分数APT中的恢复方法：我们需要计算多少个循环？
Resummation approach in Fractional APT: How many loops do we need to calculate?
摘要：
本文简要介绍了QCD中的分析微扰理论（APT），讨论了它的问题以及如何在分数微扰理论（FAPT）中解决这些问题，并简要介绍了FAPT中考虑重夸克阈值的情况。然后，我们描述了单圈APT和FAPT中的重算方法，只要已知扰动系数d_n的母函数P（t），它在欧氏区域和Minkowski区域都产生有限的解。我们考虑它在希格斯玻色子衰变宽度H ^0 \to b \bar{b}和N_f=4区域的Adler函数D（Q ^2）和比值R（s）的估计中的应用.为了提供数值答案，我们建议非常简单的微扰系数d_n的阶乘增长模型。我们看到，对于希格斯玻色子衰变的情况，当考虑d_3{ \mathcal A}_3项时，在N^3 LO近似下达到了1%量级的精度。在Adler函数D（Q^2）的情况下，我们已经在N^2LO（i）处具有0.1%的精度。例如，考虑d_2{ \mathcal A}_2项）。主要结论是：为了达到1%的精度，我们不需要计算超过四个循环，并且只需要d_4系数来估计相应的生成函数P（t）。
英文摘要：
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).
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