QCD tests from $e^+e^- \rar I=1~$ hadrons data and implication on the value of $\alpha_s$ from $\tau$-decays

Abstract

We re-examine the estimate of $\als$ and of the QCD condensates from $e^+e^-\rar I=1$ hadrons data. We conclude that $e^+e^-$ at low energies gives a value of $\Lambda$ compatible with the one from LEP and from tau inclusive decay. Using a $\tau$-like inclusive process and QCD spectral sum rules, we estimate the size of the $D$=4 to 9 condensates by a fitting procedure {\it without invoking stability criteria}. We find $\la \als G^2 \ra =(7.1\pm 0.7)10^{-2}$ GeV$^4$, $\rho\als\la \bar uu\ra^2= (5.8\pm 0.9)10^{-4}$ GeV$^6$, which confirm previous sum rules estimate based on stability criteria. The corrections due to the $D=8$ condensates and to instantons on the vector component of $\tau$-decay are respectively $\delta^{(8)}_1= -(1.5\pm 0.6)10^{-2}(1.78/M_\tau)^8$ and $\delta^{(9)}_1=-(7.0\pm 26.5)10^{-4}(1.78/M_\tau)^9 $, which indicate that the $\delta^{(8)}_1$ is one order magnitude higher than the vacuum saturation value, while the $D\geq 9$ instanton-like contribution to the the vector component of the $\tau$-decay width is a negligible correction. We also show that, due to the correlation between the $D=4$ and $1/M^2_\tau$ contributions in the ratio of the Laplace sum rules, the present value of the gluon condensate already excludes the recent estimate of the $1/M^2_\tau$-term from FESR in the axial-vector channel. Combining our non-perturbative results with the resummed perturbative corrections to the $\tau$-width $R_\tau$, we deduce from the present data $\alpha_s(M_\tau)= 0.33\pm 0.03$.