Hadron structure in high-energy collisions

TL;DR

The paper analyzes how parton distribution functions (PDFs) encode hadron structure for high-energy collisions, detailing their gauge-invariant operator definitions, renormalization, and DGLAP evolution within the factorization framework. It presents a Bayesian-Hessian approach to PDF fitting, including a rigorous set of goodness-of-fit tests and tolerance prescriptions to quantify uncertainties and address dataset tensions. The discussion extends to heavy-ion physics via nuclear PDFs (nPDFs), outlining universal factorization, A-dependence parameterizations, and cross-nucleus comparisons. Together, these methods underpin precise predictions at the LHC and inform future HL-LHC and Electron-Ion Collider efforts by clarifying uncertainties, systematics, and the role of heavy-quark masses and special kinematic regions in PDF determinations.

Abstract

Parton distribution functions (PDFs) describe the structure of hadrons as composed of quarks and gluons. They are needed to make predictions for short-distance processes in high-energy collisions and are determined by fitting to cross section data. We review definitions of the PDFs and their relations to high-energy cross sections. We focus on the PDFs in protons, but also discuss PDFs in nuclei. We review in some detail the standard statistical treatment needed to fit the PDFs to data using the Hessian method. We discuss tests that can be used to critically examine whether the assumptions are indeed valid. We also present some ideas of what one can do in the case that the tests indicate that the assumptions fail.

Figures (17)

• Figure 1: Left: The parton distribution functions $x u_v\equiv x (u-\bar{u})$, $x d_v\equiv x (d-\bar{d})$, $x S\equiv 2 x (\bar{u} +\bar{d} + \bar{s} + \bar{c})$ and $x g$ of HERAPDF2.0 NNLO at $\mu^2 = 10 \hbox{GeV}^2$. The experimental, theoretical model and parameterization uncertainties are shown separately. From Abramowicz:2015mha. Right: the PDF uncertainty bands for CT18 NNLO PDFs Hou:2019efy at $\mu^2 = 10 \hbox{GeV}^2$.
• Figure 2: The function $f(x)$ in Eq. (\ref{['eq:fexample']}) and a three parameter polynomial fit $h_3(x)$, Eq. (\ref{['eq:polynomialhn']}), to this function. Here there are not enough parameters to get a good fit.
• Figure 3: (a) Data $\{y_i,x_i\}$ generated from $f(x)$, Eq. (\ref{['eq:fexample']}), shown with a 4-parameter (dashed curve) and 13-parameter (solid curve) polynomial fits to the data. (b) Average over a large number of trials of $\chi^2(D_1,a_1)$ and $\chi^2(D_2,a_1)$ as a function of the number of parameters $N_P$. In each trial, a polynomial with $N_P$ parameters is fit to data $D_1$ generated from $f(x)$, then $\chi^2(D_2,a_1)$ is measured for that polynomial compared to an independent data sample $D_2$ generated from $f(x)$.
• Figure 4: $\chi^2$ from the fitted and control samples of data from the CT14HERA2 NLO resampling exercise.
• Figure 5: Green solid lines: the gluon PDFs from candidate fits of the CT18 NNLO analysis, obtained using alternative parameterization forms and plotted as ratios to the default CT18 NNLO $g(x,Q)$. Light blue band: the 68% C.L. uncertainty band of the published CT18 NNLO PDFs. From Hou:2019efy.