Light-state dominance from the conformal bootstrap

We derive forms of light-state dominance for correlators in CFTd, 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 Δc > 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 e2Δ−Δc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {e}^{2\varDelta -{\varDelta}_c} $$\end{document} . 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.