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

Abstract

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/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 $F_{\pi }(Q^2)$ 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 ${\Gamma }_{H\to \bar{b}b}(m_H^2)$.

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

References in corpus (9)

• Bound state approach to the QCD coupling at low energy scales
• Higher-order QCD perturbation theory in different schemes: From FOPT to CIPT to FAPT
• QCD coupling below 1 GeV from quarkonium spectrum
• Bjorken Sum Rule and pQCD frontier on the move
• Nucleon spin structure and pQCD frontier on the move
• Asymptotic freedom and IR freezing in QCD: the role of gluon paramagnetism
• Testing the Concept of Quark-Hadron Duality with the ALEPH $\tau$ Decay Data
• Resummation in (F)APT
• Higgs boson decay into bottom quarks and uncertainties of perturbative QCD predictions

Cited by in corpus (3)

• FAPT: a Mathematica package for calculations in QCD Fractional Analytic Perturbation Theory
• anQCD: a Mathematica package for calculations in general analytic QCD models
• M/F-Theory as Mf-Theory