Renormalization Group Effects in the Process $H \longrightarrow \gamma \gamma$

The partial Higgs width $\Gamma (H \longrightarrow \gamma \gamma)$ is important at the LHC for Higgs masses in the MSSM mass window up to 140 GeV as a relatively background free signal of a fundamental scalar. At the photon photon mode at the NLC it would be the Higgs production mechanism . Two loop QCD corrections exist for the fermionic contribution and in the case of the bottom loop large non-Sudakov double logarithms can be resummed to all orders and contribute up to 12 % compared to the t-quark. In more complicated Higgs sectors, such as in the MSSM, large $\tan \beta$ enhancements of bottom type Yukawa couplings can potentially dominate even the whole partial width. A main uncertainty in all existing calculations is the scale of the strong coupling as it is only renormalized at the three loop level. In this paper we include the exact two loop running coupling to all orders into the bottom contribution. We find that the effective scale is close to $\alpha_s (10 m_b^2)$.