$B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background

Lepton number as a fourth color is the intriguing theoretical idea of the famous Pati-Salam (PS) model. While in conventional PS models, the symmetry breaking scale and the mass of the resulting vector leptoquark are stringently constrained by $K_L\to \mu e$ and $K\to \pi \mu e$, the scale can be lowered to a few TeV by adding vectorlike fermions. Furthermore, in this case, the intriguing hints for lepton flavor universality violation in $b\to s{\mu }^{+}{\mu }^{-}$ and $b\to c\tau \nu$ processes can be addressed. Such a setup is naturally achieved by implementing the PS gauge group in the five-dimensional Randall-Sundrum background. The PS symmetry is broken by boundary conditions on the fifth dimension, and the resulting massive vector leptoquark automatically has the same mass scale as the vectorlike fermions and all other resonances. We consider the phenomenology of this model in the context of the hints for lepton flavor universality violation in semileptonic $B$ decays. Assuming flavor alignment in the down sector, we find that in $b\to s{\ell }^{+}{\ell }^{-}$ transitions, the observed deviations from the standard model predictions [including $R(K)$ and $R(K^{*})]$ can be explained with natural values for the free parameters of the model. Even though we find sizable effects in $R(D)$, $R(D^{*})$, and $R(J/\mathrm{\Psi })$, one cannot account for the current central values in the constrained setup of our minimal model due to the stringent constraints from $D-\overline{D}$ mixing and $\tau \to 3\mu$.

Figure 1

Left: Allowed regions from $b\to s{\mu }^{+}{\mu }^{-}$ (red) and the exclusion limits from $D-\overline{D}$ mixing (blue) and $\tau \to 3\mu$ (gray) for $M=3\mathrm{TeV}$, $s_2^{\ell }=0.2$, $s_3^{\ell }=1/\sqrt{2}$, and $s_3^q=\sqrt{3}/2$. With these values, $R(X)/R(X{)}_{\mathrm{SM}}\approx 1.07$ (with $X=D$, $D^{*}$, $J/\mathrm{\Psi }$). We see that $b\to s{\mu }^{+}{\mu }^{-}$ can be explained at the $1\sigma$ level without violating the bounds from other observables. Right: Correlations between $R(X)/R(X{)}_{\mathrm{SM}}$ and $b\to s{\mu }^{+}{\mu }^{-}$ for $M=3\mathrm{TeV}$. Here, we scanned over $0.3<s_3^q<\sqrt{3}/2$, $0<s_2^q<0.2$, $0.3<s_3^{\ell }<\sqrt{3}/2$, $0<s_2^{\ell }<0.2$, and $0<s_{\ell }<0.3$. Only the parameter points consistent with $D-\overline{D}$ mixing are shown. As we can see, the predicted branching ratio for $\tau \to 3\mu$ is large and very well within the reach of Belle II.