STATIC SYNCHRO-BETATRON BEAM-BEAM EFFECT
CAUSED BY CROSSING ANGLE
E.A. Perevedentsev
Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russia
Abstract
Beam-beam collision with a crossing angle results in astatic head-tail deformation of colliding bunches. The
static head-tail effect is cured by setting the collider working point well apart the integer resonances.

1INTRODUCTION

In multi-bunch colliders, interaction region schemes with acrossing angle at the IP are used to simplify (and, in general, to increase) the parasitic crossing separation. Thisis achieved at the expense of some geometric luminosity
reduction and, potentially, of synchro-betatron resonancesenhancement.
In this paper we consider one of the factors limiting theusable range of the crossing angle. Crossing the IP at anangle, the head and the tail of the bunch are kicked in theopposite directions by the partner bunch on each revolution. Without the synchrotron oscillation, this would resultin different closed orbits of the bunch head and tail. Obviously, the same is true for the case where the synchrotron
oscillation is adiabatically slow compared to the betatronoscillation.
For correct treatment of this static synchro-betatron
beam-beam effect with finite synchrotron tunes, we apply
the complete synchro-betatron mapping [1] of the bunch
variables over the arc passage and use the linearized beambeam interaction of the finite-length bunches [2]. Then wefind a periodic self-consistent solution to this strong-strong
beam-beam problem.

2 ONE-DIMENSIONAL CASE
First consider the case where x-y coupling is negligible.
Here the bunch deformation is confined to the crossing
plane, let it be the horizontal crossing. Following [2], we
outline the formalism for this one-dimensional x-case, thenextend it to two dimensions x-y in the next section.
We use the so-called “hollow beam” model. It assumes
that all particles of the bunch have equal synchrotron amplitudes and are evenly spread over the synchrotron phase,
forming a ring in the synchrotron phase space. The ringis divided into N mesh elements, each characterized by its
transverse dipole moment and its synchrotron phase. The
dipole moment of the ith mesh,  i  N , is proportional
to the transverse displacement x i of the centroid of the particles populating this mesh, times the portion N b =N of the1

 Visitor at KEKB.

bunch intensity, N b , per mesh. The betatron motion will
be described in terms of the normalized betatron variables,xi and pi , where pi is the respective momentum. Thus Nvariables will be needed to characterize synchro-betatronmotion in each bunch. They form a N -vector X , wherexi and pi are listed in the order corresponding to the mesh
number, according to its synchrotron phase.
The synchro-betatron oscillations of N elements forming a bunch are represented by the N  N matrix M ,which maps the above vector over the collider arc,

222M =CB; B =2sin   ;cos cos sin 

where denotes the outer product, B is the betatron oscillation matrix, C is the circulant matrix [1] with elements

Cij =sin N'ij ;N sin 'ij'ij =1 s (N2i + j)

2 ;
N

1  i; j  N; and  , s are the betatron and synchrotron phase advances. With N = 2m + 1, the eigenvectors and eigenvalues of matrix M exactly correspond to
the first m; : : : ; m synchro-betatron harmonics with thems ; : : : ;  ms ,  ;s  ;s = .
tunes Note that the synchrotron oscillation in the circulant matrix formalism transports the dipole moment values aroundthe circle formed by the mesh elements with fixed synchrotron phases (i.e., fixed longitudinal positions in thebunch), rather than performing a permutation of the meshes
themselves.
Expansion of the model to the case of two noninteracting
bunches is straightforward by using a N  N matrix,+M2 =10=01

4

24

M:The linearized beam-beam interaction is described by a4N  4N matrix Mbb consisting of consecutive short kicks

and drifts between interactions of macroparticles sitting ineach mesh, and assumed to be rigid Gaussian disks [3]. For
example let us consider the interaction of two bunches eachconsisting of 3 elements. Since matrix M 2 makes a transformation from the IP to the IP, the first step is the longitudinal unfolding of the bunch. Fig. 1 shows the position ofthe particles in their mesh elements before the first interaction. The next step is the interaction between particles 1,3,4and 6, which is linear in relative distance. For instance, thekick given to the first particle isp1 = 23 [(x1x6 ) + (x1x4 )] ;cos ~ = cos  2 sin  = const. In practice  is fixed,

