Four-loop QCD propagators and vertices with one vanishing external momentum

TL;DR

This work computes massless QCD self-energies and a comprehensive set of four-loop three-point vertex functions with one vanishing external momentum in the $\nobreak{\overline{\text{MS}}}$ scheme, using FORCER to obtain analytic results for a generic gauge group and full gauge dependence. The renormalized $D=4$ results are provided and supported by extensive checks; the authors also derive a five-loop beta function in the MiniMOM scheme for the Landau gauge, illustrating the utility of their four-loop results for scheme conversions and nonperturbative studies. The methodology enables precise perturbative analyses across MS-like and MOM-like renormalization schemes and across dimensions, with ancillary data available online. Overall, the paper advances high-precision perturbative QCD and provides a direct, non-reliant pathway to higher-loop renormalization group functions in alternative schemes.

Abstract

We have computed the self-energies and a set of three-particle vertex functions for massless QCD at the four-loop level in the MSbar renormalization scheme. The vertex functions are evaluated at points where one of the momenta vanishes. Analytical results are obtained for a generic gauge group and with the full gauge dependence, which was made possible by extensive use of the Forcer program for massless four-loop propagator integrals. The bare results in dimensional regularization are provided in terms of master integrals and rational coefficients; the latter are exact in any space-time dimension. Our results can be used for further precision investigations of the perturbative behaviour of the theory in schemes other than MSbar. As an example, we derive the five-loop beta function in a relatively common alternative, the minimal momentum subtraction (MiniMOM) scheme.

Figures (4)

• Figure 1: The gluon, ghost and quark self-energies $\Pi^{ab}_{\mu\nu}(q)$ (a), $\tilde{\Pi}^{ab}(q)$ (b) and $\Sigma^{ij}(q)$ (c).
• Figure 2: The triple-gluon vertex with one vanishing momentum, $\Gamma^{abc}_{\mu\nu\rho}(q,-q,0)$.
• Figure 3: The ghost-gluon vertex: (a) $\tilde{\Gamma}^{abc}_\mu(-q,0;q)$ with the vanishing incoming ghost momentum and (b) $\tilde{\Gamma}^{abc}_\mu(-q,q;0)$ with the vanishing gluon momentum.
• Figure 4: The quark-gluon vertex: (a) $\Lambda^a_{\mu,ij}(-q,0;q)$ with the vanishing incoming quark momentum and (b) $\Lambda^a_{\mu,ij}(-q,q;0)$ with the vanishing gluon momentum.