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arXiv 1102.2380hep-phhep-thmath-phmath.MP
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量子场论中非微扰元的必然性和重要性

Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory

Alexander P. Bakulev, Dmitry V. Shirkov

备注：
26 pages, 13 figures, invited talk delivered by the first author at the 6th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, Belgrade, Sept 15-23, 2010
发表信息：
Proceedings of the 6th Mathematical Physics Meeting, Sept. 14--23, 2010, Belgrade, Serbia (ISBN 978-86-82441-30-4), pp. 27--54
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摘要：

第一节课的主题是关于微扰理论（PT）方法的优点和缺点，即在量子理论中小参数$α$的幂的扩展。我们首先概述一个主要的量子理论工具，微扰展开方法的一般麻烦的特点。引人注目的问题是扰动级数在$α\ll 1$的幂不是一个收敛的级数。形式上的原因是量子振幅（矩阵元）$C(α)$在原点$α=0$处的本质奇异性。在许多物理上重要的情况下，人们需要某种替代的理论分析方法。特别地，这指的是低能域中的微扰QCD（pQCD）。在第二部分-讲座中，我们讨论了分析微扰理论（APT）的方法。我们从一个简短的历史序言开始，然后讨论如何结合色散关系与重整化群（RG）技术产生的APT与\myMath{\displaystyle e^{-1/α}}非分析性。接下来，我们考虑的结果APT应用到低能量QCD过程，并表明，在这种方法中的第四个回路的贡献，这似乎是在渐近边界上的pQCD方法，是为了每密耳数。然后，我们注意到，使用RG在QCD规定需要使用分数APT（FAPT）和描述其基本成分。作为FAPT在QCD中应用的一个例子，我们考虑了π介子形状因子$F_π(Q^2)$的计算。最后，讨论了{（F）APT}中非幂级数的重列，并将其应用于Higgs玻色子衰变宽度$Γ_{H\to\bar{b}b}(m_H^2)$的估计.

英文摘要：

The subject of the first section-lecture is concerned with the strength and the weakness of the perturbation theory (PT) approach, that is expansion in powers of a small parameter $α$, in Quantum Theory. We start with outlining a general troublesome feature of the main quantum theory instrument, the perturbation expansion method. The striking issue is that perturbation series in powers of $α\ll 1$ is not a convergent series. The formal reason is an essential singularity of quantum amplitude (matrix element) $C(α)$ at the origin $α=0$. In many physically important cases one needs some alternative means of theoretical analysis. In particular, this refers to perturbative QCD (pQCD) in the low-energy domain. In the second section-lecture, we discuss the approach of Analytic Perturbation Theory (APT). We start with a short historic preamble and then discuss how combining the Dispersion Relation with the Renormalization Group (RG) techniques yields the APT with \myMath{\displaystyle e^{-1/α}} nonanalyticity. Next we consider the results of APT applications to low-energy QCD processes and show that in this approach the fourth-loop contributions, which appear to be on the asymptotic border in the pQCD approach, are of the order of a few per mil. Then we note that using the RG in QCD dictates the need to use the Fractional APT (FAPT) and describe its basic ingredients. As an example of the FAPT application in QCD we consider the pion form factor $F_π(Q^2)$ calculation. At the end, we discuss the resummation of nonpower series in {(F)APT} with application to the estimation of the Higgs-boson-decay width $Γ_{H\to\bar{b}b}(m_H^2)$.

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