We analyzed the double production and the triple self-coupling of the standard model Higgs boson at future $gamma gamma$ collider energies, with the reactions $gammagamma rightarrow f bar f HH$ $(f=b, t)$. We evaluated the total cross section for $fb
ar fHH$ and calculated the total number of events considering the complete set of Feynman diagrams at tree-level and for different values of the triple coupling $kappalambda_{HHH}$. We have also analyzed the sensitivity for the considered reaction and we show the results as 95% C.L. regions in the $kappa-M_H$ plane for different values of the center of mass energy and different levels of background. The numerical computation was done for the energies which are expected to be available at a possible Future Linear $gammagamma$ Collider with a center-of-mass energy 500-3000 $GeV$ and luminosities of 1 and $5 ab^{-1}$. We found that the number of events for the process $gammagamma rightarrow t bar t HH$, taking into account the decay products of both $t$ and $H$, is small but enough to obtain information on the triple Higgs boson self-coupling in a independent way, complementing other studies on the triple vertex.
The high energy evolution equations that describe the evolution of hadronic amplitudes with energy are derived assuming eikonal interaction of the evolved hadronic wave function with the target. In this note we remark that this derivation allows a di
fferent interpretation, whereby the hadronic wave function is not evolved, but instead the evolution acts on the S - matrix operator. In this approach, analogous to the Heisenberg picture of Quantum mechanics, the scattering is not eikonal and additional boost provides for radiation of more gluons in the final state.
