Multiplicity distribution of produced gluons in deep inelastic scattering: main equations and their homotopy solutions for heavy nuclei

Abstract

In this paper we discuss the multiplicity distribution in the deep inelastic processes in the frame work of high energy QCD. We obtained three results. First, we get the new derivation of the equations for the cross sections of productions of $n$-cut Pomerons in the final states ($σ_n$). These equations coincide with the equations that have been derived using the Abramovsky, Gribov and Kancheli (AGK) cutting rules but based on the dipole approach to QCD. Second, we developed the homotopy approach for finding the solutions to these equations. It consists with the analytic solution for the first iteration and the converge procedure of calculating the next iterations using computing. Third, we found the analytical solution for $σ_n$ at large $n\,\gtrsim\,N(z) = 2 N_0 \,z\,\exp( z^2/(2 κ))$ with $z = \ln( r^2\,Q^2_s )$. Using this solution we calculate the entropy of the produced gluons at large $z$: $S_E = \ln \left( N(z)\right)$, where the saturation momentum $Q_s$ and all constants are discussed in the text.

Figures (14)

• Figure 1: The interaction of fast hadron (dipole) with the virtual photon ($\gamma^*$). The coherence of the partonic wave function of the fast hadron is destroyed at $t=0$, while the gluons can be measured at $t = \infty$.
• Figure 2: Saturation region of QCD for elastic amplitude. The critical line (z=0) is shown in red. The initial condition for scattering with the dilute system of partons (with proton) is given at $\xi_s = 0$. For heavy nuclei the initial conditions are placed at $Y_A = (1/3)\ln \,A \gg\,1$, where $A$ is the number of nucleon in a nucleus. Two blue lines show the kinematic regions for the initial conditions: the upper one for nuclei and the low one for proton.
• Figure 3: $\sigma^{\hbox{\tiny AGK}}_1$ for the scattering amplitude of Eq. (\ref{['I0']}). The vertical dashed lines show cut Pomerons.
• Figure 4: a.) The graphic form of Eq. (\ref{['ME4']}) for the interaction of one dipole which is denoted by two parallel horizontal lines with opposite directions of arrows. All notations in the caption of Fig. \ref{['eveq']}. b.) The evolution equation for $S_{in}$.
• Figure 5: The graphic form of Eq. (\ref{['ME7']}) for the interaction of one dipole which is denoted by two parallel horizontal lines with opposite directions of arrows. The first stage of the interaction is the decay of one dipole to two , which is described by the kernel $K\left( \boldsymbol{r},\boldsymbol{r}'\right)$. The vertical dashed lines denote time t=0 in Fig. \ref{['parcas']} where the interaction with the target occurs in the amplitude and complex conjugated amplitude. The vertical line denotes $t= +\infty$ where our detectors are placed. The Pomeron intercept is absorbed in the definition of $Y$. The coefficients reproduce the AGK cutting rules. Note, that the linear term has the form of Eq. (\ref{['ME70']}).