Instanton Liquid at Finite Temperature and Chemical
Potential of Quarks
S.V. Molodtsov1,3, G.M. Zinovjev2
1Joint Institute for Nuclear Research, Dubna, 141980 RUSSIA
2Bogolyubov Institute for Theoretical Physics, ul. Metrolohichna 14-b, Kiev, 03680 UKRAINE
3Institute of Theoretical and Experimental Physics, Moscow, 117259 RUSSIA
Instanton liquid in heated and strongly interacting matter is studied using the variational princi-
ple. The dependence of the instanton liquid density (gluon condensate) on the temperature and
the quark chemical potential is determined under the assumption that, at finite temperatures,
the dominant contribution is given by an ensemble of calorons. The respective one-loop effective
quark Lagrangian is used.
In current studies of strong-interacting matter under extreme conditions, primary attention is
focused on a description of its phase state at given temperature and chemical potential. For def-
initeness, we consider that T is the temperature of quarks and µ is the quark chemical potential
(it is assumed that gluons are in thermodynamical equilibrium with quarks). However, there is no
approach making it possible to describe main features of the expected phase diagram of quark-gluon
matter at least qualitatively.
In the present study, we argue that the instanton liquid model of the QCD vacuum [1] can shed
light on some important features of a full picture. It is frequently noted that this model offers a useful
tool for obtaining phenomenologically plausible estimates in spite of the fact that it is poorly justified
because the typical size of an instanton is not properly fixed. As of now, this fact is considered as
inessential because a connection has been revealed between limitations on the instanton size due
to repulsion [2] and generation of mass of the gluon field in the framework of the quasi-classical
approximation [3]. The latter mechanism is a more general property of stochastic gluon fields than
the former one. We will discuss this question later. Here we assume that the problem of instanton
size is solved in one of the following scenarios: self-stabilization of the saturating ensemble [2],[4],
freezing of the coupling constant [5], or influence of the confining component [6]. In the present
study, primary attention is focused on a plausible qualitative model describing a behavior of the
gluon condensate.
In the beginning, we recollect the variational principle proposed in [2] and the method of determi-
nation of the size of pseudoparticles and the density of the instanton liquid and introduce notation for
further considerations. In the model of instanton liquid describing the QCD vacuum, it is assumed
that the leading contribution to the QCD generating functional is given by the background fields
representing superposition of instantons in the singular gauge:
Aaµ(x; γ) =
ωabη̄bµν aν(y) , aν(y) =
y2 + ρ2
, y = x− z , µ, ν = 1, 2, 3, 4 . (1)
where ρ is the size, ω is the matrix of color rotation, and z is the position of the center of a
pseudoparticle (in the case of anti-instanton, the ’t Hooft symbol should be replaced as follows:
η̄ → η). This being so, the QCD generating functional takes the form
dγi d(ρi) e
−β Uint(γ) =
dγi e
−E(γ) , (2)
http://arxiv.org/abs/0704.0143v1
E(γ) = β Uint(γ)−
ln d(ρi) ,
where
d(ρ) =
β̃2Nc e−β(ρ) , (3)
is the instanton size distribution [7]; dγi = dzi dωi dρi, and
β(ρ) =
= −b ln(C1/bNc Λρ)
is the action of a single instanton, where (Λ = ΛMS = 0.92ΛP.V.) CNc , depends on the renormalization
scheme and, in the case under consideration, is given by CNc ≈
4.66 exp(−1.68Nc)
π2(Nc − 1)!(Nc − 2)!
, and b =
11 Nc − 2 Nf
. We assume that Nf=2 here because the leading contribution to renormalization
comes from hard massless gluons and quarks. The auxiliary function
β̃ = −b ln(Λρ̄) ,
is evaluated at the scale ρ̄ defined by an average size of pseudoparticles, Uint(γ) is considered assuming
pair interaction dominance. Its contribution has the form [2]
dω1 dω2 dz1 dz2 Uint(γ1, γ2) = V ξ
2 ρ21 ρ
where ξ2 = 27 π
N2c − 1
. The factor β that appears in the exponent in formula (2) is also evaluated
at the scale of an average size of pseudoparticles ρ̄. Assuming that the instanton liquid is topologically
neutral, we do not introduce notation to distinguish between instantons and anti-instantons, N
denotes the ovarall number of pseudoparticles in volume V .
Since the interaction is independent of coordinates or orientation in color space, it is natural to
calculate the generating functional Y on the basis of the effective one-particle distribution function
µ(ρ), which can be determined from the solution of the variational problem
dρi µ(ρ) =
dγi e
−E1(γ) , (4)
E1(γ) = −
lnµ(ρi) ,
where the factor V N in (4) is isolated in order that the result be expressed in terms of the respective
density and convenience in interpretation of the function µ(ρ). With regard to convexity of the
exponential function, the generating functional (2) for every fixed N partial contribution can be
estimated using the approximating inequality
Y ′ ≥ Ya = Y ′1 exp(−〈E − E1〉) , (5)
where an average over approximate ensemble is implied. In the case under consideration, the average
of difference 〈E − E1〉 is given by:
〈E −E1〉 =
dγi [β Uint −
ln d(ρi) +
lnµ(ρi)] e
lnµ(ρi) =
dρ µ(ρ) ln
dρ1dρ2 ξ
2 ρ21ρ
2 µ(ρ1)µ(ρ2)
where µ0 =
dρ µ(ρ).
