Ruggiero, A G

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Particle Accelerators1982 Vol. 12 pp.45-470031-2460/82/1201-0045$06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaINTEGRABILITY OF THE TWO-DIMENSIONAL BEAM-BEAMINTERACTION IN A SPECIAL CASEA. G. RUGGIEROFermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510 U.S.A.(Received March 27, 1981,. in final form July 22, 1981)It is shown that with the assumption of zero chromaticity, zero dispersion and symmetry between x and y, head-on collisions of bunched beams produce a beam-beam interaction that can be reduced to a one-dimensionalweak-strong interaction. The KAM theorem ensures that for small values of the strong beam, the motion is boundedby stable trajectories. There can therefore be no Arnol'd diffusion in these circumstances.I. INTRODUCTIONIt is often stated 1 that a non-autonomous systemwith two degrees of freedom is affected byArnol'd diffusion, an instability arising from in-tersection of nonlinear resonances in four-di-mensional phase space. The purpose of this paperis to show that under certain assumptions, thesystem admits an integral of the motion, a can-onical angular momentum Pe. It is therefore pos-sible to eliminate e as an ignorable coordinateand reduce the number of degrees of freedom toone. The KAM theorem2 then ensures that forsmall values of the beam-beam perturbation themotion is bounded by stable trajectories and doesnot exhibit Arnol'd diffusion.II. ASSUMPTIONSWe shall consider here only the so-called"weak-strong" case where a test particle in a~ ~weak" beam crosses periodically a ~ ~ strong"beam. The motion of the particles in the strongbeam is not affected by the presence of the weakbeam; therefore their charge distribution can beassumed to be constant in time.The equations of motion of a test particle inthe weak beam are3(1)y" + Ky(s)y = ~yFy(x, y)op(s),where x and yare the displacements of the par-ticle motion from a reference orbit (x, y = 0), K x45and Ky are the lattice focusing functions and ~xFx,~yFy represent the interaction with the strongbeam, with op a periodic delta-function of periodC.Equations (1) describe the motion of a particle,for instance in the proton-antiproton collidingbeam system at Fermilab,4 if the following pre-cautions are taken:(i) The two unperturbed betatron tunes Vx andV y do not depend on the particle momentum. Thisrequires chromaticity cancellation in both planesover a reasonable momentum range.(ii) The lattice parameters at the cross point(a*, B*, ~*) do not depend on the particle mo-mentum, again over some appreciable range.This might require even higher order correctionsthan those required to flatten the chromaticity.(iii) The dispersion at the crossing point van-ishes over the same momentum range. No con-dition, though is required on the derivative of thedispersion.(iv) Both beams are bunched but the interac-tion is exactly head-on.(v) The bunch length in the strong beam issmall compared with B*. In this case it is possibleto represent the interaction by a lumped kick.That is the interaction has infinitesimally smallduration which justifies the periodic delta func-tion on the right-hand side of Eqs. (1).With these assumptions, which are not too dif-ficult to meet in practice, the interaction betweenthe two beams is independent of the particle mo-mentum and therefore of the phase oscillations.In this case one only requires the integration of46 A. G. RUGGIEROIII. THE THEOREMIf the following conditionsV x = V y and cxx* = cxy* (2)are satisfied, then there exists an infinite varietyof strong-beam charge distributions for which theequations of motion (1) admit at least one integralof motion.Proof: The interaction force can be derivedfrom a potential functionX' n = - ~13~* {ax*cos(nJ-Lx + 8x) (7)+ sin(nf.Lx + 8x)}(x, X'; Y, y') ~ (r, pr,o 8, Pe), (8)withx = Vr~x* cos 8 and y == Vr~y* sin e (9)The generating function for this transformationisand similarly for Yn and Yn'. In Eqs. (7), Ex and8x are constants of motion. The first, Ex, measuresthe amplitude of the motion. We carry out thevariable transformationtically between crossings. In the limit ~x = ~y= 0 the position and angle of the test particleafter the n-th crossing are given byXn = VEx~x* cos(nf.Lx + 8x)avax~Fx(x, y) =the system of Eqs. (1) to calculate the motion ofa test particle in the weak beam.Our approach is static. That is, we are neglect-ing all sources of noise that would cause the in-teraction to fluctuate (gas scattering, intrabeamscattering, power-supply noise, etc .. etc ..).P 2 + P 2 K x 2 + K y2H= x y + x y2 2 (11 )(10)asar== x'Vr~x* cos 8+ y' Vr~y* sin 8,aspe == - - and pr ==aefrom which we derive the new momenta(3)andavayThen Eqs. (1) can be obtained from the Ham-iltonian3+ Vex, y)8p (s). (4)Let f.Lx = 21TVx and f.Ly == 21TVy be the betatronphase advances in the two planes between twoconsecutive crossings. Whether there is only oneor more interactions per revolution is here im-material, provided that the lattice repeats iden-This is a non-autonomous system with two de-grees of freedom. The independent variable is s;the canonically conjugate variables arex, Px = x' and y, py = y'. (5)According to Maxwell's equations, the poten-tial V is related to the charge distribution p(x, y)in the strong beam Laplace's equation(12)We can estimate Pe at the interaction point aftern crossings in the limit ~ == ~y == O. By insertingEqs. (7) and similar equations for y, y' in Eqs.(12) we obtainPen == VExEy sin(8y - 8x ) (13)which is a constant of the motion.Let us see now the effect of the kick with ~xpe == x' Vr~x* sin e - y' Vr~y* cos eIn particular from the first, which is an angularmomentum,(6)\/2U(X, y) == p(x, y).INTEGRABILITY OF THE TWO-DIMENSIONAL BEAM-BEAM INTERACTION IN A SPECIAL CASE 47and ~y =-1= 0 From (9) we obtain+ 2(_1_ + _1_) dU (14)f3x* f3y* drIf V is a function only of this variable, then wehave from Eq. (6) that all the charge distributionssatisfying the equationLixn = 0Lixn' = ~Fx(xn, Yn)We havelf3x* (au)-y f3y* Yn ax n~f3x* (au)+ - -f3y* n ay nwith an arbitrary V = V(r) , also satisfy the re-quirements of the theorem.In particular, if f3x* = f3v * and the strong beamis "round", as it is approximately true for the fJpproject at Fermilab,4 then a Gaussian charge dis-tribution in the strong beam is consistent withthe assumptions of the theorem.When the transformation (9) is applied,U = U(x, y) ~ U(r, e),and it is obvious that if aUlae = 0, i.e., the po-tential depends only on the "radial" coordinater,av = dVar = 2~ dVax dr ax f3x* drav = dvar = 2 L dVay dr ay f3y* drthen LiPen = 0, that is Pe remains a constant ofmotion even with beam-beam interaction.REFERENCES1. J. L. Tennyson et aI., 44 Diffusion in Near-Integrable Ham-iltonian Systems with three Degrees of Freedom," AlPConference Proceeding of Nonlinear Dynamics and theBeam-Beam Interaction, Brookhaven National Labora-tory, 1979, p. 272.2. J. Moser, HStable and Random Motions in Dynamical Sys-terns," Annals of Mathematics Studies No. 77, PrincetonUniversity Press, 1973.3. A. W. Chao, HA Summary of Some Beam-Beam Models,"AlP Conference Proceeding on Nonlinear Dynamics andthe Beam-Beam Interaction, Brookhaven National Lab-oratory, 1979, p. 42.4. Design Report Tevatron Phase 1 Project, Fermilab, Feb-ruary 1980.

Integrability Of The Two-dimensional Beam-beam Interaction In a Special case

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