Froissart Bound from Gluon Saturation

TL;DR

The paper shows that gluon saturation, as described by the non-linear BK equation within the Color Glass Condensate framework, can saturate the Froissart bound for a fixed coupling and small dipole by separating the high-energy evolution into a central, saturated black disk and an outer grey region governed by BFKL dynamics.A key mechanism is the factorization of the impact-parameter dependence in the grey area, combined with an exponential fall-off of the hadronic tail controlled by the pion mass, which yields a total cross-section growing as $\sigma \sim \ln^2 s$ with a universal coefficient.The authors compute the black-disk radius, the impact-parameter dependent saturation scale, and demonstrate two geometric-scaling regimes, highlighting the crucial role of colour neutrality of saturated gluons in suppressing long-range perturbative tails and preserving unitarity.The work connects perturbative evolution with non-perturbative confinement physics and provides a concrete, quantitative picture of how high-energy hadronic cross sections approach the Froissart bound within a CGC framework.

Abstract

We demonstrate that the dipole-hadron cross-section computed from the non-linear evolution equation for the Colour Glass Condensate saturates the Froissart bound in the case of a fixed coupling and for a small dipole (Q^2 >> Lambda_{QCD}^2). That is, the cross-section increases as the logarithm squared of the energy, with a proportionality coefficient involving the pion mass and the BFKL intercept (alpha_s N_c/pi)4 ln 2. The pion mass enters via the non-perturbative initial conditions at low energy. The BFKL equation emerges as a limit of the non-linear evolution equation valid in the tail of the hadron wavefunction. We provide a physical picture for the transverse expansion of the hadron with increasing energy, and emphasize the importance of the colour correlations among the saturated gluons in suppressing non-unitary contributions due to long-range Coulomb tails. We present the first calculation of the saturation scale including the impact parameter dependence. We show that the cross-section at high energy exhibits geometric scaling with a different scaling variable as compared to the intermediate energy regime.

Figures (5)

• Figure 1: A pictorial representation of the dipole-hadron scattering in transverse projection (only half of the hadron disk is shown). The $b$--dependence of the saturation scale illustrated by the lower plot is the one to be found in Sect. 4.3.
• Figure 2: The longitudinal profile of the hadron as it appears in a scattering at given $\tau$ and $Q^2$. The longitudinal coordinate is on the horizontal axis, and is measured in units of space-time rapidity. The tranverse coordinate is on the vertical axis. A longitudinal layer at rapidity $\eta$ is delimited for more clarity. The wavy line represents the colour field $\alpha_\eta$ created at point ${x_\perp}$ by the (saturated) source $\rho_\eta$ at $z_\perp$. The enclined dashed line represents the limit of the black disk, which increases linearly with $\eta$, as we shall see in Sect. 4.
• Figure 3: The saturation scale $Q_s^2(b)/Q_s^2(b=0)$ from eqs. (\ref{['rho_s_exact']})--(\ref{['Qs_exact']}) for $\bar{\alpha}_s\tau =3$ and the Woods-Saxon profile function of eqs. (\ref{['WS']})--(\ref{['Sb']}) with $R_0=3/2m_\pi$. On the abscisa, the radial distance is measured in units of $1/2m_\pi$.
• Figure 4: The function $\rho_s(\tau,b)$, eq. (\ref{['rho_s_exact']}), together with small--$b$ approximation, eq. (\ref{['rhosmall']}) (dotted line), and its large--$b$ approximation, cf. eq. (\ref{['Qs_asy']}) (dashed line) plotted as functions of $b/R_H(\tau)$.
• Figure 5: A pictorial representation of the expansion of the black disk with increasing rapidity. The dotted line circle of radius $\bar{b}(\tau)=R_H(\tau)/2$ separates between domains (I) and (II).