Variation of the functional 〈E −E1〉 with respect to µ(ρ) results formally in the equation µ(ρ) =
e−1 d(ρ) e−nβξ
2ρ2ρ2 (where n = N/V is the density of the instanton liquid). Here an unwanted factor
of e−1, emerges. It can be excluded due to the fact that the approximate functional Ya is independent
of the constant factor of C that can be added to the expressionfor µ(ρ). For convenience, we set
C = e, and therefore, arrive at
µ(ρ) = d(ρ) e−nβξ
2ρ2ρ2 . (6)
Substituting this solution to the approximate functional, we obtain
V N µN0
(ρ2)2 .
Defining suitable parameter ν the integral for determination µ0 can be represented in the form
µ0 = Λ
dρΛ CNcβ̃
2Nc (ρΛ)b−5 e
ρ2 . (7)
From the comparison of which with formula (6) we obtain
= βξ2nρ2 . (8)
Provided that ν is known, this formula offers a relation between the average instanton size and the
density of the instanton liquid. To find this relation, we consider the equation
dρ ρb−3 e
dρ ρb−5 e
ρ2 ν−1 Γ( b−4
Γ( b−4
which gives ν = b−4
, and therefore, µ0 = Λ
4 CNcβ̃
2Nc (ρΛ)
2 . It should be noted that the factor
of two in the denominator of this expression stems from the integration measure 2ρdρ, which, in its
turn, emerges in transformation to the Gaussian integral with respect to ρ squired. This factor was
omited in [2]; however, this fact has no noticeable consequences. The reason is that the parameter
Λ is determined from a fit to some observable, for example, to the pion decay constant. In so doing,
everything is governed by a choice of scale. Moreover, it should be remembered that the instanton
liquid model is merely a rough approximation. From the above, we derive an approximate expression
for the functional as follows:
Ya = exp
[ln(n/Λ4)− 1] +N ln
CNcβ̃
2Nc(βξ2ν)−ν/2
. (9)
Now we find the value of n at which the argument of the exponential approaches its maximum. To
do this, we should solve the equation
ln(n/Λ4) + ln
CNcβ̃
2Nc(βξ2ν)−ν/2
− n ν
= 0 . (10)
From the relation (8) we obtain
= 0 .
On the other hand,
= − bρ̄ ,
. We represent the derivative of β with respect to the density
in the form
, and obtain
4β − b ,
. (11)
Thus we derive the expressionfor the instanton liquid density
n/Λ4 =
CNcβ̃
2Nc(βξ2ν)−ν/2
ν + 2
ν − 2
ν + 2
2β − ν − 2
. (12)
The contribution of the derivatives of the functions β and β̃ with respect to the density was disre-
garded in [2]. This contribution compensates for the above-mentioned factor of 2 though, as was
noted above, this is not essential. The obtained formula for the instanton liquid density by itself
does not provide a solution to the problem because it remains to solve the transcendental equation
(8) in ρ̄, where the function β involves the logarithm of ρ̄. To solve this equation, it is convenient to
reformulate the problem without resort to the explicit formula (12) for the instanton liquid density.
By definition of the function β, the action of an isolated pseudoparticle must be positive. This gives a
limitation to the maximum size of an (anti-)instanton as follows: ρ̄ΛC
≤ 1 (actually, ρ̄Λ ≤ 1). Now
we can solve the transcendental equation (10), by bisection of the segment. In so doing, a stationary
value of ρ̄ is determined at each step and the respective instanton liquid density is determined from
equation (8). In the calculation of the generating functional, the contributions of the type (ρ̄Λ)2ν are
used rather than the expression for the instanton liquid density.
