Practical approaches to gauge theories: renormalization group functions at two-loop within 4D Implicit Regularization
The data collected at the LHC do not indicate significant deviations from the predictions of the Standard Model (SM). Taking into account that precision observables can be predicted already at two and three loops, it is evident the necessity to develop stringent tests of self-consistency of the SM. We compute the 2-loop β-function of pure Yang-Mills (YM), Quantum Electrodynamics (QED), and Quantum Chromodynamics (QCD) using the background field method (BFM) in a full setup using Implicit Regularization (IREG). We also calculate the 2-loop γ-function of QED. A thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Dirac algebra inside divergent integrals in the present of gluonic or fermionic loops as well as renormalization schemes are carefully discussed within IREG. Finally, two algorithms for the automated calculation of the β-function and the γ-function were developed in Mathematica. Some of the computational strategies applied are presented.