Application of integral equations to neutrino mass searches in beta decay

TL;DR

This paper introduces a novel mathematical approach using integral equations to analyze beta decay spectra for neutrino mass determination, enabling the identification of multiple neutrino eigenvalues with high sensitivity.

Contribution

It develops new methods based on Fredholm and Abel integral equations for extracting neutrino masses from beta decay spectra, including a matrix solution approach and a sensitivity analysis.

Findings

Solution of Fredholm equation as superposition of step-functions

Reduction to Abel integral equation for spectrum deconvolution

Numerical method achieves 0.001 fractional sensitivity

Abstract

A new mathematical method for elucidating neutrino mass from beta decay is studied. It is based upon the solutions of transformed Fredholm and Volterra integral equations. In principle, theoretical beta-particle spectra can consist of several neutrino-mass eigenvalues. Integration of the theoretical beta spectrum with a normalized instrumental response function results in the Fredholm integral equation of the first kind. This equation is transformed in such a way that the solution of it is a superposition of the Heaviside step-functions, one for each neutrino mass eigenvalue. A series expansion leading to matrix linear equations is then derived to solve the transformed Fredholm equation. Another approach is derived when the theoretical beta spectrum is obtained by a separate deconvolution of the observed spectrum. It is then proven that the transformed Fredholm equation reduces to the Abel integral equation. The Abel equation has a general integral solution, which is proven in this work by using a specific function for the beta spectrum. As an example, a numerical solution of the Abel integral equation is also provided, which has a fractional sensitivity of about 0.001 for subtle neutrino eigenvalue searches, and can distinguish from experimental beta-spectrum discrepancies, such as shape and energy nonlinearities.