Beam-beam simulations with crossing anlge + crab-waist

M. Biagini, M. Zobov, LNF-INFNP. Raimondi, SLAC/INFNI. Koop, D. Shatilov, BINP

E. Paoloni, Pisa University/INFN

SuperB III Workshop, SLAC, 14-16 June 2006

BB simulations• New “crossing angle + crab waist” idea

has solved disruption problems related to collisions with high current, small sizes beams back to two “conventional” rings

• With very small emittances and relatively low currents (comparable to present B-Factories values) a Luminosity of 1036 cm-2 s-1 is reachable without large emittance blow-up

Vertical waist has to be a function of x:Z=0 for particles at –σx (- σx/2θ at low current)Z= σx/θ for particles at + σx (σx/2θ at low current)Crabbed waist realized with a sextupole in phase with the IP in X and at π/2 in Y

2Sz

2Sx

θz

x

2Sx/θ

2Sz*θ

e-e+βY

Crabbed waist removes bb betratron couplingintroduced by the crossing angle

Luminosity considerationsIneffectiveness of collisions with large crossing angle is illusive!!!Loss due to short collision zone (say l=σz/40) is fully compensatedby denser target beam (due to much smaller vertical beam size!)

c r o s s2 2 c r o s s x

z

lN N l 2 /δ = = σ θσ

Number of particles in collision zone:

1 2 0

x y

N N fL4

⋅δ ⋅=

πσ σe 2 y

1yy x y

r N2 ( )

⋅δ ⋅βξ =

πγσ σ + σ

1y 1 0 y 1y34 36 2 1

e y x y

N f E(GeV) I(A)L 1 2.167 10 1.2 10 cm s

2r (cm)− −γξ σ ⋅ ⋅ξ⎛ ⎞

= + ⋅ ⋅⎜ ⎟β σ β⎝ ⎠

No dependence on crossing angle!Universal expression: valid for both, head-on and crossing angle collisions!

I. Koop et al, BINP

Tune shiftsRaimondi, Shatilov, Zobov:(Beam Dynamics Newsletter, 37, August 2005)

2 2 2x z xtan ( / 2)σ → σ θ + σ

( )

( )

e xx 2 2 2 2 2 2

z x z x y

yey 2 2 2

y z x y

r N2 tan ( / 2) tan ( / 2)

r N2 tan ( / 2)

βξ =

πγ σ θ + σ σ θ + σ + σ

βξ =

πγ σ σ θ + σ + σ

SuperB:

e xx 2 2

z

yey

y z

2r N 0.002

r N 0.072

βξ = =

πγ σ θβ

ξ = =πγ σ σ θ

2 2 2z x xtan ( / 2) 100 m 2.67 mσ θ + σ = μ σ = μ

2 2 2z x

y

tan ( / 2)8000 !!!

σ θ + σσ

One dimensional case for βy >>σx/θbut with crabbed waist for βy <σx/θ also!

I. Koop et al, BINP

“Crabbed” waist optics

1x x

x x x x1 1 11x x x xx x

y y y1 1 1y y y y

y y y y y

u 0 1 0u 0T T T T

F u 2u F 1F u

u F 0 F 1 0T T T T

F 0 F u 2u F 1

−

− − −−

− − −

⎛ ⎞⎛ ⎞ ⎛ ⎞= = =⎜ ⎟⎜ ⎟ ⎜ ⎟− −−⎝ ⎠ ⎝ ⎠⎝ ⎠

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟− − − −⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Appropriate transformations from first sextupole to IP and from IP to anti-sextupole:

IP

Δμx=πΔμy=π/2

Δμx=πΔμy=π/2

x,yT x,yT+ks -ks

Sextupole lens Anti-sextupole lens

Synchrotron modulation of ξy(Qualitative picture)

ξy(z-z0)

z-z0

Head-on collision.Flat beams. Tune shiftincreases for halo particles

Head-on collisionRound beams ξy=const

Crossing angle collision.Tune shiftdecreases for halo particles

Relative displacementfrom a bunch center

Conclusion: one can expect improvements of lifetime of halo-particles!

