Bootstrapping the O(N ) vector models

Open Access
Article

Abstract

We study the conformal bootstrap for 3D CFTs with O(N ) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N ) singlet and symmetric tensor operators appearing in the ϕi × ϕj OPE, where ϕi is a fundamental of O(N ). Comparing these bounds to previous determinations of critical exponents in the O(N ) vector models, we find strong numerical evidence that the O(N ) vector models saturate the bootstrap constraints at all values of N . We also compute general lower bounds on the central charge, giving numerical predictions for the values realized in the O(N ) vector models. We compare our predictions to previous computations in the 1/N expansion, finding precise agreement at large values of N .

Keywords

Conformal and W Symmetry Global Symmetries 

References

  1. [1]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].MathSciNetGoogle Scholar
  3. [3]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    F. Caracciolo and V.S. Rychkov, Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].ADSGoogle Scholar
  6. [6]
    D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    R. Rattazzi, S. Rychkov and A. Vichi, Central charge bounds in 4D conformal field theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].ADSGoogle Scholar
  8. [8]
    R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    A. Vichi, Improved bounds for CFTs with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S. Rychkov, Conformal bootstrap in three dimensions?, arXiv:1111.2115 [INSPIRE].
  12. [12]
    S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].ADSGoogle Scholar
  13. [13]
    P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    S. El-Showk and M.F. Paulos, Bootstrapping conformal field theories with the extremal functional method, arXiv:1211.2810 [INSPIRE].
  15. [15]
    C. Beem, L. Rastelli and B.C. van Rees, The \( \mathcal{N} \) = 4 superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    S. El-Showk et al., Solving the 3d Ising model with the conformal bootstrap II. c-minimization and precise critical exponents, arXiv:1403.4545 [INSPIRE].
  17. [17]
    E. Brézin, D.J. Wallace and K. Wilson, Feynman-graph expansion for the equation of state near the critical point, Phys. Rev. B 7 (1973) 232 [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    K.G. Wilson and J.B. Kogut, The renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Pelissetto and E. Vicari, Critical phenomena and renormalization group theory, Phys. Rept. 368 (2002) 549 [cond-mat/0012164] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    M. Moshe and J. Zinn-Justin, Quantum field theory in the large-N limit: a review, Phys. Rept. 385 (2003) 69 [hep-th/0306133] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N ) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    K. Lang and W. Rühl, Field algebra for critical O(N ) vector nonlinear σ-models at 2 < d < 4, Z. Phys. C 50 (1991) 285 [INSPIRE].Google Scholar
  23. [23]
    K. Lang and W. Rühl, Anomalous dimensions of tensor fields of arbitrary rank for critical nonlinear O(N ) σ-models at 2 < d < 4 to first order in 1/N , Z. Phys. C 51 (1991) 127 [INSPIRE].Google Scholar
  24. [24]
    K. Lang and W. Rühl, The critical O(N ) σ-model at dimension 2 < d < 4 and order 1/N 2 : operator product expansions and renormalization, Nucl. Phys. B 377 (1992) 371 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    K. Lang and W. Rühl, The scalar ancestor of the energy momentum field in critical σ-models at 2 < d < 4, Phys. Lett. B 275 (1992) 93 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    K. Lang and W. Rühl, The critical O(N ) σ-model at dimensions 2 < d < 4: fusion coefficients and anomalous dimensions, Nucl. Phys. B 400 (1993) 597 [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    K. Lang and W. Rühl, Critical nonlinear O(N ) σ-models at 2 < d < 4: the degeneracy of quasiprimary fields and it resolution, Z. Phys. C 61 (1994) 495 [INSPIRE].ADSGoogle Scholar
  28. [28]
    A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N ) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  29. [29]
    A.C. Petkou, C(T ) and C(J ) up to next-to-leading order in 1/N in the conformally invariant O(N ) vector model for 2 < d < 4, Phys. Lett. B 359 (1995) 101 [hep-th/9506116] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    A. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.CrossRefMathSciNetGoogle Scholar
