Higher-order corrections to beta decays and nuclear structure

Precision measurements of nuclear $\beta$ decay offer a powerful route to tests of physics beyond the Standard Model, but interpreting measurements spectra and correlation observables requires control of subtle nuclear-structure effects. This chapter develops a framework for calculating higher-order corrections–small contributions that can become important at the precision sought in searches for new interactions. After motivating their role in $\beta$-decay observables (with radiative corrections treated in the accompanying chapter), we introduce the relevant kinematics and multipole expansion of the shape factor, identify the subleading contributions, and present the impulse-approximation multipole operators. We then describe the nuclear many-body methods used to obtain the required states and transition densities, and derive the corresponding operator matrix elements in the harmonic-oscillator basis. Taken together, these ingredients enable systematic calculations of nuclear shape factors beyond the long-wavelength approximation, including higher-order multipoles where needed. By helping distinguish Standard-Model recoil effects from possible signatures of new physics, this framework supports the interpretation of increasingly precise $\beta$-decay experiments.