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Renormalization properties of a Galilean Wess-Zumino model

  • Roberto AuzziEmail author
  • Stefano Baiguera
  • Giuseppe Nardelli
  • Silvia Penati
Open Access
Regular Article - Theoretical Physics
  • 15 Downloads

Abstract

We consider a Galilean \( \mathcal{N}=2 \) supersymmetric theory with F-term couplings in 2 + 1 dimensions, obtained by null reduction of a relativistic Wess-Zumino model. We compute quantum corrections and we check that, as for the relativistic parent theory, the F-term does not receive quantum corrections. Even more, we find evidence that the causal structure of the non-relativistic dynamics together with particle number conservation constrain the theory to be one-loop exact.

Keywords

Field Theories in Lower Dimensions Space-Time Symmetries Superspaces Supersymmetric Effective Theories 

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]
    G.W. Semenoff, Condensed Matter Simulation of a Three-dimensional Anomaly, Phys. Rev. Lett. 53 (1984) 2449 [INSPIRE].CrossRefGoogle Scholar
  2. [2]
    D.P. DiVincenzo and E.J. Mele, Self-consistent effective-mass theory for intralayer screening in graphite intercalation compounds, Phys. Rev. B 29 (1985) 1685 [INSPIRE].
  3. [3]
    A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009) 109 [arXiv:0709.1163].CrossRefGoogle Scholar
  4. [4]
    D. Friedan, Z. Qiu and S.H. Shenker, Superconformal Invariance in Two-Dimensions and the Tricritical Ising Model, Phys. Lett. B 151 (1985) 37.Google Scholar
  5. [5]
    T. Grover, D.N. Sheng and A. Vishwanath, Emergent Space-Time Supersymmetry at the Boundary of a Topological Phase, Science 344 (2014) 280 [arXiv:1301.7449] [INSPIRE].CrossRefGoogle Scholar
  6. [6]
    Y. Yu and K. Yang, Simulating Wess-Zumino Supersymmetry Model in Optical Lattices, Phys. Rev. Lett. 105 (2010) 150605 [arXiv:1005.1399] [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    L. Huijse, B. Bauer and E. Berg, Emergent Supersymmetry at the Ising-Berezinskii-Kosterlitz-Thouless Multicritical Point, Phys. Rev. Lett. 114 (2015) 090404 [arXiv:1403.5565] [INSPIRE].
  8. [8]
    S.-K. Jian, Y.-F. Jiang and H. Yao, Emergent Spacetime Supersymmetry in 3D Weyl Semimetals and 2D Dirac Semimetals, Phys. Rev. Lett. 114 (2015) 237001 [arXiv:1407.4497] [INSPIRE].
  9. [9]
    A. Rahmani, X. Zhu, M. Franz and I. Affleck, Emergent Supersymmetry from Strongly Interacting Majorana Zero Modes, Phys. Rev. Lett. 115 (2015) 166401 [Erratum ibid. 116 (2016) 109901] [arXiv:1504.05192] [INSPIRE].
  10. [10]
    J. Yu, R. Roiban and C.-X. Liu, 2 + 1D Emergent Supersymmetry at First-Order Quantum Phase Transition, arXiv:1902.07407 [INSPIRE].
  11. [11]
    S.-S. Lee, Emergence of supersymmetry at a critical point of a lattice model, Phys. Rev. B 76 (2007) 075103 [cond-mat/0611658] [INSPIRE].
  12. [12]
    C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [INSPIRE].
  13. [13]
    R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D 42 (1990) 3500 [Erratum ibid. D 48 (1993) 3929] [INSPIRE].
  14. [14]
    T. Mehen, I.W. Stewart and M.B. Wise, Conformal invariance for nonrelativistic field theory, Phys. Lett. B 474 (2000) 145 [hep-th/9910025] [INSPIRE].
  15. [15]
    D.B. Kaplan, M.J. Savage and M.B. Wise, A New expansion for nucleon-nucleon interactions, Phys. Lett. B 424 (1998) 390 [nucl-th/9801034] [INSPIRE].
  16. [16]
    Y. Nishida and D.T. Son, Unitary Fermi gas, ϵ-expansion and nonrelativistic conformal field theories, Lect. Notes Phys. 836 (2012) 233 [arXiv:1004.3597] [INSPIRE].
