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Direct 95~\% C.L. reach on heavy singlet mixed with the SM Higgs doublet at various muon colliders (adapted from~\cite{Buttazzo:2018qqp}). The direct and indirect reach at other future colliders~\cite{EuropeanStrategyforParticlePhysicsPreparatoryGroup:2019qin} is also shown for comparison.

Direct (left panel) and indirect reach (right panel) on the SM plus real scalar singlet scenario for muon colliders with various center of mass energy. Dots indicate points with successful FOEWPT, while red, green and blue dots represent signal-to-noise ratio (SNR) for gravitational eave detection of $[50, +\infty)$, $[10, 50)$ and $[0, 10)$, respectively. Results are taken from \cite{Liu:2021jyc}.

Direct (left panel) and indirect reach (right panel) on the SM plus real scalar singlet scenario for muon colliders with various center of mass energy. Dots indicate points with successful FOEWPT, while red, green and blue dots represent signal-to-noise ratio (SNR) for gravitational eave detection of $[50, +\infty)$, $[10, 50)$ and $[0, 10)$, respectively. Results are taken from \cite{Liu:2021jyc}.

Cross sections versus the non-SM Higgs mass for $\sqrt{s}=3$ TeV for pair production (left panel), single production with a pair of fermions and radiative return production (right panel) for $\tan\beta=1$ under the alignment limit of $\cos(\alpha-\beta)=0$. Plot is produced by authors of Ref.~\cite{Han:2021udl}.

Cross sections versus the non-SM Higgs mass for $\sqrt{s}=3$ TeV for pair production (left panel), single production with a pair of fermions and radiative return production (right panel) for $\tan\beta=1$ under the alignment limit of $\cos(\alpha-\beta)=0$. Plot is produced by authors of Ref.~\cite{Han:2021udl}.

Signal statistical significance for various IDM benchmark points~\cite{Klamka:2022ukx} at high energy lepton collider for charged Higgs pair production and semi-leptonic final states.

Signal statistical significance for various IDM benchmark points~\cite{Klamka:2022ukx} at high energy lepton collider for charged Higgs pair production and semi-leptonic final states.

Direct reach on electroweak states in mono-$X$ signals. Left: Luminosity needed to exclude a Dirac fermion DM candidate for zero systematics~\cite{Han:2020uak} for $X=\gamma$ (solid), $X=\mu$ (dotted), $X=\mu\mu$ (dashed). Right: Mass reach on a fermionic DM candidate (assumed Majorana when $Y=0$, Dirac otherwise) at fixed 1~$\iab$ luminosity for the 3 TeV muon collider for $X=\gamma$ and $X=W$ for 0.1\% systematics~\cite{Bottaro:2021snn,Bottaro:2022}.

Minimal luminosity to exclude a thermal pure higgsino or wino dark matter (left panel) a 2.0 TeV Dirac triplet or 4.8 TeV Dirac 4-plet as function of the collider center of mass energy~\cite{RFXZ} (hyper-charge of the higgsino and Dirac $n$-plets not taken into account). Lighter color lines are for polarized beams. The thickness of the Wino and Dirac 4-plet bands covers the uncertainty on the thermal mass calculations. Diagonal lines mark the precision at which the total rate of the labeled channels are going to be measured. The shaded area indicates that at least one channel is going to be measured with 0.1\% uncertainty and systematic uncertainties need to be evaluated.

Minimal luminosity to exclude a thermal pure higgsino or wino dark matter (left panel) a 2.0 TeV Dirac triplet or 4.8 TeV Dirac 4-plet as function of the collider center of mass energy~\cite{RFXZ} (hyper-charge of the higgsino and Dirac $n$-plets not taken into account). Lighter color lines are for polarized beams. The thickness of the Wino and Dirac 4-plet bands covers the uncertainty on the thermal mass calculations. Diagonal lines mark the precision at which the total rate of the labeled channels are going to be measured. The shaded area indicates that at least one channel is going to be measured with 0.1\% uncertainty and systematic uncertainties need to be evaluated.

