title: Electroweak Couplings of the Higgs Boson at a Multi-TeV Muon Collider
authors: Tao Han Da Liu Ian Low Xing Wang
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          𝒜ℬ𝒞𝒟ℰℱ𝒢ℋℐ𝒥𝒦ℒℳ𝒩𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒳𝒴𝒵
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              εϵTr𝒪αβλλ̃σ̅⟨⟩ȧḃCSLDSLpolegutRescyclℒ𝒟≳
                  μ^+μ^-m_bb̅^κ_V^κ_V2^κ_W^κ_Z^3̨κ_3^4̨κ_4^
                  PITT-PACC-2008a]Tao Han,b]Da Liu,c,d]Ian Lowe]and Xing Wang[a]Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15217, USA[b]Center for Quantum Mathematics and Physics (QMAP),  University of California, Davis, CA 95616, USA[c]Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA[d]High Energy Physics Division, Argonne National Laboratory, Lemont, IL 60439, USA[e]Department of Physics, University of California-San Diego, 
                  La Jolla, CA 92093, USA
                  We estimate the expected precision at a multi-TeV muon collider for measuring the Higgs boson couplings with electroweak gauge bosons, HVV and HHVV (V=W^±,Z), as well as the trilinear Higgs self-coupling HHH. At very high energies both single and double Higgs productions rely on the vector-boson fusion (VBF) topology.
                  The outgoing remnant particles have a strong tendency to stay in the very forward region, 
                  leading to the configuration of the “inclusive process” and making
                   it difficult to isolate ZZ fusion events from the WW fusion. In the single Higgs channel, we perform a maximum likelihood analysis on HWW and HZZ couplings using two categories: the inclusive Higgs production and the 1-muon exclusive signal.
                  In the double Higgs channel, we consider the inclusive production and study the interplay of the trilinear HHH and the quartic VVHH couplings, by utilizing kinematic information in the invariant mass spectrum. We find that at a centre-of-mass energy of 10 TeV (30 TeV) with an integrated luminosity of 10 ab^-1 (90 ab^-1), one may reach a 95% confidence level sensitivity of 
                  0.073% (0.023%) for WWH coupling, 
                  0.61% (0.21%) for ZZH coupling,
                  0.62% (0.20%) for WWHH coupling, and
                  5.6% (2.0%) for HHH coupling. 
                  For dim-6 operators contributing to the processes, these sensitivities could probe the new physics scale Λ in the order of 1-10 (2-20) TeV at a 10 TeV (30 TeV) muon collider. 
                  Electroweak Couplings of the Higgs Boson
                   
                  at a Multi-TeV Muon Collider
                      [
                      August, 2020
                  =======================================================================
                  
                  
                  
                  
                  § INTRODUCTION
                  
                  
                  The discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) opens a new avenue in particle physics. On the one hand, the existence of the Higgs boson completes the particle spectrum in the Standard Model (SM) and provides a self-consistent mechanism in quantum field theory for mass generation of elementary particles. On the other hand, the SM does not address the underlying mechanism for the electroweak symmetry breaking (EWSB) and thus fails to understand the stability of the weak scale with respect to the Planck scale. In order to gain further insight for those fundamental questions, it is of high priority to study the Higgs boson properties to high precision in the hope to identify hints for new physics beyond the SM.
                  
                  In the SM, the Higgs sector is constructed from a complex scalar doublet Φ. After the EWSB, the neutral real component is the Higgs boson excitation H and the other three degrees of freedom become the longitudinal components of the massive gauge bosons. As such, studying the Higgs-gauge boson couplings would be the most direct probe to the underlying mechanism of the electroweak symmetry breaking. 
                  After the EWSB, the Higgs sector can be parameterized as 
                  
