# A No-Lose Theorem for Discovering the New Physics of (g-2)_μ at Muon Colliders

Rodolfo Capdevilla, David Curtin, Yonatan Kahn, Gordan Krnjaic

PPrepared for submission to JHEP

FERMILAB-PUB-21-012-T

A No-Lose Theorem for Discovering the NewPhysics of ( g − µ at Muon Colliders Rodolfo Capdevilla, a,b

David Curtin, a Yonatan Kahn, c,d

Gordan Krnjaic e,f a Department of Physics, University of Toronto, Canada b Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada c University of Illinois at Urbana-Champaign, Urbana, IL, USA d Illinois Center for Advanced Studies of the Universe, University of Illinois at Urbana-Champaign,Urbana, IL, USA e Fermi National Accelerator Laboratory, Batavia, IL, USA f Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA

E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:

We perform a model-exhaustive analysis of all possible beyond StandardModel (BSM) solutions to the ( g − µ anomaly to study production of the associatednew states at future muon colliders, and formulate a no-lose theorem for the discoveryof new physics if the anomaly is conﬁrmed and weakly coupled solutions below the GeVscale are excluded. Our goal is to ﬁnd the highest possible mass scale of new physicssubject only to perturbative unitarity, and optionally the requirements of minimum ﬂavourviolation (MFV) and/or naturalness. We prove that a 3 TeV muon collider is guaranteed todiscover all BSM scenarios in which ∆ a µ is generated by SM singlets with masses above ∼ GeV ; lighter singlets will be discovered by upcoming low-energy experiments. If new stateswith electroweak quantum numbers contribute to ( g − µ , the minimal requirements ofperturbative unitarity guarantee new charged states below O (100 TeV) , but this is stronglydisfavoured by stringent constraints on charged lepton ﬂavour violating (CLFV) decays.Reasonable BSM theories that satisfy CLFV bounds by obeying Minimal Flavour Violation(MFV) and avoid generating two new hierarchy problems require the existence of at least onenew charged state below ∼ −

20 TeV . This strongly motivates the construction of high-energy muon colliders, which are guaranteed to discover new physics: either by producingthese new charged states directly, or by setting a strong lower bound on their mass, whichwould empirically prove that the universe is ﬁne-tuned and violates the assumptions ofMFV while somehow not generating large CLFVs. The former case is obviously the desiredoutcome, but the latter scenario would perhaps teach us even more about the universe byprofoundly revising our understanding of naturalness, cosmological vacuum selection, andthe SM ﬂavour puzzle. a r X i v : . [ h e p - ph ] J a n ontents ( g − µ in Singlet Scenarios 223.2 Constraining the BSM mass scale with Perturbative Unitarity 233.3 Flavour Considerations 243.4 Muon Collider Signatures 253.4.1 Inclusive Analysis of Singlet Direct Production 253.4.2 Bhabha Scattering 283.5 UV Completion of Scalar Singlet Scenarios 30 ( g − µ in Electroweak Scenarios 324.3 Constraining the BSM Mass Scale with Perturbative Unitarity 344.4 Constraining the BSM Mass Scale with Unitarity + MFV 384.5 Constraining the BSM Mass Scale with Unitarity + Naturalness 394.6 Constraining the BSM Mass Scale with Unitarity + Naturalness + MFV 404.7 Electroweak Landau Poles 414.8 Muon Collider Signatures 42 – 1 – Introduction and Executive Summary

The magnetic moments of leptons have spurred the development of quantum ﬁeld theory(QFT) and provided the most precise comparison between theory and experiment in the his-tory of science. While the measured anomalous magnetic moment of the electron, ( g − e ,agrees with the Standard Model (SM) prediction to better than one part per billion [1] theanalogous quantity for the muon, ( g − µ , has been discrepant between theory and exper-iment at a statistically signiﬁcant level for nearly two decades [5]. Since the muon mass ismuch closer to the QCD scale than the electron mass, hadronic contributions to ( g − µ arean important part of the calculation, and a recent tour-de-force eﬀort [6] combining latticecalculations with quantities extracted from experimental data [7–26] has recently conﬁrmedthe discrepancy to be ∆ a obs µ = a exp µ − a theory µ = (2 . ± . × − , (1.1)with a statistical signiﬁcance of . σ . The Muon g − experiment at Fermilab [29] isexpected to surpass the statistics of the previous Brookhaven experiment in the comingmonths, which would further reduce the uncertainty on the experimental result. If thediscrepancy persists after this measurement (and if it is also conﬁrmed by JPARC [30]) itwould be the ﬁrst terrestrial discovery of physics beyond the Standard Model (BSM).Whenever a discrepancy is found in a low-energy precision measurement, it is imperativeto understand the implications for other experiments, both to conﬁrm the anomaly andbecause such a discrepancy could point to the existence of new particles at higher butaccessible energy scales. Direct production and observation of new states is, after all, thegold standard for discovering new physics. In the long history of the ( g − µ anomaly,many such studies were performed. Examples include investigations of complete theorieslike supersymmetry [31–33]; minimal low-energy scenarios involving only very light states[34, 35]; or various simpliﬁed model approaches to study the generation of ( g − µ at higherenergy scales [36, 37], which can include additional considerations like the existence of aviable dark matter (DM) candidate [38–43].However, in all these past investigations, a simple question was left unanswered: Whatis the highest mass that new particles could have while still generating the measured BSMcontribution to ( g − µ ? In this paper, we answer that crucial question in a precise yetmodel-exhaustive way, relying only on gauge invariance and perturbative unitarity, andoptionally on well-deﬁned tuning or ﬂavour considerations, without making any detailedassumptions about the complete underlying theory.We provide a detailed description of our model-exhaustive approach in Section 2, but itcan be brieﬂy summarized as follows. We assume that one-loop eﬀects involving BSM states While there is a ∼ . σ discrepancy between the theoretical prediction of ( g − e [1] and the experi-mental measurement [2], in this paper we proceed under the assumption that this is not evidence of newphysics. See e.g. Refs. [3, 4] for a discussion of possible BSM implications. Some lattice calculations [27] ﬁnd no discrepancy with the measured ( g − µ , but are discrepant with R -ratio measurements. The source of this tension may lie in electroweak precision observables [28], preservingthe ( g − µ anomaly. – 2 –re responsible for the anomaly, since scenarios where new contributions only appear athigher loop order require a lower BSM mass scale to generate the required new contribution.We can, thus, organize all possible one-loop BSM contributions to ∆ a µ into two classes:• Singlet Scenarios: in which each BSM ( g − µ contribution only involves a muonand a new SM singlet boson that couples to the muon (analyzed in Section 3);• Electroweak (EW) Scenarios: in which new states with EW quantum numberscontribute to ( g − µ (analyzed in Section 4).Singlet Scenarios generate ∆ a µ contributions proportional to m µ y µ v/M , where y µ ∼ − is the small SM muon Yukawa coupling. Electroweak Scenarios can generate the largestpossible ( g − µ contributions without the additional y µ suppression. In particular, wecarefully study two simpliﬁed models denoted SSF and FFS with new scalars and fermionsthat yield the largest possible BSM mass scale able to account for the anomaly. Carefulanalysis of these two EW Scenarios allows us to derive our model-exhaustive upper boundon BSM particle masses for scenarios that resolve the ( g − µ anomaly. We also account forthe possibility of many new states contributing to ∆ a µ by considering N BSM ≥ copies ofeach BSM model being present simultaneously, allowing us to understand how the maximumpossible BSM mass scales with BSM state multiplicity in each case.We ﬁnd that if ∆ a obs µ is generated in a Singlet Scenario, the maximum mass of theBSM singlet particle(s) is 3 TeV regardless of BSM multiplicity N BSM . For EW Scenarios,we ﬁnd that there must always be at least one new charged state lighter than the followingupper bound: M max , XBSM , charged ≈ (100 TeV) N / for X = (unitarity*) (20 TeV) N / for X = (unitarity + MFV) (20 TeV) N / for X = (unitarity + naturalness*) (9 TeV) N / for X = (unitarity + naturalness + MFV) , (1.2)where this upper bound is evaluated under four assumptions that the BSM solution to the ( g − µ anomaly must satisfy: perturbative unitarity only; unitarity + Minimal FlavourViolation (see e.g. [45, 46]); unitarity + naturalness (speciﬁcally, avoiding two new hierarchyproblems); and unitarity + naturalness + MFV. The unitarity-only bound represents thevery upper limit of what is possible within QFT, but realizing such high masses requiressevere alignment tuning or another unknown mechanism to avoid stringent constraints fromcharged lepton ﬂavour-violating (CLFV) decays [47, 48]. We have therefore marked every We work under the assumption that the ( g − µ anomaly is due to new physics which genuinely aﬀectsthe value of g µ in vacuum, rather than its measurement being sensitive to other BSM eﬀects on the muonspin, for example ultralight scalar dark matter [44]. The latter case is also eminently testable in upcomingexperiments. – 3 –cenario without MFV with a star (*) above, to indicate additional tuning or unknownﬂavour mechanisms that have to also be present.Our results have profound implications for the physics motivation of future muon collid-ers (MuC), which have recently garnered renewed attention as an appealing possibility forthe future of the high energy physics program [49–54]. Muon colliders still face signiﬁcanttechnical challenges [50], but are in many ways ideal BSM discovery machines: compared toelectron colliders, the suppressed synchrotron radiation loss might make it easier to reachhigh energies in excess of 10 TeV; unlike in proton collisions, the entire center-of-mass energyis available for the pair-production of new charged particles with masses up to m ∼ √ s/ [50]; and ﬁnally they collide the actual particles that exhibit the ( g − µ anomaly.These features enable us to formulate a no-lose theorem for a future muon colliderprogram. We presented our ﬁrst investigation of this issue in [55]. Here, we supply impor-tant additional details, perform detailed muon collider studies, and generalize our originalderivation to include crucial ﬂavour considerations and present all possible EW Scenar-ios that maximize BSM masses, all of which reinforce the robustness of our conclusions.Since our original study appeared, there have also been additional investigations of indirectprobes of ( g − µ at future muon colliders [53, 56]. The results of these studies, despitetheir diﬀerent technical approach, agree with our overall conclusions and strengthen themin important ways, as we explain below.We give a detailed description of this no-lose theorem in Section 5, but its most impor-tant ﬁnal points are as follows, broken down in chronological progression:1. Present day conﬁrmation:

Assume the ( g − µ anomaly is real.2. Discover or falsify low-scale Singlet Scenarios (cid:46)

GeV:

If Singlet Scenarios with BSM masses below ∼ GeV generate the required ∆ a obs µ contribution [34], multiple ﬁxed-target and B -factory experiments are projected todiscover new physics in the coming decade [35, 57–64].3. Discover or falsify all Singlet Scenarios (cid:46)

TeV:

If ﬁxed-target experiments do not discover new BSM singlets that account for ∆ a obs µ ,a 3 TeV muon collider with − would be guaranteed to directly discover thesesinglets if they are heavier than ∼

10 GeV .Even a lower-energy machine can be useful: a 215 GeV muon collider with . − could directly observe singlets as light as 2 GeV under the conservative assumptionsof our inclusive analysis, while indirectly observing the eﬀects of the singlets for allallowed masses via Bhabha scattering.Importantly, for singlet solutions to the ( g − µ anomaly, only the muon collider isguaranteed to discover these signals since the only required new coupling is to themuon.4. Discover non-pathological Electroweak Scenarios ( (cid:46)

10 TeV): – 4 –f TeV-scale muon colliders do not discover new physics, the ( g − µ anomaly must be generated by EW Scenarios. In that case, all of our results indicate that in mostreasonably motivated scenarios, the mass of new charged states cannot be higherthan few ×

10 TeV. However, such high masses are only realized by the most extremeboundary cases we consider. Therefore, a muon collider with √ s ∼

10 TeV is highlymotivated, since it will have excellent coverage for EW Scenarios in most of theirreasonable parameter space.A very strong statement can be made for future muon colliders with √ s ∼

30 TeV (40TeV): such a machine can discover via pair production of heavy new charged states all

EW Scenarios that avoid CLFV bounds by satisfying MFV and avoid generatingtwo new hierarchy problems, with N BSM (cid:46) (10).5. Unitarity Ceiling ( (cid:46)

100 TeV):

Even if such a high energy muon collider does not produce new BSM states directly,the recent investigations by [53, 56] show that a 30 TeV machine would detect devia-tions in µ + µ − → hγ , which probes the same eﬀective operator generating ( g − µ atlower energies. This would provide high-energy conﬁrmation of the presence of newphysics.In that case, our results guarantee the presence of new states below ∼

100 TeV byperturbative unitarity, and the lack of direct BSM particle production at √ s ∼

30 TeV will prove that the universe violates MFV and/or is highly ﬁne-tuned to stabilize theHiggs mass and muon mass, all while suppressing CLFV processes.Even the most pessimistic ﬁnal case would profoundly reshape our understanding of the uni-verse by providing new information about the nature of ﬁne-tuning, ﬂavour and cosmologicalvacuum selection. If no new states are discovered at 30 TeV, the renewed conﬁrmation ofthe ( g − µ anomaly at these higher energies and the associated guaranteed presence ofnew states below the unitarity bound with deep implications for naturalness and ﬂavourmeans ﬁnding the solution to all these puzzles will surely provide impetus for pushing ourknowledge of the energy frontier to even greater heights.If the ( g − µ anomaly is conﬁrmed, our analysis and the results of [53, 56] showthat ﬁnding the origin of this anomaly should be regarded as one of the most importantphysics motivations for an entire muon collider program . Indeed, a series of colliders withenergies from the test-bed-scale O (100 GeV) to the far more ambitious but still imaginable O (10 TeV) scale and beyond has excellent prospects to discover the new particles necessaryto explain this mystery. Regardless of what these direct searches ﬁnd, each will makeinvaluable contributions to allow us to understand the precise nature of the new physicsthat must be present. Therefore, this truly is a no-lose theorem for the discovery of newphysics, the greatest imaginable motivation for a heroic undertaking like the constructionof a revolutionary new type of particle collider. While we argue in this work that muon colliders are suﬃcient for discovery, they are not the only suchprobe: proton-proton colliders, electron linear colliders, and even photon colliders have strong potential for – 5 –e now present the details necessary to ﬁll out this argument. Our model-exhaustiveapproach is explained in Section 2; Singlet Scenarios and EW Scenarios are analyzed indetail in Sections 3 and 4; the implications for a future muon collider program and theno-lose theorem for discovery of new physics is fully outlined in Section 5.

