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The scale dependent nuclear effects in parton distributions for practical applications

K. J. Eskola, V. J. Kolhinen, C. A. Salgado

TL;DR

This paper demonstrates that, within leading-twist LO DGLAP evolution, nuclear modifications to parton distributions, quantified by $R_i^A(x,Q^2)$, are largely independent of the free-proton PDF set used to define the baseline. By anchoring initial nuclear inputs at $Q_0^2$ with DIS and DY data and enforcing conservation laws, the authors show that $R_i^A(x,Q^2)$ can be parametrized for all flavors and nuclei ($A>2$) over a broad kinematic range, enabling straightforward computation of hard-scattering cross sections in nuclear collisions. The proposed EKS98 parametrization, implemented in Fortran as two routines, yields a practical tool that preserves consistency across different LO PDF sets and remains robust against set-dependent variations in physical observables like $F_2^A/F_2^D$ and $R_{DY}^A$. The findings support using a universal nuclear-modification description for practical applications while noting limitations and avenues for refinement, including NLO effects and tighter constraints on nuclear gluon shadowing.

Abstract

The scale dependence of the ratios of parton distributions in a proton of a nucleus $A$ and in the free proton, $R_i^A(x,Q^2)=f_{i/A}(x,Q^2)/f_i(x,Q^2)$, is studied within the framework of the lowest order leading-twist DGLAP evolution. By evolving the initial nuclear distributions obtained with the GRV-LO and CTEQ4L sets at a scale $Q_0^2$, we show that the ratios $R_i^A(x,Q^2)$ are only moderately sensitive to the choice of a specific modern set of free parton distributions. We propose that to a good first approximation, this parton distribution set-dependence of the nuclear ratios $R_i^A(x,Q^2)$ can be neglected in practical applications. With this result, we offer a numerical parametrization of $R_i^A(x,Q^2)$ for all parton flavours $i$ in any $A>2$, and at any $10^{-6}\le x \le 1$ and any $Q^2\ge 2.25$ GeV$^2$ for computing cross sections of hard processes in nuclear collisions.

The scale dependent nuclear effects in parton distributions for practical applications

TL;DR

This paper demonstrates that, within leading-twist LO DGLAP evolution, nuclear modifications to parton distributions, quantified by , are largely independent of the free-proton PDF set used to define the baseline. By anchoring initial nuclear inputs at with DIS and DY data and enforcing conservation laws, the authors show that can be parametrized for all flavors and nuclei () over a broad kinematic range, enabling straightforward computation of hard-scattering cross sections in nuclear collisions. The proposed EKS98 parametrization, implemented in Fortran as two routines, yields a practical tool that preserves consistency across different LO PDF sets and remains robust against set-dependent variations in physical observables like and . The findings support using a universal nuclear-modification description for practical applications while noting limitations and avenues for refinement, including NLO effects and tighter constraints on nuclear gluon shadowing.

Abstract

The scale dependence of the ratios of parton distributions in a proton of a nucleus and in the free proton, , is studied within the framework of the lowest order leading-twist DGLAP evolution. By evolving the initial nuclear distributions obtained with the GRV-LO and CTEQ4L sets at a scale , we show that the ratios are only moderately sensitive to the choice of a specific modern set of free parton distributions. We propose that to a good first approximation, this parton distribution set-dependence of the nuclear ratios can be neglected in practical applications. With this result, we offer a numerical parametrization of for all parton flavours in any , and at any and any GeV for computing cross sections of hard processes in nuclear collisions.

Paper Structure

This paper contains 6 sections, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The framework and the assumptions
  3. The set-dependence of the nuclear effects
  4. The parametrization for practical applications
  5. The set-dependence in the physical quantities
  6. Discussion and conclusions

Figures (4)

  • Figure 1: The nuclear ratios $R_i^A(x,Q^2)$ for individual parton flavours $i=g, u_V, \bar{u}, \bar{d}, s, c$ of a lead nucleus $A=208$ as functions of $x$ at fixed values of $Q^2=Q_0^2=2.25$ GeV$^2$ and $Q^2=10000$ GeV$^2$ as obtained by using the GRV-LO GRVLO distributions (solid lines) and the CTEQ4L CTEQ4L distributions (dotted lines) for the free proton. The dashed lines show our numerical parametrization (EKS) to the nuclear effects obtained in the GRV-LO case. The ratios $R_{d_V}^A$ are almost identical to $R_{u_V}^A$, and are not shown. The charm ratios are presented only for $Q^2=10000$ GeV$^2$, since the charm distributions are generated only above our $Q_0^2$. The ratios $R_b^A$ at $Q^2=10000$ GeV$^2$ behave as $R_c^A$, and are not shown. For practical purposes the set-dependence of the ratios $R_i^A(x,Q^2)$ is negligible.
  • Figure 2: The nuclear ratios $R_i^A(x,Q^2)$ for individual parton flavours $i=g, u_V, \bar{u}, \bar{d}, s, c$ as functions of the mass number $A$ at a fixed value of $Q^2 = 10$ GeV$^2$ and at fixed values of $x= 10^{-4},\, 10^{-3},\, 10^{-2},\, 10^{-1}$ (see the panel for the strange quarks). The notation of the curves is the same as in Fig. \ref{['SETDEP']} and the markers show the nuclei for which we have made the computation. The set-dependence is the largest for large nuclei but keeps within $\sim$5 %.
  • Figure 3: The ratio $F_2^{\rm Sn}/F_2^{\rm C}$ as a function of $Q^2$ at several different fixed values of $x$. The data is from NMC96. The figure contains four calculated curves: two of the curves correspond to the "exact" results obtained with the nuclear ratios for the GRV-LO distributions (as in EKR98) and for the CTEQ4L-distributions separately, and two curves correspond to the results obtained with the GRV-LO and CTEQ4L distributions multiplied by our numerical parametrization of $R_i^A(x,Q^2)$. There is no significant difference between the calculated curves.
  • Figure 4: The ratio of differential Drell-Yan cross sections in p$A$ and pD from Eq. (\ref{['DRELLYAN']}) as a function of $x=x_2$ for $^{12}_{~6}$C, $^{40}_{20}$Ca, $^{56}_{26}$Fe and $^{184}_{~74}$W. The open squares show the E772-data E772, and the filled symbols stand for the calculated ratios $R_{DY}^A(x,Q^2)$ at $(x,Q^2)$ corresponding to the experimental values PMcG. The circles show $R_{DY}^A$ as computed with the nuclear ratios obtained separately for the GRV-LO set (big circles) and for the CTEQ4L set (small circles). The results obtained by using our numerical parametrization (EKS) of $R_i^A$ together with the sets GRV-LO and CTEQ4L are shown by triangles and diamonds, correspondingly. As seen from the panel for tungsten, the differencies between the two parton distribution sets used for the free proton are larger than the error from using the set-independent parametrization for the nuclear effects $R_i^A$.