Multiple Parton Scattering in Nuclei: Twist-Four Nuclear Matrix Elements and Off-Forward Parton Distributions

TL;DR

This work addresses higher-twist contributions from multiple parton scattering in large nuclei, focusing on twist-four nuclear matrix elements and the LPM interference that governs induced gluon bremsstrahlung. It develops a convolution-model framework showing these twist-four elements can be expressed as convolutions of nucleon twist-two off-forward parton distributions with two-nucleon correlations; in the asymptotic $A\to\infty$ limit they factorize into products of twist-two nucleon distributions. A compact expression for the key matrix element $K(x_1,x_2,x_L)$ is derived, revealing an $A^{4/3}$ scaling, with non-analytic behavior at $x_A=0$ and a well-defined $K_0$ limit. Numerical studies using a Gaussian two-nucleon correlation show factorization is a good approximation in many kinematic regions for large nuclei, though notable deviations occur at small $x$ or for certain momentum-sharing configurations; the formalism connects to fragmentation-function phenomenology and suggests that nuclear processes could help constrain nucleon off-forward parton distributions.

Abstract

Multiple parton scatterings inside a large nucleus generally involve higher-twist nuclear parton matrix elements. The gluon bremsstrahlung induced by multiple scattering depends not only on direct parton matrix elements but also on momentum-crossed ones, due to the Landau-Pomeranchuk-Migdal interference effect. We show that both types of twist-four nuclear parton matrix elements can be factorized approximately into the product of twist-two nucleon matrix elements in the limit of extremely large nuclei, $A\to \infty$, as assumed in previous studies. Due to the correlative nature of the twist-four matrix elements under consideration, it is actually the off-forward parton distributions that appear naturally in this decomposition, rather than the ordinary diagonal distributions probed in deeply-inelastic scattering. However, we argue that the difference between these two distribution classes is small in certain kinematic regimes. In these regions, the twist-four nuclear parton matrix elements are evaluated numerically and compared to the factorized form for different nuclear sizes within a schematic model of the two-nucleon correlation function. The nuclear size dependence is found to be $A^{4/3}$ in the limit of large $A$, as expected. We find that the factorization is reasonably good when the momentum fraction carried by the gluon field is moderate. The deviation can be more than a factor of 2, however, for small gluon momentum fractions, where the gluon distribution is very large.

Figures (7)

• Figure 1: A central-cut diagram for quark-gluon rescattering processes.
• Figure 2: An example of a crossed parton matrix element. Note the momentum transfer from gluon to quark.
• Figure 3: The dependence of the ratio of $K$ to $K_0$ on $x_L$ shown for the nuclei $^{32}$S, $^{58}$Ni, and $^{208}$Pb. The dashed lines show the saturation ratio for each nucleus ($x_L\rightarrow0$). We note that although the saturation ratios are not very close to 1, the curves are quite flat when $x_A>x_L$, and the ratio increases with nuclear size.
• Figure 4: $K/K_0$ versus $x_A$ for three different values of $x_1$. The $x_A$ dependence is approximately linear for small $x_A$, with a slope whose magnitude decreases as $x_1$ increases. Although this dependence is nominally of order $x_A$, it can lead to quite large corrections for real nuclei if $x_1$ is too small.
• Figure 5: $K/K_0$ versus $x_1$ for $^{32}$S, $^{58}$Ni, and $^{208}$Pb. Although there is strong dependence, the ratio is quite flat for moderate $x_1$. The strong dependence of the placement of these curves on the value of $x_2$ is attributed to the decrease of $G(x)$ for extremely small values of $x$.