                      ℒ   ⊃   (M_W^2W^+_μ W^-μ+1 2M_Z^2Z_μ Z^μ) ( 2H v + κ_V_2H^2 v^2) -m_H^2 2 v( κ_3^ H^3 + 1 4vκ_4^ H^4 ),
                  
                  where v=246 GeV is the vacuum expectation value of the Higgs field and κ_i =1 for the SM couplings at tree-level. This “κ-scheme” is a convenient phenomenological parameterization of  deviations from the SM expectations, which is suitable for the exploratory nature of the present study. Here it is made implicit that κ_V=κ_W=κ_Z. This is the prediction of the tree-level custodial SU(2) invariance <cit.>, which is an accidental symmetry of the SM. This has been verified to a good accuracy by precision EW measurements <cit.>. Nevertheless, in our fit we wish to be more general and will not be assuming a correlated κ_W and κ_Z.
                  
                  
                  A fully consistent and theoretically-sound framework would utilize effective field theories (EFT), by augmenting the SM Lagrangian with higher dimensional operators from integrating out the heavier states <cit.>. 
                  While a systematic account for the effects of the higher dimensional operators is much more involved and beyond the scope of the current work, we would like to consider the  following two operators for the purpose of illustration <cit.>
                      𝒪_H =c_H/2Λ^2∂_μ(Φ^†Φ) ∂^μ(Φ^†Φ) ,   𝒪_6 = -c_6 λ/Λ^2 (Φ^†Φ)^3  ,
                  
                  where Λ is the cutoff scale where new physics sets in, and 
                  λ is the quartic coupling parameter  in front of (H^† H)^2 term in the SM Higgs potential. At the dimension-six level these are the two operators that are most relevant for our study. An additional operator, Φ^†Φ (D_μΦ)^† (D^μΦ), can be removed by a suitable field-redefinition <cit.>. The resulting shifts Δκ_i ≡κ_i-1 in Eq. (<ref>) are[Interestingly, in most cases there is a positivity constraint on c_H>0, thereby reducing the VVH and VVHH coupling strengths <cit.>.]
                  
                      Δ   = -c_H 2v^2Λ^2 ,    Δ   = - 2c_H v^2Λ^2 ,
                      Δκ_3    ≈ -3c_H 2v^2Λ^2 + c_6v^2Λ^2 ,      Δκ_4   ≈  - 25/9 c_H v^2Λ^2 + 6 c_6 v^2Λ^2,
                  
                  We see that deviations in the VVH and VVHH (V=W^±,Z) couplings are correlated and controlled by the same operator O_H. However, the precision we are expecting is high and could potentially be sensitive to effects of dimension-8 operators, in which case the correlation may be modified. 
                  
                   On the other hand, the Higgs trilinear self-coupling κ_3 is among the most important interactions to be tested in the Higgs sector – it governs the shape of the Higgs potential and, consequently, the nature of the electroweak symmetry breaking. In addition, κ_3 controls the strength of the  electroweak phase transition, which is important for understanding the cosmological evolution of the early universe as well as the origin of the observed matter-anti-matter asymmetry in the current unverse <cit.>.  Precise measurements of these couplings will provide insights on how nature works at the shortest distance scale ever probed by mankind. Needless to say, should deviations from the SM predictions be observed, it would completely revolutionize  our understanding of the physical laws of nature.
                  
                  With the great success of the LHC program, we have achieved the measurement of the   VVH to  O(5%) accuracy <cit.>, which will  be further improved by roughly a factor of two with the high-luminosity LHC upgrade <cit.>. In e^+e^- collisions at the International Linear Collider (ILC) <cit.>, the proposed Higgs factories <cit.> and the CLIC <cit.>, sub-percent level accuracies for WWH of O(0.6% - 1.2%) and ZZH 
                  of O(0.2% - 0.5%) could be achievable. 
                  However, the trilinear HHH and quartic VVHH couplings are still difficult to measure to an informative level without a very high energy collider <cit.>. At a 100 TeV hadron collider such as the SPPC or FCC_hh, the trilinear Higgs self-coupling could be measured with  O(5%) uncertainty <cit.>. 
                  
