PHYSICS OPPORTUNITIES AT A MULTI-TEV MUON COLLIDER - TAO HAN UNIVERSITY OF PITTSBURGH - CERN ...

Physics Opportunities
at a Multi-TeV Muon Collider
 Tao Han
 University of Pittsburgh
 KFC Workshop
 August 26, 2021

 1

Physics @ a Multi-TeV Muon Collider
1. SM expectations:
 - QED & QCD
 - EW physics at ultra-high energies
 - Precision Higgs measurement
 - Multiple boson peoduction
2. Beyond the SM:
 - WIMP Dark Matter
 - Extended Higgs sector

 2

Lots of recent works, not covered here
 -- my apologies
D. Buttazzo, D. Redogolo, F. Sala, arXiv:1807.04743 (VBF to Higgs)
A. Costantini, F. Maltoni, et al., arXiv:2005.10289 (VBF to NP)
M. Chiesa, F. Maltoni, L. Mantani, B. Mele, F. Piccinini, and X. Zhao,
 arXiv:2005.10289 (SM Higgs)
R. Capdevilla, D. Curtin, Y. Kahn, G. Krnjaic,
 arXiv:2006.16277; arXiv:2101.10334 (g-2, flavor)
P. Bandyopadhyay, A. Costantini et al., arXiv:2010.02597 (Higgs)
D. Buttazzo, P. Paradisi, arXiv:2012.02769 (g-2)
W. Yin, M. Yamaguchi, arXiv:2012.03928 (g-2)
R. Capdevilla, F. Meloni, R. Simoniello, and J. Zurita, arXiv:2012.11292 (MD)
D. Buttazzo, F. Franceschini, A. Wulzer, arXiv:2012.11555 (general)
G.-Y. Huang, F. Queiroz, W. Rodejohann,
 arXiv:2101.04956; arXiv:2103.01617 (flavor)
W. Liu, K.-P. Xie, arXiv:2101.10469 (EWPT)
H. Ali, N. Arkani-Hamed, et al, arXiv:2103.14043 (muon smasher’s guide)
 ……

 3

µ Collider Recent technological breakthroughs:
 e+e− interactions. The small overall charge in
 µ+ the collider rings Positron Linac
 ECoM:
 µ−
 – hence, lower backgrounds in a

 Isochronous
 Rings
 production emittance allows lower

 100 KW
 target
 10s of TeV
 overall charge in the collider rings Positron collider
 Linac detector
 µ− and a higher
 – hence, lower backgrounds in a

 Isochronous
 Rings
 potential CoM energy due to Accelerators:

 100 KW
 target
 collider detector and a higher Linacs, RLA or
 Proton Driver
 potential CoM energy due Front
 to End Cooling neutrino radiation.
 Acceleration
 Accelerators:
 µ+ µ−
 Collider Ring
 neutrino radiation. Linacs, RLA or FFAG, RCS
 µ+
 ECoM:
 Higgs Factory

 Charge Separator
 Initial 6D Cooling
 Decay Channel
 MW-Class Target

 Phase Rotator
 µ− to
 J.

 Buncher
 Capture Sol.

 Final Cooling
 Combiner
 Accumulator

 Buncher

 6D Cooling

 6D Cooling
 SC Linac
 J. P. Delahaye et al.,
 ~10arXiv:1901.06150
 TeV
Driver Front End Cooling Acceleration Collider Ring

 Merge
 Bunch
 µ+ µ−
 µ+ Accelerators:

 µ Collider
 ECoM :
 Linacs, RLA or FFAG, RCS

 Muon Accelerator Program
 Higgs Factory
 Low EMittance
 Muon Accelerator ProgramMuon Accelerator
 Low EMitt
 Charge Separator
 Initial 6D Cooling
 Decay Channel
 MW-Class Target

 Phase Rotator

 µ − toAcceleration
 Buncher

 Positron Linac Positron
 Capture Sol.

 Low EMmittance Muon Collider Ring

 Final Cooling
 Combiner
 Accumulator

 Buncher

 Proton-Driver: 6D Cooling

 6D Cooling
 Ring ~10 TeV
 map.fnal.gov web.infn.it/LEMMA
 Accelerator (LEMMA):

 Merge
 Bunch
 map.fnal.gov web
 1011 µ pairs/sec from
 Protone+Driver Front End Cooling µ+ µ+ Acceleration
 µ− E :Ring
 Collider CoM
 e− interactions. The small Accelerators:
 production emittance allows lower Linacs, RLA or FFAG, RCS 10s of TeV
 overall charge in the collider rings
 µ+
 Positron Linac ECoM:
 – hence, lower backgrounds in a New results on µ cooling by MICE collaboration µ−

 Isochronous
 Rings
w EMmittance Muon Positron Linac Positron Acceleration Higgs Factory

 100 KW
 Collider Ring

 target
 Nature New results on µ cooling by MICE c

 Charge Separator
 collider detector and a higher

 Initial 6D Cooling
 508(2020)53 Rings Decay Channel
 Ring IsochronousMW-Class Target

 Phase Rotator
 µ− µ toµ

 Buncher
 + −
 celerator (LEMMA): Capture Sol.

 Final Cooling
 Combiner
 Accumulator

 Buncher

 6D Cooling

 6D Cooling
 SC Linac

 potential CoM energy due to Accelerators: ~10 TeV
011 µ pairs/sec from
 neutrino radiation. µ+ E Linacs,
 : RLA or FFAG, RCS

 Merge
 Nature 508(2020)53
 CoM

 Bunch
 interactions. The small
 µ µ + −
 on emittance allows lower
charge in the collider rings
e, lower backgrounds in a
 LEMMA: Positron Linac e e (at rest) à !Linacs,
 +
 µ−
 - + !RLA or(at
 TeV -
 10s ofAccelerators:
 threshold)
 FFAG, RCS 6 / 38
 100 KW
 target

er detector and a higher J. P. Delahaye et al., arXiv:1901.06150
 Low EMmittance Muon Positron Linac Positron µ+ Acceleration
 µ− Collider Ring
ntial CoM energy due to Accelerators: Ring
 Accelerator (LEMMA):
neutrino radiation. Linacs, RLA or FFAG, RCS
 1011 µ pairs/sec from
 µ+ ECoM:
 e+e− interactions. The small

 Muon Accelerator Program Positron Linac Low EMittance Muon Accelerator
 production emittance allows lower 10s of TeV
 overall charge in the collider rings
 µ −
 J. P. Delahaye et al., arXiv:1901.06150
 map.fnal.gov
 – hence, lower backgrounds in a
 web.infn.it/LEMMA Isochronous
 Rings
 100 KW
 target

 collider detector and a higher
 µ µ + −
 potential CoM energy due to Accelerators:
 neutrino radiation. Linacs, RLA or FFAG, RCS

erator Program New results
 Low on µ coolingMuon
 EMittance by MICE collaboration
 Accelerator
fnal.gov Nature 508(2020)53 J. P. Delahaye et al., arXiv:1901.06150
 web.infn.it/LEMMA
 J.P. Delahauge et al., arXiv:1901.06150 6 / 38
New results on µ coolingProgram
 Muon Accelerator by MICE collaboration
 4
 Low EMittance Muon Accelerator

