#  Instantaneous interquark potential in generalized Landau gauge in SU(3)
lattice QCD: a possible gauge for the quark potential model

Takumi Iritani  Department of Physics, Graduate School of Science, Kyoto
University,
Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan  Hideo Suganuma  Department
of Physics, Graduate School of Science, Kyoto University,
Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan

###### Abstract

We investigate “instantaneous interquark potential”, an interesting gauge-
dependent quantity defined from the spatial correlator
$\langle\mathrm{Tr}[U_{4}^{\dagger}(s)U_{4}(s^{\prime})]\rangle$ of the
temporal link-variable $U_{4}$, in detail in generalized Landau gauge using
SU(3) quenched lattice QCD. While the instantaneous potential has no linear
part in the Landau gauge, in the Coulomb gauge, it is expressed by the Coulomb
plus linear potential, where the slope is 2-3 times larger than the physical
string tension, and the lowest energy state is considered to be a gluon-chain
state. Using the generalized Landau gauge, we find that the instantaneous
potential can be continuously described between the Landau and the Coulomb
gauges, and it approximately reproduces the physical interquark potential in a
specific intermediate gauge, which we call “$\lambda_{C}$-gauge”. This
$\lambda_{C}$-gauge is expected to provide a quark-potential-model picture,
where dynamical gluons do not appear. We also investigate $T$-length
terminated Polyakov-line correlator and its corresponding “finite-time
potential” in generalized Landau gauge.

###### pacs:

12.38.Gc,14.70.Dj,12.39.Jh,12.39.Pn

##  I Introduction

Nowadays, quantum chromodynamics (QCD) is established as the fundamental
theory of the strong interaction, and perturbative QCD gives a standard
framework to describe high-energy reactions of hadrons. QCD is a nonabelian
gauge theory constructed from quarks and gluons, and color SU(3) gauge
symmetry is one of the guiding principles in formulating QCD Nambu66 ; GWP73 .
For actual perturbative calculations of QCD, the gauge has to be fixed to
remove gauge degrees of freedom. In the low-energy region, however, QCD
exhibits a strong-coupling nature and the resulting nonperturbative QCD is a
very difficult and complicated theory.

On the other hand, the quark potential model has been phenomenologically used
for the description of low-energy properties of hadrons in terms of their
underlying structure. The quark potential model is a successful
nonrelativistic or semi-relativistic framework with a potential
instantaneously acting among quarks, and describes many hadron properties in
terms of quark degrees of freedom. In this model, there are no dynamical
gluons, and gluonic effects indirectly appear as the instantaneous interquark
potential. (Here, the concept of “instantaneous” is to be considered at the
scale of the effective model.)

In spite of a great success of the quark potential model, its relation to QCD
is not yet so clear, and to link the quark potential model from QCD is one of
the important subjects in hadron physics Isgur8586 ; BPSV0005 ; TS0304 . In
principle, the quark model may be obtained from QCD after integrating out the
gluon degrees of freedom in the path integral formalism. Or, from the
viewpoint of “gauge” in QCD, the quark model without dynamical gluons may be
regarded as a gauge-fixed effective theory of QCD. In fact, the quark
potential model does not have local color SU(3) symmetry but has global color
SU(3) symmetry Isgur8586 ; HanNambu , since each quark has a color and
dynamical gluons are absent in this model framework.

If so, what gauge of QCD corresponds to the quark model? Since the quark model
has global color SU(3) symmetry and spatial-rotation symmetry, one may first
consider that the Landau gauge and the Coulomb gauge would be candidates of
such a gauge for the quark model. Actually, the Coulomb gauge is defined by
minimizing “spatial gluon-field fluctuations” in total, as will be shown in
Sec.II, and therefore one may expect only a small gluon-field fluctuation
appearing in the Coulomb gauge. Similarly, the Landau gauge is defined by
minimizing “total gluon-field fluctuations” in Euclidean QCD, so that only a
small gluon-field fluctuation may be expected in the Landau gauge. Such a
minimal gluonic content in these gauges seems to be preferable for the
modeling only with quarks.

In this paper, we investigate the instantaneous interquark potential in the
Landau, the Coulomb, and their intermediate gauges, i.e., “generalized Landau
gauge” (or “$\lambda$-gauge”), in SU(3) lattice QCD, from the viewpoint of the
quark potential model Iritani10 . Here, the generalized Landau gauge is a
natural general gauge to connect the Landau, the Coulomb, and the temporal
gauges, by one real parameter $\lambda$.

Besides the quark-model arguments, it is meaningful to investigate the
connection between the Landau and the Coulomb gauges, using the generalized
Landau gauge. Actually, these gauges have been often used as the typical gauge
in QCD, but their physical pictures seem to be rather different for several
important arguments in QCD.

As the typical example, the color confinement, which is an important gauge-
invariant QCD phenomenon, can be explained from various viewpoints in various
gauges. In the Landau gauge, the color confinement is mathematically
investigated by the Kugo-Ojima criterion, in terms of the BRST charge and the
inverse Higgs theorem KugoOjima . In the Coulomb gauge, the color confinement
is argued from the viewpoint of a large instantaneous Coulomb energy Gribov ;
Zwanziger98 ; Greensite03 ; Greensite04 , and its resulting gluon-chain
picture GreensiteThorn ; tHooft . Taking the maximally Abelian (MA) gauge, the
quark confinement has been discussed in terms of the dual superconductor
picture NambutHooftMandelstam .

Of course, in gauge theories, the physical quantities never depend on the
gauge choice. However, according to the gauge choice, the physical picture can
be changed, and the role of the gauge field, which is fundamental field of the
gauge theory, can be also changed. Then, it is important to link the different
gauges, and investigate the role of gluons in each gauge. The role and the
properties of gluons are expected to be clarified by the overview on the
structure of gauge dependence.

One of the aim in this paper is to investigate the gluonic properties through
a continuous view from the Landau gauge to the Coulomb gauge, using
generalized Landau gauge. In particular, we clarify the behavior of the
instantaneous potential, as an interesting gluonic correlation.

The organization of this paper is as follows. In Sec.II, we briefly review the
properties of the Landau gauge and the Coulomb gauge. In Sec.III, we give the
formalism of generalized Landau gauge ($\lambda$ gauge). In Sec.IV, we
formulate the instantaneous potential in lattice QCD. In Sec.V, we show the
lattice QCD results. In Sec.VI, we investigate Polyakov-line correlators and
its relation to the potential. Sec.VII will be devoted to Summary and
Discussions.

##  II Landau gauge and Coulomb gauge

In this section, we briefly review the properties of the Landau gauge and the
Coulomb gauge.

###  II.1 Landau gauge

The Landau gauge is one of the most popular gauges in QCD, and its gauge
fixing is given by

$\partial_{\mu}A_{\mu}=0,$ (1)

where $A_{\mu}(x)\equiv A_{\mu}^{a}(x)T^{a}\in\mathfrak{su}(N_{c})$ are gluon
fields, with $\mathfrak{su}(N_{c})$ generator $T^{a}(a=1,2,\dots
N_{c}^{2}-1)$. The Landau gauge keeps the Lorentz covariance and the global
SU($N_{c}$) color symmetry. These symmetries simplify the tensor structure of
various quantities in QCD. For example, the gluon propagator
$D_{\mu\nu}^{ab}(p)$ is simply expressed as

$D_{\mu\nu}^{ab}(p)=D(p^{2})\delta^{ab}\left(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right),$
(2)

due to the symmetries and the transverse property. Owing to this advanced
feature, the Landau gauge is often used both in the Schwinger-Dyson formalism
HigashimaMiransky ; Alkofer01  and in lattice QCD studies for quarks and
gluons Mandula99 ; Iritani09 .

In Euclidean QCD, the Landau gauge has a global definition to minimize the
global quantity,

$R_{\rm L}=\int
d^{4}x\mathrm{Tr}\left\\{A_{\mu}(x)A_{\mu}(x)\right\\}=\frac{1}{2}\int
d^{4}xA_{\mu}^{a}(x)A_{\mu}^{a}(x),$ (3)

by the gauge transformation. This global definition is more strict, and the
local form in Eq.(1) can be obtained from the minimization of $R_{\rm L}$.
Since the quantity $R_{\rm L}$ physically means the total amount of gauge-
field fluctuations, and therefore the Landau gauge maximally suppresses
artificial gauge-field fluctuations originated from the gauge degrees of
freedom.

Here, we comment on non-locality of the gauge fields. Through the gauge fixing
procedure, gauge fields have non-locality stemming from the Faddeev-Popov
determinant. In the Landau gauge, this non-locality of gauge fields is Lorentz
covariant.

Using the Landau gauge, or a covariant and globally symmetric gauge, the color
confinement has been mathematically investigated in terms of the BRST charge
and the inverse Higgs theorem, which is known as the “Kugo-Ojima criterion”
KugoOjima .