                  Recently, an attempt was made to determine the quartic Higgs self-coupling at a high-energy muon  collider <cit.>.
                  
                  In the EFT language, the precision to which one could measure the Higgs couplings can be translated into constraints on the scale suppressing dimension-6 operators, which is indicative of the scale where new physics becomes important. A figure of merit is when Λ∼ 1 TeV which, generally speaking, would induce a corresponding deviation in the Higgs couplings of the order <cit.> O(v^2/Λ^2) ∼ O(5%)      for  Λ∼ 1    TeV .
                  
                  Therefore, in order to probe new physics scale above 1 TeV, it is important to be able to reach a precision level of 5% or less. In addition, in a lepton collider a truly model-independent determination of the trilinear HHH coupling requires simultaneously measuring the 4-point VVHH coupling, which is difficult to access at low energies and without sufficiently high statistics.
                  
                  Recently, there has been a renewed interest to consider a muon collider with a very high centre-of-mass (CM) energy in the tens of TeV <cit.>. While the previous discussions for a muon collider were focused on a Higgs factory operating at the SM Higgs resonance <cit.>, a collider operating at a multi-TeV regime would certainly lead us to a new territory at the energy frontier.  
                   Such a multi-TeV  muon collider  offers a unique opportunity to probe the electroweak couplings of the Higgs boson, including VVH, HHH and VVHH couplings. The possible CM  energy under discussion ranges from 3 TeV to 30 TeV, with a representative benchmark target at 10 TeV or higher. Very high luminosities are also envisioned, with the scaling relation as <cit.> Lumi. >  5 years time (√(s) 10  TeV)^2 2 · 10^35  cm^-2 s^-1 .
                  
                  This will yield to an integrated luminosity of O(10) ab^-1 at √(s) = 10 TeV and O(90) ab^-1 at √(s) = 30 TeV, 
                  which would take us to a remarkable new energy frontier, and offer great potential to study the Higgs boson, and the Nature in general at an unprecedented short-distance scales. In this paper, we would like to explore the Higgs physics and examine the accuracies for the electroweak couplings of the Higgs boson  at the future high-energy muon collider.
                  
                  The rest of the paper is organized as follows. 
                  We first present the Higgs boson production rates via various production mechanisms at a high-energy muon collider in Sec. <ref>. We then evaluate the statistical accuracy achievable to determine the HVV couplings in Sec. <ref>. Foremost, we show the improvement for the precision measurement on the triple Higgs boson coupling as well as the VVHH coupling in Sec. <ref>. We summarize our results and conclude in Sec. <ref>. 
                  
                  
                  
                  
                  § HIGGS BOSON PRODUCTION AT A HIGH-ENERGY MUON COLLIDER
                  
                  
                  The Higgs boson couples predominantly to heavier particles. The production of a Higgs boson thus involves  other heavy particles in the SM. At high energies, gauge bosons will copiously radiate off the colliding beams.
                  Therefore, the vector boson fusion (VBF) mechanism are the dominant source for the Higgs boson  production at a high-energy muon collider <cit.>. 
                  The production processes involving the Higgs boson at a high-energy muon collider include 
                     VBF⟶   H, ZH, HH   and   tt̅ H  ,
                  
                  which are all dominantly from the VBF processes. 
                  
                  We list the production cross sections in Table <ref> for those Higgs production processes with a few representative benchmark energy choices. Cross sections are computed using the package  MadGraph<cit.>. Recently it has been advocated that, in high energy collisions, it may be appropriate to adopt the approach of electroweak parton distribution functions (EW PDF)  <cit.> to resum the potentially large collinear logarithms at high scales. For the processes under consideration, the difference is insignificant since the single Higgs production is set by a low scale m_H, while the Higgs pair production HH is dominated by the longitudinal gauge boson fusion (W_LW_L), that has no scale dependence at the leading order. 
                  
