Three-point functions at strong coupling in the BMN limit

Published for SISSA by Springer Received: January 25, 2020 Accepted: March 26, 2020 Published: April 14, 2020 Benjamin Basso and De-liang Zhong Laboratoire de physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond, 75005 Paris, France E-mail: , Abstract: We consider structure constants of single-trace operators at strong coupling in planar N = 4 SYM theory using the hexagon formalism. We concentrate on heavy-heavylight correlators where the heavy operators are BMN operators, with large R-charges and finite anomalous dimensions, and the light one is a finite-charge chiral primary operator. They describe the couplings between two highly boosted strings and a supergravity mode in the bulk dual. In the hexagon framework, two sums over virtual magnons are needed to bind the hexagons together around the light operator. We evaluate these sums explicitly at strong coupling, for a certain choice of BMN operators, and show that they factorise into a ratio of Gamma functions and a simple stringy prefactor. The former originates from giant mirror magnons scanning the AdS geometry while the latter stems from small fluctuations around the BMN vacuum. The resulting structure constants have poles at positions where an enhanced mixing with double-trace operators is expected and zeros whenever the process is forbidden by supersymmetry. We also discuss the transition to the classical regime, when the length of the light operator scales like the string tension, where we observe similitudes with the Neumann coefficients of the pp-wave String Field Theory vertex. Keywords: Conformal Field Models in String Theory, Field Theories in Higher Dimensions, Integrable Field Theories ArXiv ePrint: 1907.01534 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2020)076 JHEP04(2020)076 Three-point functions at strong coupling in the BMN limit Contents 1 2 Generalities 2.1 Operators 2.2 Hexagon sums 4 4 6 3 The one-particle integral 3.1 Classical bridge 3.2 Quantum bridge 11 11 13 4 Summing the multi-particle integrals 4.1 Free energy 4.2 Telescoping the sum 4.3 Evaluating the sum 4.4 Changing the grading 17 17 18 20 23 5 HHL structure constants 5.1 Crossing and full amplitude 5.2 Poles and zeros 5.3 Wrapping and string 24 24 27 28 6 Conclusion 31 A Hexagon amplitude and transfer matrix 33 B Analytic regularisation 36 C Near-extremal correlators at weak coupling 37 D Leading pole at finite coupling 40 1 Introduction There has been a great deal of activity recently regarding correlation functions of local operators in planar N = 4 SYM and in its holographic dual, IIB superstring theory in AdS5 × S 5 . On the one hand, relying on Mellin space techniques [1, 2] and bootstrap ideas, new approaches have been developed [3–10] to deal more efficiently with the supergravity regime, corresponding to the strong coupling limit, g 2 = λ/(4π)2  1, of the large-Nc gauge theory. They led to spectacular results, starting with a conjecture [3] for the 1/Nc2 –1– JHEP04(2020)076 1 Introduction C ◦◦• = hO1 (∞)O2 (1)Oγ (0)i , (1.1) and with L ∼ L1 ∼ g  1 and L2 , γ ∼ g 0 in the heavy-heavy-light (HHL) kinematics. This set-up is interesting in that it enables to probe correlators at low energy and still avoids bottlenecks of the hexagon approach. To understand this point, recall that the hexagon idea is to build the string vertex by attaching two hexagons together along the seams of the pair-of-pants diagram, as shown in figure 1. The picture gets more quantitative at weak coupling where the spin-chain description takes over [14]. Each seam is then identified with a bridge of planar contractions among the spin-chain sites and acquires a thickness or length. The hexagons fully decouple when the three bridge lengths in the problem (`A,B,C ) are asymptotically large, which means much larger than g at strong coupling. This requires in particular that all three operators carry extremely large charges and dimensions. For smaller `’s a sum over a complete basis of virtual excitations, which move across the seams, must be included. These excitations - dubbed mirror magnons - encode the 1 The method also applies to situations with integrable boundary as recently discussed in [23]. See also [24] for a TBA alternative to the hexagon method for structure constants involving determinant operators dual to giant gravitons in AdS5 × S 5 . –2– JHEP04(2020)076 correction to the 4pt functions of single-trace chiral primary operators of arbitrary dimensions ∼ g 0 , which generalises earlier results and proposals, see [11] and references therein. Further considerations unveiled hidden symmetries of the supergravity regime [7, 8] and yielded lots of new OPE data for double-trace operators at strong coupling [4–7, 12]. They suggest the exciting possibility that more general correlators can be found in the supergravity regime without ever using a single Witten diagram. On the other hand, in a different vein, the integrability technology, see [13] for a review, fostered the development of form-factor methods aiming at solving correlation functions, or scattering amplitudes, for any g in the large Nc limit [14–22]. Among these techniques, the hexagon method appears as the most versatile. Developed initially for the 3pt functions [17], it has been extended such as to cover higher-point functions [18, 19] and non-planar corrections [20, 21].1 The method passed all the tests at weak coupling, see [25–31] for recent examples, and has been checked at strong coupling as well, although to a lesser extent, in the semiclassical regime [32] corresponding to minimal surfaces in AdS5 × S 5 [33–38]. However, to date, the striking simplicity of the supergravity limit is still evading it. In this paper we take a step towards the low-energy regime and apply the hexagon method at strong coupling to 3pt functions of single-trace operators involving one light chiral primary operator, dual to a supergravity mode, and two heavy operators dual to highly boosted strings. The latter are the standard BMN operators, carrying large R charges and finite anomalous dimensions γ and mapping to states with finitely many magnons moving on a very large spin chain. For simplicity, we will take one of the two states to be BPS, corresponding to the spin-chain supersymmetric vacuum. The 3pt functions of interest are thus the familiar ones, with two BPS and one non-BPS single-trace operators, O1,2 and Oγ , of lengths L1,2 and L, respectively, O1 B B O2 C A A identify Oγ finite-size effects of the 3pt function geometry. Computing their sum is a difficult task in general and becomes unwieldy in the finite-length regime, which maps to a short-distance limit for the hexagon form factor series. It looks almost hopeless when the mirror magnons are given the freedom to move across many bridges. The HHL regim (...truncated)