Skip to main content
Log in

A four-dimensional approach to quantum field theories

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

I present a novel Four-Dimensional Regularization/Renormalization approach (FDR) to ultraviolet divergences in field theories which can be interpreted as a natural separation between physical and non physical degrees of freedom. Based on the observation that some infinities can be reabsorbed into the vacuum expectation value of the fields, rather than into the parameters of the Lagrangian, a new type of four-dimensional loop integral is introduced (the FDR integral) which is independent of any UV regulator and respects all properties required by gauge invariance. FDR reproduces the correct ABJ anomaly and no change in the definition of γ5 is needed. With FDR the possibility is open for an approach to UV infinities in which the renormalization program is substituted by a simple reinterpretation of the appearing loop integrals as FDR ones, leading to important consequences in the context of non-renormalizable field theories. Finally, I show how FDR can also be used to regularize infrared and collinear divergences.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Dyson, The S matrix in quantum electrodynamics, Phys. Rev. 75 (1949) 1736 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. F. Dyson, The Radiation theories of Tomonaga, Schwinger and Feynman, Phys. Rev. 75 (1949) 486 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. A. Salam, Overlapping divergences and the S matrix, Phys. Rev. 82 (1951) 217 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. A. Salam, Divergent integrals in renormalizable field theories, Phys. Rev. 84 (1951) 426 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. N. Bogoliubov and O. A. Parasiuk, On the Multiplication of the causal function in the quantum theory of fields, Acta Math. 97 (1957) 227.

    Article  MathSciNet  Google Scholar 

  6. K. Hepp, Proof of the Bogolyubov-Parasiuk theorem on renormalization, Commun. Math. Phys. 2 (1966) 301 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  7. W. Zimmermann, Convergence of Bogolyubovs method of renormalization in momentum space, Commun. Math. Phys. 15 (1969) 208 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  8. W. Pauli and F. Villars, On the Invariant regularization in relativistic quantum theory, Rev. Mod. Phys. 21 (1949) 434 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. E. Speer, On the structure of analytic renormalization, Commun. Math. Phys. 23 (1971) 23 [Erratum ibid. 25 (1972) 336] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. G. ’t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. 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].

    Article  MathSciNet  ADS  Google Scholar 

  12. F. del Aguila, A. Culatti, R. Muñoz-Tapia and M. Pérez-Victoria, Constraining differential renormalization in Abelian gauge theories, Phys. Lett. B 419 (1998) 263 [hep-th/9709067] [INSPIRE].

    ADS  Google Scholar 

  13. F. del Aguila, A. Culatti, R. Muñoz Tapia and M. Pérez-Victoria, Techniques for one loop calculations in constrained differential renormalization, Nucl. Phys. B 537 (1999) 561 [hep-ph/9806451] [INSPIRE].

    Article  ADS  Google Scholar 

  14. O. Battistel, A. Mota and M. Nemes, Consistency conditions for 4 − D regularizations, Mod. Phys. Lett. A 13 (1998) 1597 [INSPIRE].

    ADS  Google Scholar 

  15. A. Cherchiglia, M. Sampaio and M. Nemes, Systematic Implementation of Implicit Regularization for Multi-Loop Feynman Diagrams, Int. J. Mod. Phys. A 26 (2011) 2591 [arXiv:1008.1377] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  16. J. Soln, Renormalization of the scalar field theory with spontaneously broken discrete symmetry without shifting the field vacuum expectation value, Nuovo Cim. A 100 (1988) 671 [INSPIRE].

    Article  ADS  Google Scholar 

  17. Y.-L. Wu, Symmetry preserving loop regularization and renormalization of QFTs, Mod. Phys. Lett. A 19 (2004) 2191 [hep-th/0311082] [INSPIRE].

    ADS  Google Scholar 

  18. R. Pittau, A Simple method for multileg loop calculations, Comput. Phys. Commun. 104 (1997) 23 [hep-ph/9607309] [INSPIRE].

    Article  ADS  Google Scholar 

  19. G. Ossola, C.G. Papadopoulos and R. Pittau, Numerical evaluation of six-photon amplitudes, JHEP 07 (2007) 085 [arXiv:0704.1271] [INSPIRE].

    Article  ADS  Google Scholar 

  20. G. Passarino and M. Veltman, One Loop Corrections for e + e Annihilation Into μ + μ in the Weinberg Model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].

    Article  ADS  Google Scholar 

  21. G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147 [hep-ph/0609007] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: A Program implementing the OPP reduction method to compute one-loop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596] [INSPIRE].

    Article  ADS  Google Scholar 

  23. R. Pittau, Primary Feynman rules to calculate the epsilon-dimensional integrand of any 1-loop amplitude, JHEP 02 (2012) 029 [arXiv:1111.4965] [INSPIRE].

    Article  ADS  Google Scholar 

  24. G. Ossola, C.G. Papadopoulos and R. Pittau, On the Rational Terms of the one-loop amplitudes, JHEP 05 (2008) 004 [arXiv:0802.1876] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. M. Garzelli, I. Malamos and R. Pittau, Feynman rules for the rational part of the Electroweak 1-loop amplitudes, JHEP 01 (2010) 040 [Erratum ibid. 1010 (2010) 097] [arXiv:0910.3130] [INSPIRE].

    Article  ADS  Google Scholar 

  26. P. Draggiotis, M. Garzelli, C. Papadopoulos and R. Pittau, Feynman Rules for the Rational Part of the QCD 1-loop amplitudes, JHEP 04 (2009) 072 [arXiv:0903.0356] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. M. Garzelli, I. Malamos and R. Pittau, Feynman rules for the rational part of the Electroweak 1-loop amplitudes in the R x i gauge and in the Unitary gauge, JHEP 01 (2011) 029 [arXiv:1009.4302] [INSPIRE].

    Article  ADS  Google Scholar 

  28. H.-S. Shao, Y.-J. Zhang and K.-T. Chao, Feynman Rules for the Rational Part of the Standard Model One-loop Amplitudes in thet Hooft-Veltman γ5 Scheme, JHEP 09 (2011) 048 [arXiv:1106.5030] [INSPIRE].

    Article  ADS  Google Scholar 

  29. H.-S. Shao and Y.-J. Zhang, Feynman Rules for the Rational Part of One-loop QCD Corrections in the MSSM, JHEP 06 (2012) 112 [arXiv:1205.1273] [INSPIRE].

    Article  ADS  Google Scholar 

  30. S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].

    Article  ADS  Google Scholar 

  31. J. Bell and R. Jackiw, A PCAC puzzle: π0 → γγ in the σ-model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].

    Article  ADS  Google Scholar 

  32. F. Jegerlehner, Facts of life with γ5, Eur. Phys. J. C 18 (2001) 673 [hep-th/0005255] [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Pittau.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pittau, R. A four-dimensional approach to quantum field theories. J. High Energ. Phys. 2012, 151 (2012). https://doi.org/10.1007/JHEP11(2012)151

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP11(2012)151

Keywords

Navigation