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Journal of High Energy Physics

, 2013:206 | Cite as

Structure constants of β deformed super Yang-Mills

  • Justin R. David
  • Abhishake Sadhukhan
Article

Abstract

We study the structure constants of the \( \mathcal{N}=1 \) beta deformed theory perturbatively and at strong coupling. We show that the planar one loop corrections to the structure constants of single trace gauge invariant operators in the scalar sector is determined by the anomalous dimension Hamiltonian. This result implies that 3 point functions of the chiral primaries of the theory do not receive corrections at one loop. We then studythe structure constants at strong coupling using the Lunin-Maldacena geometry. We explicitly construct the supergravity mode dual to the chiral primary with three equal U(1) R-charges in the Lunin-Maldacena geometry. We show that the 3 point function of this supergravity mode with semi-classical states representing two other similar chiral primary states but with large U(1) charges to be independent of the beta deformation and identical to that found in the AdS 5 × S 5 geometry. This together with the one-loop result indicate that these structure constants are protected by a non-renormalization theorem. We also show that three point function of U(1) R-currents with classical massive strings is proportional to the R-charge carried by the string solution. This is in accordance with the prediction of the R-symmetry Ward identity.

Keywords

Supersymmetric gauge theory Gauge-gravity correspondence AdS-CFT Correspondence 

References

  1. [1]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    K. Okuyama and L.-S. Tseng, Three-point functions in N = 4 SYM theory at one-loop, JHEP 08 (2004) 055 [hep-th/0404190] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    R. Roiban and A. Volovich, Yang-Mills correlation functions from integrable spin chains, JHEP 09 (2004) 032 [hep-th/0407140] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    L.F. Alday, J.R. David, E. Gava and K. Narain, Structure constants of planar N = 4 Yang-Mills at one loop, JHEP 09 (2005) 070 [hep-th/0502186] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large-N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].MathSciNetMATHGoogle Scholar
  6. [6]
    R.A. Janik, P. Surowka and A. Wereszczynski, On correlation functions of operators dual to classical spinning string states, JHEP 05 (2010) 030 [arXiv:1002.4613] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    E. Buchbinder and A. Tseytlin, On semiclassical approximation for correlators of closed string vertex operators in AdS/CFT, JHEP 08 (2010) 057 [arXiv:1005.4516] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    K. Zarembo, Holographic three-point functions of semiclassical states, JHEP 09 (2010) 030 [arXiv:1008.1059] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    M.S. Costa, R. Monteiro, J.E. Santos and D. Zoakos, On three-point correlation functions in the gauge/gravity duality, JHEP 11 (2010) 141 [arXiv:1008.1070] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    R. Roiban and A. Tseytlin, On semiclassical computation of 3-point functions of closed string vertex operators in AdS 5 × S 5, Phys. Rev. D 82 (2010) 106011 [arXiv:1008.4921] [INSPIRE].ADSGoogle Scholar
  11. [11]
    S. Ryang, Correlators of vertex operators for circular strings with winding numbers in AdS 5 × S 5, JHEP 01 (2011) 092 [arXiv:1011.3573] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability II. Weak/strong coupling match, JHEP 09 (2011) 029 [arXiv:1104.5501] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    O. Foda, N = 4 SYM structure constants as determinants, JHEP 03 (2012) 096 [arXiv:1111.4663] [INSPIRE].ADSGoogle Scholar
  15. [15]
    N. Gromov and P. Vieira, Tailoring three-point functions and integrability IV. Theta-morphism, arXiv:1205.5288 [INSPIRE].
  16. [16]
    A. Bissi, G. Grignani and A. Zayakin, The SO(6) scalar product and three-point functions from integrability, arXiv:1208.0100 [INSPIRE].
  17. [17]
    F. Elmetti, A. Mauri, S. Penati and A. Santambrogio, Conformal invariance of the planar beta-deformed N = 4 SYM theory requires beta real, JHEP 01 (2007) 026 [hep-th/0606125] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    F. Elmetti, A. Mauri, S. Penati, A. Santambrogio and D. Zanon, Real versus complex β-deformation of the N = 4 planar super Yang-Mills theory, JHEP 10 (2007) 102 [arXiv:0705.1483] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    S. Frolov, R. Roiban and A.A. Tseytlin, Gauge-string duality for (non)supersymmetric deformations of N = 4 super Yang-Mills theory, Nucl. Phys. B 731 (2005) 1 [hep-th/0507021] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    N. Beisert and R. Roiban, Beauty and the twist: the Bethe ansatz for twisted N = 4 SYM, JHEP 08 (2005) 039 [hep-th/0505187] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    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].MathSciNetADSGoogle Scholar
