Abstract
We show that the Quantum Spectral Curve (QSC) formalism, initially formulated for the spectrum of anomalous dimensions of all local single trace operators in \( \mathcal{N}=4 \) SYM, can be extended to the generalized cusp anomalous dimension for all values of the parameters. We find that the large spectral parameter asymptotics and some analyticity properties have to be modified, but the functional relations are unchanged. As a demonstration, we find an all-loop analytic expression for the first two nontrivial terms in the small |ϕ ± θ| expansion. We also present nonperturbative numerical results at generic angles which match perfectly 4-loop perturbation theory and the classical string prediction.
The reformulation of the problem in terms of the QSC opens the possibility to explore many open questions. We attach to this paper several Mathematica notebooks which should facilitate future studies.
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].
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, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS 5 /CFT 4, JHEP 09 (2015) 187 [arXiv:1405.4857] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: A proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].
N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].
G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5 × S 5 Mirror Model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].
A. Cavaglia, D. Fioravanti and R. Tateo, Extended Y-system for the AdS 5 /CFT 4 correspondence, Nucl. Phys. B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE].
C. Marboe and D. Volin, Quantum spectral curve as a tool for a perturbative quantum field theory, Nucl. Phys. B 899 (2015) 810 [arXiv:1411.4758] [INSPIRE].
C. Marboe, V. Velizhanin and D. Volin, Six-loop anomalous dimension of twist-two operators in planar N = 4 SYM theory, JHEP 07 (2015) 084 [arXiv:1412.4762] [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, JHEP 07 (2014) 156 [arXiv:1402.0871] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4, arXiv:1504.06640 [INSPIRE].
M. Alfimov, N. Gromov and V. Kazakov, QCD Pomeron from AdS/CFT Quantum Spectral Curve, JHEP 07 (2015) 164 [arXiv:1408.2530] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Pomeron Eigenvalue at Three Loops in \( \mathcal{N}=4 \) Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 115 (2015) 251601 [arXiv:1507.04010] [INSPIRE].
A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum Spectral Curve of the \( \mathcal{N}=6 \) Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].
L. Anselmetti, D. Bombardelli, A. Cavaglià and R. Tateo, 12 loops and triple wrapping in ABJM theory from integrability, JHEP 10 (2015) 117 [arXiv:1506.09089] [INSPIRE].
N. Gromov and G. Sizov, Exact Slope and Interpolating Functions in N = 6 Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed N = 4 super-Yang-Mills, J. Phys. A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, Twisting the Mirror TBA, JHEP 02 (2011) 025 [arXiv:1009.4118] [INSPIRE].
M. de Leeuw and S.J. van Tongeren, The spectral problem for strings on twisted AdS 5 × S 5, Nucl. Phys. B 860 (2012) 339 [arXiv:1201.1451] [INSPIRE].
M. Kim, Spectral curve for γ-deformed AdS/CFT, Phys. Lett. B 735 (2014) 332 [arXiv:1401.4032] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The Quantum Deformed Mirror TBA I, JHEP 10 (2012) 090 [arXiv:1208.3478] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The Quantum Deformed Mirror TBA II, JHEP 02 (2013) 012 [arXiv:1210.8185] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS 5 × S 5) η superstring, Theor. Math. Phys. 182 (2015) 23 [arXiv:1403.6104] [INSPIRE].
Z. Bajnok et al., The spectrum of tachyons in AdS/CFT, JHEP 03 (2014) 055 [arXiv:1312.3900] [INSPIRE].
Á. Hegedüs, Extensive numerical study of a D-brane, \( \overline{D} \) -brane system in AdS5/CFT4, JHEP 04 (2015) 107 [arXiv:1501.07412] [INSPIRE].
Z. Bajnok, M. Kim and L. Palla, Spectral curve for open strings attached to the Y = 0 brane, JHEP 04 (2014) 035 [arXiv:1311.7280] [INSPIRE].
Z. Bajnok, R.I. Nepomechie, L. Palla and R. Suzuki, Y-system for Y = 0 brane in planar AdS/CFT, JHEP 08 (2012) 149 [arXiv:1205.2060] [INSPIRE].
K. Zoubos, Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].
D.H. Correa, V. Regelskis and C.A.S. Young, Integrable achiral D5-brane reflections and asymptotic Bethe equations, J. Phys. A 44 (2011) 325403 [arXiv:1105.3707] [INSPIRE].
N. MacKay and V. Regelskis, Achiral boundaries and the twisted Yangian of the D5-brane, JHEP 08 (2011) 019 [arXiv:1105.4128] [INSPIRE].
S. Caron-Huot and J.M. Henn, Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 113 (2014) 161601 [arXiv:1408.0296] [INSPIRE].
D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].
N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].
B. Fiol, B. Garolera and A. Lewkowycz, Exact results for static and radiative fields of a quark in N = 4 super Yang-Mills, JHEP 05 (2012) 093 [arXiv:1202.5292] [INSPIRE].
