  The measurements of the $\gamma\gamma$ and $b \bar b$ partial decay widths of
an intermediate-mass Standard Model Higgs boson are among the most important
goals of a future photon linear collider. While in an initially polarized
$J_z=0$ state the background process $\gamma\gamma \to b \bar b$ is suppressed
by $\frac{m^2_b}{s}$, radiative corrections remove this suppression and are
known to be very sizable at theone-loop level. In particular a new type of
purely hard double logarithmic (DL) correction can even make the cross section
negative at this order in perturbation theory. The second order term of the
series is also known and enters with a positive sign at the two loop level.
  From a theoretical as well as a practical point of view it is
clearlydesirable to resum this series and to know its high energy behavior.
  In this paper, we derive the series of these novel ``non-Sudakov'' logarithms
to all orders and calculate the high energy limit analytically. We also give
explicit three loop corrections in the DL-approximation as a check of our
results and show that the three loop structure also reveals the higher order
behavior of mixed hard and soft double logarithms. We are thus able to resum
the virtual DL-corrections to $\gamma\gamma (J_z=0) \to b \bar b$ to all
orders.