###  II.2 Coulomb gauge

The Coulomb gauge is also a popular gauge in QCD, and its gauge fixing is
given by

$\partial_{i}A_{i}=0.$ (4)

This condition resembles the Landau gauge condition (Eq.(1)), but there are no
constraints on $A_{0}$. In the Coulomb gauge, the Lorentz covariance is
partially broken, and gauge field components are completely decoupled into two
parts, $\vec{A}$ and $A_{0}$: $\vec{A}$ behave as canonical variables and
$A_{0}$ becomes an instantaneous potential.

Similarly in the Landau gauge, the Coulomb gauge has a global definition to
minimize the global quantity

$R_{\rm Coul}\equiv\int
d^{4}x\mathrm{Tr}\left\\{A_{i}(x)A_{i}(x)\right\\}=\frac{1}{2}\int
d^{4}xA_{i}^{a}(x)A_{i}^{a}(x)$ (5)

by the gauge transformation. Note here that the Euclidean metric is not
necessary for the global definition of the Coulomb gauge. Note also that there
appears no nonlocality in the temporal direction in the Coulomb gauge. Due to
this feature, a hadron mass measurement can be safely performed using a
spatially-extended quark source in the Coulomb gauge in lattice QCD
calculations CGsource .

In the Coulomb gauge, one of the advantages is the compatibility with the
canonical quantization ItzyksonZuber . The QCD Hamiltonian is expressed as

$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{3}x\left(\vec{E}^{a}\cdot\vec{E}^{a}+\vec{B}^{a}\cdot\vec{B}^{a}\right)$
(6) $\displaystyle+\frac{1}{2}\int
d^{3}xd^{3}y\rho^{a}(x)K^{ab}(x,y)\rho^{b}(y),$

where $\rho^{a}$ is the color charge density, $\vec{E}^{a}$ and $\vec{B}^{a}$
are the color electric and magnetic field, respectively. Here, $K^{ab}(x,y)$
is the instantaneous Coulomb propagator Greensite03  defined as

$K^{ab}(x,y)=[M^{-1}(-\nabla^{2})M^{-1}]_{xy}^{ab},$ (7)

with the Faddeev-Popov operator

$M^{ac}=-\partial^{2}\delta^{ac}-\varepsilon^{abc}A_{i}^{b}\partial_{i}.$ (8)

The confinement picture in the Coulomb gauge focuses on the Coulomb energy
including the inverse of $M$. Here, the Coulomb energy is the non-local second
term of the QCD Hamiltonian (6), and is regarded as the instantaneous
potential. Near the Gribov horizon, where the Faddeev-Popov operator $M$ has
zero eigenvalues Gribov , the Coulomb energy at large quark distance is
expected to be largely enhanced and leads to a confining interquark potential,
which is called as the “Gribov-Zwanziger scenario” Gribov ; Zwanziger98 .

As Zwanziger showed, the Coulomb energy (instantaneous potential) $V_{\rm
Coul}(R)$ in the Coulomb gauge gives an upper bound on the static interquark
potential $V_{\rm phys}(R)$ Zwanziger03 , i.e.,

$V_{\rm phys}(R)\leq V_{\rm Coul}(R).$ (9)

This inequality indicates that if the physical interquark potential is
confining then the Coulomb energy $V_{\rm Coul}$ is also confining. Actually,
lattice QCD calculations Greensite03  show that the Coulomb energy (the
instantaneous potential) between a quark and an antiquark leads to a linear
potential, which characterizes the confinement. However, the slope of the
instantaneous potential is too large, i.e., $2\sim 3$ times larger than the
physical string tension, and this Coulomb system turns out to be an excited
state.

As for the ground-state of the quark-antiquark system, Thorn and Greensite
proposed the “gluon-chain picture” in the Coulomb gauge GreensiteThorn . In
fact, to screen the large Coulomb energy between the quark and the antiquark,
chain-like gluons are dynamically generated between them. This gluon-chain is
expected to give the linear potential between quarks. In other words, QCD
string can be regarded as a “chain” of gluons in the Coulomb gauge.

##  III Generalized Landau gauge

In this section, we investigate the “generalized Landau gauge”, or “$\lambda$
gauge” Bernard90 , which continuously connects the Landau and the Coulomb
gauges.

Since the Landau gauge and the Coulomb gauge are useful gauges and give
different interesting pictures in QCD, it is meaningful to show the linkage of
these gauges. To link these gauges, we generalize the gauge fixing condition
(1) as

$\partial_{i}A_{i}+\lambda\partial_{4}A_{4}=0,$ (10)

by introducing one real parameter $\lambda$ Bernard90 . The case of
$\lambda=1$ corresponds to the Landau gauge fixing condition, the Coulomb
gauge is achieved at $\lambda=0$, and also temporal gauge for
$\lambda\rightarrow\infty$. Therefore, we can analyze gauge dependence of
various properties from the Landau gauge toward the Coulomb gauge by varying
$\lambda$-parameter from 1 to 0. $\lambda$-gauge keeps the global SU($N_{c}$)
color symmetry, but partially breaks the Lorentz symmetry like the Coulomb
gauge, except for $\lambda$=1.

In Euclidean QCD, the global definition of $\lambda$-gauge is expressed by the
minimization of

$R^{\lambda}\equiv\int
d^{4}x\left[\mathrm{Tr}\left\\{A_{i}(x)A_{i}(x)\right\\}+\lambda\mathrm{Tr}\left\\{A_{4}(x)A_{4}(x)\right\\}\right]$
(11)

by the gauge transformation. Here, the $\lambda$-parameter controls the ratio
of the gauge-field fluctuations of $\vec{A}$ and $A_{4}$.

Lattice QCD is formulated on the discretized Euclidean space-time, and the
theory is described with the link-variable $U_{\mu}(x)\equiv
e^{iagA_{\mu}(x)}\in{\rm SU}(N_{c})$, with the lattice spacing $a$ and the
gauge coupling constant $g$, instead of gauge fields
$A_{\mu}(x)\in\mathfrak{su}(N_{c})$. $\lambda$-gauge fixing condition is
expressed in terms of the link-variable as the maximization of a quantity

$R_{\rm
latt}^{\lambda}[U]\equiv\sum_{x}\left\\{\sum_{i}\mathrm{Re}\mathrm{Tr}U_{i}(x)+\lambda\mathrm{Re}\mathrm{Tr}U_{4}(x)\right\\}$
(12)

by the gauge transformation of the link-variables,

$U_{\mu}(x)\rightarrow\Omega(x)U_{\mu}(x)\Omega^{\dagger}(x+\hat{\mu}),$ (13)

with the gauge function $\Omega\in\mathrm{SU}(3)$. In the continuum limit of
$a\rightarrow 0$, this condition results in the minimization of $R^{\lambda}$
in Eq.(11), and satisfies the local $\lambda$-gauge fixing condition of
Eq.(10).

Note here that the gluon-field fluctuation is strongly suppressed in the
generalized Landau gauge with $\lambda\neq 0$, so that one can use the
expansion of the link-variable $U_{\mu}(x)\equiv e^{iagA_{\mu}(x)}\simeq
1+iagA_{\mu}(x)+\mathcal{O}(a^{2})$ for small lattice spacing $a$, and the
gluon field $A_{\mu}(x)$ can be defined by

$A_{\mu}(x)\equiv\frac{1}{2iag}\left[U_{\mu}(x)-U_{\mu}^{\dagger}(x)\right]_{\rm
traceless}\in\mathfrak{su}(N_{c})$ (14)

without suffering from large gluon fluctuations stemming from the gauge
degrees of freedom.

##  IV Polyakov-line correlators and instantaneous potential

In this section, we formulate Polyakov-line correlators and instantaneous
potential in lattice QCD.

First, we consider the Euclidean continuum theory to make clear the physical
interpretation of the terminated Polyakov-line correlator. Considering the
source field $J_{\mu}(x)$ which couples to the gauge field $A_{\mu}(x)$, the
generating functional is given by

$Z[J]\equiv\langle\ e^{i\int d^{4}x\mathcal{L}_{\rm
int}(J)}\rangle=\langle\exp\\{ig\int
d^{4}xA_{\mu}^{a}(x)J^{\mu}_{a}(x)\\}\rangle.$ (15)

We consider a closed-loop current such as the Wilson loop $W(R,T)$, which is
constructed in a gauge-invariant manner. From the relation $Z[J]\propto
e^{-E_{J}T}$, the ground-state energy between static quark and antiquark pair
is expressed as

$V_{\rm phys}(R)=-\lim_{T\rightarrow\infty}\frac{1}{T}\ln\langle
W(R,T)\rangle.$ (16)

Next, we consider the color current $J_{\mu}(x)$ of a quark located at
$\vec{x}=\vec{a}$ and an antiquark at $\vec{x}=\vec{b}$. For the case that
these sources are generated at $t=0$ and annihilated at $t=T$, the color
current $J_{\mu}(x)$ is expressed as

$J_{\mu}(x)=\delta_{\mu
4}\left[\delta(\vec{x}-\vec{a})-\delta(\vec{x}-\vec{b})\right]\theta(T-t)\theta(t),$
(17)

in the generalized Landau gauge. In this case, the current $J_{\mu}$ is not
conserved, and it breaks the gauge invariance. In a similar manner to the
Wilson loop $W(R,T)$, we define the “energy” of two sources in the presence of
$J_{\mu}(x)$ as

$V(R,T)=-\frac{1}{T}\ln\langle\mathrm{Tr}[L(\vec{a},T)L^{\dagger}(\vec{b},T)]\rangle$
(18)

with $R=|\vec{a}-\vec{b}|$, using the terminated Polyakov-line

$L(\vec{x},T)\equiv
P\exp\left(ig\int_{0}^{T}dx_{4}A_{4}(\vec{x},x_{4})\right),$ (19)

with $P$ being the path-ordered product. As a caution, for $\lambda\neq$ 0,
$V(R,T)$ is gauge-dependent and does not mean the energy of a physical state,
due to the temporal nonlocality of the Faddeev-Popov determinant.

