RECENT DEVELOPMENTS IN DESIGNS FOR e+ e− COLLIDERS
K. Ohmi for the KEKB Commissioning Group
KEK, Oho, Tsukuba, 305-0801, Japan
Abstract
We discuss an design of future e+ e− collider from theview point of the beam dynamics. The crossing angle permits short bunch spacing without complex design of theinteraction region. In KEKB, the crossing angle did notdegrade the collision performance. However we need to
study how the crossing angle affects the beam-beam interaction for the future high luminosity colliders. The electron
cloud instability, which limited the luminosity performancein positron ring, have been somehow recovered by using
solenoid magnets. The ion instability, which have been observed in electron rings, may be serious for higher beam
current. We have to study which particle should be stored
in high or low energy rings to minimize the instability effects. The coherent synchrotron radiation becomes serious
for short bunch length and high bunch current. We discussthese issues one by one.

INTRODUCTION
B factories in the world, PEP-II and KEKB, have beensuccessfully operated with very high luminosities. The
peak luminosities of PEP-II and KEKB are 0.61×1034 and1.06 × 1034 cm−2 s−1 , respectively, at May, 2003. The operations are somewhat different from their original designs.
In the both factories, the number of bunches is less than thedesign values, and bunch current is higher than the design
values. They are operated in a regime where beam-beam
parameter is saturated ∼ 0.05.
We are challenging to get more luminosity everyday.
The solenoid magnet coils to avoid the electron cloud effects were wound everywhere as possible as we could. The
horizontal tune was controlled to keep closed to half integer. Vertical tune and chromaticity was scanned to gethigher luminosity and useful beam life time every time.
Geometrical condition of collision, offset of two beam, vertical crossing angle and RF phase were scanned and feedbacked in every second. All of optics parameters at the col), x-ylision point, βx(y) functions, dispersions (ηx(y) , ηx(y)coupling parameters (R1 − R4 ) were scanned every time
[1, 2].
In PEP-II, 945 bunches are stored with 6 ns spacing. In
KEKB, 1000 bunches had been stored with 8 ns spacing,
because increasing of the number of bunch did not contribute luminosity until last year (2002) perhaps due to electron cloud effects, not parasitic collision effects. As theresult of addition of solenoid magnets in every long shutdown periods, the number of bunches can be made increase
gradually. Attempt to increase the number of bunches has
been continued in KEKB.

In the both of SLAC and KEK, upgrade plans toward theluminosity of 1035 − 1036 cm−2 s−1 are proposed. We discuss the design development of the upgrade plan of KEKB,
named by super KEKB, in this report. The parameters forthe present operation of KEKB and those for the design ofSuper KEKB are shown in Table 1.
We discuss three topics: beam-beam effects, instabilities
and coherent synchrotron radiation. Collision with/withouta crossing angle is reviewed in Sec.II. Electron cloud andion instabilities are discussed in Sec.III. These studies inform which particle stored in low energy or high energy
rings. Longitudinal single bunch instability due to the coherent synchrotron radiation is discussed in Sec.IV.

CROSSING ANGLE
In high luminosity e+ e− factories, KEKB and DAΦNE
adopt collision with finite crossing angle, and PEP-II
adopts head-on collision. Crossing angle makes easy a design of the interaction region for the narrow bunch spacing.
However the crossing collision scheme had been considered to be a taboo, since the unsuccessful experience of
DORIS in DESY. In KEK and INFN, many studies were
performed to decide the adoption of the crossing collision
scheme [3, 4, 5]. KEKB and PEP-II have achieved luminosities of 1.06 × 1034 and 0.61 × 1034 cm−2 s−1 , respectively at May, 2003. The beam-beam parameters are 0.05
for KEKB, respectively. Consequently, the crossing angle
does not affect the beam-beam performance in the regionof the beam-beam parameter up to 0.05.
We now target higher luminosity, L = 1035 ∼36
10 cm−2 s−1 . The beam-beam parameter should behigher to get the luminosity. We need to review the effect of crossing angle on the beam-beam interaction for ahigher beam-beam parameter.
Collision with a half crossing angle of θ is equivalent tothat between beams with z dependent dispersion (ζx = θ)at the interaction point [6]. Crab cavities generate the dispersion ζx (s). Matching the dispersion as ζx = −θ at theinteraction point, effects of the crossing angle are canceled[7].
We have studied the beam-beam effects using the weakstrong and strong-strong simulation methods [8]. In the
weak-strong method, one beam is represented by macroparticles and another beam is represented by a fixed Gaussian charge distribution. In the strong-strong method, bothbeams are represented by many macro-particles, and theirinteractions are evaluated by the particle in cell method.
The weak-strong simulation is executed for 100 macroparticles and 10 or more longitudinal slices during 40,000