Now we modify the variational principle in order to extend our description to the case of finite
temperatures. For this purpose, we employ calorons — solutions of the Yang–Mills equations periodic
in the Euclidean time. The background field should be replaced by a superposition of calorons and
anti-calorons as follows [8]:
Aaµ(x, γ) = −
ωab η̄bµν ∂ν lnΠ,
Π = 1 +
sinh(2πrT )
cosh(2πrT )− cos(2πτT ) ,
where T−1 is the period of the caloron, r = |x−z| is the distance from the center of the caloron z in
three dimensional space, and τ = x4 − z4 – is the respective interval of ”time”. As the temperature
tends to zero, such solutions go over to (anti-)instantons in the singular gauge. Yet another modifi-
cation of the variational principle is the replacement of the distribution (3) in the instanton size by
the function
d(ρ, T ) =
β̃2Nc exp[−β(ρ)− ANcT 2ρ2] , (14)
where the coefficient ANc =
6 Nc − 1
π2 accounts for the additional contribution to the action
of each individual pseudoparticle. It provides an approximation to a more exact expression
d(ρ, T ) = d(ρ, 0) exp
g2T 2
(Nc +Nf/2)
4π2ρ2
+ 12 A(πρT ) [1 + (Nc −Nf)/6]
, (15)
constructed from the respective determinants [9]. For our purposes it is sufficient to say that the
function A(πρT ) is determined by a shape of the pseudoparticle (13). This function was studied in
the cited work; however, we do not use it in the present article. It should be mentioned that the
expansion up to the terms of the order T 2 can be used as an approximate expression for the function
A(πρT ) because, within the accuracy of the variational principle, only the terms up to order ρ2
should be kept in the argument of the exponential in formula (14). The first term in formula (15)
is represented as a product of two factors; each factor was interpreted in [9]. The first factor is
the square of the electric mass, that is, the temporal component of the gluon polarization tensor
evaluated at the zero energy and momentum. It has the form
m2el = Π44(ω = 0,p = 0) = g
(Nc +Nf/2)
. (16)
The remaining components being equal to zero at zero energy-momentum. Therefore, the magnetic
mass vanishes. Note that the one-loop quark and gluon contributions to the polarization tensor
are taken into account [10], the resulting sum being rearranged in order that the quark and gluon
contributions in the medium sum up to a finite value. This, formally, gives rise to a generation of
the mass of the gluon field. The second factor is the integral of the square of the fourth component
A4 of the field in formula (13) ∫
dy Aa4(y)A
4(y) =
4π2ρ2
. (17)
It is independent of the temperature [11]. It is seen that one can take into account only one-loop
contribution 12m
4 to the Lagrangian of the gluon field and neglect other corrections. It was
demonstrated [3] that the term Uint describing the interaction of pseudoparticles can be brought in the
form 12m
2 AaµA
µ, wherem
2 = 9π2 n ρ̄2
N2c − 1
. Thus the interaction term also describes generation of
the mass of the gluon field in the instanton–anti-instanton medium in quasi-classical approximation.
This being so, chromoelectric and chromomagnetic fields are screened equally well provided that
the instanton liquid density is not equal to zero. It was shown that screening is a consequence of
stochastic character of the ensemble of gluon fields being unrelated to a specific instanton solution
of the type (1) or details of the repulsion mechanism responsible for stabilization of the ensemble
[2]. An application of these considerations to the (anti-)instanton solution (1) leads precisely to the
formula for Uint. It turns out that, in the caloron ensemble, screening of chromomagnetic fields and
the interaction term depends only weakly on the temperature. However, the anisotropy is negligible
small and the interaction term coincides with that obtained for the (anti-)instanton solution. First
it was found in [11], where the instanton liquid was studied at non zero temperature.
The one-loop contribution of Plank gluons is proportional to Nc (see formula (16)) and does not
vary as the chemical potential becomes different from zero. On the other hand, it is known that
the one-loop fermion contribution in the medium can be calculated exactly. It has no dangerous
singularities [12], [13]. The ”temporal” component of the polarization tensor generated by a quark
of definite flavor has the form
44(k4, ω) = g
dp p2
4ε2p − k2
(k2 + 2pω)2 + 4ε2pk
(k2 − 2pω)2 + 4ε2pk24
− εpk4
arctan
8pω εpk4
4ε2pk
4 − 4p2ω2 + k4
where ω = |k|, k2 = ω2 + k24, εp = (m2 + p2)1/2, where m – is the quark mass, np = n−p + n+p ,
n−p = (e
T + 1)−1, n+p = (e
T + 1)−1. After summation over all components, the polarization
tensor takes the form
Πf(k4, ω) = g
dp p2
2m2 − k2
(k2 + 2pω)2 + 4ε2pk
(k2 − 2pω)2 + 4ε2pk24
. (18)
It is seen that, at k4 = 0, and small values of ω, the first term (that is, unit) gives the dominant
contribution to the gluon mass. The spatial components are negligibly small. In particular, at ω = 0
we obtain
Πf(0, 0) = Π
44(0, 0) = g
dp p2
np , (19)
and at T = 0 we arrive at Πf (0, 0) = g2
(µ2 −m2)1/2µ
µ+ (µ2 −m2)1/2
The ultimate expression for the electric mass has the form
m2el =
g2T 2
Πf (0, 0)
 . (20)
In this approximation, the effect of the instanton liquid is completely accounted for by the quark
mass dynamically generated in the instanton medium. With such definition of mass, the formula (16)
at µ = 0 and T 6= 0 should be modified. The coefficient 16 at Nf should be replaced by
. However,
this replacement has only a little effect; self-consistency of our calculations will be discussed below.
Using the integral (17), which is also valid for the caloron solution, we derive the expression for
the distribution of pseudoparticles:
d(ρ;µ, T ) = d(ρ; 0, 0) e−η
2(µ,T ) ρ2 , η2 = 2 π2
Πf(0, 0)
 . (21)
The one-loop quark contribution to the instanton action at zero temperature, finite chemical poten-
tial, and ω 6= 0 was studied in detail in [14] (see also [15], [16]). These studies make it possible to
improve our description, however, we work within the approximation (21) and, moreover, we consider
the limit of massless quarks. A self-consistent calculation for the quark with dynamically generated
mass can be the subject of a separable study.
