Skip to main content
Log in

Higher covariant derivative regularization for calculations in supersymmetric theories

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

A variant of the higher covariant derivative regularization is used for calculation of a two-loop β-function for the general renormalizable N = 1 supersymmetric theory. It is shown that the β-function is given by integrals of total derivatives. Partially this can be explained by substituting solutions of Slavnov-Taylor identities into the Schwinger-Dyson equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. t’ Hooft and M. Veltman, “Regularization and Renormalization of Gauge Fields,” Nucl. Phys. B 44, 189–213 (1972).

    Article  Google Scholar 

  2. W. Siegel, “Supersymmetric Dimensional Regularization via Dimensional Reduction,” Phys. Lett. B 84, 193–196 (1979).

    Article  MathSciNet  Google Scholar 

  3. L. V. Avdeev and O. V. Tarasov, “The Three-Loop Beta-Function in the N = 1, 2, 4 Supersymmetric Yang-Mills Theories,” Phys. Lett. B 112, 356–358 (1982).

    Article  Google Scholar 

  4. L. F. Abbott, M. T. Grisaru, and D. Zanon, “Infrared Divergences and a Non-local Gauge for Superspace Yang-Mills Theory,” Nucl. Phys. B 244, 454–468 (1984).

    Article  MathSciNet  Google Scholar 

  5. A. Parkes and P. West, “Finiteness in Rigid Supersymmetric Theories,” Phys. Lett. B 138, 99–104 (1984).

    Article  Google Scholar 

  6. I. Jack, D. R. T. Jones, and C. G. North, “N = 1 Supersymmetry and the Three-Loop Gauge β-Function,” Phys. Lett. B 386, 138–140 (1996).

    Article  Google Scholar 

  7. I. Jack, D. R. T. Jones, and C. G. North, “N = 1 Supersymmetry and the Three-Loop Anomalous Dimension for the Chiral Superfield,” Nucl. Phys. B 473, 308–322 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Jack, D. R. T. Jones, and C. G. North, “Scheme Dependence and the NSVZ β-Function,” Nucl. Phys. B 486, 479–499 (1997).

    Article  Google Scholar 

  9. I. Jack, D. R. T. Jones, and A. Pickering, “The Connection between the DRED and NSVZ Renormalisation Schemes,” Phys. Lett. B 435, 61–66 (1998).

    Article  MathSciNet  Google Scholar 

  10. W. Siegel, “Inconsistency of Supersymmetric Dimensional Regularization,” Phys. Lett. B 94, 37–40 (1980).

    Article  MathSciNet  Google Scholar 

  11. D. Stöckinger, “Regularization by Dimensional Reduction: Consistency, Quantum Action Principle, and Supersymmetry,” J. High Energy Phys., No. 3, 076 (2005).

  12. W. Hollik and D. Stöckinger, “MSSM Higgs-Boson Mass Predictions and Two-Loop Non-supersymmetric Counterterms,” Phys. Lett. B 634, 63–68 (2006).

    Article  Google Scholar 

  13. A. Signer and D. Stöckinger, “Factorization and Regularization by Dimensional Reduction,” Phys. Lett. B 626, 127–138 (2005).

    Article  Google Scholar 

  14. J. Mas, C. Seijas, and M. Pérez-Victoria, “The Beta Function of N = 1 SYM in Differential Renormalization,” J. High Energy Phys., No. 3, 049 (2002).

  15. D. Z. Freedman, K. Johnson, and J. Latorre, “Differential Regularization and Renormalization: A New Method of Calculation in Quantum Field Theory,” Nucl. Phys. B 371, 353–414 (1992).

    Article  MathSciNet  Google Scholar 

  16. A. A. Slavnov, “Invariant Regularization of Non-linear Chiral Theories,” Nucl. Phys. B 31, 301–315 (1971).

    Article  MathSciNet  Google Scholar 

  17. A. A. Slavnov, “Invariant Regularization of Gauge Theories,” Teor. Mat. Fiz. 13(2), 174–177 (1972) [Theor. Math. Phys. 13, 1064–1066 (1972)].

