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The rare flavour changing top quark decay $t\to cZ$ is a clear sign of newphysics and experimentally very interesting due to the huge number of topquarks produced at the LHC. However, there are few (viable) models which cangenerate a sizable branching ratio for $t\to cZ$ -- in fact vector-like quarksseem to be the only realistic option. In this paper, we investigate all threerepresentations (under the Standard Model gauge group) of vector-like quarks($U$, $Q_1$ and $Q_7$) that can generate a sizable branching ratio for $t\tocZ$ without violating bounds from $B$ physics. Importantly, these are exactlythe three vector-like quarks which can lead to a sizable positive shift in theprediction for $W$ mass, via the couplings to the top quark also needed for asizable Br($t\to cZ$). Calculating and using the one-loop matching ofvector-like quarks on the Standard Model Effective Field Theory, we find thatBr($t\to cZ$) can be of the order of $10^{-6}$, $10^{-5}$ and $10^{-4}$ for$U$, $Q_1$ and $Q_7$, respectively and that in all three cases the large $W$mass measurement can be accommodated.

Large $t\to cZ$ as a Sign of Vector-Like Quarks in Light of the $W$ Mass