Expected sensitivity using 1~ab$^{-1}$ of 3~TeV $\mu^{+}\mu^{-}$ collision data as a function of the \chipm mass and mass difference with the lightest neutral state, assuming a mass-splitting equal to 344~MeV, as per a pure-higgsino scenario~\cite{Capdevilla:2021fmj}.

Expected number of events for different processes at a muon collider, as a function of the centre-of-mass energy, for integrated luminosities $L=10~\!{\rm ab}^{-1}(E_{\rm cm} [{\rm TeV}]/10~\!{\rm ~\!TeV})^2$.

Sensitivity to modified Higgs couplings in the $\kappa$ framework. We show the marginalized 68\% probability reach for each coupling modifier. For the 125 GeV muon collider, light (dark) shades correspond to a luminosity of 5 (20) fb$^{-1}$.

Global fit to the EFT operators in the Lagrangian (\ref{eq:LSilh}). We show the marginalized 68\% probability reach for each Wilson coefficient $c_i/\Lambda^2$ in Eq.~(\ref{eq:LSilh}) from the global fit (solid bars). The reach of the vertical ``T'' lines indicate the results assuming only the corresponding operator is generated by the new physics. \jb{UPDATED}

(Left) Comparison of the global reach for universal composite Higgs models at the HL-LHC and a high-energy muon collider (combined with the HL-LHC constraints). The figure compares the 2-$\sigma$ exclusion regions in the $(g_\star, m_\star)$ plane from the fit presented in Figure~\ref{fig:SILH}, using the SILH power-counting described in Eq.~(\ref{eq:SILHpc}) (Right) The same for a BSM extension with a massive replica of the $U(1)_Y$ gauge boson in the $(g_{Z^\prime}, m_{Z^\prime})$ plane from the fit presented in Figure~\ref{fig:SILH}.

(Left) Comparison of the global reach for universal composite Higgs models at the HL-LHC and a high-energy muon collider (combined with the HL-LHC constraints). The figure compares the 2-$\sigma$ exclusion regions in the $(g_\star, m_\star)$ plane from the fit presented in Figure~\ref{fig:SILH}, using the SILH power-counting described in Eq.~(\ref{eq:SILHpc}) (Right) The same for a BSM extension with a massive replica of the $U(1)_Y$ gauge boson in the $(g_{Z^\prime}, m_{Z^\prime})$ plane from the fit presented in Figure~\ref{fig:SILH}.

(Left) 2-$\sigma$ exclusion regions in the $(g_\star, m_\star)$ plane from the fit presented in Figure~\ref{fig:SILH}, using the SILH power-counting described in Eq.~(\ref{eq:SILHpc}) and below (solid regions). The solid and dashed lines denote the contributions to the constraints from different processes. The results correspond to the combination of the HL-LHC with the 3 TeV muon collider. (Right) The same for the 10 TeV muon collider.

(Left) 2-$\sigma$ exclusion regions in the $(g_\star, m_\star)$ plane from the fit presented in Figure~\ref{fig:SILH}, using the SILH power-counting described in Eq.~(\ref{eq:SILHpc}) and below (solid regions). The solid and dashed lines denote the contributions to the constraints from different processes. The results correspond to the combination of the HL-LHC with the 3 TeV muon collider. (Right) The same for the 10 TeV muon collider.

{\it Upper row:} Feynman diagrams contributing to the leptonic $g$-2 up to one-loop order in the Standard Model EFT. {\it Lower row:} Feynman diagrams of the corresponding high-energy scattering processes. Dimension-6 effective interaction vertices are denoted by a square.

Reach on the muon anomalous magnetic moment $\Delta a_\mu$ and muon EDM $d_\mu$, as a function of the $\mc$ collider center-of-mass energy $\sqrt{s}$, from the labeled processes. Figure taken from~\cite{Buttazzo:2020ibd}.

Singlet models for $g$-2 and their probes at different masses, assuming 100\% branching ratio to di-muons (top) and the minimum branching ratio to di-muon allowed by perturbativity \cite{Capdevilla:2021kcf}.

