Journal of High Energy Physics

, 2019:201 | Cite as

The large charge limit of scalar field theories, and the Wilson-Fisher fixed point at 𝜖 = 0

  • G. Arias-Tamargo
  • D. Rodriguez-GomezEmail author
  • J.G. Russo
Open Access
Regular Article - Theoretical Physics


We study the sector of large charge operators ϕn (ϕ being the complexified scalar field) in the O(2) Wilson-Fisher fixed point in 4−E dimensions that emerges when the coupling takes the critical value g ∼ 𝜖. We show that, in the limit g → 0, when the theory naively approaches the gaussian fixed point, the sector of operators with n → ∞ at fixed g n2≡ λ remains non-trivial. Surprisingly, one can compute the exact 2-point function and thereby the non-trivial anomalous dimension of the operator ϕn by a full resummation of Feynman diagrams. The same result can be reproduced from a saddle point approximation to the path integral, which partly explains the existence of the limit. Finally, we extend these results to the three-dimensional O(2)-symmetric theory with (\( \overline{\phi} \)ϕ)3 potential.


Conformal Field Theory Effective Field Theories Renormalization Group 


Open Access

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  1. [1]
    S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT operator spectrum at large global charge, JHEP12 (2015) 071 [arXiv:1505.01537] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  2. [2]
    L. Álvarez-Gaumé, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP04 (2017) 059 [arXiv:1610.04495] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone bosons and CFT data, JHEP06 (2017) 011 [arXiv:1611.02912] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A note on inhomogeneous ground states at large global charge, JHEP10 (2019) 038 [arXiv:1705.05825] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    O. Loukas, D. Orlando and S. Reffert, Matrix models at large charge, JHEP10 (2017) 085 [arXiv:1707.00710] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Banerjee, S. Chandrasekharan and D. Orlando, Conformal dimensions via large charge expansion, Phys. Rev. Lett.120 (2018) 061603 [arXiv:1707.00711] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    S. Hellerman and S. Maeda, On the large R-charge expansion in 𝒩 = 2 superconformal field theories, JHEP12 (2017) 135 [arXiv:1710.07336] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    G. Cuomo et al., Rotating superfluids and spinning charged operators in conformal field theory, Phys. Rev.D 97 (2018) 045012 [arXiv:1711.02108] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    S. Hellerman et al., Universal correlation functions in rank 1 SCFTs, arXiv:1804.01535 [INSPIRE].
  10. [10]
    O. Loukas, D. Orlando, S. Reffert and D. Sarkar, An AdS/EFT correspondence at large charge, Nucl. Phys.B 934 (2018) 437 [arXiv:1804.04151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, Observables in inhomogeneous ground states at large global charge, arXiv:1804.06495 [INSPIRE].
  12. [12]
    A. De La Fuente, The large charge expansion at large N , JHEP08 (2018) 041 [arXiv:1805.00501] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  13. [13]
    S. Favrod, D. Orlando and S. Reffert, The large-charge expansion for Schrödinger systems, JHEP12 (2018) 052 [arXiv:1809.06371] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    D. Banerjee, S. Chandrasekharan, D. Orlando and S. Reffert, Conformal dimensions in the large charge sectors at the O(4) Wilson-Fisher fixed point, Phys. Rev. Lett.123 (2019) 051603 [arXiv:1902.09542] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    D. Orlando, S. Reffert and F. Sannino, A safe CFT at large charge, JHEP08 (2019) 164 [arXiv:1905.00026] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    S.M. Kravec and S. Pal, Nonrelativistic conformal field theories in the large charge sector, JHEP02 (2019) 008 [arXiv:1809.08188] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    S.M. Kravec and S. Pal, The spinful large charge sector of non-relativistic CFTs: from phonons to vortex crystals, JHEP05 (2019) 194 [arXiv:1904.05462] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in 𝒩 = 2 theories, JHEP05 (2018) 074 [arXiv:1803.00580] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    E. Gerchkovitz et al., Correlation functions of Coulomb branch operators, JHEP01 (2017) 103 [arXiv:1602.05971] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Beccaria, On the large R-charge 𝒩 = 2 chiral correlators and the Toda equation, JHEP02 (2019) 009 [arXiv:1809.06280] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    M. Beccaria, Double scaling limit of N = 2 chiral correlators with Maldacena-Wilson loop, JHEP02 (2019) 095 [arXiv:1810.10483] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    A. Grassi, Z. Komargodski and L. Tizzano, Extremal correlators and random matrix theory, arXiv:1908.10306 [INSPIRE].
  23. [23]
    R. Rattazzi, G. Badel, G. Cuomo and A. Monin, Semiclass and multi-leg amplitudes, to appear, presented at 24thRecontres Itzykson — Effective field theory in cosmology, gravitation and particle physics , June 5–7, IPhT CEA Saclay, france (2019).Google Scholar
  24. [24]
    A. Altland and B. Simons, Condensed matter field theory, Cambridge University Press, Cambridge U.K. (2010).CrossRefGoogle Scholar
  25. [25]
    J. Zinn-Justin, Quantum field theory and critical phenomena, International Series of Monographs on Physics, Oxford University Press, U.S.A. (1996).zbMATHGoogle Scholar
  26. [26]
    S. Rychkov and Z.M. Tan, The 𝜖 -expansion from conformal field theory, J. Phys.A 48 (2015) 29FT01 [arXiv:1505.00963] [INSPIRE].
  27. [27]
    M.V. Libanov, V.A. Rubakov, D.T. Son and S.V. Troitsky, Exponentiation of multiparticle amplitudes in scalar theories, Phys. Rev.D 50 (1994) 7553 [hep-ph/9407381] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    M.V. Libanov, D.T. Son and S.V. Troitsky, Exponentiation of multiparticle amplitudes in scalar theories. 2. Universality of the exponent, Phys. Rev.D 52 (1995) 3679 [hep-ph/9503412] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    D.T. Son, Semiclassical approach for multiparticle production in scalar theories, Nucl. Phys.B 477 (1996) 378 [hep-ph/9505338] [INSPIRE].
  30. [30] the FrogGoogle Scholar
  31. [31]
    G. Badel, G. Cuomo, A. Monin and R. Rattazzi, The 𝜖 -expansion meets semiclassics, arXiv:1909.01269 [INSPIRE].
  32. [32]
    M. Watanabe, Accessing large global charge via the E-expansion, arXiv:1909.01337 [INSPIRE].
  33. [33]
    D.Z. Freedman, K. Johnson and J.I. Latorre, Differential regularization and renormalization: a new method of calculation in quantum field theory, Nucl. Phys.B 371 (1992) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    S.M. Chester and S.S. Pufu, Anomalous dimensions of scalar operators in QED3, JHEP08 (2016) 069 [arXiv:1603.05582] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • G. Arias-Tamargo
    • 1
    • 2
  • D. Rodriguez-Gomez
    • 1
    • 2
    Email author
  • J.G. Russo
    • 3
    • 4
  1. 1.Department of PhysicsUniversidad de OviedoOviedoSpain
  2. 2.Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA)OviedoSpain
  3. 3.Institució Catalana de Recerca i Estudis Avançats (ICREA)BarcelonaSpain
  4. 4.Departament de Física Cuántica i Astrofísica and Institut de Cìencies del CosmosUniversitat de BarcelonaBarcelonaSpain

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