    Google Scholar 

  18. V. K. Krivoshchekov, “Invariant Regularization for Supersymmetric Gauge Theories,” Teor. Mat. Fiz. 36(3), 291–302 (1978) [Theor. Math. Phys. 36, 745–752 (1978)].

    MathSciNet  Google Scholar 

  19. P. West, “Higher Derivative Regulation of Supersymmetric Theories,” Nucl. Phys. B 268, 113–124 (1986).

    Article  Google Scholar 

  20. C. P. Martin and F. Ruiz Ruiz, “Higher Covariant Derivative Pauli-Villars Regularization Does Not Lead to a Consistent QCD,” Nucl. Phys. B 436, 545–581 (1995).

    Article  Google Scholar 

  21. M. Asorey and F. Falceto, “Consistency of the Regularization of Gauge Theories by High Covariant Derivatives,” Phys. Rev. D 54, 5290–5301 (1996).

    Article  Google Scholar 

  22. T. D. Bakeyev and A. A. Slavnov, “Higher Covariant Derivative Regularization Revisited,” Mod. Phys. Lett. A 11, 1539–1554 (1996).

    Article  Google Scholar 

  23. D. J. Gross and F. Wilczek, “Ultraviolet Behavior of Non-Abelian Gauge Theories,” Phys. Rev. Lett. 30, 1343–1346 (1973).

    Article  Google Scholar 

  24. H. D. Politzer, “Reliable Perturbative Results for Strong Interactions?,” Phys. Rev. Lett. 30, 1346–1349 (1973).

    Article  Google Scholar 

  25. P. I. Pronin and K. V. Stepanyantz, “One-Loop Counterterms for Higher Derivative Regularized Lagrangians,” Phys. Lett. B 414, 117–122 (1997).

    Article  Google Scholar 

  26. S. Arnone, T. R. Morris, and O. J. Rosten, “Manifestly Gauge Invariant QED,” J. High Energy Phys., No. 10, 115 (2005).

  27. T. R. Morris and O. J. Rosten, “Manifestly Gauge-Invariant QCD,” J. Phys. A 39, 11657–11681 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  28. O. J. Rosten, “On the Renormalization of Theories of a Scalar Chiral Superfield,” J. High Energy Phys., No. 3, 004 (2010).

  29. S. Arnone, A. Gatti, T. R. Morris, and O. J. Rosten, “Exact Scheme Independence at Two Loops,” Phys. Rev. D 69, 065009 (2004).

    Article  Google Scholar 

  30. T. R. Morris and O. J. Rosten, “A Manifestly Gauge Invariant, Continuum Calculation of the SU(N) Yang-Mills Two-Loop β Function,” Phys. Rev. D 73, 065003 (2006).

    Article  MathSciNet  Google Scholar 

  31. S. Arnone, Y. A. Kubyshin, T. R. Morris, and J. F. Tighe, “Gauge-Invariant Regularization via SU(N|N),” Int. J. Mod. Phys. A 17, 2283–2329 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  32. A. A. Soloshenko and K. V. Stepanyantz, “Three-Loop β-Function for N = 1 Supersymmetric Electrodynamics, Regularized by Higher Derivatives,” arXiv: hep-th/0304083.

  33. A. A. Soloshenko and K. V. Stepanyantz, “Three-Loop β-Function of N = 1 Supersymmetric Electrodynamics Regularized by Higher Derivatives,” Teor. Mat. Fiz. 140(3), 437–459 (2004) [Theor. Math. Phys. 140, 1264–1282 (2004)].

    Google Scholar 

  34. A. Smilga and A. Vainshtein, “Background Field Calculations and Nonrenormalization Theorems in 4d Supersymmetric Gauge Theories and Their Low-Dimensional Descendants,” Nucl. Phys. B 704, 445–474 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  35. K. V. Stepanyantz, “Investigation of the Anomaly Puzzle in N = 1 Supersymmetric Electrodynamics,” Teor. Mat. Fiz. 142(1), 37–57 (2005) [Theor. Math. Phys. 142, 29–47 (2005)].

