Beyond Scale Variations: Perturbative Theory Uncertainties from Nuisance Parameters

We develop a new approach to estimate the uncertainty due to missing higher orders in perturbative predictions (the perturbative "theory uncertainty"), which overcomes many inherent limitations of the currently prevalent methods based on varying unphysical renormalization scales. In our approach, the true underlying sources of the theory uncertainty, namely the missing higher-order terms, are identified and parameterized in terms of mutually independent theory nuisance parameters (TNPs). The TNPs are true parameters of the calculation, i.e., they have a well-defined true value that is not or only imprecisely known. This approach affords the theory uncertainty all benefits of a truly parametric uncertainty: It provides correct correlations and allows for consistent error propagation and combination. Furthermore, the TNPs can be profiled in fits, allowing the data to reduce the theory uncertainties. On the theory side, it allows maximally exploiting all available higher-order information to reduce the theory uncertainty, such as partial higher-order results or any nontrivial knowledge of the higher-order or all-order structure. We first discuss the method in general as it can be applied across the board of perturbative calculations. As a concrete application, we then discuss the resummed transverse momentum ($q_T$) spectrum in Drell-Yan production, and how TNP-based uncertainties can correctly capture the correlations across the $q_T$ spectrum and between $Z$ and $W$ production. This application is the basis of the theory model enabling the recent precise measurement of the $W$-boson mass by the CMS experiment. In a forthcoming paper, we use it to study the theory uncertainties in extracting the strong coupling constant $\alpha_s$ from the $Z$ $q_T$ spectrum.