On mass corrections to the decay $P\rightarrow l^+l^-$

We use the Mellin–Barnes representation in order to improve the theoretical estimate of mass corrections to the width of light pseudoscalar meson decay into a lepton pair, $P\rightarrow l^+l^-$. The full resummation of the terms $\ln(m_l^2/\Lambda^2)(m_l^2/\Lambda^2)^n$ and $(m_l^2/\Lambda^2)^n$ to the decay amplitude is performed, where $m_l$ is the lepton mass and $\Lambda\approx m_{\rho}$ is the characteristic scale of the $P\rightarrow\gamma^{\ast}\gamma^{\ast}$ form factor. The total effect of mass corrections for the $e^+e^-$ channel is negligible and for the $\mu^+\mu^-$ channel its order is of a few per cent.