  We apply positivity bounds directly to a $U(1)$ gauge theory with charged
scalars and charged fermions, i.e. QED, minimally coupled to gravity. Assuming
that the massless $t$-channel pole may be discarded, we show that the improved
positivity bounds are violated unless new physics is introduced at the
parametrically low scale $\Lambda_{\rm new} \sim (e m M_{\rm pl})^{1/2}$,
consistent with similar results for scalar field theories, far lower than the
scale implied by the weak gravity conjecture. This is sharply contrasted with
previous treatments which focus on the application of positivity bounds to the
low energy gravitational Euler-Heisenberg effective theory only. We emphasise
that the low-cutoff is a consequence of applying the positivity bounds under
the assumption that the pole may be discarded. We conjecture an alternative
resolution that a small amount of negativity, consistent with decoupling
limits, is allowed and not in conflict with standard UV completions, including
weakly coupled ones.
