pQCD running couplings finite and monotonic in the infrared: when do they reflect the holomorphic properties of spacelike observables?

TL;DR

This paper studies a class of perturbative QCD couplings that remain finite and monotonic in the infrared, analyzing their singularity structure and providing an algebraic method to identify Landau singularities.

Contribution

It introduces an algebraic algorithm to locate Landau singularities of pQCD couplings with finite infrared limits, improving upon numerical methods.

Findings

The algebraic method effectively identifies Landau singularities.

Couplings with finite infrared limits can be free of Landau singularities.

Comparison shows the algebraic approach is efficient and precise.

Abstract

We investigate a large class of perturbative QCD (pQCD) renormalization schemes whose beta functions $\beta(a)$ are meromorphic functions of the running coupling and give finite positive value of the coupling $a(Q^2)$ in the infrared regime ("freezing"), $a(Q^2) \to a_0$ for $Q^2 \to 0$. Such couplings automatically have no singularities on the positive axis of the squared momenta $Q^2$ ($ \equiv -q^2$). Explicit integration of the renormalization group equation (RGE) leads to the implicit (inverted) solution for the coupling, of the form $\ln (Q^2/Q^2_{\rm in}) = {\cal H}(a)$. An analysis of this solution leads us to an algebraic algorithm for the search of the Landau singularities of $a(Q^2)$ on the first Riemann sheet of the complex $Q^2$-plane, i.e., poles and branching points (with cuts) outside the negative semiaxis. We present specific representative examples of the use of such algorithm, and compare the found Landau singularities with those seen after the 2-dimensional numerical integration of the RGE in the entire first Riemann sheet, where the latter approach is numerically demanding and may not always be precise. The specific examples suggest that the presented algebraic approach is useful to find out whether the running pQCD coupling has Landau singularities and, if yes, where precisely these singularities are.