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One-loop contributions to $\g$ from vector (left) and scalar (right) singlets.

{\bf Left:} Parameter space for which one SM singlet scalar or vector particle resolves the $\g$ anomaly. The thickness of each band represents the $\pm 2\sigma$ confidence interval and the vertical axis is the corresponding muon-singlet coupling from \Eq{singlet-couplings}. The vertical shaded region represents the bound on light relativistic species present in equilibrium during big bang nucleosynthesis. The horizontal shaded region is the bound from partial wave unitarity. For a discussion of these bounds, see Section \ref{massrange}. {\bf Right:} Minimum di-muon branching fraction for vector and scalar couplings that resolve the $\g$ anomaly, from Eqs.~(\ref{branch-min-vector}) and~(\ref{branch-min-scalar}).

{\bf Left:} Parameter space for which one SM singlet scalar or vector particle resolves the $\g$ anomaly. The thickness of each band represents the $\pm 2\sigma$ confidence interval and the vertical axis is the corresponding muon-singlet coupling from \Eq{singlet-couplings}. The vertical shaded region represents the bound on light relativistic species present in equilibrium during big bang nucleosynthesis. The horizontal shaded region is the bound from partial wave unitarity. For a discussion of these bounds, see Section \ref{massrange}. {\bf Right:} Minimum di-muon branching fraction for vector and scalar couplings that resolve the $\g$ anomaly, from Eqs.~(\ref{branch-min-vector}) and~(\ref{branch-min-scalar}).

{\bf Top:} Limits and projections on muon-philic vector (left) and scalar (right) singlets, assuming only di-muon decays where kinematically allowed. The green/orange bands represent the parameter space that resolves $\g$. Existing experimental limits are shaded in gray (Supernova constraints, not shown, can probe scalar masses up to 20 MeV and couplings up to $4\times10^{-3}$ \cite{Caputo:2021rux}). Projections are indicated with colored lines. The $M^3$ \cite{Kahn:2018cqs}, NA64$\mu$ \cite{Gninenko:2018ter}, and ATLAS fixed-target \cite{Galon:2019owl} experiments probe invisibly-decaying singlets; projections here assume a 100\% invisible branching fraction (see Sec.~\ref{s.ft}). The BABAR limits and Belle II projections are computed following the procedure described in Sec.~\ref{s.bfac}. The LHC limits and HL-LHC projections in the $3\mu$/$4\mu$ channels, along with the mass range disfavored by UV completions for scalar singlets, are discussed in Sec.~\ref{s.colliders} and \aref{appendixB}. The purple muon collider projections are based on proposed analyses~\cite{Capdevilla:2021rwo} reviewed in Sec. \ref{MuC}. For scalar singlets whose width is determined entirely by the muon coupling (top right), we also show projections for a $S \to \gamma\gamma$ beam dump search \cite{Chen:2017awl} on the minimal assumption that the scalar-photon coupling arises from integrating out the muon as discussed in Sec. \ref{beam-dump}. {\bf Bottom:} Same as the top row, only here we assume that for $m_{S,V} > 2m_\mu$, the singlets have the {\it minimum} di-muon branching fraction consistent with unitarity using {\rm Eqs.}~(\ref{branch-min-vector}) and (\ref{branch-min-scalar}). The curves which are unaffected by this change of muonic branching fraction correspond to searches that are insensitive to the singlet's decay modes. Projections for $M^3$, NA64$\mu$, and ATLAS fixed-target experiments assume a $\simeq 100\%$ invisible branching fraction for $m_{S/V} > 2m_\mu$, which is model-dependent.

