Absorptive corrections for vector mesons. Matching to complex mass scheme and longitudinal corrections

Unstable spin-1 particles are properly described by including absorptive corrections to the electromagnetic vertex and propagator, without breaking the electromagnetic gauge invariance. We show that the modified propagator can be set into a complex mass form, provided the mass and the width parameters, which are properly defined at the pole position, are replaced by energy dependent functions fulfilling the same requirements at the pole. We exemplify the case for the $K^*(892)$ vector meson, where the mass function deviates around 2 MeV from the $K\pi$ threshold to the pole position. The absorptive correction depends on the mass of the particles in the loop. For vector mesons, whose main decay is into two pseudoscalar mesons ($PP'$), the flavor symmetry breaking induces a correction to the longitudinal part of the propagator. Considering the $\tau^- \to K_S\pi^-\nu_\tau$ decay, we illustrate these corrections by obtaining the modified vector and scalar form factors. The $K_S\pi^-$ spectrum is described considering the $K^*(892)$ and $K^{'*}(1410)$ vectors and one scalar particle. Nonetheless, for this case, the correction to the scalar form factor is found to be negligible.