Fractional APT in QCD in the Euclidean and Minkowski regions

We describe the development of Analytic Perturbation Theory (APT) in QCD, called Fractional APT (FAPT), which has been suggested to apply the renormalization group evolution and QCD factorization technique in the framework of APT. 1 Basics of APT in QCD In the standard QCD Perturbation Theory (PT) we have: ✔ the Renormalization Group (RG) equation das(L)/dL = −as − c1 as − . . . for the effective coupling αs(μ ) = (4π/b0) as(L) with L = ln(μ /Λ); ✔ the one-loop solution generates Landau pole singularity: as(L) = 1/L; ✔ the two-loop solution generates square-root singularity: as(L) ∼ 1/ √ L+ c1 ln c1; ✔ PT series is a series in powers of effective coupling: D(L) = 1+d1as(L)+d2a 2 s(L)+ . . . In the Analytic Perturbation Theory (APT) we have: ✔ different effective couplings in Minkowskian (Radyushkin [1], and Krasnikov and Pivovarov [2]) and Euclidean (Shirkov and Solovtsov [3]) regions; ✔ APT is based on the RG and causality that guaranties standard perturbative UV asymptotics and spectral properties; ✔ in Euclidean domain, −q = Q, L = lnQ/Λ, APT generates the following set of images for the effective coupling and its n-th powers, {An(L)}n∈N; ✔ in Minkowskian domain, q = s, Ls = ln s/Λ , APT generates another set of images for the effective coupling and its n-th powers, {An(Ls)}n∈N; ✔ PT power series ∑ m dma m s (Q ) transforms into non-power series ∑ m dmAm(Q) in APT, where dm are numbers in MS -scheme. By the analytization in APT for an observable f(Q) we mean the “Källen–Lehman” representation