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arXiv:physics/0104022v1 [physics.acc-ph] 5 Apr 2001

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 asymmetric

collider are studied with strong-strong simulations using a

newly developed particle-in-cell (PIC) code[1]. The simu-

lated luminosity agrees with the measured one within 10%

in a large range of the beam currents. The spectra of co-

herent dipole oscillation are simulated with and without

the transparency symmetry. The simulated tune shift of

the coherent π mode agrees with the linearized Vlasov the-

ory even at large beam-beam parameters. The Poicare map

of coherent dipole is used to identify the beam-beam reso-

nances.

1INTRODUCTION

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-beameffects in this new type of e+e−collider is

one of the important physical phenomena to be studied be-

cause, with twice of more parameters, there are much more

choices of operating parameters to gain a higher luminos-

ity. Basically, there are two major choices for the operat-

ing parameters. One choice is the symmetric parameters

of lattice and beam, such as equal beta functions and beta-

tron tunes and beam sizes, in additional to maintaining the

energy transparency condition: I+E+ = I−E−[4]. The

other one is to break some unnecessary symmetry, for ex-

ample, betatron tunes. To make right choice, it is important

to understand what are the consequences when symmetry

is broken. For example, it is known that the violation of

the energy transparency condition might cause a flip-flop

of 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 spectrum

of coherent oscillation when the transparency conditions

are preserved. Then we will study the spectrum and mo-

tion of the coherent oscillation when the transparency con-

ditions are largely violated. Finally we will make compari-

son of the luminosity between the simulation and measure-

ment.

2SYMMETRIC PARAMETERS

Particle simulation is one of the important tools to study

many aspects of the beam-beam interaction such as the

beam-beam limit and the luminosity of colliders. Extend-

ing 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 to

choose muchsmaller regionof mesh and thereforeincrease

theresolutionofthesolver. Theimprovedresolutionmakes

more accurate the calculation of the dynamics in the core

of the beams.

In a typical simulation, we track 10240 macro particles

inside an area of 8σx×24σy with a rectangular mesh of

256 × 256. For a beam aspect ratio of σx: σy = 32 : 1,

we choose fifteen grids per σxand five grids per σy. This

choice of simulation parameters makes about ten particles

per cell on average within a region of 3σx×3σywhere the

most of beam reside. It is adequate to compute the quanti-

ties 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 is

ignored afterward. The loss of the particles is closely mon-

itoredunderdifferentconditionssincetoomuchloss means

thatthesimulatedresultisnotreliableanymore. Foramesh

size of 8σx×24σy, the loss of the particles is less than 1%

even at extremely high beam intensity.

Due to the limitation of the computational speed on a

computer workstation, only a single two-dimensional slice

is used to represent a bunched beam. Therefore, all lon-

gitudinal effects such as the hourglass effect and synch-

betatron 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 dis-

tributions. Then the equilibrium distributions are used to

computethe quantitieslike thebeam-beamparameters. For

extracting the power spectrum, we track additional 2048

turns after the equilibrium and save the beam centroid ev-

ery turn.

2.1Parameters

The PEP-II is an asymmetric e+e−collider with two dif-

ferentstorageringsin a 2.2kilometertunnelat theStanford

Linear Accelerator Center. The positron beam is stored in

the Low EnergyRing (LER); the electron beam in the High

Energy 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 col-

lider, it still can be operated theoretically as if it is a sym-

metrical collider if the transparency conditions: ν+

ν−

x

= β∗−

ǫ−

beam-beam interaction is the concern.

To preserve the energy transparency condition in the

simulation, we vary the beam intensity with a step of

x

=

x,ν+

y,τ+

y= ν−

t = τ−

y,β∗+

t,I+E+= I−E−, are all satisfied as far as the

x,β∗+

y

= β∗−

y,ǫ+

x= ǫ−

x,ǫ+

y=

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Parameter

E (Gev)

β∗

β∗

τt(turn)

ǫx(nm-rad)

ǫy(nm-rad)

νx

νy

Description

beam energy

beta X at the IP

beta Y at the IP

damping time

emittance x

emittance y

x tune

y tune

LER(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

x(cm)

y(cm)

Table 1: Symmetric parameters for the PEP-II

δN+= 1010and δN−= δN+E+/E−starting from zero.

