Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences

  • Daniel N. Blaschke
  • Harald Grosse
  • Jean-Christophe Wallet
Open Access
Article

Abstract

In this work we clarify some properties of the one-loop IR divergences in nonAbelian gauge field theories on non-commutative 4-dimensional Moyal space. Additionally, we derive the tree-level Slavnov-Taylor identities relating the two, three and four point functions, and verify their consistency with the divergent one-loop level results. We also discuss the special case of two dimensions.

Keywords

Non-Commutative Geometry Gauge Symmetry Renormalization Regularization and Renormalons BRST Symmetry 

References

  1. [1]
    H. Groenewold, On the Principles of elementary quantum mechanics, Physica 12 (1946) 405 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  2. [2]
    J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45 (1949) 99.MathSciNetADSMATHCrossRefGoogle Scholar
  3. [3]
    H. Grosse, G. Lechner, T. Ludwig and R. Verch, Wick Rotation for Quantum Field Theories on Degenerate Moyal Space(-Time), J. Math. Phys. 54 (2013) 022307 [arXiv:1111.6856] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    J.M. Gracia-Bondia, F. Lizzi, G. Marmo and P. Vitale, Infinitely many star products to play with, JHEP 04 (2002) 026 [hep-th/0112092] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    R.J. Szabo, Quantum field theory on noncommutative spaces, Phys. Rept. 378 (2003) 207 [hep-th/0109162] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  6. [6]
    V. Rivasseau, Non-commutative Renormalization, arXiv:0705.0705 [INSPIRE].
  7. [7]
    J.C. Wallet, Noncommutative Induced Gauge Theories on Moyal Spaces, J. Phys. Conf. Ser. 103 (2008) 012007 [arXiv:0708.2471] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    D.N. Blaschke, E. Kronberger, R.I. Sedmik and M. Wohlgenannt, Gauge Theories on Deformed Spaces, SIGMA 6 (2010) 062 [arXiv:1004.2127] [INSPIRE].MathSciNetGoogle Scholar
  9. [9]
    S. Minwalla, M. Van Raamsdonk and N. Seiberg, Noncommutative perturbative dynamics, JHEP 02 (2000) 020 [hep-th/9912072] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A. Matusis, L. Susskind and N. Toumbas, The IR/UV connection in the noncommutative gauge theories, JHEP 12 (2000) 002 [hep-th/0002075] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    H. Grosse and R. Wulkenhaar, Renormalization of ϕ 4 theory on noncommutative R 2 in the matrix base, JHEP 12 (2003) 019 [hep-th/0307017] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    H. Grosse and R. Wulkenhaar, Renormalization of ϕ 4 theory on noncommutative R 4 in the matrix base, Commun. Math. Phys. 256 (2005) 305 [hep-th/0401128] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  13. [13]
    H. Grosse and R. Wulkenhaar, The β-function in duality covariant noncommutative ϕ 4 theory, Eur. Phys. J. C 35 (2004) 277 [hep-th/0402093] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    M. Disertori and V. Rivasseau, Two and three loops β-function of non commutative \( \phi_4^4 \) theory, Eur. Phys. J. C 50 (2007) 661 [hep-th/0610224] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    M. Disertori, R. Gurau, J. Magnen and V. Rivasseau, Vanishing of β-function of Non Commutative \( \phi_4^4 \) Theory to all orders, Phys. Lett. B 649 (2007) 95 [hep-th/0612251] [INSPIRE].MathSciNetADSGoogle Scholar
  16. [16]
    E. Langmann and R.J. Szabo, Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B 533 (2002) 168 [hep-th/0202039] [INSPIRE].MathSciNetADSGoogle Scholar
  17. [17]
    H. Grosse and R. Wulkenhaar, Self-dual noncommutative ϕ 4 -theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory, arXiv:1205.0465 [INSPIRE].
  18. [18]
    H. Grosse and R. Wulkenhaar, 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory, J. Geom. Phys. 62 (2012) 1583 [arXiv:0709.0095] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  19. [19]
    J.C. Wallet, Connes distance by examples: Homothetic spectral metric spaces, Rev. Math. Phys. 24 (2012) 1250027 [arXiv:1112.3285] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    E. Cagnache, E. Jolibois and J.C. Wallet, Spectral distances: Results for Moyal plane and noncommutative torus, SIGMA 6 (2010) 026 [arXiv:0912.4185] [INSPIRE].Google Scholar