Table 1: Basic parameters of present KEKB and Super KEKB
KEKB
HER
e−

particle
C
E
N±
Nbunchβx /βyεxσzνx /νy /νsτxy /T0θc
L(cm−2 s−1 )

3016m
8 GeV
3.5 GeV
5.4 × 1010 7.3 × 1010
1280
60 cm / 7 mm
24 nm
18 nm
7 mm
0.515/0.565/0.02
4,000
4,000
11 mrad1.06 × 1034

3

L (x 1e32)

head on

xi

0.2

0.1

15 mrad

2

1

2

3

4

30

20

20

10

10
0
2

4

6

8
x/σx

10

12

0

2

4

6

8

10

12

x/σx

head on1

15 mrad

0
0

40
30

0

0
0

50
40

0

(b)

(a)

60

60
50

y/σy

We calculated the luminosity for various current keeping the transparency condition. Figure 1 shows the beambeam tune shift which is estimated by the luminosity forthe positron current, I+ . Pictures (a) and (b) are given by
the weak-strong, and (c) and (d) are by the strong-strong.
The tune shift should linearly depends on I+ , if there isno dynamical effect of the beam-beam interaction. In the
weak-strong simulation, the tune shift linearly increases up
to 0.25 for head-on collision, while it saturates around lessthan 0.08 for crossing angle 11 mrad. The tune shift isextremely high beyond all belief. We use operating tune
which is close to half integer in horizontal as is operated in
KEKB.

5000
15 cm / 3 mm
18 nm
3.5 mm0.515/0.565/0.02
4,000 4,000-8,000
0 − 15 mrad
1 × 1035 − 1036

Beam-beam halo is another limitation of the beam-beameffect [9]. The halo is evaluated by a very long term simulation using the weak-strong method. We used brute force
simulation using 500 particle times 106 turn, though there
is some technique to reduce the calculation time [9, 11].
The CPU time was a few 10 minute for 10 longitudinalslices. Figure 2 shows distribution of beam particles inx − y plane. The particle times turn numbers, 5 × 108 ,corresponds to more than 1 hour. Concerning also beambeam halo, the head-on collision is better than that withfinite crossing angle. [10].

y/σyrevolutions. The strong-strong simulation is executed for
100,000 macro-particles and 5 longitudinal slices during
20,000 revolutions.

0.3

Super KEKB
HER
LER
e±
e∓
3016m
8 GeV
3.5 GeV

LER
e+

1

2

5

3

4

5

Figure 2: Beam-beam halo for collisions without (left) andwith (right) crossing angle estimated by the weak-strong
simulation.

I+ (mA)

0.15

1
(c)

(d)
head on

0.8

xi

L (x 1e32)

0.1

0.05

11 mrad0.6

head on

0.4

PARTICLE AND ENERGY

11 mrad

0.2
0

0
0

1

2
I+ (mA)

3

4

0

1

2
I+ (mA)

3

4

Figure 1: Beam-beam parameter (a)(c) and luminosity (b)(d) function of positron current for collisions
with/without crossing angle. Pictures (a) and (b) are obtained by the weak-strong simulation, and (c) and (d) areby the strong-strong simulation.

