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Integral Equations in Neutrino Mass Searches from Beta Decay

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Document pages: 15 pages

Abstract: A new mathematical model for elucidating neutrinomass from beta decay is proposed. It is based upon the solutions of transformedFredholm and Abel integral equations. In principle, theoretical beta-particlespectra can consist of several neutrino-mass eigenstates. Integration of thebeta spectrum with a normalized instrumental response function results in theFredholm integral equation of the first kind. This equation is then transformedto yield a solution in a form of superposition of Heaviside step-functions, onefor each neutrino mass eigenstate. A series expansion leading to matrix linearequations is then derived to solve the transformed Fredholm equation. Anotherapproach is derived when the theoretical beta spectrum is obtained by aseparate deconvolution of the observed spectrum. It is then proven that thetransformed Fredholm equation reduces to the Abel integral equation. The Abelequation has a general integral solution, which is proven in this work by usinga specific equation for the beta spectrum. Several examples of numerical solutionsof the Abel equation are provided, which show a fractional sensitivity of about10-3 for subtle neutrinoeigenstate searches and can distinguish from the beta-spectrum discrepancies,such as minute shape and energy nonlinearities.

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