  21. [21]
    E. Cagnache, F. D’Andrea, P. Martinetti and J.C. Wallet, The Spectral distance on the Moyal plane, J. Geom. Phys. 61 (2011) 1881 [arXiv:0912.0906] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  22. [22]
    M. Burić and M. Wohlgenannt, Geometry of the Grosse-Wulkenhaar Model, JHEP 03 (2010) 053 [arXiv:0902.3408] [INSPIRE].ADSGoogle Scholar
  23. [23]
    M. Burić, H. Grosse and J. Madore, Gauge fields on noncommutative geometries with curvature, JHEP 07 (2010) 010 [arXiv:1003.2284] [INSPIRE].ADSGoogle Scholar
  24. [24]
    A. de Goursac, On the origin of the harmonic term in noncommutative quantum field theory, SIGMA 6 (2010) 048 [arXiv:1003.5788] [INSPIRE].Google Scholar
  25. [25]
    H. Grosse and F. Vignes-Tourneret, Quantum field theory on the degenerate Moyal space, J. Noncommut. Geom. 4 (2010) 555 [arXiv:0803.1035] [INSPIRE].MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, A Translation-invariant renormalizable non-commutative scalar model, Commun. Math. Phys. 287 (2009) 275 [arXiv:0802.0791] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  27. [27]
    A. de Goursac and J.C. Wallet, Symmetries of noncommutative scalar field theory, J. Phys. A 44 (2011) 055401 [arXiv:0911.2645] [INSPIRE].ADSGoogle Scholar
  28. [28]
    H. Grosse and M. Wohlgenannt, Induced gauge theory on a noncommutative space, Eur. Phys. J. C 52 (2007) 435 [hep-th/0703169] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    A. de Goursac, J.C. Wallet and R. Wulkenhaar, Noncommutative Induced Gauge Theory, Eur. Phys. J. C 51 (2007) 977 [hep-th/0703075] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    A. de Goursac, A. Tanasa and J.C. Wallet, Vacuum configurations for renormalizable non-commutative scalar models, Eur. Phys. J. C 53 (2008) 459 [arXiv:0709.3950] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A. de Goursac, J.C. Wallet and R. Wulkenhaar, On the vacuum states for noncommutative gauge theory, Eur. Phys. J. C 56 (2008) 293 [arXiv:0803.3035] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    D.N. Blaschke, H. Grosse and M. Schweda, Non-commutative U(1) gauge theory on R 4 (Theta) with oscillator term and BRST symmetry, Europhys. Lett. 79 (2007) 61002 [arXiv:0705.4205] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    D.N. Blaschke, H. Grosse, E. Kronberger, M. Schweda and M. Wohlgenannt, Loop Calculations for the Non-Commutative U(1) Gauge Field Model with Oscillator Term, Eur. Phys. J. C 67 (2010) 575 [arXiv:0912.3642] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    D.N. Blaschke, F. Gieres, E. Kronberger, M. Schweda and M. Wohlgenannt, Translation-invariant models for non-commutative gauge fields, J. Phys. A 41 (2008) 252002 [arXiv:0804.1914] [INSPIRE].MathSciNetADSGoogle Scholar
  35. [35]
    L.C.Q. Vilar, O.S. Ventura, D.G. Tedesco and V.E.R. Lemes, On the Renormalizability of Noncommutative U(1) Gauge Theoryan Algebraic Approach, J. Phys. A 43 (2010) 135401 [arXiv:0902.2956] [INSPIRE].MathSciNetADSGoogle Scholar
  36. [36]
    D.N. Blaschke, A. Rofner, R.I. Sedmik and M. Wohlgenannt, On Non-Commutative U (1) Gauge Models and Renormalizability, J. Phys. A 43 (2010) 425401 [arXiv:0912.2634] [INSPIRE].MathSciNetADSGoogle Scholar
  37. [37]
    D.N. Blaschke, A New Approach to Non-Commutative U (N) Gauge Fields, Europhys. Lett. 91 (2010) 11001 [arXiv:1005.1578] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    G. Marmo, P. Vitale and A. Zampini, Noncommutative differential calculus for Moyal subalgebras, J. Geom. Phys. 56 (2006) 611 [hep-th/0411223] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  39. [39]
    J.C. Wallet, Derivations of the Moyal algebra and noncommutative gauge theories, SIGMA 5 (2009) 013 [arXiv:0811.3850] [INSPIRE].MathSciNetGoogle Scholar
  40. [40]
    A. de Goursac, T. Masson and J.C. Wallet, Noncommutative epsilon-graded connections, J. Noncommut. Geom. 6 (2012) 343 [arXiv:0811.3567] [INSPIRE].MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    D.N. Blaschke et al., On the Problem of Renormalizability in Non-Commutative Gauge Field Models: A Critical Review, Fortsch. Phys. 58 (2010) 364 [arXiv:0908.0467] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  42. [42]
