linearbeam-beamresonances due to …particle accelerators, 1996, vol. 56, pp. 13-49 reprints...

Particle Accelerators, 1996, Vol. 56, pp. 13-49

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LINEAR BEAM-BEAM RESONANCESDUE TO COHERENT DIPOLE MOTION

KOHlI HIRATA1 and EBERHARD KEIL2

1KEK, National Laboratory for High Energy Physics, Tsukuba, 305, Japan2CERN, Geneva, Switzerland

(Received 24 January 1996; infinalform 8 May 1996)

We study the transverse motion of the barycentres of bunches in two beams that circulate inopposite direction in a storage ring, and are coupled by the beam-beam effect. The motion isdescribed by a linear system of oscillators, and represented by a matrix. We distinguish betweenperfect machines in which the bunch parameters and the arc and interaction point parametersare all equal, and machines with errors in which at least one of these conditions is not satisfied.We determine the regions in tune space where the motion is unstable, analytically for perfectmachines, and numerically for machines with errors. We identify these regions as resonancesrelated to the tunes of one of the two beams or to their sum. We establish which resonances areexcited under given conditions. We find that more resonances occur in machines with errors thanin perfect machines. By multi-particle tracking, we study the instability at finite amplitudes.

Keywords: Colliding beams; instabilities; resonances; storage rings.

1 INTRODUCTION

The beam-beam interaction is one of the major performance limitations ofcircular colliders and has been studied theoretically, experimentally and bycomputer simulation for a long time. Several effects of this interaction canbe studied by assuming that the bunches behave as rigid bodies: in the RigidGaussian Model (RGM)1,2, we assume that the density distributions are aGaussian with fixed beam radii, and apply the beam-beam forces betweenthe barycentres.

Carrying the simplification one step further, and limiting ourselves to thestudy of the small amplitude oscillation of the bunches around the closedorbit, we can linearize the beam-beam force and reduce the study of thebunch motion to the study of the symplectic one-tum matrix. This LinearRGM (LRGM)3 addresses the following main issues:

13

14 K. HIRATA and E. KEIL

Beam-Beam Modes: When the motion of the bunches is stable, the eigenvalues of the one-tum matrix yield the visible betatron tunes, i.e. betatronfrequencies divided by the revolution frequency, each corresponding toa coherent dipole oscillation mode of the system of oscillating bunches.They are used in setting up colliders such that the luminosity reaches amaximum.4- 7

Linear Beam-Beam Resonances: Whenever an eigenvalue exceeds unityin absolute value, the motion becomes unstable. The LRGM yields theconditions when the instability takes place. It explained why space chargecompensation with four beams in the 'DispositifaCollisions dans Ie Igloo'(DCI)8 did not work as originally planned.9,10 The LRGM also gave theargument for abandoning the idea of building high-luminosity B (andother) factories composed of two rings with different circumferences.1,11

Piwinski12 was the first to use the eigenvalue technique. He treateda whole bunch as a point particle, found an instability when integraland half-integral tunes are approached from below (above) for attractive(repulsive) beam-beam forces, and gave a closed expression for the thresholdof the instability. Our work can be thought of as a refinement and ageneralization of Piwinski's approach.

Chao and Keil13 demonstrated that half-integral resonances are not excitedin the case of N bunches in each beam colliding at 2N interaction points (IPs)in a symmetric machine, and also found more resonances in machines withphase errors between the interaction points. They identified these resonancesas complex resonances: a pair of complex eigenvalues become larger thanunity in absolute value. Keil continued the linear analysis and found thatresonances growing out of the half-integral tunes are a generic feature ofmachines with phase advance errors between the collision points. 14 Hecomputed the visible tunes, and demonstrated that resonances arise whentwo visible tunes meet. 15 In the study of coherent beam-beam effects inthe SSC, Chao and Furman16, 17 found half-integral resonances, which theycalled sail-shaped objects.

Keil used a multi-particle tracking code18 to compute the relation betweenthe beam-beam strength parameter ~ and the visible frequencies of thebarycentre modes (a and n). He found that the frequency difference wasabout a factor of two smaller than expected from Piwinski's theory. 12 Hirata3

showed that in the RGM the slope of the force between two bunches is afactor of two smaller than that between a bunch and a test particle, when the

LINEAR BEAM-BEAM RESONANCES 15

collisions are head-on. Hofmann and Myers19 obtained the same result. Itwas confirmed experimentally in the SLC.20 Our LRGM includes this effect.

Based on the LRGM, we constructed a computer code, BBMODE21 ,

which finds the eigenvalues and allows all possible errors. When we applyBBMODE to LEP with various errors, we find many narrow resonancesin surprisingly complicated pattems.22 A systematic survey of all theseresonances is one of the purposes of the present paper.

We assume throughout that the N equidistant bunches of the e+ beam andthe N equidistant bunches of the e- beam circulate in opposite directions ina single storage ring or in two rings of the same circumference. In a singlering, 2N equidistant interaction points are possible at most. When all of themexcept N IP are made inactive by separating the two beams, we denote thiscase as N E9 N = N IP . We distinguish between perfect machines in which thebunch parameters and the arc and interaction point parameters are all equal,and machines with errors in which at least one of these conditions is notsatisfied. Whenever we can relax this most restrictive definition of a perfectmachine, we shall explicitly state it. However, the phase advances of the twobeams in the arcs need not be the same. When we calculate growth ratesof the beam-beam modes, we do not include any damping mechanism, e.g.synchrotron radiation damping, feedback systems, and Landau damping.6

Our paper is organized as follows. In Section 2, we summarize the LinearRigid Gaussian Model and its simplest application to the 1 E9 1 = 1 case.In Section 3, we study perfect machines. Section 4 is devoted to machineswith errors. Section 5 contains the discussion of the complex resonances, andSection 6 our conclusions. Appendix A contains mathematical lemmas whichallow the reduction of the N E9 N = NIP case to the N' E9 N' = NIP case,where N' ~ N. Appendix B compares the results of the LRGM andmulti-particle tracking.

2 MACHINES WITH ONE INTERACTION POINT

Here, we introduce the notation, and describe the simplest case (1 E9 1 = 1)in detail, including all possible differences between bunches and the arcs.We denote the tune of the e± beam without the·beam-beam effect as Q;(z stands for either x, horizontal, or y, vertical). The revolution matrix forthe z coordinate is given by

(1)


The transformations (; through the arcs are block-diagona14 x 4 matrices:

(cos2nQ

U(Q) =- sin2nQ

Sin2n Q ) .cos2nQ

(2)

Here 0 is the null matrix of suitable dimension whose components are O.

The revolution matrix Mz operates on the dynamical variables

where (3)

Here, N± is the number of particles, y± is the relativistic Lorentz factor, Z± is

the Z coordinate of the barycentre, z± is its slope, and ai and fJt are nominalTwiss parameters of the e ± beam at the IP. The beam-beam kick matrix Ris defined by:

A(Jsts~)) ,I - A(S;-)

A(S)=(40~n~

~),(4)

with the coherent beam-beam parameter, S~,

~± _ reN~fJiYz~z - 2ny±~z(~x + ~y) ,

(5)

Here and in the following, I is the unit matrix of suitable dimension, r e is the

classical electron radius, and Yz is the Yokoya factor. This factor describes the

change of the visible tunes caused by the distortion of the beam distribution

by the beam-beam collision.a We have assumed that the force is attractive.

an should be noted that (5) is only phenomenologically correct for head-on collision. Mellerand Siemann23 and Yokoya and Kois024 calculated the beam-beam effect on the coherent tunesfor head-on beam-beam collisions and found that the visible tune difference is larger than whatwe expect from the LRGM. We include this effect by multiplying the focusing force with aphenomenological factor which we call Yokoya factor Yz • Typical values for flat beams withCfy«Cfx are Yx~1.33 and Yy~1.24. The factor is not known for cases with an offset between theaxes of the two beams of the order of the beam sizes or larger, and this simple treatment isno longer accurate. Beam-beam collisions with an offset also change the closed orbits of thebunches.2 To treat such cases, we should go beyond the linear analysis. In most of this paper,we shall confine ourselves to the case where the collision is head-on.


