Vector boson fusion at multi-TeV muon colliders

High-energy lepton colliders with a centre-of-mass energy in the multi-TeV range are currently considered among the most challenging and far-reaching future accelerator projects. Studies performed so far have mostly focused on the reach for new phenomena in lepton-antilepton annihilation channels. In this work we observe that starting from collider energies of a few TeV, electroweak (EW) vector boson fusion/scattering (VBF) at lepton colliders becomes the dominant production mode for all Standard Model processes relevant to studying the EW sector. In many cases we find that this also holds for new physics. We quantify the size and the growth of VBF cross sections with collider energy for a number of SM and new physics processes. By considering luminosity scenarios achievable at a muon collider, we conclude that such a machine would effectively be a"high-luminosity weak boson collider,"and subsequently offer a wide range of opportunities to precisely measure EW and Higgs coupling as well as to discover new particles.


Introduction
Standing out among the important results that the Large Hadron Collider (LHC) has thus far delivered are the discovery of the Higgs boson (H) and the measurements of its properties. On the other hand, long-awaited evidence of new physics based on theoretical arguments, such as the stabilization of the electroweak (EW) scale, or on experimental grounds, such as dark matter and neutrino masses, have evaded our scrutiny. Despite the fact that the LHC's physics program is far from over, with Run III and the upgrade to the high luminosity LHC (HL-LHC) already lined up, the time has come for the high-energy community to assess what could be next in exploring the energy frontier. Such a question, which has been the main theme of the activities built around the European Strategy Update for Particle Physics [1], is not an easy one: Current physics and technology challenges are formidable. The fact that we have no clear indication where the scale of new physics might reside hampers the definition of a clear target. And depending on the properties of the new phenomena, either "low-energy" precision measurements or searching for new states in "high-energy" direct production may be the most sensitive and informative strategy to follow. In any case, exploration of the energy frontier will require building a new collider.
So far, two main options have actively been discussed by the community: a very energetic hadron collider with a center-of-mass (c.m.) of about √ s = 100 TeV, and an e + e − collider, at either high energy (up to a few TeV) or ultra high luminosity. These two classes have very different characteristics. The former has a much higher discovery reach of new states while the latter is feasible on a shorter time scale and allows a precision-measurement campaign of the Higgs/EW sector. Both avenues entail incredible investments, an intense research and development program, and formidable engineering capabilities. However, as construction of such collider experiments will not start for at least another 15-20 years from now and then require up to 20-40 more years of operation to achieve tentative physics targets, the community has started to seriously consider other avenues. This includes scenarios once thought too audacious or just impossible with even foreseeable technology. In this context, both linear e + e − and circular µ + µ − machines running at energies of several-to-many TeV have recently experienced a boost of interest within the community. In the former case, novel techniques based on plasma acceleration could potentially deliver up several GeV/m acceleration gradients, thereby reaching TeV scales on the km range [2]. An outstanding challenge in this case, however, is delivering the instantaneous luminosity needed to meet physics goals. Accelerating muons, on the other hand, would allow one to merge the best of both hadron and e + e − colliders [3,4], i.e., a high energy reach on one side and a "clean" environment on the other. Such a facility could possibly reach luminosities in the range of L = 10 35 cm −2 s −1 (or 100 nb −1 Hz) [5] and, importantly, be hosted at preexisting laboratory sites and tunnel complexes. These dream-like features are counterbalanced by a number of outstanding challenges, all of which eventually originate from a simple fact: muons are unstable and decay weakly into electrons and neutrinos.
Conceptual studies of muon colliders started decades ago and recently resulted in the Muon Accelerator Program (MAP) project [6]. In the MAP proposal, muons are produced as tertiary particles in the decays of pions, which themselves are created by an intense proton beam impinging on a heavy-material target, as already achievable at acceleratorbased muon and neutrino facilities. The challenge is that muons from pion decays have relatively low energy but large transverse and longitudinal emittance. Hence, they must be "cooled" in order to achieve high beam luminosities. More recently, a different approach to muon production has been proposed: in the Low Emission Muon Accelerator (LEMMA) muons are produced in e + e − annihilation near the threshold for µ + µ − pair creation [7,8]. A novelty is that muon beams do not require cooling to reach high instantaneous luminosities. This is because when a high-energy positron beam annihilates with electrons from a target the resulting muons are highly collimated and possess very small emittance. Muons are then already highly boosted with γ ∼ 200 and reach a lifetime of τ ∼ 500 µs [7]. The low emittance of the muons may further allow high beam luminosities with a smaller number of muons per bunch. This results in a lower level of expected beam-induced background, alleviating also potential radiation hazards due to neutrinos [9].
Given the recent interest and fast progress on how to overcome technological challenges, the most urgent mission becomes to clearly identify the reach and physics possibilities that such machines could offer. Available studies at the CLIC e + e − linear collider at 3 TeV have been used to gauge the potential of a muon collider in the multi-TeV range. Earlier, dedicated studies focusing mostly on processes arising from µ + µ − annihilation are available [3][4][5]10], and indicate promising potential for finding new physics from direct searches as well as from indirect searches with precision measurements of EW physics.
The work here is motivated by the observation that at sufficiently high energies we expect EW vector boson fusion and scattering (collectedly denoted as VBF) to become the dominant production mode at a multi-TeV lepton collider. We anticipate this holding for all Standard Model (SM) final states relevant to studying the EW sector and/or the direct search of (not too heavy) new physics. To this aim, we present a systematic exploration of SM processes featuring W , Z, H bosons and top quarks t in the final state. We investigate and compare s-channel annihilation and VBF cross sections in high-energy, lepton-lepton collisions, quantifying the size and the growth of the latter with collider energy. We consider the potential utility of precision SM measurements, focusing on a few representative examples, namely in the context of the SM effective field theory (SMEFT) [11][12][13]. Finally, we consider the direct and indirect production of new, heavy states in a number of benchmark, beyond the SM (BSM) scenarios. Having in mind the luminosity scenarios envisaged for a muon collider [5], we conclude that a multi-TeV lepton collider could offer a wide range of precision measurements of EW and Higgs couplings as well as sensitivity to new resonances beyond present experiments. For example: a √ s = 10 TeV muon collider with an integrated luminosity of L = 10 ab −1 would produce about 8 · 10 6 Higgs bosons, with about 3 · 10 4 from pair production alone. This provides direct access to the trilinear coupling of the Higgs [5] and gives an excellent perspective on the quartic coupling [14]. The paper is organized in the following manner: In section 2 we briefly summarize our computational setup and SM inputs for reproducibility. We then set the stage in section 3 by presenting and critically evaluating simple methods to estimate and compare the discovery potential of a hadron collider with that of a high-energy lepton collider. In section 4 we present production cross sections for SM processes involving the Higgs bosons, top quark pairs, and EW gauge bosons in µ + µ − collisions. In particular, we report the total c.m. energies at which cross sections for VBF processes, which grow as log s, overcome the corresponding s-channel, annihilation ones, which instead decrease as 1/s. In section 5 we consider the potential of a multi-TeV muon collider to facilitate precision measurements of EW process. We do this by exploring, in detail, limits that can be obtained in the context of the SMEFT by measuring HH and HHH production as well as final states involving the top quarks and weak bosons. Section 6 presents an overview on the possibilities for direct searches for new resonances at a multi-TeV muon collider, comparing the reach with those attainable at hadron colliders. We further investigate and compare the relative importance of VBF production in BSM searches in section 7. We summarize our work in section 8.

Computational setup
We briefly summarize here our computational setup. Throughout this text the evaluation of leading order matrix elements and phase space integration are handled numerically using the general purpose event generator MadGraph5_aMC@NLO(mg5amc) v2.6.5 [15]. For SM interactions, we use the default setup, which assumes the following EW inputs: For relevant computations, we employ the NNPDF 3.0 LO parton distribution functions (PDFs) with α s (M Z ) = 0.118 [16], as maintained using the LHAPDF 6 libraries [17]. To gain confidence in our results, especially at very high energies where we find that phase space integration converges much more slowly, we employ mg5amc v2.7.2, which includes a "hard survey" option for improved phase space sampling. In addition, some SM results have been cross-checked with Whizard [18] and in-house MC computations using matrix elements provided by Recola2 [19].

Comparing proton colliders and muon colliders
In trying to assess and compare qualitatively different colliders, it is constructive to first define a translatable measure of reach. Therefore, in this section, we propose a simple methodology for comparing the reach of a hypothetical muon collider to what is attainable at a proton collider. The obvious difference between the two classes of colliders is that protons are composite particles while muons are not. This means that proton collisions involve the scattering of (primarily) QCD partons that carry only a fraction of the total available energy, whereas muon collisions, up to radiative corrections, involve the scattering of particles carrying the total available energy. Concretely, we investigate three process categories: (3.1) the annihilation of initial-state particles (partons in the pp case) into a single-particle final state at a fixed final-state invariant mass ( √ŝ ); (3.2) the two-particle final state analogue of this; and (3.3) the scattering of weak gauge boson.

