The Collider Detector at Fermilab collected a unique sample of jets originating from bottom-quark fragmentation ($b$-jets) by selecting online proton-antiproton ($pbar{p}$) collisions with a vertex displaced from the $pbar{p}$ interaction point, consistent with the decay of a bottom-quark hadron. This data set, collected at a center-of-mass energy of $sqrt{s}=$1.96 TeV, and corresponding to an integrated luminosity of $5.4~rm{fb}^{-1}$, is used to measure the $Z$-boson production cross section times branching ratio into $bbar{b}$. The number of $Zrightarrow bbar{b}$ events is determined by fitting the dijet-mass distribution while constraining the dominant $b$-jet background, originating from QCD multijet events, with data. The result, $sigma(pbar{p} rightarrow Z) times mathcal{B}(Z rightarrow bbar{b})= 1.11pm 0.08(text{stat}) pm 0.14(text{syst})~text{nb}$, is the most precise measurement of this process, and is consistent with the standard-model prediction. The data set is also used to search for Higgs-boson production. No significant signal is expected in our data and the first upper limit on the cross section for the inclusive $pbar p rightarrow Hrightarrow bbar b$ process at $sqrt{s}=$1.96 TeV is set, corresponding to 33 times the expected standard-model cross section, or $sigma = 40.6$ pb, at the 95% confidence level.
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