Download The TESLA Accelerator and Linear Collider

An Introduction to the
Physics and Technology
of e+e- Linear Colliders
Lecture 7: Beam-Beam Effects
Nick Walker (DESY)
DESY Summer thStudent
Lecture
th
USPAS, Santa Barbara, 16 -27
2003
31st June,
July 2002
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
bx
by
sx
sy
sz
Ne
GeV
mm
mm
nm
nm
mm
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
 x, y
b x, y
re Nb

2 s x , y (s x  s y )
Putting in some typical numbers (see previous table) gives:
 x  0.54
 y  1.44
Storage ring colliders try to keep  x , y  0.05
Electric Field from a Relativistic Flat Beam
• Highly relativistic beam
E+vB  2E
• Flat beam sx  sy (cf Beamstrahlung)
• Assume
• infinitely wide beam with constant density per
unit length in x ( r(x))
• Gaussian charge distribution in y:
• for now, leave r(z) unspecified
2




1
1
y
r ( y) 
exp     
 2  s y  
2s y


r ( x) 
1
2s x
Electric Field from a Relativistic Flat Beam
Use Gauss’ theorem:
q
 E  ds  
s
E y ( y, z )xz 
0
qN r ( x) r ( z )xz
0
y

r ( y ')dy '
y ' 0
 y 
qN
E y ( y, z ) 
Erf 
 r ( z)


2 2  0s x
2
s
y


Assuming Gaussian distribution for z, the peak field is given by
Eˆ y 
qN
4 0s xs z
Electric Field from a Relativistic Flat Beam
E y MV/cm
s x  500 nm
s y  5 nm
2000
s z  300 μm
1000
N  1010
10
5
5
1000
2000
Note: 2Ey plotted
Assuming a Gaussian distribution for r(z)
z 0
10
y /s y
Electric Field from a Relativistic Flat Beam
s x  500 nm
s y  5 nm
s z  300 μm
N  1010
E y MV/cm
1500
1000
effect of x width
500
z 0
10
20
Assuming a Gaussian distribution for r(z)
30
40
50
y /s y
Equation Of Motion
F = ma:
y (t )  
2qE y ( y, t )
 m0
y(t )  c 2 y( z )
Changing variable to z:
y( z )  
2qE y ( y, z )
 m0c 2
 y( z) 
q N Erf 
 r (2 z )
 2s  z
y 


2  0s x mo c 2
2
q2
re 
4 0 m0c 2
y( z )  
2 2 Nre
 sx
 y( z) 
Erf 
 r (2 z )
 2s  z
y 

why
rz(2z)?
Linear Approximation and the Disruption Parameter
Taking only the linear part of
the electric field:
y( z )  
4 Nre r (2 z )
s xs y
y( z)
k2 (z)
Take ‘weak’ approximation:
y(z) does not change during
interaction y(z) = y0
Thin-lens focal length:
y  
4 Nre
s xs y

y0  r z (2 z )dz

2 Nre
1

y0   y0
s xs y
f
2 Nre
1

f s xs y
Define Vertical Disruption
Parameter
2 Nres z
s
Dy  z 
f
s xs y
exact: Dy 
2 Nres z
 s x  s y  s y
Number of Oscillations
Equation of motion re-visited:
2 Dy
y( z )  
r z (2 z ) y( z )
sz
Approximate r(z) by rectangular distribution with same RMS
as equivalent Gaussian distribution (sz)
half-length!
Dy
y( z )  
0.4
1
2 3s z
0.3
k 
2
0.2
0.1
3s
Dy
3s z2
2
z
y( z );
3
z
sy
2
3ks z

2
1
0
3
2
 3s z
1
0
1
2
3s z
3

3 4 Dy
2
 0.21 Dy
Example of Numerical Solution
N  2  10
s x  500 nm
10
s y  5 nm
s z  300 μm
E  250 GeV
Dy  27.7
  1.1
y / nm
40
green: rectangular approximation
20
0
20
black: gaussian
40
500
0
z ( ct )/ μm
500
1000
1500
Pinch Enhancement
• Self-focusing (pinch) leads to higher luminosity
for a head-on collision.
H Dx , y
3


