= Solution
Route the <photon> <momentum> as $l$ and the internal <Scalar field>[scalar] <momentum> as $l+k$. The <Transverse projector of a vector field>[transverse projector] obeys $P_{\alpha\beta}l^\beta=0$, so the two <Interaction term>[cubic vertices] give
$$
(2k+l)^\alpha P_{\alpha\beta}(l)(2k+l)^\beta=4k^\alpha P_{\alpha\beta}(l)k^\beta.
$$
Multiplying the two <vertices> and propagators therefore gives
$$
\boxed{i\Pi_{\mathrm{ex}}(k)=4e^2\mu^\epsilon\int\frac{d^dl}{(2\pi)^d}\frac{k^2-(k\cdot l)^2/l^2}{l^2[(l+k)^2+m^2]}.}
$$
For the <Seagull vertex>[seagull], $g^{\alpha\beta}P_{\alpha\beta}=d-1$. Including the photon-loop <symmetry factor> gives instead
$$
\boxed{i\Pi_{\mathrm{sg}}=\frac12(-2ie^2)\mu^\epsilon\int\frac{d^dl}{(2\pi)^d}\frac{-i(d-1)}{l^2+m_\gamma^2}
=-(d-1)e^2\mu^\epsilon\int\frac{d^dl}{(2\pi)^d}\frac1{l^2+m_\gamma^2}.}
$$
\b[The printed <Seagull vertex>[seagull] coefficient is twice this result.] The <vertex>'s factor two differentiates the two <photon> fields; closing them into one line requires the compensating <symmetry factor>. Direct expansion of the <action> gives the same check: the <Lagrangian density>[interaction density] $-e^2A^2\phi^\dagger\phi$ has one <photon> <Wick contraction>[contraction], yielding coefficient $-e^2$, not $-2e^2$. The identical-photon origin of the <vertex> factor is also explicit in the standard scalar-electrodynamics rules.<Quantum Field Theory by David Tong>[§6.5.1, charged scalars: the seagull vertex factor two comes from the two identical photon fields]{ref} This discrepancy does not affect the massless-photon limit in part d or the <Ultraviolet divergence>[ultraviolet pole] in part f.
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