QCD condensates and αs from τ-decay

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) $R_{10}^A\left(\tau \right)$ within stability criteria and τ-like higher moments $R_{n,A}$ within stability for arbitrary τ-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 τ-decay like moment. We obtain to order ${\alpha }_s^4$ the conservative value from the $s_0$-stability until $M_{\tau }^2$: ${\alpha }_s\left(M_{\tau }\right){|}_A=0.3178\left(66\right)$ (FO) and 0.3380 (44) (CI) leading to: ${\alpha }_s\left(M_Z\right){|}_A=0.1182{\left(8\right)}_{fit}{\left(3\right)}_{evol.}$ (FO) and $0.1206{\left(5\right)}_{fit}{\left(3\right)}_{evol.}$ (CI). We extend the analysis to the V–A channel and find: ${\alpha }_s\left(M_{\tau }\right){|}_{V-A}=0.3135\left(83\right)$ (FO) and 0.3322 (81) (CI) leading to: ${\alpha }_s\left(M_Z\right){|}_{V-A}=0.1177{\left(10\right)}_{fit}{\left(3\right)}_{evol.}$ (FO) and $0.1200{\left(9\right)}_{fit}{\left(3\right)}_{evol.}$ (CI). We observe that in different channels ($e^{+}{e}^{-}\to$ Hadrons, A, V–A), the extraction of ${\alpha }_s\left(M_{\tau }\right)$ at the observed τ-mass leads to an overestimate of its value. Our determinations from these different channels lead to the mean: ${\alpha }_s\left(M_{\tau }\right)=0.3140\left(44\right)$ (FO) and 0.3346 (35) (CI) corresponding to: ${\alpha }_s\left(M_Z\right)=0.1178{\left(6\right)}_{fit}{\left(3\right)}_{evol.}$ (FO) and $0.1202{\left(4\right)}_{fit}{\left(3\right)}_{evol.}$ (CI). Comparisons with some other results are done.