Beam-beam effects in crab
crossing
K. Ohmi
MAC 07-2
29 Nov.-5 Dec. 2007

Contents• Simulation of knob scan
• Beam-beam halo with weak-strong
simulation in SAD• Emittance growth due to wake force
with offset orbit and crabbing.
• Correlation of Life time and beam size.
Touschek life time with beam-beaminteraction.

Simulation of knob scan
M. Tawada• Current 0.8/1.4 mA/bunch (HER/LER)• εx = 24/18 nm (HER/LER) 1% coupling• βx/y = 80/0.7 cm (both)• νx/y/z = 0.511/0.580/0.025Start from this initial Tilt and Dispersion error
L0=8.3x1030 cm-2s-1/bunch
LER (1unit)

HER (1unit)

r1 (mrad)-7.51 (3.17)-2.44 (0.53)

r2 (mm)1.24 (0.22)-0.1 (0.43)

r3 (/km)

367 (59.38)

423 (48.72)

r4 (mrad)-103.7 (25.02)-230 (36.85)

ey (mm)1.08 (0.36)-1.68 (0.59)

eyp (mrad)-55.0 (18.98)

61.8 (21.65)

LER r1

LER r2

LER r3

LER r4

LER eyLER eypHER r1

HER r2

HER r3

HER r4

HER eyHER ηy’

This may have two
peak at ηy’=-200.

After 1 cycle, Tilt, Dispersion error
L1=12.2x1030 cm-2s-1/bunch
LER (1unit)

HER (1unit)

r1 (mrad)-17.23 (3.17)-12.5 (0.53)

r2 (mm)0.0 (0.22)

0.0 (0.43)

r3 (/km)

0.0 (59.38)

0.0 (48.72)

r4 (mrad)-20 (25.02)

0.0 (36.85)

ey (mm)0.0 (0.36)

0.0 (0.59)

eyp (mrad)

300 (18.98)

230 (21.65)

Current dependence at this condition(after 1-st cycle)• Luminosity degradation is remarkable athigh current.

LER r1-2 (means 2nd cycle)

LER r2-2

HER ey-2

HER eyp-2

Tilt, Dispersion error after 2nd cycle
L2=18.0x1030 cm-2s-1/bunch
LER (1unit)

HER (1unit)

r1 (mrad)-14 (3.17)-17.5 (0.53)

r2 (mm)0.0 (0.22)

0.0 (0.43)

r3 (/km)

0.0 (59.38)

0.0 (48.72)

r4 (mrad)-35 (25.02)-10.0 (36.85)

ey (mm)0.0 (0.36)

0.0 (0.59)

eyp (mrad)

0 (18.98)

0 (21.65)

DownhillSimplex after 2nd cyclewith HER_r1,HER_r4,LER_r1, LER_r4
L3,simp=23.7x1030 cm-2s-1/bunch

3rd regular scan with only r1,r4 after
2nd scanHER r1-3

HER r4-3

Tilt, Dispersion error after 3rd scanL3,regular=21.5x1030 cm-2s-1/bunch
LER (1unit)

HER (1unit)

r1 (mrad)-16 (3.17)-17.5 (0.53)

r2 (mm)0.0 (0.22)

0.0 (0.43)

r3 (/km)

0.0 (59.38)

0.0 (48.72)

r4 (mrad)-20 (25.02)

10.0 (36.85)

ey (mm)0.0 (0.36)

0.0 (0.59)

eyp (mrad)

0 (18.98)

0 (21.65)

Summary for knob scansimulation• R1 and R4 mean rotation of real and momentum
space, respectively.
• R1(L)=R4(L)=R1(H)=R4(H) (others=0) means simple
rotation of both beam should result no luminosity
degradation.
• R1-R4 was not resolved and dispersion was mislead
due to error each other in 1st cycle.
• Dispersion error was corrected, and R1-R4 tend tocoincide for both ring at 2nd cycle.
• Tolerance of R4 is rough than that of R1.
• Regular scan does not seem to have problem.
• Simplex method also gave high luminosity.

Halo simulation using
Gaussian weak-strong model• Long term simulation with a small number ofparticles.
• 10 particles and 107 turns for linear arcmodel.
• 10 particles and 106 turns for SAD model.
• Aperture H ~30 σx, V ~75σy (Ax~12 µm Ay~1µm).

Simple arc transformation
using matrix trans.

No beambeam

H-axis 0-12.8σx (0.1σx/unit)
V-axis 0-64σy (0.5σy/unit)
Contour plot with log scale

0 µm+100 µm-100 µm
Symmetric for Horizontal offset
The hor. and ver. halo do not matter.