Next, we consider $N_{x}\times N_{y}\times N_{z}\times N_{t}$ lattice with the
lattice spacing $a$. Using temporal link-variables, the terminated Polyakov-
line with length $T$ is defined as

$L(\vec{x},T)=U_{4}(\vec{x},a)U_{4}(\vec{x},2a)\cdots U_{4}(\vec{x},T).$ (20)

The terminated Polyakov-line is generally gauge variant, and its expectation
value depends on the choice of gauge. Only for $T=N_{t}$, the trace of the
Polyakov-line coincides the Polyakov loop, which is gauge invariant. In the
Coulomb gauge, the expectation value of the terminated Polyakov-line is zero
due to the remnant symmetry. (See Appendix A.)

In this paper, we consider the Polyakov-line correlator in generalized Landau
gauge denoted by

$G_{\lambda}(R,T)=\langle\mathrm{Tr}[L^{\dagger}(\vec{x},T)L(\vec{y},T)]\rangle$
(21)

with $R=|\vec{x}-\vec{y}|$. From this correlator, we define “finite-time
potential”,

$V_{\lambda}(R,T)\equiv-\frac{1}{T}\ln G_{\lambda}(R,T).$ (22)

Here, for the simple expression, a normalization factor 1/3 is dropped off for
$G_{\lambda}(R,T)$, since it only gives a constant term in the potential
$V_{\lambda}(R,T)$.

Particularly for $T=a$, we call

$V_{\lambda}(R)\equiv
V_{\lambda}(R,a)=-\frac{1}{a}\ln\langle\mathrm{Tr}\big{[}U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\big{]}\rangle$
(23)

as “instantaneous potential”.

Here, these quantities depend on $\lambda$-parameter. In the Coulomb gauge,
the instantaneous potential $V_{\lambda=0}(R)$ (or the Coulomb energy) gives a
linear potential, but its slope is about $2\sim 3$ times larger than the
physical string tension Greensite03 . In the Landau gauge, the instantaneous
potential $V_{\lambda=1}(R)$ has no linear part Nakamura06 , which is also
expected from the exponential reduction of the gluon propagator Mandula99 ;
Iritani09  and the Lorentz symmetry. In Sec.VII, we will discuss the relation
between the gluon propagator and the instantaneous potential in the Landau
gauge.

##  V Lattice QCD results

We perform SU(3) lattice QCD Monte Carlo calculations at the quenched level.
We use the standard plaquette action with the lattice parameter
$\beta\equiv\frac{2N_{c}}{g^{2}}=5.8$ on a $16^{4}$-size lattice. The lattice
spacing $a$ is 0.152 fm, which is determined so as to reproduce the string
tension as $\sqrt{\sigma}=427$ MeV STI .

We use the gauge configurations, which are picked up every 1000 sweeps after a
thermalization of 20000 sweeps. After the generation of gauge configurations,
we perform gauge fixing by maximizing $R^{\lambda}_{\rm latt}[U]$. In this
paper, we use the Landau gauge ($\lambda$=1), the Coulomb gauge ($\lambda$=0),
and their intermediate gauges with
$\lambda=0.75,0.50,0.25,0.10,0.05,0.04,0.03,0.02,0.01$. We investigate in
detail the region near the Coulomb gauge ($\lambda$=0), since the behavior of
the instantaneous potential largely changes for $\lambda\sim$0, as will be
shown later. The number of gauge configurations is 50 for each $\lambda$. We
adopt the jackknife method to estimate the statistical error.

Here, we comment on $\lambda$-gauge fixing convergence. We fix the gauge by
maximizing the quantity $R^{\lambda}_{\rm latt}[U]$ in Eq.(12), which
corresponds to $\partial_{i}A_{i}+\lambda\partial_{4}A_{4}=0$. Therefore, to
check the convergence of gauge fixing, we evaluate $\epsilon_{\lambda}$
defined by

$\displaystyle\epsilon_{\lambda}$ $\displaystyle\equiv$
$\displaystyle\langle\left(\partial_{i}A_{i}^{a}+\lambda\partial_{4}A_{4}^{a}\right)^{2}\rangle$
(24) $\displaystyle\equiv$ $\displaystyle\frac{1}{(N_{c}^{2}-1){N_{\rm
site}}}\sum_{x=1}^{N_{\rm
site}}\sum_{a=1}^{N_{c}^{2}-1}\Big{\\{}\sum_{i=1}^{3}\left[A_{i}^{a}(x)-A_{i}^{a}(x-\hat{i})\right]$
$\displaystyle+$
$\displaystyle\lambda\left[A_{4}^{a}(x)-A_{4}^{a}(x-\hat{4})\right]\Big{\\}}^{2},$

with the gluon field $A_{\mu}(x)=A_{\mu}^{a}(x)T^{a}$ given in Eq.(14). We
iterate the gauge transformation to satisfy $\epsilon_{\lambda}<{10}^{-12}$
finally. As for the instantaneous potential $V_{\lambda}(R)$, this convergence
condition is very strict. Actually, we can obtain stable the lattice data of
$V_{\lambda}(R)$ even with $\epsilon_{\lambda}<{10}^{-4}$.

Also, we comment that the calculation cost of the gauge fixing is rapidly
increasing as $\lambda$ approaches to zero, while the Coulomb gauge
($\lambda=0$) itself can be easily achieved. Considering this critical slowing
down of the gauge fixing Bernard90 ; Cucchieri07 , we adopt a relatively
small-size lattice of $16^{4}$ with $\beta=5.8$, although its physical volume
of about $(2.4{\rm fm})^{4}$ is large enough to extract the relevant region
for the interquark potential.

###  V.1 “Instantaneous inter-quark potential” in generalized Landau gauge

We investigate the instantaneous potential $V_{\lambda}(R)$ defined by Eq.(23)
in generalized Landau gauge. Figure 1 shows lattice QCD results of
$V_{\lambda}(R)$ for typical values of $\lambda$. In this figure, the
statistic error is small and the error bars are hidden in the symbols.

In the Coulomb gauge ($\lambda=0$), the instantaneous potential shows linear
behavior, while there is no linear part at all in the Landau gauge
($\lambda=1$). Thus, there is a large gap between these gauges in terms of the
instantaneous potential. In our framework, however, these two gauges are
connected continuously.

By varying the $\lambda$-parameter from 1 to 0 in the generalized Landau
gauge, we find that the instantaneous potential $V_{\lambda}(R)$ changes
continuously, and the infrared slope of the potential $V_{\lambda}(R)$ at
$R\simeq 0.8{\rm fm}$ grows monotonically, from the Landau gauge to the
Coulomb gauge, as shown in Fig. 1. Note here that the growing of the infrared
slope of $V_{\lambda}(R)$ is quite rapid for $\lambda\lesssim 0.1$, near the
Coulomb gauge, while the infrared slope is rather small and almost unchanged
for $\lambda=0.1\sim 1$.

Figure 1:  “Instantaneous potential” $V_{\lambda}(R)$ in generalized Landau
gauge for typical values of $\lambda$. The symbols denote lattice QCD results,
and the curves fit-results using Coulomb plus linear form,
$V_{\lambda}(R)=-A_{\lambda}/R+\sigma_{\lambda}R+C_{\lambda}$, in the region
of $R\lesssim 0.8$ fm. For $\lambda=1\sim 0.1$, the potential has almost no
linear part. For $\lambda\lesssim 0.1$, the linear potential grows rapidly,
and $\sigma_{\lambda}\simeq 2.6\sigma_{\rm phys}$ at $\lambda=0$.

To analyze the instantaneous potential $V_{\lambda}(R)$ quantitatively, we fit
the lattice QCD results using the Coulomb plus linear Ansatz as

$V_{\lambda}(R)=-\frac{A_{\lambda}}{R}+\sigma_{\lambda}R+C_{\lambda},$ (25)

where $A_{\lambda}$ is the Coulomb coefficient and $C_{\lambda}$ a constant.
Here, $\sigma_{\lambda}$ is the infrared slope of the potential, which we call
as “instantaneous string tension”. Besides the Coulomb plus linear Ansatz, we
try several candidates of the functional form, $-A/R+\sigma(1-e^{-\varepsilon
R})/\varepsilon$, $-A\exp(-mR)/R$, $-A/R+\sigma R^{d}$, and $-A/R^{d}$ apart
from an irrelevant constant, but they are less workable. The curves in Fig. 1
are the best-fit results using Eq.(25). The Coulomb plus linear Ansatz works
well at least for $R\lesssim 0.8$fm, which is relevant region for hadron
physics. We note that the Yukawa form of $-Ae^{-mR}/R$ also works well near
the Landau gauge, which will be discussed in relation to the gluon-propagator
behavior in Sec.VI-C.