1 4253 6Figure 1: Position of macroparticles in the synchrotron
phase space.
where x and p are the particle’s coordinate and momentum,and  is the beam-beam parameter. Next follows the freedrift and interaction of particles 2,4,6 and 1,3,5; next thedrift and interaction of the “tail” particles Nos. 2 and 5,and finally return to the IP. Generalization of the algorithm
to the case of N > is evident.
The complete one-turn matrix M t is the product of thearc matrix and the beam-beam matrix, M t
Mbb M2. Itseigenvalues and eigenvectors completely characterize thesynchro-betatron modes of the beam-beam system and canbe obtained numerically using a computer algebra system.
The crossing angle is added to our variables before thebeam-beam interaction stage and subtracted after it; it is in; ; :::; ; . To findtroduced in terms of vector X 0
the static deformation of the colliding bunches, we write
the 1-turn periodicity condition

3== (0 10 1)

Mbb  (M2  X + X0 ) X0 = X :(1)
Solving this equation for X yields initial conditions foreach mesh element in the bunch corresponding to its closedorbit.
In the following part of this section we will discuss dependence of this solution on the machine parameters. In themesh divisions in each bunch,illustrations we use N
centered at the longitudinal positions=5zi = a cos (2i1)=5; 1  i  5;

numbering is cyclic starting from the bunch phead, seez , theFig. 1. With the synchrotron amplitude avariance of z is equal to the Gaussian bunch length. We
are interested in parameters close to those of the KEK Bmm maximum.
factory, particularly, we take aThe crossing angle enters proportionally in our linearmodel. We takemrad (horizontal) in what follows.
In the following set of figures the phase-space positionsof the five mesh elements in the bunch are shown joined in
the plot, the missing side of the pentagon corresponds tothe bunch head.
The synchrotron tune s determines separation of theparticles 1 and 5 (2 and 4) with the same longitudinal coordinates in the transverse phase space, see Fig. 2. We choses : in the following examples.
The beam-beam parameter  should enter almost linearly, if the dynamic tune  would be kept fixed by adjusting the machine tune  with variation of  , according to= 2=8= 11= 0 01~

and this  -dependence is shown in Fig. 3. The  -dependent
spread of the bunch distortion in fact corresponds to enhancement of the effective transverse emittance, if we havein mind the nonlinearity of the real beam-beam kick.
The bunch length represented in this “hollow beam”
model by the amplitude a also enters almost linearly whena   , Fig. 4. But for the case a   the deformationof the bunch shape becomes nontrivial.
The betatron tune is a crucial parameter for the staticsynchro-betatron effect. Since this effect is a closed orbit
distortion different over the bunch length, we can expect
 dependence. Thus, setting the machine tune far
aaway the integer, we can have a considerable reduction inthis effect. To demonstrate this, we give the top and bottom
rows in Figs. 2–4 figures. The horizontal betatron tune
: ,actually used at KEKB is f gcot ~= 0 520.000400.0002

-0.02
0

-0.04

-0.0002-0.06-0.0004

-0.4

-0.2

00.2

0.4

0.6

-0.08

-0.4

-0.2

0.08

0.1

00.2

0.4

0.14

0.16

0.18

0.03

0.00030.02

0.00020.01

0.000100-0.01-0.0001-0.0002-0.02-0.0003-0.01 -0.005

00.0050.01 0.015

-0.03

0.12

0.2

Figure 2: Transverse phase space x 0 (mrad) vs x (mcm)
past collision. The synchrotron tune is varied,  s::: , other parameters are: amm, : . Top: f g : , bottom: f g : ; left: 
cm, right: : cm.

0 01; 0 02; 0 03
0 05
= 0 08
=07

=
=
= 70

=8= 0 5200.0002-0.01

0.0001

-0.02
-0.03

0-0.04-0.0001-0.05-0.0002

-0.06
-0.4

-0.2

00.2

0.4

0.6

-0.2

0.2

0.4

0.03

0.00030.02

0.00020.01

0.000100-0.01-0.0001-0.0002-0.00030-0.02-0.01 -0.005

00.005

0.01

0.015

-0.03

0

0.05

0.1

0.15

0.2

Figure 3: Transverse phase space x 0 (mrad) vs x (mcm)
past collision. The beam-beam parameter is varied, , other parameters are:  s: ,a:::mm. Top: f g
: , bottom: f g
: ; left: 
cm, right: : cm.