2 → 1 annihilations
Despite obvious differences, our aim is to compare the reach of muon and hadron colliders in (as much as possible) a model independent manner. In all cases, we find it useful to formulate the comparison in terms of "generalized parton luminosities" [20], where a parton can be any particle in the initial state, be it a lepton, a QCD parton, or an EW boson. In this language, the total, inclusive cross section for a given process in pp collisions is Here, X is a generic final state with invariant mass m X = √ŝ = √ τ s; the parton-level cross section is given byσ(ij → X) and is kinematically forbidden for τ < τ 0 ≡ min(m 2 X )/s; and for a c.m. hadron collider energy √ s, Φ ij is the ij parton luminosity, defined as 2) The f i/p (ξ, µ f ) are the collinear parton momentum distribution functions (PDFs) for parton i carrying a longitudinal momentum p i z = ξE p , out of a hadron p with momentum p p z = E p = √ s/2, when renormalization group-evolved to a factorization scale µ f . Where applicable, we set µ f to half the partonic c.m. energy, i.e., set µ f = √ŝ /2. The Kronecker δ ij removes double counting of identical initial states in i ↔ j beam symmetrization.
Given equation 3.2, then for a muon collider process µ + µ − → Y and its cross section σ µ at a fixed muon collider energy √ s µ , we define the "equivalent proton collider energy" as the corresponding pp collider energy √ s p such that the analogous hadron-collider process pp → Y has the same hadronic cross section σ p . That is, √ s µ and √ s p such that σ p = σ µ . Now, for the case of a 1-body final state Y with mass M = √ŝ , we have where [σ ij ] p and [σ] µ are the characteristic, partonic cross sections of the two collider process. For the pp case, we make explicit the Dirac δ function arising from the 1-body phase space measure. For the µ + µ − case, we assume that this is absorbed through, for example, the use of the narrow width approximation and rescaling by a suitably chosen branching rate. By construction, s µ =ŝ = M 2 , since production can only happen on threshold.
Requiring that σ p = σ µ and evaluating the beam-threshold integral, we obtain In the last step we assume that the ij-specific partonic cross section can be approximated by a universal, ij-independent cross section [σ] p . Crucially, in the luminosity function Φ(τ, µ f ), we identify the kinematic threshold as τ = s µ /s p , and likewise the factorization scale as µ f = √ s µ /2. If one further assumes a relationship between the partonic cross sections, this identification allows us to write equation 3.6 as which can be solved * numerically for s p as a function of s µ and β. For various benchmark assumptions (β) on the partonic cross sections [σ] p and [σ] µ , and for the parton luminosity configurations ij = gg (red) and ij = qq (blue), where q ∈ {u, c, d, s} is any light quark, we plot in figure 1(a) the equivalent proton collider energy √ s p as a function of √ s µ , for a generic 2 → 1, neutral current process. In particular, for each partonic configuration, we consider the case where the ij and µ + µ − partonic rates are the same, i.e., when β = 1 (solid line) in equation 3.7, as well as when β = 10 (dash) and β = 100 (dash-dot). The purpose of these benchmarks is to cover various coupling regimes, such as when ij → Y and µ + µ − → Y are governed by the same physics (β = 1) or when ij → Y is governed by, say, QCD but µ + µ − → Y by QED (β = 10).
Overall, we find several notable features. First is the general expectation that a larger pp collider energy is needed to achieve the same partonic cross section as a µ + µ − collider. This follows from the fact that pp beam energies are distributed among many partons whereas µ + µ − collider energies are effectively held by just two incoming partons. Interestingly, we find a surprisingly simple linear scaling between the two colliders for all ij and β combinations. For the ij = qq configuration and equal partonic coupling strength, i.e., β = 1, we report a scaling relationship of √ s p ∼ 5 × √ s µ . Under the above assumptions, * Explicitly, we use the scipy function fsolve to carry out a brute force computation of this transcendental equation. We report a reasonable computation time on a 2-core personal laptop.
one would need a muon collider energy of √ s µ ∼ 10 (20) [30] to match the reach of a hadron collider with √ s p ∼ 50 (100) [150] TeV. Specifically for the √ s p = 14 TeV LHC and its potential upgrade to √ s p = 28 TeV, one needs √ s µ ∼ 3 TeV and 5.5 TeV, respectively.
For the realistic case where the µ + µ − dynamics is ultra weakly coupled but pp dynamics is strong, i.e., β = 100, and proceeds through the ij = gg partonic channel, we report a milder scaling of √ s p ∼ 3.3 × √ s µ . This translates to needing a higher √ s µ to achieve the same reach at a fixed √ s p . For example: for √ s p = 14 (28) TeV, one requires instead √ s µ ∼ 4.25 (8.5) TeV. As a cautionary note, we stress that the precise numerical values of scaling ratios reported here are somewhat accidental and can shift were one to assume alternative PDF sets or µ f . The qualitative behavior, however, should remain.

2 → 2 annihilations
Instead of comparing the two colliders' equivalent reach for 2 → 1 processes, another possibility is to compare the reach for the pair production of heavy states. Doing so accounts for the nontrivial opening of new phase space configurations and kinematic thresholds. In the 2 → 2 case, we assume that the muon collider is optimally configured, i.e., that √ s µ is chosen slightly above threshold, where production the cross section is at its maximum. For pp collisions, the situation differs from the previous consideration in that pair production cross sections do not occur at fixedŝ and, in general, are suppressed by [σ ij ] p ∼ 1/ŝ, once far above threshold. Hence, we make the approximation that the quantity [σ ijŝ ] p does not depend on √ŝ , and recast beam-level cross sections in the following way: Assuming again that [σ ij ] p can be approximated by the ij-independent [σ] p , and making analogous identifications as in equation 3.6, then after equating σ µ (s µ ) = σ p (s p ), we obtain Here, the parton luminosity ij runs over the same configurations as in the 2 → 1 case and β similarly models the relationship between the (weighted) characteristic, partonic cross sections [σŝ] p and [σŝ] µ . As in the previous case, we can solve equation 3.10 numerically (see footnote * ) for the equivalent pp collider energy √ s p as a function of s µ and β. For the same benchmark assumptions on parton luminosities and partonic cross sections [σ] p and [σ] µ as considered in figure 1(a), we plot in figure 1(b) the equivalent proton collider energy √ s p [TeV] as a function of √ s µ [TeV], for a generic, 2 → 2, neutral current process.
For concreteness, we also consider the LO production of stop squark pairstt through QCD currents in pp collisions but EW currents in µ + µ − collisions, as well as of chargino pairs χ + χ − through EW currents. For these cases, we fixed particle masses M such that 2M constitutes 90% of the total muon collider energy, i.e., M = 0.9 × √ s µ /2.
As in the previous case, we again observe that a much higher energy pp collider exhibits the same reach as lower energy µ + µ − colliders. However, we find the scaling to be more drastic, with higher equivalent proton collider energies being reached for the same muon collider energies. We attribute this difference to the fact that while a spectrum of √ŝ is sampled in pp collisions, pair production beyond threshold is kinematically suppressed; this is unlike µ + µ − collisions where √ŝ is fixed. Remarkably, we also find that the scaling relationship between √ s p and √ s µ for 2 → 2 processes retains its linear behavior for all our representative cases. In this measure of comparing colliders, we report a scaling relationship of √ s p ∼ 22 × √ s µ for the ij = qq configuration and equal partonic coupling strength, i.e., β = 1. This indicates that a muon collider of √ s µ ∼ 10 (20) [30] TeV has roughly the same reach as a proton collider at √ s p ∼ 220 (440) [660] TeV. For the physics scenario where pair production is governed by weak (strong) dynamics in muon (proton), i.e., β = 100, we find very similar behavior for both the ij = gg and qq parton configurations. As in the 2 → 1 case, we report a smaller linear scaling of about √ s p ∼ 5.5 × √ s µ , indicating that the reach of a hypothetical muon collider of √ s = 2.5 (5) [14] TeV can only exceed or match the reach of proton colliders up to √ s p = 14 (28) [80] TeV.
For the concrete cases of stop (dot) and chargino (diamond) pair production, we observe several additional trends in figure 1(b). Starkly, we see that thett scaling is in good agreement with the scenario where production is governed by ultra weak (strong) dynamics in muon (proton), i.e., β = 100, for the ij = qq configuration. The preferred agreement for ij = qq over ij = gg follows from the production of high-mass states in pp collisions being typically driven by qq annihilation, where q ∈ {u, d} is a valence quark. For χ + χ − , we find poorer agreement with naïve scaling, with √ s p ∼ 30 × √ s µ . This is about ∼ 1.5× the estimation of the ij = qq configuration with equal partonic coupling strength (β = 1). We attribute this difference to the individual EW charges carried by elementary particles: as the µµZ coupling is suppressed, µ + µ − → χ + χ − is dominated by the γ * subchannel. The uuZ and ddZ couplings in qq → χ + χ − , on the other hand, are more sizable, and interfere destructively with the γ * subchannel, which itself is suppressed due to quarks' fractional electric charge. This is unlike stop pair production since QCD and QED processes are less flavor dependent. The disagreement is hence tied to a breakdown of the assumption that [σ ij ] p are ij-independent. Nevertheless, we importantly find that our scaling relationships, as derived from equations 3.7 and 3.10, provide reliable, if not conservative, estimates for the equivalent pp collider energy for a given µ + µ − collider energy.

Weak boson fusion
We conclude this section by comparing the potential for EW VBF at high-energy µ + µ − and pp collider facilities. As we will analyze in the following sections, one of the main features of a multi-TeV lepton collider is the increased relevance of VBF over s-channel scattering as the total collider energy increases. From this perspective, a muon collider could effectively be considered a "weak boson collider". It is therefore interesting to compare the potential for VBF at a muon collider to that at a pp collider.
To make this comparison, we find it useful to continue in the language of parton luminosities and employ the Effective W Approximation (EWA) [21,22], which allows us to treat weak bosons on the same footing as QCD partons. That is to say, enabling us to consistently define V λ V λ parton luminosities in both pp and µ + µ − collisions. The validity of the EWA as an extension of standard collinear factorization in non-Abelian gauge theories [23] has long been discussed in literature [24][25][26][27][28]. More recent investigations have focused on reformulations that make power counting manifest [29][30][31] and matching prescriptions between the broken and unbroken phases of the EW theory [32][33][34].
Under the EWA, splitting functions are used to describe the likelihood of forward emissions of weak bosons off light, initial-state fermions. In the notation of equation 3.2, the helicity-dependent PDFs that describe the radiation of a weak vector boson V in helicity state λ and carrying a longitudinal energy fraction ξ from a fermion a are [21,22,28]: Here, C, g a V , and g a A represent the appropriate weak gauge couplings of a, given by At this order, the PDFs do not describe QED charge inversion, i.e., f W ∓ /µ ± = 0 + O(g 2 ).
For simplicity, we further define the spin-averaged transverse parton distribution For a lepton collider, the V λ V λ luminosities Φ V λ V λ (τ, µ f ) are defined as in equation 3.2, but with substituting the QCD parton PDFs f i/p with the weak boson PDFs f V λ /a . In particular for W + For the pp case, the V λ V λ luminosities are obtained after making the substitution which is essentially the EW gauge boson PDF of the proton. The additional convolution accounts for the fact that q in p carries a variable momentum. (For simplicity, we keep all µ f the same.) The V λ V λ luminosity at a scale τ in pp collisions is then explicitly, in both pp (hatched shading) and µ + µ − (solid shading) collisions. (b) The same but for W + λ W − λ (solid shading) and Z λ Z λ (hatched shading) in µ + µ − collisions with (λ, λ ) = (T, T ) (red), (0, T ) + (T, 0) (green), and (0, 0) (blue). Band thickness corresponds to the µ f dependency as quantified in the text.

