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

<jats:title>Abstract</jats:title><jats:p>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 <jats:italic>Z</jats:italic>-gauge boson and photon. These new couplings consist of quartic interactions of the photon with the <jats:italic>Z</jats:italic>-particle, as for example, three-photon-and-one-<jats:italic>Z</jats:italic> vertex. With that, we obtain the decay width of <jats:inline-formula><jats:alternatives><jats:tex-math>$$Z \rightarrow 3\,\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>Z</mml:mi>
<mml:mo>→</mml:mo>
<mml:mn>3</mml:mn>
<mml:mspace />
<mml:mi>γ</mml:mi>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> 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 <jats:italic>Z</jats:italic>-boson, and we obtain thereby the corresponding dispersion relations for the mixed photon-<jats:italic>Z</jats:italic>-particle system. Finally, we calculate the lowest-order modifications to the interaction energy for the anomalous coupling <jats:inline-formula><jats:alternatives><jats:tex-math>$$3\gamma -Z$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mi>γ</mml:mi>
<mml:mo>-</mml:mo>
<mml:mi>Z</mml:mi>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>, 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.</jats:p>