Integral Equations in Neutrino Mass Searches from Beta Decay

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 10 -3 for subtle neutrino eigenvalue searches, and can distinguish from experimental beta-spectrum discrepancies, such as shape and energy nonlinearities.