e, )2 Crossing Angles In The Beam-Beam Interaction R.H. Siernann* II. PERTURBATION FORMALISM SLAC-PUB-7225

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1 SLAC-PUB-7225 July, 1996 Crossing Angles In The Beam-Beam Interaction COBF- G ~ o, ~ J E I - - R.H. Siernann* Stanford Linear Accelerator Center, Stanford University,Stanford, CA USA ABSTRACT A Hamiltonian perturbation analysis of the beam- beam interaction with a horizontal crossing angle is performed. The beam-beam tune shifts and resonances that result from a crossing angle are determined. Figure 1: Beams crossing at -an angle INTRODUCTION Many storage ring colliders are being designed to reach high luminosity through the use of a large number of closely spaced bunches. This introduces a potential problem of parasitic collisions near the interaction point, but these parasitic collisions can be avoided by having the beams cross at an angle rather than head-on (Figure 1). The contributions to the tune shifts and the beambeam resonances introduced by a crossing angle are analyzed in this paper. The beam-beam interaction with a crossing angle has been studied by a number of authors, [l]- [4]. This paper is closest to that of Sagan et a1 [23 which obtained some of the results presented here. The notation and method are discussed extensively in references 151 and [61. II. PERTURBATION FORMALISM The Hamiltonian of a particle in beam 1 is where Ho is the Hamiltonian of the transverse motion in the absence of the beam-beam interaction,?bb is the beam-beam potential, N is the number of particles in beam 2, r, is the classical particle radius, and y is the energy in units of rest energy. The betatron motions in the absence of the beam-beam interaction can be written in terms of the action-angle variables (Ix, yx) (I,,, yy)of the unperturbed Hamiltonian, Ho, The sum is over all turns and the variables in this equation are: a~ = the RMS bunch length of beam 2; s = coordinate along the reference orbit; C 3 the collider circumference; c i speed of light; and z = displacement of the collision point given in terms of the synchrotron oscillation amplitude,?,and tune, Q, by z 7 = -c0~(21tnq,). A 2 The potential VF depends at the displacements of the particle from the center of beam 2 where a, and oy are the RMS transverse sizes of beam 2. Using the expressions in eq. (1) as approximations for the betatron motions and assuming that the crossing is in the x-dimension, the potential can be rewritten (ssin 29 + J2p,1, 20; +q COS e, )2 Fourier transforming the x expression with respect to s gives These expressions are used in a perturbation analysis of -- VRR. The beam-beam potential is * Work supported by the Department of Energy, contract DE-ACO3-76SFOO515. Presented at the 1996 DPFDPB Summer Study on New Directions for High-Energy Physics (Snowmass '96), Snowmass, COSJune 25-July 12,1996

2 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

3 t I Following the usual procedure of Fourier transforming VBB with respect to yx,yy and s gives an expression for VBB in terms of Fourier coefficients each of which is, related to the resonance PQx +IQy + mqs = n where p, r, m, and n are integers. Making a change of variables = o/sin2cp this expression is 5 VBB =-1 C m.n,p,r=--0 xexp(-(kp, x exp(i(pv, + y mqs)s/ C)).?dt;UH(Ix.Iy.C) -00 +csin2~p)~ot/8) x imjm((kp +csin2cp)?c/2) x exp(i(pvx + yy mqsh / C)) where The second term in the Taylor series is -, m,n,p,r=-=--o and k, = 2x(n- mqs)/ C+ p(l/ +r(l /P; & -2 7 ~ /c) ~ 0. -2xQxo /C) The quantities and P; are the p-functions at the where collision point, and Qxo and 4.0 are the tunes in the absence of the beam-beam interaction. + mqs = n occurs for The resonance pqx + values of the tune where the phase is stationary, i.e. + vy- 2Nn - mqsh / C)= 0, ds and the average value of the beam-beam potential is given by the term in the series with p = r = m = n = 0. Perform a Taylor expansion in powers of sin29 d -(pvx (5) Equations (2) - (5) contain the results. These frightening looking expressions can be inteqreted to give useful information about the beam-beam interaction. The first term in the Taylor Series is III. DISCUSSION A. Tune Shifts The tune shifts as a function of amplitude are 2

4 r. B. Beam-Beam Resonances 1 where (VBB) is given by eqs. (2) - (5) evaluated with p = r = m = n = 0. The tune shifts are the Same as for head-on collisions because there are no contributions from the second term in the Taylor series, the term proportional to sin29. This follows from the parity of the Ox integrands. In the case of AQ, the integral is Possible resonances can determined from the parity of the integrands in eqs. (3) and (5). The integrand of eq. (3) is an even function of 8, and an even function of 8,. The only allowed resonances for head-on collisions must have both p and r equal to even integers. The integrand of eq. (5) is an odd function of ex and an even function of 8,. The allowed resonances must have r equal to and even integer and p equal to an odd integer. The crossing angle has introduced new beam-beam resonances that have odd horizontal order and Fourier 2A expansion coefficients propoxtional to sin 29. There are 21 p COS^^, dqy- jd0,cosoxexp{20, x2 + q }=0 both betatron, m = 0, and synchrobetatron, m f 0, 0 resonances. The appearance of odd order betatron resonances can be understood because there is a phase because the argument is an odd function of Ix.The of IC across the interaction region. The shift horizontal tune shift, AQ,, is proportional to synchrobetatron resonances arise from modulation introduced by the synchrotron oscillations. They depend on the synchrotron amplitude and have zero Fourier expansion coefficient when? = 0 21,px cos2 8, C.Remarks d0,cos 3 8,exp 20: + q The crossing angle has not changed the tune shifts, so the beam-beam footprint, the area of the tune plane Evaluating the integrals AQ, = 0 because the arguments occupied by the beam, is the same as for head-on of both integrals are odd functions of e,. collisions. The effect of the crossing angle has been to The tune shifts from the head-on collisions have been introduce odd horizontal order resonances. These calculated in numerous references and are given here for additional resonances could lower the beam-beam limit. completeness. They are TeV33 is considering using both horizontal and vertical crossing angles. The vertical crossing angle will introduce odd order vertical resonances as well, and there is a still larger probability of a reduced beam-beam limit.. IV. REFERENCES and The functions Bx,...,A, are [l] A. Piwinski, DESY Report DESY 77/18 (1977). [2] D. Sagan. R. Siemann and S. Krishnagopal, PLQL pf the 2nd EPAC, 1649 (1990). [3] D. V. Pestrikov, KEK Preprint (1993). [4] K. Hirata, Phys Rev Lett 74,2228, (1995). [5] R. H. Siemann, Eront'iers of Particle BeamS; Factories with e+drings (Springer-Verlag, Lecture Notes in Physics, Volume 425 edited by M. Dienes, M. Month, B. Strasser and S. Turner), 327 (1994). [6] R. H. Siemann, Informal note "Extension of Beam-Beam Calculations in SLAC-PUB-6073". The variables in these equations are R = by/crx;5, and 5, are the beam-beam strength parameters; and E, and E, are the emittances. The functions 1; are related to modified Bessel functions Ik(x) = e-'i,(x). 3

5 I49 b01 Publ. Date (11) Sponsor Code (18) xf I E k, DOE