Advertisement

On the integrability of planar \( \mathcal{N}=2 \) superconformal gauge theories

  • Abhijit Gadde
  • Pedro Liendo
  • Leonardo RastelliEmail author
  • Wenbin Yan
Article

Abstract

We study the integrability properties of planar \( \mathcal{N}=2 \) superconformal field theories in four dimensions. We show that the spin chain associated to the planar dilation operator of \( \mathcal{N}=2 \) superconformal QCD fails to be integrable at two loops. In our analysis we focus on a closed SU(2|1) sector, whose two-loop spin chain we fix by symmetry arguments (up to a few undetermined coefficients). It turns out that the Yang-Baxter equation for magnon scattering is not satisfied in this sector. On the other hand, we suggest that the closed SU(2, 1|2) sector, which exists in any \( \mathcal{N}=2 \) superconformal gauge theory, may be integrable to all loops. We summarize the known results in the literature that are consistent with this conjecture.

Keywords

Supersymmetric gauge theory 1/N Expansion Integrable Field Theories 

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]
    L. Lipatov, High-energy asymptotics of multicolor QCD and exactly solvable lattice models, hep-th/9311037 [INSPIRE].
  3. [3]
    L. Faddeev and G. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311 [hep-th/9404173] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    G. Korchemsky, Bethe ansatz for QCD Pomeron, Nucl. Phys. B 443 (1995) 255 [hep-ph/9501232] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    V.M. Braun, S.E. Derkachov and A. Manashov, Integrability of three particle evolution equations in QCD, Phys. Rev. Lett. 81 (1998) 2020 [hep-ph/9805225] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    V.M. Braun, S.E. Derkachov, G. Korchemsky and A. Manashov, Baryon distribution amplitudes in QCD, Nucl. Phys. B 553 (1999) 355 [hep-ph/9902375] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    A.V. Belitsky, Renormalization of twist-three operators and integrable lattice models, Nucl. Phys. B 574 (2000) 407 [hep-ph/9907420] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    N. Beisert, G. Ferretti, R. Heise and K. Zarembo, One-loop QCD spin chain and its spectrum, Nucl. Phys. B 717 (2005) 137 [hep-th/0412029] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    A.V. Belitsky, S.E. Derkachov, G. Korchemsky and A. Manashov, Quantum integrability in super Yang-Mills theory on the light cone, Phys. Lett. B 594 (2004) 385 [hep-th/0403085] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    A. Belitsky, S.E. Derkachov, G. Korchemsky and A. Manashov, Dilatation operator in (super-) Yang-Mills theories on the light-cone, Nucl. Phys. B 708 (2005) 115 [hep-th/0409120] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    A. Belitsky, G. Korchemsky and D. Mueller, Integrability in Yang-Mills theory on the light cone beyond leading order, Phys. Rev. Lett. 94 (2005) 151603 [hep-th/0412054] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    A. Belitsky, G. Korchemsky and D. Mueller, Integrability of two-loop dilatation operator in gauge theories, Nucl. Phys. B 735 (2006) 17 [hep-th/0509121] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    G. Korchemsky, Review of AdS/CFT integrability. Chapter IV.4: Integrability in QCD and N <4 SYM, Lett. Math. Phys. 99 (2012) 425 [arXiv:1012.4000] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    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].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    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].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    N. Beisert, The SU(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487 [hep-th/0310252] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    B.I. Zwiebel, N = 4 SYM to two loops: compact expressions for the non-compact symmetry algebra of the su(1,12) sector, JHEP 02 (2006) 055 [hep-th/0511109] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    C. Sieg, Superspace computation of the three-loop dilatation operator of N = 4 SYM theory, Phys. Rev. D 84 (2011) 045014 [arXiv:1008.3351] [INSPIRE].ADSGoogle Scholar
