Crab crossing and crab waist at super KEKB

Transcription

1 Crab crossing and crab waist at super KEKB K. Ohmi (KEK) Super B workshop at SLAC 1-17, June 6 Thanks, M. Biagini, Y. Funakoshi, Y. Ohnishi, K.Oide, E. Perevedentsev, P. Raimondi, M Zobov

2 Short bunch ξ L β ξ N N ε βε β Super bunch approach K. Takaama et al, PRL, (LHC) F. Ruggiero, F. Zimmermann (LHC) P.Raimondi et al, (super B) N ε β ε βε εβ θσ z > 1 εβ ( < 1) θσ z Overlap factor Long bunch ξ N L θσ ε β ξ N θσ N θσ z > σ ε β z β > z z β ε β ε ξ is smaller due to cancellation of tune shift along bunch length θ

3 Essentials of super bunch scheme ξ N L θσ ε β N θσ z z β ε β β ε β Keep, and. ε ε β ε β ξ N θσ z β ε L β > ε β θ Bunch length is free. Small beta and small emittance are required.

4 Short bunch scheme ξ L N ε βε β N ξ ε β N ε βε β > σ z β Keep ε, β and. ε ε β Small coupling Short bunch L Another approach: Operating point closed to half integer ν +. ξ L

5 We need L=1 36 cm - s -1 Not infinit. Which approach is better? Application of lattice nonlinear force Traveling waist, crab waist

6 Nonlinear map at collision point H = a i i + bij i j + cijk i j k = = (, p,, p, z, δ ( = E/ E)) i 1 M = ep( : H :) * = [ H, ] + [ H,[ H, ]] +... = a + b + c' +... i ij j ijk j k 1 st orbit nd tune, beta, crossing angle 3 rd chromaticit, transverse nonlinearit. z- dependent chromaticit is now focused.

7 Waist control-i traveling focus M = e Μ e : H I : : H I : H H I = I = + = + azp P ap z Linear part for. z is constant during collision. H δ = δ = δ ap z az az β + β α β α t β β = T T = α γ α γ az 1 β β 1 az T = 1 α=

8 Waist position for given z Variation for s Minimum β is shifted s=-az ( ) az az s+ az s + az β + β + β β t β β M() s M () s = az 1 s+ az 1 β β β β

9 Realistic eample- I Collision point of a part of bunch with z, <s>=z/. To minimize β at s=z/, a=-1/ Required H H RFQ TM1 V I = 1 p z 1 c * ββω pc e = V~1 MV or more

10 Waist control-ii crab waist (P. Raimondi et al.) M H H = + = + I = I ap P ap Take linear part for, since is constant during collision. 1 a T = 1 = e Μ : H I : : H I : H p = p = p ap a a β + β α β α t β β = T T = α γ α γ a 1 β β e

11 Waist position for given Variation for s ( ) a a s+ a s + a β + β + β β t β β M() s M () s = a 1 s+ a 1 β β β β Minimum β is shifted to s=-a

12 Realistic eample- II To complete the crab waist, a=1/θ, where θ is full crossing angle. Required H H Setupole strength K I = 1 p θ = 1 B'' L 1 1 β K ~3- * * p/ e θ ββ β Not ver strong

13 Crabbing beam in setupole Crabbing beam in setupole can give the nonlinear component at IP Traveling waist is realized at IP. H I = ap z z * = β () s ζ ()() s s * β K 1 B'' L 1 1 β * = * p/ e θ ββ β K ~3-

14 Super B (LNF-SLAC) ε ε β β σ φ ξ F ξ = ρ( z) δ( s + z /) dzds = φz ξ

15 Luminosit for the super B Luminosit and vertical beam size as functions of K L>1e36 is achieved in this weak-strong simulation. L (cm-s-1) e e e e+36 1.e+36 1e+36 8e+3 6e+3 4e+3 e K sig (um) K

16 DAFNE upgrade ε ε β β σ φ ξ ξ

17 Luminosit for new DAFNE L (1 33 ) given b the weak-strong simulation Small ν s was essential for high luminosit ν σz=1mm σz=1mm () strong-strong, horizontal size blow-up

18 Horizontal blow-up is recovered b choice of tune. (M. Tawada) Super KEKB Crab waist ε ε β β σ ν φ ξ ξ 8.E3 4.77E3 9E3 3.94E3 4.7E3

19 Particles with z collide with central part of another beam. Hour glass effect still eists for each particles with z. No big gain in Lum.! Life time is improved. Traveling waist ε =4 nm ε =.18nm β=.m β=1mm σz=3mm e+3 4e+3 4 L 3e+3 e+3 sig (um) 3 1e Kz Kz

20 Small coupling 1e+36 8e+3 n=.3 L 6e+3 4e+3 n=.8 e+3 blue: travel waist, red: normal e (nm)

21 Traveling of positron beam

22 Increase longitudinal slice ε =18nm, ε =.9nm, β =.m β =3mm σ z =3mm Lower coupling becomes to give higher luminosit. 3 1e+36. 8e+3 1 slices sig (um) 1. L 6e+3 4e+3 slices 1 e turn turn

23 Wh the crab crossing and crab waist improve luminosit? Beam-beam limit is caused b an emittance growth due to nonlinear beambeam interaction. Wh emittance grows? Studies for crab waist is just started.

