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

Within the standard V - A theory of weak interactions, Quantum electrodynamics (QED) and the linear σ-model (LσM) of strong low-energy hadronic interactions, we analyze gauge properties of hadronic structure of the neutron and proton in the neutron radiative β--decay. We show that the Feynman diagrams, describing contributions of hadronic structure to the amplitude of the neutron radiative β--decay in the tree-approximation for strong low-energy interactions in the LσM, 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 σ-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 β--decay in agreement with Sirlin’s analysis of strong low-energy interactions in neutron β- 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 V - A weak interactions does not belong to the combined quantum field theory including the LσM and QED. We argue that gauge invariant set of Feynman diagrams of hadrons, coupled to real and virtual photons in neutron β- decays, can be obtained within the combined quantum field theory including the Standard Electroweak Model (SEM) and the LσM, where the effective V - A vertex of weak interactions is a result of the W--electroweak boson exchange.