As a function of fractional scattering scale
where √ s is the total collider energy and M V V is the V V -system invariant mass, we plot in figure 2(a) the parton luminosities for W + in both pp (hatched shading) and µ + µ − (solid shading) collisions. Due to our choice of factorization scale µ f , the curves possess a (logarithmic) dependence on the collider energy. To take this ambiguity/dependency into account, we plot the envelopes for each parton luminosity spanned by varying √ s = 14 TeV − 200 TeV (3 TeV − 30 TeV) for the proton (muon) case. The precise ranges of √ s and √ τ that we consider help ensure that the partonic fraction of energy is neither too small nor too big, and hence that the EWA remains reliable [29]. We report that this "uncertainty" as little impact on our qualitative and quantitative assessments.
In figure 2(a), we find that for each helicity polarization configuration, the W λ W λ luminosity in µ + µ − collisions unambiguously exceeds the analogous luminosity in pp collisions over the √ τ considered. At √ τ = 0.2 (0.5) [0.8], we find that the W λ W λ luminosities at a muon collider are roughly 10 2 − 10 3 (10 4 − 10 6 ) [10 8 − 10 9 ] larger than those of a proton collider. Hence, for a fixed collider energy √ s µ = √ s p , the likelihood of W W scattering in µ + µ − collisions is much higher than for pp collisions. We attribute this to several factors: First is that the emergence EW PDFs in proton beams are a subdominant phenomenon in perturbation theory whereas in muon beams they arise at lowest order. Relatedly, as muons are point particles, they carry more energy than typical partons in a proton beam with the same beam energy. This enables EW PDFs in µ + µ − collisions to access smaller momentum fractions ξ, thereby accessing a larger PDF enhancement at small-ξ.
To further explore this hierarchy, we compare in figure 2(b) the µ + µ − collider luminosities for W + λ W − λ (solid shading) and Z λ Z λ (hatched shading) pairs, for (λ, λ ) = (T, T ) (red), (0, T ) + (T, 0) (green), and (0, 0) (blue). Globally, we see that the W W and ZZ luminosities exhibit a very similar shape dependence on √ τ , which follows from the functional form of f V /a (ξ). The normalization difference is due to the SU(2) L quantum number of muons, which results in the well-known suppression of µµZ couplings in the SM. Indeed, for (λ, λ ) = (0, 0), we find that the ratio of luminosities exhibits the constant relationship While not shown, we report that the W λ Z λ luminosities also have similar shapes and are located roughly at the geometric average of the W W and ZZ curves. Furthermore, due to gauge coupling universality, we anticipate that the luminosity hierarchy observed between muon and proton colliders extends to luminosities involving γ and Z bosons.

Standard Model processes at muon colliders
In this section we investigate and present cross sections for various EW boson and top quark final states of the form X = n tt+m V +k H, where n, m and k are integers that respectively denote the number of top quark pairs, weak vector bosons V , and Higgs bosons H. One of our goals of this survey is to systematically compare s-channel annihilation channels with EW VBF production channels in µ + µ − collisions, and in particular identify the c.m. energies at which VBF rates overtake s-channel ones. Specifically, we consider VBF process V V → X as obtainable from a µ + µ − initial state. This consists of the sub-channels W + W − fusion (section 4.2), ZZ/Zγ/γγ fusion (section 4.3), and W ± Z/W ± γ fusion (section 4.4): We also consider collisions with same-sign muon pairs, µ + µ + (section 4.5). In this case, the W Z and ZZ modes give rise to the same final state X at the same rate, up to charge multiplicities. The W + W + channel, on the other hand, opens truly new kinds of signatures while possessing the same luminosity as W + W − fusion, as reported in section 3. Before presenting our survey, we briefly comment first in section 4.1 on a few technical details related to simulating many-particle final states in multi-TeV lepton collisions.

Technical nuances at high energies
An important issue in this study involves the fact that the final states above also receive contributions from non-VBF processes, like associated production of X and a W or Z boson. That is to say, from an s-channel process but with an additional V -strahlung emission that then splits into a lepton pair. In general, these contributions interfere with VBF topologies at the amplitude level and are not all separately gauge-invariant subprocesses. Therefore, in principle, they need to be considered together with VBF. However, the V boson decay contributions are dominated by regions of phase space where V are on their mass shells. Especially, at a lepton collider, where the initial-state energy of the collision is known accurately, such resonant configurations can be experimentally distinguished from the nonresonant continuum. In fact, the relative contributions of those resonant topologies as well as of their interference with the gauge-invariant VBF contributions are small when far from the on-shell region, i.e., where most of the VBF cross section is populated.
Therefore, in order to avoid double counting of results that we will present for s-channel processes, as well as to make computations more efficient, we remove contributions with instances of on-shell Z → µ + µ − decays. In general, removing diagrams would break gauge invariance and so we refrain from doing this. A simple solution, adopted for instance in Ref. [14], is to impose a minimum on the invariant mass of the final-state lepton pair. In this work, we adopt an even simpler prescription by simulating an initial state possessing a non-zero muon and electron flavor, i.e. µ − e + collisions. In so doing, s-channel annihilations are forbidden and VBF channels are automatically retained. We have checked for a few processes that indeed the numerical differences with scattering rates of the analogous µ + µ − → X channels in the far off-shell region are small at high energies.
A second technical issue that requires care is the treatment of unstable particles, and in particular the inclusion of fixed widths (Γ) in Breit-Wigner propagators. While formally suppressed by O (Γ/M ) for resonances of mass M , these terms can break gauge invariance as well as spoil delicate unitarity cancellations at high energies. Indeed, we find that these disruptions can grow with energy for some processes and spoil the correctness of our calculations. A well-known solution is to consider the complex mass scheme [35,36], an option that is available in mg5amc [15]. However, in this case, all unstable particles can only appear as internal states, not as external ones. This implies that when modeling each par-  ticle in our final state X we always must include a decay channel (or decay channel chain), complicating our work considerably. Subsequently, we have opted for the solution of simulating external, on-shell W, Z, H, t with all widths set to zero. In doing so, gauge invariance is automatically preserved. Moreover, potential singularities in W, Z, H, t propagators are also automatically regulated due to their small mass differences.

W + W − fusion
We begin our survey by considering the production of up to four heavy particles from W + W − fusion (solid lines) and s-channel, µ + µ − annihilation (dashed lines). As a function Table 1. Same as figures 3 and 4 but tabulated for representative collider energies. For the tttt processes, scenario (i) considers mixed EW-QCD production and (ii) considers pure EW production. To summarize the global picture: as expected from the different production mechanism, s-channel annihilation rates categorically scale and decrease with collider energy at least as σ ∼ 1/s, when collider energies are far beyond kinematic threshold. This is contrary to VBF processes where cross sections mildly increase with collider energy at least as a power of log(s/M 2 W ), in the high energy limit. Consequentially, we find that for all processes considered there is a √ s where VBF production overcomes s-channel production. In table 2 we report this √ s and the corresponding σ at which the s-channel and VBF cross sections are the same. In general, the larger the final state multiplicity, the larger the value of √ s where the cross section curves cross. A few more remarks are in order.
First, for processes involving a top quark pair, as shown in figure 3, the s-channel cross sections at lower energies of O(1) TeV are comparable to if not larger than those from VBF at O(30) TeV, i.e., the highest energy that we consider. In terms of statistics only, s-channel annihilations at lower energies serve as a larger source of tt events. Hence, one may wonder if there is any gain in going to higher √ s. This is addressed at length in section 5. Here, it suffices to say that sensitivity to anomalous couplings greatly improves with increasing √ s, in particular for VBF processes. For example: At lowest order, µ + µ − → γ * /Z * → tt is only sensitive to anomalous ttZ/γ * and µµtt interactions; the channel is insensitive, e.g., to unitarity cancellations in the Higgs sector. This is unlike W + W − → tt, which is also sensitive to anomalous W W H, ttH, and tW b couplings, including relative CP phases. In addition, the VBF channel features a strong, non-Abelian gauge cancellation, and therefore probes anomalous contributions that are enhanced by energy factors.
A second interesting observation is the hierarchy of ttXX production from W + W − fusion. As seen in figure 3 systematically sit about an order of magnitude higher than ttHV , which in turn is another order of magnitude higher than ttHH. In fact, the ttHH rate sits just under the mixed EW-QCD tttt rate, despite being less phase space-suppressed. We attribute the strong hierarchy to the relative minus signs among the top quark's Yukawa coupling, the Higgs boson's self-couplings, and the various weak gauge couplings, which together lead to large destructive interference.
Third, for processes involving neutral bosons in the final state, H and/or Z, VBF cross sections are systematically larger than s-channel ones already at collider energies of a few TeV. This follows from the strong suppression of the Z gauge coupling relative to the unsuppressed νW gauge interaction. (Numerically, the further power of α W in W W fusion is still larger than the vector and axial-vector couplings of electrically charged leptons to Z bosons.) Among the processes investigated, multi-Higgs production in VBF stands out.
For the specific cases of HZ annd HHZ production in figures 4(a) and 4(b), we find that VBF already exceeds s-channel annihilation at √ s = 2 TeV and 4 TeV, respectively.
Lastly, the energy growth of VBF scattering rates is in general steeper for final states with larger particle multiplicities than for lower ones. This is due to many reasons. The first is that the increase in energy crucially opens phase space. For example: ttW W and ttHH have kinematic thresholds of M min ≈ 0.5 TeV and 0.6 TeV, indicating that their VBF production rates are phase space-starved for √ s 2 − 3 TeV. The second relates to (collinear) logarithmic enhancements in processes with t-channel gauge bosons. Final states with m gauge bosons entails contributions from diagrams with the exchange of m t-channel gauge bosons. At very high energies, such contributions become dominant and give rise to cross sections that behave at least as σ ∼ log m (s/M 2 V ). Even though this largest log might not be always be dominant, we verify that the growth pattern as a function of final-state multiplicity corresponds to this expectations and is rather clearly visible in plotted curves.

