Considerations on anomalous photon and Z-boson self-couplings from the Born-Infeld weak hypercharge action

We investigate the effects of the Born-Infeld action on the Abelian sector of the electroweak model. The consequence of this approach is the emergence of anomalous couplings in the neutral sector of the $Z$-gauge boson and photon. These new couplings consist of quartic interactions of the photon with the $Z$-particle, as for example, three-photon-and-one-$Z$ vertex. With that, we obtain the decay width of $Z \to 3\,\gamma$ from which we impose a bound on the Born-Infeld parameter. Other bounds are also obtained from the photon quartic couplings. Subsequently, we consider the presence of an external uniform magnetic field in connection with this Born-Infeld weak hypercharge model. The magnetic background field yields new kinematic effects, like the kinetic mixing between the photon and the $Z$-boson, and we obtain thereby the corresponding dispersion relations for the mixed photon-$Z$-particle system. Finally, we calculate the lowest-order modifications to the interaction energy for the anomalous coupling $3\gamma-Z$, within the framework of the gauge-invariant but path-dependent variables formalism. Our results show that the interaction energy contains a linear term leading to the confinement of static probe charges. With the help of the potential that comes out, interparticle forces are estimated.