The electron energy spectrum in muon decay through 𝒪(α²)

Abstract

We compute the complete O(α 2 ) QED corrections to the electron energy spectrum in unpolarized muon decay, including the full dependence on the electron mass. Our calculation reduces the theoretical uncertainty on the electron energy spectrum well below 10 −4 , the precision anticipated by the TWIST experiment at TRIUMF, which is currently performing this measurement. For this calculation, we extend techniques we have recently developed for performing next-to-next-toleading order computations to handle the decay spectra of massive particles. Such an extension enables further applications to precision predictions for b, t, and Higgs differential decay rates.The decay of a muon into an electron and a pair of neutrinos, µ → eν µνe , occupies an important role in particle physics. The measurement of the muon lifetime [1] leads to the most accurate determination of the Fermi coupling constant, G F . The muon anomalous magnetic moment is one of the most precisely measured quantities in nature [2,3], and provides important constraints on physics beyond the Standard Model (SM) [4]. Searches for lepton flavor-violating decays of the muon, such as µ → eγ and µ → eee, constrain the flavor sector of many SM extensions [5].The calculations of radiative corrections to muon decay have a long and storied history [6]. The one-loop QED corrections were first performed within the Fermi theory of weak interactions in the 1950s [7]. The cancellation of mass singular terms such as ln(m µ /m e ) in the total rate, but not in distributions such as the electron energy spectrum, led to the development of the Kinoshita-Lee-Nauenberg theorem, which explains how to construct "infrared-safe" observables in quantum field theory where such effects cancel [8]. The calculation of the full one-loop corrections in the SU(2)×U(1) theory of the electroweak interactions was one of the first such computations performed [9]. The full two-loop corrections to the muon lifetime in the Fermi model, needed for a precision determination of G F , were completed several years ago [10]; recently, the two-loop results in the full electroweak theory were obtained [11].Muon decay continues to be of interest in particle physics. The TWIST experiment at TRIUMF [12] measures the electron energy and angular distributions in polarized muon decay; the first results were recently reported in [13]. It is anticipated that TWIST will eventually measure the Michel parameters [14], which describe muon decay for the most general form of the four-fermion interaction, to a precision of ≈ 10 −4 . This significantly increases the sensitivity of muon decay to deviations arising from new physics. For example, the lower bound on the mass of the W R in the manifest left-right symmetric model is improved from M W R > 400 GeV to M W R > 800 GeV, competitive with limits coming from the Tevatron, while the bounds on the left-right mixing parameter ζ are improved by nearly an order of magnitude [5]. Such precision requires a careful consideration of the higher order corrections. As noted ab...

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