Abstract
We study non-perturbative interpolating functions to probe the physics of anomalous dimensions associated with twist-two operators in \( \mathcal{N}=4 \) SYM of finite and infinite spin. Compared to previous studies, the novel result of this paper is to introduce single multivariate functions of both coupling g and spin j to approximate such anomalous dimensions. We provide a unified framework to study such operators in interim ranges of the parameters which so far has eluded previous results. Explicitly, we consider twist-two anomalous dimensions in two distinct scenarios using interpolating functions. For the large N case, we stick to simple Padé approximants and its generalizations. For the finite N case, \( \mathcal{N}=4 \) SYM is expected to be S-dual invariant, hence the observables are expected be modular invariant. To probe the finite N physics, we take into account the non-planar and instanton contributions by constructing modular invariant interpolating functions to approximate the cusp and twist-two anomalous dimensions. We also consider interpolating functions for the twist-four operators and study level crossing phenomenon between the twist-two and twist-four operators.
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References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett.B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys.B 72 (1974) 461 [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys.99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod. Phys. Lett.A 4 (1989) 1257 [INSPIRE].
A.M. Polyakov, Gauge Fields as Rings of Glue, Nucl. Phys.B 164 (1980) 171 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Loop Space Formalism and Renormalization Group for the Infrared Asymptotics of QCD, Phys. Lett.B 171 (1986) 459 [INSPIRE].
A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory, Nucl. Phys.B 661 (2003) 19 [Erratum ibid.B 685 (2004) 405] [hep-ph/0208220] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett.B 595 (2004) 521 [Erratum ibid.B 632 (2006) 754] [hep-th/0404092] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The Three loop splitting functions in QCD: The Nonsinglet case, Nucl. Phys.B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in N = 4 SYM, JHEP05 (2012) 082 [arXiv:1201.4170] [INSPIRE].
A. Brandhuber, M. Kostacinska, B. Penante and G. Travaglini, Higgs amplitudes from \( \mathcal{N}=4 \)super Yang-Mills theory, Phys. Rev. Lett.119 (2017) 161601 [arXiv:1707.09897] [INSPIRE].
Q. Jin and G. Yang, Analytic Two-Loop Higgs Amplitudes in Effective Field Theory and the Maximal Transcendentality Principle, Phys. Rev. Lett.121 (2018) 101603 [arXiv:1804.04653] [INSPIRE].
Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, Soft-virtual corrections to Higgs production at N 3LO, Phys. Rev.D 91 (2015) 036008 [arXiv:1412.2771] [INSPIRE].
Y. Li and H.X. Zhu, Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum Resummation, Phys. Rev. Lett.118 (2017) 022004 [arXiv:1604.01404] [INSPIRE].
A. Sen, S-duality Improved Superstring Perturbation Theory, JHEP11 (2013) 029 [arXiv:1304.0458] [INSPIRE].
C. Beem, L. Rastelli, A. Sen and B.C. van Rees, Resummation and S-duality in N = 4 SYM, JHEP04 (2014) 122 [arXiv:1306.3228] [INSPIRE].
L.F. Alday and A. Bissi, Modular interpolating functions for N = 4 SYM, JHEP07 (2014) 007 [arXiv:1311.3215] [INSPIRE].
M. Honda, On Perturbation theory improved by Strong coupling expansion, JHEP12 (2014) 019 [arXiv:1408.2960] [INSPIRE].
M. Honda and D.P. Jatkar, Interpolating function and Stokes Phenomena, Nucl. Phys.B 900 (2015) 533 [arXiv:1504.02276] [INSPIRE].
A. Chowdhury, M. Honda and S. Thakur, S-duality invariant perturbation theory improved by holography, JHEP04 (2017) 137 [arXiv:1607.01716] [INSPIRE].
H. Osborn, Topological Charges for N = 4 Supersymmetric Gauge Theories and Monopoles of Spin 1, Phys. Lett.83B (1979) 321 [INSPIRE].
