Beam-beam limit for e+e- factories K. Ohmi (KEK) e+e- Factories 03@SLAC 13-16 Oct. 2003

Introduction Luminosity Beam-beam limit is discussed. The beam-beam limit influences the design of factory machines. I: RF, Vacuum, instability by: IR Optics

Courtesy J. Rogers HALO03

DAFNE VEP2K BEPC BEPCII CESR-c E (GeV) 0.51 1 1.4 1.89 1.5 C (m) 97.69 24.38 240.4 237.5 768 ex(nm) 600 136 580 144 180 Ne(1010) 4.7/3.1 10 23 4.9 6.4 sz(cm) 3 5 bx (cm) 170 120 100 70 by(cm) 2.7 1.1 xx 0.04 0.1 0.058 0.05 Lb (1030) 1.6 (100) 11.2 (11) (6.7) xy 0.02 (0.1) 0.041 (0.04) (0.056)

PEPII SPEPII KEKB SKEKB CESR E (GeV) 3.1/9 3.5/8 5.3 C(m) 2199 3016 768 ex(nm) 33/49 19.5 18/24 24 206 Ne(1010) 9.3/5 10/4.5 7/5.2 13/5.5 12.7 sz(cm) 1.2 0.175 7 3 1.8 bx (cm) 40 15 60 20 96 by(cm) 1.1 0.15 0.7 0.3 xx 0.07 0.12 0.2 0.026 Lb (1030) 6 (145) 8 (50) 29 xy 0.048 (0.15) 0.05 (0.1)

Source of beam-beam limit Current limit, due to hardware heating etc. Collision offset. (1st order　in H) Optics error (2nd order), nonlinearity (3rd or higher) Fundamental limit We study the beam-beam limit using computer simulations.

One turn map including beam-beam interaction Hamiltonian Lattice Beam-beam See next slide. Equation of motion (one turn map)

Integration of the beam-beam interaction Lattice one turn map V0(s,0) : map of IP to collision point of the particle Collision point : s+=(z+-z-)/2 s-=- s+ integration IP (s=0)

Beam-beam potential f: potential given by solution of 2D Possion equation at every s and every z.

Numerical integration Discretize the interaction. In visual expression, slice the target bunch. It is easy to evaluate exp(-:f(z):), because it includes only x,y,z not px…. Potential kick. But …

How to estimate f The potential f have to be solved for r at every s and every z. Gaussian approximation : exact forms of r and f are given as functions of s: that is, synchro-beam mapping (Hirata). Particle in cell : it is difficult to get r and f for arbitrary s(z), because the potential has to be solved for each particle (z): that is, target particles are transferred to s=(zi-z)/2 and the potential have to be solved. To reduce the process, the potential is interpolated in the sliced region.

Other notes r is affected by preceding interaction in the strong-strong model: disruption effect. Collision should be evaluated along the right time (s) order for both beams.

Slice dependence (strong-strong) Luminosity converge at Nsl=10. Nsl=5 is 5% less, but is not bad. Nsl=5 is used in this presentation. KEKB parameter Weak strong also shows similar behavior for slicing

Convergence for the slice number By M. Tawada If we do not use this algorithm All particles in i-th slice are kicked by φcp Interpolation

Two types of beam-beam limit Particle motion and/or distribution function are determined by one turn lattice map (revolution matrix for the linear part) at the collision point, V0 and beam-beam map at IP region. Lattice dependent beam-beam limit If V0 does not include any coupling, one turn map is represented only by tune, radiation damping, excitation and the beam-beam parameters. Fundamental beam-beam limit

Lattice dependent Beam-beam limit One turn Lattice map at IP, V0 Tune b, a at IP. x-y coupling r1-r4 at IP. xy-z coupling hx, h’x, hy, h’y, zx, z’x, zy, z’y Crossing angle is regarded as zx. Chromaticity, amplitude dependent tune shift…

How do the parameters lead the beam-beam limit? Resonance Contradict of normal axis for lattice and beam-beam interaction. ….. Analytical approaches are limited to qualitative discussions. Numerical approaches are relied: weak-strong and strong-strong simulations.

Linear optics error DAFNE weak-strong codes: BBC (Hirata) & LIFETRAC (Shatilov) M. Zobov, Beam dynamics news letter 31, 2003

Optics tuning at KEKB Waist of by at IP. RF phase. Vertical dispersion at IP X-y coupling R1-R4 at IP. These are tuned everyday.

Optics parameters are controlled by local bumps

X-y coupling LER R4

Parameter dependence given by simulation

Crossing angle Crossing angle is equivalent to zx at IP. X-z coupling is induced in one turn map. Resonance. Mixing of normal mode.

Specific luminosity for various current product (I+I-). f = 0mrad and 11mrad Weak strong strong-strong

Beam-beam parameter ** The beam-beam limit in finite crossing is determined by a weak-strong effect in this parameter.

Crossing angle and luminosity Sharp peak near zero crossing angle

Effect of crossing angle We discuss later. The beam-beam parameters are >0.2 for weak-strong and 0.1 for strong-strong at the head-on collision. The weak-strong and strong-strong show similar results x~0.06 for f =11 mrad. Crossing angle degrades the luminosity.

Crossing and crab cavity Crab cavity creates zx=-q at IP. Completely agree with head-on. The difference of sx is only apparent.

