Abstract
We study three-point correlation functions of local operators in planar \( \mathcal{N}=4 \) 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].
N. Gromov and P. Vieira, Tailoring three-point functions and integrability IV. Theta-morphism, JHEP 04 (2014) 068 [arXiv:1205.5288] [INSPIRE].
N. Gromov and P. Vieira, Quantum integrability for three-point functions of maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 111 (2013) 211601 [arXiv:1202.4103] [INSPIRE].
V.M. Braun, S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Baryon distribution amplitudes in QCD, Nucl. Phys. B 553 (1999) 355 [hep-ph/9902375] [INSPIRE].
M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 01 (2007) P01021 [hep-th/0610251] [INSPIRE].
I. Kostov, Classical limit of the three-point function of N = 4 supersymmetric Yang-Mills theory from integrability, Phys. Rev. Lett. 108 (2012) 261604 [arXiv:1203.6180] [INSPIRE].
I. Kostov, Three-point function of semiclassical states at weak coupling, J. Phys. A 45 (2012) 494018 [arXiv:1205.4412] [INSPIRE].
O. Foda, N = 4 SYM structure constants as determinants, JHEP 03 (2012) 096 [arXiv:1111.4663] [INSPIRE].
O. Foda and M. Wheeler, Partial domain wall partition functions, JHEP 07 (2012) 186 [arXiv:1205.4400] [INSPIRE].
V. Kazakov and E. Sobko, Three-point correlators of twist-2 operators in N = 4 SYM at Born approximation, JHEP 06 (2013) 061 [arXiv:1212.6563] [INSPIRE].
E. Sobko, A new representation for two- and three-point correlators of operators from sl(2) sector, arXiv:1311.6957 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability III. Classical tunneling, JHEP 07 (2012) 044 [arXiv:1111.2349] [INSPIRE].
L.F. Alday and A. Bissi, Higher-spin correlators, JHEP 10 (2013) 202 [arXiv:1305.4604] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
M.S. Costa, J. Drummond, V. Goncalves and J. Penedones, The role of leading twist operators in the Regge and Lorentzian OPE limits, JHEP 04 (2014) 094 [arXiv:1311.4886] [INSPIRE].
J. Caetano and T. Fleury, Three-point functions and SU(1|1) spin chains, arXiv:1404.4128 [INSPIRE].
G.P. Korchemsky, Bethe ansatz for QCD Pomeron, Nucl. Phys. B 443 (1995) 255 [hep-ph/9501232] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
G. Arutyunov, S. Penati, A. Santambrogio and E. Sokatchev, Four point correlators of BPS operators in N = 4 SYM at order g 4, Nucl. Phys. B 670 (2003) 103 [hep-th/0305060] [INSPIRE].
G. Arutyunov and E. Sokatchev, On a large-N degeneracy in N = 4 SYM and the AdS/CFT correspondence, Nucl. Phys. B 663 (2003) 163 [hep-th/0301058] [INSPIRE].
J. Plefka and K. Wiegandt, Three-point functions of twist-two operators in N = 4 SYM at one loop, JHEP 10 (2012) 177 [arXiv:1207.4784] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Hidden symmetry of four-point correlation functions and amplitudes in N = 4 SYM, Nucl. Phys. B 862 (2012) 193 [arXiv:1108.3557] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N = 4 SYM, Nucl. Phys. B 862 (2012) 450 [arXiv:1201.5329] [INSPIRE].
B. Eden, Three-loop universal structure constants in N = 4 SUSY Yang-Mills theory, arXiv:1207.3112 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial wave expansions for N = 4 chiral four point functions, Annals Phys. 321 (2006) 581 [hep-th/0412335] [INSPIRE].
D. Chicherin and E. Sokatchev, A note on four-point correlators of half-BPS operators in N = 4 SYM, arXiv:1408.3527 [INSPIRE].
B. Eden and M. Staudacher, Integrability and transcendentality, J. Stat. Mech. 11 (2006) P11014 [hep-th/0603157] [INSPIRE].
B.I. Zwiebel, Iterative structure of the N = 4 SYM spin chain, JHEP 07 (2008) 114 [arXiv:0806.1786] [INSPIRE].