Target Energy and Luminosity
 Collider benchmark Table
 points:
 1: Main parameters of the proto
 arXiv:1901.061
 • The Higgs factory: Parameter Units Higgs
 CoM Energy TeV
 ergy:
 hh 0.126
 Ecm =mH Avg. Luminosity 1034 cm 2 s 1 0.008
 ~ 1 fb-1Exploration
 or a strikingLDirect /yr program,
 Beam after HL-LHC*, energy
 Energy Spread % should
 0.004 b
 lose or above"Ecm10~TeV
 5 MeV Higgs Production/10 7 sec 13’500&
 Circumference km 0.3
 t few TeV energy one can stillNo. exploit
 of IP’s high partonic energy for a striking 1
ndirect Exploration program,
 • Multi-TeV colliders: ⇤ by High-Energy
 Repetition Rate Precision Hz 15
We can borrow CLIC physics case x,y (see below) cm 1.7
 Lumi-scaling scheme: ! L ~ const.
 No. muons/bunch 1012 4
 minosity: Norm. Trans.
 2 Emittance, "TN
 1 ab µm-rad
 -1 /yr 200
 5 years Norm.s μ Long. Emittance, "LN µm-rad 1.5
 35 −2 −1
 L≳ Bunch Length,2 ⋅ 10
 S cm s cm 6.3
 time Proton
 10 TeV Driver Power MW 4
 Wall Plug Power MW 200
 et by asking for 100K SM “hard” SM pair-production events.
 The aggressive choices: 2 35
 ompatible
 p with other projects (e.g.A CLIC (3 TeV/10 TeV) 6 ⋅ 10
 schematic layout of a proton driven muon) collide
 =
 s = 3, 6, 10, 14, 30 and 100 TeV, L = 1, 4, 10, 20, 90, and 1000 ab 1 .
 much less, we could only betparameters
 on Direct of the enabled facilities
 Discoveries ! are summarized in Tabl
 ouldEuropean Strategy,
 be reduced arXiv:1910.11775;
 by running longerThe arXiv:1901.06150;
 functional
 than and > arXiv:2007.15684.
 5yrs elements 1ofI.P.
 the muon beam generatio
 5
 s

A Multi-TeV Muon Collider Physics at Mu

 What will happen when
you turn on a ! ! Smasher?
 + -

Leading-order "+"- annihilation:
 ↵2
 ann ⇠
 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
 s

 6

in terms of the QED gauge group. Starting at µQCD , the QCD interaction begins to enter.
 • Photon-induced QED cross sections
The QCD and QED evolutions run simultaneously until µEW , where the complete SM sector
begins to evolve according to the unbroken SM gauge group. In ↵such 2 a way,2 we need two
matchings,have larger rates
 at µQCD
 The cross sections scaleand µEW
 as 1/s, , respectively.
 with 1 As the QED
 the characteristic kinematics off the and
 usion
 invariant mass close to the collider energy mij ⇡ s. At high energies, the ISR m
 p
 ⇠ QCD
 final-state pair
 2
 jj
 e↵ects
 2 Q
 gauge( groups
 log
 m 2
 ) conserve
the charge and parity symmetry, the PDFs below µEW can be treated with no polarization,
 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

 reduce the e↵ective partonic collision energy ŝ and thus increase the cross sections ⇠ 1/ŝ. For
as illustration,
 long as the initial the
 we compare lepton
 result beams areforunpolarized.
 without ISR `+ ` ! ⌧ + ⌧ byAsthepointed
 dotted curvesoutinalready
 the in Refs. [21, 25],
thepanels.
 polarization plays
 Typically, the an important
 e↵ective reduction is(10role
 o)
 about ain theofEW
 factor 20% PDFs
 80% (10% above
 40%)the EW scale, even for the
 for an
 electron (muon) collider. The radiative returns to the Z resonant production also enhance
unpolarized
fermion
 (⌧ + ⌧ , chiral
 initial beams. Consequently, the photon and gluon become polarized due to the
 the light-particle cross sections significantly. The ISR e↵ects for light-particle production
 interactions.
 q q̄) are thus larger than the massive one (W + W ), because of the lower threshold, jj m
 i.e., ŝ > m2ij versus ŝ > (2MW )2 .
2.1 In PDF evolution
 considering the QEDin QED
 fusion and the
 processes, QCD
 initial state partons present an infrared
 enhancement at low mij and the two-parton cross section scales as
For the sake of illustration, we take ✓
 the electron
 ◆
 beam as an example. The presentation is
 2
similarly applicable to the muon ↵2 beam ↵ by Q2 recognizing a di↵erent mass. In solving the QED
 ⇠ 2 log 2 . (3.2)
 mij 2⇡ m`
and QCD DGLAP equations, it is customary to define the fermion PDFs in a basis of gauge
singlets andthe
 To separate non-singlets. The
 hadronic activities withsinglet PDFs can
 the low-momentum be defined
 transfer as processes of
 from the hard
 X
 our current interests, we X
 impose the following basic acceptance
 j
 cuts on the outgoing X
 particles
 fL momentum
 in the transverse = (f
 (p`T ),+
 i
 f`¯di-jet
 the
 i
 ), finvariant
 U = mass(fuand
 i + fūi ), fD =
 the pseudo-rapidity (⌘j ) (f
 in d + f ¯ ),
 i di (2.3)
 the lab frame
 i=e,µ,⌧ i=u,c i=d,s,b
 ✓ p ◆
 j s
where the subscripts
 p T > 4 + refer
 3 TeV to the fermion flavors and we have
 GeV, mij > 20 GeV, |⌘j | < 3.13 (2.44). excluded
 (3.3) the top quark below

