Dispersive CFT Sum Rules

TL;DR

This work unifies dispersive sum rules for four-point CFT correlators by showing the equivalence of position-space dispersion, Mellin-space dispersion, and analytic functionals through Polyakov-Regge blocks. It develops a nonperturbative Polyakov-Regge expansion that uses data from two OPE channels while maintaining good Regge behavior in the third, and constructs a rich family of subtracted, spin-aware sum rules (notably the $B_{2,v}$ family and the extremal functional $\Phi_2$) with positivity above a twist gap. These dispersive functionals provide convergent, nonperturbative constraints on the spectrum, enabling rigorous twist-gap bounds, extremal analyses for mean-field theory, and connections to lightcone bootstrap. The framework is dimension-agnostic, links to holography via AdS-Witten diagrams, and offers practical tools for analytic and numerical bootstrap efforts with controlled errors. Overall, the paper establishes dispersive sum rules as a versatile, cross-space toolkit for constraining CFT data and understanding high-energy/low-energy consistency of conformal theories.

Abstract

We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule dispersive if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of superconvergence sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably, we exhibit a candidate extremal functional for the spin-two gap problem.

Figures (16)

• Figure 1: Three natural spaces in which we consider conformal bootstrap constraints: Position space $(z,\bar{z}$), the set of double-twist labels $(n,\ell)$ and Mellin space $(\mathrm{s},\mathrm{t})$. Dispersion relations in position space and Mellin space will turn out to be equivalent to each other, and to provide generating functions for functionals dual to double-twist operators.
• Figure 2: The region of the complex plane (for the independent complex variables $z$ and $\bar{z}$) in which the s- and t- channel OPEs converge simultaneously.
• Figure 3: Relevant regions in the $u',v'$ variables. The black curve is the image of $w=\bar{w}$ and separates the Euclidean region $\mathrm{E}$, where $w$ and $\bar{w}$ are complex conjugate, from the three Lorentzian lightcones $\mathrm{L_{us}}$: $w,\bar{w}<0$, $\mathrm{L_{st}}$: $0<w,\bar{w}<1$ and $\mathrm{L_{tu}}$: $w,\bar{w}>1$. The s-channel contribution to the dispersion relation \ref{['eq:Dispersion1']} is an integral over the shaded region $\sqrt{v'}\geq\sqrt{u'}+\sqrt{u}+\sqrt{v}$. When $\sqrt{u}+\sqrt{v}\geq1$, the integration region lies inside $\mathrm{L_{us}}$. When $\sqrt{u}+\sqrt{v}<1$, it covers all of $\mathrm{L_{us}}$ and parts of $\mathrm{L_{st}}$ and $\mathrm{E}$.
• Figure 4: The integration contour used to define the $\Omega$ transform \ref{['Omega-s']}, \ref{['Omega-t']}. Note that the contour is invariant under $(w,\bar{w})\mapsto (1-\bar{w},1-w)$, i.e. it is crossing-symmetric.
• Figure 5: The Polyakov-Regge block $P_{\Delta, J}^{s|u}$ is a combination of exchange and contact Witten diagrams such that the $u$-channel Regge behavior is the best possible.