Decay ${\pi }^0\to \gamma \gamma$ to next to leading order in chiral perturbation theory

The ${\pi }^0\to \gamma \gamma$ decay width is analyzed within the combined framework of chiral perturbation theory and the ${1/N}_c$ expansion up to $\mathcal{O}{(p}^6)$ and $\mathcal{O}{(p}^4\times {1/N}_c)$ in the decay amplitude. The ${\eta }^{\prime }$ is explicitly included in the analysis. It is found that the decay width is enhanced by about 4.5% due to the isospin-breaking induced mixing of the pure $U\left(3\right)\mathrm{}$ states. This effect, which is of leading order in low energy expansion, is shown to persist nearly unchanged at next to leading order. The chief prediction with its estimated uncertainty is ${\Gamma }_{{\pi }^0\to \gamma \gamma }=8.10\mathrm{}\pm 0.08\mathrm{eV}.$ This prediction at the 1% level makes the upcoming precision measurement of the decay width even more urgent. Observations on the $\eta$ and ${\eta }^{\prime }$ can also be made, especially about their mixing, which is shown to be significantly affected by next to leading order corrections.