hitchcock, and A. Pavan",
"submitter": "John Hitchcock",
"url": "https://arxiv.org/abs/1001.0117"
}
|
1001.0156
|
# Entanglement distribution maximization over one-side Gaussian noisy channel
Xiang-Bin Wang xbwang@mail.tsinghua.edu.cn Zong-Wen Yu Department of Physics
and the Key Laboratory of Atomic and Nanosciences, Ministry of Education,
Tsinghua University, Beijing 100084, China Jia-Zhong Hu Department of
Physics and the Key Laboratory of Atomic and Nanosciences, Ministry of
Education, Tsinghua University, Beijing 100084, China
###### Abstract
The optimization of entanglement evolution for two-mode Gaussian pure states
under one-side Gaussian map is studied. Even there isn’t complete information
about the one-side Gaussian noisy channel, one can still maximize the
entanglement distribution by testing the channel with only two specific
states.
###### pacs:
03.65.Ud, 03.67.Mn, 03.65.Yz
Introduction. The study of properties about quantum entanglement has drawn
much interest for a long timenielsen ; Vedral ; Wooters ; Yutin . Although
initially quantum information processing(QIP) was studied with discrete
quantum states, it was then extended to the continuous variable (CV) quantum
statesbw . So far, many concepts and results with 2-level quantum systems have
been extended to the continuous variable case with parallel results, such as
the quantum teleportationcvqt1 , the inseparability criteionDuan , the degree
of entanglementGiedke3 ; Marian , the entanglement purificationSimon2 ; Plenio
; Fiura , the entanglement sudden deathcsd1 , the characterization of Gaussian
mapsGiedke2 , and so on. However, this does not mean all results with 2-level
quantum systems can have parallel results for Gaussian states.
Entanglement distribution is the first step towards many novel tasks in
quantum communication and QIPnielsen . In practice, there is no perfect
channel for entanglement distribution. Naturally, how to maximize the
entanglement after distribution is an important question in practical QIP. If
we distribute the quantum entanglement by sending one part of the entangled
state to a remote place through noisy channel, we can use the model of one-
side noisy channel, or one-side map.
Given the factorization law presented by Konrad et al1 , such a maximization
problem for entanglement distribution over one-side map does not exist for the
$2\times 2$ system because any one-side map will produce the same entanglement
on the output states provided that the entanglement of the input pure states
are same. The result has been experimentally testedsci and also been extended
Song recently. However, such a factorization does not hold for the continuous
variable state as shown below. In this work, we consider the following
problem: Initially we have a bipartite Gaussian pure state. Given a one-side
Gaussian map (or a one-side Gaussian noisy channel), how to maximize the
entanglement of the output state by taking a Gaussian unitary transformation
on the input mode before it is sent to the noisy channel. We find that by
testing the channel with only two different states, if a certain result is
verified, then we can find the right Gaussian unitary transformation which
optimizes the entanglement evolution for any input Gaussian pure state. That
is to say, we can maximize the output entanglement even though we don’t have
the full information of the one-side map. In what follows we shall first show
by specific example that the factorization law for $2\times 2$ system
presented by Konrad et al1 does not hold for Gaussian states. We then present
an upper bound of the entanglement evolution for initial Gaussian pure states.
Based on this, we study how to optimize the entanglement evolution over one-
side Gaussian map by taking a local Gaussian unitary transformation to the
mode before sent to the noisy channel.
Output entanglement of one-side Gaussian map and single-mode squeezing. Most
generally, a two-mode Gaussian pure state is
$|g(U,V,q)\rangle=U\otimes V|\chi(q)\rangle$ (1)
and
$|\chi(q)\rangle=\sqrt{1-q^{2}}e^{qa_{1}^{\dagger}a_{2}^{\dagger}}|00\rangle$
($-1\leq q\leq 1$) is a two-mode squeezed state (TMSS). We define map $\$$ as
a Gaussian map which acts on one mode of the state only. A Gaussian map
changes a Gaussian state to a Gaussian state only. In whatever reasonable
entanglement measure, the entanglement of a Gaussian pure state in the form of
Eq.(1) is uniquely determined by $q$. Therefore, we define the characteristic
value of entanglement of the Gaussian pure state
$\rho(q)=|g(U,V,q)\rangle\langle g(U,V,q)|$ as
$E[\rho(q)]=|q|^{2}.$ (2)
On the other hand, any bipartite Gaussian pure state is fully characterized by
its covariance matrix (CM). Suppose the CM of state $U\otimes
V|\chi(q)\rangle$ is
$\displaystyle\ \Lambda=\left(\begin{array}[]{cc}A&C\\\
C^{T}&B\end{array}\right),$ (5)
$|q|^{2}$ is uniquely determined by $|A|$ (the determinant of the matrix $A$).
So, to compare the entanglement of two Gaussian pure state, we only need to
compare $|A|$ value of their covariance matrices.
We start with the projection operator $\hat{T}_{k}(q_{\alpha})$ which acts on
mode $k$ only:
$\hat{T}_{k}(q_{\alpha})=\sum^{\infty}_{n=0}q^{n}_{\alpha}|n\rangle\langle
n|=q_{\alpha}^{a_{k}^{\dagger}a_{k}}.$ (6)
This operator has an important mathematical property
$\hat{T}_{k}(q_{\alpha})(a_{k}^{\dagger},a_{k})\hat{T}_{k}^{-1}(q_{\alpha})=(q_{\alpha}a_{k}^{\dagger},a_{k}/q_{\alpha})$
(7)
which shall be used latter in this paper. For simplicity, we sometimes omit
the subscripts of states and/or operators provided that the omission does not
affect the clarity.
Define the one-mode squeezed operator
$\mathcal{S}(r)=e^{r({a^{\dagger}}^{2}-a^{2})}$ where $r$ is a real number and
bipartite state
$|\psi_{r}(q_{0})\rangle=I\otimes\mathcal{S}(r)|\chi(q_{0})\rangle$. We have
Theorem 1. Consider the one-side map $I\otimes\hat{T}(q_{1})$ acting on the
initial state $|\psi_{r}(q_{0})\rangle$. The entanglement for the outcome
state $I\otimes\hat{T}(q_{1})|\psi_{r}(q_{0})\rangle$ is a descending function
of $|r|$. Mathematically, it is to say that if $|r_{1}|>|r_{2}|$ then
$E[I\otimes\hat{T}(q_{1})|\psi_{r_{1}}(q_{0})\rangle]<E[I\otimes\hat{T}(q_{1})|\psi_{r_{2}}(q_{0})\rangle].$
(8)
This theorem actually shows that there isn’t a factorization law similar to
that in $2\times 2$ states for the continuous variable states, in whatever
good entanglement measure. Using Backer-Compbell-Horsdorff (BCH) formula, up
to a normalization factor, we have
$|\psi_{r}(q_{0})\rangle=e^{-\frac{1}{2}{a_{1}^{\dagger}}^{2}q_{0}^{2}\tanh(2r)+\frac{1}{2}{a_{2}^{\dagger}}^{2}\tanh(2r)+\frac{q_{0}a_{1}^{\dagger}a_{2}^{\dagger}}{\cosh(2r)}}|00\rangle.$
(9)
Detailed derivation of this identity is given in the appendix. Based on
Eq.(6), the one-side map $I\otimes\hat{T}(q_{1})$ changes state
$|\psi_{r}(q_{0})\rangle$ into
$|\psi^{\prime}\rangle=e^{f_{1}{a_{1}^{\dagger}}^{2}+f_{2}{a_{2}^{\dagger}}^{2}+f_{3}a_{1}^{\dagger}a_{2}^{\dagger}}|00\rangle$
(10)
where $f_{1}=-\frac{1}{2}q_{0}^{2}\tanh(2r)$,
$f_{2}=\frac{1}{2}q_{1}^{2}\tanh(2r)$, and
$f_{3}=\frac{q_{0}q_{1}}{\cosh(2r)}$. Here we have omitted the normalization
factor. Since we only need the covariance matrix of state
$|\psi^{\prime}\rangle$, the normalization can be disregarded because it does
not change the covariance matrix. The characteristic function of state
$\rho^{\prime}=|\psi^{\prime}\rangle\langle\psi^{\prime}|$ has the form
$C(\alpha_{1},\alpha_{2})={\rm{tr}}[\rho^{\prime}\hat{D}_{1}(\alpha_{1})\hat{D}_{2}(\alpha_{2})]=e^{-\frac{1}{2}\bar{\alpha}\Lambda{\bar{\alpha}}^{T}}$
(11)
where
$\hat{D}_{k}(\alpha_{k})=e^{\alpha_{k}a^{\dagger}_{k}-\alpha^{*}_{k}a_{k}}$
and $\bar{\alpha}=(x_{1},y_{1},x_{2},y_{2})$ with
$\alpha_{k}=\frac{1}{\sqrt{2}}(x_{k}+iy_{k})$. Writing $\Lambda$ here in the
form of Eq.(5), we find $A={\rm diag}[b_{1},b_{2}]$, $C={\rm
diag}[c_{1},c_{2}]$ and $B={\rm diag}[d_{1},d_{2}]$ with
$b_{1}=-\frac{1}{2}+\frac{1+2f_{2}}{1+2f_{1}+2f_{2}+4f_{1}f_{2}-f_{3}^{2}}$,
$b_{2}=-\frac{1}{2}+\frac{1-2f_{2}}{1-2f_{1}-2f_{2}+4f_{1}f_{2}-f_{3}^{2}}$,
$d_{1}=-\frac{1}{2}+\frac{1+2f_{1}}{1+2f_{1}+2f_{2}+4f_{1}f_{2}-f_{3}^{2}}$,
$d_{2}=-\frac{1}{2}+\frac{1-2f_{1}}{1-2f_{1}-2f_{2}+4f_{1}f_{2}-f_{3}^{2}}$,
$c_{1}=\frac{-f_{3}}{1+2f_{1}+2f_{2}+4f_{1}f_{2}-f_{3}^{2}}$,
$c_{2}=\frac{f_{3}}{1-2f_{1}-2f_{2}+4f_{1}f_{2}-f_{3}^{2}}$. The entanglement
in whatever measure of state $|\psi^{\prime}\rangle$ is a rising functional of
$|A|$ and
$|A|=\frac{1}{4}+\frac{\scriptstyle{2q_{0}^{2}q_{1}^{2}}}{\scriptstyle{1-4q_{0}^{2}q_{1}^{2}+q_{1}^{4}+q_{0}^{4}(1+q_{1}^{4})+(1-q_{0}^{4})(1-q_{1}^{4})\cosh{(4r)}}}.$
(12)
This is obviously a descending functional of $|r|$.
Upper bound of entanglement evolution. Since $U\otimes I$ and $I\otimes\$$
commute, the unitary operator $U$ places no role in the entanglement evolution
under one-side map $I\otimes\$$, and hence we only need consider the initial
state $|g(I,V,q)\rangle=I\otimes V|\chi(q)\rangle=|\varphi(q)\rangle$. We also
define
$\rho^{G}(q_{\alpha})=I\otimes\$(|\varphi(q_{\alpha})\rangle\langle\varphi(q_{\alpha})|)$.
Using Eq.(7), one easily finds
$|\varphi(q=q_{a}q_{b})\rangle=\hat{T}(q_{a})\otimes I|\varphi(q_{b})\rangle$.
Since the operator $\hat{T}(q_{a})\otimes I$ and the map $I\otimes\$$ commute,
there is:
$\rho^{G}(q=q_{a}q_{b})=\hat{T}(q_{a})\otimes
I\rho^{G}(q_{b})\hat{T}^{\dagger}(q_{a})\otimes I.$ (13)
Using entanglement of formationMarian ; bennett , we can calculate the
entanglement of the state of a Gaussian state through its optimal
decomposition formMarian . Suppose $\rho^{G}(q_{b})$ has the following optimal
decompositionMarian :
$\rho^{G}(q_{b})=U_{1}\otimes U_{2}\rho^{s}(q_{0})U^{\dagger}_{1}\otimes
U^{\dagger}_{2}$ (14)
Here $U_{1},U_{2}$ are two local Gaussian unitaries and $\rho^{s}$ is in the
form
$\begin{split}\rho^{s}(q_{0})=&\int
d^{2}\beta_{1}d^{2}\beta_{2}P(\beta_{1},\beta_{2})\\\
&\hat{D}(\beta_{1},\beta_{2})|\chi(q_{0})\rangle\langle\chi(q_{0})|\hat{D}^{\dagger}(\beta_{1},\beta_{2}),\end{split}$
(15)
where $P(\beta_{1},\beta_{2})$ is positive definite,
$\hat{D}(\beta_{1},\beta_{2})=\hat{D}_{1}(\beta_{1})\otimes\hat{D}_{2}(\beta_{2})$
is a displacement operator defined as
$\hat{D}_{k}(\beta_{k})=e^{\beta_{k}a_{k}^{\dagger}-\beta_{k}^{*}a_{k}}$.
According to the definition of optimal decompositionMarian ; bennett , there
don’t exist any other $U_{1},U_{2}$ and positive definite functional
$P(\beta_{1},\beta_{2})$ which can decompose $\rho^{G}(q_{b})$ in the form of
Eq.(14) with a smaller $|q_{0}|$. The entanglement of $\rho^{G}(q_{b})$ is
equal to that of a TMSS $|\chi(q_{0})\rangle$, i.e. $q_{0}^{2}$. For the
Gaussian state $\rho^{G}(q_{b})$ with its optimal decomposition of Eq.(14), we
define the characteristic value of entanglement of $\rho^{G}(q_{b})$ as
$E[\rho^{G}(q_{b})]=|q_{0}|^{2}$.
Lemma 1. For any local Gaussian unitary $U$ and operator $\hat{T}(q_{a})$, we
can find $\theta,\theta^{\prime}$ and $\beta^{\prime\prime}$ satisfying
$\begin{split}&\hat{T}(q_{a})U_{1}\otimes
U_{2}\cdot\hat{D}(\beta_{1},\beta_{2})|\chi(q_{0})\rangle\\\
=&\mathcal{R}(\theta^{\prime})\otimes\mathcal{R}(\theta)\cdot\hat{D}(\beta^{\prime}_{1},\beta^{\prime}_{2})\cdot\hat{T}(q_{a})\mathcal{S}(r)\otimes
U_{2}|\chi(q_{0})\rangle,\end{split}$ (16)
where, $\mathcal{S}(r)$ is a squeezing operator defined earlier,
$\mathcal{R}(\theta)$ is a rotation operator defined by
$\mathcal{R}(\theta)(a^{\dagger},a)\mathcal{R}^{\dagger}(\theta)=(e^{-i\theta}a^{\dagger},e^{i\theta}a)$,
$\beta^{\prime}_{1},\beta^{\prime}_{2}$ and $\beta_{1},\beta_{2}$ are related
by a certain linear transformation.
Proof: Any local Gaussian unitary operator $U_{1}$ can be decomposed into the
product form of
$\mathcal{R}(\theta^{\prime})\mathcal{S}(r)\mathcal{R}(\theta)$. Also,
$\mathcal{S}(r)\mathcal{R}(\theta)\otimes
U_{2}\cdot\hat{D}(\beta_{1},\beta_{2})=\hat{D}(\beta^{\prime\prime}_{1},\beta_{2}^{\prime\prime})\cdot\mathcal{S}(r)\mathcal{R}(\theta)\otimes
U_{2}$. Define $\hat{d}=\hat{T}(q_{a})\otimes
I\cdot\hat{D}(\beta^{\prime\prime}_{1},\beta_{2}^{\prime\prime})\cdot\hat{T}^{-1}(q_{a})\otimes
I$, we have
$\displaystyle\hat{T}(q_{a})U\otimes
I\cdot\hat{D}(\beta_{1},\beta_{2})|\chi(q_{0})\rangle$ $\displaystyle=$
$\displaystyle\hat{T}(q_{a})\mathcal{R}(\theta^{\prime})\mathcal{S}(r)\mathcal{R}(\theta)\otimes
I\cdot\hat{D}(\beta_{1},\beta_{2})|\chi(q_{0})\rangle$ $\displaystyle=$
$\displaystyle\mathcal{R}(\theta^{\prime})\otimes
I\cdot\hat{d}\cdot\hat{T}(q_{a})\mathcal{S}(r)\mathcal{R}(\theta)\otimes
I|\chi(q_{0})\rangle$ $\displaystyle=$
$\displaystyle\mathcal{R}(\theta^{\prime})\otimes\mathcal{R}(\theta)\cdot\hat{D}(\beta^{\prime}_{1},\beta^{\prime}_{2})\cdot\hat{T}(q_{a})\mathcal{S}(r)\otimes
I|\chi(q_{0})\rangle.$
This completes the proof of Eq.(16). In the second equality above, we have
used the fact $\hat{T}(q_{a})$ and $\mathcal{R}(\theta^{\prime})$ commute.
Also, $\hat{d}$ there is not unitary. However, using BCH formula and the
vacuum state property $a_{k}|00\rangle=0$, we can always construct a unitary
operator $\hat{D}(\beta_{1}^{\prime},\beta_{2}^{\prime})$ so that the final
equality above holds. Here $\beta_{1}^{\prime},\;\beta_{2}^{\prime}$ are
certain linear functions of $\beta_{1},\;\beta_{2}$.
Using Eq.(13) and Eq.(14) with Eq.(16) we have
$\displaystyle\begin{split}&E[\rho^{G}(q=q_{a}q_{b})]\\\ =&E[I\otimes
U_{2}\cdot\hat{T}(q_{a})U_{1}\otimes
I\rho^{s}U^{\dagger}_{1}\hat{T}^{\dagger}(q_{a})\otimes I\cdot I\otimes
U^{\dagger}_{2}]\\\ =&E\left[\mathcal{R}(\theta_{1}^{\prime})\otimes
U_{2}\mathcal{R}(\theta_{1})\left(\int
d^{2}\beta_{1}d^{2}\beta_{2}P(\beta_{1},\beta_{2})\right.\right.\\\
&\hat{D}(\beta^{\prime}_{1},\beta^{\prime}_{2})\cdot\hat{T}(q_{a})\mathcal{S}(r_{1})\otimes
I|\chi(q_{0})\rangle\langle\chi(q_{0})|\mathcal{S}^{\dagger}(r_{1})\hat{T}^{\dagger}(q_{a})\\\
&\left.\left.\otimes
I\cdot\hat{D}^{\dagger}(\beta^{\prime}_{1},\beta^{\prime}_{2})\right)\mathcal{R}^{\dagger}(\theta_{1}^{\prime})\otimes\mathcal{R}^{\dagger}(\theta_{1})U^{\dagger}_{2}\right]\\\
\leq&E\left[\int
d^{2}\beta_{1}d^{2}\beta_{2}P(\beta_{1},\beta_{2})\hat{D}(\beta^{\prime}_{1},\beta^{\prime}_{2})\cdot\hat{T}(q_{a})\otimes
I\right.\\\
&\left.|\chi(q_{0})\rangle\langle\chi(q_{0})|\hat{T}^{\dagger}(q_{a})\otimes
I\cdot\hat{D}^{\dagger}(\beta^{\prime}_{1},\beta^{\prime}_{2})\right]\\\
\leq&|q_{a}q_{0}|^{2}=E[|\chi(q_{a})\rangle\langle\chi(q_{a})|]\cdot
E[\rho^{G}(q_{b})].\end{split}$ (17)
In the third step above we have used theorem 1 for the inequality sign. This
gives rise to the second theorem:
Theorem 2. Using the entanglement formation as the entanglement measure, if
the entanglement of $\rho^{G}(q_{b})$ is equal to that of TMSS
$|\chi(q_{0})\rangle$, the entanglement of $\rho^{G}(q=q_{a}q_{b})$ must be
not larger than that of TMSS $|\chi(q_{a}q_{0})\rangle$. Mathematically, it is
to say that if $|q|\leq|q_{b}|\leq 1$ we have
$\frac{E[I\otimes\$(|\varphi(q)\rangle\langle\varphi(q)|)]}{E[I\otimes\$(|\varphi(q_{b})\rangle\langle\varphi(q_{b})|)]}\leq\frac{E[|\varphi(q)\rangle\langle\varphi(q)|]}{E[|\varphi(q_{b})\rangle\langle\varphi(q_{b})|]}.$
(18)
Here $|\varphi(q)\rangle=I\otimes V|\chi(q)\rangle$ as defined earlier, $V$
can be any Gaussian unitary operator. Definitely, the inequality also holds if
we replace $|\varphi(q)\rangle$ by $|g(U,V,q)\rangle$ and replace
$|\varphi(q_{b})\rangle$ by $|g(U^{\prime},V,q_{b})\rangle$, and
$U,\;U^{\prime}$ can be arbitrary unitary operators. Theorem 2 also gives rise
to the following corollary.
Corollary 1. Given the one-side Gaussian map $I\otimes\$$, if the equality
sign holds in formula (18) for two specific values $q,\;q_{b}$ and
$0<|q|<|q_{b}|\leq 1$, then the equality sign there holds even $q,q_{b}$ there
are replaced by any $q^{\prime},q^{\prime\prime}$, respectively, as long as
$|q^{\prime}|,|q^{\prime\prime}|\in[|q|,1]$.
Proof. For simplicity, we first consider the case where $q$ is replaced by any
$q^{\prime}$. (1) suppose $|q^{\prime}|\in[|q|,|q_{b}|]$. The left side of
formula (18) is equivalent to $w^{\prime}\cdot z^{\prime}$, and
$w^{\prime}=\frac{E[I\otimes\$(|\varphi(q)\rangle\langle\varphi(q)|)]}{E[I\otimes\$(|\varphi(q^{\prime})\rangle\langle\varphi(q^{\prime})|)]}$
and
$z^{\prime}=\frac{E[I\otimes\$(|\varphi(q^{\prime})\rangle\langle\varphi(q^{\prime})|)]}{E[I\otimes\$(|\varphi(q_{b})\rangle\langle\varphi(q_{b})|)]}$.
The right side of formula (18) is equivalent to $w\cdot z$ and
$w=\frac{E[|\varphi(q)\rangle\langle\varphi(q)|]}{E[|\varphi(q^{\prime})\rangle\langle\varphi(q^{\prime})|]}$
and
$z=\frac{E[|\varphi(q^{\prime})\rangle\langle\varphi(q^{\prime})|]}{E[|\varphi(q_{b})\rangle\langle\varphi(q_{b})|]}$.
Theorem 2 itself says that $w^{\prime}\leq w$ and $z^{\prime}\leq z$. If the
equality sign holds in formula (18), we have $w^{\prime}\cdot
z^{\prime}=w\cdot z$ hence we must have $w=w^{\prime}$ and $z=z^{\prime}$
which is just corollary 1 in the case $q$ is replaced by $q^{\prime}$. (2)
Suppose $|q^{\prime}|>|q_{b}|$. As we have already known,
$\rho^{G}(q)=\hat{T}(q_{a})\otimes I\rho^{G}(q_{b})$. Consider Eq.(16).
Unitary $U_{1}$ in the optimal decomposition of Eq.(14) must be a rotation
operator only, i.e., it contains no squeezing, for, otherwise, according to
theorem 1, $E(\rho^{G}(q^{\prime}))$ is strictly less than
$q_{0}^{2}q_{a}^{2}$ which means the equality in formula (18) does not hold.
We denote $q^{\prime}=q_{b}/q_{c}$ and $|q_{c}|<1$. We have
$\displaystyle\rho^{G}(q^{\prime}=q_{b}/q_{c})$ (19) $\displaystyle=$
$\displaystyle\hat{T}^{-1}(q_{c})\otimes
I\rho^{G}(q_{b})\left(\hat{T}^{-1}(q_{c})\otimes I\right)^{\dagger}$
$\displaystyle=$ $\displaystyle\hat{T}^{-1}(q_{c})\otimes
I\cdot\mathcal{R}_{1}\otimes U_{2}\rho^{s}\mathcal{R}_{1}^{\dagger}\otimes
U_{2}^{\dagger}\cdot\hat{T}^{-1}(q_{c})\otimes I$ $\displaystyle=$
$\displaystyle\mathcal{R}_{1}\otimes U_{2}\cdot\int
d^{2}\beta_{1}d^{2}\beta_{2}P(\beta_{1},\beta_{2})\hat{D}(\beta^{\prime}_{1},\beta^{\prime}_{2})$
$\displaystyle|\chi(q_{0}/q_{c})\rangle\langle\chi(q_{0}/q_{c})|\hat{D}^{\dagger}(\beta^{\prime}_{1},\beta^{\prime}_{2})\cdot\mathcal{R}_{1}^{\dagger}\otimes
U_{2}^{\dagger}.$
Here we have used $\hat{T}^{-1}(q_{c})\otimes
I|\chi(q_{b}=q^{\prime}q_{c})\rangle=|\chi(q^{\prime})\rangle$. We have used
the optimal decomposition for $\rho^{G}(q_{b})$ in the second equality, and
lemma 1 in the last equality above. Eq.(19) is one possible decomposition of
the state $\rho^{G}(q^{\prime})$, but not necessarily the optimized
decomposition. Therefore,
$E[\rho^{G}(q^{\prime}=q_{b}/q_{c})]\leq{|q_{0}|^{2}}/{|q_{c}|^{2}}={|q^{\prime}|^{2}}/{|q_{b}|^{2}}\cdot
E[\rho^{G}(q_{b})]$. On the other hand, according to theorem 2, we further
obtain that
$E[\rho^{G}(q_{b}=q^{\prime}q_{c})]\leq{|q_{b}|^{2}}/{|q^{\prime}|^{2}}\cdot
E[\rho^{G}(q^{\prime})]$. Remark: Since here $|q^{\prime}|\geq q_{b}$, sign
$\leq$ should be replaced by sign $\geq$ in formula (18), when $q$ is replaced
by $q^{\prime}$. These two inequalities and result of (1) lead to
$\frac{E[\rho^{G}(q^{\prime})]}{E[\rho^{G}(q_{b})]}=\frac{E[|\chi(q^{\prime})\rangle\langle\chi(q^{\prime})|]}{E[|\chi(q_{b})\rangle\langle\chi(q_{b})|]}.$
(20)
for any $q^{\prime}$ provided that $|q|\leq|q^{\prime}|\leq 1$. Replacing
symbol $q^{\prime}$ above by symbol $q^{\prime\prime}$, we have another
equation. Comparing these two equations we conclude corollary 1.
Lemma 2: Given any Gaussian unitaries $U,\;V$, we have
$\displaystyle E[I\otimes\$(U\otimes
V|\phi^{+}\rangle\langle\phi^{+}|U^{\dagger}\otimes
V^{\dagger})]=E[I\otimes\$(|\phi^{+}\rangle)].$ (21)
Here $|\phi^{+}\rangle$ is the maximally entangled state defined as the
simultaneous eigenstate of position difference $\hat{x}_{1}-\hat{x}_{2}$ and
momentum sum $\hat{p}_{1}+\hat{p}_{2}$, with both eigenvalues being 0. Also,
when $q=1$, the state $|\chi(q)\rangle=|\phi^{+}\rangle$. We shall use the
following fact.
Fact 1: For any local Gaussian unitary operators $U$ and $V$, we can always
find another Gaussian unitary operator $\mathcal{V}$ so that
$U\otimes V|\phi^{+}\rangle=\mathcal{V}\otimes I|\phi^{+}\rangle.$ (22)
Proof: Any local Gaussian unitary operator can be decomposed into the product
form of $\mathcal{R}(\theta^{\prime})\mathcal{S}(r)\mathcal{R}(\theta)$. For
any TMSS $|\chi(q)\rangle$ we have
$\mathcal{R}(\theta_{1})\otimes\mathcal{R}(\theta_{2})|\chi(q)\rangle=I\otimes\mathcal{R}(\theta_{1}+\theta_{2})|\chi(q)\rangle$.
For the maximally TMSS $|\phi^{+}\rangle$ we have
$\mathcal{S}(r)\otimes\mathcal{S}(r)|\phi^{+}\rangle=|\phi^{+}\rangle$, for,
the both sides are the simultaneous eigenstates of position difference and
momentum sum, with both eigenvalues being 0. This also means
$\mathcal{S}(r)\otimes
I|\phi^{+}\rangle=I\otimes\mathcal{S}^{\dagger}(r)|\phi^{+}\rangle$. Suppose
$V=\mathcal{R}(\theta_{B}^{\prime})\mathcal{S}(r_{B})\mathcal{R}(\theta_{B})$,
then
$\displaystyle U\otimes V|\phi^{+}\rangle=\mathcal{V}\otimes
I|\phi^{+}\rangle$ (23)
where
$\mathcal{V}=U\mathcal{R}(\theta_{B})\mathcal{S}^{\dagger}(r_{B})\mathcal{R}(\theta_{B}^{\prime})$.
This completes the proof of Eq.(22). If the equality sign in formula (18)
holds, we can apply corollary 1 of theorem 2 through replacing $q_{b}$ by 1
and we obtain that $E[\rho^{G}(q^{\prime})]=|q^{\prime}|^{2}\cdot
E[I\otimes\$(|\phi^{+}\rangle)]$. On the other hand, by using theorem 2 and
lemma 2 we have $E[\rho^{G}(q^{\prime})]\leq|q^{\prime}|^{2}\cdot
E[I\otimes\$(|\phi^{+}\rangle)]$. This means
$E[\rho^{G}(q^{\prime})]=\max_{\\{V^{\prime}\\}}\\{E[I\otimes\$(|g(I,V^{\prime},q^{\prime})\rangle)]\\}$
(24)
where $\rho^{G}(q^{\prime})=I\otimes\$(|g(I,V,q^{\prime})\rangle\langle
g(I,V,q^{\prime})|)$ as defined earlier, $\\{V^{\prime}\\}$ is the set
containing all single-mode Gaussian unitary transformations. The equality
holds for any $q^{\prime}$ provided that the equality of formula(18) holds for
two specific values $q,\;q_{b}$ and $|q^{\prime}|\geq|q|$. We arrive at the
following major conclusion of this Letter:
Major conclusion: Suppose that we have a TMSS $|\chi(q^{\prime})\rangle$. We
want to maximize the entanglement distribution over a one-side Gaussian map
$I\otimes\$$ by taking local Gaussian unitary operation $I\otimes V^{\prime}$
before entanglement distribution. Although we don’t have complete information
of the map $I\otimes\$$, it’s still possible for us to find out a specific
Gaussian unitary operation $V$ so that the entanglement distribution is
maximized over all $V^{\prime}$, for an initial state
$|\chi(q^{\prime})\rangle$ with any $|q^{\prime}|\geq|q|$, as long as we can
find two specific values $|q_{b}|>|q|$, such that the equality sign in formula
(18) holds. Obviously, the conclusion is also correct for any initial state
which is a Gaussian pure state.
The conclusion actually says that, in verifying that $V$ can maximize the
entanglement distribution for all initial states
$\\{|\chi(q^{\prime})\rangle||q^{\prime}|\geq|q|\\}$, we only need to verify
the equality sign of formula (18) for two specific values.
Experimental proposal. To experimentally test our major conclusion, we can
consider the following beamsplitter channel: Initially, beams 1 and 2 are in a
TMSS, which is the initial bipartite Gaussian pure state. Beam 3 is in a
squeezed thermal state
$\rho_{3}=\tilde{S}(u_{3})\rho_{th}\tilde{S}^{\dagger}(u_{3})$ here
$\tilde{S}(u)$ is a squeezing operator defined by
$\tilde{S}(u)(\hat{x},\hat{p})\tilde{S}^{\dagger}(u)=(u\hat{x},\hat{p}/u)$ and
$\rho_{th}$ is a thermal state whose CM is ${\rm diag}[b_{3},b_{3}]$. Beam 3
together with the beamsplitter makes the one-side Gaussian channel. A
beamsplitter will transform $\hat{x}_{2},\hat{x}_{3}$ by
$U_{B}(\hat{x}_{2},\hat{x}_{3})U_{B}^{-1}\longrightarrow(\hat{x}_{2},\hat{x}_{3})\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right).$ In an experiment, we can take,
e.g., $q=0.02$ and $q_{b}=0.5$, testing with many different $V$ we should find
that the equality sign in formula (18) can hold with
$V=\tilde{S}(u_{2}=u_{3})$ . Our major conclusion is verified if we can find
that the same $V=\tilde{S}(u_{3})$ always maximizes the output entanglement
for any input state $|\chi(q^{\prime})\rangle$ provided that $|q^{\prime}|\geq
0.02$. Numerical calculation is shown in the following figure.
Figure 1: The entanglement with different squeezing factor $u_{2}$. The
maximum entanglement obtained when $u_{2}=u_{3}=3$. Here we set $u_{3}=3$ and
$q^{\prime}=2/3,\theta=\pi/6,b_{3}=1$.
In summary, we present an upper bound of the entanglement evolution of a
2-mode Gaussian pure state under one-side Gaussian map. We show that one can
maximize the entanglement distribution over an unknown one-side Gaussian noisy
channel by testing the channel with only two specific states. An experimental
scheme is proposed.
Acknowledgement. This work was supported in part by the National Basic
Research Program of China grant nos 2007CB907900 and 2007CB807901, NSFC grant
number 60725416, and China Hi-Tech program grant no. 2006AA01Z420.
Appendix. Details of the proof of Eq.(9). We will use the following lemma.
Lemma 2. If $\mathcal{A}$ and $\mathcal{B}$ are two noncommuting operators
that satisfy the conditions
$[\mathcal{A},[\mathcal{A},\mathcal{B}]]=[\mathcal{B},[\mathcal{A},\mathcal{B}]]=0,$
(25)
then
$e^{\mathcal{A}+\mathcal{B}}=e^{\mathcal{A}}e^{\mathcal{B}}e^{-\frac{1}{2}[\mathcal{A},\mathcal{B}]}.$
(26)
This is a special case of the Baker-Hausdorff theorem of group theoryLouisell
.
The squeezing operator $S(r)=e^{r({a^{\dagger}}^{2}-a^{2})}$ can be normally
ordered asBarnett
$\displaystyle S(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\cosh(2r)}}\exp\left[\frac{{a^{\dagger}}^{2}}{2}\tanh(2r)\right]$
(27)
$\displaystyle\cdot\exp\left[-a^{\dagger}a(\ln(\cosh(2r)))\right]\exp\left[-\frac{1}{2}a^{2}\tanh(2r)\right].$
We neglect the constant of normalization in all the following calculation.
$\displaystyle I\otimes S(r)|\chi(q_{0})\rangle$ $\displaystyle=$
$\displaystyle
e^{r({a_{2}^{\dagger}}^{2}-a_{2}^{2})}e^{q_{0}a_{1}^{\dagger}a_{2}^{\dagger}}|00\rangle$
$\displaystyle=$ $\displaystyle
e^{q_{0}a_{1}^{\dagger}(a_{2}^{\dagger}\cosh(2r)-a_{2}\sinh(2r))}e^{r(a_{2}^{\dagger
2}-a_{2}^{2})}|00\rangle$ $\displaystyle=$ $\displaystyle
e^{q_{0}a_{1}^{\dagger}(a_{2}^{\dagger}\cosh(2r)-a_{2}\sinh(2r))}e^{{1\over
2}{a_{2}^{\dagger}}^{2}\tanh(2r)}|00\rangle$ $\displaystyle=$ $\displaystyle
e^{{1\over
2}{a_{2}^{\dagger}}^{2}\tanh(2r)}e^{q_{0}a_{1}^{\dagger}\\{a_{2}^{\dagger}\cosh(2r)-[a_{2}+a_{2}^{\dagger}\tanh(2r)]\sinh(2r)\\}}|00\rangle$
$\displaystyle=$ $\displaystyle e^{{1\over
2}{a_{2}^{\dagger}}^{2}\tanh(2r)}e^{q_{0}a_{1}^{\dagger}({a_{2}^{\dagger}\over\cosh(2r)}-a_{2}\sinh(2r))}|00\rangle$
$\displaystyle=$ $\displaystyle e^{{1\over
2}{a_{2}^{\dagger}}^{2}\tanh(2r)}e^{q_{0}a_{1}^{\dagger}a_{2}^{\dagger}\over\cosh(2r)}e^{-{1\over
2}{a_{1}^{\dagger}}^{2}q_{0}^{2}\tanh(2r)}|00\rangle$
This is just Eq.(9). In the last equality we have used lemma 2. This completes
the proof of Eq.(9).
## References
* (1) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000).
* (2) V. Vedral, M.B. Plenio, M.A. Rippin and P.L. Knight, Phys. Rev. Lett. 78, 2275 (1997).
* (3) W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
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* (6) L. Vaidman, Phys. Rev. A 49, 1473 (1994); S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998); H.F. Hofmann, T. Ide, T. Kobayashi, and A. Furusawa, Phys. Rev. A 62, 062304 (2000); A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282,706 (1998).
* (7) L.M. Duan, G. Giedke, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 84, 2722 (2000); R. Simon, Phys. Rev. Lett. 84, 2726 (2000); R. F. Werner and M. M. Wolf, Phys. Rev. Lett. 86, 3658 (2001); G. Giedke, B. Kraus, M. Lewenstein, and J.I. Cirac, Phys. Rev. Lett. 87, 167904(2001).
* (8) G. Giedke, M. M. Wolf, O. Kr uger, R. F. Werner, and J. I. Cirac, Phys. Rev. Lett. 91, 107901 (2003).
* (9) P. Marian and T.A. Marian, Phys. Rev. Lett. 101, 220403 (2008).
* (10) Solomon Ivan and R. Simon, arXiv:0808.1658.
* (11) J. Eisert, S. Scheel, M.B. Plenio, Phys. Rev. Lett. 89, 137903 (2002).
* (12) J. Fiurasek, Phys. Rev. Lett. 89, 137904 (2002).
* (13) Juan Pablo Paz and Augusto J. Roncaglia, Phys. Rev. Lett. 100, 220401 (2008); A. S. Coelho AS, F. A. S. Barbosa, K. N. Cassemiro, A. S. Villar, M. Martinelli, and P. Nussenzveig, Science, 326, 823 (2009).
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* (15) T Konard, F.D. Melo, M, Tiersch, C. Kansztelan, A. Aragao, and A. Buchleitner, Nature Physics, 4, 99 (2008).
* (16) O. Jimenez Farias, C. Lombard Latune, S.P. Walborn, L. Davidovich, P.H. Souto Ribeiro, Science, 324, 1414, (2009).
* (17) Chang-shui Yu, X.X. Yi, and He-shan Song, Phys. Rev. A 78, 062330 (2008); Zong-Guo Li, Shao-Ming Fei, Z.D. Wang, and W.M. Liu, Phys. Rev. A 79, 024303 (2009).
* (18) C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54, 3824 (1996).
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|
arxiv-papers
| 2010-01-04T15:55:44 |
2024-09-04T02:49:07.401640
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiang-Bin Wang, Zong-Wen Yu, Jia-Zhong Hu",
"submitter": "Xiang-Bin Wang",
"url": "https://arxiv.org/abs/1001.0156"
}
|
1001.0174
|
# An intrinsic approach to invariants of framed links in 3-manifolds
Efstratia Kalfagianni
###### Abstract.
We study framed links in irreducible 3-manifolds that are
${\mathbb{Z}}$-homology 3-spheres or atoroidal ${\mathbb{Q}}$-homology
3-spheres. We calculate the dual of the Kauffman skein module over the ring of
two variable power series with complex coefficients. For links in $S^{3}$ we
give a new construction of the classical Kauffman polynomial.
Keywords. characteristic submanifold, framed links, finite type invariants,
Kauffman skein module, loop space, Seifert fibered 3-manifolds, toroidal
decompositions.
Mathematics Subject Classi cation (2010). 57N10, 57M2, 57R42, 57R56.
Supported in part by NSF grant DMS-0805942
## 1\. Introduction
The Kauffman polynomial is a 2-variable Laurent polynomial invariant for links
in $S^{3}$ [17] that has interesting applications and connections with contact
geometry. The degree in one of the variables of the Kauffman polynomial
provides an upper bound for the Thurston-Bennequin norm of Legendrian links
[8, 26]. The inequality is known to be sharp for several classes of links
(e.g. alternating links) and the proof of this sharpness has led to deeper
connections between knot polynomials and contact geometry [22].
In this paper we study framed links in oriented, irreducible 3-manifolds that
are ${\mathbb{Z}}$-homology 3-spheres or atoroidal ${\mathbb{Q}}$-homology
3-spheres. We give conditions under which an invariant that is defined on
framed singular links with one double point gives rise to an invariant of
framed links (Theorem 2.2). This allows us to construct formal power series
framed link invariants obeying the Kauffman polynomial skein relations. The
coefficients of these series are finite type framed link invariants and are
perturbative versions of the Reshetikhin-Turaev, Witten $SO(n)$-invariants
[25, 29] in the sense of Le-Murakami-Ohtsuki [20]. Using weight systems
corresponding to appropriate representations of the Lie algebras $so(n)$ and
the naturallity of the LMO invariant, one obtains a Kauffman type power series
invariant for framed links in all ${\mathbb{Q}}$-homology 3-spheres. Our
approach in this paper is quite different from this line and allows us to
solve the subtler problem of constructing power series invariants with given
values on a set of initial links. Our approach here, that exhibits the
interplay between skein framed link theory and the topology of 3-manifolds, is
inspired by the study of Vassiliev invariants (a.k.a. finite type invariants)
[27] using 3-dimensional topology techniques [12]. The precise relation of the
power series constructed here to the one obtained via the LMO invariant is not
clear to us at this point.
###### Definition 1.1.
Let $M$ be an oriented ${\mathbb{Q}}$-homology 3-sphere. A framed
$m$-component link is a collection of $m$ unordered (unoriented) circles
smoothly and disjointly embedded in $M$ and such that each component is
equipped with a continuous unit normal vector field. Two framed links are
equivalent if they are isotopic by an ambient isotopy that preserves the
homotopy class of the vector field on each component. Let
${\mathcal{\bar{L}}}:={\mathcal{\bar{L}}}(M)$ denote the set of isotopy
classes of framed links in $M$.
Figure 1. The parts of $L_{+}$, $L_{-}$ and $L_{o}$ and $L_{\infty}$ in $B$.
Figure 2. $L_{r}$ and $L_{l}$ are obtained by a full twist from $L$.
To state the main result of the paper we need some notation and conventions:
Let $L_{+}$, $L_{-}$, $L_{o}$ and $L_{\infty}$ denote four framed links that
are identical everywhere except in a 3-ball $B$ in $M$. There under a suitable
projection of the parts in $B$, $L_{+}$, $L_{-}$, $L_{o}$ and $L_{\infty}$
look as shown in Figure 1. Also for every framed link we denote by
$L_{r},L_{l}$ the framed links that are identical to $L$ everywhere except in
a 3-ball where they differ as shown in Figure 2. Here we suppose that the
orientation of $M$ agrees with the right-handed orientation of the 3-balls
containing the link parts in Figures 1 and 2 and that the framing vector for
link parts in these figures is perpendicular to the page. The framings of the
links coincide everywhere outside the parts shown in Figures 1 and 2.
Let $\hat{\Lambda}:={\mathbb{C}}[[x,\ y]]$ denote the ring of formal power
series in $x,y$ over $\mathbb{C}$ and let ${\displaystyle
t:={e}^{x}=1+x+{{x^{2}}\over{2}}+\dots}$. Let us set ${\displaystyle
a:=i{e}^{y}=i+iy+{i{y^{2}}\over{2}}+\dots}$ and set ${\displaystyle
z:=it-{(it)^{-1}}=i{e}^{x}+ie^{-x}=2i+{{ix^{2}}}+\dots}$. Note that $a$ and
$z$ are invertible in $\hat{\Lambda}$.
###### Definition 1.2.
The Kauffman skein module of $M$ over $\hat{\Lambda}$, denoted by
${\mathfrak{F}}(M)$, is the quotient of the free $\hat{\Lambda}$-module with
basis ${\mathcal{\bar{L}}}$ by its ideal generated by all the relations of the
following two types:
$L_{+}-L_{-}=z\big{[}L_{o}-L_{\infty}\big{]},$ $L_{r}=aL\ \ {\rm and}\ \
L_{l}=a^{-1}L.$
We will use
${\mathfrak{F}}^{*}(M):={\operatorname{Hom}}_{\hat{\Lambda}}{\big{(}{\mathfrak{F}}(M),\hat{\Lambda}\big{)}}$
to denote the $\hat{\Lambda}$-dual of ${\mathfrak{F}}(M)$.
###### Remark 1.1.
The usual convention in skein module theory is to allow an empty link as part
of the set ${\mathcal{\bar{L}}}$. In contrast to that, in this paper, we find
it convenient to work with non-empty links (Definition 1.1).
###### Remark 1.2.
Since the links are unoriented the declarations $L_{+}$ and $L_{-}$, when
considering a crossing, are arbitrary. However this doesn’t matter for our
purposes since the first skein relation in Definition 1.2 is invariant under
simultaneously interchanging $L_{+}$ with $L_{-}$ and $L_{o}$ with
$L_{\infty}$.
To continue let $\pi:={\pi}(M)$ denote the set of non-trivial conjugacy
classes of $\pi_{1}(M)$ and ${\hat{\pi}}$ denote the set obtained from $\pi$
by indentifying the conjugacy class of every element $1\neq x\in\pi_{1}(M)$
with that of $x^{-1}$. Also let $S({\hat{\pi}})$ denote the symmetric algebra
of the free $\hat{\Lambda}$-module, say $\hat{\Lambda}\hat{\pi}$, with basis
$\hat{\pi}$. Finally, let
$S^{*}({\hat{\pi}}):={\operatorname{Hom}}_{\hat{\Lambda}}\big{(}S({\hat{\pi}}),\hat{\Lambda}\big{)}$
denote the $\hat{\Lambda}$-dual of $S({\hat{\pi}})$.