N. Beisert and B.I. Zwiebel, On symmetry enhancement in the PSU(1, 1|2) sector of N = 4 SYM, JHEP 10 (2007) 031 [arXiv:0707.1031] [INSPIRE].
B. Sutherland, Beautiful models: 70 years of exactly solved quantum many-body problems, World Scientific Publishing Company, Singapore (2004).
A. Sever, P. Vieira and T. Wang, From polygon Wilson loops to spin chains and back, JHEP 12 (2012) 065 [arXiv:1208.0841] [INSPIRE].
K. Okuyama and L.-S. Tseng, Three-point functions in N = 4 SYM theory at one-loop, JHEP 08 (2004) 055 [hep-th/0404190] [INSPIRE].
R. Roiban and A. Volovich, Yang-Mills correlation functions from integrable spin chains, JHEP 09 (2004) 032 [hep-th/0407140] [INSPIRE].
L.F. Alday, J.R. David, E. Gava and K.S. Narain, Structure constants of planar N = 4 Yang-Mills at one loop, JHEP 09 (2005) 070 [hep-th/0502186] [INSPIRE].
S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Dual conformal symmetry on the light-cone, arXiv:1306.5951 [INSPIRE].
G.P. Korchemsky, to appear.
K.M. Watson, Some general relations between the photoproduction and scattering of pi mesons, Phys. Rev. 95 (1954) 228 [INSPIRE].
F.A. Smirnov, Form-factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) 1 [INSPIRE].
G. Mussardo, Off critical statistical models: factorized scattering theories and bootstrap program, Phys. Rept. 218 (1992) 215 [INSPIRE].
T. Klose and T. McLoughlin, Worldsheet form factors in AdS/CFT, Phys. Rev. D 87 (2013) 026004 [arXiv:1208.2020] [INSPIRE].
T. Klose and T. McLoughlin, Comments on world-sheet form factors in AdS/CFT, arXiv:1307.3506 [INSPIRE].
B. Basso, A. Sever and P. Vieira, Spacetime and flux tube S-matrices at finite coupling for N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N =4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].
R.A. Janik and A. Wereszczynski, Correlation functions of three heavy operators: the AdS contribution, JHEP 12 (2011) 095 [arXiv:1109.6262] [INSPIRE].
Y. Kazama and S. Komatsu, On holographic three point functions for GKP strings from integrability, JHEP 01 (2012) 110 [Erratum ibid. 06 (2012) 150] [arXiv:1110.3949] [INSPIRE].
Y. Kazama and S. Komatsu, Wave functions and correlation functions for GKP strings from integrability, JHEP 09 (2012) 022 [arXiv:1205.6060] [INSPIRE].
J. Caetano and J. Toledo, χ-systems for correlation functions, arXiv:1208.4548 [INSPIRE].
D. Serban and M. Staudacher, Planar N = 4 gauge theory and the Inozemtsev long range spin chain, JHEP 06 (2004) 001 [hep-th/0401057] [INSPIRE].
A. Rej, D. Serban and M. Staudacher, Planar N = 4 gauge theory and the Hubbard model, JHEP 03 (2006) 018 [hep-th/0512077] [INSPIRE].
T. Bargheer, N. Beisert and F. Loebbert, Long-range deformations for integrable spin chains, J. Phys. A 42 (2009) 285205 [arXiv:0902.0956] [INSPIRE].
L. Berdichevsky and P. Naaijkens, Four-point functions of different-weight operators in the AdS/CFT correspondence, JHEP 01 (2008) 071 [arXiv:0709.1365] [INSPIRE].
M. D’Alessandro and L. Genovese, A wide class of four point functions of BPS operators in N = 4 SYM at order g 4, Nucl. Phys. B 732 (2006) 64 [hep-th/0504061] [INSPIRE].
G. Arutyunov, F.A. Dolan, H. Osborn and E. Sokatchev, Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence, Nucl. Phys. B 665 (2003) 273 [hep-th/0212116] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1311.6404
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Vieira, P., Wang, T. Tailoring non-compact spin chains. J. High Energ. Phys. 2014, 35 (2014). https://doi.org/10.1007/JHEP10(2014)035
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)035