Horizontal offset induces
vertical halo

0 µm
20 µm

H-axis 0-12.8σx (0.1σx/unit)
V-axis 0-64σy (0.5σy/unit)
Contour plot with log scale
50 µm
100 µmiSize works for vertical halo
Experiments showed iSize did not work for life
time improvement.

The vertical halo does not matter.
εy 0.6x10-10
1.8x10-10
5.4x10-10

H-axis 0-12.8σx (0.1σx/unit)

V-axis 0-64σy (0.5σy/unit)Crossing collision with 11 mrad

Hoffset 0 µm
50 µm
100 µmNo asymmetry appeared for H offset
H-axis 0-12.8σx (0.1σx/unit)

V-axis 0-64σy (0.5σy/unit)Effect of noise• Fast horizontal
turn by turn noise,2 µm and 10 µm• Slow horizontal
noise ~30 Hz
20 µm and 100 µm• No strong effect

SAD tracking• Simple model did not explain
experiments, short life time, its
asymmetry for H offset, iSize did nothelp us.
• Lattice nonlinearity may affect KEKB
performance more than our guess.
• Beam-beam code based on weakstrong model installed in SAD (1994) isrevived.

HERNo beam-beam

H-axis 0-12.8σx (0.1σx/unit)
V-axis 0-64σy (0.5σy/unit)

0 µm+100 µm-100 µm

No remarkable asymmetry. No beam-beam case was worst.

LERNo beam-beam

H-axis 0.1σx/unit
V-axis 0.5σy/unit

0 µm+100 µm• Asymmetry is seen in vertical tail.
• No beam-beam is no problem-100 µm

HER with errors correspond to 1% εy0 µm+100 µm-100 µm

LER with errors correspond to 1% εy0 µm+100 µm-100 µm

Summary of halo simulation• Simulation for halo formation is performedwith a weak-strong simulation based on
Gaussian model.
• The halo did not seem to affect the beam Lifetime.
• Life time symmetry was not seen.
• Non Gaussian model will be tried, if this typeof halo could limit the luminosity in KEKB.

Emittance growth due to wake force
with offset orbit and crabbing• Beam with crab angle along the ringexperiences z dependent kick due to wakefield, with the result that it can be distorted tobanana shape.
• When the center of the transverse wake
deviate, similar dipole kick is induced, withthe result that the beam can be distorted tobanana shape.
• The vertical distortion is serious because of
the small size.

Tilt due to the transverse
wake force (Ikeda)• Measured by Ieiri.
# d" x &rW%( =) e 0 ,$ dI ' I = 08e*+ 0# d" & 8e)* 0
W0 = % x ($ dI ' I = 0 re +!

# d" x &)1%
( = 4A$ dI ' I = 0= 1.7 "10 6 m#2!

!

!
!

!
NreW 0% x = 7.5 &10'6$eV #eV % $"x #crab =$ z = 0 rf z = 1.25 &10'4EEc"x #2 =

*wake z=0~-0.08

Even Δν=0.006, the kickis 1/30 of crab kickz(m)νx=0.55
Spectrum by simulation withthis condition"x #crab =!

eV #eV % $$ z = 0 rf z = 1.25 &10'4
EEc

Change ofequilibrium
distribution••<xz> does notchange.
<x’z> change alittle σx’ σz/30.

The wake effectof x-z tilt beam isweak.

Asymmetric wake
• Assume beam off set shifts 1mm.
Nre &"py =W1y (z $ z') %1 (z')dz''z#"1 (z') = #y" 0 (z')!

!
!

& z2 )
1" 0 (z) =exp(% 2 +2#$ z' 2$ z *( z +Nre W1%y"py #Erfc*$ &z) 2'& z ,Check the code
• Motion of first 2 turns was consistent
with analytic estimate.
• Obtain an equilibrium distribution forradiation damping and excitation.

Beam size and y-z tilt
• They oscillate initial stage, but arrive at anequilibrium• Emittance is large compare than tilt.

* W=W0k** Threshold of strong head-tail instability, k=3

• Emittance (phase space volume)
increases.
• Equilibrium shape depends on νy.

Radiation damping/excitation off• Beam oscillates with banana shape, but
its size does not increase.

Summary of wake effect for
crabbing beam
• Asymmetry of 1mm is perhaps pessimistic.
• It does not seem to be big issue in present
KEKB, though it depends on the asymmetry
amplitude.
• The effect is more serious for Low emittancering.
• What is the mechanism of the emittance
growth? A kind of anomalous emittance
growth.