We summarize the best-fit parameters and the fit-range in Table 1. While the
Coulomb coefficient $A_{\lambda}$ has a relatively weak $\lambda$-dependence,
the instantaneous string tension $\sigma_{\lambda}$ shows a strong
$\lambda$-dependence near the Coulomb gauge, i.e., $\lambda\lesssim 0.1$.

We comment on the asymptotic value of the instantaneous potential
$\sigma_{\lambda}$. In the deep IR limit, $R\rightarrow\infty$,
$V_{\lambda}(R)$ goes to a saturated value, except for $\lambda=0$. This
feature is closely related to the property of the temporal link-variable
correlator $\langle\mathrm{Tr}\\{U_{4}^{\dagger}(x)U_{4}(y)\\}\rangle$, as
will be discussed in Sec.VI.

Table 1:  Best-fit parameters on the instantaneous potential using $V_{\lambda}(R)=-A_{\lambda}/R+\sigma_{\lambda}R+C_{\lambda}$, and the ratio of the slope $\sigma_{\lambda}$ to the physical string tension $\sigma_{\rm phys}$. The standard parameters of the physical interquark potential are $A_{\rm phys}\simeq 0.27$ and $\sigma_{\rm phys}\simeq 0.89$GeV/fm STI . The string tension $\sigma_{\lambda}$ is rather small for $\lambda=0.1\sim 1$. $\lambda$ | fit-range [fm] | $A_{\lambda}$ |  $\sigma_{\lambda}$ [GeV/fm] |  $C_{\lambda}$ [GeV] | $\sigma_{\lambda}/\sigma_{\rm phys}$
---|---|---|---|---|---
0.00 | 0.1-1.0 | 0.167(11) | 2.283(35) | -0.881(20) | 2.57(4)
0.01 | 0.1-0.8 | 0.287(27) | 1.476(78) | -0.617(46) | 1.66(9)
0.02 | 0.1-0.8 | 0.346(32) | 1.005(90) | -0.481(54) | 1.13(10)
0.03 | 0.1-0.8 | 0.372(32) | 0.728(86) | -0.416(53) | 0.82(10)
0.04 | 0.1-0.8 | 0.382(30) | 0.557(79) | -0.386(50) | 0.63(9)
0.05 | 0.1-0.8 | 0.386(29) | 0.441(73) | -0.372(47) | 0.50(8)
0.10 | 0.1-0.8 | 0.365(20) | 0.169(46) | -0.390(31) | 0.19(5)
0.25 | 0.1-0.8 | 0.281(6) | -0.005(13) | -0.544(9) | -0.01(1)
0.50 | 0.1-0.8 | 0.198(0) | -0.042(1) | -0.724(1) | -0.05(0)
0.75 | 0.1-0.8 | 0.152(1) | -0.043(3) | -0.839(2) | -0.05(0)
1.00 | 0.1-0.8 | 0.123(2) | -0.040(3) | -0.917(3) | -0.04(0)

Now, we focus on the $\lambda$-dependence of instantaneous string tension
$\sigma_{\lambda}$ in Fig. 2. For $0.1\lesssim\lambda\leq 1$, including the
Landau gauge ($\lambda$=1), $\sigma_{\lambda}$ is almost zero, so that this
region can be regarded as “Landau-like.” For $\lambda\lesssim 0.1$,
$V_{\lambda}(R)$ is drastically changed near the Coulomb gauge, and
$\sigma_{\lambda}$ grows rapidly in this small region. Finally, in the Coulomb
gauge ($\lambda$=0), one finds $\sigma_{\lambda}\simeq 2.6\sigma_{\rm phys}$,
with $\sigma_{\rm phys}\simeq 0.89$GeV/fm.

Figure 2:  Instantaneous string tension $\sigma_{\lambda}$, the slope of the
linear part in the instantaneous potential $V_{\lambda}(R)$ in the generalized
Landau gauge. The right upper figure is a close-up near the Coulomb gauge
($\lambda$=0). For $0\leq\lambda\lesssim 0.1$, $\sigma_{\lambda}$ changes
rapidly from $2.6\sigma_{\rm phys}$ to 0, while $\sigma_{\lambda}$ is rather
small for $0.1\lesssim\lambda\leq 1$. The instantaneous string tension
$\sigma_{\lambda}$ coincides with the physical value around $\lambda_{C}\simeq
0.02$.

Note that the instantaneous string tension $\sigma_{\lambda}$ continuously
changes from $0$ to $(2\sim 3)\sigma_{\rm phys}$, according to the change from
the Landau gauge to the Coulomb gauge, and therefore there exists some
specific $\lambda$-parameter of $\lambda_{C}\in[0,1]$ where the slope of the
instantaneous potential $V_{\lambda}(R)$ coincides with the physical string
tension $\sigma_{\rm phys}$. Since the instantaneous potential generally
depends on the lattice parameter $\beta$, i.e., the lattice spacing $a$
Greensite03 ; Iritani10 , the value of $\lambda_{C}$ is $\beta$-dependence,
although its dependence would be rather weak, as will be discussed in Sec.VI.
However, from the continuity between the overconfining potential in the
Coulomb gauge and the saturated potential in the Landau gauge, there must
exist $\lambda_{C}\in[0,1]$ where the instantaneous string tension
$\sigma_{\lambda}$ coincides with $\sigma_{\rm phys}$ for each lattice
spacing. We call this specific gauge as “$\lambda_{C}$-gauge”. From Fig.2, the
value of $\lambda_{C}$ is estimated to be about 0.02 at $\beta$=5.8. Note here
that $\lambda_{C}\simeq 0.02\ll 1$ is very small, and then the
$\lambda_{C}$-gauge is close to the Coulomb gauge, which indicates the small
temporal non-locality.

Figure 3 shows the instantaneous potential $V_{\lambda}(R)$ at
$\lambda=0.02\simeq\lambda_{C}$ and the physical static interquark potential
$V_{\rm phys}(R)$. In this $\lambda_{C}$-gauge, $V_{\rm phys}(R)$ is found to
be approximately reproduced by $V_{\lambda_{C}}(R)$ for $R\lesssim$ 0.8fm.
While the physical static potential $V_{\rm phys}(R)$ is derived from the
large-T behavior of the Wilson loop $W(R,T)$ Rothe  as

$V_{\rm phys}(R)=-\lim_{T\rightarrow\infty}\frac{1}{T}\ln\langle
W(R,T)\rangle,$ (26)

only instantaneous corerlation of the temporal link-variable $U_{4}$
approximately reproduces $V_{\rm phys}(R)$ in the $\lambda_{C}$-gauge, i.e.,

$\displaystyle V_{\rm phys}(R)$ $\displaystyle\simeq$ $\displaystyle
V_{\lambda_{C}}(R)$ (27) $\displaystyle=$
$\displaystyle-\frac{1}{a}\ln\langle\mathrm{Tr}U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\rangle_{\lambda_{C}},$

as is schematically illustrated in Fig.4.

Figure 3:  The comparison between the instantaneous potential $V_{\lambda}(R)$
at $\lambda=0.02(\simeq\lambda_{C})$ (black dots), and the physical interquark
potential $V_{\rm phys}(R)$ (solid line). In $\lambda_{C}$-gauge, $V_{\rm
phys}(R)=-\lim_{T\rightarrow\infty}\frac{1}{T}\ln\langle W(R,T)\rangle\simeq-
A_{\rm phys}/R+\sigma_{\rm phys}R$ ($A_{\rm phys}\simeq$ 0.27, $\sigma_{\rm
phys}\simeq$ 0.89GeV/fm) is approximately reproduced by the instantaneous
potential
$V_{\lambda_{C}}(R)=-\frac{1}{a}\ln\langle\mathrm{Tr}U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\rangle_{\lambda_{C}}$.
Figure 4:  The schematic illustration of the physical interquark potential
$V_{\rm phys}(R)$ and the instantaneous potential $V_{\rm inst}(R)$. In
$\lambda_{C}$-gauge, the instantaneous potential $V_{\lambda_{C}}(R)$
approximately reproduces the physical potential $V_{\rm phys}(R)$.

On the relation to the confinement, which is a gauge independent phenomenon,
the role of gluons generally depends on the choice of gauges, and the physical
picture of the confinement would be changed according to gauges. For example,
in the Coulomb gauge, the instantaneous Coulomb energy gives an overconfining
potential, and the ground-state of the quark-antiquark system is described as
the gluon-chain state. On the other hand, in the Landau gauge, the
instantaneous potential has no linear part, and the ghost behavior in the
deep-infrared region would be more important for the confinement.

