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. While the magnetic-field-induced anisotropy is conceptually interesting in principle, it is likely too small to be resolved in present heavy-ion measurements.