High-energy electrons from bound-muon decay

Abstract

Expressions for the electron spectrum from bound-muon decay which exist in the literature are very complex. The aim of this paper is to gain some insight into these formulas. This becomes possible when the Dirac equation is used to derive Wronskian-type relations between electron and muon wave functions. We also show how a trivial modification of the formulas accounts for the nuclear-recoil effect to a very good approximation. We present numerical results to aid order-of-magnitude estimates of the bound-muon-decay contribution to the background in $\ensuremath{\mu}e$ conversion experiments. We find that the electron spectrum seems insensitive to deviations of the weak-interaction parameters from standard values.