In data-driven determination of Parton Distribution Functions (PDFs) in global QCD analyses, uncovering the true underlying distributions is complicated by a highly convoluted inverse problem. The determination of PDFs can be understood as the inference of a function supported on [0, 1], a problem that admits multiple acceptable solutions. An ensemble of solutions exists that pass all standard goodness-of-fit criteria. In this paper, we propose algorithms for the classification, clustering, and selection of solutions to the determination of PDFs, or any functions on [0, 1], based on the characterization of their shape. We explore information-theoretic based (Rényi entropy and divergence) and optimal-transport based (Wasserstein distance) criteria. In particular, we advocate for the use of the Rényi entropy as an $\textit{absolute}$ estimator per solution, as opposed to $\textit{relative}$ estimators that compare solutions pairwise. We show that the Rényi entropy can characterize the space of solutions $\textit{w.r.t.}$ the PDF shapes. Paired with the identification of the optimal combination of solutions via Pareto fronts, it provides a plausible and minimalist selection algorithm. Moreover, Rényi entropy proves versatile for use in clustering applications.

Mexico

Aurore Courtoy

Arturo Ibsen

Information criteria for selecting parton distribution function solutions

We present Fanto10, a Hessian ensemble of next-to-leading-order error parton distribution functions (PDFs) in a charged pion, constructed to enable precision comparisons with upcoming experimental data and ab initio QCD predictions. Fanto10 estimates several experimental, theoretical, and methodological uncertainties on pion PDFs. It follows a statistical procedure, grounded in de Finetti’s theorem, to systematically determine the component of the epistemic uncertainty associated with the choice of PDF functional forms. This uncertainty is estimated by fitting the data with multiple functional forms generated in a versatile format based on polynomial universal approximators (Bézier curves). The most distinct fits are converted into a joint Monte Carlo replica ensemble and then combined into one Hessian ensemble through a major update of the metapdf methodology, expanded in this article to combine PDFs for arbitrary hadrons and partonic flavor representations. This study also goes beyond the previous pion fits by estimating, for the first time, the uncertainty from nCTEQ15 nuclear PDFs for the tungsten target in the E615 pion-nucleus Drell-Yan data—a third-party input in our analysis, also included in the expanded metapdf combination. The upgraded metapdf framework, thus, altogether provides a principled algorithm to propagate parametrization dependence and other types of the epistemic uncertainty into conveniently represented PDFs. Notably, Fanto10 modifies the key physics conclusions from the earlier, more constrained analyses of the pion’s gluon and sea-quark content. It facilitates studies of PDF-induced correlations and comparisons to lattice QCD. We summarize the methodology, structure of the Fanto10 PDFs, and experimental implications.

United States

Lucas Kotz

Pavel Nadolsky

Maximiliano Ponce-Chavez

Epistemic and nuclear uncertainties for the parton distributions of the pion

We explore the role of parametrizations for nonperturbative QCD functions in global analyses, with a specific application to extending a phenomenological analysis of the parton distribution functions (PDFs) in the charged pion realized in the xFitter fitting framework. The parametrization dependence of PDFs in our pion fits substantially enlarges the uncertainties from the experimental sources estimated in the previous analyses. We systematically explore the parametrization dependence by employing a novel technique to automate generation of polynomial parametrizations for PDFs that makes use of Bézier curves. This technique is implemented in a c++ module fantômas that is included in the xFitter program. Our analysis reveals that the sea and gluon distributions in the pion are not well disentangled, even when considering measurements in leading-neutron deep inelastic scattering. For example, the pion PDF solutions with a vanishing gluon and large quark sea are still experimentally allowed, which elevates the importance of ongoing lattice and nonperturbative QCD calculations, together with the planned pion scattering experiments, for conclusive studies of the pion structure.

