 Integrated massless leptonic tensor

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 givesIts 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.