In this paper, we aim to address a simple question: how could we discover all possible BSMsolutions to the ( g − µ anomaly? Speciﬁcally, how could we directly discover at least someof the BSM particles that play a role in generating ∆ a obs µ ? The bewildering plethora ofpossible BSM solutions to the anomaly make answering this question very challenging; byconstruction, our answer cannot depend on the particular choice of BSM model.Very light, weakly coupled solutions to ( g − µ near or below the scale of the muonmass will be exhaustively tested by low energy experiments, and we focus on all otherBSM possibilities. In that case, at the low energies at which the ( g − µ measurement isperformed, we can parameterize the deviation from the SM expectation as a BSM contribu-tion to the anomalous magnetic moment operator. Taking into account electroweak gaugeinvariance, in two-component fermion notation this is L eﬀ = C eﬀ vM ( µ L σ νρ µ c ) F νρ + h . c . , (2.1)where µ L and µ c are the two-component muon ﬁelds, v = 246 GeV is the SM Higgs vacuumexpectation value (VEV), and C eﬀ is a constant. The factor of v arises from the fact thatcoupling left- and right-handed muon ﬁelds requires a Higgs insertion, so the electroweak-symmetric operator is dimension-6, H † Lσ νρ µ c F νρ , and thus must be suppressed by twopowers of a mass scale /M . Unfortunately, such model-independent EFT analyses arelimited to indirect signatures of the new physics, making this approach unsuitable to answerthe question of how to directly discover the new states.To study high-energy direct signatures of new physics, we instead adopt a “model-exhaustive” approach. As illustrated in Figure 1, this simply involves adding the assumptionthat the new physics is perturbative, which resolves the new ( g − µ contributions intoindividual loop diagrams involving various possible BSM particles in diﬀerent SM gaugerepresentations. In principle, if all possibilities were considered, one could study directsignatures of new physics in the same full generality that model-independent EFT analysesaﬀord for indirect signatures. The idea of a model-exhaustive analysis is not, of course, a new one. However, thechallenge lies in systematically covering all possibilities of BSM particles, or at least those observing new TeV-scale EW states. That said, muon-speciﬁc singlets will likely be challenging to observeat any collider not utilizing muon beams, and discovering EW-charged states at the 10 TeV scale may notbe as straightforward with a 100 TeV pp collider due to PDF factors and a noisier detector environment [65],while reaching such energies could be challenging in an electron machine. Of course, all these cases deservea dedicated analysis. While our analysis is formally limited to perturbative BSM solutions of the ( g − µ anomaly, our resultsnonetheless end up parametrically covering the case of strongly coupled BSM scenarios as well, as we arguein Section 2.4. – 6 – µ c µ L µ c µ L µ c h H i µ L µ c µ L µ c h H i µ L µ c h H i µ L { i } µ c h H i µ L { i } µ c µ L µ c µ L µ c h H i µ L µ c µ L µ c h H i µ L µ c h H i µ L { i } µ c h H i µ L { i } µ c µ L µ c µ L µ c h H i µ L µ c µ L µ c h H i µ L µ c h H i µ L { i } µ c h H i µ L { i } µ c µ L µ c µ L µ c h H i µ L µ c µ L µ c h H i µ L µ c h H i µ L { i } µ c h H i µ L { i } Assumptions gauge invariance Δ a μ = a obs μ U (1) em SM gauge invariance Δ a μ = a obs μ SM gauge invariancePerturbativity Δ a μ = a obs μ Model-Independent “Model-Exhaustive” M H † ( Lσ νρ μ c ) F νρ M ( μ L σ νρ μ c ) F νρ Speciﬁc choices of BSM particles and their SM quantum numbers in loop diagram ( g − 2) μ How to predict new signatures

Figure 1 : The philosophy of our “model-exhaustive” analysis. Traditional model-independent anal-yses express the new physics contribution to ( g − µ as a non-renormalizable operator, either in thelow-energy theory after EW symmetry breaking (left) or in the full SM gauge invariant formulation(middle). This makes no assumptions about the new physics but is limited to indirect signaturesof the new physics produced by the same operator. Since we want to probe direct signatures of theBSM physics which solves the ( g − µ anomaly, we add the single assumption of perturbativity tothe traditional model-independent analysis, which resolves the new ∆ a µ contributions into explicitloop diagrams of new states { ψ i } carrying speciﬁc SM quantum numbers (right). If the Higgs inser-tion lies on the external muon, ∆ a µ is suppressed by y µ , while ∆ a µ can be signiﬁcantly enhancedif the Higgs couples to new particles in the loop. By exhaustively analyzing all possible choices ofnew states, we can derive predictions for direct signatures that are as universal as the traditionalmodel-independent predictions for indirect signatures. possibilities relevant to answering a speciﬁc phenomenological question. We now explainhow to perform this analysis for the ( g − µ anomaly, with an eye towards direct signaturesat future muon colliders. We limit ourselves to those perturbative BSM scenarios where the required ∆ a µ isgenerated at one-loop order . There are certainly many possibilities for BSM physics thatsolves the ( g − µ puzzle by generating only new higher-loop contributions [3, 67, 68], butthe mass scale of new physics in those scenarios is necessarily much lower (by roughly somepower of a loop factor) than the highest mass scale possible in BSM scenarios that generate ∆ a obs µ at one-loop.Our exhaustive coverage of candidate BSM theories for ( g − µ is informed by thecharacteristic experimental signatures available in each class of scenarios. For this reason,we divide up the space of possibilities into two classes, illustrated schematically in Figure 2:1. Singlet Scenarios : deﬁned as BSM solutions to the ( g − µ anomaly in which theonly new particles in the ( g − µ loop are SM gauge singlets. This selects the ﬁrsttype of diagram in Figure 1 (right box) with the Higgs VEV insertion on the external For a philosophically similar approach to the Hierarchy Problem, see [66]. – 7 – pace of BSM Theoriesthat generate Δ a μ = a obs μ B ound a r y o f p e r t u r b a ti v e un it a r it y Singlet Scenarios Electroweak Scenarios

New particles in loops: only

SM singlets ( g − 2) μ New particles in loops: not only SM singlets ( g − 2) μ Signature: direct production of

SM singlet states

Signature: direct production of new charged states

Discovery: requires inclusivesearch for singlet, with g ∝ m Discovery: discoverable at lepton collider for “all” m ≲ s /2 Figure 2 : Schematic representation of the model-exhaustive space of BSM theories that can solvethe ( g − µ anomaly, and our mutually exclusive and collectively exhaustive categorization intoSinglet Scenarios and Electroweak Scenarios. For these two classes of theories, the phenomenologicalquestions are distinct. To understand how to discover Singlet Scenarios, we have to not only ﬁndthe heaviest possible mass of the singlet(s), but also how to discover this singlet for all possiblemasses, since its phenomenology depends on its stability and decay mode, and lighter singlets haveweaker coupling. Electroweak Scenarios predict new charged states, and since those have to producevisible ﬁnal states in a collider and are eﬃciently produced at lepton colliders for m (cid:46) √ s/ , weonly have to ﬁnd the maximum mass the lightest new charged state in the BSM theory can have.(We limit ourselves to scenarios that generate ∆ a obs µ at one-loop, since higher-loop solutions havelower BSM mass scales.) muon leg, such that the chirality ﬂip and the Higgs coupling both come from themuon, and hence ∆ a µ ∝ m µ y µ v/M . Their singlet nature means these particlescould be very light ( (cid:46) GeV ) while evading present constraints [34], but they couldalso be much heavier.For Singlet Scenarios, our task is to ﬁnd the largest possible mass these singlets couldhave, and determine how a muon collider could produce and observe them for allpossible masses, regardless of how or if they decay in the detector.2.

Electroweak (EW) Scenarios : deﬁned as all BSM solutions that are not SingletScenarios. This necessarily implies that ( g − µ receives contributions from loopsinvolving BSM states with EW quantum numbers, which in turn implies the existenceof new heavy charged states with masses (cid:38)

100 GeV to evade LEP bounds. Thesecharged particles could contribute to ( g − µ directly, or be new states that mustexist due to gauge invariance. The new charged states will be our focus, since anylepton collider with √ s (cid:38) m can directly pair-produce such states of mass m , andas they have to either be detector-stable or decay into charged ﬁnal states, they– 8 –hould be discoverable in a clean detector environment regardless of their detailedphenomenology. For EW Scenarios, our task is therefore to ﬁnd the largest possiblemass that the new charged states could have.EW Scenarios can generate diagrams of both types shown in Figure 1 (right). Ofparticular interest is the second type where the Higgs insertion and chirality ﬂip belongto BSM particles in the loop, which would give ∆ a µ ∝ m µ g BSM v/M without thesuppression of the small muon Yukawa. This can result in much heavier BSM massscales than Singlet Scenarios.If we examine both of these possibilities exhaustively, we will have completed our model-exhaustive analysis.Singlet Scenarios are relatively straightforward to analyze. In the next Section 2.1we deﬁne simpliﬁed models that cover all possibilities for this singlet. These models havefew parameters, and the parameter space can be explored in full generality. ElectroweakScenarios present more of a challenge. To ﬁnd the minimum muon collider energy thatwould guarantee direct production and discovery of at least one BSM charged state, wehave to ﬁnd the heaviest possible charged state consistent with resolving the anomaly. Thisamounts to ﬁnding the following quantity: M maxBSM , charged ≡ max BSM theory space∆ a µ =∆ a obs µ (cid:26) min i ∈ BSM spectrum (cid:16) m ( i )charged (cid:17) (cid:27) . (2.2)This can be understood in the following algorithmic way. The outer maximization scansover all possible BSM theories and possible values of their parameters that give ∆ a µ = ∆ a obs µ while satisfying the constraints of perturbative unitarity. For each speciﬁc theory and givenvalues of its parameters, we ﬁnd the lightest new charged state (inner bracket) and addit to a list. The outer maximization then picks the maximum value from this list, givingthe heaviest possible mass of the lightest new charged state that must exist to resolve the ( g − µ anomaly, and therefore the minimum energy of a muon collider that is guaranteedto produce these particles. The diﬃculty obviously arises in performing the ﬁrst theoryspace maximization. In Section 2.2 we explain how this maximization can be performed,allowing our model-exhaustive analysis to determine the heaviest possible masses of newcharged states with the generality of a traditional model-independent analysis. In this case, SM singlets that could be below the GeV scale (or much heavier) generatethe new one-loop contributions to ( g − 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 relevantLagrangian terms for the real scalar case are L S ⊃ − ( g S Sµ L µ c + h . c . ) − m S S . (2.3)Note that the Yukawa coupling of the real scalar to muons g S is not gauge invariant. Thisimplies that either the interaction arises from the non-renormalizable operator c S µ L µ c HS, – 9 –

S µ c µ L F cA v F B v S/V µ c µ c µ L m F v µ c µ L A B F c Fm F v e c µ L A B F c F h µ c µ L F c F S µ c µ L F cA v F B S e c µ L F cA v F B v S µ c µ c µ L µ L v V µ c µ c µ L m F v µ c µ L A B F c Fm F v e c µ L A B F c F h µ c µ L F c F Figure 3 : Representative 1-loop contributions to ( g − µ in the simpliﬁed models we consider.Top row: Singlet Scenarios with a SM neutral vector V or scalar S that couple to the muon. Notethat the Higgs VEV on the muon line gives both the chirality ﬂip and the EW breaking insertionsin these models. Bottom left: EW Scenario of SSF type, with one BSM fermion and two BSMscalars that mix via a Higgs insertion. Bottom right: EW Scenario of FFS type, with one BSMscalar and two BSM fermions that mix via a Higgs insertion. in which case g S ∝ v/ ( √ , or the interaction comes from a singlet-Higgs mixing, in whichcase g S ∼ y µ sin θ, where θ is the mixing angle. We brieﬂy discuss the consequences ofconsistent embedding in the full electroweak theory in Section 3. For the vector case, therelevant Lagrangian terms are L V ⊃ g V V α ( µ † L ¯ σ α µ L + µ c † ¯ σ α µ c ) + m V V α V α . (2.4)These two scenarios are representative of muophilic new gauge forces or scalars that havebeen extensively studied in the literature [35, 69–71] and their contributions to ( g − µ areshown in Figure 3.As discussed in Section 3, the only viable anomaly-free vector model is gauged L µ − L τ ,which can still resolve ( g − µ for m V ∈ (10 MeV , m µ ) [72, 73]. Bounds on muon-philicsinglet scalars are more model dependent and can, in principle, resolve ( g − µ with anymass between the MeV scale and the perturbative unitarity limit ∼ few TeV. For bothscalars and vectors, the lower limit is set by cosmological constraints, most importantlybounds on ∆ N eﬀ , the eﬀective number of relativistic species at big bang nucleosynthesis– 10 –73, 74]. Thus, the scalar Singlet Scenario will be of most interest to us, but we keep thevector case in our discussions for completeness since the analyses are very similar.These Singlet Scenarios are the most minimal BSM solutions to the ( g − 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 onlyguaranteed way to probe all Singlet Scenarios. Given that ﬁxed-target experiments and B -factories will exhaustively probe Singlet Scenarios with masses below ∼ GeV [35, 57–64],we will particularly focus on Singlet Scenarios above the GeV scale in our muon colliderphysics analyses.Of course, it is possible that more than one new degree of freedom contributes to ( g − µ .We account for this possibility by considering N BSM ≥ copies of each SM Singlet Scenarioin Eqns. (2.3) or (2.4), and analyzing how the various higher-energy signatures scale withBSM multiplicity. Note that the assumption that all N BSM copies of the simpliﬁed modelhave equal masses and couplings is the most pessimistic one with regards to high-energysignatures, since non-degenerate masses and couplings always lead to larger signaturesdue to the non-linearity of the associated cross sections and amplitudes. If couplings ormasses are highly unequal, the phenomenology will be dominated by just a few new states.Considering degenerate N BSM ≥ copies therefore covers the signature space of possibilities.Finally we note that, in principle, one could also consider the case of a neutral fermion N contributing to ( g − µ . This would essentially be a right-handed-neutrino-type scenario(see e.g. [75] for a review), where the new ( g − 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 aunitary neutrino mixing matrix, such contributions would cancel up to corrections of order ∼ ( m ν /m W ) , which are inadequate to explain ∆ a obs µ . We therefore restrict our focus toscalar and vector singlets. We now move on to discuss the most general class of BSM solutions to the ( g − µ anomaly, Electroweak Scenarios . This includes an overwhelmingly large number of possibilities, butfortunately, we do not need to study all of them. To perform the maximization over all ofBSM theory space in Eqn. (2.2), we merely need to study those models which are guaranteedto give the largest possible

