Flip-flop Instability of Colliding Bunches with Round Cross Sections
D.V. Pestrikov
Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russian Federation
Abstract
Selfconsistent beam-beam interactions of colliding
bunches changes their transverse sizes due to the flip-flopinstability. Calculating the so-called dynamic β-functions
at the interaction point, we study the flip-flop instabilityof colliding bunches with variations of the beam aspectratios. Even for symmetrical unperturbed lattice conditionsof colliding bunches the flip-flop instabilities conserving
and changing the aspect ratios of colliding bunches occur
within different and non-intersecting stopbands in theworking plane of betatron tunes. In particular, this effectcan limit the beam-beam performance of the collidingbunches with round cross sections.

1

INTRODUCTION

According to observations [1], under certain conditionstransverse sizes of colliding bunches can spontaneously become very different, so that the size of one bunch may substantially exceed that of its partner thus, limiting the collider luminosity. This situation is not stable and after sometime the bunches exchange their sizes. As was reported
in Ref.[1], the variations in the transverse bunch sizes are
not accompanied by the observations of the (dipole) bunchcoherent signals. Such a phenomenon was called as theflip-flop effect. A comprehensive description of the flipflop effect is embarrassed by a necessity to take into account the self-consistent nature of the beam-beam forces
and by the fact that very frequently non-linear dependencies of the beam-beam force on the particle offsets are of
the primary importance. The self-consistent behavior of thecolliding bunches should essentially operate with simultaneous and correlative variations of sizes of both colliding
bunches. For that reason, it cannot be studied using a single
particle tracking only.
Generally, the difficulties in descriptions of the selfconsistent behavior of colliding bunches make attractive
the studies of simplified models predicting the flip-flop behavior. In Ref.[2] the flip-flop instability was found calculating self-consistent variations the bunch rms sizes at theinteraction point. The equilibrium bunch sizes ware obtained as the period-one solutions to the envelope equations
describing the transport of the bunch rms sizes in the phasespace through the ring and taking into account nonlinear
beam-beam kicks. The instability appears when the envelope equations for colliding bunches have multiple roots. It
may occur since these equations are intrinsically nonlinear.
To simplify calculations the beam-beam kick in Ref.[2] wascalculated assuming the round cross sections of bunches
and Gaussian distributions in bunches in transverse coordinates. The synchrotron radiation damping and the heatingof bunches due to the fluctuations of particle synchrotron
radiation was taken into account to provide stationary unperturbed bunch sizes. In Ref.[3] these calculations weregeneralized for the case of colliding bunches having flatcross sections at the interaction point (IP). As far as thecalculations in Refs.[2, 3] are focused on the descriptionsof nonlinear phenomena, their performances demand numerous artificial assumptions.
It was mentioned already in Ref.[2] and then in Ref.[4]that a reasonable though simplified description of theflip-flop instability can be obtained calculating in a selfconsistent way the so-called dynamic β-functions of colliding bunches at IP. Within the framework of this model theinstability is though a self-consistent, but essentially incoherent effect. The calculations of β-functions can be doneusing the linear approximation for the beam-beam kick in
the particle offsets. If we assume short bunches (σ s  β),
the linear part of the beam-beam kick depends on the transverse sizes of colliding bunches only. The periodicity conditions for β-functions enable the calculations of their selfconsistent values. Emittances of colliding bunches are notaffected by the linear part of the beam-beam kick. For thisreason, the calculated variations of the β–functions resultin the self-consistent variations of transverse bunch sizes.
Generally, spontaneous variations of β-functions at the interaction region change the lattice betatron functions along
the ring. That may result in the variations of the diffusion
coefficients and in relevant variations of the beam emittances. We neglect such variations assuming that thesevariations of the bunch emittances occur during the time intervals essentially longer than the instability growth rates.
An obvious disadvantage of the model is that due to linear
dependence of the beam-beam force on the particle offsets
the model predicts the instabilities near the parametric resonances only.
In the papers [2, 3, 4] the calculations were done assuming that the beam-beam interaction does not change the aspect ratios of the colliding bunches. That is a very strong
assumption. We report the main results of more general
study (see, e.g. in Ref.[5]), when such an assumption is not
used a priory.

