Beam-Beam Stability in Electron-Positron Storage Rings

Bjoern S. Schmekel, Joseph T. RogersCornell University, Department of Physics, Ithaca, New York 14853, USA

Abstract

At the interaction point of a storage ring each beam is sub-ject to perturbations due to the electromagnetic field of thecounter-rotating beam. These perturbations lead to a limitof the achievable luminosity in the storage ring. We in-vestigate this limit in the framework of the strong-strongpicture. Motion is considered only in the vertical direc-tion and the beams are presumed to be one-dimensional.Based on this model [1] we try to find stability criteriafor the beam in an electron-positron storage ring takinginto account the damping of the betatron oscillations bysynchrotron radiation but neglecting the discrete nature ofthe radiation process and presuming a Gaussian distribu-tion. We analyze the instabilities by solving a linearized”Fokker-Planck equation” without the quantum excitationterm.

1 BEAM EVOLUTION

Aside of dipole induced bending the beam is subject to fo-cusing, damping by synchrotron radiation and beam-beamkicks at the interaction point. The beam-beam kick fromthe first (second) beam on the second (first) one is given by

�y01;2 =

�4�NreLx

Z 1

�1

dy sgn(y � y)Z 1

�1

dy0 2;1(y; y0) (1)

where N is the number of particles in a bunch and rethe classical radius of the electron. The distributions of thebeams 1 and 2 are normalized to unity, i.e.

Z 1

�1

dy

Z 1

�1

dy0 1;2(y; y0) = 1 (2)

and are assumed to be one-dimensional with horizontalwidth Lx. Motion is considered only in the vertical direc-tion. Using the small-angle approximation we obtain forsynchrotron radiation and for focusing, respectively

dy0Kds

= �K(s)y (3)

�y0rad = �dy

dt

p

dx

dt

dx

dt

m= �y0 p

p0� ��y0 (4)

where p0 is the momentum of the particle and p the totalmomentum change after each turn. We neglect the discrete

nature of the radiation process. The dipoles have no influ-ence on the betatron oscillations. In a damped system thephase space density increases. The equations describingthe motion of 1;2 are given by

@ 1;2

@s+dy

ds

@ 1;2

@y+dy0

ds

@ 1;2

@y0=�

L 1;2 (5)

They differ from the Fokker-Planck equations by a miss-ing quantum excitation term and from the Vlasov equationsby the fact that the phase space density is not a constantanymore. This gives us

�

L 1;2 =

@ 1;2

@s+ y0

@ 1;2

@y���

Ly0 +K(s)y

�@ 1;2

@y0�

4�NreLx

@ 1;2

@y0Æp(s)

Z 1

�1

dy

sgn(y � y)Z 1

�1

dy0 2;1(y; y0; s) (6)

and a similar equation with the first and second beam be-ing interchanged where Æp(s) denotes a period delta func-tion that has singularities at all interaction points. Becausethe system is damped an equilibrium does not exist. Wedefine the distribution 0(s) such that it satisfies

�

L 0 =

@ 0

@s+ y0

@ 0

@y���

Ly0 +K(s)y

�@ 0

@y0�

4�NreLx

@ 0

@y0Æp(s)

Z 1

�1

dy

sgn(y � y)Z 1

�1

dy0 0(y; y0; s) (7)

with being a function of s if damping is present. Wecan think of 0(s) being a ”temporary equilibrium distribu-tion” if the characteristic damping time is much longer thanthe period of betatron oscillations or perturbations. We arenot interested in finding the actual distributions. All wewant to know is whether the beam gets unstable or not.Thus, we choose a perturbative ansatz

1;2 = 0 �� 1;2 (8)

Substituting eqn. 8 into eqn. 6, subtracting eqn. 7 andneglecting the term made up of two perturbations we find

@� 1;2@s

+ y0@� 1;2@y

� @� 1;2@y0

��

Ly0 + F (y; s)

��

4�NreLx

Æp(s)@ 0

@y0

Z 1

�1

dy sgn(y � y)

�Z 1

�1

dy0� 2;1(y; y0; s) =

�

L� 1;2 (9)

where

F (y; s) = K(s)y +4�NreLx

Æp(s)

