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This Letter reports a search for a narrow resonant state decaying into two$W$ bosons and a bottom-antibottom quark pair where one $W$ boson decaysleptonically and the other decays into a quark-antiquark pair. The search isparticularly sensitive to top-antitop resonant production. We use the full datasample of proton-antiproton collisions at a center-of-mass energy of 1.96 TeVcollected by the CDF II detector at the Fermilab Tevatron, corresponding to anintegrated luminosity of 9.45 fb$^{-1}$. No evidence for resonant production isfound and upper limits on the production cross section times branching ratiofor a narrow resonant state are extracted. Within a specific benchmark model,we exclude a $Z'$ boson with mass below 915 GeV/$c^2$ decaying into atop-antitop pair at the 95% credibility level assuming a $Z'$ boson decay widthof $\Gamma_{Z'} = 0.012 M_{Z'}$. This is the most sensitive search for a narrow$\ensuremath{q\bar{q}}$-initiated $\ensuremath{t\bar{t}}$ resonance in the massregion below 750 GeV/$c^2$.

Search for Resonant Top-antitop Production in the Semi-leptonic Decay Mode Using the Full CDF Data Set