Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings

The present calculations in perturbative QCD reach the order alpha(4)(s) for several correlators calculated to five loops, and the huge computational difficulties make unlikely the full six-loop calculation in the near future. This situation has practical consequences; in particular the treatment of the higher orders of the perturbation series for the current-current correlator of light quarks is one of the main sources of errors in the extraction of the strong coupling from hadronic tau decays. Several approximate estimates of the next coefficients of the corresponding Adler function have been proposed, using various arguments. In the present paper we exploit the analytic structure of the Adler function in the Borel plane, which allows the definition of an improved perturbative expansion in powers of a conformal variable which maps the cut Borel plane onto the unit disk. The new expansions converge in a larger domain of the Borel plane and, when reexpanded in powers of the strong coupling, yield definite values for the higher perturbative coefficients. We apply the method to the Adler function in the (MS) over bar scheme and to a suitable weighted integral of this function in the complex s plane, chosen such as to avoid model-dependent assumptions on analyticity. Our results c(5,1) = 287 +/- 40, c(6,1) = 2948 +/- 208 and c(7,1) = (1.89 +/- 0.75) x 10(4), for the six-, seven- and eight-loop coefficients, respectively, agree with a recent determination from Pade approximants applied to the perturbative expansion of the hadronic tau decay rate.