8/14/2019 hogan 1993 0477

1/3

Longitudinal Beam-Beam Effects in Circular CollidersM. Hogan and J. RosenzweigDepartmentof PhysicsUniversity of California, Los Angeles, CA 90024

AbstractThe longitudinal beam-beam nteraction, which can leadto incoherent heating, synchrobctatron coupling, and coherentlongitudinal instabilities in circular colliders, is examined.This analysis discusses wo types of energy kicks, those dueto the transverse particle motion coupling to the electricportion of the transverse kick, and those derived from theinductive electric field induced near the interaction, which isobtained from the transverse kick through use of ageneralization of the Panofsky-WenzelTheorem. Implicationsfor low energy e+e- colliders (cp & B factories) with beamscrossing head-on, and at finite angles, with and without crabcrossing,are discussed.

I. INTRODUCTIONTable 1: Notation used n this paper.e electronchnrger radial positionX,Y transverse oordinatesz longitudinal coordinatec speedof lights=z-ct beamcoordinateP particle momentum, d/dsi rms. beam size in i dim.P Beta-functiona Courant-SnyderamplitudeN # electrons n bunch* denotesevaluation at IPAn ultra-relativistic particle with longitudinalcoordinates, has E-M fields which are ne,arly normal to thedirection o f motion, and may be approximatedas[ ]

E, =%(s-so)rwhere the notation is given in Table 1. Since particles do notall collide head on there is some longitudinal kick givenduring the beam-beam nteraction. The net longitudinal kickcan be obtained for a single particle by taking the projection ofthe transverse ields from the opposing beam onlo the designorbit of the particle and integrating over the betntron phasespaceof the opposing beam. The resulting energy kick can bethought of as arising from two sources: a) longitudinal fields,and b) the work done by the transversemotion of the particleagainst he transverse ields.For beams that do not collide at the nominal intcrnctionpoint (IP), there is a time dependent beta funcrion,p(s), andthe beam size variation gives rise to an inductive longitudinalfield. The longitudinal momentum has been previously de&cdfrom a straight forward ret,arded elativistic calculation[2]. Inthe limit that the beams are shorter than p* (this limit isassumed Ed throughout this paper), these kicks can also be

derived from a fo rm of the Panofsky-WenzelTheorem which isgeneralized to include fields arising from free chnrges [seeAppendix], @@,.%(APr)=~ . (2)We will employ the Panofsky-Wenzel theorem method in thispaper, as it is simpler and more powerful than doing thestraight forward calculation, especially for off-axis pnrticles.

II. THE LONGITUDINAL BEAM-BEAM INTERACTIONThe longitudinal kick due to transversemotion is the sumof the individual kicks a particle receives traversing theopposing bunch, calculated by integrating over the phasespace

of the opposing bunch. The resulting differential equation forthe transverse acceleration involves complex error functionsfor elliptical beams.A focusedgaussinnbeam s describedby a time dcpendcntchargedensity and has associated ransversecurrents given bythe continuity equation. By integrating the charge and currentdensities we obtain the corresponding scalar and vectorpotentials, which intern describe associated electric andmagnetic fields. The Lorentz force law may then be used tocompute the instantaneous acceleration felt by a test particletraversing these fields. The resulting equations contain verytedious ntegrals that may be solved numerically [2].Both casesyield results that are not intuitive and do notallow for an estimate of the size of these kicks which coulddetermine he importance of this analysis for c ircular colliders.Therefore we will now consider certain limiting cases.A. Routd BecmsThe energy change (A,!?) an off-axis test particle which travelsal a nonzero angle with respect to the axis receives passingthrough an opposing beam is given by projecting the orbit ofthe particle onto the field of the opposing beam p,articles, asgiven by Eq. (1). Assuming a round beam with a gaussiandistribution, Eq. (1) can be integrated to give the total energychangeper plunge:

-=-~[i-exp[-g)]~.,+y~~. (4)Averaging over a betatron oscillation (which is assumed o bemuch shorter than a synchrotron oscillation), and expandingequation (4) for r < 0, reduces his to