{\bf Top:} Limits and projections on muon-philic vector (left) and scalar (right) singlets, assuming only di-muon decays where kinematically allowed. The green/orange bands represent the parameter space that resolves $\g$. Existing experimental limits are shaded in gray (Supernova constraints, not shown, can probe scalar masses up to 20 MeV and couplings up to $4\times10^{-3}$ \cite{Caputo:2021rux}). Projections are indicated with colored lines. The $M^3$ \cite{Kahn:2018cqs}, NA64$\mu$ \cite{Gninenko:2018ter}, and ATLAS fixed-target \cite{Galon:2019owl} experiments probe invisibly-decaying singlets; projections here assume a 100\% invisible branching fraction (see Sec.~\ref{s.ft}). The BABAR limits and Belle II projections are computed following the procedure described in Sec.~\ref{s.bfac}. The LHC limits and HL-LHC projections in the $3\mu$/$4\mu$ channels, along with the mass range disfavored by UV completions for scalar singlets, are discussed in Sec.~\ref{s.colliders} and \aref{appendixB}. The purple muon collider projections are based on proposed analyses~\cite{Capdevilla:2021rwo} reviewed in Sec. \ref{MuC}. For scalar singlets whose width is determined entirely by the muon coupling (top right), we also show projections for a $S \to \gamma\gamma$ beam dump search \cite{Chen:2017awl} on the minimal assumption that the scalar-photon coupling arises from integrating out the muon as discussed in Sec. \ref{beam-dump}. {\bf Bottom:} Same as the top row, only here we assume that for $m_{S,V} > 2m_\mu$, the singlets have the {\it minimum} di-muon branching fraction consistent with unitarity using {\rm Eqs.}~(\ref{branch-min-vector}) and (\ref{branch-min-scalar}). The curves which are unaffected by this change of muonic branching fraction correspond to searches that are insensitive to the singlet's decay modes. Projections for $M^3$, NA64$\mu$, and ATLAS fixed-target experiments assume a $\simeq 100\%$ invisible branching fraction for $m_{S/V} > 2m_\mu$, which is model-dependent.

{\bf Top:} Limits and projections on muon-philic vector (left) and scalar (right) singlets, assuming only di-muon decays where kinematically allowed. The green/orange bands represent the parameter space that resolves $\g$. Existing experimental limits are shaded in gray (Supernova constraints, not shown, can probe scalar masses up to 20 MeV and couplings up to $4\times10^{-3}$ \cite{Caputo:2021rux}). Projections are indicated with colored lines. The $M^3$ \cite{Kahn:2018cqs}, NA64$\mu$ \cite{Gninenko:2018ter}, and ATLAS fixed-target \cite{Galon:2019owl} experiments probe invisibly-decaying singlets; projections here assume a 100\% invisible branching fraction (see Sec.~\ref{s.ft}). The BABAR limits and Belle II projections are computed following the procedure described in Sec.~\ref{s.bfac}. The LHC limits and HL-LHC projections in the $3\mu$/$4\mu$ channels, along with the mass range disfavored by UV completions for scalar singlets, are discussed in Sec.~\ref{s.colliders} and \aref{appendixB}. The purple muon collider projections are based on proposed analyses~\cite{Capdevilla:2021rwo} reviewed in Sec. \ref{MuC}. For scalar singlets whose width is determined entirely by the muon coupling (top right), we also show projections for a $S \to \gamma\gamma$ beam dump search \cite{Chen:2017awl} on the minimal assumption that the scalar-photon coupling arises from integrating out the muon as discussed in Sec. \ref{beam-dump}. {\bf Bottom:} Same as the top row, only here we assume that for $m_{S,V} > 2m_\mu$, the singlets have the {\it minimum} di-muon branching fraction consistent with unitarity using {\rm Eqs.}~(\ref{branch-min-vector}) and (\ref{branch-min-scalar}). The curves which are unaffected by this change of muonic branching fraction correspond to searches that are insensitive to the singlet's decay modes. Projections for $M^3$, NA64$\mu$, and ATLAS fixed-target experiments assume a $\simeq 100\%$ invisible branching fraction for $m_{S/V} > 2m_\mu$, which is model-dependent.

