Advertisement

On renormalization group flows and the a-theorem in 6d

  • Henriette Elvang
  • Daniel Z. Freedman
  • Ling-Yan Hung
  • Michael Kiermaier
  • Robert C. Myers
  • Stefan Theisen
Article

Abstract

We study the extension of the approach to the a-theorem of Komargodski and Schwimmer to quantum field theories in d = 6 spacetime dimensions. The dilaton effective action is obtained up to 6th order in derivatives. The anomaly flow a UVa IR is the coefficient of the 6-derivative Euler anomaly term in this action. It then appears at order p 6 in the low energy limit of n-point scattering amplitudes of the dilaton for n ≥ 4. The detailed structure with the correct anomaly coefficient is confirmed by direct calculation in two examples: (i) the case of explicitly broken conformal symmetry is illustrated by the free massive scalar field, and (ii) the case of spontaneously broken conformal symmetry is demonstrated by the (2,0) theory on the Coulomb branch. In the latter example, the dilaton is a dynamical field so 4-derivative terms in the action also affect n-point amplitudes at order p 6. The calculation in the (2,0) theory is done by analyzing an M5-brane probe in AdS7 × S 4.

Keywords

Ective Action Conformal Symmetry Coulomb Branch Weyl Transformation Paneitz Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    A. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [INSPIRE].MathSciNetADSGoogle Scholar
  2. [2]
    J.L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].MathSciNetADSGoogle Scholar
  3. [3]
    D. Anselmi, J. Erlich, D. Freedman and A. Johansen, Positivity constraints on anomalies in supersymmetric gauge theories, Phys. Rev. D 57 (1998) 7570 [hep-th/9711035] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    D. Anselmi, D. Freedman, M.T. Grisaru and A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    E. Barnes, K.A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4D a-theorem, via a-maximization along RG flows, Nucl. Phys. B 702 (2004) 131 [hep-th/0408156] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N = 4 super Yang-Mills from AdS dynamics, JHEP 12 (1998) 022 [hep-th/9810126] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, The supergravity dual of N = 1 super Yang-Mills theory, Nucl. Phys. B 569 (2000) 451 [hep-th/9909047] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    D. Freedman, S. Gubser, K. Pilch and N. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].MathSciNetMATHGoogle Scholar
  10. [10]
    R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].ADSGoogle Scholar
  11. [11]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    J.T. Liu, W. Sabra and Z. Zhao, Holographic c-theorems and higher derivative gravity, Phys. Rev. D 85 (2012) 126004 [arXiv:1012.3382] [INSPIRE].ADSGoogle Scholar
  13. [13]
    Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    Z. Komargodski, The constraints of conformal symmetry on RG flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the asymptotics of 4D quantum field theory, arXiv:1204.5221 [INSPIRE].
  16. [16]
    M.J. Duff, Observations on conformal anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  18. [18]
    S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrarydimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].MathSciNetADSGoogle Scholar
  19. [19]
    D. Anselmi, Anomalies, unitarity and quantum irreversibility, Annals Phys. 276 (1999) 361 [hep-th/9903059] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  20. [20]
    D. Anselmi, Quantum irreversibility in arbitrary dimension, Nucl. Phys. B 567 (2000) 331 [hep-th/9905005] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  21. [21]
    D. Dorigoni and V.S. Rychkov, Scale invariance + unitarity ⇒ conformal invariance?, arXiv:0910.1087 [INSPIRE].
  22. [22]
    S. El-Showk, Y. Nakayama and S. Rychkov, What Maxwell theory in D ≠ 4 teaches us about scale and conformal invariance, Nucl. Phys. B 848 (2011) 578 [arXiv:1101.5385] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    R. Jackiw and S.-Y. Pi, Tutorial on scale and conformal symmetries in diverse dimensions, J. Phys. A 44 (2011) 223001 [arXiv:1101.4886] [INSPIRE].MathSciNetADSGoogle Scholar
  24. [24]
    I. Antoniadis and M. Buican, On R-symmetric fixed points and superconformality, Phys. Rev. D 83 (2011) 105011 [arXiv:1102.2294] [INSPIRE].ADSGoogle Scholar
  25. [25]
    Y. Nakayama, Comments on scale invariant but non-conformal supersymmetric field theories, Int. J. Mod. Phys. A 27 (2012) 1250122 [arXiv:1109.5883] [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    Y. Nakayama, On ϵ-conjecture in a-theorem, Mod. Phys. Lett. A 27 (2012) 1250029 [arXiv:1110.2586] [INSPIRE].MathSciNetADSGoogle Scholar
