Abstract
We study three-point functions of single-trace operators in the \( \mathfrak{s}\mathfrak{u}\left(1\Big|1\right) \) sector of planar \( \mathcal{N}=4 \) SYM borrowing several tools based on Integrability. In the most general configuration of operators in this sector, we have found a determinant expression for the tree-level structure constants. We then compare the predictions of the recently proposed hexagon program against all available data. We have obtained a match once additional sign factors are included when the two hexagon form-factors are assembled together to form the structure constants. In the particular case of one BPS and two non-BPS operators we managed to identify the relevant form-factors with a domain wall partition function of a certain six-vertex model. This partition function can be explicitly evaluated and factorizes at all loops. In addition, we use this result to compute the structure constants and show that at strong coupling in the so-called BMN regime, its leading order contribution has a determinant expression.
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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].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar N = 4 SYM theory, arXiv:1505.06745 [INSPIRE].
B. Eden and A. Sfondrini, Three-point functions in N = 4 SYM: the hexagon proposal at three loops, JHEP 02 (2016) 165 [arXiv:1510.01242] [INSPIRE].
B. Basso, V. Goncalves, S. Komatsu and P. Vieira, Gluing hexagons at three loops, Nucl. Phys. B 907 (2016) 695 [arXiv:1510.01683] [INSPIRE].
Y. Jiang and A. Petrovskii, Diagonal form factors and hexagon form factors, JHEP 07 (2016) 120 [arXiv:1511.06199] [INSPIRE].
Y. Jiang, Diagonal form factors and hexagon form factors II. Non-BPS light operator, arXiv:1601.06926 [INSPIRE].
Y. Kazama, S. Komatsu and T. Nishimura, Classical integrability for three-point functions: cognate structure at weak and strong couplings, arXiv:1603.03164 [INSPIRE].
Y. Jiang, S. Komatsu, I. Kostov and D. Serban, Clustering and the three-point function, arXiv:1604.03575 [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].
G.M. Sotkov and R.P. Zaikov, On the structure of the conformal covariant N point functions, Rept. Math. Phys. 19 (1984) 335 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
J. Caetano and T. Fleury, Three-point functions and \( \mathfrak{s}\mathfrak{u}\left(1\Big|1\right) \) spin chains, JHEP 09 (2014) 173 [arXiv:1404.4128] [INSPIRE].
V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].
N.A. Slavnov, Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz, Theor. Math. Phys. 79 (1989) 502 [Teor. Mat. Fiz. 79 (1989) 232].
M. Wheeler, Scalar products in generalized models with SU(3)-symmetry, Commun. Math. Phys. 327 (2014) 737 [arXiv:1204.2089] [INSPIRE].
Y. Kazama, S. Komatsu and T. Nishimura, Novel construction and the monodromy relation for three-point functions at weak coupling, JHEP 01 (2015) 095 [Erratum ibid. 08 (2015) 145] [arXiv:1410.8533] [INSPIRE].
G. Arutyunov, S. Frolov and M. Zamaklar, The Zamolodchikov-Faddeev algebra for AdS 5 × S 5 superstring, JHEP 04 (2007) 002 [hep-th/0612229] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 01 (2007) P01021 [hep-th/0610251] [INSPIRE].
N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 948 [hep-th/0511082] [INSPIRE].
N. Beisert, The analytic Bethe ansatz for a chain with centrally extended \( \mathfrak{s}\mathfrak{u}\left(2\Big|2\right) \) symmetry, J. Stat. Mech. 01 (2007) P01017 [nlin/0610017] [INSPIRE].
A. Garbali, The domain wall partition for the Izergin-Korepin 19-vertex model at a root of unity, arXiv:1411.2903.
O. Foda, N = 4 SYM structure constants as determinants, JHEP 03 (2012) 096 [arXiv:1111.4663] [INSPIRE].
O. Foda, M. Wheeler and M. Zuparic, Factorized domain wall partition functions in trigonometric vertex models, J. Stat. Mech. 10 (2007) P10016 [arXiv:0709.4540] [INSPIRE].
O. Foda, M. Wheeler and M. Zuparic, Two elliptic height models with factorized domain wall partition functions, J. Stat. Mech. 02 (2008) P02001 [arXiv:0711.3058].
A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, É. Ragoucy and N.A. Slavnov, Scalar products of Bethe vectors in models with \( \mathfrak{g}\mathfrak{l}\left(2\Big|1\right) \) symmetry 1. Super-analog of Reshetikhin formula, arXiv:1605.09189 [INSPIRE].
A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, É. Ragoucy and N.A. Slavnov, Scalar products of Bethe vectors in models with \( \mathfrak{g}\mathfrak{l}\left(2\Big|1\right) \) symmetry 2. Determinant representation, arXiv:1606.03573 [INSPIRE].
G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].
P. Vieira and D. Volin, Review of AdS/CFT integrability, chapter III.3: the dressing factor, Lett. Math. Phys. 99 (2012) 231 [arXiv:1012.3992] [INSPIRE].
G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].
R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].
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Caetano, J., Fleury, T. Fermionic correlators from integrability. J. High Energ. Phys. 2016, 10 (2016). https://doi.org/10.1007/JHEP09(2016)010
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DOI: https://doi.org/10.1007/JHEP09(2016)010