Quark-Antiquark Exchange in $\gamma^* \gamma^*$ Scattering

We calculate the high energy behavior of quark-antiquark exchange in $\gamma^* \gamma^*$ elastic scattering by summing, to all orders, the leading double logarithmic contributions of the QCD ladder diagrams. Motivation comes from the LEP data for $\sigma_{tot}^{\gamma^* \gamma^*}$ which indicate the need for secondary reggeon exchange. We show that, for large photon virtualities, this exchange is calculable in pQCD. This applies, in particular, to parts of the LEP kinematic region.