ZZ, Zγ, and γγ fusion
We continue our survey at a potential multi-TeV µ + µ − facility by now exploring processes mediated through the neutral gauge bosons Z and γ. For a subset of final states considered in section 4.2 for W + W − fusion that can instead proceed through ZZ, Zγ, and γγ fusion, we report in table 3 cross sections [fb] for representative collider energies. As described in section 4.1 we do not remove diagrams by hand and include γ/Z interference. To regulate phase space singularities, a p T cut of 30 GeV is applied on outgoing charged leptons. As foreseen from the simple scaling of the ZZ luminosity in section 3, the cross sections for ZZ/Zγ/γγ fusion are smaller than for W W by roughly an order of magnitude. The exceptions to this are W + W − production, which is highly comparable to the W + W − → W + W − rate, and ZZ production, which is about two orders of magnitude smaller than W + W − → ZZ. Despite being lower, these rates are not small enough to be neglected. Indeed, HH production already reaches σ ∼ 5 ab at √ s = 1 TeV and grows to be as large σ ∼ 430 (750) ab at √ s = 14 (30) TeV. Moreover, the presence of final-state charge leptons from Z/γ splittings, for example, could be exploited to obtain a full reconstruction of the event. For some particular channels it may also be useful to have charged lepton pairs to better identify a new resonance signal or increase sensitivity to an anomalous coupling.

W Z and W γ scattering
Turning away from final states with zero net electric charge, we now explore processes mediated by W γ and W Z fusion. For several representative processes, we summarize in table 4 their cross sections at our benchmark muon collider energies. We apply a p T cut of 30 GeV on outgoing charged leptons to regulate phase space singularities. Once again, following simple scaling arguments of the EWA luminosities in section 3, we expect and observe that cross sections here are somewhere between those of W W and ZZ fusion.
With the present VBF configuration, we find that the rates for  appreciable cross sections for √ s = 1 − 30 TeV. If one assumes O(1 − 100) ab −1 datasets, then in these cases, interesting, ultra rare and ultra exclusive decay channels can be studied.

W + W + fusion
Finally, we conclude our EW VBF survey by briefly exploring the case of same-sign muon collisions. This setup allows the production of doubly charged final states and therefore, as we discuss in section 6, is a natural setup where one can study lepton number-violating processes [37]. For concreteness, we consider µ + µ + collisions and in table 5 present the cross sections for representative V V and V V H processes at our benchmark collider energies. We report that the production rates for V V and V V H are highly comparable to those for W + W − fusion in table 1. We anticipate this from CP invariance, which dictates that the W + W − luminosity in µ + µ − collisions is the same at lowest order as the W + W + luminosity in µ + µ + collisions. Differences between the two sets of rates originate from differences between the W + W − → X and analogous W + W + → X matrix elements. In W + W + scattering, only t-channel exchanges of gauge and scalar bosons are allowed as there does not exist a doubly charged state in the EW sector. In W + W − scattering, these t-channel diagrams interfere (constructively and destructively) with allowed s-channel diagrams.

Precision electroweak measurements
In this section we explore the potential of a muon collider to probe new physics indirectly. As it is not realistic to try to be exhaustive, after summarizing the effective field theory formalism in which we work (section 5.1), we select a few representative examples related to the Higgs boson (section 5.2) and the top quark (section 5.3).

SMEFT formalism
Undertaking precision measurements of SM observables is of utmost importance if nature features heavy resonances at mass scales that are just beyond the kinematic reach of laboratory experiments. Be it perturbative or non-perturbative, the dynamics of such new states could leave detectable imprints through their interactions among the SM particles. This is especially the case for the heaviest SM particles if the new physics under consideration is related to the flavor sector or the spontaneous breaking of EW symmetry. Table 6. SMEFT operators at dimension-six relevant for the Higgs boson and the top quark in EW observables, in the so-called Warsaw basis [11], and where a U (3) 3 × U (2) 2 flavor symmetry is assumed. Q, t, and b denote the third generation components of q, u, and d.
Generically, two broad classes of observables, defined in different regions of phase space, can be investigated. The first are bulk, or inclusive, observables for which large statistics are available and even small deviations from the null (SM) hypothesis are detectable. The second are tail, or exclusive, observables, where the effects of new physics can be significantly enhanced by energy, say with selection cuts, and compensate for lower statistics.
A simple yet powerful approach to interpret indirect searches for new, heavy particles in low-energy observables is the SMEFT framework [11][12][13]. The formalism describes a large class of models featuring states that live above the EW scale and provides a consistent, quantum field theoretic description of deformations of SM interactions. This is done while employing a minimal set of assumptions on the underlying, ultraviolet theory. In SMEFT, new physics is parametrized through higher dimensional, i.e., irrelevant, operators that augment the unbroken SM Lagrangian, yet preserve the fundamental gauge symmetries of the SM by only admitting operators that are both built from SM fields and invariant under G SM = SU c (3) × SU L (2) × U Y (1) gauge transformations. Accidental symmetries of the SM, such as lepton and baryon number conservation, are automatically satisfied under certain stipulations [38,39]. Additional global symmetries can also be imposed on the Lagrangian. In this work, we require the flavor symmetry, This helps reduce the number of independent degrees of freedom while simultaneously singling out the top quark as a window onto new physics.
Under these assumptions, then after neglecting the Weinberg operator at dimension five and truncating the EFT expansion at dimension six, the SMEFT Lagrangian is Here, C i are the dimensionless Wilson coefficients of the dimension-six operators O i . In the absence of additional symmetries, such as the flavor symmetry S defined above, the number of independent O i stands at 59 if one considers only one generation of fermions and 2499 † The labels l, e, d, u, q refer, respectively, to the left-handed lepton doublets, the right-handed leptons, the right-handed down-type quarks, the right-handed up-type quarks, and the left-handed quark doublets.  with three generations. In practice, one usually studies only a subset of operators in order to establish the sensitivity of a measurement. Since we are mainly interested in the top quark and Higgs sectors, we consequentially retain only operators that explicitly involve top or Higgs fields and affect EW observables. The full list of operators that we consider is given in table 6, where the following conventions are adopted: Here, τ I denotes the Pauli σ matrices, and IJK is antisymmetric and normalized to unity. In the following we perform a simple sensitivity study focusing on the Higgs selfcouplings and the top quark's EW couplings. In table 7 we summarize the current constraints on Wilson coefficients corresponding to the operators in table 6.

