EFT approach to the endpoint of muon decay-in-orbit

TL;DR

This paper extends a previous EFT framework to the muon decay-in-orbit endpoint (eDIO), where a neutrino-antineutrino pair complicates the final state. By organizing a five-scale hierarchy and constructing a factorization theorem, the authors derive a systematic separation of hard, soft, and collinear dynamics, enabling RG-improved predictions and the resummation of large logs. They compute O($
d$) corrections to the matching coefficients and present the most precise eDIO endpoint spectrum to date, including a compact RG-improved expression and numerical K-factors for practical background estimates. The results substantially enhance the reliability of background predictions for future muon-conversion experiments and lay groundwork for further refinements like finite-size and recoil effects or extensions to related bound-state decays.

Abstract

As upcoming experiments aim to probe muon conversion with unprecedented precision, equally precise theoretical predictions are crucial to maximize discovery potential. This applies not only to the new physics signal, muon-electron conversion, but also to its only irreducible background, muon decay-in-orbit (DIO) near the endpoint. Accurate computation of higher-order corrections in bound states is a long-standing challenge due to the difficulty of systematically organizing contributions. In previous work, we developed an Effective Field Theory framework to address this issue and applied it to muon conversion. Here, we extend this approach to the DIO endpoint, a more complex problem due to the presence of a neutrino-antineutrino pair in the final state. We present the most precise prediction to date of the background spectrum relevant for future muon conversion searches, achieving next-to-leading logarithmic prime accuracy for QED corrections.

Figures (8)

• Figure 1: Bound muon decay. The left diagram shows the bound state $\mu_H$ as a single field, whereas the right diagram illustrates its composite structure. In both cases, the shaded region represents short-distance perturbative interactions, distinct from the long-distance Coulomb exchanges. The possible final state radiation $\mathcal{X}$ is omitted. See text for details.
• Figure 2: Differential eDIO rate in the SC approach. Left panel: LO and NLO+YFS rates, normalized to $\Gamma_0$, with inset for $\Delta E \simeq m_e$. Right: LO, NLO and NLO+YFS rates normalized to LO.
• Figure 3: Chart of the scales relevant for muon eDIO. Each scale is associated with an EFT. See text for details.
• Figure 4: LO scattering of a muon into a nucleus with the subsequent muon decay. The black circles represent the 4-Fermi interaction. The double line represents an HQET field.
• Figure 5: Differential rate for eDIO against $E_e$, normalized to the LO result, for different approaches. See text for details.