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

TL;DR

This work extends large-$N_f$ renormalon analyses from the vector to the scalar current correlator in QCD, addressing the intricate mass-renormalization structure and the resulting factorial growth of perturbative series. It develops all-orders results via Borel resummation and contour-improved techniques, and systematically studies UV and IR renormalons, scheme dependence, and analytic continuation between Euclidean and Minkowski regions. Key contributions include an exact all-orders relation for the scalar anomalous dimension, high-order coefficients $h_n$ and $g_n$, identification of a novel IR renormalon at $oldsymbol abla=1$, and detailed assessments of Naive Nonabelianization (NNA) and effective-charge methods for estimating higher-loop terms. The findings inform theoretical uncertainties in Higgs decay widths and strange-quark mass extractions from QCD sum rules, and provide a rigorous framework for renormalon analyses in scalar channels with controlled scheme-invariant resummations.

Abstract

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 $S_n/[\log(s/Λ^2)]^{n-1}$, 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.