For a repulsive force, the sign of S should be changed. When a+ = a-, we

have Sz = Yz~zj2, where ~z is the usual beam-beam parameter

(6)

As is clear from (4), the beam-beam collision is represented by S~ only: any

difference in N±, fit-, a~, y±, a;-, and a; is taken into account. As long as

the collision is head-on and the directions of the betatron modes are the same

in the e+ and e- beams, the linear motions in the x and y degrees of freedom

are independent. We assume this to be the case and drop the subscripts zhereafter.

Since M is a symplectic 4 x 4 matrix, we can use a well-known technique25

to obtain the average of the eigenvalue A and its reciprocalljA:

_± A+ 1j A cos J-l+ + cos J-l-cos J-l = = ------

2 2

,....,+. + ,....,_. _ 1 ~- 1f ~ SIn J-l - 1f ~ SIn J-l ± 2'V D ,

D = [COSJ-l+ - COSJ-l- - 21fS+ sinJ-l+ + 21fS- sinJ-l-]2

+ 161f2s- S+ sin J-l+ sin J-l- .

(7)

(8)

The mapping M is stable if and only if both cos fl+ and cos fl- are real and

fall into the region between -1 and +1. The mapping is unstable, in termsof Q± and D, if:

• D > 0 and Q+ ;S integer or Q- ;S integer, then one A is real and A > 1.We call this a positive or integral resonance.

• D > 0 and Q+ ;S half-integer or Q- ;S half-integer, then one A is realand A < -1. We call this a negative or half-integral resonance.

• D < 0 and Q+ + Q- ;S integer, then there is a complex conjugate pair

of A'S with IAI > 1. We call this a complex or sum resonance.

Here, Q+ ;S integer, for example, indicates that the instability occurs when

Q+ approaches the integer from below. The instability does not necessarily

persist until Q+ reaches the integer. More than one type of resonance canoccur at the same tunes.


I1111111111111111111111HHHHHHHHHHHHHI111111111111111Il111111111111111111CC •...•.....•.••••..... HHHHHHHH HHHHH ...••.............. I II 11111 II II II III·ccccc HHHHHHHHHHHHH .•.•..•....•.••••••.• I II III II 1111 III· .. CCCCCC .••.•.•.•..... HHH HHHHH HHHH H•..•.........•.•.....• II III II II II III· cccccc HH HHHHH HHHH H••.....•••.....•...... II III II II II III· ...•.... CCCCCCC .•.••..• HHHHHHHHHHHH ...••....••.••.•••...•. I III 1111 II III· .........•. CCCCCCC •••.• HHHH HHHHHHHH I III 111111 III· ...•..•...... CCCCCCCCC. HHHHHHHHHHHH •.•••..••••..•.•....... 111111 II II III· •..•..•.••.•.... CCCCCCCCHHHHHHHHHHH ...•................... 1111 II II II III· ...•..•.••••....•.. CCCCCCHH HHH HHHH H...•......•......... , .. I III II II I I II IHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHCHHHHHHHHHHHHHHHHHHHHHHIIIIIIIIII11I· ..•............•... HHHHHHHHHHHHHHHH. CCC I I I I I I I I I I I I I...................•• HHHHHHHHHHHHHHH .. CCCCC IIIlll!I111I.•................. ; .. HHHHHHHHHHHHHH CCCCCC 111111111111· ..............•...... HH HHHHHHH HHHH H.•..... CCCCCCC III II II II III· ...•..•....•.•.....•.• HHHHHHHHHHHH H.•. ; •..... CCCCCCC ....••. III II II II III· ...............•..•... HHH HHHHH HH HH H...•......... CCCCCCCC ... III II II II III· ..•.••..•••••...•••..• HHHHHHHHHHHHH ...•............ CCCCCCCC. 111111 II III· HHH HH HHH HHHH H•..•.............. CCCCCCCCl II II I I I II· ..•................•.• HHHHHHHH HHHH H•.•••................. CCCIl II II II III

FIGURE 1 Unstable region in tune space for the 1 EB 1=1 case with g+ == 0.1 and g- == 0.01.The abscissa is 0 :s Q+ :s 1, the ordinate is 0 :s Q- :s 1. The type of eigenvalue with thelargest growth rate is indicated: H, I and C stand for half-integral, integral and complex sumresonances, respectively. The pattern repeats itself with period 1 for Q+ and Q- .

The unstable region in the (Q+, Q-)-plane is shown in Figure 1, whichdisplays a case with 8+ =j:. 8-. All unstable regions listed above canbe seen clearly. As is clear from (1), under a replacement (Q+, Q-) -*

(Q+ + 1/2, Q- ± 1/2), M becomes -M, so that A remains the samebut with opposite sign. This is the reason why, in Figure 1, (0 ::s Q+ ::s1/2,0 ::s Q- ::s 1/2) and (1/2 ::s Q+ ::s 1, 1/2 ::s Q- ::s 1) and also(0 ::s Q+ ::s 1/2, 1/2 ::s Q- ::s 1) and (1/2 ::s Q+ ::s 1,0 ::s Q- ::s 1/2) areidentical apart from the replacement H+*I, forming a chessboard pattern. For8+ == 8-, the graph would be symmetric under the reflection with respectto the line Q+ == Q-.

3 PERFECT MACIDNES

Here, we consider cases ofmachines with equally spaced bunches and equallyspaced IPs. All bunches and IPs are equal, i.e. 8+ == 8- == S. The phaseadvances 2rrv+ for the e+ beam in the arcs connecting the IPs are all thesame, and so are the phase advances 2rrv_ of the e- beam. However, v+ andv_ may be different. The eigenvalues can be found analytically.


In some cases, the problem is reducible: The 2 E9 2 = 2 case splits intotwo mutually independent and identical 1 E9 1 = 2 cases with the sameeigenvalues. We have to examine only the irreducible cases. The number ofsuch cases is not large. We prove in Appendix A that N E9 N = N (with oddN) and N E9 N = 2N are the only irreducible cases with equally spacedIPs. We begin with the 3 E9 3 = 3 and 3 E9 3 = 6 cases, because N = 3is the smallest N which contains all essential features of cases with N ~ 3.We then treat the N E9 N = N and N E9 N = 2N cases. Finally, wederive a closed expression for the threshold of the instability for the general

N E9 N = NIP case.

3.1 The 3 E9 3 = 3 Case

We use the Eulerian view26 ; instead of labelling the bunches and followingthem around the ring, we label the IPs, and call them IPI, IP2, and IP3in a clockwise manner, and we give the bunches, which collide there ata particular instant in time, the label of the IP. We define the state vectorY = (YI, Y2, ... Y6)t, where,

YI : e+ at IPIY2 : e+ at IP2Y3 : e+ at IP3

Y4 : e- at IPIYs : e- at IP2Y6 : e- at IP3 .

Let YI represent Z+ for the first e+ bunch ei at a particular instant of time.One-third of a tum later, this bunch has moved to IP2 and is representedby Y2, while the third e+ bunch ej has moved from IP3 to IPI and isnow represented by YI. The e+ bunches pass through IPI in the order1,3, 2, through IP2 in the order 2, 1, 3, and through IP3 in the order 3, 2, 1.Similarly, the e- bunches pass through IPI in the order 1, 2,3, through IP2in the order 2,3, 1, and through IP3 in the order 3, 1, 2. The correspondencebetween state vectors Y i and e± bunch numbers for three successive collisionsis shown in Table I.