Higgs self-couplings at muon colliders
A precise determination of the Higgs boson's properties is one of the foremost priorities of the high-energy physics community [1]. At the moment, measurements of the Higgs's couplings to the heaviest fermions and gauge bosons are in full agreement with the SM predictions. However, there exists several couplings that have yet to be measured, and in some cases bounds are only weakly constraining. Among these are the Yukawa couplings to the first and second generation of fermions as well as the shape of the SM's scalar potential. Subsequently, a determination of the Higgs's trilinear and the quartic self-couplings, which are now fully predicted in the SM, would certainly help elucidate the EW symmetry breaking mechanism [44] and its role in the thermal history of the universe.
Despite this motivation, measurements of the Higgs's self-interactions appears to be too challenging for the LHC, unless substantial deviations from the SM exist [45][46][47][48][49][50][51][52][53][54][55][56][57]. As such, conclusively measuring the Higgs's properties is among the most compelling motivations for constructing a lepton collider at a c.m. energy of a few hundred GeV. ‡ The case for higher energies is also well-founded. For example: Higgs sensitivity studies for CLIC up to √ s = 3 TeV [58][59][60][61][62][63] support the expectation that increasing collider energy provides additional leverage for precision measurements through VBF channels. Indeed, as already shown in Fig. 4, VBF processes emerge as the dominant vehicles for H, HH, and even HHH production at high-energy lepton colliders and surpasses s-channel processes below √ s = 3 TeV. For instance: at √ s = 10 TeV and with a benchmark luminosity of L = 10 ab −1 , one anticipates 8 · 10 6 Higgs bosons in the SM [5]. As backgrounds are expected to be under good control, multi-TeV muon colliders essentially function as de facto Higgs factories. Limitations to determining the Higgs's self-couplings at the LHC and e + e − colliders certainly motivate other opportunities, particularly those offered by muon colliders. We therefore perform in this section a first exploratory investigation with such a machine.
Within the SMEFT framework, three operators directly modify the Higgs potential: The first contributes to the Higgs potential's cubic and quartic terms and shifts the field's (ϕ's) vev v. The latter two modify the Higgs boson's kinetic term and a field redefinition is necessary to recover the canonical normalization. All of these operators give a contribution to VBF production of H, HH, and HHH through the following Lagrangian terms: For conciseness, we investigate only the impact of O ϕ and O ϕd on prospective Higgs's self-coupling measurements. We neglect O ϕD since it also modifies couplings to gauge bosons and hence is already well-constrained by precision EW measurements. (See table 7 for details.) In the following, we consider a high-energy µ + µ − collider at a c.m. energy of √ s = 3, 14, and 30 TeV, with respective benchmark luminosities L = 6, 20, and 100 ab −1 .
For the processes under consideration, we first discuss the impact of a single operator on inclusive cross sections while fixing all other higher dimensional Wilson coefficients to zero. Within the SMEFT, the total cross section (σ) of a process can be expressed by Here, the σ i Int are the leading corrections in the Λ power counting to the SM cross sections (σ SM ) and are given by the interference between SM amplitudes and SMEFT amplitudes at O(Λ −2 ). The σ i,j Sq corrections are the square contributions at O(Λ −4 ), and come purely from SMEFT amplitudes at O(Λ −2 ). The indices i, j run through the set of operators that ‡ It is remarkable that a 100 m radius circular muon collider could reach this energy [5]. directly affect the process. We work under the assumption that the Wilson coefficients C i for operators in equation 5.5 are real. This indicates that the coefficients in σ correspond to c i = C i and c i,j = C i C j . As a naïve measure of the sensitivity to the dimension-six operators O i and considering only one operator at the time, we define the ratio 14 (blue) and 30 TeV (green). Immediately, one sees that two operators affect the ratio R(c i ) = σ/σ SM , and hence prospects for measuring the Higgs's self couplings, in qualitatively different ways. To explore this, we first note that O ϕ in equation 5.6 only shifts the Higgs's trilinear and quartic couplings. The operator does not generate an additional energy dependence in the squared matrix element, apart from that which could be obtained by spoiling SM unitarity cancellations. As a result, the highest sensitivity to O ϕ is reached near threshold production. Increasing √ s actually results in losing sensitivity to HH production. Similarly for HHH production, no significant impact on cross section ratio is observed with increasing the collider energy, only a gain in the total number of events stemming from an increasing production rate. For the particular case of HHH production at √ s = 3 TeV, the cross section is negligible and no measurement for this process can be undertaken. Independent of shifts to R(c i ), it is important to point out that the higher the event rate the more feasible it becomes to study differential distributions of the above processes. Generically, an increased number of events allow us to more fully explore, and therefore exploit, regions of phase space that are more sensitive to BSM. Contrary to O ϕ , the operator O ϕd introduces a kinematical p 2 dependence in interaction vertices. As a consequence, the impact of O ϕd grows stronger and stronger as collider energy increases, potentially leading to a substantial gain in sensitivity. The imprint of this behavior is visible in the fact that the c i interference term between the SM and new physics becomes negligible as probing energy goes up. In this limit, the squared c ii term dominates as naïvely expected from power counting at higher energies. This follows from the purely new physics contributions in SMEFT forcing R(c i ) to grow at most as (E/Λ) 4 , while the linear c i contribution force R(c i ) to grow at most as (E/Λ) 2 . Leaving aside questions of the EFT's validity when (E/Λ) 4 corrections exceed those at (E/Λ) 2 , our point is that it is clear that sensitivity to O ϕ and O ϕd are driven by complementary phase space regions.
As a final comment, we would like to note that while the study of individual SMEFT operators can give important and useful information, in a realistic BSM scenario, multiple operators would simultaneously contribute to a given observable. In this more complicated scenario, correlations and numerical cancellations among operators appear, and phenomenological interpretations becomes more nuanced, more difficult. If we nevertheless put ourselves in the scenario where a measured cross section (σ) is consistent with the SM, then we can still define a simplified estimate of the experiment's constraining power. In particular, we define the space of Wilson coefficients that predicts a cross section indistinguishable from SM expectation at the 95% confidence level (CL) by the following: Here, σ is the same SMEFT cross section as defined in equation 5.9. The number of background events B is the SM expectation (σ SM ) at a given luminosity L, and the number of signal events S is determined from the net difference between SMEFT and SM expectations. If we restrict ourselves to the two aforementioned operators, then equation 5.11 identifies an annulus or a disk in the 2D parameter space of Wilson coefficients. Hence, by combining observables one can gain a lot in constraining power by breaking such degeneracies. To see this explicitly, we show in Fig. 7 the 2D contour of allowed Wilson coefficients for O ϕ and O ϕd at (a) √ s = 3 TeV via HH production and (b) 14 TeV via both HH and HHH production. (This difference is due to the smallness of the HHH rate at 3 TeV as mentioned above.) Solutions to 5.11 are provided under the assumption that only linear (c i ) corrections to σ are retained (red band) as well as when quadratic (c i,j ) corrections are included (blue band). We also report the projected, marginalized limits on the two Wilson coefficients in table 8. Clearly, the lower energy machine leaves a much larger volume of parameter space unconstrained. In the 3 TeV case, the absence of a second measurement leads to a flat constraint in the linear case, which suggests an impossibility of conclusively constraining the parameter space. Moreover, this represents a strong case for measuring the triple Higgs production at lepton colliders in order to pin down the Higgs's self-couplings. From this perspective, we argue that a √ s = 14 TeV lepton (muon or electron) collider would be ideal over lower energy scenarios. Such a machine allows us to take advantage of both double and triple Higgs production, and at last measure the SM's scalar potential.
In order to offer a comparison with other hypothetical future collider proposals, we quote here the projections from combined results at FCC-ee 240 , FCC-ee 365 , FCC-eh and FCC-hh, as reported in Ref. [63]. The first two are e + e − colliders with L = 5, 1.5 ab −1 at √ s = 240, 365 GeV respectively. The third is an e ± p collider with L = 2 ab −1 at √ s = 3.5 TeV, while the last is a pp collider with L = 30 ab −1 at √ s = 100 these scenarios, the projected individual bounds at 68% CL for operators we consider are The difference is roughly a factor of 40. In the absence of HHH production, the results here are comparable to those reported elsewhere [14]. This naïve comparison again shows the potential of a high-energy lepton collider in studying EW physics, allowing us to reach a precision that is certainly competitive with what attainable at other proposed colliders.

Top electroweak couplings at muon colliders
Due to its ultra heavy mass and complicated decay topologies, the era of precision top quark physics has only recently begun in earnest at the LHC. This is despite the particle's discovery decades ago and rings particularly true for the quark's neutral, EW interactions [64]. For example: The associated production channel ttZ was only first observed using the entirety of the LHC's Run I dataset [65,66]. Likewise, the single top channel tZ was observed only for the first time during the Run II program [67,68]. And importantly, only recently has the direct observation of ttH production process confirmed that the top quark's Yukawa coupling to the Higgs boson is O(1) [69][70][71][72][73]. Since a precision program for measuring the top quark's EW couplings is still in its infancy, there exists a margin for O(10%) deviations from SM expectations. This makes it of stark importance to understand how to best measure these couplings, as searching for deviations could lead to new physics. On this pretext, Ref. [74] studied a class of 2 → 2 scattering processes involving the top quark and the EW sector within the SMEFT framework. There, the authors performed a systematic analysis of unitarity-violating effects induced by higher dimensional operators. By considering 2 → 2 scattering amplitudes embedded in physical processes at present and future colliders, specific processes were identified that exhibited a distinct sensitivity to new physics. Among these processes, VBF at future lepton colliders stands out. The Wilson coefficients belonging to the operators in table 6 that impact VBF processes involving the top quark are not strongly constrained. Hence, an improved measurement of these channels is important for the indirect tests of a plethora of BSM models.
In the context of a multi-TeV muon collider and following the proposal of Ref. [75], in this subsection we consider and compare the constraining potential of 2 → 3 processes on anomalous couplings of the top quark. Even though such processes feature more complex Feynman diagram topologies and additional phase space suppression, their utility within the SMEFT framework stems from also featuring higher-point (higher-leg) contact interactions with a stronger power-law energy dependence at tree-level. In addition, a larger number of diagrammatic topologies translates into more possibilities to insert dimension-six operators, which, roughly speaking, may trigger larger deviations from the SM. (Though arguably larger cancellations are also possible.) For rather understandable limitations, such as finite computing resources, such considerations were not widely investigated before.
As an example, we consider the operator O tW from table 6. For the case of W + W − → tt scattering, this operator generates the four-point contact vertex Here, one has to pay a vev penalty of (v/Λ), where the v originates from the Higgs doublet ϕ, and thereby makes the term effectively a dimension-five contact term. On the other hand, by extending the final state with a Higgs field one can saturate the operator: Remarkably, instead of (v/Λ), one is "penalized" by a factor of (E/Λ), where the energy dependence originates from the three-body, phase space volume. This mechanism is rather generic and hence can be exploited for other operators and multiplicities in order to maximize the energy growth of amplitudes, and therefore the sensitivity to new physics. For concreteness, we compare the 2 → 2 production of tt from VBF to the 2 → 3 associated production of ttH and ttZ from VBF. For each process we present in Fig. 8  On the other hand, the quadratic terms exhibit an overall growth, just not a dramatic one. The smallness of |r i | contributions suggests that considering higher multiplicity processes, such as ttH and ttZ, could prove more sensitive to new physics, despite naïve phase space suppression. Adding a Higgs boson (Fig. 8(b)) or a Z boson (Fig. 8(c)) in the final state has a noticeable, quantitative impact on the overall behavior of ratio coefficients in the radar plots. When looking at the linear interference terms, it is surprising to see that many of the operators' contributions decrease when going to higher energies. On the other hand, a sensitivity gain is unambiguous for all the operators in the quadratic case, which reach as much as O (100). The behavior of interference is often more subtle to predict. Being non-positive definite, cancellations can and do readily take place depending on the specific phase space region that is considered. In particular, we infer that at higher energies these cancellations are enhanced, leading effectively to a lower sensitivity at the inclusive level. Generically, each operator and process has a cancellation pattern of its own, which is also reinforced by the linear independence of SMEFT operators. Hence, designing a single recipe for every operator to invert cancellations with the aim of fully exploiting the increased sensitivity to energy is complicated. On the other hand, dedicated studies could lead to the discovery of a most sensitive (or a highly optimized) phase space region for a specific set of operators, enhancing the possibility to detect new physics.
While being more difficult to measure, these 2 → 3 processes offer an overall improvement to sensitivity with respect to 2 → 2 production of tt. This is both from the energy-growing perspective and from an absolute one. In essence, our very preliminary study here suggests that having a multi-TeV muon collider would benefit us for at least two reasons: (i) Due to phase space enhancements (E/Λ), a higher energy collider would allow us to take advantage of larger deviations from SM expectations, and hence higher sensitivity to SMEFT operators. (ii) The growth in the inclusive VBF cross section would allow us to have enough statistics to precisely measure higher multiplicity final states that would otherwise be infeasible even at √ s = 3 TeV. For example: we compare the ∼ 100 ttH events at 3 TeV to the ∼ 3000 at 14 TeV, assuming the benchmark luminosities considered (L = 6 ab −1 and 20 ab −1 , respectively). The program to precisely determine the top quark's EW interactions would therefore benefit greatly from a potential future muon collider by allowing us to take into account new processes that could help break degeneracies among SMEFT operators and learn about the dynamics of EW symmetry breaking.

