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g-functions and gluon scattering amplitudes at strong coupling

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Abstract

We study gluon scattering amplitudes/Wilson loops in \( \mathcal{N} = 4 \) super Yang-Mills theory at strong coupling by calculating the area of the minimal surfaces in AdS 3 based on the associated thermodynamic Bethe ansatz system. The remainder function of the amplitudes is computed by evaluating the free energy, the T- and Y-functions of the homogeneous sine-Gordon model. Using conformal field theory (CFT) perturbation, we examine the mass corrections to the free energy around the CFT point corresponding to the regular polygonal Wilson loop. Based on the relation between the T-functions and the g-functions, which measure the boundary entropy, we calculate corrections to the T-and Y-functions as well as express them at the CFT point by the modular S-matrix. We evaluate the remainder function around the CFT point for 8 and 10-point amplitudes explicitly and compare these analytic expressions with the 2-loop formulas. The two rescaled remainder functions show very similar power series structures.

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References

  1. L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  2. J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  3. A. Brandhuber, P. Heslop and G. Travaglini, MHV Amplitudes in N = 4 Super Yang-Mills and Wilson Loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  7. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. Z. Bern et al., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  9. 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] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  10. L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  11. L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, arXiv:0911.4708 [SPIRES].

  12. L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [SPIRES].

    MathSciNet  Google Scholar 

  13. Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, Thermodynamic Bethe Ansatz Equations for Minimal Surfaces in AdS 3, JHEP 04 (2010) 108 [arXiv:1002.2941] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  14. A. Kuniba, T. Nakanishi and J. Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A 44 (2011) 103001 [arXiv:1010.1344] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  15. Al.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  16. Al.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. Scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  17. Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, Six-point gluon scattering amplitudes from Z 4 -symmetric integrable model, JHEP 09 (2010) 064 [arXiv:1005.4487] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  18. I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. P. Dorey, A. Lishman, C. Rim and R. Tateo, Reflection factors and exact g-functions for purely elastic scattering theories, Nucl. Phys. B 744 (2006) 239 [hep-th/0512337] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  20. P. Dorey, D. Fioravanti, C. Rim and R. Tateo, Integrable quantum field theory with boundaries: The exact g-function, Nucl. Phys. B 696 (2004) 445 [hep-th/0404014] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  21. C.R. Fernandez-Pousa, M.V. Gallas, T.J. Hollowood and J.L. Miramontes, The symmetric space and homogeneous sine-Gordon theories, Nucl. Phys. B 484 (1997) 609 [hep-th/9606032] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  22. D. Gepner, New Conformal Field Theories Associated with Lie Algebras and their Partition Functions, Nucl. Phys. B 290 (1987) 10 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. V. Del Duca, C. Duhr and V.A. Smirnov, A Two-Loop Octagon Wilson Loop in N = 4 SYM, JHEP 09 (2010) 015 [arXiv:1006.4127] [SPIRES].

    Article  ADS  Google Scholar 

  24. P. Heslop and V.V. Khoze, Analytic Results for MHV Wilson Loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  25. D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Bootstrapping Null Polygon Wilson Loops, JHEP 03 (2011) 092 [arXiv:1010.5009] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  26. J. Maldacena and A. Zhiboedov, Form factors at strong coupling via a Y-system, JHEP 11 (2010) 104 [arXiv:1009.1139] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  27. I. Bakas, Conservation laws and geometry of perturbed coset models, Int. J. Mod. Phys. A9 (1994) 3443 [hep-th/9310122] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  28. Q.-H. Park, Deformed coset models from gauged WZW actions, Phys. Lett. B 328 (1994) 329 [hep-th/9402038] [SPIRES].

    ADS  Google Scholar 

  29. I. Bakas, Q.-H. Park and H.-J. Shin, Lagrangian Formulation of Symmetric Space sine-Gordon Models, Phys. Lett. B 372 (1996) 45 [hep-th/9512030] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  30. D. Gepner and Z. Qiu, Modular Invariant Partition Functions for Parafermionic Field Theories, Nucl. Phys. B 285 (1987) 423 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. V.A. Fateev and A.B. Zamolodchikov, Parafermionic Currents in the Two-Dimensional Conformal Quantum Field Theory and Selfdual Critical Points in Z(n) Invariant Statistical Systems, Sov. Phys. JETP 62 (1985) 215 [Zh. Eksp. Teor. Fiz. 89 (1985) 380] [SPIRES].

    MathSciNet  Google Scholar 

  32. Al.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [SPIRES].

    ADS  Google Scholar 

  33. J.L. Miramontes and C.R. Fernandez-Pousa, Integrable quantum field theories with unstable particles, Phys. Lett. B 472 (2000) 392 [hep-th/9910218] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  34. O.A. Castro-Alvaredo and A. Fring, Decoupling the SU(N)2 homogeneous sine-Gordon model, Phys. Rev. D 64 (2001) 085007 [hep-th/0010262] [SPIRES].

    ADS  Google Scholar 

  35. O.A. Castro-Alvaredo, A. Fring, C. Korff and J.L. Miramontes, Thermodynamic Bethe ansatz of the homogeneous sine-Gordon models, Nucl. Phys. B 575 (2000) 535 [hep-th/9912196] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  36. P. Dorey and J.L. Miramontes, Mass scales and crossover phenomena in the homogeneous sine-Gordon models, Nucl. Phys. B 697 (2004) 405 [hep-th/0405275] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  37. I.V. Cherednik, Factorizing Particles on a Half Line and Root Systems, Theor. Math. Phys. 61 (1984) 977 [Teor. Mat. Fiz. 61 (1984) 35] [SPIRES].

