Light-state Dominance from the Conformal Bootstrap

TL;DR

This work establishes a concrete, nonperturbative form of light-state dominance in d-dimensional CFTs by adapting a modular-invariance–style crossing argument to four-point functions. It proves that correlators with external dimension $\Delta$ are dominated by exchanges of operators with dimension below a cutoff $\Delta_c>2\Delta$, with the heavy-state contribution bounded by $\mathcal{E}(z) \le \frac{R(z)}{1-R(z)}$ and $R(z)=(\frac{z}{1-z})^{\Delta_c-2\Delta}$, and it extends these bounds to the first sheet and derivatives. The results generalize to $2d$ partition functions and are connected to OPE inversion, showing that, under appropriate gap assumptions, integrating out heavy exchanges gives higher-derivative bulk couplings suppressed by the gap, consistent with EFT expectations in AdS/CFT. The paper also analyzes implications for gapped theories, non-identical external operators, and the potential utility of derivative bounds in the numerical/bootstrap program. Overall, light-state dominance provides a robust, duality-based mechanism to bound heavy data from light CFT data and informs bulk locality ideas in holographic theories.

Abstract

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 $Δ$ is approximated, with bounded error, by the contribution of operators with scaling dimension below $Δ_c > 2Δ$ 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 $e^{2Δ-Δ_c}$. 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 $Δ_{gap}$.