Fredrick Olness

Analysis of parton distributions in a pion with Bézier parametrizations

Parton distribution functions (PDFs) play a central role in calculations for the LHC. To gain a deeper understanding of the emergence and interplay of constraints on the PDFs in the global QCD analyses, it is important to examine the relative significance and mutual compatibility of the experimental datasets included in the PDF fits. Toward this goal, we discuss the <math><msub><mi>L</mi><mn>2</mn></msub></math> sensitivity, a convenient statistical indicator for exploring the statistical pulls of individual datasets on the best-fit PDFs and identifying tensions between competing datasets. Unlike the Lagrange multiplier method, the <math><msub><mi>L</mi><mn>2</mn></msub></math> sensitivity can be quickly computed for a range of PDFs and momentum fractions using the published Hessian error sets. We employ the <math><msub><mi>L</mi><mn>2</mn></msub></math> sensitivity as a common metric to study the relative importance of datasets in the recent ATLAS, CTEQ-TEA, MSHT, and reduced PDF4LHC21 PDF analyses at next-to-next-to-leading-order and approximate next-to-next-to-next-to-leading-order. We illustrate how this method can aid the users of PDFs to identify datasets that are important for a PDF at a given kinematic point, to study quark flavor composition and other detailed features of the PDFs, and to compare the data pulls on the PDFs for various perturbative orders and functional forms. We also address the feasibility of computing the sensitivities using Monte Carlo error PDFs. Together with the article, we present a companion interactive website with a large collection of plotted <math><msub><mi>L</mi><mn>2</mn></msub></math> sensitivities for eight recent PDF releases and a C++ program to plot the <math><msub><mi>L</mi><mn>2</mn></msub></math> sensitivities.

Xiaoxian Jing

United Kingdom

Amanda Cooper-Sarkar

Germany

Thomas Cridge

CERN

Francesco Giuli

Lucian Harland-Lang

T. J. Hobbs

Joey Huston

Robert S. Thorne

Keping Xie

C.P. Yuan

Quantifying the interplay of experimental constraints in analyses of parton distributions

In global QCD fits of parton distribution functions (PDFs), a large part of the estimated uncertainty on the PDFs originates from the choices of parametric functional forms and fitting methodology. We argue that these types of uncertainties can be underestimated with common PDF ensembles in high-stake measurements at the Large Hadron Collider and Tevatron. A fruitful approach to quantify these uncertainties is to view them as arising from sampling of allowed PDF solutions in a multidimensional parametric space. This approach applies powerful insights gained in recent statistical studies of large-scale population surveys and quasi-Monte Carlo integration methods. In particular, PDF fits may be affected by the big data paradox, which stipulates that more experimental data do not automatically raise the accuracy of PDFs—close attention to the data quality and sampling of possible PDF solutions is as essential. To test if the sampling of the PDF uncertainty of an experimental observable is truly representative of all acceptable solutions, we introduce a technique (“a hopscotch scan”) based on a combination of parameter scans and stochastic sampling. With this technique, we examine the PDF uncertainty on key LHC cross sections at 13 TeV obtained with the public NNPDF4.0 fitting code, while accounting for the likelihood distribution. We show that the uncertainties on the charm distribution at a large momentum fraction <math><mrow><mi>x</mi></mrow></math> and gluon PDF at small <math><mi>x</mi></math> are enlarged. In PDF ensembles obtained in the analytic minimization (Hessian) formalism, the tolerance on the PDF uncertainty must be based on sufficiently complete sampling of PDF functional forms and choices of the experiments.

China

Mengshi Yan

Parton distributions need representative sampling

We present the first point-by-point extraction of a twist-3 PDF. The scalar PDF, <math><mi>e</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, is accessed through the analysis of the data for the <math><mi>sin</mi><msub><mi>ϕ</mi><mi>R</mi></msub></math>-moment of the beam-spin asymmetry for dihadron production in semi-inclusive DIS off proton target at CLAS and CLAS12. The dihadron formalism allows for use of collinear framework, hence calling for a minimal set of approximations and hypotheses. The extracted PDF <math><mi>e</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> carries insights into the physics of the largely-unexplored quark-gluon correlations, and its first Mellin moment is related to the marginally known scalar charge of the nucleon. We show that the proton flavor combination of the scalar PDF is nonzero at more than 74% probability.