2.2Beam-Beam Parameters

Given equilibriumdistributionsthat are close enoughto the

Gaussian, we introduce the beam-beam parameters

ξ±

y=

reN∓β±

y(σ∓

y

2πγ±σ∓

x + σ∓

y),

(1)

where reis the classical electron radius, γ is the energy of

the beam in unit of the rest energy, and N is total num-

ber of the charge in the bunch. Here the superscript “+”

denotes quantities corresponding to the positron beam and

“−” quantities corresponding to the electron beam.

01234567

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

I(mA)

ξy

Figure1: Thebeam-beamparameterasa functionof single

bunch current. The circles represent the positron beam and

the 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 is

saturated at high beam intensity. That is consistent with

many experimental observations [8]. Moreover, the depen-

dence of the beam-beam parameter upon the single bunch

current can be fitted rather well with two parameters,

ξy(I) = ξy(∞)[1 − exp(−αI)],

(2)

where ξy(∞) is the beam-beam limit and α is the decay

rate with respect to the beam current I. For this particular

case, ξy(∞) = 0.0422. At I+ = 1.26mA, which is the

nominal value of the design single bunch current, ξy =

0.025. That is about 15% less than the design value of

the beam-beam parameter. At I+= 2.31mA, which is the

single bunch current at the top of each filling last October,

ξy= 0.033, which is less than ξy(∞). 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

νx= 0.2962 and νy= 0.2049, we found the similar phe-

nomenon. The same is true in the horizontal plane. It is

intriguingthat such simple parameterizationcan be applied

to the beam-beam parameter.

2.3Spectrum

Using the fast Fourier transformation (FFT), we computed

the power spectrum with the beam centroids which were

recorded in 2048 consecutive turns after the equilibrium

distributions were established. The several spectra with at

different beam intensities are shown in Fig. 2. There are

two peaks clear seen in each spectrum. They are σ and π

modes of the coherent dipole oscillations. The tune shift of

the π mode increases with respect to the beam intensity.

0.30.350.40.450.5

0

0.5

1

1.5

0.350.40.450.5

0

0.5

1

1.5

0.350.4

νx

0.450.5

0

0.5

1

1.5

0.350.4 0.450.5

0

0.5

1

1.5

0.35 0.40.45 0.5

0

0.5

1

1.5

0.350.4

νx

0.45 0.5

0

0.5

1

1.5

N = 1E10

N = 5E10

N = 9E10

N = 13E10

N = 17E10

N = 20E10

Figure 2: Power spectra of coherent dipole oscillation at

different beam intensities. The dashed line represents the

machine tune νx= 0.39.

However, the tune shift of the σ mode also increases

though less than the π mode as the intensity increases.

This phenomenonis beyondthe capabilityof the linearized

Vlasov theorysince it predicts no tune shift for the σ mode.

2.4The Yokoya Factor

Studying the power spectrum of colliding beams is a pow-

erful way to investigate and understand the beam-beam in-

teraction. Historically, in symmetric colliders where two

beams are identical, the tune shift of the coherent π mode

has provided many useful insights into the dynamics of the

beam-beam interaction. It has been shown analytically that

Page 3

this tune shift is proportional to the beam-beam parameter

ξ, namely δνπ = Λξ [9, 10, 11, 12]. The coefficient Λ is

between 1 and 2 depending on the beam distribution. For a

self-consistent beam distribution [12],

δνx,π= Λξx,δνy,π= Λ(1 − r)ξy

(3)

where Λ = 1.330−0.370r+0.279r2, r = σy/(σx+σy),

and σxand σy are the horizontal and vertical beam size

respectively.

Experimentally, this relation has been observed in many

different colliders [13, 14]. The results of measurements

are consistent with the calculation based on the Vlasov the-

ory. 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 π mode away from the machine tune

are extracted from the spectra as shown Fig. 2 at different

beamintensities. The shifts as a functionofthe beam-beam

parameter, which is shown in Fig. 1, are summarized in

Fig. 3.

