  Dispersion relations let us leverage the analytic structure of scattering
amplitudes to derive constraints such as bounds on EFT coefficients.
  An important input is the large-energy behavior of the amplitude. In this
paper, we systematically study how different large-energy behavior affects EFT
bounds for the $2 \to 2$ amplitude of complex scalars coupled to photons,
gravity, both, or neither. In many cases we find that singly-subtracted
dispersion relations (1SDRs) yield exactly the same bounds as doubly subtracted
relations (2SDRs). However, we identify another assumption, which we call
"$t$-channel dominance," that significantly strengthens the EFT bounds. This
assumption, which amounts to the requirement that the $++ \to ++$ amplitude has
no $s$-channel exchange, is justified in certain cases and is analogous to the
condition that the isospin-2 channel does not contribute to the pion amplitude.
Using this assumption in the absence of massless exchanges, we find that the
allowed region for the complex scalar EFT is identical to one recently
discussed for pion scattering at large-$N$. In the case of gravity and a gauge
field, we are able to derive a number of interesting bounds. These include an
upper bound for $G$ in terms of the gauge coupling $e^2$ and the leading
dispersive EFT coefficient, which is reminiscent of the weak gravity
conjecture. In the $e \to 0$ limit, we find that assuming smeared 1SDRs plus
$t$-channel dominance restores positivity on the leading EFT coefficient whose
positivity was spoiled by the inclusion of gravity. We interpret this to mean
that the negativity of that coefficient in the presence of gravity would imply
that the global $U(1)$ symmetry must be gauged.
