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Example of relevant processes for the LHC limits on our $SU(2)_f$ model for both the $\Zt$ and $\Wt$ boson coupling to first and second generations only.

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: LHC limits on our $SU(2)_f$ model with $12_Q$ and $12_\ell$ flavour alignment for (a) scenario (LH) (b) scenario (RH) (c) scenario (M1) and (d) scenario (M2). Aquamarine regions indicate di-jets driven limits~\cite{ATLAS:2019fgd,CMS:2019gwf,ATLAS:2018qto,CMS:2016ltu,ATLAS:2019itm,CMS:2019emo,CMS:2019xai}, the blue regions are for di-muon limits~\cite{CMS:2019buh,CMS:2021ctt} and the brown exclusions from LFV $e\mu$ search~\cite{CMS:2022fsw}. The orange exclusion regions at low mass are recast from LFV $H$ and $Z$ decay searches~\cite{ATLAS:2019old,ATLAS:2021bdj,CMS:2023pte}. We overlaid the limit from pure flavour-transfer processes in dashed grey: $K_L \to \mu e$~\cite{BNL:1998apv}, $CR(\mu \to e)_{Au}$ ~\cite{SINDRUMII:2006dvw} and $K\to \pi \nu_e \nu_\mu$~\cite{NA62:2021zjw}. : Caption not extracted

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: LHC limits on our $SU(2)_f$ model for the scenario (LH) with (a) $(12)_Q$ and $(13)_\ell$ flavour alignment (b) $(23)_Q$ and $(23)_\ell$ flavour alignment. Aquamarine regions indicate di-jets driven limits~\cite{ATLAS:2019fgd,CMS:2019gwf,ATLAS:2018qto,CMS:2016ltu,ATLAS:2019itm,CMS:2019emo,CMS:2019xai}, the blue regions are for di-muon limits~\cite{CMS:2019buh,CMS:2021ctt} and the brown exclusions from LFV $ e\tau$ (a) and $\mu\tau$ searches (b) from~\cite{CMS:2022fsw}. The orange exclusion regions at low mass are recast from LFV $H$ and $Z$ decay searches~\cite{CMS:2021rsq,ATLAS:2022uhq}. The purple exclusion at low masses is from the earlier $e^+e^-$ colliders as obtained in~\cite{Leike:1998wr}. We overlaid selected limits from pure flavour-transfer processes in dashed grey: $\tau \to e K_S$~\cite{Belle:2010rxj}, $K\to \pi \nu_e \nu_\mu$~\cite{NA62:2021zjw}, $B\to K^* \nu_\tau \nu_\mu$~\cite{Belle:2017oht} and $D^0$ oscillations~\cite{ParticleDataGroup:2020ssz}. : Caption not extracted

: : Constraints on the $(12)_Q (12)_\ell$ scenario, with a focus on the small mass region. The green points pass the full flavour fit described in Sec.~\ref{sec:flavourobs} at $2 \sigma$. The blue regions are for di-muon limits~\cite{CMS:2019buh,CMS:2021ctt} and the brown exclusions from LFV $e\mu$ search~\cite{CMS:2022fsw}. The orange, purple and red exclusion regions at low mass are respectively obtained from recasted $Z\to e \mu$ ATLAS search~\cite{ATLAS:2021bdj}, recasted $H\to e \mu$ ATLAS search~\cite{ATLAS:2019old} and a $X\to e \mu$ dedicated NP search at CMS~\cite{CMS:2023pte}.

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: Overview of the distribution of the scan points passing the global fit within $1\sigma$ of the SM $\chi^2$ from \superiso~ as a function of the leptonic angles $\thLmutau$ and $\thLetau$ (a) and of the quark angles $\thQdb$ and $\thQsb$ (b). The colour coding refers to the ratio $M_V/g_f$ with blue points corresponding to lower NP scales. : Caption not extracted

: Parameter points passing all intensity frontier limits at $1\sigma$ in our global fit (green points) as a function of $M_V/ g_f$ on the x-axis and of the parameter: (a) $\theta_{\ell 13}$ , (b) $\theta_{\ell 23}$, (c) $\theta_{Q 13}$ , (d) $\theta_{Q 23}$ on the y-axis. We fix the quark rotation angles as indicated on the plots. : Caption not extracted

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LHC and intensity frontier limits on our scenario (LH) with $12_Q$ and $12_\ell$ flavour alignments as a function of the scale $M_V / g_f$. Aquamarine regions indicate di-jets driven limits~\cite{ATLAS:2019fgd,CMS:2019gwf,ATLAS:2018qto,CMS:2016ltu,ATLAS:2019itm,CMS:2019emo,CMS:2019xai}, the blue regions are for di-muon limits~\cite{CMS:2019buh,CMS:2021ctt} and the brown exclusions from the LFV $e\mu$ search~\cite{CMS:2022fsw}. The orange and red exclusion regions at low mass are re-interpreted from LFV $H$ and $Z$ decay searches~\cite{ATLAS:2019old,ATLAS:2021bdj,CMS:2023pte}. Filled region indicates current constraints, while dashed lines are simple rescaling projections at HL-LHC full luminosity of 3~$\textrm{ab}^{-1}$. We overlaid the points passing all intensity frontier limits at $1\sigma$ in our global fit with the cut $M_V / g_f < 600 $ TeV.

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: : Same as in the previous plots, but with (a) $(12)_Q$ and $(13)_\ell$ flavour alignment (b) $(23)_Q$ and $(23)_\ell$ flavour alignment. We overlaid the points passing all intensity frontier limits at $1\sigma$ in our global fit with the cut $M_V / g_f < 600 $ TeV.

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