Calculation of lepton magnetic moments in quantum electrodynamics: a justification of the flexible divergence elimination method

Abstract

The flexible method of reduction to finite integrals, briefly described in earlier publications of the author, is described in detail. The method is suitable for the calculation of all quantum electrodynamical contributions to the magnetic moments of leptons. It includes mass-dependent contributions. The method removes all divergences (UV, IR and mixed) point-by-point in Feynman parametric space without any usage of limit-like regularizations. It yields a finite integral for each individual Feynman graph. The subtraction procedure is based on the use of linear operators applied to the Feynman amplitudes of UV-divergent subgraphs; a placement of all terms in the same Feynman parametric space is implied. The final result is simply the sum of the individual graph contributions; no residual renormalization is required. The method also allows us to split the total contribution into the contributions of small gauge-invariant classes. The procedure offers a great freedom in the choice of the linear operators. This freedom can be used for improving the computation speed and for a reliability check. The mechanism of divergence elimination is explained, as well as the equivalence of the method and the on-shell renormalization. For illustrative purposes, all 4-loop contributions to the anomalous magnetic moments of the electron and muon are given for each small gauge-invariant class, as well as their comparison with previously known results. This also includes the contributions that depend on the ratios of the tau-lepton mass to the electron and muon mass.