Light-state Dominance from the Conformal Bootstrap

We derive forms of light-state dominance for correlators in CFT$_d$, making precise the sense in which correlators can be approximated by the contribution of light operator exchanges. Our main result is that the four-point function of operators with dimension $\Delta$ is approximated, with bounded error, by the contribution of operators with scaling dimension below $\Delta_c > 2\Delta$ in the appropriate OPE channel. Adapting an existing modular invariance argument, we use crossing symmetry to show that the heavy-state contribution is suppressed by a relative factor of $e^{2\Delta-\Delta_c}$. We extend this result to the first sheet and derivatives of the correlator. Further exploiting technical similarities between crossing and modular invariance, we prove analogous results for the $2d$ partition function along the way. We then turn to effective field theory in gapped theories and AdS/CFT, and make some general comments about the effect of integrating out heavy particles in the bulk. Combining our bounds with the Lorentzian OPE inversion formula we show that, under certain conditions, light-state dominance implies that integrating out heavy exchanges leads to higher-derivative couplings suppressed at large $\Delta_{gap}$.