Numerator seagull and extended Symmetries of Feynman Integrals

Abstract

The Symmetries of Feynman Integrals (SFI) method is extended for the first time to incorporate an irreducible numerator. This is done in the context of the so-called vacuum and propagator seagull diagrams, which have 3 and 2 loops, respectively, and both have a single irreducible numerator. For this purpose, an extended version of SFI (xSFI) is developed. For the seagull diagrams with general masses, the SFI equation system is found to extend by two additional equations. The first is a recursion equation in the numerator power, which has an alternative form as a differential equation for the generating function. The second equation applies only to the propagator seagull and does not involve the numerator. We solve the equation system in two cases: over the singular locus and in a certain 3 scale sector where we obtain novel closed-form evaluations and epsilon expansions, thereby extending previous results for the numerator-free case.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    B. Kol, Symmetries of Feynman integrals and the integration by parts method, arXiv:1507.01359 [INSPIRE].

  2. [2]

    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 

  3. [3]

    A.V. Kotikov, Differential equations method: the calculation of vertex type Feynman diagrams, Phys. Lett. B 259 (1991) 314 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  4. [4]

    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].

    ADS  Google Scholar 

  5. [5]

    M. Caffo, H. Czyz, S. Laporta and E. Remiddi, The master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cim. A 111 (1998) 365 [hep-th/9805118] [INSPIRE].

    ADS  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]

    K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].

    ADS  Article  Google Scholar 

  8. [8]

    P. Burda, B. Kol and R. Shir, Vacuum seagull: evaluating a three-loop Feynman diagram with three mass scales, Phys. Rev. D 96 (2017) 125013 [arXiv:1704.02187] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. [9]

    B. Kol, The algebraic locus of Feynman integrals, arXiv:1604.07827 [INSPIRE].

  10. [10]

    B. Kol, Bubble diagram through the symmetries of Feynman integrals method, arXiv:1606.09257 [INSPIRE].

  11. [11]

    B. Kol, Algebraic aspects of when and how a Feynman diagram reduces to simpler ones, arXiv:1804.01175 [INSPIRE].

  12. [12]

    B. Kol, Two-loop vacuum diagram through the symmetries of Feynman integrals method, arXiv:1807.07471 [INSPIRE].

  13. [13]

    B. Kol and S. Mazumdar, Kite diagram through symmetries of Feynman integrals, Phys. Rev. D 99 (2019) 045018 [arXiv:1808.02494] [INSPIRE].

  14. [14]

    B. Kol and R. Shir, The propagator seagull: general evaluation of a two loop diagram, JHEP 03 (2019) 083 [arXiv:1809.05040] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    B. Kol and S. Mazumdar, Triangle diagram, distance geometry and symmetries of Feynman integrals, JHEP 03 (2020) 156 [arXiv:1909.04055] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  16. [16]

    R.J. Gonsalves, Dimensionally regularized two loop on-shell quark form-factor, Phys. Rev. D 28 (1983) 1542 [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    A.I. Davydychev, A simple formula for reducing Feynman diagrams to scalar integrals, Phys. Lett. B 263 (1991) 107 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  18. [18]

    N.I. Usyukina and A.I. Davydychev, Two loop three point diagrams with irreducible numerators, Phys. Lett. B 348 (1995) 503 [hep-ph/9412356] [INSPIRE].

  19. [19]

    A.I. Davydychev and J.B. Tausk, Two loop vacuum diagrams and tensor decomposition, in 4th international workshop on software engineering and artificial intelligence for high-energy and nuclear physics, (1995), pg. 0155 [hep-ph/9504432] [INSPIRE].

  20. [20]

    K.G. Chetyrkin, M. Misiak and M. Münz, β-functions and anomalous dimensions up to three loops, Nucl. Phys. B 518 (1998) 473 [hep-ph/9711266] [INSPIRE].

  21. [21]

    O.V. Tarasov, Generalized recurrence relations for two loop propagator integrals with arbitrary masses, Nucl. Phys. B 502 (1997) 455 [hep-ph/9703319] [INSPIRE].

  22. [22]

    C. Anastasiou, J.B. Tausk and M.E. Tejeda-Yeomans, The on-shell massless planar double box diagram with an irreducible numerator, Nucl. Phys. B Proc. Suppl. 89 (2000) 262 [hep-ph/0005328] [INSPIRE].

  23. [23]

    S. Groote, J.G. Korner and A.A. Pivovarov, Laurent series expansion of sunrise type diagrams using configuration space techniques, Eur. Phys. J. C 36 (2004) 471 [hep-ph/0403122] [INSPIRE].

  24. [24]

    V.A. Smirnov, Evaluating multiloop Feynman integrals by Mellin-Barnes representation, Nucl. Phys. B Proc. Suppl. 135 (2004) 252 [hep-ph/0406052] [INSPIRE].

  25. [25]

    A.I. Davydychev and M.Y. Kalmykov, New results for the ϵ-expansion of certain one, two and three loop Feynman diagrams, Nucl. Phys. B 605 (2001) 266 [hep-th/0012189] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  26. [26]

    S.P. Martin, Evaluation of two loop selfenergy basis integrals using differential equations, Phys. Rev. D 68 (2003) 075002 [hep-ph/0307101] [INSPIRE].

  27. [27]

    S.P. Martin and D.G. Robertson, TSIL: a program for the calculation of two-loop self-energy integrals, Comput. Phys. Commun. 174 (2006) 133 [hep-ph/0501132] [INSPIRE].

  28. [28]

    S.P. Martin and D.G. Robertson, Evaluation of the general 3-loop vacuum Feynman integral, Phys. Rev. D 95 (2017) 016008 [arXiv:1610.07720] [INSPIRE].

  29. [29]

    A. Freitas, Three-loop vacuum integrals with arbitrary masses, JHEP 11 (2016) 145 [arXiv:1609.09159] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  30. [30]

    S.P. Martin, Effective potential at three loops, Phys. Rev. D 96 (2017) 096005 [arXiv:1709.02397] [INSPIRE].

  31. [31]

    S.P. Martin and D.G. Robertson, Standard Model parameters in the tadpole-free pure \( \overline{MS} \) scheme, Phys. Rev. D 100 (2019) 073004 [arXiv:1907.02500] [INSPIRE].

  32. [32]

    O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  33. [33]

    P.A. Baikov, Explicit solutions of n loop vacuum integral recurrence relations, hep-ph/9604254 [INSPIRE].

  34. [34]

    M.Y. Kalmykov and B.A. Kniehl, Counting master integrals: integration by parts versus differential reduction, Phys. Lett. B 702 (2011) 268 [arXiv:1105.5319] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    B.A. Kniehl and O.V. Tarasov, Counting master integrals: integration by parts vs. functional equations, arXiv:1602.00115 [INSPIRE].

  36. [36]

    R.N. Lee and V.A. Smirnov, The dimensional recurrence and analyticity method for multicomponent master integrals: using unitarity cuts to construct homogeneous solutions, JHEP 12 (2012) 104 [arXiv:1209.0339] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  37. [37]

    E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].

  38. [38]

    A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys. B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  39. [39]

    R.N. Lee, Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals, JHEP 07 (2008) 031 [arXiv:0804.3008] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ruth Shir.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2009.04947

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kol, B., Schiller, A. & Shir, R. Numerator seagull and extended Symmetries of Feynman Integrals. J. High Energ. Phys. 2021, 165 (2021). https://doi.org/10.1007/JHEP01(2021)165

Download citation

Keywords

  • Perturbative QCD
  • Scattering Amplitudes
  • Quark Masses and SM Parameters