Lattice nonlinearity DAFNE M. Zobov, Beam dynamics news letter 31,2003

Fundamental beam-beam limit Coherent motion p mode or higher order instability Studies coherent mode spectrum (Yokoya, Alexahin et al.). Two stream type (Perevedentsev, Valishev) Incoherent effects Quantitative analysis is relied on numerical approaches.

Beam-beam limit due to coherent motion In strong-strong simulations, coherent p mode limits luminosity for short bunch and 2D model. Coherent motion is observed for sz<by/2. No coherent motion Coherent motion

Coherent motion seen in some codes J. Qiang (LBL) Y. Cai (SLAC) K.Ohmi (KEK)

Coherent beam-beam limit for short bunch The beam-beam limit is about 0.06-0.07 for rings with the damping time ~5000 turns (B factory level). In actual machine, sz~by, the coherent motion may be smeared by tune spread along the bunch. Even for short bunch, Landau damping caused by many unknown sources may smear the coherent motion. Tune and/or intensity difference of two rings also suppress the coherent motion.

Other notes Strong-strong simulation with Gauss approximation is insensitive for the p mode instability. p mode instability s mode always dominate. ??? PIC xlim~0.06-7 Gauss xlim>0.2

Different tune No coherent motion. Luminosity is recovered to that by 3D-PIC level. The luminosity is closed to that obtained by 3D PIC strong-strong.

Two stream type of instability (Perevedentsev, Valishev) For high disruption parameter and low beta, two stream type of instability is caused by the beam-beam interaction. Disruption parameter w: beam-beam oscillation freq. It is serious for ny closed to half integer.

Example of beam-beam two-stream instability nx=0.58 ny=0.51 ns=0.02 sz <yz>

Rogers, beam-beam 03

Beam-beam limit due to incoherent effects Tune scan using weak-strong. How large beam-beam limit can we achieved if no coherent instability occurs. Are weak-strong and Gaussian strong-strong simulations accurate to predict the beam-beam limit (x>0.2). Do all type of simulations show the unique beam-beam limit? Can we understand the beam-beam limit?

Tune scan for BEPC II P.D.Gu et al., Beam-dynamics News Letter 31, 2003 BBC (Hirata)

Tune survey by strong-strong at KEKB (Tawada)

Golden operating point In many machines, tune operating point around horizontal, nx=0.50-0.52, and vertical, ny=0.55-0.60, is the best region of the tune space. Horizontal beam size is squeezed by the dynamic beta effect. ny=0.50-0.52 is bad due to the two-stream effect.

How high beam-beam parameter can be achieved Strong-strong simulation with Gauss approximation and Particle In Cell method Remarkable difference. Gauss x>0.2 PIC x~0.1

Check using some PIC codes LBL (J. Qiang)

SLAC (Cai) and KEK (Ohmi) codes 50

Luminosity evolution by PIC What is the sudden dip? Such dips are observed randomly for high beam-beam parameter. Once or none in a simulation

An example with a sudden dip: particles are initialized with final distribution of Gaussian simulation Only sy Does this behavior have a hint of the beam-beam limit?

Change of particle distribution No coherent motion during the growth 150 170 200 final

Is any small coherent motion contribute? Weak-strong simulation with PIC. The strong-beam was given the final distribution of the strong-strong simulation. Basically the beam-beam limit is caused by weak-strong effect due to distorted beam. There is a luminosity difference 15% between them.

weak-strong with PIC strong-strong Vertical beam size weak-strong with PIC strong-strong Weak-strong with PIC & Gauss distr. Incoherent effect dominates for the beam-beam limit. There are small other effects (DL=15%): coherent motion?

Equilibrium distribution of two beams Vlasov equation Fokker Plank equation D:rad. damping B:rad. excit.

Does an equilibrium distribution exist? If the Hamiltonian system is solvable, an equilibrium distribution exists. Function property of the distribution for J is determined by Fokker Plank term. The distribution depends only on emittance, B/D, basically.

Nonlinear diffusion of beam-beam system If the Hamiltonian system is not solvable, equilibrium distribution does not exist. Arnord’s diffusion is caused by the nonlinearity (Chirikov, Tennyson et al.). The beam-beam interaction is not solvable, even in weak-strong model. Many discussions have been done for halo formation. In this strong-strong case, diffusion for coupled two beams seems to be important.

Quasi-strong-strong simulation The equilibrium distribution can be given by more simple way. Switch the PIC weak-strong simulation with a certain period. Talk at Working group

Round beam (VEPP2K) Beam-beam limit, ~0.2, is higher than that of the flat beam. To gain the luminosity for xr=2xy, br<4by is required. It is not easy.

Summary Luminosity beam-beam limit is caused by various reasons. Linear coupling including crossing angle and nonlinearity of lattice degrade the luminosity. Coherent motion may limit the luminosity, but Landau damping suppress the motion. Whether the coherent motion limits, depends on the bunch length and the operating point. Tune scan shows the best operating point, nx=0.50-0.52, ny=0.55-0.60.

Summary Two stream type of beam-beam effect is serious for tune closed to half integer in the plane with high disruption (vertical). Incoherent beam-beam limit caused by equilibrium distribution of two beams appear at ~0.1 for B factories. The limit depends on the damping time. The beam-beam limit does not appear in simulations with Gauss approximation.　Gauss Approximation is not suitable for study of the beam-beam limit.

Thanks Y Cai Y. Funakoshi J. Qiang K. Oide J. Rogers M. Tawada A. Valishev M. Zobov