D
x, y
1/ 4
 1  Dx , y 
ln
 1  D3  
x, y  


Dx , y
 0.8b x , y
 1  2 ln 
 sz




‘hour glass’ effect
Empirical fit to beam-beam simulation results
Only a function of disruption parameter Dx,y
3
3
2
2
1
1
Y
Y
The Luminosity Issue: Hour-Glass
0
0
1
1
2
2
3
3
2
1
0
Z
1
b = “depth of focus”
reasonable lower limit for
b is bunch length sz
2
2
1
0
Z
1
2
Luminosity as a function of by
L (cm2s1)
34
5 10
 BS  s1
s z  100m m
z
34
4 10
34
s z  300m m
2 1034
500m m
3 10
nb N 2 f
L
4s xs y
700m m
900m m
1 1034
200
400
600
b y (m m)
800
1000
Beating the hour glass effect
Travelling focus (Balakin)
• Arrange for finite chromaticity at IP (how?)
• Create z-correlated energy spread along the bunch (how?)
s z  b y*
Beating the hour glass effect
Travelling focus
• Arrange for finite chromaticity at IP (how?)
• Create z-correlated energy spread along the bunch (how?)
s z  b y*
Beating the hour glass effect
Travelling focus
• Arrange for finite chromaticity at IP (how?)
• Create z-correlated energy spread along the bunch (how?)
ct
Beating the hour glass effect
foci ‘travel’ from z = 0 to z =  3s z
chromaticity:
3s z
3s z
f y   y
travelling focus:
t=0
3s z  2ct
NB: z correlated!
 RMS
f = ct
t>0
sz

2 y
2
z  2 y RMS
s z2

2 y
1
2
3
6
4
5
The arrow shows position of focus for the read
beam during travelling focus collision
Kink Instability
Simple model: ‘sheet’ beams with:
r x, z 
1
2 3s x , z
Linear equation of motion becomes
2 Dy
y( z ) 
y ( z );
2
6 sz
3
z 
sz
2
Need to consider relative motion of both beams in t and z:


2

c
y
(
t
,
z
)



0 ( y1  y2 )

 1
z 
 t
2


2

c
y
(
t
,
z
)



0 ( y1  y2 )

 2
z 
 t
2
Classic coupled EoM.
2 2 Dy
 
c 2
6
sz
2
0
Kink Instability
Assuming solutions of the form
y1(2)  a1(2) exp i  kz  t 
and substituting into EoM leads to the dispersion relation:
 2  c 2 k 2   02  4 02 c 2 k 2   04
Motion becomes unstable when 2 0, which occurs when
k 
2 0
c
Kink Instability
Exponential growth rate:
 4 2 c 2 k 2   4   2  c 2 k 2 
0
0
0


Maximum growth rate when
Remember!
1
2

3 0
;  i 0
2 c
2
k 
3 sz
3 sz

t 
2 c
2 c
 t  3s z / c
Thus ‘amplification’ factor of an initial offset is:
1   
 0 t 
exp 
  exp   
 2 
 2  2 
1
4
 0.6
Dy   e

For our previous example with Dy ~28, factor ~ 3
Dy
Kink Instability
y  0.1s y
Dy  20
s z  300 μm
Pinch Enhancement
L  Lgeom H
H= enhancement factor
results of
simulations:
  2y 
exp   2 
 4s 
y 

High Dy example: TESLA 500
Dy  24
Disruption Angle
Dy 
Remembering definition of Dy
sz
f
The angles after collision are characterised by
0 
Dxs x
sz