  27. [27]
    J.-F. Fortin, B. Grinstein and A. Stergiou, Scale without conformal invariance: an example, Phys. Lett. B 704 (2011) 74 [arXiv:1106.2540] [INSPIRE].MathSciNetADSGoogle Scholar
  28. [28]
    J.-F. Fortin, B. Grinstein and A. Stergiou, Scale without conformal invariance: theoretical foundations, JHEP 07 (2012) 025 [arXiv:1107.3840] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    J.-F. Fortin, B. Grinstein and A. Stergiou, Scale without conformal invariance at three loops, JHEP 08 (2012) 085 [arXiv:1202.4757] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    A. Logunov et al., Dispersion relation for the 3 → 3 forward amplitude and generalized optical theorem, Theor. Math. Phys. 33 (1978) 935 [Teor. Mat. Fiz. 33 (1977) 149] [INSPIRE].Google Scholar
  32. [32]
    R.J. Eden et al., The analytic S-matrix, Cambridge University Press, Cambridge U.K. (1966).Google Scholar
  33. [33]
    T. Maxfield and S. Sethi, The conformal anomaly of M 5-branes, JHEP 06 (2012) 075 [arXiv:1204.2002] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    F. Bastianelli, S. Frolov and A.A. Tseytlin, Conformal anomaly of (2, 0) tensor multiplet in six-dimensions and AdS/CFT correspondence, JHEP 02 (2000) 013 [hep-th/0001041] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    A. Schwimmer and S. Theisen, Spontaneous breaking of conformal invariance and trace anomaly matching, Nucl. Phys. B 847 (2011) 590 [arXiv:1011.0696] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    L. Bonora, P. Pasti and M. Bregola, Weyl cocycles, Class. Quant. Grav. 3 (1986) 635 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  39. [39]
    C.R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003) 89 [math/0109089].MathSciNetADSMATHCrossRefGoogle Scholar
  40. [40]
    S.B. Giddings and M. Srednicki, High-energy gravitational scattering and black hole resonances, Phys. Rev. D 77 (2008) 085025 [arXiv:0711.5012] [INSPIRE].ADSGoogle Scholar
  41. [41]
    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    R. Penrose and W. Rindler, Spinors and spacetime, volume 2, Cambridge University Presss, Cambridge U.K. (1986).CrossRefGoogle Scholar
  43. [43]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  44. [44]
    C. Fefferman and C. R. Graham, Conformal Invariants, in Elie Cartan et les Mathématiques daujourd hui Astérisque (1985) 95.Google Scholar
  45. [45]
    C. Fefferman and C.R. Graham, The ambient metric, arXiv:0710.0919.
  46. [46]
    C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  47. [47]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  48. [48]
    I. Buchbinder, A.Y. Petrov and A.A. Tseytlin, Two loop N = 4 super Yang-Mills effective action and interaction between D3-branes, Nucl. Phys. B 621 (2002) 179 [hep-th/0110173] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    A.A. Tseytlin, R 4 terms in 11 dimensions and conformal anomaly of (2, 0) theory, Nucl. Phys. B 584 (2000) 233 [hep-th/0005072] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Entanglement entropy of 3D conformal gauge theories with many flavors, JHEP 05 (2012) 036 [arXiv:1112.5342] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    L.Y. Hung and R.C. Myers, unpublished.Google Scholar
  56. [56]
    C.R. Graham, R. Jenne, L.J. Mason and G.A.J. Sparling, Conformally invariant powers of the Laplacian, I: existence, J. London Math. Soc. 46 (1992) 557.MathSciNetMATHCrossRefGoogle Scholar
  57. [57]
    T. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995) 3671.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    P. Kraus, Lectures on black holes and the AdS 3 /CF T 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].MathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Henriette Elvang
    • 1
  • Daniel Z. Freedman
    • 2
    • 3
    • 4
  • Ling-Yan Hung
    • 5
  • Michael Kiermaier
    • 6
  • Robert C. Myers
    • 5
  • Stefan Theisen
    • 7
  1. 1.Randall Laboratory of Physics, Department of PhysicsUniversity of MichiganAnn ArborU.S.A.
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeU.S.A.
  3. 3.Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeU.S.A.
  4. 4.Stanford Institute for Theoretical Physics, Department of PhysicsStanford UniversityStanfordU.S.A.
  5. 5.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  6. 6.Joseph Henry LaboratoriesPrinceton UniversityPrincetonU.S.A.
  7. 7.Max-Planck-Institut für GravitationsphysikGolmGermany

Personalised recommendations