Longitudinal-vector annihilation through a scalar contact interaction (${\sigma }_{LL}\sim g^{4}s/(32\pi m_A^4)$)

For a vertex $2i{g}^2{\eta }_{\mu \nu }$ and two identical outgoing real scalar particles, the longitudinal-pair amplitude is $g^2(s-2m_A^2)/m_A^2$. The high-energy cross section is the displayed expression, including the final-state factor $1/2!$. An unpolarized incoming pair averages over nine spin states and has leading cross section $g^{4}s/(288\pi m_A^4)$. A written interaction $g^2{A}_{\mu }{A}^{\mu }{h}^2$ instead gives vertex $4i{g}^2{\eta }_{\mu \nu }$ and four times these cross sections. The distinction separates a vertex normalization from polarization averaging and identical-particle phase-space counting.