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

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 $\alpha$, 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 $\alpha \ll 1$ is not a convergent series. The formal reason is an essential singularity of quantum amplitude (matrix element) $C(\alpha)$ at the origin $\alpha=0$. 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/\alpha}} 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 $F_\pi(Q^2)$ 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 $\Gamma_{H\to\bar{b}b}(m_H^2)$.