Simulation of Beam-Beam Effects in e+ e; Storage Rings 
Yunhai Cai, SLAC, Stanford, CA 94309, USA
Abstract
The beam-beam effects of the PEP-II as an asymmetriccollider are studied with strong-strong simulations using a
newly developed particle-in-cell (PIC) code[1]. The simulated luminosity agrees with the measured one within 10%
in a large range of the beam currents. The spectra of coherent dipole oscillation are simulated with and without
the transparency symmetry. The simulated tune shift ofthe coherent  mode agrees with the linearized Vlasov theory even at large beam-beam parameters. The Poicare mapof coherent dipole is used to identify the beam-beam resonances.

1

INTRODUCTION

The PEP-II [2] and KEKB [3] as asymmetric collider,which consists of two different rings at different energy,
have been successfully constructed and fully operational.
The beam-beam effects in this new type of e + e; collider is
one of the important physical phenomena to be studied because, with twice of more parameters, there are much morechoices of operating parameters to gain a higher luminosity. Basically, there are two major choices for the operating parameters. One choice is the symmetric parametersof lattice and beam, such as equal beta functions and betatron tunes and beam sizes, in additional to maintaining theenergy transparency condition: I + E+ = I; E; [4]. The
other one is to break some unnecessary symmetry, for example, betatron tunes. To make right choice, it is importantto understand what are the consequences when symmetryis broken. For example, it is known that the violation of
the energy transparency condition might cause a flip-flopof the colliding beams [5]. The main subject of this paper
is to study the symmetry using the PEP-II as an example.
First we simulate the beam-beam limit and the spectrumof coherent oscillation when the transparency conditions
are preserved. Then we will study the spectrum and motion of the coherent oscillation when the transparency conditions are largely violated. Finally we will make comparison of the luminosity between the simulation and measurement.

2

SYMMETRIC PARAMETERS

Particle simulation is one of the important tools to study
many aspects of the beam-beam interaction such as thebeam-beam limit and the luminosity of colliders. Extending the work [6, 7] of solving the Poisson equation, we
reduce the region of mesh by assigning inhomogeneous Work supported by the Department of Energy under Contract No.
DE-AC03-76SF00515

potential on the boundary [1]. The method allows us tochoose much smaller region of mesh and therefore increase
the resolution of the solver. The improved resolution makes
more accurate the calculation of the dynamics in the coreof the beams.
In a typical simulation, we track 10240 macro particles
inside an area of 8x 24y with a rectangular mesh of
256  256. For a beam aspect ratio of x : y = 32 : 1,
we choose fifteen grids per x and five grids per y . This
choice of simulation parameters makes about ten particles
per cell on average within a region of 3 x 3y where themost of beam reside. It is adequate to compute the quantities that are mostly determined by the core of the beam.
The particles lost outside the meshed region are kept
where they are lost and their contribution to the force isignored afterward. The loss of the particles is closely monitored under different conditions since too much loss meansthat the simulated result is not reliable anymore. For a meshsize of 8x 24y , the loss of the particles is less than 1%even at extremely high beam intensity.
Due to the limitation of the computational speed on acomputer workstation, only a single two-dimensional slice
is used to represent a bunched beam. Therefore, all longitudinal effects such as the hourglass effect and synchbetatron coupling are neglected in the simulations.
At each beam intensity, we track the particles up to three
damping time till the beams reach their equilibrium distributions. Then the equilibrium distributions are used tocompute the quantities like the beam-beam parameters. For
extracting the power spectrum, we track additional 2048
turns after the equilibrium and save the beam centroid every turn.

