QCD condensates and $\alpha_s$ from $e^+e^-$ and $\tau$-decays

TL;DR

This paper reviews determinations of QCD condensates and the strong coupling constant _s from e+e- annihilation and -decay data, confirming previous results and excluding significant duality violations in the Euclidean region.

Contribution

It provides updated and consistent extractions of QCD condensates and _s using Laplace sum rules and -like moments, correcting previous misprints and analyzing duality violations.

Findings

Gluon condensate value agrees with quarkonia sum rules.

Four-quark condensate factorization is violated by a factor of about 6.

No exponential growth of condensates up to dimension 20, excluding significant duality violation.

Abstract

In this talk, I review the determinations of the QCD condensates and $\alpha_s$ within the SVZ expansion using the ratio of Laplace sum rule (LSR) and $\tau$-like moments in $e^+e^-\to I=1$ Hadrons and in $\tau\to \nu_\tau+$Hadrons Axial-vector (A) and V-A channels. Some misprints in the original papers [1-3] have been corrected. We found that the value of the gluon condensate agrees with the one $\langle \alpha_s G^2\rangle=(6.35\pm 0.35)\times 10^{-2}$ GeV$^4$ from quarkonia and some other sum rules but less accurate, while the factorization of the four-quark condensate is violated by a factor about 6: $\rho\langle\bar\psi\psi\rangle^2=(5.98\pm 0.64)\times 10^{-4}$ GeV$^6$ which confirms previous findings. Extracting the QCD condensates up to dimension D=20 from $e^+e^-$ and the axial-vector channel of $\tau$-decay, we do not find any exponential growth of their values in the Euclidian region, thus excluding (by duality) any significant effect of the so-called Duality Violation (DV) in the Time-like one. The optimal values of $\alpha_s(M_\tau)$ from $e^+e^-\to$ Hadrons and $\tau$-decays agree each others and lead to the average: $\alpha_s(M_\tau)=0.3128(51)$ [resp.0.3330(57)] $ \longrightarrow \alpha_s(M_Z) = 0.1176$ [resp. 0.1201] $(7)_{fit}(3)_{evol.}$ for Fixed Order (FO) [resp. Contour improved (CI)] PT series to be compared with the PDG24 average (without Lattice calculations): $ \alpha_s(M_Z) = 0.1175(10)$.