Advertisement

LargeN expansion of QED: asymptotic photon propagator and contributions to the muon anomaly, for any number of loops

  • D. J. Broadhurst
Article

Abstract

We calculate analytical contributions to then-loop asymptotic photon propagator from diagrams withn−1 electron loops, i.e. theO(1/N) terms in the largeN limit. The corresponding contributions to the on-shell β-function, β(α)=τ6 log α/∂ logm reduced to rational combinations of ζ s =∑ p ps. For the β-function of the MOM scheme (i.e. the Gell-Man-Low function) we obtain theO(1/N) terms of
$$\Psi (\alpha _{MOM} ) = \beta (\alpha )\partial \log \alpha _{MOM} /\partial \log \alpha = \sum\nolimits_n {\Psi _n (\alpha _{MOM} /\pi )^n } $$
in closed form:
$$\begin{gathered} \Psi _n^{[n - 1]} = \frac{{(n - 1)!}}{{( - 3)^{n - 1} }}\left[ { - 2n + 4 - \frac{{n + 4}}{{2^n }}} \right. \hfill \\ \left. { + \frac{{16}}{{n - 1}}\sum\limits_{n/2 > s > 0} {s(1 - 2^{ - 2s} )(1 - 2^{2s - n} )} \zeta _{2s + 1} } \right]. \hfill \\ \end{gathered} $$
Neglecting terms of orderm e /mμ, we calculate (n+1-loop muon-anomaly contributions, obtained by inserting, in the one-loop diagram, alln-loop photon-propagator diagrams with at least (n−1) electron loops. The methods are applicable to any number of loops. Results have been obtained analytically up to 20 loops and numerically up to 100 loops.

Keywords

Field Theory Elementary Particle Quantum Field Theory Closed Form Particle Acceleration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T. Kinoshita, H. Kawai, Y. Okamoto: Phys. Lett. B254 (1991) 235Google Scholar
  2. 2.
    R.N. Faustov, A.L. Kataev, S.A. Larin, V.V. Starshenko: Phys. Lett. B254 (1991) 241Google Scholar
  3. 3.
    A. Palanques-Mestre, P. Pascual: Comm. Math. Phys. 95 (1984) 277Google Scholar
  4. 4.
    D.J. Broadhurst, A.L. Kataev, O.V. Tarasov: Phys. Lett. B298 (1993) 445; D.J. Broadhurst: OUT-4102-40 (1992)Google Scholar
  5. 5.
    T. Kinoshita: CLNS 93/1187 (January 1993)Google Scholar
  6. 6.
    S.G. Gorishny, A.L. Kataev, S.A. Larin: Phys. Lett. B273 (1991) 141; B275 (1992) 512 (Erratum)Google Scholar
  7. 7.
    D.J. Broadhurst: Z. Phys. C54 (1992) 599Google Scholar
  8. 8.
    T. Kinoshita, B. Nizic, Y. Okamoto: Phys. Rev. D41 (1990) 593Google Scholar
  9. 9.
    A.L. Kataev: Phys. Lett. B284 (1992) 401Google Scholar
  10. 10.
    E. de Rafael, J.L. Rosner: Ann. Phys. 82 (1974) 369Google Scholar
  11. 11.
    H. Kawai, T. Kinoshita, Y. Okamoto: Phys. Lett. B260 (1991) 193Google Scholar
  12. 12.
    D.J. Broadhurst: Z. Phys. C32 (1986) 249Google Scholar
  13. 13.
    D.T. Barfoot, D.J. Broadhurst: Z. Phys. C41 (1988) 81Google Scholar
  14. 14.
    A.N. Vasil'ev, Yu.M. Pis'mak, Yu.R. Khonkonen: Theor. Math. Phys. 46 (1981) 104; 47 (1982) 465; 50 (1982) 127Google Scholar
  15. 15.
    J.A. Gracey: Phys. Lett. B246 (1990) 114; B262 (1991) 49; J. Phys. A23 (1990) 2183; J. Mod. Phys. A6 (1991) 395; Nucl. Phys. 348 (1991) 737; B352 (1991) 183Google Scholar
  16. 16.
    J.A. Gracey: J. Phys. A24 (1991) L431; Mod. Phys. Lett A7 (1992) 1945Google Scholar
  17. 17.
    S.G. Gorishny, A.L. Kataev, S.A. Larin, L.R. Surguladze: Phys. Lett. B256 (1991) 81Google Scholar
  18. 18.
    R. Coquereaux: Phys. Rev. D23 (1981) 2276Google Scholar
  19. 19.
    J. Calmet, E. de Rafael: Phys. Lett. B56 (1975) 181Google Scholar
  20. 20.
    A.C. Hearn: REDUCE user's manual, version 3.4, Rand publication CP78 (1991)Google Scholar
  21. 21.
    B. Lautrup, E. de Rafael: Nucl. Phys. B 70 (1974) 317; B78 (1974) 576 (Erratum)Google Scholar
  22. 22.
    R. Barbieri, E. Remidd: Nucl. Phys. B90 (1975) 233Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • D. J. Broadhurst
    • 1
  1. 1.Physics DepartmentOpen UniversityMilton KeynesUK

Personalised recommendations