Dipole-dipole scattering: summing large Pomeron loops in non-linear evolution with leading twist kernel

TL;DR

The paper develops a QCD-based framework to sum large BFKL Pomeron loops in dipole-dipole scattering by deriving dipole densities from the BK equation with a leading-twist kernel and recasting the evolution as fan Pomeron diagrams. It combines t-channel unitarity with the BFKL Pomeron calculus and AGK cutting rules to obtain multi-Pomeron amplitudes, multiplicity distributions, and the entropy of produced gluons. Key results show KNO-scaling for gluon multiplicities and an entropy relation S_E = $\ln(xG(x,Q^2))$, aligning with prior Kharzeev-Levin predictions while emphasizing the leading-twist regime. The work provides a coherent method to sum large Pomeron loops across DIS and dipole-dipole scattering, with implications for dilute-dense and high-energy collisions and a path toward incorporating enhanced diagrams in future studies.

Abstract

It is shown in this paper that the QCD equations for dipole density have the natural solution: the 'fan' diagrams of the Pomeron calculus. We found the dipole densities comparing the analytic solution to the Balitsky-Kovchegov (BK) equation for the simplified leading twist kernel with the $t$ channel unitarity. Using these densities we calculate the contributions of large Pomeron loops to dipole-dipole scattering at high energies. Applying the Abramovsky,Gribov and Kancheli cutting rules we found that the produced gluons are distributed accordingly the KNO (Koba, Nielsen and Olesen) law which leads to the entropy $S_E = \ln(x G(x,Q^2))$ in an agreement with Kharzeev - Levin predictions.

Figures (12)

• Figure 1: Summing large Pomeron loops. The wavy lines denote the BFKL Pomeron exchanges. The circles denote the amplitude $\gamma$.
• Figure 2: The non-linear BK equation. The wavy lines denote the BFKL Pomeron exchanges. The circles denote the amplitude $\gamma$ for interaction of the dipole with the target.
• Figure 3: $\Omega\left( z\right)$ versus $z$. $\Omega_{exact}$ is the numerical solution of Eq. (\ref{['BKS5']}) with $\Omega_0 =0.1$. Fig. \ref{['om']}-a shows the large range of $z$, while Fig. \ref{['om']}-b is concentrated at small $z< 10$ to illustrate how the exchange of the BFKL Pomeron describes $\Omega\left( z\right)$.
• Figure 4: The dipole densities in the BFKL Pomeron calculus. Fig. \ref{['rho']}-a shows that $\rho_1$ is given by the exchange of the BFKL Pomeron. Fig. \ref{['rho']}-b demonstrates that $\rho_2$ is the contribution of the first 'fan' diagram of the BFKL Pomeron calculus. The wavy lines denote the BFKL Pomeron exchanges. The circle denotes the triple Pomeron vertex, given by the BFKL kernel.
• Figure 5: The graphic form of the equation for $\rho_n$ in the BFKL Pomeron calculus. in the BFKL Pomeron calculus. The wavy lines denote the BFKL Pomeron exchanges. The black circle denotes the triple Pomeron vertex, given by the BFKL kernel. The blue circles stand for the BFKL kernels