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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].
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].
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].
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].
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].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [SPIRES].
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].
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].
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].
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].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, arXiv:0911.4708 [SPIRES].
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].
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].
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].
Al.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [SPIRES].
Al.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. Scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [SPIRES].
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].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [SPIRES].
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].
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].
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].
D. Gepner, New Conformal Field Theories Associated with Lie Algebras and their Partition Functions, Nucl. Phys. B 290 (1987) 10 [SPIRES].
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].
P. Heslop and V.V. Khoze, Analytic Results for MHV Wilson Loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [SPIRES].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Bootstrapping Null Polygon Wilson Loops, JHEP 03 (2011) 092 [arXiv:1010.5009] [SPIRES].
J. Maldacena and A. Zhiboedov, Form factors at strong coupling via a Y-system, JHEP 11 (2010) 104 [arXiv:1009.1139] [SPIRES].
I. Bakas, Conservation laws and geometry of perturbed coset models, Int. J. Mod. Phys. A9 (1994) 3443 [hep-th/9310122] [SPIRES].
Q.-H. Park, Deformed coset models from gauged WZW actions, Phys. Lett. B 328 (1994) 329 [hep-th/9402038] [SPIRES].
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].
D. Gepner and Z. Qiu, Modular Invariant Partition Functions for Parafermionic Field Theories, Nucl. Phys. B 285 (1987) 423 [SPIRES].
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].
Al.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [SPIRES].
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].
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].
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].
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].
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].
Al.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [SPIRES].
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].
T.R. Klassen and E. Melzer, The Thermodynamics of purely elastic scattering theories and conformal perturbation theory, Nucl. Phys. B 350 (1991) 635 [SPIRES].
Al.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [SPIRES].
G. Mussardo, Statistical Field Theory, Oxford Univ. Press, Oxford U.K. (2010).
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].
F. Ravanini, R. Tateo and A. Valleriani, Dynkin TBAs, Int. J. Mod. Phys. A 8 (1993) 1707 [hep-th/9207040] [SPIRES].
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].
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].
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].
A. Fring and R. Koberle, Factorized scattering in the presence of reflecting boundaries, Nucl. Phys. B 421 (1994) 159 [hep-th/9304141] [SPIRES].
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].
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].
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].
F. Woynarovich, On the normalization of the partition function of Bethe Ansatz systems, arXiv:1007.1148 [SPIRES].
P. Dorey and R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (1998) 575 [hep-th/9706140] [SPIRES].
V.G. Kac and D.H. Peterson, Infinite dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984) 125 [SPIRES].
T. Gannon, Algorithms for affine Kac-Moody algebras, hep-th/0106123 [SPIRES].
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].
W. Nahm, A. Recknagel and M. Terhoeven, Dilogarithm identities in conformal field theory, Mod. Phys. Lett. A 8 (1993) 1835 [hep-th/9211034] [SPIRES].
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].
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].
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].
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].
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].
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].
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, Series and Products, Fifth edition, Academic Press, (1980).
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].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1102.2477
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2011)100