Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension

  • D. J. Broadhurst
  • J. Fleischer
  • O. V. Tarasov


In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and smallq2, ind dimensions, using identities derived from integration by parts. Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations. Padé approximants to truncations of the expansions are shown to be of great utility. As an application, we obtain the two-loop photon self-energy, for alld, and achieve highly accelerated convergence of its expansions in powers ofq2/m2 orm2/q2, ford=4.


Differential Equation Field Theory Elementary Particle Quantum Field Theory Asymptotic Expansion 
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  1. 1.
    LEP Coll., ALEPH, DELPHI, L3, OPAL: Phys. Lett. B276 (1992) 247Google Scholar
  2. 2.
    L. Rolandi: CERN-PPE-92-175, Precision tests of the electroweak interaction, talk at ICHEP 92 conference, Dallas (1992); G. Altarelli: CERN-TH.6525/92, Precision electroweak data and constraints on new physics, talk at XXVIIth Recontres de Moriond, Les Arcs (1992); P. Renton: Z. Phys. C56 (1992) 355; M. Cvetič, P. Langacker: Testing the Standard Model (World Scientific, Singapore, 1991)Google Scholar
  3. 3.
    A.L. Kataev, V.T. Kim: Annecy preprint ENSLAPP-A-407/92 (1992)Google Scholar
  4. 4.
    D.J. Broadhurst: Z. Phys. C47 (1990) 115Google Scholar
  5. 5.
    B.A. Kniehl: Nucl. Phys. B347 (1990) 86; F. Halzen, B.A. Kniehl, M.L. Stong: Z. Phys. C58 (1993) 119Google Scholar
  6. 6.
    N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher: Z. Phys. C48 (1990) 673; D.J. Broadhurst, N. Gray, K. Schilcher: Z. Phys. C52 (1991) 111Google Scholar
  7. 7.
    J. Fleischer, O.V. Tarasov: Comp. Phys. Commun. 71 (1992) 193Google Scholar
  8. 8.
    D. J. Broadhurst: OUT-4102-40 (1992), to appear in Proceedings of QUARKS-92Google Scholar
  9. 9.
    D.J. Broadhurst: in New computing techniques in physics research II, p. 579; D. Perret-Gallix, ed. Singapore: World Scientific, 1992; G. Weiglein, R. Mertig, R. Scharf, M. Böhm: ibid in New computing techniques in physics research II, p. 579; D. Perret-Gallix, ed. Singapore: World Scientific, 1992; p. 617; J. Fujimoto, Y. Shimizu, K. Kato, Y. Oyanagi: ibid in New computing techniques in physics research II, p. 579; D. Perret-Gallix, ed. Singapore: World Scientific, 1992; p. 625Google Scholar
  10. 10.
    D. Kreimer: Phys. Lett. B273 (1991) 277; Mainz preprint MZ-TH-92-50 (1992)Google Scholar
  11. 11.
    A.I. Davydychev, J.B. Tausk: Nucl. Phys. B397 (1993) 123Google Scholar
  12. 12.
    A.I. Davydychev, V.A. Smirnov, J.B. Tausk: INLO-PUB-5/93 (1993)Google Scholar
  13. 13.
    K.G. Chetyrkin, F.V. Tkachov: Nucl. Phys. B192 (1981) 159; F.V. Tkachov: Phys. Lett. B100 (1981) 65Google Scholar
  14. 14.
    M.A. Shifman, A.I. Vainshtein, V.I. Zakharov: Nucl. Phys. B147 (1979) 385, 448, 519.Google Scholar
  15. 15.
    P.A. Baikov et al.: Phys. Lett. B263 (1991) 481; K. G. Chetyrkin et al.: Phys. Lett. B225 (1989) 411Google Scholar
  16. 16.
    G. Källén, A. Sabry: K. Dan. Vidensk. Selsk. Mat.-Fys Medd. 29 (1955) No. 17Google Scholar
  17. 17.
    J. Schwinger: Particles, sources and fields Vol. 2, p. 407, Reading, Mass. Addison-Wesley, 1973Google Scholar
  18. 18.
    R. Barbieri, E. Remiddi: Nuovo Cimento 13 (1973) 99Google Scholar
  19. 19.
    D.J. Broadhurst: Phys. Lett. B101 (1981) 423Google Scholar
  20. 20.
    S.C. Generalis: Open University thesis OUT-4102-13 (1984); J. Phys. G15 (1989) L225; G16 (1990) 367, 785, L115Google Scholar
  21. 21.
    P. Ditsas, G. Shaw: Nucl. Phys. B229 (1983) 29; A Bradley, C.S. Langensiepen, G. Shaw: Phys. Lett. B102 (1981) 180, 359Google Scholar
  22. 22.
    L. Lewin: Polylogarithms and associated functions, New York: North Holland 1981Google Scholar
  23. 23.
    D. Shanks: J. Math. Phys. 34 (1955) 1; P. Wynn: Math. Comp. 15 (1961) 151; G.A. Baker, P. Graves-Morris: Padé approximants, in Encyclopedia of mathematics and its applications, Vol. 13, pp. 76–78 G.-C. Rota ed. Reading, Mass. Addison-Wesley. (1981)Google Scholar
  24. 24.
    A.V. Kotikov: Phys. Lett. B254 (1991) 185; B259 (1991) 314; B267 (1991) 123Google Scholar
  25. 25.
    A.I. Davydychev: J. Math. Phys. 32 (1991) 1052; 33 (1992) 358; E.E. Boos, A.I. Davydychev: Teor. Mat. Fiz. 89 (1991) 56Google Scholar
  26. 26.
    D.J. Broadhurst: Z. Phys. C54 (1992) 599Google Scholar
  27. 27.
    C. Ford, D.R.T. Jones: Phys. Lett. B274 (1992) 409; B285 (1992) 399(E); C. Ford, I. Jack, D.R.T. Jones: Nucl. Phys. B387 (1992) 373; E. Mendels: Nuovo Cimento A45 (1978) 87Google Scholar
  28. 28.
    A.C. Hearn: REDUCE user's manual, version 3.4, Rand publication CP78 (1991); J.A.M. Vermaseren: Symbolic manipulation with FORM. Amsterdam, Computer Algebra Nederland, 1991Google Scholar
  29. 29.
    M. Baranger, F.J. Dyson, E.E. Salpeter: Phys. Rev. 88 (1952) 680Google Scholar
  30. 30.
    D.J. Broadhurst, A.L. Kataev, O.V. Tarasov: Phys. Lett. B298 (1993) 445Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • D. J. Broadhurst
    • 1
  • J. Fleischer
    • 2
  • O. V. Tarasov
    • 2
  1. 1.Physics DepartmentOpen UniversityMilton KeynesUK
  2. 2.Fakultät für PhysikUniversität BielefeldBielefeld 1Germany

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