Necessary modifications in the variational principle are as follows. It was revealed that only
the distribution function d(ρ;µ, T ) of pseudoparticles changes, whereas the repulsion interaction
Uint between pseudoparticles remains as before. Similar to the case of instantons, we introduce the
parameter ν satisfying the relation
= η2 + βξ2nρ2 , (22)
instead of (8). Since the instanton liquid density is greater than zero, a new limitation on the average
size of pseudoparticle emerges ρ̄Λ ≤ ν
η . If this limit is smaller than the limit descussed above,
then it must be the starting point for the determination of the equilibrium size of pseudoparticles by
the bisection method. The derivative of the function β with respect to the density of the instanton
liquid can be determined from the relation (22). The result is
4β − b+ 2η2 ρ̄2β
ν−η2 ρ̄2
, (23)
it should be substituted in Eq. (10) that determines the saddle-point. The integral (19) should be
evaluated numerically because it cannot be calculated analytically at arbitrary temperatures even
though the quark mass equals zero. Thus we are ready to determine the parameters of the instanton
liquid everywhere over the µ – T plane. For simplicity, the calculations are performed at zero quark
masses. We neglect the light-quark contribution to the respective determinants [17] (see also [18]).
We also disregard a possible temperature molecular behavior of instanton–anti-instanton pairs [19].
The results of the calculations are shown in Fig. 1 by the lines of constant density. The instanton
liquid density is plotted in Fig 2. versus the temperature (at zero chemical potential) and versus the
chemical potential (at zero temperature). Though the conventional natation for the instanton liquid
density at nonzero temperature is n = TN/V3, we use the label n which is more simple. At T 6= 0,
and µ = 0, our results coincide with the results obtained in [11] and [9]. It sould be noted that our
results are consistent with recent calculations on a lattice at finite temperatures [20], [21], where a
rapid decrease of the chromoelectric components in the respective correlation functions was found.
In our model, such suppression is due to the term 12m
4 in the effective action; with neglect of
this factor, the chromoelectric and chromomagnetic correlators coincide. From this point of view, our
calculations may seem inconsistent. We use the caloron solution (13), which is symmetric under an
interchange of chromoelectric and chromomagnetic fields. However, the caloron components manifest
themthelves in the observables differently because of the anisotropy of the weight function. In fact,
our method of taking the gluon mass term into account is consistent only in perturbation theory. In
a complete study, one must find an analogue of the solution (13) for the effective Lagrangian with the
Figure 1: Lines of equal density of the instanton liquid in the temperature-chemical potential plane.
Curve 1 corresponds to the density n = 0.75 n0, where n0 is the density at zero temperature and
chemical potential. Also shown are the densities from (curve 2) n = 0.5 n0 to (curve 6) n = 0.1 n0.
Curves 3–5 correspond to intermediate densities at intervals of 0.1.
Figure 2: Instanton liquid density versus (curve 1) temperature and (curve 2) chemical potential.
gluon mass generated for the chromoelectric field and gain a self-consistent description of ensemble
of pseudoparticles in the long-wave approximation [3].
It is of interest that the data on correlation functions for cooled configurations [20] are fitted
well by the instanton ensemble [22]. In so doing, the contribution of the terms of the second order
in the instanton liquid density (∼ n2) is in excellent agreement with the effect of the standard
instanton ensemble with the respective admixture of the perturbative component everywhere over the
distance range chosen for a fit [23]. This agreement indicates that the confining component is absent
from the lattice configurations isolated by cooling. It is surprising because lattice simulations with
cooling were aimed at the searches for a long-wave confining component. However, an interpretation
of lattice simulations at finite temperature presents difficulties because it is not clear what scale
corresponds to the configarations used for the measurements. The magnitude of deformation of the
chromoelectric component of the solution for the effective Lagrangian with the mass term is also
poorly known. The scale of lattice configurations can, in principle, be estimated using the scale at
which the chromoelectric field decrease since only this scale has emerged in our calculations.
In conclusion we note that, though we used only a rough approximation, the most important
features of the behavior of the instanton liquid density (gluon condensate) in the medium have been
revealed. The lines of equal density are markedly extended along the µ axis because, according to
the formula (20), the most substantial gluon component of screening vanishes at small temperatures.
Typical values of T and µ at which the effects of the medium become significant are related to each
other by the formula
(Nc +Nf/2)
(T/Λ)2 ∼ Nf
(µ/Λ)2
∼ 1, which leads to a plausible coefficient
of oblongness along the µ axis
2π Tc ,
(at Nc = 3 and Nf = 2). A fall in density evaluated with allowance for the dynamically generated
quark mass should begin at a greater value and be more steep. The reason is that, at chemical
potentials less than the quark mass, the quark contribution to screening is reduced. This gives rise
to formation of a plateau and concentration of the lines of equal density. The dependence of the
dynamical quark mass on the momentum ω is significant at small temperatures leading to a decrease
of screening approximately by a factor of two [14].
We are grateful to A.E. Dorokhov for helful discussions.
This work was supported in part by grants STCU #P015c, CERN-INTAS 2000-349, NATO
2000-PST.CLG 977482.
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  Instanton liquid in heated and strongly interacting matter is studied using
the variational principle. The dependence of the instanton liquid density
(gluon condensate) on the temperature and the quark chemical potential is
determined under the assumption that, at finite temperatures, the dominant
contribution is given by an ensemble of calorons. The respective one-loop
effective quark Lagrangian is used.

