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The R *-operation for Feynman graphs with generic numerators

A preprint version of the article is available at arXiv.

Abstract

The R *-operation by Chetyrkin, Tkachov, and Smirnov is a generalisation of the BPHZ R-operation, which subtracts both ultraviolet and infrared divergences of euclidean Feynman graphs with non-exceptional external momenta. It can be used to compute the divergent parts of such Feynman graphs from products of simpler Feynman graphs of lower loops. In this paper we extend the R *-operation to Feynman graphs with arbitrary numerators, including tensors. We also provide a novel way of defining infrared counterterms which closely resembles the definition of its ultraviolet counterpart. We further express both infrared and ultraviolet counterterms in terms of scaleless vacuum graphs with a logarithmic degree of divergence. By exploiting symmetries, integrand and integral relations, which the counterterms of scaleless vacuum graphs satisfy, we can vastly reduce their number and complexity. A FORM implementation of this method was used to compute the five loop beta function in QCD for a general gauge group. To illustrate the procedure, we compute the poles in the dimensional regulator of all top-level propagator graphs at five loops in four dimensional ϕ 3 theory.

References

  1. [1]

    T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  2. [2]

    T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549.

    ADS  MathSciNet  Article  Google Scholar 

  3. [3]

    C.G. Bollini and J.J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].

    Google Scholar 

  4. [4]

    G. ’t Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].

  5. [5]

    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  6. [6]

    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].

  7. [7]

    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].

    ADS  Article  Google Scholar 

  8. [8]

    V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].

  9. [9]

    J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].

  10. [10]

    C. Anastasiou and A. Daleo, Numerical evaluation of loop integrals, JHEP 10 (2006) 031 [hep-ph/0511176] [INSPIRE].

  11. [11]

    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].

  12. [12]

    T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B 585 (2000) 741 [hep-ph/0004013] [INSPIRE].

  13. [13]

    M. Roth and A. Denner, High-energy approximation of one loop Feynman integrals, Nucl. Phys. B 479 (1996) 495 [hep-ph/9605420] [INSPIRE].

  14. [14]

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

    ADS  Article  MATH  Google Scholar 

  15. [15]

    E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071 [arXiv:1401.4361] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    A. von Manteuffel, E. Panzer and R.M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals, JHEP 02 (2015) 120 [arXiv:1411.7392] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].

  18. [18]

    C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 07 (2004) 046 [hep-ph/0404258] [INSPIRE].

  19. [19]

    C. Studerus, Reduze-Feynman integral reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].

  21. [21]

    A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  22. [22]

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

    MathSciNet  Article  Google Scholar 

  23. [23]

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

    ADS  Article  MATH  Google Scholar 

  24. [24]

    G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].

  25. [25]

    W.E. Caswell and A.D. Kennedy, A simple approach to renormalization theory, Phys. Rev. D 25 (1982) 392 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  26. [26]

    D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 303 [q-alg/9707029] [INSPIRE].

  27. [27]

    A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. 1. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210 (2000) 249 [hep-th/9912092] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    K.G. Chetyrkin and F.V. Tkachov, Infrared R operation and ultraviolet counterterms in the MS scheme, Phys. Lett. B 114 (1982) 340 [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

    K.G. Chetyrkin and V.A. Smirnov, R * operation corrected, Phys. Lett. B 144 (1984) 419 [INSPIRE].

    ADS  Article  Google Scholar 

  30. [30]

    V.A. Smirnov and K.G. Chetyrkin, R * operation in the minimal subtraction scheme, Theor. Math. Phys. 63 (1985) 462 [INSPIRE].

    MathSciNet  Article  Google Scholar 

  31. [31]

    A.A. Vladimirov, Method for computing renormalization group functions in dimensional renormalization scheme, Theor. Math. Phys. 43 (1980) 417 [INSPIRE].

    Article  Google Scholar 

  32. [32]

    K.G. Chetyrkin, Combinatorics of R -, R −1 - and R * -operations and asymptotic expansions of Feynman integrals in the limit of large momenta and masses, arXiv:1701.08627 [INSPIRE].

  33. [33]

    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118 (2017) 082002 [arXiv:1606.08659] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].

    ADS  Article  Google Scholar 

  35. [35]

    K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Five loop calculations in the gϕ 4 model and the critical index η, Phys. Lett. B 99 (1981) 147 [Erratum ibid. B 101 (1981) 457] [INSPIRE].

