Two-loop renormalisation of gauge theories in 4D implicit regularisation and connections to dimensional methods

Abstract

We compute the two-loop $$\beta $$ β -function of scalar and spinorial quantum electrodynamics as well as pure Yang–Mills and quantum chromodynamics using the background field method in a fully quadridimensional setup using implicit regularization (IREG). Moreover, a thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Lorentz algebra contractions/symmetric integrations inside divergent integrals as well as renormalisation schemes are carefully discussed within IREG where the renormalisation constants are fully defined as basic divergent integrals to arbitrary loop order. Moreover, we confirm the hypothesis that momentum routing invariance in the loops of Feynman diagrams implemented via setting well-defined surface terms to zero deliver non-abelian gauge invariant amplitudes within IREG just as it has been proven for abelian theories.

Two-loop gauge coupling $β$-function in a four-dimensional framework: the Standard Model case

“…As a non-trivial example of the application of IREG we will pursue the computation of the twoloop coefficients for the gauge couplings β-function in the Standard Model. This improves the analysis performed in [13] where the β-function for an abelian (QED) and non-abelian (QCD) theory was studied.…”

Two-loop gauge coupling $β$-function in a four-dimensional framework: the Standard Model case

“…As a non-trivial example of the application of IREG we will pursue the computation of the two-loop coefficients for the gauge couplings β-function in the Standard Model. This improves the analysis performed in [13] where the β-function for an abelian (QED) and non-abelian (QCD) theory was studied.…”

May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations

“…ST's vanish if and only if momentum routing invariance (MRI) holds in the loops of Feynman diagrams. Moreover these requirements automatically deliver gauge invariant amplitudes [76][77][78][79][80] which has been demonstrated for abelian gauge theories to arbitrary loop order [68,70] and verified for non-abelian gauge models [81][82][83]. Rephrasing it, unless MRI is verified, a symmetric integration leads to a finite definite value for the arbitrary surface term which potentially breaks (gauge) symmetry.…”

“…Such program is compatible with the BPHZ theorem, assuring locality, causality and Lorentz invariance. Further understanding of the relation between ireg and dimensional schemes was achieved recently [83], with prospects to map BDI's into poles in −n for a UV n-loop calculation in general.…”