BSM mass scales. This will be suﬃcient to model-exhaustivelydetermine the heaviest possible mass for new charged states.Which EW Scenarios maximize the BSM mass scale? Consider the most general newone-loop diagrams that could contribute to ( g − µ . To make sure the relevant masses andcouplings are maximally unconstrained, we consider the cases where all ﬁelds in the loopare BSM ﬁelds. Furthermore, the chirality ﬂip and the Higgs VEV insertion necessary togenerate Eq. (2.1) should both come from these BSM ﬁelds to avoid additional suppressionby 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 twofermions; – 11 –. a pair of these ﬁelds undergo mass-mixing with each other via a Higgs coupling afterelectroweak symmetry breaking (EWSB);3. all new fermions are vector-like under the SM to maximize allowed masses and avoidconstraints on new 4th generation fermions [76];4. no VEVs for any new scalars with EW charge. Since we are primarily interested inBSM states above the TeV scale, any new VEVs that break electroweak symmetrywill exceed the measured value v ≈ GeV for perturbative scalar self couplings.As in our analysis for Singlet Scenarios, our default focus is on the most experimentallypessimistic 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 ∆ a µ contributions than the analogous scenario with a new scalar, and likewise for Majoranafermions or real scalars. Since this results in a lower BSM mass scale that would be easierto probe, we focus on EW Scenarios with new complex scalars and vector-like 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 Φ A , Φ B in SU (2) L representations R A , R B with hypercharges Y A , Y B and a single vector-like fermion pair F ( F c ) in SU (2) L representation R ( ¯ R ) with hypercharge Y ( − Y ): L SSF ⊃ − y F c L ( µ ) Φ ∗ A − y F µ c Φ B − κH Φ ∗ A Φ B − m A | Φ A | − m B | Φ B | − m F F F c + h . c . . (2.5)Here y , y are new Yukawa couplings and κ is a trilinear coupling with dimensions of mass. L ( µ ) = ( ν L , µ L ) and µ c are the two 2-component second-generation SM lepton ﬁelds, and H is the Higgs doublet. A typical SSF contribution to ( g − µ is shown in Figure 3 (b).Note that the chirality ﬂip comes from the heavy vector-like fermion F while the HiggsVEV insertion arises due to mixing of the new scalars.The FFS simpliﬁed model is analogously deﬁned but reverses the role of fermions andscalars, featuring two vector-like fermion pairs F A , F B ( F cA , F cB ) in SU (2) L representations R A , R B ( ¯ R A , ¯ R A ) with hypercharges Y A , Y B ( − Y A , − Y B ) and a single complex scalar S in SU (2) L representation R with hypercharge Y : L FFS ⊃ − y F cA L ( µ ) Φ ∗ − y F B µ c Φ − y HF cA F B − y (cid:48) H † F A F cB − m A F A F cA − m B F B F cB − m S | Φ | + h . c . (2.6)There are now two renormalizable Yukawa couplings y , y (cid:48) which control the mixing ofthe A and B fermions via the Higgs. A typical FFS contribution to ( g − µ is shown inFigure 3 (c). The chirality ﬂip and Higgs VEV insertion both arise in the loop due to theHiggs couplings of the new fermions.These two simpliﬁed models generate the largest possible BSM particle masses thatcould account for ∆ a obs µ . Therefore, the maximization over theory space in Eqn. (2.2) canbe replaced by a maximization over the SSF and FFS parameter spaces: M maxBSM , charged ≡ max SSF , FFS models (cid:26) min i ∈ BSM spectrum (cid:16) m ( i )charged (cid:17) (cid:27) . (2.7)– 12 –ote that one could in principle consider extensions of the SM Higgs sector with additionalscalar contributing to EWSB. In that case, the κ and y , terms in the above Lagrangianscould arise from coupling to these new scalars rather than a SM-like Higgs doublet, whichmight change the allowed EW representations of the BSM states. However, current con-straints already dictate that most of the observed EWSB arises from the VEV of a singledoublet [77, 78], which means that relying only on BSM scalars to generate the requiredEWSB insertions in the above Lagrangians would lead to smaller eﬀective mixings andhence smaller ∆ a µ and BSM masses. We therefore do not have to consider such extendedscenarios to perform the maximization of the lightest new charged particle mass over BSMtheory space.In both SSF and FFS models, the choices of representations must satisfy ⊂ R A ⊗ R ⊗ (2.8) R B = ¯ RY A = − − YY B = − − Y, with Y chosen to make the electric charges integer-valued. We will explore all choices ofrepresentations involving SU (2) L singlets, doublets and triplets, and all choices of Y thatensure that all electric charges satisfy | Q | ≤ . As we discuss, this is suﬃcient to performthe above maximization. The possibility of a high multiplicity of new BSM states is againtaken into account by considering the trivial generalizations where there are N BSM identicalcopies of the above ﬁelds contributing to ∆ a µ .The Lagrangians in Eq. (2.5) and Eq. (2.6) only show the interactions necessary toform new one-loop contributions to ( g − µ . Depending on the choice of SU (2) L ⊗ U (1) Y representations, additional couplings between the new fermions/scalars and the muon orHiggs may be allowed by gauge invariance. However, these couplings will not contributeto ( g − µ at leading order, at most supplying a small correction to the leading termsgenerated by the couplings in Eqns. (2.5) and (2.6), or slightly modifying the mass spectrumof the fermions/scalars that couple to the Higgs after EWSB by (cid:46) TeV , which does notmeaningfully aﬀect our results or discussion. We can therefore neglect these additionalcouplings in our analysis. We also assume that the new BSM states do not couple toany other SM fermions (except when discussing leptonic ﬂavour violation bounds). Both ofthese assumptions are conservative in that they minimize additional experimental signaturesarising from the new physics responsible for the ( g − µ anomaly.Depending on the choice of representations, some of the EW Scenarios we considerwere previously studied in Refs. [39–42, 79–81]. There have also been previous attemptsto deﬁne simpliﬁed model dictionaries for generating ∆ a obs µ [36, 37, 39, 41, 42, 82–85], butnone took our completely model-exhaustive approach and none aimed to ﬁnd the highestpossible mass of new BSM charged states that could account for ∆ a obs µ . We also make noassumptions about e.g. the existence of a viable DM candidate, or any couplings of thenew degrees of freedom (dof’s) that are not required for resolving the ( g − µ anomaly(except optionally considering ﬂavour). Other possible simpliﬁed models for ( g − µ that– 13 –e do not examine do not change the outcome of the maximization over theory space ofEqn. (2.2). For example, one could add fewer than 3 new BSM dof’s with non-trivial EWrepresentations, but in that case the largest possible BSM mass scale is much lower thanwhat we ﬁnd in the SSF and FFS models (see e.g. [36]). The size of the ( g − µ contribution is controlled by BSM couplings and masses, and thelargest possible BSM masses that can account for the anomaly depend on the largest pos-sible BSM couplings. In Section 2.3.1 we describe ﬁrst how perturbative unitarity suppliesan absolute upper bound on the new couplings. This will inform our baseline analysis, butmore careful consideration of how these simpliﬁed models must arise as part of a more com-plete BSM theory suggests that an upper bound based on unitarity alone is likely far tooconservative, especially in light of stringent CLFV bounds. In Sections 2.3.2 and 2.3.3 wetherefore consider the additional constraints on the new muon couplings arising by assum-ing either Minimal Flavour Violation (MFV) or requiring the absence of large, explicitlycalculable new tunings. To deﬁne the boundaries of parameter space in our simpliﬁed models we appeal to tree-levelpartial-wave unitarity, expressed in terms of helicity amplitudes so that we can apply theconstraints to fermions as well as bosons [86]. (See e.g. [87–92] for more recent studies.) Webegin from the partial-wave expansion of the (azimuthally symmetric) scattering amplitudefor the → process i → f ≡{ a, b } → { c, d } : M i → f ( θ ) = 8 π ∞ (cid:88) j =0 (2 j + 1) T ji → f d jλ f λ i ( θ ) , (2.9)where d jλ f λ i ( θ ) are the Wigner d-functions, T ji → f is the j -th partial wave of the tree-levelscattering amplitude, λ i = λ a − λ b and λ f = λ c − λ d are the helicities of the initial andﬁnal states, and j is the eigenvalue of the total angular momentum. The coeﬃcients T ji → f can be found by using the orthogonality condition of the d-functions π (cid:90) M i → f ( θ ) d jλ f λ i ( θ ) d (cos θ ) = T ji → f . (2.10)From the optical theorem one can get the partial-wave unitarity condition of an inelasticprocess for each j β i β f | T ji → f | ≤ , (2.11)where the phase space factors for states of mass m and m are β ( m , m ) = 1 s (cid:112) [ s − ( m + m ) ][ s − ( m − m ) ] , (2.12)and s is the squared center of mass energy. For a given set of mass eigenstates whichappear in our theory, we will require that the lowest partial-wave tree-level 2-to-2 scattering– 14 –calar-Scalar T j =00 → d ( θ ) = 1 Scalar-Fermion T j =1 / → + d /

12 12 ( θ ) = cos( θ/ T j =1 / →− d / − ( θ ) = sin( θ/ Fermion-Fermion T j =0++ →±± d ( θ ) = 1 T j =1+ −→±± d ( θ ) = −√ θ/

2) cos( θ/ T j =1+ −→ + − d ( θ ) = cos ( θ/ Table 1 : The Wigner d-functions used in our partial-wave unitarity calculations. amplitudes between initial states i and ﬁnal states f satisfy the unitarity condition (2.11).We will consider boson-boson ( j = 0 ), boson-fermion ( j = ± / ), and fermion-fermion ( j =0 , ) scattering; fermion-vector scattering ( j = 1 / ) will always lead to weaker constraintsfor large N BSM . The relevant Wigner d-functions are given in Table 1.Note that the partial wave decomposition in Eq. (2.9) requires specifying the angularmomenta of the initial and ﬁnal states, so in principle the diﬀerent helicity amplitudes for j = 1 / can give independent constraints. Note also that these partial-wave constraints arevalid at any kinematically allowed value of s , as the phase space factors vanish at kinematicthresholds and enforce physical kinematics.The constraints obtained from (2.11) amount to the requirement that loop contribu-tions to scattering amplitudes are smaller than tree-level contributions at scales up to afactor of a few above m max , where m max is the largest mass eigenvalue in the model underconsideration. The violation of these constraints would require nonperturbative physics toappear at an energy scale close to m max to unitarize the theory, so restricting to parameterspace which satisﬁes tree-level unitarity amounts to the following statement: either a theorywith masses up to m max is perturbatively calculable, or new physics appears at the scale s max .In some processes, we may encounter singularities either in the scattering amplitudeitself in the form of s -channel poles, or after integrating the amplitude as demanded by Eqn.(2.11). The latter appear in t - and u -channel diagrams. In Ref. [89], these singularities aretreated by removing values of the CM energy √ s around the singularities. We avoid such acomplication by studying processes where t - and u -channel amplitudes do not appear, andwhere s -channel singularities correspond to poles at energies below the threshold wherethe cross section is nonvanishing. This will become clear when we discuss the perturbativeunitarity constraints for speciﬁc processes in the sections below.Note that somewhat stronger constraints could be achieved by considering a coupled-channel analysis where the full scattering matrix between all initial and ﬁnal states isdiagonalized, by considering higher partial waves, and/or by relaxing the constraints onpoles; our constraints are thus conservative, but will suﬃce for the statement of our no-losetheorem. Speciﬁcally, in some processes we take the s → ∞ limit to obtain our constraint, but numerically theconstraint asymptotes rapidly for energies a factor of a few times above threshold. – 15 – .3.2 Unitarity and Minimal Flavour Violation Proposing new scalars with Yukawa couplings to the muon prompts us to ask how these newdegrees of freedom couple to the other lepton generations. The physics which solves the ( g − µ anomaly would have to be embedded in whichever UV-complete framework explains theﬂavour structure of the SM fermions. From a bottom-up perspective, this is most relevantsince ﬂavour-changing neutral currents (FCNCs) in the lepton sector, most importantlycharged-lepton ﬂavour violating (CLFV) decays (cid:96) i → (cid:96) j γ , are tightly constrained [47, 48]: Br( µ → eγ ) < . × − (2.13) Br( τ → µγ ) < . × − (2.14) Br( τ → eγ ) < . × − (2.15)It is well known that CLFV constraints impose stringent requirements on BSM solutionsto the ( g − µ anomaly (see e.g. [36, 82]). We can demonstrate this by considering aﬂavour-anarchic version of the scalar Singlet Scenario: −L ⊃ S ( g eeS e L e c + g µµS µ L µ c + g ττS τ L τ c + g eµS µ L e c + g µeS e L µ c . . . ) . (2.16)where “. . . ” indicates the additional oﬀ-diagonal terms. This would generate ﬂavour-violating versions of the low-energy operator Eqn. (2.1) L eﬀ = C ( ij )eﬀ vM ( (cid:96) ( j ) L σ νρ (cid:96) ( i ) c ) F νρ + h . c . , (2.17)where i, j are lepton generation indices. The assumption that the above scalar SingletScenario resolves the ( g − µ anomaly ﬁxes the C µµ eﬀ Wilson coeﬃcient. Assuming forsimplicity that C µµ eﬀ is fully determined by g µµS , this determines all the other operators upto ratios of g ijS couplings: C ij eﬀ ≈ max( m (cid:96) i , m (cid:96) j ) m µ (cid:88) k g ikS g µµS g kjS g µµS , (2.18)where we have set g ijS = g jiS , again for simplicity. It is straightforward to obtain CLFVbranching ratios from this low-energy description, which can be used to constrain ratios ofthe singlet scalar couplings to diﬀerent fermion generations: (cid:88) (cid:96) g µ(cid:96)S g µµS g (cid:96)eS g µµS (cid:46) × − , (cid:88) (cid:96) g τ(cid:96)S g µµS g (cid:96)µS g µµS (cid:46) × − , (cid:88) (cid:96) g τ(cid:96)S g µµS g (cid:96)eS g µµS (cid:46) × − , (2.19)from µ → eγ , τ → µγ and τ → eγ decays respectively. We emphasize that these boundsassume that g µµS is ﬁxed to generate ∆ a obs µ . Clearly, ﬂavour-universal couplings of thesinglet scalar are excluded, and ﬂavour-anarchic couplings are severely disfavoured by CLFVbounds.The situation is similar for EW Scenarios. Consider ﬂavour anarchic versions of theSSF and FFS models: −L SSF ⊃ y i F c L i Φ ∗ A + y i F (cid:96) ci Φ B + κH Φ ∗ A Φ B (2.20) −L FFS ⊃ y i F cA L i S ∗ + y i F B (cid:96) ci S + y HF A F cB . (2.21)– 16 –gain, in this anarchic ansatz, the same new fermions and scalars that account for the ( g − µ anomaly generate the ﬂavour violating operators in Eqn. (2.17), and C ij eﬀ is determinedby ∆ a obs µ up to coupling ratios: C ij eﬀ ≈ g iS g µS g jS g µS , (2.22)where we again assumed for simplicity that y i = y i and that C µµeff is fully determined by y µ , . The only diﬀerence to scalar Singlet Scenarios is the absence of the lepton mass ratioin Eqn. (2.18), since for FFS and SSF models, the chirality ﬂip and Higgs coupling insertionnow lie on the propagators of the BSM particles in the loop. Repeating the estimates forCLFV decay branching ratios, we obtain the following bounds on the lepton coupling ratios: y e , y µ , (cid:46) − , y τ , y µ , (cid:46) − , y τ , y µ , y e , y µ , (cid:46) − , (2.23)from µ → eγ , τ → µγ and τ → eγ decays respectively if y µ , is ﬁxed by resolving the ( g − µ anomaly.Clearly CLFV constraints, in particular µ → eγ , exclude ﬂavour-universal BSM solu-tions to the ( g − µ anomaly (that involve new scalars), and severely constrain ﬂavour-anarchic ones. It is of course possible that a ﬂavour anarchic model evade the above con-straints by some coincidence (perhaps all the more unlikely given that the above couplingratio constraints have to be satisﬁed in the lepton mass basis after PMNS diagonalization,not the lepton gauge basis). However, it seems much more reasonable to take the absence ofobserved CLFVs as evidence of some protection against FCNCs in whatever UV-completetheory solves the SM ﬂavour puzzle, and that the physics of ( g − µ has to respect thatprotection.A robust model-independent framework that encompasses many possible ﬂavour em-beddings and provides strong protection against FCNCs is the Minimal Flavour Violation(MFV) ansatz (see e.g. [45, 46]). In MFV, the SM Higgs Yukawa matrices couplings areassumed to be the only spurions of global U (3) L × U (3) (cid:96) c → U (1) lepton ﬂavour breaking, sothat all BSM ﬂavour violation is aligned with the SM Yuwakas. Such a structure naturallyemerges if the SM Yukawa matrices arise as the VEVs of heavy UV ﬁelds responsible forbreaking a larger ﬂavour group.The MFV ansatz does not specify the representations of BSM ﬁelds under the ﬂavourgroup, but it does require all Lagrangian terms to be ﬂavour-singlets (with the Yukawamatrices as spurions). This would, for example, forbid oﬀ-diagonal terms in Eqn. (2.16),avoiding large CLVFs while still providing a viable explanation for ( g − µ over a wide rangeof scalar masses [93]. For EW Scenarios, the muon-scalar-fermion index has to involve aYukawa coupling factor and the scalar and fermion together have to contract into tripletsof U (3) L or U (3) (cid:96) c . This automatically forbids interactions of the form Eqn. (2.20) sincethere would have to be at least one separate BSM fermion (or scalar) for each lepton ﬂavourand the CLFV diagrams are not generated. This statement is strictly true only for massless neutrinos, in which case the lepton Yukawa matrices arespurions of U (3) L × U (3) (cid:96) c → U (1) e × U (1) µ × U (1) τ ﬂavor breaking and lepton ﬂavors are are separately – 17 –mposing MFV has several important consequences. First, non-trivial ﬂavour repre-sentations of BSM ﬁelds in EW Scenarios can give rise to more than one set of BSM statescoupling to the muon and contributing to ( g − µ . In eﬀect, this corresponds to N BSM > ,which is covered by our analysis. Second, MFV requires that some of the muonic BSMcouplings in the scalar singlet, SSF and FFS models have a tau-like equivalent that is atleast a factor m τ /m µ ≈ larger. This larger tau-like coupling will therefore have to satisfythe bounds of perturbative unitarity, eﬀectively lowering the upper bound from unitarityon the relevant muonic coupling that generates ∆ a obs µ by a factor of ≈ . This leadsto a dramatic reduction in the maximum allowed BSM mass scale compared to imposingunitarity alone (and implicitly assuming that CLFV decays are suppressed by accidentallysmall ﬂavour-anarchic BSM couplings in the lepton mass basis).Precisely which muonic BSM couplings have a tau-equivalent can depend on the U (3) L × U (3) (cid:96) c → U (1) lepton representation of the BSM ﬁelds. The situation is simple for the scalarSinglet Scenario, since the g S coupling must be in the same representation as the SMYukawas, and therefore g µS /g τS = m µ /m τ . For EW Scenarios there is more ambiguity. Anexample of a minimal choice for the ﬂavour representation of the BSM ﬁelds in the SSFmodel (the discussion is similar for FFS) is F ∼ (3 , , F c ∼ (¯3 , , S A,B ∼ (1 , (2.24)Since L ∼ (3 , and e c ∼ (1 , ¯3) this implies that y must transform like the SM electronYukawa while y can be a ﬂavour singlet: y ∼ y e ∼ (¯3 , , y ∼ (1 , (2.25)Therefore, the MFV assumption implies y µ /y τ = m µ /m τ and the y µ coupling eﬀectivelyhas a smaller perturbativity bound, while the upper bound for y is unaﬀected since thatcoupling is ﬂavour-universal. Other minimal choices can make y a ﬂavour singlet and y a bifundamental, but at least one of the two muonic y , couplings has its perturbativitybound reduced by m µ /m τ . Non-minimal ﬂavour representations for the BSM ﬁelds mayintroduce additional coupling ratios and hence even tighter perturbativity bounds, but forthe purposes of our conservative estimates, we only make the minimal assumption. The hierarchy problem in the SM is often formulated using an estimate of loop correctionsto the Higgs mass regulated with a ﬁnite momentum cutoﬀ Λ UV : ∆ m H ∼ y t π Λ , (2.26)where y t is the SM top Yukawa, which dominates this estimate. Avoiding ﬁne tuningof the Higgs mass parameter in the Lagrangian requires either cancellation of the above conserved. However, for nonzero neutrino masses, there will still be some CLFV contributions from thesemodels, but they involve diagrams with virtual W exchange and are further suppressed by powers of m ν /m W relative to the leading diagrams that resolve ( g − µ , so we do not consider them here. – 18 –uadratically divergent correction (SUSY) or new physics far below the Planck or GUT scale(i.e. a low UV cutoﬀ). This is simple and intuitive, appealing to the physical interpretationof unknown physics at some high scale in a Wilsonian picture. The cutoﬀ argument is also“morally correct” in that it accurately indicates the quadratic sensitivity of the Higgs massto UV corrections, whatever they may be. However, without knowledge of what the newphysics is, one could argue that the speciﬁc cutoﬀ-dependent quantity in Eqn. (2.26) hasno physical meaning. While it might seem unlikely or even absurd that quantum gravitycorrections at the Planck scale contribute nothing to ∆ m H , without explicit knowledge of(1) new physics between the weak scale and the Planck scale, and (2) the precise nature ofquantum gravity, one cannot be absolutely sure that the hierarchy problem does, in fact,refer to a real tuning of our universe’s parameters.The situation is entirely diﬀerent when explicit new states with high mass and sizeablecouplings to the Higgs are introduced, as is the case for the EW Scenarios we examine.These models have been engineered to account for the ( g − µ anomaly with the high-est possible BSM particle masses in order to perform the theory-space maximization ofEqn. (2.2) and identify the experimental worst-case scenario and the minimum energy offuture colliders required for discovery. Realizing these high-mass scenarios requires un-avoidably large couplings to the Higgs, which in turn leads to large but ﬁnite and calculable corrections to the Higgs mass; this makes the hierarchy problem explicit.Speciﬁcally, we can calculate the one-loop contributions of the new S, F ﬁelds to theHiggs mass using dimensional regularization (DR) as a regulator in the