In asymmetric B factories, PEP-II and KEKB, energies
of electrons and positrons are chosen to be 8-9 GeV and
3-3.5 GeV. The choice is reasonable by considering an injector Linac, in which positrons are created as secondary
particles by electrons accelerated to be several GeV. Electron beam can be accelerated to higher energy, while its
intensity, which is primary, is stronger than that of positron
beam.
From the viewpoint of the beam-beam interaction, thenumber of particles with the low energy in a bunch is required to be more than those with the high energy. The socalled transparency condition is NH γH = NL γL , where
NH(L) and γH(L) are the number of particles and the relativistic factor of the high (low) energy, respectively. Thereis a merit to choose that electron beam has a lower energy.
In the B factories, ion and electron cloud instabilities areobserved in electron and positron rings, respectively, andthey affect the performances of the rings. We survey these
instabilities and study which beam should have high or low
energy.

the second regime. Since these behaviors depend on parameters as electron production and secondary rates etc.,
we need more study whether these results are always obtained.

Electron cloud effectFigure 3: Electron cloud density for 3.5 GeV and 8 GeV
cases.

Y2 (E)= Y0 exp(−5E/Emax )E
1.44+ Ymax. (1)Emax 0.44 + (E/Emax )1.44

Emax = 200 eV, Y0 = 0.5 and Ymax = 1.2 are used here.
Figure 3 shows the electron cloud density for the beamenergy of the cases of 3.5 GeV and 8 GeV. The density for
8 GeV is higher than that for 3.5 GeV: ρe (8GeV)> ρe (3.5
GeV).
In a regime for a lower cloud density than neutralization,
electrons are strongly overfocused by the beam, therefore
cloud density is determined by dynamical balance of creation and absorption. In another regime for a high clouddensity close to neutralization, the overfocusing force isweakened, therefore electrons tend to be accumulated up to
the neutralization. If these two cases of 3.5 ad 8 GeV are in
the first regime, the cloud density of 8 GeV case should be
more than that of 3.5 GeV: ρe (8 GeV)> ρe (3.5 GeV). Because the transparency condition, NH γH = NL γL , gives
an equal number of photoelectrons production due to thenumber of photon scaling to N+ γ+ , and the overfocusing
force of low energy (high intensity) beam is stronger. Ifthey are in the second regime, the electron line density isnear the beam line density: that is, 3.5 GeV, which is higherline density, should be higher density, ρe (8 GeV)< ρe (3.5
GeV).
The simulation result in Figure 3, ρe (8 GeV)> ρe (3.5
GeV), shows that the both cases are in the first regime. The
case of 8 GeV has so slow build-up time that it may be in

4e+12(b) 8 GeV(a) 3.5 GeV

1.5e+123e+12rhoe (m^-3)

rhoe (m^-3)

2e+12near beam

1e+12
averagenear beam

2e+12
average
1e+12

5e+11
0

0
0

50

100
150
bunch

200

250

0

5e+06

100

200

300 400
bunch

500

600

1e+07

4e+06

(a) 3.5 GeV

8e+06

3e+06

(b) 8 GeV

6e+06
Wy

Wy

We first estimate build-up of an electron cloud for twocase of the positron beam energies, E = 3.5 GeV and 8
GeV. The photoelectron instability model [12] is used forthe estimation of the build-up. There are some codes toestimate the build-up [12, 13, 14]. We used the code PEI[12].
The primary electrons are created by synchrotron radiation with a production rate, ne = 0.0015/(m·e+ ), for apositron passage per a mater. This production rate corresponds that 100 photons create 1 photoelectron at thechamber surface. This is an empirical value for a test antechamber at KEKB. The photons are assumed to be createduniformly at the chamber.
The secondary yield is used a formula given by ref.[13],2.5e+12

2e+06
1e+06

4e+06
2e+06

0

0

-1e+06100-2e+06
500

120

140
160
bunch

180

200

520

540
560
bunch

580

600

Figure 4: Wake field caused by the electron cloud for 3.5
GeV and 8 GeV cases.