Searches for new physics
Like hadron beams, muon beams emit significantly less synchrotron radiation than their electronic counterpart due to the muon's much larger mass. As a result, µ + µ − colliders can reach partonic c.m. energies that far exceed conventional e + e − facilities, such as LEP II, and potentially even pp colliders; see section 3 for further details. Thus, in addition to the abundance of achievable SM measurements described in sections 4 and 5, a muon collider is able to explore new territory in the direct search for new physics.
In this section, we present a survey of BSM models and the potential sensitivity of a µ + µ − collider. Explicitly, we consider the s-channel annihilation and VBF processes µ + µ − → X and µ + µ − → X . (6.1) Here, ∈ {µ ± , (−) ν µ } and X is some BSM final state, which may include SM particles. We focus on the complementarity of the two processes because while s-channel annihilation grants accesses to the highest available c.m. energies, it comes at the cost of a cross section suppression that scales as σ ∼ 1/s when far above production threshold. On the other hand, in VBF, the emission of transversely polarized, t-channel bosons gives rise to logarithmic factors that grow with the available collider energy. Thus, VBF probes a continuum of mass scales while avoiding a strict 1/s-suppression, but at the cost of EW coupling suppression.
To investigate this interplay, for each scenario, we consider the mass and collider ranges: These logarithms can potentially spoil the perturbative reliability of cross sections at LO and necessitate resummation of EW Sudakov factors [32,[76][77][78]. For the various BSM scenarios, we assume benchmark values for relevant couplings. We omit generator-level phase space cuts where possible but stipulate them when needed to regulate matrix elements. In the following, we present the production rates of new processes. As a standard candle reference, in most scenarios, we also plot SM H production via W + W − fusion (black, solid curve).
We start our survey in section 6.1 with minimally extending the SM by a scalar that is a singlet under the SM's gauge symmetries. We then move onto the production of scalars in the context of the Two Higgs Doublet Model in section 6.2, and the Georgi-Machacek Model in section 6.3. In section 6.4, we investigate the production of sparticles in the context of the Minimal Supersymmetric Standard Model. We also consider representative phenomenological models describing the production of leptoquarks in section 6.5, heavy neutrinos in section 6.6, and vector-like quarks in section 6.7. We give an overview of this survey in section 6.8. A detailed comparison of s-channel and VBF production mechanisms in BSM searches at multi-TeV muon colliders is deferred to section 7.

Scalar singlet extension of the Standard Model
The scalar sector of the SM consists of a single scalar SU (2) L doublet with a nonzero U (1) Y charge. While this is the minimal scalar content that supports the generation of fermion and weak boson masses through EWSB, the measured couplings of the M H ≈ 125 GeV Higgs bosons uphold this picture [79,80]. Theoretical motivation for extending this scalar sector, however, is well-established and the phenomenology of these scenarios have been studied extensively. For reviews, see Refs. [81][82][83][84][85][86][87] and references therein.
One of the simplest extensions that respects the SM symmetries is the addition of a single, real scalar (σ) that is neutral under all SM charges but carries an odd Z 2 parity. In light of LHC data, the phenomenology of such a singlet scenario is categorized by whether σ acquires a nonzero vev: In the so-called inert scenario, σ does not acquire a vev, interacts at tree level only with the SM Higgs boson (H), and impacts H's coupling to fermions and bosons at loop level [88]. If instead the singlet σ acquires a vev, then it mixes with the SM Higgs, which in turn modifies H's coupling to SM particles at tree-level.
We investigate the muon collider sensitivity to the SM with an extra scalar singlet by considering the case where the vev of σ is nonzero, i.e., σ ≈ v σ + σ 0 . The (unbroken) Lagrangian that describes such a scenario, including the Z 2 symmetry, is given by where L SM is the full SM Lagrangian. After both the SM doublet ϕ and σ acquire their respective vevs, v and v σ , a mass-mixing term between σ 0 and the neutral part of the doublet ϕ 0 , and proportional to δm 2 ∝ κ σ vv σ , is generated. Rotating ϕ 0 and σ 0 from the gauge basis and into the mass basis by an angle θ, we obtain the mass eigenstates H and S with mass eigenvalues M H and M S . The coupling of the lightest neutral scalar, which we assume is H, to SM fermions and gauge bosons is suppressed relative to the SM by a factor of cos θ. Owing to strong constraints on anomalous Higgs couplings [79,80], one scalar is aligned closely with the SM Higgs, which we assign to H, implying cos θ 1. The bare parameters m σ , λ σ , κ σ , can subsequently be exchanged for the physical parameters M S , v σ , θ, which therefore permits us to express the scalar trilinear scalar interactions as: The non-inert singlet scenario § is implemented in the Minimal Dilaton Model UFO libraries by Ref. [90], and hence can be simulated using general purpose event generators. S production in µ + µ − collisions can proceed through several mechanisms, including W + W − fusion, as shown in figure 9(a), which is mediated by an s-channel H boson. As § Similarly, the inert singlet scenario is available using the SM_Plus_Scalars_UFO UFO libraries [89]. shown above, for a given v σ and θ, the λ hss coupling is related to the H − S mass difference. Assuming the fixed, baseline mass splitting of Ref. [90], we show in figure 9(b) the SS pair production cross section [fb] via EW VBF as a function of collider energy √ s [TeV].
For M S = 0.4 − 0.8 TeV, we see that the VBF process rate spans roughly σ ∼ 10 −3 − 10 −2 fb for √ s = 5 − 30 TeV. For M S = 2 − 4 TeV, we observe that the corresponding rates reach the order of 10 −4 − 10 −3 fb at √ s = 30 TeV. By comparing these numbers with the SM productions of H via VBF over the whole range of collider energies, we find that the latter are several order of magnitude larger, spanning σ ∼ 100 − 1000 fb.

Two Higgs Doublet Model
If a new neutral scalar does indeed exist, rather than being a SM singlet as posed in section 6.1, it may actually be a component of a second scalar SU (2) L doublet. Such scenarios, known in the literature as Two Higgs Doublet Models (2HDMs), have been extensively reviewed [81,82,91,92], particularly for their necessity to realize Supersymmetry in nature. We consider the benchmark, CP-conserving 2HDM, the scalar potential of which is Here, the couplings λ i are real and the scalar SU (2) L doublets ϕ 1 and ϕ 2 are given by After ϕ 1 and/or ϕ 2 acquire vacuum expectation values, EW is broken and fields with identical quantum numbers mix. More specifically, the charged scalars and neutral, CPodd scalars mix into the EW Goldstone bosons G ± , G 0 and the physical states H ± , A 0 . Likewise, the neutral, CP-even scalars mix by an angle θ into the physical states H 1 and H 2 . Here, H 1 is identified as the observed, SM-like Higgs with M H 1 ≈ 125 GeV and H 2 is heavier with M H 2 > M H 1 . In terms of mass eigenstates, h 0 1 and h 0 2 are given explicitly by Among the simplest processes we can analyze at a muon collider is resonant production of H 2 from W + W − fusion, which we show diagrammatically in figure 10(a). To estimate the sensitivity to this process, we consider the 2HDM in its CP-conserving scenario, as implemented in the 2HDM model file [93]. We show in figure 10(b)  Over the entire range of collider energies, we see that the SM production of H is over an order of magnitude larger, reaching σ ∼ 100 − 1000 fb.

Georgi-Machacek Model
Another possibility at a future muon facility is the VBF production of electrically charged scalars. These, of course, do not exist in the SM nor in the simplest, naïve extensions of the SM scalar sector. In models such as the Georgi-Machacek (GM) model [94] and the Type II Seesaw model for neutrino masses [95][96][97][98][99], VBF production of singly charge (H ± ) and doubly charged (H ±± ) charged scalars is possible due to the existence of scalar triplet representations of SU (2) L with nonzero hypercharge. (Higher SU (2) L ⊗ U (1) Y representations also permit scalars with even larger electric charges.) For present purposes, we focus on the feasibility of seeing exotically charged scalars from the GM mode ¶ . Broadly speaking, the model extends the SM with a real and a complex triplet with hypercharge Y = 0 and 1, respectively. If the vevs of the triplets' neutral components are aligned, then tree-level, custodial symmetry is respected and strong constraints on the ρ parameter are alleviated [81,[101][102][103][104][105][106][107]. More specifically, the GM scalar sector consists of the usual SM complex doublet (ϕ + , ϕ 0 ) with Y = 1/2, a real SU (2) L triplet (ξ + , ξ 0 , ξ − ) with Y = 0, and a complex SU (2) L triplet (χ ++ , χ + , χ 0 ) with Y = 1. Writing the doublet and triplets in the form of a bi-doublet (Φ) and bi-triplet (X), we have  For our numerical results, we consider the decoupling limit of the GM model as implemented in the GM_UFO UFO [108,109]. The (unbroken) scalar potential is given by After aligning all states into their mass eigenstates, we are left with H ± and H ±± , in addition to a number of neutral scalar and pseudoscalar states that we do not consider. In order to keep a consistent measure of collider sensitivity, we restrict ourselves to EW VBF production of H ± and H ±± . In figure 11(a), we show a diagrammatic representation of the singly charged scalar H ± produced resonantly through EW boson fusion, and present in figure 11(b) the production cross section [fb] as a function of collider energy √ s [TeV] for representative masses (M H ± ). For relatively light M H ± < 1 TeV, we find that resonant production rates are as low as σ ∼ 0.01 − 1 fb at √ s = 2 TeV and can reach as high as σ ∼ 5 − 10 fb at √ s = 30 TeV. For the relatively heavy M H ± = 2 − 4 TeV, rates can reach up to several fb at the largest √ s we consider. In figures 11(c) and 11(d), we show the same for H ++ in µ + µ + collisions. For the same mass and collider scales, we find that the resonant production rates of H ++ are a factor of a few larger than for H ± . We attribute this to the fact that the W ν coupling in the SM is larger than the Z coupling.

