Dynamical gluon mass at non-zero temperature in instanton vacuum model

In the framework of the instanton liquid model (ILM), we consider thermal modifications of the gluon properties in different scenarios of temperature $T$ dependence of the average instanton size $\bar{\rho}(T)$ and the instanton density $n(T)$ known from the literature. Due to interactions with instantons, the gluons acquire the dynamical temperature dependent"electric"gluon mass $M_{el}(q,T).$ We found that at small momenta and zero temperature $M_{el}(0,0)\approx362\,{\rm MeV}$ at the phenomenological values of $\bar{\rho}(0)=1/3\,{\rm fm}$ and $n(0)=1\,{\rm fm}^{-4}$, however the $T$-dependence of the mass is very sensitive to the temperature dependence of the instanton vacuum parameters $\bar{\rho}(T),\,n(T)$: it is very mild in case of the lattice-motivated dependence and decreases steeply in the whole range with theoretical parametrization. We see that in region $0

In fact, the accuracy of the approximation (4) is about one per cent up to r, t ∼ β. The extension of the instanton vacuum liquid model (ILM) [5,6] to non-zero temperature in this regime is straightforward and might be encoded in the temperature dependencies of main parameters of the model, the average instanton sizeρ(T ) and average instanton density n(T ) = N T /V 3 = 1/R 4 (T ), where N is the total number of instantons [7]. Bothρ(T ) and n(T ) in ILM are homogeneously decreasing functions of T .
But the simplified approximation (4) does not describe nontrivial phase transition near the critical temperature T c ∼ Λ QCD . Indeed, for T < T c all color objects are bound into colorless hadrons. The heat bath predominantly consists of weakly interacting pions, so the T -dependence of instanton density n(T ) should be rather mild, which agrees with expec-tation of almost constant T -dependence n(T ) = n 0 (1 + O(T 2 /(6f 2 π ))) [6]. However, this behavior changes during phase transition, and the expected instanton density n(T ) should be exponentially suppressed at large temperature T T c [6].
The extension of ILM which is able to describe the phase transition from confined to deconfined phase near the critical temperature T c is the so-called dyon-instanton liquid model (DLM) [10]. The authors of [10] concluded later in [11] that at very low temperature, the semi-classical description of the Yang-Mills state reconciles the instanton liquid model without confinement, with the t'Hooft-Mandelstam proposal of confinement. In the former, the low temperature thermal state is composed of a liquid of instantons and anti-instantons, while in the latter it is a superfluid of monopoles and anti-monopoles.
The temperature dependence of the QCD vacuum model might be tested by comparison with results of lattice simulations. For example, for the "electric" gluon mass M el (T ) in the framework of lattice QCD [12,13], it was found that for T ≥ T c the linear dependence M el (T ) ∼ T is consistent with Debye screening and has a minimum at T ∼ T c , whereas for T ≤ T c the electric mass M el (T ) is a rather slowly decreasing function of temperature T . This behavior might be naturally explained in the framework of the ILM, which predicts the temperature dependence like M el ∼ (packing parameter(T )) 1/2ρ−1 (T ) =ρ(T )n 1/2 (T ), a decreasing function of temperature at T ≤ T c . Combined with perturbative one-loop thermal gluon contribution to the gluon propagator, which is rising with temperature as M pert,el (T ) ∼ T , this model is able to reproduce lattice results for the dynamical gluon mass [12,13].
There are two major technical challenges in calculation of the gluon propagator in ILM framework: the account of zero-modes (fluctuations along of instanton collective coordinates), and the averaging over the collective coordinates of all instantons. We address the former using the approach of [18], while for the latter we extend Pobylitsa's approach [15], applied earlier by us for the gluons at T = 0 [16], and consider in this paper its further extension for the ILM averaged gluon propagator at T = 0.
The paper is structured as follows. In the Section II we review briefly the formulation of ILM at nonzero temperature T = 0 and discuss the temperature dependence of main instanton vacuum parameters. In Section III we consider a simplified case and evaluate the propagator of scalar color octet particle (which we call "scalar gluon") in the instanton background at nonzero temperature. It allows us to get several important results which will be used later. In the Section IV we consider the case of real gluon and evaluate the propagator at nonzero temperature. We extract the electric mass M el and compare it with lattice results. Finally in Section V we draw conclusions.

