$B$ decay anomalies from non-Abelian local horizontal symmetry

Recent anomalies in $B\to K^{(*)}\ell \ell$ meson decays are consistent with exchange of a heavy $Z^{\prime }$ vector boson. Here we try to connect such new physics to understanding the origin of flavor, by gauging generation number. Phenomenological and theoretical considerations suggest that the smallest viable flavor symmetry [not including any extra U(1) factors] is chiral ${\mathrm{SU}(3)}_L\times {\mathrm{SU}(3)}_R$, which acts only on generation indices and does not distinguish between quarks and leptons. Spontaneous breaking of the symmetry gives rise to the standard model Yukawa matrices, and masses for the 16 $Z^{\prime }$-like gauge bosons, one of which is presumed to be light enough to explain the $B\to K^{(*)}\ell \ell$ anomalies. We perform a bottom-up study of this framework, showing that it is highly constrained by LHC dilepton searches, meson mixing, $Z$ decays and Cabibbo-Kobayashi-Maskawa unitarity. Similar anomalies are predicted for semileptonic decays of $B$ to lighter mesons, with excesses in the $ee$, $\tau \tau$ channels and deficits in $\mu \mu$, but no deviation in $\nu \nu$. The lightest $Z^{\prime }$ mass is $\lesssim 6\mathrm{TeV}$ if the gauge coupling is $\lesssim 1$.

Figure 1

$1\sigma$ and $3\sigma$ bounds given by the measurements of $R_{K^{*}}$, $R_K$ and the branching fraction of $B_s\to \mu \mu$ in the ($C_{b_L{\mu }_L}$, $C_{b_L{e}_L}$) plane. The line corresponds to the trajectory of the contributions of the $Z^{\prime }$ to these Wilson coefficients in our model, where we have indicated benchmark points of $m_{Z^{\prime }}/g_L$, whereas the embedded plot shows the projected ${\chi }^2$. The absolute minimum in the plane, indicated by a red cross corresponds to ${\chi }_{\min }^2=2.64$.

Figure 2

Predictions and experimentally allowed ranges for the neutral meson mixing parameters (23), (24).

Figure 3

Loop contribution to neutral meson mixing.

Figure 4

Production cross section for $Z^{\prime }$ times branching ratio into muons, versus $m_{Z^{\prime }}$, as predicted by model and as constrained by ATLAS [86]. The rescaling of the limit by 1.8 corrects for the low branching ratio into $ee$ in our model; see text.