In the $\lambda_{C}$-gauge, the physical interquark potential $V_{\rm
phys}(R)$ is approximately reproduced by the instantaneous potential
$V_{\lambda_{C}}(R)$. This physically means that all other complicated effects
including dynamical gluons and ghosts are approximately cancelled in the
$\lambda_{C}$-gauge, and therefore we do not need to introduce any redundant
gluonic degrees of freedom. The absence of dynamical gluon degrees of freedom
would be a desired property for the quark model picture.

###  V.2 “Finite-time potential” and “finite-time string tension”

In the previous subsection, we investigated the instantaneous potential
$V_{\lambda}(R)$, which is defined by the Polyakov-line correlator with a
minimum length on the lattice. For the quark-potential model, it is desired
that the interquark potential does not have large dependence on the temporal
length $T$ of the typical reaction scale. From this viewpoint, we investigate
the “finite-time potential” $V_{\lambda}(R,T)$ defined by Eq.(18) in Sec.IV,
and its temporal-length dependence. Here, $V_{\lambda}(R,T)$ is expressed by
$T$-length terminated Polyakov-line $L(\vec{x},T)$ in Eq.(19), and a
generalization of the instantaneous potential $V_{\lambda}(R)$.

First, we consider the Coulomb gauge Greensite03 ; Iritani10 . Figure 5 shows
the lattice QCD result for $V_{\lambda}(R,T)$ in the Coulomb gauge. Similar to
the instantaneous potential, $V_{\lambda}(R,T)$ is well reproduced by the
Coulomb plus linear form. However, the parameter values are changed according
to $T$-length. In particular, the slope of the potential becomes smaller as
$T$ becomes larger, which shows an “instability” of $V_{\lambda}(R,T)$ in
terms of $T$ in the Coulomb gauge.

For general $\lambda$, finite-time potential $V_{\lambda}(R,T)$ is found to be
reproduced by the Coulomb plus linear form as

$V_{\lambda}(R,T)=-\frac{A_{\lambda}(T)}{R}+\sigma_{\lambda}(T)R+C_{\lambda}(T),$
(28)

at least for $R\lesssim 0.8$fm, similarly for the instantaneous potential.

Figure 5:  “Finite-time potential” $V_{\lambda}(R,T)$ in the Coulomb gauge
($\lambda=0$) for $T$=1,2,3,4,5. Here, for the comparison, an irrelevant
constant is shifted for each $T$. The curves denote the fit-results using the
Coulomb plus linear form. The slope of $V_{\lambda}(R,T)$ is clearly changed
according to $T$.

Since our main interest is linear part of the potential, we focus on “finite-
time string tension” $\sigma_{\lambda}(T)$, the slope of $V_{\lambda}(R,T)$.
Figure 6 shows $\sigma_{\lambda}(T)$ in generalized Landau gauge for typical
values of $\lambda$. In Table 2, we summarize the best-fit parameters of
$\sigma_{\lambda}(T)$ at $T=1,2,\dots,6$, and the ratio of
$\sigma_{\lambda}(1)/\sigma_{\lambda}(6)$, $\sigma_{\lambda}(1)/\sigma_{\rm
phys}$, and $\sigma_{\lambda}(6)/\sigma_{\rm phys}$, respectively.

Figure 6:  $T$-length dependence of “Finite-time string tension” $\sigma_{\lambda}(T)$, the infrared slope of finite-time potential $V_{\lambda}(R,T)$, in generalized Landau gauge for several typical $\lambda$-values. Near the Coulomb gauge, e.g., for $\lambda\lesssim 0.03$, $\sigma_{\lambda}(T)$ goes to the same value for large $T\sim$ 1fm. For $\lambda\gtrsim 0.1$, $\sigma_{\lambda}(T)$ is an increasing function of $T$. In fact, even though the instantaneous potential has no linear part, the linear part of $V_{\lambda}(R,T)$ appears and gradually grows, as the Polyakov-line grows.  Table 2:  Finite-time string tension $\sigma_{\lambda}(T)$ in generalized Landau gauge for $T=1,2,\dots,6$, together with the ratio, $\sigma_{\lambda}(1)/\sigma_{\lambda}(6)$, $\sigma_{\lambda}(1)/\sigma_{\rm phys}$, and $\sigma_{\lambda}(6)/\sigma_{\rm phys}$. The fit-range is the same as that listed in Table 1.  |  $\sigma_{\lambda}(T)$ [GeV/fm] |  |  |
---|---|---|---|---
$\lambda$ | $T=1$ | $T=2$ | $T=3$ | $T=4$ | $T=5$ | $T=6$ | $\sigma_{\lambda}(1)/\sigma_{\lambda}(6)$ | $\sigma_{\lambda}(1)/\sigma_{\rm phys}$ | $\sigma_{\lambda}(6)/\sigma_{\rm phys}$
0.00 | 2.283(35) | 1.704(11) | 1.463(8) | 1.322(16) | 1.244(24) | 1.147(75) | 1.99(13) | 2.57(4) | 1.29(8)
0.01 | 1.476(78) | 1.466(30) | 1.348(17) | 1.252(16) | 1.191(27) | 1.181(43) | 1.25(8) | 1.66(9) | 1.33(5)
0.02 | 1.005(90) | 1.225(55) | 1.225(31) | 1.176(20) | 1.135(25) | 1.119(53) | 0.90(9) | 1.13(10) | 1.26(6)
0.03 | 0.728(86) | 1.034(67) | 1.119(43) | 1.113(30) | 1.104(30) | 1.086(40) | 0.67(8) | 0.82(10) | 1.22(5)
0.04 | 0.557(79) | 0.896(72) | 1.030(51) | 1.057(35) | 1.053(30) | 1.019(53) | 0.55(8) | 0.63(9) | 1.15(6)
0.05 | 0.441(73) | 0.785(76) | 0.947(59) | 1.002(43) | 1.021(35) | 1.020(44) | 0.43(7) | 0.50(8) | 1.15(5)
0.10 | 0.169(46) | 0.467(71) | 0.684(70) | 0.811(58) | 0.872(50) | 0.875(56) | 0.19(5) | 0.19(5) | 0.98(6)
0.25 | -0.005(13) | 0.152(42) | 0.324(60) | 0.474(65) | 0.586(62) | 0.690(48) | -0.01(2) | -0.01(1) | 0.78(5)
0.50 | -0.042(1) | 0.015(21) | 0.111(41) | 0.218(57) | 0.320(69) | 0.423(79) | -0.10(2) | -0.05(0) | 0.48(9)
0.75 | -0.043(3) | -0.025(11) | 0.028(27) | 0.100(44) | 0.181(58) | 0.275(69) | -0.16(4) | -0.05(0) | 0.31(8)
1.00 | -0.040(3) | -0.041(5) | -0.012(18) | 0.039(32) | 0.104(45) | 0.180(56) | -0.22(7) | -0.04(0) | 0.20(6)

On the $T$-dependence of the finite-time string tension $\sigma_{\lambda}(T)$,
there are three groups of the $\lambda$-parameter region: (i) Coulomb-like
region ($0\leq\lambda\ll\lambda_{C}$), (ii) Landau-like region
($\lambda_{C}\ll\lambda\leq 1$), and (iii) $\lambda_{C}$-like region
($\lambda\sim\lambda_{C}$).

(i) The first category is the Coulomb-like region of
$0\leq\lambda\ll\lambda_{C}$, i.e., $0\leq\lambda\lesssim 0.01$. In this
region, the instantaneous string tension
$\sigma_{\lambda}\equiv\sigma_{\lambda}(T=1)$ is larger than the physical
string tension $\sigma_{\rm phys}$, and the instantaneous potential
$V_{\lambda}(R)\equiv V_{\lambda}(R,T=1)$ gives an overconfining potential.
The ground-state of the quark-antiquark system is considered as the gluon-
chain state. As the temporal length $T$ of the Polyakov-line increases,
finite-time string tension $\sigma_{\lambda}(T)$ decreases and approaches to
the physical string tension $\sigma_{\rm phys}$. This decreasing behavior is
interpreted that the component of the ground-state, i.e., the gluon-chain
state, becomes dominant as $T$-length becomes large. Thus, this region would
not be compatible with the quark potential picture, because of the large
$T$-dependence of the confining force $\sigma_{\lambda}(T)$ in addition to the
dynamical generation of the gluon-chain.

(ii) The second category is the Landau-like region of
$\lambda_{C}\ll\lambda\leq 1$, i.e. $0.1\lesssim\lambda\leq 1$. In this
region, the instantaneous string tension is almost zero, i.e.,
$\sigma_{\lambda}\simeq 0$, and finite-time string tension
$\sigma_{\lambda}(T)$ is an increasing function of $T$. Although its
asymptotic value is unclear for $T\sim 0.8$ fm, $\sigma_{\lambda}(T)$ seems to
approach to the physical string tension $\sigma_{\rm phys}$, which will be
discussed in Sec.VI.

