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Finite Z-Less Integral Expressions for β-Functions in the MS4 Scheme

  • PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
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Abstract

The generalized minimal subtraction scheme for ultraviolet renormalization (Kuznetsov and Tkachov, 1988) is fine-tuned with applications in mind. The resulting MS4 scheme obviates extraneous regularizations and renders momentum integrands integrable by subtracting troublesome asymptotic terms in a physically correct fashion due to the use of special minimal subtraction operators defined congruously with the physically natural Polchinski cutoffs. A direct derivation of the Callan–Symanzik equations avoids divergent renormalization factors or counterterms, and automatically yields explicit exact finite integral expressions for renormalization group functions.

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REFERENCES

  1. N. D. Lenshina, A. A. Radionov, and F. V. Tkachov, “MS4: An alternative to the Bogolyubov–Parasiuk–Hepp–Zimmermann (BPHZ) theory,” Phys. Part. Nucl. 51, 567 (2020); arXiv:1911.10402.

  2. N. D. Lenshina, A. A. Radionov, and F. V. Tkachov, “Renormalization group equations and functions in the MS4 scheme,” in preparation.

  3. F. V. Tkachov, “Factorization via R operation: The extension principle,” in Proceedings of the Quarks-82 Conference (1982), p. 196; inspirehep: 186256.

  4. A. N. Kuznetsov and F. V. Tkachov, “R-operation via asymptotic expansions,” in Proceedings of the Conference on Renormalization Group'91, Dubna, Sept. 3–6, 1991; inspirehep:327958; F. V. Tkachov and V. V. Vlasov, “New methods for Feynman diagrams,” in Proceedings of the Quarks-88 Conference, Tbilisi (World Scientific, 1988); inspirehep:274597.

  5. F. V. Tkachov, “Theory of asymptotic operation. a summary of basic principles,” Sov. J. Part. Nucl. 25, 649 (1994); arXiv:9701272.

  6. F. V. Tkachov, “Distribution-theoretic methods in quantum field theory,” Sov. J. Part. Nucl. 31, 200 (2000); arXiv:9911236.

  7. F. V. Tkachov, “Loops, singularities, and distributions. A byzantine history of asymptotic operation,” in preparation.

  8. A. N. Kuznetsov, F. V. Tkachov, and V. V. Vlasov, “Techniques of distributions in perturbative quantum field theory. I Euclidean asymptotic operation for products of singular functions,” arXiv: 9612037.

  9. A. N. Kuznetsov and F. V. Tkachov, “Techniques of distributions in perturbative quantum field theory. II. Applications to theory of multiloop diagrams,” arXiv:9612038.

  10. C. Bollini and J. J. Giambiagi, “Dimensional renormalization: The number of dimensions as a regularizing parameter,” Nuovo Cim., B 12, 20 (1972); inspirehep: 74881; G. ‘t Hooft and M. Veltman, “Regularization and renormalization of Gauge fields,” Nucl. Phys. B 44, 189 (1972); inspirehep:74886.

  11. G. ‘t Hooft, “Dimensional regularization and the renormalization group,” Nucl. Phys. B 61, 455 (1973); inspirehep:85150.

  12. J. Polchinski, “Renormalization and effective lagrangians,” Nucl. Phys. B 231, 269 (1984); inspirehep:189574.

  13. L. Dixon et al., “All tree-level amplitudes in massless QCD,” J. High Energy Phys. 1, 035 (2011); arXiv: 1010.3991; L. Dixon, “A brief introduction to modern amplitude methods,” CERN-2014-008; arXiv: 1310.5353.

  14. E. Brezin, J. C. le Guillou, and J. Zinn-Justin, “Addendum to Wilson’s theory of critical phenomena and Callan–Symanzik equations in 4-ε dimensions,” Phys. Rev. D 9, 1121 (1974).

    Article  ADS  Google Scholar 

  15. O. Schnetz, “Natural renormalization,” J. Math. Phys. 38, 738–758 (1997); inspirehep:424193.

  16. T. Aoyama, T. Kinoshita, and M. Nio, “Revised and improved value of the QED tenth-order electron anomalous magnetic moment,” Phys. Rev. D: Part. Fields 97, 036001 (2018); arXiv:1712.06060; S. Volkov, “Calculating the five-loop QED contribution to the electron anomalous magnetic moment: graphs without lepton loops,” Phys. Rev. D: Part. Fields 100, 096004 (2019); arXiv:1909.08015.

  17. S. Volkov, “Infrared and ultraviolet power counting on the mass shell in quantum electrodynamics,” arXiv: 1912.04885.

  18. J. C. Collins and F. V. Tkachov, “Breakdown of dimensional regularization in the sudakov problem,” Phys. Lett. B 294, 403 (1992); arXiv:9208209; F. V. Tkachov, “On dimensional regularization and mathematical rigour,” in Proceedings of the VIII QFTHEP Conference, Zvenigorod, Russia, 1993; arXiv: 9612288.

  19. 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, 353 (1992); inspirehep:316732.

  20. G. F. R. Sborlini, “Loop-tree duality and quantum field theory in four dimensions,” PoS (RADCOR2015), 082 (2015); arXiv:1601.04634.

  21. R. Pittau, “A four-dimensional approach to quantum field theories,” J. High Energy Phys. 1211, 151 (2012); arXiv:1208.5457.

    Article  ADS  MathSciNet  Google Scholar 

  22. P. Grangé, J.-F. Mathiot, B. Mutet, and E. Werner, “Taylor–Lagrange renormalization scheme, Pauli–Villars subtraction, and light-front dynamics,” Phys. Rev. D: Part. Fields 80, 105012 (2009); arXiv: 0909.1715.

