ChiliPDF: Chebyshev Interpolation for Parton Distributions

Abstract

Parton distribution functions (PDFs) are an essential ingredient for theoretical predictions at colliders. Since their exact form is unknown, their handling and delivery for practical applications relies on approximate numerical methods. We discuss the implementation of PDFs based on a global interpolation in terms of Chebyshev polynomials. We demonstrate that this allows for significantly higher numerical accuracy at lower computational cost compared with local interpolation methods such as splines. Whilst the numerical inaccuracy of currently used local methods can become a nontrivial limitation in high-precision applications, in our approach it is negligible for practical purposes. This holds in particular for differentiation and for Mellin convolution with kernels that have end point singularities. We illustrate our approach for these and other important numerical operations, including DGLAP evolution, and find that they are performed accurately and fast. Our results are implemented in the C++ library ChiliPDF.

arXiv comment38 pages, 15 figures; v2: added discussion and references, mainly in the introduction and conclusions

References in corpus (52)

• Parton distributions for the LHC Run II
• LHAPDF6: parton density access in the LHC precision era
• Parton distributions from high-precision collider data
• New parton distribution functions from a global analysis of quantum chromodynamics
• Parton distributions in the LHC era: MMHT 2014 PDFs
• The Three-Loop Splitting Functions in QCD: The Non-Singlet Case
• The Three-Loop Splitting Functions in QCD: The Singlet Case
• 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
• Parton Distribution Functions, ${\alpha }_s$ and Heavy-Quark Masses for LHC Run II
• APFEL: A PDF Evolution Library with QED corrections
• Strong Coupling Constant with Flavour Thresholds at Four Loops in the MS-bar Scheme
• A Higher Order Perturbative Parton Evolution Toolkit (HOPPET)
• QCDNUM: Fast QCD Evolution and Convolution
• Charm electroproduction viewed in the variable-flavour number scheme versus fixed-order perturbation theory
• Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS
• Delineating parton distributions and the strong coupling
• Subleading Power Corrections for N-Jettiness Subtractions
• Subleading Power Rapidity Divergences and Power Corrections for $q_T$
• Numerical solution of $Q^2$ evolution equations in a brute-force method
• Power Corrections in the N-jettiness Subtraction Scheme
• Neural network determination of parton distributions: the nonsinglet case
• Resummed Higgs cross section at N${}^3$LL
• Resummation of Double-Differential Cross Sections and Fully-Unintegrated Parton Distribution Functions
• Next-to-leading-logarithmic power corrections for $N$-jettiness subtraction in color-singlet production
• Resummation and Matching of $b$-quark Mass Effects in $b\bar{b}H$ Production
• The asymptotic behaviour of parton distributions at small and large $x$
• N-Jettiness Subtractions for $gg\to H$ at Subleading Power
• Extended Parameterisations for MSTW PDFs and their effect on Lepton Charge Asymmetry from W Decays
• Power Corrections for N-Jettiness Subtractions at $\mathcal{O}({\alpha }_s)$
• The Logarithmic Contributions to the O(α_s^3) Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering
• Soft gluon resummation of Drell-Yan rapidity distributions: theory and phenomenology
• The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2,N_F)}$
• Differential Higgs production at N3LO beyond threshold
• EKO: Evolution Kernel Operators
• Resummation of the jet broadening in DIS
• Factorization and Resummation for Massive Quark Effects in Exclusive Drell-Yan
• The QCD/SM Working Group: Summary Report
• Joint Two-Dimensional Resummation in $q_T$ and $0$-Jettiness at NNLL
• FORTRAN program for a numerical solution of the nonsinglet Altarelli-Parisi equation
• Precision Studies of the NNLO DGLAP Evolution at the LHC with CANDIA
• The Fully-Differential Quark Beam Function at NNLO
• PDFFlow: parton distribution functions on GPU
• Helicity Methods for High Multiplicity Subleading Soft and Collinear Limits
• Jet Veto Resummation with Jet Rapidity Cuts
• Parametrization dependence and Delta Chi-squared in parton distribution fitting
• Fast evolution of parton distributions
• Transverse Vetoes with Rapidity Cutoff in SCET
• Parton Distribution Uncertainties using Smoothness Prior
• The Fully-Differential Gluon Beam Function at NNLO
• A Matrix Approach to Numerical Solution of the DGLAP Evolution Equations
• A fast and precise method to solve the Altarelli-Parisi equations in x space

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• On sum rules for double and triple parton distribution functions and Pythia's model of multiple parton interactions
• A semi-analytical $x$-space solution for parton evolution -- Application to non-singlet and singlet DGLAP equation
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• Higher-order structure of Hamiltonian truncation effective theory