In this case, SM singlets that could be below the GeV scale (or much heavier) generate the new one-loop contributions to (g-2)μ. The singlet could either be a scalar, vector, or fermion. Our focus will be the case of a new real scalar S or vector V. The relevant Lagrangian terms for the real scalar case are LS⊃-(gSSμLμc+H.c.)-12mS2S2.(5)Note that the Yukawa coupling of the real scalar to muons gS is not gauge invariant. This implies that either the interaction arises from the nonrenormalizable operator 1ΛcSμLμcHS, in which case gS∝v/(2Λ), or the interaction comes from a singlet-Higgs mixing, in which case gS∼yμsinθ, where θ is the mixing angle. We briefly discuss the consequences of consistent embedding in the full electroweak theory in Sec. III. For the vector case, the relevant Lagrangian terms are LV⊃gVVα(μL†σ¯αμL+μcσαμc†)+mV22VαVα.(6)These two scenarios are representative of muophilic new gauge forces or scalars that have been extensively studied in the literature [39,77–79] and their contributions to (g-2)μ are shown in Fig. 3.

As discussed in Sec. III, the only viable anomaly-free vector model is gauged Lμ-Lτ, which can still resolve (g-2)μ for mV∈(10  MeV,2mμ) [73,80]. Bounds on muonphilic singlet scalars are more model dependent and can, in principle, resolve (g-2)μ with any mass between the MeV scale and the perturbative unitarity limit ∼ few TeV. For both scalars and vectors, the lower limit is set by cosmological constraints, most importantly bounds on ΔNeff, the effective number of relativistic species at big bang nucleosynthesis [73,81]. Thus, the scalar singlet scenario will be of most interest to us, but we keep the vector case in our discussions for completeness since the analyses are very similar.

These singlet scenarios are the most minimal BSM solutions to the (g-2)μ anomaly, featuring new particles required only to couple to the muon and no other SM particles. Consequently, muon colliders and muon-beam fixed-target experiments might be the only guaranteed way to probe all singlet scenarios. Given that fixed-target experiments and B factories will exhaustively probe singlet scenarios with masses below ∼GeV [39,66–73], we will particularly focus on singlet scenarios above the GeV scale in our muon collider physics analyses.

Of course, it is possible that more than one new degree of freedom contributes to (g-2)μ. We account for this possibility by considering NBSM≥1 copies of each SM singlet scenario in Eqs. (5) or (6), and analyzing how the various higher-energy signatures scale with BSM multiplicity. Note that the assumption that all NBSM copies of the simplified model have equal masses and couplings is the most pessimistic one with regards to high-energy signatures, since nondegenerate masses and couplings always lead to larger signatures due to the nonlinearity of the associated cross sections and amplitudes. If couplings or masses are highly unequal, the phenomenology will be dominated by just a few new states. Considering degenerate NBSM≥1 copies therefore covers the signature space of possibilities.

In principle, one could also consider the case of a neutral fermion N contributing to (g-2)μ. This would essentially be a right-handed-neutrino-type scenario (see e.g., [82] for a review), where the new (g-2)μ contribution consists of a loop of a W boson and the neutral N that mixes with the muon neutrino. However, in the presence of a unitary neutrino mixing matrix, such contributions would cancel up to corrections of order ∼(mν/mW)2, which are inadequate to explain Δaμobs. We therefore restrict our focus to scalar and vector singlets.

Finally, we point out a peculiar but interesting edge case. It is possible to define versions of these singlet models where the virtual fermion in Fig. 3(top) is replaced by a virtual tau lepton. This appears quite pathological, since they lead to large charged-lepton flavor-violating tau decays unless singlet couplings to two muons [as in Eqs. (5) and (6)] are completely negligible or absent. However, insofar as they are a theoretical possibility, we classify them as part of the electroweak scenarios that we discuss below, since the contribution to Δaμ is enhanced by the larger tau mass. Their experimental probes at muon colliders is discussed in Sec. IV H.

We now move on to discuss the most general class of BSM solutions to the (g-2)μ anomaly, electroweak scenarios. This includes an overwhelmingly large number of possibilities, but fortunately, we do not need to study all of them. To perform the maximization over all of BSM theory space in Eq. (4), we merely need to study those models which are guaranteed to give the largest possible BSM mass scales. This will be sufficient to model-exhaustively determine the heaviest possible mass for new charged states.

