Contributions of the W-boson propagator to muon and tau leptonic decay rates

TL;DR

This paper derives precise formulas for how the W-boson propagator affects muon and tau decay rates, including corrections and discrepancies with recent literature, with implications for fundamental constants.

Contribution

It provides new, exact expressions and expansions for W-boson propagator contributions to leptonic decay rates, clarifying previous discrepancies and extending validity.

Findings

Leading corrections match the canonical (3/5) M^2/M_W^2 value.

Subleading correction coefficients differ from recent reports.

Numerical effects of corrections are briefly discussed.

Abstract

We derive closed expressions and useful expansions for the contributions of the tree-level W-boson propagator to the the muon and tau leptonic decay rates. Calling M and m the masses of the initial and final charged leptons, our results in the limit m=0 are valid to all orders in M^2/M_W^2. In the terms of O(m_j^2/M_W^2) (m_j=M,m), our leading corrections, of O(M^2/M_W^2), agree with the canonical value (3/5) M^2/M_W^2, while the coefficient of our subleading contributions, of O(m^2/M_W^2), differs from that reported in the recent literature. A possible explanation of the discrepancy is presented. The numerical effect of the O(m_j^2/M_W^2) corrections is briefly discussed. A general expression, valid for arbitrary values of M_W, M and m in the range M_W>M>m, is given in the Appendix. The paper also contains a review of the traditional definition and evaluation of the Fermi constant.