Interaction field strength between a scalar particle and two massless vector bosons in presence of an external magnetic field

In this work we study the interaction strength among a neutral scalar boson and two massless vector bosons in presence of an external magnetic field. Based on global symmetries, we build the general tensor structure amplitude ${\mathcal{M}}^{\mu \nu }$, for the process $V^{\mu }+V^{\nu }⟶\phi$, in terms of the vector bosons polarization states. Then, we present a novel methodology to compute the one-loop amplitude contributions for an homogeneous magnetic field with arbitrary strength. With the obtained results, expressed in terms of integrals over Schwinger parameters, we explore its behavior in two regions, widely used in the literature, the strong and weak field strength regions. The methodology presented in this work can be employed to compute an arbitrary process in presence of an external magnetic field where the initial and final states are neutral.