Minimal Supersymmetric Standard Model
In the SM, the Higgs boson possesses no symmetry that protects or stabilizes its mass against quantum corrections that naturally drive the mass away from the EW scale and toward the scale of new physics. As such, supersymmetric extensions of the SM (SUSY) are well-motivated theoretical scenarios. Under SUSY, the so-called hierarchy problem is softened or removed by hypothesizing that the Lorentz invariance obeyed by SM particles is more generally a manifestation Poincaré invariance [110][111][112][113]. Consequentially, a new degree of freedom for each SM one, but with opposite spin-statistics, must exist and orderby-order contribute oppositely to quantum corrections of the Higgs's mass. A further consequence is that these particle pairs belong to a "supermultiplet," and therefore are degenerate in mass. The lack of experimental evidence for superpartners [113][114][115][116][117][118][119][120][121][122], however, imply that if SUSY is realized at a certain scale it is broken at the EW scale. While many variations of SUSY exists and are actively investigated, the Minimal Supersymmetric Standard Model (MSSM) is the simplest supersymmetric model supported by phenomenology [110][111][112][113]. In it, the holomorphicity of the superpotential and anomaly cancellation require that two Higgs superfields be present (implying also that the MSSM is a supersymmetric extension of the 2HDM). The superpotential of the MSSM is given by where H u , H d , Q, L,ū,d,ē, are the chiral superfields to which the Higgs and fermions belong. Apart from these terms are the vector superfields containing gauge bosons and gauginos as well as the Kähler potential, which describes particles' kinetic terms. In studies and tests of the MSSM, one often also considers R-parity, defined for each particle as where B, L, and s are the baryon number, lepton number, and spin of the particle. By construction, all SM particles (and 2HDM scalars) have P R = +1, whereas their superpartners have P R = −1. A consequence of R parity is that the lightest supersymmetric particle is stable and, if it is electrically neutral, it is a good DM candidate. Generically, scalar superpartners of quarks and leptons (squark and sleptons) with the same electric charge and color quantum numbers mix. In the MSSM, this results in two 6 × 6 mixing matrices for the squarks (one each for the up and down sectors) and a 3 ×   [110][111][112][113][114], including direct searches at the LHC [115][116][117][118][119][120][121][122], evidence for the MSSM at the weak scale has yet be established. If the MSSM, or any variation of SUSY, is realized at the EW-or TeV-scale, then a multi-TeV muon collider could be an optimal machine to discover missing superparticles or study the spectrum properties.
To investigate the sensitivity of muon colliders to the MSSM, we consider the benchmark, simplified scenario where generation-1 and -2 sfermions decouple while generation-3 squarks mix in pairs, (t R ,t L ) and (b R ,b L ). We use the MSSM UFO libraries as developed by Ref. [123], and vary masses while keeping mass-splittings and couplings fixed.
In figure 12 we show diagrammatically and numerically pair production of (a,b) top squarks, (c,d) neutralinos, and (c,d) charginos through VBF in µ + µ − collisions. Starting with figure 12( In figure 12(d), we show the same information but forχ 0χ0 . Overall, the picture is bleaker. For lighter neutralinos with mχ0 1 TeV, pair production rates through weak boson fusion remain below σ ∼ 0.01 fb for collider energies below √ s ∼ 7 − 10 TeV. They reach just below σ ∼ 0.2 fb at √ s ∼ 30 TeV. For heavier neutralinos with mχ0 = 2−4 TeV, we see that cross sections remain below σ ∼ 0.1 fb for √ s 30 TeV.

Vector leptoquarks
The existence of leptoquarks, i.e., scalar and vector bosons with nonzero baryon and lepton numbers, that also possess SM gauge charges have long been predicted due to their necessity in certain grand unified theories (GUTs) [124][125][126][127][128][129][130][131]. Though not conclusively established [114,132], the possibility of leptoquarks is a viable solution to longstanding anomalies observed across several flavor experiments [133][134][135][136][137][138]. These anomalies suggest a violation of lepton flavor universality beyond what is allowed by neutrino oscillations. Hence, discovering and measuring properties of leptoquarks constitute an intriguing prospect at current and future experiments. For reviews on the topic, see Refs. [132,139] and references therein. While the spectrum of leptoquark models is vast, especially interesting options are those featuring vector leptoquarks due to their direct role in GUTs and recent demonstrations of their ultraviolet completions [140][141][142][143]. For our purposes, we consider the concrete example [144] where the vector leptoquark U µ 1 arises from the enlarged gauge group which itself is a subgroup of the Pati-Salam group G PS = SU (4) × SU (2) L × SU (2) R [124].
In this case, U µ 1 is in the (3, 1, 2/3) representation of the SM gauge group. At low energies, the relevant Lagrangian (before EWSB) can be described phenomenologically by [144]: Here, G µν = T a G aµν and B µν are the QCD and hypercharge field strengths, with associated gauge couplings g s and g Y . U µν 1 and M U are the field strength and mass of U 1 . κ U and κŨ are anomalous couplings that vanish in gauged leptoquark models. q L , L , d R , e R are the SM chiral fermion fields in the flavor basis, and g U is a flavor-universal q − − U coupling strength while β ij absorbs possible flavor dependencies.
In view of the aforementioned flavor anomalies, we assume that leptoquarks, if they indeed exist, couple mainly to generation-3 fermions with the possible extension to muons. Hence, to explore the sensitivity of multi-TeV muon colliders, we consider the process mediated by a t-channel exchange of the vector leptoquark U µ 1 , as as shown in figure 13(a). We work in the framework of equation 6.16 as implemented into the LeptoQuark FeynRules UFO model [144]. For our purposes, the relevant parameters are g U and β ij L/R and we assume the default values of the model file. We report our results in figure 13(

Heavy Dirac and Majorana neutrinos
In the SM, neutrinos are massless fermions. Neutrino oscillation data [145,146], however, unambiguously demonstrate that they in fact possess exceptionally tiny masses, with m ν k < O(1) eV [147]. If neutrinos are Majorana fermions, then their mass are also related to the breaking of lepton number conservation [148][149][150][151], an accidental symmetry in the SM. In order to reconcile these observations with the SM paradigm, neutrino mass models, collectively known as Seesaw models, hypothesize the existence of new particles that necessarily [152] couple to SM leptons and the Higgs. If kinematically accessible, such states may be discovered at collider experiments through spectacular processes that violate lepton flavor and lepton number conservation; for comprehensive reviews, see Refs. [37,153]. A commonality of many Seesaw models is the existence of heavy neutrino mass eigenstates N m that can be either (pseudo-)Dirac or Majorana. These states couple to the SM sector through mixing with SM neutrinos and/or new gauge couplings. For our purposes, we neglect subtleties related to decoupling of lepton number-violating processes in simplified models with only heavy neutrinos [151,154,155], and consider the well-studied [156][157][158] Phenomenological Type I Seesaw benchmark model, as implemented in the HeavyN UFO libraries of Ref. [159,160]. In this model, neutrino flavor eigenstates ν can be expressed generically [157] in terms of light and heavy mass eigenstates by the decomposition In the last expression we assumed that active-sterile mixing V N is dominated by the lightest, heavy mass eigenstate (m = 4), which we relabel as N ≡ N m =4 . The relevant interaction Lagrangian coupling N to the SM Weak bosons after EWSB is Here, g ≈ 0.65 is the SU (2) coupling constant, θ W is the weak mixing angle, and P L/R = (1 ∓ γ 5 )/2 are the usual chiral projection operators for four-component fermions. While there exists a number of processes in which heavy neutrinos can participate, we focus on the production of oppositely charged lepton pairs through W + W − scattering: as show in figure 14(a). This signature complements conventional channel, including the s-channel N and N ν processes and W ± γ → N ± fusion, due to its particular sensitivity to active-sterile mixing, which scales as σ µµ ∼ |V i N V * j N | 2 , and not requiring that N be on-shell. Furthermore, observing this process for i = j would give a clear indication of charged lepton flavor violation and provide guidance on the structure of neutrino mixing.
In figure 14(b), we show the cross section [fb] for the flavor-conserving process,

Vector-like quarks
A more curious aspects of the SM is the existence of three copies, or generations, of matter. While at least three generations are necessary for CP violation in the quark sector [161], no first-principle argument establishes this to be the case. Moreover, as additional chiral generations are constrained by flavor and Higgs data [162][163][164], if more copies do exist, such matter particles likely belong to different EW representations or possess new quantum numbers. One such example: vector-like fermions, which are characterized by their leftand right-handed chiral components possessing identical gauge transformations but may nonetheless carry the same gauge charges as SM particles after EWSB. As discussed in section 6.6, vector-like electrons and neutrinos are key ingredients of neutrino mass models. In addition, vector-like quarks (VLQ) offer viable, non-supersymmetric solutions to the Higgs mass hierarchy and dynamical EWSB [165][166][167]. The phenomenology of such models is rich, well-documented [168][169][170][171][172], and has led to LHC searches for vector-like top and bottom quarks in a variety of final states [173][174][175][176][177][178]. For fermionic top partners t , i.e., a VLQ with the same quantum numbers as the top quark after EWSB, the effective Lagrangian describing t can be parametrized by "decomposing" the top quark further into two mass eigenstate: Here, κ is a small, model-dependent mixing parameter and the abuse of notation is obvious. While the Lorentz structure of the gluon and photon interactions with t are dictated by gauge invariance, those of the EW bosons are less restricted. Generically, the EW couplings of a single t with uand d-flavored, SM quarks can be written as [172]: Here, M t is the mass of the VLQ, V 4i L/R is model-dependent and accounts for any potential flavor mixing, the index i runs over the three SM generations, and the parameters κ V (V = W , Z, H) encode anomalous couplings to the EW bosons.
To investigate the sensitivity of multi-TeV muon colliders to VLQs, we consider t t pair production from W + W − fusion, as shown in figure 15(a) and given by (6.24) Using equation 6.23 as implemented in the VLQ UFO libraries by Ref. [172], we show in figure 15(b) the W + W − → t t cross section [fb] in µ + µ − collisions as a function of collider energy √ s [TeV], for representative M t . We assume the default couplings of Ref. [172].
Overall, we observe a large variation of production rates as a function of mass and collider energy. For lighter t , with M t = 0.4 − 0.8 TeV, we find that cross sections remain below the σ ∼ 10 −4 fb level for √ s = 2 − 3 TeV, but quickly grow with increasing √ s. For the same mass range, rates reach roughly σ ∼ 0.5 − 5 fb by √ s = 30 TeV. For heavier t with M t = 2−4 TeV, we see that the rate growth is milder, with the σ ∼ 10 −4 fb threshold achieved at √ s ∼ 7 − 15 TeV. By √ s = 30 TeV, rates reach up to σ ∼ 10 −3 − 10 −2 fb.

