Geodesic
Eden–Staudacher and Beisert–Eden–Staudacher equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 1, pp. 117-129 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate the Eden–Staudacher and Beisert–Eden–Staudacher equations for the anomalous dimension of twist-$2$ operators at a large spin $s$ in the $\mathcal{N}{=}4$ supersymmetric gauge theory. We reduce these equations to a set of linear algebraic equations and calculate their kernels analytically. We demonstrate that in the perturbation theory, the anomalous dimension is a sum of products of the Euler functions $\zeta(k)$ having the maximum transcendentality property. We also show that at a large coupling, the "singular" solution of the Beisert–Eden–Staudacher equation reproduces the anomalous dimension constants predicted from the string side of the AdS/CFT correspondence.
Mots-clés : anomalous dimension
Keywords: $\mathcal{N}{=}4$ supersymmetric gauge theory. 
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A. V. Kotikov; L. N. Lipatov. Eden–Staudacher and Beisert–Eden–Staudacher equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 1, pp. 117-129. http://geodesic.mathdoc.fr/item/TMF_2008_155_1_a9/

[1] V. N. Gribov, L. N. Lipatov, YaF, 15 (1972), 781–807; 1218–1237; Л. Н. Липатов, ЯФ, 20 (1974), 181–198; G. Altarelli, G. Parisi, Nucl. Phys. B, 126:2 (1977), 298–318 ; Ю. Л. Докшицер, ЖЭТФ, 73 (1977), 1216–1240 | DOI

[2] S. Moch, J. A. M. Vermaseren, A. Vogt, Nucl. Phys. B, 688:1–2 (2004), 101–134 | DOI | Zbl

[3] A. V. Kotikov, L. N. Lipatov, Nucl. Phys. B, 661:1–2 (2003), 19–61 ; hep-ph/0112346 | DOI | Zbl

[4] L. N. Lipatov, YaF, 23 (1976), 642–656; V. S. Fadin, E. A. Kuraev, L. N. Lipatov, Phys. Lett. B, 60:1 (1975), 50–52 ; Е. А. Кураев, Л. Н. Липатов, В. С. Фадин, ЖЭТФ, 71 (1976), 840–855; ЖЭТФ, 72:2 (1977), 377–389 ; И. И. Балицкий, Л. Н. Липатов, ЯФ, 28 (1978), 1597–1611; Письма в ЖЭТФ, 30 (1979), 383 | DOI | MR

[5] V. S. Fadin, L. N. Lipatov, Phys. Lett. B, 429:1–2 (1998), 127–134 | DOI

[6] A. V. Kotikov, L. N. Lipatov, Nucl. Phys. B, 582:1–3 (2000), 19–43 | DOI

[7] L. N. Lipatov, “Effective action for high energy processes in QCD”, Very High Multiplicity Physics, Proc. Int. Workshop (Dubna 2000), eds. J. Manjavidze, A. Sissakian, JINR, Dubna, 2000, 159; Nucl. Phys. Proc. Suppl., 99 (2001), 175–179 | DOI

[8] A. V. Kotikov, L. N. Lipatov, V. N. Velizhanin, Phys. Lett. B, 557:1–2 (2003), 114–120 | DOI | MR | Zbl

[9] A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko, V. N. Velizhanin, Phys. Lett. B, 595:1–4 (2004), 521–529 ; Phys. Lett. B, 632:5–6 (2006), 754–756 | DOI | MR | Zbl | DOI | MR | Zbl

[10] B. Eden, C. Jarczak, E. Sokatchev, Nucl. Phys. B, 712:1–2 (2005), 157–195 | DOI | MR | Zbl

[11] A. V. Kotikov, L. N. Lipatov, A. Rey, M. Staudacher, V. N. Velizhaninet, Dressing and Wrapping, arXiv: 0704.3586

[12] N. Beisert, M. Staudacher, Nucl. Phys. B, 727:1–2 (2005), 1–62 | DOI | MR | Zbl

[13] J. Maldacena, Adv. Theor. Math. Phys., 2:2 (1998), 231–252 ; Internat. J. Theoret. Phys., 38 (1999), 1113–1133 ; S. S. Gubser, I. R. Klebanov, A. M. Polyakov, Phys. Lett. B, 428:1–2 (1998), 105–114 ; E. Witten, Adv. Theor. Math. Phys., 2:2 (1998), 253–291 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[14] B. Eden, M. Staudacher, J. Stat. Mech., 11 (2006), P11014 | MR

