$\mathrm{Sp}(6)$ Unifying Deconstructed $\mathrm{SU}(2)$'s

Abstract

Understanding the origin of flavour hierarchies in the Standard Model remains an open problem, motivating extensions with non-trivial flavour symmetries. We unify deconstructed weak isospin $\mathrm{SU}(2)_\mathrm{L}^3$ into an $\mathrm{Sp}(6)_\mathrm{L}$ symmetry at a high scale $v_S$. The three generations of Standard Model (SM) left-handed doublets are unified into a single fundamental representation of $\mathrm{Sp}(6)_\mathrm{L}$. In addition to two BSM triplets from the breaking of $\mathrm{SU}(2)_\mathrm{L}^3$, the enlarged symmetry predicts six additional gauge bosons below the unification scale: three $\mathrm{SU}(2)_\mathrm{L}$ triplets and three singlets, which induce flavour transitions in both the quark and lepton sectors, even in the presence of mass-gauge alignment. We derive updated bounds on the intermediate breaking scale $v_{12}$ of $\mathrm{SU}(2)_\mathrm{L}^3$ in the presence of the unification scale $v_S$, mapping exclusions in the $(v_{12}, r)$ parameter space with $r = v_S^2 / v_{12}^2$. The most stringent constraints arise from precision flavour observables involving first- and second-generation transitions, including neutral meson mixing ($K^0 - \bar{K}^0$, $D^0 - \bar{D}^0$), $\mu \to 3e$ and $\mu \to e$ conversion in nuclei. These measurements probe scales well beyond direct collider reach and imply $v_{12} \gtrsim 550 ~\mathrm{TeV}$ for small $r \approx 1$, while for $r\gtrsim100$ the bound relaxes to $v_{12} \gtrsim 150~\mathrm{TeV}$, recovering the limits of the $\mathrm{SU}(2)_\mathrm{L}^3$ model. Additionally we include projections from Mu3e and COMET-I and -II experiments which show promising further reach into the parameter space.