Large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>t</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>c</mml:mi><mml:mi>Z</mml:mi></mml:math> as a sign of vectorlike quarks in light of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inl

The rare flavour changing top quark decay $t\to cZ$ is a clear sign of new physics and experimentally very interesting due to the huge number of top quarks produced at the LHC. However, there are few (viable) models which can generate a sizable branching ratio for $t\to cZ$ -- in fact vector-like quarks seem to be the only realistic option. In this paper, we investigate all three representations (under the Standard Model gauge group) of vector-like quarks ($U$, $Q_1$ and $Q_7$) that can generate a sizable branching ratio for $t\to cZ$ without violating bounds from $B$ physics. Importantly, these are exactly the three vector-like quarks which can lead to a sizable positive shift in the prediction for $W$ mass, via the couplings to the top quark also needed for a sizable Br($t\to cZ$). Calculating and using the one-loop matching of vector-like quarks on the Standard Model Effective Field Theory, we find that Br($t\to cZ$) 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.