The cyclic permutation of the e- bunches from (1, 2, 3) to (2, 3, 1) isdescribed by the 6 x 6 matrix P:

Ioo

(9)


TABLE I Correspondence between state vectors Yi and e± bunches for the3 E9 3 == 3 case

Y I Y2 Y3 Y4 Ys Y6

e+ e+ e+ e~ e~ e31 2 3

2 e+ e+ e+ e~ e3 e~3 1 2

3 e+ e+ e+ e3 e~ e~2 3 1

with p3 = I. Because of the Eulerian view, the 6 x 6 one-tum matrix Ml

can be written as the cube of a matrix Ml/3 for 1/3 of a tum:

(P2 ) A

Ml/3 = P UR,

(

COS 2rr v± sin 2rr v± )U±= ,

- sin 2rrv± cos 2rrv±

I-A AI-A A

R=I-A A

A I-AA I-A

A I-A

(10)

(11)

(12)

(13)

(14)

Here, diag (A, B, ... ) is a matrix which has the submatrices A, B, ... alongthe main diagonal, A = A(8) and v± = Q±/3. To bring Ml/3 into blockdiagonal form, we introduce X = exp(2rr i /3) and the matrix V

1 (I I I)V = h I Xl X*I .-v3 I X*I Xl

(15)


It is easily found that VtpV = diag(I,xI,X*I), where vt is theHermitian conjugate of V, and VVt = I holds. The first step towards blockdiagonalizing MI/3 is multiplying with diag (V, V t ) from the left and withdiag (V t , V) from the right:

Ml/3 rv diag (I, XI, X* I, I, XI, X* I)U R' ,

I-A AI-A A

R'= I-A AA I-A

A I-AA I-A

(16)

(17)

When all eigenvalues of any two matrices A and B are identical, we say thatA and B are equivalent, and write A rv B. By reordering of the basis vectors,we can bring MI/3 into block diagonal form:

Here

(18)

O)(I-A A)U_ A I - A '

° )(I - A A)X*U- A I - A '

(19)

(20)

and M x* is the complex conjugate of M x. Since the eigenvalues of a blockdiagonal matrix are equal to the set of eigenvalues of the blocks, we candiscuss the blocks one by one.

As its name implies, Man is identical with the one-turn matrix for the1 EB 1 = 1 case, (1). It is symplectic. Although we have given the exacteigenvalues earlier in (8), we start, for later convenience, the discussion ofMan with 8 « 1 where the eigenvalues of Man are:

A+ ~ exp[2ni(+v+ + 8)], A~ ~ exp[2ni(-v+ - 8)] ,

A_ ~ exp[2ni(+v_ + 8)], A~ ~ exp[2ni(-v_ - 8)] .(21)


TABLE II Mode coupling pattern of M(57r for the 3 EB 3 = 3 case. Thetype -, + and c refers to negative, positive and complex eigenvalues

Encounter Unstable region Type

A+ vs. A~ v+;:: 1/2

v+;:: 1 +

A_ vs. A~ v_ ;:: 1/2

v_;:: 1 +

A+ vs. A~ v+ + v_ ;:: integer /2 c

A_ vs. A~ c

As S grows, the eigenvalues move on the unit circle until two of themmeet. We have observed that then one of the eigenvalues A of the matrixMaJ'( becomes larger than unity in absolute value. We summarize the modecoupling pattern in Table II.

When Ais an eigenvalue of M x' A* is an eigenvalue ofM x* , and vice versa.Thus only Mx or Mx* needs to be studied. Also for the discussion of Mx'we start with S « 1 where the eigenvalues of M x are:

Al ~ exp[2Jri(+1/3 + v+ + S)] ,

A2 ~ exp[2Jri(+1/3 - v+ - S)] ,

A3 ~ exp[2Jri(-1/3 + v_ + S)] ,

A4 ~ exp[2Jri(-1/3 - v_ - S)] .

(22)

As S grows, the eigenvalues move on the unit circle until two of them meet.We have observed numerically that then one of the eigenvalues Aof the matrixMx becomes larger than unity in absolute value. All possible encounters ofthe eigenvalues and corresponding instabilities can thus be understood and areshown in Table III. The unstable regions of M~, M~Jr and MI are comparedin Figure 2. We can limit the ranges of v+ and v_ because (13) shows thatU± is invariant to a change of v± by one unit. Furthermore, a simultaneouschange v+ ---+ v+ ± 1/2 and v_ ---+ v_ ± 1/2 changes only the sign of fJbut not the absolute values of the eigenvalues, resulting in the chessboardpattern. Figure 2 and Tables II and III summarize the 3 EB 3 == 3 case.

(A)

(B)

(C)


FIGURE 2 The largest absolute value of the eigenvalues of M~ (A), M~7T (B) and M1 (C) asfunctions of (v+, v_) = (Q+ /3, Q- /3) for the 3 ED 3 = 3 case with 8 = 0.025.


TABLE III Mode coupling pattern of Mx' The type -, + and c refers tonegative, positive and complex eigenvalues for M~


Al vs. A2 V+;S 1/2V+;S 1 +

A3 vs. A4 v-;S 1/2v-;S 1 +

Al vs. A4 v+ + v-;S 1/3 C

v+ + v_ ;S4/3 C

A2 vs. A3 v+ + v_ ;S2/3 C

v+ + v_ ;SS/3 c

3.2 The 3 E9 3 = 6 Case

This case is the simplest non-trivial N E9 N = 2N case. We add three primedIPs, labelled IP~, IP~ and IP; to the IPs of the 3 E9 3 = 3 case, such thatprimed and unprimed IPs alternate. Let all bunches collide at the unprimedIPs at a particular instant of time. They will then collide at the primed IPs onecollision later. We still use the state vectors Y for the unprimed IPs. For theprimed IPs, we define new state vectors W = (WI, W2, ... W6)t , where,

WI : e+ at IP~ W4: e- at IP~

W2 : e+ at IP~ Ws: e- at IP;W3 : e+ at IP; W6: e- at IP; .

The correspondence between state vectors Wi and Yi and e± bunch numbersfor six successive collisions is shown in Table IV.

TABLE IV Correspondence between state vectors Wi and Yi and e± bunches for six successiveinstants i during a tum in the 3 EB 3 = 6 case

YI Y2 Y3 Y4 Ys Y6 WI W 2 W 3 W4 Ws W 6

1 e+ e+ e+ et e2" e3I 2 3

2 e+ e+ e+ et e2" e3I 2 3

3 e+ e+ e+ e2" e3 et3 I 2

4 e+ e+ e+ e2" e3 et3 I 2

5 e+ e+ e+ e3 et e2"2 3 I

6 e+ e+ e+ e3 et e2"2 3 I

LINEAR BEAM-BEAM RESONANCES

The W is related to the Y at the previous collision by

w = (~ ~ ) URY previous .

This W is related to the next Y by

25

(23)

(P2

Y new = 0 0) "I URW. (24)

Thus, we have

MI = Mi/3 ' MI/3 = (~ ~ ) UR (~ ~ ) UR . (25)

As before, by multiplying with diag (Y, yt) from the left and withdiag (yt, Y) from the right and by reordering the basis vectors, we havethree mutually decoupled systems:

MI/3 '" diag [M;lf' M; 1/2, M; 1/2*] • (26)

Here Man is defined by (19), and Mxl/2 is the same as that defined by (20)but X is replaced by X 1/2 = exp(Jri /3). We thus arrive at

(27)

For 8 « 1, the eigenvalues of Mxl/2 can be predicted as shown in (22) with1/3 replaced by 1/6. Table III also applies to the 3 E9 3 = 6 case if weinterchange "A1 vs. A4" and "A2 vs. A3".

Thus we conclude that, in the (v+, v_)-plane, the 3 E9 3 = 6 case hasexactly the same instability pattern as the 3 E9 3 = 3 case in Tables TI and TILWe will show in Section 3.5 that this is not an accident. In the (Q+, Q-)

plane, the resonances are twice as far apart, compared to the 3 E9 3 = 3case.

