Speaker
Gorazd Cvetic
(Santa Maria University)
Description
Perturbative QCD in the usual mass independent schemes gives us running coupling $a(Q^2) \equiv \alpha_s(Q^2)/\pi$ which has unphysical (Landau) singularities at low squared momenta $|Q^2| < 1 \ {\rm GeV}^2$ (where $Q^2 \equiv -q^2$). Such singularities do not reflect correctly the analytic (holomorphic) properties of spacelike observables ${\cal D}(Q^2)$ such as current correlators or structure function sum rules, the properties dictated by the general principles of (local) quantum field theory. Therefore, evaluating ${\cal D}(Q^2)$ in perturbative QCD in terms of the coupling $a(\kappa Q^2)$ (where $\kappa \sim 1$ is the renormalization scale parameter) cannot give us correct results at low $|Q^2|$. As an alternative, analytic (holomorphic) models of QCD have been constructed in the literature, where $A_{1}(Q^2)$ [the holomorphic analog of the underlying perturbative $a(Q^2)$] has the desired properties. We present our programs, written in Mathematica and in Fortran, for the evaluation of the $A_{\nu}(Q^2)$ coupling, a holomorphic analog of the powers $a(Q^2)^{\nu}$ where $\nu$ is a real power index, for various versions of analytic QCD:
(A) (Fractional) Analytic Perturbation Theory ((F)APT) model of Shirkov, Solovtsov et al. (extended by Bakulev, Mikhailov and Stefanis to noninteger $\nu$); in this model, the discontinuity function $\rho_{\nu}(\sigma) \equiv {\rm Im} A_{\nu}(-\sigma - i \epsilon)$, defined at $\sigma>0$, is set equal to its perturbative counterpart: $\rho_{\nu}(\sigma) = {\rm Im} a(-\sigma - i \epsilon)^{\nu}$ for $\sigma>0$, and zero for $\sigma<0$.
(B) Two-delta analytic QCD model (2$\delta$anQCD) of Ayala, Contreras and Cvetic; in this model, the discontinuity function $\rho_1(\sigma) \equiv {\rm Im} A_{1}(-\sigma - i \epsilon)$ is set equal to its perturbative counterpart for high $\sigma > M_0^2$ (where $M_0 \sim 1$ GeV), and at low postive $\sigma$ the otherwise unknown behavior of $\rho_1(\sigma)$ is parametrized as a linear combination of two delta functions.
(C) The massive QCD of Shirkov, where $A_{1}(Q^2) = a(Q^2+M^2)$ with $M \sim 1$ GeV.
Summary
We present programs, in Mathematica and in Fortran, for calculation of the general power analogs of the coupling in three different analytic (holomorphic) models of QCD.
Author
Gorazd Cvetic
(Santa Maria University)
Co-author
Dr
Cesar Ayala
(Santa Maria University)
