Radiative corrections to the muon decay and the analytic regularization method

Abstract

In the present paper we analyze the analytic behavior of the amplitude corresponding to the reaction μ → eν ν with second-order electromagnetic radiative corrections as a function of the complex parameters characteristic of the analytic regularization methods [1, 2] ∗ ∗ Ref. [1] will be referred to as I. The method of analytic regularization indicated in this paper will be referred to as method I. . We regularized the amplitude first, as it is indicated by Bollini, Giambiagi and Gonzalez Dominguez [1] and second following the more elaborated method given by Speer [2] ∗∗ ∗∗ This paper will be referred to as II. . The finite part of the amplitude obtained with each of the above mentioned methods is the same and agrees with the result given for it by Behrends, Finkelstein and Sirlin [3] ∗∗∗ ∗∗∗ This paper will be referred to as III. using the cut off method. However, the method I, when applied to the complete amplitude and not to each particular Feynman diagram, gives an analytic function of the complex parameter which is regular at the physical point, whereas with the Speer's method [2] an undetermined constant appears in the final result which has to be absorbed by weak coupling constant renormalization. This last result contradicts the one given in III. Now, the method I has not been extended for higher orders of perturbation theory whereas the Speer's method [2, 4] ‡ ‡ Ref. [4] will be referred to as IV. is quite general and gives a well defined prescription to regularize any Feynman diagram. Then it seems to us that the Speer's method of analytic regularization [2] should be complemented in some way in order to take into account those cases in which the singularities of all Feynman diagrams, corresponding to a given order perturbation theory, cancel out exactly among themselves giving, in that order of perturbation, a regular amplitude.