  The experimental determination of the partial width $\Gamma (H
\longrightarrow \gamma \gamma)$ of an intermediate mass Higgs is among the most
important measurements at a future photon photon collider. Recently it was
shown that large non-Sudakov as well as Sudakov double logarithmic (DL)
corrections can be summed to all orders in the background process $\gamma
\gamma (J_z=0) \longrightarrow q \bar{q}$. It was found that positivity and
stability of the cross section was only restored at the four-loop level. One
remaining large source of uncertainty stems from the fact that the scale of the
strong coupling is unspecified within the double logarithmic approximation. In
this paper we include the leading and next-to-leading order running coupling to
all orders. We thus remove the inherent scale uncertainty of both the exact
one-loop and all-orders DL result without encountering any Landau-pole
singularities. The effect is significant and, for the non-Sudakov form factor,
is found to correspond to an effective scale of roughly $\alpha_s(9m_q^2)$.