We first estimate the coupled bunch instability causedby the electron cloud [12]. A bunch train which passes
through the electron cloud experiences a wake force. Figure 4 shows the wake field for each case. The growth time
can be estimated by the wake force induced by the electron
cloud. The growth times were obtained as τ = 0.3T0 and
0.4T0 for the case of 8 GeV and 3.5 GeV, respectively. The
growth time for 8 GeV was a little faster than that for 3.5
GeV. The strongest mode was around m ≈ 4500, whose
frequency was ω ≈ (5120 − 4500)ω0 − ωβ for either case.
The growth rates are too fast to recover a bunch by bunch
feedback system. Actually the electron cloud should be removed by solenoid magnets, ante-chamber, electrodes andother measures. We here focus the relative value of thegrowth: that is, 3.5 GeV is a little slower. This result may
be due to that the cloud density for 3.5 GeV was further
weaker, though the gamma factor is smaller.
We next study single bunch instability due to the electron
cloud. The positron beam, which passes through the electron cloud, experiences a short range wake force. The wakeforce causes the strong head-tail instability. The instability
can be simulated by some simulation codes [15, 16, 17, 18].
Figure 5 shows the growth of the vertical beam size for thecases of 3.5 and 8 GeV obtained by the code PEHTS [17].
The threshold of the strong head-tail instability due tothe electron cloud is ρe,th = 0.5 − 1 × 1012 m−3 for 3 GeV
and 1−2×1012 m−3 for 8 GeV. The threshold densities forthe cases of 3.5 and 8 GeV are about half of those estimatedby the build up code in Figure 3. These results show that

300

(b)

200

sigy (um)sigy (um)given by300(a)1e12 m^-3
100

200|Z /n| = 0.66Z02e12 m^-3

100

rhoe=0.5e12 m^-3

rhoe=0.5e12, 1e12 m^-3

0

0
0

200

400

600

800

1000

0

200

400

turn

600

800

1000

turn

Figure 5: Growth of the single bunch instability caused by
the electron cloud for (a) 3.5 GeV and (b) 8 GeV cases.

difficulties for the two cases are not big difference, because
effect of γ is cancelled by the density difference.

Ion effectWe study which should choose 3.5 or 8 GeV for the electron beam energy. In an electron ring, ions, which producedby ionization of residual gas due to the beam, trapped in
the bunch train, and causes a two-stream instability [19].
We studied this instability using a simulation model shown
in Ref.[20, 21]. Electron bunch is assumed to be Gaussian
distribution in the transverse plane and ions are represented
by point-like macro-particles. In the simulation, the instability is caused by the oscillation of ions bounded in thebunch train. Ion species and yield are assumed as CO+
and 4 × 10−9 /(m·e). This yield corresponds to the pressure−7
Pa. Figure 6 shows growth of transverse amof 1 × 10
plitudes, Jy(x) . Vertical growth is faster than horizontal
0.001

0.001
3.5 GeV

0.00011e-05

sqrt(Jy)sqrt(Jy)

0.0001

8 GeV

1e-06
1e-07

(a) Vertical

3.5 GeV

1e-05
8 GeV

1e-06
1e-07

1e-08

(b) Horizontal

1e-08
0

100

200

300
turn

400

500

0

200

400

600

800

1000

turn

Figure 6: Growth of the fast ion instability. Blue and redlines are evolutions of Jy(x) m1/2 for E = 8 GeV and 3.5
GeV, respectively. Pictures (a) and (b) depict the verticaland horizontal growth, respectively.
one. The growth time is ∼ 50 and < 10 turns for 8 GeV and
3.5 GeV cases, respectively. The growth time of each energy case is reasonable for the consideration of the γ factor
and the beam line density. Perhaps growth time of 50 turnis limit for bunch by bunch feedback system. Concerning
ion issue, the option of 8 GeV is better.