Angel S. Miramontes

Harut Avakian

Italy

Marco Mirazita

Silvia Pisano

Extraction of the higher-twist parton distribution <math><mi>e</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> from CLAS data

We discuss strategies for comparisons of nonperturbative QCD predictions for parton distribution functions (PDFs) with high-energy experiments in the region of large partonic momentum fractions <math><mi>x</mi></math>. Analytic functional forms for PDFs cannot be uniquely determined solely based on discrete experimental measurements because of a mathematical property of mimicry of PDF parametrizations that we prove using a representation based on Bézier curves. Predictions of nonperturbative QCD approaches for the <math><mi>x</mi></math> dependence of PDFs instead should be cast in a form that enables decisive comparisons against experimental measurements. Predictions for effective power laws of (<math><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></math>) dependence of PDFs may play this role. Expectations for PDFs in a proton based on quark counting rules are compared against the effective power laws of (<math><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></math>) dependence satisfied by CT18 next-to-next-to-leading-order parton distributions. We comment on implications for studies of PDFs in a pion, in particular on the comparison of nonperturbative approaches with phenomenological PDFs.

Testing momentum dependence of the nonperturbative hadron structure in a global QCD analysis

We present an analysis of two-current correlations for the pion in the Nambu–Jona-Lasinio model, with Pauli–Villars regularization. We provide explicit expressions in momentum space for two-current correlations corresponding to the zeroth component of the vector Dirac bilinear in the quark vertices, which has been evaluated on the lattice, thinking to applications in a high energy framework, as a step towards the calculation of pion double parton distributions. The numerical results show a remarkable qualitative agreement with recent lattice data. The factorization approximation into one-body currents is discussed based on previous evaluation of the relevant low energy matrix elements in the Nambu–Jona-Lasinio model, confirming the lattice result.

Spain

Santiago Noguera

Sergio Scopetta

Two-current correlations in the pion in the Nambu and Jona-Lasinio model

Two-parton correlations in the pion, a non perturbative information encoded in double parton distribution functions, are investigated in the Nambu–Jona-Lasinio model. It is found that double parton distribution functions expose novel dynamical information on the structure of the pion, not accessible through one-body parton distributions, as it happens in several estimates for the proton target and in a previous evaluation for the pion, in a light-cone framework. Expressions and predictions are given for double parton distributions corresponding to leading-twist Dirac operators in the quark vertices, and to different regularization methods for the Nambu–Jona-Lasinio model. These results are particularly relevant in view of forthcoming lattice data.

Double parton distributions in the pion in the Nambu–Jona-Lasinio model

We present a thorough analysis of unpolarized Drell–Yan (DY) pair production in pion–nucleus scattering. On the nucleus side, we use nuclear parton distributions along with parametrisations of the nucleon partonic transverse distribution available in the literature. Partonic longitudinal and transverse distributions of the pion are those obtained in a recent calculation in a Nambu–Jona Lasinio (NJL) framework, with Pauli–Villars regularization. The scale of the NJL model is determined with a minimisation procedure comparing NLO predictions based on NJL evolved pion distributions to rapidity differential DY cross sections data. The resulting distributions are then used to describe, up to next-to-leading logarithmic accuracy, the transverse momentum spectrum of dilepton pairs up to a transverse momentum of 2 GeV. With no additional parameters, fair agreement is found with available pion–nucleus data, confirming the virtues of the NJL description of pion parton structure. We find sizable evolution effects on the shape of the distributions and on the generated average transverse momentum of the dilepton pair. We furthermore discuss the possibility of gaining information about the behavior of the pion unpolarized transverse momentum dependent parton distribution from pion nucleus DY data.

Belgium

Federico Ceccopieri

Pion nucleus Drell–Yan process and parton transverse momentum in the pion