Nonperturbative path integral quantization of the electroweak model: The Maxwell integration

The nonperturbative path integral quantization of the electroweak model is confronted with an apparent instability when integrating over the Maxwell potential Aμ due to the fast growth of the box graphs AAAA and AAAZ for large amplitude variations of Aμ. Zμ is from the vector part of the weak neutral current. These graphs are unavoidable because they are conditionally convergent and have to be isolated in the model's exact Euclidean one-loop effective action arising from its fermion determinants. A previous QED calculation of the large amplitude variation of its fermion determinant for a class of random potentials showed that the AAAA box graph cancels in this limit. Using this result it is shown that within the electroweak model large amplitude variations of Aμ for fixed Zμ in a superposition of these fields cancel the AAAA and AAAZ graphs, thereby removing an apparent obstacle to the model's nonperturbative quantization. A negative paramagnetic term in the remainder opposes the effective action's growth for such variations. Its calculation requires knowledge of the degeneracy of the bound states of a charged fermion in the four-dimensional magnetic fields generated by the functional measure of Aμ.