Abstract
We present a methodology for the construction of parton distribution functions (PDFs) designed to provide an accurate representation of PDF uncertainties for specific processes or classes of processes with a minimal number of PDF error sets: specialized minimal PDF sets, or SM-PDFs. We construct these SM-PDFs in such a way that sets corresponding to different input processes can be combined without losing information, specifically as regards their correlations, and that they are robust upon smooth variations of the kinematic cuts. The proposed strategy never discards information, so that the SM-PDF sets can be enlarged by the addition of new processes, until the prior PDF set is eventually recovered for a large enough set of processes. We illustrate the method by producing SM-PDFs tailored to Higgs, top-quark pair, and electroweak gauge boson physics, and we determine that, when the PDF4LHC15 combined set is used as the prior, around 11, 4, and 11 Hessian eigenvectors, respectively, are enough to fully describe the corresponding processes.
Highlights
This Singular-Value Decomposition (SVD)+Principal Component Analysis (PCA) strategy achieves the twofold goal of obtaining a multigaussian representation of a starting Monte Carlo parton distribution functions (PDFs) set, and of allowing for an optimization of this representation for a specific set of input cross sections, which uses the minimal number of eigenvectors required in order to reach a desired accuracy goal
We present the validation of the Specialized Minimal PDFs (SM-PDFs) algorithm described in the previous section
The SM-PDFs for W, Z production could be relevant for the determination of the W boson mass [18,19,20], which is a extremely CPUtime consuming task
Summary
The SM-PDF methodology is built upon the strategy based on Singular-Value Decomposition (SVD) followed by Principal Component Analysis (PCA) described in the appendix of Ref. [11], in which the MCH method was presented. The optimized representation of the original covariance matrix, Eq (2), is found by replacing V with its principal submatrix P in Eq (5) This principal matrix P is the output of the SVD+PCA method: it contains the coefficients of the linear combination of the original replicas or error sets which correspond to the principal components, which can be used to compute PDF uncertainties using the Hessian method. The subsequent PCA projection may depend on scale if there are level crossings, but this is clearly a minor effect if a large enough number of principal components is retained Because of this property, the SVD+PCA methodology can be used for the efficient construction [9] of a Hessian representation of combined PDF sets, even when the sets which enter the combination satisfy somewhat different evolution equations, e.g., because of different choices in parameters such as the heavy quark masses, or in the specific solution of the DGLAP equations
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