= Solution
{c}
There are exactly four <tree-level Feynman diagrams>: one four-<photon> contact graph and one exchange graph for each of the three partitions $12|34$, $13|24$ and $14|23$ of the external legs. The exchange graphs contain two cubic <Feynman vertices>. A <fermion loop> or a closed <Faddeev-Popov ghost> line has positive <loop order>, so neither enters here.
\Image[/past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2021-nonlinear-gauge-photon-scattering.png]
{title=The contact graph and three exchange graphs for physical photon scattering in the nonlinear gauge; all external momenta are incoming}
Write $p_1+p_2+p_3+p_4=0$ and take physical <photon polarization vectors> $\varepsilon_i$ with $p_i\cdot\varepsilon_i=0$ and $p_i^2=0$. At the cubic <Feynman vertex> attached to legs $i,j$, the two terms containing $p_i\cdot\varepsilon_i$ or $p_j\cdot\varepsilon_j$ vanish. If $q=p_i+p_j$ flows into the internal line, the remaining factor at that end is
$$
-2\lambda q^\mu(\varepsilon_i\cdot\varepsilon_j).
$$
At the other end the internal incoming <four-momentum> is $q$ rather than $-q$, giving $+2\lambda q^\nu(\varepsilon_k\cdot\varepsilon_l)$. Multiplication by the <photon propagator> gives
$$
i\mathcal M_{ij|kl}=(-2\lambda q^\mu)\frac{-ig_{\mu\nu}}{q^2}(2\lambda q^\nu)
(\varepsilon_i\cdot\varepsilon_j)(\varepsilon_k\cdot\varepsilon_l)
=4i\lambda^2(\varepsilon_i\cdot\varepsilon_j)(\varepsilon_k\cdot\varepsilon_l).
$$
The cancellation holds first at generic nonzero channel <Mandelstam invariants> and then by continuation at degenerate kinematics. Summing the three exchange graphs 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 $-4i\lambda^2P$. Therefore
$$
\boxed{\mathcal M_{\gamma\gamma\to\gamma\gamma}^{\mathrm{tree}}=0.}
$$
This is a cancellation for physical <on shell> <photon polarization vectors>. The <off shell> four-<photon> correlation function need not vanish. Physical <light-by-light scattering> first occurs through charged-particle <loop Feynman diagrams>.
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