Abstract
We improve the determinations of the QCD condensates within the SVZ expansion in the axial-vector (A) channel using the ratio of Laplace sum rule (LSR) ${\cal R}_{10}^A(\tau)$ within stability criteria and $\tau$-like higher moments ${R}_{n,A}$ within stability for arbitrary $\tau$-mass squared $s_0$. We find the same violation of the factorization by a factor 6 of the four-quark condensate as from $e^+e^- \to$ Hadrons data. One can notice a systematic alternate sign and no exponential growth of the size of these condensates. Then, we extract $\alpha_s$ from the lowest $\tau$-decay like moment and find: $\alpha_s(M_\tau)\vert_{V+A}=0.3135(83)$ (FO) and 0.3322 (81) (CI) leading to: $\alpha_s(M_Z)\vert_{V+A}=0.1177(10)_{fit}(3)_{evol.}$ (FO) and $0.1200(9)_{fit}(3)_{evol.}$ (CI). We extend the analysis to the V+A channel and find: $ \alpha_s(M_\tau)\vert_{V+A}=0.3135(83)$ (FO) and 0.3322 (81) (CI) leading to: $\alpha_s(M_Z)\vert_{V+A}=0.1177(10)_{fit}(3)_{evol.}$ (FO) and $0.1200(9)_{fit}(3)_{evol.}$ (CI). We observe that in different channels ($e^+e^-\to$ Hadrons, A, V+A), the extraction of $\alpha_s(M_\tau)$ at the observed $\tau$-mass leads to an overestimate of its value. Our determinations from these different channels lead to the mean: $ \alpha_s(M_\tau)=0.3140(44)$ (FO) and 0.3346 (35) (CI) leading to: $\alpha_s(M_Z)=0.1178(6)_{fit}(3)_{evol.}$ (FO) and $0.1202(4)_{fit}(3)_{evol.}$(CI). Comparisons with some other results are done.