I. Koop et al, BINP

ξy increase caused by hourglass effect

For Super-B parameters set: Increase of ξy only by 26%

Dependence of ξy on βy for constant beam sizes at IP

2

y yy

1 zg(x(z)) dz4

⎛ ⎞ξ = β +⎜ ⎟⎜ ⎟π β⎝ ⎠

∫

I. Koop et al, BINP

Collisions with uncompressed beamsCrossing angle = 2*25mradRelative Emittance growth per collision about 1.5*10-3

εyout/εy

in=1.0015

Horizontal Plane Vertical Plane

SuperB parameters

GuineaPig modifications• With the large crossing angle scheme and long

bunches the actual collision region is very short

• The code solves Poisson equation for all the volume occupied by the particles very long computing time, not needed !

• Modification of the code to perform fields calculation in the collision region only

• Computing time was reduced by a factor 10!!

GuineaPig modifiedE. Paoloni, Pisa

Luminosity vs Number of particles /bunch

Crab-waist simulations• The new idea is being checked by

several beam-beam codes:– Guinea-Pig: strong-strong , ILC centered– BBC (Hirata): weak-strong – Lifetrack (Shatilov): weak-strong with tails

growths calculation– Ohmi: weak-strong (strong-strong to be

modified for long bunches and large angles)

Storage rings

Ohmi’s weak-strong code

K2 is the strength of the sextupolar nonlinearity introduced to have crab waist

Luminosity Vertical blow-up

DAΦNE (M.Zobov, LNF)• Hirata’s BBC code simulation

(weak-strong, strong beam stays gaussian, weak beam has double crossing angle)

• Np = 2.65x1010, 110 bunches• Ib = 13 mA (present working current)• σx = 300 μm, σy = 3 μm• βx = 0.3 m, by = 6.5 mm• σz = 25 mm (present electron bunch length)• θ = 2x25 mrad• YIP = y+0.4/(θ * x * y’) crabbed waist shift• Lo=2.33x1024 (geometrical)• L(110 bunches,1.43A) = 7.7x1032

• Lequil=6x1032

(Geometric) Luminosity

Takes into account both bb interactions and geometric factor due to crab waist

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

0 0,2 0,4 0,6 0,8 1

L/Lo

n x [1/angle]

n/θ

Peak around 0.3/θ

M.Zobov, LNF

Scan vs crab waist n/θ

Vertical Tails(max amplitude

after 10 damping times)

Vertical Size Blow-up

2

4

6

8

10

12

14

16

0 0,2 0,4 0,6 0,8 1

Ay/σy0

n x [1/angle]

n/θ

0,5

1

1,5

2

2,5

3

0 0,2 0,4 0,6 0,8 1

σy/σy0

n x [1/angle]

n/θ

M.Zobov, LNF

Present WP:νx = 0.11νy = 0.19

Possible WP:νx = 0.057 νy = 0.097

0

0,5

1

1,5

2

2,5

0 5 10 15 20 25

(0.11,0.19)(0.057,0.097)Theory

I [mA]

L [10^33]

Luminosity vs bunch currentfor 2 different working points

M.Zobov, LNF

0

2

4

6

8

10

12

14

0 10 20 30 40 50

200um,20mm200um,15mm100um,15mm

I [mA]

L [10^33]

110 bunches

M.Zobov, LNF

Luminosity with shorter bunch, smaller σx

With the present achieved beam parameters (currents, emittances, bunchlenghts etc) a luminosity inexcess of 1033 is predicted.With 2A+2A L> 2*1033 is possibleBeam-Beam limit is way above the reachable currents

Luminosity scan

Without Crab Focus

Vertical Size blow-up scan

M. Zobov

With Crab Focus

Beam-Beam TailsWithout Crab Waist With Crab Waist

Bunch core blowup

also reduced

Ay = 90

Ay = 45

Vertical tails growth Greatly reduced

(A is the amplitude in number of beamsize σ)

