Mellin bootstrap for scalars in generic dimension

  • John Golden
  • Daniel R. MayersonEmail author
Open Access
Regular Article - Theoretical Physics


We use the recently developed framework of the Mellin bootstrap to study perturbatively free scalar CFTs in arbitrary dimensions. This approach uses the crossing-symmetric Mellin space formulation of correlation functions to generate algebraic bootstrap equations by demanding that only physical operators contribute to the OPE. We find that there are no perturbatively interacting CFTs with only fundamental scalars in d > 6 dimensions (to at least second order in the perturbation). Our results can be seen as a modest step towards understanding the space of interacting CFTs in d > 6 and are consistent with the intuition that no such CFTs exist.


Conformal Field Theory Field Theories in Higher Dimensions 


Open Access

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  1. [1]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].Google Scholar
  2. [2]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].ADSMathSciNetCrossRefGoogle 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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, Springer Briefs in Physics, Springer, Germany (2016).Google Scholar
  5. [5]
    D. Simmons-Duffin, The conformal bootstrap, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), June 1-26, Boulder, U.S.A. (2015), arXiv:1602.07982 [INSPIRE].
  6. [6]
    D. Poland and D. Simmons-Duffin, The conformal bootstrap, Nature Phys. 12 (2016) 535.ADSCrossRefGoogle Scholar
  7. [7]
    S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
  8. [8]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal bootstrap in Mellin space, Phys. Rev. Lett. 118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
  9. [9]
    R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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].
  11. [11]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
  12. [12]
    M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Dey and A. Kaviraj, Towards a bootstrap approach to higher orders of ϵ-expansion, JHEP 02 (2018) 153 [arXiv:1711.01173] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    P. Dey, A. Kaviraj and A. Sinha, Mellin space bootstrap for global symmetry, JHEP 07 (2017) 019 [arXiv:1612.05032] [INSPIRE].
  15. [15]
    P. Dey, K. Ghosh and A. Sinha, Simplifying large spin bootstrap in Mellin space, JHEP 01 (2018) 152 [arXiv:1709.06110] [INSPIRE].
  16. [16]
    W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    F. Gliozzi, A. Guerrieri, A.C. Petkou and C. Wen, Generalized Wilson-Fisher critical points from the conformal operator product expansion, Phys. Rev. Lett. 118 (2017) 061601 [arXiv:1611.10344] [INSPIRE].
  18. [18]
    F. Gliozzi, A.L. Guerrieri, A.C. Petkou and C. Wen, The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points, JHEP 04 (2017) 056 [arXiv:1702.03938] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].
  20. [20]
    A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of Physics and Leinweber Center for Theoretical PhysicsUniversity of MichiganAnn ArborU.S.A.

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