II. VARIATIONAL ESTIMATES IN ILM AT T = 0
The application of the Feynman variational principle to the QCD vacuum filled with instanton gas leads to the ILM [4], which was generalized to non-zero temperatures in [7].
Main variational ingredients of this approach are the instanton size distribution function µ(ρ, T, n) and the density of instantons n(T ) [6,7,9]. The ILM instanton size distribution function is closely related to the thermal single instanton one-loop distribution function [3] , The result (8) might be obtained maximizing the variational ILM partition function [7] Z with respect to parameter µ.The minimization of the free energy F = −T /V 3 log Z by variation over n leads to the equation for the density The variational estimates demonstrated thatρ(T ) and n(T ) are decreasing functions of temperature T (see fig. 1) due to the exponential factor in d(ρ, T ) ∼ A Nc . On the other hand, lattice data show that the instanton density n is not modified by temperature up to critical temperature T c [8]. In numerical simulations of ILM [6] it was suggested to interpolate between no suppression (A Nc = 0 in Eq. (9)) below T c and full suppression (A Nc = 0 in Eq. (9)) above T c ∼ 150 MeV , with a width ∆T = 0.3T c to be in the correspondence with lattice result [8]. We are following this suggestion and are repeating the calculations with the modification in the Eq.
3T c , and we introduced a smooth interpolation with a step-like function of width ∆y, The temperature dependence ofρ 2 (y)/ρ 2 (0) and n(y)/n(0) is shown in the Fig The (normalized to unity) temperature dependence of the instanton sizeρ 2 (y)/ρ 2 (0) (left plot ) and (normalized to unity) temperature dependence of instanton density n(y)/n(0) (right plot, obtained with variational estimates from Refs. [6,7,9]). In both plots we use y =ρ 0 T notations and phenomenological valuesρ(0) = 1/3 fm and n(0) = 1 fm −4 for estimates. The solid line corresponds to evaluation with the modification A Nc → A Nc Θ ∆y (y − y c ) in (9), where Θ ∆y (y − y c ) is a smooth interpolating step-like function Eq.(14) with a width ∆T = 0.3T c [6]. Dashed lines correspond to the evaluation with A Nc = const in (9) (full suppression) for all T .

III. COLOR OCTET SCALAR PROPAGATOR AT NON-ZERO TEMPERATURE
We start from the scalar massless field φ belonging to the adjoint representation, the same as a physical gluon. We have to find its propagator in the external classical gluon field of instanton gas A µ = I A I µ (γ I ), where A I µ (γ I ) is a generic notation for the QCD (anti-) instanton, and γ I stands for all the relevant collective coordinates: the position in Euclidean 4D space z I , the size ρ I and the SU (N c ) color orientation U I (4N c collective coordinates in total). The averaging over the instanton collective coordinates includes the integral over the instanton position 4 . In view of periodicity of the fields φ( x, t + β) = φ( x, t) at nonzero temperature, we may restrict the integration over t to the period β, so the effective action takes a form where P µ = p µ + A µ (in the coordinate representation p µ = i∂ µ ). The color octet scalar propagator in the field of instanton gas is given by It is convenient to introduce also the propagators "gluon" in the field of individual instantons, and in the instanton gas background when the overlaps There are no zero modes in ∆ −1 i = P 2 i nor in ∆ −1 = P 2 , which means the existence of the inverse operators ∆ i and ∆. Our aim is to find the propagator averaged over instanton collective coordinates∆ ≡ ∆ = Dγ ∆. In coordinate space the propagator ∆ and free propagator ∆ 0 must be periodical functions of time with period β. In what follows we will use notation ∆(x, x ) ≡ x|∆|x for the matrix element of the operator ∆ between Fock states labeled by space-time coordinates |x ≡ | x, t = | x |t (the same for |x ) and [x µ , p ν ] = iδ µν [14]. These states form a complete orthonormalized set, t |t t| = 1, t |t = δ(t − t).