00.010.02

ξx

0.030.04

0

0.01

0.02

0.03

0.04

0.05

0.06

δνπ x

00.010.02

ξy

0.030.04

0

0.01

0.02

0.03

0.04

0.05

0.06

δνπ y

λ1 = 1.319

λ1 = 1.280

νy = 0.37

νx = 0.39

Figure 3: The tune shift of coherent π mode as a function

of the beam-beam parameter. The left plot is for the hor-

izontal plane and right plot is for the vertical plane. The

circles represent the simulated tune shifts. The solid lines

represent δνπ= λξ.

The predictedlinearrelationbasedonEqn.3 is alsoplot-

ted in the figure. One can see that the agreement between

the theory and simulation is rather good even at very high

beam-beam parameter.

3ASYMMETRIC PARAMETERS

In October 2000,PEP-II has achievedits design luminosity

of 3.0 × 1033cm−2s−1. The parameters of the lattices and

beams when the design luminosity was reached are tabu-

lated in Tab. 2. It is clear that many transparency con-

ditions are violated. Among them, the betatron tunes are

very different and well separated compared to the beam-

beam 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 parame-

ters, thePEP-IIhasbeenoperatedata regionofasymmetry.

To simulate the beam-beam effects under the nominal

running condition of the PEP-II, we vary the beam inten-

Parameter

E (Gev)

β∗

β∗

τt(turn)

ǫx(nm-rad)

ǫy(nm-rad)

σz(cm)

νx

νy

νs

Description

beam energy

beta x at the IP

beta y at the IP

damping time

emittance X

emittance Y

bunch length

x tune

y tune

z tune

LER(e+)

3.1

50.0

1.25

9740

24.0

1.50

1.30

0.649

0.564

0.025

HER(e-)

9.0

50.0

1.25

5014

48.0

1.50

1.30

0.569

0.639

0.044

x(cm)

y(cm)

Table 2: Operating parameters for PEP-II

sity witha step of δN+= 1010andδN−= δN+/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 distribu-

tions.

3.1Dipole Motion

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.

−10010

−20

−10

0

10

20

Px(µrad)

−10010

−20

−10

0

10

20

−10010

−20

−10

0

10

20

−100 10

−20

−10

0

10

20

−10010

−20

−10

0

10

20

X(µm)

Px(µrad)

−100 10

−20

−10

0

10

20

X(µm)

−10010

−20

−10

0

10

20

X(µm)

−10010

−20

−10

0

10

20

X(µm)

e+

e−

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 oscil-

lation. The amplitudes are very small and within σx,y/30

when the beam intensities stay bellow the peak operating

intensity N+= 10.6 × 1010.

In Fig. 4, some structures of the resonancecan be clearly

seen. For instance, at N+= 1×1010, we see seven islands

in the Poincare map for the electron beams. This seventh

order resonance can be identified as 7ν−

positron beam, we see a triangle shape which is consistent

with the third order resonance 3ν+

1010.

It is worthwhile to note that the resonance structure dis-

played in the figure is near σx/30, which is a factor of two

smaller than the size of the grid (σx/15). This does not nec-

essarily 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 co-

herent dipole, the resolution can be smaller than the size

x= 172. For the

x= 116 at N+= 5 ×

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of the grid. Because they are an average over the distribu-

tion of the particles, the noises from the finite size of the

mesh and the representation of beam distribution as a finite

number of particles are much reduced.

Below the peak operating intensity, the amplitude of the

vertical oscillation is about σy/30. But no higherorderres-

onances are identifiable. One of possible reasons is that the

vertical grid size σy/5 is too large to resolve the structure

of resonance within σy/30.

Above the peak operating intensity, the amplitude of

the vertical oscillation increases more than ten times and

reaches half of the beam size for the electron beam. The

oscillationacts coherentlyas a singleparticle. It is notclear

whythedipolemodeofthe electronbeamis excitedtosuch

large amplitude at the higher intensity.

3.2Spectrum

In the vertical plane, the dipole mode at different beam in-

tensities are plotted in Fig. 5. Unlike the symmetric col-

liding beams, there are no visible π modes in the spectra.

