DESY Summer Student Lecture 31 st July 2002

An Introduction to thePhysics and Technologyof e+e- Linear Colliders

Lecture 7: Beam-Beam Effects

Nick Walker (DESY)

DESY Summer Student Lecture31st July 2002USPAS, Santa Barbara, 16th-27th June, 2003

Introduction

• Beam-beam interaction in a linear collider is basically the same Coulomb interaction as in a storage ring collider. But:– Interaction occurs only once for each bunch (single

pass); hence very large bunch deformations permissible.– Extremely high charge densities at IP lead to very

intense fields; hence quantum behaviour becomes important

• Consequently, can divide LC beam-beam phenomena into two categories:– classical– quantum

Introduction (continued)

• Beam-Beam Effects– Electric field from a “flat” charge bunch– Equation of motion of an electron in flat bunch– The Disruption Parameter (Dy)– Crossing Angle and Kink Instability– Beamstrahlung – Pair production (background)

Ebeam x y x y z Ne

GeV mm mm nm nm m 1010

NLC 250 8 0.1 245 2.7 110 0.75

TESLA 250 14 0.4 550 5 300 2.0

CLIC 1500 8 0.15 43 1 30 0.42

Storage Ring Collider Comparison

Linear beam-beam tune shift ,,

,2 ( )x ye b

x yx y x y

r N

Putting in some typical numbers (see previous table) gives:

0.541.44

x

y

Storage ring colliders try to keep , 0.05x y

Electric Field from a Relativistic Flat Beam

• Highly relativistic beam E+vB 2E• Flat beam x y (cf Beamstrahlung)• Assume

• infinitely wide beam with constant density per unit length in x ( (x))

• Gaussian charge distribution in y:• for now, leave (z) unspecified

1( )2 x

x

2

1 1( ) exp22 yy

yy


0 ' 0

0

( ) ( )( , ) ( ') '

( , ) Erf ( )2 2 2

y

yy

yx y

qN x z x zE y z x z y dy

qN yE y z z

Use Gauss’ theorem:0s

q

E ds

Assuming Gaussian distribution for z, the peak field is given by

0

ˆ4y

x z

qNE


10 5 5 10

2000

1000

1000

2000

10

500 nm5 nm

300μm

10

x

y

z

N

/ yy

MV/cmyE

0z

Note: 2Ey plotted

Assuming a Gaussian distribution for (z)


10 20 30 40 50

500

1000

1500

10

500 nm5 nm

300μm

10

x

y

z

N

/ yy

MV/cmyE

0z

Assuming a Gaussian distribution for (z)

effect of x width

Equation Of Motion

F = ma:

Changing variable to z:

0

2 ( , )( ) yqE y t

y tm

20

2

20

2 ( , )( )

( )Erf (2 )2

2

y

zy

x o

qE y zy z

m c

y zq N z

m c

2

20 04eqrm c

2 2 ( )( ) Erf (2 )2

ez

x y

Nr y zy z z

2( ) ( )y t c y z

whyz(2z)?

Linear Approximation and the Disruption Parameter

Taking only the linear part of the electric field:

Take ‘weak’ approximation:y(z) does not change during interaction y(z) = y0

Thin-lens focal length:

Define Vertical Disruption Parameter

2 ( )

4 (2 )( ) ( )e

x y

k z

Nr zy z y z

0

0 0

4 (2 )

2 1

ez

x y

e

x y

Nry y z dz

Nr y yf

21 e

x y

Nrf

2 e zzy

x y

NrDf

2 e z

yx y y

NrD

exact:

Number of Oscillations

Equation of motion re-visited:

Approximate (z) by rectangular distribution with same RMS as equivalent Gaussian distribution (z)

2( ) (2 ) ( )y

zz

Dy z z y z

3 2 1 0 1 2 30

0.1

0.2

0.3

0.4

3 z3 z

12 3 z

2

3( ) ( );23

yy

z

Dy z y z z

14

22

323

30.21

2

y z

z

yy

D kk

DD

half-length!

Example of Numerical Solution

500 0 500 1000 1500

40

20

0

20

40102 10

500 nm5 nm

300 μm250 GeV

x

y

z

N

E

27.7

1.1yD

( )/ μmz ct

/ nmygreen: rectangular approximation

black: gaussian

Pinch Enhancement

• Self-focusing (pinch) leads to higher luminosity for a head-on collision.

3, ,1/ 4

, , ,3,

0.81 ln 1 2ln

1x y x y

Dx y x y x yx y z

DH D D

D

‘hour glass’ effect

Empirical fit to beam-beam simulation results

Only a function of disruption parameter Dx,y

The Luminosity Issue: Hour-Glass

= “depth of focus”

reasonable lower limit for is bunch length z

2 1 0 1 2Z

3

2

1

0

1

2

3

Y

2 1 0 1 2Z

3

2

1

0

1

2

3

Y

Luminosity as a function of y

200 400 600 800 1000

11034

21034

31034

41034

51034

300z m

100z m

500 m

700 m

900 m

( )y m

2 1( )L cm s

2

4b

x y

n N fL

1BS z

Beating the hour glass effect

Travelling focus (Balakin)

• Arrange for finite chromaticity at IP (how?)

