Gauge-independent Thermal β Function In Yang-Mills Theory

It is proposed to use the pinch technique (PT) to obtain the gaugeindependent thermal β function in a hot Yang-Mills gas. Calculations of the thermal β function are performed at one-loop level in four different gauges, (i) the background field method with an arbitrary gauge, (ii) the Feynman gauge, (iii) the Coulomb gauge, and (iv) the temporal axial gauge, and they yield the same result in all four cases. ∗e-mail address: [email protected] or [email protected] It is important for the study of the quark-gluon plasma and/or the evolution of the early Universe to fully understand the behaviour of the effective coupling constant αs(= g /4π) in QCD at high temperature. The running of αs with the temperature T and the external momentum κ(= |~k|) is governed by the thermal β function βT [1]. However, the previous calculations of βT have exposed various problems [2], a serious one of which is that the results are gauge-fixing dependent [3]. The background field method (BFM) has been applied to the calculation of βT at one-loop [2][4][5]. First introduced by DeWitt [6], BFM is a technique for quantizing gauge field theories while retaining explicit gauge invariance for the background fields. Since the Green’s functions constructed by BFM manifestly maintain gauge invariance, they obey the naive QED-like Ward identities. As a result, the spatial part of the three-gluon vertex, for static and symmetric external momenta, is related to the transverse function ΠT (T, k0 = 0, κ = |~k|) of the polarization tensor Πμν , and thus βT is obtained in BFM from [2] βT ≡ T dg(T, κ) dT = g 2κ2 T dΠT (T, κ) dT . (1) Due to the O(3) invariance, the spatial part of the gluon plarization tensor Πij is expressed as follows: Πij(k) = ΠT (δij − kikj ~k2 ) + ΠL kikj ~k2 (2) and ΠT can be extracted by applying the projection operator tij = 1 2 (δij − kikj ~k2 ) (3) to Πij. The thermal β function has been calculated in BFM at one-loop level for the cases of the gauge parameter ξQ = 0 [4], ξQ = 1 [5] and ξQ =an arbitrary number [2][7]. The results are expressed in a form, β T = gN 128 { 7 16 − 1 8 (1− ξQ) + 1 64 (1− ξQ) } T κ , (4) where N is the number of colors. Contrary to the case of the QCD β function at zero temperature, β T is dependent on the gauge-parameter ξQ. The reason