BEAM-BEAM INTERACTION IN A MUON COLLIDER RING
Eun-San Kim and Moohyun Yoon, PAL, POSTECH, Pohang, Kyungbuk, Korea, 790-784
Abstract
The effects of transverse and longitudinal beam-beam interaction in a 50 GeV50 GeV muon collider ring areinvestigated by a weak-strong simulation. In the present
design of the muon-collider storage ring, the frequency 7 so that theslippage factor  1 is approximately
synchrotron motion is nearly frozen. This may lead to anincrease in the energy spread due to the beam-beam interaction. Through the weak-strong simulation, we have investigated the parametric dependencies of the energy spreadinduced by the beam-beam interaction. We show that theenergy spread due to the beam-beam interaction is large
and can be compensated for by employing one rf cavity inthe ring with a modest rf voltage.

5 101INTRODUCTION

It is well known that the beam-beam interaction can limit
the performance of collider rings. A particle in a bunchfeels a nonlinear force due to electric fields produced byparticles in an opposite beam. In the transverse direction, the beam-beam force can induce a tune shift [1]. If
the beam size varies significantly within the bunch length,
however, the longitudinal effect which the beam energychanges depending on the transverse position should beconsidered [2]. This is the case for the proposed 50GeV50 GeV muon-collider ring [3].
We present investigations of the beam-beam interaction
in a 50 GeV50 GeV muon-collider ring. We note thatthe muon-collider ring has several remarkable features: (i) 12), (ii) the natuthe bunch has a large charge (N5 ),ral beam energy spread is very small ( E=E 0 = (iii) the beta function ( IP ) at IP is comparable to the nominal bunch length of 14.1cm, (iv) the muon has a lifetime, ' : ms at 50 GeV, corresponding to 1000 turns in aring with a circumference (C ) of 350 meters, and (v) theneed to minimize the rf voltage leads to a small slippage
 7 and the synchrotron-oscillation pefactor 1
riod is much longer than the storage time. The small  1
freezes the longitudinal positions of particles in the beam.
Accordingly, the energy variation of a particle in the beam
due to the beam-beam interaction accumulates from turn toturn, and may increase the energy spread of the beam. It isnecessary to maintain a small energy spread of the beams
circulating in a muon collider ring because this makes it
possible for a direct measurement of the resonance crosssection of a low-mass Higgs boson.
For this study, a model of the weak-strong beam is employed. In the simulation, both transverse and longitudinalmotions are included. The dependence of the bunch lengthof a beam on the energy spread is included because it is an= 4 10

11= 5 103 10important consideration since the transverse beam size ofthe strong beam is a function of the longitudinal positionin the interaction region. After we show that the beam energy spread is increased by the beam-beam interaction, wepresent the parametric dependencies of the energy spreadof a beam on various parameters, such as the beam-beamparameter, bunch length, beam size, emittance and betatron function at IP. We then provide a means of controllingthe energy spread by showing that the increase in the energy spread can be reduced by adding one rf cavity to thering with a modest rf voltage. Since the beam intensity decreases due to the muon decay, the rf voltage must vary intime. For completeness, we take design option in the muon
collider proposal [3].
Hirata et al. [2, 4] showed that the energy of a particle
in a beam can be changed by the beam-beam interaction
caused by the electric field due to particles in an opposite
beam. Kim et al. [5] showed that the energy spread of abeam induced by coupling impedances in the 50 GeV50GeV muon-collider ring could be compensated for by employing two additional rf cavities. In our study, we consider
these works and extend them to explore the compensationof the energy spread induced due to the beam-beam interaction in a muon collider ring.

2EQUATIONS OF MOTION

Equations of motion for a particle can be conveniently described by introducing scaled coordinates in transverse andlongitudinal phase spaces. With s denoting the longitudinalcoordinate, they are given byy(s)y0 (s); Y 0 (s) = IP;y0
y0
(s)z (s); E (s) =;Z (s) =z 0
0Y (s) =(1)where IP represents the betatron function at the IP andy0 , z0 and 0 are the nominal rms values at the IP of thetransverse beam size, the bunch length and the relative energy spread, respectively. y and y 0 are the position and theslope of a macroparticle in a weak beam in the transverse
direction, respectively, and  is the relative energy deviationfrom the nominal energy [E E 0 =E0 ]. We use=(z = s ct(s);)(2)

where c is the speed of light, and t is the difference in thearrival time at s between the particle under considerationand the reference particle. Thus, z s > indicates thatthe particle arrives at s earlier than the reference particle.
Here, we are assuming a round beam, as is the case for abeam in the muon collider ring.

