Cronin effect and high-p_T suppression in the nuclear gluon distribution at small x

TL;DR

This paper provides a comprehensive analytic treatment of the Cronin effect and the subsequent suppression of gluon distributions in nuclei at small x within the Color Glass Condensate framework. By starting from McLerran–Venugopalan initial conditions and comparing fixed versus running coupling, the authors isolate how saturation reshapes the unintegrated gluon distribution and drives the Cronin peak near the saturation scale $Q_s(A)$. They develop a unified picture of the evolution: nonlinear saturation controls low-$k_\perp$ behavior, while linear BFKL dynamics governs high-$k_\perp$ regions, with running coupling slowing evolution and enhancing suppression at high energies. A key finding is that the rapid early suppression of $R_{pA}$ arises from faster DGLAP-like evolution in the proton, and the Cronin peak flattens as the nucleus evolves, an effect amplified at higher energies when running coupling is included. The results highlight saturation as the underlying mechanism for both peak formation and its eventual disappearance, with universal long-distance behavior emerging at very large rapidities.

Abstract

We present a systematic, and fully analytic, study of the ratio R_{pA} between the gluon distribution in a nucleus and that in a proton scaled up by the atomic number A. We consider initial conditions of the McLerran-Venugopalan type, and quantum evolution in the Color Glass Condensate, with both fixed and running coupling. We perform an analytic study of the Cronin effect in the initial conditions and point out an interesting difference between saturating effects and twist effects in the nuclear gluon distribution. We show that the distribution of the gluons which make up the condensate in the initial conditions is localized at low momenta, but this particular feature does not survive after the quantum evolution. We demonstrate that the rapid suppression of the ratio R_{pA} in the early stages of the evolution is due to the DGLAP-like evolution of the proton, whose gluon distribution grows much faster than that in the nucleus because of the large separation between the respective saturation momenta. The flattening of the Cronin peak, on the other hand, is due to the evolution of the nucleus. We show that the running coupling effects slow down the evolution, but eventually lead to a stronger suppression in R_{pA} at sufficiently large energies.

Figures (11)

• Figure 1: Physical regimes for evolution in the kinematic plane $y-\ln k_\perp^2$ (for fixed coupling, for definiteness; see Fig. \ref{['EVOL-MAP-run']} for the corresponding picture with a running coupling).Both the saturation momentum $Q_s(A)$ and the 'geometric scale' momentum $Q_g(y)$ rise exponentially with $y$ (for both the proton and the nucleus), and thus are represented by straight lines in the logarithmic scale of the plot. As visible on this plot, the geometric scale rises faster (its logarithmic slope is roughly twice as large as that of the saturation momentum).
• Figure 2: The gluon occupation factor $\varphi_A(z)$ as a function of the scaled momentum variable $z=k^2/Q_s^2(A)$ in the MV model with either fixed (figure above) or running (below) coupling and $\rho_A=6$.With respect to the text definitions, we plot the quantities $\rho_A\alpha_s N_c\times\varphi_A$ for fixed coupling, and $b_0 N_c\times\varphi_A$ for running coupling. The black (thick) line corresponds to $\varphi_A(z)$; the blue (solid) line shows the saturation contribution $\varphi^{\rm sat}_A(z)$; the magenta (dotted) line shows the twist contribution $\varphi^{\rm twist}_A(z)$; the green (dashed) line represents the bremsstrahlung spectrum.
• Figure 3: Logarithmic plot of the gluon occupation factor $\varphi_A(z)$ as a function of the scaled momentum variable $z=k^2/Q_s^2(A)$ in the MV model with either fixed (figure above) or running (below) coupling and $\rho_A=6$.The plotted quantities are rescaled as explained in the caption to Fig. \ref{['phi']}. The black (thick) line corresponds to the gluon occupation factor $\varphi_A(z)$; the blue (solid) line shows the saturation contribution $\varphi^{\rm sat}_A(z)$; the green (dashed) line represents the bremsstrahlung spectrum.
• Figure 4: The Cronin ratio $\mathcal{R}_{pA}(z)$ as a function of the scaled momentum variable $z=k^2/Q_s^2(A)$ in the fixed (above) and running (below) coupling McLerran-Venugopalan model for $\rho_A=6$.The black (thick) line corresponds the ratio $\mathcal{R}_{pA}(z)$; the blue (solid) line shows the saturation contribution $\mathcal{R}^{\rm sat}_{pA}(z)$; the magenta (dotted) line shows the twist contribution $\mathcal{R}^{\rm twist}_{pA}(z)$.
• Figure 5: The integrated gluon distribution function $\mathcal{G}_A(Z)$ as a function of the scaled momentum variable $Z=Q^2/Q_s^2(A)$ in fixed (above) and running (below) coupling McLerran-Venugopalan model for $\rho_A=6$.With respect to the text definitions, we plot the quantities $[\rho_A\alpha_s N_c/Q_s^2(A)]\times \mathcal{G}_A$ for fixed coupling, and $[b_0 N_c/Q_s^2(A)]\times\mathcal{G}_A$ for running coupling. The black (thick) line corresponds to the integrated gluon distribution function $\mathcal{G}_A(Z)$; the blue (solid) line shows the saturation contribution $\mathcal{G}^{\rm sat}_A(Z)$; the green (dashed) line represents the bremsstrahlung distribution function.