2.1

Parameters

The PEP-II is an asymmetric e+ e; collider with two different storage rings in a 2.2 kilometer tunnel at the Stanford
Linear Accelerator Center. The positron beam is stored in
the Low Energy Ring (LER); the electron beam in the HighEnergy Ring (HER). The two rings are vertically separated
and brought into the collision at an interaction point (IP).
In Tab. 1, we list a possible set of symmetric parameters
for the PEP-II.
It has been shown [4] that even for an asymmetric collider, it still can be operated theoretically as if it is a symmetrical collider if the transparency conditions: x+ =x; ; y+ = y; ; x+ = x; ; y+ = y; ; +x = ;x ; +y =
;y ; t+ = t; ; I+ E+ = I; E; , are all satisfied as far as thebeam-beam interaction is the concern.
To preserve the energy transparency condition in thesimulation, we vary the beam intensity with a step of

Parameter
E (Gev)x (cm)
y (cm)
t (turn)
x (nm-rad)y (nm-rad)xyDescription
beam energy
beta X at the IPbeta Y at the IP
damping timeemittance x
emittance yx tuney tuneLER(e+)
3.1
50.0
1.5
5014
48.0
1.50
0.390
0.370

HER(e-)
9.0
50.0
1.5
5014
48.0
1.50
0.390
0.370

Table 1: Symmetric parameters for the PEP-IIN + = 1010 and N ; = N + E+ =E; starting from zero.
2.2

where y (1) is the beam-beam limit and is the decayrate with respect to the beam current I . For this particularcase, y (1) = 0:0422. At I+ = 1:26mA, which is thenominal value of the design single bunch current,  y =0:025. That is about 15% less than the design value ofthe beam-beam parameter. At I+ = 2:31mA, which is the
single bunch current at the top of each filling last October,y = 0:033, which is less than y (1). That indicates that
there is still room to improve if the symmetric parameters
could be implemented in the machines of the PEP-II.
At a few other tunes that we have studied, for example
x = 0:2962 and y = 0:2049, we found the similar phenomenon. The same is true in the horizontal plane. It is
intriguing that such simple parameterization can be applied
to the beam-beam parameter.

Beam-Beam Parameters

Given equilibrium distributions that are close enough to theGaussian, we introduce the beam-beam parametersy =re N  y;
2  y (x + y )(1)where re is the classical electron radius, is the energy ofthe beam in unit of the rest energy, and N is total number of the charge in the bunch. Here the superscript “+”denotes quantities corresponding to the positron beam and“;” quantities corresponding to the electron beam.

2.3

Spectrum

Using the fast Fourier transformation (FFT), we computed
the power spectrum with the beam centroids which wererecorded in 2048 consecutive turns after the equilibrium
distributions were established. The several spectra with atdifferent beam intensities are shown in Fig. 2. There are
two peaks clear seen in each spectrum. They are  and 
modes of the coherent dipole oscillations. The tune shift ofthe  mode increases with respect to the beam intensity.
1.5
1

1.5
N = 1E101

N = 13E10

0.045
0.5

0.5

0.04
0
0.3
0.035

1.5

0.03

1

0.35

0.4

0.45

0.5

0

0.35

0.4

0.45

0.5

0.4

0.45

0.5

0.4
νx

0.45

0.5

1.5
N = 5E101

0.5

N = 17E10

0.5

ξy

0.025
0

0.35

0.4

0.45

0.5

0

0.35

0.02
1.5
N = 9E10

0.015

0.01

1

1

0.5

0.5

0

0.005

0

1.5

0

1

2

3

4

5

6

7

I(mA)

Figure 1: The beam-beam parameter as a function of single
bunch current. The circles represent the positron beam andthe cross represent the electron beam. The solid lines are
fitted curves.
At every the intensity of beams, we computed the beam
sizes and the beam-beam parameters of the equilibrium
beam distribution. The result is summarized in Fig. 1.
Clearly, the beam-beam parameter in the vertical plane issaturated at high beam intensity. That is consistent withmany experimental observations [8]. Moreover, the dependence of the beam-beam parameter upon the single bunch
current can be fitted rather well with two parameters,y (I ) = y (1)[1 ; exp(; I )];

(2)

0.35

0.4
νx

0.45

0.5

0

N = 20E10

0.35

Figure 2: Power spectra of coherent dipole oscillation at
different beam intensities. The dashed line represents themachine tune x = 0:39.
However, the tune shift of the  mode also increases
though less than the  mode as the intensity increases.
This phenomenon is beyond the capability of the linearized
Vlasov theory since it predicts no tune shift for the  mode.

