Search for the Neutral Current Top Quark Decay t-->Zc Using the Ratio of Z-Boson + 4 Jets to W-Boson + 4 Jets Production

We have used the Collider Detector at Fermilab (CDF II) to search for the flavor-changing neutral-current (FCNC) top quark decay ${t\goes Zc}$ using a technique employing ratios of $W$ and $Z$ production, measured in \ppbar data corresponding to an integrated luminosity of 1.52 $fb^{-1}$. The analysis uses a comparison of two decay chains, ${p\bar{p}\goes t\bar{t}\goes WbWb\goes\ell\nu bjjb}$ and ${p\bar{p}\goes t\bar{t}\goes ZcWb\goes\ell\ell cjjb}$, 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 $t\bar{t}$ 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 $Br({t\goes Zc})$ 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 assumptions regarding the theoretical top quark pair production cross-section.