Determination of perturbative QCD coupling from ALEPH $τ$ decay data using pinched Borel-Laplace and Finite Energy Sum Rules

Abstract

We present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH τ -decay data. The determination involves the double-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules. The theoretical basis is the Operator Product Expansion (OPE) of the V+A channel Adler function in which the higher order terms of the leading-twist part originate from a model based on the known structure of the leading renormalons of this quantity. The applied evaluation methods are contour-improved perturbation theory (CIPT), fixed-order perturbation theory (FOPT), and Principal Value of the Borel resummation (PV). All the methods involve truncations in the order of the coupling. In contrast to the truncated CIPT method, the truncated FOPT and PV methods account correctly for the suppression of various renormalon contributions of the Adler function in the mentioned sum rules. The extracted value of the MS coupling is αs(m 2 τ ) = 0.3116±0.0073 [αs(M 2 Z ) = 0.1176±0.0010] for the average of the FOPT and PV methods. Inclusion of the CIPT method would increase these average values and uncertainties significantly.

Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $τ$ decay data, and determination of pQCD coupling

“…If we included in the average also CI extraction, the value would be αs(m 2 τ ) = 0.3270 +0.0244 −0.0241 [αs(M 2 Z ) = 0.1195 ± 0.0030]. This work is an extension and improvement of our previous work [1] where we used for the truncated OPE a more naive (and widely used) form and where the extracted values for αs(M 2 Z ) were somewhat lower.…”

“…This implies that the sum rules have the perturbative part (dimension D = 0), and the nonperturbative corrections (D > 0), and the latter are often small such as in the case of R τ (r τ ) [9,18]. One of the novel aspects covered in the present work, in comparison with the previous one [1], is that we will deal with the nonperturbative contributions more carefully. In particular here the D (≡ 2k) ≥ 6 condensate contributions to the Adler function d(Q 2 ) have two parts, which eliminate the renormalon ambiguity originating from the two infrared (IR) renormalon contributions of the Adler function at u = k (k = 3, 4, .…”

“…Beyond LO (bLO), such cancellations were demonstrated in an approach with a modified Borel transform in a specific renormalisation scheme [28]. In our previous work [1], we presented arguments that such cancellations take place in sum rules at the bLO level in any renormalisation scheme when the D = 0 contribution to the sum rule is written in terms of the series in logarithmic derivatives of a(σ max ) (cf. Appendix of [1]).…”

“…Specific contour integrals involving the Adler function give us various τ -decay sum rules, including the inclusive strangeless decay ratio R τ = Γ(τ → ν τ hadrons)/Γ(τ → ν τ e − νe ) 1 . The Adler function was calculated by perturbation techniques of QCD up to O(α 4 s ), cf.…”

“…In our previous work [1], we presented arguments that such cancellations take place in sum rules at the bLO level in any renormalisation scheme when the D = 0 contribution to the sum rule is written in terms of the series in logarithmic derivatives of a(σ max ) (cf. Appendix of [1]).…”