Gauge properties of hadronic structure of nucleon in neutron radiative beta decay to order O(α/π) in standard V − A effective theory with QED and linear sigma model of strong low-energy interactions

Abstract

Within the standard [Formula: see text] theory of weak interactions, Quantum electrodynamics (QED) and the linear [Formula: see text]-model (L[Formula: see text]M) of strong low-energy hadronic interactions, we analyze gauge properties of hadronic structure of the neutron and proton in the neutron radiative [Formula: see text]-decay. We show that the Feynman diagrams, describing contributions of hadronic structure to the amplitude of the neutron radiative [Formula: see text]-decay in the tree-approximation for strong low-energy interactions in the L[Formula: see text]M, are gauge invariant. In turn, the complete set of Feynman diagrams, describing the contributions of hadron–photon interactions in the one-hadron-loop approximation, is not gauge invariant. In the infinite limit of the scalar [Formula: see text]-meson, reproducing the current algebra results (S. Weinberg, Phys. Rev. Lett. 18, 188 (1967)), and to leading order in the large nucleon mass expansion the Feynman diagrams, violating gauge invariance, do not contribute to the amplitude of the neutron radiative [Formula: see text]-decay in agreement with Sirlin’s analysis of strong low-energy interactions in neutron [Formula: see text] decays. We assert that the problem of appearance of gauge noninvariant Feynman diagrams of hadronic structure of the neutron and proton is related to the following. The vertex of the effective [Formula: see text] weak interactions does not belong to the combined quantum field theory including the L[Formula: see text]M and QED. We argue that gauge invariant set of Feynman diagrams of hadrons, coupled to real and virtual photons in neutron [Formula: see text] decays, can be obtained within the combined quantum field theory including the Standard Electroweak Model (SEM) and the L[Formula: see text]M, where the effective [Formula: see text] vertex of weak interactions is a result of the [Formula: see text]-electroweak boson exchange.