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Four-loop corrections with two closed fermion loops to fermion self energies and the lepton anomalous magnetic moment

  • Roman Lee
  • Peter Marquard
  • Alexander V. Smirnov
  • Vladimir A. Smirnov
  • Matthias Steinhauser
Article

Abstract

We compute the eighth-order fermionic corrections involving two and three closed massless fermion loops to the anomalous magnetic moment of the muon. The required four-loop on-shell integrals are classified and explicit analytical results for the master integrals are presented. As further applications we compute the corresponding four-loop QCD corrections to the mass and wave function renormalization constants for a massive quark in the on-shell scheme.

Keywords

Electromagnetic Processes and Properties QCD Standard Model 

References

  1. [1]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kuhn, Scalar correlator at \( O\left( {\alpha_s^4} \right) \) , Higgs decay into b-quarks and bounds on the light quark masses, Phys. Rev. Lett. 96 (2006) 012003 [hep-ph/0511063] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kuhn, Order \( \left( {\alpha_s^4} \right) \) QCD corrections to Z and τ decays, Phys. Rev. Lett. 101 (2008) 012002 [arXiv:0801.1821] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn and J. Rittinger, Complete \( O\left( {\alpha_s^4} \right) \) QCD corrections to hadronic Z-decays, Phys. Rev. Lett. 108 (2012) 222003 [arXiv:1201.5804] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    K.G. Chetyrkin, J.H. Kuhn and C. Sturm, Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD, Eur. Phys. J. C 48 (2006) 107 [hep-ph/0604234] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    R. Boughezal, M. Czakon and T. Schutzmeier, Charm and bottom quark masses from perturbative QCD, Phys. Rev. D 74 (2006) 074006 [hep-ph/0605023] [INSPIRE].ADSGoogle Scholar
  6. [6]
    C. Sturm, Moments of heavy quark current correlators at four-loop order in perturbative QCD, JHEP 09 (2008) 075 [arXiv:0805.3358] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    A. Maier, P. Maierhofer and P. Marqaurd, The second physical moment of the heavy quark vector correlator at \( O\left( {\alpha_s^3} \right) \), Phys. Lett. B 669 (2008) 88 [arXiv:0806.3405] [INSPIRE].ADSGoogle Scholar
  8. [8]
    A. Maier, P. Maierhofer, P. Marquard and A.V. Smirnov, Low energy moments of heavy quark current correlators at four loops, Nucl. Phys. B 824 (2010) 1 [arXiv:0907.2117] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    K.G. Chetyrkin, J.H. Kühn, A. Maier, P. Maierhöfer, P. Marquard, M. Steinhauser and C. Sturm, Charm and bottom quark masses: An update, Phys. Rev. D 80 (2009) 074010 [arXiv:0907.2110] [INSPIRE].ADSGoogle Scholar
  10. [10]
    K. Kajantie, M. Laine, K. Rummukainen and Y. Schröder, Pressure of hot QCD up to \( {g^6}ln\left( {{1 \left/ {g} \right.}} \right) \), Phys. Rev. D 67 (2003) 105008 [hep-ph/0211321] [INSPIRE].ADSGoogle Scholar
  11. [11]
    N. Gray, D.J. Broadhurst, W. Grafe and K. Schilcher, Three loop relation of quark (modified) MS and pole masses, Z. Phys. C 48 (1990) 673 [INSPIRE].ADSGoogle Scholar
  12. [12]
    D.J. Broadhurst, N. Gray and K. Schilcher, Gauge invariant on-shell Z 2 in QED, QCD and the effective field theory of a static quark, Z. Phys. C 52 (1991) 111 [INSPIRE].MathSciNetADSGoogle Scholar
  13. [13]
    S. Laporta and E. Remiddi, The Analytical value of the electron \( \left( {g-2} \right) \) at order α 3 in QED, Phys. Lett. B 379 (1996) 283 [hep-ph/9602417] [INSPIRE].ADSGoogle Scholar
  14. [14]
    K. Melnikov and T.v. Ritbergen, The three loop relation between the MS-bar and the pole quark masses, Phys. Lett. B 482 (2000) 99 [hep-ph/9912391] [INSPIRE].ADSGoogle Scholar
  15. [15]
