Universality and tree structure of high energy QCD

Abstract

Using non-trivial mathematical properties of a class of nonlinear evolution equations, we obtain the universal terms in the asymptotic expansion in rapidity of the saturation scale and of the unintegrated gluon density from the Balitsky-Kovchegov equation. These terms are independent of the initial conditions and of the details of the equation. The last subasymptotic terms are new results and complete the list of all possible universal contributions. Universality is interpreted in a general qualitative picture of high energy scattering, in which a scattering process corresponds to a tree structure probed by a given source.

Figures (3)

• Figure 1: The effective dependence $\partial\log{Q_s^2(Y)}/\partial Y$ of the saturation scale upon the rapidity $Y$.$\bar{\alpha}$ was set to 0.2. The truncations of the asymptotic expansion Eq. (\ref{['satscal']}) to order $1$, $1/Y$ and $1/Y^{3/2}$ are shown.
• Figure 2: The reduced front profile ${\cal N}(k,Y)\times (k^2/Q_s^2(Y))^{\gamma_c}$ in the leading edge . The curves are for $Y=10$ (lower curves) to $Y=50$ (upper curves). $\bar{\alpha}=0.2$ and the constants in Eq. (\ref{['front']}) were set to $C_1=1$, $C_2=0$. The leading edge expansion (\ref{['front']}) is truncated at level ${\cal O}(\sqrt{Y})$ (dashed lines) and ${\cal O}(1)$ (solid lines). The reduced asymptotic front, i.e. ${\cal N}_\infty(k,Y)\times (k^2/Q_s^2(Y))^{\gamma_c}$ (see Eq. (\ref{['ninf']})), is the straight line.
• Figure 3: The tree structure. The parton branching is represented along the rapidity axis (upper part) and in the transverse coordinate space at two different rapidities (lower part). The probe is represented by a shaded disk of size $1/Q$. At rapidity $Y_1$, the probe counts the partons (linear regime), at $Y_2$, the probe counts groups of partons (transition to saturation). This effect, and also other effects such as recombination of partons inside the tree (r.h.s. of the plot), generate nonlinear damping terms in the evolution equations.