Which EW scenarios maximize the BSM mass scale? Consider the most general new one-loop diagrams that could contribute to (g-2)μ. To make sure the relevant masses and couplings are maximally unconstrained, we consider the cases where all fields in the loop are BSM fields. Furthermore, the chirality flip and the Higgs VEV insertion necessary to generate Eq. (3) should both come from these BSM fields to avoid additional suppression by the small muon Yukawa. The minimal ingredients are, therefore, (1)

at least 3 BSM fields, either two bosons and one fermion or one boson and two fermions;

As in our analysis for singlet scenarios, our default focus is on the most experimentally pessimistic case in which these new BSM states only couple to the SM through their muonic (and gauge) interactions. We find that scenarios with new vectors generate smaller Δaμ contributions than the analogous scenario with a new scalar, and likewise for Majorana fermions or real scalars. Since this results in a lower BSM mass scale that would be easier to probe, we focus on EW scenarios with new complex scalars and vectorlike fermions only. This leaves just two classes of models, which we label SSF and FFS by their field content.

The SSF simplified model is defined by two complex scalars ΦA, ΦB in SU(2)L representations RA, RB with hypercharges YA, YB and a single vectorlike fermion pair F(Fc) in SU(2)L representation R(R¯) with hypercharge Y (-Y): LSSF⊃-y1FcL(μ)ΦA*-y2FμcΦB-κHΦA*ΦB-mA2|ΦA|2-mB2|ΦB|2-mFFFc+H.c.(7)Here y1, y2 are new Yukawa couplings and κ is a trilinear coupling with dimensions of mass. L(μ)=(νL,μL) and μc are the two 2-component second-generation SM lepton fields, and H is the Higgs doublet. A typical SSF contribution to (g-2)μ is shown in Fig. 3(b). Note that the chirality flip comes from the heavy vectorlike fermion F while the Higgs VEV insertion arises due to mixing of the new scalars.

The FFS simplified model is analogously defined but reverses the role of fermions and scalars, featuring two vectorlike fermion pairs FA, FB (FAc,FBc) in SU(2)L representations RA, RB (R¯A,R¯A) with hypercharges YA, YB (-YA,-YB) and a single complex scalar S in SU(2)L representation R with hypercharge Y: LFFS⊃-y1FAcL(μ)Φ*-y2FBμcΦ-y12HFAcFB-y12′H†FAFBc-mAFAFAc-mBFBFBc-mS2|Φ|2+H.c.(8)There are now two renormalizable Yukawa couplings y12,y12′ which control the mixing of the A and B fermions via the Higgs. A typical FFS contribution to (g-2)μ is shown in Fig. 3(c). The chirality flip and Higgs VEV insertion both arise in the loop due to the Higgs couplings of the new fermions.

These two simplified models generate the largest possible BSM particle masses that could account for Δaμobs. Therefore, the maximization over theory space in Eq. (4) can be replaced by a maximization over the SSF and FFS parameter spaces: MBSM,chargedmax≡maxSSF,FFS models{mini∈BSM spectrum(mcharged(i))}.(9)Note that one could in principle consider extensions of the SM Higgs sector with additional scalar contributing to EWSB. In that case, the κ and y1,2 terms in the above Lagrangians could arise from coupling to these new scalars rather than a SM-like Higgs doublet, which might change the allowed EW representations of the BSM states. However, current constraints already dictate that most of the observed EWSB arises from the VEV of a single doublet [84,85], which means that relying only on BSM scalars to generate the required EWSB insertions in the above Lagrangians would lead to smaller effective mixings and hence smaller Δaμ and BSM masses. We therefore do not have to consider such extended scenarios to perform the maximization of the lightest new charged particle mass over BSM theory space.

In both SSF and FFS models, the choices of representations must satisfy 1⊂RA⊗R⊗2,RB=R¯,YA=-12-Y,YB=-1-Y,(10)with Y chosen to make the electric charges integer valued. We will explore all choices of representations involving SU(2)L singlets, doublets, and triplets, and all choices of Y that ensure that all electric charges satisfy |Q|≤2. As we discuss, this is sufficient to perform the above maximization. The possibility of a high multiplicity of new BSM states is again taken into account by considering the trivial generalizations where there are NBSM identical copies of the above fields contributing to Δaμ.

