Diffusive scaling and the high-energy limit of deep inelastic scattering in QCD at large N_c

TL;DR

The paper demonstrates that at high energy and large N_c, gluon-number fluctuations in the target produce a diffusive scaling regime in deep inelastic scattering, replacing the geometric scaling of mean-field approaches. By embedding Mueller’s dipole picture within the Good-Walker diffraction framework and employing Pomeron-loop evolution, it shows that rare saturated configurations ('black spots') dominate cross-sections even for Q^2 well above the average saturation momentum. The analysis reveals that inclusive and diffractive DIS cross-sections are governed by the same saturated configurations and that elastic diffraction dominates in the high-energy diffusive window, with cross-sections obeying diffusive scaling in the variable (ln(Q^2/⟨Q_s^2⟩))/σ. These results highlight the critical role of fluctuations in high-energy QCD evolution and offer a framework for exploring saturation phenomena beyond mean-field theories. The work sets the stage for quantitative assessments of λ and D_fr via Pomeron-loop dynamics and suggests potential experimental signatures at future colliders.

Abstract

Within the limits of the large-N_c approximation (with N_c the number of colors), we establish the high-energy behaviour of the diffractive and inclusive cross-sections for deep inelastic scattering at fixed impact parameter. We demonstrate that for sufficiently high energies and up to very large values of Q^2, well above the proton average saturation momentum <Q_s^2>, the cross-sections are dominated by dense fluctuations in the target wavefunction, that is, by the relatively rare gluon configurations which are at saturation on the resolution scale Q^2 of the virtual photon. This has important physical consequences, like the emergence of a new, diffusive, scaling, which replaces the `geometric scaling' property characteristic of the mean field approximation. To establish this, we shall rely on a dipole version of the Good-Walker formula for diffraction (that we shall derive here in the context of DIS), together with the high-energy estimates for the dipole scattering amplitudes which follow from the recently established evolution equations with Pomeron loops and include the relevant fluctuations. We also find that, as a consequence of fluctuations, the diffractive cross-section at high energy is dominated by the elastic scattering of the quark-antiquark component of the virtual photon, up to relatively large virtualities Q^2 >> <Q_s^2>.

Figures (20)

• Figure 1: "Phases" of the hadronic wavefunction in the kinematical plane $Y$--$\,\ln(Q^2/Q_0^2)$. When $Y \lesssim Y_{\rm DS}$ the gluon distribution scales with momentum in a 'geometric' way, i.e. it is a function of $Q^2/Q_s^2$. When $Y \gtrsim Y_{\rm DS}$ it scales in a 'diffusive' way, i.e. it becomes a function of $\,\ln(Q^2/Q_s^2)/\sqrt{Y}$.
• Figure 2: Kinematics for diffractive DIS at high energy, or small Bjorken--$x$: $Q^2$ is the virtuality of $\gamma^*$; $W$ is the center--of--mass energy of the $\gamma^* h$ system (with $W^2\gg Q^2$); $M_X^2$ is the invariant mass squared of the diffractive system.
• Figure 3: Typical diagram contributing to the diffractive process $\gamma^* h \to X h$ in the frame where the target $Y_0$ coincides with the rapidity gap. For the projectile, we illustrate the gluon dynamics before and at the time of scattering. The final hadronic state $X$ can be formed with an arbitrary number of gluons produced via 'final state interactions' (see the discussion in Sect. \ref{['SECT_PDIFF']}). The gluons in the target recombine back before the final state, so that the hadron emerges intact from the collision. For simplicity, we exhibit only two--gluon exchanges.
• Figure 4: The same as in Fig. 3 but in the large--$N_c$ limit. Each gluon in the wavefunction of the virtual photon has been replaced with a pointlike quark--antiquark pair in a color octet state. The gluons inside the target wavefunction are not shown explicitly anymore. The relative simplicity of the $q\bar{q}$ representation allows us to also exhibit some multiple gluon exchanges, corresponding to unitarity corrections.
• Figure 5: Pomeron loops in the forward amplitude for onium-hadron scattering.