OPE of the energy-momentum tensor correlator and the gluon condensate operator in massless QCD to three-loop order

Abstract

The correlator of two gluonic operators plays an important role for example in transport properties of a Quark Gluon Plasma (QGP) or in sum rules for glueballs.In [1] an operator product expansion (OPE) at zero temperature was performed for the correlators of two scalar operators O 1 = − 1 4 G µν G µν and two QCD energy-momentum tensors T µν . There we presented analytical two-loop results for the Wilson coefficient C 1 in front of the gluon condensate operator O 1 . In this paper these results are extended to three-loop order.The three-loop Wilson coefficient C 0 in front of the unity operator O 0 = 1 was already presented in [1] for the T µν -correlator. For the O 1 -correlator the coefficient C 0 is known to four loop order from [2]. For the correlator of two pseudoscalar operators O 1 = ε µνρσ G µν G ρσ both coefficients C 0 and C 1 were computed in [3] to three-loop order. At zero temperature C 0 and C 1 are the leading Wilson coefficients in massless QCD.

OPE and a low-energy theorem in QCD-like theories

“…We have also observed that the O(g 4 ) log-divergent contact term in C (S) 1 [11][12][13] mixes in Eq. (3.10) with the scheme-dependent divergences to the order of g 4 due to the second coefficient of the anomalous dimension of F 2 (Sec.…”

“…A previous perturbative lower-order computation appeared in [14]. We pass from the perturbative OPE in the momentum representation [11][12][13] in Apps. B.1 and B.4 to the OPE in the coordinate representation by the Fourier transform in Apps.…”

“…are the first-three coefficients of the QCD beta function, ∂g ∂ log Λ = β(g) = −β 0 g 3 − β 1 g 5 − β 2 g 7 + · · · , and A (S) 0,j , A (S) 1,j are finite coefficients computed in [11,12]. δ (4) (x) and ∆ 2 δ (4) (x) are contact terms in the coordinate representation, i.e., distributions supported at coinciding points.…”

“…(1.3), for n = 2 and O k = F 2 (Sec. 4), by means of the OPE of the operator F 2 with itself worked out in [11][12][13]. The contact terms in C (S) 0 have been, in fact, inessential to verify the low-energy theorem, since they can be skipped by choosing z = 0 in Eq.…”

On the infrared behavior of the shear spectral function in hot Yang-Mills theory

“…This challenge has indeed been addressed first on the level of the Operator Product Expansion (OPE) in [23], and later by considering the full NLO spectral function in [24] (see also the related T = 0 work of [25,26]). While consistent with known sum rules [27,28] as well as the arguments of [29] concerning the UV behavior of various Green's functions, the latter of these calculations delivered somewhat surprising results.…”