The unique Polyakov blocks

167

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.

Section: Jhep05(2021)243mentioning

confidence: 93%

Section: Jhep05(2021)243mentioning

confidence: 99%

Dispersive CFT sum rules

Caron-Huot

Mazáč

Rastelli

et al. 2021

167

“…On a similar note, how do the factorisation properties of momentum-space correlators envisaged by Polyakov [79] arise? Related recent discussions include [46,50,[80][81][82][83][84]. Since the simplex representation is a generalised Feynman integral, it should be especially well-suited for extracting the discontinuities needed to understand the implications of unitarity.…”

Section: Jhep01(2021)192mentioning

confidence: 99%

“…Momentum-space methods are moreover useful for the analysis of renormalisation [41][42][43], where they offer a simple extraction of divergences plus an elegant decomposition of tensorial structure. Looking ahead, a particularly exciting future application -for which the present work is a necessary first step -is the formulation of momentum-space approaches to the conformal bootstrap (see [44,45] for reviews and [46][47][48][49][50] for related ideas). With the form of general n-point correlators to hand, understanding the partial wave decomposition and singularity structure of correlators is now within reach.…”

Section: Introductionmentioning

confidence: 99%

Conformal correlators as simplex integrals in momentum space

Bzowski

McFadden

Skenderis

2021

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

“…Although s-channel exchange diagrams have a vanishing t-and u-channel dDisc, they cannot be used to improve the CFT s-channel Regge growth. This is clearest to see in Mellin space[98,103].…”

mentioning

confidence: 99%

Dispersion formulas in QFTs, CFTs and holography

Meltzer

2021

We study momentum space dispersion formulas in general QFTs and their applications for CFT correlation functions. We show, using two independent methods, that QFT dispersion formulas can be written in terms of causal commutators. The first derivation uses analyticity properties of retarded correlators in momentum space. The second derivation uses the largest time equation and the defining properties of the time-ordered product. At four points we show that the momentum space QFT dispersion formula depends on the same causal double-commutators as the CFT dispersion formula. At n-points, the QFT dispersion formula depends on a sum of nested advanced commutators. For CFT four-point functions, we show that the momentum space dispersion formula is equivalent to the CFT dispersion formula, up to possible semi-local terms. We also show that the Polyakov-Regge expansions associated to the momentum space and CFT dispersion formulas are related by a Fourier transform. In the process, we prove that the momentum space conformal blocks of the causal double-commutator are equal to cut Witten diagrams. Finally, by combining the momentum space dispersion formulas with the AdS Cutkosky rules, we find a complete, bulk unitarity method for AdS/CFT correlators in momentum space.