MUON COLLIDER: CONCEPTION AND SOME DISTINCTIVEFEATURES OF PHYSICS POTENTIAL
Feodor F. Tikhonin,∗ Institute of High Energy Physics, Protvino, 142284, Russia
Abstract+ −Machines with µ µ colliding beams have a number ofdistinctive features compared to the e + e− colliders. A fewof them are briefly outlined.

1 CONCEPTION
Concept of a muon collider has been around for a while:[1] [2], [3], [4]. The main impetus for such an idea was thedesire to get rid of the unpleasant synchrotron radiation, because of which so suffer the cycling electron-positron colliders. The muons being as much as  200 times heavier
than electrons undergo this disease to a much less extent
because the intensity of synchrotron radiation is proportional to the mass−4 of radiating particle. The interest of
last several years to the muon collider hugely growths due
to several reasons. From the point of view of machine design consideration it was revealed, that ionization coolingconcept [3, 4] offers the possibility of making a high luminosity accelerator. From the other hand the physics potential of muon collider recently was enriched by possibility tobuild the ”Higgs boson factory”. We know what rich harvest was gathered from the ”Z boson factories” at CERN(LEP-I) and at SLAC (SLC) and therefore such facility
might provide a unique instrument for particles physics research.

2 PHYSICS MOTIVATION
Apparently one of the main objectives of present-days elementary particles physics are to find the Higgs boson andinvestigate its properties with as high precision as possible.
It is hoped that this particle will be revealed at the forthcoming LHC machine. However there exists the so calledintermediate mass region, extended from the m H  95
GeV to mH < 2mZ , which is the most difficult for experimental study. In spite of this difficulty it is hoped that
LHC will allow to know the Higgs mass with the precision enough to tune at the resonance with muon collider.
Happily, just at this region the SM Higgs boson has a narrow width in the channel µ + µ− → bb̄. Therefore, theµ+ µ− collisions would not merely fill the gap but improve
the precision of studying to a greit extent. Effect disappeares beyond this region, more exactly, if the Higgs mass
will happens to be bigger then threshold of W boson pair
production. To see this explicitly we calculate the crosssection of the process µ + µ− → W + W − with the Higgs
boson exchange only. Using the standard technique the re∗ e-mail: tikhonin@mx.ihep.su

sult for this cross section reads as follows:
m2µ+ −+−πα2
σ (µ µ →H→W W ) =16 sin4 θEW m2Ws − 4m2µ1 s2s· 2−2+6(mH − s)2 + Γ2H m2H 2 m4Wm2W(1)In this expression s is the energy squared of collision. Alsomµ , mH and mW are the masses of the µ - meson,Higgs scalar and W boson, respectevely, and Γ H is thedecay width of Higgs. For the fine structure constant, α,
we use an approximate value 1/128 and for the sine ofelectroweak mixing angle the value of sin 2 θEW  0.23.
As can be seen from this equation, the cross section due
to the Higgs boson exchange alone reaches√ its maximum,σmax  0.067pb at the c.m. energy s  2mZ whilethe ”conventional” cross-section (due to γ, Z 0 and ν exchange) reaches at this point the value of  15 pb. Therefore the ”Higgs boson factory” is feasible only in the region mH < 2mW . Partly in view of this it is expedient tosearch for other processes, where the interaction of Higgs
within the lepton sector would be involved. In this talk werepresent two examples of such processes, µ + µ− → HZ 0and µ+ µ− → Hγ, and will consider them as complementary to that of resonant Higgs scalar production. Note, that
second of the processes above is negligibly small at the treelevel in the case of electron-positron collisions, justifying
word ”distinctive” in the title of present talk.

