Integral Equations in Neutrino Mass Searches from Beta Decay  

作　　者：Thomas M. Semkow Xin Li 

机构地区：[1]Wadsworth Center, New York State Department of Health, Albany, NY, USA [2]School of Public Health, University at Albany, State University of New York, Rensselaer, NY, USA

出　　处：《Journal of Applied Mathematics and Physics》2019年第1期31-45,共15页应用数学与应用物理（英文）

摘　　要：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.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.

关 键 词：FREDHOLM EQUATION ABEL EQUATION Heaviside Function HEAVY NEUTRINO 

分 类 号：O1[理学—数学]