###### Theorem 1.3.
Let $M$ be an oriented ${\mathbb{Q}}$-homology 3-sphere with $\pi_{2}(M)=0$
and such that if $H_{1}(M)\neq 0$ then $M$ is atoroidal. Then there is a
$\hat{\Lambda}$-module isomorphism
${\mathfrak{F}}^{*}(M)\cong S^{*}({\hat{\pi}}).$
For components that are homologically trivial in $M$ the homotopy class of the
framing vector field is determined by an integer: the algebraic intersection
number of a push-out of the component in the direction of the framing vector
field with a Seifert surface bounded by the component. This algebraic
intersection number is the self-linking number of the component. There is a
canonical framing defined by the Seifert surface that corresponds to the
integer zero. This implies that in a ${\mathbb{Z}}$-homology sphere, for every
underlying (unframed) isotopy class of knots the framed knot types correspond
to integers. The self-linking number can also be defined in terms of
Vassiliev-Gusarov axioms; it is a finite type framed link invariant of order
one. As shown by Chernov [3] this point of view generalizes to all framed
knots in 3-manifolds; in particular for knots in irreducible
${\mathbb{Q}}$-homology 3-spheres that we study here. For $M$ as above, given
a conjugacy class $c$ in $\pi_{1}(M)$ and a fixed framed knot $CK$
representing $c$, Chernov shows that there is a unique $\mathbb{Z}$-valued
invariant for all framed knots representing $c$ with given value on $CK$
(Theorem 2.2 of [3]). His work implies that, with a chosen set of initial
knots, for every underlying (unframed) isotopy class of knots the framed knot
types correspond to integers. This point will be useful to us in the next
sections.
The isomorphism in Theorem 1.3 also depends on a choice of initial links which
we now discuss: For every unordered sequence of elements in
$\hat{\pi}\cup\\{1\\}$ we choose a framed link $CL$ that realizes it and call
it an initial link. For elements in $\hat{\pi}\cup\\{1\\}$ that are trivial in
$H_{1}(M)$ we choose the canonical framing. This means that the integer
describing the framing on each component of an initial link is zero. For an
initial link $CL$ with $k$ homotopically trivial components we choose
$CL=CL^{*}\sqcup U^{k}$, where $CL^{*}$ is an initial link with no
homotopically trivial components and $U^{k}$ is the standard unlink in a
3-ball disjoint from $CL^{*}$. The one component unlink $U^{1}$ will be
abbreviated to $U$. In general we will assume that each component of an
initial link $CL$ is the chosen initial knot for the corresponding element in
$\hat{\pi}\cup\\{1\\}$. We will also assume that each component is the initial
knot required to define Chernov’s self-linking invariant. We will denote by
$\mathcal{C}\mathcal{L}^{*}$ the set of all initial links with no
homotopically trivial components.
The elements in the set $\mathcal{C}\mathcal{L}^{*}\cup\\{U\\}$ are in one-to-
one correspondence with a basis of $S({\hat{\pi}})$. An element
$R_{M}\in{\mathfrak{F}}^{*}(M)$ gives rise to one in $S^{*}({\hat{\pi}})$ by
restriction on the set $\mathcal{C}\mathcal{L}^{*}\cup\\{U\\}$. Theorem 1.3
will follow easily once we have proven the following result (see Section 4 for
details).
###### Theorem 1.4.
Let $M$ be an oriented ${\mathbb{Q}}$-homology 3-sphere with $\pi_{2}(M)=0$,
and such that if $H_{1}(M)\neq 0$ then $M$ is atoroidal. Given a map
${\mathcal{R}}_{M}:\mathcal{C}\mathcal{L}^{*}\cup\\{U\\}\rightarrow\hat{\Lambda}$
there exists a unique map $R_{M}:{\bar{\mathcal{L}}}\rightarrow\hat{\Lambda}$
such that:
1. (1)
The restriction of $R_{M}$ on ${\mathcal{C}{{\mathcal{L}^{*}}}}\cup\\{U\\}$ is
equal to ${\mathcal{R}}_{M}$.
2. (2)
$R_{M}$ satisfies the Kauffman skein relation
$R_{M}(L_{+})-R_{M}(L_{-})=z\big{[}R_{M}(L_{o})-R_{M}(L_{\infty})\big{]},$
for every skein quadruple of links $L_{+}$, $L_{-}$, $L_{o}$ and $L_{\infty}$
as in Figure 1.
3. (3)
$R_{M}(L_{r})=aR_{M}(L)$ and $R_{M}(L_{l})=a^{-1}R_{M}(L)$ for every
$L\in{\bar{\mathcal{L}}}$.
Let $\Lambda:={\mathbb{C}}[a^{\pm 1},z^{\pm 1}]$ denote the ring of Laurent
polynomials in $a$ and $z$. We can define the Kauffman skein module of $M$
over $\Lambda$, denoted by ${\mathfrak{F}}_{\Lambda}(M)$, and consider its
$\Lambda$-dual,
${\mathfrak{F}}_{\Lambda}^{*}(M):={\operatorname{Hom}}_{\Lambda}{\big{(}{\mathfrak{F}}_{\Lambda}(M),\Lambda\big{)}}$.
As we will discuss in Section 4, for links in $S^{3}$, if we choose the value
$R_{S^{3}}(U)$ to lie in $\Lambda$ then $R_{S^{3}}(L)\in\Lambda$, for every
$L\in{\bar{\mathcal{L}}}$. This implies that
${\mathfrak{F}}_{\Lambda}^{*}(S^{3})\cong\Lambda$ and leads to the following
question:
###### Question.
Let $M$ be as in Theorem 1.3. Can we choose the initial links
$CL^{*}\in\mathcal{C}\mathcal{L}^{*}$ so that we have a $\Lambda$-module
isomorphism
${\mathfrak{F}}_{\Lambda}^{*}(M)\cong S_{\Lambda}^{*}({\hat{\pi}})?$
Here, $S_{\Lambda}^{*}({\hat{\pi}})$ denotes the $\Lambda$-dual of the
symmetric algebra of the free $\Lambda$-module with basis $\hat{\pi}$.
In [13] we constructed formal power series invariants that satisfy the HOMFLY
skein change formula for unframed oriented links in large classes of
${\mathbb{Q}}$-homology 3-spheres. Cornwell [4, 5, 6] shows that for lens
spaces both the question above and its analogue for the HOMFLY skein module of
[14] have a positive answer. As a result he obtains analogues of the
aforementioned results of [8, 26] for Legendrian links in contact lens spaces.
Theorem 2.2 of this paper is the framed link analogue of the “integrability of
singular link invariants” results proved in [12, 13]. Theorem 2.2 doesn’t
follow from the results in these papers: In [12] we only treat knots while in
[13] we treat links in some classes of irreducible $\mathbb{Z}$-homology
3-spheres. In this paper we are able to remove those restrictions and deal
with all irreducible $\mathbb{Z}$-homology 3-spheres; see Theorem 3.1 and
Remark 3.2. If one forgets the framing, Theorem 3.1 generalizes the
integrability results and Theorem A of [13] for links in all irreducible
$\mathbb{Z}$-homology 3-spheres.
Framed links in general 3-manifolds and their skein modules were studied by
several authors before; see [24] and references therein. In particular,
Przytycki [23] introduced a two term homotopy skein module of framed links in
oriented 3-manifolds as quantum deformation of the fundamental group. In [16]
Kaiser calculated this module over the ring of Laurent polynomials with
${\mathbb{Z}}$-coefficients. He showed that if a 3-manifold contains no non-
separating 2-spheres or tori then Przytycki’s module is a symmetric algebra of
the free module with basis the set of non-trivial conjugacy classes of
$\pi_{1}(M)$. Kaiser also studied several variations of two term skein modules
and put the classical self-linking number for null homologous knots as well as
Chernov’s generalization of it in the skein module theory framework. For
details the reader is referred to [16].
The paper is organized as follows: In Section 2 we formulate the problem of
integrating framed singular link invariants to invariants of framed links.
Then we state an integrability theorem and prove it for atoroidal
${\mathbb{Q}}$-homology spheres. In Section 3 we treat manifolds containing
essential tori and in Section 4 we construct the Kauffman power series
invariants and prove Theorems 1.3 and 1.4.
Throughout the paper we will work in the smooth category.
Acknowledgment: I thank Chris Cornwell for his interest in this work and for
several stimulating questions about link theory in 3-manifolds that motivated
me to go back and work on this project. I thank Vladimir Turaev for suggesting
that I formulate the main result of the paper in terms of skein modules. I am
grateful to the anonymous referees for reading the paper carefully and making
thoughtful comments and suggestions that helped me improve the exposition.
## 2\. Framed oriented Singular Link Invariants
Throughout this section we will work with oriented links in oriented
3-manifolds. Theorem 2.2, as well as its unframed counterparts [12, 13, 21],
are proved for oriented links in oriented 3-manifolds. For example, the
definitions of the signs of resolutions of double points below use the
orientation of links as well as that of the ambient 3-manifolds.
### 2.1. Framed oriented singular links and resolutions
Let $M$ be an oriented ${\mathbb{Q}}$-homology 3-sphere. An $m$-component
oriented framed singular link of order $n$ is a collection of unordered
oriented circles, smoothly immersed in $M$ such that (i) the only
singularities are exactly $n$ transverse double points; and (ii) the image of
each component is equipped with a continuous unit normal vector field. We
consider framed singular links up to ambient isotopy that preserves the
orientations, the transversality of the double points and the homotopy class
of the vector field on each component. For $n=0$ we have an oriented framed
link. We will denote by ${\mathcal{L}}^{(n)}:={\mathcal{L}}^{(n)}(M)$ (resp.
${\mathcal{L}}:={\mathcal{L}}(M)$) the set of isotopy classes of oriented
framed singular links of order $n$ (resp. links) in $M$.
Convention: To simplify the exposition, for the remaining of the section and
the next section, we will say a framed link (resp. singular link) to mean an
oriented framed link (resp. singular link). Also when we say a 3-manifold, we
will mean an oriented 3-manifold.
Let $P$ denote a disjoint union of oriented circles and consider a framed
singular link represented by a smooth immersion $L:P\longrightarrow M$. Let
$p\in M$ be a double point of $L$; the inverse image consists of two points
$p_{1},p_{2}\in P$. There are disjoint intervals $\sigma_{1}$ and $\sigma_{2}$
on $P$ with $p_{i}\in\hbox{int}(\sigma_{i})$, $i=1,\ 2$, such that for a
neighborhood $B$ of $p$ we have $L\cap B=L(\sigma_{1})\cup L(\sigma_{2})$.
Moreover, there is a proper 2-disc $D$ in $B$ such that $L(\sigma_{1})$,
$L(\sigma_{2})\subset D$ intersect transversally at $p$. Now
$L(\sigma_{1})\cup L(\sigma_{2})$ intersects $\partial D$ at four points and,
since $\sigma_{i}$ inherits an orientation from that of $P$, we can talk of
the initial and terminal point of $L(\sigma_{i})$. Choose arcs $a_{1}$,
$a_{2}$, $b_{1}$, $b_{2}$ with disjoint interiors such that
1. (1)
$a_{1}$ and $a_{2}$ go from the initial point of $L(\sigma_{1})$ to the
terminal point of $L(\sigma_{1})$ and lie in distinct components of $\partial
B\setminus\partial D$; and
2. (2)
$b_{1}$ and $b_{2}$ lie on $\partial D$ with $b_{1}$ going from the initial
point of $L(\sigma_{1})$ to the terminal point of $L(\sigma_{2})$ and $b_{2}$
from the initial point of $L(\sigma_{2})$ to the terminal point of
$L(\sigma_{1})$. The complement of $b_{1}\sqcup b_{2}$ in $\partial D$
consists of two arcs, say $c_{1},c_{2}$.
The orientation of $M$ and that of $L(\sigma_{2})$ define an orientation of
$a_{1}\sqcup a_{2}$; suppose that this induced orientation agrees with the one
of $a_{1}$ and is opposite to that of $a_{2}$. Define the positive resolution
of $L$ at $p$ to be
$L_{+}=\overline{L\setminus L(\sigma_{2})}\cup a_{1},$
and the negative resolution to be
$L_{-}=\overline{L\setminus L(\sigma_{2})}\cup a_{2}.$
In the case that $n=1$ we also define
$L_{o}=\overline{L\setminus(L(\sigma_{2})\cup L(\sigma_{1}))}\cup(b_{1}\sqcup
b_{2})$ $L_{\infty}=\overline{L\setminus(L(\sigma_{2})\cup
L(\sigma_{1}))}\cup(c_{1}\sqcup c_{2})$
Note that $L_{\infty}$ only makes sense as an unoriented link.
###### Definition 2.1.
A framed singular link $L$ is called inadmissible if there is a 2-disc
$D\subset M$ such that $L\cap D=\partial D$ and exactly one double point of
$L$ lies on $\partial D$. Otherwise the singular link is called admissible. A
crossing change on a link that produces an inadmissible singular link as
intermediate step will be called an inadmissible crossing change.
In the proof of Theorem 2.2 it will be convenient for us to work with framed
links with ordered components: Let ${\tilde{\mathcal{L}}}$ denote the set of
isotopy classes of such framed links in $M$. Similarly, let
${\tilde{\mathcal{L}}}^{(n)}$ denote the set of isotopy classes of ordered
framed singular links with $n$-double points. There is an obvious map
${\mathfrak{r}}:{\tilde{\mathcal{L}}}\longrightarrow{\mathcal{L}}$ that
forgets the ordering of the components of links; similarly we have forgetful
maps
${\mathfrak{r}}_{n}:{\tilde{\mathcal{L}}}^{(n)}\longrightarrow{\mathcal{L}}^{(n)}$,
for all $n\in{{\mathbb{N}}}$. Recall from the Introduction that the framing of
a knot is determined by an integer, where in the case of not homologically
trivial knots this integer is provided by Chernov’s work. Thus the framing of
an $m$-component link in ${\tilde{\mathcal{L}}}$ is determined by an ordered
sequence $\\{{\bf f}_{1},\ldots,{\bf f}_{m}\\}$ of $m$ integers; one assigned
to each component of the link. Every entry of the sequence is the affine self-
linking number of a link component and it changes by $2$ under an inadmissible
crossing change while it remains unchanged under admissible crossing changes
(Theorems 2.2, [3]). Then, via ${\mathfrak{r}}$, an unordered link
$L\in{\mathcal{L}}$ inherits an unordered sequence of integers: More
specifically, given $L\in{\mathcal{L}}$, there is a set of ordered integer
sequences, say ${\bf f}$, corresponding to elements in
${\mathfrak{r}}^{-1}(L)$. We assign to $L$ the map
${\mathfrak{r}}^{-1}(L)\longrightarrow{\bf f},$
sending each element to its corresponding ordered sequence. We will often
abuse the terminology and refer to ${\bf f}$ as the framing of the link $L$.
###### Definition 2.2.
The total framing of a link $L\in{\mathcal{L}}$ is defined to be
${\tau}(L):=\sum_{i=1}^{m}{\bf f_{i}}$ where $\\{{\bf f}_{1},\ldots,{\bf
f}_{m}\\}$ is the ordered sequence corresponding to an appropriate lift
${\tilde{L}}\in{\mathfrak{r}}^{-1}(L)$ of $L$.
###### Definition 2.3.
For an ordered, framed singular link
${\tilde{L}}_{\times}\in{\tilde{\mathcal{L}}}^{(1)}$ we define a sequence of
integers $\\{{\bf f}_{1},\ldots,{\bf f}_{m}\\}$ by
${\bf f}_{i}({\tilde{L}}_{\times}):=\;\left\\{\begin{array}[]{cl}{\bf
f}_{i}({\tilde{L}_{+}})-{\bf f}_{i}({\tilde{L}}_{-}),\mbox{
if}\quad\times\in{\tilde{L}}_{i}\\\ \mbox{\quad}\\\ {\bf
f}_{i}({\tilde{L}_{+}})={\bf f}_{i}({\tilde{L}_{-}}),\mbox{\quad}{\rm
otherwise.}\\\ \end{array}\right.$
Note that, in the first case, ${\bf f}_{i}({\tilde{L}}_{\times})$ is non-zero
only if $L_{\times}$ is inadmissible, in which case it is equal to 2. For an
unordered singular link $L_{\times}\in{{\mathcal{L}}}^{(1)}$ we have a set of
ordered integer sequences, say ${\bf f}$, corresponding to elements in
${\mathfrak{r}}^{-1}(L_{\times})$. The map
${\mathfrak{r}}^{-1}(L_{\times})\longrightarrow{\bf f}$, assigning to every
ordered link in that preimage its corresponding sequence, gives an unordered
sequence of integers for $L_{\times}$.
### 2.2. Integration of singular link invariants.
Given an abelian group $\mathbb{A}$ and a framed link invariant
$F:{\mathcal{L}}\longrightarrow\mathbb{A}$, we can extend it to an invariant
of framed singular links by defining
$None$ $f(L_{\times}):=F(L_{+})-F(L_{-}),$
for every $L_{\times}\in{\mathcal{L}}^{(1)}$. Continuing inductively we can
extend the invariant on singular links in ${\mathcal{L}}^{(n)}$ for all
$n\in{\bf N}$. We are interested in reversing this process; the reverse
process is usually referred to as integration of the singular link invariant
to an invariant of links [1, 12, 13, 21]. In this section we deal with the
following question: Suppose that we are given an invariant of framed singular
links $f:{\mathcal{L}}^{(1)}\longrightarrow\mathbb{A}$. Under what conditions
is there a framed link invariant $F:{\mathcal{L}}\longrightarrow\mathbb{A}$ so
that (1) holds for all singular links $L_{\times}\in{\mathcal{L}}^{(1)}$? We
will address this question for links in ${\mathbb{Q}}$-homology $3$-spheres
with trivial $\pi_{2}$.
###### Definition 2.4.
Let $N$ be an oriented compact 3-manifold with or without boundary. A map
$\Phi:S^{1}\times S^{1}\longrightarrow N$ is called essential if it induces an
injection on $\pi_{1}$ and it cannot be homotoped to a map
$\Phi^{\prime}:S^{1}\times S^{1}\longrightarrow\partial N$. Otherwise $\Phi$
is called inessential. The manifold $N$ is called atoroidal if there are no
essential maps $S^{1}\times S^{1}\longrightarrow N$.
###### Remark 2.1.
Let $L_{\times\times}\in{\mathcal{L}}^{(2)}$ be a framed singular link with
two inadmissible singular points. By resolving the singular points, one at a
time, we obtain four singular links in ${\mathcal{L}}^{(1)}$. These are shown
in Figure 3, where the notation is consistent with that of Figure 2.
Figure 3. From left to right: $L_{\times r}$, $L_{r\times}$, $L_{\times l}$,
$L_{l\times}$
We note that $L_{{\times}r}$ is equivalent to $L_{{r\times}}$. Similarly,
$L_{{\times}l}$ is equivalent to $L_{{l\times}}$. Thus if
$f:{\mathcal{L}}^{(1)}\longrightarrow{\mathbb{A}}$ is an invariant of framed
singular links then we have
$None$ $f(L_{\times r})=f(L_{r\times})\ {\rm and}\ f(L_{\times
l})=f(L_{l\times}).$
Now (2) implies that the signed sum of $f$ on the four singular links in
Figure 3, where signs are determined by (1), is equal to zero. Next we will
show that if this holds true for all $L_{\times\times}\in{\mathcal{L}}^{(2)}$,
then $f$ can be integrated to a framed link invariant.
###### Theorem 2.2.
Suppose that $M$ is a ${\mathbb{Q}}$-homology sphere with $\pi_{2}(M)=0$ and
such that that if $H_{1}(M)\neq 0$ then $M$ is atoroidal. Let
$f:{\mathcal{L}}^{(1)}\longrightarrow{\mathbb{A}}$ be an invariant of framed
singular links with one double point. Suppose that ${\mathbb{A}}$ is torsion
free and that the invariant $f$ satisfies the relation
$None$ $f(L_{{\times}+})-f(L_{{\times}-})=f(L_{+{\times}})-f(L_{-{\times}}),$
for every $L_{\times\times}\in{\mathcal{L}}^{(2)}$. Then there exists a framed
link invariant $F$ such that $f$ is derived from $F$ via equation (1). Here,
the four singular links appearing in (3) are obtained by resolving the
singular points of $L_{\times\times}$ one at a time.
Theorem 2.2 is the framed link analogue of Theorem 3.16 of [12] and Theorem
3.1.2 of [13]. As explained in the Introduction, however, here we work in a
more general class of manifolds. Also the presence of framing requires an
adaptation of the arguments: to formulate the correct “global integrability
condition” (equation (6) below) we need a notion of global framing around
homotopies of links. The definition of such a notion is facilitated by the
works of Chernov and Kaiser [3, 16] (Definition 2.5). For arguments that are
very similar to these in [12, 13] we will refer the reader to these articles
for details.
### 2.3. Loop space and framing control
Because in this section we work with oriented links we need to slightly modify
the set of initial links $\mathcal{C}\mathcal{L}^{*}\cup\\{U\\}$ chosen in the
Introduction. Recall that ${\mathcal{L}}$ (resp. ${\bar{\mathcal{L}}}$)
denotes the set of isotopy classes of framed oriented (resp. unoriented) links
in $M$. Consider the set of oriented links
${\mathcal{C}}{\mathcal{L}}:={\mathfrak{o}}^{-1}(\mathcal{C}\mathcal{L}^{*}\cup\\{U\\})$,
where ${\mathfrak{o}}:{\mathcal{L}}\longrightarrow{\bar{\mathcal{L}}}$ is the
obvious forgetful map. Also recall that ${\tilde{\mathcal{L}}}$ denotes the
set of isotopy classes of ordered framed links in $M$ and that we defined a
forgetful map
${\mathfrak{r}}:{\tilde{\mathcal{L}}}\longrightarrow{\mathcal{L}}$. Given
$CL\in{\mathcal{C}}{\mathcal{L}}$, we pick
$L\in{\mathfrak{r}}^{-1}({\mathcal{C}}{\mathcal{L}})$. We will also use $L$ to
denote a representative $L:P\longrightarrow M$ of $L$, where $P$ is a disjoint
union of oriented circles. Let ${{\mathcal{M}}}^{L}(P,M)$ denote the space of
ordered smooth framed immersions $P\longrightarrow M$ homotopic to $L$,
equipped with the compact-open topology. For every
$L^{\prime}\in{\tilde{\mathcal{L}}}$ and representative
$L^{\prime}\in{\mathcal{M}}^{L}(P,M)$, let $\Phi:P{\times}[0,1]\longrightarrow
M$ be a homotopy with $\Phi(P\times\\{0\\})=L^{\prime}$ and
$\Phi(P\times\\{1\\})=L$. After a small perturbation we can assume that for
only finitely many points $0<t_{1}<t_{2}<\cdots<t_{n}<1$,
${\phi}_{t}:=\Phi(P\times\\{t\\})$ is not an embedding and it is a singular
framed link of order $1$. For different $t^{\prime}s$ in an interval of $[0,\
1]\setminus\\{t_{1},\ t_{2},\ \dots,t_{n}\\}$ the corresponding framed links
are equivalent and when $t$ passes through $t_{i}$, ${\phi}_{t}$ changes from
one resolution of ${\phi}_{t_{i}}$ to the other.
For $CL\in{\mathcal{C}}{\mathcal{L}}$, let ${\mathcal{M}}^{CL}(M)$ denote the
space of unordered smooth framed immersions homotopic to $CL$, equipped with
the compact-open topology. The projection
${\mathfrak{q}}:{\mathcal{M}}^{L}(P,M)\longrightarrow{\mathcal{M}}^{CL}(M)$ is
a covering map away from points that are fixed under permutation of
components.
###### Definition 2.5.
Let $\Phi$ be a homotopy between ordered links
$L_{1},L_{2}\in{{\mathcal{M}}}^{L}(P,M)$ with points $0<t_{1}<\cdots<t_{n}<1$
such that ${\phi}_{t_{j}}\in{\tilde{\mathcal{L}}}^{(1)}$. For each singular
link ${\phi}_{t_{j}}$ we have a sequence $\\{{\bf f}^{j}_{i}|i=1,\ldots,m\\}$
as in Definition 2.3. We define the _t_ otal framing of $\Phi$ to be the
sequence of integers $\\{\Delta{\bf f}_{i}|i=1,\dots,m\\}$, where
$None$ $\Delta{\bf f}_{i}:=\sum_{j=1}^{n}\delta_{j}^{i}{\epsilon}_{j}{\bf
f}^{j}_{i}({\phi}_{t_{j}}).$
Here $\delta_{j}^{i}=1$ if the $i$-th component of ${\phi}_{t_{j}}$ contains
the double point and 0 otherwise. Also ${\epsilon}_{j}=1$ if
${\phi}_{t_{j}+\delta}$, for $\delta>0$ sufficiently small, is a positive
resolution of ${\phi}_{t_{j}}$ and ${\epsilon}_{j}={-1}$ otherwise. We will
say that the total framing is zero iff $\Delta{\bf f}_{i}=0$, for all
$1,\dots,m$.
Given a loop $\Phi\in{\mathcal{M}}^{CL}(M)$ we obtain a set of ordered
sequences $\Delta{\bf f}_{\Phi}$ associated to the set of all lifts of $\Phi$
in ${\mathcal{M}}^{L}(P,M)$. The map
${\mathfrak{q}}^{-1}(\Phi)\longrightarrow\Delta{\bf f}_{\Phi}$ defines an
unordered sequence of integers for $\Phi$. The homotopy $\Phi$ is called
framing preserving iff the total framing of every element in
${\mathfrak{q}}^{-1}(\Phi)$ is zero. We will write $\Delta{\bf f}_{\Phi}={\bf
0}$.
### 2.4. Beginning the proof of Theorem 2.2
We want to define an invariant $F:{\mathcal{L}}\longrightarrow{\mathbb{A}}$
that is obtained from the given
$f:{\mathcal{L}}^{(1)}\longrightarrow{\mathbb{A}}$ via (1). First we assign
values of $F$ on the set of initial links ${\mathcal{C}}{\mathcal{L}}$. Now
fix $CL\in{\mathcal{C}}{\mathcal{L}}$ and let
$L^{\prime}\in{\mathcal{M}}^{CL}(M)$ be a framed link. Choose a generic
homotopy $\Phi$ from $L^{\prime}$ to $CL$. Let $0<t_{1}<t_{2}<\cdots<t_{n}<1$
denote the points where ${\phi}_{t}$ is not an embedding. Recall that
${\phi}_{t_{i}}\in{\mathcal{L}}^{(1)}$ such that for different $t^{\prime}s$
in an interval of $[0,\ 1]\setminus\\{t_{1},\ t_{2},\ \dots,t_{n}\\}$, the
corresponding framed links are equivalent. When $t$ passes through $t_{i}$,
${\phi}_{t}$ changes from one resolution of ${\phi}_{t_{i}}$ to another. We
define
$None$ ${F(L^{\prime})=F(CL)+\sum_{i=1}^{n}{\epsilon}_{i}f({\phi}_{t_{i}})}$
Here ${\epsilon}_{i}={\pm}1$ is determined as follows: If
${\phi}_{t_{i}+\delta}$, for $\delta>0$ sufficiently small, is a positive
resolution of ${\phi}_{t_{i}}$ then ${\epsilon}_{i}=1$. Otherwise
${\epsilon}_{i}={-1}$.
To prove that $F$ is well defined we have to show that modulo “the integration
constant” $F(CL)$, the definition of $F(L^{\prime})$ is independent of the
choice of the homotopy. For this we consider a closed homotopy $\Psi$ from
$CL$ to itself. After a small perturbation, we can assume that there are only
finitely many points $x_{1},x_{2},\dots,x_{n}\in S^{1}$, ordered cyclicly
according to the orientation of $S^{1}$, so that
${\psi}_{x_{i}}\in{\mathcal{L}}^{1}$ and $\psi_{x}$ is equivalent to
$\psi_{y}$ for all $x_{i}<x,y<x_{i+1}$. To prove that $F$ is well defined we
need to show that
$None$ ${X_{\Psi}:=\sum_{i=1}^{n}{\epsilon}_{i}f({\psi}_{t_{i}})=0}$
where ${\epsilon}_{i}={\pm}1$ is determined by the same rule as above.
Independence of link component orderings: To prove (6) we will turn our
attention to ordered links: First we note that the invariant $f$ pulls back to
an invariant on ${\tilde{\mathcal{L}}}^{(1)}$ via the forgetful map
${\mathfrak{r}}$. After iterating $\Phi$ several times if necessary we can
assume that it lifts to a loop in ${\mathcal{M}}^{L}(P,M)$ based at $L$
(compare, page 3874 of [16]). Given a self-homotopy $\Phi$ of $CL$ and the
associated quantity $X_{\Phi}$, lift $\Phi$ to a closed homotopy $\Psi$ in
${\mathcal{M}}^{L}(P,M)$ and let $X_{\Psi}$ denote the lift of $X_{\Phi}$.
Note that $X_{\Psi}=aX_{\Phi}$, for some integer $a\in{{\mathbb{Z}}}$. Since
${\mathbb{A}}$ is torsion free we have $X_{\Phi}=0$ exactly when $X_{\Psi}=0$.
Thus, it is enough to check (6) for homotopies that preserve the ordering of
components.
Restriction to framing preserving homotopies: Next we observe that it is
enough to check (6) for homotopies that are framing preserving in the sense of
Definition 2.5: To see that we recall that given a framed link
$L^{\prime}\in{\mathcal{M}}^{CL}(M)$ we need to check that (5) does not depend
on the homotopy from $L^{\prime}$ to the framed link $CL$ used to define it.
Thus the closed homotopies $\Phi$ that we need (6) to hold for, are those
obtained by composing two homotopies from $L^{\prime}$ to $CL$. Each component
of $CL$ is equipped with a vector field and going around $\Phi$ does not
change the homotopy class of this vector field (that is the equivalence class
of $CL$ as a framed link). We can think that the framing of $CL$ transports to
a “new” framing around $\Phi$. The two framings might differ by twists on the
components of $CL$ but the total singed number of the twists must be zero. The
total sum of such twists is captured exactly by the quantity $\Delta{\bf
f}_{\Phi}$ (compare, Theorem 6 of [16]). The framing of $CL$ lifts to one on
$L$ and going around the self-homotopy of $L$ that lifts $\Phi$ also preserves
the homotopy class of the framing vector field.
The proof of (6), which occupies the remaining of Section 2 and Section 3,
will be divided into several steps. In this section we will give the proof of
(6) for closed homotopies in atoroidal 3-manifolds and in the next section we
deal with essential tori.
To continue, suppose that $P$ has $m$ components $P=\sqcup_{i=1}^{m}P_{i}$,
where each $P_{i}$ is an oriented circle. Let $L:P\longrightarrow M$ be a
link. Pick a base point $p_{i}\in P_{i}$ and let $a_{i}$ denote the homotopy
class of $L(P_{i})$ in $\pi_{1}(M,L(p_{i}))$. We denote by $Z(a_{i})$ the
centralizer of $a_{i}$ in $\pi_{1}(M,L(p_{i}))$. We begin with the following
lemma (see, for example, the proof of Proposition 4.3 of [21]).
###### Lemma 2.3.
Suppose that $M$ is an orientable 3-manifold with $\pi_{2}(M)=0$ and let the
notation be as above. Then
$\pi_{1}({\mathcal{M}}^{L}(P,M),L)\cong\oplus_{i=1}^{m}Z(a_{i}).$
### 2.5. Integrating around inessential tori
Here we show how to derive (6) in the case where the closed homotopy $\Phi$
represents a collection of inessential tori in $M$. Since $\partial
M=\emptyset$ this means that the induced map
$({\Phi_{i}})_{*}:\pi_{1}(P_{i}\times S^{1})\longrightarrow\pi_{1}(M)$ has
non-trivial kernel. Here $\Phi_{i}:=\Phi|P_{i}\times S^{1}$, for
$i=1,\dots,m$.
###### Lemma 2.4.
Let $\Phi$ be a loop in ${\mathcal{M}}^{L}(P,M)$ representing a framing
preserving self-homotopy of $L$. Suppose that $\Phi$ can be extended to a map
${\hat{\Phi}}:P\times D^{2}\longrightarrow M$ where $D^{2}$ is a 2-disc with
$\partial D^{2}=\\{*\\}\times S^{1}$. Then $X_{\Phi}=0$.
###### Proof.
We perturb ${\hat{\Phi}}$, relatively $\partial D^{2}$, so that it is in
general position in the sense of Proposition 1.1 of [12]. Then the set
$S_{{\hat{\Phi}}}:=\\{x\in D^{2}\ |\
{\hat{\phi}}_{x}:={\hat{\Phi}}(P\times\\{x\\})\ {\rm is\ not\ an\
embedding}\\},$
is a graph in $D^{2}$ with properties (1)-(5) given in Proposition 1.1 of
[12]. The vertices of $S_{{\hat{\Phi}}}$ in the interior of $D^{2}$ are of
valence one or four (see Figure 4).
Figure 4. The set of singularities $S_{{\hat{\Phi}}}$ with the types of double
points they represent.
The invariant $f$ assigns an element of $\mathbb{A}$ to every edge of
$S_{{{\hat{\Phi}}}}$. We observe that condition (3) in the statement of
Theorem 2.2 implies that $X_{{\Phi}}$ is independent on the order in which the
crossing changes around $\Phi:={\hat{\Phi}}|P\times\partial D^{2}$ occur.
Thus, without loss of generality, we may assume that the valence one vertices
of $S_{{{\hat{\Phi}}}}$ in the interior of $D^{2}$ correspond to inadmissible
crossing changes on $\partial D^{2}$. With the notation as above, we will
assume that the framed singular link ${\phi}_{x_{i}}\in{\mathcal{L}}^{1}$ is
inadmissible for $i=1,\ldots,s$ and admissible for $i=s,\ldots,n$. In
particular, there are $s$ edges of $S_{{{\hat{\Phi}}}}$ emanating from
$x_{1},\dots,x_{s}$ respectively and ending at an interior vertex of valence
one, and these are the only valence one vertices of $S_{{{\hat{\Phi}}}}$.
Figure 5. The singular links ${\phi}_{x_{1}}$, ${\phi}_{x_{2}}$ form a pair of
type $L_{\times r},L_{r\times}$ (or $L_{\times l},L_{l\times}$) shown in
Figure 3. The framed links corresponding to the components $e,e^{\prime}$ of
${\partial D^{2}}\setminus\\{x_{1},x_{2},\dots\\}$ are isotopic.
For every interior vertex of $S_{{{{\hat{\Phi}}}}}$ we draw a small circle $C$
around it so that the number of points in $C\cap S_{{{{\hat{\Phi}}}}}$ is
equal to the valence of the vertex. See Figure 5. Let $C_{1},\ldots,C_{s}$
denote the circles surrounding the valence one vertices of
$S_{{{{\hat{\Phi}}}}}$ and let ${\Gamma}$ denote the disjoint union of the
circles surrounding the vertices of valence four. For a vertex of valence four
the four points in $C\cap S_{{\hat{\Phi}}}$ correspond exactly to these
appearing in equation (3). Thus by (3) we have
$None$ ${\sum_{x\in\Gamma\cap
S_{{{{\hat{\Phi}}}}}}{\epsilon_{x}}f({{\hat{\phi}}}_{x})=0,}$
where ${\hat{\phi}}_{x}:={{{\hat{\Phi}}}(P\times\\{x\\})}.$ Now observe that
$\sum_{i=s+1}^{n}{\epsilon}_{i}f({{\phi}}_{x_{i}})=\sum_{x\in\Gamma\cap
S_{{{{\hat{\Phi}}}}}}{\epsilon_{x}}f({{\hat{\phi}}}_{x})=0.$
The last equation and (7) imply that
$None$ $X_{\Phi}=\sum_{i=1}^{s}{\epsilon}_{i}f({{\phi}}_{x_{i}}).$
Since $\Phi$ is framing preserving we have $\Delta{\bf f}_{\Phi}={\bf 0}$. By
Definitions 2.3 and 2.5 and the fact that ${\bf f}$ remains unchanged under
admissible crossing changes we have $\Delta{\bf f}_{C}={\bf 0}$, for every
loop $C\in\Gamma$. This in turn implies that
$\Delta{\bf f}_{\Gamma}:=\sum_{C\in\Gamma}\Delta{\bf f}_{C}={\bf 0}$
Since we have
$\Delta{\bf f}_{\Phi}=\sum_{i=1}^{s}{\epsilon}_{i}{\bf
f}({{\phi}}_{x_{i}})+\Delta{\bf f}_{\Gamma}={\bf 0}$
we conclude that $\sum_{i=1}^{s}{\epsilon}_{i}{\bf f}({{\phi}}_{x_{i}})={\bf
0}$. This in turn implies that the inadmissible singular links
${{\phi}}_{x_{i}}$ can be partitioned into pairs of the forms shown in Figure
3. Relation (2) in Remark 2.1 shows that the right hand side of (8) is
identically zero. Thus $X_{\Phi}=0$, as desired. ∎
###### Remark 2.5.
Let ${\bar{X}}_{\Phi}$ denote the contribution of the admissible singular
links around $\Phi$ to $X_{\Phi}$. The proof of Lemma 2.4 shows that
regardless of whether $\Phi$ is framing preserving, relation (3) implies that
${\bar{X}}_{\Phi}=0$.
###### Remark 2.6.
Proposition 1.1 of [12], referenced in the proof of Lemma 2.4, is stated in
there for the PL-category. However, as explained by Kaiser in Section 3 of
[15], the statement is true in the smooth category which is actually what we
need here. We should also remark that, as explained by Lin in [21], the
conclusion holds if the disc $D^{2}$ is replaced by any planar surface $F$.
Furthermore, if $\Phi|\partial F$ is already in general position then the
modifications that put $\Phi$ into general position on $F$ can be performed
relatively $\partial F$.
A slight variation of the proof of Lemma 2.4 shows the following:
###### Lemma 2.7.
Let $\Phi$ be a loop in ${\mathcal{M}}^{L}(P,M)$ representing a framing
preserving self-homotopy of a framed link $L$. Let $P^{\prime}:=P\setminus
P_{1}$. Suppose that $\Phi|P^{\prime}$ can be extended to a map
${\hat{\Phi}}:P^{\prime}\times D^{2}\longrightarrow M$ where $D^{2}$ is a
2-disc with $\partial D^{2}=\\{*\\}\times S^{1}$. Suppose moreover that
$\Phi|(P_{1}\times S^{1})$ is an embedding. Then $X_{\Phi}=0$.
The proof of the next lemma is given in the proof of Lemma 3.3.4 of [13].
###### Lemma 2.8.
Let $M$ be a ${\mathbb{Q}}$-homology 3-sphere with $\pi_{2}(M)=0$. Suppose
that $\pi_{1}(M)$ is infinite and that $L$ has no homotopically trivial
components. Let $\Phi\subset{\mathcal{M}}^{L}(P,M)$ be a framing preserving
closed homotopy such that the restriction $\Phi|P_{i}\times
S^{1}\longrightarrow M$ is inessential, for all $i=1,\dots,m$. There exists a
2-disc $D^{2}$ and a map ${\tilde{\Phi}}:P\times D^{2}\longrightarrow M$ such
that
$None$ $X_{\partial\tilde{\Phi}}=aX_{{\Phi}},$
for some $a\in{\mathbb{Z}}$. Here
${\partial\tilde{\Phi}}={\tilde{\Phi}}|P\times{\partial D^{2}}$.
### 2.6. Theorem 2.2 for atoroidal manifolds
Before we can proceed with the proof of the theorem we need two additional
lemmas.
###### Lemma 2.9.
Consider ${\Phi},{\Phi^{\prime}}:S^{1}\longrightarrow{{\mathcal{M}}^{L}(P,M)}$
two self-homotopies of $L$. Let ${\bar{X}}_{\Phi}$ and
${\bar{X}}_{\Phi^{\prime}}$ denote the contribution to $X_{\Phi}$ and
${X}_{\Phi^{\prime}}$ coming from admissible singular links around $\Phi$ and
$\Phi^{\prime}$, respectively. Suppose that ${\Phi},{\Phi^{\prime}}$ are
freely homotopic as loops in ${\mathcal{M}}^{L}(P,M)$. Then we have
${\bar{X}}_{\Phi^{\prime}}={\bar{X}}_{\Phi}$. Furthermore, there is a group
homomorphism $\psi:\pi_{1}({\mathcal{M}}^{L}(P,M),\
L)\longrightarrow{\mathbb{A}}$ defined by $\psi([\Phi]):={\bar{X}}_{\Phi}$.
###### Proof.
By a slight variation of the argument in the proof of Lemma 3.3.2 of [13] we
have the following: There exists a map
${\hat{\Psi}}:D^{2}\longrightarrow{{\mathcal{M}}^{L}(P,M)}$ such that if we
set $\Psi:={\hat{\Psi}}|{\partial D^{2}}$ then
$\Psi:S^{1}\longrightarrow{{\mathcal{M}}^{L}(P,M)}$ is a self-homotopy of $L$
with
$X_{\Psi}=X_{\Phi}-X_{\Phi^{\prime}}.$
Lemma 2.4 and Remark 2.5 imply ${\bar{X}}_{\Psi}=0$; thus
${\bar{X}}_{\Phi}={\bar{X}}_{\Phi^{\prime}}$.
For the remaining of the claim define $\psi:\pi_{1}({\mathcal{M}}^{L}(P,M),\
L)\longrightarrow{\mathbb{A}}$ as follows: Given
$\alpha\in\pi_{1}({\mathcal{M}}^{L}(P,M),\ L)$, let $\Phi$ is be a self-
homotopy of $L$ representing $\alpha$. Define $\psi(\alpha)={\bar{X}}_{\Phi}$.
By our earlier arguments $\psi(\alpha)$ is independent on the representative
$\Phi$. The fact that $\psi$ is a group homomorphism follows easily. ∎
The next lemma is Lemma 3.2.5 in [13]. We point out that the proof of this
lemma uses the hypothesis that the group $\mathbb{A}$ in which the invariants
take values is torsion free.
###### Lemma 2.10.
Suppose that $M$ is a ${\mathbb{Q}}$-homology 3-sphere with $\pi_{2}(M)=0$.
Let $L:P\longrightarrow M$ be a framed link and let
$\Phi:P{\times}S^{1}\longrightarrow M$ be a framing preserving self-homotopy
of $L$. Assume that, for some $i=1,\ldots,m$, we have $a_{i}=1$. Set
$P^{\prime}:=P\setminus P_{i}$ and $\Phi^{\prime}:=\Phi|P^{\prime}$. If
$X_{\Phi^{\prime}}=0$ then $X_{\Phi}=0$.
We are now ready to give the proof of Theorem 2.2 in the case where $M$ is an
atoroidal ${\mathbb{Q}}$-homology 3-sphere.
###### Theorem 2.11.
Suppose that $M$ is an atoroidal ${\mathbb{Q}}$-homology 3-sphere with
$\pi_{2}(M)=0$. Then the conclusion of Theorem 2.2 is true for $M$.
###### Proof.
Let $f:{\mathcal{L}}^{1}\longrightarrow{\mathbb{A}}$ be a framed singular link
invariant satisfying (3) of the statement of Theorem 2.2 and let $\Phi:P\times
S^{1}\longrightarrow M$ be a framing preserving self-homotopy of a framed link
$L:P\longrightarrow M$. We have to show that
$X_{\Phi}=0$
where $X_{\Phi}$ is the signed sum of values of $f$ around $\Phi$ defined in
(6).
First suppose that $\pi_{1}(M)$ is finite. Then, by Lemma 2.3,
$\pi_{1}({\mathcal{M}}^{L}(P,M),\ L)$ is finite. Since ${\mathbb{A}}$ is
torsion free the homomorphism $\psi$ of Lemma 2.9 must be the trivial one.
Thus, in particular, $X_{\Phi}=0$.
Now suppose that $\pi_{1}(M)$ is infinite. If the link $L$ to begin with
contains no homotopically trivial components, then since $M$ is atoroidal,
Lemma 2.8 applies to conclude that $X_{\partial\tilde{\Phi}}=aX_{{\Phi}}$, for
a map ${\tilde{\Phi}}:P\times D^{2}\longrightarrow M$. By Lemma 2.4,
$X_{\partial\tilde{\Phi}}=aX_{{\Phi}}=0$ and thus, since ${\mathbb{A}}$ is
torsion free, $X_{{\Phi}}=0$.
Next suppose that all the components of $L$ are homotopically trivial; that is
$a_{i}=1$, for $i=1,\ldots,m$. Then, by Lemma 2.3,
$\pi_{1}({\mathcal{M}}^{L}(P,M),L)\cong\oplus_{i}^{m}\pi_{1}(M,L(p_{i})).$
Since $H_{1}(M)$ is finite the above equality implies that the abelianization
of the group $\pi_{1}({\mathcal{M}}^{L}(P,M),L)$ is a finite group. By Lemma
2.9 we have a homomorphism
$\psi:\pi_{1}({\mathcal{M}}^{L}(P,M),L)\longrightarrow{\mathbb{A}}$ with
$\psi([\Phi])=X_{\Phi}$. Since ${\mathbb{A}}$ is abelian $\psi$ factors
through the abelianization of $\pi_{1}({\mathcal{M}}^{L}(P,M),L)$; a finite
group. But since ${\mathbb{A}}$ is torsion free $\psi$ is the trivial
homomorphism. Thus $X_{\Phi}=0$.
