Abstract
Given the recent progress in computing three-point functions in \( \mathcal{N} \) = 4 SYM via integrability, I provide here a novel direct calculation of some structure constants at weak coupling. The main focus is on correlators involving more than one unprotected operator, at two-loop order in the perturbative expansion.
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].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP 10 (2017) 098 [arXiv:1611.05436] [INSPIRE].
B. Basso, F. Coronado, S. Komatsu, H.T. Lam, P. Vieira and D.-l. Zhong, Asymptotic Four Point Functions, arXiv:1701.04462 [INSPIRE].
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling Handles: Nonplanar Integrability in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 121 (2018) 231602 [arXiv:1711.05326] [INSPIRE].
B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, Colour-dressed hexagon tessellations for correlation functions and non-planar corrections, JHEP 02 (2018) 170 [arXiv:1710.10212] [INSPIRE].
N. Beisert, C. Kristjansen, J. Plefka, G.W. Semenoff and M. Staudacher, BMN correlators and operator mixing in N = 4 superYang-Mills theory, Nucl. Phys. B 650 (2003) 125 [hep-th/0208178] [INSPIRE].
C.-S. Chu, V.V. Khoze and G. Travaglini, Three point functions in N = 4 Yang-Mills theory and pp waves, JHEP 06 (2002) 011 [hep-th/0206005] [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].
L.F. Alday, J.R. David, E. Gava and K.S. Narain, Towards a string bit formulation of N = 4 super Yang-Mills, JHEP 04 (2006) 014 [hep-th/0510264] [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].
G. Georgiou, V.L. Gili and R. Russo, Operator Mixing and the AdS/CFT correspondence, JHEP 01 (2009) 082 [arXiv:0810.0499] [INSPIRE].
A. Grossardt and J. Plefka, One-Loop Spectroscopy of Scalar Three-Point Functions in planar N = 4 super Yang-Mills Theory, arXiv:1007.2356 [INSPIRE].
G. Georgiou, V. Gili, A. Grossardt and J. Plefka, Three-point functions in planar N = 4 super Yang-Mills Theory for scalar operators up to length five at the one-loop order, JHEP 04 (2012) 038 [arXiv:1201.0992] [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].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring Three-Point Functions and Integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring Three-Point Functions and Integrability II. Weak/strong coupling match, JHEP 09 (2011) 029 [arXiv:1104.5501] [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].
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].
N. Gromov and P. Vieira, Tailoring Three-Point Functions and Integrability IV. Theta-morphism, JHEP 04 (2014) 068 [arXiv:1205.5288] [INSPIRE].
P. Vieira and T. Wang, Tailoring Non-Compact Spin Chains, JHEP 10 (2014) 35 [arXiv:1311.6404] [INSPIRE].
B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Four point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Nucl. Phys. B 557 (1999) 355 [hep-th/9811172] [INSPIRE].
B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Simplifications of four point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Phys. Lett. B 466 (1999) 20 [hep-th/9906051] [INSPIRE].
B. Eden, C. Schubert and E. Sokatchev, Three loop four point correlator in N = 4 SYM, Phys. Lett. B 482 (2000) 309 [hep-th/0003096] [INSPIRE].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, On the logarithmic behavior in N = 4 SYM theory, JHEP 08 (1999) 020 [hep-th/9906188] [INSPIRE].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Anomalous dimensions in N = 4 SYM theory at order g 4, Nucl. Phys. B 584 (2000) 216 [hep-th/0003203] [INSPIRE].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Properties of the Konishi multiplet in N = 4 SYM theory, JHEP 05 (2001) 042 [hep-th/0104016] [INSPIRE].
F.A. Dolan and H. Osborn, Superconformal symmetry, correlation functions and the operator product expansion, Nucl. Phys. B 629 (2002) 3 [hep-th/0112251] [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].
D. Chicherin, J. Drummond, P. Heslop and E. Sokatchev, All three-loop four-point correlators of half-BPS operators in planar \( \mathcal{N} \) = 4 SYM, JHEP 08 (2016) 053 [arXiv:1512.02926] [INSPIRE].
D. Chicherin, A. Georgoudis, V. Gonçalves and R. Pereira, All five-loop planar four-point functions of half-BPS operators in \( \mathcal{N} \) = 4 SYM, JHEP 11 (2018) 069 [arXiv:1809.00551] [INSPIRE].
