Determination of proton PDF uncertainties with Markov chain Monte Carlo

Abstract

We present an analysis of parton distribution functions (PDFs) of the proton using Markov Chain Monte Carlo (MCMC) methods. The MCMC approach naturally implements Bayes' theorem and thus provides a means to directly sample the underlying probability distribution - in this case the probability distribution of the PDF parameters. This allows for a straightforward propagation of the resulting uncertainties into any PDF-dependent observable, preserving their simple probabilistic interpretation. In our analysis we include a broad set of deep inelastic scattering data from HERA, BCDMS and NMC experiments along with the Drell-Yan, $W$ and $Z$ boson data from LHC and Tevatron experiments, which combined with theoretical calculations at next-to-next-to-leading order in QCD allow for realistic determination of PDFs. The main focus of this analysis is to explore alternative methods for PDF uncertainty estimation that are more firmly grounded in statistical principles. We show that the flexibility of the Bayes framework, allowing e.g. to account for non-Gaussianity or inconsistencies of data sets, is crucial to extract realistic uncertainties when such assumptions are not fulfilled. We also demonstrate that MCMC allows one to determine the $\Delta {\chi }^2$ value corresponding to a given confidence level in the sample, which can in turn be used as a statistically well-founded tolerance criterion used in the Hessian method, thus addressing one of its main long-standing drawbacks.

arXiv comment20 pages, 9 figures, matches published version: Phys. Rev. D 113, 054049

References in corpus (60)

• Parton distributions for the LHC
• New Generation of Parton Distributions with Uncertainties from Global QCD Analysis
• Parton distributions from high-precision collider data
• Vector boson pair production at the LHC
• New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC
• Parton distributions from LHC, HERA, Tevatron and fixed target data: MSHT20 PDFs
• Leptoproduction of Heavy Quarks II -- A Unified QCD Formulation of Charged and Neutral Current Processes from Fixed-target to Collider Energies
• Parton Distribution Functions, ${\alpha }_s$ and Heavy-Quark Masses for LHC Run II
• The Path to Proton Structure at One-Percent Accuracy
• Uncertainties of predictions from parton distribution functions II: the Hessian method
• Measurement of the proton and deuteron structure functions, F2p and F2d, and of the ratio sigma(L)/sigma(T)
• Uncertainties of predictions from parton distributions. I: experimental errors
• APFEL: A PDF Evolution Library with QED corrections
• Monte Carlo errors with less errors
• Constraints on large-$x$ parton distributions from new weak boson production and deep-inelastic scattering data
• A Higher Order Perturbative Parton Evolution Toolkit (HOPPET)
• A posteriori inclusion of parton density functions in NLO QCD final-state calculations at hadron colliders: The APPLGRID Project
• Uncertainties of Predictions from Parton Distribution Functions I: the Lagrange Multiplier Method
• The ABM parton distributions tuned to LHC data
• Parton distributions and lattice QCD calculations: a community white paper
• The Structure of the Proton in the LHC Precision Era
• Leptoproduction of Heavy Quarks I-- General Formalism and Kinematics of Charged Current and Neutral Current Production Processes
• Treatment of Heavy Quarks in Deeply Inelastic Scattering
• Dynamical parton distributions of the nucleon and very small-x physics
• Measurement of the transverse momentum and ${\varphi }_{\eta }^{\ast }$ distributions of Drell-Yan lepton pairs in proton-proton collisions at $\sqrt{s}=8$ TeV with the ATLAS detector
• A guide to light-cone PDFs from Lattice QCD: an overview of approaches, techniques and results
• Measurement of the shape of the boson rapidity distribution for p\bar{p}-> Z/gamma*-> e+e- +X events produced at sqrt(s) of 1.96 TeV
• Measurement of the forward $Z$ boson production cross-section in $pp$ collisions at $\sqrt{s}$ = 7 TeV
• Measurement of forward $W$ and $Z$ boson production in $pp$ collisions at $\sqrt{s}=8\mathrm{Te}\mathrm{V}$
• Multivariate Fitting and the Error Matrix in Global Analysis of Data
• Parton Distributions in Nucleons and Nuclei
• Fitting Parton Distribution Data with Multiplicative Normalization Uncertainties
• Measurement of the differential and double-differential Drell-Yan cross sections in proton-proton collisions at sqrt(s) = 7 TeV
• Meaningful characterisation of perturbative theoretical uncertainties
• Measurement of the Z boson differential cross section in transverse momentum and rapidity in proton-proton collisions at 8 TeV
• Neural network determination of parton distributions: the nonsinglet case
• Hadron structure in high-energy collisions
• Approximate N${}^3$LO Parton Distribution Functions with Theoretical Uncertainties: MSHT20aN${}^3$LO PDFs
• APFEL Web: a web-based application for the graphical visualization of parton distribution functions
• Heavy Quark Initiated Contributions to Deep Inelastic Structure Functions
• APFELgrid: a high performance tool for parton density determinations
• A Markov Chain Monte Carlo for Galactic Cosmic Ray physics: I. Method and results for the Leaky Box Model
• An extensive survey of the estimation of uncertainties from missing higher orders in perturbative calculations
• Impact of heavy quark and quarkonium data on nuclear gluon PDFs
• Probabilistic definition of the perturbative theoretical uncertainty from missing higher orders
• Markov Chain Monte Carlo Estimation of Quantiles
• Charm quark mass dependence in a global QCD analysis
• Why ${\alpha }_s$ Cannot be Determined from Hadronic Processes without Simultaneously Determining the Parton Distributions
• Practical Sparse Matrices in C++ with Hybrid Storage and Template-Based Expression Optimisation
• Compatibility of Neutrino DIS Data and Its Impact on Nuclear Parton Distribution Functions
• Parton distributions with scale uncertainties: a MonteCarlo sampling approach
• Determination of alpha_s from scaling violations of truncated moments of structure functions
• Heavy Quark Production in the ACOT Scheme at NNLO and N3LO
• Modelling theoretical uncertainties in phenomenological analyses for particle physics
• On the determination of uncertainties in parton densities
• Machine learning of log-likelihood functions in global analysis of parton distributions
• Systematic uncertainties from higher-twist corrections in DIS at large x
• Robust estimates of theoretical uncertainties at fixed-order in perturbation theory
• Markov Chain Monte Carlo technics applied to Parton Distribution Functions determination: proof of concept
• PartonDensity.jl: a novel parton density determination code