To handle the general case let $h(L)$ denote the number of components of $L$
that are homotopically trivial. The proof is by induction on $h(L)$. In the
light of our discussion above, the conclusion is true if $h(L)=0$ or $h(L)=m$.
Thus we may assume that $h(L)\neq 0,m$. Let $L_{i}\subset L$ be a component
that is homotopically trivial and let $L^{\prime}:=L\setminus L_{i}$. Also let
$\Phi$ be a self-homotopy of $L$ and let $\Phi^{\prime}$ denote the
restriction of $\Phi$ on $P^{\prime}$, where $P^{\prime}:=P\setminus P_{i}$.
Since $h(L^{\prime})<h(L)$, by induction, $X_{\Phi^{\prime}}=0$. Then, by
Lemma 2.10, $X_{\Phi}=0$. ∎
## 3\. Integration of invariants in toroidal 3-manifolds
To study the question of integrability of singular link invariants in toroidal
3-manifolds we need several results from the theory of the characteristic
submanifold of Jaco-Shalen [10] and Johannson [11]. The statements of the
results from these theories, in the form needed in our setting, are summarized
in Section 2 of [12] and in Section 2 of [13]. It will be convenient for us to
recall the statements we need below from therein, instead from the original
references. In particular we will need the Enclosing Theorem and the Torus
Theorem both stated on pp. 679 of [12]. The later, in the form needed for our
purposes, follows from work of Scott, Casson-Jungreis and Gabai.
###### Theorem 3.1.
Let $M$ be a ${\mathbb{Z}}$-homology 3-sphere with $\pi_{2}(M)=0$ and let
${\mathbb{A}}$ be a torsion free abelian group. Suppose that a map
$f:{\mathcal{L}}^{(1)}\longrightarrow{\mathbb{A}}$ satisfies (3) of Theorem
2.2. Then there exists a framed link invariant $F$ such that $f$ is derived
from $F$ via equation (1).
###### Remark 3.2.
The restriction to ${\mathbb{Z}}$-homology 3-spheres in Theorem 3.1 is
necessary. As explained in Remark 3.13 of [12] and the discussion at the end
of Section 3 in [13], in general, local conditions are not sufficient for the
integration of singular link invariants. When the characteristic submanifold
contains Seifert fibered components over non-orientable surfaces one needs to
impose extra non-local conditions. Specific constructions demonstrating these
phenomena are given by Kirk and Livingston in [19]. The necessity of working
with irreducible 3-manifolds is demonstrated by [19] as well as the work of
Eiserman [7].
The proof of Theorem 2.2 will be completed once we have proved Theorem 3.1.
For the proof of Theorem 3.1 we will need the following:
###### Lemma 3.3.
Let $M$ be a $\mathbb{Z}$-homology 3-sphere with $\pi_{2}(M)=0$. Suppose that
$\Phi:T=S^{1}\times S^{1}\longrightarrow M$ is an essential map. Then there
exists a map $\Psi:T\longrightarrow M$ homotopic to $\Phi$ and such that one
of the following holds:
1. (1)
$\Psi(T)$ lies on an essential embedded torus in $M$.
2. (2)
There exists an oriented surface $F$ with $\partial F\neq\emptyset$, and a
trivial fiber bundle $Y=S^{1}\times F$, with the following property: $\Psi$
extends to a map ${\hat{\Psi}}:Y\longrightarrow M$ so that the image
${\hat{\Psi}(\partial Y\setminus T})$ is contained on a collection of embedded
tori in $M$.
###### Proof.
By the Torus Theorem and the discussion at the end of Section 2 of [13],
either $M$ is Haken or it is a Seifert fibered 3-manifold that fibers over
$S^{2}$ with three or less exceptional fibers.
First suppose that $M$ is Haken. Then by the Enclosing Theorem there is a
Seifert fibered submanifold $S\subset M$ and a homotopy
$\Phi^{\prime}_{t}:T\longrightarrow M$ such that $\Phi^{\prime}_{0}=\Phi$ and
$\Phi^{\prime}_{1}(T)\subset S$. If $\Phi^{\prime}_{1}(T)$ can be further
homotoped in $S$ so that it lies on a component of $\partial S$ then we have
conclusion (1). Otherwise, by the classification of essential tori in Haken
Seifert fibered spaces (Proposition 2.11 of [12]) we can homotope
$\Phi^{\prime}_{1}$ in $S$ to a map $\Psi:T\longrightarrow S$ which is
vertical with respect to the fibration.
Next suppose that $M$ is a Seifert fibered space. By Proposition 2.2.5 of
[13], $\Phi$ is homotopic to a map $\Psi:T\longrightarrow M$ which is vertical
with respect to the fibration of $M$.
Thus, in both cases, either (1) holds or we have a Seifert fibered manifold
$S\subseteq M$, with orbit space $B$ and fiber projection $p$, such that
$\Phi$ is homotopic to a map $\Psi:T\longrightarrow M$ that is vertical with
respect to the fibration of $S$. This means that $\Psi$ is a composition
$\Phi_{1}\circ q$, where $q$ is a covering map from the torus $T$ to itself
and $\Phi_{1}:T\longrightarrow S$ is an immersion without triple points. Then,
there exists a decomposition $T=S^{1}\times S^{1}$ such that
a) $\Phi_{1}(S^{1}\times\\{*\\})$ maps onto a regular fiber $h$ of $S$;
b) we have $p({\Phi_{1}}(\\{*\\}\times S^{1}))=p(\Phi_{1}(T))$ on the orbit
surface $B$ of $S$.
Let $H$ (resp. $Q$) denote the curve $S^{1}\times\\{*\\}$ (resp.
$\\{*\\}\times S^{1}$) on $T$. Now $\alpha:=p(\Phi_{1}(T))$ is an immersed
closed curve on $B$ with singularities finitely many transverse double points.
A neighborhood $N:=N(\alpha)\subset B$ of $\alpha$ on $B$ is an oriented
planar surface. Choose $N$ small enough so that $Y:=p^{-1}(N)$ contains no
exceptional fibers of $S$. Now $p:Y\longrightarrow N$ is an $S^{1}$-bundle and
since $H^{2}(N)=0$ this bundle is trivial. Choose a trivialization $Y\cong
S^{1}\times N$ so that $N$ is embedded as a cross-section. Pick a base point
$b\in N$ and arcs from $b$ to the components of $\partial N$ whose homotopy
classes freely generate $\pi_{1}(N)$; we pick one arc for each such component.
Assume that these arcs intersect $\alpha$ only at its double points; let
$x_{1},\ldots,x_{s}$ denote the resulting generators of $\pi_{1}(N,b)$. Write
$\alpha$ as a word in these generators; say
$[\alpha]=x_{i_{1}}^{k_{1}}x_{i_{2}}^{k_{2}}\cdots x_{i_{r}}^{k_{r}}.$
We can extend the restriction $\Psi|\\{*\\}\times S^{1}$ to a map
${\hat{\Psi}}:(F,\ \partial F)\longrightarrow(N,\ \partial N)$, where $F$ is a
planar surface, such that: (i) the induced map
${\hat{\Psi}}_{*}:\pi_{1}(F)\longrightarrow\pi_{1}(N)$ is onto; (ii)
$\pi_{1}(F)$ is freely generated by elements
$a^{1}_{1},\cdots,a^{1}_{k_{1}},a^{2}_{1},\cdots,a^{2}_{k_{2}},a^{r}_{1},\cdots,a^{r}_{k_{r}}$;
(iii) ${\hat{\Psi}}_{*}([Q])=x_{i_{1}}^{k_{1}}x_{i_{2}}^{k_{2}}\cdots
x_{i_{r}}^{k_{r}}$ (see proof of Lemma 3.11 of [12]). We pull back the fiber
bundle structure by ${\hat{\Psi}}$ to obtain a fiber bundle
${\hat{\Psi}}^{*}(Y)\longrightarrow F$, over $F$. The pull-back of the cross-
section $\alpha$ is a cross-section of ${\hat{\Psi}}^{*}(Y)$. Extending this
cross-section over $F$, and conclusion. ∎
We now recall that the proof of Theorem 3.1 is reduced to showing (6) for
every framing preserving self-homotopy of $L$. Using Lemmas 2.3, 2.9, and 2.10
we will see that the general case is essentially reduced to the case of knots.
Before we continue with the proof Theorem 3.1 some remarks are in order.
###### Remark 3.4.
Let $\Phi:P\times S^{1}\longrightarrow M$ denote a framing preserving self-
homotopy of a framed link $L$ and let $\Phi^{\prime}$ be obtained by a free
homotopy of $\Phi$ in $M$. Consider the homotopy from $\Phi$ to
$\Phi^{\prime}$ as a map ${\mathcal{H}}:P\times
S^{1}\times[0,1]\longrightarrow M$. We can smoothly approximate
${\mathcal{H}}$ by a homotopy in general position as in the proof of Lemma 2.4
(see Remark 2.6). Then we can view $\mathcal{H}$ as a family of smooth framed
immersions $S^{1}\longrightarrow M$ parametrized by an annulus. We note that
the closed homotopy $\Phi^{\prime}$ is not necessarily framing preserving.
###### Remark 3.5.
Suppose that we have a map $\Phi:Y:=S^{1}\times F\longrightarrow M$, such that
$F$ is a planar surface so that there is a component $\alpha\subset\partial F$
such that the restriction $\Phi|S^{1}\times\alpha$ is a loop in $M^{L}(P,M)$.
We can view $\Phi$ as a family of framed immersions in $M$, parametrized by
$F$. We can cut $Y:=S^{1}\times F\longrightarrow M$ along a collection of
properly embedded annuli (the projection of which on $F$ decomposes $F$ into a
disc) into a product $S^{1}\times D^{2}$. By considering the pull back of
$\Phi$ on $S^{1}\times D^{2}$ we obtain a family of framed immersions in $M$
parametrized by $D^{2}$.
In the next lemma we treat homotopies that involve essential tori. The proof
treats separately the case of knots and that of links. In the case of knots
($m=1$ below) the proof is very similar to that of Case 1 of Lemma 3.3.3 in
[13]. The starting ingredient in the proof of [13] is Lemma 3.3.2 therein.
Here we replace that ingredient with Lemma 3.3 and we outline the argument
below.
###### Lemma 3.6.
Let $M$ be a $\mathbb{Z}$-homology 3-sphere with $\pi_{2}(M)=0$ and let
$\Phi:P\times S^{1}\longrightarrow M$ be a framing preserving self-homotopy of
a framed link $L$. Suppose that $\Phi_{i}:=\Phi|P_{i}\times S^{1}$ is an
essential map, for some $i=1,\dots,m$. Suppose, moreover, that $\Phi_{i}$
cannot be homotoped so that its image lies on an essential embedded torus in
$M$. Then we have $X_{\Phi}=0$.
###### Proof.
Let $m$ be the number of components of $L$. We distinguish two cases according
to whether $m=1$ or $m>1$.
We have $m=1$: Since $\Phi$ is framing preserving, relation (2) implies that
the total contribution of the inadmissible singular links along $\Phi$ to
$X_{\Phi}$ is zero (proof of Lemma 2.4). Thus, without loss of generality, we
can assume that no inadmissible crossing changes occur along $\Phi$. Now let
$\Psi:P\times S^{1}\longrightarrow M$ be a map that is freely homotopic to
$\Phi$ in $M$. By Lemma 2.9, and our earlier assumption on $\Phi$, we have
${\bar{X}}_{\Psi}=X_{\Phi}$.
Set $T:=P\times S^{1}$, $l:=P\times\\{*\\}$ and $m:=\\{*\\}\times S^{1}$. By
assumption $\Phi|P\times S^{1}\longrightarrow M$ is an essential map and it
cannot be homotoped so that its image lies on an essential embedded torus in
$M$. By Lemma 3.3 we can homotope $\Phi$ to a map $\Psi:P\times
S^{1}\longrightarrow M$ so that: There is a trivial fiber bundle
$Y=S^{1}\times F$, over a planar surface $F$, such that $\Psi$ extends to a
map ${\hat{\Psi}}:S^{1}\times F\longrightarrow M$ and the image
${\hat{\Psi}}{(\partial Y\setminus T})$ is contained on a collection of
embedded tori in $M$. Let $H$ denote a simple closed curve $T$ representing a
fiber of $Y$ and let $Q$ denote the component of $\partial F$ (embedded as a
cross-section of the bundle) on $T$. In $\pi_{1}(T)$ we have $[l]=a[H]+b[Q]$,
for some $a,b\in{\mathbb{Z}}$.
First suppose that $a=0$. Then Lemma 3.12 of [12] (or Lemma 3.3.1 of [13])
applies to conclude that ${\bar{X}}_{\Psi}=0$. By our discussion above,
$X_{\Phi}={\bar{X}}_{\Psi}=0$ and the conclusion in this case follows.
Suppose now that $a\neq 0$. Let $q:{\tilde{Y}}\longrightarrow Y$ be the
covering of $Y$ corresponding to the subgroup
$a{{\mathbb{Z}}}\times\pi_{1}(F)$ of
$\pi_{1}(Y)={\mathbb{Z}}\times\pi_{1}(F)$. Lift $l$, $H$, and $Q$ to curves
${\tilde{l}}$, ${\tilde{H}}$, ${\tilde{Q}}$, respectively, on the torus
${\tilde{T}}:=q^{-1}(T)$. Now $\tilde{Y}$ is a trivial fiber bundle over a
surface ${\tilde{F}}$ with fiber $\tilde{l}$; we will write
$\tilde{Y}=\tilde{l}\times{\tilde{F}}$. Consider the composition
${\tilde{\Psi}}:={\hat{\Psi}}\circ q$ and its restriction on
$\tilde{T}\cong{\tilde{l}}\times{\tilde{Q}}$. Since
${\tilde{\Psi}}({\tilde{l}}\times\\{x\\})=\Psi(l\times q(\\{x\\}))$, for all
$x\in{\tilde{Q}}$, the restriction
${\tilde{\Psi}}|{\tilde{l}}\times{\tilde{Q}}$ is a self-homotopy of a framed
knot; the parameter space is ${\tilde{Q}}$. As in Remark 3.5 we will think of
${\tilde{\Psi}}$ as a family of framed immersions parametrized by a disc
$D^{2}$. Then we can consider $X_{\partial\tilde{\Psi}}$. As in the proof of
Lemma 3.14 of [12] we obtain that $X_{\partial\tilde{\Psi}}=cX_{\Psi}$, for
some $c\in{{\mathbb{Z}}}$. Since, as discussed at the beginning of this proof
we have ${{\bar{X}}_{\Psi}}=X_{{\Phi}}$, it follows that
${\bar{X}}_{\partial\tilde{\Psi}}=cX_{{\Phi}}$. By Remark 2.5, we have
${\bar{X}}_{\partial\tilde{\Psi}}=0$. Hence we conclude that we have
$cX_{{\Phi}}=0$ for some $c\in{\mathbb{Z}}$. Since $\mathbb{A}$ is torsion
free this implies that $X_{{\Phi}}=0$; finishing thereby the proof of the
Lemma in the case $m=1$.
We have $m>1$. By Lemma 2.3, $\pi_{1}({\mathcal{M}}^{L}(P,M),L)$ is isomorphic
to a direct product of the groups $\pi_{1}({\mathcal{M}}^{L}(P_{i},M),L_{i})$
for $i=1,\ldots,m$. By Lemma 2.9 it is enough to verify (6) only for
homotopies $\Phi$ that are fixed on all but one component of $L$. To that end,
let $\Psi$ be a homotopy in general position that only moves one component;
say $L_{1}$. Suppose, without loss of generality, that $\Psi|P_{1}\times
S^{1}\longrightarrow M$ is an essential map that cannot be homotoped so that
its image lies on an essential embedded torus in $M$. By (3), we may decompose
$\Psi$ into two homotopies $\Psi_{1}$ and $\Psi_{2}$ such that during
$\Psi_{1}$ we only have self-crossing changes on $L_{1}$, while during
$\Psi_{2}$ we only have crossing changes between $L_{1}$ and the rest of the
components. The argument of Case 1 applies to $\Psi_{1}$ to conclude that
$X_{\Psi_{1}}=0$. Since the restriction of $\Psi_{2}$ on $P^{\prime}\times
S^{1}$, where $P^{\prime}=P\setminus P_{1}$, is constant; it extends to a map
$P^{\prime}\times D^{2}\longrightarrow M$. Then by Lemma 2.7 we have
$X_{\Psi_{2}}=0$. ∎
### 3.1. The completion of the proof of Theorem 3.1
Let $\Phi$ be a framing preserving loop in $M^{L}(P,M)$. Suppose that
$\Phi|P_{i}\times S^{1}\longrightarrow M$ represents an essential torus for
some $i=1,\ldots,m$. First suppose that some component, say
$\Phi_{i}:=\Phi|P_{i}\times S^{1}\longrightarrow M$, can be homotoped to lie
on an embedded essential torus in $M$. Then a theorem of Nielsen ([9], theorem
13.1) implies that after further homotopy, we may assume that $\Phi_{i}$ is a
covering map of an embedded torus. It follows that the contribution of
$\Phi_{i}$ to $X_{\Phi}$ is zero. Thus, for our purposes, we can assume that
if $\Phi_{i}$ induces an injection on $\pi_{1}$ then it cannot be homotoped to
lie on an embedded torus. Then by Lemma 3.6 we obtain $X_{\Phi}=0$.
As in the proof of Lemma 3.6 we may assume that $\Phi$ fixes all but one
component of $L$; say $L_{1}$. If $\Phi:P_{1}\times S^{1}\longrightarrow M$ is
inessential the argument in the proof of Theorem 2.11 applies to conclude that
$X_{\Phi}=0$. Assume that $\Phi:P_{1}\times S^{1}\longrightarrow M$ is
essential. Then $X_{\Phi}=0$ by Lemma 3.6.
## 4\. Kauffman power series
### 4.1. Links in oriented ${\mathbb{Q}}$-homology 3-spheres
For framed links in $S^{3}$ the Kauffman polynomial is equivalent to a
sequence of 1-variable Laurent polynomials ${\\{R_{n}=R_{n}(t)\\}}_{n\in\bf
Z}$ determined by relations:
$R_{n}(U)=1$ $R_{n}(L_{r})=t^{{{{-(n+1)}}}}R_{n}(L)$
$R_{n}(L_{l})=t^{{{{(n+1)}}}}R_{n}(L)$
$R_{n}({L_{+}})-R_{n}({L_{-}})=(t-t^{-{1}})[R_{n}(L_{o})-R_{n}(L_{\infty})]$
where $L_{+}$, $L_{-}$, $L_{o}$, $L_{\infty}$ are as in Figure 1 and
$L_{r},L_{l}$ are as in Figure 2. Notice that the initial value $R_{n}(U)=1$
is just a normalization. Any choice of the initial value together with the
rest of the relations will determine a unique $R_{n}$. Set
$None$ $u_{n}(t):={{t^{n+1}-t^{-{(n+1)}}}\over{t-t^{-1}}}+1.$
By the relations above one obtains $R_{n}(L\sqcup U)=u_{n}(t)\ R_{n}(L)$,
where the link $L\sqcup U$ is obtained from $L$ by adding an unknotted and
unlinked component $U$. The coefficients of the power series $R_{n}(x)$
obtained from $R_{n}(t)$ by substituting $t=e^{x}$ are invariants of finite
type [1, 2]. In the theorem below we reverse this procedure and guided by the
axioms above we will construct power series invariants generalizing the
$R_{n}(x)$’s: Suppose that $M$ is a ${\mathbb{Q}}$-homology sphere with
$\pi_{2}(M)=0$ and such that if $H_{1}(M)\neq 0$ then $M$ is atoroidal. For
every $n\in{\mathbb{Z}}$ we will construct a sequence of framed link
invariants $\\{v_{n}^{m}|m\in{\mathbb{N}}\\}$ such that the formal power
series
$R_{\\{M,n\\}}=\sum_{m=0}^{\infty}v_{n}^{m}x^{m}$
satisfy the axioms above under the change of variable $t={e}^{x}$: We will
construct our invariants inductively (induction on $m$) by using Theorem 2.2.
Each $v_{n}^{m}$ is going to be obtained by integrating a suitable singular
link invariant determined by the $v_{n}^{j}$’s with $j<m$. Although the
resulting invariants will be invariants of unoriented framed links, for their
construction we need to work with oriented links. The reason is that Theorem
2.2 applies to oriented framed links. Recall that ${\mathcal{L}}$ (resp.
${\bar{\mathcal{L}}}$) denotes the set of isotopy classes of framed oriented
(resp. unoriented) links in $M$. Also recall the set of oriented initial links
${\mathcal{C}}{\mathcal{L}}:={\mathfrak{o}}^{-1}(\mathcal{C}\mathcal{L}^{*}\cup\\{U\\})$,
defined in the beginning of subsection $\S 2.3$. By Theorem 2.2 and its proof
the invariant $v_{n}^{m}$ is unique once the values on the set
${\mathcal{C}}{\mathcal{L}}$ are specified.
###### Theorem 4.1.
Assume that $M$ is a ${\mathbb{Q}}$-homology 3-sphere with $\pi_{2}(M)=0$ and
such that if $H_{1}(M,{\mathbb{Z}})\neq 0$ then $M$ is atoroidal. Fix
$n\in{\mathbb{Z}}$. Given maps
${\mathcal{V}}_{n}^{m}:\mathcal{C}{\mathcal{L}}^{*}\cup\\{U\\}\longrightarrow{\mathbb{C}}$,
$m\in{\mathbb{N}}$, there exists a unique sequence of complex valued link
invariants $\\{v_{n}^{m}|m\in{\mathbb{N}}\\}$ with the following properties:
1. (1)
$v_{n}^{m}(CL)={\mathcal{V}}_{n}^{m}(\mathfrak{o}(CL))$ for all
$CL\in{\mathcal{C}\mathcal{L}}$ and $m\in{\mathbb{N}}$.
2. (2)
$v_{n}^{m}(L)=v_{n}^{m}({\mathfrak{o}}(L))$ for all $L\in{\mathcal{L}}$ and
$m\in{\mathbb{N}}$. Thus the values of the invariants are independent of the
link orientation.
3. (3)
If we define a formal power series
$R_{n}:=R_{\\{M,n\\}}(L)=\sum_{m=0}^{\infty}v_{n}^{m}(L)x^{m}$
then we have
$None$ $R_{n}(U)=1$ $None$ $R_{n}(L_{r})=t^{{{{-(n+1)}}}}R_{n}(L)$ $None$
$R_{n}(L_{l})=t^{{{{(n+1)}}}}R_{n}(L)$ $None$
$R_{n}({L_{+}})-R_{n}({L_{-}})=(t-t^{-{1}})[R_{n}(L_{o})-R_{n}(L_{\infty})]$
where ${\displaystyle t:={e}^{x}=1+x+{{x^{2}}\over{2}}+\dots}$.
###### Proof.
Define $v_{n}^{m}(CL)={\mathcal{V}}_{n}^{m}(\mathfrak{o}(CL))$ for all
$CL\in{\mathcal{C}\mathcal{L}}$ and $m\in{\mathbb{N}}$. Now we can form the
power series $R_{n}(CL)$. Guided by (12)-(13) we define
$None$ $R_{n}(CL_{r})=t^{{{{-(n+1)}}}}R_{n}(CL)\ {\rm and}\
R_{n}(CL_{l})=t^{{{{(n+1)}}}}R_{n}(CL).$
Now guided by these we can define the values of $R_{n}$ on all framed links
whose underlying unframed isotopy class is $CL$. To explain this suppose that
$CL$ has $s$ components. Let $CL({\bf f})$ be a framed link in the same
(unframed) isotopy class with $CL$ with framing unordered sequence ${\bf f}$
(see Definition 2.2 and preceding discussion). Then define
$R_{n}(CL({\bf f}))=t^{(n+1)\tau}R_{n}(CL),$
where ${\tau}:={\tau}(CL({\bf f}))$ is the total framing of $CL({\bf f})$.
Using (14)-(15), and inducting on $k$, we can check that
$None$ $R_{n}(CL\sqcup U^{k})=[u_{n}(t)]^{k-1}\ R_{n}(CL)$
where $u_{n}(t)$ is given by (10). Now $R_{n}$ has been defined on all framed
links in the unframed isotopy classes of the links in
$\mathcal{C}\mathcal{L}$.
To continue for every $L({\bf f})\in{\mathcal{L}}$ with framing sequence ${\bf
f}$ we define
$v_{n}^{0}(L({\bf{f}}))=v_{n}^{0}(CL({\bf f})),$
where $CL$ is the initial link homotopic to $L$. Inductively, suppose that the
invariants $v_{n}^{0},v_{n}^{1},\dots,v_{n}^{m-1}$ have been defined such that
if we let
$R_{n}^{(m-1)}(L):=\sum_{i=1}^{m-1}v_{n}^{i}(L)x^{i},$
then we have
$None$ $R_{n}^{(m-1)}(L_{r})=t^{{{{-(n+1)}}}}R^{(m-1)}_{n}(L)\ {\rm mod}\
x^{m}$ $None$ $R_{n}^{(m-1)}(L_{l})=t^{{{{(n+1)}}}}R_{n}^{(m-1)}(L)\ {\rm
mod}\ x^{m}$ $None$ $R_{n}^{(m-1)}(L\sqcup U)=u_{n}(t)\ R_{n}(L)\ {\rm mod}\
x^{m}$
and
$R_{n}^{(m-1)}({L_{+}})-R_{n}^{(m-1)}({L_{-}})=(t-t^{-1})[R_{n}^{(m-1)}(L_{o})-R_{n}^{(m-1)}(L_{\infty})]\
{\rm mod}\ x^{m}.$
Furthermore, suppose that these invariants do not depend on the orientation of
the links. The last equation leads us to define
$None$
$R_{n}^{(m)}({L_{\times}}):=(t-t^{-{1}})[R_{n}^{(m-1)}(L_{o})-R_{n}^{(m-1)}(L_{\infty})]\
{\rm mod}\ x^{m+1}$
We want to define the invariant $v_{m}^{n}$: Recall that it is already defined
on the initial links. Next we examine the right hand side of (20). It is a
polynomial of degree $m$ such that the coefficient of $x^{m}$ comes from
$(t-t^{-{1}})[R_{n}^{(m-1)}(L_{o})-R_{n}^{(m-1)}(L_{\infty})].$
The expression above has no constant term and thus the coefficient of $x^{m}$
depends on the inductively well defined invariants $v_{n}^{i}$, $i=1$, $2$,
$\ldots$,$m-1$. Thus the coefficient of $x^{m}$ in (20) is a “new” framed
singular link invariant. We are going to prove that it is derived from a
framed link invariant by using Theorem 2.2. For that we need to check that
condition (3) in the statement 2.2 is satisfied. It is enough to check it
modulo $x^{m+1}$. In what follows the symbol “$\equiv$” will denote
calculation modulo $x^{m+1}$.
Let $L_{\times+}$ and $L_{\times-}\in{\mathcal{L}}^{(1)}$ be two singular
framed links as in the left hand side of (3) in the statement 2.2. From (20)
we have
$\displaystyle{R}_{n}^{(m)}(L_{{\times}+})-R_{n}^{(m)}(L_{{\times}-})\equiv$
$\displaystyle\equiv$
$\displaystyle(t-t^{-{1}})\big{[}R_{n}^{(m-1)}(L_{o+})-R_{n}^{(m-1)}(L_{\infty+})\big{]}-$
$\displaystyle-$ $\displaystyle(t-t^{-{1}})\
\big{[}R_{n}^{(m-1)}(L_{o-})-R_{n}^{(m-1)}(L_{\infty-})\big{]}\equiv$
$\displaystyle\equiv$
$\displaystyle(t-t^{-{1}})\big{[}R_{n}^{(m-1)}(L_{o+})-R_{n}^{(m-1)}(L_{o-})\big{]}-$
$\displaystyle-$
$\displaystyle(t-t^{-{1}})\big{[}R_{n}^{(m-1)}(L_{\infty+})-R_{n}^{(m-1)}(L_{\infty-})\big{]}\equiv$
$\displaystyle\equiv$
$\displaystyle{(t-t^{-1})}^{2}\big{[}R_{n}^{(m-1)}(L_{oo})-R_{n}^{(m-1)}(L_{o\infty})\big{]}-$
$\displaystyle-$
$\displaystyle{(t-t^{-{1}})}^{2}\big{[}R_{n}^{(m-1)}(L_{\infty
o})-R_{n}^{(m-1)}(L_{\infty\infty})\big{]}\equiv$ $\displaystyle\equiv$
$\displaystyle{(t-t^{-1})}^{2}\big{[}R_{n}^{(m-1)}(L_{oo})+R_{n}^{(m-1)}(L_{\infty\infty})\big{]}-$
$\displaystyle-$
$\displaystyle{(t-t^{-{1}})}^{2}\big{[}R_{n}^{(m-1)}(L_{\infty
o})+R_{n}^{(m-1)}(L_{o\infty})\big{]}.$
Since the result is symmetric with respect to the two double points we deduce
that
$R_{n}^{(m)}(L_{{\times}+})-R_{n}^{(m)}(L_{{\times}-})\equiv
R_{n}^{(m)}(L_{+{\times}})-R_{n}^{(m)}(L_{-{\times}}).$
Thus the framed singular link invariant defined above is induced by a framed
link invariant. Recall that we have already defined the values of $v_{n}^{m}$
on all framed links with unframed underlying isotopy classes in
$\mathcal{C}\mathcal{L}$. Using this values we can define a link invariant
$v_{n}^{m}$ for all links in $\mathcal{L}$ such that if we let
$R_{n}^{(m)}(L)=\sum_{i=1}^{m}v_{n}^{m}(L)x^{i}$
we have
$None$ $R_{n}^{(m)}({L_{+}})-R_{n}^{(m)}({L_{-}})=R_{n}^{(m)}({L_{\times}}),$
for every $L_{\times}\in{\mathcal{L}}^{(1)}$. Now it is a straightforward
calculation to check that the inductive hypotheses hold mod $x^{m+1}$. For
example let us check (18); the others are similar. Consider a framed link
$L_{r}$. Keeping the kink intact in a small 3-ball, make a sequence of
crossing changes to transform $L_{l}$ to an initial link say $CL_{l}$. Over
all such sequences of crossing changes, and initial links $CL_{l}$, choose one
that minimizes the number of the required crossing changes. Suppose, without
loss of generality, that the first crossing to be changed in that sequence is
a positive crossing. By (20) and (21) we have
$R_{n}^{(m)}({L_{l+}})\equiv
R_{n}^{(m)}({L_{l-}})+(t-t^{-1})[R_{n}^{(m-1)}(L_{lo})-R_{n}^{(m-1)}(L_{l\infty})]\
{\rm mod}\ x^{m+1}.$
By (15) and induction on the number of crossing changes needed to go from
$L_{l+}$ to $CL_{l}$ we can assume that
$R_{n}^{(m)}(L_{l-})\equiv t^{{{{(n+1)}}}}R_{n}^{(m)}(L)$
By (18) we have
$R_{n}^{(m-1)}(L_{lo})=t^{{{{(n+1)}}}}R_{n}^{(m-1)}(L_{o})\ {\rm mod}\ x^{m},$
and
$R_{n}^{(m-1)}(L_{l\infty})=t^{{{{(n+1)}}}}R_{n}^{(m-1)}(L_{\infty})\ {\rm
mod}\ x^{m}.$
Combining the last four equations we have
$R_{n}^{(m)}({L_{l+}})\equiv t^{{{{(n+1)}}}}R_{n}^{(m)}({L_{+}}),$
as desired. To finish the proof we need to show that $v_{n}^{m}$ is
independent of the link orientation. Inductively, we assume that
$v_{n}^{0},v_{n}^{1},\dots,v_{n}^{m-1}$ are uniquely determined by their
values on ${\mathcal{C}L}$ and independent of the (singular) link orientation.
We have that
$v_{n}^{m}(L)=v_{n}^{m}(CL)+\sum_{i=1}^{s}{\pm}v_{n}^{m}(L_{i})$
where $L_{1},\dots,L_{s}\in{\mathcal{L}}^{(1)}$ are singular links appearing
in a homotopy from $L$ to $CL$, where $CL$ is the representative of $L$ in
${\mathcal{C}L}$ (compare relation (5)). Recall that we defined
$v_{n}^{m}(CL)$ to be independent of the orientation for $CL$. The proof of
Theorem 2.2 establishes that $v_{n}^{m}(L)$ doesn’t depend on the homotopy
from $L$ to $CL$ chosen. By induction $v_{n}^{0},v_{n}^{1},\dots,v_{n}^{m-1}$
do not depend on orientations. It follows that $v_{n}^{m}(L)$ is unique once
$v_{n}^{m}(CL)$ is chosen and independent on link orientation. ∎
Theorem 1.4 stated in the Introduction is obtained from Theorem 4.1 if we set
$z:=it-(it)^{-1}=ie^{x}+ie^{-x}$ and $a:=ie^{y}$, where $y=(n+1)x$. Now we
derive Theorem 1.3 stated in the Introduction.
###### Proof.
The elements in the set $\mathcal{C}\mathcal{L}^{*}\cup\\{U\\}$ are in one-to-
one correspondence with a basis of $S({\hat{\pi}})$. An element
$R\in{\mathfrak{F}}^{*}(M)$ gives rise to one in $S^{*}({\hat{\pi}})$ by
restriction on the set $\mathcal{C}\mathcal{L}^{*}\cup\\{U\\}$. Thus one
direction of the theorem follows. For the other direction, we note that an
element in $S^{*}({\hat{\pi}})$ defines a map
${\mathcal{R}}_{M}:\mathcal{C}\mathcal{L}^{*}\cup\\{U\\}\rightarrow\hat{\Lambda}$.
Then by Theorem 1.4 there is a unique map
$R_{M}:{\bar{\mathcal{L}}}\rightarrow\hat{\Lambda}$ with properties (1)-(3).
These properties guarantee that $R_{M}$ factors through the Kauffman module
${\mathfrak{F}}(M)$ to give an element in ${\mathfrak{F}}^{*}(M)$ (see
Definition 1.2). ∎
### 4.2. Links in $S^{3}$
Links in $S^{3}$ are studied via projections on a sphere $S^{2}\subset S^{3}$.
Let $U^{m}$ denote the standard $m$-component unlink and $U^{m}({\bf f})$
denote the one with framing $\bf f$. Every $m$-component link projection
$L\subset S^{2}$ is transformed to a framed unlink by finitely many crossing
changes and regular isotopy moves on $S^{2}$ (i.e. isotopy using the
Reidemeister moves of type II and III only). For a link projection $L\subset
S^{2}$, we define a complexity
$s(L):=(u(L),\ c(L)),$
as follows: $c(L)$ is the number of crossings of $L$ and $u(L)$ is the number
of admissible crossing changes required to transform $L$ into a diagram of a
framed unlink that that admits a type I Reidemeister move that reduces its
crossing number. We order the complexities lexicographically. Let
$R:=R_{S^{3}}:{\mathcal{L}}\rightarrow\hat{\Lambda}$ be a map constructed as
in Theorem 1.4 and recall that $\Lambda:={\mathbb{C}}[a^{\pm 1},z^{\pm 1}]$.
Note that the complexity $s(L)$ defined above has the properties that
$s(L_{r}),s(L_{l})>s(L)$.
###### Proposition 4.2.
Define $R(U({\bf f}))=a^{-{\tau}}(a+a^{-1})z^{-1}+1$, where
${\tau}:=\sum_{i=1}^{m}{\bf f_{i}}$. Then, $R(L)\in\Lambda$ for every link. In
fact, $R(L)$ is the two variable Kauffman polynomial.
###### Proof.
Given a diagram $L$ first perform all type I Reidemeister moves that reduce
the number of crossings of $L$. If there are no such moves and $L$ is non-
trivial then there is a crossing change such that three of the terms
$s(L_{-}),s(L_{o}),s(L_{\infty}),s(L_{+})$ are strictly less that the
remaining fourth one. Thus the skein relations
$R(L_{+})-R(L_{-})=z\big{[}R(L_{o})-R(L_{\infty})\big{]},$ $R(L_{r})=aR(L)\ \
{\rm and}\ \ R(L_{l})=a^{-1}R(L)$
allow us to write the invariant $R(L)$ of every link $L$ as a finite sum of
the invariants of links of strictly less complexity than $s(L)$ and with
coefficients in $\Lambda$. The result follows by induction on $s(L)$ and the
observation that $R(U({\bf f}))\in\Lambda$. The last claim in the statement of
the proposition follows by the uniqueness properties of $R$. ∎
## References
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|
arxiv-papers
| 2010-01-01T00:01:53 |
2024-09-04T02:49:07.410160
|
{
"license": "Public Domain",
"authors": "Efstratia Kalfagianni",
"submitter": "Efstratia Kalfagianni",
"url": "https://arxiv.org/abs/1001.0174"
}
|
1001.0190
|
# Kepler-7b: A Transiting Planet with Unusually Low Density††affiliationmark:
David W. Latham11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138 , William J. Borucki22affiliation: NASA
Ames Research Center, Moffett Field, CA 94035 , David G. Koch22affiliation:
NASA Ames Research Center, Moffett Field, CA 94035 , Timothy M.
Brown33affiliation: Las Cumbres Observatory Global Telescope, Goleta, CA 93117
, Lars A. Buchhave11affiliation: Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge, MA 02138 44affiliation: Niels Bohr Institute,
Copenhagen University, DK-2100 Copenhagen, Denmark , Gibor Basri55affiliation:
University of California, Berkeley, Berkeley, CA 94720 , Natalie M.
Batalha66affiliation: San Jose State University, San Jose, CA 95192 , Douglas
A. Caldwell77affiliation: SETI Institute, Mountain View, CA 94043 , William D.
Cochran88affiliation: University of Texas, Austin, TX 78712 , Edward W.
Dunham99affiliation: Lowell Observatory, Flagstaff, AZ 86001 , Gabor
Fűrész11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138 , Thomas N. Gautier III1010affiliation: Jet
Propulsion Laboratory/California Institute of Technology, Pasadena, CA 91109 ,
John C. Geary11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138 , Ronald L. Gilliland1111affiliation: Space
Telescope Science Institute, Baltimore, MD 21218 , Steve B.
Howell1212affiliation: National Optical Astronomy Observatory, Tucson, AZ
85719 , Jon M. Jenkins77affiliation: SETI Institute, Mountain View, CA 94043 ,
Jack J. Lissauer22affiliation: NASA Ames Research Center, Moffett Field, CA
94035 , Geoffrey W. Marcy55affiliation: University of California, Berkeley,
Berkeley, CA 94720 , David G. Monet1313affiliation: US Naval Observatory,
Flagstaff Station, Flagstaff, AZ 86001 , Jason F. Rowe1414affiliation: NASA
Postdoctoral Program Fellow 22affiliation: NASA Ames Research Center, Moffett
Field, CA 94035 , Dimitar D. Sasselov11affiliation: Harvard-Smithsonian Center
for Astrophysics, 60 Garden Street, Cambridge, MA 02138
###### Abstract
We report the discovery and confirmation of Kepler-7b, a transiting planet
with unusually low density. The mass is less than half that of Jupiter,
$M_{\rm P}=0.43\,M_{\rm J}$, but the radius is fifty percent larger, $R_{\rm
P}=1.48\,R_{\rm J}$. The resulting density, $\rho_{\rm P}=0.17\,\rm
g\,cm^{-3}$, is the second lowest reported so far for an extrasolar planet.
The orbital period is fairly long, $P=4.886$ days, and the host star is not
much hotter than the Sun, $T_{\rm eff}=6000$ K. However, it is more massive
and considerably larger than the sun, $M_{\star}=1.35\,M_{\sun}$ and
$R_{\star}=1.84\,R_{\sun}$, and must be near the end of its life on the Main
Sequence.
planetary systems — stars: individual (Kepler-7, KIC 5780885, 2MASS
19141956+4105233) — techniques: spectroscopic
††slugcomment: Version resubmitted — 26 December
2009$\dagger$$\dagger$affiliationtext: Based in part on observations obtained
at the W. M. Keck Observatory, which is operated by the University of
California and the California Institute of Technology.
## 1 INTRODUCTION
The final test of the Kepler photometer at the end of commissioning was a run
of 9.7 continuous days in science mode, to evaluate the noise performance of
the instrument. The Kepler Input Catalog (KIC) was used to select fifty
thousand isolated targets, all with magnitudes brighter than 13.8 in the
Kepler passband, and with no nearby companions that would contaminate the
photometry. The preliminary light curves from this test run were inspected by
team members with great excitement, and a few dozen obvious planet candidates
were quickly identified and passed on to the team responsible for ground-based
follow-up observations. Kepler-7 was observed but was not identified among the
sample of initial candidates.
After a gap of 1.3 days, normal science observations began for a full list of
more than 150,000 planet-search targets and continued for 33.5 days until
interrupted on 15 June 2009, followed by a data download and roll of the
spacecraft to the summer orientation. By the middle of July the preliminary
light curves were available for inspection, and dozens of additional
candidates were identified and passed on to the follow-up team. This time
Kepler-7 was included. Along with the other candidates, Kepler-7 was
scrutinized for evidence of astrophysical false positives involving eclipsing
binaries. It survived this stage of the follow up and was then observed
spectroscopically for very precise radial velocities using the FIber-fed
Echelle Spectrograph (FIES) on the Nordic Optical Telescope (NOT) during a ten
night run in early October. These observations yielded a spectroscopic orbit
that confirmed that an unseen companion with a planetary mass was responsible
for the dips in the light curve observed by Kepler.
The KIC used ground-based multi-band photometry to assign an effective
temperature and surface gravity of $T_{\rm eff}=5944$ K and $\log{g}=4.27$
(cgs) to Kepler-7, corresponding to a late F or early G dwarf. Stellar
gravities in this part of the H-R Diagram are notoriously difficult to
determine from photometry alone, and one of the conclusions of this paper is
that the star is near the end of its Main Sequence lifetime, with a radius
that has expanded to $R_{\star}=1.843^{+0.048}_{-0.066}\,R_{\sun}$ and a
surface gravity that has weakened to $\log{g}=4.030^{+0.018}_{-0.019}$ (cgs).
In turn this implies an inflated radius for the planet, resulting in an
unusually low density of $\rho_{\rm P}=0.17\,\rm g\,cm^{-3}$. This conclusion
is hard to avoid, because the relatively long duration of the transit, more
than 5 hours from first to last contact, demands a low density and expanded
radius for the star.
## 2 KEPLER PHOTOMETRY
The light curve for Kepler-7 (= KIC 5780885,
$\alpha=19^{\mathrm{h}}14^{\mathrm{m}}19\fs{56},\delta=+41^{\circ}05^{\prime}23\farcs{3}$,
J2000, KIC $r=12.815$ mag) is plotted in Figure 1. The numerical data are
available electronically from the Multi Mission Archive at the Space Telescope
Science Institute (MAST) High Level Science Products (HLSP)
website111http://archive.stsci.edu/prepds/kepler_hlsp. Only a modest amount of
detrending has been applied (Koch et al., 2010) to this time series of long
cadence data (29.4-minute accumulations). There is no evidence for any
systematic difference between alternating events, which are plotted with $+$
and $\times$ symbols, supporting the interpretation that all the events are
primary transits. Indeed, there is weak evidence for a secondary eclipse
centered at phase 0.5, as would be expected for a circular orbit, but the
significance is only about $2.4\sigma$. If this detection is real, it is not
inconsistent with the thermal emission expected from the planet for reasonable
assumptions (Koch et al., 2010).
## 3 FOLLOW-UP OBSERVATIONS
As described in more detail by Gautier et al. (2010), the initial follow-up
observations of Kepler planet candidates involved reconnaissance spectroscopy
to look for evidence of a stellar companion or a nearby eclipsing binary
responsible for the observed transits. However, the follow-up team soon
learned that the astrometry derived from the Kepler images themselves, when
combined with high-resolution images of the target neighborhood, could provide
a very powerful tool for identifying background eclipsing binaries blended
with and contaminating the target images (Batalha et al., 2010; Monet et al.,
2010). The astrometry of Kepler-7 indicated a very slight image centroid shift
during transits of +0.1 millipixels in its CCD row direction only.
The only star listed in the KIC that is closer than $30\arcsec$ to Kepler-7
and that can contribute significant light to the Kepler-7 photometry is KIC
5780899, which is 4.4 mag fainter and lies at a separation of $15.5\arcsec$.
KIC 5780899 cannot be the source of the observed dips, because that would
induce centroid shifts of about 25 millipixels. If KIC 5780899 is constant and
Kepler-7 is the source of the transits, the predicted shifts are in the right
direction and have an amplitude of roughly 0.1 millipixels if a quarter of KIC
5780899’s light leaks into the Kepler-7 aperture. Thus KIC 5780899 provides a
satisfactory explanation for the observed shifts.
To check for very close companions, a speckle observation of Kepler-7 was
obtained by S. Howell with the WIYN 3.5-m telescope on Kitt Peak. It showed no
companions in a $2\arcsec$ box centered on Kepler-7. Subsequently, images
obtained by H. Isaacson with the HIRES guider on Keck 1, and independently by
G. Mandushev with the 1.8-m Perkins telescope and PRISM camera at the Lowell
Observatory and by N. Baliber with the LCOGT Faulkes Telescope North on
Haleakala, Maui, all detected a companion at a separation of $1.8\arcsec$
(just outside the WIYN speckle window) and about 4.4 mag fainter in the red.