M S renormalizationscheme. This gives contributions of the schematic form ∆ m H ∼ π (cid:18) c M + c M log µ R M (cid:19) , (2.27)where in this instance M BSM stands for various combinations of BSM masses in each term,and µ R is the renormalization scale. The quadratic UV sensitivity of the Higgs mass isillustrated by the ﬁrst term, with the size of the correction given by the scale of newphysics as expected.Naively, one might worry that the dependence of the second term on the renormalizationscale invalidates such a straightforward physical interpretation. One might in principlechoose µ R to set the above correction to zero. However, this would not be physicallymeaningful, since for such a choice of µ R , the perturbative expansion would be invalid.Restoration of perturbativity by inclusion of higher-loop diagrams would restore the largesize of ∆ m H . Therefore, the most reasonable physical interpretation of this correction isobtained setting µ R to optimize the validity of the perturbative expansion , in which casethe above one-loop result is the best possible approximation for the total size of the Higgsmass correction to all orders. This is why one typically choose µ R ∼ m in M S calculationsthat are dominated by physics at scale m . In that case, the µ R dependence becomes minorand simply corresponds to the fact that in a truncated perturbative expansion, there areunknown higher-order terms that could slightly modify the one-loop result.With this in mind, we ﬁx µ R ∼ O ( M BSM ) to the value that sets the log terms to zero.– 19 –his gives the following expressions for the Higgs mass corrections in SSF and FFS models: ∆ m H = C N BSM κ π (SSF) (2.28) ∆ m H = C N BSM π (cid:16) y y (cid:48) ( m A + m B ) + ( y + y (cid:48) ) m A m B (cid:17) (FFS) (2.29)where C , ∼ O (1) depend on the gauge representations of the new scalars and fermionsin the SSF/FFS model. The required presence of such corrections in BSM theories thatsolve the ( g − µ anomaly with the highest possible BSM mass scale makes the hierarchyproblem explicit.What is more surprising, if not entirely unfamiliar [4, 55, 80], is that these same theoriesactually lead to a second hierarchy problem for the muon mass . Fermion masses are usuallytechnically natural, but the required muon coupling to new heavy fermions F means theirchiral symmetry is shared in the limit where both are massless. Corrections to the muonmass therefore no longer scale with the muon Yukawa y µ . Following the same procedure asthe calculation of Higgs mass corrections we obtain corrections to the muon Yukawa due toloops of heavy fermions and scalars in EW Scenarios: ∆ y µ ∼ N BSM y y π κ m F M (SSF) (2.30) ∆ y µ ∼ N BSM y y y ( (cid:48) )12 π (FFS) (2.31)For large BSM couplings and masses, | ∆ y µ | (cid:29) y µ , necessitating tuning of the Lagrangianparameters. This hierarchy problem of the muon Yukawa arises due to large, calculablecorrections from new states present in the theory, making it just as explicit as the Higgshierarchy problem above.It is therefore reasonable to consider BSM scenarios that avoid adding two explicithierarchy problems to the SM by keeping such a dual-ﬁne-tuning to a reasonable minimum, e.g. each for the muon and Higgs mass. Similar to the MFV ansatz, this shrinks theviable parameter space by reducing the maximum allowed size of BSM couplings, therebyreducing the maximum BSM mass scale. The analysis of Singlet and EW Scenarios is discussed in detail in Sections 3 and 4. Ineach scenario, the viable parameter space of BSM masses and couplings is compact, sincewe require the new states to explain the ( g − µ anomaly, and the couplings cannot exceedthe limit set by perturbative unitarity, or unitarity + MFV, or unitarity + naturalness.Therefore, each scenario has well-deﬁned maximum BSM particle masses for a given BSMmultiplicity N BSM . 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 collidersignatures. Any lower-scale new physics that somehow cancels this ﬁne-tuning would lead to new experimentalsignatures and hence also lead to a discovery. – 20 –inglet Scenarios feature new SM singlets which can be invisible. Lighter singlets aremore weakly coupled to account for the ( g − µ anomaly, so the scenarios with the heaviestBSM particles are not necessarily the hardest to discover. Furthermore, the sensitivity ofcollider searches can depend on whether the new singlets are stable or how they decay. Wetherefore have to map out the complete parameter space of the simpliﬁed Singlet Scenarios.Fortunately, with the muon coupling g S,V determined by the requirement of accounting forthe observed ∆ a obs µ , the model has just two parameters, singlet mass m S,V and multiplicity N BSM (as well as the choice of singlet being a scalar or vector). As a function of massand multiplicity we then analyze the sensitivity of a completely inclusive search for theproduction of the BSM singlets at muon colliders regardless of their decays. We also analyzethe reach of an indirect search based on deviations in Bhabha scattering to explore thephysics potential of a muon collider Higgs factory. We ﬁnd that the singlet BSM statescannot be heavier than about 3 TeV, and can be directly discovered at a 3 TeV muoncollider with − for masses (cid:38)

10 GeV in singlet + photon production processes. A 215GeV muon collider that might be used as a Higgs factory can directly discover singlets aslight as 2 GeV in our conservative inclusive analysis with . − of luminosity. Heaviersinglets 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 performthe EW Scenario theory space maximization of BSM charged particle mass in Eqn. (2.2) ismuch more complex, featuring three masses, several BSM couplings, the number of BSMﬂavours N BSM , 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 witha given choice of N BSM and EW gauge representation we can simply ﬁnd the boundaries ofthe parameter space deﬁned by the maximum possible BSM masses that still allow BSMcouplings below the unitarity (or unitarity + MFV/naturalness) limit to account for the ( g − µ anomaly.For EW Scenarios we ﬁnd that requiring only perturbative unitarity allows the lightestcharged states to sit at the 100 TeV scale, but this assumption is disfavoured by CLFVbounds. Requiring either consistency with MFV to avoid CLFVs, or avoiding two explicitnew tunings worse than 1%, predicts new charged states at the 10 TeV scale or below.Encouragingly, these states are in reach of some muon collider proposals.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 strictlyspeaking 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 analysistools. This is suitable for deriving upper bounds on the BSM mass scale, but it is interestingto note these bounds could actually be saturated by strongly coupled BSM solutions to the ( g − 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 byconsidering N BSM > , with N BSM = 10 serving as a “high-multiplicity benchmark” for ouranalyses. Therefore, while our quantitative predictions are unlikely to apply precisely tostrongly coupled BSM solutions of the ( g − µ anomaly, by including couplings up to theunitarity limit and considering large numbers of BSM ﬂavours we parametrically include– 21 –he signature space swept out by these strongly coupled theories. The statements we makeabout the discoverability of new physics should, broadly speaking, apply to those scenariosas well. That being said, it would be interesting to undertake a dedicated investigation ofhigh-scale composite BSM solutions to the ( g − µ anomaly within our framework. Weleave this for future work.While CLFV constraints strongly favour the existence of some kind of ﬂavour protectionmechanism, the degree to which the precise assumptions of MFV would have to be satisﬁedis obviously up for debate. Similarly, the precise degree of tuning depends on the tuningmeasure, and it is diﬃcult to deﬁne exactly at what point a theory becomes “un-natural” ina meaningful sense. However, our model-exhaustive approach has the advantage of throwingthese issues into stark relief: Solving the ( g − µ anomaly with BSM masses up to ∼ and violation of MFV while somehow suppressing CLFV decays. In particular, if the 10 TeV scale were exhaustively probed without direct detectionof new states while the ( g − µ anomaly is conﬁrmed, this would conﬁrm empiricallythat nature is ﬁne-tuned and does not obey the assumptions of the MFV ansatz butstill suppresses CLFV decays in some way. An analogy would be the discovery of splitsupersymmetry [94, 95], where the lightest new physics states are heavy and couple to theHiggs; in our case, the situation is even more severe since heavy states in EW Scenariosmake the muon mass radiatively unstable as well, and very heavy BSM states also precludeMFV solutions to the SM ﬂavour puzzle.Our analysis generalizes and reinforces our earlier results in [55] by including a morecomplete basis for the relevant EW Scenarios, considering consistent electroweak embed-dings of Singlet Scenarios, addressing ﬂavour physics considerations, and supplying impor-tant technical details. Subsequent studies have employed an eﬀective ﬁeld theory (EFT)approach to explore indirect signatures of the new physics causing the ( g − µ anomaly atmuon colliders [53, 56]. While this EFT approach would not allow us to ask detailed ques-tions about the BSM physics – like studying direct particle production, tuning, and ﬂavourconsiderations – it is nonetheless extremely useful due to its maximal model-independenceand simplicity. As we discuss in Section 5, the results of these analyses are highly comple-mentary to our own and help ﬂesh out the muon collider no-lose theorem. ( g − µ in Singlet Scenarios As deﬁned in Eqns. (2.3) and (2.4), if BSM singlet scalars or vectors are responsible for the ( g − µ anomaly, the relevant muonic interactions are ( g S µ L µ c S + h.c. ) , g V V α ( µ † L ¯ σ α µ L + µ c † ¯ σ α µ c ) . (3.1) A similar observation was made in connection with electron EDM measurements [4] and also in [80] – 22 –he contribution of N BSM scalar singlets to ( g − µ is ∆ a Sµ = N BSM g S π (cid:90) dz m µ (1 − z )(1 − z ) m µ (1 − z ) + m S z ≈ × − N BSM g S (cid:18)

