$Z$ boson production in bottom-quark fusion: a study of $b$-mass effects beyond leading order

We compute the total cross-section for $Z$ boson production in bottom-quark fusion, applying to this case the method we previously used for Higgs production in bottom fusion. Namely, we match, through the FONLL procedure, the next-to-next-to-leading-log five-flavor scheme result, in which the $b$~quark is treated as a massless parton, with the next-to-leading-order $\order{\alpha_s^3}$ four-flavor scheme computation in which bottom is treated as a massive final-state particle. Our computation provides a test-case for the discussion of issues of scale dependence and treatment of heavy quarks, which we discuss in light of our results.