Two-loop renormalisation of gauge theories in $4D$ Implicit Regularisation and connections to dimensional methods

Abstract

We compute the two-loop 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. Particularly, for our calculations we show that the inclusion of evanescent -scalar particle contributions needed in quasi-dimensional methods such as DRED and Four Dimensional Helicity (FDH) cancel out in the determination of the ultraviolet (UV) structure of the models we study. 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 diagramas 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.

A step towards a consistent treatment of chiral theories at higher loop order: the abelian case

“…As a non-trivial example of the application of IREG we will pursue the computation of the two-loop coefficients for the β function gauge couplings in the standard model which, using the same conventions of [25].We emphasize that all the standard model contributions were included, implying that contributions related to the Higgs sector (Yukawa couplings) were also considered. This improves the analysis performed in [26] where the β function for an abelian (QED) and non-abelian (QCD) theory was studied.…”

“…Some comments are in order: 1) when extracting the renormalization constants, we obtained that the end result is gauge-invariant (transverse) which illustrate the importance of applying the normal form as first proposed in [7,26]; 2) DREG and DRED differs in intermediate terms, but the final result is identical to each other as it should be (both subtraction schemes DR and MS are mass-independent [32]); 3) counterterms for quantum fields are not needed as first discussed in [27], however, since we adopted the Feynman gauge, the renormalization for the gauge fixing was required.…”

“…For comparison, we reproduce in tab 2 the results in standard QED [26] Diagram A B I By comparing tables 1 and 2 one notices that the total divergent part differs by a factor 1/2 which can be easily understood. In the Left-model we are considering, the lagrangian can be written as in eq.…”

Universal opening of four-loop scattering amplitudes to trees

“…This knowledge was crucial for developing the four dimensional unsubtraction (FDU) [26][27][28][29][30], which allows to combine real and virtual corrections into a single numerically-stable integral. As other methods proposed in the literature [31][32][33][34][35][36][37][38], FDU is aimed at performing most of the calculations directly in the four physical dimensions of the space-time. Besides these properties, the LTD formalism posses many others features that convert it into a promising technique for tackling higher-order computations.…”

“…Besides these properties, the LTD formalism posses many others features that convert it into a promising technique for tackling higher-order computations. For instance, the number of integration variables in numerical implementations is independent of the number of external legs [38][39][40][41][42][43], since it does not rely on Feynman parameters. On top of that, LTD efficiently provides asymptotic expansions [44], [45][46][47] and it also constitutes a promising strategy towards local renormalization approaches [48].…”