    P. Marquard, L. Mihaila, J.H. Piclum and M. Steinhauser, Relation between the pole and the minimally subtracted mass in dimensional regularization and dimensional reduction to three-loop order, Nucl. Phys. B 773 (2007) 1 [hep-ph/0702185] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    K. Melnikov and T. van Ritbergen, The three loop on-shell renormalization of QCD and QED, Nucl. Phys. B 591 (2000) 515 [hep-ph/0005131] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    K.G. Chetyrkin and M. Steinhauser, Short distance mass of a heavy quark at order \( \left( {\alpha_s^3} \right) \), Phys. Rev. Lett. 83 (1999) 4001 [hep-ph/9907509] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    K.G. Chetyrkin and M. Steinhauser, The relation between the MS-bar and the on-shell quark mass at order \( \left( {\alpha_s^3} \right) \), Nucl. Phys. B 573 (2000) 617 [hep-ph/9911434] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    K. Melnikov and T. van Ritbergen, The three loop slope of the Dirac form-factor and the S Lamb shift in hydrogen, Phys. Rev. Lett. 84 (2000) 1673 [hep-ph/9911277] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A.G. Grozin, P. Marquard, J.H. Piclum and M. Steinhauser, Three-loop chromomagnetic interaction in HQET, Nucl. Phys. B 789 (2008) 277 [arXiv:0707.1388] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    B.E. Lautrup and E. De Rafael, The anomalous magnetic moment of the muon and short-distance behaviour of quantum electrodynamics, Nucl. Phys. B 70 (1974) 317 [Erratum ibid. B 78 (1974) 576] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    T. Kinoshita, H. Kawai and Y. Okamoto, Asymptotic photon propagator in massive QED and the muon anomalous magnetic moment, Phys. Lett. B 254 (1991) 235 [INSPIRE].ADSGoogle Scholar
  23. [23]
    H. Kawai, T. Kinoshita and Y. Okamoto, Asymptotic photon propagator and higher order QED Callan-Symanzik β-function, Phys. Lett. B 260 (1991) 193 [INSPIRE].ADSGoogle Scholar
  24. [24]
    R.N. Faustov, A.L. Kataev, S.A. Larin and V.V. Starshenko, The analytical contribution of the three loop diagrams with two fermion circles to the photon propagator and the muon anomalous magnetic moment, Phys. Lett. B 254 (1991) 241 [INSPIRE].ADSGoogle Scholar
  25. [25]
    D.J. Broadhurst, A.L. Kataev and O.V. Tarasov, Analytical on-shell QED results: Three loop vacuum polarization, four loop β-function and the muon anomaly, Phys. Lett. B 298 (1993) 445 [hep-ph/9210255] [INSPIRE].ADSGoogle Scholar
  26. [26]
    P.A. Baikov and D.J. Broadhurst, Three loop QED vacuum polarization and the four loop muon anomalous magnetic moment, hep-ph/9504398 [INSPIRE].
  27. [27]
    P.A. Baikov et al., The relation between the QED charge renormalized in MSbar and on-shell schemes at four loops, the QED on-shell β-function at five loops and asymptotic contributions to the muon anomaly at five and six loops, Nucl. Phys. B 867 (2013) 182 [arXiv:1207.2199] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    S. Laporta, The Analytical contribution of some eighth order graphs containing vacuum polarization insertions to the muon \( \left( {g-2} \right) \) in QED, Phys. Lett. B 312 (1993) 495 [hep-ph/9306324] [INSPIRE].ADSGoogle Scholar
  29. [29]
    J.-P. Aguilar, D. Greynat and E. De Rafael, Muon anomaly from lepton vacuum polarization and the Mellin-Barnes representation, Phys. Rev. D 77 (2008) 093010 [arXiv:0802.2618] [INSPIRE].ADSGoogle Scholar
  30. [30]
    T. Kinoshita and M. Nio, Improved α 4 term of the muon anomalous magnetic moment, Phys. Rev. D 70 (2004) 113001 [hep-ph/0402206] [INSPIRE].ADSGoogle Scholar
  31. [31]
    T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron, Phys. Rev. D 77 (2008) 053012 [arXiv:0712.2607] [INSPIRE].ADSGoogle Scholar
  32. [32]
    T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Tenth-order QED contribution to the electron \( g-2 \) and an improved value of the fine structure constant, Phys. Rev. Lett. 109 (2012) 111807 [arXiv:1205.5368] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Complete tenth-order QED contribution to the muon \( g-2 \), Phys. Rev. Lett. 109 (2012) 111808 [arXiv:1205.5370] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    F. Jegerlehner, The anomalous magnetic moment of the muon, Springer Tracts Mod. Phys. 226 (2008) 1 [INSPIRE].Google Scholar
  35. [35]
    F. Jegerlehner and A. Nyffeler, The muon \( g-2 \), Phys. Rept. 477 (2009) 1 [arXiv:0902.3360] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  37. [37]
    R. Harlander, T. Seidensticker and M. Steinhauser, Complete corrections of order αα s to the decay of the Z boson into bottom quarks, Phys. Lett. B 426 (1998) 125 [hep-ph/9712228] [INSPIRE].ADSGoogle Scholar
