Expansion functions in perturbative QCD and the determination ofαs(Mτ2)

Abstract

The conventional series in powers of the coupling in perturbative QCD have zero radius of convergence and fail to reproduce the singularity of the QCD correlators like the Adler function at αs = 0. Using the technique of conformal mapping of the Borel plane, combined with the "softening" of the leading singularities, we define a set of new expansion functions that resemble the expanded correlator and share the same singularity at zero coupling. Several different conformal mappings and different ways of implementing the known nature of the first branch-points of the Adler function in the Borel plane are investigated, in both the contour-improved (CI) and fixed-order (FO) versions of renormalization-group resummation. We prove the remarkable convergence properties of a set of new CI expansions and use them for a determination of the strong coupling from the hadronic τ decay width. By taking the average upon this set, with a conservative treatment of the errors, we obtain αs(M 2 τ ) = 0.3195 +0.0189 −0.0138 . PACS numbers: 12.38.Bx, 12.38.Cy 1 In the so-called "order-dependent" conformal mappings, which were defined also in the coupling plane [12,13], the singularity is shifted away from the origin by a certain amount at each finiteorder, and tends to the origin only when an infinite number of terms are considered.

Review of alpha_s determinations

“…However, assuming that the known n ≤ 4 terms of the Adler series are already governed by low-lying infrared renormalons (i.e., already sensitive to the asymptotic series regime), it has been argued that CIPT could miss cancelations induced by the renormalonic behaviour [28]; in that case, FOPT could approach faster the Borel summation of the full renormalon series. Making optimal conformal mappings in the Borel plane and properly implementing the CIPT procedure within the Borel transform, one also obtains numerical results close to the FOPT value [29,30].…”

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

“…1, we show the behaviour of the modulus of the Adler function along the circle given by the first N = 5 terms in the expansions (4), (11) and (12), respectively. In this calculation and below we used the standard value α s (M 2 τ ) = 0.34, adopted also in previous studies [9,31]. One may see that the behaviour of the new RGS expansion is similar to that of the CIPT expansion.…”

“…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.…”

Constraints on the omega-pi form factor from analyticity and unitarity

“…Using as input the value α s (m 2 τ ) = 0.320 ± 0.020, which covers most of the recent determinations from hadronic τ decays (see for instance [35,36,37,38,39]), and the coupling's running we obtain α s = 0.357 ± 0.025. This yields for Π ′ pert the central value 0.0073 GeV −2 with an error of about 1.3%.…”