Abstract
We compute the general form of the six-loop anomalous dimension of twist-two operators with arbitrary spin in planar \( \mathcal{N}=4 \) SYM theory. First we find the contribution from the asymptotic Bethe ansatz. Then we reconstruct the wrapping terms from the first 35 even spin values of the full six-loop anomalous dimension computed using the quantum spectral curve approach. The obtained anomalous dimension satisfies all known constraints coming from the BFKL equation, the generalised double-logarithmic equation, and the small spin expansion.
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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].
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].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
L.N. Lipatov, High-energy asymptotics of multicolor QCD and exactly solvable lattice models, hep-th/9311037 [INSPIRE].
L.N. Lipatov, Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models, JETP Lett. 59 (1994) 596 [INSPIRE].
L.D. Faddeev and G.P. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311 [hep-th/9404173] [INSPIRE].
V.M. Braun, S.E. Derkachov and A.N. Manashov, Integrability of three particle evolution equations in QCD, Phys. Rev. Lett. 81 (1998) 2020 [hep-ph/9805225] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS5 × S5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
N. Beisert, C. Kristjansen and M. Staudacher, The Dilatation operator of conformal N = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [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, The complete one loop dilatation operator of N = 4 super Yang-Mills theory, Nucl. Phys. B 676 (2004) 3 [hep-th/0307015] [INSPIRE].
N. Beisert, The SU(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487 [hep-th/0310252] [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].
V.A. Kazakov, A. Marshakov, J.A. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [INSPIRE].
N. Beisert, V. Dippel and M. Staudacher, A novel long range spin chain and planar N = 4 super Yang-Mills, JHEP 07 (2004) 075 [hep-th/0405001] [INSPIRE].
G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].
M. Staudacher, The Factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].
N. Beisert and M. Staudacher, Long-range P SU (2, 2|4) Bethe Ansatze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [INSPIRE].
N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].
R.A. Janik, The AdS5 × S5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].
G. Arutyunov and S. Frolov, The dressing Factor and Crossing Equations, J. Phys. A 42 (2009) 425401 [arXiv:0904.4575] [INSPIRE].
D. Volin, Minimal solution of the AdS/CFT crossing equation, J. Phys. A 42 (2009) 372001 [arXiv:0904.4929] [INSPIRE].
R. Hernandez and E. Lopez, Quantum corrections to the string Bethe ansatz, JHEP 07 (2006) 004 [hep-th/0603204] [INSPIRE].
G. Arutyunov and S. Frolov, On AdS5 × S5 string S-matrix, Phys. Lett. B 639 (2006) 378 [hep-th/0604043] [INSPIRE].
N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS5 × S5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [INSPIRE].
B. Eden and M. Staudacher, Integrability and transcendentality, J. Stat. Mech. (2006) P11014 [hep-th/0603157] [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].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [INSPIRE].
S. Schäfer-Nameki, M. Zamaklar and K. Zarembo, How accurate is the quantum string Bethe ansatz?, JHEP 12 (2006) 020 [hep-th/0610250] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A. Rej, M. Staudacher and V.N. Velizhanin, Dressing and wrapping, J. Stat. Mech. (2007) P10003 [arXiv:0704.3586] [INSPIRE].
Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B 807 (2009) 625 [arXiv:0807.0399] [INSPIRE].
Z. Bajnok, R.A. Janik and T. Lukowski, Four loop twist two, BFKL, wrapping and strings, Nucl. Phys. B 816 (2009) 376 [arXiv:0811.4448] [INSPIRE].
Z. Bajnok, A. Hegedus, R.A. Janik and T. Lukowski, Five loop Konishi from AdS/CFT, Nucl. Phys. B 827 (2010) 426 [arXiv:0906.4062] [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].
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].
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].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of planar \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory: Konishi dimension at any coupling, Phys. Rev. Lett. 104 (2010) 211601 [arXiv:0906.4240] [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].