 Quarks/gluons come into the picture via SM DGLAP:
the EW scale. The DGLAP equationsj in Eq. (2.1), involving the photon and gluon, can be
 The energy-dependent⇣cut on ⌘the final state pT is to uniformly control the collinear logs
written as p0 1 0 1 with0 1
 of the form (↵s /⇡) log pjT / s f , and the pseudo-rapidity
 P`` 0 0 2N` P` cut corresponds to an
 0 angle fL
 L
 respect to the beam in the lab Bframe
 f U C✓j ⇠ B
 5° (10°),
 0 P in accordance
 uu 0 2Nwith
 u P u
 the2N
 detector
 u P ug C B fU C
 coverage.
 d B the
 For an equal-footing comparison, CsameBacceptance cuts have been applied to the C BhabhaB C
 B f C = B 0 0 P 2N P 2N P C ⌦ Bf D C , (2.4)
 B C B
 D
 scattering and annihilation2 processes
 d log Q in Fig. 3 as well. dd d d d dg C B C
 In Fig. 3, the solid lines @ A Compton
 f the
 show @P `scattering
 P u P and d the 0 A @f A
 P fusion processes
 ` ! `; fg! `+ ` , qq̄ 0(u, d,Pc,gus, b),
 Pgdand W 0+ W , Pgg (3.4) fg
where the active flavors below the EW scale are 7
 by exploiting the EPA in Eq. (2.16). The upper panels and lower panels are with a di↵erent

fusion mechanism, which would ⇠ 2be the
 ↵ ↵
 log dominant
 Q
 2 . phenomena at(3.2) low energies.
 mij 2⇡ m`
 channels include
 Di-jetToourproduction: !theq q̄,
 separate the hadronic activities with g ! q q̄,
 low-momentum transfer ! gq,
 q from the hard processes of
 current interests, we impose the following basic acceptance cuts on the outgoing particles
 in the transverse momentum (pqq j
 ! qq(gg), gq ! gq, and gg ! gg(q q̄),
 T ), the di-jet invariant mass and the pseudo-rapidity (⌘j ) in
 the lab frame
 where q includes j
 d,✓ u, s, p c, ◆
 s b and the possible anti-quarks as well. The PDFs
 pT > 4 + GeV, mij > 20 GeV, |⌘j | < 3.13 (2.44). (3.3)
 sponding luminosity are
 3 TeV already shown in Fig. 1-2. We present the cross se
 +
 production versus the collider c.m. energy
 j
 The energy-dependent⇣cut on ⌘the final state pT is to uniformly control at an e e collider
 the collinear(left
 logs panel) and
 j p
 (rightofpanel) ins/⇡)
 the form (↵ Fig.
 log p4.T / s , and the pseudo-rapidity cut corresponds to an angle with

 Forrespect
 theto quark
 the beam in pair
 the lab through
 frame ✓j ⇠ 5° (10°), fusion ( with
 in accordance ! theq q̄), Event
 obtainrate
 wecoverage.
 detector the cros
 For an equal-footing comparison, the same acceptance cuts have been applied to the~Bhabha a few Hzcollider
 20%(10%)
 scatteringless than the
 and annihilation EPAin results
 processes Fig. 3 as well. in Fig. 3 for electron (muon)
 contribute to3,this
 In Fig. di↵erence.
 the solid lines show theFirst,
 Compton inscattering
 the EPA calculation,
 and the fusion processeswe take a the fixed

 ↵e = 1/132.5, which ` !corresponds
 `; ! `+ ` , to the
 q q̄ (u, d, c,running
 s, b), and W + coupling
 W , around (3.4) Q = 10 GeV
 photon PDF istheobtained
 by exploiting through
 EPA in Eq. (2.16). evolving
 The upper panels and the DGLAP
 lower equation,
 panels are with a di↵erent in which th
 with scale
 rapidity as well.
 (angle)
 2 p2
 Due
 cut as in Eq.to the
 (3.3). sharp
 The peak
 cross section forofthem distribution
 Compton
 jj scattering ( `)around
 also the inva
 falls as ↵ /(s ✓ ), as evidenced from the figures. The cross sections for the other fusion
 the scale Q increase
 processes = ŝ/2 for most
 with energy eventsandare
 logarithmically around
 decreases with pThalf
 (or mijof
 ) asm Eq.cut,
 injj (3.2). which are 1
 The angularrunning
 dependence from is much m weaker than 1/✓ 2 and becomes roughly like ⌘ 2 due to the
 the DGLAP ` up to 15 (25) GeV, the
 boost factor. We see that the fermion pair production can be larger than that of the W W
 couplings at low energ
 more channel,
 weight, whichand,
 is known therefore,
 to be one of the the averaged
 leading channels for one is smaller
 high-energy than the fixed va
 leptonic collisions.
 For the sakeAnother
 calculation. of illustration, we havecontributes
 factor only included theis leading
 that contributions
 the higherfrom order
 fusion splitting
 in Fig. 3. We remind the reader that for the W + W production at these energies, the sub-
 take away a partZ of
 leading channel !W the
 + W momentum
 contributes to about fraction
 20% (40%), from the
 and ZZ, W +photon.
 W ! W +W
 about 10% (30%) with respect to the contribution at an e+ e (µ+ µ ) collider. They are
 More interestingly, we see that the QCD parton initialed subprocesses ex
à Jet production dominates at low energies
 neglected in our comparison for simplicity, which does not change the conclusion [42].
 fusion by 3(2) magnitudes for the electron (muon) beam. Even the g
 larger than the fusion for the electron collider. For the muon collid
 TH,
 fusion Yangcan
 process Ma, reach Keping the same Xie,size arXiv:2103.09844.
 as the fusion, depending on the
 This indicate the importance 8of quark – 10 – and gluon PDFs for a high-energy le

Di-jet kinematical features

To effectively separate the QCD backgrounds:
 pT > 60 GeV
 9

• EW physics at ultra-high energies:
 v v (250 GeV) ⇤QCD (300 MeV)
 : ⇡
 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
 E 10 TeV 10 GeV
 v/E, mt /E, MW /E ! 0!

 • A massless theory:
 à splitting phenomena dominate!
 • EW symmetry restored:
 à SU(2)L x U(1)Y unbroken gauge theory
 • v/E as power corrections
 à Higher twist effects.
 J. Chen, TH, B. Tweedie, arXiv:1611.00788;
 G. Cuomo, A. Wulzer, arXiv:1703.08562; 1911.12366.
Ciafaloni et al., hep-ph/0004071; 0007096; A. Manohar et al., 1803.06347.
C. Bauer, Ferland, B. Webber et al., arXiv:1703.08562; 1808.08831.
 10

this level have been discussed, e.g., in [38].
 EW splitting functions: Anticipating electroweak symmetry breaking, we adopt the
 isospin space. The corresponding SU (2)L bosons are W ± an
 Start from theSM
 The unbroken phase – all massless.
 EW sector: gauge boson we denote as B 0 . Gauge boson helicities are pu
 averaged.8 For the scalar doublet, we decompose as
 ! " ! "
 + φ +
 H
 Chiral fermions: fs, gauge bosons: B,W0,W ; H = H 0 = √1 (h − iφ0 ) ,
 2

 e.g.:where
 fermion ± 0 splitting:
 φ , φ will later become the electroweak Goldstone bos
 However, at this stage, we will keep the neutral bosons h and
 The Higgs: ⇐
 scalar H 0 , as they are produced and showered together coher
 ⇐
 fermion of a given helicity by f⇒
 ⇐ s with s = L, R (or equivalentl
 !specify2the
 " explicit isospin components of f at this stage, but
 1 1 1 + z̄ 1 1 #z $
 8π 2 kT2 (u, d)/(ν,
 z e) basis. Isospin-flips
 8π 2 kT2 # 2 (including RH-chiral isospin
 Ciafaloni et al.,
 indicated by a prime, e.g. u = d. Effects of flavor mixing are
 (!) 0 0(∗) ± ! Hep-ph/0505047.
 → VT f s [BWThe]TUf(1)
 s Y Hand SU f-s(2)or φ f-scouplings are respectively
 L gauge

