= Integrated massless leptonic tensor
{title2=$\int d\Phi_2\,L_{\mu\nu}=(q_\mu q_\nu-g_{\mu\nu}q^2)/(3\pi)$}
For the current with $(1-\gamma^5)$ rather than $P_L$, the spin-summed symmetric <leptonic tensor> is $8(q_{1\mu}q_{2\nu}+q_{1\nu}q_{2\mu}-g_{\mu\nu}q_1\cdot q_2)$. Integrating the standard <two-body Lorentz-invariant phase space> of two massless particles gives
$$
\int d\Phi_2(q;q_1,q_2)\,L_{\mu\nu}=\frac1{3\pi}(q_\mu q_\nu-g_{\mu\nu}q^2).
$$
Its antisymmetric <Levi-Civita symbol> term integrates to zero. The normalization changes by a factor of four if the current uses $P_L$ instead. The result separates the leptonic integration from hadronic <pseudoscalar-to-pseudoscalar form factors> in a three-body <decay rate>.
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