3.3 The N E9 N =N Case

For the discussion of the N E9 N = N case, we can restrict ourselves tothe case where N is odd. It follows from the family theorem in Appendix Athat the N E9 N = N case is irreducible when N is odd. When N is even,the N E9 N = N case splits into two mutually independent and identicalN /2 E9 N /2 = N cases with the same eigenvalues. The latter case will bestudied later when we discuss the N E9 N = 2N case.


As before, we define the state vector Y = (YI, ... Y2N )t , where Yi denotesthe state vector of the e+ bunch at the i-th IP and YN+i that of the e- bunchat the i-th IP (i = 1, 2, ... , N). With XN == exp(2ni IN), the matrices P in(9) and V in (15) are replaced by

0 I 0 0

0P= 0 0 0

0 0 0 0 II 0 0

I I I II XNI X~I N-I IXN

1 I X~I xtI2(N-l)

V=- XN~

I N-I I 2(N-I) (N-I)2XN XN XN

Then we have MI rv M~N' where

(28)

(29)

MI/N rv diag [m(O), m(11N), m(-IIN), ... ,m(nlN), m(-nlN),

... ,m«N - l)j(2N)), m(-(N - l)j(2N))]. (30)

Here m (n j N) is the same as M x (20), with X = exp(2n in IN), and m (0)

is identical to Man (19). Clearly, m(nlN) and m(-nlN) have the sameinstability property. The generalization of (22) to m(n/N) for 8 « 1 yields:

Al ~ exp[2ni(+nIN + v+ + 8)] ,

A2 ~ exp[2ni(+nIN - v+ - 8)] ,

A3 ~ exp[2ni(-n/N + v_ + 8)] ,

A4 ~ exp[2ni(-njN - v_ - 8)].

(31)

Note that n j N = 1/2 never happens because N is odd. Also note that we canassume 0 :::; n < N /2 without losing generality. All mode-coupling patternsfor the N EB N = N case are listed in Table V.


TABLE V Mode coupling pattern of m (n I N) for the N ffi N = Nease.Note that v± = Q±IN (mod 1). The type -, + and c refers to the negative,positive and complex eigenvalues for m (nI N)N. We have 0 :s n < N 12.The case n = 0 corresponds to MUIr


Al vs. A2 V+;S 1/2

V+;S 1 +

A3 vs. A4 v_ ;S 1/2

v-;S 1 +

Al vs. A4 v+ + v-;S (N - 2n)IN c

v+ + v_ ;S2(N - n)IN c

A2 vs. A3 v+ + v_ ;S2n1N c

v+ + v-;S (N + 2n)IN c

3.4 The N ED N =2N Case

27

In the general N EB N == 2N case, N can either be odd or even. Both casesare irreducible. By the same transformations as before, using diag (Y, yt)

with Y defined by (29) and X == exp(21l'i fN) as before, we get

Ml == Ml~2N)'

where

Ml/(2N) ~ diag [m(O), m(lf(2N», m(-If(2N)),

... ,m(nf(2N», m(-nf(2N»,

(32)

(33)

... ,m((N - 1)f(4N», m(-(N - 1)f(4N»] (N odd)

Ml/(2N) ~ diag [m(O), m(lf(2N», m(-If(2N)),

... ,m(nf(2N», m(-nf(2N», (34)

... ,m((N - 2)f(4N», m(-(N - 2)f(4N», m(lf4)] (N even)


TABLE VI Mode-coupling pattern of m(n/(2N)). Note that v± == Q±/(2N)( mod 1). For the N EB N == 2N case, the type -, + and c refers to the negative,positive and complex eigenvalues for m (n / (2N) )2N. We have 0 :s n :s N /2


At vs. A2 v+;:: 1/2 +v+;:: 1 +

A3 vs. ).,4 v_ ;:: 1/2 +v_;:: 1 +

At vs. ).,4 v++v_;::(N-n)/N c

v++v_;::(2N-n)/N c

).,2 vs. ).,3 v+ + v_ ;::n/N c

v++v_;::(N+n)/N c

For 8 « 1, the eigenvalues of m (n / (2N)) are

At ~ exp[2ni(+n/(2N) + v+ + 8)]

A2 ~ exp[2ni(+n/(2N) - v+ - S)]

A3 ~ exp[2ni(-n/(2N) + v_ + S)]

A4 ~ exp[2ni(-n/(2N) - v_ - S)]

(35)

Here 0 ::; n ::; N /2 is assumed and the n = 0 case corresponds to Man. Thecase n = N /2 occurs only when N is even. Table VI shows the types of theunstable regions, and Figure 3 the schematic instability lines.

3.5 Equivalence of N ED N = Nand N ED N = 2N Cases for Odd N

So far, we have examined the cases N E9 N = N for odd Nand N E9 N = 2N

for arbitrary N, and we show in the Appendix A that they are the onlyirreducible cases for equally spaced bunches and interaction points. We nowshow that the N E9 N = N and N E9 N = 2N cases are equivalent when Nis odd. We observe that there are identical terms for even n in Equations (30)and (33). For odd n, we use mew) = -mew ± 1/2), and change the typicalterm in (33) as follows:

m(2:) = -m (2: +~) = -m (- N2~n). (36)


FIGURE 3 The edges of the unstable region in (v+, v_) plane for 0 :::: v± :::: 1. For theN EB N = 2N case, these graphs can be taken as graphs in the (Q+, Q-)-plane for 0 :::: Q±(mod 2N):::: 2N.FortheN EB N = NcasewithoddN,theycanbeusedinthe(Q+, Q-)-planefor 0 :::: Q± (mod N) :::: N. N is indicated in each graph.

We then notice that the modified terms in (33) are identical to the remainingterms in (30), apart from the sign which is irrelevant for the stability. Hence,we have demonstrated that, apart from the sign, the matrices

Ml/N(N E9 N == N) r-v Ml/(2N)(N E9 N == 2N), (37)

are equivalent, i.e. have the same absolute eigenvalues when N is odd.Therefore, the graphs in Figure 3 can be used both for N E9 N == N caseswith odd Nand N E9 N == 2N cases.

3.6 The N E9 N = N IP Case for Arbitrary N IP

We now consider the case of equally spaced bunches and IPs for arbitraryN E9 N == N IP , such that N IP is an integral fraction of 2N. The algorithmfor reducing N is as follows:


1. calculate the number of families NF = gcd(N,2N/NIP), (see Appendix A),

2. reduce the N E9 N = N IP case to the N / N F E9 N / N F = N IP case,arriving at either N' E9 N' = N' (N' odd) or at N' E9 N' = 2N' forN' = N / N F , which are both irreducible,

3. find a graph in Figure 3 with index N', which shows the unstable regionin the (Q+, Q-)-plane for 0 ~ Q± (mod NIP ) ~ NIP.

Let us consider cases with N = 6 and all possible values of N IP , startingat the highest value NIP = 12, and taking all values which are divisors of12, as shown in Table VII. For example, we find the instability pattern of the6 E9 6 = 3 case by looking at the index 3 in Figure 3 where both axes arefrom 0 to 3 in Q±.

TABLE VII Table of all possible cases for 6 bunches in each beam

N E9 N == NIP S N p N' reduced case

6 E9 6 == 12 1 1 6 irreducible

6E96==6 2 2 3 3E93==6

6E96==4 3 3 2 2E92==4

6E96==3 4 2 3 3E93==3

6E96==2 6 6 1 1E91==2

6E96==1 12 6 1 1E91==1

3.7 Instability Threshold for Equal Thnes

In all N E9 N = NIP cases, where v+ = v_ == v holds in addition to8+ = 8- == 8, we can analytically calculate the threshold of the instability,Le. the minimum value of 8 that gives the instability.

