Interplay of Hard and Soft Physics in Small $x$ Deep Inelastic Processes

TL;DR

This paper argues that high-energy small-x DIS entails a coherent interplay between hard and soft QCD dynamics, making color coherence a guiding principle for understanding diffractive processes. By developing a QCD-based framework that connects light-cone wave functions, gluon distributions xG(x,Q^2), and multiple scattering in nuclei, it derives calculable predictions for vector-meson electroproduction, color transparency, and forward jet diffraction. It highlights the non-universality of the Pomeron across soft and hard regimes, discusses unitarity constraints, and introduces evolution equations for diffraction that connect to the Ingelman-Schlein model while allowing for non-diagonal transitions. The work emphasizes that diffractive and coherent phenomena offer a three-dimensional image of hadrons and a testing ground for PQCD, nonperturbative QCD, and the origin of the Pomeron, with clear experimental implications for HERA and beyond.

Abstract

Coherence phenomena, the increase with energy of coherence length and the non-universality of parton structure of the effective Pomeron are explained. New hard phenomena directly calculable in QCD such as diffractive electroproduction of states with $M^2\ll Q^2$ and the color transparency phenomenon as well as new options to measure the light-cone wave functions of various hadrons are considered. An analogue of Bjorken scaling is predicted for the diffractive electroproduction of $ρ$ mesons at large momentum transfers and for the production of large rapidity gap events, as observed at HERA. A phenomenological QCD evolution equation is suggested to calculate the basic characteristics of the large rapidity gap events. The increase of parton densities at small $x$ as well as new means to disentangle experimentally soft and hard physics are considered. We discuss constraints on the increase of deep inelastic amplitudes with $Q^2$ derived from the inconsistency of QCD predictions for inclusive and exclusive processes and from unitarity of the S matrix for collisions of wave packets. New ways to probe QCD physics of hard processes at large longitudinal distances and to answer the long standing problems on the origin of the Pomeron are suggested. Unresolved problems and perspectives of small $x$ physics are also outlined.

Figures (6)

• Figure 1: A typical two-gluon exchange contribution to the amplitude $\gamma ^*p\rightarrow Vp$.
• Figure 2: The total longitudinal cross section, $\sigma_{\gamma^* N \rightarrow \rho N}^L$, calculated from Eq. (\ref{['eq:14c']}) for several recent parameterizations of the gluon density in comparison with experimental data from ZEUS ZEUSb (full circles) and NMC NMC1 (squares). Typical parameters for the $\rho$-meson wave functions as discussed above are taken ($\left<k_t^2\right>^{1/2} =0.45 GeV/c$). We set $\eta_V=3$ and parameterize the dependence of the differential cross section on the momentum transfer in exponential form with $B \approx 5$ GeV$^{-2}$. Note that a change of $T^2(Q^2)$ in the range corresponding to $\left<k_t^2\right>^{1/2}$ between 0.3 GeV/c and 0.6 GeV/c introduces an extra scale uncertainty of $0.7 \div 1.4$.
• Figure 3: The ratios $R_S$, $R_V$, and $R_G$ of sea, valence and gluon distributions for A=40 and A=2 and (left) their logarithmic derivatives, $d R_{S(V,G)}(x,Q^2) /d (\ln Q^2)$ as a function of $x$ for $Q^2 = 4$ GeV$^2$ (full line), $Q^2 = 25 {\rm~GeV}^2$ (dashed line) and $Q^2 = 100 {\rm~GeV}^2$ (dotted line).
• Figure 4: The square of the $xG(x,Q^2)$ distribution as a function of $Q^2$ for a fixed $x=10^{-4}$ and as a function of $x$ for a fixed $Q^2=10$ GeV$^2$ for the CTEQ3L parameterization.
• Figure 5: The ratio $R_D \sim \frac{(xG(x,Q^2))^2}{Q^4F_2}$ normalized to be 0.1 at $x=10^{-3}$ at $Q^2=10$ GeV$^2$ for two different parameterizations of parton distributions and obtained assuming $xG(x,Q^2) \sim x^n$ as indicated in the figure.