Simulation of Beam-Beam Effects in $e^+e^-$ Storage Rings
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. 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 of the coherent $\pi$ mode agrees with the linearized Vlasov theory even at large beam-beam parameters. The Poicare map of coherent dipole is used to identify the beam-beam resonances.
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ABSTRACT: A simple consideration is given of the eigenmodes of the coherent betatron oscillation excited by a series of sinusoidal kicks and influenced by the beam-beam interaction in e+e− colliding storage rings. The differences between tunes corresponding to the eigenmodes are just half of what is expected from rigid point like bunches. The discussion is mainly limited to the case where the beam-beam force is not so strong so that the bunches remain Gaussian and the linear approximation of the force is valid. The nonlinear effect is also discussed briefly.Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment 06/1988; · 1.32 Impact Factor
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ABSTRACT: The coherent normal modes of the nonlinear colliding beam system are calculated using averaging methods to find an equivalent infinite system of linear coupled oscillators. The condition of a consistent, stationary state leads to an eigenvalue equation for the characteristic frequencies and transverse exitation distributions of the modes. Results are presented for modes along the narrow axis of a Gaussian ribbon beam. The tune split is related to L/Â¿(I+I-) for different values of I+/I- and of Â¿+/Â¿- .IEEE Transactions on Nuclear Science 07/1981; · 1.46 Impact Factor
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ABSTRACT: We have modified the beam-beam simulation code CBI to handle asymmetric beams and used it to look at energy transparency and symmetries in the beam-beam interaction. We find that even a small violation of energy transparency, or of the symmetry between the two beams, changes the character of the collective (coherent) motion; in particular, period-n oscillations are no longer seen. We speculate that the one-time observation of these oscillations at LEP, and the more ubiquitous observation of the flip-flop instability in colliders around the world, may be a consequence of breaking the symmetry between the electron and positron beams. We also apply this code to the asymmetric collider PEP-II, and find that for the nominal parameters of PEP-II, in particular, the nominal tune-shift parameter of xi0=0.03, there are no collective beam-beam issues. Collective quadrupole motion sets in only at xi0=0.06 and above, consistent with earlier observations for symmetric beams.Physical Review Special Topics - Accelerators and Beams 02/2000; 3(2). · 1.52 Impact Factor
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
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. 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-
The PEP-II  and KEKB  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−. 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 . 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-
2 SYMMETRIC 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.
potential on the boundary . The method allows us to
choose muchsmaller regionof mesh and thereforeincrease
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-
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-
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  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+
beam-beam interaction is the concern.
To preserve the energy transparency condition in the
simulation, we vary the beam intensity with a step of
t = τ−
t,I+E+= I−E−, are all satisfied as far as the
beta X at the IP
beta Y at the IP
Table 1: Symmetric parameters for the PEP-II
δN+= 1010and δN−= δN+E+/E−starting from zero.
Given equilibriumdistributionsthat are close enoughto the
Gaussian, we introduce the beam-beam parameters
x + σ∓
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.
Figure1: Thebeam-beamparameterasa functionof single
bunch current. The circles represent the positron beam and
the cross represent the electron beam. The solid lines are
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 . 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)],
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.
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.
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
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 ,
δνx,π= Λξx,δνy,π= Λ(1 − r)ξy
where Λ = 1.330−0.370r+0.279r2, r = σy/(σx+σy),
and σxand σy are the horizontal and vertical beam size
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
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
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
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-
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-
beta x at the IP
beta y at the IP
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-
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.
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+
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 ×
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.
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 .
0.540.56 0.580.6 0.620.640.660.68
0.54 0.560.58 0.60.620.640.660.68
N+ = 1010
N+ = 5x1010
N+ = 9x1010
N+ = 13x1010
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
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.
3.3 Tune 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 . 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.
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-
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
0 50100150 200250
0 50100 150200250
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
When the transparency conditions are violated, especially
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.
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
will be directly compared to the controlled experiment.
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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