On the Scale Variation of the Total Cross Section for Higgs Production at the LHC and at the Tevatron

We present a detailed study of the total $p p$ cross section for scalar Higgs production to next-to-next-to leading order in $\alpha_s$ at LHC energies, and of the $p\bar p$ cross section at the Tevatron, combining an implementation of the solutions of the parton evolution equations at the 3-loop order with the corresponding hard scatterings, evaluated at the same perturbative order. Our solutions of the DGLAP equations are implemented directly in $x$-space and allow the study of the dependence of the results on the factorization ($\mu_F$) and renormalization scales $(\mu_R)$ typical of a given process, together with the stability of the perturbative expansion. The input sets for the parton evolutions are those given by Martin, Roberts, Stirling and Thorne and by Alekhin. Results for K-factors are also presented. The NNLO corrections can be quite sizeable at typical collider energies. The stability region of the perturbative expansion is found when $\mu_R > m_H\sim\mu_F$.