700 GeV m S (cid:19) , (3.2)where in the last step we have taken the m S (cid:29) m µ limit. For vectors, the corresponding ( g − µ contribution is ∆ a Vµ = N BSM g V π (cid:90) dz m µ z (1 − z ) m µ (1 − z ) + m V z ≈ × − N BSM g V (cid:18)

200 GeV m V (cid:19) , (3.3)where again we have taken the m V (cid:29) m µ limit. It is known in the literature that pseudo-scalar or pseudo-vector contributions to ( g − µ have the wrong sign to explain the anomaly[36], so we do not consider these scenarios here. Note also that in both cases ∆ a µ ∝ m µ ,which implies a low ( (cid:46) TeV) mass scale for any choice of perturbative couplings that yield ∆ a µ ∼ − required to explain the anomaly (see discussion in Sec. 3.2). Therefore, anyTeV-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 not just to identify thehighest singlet masses of interest, but rather to demonstrate that a plausible muon colliderwould unambiguously discover the signatures associated with these states regardless of theirmass or how they decay. In our analysis, we ﬁrst calculate the perturbative unitarity constraints on singlet couplings g S and g V that arise from the amplitude µ − µ + → µ − µ + with an intermediate S or V . Wethen calculate how the singlet mass is determined by the coupling to explain ( g − µ , up tothe maximum allowed values of these couplings. This will give a maximum possible massfor the singlet(s).The amplitude for the process µ − ( p ) µ + ( p ) → S/V → µ − ( p ) µ + ( p ) is given by (notethat we have temporarily switched to 4-component fermion notation for convenience) M S = ¯ u ( − ig S ) v is − m s ¯ v ( − ig S ) u − ¯ u ( − ig S ) u it − m s ¯ v ( − ig S ) v , (3.4) M V = ¯ u ( − ig V γ α ) v is − m V (cid:20) − g αβ + ( p + p ) α ( p + p ) β m V (cid:21) ¯ v ( − ig V γ β ) u − ¯ u ( − ig V γ α ) u it − m V (cid:20) − g αβ + ( p − p ) α ( p − p ) β m V (cid:21) ¯ v ( − ig V γ β ) v . (3.5)We calculated the constraints on the scalar and vector singlets by calculating Eqn. 2.11 fordiﬀerent 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 strongestconstrain was obtained for the process µ − ( λ + ) µ + ( λ − ) → µ − ( λ − ) µ + ( λ + ) . Using the proce-dures outlined in Section 2.3.1 we get the following constraints: g S ≤ πN BSM , g V ≤ πN BSM , (3.6)– 23 – igure 4 : The coupling of the singlet scalar ( g S ) and vector ( g V ) required to account for the ( g − µ anomaly as a function of its mass m S,V and multiplicity. For N BSM = 1 , perturbative unitarityimposes g S ≤ . and g V ≤ . , which implies an upper bound on the masses needed for ( g − µ of m s ≤ . and m V ≤ . , respectively. If one imposes MFV in the scalar couplings, theupper bounds for scalars become ( g S , m s ) ≤ (0 . ,

155 GeV) . Note that the N BSM -dependence ofthe singlet mass drops out by requiring ∆ a µ = ∆ a obs µ . where N BSM is the number of singlets with common masses and couplings in the theory.For N BSM = 1(10) the upper bound on the scalar singlet coupling is g S ≤ .

54 (1 . andon the vector singlet coupling is g V ≤ .

14 (1 . . In Figure 4 we show the singlet scalar or vector coupling required for a given mass toaccount for the ( g − µ anomaly. The upper bounds are m s ≤ . and m V ≤ . ,for scalar and vector singlets respectively. Even though the upper bound on the singletcouplings decreases as the number of BSM ﬂavours increases, the upper bound on the singletmasses does not change, since the N BSM dependence drops out by imposing ∆ a µ = ∆ a obs µ . As discussed in Section 2.3.2, CLFV constraints exclude ﬂavour-universal couplings of thescalar to leptons, and severely disfavour anarchic ones. This serves as strong motivation forthe MFV ansatz in scalar Singlet Scenarios, resulting in a lower maximum mass scale thanunitarity alone. Figure 4 shows that the scalar should be no heavier than 200 GeV if MFVis satisﬁed.The vector interaction V α ( µ † L ¯ σ α µ L + µ c † ¯ σ α µ c ) must arise from a new U (1) gauge ex-tension 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, the parameter space for explaining ( g − µ has been fully excluded for both visibly and invisibly decaying V [70, 96]; someviable parameter space still exists for semi-visible cascade decays, but this will be tested inwith upcoming low energy experiments [63]. If, instead, V couples directly to muons, the The process µ − ( λ + ) V ( λ + ) → µ − ( λ + ) V ( λ + ) can provide stronger constraints for singlet vectors with N BSM = 1 . However, because this process is N BSM independent, for larger values of N BSM the strongestconstraint is provided by Eqn. 3.6. We omit this constraint from our analysis for simplicity since it doesnot change our ﬁnal result. – 24 –nly anomaly free options for this gauge group are U (1) B − L , U (1) L i − L j , U (1) B − L i , (3.7)where B and L are baryon and lepton number respectively, and L i is a lepton ﬂavourwith i = e, µ, τ . Importantly, all of these options require couplings to ﬁrst generation SMparticles and are, therefore, excluded as explanations for ( g − µ by the same bounds thatrule out dark photons [70, 96], see also [99]. The sole exception is gauged L µ − L τ whichcan still explain the anomaly for m V , but in that case the vector mass is constrained tolie in the narrow range ∼ (1-200) MeV. This scenario will soon be tested with a variety oflow-energy and cosmological probes [72, 73, 100, 101]. Therefore, singlet vector scenariosare less relevant to our discussion of high energy muon collider signatures, but we includethem since their phenomenology is nearly identical to that of singlet scalars. We now discuss the collider signatures of Singlet Scenario explanations for the ( g − µ anomaly. In particular, here we focus on the region of masses above ∼ GeV , with theunderstanding that low energy experiments will cover the lower mass region. The ﬁrstsignal we discuss is direct production of the singlets in association with a photon. Thepresence of a photon is important because we will consider the possibility that the singletsdecay 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 isBhabha 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. Animportant question that we want to address is at which luminosity a given signal can bedetected at σ signiﬁcance for a given collider energy.We consider two possible muon colliders: a high energy 3 TeV collider with − ofintegrated luminosity and a low energy 215 GeV collider (a potential Higgs factory) with . − of luminosity. These benchmark luminosities are discussed by the internationalmuon collider collaboration at CERN [102]. As opposed to conventional colliders, MuC hasthe extra complication of Beam-Induced Background (BIB) due to muon decay-in-ﬂight.For this reason the detector design includes two tungsten shielding cones along the directionof the beam. The opening angle of these cones should be optimized as a function of theenergy of the MuC. In order to be conservative, our simulations assume that the detectorcannot reconstruct particles with angles to the beamline below ◦ ( ◦ ) for the higher(lower) energy muon collider [103]. Here we focus on single production of the singlets in association with a photon. In principle,to study direct production of the singlets one would need to make an assumption about Other U (1) options may also be viable if additional electroweak charged BSM states are included tocancel anomalies, but these models are phenomenologically similar for the purpose of our ( g − µ analysisand are further subject to strong bounds at scales below the masses of these new particles [97, 98]. – 25 – − µ + ? γS/V Figure 5 : Single production of the singlet in association with a photon at a muon collider. Thesinglets can be stable and constitute missing energy, or decay to any SM ﬁnal states. The searchis deﬁned by the search for the recoiling photon, as well as any possible SM ﬁnal states (includingmissing energy) inside the singlet decay cone. how they decay to optimally search for them at the collider. We want to avoid such a modeldependence by implementing an inclusive analysis for singlet + photon production with thefollowing signal topology for a given singlet mass m S , illustrated in Figure 5:1. A nearly monochromatic photon with E γ ∼ √ s/ (with some mild dependence onthe singlet mass) in one half of the detector.2. No other activity anywhere else in the detector, except inside of a “singlet decay cone”of angular size φ max around the assumed singlet momentum vector (cid:126)p S = − (cid:126)p γ .3. For each singlet mass, φ max is deﬁned as the opening angle within which ∼ ofsinglet decay products must lie, regardless of decay mode. This is determined fromsimulation under the assumption that the singlet decays to two massless particles,which gives the largest possible opening angle of any decay mode.4. There are no requirements of any kind on what ﬁnal states are found inside the singletdecay cone. This gives near-unity signal acceptance for stable singlets (resulting inmissing energy) as well as all possible visible or semi-visible decay modes.The veto on detector activity anywhere except the monochromatic photon and inside thesinglet decay cone would have to be adjusted for a realistic analysis due to the presence ofBIB and initial- and ﬁnal-state radiation. However, the former is likely to be subtractableand the latter are small corrections at a lepton collider, not greatly reducing signal accep-tance. We therefore ignore this complication with the understanding that a more completetreatment would not signiﬁcantly change our results.This inclusive analysis allows us to remain as model-independent as possible, somethingthat is necessary when scanning over a large range of singlet masses with only the coupling tothe muon known, without paying any branching fraction penalty that would arise by perhapstrying to exploit some minimum decay rate to muons. For instance, for m S (cid:38)

200 GeV , themuon coupling is > , making it natural for the dominant decay mode to yield two muons,although other visible or invisible decay modes could be co-dominant. For smaller masses,– 26 –ass(GeV) E γ bin(GeV) ∆ E γ (Γ singlet ) ∆ E γ (ECAL) Background(fb) Signal (fb)Scalar Vector10 (1492, 1508) 0.02 16.17 3.23 0.22 4.31100 (1490, 1506) 2.0 16.15 3.65 14.1 391500 (1433, 1483) 50 15.75 2.51 372 11,1771000 (1233, 1433) 200 14.50 3.18 1,636 52,074Muon Collider Energy: 3 TeV Table 2 : Photon energy bins as well as background and signal cross sections for diﬀerent singletmasses. The width of the energy bin corresponds to the maximum of the third and fourth columnsfor a given row. Values in this table correspond to a MuC with √ s = 3 TeV.

Mass(GeV) E γ bin(GeV) ∆ E γ (Γ singlet ) ∆ E γ (ECAL) Background(fb) Signal (fb)Scalar Vector1 (106, 108) 0.01 2.07 2.56 0.247 1.5810 (106, 108) 0.28 2.07 9.14 10.86 147.450 ( 98, 105) 6.98 2.02 77.9 172.7 3356100 ( 90, 96) 28 1.96 5.78 6.821 100.8Muon Collider Energy: 215 GeV Table 3 : Similar to Table 2 but for a 215 GeV MuC. e.g. close to 1 GeV, the muon coupling is 2-3 orders of magnitude smaller, and the singletcould decay to invisible particles, electrons, quarks, or photons.Note that instead of searching for bumps in the invariant mass distribution of candidatesinglet decay products inside the decay cone, we analyze the photon energy distribution.This takes advantage on the fact that producing an on-shell particle in association with aphoton forces the latter to be nearly monochromatic in a lepton collider. For a given singletmass, the photon energy is determined within a bin ( ∆ E bin γ ) whose width is correlated withthe decay width of the singlet. We calculated ∆ E bin γ assuming a decay width of around the mass of the singlet, which is near the upper bound from perturbative unitarityand very conservative. For small singlet masses that result in a very narrow photon energydistribution, we instead deﬁne the bin size ∆ E bin γ to be equal to the energy resolution of theelectromagnetic calorimeter (ECAL). We assume an ECAL resolution similar to that of theLarge Hadron Collider (LHC) main detectors [104], again a very conservative assumptionthat takes into account the most important detector eﬀects. Tables 2 and 3 show theassumed photon energy bins ∆ E bin γ for a few values of the singlet mass at a 3 TeV and 215GeV MuC.We assume singlet production for each possible scalar or vector mass is determinedonly by the coupling g S , g V to the muon, which is in turn ﬁxed by ∆ a µ = ∆ a obs µ . We thencalculated the production cross section by coding up the Singlet Scenarios as simpliﬁedmodels in FeynRules [105] and generating tree-level signal events with [email protected] [106]. We conﬁrmed that with the above cuts, signal acceptance for singlet decays is close to– 27 – igure 6 : Luminosity needed for σ discovery signiﬁcance of inclusive Singlet Scenario searches ata 215 GeV and 3 TeV muon collider for singlet scalars (green) and singlet vectors (orange). Thisis shown for singlet masses up to the perturbativity limit calculated in Section 3.2. Dashed lines(solid lines) show the results from the inclusive direct γ + X analysis (Bhabha scattering analysis).Note that these sensitivities do not depend on N BSM . γ + ¯ f f (includingneutrinos) and γ + γγ ﬁnal states at tree-level and imposing the above cuts in an oﬄineanalysis. Background contributions involving additional SM states would either fail one ofthe vetos or cut on additional states outside of the decay cone, or supply small correctionsto the lowest-order background rates we calculate in our signal region. Our analysis shouldtherefore reliably estimate the sensitivity of a realistic inclusive singlet search. Table 2shows the total background cross section after imposing analysis cuts for a few values ofthe singlet mass and compares them to signal.In the right panel of Fig. 6, dashed lines show that a 3 TeV MuC with − ofluminosity will be able to probe singlet masses above 11 GeV for scalars and 2.4 GeV forvectors through γ + X events. Note that these sensitivities do not depend on N BSM , sincesignal rates at the MuC and ∆ a µ both scale as N BSM g S,V .In order to probe smaller masses, one could use a lower energy MuC. In the left panelof Fig. 6 we see that a 215 GeV MuC with . − will probe masses above 1.4 GeV forscalars and sub-GeV masses for vectors, owing to the larger production rate for light statesat lower collider energies. Such a lower-energy collider might be built as a MuC test-bedor Higgs factory, and while it would not be able to directly produce singlets at the heaviestpossible masses allowed by unitarity, it would cover most of the scalar parameter spaceallowed under the most motivated MFV assumption. Furthermore, as we show in the nextsection, it will be able to indirectly discover the eﬀects of the Singlet Scenarios by detectingdeviations in Bhabha scattering. In the Standard Model, Bhabha scattering is mediated by s - and t -channel exchange of botha photon and a Z boson (Fig. 7, top). New physics contributions from singlet scalars andvectors have a similar topology (Fig. 7, bottom) and can produce measurable deviations.– 28 – + µ − µ − µ + γ/Z µ + µ + µ − µ − γ/Z µ + µ − µ − µ + S/V µ + µ + µ − µ − S/V

Figure 7 : Feynman diagrams for Bhabha scattering in the SM (top) and contributions fromsinglet scalars or vectors (bottom). (Note that the arrows in this diagram represent charge ﬂow,not helicity.)