Singlet models for $g$-2 and their probes at different masses, assuming 100\% branching ratio to di-muons (top) and the minimum branching ratio to di-muon allowed by perturbativity \cite{Capdevilla:2021kcf}.

Singlet models for $g$-2 and their probes at different masses, assuming 100\% branching ratio to di-muons (top) and the minimum branching ratio to di-muon allowed by perturbativity \cite{Capdevilla:2021kcf}.

Singlet models for $g$-2 and their probes at different masses, assuming 100\% branching ratio to di-muons (top) and the minimum branching ratio to di-muon allowed by perturbativity \cite{Capdevilla:2021kcf}.

{\it Left:} Cross sections for $hh$ (cyan) and $hhh$ (green) production as a function of $\sqrt{s}$ in models with VLF. {\it Right:} Cross sections for $hh$ (left axis) and $hhh$ (right axis) production as a function of $\tan\beta$ in models with VLF and 2HDM for $M_{L,E}\simeq m_{H,A,H^{\pm}}$. The dot-dashed and dashed lines correspond to the predictions corresponding to the central value of $\Delta a_{\mu}$ and $m_{H,A,H^{\pm}}=3\times M_{L,E}$ and $m_{H,A,H^{\pm}}=5\times M_{L,E}$, respectively. Both panels assume $\Delta a_{\mu}$ is within $1\sigma$ of the measured value (shaded ranges)~\cite{Dermisek:2021mhi}.

Sensitivity reach (at 95\% CL) for the $(\bar{s}_L \gamma_\alpha b_L) (\bar{\mu}_L \gamma^\alpha \mu_L)$ (top) and $(\bar{b}_L \gamma_\alpha b_L) (\bar{\mu}_L \gamma^\alpha \mu_L)$ (bottom) contact interactions as a function of the upper cut on the final-state invariant mass for various $\mc$, HL-LHC, FCC-hh, and the present LHC bounds. These are compared with values required to fit $b \to s \mu^+ \mu^-$ anomalies without (dashed orange line) or with (dotted orange line) a flavor enhancement of the $bb$ operator compared to the $bs$ one. For the bottom plot solid (dashed) lines represent the limit for positive (negative) values of $C_{bb\mu\mu}$. The gray area represents a region where the EFT bounds are not valid (for a strongly coupled UV completion, for weakly coupled ones the area is larger).

Sensitivity reach (at 95\% CL) for the $(\bar{s}_L \gamma_\alpha b_L) (\bar{\mu}_L \gamma^\alpha \mu_L)$ (top) and $(\bar{b}_L \gamma_\alpha b_L) (\bar{\mu}_L \gamma^\alpha \mu_L)$ (bottom) contact interactions as a function of the upper cut on the final-state invariant mass for various $\mc$, HL-LHC, FCC-hh, and the present LHC bounds. These are compared with values required to fit $b \to s \mu^+ \mu^-$ anomalies without (dashed orange line) or with (dotted orange line) a flavor enhancement of the $bb$ operator compared to the $bs$ one. For the bottom plot solid (dashed) lines represent the limit for positive (negative) values of $C_{bb\mu\mu}$. The gray area represents a region where the EFT bounds are not valid (for a strongly coupled UV completion, for weakly coupled ones the area is larger).

{\it Left:} Sensitivities to the $Z'$ model with $\lambda^{\rm L}_{22} = 1$ (upper panel) and $\lambda^{\rm L}_{22} = \sqrt{4\pi}$ (lower panel) via $\mu^{+}\mu^{-} \to b \bar s$ at a $\mc$ with $\sqrt{s} = 3,6,10~{\rm TeV}$ (red, blue, green). Other limits include the neutrino trident production~\cite{Altmannshofer:2014pba}, LHC~\cite{Allanach:2015gkd}, HL-LHC~\cite{delAguila:2014soa}, and $B^{}_{s}$ mixing~\cite{DiLuzio:2019jyq}. {\it Right:} Sensitivities to the LQ model via $\mu^{+}\mu^{-} \to b \bar s$ at a $\mc$ for scalar (upper) and vector (lower) LQ. Figures from Ref.~\cite{Huang:2021biu}.

