The remarkable role of the vector-like quark doublet in the Cabibbo angle and $W$-mass anomalies

Abstract: The combination of precise determinations of $V_{us}$ and $V_{ud}$ hints towards a violation of the CKM first row unitarity at about $3\sigma$ level. Conversely, the recent measurement of the $W$-boson mass by the CDF Collaboration exhibits significant tension with the SM prediction, intensifying the conflict that may arise in models addressing the unitarity violation. We demonstrate that one vector-like $SU(2)L$ doublet with mass of a few TeV mixing with light quarks and the top quark can simultaneously account for the two anomalies, without conflicting with flavour-changing phenomena and electroweak observables. Moreover, another tension in the value of the Cabibbo angle is reported also at $3\sigma$ level, between two determinations of $V{us}$ obtained from semi-leptonic $K\ell3$ and leptonic $K\mu 2$ kaon decays. We show that the vector-like doublet can be at the origin of this discrepancy and the substantial positive shift in the $W$-boson mass. This unique feature of the vector-like quark doublet may render it a crucial puzzle piece in new physics scenarios addressing the CKM unitarity problem. The model can be potentially probed in future colliders.