= Leptonic pseudoscalar decay width
{title2=$\Gamma=G_F^2F_P^2m_\ell^2M_P(1-m_\ell^2/M_P^2)^2/(4\pi)$}
In the convention $\langle0|J^\mu|P(p)\rangle=-i\sqrt2F_Pp^\mu$, the <Fermi interaction> gives $\mathcal M=G_FF_Pp_\mu\bar u_\ell\gamma^\mu(1-\gamma_5)v_\nu$. With a massless <antineutrino>, the external <Dirac equations> reduce this to $G_FF_Pm_\ell\bar u_\ell(1-\gamma_5)v_\nu$. The spin sum is $8G_F^2F_P^2m_\ell^2 k\cdot q=4G_F^2F_P^2m_\ell^2(M_P^2-m_\ell^2)$. The <two-body Lorentz-invariant phase space> has integrated value $(1-m_\ell^2/M_P^2)/(8\pi)$, and the initial flux factor is $1/(2M_P)$, giving the displayed width. If the decay constant is instead defined without the square root of two, $f_P=\sqrt2F_P$ and the denominator is $8\pi$. A weak mixing coefficient multiplies the amplitude and its modulus squared multiplies the width.
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