The instability threshold San for the Man block, (19), is well known.Applying a similarity transformation, we can reduce Man to blockwisediagonal form:

MUIr "-' TU(v)TTR(S, S)T "-' U(v) (~ I~2A)

rv (Uo(V) 0 )U(v)(I - 2A)

(38)


where U is defined by (2) and T is the symplectic 4 x 4 matrix

1 (IT=,.j2 I

31

(39)

which satisfies T 2 == I. The upper half of (38) corresponds to the so-calleda mode whose tune is not shifted by the beam-beam interaction. The lowerhalf is the Jr mode. Its perturbed tune Vn is

cos 2Jrvn == cos 2Jrv - 4Jr 8 sin 2Jrv, (40)

(41)

which can also be derived from (8). For 8 « 1, Vn ~ v + 28. The Jr modebecomes unstable if and only if cos 2Jrvn becomes ±1. Solving (40) withthis condition yields for the instability threshold

~ _ cos 2Jr V =f 1uan - 4Jr sin 2Jr v .

The 8an is shown in the graph in Figure 4, labelled 1.To find the threshold 8x of the M x block, (20), for arbitrary X, explicit

expressions for the eigenvalues are not needed, because we know fromTables V and VI that the instability develops if and only if an eigenvalueAbecomes ±1. The eigenvalue equation for M x yields:

~ (1 - 2AX cos 2Jrv + A2 X2)(X 2 + A2- 2AX cos 2Jrv)

b == (4Jr AX sin 2Jrv) [4AX cos 2Jrv - (A2 + 1)(X 2 + 1)] ,

Hence, we find the threshold for arbitrary X by substituting A == ±1:

cos 2Jr V =f cos 2JrW8 x == -------

4Jr sin 2Jrv

(42)

(43)

Here X == exp(2Jriw). When w == 0, this equation is identical to (41).In order to obtain the instability threshold S for the one-tum map Ml of

the N EB N == NIP case we first find the irreducible N' EB N' == NIP case,cf. Section 3.6. Then we evaluate (43) for all values of w appearing inEquations (33) and (34), and obtain the smallest positive Sx which is 8.Since we know the branch which leads to the lowest value of 8, we canavoid looking for the minimum Sx' and find:

32

0.1

0.08

0.06

0.04

0.02

0.04

K. HIRATA and E. KEIL

0.1

0.08

0.06

0.04

0.02

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

0.1

0.08

0.06

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

FIGURE 4 The threshold value of 8 for N EB N = 2N cases as a function of v. The N isindicated in the graph. Each graph gives the 8 for the case given in Figure 3 with the sameindex. It is thus also usable for general cases if N in the graph is interpreted as N' = N / N F

and the horizontal axis as Q (modulo NIP) from 0 to NIP.

cos {2rrQ/NIP} - cos{2rr([Q + l])/(NIP )}8=

4rr sin {2rr Q/NIP}(44)

Here we use the tune Q == NIP v, and [a] is the largest positive integer thatdoes not exceed a. The value of 8 given by (44) is shown as a function ofv in Figure 4 with 1 ~ N ~ 6 as a parameter. The graphs are valid both forN' EB N' = N' with odd N' and N' EB N' = 2N' cases. The horizontalaxes can also be taken as 0 ~ Q (mod NIP ) ~ NIP.


Piwinski's result12 applies to the N E9 N = 2N case and looks similarbut is different even in this case in two important respects: (i) Piwinski findsresonances at half-integral values of Q which do not exist, (ii) a factor of twois caused by the ratio of 8 and ~. Our result applies to all N E9 N = N IP

cases.It is interesting to note that the threshold 8 can be obtained by requiring

that the rr mode eigenvalues of Ml become ±1, although it is not always therr mode which causes the instability. (The eigenvalue can pass ±1 along theunit circle without causing instability.) This is so because (42) is symmetricin X and A, and therefore the eigenvalue An of Man becomes ± exp(2rriw)when the eigenvalue Aw of Mx becomes ±1. Thus, (43) can also be derivedby putting cos 2rr Vn = ± cos w into (40).

3.8 Summary for Perfect Machines

We have studied the coherent beam-beam effects in the framework ofthe LRGM where we consider only the linear focusing force between thebarycentres of two bunches colliding head-on. The dominant effect in thiscase is a change of focusing, parametrized by the beam-beam strengthparameter 8. We study the stability of the motion of the barycentres usingthe eigenvalues of the one-tum map. We call the case of N electron bunchescolliding with N positron bunches in N IP interaction points N E9 N = NIP,

and give closed analytical solutions for it.In Section 3.6, we give the algorithm for finding the irreducible N' E9 N' =

N IP case for the arbitrary N E9 N = N IP case. All irreducible cases are eitherN E9 N = N with odd N or N E9 N = 2N. Furthermore, these two casesshow exactly the same instability patterns in the (Q+, Q-)-plane. The periodof the resonances in Q±.is N for the N E9 N = N case with odd N, and 2Nfor the N E9 N = 2N case with odd N, and the (Q+, Q-)-plane is filled ina chess board pattern. In all cases with even N, the period in Q± is N, andthe resonance pattern repeats itself in the (Q+, Q-)-plane every N units oftune. For all cases, the edges of the resonances in tune space can be found inFigure 3. The case with equal tunes, Q+ = Q-, corresponds to the diagonalfrom lower left to upper right, the threshold is given by (44) and is found inFigure 4.

4 MACHINES WITH ERRORS

We have neglected up to now the errors in real machines which occur therefor several reasons:

34 K. HIRATA and E.KEIL

1. The currents of the bunches in the two beams are not exactly equal.2. The betatron functions a and fJ differ from IP to IP and between the two

beams.3. The phase advances J.L between IPs differ from arc to arc and between the

two beams.4. Electrons and positrons have different energies at all IPs because of

asymmetries of the RF accelerations between the IPs, either by designor by errors in the RF system.27

5. The emittances of the two beams are different.

In this section, we present the results of numerical computations of theconsequences of these errors, which cause N E9 N = 2N machines witherrors to behave to a large extent like 1 E9 1 = 1 machines. In particular,the half-integral resonances appear, and the complex sum resonances occurwhen the sum of the tunes in the two beams is just below an integer. We havealready discussed errors in 1 E9 1 = 1 machines in Section 2.

4.1 1 ED 1 = 2 Machines with Errors

We simultaneously put errors on the phase advances vt = Q±/2 + 8v±r,bunch currents Ii± = I + 8I±r, and fJ-functions fJ~ = fJ~ + 8fJ;-r,fJ~ = fJ~ + 8fJ;r, where the r's are all independent random Gaussianvariables with zero average and unit variance. In Figure 5 we show theaverage growth rates in the (Q+, Q-)-plane for ten sets of random errors.The excitation of the error-driven resonances depends strongly on the randomerrors, while the resonances already present in the perfect machine remainabout the same. In contrast to the perfect 1 E9 1 = 2 machine, the errorscause the following error-driven resonances: (i) half-integral resonanceswhen either Q+ or Q- is equal to an integer and one-half, (ii) complex sumresonances when the sum of the tunes (Q+ + Q-) is just below an integer.These resonances are already present in perfect 1 E9 1 = 1 machines.Figure 5 shows that the histogram channels just below the integral andhalf-integral tunes are filled with the integral and half-integral resonances.Hence, the upper edge of these resonances coincides with the exact integral orhalf-integral tunes within the accuracy of the histograms. However, the upperedge of the error-driven sum resonances is one channel or more below theintegral value of the sum (Q+ + Q-). We will come across other examplesof this shifting of error-driven sum resonances later, and believe that it is


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FIGURE 5 .Vertical growth rate for the 1 EB 1 = 2 case with random errors on phaseadvances 8v:, bunch currents 8J±/ J±, ,8-functions at the IP 8,8;- /,8;- and 8,8; /,8;, averagedover 10 sets of random errors. The standard deviations are 8v: = 0.16, 8Ji±/ Ji± = 0.2, and8,8;- / fJ;- = 8,8;/,8; = 0.2. The abscissa and ordinate are the tunes of the two beams in theintervals 0 < Q+ < 2 and 0 < Q- < 1, respectively. The nominal beam-beam parameter is8 = 0.015. The resonances have period 2 in Q±. The left and right half fill the (Q+, Q-)-planein a chessboard pattern (cf. Sections 2 and 3.5). The value of the growth rate is indicated by thecode at the bottom.

a generic feature. Energy differences of the two beams at the IPs do notproduce qualitatively new features.

