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Total Cross section

The $\gamma$$\gamma$-collision processes have for LEP II energies typically cross sections that are two to three orders of magnitude larger than e⁺e^- annihilation processes [28]. The cross section for the process $\gamma$$\gamma$ $\rightarrow$ hadrons is extracted from the measurement of e⁺e^- $\rightarrow$ e⁺e^- + hadrons [29]:

$${\frac{d\sigma_{ee}}{dW_{\gamma\gamma}^2/s}}$$ = $$\int$$$${\frac{dQ_1^2}{Q_1^2}}$$$${\frac{dQ_1^2}{Q_1^2}}$$$$\sum_{a,b=T,S}^{}$$$$\cal {L}$$_ab$$\sigma_{ab}^{\gamma\gamma}$$(W_{$\gamma$$\gamma$}², Q₁², Q₂²)

Here d$\sigma_{ee}^{}$ is the measured cross section for the e⁺e^- $\rightarrow$ e⁺e^- + hadrons reaction for a certain dW interval. The center of mass energy is given by $\sqrt{s}$ and $\cal {L}$_ab is the two photon luminosity function, which describes the photon flux. Q_i² are the virtualities (momentum transfer in the e$\gamma$ vertice) of the radiated photons. The hadronic cross sections $\sigma_{ab}^{}$ correspond to specific helicity states (T=transverse and S=Scalar) of the interacting photons. If W² > > Q_i² it is possible, to a very good approximation, assume factorization of the Q and W dependencies of $\sigma_{ab}^{}$ [30]:

$$\sigma_{ab}^{}$$(W_{$\gamma$$\gamma$}², Q₁², Q₂²) = h_a(Q₁²)h_b(Q₂²)$$\sigma_{\gamma\gamma}^{}$$(W_{$\gamma$$\gamma$}²)

The functions h_{a, b} are model dependent and describe the Q² behavior of the hadronic cross section. If this is known it is possible to extrapolate the $\sigma_{ab}^{}$(W_{$\gamma$$\gamma$}², Q₁², Q₂²) to Q_i² = 0 without any loss of the W dependence. It is clear that the extrapolation to Q² = 0 is better for small values of Q_i, which favor notag data. At low Q² most of the hadronic system is however lost in the beampipe and the W measurement of notag data needs to be unfolded with different MC simulations.

Double tag data do on the other hand provide an excellent W measurement from the tagged electrons, and no model dependent unfolding is needed. The extrapolated to Q² = 0 is however strongly model dependent for large Q² values and this can result in large uncertainties in the $\sigma_{\gamma \gamma }^{}$(W²) measurement. The VSAT detector therefore has a unique advantage, as is it can measure double tag events at low Q² values.