The Cauchy-singular integral equation appearing in the theory of chiral spontaneous symmetry breaking

Summary

It is shown how the combination of associated hypergeometric functions, in the case of equal vector and axial couplings, satisfies an analytic integral equation on the right-hand cut with Cauchy-singular kernel having the same discontinuity as the original integral equation. This is obtained by using an extension of the quadratic transformation of the Gauss function. A projective quasi-solution is found, fixing an accurate value of the ratioG _A/G_V=1.154, due to radiative corrections, once favored by Schwinger as twice the Euler numerical constant. Also noticed is a quasi-solution withɛ=d−4=15% andG _A=G _V indicating possible functioning between Minkowski and de Sitter spaced=5. Finally, evaluation of the conjugate function by the initial—discontinuous—integral operator allows one to understand both the massless confluent limit obtained by the author (J.-P. Lebrun,Nuovo Cimento A,103, 1735 (1990);104, 1071 (1991)) and the one-half the expected rate mentioned in solar-neutrino observations.