G.P. Korchemsky, On level crossing in conformal field theories, JHEP03 (2016) 212 [arXiv:1512.05362] [INSPIRE].
H. Kleinert and V. Schulte-Frohlinde, Critical properties of ϕ 4-theories, World Scientific (2001).
R. Pius and A. Sen, S-duality improved perturbation theory in compactified type-I/heterotic string theory, JHEP06 (2014) 068 [arXiv:1310.4593] [INSPIRE].
O. Klevang, Automorphic Forms in String Theory, MSc Thesis, Chalmers University of Technology, Goteborg, Sweden (2010).
V. Gonçalves, Four point function of \( \mathcal{N}=4 \)stress-tensor multiplet at strong coupling, JHEP04 (2015) 150 [arXiv:1411.1675] [INSPIRE].
L.F. Alday, A. Bissi and E. Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP06 (2019) 010 [arXiv:1809.10670] [INSPIRE].
Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev.D 72 (2005) 085001 [hep-th/0505205] [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].
M. Kruczenski, A Note on twist two operators in N = 4 SYM and Wilson loops in Minkowski signature, JHEP12 (2002) 024 [hep-th/0210115] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech.0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev.D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
F. Cachazo, M. Spradlin and A. Volovich, Four-loop cusp anomalous dimension from obstructions, Phys. Rev.D 75 (2007) 105011 [hep-th/0612309] [INSPIRE].
J.M. Henn and T. Huber, The four-loop cusp anomalous dimension in \( \mathcal{N}=4 \)super Yang-Mills and analytic integration techniques for Wilson line integrals, JHEP09 (2013) 147 [arXiv:1304.6418] [INSPIRE].
R. Roiban and A.A. Tseytlin, Strong-coupling expansion of cusp anomaly from quantum superstring, JHEP11 (2007) 016 [arXiv:0709.0681] [INSPIRE].
R. Roiban and A.A. Tseytlin, Spinning superstrings at two loops: Strong-coupling corrections to dimensions of large-twist SYM operators, Phys. Rev.D 77 (2008) 066006 [arXiv:0712.2479] [INSPIRE].
R.H. Boels, T. Huber and G. Yang, Four-Loop Nonplanar Cusp Anomalous Dimension in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett.119 (2017) 201601 [arXiv:1705.03444] [INSPIRE].
R.H. Boels, T. Huber and G. Yang, The Sudakov form factor at four loops in maximal super Yang-Mills theory, JHEP01 (2018) 153 [arXiv:1711.08449] [INSPIRE].
B. Basso, G.P. Korchemsky and J. Kotanski, Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. Lett.100 (2008) 091601 [arXiv:0708.3933] [INSPIRE].
G.P. Korchemsky, Instanton effects in correlation functions on the light-cone, JHEP12 (2017) 093 [arXiv:1704.00448] [INSPIRE].
L.F. Alday and G.P. Korchemsky, Revisiting instanton corrections to the Konishi multiplet, JHEP12 (2016) 005 [arXiv:1605.06346] [INSPIRE].
N. Dorey, V.V. Khoze, M.P. Mattis and S. Vandoren, Yang-Mills instantons in the large N limit and the AdS/CFT correspondence, Phys. Lett.B 442 (1998) 145 [hep-th/9808157] [INSPIRE].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, J. Phys.A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N}=4 \)Super-Yang-Mills Theory, Phys. Rev. Lett.112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk, G. Sizov and S. Valatka, Quantum spectral curve at work: from small spin to strong coupling in \( \mathcal{N}= 4 \)SYM, JHEP07 (2014) 156 [arXiv:1402.0871] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS 5/CFT 4, JHEP09 (2015) 187 [arXiv:1405.4857] [INSPIRE].