3ASSOCIATED HZ 0 PRODUCTION [5]

We begin with the so called Bjorken process having muons
as the initial state particles, µ+ µ− → HZ 0 .
Usually in the course of cross-section calculations one
uses the the s-channel diagram alone. Let us do it webut in addition take into account masses of initial muons.
Then√ we obtain the following asymptotics of this process
at s → ∞:2m2µ2π α2 gA(s−channel),asympt
σµ+ µ− → HZ 0|mµ =0 =(2)sin4 (2θEW ) m4Z
It can be seen that despite the fact that this diagram ispure s–channel one, the corresponding cross section is not
falling with energy, but approaches a constant limit, whose
value is equal to ∼= 1.2 · 10−2 f b. Concerning the angulardependence of this cross section, it could be seen, that thisdistribution is flat, indicating that it comes entirely fromthe J = 0 partial wave. Obviously, this behaviour contradicts unitarity condition, which requires σ J=0 ≤ s−1 athigh energies. Now we calculate the contributions to thiscross section, stemming from the cross channels diagrams,

t- and u- one. It turns out that corresponding contribution
is again equal exactly to the value of ∼= 1.2 · 10−2 f b. Corresponding angular distribution is also flat. At last, let us
take into account interference term between the two classesof the diagrams above. We found that it is equal exactly to∼= −2.4 · 10−2f b. Adding all three contributions we obtainresult, which removes a seeming contradiction. As it mustbe, the asymptotic form
√ of cross section for the process under consideration at s → ∞ acquires ”desired” form,

i.e. it falls with energy.
222g+gπαVA 11(all diagrams),asympt(3)=σµ+ µ− → HZ 03 sin4 (2θEW ) sThe cancellation obtained reflects the most fundamental
property of electroweak theory. This is a consequence of
unitarity condition at tree level which must be fulfilled inany non-Abelian gauge theory with the symmetry broken
in a manner like the Higgs mechanism [6].
In order to extract information about the Higgs - lepton
sector interplay let us look once more at the individual contributions to the discussed cross section. In this respect
it is worthwhile to note that all three contributions reach
their high energy asymptotic (constant) values not simultaneously. Those stemming from the sum of the t-channel
and u-channel go to the plateau at the energy around 1TeV.
√ The negative contribution reaches its minimum value
at s ∼ 2.5 TeV, while the part of cross section, corresponding to the s-channel becomes constant (at nonnegligible muon mass) far away from 1–2 TeV region. However,in spite of different characters of the behaviour of thesecontributions, it seems unlikely that this difference will beobserved experimentally owing to the small value of the µmeson mass. Partly because of that, in the next paragraphwe will consider the process to which the Higgs-GaugeBoson vertex is not involved.

4

ASSOCIATED Hγ PRODUCTION [5]

We now turn to process analogous to those just described,
but which is free from the s-channel diagram complication.
The differential cross section of the process µ + µ− → Hγfor the case when photon is hitting a non-forward detector,
and when we neglect the muon mass (apart from where itaffects the muon - Higgs boson coupling constant), reads
as follows 1 :m2µ s2 + m4H
1πα=,2222d cos ϑγ2 sin θEW mW s (s − mH ) 1 − cos2 ϑγ
dσ0Hγ2m2µm2H

1 − | cos ϑγ |,(4)

1 It is worthwile to note that in the case of high energy limit the result isindependent of whether the Higgs boson is scalar or pseudoscalar, but inthe low energy limit when the mass of the µ - meson can not be neglected
the two cases are sharply different. It is the γ5 non-invariance, whichis responsible for this difference. Formulae obtained for the case of low
energies [5] are very important, for example, in searching for the lightaxion or light Higgs at Υ decays to the γ + nothing, i.e. in the bb̄
annihilation, in which case the cross section is greately enhanced due tolarge mass of b-quark.

where ϑγ is the angle emission of the photon with respect
to the beam of initial µ − . Subscript ”zero” means that thecross section has been calculated at tree level. With theyearly integrated luminosity of L ∼= 103 fb−1 expected
at future µ+ µ− colliders, one could collect 20 to 30 Hγevents (detector efficiency is supposed equal to 1, and acceptance 4π). The signal which mainly consists of a photon and bb̄ pairs in the low Higgs mass range or W W/Z 0 Z 0pairs for Higgs masses larger than ∼ 200 GeV, is extremelyclean. The background should be very small since the photon must be very energetic and the b b̄ or W W/Z 0 Z 0 pairs
should peak at an invariant mass m H . Therefore, despite
the low rates, a clean signal gives a good possibility to detect these events.
As can be seen from√ the Eq.(4) the tree - level cross section is singular 2 at s → mH . In order to overcome thisdifficulty we need to calculate the Quantum Electrodynamics (QED) corrections to the tree-level amplitude. That is
the aim of the next paragraph.

