Abstract
We formulate a general approach to the inclusion of theoretical uncertainties, specifically those related to the missing higher order uncertainty (MHOU), in the determination of parton distribution functions (PDFs). We demonstrate how, under quite generic assumptions, theory uncertainties can be included as an extra contribution to the covariance matrix when determining PDFs from data. We then review, clarify, and systematize the use of renormalization and factorization scale variations as a means to estimate MHOUs consistently in deep inelastic and hadronic processes. We define a set of prescriptions for constructing a theory covariance matrix using scale variations, which can be used in global fits of data from a wide range of different processes, based on choosing a set of independent scale variations suitably correlated within and across processes. We set up an algebraic framework for the choice and validation of an optimal prescription by comparing the estimate of MHOU encoded in the next-to-leading order (NLO) theory covariance matrix to the observed shifts between NLO and NNLO predictions. We perform a NLO PDF determination which includes the MHOU, assess the impact of the inclusion of MHOUs on the PDF central values and uncertainties, and validate the results by comparison to the known shift between NLO and NNLO PDFs. We finally study the impact of the inclusion of MHOUs in a global PDF determination on LHC cross-sections, and provide guidelines for their use in precision phenomenology. In addition, we also compare the results based on the theory covariance matrix formalism to those obtained by performing PDF determinations based on different scale choices.
Highlights
An accurate estimate of the uncertainty in Standard Model (SM) predictions is a crucial ingredient for precision phenomenology at the Large Hadron Collider (LHC)
When computing factorized observables of the form Eqs. (3.14, 3.22), an entirely independent source of missing higher order uncertainty (MHOU) arises from the truncation of the perturbative expansion of the splitting functions that govern the parton distribution functions (PDFs) evolution equations. We show that this MHOU can again be estimated by scale variation; we will show that this scale variation can be performed in different ways: either at the level of the anomalous dimension; or at the level of the PDFs themselves; or at the level of the hard-scattering partonic coefficient functions, by exploiting the fact that physical results cannot depend on the scale at which the PDF is evaluated, and so one may trade the effect of scale variation between the PDF and the hard coefficient function
This is in principle required for consistency, given that MHOU are routinely part of the theoretical predictions for hadron collider processes, and likely to become a requirement for precision collider phenomenology as other sources of uncertainties decrease
Summary
An accurate estimate of the uncertainty in Standard Model (SM) predictions is a crucial ingredient for precision phenomenology at the Large Hadron Collider (LHC). The neglect of MHOUs might be biasing current global PDF fits It is the purpose of this paper to set up a general formalism for the inclusion of theoretical uncertainties, MHOUs, in PDF determinations, and to perform a first exploration of their impact on LHC phenomenology. In estimating MHOUs for a given process, the most commonly adopted option is the so-called seven-point envelope prescription, in which μr and μ f are independently varied by a factor of two about the central choice while ensuring that 1/2 ≤ μr /μ f ≤ 2, and the MHOU is taken as the envelope of the results For our purposes this is insufficient: rather than taking an envelope, we wish to contruct a covariance matrix out of the scale variations. A concise discussion of the main results of this work was presented in Ref. [16], of which this paper represents the extended companion
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