  17. [17]
    M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime Symmetries of the Quantum Hall Effect, Phys. Rev. D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].
  18. [18]
    D.T. Son, Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
  19. [19]
    K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
  20. [20]
    C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann Structures and Newton-cartan Theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
  21. [21]
    R. Puzalowski, Galilean Supersymmetry, Acta Phys. Austriaca 50 (1978) 45 [INSPIRE].
  22. [22]
    T.E. Clark and S.T. Love, Nonrelativistic supersymmetry, Nucl. Phys. B 231 (1984) 91 [INSPIRE].
  23. [23]
    J.A. de Azcarraga and D. Ginestar, Nonrelativistic limit of supersymmetric theories, J. Math. Phys. 32 (1991) 3500 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  24. [24]
    A. Meyer, Y. Oz and A. Raviv-Moshe, On Non-Relativistic Supersymmetry and its Spontaneous Breaking, JHEP 06 (2017) 128 [arXiv:1703.04740] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M. Leblanc, G. Lozano and H. Min, Extended superconformal Galilean symmetry in Chern-Simons matter systems, Annals Phys. 219 (1992) 328 [hep-th/9206039] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    O. Bergman and C.B. Thorn, SuperGalilei invariant field theories in (2 + 1)-dimensions, Phys. Rev. D 52 (1995) 5997 [hep-th/9507007] [INSPIRE].
  27. [27]
    J. Beckers and V. Hussin, Dynamical Supersymmetries of the Harmonic Oscillator, Phys. Lett. A 118 (1986) 319 [INSPIRE].
  28. [28]
    J.P. Gauntlett, J. Gomis and P.K. Townsend, Particle Actions as Wess-Zumino Terms for Space-time (Super)symmetry Groups, Phys. Lett. B 249 (1990) 255 [INSPIRE].
  29. [29]
    C. Duval and P.A. Horvathy, On Schrödinger superalgebras, J. Math. Phys. 35 (1994) 2516 [hep-th/0508079] [INSPIRE].
  30. [30]
    S. Chapman, Y. Oz and A. Raviv-Moshe, On Supersymmetric Lifshitz Field Theories, JHEP 10 (2015) 162 [arXiv:1508.03338] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S.Y. Yong and D.T. Son, Effective field theory for one-dimensional nonrelativistic particles with contact interaction, Phys. Rev. A 97 (2018) 043630 [arXiv:1711.10517] [INSPIRE].
  32. [32]
    E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan (super)gravity as a non-relativistic limit, Class. Quant. Grav. 32 (2015) 205003 [arXiv:1505.02095] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
  34. [34]
    Y. Nakayama, Superfield Formulation for Non-Relativistic Chern-Simons-Matter Theory, Lett. Math. Phys. 89 (2009) 67 [arXiv:0902.2267] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    J. Wess and B. Zumino, Supergauge Transformations in Four-Dimensions, Nucl. Phys. B 70 (1974) 39 [INSPIRE].
  36. [36]
    S.J. Gates, M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].zbMATHGoogle Scholar
  37. [37]
    M.T. Grisaru, W. Siegel and M. Roček, Improved Methods for Supergraphs, Nucl. Phys. B 159 (1979) 429 [INSPIRE].
  38. [38]
    N. Seiberg, Naturalness versus supersymmetric nonrenormalization theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [INSPIRE].
  39. [39]
    S. Weinberg, The quantum theory of fields. Vol. 3: Supersymmetry, Cambridge University Press, Cambridge U.K. (2005).Google Scholar
  40. [40]
    R. Auzzi, S. Baiguera and G. Nardelli, Trace anomaly for non-relativistic fermions, JHEP 08 (2017) 042 [arXiv:1705.02229] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    L.F. Abbott and M.T. Grisaru, The Three Loop β-function for the Wess-Zumino Model, Nucl. Phys. B 169 (1980) 415 [INSPIRE].