    Google Scholar 

  36. A. B. Pimenov, A. A. Soloshenko, K. V. Stepanyantz, and E. S. Shevtsova, “Regularization by Higher Derivatives and Quantum Correction for N = 1 Supersymmetric Theories,” Izv. Vyssh. Uchebn. Zaved., Fiz. 51(5), 5–35 (2008) [Russ. Phys. J. 51, 444–479 (2008)].

    MathSciNet  Google Scholar 

  37. A. B. Pimenov, E. S. Shevtsova, and K. V. Stepanyantz, “Calculation of Two-Loop β-Function for General N = 1 Supersymmetric Yang-Mills Theory with the Higher Covariant Derivative Regularization,” Phys. Lett. B 686, 293–297 (2010).

    Article  MathSciNet  Google Scholar 

  38. P. West, Introduction to Supersymmetry and Supergravity (World Sci., Singapore, 1986).

    Google Scholar 

  39. I. L. Buchbinder and S. V. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity (Taylor & Francis, Boca Raton, FL, 1998).

    MATH  Google Scholar 

  40. A. A. Slavnov, “Universal Gauge Invariant Renormalization,” Phys. Lett. B 518, 195–200 (2001).

    Article  MATH  Google Scholar 

  41. A. A. Slavnov, “Regularization-Independent Gauge-Invariant Renormalization of the Yang-Mills Theory,” Teor. Mat. Fiz. 130(1), 3–14 (2002) [Theor. Math. Phys. 130, 1–10 (2002)].

    MathSciNet  Google Scholar 

  42. A. A. Slavnov and K. V. Stepanyantz, “Universal Invariant Renormalization for Supersymmetric Theories,” Teor. Mat. Fiz. 135(2), 265–279 (2003) [Theor. Math. Phys. 135, 673–684 (2003)].

    Google Scholar 

  43. A. A. Slavnov and K. V. Stepanyantz, “Universal Invariant Renormalization for the Supersymmetric Yang-Mills theory,” Teor. Mat. Fiz. 139(2), 179–191 (2004) [Theor. Math. Phys. 139, 599–608 (2004)].

    Google Scholar 

  44. L. D. Faddeev and A. A. Slavnov, Gauge Fields: An Introduction to Quantum Theory (Westview Press, Boulder, CO, 1991).

    Google Scholar 

  45. N. Arkani-Hamed and H. Murayama, “Holomorphy, Rescaling Anomalies and Exact β Functions in Supersymmetric Gauge Theories,” J. High Energy Phys., No. 6, 030 (2000).

  46. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus,” Nucl. Phys. B 229, 381–393 (1983).

    Article  Google Scholar 

  47. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “The Beta Function in Supersymmetric Gauge Theories: Instantons versus Traditional Approach,” Phys. Lett. B 166, 329–333 (1986).

    Article  Google Scholar 

  48. M. A. Shifman and A. I. Vainshtein, “Solution of the Anomaly Puzzle in SUSY Gauge Theories and the Wilson Operator Expansion,” Nucl. Phys. B 277, 456–486 (1986).

    Article  Google Scholar 

  49. A. B. Pimenov and K. V. Stepanyantz, “Four-Loop Verification of an Algorithm for Summing Feynman Diagrams in the N = 1 Supersymmetric Electrodynamics,” Teor. Mat. Fiz. 147(2), 290–302 (2006) [Theor. Math. Phys. 147, 687–697 (2006)].

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. V. Stepanyantz.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Vol. 272, pp. 266–276.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stepanyantz, K.V. Higher covariant derivative regularization for calculations in supersymmetric theories. Proc. Steklov Inst. Math. 272, 256–265 (2011). https://doi.org/10.1134/S008154381101024X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154381101024X

Keywords

Navigation