We have used the Collider Detector at Fermilab (CDF II) to search for the flavor-changing neutral-current (FCNC) top quark decay t --> Zc using a technique employing ratios of W and Z production, measured in p anti-p data corresponding to an integrated luminosity of 1.52 fb-1. The analysis uses a comparison of two decay chains, p anti-p --> tt-bar --> WbWb --> el nu bjjb and p anti-p --> tt-bar --> ZcWb --> ell ell c jjb, to cancel systematic uncertainties in acceptance, efficiency, and luminosity. We validate the modeling of acceptance and efficiency for lepton identification over the multi-year dataset using another ratio of W and Z production, in this case the observed ratio of inclusive production of W to Z bosons. To improve the discrimination against standard model backgrounds to top quark decays, we calculate the top quark mass for each event with two leptons and four jets assuming it is a tt-bar event with one of the top quarks decaying to Zc. For additional background discrimination we require at least one jet to be identified as originating from a b-quark. No significant signal is found, and we set an upper limit on the FCNC branching ratio Br(t --> Zc) using a likelihood constructed from the ell ell cjjb top quark mass distribution and the number of ell nu bjjb events. Limits are set as a function of the helicity of the Z boson produced in the FCNC decay. For 100% longitudinally polarized Z bosons we find limits of 8.3% and 9.3% (95% C.L.) depending on the assumption regarding the theoretical top quark pair production cross-section.
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