Fermion-loop Higgs diphoton amplitude ($T_{\mu \nu }=(p_1\cdot p_2)g_{\mu \nu }-p_{2,\mu }{p}_{1,\nu }$)

For a charged Dirac field excitation of mass $m$, multiplicity $N$, electric coupling $eQ$ and scalar Yukawa interaction $-yh\bar{\psi }\psi$, two orientations of a fermion loop induce a Higgs boson coupling to two photons. With metric $(+---)$ and the usual $S=1+iT$ convention, their physical on-shell scattering amplitude isThe Ward identity holds on both photon legs. At low external energy, the integral tends to $1/(3m^2)$. Thus fixed $y$ gives an inverse-mass coefficient, whereas $y=m/v$ gives a constant coefficient as $m$ increases at fixed $v$. The distinction is essential when applying heavy-field decoupling.