 fs=L,R 2 V 2 g2 ≈ 0.65 (evaluated
 gV (Qfs ) 3
 g1 g2 Yfs Tfs
 near 2the weak scale), and for compactness
 y (!)
 Unitary gauge: 8 fR
 While the gauge helicity averaging is not strictly necessary, especiall
 Infrared & collinear Collinear singularity,
 distinction between 2transverse and longitudinal polarizations, it does sim
e 1: Chiral fermion splitting functions
 do not dP/dz dkTazimuthal
 incorporate in the interference
 massless effects,
 limit, though
 with zthis(z̄would
 ≡ be
 singularities (P )
) labeling the energy fraction of QCD
 thegqfirst
 9
 Chirality-flip, Yukawa
 [5]. (second) produced particle. The fermion
 We have expanded the neutral scalar field as H 0 ∝ h − iφ0 , adoptin
ty is labelled by s. Double-arrows in Feynman diagrams indicate example fermion
 and φ0 fields create/annihilate their respective one-particle states with t
 11!
ty flows. Prime indicates isospin the
 partner (u = ds|H
 one-particle state , 0etc, independent
 # ∝ |h# of hs).
 + i|φ0 #. Treating and Yukawa
 φ0 as independ

EW Symmetry breaking &
Goldstone-boson Equivalence Theorem (GET):
 Lee, Quigg, Thacker (1977); Chanowitz & Gailard (1984)
At high energies E>>MW, the longitudinally polarized gauge bosons
behave like the corresponding Goldstone bosons.
(They remember their origin!)
 “Scalarization” to implement the Goldstone-boson
 Equivalence Theorem (GET):
 E kµ
 µ
 ✏(k)L = ( W , k̂) ⇡ + O(mW/E)
 mW mW
 GET violation as power corrections v/E.
 Like in QCD: higher-twist effects !QCD/E.

 J. Chen, TH, B. Tweedie, arXiv:1611.00788;
 G. Cuomo, A. Wulzer, arXiv:1703.08562; 1911.12366.
 12

Splitting in a broken gauge theory:
 v 2 dkT2 v2
 New fermion splitting: 2 2 ⇠ (1 2
 )
 kT kT Q
 VL is of IR, h no IR
 φ/VL h

 ⇐ ⇐ ⇐ ⇐ ⇐ ⇒
 2
 1 v2 1 v2
 ! "
 1 v 1
 16π 2 k̃T4 z 16π 2 k̃T4 16π 2 k̃T4
 (!) (!)
 → VL fs (V "= γ) h fs VT f-s

 Chirality− conserving:
 $2 $2
 (yf2 z̄ yf2 (!) )z QVfL gV2 1 4 2
 gV2 Chirality flipping:
 # #
 fs=L IfV − z̄ + z̄) 4 yf z(1 z QVfR yf z̄
 − V
 QfL yf (!)
 IfNon-zero
 yf yf z − Qf for
 gV z̄ massless f
 $2 $2
 2 V 2 1 4 2
 gV z Qf yf z̄ − Q~m
 2
 # V # V V
 fs=R (!) y z(1 + z̄) f yf (!)
 4 f
 R L
 f R

 The DPFs for W thus don’t run at leading log:
Table 4: Ultra-collinear fermion splitting functions dP/dz dkT2 in the broken phase. Wavy
 L
lines represent transverse gauge bosons, while the longitudinals/Goldstones and Higgs
 “Bjorken scaling” restored (higher-twist effects)!
bosons are represented by dashed lines. The k̃T4 symbol is defined in Eq. (4.6). The
IfV symbol is a shorthand for the “charge” of a fermion in its Yukawa coupling to the eaten
Goldstone boson, or equivalently the fermion’s 13
 axial charge under the vector V . These

The EW PDFs evolve according to the full E
 ated by the electroweak and Yukawa interactions. T
 ns.
 • EW PDFsThe and their
 at a muon
 longitudinally RGE running
 equations [16, 25]
 collider:
 polarized are
 gauge bosonsXcapture fully the rta
 dfi ↵I X
the rem- cluding
 nants
 “partons” dynamically
 of the EWthe correlation
 symmetry breaking. between
 = The
 generatedof the order MZ2 /Q2 [22,
 d ln Q 2
 I
 e↵ects
 2⇡ j
 P i,j the
 I
 are
 ⌦ fj , g
 erned by power corrections
 are gov- a measure the corresponding
 of the Goldstone-Boson polarized scatteri
 Equivalence violat
 1.5 3.0 15 30
 2

 [22, 23],
 10
 one-loop
 [15, 24], virtual
 analogous to10
 corrections,
 1
 higher-twist e↵ects in our
 QCD. re
 II. Electroweak
 thethe leading-log Parton (LL)
 0
 Distribution
 order. Functions
 In Fig.
violation Below
 10 1
 10

 EW scale Q < MZ , the e↵ects of the SU(
 D.10 0 PDFs
 gauge bosonsforare the10
 states
 suppressed
 -1

 by in
 g 2 Eq.
 /M 2 (4) for
 Z . The gauge
 ± ±
 sonQradiation
 =3 TeV o↵ a
 andcharged
 10
 5 lepton
 TeV.
 -2
 Forbeam (`
 completen= e ,
 tions
 10-1 is essentially purely electromagnetic. P t At the EW s
e SU(2) andcluded
 above, allthe quarks
 10
 electroweak
 -3
 q
 states = in the i=d (q
 unbrokeni + q̄
 SM i )
 L dynamically activated. The massless states involved
 higher-order splittings. We 0.5give 1 th
 -2
 10 10 -4

auge bo- the leading order are
 10 -3
 10 -2
 10 -1
 10 0 0.05 0.1
 R
 ± tum
 ± fractions hxf i i = xf i
 e , µ :)the valance. `R , `L , ⌫L and B, W :. LO sea.
 ±,3
 (x)dx ca
 ton
EW scale WeQuarks: species in Table I. The two scal
 will notNLO; gluons:
 include NNLO.
 the Higgs sector in the initial st
 than 20%
 since thedi↵erence
 Yukawa for
 couplings the
n SM are TH, Yang Ma, Keping Xie, arXiv:2007.14300
 partons to e, EW
 µ are PD
 not r
 fermionic
 vant states
 for the current sharplyHowever,
 consideration. peak at
 14
 we x ⇡
 must

• “Semi-inclusive” processes
 Just like in hadronic collisions:
! ! à exclusive particles + remnants
 + -