N. Gromov, Introduction to the Spectrum of N = 4 SYM and the Quantum Spectral Curve, arXiv:1708.03648 [INSPIRE].
A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory, in 35th Annual Winter School on Nuclear and Particle Physics, Repino, Russia, February 19-25, 2001 (2001) [hep-ph/0112346] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A. Rej, M. Staudacher and V.N. Velizhanin, Dressing and wrapping, J. Stat. Mech.0710 (2007) P10003 [arXiv:0704.3586] [INSPIRE].
A.V. Kotikov and L.N. Lipatov, NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories, Nucl. Phys.B 582 (2000) 19 [hep-ph/0004008] [INSPIRE].
A.V. Kotikov, L.N. Lipatov and V.N. Velizhanin, Anomalous dimensions of Wilson operators in N = 4 SYM theory, Phys. Lett.B 557 (2003) 114 [hep-ph/0301021] [INSPIRE].
T. Lukowski, A. Rej and V.N. Velizhanin, Five-Loop Anomalous Dimension of Twist-Two Operators, Nucl. Phys.B 831 (2010) 105 [arXiv:0912.1624] [INSPIRE].
V.N. Velizhanin, Six-Loop Anomalous Dimension of Twist-Three Operators in N = 4 SYM, JHEP11 (2010) 129 [arXiv:1003.4717] [INSPIRE].
C. Marboe, V. Velizhanin and D. Volin, Six-loop anomalous dimension of twist-two operators in planar \( \mathcal{N}=4 \)SYM theory, JHEP07 (2015) 084 [arXiv:1412.4762] [INSPIRE].
C. Marboe and V. Velizhanin, Twist-2 at seven loops in planar \( \mathcal{N}=4 \)SYM theory: full result and analytic properties, JHEP11 (2016) 013 [arXiv:1607.06047] [INSPIRE].
V.N. Velizhanin, The Non-planar contribution to the four-loop universal anomalous dimension in N = 4 Supersymmetric Yang-Mills theory, JETP Lett.89 (2009) 593 [arXiv:0902.4646] [INSPIRE].
V.N. Velizhanin, The Non-planar contribution to the four-loop anomalous dimension of twist-2 operators: First moments in N = 4 SYM and non-singlet QCD, Nucl. Phys.B 846 (2011) 137 [arXiv:1008.2752] [INSPIRE].
V.N. Velizhanin, Non-planar anomalous dimension of twist-2 operators: higher moments at four loops, Nucl. Phys.B 885 (2014) 772 [arXiv:1404.7107] [INSPIRE].
B. Basso, An exact slope for AdS/CFT, arXiv:1109.3154 [INSPIRE].
N. Gromov, D. Serban, I. Shenderovich and D. Volin, Quantum folded string and integrability: From finite size effects to Konishi dimension, JHEP08 (2011) 046 [arXiv:1102.1040] [INSPIRE].
R. Roiban and A.A. Tseytlin, Semiclassical string computation of strong-coupling corrections to dimensions of operators in Konishi multiplet, Nucl. Phys.B 848 (2011) 251 [arXiv:1102.1209] [INSPIRE].
B.C. Vallilo and L. Mazzucato, The Konishi multiplet at strong coupling, JHEP12 (2011) 029 [arXiv:1102.1219] [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].
E. D’Hoker, S.D. Mathur, A. Matusis and L. Rastelli, The Operator product expansion of N =4 SYM and the 4 point functions of supergravity, Nucl. Phys.B 589 (2000) 38 [hep-th/9911222] [INSPIRE].
G. Arutyunov, S. Frolov and A.C. Petkou, Operator product expansion of the lowest weight CPOs in \( \mathcal{N}=4 \)SY M 4at strong coupling, Nucl. Phys.B 586 (2000) 547 [Erratum ibid.B 609 (2001) 539] [hep-th/0005182] [INSPIRE].
L.F. Alday and A. Bissi, Loop Corrections to Supergravity on AdS 5 × S 5, Phys. Rev. Lett.119 (2017) 171601 [arXiv:1706.02388] [INSPIRE].
