Analysis of parton distributions in a pion with Bézier parametrizations

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.

Figure 1

Illustration of a fit by a metamorph. (a) An ensemble of data points is generated by random fluctuations around a “truth” function constructed from piecewise polynomial interpolations. (b) After minimization of the ${\chi }^2$ between data and theory, the carrier function (short-dashed red curve) has varied and the values of all control points have been shifted, helped by the modulator, i.e., the Bézier curve. The fixed CPs (blue crosses) lie on the updated carrier function. The free CPs (golden arrows) can deviate from it. The final result is the long-dashed cyan curve, labeled Metamorph. It better approximates the data than the carrier only. This example is given for $N_m=4$, ${\alpha }_x=0.45$.

Figure 2

The fantômas technique illustrated by applying the bootstrap (or importance) sampling on the data (upper plot) or the Fântomas methodology, that consists in sampling over representative choices for the CPs and the scaling factor ${\alpha }_x$ (lower plot). The resulting uncertainties are displayed in cyan (bootstrap) and green (parameter-space sampling).

Figure 3

Kinematic coverage in $x_{\pi }$ and $Q^2$ of the Drell-Yan, prompt-photon, and leading-neutron DIS data used in this work.

Figure 4

Ratios of the uncertainties to the respective central PDFs for the sea $xS(x,Q)$ (upper) and gluon $g(x,Q)$ (lower): xFitter published results [9] using Hesse evaluation of the uncertainties (without uncertainty from scale variations) in mustard; the same “norm 1” fit with Iterate uncertainties in medium-dashed red; and “norm 2” fit with Iterate uncertainties in long-dashed cyan. Then, for a fixed $B_S=0.47$, norm 1 and norm 2 fits are compared as overlapping dark blue and green curves.

Figure 5

The fantômas PDFs for $N_m=0$, based on $\mathrm{DY}+\gamma$ data, shown for various values of the fixed parameter $B_S$. The top (bottom) shows the quark sea (the gluon) PDF and its uncertainty.

Figure 6

The sea and gluon pion PDFs obtained with the fantômas environment for an $N_m=1$ and ${\alpha }_x=0.75$ setting, shown at $Q_0$. The control points are indicated for each curve with a star and a triangle for a fixed and free $CP$, respectively. Notice that all but two curves share a fixed $CP$ at $x=0.7$.

Figure 7

Central values for pion PDFs obtained with the fantômas environment for various $N_m$, ${\alpha }_x$, and $CP$ settings, shown at $Q_0=\sqrt{1.9}\mathrm{GeV}$. The baseline fits chosen for the final combination are shown in color. Light gray curves illustrate the many possible shapes that were obtained.

Figure 8

Five final sets of the fantômas PDF combination, shown at $Q_0=\sqrt{1.9}\mathrm{GeV}$, with their respective Hessian uncertainty band as obtained from the $\mathsf{xFitter}$ framework.

Figure 9

The FantoPDF sets at $Q_0$, shown for a ${\pi }^{+}$ flavor configuration.

Figure 10

Momentum fractions for the sea and gluon PDFs obtained in the five baseline fits (orange/brown) and the MC FantoPDF combination (green ellipse) at $Q_0$. JAM21 and xFitter results, to be discussed in Sec. 6, are displayed in red and blue, respectively.

Figure 11

The histograms represent momentum fractions for the valence (red), gluon (green), and sea (blue) PDFs from 500 MC FantoPDF distributions generated from five candidate fits. The histograms are not symmetric as a consequence of parametrization dependence. Vertical boundaries represent the extrema of momentum fractions for pre-fantômas fits with $\mathrm{DY}+\gamma$ data only (Fig. 5). These results are at the initial scale $Q_0$.

Figure 12

The effective ($1-x$) exponent of the valence PDF in the FantoPDF ensemble—the definition is given in [48]. In green, the effective exponent at $Q_0=\sqrt{1.9}\mathrm{GeV}$ and, in blue, at $\sqrt{10}\mathrm{GeV}$. The plot is cut at $x=0.94$ for grid-extrapolation reasons. We have verified analytically that the highlighted Bézier curves of Fig. 7 converge to $C_V^{\mathrm{eff}}=1$ at most for $x\to 1$ at $Q_0$.

Figure 13

The valence PDF of the final fantômas ensemble compared to JAM21 and xFitter results, at $Q=1.4\mathrm{GeV}$. For the FantoPDF set, the 68% C.L. of the MC output is shown. xFitter’s results are plotted without accounting for the uncertainty coming from the scale variation. The inner frame shows the ratio to the central value of each set—symmetric uncertainties are used for all three sets.

Figure 14

Same as in Fig. 13, for the quark sea PDF (top) and gluon PDF (bottom).

Figure 15

Valence pion PDF $xV(x,2\mathrm{GeV})$ for the FantoPDF in dark cyan, as well as for the lattice result of [25] in red [deep neural network (DNN) version] and green (four-parameter version), and BNL-ANL21 [98] in purple.

Figure 16

Gluon pion PDF normalized to its momentum fraction $xg(x,2\mathrm{GeV})/⟨xg(x,2\mathrm{GeV})⟩$ for the FantoPDF in dark cyan and for the lattice result of [23] in yellow mustard. Only statistical uncertainties are quoted for the lattice prediction.

Figure 17

Contributions of individual data points to ${\chi }^2$ of respective experiments, shown in the plane of $x_{\pi }$ and $Q^2$ and using the same shapes of symbols for the individual experiments as in Fig. 3 (circle for NA10, square for WA70, rhombus E615, triangle for HERA leading-neutron data). (a) The color code indicates the partial ${\chi }^2$ value for each data point for the zero-gluon scenario, $⟨xg(Q_0^2)⟩=0$, according to the ${\chi }^2$ intervals indicated in the legend. (b) The color indicates the differences in the partial ${\chi }^2$ of the points in the scenarios with $⟨xg(Q_0^2)⟩=0$ and $⟨xg(Q_0^2)⟩=0.32$. Most data points show a negligible difference in ${\chi }^2$ between the two scenarios (orange points).

Figure 18

Comparisons of data and theory (without or with systematic) shifts for typical bins of (a) E615, (b) NA10, (c) WA70, and (d) H1 leading-neutron datasets. The legend indicates the quark sea and gluon momentum fractions for the used PDFs.