4.2 2 ED 2 = 4 Machines with Errors

In practice, 2 EB 2 = 4 machines with errors are important since theyrepresent the case ofTRISTAN. The 4 EB 4 = 4 and 8 EB 8 = 4 cases ofLEPcan be reduced to it when we neglect the beam-beam interactions in everysecond IP where the beams are vertically separated and at the centres of all


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FIGURE 6 Vertical growth rate for the 2 EB 2 = 4 case with random errors on phaseadvances 8v;=, bunch currents 8Ii± / Ii±, ,8-functions at the IP 8,8;- /,8;- and 8,8i /,8i, averaged

over 10 sets of random errors. The standard deviations are 8v;= = 0.16, 8Ii± /Ii± = 0.2, and8,8;- /,8;- = 8,8i /,8i = 0.2. The abscissa and ordinate are the tunes of the two beams in theintervals 0 < Q+ < 2 and 0 < Q- < 2, respectively. The nominal beam-beam parameter isS = 0.015. The resonances have period 2 in Q±. The value of the growth rate is indicated bythe code at the bottom.

eight arcs where the beams are horizontally separated. We have studied thiscase numerically, including random errors on the phases v~ == Q± j4+8v±r,on the bunch currents Ii± == J± + 8J±r, and on the f3-functions at the IPs

f3:t == rff- + 8fJ;-r and fJ~ == fJi + 8fJir , where the r's are all independentrandom Gaussian variables with zero average and unit variance. In Figure 6we show the average growth rate over ten different seeds of the randomnumbers. We observe instability when either Q+ or Q- are just below ahalf-integer or an integer, and when 'the sum (Q+ + Q-) is below an integer.We should compare Figure 6 with the lower half of the N == 2 sketch inFigure 3 for the perfect 2 Ee 2 == 4 case. The number of sum resonances is


now twice as large, and the number of resonances in Q+ or Q- is four timesas large. The increased number of resonances divides the (Q+, Q-)-planeinto much smaller pieces, and reduces the fraction not covered by resonancesfrom 0.696 to 0.582 ± 0.045.

4.3 Machines with N ~ 3

Further numerical studies with random errors on the bunch currents, phaseadvances and fJ-functions at the IP, in both N E9 N = N for N = 3 andN E9 N = 2N cases with N = 3,4, show that resonances occur whenQ± (mod 1) ~ 1/2, Q± (mod 1) ~ 1, and (Q+ + Q-) (mod 1) ~ 1, i.e.at the same tunes where resonances occur in the 1 E9 1 = 1 case. Figure 7shows the 3 E9 3 = 3 case as an example. Compared to the perfect N E9 N = N

case with odd N, the number of resonances in Q± increases by a factor ofN, and the number of sum resonances remains the same. Compared to theperfect N E9 N = 2N case with any N, the number of resonances in Q±increases by a factor of 2N, and the number of sum resonances doubles. Theperiodicity of the resonances in the (Q+, Q-)-plane is the same as in perfectmachines, dicussed in Section 3.8.

5 DISCUSSION OF COMPLEX RESONANCES

In this section, we first calculate analytically the complex resonances forthe 1 E9 1 = 2 case which are excited by the differences of the phaseadvances in the arcs. Next we discuss LEP, where the separation in half ofthe interaction points is not so large that the beam-beam tune shifts can beneglected completely.

5.1 Complex Resonances in Asymmetric 1 E9 1 = 2 Machines

Chao and Keil l3 studied the case ofa conventional single-ring machine wherethe phase advances of the e+ and the e- beams are identical in the same arc,

vi = VI == Q/2 + 8 and vi = vi == Q/2 - 8, with 8 # 0, and where all8t2 = 8 are identical. We use this case for a demonstration of the behaviourof the complex resonances. The one-tum matrix Ml is:

Ml = U(Q/2+8, Q/2-8)R(8, 8)U(Q/2-8, Q/2+8)R(8, 8) . (45)

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FIGURE 7 Vertical growth rate for the 3 E9 3 == 3 case with random errors on phaseadvances 8vi±, bunch currents 8Ii±/Ii±, ,8-functions at the IP 8,8; /,8; and 8,8i /,8i, averagedover 10 sets of random errors. The standard deviations are 8v~ == 0.16, 8/.± /I.± == 0.2, and8,8; /,8; == 8,8i / fJi == 0.2. The abscissa and ordinate are th~ tunes of the \wo

l

beams in theintervals 0 < Q+ < 3 and 0 < Q- < 1.5, respectively. The nominal beam-beam parameter isS == 0.015. The resonances have period 3 in Q±. The left and right half fill the (Q+, Q-)-planein a chessboard pattern (cf. Sections 2 and 3.5). The value of the growth rate is indicated by thecode at the bottom.

Here Uis defined in (2) and R in (4). The eigenvalues can be obtained from

A+A- 1

cosji, = 2 = cos2nQ - a sin2nQ + (a sinnQcosno)2 ±.Jij,(46)

D = (2a cos no sin n Q)2 [(2 cos n Q - a sin n Q)2 - (a sin n Q sin no)2] ,

(47)


o. 05 ••••••••••••••••· .· .· .0.04 .:::::::::::::::::::: ••

39

0.03

0.02

0.01

0.4 0.42 0.44 0.46 0.48 0.5

FIGURE 8 Half-integral stopband in a 1 Eel = 2machinewithphaseadvances2n(Q/2+0.l)and 2n(Q/2 - 0.1). The abscissa is the tune Q, the ordinate is S.

where a = 41l' S. The complex resonance with D < 0 can only occur whenthe phases in the two arcs are different, Le. 8 =j=. 0 holds. It is centred wherethe first term in the square bracket of D vanishes, at S = (1 /21l') cot 1l' Q,

i.e. at Q < 1/2 for attractive beam-beam forces with a > O. Its width in theS direction is determined by the second term in the square bracket for D:

cot 1l' Q cot 1l' Q------ < S < ---------.,...21l' (1 + Isin 1l' 81) 21l' (1 - Isin 1l' 81)

(48)

The region of instability in the (Q, S)-plane is shown in Figure 8. Forsmall S, the instability occupies a small range of Q just below Q = 0.5.With increasing S, the instability becomes wider and shifts towards smallervalues of Q. For a fixed value of Q < 1/2, the instability occupies a rangeof S values, with stability above and below this range.

Another simple case is that of DA<l>NE28,29 which has two rings, eachconsisting of two rather different arcs with phase advances 21l' (Q/2 + 8)and 2TC (Q /2 - 8), respectively, and collisions in two interaction points. Therevolution matrix is different from (45):

M = U(Q/2 +8, Q/2 +8)R(S, S)U(Q/2 - 8, Q/2 - 8)R(S, S) . (49)


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(B)

FIGURE 9 (A) Tune space (0 < Q+ < 2, 0 < Q- < 1). Parameters: 8 = 0.05,vt = Q±/2 + 0.1, v~ = Q±/2 - 0.1, corresponding to the conventional 1 EB 1 = 2 type. (B)Tune space (0 < Q+ < 2,0 < Q- < 1). Parameters: '8 = 0.05, vi = v; = Q+ /2 + 0.1,v~ = vi = Q- /2 - 0.1, corresponding to the DA<I>NE 1 EB 1 = 2 type.