The Lagrangians in Eqs. (7) and (8) only show the interactions necessary to form new one-loop contributions to (g-2)μ. Depending on the choice of SU(2)L⊗U(1)Y representations, additional couplings between the new fermions/scalars and the muon or Higgs may be allowed by gauge invariance. However, these couplings will not contribute to (g-2)μ at leading order, at most supplying a small correction to the leading terms generated by the couplings in Eqs. (7) and (8), or slightly modifying the mass spectrum of the fermions/scalars that couple to the Higgs after EWSB by ≲TeV, which does not meaningfully affect our results or discussion. We can therefore neglect these additional couplings in our analysis. We also assume that the new BSM states do not couple to any other SM fermions (except when discussing leptonic flavor violation bounds). Both of these assumptions are conservative in that they minimize additional experimental signatures arising from the new physics responsible for the (g-2)μ anomaly.

Depending on the choice of representations, some of the EW scenarios we consider were previously studied in Refs. [45–48,86–88]. There have also been previous attempts to define simplified model dictionaries for generating Δaμobs [41,42,45,47,48,89–92], but none took our completely model-exhaustive approach and none aimed to find the highest possible mass of new BSM charged states that could account for Δaμobs. We also make no assumptions about e.g., the existence of a viable DM candidate, or any couplings of the new degrees of freedom that are not required for resolving the (g-2)μ anomaly (except optionally considering flavor). Other possible simplified models for (g-2)μ, such as adding fewer than three new BSM particles with nontrivial EW representations (see e.g., [41]), require smaller masses for the new charged particles than the SSF and FFS models, and their inclusion does not affect the outcome of the maximization over theory space of Eq. (4). We demonstrate this explicitly in Sec. IV H.

The size of the (g-2)μ contribution is controlled by BSM couplings and masses, and the largest possible BSM masses that can account for the anomaly depend on the largest possible BSM couplings. In Sec. II C 1 we describe first how perturbative unitarity supplies an absolute upper bound on the new couplings. This will inform our baseline analysis, but more careful consideration of how these simplified models must arise as part of a more complete BSM theory suggests that an upper bound based on unitarity alone is likely far too conservative, especially in light of stringent CLFV bounds. In Secs. II C 2 and II C 3 we therefore consider the additional constraints on the new muon couplings arising by assuming either MFV or requiring the absence of large, explicitly calculable new tunings.

The analysis of singlet and EW scenarios is discussed in detail in Secs. III and IV. In each scenario, the viable parameter space of BSM masses and couplings is compact, since we require the new states to explain the (g-2)μ anomaly, and the couplings cannot exceed the limit set by perturbative unitarity, or unitarity+MFV, or unitarity+naturalness. Therefore, each scenario has well-defined maximum BSM particle masses for a given BSM multiplicity NBSM. We then analyze the signatures of these models at future muon colliders. The details are slightly different for singlet and EW scenarios due to their different collider signatures.

Singlet scenarios feature new SM singlets which can be invisible. Lighter singlets are more weakly coupled to account for the (g-2)μ anomaly, so the scenarios with the heaviest BSM particles are not necessarily the hardest to discover. Furthermore, the sensitivity of collider searches can depend on whether the new singlets are stable or how they decay. We therefore have to map out the complete parameter space of the simplified singlet scenarios. Fortunately, with the muon coupling gS,V determined by the requirement of accounting for the observed Δaμobs, the model has just two parameters, singlet mass mS,V and multiplicity NBSM (as well as the choice of singlet being a scalar or vector). As a function of mass and multiplicity we then analyze the sensitivity of a completely inclusive search for the production of the BSM singlets at muon colliders regardless of their decays. We also analyze the reach of an indirect search based on deviations in Bhabha scattering to explore the physics potential of a muon collider Higgs factory. We find that the singlet BSM states cannot be heavier than about 3 TeV, and can be directly discovered at a 3 TeV muon collider with 1  ab-1 for masses ≳10  GeV in singlet+photon production processes. A 215 GeV muon collider that might be used as a Higgs factory can directly discover singlets as light as 2 GeV in our conservative inclusive analysis with 0.4  ab-1 of luminosity. Heavier singlets up to the 3 TeV maximum can be probed with Bhabha scattering.

The parameter space of the SSF and FFS simplified models that allow us to perform the EW scenario theory space maximization of BSM charged particle mass in Eq. (4) is much more complex, featuring three masses, several BSM couplings, the number of BSM flavors NBSM, and the choice of EW gauge representations for the BSM states. However, since we only need to find the heaviest possible BSM masses, for each SSF/FFS model with a given choice of NBSM and EW gauge representation we can simply find the boundaries of the parameter space defined by the maximum possible BSM masses that still allow BSM couplings below the unitarity (or unitarity+MFV/naturalness) limit to account for the (g-2)μ anomaly.

