Next-to-leading order thermal spectral functions in the perturbative domain

Abstract

Motivated by applications in thermal QCD and cosmology, we elaborate on a general method for computing next-to-leading order spectral functions for composite operators at vanishing spatial momentum, accounting for real, virtual as well as thermal corrections. As an example, we compute these functions (together with the corresponding imaginary-time correlators which can be compared with lattice simulations) for scalar and pseudoscalar densities in pure Yang-Mills theory. Our results may turn out to be helpful in non-perturbative estimates of the corresponding transport coefficients, which are the bulk viscosity in the scalar channel and the rate of anomalous chirality violation in the pseudoscalar channel. We also mention links to cosmology, although the most useful results in that context may come from a future generalization of our methods to other correlators.

The shear channel spectral function in hot Yang-Mills theory

“…The setting of our calculation is very similar to that introduced in detail in [17,20], and for simplicity we adopt our notation from these references. We work within pure SU(N ) Yang-Mills theory at a nonzero temperature T , defined by the dimensionally regularized Euclidean action…”

“…The primary computational task of the present paper will be to derive explicit results for these quantities, evaluating the divergent terms analytically and the most complicated finite parts numerically. The general techniques required in the computation of the above masters were developed and rather thoroughly explained in [17]; in fact, many of the results needed in the present calculation can be directly read off from this reference. The main complication involved in the evaluation of the masters only appearing in the shear channel is related to how one should deal with the projection operators P T (Q) ≡ Q µ Q ν P T µν (P ), which appear frequently in the definitions of appendix A.1.…”

“…Next, we introduce the notation 12) in which the one-and two-loop functions f J 0 x and f I 0 x , corresponding to the integrals encountered in the bulk case, can be read off from [17] and are for completeness reproduced in appendix A.2. Using eq.…”

“…The latter quantities are most conveniently obtained from the behavior of the corresponding spectral functions at ω ∼ πT , computable using finite-temperature perturbation theory. For the bulk channel of SU(N ) Yang-Mills theory, including operators such as the trace of the energy momentum tensor, the task of perturbatively determining the spectral function was undertaken in [17]. There, the quantity was evaluated to next-to-leading order (NLO) in the limit of vanishing external three-momentum, building on the earlier work of [18,19].…”

“…There, the quantity was evaluated to next-to-leading order (NLO) in the limit of vanishing external three-momentum, building on the earlier work of [18,19]. In the paper at hand, our goal is to extend this result to the perhaps even more interesting but also technically more complicated shear channel, which necessitates the generalization of the methods of [17] to a number of new master integrals, involving components of the loop three-momenta in the numerator. As a preparation for this, the UV limit of the shear correlator was analyzed in terms of an Operator Product Expansion (OPE) in [20], leading to results compatible with known sum rules [21,22] as well as the arguments of [23].…”