When the energy of the collisions is close to the mass of the singlets, the distinctive signatureof Bhabha scattering is a resonance peak at the mass of the singlet. However, when theenergy of the collisions is lower, one could instead can look for deviations in the total crosssection of the process due to contributions from oﬀ-shell singlets. The potential problemwith this approach is that measurements of total rates for Bhabha scattering are sometimesused to calibrate beams and measure instantaneous luminosity [107]. To avoid possiblecomplications in that regard, one can measure deviations in ratio variables similar to aforward-backward asymmetry in parity-violating observables. Ratio variables also havethe advantage of mitigating the eﬀect of systematics. We therefore deﬁne the ratio of thenumber of forward to backward µ + µ − → µ + µ − events: r FB ≡ (cid:90) c θ dσdc θ dc θ (cid:90) − c θ dσdc θ dc θ , (3.8)where c θ is the cosine of the muon scattering angle, dσ/dc θ is the diﬀerential cross section ofthe process µ − µ + → µ − µ + , and the minimum angle θ is given by the angular acceptanceof the MuC detector. The dependence of this variable on singlet mass is illustrated inFig. 8 for a 215 GeV (left) and 3 TeV (right) MuC. For a given mass, the singlet coupling isdetermined by the value of ( g − µ . Note that this result again does not depend on N BSM since it depends only on g S,V N BSM , which is ﬁxed by ∆ a µ = ∆ a obs µ .In Figure 8, blue lines represent the SM result. As expected, the number of forwardevents exceeds that of the backward events by orders of magnitude in the SM. This istypical for Bhabha scattering due to t -channel enhancements. The contribution of singlets– 29 – MScalar

Vector

SMScalar

Vector

Figure 8 : Prediction for the forward-backward asymmetry variable r FB in Bhabha scattering forSinglet Scenarios at a 215 GeV and 3 TeV MuC. This is independent of N BSM . interferes with the SM contribution and reshapes the angular distribution, resulting indeviations from the SM expectation for r FB . In particular, near an s -channel resonance, r FB → , as expected because the singlet-muon coupling is parity-conserving. To addressthe question of how much luminosity is needed to discover deviations from the expectedSM behaviour of Bhabha scattering with σ statistical signiﬁcance, we calculate r FB forthe background-only hypothesis r SMFB and compare it with the background+signal hypothesis r SM+NPFB , obtaining the corresponding χ , χ = (cid:16) r SM+NPFB − r SMFB (cid:17) (cid:16) ∆ r SM+NPFB (cid:17) + (cid:0) ∆ r SMFB (cid:1) . (3.9)The uncertainties in the denominator arise from Poisson statistics in the number of forwardand backward events expected at each mass and luminosity.In the right panel of Fig. 6, solid lines show that a 3 TeV ( − ) MuC will be ableto probe singlet masses above 58 GeV for scalars and 14 GeV for vectors through Bhabhascattering. More importantly, a 215 GeV ( . − ) MuC will probe masses above 17.5 GeVfor scalars and 5.5 GeV for vectors. The most important role of Bhabha scattering is inenabling a lower-energy 215 GeV muon collider to discover the eﬀects of Singlet Scenariosthat solve the ( g − µ anomaly over the entire allowed mass range of the singlets (incombination with the inclusive direct search). We close this section by commenting on possible UV completions of Singlet Scenarios. Itis important to keep in mind that the scalar-muon coupling in the singlet scalar modelhas to be generated by the non-renormalizable operator c S Λ Hµ L µ c S after electroweak sym-metry breaking. There are only a few ways of generating this operator at tree-level usingrenormalizable interactions. – 30 –he simplest possibility involves the SH † H operator, which introduces S - H mass mix-ing after electroweak symmetry breaking. Diagonalizing away this mixing induces the Sµ L µ c 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 explanationfor ( g − µ [108] and similar arguments sharply constrain models in which S mixes withthe 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 vector-like fourth generation of leptons ψ i . If the ψ i undergo mass mixingwith L and µ c , the requisite operator Sµ L µ c can arise upon diagonalizing the full leptonicmass matrix after electroweak symmetry breaking. In such models, these states inheritthe ﬂavour 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 ψ i states aresuﬃciently light ( (cid:46) few TeV), they may be accessible at future proton and electron colliders,e.g. via established search strategies for heavy new vector-like leptons [76]. However, giventhe multiple dimensionless and dimensionful couplings that these models allow (each withpotentially non-trivial ﬂavour structure), it is also possible for these additional states to befar 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, but wemerely point out that the existence of charged states at or below the TeV scale is not strictlynecessary to realize the scalar Singlet Scenario. On the other hand, discovering these scalarsinglets at a muon collider only relies on the coupling g S that is determined by solving the ( g − µ anomaly. In Section 2.2, we deﬁned the SSF and FFS simpliﬁed models, with Lagrangians given inEqns. (2.5) and (2.6), which we repeat here for convenience L SSF ⊃ − y F c L ( µ ) Φ ∗ A − y F µ c Φ B − κH Φ ∗ A Φ B − m A | Φ A | − m B | Φ B | − m F F F c + h.c. (4.1) L FFS ⊃ − y F cA L ( µ ) Φ ∗ − y F B µ c Φ − y HF cA F B − y (cid:48) H † F A F cB − m A F A F cA − m B F B F cB − m S | Φ | + h.c. . (4.2)For N BSM > , we simply consider multiple degenerate copies of the above ﬁeld content. InSSF (FFS) models, the fermion F (complex scalar S ) is in SU (2) L representation R with hy-percharge Y , while the two complex scalars Φ A,B (two fermions F A,B ) are in representation R A,B with hypercharges Y A,B .As we discussed in Section 2, these two simpliﬁed models include the most generalform of new one-loop contributions to ( g − µ , see Figure 3 (bottom). In particular, sinceevery particle in the loop is assumed to be a BSM ﬁeld, the new couplings y , y , y , y (cid:48) , κ – 31 –re experimentally unconstrained for BSM masses above a TeV or so, and can be chosento maximize ∆ a µ subject only to perturbative unitarity (and optionally imposing MFV ornaturalness), which in turn allows ∆ a obs µ to be generated by the heaviest possible BSMstates under the assumption of perturbative unitarity and electroweak gauge invariance.This allows us to perform the theory space maximization in Eqn. (2.2) by only performingthe maximization over the parameter space of all possible SSF and FFS models, as inEqn. (2.7). The possibilities not covered by these scenarios, like Majorana fermions orreal scalars, give smaller ( g − µ contributions and hence must feature lighter BSM statesthan the SSF and FFS scenarios, which does not change the outcome of the theory spacemaximization.Analyzing these two SSF and FFS simpliﬁed model classes therefore allows us to ﬁndthe heaviest possible mass of the lightest new charged state in the theory. This dictatesthe minimum center-of-mass energy a future collider must have to guarantee discovery ofnew physics by direct Drell-Yan production and visible decay of heavy new states. Inparticular, the discovery of charged states with mass m (cid:46) √ s/ at lepton colliders is highlyrobust [109], since they have sizeable production rates given by their gauge charge and haveto lead to visible ﬁnal states in the detector. This is why our results allow us to formulatea no-lose theorem for future muon colliders.Each individual SSF or FFS model is deﬁned by the choice of electroweak representa-tions for the new scalars and fermions. In principle there are inﬁnitely many possibilitiesthat satisfy the requirements in Eqn. (2.8), but theories with very large EW representationslead to issues such as low-energy Landau poles (see Section 4.7) or multiply-charged stablecosmological relics. We therefore restrict ourselves to models where all new particles haveelectric charge | Q | ≤ . Table 4 shows a summary of all the EW Scenarios we explicitlyanalyzed as part of our study, showing the SU (2) L ⊗ U (1) Y representation of the BSMﬁelds, which are all the unique possibilities with electric charges of 2 or below and represen-tations up to and including triplets of SU (2) L . This table also lists the highest mass thatthe lightest charged BSM state in the spectrum can have subject to unitarity, unitarity +MFV, unitarity + naturalness and unitarity + naturalness + MFV constraints. For eachassumption, the last row contains M maxBSM , charged . This constitutes our main result, which weexplain in the sections below. Crucially, in some scenarios the lightest charged state doesnot actually participate in the loop that generates ∆ a obs µ , but its existence is nonethelessrequired by electroweak gauge invariance.The requirement of | Q | ≤ in principle allows for theories featuring SU (2) L representa-tions up to and including the . However, we ﬁnd that the largest possible BSM mass doesnot appear to increase for higher-rank representations. Therefore, we believe our results for M maxBSM , charged to be robust even though we do not explicitly analyze scenarios involving and representations. ( g − µ in Electroweak Scenarios It is straightforward to compute the general BSM one-loop contribution to ( g − µ , repro-ducing results from the literature [36, 39]. It is convenient to work in the low-energy theorybelow the scale of electroweak symmetry breaking. Consider an eﬀective Lagrangian with a– 32 – ighest possible mass (TeV)of lightest charged BSM state Unitarity Unitarity + Unitarity + Unitarity +only MFV Naturalness Naturalness +MFV N BSM : N BSM : N BSM : N BSM :Model

R R A R B − / − / − / − − / − − / − / − / / / − − / − / − / − / / / − − / − / − / − − / − / − / − − / − − / − /

40 126 9.38 29.7 10 14.3 3.51 8.04 − / / / − − / − / − / − / / / − − / − /

71 225 17 53.6 15.6 22.2 5.72 12.8 − / − M maxBSM , charged (max in each column) Table 4 : Summary of all the EW Scenarios we analyze as part of our study. In SSF models, F ∼ R, Φ A,B ∼ R A,B . In FFS models, S ∼ R, F

