Small-x Physics: From HERA to LHC and beyond

TL;DR

This work synthesizes small-$x$ QCD insights from HERA and Tevatron to predict rich phenomena at the LHC, centering on QCD factorization, the dipole picture, and the black-disk limit. It argues that rapid gluon density growth drives unitarity constraints that push hard interactions toward a saturated, central-collision regime, with profound implications for particle production, diffraction, and heavy-ion dynamics. The authors emphasize a two-scale transverse nucleon structure, where hard gluons occupy a compact area while soft interactions dominate at large impact parameters, and they outline experimental strategies—vector-mmeson production, DVCS, ultraperipheral collisions, and central pp/pA measurements—to probe these effects. The paper also connects leading-twist diffraction to nuclear shadowing, highlights the potential of exclusive diffractive Higgs production, and charts a path for future facilities (EIC) to explore small-$x$ dynamics across a broad kinematic range.

Abstract

We summarize the lessons learned from studies of hard scattering processes in high-energy electron-proton collisions at HERA and antiproton-proton collisions at the Tevatron, with the aim of predicting new strong interaction phenomena observable in next-generation experiments at the Large Hadron Collider (LHC). Processes reviewed include inclusive deep-inelastic scattering (DIS) at small x, exclusive and diffractive processes in DIS and hadron-hadron scattering, as well as color transparency and nuclear shadowing effects. A unified treatment of these processes is outlined, based on factorization theorems of quantum chromodynamics, and using the correspondence between the "parton" picture in the infinite-momentum frame and the "dipole" picture of high-energy processes in the target rest frame. The crucial role of the three-dimensional quark and gluon structure of the nucleon is emphasized. A new dynamical effect predicted at high energies is the unitarity, or black disk, limit (BDL) in the interaction of small dipoles with hadronic matter, due to the increase of the gluon density at small x. This effect is marginally visible in diffractive DIS at HERA and will lead to the complete disappearance of Bjorken scaling at higher energies. In hadron-hadron scattering at LHC energies and beyond (cosmic ray physics), the BDL will be a standard feature of the dynamics, with implications for (a) hadron production at forward and central rapidities in central proton-proton and proton-nucleus collisions, in particular events with heavy particle production (Higgs), (b) proton-proton elastic scattering, (c) heavy-ion collisions. We also outline the possibilities for studies of diffractive processes and photon-induced reactions (ultraperipheral collisions) at LHC, as well as possible measurements with a future electron-ion collider.

Figures (20)

• Figure 1: The kinematics of deep--inelastic lepton--hadron scattering (DIS). The interaction proceeds by exchange of a virtual photon, whose four--momentum is given by the difference of the lepton momenta, $q = k' - k$. The hadronic scattering process is characterized by two kinematic invariants, the photon virtuality, $q^2 \equiv - Q^2 < 0$, and the photon--proton center--of--mass energy, $W$, or, alternatively, the Bjorken scaling variable, $x$. (We neglect the target mass.)
• Figure 2: (a) The total cross section for $\gamma^\ast p$ scattering is given by the imaginary part of the forwad scattering amplitude. (b) The total cross section in the parton model (Bjorken scaling). (c,d) QCD radiative corrections, giving rise to the leading scaling violations in $\alpha_s \ln (Q^2 / Q_0^2)$.
• Figure 3: (a) A typical Feynman diagram for inclusive $\gamma^\ast p$ scattering at small $x$. (b) Time evolution of small--$x$ scattering in the target rest frame. (c) The decay of a fast parton in the $q\bar{q}$ dipole. The degradation of longitudinal momenta is accompanied by a random walk in the transverse coordinate.
• Figure 4: The proton structure function, $F_2 (x)$, as measured by the H1 and ZEUS experiments at HERA Moriond2004. Also included are data from fixed--target experiments. The lines show a QCD fit based on the NLO DGLAP approximation.
• Figure 5: The exponents characterizing the $x$--dependence of $F_2$, $\lambda_2$, the gluon distribution, $\lambda_g$ (black squares), and the sea quark distributions, $\lambda_q$ (green triangles), cf. Eq. (\ref{['power']}), as extracted from the NLO DGLAP fit to the H1 and ZEUS data (A.Levy, private communication).