F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Quantum Gravity from Conformal Field Theory, JHEP01 (2018) 035 [arXiv:1706.02822] [INSPIRE].
T. Banks and T.J. Torres, Two Point Pade Approximants and Duality, arXiv:1307.3689 [INSPIRE].
N. Gromov and S. Valatka, Deeper Look into Short Strings, JHEP03 (2012) 058 [arXiv:1109.6305] [INSPIRE].
A. Tirziu and A.A. Tseytlin, Quantum corrections to energy of short spinning string in AdS 5, Phys. Rev.D 78 (2008) 066002 [arXiv:0806.4758] [INSPIRE].
R. Roiban and A.A. Tseytlin, Quantum strings in AdS 5 × S 5: Strong-coupling corrections to dimension of Konishi operator, JHEP11 (2009) 013 [arXiv:0906.4294] [INSPIRE].
E. Floratos, G. Georgiou and G. Linardopoulos, Large-Spin Expansions of GKP Strings, JHEP03 (2014) 018 [arXiv:1311.5800] [INSPIRE].
N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP02 (2005) 010 [hep-th/0501109] [INSPIRE].
B. Chen and W. He, On 1/2-BPS Wilson-’t Hooft loops, Phys. Rev.D 74 (2006) 126008 [hep-th/0607024] [INSPIRE].
R.H. Boels, B.A. Kniehl, O.V. Tarasov and G. Yang, Color-kinematic Duality for Form Factors, JHEP02 (2013) 063 [arXiv:1211.7028] [INSPIRE].
L.F. Alday and G.P. Korchemsky, Instanton corrections to twist-two operators, JHEP06 (2017) 008 [arXiv:1609.08164] [INSPIRE].
N. Beisert, C. Kristjansen and M. Staudacher, The Dilatation operator of conformal N = 4 superYang-Mills theory, Nucl. Phys.B 664 (2003) 131 [hep-th/0303060] [INSPIRE].
G. Arutyunov, S. Penati, A.C. Petkou, A. Santambrogio and E. Sokatchev, Nonprotected operators in N = 4 SYM and multiparticle states of AdS 5SUGRA, Nucl. Phys.B 643 (2002) 49 [hep-th/0206020] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=4 \)Superconformal Bootstrap, Phys. Rev. Lett.111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
L.F. Alday and A. Bissi, The superconformal bootstrap for structure constants, JHEP09 (2014) 144 [arXiv:1310.3757] [INSPIRE].
L.F. Alday and A. Bissi, Generalized bootstrap equations for \( \mathcal{N}=4 \)SCFT, JHEP02 (2015) 101 [arXiv:1404.5864] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Lessons from crossing symmetry at large N, JHEP06 (2015) 074 [arXiv:1410.4717] [INSPIRE].
B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, Partial nonrenormalization of the stress tensor four point function in N = 4 SYM and AdS/CFT, Nucl. Phys.B 607 (2001) 191 [hep-th/0009106] [INSPIRE].
G. Arutyunov, B. Eden, A.C. Petkou and E. Sokatchev, Exceptional nonrenormalization properties and OPE analysis of chiral four point functions in N = 4 SY M 4, Nucl. Phys.B 620 (2002) 380 [hep-th/0103230] [INSPIRE].
B. Eden and E. Sokatchev, On the OPE of 1/2 BPS short operators in N = 4 SCF T 4, Nucl. Phys.B 618 (2001) 259 [hep-th/0106249] [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP11 (2007) 019 [arXiv:0708.0672] [INSPIRE].
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Banerjee, A., Chowdhury, A., Thakur, S. et al. On interpolating anomalous dimension of twist-two operators with general spins. J. High Energ. Phys. 2019, 86 (2019). https://doi.org/10.1007/JHEP07(2019)086
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DOI: https://doi.org/10.1007/JHEP07(2019)086