    D.N. Blaschke et al., On the Renormalization of Non-Commutative Field Theories, Eur. Phys. J. C 73 (2013) 2262 [arXiv:1207.5494] [INSPIRE].ADSGoogle Scholar
  43. [43]
    A. Armoni, Comments on perturbative dynamics of noncommutative Yang-Mills theory, Nucl. Phys. B 593 (2001) 229 [hep-th/0005208] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    F.R. Ruiz, Gauge fixing independence of IR divergences in noncommutative U(1), perturbative tachyonic instabilities and supersymmetry, Phys. Lett. B 502 (2001) 274 [hep-th/0012171] [INSPIRE].ADSGoogle Scholar
  45. [45]
    V. Gribov, Quantization of Nonabelian Gauge Theories, Nucl. Phys. B 139 (1978) 1 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    D. Zwanziger, Local and renormalizable action from the Gribov horizon, Nucl. Phys. B 323 (1989) 513 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    D. Zwanziger, Renormalizability of the critical limit of lattice gauge theory by BRS invariance, Nucl. Phys. B 399 (1993) 477 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    L. Baulieu and S. Sorella, Soft breaking of BRST invariance for introducing non-perturbative infrared effects in a local and renormalizable way, Phys. Lett. B 671 (2009) 481 [arXiv:0808.1356] [INSPIRE].MathSciNetADSGoogle Scholar
  49. [49]
    J. Ader and J.C. Wallet, Gauged BRST symmetry for the free bosonic string, Phys. Lett. B 192 (1987) 103 [INSPIRE].MathSciNetADSGoogle Scholar
  50. [50]
    M. Abud, J. Ader and J.C. Wallet, The gauged BRST symmetry, Annals Phys. 203 (1990) 339 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  51. [51]
    J.C. Wallet, Algebraic setup for the gauge fixing of BF and super BF systems, Phys. Lett. B 235 (1990) 71 [INSPIRE].MathSciNetADSGoogle Scholar
  52. [52]
    L. Baulieu, M.P. Bellon, S. Ouvry and J.C. Wallet, Balatin-Vilkovisky analysis of supersymmetric systems, Phys. Lett. B 252 (1990) 387 [INSPIRE].MathSciNetADSGoogle Scholar
  53. [53]
    R. Stora, F. Thuillier and J.C. Wallet, Algebraic structure of cohomological field theory models and equivariant cohomology, in Infinite dimensional geometry, non commutative geometry, operator algebras, fundamental interactions, p.266-297, Cambridge Press (1995)Google Scholar
  54. [54]
    D.N. Blaschke, Towards Consistent Non-Commutative Gauge Theories, Ph.D. thesis, Vienna University of Technology (2008), http://media.obvsg.at/AC05036560.
  55. [55]
    E. Cagnache, T. Masson and J.C. Wallet, Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus, J. Noncommut. Geom. 5 (2011) 39 [arXiv:0804.3061] [INSPIRE].MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    Y. Frishman and J. Sonnenschein, Bosonization and QCD in two-dimensions, Phys. Rept. 223 (1993) 309 [hep-th/9207017] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    M. Attems et al., Gauge independence of IR singularities in non-commutative QFT: And interpolating gauges, JHEP 07 (2005) 071 [hep-th/0506117] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    J.S. Schwinger, Gauge Invariance and Mass. 2., Phys. Rev. 128 (1962) 2425 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  59. [59]
    F. Ardalan, M. Ghasemkhani and N. Sadooghi, On the mass spectrum of noncommutative Schwinger model in Euclidean \( \mathbb{R} \) 2 space, Eur. Phys. J. C 71 (2011) 1606 [arXiv:1011.4877] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    A. Armoni, Noncommutative Two-Dimensional Gauge Theories, Phys. Lett. B 704 (2011) 627 [arXiv:1107.3651] [INSPIRE].MathSciNetADSGoogle Scholar

Copyright information

© SISSA 2013

Authors and Affiliations

  • Daniel N. Blaschke
    • 1
  • Harald Grosse
    • 2
  • Jean-Christophe Wallet
    • 3
  1. 1.Theory DivisionLos Alamos National LaboratoryLos AlamosUSA
  2. 2.Faculty of PhysicsUniversity of ViennaViennaAustria
  3. 3.Laboratoire de Physique ThéoriqueBât. 210 CNRS and Université Paris-Sud 11Orsay CedexFrance

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