The real parts of the eigenvalues are:

cos jla = cos 2rr Q ,

cos jlJr = cos 2rr Q - 2a sin 2rr Q + a2 (cos 4rrd - cos 2rr Q) .

41

(50)

Hence, the tune ofthe a-mode does not move, cos jlJr is real, and the complexresonance does not occur.

In Figure 9 we show the instability region in the tune space for bothcases. When moving along the diagonal line (v+ = v_) in the left halvesof Figures 9A and 9B, instability occurs at half-integers. It is of complextype for the conventional case (45), and of half-integral type for DA<I>NE(49). If the phase advances in the arcs are different in the 1 E9 1 = 2 case,we can predict the existence but not the type of the instability originating athalf-integral tunes.

Chao and Furman16,17 used a tracking method for the beam-beam modes in

the SSC where the interaction points are arranged in two different 'clusters'.They found half-integral resonances of a characteristic sail shape very similarto our Figure 8. This result is not surprising, given the fact that the two halvesof the SSC are different.

5.2 The LEP Case

In LEP, the two beams are vertically separated at the odd-numbered pits byabout 3 mID. The beam radii there are ax ~ 0.57 mm and ay ~ 50 JLm. Thisseparation is not quite enough to make the beam-beam focusing completelynegligible. In both even- and odd-numbered pits in LEP, the ratio fix / fiy » 1.Therefore, the ratio 8 x / 8 y » 1 in the odd-numbered pits where the beamsare separated, while in the even-numbered pits where the beams collidehead-on, the ratio 8 x / 8 y ~ 1. Hence, the effects of the residual beam-beamtune shifts in the odd-numbered pits are stronger in the horizontal plane. Whenwe apply our formalism to LEP, we are confronted with two difficulties:

1. The beam-beam force in the odd-numbered pits modifies the closed orbitsof all bunches in both beams around the whole circumference of LEP.2

We overcome this difficulty by assuming that the closed orbit in theeven-numbered pits, where the head-on beam-beam collisions occur, iscorrected locally by fine adjustments ofthe vertical electrostatic separatorsthere, as is actually the case to optimize the luminosity.


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FIGURE 10 Horizontal growth rate for the 4 E9 4 == 8 LEP case with vertical separation inevery second IP and random errors on the phase advances OlJ~ == 0.01, averaged over 10 setsof random errors. The abscissa and ordinate are the horizontal tunes of the two beams in theintervals 94.25 < Q; < 94.5. The nominal beam-beam parameter is 8 == 0.015. The value ofthe growth rate is indicated by the code at the bottom of the graph, in steps of 0.002.

2. The Yokoya factor24 is not known for collisions with offsets. Since itis typically 4/3 for head-on collisions, and may tend towards unity forseparated beams, we leave it at the typical value. The difference is smallenough to be ignored.

We studied this case numerically with phase advance errors with standarddeviations 8vx = 8vy = 0.01 in the arcs, and observed stopbands. Theaverage growth rate obtained for ten seeds of the phase errors are shownin Figure 10 for the horizontal tune range 94.25 < Q; < 94.5 in the twobeams which was used in actual LEP operation in 1992. Two systems ofcomplex instabilities cross over when the two horizontal tunes are equal.Between seeds there are large variations in the widths and growth rates ofthese stopbands. However, their positions in the tune plane are always thesame, which makes it possible to display a meaningful average.


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FIGURE 11 Horizontal growth rate for the 4 E9 4 = 8 LEP case with vertical separationin every second IP and random errors on the phase advances 8v;- = 0.01, averaged over 10sets of random errors. The abscissa is the horizontal tune of the two beams in the interval94.25 < Q; = Q; < 94.5. The ordinate is the bunch current in the interval 0 :::s I :::s 0.0005 A.The beam-beam parameter is S = 0.015 at the maximum I. The value of the growth rate isindicated by the code at the bottom.

Figure 11 shows these stopbands as a function of the bunch current Ifor the case with equal horizontal tunes in the two beams. The stopbandsstart at the half-integral tune Q~ = 94.5 at small bunch current I, andmove towards smaller fractional tunes as I increases. This behaviour isvery similar to Figure 8 and the sse results.16,17 It is very plausible thatcoherent horizontal oscillations, which are often observed in LEP when thetwo beams are colliding in the even-numbered pits, are indeed related to thesestopbands.3o

6 CONCLUSIONS

We studied the coherent beam-beam effects in storage rings in the frameworkof the linear rigid Gaussian model where we consider only the dipole


forces between the barycentres of two bunches, assumed to be rigid bodiescolliding head-on. The dominant effect in this case is a change of focusing,parametrized by the beam-beam strength parameter S. We study thestability of the motion of the barycentres in linear approximation, using theeigenvalues of the one-turn map. We call the case of N equidistant electronbunches colliding with N equidistant positron bunches at N IP equidistant

interaction points N E9 N == NIP. In Section 3.6, we give the procedurewhich reduces the general case to its associated irreducible case. Figure 3shows schematic diagrams ofthe resonances in perfect machines up to N == 6,while Figure 4 shows and (44) gives a closed expression for the threshold interms of the coherent beam-beam parameter S for the case of equal tunes inthe two beams.

We find that errors in the machine, in particular in the bunch currents, the,B-functions at the IPs and the phase advances through the arcs, which maybe caused by the design or by inevitable errors in the construction, increasethe number of beam-beam resonances. In machines with errors, integralresonances occur when one of the tunes Q± (mod 1) < 1, half-integralresonances occur when one of the tunes Q± (mod 1) < 1/2, and complexresonances occur when the sum of the tunes (Q+ + Q-) (mod 1) < 1. Theschematic resonance diagram for N == 1 in Figure 3 shows the resonancesin machines with errors for all N. Figures 1,5-7, show examples of these

resonances in the cases 1 E9 1 == 1, 1 E9 1 == 2, 2 E9 2 == 4, and 3 E9 3 == 3,respectively. The growth rates displayed correspond to e-folding times of afew turns, making it rather unlikely that the resonances can be cured by afeedback system at present or in the near future.

APPENDIX A BEAM-BEAM FAMILIES

In a machine of type N E9 N == NIP, NIP is at most 2N. When some ofthese IPs are inactivated by separators, for example, it might happen that theone-turn matrix Ml can be blockwise diagonalized by reordering the basisvectors. When this happens, we call this case reducible. If the one-tum matrixcannot be blockwise diagonalized by any permutation of basis vectors, wecall this case irreducible. The reducible one-tum matrix can be reduced toNF, the number of families, irreducible diagonal blocks. When Np == 1, thecase is irreducible.

Each block of bunches, corresponding to an irreducible diagonal block,is called a (beam-beam) family. Bunches belonging to different families do


not interact with each other even indirectly: if we kick a bunch, the bunchesbelonging to the same family of the kicked bunch are eventually affected bythe kick but those belonging to other families are not affected. Conversely,bunches belonging to different families do not collide with each other inany· active IP. In this appendix, we show how to calculate N F . A family iscomposed of one or more e+ bunches and the same number of e- bunches.We consider the e+ bunches only because e- bunches can be treated in thesame manner.

We label IPs and bunches in a different way from the main text. Each beamis composed of N equidistant bunches. The e+ beam rotates clockwise andthe e- counterclockwise in the ring. Possible IPs are labelled clockwise asIP i, 0 :::; i :::; 2N - 1. Bunches are called clockwise et ==, 0 :::; j :::; N - 1.

