We measure the forward--backward asymmetry of the production of top quark and antiquark pairs in proton-antiproton collisions at center-of-mass energy $sqrt{s} = 1.96~mathrm{TeV}$ using the full data set collected by the Collider Detector at Fermilab (CDF) in Tevatron Run II corresponding to an integrated luminosity of $9.1~rm{fb}^{-1}$. The asymmetry is characterized by the rapidity difference between top quarks and antiquarks ($Delta y$), and measured in the final state with two charged leptons (electrons and muons). The inclusive asymmetry, corrected to the entire phase space at parton level, is measured to be $A_{text{FB}}^{tbar{t}} = 0.12 pm 0.13$, consistent with the expectations from the standard-model (SM) and previous CDF results in the final state with a single charged lepton. The combination of the CDF measurements of the inclusive $A_{text{FB}}^{tbar{t}}$ in both final states yields $A_{text{FB}}^{tbar{t}}=0.160pm0.045$, which is consistent with the SM predictions. We also measure the differential asymmetry as a function of $Delta y$. A linear fit to $A_{text{FB}}^{tbar{t}}(|Delta y|)$, assuming zero asymmetry at $Delta y=0$, yields a slope of $alpha=0.14pm0.15$, consistent with the SM prediction and the previous CDF determination in the final state with a single charged lepton. The combined slope of $A_{text{FB}}^{tbar{t}}(|Delta y|)$ in the two final states is $alpha=0.227pm0.057$, which is $2.0sigma$ larger than the SM prediction.
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