Also, we define the step-operator, t |Θ|t = Θ(t − t). In view of d dt Θ(t − t) = δ(t − t), we may conclude that Θ −1 ≡ d dt . The time-periodic state with period β may be represented in terms of states |t as Now the evaluation of the propagator ∆ = ( , we have the equation in the form where ∆( x , t | x, t − nβ) is a usual zero-temperature (T = 0) aperiodical propagator. For physical applications we need to average the propagator ∆ over collective coordinates of all instantons∆ = ∆ = Dγ ∆. We are following the procedure developed in our previous paper [16], where the approach [15] derived for the quark correlators, was extended to the gluon case. We start first from averaging over collective coordinates of the operator∆ = ∆ (see Eq. (16)). Since Pobylitsa Eqs. [15,16] are written in operator form, they can be easily extended to T = 0 case just by calculating of matrix elements of propagator∆ with periodical states |t β on the right side.
Since the instanton gas is dilute, namely the packing parameter ρ 4 n ∼ (1/3) 4 = 1.2 × 10 −1 1, we may develop a systematic expansion over the parameter n. The expansion of the inverse propagator up the first-order O(n) terms has a form where∆ I = dγ I ∆ I is the propagator in the field of individual instanton averaged over its collective degrees of freedom. In the same order of expansion we may approximate the inverse propagator as∆ −1 =∆ −1 = p 2 + M 2 s , where we introduced squared dynamical color octet scalar mass operator M 2 s whose matrix elements are given by According to [3], the periodic color octet scalar propagator in instanton field (44) is given by ∆ ab At short distances r ∼ t ≤ β the caloron field becomes instanton-like (5) with modified instanton radius ρ 2 = ρ 2 /(1 + 1/3 λ 2 ), and λ = πρ/β. In this region we may simplify the first term in (23) as F 0 (x, y) = 1 + ρ 2 (τ x)(τ + y) x 2 y 2 = 1 + ρ 2 (xy) whereη aµν = −η aνµ is the 'tHooft symbol. As will be shown below, the contribution of the terms ∆ ab 1 (x, y), ∆ ab 2 (x, y) in (23) is small and might be neglected. The collective degrees of freedom (center of instanton position z and orientation U ) in (31) might be introduced shifting the arguments x → x − z, y → y − z and color rotation factors ∆ ab where O ab are the matrices of color rotation in adjoint representation ( O ab are related to color rotation matrices in fundamental repre- where t a are SU (N c )-matrices.). The averaging over the collective coordinates reduces to integration over the instanton center, β 0 dz 4 V 3 d 3 z, and color orientation, dO. The latter integral might be evaluated analytically using the well-known identities [6]: The contribution of the ∆ ab I,0 to the color octet scalar dynamical mass operator (22) is given by so for the expression in parenthesis in (34) we may obtain after collective coordinate averaging in the coordinate representation where we introduced notations In what follows we will use notation M 2 s ( q, m) for the dynamical color octet scalar mass corresponding to the Matsubara mode m with frequency ω m = 2πmT in the 3momentum q representation. We are especially interesting with the m = 0 Matsubara mode, M 2 s ( q, m = 0) ≡ M 2 s ( q, T ).
If we define the Fourier transformation to coordinate space as then, the contribution of the first term in Eq. (35) might be rewritten as where and K 1 (z) is a modified Bessel function of the second kind, with lim z→0 z K 1 (z) = 1. Since temperature mildly affects dynamical mass form-factor, we may neglect this modification at small temperatures T ≤ T c . Careful analysis shows that second term in Eq. (35) and all of other terms including ∆ ab 1 and ∆ ab 2 give zero or negligible contribution, so we finally obtain where F (0, 0) = 1, F (q, T ) ≤ F (q, 0) = qρK 1 (qρ).