[15] A. V. Belitsky, Phys. Lett. B, 643:6 (2006), 354–361 | DOI | MR | Zbl

[16] N. Beisert, B. Eden, M. Staudacher, J. Stat. Mech., 1 (2007), P01021

[17] Z. Bern, L. J. Dixon, V. A. Smirnov, Phys. Rev. D, 72 (2005), 085001 ; F. Cachazo, M. Spradlin, A. Volovich, Phys. Rev. D, 75 (2007), 105011 | DOI | MR | DOI | MR

[18] Z. Bern, M. Czakon, L. Dixon et al., Phys. Rev. D, 75 (2007), 085010 ; Z. Bern, J. J. M. Carrasco, H. Johansson, D. A. Kosower, Maximally Supersymmetric Planar Yang–Mills Amplitudes at Five Loops, arXiv: 0705.1864 | DOI | MR | MR

[19] A. V. Kotikov, L. N. Lipatov, Nucl. Phys. B, 769:3 (2007), 217–255 | DOI | MR | Zbl

[20] L. N. Lipatov, “Transcendentality and the Eden–Staudacher equation”, Talk at the Workshop on Integrability in Gauge and String Theory (AEI, Potsdam, Germany, July 24–28, 2006) | MR

[21] N. Beisert, R. Hernandez, E. Lopez, JHEP, 11 (2006), 070 | DOI | MR

[22] N. Beisert, T. Klose, J. Stat. Mech., 07 (2006), P07006 ; C. Gomez, R. Hernandez, Phys. Lett. B, 644:5–6 (2007), 375–378 | DOI | MR | DOI | MR | Zbl

[23] G. Arutyunov, S. Frolov, M. Staudacher, JHEP, 10 (2004), 016 | DOI | MR

[24] N. Beisert, A. A. Tseytlin, Phys. Lett. B, 629:2–4 (2005), 102–110 ; R. Hernandez, E. Lopez, JHEP, 07 (2006), 004 ; L. Freyhult, C. Kristjansen, Phys. Lett. B, 638:2–3 (2006), 258–264 | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl

[25] R. A. Janik, Phys. Rev. D, 73:8 (2006), 086006 | DOI | MR

[26] S. Gubser, I. Klebanov, A. Polyakov, Nucl. Phys. B, 636:1–2 (2002), 99–114 | DOI | MR | Zbl

[27] M. K. Benna, S. Benvenuti, I. R. Klebanov, A. Scardicchio, Phys. Rev. Lett., 98:13 (2007), 131603 | DOI

[28] R. Roiban, A. Tirziu, A. A. Tseytlin, Two-loop world-sheet corrections in $\mathrm{Ad}S_5\times S^5$ superstring, arXiv: 0704.3638 | MR

[29] L. F. Alday, G. Arutyunov, M. K. Benna, B. Eden, I. R. Klebanov, JHEP, 04 (2007), 082 | DOI | MR

[30] P. Y. Casteill, C. Kristjansen, The Strong Coupling Limit of the Scaling Function from the Quantum String Bethe Ansatz, arXiv: 0705.0890 | MR

[31] N. Beisert, T. McLoughlin, R. Roiban, Phys. Rev. D, 76 (2007), 046002 ; ; N. Beisert, PoS (Solvay), 002 (2007); ; S. Ryang, Analytic Bethe Ansatz Solutions for Highest States in the $su(1|1)$ and $su(2)$ Sectors, ; A. V. Belitsky, Baxter equation for long-range $SL(2|1)$ magnet, ; I. Kostov, D. Serban, D. Volin, Strong coupling limit of Bethe Ansatz equations, ; A. Rej, M. Staudacher, S. Zieme, Nesting and Dressing, ; A. Torrielli, Classical $r$-matrix of the $su(2|2)$ SYM spin-chain, ; G. Arutyunov, S. Frolov, M. Zamaklar, JHEP, 04 (2007), 002 ; S. Frolov, A. Tirziu, A. A. Tseytlin, Nucl. Phys. B, 766:1–3 (2007), 232–245 arXiv: 0705.0321arXiv: 0704.0400arXiv: hep-th/0703241arXiv: hep-th/0703058arXiv: hep-th/0703031arXiv: hep-th/0702151arXiv: hep-th/0701281 | DOI | MR | MR | MR | DOI | MR | DOI | MR | Zbl