  20. [20]
    B.I. Zwiebel, From scattering amplitudes to the dilatation generator in N = 4 SYM, J. Phys. A 45 (2012) 115401 [arXiv:1111.0083] [INSPIRE].MathSciNetADSGoogle Scholar
  21. [21]
    J.A. Minahan, Review of AdS/CFT integrability. Chapter I.1: Spin chains in N = 4 super Yang-Mills, Lett. Math. Phys. 99 (2012) 33 [arXiv:1012.3983] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. [22]
    C. Sieg, Review of AdS/CFT integrability. Chapter I.2: The spectrum from perturbative gauge theory, Lett. Math. Phys. 99 (2012) 59 [arXiv:1012.3984] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  23. [23]
    K. Zoubos, Review of AdS/CFT integrability,. Chapter IV.2: Deformations, orbifolds and open boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Gadde, E. Pomoni and L. Rastelli, The Veneziano limit of N = 2 superconformal QCD: towards the string dual of N = 2 SU(N (c)) SYM with N (f) = 2N (c), arXiv:0912.4918 [INSPIRE].
  25. [25]
    A. Gadde, E. Pomoni and L. Rastelli, Spin chains in N = 2 superconformal theories: from the Z2 quiver to superconformal QCD, JHEP 06 (2012) 107 [arXiv:1006.0015] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    P. Liendo, E. Pomoni and L. Rastelli, The complete one-loop dilation operator of N = 2 SuperConformal QCD, JHEP 07 (2012) 003 [arXiv:1105.3972] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    P. Liendo and L. Rastelli, The complete one-loop spin chain of N = 1 SQCD, JHEP 10 (2012) 117 [arXiv:1111.5290] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    N. Beisert, The dilatation operator of N = 4 super Yang-Mills theory and integrability, Phys. Rept. 405 (2005) 1 [hep-th/0407277] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    P. Liendo, E. Pomoni and L. Rastelli, work in progress.Google Scholar
  30. [30]
    N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  31. [31]
    E. Pomoni and C. Sieg, From N = 4 gauge theory to N = 2 conformal QCD: three-loop mixing of scalar composite operators, arXiv:1105.3487 [INSPIRE].
  32. [32]
    M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    S. Kachru and E. Silverstein, 4 − D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855 [hep-th/9802183] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  35. [35]
    A.E. Lawrence, N. Nekrasov and C. Vafa, On conformal field theories in four-dimensions, Nucl. Phys. B 533 (1998) 199 [hep-th/9803015] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    N. Beisert and R. Roiban, The Bethe ansatz for Z(S) orbifolds of N = 4 super Yang-Mills theory, JHEP 11 (2005) 037 [hep-th/0510209] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    A. Solovyov, Bethe Ansatz equations for general orbifolds of N = 4 SYM, JHEP 04 (2008) 013 [arXiv:0711.1697] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    A. Gadde and L. Rastelli, Twisted magnons, JHEP 04 (2012) 053 [arXiv:1012.2097] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    W. Yan, work in progress.Google Scholar
  40. [40]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  41. [41]
    S.-J. Rey and T. Suyama, Exact results and holography of Wilson loops in N = 2 superconformal (quiver) gauge theories, JHEP 01 (2011) 136 [arXiv:1001.0016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    F. Passerini and K. Zarembo, Wilson loops in N = 2 super-Yang-Mills from matrix model, JHEP 09 (2011) 102 [Erratum ibid. 1110 (2011) 065] [arXiv:1106.5763] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    D. Poland and D. Simmons-Duffin, N = 1 SQCD and the Transverse Field Ising Model, JHEP 02 (2012) 009 [arXiv:1104.1425] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Abhijit Gadde
    • 1
    • 2
  • Pedro Liendo
    • 2
  • Leonardo Rastelli
    • 2
    Email author
  • Wenbin Yan
    • 1
    • 2
  1. 1.California Institute of TechnologyPasadenaU.S.A.
  2. 2.C.N. Yang Institute for Theoretical PhysicsStony Brook UniversityStony BrookU.S.A.

Personalised recommendations