Crab Waist Studies for Super B and KEKB

Crab Waist Studies for Super. B and KEKB Y. Ohnishi/KEK Super. B Workshop V Paris 10/May/2007 1

Crab Crossing and Crab Waist • Head-on collision is effectively realized by crab cavity, while crab waist controls nonlinear interaction induced by crossing angle. • For high current scheme, either scheme will work, but not both at once. • For low emittance scheme, only crab waist is applicable because beam-beam tune shift becomes too high. • Crab waist can make luminosity comparable to that of crab crossing. 2

Crab Waist Scheme • Hamiltonian can be considered at IP as follows: This Hamiltonian can be made by sextupoles. • Then, the vertical position changes as: • Transformation of phase space can be written by: • If drift space is also considered, total transformation: 3

Crab Waist Scheme (cont'd) • Twiss parameters at IP can be transformed by M: • The waist moves along the x position: • When an appropriate K is chosen as follows, particle collides with other beam at the waist point. x 2 fx s The strength of sextupoles are determined by above formula. 4

Crab Waist Scheme (cont'd) • In order to make H=xpy 2 at IP using sextupoles, an appropriate betatron phase advance is needed. This term makes crab waist. • The Hamiltonian of sextupoles is: • X and Y are reference coordinate (x and y are physical coordinate): • Transformation of sextupoles(SX 1 → IP and SX 2 ← IP): • m, n, k, l are integer and arbitrary. 5

Crab Waist Scheme (cont'd) • When Dyx=p, Dyy=p/2 is chosen, effect of the crab waist becomes: • In order to make crab waist, strength of sextupoles is: 2 fx=34 mrad, bx*=20 mm, by*=0. 3 mm, bx, sext=14 m, by, sext=140 m → K=26. 5 m-2 • On the other hand, x 3 term becomes: F 3, x = ~80000 3 F 3, crab= 29 Very large effect ! 6

Crab Waist Studies for Super. B 7

Super. B-HER new lattice by Pantaleo 8

Lattice Parameter betatron tunes 9

Final Focus - Old(left) & New(right) -I' -I' DECY 4(K 3=-6) OCTX 4(K 3=-23) DECY 0(K 3=1) OCTX 0(K 3=-8) SFX 5 K 2=0 crab waist sextupole 10

Final Focus with Crab Waist Dyx = 6. 0 p Dyy = 5. 5 p bx*/by* = 20 mm/200 mm bxcrab/bycrab=14 m/140 m Sextupole(thin) for crab waist 11

Interaction Region close-up of IR IP 30 cm OCT 0 magnet (multipole) 12

Chromaticity Correction 13

DDX WX, WY Super. B-HER Chromatic Functions DDX WY WX IP s (m) 14

DDX WX, WY Chromatic Functions (cont'd) DDX WX WY IP s (m) 15

Dynamic Aperture for Super. B-HER 7 Ge. V nx=x/sx Jy/Jx=0. 25% (fixed) Crab waist: K 2=0, Fringe effect: OFF, Synch. oscillation: OFF nx/ny=48. 57/23. 60 bx*/by*=20 mm/200 mm bxcrab/bycrab=14 m/140 m (thin-lens sextupole) Synchrotron oscillation: ON Radiation damping: OFF Quantum excitation: OFF #turns: 1000 Crab waist K 2=+20/-20 Fringe effect for all magnets Dashed line: OFF Solid line: ON Dp/p 0 Fringe field reduces dynamic aperture. 16

Effects of Fringe Field(Maxwellian Fringe) (2 Jx/ex)1/2 Jy/Jx=0. 25% Dp/p 0 All magnets fringe off QD 0 only fringe on FF region fringe on All magnets fringe on Blue: Crab waist off/Red: Crab waist on Fringe of QD 0 strongly affects dynamic aperture. 17

Local Compensation of Final Focus Fringe Jy/Jx=0. 25% K 3=24 K 3=16 K 3=12 K 3=8 K 3=4 K 3=0 K 3=-4 Crab waist ON Dp/p 0 Score(are of dynamic aperture) (2 Jx/ex)1/2 OCT 0 is turned on OCT 0 K 3 üAll magnets include nonlinear fringe effects üOctupole near QD 0 increases dynamic aperture(on-momentum) by 30%. 18

Summary of Studies at Super. B • SAD is consistent with MAD for the case of no fringe field. • If the fringe field is ignored, the 50 sx(70 sx for K 2=0) can be achieved in the dynamic aperture for onmomentum. • However, fringe field is turned on, the dynamic aperture decreases to 18 sx. • Final focus doublet(QD 0) affects the dynamic aperture significantly. • Octupoles near QD 0 increase the dynamic aperture(onmomentum) by 30%. • Cure of x 3 term should be considered ? 19