5RADIATIVE CORRECTIONS TO THE
PROCESS µ+µ – → Hγ

Due of the lack of the space here only very concise sketchof RC will be given. For the comprehensive information
I refer to [7]. For the process under study 1) the first order leading logarithmic correction were obtained and 2) thecontribution of the higher order perturbation theory weretaken into account in the leading logarithmic approximation. It is expedient to give here expression for the case 2).
The master formula for radiative corrections has the formof the Drell–Yan cross section. So, we suggest to write
the result as a convolution of the lepton structure functions
with the shifted cross section of the hard subprocess. Itreads
dσ Hγ=d cos ϑγ

1z1min1dz1 D(z1 )dz2 D(z2 )z2min

dσ0Hγ (z1 p1 , z2 p2 ),d cos ϑγ(5)where p1 and p2 are the momenta of µ − and µ+ and ωth
is the experimental energy threshold of the photon registration. D–function has the following form:D(z, L) =β1β(1 − z) 2 −1 (1 + z 2 )(1 + O(β 2 )),2β=s2α(ln 2 − 1),πmµ(6)and the lower limits for integration over z 1,2 are as follows:√m2H + sωth (1 + cos ϑγ )min√
,z1 =s − sωth (1 − cos ϑγ )√m2 + sz1 ωth (1 − cos ϑγ )√.
(7)z2min = H
sz1 − sωth (1 + cos ϑγ )2 Another factor, 1/(1 − cos2 ϑ ), is seemingly singular but it is notγ
dangerous when masses of initial muons are taken into account.

One can easily see by direct inspection that the expression of Eq.(5) is smooth.√It achieves its maximal values inthe√vicinity of the point s ∼ mH and falls rapidly to zeroas√ s → mH , Cross section slowly falls with increasings.
The associated Hγ pair can be produced through loop diagrams of massive particles (gauge bosons, heavy quarks)
also [8]. In this case the cross section behaviour is quite
different from that of tree-level with QED corrections justconsidered. Contrary to this last case the loop induced process has the negligible cross section at the threshold andslowly rises as the energy√ grows. Both cross section values
coincide at the point s/2 ∼ mH . Before this point thetree-level QED corrected process dominates while after thispoint the loop induced process dominates. So, this processdelivers the unique possibility of studying simultaneously√
the (µµ̄H) vertex (in the region of ∼ m H ≤ s < 2mH )+−and, for√instance, (t t̄H) or (W W H)vertices (in the region of s > 2mH ).

6 CONCLUSION
Muon colliders will offer an excellent opportunity for theparticle physics investigations.

7REFERENCES[1] Tikhonin F.F., On the Effects with Muon Colliding
Beams. Preprint JINR P2–4120, Dubna, 1968.
[2] Budker G.I., Proceedings of the Intern. Conf. on HighEnergy Physics, Kiev, 1970, Dubna, 1970, p.1017.
[3] Skrinsky A.N. and Parkhomchuk V.V., Sov.J. Nucl.
Physics 12, 3 (1981).
[4] Neuffer D., Particle Accelerators 14, 75 (1983).
[5] Litvin V.A. and Tikhonin F.F., Associated Production of H 0 γ-Pair and H 0 Z 0 -Pair in µ+ µ− Collisions.
Preprint IFVE-97-24, Protvino 1997, e-Print Archive:
hep-ph/9704417.
[6] Llewellyn-Smith C.H., Phys.Lett. B46, 233 (1973);
Cornwall J.F. et all., Phys.Rev.Lett. 30, 1268 (1973).
[7] Arbuzov A.B., Kuraev E.A., Tikhonin F.F. andShaikhatdenov B.G., Phys. Atom. Nucl. 62, 1393
(1999) ( Yad. Fiz. 62,1477 (1999)).
[8] Abbasabadi A., Bowser-Chao D., Dicus D. and Repko
W., Higgs - Photon Production at µµ̄ colliders.
Phys.Rev.D57, 550 (1998 ), e-Print Archive: hepph/9708328.