Thresholds in FAPT: Euclid vs Minkowski

We give a short introduction to the Analytic Perturbation Theory (APT) [1, 2, 3, 4, 5] in QCD, describe its problems and suggest as a tool for their resolution the Fractional APT (FAPT) [6, 7, 8, 9]. We also describe shortly how to treat heavy-quark thresholds in FAPT. As an applications of this technique we discuss (i) the pion form factor calculation in the Euclidean FAPT and (ii) the Higgs boson decay H → bb̄ in Minkowskian FAPT. We conclude with comparison of both approaches, Euclidean and Minkowskian FAPT. 1 Analytic Perturbation Theory in QCD In the standard QCD Perturbation Theory (PT) we know that the Renormalization Group (RG) equation das[L]/dL = −as−. . . for the effective coupling αs(Q) = a[L]/βf with L = ln(Q2/Λ2), βf = b0(Nf )/(4π) = (11 − 2Nf/3)/(4π) . Then the one-loop solution generates Landau pole singularity, a(1)[L] = 1/L with subscript (l) meaning l-loop order. In the Analytic Perturbation Theory (APT) we have different effective couplings in Minkowskian (Radyushkin [10] and Krasnikov&Pivovarov [11]) and Euclidean (Shirkov&Solovtsov [3]) regions. In the Euclidean domain, −q = Q, L = lnQ/Λ, APT generates one set of images for the effective coupling and its n-th powers, {An[L]}n∈N, whereas in the Minkowskian domain, q = s, Ls = ln s/Λ , it generates another set, {An[Ls]}n∈N. APT is based on the RG and causality that guaranties standard perturbative UV asymptotics and spectral properties. Power series ∑ m dma m (1)[L] transforms into non-power series ∑ m dmAm[L] in APT. By the analytization in APT for an observable f(Q2) we mean the “Källen–Lehman” representation [ f(Q) ] an = ∫ ∞ 0 ρf (σ) σ +Q2 − iǫ dσ (1) with the spectral density defined through the perturbative result, ρf (σ) = (1/π)Im [ fpert(−σ) ] . This results in different analytic images in Euclidean and Minkowski regions An[L] ≡ AE [a[L]] = ∫ ∞ 0 ρn(σ) σ +Q2 dσ , (2a) An[Ls] ≡ AM [a[L]] = ∫ ∞