Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alastair Gosnay, David J Miller

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 ($K^0 - \bar{K}^0$, $D^0 - \bar{D}^0$), $μ\to 3e$ and $μ\to e$ 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.

$\mathrm{Sp}(6)$ 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 into an symmetry at a high scale . The three generations of Standard Model (SM) left-handed doublets are unified into a single fundamental representation of . In addition to two BSM triplets from the breaking of , the enlarged symmetry predicts six additional gauge bosons below the unification scale: three 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 of in the presence of the unification scale , mapping exclusions in the parameter space with . The most stringent constraints arise from precision flavour observables involving first- and second-generation transitions, including neutral meson mixing (, ), and conversion in nuclei. These measurements probe scales well beyond direct collider reach and imply for small , while for the bound relaxes to , recovering the limits of the model. Additionally we include projections from Mu3e and COMET-I and -II experiments which show promising further reach into the parameter space.

Paper Structure

This paper contains 25 sections, 152 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Model
  3. Gauge Group and Field Content
  4. Antisymmetric Scalars
  5. Symmetry Breaking Pattern
  6. SSB (i)
  7. SSB (ii)-(iii)
  8. Mass eigenstate Gauge Fields
  9. Gauge-fermion interactions
  10. Gauge-Higgs interactions
  11. Summary
  12. SMEFT Matching
  13. Four-Fermion Operators
  14. Higgs-Fermion Operators
  15. Phenomenology
  16. ...and 10 more sections

Figures (7)

  • Figure 1: Breaking pattern from $\mathrm{Sp}(6)_\text{L}$ down to a flavour universal $\mathrm{SU}(2)_\text{L}$. Marked on the right are the lowest orders of magnitude of scales allowed by observation as determined in Davighi:2023xqn.
  • Figure 2: The mass of the $W_{23}$ as a function of $r$. The mass $M_{W_{23}}/g_\text{SM} v_{23} = 3/\sqrt{2}$ as predicted by the $\mathrm{SU}(2)_\text{L}^3$ model Davighi:2023xqn is marked with the grey dashed (horizontal) line.
  • Figure 3: Exclusion on the parameter space (at $95\%$ C.L) from $K$ and $D$ meson mixing. Here $u$ and $d$ aligned refer to the choice of $V_u$ (or $V_d$) as defined in (\ref{['eq:Vu_def']}), with $V_u = \mathbb{I}$ referring to $u$ aligned and $V_u = V$ referring to $d$ aligned.
  • Figure 4: Constraints on the $(v_{23}, \alpha)$ parameter space with $v_{12}=10 v_{23}$ for current bound $\text{BR} \lesssim 1 \times 10^{-12}$ParticleDataGroup:2024cfk and target future bound $\text{BR} \lesssim 1 \times 10^{-16}$Hesketh:2022wgw Figure \ref{['fig:alpha_ckm']} shows the current and future bounds with choice of $V_l$ and hierarchy (\ref{['eq:Vl_1']}), meanwhile \ref{['fig:alpha_other']} shows the same bounds with $V_l$ and hierarchy (\ref{['eq:Vl_2']}).
  • Figure 5: Exclusion on the parameter space from $\text{BR}(\mu \rightarrow 3e)$. Figure \ref{['fig:Mu3e_current']} shows the current bound $\text{BR}<1\times10^{-12}$ParticleDataGroup:2024cfk, meanwhile \ref{['fig:Mu3e_future']} shows the target precision of the proposed Mu3e experiment $\text{BR} \lesssim 1.0 \times 10^{-16}$Hesketh:2022wgw All plots are drawn with $v_{23} = 25\text{TeV}$.
  • ...and 2 more figures