This companion cannot be the source of the observed centroid shifts. If it is
the source of the dips in the light curve, the centroid shifts would have to
be larger than 1 millipixel, and in the wrong direction. If it is constant,
the shifts would be much too small to detect. However, this companion does
dilute the photometry of Kepler-7 with a contribution of about 2.1%. Adding in
a quarter of the light from the more distant companion gives a total dilution
of about $2.5\pm 0.4\%$. This dilution has been included in the analysis of
the light curve.
Reconnaissance spectra obtained by M. Endl and W. Cochran with the coudé
echelle spectrograph on the 2.7-m Harlan J. Smith Telescope at the McDonald
Observatory showed that there was no significant velocity variation at the
level of 1 $\rm km\,s^{-1}$, and therefore that an orbiting stellar companion
could not be responsible for the observed transits. Furthermore, there was no
sign of a composite spectrum or contamination by the spectrum of an eclipsing
binary. The McDonald spectra were classified by L. Buchhave by finding the
best match between the observed spectra and a library of synthetic spectra
calculated by J. Laird for an extensive grid of stellar models (Kurucz, 1992)
using a line list developed by J. Morse. This yielded $T_{\rm eff}=6000\pm
125$ K, $\log{g}=4.0\pm 0.2$ (cgs), and $v\sin{i}=4\,\rm km\,s^{-1}$, very
close to the final values reported in Table 2.
## 4 FIES SPECTROSCOPY
The FIbre-fed Echelle Spectrograph (FIES) on the 2.5-m Nordic Optical
Telescope (NOT) at La Palma, was not originally designed with very precise
radial velocities in mind. In particular, the fiber feed does not incorporate
a scrambler, there is no attempt to control the atmospheric pressure (e.g. by
housing the optics in a vacuum enclosure), and there is no correction of the
images for atmospheric dispersion. However, the spectrograph does reside in
its own well-insulated room with active control of the temperature to a few
hundreths K, with the result that the optics are quite stable. Furthermore,
FIES has good throughput, partly because the seeing is often excellent at the
NOT site, and an automatic guider keeps the image well centered on a fiber
$1.3\arcsec$ in diameter. These advantages encouraged us to develop
specialized observing procedures and a new data reduction pipeline with the
goal of measuring radial velocities to better than $10\,\rm m\,s^{-1}$ for the
relatively faint planet candidates identified by Kepler.
To establish a wavelength calibration that tracks slow drifts during a long
exposure, we adopted the strategy of obtaining strong exposures of a Thorium-
Argon hollow cathode lamp through the science fiber immediately before and
after each science exposure. Long science exposures are divided into three or
more sub-exposures, to allow detection of and correction for radiation events.
Contamination by scattered moonlight can be a serious problem for very precise
velocities of faint targets. FIES does not yet have a separate fiber for
monitoring the sky brightness, so care is needed to avoid the moon, especially
if there are thin clouds.
A new reduction and analysis pipeline optimized for measuring precise radial
velocities was developed by L. Buchhave. After extraction of intensity- and
wavelength-calibrated spectra, relative velocities are derived for each
echelle order by cross correlation against a combined template created by
shifting all the observed spectra of the same star to a common velocity scale
and co-adding them. The final velocity for each observation is the mean of the
results for the individual orders, weighted by the number of detected photons
but not by the velocity information content. Orders with very low signal
levels and orders contaminated by telluric lines are not used. The internal
error of the mean is estimated from the scatter over the orders.
We observed Kepler-7 with FIES for an hour on each of ten consecutive nights
in October 2009. On every night we observed a standard star, HD 182488, soon
before Kepler-7 and also soon after on half the nights. HD 182488 is
conveniently located close to the Kepler field of view and is known from HIRES
observations over several years to be stable to better than $3\,\rm
m\,s^{-1}$, and thus was adopted as the primary velocity standard by the
follow-up team. Our 15 velocities for HD 182488 show an rms of $7\,\rm
m\,s^{-1}$, with a slow drift pattern with an amplitude of several $\rm
m\,s^{-1}$. Therefore we interpolated a correction to our velocity zero point
for each observation of Kepler-7 by assuming that HD 182488 should not vary.
One of the 10 observations was obtained through clouds and clearly showed a
distortion of the correlation peak due to contamination by scattered moonlight
for several of the blue orders. This observation was rejected. The results for
the other 9 observations are reported in Table 1, including the variations in
the line bisectors and errors.
We fit a circular orbit to the 9 velocities reported in Table 1, adopting the
photometric ephemeris, which leaves the orbital semi-amplitude, $K$, and
center-of-mass velocity, $\gamma$, as the only free parameters. A plot of this
orbital solution is shown in Figure 2, together with the velocity residuals
and the line bisector variations. There is no evidence of a correlation
between the velocities and the bisectors, which supports the interpretation
that the velocity variations are due to a planetary companion. The orbital
parameters are listed in Table 2. Allowing the eccentricity to be a free
parameter reduced the velocity residuals by only a small amount and yielded an
eccentricity that was not significantly different from circular. A solution
for a circular orbit using the velocities uncorrected for the drifts exhibited
by the standard star gave similar velocity residuals, but a smaller value of
$K$ by $7.6\,\rm m\,s^{-1}$, corresponding to an 18% smaller mass.
The combined template spectrum for Kepler-7 from FIES was analyzed by A.
Sozzetti using MOOG222http://verdi.as.utexas.edu/moog.html, to provide the
stellar parameters needed to estimate the mass and radius of the host star
using stellar evolution tracks. The critical input parameters to the models
are $T_{\rm eff}$ and [Fe/H], but the spectroscopic $\log{g}$ is also of
interest for a consistency check. A spectrum of Kepler-7 obtained by H.
Isaacson and G. Marcy with HIRES on Keck 1 was analyzed by D. Fischer using
SME (Valenti & Piskunov, 1996), with very similar results: $T_{\rm
eff}=5933\pm 44$ vs. $6000\pm 75$ K, $\rm{[Fe/H]}=+0.11\pm 0.03$ vs. $+0.13\pm
0.07$, and $\log{g}=3.98\pm 0.10$ vs. $4.00\pm 0.10$ (cgs), for SME and MOOG,
respectively. For the results reported in Table 2, we used the SME values. The
mean absolute velocity of Kepler-7, $+0.40\pm 0.10\,\rm km\,s^{-1}$, was
determined from the FIES observations by adopting $-21.508\,\rm km\,s^{-1}$ as
the velocity for the standard star HD 182488.
## 5 DISCUSSION
The analysis of the Kepler photometry and the determination of the stellar and
planetary parameters for Kepler-7 followed exactly the procedures reported in
Koch et al. (2010) and Borucki et al. (2010). The results are reported in
Table 2. These results were checked and confirmed by independent analyses
carried out by C. Burke and G. Torres.
The Kepler-7 host star is not much hotter than the Sun, $T_{\rm eff}=6000\pm
75$ K. However, it is more massive and considerably larger than the sun,
$M_{\star}=1.347^{+0.072}_{-0.054}\,M_{\sun}$ and
$R_{\star}=1.843^{+0.048}_{-0.066}\,R_{\sun}$, which puts it in a region of
the H-R Diagram near the end of its Main Sequence lifetime. Indeed, the Yale-
Yonsei evolutionary tracks have hooks that cross at the position of Kepler-7,
and the probability distribution for the stellar mass has two peaks. The
stronger peak is for an evolutionary state not long before Hydrogen burning in
the core is exhausted with $M_{\star}=1.362\pm 0.040\,M_{\sun}$ and
$R_{\star}=1.857\pm 0.047\,R_{\sun}$, while the weaker peak corresponds to a
state soon after the star starts to evolve rapidly, with $M_{\star}=1.204\pm
0.035\,M_{\sun}$ and $R_{\star}=1.781\pm 0.042\,R_{\sun}$. The mass for the
evolved peak is 12% smaller, and the radius is 4% smaller (as it must be to
yield the same stellar density). The corresponding planetary radius is also 4%
smaller, while the planetary mass is 8% smaller (because of the dependence on
the 2/3 power of the system mass). As our best guess for the mass and radius
of the host star and for the mass, radius, and density of the planet, in Table
2 we report the mode and errors for the corresponding probability
distributions. This takes into account all the possible evolutionary states
for the host star that are consistent with the observations.
The planetary radius is fifty percent larger than that of Jupiter, $R_{\rm
P}=1.478^{+0.050}_{-0.051}\,R_{\rm J}$, but the mass is less than half,
$M_{\rm P}=0.433^{+0.040}_{-0.041}\,M_{\rm J}$, which leads to an unusually
low density of $\rho_{\rm P}=0.166^{+0.019}_{-0.020}\,\rm g\,cm^{-3}$. Among
the known planets, only WASP-17b appears to have a lower density (Anderson et
al., 2009), although the actual value for that planet is not yet well
determined. The position of Kepler-7b on the mass/radius diagram is
illustrated in Figure 3, which plots all of the transiting planets with known
parameters as of 5 November 2009. Because of possible systematic errors in the
radial velocities measured using FIES, the mass of Kepler-7b may be smaller
than we report by as much as 20% or even more. However, the systematic error
in the mass on the high side is unlikely to be this large, because a larger
orbital amplitude is less vulnerable to systematic velocity errors. For the
planetary radius, it is hard to avoid the conclusion that the planet is
strongly inflated, because the relatively long duration of the transit demands
a low density and expanded radius for the star. A robust measure of the
transit duration is the time between the moment when the center of the planet
crosses the limb of the star during ingress and the corresponding moment
during egress. A general formula for this duration including the effect of
orbital eccentricity is given by Pál et al. (2010), leading to a value of
$4.63\pm 0.06$ hours for Kepler-7. We conclude that future observational
refinements to the characteristics of Kepler-7b are more likely to decrease
the density than increase it, with a significant uncertainty remaining as long
as the evolutionary state of the host star is uncertain.
Many people have contributed to the success of the Kepler Mission, and it is
impossible to acknowledge them all by name. We offer our special thanks to the
team of scientists and programmers working with J. M. Jenkins to create the
photometric pipeline - H. Chandrasekaran, S. T. Bryson, J. Twicken, E
Quintana, B. Clarke, C. Allen, J. Li, P. Tenenbaum, and H. Wu; to C. J. Burke
and G. Torres for running independent checks of the analysis of the Kepler-7
light curve and system parameters; to J. Andersen for help with the FIES
observations and unwavering moral support; to M. Endl, H. Isaacson, D. Ciardi,
G. Mandushev, N. Baliber, and M. Crane for important contributions to the
follow-up work; to A. Sozzetti for his analysis of the FIES combined template
spectrum and to D. Fischer for her analysis of the HIRES template spectrum; to
M. Everett and G. Esquerdo for critical contributions to the KIC; to E.
Bachtel and his team at Ball Aerospace for their work on the Kepler
photometer; to R. Duren and R. Thompson for key contributions to engineering;
and to C. Botosh, M. Haas, and J. Fanson, for able management. DWL gratefully
acknowledges partial support from NASA Cooperative Agreement NCC2-1390 and the
help of S. Cahill and L. McArthur-Hines. Funding for this Discovery mission is
provided by NASA’s Science Mission Directorate. Facilities: The Kepler
Mission, NOT (FIES), Keck:I (HIRES), WIYN (Speckle)
## References
* Anderson et al. (2009) Anderson, D. R., et al. ApJ, submitted, arXiv:0908.1553
* Batalha et al. (2010) Batalha, N. M., et al. 2010, ApJ, this issue
* Borucki et al. (2010) Borucki, W. J., et al. 2010, ApJ, this issue
* Dunham et al. (2010) Dunham, E. W., et al 2010, ApJ, this issue
* Gautier et al. (2010) Gautier, T. N., et al. 2010, ApJ, this issue
* Koch et al. (2010) Koch, D. G., et al. 2010, ApJ, this issue
* Kurucz (1992) Kurucz, R. L. 1992, in The Stellar Populations of Galaxies, IAU Symp. No. 149, ed. B. Barbuy and A. Renzini (Kluwer Acad. Publ.: Dordrecht), 225
* Monet et al. (2010) Monet, D. G., et al. 2010, ApJ, this issue
* Pál et al. (2008) Pál, A., et al. 2008, ApJ, 680, 1450
* Pál et al. (2010) Pál, A., et al. 2010, MNRAS in press (arXiv:0908.1705)
* Skrutskie et al. (2006) Skrutskie, M. J., et al. 2006, AJ, 131, 1163
* Valenti & Piskunov (1996) Valenti, J. A., & Piskunov, N. 1996, A&AS, 118, 595
* Yi et al. (2001) Yi, S. K., Demarque, P., Kim, Y.-C., Lee, Y.-W., Ree, C. H., Lejeune, T., & Barnes, S. 2001, ApJS, 136, 417
Table 1: Relative Radial-Velocity Measurements of Kepler-7 HJD | Phase | RV | $\sigma_{\rm RV}$ | BS | $\sigma_{\rm BS}$
---|---|---|---|---|---
(days) | (cycles) | ($\rm m\,s^{-1}$) | ($\rm m\,s^{-1}$) | ($\rm m\,s^{-1}$) | ($\rm m\,s^{-1}$)
2455107.37937 | 28.677 | $+43.7$ | $\pm 6.8$ | $+19.9$ | $\pm 7.3$
2455108.36845 | 28.879 | $+32.7$ | $\pm 7.1$ | $+1.5$ | $\pm 5.4$
2455110.50735 | 29.317 | $-34.2$ | $\pm 9.8$ | $+4.8$ | $\pm 17.9$
2455111.40251 | 29.500 | $-11.5$ | $\pm 6.7$ | $-4.0$ | $\pm 7.2$
2455112.41378 | 29.707 | $+33.2$ | $\pm 8.2$ | $-4.6$ | $\pm 5.4$
2455113.40824 | 29.911 | $+27.9$ | $\pm 6.1$ | $-12.0$ | $\pm 8.2$
2455114.44632 | 30.123 | $-31.1$ | $\pm 8.1$ | $-5.5$ | $\pm 8.9$
2455115.44411 | 30.328 | $-29.2$ | $\pm 10.7$ | $-14.8$ | $\pm 8.9$
2455116.37077 | 30.517 | $-0.1$ | $\pm 9.4$ | $+13.8$ | $\pm 10.6$
Table 2: System Parameters for Kepler-7
Parameter | Value | Notes
---|---|---
Transit and orbital parameters
Orbital period $P$ (d) | $4.885525\pm 0.000040$ | A
Midtransit time $E$ (HJD) | $2454967.27571\pm 0.00014$ | A
Scaled semimajor axis $a/R_{\star}$ | $7.22^{+0.16}_{-0.13}$ | A
Scaled planet radius $R_{\rm P}$/$R_{\star}$ | $0.08241^{+0.00030}_{-0.00043}$ | A
Impact parameter $b\equiv a\cos{i}/R_{\star}$ | $0.445^{+0.032}_{-0.044}$ | A
Orbital inclination $i$ (deg) | $86\fdg{5}\pm 0.4$ | A
Orbital semi-amplitude $K$ ($\rm m\,s^{-1}$) | $42.9\pm 3.5$ | A,B
Orbital eccentricity $e$ | 0 (adopted) | A,B
Center-of-mass velocity $\gamma$ ($\rm m\,s^{-1}$) | $0$ | A,B
Observed stellar parameters
Effective temperature $T_{\rm eff}$ (K) | $5933\pm 44$ | C
Spectroscopic gravity $\log{g}$ (cgs) | $3.98\pm 0.10$ | C
Metallicity [Fe/H] | $+0.11\pm 0.03$ | C
Projected rotation $v\sin{i}$ ($\rm km\,s^{-1}$) | $4.2\pm 0.5$ | C
Mean radial velocity ($\rm km\,s^{-1}$) | $+0.40\pm 0.10$ | B
Derived stellar parameters
Mass $M_{\star}$($M_{\sun}$) | $1.347^{+0.072}_{-0.054}$ | C,D
Radius $R_{\star}$($R_{\sun}$) | $1.843^{+0.048}_{-0.066}$ | C,D
Surface gravity $\log{g}_{\star}$ (cgs) | $4.030^{+0.018}_{-0.019}$ | C,D
Luminosity $L_{\star}$ ($L_{\sun}$) | $4.15^{+0.63}_{-0.54}$ | C,D
Age (Gyr) | $3.5\pm 1.0$ | C,D
Planetary parameters
Mass $M_{\rm P}$ ($M_{\rm J}$) | $0.433^{+0.040}_{-0.041}$ | A,B,C,D
Radius $R_{\rm P}$ ($R_{\rm J}$, equatorial) | $1.478^{+0.050}_{-0.051}$ | A,B,C,D
Density $\rho_{\rm P}$ ($\rm g\,cm^{-3}$) | $0.166^{+0.019}_{-0.020}$ | A,B,C,D
Surface gravity $\log{g}_{\rm P}$ (cgs) | $2.691^{+0.038}_{-0.045}$ | A,B,C,D
Orbital semimajor axis $a$ (AU) | $0.06224^{+0.00109}_{-0.00084}$ | E
Equilibrium temperature $T_{\rm eq}$ (K) | $1540\pm 200$ | F
Note. —
A: Based on the photometry.
B: Based on the radial velocities.
C: Based on a MOOG analysis of the FIES spectra.
D: Based on the Yale-Yonsei stellar evolution tracks.
E: Based on Newton’s version of Kepler’s Third Law and total mass.
F: Assumes Bond albedo = 0.1 and complete redistribution.
Figure 1: The detrended light curve for Kepler-7. The time series for the
entire data set is plotted in the upper panel. The lower panel shows the
photometry folded by the period $P=4.885525$ days. The model fit to the
primary transit is plotted in red, and our attempt to fit a corresponding
secondary eclipse for a circular orbit is shown in green with an expanded and
offset scale. Figure 2: a) The orbital solution for Kepler-7. The observed
radial velocities obtained with FIES on the Nordic Optical Telescope are
plotted together with the velocity curve for a circular orbit with the period
and time of transit fixed by the photometric ephemeris. The $\gamma$ velocity
has been subtracted from the relative velocities here and in Table 1, and thus
the center-of-mass velocity for the orbital solution is 0 by definition. b)
The velocity residuals from the orbital solution. The rms of the velocity
residuals is 7.4 $\rm m\,s^{-1}$. c) The variation in the bisector spans for
the 9 FIES spectra. The mean value has been subtracted. Figure 3: The
Mass/Radius diagram for all the transiting planets with known parameters as of
5 November 2009. The four new Kepler planets are labeled and plotted as
diamonds. Kepler-7 has an unusually low density.
|
arxiv-papers
| 2009-12-31T23:14:20 |
2024-09-04T02:49:07.422492
|
{
"license": "Public Domain",
"authors": "David W. Latham (Harvard-Smithsonian Center for Astrophysics), William\n J. Borucki (NASA Ames Research Center), David G. Koch (NASA Ames Research\n Center), Timothy M. Brown (Las Cumbres Observatory Global Telescope), Lars A.\n Buchhave (Harvard-Smithsonian Center for Astrophysics), Gibor Basri\n (University of California, Berkeley), Natalie M. Batalha (San Jose State\n University), Douglas A. Caldwell (SETI Institute), William D. Cochran\n (University of Texas, Austin), Edward W. Dunham (Lowell Observatory,\n Flagstaff), Gabor Furesz (Harvard-Smithsonian Center for Astrophysics),\n Thomas N. Gautier III (Jet Propulsion Laboratory), John C. Geary\n (Harvard-Smithsonian Center for Astrophysics), Ronald L. Gilliland (Space\n Telescope Science Institute), Steve B. Howell (National Optical Astronomy\n Observatory), Jon M. Jenkins (SETI Institute), Jack J. Lissauer (NASA Ames\n Research Center), Geoffrey W. Marcy (University of California, Berkeley),\n David G. Monet (US Naval Observatory, Flagstaff Station), Jason F. Rowe (NASA\n Ames Research Center), Dimitar D. Sasselov (Harvard-Smithsonian Center for\n Astrophysics)",
"submitter": "David Latham PhD",
"url": "https://arxiv.org/abs/1001.0190"
}
|
1001.0299
|
# The solutions of four $q$-functional equations
Jun-Ming Zhu 111E-mail address: jm_zh@sohu.com, junming_zhu@163.com
Abstract In this note we obtain the solutions of four $q$-functional equations
and express the solutions in $q$-operator forms. These equations give
sufficient conditions for $q$-operator methods
Key words: Basic hypergeometric series, $q$-Series, $q$-Functional equation,
$q$-Difference equation, q-Exponential operator
## 1 Introduction
We follow the notation and terminology in [6] , and for convenience, we always
assume that $0<|q|<1$. The $q$-shifted factorials are defined by
${(a;q)_{n}=}\left\\{\begin{array}[]{ll}1,&n=0,\\\
(1-a)(1-aq)\cdots(1-aq^{n-1}),&n=1,2,3,\cdots\cdots,\end{array}\right.$
The $q$-derivative operator is defined by
$D_{q}\\{f(a)\\}=\frac{f(a)-f(aq)}{a},~{}~{}D_{q}^{n}\\{f(a)\\}=D_{q}\\{D_{q}^{n-1}\\{f(a)\\}\\}.$
The operator $\theta$ is defined by
$\theta=\eta^{-1}D_{q},~{}~{}\theta^{n}\\{f(a)\\}=\theta\\{\theta^{n-1}\\{f(a)\\}\\},$
where $\eta^{-1}\\{f(a)\\}=f(q^{-1}a).$ Both $D_{q}$ and $\theta$ are
obviously linear transforms, and by convention, $D_{q}^{0}$ and $\theta^{0}$
are both understood as the identity.
The two $q$-exponential operators (see, [2, 3, 7, 8, 10]) are defined by
$T(bD_{q})=\sum_{n=0}^{\infty}\frac{(bD_{q})^{n}}{(q;q)_{n}},$
and
$E(b\theta)=\sum_{n=0}^{\infty}\frac{q^{n(n-1)/2}(b\theta)^{n}}{(q;q)_{n}},$
respectively.
The following operator was first introduced by Fang [5]. But we follow Chen
and Gu’s notation [1]. It seems to be more convenient.
$T(a,b;D_{q})=\sum_{n=0}^{\infty}\frac{(a,q)_{n}}{(q;q)_{n}}(bD_{q})^{n}.$
Chen and Gu [1] named $T(a,b;D_{q})$ as Cauchy operator. But compared with the
following operator, we called $T(a,b;D_{q})$ the first Cauchy operator in this
paper.
The operator $T(a,b;\theta)$, introduced by Fang [5], is defined by
$T(a,b;\theta)=\sum_{n=0}^{\infty}\frac{(a,q)_{n}}{(q;q)_{n}}(b\theta)^{n}.$
We called $T(a,b;\theta)$ the second Cauchy operator.
All the papers using the $q$-operator methods imply the following definition
but do not state it explicitly. Unless otherwise stated, all the operators are
applied with respect to the parameter $a$.
Definition 1.1
$T(bD_{q})\\{f(a)\\}=\sum_{n=0}^{\infty}\frac{b^{n}}{(q;q)_{n}}\\{{D_{q}}^{n}\\{f(a)\\}\\},$
and
$E(b\theta)\\{f(a)\\}=\sum_{n=0}^{\infty}\frac{q^{n(n-1)/2}b^{n}}{(q;q)_{n}}\\{{\theta}^{n}\\{f(a)\\}\\}.$
The first and the second Cauchy operators and the operators $T(b\theta)$ and
$E(bD_{q})$ below also act in this way.
When using the methods of operators, rearrangememts of series are often
emploied. We know that rearrangememts of series must be under the condition
that the series are absolutely convergent, which may be not very easy to check
sometimes. In his paper [9] , Liu obtained the following two theorems using
the $q$-functional equation method.
###### Theorem 1.1
Let $f(a,b)$ be a two variables analytic function in a neighborhood of
$(a,b)=(0,0)\in\mathbb{C}^{2}$, satisfying the $q$-difference equation
$bf(aq,b)-af(a,bq)=(b-a)f(a,b).$
Then we have
$f(a,b)=T(bD_{q})\\{f(a,0)\\}.$
###### Theorem 1.2
Let $f(a,b)$ be a two variables analytic function in a neighborhood of
$(a,b)=(0,0)\in\mathbb{C}^{2}$, satisfying the $q$-difference equation
$af(aq,b)-bf(a,bq)=(a-b)f(aq,bq).$
Then we have
$f(a,b)=E(b\theta)\\{f(a,0)\\}.$
These two theorems give us a method to use operator $T(bD_{q})$ and
$E(b\theta)$ without having to check the absolute convergence of the series.
Originated by Liu’s work, we give the above two theorems more general forms in
the following section.
## 2 Four $q$-functional equations and the solutions
###### Theorem 2.1
Let $f(a,b,c)$ be a three variables analytic function in a neighborhood of
$(a,b,c)=(0,0,0)\in\mathbb{C}^{3}$, satisfying the $q$-functional equation
$c(f(a,b,c)-f(a,bq,c))=b(f(a,b,c)-f(a,b,cq)-af(a,bq,c)+af(a,bq,cq)).$ (1)
Then we have
$f(a,b,c)=T(a,b;D_{q})\\{f(a,0,c)\\},$
where $T(a,b;D_{q})$ is applied with respect to the parameter $c$.
###### Theorem 2.2
Let $f(a,b,c)$ be a three variables analytic function in a neighborhood of
$(a,b,c)=(0,0,0)\in\mathbb{C}^{3}$, satisfying the $q$-functional equation
$c(f(a,bq,cq)-f(a,b,cq))=b(f(a,bq,cq)-f(a,bq,c)-af(a,b,c)+af(a,b,cq)).$ (2)
Then we have
$f(a,b,c)=T(-\frac{1}{a},ab;\theta)\\{f(a,0,c)\\},$
where $T(-\frac{1}{a},ab;\theta)$ is also applied with respect to the
parameter $c$.
When letting $a\rightarrow 0$ in theorem 2.1 and 2.2, we get theorem 1.1 and
1.2, respectively. The proof of theorem 2.2 is similar to that of theorem 2.1
and so is omitted.
Proof of Theorem 2.1 We now begin to solve this equation. From the theory of
several complex variables, we assume that
$f(a,b,c)=\sum_{n=0}^{+\infty}A_{n}(a,c)b^{n}.$ (3)
We substitute the above equation into (1) to get
$\displaystyle c\sum_{n=0}^{+\infty}(1-q^{n})A_{n}(a,c)b^{n}$ $\displaystyle=$
$\displaystyle(\sum_{n=0}^{+\infty}A_{n}(a,c)-\sum_{n=0}^{+\infty}A_{n}(a,cq)-a\sum_{n=0}^{+\infty}A_{n}(a,c)q^{n}$
$\displaystyle+a\sum_{n=0}^{+\infty}A_{n}(a,cq)q^{n})b^{n+1}.$
This is
$\displaystyle
c\sum_{n=1}^{+\infty}(1-q^{n})A_{n}(a,c)b^{n}=\sum_{n=1}^{+\infty}(1-aq^{n-1})(A_{n-1}(a,c)-A_{n-1}(a,cq))b^{n}.$
Comparing the coefficients of $b^{n}$ gives
$c(1-q^{n})A_{n}(a,c)=(1-aq^{n-1})(A_{n-1}(a,c)-A_{n-1}(a,cq)).$
Then we have
$\displaystyle A_{n}(a,c)$ $\displaystyle=$
$\displaystyle\frac{(1-aq^{n-1})}{(1-q^{n})}\frac{(A_{n-1}(a,c)-A_{n-1}(a,cq))}{c}$
$\displaystyle=$
$\displaystyle\frac{(1-aq^{n-1})}{(1-q^{n})}D_{q}\\{A_{n-1}(a,c)\\},$
where $D_{q}$ is applied with respect to the parameter $c$. Iterate the above
equation to get
$\displaystyle A_{n}(a,c)=\frac{(a;q)_{n}}{(q,q)_{n}}D_{q}\\{A_{0}(a,c)\\}.$
(4)
It remains to calculate $A_{0}(a,c)$. Putting $b=0$ in (3), we immediately
deduce that $A_{0}(a,c)=f(a,0,c)$. Substituting (4) back into (3) gives
$f(a,b,c)=\sum_{n=0}^{\infty}\frac{(a,q)_{n}}{(q;q)_{n}}(bD_{q})^{n}\\{f(a,0,c)\\}=T(a,b;D_{q})\\{f(a,0,c)\\}.$
This completes the proof.
Using the $q$-functional equation method we easily obtain the following two
theorems.
###### Theorem 2.3
Let $f(a,b)$ be a two variables analytic function in a neighborhood of
$(a,b)=(0,0)\in\mathbb{C}^{2}$, satisfying the $q$-functional equation
$af(a,b)+bf(aq,bq)=(a+b)f(a,bq).$
Then we have
$f(a,b)=E(bD_{q})\\{f(a,0)\\},$
where the operator $E(bD_{q})$ is defined by
$E(bD_{q})=\sum_{n=0}^{\infty}\frac{q^{n(n-1)/2}(bD_{q})^{n}}{(q;q)_{n}}.$
###### Theorem 2.4
Let $f(a,b)$ be a two variables analytic function in a neighborhood of
$(a,b)=(0,0)\in\mathbb{C}^{2}$, satisfying the $q$-functional equation
$af(aq,bq)+bf(a,b)=(a+b)f(aq,b).$
Then we have
$f(a,b)=T(-b\theta)\\{f(a,0)\\},$
where the operator $T(b\theta)$ is defined by
$T(b\theta)=\sum_{n=0}^{\infty}\frac{(b\theta)^{n}}{(q;q)_{n}}.$
From all the theorems in this little paper, we can look at the q-operators
from a different standpoint.
## References
* [1] V. Y. B. Chen, N. S. S. Gu, The Cauchy operator for basic hypergeometric series, Adv. in Appl. Math, 41(2008), 177–196.
* [2] W. Y. C. Chen, Z.-G. Liu, Parameter augmentation for basic hypergeometric series, II, J. Combin. Theory Ser. A 80(1997), 175–195.
* [3] W. Y. C. Chen, Z.-G. Liu, Parameter augmentation for basic hypergeometric series, I, in: B. E. Sagan, R. P. Stanley (Eds.), Mathematical Essays in honor of Gian-Carlo Rota, Birkäuser, Basel, 1998, pp. 111–129.
* [4] J.-P. Fang, $q$-Differential operator identities and applications, J. Math. Anal. Appl. 332(2007), 1393–1407.
* [5] J.-P. Fang, $q$-Operator identities and its applications, Journal of East China Normal University, 1(2008), 20–24 (in Chinese).
* [6] G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, MA, 2003.
* [7] Z.-G. Liu, A new proof of the Nassrallah-Rahman integral, Acta Mathematica Sinica 41(1998), 405–410 (in Chinese).
* [8] Z.-G. Liu, Some operator identities and $q$-series transformation formulas, Discrete Math., 265(2003), 119–139.
* [9] Z.-G. Liu, Two $q$-difference equations and $q$-operator indetities, Journal of Difference Equations and Applications (in press).
* [10] L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc., 24(1894), 337-352.
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arxiv-papers
| 2010-01-02T12:20:02 |
2024-09-04T02:49:07.430704
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun-Ming Zhu",
"submitter": "Jun-Ming Zhu",
"url": "https://arxiv.org/abs/1001.0299"
}
|
1001.0305
|
# Preliminary Astrometric Results from Kepler
David G. Monet U. S. Naval Observatory, Flagstaff, AZ 86001 Jon M. Jenkins
SETI Institute, Mountain View, CA 94043 Edward Dunham Lowell Observatory,
Flagstaff, AZ 86001 Stephen T. Bryson NASA/Ames Research Center, Moffett
Field, CA 94035 Ronald L. Gilliland Space Telescope Science Institute,
Baltimore, MD 21218 David W. Latham Harvard-Smithsonian Center for
Astrophysics, Cambridge MA 02138 William J. Borucki NASA/Ames Research
Center, Moffett Field, CA 94035 David G. Koch NASA/Ames Research Center,
Moffett Field, CA 94035
###### Abstract
Although not designed as an astrometric instrument, Kepler is expected to
produce astrometric results of a quality appropriate to support many of the
astrophysical investigations enabled by its photometric results. On the basis
of data collected during the first few months of operation, the astrometric
precision for a single 30 minute measure appears to be better than 4
milliarcseconds (0.001 pixel). Solutions for stellar parallax and proper
motions await more observations, but the analysis of the astrometric residuals
from a local solution in the vicinity of a star have already proved to be an
important tool in the process of confirming the hypothesis of a planetary
transit.
astrometry — stars: fundamental parameters
## 1 Introduction
The measurement of astrometric parameters, particularly the parallax, for
Kepler stars is a critical component of computing the physical values for
various stellar parameters using the relative values that are computed from
the photometric analysis. If the Kepler data can be shown to have the
necessary astrometric accuracy, then such a conversion can be included in the
processing for most if not all Kepler stars. Although the discussion of Kepler
astrometric accuracy must await more data and modeling, the very high
precision of Kepler positions is already a powerful tool for understanding the
photometric variations of stars and the possible presence of planetary
companions. Detailed discussions of the Kepler spacecraft and mission are
presented by Borucki et al. (2010) and Koch et al. (2010). A short overview
for the astrometric discussion is the following. The Schmidt telescope has a
1.4-m primary and a 0.95-m corrector, and the photometer is a mosaic of 42
charge coupled devices (CCDs). The boresight of the telescope remains constant
for the mission, but the spacecraft rolls 90 degrees every three months. Due
to restrictions in memory and bandwidth, only the pixels associated with the
target stars are sent to the ground for processing. Target stars are defined
by software, and pixels not associated with targets are saved only
infrequently. The basic integration time is 6.02 seconds, and the long cadence
(LC) sequence (Jenkins et al., 2010) co-adds the target pixels for 29.4
minutes. The co-added pixel data are sent to the ground every month for
processing and analysis. This strategy enables the extremely high signal-to-
noise (SNR) observations needed to achieve the planetary detection mission.
To obtain the large field of view, the image sampling is very coarse compared
to other astrometric assets. The Pixel Response Function (PRF) is described in
great detail by Bryson et al. (2010), but a quick summary is as follows. The
images contain three components, a sharp spike in the middle that comes from
optical diffraction (about 0.1 arcsecond), a wider component with a
characteristic size of 5 or 6 arcseconds set primarily by mechanical alignment
tolerances of the CCD mosaic, and a much broader scattering profile. The
intermediate component of the image profile produces the majority of the
astrometric signal, and it contains about 70% of the light. The 43 days of
data available so far 111As described more fully by Caldwell at al. (2010),
the spacecraft data are grouped by observing quarters as defined by the
mandatory rolls of the spacecraft itself. So far, data from Quarters 0 and 1
are available on the ground. have shown a remarkable astrometric precision,
but the demonstration of astrometric accuracy is still a work in progress.
Even if the centroiding process was fully understood, the short interval of
available data precludes the lifting of the degeneracies between effects of
proper motion, parallax, and velocity aberration. No measured astrometric
parameters are given here. Indeed, the entire range of observed image motion
is about 0.2 pixels, and this is dominated by the spacecraft guiding
precision.
Various authors, including King (1983) and Kaiser et al. (2000), have
developed theoretical expectations for the astrometric precision of an image.
A simple approximation is
$precision=FWHM/(2*SNR)$ (1)
where FWHM is the image full width at half maximum, and SNR is the photometric
signal-to-noise ratio of that star image. The differences in the theoretical
derivations concern the exact value for which the approximate value of 2 is
used above. The observational confirmation of this relationship has yet to be
done, but essentially all ground- and space-based astrometric studies have
demonstrated the validity of the scaling of this relationship. Improved
astrometric precision is obtained for smaller image FWHM, higher SNR, or both
assuming that adequate image sampling is available.
Kepler operates in a heretofore unstudied astrometric domain. The pixels are
very large, 3.98 arcseconds, as compared to other ground- and space-based
astrometric assets, and the observed FWHM is approximately 5 to 6 arcseconds
and depends on the location in the field of view. (See Bryson et al. (2010)
for further discussion and examples.) The effects of undersampled image
components are not captured by Eq. 1. However, Kepler was designed for
extremely high SNR observations. The well capacity of the 27-micron CCD pixels
is more than a million electrons, and most of the stars are bright. As more
fully discussed by Caldwell at al. (2010), the onset of saturation in the
basic 6.02 second integration cycle is near the magnitude Kp = 11.3. 222The
Kepler magnitude Kp includes a very wide passband and is similar to an
astronomical R magnitude in central wavelength. A single LC co-addition
produces a SNR of about 10,000 for an bright, unsaturated, uncrowded star.
There is no atmosphere to degrade the image quality, and the flux is so large
that effects such as sensor readout noise and dark current are unimportant for
the brightest 2-3 magnitudes of unsaturated stars.
## 2 Preliminary Astrometric Investigations
The astrometric processing of Kepler data is conceptually no different than
the traditional differential astrometric process of data from other ground-
and space-based assets. There is no need to worry about the actual coordinates
(i.e., J2000 RA and Dec) of the stars. Essentially all Kepler stars are in the
2MASS catalog (Skrutskie et al., 2006), and most are in the UCAC-2 catalog
(Zacharias, et al., 2004). Rather, the goal of the analysis is to measure the
small changes in position associated with proper motion, parallax,
perturbations from unseen companions, and blending with photometrically
variable stars. The current astrometric pipeline involves three distinct
steps: centroids are computed from the pixel data, transformations are
computed from each channel 333As described more fully by Jenkins et al.
(2010), each CCD is split into two channels by the flight electronics. The
astrometric verification of the stability between the two channels of a single
CCD has yet to be performed. of each LC co-addition into an intermediate
coordinate system, and solutions for each star are computed using the
intermediate coordinates and terms such as time, parallax factor, etc. Steps
two and three are iterated a few times, and convergence is quite rapid.
Details of each of these steps are presented in the following subsections.
### 2.1 Centroids
Most modern centroiding algorithms fall into three classes: moment analysis,
fits to analytic functions, and fits to the instrumental point spread function
(PSF). So far, various algorithms from the first two classes have been
implemented and tested. The data processing pipeline of the Science Operations
Center (SOC; Jenkins et al. (2010)) compute flux-weighted means for all stars
and Gaussian fits and PSF fits for a few stars. PSF fitting of all stars
remains a task for the future. In a separate effort, several other centroiding
algorithms based on fitting the images to analytic functions have been
evaluated, and centroids for all stars have been computed for many of these.
The choice of centroiding algorithm requires special attention because of the
properties of the Kepler images discussed above. The effect of the
undersampled components of the images has not been fully evaluated, and the
number of pixels for each star has been minimized by the Optimal Aperture
algorithm (Bryson et al., 2010) so that the maximum number of stars can be
transmitted in the fixed spacecraft bandwidth. Because the best astrometric
centroiding algorithm has not been identified yet, analysis is proceeding with
parallel tracks to evaluate a few of the most promising algorithms.
### 2.2 Transformation Coefficients
The current astrometric pipeline supports two different coordinate systems.
For some investigations, working in a sky-based system seems appropriate. The
tangent point is taken to be the nominal Kepler boresight
($\alpha=19^{\mathrm{h}}22^{\mathrm{m}}40^{\mathrm{s}},\delta={+44^{\circ}30^{\prime}}$,
J2000) and a simple tangent plane projection is computed from the nominal
positions of the stars listed in the Kepler Input Catalog (KIC).
444http://archive.stsci.edu/kepler/kepler_fov/search.php In this coordinate
system, effects such as differential velocity aberration and parallax are easy
to visualize. For other investigations, a channel-based coordinate system
seems appropriate, and effects arising from the structure and behavior of CCD
pixels are easier to visualize. Of particular importance are effects based on
where the star falls with respect to the pixel grid, a term called “pixel
phase”. In either coordinate system, a separate set of transformation
coefficients is computed for the measures from each channel for each cadence.
A sample of 1000 known distant giant stars was included in the Kepler star
list, and are used during the first iteration to generate a transformed
coordinate system that is as close to inertial as possible.
Figure 1: Astrometric error as a function of Kp magnitude for stars on Channel
2 in the Quarter 1 data collection.
### 2.3 Astrometric Coefficients
The tasks of computing the centroids and the transformation coefficients for
the Kepler data are similar to their counterparts for traditional ground- and
space-based assets. It is the modeling of the measured positions for each star
that involves special attention, and this flows from the extremely high SNR of
the data. The simplest solutions based on computing only the mean positions
from the data currently available show an astrometric precision near 20
milliarcseconds (about 0.005 pixels). This value and those presented below
refer to the uncertainty in a single measure of a single axis (row or column)
for a single star from a single LC co-addition. Because these solutions are
local and contain only a small number of measures, they should be construed as
estimators of the astrometric precision and not of the overall astrometric
accuracy of the spacecraft and photometer. Adding terms that model the
differential velocity aberration across the Kepler field reduces this error to
about 4 milliarcseconds (about 0.001 pixels), and adding terms arising from
the pixel phase reduce the errors to about 2 milliarcseconds (about 0.0005
pixels). These pixel phase terms are empirical fits, and are not derived from
detailed modeling of the image formation and sampling processes.
During the first 33.5 days of science operations that followed the end of
commissioning, the number of stars was increased to 156,000. The volume of
data for this number of stars observed every 29.4 minutes is large, and the
data from the 84 channels are diverse. Thus it is difficult to characterize
the entirety of the astrometric solution with a single number. Again, much
development in the astrometric processing is needed because every Kepler star
is important. Although only preliminary measures of astrometric precision have
been obtained, it is reassuring to see that the observed errors follow the
prediction based on the flux of the stars involved. Fig. 1 shows the errors
for stars in a single channel as a function of the measured Kp, and
demonstrates that the error rises as the SNR decreases.
## 3 Local Astrometric Solutions and the “Rain” Plots
A special case of the astrometric solutions described above can be computed in
the vicinity of individual stars. Under the assumptions of small parallax and
adequate removal of differential velocity aberration, the equations for the
apparent place of a star can be linearized. Simple trend analysis produces a
robust estimator for the mean position of a star, and residuals from each
measurement are computed. This enables astrometric processing to contribute to
the understanding of the Kepler stars. As more fully discussed by Batalha et
al. (2010), what appears to be a single object can be two or more stars, and
each can have photometric variability. Such astrations can mimic the
photometric properties of a transiting planet, and adding astrometry to the
vetting procedure can assist in the confirmation or denial process. Fig. 2
shows astrometric residuals for two stars. The upper star is a blend of a
variable star and one or more constant stars while the lower shows residuals
that are typical for a bright, constant star. The astrometric amplitude of the
variable star is huge - almost 0.02 pixels.
The astrometric behavior of an image composed of an unknown number of stars
each of which having unknown photometric variations is complicated. However,
the simple case of two stars, a small photometric variation, and centroids
computed from flux-weighted means provides much insight. Where $\Delta s$ is
the true separation of the stars, $\delta s$ is the small measured astrometric
shift, $F$ is the small relative brightness of the fainter B component
compared to the brighter A component, and $f$ is the small relative change in
total brightness due to a transit or stellar variability, the observed
astrometric shift assuming that the A component is variable is given by
$\delta s=Ff\Delta s$ (2)
If the B component has the photometric variation, the observed shift is
$\delta s=f\Delta s$ (3)
When the Kepler observations of $\delta s$ and $f$ are combined with high
resolution imagery that can measure $\Delta s$, $F$, and other characteristics
such as stellar colors, then the model of the composite image can be improved.
On the basis of its photometric signature alone, the Kepler Object of Interest
(KOI-) 15 might involve a transiting planet. The analysis of the combined
photometric and astrometric residuals denies this hypothesis. Fig. 3. shows
the time series astrometric and photometric residuals for KOI-15 after they
have been high-pass filtered so as to emphasize signals with shorter
timescales. A different visualization of the same data is called a “Rain
Plot”, and is shown in Fig. 4. Clearly, the astrometric and photometric
residuals for KOI-15 are strongly correlated, and these correlations are the
signature of an astration that includes one or more relatively constant stars
and a background eclipsing binary. Indeed, the secondary eclipse is more
apparent in the astrometric residuals than in the photometric residuals. A
true transiting planet system should not show these correlations. As suggested
by the Rain Plot, the pixel data were re-examined and the offending variable
star was identified as being about 11 arcseconds away from and about 4.8 Kp
magnitudes fainter than the brighter star.
## 4 Conclusions
Although based on just a small fraction of the data expected from the entire
mission, the following conclusions can be drawn.
a) Both the preliminary version of the full astrometric solution and the
locally linearized astrometric solution indicate that the precision of a
single measure of a typical star is about 0.004 arcsecond (about 0.001 pixel).
Results such as those shown in Figs. 2 and 3 suggest that the precision for
some stars might be substantially better.
b) On the basis of the limited data available so far, many astrometric effects
cannot be separated. The astrometric precision should enable the measurement
of the proper motions of the known large motion stars in the star list, but as
yet proper motion is indistinguishable from differential velocity aberration,
pixel phase, and similar effects.
c) Because of the large pixel size, Kepler images can often be blends of
multiple stars. If one or more components of such a blend is a photometrically
variable star, the astrometric position of the image can have a significant
motion. Tools such as the Rain Plot have already demonstrated the utility of
combining astrometric and photometric processing in the evaluation of
planetary transit candidates.
d) Whereas the astrometric precision of Kepler data has been demonstrated, the
astrometric accuracy has yet to be evaluated.
