Dispersive formalism for the nuclear structure correction $\delta_\mathrm{NS}$ to the $\beta$ decay rate

TL;DR

This paper develops a dispersive formalism to accurately calculate the nuclear structure correction $\,\delta_{NS}$ in beta decay, crucial for precise determination of the CKM matrix element $|V_{ud}|$ from nuclear experiments.

Contribution

It introduces a dispersion representation of $\,\delta_{NS}$ including energy dependence and summarizes isospin rotation formulas to aid future high-precision calculations.

Findings

Provides a rigorous theoretical framework for $\,\delta_{NS}$ calculation.

Summarizes isospin rotation formulas for nuclear theory.

Facilitates future precise determination of $|V_{ud}|$.

Abstract

We analyze the axial $\gamma W$-box diagram for $I(J^P)=1(0^+)$ nuclei and provide a dispersion representation of the nuclear-structure correction $\delta_\text{NS}$ including its energy-dependent part. We also summarize useful isospin rotation formula and representations in nuclear theory that could facilitate the calculation of the parity-odd nuclear structure function $F_3(\nu,Q^2)$. They provide a rigorous theory framework for the future, high-precision calculation of the nuclear structure correction $\delta_\text{NS}$ necessary for the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ud}|$ from superallowed nuclear $\beta$ decays.