Dys y
sz
2 Nre
2 Nre


 (s x  s y ) s x
Numbers from our previous example give  0  467 μrad
OK for horizontal plane where Dx<1
For vertical plane (strong focusing Dy > 1), particles oscillate:
previous linear approximation:
y( z )  
Dy
3s
2
z
y( z );
3
z
sy
2

Dy
3 sz
1
4
sy 
0
1
1
3 Dy
4
 67 μrad
2
Disruption Angle: simulation results
250
1000
200
800
150
600
100
400
50
200
0
0
- 400
- 200
0
200
400
horizontal angle (mrad)
- 400
- 200
0
200
vertical angle (mrad)
Important in designing IR (spent-beam extraction)
400
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
parameters
Dy~12
Nx2
Dy~24
Examples of GUINEAPIG Simulations
NLC parameters
Dy~12
Luminosity
enhancement
HD ~ 1.4
Not much of an
instability
Examples of GUINEAPIG Simulations
Nx2
Dy~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 deflection
allow to control collisions
Examples of GUINEAPIG Simulations
Examples of GUINEAPIG Simulations
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 ( D1) and offset (y/sy<<1)
1
y
 bb   0
2 sy
For large disruption or
offset, we introduce the
form factor F:
1
 bb   0 F  y / s y 
2
F ()  
Long Range Kink Instability & Crossing Angle
To avoid parasitic bunch interactions in the IR, a horizontal
crossing angle is introduced:
ctb c
X
2
angle fc
ctb
l
2
parasitic beam-beam kick: r  
2 Nre
r
r
s x, y
Long Range Kink Instability & Crossing Angle
y
IP
 

ip
e+
1
 0 F (     )
2
small vertical offset

e
( = y/s ) gives
y
rise to beam-beam
kick
y
 
l

l

2 Nre ip
  

2
 X

Nre  0 F (     )

lfc2
resulting vertical
offset at parasitic
crossing gives next
incoming bunches
additional vertical
kick: IP offset
increases
instability
Long Range Kink Instability & Crossing Angle
offset at IP of k-th bunch:
 k   k   k
(k  l )tb c
distance from IP to encounter with l-th bunch (l < k) llk 
2
contribution from encounter with l-th bunch:
 k   k ,0  llk,lk / s y


k ,0

Nre  0
 f sy
2
c
F ( l )
 k   k   k
  k ,0 
fc2s y
F ( l )
NB. independent
of llk
s x / s z 
 k   k ,0  C  F ( i ); C  2
 Dx Dy 

f
s
f
i 1
c
y
c


k 1
total offset:
2 Nre 0
2 Nre 0
2
Long Range Kink Instability & Crossing Angle
Assuming all bunches have same initial offset:  k ,0  0
For small disruption and small offsets, F (i )  i
 k  (1  C )k 1  0
s x / s z 
since C  Dx Dy 

f
c


thus (mb  1)C  1
example:
Dx,y = 0.1, 10
sx = 220 nm
sz = 100 mm
fc = 20 mrad
2
1
k
 1  (k  1)C
0
where mb = number of interacting bunches
C = 0.012  mb < 80 for factor of 2 increase
Crab Crossing
x
s x , projected  s x2  fc2s z2
 fcs z
 20mr 100μm  2μm
factor 10 reduction in L!
use transverse (crab) RF
cavity to ‘tilt’ the bunch at IP
x
RF kick
ˆ z
2

V
2

z


RF
V ( z )  VˆRF sin 






VˆRF
fc  4 b cav b ip
sz
RF
Beamstrahlung
Magnetic field of bunch B = E/c
2 Emax
qN
Bmax 

 1160Tesla
Peak field:
c
2 0 cs xs z
From classical theory, power radiated is given by
cC E 4
5
-3
P 
;
C

8.85

10
m
..GeV

2 r 2
For E = 250 GeV and sz = 300mm:
1
Erad  P t  P

r
sz
c
cB
 1.4 m-1
E
 32 GeV
E/E ~ 12%
note:
s z / r  200μrad
1/   2μrad
Beamstrahlung
Most important parameter is 
2  c e 2
2B
e