�Z 1

�1

dy sgn(y � y)Z 1

�1

dy0 0(y; y0) (10)

With the approximation

F (y; s) � F (s)y (11)

we can treat the perturbation as a part of the perturbed fo-cusing function F (s). The drawback is that this step makesthe study of phenomena like Landau damping impossible.In the next step we transform eqn. 9 to action-angle coor-dinates

y =p2�J cos� (12)

y0 = �s

2J

�

�sin�� �0

2cos�

�(13)

Only the phase of the two beams at the interaction pointis of importance for our problem, but not the particular lat-tice design. Therefore, we can set � 0 = 0. This gives us anexplicit representation of J

J =1

2y02� +

1

2

y2

�(14)

where � denotes the perturbed instead of the unperturbedbetatron function now. Forming the linear combinations

f� = � 1 �� 2 (15)

and using the envelope equation eqn. 9 can be rewrittenin action-angle coordinates as

�

Lf� =

@f�

@s+

1

�

@f�

@�� 2�J

�Lsin2 �

@f�

@J+

�

Lcos� sin�

@f�

@��p2�J sin�

4�NreLx

Æp(s)@ 0

@J

�Z 1

�1

dy sgn(y � y)Z 1

�1

dy0f�(y; y0; s) (16)

assuming that 0 = 0(J). In the following discussionwe omit the label �.

2 SOLVING THE EQUATIONS OFMOTION

The distribution for electron bunches is approximatelyGaussian.

0(J) =1p2���

e�J2

2�2 (17)

Note that the Jacobi determinant of the transformation13 is 1. Although our 0 as it is used in eqn. 7 depends ons we choose an s-independent 0. This can be done if thedamping after each turn is small and if we keep adjusting �or the beam-beam parameter �, respectively, which will beintroduced later. In the final result we simply have to con-sider � in an appropriate range. Considering the problemon a turn-by-turn basis also justifies the usage of 13 whereJ is a constant of motion. f must be periodic in �. Thus,we choose the ansatz

f(J; �; s) = Je�J2

2�2

1Xl=�1

gl(s)eil� (18)

Substituting eqn. 17 and eqn. 18 into eqn. 16 and mak-ing use of the symmetry of the integrand we obtain

@gl

@s+il

�gl �

2Nrep�

Lx �3=2�5=2Æp(s)

�1X

k=�1

gk(�i)Z 2�

0

d�

Z 2�

0

d��

Z 1

0

d �J

sin(�) sin(l�) cos(k ��) �Je��J2

2�2

� sgn(p� cos(�) �

p�J cos(��)) =

�

Lgl (19)

having averaged out the sin� cos�-term which is dueto synchrotron radiation (slow process). We evaluate theequation at J = � where we expect the perturbations to belargest since J 0(J) has a maximum at J = �. However,doing so removes all radial modes. The d �J - integrationcan be evaluated by integrating by parts. The Heavisidestep function is denoted by H0.

Z 1

0

�Je��J2

2�2 sgn(p� cos��

p�J cos ��) =

�2 sgn(cos�)� 2�2 sgn(cos ��)H0

�cos�

cos ��

�e�

cos4 �

cos4 �� (20)

Thus, the equation determining the g l’s becomes

@gl

@s+il

�gl �

1p�

4Nre�Lx

r�

�Æp(s)

1Xk=�1

gkMlk =�

Lgl (21)

where

Mlk � i

Z 2�

0

d�

Z 2�

0

d��

sgn(cos ��)H0

�cos�

cos ��

�sin� sin l� cos k ��e

�cos4 �

cos4 �� (22)

The matrix M has the following properties

Mlk =

�2iZ 3�=2

�=2

Z 3�=2

�=2

d�d�� sin� sin l� cos k ��e�

cos4 �

cos4 ��

for l + k = even and

Mlk = 0 for l + k = odd

Ml;�k =Mlk

M�l;k = �Mlk (23)

No attempt is made to solve the remaining double in-tegral analytically. Introducing a beam-beam strength pa-rameter

� � 4Nre�Lx

r��

�(24)

where �� denotes the beta function at the interactionpoint one obtains the following relation for the g l’s imme-diately before and immediately after the interaction pointby integrating through the interaction point.

