Unitarity and the QCD-improved dipole picture

TL;DR

This work develops a QCD-improved dipole picture that unifies perturbative small-dipole dynamics with nonperturbative large-dipole physics by constructing a realistic dipole cross section (DCS) that interpolates between the perturbative region and the soft, hadronic regime using the pion–proton cross section as a constraint. It demonstrates how unitarity corrections emerge at small $x$ and moderate $Q^2$, with central collisions tending toward a black limit while peripheral collisions continue to contribute, thereby challenging standard leading-twist DGLAP expectations. The model yields structure functions $F_L$ and $F_2$ in good agreement with HERA data without additional tuning and reveals substantial large-$b$ contributions at moderate $Q^2$, signaling higher-twist and nonperturbative effects that become important near HERA kinematics. The analysis also provides a framework to study the onset of a new QCD regime where unitarity and diffusional (peripheral) dynamics shape the energy evolution of DIS, diffraction, and exclusive processes, with diffractive channels offering early probes of this regime.

Abstract

As a consequence of QCD factorization theorems, a wide variety of inclusive and exclusive cross sections may be formulated in terms of a universal colour dipole cross section at small $x$. It is well known that for small transverse size dipoles this cross section is related to the leading-log gluon density. Using the measured pion-proton cross section as a guide, we suggest a reasonable extrapolation of the dipole cross section to the large transverse size region. We point out that the observed magnitude and small $x$ rise of the gluon density from conventional fits implies that the DGLAP approximation has a restricted region of applicability. We found that `higher twist' or unitarity corrections are required in, or close to, the HERA kinematic region, even for small `perturbative' dipoles for scattering at central impact parameters. This means that the usual perturbative leading twist description, for moderate virtualities, $1 < Q^2 < 10$ GeV$^2$, has rather large `higher twist' corrections at small $x$. In addition, for these virtualities, we also find sizeable contributions from large non-perturbative dipoles ($b \gsim 0.4$ fm) to $F_2$, and also to $F_L$. This also leads to deviations from the standard leading twist DGLAP results, at small $x$ and moderate $Q^2$. Our model also describes the low $Q^2$ data very well without any further tuning. We generalize the Gribov unitarity limit for the structure functions of a hadron target to account for the blackening of the interaction at central impact parameters and to include scattering at peripheral impact parameters which dominate at extremely large energies.

Figures (16)

• Figure 1: Dipole cross section in mb for fixed $\lambda = 10$, with the toy model ansatz at large $b$. For an input scale of $Q_0 = 1.6$ GeV, $b_{Q0} = 0.39$ fm marks the boundary of the perturbative region.
• Figure 2: Longitudinal photon wavefunction squared, integrated over z (in units of fm$^{-2}$), for $Q^2=4,10,40$ GeV$^2$.
• Figure 3: Integrand of $\sigma_L$, in units of fm, for fixed $\lambda = 10$, $Q^2=4$ GeV$^2$.
• Figure 4: Integrand of $\sigma_L$, in units of fm, for fixed $\lambda = 10$, $Q^2=40$ GeV$^2$.
• Figure 5: The function $f({\bar{Q}^2}) = \alpha_s x g$ at fixed $x$ for various parton sets