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Monte Carlo models

The different processes above have to be simulated using theoretical assumptions, where the physical properties of the reaction is translated into a mathematical form in a Monte Carlo model (see chapter 7). The process in Fig. 6.1 is simulated by a non-perturbative phemenological model called the Vector Dominance Model (VDM). This was developed about 25 years ago in order to relate $\gamma$$\gamma$-collisions with hadron-hadron physics.

The VDM model assumes that the virtual fluctuation of a photon into a quark-antiquark pair has a lifetime long enough to form a bound vector meson state. This is possible when the transverse momenta (p_T) of the branching ( $\gamma$ $\rightarrow$ q$\bar{q}$) is small, and the model is thus describing the low p_T part of a $\gamma$$\gamma$-collision.

At higher p_T the photons can give a q$\bar{q}$-pair, which is described by the QPM model. The direct and pointlike coupling of photons to quarks is described by perturbative calculations and is treated in similar way as the QED reaction e⁺e^- $\rightarrow$ e⁺e^-$\mu^{+}_{}$$\mu^{-}_{}$. The QPM model is only applicable at large Q², high p_T or large quark masses. The contribution to the total cross section of the relatively low Q² of most two-photon events is therefore quite small, but grows to become more important with increasing Q².

The q$\bar{q}$ pair from a photon can fluctuate further into more complicated states, as shown in Fig. 6.3 and 6.4. The photon can therefore be described by a set of Parton Density Functions (PDF's). These functions give, for each parton, the parton density for a given x and Q² value. Here x is the fraction of the total longitudinal momentum of the photon carried by the parton.

The PDF's are used in the QCD Resolved Photon Contribution model (QCD-RPC), which is needed in order to describe data at low Q². The QCD-RPC model considers hadronic scattering subprocesses to be perturbative and has therefore a p_T^min cut on all PDF's, setting the lower transverse momentum limit on the outgoing partons. This cut separates the model from contributions from the non-perturbative region at low p_T. There are several different parameterizations available to describe the parton density of the photon. The difference in their behavior is mainly seen at low values of x [27].