IV. GLUON PROPAGATOR AT NON-ZERO TEMPERATURE
In this section we will extend the calculations of averaged full gluon propagatorS µν , considered in [16], to non-zero temperature case. It is a rather straightforward task, since there all principal equations and their solutions were found in operator form. First, we have the solution of Pobylitsa equation in operator form as where free and single instanton gluon propagators [17] are given by and we introduced notation q µνρσ = δ µν δ ρσ + δ µρ δ νσ − δ µσ δ νρ + µνρσ (for the antiinstanton case + µνρσ ⇒ − µνρσ ). The Eq. (40) can be rewritten (compare with the Eq. (21)) as At non-zero temperature, the most essential point is the lack of the relativistic covariance, k µ ≡ k, k 4 , k 2 = k 2 + k 2 4 ). We expect that the the dominant contribution to Π νµ will come from the large-distance asymptotics of the matrix elements of S I νµ − S 0 νµ . In coordinate space, comparing the effects from i∂ µ and from multiplication by A I µ in (42), we conclude that the dominant asymptotic contribution to S I νµ − S 0 νµ in Eq. (43) comes from the term p ρ (the most slowly decreasing part of ( and only this term will contribute to Π νµ . So, the Eq. (43) reduces to By definition the square of "electric" gluon mass M 2 el (| k|, T ) is related to Π µν as M 2 el (| k|, T ) = Π 44 ( k, k 4 = 0). Comparing it with Eq. (34), we conclude that M 2 el (| k|, T ) = 2M 2 s (| k|, T ), is gauge invariant (ξ independent) and its T and q dependencies are represented by Fig.2. It is obvious that M 2 el (| k|, T = 0) = M g (| k|), where the gauge invariant dynamical gluon mass M g was obtained before [16]. Using the phenomenological values ofρ and n at T = 0, we obtain M el (0, 0) = 362 MeV.   Fig. 2). Right: Lattice measurements results for the same quantity taken at the scale 2 GeV [12,13] (Right plot is a part of Fig.12 from [12]. See the caption of this Fig.).
From Fig. 3 we see that the most natural explanation of non-zero "electric" gluon dynamical mass at T < T c region seen in lattice measurements [12,13] is given by ILM, since ILM is able at least qualitatively reproduce its value at T = 0 and its T dependencies.

V. SUMMARY AND DISCUSSION
In this paper we extended the calculations of the dynamical gluon mass in ILM [16] to non-zero temperature and studied the so-called "electric" gluon mass M el (q, T ), which corresponds to Π 44 -component of polarization operator. We also analyzed the temperature T dependence of the main parameters of the ILM, the average instanton sizeρ(T ) and instanton density n(T ). We found that they are homogeneously decreasing functions of temperature due to influence of thermal gluon fluctuations [7]. Our findings agree with lattice investigations [8], which demonstrated thatρ(T ), n(T ) are decreasing rapidly for T ≥ T c , where T c is the critical temperature. For temperatures below the critical temperature T c , these functions are almost constant, and we took into account this scenario by neglecting the contributions of thermal gluon fluctuations at low temperature T ≤ T c [6]. The comparison of both of these scenarios is presented at the Fig. 1.
In order to find gluon propagator in the ILM background field at nonzero temperature T = 0, we have solved the gluon zero-modes problem and averaged full gluon propagator over collective coordinates of all instantons. This was done in the framework developed in our previous paper [16] and extended to non-zero temperature case. First, we evaluated the "electric" color octet scalar dynamical mass M s (q, T ) as a function of the three-momentum q and temperature T . The solution of zero-modes problem yields M 2 el (q, T ) = 2M 2 s (q, T ), which allowed us to relate. The final results for the "electric" gluon dynamical mass M el (q, T ) are presented in the Fig.2.
It is interesting to compare our result for the dynamical "electric" gluon mass M el with the result of lattice calculations (see Fig. 3), which observed that M el (0, T ) is a decreasing function of T for T ≤ T c in the correspondence with ILM, and is an increasing function of T above the confinement-deconfinement phase transition [12,13]. The grows of M el (0, T ) for T ≥ T c may be explained by perturbative thermal gluon correction and is expected to have an almost linear functional dependence M pert,el (0, T ) ∼ T . Since thermal gluons are incorporated in our framework, probably it is easy to reproduce within ILM model the lattice measurements of the dynamical "electric" gluon mass in whole region of temperature.
We assume to apply our result to the calculations of temperature dependence of the heavy quarkonium properties.