Abstract
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 $0<T<T_{c}$ ILM is able to reproduce lattice results for the dynamical "electric" gluon mass.
Highlights
Gluodynamics at nonzero temperature Tð≡1=βÞ is described by the partition function Z Z1⁄4 DAμ exp − Z β dx4d3xtrFμνFμν ; ð1Þ where Fμν 1⁄4 ∂μAν − ∂νAμ − i1⁄2Aμ; Aν, and the gauge field Aμ satisfies the periodic condition Aμðx⃗ ;x4 þβÞ1⁄4Aμðx⃗ ;x4Þ
The extension of the instanton vacuum liquid model (ILM) [3,4,5] to nonzero temperature in this regime is straightforward and might be encoded in the temperature dependences of the main parameters of the model: the average instanton size ρðTÞ and average instanton density nðTÞ 1⁄4 NT=V3 1⁄4 1=R4ðTÞ, where N is the total number of instantons [6]
There are two major technical challenges in the calculation of the gluon propagator in the instanton liquid model (ILM) framework: the zero-mode problem, and averaging over the collective coordinates of all instantons. We address the former using the approach of Ref. [14], while for the latter we extend Pobylitsa’s approach [15], applied earlier by us for the gluons at T 1⁄4 0 [16], and consider in this paper its further extension for the ILM averaged gluon propagator at T ≠ 0
Summary
Gluodynamics at nonzero temperature Tð≡1=βÞ is described by the partition function. d3xtrFμνFμν ; ð1Þ where Fμν 1⁄4 ∂μAν − ∂νAμ − i1⁄2Aμ; Aν, and the gauge field Aμ satisfies the periodic condition Aμðx⃗ ;x4 þβÞ1⁄4Aμðx⃗ ;x4Þ. The extension of the instanton vacuum liquid model (ILM) [3,4,5] to nonzero temperature in this regime is straightforward and might be encoded in the temperature dependences of the main parameters of the model: the average instanton size ρðTÞ and average instanton density nðTÞ 1⁄4 NT=V3 1⁄4 1=R4ðTÞ, where N is the total number of instantons [6]. III we consider a simplified case and evaluate the propagator of the scalar color-octet particle (which we call the “scalar gluon”) in the instanton background at nonzero temperature This allows us to get several important results which will be used later.
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