On the decay of a light spinless particle into two photons

We analyze the effective couplings of a light, spinless, gauge-singlet particle $\phi$ to on-shell photons. Starting from the most general theory at the electroweak scale, which allows for CP-violating interactions suppressed by inverse powers of an ultraviolet scale $\Lambda$, we derive the corresponding low-energy effective theory valid below the GeV scale. Within this framework, we systematically expand the effective couplings of $\phi$ to on-shell photons in powers of small parameters. Working at the one-loop level, we retain terms at first order in $1/\Lambda$. We incorporate both isospin-breaking effects and $\eta$-$\eta'$ mixing, and provide explicit expressions for the couplings up to first order in $m_\phi^2/m_\eta^2$ and $m_\pi^2/m_\eta^2$. As applications, we compute the decay rate of $\phi$ into two photons and illustrate our results in several physically motivated scenarios.