The chronological step is numbered as tk, k == ... , -2,0,2, .... At to,et and ej sit at IP2j. It is convenient to define these numbers in a cyclic

manner: et is identical with et+nN' IPi with IPi+2nN and tk with tk+2nN,

where n is any integer. The et bunch comes to IP i at tk as

i( IP) == 2j( bunch) ± k( time) mod 2N. (51)

Two .bunches et and et are in the same family when et can influence etthrough a beam-beam interaction. We write

If only one IP, IPo, is active, each e+ bunch forms a family so that N F is N.Let us first consider the case where two IPs ( IP0, IPs) are active. We can

assume that 1 :::; s :::; N without loss of generality. From (51), eo collideswith et at IPs at t-s and with et at IPo at to. Repetition of this processyields

By repeating the same argument, we finally arrive at

e+ rv e+o ns'

for any value of the integer n. Thus, the collisions at IP0 and IPs cause allet with j == 0, s, 2s, ... to be in one family. It follows that the number offamilies is


NF == gcd(N, s), (52)

where gcd stands for the greatest common divisor. Each family containsN / N F bunches in each beam. In particular, N F == 1 when s == 1 or whenN is a prime number and N F == N when s == N (two diametrically oppositeIPs). The family due to two diametrically opposite IPs (i.e. IPo and IP N) isidentical with the family due to IP0 (or IPN ) alone.

A simple consequence is that the family due to IPs of ( IP0, IP s, IP N ) isidentical with the family due to (IPo, IPs). The proof is as follows: Considera family due to ( IP0, IP s ). This family might interact with other families(so that families will merge) by introducing a new IP. By IPN, however, thismerging of families does not happen because IP N defines the same familyas IPo. Hence, we need not consider the diametrically opposite IP. Anothersimple consequence is that the family due to ( IP0, IPs, IP2s, . .. , IP K) isthe same as the family due to ( IP0, IP s ). The equidistant (K + 1) IPs definethe same family so that the number of families N F is also N F == gcd(N, s).

Let us now discuss machines with equidistant periodic IPs, in whichN IP must be a divisor of 2N. We can apply the discussion above usings == 2N/ NIP. All possibilities for N F == gcd(N, s) are listed in Table VIII.

For LEP with complete separation at the odd pits, the IPs are (IPo, IP2,IP4, IP6) and N == 4. Thus NF == gcd(4, 2) == 2. For Furman's case16,17N == 26 and active IPs are (IPo, IP4, IP26, IP30). The IPo and IP26 are adiametrically opposite pair and so are IP4and IP30. Thus the family structureis the same as that due to (IPo, IP4), and we have N F == gcd(26, 4) == 2.More general and complicated cases can be discussed in the same manner.

The discussion given here applies to machines with errors as well aswithout errors because it depends only on the geometrical configuration ofthe collisions.

TABLE VIII Number of families N p for perfectly periodic N ED N = NIP cases, where m isa divisor of N (if any) but not 1, 2 or N

N p

when

N

always

2

N

always

m

N/mNodd

m

2N/mN even

N

1

Nodd

N

2

N even

2N

always


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FIGURE 12 Vertical growth rate from the eigenvalue analysis. The abscissa is the bunchpopulation in the interval 0 < N < 5 X 1011. The maximum N corresponds to 8 == 0.027. Theordinate is the tune in the interval 0.4 < Q~ < 0.5. The phase advance error is 8 == ±0.16. Thevalue of the growth rate per tum is indicated by the code at the bottom.

APPENDIX B COMPARISON WITH MULTI-PARTICLETRACKING

The eigenvalue analysis in the LRGM does not allow us to state what happenswhen the instability has grown to amplitudes comparable to the r.m.s. beamradius or larger, where the beam-beam tune shifts drop in proportion tothe square of the separation. We have therefore compared the results ofthe eigenvalue analysis of BBMODE for the half-integral resonance in the1 E9 1 = 2 case with the results of the multi-particle tracking program BB.18

Figure 12 shows the growth rate obtained as a function of the tune Q andthe bunch current I, and the characteristic sail shape of the stopband. Whenthe tune is somewhat below the half-integer and the bunch current varies, theinstability occurs for a range of the bunch current, with stability above andbelow that range.


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FIGURE 13 Absolute value of the vertical beam separation in units of O"y, averaged over1000 turns, from multi-particle simulation. The abscissa is the bunch population in the intervalo < N < 5 X 1011. The ordinate is the vertical tune in the interval 0.4 < Q~ < 0.5. Thehorizontal tune is Q~ = 0.3. The maximum N corresponds to S = 0.027. The phase advanceerror is 8 = ±0.16. The value of the separation is indicated by the code at the bottom.

Figure 13 shows the most dramatic effect seen in the multi-particletracking. The two bunches separate spontaneously in the vertical plane. Anincrease in the vertical r.m.s. beam radius is also observed. Figure 13 alsoshows that the vertical separation at a given tune below the half-integer doesnot stop above a certain bunch current as was the case for the growth rateshown in Figure 12. However, the agreement between the lower edge of theinstability stopband in Figure 12 and the onset of the spontaneous separationin Figure 13 is better than about ±10%.

References

[1] Hirata, K. and Keil, E. (1990). Nue!. Instrum. Methods, A292, 156.[2] Hirata, K. (1993). Part. Aeee!., 41, 93.[3] Hirata, K. (1988). Nue!. Instrum. Methods, A269, 7.


[4] Piwinski, A. (1979). IEEE Trans. Nucl. Sci., NS-26, 4268.[5] Satoh, K. (1987). Proc. 6th Symp. Accel. Sci. Tech. (Tokyo), 18.[6] Talman, R. (1987). AlP Conf. Proc., 153, 789.[7] Ieiri, T., Kawamoto, T. and Hirata, K. (1988). Nucl. Instrum. Methods, A265, 364.[8] Arzelier, G., et al. (1971). Proe. 8th Int. ConI. High-Energy Aeeel. (CERN), 150.[9] Derbenev, Ya.S. (1972). 3rd All-Union Conf. on Accel. (Moscow, 1972) 382, cf. SLAC

TRANS 151.[10] Nguyen Ngoc Chau and Potaux, D., Laboratoire de I'Accelerateur Lineaire, Orsay, Rapport

Technique 5-74 (1974) and 2-75 (1975).[11] Hirata, K. and Keil, E. (1989). Phys. Lett., B232, 413.[12] Piwinski, A. (1971). Proc. 8th Int. ConI. High-Energy Aecel. (CERN), 357.[13] Chao, A. and Keil, E. (1979). CERN/ISR-TH/79-31.[14] Keil, E. (1980). LEP Note 226.[15] Keil, E. (1980). LEP Note 268.[16] Furman, M.A. and Chao, A.W. (1985). IEEE Trans. Nucl. Sci., NS-32, 2297.[17] Furman, M.A. (1986). SSC-62.[18] Keil, E. (1981). Nucl. Instrum. Methods, 188, 9.[19] Hofmann, A. and Myers, S. (1988). LEP Note 604.[20] Bambade, P. et.al. (1988). Phys. Rev. Lett., 62, 2949.[21] Hirata, K. and Keil, E. (1989). CERN-LEP-TH/89-57.[22] Keil, E. (1992). CERN SL/92-29, 373.[23] Meller, R.E. and Siemann, R.H. (1981). IEEE Trans. Nucl. Sci., NS-28, 2431.[24] Yokoya, K. and Koiso, H. (1989). Part. Aecel., 27, 181.[25] Courant, E.D. and Snyder, H.S. (1958). Ann. Phys. (N.Y.), 3, 1.[26] Forest, E. (1987). Part. Accel., 21, 133.[27] Bassetti, M. (1989). Proe. 11th Int. Conf. High-Energy Aceel. (Geneva), 650.[28] Vignola, G., Talk at workshop on physics and detectors for DA<I>NE (Frascati, Italy).[29] Hirata, K. and Keil, E. (1991). IEEE Part. Accel. Conf. (San Francisco), 482.[30] Keil, E., Cornelis, K. and Hirata, K. (1992). Proc. 15th Int. ConI. High-Energy Aeeel.

(Hamburg 1992) 1106; cf. CERN SL 92-33 (DI).

linearbeam-beamresonances due to …particle accelerators, 1996, vol. 56, pp. 13-49 reprints...

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