Renorm-group, Causality and Non-power Perturbation Expansion in QFT

TL;DR

This paper introduces an invariant analytic approach to perturbative QCD, transforming the coupling to eliminate ghost singularities and exploring non-power expansions for observables, which may improve the understanding of nonperturbative effects.

Contribution

It develops a new invariant analytic method for QCD perturbation series, replacing power expansions with non-power sets to better incorporate nonperturbative structures and reduce scheme dependence.

Findings

The invariant analytization removes ghost singularities from the coupling.

Non-power expansions can better capture nonperturbative effects.

The approach reduces scheme dependence of QCD observables.

Abstract

The structure of the QFT expansion is studied in the framework of a new "Invariant analytic" version of the perturbative QCD. Here, an invariant (running) coupling $a(Q^2/\Lambda^2)=\beta_1\alpha_s(Q^2)/4\pi$ is transformed into a "$Q^2$--analytized" invariant coupling $a_{\rm an}(Q^2/\Lambda^2) \equiv {\cal A}(x)$ which, by constuction, is free of ghost singularities due to incorporating some nonperturbative structures. Meanwhile, the "analytized" perturbation expansion for an observable $F$, in contrast with the usual case, may contain specific functions ${\cal A}_n(x)= [a^n(x)]_{\rm an}$, the "n-th power of $a(x)$ analytized as a whole", instead of $({\cal A}(x))^n$. In other words, the pertubation series for $F(x)$, due to analyticity imperative, may change its form turning into an {\it asymptotic expansion \`a la Erd\'elyi over a nonpower set} $\{{\cal A}_n(x)\}$. We analyse sets of functions $\{{\cal A}_n(x)\}$ and discuss properties of non-power expansion arising with their relations to feeble loop and scheme dependence of observables. The issue of ambiguity of the invariant analytization procedure and of possible inconsistency of some of its versions with the RG structure is also discussed.