Structure function of the nucleus in the perturbative QCD with $N_c\to\infty$ (BFKL pomeron fan diagrams)

TL;DR

Braun rederives and solves the BFKL pomeron fan-diagram equation in the large-$N_c$ limit using direct summation and a numerical scheme up to rapidity $y\le 50$. The results demonstrate unitarity restoration with cross-sections saturating to a universal value $0.1768\,R_A^2$, corresponding to a colour dipole hitting a black disk and imply a simple pole at $j=1$ in the $t$-channel. The longitudinal structure function saturates as $F_{2L}(x,Q^2)\propto Q^2$ at small $x$, while the transverse part continues to grow roughly as $\ln(1/x)$. The gluon density develops a soliton-like form in $\log k$, moving to higher momenta with rapidity while preserving its shape, leading to a phenomenon termed supersaturation where the density at fixed $k$ vanishes as $y\to\infty$.

Abstract

Equation for the sum of BFKL pomeron fan diagrams is rederived by direct summation and solved numerically for rapidities $y\leq 50$. At high rapidities y>20 the resulting cross-sections for the scattering of a longitudinally polarized $q\bar q$ pair on the nucleus cease to depend on its transverse dimension and tend to a constant limit 0.1768 $R_A^2$, which corresponds to scattering of a colour dipole on a black disk. Thus the unitarity is restored and the singularity in the j plane is reduced to a simple pole at j=1.The nuclear structure function at small x behaves as $Q^2\ln(1/x)$. The found gluon density has a soliton-like form in the $\log k$ space: its form is close to Gaussian, independent of rapidity, the centermoving towards higher $\log k$ with a nearly constant velocity as rapidity increases.