COHERENT SYNCHROTRONRADIATION
Bunch lengthening is important issue for keeping thelength with high population. The threshold of longitudinal single bunch instability is discussed using Keil-SchnellBoussard relation [22]. The impedance threshold value isαγ 2σ σz .
Ne re δ(2)where n = ω/ω0 and α is the momentum compaction factor. The threshold of the impedance Z/n is ∼0.05 Ω forpresent KEKB.
Coherent synchrotron radiation affects the bunch lengthening [23, 24]. The wake force of the coherent radiation is
given by [25, 26]∂ −1/32z2
1/3(3R ) ∂z

W0 (z) =(3)

where R is the bending radius. The impedance integrated
over the circumference is given asZ (ω) =

iAZ0
2



ωR
c

1/3
(4)

√
where A = 3−1/3 Γ(2/3)( 3i − 1) = 1.63i − 0.94. Since
the impedance scales as Z/n ∝ ω −2/3 , the Keil-SchnellBoussard criteria gives threshold frequency (ωth ) of the instability,(5)ωth R/c = 2.0Λ3/2 ,where 1
Λ=

R
Ne re
√
2
αγσδ 2πσz R̄

(6)and R̄ = L/2π = c/ω0 .
We have to consider that lower frequency component of
the impedance, which satisfies a condition of ωσz /c < 1,does not contribute the instability. The frequency component, which satisfies ωR/c > (πR/2b)3/2 , is shieldedby the chamber wall, where b is the chamber radius [27].
This gives another necessary condition for the instability,bΛ/R > 1.
The length of the bending magnet should be remember
to be short for LER, LBend = 0.76 m [24]. If the correlation is determined by interaction in a bending magnet, thecorrelation length is estimated by the overtaking length forthe radiation.
1
cor =
3R2



LBend
2

3
.

(7)

It is difficult to understand whether the synchrotron radiation emitted in a bending magnet affects the motion in another bending magnet. If it does not affect, the wake force
should be truncated with the correlation length. Otherwise,
a numerical simulation of motion, in which the correlation
across bending magnets is included, should be performed[24].
These values for KEKB and super KEKB are summarized in Table 2.
1 The coefficient of Λ3/2 was followed by Ref.[23].

Table 2: Coherent synchrotron radiation effect for KEKB and Super KEKB

Ne
σz
α
σδ
Λ
ωth
ωth σz /c
bΛ/R
Lbend
cor

KEKB
LER
HER
10
7.3 × 10
5.4 × 1010
5 mm
5 mm
10−4
10−4
−4
7 × 10
7 × 10−41674
2905
2.5 × 1012 s−1 1.1 × 1012 s−1
42
17
5.1
1.7
0.76 m
5.8 m
69 µm
1 mm

SUMMARY
We discussed the design development of future B factories based on parameters of Super KEKB concerning thethree issues (1) beam-beam effect with/without crossing
angle, (2) energy choice, electron cloud and ion instabilities, and (3) coherent synchrotron radiation. The collision performance was improved for the head-on collision
at high beam-beam parameter ξ > 0.05 in our simulations.
Crab cavities make the condition of the head-on collisioneffectively for the finite crossing angle scheme of the beamline arrangement. The crab cavity is expected to boost up
the luminosity. Studies of the electron cloud and ion instabilities gave information of energy choice, which particles
e+ or e− should be accumulated in high/low energy ring.
In the present simulation, LER-e+ option was slightly better than HER-e+ option. Since this result may be delicate
for the electron production parameters, careful studies areneeded. In either case, it is important to be cured by antechamber, solenoids, etc. Coherent synchrotron radiation
affects the bunch lengthening in the B factories. We haveto solve how to treat the correlation length.
The author thanks fruitful discussions with S. Heifets,
Y. Ohnishi, K. Oide, E. Perevedentsev, G. Stupakov,
M. Tawada, K. Yokoya and F. Zimmermann.

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Super KEKB
LER
HER
11
1.2 × 10
5.4 × 1011
3.5 mm
3.5 mm10−4
10−4−4
7 × 10
7 × 10−44087
4150
8.9 × 1012 s−1 2.2 × 1013 s−1
104
21
11.9
2.4
0.76 m
5.8 m
69 µm
1 mm

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