2.4

The Yokoya Factor

Studying the power spectrum of colliding beams is a powerful way to investigate and understand the beam-beam interaction. Historically, in symmetric colliders where two
beams are identical, the tune shift of the coherent  mode
has provided many useful insights into the dynamics of thebeam-beam interaction. It has been shown analytically that

this tune shift is proportional to the beam-beam parameter , namely  =  [9, 10, 11, 12]. The coefficient  is
between 1 and 2 depending on the beam distribution. For a
self-consistent beam distribution [12],x; = x ; y; = (1 ; r)y(3)2
where  = 1:330 ; 0:370r + 0:279r , r = y =(x + y ),and x and y are the horizontal and vertical beam size
respectively.
Experimentally, this relation has been observed in manydifferent colliders [13, 14]. The results of measurements
are consistent with the calculation based on the Vlasov theory. In simulations[15, 16] using the PIC this relation was
also confirmed. Now, this well-established relation is often
used to measure the beam-beam parameter or test a newly
developed code.
The shifts of the  mode away from the machine tune
are extracted from the spectra as shown Fig. 2 at different
beam intensities. The shifts as a function of the beam-beam
parameter, which is shown in Fig. 1, are summarized inFig. 3.
0.06

0.06

0.05

0.05

λ1 = 1.319

Description
beam energy
beta x at the IPbeta y at the IP
damping timeemittance X
emittance Y
bunch length
x tuney tunez tunexys

Table 2: Operating parameters for PEP-IIsity with a step of N + = 1010 and N ; = N + =2. Since
the LER has longer damping time than the one in HER,
we track the particles with three damping time of the LER
to ensure that both beams reach their equilibrium distributions.

3.1

Dipole Motion

λ1 = 1.280

0.03
0.02

0.01

0.01
νx = 0.39
0

0.01

0.02
ξx

νy = 0.37
0.03

0.04

0

0

0.01

0.02
ξy

20

20

20

10

10

10

0

0

0

0

−10

−10

−10

−10

Px(µrad)

20
0.02

0.03

0.04

10

e+

−20
−10

0

10

20

The predicted linear relation based on Eqn. 3 is also plotted in the figure. One can see that the agreement between
the theory and simulation is rather good even at very highbeam-beam parameter.

ASYMMETRIC PARAMETERS

In October 2000, PEP-II has achieved its design luminosityof 3:0  1033 cm;2 s;1 . The parameters of the lattices andbeams when the design luminosity was reached are tabulated in Tab. 2. It is clear that many transparency conditions are violated. Among them, the betatron tunes arevery different and well separated compared to the beambeam parameter.
During the last run, the ratio of the beam current I + : I;
is about 2:1. As a result, the energy transparency condition
I+ E+ = I; E; is also violated. With this set of parameters, the PEP-II has been operated at a region of asymmetry.
To simulate the beam-beam effects under the nominal
running condition of the PEP-II, we vary the beam inten-

10

e−

−20
−10

0

10

−20
−10

0

10

−20
−10

20

20

20
10

10

10

0

0

0

0

−10

−10

−10

−10

Px(µrad)Figure 3: The tune shift of coherent  mode as a functionof the beam-beam parameter. The left plot is for the horizontal plane and right plot is for the vertical plane. The
circles represent the simulated tune shifts. The solid lines
represent  =  .

3

HER(e-)
9.0
50.0
1.25
5014
48.0
1.50
1.30
0.569
0.639
0.044

δνπ y

0.03

0

LER(e+)
3.1
50.0
1.25
9740
24.0
1.50
1.30
0.649
0.564
0.025

The Poincare maps of self-excited coherent dipole in
the plane with four different beam intensities: N + =(1; 5; 9; 13)  1010 (from left to right) are shown in Fig. 4.