Instanton Liquid at Finite Temperature and Chemical
Potential of Quarks
S.V. Molodtsov1,3, G.M. Zinovjev2
1Joint Institute for Nuclear Research, Dubna, 141980 RUSSIA
2Bogolyubov Institute for Theoretical Physics, ul. Metrolohichna 14-b, Kiev, 03680 UKRAINE
3Institute of Theoretical and Experimental Physics, Moscow, 117259 RUSSIA
Instanton liquid in heated and strongly interacting matter is studied using the variational princi-
ple. The dependence of the instanton liquid density (gluon condensate) on the temperature and
the quark chemical potential is determined under the assumption that, at finite temperatures,
the dominant contribution is given by an ensemble of calorons. The respective one-loop effective
quark Lagrangian is used.
In current studies of strong-interacting matter under extreme conditions, primary attention is
focused on a description of its phase state at given temperature and chemical potential. For def-
initeness, we consider that T is the temperature of quarks and µ is the quark chemical potential
(it is assumed that gluons are in thermodynamical equilibrium with quarks). However, there is no
approach making it possible to describe main features of the expected phase diagram of quark-gluon
matter at least qualitatively.
In the present study, we argue that the instanton liquid model of the QCD vacuum [1] can shed
light on some important features of a full picture. It is frequently noted that this model offers a useful
tool for obtaining phenomenologically plausible estimates in spite of the fact that it is poorly justified
because the typical size of an instanton is not properly fixed. As of now, this fact is considered as
inessential because a connection has been revealed between limitations on the instanton size due
to repulsion [2] and generation of mass of the gluon field in the framework of the quasi-classical
approximation [3]. The latter mechanism is a more general property of stochastic gluon fields than
the former one. We will discuss this question later. Here we assume that the problem of instanton
size is solved in one of the following scenarios: self-stabilization of the saturating ensemble [2],[4],
freezing of the coupling constant [5], or influence of the confining component [6]. In the present
study, primary attention is focused on a plausible qualitative model describing a behavior of the
gluon condensate.
In the beginning, we recollect the variational principle proposed in [2] and the method of determi-
nation of the size of pseudoparticles and the density of the instanton liquid and introduce notation for
further considerations. In the model of instanton liquid describing the QCD vacuum, it is assumed
that the leading contribution to the QCD generating functional is given by the background fields
representing superposition of instantons in the singular gauge:
Aaµ(x; γ) =
ωabη̄bµν aν(y) , aν(y) =
y2 + ρ2
, y = x− z , µ, ν = 1, 2, 3, 4 . (1)
where ρ is the size, ω is the matrix of color rotation, and z is the position of the center of a
pseudoparticle (in the case of anti-instanton, the ’t Hooft symbol should be replaced as follows:
η̄ → η). This being so, the QCD generating functional takes the form
dγi d(ρi) e
−β Uint(γ) =
dγi e
−E(γ) , (2)
http://arxiv.org/abs/0704.0143v1
E(γ) = β Uint(γ)−
ln d(ρi) ,
where
d(ρ) =
β̃2Nc e−β(ρ) , (3)
is the instanton size distribution [7]; dγi = dzi dωi dρi, and
β(ρ) =
= −b ln(C1/bNc Λρ)
is the action of a single instanton, where (Λ = ΛMS = 0.92ΛP.V.) CNc , depends on the renormalization
scheme and, in the case under consideration, is given by CNc ≈
4.66 exp(−1.68Nc)
π2(Nc − 1)!(Nc − 2)!
, and b =
11 Nc − 2 Nf
. We assume that Nf=2 here because the leading contribution to renormalization
comes from hard massless gluons and quarks. The auxiliary function
β̃ = −b ln(Λρ̄) ,
is evaluated at the scale ρ̄ defined by an average size of pseudoparticles, Uint(γ) is considered assuming
pair interaction dominance. Its contribution has the form [2]
dω1 dω2 dz1 dz2 Uint(γ1, γ2) = V ξ
2 ρ21 ρ
where ξ2 = 27 π
N2c − 1
. The factor β that appears in the exponent in formula (2) is also evaluated
at the scale of an average size of pseudoparticles ρ̄. Assuming that the instanton liquid is topologically
neutral, we do not introduce notation to distinguish between instantons and anti-instantons, N
denotes the ovarall number of pseudoparticles in volume V .
Since the interaction is independent of coordinates or orientation in color space, it is natural to
calculate the generating functional Y on the basis of the effective one-particle distribution function
µ(ρ), which can be determined from the solution of the variational problem
dρi µ(ρ) =
dγi e
−E1(γ) , (4)
E1(γ) = −
lnµ(ρi) ,
where the factor V N in (4) is isolated in order that the result be expressed in terms of the respective
density and convenience in interpretation of the function µ(ρ). With regard to convexity of the
exponential function, the generating functional (2) for every fixed N partial contribution can be
estimated using the approximating inequality
Y ′ ≥ Ya = Y ′1 exp(−〈E − E1〉) , (5)
where an average over approximate ensemble is implied. In the case under consideration, the average
of difference 〈E − E1〉 is given by:
〈E −E1〉 =
dγi [β Uint −
ln d(ρi) +
lnµ(ρi)] e
lnµ(ρi) =
dρ µ(ρ) ln
dρ1dρ2 ξ
2 ρ21ρ
2 µ(ρ1)µ(ρ2)
where µ0 =
dρ µ(ρ).
