Convergence of the expansion of the Laplace-Borel integral in perturbative QCD improved by conformal mapping

Abstract

The optimal conformal mapping of the Borel plane was recently used to accelerate the convergence of the perturbation expansions in QCD. In this work we discuss the relevance of the method for the calculation of the Laplace-Borel integral expressing formally the QCD Green functions. We define an optimal expansion of the Laplace-Borel integral in the principal value prescription and establish conditions under which the expansion is convergent.PACS number͑s͒: 13.35.Dx, 12.38.Bx, 12.38.Cy 1 The conformal mapping that maps the whole holomorphy domain of the expanded function onto the unit disk will be called optimal. In this case, the singularities are mapped onto the boundary circle, and the requirement of holomorphy implies convergence of the power series at every point of the disk, which is the map of the holomorphy domain.

Expansions ofτhadronic spectral function moments in a nonpower QCD perturbation theory with tamed large order behavior

“…In this work we have investigated several spectral function moments of the massless Adler function in the frame of a new class of "nonpower" perturbative expansions in QCD, where the powers of the coupling are replaced by more adequate functions [11,14,16,[31][32][33]. The new expansions simultaneously implement RG summation, either in the "contour-improved" or in the "renormalization-group-summed" form, and the known location and nature of the first singularities of the expanded function in the Borel plane.…”

“…The new expansions simultaneously implement RG summation, either in the "contour-improved" or in the "renormalization-group-summed" form, and the known location and nature of the first singularities of the expanded function in the Borel plane. Mathematically, the definition is based on the acceleration of series convergence by the technique of conformal mappings [34] applied in the Borel plane [31][32][33]. When reexpanded in powers of a s , the new series reproduce order by order the perturbative coefficients known from Feynman diagrams.…”

“…More detailed arguments are given in two lemmas formulated and proved in [13,14]. For the Adler function in QCD, the optimal mapping w(u) and the corresponding perturbative expansion were defined and investigated in [31][32][33].…”

Determination of the Strong Coupling From Hadronic Tau Decays Using Renormalization Group Summed Perturbation Theory

“…In this model, FOPT expansion approaches better the 'true value'. A method for taming the divergent behaviour of the QCD perturbative expansions was proposed in [27,28,29,30,31,32], using the series acceleration by the conformal mappings of the Borel plane and the implementation of the known nature of the leading singularities in this plane.…”

“…Consequently one can study the Borel transform of the QCD Adler function which has ultraviolet (UV) and infrared (IR) renormalon singularities in the Borel plane. The divergent behaviour can be considerably tamed by using techniques of series acceleration based on conformal mappings and 'singularity softening' [27,28,29,30,31,32]. The method is not applicable to the perturbative series in powers of α s since the expanded correlators are singular at α s = 0, but can be applied in the Borel plane.…”