High precision analytical description of the allowed $\beta$ spectrum shape

TL;DR

This paper provides a comprehensive analytical description of the allowed beta spectrum shape, incorporating various corrections, to enhance precision in searches for new physics beyond the standard model.

Contribution

It extends previous analytical models by nearly an order of magnitude in precision, including atomic, nuclear, and QCD effects, for low to medium Z nuclei.

Findings

Analytical formulas agree with numerical results within 10^{-4} precision.

Extended description includes atomic, nuclear, and QCD recoil effects.

Highlights importance of weak magnetism and induced pseudoscalar contributions.

Abstract

A fully analytical description of the allowed $\beta$ spectrum shape is given in view of ongoing and planned measurements. Its study forms an invaluable tool in the search for physics beyond the standard electroweak model and the weak magnetism recoil term. Contributions stemming from finite size corrections, mass effects, and radiative corrections are reviewed. A particular focus is placed on atomic and chemical effects, where the existing description is extended and analytically provided. The effects of QCD-induced recoil terms are discussed, and cross-checks were performed for different theoretical formalisms. Special attention was given to a comparison of the treatment of nuclear structure effects in different formalisms. Corrections were derived for both Fermi and Gamow-Teller transitions, and methods of analytical evaluation thoroughly discussed. In its integrated form, calculated $f$ values were in agreement with the most precise numerical results within the aimed for precision. We stress the need for an accurate evaluation of weak magnetism contributions, and note the possible significance of the oft-neglected induced pseudoscalar interaction. Together with improved atomic corrections, we then present an analytical description of the allowed $\beta$ spectrum shape accurate to a few parts in $10^{-4}$ down to 1\,keV for low to medium $Z$ nuclei, thereby extending the work by previous authors by nearly an order of magnitude.