Tailoring non-compact spin chains

Abstract

We study three-point correlation functions of local operators in planar N = 4 SYM at weak coupling using integrability. We consider correlation functions involving two scalar BPS operators and an operator with spin, in the so called SL(2) sector. At tree level we derive the corresponding structure constant for any such operator. We also conjecture its one loop correction. To check our proposals we analyze the conformal partial wave decomposition of known four-point correlation functions of BPS operators. In perturbation theory, we extract from this decomposition sums of structure constants involving all primaries of a given spin and twist. On the other hand, in our integrable setup these sum rules are computed by summing over all solutions to the Bethe equations. A perfect match is found between the two approaches.

A note on four-point correlators of half-BPS operators in N = 4 $$ \mathcal{N}=4 $$ SYM

“…Here we present the calculation of one such correlator whose OPE gives rise to non-symmetric sum rules. Our result has already been communicated to the authors of [1] who used it and found perfect agreement with their integrability prediction.…”

“…The functions Φ (1) , Φ (2) correspond to the so-called one-and two-loop ladder integrals. Their explicit expressions in terms of polylogarithms can be found in [7,8].…”

“…In the paper [1], which generalizes the results of [2,3] to the non-compact case, the threepoint correlators of two half-BPS operators and one unprotected operator in the SL (2) sector were studied in the one-loop approximation. A conjecture was made, based on integrability, for the values of the corresponding structure constants.…”

“…In the absence of direct calculations of the relevant three-point correlators with unprotected operators, use was made of the OPE of the two-loop four-point correlators of half-BPS operators, which produces sum rules for the structure constants. At the time when the paper [1] was written, only results on four-point correlators of equal weights were available [4]. They allow one to test only sum rules that contain the squares of the structure constants.…”

Quark-anti-quark potential in N $$ \mathcal{N} $$ = 4 SYM

“…Finally, as the ladders limit allows for a simpler access to the wrapping corrections, it could also serve as a useful ground to attempt a finite-size resummation of perturbation theory for 3-point correlators [43][44][45][46][47], using Q-functions as building blocks.…”