    Article  MathSciNet  MATH  Google Scholar 

  38. Al.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  39. H. Itoyama and P. Moxhay, Neutral excitations and the massless limit of the sine-Gordon massive Thirring theory, Phys. Rev. Lett. 65 (1990) 2102 [SPIRES].

    Article  ADS  Google Scholar 

  40. T.R. Klassen and E. Melzer, The Thermodynamics of purely elastic scattering theories and conformal perturbation theory, Nucl. Phys. B 350 (1991) 635 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  41. Al.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  42. G. Mussardo, Statistical Field Theory, Oxford Univ. Press, Oxford U.K. (2010).

    MATH  Google Scholar 

  43. Al.B. Zamolodchikov, TBA equations for integrable perturbed SU(2) k × SU(2) l /SU(2) k+l coset models, Nucl. Phys. B 366 (1991) 122 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  44. F. Ravanini, R. Tateo and A. Valleriani, Dynkin TBAs, Int. J. Mod. Phys. A 8 (1993) 1707 [hep-th/9207040] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  45. V.A. Fateev, The Exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324 (1994) 45 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  46. V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. P. Dorey, I. Runkel, R. Tateo and G. Watts, g-function flow in perturbed boundary conformal field theories, Nucl. Phys. B 578 (2000) 85 [hep-th/9909216] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  48. A. Fring and R. Koberle, Factorized scattering in the presence of reflecting boundaries, Nucl. Phys. B 421 (1994) 159 [hep-th/9304141] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  49. S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. A 9 (1994) 4353] [hep-th/9306002] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  50. R. Sasaki, Reflection Bootstrap equations for Toda field theory, in the proceedings of the conference Interface between physics and mathematics, W. Nahm and J.-M. Shen eds., hep-th/9311027 [SPIRES].

  51. B. Pozsgay, On O(1) contributions to the free energy in Bethe Ansatz systems: the exact g-function, JHEP 08 (2010) 090 [arXiv:1003.5542] [SPIRES].

    Article  ADS  Google Scholar 

  52. F. Woynarovich, On the normalization of the partition function of Bethe Ansatz systems, arXiv:1007.1148 [SPIRES].

  53. P. Dorey and R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (1998) 575 [hep-th/9706140] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. V.G. Kac and D.H. Peterson, Infinite dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984) 125 [SPIRES].

    Article  MathSciNet  MATH  Google Scholar 

  55. T. Gannon, Algorithms for affine Kac-Moody algebras, hep-th/0106123 [SPIRES].

  56. A.N. Kirillov, Identities for the Rogers dilogarithm function connected with simple Lie algebras, Zap. Nauchn. Semin. Leningr. Otdel. Mat. Inst. 164 (1987) 121 [J. Math. Sci. 47 (1989) 2450].

    Google Scholar 

  57. W. Nahm, A. Recknagel and M. Terhoeven, Dilogarithm identities in conformal field theory, Mod. Phys. Lett. A 8 (1993) 1835 [hep-th/9211034] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  58. A. LeClair, G. Mussardo, H. Saleur and S. Skorik, Boundary energy and boundary states in integrable quantum field theories, Nucl. Phys. B 453 (1995) 581 [hep-th/9503227] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  59. R. Chatterjee, Exact Partition Function and Boundary State of 2-D Massive Ising Field Theory with Boundary Magnetic Field, Nucl. Phys. B 468 (1996) 439 [hep-th/9509071] [SPIRES].

    Article  ADS  Google Scholar 

  60. A. Brandhuber, P. Heslop, V.V. Khoze and G. Travaglini, Simplicity of Polygon Wilson Loops in N = 4 SYM, JHEP 01 (2010) 050 [arXiv:0910.4898] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  61. L.F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, An Operator Product Expansion for Polygonal null Wilson Loops, arXiv:1006.2788 [SPIRES].

  62. J. Bartels, J. Kotanski and V. Schomerus, Excited Hexagon Wilson Loops for Strongly Coupled N = 4 SYM, JHEP 01 (2011) 096 [arXiv:1009.3938] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  63. V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation, Commun. Math. Phys. 190 (1997) 247 [hep-th/9604044] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, Series and Products, Fifth edition, Academic Press, (1980).

  65. V.S. Dotsenko and V.A. Fateev, Operator Algebra of Two-Dimensional Conformal Theories with Central Charge C ≤ 1, Phys. Lett. B 154 (1985) 291 [SPIRES].

    MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwake-Cho, Kyoto, 606–8502, Japan

    Yasuyuki Hatsuda

  2. Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152–8551, Japan

    Katsushi Ito

  3. Research and Education Center for Natural Sciences and Hiyoshi Department of Physics, Keio University, 4-1-1 Hiyoshi, Kouhoku-ku, Yokohama, 223–8521, Japan

    Kazuhiro Sakai

  4. Institute of Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305–8571, Japan

    Yuji Satoh

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  1. Yasuyuki Hatsuda
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Correspondence to Yasuyuki Hatsuda.

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Hatsuda, Y., Ito, K., Sakai, K. et al. g-functions and gluon scattering amplitudes at strong coupling. J. High Energ. Phys. 2011, 100 (2011). https://doi.org/10.1007/JHEP04(2011)100

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