Advertisement

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
First Online:

Abstract

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.

Keywords

Conformal Field Theory Effective Field Theories Renormalization Group 
Download to read the full article text

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

References

  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]
    https://en.wikipedia.org/wiki/Kermit 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

Personalised recommendations