A,B ∼ R A,B , and the choices of representationsare shown in columns 2–4, which covers all unique possibilities satisfying | Q | ≤ involving SU (2) L representations up to and including triplets. Columns 5–6, 7–8, 9–10 and 11–12 show the highestpossible mass in TeV of the lightest BSM state in the spectrum, with the BSM couplings constrainedonly by unitarity, unitarity + MFV, unitarity + naturalness and unitarity + naturalness + MFVrespectively. For illustration of the N BSM dependence, we show results for a single copy of theBSM states N BSM = 1 , or for N BSM = 10 . The highest possible BSM mass scale for unitarityand unitarity + MFV constrained couplings scales as ∼ N / , while the naturalness constraintof less than tuning of both the Higgs and muon mass softens this dependence to ∼ N / .The combined MFV + naturalness constraint scales as ∼ N / Note that in some scenarios, thelightest charged state does not directly contribute to ( g − µ , but its existence is nonetheless arequirement of EW gauge invariance. The largest possible mass of the lightest new charged stateacross all the scenarios we examine is shown in the last row, which corresponds to the theory-spacemaximization in Eqn. (2.7) and hence Eqn. (2.2). We do not expect the inclusion of higher SU (2) L representations to meaningfully increase this upper bound. – 33 –ingle new Dirac fermion Ψ F with mass m F and charge Q F , and a complex scalar Φ S withmass m S and charge Q S interacting with the muon as follows: L ⊃ − ¯Ψ F ( aP L + bP R ) µ Φ ∗ S + h.c. (4.3)Note we have temporarily switched to 4-fermion notation for this low-energy calculation; µ is the muon spinor, and P L,R are the left- and right-chirality projectors. The contributionof particles Ψ F , Φ S to ( g − µ is given by: ∆ a µ ( a, b, m F , m S , Q F , Q S ) = − m µ m F π m S (cid:26) Q F (cid:20) Re( a ∗ b ) I F ( (cid:15), x ) + ( | a | + | b | ) m µ m F ˜ I F ( (cid:15), x ) (cid:21) − Q S (cid:20) Re( a ∗ b ) I S ( (cid:15), x ) + ( | a | + | b | ) m µ m F ˜ I S ( (cid:15), x ) (cid:21)(cid:27) (4.4)where (cid:15) = m µ /m S , x = m F /m S and the loop integrals are: I F ( (cid:15), x ) = (cid:90) dz (1 − z ) (1 − z )( x − z(cid:15) ) + z (4.5) ˜ I F ( (cid:15), x ) = 12 (cid:90) dz z (1 − z ) (1 − z )( x − z(cid:15) ) + z (4.6) I S ( (cid:15), x ) = (cid:90) dz z (1 − z )(1 − z )(1 − z(cid:15) ) + zx (4.7) ˜ I S ( (cid:15), x ) = 12 (cid:90) dz z (1 − z ) (1 − z )(1 − z(cid:15) ) + zx (4.8)Eqn. (4.4) makes it straightforward to calculate ( g − µ for all the EW Scenarios inTable 4 (which may involve several scalar-fermion combinations coupling to the muon andcontributing to ∆ a µ ), after solving for the BSM spectrum after EWSB. In FFS models, ∆ a µ ∼ N BSM y , y ( (cid:48) )12 v m µ m , (4.9)where m BSM is some combination of the BSM particle masses, while for FFS models, ∆ a µ ∼ N BSM y , κv m µ m . (4.10)Once upper bounds on the BSM couplings from unitarity or other considerations are deter-mined, we can therefore ﬁnd upper bounds on the BSM mass scale under the assumptionthat ∆ a µ = ∆ a obs µ . As discussed in Section 2.3.1, the BSM couplings in SSF and FFS theories have to sat-isfy perturbative unitarity. Deriving the upper bounds for the new Yukawa couplings is– 34 –traightforward. We constrain the Yukawa couplings y and y in the SSF models fromthe process µ − ( λ ± ) F ( λ ∓ ) → µ − ( λ ± ) F ( λ ∓ ) . The same Yukawas in the FFS models wereconstrained from processes µ − ( λ ± ) S → µ − ( λ ± ) S , whereas for the extra Yukawas y and y (cid:48) we used the processes f i ( λ ± ) f j ( λ ± ) → f k ( λ ± ) f l ( λ ± ) , where f i are the mass eigenstatesof the two fermions in the model after mixing. For scalar-fermion scattering, the inter-mediate fermion propagator scales at large s as / √ s for the + → + helicity-preservingamplitude, and M/s for the helicity-violating + → − amplitude, where M is the mass ofthe intermediate fermion. After taking into account the normalization of the initial-andﬁnal-state spinors, we ﬁnd that the + → + amplitudes are independent of energy (and giveconstraints y (cid:39) O (1) × √ π where y is a Yukawa coupling), while the + → − amplitudesare largest at small s . For the SSF and FFS model respectively, the constraints are: | y | , | y | ≤ √ π ≈ . (SSF unitarity bound) (4.11) | y | , | y | ≤ √ π ≈ . (FFS unitarity bound) (4.12) | y | , | y (cid:48) | ≤ √ π ≈ . independent of N BSM .Obtaining a unitarity bound for the dimensionful coupling κ in SSF models is slightlymore involved. It has to satisfy | κ | < κ max , where parametrically, κ max = d ( m A , m B , m F ) m A m B v . (4.13)The dimensionless factor d is a function of BSM mass parameters with size d ∼ O (0 . − if there is large hierarchy between m A and m B , asymptoting to d (cid:28) as m A → m B . Thisupper bound on the size of κ is far more restrictive than the requirement that none of thenew scalars acquire VEVs. The derivation is as follows. Scalar-scalar amplitudes are asum of 3-point and 4-point diagrams; the latter are independent of energy, but the formerscale as κ /s . Thus the amplitude will be largest, and hence the strongest constraintson κ will generally be obtained, at the smallest s which is kinematically accessible, whichin principle motivates focusing on the scattering channels with the smallest initial- andﬁnal-state masses, namely hS i → hS j . However, these processes include cases where s -, t -, and u -channel singularities appear. The s -channel poles appear due to the exchange ofa scalar S k whose mass is above the threshold s = ( m h + m S i ) . We can avoid dealingwith such poles by considering the scattering of the lightest scalars S i through s - and t -channel exchange of a Higgs boson. This way, neither of the s, t, u channel singularitiesappear when calculating the constraints given by Eqn. (2.11). In this sense, our constraintsare conservative, but they avoid deﬁning arbitrary ways to deal with singularities (a fullycorrect treatment would be model-dependent), and is suﬃcient to ﬁnd a conservative butuseful estimate of M maxBSM , charged .The scattering amplitude for the process S i S i → S i S i is given by M = − λ eﬀ − κ (cid:18) s − m h + 1 t − m h (cid:19) , (4.14)– 35 –here the coeﬃcients λ eﬀ and κ eﬀ are functions of mixing angles, self-quartics for the scalars S A , S B , quartics between diﬀerent scalars and/or the Higgs (indicated by subscripts): λ eﬀ = cos θ λ A + cos θ sin θ λ AB + sin θ λ B , (4.15) κ eﬀ = −√ θ sin θκ + cos θ vλ AH + sin θ vλ BH . (4.16)From this process, the lowest-order partial wave is given by a = − π (cid:115) − m S i s (cid:18) λ eﬀ + 2 κ s − m h (cid:19) + 2 κ (cid:113) s ( s − m S i ) log (cid:34) m h m h + ( s − m S i ) (cid:35) . (4.17)The unitarity bound on κ < κ max corresponds to the maximum value that for a given setof parameters (couplings, masses, etc.), satisﬁes the condition | Re( a ) | ≤ , (4.18)for large s , but since the constraint asymptotes rapidly above threshold, this correspondsto requiring consistency of the theory close to (a factor of a few above) threshold s (cid:38) m S i .To marginalize over the dependence of scalar quartic couplings, we maximized κ max withrespect to the unknown quartics, subject to these quartics themselves obeying perturbativeunitarity.We can now ﬁnd the upper bound on the BSM particle masses in each model, underthe assumption that ∆ a µ = ∆ a obs µ . For each SSF (FFS) model in Table 4 the explicit stepsin the calculation are the following:1. For a given choice of scalar (fermion) mass parameters m A , m B and coupling κ ( y , y (cid:48) ), ﬁnd the masses and eﬀective muon couplings of all the mass eigenstates.The ∆ a µ contribution can then be found using Eqn. (4.4).2. Find largest fermion mass m F (scalar mass m S ) that can still generate ∆ a obs µ , un-der the assumption that the BSM couplings y , y , κ ( y , y , y , y (cid:48) ) are chosen tomaximize ∆ a µ subject only to the above unitarity bounds.3. With the fermion (scalar) mass ﬁxed to this maximum value and the couplings chosento maximize ∆ a µ , the entire BSM spectrum of the theory is fully determined as afunction of just the two scalar (fermion) masses m A , m B . As expected, we ﬁnd that ∆ a obs µ can be generated only in a compact region of the ( m A , m B ) -plane.4. We can then ask at each point in this plane what the mass of the lightest chargedBSM state is. This is shown in Figure 9 (1st row) for two representative SSF models.Importantly, in some theories, the lightest charged state does not contribute to ( g − µ , but its existence and mass is determined by gauge invariance in the given SSFor FFS model. – 36 – SF, all BSM ﬁelds charged SSF, charged and neutral ﬁelds U n i t a r i t y o n l y U n i t a r i t y + M F VU n i t a r i t y + N a t u r a l n e ss U n i t a r i t y + N a t u r a l n e ss + M F V Figure 9 : Contours show mass in TeV of lightest charged state in two representative SSF modelswith N BSM = 1 as a function of scalar masses m A , m B . The largest possible fermion mass m F was determined by ∆ a BSM = ∆ a obs µ , with the couplings y , y , κ chosen to maximize ( g − µ whileobeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity+ naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, ( R, R A , R B ) =(1 − , / , ) , and all ﬁelds contributing to ( g − µ are charged. On the right, ( R, R A , R B ) =(1 − , / , ) , and the scalars in the ( g − µ loop are neutral but since Φ A is an EW doublet, thereis a charged scalar with mass m A . – 37 –. Since the region of parameter space that can account for ∆ a obs µ is compact, we candetermine the highest possible mass of the lightest charged BSM state that is consistentwith this particular EW Scenario accounting for the ( g − µ anomaly.In eﬀect, this procedure allows us to explore the “maximum-BSM-mass boundary” of eachEW Scenario’s parameter space, subject to the requirement that ∆ a µ = ∆ a obs µ and the BSMcouplings obey perturbative unitarity. The resulting highest possible mass of the lightestBSM state in the spectrum for each EW Scenario we examine is listed in columns 5 and 6of Table 4 for N BSM = 1 and 10 respectively.Obviously, the result for a given model in Table 4 is not particularly illuminating, sinceit is by deﬁnition model-dependent. However, obtaining this maximum allowed mass of thelightest new charged state for diﬀerent possible choices of EW gauge representations in bothSSF and FFS models allows us to perform the theory space maximization in Eqn. (2.7), andhence obtain M maxBSM , charged for all possible perturbative solutions of the ( g − µ anomaly: M max , unitarityBSM , charged ≡ max ∆ a µ =∆ a obs µ , perturbative unitarity (cid:26) min i ∈ BSM spectrum (cid:16) m ( i )charged (cid:17) (cid:27) (4.19)where we have added the ‘unitarity’ superscript to distinguish this bound from subsequentresults with additional assumptions. We can perform this maximization by taking thelargest values from columns 5 and 6 in Table 4, which are shown in the last row. Wetherefore present the ﬁnal result of our perturbative unitarity analysis of EW Scenarios: M max , unitarityBSM , charged ≈ (cid:40)

100 TeV for N BSM = 1360 TeV for N BSM = 10 (cid:41) ≈ (100 TeV) · N / . (4.20)The N BSM scaling arises due to the linear dependence of ∆ a µ on N BSM . For FFS models,this is clearly seen from Eqn. (4.9), while for SSF models this relationship is obscured bythe detailed form of the unitarity bound on κ , but we veriﬁed the approximate √ N BSM scaling empirically. New charged states therefore have to appear at or below the 100 TeVscale unless N BSM is truly enormous, a scenario which is disfavoured not just by theoreticalparsimony but also by avoiding Landau Poles close to the BSM mass scale, see Section 4.7.It is important to keep in mind that realizing this upper bound from unitarity would alsorequire extreme alignment of the non-muonic BSM couplings to avoid CLFV decay bounds,see Section 2.3.2. This can be regarded as a severe form of tuning of the BSM leptoncouplings before mass diagonalization, which disfavours the unitarity-only assumption.

As discussed in Section 2.3.2, the MFV assumption is motivated for EW Scenarios bysevere experimental bounds on CLFV decays. Adopting this “Unitarity + MFV” assump-tion signiﬁcantly reduces the maximum allowed BSM mass scale. We repeat verbatim theunitarity-only analysis from Section 4.3, with the additional step of lowering the pertur-bativity bound on either y or y by m µ /m τ , whichever gives higher BSM masses at thatpoint in parameter space. (In practice there is almost no diﬀerence between these twopossibilities since ∆ a µ ∝ y y up to tiny corrections.) The resulting largest possible mass– 38 –f the lightest BSM charged state for two representative SSF models is shown in Figure 9(2nd row), with the results for all EW Scenarios we examine summarized in columns 7 and8 of Table 4 for N BSM = 1 and 10 respectively. We can therefore deﬁne, for all possibleperturbative solutions of the ( g − µ anomaly that obey MFV: M max , MFVBSM , charged ≡ max ∆ a µ =∆ a obs µ , unitarity , MFV (cid:26) min i ∈ BSM spectrum (cid:16) m ( i )charged (cid:17) (cid:27) (4.21)where the outer theory-space maximization is now constrained by unitarity as well as MFV,and can again be performed by taking the largest values from columns 7 and 8 in Table 4,which are shown in the last row. This gives: M max , MFVBSM , charged ≈ (cid:40)

20 TeV for N BSM = 173 TeV for N BSM = 10 (cid:41) ≈ (20 TeV) · N / . (4.22)The reduction in BSM mass scale compared to the unitarity-only assumption is very sig-niﬁcant, and could be within reach of future muon collider proposals. The physical concreteness of the Higgs and muon mass corrections in EW Scenarios, seeEqns. (2.28) - (2.31), means that conﬁrmation of the ( g − µ anomaly and conﬁrmednon-existence of the required new charged states up to some scale M exp means that thesestates must exist at some scale M BSM > M exp , which implies a certain amount of tuningin the Lagrangian. Such an empirical conﬁrmation of ﬁne-tuning would have profoundconsequence for our thinking about the hierarchy problem or cosmological vacuum selection.It is therefore worth quantifying how heavy the new charged states could be without inducingsuch physical ﬁne-tuning.We therefore deﬁne a very conservative “naturalness” criterion by requiring the tuningin both the Higgs mass and the muon Yukawa coupling to not exceed , which amountsto imposing ∆ ≡ max (cid:18) ∆ m H m H , ∆ y µ y µ (cid:19) < . (4.23)We repeat verbatim the unitarity-only analysis from Section 4.3, with the above naturalnessbound applied in addition to the unitarity bound. In practice, this means that both theHiggs and muon masses are tuned at the level for the largest BSM masses we ﬁnd, sincemaximizing all couplings relevant for ∆ a µ saturates both tuning bounds.The largest possible mass of the lightest BSM charged state for two representativeSSF models under this “unitarity + naturalness” assumption is shown in Figure 9 (3rdrow), with the results for all EW Scenarios we examine summarized in columns 9 and 10of Table 4 for N BSM = 1 and 10 respectively. We can therefore deﬁne, for all possibleperturbative solutions of the ( g − µ anomaly that obey our conservative naturalnessrequirement Eqn. (4.23), the largest possible mass of the lightest BSM states: M max , naturalnessBSM , charged ≡ max ∆ a µ =∆ a obs µ , unitarity , ∆ < (cid:26) min i ∈ BSM spectrum (cid:16) m ( i )charged (cid:17) (cid:27) (4.24)– 39 –here again the superscript indicates the additional naturalness constraint on the theoryspace maximization, and we can perform this maximization by taking the largest valuesfrom columns 9 and 10 in Table 4, which are shown in the last row. This gives: M max , naturalnessBSM , charged ≈ (cid:40)

20 TeV for N BSM = 130 TeV for N BSM = 10 (cid:41) ≈ (20 TeV) · N / . (4.25)The reduction in BSM mass scale compared to the unitarity-only analysis is even moredramatic than for the MFV assumption. The unusual N BSM scaling was empirically de-termined, but arises because unlike the unitarity constraint, the tuning constraint on thecouplings becomes more severe with increasing BSM multiplicity, which mostly cancels theincreased contribution to ∆ a µ . Given how strongly CLFV decay bounds motivate the MFV ansatz, it is reasonable toask how high the BSM mass scale could be if solutions to the ( g − µ anomaly haveto respect both naturalness and MFV. We investigate this by imposing both constraintssimultaneously in our analysis. The largest possible mass under this combined assumptionfor two representative SSF models is shown in Figure 9 (4th row), with the results for allEW Scenarios we examine summarized in columns 11 and 12 of Table 4 for N BSM = 1 and10 respectively.This allows us to deﬁne, for all possible perturbative, natural and MFV-respectingsolutions of the ( g − µ anomaly, the largest possible mass of the lightest BSM states: M max , naturalness , MFVBSM , charged ≡ max ∆ a µ =∆ a obs µ , unitarity , ∆ < , MFV (cid:26) min i ∈ BSM spectrum (cid:16) m ( i )charged (cid:17) (cid:27) (4.26)We can perform this maximization by taking the largest values from columns 11 and 12 inTable 4, which are shown in the last row. This gives our strongest constraint: M max , naturalness , MFVBSM , charged ≈ (cid:40) for N BSM = 119 TeV for N BSM = 10 (cid:41) ≈ (9 TeV) · N / . (4.27)The N BSM scaling, which lies between that of the unitarity or MFV constrained scenariosand that of the naturalness-constrained scenarios, was empirically determined and is obeyedto very good precision for N BSM (cid:46) . This result strongly reinforces the notion that any“theoretically reasonable” BSM solution to the ( g − µ anomaly must give rise to chargedstates at or below the 10 TeV scale. In fact, for many SSF models the maximum BSM mass is realized in regions of parameter space wherethe maximum allowed value for all

BSM couplings is set by the naturalness constraint. In that case the N BSM dependence cancels exactly, but this does not aﬀect the model-exhaustive upper bound, since it isnot the case for all SSF models, and is never the case for FFS models (which have an additional BSMcoupling, meaning that there is always a coupling combination that can saturate unitarity). Note that under the MFV assumption, there may be additional states generating contributions to theHiggs mass or the other lepton Yukawas. Since these depend on the representations chosen under the ﬂavourgroup we do not include them in our tuning measure, making our analysis conservative. – 40 – .7 Electroweak Landau Poles

Apart from ﬂavour and naturalness considerations, the parameter space for ElectroweakScenarios may be restricted by imposing the requirement that the SU(2) L and U(1) Y gaugecouplings do not hit low-lying Landau poles. In this section, we demonstrate parametricallythat such considerations disfavour truly enormous values of the BSM multiplicity N BSM ,which is relevant since our upper bounds on the BSM scale increase with N BSM .Since new matter of mass M BSM with electroweak charges only contributes to therunning of gauge couplings at scales µ > M

BSM , a muon collider which is only barelyable to produce new states on-shell cannot easily probe the threshold corrections to thegauge coupling. However, in the spirit of our ﬂavour and naturalness discussions to ﬁndthe most “reasonably theoretically motivated” parts of parameter space, we will impose themodest requirement that both of the electroweak gauge couplings remain ﬁnite up to a scale

Λ = 10 M BSM , where M BSM here represents the largest mass of all the new states. For thissimple estimate, we set M BSM = 100 TeV , inspired by the upper bounds from unitarity.This allows us to obtain approximate bounds on N BSM which depend on the electroweakrepresentations of the new states in SSF and FFS models.The 1-loop SU(2) L and U(1) Y beta functions are β Y,L = π b Y,L g Y,L , where b Y = 416 + 13 (cid:88) S Y S + 23 (cid:88) F Y F , (4.28) b L = −