10 5

10 4
 VBF

10 2

10 0

 !+!-
10 -2
 annihilations
 5 10 15 20 25 30

 15

Underlying sub-processes:
 10 -2

10 2

 10 -3 VBF
10 1

 10 -4 !+!-anni
10 0

 10 -5
 -1 10 0 10 1
10
 5 10 15 20 25 30

 26
 !+!-anni
 Partonic contributions
 0.3

 !+!- Collider: 0.2 VBF
“Buy one, get one free” 0.1

 Annihilation +VBF
 0
 -4 -2 0 2 4
 16

g µ µ ! ZH falls as 1/s. The high statistics channels for measurements of V V H
 • Precision Higgs Physics
s rely on the W W and ZZ fusion via the VBF topology:

 µ+ µ ! ⌫µ ⌫¯µ H (W W fusion), (3.1)
 µ+ µ ! µ+ µ H (ZZ fusion). (3.2)
 FigureWWH / ZZH
 1: VBF production couplings
 of a single Higgs boson at a high energy muon collider via W W
 1 for a representative Feynman diagram. It would be desirable to separate these
 fusion. For ZZ fusion, replace the W propagator by the Z propagator and the outgoing
ses of events by tagging by the outgoing muons and achieve independent measurements
 neutrinos
 HHH / WWHH couplings:
 muons.
H and ZZH couplings. However, for the VBF topology, the outgoing muons have a
y to stay in the forward region due to the t-channel propagator shown in Fig. 2(a).
h the transverse momentum of the outgoing muons is sizable and governed by the
tor mass pµT ⇠ MZ , at very high energies the muons are all extremely forward with
angle typically ✓µ ⇡ MZ /Eµ . In Fig. 3(a), we show the angular distributions of the
 p
 muons at s = 3, 10, 30 TeV. (a) One can see that, for (b) example, the scattering(c) angle for
 is peaked nearFigure
 ✓µ p⇠s 2:
 0.02
 (TeV) ⇡ 1.2 at3 10 TeV.
 Double Higgs production 6 atThese very
 10 energy
 a high 14 forward muons
 30 collider
 muon
 would most
 via W W fusion. The
cape the detection
 benchmarkin lumi
 production agoes
 detector
 (ab 1 ) at 1a few 4degrees
 through the VBF topology,10 as away from
 20 1.
 in Fig. 90 colliding beams. This
 (fb): W W ! H 490 700 830 950 1200 10M H
 ZZ ! Boson
 2 Higgs H 51
 Production 72 89
 at a High-energy96 Muon 120 Collider
 W W ! HH 0.80 1.8 3.2 4.3 6.7
 TheZZHiggs
 !boson
 HH couples predominantly
 0.11 0.24 to 0.43
 heavier 0.57
 particles.
 0.91 500k HH
 The production of a Higgs boson
 thusWinvolves
 W ! ZH – 5 – 22
 other heavy particles
 9.5 in the SM.
 33 At high
 42 energies,
 67 gauge bosons will copiously
 radiate
 W Woff!the colliding beams.
 tt̄H 0.012 Therefore, the vector
 0.046 0.090 boson
 0.14 fusion (VBF) mechanism are the
 0.28
 dominant source for the Higgs boson production at a high-energy muon collider [29, 30]. The
 WW ! Z 2200 3100 3600 4200 5200
 420 TH, D. Liu, I. Low,
 production processes involving the Higgs boson at a high-energy muon collider include
 W W ! ZZ 57 130 200 260
 and tt̄HX. Wang, arXiv:2008.12204
 VBF
 µ+ µ ! H, 17
 ZH, HH , (2.1)
able 1: SM Higgs boson production cross sections in units of fb at a muon collider for

T

 gain focusand onthus
 thefails
 leading Achievable accuracies
 does not address the underlying mechanism for the electroweak symmetry breaking (EWSB)
 decay channel HH ! bb̄ bb̄,
 to understand the stability of the weak scale with respect to the Planck scale. In
 R(4b) ' 34%. order We impose
 to gain furtherbasic acceptance
 insight cuts
 for those fundamental questions, it is of high priority to study the
 µ+ µ ! ZZ ! µ + µ Z with Z ! bb̄. There is no W W fusion analogue for
 Leading channel H à bb:
 Higgs boson properties to high precision in the hope to identify hints for new physics beyond
 10 < ✓b < 170 ,
We adopt the same basic
 Rbb > 0.4.
 cuts as in Eqs. (3.3), (3.4)
 (4.4)
 and (3.6). The background
 the SM.
 essed.
 energy In addition,
 In the
 resolution toweberequire
 SM, theE/E the
 Higgs presence
 sector
 = 10%. of at leastfrom
 is constructed one muon
 a complexto be in doublet . After
 scalar
 ructed from the the
 EWSB, fourthemostneutral real component
 energetic jets. The is the Higgs
 four jetsboson excitation H and the other three
 degrees of freedom 10 become
 < ✓µ±the < longitudinal
 170 . components of the massive gauge (3.8)
 bosons. As such,
 Figurethe
 studying Summary of the
 8: Higgs-gauge expected
 boson accuracies
 couplings would at be
 95%
 theC.L. for direct
 most the Higgs couplings
 probe to the at a
 underlying
 2 2 collider collider energies and luminosities. The upper horizontal axis marks
 of(4.5)
 m ) + (mvariety ofmmuon ) .electroweak
 to be verymechanism
 H costly
 j 3 4 to of
 j the
 H thesignal, since symmetry
 the majoritybreaking. the muons
 After have ✓µthe 0.17E and,
 intervals
 µ consequently, induces a strong recoil in the Higgs
d in the finalTable
 state. In Fig. table
 7: Summary 5 weofshow the pTaccuracies
 the expected distribution of the
 at 95% C.L. Higgs
 for the Higgsboson
 couplings at a
0,
 µ 550, 650, as
 channel 750, 950,
 variety
 well of 1350,
 asmuon 5000]
 Rbb collider GeV.
 in (b),collider (4.8)b-jets from H ! bb̄. In
 energies and luminosities.
 the separation of the
p 18 TH, D. Liu, I. Low, X. Wang, arXiv:2008.12204
 oduction in hadron colliders can be found in Ref. [40].

• Multiple Boson Production
 Pair Boson Production
 s-channel

 106 events

 Radiative return

 VBFs take over ~ 2 - 3 TeV.
TH, W. Kilian, N. Kreher, Y.Ma, J. Reuter, T. Striegl and K. Xie: arXiv:2108.05362
 19

Triple Boson Production

 104 events

 are difficult to observe at lower energy. As our results indicat
 in the analysis of exclusive multi-boson final states, which sh
 discrimination power regarding more detailed models of the Hi
 SM. If we translate an experimental bound on µ to the SM
 we obtain a bound on the scale of new physics as
 • VBFs take over ~ 4 - 6 TeV. r
 g
 • Sensitive to Higgs-! coupling: ⇤ > 10 TeV  .
 µ
 "#! ~ 1% - 10%
 2.2.3 Unitarity bounds on a nonstandard Yukawa sec
TH, W. Kilian, N. Kreher, Y.Ma, J. Reuter, T. Striegl and K. Xie: arXiv:2108.05362
 In the SM, the 20
 high-energy asymptotics of the multi-boson pro

Quadruple Boson Production

 103 events

• VBFs take over ~ 8 - 10 TeV.
• Sensitive to H-mu coupling.