0 05; 0 025; 0 0125
8= 0 0870=07== 0 01 == 0 52=

00.0002
-0.02

0.05

0.02

0.025

0.015

-0.03

00

-0.04

-0.0001

-0.2

00.2

0.4

0.005

-0.05

-0.06
-0.4

0.01

-0.025-0.05-0.0002

0.025

0.075

-0.01
0.0001-0.075

0.6

-0.2

00.2

0.4

0.15

0.2

0.25

0.3

0.35

0.4

-0.4

-0.2

00.2

0.4

0.1
0.00030.00020.0001

0.03

0.075

0.02

0.05

0.01

0.025
00-0.0001-0.01-0.0002-0.02-0.0003-0.01 -0.005

00.005

0.01

0.015

-0.03

30.05

0.1

0.15

0.2

With the account of the local x-y coupling at the IP thehorizontal crossing angle can cause vertical bunch deformations which are critical for a flat-beam collider.
For a coupled lattice we replace the  betatron matrix
B of the previous section by the  matrix B 4 expressed
via the Twiss matrices of normal modes u, v ,4 4

B4 = R4 1 N R4 ; N =2 2

U0 ;

0 V

where the decoupling matrix R 4 is symplectic. Following
Edwards and Teng [4], we write R 4 in terms of a coupling matrix R and its symplectic conjugate R + :2 2

R4 =R+I
; R=

r1 r2r3 r4;r4r20 1 ;r3 r1 ; S =1 0p
where  = 1 det R and superscript T means matrix
transpose. We will refer to r 1 4 as coupling parameters.
R+ = SRT S =To understand the role of different coupling parameters
we can consider the case where the horizontal orbit kick isapplied at the distance l past the IP, and let L 4 be this driftmatrix. The vertical closed orbit distortion projected ontoI B 4 1 L4 1,
the IP is determined by the matrix QQ32=

0-0.0002-0.05
0TWO-DIMENSIONAL CASE

IR

0.00040.0002-0.025= 2; 4; 8= 0 01 = 0 05
= 0 52= 70



0.0006

0Figure 4: Transverse phase space x 0 (mrad) vs x (mcm)
past collision. The bunch length is varied, amm, other parameters are:  s: ,
: . Top:f g : , bottom: f g : ; left: cm, right:
: cm.

= 0 08=070.0008=()
 r1 u (r2 + r1 l) u + r2 l u cot u(r4 r3 l) v + (r2 r1 l) v  cot v :The exact solution of the self-consistent problem is available from the 2-dimensional analog of Eq. (1).
In practice, r1 and r2 are minimized by monitoring thevertical beam size at the IP, e.g. from the specific luminosity data, while r3 and r4 are the most difficult for measurement. In Fig. 5 we give an illustration of their effect on
the vertical bunch distortion at the IP caused by the horizontal crossing angle. We take values of r 3 and r4 which-0.0004-0.075-0.02 -0.018 -0.016 -0.014 -0.012 -0.01 -0.008 -0.006

-0.02

-0.01

00.01

0.02

0.03

Figure 5: Vertical phase space y 0 (mrad) vs y (mcm) pastcollision. Top: fy g: , bottom: fy g: ; left:r3: cm 1, right: r4: Other parameters are:fx g
: , s: , x;y: , xcm,
y: cm, amm.

= 0 08= 0 56= 01=1= 0 52 = 0 01
= 0 05 = 60
=07
=8are not observable from the beam size monitoring at the IP.
With the KEKB parameters we get the effect comparable tothe design vertical size. Again setting the vertical tune far
from the integer resonances results in suppression of thisunwanted effect by almost one order of magnitude.

4CONCLUSION

Beam-beam collision with a crossing angle results in astatic head-tail deformation of colliding bunches, proportional to the crossing angle and dependent on the bunch
length, synchrotron and betatron tunes, and the beam-beam
parameter. Because of residual x-y coupling at the interaction point, this deformation is not confined to the crossing plane.With the account of nonlinearity of the beambeam kick, this effect leads to an increase in the effective
emittances in the both transverse planes and potentially degrades the luminosity, especially for flat colliding beams.
The static head-tail effect is strongly reduced by setting the collider working point well apart the integer resonances, this is a way to relax the requirements on residual
coupling at the IP. In practice this means that both betatrontunes should be slightly above a half-integer, and reasonably far from the main coupling resonance.
I thank K. Oide, Y. Funakoshi and H. Koiso for useful
discussions. I am grateful to Prof. S.-i. Kurokawa and theKEKB Division for their hospitality and support during thiswork.

5REFERENCES[1] V.V. Danilov, E.A. Perevedentsev, Nucl. Instr. Meth. A 391,77 (1997).
[2] E.A. Perevedentsev and A.A. Valishev, Phys. Rev. ST-AB 4,024403 (2001).
[3] K. Hirata, “Coherent betatron oscillation modes due to beambeam interaction,” Nucl. Instrum. Methods A 269, 7 (1988).
[4] D. Edwards and L. Teng, IEEE Trans. Nucl. Sci., 20, No. 3,885 (1973).