For EW scenarios we find that requiring only perturbative unitarity allows the lightest charged states to sit at the 100 TeV scale,¹¹

Leptoquark (LQ) models provide one example that realizes these extremal cases. LQ models that couple to the first generations of fermions are constrained by rare meson decays to have masses ≳O(100  TeV), depending on their particular flavor structure [101]. If these LQ models couple to muons and the top quark, then they can explain the (g-2)μ anomaly even for such heavy masses [102].

It is worth noting that at the very boundaries of the BSM parameter spaces we explore, with couplings set at the upper limit set by perturbative unitarity, the theory itself strictly speaking has already lost predictivity, by definition. If the couplings actually had this value, we would have to regard the theory as a strongly coupled one, requiring different analysis tools. This is suitable for deriving upper bounds on the BSM mass scale, but it is interesting to note these bounds could actually be saturated by strongly coupled BSM solutions to the (g-2)μ anomaly (which would still have to feature new states with EW gauge charges). One feature of composite theories is a large multiplicity of states, which we include by considering NBSM>1, with NBSM=10 serving as a “high-multiplicity benchmark” for our analyses. Therefore, while our quantitative predictions are unlikely to apply precisely to strongly coupled BSM solutions of the (g-2)μ anomaly, by including couplings up to the unitarity limit and considering large numbers of BSM flavors we parametrically include the signature space swept out by these strongly coupled theories. The statements we make about the discoverability of new physics should, broadly speaking, apply to those scenarios as well. That being said, it would be interesting to undertake a dedicated investigation of high-scale composite BSM solutions to the (g-2)μ anomaly within our framework. We leave this for future work.

While CLFV constraints strongly favor the existence of some kind of flavor protection mechanism, the degree to which the precise assumptions of MFV would have to be satisfied is obviously up for debate. Similarly, the precise degree of tuning depends on the tuning measure, and it is difficult to define exactly at what point a theory becomes “un-natural” in a meaningful sense. However, our model-exhaustive approach has the advantage of throwing these issues into stark relief: solving the (g-2)μ anomaly with BSM masses up to ∼10  TeV is apparently relatively “easy,” while pushing the masses of new states to the maximum 100 TeV scale limited only by unitarity appears to require some extreme form of tuning and violation of MFV while somehow suppressing CLFV decays.

In particular, if the 10 TeV scale were exhaustively probed without direct detection of new states while the (g-2)μ anomaly is confirmed, this would confirm empirically that nature is fine-tuned¹²

A similar observation was made in connection with electron EDM measurements [5] and in [87]. On a similar ground, see [103] for the implications of tuning in the context of models with radiative leptonic mass generation.

Our analysis generalizes and reinforces our earlier results in [64] by including a more complete basis for the relevant EW scenarios, considering consistent electroweak embeddings of singlet scenarios, addressing flavor physics considerations, and supplying important technical details. Subsequent studies have employed an effective field theory (EFT) approach to explore indirect signatures of the new physics causing the (g-2)μ anomaly at muon colliders [62,65]. While this EFT approach would not allow us to ask detailed questions about the BSM physics—like studying direct particle production, tuning, and flavor considerations—it is nonetheless extremely useful due to its maximal model-independence and simplicity. As we discuss in Sec. V, the results of these analyses are highly complementary to our own and help flesh out the muon collider no-lose theorem.

As defined in Eqs. (5) and (6), if BSM singlet scalars or vectors are responsible for the (g-2)μ anomaly, the relevant muonic interactions are (gSμLμcS+H.c.),gVVα(μL†σ¯αμL+μcσαμc†).(34)The contribution of NBSM scalar singlets to (g-2)μ is ΔaμS=NBSMgS216π2∫01dzmμ2(1-z)(1-z2)mμ2(1-z)2+mS2z,≈2×10-9NBSMgS2(700  GeVmS)2,(35)where in the last step we have taken the mS≫mμ limit. For vectors, the corresponding (g-2)μ contribution is ΔaμV=NBSMgV24π2∫01dzmμ2z(1-z)2mμ2(1-z)2+mV2z,≈2×10-9NBSMgV2(200  GeVmV)2,(36)where again we have taken the mV≫mμ limit. It is known in the literature that pseudoscalar or pseudovector contributions to (g-2)μ have the wrong sign to explain the anomaly [41], so we do not consider these scenarios here. Note also that in both cases Δaμ∝mμ2, which implies a low (≲TeV) mass scale for any choice of perturbative couplings that yield Δaμ∼10-9 required to explain the anomaly (see discussion in Sec. III B). Therefore, any TeV-scale collider with sufficient luminosity will produce the S or V states on shell via μ+μ-→γS/V. Our challenge in the remainder of this section is to identify the highest singlet masses of interest and to demonstrate that a plausible muon collider would unambiguously discover the signatures associated with these states regardless of their mass and decay channels.