    Article  ADS  Google Scholar 

  23. L. Schwartz, Théorie des distributions (Hermann, Paris, 1966).

    MATH  Google Scholar 

  24. E. C. G. Stueckelberg and D. Rivier, “Causalite et structure de la matrices,” Helv. Phys. Acta 23, 215 (1950);

    MathSciNet  MATH  Google Scholar 

  25. E. C. G. Stueckelberg and T. A. Green, “Elimination des constantes arbitraires dans la theorie relativiste des quanta,” Helv. Phys. Acta 24, 153 (1951).

    MATH  Google Scholar 

  26. N. N. Bogolyubov, Dokl. Akad. Nauk USSR 82, 217 (1952).

    Google Scholar 

  27. H. Epstein and V. Glaser, “The role of locality in perturbation theory,” Ann. Inst. H. Poincare A 19, 211 (1973).

    MathSciNet  MATH  Google Scholar 

  28. N. N. Bogoliubow and O. S. Parasiuk, Acta Math. 97, 227 (1957); arXiv:1112.1180.

    Article  MathSciNet  Google Scholar 

  29. F. Brown and D. Kreimer, “Angles, scales and parametric renormalization,” Lett. Math. Phys. 103, 933 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Hairer, “An analyst’s take on the BPHZ theorem,” arXiv:1704.08634.

  31. K. G. Chetyrkin and F. V. Tkachov, “Infrared R-operation and ultraviolet counterterms in the mS-scheme,” Phys. Lett. B 114, 340 (1982); inspirehep:182141.

  32. F. V. Tkachov, “On the operator product expansion in the MS scheme,” Phys. Lett. B 124, 212 (1983); inspirehep:193146.

  33. S. G. Gorishnii, S. A. Larin, and F. V. Tkachov, “The algorithm for OPE coefficient functions in the mS scheme,” Phys. Lett. B 124, 217 (1983); inspirehep:193147.

  34. F. V. Tkachov, Preprint IYaI-P-0358 (1984); “Euclidean asymptotic expansions of green functions of quantum fields (I) expansions of products of singular functions,” Int. J. Mod. Phys. A 8, 2047 (1993); arXiv: 9612284.

  35. G. B. Pivovarov and F. V. Tkachov, Preprint No. IYaI-P-0370 (1984); “Euclidean asymptotic expansions of green functions of quantum fields (II) combinatorics of the asymptotic operation,” Int. J. Mod. Phys. A 8, 2241 (1993); arXiv:9612287.

  36. F. V. Tkachov, “Towards systematic near-threshold calculations in perturbative QFT,” Phys. Lett. B 412, 350 (1997); arXiv:9703424.

  37. F. V. Tkachov, “Perturbation theory with unstable fundamental fields,” in Proceedings of the 32nd PNPI Winter School, 1999; arXiv:9802307.

  38. F. V. Tkachov, “A comment on the LLA method, the KT jet algorithm and the BFKL theory,” in Proceedings of the Conference on Hadron Structure and QCD, Gatchina, 2008; arXiv:0807.3090.

  39. D. I. Kazakov, O. V. Tarasov, and A. A. Vladimirov, “Calculation of critical exponents by quantum field theory methods,” Sov. Phys. JETP 50, 521 (1979); inspirehep:140661.

  40. M. Kompaniets and E. Panzer, “Renormalization group functions of Φ4 theory in the MS-scheme to six loops,” PoS LL2016, 038 (2016); inspirehep:1472996.

  41. A. A. Vladimirov, “Methods of multiloop calculations and the renormalization group analysis of phi**4 theory,” Sov. J. Theor. Math. Phys. 36, 732 (1978); inspirehep:122647.

  42. K. G. Chetyrkin, A. L. Kataev, and F. V. Tkachov, “Higher order corrections to sigma-t (e + e–→ hadrons) in quantum chromodynamics,” Phys. Lett. B 85, 277 (1979); inspirehep:141747.

  43. L. Schwartz, Méthodes mathématiques pour les sciences physiques (Hermann, Paris, 1961).

    MATH  Google Scholar 

  44. I. M. Gel’fand and G. E. Shilov, Generalized Functions (AMS, Philadelphia, 1964), Vol. 1.

    MATH  Google Scholar 

  45. F. V. Tkachov, “A theorem on analytical calculability of four loop renormalization group functions,” Phys. Lett. B 100, 65 (1981); inspirehep:167175.

  46. E. Panzer, “Feynman integrals and hyperlogarithms,” PhD Thesis (2015); arXiv:1506.07243.

  47. O. Schnetz, “Geometries in perturbative quantum field theory,” arXiv: 1905.08083.

  48. H. Elvang and Yu-Tin Huang, Scattering Amplitudes in Gauge Theory and Gravity (Cambridge Univ. Press, Cambridge, 2015); arXiv:1308.1697.

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ACKNOWLEDGMENTS

We thank A. Lokhov for bringing the team together, S. Volkov for discussions and references, M. Kalmykov for a correction, and Rinat Menyashev for supporting this work.

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  1. Institute for Nuclear Research RAS, 117312, Moscow, Russia

    N. D. Lenshina, A. A. Radionov & F. V. Tkachov

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  1. N. D. Lenshina
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  3. F. V. Tkachov
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Correspondence to F. V. Tkachov.

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Lenshina, N.D., Radionov, A.A. & Tkachov, F.V. Finite Z-Less Integral Expressions for β-Functions in the MS4 Scheme. Phys. Part. Nuclei Lett. 18, 131–140 (2021). https://doi.org/10.1134/S1547477121020102

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