In summary, it is the extremely high SNR of the Kepler photometer that enables
the astrometric analysis of Kepler data. Even with just the preliminary data,
astrometric analysis is providing an important tool for the physical
understanding of the observations. Should the astrometric results continue to
follow the theoretical expectation of improving with the SNR, then solutions
for the parallax and proper motions of all stars will be computed.
Figure 2: Astrometric residuals from a blended variable (top) and a non-
variable star (bottom) taken from a solution for a single channel. Red symbols
are from the columns and blue symbols are from the rows. Figure 3: Time series
residuals for the relative photometric flux (A) in parts per thousand, and the
relative astrometric row (B) and column (C) residuals in millipixels (1 pixel
= 3.98 arcseconds) for KOI-15. The observations have been filtered to remove
long-period trends. The strong correlation between these residuals is taken as
evidence that this Kepler star is actually an astration of one or more
relatively constant stars and a background eclipsing binary. Figure 4: The
Rain Plot for KOI-15. This visualization shows the correlation between the
photometric and astrometric row (red X) and column (blue +) residuals. The
strong correlation between the position and brightness is evidence that this
is an astration of one or more relatively constant stars and a background
eclipsing binary.
Funding for this Discovery mission is provided by NASA’s Science Mission
Directorate. Many people have contributed to the success of the Kepler
Mission, and the authors wish to express their profound thanks to all.
Facilities: The Kepler Mission.
## References
* Batalha et al. (2010) Batalha, N. M., et al. 2010, this issue
* Borucki et al. (2010) Borucki, W. J. et al. 2010, Science, submitted
* Bryson et al. (2010) Bryson, S. T., et al. 2010, this issue
* Caldwell at al. (2010) Caldwell, D. A., et al. 2010, this issue
* Jenkins et al. (2010) Jenkins, J. M., et al. 2010, this issue
* Kaiser et al. (2000) Kaiser, N., Tonry, J. L., Luppino, J. L. 2000, PASP112, 768
* King (1983) King, I. R. 1983, PASP95, 163
* Koch et al. (2010) Koch, D. G., et al. 2010, this issue
* Skrutskie et al. (2006) Skrutskie, M. J., et al. 2006, AJ131, 1163
* Zacharias, et al. (2004) Zacharias N., et al. 2004, AJ127, 3043
|
arxiv-papers
| 2010-01-02T14:26:04 |
2024-09-04T02:49:07.435360
|
{
"license": "Public Domain",
"authors": "David G. Monet (US Naval Observatory, Flagstaff Station), Jon M.\n Jenkins (SETI Institute), Edward W. Dunham (Lowell Observatory, Flagstaff),\n Stephen T. Bryson (NASA Ames Research Center), Ronald L. Gilliland (Space\n Telescope Science Institute), David W. Latham (Harvard-Smithsonian Center for\n Astrophysics), William J. Borucki (NASA Ames Research Center), David G. Koch\n (NASA Ames Research Center)",
"submitter": "David G. Monet",
"url": "https://arxiv.org/abs/1001.0305"
}
|
1001.0315
|
# Computation of the coefficients for the order $p^{6}$ anomalous chiral
Lagrangian
Shao-Zhou Jiang1,2111Email:jsz@mails.tsinghua.edu.cn., and Qing
Wang1,2222Email: wangq@mail.tsinghua.edu.cn.333corresponding author
1Center for High Energy Physics, Tsinghua University, Beijing 100084, P.R.
China
2Department of Physics, Tsinghua University, Beijing 100084, P.R.
China444mailing address
###### Abstract
We present the results of computing the order $p^{6}$ low energy constants in
the anomalous part of the chiral Lagrangian for both two and three flavor
pseudoscalar mesons. This is a generalization of our previous work on
calculating the order $p^{6}$ coefficients for the normal part of the chiral
Lagrangian in terms of the quark self energy $\Sigma(p^{2})$. We show that
most of our results are consistent with those we have found in the literature.
###### pacs:
12.39.Fe, 11.30.Rd, 12.38.Aw, 12.38.Lg
††preprint: TUHEP-TH-09170
## I Introduction and Background
It is well known that the chiral symmetry in quantum chromodynamics (QCD)
suffers anomalies due to the non-invariance of the path integral measure of
the quark fields under the chiral symmetry transformation. The anomaly
reflects the fact that the classical chiral symmetry may be violated by
quantum corrections. At the level of the effective chiral Lagrangian for the
pseudoscalar meson field $U$, anomaly no longer comes from the path integral
measure. Instead it is due to the non-invariance of the effective chiral
Lagrangian. If we denote by $\Gamma_{\mathrm{eff}}[U,J]$ the effective action
for the pseudoscalar meson field $U$ and the external source $J$, then this
non-invariance can be expressed as
$\displaystyle\Gamma_{\mathrm{eff}}[U,J]-\Gamma_{\mathrm{eff}}[U_{\Omega},J_{\Omega}]=\Gamma[\Omega,J]\;,$
(1)
where $U_{\Omega}\equiv\Omega^{{\dagger}}U\Omega^{{\dagger}}$ and
$J_{\Omega}\equiv[\Omega
P_{R}+\Omega^{{\dagger}}P_{L}][J+\not{\partial}][\Omega
P_{R}+\Omega^{\dagger}P_{L}]$. $\Gamma[\Omega,J]$ is the anomaly from the
light quark path integral measure
$\mathcal{D}\bar{\psi}_{\Omega}\mathcal{D}\psi_{\Omega}=\mathcal{D}\bar{\psi}\mathcal{D}\psi~{}e^{\Gamma}$
or the well known Wess-Zumino-Witten term. We can formally express it as
$\displaystyle\Gamma[\Omega,J]$ $\displaystyle=$
$\displaystyle-\ln\mathrm{Det}[(\Omega P_{R}+\Omega^{{\dagger}}P_{L})(\Omega
P_{R}+\Omega^{{\dagger}}P_{L})]=-\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}]+\mathrm{Tr}\ln[\not{\partial}+J]\;,~{}~{}~{}~{}$
(2)
Because for $N_{f}$ light quarks, each generator of the chiral symmetry
$SU(N_{f})_{L}\otimes SU(N_{f})_{R}/SU(N_{f})_{V}$ corresponds to a Goldstone
boson, which is treated phenomenologically as the physical pseudoscalar meson
field, the phase angle of the chiral rotation group element $\Omega$ can be
treated as the pseudoscalar meson field, i.e. $U=\Omega^{2}$. Then comparing
(1) and (2), we can rewrite the effective action $\Gamma_{\mathrm{eff}}$ as
$\displaystyle\Gamma_{\mathrm{eff}}[U,J]=-\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}]+\mathrm{Tr}\ln[\not{\partial}+J]+F[U,J]\hskip
56.9055ptF[U,J]=F[U_{\Omega},J_{\Omega}]$ (3)
The $U$ and $J$ dependence for $F[U,J]$ is not fixed by (1), but $F[U,J]$ is
invariant on $U\rightarrow U_{\Omega}$ and $J\rightarrow J_{\Omega}$. Hence
$F[U,J]$ represents those chiral invariant terms. In fact
$U_{\Omega}=\Omega^{\dagger}\Omega^{2}\Omega^{\dagger}=1$, and
$\Gamma_{\mathrm{eff}}[U_{\Omega},J_{\Omega}]=\Gamma_{\mathrm{eff}}[1,J_{\Omega}]=-\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}]+\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}]+F[U_{\Omega},J_{\Omega}]=F[U,J]$.
Note that the effective action is the path integration result for
$S_{\mathrm{eff}}[U,J]$, the action of the effective chiral Lagrangian for the
pseudoscalar meson field $U$ and the external source $J$,
$\displaystyle
e^{-\Gamma_{\mathrm{eff}}[U_{cl},J]}=\int\mathcal{D}U~{}e^{-S_{\mathrm{eff}}[U,J]}\hskip
56.9055ptU_{cl}(x)\equiv\int\mathcal{D}U~{}U(x)~{}e^{-S_{\mathrm{eff}}[U,J]}\;,$
(4)
where the second equation gives the definition of $U_{cl}$ which fixes
$U_{cl}$ as the functional of the external source $J$. With (3), (4) becomes
$\displaystyle
e^{\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}]-\mathrm{Tr}\ln[\not{\partial}+J]-F(U_{cl},J)}=\int\mathcal{D}U~{}e^{-S_{\mathrm{eff}}[U,J]}\;.$
(5)
Ref.AnomApproach ,AnomApproach1 ,AnomApproach2 choose as an approximation
$\displaystyle
S_{\mathrm{eff,0}}[U,J]=-\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}]+\mathrm{Tr}\ln[\not{\partial}+J]\;,$
(6)
where subscript 0 is used to denote the approximation. From (1), (3) and (5),
we find that under the chiral symmetry transformation,
$S_{\mathrm{eff,0}}[U,J]$, defined in (6), is not invariant. Substituting (6)
back into (5) and using standard loop expansion as developed in Ref.LoopExp ,
we find $F[U_{cl},J]$ is the pure loop correction from the action
$S_{\mathrm{eff,0}}[U,J]$. From the action (6), one can calculate various low
energy constants (LECs) of the effective chiral Lagrangian for pseudoscalar
mesons. In Ref.WQ3 , we call (6) the anomaly approach. In our previous paper
WQ4 , we have shown that the finite order $p^{4}$ LECs of the normal part of
$S_{\mathrm{eff,0}}[U,J]$ are exactly canceled by the summation of all the
$p^{6}$ and higher order terms. Eq.(2) further shows that even for the
anomalous part, $S_{\mathrm{eff,0}}[U,J]$ only contributes the Wess-Zumino-
Witten term; it cannot produce the $p^{6}$ and higher order anomaly terms.
This absence of the normal part and the $p^{6}$ and more higher order
anomalous part reflects the fact that the choice of (6) is not correct,
although it offers the correct Wess-Zumino-Witten term. Further, (6) is
independent of the strong interaction dynamics, i.e., even we switch off the
quark-gluon interaction by deleting the strong interaction coupling constant,
(6) is not changed. These facts imply that we need to add some strong dynamics
dependent correction term $\Delta S_{\mathrm{eff}}[U,J]$ to
$S_{\mathrm{eff,0}}[U,J]$ as given in (6),
$\displaystyle S_{\mathrm{eff}}[U,J]=S_{\mathrm{eff,0}}[U,J]+\Delta
S_{\mathrm{eff}}[U,J]\;.$ (7)
From (5) and (6), we find that $\Delta S_{\mathrm{eff}}[U,J]$, introduced in
(7), must be invariant under chiral symmetry transformations. In Refs.WQ1 and
WQ2 , $\Delta S_{\mathrm{eff}}[U,J]$ is taken to be
$\displaystyle\Delta
S_{\mathrm{eff}}[U,J]=\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}+\Sigma(-\bar{\nabla}^{2})]$
(8)
with $\Sigma$ being the quark self energy satisfying the Schwinger-Dyson
equation (SDE) and $\bar{\nabla}^{\mu}$ is defined as
$\bar{\nabla}^{\mu}\equiv\partial^{\mu}-iv^{\mu}_{\Omega}$. This expression
for $\Delta S_{\mathrm{eff}}[U,J]$ encodes the dynamics of the underlying QCD
through quark self energy $\Sigma$ and in Ref.WQma , we have shown that (8)
does not produce the Wess-Zumino-Witten term ensuring the correctness of (1).
In Ref.WQ1 , we have calculated the orders $p^{2}$ and $p^{4}$ normal part
LECs in terms of the action (7) and (8). The importance of knowledge of LECs
of the chiral Lagrangian, especially for order $p^{6}$ LECs was emphasized in
Ref.Review . Recently, in Ref.WQ4 , we improved the computation procedure and
generalized the calculations up to the order $p^{6}$ normal part LECs. In
Ref.WQma , we have calculated the $p^{4}$ order anomalous part and shown that
the $\Sigma$ dependent coefficient generates the correct coefficient $N_{c}$
for the Wess-Zumino-Witten term. It is the purpose of this paper to calculate
all order $p^{6}$ LECs for the anomalous part of the chiral Lagrangian (7). In
fact the general structure of the $p^{6}$ order anomalous part chiral
Lagrangian was first given by Refs.anom-1 and anom0 and later clarified by
Refs.anom1 and anom2 . Ref.p6anomLEC estimates the values of several of the
order $p^{6}$ LECs for the anomalous part of the chiral Lagrangian. Although
order $p^{6}$ LECs for the normal part of the chiral Lagrangian seem attract
more attentions in the literature (see references given in WQ4 ), they are the
next next to leading order terms. The order $p^{6}$ LECs for the anomalous
part of the chiral Lagrangian are belong to the next leading order terms.
This paper is organized as follows: in Sec.II, we review the calculation of
the order $p^{4}$ anomalous part of the chiral Lagrangian in terms of the
action (7). With the method used in section II, in Sec.III, we compute the
order $p^{6}$ LECs for the anomalous part of the chiral Lagrangian, and obtain
the analytical expression for the LECs in terms of quark self energy $\Sigma$.
We further compute the numerical values for these LECs. We compare our results
with those obtained in literature. Sec.IV is the summary and future directions
of our work. We list some necessary tables and formulae in appendices.
## II Review the order $p^{4}$ anomalous part of the chiral Lagrangian
For the anomalous part of the chiral Lagrangian, the leading nontrivial order
is $p^{4}$ and it is the well known Wess-Zumino-Witten term. In Ref.WQma , we
have calculated the action (7) by several different methods and all obtain the
same Wess-Zumino-Witten term. If we naively apply these methods to the next to
leading order $p^{6}$ computations, we will find that they are too complex to
be achieved even with the help of the computer. In this section, we build a
method which is suitable to be generalized to the order $p^{6}$ calculations.
The order $p^{4}$ of the anomalous chiral Lagrangian is here only to be used
to explain our method. Ref.WQma only expresses the Wess-Zumino-Witten term in
terms of a parameter integration. In this section, we will explicitly finish
this parameter integration and show that it does recover the Wess-Zumino-
Witten term.
Since we are only interested in the $U$ field dependent part of the anomalous
part of the chiral Lagrangian, we can drop out the pure source terms. Then our
choice of $\Delta S_{\mathrm{eff}}[U,J]$ in (8) gives the result that only
$\Sigma$ dependent terms in $\Delta S_{\mathrm{eff}}[U,J]$ contribute to the
chiral Lagrangian, while the $\Sigma$ independent terms in $\Delta
S_{\mathrm{eff}}[U,J]$ are completely canceled by the term
$-\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}]$ in $S_{\mathrm{eff,0}}[U,J]$,
leaving a pure $U$ field independent term $\mathrm{Tr}\ln[\not{\partial}+J]$.
So what we need to compute is
$\displaystyle
S_{\mathrm{eff}}[U,J]=\bigg{[}\mathrm{Tr}\ln[\not{\partial}+J_{\Omega}+\Sigma(-\bar{\nabla}^{2})]-\mathrm{Tr}\ln[\not{\partial}+J+\Sigma(-\nabla^{2})]\bigg{]}_{\Sigma~{}\mbox{\tiny
dependent}}\;,$ (9)
in which we have added in $S_{\mathrm{eff}}[U,J]$ an extra pure source term
$-\mathrm{Tr}\ln[\not{\partial}+J+\Sigma(-\nabla^{2})]\bigg{|}_{\Sigma~{}\mbox{\tiny
dependent}}$ for later use, and we define
$\nabla^{\mu}\equiv\partial^{\mu}-iv^{\mu}$. Now we write $\Omega$ as
$\Omega=e^{-i\beta}$ and further introduce a parameter $t$ dependent rotation
element $\Omega(t)=e^{-it\beta}$. With the help of the relation
$\Omega(1)=\Omega$ and $\Omega(0)=1$, (9) becomes
$\displaystyle
S_{\mathrm{eff}}[U,J]=\mathrm{Tr}\ln[\not{\partial}+J_{\Omega(t)}+\Sigma(-\nabla_{t}^{2})]\bigg{|}^{t=1}_{t=0,~{}\Sigma~{}\mbox{\tiny
dependent}}\hskip
56.9055pt\nabla_{t}^{\mu}=\overline{\nabla}^{\mu}\bigg{|}_{\Omega\rightarrow\Omega(t)}$
(10)
with $\nabla^{\mu}_{t}=\partial^{\mu}-iv^{\mu}_{\Omega(t)}$. $J_{\Omega(t)}$
is $J_{\Omega}$ with $\Omega$ replaced by $\Omega(t)$. We decompose
$J_{\Omega}$ as
$J_{\Omega}=-i\not{v}_{\Omega}-i\not{a}_{\Omega}\gamma_{5}-s_{\Omega}+ip_{\Omega}\gamma_{5}$,
so we can also decompose $J_{\Omega(t)}$ as
$J_{\Omega(t)}=-i\not{v}_{t}-i\not{a}_{t}\gamma_{5}-s_{t}+ip_{t}\gamma_{5}$.
Result (10) implies that our chiral Lagrangian can be expressed as the
difference of Trln($\cdots$) at $t$ dependent chiral rotation between $t=1$
and $t=0$. Since the $t$ dependent rotated source $J_{\Omega(t)}$ satisfies
$\displaystyle\frac{\partial J_{\Omega(t)}}{\partial
t}=\frac{1}{2}[\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}\gamma_{5},\not{\partial}+J_{\Omega(t)}]_{+}\hskip
56.9055ptU_{t}=\Omega^{2}(t)\;,$ (11)
we can further proceed to express the chiral Lagrangian in terms of
integration over the parameter $t$:
$\displaystyle S_{\mathrm{eff}}[U,J]$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dt~{}\frac{d}{dt}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega(t)}+\Sigma(-\nabla_{t}^{2})]\bigg{|}_{\Sigma~{}\mbox{\tiny
dependent}}$ (12) $\displaystyle=$
$\displaystyle\int_{0}^{1}dt~{}\mathrm{Tr}\bigg{[}[\frac{\partial
J_{\Omega(t)}}{\partial t}+\frac{\partial\Sigma(-\nabla_{t}^{2})}{\partial
t}][i\not{\partial}+J_{\Omega(t)}+\Sigma(-\nabla_{t}^{2})]^{-1}\bigg{]}_{\Sigma~{}\mbox{\tiny
dependent}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dt~{}\mathrm{Tr}\bigg{[}\bigg{(}\frac{1}{2}[\frac{\partial
U_{t}}{\partial
t}U^{\dagger}_{t}\gamma_{5},\not{\partial}+J_{\Omega(t)}]_{+}+\frac{\partial\Sigma(-\nabla_{t}^{2})}{\partial
t}\bigg{)}[i\not{\partial}+J_{\Omega(t)}+\Sigma(-\nabla_{t}^{2})]^{-1}\bigg{]}_{\Sigma~{}\mbox{\tiny
dependent}}\;.$
(12) is the main formula we rely on to calculate LECs. Ref.WQma explicitly
calculates the order $p^{4}$ anomalous part of the r.h.s. of (12) and finds
the result
$\displaystyle S_{\mathrm{eff}}[U,J]\bigg{|}_{\mbox{\tiny anomalous~{}}p^{4}}$
$\displaystyle=$ $\displaystyle-2N_{c}\epsilon_{\mu\nu\alpha\beta}\int
d^{4}x\int_{0}^{1}dt\int\frac{d^{4}k}{(2\pi)^{4}}~{}\mathrm{tr}_{f}\bigg{[}\frac{\partial
U_{t}}{\partial
t}U^{\dagger}_{t}\bigg{(}\frac{\Sigma(k^{2})[\Sigma^{2}(k^{2})-k^{2}][\Sigma(k^{2})-2k^{2}\Sigma^{\prime}(k^{2})]}{[\Sigma^{2}(k^{2})+k^{2}]^{4}}$
(13)
$\displaystyle\times(2\nabla^{\mu}_{t}\nabla^{\nu}_{t}\nabla^{\alpha}_{t}\nabla^{\beta}_{t}+2a^{\mu}_{t}a^{\nu}_{t}\nabla^{\alpha}_{t}\nabla^{\beta}_{t}-2\nabla^{\mu}_{t}a^{\nu}_{t}\nabla^{\alpha}_{t}a^{\beta}_{t}+2\nabla^{\mu}_{t}a^{\nu}_{t}a^{\alpha}_{t}\nabla^{\beta}_{t}+2a^{\mu}\nabla^{\nu}_{t}\nabla^{\alpha}_{t}a^{\beta}_{t}-2a^{\mu}_{t}\nabla^{\nu}_{t}a^{\alpha}_{t}\nabla^{\beta}_{t}$
$\displaystyle+2\nabla^{\mu}_{t}\nabla^{\nu}_{t}a^{\alpha}_{t}a^{\beta}_{t}+2a^{\mu}_{t}a^{\nu}_{t}a^{\alpha}_{t}a^{\beta}_{t})+\frac{k^{2}\Sigma(k^{2})[\Sigma(k^{2})-2k^{2}\Sigma^{\prime}(k^{2})]}{[\Sigma^{2}(k^{2})+k^{2}]^{4}}(4\nabla^{\mu}_{t}\nabla^{\nu}_{t}\nabla^{\alpha}_{t}\nabla^{\beta}_{t}+2a^{\mu}_{t}a^{\nu}_{t}\nabla^{\alpha}_{t}\nabla^{\beta}_{t}$
$\displaystyle-2\nabla^{\mu}_{t}a^{\nu}_{t}\nabla^{\alpha}_{t}a^{\beta}_{t}+4a^{\mu}_{t}\nabla^{\nu}_{t}\nabla^{\alpha}_{t}a^{\beta}_{t}-2a^{\mu}_{t}\nabla^{\nu}_{t}a^{\alpha}_{t}\nabla^{\beta}_{t}+2\nabla^{\mu}_{t}\nabla^{\nu}_{t}a^{\alpha}_{t}a^{\beta}_{t})\bigg{)}\bigg{]}\;.$
The momentum integration can be calculated analytically, because the integrand
is a total derivative. The result is
$\displaystyle S_{\mathrm{eff}}[U,J]\bigg{|}_{\mbox{\tiny anomalous~{}}p^{4}}$
$\displaystyle=$
$\displaystyle\frac{1}{32\pi^{2}}\epsilon_{\mu\nu\alpha\beta}\int
d^{4}x\int_{0}^{1}dt~{}\mathrm{tr}_{f}\bigg{[}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}~{}\bigg{(}V_{t}^{\mu\nu}V_{t}^{\alpha\beta}+\frac{2i}{3}[a^{\mu}_{t}a_{t}^{\nu},V^{\alpha\beta}_{t}]_{+}+\frac{4}{3}d_{t}^{\mu}a^{\nu}_{t}d_{t}^{\alpha}a^{\beta}_{t}$
(14)
$\displaystyle+\frac{8i}{3}a^{\mu}_{t}V_{t}^{\nu\alpha}a_{t}^{\beta}+\frac{4}{3}a^{\mu}_{t}a^{\nu}_{t}a^{\alpha}_{t}a^{\beta}_{t}\bigg{)}\bigg{]}\;,$
where
$V^{\mu\nu}_{t}=\partial^{\mu}v^{\nu}_{t}-\partial^{\nu}v^{\mu}_{t}-i[v^{\mu}_{t},v^{\nu}_{t}]$
and
$d^{\mu}_{t}a^{\nu}_{t}=\partial^{\mu}a^{\nu}_{t}-i[v^{\mu}_{t},a^{\nu}_{t}]$.
Ref.WQma only gives the above result (14) without finishing the integration
over parameter $t$. Now we continue to achieve this integration, with the help
of following relations:
$\displaystyle\frac{\partial a_{t}^{\mu}}{\partial
t}=\frac{i}{2}[\nabla^{\mu}_{t}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}-\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}\nabla^{\mu}_{t}]\hskip 113.81102pt\frac{\partial
v_{t}^{\mu}}{\partial t}=\frac{1}{2}[a^{\mu}_{t}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}-\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}a^{\mu}_{t}]$ $\displaystyle\frac{\partial s_{t}}{\partial
t}=-\frac{i}{2}[p_{t}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}+\frac{\partial U_{t}}{\partial t}U^{\dagger}_{t}p_{t}]\hskip
119.50148pt\frac{\partial p_{t}}{\partial t}=\frac{i}{2}[s_{t}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}+\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}s_{t}]$ (15) $\displaystyle\frac{\partial
d^{\mu}a_{t}^{\nu}}{\partial
t}\\!=\frac{i}{2}[(\nabla^{\mu}_{t}\nabla^{\nu}_{t}\\!\\!+\\!a^{\nu}_{t}a^{\mu}_{t})\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}\\!-\\!\nabla^{\nu}_{t}\frac{\partial
U_{t}}{\partial
t}U^{\dagger}_{t}\nabla^{\mu}_{t}\\!-\\!\nabla^{\mu}_{t}\frac{\partial
U_{t}}{\partial
t}U^{\dagger}_{t}\nabla^{\nu}_{t}\\!-\\!a^{\nu}_{t}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}a^{\mu}_{t}\\!-\\!a^{\mu}_{t}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}a^{\nu}_{t}\\!+\\!\frac{\partial
U_{t}}{\partial
t}U^{\dagger}_{t}(\nabla^{\nu}_{t}\nabla^{\mu}_{t}\\!\\!+\\!a^{\mu}_{t}a^{\nu}_{t})]~{}~{}~{}~{}$
$\displaystyle\frac{\partial V^{\mu\nu}_{t}}{\partial
t}=\frac{1}{2}[-\nabla^{\mu}_{t}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}a^{\nu}_{t}+\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}\nabla^{\mu}_{t}a^{\nu}_{t}-\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}d_{t}^{\mu}a^{\nu}_{t}+d_{t}^{\mu}a^{\nu}_{t}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}+a^{\nu}_{t}\nabla^{\mu}_{t}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}-a^{\nu}_{t}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}\nabla^{\mu}_{t}$ $\displaystyle\hskip
28.45274pt+\nabla^{\nu}_{t}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}a^{\mu}_{t}-\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}\nabla^{\nu}_{t}a^{\mu}_{t}+\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}d_{t}^{\nu}a^{\mu}_{t}-d_{t}^{\nu}a^{\mu}_{t}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}-a^{\mu}_{t}\nabla^{\nu}_{t}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}+a^{\mu}_{t}\frac{\partial U_{t}}{\partial
t}U^{\dagger}_{t}\nabla^{\nu}_{t}]$
and by lengthy calculations, we can rewrite (14) as
$\displaystyle S_{\mathrm{eff}}[U,J]\bigg{|}_{\mbox{\tiny anomalous~{}}p^{4}}$
$\displaystyle=$ $\displaystyle-\frac{N_{c}}{48\pi^{2}}\int
d^{4}x\int_{0}^{1}dt~{}\epsilon_{\mu\nu\lambda\rho}\mathrm{tr}_{f}\bigg{[}\frac{\partial
U_{t}}{\partial
t}U^{\dagger}_{t}R^{\mu}_{t}R^{\nu}_{t}R^{\lambda}_{t}R^{\rho}_{t}+\frac{d}{dt}W^{\mu\nu\lambda\rho}(U_{t},l,r)\bigg{]}$
(16)
with $l^{\mu}=v^{\mu}-a^{\mu},~{}r^{\mu}=v^{\mu}+a^{\mu}$,
$R^{\mu}_{t}=U^{\dagger}_{t}\partial^{\mu}U_{t}$,
$L^{\mu}_{t}=(\partial^{\mu}U_{t})U^{\dagger}_{t}$ and
$\displaystyle W^{\mu\nu\lambda\rho}(U_{t},l,r)$ $\displaystyle=$
$\displaystyle
iR^{\mu}_{t}R^{\nu}_{t}R^{\lambda}_{t}l^{\rho}+l^{\mu}\partial^{\nu}l^{\lambda}R_{t}^{\rho}+\partial^{\mu}l^{\nu}l^{\lambda}R_{t}^{\rho}-\frac{1}{2}R_{t}^{\mu}l^{\nu}R_{t}^{\lambda}l^{\rho}+r^{\mu}U_{t}l^{\nu}R_{t}^{\lambda}R^{\rho}U_{t}^{\dagger}+iR^{\mu}_{t}l^{\nu}l^{\lambda}l^{\rho}+iU^{\dagger}_{t}r^{\mu}U_{t}\partial^{\nu}l^{\lambda}l^{\rho}$
(17)
$\displaystyle+iU^{\dagger}_{t}r^{\mu}\partial^{\nu}r^{\lambda}U_{t}l^{\rho}-il^{\mu}U^{\dagger}_{t}r^{\nu}U_{t}l^{\lambda}R^{\rho}_{t}-R^{\mu}_{t}U^{{\dagger}}_{t}\partial^{\nu}r^{\lambda}U_{t}l^{\rho}+U^{\dagger}_{t}r^{\mu}U_{t}l^{\nu}l^{\lambda}l^{\rho}+\frac{1}{4}U^{\dagger}_{t}r^{\mu}U_{t}l^{\nu}U^{\dagger}_{t}r^{\lambda}U_{t}l^{\rho}$
$\displaystyle-(U_{t}\leftrightarrow U^{\dagger}_{t},l^{\mu}\leftrightarrow
r^{\mu},L^{\mu}_{t}\leftrightarrow-R^{\mu}_{t})\;.$
In Ref.WQma , we already show that the first term of the r.h.s. of Eq.(16) is
just the Wess-Zumino-Witten term of the form defined on a four dimensional
boundary disc $Q$ in five dimensional space-time
$\displaystyle-\frac{N_{c}}{48\pi^{2}}\int
d^{4}x\int_{0}^{1}dt~{}\epsilon_{\mu\nu\lambda\rho}\mathrm{tr}_{f}\bigg{[}\frac{\partial
U_{t}}{\partial
t}U^{\dagger}_{t}R^{\mu}_{t}R^{\nu}_{t}R^{\lambda}_{t}R^{\rho}_{t}\bigg{]}=-\frac{N_{c}}{240\pi^{2}}\int_{Q}d\Sigma_{ijklm}\mathrm{tr}_{f}[R^{i}R^{j}R^{k}R^{l}R^{m}]$
(18)
with $R^{i}\equiv U^{\dagger}\partial^{i}U$. For the second term of the r.h.s.
of Eq.(16), the integration over parameter $t$ can be calculated explicitly,
$\displaystyle-\frac{N_{c}}{48\pi^{2}}\int
d^{4}x\int_{0}^{1}dt~{}\epsilon_{\mu\nu\lambda\rho}\mathrm{tr}_{f}\bigg{[}\frac{d}{dt}W^{\mu\nu\lambda\rho}(U_{t},l,r)\bigg{]}=-\frac{N_{c}}{48\pi^{2}}\int
d^{4}x~{}\epsilon_{\mu\nu\lambda\rho}\mathrm{tr}_{f}\bigg{[}W^{\mu\nu\lambda\rho}(U,l,r)-W^{\mu\nu\lambda\rho}(1,l,r)\bigg{]}\;,$
(19)
which the just the gauge part of the Wess-Zumino-Witten term given by
Ref.anom1 and Zhou . This finishes the explicit calculation of the order
$p^{4}$ anomalous part of the chiral Lagrangian starting from formula (12). We
leave the order $p^{6}$ part to the next section.
## III Calculation of the order $p^{6}$ anomalous part of the chiral
Lagrangian
In this section, we start from Eq.(12) to calculate its order $p^{6}$
anomalous part of the chiral Lagrangian. For convenience, we change to the
Minkowski space to perform our calculations. Direct computation gives the
result
$\displaystyle S_{\mathrm{eff}}[U,J]\bigg{|}_{\mbox{\tiny
anomalous~{}}p^{6}}={\displaystyle\sum_{m=1}^{210}}\int
d^{4}x~{}\bar{K}_{m}^{W}\int_{0}^{1}dt~{}\mathrm{tr}_{f}[\bar{O}_{m}^{W}(x,t)]\;,$
(20)
where $\bar{K}_{m}^{W}$ is the coefficient in front of the operator
$\bar{O}_{m}^{W}(x,t)$, which depends on quark self energy $\Sigma(k^{2})$.
The 210 parameter $t$ dependent operators $\bar{O}_{m}^{W}(x,t)$ all have the
structure of $\bar{O}_{m}^{W}(x,t)=\epsilon_{\mu\nu\lambda\rho}\frac{\partial
U_{t}}{\partial t}U^{\dagger}_{t}\bar{O}^{\mu\nu\lambda\rho}_{m}(x,t)$ and
$\bar{O}^{\mu\nu\lambda\rho}_{m}(x,t)$ are order $p^{6}$ operators consisting
of multiplications of various compositions of $a^{\mu}_{t}$,
$\nabla^{\nu}_{t}$, $s_{t}$ and $p_{t}$. In Appendix A we list all these
operators in Table V. In obtaining (20), we have applied the Schouten
identity, which reduces the original total 294 operators to the present 210
operators. In the literature, the general $p^{6}$ order anomalous part of the
chiral Lagrangian given in Ref.anom1 has only 24 independent operators. For
$N_{f}=3,2$ this number reduces to 23 and five respectively. Specially for the
case of $N_{f}=2$, to incorporate the electro-magnetic field into the external
source $v^{\mu}$, the original traceless property of $v^{\mu}$ must be
dropped, this changes the original five independent $p^{6}$ order anomalous
operators into thirteen. If we denote the independent operators by
$O_{n}^{W}(x)$ ($o_{n}^{W}(x)$ for $N_{f}=2$) and corresponding coefficients
in front of the operators by $C_{n}^{W}$ ($c_{n}^{W}(x)$ for $N_{f}=2$)
respectively, then (20) becomes
$\displaystyle S_{\mathrm{eff}}[U,J]\bigg{|}_{\mbox{\tiny
anomalous~{}}p^{6}}={\displaystyle\sum_{n=1}^{24}}\int
d^{4}x~{}C_{n}^{W}O_{n}^{W}(x)\stackrel{{\scriptstyle
N_{f}=2}}{{=====}}{\displaystyle\sum_{n=1}^{13}}\int
d^{4}x~{}c_{n}^{W}o_{n}^{W}(x)\;.$ (21)
Note that our starting chiral Lagrangian (7) only involves one trace for
flavor indices. If we further apply the equation of motion to (21), there will
appear some operators with two flavor traces. Our result prohibits the
appearance of three operators $O_{3}^{W},O_{18}^{W},O_{24}^{W}$, leaving 21
independent operators. This implies that our formulation gives
$C_{3}^{W}=C_{18}^{W}=C_{24}^{W}=0$. If we do not apply the equation of
motion, there will be more independent operators and now this number is 23. To
make our calculation more convenient, we denote these operators before
applying the equation of motion by $\tilde{O}_{n}^{W}(x)$ and the
corresponding coefficients in front of the operators by $\tilde{K}_{n}^{W}$.
We list all possible $\tilde{O}_{n}^{W}(x)$ in the Table VI of Appendix A.
With these operators, (21) can also be written as
$\displaystyle S_{\mathrm{eff}}[U,J]\bigg{|}_{\mbox{\tiny
anomalous~{}}p^{6}}={\displaystyle\sum_{n=1}^{23}}\int
d^{4}x~{}\tilde{K}_{n}^{W}~{}\tilde{O}_{n}^{W}(x)\;.$ (22)
Through using the equation of motion, we can obtain the relations among the
two sets of operators $\tilde{O}_{n}^{W}(x)$ and $O_{n}^{W}(x)$ as follows
$\displaystyle\tilde{O}_{1}^{W}=O_{1}^{W}/B_{0}\hskip
22.76228pt\tilde{O}_{2}^{W}=O_{2}^{W}/B_{0}\hskip
22.76228pt\tilde{O}_{3}^{W}=O_{4}^{W}/B_{0}\hskip
22.76228pt\tilde{O}_{4}^{W}=O_{5}^{W}/B_{0}\hskip
22.76228pt\tilde{O}_{5}^{W}=O_{7}^{W}/B_{0}\hskip
22.76228pt\tilde{O}_{6}^{W}=O_{9}^{W}/B_{0}$
$\displaystyle\tilde{O}_{7}^{W}=O_{11}^{W}/B_{0}\hskip
25.6073pt\tilde{O}_{8}^{W}=O_{12}^{W}\hskip
25.6073pt\tilde{O}_{9}^{W}=O_{1}^{W}\hskip
25.6073pt\tilde{O}_{10}^{W}=O_{16}^{W}\hskip
25.6073pt\tilde{O}_{11}^{W}=O_{17}^{W}\hskip
25.6073pt\tilde{O}_{12}^{W}=O_{13}^{W}\hskip
25.6073pt\tilde{O}_{13}^{W}=O_{14}^{W}$
$\displaystyle\tilde{O}_{14}^{W}=O_{15}^{W}\hskip
25.6073pt\tilde{O}_{15}^{W}=-O_{4}^{W}\\!\\!+\\!\frac{2}{N_{f}}O_{6}^{W}\hskip
25.6073pt\tilde{O}_{16}^{W}=-O_{5}^{W}\\!\\!-\\!\frac{1}{N_{f}}O_{6}^{W}\hskip
25.6073pt\tilde{O}_{17}^{W}=O_{19}^{W}\hskip
25.6073pt\tilde{O}_{18}^{W}=O_{20}^{W}\hskip
25.6073pt\tilde{O}_{19}^{W}=O_{21}^{W}$
$\displaystyle\tilde{O}_{20}^{W}=O_{22}^{W}\hskip
28.45274pt\tilde{O}_{21}^{W}=O_{23}^{W}\hskip
28.45274pt\tilde{O}_{22}^{W}=O_{7}^{W}\\!\\!-\\!\frac{1}{N_{f}}O_{8}^{W}\hskip
28.45274pt\tilde{O}_{23}^{W}=O_{9}^{W}\\!\\!-\\!\frac{1}{N_{f}}O_{10}^{W}\;,~{}~{}~{}~{}$
(23)
where $B_{0}$ is the order $p^{2}$ LEC in the normal part of the chiral
Lagrangian. Here we divide $O^{W}_{1},\cdots,O^{W}_{7}$ by $B_{0}$, making the
matrices $A_{mn}$ introduced later in Eq.(27) independent of $B_{0}$. For
$N_{f}=2$, (23) is changed to
$\displaystyle\tilde{O}_{1}^{W}=0\hskip
19.91684pt\tilde{O}_{2}^{W}=o_{1}^{W}/B_{0}\hskip
19.91684pt\tilde{O}_{3}^{W}=o_{2}^{W}/B_{0}\hskip
19.91684pt\tilde{O}_{4}^{W}=-o_{2}^{W}/(2B_{0})\\!+\\!o_{6}^{W}/B_{0}\hskip
19.91684pt\tilde{O}_{5}^{W}=o_{3}^{W}/B_{0}\hskip
19.91684pt\tilde{O}_{6}^{W}=o_{4}^{W}/B_{0}$
$\displaystyle\tilde{O}_{7}^{W}=o_{5}^{W}/B_{0}\hskip
22.76228pt\tilde{O}_{8}^{W}=\tilde{O}_{9}^{W}=\tilde{O}_{10}^{W}=\tilde{O}_{11}^{W}=0\hskip
22.76228pt\tilde{O}_{12}^{W}=-o_{9}^{W}\hskip
22.76228pt\tilde{O}_{13}^{W}=\tilde{O}_{14}^{W}=-\frac{1}{2}o_{6}^{W}\\!+o_{9}^{W}\hskip
22.76228pt\tilde{O}_{15}^{W}=-o_{6}^{W}$
$\displaystyle\tilde{O}_{16}^{W}=-\frac{1}{2}o_{6}^{W}\hskip
25.6073pt\tilde{O}_{17}^{W}=o_{10}^{W}\hskip
25.6073pt\tilde{O}_{18}^{W}=\tilde{O}_{19}^{W}=-o_{10}^{W}\hskip
25.6073pt\tilde{O}_{20}^{W}=\frac{1}{4}o_{7}^{W}-\frac{1}{8}o_{8}^{W}-o^{W}_{10}+o^{W}_{11}-2o^{W}_{13}\hskip
25.6073pt\tilde{O}_{21}^{W}=0$
$\displaystyle\tilde{O}_{22}^{W}=o_{7}^{W}-\frac{1}{2}o_{8}^{W}\hskip
28.45274pt\tilde{O}_{23}^{W}=0\;.~{}~{}~{}~{}$ (24)
Direct comparison between (20) and (22) is difficult, since in (20) we have an
extra integration over parameter $t$ and the number of operators in (20) is
much larger than it is in (22). Instead of finishing the integration over
parameter $t$ in (20), we introduce an integration of parameter $t$ in (22).
Since we are only interested in the $U$ dependent part of the chiral
Lagrangian, adding some $U$ field independent pure source terms in (22) will
not change our result; therefore we can rewrite (22) as
$\displaystyle S_{\mathrm{eff}}[U,J]\bigg{|}_{\mbox{\tiny anomalous~{}}p^{6}}$
$\displaystyle=$ $\displaystyle{\displaystyle\sum_{n=1}^{23}}\int
d^{4}x~{}\tilde{K}_{n}^{W}[\tilde{O}_{n}^{W}(x)-\tilde{O}_{n}^{W}(x)\bigg{|}_{U=1}]={\displaystyle\sum_{n=1}^{23}}\int
d^{4}x~{}\tilde{K}_{n}^{W}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t=1}}-\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow U_{t=0}}]$ (25)
$\displaystyle=$ $\displaystyle{\displaystyle\sum_{n=1}^{23}}\int
d^{4}x~{}\tilde{K}_{n}^{W}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t}}]\bigg{|}^{t=1}_{t=0}={\displaystyle\sum_{n=1}^{23}}\int
d^{4}x~{}\tilde{K}_{n}^{W}\int_{0}^{1}dt~{}\frac{d}{dt}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t}}]\;.$
In expression (25), integration of parameter $t$ is already present in the
formula, then the only remaining problem is that in (25) there are only 23
independent terms acted on by the differential of $t$, while in (20) there are
210 terms. comparing (25) and (20), we obtain
$\displaystyle{\displaystyle\sum_{n=1}^{23}}~{}\tilde{K}_{n}^{W}~{}\frac{d}{dt}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t}}]={\displaystyle\sum_{m=1}^{210}}~{}\bar{K}_{m}^{W}~{}\bar{O}_{m}^{W}(x,t)\;.$
(26)
Note that with the help of relation (15),
$\frac{d}{dt}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow U_{t}}]$ appearing in
the above equation can be reduced to linear composition of
$\bar{O}_{m}^{W}(x,t)$, i.e.
$\displaystyle\frac{d}{dt}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t}}]={\displaystyle\sum_{m=1}^{210}}~{}A_{nm}\bar{O}_{m}^{W}(x,t)$ (27)
with the $23\times 210$ matrix $A_{nm}$ given by Table VII in Appendix B, Then
we rearrange (27) by multiplying both sides of the equation by some $23\times
23$ matrix elements $C_{n^{\prime}n}$,
$\displaystyle{\displaystyle\sum_{n=1}^{23}}C_{n^{\prime}n}\frac{d}{dt}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t}}]={\displaystyle\sum_{n=1}^{23}\sum_{m=1}^{210}}~{}C_{n^{\prime}n}A_{nm}\bar{O}_{m}^{W}(x,t)={\displaystyle\sum_{m=1}^{210}}~{}R_{n^{\prime}m}\bar{O}_{m}^{W}(x,t)\hskip
42.67912ptR_{n^{\prime}m}\equiv{\displaystyle\sum_{n=1}^{23}}C_{n^{\prime}n}A_{nm}~{}~{}~{}~{}$
(28)
and tune $C_{n^{\prime}n}$ in such a way that a $23\times 23$ submatrix
$R^{\prime}$ is a unit matrix, i.e.
$R^{\prime}_{n^{\prime}m^{\prime}}=\delta_{n^{\prime}m^{\prime}}$ with
$n^{\prime},m^{\prime}\\!=\\!1,3,4,5,6,7,20,43,44,49,50,51,52$,
$54,57,59,62,63,64,127,128,133,134$. The $C$ matrix is found to be of the form
$\left(\begin{array}[]{cc}\bar{C}_{7\times 7}&0_{7\times 15}\\\ 0_{15\times
7}&\tilde{C}_{15\times 15}\end{array}\right)$ where $\bar{C}$ and $\tilde{C}$
are $7\times 7$ and $15\times 15$ matrices respectively. The off diagonal
parts are two matrices with null matrix elements and the dimensions are
$7\times 15$ and $15\times 7$. We label the dimension of the sub-matrices as
their subscripts. $\bar{C}$ and $\tilde{C}$ matrices are given in Table VIII
and Table IX in Appendix B. We call the remaining part of $R_{n^{\prime}m}$
the matrix $R_{n^{\prime}m^{\prime\prime}}~{}m^{\prime\prime}\neq m^{\prime}$.