  38. [38]
    T. Seidensticker, Automatic application of successive asymptotic expansions of Feynman diagrams, hep-ph/9905298 [INSPIRE].
  39. [39]
    J. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
  40. [40]
    P. Marquard and D. Seidel, unpublished.Google Scholar
  41. [41]
    A. Smirnov, Algorithm FIREFeynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].MathSciNetADSGoogle Scholar
  43. [43]
    K. Chetyrkin and F. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    R. Lee, Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B 830 (2010) 474 [arXiv:0911.0252] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    A.V. Smirnov and M.N. Tentyukov, Feynman integral evaluation by a sector decomposition approach (FIESTA), Comput. Phys. Commun. 180 (2009) 735 [arXiv:0807.4129] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  46. [46]
    A.V. Smirnov, V.A. Smirnov and M.N. Tentyukov, FIESTA 2: parallelizeable multiloop numerical calculations, Comput. Phys. Commun. 182 (2011) 790 [arXiv:0912.0158] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  47. [47]
    V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].ADSGoogle Scholar
  48. [48]
    J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].MathSciNetADSGoogle Scholar
  49. [49]
    V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2013) 1 [INSPIRE].CrossRefGoogle Scholar
  50. [50]
    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  51. [51]
    A.V. Smirnov and V.A. Smirnov, On the resolution of singularities of multiple Mellin-Barnes integrals, Eur. Phys. J. C 62 (2009) 445 [arXiv:0901.0386] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    H.R.P. Ferguson and D.H. Bailey, A polynomial time, numerically stable integer relation algorithm, RNR Techn. Rept. RNR-91-032 (1992).Google Scholar
  53. [53]
    S. Laporta, High precision ϵ-expansions of three loop master integrals contributing to the electron \( g-2 \) in QED, Phys. Lett. B 523 (2001) 95 [hep-ph/0111123] [INSPIRE].MathSciNetADSGoogle Scholar
  54. [54]
    K.G. Chetyrkin, Quark mass anomalous dimension to \( O\left( {\alpha_s^4} \right) \), Phys. Lett. B 404 (1997) 161 [hep-ph/9703278] [INSPIRE].ADSGoogle Scholar
  55. [55]
    J.A.M. Vermaseren, S.A. Larin and T. van Ritbergen, The four loop quark mass anomalous dimension and the invariant quark mass, Phys. Lett. B 405 (1997) 327 [hep-ph/9703284] [INSPIRE].ADSGoogle Scholar
  56. [56]
    K.G. Chetyrkin, Four-loop renormalization of QCD: full set of renormalization constants and anomalous dimensions, Nucl. Phys. B 710 (2005) 499 [hep-ph/0405193] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    M. Beneke and V.M. Braun, Naive non-abelianization and resummation of fermion bubble chains, Phys. Lett. B 348 (1995) 513 [hep-ph/9411229] [INSPIRE].ADSGoogle Scholar
  58. [58]
    B. Krause, Zwei- und Dreischleifen-Berechnungen zu elektrischen und magnetischen Dipolmomenten von Elementarteilchen, Ph.D. thesis, University of Karlsruhe, Germany (1997).Google Scholar
  59. [59]
    D.J. Broadhurst, Three loop on-shell charge renormalization without integration: Λ-MS (QED) to four loops, Z. Phys. C 54 (1992) 599 [INSPIRE].ADSGoogle Scholar
  60. [60]
    P.J. Mohr, B.N. Taylor and D.B. Newell, CODATA recommended values of the fundamental physical constants: 2010, Rev. Mod. Phys. 84 (2012) 1527 [arXiv:1203.5425] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun. 83 (1994) 45 [INSPIRE].ADSMATHCrossRefGoogle Scholar
  62. [62]
    D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Roman Lee
    • 1
  • Peter Marquard
    • 2
  • Alexander V. Smirnov
    • 3
  • Vladimir A. Smirnov
    • 4
    • 5
  • Matthias Steinhauser
    • 2
  1. 1.Budker Institute of Nuclear Physics and Novosibirsk State UniversityNovosibirskRussia
  2. 2.Institut für Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  3. 3.Scientific Research Computing CenterMoscow State UniversityMoscowRussia
  4. 4.Skobeltsyn Institute of Nuclear PhysicsMoscow State UniversityMoscowRussia
  5. 5.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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