Convergence of Perturbative Expansion Series in QED and the Muon g-2: One of the Oldest Problems in Quantum Field Theory and of the Latest Problems in the Standard

Abstract

Whether perturbative expansion series in quantum electrodynamics(QED) are convergent or not is discussed in detail, by taking the radiative corrections to the magnetic moment of a charged Dirac fermion such as the muon as an example. It is shown that they are asymptotic expansion, convergent, or divergent series, depending on the fine-structure constant and the total number and masses of the charged particles. It is remarkable that their convergence not only constraints the fine-structure constant and the total number and masses of the fermions but also forbids the existence of the magnetic monopole. It is also pointed out that the disagreement between the experimental value of the muon g-2 and the theoretical one predicted in the Standard Model may be due to an unexpectedly large sum of the perturbative expansion series.