Resummation approach in Fractional APT: How many loops do we need to calculate?

Abstract

We give a short introduction to the Analytic Perturbation Theory (APT) in QCD, discuss its problems and how they can be resolved in Fractional APT (FAPT), and give a brief report about taking into account heavy-quark thresholds in FAPT. Then we describe the resummation approach in the one-loop APT and FAPT, which produces finite answers in both Euclidean and Minkowski regions, provided the generating function P(t) of perturbative coefficients d_n is known. We consider its applications in estimations of the width of Higgs boson decay H^0\to b\bar{b} and of the Adler function D(Q^2) and the ratio R(s) in the N_f=4 region. In order to provide numerical answers we suggest very simple factorially growing models for perturbative coefficients d_n. We see that for the case of Higgs boson decay an accuracy of the order of 1% is reached at N^3LO approximation, when term d_3{\mathcal A}_3 is taken into account. In the case of Adler function D(Q^2) we have an accuracy of the order of 0.1% already at N^2LO (i. e., with taking into account d_2{\mathcal A}_2 term). The main conclusion is: In order to achieve an accuracy of the order of 1%, we do not need to calculate more than four loops and d_4 coefficients are needed only to estimate corresponding generating functions P(t).