{\bf Top:} Limits and projections on muon-philic vector (left) and scalar (right) singlets, assuming only di-muon decays where kinematically allowed. The green/orange bands represent the parameter space that resolves $\g$. Existing experimental limits are shaded in gray (Supernova constraints, not shown, can probe scalar masses up to 20 MeV and couplings up to $4\times10^{-3}$ \cite{Caputo:2021rux}). Projections are indicated with colored lines. The $M^3$ \cite{Kahn:2018cqs}, NA64$\mu$ \cite{Gninenko:2018ter}, and ATLAS fixed-target \cite{Galon:2019owl} experiments probe invisibly-decaying singlets; projections here assume a 100\% invisible branching fraction (see Sec.~\ref{s.ft}). The BABAR limits and Belle II projections are computed following the procedure described in Sec.~\ref{s.bfac}. The LHC limits and HL-LHC projections in the $3\mu$/$4\mu$ channels, along with the mass range disfavored by UV completions for scalar singlets, are discussed in Sec.~\ref{s.colliders} and \aref{appendixB}. The purple muon collider projections are based on proposed analyses~\cite{Capdevilla:2021rwo} reviewed in Sec. \ref{MuC}. For scalar singlets whose width is determined entirely by the muon coupling (top right), we also show projections for a $S \to \gamma\gamma$ beam dump search \cite{Chen:2017awl} on the minimal assumption that the scalar-photon coupling arises from integrating out the muon as discussed in Sec. \ref{beam-dump}. {\bf Bottom:} Same as the top row, only here we assume that for $m_{S,V} > 2m_\mu$, the singlets have the {\it minimum} di-muon branching fraction consistent with unitarity using {\rm Eqs.}~(\ref{branch-min-vector}) and (\ref{branch-min-scalar}). The curves which are unaffected by this change of muonic branching fraction correspond to searches that are insensitive to the singlet's decay modes. Projections for $M^3$, NA64$\mu$, and ATLAS fixed-target experiments assume a $\simeq 100\%$ invisible branching fraction for $m_{S/V} > 2m_\mu$, which is model-dependent.

Radiative singlet production via coherent muon-nucleus scattering at muon beam fixed target experiments $M^3$ \cite{Kahn:2018cqs}, NA64$\mu$ \cite{Gninenko:2018ter}, and proposed beam dump searches \cite{Chen:2017awl}.

Representative Feynman diagrams that yield singlet scalar (left) and vector (right) production at a $B$-factory via $e^+e^-$ annihilation. The BABAR and Belle II search strategies discussed in Sec. \ref{s.bfac} involve the $4\mu$ channel in which $S/V \to \mu^+\mu^-$ decays yield $4\mu$ final states with an opposite-sign di-muon resonance that reconstructs the singlet's mass.

Singlet production at a hadron collider via the neutral ($W$-mediated) and charged ($\gamma/Z$-mediated) Drell-Yan processes.

Singlet production at a hadron collider via the neutral ($W$-mediated) and charged ($\gamma/Z$-mediated) Drell-Yan processes.

LHC production cross sections for scalar and vector singlets in $pp \to 3\mu$ (left) and $pp \to 4\mu$ (right) final states that include $S/V \to \mu^+\mu^-$ contributions assuming $100\%$ branching ratios to di-muons. In all cases we assume $\sqrt{s} = 13$ TeV and the singlet couplings are chosen to resolve the $\g$ anomaly as shown in Fig. \ref{f.g-2contours}. The dashed gray line in each plot corresponds to the SM background prediction in each channel.

LHC production cross sections for scalar and vector singlets in $pp \to 3\mu$ (left) and $pp \to 4\mu$ (right) final states that include $S/V \to \mu^+\mu^-$ contributions assuming $100\%$ branching ratios to di-muons. In all cases we assume $\sqrt{s} = 13$ TeV and the singlet couplings are chosen to resolve the $\g$ anomaly as shown in Fig. \ref{f.g-2contours}. The dashed gray line in each plot corresponds to the SM background prediction in each channel.

LHC luminosity required to exclude vector (left) and scalar (right) singlet models at 2 sigma via a resonant singlet search in the 3+4$\mu$ channel, for singlet couplings set to resolve the $(g-2)_\mu$ anomaly. The upper dashed curves corresponds to the minimum singlet branching fraction to muons, see Eqs.~(\ref{branch-min-vector}) and~(\ref{branch-min-scalar}). The lower solid curves correspond to singlets decaying entirely to muons.

