Three-loop vertex integrals at symmetric point

Andrey Pikelner

doi:10.1007/jhep06(2021)083

Abstract

This paper provides details of the massless three-loop three-point integrals calculation at the symmetric point. Our work aimed to extend known two-loop results for such integrals to the three-loop level. Obtained results can find their application in regularization-invariant symmetric point momentum-subtraction (RI/SMOM) scheme QCD calculations of renormalization group functions and various composite operator matrix elements. To calculate integrals, we solve differential equations for auxiliary integrals by transforming the system to the ε-form. Calculated integrals are expressed through the basis of functions with uniform transcendental weight. We provide expansion up to the transcendental weight six for the basis functions in terms of harmonic polylogarithms with six-root of unity argument.

Highlights

Q numerical results for renormalization of different operators matrix elements appeared in the series of papers [14, 15]
Obtained results can find their application in regularization-invariant symmetric point momentum-subtraction (RI/SMOM) scheme QCD calculations of renormalization group functions and various composite operator matrix elements
In subsequent two-loop calculations [7,8,9,10,11], authors adopted Integration-By-Parts(IBP) [17, 18] reduction to the minimal set of two-loop master integrals known for a long time, even in general kinematics not restricted to the symmetric point [19,20,21]

Summary

Notations for integrals topologies

To reduce many integrals in the course of calculations [12, 13], we apply IBP reduction to the small set of master integrals considered in the present paper. To uniquely identify an integral inside topology, we use a vector of integer numbers corresponding to propagators’ powers. For a vertex integral at the symmetric point, we assign external momenta according to the left part of the figure 1 and set p21 = p22 = q2 = −1. This definition makes all integrals real, and self-energy integrals are in one-to-one correspondence with integrals from the MINCER package [29, 30]. The number of unique master integrals after IBP reduction are given in table 3. We present all master integrals graphs in tables 4, 5 and 6

Differential equations for auxiliary integrals

Fixing boundary conditions

Basis of integrals with uniform transcendentality weight

Results and conclusion