QCD Factors a1 and a2 Beyond Leading Logarithms Versus Factorization in Non-Leptonic Heavy Meson Decays

We calculate the QCD factors $a_1$ and $a_2$, entering the tests of factorization in non-leptonic heavy meson decays, beyond the leading logarithmic approximation in three renormalization schemes (NDR, HV, DRED). We investigate their $\mu$--dependence and the renormalization scheme dependence. We point out that $a_1$ for B-decays depends very weakly on $\Lms$, $\mu$ and the choice of the renormalization scheme. For $\Lms^{(5)}=225\pm85~MeV$ and $4~GeV\leq\mu\leq 8~GeV$ we find $a_1=1.01\pm0.02$ in accordance with phenomenology. The $\Lms$, $\mu$ and scheme dependences of $a_2$ are on the other hand sizable. Interestingly, for the NDR scheme we find $a_2^{NDR}=0.20\pm0.05$, in the ball park of recent phenomenological results and substantially larger than leading order estimates. However $a_2^{HV}=0.16\pm0.05$ and $a_2^{DRED}=0.15\pm0.05$. Implications of these findings for the tests of factorization in B-decays and D-decays are critically discussed.