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

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 p ^−s. 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 /m _μ, 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.