 TH, W. Kilian, N. Kreher, Y.Ma, J. Reuter, T. Striegl and K. Xie: arXiv:2108.05362
 21

BSM In a New Territory
 • WIMP Dark Matter
 (a conservative SUSY scenario)
Consider the “minimal EW dark matter”: an EW multi-plet
• The lightest neutral component as DM
 Thermal targets
• Interactions well defined à pure gauge
• Mass upper limit predicted à thermal relic abundance
 Model Therm. 5σ discovery coverage (TeV)
 (color, n, Y ) target mono-γ mono-µ di-µ’s disp. tracks
 (1,2,1/2) Dirac 1.1 TeV — 2.8 — 1.8 − 3.7
 (1,3,0) Majorana 2.8 TeV
 Cirelli,
 —
 Fornengo
 3.7
 and
 —
 Strumia:
 13 − 14
 (1,3,#) Dirac 2.0 TeV
 hep-ph/0512090,
 0.9 4.6
 0903.3381;
 — 13 − 14
 TH, Z. Liu, L.T. Wang, X. Wang:
 (1,5,0) Majorana 14 TeV 3.1 7.0 3.1 10 − 14
 arXiv:2009.11287
 (1,5,#) Dirac 6.6 TeV 6.9 7.8 4.2 11 − 14
 (1,7,0) Majorana 23 TeV 11 Mitridate,
 Figure 5: Thermal relic DM abundance computed taking
 8.6 Redi,
 into account
 6.1 Smirnov,8.1
 tree-level Strumia,
 scatterings 1702.0
 (blue
 → 40 TeV? − 12
 curve), adding Sommerfeld corrections (red curve), and adding bound state formation (ma-
 (1,7,#) Dirac
 genta). 16DM
 We consider TeV 13SU(2)L triplet
 as a fermion 9.2(left panel)7.4 8.6 −quintuplet
 and as a fermion 13
 (right panel). In the first case the SU(2)L -invariant approximation is not good, but it’s enough
 to show that bound states have22a negligible impact. In the latter case the SU(2)L -invariant

Mono-photon channel
 1. Mono-photon signal:
 A single photon against missing particles
 µ µ + W
 Signal production: Backgrounds:

 Signal:

 µ+ µ
 (a)2. Mono-muon W
 (b) signal:
 WW
 (c) (d)
 p W Z ! ⌫ ⌫¯
 As =single muon against
 14 1: Representative
 Figure missing
 Feynman diagrams for theparticles
 mono-photon signal from a variety
 χχ production channels (a) µ µ
 + − annihilation, (b) γγ fusion, (c) γW fusion, and (d) W
 fusion.
 Dominated by the production of charged members of the multiplet.

 µ+ µ ! via annihilation µ+ µ ! ,
 Consider the delayed decay and the decay products
 ! via from,
 ! , e.g.,
W Z ! ⌫ ⌫¯ µ± ! ⌫ via W ! ,
 ±
 χ → χ0 + soft particles
 +
 Z µ µ ! ⌫⌫ via W W ! and µ+ µ !
W Z/W (a) (b)
 W ! µ¯
 ⌫ Z ! ⌫ ⌫¯ n
 TH,
 as invisible Z. Liu,
 here.
 Figure L.T. Wang,
 (More on this
 2: Representative X. Wang:
 later)
 Feynman arXiv:2009.11287
 diagrams for the SM mono-photon background (a) fro
 W -exchange, and (b) from Z → ν23
 ν̄.

Mono-photon channel: Mono-muon channel:
 103 0% syst (1, 2, 1/2) 103 (1, 2, 1/2)
 0.1% syst (1, 3, ≤) (1, 3, ≤)
 102 (1, 5, ≤) 102 (1, 5, ≤)
 (1, 7, ≤) (1, 7, ≤)
 101 101
NSD

 NSD
 100
 100

 10°1
 p 10°1 p
 0% syst
 s = 14 TeV s = 14 TeV
 10°2 L = 20 ab°1 L = 20 ab°1 0.5% syst
 10°2
 1 2 3 4 5 6 1 2 3 4 5
 m¬ [TeV] m¬ [TeV]
Required luminosity at 95% C.L. [ab 1]

 Required luminosity at 95% C.L. [ab 1]
 103 Mono-photon 103 pMono-muon
 s = 14 TeV
 2 2
 10 10
 101 101
 100 100
 1
 10 10 1
 (1, 2, 1/2) (1, 2, 1/2)
 2
 10 (1, 3, ) 10 2
 (1, 3, )
 3 p (1, 5, ) (1, 5, )
 10 s = 14 TeV 10 3
 (1, 7, ) (1, 7, )
 4
 10 10 4
 1 2 3 4 5 6 1 2 3 4 5
 m [TeV] m [TeV]

 TH, Z. Liu, L.T. Wang, X. Wang: arXiv:2009.11287
 pp 24

short lifetimes, especially for Higgsino-like signals, and the moderate-to-low boost
 Higherdark
 inimal Signal acceptance
 center of mass
 matter energy
 particles when
 Missing-track
 helps to masses
 their increaseare theclose
 signal:
 efficiency for a fixedboundary
 to the kinematic mass by orders of
 magnitude
/2. Hence, as wecritical
 it is Atosingle
 are catching pushmorethephoton
 events
 limit in the plus
 the missing
 exponential
 detector tracks
 decay
 design totails from the
 enhance suchboost.