0.04

δνπ x

0.04

Parameter
E (Gev)x (cm)
y (cm)
t (turn)
x (nm-rad)y (nm-rad)z (cm)

−20
−10

0
X(µm)

10

−20
−10

0
X(µm)

10

−20
−10

0
X(µm)

10

−20
−10

0

10

0
X(µm)

10

Figure 4: Poincare map of coherent dipole at different
beam intensities in the horizontal plane.
It is clear that there is maximum amplitude of the oscillation. The amplitudes are very small and within x;y =30
when the beam intensities stay bellow the peak operating
intensity N + = 10:6  1010 .
In Fig. 4, some structures of the resonance can be clearly
seen. For instance, at N + = 1  1010 , we see seven islandsin the Poincare map for the electron beams. This seventhorder resonance can be identified as 7x; = 172. For the
positron beam, we see a triangle shape which is consistent
with the third order resonance 3 x+ = 116 at N + = 5 1010 .
It is worthwhile to note that the resonance structure displayed in the figure is near x =30, which is a factor of two
smaller than the size of the grid (x /15). This does not necessarily mean that the resolution of resolving the dynamics
of individual particle is less than the size of the grid. But
for the collective motion, such as the oscillation of the coherent dipole, the resolution can be smaller than the size

of the grid. Because they are an average over the distribution of the particles, the noises from the finite size of themesh and the representation of beam distribution as a finite
number of particles are much reduced.
Below the peak operating intensity, the amplitude of thevertical oscillation is about y =30. But no higher order resonances are identifiable. One of possible reasons is that thevertical grid size y =5 is too large to resolve the structureof resonance within y =30.
Above the peak operating intensity, the amplitude of
the vertical oscillation increases more than ten times andreaches half of the beam size for the electron beam. The
oscillation acts coherently as a single particle. It is not clear
why the dipole mode of the electron beam is excited to suchlarge amplitude at the higher intensity.

3.2

Spectrum

3.3

Tune Shift

Similar to the symmetric case, the spectrum shifts as the intensity increases. The tune shifts as a function of the beam
intensities is shown in Fig. 6. The center of the spectrum
is the fitted result of the Lorentz spectrum. In general, thecenter does not coincide with the peak in the spectrum dueto the asymmetric nature of the spectrum. Therefore, thetune shifts as plotted in the figure should be considered as
the average values. The tune shifts saturate around 0.015
in the both planes. In particular, the vertical shifts actually
start to decrease near the peak operating intensity. Similar behavior had been observed in the measurements of thepower spectrum for the PEP-II. This behavior is certainly
very different with the behavior of the beam-beam parameters as simulated [1]. For the PEP-II, which is operated
at very asymmetric parameters, we do not have a simple
linear relation between the beam-beam parameter and thetune shift of the dipole spectrum.

In the vertical plane, the dipole mode at different beam intensities are plotted in Fig. 5. Unlike the symmetric colliding beams, there are no visible  modes in the spectra.
An asymmetric shape of the spectrum is clear visible especially at the high intensity. Similar spectrum has been
observed experimentally [17].

20δνy(10−3)

15

10

5

0

0

2

4

6

8

10
N+(1010)

12

14

16

18

20

0

2

4

6

8

10
N+(1010)

12

14

16

18

20

N+ = 1010

0.56

−5

10

0.58

0.6

0.62

0.64

0.66

+

e

2

0.68

e−

5
0
0.54

−5

4
10

20

10

N+ = 5x1010

0.56

0.58

+
10
N = 9x10

0.6

0.62

0.64

0.66

+

e

0.56
+

0.58

0.6

0.62

0.64

0.66

0.68

0.66

0.68

10
−

e

2

0.56

0.58

0

e

N = 13x10

0
0.54

0.68

−

2
0
0.54
4

−3

e

15

0
0.54
4

10

−

+

e

2

δνx(10−3)

10

−5

4

0.6

νy

0.62

0.64

Figure 5: The vertical power spectra at different beam intensities for the PEP-II.
At N + = 13  1010 , a single mode is coherently excited in the electron beam. Correspondingly, the excitedmode is shown as an ellipse in the Poincare map. And thepositron beam is blowup vertically at the same time. As
a result, the luminosity decreases. In order to check if thishighly excited mode is the cause of the rapid increase in thevertical size of the positron beam, we eliminate the dipole
oscillation every turn in simulation. But the peak luminosity remains the same. So we conclude, in this case, thatthe collective dipole motion is not the main reason for the
beam-beam blowup.
In the horizontal plane, the spectra are broader than thevertical ones largely because the resonances, which wehave shown in the previous section. The spectra are notso asymmetric as the vertical ones.