LHC luminosity required to exclude vector (left) and scalar (right) singlet models at 2 sigma via a resonant singlet search in the 3+4$\mu$ channel, for singlet couplings set to resolve the $(g-2)_\mu$ anomaly. The upper dashed curves corresponds to the minimum singlet branching fraction to muons, see Eqs.~(\ref{branch-min-vector}) and~(\ref{branch-min-scalar}). The lower solid curves correspond to singlets decaying entirely to muons.

The singlet scalar scenario requires a UV completion to generate the dimension-5 operator $ \frac{1}{\Lambda} S H^\dagger L \mu^c$ (left), see \eref{singlet-yukawa-higher-dim}. There are three possibilities to generate this operator via tree-level exchange of a mediator field, shown on the right. Assuming a minimal particle content for the additional fields, this corresponds to integrating out a fermion singlet ($t$-channel), a fermion doublet ($t$-channel), or a scalar doublet ($s$-channel).

SM expectation for the ratio $R_{\mu e}$ and its \emph{minimum} deviation due to muon mixing if a singlet scalar model UV completed by fermion mediators resolves the $(g-2)\mu$ anomaly. The gray solid line represents the leading order-calculation, and the gray dashed line with shaded band shows the experimental result from LEP with an error of $0.3\%$~\cite{ParticleDataGroup:2020ssz}. The solid (dashed) green line represents the fermion UV completion $\mathcal{L}_I$ ($\mathcal{L}_{II}$) from Eq.~(\ref{e.uv_lag}).

Constraints on the parameter space of the scalar UV completion for the singlet scalar model. \emph{Note that $m_S, m_\Phi$ are now mass parameters in $\mathcal{L}_{III}$ (Eq.~(\ref{e.uv_lag})), not mass eigenvalues.} The parameter $\xi = \kappa v/m_S m_\Phi $ controls the mixing between the singlet $S$ and the scalar doublet mediator $\Phi$. At each point in this parameter space, the coupling $y$ is chosen so that $a_\mu^{\rm BSM} = \Delta a_\mu$. The bottom-right region requires $y$ couplings that violate unitarity. For large values of $\xi$, the trilinear coupling $\kappa$ violates perturbative unitarity. The region below the black solid line is excluded by EWPT as described in the text. The regions to the left of the dashed (solid) red lines would be excluded by a conservative search in the 3+4$\mu$ channels at the (HL-)LHC, see text. The blue (orange) contours represent the mass of the $\varphi_1$ eigenstate that is mostly the singlet $S$ (the coupling of this eigenstate to muons), corresponding to the parameters $m_S$ ($g_S$) in the singlet scalar effective theory~(\ref{singlet-couplings}).

Constraints on the parameter space of the scalar UV completion for the singlet scalar model. \emph{Note that $m_S, m_\Phi$ are now mass parameters in $\mathcal{L}_{III}$ (Eq.~(\ref{e.uv_lag})), not mass eigenvalues.} The parameter $\xi = \kappa v/m_S m_\Phi $ controls the mixing between the singlet $S$ and the scalar doublet mediator $\Phi$. At each point in this parameter space, the coupling $y$ is chosen so that $a_\mu^{\rm BSM} = \Delta a_\mu$. The bottom-right region requires $y$ couplings that violate unitarity. For large values of $\xi$, the trilinear coupling $\kappa$ violates perturbative unitarity. The region below the black solid line is excluded by EWPT as described in the text. The regions to the left of the dashed (solid) red lines would be excluded by a conservative search in the 3+4$\mu$ channels at the (HL-)LHC, see text. The blue (orange) contours represent the mass of the $\varphi_1$ eigenstate that is mostly the singlet $S$ (the coupling of this eigenstate to muons), corresponding to the parameters $m_S$ ($g_S$) in the singlet scalar effective theory~(\ref{singlet-couplings}).