On the ultraviolet behavior of the invariant charge in quantum electrodynamics

TL;DR

This paper investigates the ultraviolet behavior of the invariant charge in QED, showing it lacks Landau pole singularity for complex momenta and exploring its asymptotics using $1/N$ perturbation theory and nonphysical models.

Contribution

It introduces a new invariant charge definition, applies $1/N$ perturbation theory to QED with imaginary charge, and proposes a modified $1/N$ expansion for better ultraviolet analysis.

Findings

Invariant charge has no Landau pole for complex momenta.

Ultraviolet asymptotics of corrections follow specific logarithmic patterns.

Modified $1/N$ expansion is ultraviolet finite.

Abstract

In this paper we study the ultraviolet behavior of the invariant charge in QED. We show that for complex momenta the invariant charge does not have Landau pole singularity. We can define new invariant charge as real part of standard invariant charge. New invariant charge is limited from above and does not have Landau pole singularity. Also we use the $1/N$ perturbation theory for the investigation of the ultraviolet behavior of the invariant charge. To this aim we consider QED with imaginary charge which is asymptotically free but nonphysical model. In QED with nonphysical imaginary charge we can reliably calculate the ultraviolet asymptotics for the $(1/N)^k$ correction to the invariant charge, namely: $\alpha_k(\frac{p^2}{\mu^2}, \alpha) \sim (\ln(\frac{p^2}{\mu^2}))^{-k-1}$ at $k > 1$ and $\alpha_1(\frac{p^2}{\mu^2}, \alpha) \sim (\frac{\ln(\ln(\frac{p^2}{\mu^2})}{\ln^2(\frac{p^2}{\mu^2})})$ at $k =1$. The $1/N$ perturbation theory coincides for QED with imaginary charge and standard QED with real charge. It means in particular that ultraviolet behavior of the $(1/N)^k$ correction $\alpha_k(\frac{p^2}{\mu^2}, \alpha)$ in real QED coincides with the corresponding asymptotics for QED with imaginary charge. We propose also to use the modified $1/N$ expansion which is ultraviolet finite. The comparison of the standard QED and nonphysical QED with imaginary charge gives hint that in other non asymptotically free models like supersymmetric QED, scalar QED or Wess-Zumino model ultraviolet asymptotics of the invariant charge coincides with leading log approximtion.