[Paper Review] Unitarity Cutting Rules for Hard Processes on Nuclear Targets

Wolfgang Schäfer|ArXiv.org|Sep 4, 2008

Fusion materials and technologies3 references3 citations

TL;DR

This paper develops unitarity cutting rules for hard processes on nuclear targets within nonlinear $k_{\perp}$ factorization, showing that color-coupled channel effects lead to two distinct types of cut Pomerons—those inducing transitions between color multiplets and those rotating within the same multiplet. The key result is a nontrivial, non-hierarchical distribution of topological cross sections across cut Pomeron multiplicities, contrasting sharply with the simplistic Glauber–AGK model and revealing strong impact-parameter dependence in deep inelastic scattering on heavy nuclei.

ABSTRACT

Unitarity cutting rules for the multiplicity of cut Pomerons, or topological cross sections, have been obtained within the framework of nonlinear k_t factorisation. Proper account for the color coupled channel aspects of the problem leads to the emergence of two types of cut Pomerons. This is illustrated on the example of topological cross sections in deep inelastic scattering on a nucleus.

Motivation & Objective

• To derive unitarity cutting rules for topological cross sections in hard processes on nuclear targets, particularly in deep inelastic scattering (DIS) on heavy nuclei.
• To account for color-coupled channel effects in multi-gluon exchange processes, which are critical in dense nuclear environments.
• To replace the simplistic Glauber–AGK model with a QCD-based framework that properly handles infrared-sensitive, nonperturbative dynamics via the $k_{\perp}$ factorization approach.
• To predict observable deviations in impact-parameter-dependent structure functions and topological cross sections due to distinct cut Pomeron types.

Proposed method

• The study employs nonlinear $k_{\perp}$ factorization to compute dijet and inclusive DIS cross sections, using multiparton S-matrices for virtual parton transitions such as $\gamma^* \to q\bar{q}$, $q \to qg$, $g \to gg$, and $g \to Q\bar{Q}$.
• The S-matrices are decomposed into elastic (uncut Pomeron) and excitation (cut Pomeron) parts, with the excitation part defined via the color-dipole cross section operator $\Sigma^{(4)}_{\text{ex}}$.
• The cut Pomeron contributions are expanded in powers of $\Sigma^{(4)}_{\text{ex}}$, leading to a topological cross section decomposition in terms of $k$-cut Pomeron states.
• The profile function for $k$-cut Pomerons is derived using a modified Bessel function expansion and the incomplete gamma function $\gamma(k, \lambda)$, incorporating the nonperturbative parameter $\sigma_0$.
• The method distinguishes two types of cut Pomerons: those inducing color multiplet transitions ($R_i \to R_j$) and those causing internal rotations within a multiplet ($R_i \to R_i$), arising from the color-coupled channel structure.
• The results are compared with the Glauber–AGK model, which assumes a simple Poisson-like hierarchy, by computing the $k$-cut Pomeron contribution to the nuclear structure function $F_2$ as a function of impact parameter.

Experimental results

Research questions

• RQ1 How do color-coupled channel effects modify the standard Glauber–AGK picture of multiple Pomeron cuts in nuclear DIS?
• RQ2 What is the correct QCD-based formulation of topological cross sections in hard processes on nuclei, accounting for nonperturbative, infrared-sensitive dynamics?
• RQ3 Why does the Glauber–AGK model fail to describe the true distribution of cut Pomeron multiplicities in heavy-ion collisions?
• RQ4 How do the two distinct types of cut Pomerons—those inducing color multiplet transitions and those causing internal rotations—affect the impact-parameter dependence of topological cross sections?
• RQ5 What quantitative differences emerge between the Glauber–AGK prediction and the QCD-based cutting rules in the structure function $F_2$ at small $x$?

Key findings

• The inclusion of color-coupled channel effects leads to two distinct types of cut Pomerons: those that induce transitions between different color multiplets ($R_i \to R_j$) and those that rotate the system within the same multiplet ($R_i \to R_i$), which are not captured by the Glauber–AGK model.
• The topological cross section profile function for $k$-cut Pomerons is given by $\Gamma^{(k)}(\mathbf{b}, \mathbf{r}) = \sigma(\mathbf{r})T(\mathbf{b}) \cdot w_{k-1}(2\nu_A(\mathbf{b})) \cdot \frac{e^{-2\nu_A(\mathbf{b})}}{\lambda^k} \gamma(k, \lambda)$, where $\lambda = 2\nu_A(\mathbf{b}) - \sigma(\mathbf{r})T(\mathbf{b})$, $\nu_A(\mathbf{b}) = \frac{1}{2}\sigma_0 T(\mathbf{b})$, and $\gamma(k, \lambda)$ is the incomplete gamma function.
• The QCD-based cutting rules produce a non-hierarchical distribution of topological cross sections, in stark contrast to the Glauber–AGK model, which predicts a strong hierarchy with decreasing contributions for higher $k$.
• For a dipole size of $r = 0.6$ fm and $A = 208$, the QCD result shows significant contributions from multiple cut Pomerons even at large impact parameters, where the Glauber–AGK model predicts negligible higher-order terms.
• The $k$-cut Pomeron contribution to the nuclear structure function $F_2$ exhibits strong impact-parameter dependence, with a dramatic deviation from the Glauber–AGK prediction, especially at mid-rapidity and small $x$, indicating a nontrivial role of color entanglement.
• The results depend crucially on the nonperturbative parameter $\sigma_0$, the large-dipole color-dipole cross section, which renders the cut Pomeron contributions infrared sensitive and physically meaningful.