Low-Energy Electron Spectrum from the Decay of Unpolarized Muons

An exact expression for the electron spectrum from the decay of unpolarized muons, including first-order radiative corrections, was previously calculated by Kinoshita and Sirlin assuming a $V-A$ theory of weak interactions. Most results in the literature, however, have been expressed in the limit of small electron mass, this approximation being valid if the electron's energy is greater than about $10m_e$. In this paper the exact spectrum is presented, and other approximations are used to derive simple formulas useful in the ranges $E_e>3\mathrm{}{m}_e$ and $E_e<3\mathrm{}{m}_e$. In the low-energy region, it is shown that the spectrum is dominated by the emission of noninfrared photons from the electron. Our results are compared with the Michel formula (no radiative corrections) for $E_e\le 0.1M_{\mu }$.