                  
                  
                  
                  We will examine the precision measurements of the Higgs boson couplings via the production processes as depicted in Figs. <ref> and <ref>. 
                  For instance, at a 10 TeV muon collider with an integrated luminosity of 10 ab^-1, we may expect the production of about 10^7 Higgs bosons and 3.6× 10^4 Higgs pairs. 
                  For comparison, we have also included in Table <ref> the SM irreducible backgrounds VBF→ Z, ZZ, which are also largely from the VBF mechanism, in Table <ref>. Although the background rates are larger than the signals by a factor of 4 (55) for the H (HH) process, they populate different kinematical regions from the signals and can be reduced by appropriate kinematic cuts. 
                  
                  
                  
                  
                  § VVH COUPLINGS
                  
                  
                  At high energy lepton colliders, the cross section for single H production via the Higgs-strahlung → ZH falls as 1/s. The high statistics channels for measurements of VVH couplings rely on the WW and ZZ fusion via the VBF topology:
                     →   ν_μν̅_μ H     (WW fusion), 
                     →    H     (ZZ fusion).  
                  See Fig. <ref> for a representative Feynman diagram. 
                  It would be desirable to separate these two classes of events by tagging the outgoing muons and achieve independent measurements on WWH and ZZH couplings. However, 
                  for the VBF topology, the outgoing muons have a tendency to stay in the forward region
                  due to the t-channel propagator shown in Fig. <ref>(a). Although the transverse momentum of the outgoing muons is sizable and governed by the propagator mass p_T^μ∼ M_Z, 
                  at very high energies the muons are all extremely forward with a polar angle typically θ_μ≈ M_Z/E_μ. In Fig. <ref>(a), we show the  angular distributions of the outgoing muons at √(s)=3, 10, 30 TeV. One can see that, for example, the scattering angle for a muon is peaked near θ_μ∼ 0.02≈ 1.2^∘ at 10 TeV. These very forward muons would most likely escape the detection in a detector at a few degrees away from colliding beams. 
                  This feature makes it increasingly difficult to distinguish the processes of the neutral currents 
                  (ZZ fusion <cit.>) from the charged currents (WW fusion) at higher energies. 
                  Therefore, separating these two classes of events would require the capability of detecting very energetic muons in the forward region of a few degrees with respect to the beam. Without this, we would have to focus on
                  the “inclusiveness,” a dominant behavior of the collinear splitting physics recently emphasized in Ref. <cit.>.
                  As a consequence, we will consider two classes of events for VBF production of single H:
                  
                    *  Inclusive channel: events from WW fusion and from ZZ fusion without detecting muons;
                  
                    *  Exclusive 1μ channel: events from ZZ fusion with at least one muon detected.
                  
                  
                  The inclusive channel is populated predominantly by events from the WW fusion, but also contains  events from ZZ fusion when the outgoing muons go down the beam pipe and escape detection. 
                  However, as seen from Table <ref>, ZZ-fusion cross section is roughly 10% of the WW fusion cross section, and thus a small contamination for the WWH measurement. 
                  
                  The 1μ channel, on the other hand, comes from the ZZ fusion and is uniquely sensitive to the ZZH coupling, although  it suffers from poor selection efficiency after requiring a muon identification.  In Fig. <ref>(b), we illustrate the  fiducial cross section after the angular acceptance cut θ_μ^-^ cut. At a fixed angular acceptance, the cross section falls as σ∼ 1/E_μ^2. 
                  
                  
                  
                  
                  
                   §.§ Inclusive channel
                  
                  
                  
                  Processes contributing to the inclusive channel are shown in Eqs. (<ref>) and (<ref>).
                  We focus on the leading decay channel H → b b̅. 
                  The Higgs boson signal will be b b̅ pair near the Higgs mass m_H plus large missing energy, resulting from the missing neutrinos and the undetected muons. 
                  We impose the basic acceptance cuts on the b jets
                  
                  p_T(b) > 30  GeV ,      10^∘ < θ_b  < 170^∘ ,
                  
                  where θ_b is the polar angle of the b (b̅) jet in the lab frame. 
                  