 2
3 E
r
Bs m0
c
e
r
critical photon frequency
B
Bs
beam magnetic field
Fm
em field tensor
p
electron 4-momentum
F
m
Compton wavelength
local bending radius
Schwinger’s critical field (= 4.4 GTesla)

 2
p
Beamstrahlung: photon spectrum
NOT classical synchrotron radiation spectrum!
Need to use Sokolov-Ternov formula:
10
FBS  


y2
K 5 3 ( )d  
K 2 3 ( )
1 y
5
  10
 1
1
0.5
classical
y
quantum theoretical

E
2  y     1 


   

3  1  y    c   1  y 
  1/ 3
0.1
0.05
0.01
0.001
  0.1
0.005 0.01
0.05 0.1
y /E
0.5
1
Beamstrahlung Numbers
average and maximum 
 avr
5 Nre e

6 s z (s x  s y )
 max  2.4 avr
photons per electron:
average energy loss:  BS
s z   avr
n  2.54 

2/3


1


e


avr
s z 
 2avr
E
 
 1.24 

2/3 2
E


 e  1  (1.5 avr ) 
BS and n (and hence ) talk directly to luminosity spectrum and
backgrounds
Beamstrahlung energy loss
In lecture 1, we used the following equation to derive our
luminosity scaling law
 BS
ere3  Ecm 
N2
 0.86


2m0c2  s z  (s x  s y )2
Now we have this:
 BS
with
s z 
 2avr
 1.24 

2/3 2


 e  1  (1.5 avr ) 
 avr 
5 Nre e
6 s z (s x  s y )
under what regime is our original expression valid?
Luminosity scaling revisited
b y*  s z
 BS
 y ,n
low beamstrahlung regime 1:
L  Pbeam
high beamstrahlung regime 1:
 BS
L  Pbeam
s z  y ,n
3
homework: derive high BS scaling law
2
Beamstrahlung Numbers: example
E  500GeV
s x  220 nm
 avr  0.28
s y  2 nm
n  1.3
s x  110μm
 BS  7.5%
 max  0.66
N  0.75 1010
TESLA
E
 ,c
 210 GeV
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 (0.6100), coherent pairs are generated by interaction of
photons with macroscopic field of bunch.
– Very high energies (U100), coherent direct trident production
e  e e+ e
TESLA
NLC
CLIC
0.5TeV  ~ 0.06
1TeV  ~ 0.28
3TeV  ~ 9
• Beamstrahlung degrades Luminosity Spectrum
Pair Production
• Incoherent e+e pairs 0.6
– Breit-Wheeler:
– Bethe-Heitler:
– Landau-Lifshitz:
  e e
e  e e e
e e  e e e e
• Coherent e+e pairs 0.6100
– threshold defined by

10
5
  10

 1
1
0.5
for >1
  1/ 3
0.1
0.05
0.01
0.001
  0.1
0.005 0.01
0.05 0.1
y /E
0.5
1
 2B
m0 c 2 Bs

E
 1
 / E ~ O(1)    1
for intermediate colliders (Ecm<1TeV),
incoherent pairs dominate
Pair Production
e+e pairs are a
potential major source
of background
PT (GeV/c)
Most important: angle
with beam axis () and
transverse momentum
PT.
1
0.1
0.01
0.1
l
d
rd
d
vertex
detector
 (rad)
1
pairs curl-up (spiral) in
PT
solenoid field of
r
 rd
cBz
detector
Summary
• Single pass collider allows us to use very strong
beam-beam to increase luminosity
• beam-beam is characterised by following
important parameters:
– Dy = sz/f
– 
– BS [=f(av)]
defines pinch effect (HD), kink
instability, dynamics
QM effects, backgrounds,
energy loss, lumi spectrum
• strong-strong regime requires simulation
(e.g. GUINEAPIG). Analytical treatments limited.