The electron energy spectrum in muon decay through O(alpha^2)

TL;DR

This work delivers the complete $O(\alpha^2)$ QED corrections to the electron energy spectrum in muon decay while retaining full electron-mass dependence, achieving a theoretical uncertainty well below $10^{-4}$ to meet TWIST’s precision goals. The authors extend NNLO techniques to massive-particle spectra, combining real radiation, real-virtual, and two-loop virtual corrections within dimensional regularization and employing sector decomposition and master-integral methods for numerical evaluation. They find the constant term $f^{(2)}_0(x)$—the nonlogarithmic part of the NNLO correction—contributes at the level of roughly $0.5\times10^{-4}$, with the total NNLO effect spanning about $-5\times10^{-4}$ to $8\times10^{-4}$ and residual uncertainties around $5\times10^{-6}$. The results validate the approach and provide a robust, general framework for precision predictions of decay spectra for massive particles in other contexts, including heavy-quark decays and processes at the LHC or future colliders.

Abstract

We compute the complete O(alpha^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-to-leading 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.

Figures (7)

• Figure 1: A sample of LO and NLO diagrams which appear for the effective $\mu-e$ current after the neutrinos are integrated out. The factor to be associated with the effective $\mu-e$ vertex after squaring the matrix elements is given in Eq. \ref{['neutproj']}.
• Figure 2: Sample diagrams which contribute to $\mu \rightarrow e\nu\nu+\gamma\gamma$ (left) and $\mu \rightarrow e\nu\nu+e^+e^-$ (right)
• Figure 3: Sample diagrams which contribute to $\mu \rightarrow e\nu\nu+\gamma$
• Figure 4: A sample of two-loop diagrams which contribute to $\mu \rightarrow e\nu\nu$.
• Figure 5: A two-loop diagram with a self-energy insertion.