Nonpower ‘Spectral’ Perturbation Expansion in QFT

Abstract

We briefly report on new results concerning a perturbation expansion structure within the framework of an ‘analytic version’ of perturbative quantum chromodynamics (pQCD). This approach combines the RG symmetry with the Kallen–Lehmann analyticity in the Q2 variable. The procedure of analytization matches this analyticity with the RG invariance by adding to the analytized invariant coupling \(\overline \alpha _{s,{\text{an}}}\) some nonperturbative contributions containing no adjustable parameters. In turn, the new perturbative expansion (the APT expansion) for an observable represents asymptotic expansion over a nonpower set of specific functions \(\left\{ {A_n (x)} \right\}\) rather than in powers of \(\bar \alpha _{s,{\text{an}}} (x = Q^2 /\Lambda ^2 )\). We analyze this set and show that it obeys different properties in various ranges of the Q2 variable. In the UV, it is close to the power set \(\left\{ {\left[ {\bar \alpha _s (x)} \right]^n } \right\}\) used in the pQCD calculation. However, generally, this set is of a more complicated nature. In the ‘low Q2 region’ the behavior of \(A_n (x){\text{ at }}n >2\) is oscillating. Here, the APT expansion has a feature of asymptotic expansion a la Erdelyi. The issue of the consistency of an analytization procedure with the RG structure of observables is also discussed.