B. Eden, Three-loop universal structure constants in N = 4 SUSY Yang-Mills theory, arXiv:1207.3112 [INSPIRE].
B. Eden and A. Sfondrini, Three-point functions in \( \mathcal{N} \) = 4 SYM: the hexagon proposal at three loops, JHEP 02 (2016) 165 [arXiv:1510.01242] [INSPIRE].
B. Basso, V. Gonçalves, S. Komatsu and P. Vieira, Gluing Hexagons at Three Loops, Nucl. Phys. B 907 (2016) 695 [arXiv:1510.01683] [INSPIRE].
B. Eden and F. Paul, Half-BPS half-BPS twist two at four loops in N = 4 SYM, arXiv:1608.04222 [INSPIRE].
V. Gonçalves, Extracting OPE coefficient of Konishi at four loops, JHEP 03 (2017) 079 [arXiv:1607.02195] [INSPIRE].
B. Basso, V. Gonçalves and S. Komatsu, Structure constants at wrapping order, JHEP 05 (2017) 124 [arXiv:1702.02154] [INSPIRE].
A. Georgoudis, V. Gonçalves and R. Pereira, Konishi OPE coefficient at the five loop order, JHEP 11 (2018) 184 [arXiv:1710.06419] [INSPIRE].
A.V. Belitsky, J. Henn, C. Jarczak, D. Mueller and E. Sokatchev, Anomalous dimensions of leading twist conformal operators, Phys. Rev. D 77 (2008) 045029 [arXiv:0707.2936] [INSPIRE].
G.M. Sotkov and R.P. Zaikov, Conformal Invariant Two Point and Three Point Functions for Fields with Arbitrary Spin, Rept. Math. Phys. 12 (1977) 375 [INSPIRE].
D. Young, ABJ(M) Chiral Primary Three-Point Function at Two-loops, JHEP 07 (2014) 120 [arXiv:1404.1117] [INSPIRE].
D. Young, An Extremal Chiral Primary Three-Point Function at Two-loops in ABJ(M), JHEP 12 (2014) 141 [arXiv:1411.0626] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. 100B (1981) 65 [INSPIRE].
S. Laporta and E. Remiddi, The analytical value of the electron (g − 2) at order α 3 in QED, Phys. Lett. B 379 (1996) 283 [hep-ph/9602417] [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A.V. Smirnov and V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial × Space Technique, Nucl. Phys. B 174 (1980) 345 [INSPIRE].
W. Siegel, Supersymmetric Dimensional Regularization via Dimensional Reduction, Phys. Lett. 84B (1979) 193 [INSPIRE].
L.V. Avdeev, O.V. Tarasov and A.A. Vladimirov, Vanishing of the three loop charge renormalization function in a supersymmetric gauge theory, Phys. Lett. 96B (1980) 94 [INSPIRE].
L.V. Avdeev, G.A. Chochia and A.A. Vladimirov, On the Scope of Supersymmetric Dimensional Regularization, Phys. Lett. 105B (1981) 272 [INSPIRE].
L.V. Avdeev and O.V. Tarasov, The Three Loop β-function in the N = 1, N = 2, N = 4 Supersymmetric Yang-Mills Theories, Phys. Lett. 112B (1982) 356 [INSPIRE].
V.N. Velizhanin, Three-loop renormalization of the N = 1, N = 2, N = 4 supersymmetric Yang-Mills theories, Nucl. Phys. B 818 (2009) 95 [arXiv:0809.2509] [INSPIRE].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].
B. Eden, P.S. Howe and P.C. West, Nilpotent invariants in N = 4 SYM, Phys. Lett. B 463 (1999) 19 [hep-th/9905085] [INSPIRE].
G. Arutyunov, B. Eden and E. Sokatchev, On nonrenormalization and OPE in superconformal field theories, Nucl. Phys. B 619 (2001) 359 [hep-th/0105254] [INSPIRE].
P.J. Heslop and P.S. Howe, OPEs and three-point correlators of protected operators in N = 4 SYM, Nucl. Phys. B 626 (2002) 265 [hep-th/0107212] [INSPIRE].
M. Bianchi, B. Eden, G. Rossi and Y.S. Stanev, On operator mixing in N = 4 SYM, Nucl. Phys. B 646 (2002) 69 [hep-th/0205321] [INSPIRE].
S. Penati, A. Santambrogio and D. Zanon, Two point functions of chiral operators in N = 4 SYM at order g 4, JHEP 12 (1999) 006 [hep-th/9910197] [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: 1809.04376
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bianchi, M.S. A note on three-point functions of unprotected operators. J. High Energ. Phys. 2019, 154 (2019). https://doi.org/10.1007/JHEP03(2019)154
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2019)154