Impact-parameter dependent nuclear parton distribution functions: EPS09s and EKS98s and their applications in nuclear hard processes

TL;DR

This work introduces spatially dependent nuclear PDFs by modeling the per-nucleon modification $r_i^A(x,Q^2,\mathbf{s})$ as a thickness-function expansion in $T_A(\mathbf{s})$, retaining terms up to $[T_A]^4$ to reproduce the $A$-dependence of the globally fitted $R_i^A(x,Q^2)$. The authors publish EPS09s and EKS98s, providing a practical framework and numerical routines to compute centrality-dependent hard-process cross sections in A+A, p+A, and d+A collisions in a way that is fully consistent with the underlying global nPDF analyses. They validate the approach by computing $R_{AA}^{1\rm jet}$ and $R_{CP}^{1\rm jet}$ for jet production, as well as $R_{dAu}^{\pi^0}$ across centralities at RHIC and $R_{pPb}^{\pi^0}$ at the LHC, showing compatibility with PHENIX data within normalization uncertainties and offering predictions for forthcoming p+Pb runs. The 4th-order thickness-expansion approach improves over 1-parameter models and enables consistent centrality-dependent studies, with broad applicability to Monte Carlo simulations and future global nPDF fits that incorporate spatial dependence from the outset.

Abstract

We determine the spatial (impact parameter) dependence of nuclear parton distribution functions (nPDFs) using the $A$-dependence of the spatially independent (averaged) global fits EPS09 and EKS98. We work under the assumption that the spatial dependence can be formulated as a power series of the nuclear thickness functions $T_A$. To reproduce the $A$-dependence over the entire $x$ range we need terms up to $[T_A]^4$. As an outcome, we release two sets, EPS09s (LO, NLO, error sets) and EKS98s, of spatially dependent nPDFs for public use. We also discuss the implementation of these into the existing calculations. With our results, the centrality dependence of nuclear hard-process observables can be studied consistently with the globally fitted nPDFs for the first time. As an application, we first calculate the LO nuclear modification factor $R^{1jet}_{AA}$ for primary partonic-jet production in different centrality classes in Au+Au collisions at RHIC and Pb+Pb collisions at LHC. Also the corresponding central-to-peripheral ratios $R_{CP}^{1jet}$ are studied. We also calculate the LO and NLO nuclear modification factors for single inclusive neutral pion production, $R_{dAu}^{π^0}$, at mid- and forward rapidities in different centrality classes in d+Au collisions at RHIC. In particular, we show that our results are compatible with the PHENIX mid-rapidity data within the overall normalization uncertainties given by the experiment. Finally, we show our predictions for the corresponding modifications $R_{pPb}^{π^0}$ in the forthcoming p+Pb collisions at LHC.

Figures (23)

• Figure 1: The nuclear modifications and their uncertainties in a lead nucleus ($A=208$) for different parton flavors from EPS09NLO at the EPS09 initial scale $Q_0^2=1.69$ GeV$^2$ (upper panel), and from EPS09LO and EKS98 at the EKS98 initial scale $Q_0^2=2.25$ GeV$^2$ (lower panel).
• Figure 2: The problematic $A$ dependence of the parameter $c^g(x,Q^2) = [R_g^A(x,Q^2)-1]A/T_{AA}(0)$ for EPS09NLO1 and EPS09LO1 (EKS98) gluons at $x = 0.01$ and $Q^2= 1.69\, (2.25) \textrm{ GeV}^2$ in the 1-parameter approach where one includes only the first nontrivial term in the power series in Eq. (\ref{['eq:ta_series']}).
• Figure 3: Left: The spatially averaged nuclear modification $R_g^A(x,Q^2)$ for a lead nucleus ($A=208$) from the NLO set EPS09NLO1 (dotted lines) and from the EPS09sNLO1 spatial fit presented here (solid lines) at four different scales. Right: The same with the LO sets EKS98 and EPS09LO1 (dotted) and with the spatial fits EKS98s (dashed) for three different scales and EPS09sLO1 (solid) for four different scales.
• Figure 4: Left: The $A$ dependence of the spatially averaged nuclear modification $R_g^A(x,Q^2)$ at fixed values $x=0.001$ and $Q^2=1.69$ GeV$^2$ from the sets EPS09NLO1 (crosses) and EPS09LO1 (pluses) and from the corresponding spatial fits EPS09sNLO1 (solid green line) and EPS09sLO1 (solid blue line). Right: The same but with the LO set EKS98 (circles) and the corresponding spatial fit EKS98s (solid red) at $Q^2=2.25$ GeV$^2$. The small nuclei shown with gray markers in both panels were not used in our spatial fits.
• Figure 5: The spatially dependent modification of gluon distribution in a lead nucleus, $r_g^{\rm Pb}(x,Q^2,s)$, from EPS09sNLO1 (upper left), EPS09sLO1 (upper right) and EKS98s(lower plot) as a function of $x$ and $s$ at the initial scale $Q^2 = 1.69 (2.25) \textrm{ GeV}^2$ of EPS09 (EKS98). For examples of the corresponding plots of other parton flavors, see App. \ref{['sec:ruvus']}.