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The `optimal' factorization scale $\mu_0$ is calculated for open heavy quarkproduction. We find that the optimal value is $\mu_F=\mu_0\simeq0.85\sqrt{p^2_T+m_Q^2} $; a choice which allows us to resum thedouble-logarithmic, $(\alpha_s\ln\mu^2_F\ln(1/x))^n$ corrections (enhanced atLHC energies by large values of $\ln(1/x)$) and to move them into the incomingparton distributions, PDF$(x,\mu_0^2)$. Besides this result for the singleinclusive cross section (corresponding to an observed heavy quark of transversemomentum $p_T$), we also determined the scale for processes where theacoplanarity can be measured; that is, events where the azimuthal angle betweenthe quark and the antiquark may be determined experimentally. Moreover, wediscuss the important role played by the $2\to 2$ subprocesses, $gg\toQ\bar{Q}$ at NLO and higher orders. In summary, we achieve a better stabilityof the QCD calculations, so that the data on $c\bar{c}$ and $b\bar{b}$production can be used to further constrain the gluons in the small $x$,relatively low scale, domain, where the uncertainties of the global analysesare large at present.

Scale dependence of open $c\bar{c}$ and $b\bar{b}$ production in the low $x$ region