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Crab crossing and crab waist at super KEKB K. Ohmi (KEK) Super B workshop at SLAC 15-17, June 2006 Thanks, M. Biagini, Y. Funakoshi, Y. Ohnishi, K.Oide, E. Perevedentsev, P. Raimondi, M Zobov

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Super bunch approach K. Takayama et al, PRL, (LHC) F. Ruggiero, F. Zimmermann (LHC) P.Raimondi et al, (super B) Short bunch Long bunch x is smaller due to cancellation of tune shift along bunch length Overlap factor

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Essentials of super bunch scheme Bunch length is free. Small beta and small emittance are required.

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Short bunch scheme Small coupling Short bunch Another approach: Operating point closed to half integer

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We need L=10 36 cm -2 s -1 Not infinity. Which approach is better? Application of lattice nonlinear force Traveling waist, crab waist

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Nonlinear map at collision point 1 st orbit 2 nd tune, beta, crossing angle 3 rd chromaticity, transverse nonlinearity. z- dependent chromaticity is now focused.

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Waist control-I traveling focus Linear part for y. z is constant during collision. =0

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Waist position for given z Variation for s Minimum is shifted s=-az

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Realistic example- I Collision point of a part of bunch with z, =z/2. To minimize at s=z/2, a=-1/2 Required H RFQ TM210 V~10 MV or more

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Waist control-II crab waist (P. Raimondi et al.) Take linear part for y, since x is constant during collision.

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Waist position for given x Variation for s Minimum is shifted to s=-ax

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Realistic example- II To complete the crab waist, a=1/ , where is full crossing angle. Required H Sextupole strength K 2 ~30-50 Not very strong

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Crabbing beam in sextupole Crabbing beam in sextupole can give the nonlinear component at IP Traveling waist is realized at IP. K 2 ~30-50

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Super B (LNF-SLAC) BasePEP-III C xx 4.00E E-08 yy 2.00E E-10 x (mm) y (mm) z (mm) 410 ne2.00E E+10 np4.40E E+10 /2 (mrad) 2514 xx yy 0.1

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Luminosity for the super B Luminosity and vertical beam size as functions of K2 L>1e36 is achieved in this weak-strong simulation.

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DAFNE upgrade DAFNE C97.7 xx 3.00E-07 yy 1.50E-09 x (mm) 133 y (mm) 6.5 z (mm) 15 ne1.00E+11 np1.00E+11 /2 (mrad) 25 xx yy

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Luminosity for new DAFNE L (x10 33 ) given by the weak-strong simulation Small s was essential for high luminosity netune s L(K2=10) E , E , E , E , z=10mm E , z=10mm E , (3.3) 1.00E , (4.8) () strong-strong, horizontal size blow-up

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Super KEKB Crab waist xx 9.00E E-09 yy 4.50E E-11 x (mm) y (mm) z (mm) s ne5.50E E+1 0 np1.26E E E+1 0 /2 (mrad) 015 xx yy > Lum (W.S.)8E E E E E+35 Lum (S.S.)8.25E354.77E359E353.94E354.27E35 Horizontal blow-up is recovered by choice of tune. (M. Tawada)

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Particles with z collide with central part of another beam. Hour glass effect still exists for each particles with z. No big gain in Lum.! Life time is improved. x =24 nm y =0.18nm x=0.2m y =1mm z=3mm Traveling waist

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Small coupling

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Traveling of positron beam

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Increase longitudinal slice x =18nm, y =0.09nm, x =0.2m y =3mm z =3mm Lower coupling becomes to give higher luminosity.

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Why the crab crossing and crab waist improve luminosity? Beam-beam limit is caused by an emittance growth due to nonlinear beam- beam interaction. Why emittance grows? Studies for crab waist is just started.

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Weak-strong model 3 degree of freedom Periodic system Time (s) dependent

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Solvable system Exist three J’s, where H is only a function of J’s, not of ’s. For example, linear system. Particles travel along J. J is kept, therefore no emittance growth, except mismatching. Equilibrium distribution

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y pypy

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

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Schematic view of equilibrium distribution Limited emittance growth

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More degree of freedom Gaussian weak-strong beam-beam model Diffusion is seen even in sympletic system.

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Diffusion due to crossing angle and frequency spectra of Only 2 y signal was observed.

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Linear coupling for KEKB Linear coupling (r’s), dispersions, ( , =crossing angle for beam-beam ) worsen the diffusion rate. M. Tawada et al, EPAC04

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Two dimensional model Vertical diffusion for =0.136 D C <

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3-D simulation including bunch length ( z ~ y ) Head-on collision Global structure of the diffusion rate. Fine structure near x =0.5 Contour plot (0.51,0.51) (0.7,0.51) Good region shrunk drastically. Synchrobeta effect near y~0.5. x~0.5 region remains safe.

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Crossing angle ( z / x ~1, z ~ y ) Good region is only ( x, y )~(0.51,0.55). (0.51,0.51) (0.7,0.51) (0.51,0.7)

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Reduction of the degree of freedom. For x~0.5, x-motion is integrable. (work with E. Perevedentsev) if zero-crossing angle and no error. Dynamic beta, and emittance Choice of optimum x

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Crab waist for KEKB H=25 x p y 2. Crab waist works even for short bunch.

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Sextupole strength and diffusion rate K2=

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Why the sextupole works? Nonlinear term induced by the crossing angle may be cancelled by the sextupole. Crab cavity Crab waist Need study

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Conclusion Crab-headon Lpeak=8x Bunch length 2.5mm is required for L= Small beam size (superbunch) without crab-waist. Hard parameters are required x =0.4nm x =1cm y =0.1mm. Small beam size (superbunch) with crab-waist. If a possible sextupole configuration can be found, L=10 36 may be possible. Crab waist scheme is efficient even for shot bunch scheme.

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