QCD condensates and ${\alpha }_s$ from $\tau$-decay

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) ${\mathcal{R}}_{10}^A(\tau )$ 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: ${\alpha }_s(M_{\tau }){|}_{V-A}=0.3135(83)$ (FO) and 0.3322 (81) (CI) leading to: ${\alpha }_s(M_Z){|}_{V-A}=0.1177(10{)}_{fit}(3{)}_{evol.}$ (FO) and $0.1200(9{)}_{fit}(3{)}_{evol.}$\,(CI). We extend the analysis to the V--A channel and find: ${\alpha }_s(M_{\tau }){|}_{V-A}=0.3135(83)$ (FO) and 0.3322 (81) (CI) leading to: ${\alpha }_s(M_Z){|}_{V-A}=0.1177(10{)}_{fit}(3{)}_{evol.}$ (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.

Comments: Latex file 18 pages, 3 Tables, 14 figures. Version published in Nucl.Phys. A after inclusion of the referee's suggestions