2 ENVELOPE EQUATIONS
We assume head-on collisions, Gaussian distributions incolliding bunches, a short bunch length (σ s ) as comparedto the value of β-function at the interaction point (IP) andzero dispersion function at IP. Then, for small betatron particle offsets from the closed orbit (|x|  σ x and |z|  σz )the beam-beam kicks read= −kz,± z± ,
δz±kz,± =2N∓ re,γ± σz∓ (σz∓ + σx∓ )(1)

For colliding bunches with round unperturbed
cross secz/x, we rewrite
tions at IP (r0 = 1) and using r =Eqs.(9) and (10) in the following form
22z±= 1 + 2B cot µ0 w∓ − B 2 w∓
,andkx,± =2N∓ re. (2)γ± σx∓ (σz∓ + σx∓ )Here, suffixes ± mark the values referring to the electron(mark −) and the positron bunches, σ x,z are the rms bunch
width and height, N ± are the numbers of particles in thebunches, γ± = E± /mc2  1 are the relativistic factors ofparticles, re = e2 /mc2 is the classical radius of the electron; an independent variable is the path along the closed
orbit (s), so that x = dx/ds. For simplicity, we neglect thecoupling of betatron oscillations in the ring arcs. Assuming also zero dispersion and the β-function slopes at IP as
well as a linear map in the arcs without with equal betatrontunes (µ0 = 2πν0 ) and betatron functions at IP for both colliding bunches we map the betatron oscillations though IP
and through the arcs. Then, simple matrix multiplications
result in the following expressions for the self-consistenttune shifts and β-functions:cos µz±=βz± sin µz±=

cos µx+βx± sin µx±kz± βz0
sin µ0 ,2βz0 sin µ0 ,cos µ0 −kx,± βx0
sin µ0 ,2= βx0 sin µ0 ,= cos µ0 −(3)(kx,± βx )0(kz,± βz )0=.
22z±2r±2= 1 + 2B cot µ0w∓
− B2r∓

w∓r∓

2,(13)where w± = 2z± /(1 + r± ). Among others, these envelope
equations have the roots corresponding to r + = r− = 1,
when the beam-beam interaction does not change the bunch
aspect ratios. In this case, the blowup in, say, vertical direction results in a proportional increase in the horizontal
bunch size keeping the aspect ratio unaltered.

3NON-SYMMETRICAL ASPECTRATIOS

More general cases were studied solving Eqs.(12) and (13)numerically. Typical results of these calculations are shownin Figs.1 and 2. These graphs contain only the points corresponding to stable betatron oscillations. As is seen from

1E+2

(4)

(5)
(6)

We assume that parameters for all bunches are chosen in
the way that the beam-beam parameters are equal for theunperturbed bunch sizes:
B = 2πξ0 =β/β0

δx± = −kx,± x± ,(12)1E+1(7)Defining r = σx /σz andz± =βz0,βz±x± =βx0,βx±q± =1E+00 .4 0

1 + r0,
1 + r±0 .4 20 .440 .4 60 .4 8

B e tatron T une

we rewrite e.g. Eqs.(3) and (4) like follows

andcos µz± = cos µ0 − Bz∓ q∓ sin µ0 ,(8)22 2z+= 1 + 2B cot µ0 z− q− − B 2 z−q− .

(9)Figure 1: Dependence of the β–functions at the interaction
point on the betatron tune. The cross sections of the unperturbed bunches are round (r 0 = 1); ξ = 0.05; open
dots – roots from numerical solution, solid line – flip-flop
β–functions assuming unaltered bunch aspect ratios.