()0At the IP, the change in scaled coordinates due to thebeam-beam interaction is given by [2, 4]! Y R1 (Z Z )ÆP;! Y 0 + ÆP;
ÆPE ! E + R2 ÆP (Y 0 + ) G:2YY0(3)Here, the beam-beam interaction is modeled by the interaction between a macroparticle in a weak beam and one in astrong beam with several slices. The center of each slice isrepresented by the longitudinal position z  . In eq. (3), wehave definedA==
R2 =Z =R1

The strong beam was divided into slices to take into account the effect of the initial bunch length on the motion ofa beam. To examine this effect, the total number of slices nwas varied from one to five. Figure 1(a) shows the resulting normalized rms energy spread as a function of the turnnumber with different numbers of slices (n=1, 2, 3, 5) inthe strong beam. The figure clearly shows that the energy
spread depends on n if it is too small. It is seen that the energy spreads for n=3 and n=5 are almost identical, leading
to a factor of four increase in the energy spread. Accordingly, n=5 has been employed in all the simulations.

(a)

Z
E



n =14

n =532
1100(5)

Y
Y0



; (6)

Neglecting the synchrotron oscillation, as is the case forthe muon-collider ring under consideration, the longitudinal phase-space transformation is given by532 
Z= 01 11E ;(7)=where 1 is the frequency-slippage factor defined by  1
1 . Here, is the momentum-compaction factor of the2ring.

200

400

600

800

1000

Turn number

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

σ z 0/β IP

(c)
1.014
1.012
1.010
1.008

σy/σy0where 0 is the transverse nominal emittance of the strong
beam, re is the classical electron radius and is the usual
relativistic factor.
As mentioned above, the strong beam is divided longitudinally into several slices, and a beam-beam kick given ineq. (3) for each slice is delivered at the barycenter Z  [6].
Here, Z is the longitudinal barycenter of each slice in thestrong beam and each barycenter represents n  =N of allparticles in the strong beam where N  and n are the numbers of particles in the strong beam and in each slice, respectively.
The transformation in the transverse plane from IP to IP
around the ring is simply a rotation:

n=1
n=2
n=3n=54(4)Nr=  e;4 0Ycos2y sin 2yY0 =sin2y cos2ywhere y is the betatron tune of a ring.

(b )65Here,  is the nominal beam-beam parameter defined byBunch-length effect

ε0=

1 ;

3.1εG

 2e A0Z )Y8 Nn 1 +R1RR22(ZZ ZZ)2 e A2 ;1Y + R1 (Z Z )Y 0p;
2(1 + R12(Z Z)2 )z02 IP ;020 IP ;z:z011σ /σ= 8 Nn Y + R1 (Z1For the initial parameters of eq. (1), a total number of 20000
macroparticles are distributed with a Gaussian distribution
with an average of zero and a root-mean square (rms) valueof one. Tracking up to 1000 turns is sufficient because itcorresponds to the muon lifetime    : ms.

σε/σε0ÆP

3 APPLICATION AND RESULTS

1.006
1.004
1.002
1.000
0.998
0.996
0200

400

600

800

1000

Turn numberFigure 1: (a) Normalized rms energy spread (  =0 ) asa function of turn number and number of slices (n): n=1,n=2, n=3 and n=5 are shown from top to bottom. (b) Normalized rms energy spread as a function of the normalized
nominal bunch length ( z0 = IP ) at IP for n=1 and n=5. (c)Normalized rms transverse beam size ( y =y0 ) as a function of turn number.  y =6.239 and  yIP =294m.
Figure 1(b) shows the normalized rms energy spread as afunction of the normalized bunch length ( z0 = IP ), where
IP =14.1 cm for n=1 and n=5. As the bunch length increases the energy spread increases. The figure shows thatas the bunch length increases the number of slices shouldbe sufficiently large to obtain a correct result.
Figure 1(c) describes the normalized rms transverse
beam size (y =y0 ) as a function of the turn. The beambeam parameter  , defined in eq. (5), is 0.015. Figure 1(c)indicates that the beam size increases by only 1 , whichis small. From this we can conclude that the beam-beam%interaction in a 50 GeV50 GeV muon-collider ring does
not affect the transverse beam size.
-44x 10-4