  34. [34]
    S. Ferrara, R. Gatto and A.F. Grillo, Positivity restrictions on anomalous dimensions, Phys. Rev. D 9 (1974) 3564 [INSPIRE].ADSGoogle Scholar
  35. [35]
    G. Mack, All unitary ray representations of the conformal group SU(2,2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  36. [36]
    R.R. Metsaev, Massless mixed symmetry bosonic free fields in d-dimensional anti-de Sitter space-time, Phys. Lett. B 354 (1995) 78 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].Google Scholar
  38. [38]
    B. Grinstein, K.A. Intriligator and I.Z. Rothstein, Comments on unparticles, Phys. Lett. B 662 (2008) 367 [arXiv:0801.1140] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  41. [41]
    M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].ADSGoogle Scholar
  42. [42]
    F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [INSPIRE].
  43. [43]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  44. [44]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    M. Hasenbusch, Finite size scaling study of lattice models in the three-dimensional Ising universality class, Phys. Rev. B 82 (2010) 174433 [arXiv:1004.4486].ADSCrossRefGoogle Scholar
  47. [47]
    M. Campostrini, A. Pelissetto, P. Rossi and E. Vicari, 25th order high temperature expansion results for three-dimensional Ising like systems on the simple cubic lattice, Phys. Rev. E 65 (2002) 066127 [cond-mat/0201180] [INSPIRE].ADSGoogle Scholar
  48. [48]
    M. Campostrini, M. Hasenbusch, A. Pelissetto and E. Vicari, Theoretical estimates of the critical exponents of the superfluid transition in 4 He by lattice methods, Phys. Rev. B 74 (2006) 144506 [cond-mat/0605083] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    P. Calabrese and P. Parruccini, Harmonic crossover exponents in O(n) models with the pseudo-ϵ expansion approach, Phys. Rev. B 71 (2005) 064416 [cond-mat/0411027] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi and E. Vicari, Critical exponents and equation of state of the three-dimensional Heisenberg universality class, Phys. Rev. B 65 (2002) 144520 [cond-mat/0110336] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    M. Hasenbusch, Eliminating leading corrections to scaling in the three-dimensional O(N ) symmetric ϕ 4 model: N = 3 and N = 4, J. Phys. A 34 (2001) 8221 [cond-mat/0010463] [INSPIRE].ADSMathSciNetGoogle Scholar
  52. [52]
    A. Butti and F. Parisen Toldin, The critical equation of state of the three-dimensional O(N ) universality class: N > 4, Nucl. Phys. B 704 (2005) 527 [hep-lat/0406023] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  53. [53]
    J.A. Gracey, Crossover exponent in O(N ) ϕ 4 theory at O(1/N 2), Phys. Rev. E 66 (2002) 027102 [cond-mat/0206098] [INSPIRE].ADSGoogle Scholar
  54. [54]
    A.L. Fitzpatrick, J. Kaplan and D. Poland, Conformal blocks in the large D limit, JHEP 08 (2013) 107 [arXiv:1305.0004] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  55. [55]
    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  56. [56]
    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    G. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
  58. [58]
    G. Mack, D-dimensional conformal field theories with anomalous dimensions as dual resonance models, Bulg. J. Phys. 36 (2009) 214 [arXiv:0909.1024] [INSPIRE].MATHMathSciNetGoogle Scholar
  59. [59]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  60. [60]
    A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A natural language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  62. [62]
    A.L. Fitzpatrick and J. Kaplan, Analyticity and the holographic S-matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    A.L. Fitzpatrick and J. Kaplan, Unitarity and the holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    M.F. Paulos, M. Spradlin and A. Volovich, Mellin amplitudes for dual conformal integrals, JHEP 08 (2012) 072 [arXiv:1203.6362] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  65. [65]
    A.L. Fitzpatrick and J. Kaplan, AdS field theory from conformal field theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  66. [66]
    L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38 (1996) 49.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of PhysicsYale UniversityNew HavenU.S.A
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

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