Natural Suppression of FCNCs at the One-Loop Level in $Z^{\mathrm{\prime }}$ Models with Implications for K, D and B Decays

We analyse $Z^{\mathrm{\prime }}$ contributions to FCNC processes at the one-loop level. In analogy to the CKM matrix we introduce two $3\times 3$ unitary matrices ${\widehat{\mathrm{\Delta }}}_d(Z^{\mathrm{\prime }})$ and ${\widehat{\mathrm{\Delta }}}_u(Z^{\mathrm{\prime }})$ which are also hermitian. They govern the flavour interactions mediated by $Z^{\mathrm{\prime }}$ between down-quarks and up-quarks, respectively, with ${\widehat{\mathrm{\Delta }}}_d(Z^{\mathrm{\prime }})={\widehat{\mathrm{\Delta }}}_u(Z^{\mathrm{\prime }})\equiv {\widehat{\mathrm{\Delta }}}_L(Z^{\mathrm{\prime }})$ for left-handed currents due to the unbroken ${\text{SU(2)}}_L$ gauge symmetry. This assures the suppression of these contributions to all $Z^{\mathrm{\prime }}$ mediated FCNC processes at the one-loop level. As, in contrast to the GIM mechanism, one-loop $Z^{\mathrm{\prime }}$ contributions to flavour observables in $K$ and $B_{s,d}$ systems are governed by down-quark masses, they are $\mathcal{O}(m_b^2\mathrm{/}{M}_{Z^{\mathrm{\prime }}}^2)$ and negligible. With the $\mathcal{O}(m_t^2\mathrm{/}{M}_{Z^{\mathrm{\prime }}}^2)$ suppression they are likely negligible also in the $D$ system. We present an explicit parametrization of ${\widehat{\mathrm{\Delta }}}_L(Z^{\mathrm{\prime }})$ in terms of two mixing angles and two complex phases that distinguishes it profoundly from the CKM matrix. This framework can be generalized to purely leptonic decays with matrices analogous to the PMNS matrix but profoundly different from it. Interestingly, the breakdown of flavour universality between the first two generations and the third one, both for quark and lepton couplings to $Z^{\mathrm{\prime }}$, is identified as a consequence of ${\widehat{\mathrm{\Delta }}}_L(Z^{\mathrm{\prime }})$ being hermitian. The importance of the unitarity for both ${\widehat{\mathrm{\Delta }}}_L(Z^{\mathrm{\prime }})$ and the CKM matrix in the light of the Cabibbo anomaly is emphasized.