                  The irreducible backgrounds, →ν_μν̅_μ Z, from either WW fusion shown in Table <ref> or → ZZ →ν_μν̅_μ Z,
                  which can be readily removed due to the on-shell Z decay Z→ν_μν̅_μ, by a “recoil mass” cut
                  
                  M_ recoil = (p_μ^+ + p_μ^+ - p_H^)^2 > 200  GeV.
                  
                  The key aspect to identify the Higgs signal lies in the resolution to effectively select the bb̅ at the resonant m_H. 
                  In Fig. <ref>(a) we  plot the invariant mass distribution for the H signal for √(s)=10 TeV, after the acceptance cuts and assuming a jet energy resolution of 
                  Δ E/E= 10%.
                    
                  For comparison, we have also shown in the same plot the distribution from the Z background. Here we have included all quarks flavors b,c,s,d,u. If we demanded a b-tagging for our signal selection, we would be able to reduce the Z→ jj background by a factor of 5. However, we do not find the b-tagging necessary due to the highly efficient kinematical constraint on m_bb̅. In estimating the statistical accuracy for the coupling measurement, we  impose the a mass cut 
                   = m_H ± 15  GeV .
                  
                  With those cuts, the Z background is essentially removed and we retain the majority of the signal. 
                  The event selection efficiencies (ϵ_ in) and the resulting cross sections at different collider energies are summarized in Table <ref> in the top rows.  
                  
                  
                  It is worth noting that, at higher CM energies, the b jets have increasingly small polar angles in the Lab frame and become more forward. The angular distributions for various energies are shown in the right panel of Fig. <ref>, where we see the majority of b jets have θ_b < 10^∘ at √(s)=30 TeV.  This is the reason for the worsening selection efficiencies in Table <ref> as we go to higher CM energies. Obviously, extending the detector angular coverage would significantly increase the signal acceptance. 
                  If the angular cut on θ_b in Eq. (<ref>) is tightened up to 20^∘ - 160^∘ instead, the signal reconstruction efficiency will be scaled down by about 10%. 
                  
                  
                  
                  
                  
                  The total cross section in the inclusive channel can be written, at the leading order, as
                  σ_ in   =    (1+Δκ_W)^2 σ^ SM_W + (1+Δκ_Z)^2 σ^ SM_Z
                  
                  where σ^ SM_W and σ^ SM_Z are the SM cross sections for the WW/ZZ fusion processes. In cases where Δκ_W/Z≪ 1, the linear terms dominate which, in the EFT language, is equivalent to keeping only the interference term from the dim-6 operators. We do not make such an assumption in the κ-scheme adopted in this work.
                  
                  In this subsection we will vary  and  one at a time, and consider a simultaneous fit to both parameters later in this section. The 95% confidence level (C.L.) sensitivities in the relative errors Δκ_W/Z are shown in Table <ref>. 
                  The achievable accuracies are impressive, comparing with the anticipated best results Δ∼ 0.6% from the ILC/CLIC and Δ∼ 0.2% from the expectations at the Higgs factories <cit.>. 
                  
                  
                  
                  
                  
                   §.§ Exclusive 1μ channel
                  
                  
                  The leading process contributing to the exclusive 1μ channel is ZZ fusion in Eq. (<ref>), whose rate is shown in Table <ref>. 
                  Again, with the same decay mode, the Higgs boson signal will be b b̅ pair near the Higgs mass m_H plus  in the forward-backward regions. 
                  
                  The leading background is → ZZ → Z with Z→ b b̅. There is no WW fusion analogue for this channel. We adopt the same basic cuts as in Eqs. (<ref>), (<ref>) and (<ref>). The background is highly suppressed.  In addition, we require the presence of at least one muon to be in
                     10^∘ < θ_μ^±  < 170^∘ .
                  
