# Dispersive Evaluation of the Inner Radiative Correction in Neutron and Nuclear $\beta$-decay

## TL;DR

This paper introduces a dispersive approach to evaluate the inner radiative correction in neutron and nuclear beta decay, leading to a more precise determination of the CKM matrix element V_{ud} and highlighting tensions with unitarity constraints.

## Contribution

It presents a novel data-driven dispersive method to evaluate the inner radiative correction, improving precision and linking hadronic and nuclear corrections within a unified framework.

## Key findings

- New radiative correction value: Δ_R^{V, new} = 0.02467(22)
- Shift in V_{ud} affects CKM unitarity tests
- Potential to unify hadronic and nuclear corrections using dispersion relations

## Abstract

We propose a novel dispersive treatment of the so-called inner radiative correction to the neutron and nuclear $\beta$-decay. We show that it requires knowledge of the parity-violating structure function $F_3^{(0)}$ that arises from the interference of the axial vector charged current and the isoscalar part of the electromagnetic current. By isospin symmetry, we relate this structure function to the charged current inelastic scattering of neutrinos and antineutrinos. Applying this new data-driven analysis we obtain a new, more precise evaluation for the universal radiative correction $\Delta_{R}^{V,\,new}=0.02467(22)$ that supersedes the previous estimate by Marciano and Sirlin, $\Delta_R^V=0.02361(38)$. The substantial shift in the central value of $\Delta_R^V$ reflects in a respective shift of $V_{ud}$ and a considerable tension in the unitarity constraint on the first row of the CKM matrix which is used as one of the most stringent constraints on New Physics contributions in the charged current sector. We also point out that dispersion relations offer a unifying tool for treating hadronic and nuclear corrections within the same framework. We explore the potential of the dispersion relations for addressing the nuclear structure corrections absorbed in the ${\cal F}t$ values, a crucial ingredient alongside $\Delta_R^V$ in extracting $V_{ud}$ from superallowed nuclear decays.