Renormalons and multi-loop estimates in scalar correlators, Higgs decay and quark-mass sum rules

The single renormalon-chain contribution to the correlator of scalar currents in QCD is calculated in the (MS) over bar -scheme in the limit of a large number of fermions, N-f. At n-loop order we find that in the (MS) over bar -scheme the factorial growth of the perturbative coefficients due to renormalons takes over almost immediately in the Euclidean region. The essential differences between the large-order growth of perturbative coefficients in the present scalar case, and in the previously-studied vector case are analysed. In the timelike region a stabilization of the corresponding perturbative series for the imaginary part, with n-loop behaviour S-n/[log(s/Lambda (2))](n-1), where S-n is essentially constant for n less than or equal to 6, is observed. Only for n greater than or equal to 7 does one discern the factorial growth and alternations of sign. We use the new all-orders results to scrutinize the performance of multi-loop estimates, using a large-beta (0) = (11N(c) - 2N(f))/12 approximation, the so-called naive nonabelianization procedure, and within the effective charges approach. The asymptotic behaviour of perturbative coefficients, in both large-N-f and large-N-c limits, is analysed both in the spacelike and timelike regions. A contour-improved resummation technique in the timelike region is developed. Some subtleties connected with scheme-dependence are analysed, and illustrated using results in the (MS) over bar- and V-schemes. The all-orders series under investigation are summed up with the help of the Borel resummation method. The results obtained are relevant to the analysis of the theoretical uncertainties in the 4-loop extractions of the running and invariant s-quark masses from QCD sum rules, and in calculations of the Higgs boson decay width into a quark-antiquark pair. (C) 2001 Elsevier Science B.V. All rights reserved.