G. Arutyunov, S. Frolov and R. Suzuki, Exploring the mirror TBA, JHEP 05 (2010) 031 [arXiv:0911.2224] [INSPIRE].
G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS5 × S5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].
G. Arutyunov, S. Frolov and R. Suzuki, Five-loop Konishi from the mirror TBA, JHEP 04 (2010) 069 [arXiv:1002.1711] [INSPIRE].
J. Balog and A. Hegedus, 5-loop Konishi from linearized TBA and the XXX magnet, JHEP 06 (2010) 080 [arXiv:1002.4142] [INSPIRE].
J. Balog and A. Hegedus, The Bajnok-Janik formula and wrapping corrections, JHEP 09 (2010) 107 [arXiv:1003.4303] [INSPIRE].
Z. Bajnok and O. el Deeb, 6-loop anomalous dimension of a single impurity operator from AdS/CFT and multiple zeta values, JHEP 01 (2011) 054 [arXiv:1010.5606] [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Wrapping at four loops in N = 4 SYM, Phys. Lett. B 666 (2008) 100 [arXiv:0712.3522] [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Anomalous dimension with wrapping at four loops in N = 4 SYM, Nucl. Phys. B 805 (2008) 231 [arXiv:0806.2095] [INSPIRE].
V.N. Velizhanin, The four-loop anomalous dimension of the Konishi operator in N = 4 supersymmetric Yang-Mills theory, JETP Lett. 89 (2009) 6 [arXiv:0808.3832] [INSPIRE].
V.N. Velizhanin, Leading transcedentality contributions to the four-loop universal anomalous dimension in N = 4 SYM, Phys. Lett. B 676 (2009) 112 [arXiv:0811.0607] [INSPIRE].
L.N. Lipatov, Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories, Sov. J. Nucl. Phys. 23 (1976) 338 [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk singularity in nonabelian gauge theories, Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72 (1977) 377].
I.I. Balitsky and L.N. Lipatov, The Pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [INSPIRE].
V.G. Gorshkov, V.N. Gribov, L.N. Lipatov and G.V. Frolov, Doubly logarithmic asymptotic behavior in quantum electrodynamics, Sov. J. Nucl. Phys. 6 (1968) 95 [INSPIRE].
V.G. Gorshkov, V.N. Gribov, L.N. Lipatov and G.V. Frolov, Backward electron-positron scattering at high-energies, Sov. J. Nucl. Phys. 6 (1968) 262 [INSPIRE].
V.G. Gorshkov, V.N. Gribov, L.N. Lipatov and G.V. Frolov, Double logarithmic asymptotics of quantum electrodynamics, Phys. Lett. 22 (1966) 671 [INSPIRE].
R. Kirschner and L.N. Lipatov, Double logarithmic asymptotics of quark scattering amplitudes with flavor exchange, Phys. Rev. D 26 (1982) 1202 [INSPIRE].
R. Kirschner and L.n. Lipatov, Doubly logarithmic asymptotic of the quark scattering amplitude with nonvacuum exchange in the t channel, Sov. Phys. JETP 56 (1982) 266 [INSPIRE].
R. Kirschner and L.n. Lipatov, Double logarithmic asymptotics and Regge singularities of quark amplitudes with flavor exchange, Nucl. Phys. B 213 (1983) 122 [INSPIRE].
V.N. Velizhanin, Double-logs, Gribov-Lipatov reciprocity and wrapping, JHEP 08 (2011) 092 [arXiv:1104.4100] [INSPIRE].
M. Alfimov, N. Gromov and V. Kazakov, QCD Pomeron from AdS/CFT Quantum Spectral Curve, arXiv:1408.2530 [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for planar \( \mathcal{N} \) = 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, arXiv:1405.4857 [INSPIRE].
V.N. Velizhanin, Twist-2 at five loops: Wrapping corrections without wrapping computations, JHEP 06 (2014) 108 [arXiv:1311.6953] [INSPIRE].
