= Fermion-loop Higgs diphoton amplitude
{title2=$T_{\mu\nu}=(p_1\cdot p_2)g_{\mu\nu}-p_{2,\mu}p_{1,\nu}$}
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> is
$$
\mathcal M=-\frac{Nye^2Q^2m}{2\pi^2}\epsilon_1^\mu\epsilon_2^\nu T_{\mu\nu}
\int_0^1dx\int_0^{1-x}dy\,\frac{1-4xy}{m^2-(p_1+p_2)^2xy-i0}.
$$
The <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>.
Back to article page