D.Shatilov, BINP

Beam size and tails vs Crab-waistSimulations with beam-beam code LIFETRAC

Beam parameters for DAΦNE2

An effective “crabbed” waist map at IP:0 0

0

Vy y xy

y y

′= +θ

′ ′=

Optimum is shifted from the “theoretical” value V=1 to V=0.8,since it scales like σzθ/sqrt((σzθ)2+σx

2) D.N. Shatilov, BINP

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

0,06 0,08 0,1 0,12 0,14 0,16

Qx

L/L0 at Qy = 0.09

Some resonances

0

0,2

0,4

0,6

0,8

1

1,2

0,06 0,08 0,1 0,12 0,14 0,16

no crab focus0.25/teta0.40/teta

Qx

L/L0 at Qy = 0.05

M.Zobov, LNF

2Qx = 2Qy1Qx = 2Qy

(present with crossing angle only)

No crab waist

Crab waist

Very weak luminosity dependence from damping time given the very small beam-beam blow-up

0

0,2

0,4

0,6

0,8

1

0 5 104 1 105 1,5 105 2 105 2,5 105

x^(-0.37)

x^(-0.48)

x^(-0.56)

x^(-0.50)

σy0/σy

turns

(0.057,0.097,-0.01)(0.057,0.097,+0.01)(0.11,0.19,-0.01)(0.11,0.19,+0.01)

Wigglers offDAΦNE Wigglers

SC Wigglers

M. Zobov,LNF

0,5

1

1,5

2

2,5

3

3,5

5 104 1 105 1,5 105 2 105

(0.057,0.097,-0.01)(0.057,0.097,+0.01)(0.11,0.19,-0.01)(0.11,0.19,+0.01)

turns

L, 10^33

Wigglers offSC Wigglers

DAΦNE Wigglers

LuminosityVertical blow-up

Preliminary results on Super PEPIIεx = 20 nmεy = 0.2 nmσx = 14.4 μmσy = 0.4 μmσz = 10 mmσE = 7x10-4

βx = 10 mmβy = 0.8 mmνs = 0.03C = 2.2 kmfcol = 238 MHzθ = 2 x 14 mradτx = 35 msN1 = 1.3x1011

N2 = 4.4x1010

I1 = 5 AI2 = 1.7 A

First approach with new parameters,weak-strong code

M. Zobov, D. Shatilov

L = 1.65x1035 cm-2s-1

Tune scan for Super-PEPII

M.Zobov, D.ShatilovSynchrobetatron resonances

Coupling resonance

crab focus on

crab focus off

Coupling resonance disappears

No dependence on tunes !!

Tails growth

M.Zobov, D.Shatilovn/θ = 0 n/θ = 0.6 n/θ = 1

νx = 0.5325, νy = 0.5775

νx = 0.54, νy = 0.5825

WP \ crab 0.0 0.6 1.0

(0.5325,0.5775)

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

Ay

Ax

pep2_nx5325_ny5775_xiy045_cw00

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

Ay

Ax

pep2_nx5325_ny5775_xiy045_cw06

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

Ay

Ax

pep2_nx5325_ny5775_xiy045_cw10

L=1.51E+35 L=1.65E+35 L=1.64E+35

(0.5400,0.5825)

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

Ay

Ax

pep2_nx5400_ny5825_xiy045_cw00

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

Ay

Ax

pep2_nx5400_ny5825_xiy045_cw06

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

Ay

Ax

pep2_nx5400_ny5825_xiy045_cw10

L=1.49E+35 L=1.65E+35 L=1.64E+35

Crab OnCrab Off

30 σy

10 σy15 σy

40 σy

15 σy

20 σy

L=1.5x1035 L=1.65x1035 L=1.64x1035

Conclusions• The “crossing angle with crab waist” scheme

has shown big potentiality and exciting results LNF, Pisa, BINP and KEKB physicists are

working on the bb simulation with different codes to explore its properties and find the best set of parameters

• This scheme is promising also for increasing luminosity at existing factories, as DAΦNE, KEKB and possibly PEPII