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 $fbar fHH$ 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.
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