In our analysis, we first calculate the perturbative unitarity constraints on singlet couplings gS and gV that arise from the amplitude μ-μ+→μ-μ+ with an intermediate S or V. We then calculate how the singlet mass is determined by the coupling to explain (g-2)μ, up to the maximum allowed values of these couplings. This will give a maximum possible mass for the singlet(s).

The amplitude for the process μ-(p1)μ+(p2)→S/V→μ-(p3)μ+(p4) is given by (note that we have temporarily switched to 4-component fermion notation for convenience) MS=u¯3(-igS)v4is-ms2v¯2(-igS)u1-u¯3(-igS)u1it-ms2v¯2(-igS)v4,(37)MV=u¯3(-igVγα)v4is-mV2[-gαβ+(p1+p2)α(p1+p2)βmV2]v¯2(-igVγβ)u1-u¯3(-igVγα)u1it-mV2[-gαβ+(p1-p3)α(p1-p3)βmV2]v¯2(-igVγβ)v4.(38)We calculated the constraints on the scalar and vector singlets by calculating Eq. (13) for different j. For scalars, the strongest constraint was obtained from the process μ-(λ+)μ+(λ+)→μ-(λ-)μ+(λ-), where λ± represents positive/negative helicities. For vectors, the strongest constrain was obtained for the process μ-(λ+)μ+(λ-)→μ-(λ-)μ+(λ+). Using the procedures outlined in Sec. II C 1 we get the following constraints: gS2≤4πNBSM,gV2≤12πNBSM,(39)where NBSM is the number of singlets with common masses and couplings in the theory. For NBSM=1(10) the upper bound on the scalar singlet coupling is gS≤3.54(1.12) and on the vector singlet coupling is gV≤6.14(1.94).¹³

The process μ-(λ+)V(λ+)→μ-(λ+)V(λ+) can provide stronger constraints for singlet vectors with NBSM=1. However, because this process is NBSM independent, for larger values of NBSM the strongest constraint is provided by Eq. (39). We omit this constraint from our analysis for simplicity since it does not change our final result.

In Fig. 4 we show the singlet scalar or vector coupling required for a given mass to account for the (g-2)μ anomaly. The upper bounds are ms≤2.7  TeV and mV≤1.1  TeV, for scalar and vector singlets, respectively. Even though the upper bound on the singlet couplings decreases as the number of BSM flavors increases, the upper bound on the singlet masses does not change, since the NBSM dependence drops out by imposing Δaμ=Δaμobs.

As discussed in Sec. II C 2, CLFV constraints exclude flavor-universal couplings of the scalar to leptons, and severely disfavor anarchic ones. This serves as strong motivation for the MFV ansatz in scalar singlet scenarios, resulting in a lower maximum mass scale than unitarity alone. Figure 4 shows that the scalar should be no heavier than 200 GeV if MFV is satisfied.

The vector interaction Vα(μL†σ¯αμL+μc†σ¯αμc) must arise from a new U(1) gauge extension to the SM, which is spontaneously broken at low energies. If V is a “dark photon” whose SM interactions arise from V-γ kinetic mixing, then the parameter space for explaining (g-2)μ has been fully excluded for both visibly and invisibly decaying V [77,78]; some viable parameter space still exists for semivisible cascade decays, but this will be tested in with upcoming low energy experiments [72]. If, instead, V couples directly to muons, then the only¹⁴

Other U(1) options may also be viable if additional electroweak charged BSM states are included to cancel anomalies, but these models are phenomenologically similar for the purpose of our (g-2)μ analysis and are further subject to strong bounds at scales below the masses of these new particles [106,107].