196 + 13 (cid:88) S T ( R S ) + 23 (cid:88) F T ( R F ) . (4.29)The ﬁrst term in b Y and b L represents the SM contribution, and the remaining termsgive the contributions from complex scalars S and 2-component fermions F , respectively.In b L , T ( R ) is the index of the representation, equal to ( d + 1)( d )( d − / for the d -dimensional representation of SU(2). A positive b L or b Y indicates a coupling which growswith increasing energy, hitting a Landau pole at the scale Λ when ln (cid:18) Λ Y,L µ (cid:19) = 2 πα Y,L ( µ ) b Y,L . (4.30)Using the measured values of the couplings at µ = m Z , evolving them with the SM betafunctions up to µ = 100 TeV, and imposing the absence of a Landau pole at 1 PeV requires b Y < , b L < . (4.31)Since the BSM states do not all have the same mass, these bounds are approximate butsuﬃcient for a useful estimate. Applying these constraints to the 24 models in Table 4,we ﬁnd the maximum values of N BSM shown in Table 5. The maximum allowed BSMmultiplicity decreases for larger electroweak representations, with the strongest constraintbeing N BSM ≤

27 (23) for the highest-representation SSF (FFS) models.Given the very modest scaling of our mass bounds with N BSM , this suggests that N BSM (cid:46) O (10) (4.32)– 41 – odel R R A R B N BSM (U(1) Y ) N BSM (SU(2) L ) N BSM (min)

SSF − /

170 571 − /

37 571 − / −

580 571 − / −

70 571 − / − /

580 114 − / /

70 114 / − − /

170 114 − / − /

580 63 − / /

70 63 / − − /

170 63 − /

170 27 − / −

580 27 FFS − /

362 142 − /

42 142 − / −

145 142 − / −

27 142 − / − /

580 114 − / /

100 114 / − − /

54 114 − / − /

580 27 − / /

100 27 / − − /

54 27 − /

362 23 − / −

145 23 Table 5 : Approximate maximum values of N BSM for each of the models in Tab. 4, obtained usingEqn. (4.31) by requiring that each model avoid a Landau pole below 1 PeV in the hypercharge (4thcolumn) and SU(2) L (5th column) gauge coupling. The last column is the minimum of the two N BSM values for the two EW gauge groups. represents the most reasonably motivated BSM parameter space. It also justiﬁes our choiceto restrict our numerical model-exhaustive analysis of SSF/FFS models to representationsup to and including triplets. Models with larger representations hit Landau poles for muchlower BSM multiplicities, lowering the maximum possible BSM mass compared to modelsthat account for the ( g − µ anomaly with smaller EW representations. We focus on the simplest and most robust signature of EW Scenarios at muon collid-ers: direct production of new heavy charged states. Such a state X would be pair-produced in Drell-Yan processes independent of its direct couplings to muons, with a pairproduction cross section similar to SM EW → processes above threshold, σ XX ∼ fb (10 TeV / √ s ) [110], as long as √ s > m X . At high energies far above a TeV, the sameis true of electrically neutral states carrying weak quantum numbers, which are also presentin EW Scenarios. However, charged states must either decay to visible SM ﬁnal states orare visible if detector stable. As a result, conclusive discovery of such heavy states should– 42 –e possible in the clean environment and known center-of-mass-frame of a lepton colliderregardless of their detailed phenomenology.In the discussions of the next section, we can therefore simply assume a muon colliderwill be able to discover any heavy BSM charged state with m X (cid:46) √ s . As we have seen,for reasonable BSM solutions to the ( g − µ anomaly, this will call for an O (10 TeV) muoncollider (or an electron collider, if it could be built at such high energies).The complications particular to a muon collider, like the shielding cone necessary toreduce beam-induced background, do not aﬀect this argument for heavy charged states. Ofcourse, it is always possible to imagine very unusual scenarios where details of the modelconspire to make discovery much harder than generically expected. However, such edgecases do not invalidate a no-lose theorem. For example, while models that could hide theHiggs boson at the LHC were certainly considered prior to its discovery (see e.g. [111]),this did not invalidate the fact that the combination of EWSB and basic unitarity requiresthe production of new states at the LHC. Indeed, if such a scenario had come to pass, theno-lose theorem for the Higgs would have motivated herculean analysis eﬀorts to tease thehidden signals out of the data. (Furthermore, production and observation of new chargedstates via gauge couplings is much more robust than production of neutral scalars.) Ourno-lose theorem serves a similar function: it motivates the construction of colliders that canproduce the predicted new charged states, and in case those states are not found right away,it will hopefully provide similar emotional fortiﬁcation for future experimentalists lookingto uncover the new physics behind the by then well-established ( g − µ anomaly.In some EW Scenarios there is an electrically neutral or even complete SM singlet statethat is lighter than the lightest charged state. As we discussed in our ﬁrst study [55], thiscan also be discovered in a mono-photon search if the new state escapes as missing energy,where VBF-enhanced SM backgrounds can be eﬀectively vetoed with a high-momentum-cuton the recoiling photon. However, while this signature is interesting in its own right, it isnot our focus in this study. Across the whole space of possible EW Scenarios and hence alltheories that solve the ( g − µ anomaly, assuming that all kinematically accessible BSMstates can be discovered versus only assuming that charged states can be discovered doesnot actually lower the resulting minimum required energy of the muon collider necessary toguarantee discovery of new physics. We can therefore focus on charged BSM states withoutbeing unduly conservative. We now synthesize the results of our model-exhaustive analysis to understand the concreteimplications for a future muon collider program, and use them to derive our no-lose theoremfor the discovery of new physics.One-loop perturbative solutions to the ( g − µ anomaly can be classiﬁed as eitherSinglet Scenarios or EW Scenarios, based simply on whether the new physics contributionsin the loop are only SM singlets or if there are any particles with SM gauge quantumnumbers. Direct discovery of Singlet Scenarios requires observation of the SM singlet,– 43 –hile EW Scenarios can be discovered by producing the lightest new charged state atlepton colliders.BSM theories that only generate ( g − µ at higher-loop order necessarily feature lowermass scales relative to those found in one-loop models and are thus easier to discover.Furthermore, strongly coupled BSM scenarios involving composite new states in the ( g − µ loop are parametrically covered by our analysis, since we consider BSM multiplicity of states N BSM > and large couplings at the unitarity limit. If Singlet Scenarios explain the ( g − µ anomaly, the maximum possible mass of BSMstates based on perturbative unitarity only is 3 TeV, and only 200 GeV if we impose MFV,as motivated by CLFV decay bounds. We performed a careful analysis of direct singletproduction at muon colliders via the same coupling that generates ∆ a µ , which is completelyinclusive with respect to the singlet stability or decay mode. We ﬁnd that a 3 TeV muoncollider with − integrated luminosity would be able to discover all Singlet Scenariosthat solve the ( g − µ anomaly, provided the mass of the singlet is larger than ∼

10 GeV . A215 GeV muon collider with . − would not be able to probe the highest possible singletmasses, but could discover singlets heavier than 2 GeV. However, such a lower-energy muoncollider would also be able to observe deviations in Bhabha scattering µ + µ − → µ + µ − atthe 5 σ level to indirectly discover the eﬀects of these singlets with masses as high as theunitarity limit. These results are independent of N BSM because all observables scale with g BSM N BSM , the same combination of parameters that determines ∆ a µ .On the other hand, EW Scenarios are the most general way to solve the ( g − µ anomaly at one-loop, hence resulting in much higher possible BSM mass scales. We deﬁnedthe following highest possible mass for the lightest BSM charged state in the spectrum: M max , XBSM , charged ≡ max ∆ a µ =∆ a obs µ , X (cid:26) min i ∈ BSM spectrum (cid:16) m ( i )charged (cid:17) (cid:27) . (5.1)The outer max represents a maximization over theory space subject to assumptions X ,where we examined four possibilities: X = perturbative unitarity*unitarity + MFVunitarity + naturalness*unitarity + naturalness + MFV . (5.2)The last three assumptions include perturbative unitarity but are more restrictive. MFVavoids CLFV decay bounds and assumes that the SM Yukawas are the only source of ﬂavourviolation in whatever new physics solves the ﬂavour puzzle, which lowers the unitarity boundon some of the BSM muon couplings, since the corresponding BSM tau coupling must obeyperturbative unitarity. Naturalness is deﬁned to require that both the muon and Higgsmass, which both become technically unnatural in EW Scenarios due to calculable new loopcorrections, are tuned to no more than 1%. The star (*) indicates that assumptions without While we considered large BSM couplings that are borderline non-perturbative to derive upper boundson new particle masses, the existence and production of the new EW states at colliders is a consequence ofgauge invariance and only involves perturbative couplings, making our signal predictions robust. – 44 –FV implicitly rely on some coincidence or unknown mechanism to suppress CLFVs whileallowing the muonic BSM couplings to be pushed up to the unitarity (or naturalness) limit.We can perform this theory space maximization using our SSF and FFS simpliﬁedmodels to obtain the highest possible mass of the lightest new charged state as a consequenceof resolving the ( g − µ anomaly: M max , XBSM , charged ≈ (100 TeV) N / for X = (unitarity*) (20 TeV) N / for X = (unitarity + MFV) (20 TeV) N / for X = (unitarity + naturalness*) (9 TeV) N / for X = (unitarity + naturalness + MFV) , (5.3)The presence of required CLFV suppression is again indicated with a star. In light ofCLFV decay bounds, the two MFV results are the most theoretically and experimentallymotivated. Furthermore, avoiding relatively low-lying Landau poles motivates N BSM (cid:46) O (10) .Since charged states of mass m are eﬃciently produced by a lepton collider with √ s (cid:38) m and have to leave visible signals in the detector, we assume that any such BSM statewould be discovered at a suﬃciently energetic muon collider. Speciﬁcally, a √ s ∼

30 TeV (40 TeV) muon collider would be able to discover any high-scale, MFV-respecting solutionto the ( g − µ anomaly that avoids introducing two new hierarchy problems and has BSMmultiplicity up to N BSM (cid:46) (10). A 30 TeV collider would also be able to indirectlyconﬁrm the existence of the eﬀective BSM operator responsible for generating ∆ a µ via hγ measurements [53, 56]. This makes a 30 TeV muon collider a highly ambitious but highlymotivated benchmark goal for the discovery of new physics.High-scale solutions to the ( g − µ anomaly which evade discovery at a 30 - 40 TeVmachine are extremely strange: they would have to have a high BSM multiplicity, resultingin possible Landau poles below 1 PeV, or violate the assumptions of MFV while avoidingCLFV decay bounds, or be highly tuned in an explicitly calculable way. Therefore, non-observation of new states at a 30 TeV muon collider (alongside conﬁrmation of the newBSM operator via hγ measurement) would force the ( g − µ solution into theoreticallyextreme territory, which still has to satisfy the bounds of unitarity with charged statesbelow a few hundred TeV. Such a scenario would constitute empirical proof that nature isﬁne-tuned, and/or refute the MFV ansatz for the solution of the ﬂavour puzzle, which wouldnow be much more severe since unknown mechanisms have to suppress naively large CLFVcontributions. This in itself would be highly meaningful and new information about thefundamental nature of our universe, the selection of its vacuum, and the origin of ﬂavour.These results allow us to formulate the no-lose theorem for future muon colliders,which we already stated in Section 1, but we repeat the chronological progression here forcompleteness:1. Present day conﬁrmation: – 45 –ssume the ( g − µ anomaly is real.2. Discover or falsify low-scale Singlet Scenarios (cid:46)

GeV:

If Singlet Scenarios with BSM masses below ∼ GeV generate the required ∆ a obs µ contribution [34], multiple ﬁxed-target and B -factory experiments are projected todiscover new physics in the coming decade [35, 57–64].3. Discover or falsify all Singlet Scenarios (cid:46)

TeV:

If ﬁxed-target experiments do not discover new BSM singlets that account for ∆ a obs µ ,a 3 TeV muon collider with − would be guaranteed to directly discover thesesinglets if they are heavier than ∼

10 GeV .Even a lower-energy machine can be useful: a 215 GeV muon collider with . − could directly observe singlets as light as 2 GeV under the conservative assumptionsof our inclusive analysis, while indirectly observing the eﬀects of the singlets for allallowed masses via Bhabha scattering.Importantly, for singlet solutions to the ( g − µ anomaly, only the muon collider isguaranteed to discover these signals since the only required new coupling is to themuon.4. Discover non-pathological Electroweak Scenarios ( (cid:46)

10 TeV):

If TeV-scale muon colliders do not discover new physics, the ( g − µ anomaly must be generated by EW Scenarios. In that case, all of our results indicate that in mostreasonably motivated scenarios, the mass of new charged states cannot be higherthan few ×

10 TeV. However, such high masses are only realized by the most extremeboundary cases we consider. Therefore, a muon collider with √ s ∼

10 TeV is incredi-bly motivated, since it will have excellent coverage for EW Scenarios in most of theirreasonable parameter space.A very strong statement can be made for future muon colliders with √ s ∼

30 TeV (40TeV): such a machine can discover via pair production of heavy new charged states all

EW Scenarios that avoid CLFV bounds by satisfying MFV and avoid generatingtwo new hierarchy problems, with N BSM (cid:46) (10).5. Unitarity Ceiling ( (cid:46)

100 TeV):

Even if such a high energy muon collider does not produce new BSM states directly,the recent investigations by [53, 56] show that a 30 TeV machine would detect devia-tions in µ + µ − → hγ , which probes the same eﬀective operator generating ( g − µ atlower energies. This would provide high-energy conﬁrmation of the presence of newphysics.In that case, our results guarantee the presence of new states below ∼

100 TeV byperturbative unitarity, and the lack of direct BSM particle production at √ s ∼

30 TeV will prove that the universe violates MFV and/or is highly ﬁne-tuned to stabilize theHiggs mass and muon mass, all while suppressing CLFV processes.– 46 –s we already argued in Section 1, if the ( g − µ anomaly is conﬁrmed, this should serve assupremely powerful motivation for an ambitious muon collider program, from the test-bedor Higgs-factory scale of O (100 GeV) to energies in excess of 10 TeV. It would of coursealso be interesting to understand if and how proposed future hadron or electron colliderscould explore the same physics. Acknowledgements:

We thank Pouya Asadi, Jared Barron, Brian Batell, NikitaBlinov, Ayres Freitas, Chris Tully, Aida El-Khadra, Tao Han, Shirley Li, Patrick Meade,Federico Meloni, Jessie Shelton, Raman Sundrum, and José Francesco Zurita for helpfulconversations. The research of RC and DC was supported in part by a Discovery Grantfrom the Natural Sciences and Engineering Research Council of Canada, and by the CanadaResearch Chair program. The work of RC was supported in part by the Perimeter Institutefor Theoretical Physics (PI). Research at PI is supported in part by the Government ofCanada through the Department of Innovation, Science and Economic Development Canadaand by the Province of Ontario through the Ministry of Colleges and Universities. Thework of YK was supported in part by US Department of Energy grant DE-SC0015655.This manuscript has been authored by Fermi Research Alliance, LLC under Contract No.DE-AC02-07CH11359 with the U.S. Department of Energy, Oﬃce of High Energy Physics.

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