An extensive survey of the estimation of uncertainties from missing higher orders in perturbative calculations

Abstract

We consider two approaches to estimate and characterise the theoretical uncertainties stemming from the missing higher orders in perturbative calculations in Quantum Chromodynamics: the traditional one based on renormalisation and factorisation scale variation, and the Bayesian framework proposed by Cacciari and Houdeau. We estimate uncertainties with these two methods for a comprehensive set of more than thirty different observables computed in perturbative Quantum Chromodynamics, and we discuss their performance in properly estimating the size of the higher order terms that are known. We find that scale variation with the conventional choice of varying scales within a factor of two of a central scale gives uncertainty intervals that tend to be somewhat too small to be interpretable as 68% confidence-level-heuristic ones. We propose a modified version of the Bayesian approach of Cacciari and Houdeau which performs well for non-hadronic observables and, after an appropriate choice of the relevant expansion parameter for the perturbative series, for hadronic ones too.

On the Higgs cross section at N3LO+N3LL and its uncertainty

“…However, in ref. [21] this factorial growth is instead included in the definition of the series expansion,…”

“…We included an explicit offset k 0 in the factorial: this is useful since different observables will have the factorial growth starting at different orders. Note that in the original works [19,21] there is a distinction between observables starting at different orders α l s . In our case, having factored out the LO, the complications arising from this distinction go away.…”

“…Section 4 contains the main results of this study, namely the matched N 3 LO+N 3 LL cross section and a detailed study of the perturbative uncertainty. Finally, in section 5 we compare our findings to different methods to assess the size of missing higher orders, namely the Cacciari-Houdeau method [19][20][21] and the application of convergence acceleration algorithms to the perturbative series, as suggested by David and Passarino [22].…”

“…In this section we want to explore alternative strategies to estimate the uncertainty from missing higher orders which do not rely on arbitrary variations of the perturbative ingredients. In particular, we will consider in section 5.1 the Cacciari-Houdeau method [19][20][21] for estimating the theory uncertainty using a Bayesian approach to infer the uncertainty from the progression of the perturbative expansion. Then, in section 5.2, we will follow the idea of ref.…”

The Higgs legacy of the LHC Run I

“…Note that a similar problem of choosing a prior for the theoretical uncertainty arises in the Bayesian approach [92,93]. On the one hand the Bayesian approach allows for a choice of priors in general, including the theoretical uncertainty, without having to ask for a statistical interpretation.…”