Three-loop matching coefficients for hot QCD: reduction and gauge independence

Abstract

Abstract:We perform an integral reduction for the 3-loop effective gauge coupling and screening mass of QCD at high temperatures, defined as matching coefficients appearing in the dimensionally reduced effective field theory (EQCD). Expressing both parameters in terms of a set master (sum-) integrals, we show explicit gauge parameter independence. The lack of suitable methods for solving the comparatively large number of master integrals forbids the complete evaluation at the moment. Taking one generic class of masters as an example, we highlight the calculational techniques involved. The full result would allow to improve on one of the classic probes for the convergence of the weak-coupling expansion at high temperatures, namely the comparison of full and effective theory determinations of the spatial string tension. Furthermore, the full result would also allow to determine one new contribution of order O g 7 to the pressure of hot QCD.

3-loop coupling for hot gauge theory

“…Secondly, symbolic manipulation in FORM [16] projects the calculation onto ∼ 10 7 vacuum sum-integrals. Third, systematic use of linear integration-by-parts (IBP) relations [17] applied to the 3d piece of the sum-integrals achieves a reduction to ∼ 10 2 so-called 'master' sum-integrals, of which ∼ 10 1 are bosonic [18]. Using the IBP tables, a basis transformation of the bosonic masters can be performed in order to render the actual polynomial pre-factors of non-trivial masters finite as d → 4, such that it suffices to evaluate them up to their constant parts.…”

Debye screening mass of hot Yang-Mills theory to three-loop order

“…The detailed framework of performing the matching computation has been presented in [15,22]. Here, we merely provide a concise version of it and generalize the matching condition in order to account for higher-order operators.…”

“…Full d-dimensional representations of the various coefficients Π En can be found in App. C of [15]. For the reader's convenience, we have collected the corresponding one-and two-loop self-energies in App.…”

“…2 we provide a brief overview of the theoretical setup and of the matching relations between full QCD and EQCD and sketch the status of the Debye mass computation reached in Ref. [15]. In Sec.…”

Automated computation meets hot QCD

“…[22], it has been shown that the computation of NNLO corrections to the spatial string tension of pure Yang-Mills theory (mapped to Taylor coefficients of background-gauge-field selfenergies [21]) can be reduced to 3-loop basketball-type master sum-integrals K i (and products of simpler 1-loop ones I i ), giving the gauge-invariant expression which, much in the spirit of the epsilon-finite basis advocated in Ref. [25], does not contain divergences in the coefficients r i (d) as d → 3 (note that V 6 = K 2 was already contained in the old basis listed in Eq.…”