*E = NcP(f) ax + a,[ 1/q+) 2E (5)where we have introduced Ihe Courant-Snyder amplitudes ofthe particle. Note that if rhe beams do not collide at thenominal IP, implying that0 f 0, the energy kick averagedover a betalron oscillation is non-vanishing.0-7803-1203-l/93$03.00 0 1993 IEEE 3494

1993 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material

for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers

or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

PAC 1993

8/14/2019 hogan 1993 0477

2/3

The energy kick due to the inductive field is computedusing the Panofsky-WenzelTheorem:4h) = 4*df3r as *For round beams he transversekick is, from equation (4),Ap, =2fff-[l-exp(-$)J.

Since

(G)

(7)

(8)has an s dependence, he longitudinal kick is obtained usingEqs. (7) and (8), and integrating the Panofsky-Wenzelexpression (taking the boundary condition Al>lr=- = 0) toyield

AE = Ap,c = N;;w:i exp(-$1; rAveraging over a betatron oscillation for r < 0, we have

LiE = Ne*P (f>- 2/q+) [i-y].

(9)

and the amplitude dependenceof this kick is canceled by theterm found in Eq. (5). Previous ,analysis[l] has not includedthis cancellation, which makes he energy kick more uniformas a function of position. The resulting total energy kick forround beams sAE = Apzc = Ne*p ($)W($) (10)This energy kick, which is now correlated only to the relativelongitudinal position of the oncoming bunch, can contributeto a coherent nstability, as discussed n section III.

B. Flat BeamsOther relevant aspectsof the beam-beam nteraction canbe analyzed in the limit that 0, >> oy. In this quasi-one

dimensionaJ ase he Panofsky-Wenzel heorem eads4~~~144dy =-r-The transversekick in a gaussianbenm s given by (11)ApY (12)

where1

(13)The contribution to the energy kick is now due to transversemotion can be found in analogy to Eq. 4. Also, the inductiveenergy kick is, using Eq. (1 ),AE=A&c=e*o,P;$)2crxpy (i) (14)

Again, the transversedependence s proportional to the currentdensity. a variation which is canceled exactly by the energykick due to transversemotion derivable from Eq. 12 for smallamplitude (y < 0;) particles. Note that this expression issmaller than he equivalent round beam ormula (Eq. (10)) by afactor of R = cry/o,. This factor is due to larger averagedistancesbetweenparticles (R/2) and weaker focusing in the x-dimension (2).

III. COHERENT BEAM-BEAM OSCILLATIONSLongitudinal beam-beam effects can drive a coherentlongitudinal oscillation. While this subject has been analyzedbefore[l], it has never been understood that the longitudinalbeam-beam ick is nearly independentof x and y. The coupledequationsof motion for the be,am entroids (s,,~)are

s, + c&s, = (k)k,,(s, -s*)s* + w,2s2= (+-)k[,b& -s,) (1%

Here, &, E VL, / V;, the rf gradient V; = krfVrf, and V&,(the effective beam-beam radient) ncludes componentsdue toboth pnrallel and transversemotion of the beam particles:Ne*G, = m. w-3The + (-) sign rcfcrs to operation above (below) transition.Above transition, we obtain the dispersion relation

w = w,(l- 2k,;,)f. (17)Thus the instability threshold, which occurs when w=O, isgiven by 2V&, = ,,-. If the vertical beta function is loweredby a factor TJ, hen VLh will increnseby a factor of q*. Thisis a strong dependenceand may indicate trouble with higherluminosity designs.If the machine is run below transition, the frequency ofthe coupled modebecomes

w= w,(1+2k,,)+ (18)and there is no possibility of this coherent longitudinalinstability.IV. IMPLICATIONS FOR LOW ENERGY COLLIDERS

A. (p FactoriesFor the UCLA rp factory design parameters, he expectedenergy kick for a one CT,particle is - 1 keV. The rf voltage

gradient V,) = 1 MeV/m is quite small due to the quasi-isochronous condition being employed. With N=1.6 x loll,R=7, /3;= 4 mm, and emittances of 1.1 x 10m6m-rad in bothx and y, the effective beam-beam radient is