  23. [23]
    F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Finite-size effects in the superconformal beta-deformed N = 4 SYM, JHEP 08 (2008) 057 [arXiv:0806.2103] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    G. Arutyunov, M. de Leeuw and S.J. van Tongeren, Twisting the mirror TBA, JHEP 02 (2011) 025 [arXiv:1009.4118] [INSPIRE].ADSGoogle Scholar
  25. [25]
    C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP 12 (2011) 059 [arXiv:1108.4914] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    D. Berenstein and R.G. Leigh, Discrete torsion, AdS/CFT and duality, JHEP 01 (2000) 038 [hep-th/0001055] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    D. Berenstein, V. Jejjala and R.G. Leigh, Marginal and relevant deformations of N = 4 field theories and noncommutative moduli spaces of vacua, Nucl. Phys. B 589 (2000) 196 [hep-th/0005087] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    D.Z. Freedman and U. Gürsoy, Comments on the beta-deformed N = 4 SYM theory, JHEP 11 (2005) 042 [hep-th/0506128] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    S. Penati, A. Santambrogio and D. Zanon, Two-point correlators in the beta-deformed N = 4 SYM at the next-to-leading order, JHEP 10 (2005) 023 [hep-th/0506150] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    A. Mauri, S. Penati, A. Santambrogio and D. Zanon, Exact results in planar N = 1 superconformal Yang-Mills theory, JHEP 11 (2005) 024 [hep-th/0507282] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    G. Georgiou, B.-H. Lee and C. Park, Correlators of massive string states with conserved currents, JHEP 03 (2013) 167 [arXiv:1301.5092] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γ i -deformed N = 4 SYM theory, arXiv:1308.4420 [INSPIRE].
  34. [34]
    G. Georgiou, V.L. Gili and R. Russo, Operator mixing and three-point functions in N = 4 SYM, JHEP 10 (2009) 009 [arXiv:0907.1567] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    G. Georgiou, V. Gili, A. Grossardt and J. Plefka, Three-point functions in planar N = 4 super Yang-Mills theory for scalar operators up to length five at the one-loop order, JHEP 04 (2012) 038 [arXiv:1201.0992] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    J. Plefka and K. Wiegandt, Three-point functions of twist-two operators in N = 4 SYM at one loop, JHEP 10 (2012) 177 [arXiv:1207.4784] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    P.S. Howe, E. Sokatchev and P.C. West, Three point functions in N = 4 Yang-Mills, Phys. Lett. B 444 (1998) 341 [hep-th/9808162] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    A. Basu, M.B. Green and S. Sethi, Some systematics of the coupling constant dependence of N = 4 Yang-Mills, JHEP 09 (2004) 045 [hep-th/0406231] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    K. Madhu and S. Govindarajan, Chiral primaries in the Leigh-Strassler deformed N = 4 SYM: a perturbative study, JHEP 05 (2007) 038 [hep-th/0703020] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    C. Ahn and P. Bozhilov, Three-point correlation function of giant magnons in the Lunin-Maldacena background, Phys. Rev. D 84 (2011) 126011 [arXiv:1106.5656] [INSPIRE].ADSGoogle Scholar
  42. [42]
    D. Arnaudov and R. Rashkov, Three-point correlators: examples from Lunin-Maldacena background, Phys. Rev. D 84 (2011) 086009 [arXiv:1106.4298] [INSPIRE].ADSGoogle Scholar
  43. [43]
    P. Bozhilov, Leading finite-size effects on some three-point correlators in TsT-deformed AdS 5 × S 5, Phys. Rev. D 88 (2013) 026017 [arXiv:1304.2139] [INSPIRE].ADSGoogle Scholar
  44. [44]
    S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    P. Meessen and T. Ortín, An SL(2, Z) multiplet of nine-dimensional type-II supergravity theories, Nucl. Phys. B 541 (1999) 195 [hep-th/9806120] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    H. Kim, L. Romans and P. van Nieuwenhuizen, The mass spectrum of chiral N = 2 D = 10 supergravity on S 5, Phys. Rev. D 32 (1985) 389 [INSPIRE].ADSGoogle Scholar
  47. [47]
    D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT(d)/AdS(d+1) correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    J. Minahan and C. Sieg, Four-loop anomalous dimensions in Leigh-Strassler deformations, J. Phys. A 45 (2012) 305401 [arXiv:1112.4787] [INSPIRE].MathSciNetGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia

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