= Solution
{c}
Use the <Feynman parameters> and $J_3$ defined in <1/iv/Solution>. With $A=4C$, the finite term just found is $-C\Delta J_3$. Combining it with the remaining $g_{\mu\nu}$ coefficient gives
$$
C[-\Delta+m^2-s/2+sxy]J_3
=-\frac{sC}{2}(1-4xy)J_3.
$$
The other coefficient is $(B-4Cxy)J_3$. Thus, writing $\mathscr I_{\mu\nu}$ for the <Feynman integral> without the overall prefactor,
$$
\mathscr I_{\mu\nu}=2\int_{\mathcal T}dx\,dy\,J_3(\Delta)
\left[-\frac{sC}{2}(1-4xy)g_{\mu\nu}+(B-4Cxy)p_{2,\mu}p_{1,\nu}\right].
$$
For $T_{\mu\nu}=\frac s2g_{\mu\nu}-p_{2,\mu}p_{1,\nu}$, the two coefficients must have the corresponding ratio. The mismatch is simply $(B-C)J_3$, independent of $xy$. Therefore the required relation is
$$
\boxed{B=C,}
$$
which is satisfied. In the primitive normalization the result is
$$
\mathscr I_{\mu\nu}=\frac{i}{16\pi^2}\int_0^1dx\int_0^{1-x}dy\,
\frac{1-4xy}{m^2-sxy-i0}\,T_{\mu\nu},
$$
and the full <top quark> <scattering amplitude>, in our stated $S$ convention, is
$$
\boxed{\mathcal M=-\frac{N_cye^2Q^2m}{2\pi^2}\,
\epsilon_1^\mu\epsilon_2^\nu T_{\mu\nu}
\int_0^1dx\int_0^{1-x}dy\,\frac{1-4xy}{m^2-sxy-i0}.}
$$
Indeed $p_1^\mu T_{\mu\nu}=p_2^\nu T_{\mu\nu}=0$: replacing either <polarization vector> by its <photon> <momentum> makes the <scattering amplitude> vanish. This is the <Ward identity>, expressing electromagnetic <gauge invariance>. The result also is symmetric under exchange of the two <photons>. The finite term in <1/iv/Solution> is essential for this <Ward identity>.
As an independent low-<energy> check, for $s/m^2\to0$ the <Feynman parameter> integral is
$$
\frac1{m^2}\int_{\mathcal T}(1-4xy)\,dx\,dy+O(s/m^4)
=\frac1{3m^2}+O(s/m^4).
$$
The coefficient is consequently $-N_cye^2Q^2/(6\pi^2m)$ at leading order. It has the inverse-<mass> dimension required for an effective $hF_{\mu\nu}F^{\mu\nu}$ coupling. If $y=m/v$, this coefficient stays finite in the large-$m$ limit at fixed $v$; that limit increases the <Yukawa coupling>, so it is not a fixed-coupling <heavy-field decoupling> limit.
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