Correlation of life time and beam size(emittance)• We made effort to realize a small beam size to get highluminosity. The small beam size seems to give a short life
time in many cases.
• We can not achieve high luminosity due to the short life
time.
• The life time is related to asymmetry in the horizontaloffset.
• Large angle scattering can be a reason for the beambeam limit.
• Ohnishi had showed dynamic aperture shrink due to thebeam-beam force.

iSize scan ~ V emittance scan
HER/LER V size vs. life
HER iSize bump 0.4 -> 0 mm
Short life time at small size foriSize scan. LER, which respond
HER, also have the samefeature.
HER

Short life time at small sizefor R4crab scan.

LERHER/LER V size vs. life - II
HER iSize bump 0.4 -> 0 mm
Short life time at small sizefor iSize scan.
Short life time at large beam
size LER ηx

HERLERShift Summary 29Nov day shift
Size-minimum simplex for HER

AR injection

D7A/B CRB
Beam phase
D8-D ArcCoupler

Measurement higher order momentof Horizontal beam distribution
J. Flanagan

4-th2nd3rd

Pure emittance iSize bump
N. Iida & H. Koiso• Present iSize bump gives dispersion
whole of ring, even IP.
• Pure emittance bump without
dispersion at IP will be tried to study thecorrelation of the beam life andemittance.

Intrabeam scattering• Cross-section of e-e scattering),d"4re2 +
43.
= 2 4&222
d# $ p% +(1& sin 2' cos 2 ( ) 1& sin ' cos ( .
*-• Touschek life
! 1 = 2N d# dx dx A(x ,x ,$,% ) d& v ) (x )) (x )' ' '"
d#1212(12# dx" (x) = 1! • A=1 or 0 for outside/inside of aperture.
!

Longitudinal kick due to a large
angle scattering• Longitudinal component of the kick is enhanced by
the relativistic factor, γ.
• Betatron oscillation is induced by the longitudinalkick via dispersion.
• Transverse component of the kick is neglected.
• The cross-section is! integrated for ϕ .
• Aperture is function of pt and ϕ.
d" 8$re2 ( 21 += 2 4* 3 'sin#d# % p& )cos # cos# -,A(x1,x 2 ,",# ) = A( p$ ," )!

Touschek life, integration• Trivial integrations are performed.
(x1=x2,y1=y2,z1=z2)
1d&= 2N ' d# ' dp1 ' dp2 A( p$ ,#,% ) v $ ( P (p1 ) ( P (p2 )"d#d& ( p% ,$ )= 2N # dr" (r) # d$ # dpA( p% ,$ )cp% " P (p)d$2!

• This integral is performed for particles out of! aperture. Generally θ is integrated from 0 to δbucket/γpt; A=1 for θ < δbucket/γpt, otherwise A=0.
• We integrate it for all phase space variable withconsidering aperture for all directions.

Numerical integration• Outside of the aperture area, A(pt,θ), is
found by SAD.
1
1 32N#cre2
1* 2
1 2=dr%(r) & dp & 0 d cos'A( p( ,' ) 3 , 3 )% P (p)&/2"$p( +cos ' cos' .

!

!

3 3*#p32N"cre2
12#p( #p( (2=dr$(r) % dp % 0 d&'A( p( ,&' ) 4 ,)%/$ P (p)33#p( + &'&' .

• Monte Carlo integration with realistic
1events.
"# = $p cos &dr"(r)$#V2%"x = #x "$!
!

"px = #'x "$!

V: Volume of the beam. The
exact value is possible tocalculate numerically.

Monte Carlo integration• Distribution of p is realistic one (ρp).
" (p) = %# (p $ p )• Uniform distribution for Δδ<δbucket. The
integral Δδ>δbucket is given.
npii=1!

nf (" ) = %" (#" $ #"i )i=1

3$p',i ,
1 32N#cr %&bucket1 )2$ 3 p',i=A( p',i ,%&i ) 4 +(/.
33"$V
n i=1p',i * %&i%&i !
2enp" >!

#$%

Life time estimation• The life time estimation is underprogress.
• A preliminary result did not show
asymmetry of the life time. Consistentwith Ohnishi’s result.

How do we solve?

Low emittance parameter• This effect does not depend on designemittance.
• SuperBは低電流なのでthreholdはk=5

Beam size and tilt
• A large emittance growthTune dependence