Then (28) is changed to
$\displaystyle{\displaystyle\sum_{n=1}^{23}}C_{m^{\prime}n}\frac{d}{dt}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t}}]=\bar{O}_{m^{\prime}}^{W}(x,t)+{\displaystyle\sum_{m^{\prime\prime}}}~{}R_{m^{\prime}m^{\prime\prime}}\bar{O}_{m^{\prime\prime}}^{W}(x,t)\;.~{}~{}~{}~{}$
(29)
Multiplying both sides of the above equation by $\bar{K}^{W}_{m^{\prime}}$,
$\displaystyle{\displaystyle\sum_{m^{\prime}}\sum_{n=1}^{23}}\bar{K}^{W}_{m^{\prime}}C_{m^{\prime}n}\frac{d}{dt}[\tilde{O}_{n}^{W}(x)\bigg{|}_{U\rightarrow
U_{t}}]={\displaystyle\sum_{m^{\prime}}}\bar{K}^{W}_{m^{\prime}}\bar{O}_{m^{\prime}}^{W}(x,t)+{\displaystyle\sum_{m^{\prime}}\sum_{m^{\prime\prime}}}~{}\bar{K}^{W}_{m^{\prime}}R_{m^{\prime}m^{\prime\prime}}\bar{O}_{m^{\prime\prime}}^{W}(x,t)\;.~{}~{}~{}~{}$
(30)
Comparing (30) and (26), to make these two equations consistent with each
other, we must have conditions,
$\displaystyle\tilde{K}_{n}^{W}={\displaystyle\sum_{m^{\prime}}}\bar{K}^{W}_{m^{\prime}}C_{m^{\prime}n}\hskip
56.9055pt\bar{K}^{W}_{m^{\prime\prime}}={\displaystyle\sum_{m^{\prime}}}~{}\bar{K}^{W}_{m^{\prime}}R_{m^{\prime}m^{\prime\prime}}\;,$
(31)
in which the second equation is a consistency check for the coefficients
$\bar{K}^{W}_{m^{\prime\prime}}$ of the dependent operators
$\bar{O}_{m^{\prime\prime}}^{W}(x,t)$. We have checked analytically that these
constraints are all automatically satisfied and this can be seen as a
consistency check of our formulation. The first equation gives
$\tilde{K}_{n}^{W}$ in terms of $\bar{K}^{W}_{m^{\prime}}$ and
$C_{m^{\prime}n}$. Substituting it in the expressions obtained for
$\bar{K}_{m^{\prime}}^{W}$ and $C_{m^{\prime}n}$, we finally obtain the 23
order $p^{6}$ LECs for the three and more flavors anomalous part of chiral
Lagrangian.
The resulting analytical expressions for $\tilde{K}_{n}^{W}$ as functions of
quark self energy $\Sigma$ are given in Appendix C. With $\tilde{K}_{n}^{W}$
given in Appendix C, we can choose a suitable running coupling constant
$\alpha_{s}(p^{2})$, solve the Schwinger-Dyson equation numerically, obtaining
the quark self energy $\Sigma$, and then calculate the numerical values of all
order $p^{6}$ anomalous LECs. To obtain the final numerical result, we have
assumed $F_{0}=87$MeV as input to fix the dimensional parameter
$\Lambda_{\mathrm{QCD}}$ appearing in the running coupling constant
$\alpha_{s}(p^{2})$ and taken momentum cutoff
$\Lambda=1.00^{+0.10}_{-0.10}$GeV. Because of the appearance of the divergent
order $p^{2}$ LEC $B_{0}$ in Eqs.(23) and (24), we need a momentum cutoff
$\Lambda$ to make $B_{0}$ finite as we did previously in Ref.WQ4 . In Table I,
we give the numerical values for all 21 nonzero LECs for three
flavors($C_{3}^{W}=C_{18}^{W}=0$ in our formulation).
Combined with our numerical result, we also list the numerical estimates for
some of the LECs from five different models and different processes given in
Ref.p6anomLEC ,p6anomLEC1 ,p6anomLEC2 ,p6anomLEC3 and p6anomLEC4 . In
Ref.p6anomLEC , model I and III are all from direct chiral perturbation(ChPT)
computations, except that model I is the full ChPT result, while in model III,
low energy experiment data are extrapolated to the high energy region; model
II is the vector meson dominance model (VMD); model IV and V are the chiral
constituent quark model (CQM) with some extrapolations included in model V.
For a fixed model, different processes may give different results. For
example, in model I for $C_{7}^{W}$ and models I and IV for $C_{22}^{W}$, we
all obtain two results from two different processes. Further, Ref.p6anomLEC
,p6anomLEC3 and p6anomLEC4 also give estimations on some combinations or
ratios of LECs. We list our and their results in Table II. For $N_{f}=2$, in
Table III, we give the numerical values of all 12 nonzero LECs ($c_{12}^{W}=0$
in our formulation) which are actually of the very same structure as that
given by anom1 .
TABLE I. The nonzero values of the order $p^{6}$ anomalous LECs
$C_{1}^{W},C_{2}^{W},C_{4}^{W},\ldots,C_{17}^{W},C_{19}^{W},\ldots,C_{23}^{W}$
for three flavors. The LECs are in units of $10^{-3}\mathrm{GeV}^{-2}$. The
2nd column is our result LECs with the values at $\Lambda=1$GeV with
superscript the difference caused at $\Lambda=1.1$GeV (i.e.
$C_{i}^{W}\big{|}_{\Lambda=1.1\mathrm{GeV}}-C_{i}^{W}\big{|}_{\Lambda=1\mathrm{GeV}}$)
and subscript the difference caused at $\Lambda=0.9$GeV (i.e.
$C_{i}^{W}\big{|}_{\Lambda=0.9\mathrm{GeV}}-C_{i}^{W}\big{|}_{\Lambda=1\mathrm{GeV}}$).
The 3rd to 7th columns are results given in Ref.p6anomLEC : (I)–ChPT,
(II)–VMD, (III)–ChPT(extrapolation), (IV)–CQM, (V)–CQM(extrapolation). The 8th
column shows results from Ref.p6anomLEC1 ,p6anomLEC2 ,p6anomLEC3 ,p6anomLEC4 .
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|}\hline\cr
n&C^{W}_{n}~{}\mbox{ours}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}(I)}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}(II)}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}(III)}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}(IV)}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}(V)}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC1}{\@@citephrase{(}}{\@@citephrase{)}}},~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC2}{\@@citephrase{(}}{\@@citephrase{)}}},~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC3}{\@@citephrase{(}}{\@@citephrase{)}}},~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC4}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr 1&4.97^{+0.55}_{-0.79}&&&&&&\\\ 2&-1.43^{+0.10}_{-0.12}&-0.32\pm
10.4&&0.78\pm 12.7&4.96\pm 9.70&-0.074\pm 13.3&\\\
4&-0.96^{+0.22}_{-0.29}&0.28\pm 9.19&&0.67\pm 10.9&6.32\pm 6.09&-0.55\pm
9.05&\\\ 5&3.26^{+0.34}_{-0.49}&28.50\pm 28.83&&9.38\pm 152.2&33.05\pm
28.66&34.51\pm 41.13&\\\ 6&0.91^{+0.03}_{-0.04}&&&&&&\\\
7&1.68^{-0.24}_{+0.31}&0.013\pm 1.17&&&0.51\pm 0.06&&0.1\pm 1.2\\\ &&20.3\pm
18.7&&&&&0.1^{*}\\\ 8&0.41^{+0.01}_{-0.02}&0.76\pm 0.18&&&&&0.58\pm 0.20\\\
&&&&&&&0.5^{*}\\\ 9&1.15^{-0.03}_{+0.03}&&&&&&\\\
10&-0.18^{-0.01}_{+0.01}&&&&&&\\\ 11&-1.15^{+0.08}_{-0.10}&-6.37\pm
4.54&&&-0.00143\pm 0.03&&0.68\pm 0.21\\\ 12&-5.13^{-0.15}_{+0.25}&&&&&&\\\
13&-6.37^{-0.18}_{+0.31}&-74.09\pm 55.89&-20.00&-8.44\pm 69.9&14.15\pm
15.22&-7.46\pm 19.62&\\\ 14&-2.00^{-0.06}_{+0.10}&29.99\pm 11.14&-6.01&0.72\pm
15.3&10.23\pm 7.56&-0.58\pm 9.77&\\\ 15&4.17^{+0.12}_{-0.20}&-25.30\pm
23.93&2.00&-3.10\pm 28.6&19.70\pm 7.49&8.89\pm 9.72&\\\
16&3.58^{+0.10}_{-0.17}&&&&&&\\\ 17&1.98^{+0.06}_{-0.10}&&&&&&\\\
19&0.29^{+0.01}_{-0.01}&&&&&&\\\ 20&1.83^{+0.05}_{-0.09}&&&&&&\\\
21&2.48^{+0.07}_{-0.12}&&&&&&\\\ 22&5.01^{+0.14}_{-0.24}&6.52\pm
0.78&8.01&&3.94\pm 0.43&&5.4\pm 0.8\\\ &&5.07\pm 0.71&&&3.94\pm 0.43&&\\\
23&2.74^{+0.08}_{-0.13}&&&&&&\\\ \hline\cr\end{array}$ (57)
∗ This result is just the absolute value given in Ref.p6anomLEC3 .
TABLE II. Some combinations or ratios of LECs in units of
$10^{-3}\mathrm{GeV}^{-2}$. The 2nd column is our result LECs with the values
at $\Lambda=1$GeV, and with superscript the difference caused at
$\Lambda=1.1$GeV and subscript the difference caused at $\Lambda=0.9$GeV. The
3rd to 5th columns are results given in Ref.p6anomLEC : (I)–ChPT, (II)–VMD,
(III)–ChPT (extrapolation), (IV)–CQM, (V)–CQM (extrapolation). The 6th and 7th
columns are results given in Ref.p6anomLEC3 and p6anomLEC4 respectively.
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr&\mbox{ours}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC3}{\@@citephrase{(}}{\@@citephrase{)}}}}&\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{p6anomLEC4}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr C^{W}_{3}-C^{W}_{6}&-0.91^{-0.03}_{+0.04}&21.67\pm
17.41~{}\mbox{(I)}&5.07\pm 5.07~{}\mbox{(IV)}&-2.14\pm 6.54~{}\mbox{(V)}&&\\\
2C^{W}_{15}-4C^{W}_{14}+C^{W}_{13}&9.95^{+0.29}_{-0.48}&-244.7\pm
148.4~{}\mbox{(I)}&\approx 8.0~{}\mbox{(II)}&-17.52\pm
188.3~{}\mbox{(III)}&&\\\ 2C^{W}_{14}-C^{W}_{13}&2.38^{+0.07}_{-0.12}&134.1\pm
78.17~{}\mbox{(I)}&\approx 8.0~{}\mbox{(II)}&9.88\pm 100.5~{}\mbox{(III)}&&\\\
|C_{7}^{W}|/|C_{8}^{W}|&4.12^{-0.69}_{+1.01}&&&&0.2&<0.1\\\
\hline\cr\end{array}$ (63) TABLE III. The nonzero values of the $p^{6}$ order
anomalous LECs $c_{1}^{W},\ldots,c_{11}^{W},c_{13}^{W}$ for two flavor in
units of $10^{-3}\mathrm{GeV}^{-2}$.
$\displaystyle{\footnotesize\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
c_{1}^{W}&c_{2}^{W}&c_{3}^{W}&c_{4}^{W}&c_{5}^{W}&c_{6}^{W}&c_{7}^{W}&c_{8}^{W}&c_{9}^{W}&c_{10}^{W}&c_{11}^{W}&c_{13}^{W}\\\
\hline\cr-1.46^{+0.10}_{-0.12}&-1.25^{+0.09}_{-0.11}&2.96^{-0.20}_{+0.25}&0.63^{-0.04}_{+0.05}&-1.17^{+0.08}_{-0.10}&0.77^{+0.26}_{-0.36}&-0.04^{-0.00}_{+0.00}&0.02^{+0.00}_{-0.00}&8.19^{+0.23}_{-0.38}&-8.73^{-0.24}_{+0.41}&4.85^{+0.13}_{-0.23}&-9.70^{-0.27}_{+0.45}\\\
\hline\cr\end{array}}$ (66)
We see that most of our results are consistent with those we have found in the
literature.
As a phenomenological check for two flavor anomalous LECs, we discuss the
$\pi^{0}\rightarrow\gamma\gamma$ process. Ref.p6anomLEC4 gives the amplitude
of this process by
$\displaystyle T_{\mathrm{LO+NLO}}$ $\displaystyle=$
$\displaystyle\frac{1}{F}\bigg{\\{}\frac{1}{4\pi^{2}}+\frac{16}{3}m_{\pi}^{2}(-4c_{3}^{Wr}-4c_{7}^{Wr}+c_{11}^{Wr})+\frac{64}{9}B(m_{d}-m_{u})(5c_{3}^{Wr}+c_{r}^{Wr}+2c_{8}^{Wr})\bigg{\\}}\;.$
(67)
In our calculation, we choose the center value
$B(m_{d}-m_{u})=0.32m^{2}_{\pi^{0}}$ given in Ref.p6anomLEC4 . Experimentally,
the $\pi^{0}\rightarrow\gamma\gamma$ process dominates the life time of
$\pi^{0}$ to $98.79\%$, and if we ignore that small fraction from other
processes, then the life time of $\pi^{0}$ can be expressed in terms of
amplitude $T$ as $1/\tau=\pi\alpha m_{\pi}^{3}T^{2}/4$. In Table IV we give
our result for $\tau_{\mathrm{LO}}$ up to the leading order $p^{4}$, which
corresponds to the first term of the r.h.s of Eq.(67), and
$\tau_{\mathrm{NLO}}$ up to the next leading order $p^{6}$ of the low energy
expansion. Experimental result from particle data groupPDG is also included
in the table for comparison.
TABLE IV. $\pi^{0}$ life time in units of $10^{-17}\mathrm{s}$.
$\displaystyle\begin{array}[]{|c|c|c|}\hline\cr&\tau_{\rm LO}&\tau_{\rm
NLO}\\\ \hline\cr F=87\mathrm{MeV}&7.56&7.59^{-0.03}_{+0.04}\\\
F=93\mathrm{MeV}&8.63&8.67^{-0.03}_{+0.04}\\\
\hline\cr\mbox{Exp.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}&\lx@intercol\hfil
8.4\pm 0.6\hfil\lx@intercol\vline\\\ \hline\cr\end{array}$ (72)
Our result roughly matches the experimental value and we see that the order
$p^{6}$ results have less effect on the life time of $\pi^{0}$.
## IV Summary and Future Work
In this work, we review the general anomaly structure of the effective chiral
Lagrangian and then generalize our order $p^{6}$ calculation in Ref.WQ4 from
the normal part to the anomalous part of the chiral Lagrangian for
pseudoscalar mesons. The result is obtained by computing the imaginary
$\Sigma$ dependent part of
Tr$\ln[\not{\partial}+J_{\Omega}+\Sigma(-\bar{\nabla}^{2})]$. To match the
calculation of the order $p^{4}$ anomalous part, in practice we calculate the
integration of parameter $t$ over
$\frac{d}{dt}\mathrm{Tr}\ln[i\not{\partial}+J_{\Omega(t)}+\Sigma(-\nabla_{t}^{2})]$.
The conventional chiral Lagrangian is also reformulated to an integration of
$t$ and through comparison of it with our result, we read out all order
$p^{6}$ anomalous LECs expressed in terms of quark self energy $\Sigma$.
Inputting the SDE solution of $\Sigma(k^{2})$, we obtain numerical values and
compare them with those we can find in literature. Some of them are
consistent, some are not. We leave those inconsistent results to future
investigations. Combined with the previous result on the order $p^{6}$ normal
LECs given in Ref.WQ4 , we have now completed all the order $p^{6}$ LECs
computations. Based on them, one direction of the further research is to apply
the order $p^{6}$ chiral Lagrangian to various pseudoscalar meson processes
and discuss the corresponding physics. Another direction is to improve the
precision of (7) and our ladder approximation SDE. With these improvements, we
expect a more precise estimation on all LECs in future.
## Acknowledgments
This work was supported by National Science Foundation of China (NSFC) under
Grant No.10875065.
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## Appendix A List of All Operators $\bar{O}^{\mu\nu\lambda\rho}_{n}$ and
$\tilde{O}^{W}_{n}$
In this appendix, we first explicitly write down all 210
$\bar{O}^{\mu\nu\lambda\rho}_{n}$ operators. To save the space, we use some
simplified symbols to represent the original symbols in the text. Our
$\bar{O}^{\mu\nu\lambda\rho}_{n}$s are constructed in such a way that they are
invariant under charge conjugation transformation. This causes the result that
most of $\bar{O}^{\mu\nu\lambda\rho}_{n}$s consist of two terms which are
charge conjugates to each other.
$\displaystyle\mbox{\small\bf TABLE
V.}~{}~{}~{}~{}~{}~{}~{}\mu\equiv\nabla_{t}^{\mu},~{}~{}\nu\equiv\nabla_{t}^{\nu},~{}~{}\lambda\equiv\nabla_{t}^{\lambda},~{}~{}\rho\equiv\nabla_{t}^{\rho},~{}~{}\bar{\mu}\equiv
a_{t}^{\mu},~{}~{}\bar{\nu}\equiv a_{t}^{\nu},~{}~{}\bar{\lambda}\equiv
a_{t}^{\lambda},~{}~{}\bar{\rho}\equiv a_{t}^{\rho},~{}~{}s\equiv
s_{t},~{}~{}p\equiv p_{t}$
$\displaystyle\cdots\sigma\cdots\sigma\equiv\cdots\nabla_{t}^{\sigma}\cdots\nabla_{t,\sigma},~{}~{}\cdots\bar{\sigma}\cdots\bar{\sigma}\equiv\cdots
a_{t}^{\sigma}\cdots
a_{t,\sigma},~{}~{}\cdots\sigma\cdots\bar{\sigma}\equiv\cdots\nabla_{t}^{\sigma}\cdots
a_{t,\sigma},~{}~{}\cdots\bar{\sigma}\cdots\sigma\equiv\cdots
a_{t}^{\sigma}\cdots\nabla_{t,\sigma}$ $\displaystyle\hskip
14.22636pt{\scriptsize\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|}\hline\cr
n&\bar{O}^{\mu\nu\lambda\rho}_{n}&n&\bar{O}^{\mu\nu\lambda\rho}_{n}&n&\bar{O}^{\mu\nu\lambda\rho}_{n}&n&\bar{O}^{\mu\nu\lambda\rho}_{n}&n&\bar{O}^{\mu\nu\lambda\rho}_{n}\\\
\hline\cr 1&s\mu\nu\lambda\rho+\mu\nu\lambda\rho
s&43&\mu\nu\lambda\rho\sigma\sigma+\sigma\sigma\mu\nu\lambda\rho&85&\mu\sigma\bar{\nu}\lambda\rho\bar{\sigma}+\bar{\sigma}\mu\nu\bar{\lambda}\sigma\rho&127&\mu\nu\bar{\lambda}\bar{\rho}\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\bar{\mu}\bar{\nu}\lambda\rho&169&\mu\bar{\nu}\bar{\lambda}\bar{\sigma}\bar{\sigma}\rho+\mu\bar{\sigma}\bar{\sigma}\bar{\nu}\bar{\lambda}\rho\\\
2&\mu s\nu\lambda\rho+\mu\nu\lambda
s\rho&44&\mu\nu\lambda\sigma\rho\sigma+\sigma\mu\sigma\nu\lambda\rho&86&\mu\sigma\bar{\nu}\lambda\sigma\bar{\rho}+\bar{\mu}\sigma\nu\bar{\lambda}\sigma\rho&128&\mu\nu\bar{\lambda}\bar{\sigma}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\sigma}\bar{\nu}\lambda\rho&170&\mu\bar{\nu}\bar{\sigma}\bar{\lambda}\bar{\rho}\sigma+\sigma\bar{\mu}\bar{\nu}\bar{\sigma}\bar{\lambda}\rho\\\
3&\mu\nu
s\lambda\rho&45&\mu\nu\lambda\sigma\sigma\rho+\mu\sigma\sigma\nu\lambda\rho&87&\mu\sigma\bar{\nu}\sigma\lambda\bar{\rho}+\bar{\mu}\nu\sigma\bar{\lambda}\sigma\rho&129&\mu\nu\bar{\lambda}\bar{\sigma}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\sigma}\bar{\nu}\lambda\rho&171&\mu\bar{\nu}\bar{\sigma}\bar{\lambda}\bar{\sigma}\rho+\mu\bar{\sigma}\bar{\nu}\bar{\sigma}\bar{\lambda}\rho\\\
4&s\mu\nu\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\lambda\rho
s&46&\mu\nu\sigma\lambda\rho\sigma+\sigma\mu\nu\sigma\lambda\rho&88&\mu\sigma\bar{\sigma}\nu\lambda\bar{\rho}+\bar{\mu}\nu\lambda\bar{\sigma}\sigma\rho&130&\mu\nu\bar{\sigma}\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\bar{\sigma}\lambda\rho&172&\mu\bar{\sigma}\bar{\nu}\bar{\lambda}\bar{\rho}\sigma+\sigma\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\sigma}\rho\\\
5&s\mu\bar{\nu}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\lambda}\rho
s&47&\mu\nu\sigma\lambda\sigma\rho+\mu\sigma\nu\sigma\lambda\rho&89&\mu\nu\bar{\lambda}\rho\bar{\sigma}\sigma+\sigma\bar{\sigma}\mu\bar{\nu}\lambda\rho&131&\mu\nu\bar{\sigma}\bar{\lambda}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\nu}\bar{\sigma}\lambda\rho&173&\bar{\mu}\nu\lambda\bar{\rho}\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\bar{\mu}\nu\lambda\bar{\rho}\\\
6&s\mu\bar{\nu}\bar{\lambda}\rho+\mu\bar{\nu}\bar{\lambda}\rho
s&48&\mu\sigma\nu\lambda\rho\sigma+\sigma\mu\nu\lambda\sigma\rho&90&\mu\nu\bar{\lambda}\sigma\bar{\rho}\sigma+\sigma\bar{\mu}\sigma\bar{\nu}\lambda\rho&132&\mu\nu\bar{\sigma}\bar{\sigma}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\sigma}\bar{\sigma}\lambda\rho&174&\bar{\mu}\nu\lambda\bar{\sigma}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\sigma}\nu\lambda\bar{\rho}\\\
7&s\bar{\mu}\nu\lambda\bar{\rho}+\bar{\mu}\nu\lambda\bar{\rho}s&49&\mu\nu\lambda\rho\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\mu\nu\lambda\rho&91&\mu\nu\bar{\lambda}\sigma\bar{\sigma}\rho+\mu\bar{\sigma}\sigma\bar{\nu}\lambda\rho&133&\mu\sigma\bar{\nu}\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\bar{\lambda}\sigma\rho&175&\bar{\mu}\nu\lambda\bar{\sigma}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\sigma}\nu\lambda\bar{\rho}\\\
8&s\bar{\mu}\nu\bar{\lambda}\rho+\mu\bar{\nu}\lambda\bar{\rho}s&50&\mu\nu\lambda\sigma\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\sigma\nu\lambda\rho&92&\mu\nu\bar{\sigma}\lambda\bar{\rho}\sigma+\sigma\bar{\mu}\nu\bar{\sigma}\lambda\rho&134&\mu\sigma\bar{\nu}\bar{\lambda}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\nu}\bar{\lambda}\sigma\rho&176&\bar{\mu}\nu\sigma\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\sigma\lambda\bar{\rho}\\\
9&s\bar{\mu}\bar{\nu}\lambda\rho+\mu\nu\bar{\lambda}\bar{\rho}s&51&\mu\nu\lambda\sigma\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\sigma\nu\lambda\rho&93&\mu\nu\bar{\sigma}\lambda\bar{\sigma}\rho+\mu\bar{\sigma}\nu\bar{\sigma}\lambda\rho&135&\mu\sigma\bar{\nu}\bar{\sigma}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\sigma}\bar{\lambda}\sigma\rho&177&\bar{\mu}\nu\sigma\bar{\lambda}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\nu}\sigma\lambda\bar{\rho}\\\
10&\mu s\nu\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\lambda
s\rho&52&\mu\nu\sigma\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\sigma\lambda\rho&94&\mu\nu\bar{\sigma}\sigma\bar{\lambda}\rho+\mu\bar{\nu}\sigma\bar{\sigma}\lambda\rho&136&\mu\sigma\bar{\sigma}\bar{\nu}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\sigma}\sigma\rho&178&\bar{\mu}\nu\sigma\bar{\sigma}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\sigma}\sigma\lambda\bar{\rho}\\\
11&\mu
s\bar{\nu}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\lambda}s\rho&53&\mu\nu\sigma\lambda\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\nu\sigma\lambda\rho&95&\mu\sigma\bar{\nu}\lambda\bar{\rho}\sigma+\sigma\bar{\mu}\nu\bar{\lambda}\sigma\rho&137&\mu\bar{\nu}\lambda\bar{\rho}\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\bar{\mu}\nu\bar{\lambda}\rho&179&\bar{\mu}\sigma\nu\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\lambda\sigma\bar{\rho}\\\
12&\mu
s\bar{\nu}\bar{\lambda}\rho+\mu\bar{\nu}\bar{\lambda}s\rho&54&\mu\nu\sigma\sigma\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\sigma\sigma\lambda\rho&96&\mu\sigma\bar{\nu}\lambda\bar{\sigma}\rho+\mu\bar{\sigma}\nu\bar{\lambda}\sigma\rho&138&\mu\bar{\nu}\lambda\bar{\sigma}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\sigma}\nu\bar{\lambda}\rho&180&\bar{\mu}\sigma\nu\bar{\lambda}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\nu}\lambda\sigma\bar{\rho}\\\
13&\bar{\mu}s\nu\lambda\bar{\rho}+\bar{\mu}\nu\lambda
s\bar{\rho}&55&\mu\sigma\nu\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\lambda\sigma\rho&97&\mu\sigma\bar{\nu}\sigma\bar{\lambda}\rho+\mu\bar{\nu}\sigma\bar{\lambda}\sigma\rho&139&\mu\bar{\nu}\lambda\bar{\sigma}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\sigma}\nu\bar{\lambda}\rho&181&\bar{\mu}\sigma\nu\bar{\sigma}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\sigma}\lambda\sigma\bar{\rho}\\\
14&\bar{\mu}s\nu\bar{\lambda}\rho+\mu\bar{\nu}\lambda
s\bar{\rho}&56&\mu\sigma\nu\lambda\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\nu\lambda\sigma\rho&98&\mu\sigma\bar{\sigma}\nu\bar{\lambda}\rho+\mu\bar{\nu}\lambda\bar{\sigma}\sigma\rho&140&\mu\bar{\nu}\sigma\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\sigma\bar{\lambda}\rho&182&\bar{\mu}\sigma\sigma\bar{\nu}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\lambda}\sigma\sigma\bar{\rho}\\\
15&\bar{\mu}s\bar{\nu}\lambda\rho+\mu\nu\bar{\lambda}s\bar{\rho}&57&\mu\sigma\nu\sigma\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\sigma\lambda\sigma\rho&99&\mu\nu\bar{\lambda}\bar{\rho}\sigma\sigma+\sigma\sigma\bar{\mu}\bar{\nu}\lambda\rho&141&\mu\bar{\nu}\sigma\bar{\lambda}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\nu}\sigma\bar{\lambda}\rho&183&\bar{\mu}\nu\bar{\lambda}\rho\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\mu\bar{\nu}\lambda\bar{\rho}\\\
16&\mu\nu
s\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}s\lambda\rho&58&\mu\sigma\sigma\nu\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\lambda\sigma\sigma\rho&100&\mu\nu\bar{\lambda}\bar{\sigma}\rho\sigma+\sigma\mu\bar{\sigma}\bar{\nu}\lambda\rho&142&\mu\bar{\nu}\sigma\bar{\sigma}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\sigma}\sigma\bar{\lambda}\rho&184&\bar{\mu}\nu\bar{\lambda}\sigma\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\sigma\bar{\nu}\lambda\bar{\rho}\\\
17&\mu\bar{\nu}s\lambda\bar{\rho}+\bar{\mu}\nu
s\bar{\lambda}\rho&59&\mu\nu\lambda\bar{\rho}\sigma\bar{\sigma}+\bar{\sigma}\sigma\bar{\mu}\nu\lambda\rho&101&\mu\nu\bar{\lambda}\bar{\sigma}\sigma\rho+\mu\sigma\bar{\sigma}\bar{\nu}\lambda\rho&143&\mu\bar{\sigma}\nu\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\lambda\bar{\sigma}\rho&185&\bar{\mu}\nu\bar{\lambda}\sigma\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\sigma\bar{\nu}\lambda\bar{\rho}\\\
18&\mu\bar{\nu}s\bar{\lambda}\rho&60&\mu\nu\lambda\bar{\sigma}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\sigma}\nu\lambda\rho&102&\mu\nu\bar{\sigma}\bar{\lambda}\rho\sigma+\sigma\mu\bar{\nu}\bar{\sigma}\lambda\rho&144&\mu\bar{\sigma}\nu\bar{\lambda}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\nu}\lambda\bar{\sigma}\rho&186&\bar{\mu}\nu\bar{\sigma}\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\bar{\sigma}\lambda\bar{\rho}\\\
19&\bar{\mu}\nu
s\lambda\bar{\rho}&61&\mu\nu\lambda\bar{\sigma}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\sigma}\nu\lambda\rho&103&\mu\nu\bar{\sigma}\bar{\lambda}\sigma\rho+\mu\sigma\bar{\nu}\bar{\sigma}\lambda\rho&145&\mu\bar{\sigma}\nu\bar{\sigma}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\sigma}\lambda\bar{\sigma}\rho&187&\bar{\mu}\nu\bar{\sigma}\lambda\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\nu\bar{\sigma}\lambda\bar{\rho}\\\
20&s\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\rho}s&62&\mu\nu\sigma\bar{\lambda}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\nu}\sigma\lambda\rho&104&\mu\sigma\bar{\nu}\bar{\lambda}\rho\sigma+\sigma\mu\bar{\nu}\bar{\lambda}\sigma\rho&146&\mu\bar{\sigma}\sigma\bar{\nu}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\lambda}\sigma\bar{\sigma}\rho&188&\bar{\mu}\nu\bar{\sigma}\sigma\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\sigma\bar{\sigma}\lambda\bar{\rho}\\\
21&\bar{\mu}s\bar{\nu}\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\bar{\lambda}s\bar{\rho}&63&\mu\nu\sigma\bar{\lambda}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\nu}\sigma\lambda\rho&105&\mu\bar{\nu}\lambda\rho\sigma\bar{\sigma}+\bar{\sigma}\sigma\mu\nu\bar{\lambda}\rho&147&\mu\bar{\nu}\bar{\lambda}\rho\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\mu\bar{\nu}\bar{\lambda}\rho&189&\bar{\mu}\sigma\bar{\nu}\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\bar{\lambda}\sigma\bar{\rho}\\\
22&\bar{\mu}\bar{\nu}s\bar{\lambda}\bar{\rho}&64&\mu\nu\sigma\bar{\sigma}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\sigma}\sigma\lambda\rho&106&\mu\bar{\nu}\lambda\sigma\rho\bar{\sigma}+\bar{\sigma}\mu\sigma\nu\bar{\lambda}\rho&148&\mu\bar{\nu}\bar{\lambda}\sigma\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\sigma\bar{\nu}\bar{\lambda}\rho&190&\bar{\mu}\sigma\bar{\nu}\lambda\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\nu\bar{\lambda}\sigma\bar{\rho}\\\
23&p\mu\nu\lambda\bar{\rho}-\bar{\mu}\nu\lambda\rho
p&65&\mu\sigma\nu\bar{\lambda}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\nu}\lambda\sigma\rho&107&\mu\bar{\nu}\lambda\sigma\sigma\bar{\rho}+\bar{\mu}\sigma\sigma\nu\bar{\lambda}\rho&149&\mu\bar{\nu}\bar{\lambda}\sigma\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\sigma\bar{\nu}\bar{\lambda}\rho&191&\bar{\mu}\sigma\bar{\nu}\sigma\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\sigma\bar{\lambda}\sigma\bar{\rho}\\\
24&p\mu\nu\bar{\lambda}\rho-\mu\bar{\nu}\lambda\rho
p&66&\mu\sigma\nu\bar{\lambda}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\nu}\lambda\sigma\rho&108&\mu\bar{\nu}\sigma\lambda\rho\bar{\sigma}+\bar{\sigma}\mu\nu\sigma\bar{\lambda}\rho&150&\mu\bar{\nu}\bar{\sigma}\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\bar{\sigma}\bar{\lambda}\rho&192&\bar{\mu}\sigma\bar{\sigma}\nu\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\lambda\bar{\sigma}\sigma\bar{\rho}\\\
25&p\mu\bar{\nu}\lambda\rho-\mu\nu\bar{\lambda}\rho
p&67&\mu\sigma\nu\bar{\sigma}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\sigma}\lambda\sigma\rho&109&\mu\bar{\nu}\sigma\lambda\sigma\bar{\rho}+\bar{\mu}\sigma\nu\sigma\bar{\lambda}\rho&151&\mu\bar{\nu}\bar{\sigma}\lambda\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\nu\bar{\sigma}\bar{\lambda}\rho&193&\bar{\mu}\nu\bar{\lambda}\bar{\rho}\sigma\bar{\sigma}+\bar{\sigma}\sigma\bar{\mu}\bar{\nu}\lambda\bar{\rho}\\\
26&p\bar{\mu}\nu\lambda\rho-\mu\nu\lambda\bar{\rho}p&68&\mu\sigma\sigma\bar{\nu}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\lambda}\sigma\sigma\rho&110&\mu\bar{\nu}\sigma\sigma\lambda\bar{\rho}+\bar{\mu}\nu\sigma\sigma\bar{\lambda}\rho&152&\mu\bar{\nu}\bar{\sigma}\sigma\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\sigma\bar{\sigma}\bar{\lambda}\rho&194&\bar{\mu}\nu\bar{\lambda}\bar{\sigma}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\sigma}\bar{\nu}\lambda\bar{\rho}\\\
27&\mu p\nu\lambda\bar{\rho}-\bar{\mu}\nu\lambda
p\rho&69&\mu\nu\lambda\bar{\rho}\bar{\sigma}\sigma+\sigma\bar{\sigma}\bar{\mu}\nu\lambda\rho&111&\mu\bar{\sigma}\nu\lambda\rho\bar{\sigma}+\bar{\sigma}\mu\nu\lambda\bar{\sigma}\rho&153&\mu\bar{\sigma}\bar{\nu}\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\bar{\lambda}\bar{\sigma}\rho&195&\bar{\mu}\nu\bar{\lambda}\bar{\sigma}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\sigma}\bar{\nu}\lambda\bar{\rho}\\\
28&\mu p\nu\bar{\lambda}\rho-\mu\bar{\nu}\lambda
p\rho&70&\mu\nu\lambda\bar{\sigma}\bar{\rho}\sigma+\sigma\bar{\mu}\bar{\sigma}\nu\lambda\rho&112&\mu\bar{\sigma}\nu\lambda\sigma\bar{\rho}+\bar{\mu}\sigma\nu\lambda\bar{\sigma}\rho&154&\mu\bar{\sigma}\bar{\nu}\lambda\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\nu\bar{\lambda}\bar{\sigma}\rho&196&\bar{\mu}\nu\bar{\sigma}\bar{\lambda}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\nu}\bar{\sigma}\lambda\bar{\rho}\\\
29&\mu
p\bar{\nu}\lambda\rho-\mu\nu\bar{\lambda}p\rho&71&\mu\nu\lambda\bar{\sigma}\bar{\sigma}\rho+\mu\bar{\sigma}\bar{\sigma}\nu\lambda\rho&113&\mu\bar{\sigma}\nu\sigma\lambda\bar{\rho}+\bar{\mu}\nu\sigma\lambda\bar{\sigma}\rho&155&\mu\bar{\sigma}\bar{\nu}\sigma\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\sigma\bar{\lambda}\bar{\sigma}\rho&197&\bar{\mu}\nu\bar{\sigma}\bar{\lambda}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\nu}\bar{\sigma}\lambda\bar{\rho}\\\
30&\bar{\mu}p\nu\lambda\rho-\mu\nu\lambda
p\bar{\rho}&72&\mu\nu\sigma\bar{\lambda}\bar{\rho}\sigma+\sigma\bar{\mu}\bar{\nu}\sigma\lambda\rho&114&\mu\bar{\sigma}\sigma\nu\lambda\bar{\rho}+\bar{\mu}\nu\lambda\sigma\bar{\sigma}\rho&156&\mu\bar{\sigma}\bar{\sigma}\nu\bar{\lambda}\bar{\rho}+\bar{\mu}\bar{\nu}\lambda\bar{\sigma}\bar{\sigma}\rho&198&\bar{\mu}\sigma\bar{\nu}\bar{\lambda}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\nu}\bar{\lambda}\sigma\bar{\rho}\\\
31&\mu\nu p\lambda\bar{\rho}-\bar{\mu}\nu
p\lambda\rho&73&\mu\nu\sigma\bar{\lambda}\bar{\sigma}\rho+\mu\bar{\sigma}\bar{\nu}\sigma\lambda\rho&115&\mu\bar{\nu}\lambda\rho\bar{\sigma}\sigma+\sigma\bar{\sigma}\mu\nu\bar{\lambda}\rho&157&\mu\bar{\nu}\bar{\lambda}\bar{\rho}\sigma\bar{\sigma}+\bar{\sigma}\sigma\bar{\mu}\bar{\nu}\bar{\lambda}\rho&199&\bar{\mu}\bar{\nu}\lambda\rho\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\mu\nu\bar{\lambda}\bar{\rho}\\\
32&\mu\nu
p\bar{\lambda}\rho-\mu\bar{\nu}p\lambda\rho&74&\mu\nu\sigma\bar{\sigma}\bar{\lambda}\rho+\mu\bar{\nu}\bar{\sigma}\sigma\lambda\rho&116&\mu\bar{\nu}\lambda\sigma\bar{\rho}\sigma+\sigma\bar{\mu}\sigma\nu\bar{\lambda}\rho&158&\mu\bar{\nu}\bar{\lambda}\bar{\sigma}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\sigma}\bar{\nu}\bar{\lambda}\rho&200&\bar{\mu}\bar{\nu}\lambda\sigma\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\sigma\nu\bar{\lambda}\bar{\rho}\\\
33&p\mu\bar{\nu}\bar{\lambda}\bar{\rho}-\bar{\mu}\bar{\nu}\bar{\lambda}\rho
p&75&\mu\sigma\nu\bar{\lambda}\bar{\rho}\sigma+\sigma\bar{\mu}\bar{\nu}\lambda\sigma\rho&117&\mu\bar{\nu}\lambda\sigma\bar{\sigma}\rho+\mu\bar{\sigma}\sigma\nu\bar{\lambda}\rho&159&\mu\bar{\nu}\bar{\lambda}\bar{\sigma}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\sigma}\bar{\nu}\bar{\lambda}\rho&201&\bar{\mu}\bar{\nu}\lambda\sigma\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\sigma\nu\bar{\lambda}\bar{\rho}\\\
34&p\bar{\mu}\nu\bar{\lambda}\bar{\rho}-\bar{\mu}\bar{\nu}\lambda\bar{\rho}p&76&\mu\sigma\nu\bar{\lambda}\bar{\sigma}\rho+\mu\bar{\sigma}\bar{\nu}\lambda\sigma\rho&118&\mu\bar{\nu}\sigma\lambda\bar{\rho}\sigma+\sigma\bar{\mu}\nu\sigma\bar{\lambda}\rho&160&\mu\bar{\nu}\bar{\sigma}\bar{\lambda}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\nu}\bar{\sigma}\bar{\lambda}\rho&202&\bar{\mu}\bar{\nu}\sigma\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\sigma\bar{\lambda}\bar{\rho}\\\
35&p\bar{\mu}\bar{\nu}\lambda\bar{\rho}-\bar{\mu}\nu\bar{\lambda}\bar{\rho}p&77&\mu\sigma\nu\bar{\sigma}\bar{\lambda}\rho+\mu\bar{\nu}\bar{\sigma}\lambda\sigma\rho&119&\mu\bar{\nu}\sigma\lambda\bar{\sigma}\rho+\mu\bar{\sigma}\nu\sigma\bar{\lambda}\rho&161&\mu\bar{\nu}\bar{\sigma}\bar{\lambda}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\nu}\bar{\sigma}\bar{\lambda}\rho&203&\bar{\mu}\bar{\nu}\sigma\lambda\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\nu\sigma\bar{\lambda}\bar{\rho}\\\
36&p\bar{\mu}\bar{\nu}\bar{\lambda}\rho-\mu\bar{\nu}\bar{\lambda}\bar{\rho}p&78&\mu\sigma\sigma\bar{\nu}\bar{\lambda}\rho+\mu\bar{\nu}\bar{\lambda}\sigma\sigma\rho&120&\mu\bar{\sigma}\nu\lambda\bar{\rho}\sigma+\sigma\bar{\mu}\nu\lambda\bar{\sigma}\rho&162&\mu\bar{\nu}\bar{\sigma}\bar{\sigma}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\sigma}\bar{\sigma}\bar{\lambda}\rho&204&\bar{\mu}\bar{\sigma}\nu\lambda\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\nu\lambda\bar{\sigma}\bar{\rho}\\\
37&\mu
p\bar{\nu}\bar{\lambda}\bar{\rho}-\bar{\mu}\bar{\nu}\bar{\lambda}p\rho&79&\mu\nu\bar{\lambda}\rho\sigma\bar{\sigma}+\bar{\sigma}\sigma\mu\bar{\nu}\lambda\rho&121&\bar{\mu}\nu\lambda\rho\sigma\bar{\sigma}+\bar{\sigma}\sigma\mu\nu\lambda\bar{\rho}&163&\mu\bar{\sigma}\bar{\nu}\bar{\lambda}\rho\bar{\sigma}+\bar{\sigma}\mu\bar{\nu}\bar{\lambda}\bar{\sigma}\rho&205&\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\rho}\bar{\sigma}\bar{\sigma}+\bar{\sigma}\bar{\sigma}\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\rho}\\\
38&\bar{\mu}p\nu\bar{\lambda}\bar{\rho}-\bar{\mu}\bar{\nu}\lambda
p\bar{\rho}&80&\mu\nu\bar{\lambda}\sigma\rho\bar{\sigma}+\bar{\sigma}\mu\sigma\bar{\nu}\lambda\rho&122&\bar{\mu}\nu\lambda\sigma\rho\bar{\sigma}+\bar{\sigma}\mu\sigma\nu\lambda\bar{\rho}&164&\mu\bar{\sigma}\bar{\nu}\bar{\lambda}\sigma\bar{\rho}+\bar{\mu}\sigma\bar{\nu}\bar{\lambda}\bar{\sigma}\rho&206&\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\sigma}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\sigma}\bar{\nu}\bar{\lambda}\bar{\rho}\\\
39&\bar{\mu}p\bar{\nu}\lambda\bar{\rho}-\bar{\mu}\nu\bar{\lambda}p\bar{\rho}&81&\mu\nu\bar{\lambda}\sigma\sigma\bar{\rho}+\bar{\mu}\sigma\sigma\bar{\nu}\lambda\rho&123&\bar{\mu}\nu\lambda\sigma\sigma\bar{\rho}+\bar{\mu}\sigma\sigma\nu\lambda\bar{\rho}&165&\mu\bar{\sigma}\bar{\nu}\bar{\sigma}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\sigma}\bar{\lambda}\bar{\sigma}\rho&207&\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\sigma}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\sigma}\bar{\nu}\bar{\lambda}\bar{\rho}\\\
40&\bar{\mu}p\bar{\nu}\bar{\lambda}\rho-\mu\bar{\nu}\bar{\lambda}p\bar{\rho}&82&\mu\nu\bar{\sigma}\lambda\rho\bar{\sigma}+\bar{\sigma}\mu\nu\bar{\sigma}\lambda\rho&124&\bar{\mu}\nu\sigma\lambda\rho\bar{\sigma}+\bar{\sigma}\mu\nu\sigma\lambda\bar{\rho}&166&\mu\bar{\sigma}\bar{\sigma}\bar{\nu}\lambda\bar{\rho}+\bar{\mu}\nu\bar{\lambda}\bar{\sigma}\bar{\sigma}\rho&208&\bar{\mu}\bar{\nu}\bar{\sigma}\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\bar{\sigma}\bar{\lambda}\bar{\rho}\\\
41&\mu\bar{\nu}p\bar{\lambda}\bar{\rho}-\bar{\mu}\bar{\nu}p\bar{\lambda}\rho&83&\mu\nu\bar{\sigma}\lambda\sigma\bar{\rho}+\bar{\mu}\sigma\nu\bar{\sigma}\lambda\rho&125&\bar{\mu}\nu\sigma\lambda\sigma\bar{\rho}+\bar{\mu}\sigma\nu\sigma\lambda\bar{\rho}&167&\mu\bar{\nu}\bar{\lambda}\bar{\rho}\bar{\sigma}\sigma+\sigma\bar{\sigma}\bar{\mu}\bar{\nu}\bar{\lambda}\rho&209&\bar{\mu}\bar{\nu}\bar{\sigma}\bar{\lambda}\bar{\sigma}\bar{\rho}+\bar{\mu}\bar{\sigma}\bar{\nu}\bar{\sigma}\bar{\lambda}\bar{\rho}\\\
42&\bar{\mu}\nu
p\bar{\lambda}\bar{\rho}-\bar{\mu}\bar{\nu}p\lambda\bar{\rho}&84&\mu\nu\bar{\sigma}\sigma\lambda\bar{\rho}+\bar{\mu}\nu\sigma\bar{\sigma}\lambda\rho&126&\bar{\mu}\sigma\nu\lambda\rho\bar{\sigma}+\bar{\sigma}\mu\nu\lambda\sigma\bar{\rho}&168&\mu\bar{\nu}\bar{\lambda}\bar{\sigma}\bar{\rho}\sigma+\sigma\bar{\mu}\bar{\sigma}\bar{\nu}\bar{\lambda}\rho&210&\bar{\mu}\bar{\sigma}\bar{\nu}\bar{\lambda}\bar{\rho}\bar{\sigma}+\bar{\sigma}\bar{\mu}\bar{\nu}\bar{\lambda}\bar{\sigma}\bar{\rho}\\\
\hline\cr\end{array}}$ (116)
Next, we list all 23 $\widetilde{O}^{W}_{n}$ operators.
