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

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