C. Marboe and D. Volin, Quantum spectral curve as a tool for a perturbative quantum field theory, arXiv:1411.4758 [INSPIRE].
V. Dippel, unpublished.
A.V. Kotikov, A. Rej and S. Zieme, Analytic three-loop solutions for N = 4 SYM twist operators, Nucl. Phys. B 813 (2009) 460 [arXiv:0810.0691] [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 and V.N. Velizhanin, Anomalous dimensions of Wilson operators in N = 4 SYM theory, Phys. Lett. B 557 (2003) 114 [hep-ph/0301021] [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 non-singlet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
A. Vogt, S. Moch and J.A.M. Vermaseren, The three-loop splitting functions in QCD: the singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].
J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].
Yu. L. Dokshitzer, G. Marchesini and G.P. Salam, Revisiting parton evolution and the large-x limit, Phys. Lett. B 634 (2006) 504 [hep-ph/0511302] [INSPIRE].
Yu. L. Dokshitzer and G. Marchesini, N = 4 SUSY Yang-Mills: three loops made simple(r), Phys. Lett. B 646 (2007) 189 [hep-th/0612248] [INSPIRE].
B. Basso and G.P. Korchemsky, Anomalous dimensions of high-spin operators beyond the leading order, Nucl. Phys. B 775 (2007) 1 [hep-th/0612247] [INSPIRE].
V.N. Gribov and L.N. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [INSPIRE].
M. Beccaria, V. Forini, T. Lukowski and S. Zieme, Twist-three at five loops, Bethe Ansatz and wrapping, JHEP 03 (2009) 129 [arXiv:0901.4864] [INSPIRE].
M. Beccaria, Yu. L. Dokshitzer and G. Marchesini, Twist 3 of the sl(2) sector of N = 4 SYM and reciprocity respecting evolution, Phys. Lett. B 652 (2007) 194 [arXiv:0705.2639] [INSPIRE].
M. Beccaria and V. Forini, Four loop reciprocity of twist two operators in N = 4 SYM, JHEP 03 (2009) 111 [arXiv:0901.1256] [INSPIRE].
A. Lenstra, H. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982) 515.
V.N. Velizhanin, Six-loop anomalous dimension of twist-three operators in N = 4 SYM, JHEP 11 (2010) 129 [arXiv:1003.4717] [INSPIRE].
V.N. Velizhanin, Three loop anomalous dimension of the non-singlet transversity operator in QCD, Nucl. Phys. B 864 (2012) 113 [arXiv:1203.1022] [INSPIRE].
C. Marboe and D. Volin, MATHEMATICA notebook for the paper: “Quantum spectral curve as a tool for a perturbative quantum field theory”, http://www.maths.tcd.ie/~dvolin/QSC/loop10sl2.zip.
M. Albrech, D. Cadé, X. Pujol and D. Stehlé, fplll-4.0, a floating-point LLL implementation, https://github.com/dstehle/fplll.
V.S. Fadin and L.N. Lipatov, BFKL Pomeron in the next-to-leading approximation, Phys. Lett. B 429 (1998) 127 [hep-ph/9802290] [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].
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, New features of FORM, math-ph/0010025 [INSPIRE].
J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The multiple zeta value data mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].
B. Basso, An exact slope for AdS/CFT, arXiv:1109.3154 [INSPIRE].
B. Basso, Scaling dimensions at small spin in N = 4 SYM theory, arXiv:1205.0054 [INSPIRE].
N. Gromov, On the derivation of the exact slope function, JHEP 02 (2013) 055 [arXiv:1205.0018] [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].
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Marboe, C., Velizhanin, V. & Volin, D. Six-loop anomalous dimension of twist-two operators in planar \( \mathcal{N}=4 \) SYM theory. J. High Energ. Phys. 2015, 84 (2015). https://doi.org/10.1007/JHEP07(2015)084
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DOI: https://doi.org/10.1007/JHEP07(2015)084