(iii) The third category is the $\lambda_{C}$-like region of
$\lambda\sim\lambda_{C}$, i.e., $0.01\lesssim\lambda\lesssim 0.1$. In this
region, the instantaneous string tension $\sigma_{\lambda}$ is approximately
equal to the physical string tension, i.e., $\sigma_{\lambda}\simeq\sigma_{\rm
phys}(\simeq$ 0.89GeV/fm), and the instantaneous potential $V_{\lambda}(R)$
approximately reproduces the physical static potential $V_{\rm phys}(R)$, as
shown in Fig. 3. As the temporal length $T$ increases, finite-time string
tension $\sigma_{\lambda}(T)$ is slightly changed and takes a little larger
value ($\simeq$ 1.1GeV/fm) around $T\simeq$ 0.8fm. In particular, near
$\lambda_{C}\simeq 0.02$, $\sigma_{\lambda}(T)$ shows only a weak
$T$-dependence, while $\sigma_{\lambda}(T)$ largely changes as $T$ in the
Coulomb gauge. As a whole, finite-time potential $V_{\lambda}(R,T)$ has small
$T$-dependence, as shown in Fig. 7. This is also a desired feature for the
linkage to the quark potential model.

Figure 7:  Finite-time potential $V_{\lambda}(R,T)$ at
$\lambda=0.02(\simeq\lambda_{C})$. For the comparison, an irrelevant constant
is shifted for each $T$. The slope of $V_{\lambda}(R,T)$ is almost the same
for $T=1,2,\cdots 5$, and thus the shape of $V_{\lambda}(R,T)$ is rather
stable against the temporal length $T$.

When the length $T$ of the Polyakov-line increases, $\lambda$-dependence of
$\sigma_{\lambda}(T)$ is weakened, and $\sigma_{\lambda}(T)$ seems to converge
to the physical string tension $\sigma_{\rm phys}$ for enough large $T$, as
indicated in Fig.6.

There are two ingredients on the above gauge-dependence
($\lambda$-dependence): one is the large excess of the Coulomb energy in the
Coulomb gauge, the other is the non-locality from the Faddeev-Popov
determinant. From the fixing condition of generalized Landau gauge in Eq.(10),
one finds that the $\lambda$-parameter controls the non-locality in the
temporal direction. In the Landau gauge, the non-locality appears equally in
spatial and temporal directions, while temporal non-locality disappears in the
Coulomb gauge. If the Coulomb-energy excess can be neglected, e.g., for large
$\lambda$, $V_{\lambda}(R,T)$ is expected to reproduce the static potential
$V_{\rm phys}(R)$, when $T$-length and $R$ are large enough to neglect the
non-locality scale. Owing to the $\lambda$-dependence of the non-locality,
such a $T$-length exceeding the non-locality is to be larger for larger
$\lambda$ in the Landau-like region.

Near the $\lambda_{C}$-gauge, finite-time potential $V_{\lambda}(R,T)$ has
only weak $T$-length dependence. In other words, this can be regarded as an
approximate “fixed point” against $T$ around $\lambda_{C}\simeq 0.02$. We
conjecture that this is due to the approximate cancellation between the
Coulomb-energy excess and the non-locality. Actually, in contrast to the large
$T$-dependence of $V_{\lambda}(R,T)$ in the Coulomb gauge as shown in Fig.5,
$V_{\lambda_{C}}(R,T)$ is rather stable against $T$-length in the
$\lambda_{C}$-gauge, as shown in Fig.7.

##  VI Terminated Polyakov-line correlator and potentials

In the previous section, we investigated instantaneous potential
$V_{\lambda}(R)$ and finite-time potential $V_{\lambda}(R,T)$, which are
derived from the correlation of terminated Polyakov-line $L(\vec{x},T)$. In
this section, we investigate properties of the Polyakov-line correlator, and
clarify its relation to $V_{\lambda}(R)$ and $V_{\lambda}(R,T)$.

###  VI.1 Asymptotic behavior of link-variable correlator and instantaneous
potential

First, we investigate the spatial correlator $G_{\lambda}(R)$ of the temporal
link-variable $U_{4}$, and the relation to the instantaneous potential
$V_{\lambda}(R)\equiv-\frac{1}{a}{\rm ln}G_{\lambda}(R)$. For the large
spatial separation of $R\equiv|\vec{x}-\vec{y}|\rightarrow\infty$,
$G_{\lambda}(R)$ behaves asymptotically as

$\displaystyle G_{\lambda}(R)$ $\displaystyle\equiv$
$\displaystyle\langle\mathrm{Tr}\
[U_{4}^{\dagger}(\vec{x},t)U_{4}(\vec{y},t)]\rangle$ (29)
$\displaystyle\rightarrow$
$\displaystyle\langle(U_{4})_{ij}^{*}\rangle\langle(U_{4})_{ij}\rangle=\frac{1}{3}\langle\mathrm{Tr}\
U_{4}\rangle^{2},$

where $\langle(U_{4})_{ij}\rangle=\frac{1}{3}\langle\mathrm{Tr}\
U_{4}\rangle\delta_{ij}\in\mathbf{R}$ from the global color symmetry. Here,
$\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle^{2}$ is found to give the lower
bound of $G_{\lambda}(R)$. If $\langle\mathrm{Tr}\ U_{4}\rangle$ takes some
finite value, $V_{\lambda}(R)$ inevitably saturates for large $R$. Then,
$\langle\mathrm{Tr}\ U_{4}\rangle=0$ is a necessary condition for the deep-
infrared confinement feature of $V_{\lambda}(R=\infty)=\infty$.

In the Coulomb gauge, $\langle\mathrm{Tr}\ U_{4}\rangle$ is zero due to the
remnant symmetry, as is shown in Appendix. Therefore, as $R\rightarrow\infty$,
the correlator $G_{\lambda}(R)$ converges to zero, and
$V_{\lambda}(R)\equiv-\frac{1}{a}{\rm ln}G_{\lambda}(R)\rightarrow+\infty$,
which corresponds to the deep-infrared confinement.

For the general case of $\lambda\neq 0$, however, $\langle\mathrm{Tr}\
U_{4}\rangle$ has a non-zero value, and $G_{\lambda}(R)$ approaches to some
finite constant. The finiteness of $\langle\mathrm{Tr}\ U_{4}\rangle$ gives a
saturation of $V_{\lambda}(R)$, which leads to the absence of its linear part
in the case of rapid convergence.

Figure 8 shows $G_{\lambda}(R)$ and its asymptotic value
$\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle^{2}$ in the Landau and the
Coulomb gauges. In the Landau gauge, $\langle\mathrm{Tr}\ U_{4}\rangle$ has a
large expectation value, according to the maximization of
$\sum_{x}\mathrm{Re}\ \mathrm{Tr}\ U_{\mu}(x)$, and $G_{\lambda}(R)$ rapidly
converges to a finite constant for $R\gtrsim$ 0.4fm, which leads to a rapid
saturation of the instantaneous potential $V_{\lambda}(R)$. In the Coulomb
gauge, we find $\langle\mathrm{Tr}\ U_{4}\rangle=0$, and $G_{\lambda}(R)$
decreases monotonically to zero as $R$, which leads to
$V_{\lambda}(R)=+\infty$ for $R=\infty$.

Figure 8:  The spatial correlator $G_{\lambda}(R)\equiv\langle\mathrm{Tr}\
U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\rangle$ $(R=|\vec{x}-\vec{y}|)$ in
the Landau gauge (open circles) and the Coulomb gauge (open triangles),
together with its asymptotic value of $\frac{1}{3}\langle\mathrm{Tr}\
U_{4}\rangle^{2}$ (solid lines and cross symbols). In the Landau gauge,
$\langle\mathrm{Tr}\ U_{4}\rangle\neq 0$, and $G_{\lambda}(R)$ rapidly
converges to a constant for $R\gtrsim$ 0.4fm, which leads to a rapid
saturation of the instantaneous potential
$V_{\lambda}(R)\equiv-\frac{1}{a}{\rm ln}G_{\lambda}(R)$. In the Coulomb
gauge, $\langle\mathrm{Tr}\ U_{4}\rangle=0$, and $G_{\lambda}(R)$ decreases
monotonically to zero as $R$, which leads to $V_{\lambda}(R)=+\infty$ for
$R=\infty$.

Figure 9 shows $\lambda$-dependence of $\frac{1}{3}\langle\mathrm{Tr}\
U_{4}\rangle$ in generalized Landau gauge. For $\lambda\neq 0$,
$\langle\mathrm{Tr}\ U_{4}\rangle$ takes a non-zero real value, and it
approaches to zero continuously as $\lambda\rightarrow 0$. Here, it largely
changes in the small region of $0\leq\lambda\lesssim 0.1$. The finiteness of
$\langle\mathrm{Tr}\ U_{4}\rangle$ is directly related to the infrared damping
of the correlator $G_{\lambda}(R)$ and the infrared form of the instantaneous
potential $V_{\lambda}(R)$.