The $5\sigma$ discovery reach of 3 (14) TeV $\mc$ with 1 (20) ab$^{-1}$ of data. The reach is calculated using the flavor scenario described in Eq.~\ref{eq:betas}. The straight-line boundary of the pair-production channel corresponds to pure EW production, and is therefore independent of $\beta_L$. Figure taken from Ref.~\cite{Asadi:2021gah}.

The $5\sigma$ discovery reach of 3 (14) TeV $\mc$ with 1 (20) ab$^{-1}$ of data. The reach is calculated using the flavor scenario described in Eq.~\ref{eq:betas}. The straight-line boundary of the pair-production channel corresponds to pure EW production, and is therefore independent of $\beta_L$. Figure taken from Ref.~\cite{Asadi:2021gah}.

Summary of $\mc$ and low-energy constraints on flavor-violating 3-body lepton decays. The colored horizontal lines show the sensitivity to the $\tau 3\mu$ operator at various energies, all assuming $1\text{ ab}^{-1}$ of data. The dashed horizontal (vertical) lines show the current or expected sensitivity from $\tau \to 3\mu$ ($\mu \to 3e$) decays for comparison. The diagonal black lines show the expected relationship between different Wilson coefficients with various ansatz for the scaling of the flavor-violating operators (e.g., ``Anarchy'' assumes that all Wilson coefficients are $\mathcal{O}(1)$).

Constraints on lepton flavor violation in the MSSM in the $\Delta m^2 / \bar{m}^2$ vs. $\sin 2\theta_R$ plane (left) and the $\sin 2\theta_R$ vs. $M_1$ plane (right) from measurements of the slepton pair production process with flavor-violating final states (red band) at a 3 TeV $\mc$, assuming $1\,\textrm{ab}^{-1}$ of luminosity. The width of the band represents the uncertainty on the reach from the measurement of the slepton and neutralino masses in flavor-conserving channels. The purple and blue shaded lightly shaded regions indicate parameters preferred in Gauge-Mediated Supersymmetry Breaking scenarios and flavor-dependent mediator scenarios, respectively. Both plots assume a mean slepton mass of $1\,\textrm{TeV}$. In the left plot we fix the neutralino mass $M_1 = 500\,\textrm{GeV}$, while in the right figure $\Delta m^2 / \bar{m}^2$ is fixed to 0.1. The current (solid) and expected (dashed, dotted) limits from low-energy lepton flavor violation experiments are indicated by the blue, purple and green lines.

Constraints on lepton flavor violation in the MSSM in the $\Delta m^2 / \bar{m}^2$ vs. $\sin 2\theta_R$ plane (left) and the $\sin 2\theta_R$ vs. $M_1$ plane (right) from measurements of the slepton pair production process with flavor-violating final states (red band) at a 3 TeV $\mc$, assuming $1\,\textrm{ab}^{-1}$ of luminosity. The width of the band represents the uncertainty on the reach from the measurement of the slepton and neutralino masses in flavor-conserving channels. The purple and blue shaded lightly shaded regions indicate parameters preferred in Gauge-Mediated Supersymmetry Breaking scenarios and flavor-dependent mediator scenarios, respectively. Both plots assume a mean slepton mass of $1\,\textrm{TeV}$. In the left plot we fix the neutralino mass $M_1 = 500\,\textrm{GeV}$, while in the right figure $\Delta m^2 / \bar{m}^2$ is fixed to 0.1. The current (solid) and expected (dashed, dotted) limits from low-energy lepton flavor violation experiments are indicated by the blue, purple and green lines.

The sensitivity of the $\mc$ with the COM energy $\sqrt{s}=3~{\rm TeV}$ and luminosity $L=1~{\rm ab^{-1}}$, given as orange regions. Other limits and projections are also shown for comparison. Our concerned parameter regions explaining the $(g-2)^{}_{\mu}$ and $B$ anomalies are given as yellow and blue bands, respectively. Figure from Ref.~\cite{Huang:2021nkl}.

The kinematic distributions $\theta_B,R_{BB},M_{3B}(B=Z,H)$ of $ZZH$ (left) and $ZZZ$ (right) production at a $\sqrt{s}=10$ TeV $\mc$.

