Abstract
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.
Highlights
We start by summarizing the physics of neutrino-mass searches
It is proven that the transformed Fredholm equation reduces to the Abel integral equation
The transformed Fredholm equation was derived (Equation (14a) with the kernel given by Equation (14b)), linking the observed beta spectrum with the theoretical one as well as with the instrumental response
Summary
We start by summarizing the physics of neutrino-mass searches. Since the neutrino was conceptually proposed by Pauli in 1930 and given its name by Fermi, followed by his development of the beta decay theory in 1933-1934 [1] [2] [3]. The existing methods of identification of the neutrino mass in beta decay, whether close to or away from the endpoint energy, employ two approaches: 1) convolution of the theoretical beta spectrum with the normalized instrumental response function and comparison of the convolution with the observed spectrum, and 2) deconvolution of the observed spectrum and comparison with the theoretical spectrum [6]-[11] Statistical measures, such as χ 2 minimization or Bayesian likelihood, are used for the above-mentioned comparisons. We summarize the advantages and limitations of the proposed new mathematical method of elucidating neutrino mass in beta decay in Section 7, as well as propose future directions of this work
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