Integral Equations in Neutrino Mass Searches from Beta Decay
2019
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.
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