  42. [42]
    A. Sen and M.K. Sundaresan, The Four Loop Beta Function for the Wess-Zumino Model, Phys. Lett. B 101 (1981) 61.Google Scholar
  43. [43]
    O. Bergman, Nonrelativistic field theoretic scale anomaly, Phys. Rev. D 46 (1992) 5474 [INSPIRE].
  44. [44]
    K. Jensen, Anomalies for Galilean fields, SciPost Phys. 5 (2018) 005 [arXiv:1412.7750] [INSPIRE].CrossRefGoogle Scholar
  45. [45]
    I. Arav, S. Chapman and Y. Oz, Non-Relativistic Scale Anomalies, JHEP 06 (2016) 158 [arXiv:1601.06795] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    R. Auzzi, S. Baiguera and G. Nardelli, On Newton-Cartan trace anomalies, JHEP 02 (2016) 003 [Erratum ibid. 1602 (2016) 177] [arXiv:1511.08150] [INSPIRE].
  47. [47]
    R. Auzzi and G. Nardelli, Heat kernel for Newton-Cartan trace anomalies, JHEP 07 (2016) 047 [arXiv:1605.08684] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    S. Pal and B. Grinstein, Heat kernel and Weyl anomaly of Schrödinger invariant theory, Phys. Rev. D 96 (2017) 125001 [arXiv:1703.02987] [INSPIRE].
  49. [49]
    S. Pal and B. Grinstein, Weyl Consistency Conditions in Non-Relativistic Quantum Field Theory, JHEP 12 (2016) 012 [arXiv:1605.02748] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    R. Auzzi, S. Baiguera, F. Filippini and G. Nardelli, On Newton-Cartan local renormalization group and anomalies, JHEP 11 (2016) 163 [arXiv:1610.00123] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    Y. Nakayama, S. Ryu, M. Sakaguchi and K. Yoshida, A Family of super Schrödinger invariant Chern-Simons matter systems, JHEP 01 (2009) 006 [arXiv:0811.2461] [INSPIRE].CrossRefzbMATHGoogle Scholar
  52. [52]
    Y. Nakayama, M. Sakaguchi and K. Yoshida, Interacting SUSY-singlet matter in non-relativistic Chern-Simons theory, J. Phys. A 42 (2009) 195402 [arXiv:0812.1564] [INSPIRE].
  53. [53]
    K.-M. Lee, S. Lee and S. Lee, Nonrelativistic Superconformal M2-Brane Theory, JHEP 09 (2009) 030 [arXiv:0902.3857] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  54. [54]
    C. Lopez-Arcos, J. Murugan and H. Nastase, Nonrelativistic limit of the abelianized ABJM model and the AdS/CMT correspondence, JHEP 05 (2016) 165 [arXiv:1510.01662] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    N. Doroud, D. Tong and C. Turner, On Superconformal Anyons, JHEP 01 (2016) 138 [arXiv:1511.01491] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    O. Bergman and G. Lozano, Aharonov-Bohm scattering, contact interactions and scale invariance, Annals Phys. 229 (1994) 416 [hep-th/9302116] [INSPIRE].CrossRefGoogle Scholar
  57. [57]
    N. Doroud, D. Tong and C. Turner, The Conformal Spectrum of Non-Abelian Anyons, SciPost Phys. 4 (2018) 022 [arXiv:1611.05848] [INSPIRE].CrossRefGoogle Scholar
  58. [58]
    C. Turner, Bosonization in Non-Relativistic CFTs, arXiv:1712.07662 [INSPIRE].
  59. [59]
    S.P. Martin, A Supersymmetry primer, Adv. Ser. Direct. High Energy Phys. 21 (2010) 1 [Adv. Ser. Direct. High Energy Phys. 18 (1998) 1] [hep-ph/9709356].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità Cattolica del Sacro CuoreBresciaItaly
  2. 2.INFN Sezione di PerugiaPerugiaItaly
  3. 3.Università degli studi di Milano Bicocca and INFN, Sezione di Milano - BicoccaMilanoItaly
  4. 4.TIFPA - INFN, c/o Dipartimento di FisicaUniversità di TrentoPovoItaly

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