Variation of the functional 〈E −E1〉 with respect to µ(ρ) results formally in the equation µ(ρ) =
e−1 d(ρ) e−nβξ
2ρ2ρ2 (where n = N/V is the density of the instanton liquid). Here an unwanted factor
of e−1, emerges. It can be excluded due to the fact that the approximate functional Ya is independent
of the constant factor of C that can be added to the expressionfor µ(ρ). For convenience, we set
C = e, and therefore, arrive at
µ(ρ) = d(ρ) e−nβξ
2ρ2ρ2 . (6)
Substituting this solution to the approximate functional, we obtain
V N µN0
(ρ2)2 .
Defining suitable parameter ν the integral for determination µ0 can be represented in the form
µ0 = Λ
dρΛ CNcβ̃
2Nc (ρΛ)b−5 e
ρ2 . (7)
From the comparison of which with formula (6) we obtain
= βξ2nρ2 . (8)
Provided that ν is known, this formula offers a relation between the average instanton size and the
density of the instanton liquid. To find this relation, we consider the equation
dρ ρb−3 e
dρ ρb−5 e
ρ2 ν−1 Γ( b−4
Γ( b−4
which gives ν = b−4
, and therefore, µ0 = Λ
4 CNcβ̃
2Nc (ρΛ)
2 . It should be noted that the factor
of two in the denominator of this expression stems from the integration measure 2ρdρ, which, in its
turn, emerges in transformation to the Gaussian integral with respect to ρ squired. This factor was
omited in [2]; however, this fact has no noticeable consequences. The reason is that the parameter
Λ is determined from a fit to some observable, for example, to the pion decay constant. In so doing,
everything is governed by a choice of scale. Moreover, it should be remembered that the instanton
liquid model is merely a rough approximation. From the above, we derive an approximate expression
for the functional as follows:
Ya = exp
[ln(n/Λ4)− 1] +N ln
CNcβ̃
2Nc(βξ2ν)−ν/2
. (9)
Now we find the value of n at which the argument of the exponential approaches its maximum. To
do this, we should solve the equation
ln(n/Λ4) + ln
CNcβ̃
2Nc(βξ2ν)−ν/2
− n ν
= 0 . (10)
From the relation (8) we obtain
= 0 .
On the other hand,
= − bρ̄ ,
. We represent the derivative of β with respect to the density
in the form
, and obtain
4β − b ,
. (11)
Thus we derive the expressionfor the instanton liquid density
n/Λ4 =
CNcβ̃
2Nc(βξ2ν)−ν/2
ν + 2
ν − 2
ν + 2
2β − ν − 2
. (12)
The contribution of the derivatives of the functions β and β̃ with respect to the density was disre-
garded in [2]. This contribution compensates for the above-mentioned factor of 2 though, as was
noted above, this is not essential. The obtained formula for the instanton liquid density by itself
does not provide a solution to the problem because it remains to solve the transcendental equation
(8) in ρ̄, where the function β involves the logarithm of ρ̄. To solve this equation, it is convenient to
reformulate the problem without resort to the explicit formula (12) for the instanton liquid density.
By definition of the function β, the action of an isolated pseudoparticle must be positive. This gives a
limitation to the maximum size of an (anti-)instanton as follows: ρ̄ΛC
≤ 1 (actually, ρ̄Λ ≤ 1). Now
we can solve the transcendental equation (10), by bisection of the segment. In so doing, a stationary
value of ρ̄ is determined at each step and the respective instanton liquid density is determined from
equation (8). In the calculation of the generating functional, the contributions of the type (ρ̄Λ)2ν are
used rather than the expression for the instanton liquid density.