We now discuss the collider signatures of singlet scenario explanations for the (g-2)μ anomaly. In particular, here we focus on the region of masses above ∼GeV, with the understanding that low energy experiments will cover the lower mass region. The first signal we discuss is direct production of the singlets in association with a photon. The presence of a photon is important because we will consider the possibility that the singlets decay invisibly, in which case the MUC can look for monophoton signatures. This γ+X signal is particularly important for low masses. The second signal that we will discuss is Bhabha scattering. The process μ-μ+→μ-μ+ receives contributions via singlet exchange. This process is particularly important for high singlet masses in a low-energy collider. An important question that we want to address is at which luminosity a given signal can be detected at 5σ significance for a given collider energy.

We consider two possible muon colliders: a high energy 3 TeV collider with 1  ab-1 of integrated luminosity and a low energy 215 GeV collider (a potential Higgs factory) with 0.4  ab-1 of luminosity. These benchmark luminosities are discussed by the international muon collider collaboration at CERN [111]. As opposed to conventional colliders, a MUC has the extra complication of beam-induced background due to muon decay in flight. For this reason the detector design includes two tungsten shielding cones along the direction of the beam. The opening angle of these cones should be optimized as a function of the energy of the MUC. In order to be conservative, our simulations assume that the detector cannot reconstruct particles with angles to the beam line below 10° (20°) for the higher (lower) energy muon collider [112].

We close this section by commenting on possible UV completions of singlet scenarios. It is important to keep in mind that the scalar-muon coupling in the singlet scalar model has to be generated by the nonrenormalizable operator (cS/Λ)HμLμcS after electroweak symmetry breaking. There are only a few ways of generating this operator at tree level using renormalizable interactions.

The simplest possibility involves the SH†H operator, which introduces S-H mass mixing after electroweak symmetry breaking. Diagonalizing away this mixing induces the SμLμc operator, which is proportional to both the SM muon Yukawa coupling and S-H mixing angle. However, this scenario is experimentally excluded as a candidate explanation for (g-2)μ [117] and similar arguments sharply constrain models in which S mixes with the scalar states in a two-Higgs doublet model.

The singlet-muon Yukawa interaction can also be induced in models where the singlet S couples to a vectorlike fourth generation of leptons ψi. If the ψi undergo mass mixing with L and/or μc, then the requisite operator SμLμc can arise upon diagonalizing the full leptonic mass matrix after electroweak symmetry breaking. In such models, these states inherit the flavor structure of their UV mixing interactions, whose form must be restricted (e.g., by MFV) to ensure that FCNC bounds are not violated. If these additional ψi states are sufficiently light (≲ few TeV), they may be accessible at future proton and electron colliders, e.g., via established search strategies for heavy new vectorlike leptons [83]. As an example, the Lagrangian L⊃-yLH†Lψc-yRμcψS,(43)can generate the operator (cS/Λ)HμLμcS after integrating out the vectorlike lepton ψ. In such a case we identify gS=yLyRv2mψ.(44)For NBSM=1, gS≈1×(mS/TeV) to generate the required Δaμ contribution, see Fig. 4. For perturbative couplings yL,R≲4π, the vectorlike lepton must therefore obey mψ<mψmax≈(2  TeV)·(1  TeVmS),(45)which means that these states may be discoverable at the HL-LHC or future proton, electron or muon colliders if the singlet mass is near its upper bound allowed by perturbativity. However, for lighter singlets these states can be far heavier than the TeV scale, and therefore inaccessible at traditional colliders.

A detailed study of these UV completions is beyond the scope of this paper. We merely emphasize that the existence of charged states at or below the TeV scale is not necessary to realize the scalar singlet scenario. On the other hand, discovering these scalar singlets at a muon collider only relies on the coupling gS that is determined by solving the (g-2)μ anomaly.

In Sec. II B, we defined the SSF and FFS simplified models, with Lagrangians given in Eqs. (7) and (8), which we repeat here for convenience LSSF⊃-y1FcL(μ)ΦA*-y2FμcΦB−κHΦA*ΦB−mA2|ΦA|2−mB2|ΦB|2−mFFFc+H.c.,(46)LFFS⊃-y1FAcL(μ)Φ*−y2FBμcΦ−y12HFAcFB−y12′H†FAFBc−mAFAFAc−mBFBFBc−mS2|Φ|2+H.c.(47)For NBSM>1, we simply consider multiple degenerate copies of the above field content. In SSF (FFS) models, the fermion F (complex scalar S) is in SU(2)L representation R with hypercharge Y, while the two complex scalars ΦA,B (two fermions FA,B) are in representation RA,B with hypercharges YA,B.