                  This turns out to be very costly to the signal, since the majority of the muons have θ_μ < 10^∘, as already seen in Fig. <ref>. As such, the signal reconstruction efficiencies for this channel are very low and are shown in Table <ref>, together with the predicted cross sections in the middle rows. 
                  With the high luminosity expected, the 95% C.L. on the coupling measurements is shown also in Table <ref> for the exclusive 1μ channel.
                  Although the result at a 3 TeV collider is comparable to that from the inclusive channel, at higher energies the estimated precision is worse than  the inclusive channel despite the higher energies and more luminosities. This is mainly due to the significantly reduced number of events from the tagging requirement for a forward-backward muon.  
                  
                  It is important to note another significant consequence of requiring one muon in the range of 10^∘ < θ_μ^±  < 170^∘. For highly energetic muons, this large scattering angle leads to a high transverse momentum p_T^μ > 0.17E_μ and, consequently, induces a strong recoil in the Higgs boson produced in the final state. In Fig. <ref> we show the p_T distribution of the Higgs boson in (a) for the 1μ channel as well as R_bb in (b), the separation of the b-jets from H→ bb̅. In particular, at √(s)=30 TeV, the Higgs boson tend to have a large p_T, in the order of 2.5 TeV, and the resulting decay is boosted with R_bb∼ 0.2. Care needs to be taken when reconstructing such  boosted events. 
                  
                  
                  
                  
                  
                   §.§ Two-parameter likelihood fit of κ_W and κ_Z
                  
                  
                  
                  In this subsection we perform a two-bin likelihood fit of κ_W and κ_Z making use of  the inclusive and exclusive 1μ channels. 
                  We construct a Poisson log-likelihood function 
                  
                      LL = lne^-N(Δκ_W, Δκ_Z)[N(Δκ_W, Δκ_Z)]^N_ SM/N_ SM!,
                  
                  where the numbers of events are
                  
                      N(Δκ_W, Δκ_Z) = σ(Δκ_W,Δκ_Z) ℒ_ lumi ,   N_ SM = σ(Δκ_W =0, Δκ_Z =0) ℒ_ lumi ,
                  
                  and ℒ_ lumi is the integrated luminosity.
                  We compute such likelihood function for each channel and a global likelihood as the product of the individual ones. Then we compute the 68% and 95% C.L. regions on the 
                  Δ-Δ plane, corresponding to LL =  LL_ max - 1.15 and LL =  LL_ max - 3.10, respectively. The resulting contours are shown in Fig. <ref>.
                  As expected, the precision for Δ is better than Δ by about an order of magnitude at high energies. The projection of the elipses onto the Δκ_W-axis in Fig. <ref> gives the uncertainty marginalized over Δκ_Z, and vice versa. The resulting errors are larger than those in the single parameter fit, which varies one parameter at a time and assumes SM values for the rest.
                  
                  
                  
                  
                  
                  
                  § HHH AND WWHH COUPLINGS
                  
                  
                  
                  Pair production of the Higgs boson provides a direct measurement on the trilinear HHH and quartic VVHH couplings. The main advantage of a high-energy collider, with √(s)≫2 m_H, lies in the capability to copiously produce Higgs boson pairs. At a high-energy muon collider, as shown in Sec. <ref>, one would expect about 36,000 (68,000) HH at 10 TeV (30 TeV).
                  To probe the Higgs self-coupling, we utilize the VBF mechanism for the inclusive double Higgs production 
                  
                      μ^+μ^- VBF⟶ HH + X ,
                  
                  where X=νν̅ for WW fusion and μ^+μ^- for the ZZ fusion. As can be seen from the Feynman diagrams in Fig. <ref>, the HH production involves three classes of couplings: κ_W, κ_3 and κ_W2. Since κ_W can be measured very precisely from the single Higgs production, as shown in Section <ref>, we will assume in the current section that κ_W=1 as in the SM and study the interplay of κ_3 and κ_W2 in the HH production. As discussed in Section <ref>, the outgoing remnant particles tend to stay in the forward region and escape detection. Therefore, similar to the single Higgs production, we will consider the inclusive channel in Eq. (<ref>), which is populated dominantly by the WW fusion and, to a less extent, by the ZZ fusion events when the outgoing muons are too forward to be detected.
                  
