Renormalons and multiloop 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 $\bar{MS}$-scheme in the limit of a large $N_f$. We find that in the factorial growth of the coefficients due to renormalons takes over almost immediatelly in the euclidean region. The essential differences between the large-order growth of coefficients in the scalar case, and in the vector case are analysed. In the timelike region a stabilization of the perturbative series for the imaginary part, with $n$-loop behaviour $S_n/[\log(s/\Lambda^2)]^{n-1}$, where $S_n$ is essential constant for $n\le{6}$, is observed. Only for $n\ge{7}$ does one discern the factorial growth and alternations of sign. Out all-order results are used to scrutinize the performance of multiloop estimates, within the ``naive nonabelianization'' procedure, and the effective charges approach. The asymtotic behaviour of perturbative coefficients, in both large $N_f$ and large $N_c$ limits, is analysed. A contour-improved resummation technique in the time-like region is developed. Some subtleties of scheme-dependence are illustrated using results in the $\bar{MS}$ and $V$-schemes. The all-order 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.