Overview of vector boson fusion sensitivity
In this section we investigated the sensitivity of EW VBS to a variety of BSM scenarios at multi-TeV muon colliders. In order to give an overview picture of this reach, we present in figure 16 the requisite integrated luminosity L [fb −1 ] for a 5σ discovery as a function of new particle mass in √ s = 14 TeV (solid) and 30 TeV (dashed) muon collisions. We consider specifically the doubly charged Higgs H ++ (red) from the GM model (see section 6.3); tt (blue),χ +χ− (purple), andχ 0χ0 (yellow) pairs from the MSSM (see section 6.4). As dedicated signal and background analyses are beyond the scope of this document, we crudely assume a zero background hypothesis and full signal acceptance. We therefore also use as a simple measure of statistical significance (S) the formula, S = √ L × σ. As a general feature, we see that less integrated luminosity is needed to achieve the same discovery at higher collider energies (dashed lines) than is needed at lower collider energies (solid lines). While highly intuitive for pp colliders, this behavior is somewhat a novelty for lepton colliders because typical, s-channel annihilation processes exhibit cross sections that decrease with increasing collider energy. Hence, for s-channel annihilations, one typically needs more data at higher collider energies to achieve the same discovery potential. We attribute this improved outlook to the increasing likelihood for forward, initial-state EW boson radiation at higher collider energies. That is to say, the opening and increasing importance of EW vector boson fusion channels. In terms of the parton luminosity language of section 3, a higher collider energy translates to a larger EW boson parton luminosity. For a fixed "partonic" scattering rate, this leads to an increased, beam-level cross section, and therefore higher sensitivity. In this sense, multi-TeV lepton colliders start resembling proton colliders, and effectively become high-luminosity, weak boson colliders. While remaining in the context of the above BSM scenarios, we now explore this perspective further.

New physics processes at muon colliders: annihilation vs fusion
As we have shown here and throughout previous sections, VBF production cross sections (σ VBF ) grow with increasing √ s, a phenomenon that follows from the propensity for forward emission of transverse gauge bosons at increasing collider energies. While the precise dependence of σ VBF on collider energies of course depends on the precise BSM signature, for example on the particles involved, their underlying dynamics, and their kinematics, it nevertheless contrasts with s-channel, annihilation processes. These processes feature cross sections (σ s−ch. ) that instead decrease with collider energy as σ s−ch. ∼ 1/s, when well above kinematic thresholds. Hence, just as in the SM, we find a commonality in all VBF process here: assuming fixed model inputs, then for sufficiently high energy collider energies, VBF cross sections exceed those of analogous, s-channel production modes. As in the SM case studies of section 4, there is not a definite energy beyond which schannel, µ + µ − annihilations are categorically subdominant. The situation is more nuanced. For example: as in our SM cases, the more final-state particles involved, the larger the √ s needed for σ VBF to exceed σ s−ch. . For the resonant production of BSM states, there is of course another important parameter that plays a role: the mass scale of new, final state particle(s). New mass scales complicates the naïve scaling for VBS in two ways. First is the aforementioned propensity for collinear emission of transverse gauge bosons, which, more precisely, grows with the invariant mass of the VBF system. Second is the possible enhancement of "soft" EW boson emissions at small momentum fractions. Third is the role of matrix elements featuring large longitudinal gauge boson couplings that nevertheless possess a relatively suppressed V 0 V 0 parton luminosity (see section 3.3).
To explore how the mass scale of new particles impacts the threshold at which σ VBF surpasses σ s−ch. , and working in the context of the BSM scenarios of section 6, we compare in figure 17 a variety of VBF and analogous s-channel, annihilation processes in multi-TeV µ + µ − collisions. Assuming representative input parameters and as a function of muon collider energy [TeV], we show the VBF (solid lines) and s-channel (dashed lines) cross sections for: 17(a) SZ associated production in a singlet-scalar extension of the SM (see section 6.1); 17(b) HZ associated production in the 2HDM (see section 6.2); 17(c)tt, 17(e) χ 0χ0 , and 17(f) χ + χ − pair production in the MSSM (see section 6.4); as well as 17(d) t t pair production in a vector-like quark scenario (see section 6.7). Estimated collider energies √ s at which the VBF rates surpass the s-channel rates are summarized in Table 9.   Table 9. For representative processes and inputs, the required muon collider energy √ s [TeV] at which the VBF production cross section surpasses the s-channel, annihilation cross section, as shown in figure 17. Also shown are the cross over energies as estimated from the scaling relationship in equation (7.7) assuming a mass scale M X (2M X ).
From this exercise we observe several trends. We start by noting that the VBF production rates supersede s-channel rates at relatively lower collider energies for SZ, HZ, tt, andχ 0χ0 production than for t t and χ + χ − pair production. In particular, for SZ and HZ, we report that σ VBF becomes larger than σ s−ch. at around For pair production of t t and χ + χ − , we find that the VBF channels become more important than s-channel production at much later collider energies than the previously discussed processes. More specifically, for χ + χ − , we find that collider energies must exceed √ s ∼ 7.5 − 8.5 (11 − 15) for lighter (heavier) mass scales. For t t , the outlook is even worse.
We find that VBF production only becomes important for √ s ∼ 11 − 22 TeV for relatively light masses of M t = 0.4 − 0.8 TeV, whereas for heavier masses of M t = 2 − 4 TeV, one requires collider energies that exceed √ s = 30 TeV.
We attribute the qualitative differences between these two processes and the previous four processes to differences between subprocesses in the s-channel and VBF mechanisms. In the first four cases, both s-channel and VBF proceed largely through the same EW gauge interactions. In the latter two cases, the s-channel and VBF channels differ by additional t-channel exchanges that are governed not by gauge couplings but by mixing factors and Yukawa couplings. The crossover, then, exhibits a stronger model dependency when VBF and s-channel diagrams adhere to different dynamics or interaction strengths.
As already stated, for the SZ, HZ,tt, andχ 0χ0 channels, we observe a suggestive, universal-like behavior at which the VBF cross sections surpass their s-channel counter parts for a given final-state mass scale M X . This behavior can be roughly estimated by noting that the kinematic scaling for µ + µ − → X, s-channel cross sections are of the form We observe that the scaling behavior for VBF processes exhibits a double collinear logarithmic dependence on s, which stems from two collinear, EW PDFs, but remarkably only a single soft logarithm. The double soft logarithm does not arise as the V V → X hard process is power-suppressed by a relative factor of 1/(z 1 z 2 s) = 1/(τ s). This in turn manifests as a power-law factor that grows linearly with (s/M 2 X ). Altogether, this enables us to roughly estimate the collider energy √ s at which σ VBF surpasses σ s−ch. for a given final-state mass M X . Essentially, one must solve for when While the result is transcendental, the solution can easily be extracted numerically for the representative M X , which we report in the rightmost column of Table 9 assuming a mass scale M X (2M X ). For sub-TeV masses, the scaling behavior systematically underestimates the true crossover by roughly a factor of two. This is unsurprising as equation 7.7 assumes a large hierarchy between relevant scales. For TeV masses and above, however, we find good agreement between equation 7.7 and explicit computation from Monte Carlo computations. We report differences ranging from the percent level to the 20% level.

Conclusions
The next generation of particle accelerators needed to explore the energy frontier will offer tremendous challenges. Among these is a muon collider running at energies up to several TeVs and luminosities in the tens of inverse attobarns, a dream machine both from the technology and physics points of view. Overcoming the challenges posed by producing, storing, and colliding high-intensity beams of high-energy muons will take years of further research and development. Exploring the physics potential of such machines, on the other hand, is a relatively easy task that can be undertaken on a short time scale. In this paper, we have moved a small step forward in the latter direction by considering electroweak vector boson fusion/scattering (VBF) processes at a future multi-TeV lepton collider in a rather systematic way. Our study is motivated by the simple observation that while s-channel production rates decreases with increasing collider energy as 1/s, VBF rates grow as a power of log s, and therefore, for any final state, VBF is consigned to eventually emerge as the leading production mechanism.
In this context, we have investigated and show in section 3 that, compared to hadron colliders, VBF is a much more relevant production mechanism at a high-energy lepton collider. We continue in section 4 and present for a rather large set of SM final states involving EW vector bosons, Higgs bosons, and top quarks the corresponding VBF cross sections and at what collider energy they surpass s-channel production modes. We find that VBF becomes the dominant production mechanism at relatively low collider energies, starting at just a few TeV for low final-state multiplicities and increases for higher multiplicities.
In order to further illustrate what could be attainable in terms of new physics reach, we then moved in two directions, focusing mostly on luminosity scenarios envisaged for a Explicitly, we use the Mathematica function NSolve and report very quick runtime on a personal laptop. muon collider. First, in section 5, we considered prospects for precision measurements of the Higgs's self-couplings and the top quark's EW couplings, and interpreted sensitivity in terms of Wilson coefficients within the SMEFT framework. Second, in section 6, we explored a variety of simplified extensions of the SM and how large VBF luminosities can maximize the direct search for new physics. In particular we find evidence that in several instances the reach of a multi-TeV muon collider is comparable or better than that attainable at a 100 TeV proton-proton collider. A detailed comparison of VBF's utility over s-channel annihilations in BSM searches was then summarized in section 7.
The results presented in this work are meant to provide a first glimpse of what could be achieved at a multi-TeV muon collider in VBF channels, and certainly motivates further and more refined investigations. We close by stressing that while we focus on the specific prospects of a muon collider, our conclusions hold equally for other lepton colliders.