TABLE VI. List of $\widetilde{O}^{W}_{n}$ operators, where we divide
$\tilde{O}^{W}_{1},...,\tilde{O}^{W}_{7}$ by $B_{0}$ making the matrices
$A_{mn}$ introduced in Eq.(36) independent of $B_{0}$. The symbols are
introduced in Ref.p6-1 . The comparisons between the symbols introduced in
Ref.p6-1 and ours are given in Table XV. of Ref.WQ4 .
$\displaystyle\begin{array}[]{|c|c|c|c|}\hline\cr
n&\widetilde{O}^{W}_{n}&n&\widetilde{O}^{W}_{n}\\\ \hline\cr 1&\langle
iu^{\mu}u^{\nu}u^{\lambda}u^{\rho}\chi_{-}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}&13&-i\langle
f_{+}^{\mu\nu}u_{\sigma}u^{\lambda}h^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}-i\langle
f_{+}^{\mu\nu}h^{\lambda\sigma}u^{\rho}u_{\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 2&\langle
u^{\mu}u^{\nu}\chi_{+}f_{-}^{\lambda\rho}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}-\langle
u^{\mu}u^{\nu}f_{-}^{\lambda\rho}\chi_{+}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}&14&-i\langle
f_{+}^{\mu\nu}u^{\lambda}h^{\rho\sigma}u_{\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}-i\langle
f_{+}^{\mu\nu}u_{\sigma}h^{\lambda\sigma}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 3&\langle
f_{+}^{\mu\nu}u^{\lambda}u^{\rho}\chi_{-}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}+\langle
f_{+}^{\mu\nu}\chi_{-}u^{\lambda}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}&15&i\langle
f_{+}^{\mu\nu}u^{\lambda}u^{\rho}h^{\sigma}_{~{}\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}+i\langle
f_{+}^{\mu\nu}h^{\sigma}_{~{}\sigma}u^{\lambda}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 4&-\langle
f_{+}^{\mu\nu}u^{\lambda}\chi_{-}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}&16&-i\langle
f_{+}^{\mu\nu}u^{\lambda}h^{\sigma}_{~{}\sigma}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 5&i\langle
f_{+}^{\mu\nu}f_{+}^{\lambda\rho}\chi_{-}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}&17&i\langle{f_{+}^{\mu}}_{\sigma}u^{\nu}u^{\lambda}f_{-}^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}+i\langle{f_{+}^{\mu}}_{\sigma}f_{-}^{\nu\sigma}u^{\lambda}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 6&i\langle
f_{-}^{\mu\nu}f_{-}^{\lambda\rho}\chi_{-}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}&18&i\langle
f_{+}^{\mu\sigma}u^{\nu}u_{\sigma}f_{-}^{\lambda\rho}\rangle\epsilon_{\mu\nu\lambda\rho}-i\langle
f_{+}^{\mu\sigma}f_{-}^{\nu\lambda}u_{\sigma}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 7&i\langle
f_{+}^{\mu\nu}f_{-}^{\lambda\rho}\chi_{+}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}-i\langle
f_{+}^{\mu\nu}\chi_{+}f_{-}^{\lambda\rho}\rangle\epsilon_{\mu\nu\lambda\rho}/B_{0}&19&-i\langle
f_{+}^{\mu\sigma}u^{\nu}f_{-}^{\lambda\rho}u_{\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}+i\langle
f_{+}^{\mu\sigma}u_{\sigma}f_{-}^{\nu\lambda}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 8&-\langle
u_{\sigma}u^{\mu}u^{\nu}u^{\lambda}h^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}+\langle
u^{\mu}u^{\nu}u^{\lambda}u_{\sigma}h^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}&20&-\langle
f_{+}^{\mu\nu}u^{\lambda}\nabla_{\sigma}f_{+}^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}+\langle
f_{+}^{\mu\nu}\nabla_{\sigma}f_{+}^{\lambda\sigma}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 9&\langle
u^{\mu}u^{\nu}u^{\lambda}u^{\rho}h^{\sigma}_{~{}\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}&21&-\langle
u^{\mu}\nabla_{\sigma}f_{-}^{\nu\sigma}f_{-}^{\lambda\rho}\rangle\epsilon_{\mu\nu\lambda\rho}-\langle
u^{\mu}f_{-}^{\nu\lambda}\nabla_{\sigma}f_{-}^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 10&\langle
u_{\sigma}u^{\mu}u^{\nu}u^{\lambda}f_{-}^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}-\langle
u^{\mu}u^{\nu}u^{\lambda}u_{\sigma}f_{-}^{\rho\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}&22&\langle
f_{+}^{\mu\nu}f_{+}^{\lambda\rho}h^{\sigma}_{~{}\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 11&\langle
u^{2}u^{\mu}u^{\nu}f_{-}^{\lambda\rho}\rangle\epsilon_{\mu\nu\lambda\rho}-\langle
u^{\mu}u^{\nu}u^{2}f_{-}^{\lambda\rho}\rangle\epsilon_{\mu\nu\lambda\rho}&23&\langle
h^{\sigma}_{~{}\sigma}f_{-}^{\mu\nu}f_{-}^{\lambda\rho}\rangle\epsilon_{\mu\nu\lambda\rho}\\\
\hline\cr 12&i\langle
f_{+}^{\mu\sigma}u^{\nu}u^{\lambda}h^{\rho}_{~{}\sigma}\rangle\epsilon_{\mu\nu\lambda\rho}+i\langle
f_{+}^{\mu\sigma}h^{\nu}_{~{}\sigma}u^{\lambda}u^{\rho}\rangle\epsilon_{\mu\nu\lambda\rho}&&\\\
\hline\cr\end{array}$ (130)
## Appendix B $A$, $C$ and $R$ matrices
In this appendix, we give matrix $A_{nm}$, $C_{m^{\prime}n}$ and
$R_{m^{\prime}m}$. For convenience of writing, in practice, we do not write
$A$ matrix, but its transverse $A^{T}$ multiplied by $-i$.
$\displaystyle\hskip 128.0374pt\mbox{\small\bf TABLE
VII.}~{}~{}~{}~{}~{}~{}~{}-i(A^{T})_{mn}~{}\mbox{matrix}$
$\displaystyle{\scriptsize\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
m,n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&23\\\ \hline\cr
1&0&0&0&0&32&0&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
2&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
3&0&0&0&0&0&0&-64&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
4&0&0&-32&0&-32&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
5&0&0&0&0&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
6&0&-32&0&0&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
7&0&0&0&-32&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
8&0&-32&0&0&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
9&0&32&-32&0&-32&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
10&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
11&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
12&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
13&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
14&0&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
15&0&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
16&0&-32&0&0&0&0&64&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
17&0&32&0&0&0&0&-64&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
18&0&64&0&0&0&0&-64&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
19&0&0&0&0&0&0&-64&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
20&-32&-32&64&32&32&0&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
21&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
22&0&64&0&0&0&0&-64&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
23&0&0&0&0&-32&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
24&0&0&-32&0&-32&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
25&0&0&0&0&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
26&0&0&0&-32&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
27&0&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
28&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
29&0&0&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
30&0&0&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
31&0&0&0&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
32&0&0&-32&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
33&0&32&0&0&0&32&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
34&0&32&0&0&0&32&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
35&0&-32&32&0&32&0&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
36&-32&-32&64&32&32&0&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
37&-32&0&32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
38&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
39&0&0&0&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
40&32&0&-32&-32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
41&32&0&-64&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
42&0&0&-32&0&-32&32&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
43&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&-64&0\\\
44&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-24&0&32&0\\\
45&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&24&0&0&0\\\
46&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&-32&0\\\
47&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&0&0\\\
48&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&32&0\\\
49&0&0&0&0&0&0&0&0&0&0&0&0&32&0&0&0&0&0&0&8&0&-32&0\\\
50&0&0&0&0&0&0&0&0&0&0&0&-8&16&0&0&0&8&-16&0&0&0&0&0\\\
51&0&0&0&0&0&0&0&0&0&0&0&8&-48&0&-32&0&-8&0&0&-8&0&0&0\\\
52&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&-8&0&0&8&8&0&0\\\
53&0&0&0&0&0&0&0&0&0&0&0&-16&32&0&0&0&16&0&0&0&-16&0&0\\\
54&0&0&0&0&0&0&0&0&0&0&0&-8&16&0&0&0&-8&0&0&-8&16&0&0\\\
55&0&0&0&0&0&0&0&0&0&0&0&0&0&0&32&0&0&16&0&0&0&-32&0\\\ \hline\cr\end{array}}$
(187)
$\displaystyle\hskip 99.58464pt\mbox{\small\bf TABLE
VII~{}(continued).}~{}~{}~{}~{}~{}~{}~{}-i(A^{T})_{mn}~{}\mbox{matrix}$
$\displaystyle{\scriptsize\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
m,n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&23\\\ \hline\cr
56&0&0&0&0&0&0&0&0&0&0&0&8&-48&0&-32&0&-8&0&0&0&0&32&0\\\
57&0&0&0&0&0&0&0&0&0&0&0&24&-16&0&32&0&8&0&0&16&0&-32&0\\\
58&0&0&0&0&0&0&0&0&0&0&0&-16&0&0&-32&0&0&0&0&-8&0&32&0\\\
59&0&0&0&0&0&0&0&0&0&0&0&0&-16&32&0&32&0&0&0&0&0&0&-32\\\
60&0&0&0&0&0&0&0&0&0&0&0&0&-16&0&0&0&0&0&0&8&0&0&0\\\
61&0&0&0&0&0&0&0&0&0&0&0&0&32&0&32&0&0&0&0&-8&8&0&0\\\
62&0&0&0&0&0&0&0&0&0&0&0&0&0&16&0&0&0&-16&-16&-24&16&0&0\\\
63&0&0&0&0&0&0&0&0&0&0&0&8&-16&16&0&0&8&0&16&24&-32&0&0\\\
64&0&0&0&0&0&0&0&0&0&0&0&8&-16&0&0&0&-8&0&0&-16&24&32&-32\\\
65&0&0&0&0&0&0&0&0&0&0&0&0&0&-16&0&0&0&16&16&16&0&0&-32\\\
66&0&0&0&0&0&0&0&0&0&0&0&-8&16&-16&0&0&-8&0&-16&-16&0&0&32\\\
67&0&0&0&0&0&0&0&0&0&0&0&-8&16&0&0&0&8&0&0&8&8&0&-32\\\
68&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&32\\\
69&0&0&0&0&0&0&0&0&0&0&0&0&0&-32&-64&-32&0&-16&0&-8&-8&64&32\\\
70&0&0&0&0&0&0&0&0&0&0&0&-8&80&0&64&0&8&0&0&16&8&-64&0\\\
71&0&0&0&0&0&0&0&0&0&0&0&8&-32&0&0&0&-8&16&0&-8&0&0&0\\\
72&0&0&0&0&0&0&0&0&0&0&0&-40&16&-16&-64&0&-8&0&-16&-40&0&64&0\\\
73&0&0&0&0&0&0&0&0&0&0&0&24&0&-16&0&0&8&-16&16&48&-8&0&0\\\
74&0&0&0&0&0&0&0&0&0&0&0&-32&32&0&-32&0&0&0&0&-40&8&32&-32\\\
75&0&0&0&0&0&0&0&0&0&0&0&32&0&16&64&0&0&0&16&24&0&-64&-32\\\
76&0&0&0&0&0&0&0&0&0&0&0&-16&0&16&0&0&0&0&-16&-24&0&0&32\\\
77&0&0&0&0&0&0&0&0&0&0&0&16&-32&0&0&0&0&0&0&16&8&0&-32\\\
78&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&32\\\
79&0&0&0&0&0&0&0&0&0&0&0&0&0&-16&0&0&0&16&16&16&0&0&-32\\\
80&0&0&0&0&0&0&0&0&0&0&0&0&0&16&0&0&0&-16&-16&-24&16&0&0\\\
81&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&-8&0&0\\\
82&0&0&0&0&0&0&0&0&0&0&0&0&-16&0&0&0&0&0&0&8&0&0&0\\\
83&0&0&0&0&0&0&0&0&0&0&0&8&0&-16&0&0&-8&16&16&0&-8&0&0\\\
84&0&0&0&0&0&0&0&0&0&0&0&-8&0&16&0&0&8&-16&-16&0&-8&-32&32\\\
85&0&0&0&0&0&0&0&0&0&0&0&0&-16&32&0&32&0&0&0&0&0&0&-32\\\
86&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&-32&8&-16&0&0&0&0&32\\\
87&0&0&0&0&0&0&0&0&0&0&0&-8&0&0&0&32&-8&16&0&0&16&0&-32\\\
88&0&0&0&0&0&0&0&0&0&0&0&0&0&-32&-32&-32&0&0&0&0&-8&0&32\\\
89&0&0&0&0&0&0&0&0&0&0&0&0&0&16&0&0&0&-16&-16&0&-32&0&96\\\
90&0&0&0&0&0&0&0&0&0&0&0&8&-16&16&0&0&8&0&16&-8&48&0&-64\\\
91&0&0&0&0&0&0&0&0&0&0&0&-8&0&0&0&32&-8&16&0&8&-16&0&0\\\
92&0&0&0&0&0&0&0&0&0&0&0&8&-16&0&0&0&-8&0&0&0&-24&0&64\\\
93&0&0&0&0&0&0&0&0&0&0&0&-8&0&16&0&0&8&-16&-16&0&8&0&0\\\
94&0&0&0&0&0&0&0&0&0&0&0&16&0&16&32&0&0&0&16&0&-16&-32&32\\\
95&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&-96\\\
96&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&32\\\
97&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&24&0&-32\\\
98&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&32\\\
99&0&0&0&0&0&0&0&0&0&0&0&-16&0&-16&-32&0&0&0&-16&0&-24&32&32\\\
100&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&-8&0&0&0&24&0&-32\\\
101&0&0&0&0&0&0&0&0&0&0&0&8&0&-16&0&-32&8&0&16&-8&0&0&0\\\
102&0&0&0&0&0&0&0&0&0&0&0&-8&16&16&0&0&8&-16&-16&0&-8&0&32\\\
103&0&0&0&0&0&0&0&0&0&0&0&-8&16&-32&0&0&-8&16&0&8&0&0&0\\\
104&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-32\\\
105&0&0&0&0&0&0&0&0&0&0&0&0&0&0&32&0&0&16&0&0&0&-32&0\\\
106&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&-8&0&0&8&8&0&0\\\
107&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&8&-16&0&0&-8&0&0\\\
108&0&0&0&0&0&0&0&0&0&0&0&-8&16&0&0&0&8&-16&0&0&0&0&0\\\
109&0&0&0&0&0&0&0&0&0&0&0&-16&0&0&0&0&-16&32&0&-16&0&0&0\\\
110&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&8&-16&0&16&-8&0&0\\\
111&0&0&0&0&0&0&0&0&0&0&0&0&32&0&0&0&0&0&0&8&0&-32&0\\\
112&0&0&0&0&0&0&0&0&0&0&0&-8&16&0&0&0&8&-16&0&-8&0&32&0\\\
113&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&-8&0&0&0&8&-32&0\\\
114&0&0&0&0&0&0&0&0&0&0&0&0&0&0&32&0&0&16&0&0&0&32&0\\\ \hline\cr\end{array}}$
(248)
$\displaystyle\hskip 128.0374pt\mbox{\small\bf TABLE
VII~{}(continued).}~{}~{}~{}~{}~{}~{}~{}-i(A^{T})_{mn}~{}\mbox{matrix}$
$\displaystyle{\scriptsize\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
m,n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&23\\\ \hline\cr
115&0&0&0&0&0&0&0&0&0&0&0&0&-16&-32&-32&-32&0&0&0&-8&-8&32&32\\\
116&0&0&0&0&0&0&0&0&0&0&0&-8&0&0&0&32&-8&16&0&8&24&0&-32\\\
117&0&0&0&0&0&0&0&0&0&0&0&-8&16&0&0&0&8&0&0&0&-16&0&0\\\
118&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&-32&8&-16&0&-8&8&0&32\\\
119&0&0&0&0&0&0&0&0&0&0&0&8&-16&0&0&0&-8&0&0&0&0&0&0\\\
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123&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&24&0&0&0\\\
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125&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&0&0\\\
126&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&32&0\\\
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129&0&0&0&0&0&0&0&0&0&0&-32&8&0&0&0&0&-8&16&0&0&8&0&0\\\
130&0&0&0&0&0&0&0&-16&0&-16&32&-8&32&-16&0&0&-8&0&16&-16&16&0&0\\\
131&0&0&0&0&0&0&0&32&0&32&0&-16&0&32&0&0&16&-32&-32&0&-16&0&0\\\
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142&0&0&0&0&0&0&0&0&-32&0&0&-24&-32&0&-64&0&-8&16&0&-24&16&32&-32\\\
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147&0&0&0&0&0&0&0&16&0&16&0&0&-16&16&0&0&0&-16&-16&0&-24&0&32\\\
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152&0&0&0&0&0&0&0&16&32&-16&0&24&16&16&64&0&8&0&16&8&0&-32&32\\\
153&0&0&0&0&0&0&0&0&0&0&64&0&0&0&0&0&0&-32&0&0&-8&0&32\\\
154&0&0&0&0&0&0&0&0&0&0&-32&-8&16&0&0&0&8&16&0&0&8&0&-32\\\
155&0&0&0&0&0&0&0&16&0&-16&32&-8&0&0&0&0&-8&0&0&-8&0&0&32\\\
156&0&0&0&0&0&0&0&16&0&16&0&0&-16&0&0&0&0&0&0&0&0&0&-32\\\
157&0&0&0&0&0&0&0&-48&-32&16&-32&-16&16&-48&-64&-32&0&0&-16&0&-16&32&0\\\
158&0&0&0&0&0&0&0&-16&0&-16&32&-8&32&-16&0&0&-8&0&16&-16&16&0&0\\\
159&0&0&0&0&0&0&0&0&-32&0&0&8&-48&0&-32&-32&8&0&0&8&0&0&0\\\
160&0&0&0&0&0&0&0&-16&0&-16&-32&8&16&-16&0&0&8&16&16&24&-8&0&0\\\
161&0&0&0&0&0&0&0&-32&0&32&0&-16&32&-32&0&0&-16&0&-32&-16&0&0&0\\\
162&0&0&0&0&0&0&0&-16&0&-16&-32&-8&32&-16&0&0&8&16&16&8&0&0&0\\\
163&0&0&0&0&0&0&0&16&0&16&0&-16&-16&16&-32&0&0&-16&-16&-24&0&32&0\\\
164&0&0&0&0&0&0&0&16&0&-16&32&24&-32&16&32&0&8&0&16&24&0&-32&0\\\
165&0&0&0&0&0&0&0&0&0&0&-32&8&-48&0&-32&0&-8&16&0&-8&0&32&0\\\
166&0&0&0&0&0&0&0&0&0&0&64&0&0&0&32&0&0&-16&0&0&0&-32&0\\\
167&0&0&0&0&0&0&0&48&64&-16&32&32&16&48&128&64&0&0&16&32&0&-64&0\\\
168&0&0&0&0&0&0&0&0&-32&0&32&-16&-32&0&-64&-32&0&-16&0&-32&0&32&0\\\
169&0&0&0&0&0&0&0&16&0&16&-32&0&-16&16&0&0&0&0&-16&0&0&0&0\\\
170&0&0&0&0&0&0&0&16&32&-16&32&16&16&16&64&32&0&0&16&16&0&-32&0\\\
171&0&0&0&0&0&0&0&0&0&0&-32&0&0&0&0&0&0&16&0&0&0&0&0\\\
172&0&0&0&0&0&0&0&-64&-32&0&0&-16&32&-64&-64&-32&0&0&0&0&0&32&0\\\
173&0&0&0&0&0&0&0&0&0&0&0&0&0&-32&-64&-32&0&-16&0&-8&-8&64&32\\\
\hline\cr\end{array}}$ (309)
$\displaystyle\hskip 128.0374pt\mbox{\small\bf TABLE
VII~{}(continued).}~{}~{}~{}~{}~{}~{}~{}-i(A^{T})_{mn}~{}\mbox{matrix}$
$\displaystyle{\scriptsize\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
m,n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&23\\\ \hline\cr
174&0&0&0&0&0&0&0&0&0&0&0&-8&80&0&64&0&8&0&0&16&8&-64&0\\\
175&0&0&0&0&0&0&0&0&0&0&0&8&-32&0&0&0&-8&16&0&-8&0&0&0\\\
176&0&0&0&0&0&0&0&0&0&0&0&-40&16&-16&-64&0&-8&0&-16&-40&0&64&0\\\
177&0&0&0&0&0&0&0&0&0&0&0&24&0&-16&0&0&8&-16&16&48&-8&0&0\\\
178&0&0&0&0&0&0&0&0&0&0&0&-32&32&0&-32&0&0&0&0&-40&8&32&-32\\\
179&0&0&0&0&0&0&0&0&0&0&0&32&0&16&64&0&0&0&16&24&0&-64&-32\\\
180&0&0&0&0&0&0&0&0&0&0&0&-16&0&16&0&0&0&0&-16&-24&0&0&32\\\
181&0&0&0&0&0&0&0&0&0&0&0&16&-32&0&0&0&0&0&0&16&8&0&-32\\\
182&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&32\\\
183&0&0&0&0&0&0&0&0&0&0&0&0&0&16&0&0&0&-16&-16&0&-32&0&96\\\
184&0&0&0&0&0&0&0&0&0&0&0&8&-16&16&0&0&8&0&16&-8&48&0&-64\\\
185&0&0&0&0&0&0&0&0&0&0&0&-8&0&0&0&32&-8&16&0&8&-16&0&0\\\
186&0&0&0&0&0&0&0&0&0&0&0&8&-16&0&0&0&-8&0&0&0&-24&0&64\\\
187&0&0&0&0&0&0&0&0&0&0&0&-8&0&16&0&0&8&-16&-16&0&8&0&0\\\
188&0&0&0&0&0&0&0&0&0&0&0&16&0&16&32&0&0&0&16&0&-16&-32&32\\\
189&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&-96\\\
190&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&32\\\
191&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&24&0&-32\\\
192&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-8&0&32\\\
193&0&0&0&0&0&0&0&0&0&0&0&-16&0&-16&-32&0&0&0&-16&0&-24&32&32\\\
194&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&-8&0&0&0&24&0&-32\\\
195&0&0&0&0&0&0&0&0&0&0&0&8&0&-16&0&-32&8&0&16&-8&0&0&0\\\
196&0&0&0&0&0&0&0&0&0&0&0&-8&16&16&0&0&8&-16&-16&0&-8&0&32\\\
197&0&0&0&0&0&0&0&0&0&0&0&-8&16&-32&0&0&-8&16&0&8&0&0&0\\\
198&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-32\\\
199&0&0&0&0&0&0&0&0&0&0&0&0&-16&-32&-32&-32&0&0&0&-8&-8&32&32\\\
200&0&0&0&0&0&0&0&0&0&0&0&-8&0&0&0&32&-8&16&0&8&24&0&-32\\\
201&0&0&0&0&0&0&0&0&0&0&0&-8&16&0&0&0&8&0&0&0&-16&0&0\\\
202&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&-32&8&-16&0&-8&8&0&32\\\
203&0&0&0&0&0&0&0&0&0&0&0&8&-16&0&0&0&-8&0&0&0&0&0&0\\\
204&0&0&0&0&0&0&0&0&0&0&0&0&-32&32&0&32&0&0&0&0&0&0&-32\\\
205&0&0&0&0&0&0&0&48&64&-16&32&32&16&48&128&64&0&0&16&32&0&-64&0\\\
206&0&0&0&0&0&0&0&0&-32&0&32&-16&-32&0&-64&-32&0&-16&0&-32&0&32&0\\\
207&0&0&0&0&0&0&0&16&0&16&-32&0&-16&16&0&0&0&0&-16&0&0&0&0\\\
208&0&0&0&0&0&0&0&16&32&-16&32&16&16&16&64&32&0&0&16&16&0&-32&0\\\
209&0&0&0&0&0&0&0&0&0&0&-32&0&0&0&0&0&0&16&0&0&0&0&0\\\
210&0&0&0&0&0&0&0&-64&-32&0&0&-16&32&-64&-64&-32&0&0&0&0&0&32&0\\\
\hline\cr\end{array}}$ (348)
From (28), $C$ matrix is consist of two sub-matrices $\bar{C}$ and
$\tilde{C}$. $\bar{C}$ matrix is
$\displaystyle\hskip 28.45274pt\mbox{\small\bf TABLE.
VIII}~{}~{}~{}~{}~{}~{}~{}\bar{C}_{m^{\prime}n}~{}\mbox{matrix}$
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|}\hline\cr
m^{\prime},n&1&2&3&4&5&6&7\\\ \hline\cr
1&-\frac{i}{32}&0&-\frac{i}{32}&0&\frac{i}{32}&0&0\\\
3&0&0&0&0&\frac{i}{64}&-\frac{i}{64}&-\frac{i}{64}\\\
4&-\frac{i}{16}&0&-\frac{i}{32}&0&0&0&0\\\
5&0&\frac{i}{32}&0&\frac{i}{32}&0&-\frac{i}{32}&0\\\
6&\frac{i}{32}&-\frac{i}{32}&0&0&0&0&0\\\
7&-\frac{i}{32}&0&0&-\frac{i}{32}&0&0&0\\\ 20&-\frac{i}{32}&0&0&0&0&0&0\\\
\hline\cr\end{array}$ (357)
$\tilde{C}$ matrix is
$\displaystyle\hskip 113.81102pt\mbox{\small\bf TABLE.
IX}~{}~{}~{}~{}~{}~{}~{}\tilde{C}_{m^{\prime}n}~{}\mbox{matrix}$
$\displaystyle\begin{array}[]{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}\hline\cr
m^{\prime},n&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&23\\\ \hline\cr
43&\frac{19i}{160}&-\frac{11i}{80}&-\frac{19i}{160}&-\frac{13i}{160}&-\frac{3i}{80}&-\frac{i}{80}&-\frac{7i}{160}&\frac{9i}{160}&-\frac{i}{160}&-\frac{13i}{80}&-\frac{3i}{40}&-\frac{i}{32}&-\frac{i}{40}&-\frac{i}{10}&-\frac{3i}{160}&-\frac{7i}{160}\\\
44&\frac{31i}{320}&-\frac{3i}{20}&-\frac{21i}{320}&-\frac{17i}{320}&\frac{3i}{160}&\frac{i}{160}&-\frac{13i}{320}&\frac{11i}{320}&\frac{11i}{320}&-\frac{17i}{160}&-\frac{9i}{160}&\frac{i}{64}&-\frac{i}{20}&-\frac{3i}{40}&-\frac{i}{160}&-\frac{3i}{320}\\\
49&-\frac{i}{16}&\frac{3i}{32}&\frac{3i}{32}&\frac{i}{16}&\frac{i}{16}&\frac{i}{32}&\frac{i}{32}&-\frac{i}{16}&0&\frac{i}{8}&\frac{i}{16}&\frac{i}{32}&0&\frac{i}{16}&0&\frac{i}{64}\\\
50&0&0&0&0&0&0&-\frac{i}{32}&0&\frac{i}{32}&0&-\frac{i}{16}&\frac{i}{32}&0&0&0&0\\\
51&-\frac{5i}{64}&\frac{3i}{32}&\frac{5i}{64}&\frac{3i}{64}&\frac{i}{32}&0&\frac{i}{64}&-\frac{3i}{64}&\frac{i}{64}&\frac{3i}{32}&\frac{i}{32}&\frac{3i}{64}&0&\frac{i}{16}&0&\frac{i}{32}\\\
52&\frac{5i}{32}&-\frac{3i}{16}&-\frac{5i}{32}&-\frac{3i}{32}&0&0&-\frac{i}{16}&\frac{i}{16}&\frac{i}{32}&-\frac{i}{4}&-\frac{i}{8}&-\frac{i}{16}&0&-\frac{i}{8}&0&-\frac{i}{32}\\\
54&-\frac{3i}{32}&\frac{i}{32}&\frac{5i}{32}&\frac{i}{32}&0&0&\frac{i}{16}&-\frac{i}{32}&0&\frac{i}{8}&\frac{i}{16}&\frac{i}{8}&0&\frac{i}{8}&0&\frac{i}{16}\\\
57&-\frac{3i}{64}&\frac{i}{32}&\frac{7i}{64}&\frac{i}{64}&\frac{i}{32}&0&\frac{i}{64}&-\frac{i}{64}&\frac{i}{64}&\frac{3i}{32}&\frac{i}{32}&\frac{3i}{64}&0&\frac{i}{16}&0&\frac{i}{32}\\\
59&0&-\frac{i}{32}&0&0&0&0&0&0&\frac{i}{32}&0&0&0&0&0&0&0\\\
62&-\frac{i}{32}&\frac{i}{32}&-\frac{i}{32}&0&0&0&\frac{i}{32}&0&-\frac{i}{32}&0&0&-\frac{i}{32}&0&0&0&0\\\
63&-\frac{i}{32}&\frac{i}{32}&\frac{i}{32}&0&0&0&\frac{i}{32}&0&-\frac{i}{32}&0&0&\frac{i}{32}&0&0&0&0\\\
64&0&\frac{i}{32}&0&0&0&0&0&0&-\frac{i}{32}&0&0&0&0&0&0&-\frac{i}{32}\\\
127&\frac{i}{32}&0&\frac{i}{32}&-\frac{i}{32}&0&0&0&0&0&0&0&0&0&0&0&0\\\
128&\frac{i}{32}&-\frac{i}{32}&-\frac{i}{32}&-\frac{i}{32}&0&0&0&0&0&0&0&0&0&0&0&0\\\
133&-\frac{i}{32}&\frac{i}{32}&\frac{i}{32}&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
134&-\frac{i}{32}&\frac{i}{16}&\frac{i}{32}&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
\hline\cr\end{array}$ (375)
## Appendix C Final Analytical Result on $\tilde{K}^{W}_{n}$
In this appendix, we list our analytical result on 23 LECs for $p^{6}$ order
anomalous part of the chiral Lagrangian,
$\displaystyle\tilde{K}^{W}_{1}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}-\frac{1}{2}k^{2}\Sigma_{k}^{3}X^{5}-\frac{1}{2}\Sigma_{k}^{5}X^{5}+\frac{3}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime}X^{5}+\frac{3}{4}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime}X^{5}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{2}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}-\frac{1}{4}\Sigma_{k}^{5}X^{5}+\frac{1}{8}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime}X^{5}+\frac{5}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime}X^{5}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{3}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}-\frac{1}{4}k^{2}\Sigma_{k}^{3}X^{5}-\frac{1}{4}\Sigma_{k}^{5}X^{5}+\frac{1}{2}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime}X^{5}+\frac{1}{2}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime}X^{5}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{4}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}-\frac{1}{4}k^{2}\Sigma_{k}^{3}X^{5}-\frac{1}{4}\Sigma_{k}^{5}X^{5}+\frac{1}{2}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime}X^{5}+\frac{1}{2}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime}X^{5}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{5}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}\frac{3}{16}k^{2}\Sigma_{k}^{3}X^{5}+\frac{3}{16}\Sigma_{k}^{5}X^{5}-\frac{1}{16}k^{6}\Sigma_{k}^{\prime}X^{5}-\frac{1}{2}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime}X^{5}-\frac{7}{16}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime}X^{5}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{6}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}\frac{1}{16}k^{2}\Sigma_{k}^{3}X^{5}+\frac{1}{16}\Sigma_{k}^{5}X^{5}-\frac{1}{8}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime}X^{5}-\frac{1}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime}X^{5}]$
$\displaystyle\tilde{K}^{W}_{7}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}-\frac{1}{16}k^{2}\Sigma_{k}^{3}X^{5}-\frac{1}{16}\Sigma_{k}^{5}X^{5}+\frac{1}{32}k^{6}\Sigma_{k}^{\prime}X^{5}+\frac{3}{16}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime}X^{5}+\frac{5}{32}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime}X^{5}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{8}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(\frac{9}{40}k^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{40}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}+\frac{3}{40}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{1}{180}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(-\frac{29}{80}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{17}{80}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{3}{40}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle+\frac{7}{16}k^{6}\Sigma_{k}^{\prime
2}-\frac{3}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{3}{16}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{67}{240}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}+\frac{31}{120}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{3}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}+\frac{27}{80}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{1}{2}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle+\frac{13}{80}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(-\frac{17}{80}k^{4}\Sigma_{k}^{2}-\frac{1}{8}k^{2}\Sigma_{k}^{4}-\frac{3}{80}\Sigma_{k}^{6}+\frac{151}{240}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{5}{8}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{13}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}-\frac{59}{120}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle+\frac{893}{480}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{217}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{39}{160}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}-\frac{293}{240}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{5}{8}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{39}{80}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{9}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{7}{40}k^{2}\Sigma_{k}^{\prime\prime}+\frac{3}{40}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{7}{120}k^{4}\Sigma_{k}^{\prime\prime\prime}-\frac{1}{360}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{37}{80}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{80}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{1}{40}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{11}{16}k^{6}\Sigma_{k}^{\prime
2}+\frac{1}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{1}{16}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{27}{80}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{1}{15}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{1}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{8}{15}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{1}{6}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{7}{15}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{8}{15}k^{4}\Sigma_{k}^{2}+\frac{1}{4}k^{2}\Sigma_{k}^{4}+\frac{3}{10}\Sigma_{k}^{6}-\frac{139}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{7}{12}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{17}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{17}{20}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{317}{240}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{23}{120}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{51}{80}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{197}{120}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{11}{12}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{51}{40}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{10}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{33}{80}k^{2}\Sigma_{k}^{\prime\prime}+\frac{27}{80}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{11}{80}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{71}{720}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{17}{40}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{31}{40}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{3}{10}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{4}k^{6}\Sigma_{k}^{\prime
2}+2k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{3}{4}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{107}{240}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{217}{240}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{3}{20}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{7}{30}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{23}{12}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{17}{20}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{11}{240}k^{4}\Sigma_{k}^{2}-\frac{7}{8}k^{2}\Sigma_{k}^{4}+\frac{43}{80}\Sigma_{k}^{6}-\frac{37}{80}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{71}{24}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{213}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{29}{120}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{641}{160}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1127}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{89}{160}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{283}{240}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{97}{24}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{89}{80}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{11}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{17}{160}k^{2}\Sigma_{k}^{\prime\prime}+\frac{3}{160}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{7}{180}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{1}{360}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{9}{40}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{20}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{1}{40}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{5}{16}k^{6}\Sigma_{k}^{\prime
2}+\frac{1}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{16}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{49}{240}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{7}{120}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}-\frac{1}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{31}{120}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{1}{8}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{7}{60}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{9}{80}k^{4}\Sigma_{k}^{2}+\frac{1}{80}\Sigma_{k}^{6}-\frac{3}{10}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{2}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{1}{20}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{2}{5}k^{8}\Sigma_{k}^{\prime
2}-\frac{541}{480}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime 2}$
$\displaystyle-\frac{41}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{23}{160}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{221}{240}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{5}{24}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{23}{80}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{12}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{1}{40}k^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{40}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{120}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{7}{360}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(-\frac{1}{20}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{10}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{1}{20}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle+\frac{1}{8}k^{6}\Sigma_{k}^{\prime
2}X^{5}+\frac{1}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{1}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{1}{5}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}-\frac{1}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}+\frac{7}{80}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{7}{24}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{31}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(-\frac{53}{120}k^{4}\Sigma_{k}^{2}-\frac{1}{2}k^{2}\Sigma_{k}^{4}+\frac{1}{40}\Sigma_{k}^{6}+\frac{11}{15}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{5}{6}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{2}{5}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}-\frac{7}{60}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{37}{80}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{59}{120}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{13}{80}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}-\frac{3}{40}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{5}{12}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{13}{40}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{13}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(\frac{1}{80}k^{2}\Sigma_{k}^{\prime\prime}+\frac{1}{80}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}+\frac{1}{240}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{1}{240}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(-\frac{1}{80}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{3}{80}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{40}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle+\frac{1}{16}k^{6}\Sigma_{k}^{\prime
2}-\frac{1}{16}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{1}{60}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{1}{240}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{1}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}+\frac{7}{240}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{7}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}$
$\displaystyle+(-\frac{7}{40}k^{4}\Sigma_{k}^{2}-\frac{1}{8}k^{2}\Sigma_{k}^{4}+\frac{1}{20}\Sigma_{k}^{6}+\frac{41}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{6}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{7}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}-\frac{1}{15}k^{8}\Sigma_{k}^{\prime
2}+\frac{13}{240}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime 2}$
$\displaystyle+\frac{13}{120}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{1}{80}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}-\frac{13}{120}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{1}{12}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{1}{40}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{14}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{9}{80}k^{2}\Sigma_{k}^{\prime\prime}+\frac{11}{80}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{2}{45}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{1}{40}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{7}{80}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{21}{80}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{3}{20}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle+\frac{5}{8}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{3}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{13}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{19}{60}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{3}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{1}{30}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{13}{24}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{17}{40}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{13}{240}k^{4}\Sigma_{k}^{2}+\frac{19}{80}\Sigma_{k}^{6}+\frac{1}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{7}{12}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{47}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}-\frac{1}{40}k^{8}\Sigma_{k}^{\prime
2}-\frac{113}{160}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime 2}$
$\displaystyle+\frac{197}{80}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{57}{160}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}-\frac{7}{80}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{41}{24}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{57}{80}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{15}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(\frac{11}{160}k^{2}\Sigma_{k}^{\prime\prime}-\frac{9}{160}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}+\frac{19}{720}k^{4}\Sigma_{k}^{\prime\prime\prime}-\frac{11}{720}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(-\frac{9}{80}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{7}{80}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{20}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{1}{8}k^{6}\Sigma_{k}^{\prime
2}-\frac{1}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{2}{15}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}+\frac{11}{120}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}-\frac{1}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}+\frac{19}{160}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{3}{16}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{31}{160}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{3}{80}k^{4}\Sigma_{k}^{2}-\frac{13}{80}\Sigma_{k}^{6}+\frac{1}{40}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{4}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{19}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{5}k^{8}\Sigma_{k}^{\prime
2}+\frac{343}{480}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{97}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{29}{160}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}-\frac{103}{240}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}+\frac{5}{24}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{29}{80}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{16}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{7}{80}k^{2}\Sigma_{k}^{\prime\prime}-\frac{7}{80}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{45}k^{4}\Sigma_{k}^{\prime\prime\prime}-\frac{1}{120}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{3}{40}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{10}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{7}{40}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{16}k^{6}\Sigma_{k}^{\prime
2}+\frac{3}{8}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{7}{16}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{23}{240}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}+\frac{1}{120}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}-\frac{7}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{3}{80}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{5}{12}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle+\frac{109}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(-\frac{61}{240}k^{4}\Sigma_{k}^{2}-\frac{3}{8}k^{2}\Sigma_{k}^{4}-\frac{13}{80}\Sigma_{k}^{6}+\frac{23}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{2}{3}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{29}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{1}{20}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{209}{160}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{109}{80}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{79}{160}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{49}{80}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{1}{24}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}-\frac{79}{80}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{17}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{2}{5}k^{2}\Sigma_{k}^{\prime\prime}+\frac{1}{10}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{2}{15}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{11}{180}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{23}{40}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{19}{40}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{1}{20}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{5}{8}k^{6}\Sigma_{k}^{\prime
2}+\frac{3}{2}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{1}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{8}{15}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{79}{120}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}-\frac{1}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{119}{240}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle+\frac{29}{24}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}-\frac{151}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{37}{120}k^{4}\Sigma_{k}^{2}-\frac{1}{2}k^{2}\Sigma_{k}^{4}-\frac{9}{40}\Sigma_{k}^{6}-\frac{14}{15}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{8}{3}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{1}{10}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle+\frac{43}{60}k^{8}\Sigma_{k}^{\prime
2}-\frac{1031}{240}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{53}{40}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{53}{80}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{271}{120}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{13}{12}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{53}{40}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{18}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{9}{80}k^{2}\Sigma_{k}^{\prime\prime}+\frac{11}{80}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{2}{45}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{7}{180}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{3}{20}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{5}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{3}{20}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{8}k^{6}\Sigma_{k}^{\prime
2}+\frac{1}{2}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{3}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{13}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{19}{60}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{3}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{7}{60}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{1}{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{19}{30}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(-\frac{1}{20}k^{4}\Sigma_{k}^{2}-\frac{1}{8}k^{2}\Sigma_{k}^{4}+\frac{7}{40}\Sigma_{k}^{6}+\frac{2}{15}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{7}{12}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{21}{20}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{1}{10}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{97}{240}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{223}{120}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{51}{80}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{17}{120}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{7}{12}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{51}{40}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{19}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{1}{4}k^{2}\Sigma_{k}^{\prime\prime}+\frac{1}{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{12}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{5}{72}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{3}{16}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{9}{16}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{1}{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle+\frac{11}{8}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{5}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{4}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{5}{8}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{1}{8}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{1}{24}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{37}{24}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{5}{12}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(-\frac{1}{48}k^{4}\Sigma_{k}^{2}-\frac{1}{4}k^{2}\Sigma_{k}^{4}+\frac{5}{16}\Sigma_{k}^{6}-\frac{1}{6}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{4}{3}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{7}{4}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}-\frac{1}{24}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{91}{32}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{173}{48}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{3}{32}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{25}{48}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{29}{8}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{3}{16}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{20}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(\frac{1}{40}k^{2}\Sigma_{k}^{\prime\prime}+\frac{1}{40}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{180}k^{4}\Sigma_{k}^{\prime\prime\prime}-\frac{1}{180}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{1}{20}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{10}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{1}{20}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{8}k^{6}\Sigma_{k}^{\prime
2}-\frac{1}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{1}{8}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{11}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}+\frac{7}{60}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{1}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{13}{120}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{1}{12}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{1}{40}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{3}{20}k^{4}\Sigma_{k}^{2}+\frac{1}{4}k^{2}\Sigma_{k}^{4}+\frac{1}{10}\Sigma_{k}^{6}-\frac{2}{5}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{1}{2}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{1}{10}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle+\frac{1}{5}k^{8}\Sigma_{k}^{\prime
2}-\frac{1}{10}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{1}{5}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{1}{10}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{1}{5}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{1}{5}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{21}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{11}{40}k^{2}\Sigma_{k}^{\prime\prime}+\frac{9}{40}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{11}{120}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{3}{40}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{13}{40}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{19}{40}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{1}{5}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{4}k^{6}\Sigma_{k}^{\prime
2}+\frac{5}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{1}{2}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{13}{40}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{23}{40}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{1}{10}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{11}{60}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{5}{4}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{17}{30}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(\frac{1}{10}k^{4}\Sigma_{k}^{2}-\frac{1}{4}k^{2}\Sigma_{k}^{4}+\frac{3}{20}\Sigma_{k}^{6}-\frac{31}{60}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{4}{3}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{23}{20}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{3}{10}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{169}{60}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{38}{15}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{7}{20}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{29}{30}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{7}{3}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{7}{10}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{22}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{1}{80}k^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{80}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{1}{240}k^{4}\Sigma_{k}^{\prime\prime\prime}-\frac{1}{240}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{3}{80}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{80}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{1}{40}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
$\displaystyle-\frac{1}{16}k^{6}\Sigma_{k}^{\prime
2}+\frac{1}{16}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{3}{80}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}+\frac{1}{40}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}-\frac{1}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{19}{480}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{1}{48}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle+\frac{29}{480}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(-\frac{1}{80}k^{4}\Sigma_{k}^{2}-\frac{1}{16}k^{2}\Sigma_{k}^{4}-\frac{1}{20}\Sigma_{k}^{6}+\frac{1}{80}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{1}{4}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{19}{80}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{1}{40}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{13}{40}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{11}{40}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{3}{40}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{3}{20}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{3}{20}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
$\displaystyle\tilde{K}^{W}_{23}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}(-\frac{23}{160}k^{2}\Sigma_{k}^{\prime\prime}+\frac{17}{160}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime}-\frac{23}{480}k^{4}\Sigma_{k}^{\prime\prime\prime}+\frac{17}{480}k^{2}\Sigma_{k}^{2}\Sigma_{k}^{\prime\prime\prime})k^{2}\Sigma_{k}X^{4}+(\frac{19}{160}k^{4}\Sigma_{k}\Sigma_{k}^{\prime}-\frac{47}{160}k^{2}\Sigma_{k}^{3}\Sigma_{k}^{\prime}+\frac{7}{80}\Sigma_{k}^{5}\Sigma_{k}^{\prime}$
(376) $\displaystyle-\frac{1}{32}k^{6}\Sigma_{k}^{\prime
2}+\frac{3}{4}k^{4}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}-\frac{7}{32}k^{2}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}+\frac{31}{240}k^{6}\Sigma_{k}\Sigma_{k}^{\prime\prime}-\frac{157}{480}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime\prime}+\frac{7}{160}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime\prime}-\frac{11}{480}k^{8}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}+\frac{19}{24}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime}$
$\displaystyle-\frac{89}{480}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime}\Sigma_{k}^{\prime\prime})X^{5}+(-\frac{9}{80}k^{4}\Sigma_{k}^{2}-\frac{3}{16}k^{2}\Sigma_{k}^{4}+\frac{7}{40}\Sigma_{k}^{6}+\frac{1}{120}k^{6}\Sigma_{k}\Sigma_{k}^{\prime}+\frac{7}{12}k^{4}\Sigma_{k}^{3}\Sigma_{k}^{\prime}-\frac{37}{40}k^{2}\Sigma_{k}^{5}\Sigma_{k}^{\prime}+\frac{1}{60}k^{8}\Sigma_{k}^{\prime
2}$ $\displaystyle-\frac{223}{160}k^{6}\Sigma_{k}^{2}\Sigma_{k}^{\prime
2}+\frac{371}{240}k^{4}\Sigma_{k}^{4}\Sigma_{k}^{\prime
2}-\frac{7}{160}k^{2}\Sigma_{k}^{6}\Sigma_{k}^{\prime
2}+\frac{23}{80}k^{8}\Sigma_{k}\Sigma_{k}^{\prime
3}-\frac{13}{8}k^{6}\Sigma_{k}^{3}\Sigma_{k}^{\prime
3}+\frac{7}{80}k^{4}\Sigma_{k}^{5}\Sigma_{k}^{\prime 3})X^{6}\bigg{]}$
where $B_{0}$ is the LEC appear in $p^{2}$ order normal part of the chiral
Lagrangian. $\Sigma_{k}\equiv\Sigma(k^{2})$ and
$X\equiv\frac{1}{k^{2}+\Sigma_{k}^{2}}$.
|
arxiv-papers
| 2010-01-02T16:02:27 |
2024-09-04T02:49:07.442474
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shao-Zhou Jiang, Qing Wang",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/1001.0315"
}
|
1001.0464
|
yearnumberscity
Michael Kowalczyk
Jin-Yi Cai
# Holant Problems for Regular Graphs with Complex Edge Functions
M. Kowalczyk Department of Mathematics and Computer Science, Northern
Michigan University
Marquette, MI 49855, USA mkowalcz@nmu.edu and J-Y. Cai Computer Sciences
Department, University of Wisconsin, Madison, WI 53706, USA jyc@cs.wisc.edu
###### Abstract.
We prove a complexity dichotomy theorem for Holant Problems on $3$-regular
graphs with an arbitrary complex-valued edge function. Three new techniques
are introduced: (1) higher dimensional iterations in interpolation; (2)
Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial
gadgets _in combination_ succeed in proving #P-hardness; and (3) algebraic
symmetrization, which significantly lowers the _symbolic complexity_ of the
proof for computational complexity. With _holographic reductions_ the
classification theorem also applies to problems beyond the basic model.