The kinematic distributions $\theta_B,R_{BB},M_{3B}(B=Z,H)$ of $ZZH$ (left) and $ZZZ$ (right) production at a $\sqrt{s}=10$ TeV $\mc$.

The kinematic distributions $\theta_B,R_{BB},M_{3B}(B=Z,H)$ of $ZZH$ (left) and $ZZZ$ (right) production at a $\sqrt{s}=10$ TeV $\mc$.

The kinematic distributions $\theta_B,R_{BB},M_{3B}(B=Z,H)$ of $ZZH$ (left) and $ZZZ$ (right) production at a $\sqrt{s}=10$ TeV $\mc$.

The kinematic distributions $\theta_B,R_{BB},M_{3B}(B=Z,H)$ of $ZZH$ (left) and $ZZZ$ (right) production at a $\sqrt{s}=10$ TeV $\mc$.

The kinematic distributions $\theta_B,R_{BB},M_{3B}(B=Z,H)$ of $ZZH$ (left) and $ZZZ$ (right) production at a $\sqrt{s}=10$ TeV $\mc$.

(a) A high-energy MC's statistical sensitivity contour $\mathcal{S}=2$ to probe the muon-Higgs coupling $\kappa_\mu$ based on the measurement of three-boson production. (b) The probe of new physics scale with the assumption in Eq.~(\ref{eq:k2L}).

(a) A high-energy MC's statistical sensitivity contour $\mathcal{S}=2$ to probe the muon-Higgs coupling $\kappa_\mu$ based on the measurement of three-boson production. (b) The probe of new physics scale with the assumption in Eq.~(\ref{eq:k2L}).

Recoil mass distribution for heavy Higgs mass of 0.5, 1, 1.5, 2, 2.5, 2.9 TeV with a total width 1 (red), 10 (blue), and 100 (green) GeV at a 3 TeV $\mc$. ISR and FSR are included in this calculation. Background (black shaded region) includes all events with a photon of $p_T>10~$GeV. Note that signal and background have different re-scale factors for clarity. This figure is obtained from~\cite{Chakrabarty:2014pja} and more detailed discussion can be found there.

Comparison of sensitivities between different production mechanisms in the parameter plane $\kappa_\mu$\nobreakdash-$\kappa_Z$ for different masses of the heavy Higgs boson at the $3~\TeV$ $\mc$. The shaded regions show a higher direct signal rate from the RR process than the $ZH$ associated production and $HA$ pair production channels. One can also see the allowed parameter regions (extracted from Ref.~\cite{Barger:2013ofa}). This figure is obtained from~\cite{Chakrabarty:2014pja} and more detailed discussion can be found there.

\small Limits on $1/\Lambda$ scale for the dark-photon as a function of the dark-photon mass $m_{DP}$: for SN the scale of the coupling to muons has been set at $10^{7.4}$ GeV \cite{Bollig:2020xdr} by the effect of dark radiation on Supernovae dynamics. For CMB see~\cite{DEramo:2018vss}. For $g-2$ see~\cite{Pospelov:2008zw,Escudero:2019gzq}. Comparable bounds hold for the ALP to muons because of the similar structure of the interaction vertex. For masses up to 100 GeV the $\mu$Collider limits are for all practical purposes mass independent.

\small Limits on $g_{a\gamma}=4/ \Lambda$ as a function of the ALP mass $m_a$: NA64a~\cite{NA64:2020qwq}, Delphi~\cite{DELPHI:2008uka} and BaBar~\cite{BaBar:2017tiz} are actual limits. Belle-II~\cite{Belle-II:2018jsg,Dolan:2017osp}, NA64b~\cite{NA64:2020qwq} and $\mu$Collider~\cite{Casarsa:2021rud} are future estimates. The limit indicated by E137 is the one from \cite{Bjorken:1988as} as modified for a small ($10^{-4}$) visible branching fraction~\cite{Darme:2020sjf}. For masses up to 100 GeV the $\mu$Collider limits are for all practical purposes mass independent.