Crab Waist Studies for KEKB 20

Crab Waist Optics at KEKB Beam-beam simulation by K. Ohmi has shown that the crab waist scheme will boost the luminosity of the present KEKB, as well as the crab cavity. waist sextupole Several lattice design was tried on the computer. sextupole As the result, drastic degradation of the dynamic aperture was found. No good solution has been obtained so far. 21

LER Crab Waist Oho-Nikko Version H. Koiso LER NX 25 NY 23 I - transformation K 2 9. 3 y. X 12. 5*2 p yy 11. 25*2 p Oho y. X 12. 5*2 p yy 11. 75*2 p Tsukuba K 2 -9. 3 Nikko 22

x/sx Crab Waist Oho-Nikko Version Dp/p 0 • The dynamic aperture is estimated by tracking in the 6 D phase space. • 1000 turns (1/4 of the longitudinal damping time) • Crab waist sextupoles significantly decrease the dynamic aperture. 23

Crab Waist Tsukuba Version 1 üCrab waist sextupoles (SCWTR/L) are located in IR. ü-I’ transformation between the sextupoles üb. X/b. Y=21/150 m at SCWTR/L üThose sextupoles decrease the dynamic aperture that is the same as Oho-Nikko version. K 2=0 y. X 1. 5*2 p yy 1. 75*2 p SX K 2 9. 3 y. X 2. 0*2 p yy 1. 75*2 p SX K 2 9. 3 RF ON Two additional quadrupoles to match the condition for the crab waist. K 2 ± 9. 3 SL 0をoff 24

Crab Waist Tsukuba Version 2 üSextupoles both in IR and in other straight sections to cancel x 3 term at IP. übx/by=10/300 m @ SCWTR/L übx/by=60/15 m @ SCWN/O üRemoved solenoid and multipole components of the final quadrupoles (QCS, QC 2) y. X 1. 5*2 p yy 1. 75*2 p y. X 2. 0*2 p yy 1. 75*2 p SX K 2 6. 6 üThin sextupoles y. X 11. 5*2 p yy 10. 5*2 p y. X 11*2 p yy 10*2 p SCWTR/L : K 2=+6. 6 SCWN : K 2=-. 449 SCWO : K 2=+. 449 25

Crab Waist Tsukuba Version 2 üSextupoles(SCWTR/L) for the crab waist are located in the Tsukuba straight section(IR). üThe sextupoles in the Oho and Nikko section cancel out the x 3 term. SCWTR/L: K 2=+6. 6 SCWN : K 2=-. 449 SCWO : K 2=+. 449 übx/by=10/300 @ SCWTR/L übx/by=60/15 @ SCWN/O To make cancellation of x 3 term (K. Oide) x 3 xpy 2 300 - 10/60× 15 26

Crab Waist Tsukuba Version 2 RF ON K 2=0 Jy/Jx=10 % Jy=0 Jy/Jx=10 % Optics: 27 JAN 07 C 　No solenoid, skew quads SCWTR/L: bx/by = 10/300 m |K 2|=6. 6 SCWN/O : bx/by = 60/15 m |K 2|=0. 449 27

Thin SCWN/O NZ=0 NZ=+2 x/sx ・Optics: 27 JAN 07 C ・RF OFF ・Jy/Jx=10 % ・SCWTR/L K 2 =0, 0. 66, 1. 1, 2. 2, 3. 3, 6. 6 ・SCWN/O K 2=0 x/sx Crab Waist 　 Tsukuba Version 2 28

Crab Waist Simple IP Version K 2 6. 6 Quadruples only for the region between SCWTR and SCWTL SCWTR/L: thin lens. 　 29

Crab Waist Simple IP Version Jy/Jx=10 % kinematical terms of drift spaces are removed. RF OFF SCWTR/L K 2= 6. 6, 0 x/sx Quadrupoles between two sextupoles for the crab waist are thin lens. OFF crab waist sextupoles Elements between two sextupoles are investigated. x/sx Nonlinear fringe fields of quadrupoles between crab waist sextuples are removed. ON 30

HER Model Optics for Crab Waist A. Morita yx yy *2 p *2 p Sextupoles are installed in the Nikko and Oho section. 31

Dynamic Aperture of HER Jy/Jx = 10 % Octupoles(more than 30) are located at RF sections. These octupoles can not recover the dynamic aperture. 32

Summary of Studies at KEKB Several kinds of the optics for the crab waist at KEKB are studied: • Sextupoles in Oho and Nikko for the crab waist. • Sextupoles in Tsukuba for the crab waist. 　 • Sextupoles for the crab waist in Tsukuba and sextupoles to cancel the x 3 term in Oho and Nikko section. • Solenoid, QCS offset, QCS & QC 2 mutlipole elements are removed from the realistic optics. • Replace the realistic IR with the simple low-beta IR. • Optimization of octupoles for the crab waist optics in HER Maxwellian fringe of quadrupoles, kinematics terms of drift space etc. reduce the dynamic aperture. There is no good solution found so far. 33