Dispersive CFT Sum Rules

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 doubletwist 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 doubletwist 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.

Mixed Correlator Dispersive CFT Sum Rules

“…Moreover, the presence of the double discontinuity suppresses doubletwist operators thereby allowing us to probe non-perturbative features of CFTs. Finally, they enjoy positivity properties due to the double-zeros that make them desirable for the numerical 2 Our convention differs from that of [29,30]: for equal operators, the u-channel identity is always present. Therefore, they define their sum rules to be normalized as follows:…”

“…In recent years, with the introduction of analytic functionals [20][21][22][23][24][25][26][27] and dispersion relation methods [28][29][30][31][32][33][34], the gap between numerics and analytics has greatly narrowed revealing new horizons for the bootstrap program. This paper builds upon this bridge by introducing new analytic dispersive functionals for correlators with unequal external scalar operators.…”

“…The derivative functional has long served as the cornerstone for the numerical bootstrap, but progress in the field has encouraged the pursuit of new methods with more desirable positivity properties and computational efficiency. A potential candidate with such features is a so-called dispersive CFT functional first introduced in [29], which reconstructs functions in terms of their absorptive part [28]. Such a functional can be written as a dispersive transform acting on a stripped correlator G(u, v):…”

“…For equal scalar operators of scaling dimension ∆ φ , the authors in [29] introduced the collinear B k,v dispersive functional which enjoys the following three key properties:…”

“…Unfortunately, evaluating these B k,v functionals in position-space can become expensive in certain limits such as for large number of derivatives, or for low twist operators (near the unitarity bound). An alternative approach is to construct them in Mellin-space [37][38][39][40][41] where they can be computationally more efficient, and where projected functionals such as the Φ functional introduced in [29] are easier to construct. This paper builds upon the construction of dispersion relations in both position-and Mellin-space by extending previous results to mixed correlators.…”