Similarly, we findz∓ q∓ r0− B2x2± = 1 + 2B cot µ0r∓

z∓ q∓ r0r∓

2, (10)The oscillations are stable (| cos µ| ≤ 1) provided that µ µ r000Bz± q± ≤ cot, Bz± q±.
≤ cot2r+2(11)these figures, with an increase in the unperturbed betatrontunes the flip-flop instability occurs in two regions. In thefirst, the instability does not change the aspect ratios of thebunches. So that after the blowup the initially round crosssection of the bunch remains to be round. However, forhigher betatron tunes appears a different flip-flop region,
where the self-consistent beam-beam interaction apart fromseparation of β–functions of colliding bunches may result

A spe ct R a tio

1E+1

1E+0

1E -1
0.40

0 .4 2

0 .44

0.46

0 .4 8B e tatron T u ne
Figure 2: Dependence of the bunch aspect ratios on betatron tune. The cross sections of the unperturbed bunches
are round (r 0 = 1); ξ = 0.05.

stopbands occur below parametric resonances. The nonsymmetrical solutions may correspond to separation of theβ–functions either with the conservation of the bunch aspect ratios, or the separation in the β–functions can be accompanied by the variations in the bunch aspect ratios of
the colliding bunches. The stopbands for these instabilities
are separated in the betatron tunes. Solutions with nonsymmetrical aspect ratios occur closer to the upper edge ofthe parametric beam-beam resonance stopband. This circumstance can be used to control the self-consistent aspectratios of the colliding bunches. In the case of the flip-flop
instability with the variation of the bunch aspect ratios, oneof the colliding bunches becomes flat in the horizontal direction, while its partner becomes flat in the vertical direction.
For more realistic cases, we may expect similar phenomena within the stopbands of the non-linear beam-beam resonances.

5REFERENCES[1] M.H.R. Donald and J.M. Paterson. IEEE Trans. on Nucl. Sci.,
NS-26, p.3580, 1979.
[2] K. Hirata. Phys. Rev. D 37, 1307, 1988.

in the spontaneous separation of their aspect ratios. Because of this new instability, if one of the bunches becomes
more flat in, say, the horizontal direction, then its partner
becomes flat in the vertical direction and vice versa. In ourmodel and for ordinary values of the beam-beam parameter(ξ = 0.05) such an instability occurs below the parametricresonances. We may expect that in more realistic cases,
when the beam-beam force is a non-linear function of theparticle offsets, similar splitting of beta-functions and aspect parameters will take place within the stopbands of
non-linear beam-beam resonances. Note, that the region
where the flip-flop instability changes the bunch aspect ratios has a different orientation in the tune diagram than that,
where the bunch aspect ratios remain unaltered. In the case,when aspect ratios are altered, the differences in the bunch
β–functions and of aspect ratios increase, when tune approaches the upper edge of the stopband. A decrease in thebeam-beam parameter (ξ) decreases the widths of the instability stopbands, but does not eliminate these instabilities.

4 CONCLUSION
The self-consistent beam-beam interaction may result ina spontaneous breaking of the symmetry of the colliding bunches. Within the framework of used here simplified model we have found that self-consistent envelope
equations may have multiple roots. These roots can describe both the symmetrical self-consistent variations in
the betatron functions of the colliding bunches and thenon-symmetrical ones. Symmetrical solutions exist for alltunes. The non-symmetrical solutions exist within definite stopbands of betatron tunes. For usual values of thebeam-beam parameter and for our simplified model, these[3] A. Chao, M. Furman and K. Ng, Proc. EPAC, 1988, Rome, p.
684.
[4] N.S. Dikansky, A.A. Didenko, et al, Conceptual Design
Study of the GSI Electron-Nucleon Collider. GSI Report 9707, GSI, 1997; Budker INP 97-51, Novosibirsk 1997.
[5] D.V. Pestrikov, Flip-flop Effect with Variations of Bunch Aspect Ratios, KEK Preprint 98-230, 1999.