Minimal model for the $W$-boson mass, $(g-2)_\mu$, $h\to\mu^+\mu^-$ and quark-mixing-matrix unitarity

The $SU(2)_L$ triplet scalar with hypercharge $Y=0$ predicts a positive definite shift in the $W$ mass, w.r.t.~the Standard Model prediction, if it acquires a vacuum expectation value. As this new field cannot couple directly to SM fermions (on its own), it has no significant impact on other low-energy precision observables and is weakly constrained by collider searches. In fact, the multi-lepton anomalies at the LHC even point towards new scalars that decay dominantly to $W$ bosons, as the neutral component of the triplet naturally does. In this article, we show that with a minimal extension of the scalar triplet model by a heavy vector-like lepton, being either I) an $SU(2)_L$ doublet with $Y=-1/2$ or II) an $SU(2)_L$ triplet with $Y=-1$, couplings of the triplet to Standard Model leptons are possible. This minimal extension can then provide, in addition to the desired positive shift in the $W$ mass, a chirally enhanced contribution to $(g-2)_\mu$. In addition version I) and II) can improve on $Z\to\mu^+\mu^-$ and alleviate the tension in first-row CKM unitarity (known as the Cabibbo angle anomaly), respectively. Finally, both options, in general, predict sizable changes of $h\to\mu^+\mu^-$, i.e.,~much larger than most other $(g-2)_\mu$ explanations where only $O(\%)$ effects are expected, making this channel a smoking gun signature of our model.