= Solution
{c}
There are four <tree-level Feynman diagrams>: three <photon> exchange graphs with pairings $12|34$, $13|24$, $14|23$, and the four-photon <contact interaction>. They are drawn in the bottom row of the figure. With no external <fermions> or <Faddeev-Popov ghost fields>, an internal matter or ghost line would have to close into a loop; it cannot add a <tree-level Feynman diagram> here.
Use all-incoming null <four-momenta> $k_i$ with $\sum_i k_i=0$. Let $\varepsilon_i$ be the external <photon polarization vectors>, including the complex conjugation for physical outgoing <photons>, and impose $k_i\cdot\varepsilon_i=0$. For one pairing $ij|kl$, put $K=k_i+k_j=-(k_k+k_l)$. At the first cubic <Feynman vertex>, the terms proportional to $k_i\cdot\varepsilon_i$ and $k_j\cdot\varepsilon_j$ vanish, leaving
$$
\varepsilon_{i\alpha}\varepsilon_{j\beta}V_3^{\alpha\beta\rho}(k_i,k_j,-K)=-2\lambda(\varepsilon_i\cdot\varepsilon_j)K^\rho.
$$
At the other end, the incoming internal <four-momentum> is $+K$, giving $+2\lambda(\varepsilon_k\cdot\varepsilon_l)K^\sigma$. The <Feynman gauge> <photon propagator> then gives
$$
i\mathcal M_{ij|kl}=(-2\lambda)(2\lambda)(\varepsilon_i\cdot\varepsilon_j)(\varepsilon_k\cdot\varepsilon_l)K^\rho\frac{-ig_{\rho\sigma}}{K^2+i0}K^\sigma
=4i\lambda^2(\varepsilon_i\cdot\varepsilon_j)(\varepsilon_k\cdot\varepsilon_l).
$$
The last expression is evaluated at generic kinematics, with singular channel limits taken after cancellation. Summing the three pairings produces $+4i\lambda^2P$, where
$$
P=(\varepsilon_1\cdot\varepsilon_2)(\varepsilon_3\cdot\varepsilon_4)+(\varepsilon_1\cdot\varepsilon_3)(\varepsilon_2\cdot\varepsilon_4)+(\varepsilon_1\cdot\varepsilon_4)(\varepsilon_2\cdot\varepsilon_3).
$$
The quartic <Feynman vertex> contributes $i\mathcal M_{\mathrm{contact}}=-4i\lambda^2P$. Thus
$$
\boxed{i\mathcal M_{\mathrm{tree}}=4i\lambda^2P-4i\lambda^2P=0.}
$$
This is the <tree photon-scattering cancellation in a quadratic Abelian gauge>: the individual <gauge fixing> interactions cancel in the physical <scattering amplitude>. The argument uses transverse external <photon polarization vectors>; it does not assert that an arbitrary <off shell> four-point <correlation function> vanishes. Physical <light-by-light scattering> first occurs through charged-particle <loop Feynman diagrams>.
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