gl(0+)� gl(0�) = �

�p�

1Xk=�1

Mlkgk(0�) (25)

There is no coupling among different Fourier compo-nents between collisions. In this case eqn. 21 simplifiesto

@gl

@s+

il

�(s)gl =

�

Lgl (26)

which is solved by

gl(L�) = gl(0

+)e��ilRL

01

�(s)ds = gl(0

+)e��il� (27)

3 DYNAMIC TUNE

We calculate the tune � in terms of the unperturbed tune �0which allows us to express our final result in terms of �0.This is more convenient since we usually neglect beam-beam effects when doing the lattice design. The tune shiftdue to synchrotron radiation is many orders of magnitudelower and is neglected. Eqn. 28 expresses the perturbedbetatron tune � in terms of the unperturbed tune � 0 and thedifference in the focusing structure.

cos 2�� = cos 2��0 �1

2sin 2��0

I�(s)(F (s) �K(s))ds (28)

Evaluating the integral in eqn 10 gives

1p2���

Z 1

�1

d�y sgn(y � y)Z 1

�1

dy0e��J2

2�2 =

2p2

2�p2���

�Z y

0

dye�

y4

16�2�2K1=4

�y4

16�2�2

�y =

23=4p����( 3

4)y +O(y2) (29)

where we have used the linearization eqn. 11 andexpanded the integrand keeping only the constant term.K�(x) denotes the modified Bessel function. Thus,

cos 2�� = cos 2��0 �23=4�3=2

2�( 34)� sin 2��0 (30)

4 COHERENT BEAM-BEAMINSTABILITY

We can summarize the solution of the equations of motionby introducing a matrix that acts on a coloumn vector Gwhich contains all gl’s. Therefore,

G(L) = TG(0) (31)

where

T = R

�1� �p

�M

�(32)

and R is a diagonal matrix which has entries e�2�il�

(with � = 2��) for all l 2 Z on its diagonal. Beam-beaminstability occurs if one of the eigenvalues of T has eigen-values whose absolute value is larger than 1.

5 RESULTS

In the following plots we have calculated the matrix to theindicated order for both signs and drawn a point at (� 0; �)if all eigenvalues of T have an absolute value smaller orequal 1.

6 DISCUSSION

The coarse structure of all plots is shown in fig. 4 where weplotted the region in which the dynamic tune becomes com-plex. In this region the accelerator cannot maintain a stablebeam. All plots have this basic structure in common. In-cluding resonances up to higher and higher orders the plots

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

(A)

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

(B)

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

(C)

Figure 1: Stability diagrams for � = 0:00. Resonances upto second (A), fourth (B) and sixth order (C), respectively,have been included. The horizontal axis refers to �0 andthe vertical axis refers to �.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

(A)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

(B)

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

(C)

Figure 2: Stability diagrams for � = 0:05. Resonances upto second (A), fourth (B) and sixth order (C), respectively,have been included. The horizontal axis refers to �0 andthe vertical axis refers to �.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

(A)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

(B)

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

(C)

Figure 3: Stability diagrams for � = 0:20. Resonances upto second (A), fourth (B) and sixth order (C), respectively,have been included. The horizontal axis refers to �0 andthe vertical axis refers to �.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.2 0.4 0.6 0.8 1

Figure 4: Area in which the dynamic tune is real. Thehorizontal axis refers to �0 and the vertical axis refers to �.

get more and more complicated. Small ”joints” enter thediagrams which disappear again when synchrotron radia-tion damping is increased. As a rule of thumb the beamis stable if the operating point lies in the shaded area offig. 4 and satisfies � < �. However, this rule is a bit toorestrictive.

7 ACKNOWLEDGMENTS

We would like to express our thanks to Alex Chao fromSLAC for his helpful criticism. This work was supportedby the National Science Foundation.

8 REFERENCES

[1] A. W. Chao, R. D. Ruth, ”Coherent Beam-Beam Instability inColliding-Beam Storage Rings”, Particle Accelerators, 1985,Vol. 16, pp. 201-216, Gordon and Breach