Implicit Regularisation Technique: Calculation of the Two-loop $ϕ^4_4$-theory $β$-function

Abstract

We propose an implicit regularisation scheme. The main advantage is that since no explicit use of a regulator is made, one can in principle avoid undesirable symmetry violations related to its choice. The divergent amplitudes are split into basic divergent integrals which depend only on the loop momenta and finite integrals. The former can be absorbed by a renormalisation procedure whereas the latter can be evaluated without restrictions. We illustrate with the calculation of the $QED$ and $ϕ^4_4$-theory $β$-function to one and two-loop order, respectively.