Quantum Chromodynamics 2026, 10 - Small-x Physics
Beyond leading-log approximations lie the kinematical ranges of deep inelastic scattering at very small x, so that (αs/π)log(1/x) > 1. Q2 is relatively large, so perturbation theory is applicable. Beginning with the DIS process with 1 lepton scattering off a nucleon and producing a final hadronic state. The interaction is mediated by exchange of a vector boson, γ, or Z0 for the neutral current, and W± for charged current interactions. Choose the usual Bjorken parameters. Small-x behavior of structure functions for fixed Q2 reflects behavior at high-energy (p. 431). The small-x limit for deep inelastic scattering derives from the Bjorken limit, and has the scattering energy being kept much larger than all external masses and momentum transfer. This is the Regge limit (p. 431).
Two-body hadron scattering is strongly dominated by small squared momentum transfer (i.e. small scattering angles), usually modelled further by assuming particle exchange treating quantum numbers as invariant quantities. Regge exchange generalizes this. Instead of particles, the objects exchanged are Regge poles, which are characterized by quantum numbers. Regge poles carrying the vacuum quantum numbers are referred to as pomerons. They dominate the total cross sections of proton-proton and proton-antiproton scattering. Others are reggeons. Together, they describe the exchange of states with appropriate quantum numbers and different virtuality and spin. Their relationship is the Regge trajectory α(t). When it passes through an integer (bosonic pole) or half-integer (fermionic pole), there should be a particle with that number spin, and associated mass. It interpolates between particles of different spins (p. 432). At high energy, asymptotic behavior of a 2-body amplitude is parametrized as A(s, t) ~ sα(t). The optical theorem states Im A(s, t = 0) sσtot, and Regge theory predicts the full behavior of the cross section as σtot = sα(0) - 1 where α(t = 0) is the intercept. The total cross section decreases with increasing energy (p. 433).
In LLA, the evolution equation for Q2 yields the GLAP equations. If a physical gauge is chosen, the approximation corresponds to resumming ladder diagrams with gluon and quark exchange. Terms with higher powers of the coupling lead merely to next-order leading-log approximations, but not the small-x limit of the GLAP evolution. The gluon-splitting function behaves as 6/z at small z, so addressing only these terms of the GLAP equations yields the maximal powers of both large ln(Q2), and ln(1/x). This is the double logarithmic approximation. The resulting n-th iterations behave as
(Eq. 6.84 - 85). Beyond the double-logarithmic approximation is the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation, which sums the diagrams into an expression for the nonintegrated gluon distribution f(x, k2) (6.93 - 94). Including shadowing corrections, they turn into the Gribov-Levin-Ryskin (GLR) equation (Eq. 6.97).