Integral Equations in Neutrino Mass Searches from Beta Decay

A new mathematical model for elucidating neutrino mass from beta decay is proposed. It is based upon the solutions of transformed Fredholm and Abel integral equations. In principle, theoretical beta-particle spectra can consist of several neutrino-mass eigenstates. Integration of the beta spectrum with a normalized instrumental response function results in the Fredholm integral equation of the first kind. This equation is then transformed to yield a solution in a form of superposition of Heaviside step-functions, one for each neutrino mass eigenstate. 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 equation for the beta spectrum. Several examples of numerical solutions of the Abel equation are provided, which show a fractional sensitivity of about 10-3 for subtle neutrino eigenstate searches and can distinguish from the beta-spectrum discrepancies, such as minute shape and energy nonlinearities.