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 presaturation 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 ${\mathrm{AdS}}_5$ curvature on the interacting string may be 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 $1/x^{0.31}$.

Figure 1

Free transverse string shape at a resolution $x=1/10$ and $b=5$.

Figure 2

Free transverse string shape at a resolution of $x=1/50$ and $b=5$.

Figure 3

Transverse string bit distributions: at $x=1/10$ sampling 200 strings (left) and at $x=1/50$ sampling 40 strings (right).

Figure 4

Attractive interaction: $g=0.3$ (left). No interaction: $g=\tilde{g}=0$ (center). Repulsive interaction: $\tilde{g}=0.3$ (right).

Figure 5

Attractive interaction: $g=0.3$ (left). No interaction: $g=\tilde{g}=0$ (center). Repulsive interaction: $\tilde{g}=0.1$ (right).

Figure 6

Attractive interaction: $g=0.1,0.2,0.3$ (left). Repulsive interaction: $\tilde{g}=0.1,0.2,0.3$ (right).

Figure 7

Attractive interaction: $g=0.1,0.2,0.3$ (left). Repulsive interaction: $\tilde{g}=0.1,0.2,0.3$ (right).

Figure 8

Solid line is the analytical result in (2.14). Numeric data are for attractive interaction: $g=0.1,0.2,0.3$ (left) and repulsive interaction: $\tilde{g}=0.1,0.2,0.3$ (right).

Figure 9

Attractive interaction: $g=0.1,0.2,0.3$ (left). Repulsive interaction: $\tilde{g}=0.1,0.2,0.3$ (right).

Figure 10

$R_{\perp }[g]$ for $g=0.1,0.2,0.3$ (left). $R_{\perp }[g=0.1]/R_{\perp }[g=0]$ for large $N$ (right).

Figure 11

Saturation (red), presaturation (blue) and diffusive (green) regimes for a transverse string with decreasing resolution in $D_{\perp }=3$. See text.

Figure 12

3D configuration of the string with $N=200$ and $g=0.3$ using $D_{\perp }(\lambda )$ in the interaction. See text.

Figure 13

$R_{\perp }[g]$ for $g=0.1,0.2,0.3$ (left). $R_{\perp }[g=0.3]$ and Eq. (6.9) (right).

Figure 14

Saturation (red), presaturation (blue) and diffusive (green) regimes for a transverse string with decreasing resolution in $D_{\perp }(\lambda )$. See text.

Figure 15

Attractive interaction: $g=0.1,0.2,0.3$ (left). Repulsive interaction: $\tilde{g}=0.1,0.2,0.3$ (right).

Figure 16

Attractive interaction: $g=0.1,0.2,0.3$ (left). Repulsive interaction: $\tilde{g}=0.1,0.2,0.3$ (right).

Figure 17

3D Histograms, 1000 random generated strings. $N=200$.

Figure 18

3D Histograms, 1000 random generated strings. $N=200$. Attractive interaction $g=0.3$.

Figure 19

3D Histograms, 1000 random generated strings. $N=200$. Repulsive interaction $\tilde{g}=0.3$.

Figure 20

Attractive interaction $g=0.3$.

Figure 21

Noninteracting $g/\tilde{g}=0$.

Figure 22

Repulsive interaction $\tilde{g}=0.3$.

Figure 23

Attractive interaction $g=0.3$.

Figure 24

Noninteracting $g/\tilde{g}=0$.

Figure 25

Repulsive interaction $\tilde{g}=0.3$.

Figure 26

Attractive interaction $g=0.3$.

Figure 27

Noninteracting $g/\tilde{g}=0$.

Figure 28

Repulsive interaction $\tilde{g}=0.3$.