                  
                  
                  The cross section for the inclusive μ^+μ^-→  HH+X can be parametrized as <cit.>
                      σ = σ_ SM[1 + R_1 Δκ_W_2 + R_2Δκ_3 + R_3 Δκ_W_2Δκ_3 + R_4 (Δκ_W_2)^2 + R_5 (Δκ_3)^2],
                  
                  where the σ_ SM is the SM cross section. The SM cross section σ_ SM and coefficients R_i, before any cuts, are given in Table <ref>.
                  
                  It is instructive to consider the energy dependence of different classes of Feynman diagrams contributing to HH production, by studying the partonic scattering W^+W^-→ HH. As the dominant contribution comes from the longitudinal W scattering W^+_LW^-_L→ HH, the scattering amplitude can be written as
                  
                      𝒜(W^+_LW^-_L→ HH) = 𝒜_ SM + 𝒜_1 Δκ_W_2 + 𝒜_2 Δκ_3,
                  
                  where 𝒜_ SM, 𝒜_2 ∼ constant, and 𝒜_1 ∼ E^2 at high energies E≫ M_W. Because of the energy growing behavior of 𝒜_1, the cross section has a strong dependence on Δκ_W_2 over a large range of phase space. As a result, we expect to be able to constrain κ_W_2 better than κ_3. This argument also shows, when extracting the trilinear Higgs self-coupling it is important to consider the impact from the quartic VVHH coupling.  In this study, we have assumed the HHVV vertex is modified only in its strength for simplicity, while in many well-motivated new physics models the tensor structure of the quartic coupling could also be corrected <cit.>. It will be interesting to further assess the impact of these additional modifications on the extraction of κ_3<cit.>. 
                  
                  For the Higgs decays, we once again focus on the leading decay channel HH → b b̅ b b̅, which has a SM branching fraction BR(4b)≃ 34%. We impose basic acceptance cuts
                  
                      p_T(b) > 30  GeV,   10^∘ < θ_b < 170^∘,  Δ R_bb > 0.4.
                  
                  As before, we further assume the jet energy resolution to be Δ E/E = 10%.
                  
                  The Higgs candidates are reconstructed from the four most energetic jets. The four jets are paired by minimizing
                  
                      (m_j_1j_2 - m_H)^2 + (m_j_3j_4 - m_H)^2.
                  
                  And for each Higgs candidate, we impose 
                  
                      |m_jj - m_H| < 15  GeV
                  
                  to reject background from Z and W resonances. We also require the recoil mass
                  
                      M_ recoil = √((p_μ^+ + p_μ^- - p_H_1 - p_H_2)^2) > 200  GeV.
                  
                  The signal selection efficiencies and the corresponding cross sections are listed in Table <ref>.
                  If we tighten the angular cut to 20^∘, the efficiencies would drop by a factor of 3–4. 
                  
                  
                  
                  We again perform a simultaneous fit to κ_3 and κ_W_2 using binned maximum likelihood fit. Given the different energy dependence in the subamplitudes controlled by κ_3 and κ_W_2, we decided to bin the m_HH distribution into the following intervals[A similar procedure for double Higgs production in hadron colliders can be found in Ref. <cit.>.]
                      m_HH^ = [0, 350, 450, 550, 650, 750, 950, 1350, 5000]  GeV.
                  
