All-orders resummation of leading logarithmic contributions to heavy quark production in polarized γγ collisions

The measurements of the γγ and $b\overline{b}$ 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 $J_z=0\mathrm{}$ state the background process $\gamma \overrightarrow{\gamma }b\overline{b}$ is suppressed by $m_b^2/s;$ radiative corrections remove this suppression and are known to be very sizable at the one-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 clearly desirable 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 $\gamma \gamma {(J}_z=0)\to b\overline{b}$ to all orders.