 |
 ⟶
 In the following,
 particular, the number we will discusslayers
 of tracker the experimental
 close to theidentification
 interaction point of the would
 disappearing
 be track
 n
 d mi
 signals.
rrent disappearingT
 The disappearing track signals
 track searches at the of LHC
 the minimal
 requires dark3 tomatter
 4 hitsare [42,particularly
 43] in order challenging
 |⟵

ydue to the the
 suppress short lifetimes, especially
 backgrounds. for Higgsino-like
 New proposals signals, andLHC
 for high-luminosity the moderate-to-low
 and FCC-hh boost
 for heavy
ning two-hitminimal
 signals dark matter
 while the particles
 background whenistheir still masses
 under are close [44].
 control to the Hence,
 kinematic we boundary
 √
 mdm ∼
 ipate s/2. Hence,
 the needs ✓itto170
 track push or
 twice thethree
 limittimesin thefordetector design totrack
 azdisappearing enhance such

 |
esignals. In particular,
 identified. Some of the ⟶
 the current
 number studies
 of tracker of layers close toO(10%)
 the detector the interaction
 performance for point
 muon would be
 n
 d mi
 |⌘| > 1.5
 T

 crucial.
 5–47] haveCurrent
 Toused adisappearing
 reconstruct
 setup in trackthere
 a disappearing
 which searches
 track:
 are 5attracker
 the hits
 2-3 LHC requires
 needed.
 layers from 33 to cm4 to hits [42,cm.
 12.9 43] in order
 |⟵

 to effectively
 rformance suppress
 target neededthe backgrounds.
 for the searchNew of theproposals
 minimal for dark
 high-luminosity
 matter, and LHC and FCC-hh
 in the
 are Focusing
 envisioning two-hiton the barrel
 signals region,
 while the endcap
 background further is away.under control [44]. Hence, we
 still
a concrete design, we will adopt
 would anticipate the needs for hitting the track twice or three times for a disappearing track
 signal to be identified. dmin
 T Some
 = 5 of
 cm the
 with current
 |η χ | < studies
 1.5 of the
 Single detector
 Disappearing performance (3.25) for muon
 Track Efficiency
 ( = 3, 6, 10, 14, 30, 100 TeV)
 To reconstruct a
 colliders [45–47] have used a setup in which there 10 disappearing track: 2-3
 are 5 tracker
 0
 s
 hitslayers
 needed.from 3 cm to 12.9 cm.
 mal
 To settransverse distance
 a performance for aneeded
 target charged for partner
 the search of theof the dark matterdark
 minimal to travel
 matter, andand in the
 Focusing on the barrel region,10-1 endcap further away.
 identified
 absence ofasa concrete
 a disappearing
 design, track
 we will (with
 ✓ adopt a◆ minimal of 2–3 hits, depending on the
 Efficiency

 d min
sign). The dependence
 ✏ (cos ✓, , of the
 dmin ) =signal
 exp efficiency
 T
 , on the10-2 dT is shown in the right panel
 T min
3. d T =T 5c⌧cm with |ηχ | < 1.5 (3.25)
ue challenge for p
 a 1muon collider 10-3 Wino-like
 m = 1/ 2 sin ✓ in identifying the disappearing track signal is the
 as the minimal transverse distance for a charged partner of the dark matter
 T to travel and
 Higgsino-like
f the beam-induced background. The disappearing10-4tracks would be identified with
 then to be identified as a disappearing track (with a0.5minimal 1 of 2–35 hits, 10 depending 50 on the
 detector design). The dependence of the signal 25efficiency on the dTmis(TeV) shown in the right panel

Muon Collider 2 Reach ( s = 3, 6, 10, 14, 30, 100 TeV)

7, )
 The mass reach for minimal WIMP DM:
7,0)

5, )
 Muon Collider 2 Reach ( s = 3, 6, 10, 14, 30, 100 TeV) Muon Collider 5 Reach ( s = 3, 6, 10, 14, 30, 100 TeV)

5,0)
 (1,7, ) (1,7, )

3, )(1,7,0) (1,7,0)

 (1,5, ) (1,5, )
3,0)(1,5,0)Wino-like (1,5,0)
 | Thermal Target
2, 1 )(1,3, )Higgsino-like (1,3, )
 2
 (1,3,0) Wino-like (1,3,0) Wino-like
 | Thermal Target | Thermal Target
 0.5
 (1,2,
 2
 1
 ) 1
 Higgsino-like 5 10 (1,2, 1 )
 2
 50
 Higgsino-like

 0.5 1 5
 m
 10
 (TeV) 50 0.5 1 5 10 50
 Muon Collider 5 Reach
 m (TeV) ( s = 3, 6, 10, 14, 30, 100 TeV) m (TeV)

clusive
7, ) signal: ECM ≈ 14 TeV enough to cover n≤3 multiplets.

7,0) Higher energy needed to cover higher multiplets.

5, )
 TH, Z. Liu, L.T. Wang, X. Wang: arXiv:2009.11287
sappearing
 5,0) track: potential to reach almost m ≈ 1/2 ECM
 26

• Heavy Higgs Bosons Production
 annihilation VBF
 100 104 100 104
 m = 1 TeV
 m = 2 TeV

 10 1 H +H 103 10 1
 103

 1

 1
 Events/10 ab

 Events/10 ab
 HA
 [fb]

 [fb]
 H +H
 H ± H/H ± A
 2
 10 102 10 2
 HH/AA 102
 m = 1 TeV
ntributions from di↵erent fusion sub-processes mto =the
 (left panel) and H +H
 2 TeV
 m = 5 TeV HA
ctions. Cross sections
 10 3
 are calculated with the degenerate 10
 heavy
 1 Higgs
 10 3
 mass of 101
 5 10 p15 20 25 30 5 10 p15 20 25 30
 s [TeV] s [TeV]

 0
 µ+ µ ! H + H production Type-I Type-II Type-F Type-L
 10
 Figure 3. Cross sections ofmHµ+=µ1 TeV! H + H (red), and HA
 ± H H+(green) through µ +
 µ annihilation tb̄, t̄b
 (left
 ± 2 TeV ±
 panel), and in addition andmH
 H = H/H A (blue),
 small tanHH/AA
 10 HA/HH/AA tt̄, tt̄ bb̄, bb̄(⌧ + ⌧ ) bb̄, bb̄ ⌧ +⌧ , ⌧ +⌧
 H ± H/A tb, tt̄ tb(⌧ ⌫⌧ ), bb̄(⌧ + ⌧ ) tb, bb̄ ⌧ ± ⌫⌧ , ⌧ + ⌧

 Table 6. leading signal channels of Higgs pair production for various 2HDMs in di↵erent regions of
 TH, S. Li, S. Su, W. Su, Y. Wu, arXiv:2102.08386.
 3
 10
 0 2 4 6 8 10 12 14 small, intermediate and large tan . Channels in the parenthesis are the sub-leading channels.
 mH +H [TeV]
 27

acceptance.
 for mH = For 1, 5,a 15single
 TeVphoton
 at tanproduction,
 = 1. 10 its < ✓energy < 170is mono-chromatic
 is imposed for the Ep
 This mechanism can be characterizedµ by the process
 Theacceptance.
 results areFor a single
 given photon
 in the production,
 left panel of Fig.its17energy by theis dashed µ curves. E =
 mono-chromatic

