Stretched String with Self-Interaction at High Resolution: Spatial Sizes and Saturation

We model the (holographic) QCD Pomeron as a long and stretched (fixed impact parameter) transverse quantum string in flat $D_\perp=3$ dimensions. After discretizing the string in $N$ string bits, we analyze its length, mass and spatial distribution for large $N$ or low-x ($x=1/N$), and away from its Hagedorn point. The string bit distribution shows sizable asymmetries in the transverse plane that may translate to azimuthal asymmetries in primordial particle production in the Pomeron kinematics, and the flow moments in minimum bias $pp$ and $pA$ events. At moderately low-x and relatively small string self-interactions $g_s\approx \alpha_s$ (the gauge coupling), a pre-saturation phase is identified whereby the string transverse area undergoes a sharp transition from a large diffusive growth to a small fixed size area set by few string lengths $l_s$. For lower values of $x$ the transverse string bit density is shown to increase as $1/x$ before saturating at the Bekenstein bound of one bit per Planck area with the Planck length $l_P/l_s\approx \alpha_s^{2/3}$. We argue that the effects of the AdS$_5$ curvature on the interacting string maybe estimated using an effective transverse dimension between the interacting string bits. The result is a smoother transition with a transverse string bit density increasing as $1/x^{0.31}$.