Compensation for the energy spread-1.5x10ε-4-0.6

0.00.6-4x1 00-4-0.6z (m)0.00.6z (m )(C )1x10(d )-6

2 x10-4-1 x 1 00εε)-6-0 .6

0 .0

0 .6-2x 100-4-0.6z (m )0 .0

0 .6z (m )(e )1 .0 4ε0

1 .0 3ε=(0.0εIt was shown in Fig. 1(a) that the energy spread increased
due to the beam-beam interaction. It is therefore expectedthat the longitudinal phase space would be distorted. Figures 2(a) and 2(b) show the beam-phase spaces in the longitudinal direction after one turn in 2(a) and after 1000 turnsin 2(b), respectively. In these figures, the horizontal axis
denotes the longitudinal position z and the vertical axis represents the relative energy deviation, E E 0 =E0 , ofa particle. Distortion of the phase portrait is apparent in
Fig. 2(b).
Figure 2(c) shows the increment of the energy spread ofa beam due to the beam-beam interaction in the longitudinal direction after one turn. In order to compensate for theincrease in the energy spread, a rf cavity can be utilized insuch a way that the rf voltage almost cancels the induced
energy spread by applying an opposite kick, as evident in
the energy spread due to the additional rf in Fig. 2(c). Since
the muon beam decays with time, the magnitude of the applied rf voltage also decreases with time according to

σ /σ3.2(b )

(a)1.5x10

1 .0 2

1 .0 1

1 .0 00200

400600

800

1000

T u rn n u m b e r

Vrf = V0 eNT0sin(!rf t + );(8)where V0 is the initial amplitude of the applied rf voltage,
N is the number of turns, T 0 is the revolution period of amuon beam,   is the lifetime of a muon particle, and  isthe initial rf phase. In eq. (8), the angular frequency ! rfof the rf is determined by the bunch length, and if its value
is appropriately chosen together with that of the voltage
amplitude V0 , the induced energy spread due to the beambeam interaction can be minimized.
Figure 2(d) shows the longitudinal beam-phase space afMHz, V0 =16.4 kV and theter 1000 turns when f rf
phase offset . Upon comparing this figure withFig. 2(b), it is clear that in the presence of the rf cavity theinduced energy spread due to the beam-beam interaction issignificantly reduced. The energy spread due to the beambeam interaction becomes smaller as the number of turns
increases because muons decay with time, and therefore
the rf voltage has been reduced in accordance with eq. (8).
Figure 2(e) shows the normalized rms energy spread as afunction of turn number with the applied rf voltage. This
figure is compared with Fig. 1(a), which shows the energy
spread in the absence of the rf. We see that with the rf thenormalized rms energy spread is increased by only 3% after 1000 turns. Figures 2(d) and 2(e) clearly show that theenergy spread due to the beam-beam interaction in the 50GeV50 GeV muon-collider ring can be compensated forby one rf cavity with a low rf voltage.

== 255

4 CONCLUSIONSThe effect of the beam-beam interaction in a 50 GeV50
GeV muon-collider ring has been investigated with a multi-Figure 2: Longitudinal phase space distribution (a) afterone turn and (b) after 1000 turns. (c) increments of theenergy spread due to the beam-beam interaction after one
turn. Energy spread with one rf voltage after 1 turn is also
shown. (d) longitudinal phase space after 1000 turns whenthe energy spread due to the beam-beam interaction is compensated for by one rf voltage. (e) Normalized rms energy
spread verse turn number with an rf voltage applied.
particle tracking simulation. The simulation demonstrated
the necessity of including the bunch-length effect. It was
shown that the transverse beam size was not affected significantly by the beam-beam interaction. On the other hand,the energy spread due to the beam-beam interaction causeda large energy spread, but the growth of the energy spread
could be controlled by employing an additional rf cavity
with a modest rf voltage.

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[4] K. Hirata, H. Moshammaer and F. Ruggiero: CERN SLAP/90-02, 1990.
[5] E.-S. Kim, A.M. Sessler and J.S. Wurtele: Phys. Rev. Special Topics: Accelerators and Beams 2 (1999) 051001.
[6] S. Krishnagopal and R. Siemann: Phys. Rev. D 41 (1990)
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