• Create z-correlated energy spread along the bunch (how?)

*z y


Travelling focus



*z y


Travelling focus



ct


foci ‘travel’ from z = 0 to z =

t = 0

3 z

f = c t t > 0

3 z

3 2z c t

3 z

RMS

2RMS

2

2

2

2

z

y

y

z

y

z

chromaticity:

y yf

travelling focus:

NB: z correlated!

1

654

32

The arrow shows position of focus for the read beam during travelling focus collision

Kink Instability

Simple model: ‘sheet’ beams with:

Linear equation of motion becomes

Need to consider relative motion of both beams in t and z:

Classic coupled EoM.

,,

12 3x z

x z

2

2 3( ) ( );6 2

yz

z

Dy z y z z

22

1 0 1 2

22

2 0 1 2

( , ) ( )

( , ) ( )

c y t z y yt z

c y t z y yt z

2 20 2

26

y

z

Dc

Kink Instability

and substituting into EoM leads to the dispersion relation:

1(2) 1(2) expy a i kz t

2 2 2 2 2 2 2 40 0 04c k c k

Assuming solutions of the form

Motion becomes unstable when 2 0, which occurs when

02kc

Kink Instability

Exponential growth rate:

Maximum growth rate when

Remember!

Thus ‘amplification’ factor of an initial offset is:

122 2 2 4 2 2 2

0 0 04 c k c k

0 03 ;2 2

k ic

3 3 3 /2 2

z zzt t c

c c

14

0.60 1exp exp2 2 2

yDy

t D e

For our previous example with Dy ~28, factor ~ 3

Kink Instability

0.1

20

300μm

y

y

z

y

D

Pinch Enhancement

geomL L H H= enhancement factor

2

2exp4

y

y

results ofsimulations:

High Dy example: TESLA 500

24yD

Disruption Angle

Remembering definition of Dyz

yDf

The angles after collision are characterised by

02 2

( )y yx x e e

z z x y x

DD Nr Nr

Numbers from our previous example give

OK for horizontal plane where Dx<1

For vertical plane (strong focusing Dy > 1), particles oscillate:

0 467 μrad

2

3( ) ( );23

yy

z

Dy z y z z

1 1 14 4 2

0 67 μrad3 3

yy

z y

D

D

previous linear approximation:

Disruption Angle: simulation results

- 400 - 200 0 200 400

0

50

100

150

200

250

- 400 - 200 0 200 4000

200

400

600

800

1000

horizontal angle (rad) vertical angle (rad)

Important in designing IR (spent-beam extraction)

Beam-Beam Animation Wonderland

Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code

(written by D. Schulte, CERN).

Examples of GUINEAPIG Simulations

NLC parametersDy~12

Nx2Dy~24

NLC parametersDy~12

Luminosity enhancement HD ~ 1.4

Not much of an instability


Nx2Dy~24

Beam-beam instability is clearly pronounced

Luminosity enhancement is compromised by higher sensitivity to initial offsets


Beam-beam deflection

Sub nm offsets at IP cause large well detectable offsets (micron scale) of the beam a few meters downstream

Beam-beam deflectionallow to control collisions

Beam-Beam Kick

Beam-beam offset gives rise to an equal and opposite mean kick to the bunches – important signal for feedback!

For small disruption ( D1) and offset (y/y<<1)

012bb

y

y

For large disruption or offset, we introduce the form factor F:

01 /2bb yF y

( )F

Long Range Kink Instability & Crossing Angle

To avoid parasitic bunch interactions in the IR, a horizontal crossing angle is introduced:

angle c

2bctl

2b cctX

parasitic beam-beam kick: ,2 e

x yNrr rr

Long Range Kink Instability & Crossing Angley

y

IP

small vertical offset ( = y/y) gives rise to beam-beam kick

resulting vertical offset at parasitic crossing gives next incoming bunches additional vertical kick: IP offset increases

instability

01 ( )2ip F

e+e

02 2

2 ( )ipe e

c

lNr Nr FX l

l


offset at IP of k-th bunch:

distance from IP to encounter with l-th bunch (l < k)k k k

,0 ,

0,0 2

/

( )

k k lk lk y

ek l

c y

l

Nr F

0,0 2

2 ( )

k k k

ek l

c y

Nr F

( )2

blk

k l t cl

contribution from encounter with l-th bunch:

NB. independent of llk

total offset:21

0,0 2

1

2 /( );k

e x zk k i x y

i c y c

NrC F C D D


Assuming all bunches have same initial offset:

For small disruption and small offsets,,0 0k

( )i iF

10(1 )k

k C

since2

/ 1x zx y

c

C D D

0

1 ( 1)k k C

thus ( 1) 1bm C where mb = number of interacting bunches

example:Dx,y = 0.1, 10x = 220 nmz = 100 mc = 20 mrad

C = 0.012 mb < 80 for factor of 2 increase

Crab Crossing

2 2 2,

20mr 100μm 2μm

x projected x c z

c z

factor 10 reduction in L!

x

xuse transverse (crab) RF cavity to ‘tilt’ the bunch at IP

ˆ22ˆ( ) sin

ˆ4

RFRF

RFc cav ip z

RF

V zzV z V

V

RF kick

Beamstrahlung

Magnetic field of bunch B = E/c

Peak field:

From classical theory, power radiated is given by

For E = 250 GeV and z = 300m:

maxmax

0

2 1160Tesla2 x z

E qNBc c

45 -3

2 ; 8.85 10 m..GeV2cC EP C

-11 1.4 mcBE

32 GeVzradE P t P

c

E/E ~ 12%note: / 200μrad 1/ 2μradz

Beamstrahlung

Most important parameter is

2 2

20

2 23

c e

s

B e F pE B m

c

e

s

BBF

p

critical photon frequencyCompton wavelengthlocal bending radiusbeam magnetic fieldSchwinger’s critical field (= 4.4 GTesla)em field tensorelectron 4-momentum

Beamstrahlung: photon spectrum

NOT classical synchrotron radiation spectrum!Need to use Sokolov-Ternov formula:

5 233

2

( ) ( )1

2 13 1 1

BS

c

yF K d Ky

yE

yy y

classical quantum theoretical

0.001 0.0050.01 0.05 0.1 0.5 10.01

0.05

0.1

0.5

1

5

10

0.1

1

10

1/3

/y E

Beamstrahlung Numbers

56 ( )

2.4

e eavr

z x y

max avr

Nr

average and maximum

photons per electron: 2/32.54

1avrz

e avr

n

2

22/31.24

1 (1.5 )avrz

BSe avr

EE

average energy loss:

BS and n (and hence ) talk directly to luminosity spectrum and backgrounds

Beamstrahlung energy loss

56 ( )

e eavr

z x y

Nr

2

22/31.24

1 (1.5 )avrz

BSe avr

3 2

2 20

0.862 ( )

e cmBS

z x y

er E Nm c

In lecture 1, we used the following equation to derive our luminosity scaling law

Now we have this:

with

under what regime is our original expression valid?

Luminosity scaling revisited

BSbeam

,y nL P

low beamstrahlung regime 1:

32

BSbeam

,z y nL P

high beamstrahlung regime 1:

*y z

homework: derive high BS scaling law

Beamstrahlung Numbers: example

10

500GeV220nm2nm

110μm

0.75 10

x

y

x

E

N

,0.28 210GeV

0.661.3

7.5%

avr c

max

BS

E

n

TESLA

Why Beamstrahlung is bad

• Large number of high-energy photons interact with electron (positron) beam and generate e+e pairs– Low energies (0.6), pairs made by incoherent process

(photons interact directly with individual beam particles)

– High energies (0.6100), coherent pairs are generated by interaction of photons with macroscopic field of bunch.

– Very high energies (100), coherent direct trident production e e e+ e

TESLA 0.5TeV ~ 0.06NLC 1TeV ~ 0.28CLIC 3TeV ~ 9

• Beamstrahlung degrades Luminosity Spectrum

Pair Production

• Incoherent e+e pairs 0.6– Breit-Wheeler:– Bethe-Heitler:– Landau-Lifshitz:

• Coherent e+e pairs 0.6100

– threshold defined by

e e e e e e

e e e e e e

20

2

1

s

Bm c B

E

0.001 0.005 0.01 0.05 0.1 0.5 10.01

0.05

0.1

0.5

1

5

10

0.1

1

10

1/3

/y E

for >1 / ~ (1)E O 1

for intermediate colliders (Ecm<1TeV), incoherent pairs dominate

Pair Production

0.1 1

0.01

0.1

1

PT (

GeV

/c)

(rad)

Td

z

Pr rcB

pairs curl-up (spiral) in solenoid field of detector

rd

ld

d

vertexdetector

e+e pairs are a potential major source of background

Most important: angle with beam axis () and transverse momentum PT.

Summary

• Single pass collider allows us to use very strong beam-beam to increase luminosity

• beam-beam is characterised by following important parameters:– Dy = z/f defines pinch effect (HD), kink

instability, dynamics QM effects, backgrounds, – BS [=f(av)] energy loss, lumi spectrum

• strong-strong regime requires simulation(e.g. GUINEAPIG). Analytical treatments limited.

DESY Summer Student Lecture 31 st July 2002

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