Color Neutrality and the Gluon Distribution in a Very Large Nucleus

TL;DR

This work refines the MV description of gluon distributions in very large nuclei by enforcing color neutrality of nucleons, thereby curing infrared divergences in the transverse structure. The authors derive an infrared-finite gluon number density whose small-$x_F$ behavior remains $~1/x_F$ in the classical limit, while saturating at small transverse momentum; a transverse momentum sum rule shows non-Abelian corrections merely shift gluons in $|oldsymbol{q}|$ without changing the total count, connecting naturally to DGLAP evolution at large $Q^2$. A Kovchegov-based color-neutral nucleon model provides explicit forms for the correlator function ${ m D}$, demonstrating how color screening and a fixed scale $oxed{oldsymbol{ m oldsymbol{ angle}}}$ control the infrared and ultraviolet behavior, and highlighting the interplay between confinement-scale structure and high-density gluon dynamics. Overall, the paper offers a coherent framework linking MV, saturation physics, and DGLAP, with practical implications for understanding gluon distributions in large nuclei.

Abstract

We improve the McLerran-Venugopalan model for the gluon distribution functions in very large nuclei by imposing the condition that the nucleons should be color neutral. We find that enforcing color neutrality cures the infrared divergences in the transverse coordinates which are present in the McLerran-Venugopalan model. Since we obtain well-defined expressions for the distribution functions, we are able to draw unambiguous conclusions about various features of the model. In particular, we show that the gluon distribution functions in the absence of quantum corrections behave as 1/x to all orders in the coupling constant. Furthermore, our distribution functions exhibit saturation at small transverse momenta. The normalization of the distribution function we obtain is not arbitrary but specified in terms of the nucleon structure. We derive a sum rule for the integral of the gluon distribution function over transverse momenta, and show that the non-Abelian contributions serve only to modify the shape of the transverse momentum distribution. We obtain a relatively simple expression for the mean value of the transverse momentum-squared. The connection between the McLerran-Venugopalan model and the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation is discussed quantitatively. Finally, we illustrate our results in terms of a simple nuclear model due to Kovchegov.

Figures (9)

• Figure 1: The smooth part of the two-point charge density correlation function in Kovchegov's model paper12, as given in Eqs. (\ref{['IHATu']}) (uniform quark/nucleon distribution) and (\ref{['IHATg']}) (Gaussian quark/nucleon distribution). The nucleon size parameters $a$ have been chosen according to Eq. (\ref{['aGdef']}) so that resulting gluon number densities match in the ultraviolet limit. In the uniform case, $C(\Delta)$ vanishes for $\Delta > 2a$. In the Gaussian case, there is a Gaussian tail in this region.
• Figure 2: The trace of the two-point vector potential correlation function in position space. The Fourier transform of this function is proportional to the gluon number density [ c.f. Eq. (\ref{['GluonDensity']})]. The nucleon size parameters $a$ and $\Lambda_{\rm QCD}$ have been chosen according to Eqs. (\ref{['aGdef']}) and (\ref{['LQCDdef']}) so that these functions match in the ultraviolet ($\vert\bbox{x}-\bbox{x}^\prime\vert\rightarrow 0$) limit. The longitudinal coordinates $(x^{-},x^{\prime -})$ have been fixed at a place where $a^2 \chi = 20$. Plotted are the results of Ref. paper9 (labelled "MV") as well as results using Kovchegov's model paper12 with a uniform or Gaussian distribution of quarks and nucleons.
• Figure 3: Plot of the gluon number density (\ref{['GlueEnd']}) at fixed $q^{+}$. The nucleon size parameters $a$ and $\Lambda_{\rm QCD}$ have been chosen according to Eqs. (\ref{['aGdef']}) and (\ref{['LQCDdef']}) so that these functions match in the ultraviolet limit. The total charge squared per unit area has been set to $a^2 {\cal X}_\infty = 20$. Plotted are the results of Ref. paper9 (labelled "MV") as well as results using Kovchegov's modelpaper12 with a uniform or Gaussian distribution of quarks and nucleons.
• Figure 4: Plot of $(aq)^2$ times the gluon number density (\ref{['GlueEnd']}) at fixed $q^{+}$. This combination has been chosen to enhance the high momentum part of the distribution and show the approach to the predicted $1/\bbox{q}^2$ ultraviolet behavior. The nucleon size parameters $a$ and $\Lambda_{\rm QCD}$ have been chosen according to Eqs. (\ref{['aGdef']}) and (\ref{['LQCDdef']}) so that these functions match in this limit. The total charge squared per unit area has been set to $a^2 {\cal X}_\infty = 20$. Plotted are the results of Ref. paper9 (labelled "MV") as well as results using Kovchegov's model paper12 with a uniform or Gaussian distribution of quarks and nucleons.
• Figure 5: The effect of the non-Abelian terms on the trace of the correlation function. Plotted is the ratio of the all-orders result to the lowest-order (purely Abelian) result using the uniform version of Kovchegov's model paper12. The different curves are for fixed values of the longitudinal coordinates $(x^{-},x^{\prime -})$ such that $a^2 \chi = 2$, 5, 10, 20, and 40.