Now we modify the variational principle in order to extend our description to the case of finite
temperatures. For this purpose, we employ calorons — solutions of the Yang–Mills equations periodic
in the Euclidean time. The background field should be replaced by a superposition of calorons and
anti-calorons as follows [8]:
Aaµ(x, γ) = −
ωab η̄bµν ∂ν lnΠ,
Π = 1 +
sinh(2πrT )
cosh(2πrT )− cos(2πτT ) ,
where T−1 is the period of the caloron, r = |x−z| is the distance from the center of the caloron z in
three dimensional space, and τ = x4 − z4 – is the respective interval of ”time”. As the temperature
tends to zero, such solutions go over to (anti-)instantons in the singular gauge. Yet another modifi-
cation of the variational principle is the replacement of the distribution (3) in the instanton size by
the function
d(ρ, T ) =
β̃2Nc exp[−β(ρ)− ANcT 2ρ2] , (14)
where the coefficient ANc =
6 Nc − 1
π2 accounts for the additional contribution to the action
of each individual pseudoparticle. It provides an approximation to a more exact expression
d(ρ, T ) = d(ρ, 0) exp
g2T 2
(Nc +Nf/2)
4π2ρ2
+ 12 A(πρT ) [1 + (Nc −Nf)/6]
, (15)
constructed from the respective determinants [9]. For our purposes it is sufficient to say that the
function A(πρT ) is determined by a shape of the pseudoparticle (13). This function was studied in
the cited work; however, we do not use it in the present article. It should be mentioned that the
expansion up to the terms of the order T 2 can be used as an approximate expression for the function
A(πρT ) because, within the accuracy of the variational principle, only the terms up to order ρ2
should be kept in the argument of the exponential in formula (14). The first term in formula (15)
is represented as a product of two factors; each factor was interpreted in [9]. The first factor is
the square of the electric mass, that is, the temporal component of the gluon polarization tensor
evaluated at the zero energy and momentum. It has the form
m2el = Π44(ω = 0,p = 0) = g
(Nc +Nf/2)
. (16)
The remaining components being equal to zero at zero energy-momentum. Therefore, the magnetic
mass vanishes. Note that the one-loop quark and gluon contributions to the polarization tensor
are taken into account [10], the resulting sum being rearranged in order that the quark and gluon
contributions in the medium sum up to a finite value. This, formally, gives rise to a generation of
the mass of the gluon field. The second factor is the integral of the square of the fourth component
A4 of the field in formula (13) ∫
dy Aa4(y)A
4(y) =
4π2ρ2
. (17)
It is independent of the temperature [11]. It is seen that one can take into account only one-loop
contribution 12m
4 to the Lagrangian of the gluon field and neglect other corrections. It was
demonstrated [3] that the term Uint describing the interaction of pseudoparticles can be brought in the
form 12m
2 AaµA
µ, wherem
2 = 9π2 n ρ̄2
N2c − 1
. Thus the interaction term also describes generation of
the mass of the gluon field in the instanton–anti-instanton medium in quasi-classical approximation.
This being so, chromoelectric and chromomagnetic fields are screened equally well provided that
the instanton liquid density is not equal to zero. It was shown that screening is a consequence of
stochastic character of the ensemble of gluon fields being unrelated to a specific instanton solution
of the type (1) or details of the repulsion mechanism responsible for stabilization of the ensemble
[2]. An application of these considerations to the (anti-)instanton solution (1) leads precisely to the
formula for Uint. It turns out that, in the caloron ensemble, screening of chromomagnetic fields and
the interaction term depends only weakly on the temperature. However, the anisotropy is negligible
small and the interaction term coincides with that obtained for the (anti-)instanton solution. First
it was found in [11], where the instanton liquid was studied at non zero temperature.
The one-loop contribution of Plank gluons is proportional to Nc (see formula (16)) and does not
vary as the chemical potential becomes different from zero. On the other hand, it is known that
the one-loop fermion contribution in the medium can be calculated exactly. It has no dangerous
singularities [12], [13]. The ”temporal” component of the polarization tensor generated by a quark
of definite flavor has the form
44(k4, ω) = g
dp p2
4ε2p − k2
(k2 + 2pω)2 + 4ε2pk
(k2 − 2pω)2 + 4ε2pk24
− εpk4
arctan
8pω εpk4
4ε2pk
4 − 4p2ω2 + k4
where ω = |k|, k2 = ω2 + k24, εp = (m2 + p2)1/2, where m – is the quark mass, np = n−p + n+p ,
n−p = (e
T + 1)−1, n+p = (e
T + 1)−1. After summation over all components, the polarization
tensor takes the form
Πf(k4, ω) = g
dp p2
2m2 − k2
(k2 + 2pω)2 + 4ε2pk
(k2 − 2pω)2 + 4ε2pk24
. (18)
It is seen that, at k4 = 0, and small values of ω, the first term (that is, unit) gives the dominant
contribution to the gluon mass. The spatial components are negligibly small. In particular, at ω = 0
we obtain
Πf(0, 0) = Π
44(0, 0) = g
dp p2
np , (19)
and at T = 0 we arrive at Πf (0, 0) = g2
(µ2 −m2)1/2µ
µ+ (µ2 −m2)1/2
The ultimate expression for the electric mass has the form
m2el =
g2T 2
Πf (0, 0)
 . (20)
In this approximation, the effect of the instanton liquid is completely accounted for by the quark
mass dynamically generated in the instanton medium. With such definition of mass, the formula (16)
at µ = 0 and T 6= 0 should be modified. The coefficient 16 at Nf should be replaced by
. However,
this replacement has only a little effect; self-consistency of our calculations will be discussed below.
Using the integral (17), which is also valid for the caloron solution, we derive the expression for
the distribution of pseudoparticles:
d(ρ;µ, T ) = d(ρ; 0, 0) e−η
2(µ,T ) ρ2 , η2 = 2 π2
Πf(0, 0)
 . (21)
The one-loop quark contribution to the instanton action at zero temperature, finite chemical poten-
tial, and ω 6= 0 was studied in detail in [14] (see also [15], [16]). These studies make it possible to
improve our description, however, we work within the approximation (21) and, moreover, we consider
the limit of massless quarks. A self-consistent calculation for the quark with dynamically generated
mass can be the subject of a separable study.