Figure 9:  The expectation value of $\frac{1}{3}\langle\mathrm{Tr}\
U_{4}\rangle$ in generalized Landau gauge. For $\lambda\neq 0$,
$\langle\mathrm{Tr}\ U_{4}\rangle$ takes a non-zero value, and it approaches
to zero continuously as $\lambda\rightarrow 0$. The value of
$\langle\mathrm{Tr}\ U_{4}\rangle$ relates to the infrared behavior of the
correlator $G_{\lambda}(R)$ and the instantaneous potential $V_{\lambda}(R)$.

###  VI.2 Asymptotic behavior of Polyakov-line correlator and $T$-length
potential

Next, we consider $T$-length terminated Polyakov-line correlator
$G_{\lambda}(R,T)$, which behaves asymptotically as

$\displaystyle G_{\lambda}(R,T)$ $\displaystyle\equiv$
$\displaystyle\langle\mathrm{Tr}L^{\dagger}(\vec{x},T)L(\vec{y},T)\rangle$
(30) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{3}\langle\mathrm{Tr}\
L(T)\rangle^{2}$

for large separation of $R=|\vec{x}-\vec{y}|$. As well as the instantaneous
potential, $\frac{1}{3}\langle\mathrm{Tr}\ L(T)\rangle^{2}$ is found to give
the lower bound of the correlator $G_{\lambda}(R,T)$, and the finiteness of
$\langle\mathrm{Tr}\ L(T)\rangle$ is responsible for the infrared saturation
of the finite-time potential $V_{\lambda}(R,T)$.

Figure 10 shows $T$-dependence of the terminated Polyakov-line
$\frac{1}{3}\langle\mathrm{Tr}\ L(T)\rangle$ in generalized Landau gauge for
typical values of $\lambda$. In the Coulomb gauge ($\lambda$=0),
$\langle\mathrm{Tr}\ L(T)\rangle$ is always zero as well as
$\langle\mathrm{Tr}\ U_{4}\rangle$, which means that
$V_{\lambda}(R=\infty,T)=-\frac{1}{T}{\rm ln}G_{\lambda}(R=\infty,T)=+\infty$
for any values of $T$. Actually, finite-time potential $V_{\lambda}(R,T)$
always has a linear part in the Coulomb gauge, as shown in Fig.5.

For $\lambda\neq 0$, $\langle\mathrm{Tr}\ L(T)\rangle$ is a decreasing
function of $T$, and it converges to zero in large-$T$ limit. At $T=N_{t}$,
the $T$-length terminated Polyakov-line $\langle\mathrm{Tr}\ L(T)\rangle$
results in the Polyakov loop, and $\langle\mathrm{Tr}\ L(N_{t})\rangle=0$ in
the confinement phase. Therefore, $\langle\mathrm{Tr}\ L(T)\rangle$ converges
to zero as $T\rightarrow N_{t}$, and then one finds

$\displaystyle G_{\lambda}(R=\infty,N_{t})$ $\displaystyle=$ $\displaystyle
0,$ (31) $\displaystyle V_{\lambda}(R=\infty,N_{t})$ $\displaystyle=$
$\displaystyle-\frac{1}{T}{\rm ln}G_{\lambda}(\infty,N_{t})=+\infty,$ (32)

which gives a confinement potential. From Fig.10, this convergence is found to
be fast for smaller $\lambda$-value, and such a convergence is closely related
to the growing of finite-time string tension $\sigma_{\lambda}(T)$.

Figure 10:  $T$-dependence of $\frac{1}{3}\langle\mathrm{Tr}\ L(T)\rangle$ in
generalized Landau gauge. In the Coulomb gauge, $\langle\mathrm{Tr}\
L(T)\rangle$ is always zero. For $\lambda\neq 0$, $\langle\mathrm{Tr}\
L(T)\rangle$ is a decreasing function of $T$, and it converges to zero in
large-$T$ limit.

###  VI.3 Gluon propagator and instantaneous potential in the Landau gauge

In this subsection, we discuss the relation between the gluon propagator and
the instantaneous potential in the Landau gauge.

The gluon propagator is a two-point function of the gauge field $A_{\mu}(x)$,
and is defined in Euclidean QCD as

$D_{\mu\nu}(x,y)\equiv\langle\mathrm{Tr}A_{\mu}(x)A_{\nu}(y)\rangle,$ (33)

where the trivial color structure is dropped off by taking the trace. In the
Landau gauge, we use the expression of $A_{\mu}$ in terms of $U_{\mu}$ as

$A_{4}(x)=\frac{1}{2iag}[U_{4}(x)-U_{4}^{\dagger}(x)]+\mathcal{O}(a^{2}).$
(34)

Note that this expression is only justified in the Landau gauge, or more
generally in large-$\lambda$ gauges, where the fluctuation of $A_{4}$ is
highly suppressed.

Then, the gluon propagator $D_{\mu\nu}$ is expressed using link-variables,
e.g.,

$\displaystyle
a^{2}g^{2}D_{44}(x,y)=a^{2}g^{2}\langle\mathrm{Tr}A_{4}(x)A_{4}(y)\rangle$
(35) $\displaystyle\simeq$
$\displaystyle-\frac{1}{4}\langle\mathrm{Tr}[U_{4}(x)-U_{4}^{\dagger}(x)][U_{4}(y)-U_{4}^{\dagger}(y)]\rangle$
$\displaystyle=$
$\displaystyle\langle\mathrm{Tr}[U_{4}(x)U_{4}^{\dagger}(y)]\rangle$
$\displaystyle\quad-\frac{1}{4}\langle\mathrm{Tr}[U_{4}(x)+U_{4}^{\dagger}(x)][U_{4}(y)+U_{4}^{\dagger}(y)]\rangle.$

The last term in Eq.(35) has only $O(a^{4})$-order $(x-y)$-dependence, and
actually it changes only a few $\%$ in the Landau gauge at $\beta$=5.8, so
that we here approximate this term as a constant $C$. Thus, Eq.(35) reduces

$a^{2}g^{2}D_{44}(x,y)\simeq\langle\mathrm{Tr}[U_{4}(x)U_{4}^{\dagger}(y)]\rangle-C,$
(36)

and we calculate the instantaneous potential $V_{\rm inst}(R)$ as

$\displaystyle V_{\rm inst}(R)$ $\displaystyle=$
$\displaystyle-\frac{1}{a}\ln\langle\mathrm{Tr}[U_{4}(\vec{x},a)U_{4}^{\dagger}(\vec{y},a)]\rangle$
(37) $\displaystyle=$
$\displaystyle-\frac{1}{a}\ln\left[C+a^{2}g^{2}D_{44}(R)\right]$
$\displaystyle\simeq$
$\displaystyle-\frac{ag^{2}}{C}D_{44}(R)+\mathrm{const.}$

In this way, the instantaneous potential $V_{\rm inst}(R)$ is expressed by
using the $44$-component of the gluon propagator in the Landau and
large-$\lambda$ gauges.

In the previous work Iritani09 , we have found that the Landau-gauge gluon
propagator is well reproduced by the four-dimensional Yukawa-function as

$D(r)\equiv D_{\mu\mu}(r)\propto\frac{1}{r}e^{-mr},$ (38)

with the Yukawa mass-parameter $m\simeq$ 0.6GeV, in the region of $r=0.1\sim
1$ fm. Apart from a pre-factor from the tensor factor, $D_{44}(R)$
approximately behaves as the Yukawa-function, and therefore the instantaneous
potential is expressed as

$V_{\rm inst}(R)\simeq-\frac{A}{R}e^{-mR}+\mathrm{const.}$ (39)

in the Landau gauge.

Figure 11:  Fit-result of the instantaneous potential $V_{\rm inst}(R)$ in the
Landau gauge using Coulomb plus linear form (solid line) and Yukawa-function
form (dashed line). Both forms well reproduce lattice QCD result. The best-fit
parameter of the Yukawa mass is $m$ = 0.634(3)GeV, which coincides with the
infrared effective gluon mass obtained from the Landau-gauge gluon propagator
Iritani09 .

Figure 11 shows the two fit-results of instantaneous potential in the Landau
gauge, using the Coulomb plus linear form $V_{\rm Coul.+lin.}(R)=-A/R+\sigma
R$ and the Yukawa-function form $V_{\rm Yukawa}(R)=-A\exp(-mR)/R$. Both
functions well reproduce the lattice QCD result. The best-fit Yukawa mass-
parameter is $m$=0.634(3)GeV, and this value coincides with the infrared
effective gluon mass obtained from the Landau-gauge gluon propagator Iritani09
.

We note again that this relation is only valid in the Landau and
large-$\lambda$ gauges, where the temporal link-variable $U_{4}$ can be
expanded in terms of lattice spacing $a$, and the last-term in Eq.(35) is
almost constant.

