Photon emission from weakly magnetized neutral pions

Using a hadronic framework, we derive an explicit expression for photon production from neutral pions in a weak background magnetic field. Our calculation is built on the proton triangle diagram with an effective Yukawa $\pi^0$-proton coupling, offering an alternative to quark-level descriptions that is advantageous when the magnetic length greatly exceeds the proton size. Corrections to the pion decay constant are computed up to second order in the magnetic field strength, revealing that the field generally suppresses the decay rate. Quantitatively, however, the effect remains modest even for fields as strong as $|eB|\simeq m_\pi^2$. The differential photon emission rate exhibits anisotropy, with the strongest suppression occurring when the pion momentum is perpendicular to the magnetic field. Overall, the modification of the $\pi^0 \to \gamma\gamma$ rate is parametrically small, scaling as $|eB|^2/m_P^4$, where $m_P$ is the proton mass. Nevertheless, under favorable conditions, the resulting angular anisotropy could lead to observable signatures in heavy-ion collisions.