 Radiative returns:
 +
 The Asresults are givenwe
 a comparison, in the
 calculate µ
 theµof
 left panel µ+! µ H,
 Fig. !17 H by process
 the dashed withcurves.
 ISR spectru
 H/A
 As a comparison, we calculate the µ+ µ ! H 2process with ISR spectru
 where can be a mono-photon observed in the ↵ 1detector,
 +H/Ax orsunobserved along
 f`/` (x) = ↵ 1 + x2 log s2
 10 3 as the collinear10radiation.1
 We first calculate the
 2⇡ cross
 1 xsection mµof the mono-photo
 ISR spectrum f`/` (x)
 µ + = log 2 µ+
 for mHµ µ=!1,
 +
 H 5, 15 TeV at tan = 1. 10 < ✓ 2⇡
 < 1
 170 xis m
 imposed µ for the photon
 applied to the muon beam. The partonic cross section is
 5 TeV acceptance.
 15 TeV
 applied For
 to a single
 the muon photon
 beam. production,
 The partonic its energy
 cross
 (a) H/A
 is mono-chromatic
 section is
 “Radiative Return”
 E = (s
 10 4 0 2 2 2
 The results are 10given in the+left panel of Fig. ⇡Yµ17 ⇡Y

 1
 by the dashed curves.
 µ m

 Events/10 ab
7. Left panel shows the cross section of single 2 2H ).
 ˆ (µheavy
 µ ! Higgs
 H) = production
 +µ ! ⇡Y 2 (ŝ through
 m H ) = ⇡Y radiative
 2 (⌧
 m
 [fb]

 As a comparison, we ˆcalculate (µ +
 µ ! theH)µ= 4µ H (ŝ process
 FIG. m 2
 )
 1: with
 =
 Main 4s µ
 ISR (⌧spectrum
 production sH mecha
 ).
 mH = 1, 2 1and TeV 15 TeV at tan = 1. Solid curves are the convoluted 4 cross section
 H with
 4s ISR s
 Toµcompare
 + with processpanel
 in Eq. (5.1),
 ↵ 1the wextan
 + 2
 calculate the cross of
 sdependence section to
 while the
 10 5 dashed curves are for µ 10! 1
 H . Right shows
 p To compare with process f`/`in(x)
 Eq.=(5.1), we calculate log 2 the cross section to t
 ection for s = 14 TeV and ↵ bym convoluting
 = 12 TeV. the ISR spectrum to2⇡ one1 muon x beam, mµ Coupling  ⌘ g/gSM T
 ↵ byH convoluting the ISR spectrum to one muon beam,
 Z 2 gHµ+ µ4 µ
 applied to the muon beam. The Z partonic cross section is↵Yµ2 s + m4H /s s
 10 6
 10 2
 = 2 dx1 f`/` (x1 )ˆ (⌧ = x1 ) = ↵Y sg +
 µ Aµ µH2 +m /s  s
 log 2 .
 µ
 5 10 p15 20 25 30
 = 2 dx1 f`/` (x
 ⇡Y 21 )ˆ (⌧ = x 1 ) = ⇡Y 4sµ sg m
 2 p m 22
 H
 log m2µ.
 s [TeV]
right panel of Fig. 17 shows the tan dependence ˆ (µ+ µ ! H)of = the cross
 µ
 (ŝ msection
 2 4sfor (⌧s HZZ
 sm =HH14 ). Z  mµ
 H) =
 The results are given in the left 4
 panel of Fig. 17 by 4sthe gsolid
 HAZ scurves. 1  2
 ZAs w
mHFigure
 =10121 TeV.
 17. ISRLeft
 While
 panel
 the
 shows
 cross
 The
 the
 section
 results
 cross
 103 are at
 section
 given
 of
 tan
 single
 =
 in the
 heavy
 1 is
 left
 Higgs
 much
 panel ofsmaller
 production
 Fig. 17 than
 through
 by thethe
 radiative
 solidother
 curves. As w
 spectrum To compare
 section with process
 is isincreasing with in Eq.Higgs
 heavy (5.1), mass
 we calculate
 m H
 the cross
 ,2, which benefitssection
 fromtothe
 thetherich
 fir
 n channels
 return for mwe µ µ=considered
 H
 +
 !
 section
 1,H2 and 15 TeVearlier,at tan = increasing
 the cross
 1. Solid with
 curves heavy
 section Higgs
 are thescales
 convoluted mass
 like m
 tan
 cross H
 sectionwhichin
 with benefits
 Type-II/L,
 TABLE from
 ISR I: Parametrizat richn
 ↵ by convoluting
 space. the ISR spectrum to one muon beam,
 spectrum, while the dashed curves space.are for µ+2 µ ! H . Right panel shows the tan dependence of
uld be10sizable at large
 2
 p tan . It could 10 evenZ be the dominant production 2 for
 4
 heavy
 ↵Y
 the cross section for s = 14 TeV and m = 12 TeV.
 Depending on1 wethe coupling,
 µ s + mH /s s
 1

 Type-I/F Type-II/L H
the large tan region of Type-L, when pair = 2 dx f (x
 production )ˆ (⌧ =
 1 `/` In1 Sec. is x ) =
 kinematically log
 forbidden .
 Events/10 ab

 II A, first
 4s present
 s m 2 the radiative
 m 2 retu
 [fb]

 H µ
 M ~ E
 3 1
 10 10 +
k associated productions are suppressed. also consider the production
 p l l ! ZH and l
 The right panel of Fig. The 17 showsare
 results thegiven
 tan independence
 the leftconcrete,
 panel Hcompare
 of theofcross
 Fig.
 we 17 bycm
 section for
 the s =production
 solid
 these 14
 curves. Asmodeswe see,
 in
 TeV10 and mH = 12 TeV.section
 While the cross section at tan = 1 is much –– 25
 smaller 25 –– the other
 than
 4 is increasing
 10 0 experimental
 with heavy Higgs mass mHenvironment andfrom
 , which benefits the the
 model-indepe
 richness of
 production channels wespace.considered earlier, the cross 2
 TH, section
 S. Li, scales
 we S.
 also like
 Su,study tan
 W. Su,theY. in Type-II/L,
 sensitivity
 Wu, of the invisible decay
 arXiv:2102.08386;
 mary
 which could be sizable at large tan . It could even be the dominant
 ouretresults production
 and concludeforinheavy
 Sec. IV.
 10 5
 10 1 TH, Z.Liu al., arXiv:1408.5912.
 Higgs10in1 the large10tan
 0
 101 of Type-L, when pair production is kinematically forbidden
 region
 tan
 and quark
 gy muon associated
 colliders productions
 o↵ers are suppressed.
 new opportunity for the
 28 direct production of heavy particles.
 II. PROD

Summary
• Multi-TeV colliders:
 - Unprecedented accuracies for WWH, WWHH, H3, H4
 - Bread & butter SM EW physics in the new territory
 - Multiple boson processes sensitive to new physics:
 muon-Higgs coupling
 - New particle (Q,H…) mass coverage MH ~ (0.5 – 1)Ecm
 - Decisive coverage for minimal WIMP DM M ~ 0.5 Ecm
 - Complementary to Astro/Cosmo/GW & to FCC-hh:

 Exciting journey
 ahead!