###### Key words and phrases:
Computational complexity
###### 1991 Mathematics Subject Classification:
F.2.1
The second author is supported by NSF CCF-0830488 and CCF-0914969.
## 1\. Introduction
In this paper we consider the following subclass of Holant Problems [FOCS08,
TAMC]. An input regular graph $G=(V,E)$ is given, where every $e\in E$ is
labeled with a (symmetric) edge function $g$. The function $g$ takes 0-1
inputs from its incident nodes and outputs arbitrary values in $\mathbb{C}$.
The problem is to compute the quantity ${\rm
Holant}(G)=\sum_{\sigma:V\rightarrow\\{0,1\\}}\prod_{\\{u,v\\}\in
E}g(\\{\sigma(u),\sigma(v)\\})$.
Holant Problems are a natural class of counting problems. As introduced in
[FOCS08, TAMC], the general Holant Problem framework can encode all Counting
Constraint Satisfaction Problems (#CSP). This includes special cases such as
weighted Vertex Cover, Graph Colorings, Matchings, and Perfect Matchings. The
subclass of Holant Problems in this paper can also be considered as (weighted)
$H$-homomorphism (or $H$-coloring) problems [BulatovG05, Homomorphisms,
acyclic, DyerG00, Goldberg-4, Hell] with an arbitrary $2\times 2$ symmetric
complex matrix $H$, however _restricted to_ regular graphs $G$ as input. E.g.,
Vertex Cover is the case when $H={\left[\begin{array}[]{cc}0&1\\\
1&1\end{array}\right]}$. When the matrix $H$ is a 0-1 matrix, it is called
unweighted. Dichotomy theorems (i.e., the problem is either in ${\rm{P}}$ or
#P-hard, depending on $H$) for unweighted $H$-homomorphisms with undirected
graphs $H$ and directed acyclic graphs $H$ are given in [DyerG00] and
[acyclic] respectively. A dichotomy theorem for any symmetric matrix $H$ with
non-negative real entries is proved in [BulatovG05]. Goldberg et al.
[Goldberg-4] proved a dichotomy theorem for all real symmetric matrices $H$.
Finally, Cai, Chen, and Lu have proved a dichotomy theorem for all complex
symmetric matrices $H$ [Homomorphisms].
The crucial difference between Holant Problems and #CSP is that in #CSP,
Equality functions of arbitrary arity are _presumed_ to be present. In terms
of $H$-homomorphism problems, this means that the input graph is allowed to
have vertices of arbitrarily high degrees. This may appear to be a minor
distinction; in fact it has a major impact on complexity. It turns out that if
Equality gates of arbitrary arity are freely available in possible inputs then
it is technically easier to prove #P-hardness. Proofs of previous dichotomy
theorems make extensive use of constructions called thickening and stretching.
These constructions require the availability of Equality gates of arbitrary
arity (equivalently, vertices of arbitrarily high degrees) to carry out.
Proving #P-hardness becomes more challenging in the degree restricted case.
Furthermore there are indeed cases within this class of counting problems
where the problem is #P-hard for general graphs, but solvable in ${\rm{P}}$
when restricted to 3-regular graphs.
We denote the (symmetric) edge function $g$ by $[x,y,z]$, where $x=g(0,0)$,
$y=g(0,1)=g(1,0)$ and $z=g(1,1)$. Functions will also be called gates or
signatures. (For Vertex Cover, the function corresponding to $H$ is the Or
gate, and is denoted by the signature $[0,1,1]$.) In this paper we give a
dichotomy theorem for the complexity of Holant Problems on 3-regular graphs
with arbitrary signature $g=[x,y,z]$, where $x,y,z\in\mathbb{C}$. First, if
$y=0$, the Holant Problem is easily solvable in ${\rm{P}}$. Assuming $y\not=0$
we may normalize $g$ and assume $y=1$. Our main theorem is as follows:
###### Theorem 1.1.
Suppose $a,b\in\mathbb{C}$, and let $X=ab$, $Z=(\frac{a^{3}+b^{3}}{2})^{2}$.
Then the Holant Problem on 3-regular graphs with $g=[a,1,b]$ is #P-hard except
in the following cases, for which the problem is in ${\rm{P}}$.
1. (1)
$X=1$.
2. (2)
$X=Z=0$.
3. (3)
$X=-1$ and $Z=0$.
4. (4)
$X=-1$ and $Z=-1$.
If we restrict the input to planar 3-regular graphs, then these four
categories are solvable in ${\rm{P}}$, as well as a fifth category $X^{3}=Z$.
The problem remains #P-hard in all other cases. 111Technically, computational
complexity involving complex or real numbers should, in the Turing model, be
restricted to computable numbers. In other models such as the Blum-Shub-Smale
model [BSS] no such restrictions are needed. Our results are not sensitive to
the exact model of computation. ∎
These results can be extended to $k$-regular graphs (we detail how this is
accomplished in a forthcoming work). One can also use holographic reductions
[HA_FOCS] to extend this theorem to more general Holant Problems.
In order to achieve this result, some new proof techniques are introduced. To
discuss this we first take a look at some previous results. Valiant
[Valiant79b, Valiant:sharpP] introduced the powerful technique of
_interpolation_ , which was further developed by many others. In [FOCS08] a
dichotomy theorem is proved for the case when $g$ is a Boolean function. The
technique from [FOCS08] is to provide certain algebraic criteria which ensure
that _interpolation_ succeeds, and then apply these criteria to prove that (a
large number yet) finitely many individual problems are #P-hard. This involves
(a small number of) gadget constructions, and the algebraic criteria are
powerful enough to show that they succeed in each case. Nonetheless this
involves a case-by-case verification. In [TAMC] this theorem is extended to
all real-valued $a$ and $b$, and we have to deal with infinitely many
problems. So instead of focusing on one problem, we devised (a large number
of) recursive gadgets and analyzed the regions of $(a,b)\in\mathbb{R}^{2}$
where they fail to prove #P-hardness. The algebraic criteria from [FOCS08] are
not suitable (Galois theoretic) for general $a$ and $b$, and so we formulated
weaker but simpler criteria. Using these criteria, the analysis of the failure
set becomes expressible as containment of semi-algebraic sets. As semi-
algebraic sets are decidable, this offers the ultimate possibility that _if_
we found enough gadgets to prove #P-hardness, _then_ there is a
_computational_ proof (of computational intractability) in a finite number of
steps. However this turned out to be a tremendous undertaking in symbolic
computation, and many additional ideas were needed to finally carry out this
plan. In particular, it would seem hopeless to extend that approach to all
complex $a$ and $b$.
In this paper, we introduce three new ideas. (1) We introduce a method to
construct gadgets that carry out iterations at a higher dimension, and then
collapse to a lower dimension for the purpose of constructing unary
signatures. This involves a starter gadget, a recursive iteration gadget, and
a finisher gadget. We prove a lemma that guarantees that among polynomially
many iterations, some subset of them satisfies properties sufficient for
interpolation to succeed (it may not be known _a priori_ which subset worked,
but that does not matter). (2) Eigenvalue Shifted Pairs are coupled pairs of
gadgets whose transition matrices differ by $\delta I$ where $\delta\neq 0$.
They have shifted eigenvalues, and by analyzing their failure conditions, we
can show that except on very rare points, one or the other gadget succeeds.
(3) Algebraic symmetrization. We derive a new expression of the Holant
polynomial over 3-regular graphs, with a crucially reduced degree. This
simplification of the Holant and related polynomials condenses the problem of
proving ${\\#\rm{P}}$-hardness to the point where all remaining cases can be
handled by symbolic computation. We also use the same expression to prove
tractability.
The rest of this paper is organized as follows. In Section 2 we discuss
notation and background information. In Section 3 we cover interpolation
techniques, including how to collapse higher dimensional iterations to
interpolate unary signatures. In Section LABEL:complexSignatures we show how
to perform algebraic symmetrization of the Holant, and introduce Eigenvalue
Shifted Pairs (ESP) of gadgets. Then we combine the new techniques to prove
Theorem 1.1.
## 2\. Notations and Background
We state the counting framework more formally. A signature grid
$\Omega=(G,{\mathcal{F}},\pi)$ consists of a labeled graph $G=(V,E)$ where
$\pi$ labels each vertex $v\in V$ with a function $f_{v}\in{\mathcal{F}}$. We
consider all edge assignments $\xi:E\rightarrow\\{0,1\\}$; $f_{v}$ takes
inputs from its incident edges $E(v)$ at $v$ and outputs values in
$\mathbb{C}$. The counting problem on the instance $\Omega$ is to
compute222The term Holant was first introduced by Valiant in [HA_FOCS] to
denote a related exponential sum.
${\rm Holant}_{\Omega}=\sum_{\xi}\prod_{v\in V}f_{v}(\xi\mid_{E(v)}).$
Suppose $G$ is a bipartite graph $(U,V,E)$ such that each $u\in U$ has degree
2. Furthermore suppose each $v\in V$ is labeled by an Equality gate $=_{k}$
where $k={\rm deg}(v)$. Then any non-zero term in ${\rm Holant}_{\Omega}$
corresponds to a 0-1 assignment $\sigma:V\rightarrow\\{0,1\\}$. In fact, we
can merge the two incident edges at $u\in U$ into one edge $e_{u}$, and label
this edge $e_{u}$ by the function $f_{u}$. This gives an edge-labeled graph
$(V,E^{\prime})$ where $E^{\prime}=\\{e_{u}:u\in U\\}$. For an edge-labeled
graph $(V,E^{\prime})$ where $e\in E^{\prime}$ has label $g_{e}$, ${\rm
Holant}_{\Omega}=\sum_{\sigma:V\rightarrow\\{0,1\\}}\prod_{e=(v,w)\in
E^{\prime}}g_{e}(\sigma(v),\sigma(w))$. If each $g_{e}$ is the same function
$g$ (but assignments $\sigma:V\rightarrow[q]$ take values in a finite set
$[q]$) this is exactly the $H$-coloring problem (for undirected graphs $g$ is
a symmetric function). In particular, if $(U,V,E)$ is a $(2,k)$-regular
bipartite graph, equivalently $G^{\prime}=(V,E^{\prime})$ is a $k$-regular
graph, then this is the $H$-coloring problem restricted to $k$-regular graphs.
In this paper we will discuss 3-regular graphs, where each $g_{e}$ is the same
symmetric complex-valued function. We also remark that for general bipartite
graphs $(U,V,E)$, giving Equality (of various arities) to all vertices on one
side $V$ defines #CSP as a special case of Holant Problems. But whether
Equality of various arities are present has a major impact on complexity, thus
Holant Problems are a refinement of #CSP.
A symmetric function $g:\\{0,1\\}^{k}\rightarrow\mathbb{C}$ can be denoted as
$[g_{0},g_{1},\ldots,g_{k}]$, where $g_{i}$ is the value of $g$ on inputs of
Hamming weight $i$. They are also called signatures. Frequently we will revert
back to the bipartite view: for $(2,3)$-regular bipartite graphs $(U,V,E)$, if
every $u\in U$ is labeled $g=[g_{0},g_{1},g_{2}]$ and every $v\in V$ is
labeled $r=[r_{0},r_{1},r_{2},r_{3}]$, then we also use
$\\#[g_{0},g_{1},g_{2}]\mid[r_{0},r_{1},r_{2},r_{3}]$ to denote the Holant
Problem. Note that $[1,0,1]$ and $[1,0,0,1]$ are Equality gates $=_{2}$ and
$=_{3}$ respectively, and the main dichotomy theorem in this paper is about
$\\#[x,y,z]\mid[1,0,0,1]$, for all $x,y,z\in\mathbb{C}$. We will also denote
$\mathrm{Hol}(a,b)=\\#[a,1,b]\mid[1,0,0,1]$. More generally, If
${\mathcal{G}}$ and ${\mathcal{R}}$ are sets of signatures, and vertices of
$U$ (resp. $V$) are labeled by signatures from ${\mathcal{G}}$ (resp.
${\mathcal{R}}$), then we also use $\\#{\mathcal{G}}\mid{\mathcal{R}}$ to
denote the bipartite Holant Problem. Signatures in ${\mathcal{G}}$ are called
generators and signatures in ${\mathcal{R}}$ are called recognizers. This
notation is particularly convenient when we perform holographic
transformations. Throughout this paper, all $(2,3)$-regular bipartite graphs
are arranged with generators on the degree 2 side and recognizers on the
degree 3 side.
We use ${\mathrm{Arg}}$ to denote the principal value of the complex argument;
i.e., ${\mathrm{Arg}}(c)\in(-\pi,\pi]$ for all nonzero $c\in\mathbb{C}$.
### 2.1. ${\mathcal{F}}$-Gate
Any signature from ${\mathcal{F}}$ is available at a vertex as part of an
input graph. Instead of a single vertex, we can use graph fragments to
generalize this notion. An ${\mathcal{F}}$-gate $\Gamma$ is a pair
$(H,{\mathcal{F}})$, where $H=(V,E,D)$ is a graph with some dangling edges $D$
(Figure 1 contains some examples). Other than these dangling edges, an
${\mathcal{F}}$-gate is the same as a signature grid. The role of dangling
edges is similar to that of external nodes in Valiant’s notion
[Valiant:Qciricuit], however we allow more than one dangling edge for a node.
In $H=(V,E,D)$ each node is assigned a function in ${\mathcal{F}}$ (we do not
consider “dangling” leaf nodes at the end of a dangling edge among these), $E$
are the regular edges, and $D$ are the dangling edges. Then we can define a
function for this ${\mathcal{F}}$-gate $\Gamma=(H,{\mathcal{F}})$,
$\Gamma(y_{1},y_{2},\ldots,y_{q})=\sum_{(x_{1},x_{2},\ldots,x_{p})\in\\{0,1\\}^{p}}H(x_{1},x_{2},\ldots,x_{p},y_{1},y_{2},\ldots,y_{q}),$
where $p=|E|$, $q=|D|$, $(y_{1},y_{2},\ldots,y_{q})\in\\{0,1\\}^{q}$ denotes
an assignment on the dangling edges, and
$H(x_{1},x_{2},\ldots,x_{p},y_{1},y_{2},\ldots,y_{q})$ denotes the value of
the partial signature grid on an assignment of all edges, i.e., the product of
evaluations at every vertex of $H$, for
$(x_{1},x_{2},\ldots,x_{p},y_{1},y_{2},\ldots,y_{q})\in\\{0,1\\}^{p+q}$.
(a) A starter gadget
(b) A recursive gadget
(c) A finisher gadget
(d) A planar embedding of a single iteration
Figure 1. Examples of binary starter, recursive, and finisher gadgets
We will also call this function the signature of the ${\mathcal{F}}$-gate
$\Gamma$. An ${\mathcal{F}}$-gate can be used in a signature grid as if it is
just a single node with the same signature. We note that even for a very
simple signature set ${\mathcal{F}}$, the signatures for all
${\mathcal{F}}$-gates can be quite complicated and expressive. Matchgate
signatures are an example [Valiant:Qciricuit].
The dangling edges of an $\mathcal{F}$-gate are considered as input or output
variables. Any $m$-input $n$-output $\mathcal{F}$-gate can be viewed as a
$2^{n}$ by $2^{m}$ matrix $M$ which transforms arity-$m$ signatures into
arity-$n$ signatures (this is true even if $m$ or $n$ are 0). Our construction
will transform symmetric signatures to symmetric signatures. This implies that
there exists an equivalent $n+1$ by $m+1$ matrix $\widetilde{M}$ which
operates directly on column vectors written in symmetric signature notation.
We will henceforth identify the matrix $\widetilde{M}$ with the
$\mathcal{F}$-gate itself. The constructions in this paper are based upon
three different types of bipartite $\mathcal{F}$-gates which we call starter
gadgets, recursive gadgets, and finisher gadgets. An arity-$r$ starter gadget
is an $\mathcal{F}$-gate with no input but $r$ output edges. If an
$\mathcal{F}$-gate has $r$ input and $r$ output edges then it is called an
arity-$r$ recursive gadget. Finally, an $\mathcal{F}$-gate is an arity-$r$
finisher gadget if it has $r$ input edges 1 output edge. As a matter of
convention, we consider any dangling edge incident with a generator as an
output edge and any dangling edge incident with a recognizer as an input edge;
see Figure 1.
## 3\. Interpolation Techniques
### 3.1. Binary recursive construction
In this section, we develop our new technique of higher dimensional iterations
for interpolation of unary signatures.
###### Lemma 3.1.
Suppose $M\in\mathbb{C}^{3\times 3}$ is a nonsingular matrix,
$s\in\mathbb{C}^{3}$ is a nonzero vector, and for all integers $k\geq 1$, $s$
is not a column eigenvector of $M^{k}$. Let $F_{i}\in\mathbb{C}^{2\times 3}$
be three matrices, where ${\rm rank}(F_{i})=2$ for $1\leq i\leq 3$, and the
intersection of the row spaces of $F_{i}$ is trivial $\\{0\\}$. Then for every
$n$, there exists some $F\in\\{F_{i}:1\leq i\leq 3\\}$, and some
$S\subseteq\\{FM^{k}s:0\leq k\leq n^{3}\\}$, such that $|S|\geq n$ and vectors
in $S$ are _pairwise_ linearly independent.
###### Proof 3.2.
Let $k>j\geq 0$ be integers. Then $M^{k}s$ and $M^{j}s$ are nonzero and also
linearly independent, since otherwise $s$ is an eigenvector of $M^{k-j}$. Let
$N=[M^{j}s,M^{k}s]\in\mathbb{C}^{3\times 2}$, then ${\rm rank}(N)=2$, and
$\mathrm{ker}(N^{\mathrm{T}})$ is a 1-dimensional linear subspace. It follows
that there exists an $F\in\\{F_{i}:1\leq i\leq 3\\}$ such that the row space
of $F$ does not contain $\mathrm{ker}(N^{\mathrm{T}})$, and hence has trivial
intersection with $\mathrm{ker}(N^{\mathrm{T}})$. In other words,
$\mathrm{ker}(N^{\mathrm{T}}F^{\mathrm{T}})=\\{0\\}$. We conclude that
$FN\in\mathbb{C}^{2\times 2}$ has rank 2, and $FM^{j}s$ and $FM^{k}s$ are
linearly independent.
Each $F_{i}$, where $1\leq i\leq 3$, defines a coloring of the set
$K=\\{0,1,\dots,n^{3}\\}$ as follows: color $k\in K$ with the linear subspace
spanned by $F_{i}M^{k}s$. Thus, $F_{i}$ defines an equivalence relation
$\approx_{i}$ where $k\approx_{i}k^{\prime}$ iff they receive the same color.
Assume for a contradiction that for each $F_{i}$, where $1\leq i\leq 3$, there
are not $n$ pairwise linearly independent vectors among $\\{F_{i}M^{k}s:k\in
K\\}$. Then, including possibly the 0-dimensional space $\\{0\\}$, there can
be at most $n$ distinct colors assigned by $F_{i}$. By the pigeonhole
principle, some $k$ and $k^{\prime}$ with $0\leq k<k^{\prime}\leq n^{3}$ must
receive the same color for all $F_{i}$, where $1\leq i\leq 3$. This is a
contradiction and we are done. ∎
The next lemma says that under suitable conditions we can construct all unary
signatures $[x,y]$. The method will be interpolation at a higher dimensional
iteration, and finishing up with a suitable _finisher_ gadget. The crucial new
technique here is that when iterating at a higher dimension, we can guarantee
the existence of _one_ finisher gadget that succeeds on polynomially many
steps, which results in overall success. Different finisher gadgets may work
for different initial signatures and different input size $n$, but these need
not be known in advance and have no impact on the final success of the
reduction.
###### Lemma 3.3.
Suppose that the following gadgets can be built using complex-valued
signatures from a finite generator set $\mathcal{G}$ and a finite recognizer
set $\mathcal{R}$.
1. (1)
A binary starter gadget with nonzero signature $[z_{0},z_{1},z_{2}]$.
2. (2)
A binary recursive gadget with nonsingular recurrence matrix $M$, for which
$[z_{0},z_{1},z_{2}]^{\mathrm{T}}$ is not a column eigenvector of $M^{k}$ for
any positive integer $k$.
3. (3)
Three binary finisher gadgets with rank 2 matrices
$F_{1},F_{2},F_{3}\in\mathbb{C}^{2\times 3}$, where the intersection of the
row spaces of $F_{1}$, $F_{2}$, and $F_{3}$ is the zero vector.
Then for any $x,y\in\mathbb{C}$,
$\\#\mathcal{G}\cup\\{[x,y]\\}\mid\mathcal{R}\leq_{T}\\#\mathcal{G}\mid\mathcal{R}$.
|
arxiv-papers
| 2010-01-04T12:33:25 |
2024-09-04T02:49:07.467754
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Kowalczyk and Jin-Yi Cai",
"submitter": "Michael Kowalczyk",
"url": "https://arxiv.org/abs/1001.0464"
}
|
1001.0527
|
# Squeezed correlations of strange particle-antiparticles
Sandra S Padula1, Danuce M Dudek1 and O Socolowski Jr2 1 Inst. Física Teórica
- UNESP, C. P. 70532-2, 01156-970 São Paulo, SP, Brazil 2 Departamento de
Física, FURG, C. P. 474, 96201-900 Rio Grande, RS, Brazil padula@ift.unesp.br
###### Abstract
Squeezed correlations of hadron-antihadron pairs are predicted to appear if
their masses are modified in the hot and dense medium formed in high energy
heavy ion collisions. If discovered experimentally, they would be an
unequivocal evidence of in-medium mass shift found by means of hadronic
probes. We discuss a method proposed to search for this novel type of
correlation, illustrating it by means of $D_{s}$-mesons with in-medium shifted
masses. These particles are expected to be more easily detected and identified
in future upgrades at RHIC.
###### pacs:
25.75.Gz, 25.75.-q, 21.65.Jk
## 1 Introduction: The hadronic squeezed states
About ten years ago, M. Asakawa, T. Csörgő and M. Gyulassy[1] finalized a
model description for the effects of in-medium hadronic mass modification
leading to correlations of boson-antiboson pairs, also known as Back-to-Back
Correlations (BBC). This type of correlation between a particle and its own
antiparticle was first noted by R. Weiner et al.[2]. Within a short period
after the final proposition in Ref.[1], P. K. Panda et al.[3] showed that
similar correlations between a fermion and a antifermion should appear, if
their masses were shifted in the hot and dense media formed in high energy
heavy ion collisions.
In both the bosonic and the fermionic cases, the in-medium quasi-particles
produced in those collisions are related to the asymptotic, observed
particles, by means of a Bogoliubov-Valatin (BV) transformation. This
transformation links the creation and annihilation operators in both
environments, i.e., the asymptotic operators $a$ and $a^{\dagger}$, to their
in-medium counterparts, $b$ and $b^{\dagger}$. The corresponding Hamiltonians
are given by $H_{0}$ and $H_{m}=H_{0}+H^{\prime}$, where $H^{\prime}$ contains
the parameter expressing the mass-shift.
The BV transformation relating the operators $a$ ($a^{\dagger}$) to $b$
($b^{\dagger}$) are given by
$a_{k}=c_{k}b_{k}+s^{*}_{-k}b^{\dagger}_{-k}\;;\;a^{\dagger}_{k}=c^{*}_{k}b^{\dagger}_{k}+s_{-k}b_{-k}$,
where $c_{k}=\cosh(f_{k})$ and $s_{k}=\sinh(f_{k})$. The argument is the
squeezing parameter, named in this way because the BV transformation is
equivalent to a squeezing operation. It is written as
$f_{k}=\frac{1}{2}\log\left(\frac{\omega_{k}}{\Omega_{k}}\right)$, with
$\omega_{k}^{2}={\mathbf{k}}^{2}+m^{2}$, $m$ being the asymptotic mass,
$\Omega_{k}^{2}={\mathbf{k}}^{2}+m_{*}^{2}$, $m_{*}$ being the in-medium
modified mass, and $\mathbf{k}$ is the momentum. A constant mass-shift is
considered here, homogeneously distributed over all the system, and related to
the asymptotic mass by $m_{*}=m\pm\delta M$. More generally, however, it could
be a function of the coordinates inside the system and the momenta, $\delta
M=\delta M(|{\mathbf{r}}|,|{\mathbf{k}}|$). Both the bosonic and the fermionic
squeezed correlations are positive, have unlimited intensity, and are also
described by similar formalisms [1, 3]. This is illustrated in Ref. [3] for
the case of $\phi\phi$ and $\bar{p}p$ pairs, evidencing the resemblance of the
correlations for bosons and for fermions of similar asymptotic masses. In the
remainder of this paper, we will focus on the bosonic case.
The effects of the shifted mass on the squeezed particle-antiparticle
correlations can be understood by analyzing the the joint probability for
observing two particles,
$N_{2}(\mathbf{k}_{1},\mathbf{k}_{2})=\omega_{\mathbf{k}_{1}}\omega_{\mathbf{k}_{2}}\Bigl{[}\langle
a^{\dagger}_{\mathbf{k}_{1}}a_{\mathbf{k}_{1}}\rangle\langle
a^{\dagger}_{\mathbf{k}_{2}}a_{\mathbf{k}_{2}}\rangle+\langle
a^{\dagger}_{\mathbf{k}_{1}}a_{\mathbf{k}_{2}}\rangle\langle
a^{\dagger}_{\mathbf{k}_{2}}a_{\mathbf{k}_{1}}\rangle+\langle
a^{\dagger}_{\mathbf{k}_{1}}a^{\dagger}_{\mathbf{k}_{2}}\rangle\langle
a_{\mathbf{k}_{2}}a_{\mathbf{k}_{1}}\rangle\Bigr{]}$, which results from the
application of a generalization of Wick’s theorem for locally equilibrated
systems[4, 6]; $\langle...\rangle$ means thermal averages. In this expression,
all three contributions could survive when the boson is its own antiparticle,
as is the case of $\phi$-mesons or $\pi^{0}$’s. The last term is in general
identically zero. However, if the particle’s mass is shifted in-medium, it can
contribute significantly. It is identified with the square modulus of the
squeezed amplitude,
$G_{s}({\mathbf{k}_{1}},{\mathbf{k}_{2}})=\sqrt{\omega_{\mathbf{k}_{1}}\omega_{\mathbf{k}_{2}}}\;\langle
a_{\mathbf{k}_{1}}a_{\mathbf{k}_{2}}\rangle$. The first term corresponds to
the product of the spectra of the two identical bosons,
$N_{1}(\mathbf{k}_{i})\\!=\\!\omega_{\mathbf{k}_{i}}\frac{d^{3}N}{d\mathbf{k}_{i}}\\!=\\!\omega_{\mathbf{k}_{i}}\,\langle
a^{\dagger}_{\mathbf{k}_{i}}a_{\mathbf{k}_{i}}\rangle$, and the second term,
to the identical particle contribution, identified with the square modulus of
the chaotic amplitude,
$G_{c}({\mathbf{k}_{1}},{\mathbf{k}_{2}})=\sqrt{\omega_{\mathbf{k}_{1}}\omega_{\mathbf{k}_{2}}}\;\langle
a^{\dagger}_{\mathbf{k}_{1}}a_{\mathbf{k}_{2}}\rangle$.
The full two-particle correlation function for $\phi\phi$ or $\pi^{0}\pi^{0}$
can be written as
$C_{2}({\mathbf{k}_{1}},{\mathbf{k}_{2}})=1+\frac{|G_{c}({\mathbf{k}_{1}},{\mathbf{k}_{2}})|^{2}}{G_{c}({\mathbf{k}_{1}},{\mathbf{k}_{1}})G_{c}({\mathbf{k}_{2}},{\mathbf{k}_{2}})}+\frac{|G_{s}({\mathbf{k}_{1}},{\mathbf{k}_{2}})|^{2}}{G_{c}({\mathbf{k}_{1}},{\mathbf{k}_{1}})G_{c}({\mathbf{k}_{2}},{\mathbf{k}_{2}})}.$
(1)
In case of charged mesons, such as $D_{s}^{\pm}$ , the terms in Eq.(1) would
act independently, i.e., either the first and the second terms together would
contribute to the HBT effect ($D_{s}^{\pm}D_{s}^{\pm}$), or the first and the
last terms, to the BBC effect ($D_{s}^{+}D_{s}^{-}$).
In Refs.[1, 3], an infinite system was considered. Later, a finite expanding
system was studied, within a non-relativistic approach, assuming flow-
independent squeezing parameter, which allowed for obtaining analytical
solutions to the problem[7]. Considering a hydrodynamical ensemble[1, 5, 7]
the squeezed amplitude results in
$\displaystyle G_{s}(\mathbf{k}_{1},\mathbf{k}_{2})$ $\displaystyle=$
$\displaystyle\frac{E_{{}_{1,2}}}{(2\pi)^{\frac{3}{2}}}|c_{{}_{0}}||s_{{}_{0}}|\Bigl{\\{}R^{3}\exp\Bigl{[}-\frac{R^{2}}{2}(\mathbf{k}_{1}+\mathbf{k}_{2})^{2}\Bigr{]}+2n^{*}_{0}R_{*}^{3}\exp\Bigl{[}-\frac{(\mathbf{k}_{1}-\mathbf{k}_{2})^{2}}{8m_{*}T}\Bigr{]}$
(2) $\displaystyle\times\exp\Big{[}\Bigl{(}-\frac{im\langle u\rangle
R}{2m_{*}T_{*}}-\frac{1}{8m_{*}T_{*}}-\frac{R_{*}^{2}}{2}\Bigr{)}(\mathbf{k_{1}}+\mathbf{k_{2}})^{2}\Big{]}\Bigl{\\}},$
and the spectrum in
$G_{c}(\mathbf{k}_{i},\mathbf{k}_{i})=\frac{E_{i,i}}{(2\pi)^{\frac{3}{2}}}\Bigl{\\{}|s_{{}_{0}}|^{2}R^{3}+n^{*}_{0}R_{*}^{3}(|c_{{}_{0}}|^{2}+|s_{{}_{0}}|^{2})\exp\Bigl{(}-\frac{\mathbf{k}_{i}^{2}}{2m_{*}T_{*}}\Bigr{)}\Bigr{\\}},$
(3)
where $R_{*}=R\sqrt{T/T_{*}}$ and $T_{*}=T+\frac{m^{2}\langle
u\rangle^{2}}{m_{*}}$ [7]. For the sake of simplicity, the system was supposed
to be Gaussian in shape, with a cross-sectional area of radius $R$, and $T$ is
the freeze-out temperature; $R_{*}$ and $T_{*}$ are, respectively, the flow-
modified radius and temperature. The flow velocity, introduced before
estimating the results in Eqs. (2) and (3), was written as $\mathbf{v}=\langle
u\rangle\mathbf{r}/R$, where the values $\langle u\rangle=0,0.23$ or $0.5$
were used in the present work. For finite particle emissions, we considered a
Lorentzian distribution, $F(\Delta t)=[1+(\omega_{1}+\omega_{2})^{2}\Delta
t^{2}]^{-1}$, multiplying the third term in Eq. (1). We adopt here
$\hbar=c=1$. The results in Eq.(2) and (3) are then introduced, respectively,
in the third and first terms of Eq.(1), leading to the squeezed correlation
function.
## 2 Illustrative results
We previously applied the analytical results shown above to $\phi$-meson
squeezed correlations, and later, to $K^{+}K^{-}$ pairs. We investigated the
behavior of the squeezed correlation function for precise back-to-back pairs,
i.e., for particle-antiparticle pairs emitted with exactly opposite momenta,
studying $C_{s}(\vec{k},-\vec{k},m_{*})$ as a function of $|\vec{k}|$ and
$m_{*}$. Preliminary results for those particles where shown in previous
meetings[7]-[13]. Recent results considering $\phi\phi$ pairs from simulation
are in Ref. [14].
Figure 1: Squeezed correlation function versus the shifted mass and the
momenta of the particles for back-to-back $D_{s}^{+}D_{s}^{-}$ pairs.
The analytical results can be applied to any other particles subjected to in-
medium mass-shift and compatible with the non-relativistic limit considered in
the formulation. Recently, STAR reconstructed $D^{0}+\bar{D^{0}}$’s through
their decay into $K^{\mp}\pi^{\pm}$, by measuring the invariant mass
distribution of those decay products[15]. The identification of $D$-mesons
could be improved after the detector’s upgrade. Motivated by this possibility,
and considering that charged mesons may be easier to observe, we apply that
formulation to the case of $D_{s}^{+}D_{s}^{-}$ pairs. Similar to what was
previously done for $\phi\phi$ and $K^{+}K^{-}$, we analyze the behavior of
$C_{s}(\vec{k},-\vec{k},m_{*})$ for $D_{s}^{+}D_{s}^{-}$ pairs in the
$(|\vec{k}|,m_{*})$-plane. Fig. 1 shows that the intensity of
$C_{s}({\mathbf{k}},-{\mathbf{k}},m_{*})$ vs. $|{\mathbf{k}}|$ vs. $m_{*}$
increases for decreasing freeze-out temperatures, which can be seen by
comparing the two left plots with $T=220$ MeV, with the bottom right plot with
$T=140$ MeV, in all of which $\Delta t=2$ fm/c. Fig. 1 also illustrates
another striking feature, i.e., finite emission intervals can dramatically
reduce the strength of the squeezed correlation function. This can be seen by
comparing the two plots in the right panel: a Lorentzian emission distribution
with $\Delta t=2$ fm/c has the effect of reducing the signal by about three
orders of magnitude, as compared to the instantaneous emission ($\Delta t=0$).
Finally, the left panel shows how the behavior of
$Cs({\mathbf{k}},-{\mathbf{k}},m_{*})$ is affected by the presence of flow. We
see that the growth of the signal’s intensity for increasing values of
$|\mathbf{k}|$ is faster in the static case than when $<u>\neq 0$.
Nevertheless, flow seems to enhance the overall intensity of
$Cs({\mathbf{k}},-{\mathbf{k}},m_{*})$ in the whole region of $|\mathbf{k}|$
investigated. Naturally, at $m_{*}=m_{D_{s}}\sim 1969$ MeV, the squeezing
disappears, i.e., $C_{s}({\mathbf{k}},-{\mathbf{k}},m_{*})\equiv 1$.
A panoramic view of the variation of $Cs({\mathbf{k}},-{\mathbf{k}},m_{*})$ in
the $(|\mathbf{k}|,m_{*})$-plane can be better appreciated by contour plots of
the previous results, as shown in Fig. 2.
Figure 2: Contour plots of the results of Fig. 1, corresponding to the
squeezed correlation functions vs. $|\mathbf{k}|$ and $m_{*}$.
We can see that the location of the maxima of
$C_{s}({\mathbf{k}},-{\mathbf{k}},m_{*})$ for $D_{s}^{+}D_{s}^{-}$ pairs is
shifted about $\approx 2-2.5\%$ from the value of the particle’s asymptotic
mass, either above or below it. A similar behavior could be observed in the
case of $\phi\phi$ pairs [8]-[14]. However, for $K^{+}K^{-}$ pairs, the
maximum was shifted about $30\%$ from the $K$’s asymptotic mass [12, 13],
perhaps signaling to the limit of applicability of the non-relativistic
approximations considered in the model, in this case.
The outcome properties shown above are important for understanding the
expected behavior of the squeezed correlation function, within the
approximations of the proposed model. However, for practical purposes of
searching for it experimentally, the approach analyzed so far is not very
helpful, since the modified mass of particles is not a measurable quantity,
existing only inside the hot and dense medium. Besides, the measurement of
particle-antiparticle pairs with exactly back-to-back momenta is unrealistic.
Thus, a promising form to empirically search for the hadronic squeezed
correlation was proposed, in analogy with HBT [11]-[14], i.e., to measure the
squeezed correlation function in terms of the momenta of the particles
combined as
${\mathbf{K}}_{12}=\\!\frac{1}{2}({\mathbf{k}_{1}}+{\mathbf{k}_{2}})$, and
${\mathbf{q}_{12}}=({\mathbf{k}_{1}}-{\mathbf{k}_{2}})$. In a relativistic
treatment, as proposed by M. Nagy [11], we should construct the momentum
variable as
$Q^{\mu}_{back}=(\omega_{1}-\omega_{2},\mathbf{k}_{1}+\mathbf{k}_{2})=(q^{0},2\mathbf{K})$.
In fact, it is preferable to redefine this variable as
$Q^{2}_{bbc}=-(Q_{back})^{2}=4(\omega_{1}\omega_{2}-K^{\mu}K_{\mu})$, whose
non-relativistic limit is $Q^{2}_{bbc}\rightarrow(2{\mathbf{K}_{12}})^{2}$,
correctly recovering that variable.
The results for $C_{s}({2{\mathbf{K}_{12}}},{\mathbf{q}_{12}})$ are shown in
Fig. 3. In the plots on the left panel, the radius of the system was fixed to
be $R=4$ fm, and on the right, to $R=7$ fm. In both, it was assumed that the
mass was shifted by the amount corresponding to the lower maximum in Figs. 1
and 2, a relative reduction in the mass of about $2\%$.
Figure 3: Squeezed correlation functions for $R=4$ fm (left panel) and $R=7$
fm (right panel), considering a reduction of the in-medium mass to
$m_{*}=1930$ MeV.
From Fig. 3 we can see that radial flow ($\langle u\rangle=0.23$) does have an
effect on $C_{s}(2{\mathbf{K}_{12}},{\mathbf{q}_{12}})$, making it more
intense if compared to $\langle u\rangle=0$, in all the investigated region of
the $({\mathbf{2}K_{12}},{\mathbf{q}_{12}})$-plane, from about $40\%$, at low
$|\mathbf{q}_{12}|$, to roughly $15\%$, at high $|\mathbf{q}_{12}|$. Fig. 3
also shows that the inverse width of
$C_{s}(2{\mathbf{K}_{12}},{\mathbf{q}_{12}})$ reflects the size of the
squeezing region, being narrower for larger systems ($R=7$ fm) than for
smaller ones ($R=4$ fm). We also note that the intensity of the squeezed
correlation function is high enough for stimulating its experimental search,
even after applying the time reduction factor corresponding to an emission
during a finite interval of $\Delta t=2$ fm/c.
## 3 Summary and conclusions
The results of Fig. 1 and 2 showed that the squeezed correlation function
survives both finite system sizes and flow, with measurable intensity. The
finite emission process has a strong reduction effect in its strength, even if
it happens in a sudden manner, as considered. The plots in the right panel of
Fig. 1 show that the time multiplicative factor reduces
$Cs({\mathbf{k}},-{\mathbf{k}},m_{*})$ about three orders of magnitude, for
$D_{s}^{+}D_{s}^{-}$ pairs. The left panel in Fig. 1 shows that, if the system
is subjected to flow, this could enhance the squeezed correlation signal,
facilitating its potential discovery in experiments. The freeze-out
temperature is an essential ingredient, the lower the temperature, the higher
the intensity of $Cs({\mathbf{k}},-{\mathbf{k}},m_{*})$. The expectations are
that $D_{s}$’s would decouple early. From Fig. 1 we can see that, even in a
high temperature limit and subjected to the time-reducing factor, the squeezed
correlation function still has measurable intensity.
We should remember that, if the shift in the mass is away from the value
considered in the calculation leading to Fig. 3,
$C_{s}(2{\mathbf{K}_{12}},{\mathbf{q}_{12}})$ would attain smaller values than
the ones shown, but the signal could still be high enough to be searched for.
Another important point to emphasize is that it is crucial to accumulate high
statistics and look for the effect in the $3-D$ configuration shown in the
plots, since projecting it into the ${\mathbf{K}_{12}}$-axis can drastically
reduce the signal, making it more difficult to discover the hadronic squeezing
effect. Finally, comparing with previous results for $\phi\phi$ [8]-[14] and
$K^{+}K^{-}$[11, 12, 13]-[16, 17], we can conclude that, within this model,
the strength of squeezed correlation function seems to grow with the
asymptotic mass of the particles involved, making it even more promising to
look for BBC’s for heavier particles.
We are grateful to FAPESP and CAPES for the support to participate in the SQM
’09.
## References
## References
* [1] Asakawa M, Csörgő T and Gyulassy M 1999 Phys. Rev. Lett. 83 4013
* [2] Andreev I V, Plümer M and Weiner R M 1991 Phys. Rev. Lett. 67 3475
* [3] Panda P K, Csörgő T, Hama Y, Krein G and Padula Sandra S 2001 Phys. Lett B 512 49
* [4] Gyulassy M, Kauffmann S K and Wilson L W 1979 Phys. Rev. C 20 2267
* [5] Makhlin A and Sinyukov Yu 1987 Sov. J. Nucl. Phys. 46 354
* [6] Sinyukov Yu 1994 Nucl. Phys. A 566 589c
* [7] Padula Sandra S, Hama Y, Krein G, Panda P K and Csörgő T 2006 Phys. Rev. C 73 044906
* [8] Padula Sandra S, Hama Y, Krein G, Panda P K and Csörgő T 2006 Proc. Quark Matter ’05 (Budapest) Nucl. Phys. A774, 615
* [9] Padula Sandra S, Hama Y, Krein G, Panda P K and Csörgő T 2006 Proc. Workshop on Particle Correlations and Femtoscopy (Kromeriz) AIP Conf. Proc. 828, 645
* [10] Csörgő T and Padula Sandra S 2007 Proc. WPCF 2006 (São Paulo) Braz. J. Phys. 37 949
* [11] Padula Sandra S, Socolowski Jr O, Csörgő T and Nagy M 2008 Proc. Quark Matter 2008 (Jaipur) J. Phys. G: Nucl. Part. Phys. 35, 104141
* [12] Padula Sandra S, Dudek Danuce M and Socolowski Jr O, 2009 Proc. WPCF 2008 (Krakow) A. Phys. Pol. 40, N. 4, 1225
* [13] Padula Sandra S, Socolowski Jr O and Dudek Danuce M 2009 Proc. ISMD 2008 (Hamburg) DESY PROC 2009 01 271 (ArXiv:0812.1784v1 (nucl-th) and ArXiv:0902.0377 (hep-ph) p271)
* [14] Padula Sandra S, Socolowski Jr O 2009 Searching for squeezed particle-antiparticle correlations in high energy heavy ion collisions ArXiv:1001.0126-v1 (nucl-th).
* [15] Abelev B I et al., STAR Collaboration ArXiv:0805.0364 (nucl-ex).
* [16] Dudek Danuce M 2009 Master Dissertation
* [17] Padula Sandra S and Dudek Danuce M, in preparation.
|
arxiv-papers
| 2010-01-03T00:57:58 |
2024-09-04T02:49:07.474291
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sandra S Padula, Danuce M Dudek and O Socolowski Jr",
"submitter": "Otavio Socolowski Jr.",
"url": "https://arxiv.org/abs/1001.0527"
}
|
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