Necessary modifications in the variational principle are as follows. It was revealed that only
the distribution function d(ρ;µ, T ) of pseudoparticles changes, whereas the repulsion interaction
Uint between pseudoparticles remains as before. Similar to the case of instantons, we introduce the
parameter ν satisfying the relation
= η2 + βξ2nρ2 , (22)
instead of (8). Since the instanton liquid density is greater than zero, a new limitation on the average
size of pseudoparticle emerges ρ̄Λ ≤ ν
η . If this limit is smaller than the limit descussed above,
then it must be the starting point for the determination of the equilibrium size of pseudoparticles by
the bisection method. The derivative of the function β with respect to the density of the instanton
liquid can be determined from the relation (22). The result is
4β − b+ 2η2 ρ̄2β
ν−η2 ρ̄2
, (23)
it should be substituted in Eq. (10) that determines the saddle-point. The integral (19) should be
evaluated numerically because it cannot be calculated analytically at arbitrary temperatures even
though the quark mass equals zero. Thus we are ready to determine the parameters of the instanton
liquid everywhere over the µ – T plane. For simplicity, the calculations are performed at zero quark
masses. We neglect the light-quark contribution to the respective determinants [17] (see also [18]).
We also disregard a possible temperature molecular behavior of instanton–anti-instanton pairs [19].
The results of the calculations are shown in Fig. 1 by the lines of constant density. The instanton
liquid density is plotted in Fig 2. versus the temperature (at zero chemical potential) and versus the
chemical potential (at zero temperature). Though the conventional natation for the instanton liquid
density at nonzero temperature is n = TN/V3, we use the label n which is more simple. At T 6= 0,
and µ = 0, our results coincide with the results obtained in [11] and [9]. It sould be noted that our
results are consistent with recent calculations on a lattice at finite temperatures [20], [21], where a
rapid decrease of the chromoelectric components in the respective correlation functions was found.
In our model, such suppression is due to the term 12m
4 in the effective action; with neglect of
this factor, the chromoelectric and chromomagnetic correlators coincide. From this point of view, our
calculations may seem inconsistent. We use the caloron solution (13), which is symmetric under an
interchange of chromoelectric and chromomagnetic fields. However, the caloron components manifest
themthelves in the observables differently because of the anisotropy of the weight function. In fact,
our method of taking the gluon mass term into account is consistent only in perturbation theory. In
a complete study, one must find an analogue of the solution (13) for the effective Lagrangian with the
Figure 1: Lines of equal density of the instanton liquid in the temperature-chemical potential plane.
Curve 1 corresponds to the density n = 0.75 n0, where n0 is the density at zero temperature and
chemical potential. Also shown are the densities from (curve 2) n = 0.5 n0 to (curve 6) n = 0.1 n0.
Curves 3–5 correspond to intermediate densities at intervals of 0.1.
Figure 2: Instanton liquid density versus (curve 1) temperature and (curve 2) chemical potential.
gluon mass generated for the chromoelectric field and gain a self-consistent description of ensemble
of pseudoparticles in the long-wave approximation [3].
It is of interest that the data on correlation functions for cooled configurations [20] are fitted
well by the instanton ensemble [22]. In so doing, the contribution of the terms of the second order
in the instanton liquid density (∼ n2) is in excellent agreement with the effect of the standard
instanton ensemble with the respective admixture of the perturbative component everywhere over the
distance range chosen for a fit [23]. This agreement indicates that the confining component is absent
from the lattice configurations isolated by cooling. It is surprising because lattice simulations with
cooling were aimed at the searches for a long-wave confining component. However, an interpretation
of lattice simulations at finite temperature presents difficulties because it is not clear what scale
corresponds to the configarations used for the measurements. The magnitude of deformation of the
chromoelectric component of the solution for the effective Lagrangian with the mass term is also
poorly known. The scale of lattice configurations can, in principle, be estimated using the scale at
which the chromoelectric field decrease since only this scale has emerged in our calculations.
In conclusion we note that, though we used only a rough approximation, the most important
features of the behavior of the instanton liquid density (gluon condensate) in the medium have been
revealed. The lines of equal density are markedly extended along the µ axis because, according to
the formula (20), the most substantial gluon component of screening vanishes at small temperatures.
Typical values of T and µ at which the effects of the medium become significant are related to each
other by the formula
(Nc +Nf/2)
(T/Λ)2 ∼ Nf
(µ/Λ)2
∼ 1, which leads to a plausible coefficient
of oblongness along the µ axis
2π Tc ,
(at Nc = 3 and Nf = 2). A fall in density evaluated with allowance for the dynamically generated
quark mass should begin at a greater value and be more steep. The reason is that, at chemical
potentials less than the quark mass, the quark contribution to screening is reduced. This gives rise
to formation of a plateau and concentration of the lines of equal density. The dependence of the
dynamical quark mass on the momentum ω is significant at small temperatures leading to a decrease
of screening approximately by a factor of two [14].
We are grateful to A.E. Dorokhov for helful discussions.
This work was supported in part by grants STCU #P015c, CERN-INTAS 2000-349, NATO
2000-PST.CLG 977482.
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