Hadronic structure in high-energy collisions

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. Definitions of the PDFs and their relations to high-energy cross sections are reviewed. The focus is on the PDFs in protons, but PDFs in nuclei are also discussed. The standard statistical treatment needed to fit the PDFs to data using the Hessian method is reviewed in some detail. Tests are discussed that critically examine whether the needed assumptions are indeed valid. Also presented are some ideas of what one can do in case tests indicate that the assumptions fail.

Figure 1

Left panel: the parton distribution functions $x{u}_v\equiv x(u-\overline{u})$, $x{d}_v\equiv x(d-\overline{d})$, $xS\equiv 2x(\overline{u}+\overline{d}+\overline{s}+\overline{c})$, and $xg$ of HERAPDF2.0 NNLO at ${\mu }^2=10{\mathrm{GeV}}^2$. The experimental, theoretical model, and parametrization uncertainties are shown separately. From [10]. Right panel: the PDF uncertainty bands for CT18 NNLO PDFs ([140]) at ${\mu }^2=10{\mathrm{GeV}}^2$.

Figure 2

The function $f(x)$ in Eq. (142) and a three-parameter polynomial fit $h_3(x)$ [Eq. (143)] 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. (142)], shown with a four-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: gluon PDFs from candidate fits of the CT18 NNLO analysis, obtained using alternative parametrization forms and plotted as ratios to the default CT18 NNLO $g(x,Q)$. Light blue band: 68% C.L. uncertainty band of the published CT18 NNLO PDFs. From [140].

Figure 6

Distribution of nuisance parameters ${\lambda }_J^{\mathrm{fit}}$ for the CT14 HERA2 NNLO parton distribution fit.

Figure 7

Distribution of residuals for the CT14 HERA2 NNLO parton distribution fit.

Figure 8

Probability distributions of $S_E=\sqrt{2{\chi }^2\mathbf{(}D(E),a_{\mathrm{fit}}\mathbf{)}}-\sqrt{2N_E-1}$ for 35 and 500 random pseudoexperiments, each of which has the number $N_E$ of data chosen at random in the range $0\le N_E\le 3000$. The red dashed line shows the $\mathcal{N}(0,1)$ Gaussian distribution, which describes the observed probabilities well.

Figure 9

Probability distributions in the effective Gaussian variable $S_E$ for ${\chi }^2$ values of the fitted datasets from the NNLO fits CT14HERA2, MMHT2014, NNPDF3.0, and NNPDF3.1.

Figure 10

(a) Dependence of ${\chi }^2$ in a CT18 NNLO fit as a function of distance $t$ in parameter space corresponding to changes in $g\mathbf{(}0.01,(125\mathrm{GeV}{)}^2\mathbf{)}$. The black solid curve shows the total ${\chi }^2$, while the remaining three curves show ${\chi }^2$ as a function of $t$ with particular experiments removed from the dataset. (b) ${\chi }^2$ is as before, but along a line corresponding to changes in $g\mathbf{(}0.3,(125\mathrm{GeV}{)}^2\mathbf{)}$.

Figure 11

Dependence of ${\chi }^2$ in a CT18 NNLO fit as a function of distance $t$ in parameter space corresponding to changes in $g\mathbf{(}0.3,(125\mathrm{GeV}{)}^2\mathbf{)}$. The black solid curve shows the total ${\chi }^2$, while the remaining curves show ${\chi }^2$ as a function of $t$ in certain experiments $E$ alone.

Figure 12

Same as Figs. 10 and 11 if the estimated errors for the experiment B1 are multiplied by $\sqrt{2}$.

Figure 13

${\chi }^2$ curves for individual experiments as in Fig. 11 but with an extra systematic error added for experiment B1 according to Eqs. (170) and (171). The fit is repeated with the new systematic error for experiment B1. The new fit gives a new best-fit choice $a_{\mathrm{fit}}$. Now the observable $g\mathbf{(}0.3,(125\mathrm{GeV}{)}^2\mathbf{)}$ defines a new direction $e(\sigma )$ in parameter space. This plot uses the new $a_{\mathrm{fit}}$, $e(\sigma )$, and the total ${\chi }^2$ values after the fit.

Figure 14

Interactions of the nucleus with the initial- and final-state partons. From [183].

Figure 15

Probability distributions in the effective Gaussian variable $S_E$ for ${\chi }^2$ values of the fitted datasets from the NLO nuclear PDF fits EPPS16, nCTEQ15, DSSZ, and HKN07.

Figure 16

Nuclear-correction factor $R_i(x,\mathtt{A})=f_{i/p}^{\mathtt{A}}(x,\mathtt{A},Q^2)/f_{i/p}(x,Q^2)$ for lead ($\mathtt{A}=208$) and the partons $i=g,s,u_v,d_v,\overline{u},\overline{d}$ and at the momentum transfer $Q=10\mathrm{GeV}$.