  36. [36]

    S.G. Gorishnii, S.A. Larin, F.V. Tkachov and K.G. Chetyrkin, Five loop renormalization group calculations in the gϕ 4 in four-dimensions theory, Phys. Lett. B 132 (1983) 351 [INSPIRE].

    ADS  Article  Google Scholar 

  37. [37]

    H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin and S.A. Larin, Five loop renormalization group functions of O(n) symmetric ϕ 4 theory and ϵ-expansions of critical exponents up to ϵ5, Phys. Lett. B 272 (1991) 39 [Erratum ibid. B 319 (1993) 545] [hep-th/9503230] [INSPIRE].

  38. [38]

    M. Kompaniets and E. Panzer, Renormalization group functions of ϕ 4 theory in the MS-scheme to six loops, PoS(LL2016)038 [arXiv:1606.09210] [INSPIRE].

  39. [39]

    D.V. Batkovich, K.G. Chetyrkin and M.V. Kompaniets, Six loop analytical calculation of the field anomalous dimension and the critical exponent η in O(n)-symmetric φ 4 model, Nucl. Phys. B 906 (2016) 147 [arXiv:1601.01960] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. [40]

    P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn and J. Rittinger, Vector correlator in massless QCD at order O(α 4 s ) and the QED β-function at five loop, JHEP 07 (2012) 017 [arXiv:1206.1284] [INSPIRE].

    ADS  Article  Google Scholar 

  41. [41]

    P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn and J. Rittinger, Complete \( \mathcal{O}\left({\alpha}_s^4\right) \) QCD corrections to hadronic Z-decays, Phys. Rev. Lett. 108 (2012) 222003 [arXiv:1201.5804] [INSPIRE].

    ADS  Article  Google Scholar 

  42. [42]

    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Quark mass and field anomalous dimensions to \( \mathcal{O}\left({\alpha}_s^5\right) \), JHEP 10 (2014) 076 [arXiv:1402.6611] [INSPIRE].

    ADS  Article  Google Scholar 

  43. [43]

    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-loop fermion anomalous dimension for a general gauge group from four-loop massless propagators, JHEP 04 (2017) 119 [arXiv:1702.01458] [INSPIRE].

    Article  Google Scholar 

  44. [44]

    P.A. Baikov, K.G. Chetyrkin and J.H. Kuhn, Scalar correlator at O(α 4 s ), Higgs decay into b-quarks and bounds on the light quark masses, Phys. Rev. Lett. 96 (2006) 012003 [hep-ph/0511063] [INSPIRE].

  45. [45]

    H. Kleinert and V. Schulte-Frohlinde, Critical properties of ϕ 4 -theories, World Scientific, Singapore (2001).

    Book  MATH  Google Scholar 

  46. [46]

    S. Larin and P. van Nieuwenhuizen, The infrared R * operation, hep-th/0212315 [INSPIRE].

  47. [47]

    D.V. Batkovich and M. Kompaniets, Toolbox for multiloop Feynman diagrams calculations using R * operation, J. Phys. Conf. Ser. 608 (2015) 012068 [arXiv:1411.2618] [INSPIRE].

    Article  Google Scholar 

  48. [48]

    E.R. Speer, Contraction anomalies in dimensional renormalization, Nucl. Phys. B 134 (1978) 175 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  49. [49]

    J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].

  50. [50]

    T. Ueda, B. Ruijl and J.A.M. Vermaseren, Forcer: a FORM program for 4-loop massless propagators, PoS(LL2016)070 [arXiv:1607.07318] [INSPIRE].

  51. [51]

    T. Ueda, B. Ruijl and J.A.M. Vermaseren, Calculating four-loop massless propagators with Forcer, J. Phys. Conf. Ser. 762 (2016) 012060 [arXiv:1604.08767] [INSPIRE].

    Article  Google Scholar 

  52. [52]

    B. Ruijl, T. Ueda and J.A.M. Vermaseren, Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams, arXiv:1704.06650.

  53. [53]

    P.A. Baikov and K.G. Chetyrkin, Four loop massless propagators: an algebraic evaluation of all master integrals, Nucl. Phys. B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Ben Ruijl.

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ArXiv ePrint: 1703.03776

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Herzog, F., Ruijl, B. The R *-operation for Feynman graphs with generic numerators. J. High Energ. Phys. 2017, 37 (2017). https://doi.org/10.1007/JHEP05(2017)037

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Keywords

  • Perturbative QCD
  • Renormalization Group