Figure 6: Tune shift as a function of bunch intensity. The
circles represent the tune shift of the positron bunch. The
crosses represent the tune shift of the electron bunch. The
dashed lines represent the peak bunch intensity of the PEPII operation.

3.4

Luminosity

To make a direct comparison between simulation and experimental observation, we have recorded the luminosity
during a period of four hours on October 1, 2000. The
data are shown in Fig. 7. Duration of each measurement
was three minutes. The first and second plots in the figure present the total decaying beam current of positron andelectron beams respectively. The third plot shows the measured and simulated luminosities at the same beam current
displayed in the figure. The other parameters used in thesimulation are the same as in Tab. 2.
The agreement of the simulation and measurement waswithin 10%. Since the longitudinal effects of the beambeam interaction are not yet included in the simulations,
three-dimensional simulation could reduce the simulated

6

1400

[1] Y. Cai, A. W. Chao, S. I. Tzenov, and T. Tajima, “Simulation of the Beam-Beam Effects in e+ e; Storage Rings with a
Method of Reduced Region of Mesh,” Phys. Rev. ST Accel.
Beams 4, 011001 (2001).

1200
I+(mA)

1000
800
600
400

REFERENCES

650[2] “PEP-II: An Asymmetric B Factory”, Conceptual Design Report, SLAC-418, June 1993.

600[3] KEK B-Factory Design Report No. KEK 95-7, 1995.

50

100

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200

250

I−(mA)

0[4] S. Krishnagopal and R. Siemann, “Beam-Energy Inequality in the Beam-Beam Interaction,” Phys. Rev. D 41, 1741
(1990).

550

500
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0

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Luminosity(1033cm−2s−1)

3
2.5
simulation
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67, 2461(1991).

1.5
measurement

1
0.5

0

50

100

150

200

250

MinuteFigure 7: Luminosity of a routine operation of PEP-II. The
crosses represent measurement and the circles represent
simulation. The number of bunches was 605.
luminosity. For example, the hourglass effect should reduce the simulated luminosity by 12% given  z = 1:3cm
and y = 1:25cm.

4 CONCLUSION
When the transparency conditions are violated, especiallythe betatron tunes are well separated, the strong-strong simulations show the spectra of coherent oscillation in the
beam-beam interaction is very different from the spectraseen in the symmetric collider. In particular, there is no mode seen in the spectrum. The simple linear relation
between the beam-beam parameter and the tune shift ofthe  mode is no longer existing. Given the operating parameters of the PEP-II, we do not see any simple relation
between the tune shift of the continuum spectrum and thebeam-beam parameter. Therefore, the beam-beam parameter would not be estimated using the spectrum.
The agreement of luminosity between the simulation andmeasurement is surprising and remarkable considering thesimplicity of the two-dimensional model. In general, thethree-dimensional effects such as the hourglass effects andthe synch-betatron resonance could become very important. The code is being extended to include the bunch
length and synchrotron oscillation. More simulation results
will be directly compared to the controlled experiment.

5 ACKNOWLEDGMENTS
I would like to thank A. Chao, S. Tzenov, and T. Tajima forthe collaboration. I would also like to thank S. Heifets, W.
Kozanecki, M. Minty, I. Reichel, J. Seeman, R. Warnock,
U. Wienands, and Y. Yan for many helpful discussions.

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[14] H. Koiso, et al, “Measurement of the Coherent Beam-Beam
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[15] E. B. Anderson, T.I Banks, J.T. Rogers, “ODYSSEUS: Description of Results from a Strong-Strong Beam-Beam Simulation For Storage Rings,” Proceedings of Particle Accelerator Conference, New York, 1999.
[16] K. Ohmi, “Simulation of Beam-Beam Effects in a Circulare+ e; Collider,” Phys. Rev. E 62, 7287 (2000).
[17] S. Heifets and H. U. Wienands, “On the Shape fo the Betatron Side-Band,”Proceedings of the Seventh European Particle Accelerator Conference, Vienna, Austria, June 2000.