N. Gromov and A. Sever, Analytic Solution of Bremsstrahlung TBA, JHEP 11 (2012) 075 [arXiv:1207.5489] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic Solution of Bremsstrahlung TBA II: Turning on the Sphere Angle, JHEP 10 (2013) 036 [arXiv:1305.1944] [INSPIRE].
G. Sizov and S. Valatka, Algebraic Curve for a Cusped Wilson Line, JHEP 05 (2014) 149 [arXiv:1306.2527] [INSPIRE].
M. Beccaria and G. Macorini, On a discrete symmetry of the Bremsstrahlung function in N = 4 SYM, JHEP 07 (2013) 104 [arXiv:1305.4839] [INSPIRE].
A. Dekel, Algebraic Curves for Factorized String Solutions, JHEP 04 (2013) 119 [arXiv:1302.0555] [INSPIRE].
R.A. Janik and P. Laskos-Grabowski, Surprises in the AdS algebraic curve constructions: Wilson loops and correlation functions, Nucl. Phys. B 861 (2012) 361 [arXiv:1203.4246] [INSPIRE].
Z. Bajnok, J. Balog, D.H. Correa, Á. Hegedüs, F.I. Schaposnik Massolo and G. Zsolt Tóth, Reformulating the TBA equations for the quark anti-quark potential and their two loop expansion, JHEP 03 (2014) 056 [arXiv:1312.4258] [INSPIRE].
Y. Makeenko, P. Olesen and G.W. Semenoff, Cusped SYM Wilson loop at two loops and beyond, Nucl. Phys. B 748 (2006) 170 [hep-th/0602100] [INSPIRE].
N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, The cusp anomalous dimension at three loops and beyond, JHEP 05 (2012) 098 [arXiv:1203.1019] [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, JHEP 09 (2013) 147 [arXiv:1304.6418] [INSPIRE].
N. Gromov and P. Vieira, The AdS 5 × S 5 superstring quantum spectrum from the algebraic curve, Nucl. Phys. B 789 (2008) 175 [hep-th/0703191] [INSPIRE].
S. Leurent and D. Volin, Multiple zeta functions and double wrapping in planar N = 4 SYM, Nucl. Phys. B 875 (2013) 757 [arXiv:1302.1135] [INSPIRE].
D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].
J. Ablinger, A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics, arXiv:1011.1176 [INSPIRE].
J. Ablinger, Computer Algebra Algorithms for Special Functions in Particle Physics, Ph.D. Thesis, Johannes Kepler University, Linz, Austria (2012).
J. Ablinger, J. Blümlein and C. Schneider, Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms, J. Math. Phys. 54 (2013) 082301 [arXiv:1302.0378] [INSPIRE].
J. Ablinger, J. Blumlein and C. Schneider, Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials, J. Math. Phys. 52 (2011) 102301 [arXiv:1105.6063] [INSPIRE].
J. Blumlein, Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5, Comput. Phys. Commun. 180 (2009) 2218 [arXiv:0901.3106] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].
V. Kazakov, S. Leurent and D. Volin, T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve, arXiv:1510.02100 [INSPIRE].
D. Bykov and K. Zarembo, Ladders for Wilson Loops Beyond Leading Order, JHEP 09 (2012) 057 [arXiv:1206.7117] [INSPIRE].
J.M. Henn and T. Huber, Systematics of the cusp anomalous dimension, JHEP 11 (2012) 058 [arXiv:1207.2161] [INSPIRE].
V. Forini, Quark-antiquark potential in AdS at one loop, JHEP 11 (2010) 079 [arXiv:1009.3939] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Quark-anti-quark potential in N = 4 SYM, arXiv:1601.05679 [INSPIRE].
C. Duhr, Mathematical aspects of scattering amplitudes, arXiv:1411.7538 [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: 1510.02098
Electronic supplementary material
Below is the link to the electronic supplementary material.
ESM 1
Numerical data for the cusp anomalous dimension. Description: This Mathematica notebook contains numerical data for the generalized cusp anomalous dimension, which is shown in the paper in Table 1 (where precision is lowered to fit the page) and on Figure 3. (NB 22 kb)
ESM 2
Weak coupling expansion of the near-BPS result. Description: This Mathematica notebook contains the tools needed to generate the weak coupling expansion of our all-loop result (3.79) for the anomalous dimension at order (phi-theta)^2 in the near-BPS limit. (NB 94 kb)
ESM 3
Tools for weak coupling solution. Description: This Mathematica notebook contains tools which should be useful for a perturbative weak-coupling solution of the Quantum Spectral Curve. In particular it implements various operations on generalized eta-functions (see Appendix F). (M 5 kb)
ESM 4
Usage examples for several tools from the file “TwistTools.m”. Description: Demonstrates usage of various functions from TwistTools.m. (NB 23 kb)
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
Gromov, N., Levkovich-Maslyuk, F. Quantum Spectral Curve for a cusped Wilson line in \( \mathcal{N}=4 \) SYM. J. High Energ. Phys. 2016, 134 (2016). https://doi.org/10.1007/JHEP04(2016)134
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2016)134