An asymmetric shape of the spectrum is clear visible es-

pecially at the high intensity. Similar spectrum has been

observed experimentally [17].

0.54

4

0.560.580.60.620.640.66 0.68

0

2

4

10−5

0.540.560.580.60.620.64 0.660.68

0

2

10−5

0.54

4

0.560.580.60.620.640.660.68

0

2

4

10−5

0.540.560.58 0.60.620.640.660.68

0

2

10−3

νy

N+ = 1010

N+ = 5x1010

N+ = 9x1010

N+ = 13x1010

e+

e+

e+

e−

e−

e−

e−

Figure 5: The vertical power spectra at different beam in-

tensities for the PEP-II.

At N+= 13 × 1010, a single mode is coherently ex-

cited in the electron beam. Correspondingly, the excited

mode is shown as an ellipse in the Poincare map. And the

positron beam is blowup vertically at the same time. As

a result, the luminosity decreases. In order to check if this

highlyexcitedmodeis the cause of the rapidincrease in the

vertical size of the positron beam, we eliminate the dipole

oscillation every turn in simulation. But the peak luminos-

ity remains the same. So we conclude, in this case, that

the collective dipole motion is not the main reason for the

beam-beam blowup.

In the horizontal plane, the spectra are broader than the

vertical ones largely because the resonances, which we

have shown in the previous section. The spectra are not

so asymmetric as the vertical ones.

3.3Tune Shift

Similar to the symmetriccase, the spectrumshifts as the in-

tensity 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, the

center does not coincide with the peak in the spectrum due

to the asymmetric nature of the spectrum. Therefore, the

tune 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. Simi-

lar behavior had been observed in the measurements of the

power spectrum for the PEP-II. This behavior is certainly

very different with the behavior of the beam-beam param-

eters 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 the

tune shift of the dipole spectrum.

0246810121416 1820

0

5

10

15

20

δνy(10−3)

N+(1010)

02468101214 16 1820

0

5

10

15

20

δνx(10−3)

N+(1010)

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 bunchintensity of the PEP-

II operation.

3.4 Luminosity

To make a direct comparison between simulation and ex-

perimental 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 fig-

ure present the total decaying beam current of positron and

electron beams respectively. The third plot shows the mea-

sured and simulated luminosities at the same beam current

displayed in the figure. The other parameters used in the

simulation are the same as in Tab. 2.

The agreement of the simulation and measurement was

within 10%. Since the longitudinal effects of the beam-

beam interaction are not yet included in the simulations,

three-dimensional simulation could reduce the simulated

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0 50 100150200250

400

600

800

1000

1200

1400

I+(mA)

050 100150 200 250

450

500

550

600

650

I−(mA)

050100 150200250

0.5

1

1.5

2

2.5

3

Luminosity(1033cm−2s−1)

Minute

measurement

simulation

Figure 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 re-

duce the simulated luminosity by 12% given σz = 1.3cm

and β∗

y= 1.25cm.

4CONCLUSION

When the transparency conditions are violated, especially

thebetatrontunesarewellseparated,thestrong-strongsim-

ulations show the spectra of coherent oscillation in the

beam-beam interaction is very different from the spectra

seen in the symmetric collider. In particular, there is no

π mode seen in the spectrum. The simple linear relation

between the beam-beam parameter and the tune shift of

the π mode is no longer existing. Given the operating pa-

rameters of the PEP-II, we do not see any simple relation

between the tune shift of the continuum spectrum and the

beam-beam parameter. Therefore, the beam-beam parame-

ter would not be estimated using the spectrum.

Theagreementofluminositybetweenthesimulationand

measurement is surprising and remarkable considering the

simplicity of the two-dimensional model. In general, the

three-dimensional effects such as the hourglass effects and

the synch-betatron resonance could become very impor-

tant. The code is being extended to include the bunch

lengthandsynchrotronoscillation. Moresimulationresults

will be directly compared to the controlled experiment.

5ACKNOWLEDGMENTS

I would like to thank A. Chao, S. Tzenov,and T. Tajima for

the 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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