Skip to main content

Massive fermion model in 3d and higher spin currents

A preprint version of the article is available at arXiv.

Abstract

We analyze the 3d free massive fermion theory coupled to external sources. The presence of a mass explicitly breaks parity invariance. We calculate two- and three-point functions of a gauge current and the energy momentum tensor and, for instance, obtain the well-known result that in the IR limit (but also in the UV one) we reconstruct the relevant CS action. We then couple the model to higher spin currents and explicitly work out the spin 3 case. In the UV limit we obtain an effective action which was proposed many years ago as a possible generalization of spin 3 CS action. In the IR limit we derive a different higher spin action. This analysis can evidently be generalized to higher spins. We also discuss the conservation and properties of the correlators we obtain in the intermediate steps of our derivation.

References

  1. [1]

    J.M. Maldacena and G.L. Pimentel, On graviton non-gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  2. [2]

    X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality constraints on corrections to the graviton three-point coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].

    ADS  Article  Google Scholar 

  3. [3]

    Y. Huh, P. Strack and S. Sachdev, Conserved current correlators of conformal field theories in 2 + 1 dimensions, Phys. Rev. B 88 (2013) 155109 [Phys. Rev. B 90 (2014) 199902] [arXiv:1307.6863] [INSPIRE].

  4. [4]

    S.F. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3 − D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  6. [6]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    C. Closset et al., Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091.

    ADS  MathSciNet  Article  Google Scholar 

  8. [8]

    S. Giombi, S. Prakash and X. Yin, A note on CFT correlators in three dimensions, JHEP 07 (2013) 105 [arXiv:1104.4317] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. [9]

    S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wadia and X. Yin, Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].

    ADS  Article  Google Scholar 

  10. [10]

    K.S. Babu, A.K. Das and P. Panigrahi, Derivative expansion and the induced Chern-Simons term at finite temperature in (2 + 1)-dimensions, Phys. Rev. D 36 (1987) 3725 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  11. [11]

    G.V. Dunne, Aspects of Chern-Simons theory, hep-th/9902115 [INSPIRE].

  12. [12]

    F.S. Gama, J.R. Nascimento and A. Yu. Petrov, Derivative expansion and the induced Chern-Simons term in N = 1, D = 3 superspace, Phys. Rev. D 93 (2016) 045015 [arXiv:1511.05471] [INSPIRE].

    ADS  Google Scholar 

  13. [13]

    X. Bekaert, E. Joung and J. Mourad, Effective action in a higher-spin background, JHEP 02 (2011) 048 [arXiv:1012.2103] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    E. Witten, Anomalies revisited, lecture at Strings 2015, June 22, ICTS-TITR, Bangalore, India (2015).

  15. [15]

    E. Witten, Fermion path integrals and topological phases, arXiv:1508.04715 [INSPIRE].

  16. [16]

    L. Bonora and B.L. de Souza, Pure contact term correlators in CFT, arXiv:1511.06635 [INSPIRE].

  17. [17]

    L. Bonora, S. Giaccari and B. Lima de Souza, Trace anomalies in chiral theories revisited, JHEP 07 (2014) 117 [arXiv:1403.2606] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  18. [18]

    L. Bonora, S. Giaccari and B.L.D. Souza, Revisiting trace anomalies in chiral theories, Springer Proc. Math. Stat. 111 (2014) 3.

    Article  MATH  Google Scholar 

  19. [19]

    L. Bonora, A.D. Pereira and B. Lima de Souza, Regularization of energy-momentum tensor correlators and parity-odd terms, JHEP 06 (2015) 024 [arXiv:1503.03326] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  20. [20]

    I. Vuorio, Parity violation and the effective gravitational action in three-dimensions, Phys. Lett. B 175 (1986) 176 [INSPIRE].

    ADS  Article  Google Scholar 

  21. [21]

    C.N. Pope and P.K. Townsend, Conformal higher spin in (2 + 1)-dimensions, Phys. Lett. B 225 (1989) 245 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  22. [22]

    E.E. Boos and A.I. Davydychev, A method of evaluating massive Feynman integrals, Theor. Math. Phys. 89 (1991) 1052 [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    A.I. Davydychev, A simple formula for reducing Feynman diagrams to scalar integrals, Phys. Lett. B 263 (1991) 107 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. [24]

    A.I. Davydychev. Recursive algorithm of evaluating vertex type Feynman integrals, J. Phys. A 25 (1992) 5587.

    ADS  MathSciNet  MATH  Google Scholar 

  25. [25]

    R. Argurio et al., Higher-spin gauge theories. Proceedings of the First Solvay Workshop, held in Brussels on May 12-14, 2004, Int. Solvay Institutes (2006).

  26. [26]

    D. Sorokin, Introduction to the classical theory of higher spins, AIP Conf. Proc. 767 (2005) 172 [hep-th/0405069] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    D. Francia and A. Sagnotti, Higher-spin geometry and string theory, J. Phys. Conf. Ser. 33 (2006) 57 [hep-th/0601199].

    ADS  Article  Google Scholar 

  28. [28]

    A. Fotopoulos and M. Tsulaia, Gauge invariant lagrangians for free and interacting higher spin fields. A review of the BRST formulation, Int. J. Mod. Phys. A 24 (2009) 1 [arXiv:0805.1346] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. [29]

    C. Iazeolla, On the algebraic structure of higher-spin field equations and new exact solutions, arXiv:0807.0406 [INSPIRE].

  30. [30]

    A. Campoleoni, Metric-like lagrangian formulations for higher-spin fields of mixed symmetry, Riv. Nuovo Cim. 033 (2010) 123 [arXiv:0910.3155].

    Google Scholar 

  31. [31]

    A. Sagnotti, Higher spins and current exchanges, arXiv:1002.3388.

  32. [32]

    D. Francia, Low-spin models for higher-spin Lagrangians, Prog. Theor. Phys. Suppl. 188 (2011) 94 [arXiv:1103.0683] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  33. [33]

    A. Campoleoni, Higher spins in D = 2 + 1, Subnucl. Ser. 49 (2013) 385 [arXiv:1110.5841] [INSPIRE].

    MATH  Google Scholar 

  34. [34]

    M.P. Blencowe, A consistent interacting massless higher spin field theory in D = (2 + 1), Class. Quant. Grav. 6 (1989) 443 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].

  36. [36]

    B. de Wit and D.Z. Freedman, Systematics of higher spin gauge fields, Phys. Rev. D 21 (1980) 358 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  37. [37]

    T. Damour and S. Deser, ‘Geometry’ of spin 3 gauge theories, Annales Poincaré Phys. Theor. 47 (1987) 277 [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  38. [38]

    L. Parker and D. Toms, Quantum field theory in curved spacetime, Cambridge University Press, Cambridge U.K. (2009).

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to L. Bonora.

Additional information

ArXiv ePrint: 1602.07178

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bonora, L., Cvitan, M., Prester, P.D. et al. Massive fermion model in 3d and higher spin currents. J. High Energ. Phys. 2016, 72 (2016). https://doi.org/10.1007/JHEP05(2016)072

Download citation

Keywords

  • Higher Spin Gravity
  • Chern-Simons Theories
  • Field Theories in Lower Dimensions