Extraction of ${\alpha }_s$ using Borel-Laplace sum rules for tau decay data

Double-pinched Borel-Laplace sum rules are applied to ALEPH $\tau$-decay data. For the leading-twist ($D=0$) Adler function a renormalon-motivated extension is used, and the 5-loop coefficient is taken to be $d_4=275\pm 63$. Two $D=6$ terms appear in the truncated OPE ($D\le 6$) to enable cancellation of the corresponding renormalon ambiguities. Two variants of the fixed order perturbation theory, and the inverse Borel transform, are applied to the evaluation of the $D=0$ contribution. Truncation index $N_t$ is fixed by the requirement of local insensitivity of the momenta $a^{(2,0)}$ and $a^{(2,1)}$ under variation of $N_t$. The averaged value of the coupling obtained is ${\alpha }_s(m_{\tau }^2)=0.3235_{-0.0126}^{+0.0138}$ [$\alpha_s(M_Z^2)=0.1191 \pm 0.0016$]. The theoretical uncertainties are significantly larger than the experimental ones.