##  VII Summary and Discussion

In this paper, aiming to grasp the gauge dependence of gluon properties, we
have investigated generalized Landau gauge and applied it to instantaneous
interquark potential in SU(3) quenched lattice QCD at $\beta$=5.8. In the
Coulomb gauge, the instantaneous potential is expressed by the sum of Coulomb
potential and linear potential with 2-3 times larger string tension. In
contrast, the instantaneous potential has no linear part in the Landau gauge.
Thus, there is a large gap between these two gauges. Using generalized Landau
gauge, we have found that the instantaneous potential $V_{\lambda}(R)$ is
connected continuously from the Landau gauge towards the Coulomb gauge, and
the linear part in $V_{\lambda}(R)$ grows rapidly in the neighborhood of the
Coulomb gauge.

Since the slope $\sigma_{\lambda}$ of the instantaneous potential
$V_{\lambda}(R)$ grows continuously from 0 to 2-3$\sigma_{\rm phys}$, there
must exist some specific intermediate gauge where the slope $\sigma_{\lambda}$
coincides with the physical string tension $\sigma_{\rm phys}$. From the
lattice QCD calculation, the specific $\lambda$-parameter, $\lambda_{C}$, is
estimated to be about $0.02$. In this $\lambda_{C}$-gauge, the physical static
interquark potential $V_{\rm phys}(R)$ is approximately reproduced by the
instantaneous potential $V_{\lambda}(R)$.

We have also investigated $T$-length terminated Polyakov-line correlator, and
its corresponding finite-time potential $V_{\lambda}(R,T)$, which is a
generalization of the instantaneous potential $V_{\lambda}(R)$, in generalized
Landau gauge. The behavior of the slope $\sigma_{\lambda}(T)$ of the finite-
time potential is classified into three groups: the Coulomb-like gauge
($0\leq\lambda\lesssim 0.01$), the Landau-like gauge ($0.1\lesssim\lambda\leq
1$), and neighborhood of $\lambda_{C}$-gauge ($\lambda\sim\lambda_{C}\simeq
0.02$). In the Coulomb-like gauge, the slope $\sigma_{\lambda}(T)$ is a
decreasing function of $T$, and seems to approach to physical string tension
$\sigma_{\rm phys}$ for large $T$. In the Landau-like gauge,
$\sigma_{\lambda}(T)$ is an increasing function. Around the
$\lambda_{C}$-gauge, $\sigma_{\lambda}(T)$ has a weak $T$-length dependence.
We have also investigated $T$-length terminated Polyakov-line correlator and
its relation to the finite-time potential.

Finally, we consider a possible gauge of QCD to describe the quark potential
model from the viewpoint of instantaneous potential. The quark potential model
is a successful nonrelativistic framework with a potential instantaneously
acting among quarks, and describes many hadron properties in terms of quark
degrees of freedom. In this model, there are no dynamical gluons, and gluonic
effects indirectly appear as the instantaneous interquark potential.

As for the Coulomb gauge, the instantaneous potential has too large linear
part, which gives an upper bound on the static potential Zwanziger03 . It has
been suggested by Greensite et al. that the energy of the overconfining state
is lowered by inserting dynamical gluons between (anti-)quarks, which is
called “gluon-chain picture”. This gluon-chain state is considered as the
ground-state in the Coulomb gauge Greensite03 ; Greensite04 ; GreensiteThorn .
Therefore, dynamical gluon degrees of freedom must be also important to
describe hadron states in the Coulomb gauge.

For $\lambda_{C}$-gauge, the physical interquark potential $V_{\rm phys}(R)$
is approximately reproduced by the instantaneous potential
$V_{\lambda_{C}}(R)$ unlike the Coulomb gauge, as schematically shown in
Fig.12. This physically means that all other complicated effects including
dynamical gluons and ghosts are approximately cancelled in the
$\lambda_{C}$-gauge, and therefore we do not need to introduce any redundant
gluonic degrees of freedom. The absence of dynamical gluon degrees of freedom
would be a desired property for the quark model picture. The weak $T$-length
dependence of $\sigma_{\lambda}(T)$ around the $\lambda_{C}$-gauge ($T$-length
stability) is also a suitable feature for the potential model.

Figure 12:  The schematic illustration of the Coulomb gauge and the
$\lambda_{C}$-gauge. In the Coulomb gauge, the instantaneous potential gives
an “overconfining state” as an excited-state for the quark-antiquark system.
The ground-state is considered as the gluon-chain state, which contains
dynamical gluons between static sources. In the $\lambda_{C}$-gauge, the
instantaneous potential gives the physical interquark potential approximately,
and dynamical gluons need not appear.

In this way, as an interesting possibility, the $\lambda_{C}$-gauge is
expected to be a useful gauge in considering the linkage from QCD to the quark
potential model. In addition, since $\lambda_{C}\simeq 0.02\ll 1$ is a very
small parameter in this framework, it is interesting to apply the perturbative
technique in terms of $\lambda_{C}$ for the calculation of the Faddeev-Popov
determinant and so on.

###### Acknowledgements.

The authors thank Dr. H. Iida for his useful arguments. H.S. is supported in
part by the Grant for Scientific Research [(C) No.19540287, Priority Areas
“New Hadrons” (E01:21105006)] from the Ministry of Education, Culture, Science
and Technology (MEXT) of Japan. This work is supported by the Global COE
Program, “The Next Generation of Physics, Spun from Universality and
Emergence”. The lattice QCD calculations have been done on NEC-SX8 at Osaka
University.

##  Appendix A On the Coulomb gauge

In this appendix, we briefly discuss the difference between the Coulomb gauge
and the $\lambda\rightarrow 0$ limit of generalized Landau gauge.

###  A.1 Residual gauge degrees of freedom in the Coulomb gauge

We comment on the residual gauge degrees of freedom in the Coulomb gauge. In
lattice QCD, the Coulomb gauge fixing condition is expressed by the
maximization of the quantity

$R_{\rm Coul}[U]\equiv\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\
U_{i}(\vec{x},t)$ (40)

by the gauge transformation.

Now, we consider the spatially-global gauge transformation as

$\displaystyle U_{i}(\vec{x},t)$ $\displaystyle\rightarrow$
$\displaystyle\Omega(t)U_{i}(\vec{x},t)\Omega^{\dagger}(t),$ (41)
$\displaystyle U_{4}(\vec{x},t)$ $\displaystyle\rightarrow$
$\displaystyle\Omega(t)U_{4}(\vec{x},t)\Omega^{\dagger}(t+1),$ (42)

with the gauge function $\Omega(t)\in{\rm SU}(N_{c})$. $R_{\rm Coul}[U]$ is
invariant under this transformation,

$\displaystyle R_{\rm Coul}[U]$ $\displaystyle\equiv$
$\displaystyle\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\
U_{i}(\vec{x},t)$ (43) $\displaystyle\rightarrow$
$\displaystyle\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\
\Omega(t)U_{i}(\vec{x},t)\Omega^{\dagger}(t)$ $\displaystyle=$
$\displaystyle\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\
U_{i}(\vec{x},t).$

Therefore, the Coulomb gauge has the corresponding residual gauge degrees of
freedom.

Under this residual symmetry, however, $\mathrm{Tr}\ U_{4}$ is gauge-variant
as

$\displaystyle\mathrm{Tr}\ U_{4}(\vec{x},t)$ $\displaystyle\rightarrow$
$\displaystyle\mathrm{Tr}\ \Omega(t)U_{4}(\vec{x},t)\Omega^{\dagger}(t+1)$
(44)

so that the expectation value of $\mathrm{Tr}\ U_{4}$ is to be zero in the
Coulomb gauge.

In the generalized Landau gauge with non-zero $\lambda$-parameter, this
residual symmetry does not exist, and hence $\mathrm{Tr}\ U_{4}$ has a finite
expectation value. (See Fig.9.)

###  A.2 Convergence into Coulomb gauge

We here investigate the convergence of the generalized Landau gauge into the
Coulomb gauge in the limit of $\lambda\rightarrow 0$. To check the
convergence, we evaluate the quantity
$\langle(\partial_{i}A_{i}^{a})^{2}\rangle$,

Figure 13:  The lattice QCD result of
$\langle(\partial_{i}A_{i}^{a})^{2}\rangle$ in the generalized Landau gauge.
As $\lambda\rightarrow 0$, this quantity goes to zero monotonically.

$\displaystyle\langle\left(\partial_{i}A_{i}^{a}\right)^{2}\rangle$
$\displaystyle\equiv$ $\displaystyle\frac{1}{(N_{c}^{2}-1){N_{\rm site}}}$
(45) $\displaystyle\times$ $\displaystyle\sum_{x=1}^{N_{\rm
site}}\sum_{a=1}^{N_{c}^{2}-1}\Big{\\{}\sum_{i=1}^{3}\left[A_{i}^{a}(x)-A_{i}^{a}(x-\hat{i})\right]\Big{\\}}^{2}.~{}~{}~{}~{}$

In the Coulomb gauge, $\langle(\partial_{i}A_{i}^{a})^{2}\rangle$ is equal to
zero.

Figure 13 shows the lattice QCD result of
$\langle(\partial_{i}A_{i}^{a})^{2}\rangle$, which is monotonically decreasing
toward zero by varying $\lambda$ from 1 to 0. This result supports that the
generalized Landau gauge approaches to the Coulomb gauge in the
$\lambda\rightarrow 0$ limit.

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