24 Weak-strong model 3 degree of freedom Periodic sstem Time (s) dependent H (, p,, p, z, p ; s) = H'( J, ϕ, J, ϕ, J, ϕ ; s) z ϕ( s+ L) = ϕ( s) + πν

25 Solvable sstem Eist three J s, where H is onl a function of J s, not of ϕ s. For eample, linear sstem. Particles travel along J. J is kept, therefore no emittance growth, ecept mismatching. dj ds H = = J = const ϕ dϕ H( J) = dϕ = πν( J) ds J Equilibrium distribution ψ J J J J ε1 ε ε3 1 3 ( ) ep ε: emittance

26 p

27 One degree of freedom Eistence of KAM curve Particles can not across the KAM curve. Emittance growth is limited. It is not essential for the beam-beam limit.

28 Schematic view of equilibrium distribution Limited emittance growth 4 3 (um) φ/π ρ

29 More degree of freedom Gaussian weak-strong beam-beam model Diffusion is seen even in smpletic sstem.

30 Diffusion due to crossing angle and frequenc spectra of < > 4 Amplitude 1e-16 1e-18 1e- 1e- 1e-4 1e tune 1e-16 1e-18 <> (um^) mrad 4 mrad 1 mrad mrad turn Onl ν signal was observed. Amplitude Amplitude 1e- 1e- 1e-4 1e tune 1e-18 1e- 1e- 1e-4 1e-6 1e

31 Linear coupling for KEKB Linear coupling (r s), dispersions, (η, ζ=crossing angle for beam-beam) worsen the diffusion rate. M. Tawada et al, EPAC4 <> (µm ) r=1 mm turn r= mm r= mm r= mm <> (µm ) η * =3. mm turn 3 η * =1. mm η * =. mm η * = mm 4 1

32 Two dimensional model Vertical diffusion for ξ=.136 D C <<D γ for wide region (painted b black). No emittance growth, if no interference. Actuall, simulation including radiation shows no luminosit degradation nor emittance growth in the region. Note KEKB D γ =1-4 /turn DAFNE D γ = /turn (.7,.1) (.1,.7) (.7,.7) ,.7) 1 1 nu 1 1 nu KEKB D γ =. (.1,.1) (.1,.1) (.7,.1)

33 3-D simulation including bunch length (σ z ~β ) Head-on collision Global structure of the diffusion rate. Fine structure near ν =. Contour plot (.7,.1) Good region shrunk drasticall. Snchrobeta effect near ν~.. ν~. region remains safe nu 1 1 nu (.1,.1)

34 Crossing angle (φσ z /σ ~1, σ z ~β ) Good region is onl (ν, ν )~(.1,.). (.7,.1) (.1,.7) 1 1 nu 1 1 nu (.1,.1)

35 Reduction of the degree of freedom. For ν~., -motion is integrable. (work with E. Perevedentsev) lim = ν.+ D C, if zero-crossing angle and no error. L ν + Dnamic beta, and emittance lim ν.+ Choice of optimum ν 1 (crossing angle)+(coupling)+(fast noise) < σ, lim ν.+ p =

36 H= p. Crab waist for KEKB Crab waist works even for short bunch...1 i.1. mrad 11 mrad & K= 11 mrad turn

37 Setupole strength and diffusion rate K= nu 1 1 nu nu 1 1 nu 1 1 nu nu 1 1 nu 1 1 nu nu

38 Wh the setupole works? Nonlinear term induced b the crossing angle ma be cancelled b the setupole. Crab cavit ep( : F:) = ep( : θ p z:)ep(: θ p z:) = 1 Crab waist ep( : F :) = ep( : θ pz:)ep( K : p :) =... Need stud ep(: F :) M ep( : F:) BB

39 Conclusion Crab-headon Lpeak=81 3. Bunch length.mm is required for L=1 36. Small beam size (superbunch) without crab-waist. Hard parameters are required ε =.4nm β =1cm β =.1mm. Small beam size (superbunch) with crab-waist. If a possible setupole configuration can be found, L=1 36 ma be possible. Crab waist scheme is efficient even for shot bunch scheme.