The Fermi function and the neutron's lifetime

TL;DR

This paper develops a quantum field theory-based approach to accurately compute neutron beta decay corrections, providing a two-loop analysis and resummation of large logarithms, which refines the neutron lifetime prediction and impacts Standard Model tests.

Contribution

It introduces a new factorization formula for neutron beta decay beyond the traditional Fermi function, including two-loop calculations and resummation of large logarithmic corrections.

Findings

First two-loop input to long-distance corrections for neutron decay.

Shift in long-distance radiative corrections compared to previous estimates.

Refined neutron lifetime calculation with implications for CKM matrix element |V_{ud}|.

Abstract

The traditional Fermi function ansatz for nuclear beta decay describes enhanced perturbative effects in the limit of large nuclear charge $Z$ and/or small electron velocity $\beta$. We define and compute the quantum field theory object that replaces this ansatz for neutron beta decay, where neither of these limits hold. We present a new factorization formula that applies in the limit of small electron mass, analyze the components of this formula through two loop order, and resum perturbative corrections that are enhanced by large logarithms. We apply our results to the neutron lifetime, supplying the first two-loop input to the long-distance corrections. Our result can be summarized as \begin{equation*} \tau_n \times |V_{ud}|^2\big[1+3\lambda^2\big]\big[1+\Delta_R\big] = \frac{5263.284(17)\,{\rm s}} {1 + 27.04(7)\times 10^{-3} }~, \end{equation*} with $|V_{ud}|$ the up-down quark mixing parameter, $\tau_n$ the neutron's lifetime, $\lambda$ the ratio of axial to vector charge, and $\Delta_R$ the short-distance matching correction. We find a shift in the long-distance radiative corrections compared to previous work, and discuss implications for extractions of $|V_{ud}|$ and tests of the Standard Model.