A new method for estimating unknown one-order higher QCD corrections to the perturbative series using the linear regression through the origin

Abstract

Quantum chromodynamics (QCD) is the fundamental theory describing strong interactions. Owing to asymptotic freedom at short distances, high-energy physical observables can be predicted using perturbative QCD (pQCD) following proper factorization. It has been demonstrated that the conventional renormalization scheme-and-scale ambiguities that appear in fixed-order pQCD series can be eliminated by recursively applying the renormalization group equation, aided by the principle of maximum conformality (PMC). To extend the predictive power of pQCD, we still face the challenge of reliably estimating contributions from unknown higher-order (UHO) terms. In this paper, we propose a novel method for estimating one-order higher QCD corrections to the perturbative series: using linear regression through the origin (LRTO) to determine the asymptotic form of the pQCD series below the optimal truncation order \(N^*\). When the given \(\alpha _s\)-order is below \(N^*\), its perturbative behavior will be dominated by the usual \(\alpha _s\)-power suppression and the sub-leading corrections are treated as a source of theoretical uncertainty. This approach enables a quantitative assessment of the series convergence and derives estimate for unknown higher-order contributions. To illustrate this method, we apply it to the important ratio \(R_\tau \) which has been calculated up to four-loop QCD corrections. Our results show that the LRTO method yields reliable estimates of the UHO terms, demonstrating its own reliability and significant predictive power for such estimations. In particular, we find that the scale-invariant, more rapidly convergent PMC series exhibits better predictive power – along with greater stability and reliability – compared to the initial scale-dependent pQCD series.