Ne*eV&, = - =2RP** 1 MeV/m . (19)If the machine is run above transition, this gradient islongitudinally defocusing, and one would expect seriousbunchlengthening, since to fist order there is complete longitudinaldefocusing,

3495PAC 1993

8/14/2019 hogan 1993 0477

3/3

i 1-3a,= &!!b?. =Mo- 0 Tf (20)This system is also above threshold for the longitudinalinstability by a factor of two in beam charge.If the machine s designed o operatebelow transition, thebeams have stable coherent longitudinal motion. The benm-beameffects einforce the rf focusing because

o* - l-v,:,t 1-40 0 ?f =&. (21)This implies that the bunches could be shortened allowingshorterbunches han presentdesigns ndicate.B. B Factories with Crab Crossing

Crab crossing schemes (Figure 1) may bc necessnry toprovide the high luminositys (-3 x 103j) required for B-factories.I 1electrons positrons

c IFigure 1: Schematic representation of crab crossing. Crabcavities apply time dependent f kicks which tilt the bunches.After the collision, another set of crab cavities kick the beamsback to there original orientations.

*p=;; pb$+r@bA;-@)[ 1z. (27)zOutside of region R. i = 0, leaving

Ap= t[s$l; -.!,].The longitudinal beam-beam nteraction could becomeimportant in this type of scheme.Recall that the energy kickdue to transverse motion for round beams contains termsproportional to xx,yy. For crab crossing the angle X isnow essentially to the crossing angle. The resulting energykick may again lead to large longitudinal effects. This issueneeds o be investigated urther.

Since it can be derived from a potential, Ap satisfies therelation vx(Ajj)=O (29)In shorthand notation, this can be written as?,(A/,Z) = y. 00)i

V. CONCLUSION REFERENCESThis paper has analyzed two types of longitudinal kicksarising from the longitudinal benm-beam nteraction: those due

to the transverse particle motion coupling to the transverseportion of the electric k ick, and those derived from theinductive electric field induced near the interaction point.These effects may become important in low energy or highluminosity colliders (cp& B factories) since they may lead tocoherent longitudinal instabilities. These effec ts can beminimized by the use of fla t beams. In addition, if coherentinstabilities become a problem, it may be necessary o operatebelow transition. The effec t on low energy colliders such as B-factories which may utilize crab crossing to improveluminosity ncedr to be addressedurther.

Appendix: Generalized Partofsky-Wenzel TheoremThe Panofsky-Wenzel theorem gives a relationship betweenthe integrated longitudinal and transverse momentum kicks ap&articleeceives as t traversesa medium or device excited inthe wnke of another particle [4]. This appendix will generalizethis theorem to include fields arising from free charges.assuming that fields and potentials vanish at infinity, and thatthe particle receiving the kick travels p,arallel to the z axis. Ingeneral. an electric field E may be described in terms of ascalnr potential (0 ) and vector potential ( 2 ):

&-I&-pj)c arInserting (22) into the Lorentz force equation

and noting that for a particle traveling parallel to the z-axis~l)xB=Poix(Vx~)=~(PbA;)-Pb~ (24)Iwe obtain the following expression for W . the force per unitchargeq: w=-f~-~~-[p(B~a,)+ip,~]. (25)

By noting that~[~+vb$p$$$, (26)equation (25) can be rewritten as

(28)

[l] V.V.Danilov, et. al., Longitudind Effects in Beam-BeamItltercrctiorl for an Ultru-High Luminosity Regime, in Be&am-Beam and Beam-Radiation nteractions p.l-10, Eds. C. Pelleg-rini, J. Rosenzweigand T. Katsouleas, World ScientificPublishing Co., Singapore, 1991)[2] G. Jacksonand R. H. Seimann, Nucl. Instr. and Meth.A286 (1990) p. 17-31.[3] W. K. H. Panofsky and W. A. Wcnzcl, Rev. Sci. Inst.,27. 967, 1956.

PAC 1993