                  The binned cross section of μ^+μ^-→  HH + X → bb̅ bb̅ + X after the selection cuts can be parametrized, in a similar fashion, as 
                  
                      σ = σ_ SM[1 + r_1 Δκ_W_2 + r_2Δκ_3 + r_3 Δκ_W_2Δκ_3 + r_4 (Δκ_W_2)^2 + r_5 (Δκ_3)^2],
                  
                  where the values are given in Table <ref> for √(s)=10 TeV for illustration. 
                  It is important to note again the increasing sensitivity on κ_W_2 at higher values of m_HH^. The resulting contours are shown in Fig. <ref>. In Table <ref> we also provide the 95% C.L. from the single parameter fit, by allowing κ_3 and κ_W_2 to vary only one at a time.
                  
                  
                  
                  
                  
                  
                  § DISCUSSION AND CONCLUSION
                  
                  
                  As we have shown in this work, a multi-TeV high energy muon collider will have a tremendous potential to constrain the electroweak Higgs couplings with  unprecedented accuracy. It will offer a unique probe into the nature of the Higgs boson as well as the scale of possible new physics beyond the SM. 
                  In Table <ref>, we present a summary of the estimated sensitivities  at different collider energies and luminosities. In the last column of the table, we compare with the expected precision from other proposed colliders. It is clear that a multi-TeV muon collider could improve the measurements substantially. 
                  
                  It is possible to translate the bound in the κ-scheme into the constraint on Λ, the scale of new physics associated with the dim-6 operators in Eq. (<ref>),
                  Λ∼√(c^_H,6Δκ) v.
                  
                  Assuming c_6,H∼ O(1), the scale is estimated to be Λ∼ 1  TeV/√(16Δκ), as shown in Table <ref>. 
                  A summary figure, which combines our results for the coupling measurements, is given in Fig. <ref>, with the upper horizontal axis marking the estimated scale Λ in TeV. With Λ / √(c)_i ∼ (10-16) TeV at a collider of (10-30) TeV, we would be probing new physics at very high scales or deeply into quantum effects.
                  
                  
                  
                  
                  
                  In our analyses, we only focused on the leading decay channel H→ bb̅. A more comprehensive study could include the other decay channels as well, such as H→, WW, ZZ,ττ and γγ, to further improve the precision. On the other hand, due to the lack of knowledge of the specifics of the detector design, we have not made any attempts for experimental detector simulations. Further work may be needed to draw a more complete conclusion for the expected sensitivity reach. 
                  
                  In summary, we estimated the expected precision at a multi-TeV muon collider for measuring the Higgs boson couplings with electroweak gauge bosons, HVV and HHVV, as well as the trilinear Higgs self-coupling HHH. With the anticipated high CM energies and high luminosities,  a multi-TeV muon collider could provide us with unparalleled precision for Higgs physics and, consequently, offer some of the most stringent experimental tests of the SM Higgs sector. 
                  As we have shown in this study, the outgoing remnant particles have a strong tendency to stay in the very forward region.
                  The enhanced collinear behavior of the final state particles results in the dominant configuration of  “inclusive” processes, a notion usually reserved for hadron colliders, unless there is a device to detect the very forward muons of a few degrees from the beam. 
                  
                  These features add new subtlety to Higgs coupling measurements, since it is now difficult to isolate WW fusion from ZZ fusion events in the Higgs production. We addressed the subtlety by performing binned maximum likelihood analyses to simultaneously fit two parameters involved in the inclusive processes. The approach and  methodology adopted in this study could be applicable to new physics searches at a high energy muon collider. 
                  
                  
                  
                  The work of TH was supported in part by the U.S. Department of Energy under grant No. DE-FG02- 95ER40896 and in part by the PITT PACC. The work of DL was supported in part by the U.S. Department of Energy under grant DE-SC-0009999.  IL is supported in part by the U.S. Department of Energy under contracts No. DE- AC02-06CH11357 at Argonne and No. DE-SC0010143 at Northwestern.  XW was supported by the National Science Foundation under Grant No. PHY-1915147.
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