  We bound EFT coefficients appearing in $2 \to 2$ photon scattering amplitudes
in four dimensions. After reviewing unitarity and positivity conditions in this
context, we use dispersion relations and crossing symmetry to compute sum rules
and null constraints. This allows us to derive new rigorous bounds on operators
with four, six, and eight derivatives, including two-sided bounds on their
ratios. Comparing with a number of partial UV completions, we find that some of
our bounds are saturated by the amplitudes that arise from integrating out a
massive scalar or axion, while others suggest the existence of unknown
amplitudes.
