Anisotropic photon emission from gluon fusion and splitting in a strong magnetic background I: The two-gluon one-photon vertex

Having in mind the pre-equilibrium stage in peripheral heavy-ion collisions as a possible scenario for the production of electromagnetic radiation, we compute the two-gluon one-photon vertex in the presence of an intense magnetic field at one-loop order. The quarks in the loop are taken such that two of them occupy the lowest Landau level, with the third one occupying the first exited Landau level. When the field strength is the largest of the energy (squared) scales, the tensor basis describing this vertex corresponds to two of the three vector particles polarized in the longitudinal direction whereas the third one is polarized in the transverse direction. However, when the photon energy is of order or larger than the field strength, the explicit one-loop computation contains extra tensor structures that spoil the properties of the basis, compared to the case when the field strength is the largest of the energy scales, which signals that the calculation is incomplete. Nevertheless, by projecting the result onto the would-be basis, we show that the squared amplitude for processes involving two gluons and one photon exhibits the expected properties such as a preferred in-plane photon emission and a slightly decreasing strength for an increasing magnetic field strength. We comment on possible venues to improve the one-loop calculation that include accounting for progressive occupation of the three quarks of the lowest and first excited Landau levels such that, still working in the large field limit, a more complete description can be achieved when the photon energy increases.