DIS Kinematics

The event kinematics of deep inelastic scattering, DIS, are determined by the negative square of the four-momentum transfer at the positron vertex, $Q^2\equiv-q^2$, and the Bjorken scaling variable, $x=Q^2/2P \cdot q$,where P is the four-momentum of the proton. In the Quark Parton Model (QPM), the interacting quark from the proton carries the four-momentum xP. The variable y, the fractional energy transfer to the proton in its rest frame, is related to x and Q² by $y\simeq Q^2/xs$, where $\sqrt s$is the positron-proton centre of mass energy. Because the H1 and ZEUS detectors are almost hermetic the kinematic variables x and Q² can be reconstructed in a variety of ways using combinations of electron and hadronic system energies and angles [8].

  

Figure 2: (a) QPM (b) QCDC and (c) BGF diagrams

In QPM there is a 1+1 parton configuration, fig. 2a, which consists of a single struck quark and the proton remnant, denoted by ``+1''. At HERA energies there are significant higher-order Quantum Chromodynamic (QCD) corrections: to leading order in the strong coupling constant, $\alpha_{\rm s},$these are QCD-Compton scattering (QCDC), where a gluon is radiated by the scattered quark and Boson-Gluon-Fusion (BGF), where the virtual boson and a gluon fuse to form a quark-antiquark pair. Both processes have 2+1 partons in the final state, as shown in fig. 2. There also exists calculations for the higher, next-to-leading (NLO) processes.

Perturbative QCD does not predict the absolute value of the parton densities within the proton but determines how they vary from a given input. For a given initial distribution at a particular scale Altarelli-Parisi (DGLAP) evolution [9] enables the distributions at higher Q² to be determined. DGLAP evolution resums the leading $\log(Q^2)$ contributions associated with a chain of gluon emissions. At large enough electron-proton centre-of-mass energies there is a second large variable 1/x and, therefore, it is also necessary to resum the $\log(1/x)$ contributions. This is acheived by using the BFKL equation [10].