Small-x Processes in Perturbative Quantum Chromodynamics

TL;DR

The work surveys how perturbative QCD describes hard and semi-hard processes, emphasizing the DGLAP and BFKL formalisms and their limits. It introduces an RG-improved small-$x$ framework that resums collinear and high-energy logarithms via an $\,\omega$-expanded kernel, addressing large NL$x$ corrections and scale ambiguities. A central contribution is the construction of an RG-consistent small-$x$ equation and its NL$x$ corrections, supported by a toy collinear model that demonstrates improved stability and a hard pomeron intercept around $\,\omega_s(\alpha_s) \approx 0.27-0.32$ for realistic couplings. The work also clarifies the division between perturbative and non-perturbative effects, the role of impact factors, and the dependence on energy-scale choices, providing a framework potentially applicable to two-scale hard processes while highlighting residual uncertainties in single-scale DIS. Altogether, it presents a coherent strategy to unify DGLAP and BFKL dynamics and to quantify small-$x$ behaviour with RG-guided resummations, improving predictive power in perturbative QCD for high-energy processes.

Abstract

Starting from a rewiev of DGLAP and BFKL evolution equations for small-x processes, a sistematic study is performed in order to understand the limits of both the formulations and to improve them in a unique framework, which aims to cover the whole range of applicability of perturbative QCD and which describes the transition mechanism from perturbative to non-perturbative physics in the region where unitarity contributions are expected not to be important.

Figures (49)

• Figure 1: DIS: $p_1$ ($p_1'$) represents the incoming (scattered) electron, $p_2$ the incoming proton, $q$ the exchanged virtual photon and $X$ the hadronic shower.
• Figure 2: Elastic scattering of a quark by absorption of a virtual photon.
• Figure 3: The longitudinal proton structure function $F_L$ data compared with the $F_2$ structure function fit (the black line).
• Figure 4: The proton structure function $F_2$ versus $Q^2$ for various values of $x$.
• Figure 5: The diagram representing the partonic tensor $\hat{W}{}_{\mu\nu}^{(\text{\sf a},0)}$ at Born level. $\text{\sf a}$ labels the incoming quark flavour an $\hat{p}{}$ its momentum.