$Z^{\prime }$-Tandem Mechanism for the Suppression of New Physics in Quark Mixing with Implications for K, D and B Decays

Abstract

$Z^{\prime }$ models belong to the ones that can most easily explain the anomalies in $b\to s{\mu }^{+}{\mu }^{-}$ transitions. However, such an explanation by a single $Z^{\prime }$ gauge boson, as done in the literature, is severly constrained by the $B_s^0-{\bar{B}}_s^0$ mixing. Also the recent finding, hat the mass differences $\Delta M_s$, $\Delta M_d$, the CP-violating parameter ${\epsilon }_K$, and the mixing induced CP-asymmetries $S_{\psi K_S}$ and $S_{\psi \varphi }$ can be simultaneously well described within the SM without new physics (NP) contributions, is a challenge for $Z^{\prime }$ models with a single $Z^{\prime }$ contributing at tree-level to quark mixing. We point out that including a second $Z^{\prime }$ in the model allows to eliminate simultaneously tree-level contributions to the five $\Delta F=2$ observables used in the determination of the CKM parameters while leaving the room for NP in $\Delta M_K$ and $\Delta M_D$. The latter one can be removed at the price of infecting $\Delta M_s$ or $\Delta M_d$ by NP which is presently disfavoured. This pattern is transparently seen using the new mixing matrix for $Z^{\prime }$ interactions with quarks. This strategy allows significant tree-level contributions to $K$, $B_s$ and $B_d$ decays thereby allowing to explain the existing anomalies in $b\to s{\mu }^{+}{\mu }^{-}$ transitions and the anticipated anomaly in the ratio ${\epsilon }^{\prime }/\epsilon$ much easier than in $Z^{\prime }$-Single scenarios. The proposed $Z^{\prime }$-Tandem mechanism bears some similarities to the GIM mechanism for the suppression of the FCNCs in the SM with the role of the charm quark played here by the second $Z^{\prime }$. However, it differs from the latter profoundly in that only NP contributions to quark mixing are eliminated at tree-level. We discuss briefly the implied flavour patterns in $K$ and $B$ decay observables in this NP scenario.

Comments: 10 pages, no figures. V2: the use of the new mixing matrix for $Z^{\prime }$ interactions allows a much more transparent description of this tandem and provides new insights into this framework