Italo-Hellenic School of Physics 2006
The Physics of LHC: Theoretical Tools and Experimental Challenges
An Introduction to
Charged Particles Tracking
Francesco Ragusa
Università di Milano
francesco.ragusa@mi.infn.it
June 12-18 2006, Martignano (Lecce, Italy)
Contents
LECTURE 1
LECTURE 2
Introduction
Transport of parameters
Motion in magnetic field
Multiple Scattering
Helical trajectories
covariance matrix
Magnetic spectrometers
momentum resolutions at low
momenta
Tracking Systems: ATLAS & CMS
Straight line fit
Error on the slope
Error on the impact parameter
Vertex detector and central
detector
Momentum measurement
impact parameter resolution at low
momenta
Measurement of the sign of charge
Systematic effects
misalignments and distortions
Kalman Filter
Sagitta
Tracking in magnetic field
Quadratic (parabola) fit
Momentum resolution
Extrapolation to vertex
An Introduction to Charged Particles Tracking – Francesco Ragusa
1
Introduction
Tracking is concerned with the reconstruction
of charged particles trajectory ( tracks ) in
experimental particle physics
the sign of the charge
F = q v×B
the aim is to measure ( not a full list )
momentum (magnetic field)
v
B
p = 0.3·B·R
particle ID (mass), not necessarily
with the same detector
lifetime tag
p = mo γ β
secondary vertex
primary vertex
impact parameter
An Introduction to Charged Particles Tracking – Francesco Ragusa
2
Motion in Magnetic Field
In a magnetic field the motion of a charged particle is determined by the Lorentz
Force
dp
= ev×B
dt
Since magnetic forces do not change the
energy of the particle
mo γ
dv
= ev×B
dt
and finally
d 2r
e dr
=
×B
p ds
ds 2
In case of inhomogeneus magnetic field,
B(s) varies along the track and to find
the trajectory r(s) one has to solve a
differential equation
In case of homogeneus magnetic field the
trajectory is given by an helix
d 2r
dr
mo γ 2 = e × B
dt
dt
vz
using the path length s along the track
instead of the time t
ds = vdt
we have
d 2r
dr
mo γv 2 = e × B
ds
ds
B
An Introduction to Charged Particles Tracking – Francesco Ragusa
vs
v
3
Magnetic Spectrometers
Almost all High Energy experiments done
at accelerators have a magnetic spectrometer to measure the momentum of
charged particles
2 main configurations:
solenoidal magnetic field
tracking detectors arranged in
cylindrical shells
measurement of curved trajectories
in ρ - φ planes at fixed ρ
Dipole field
dipole field
B
Solenoidal field
x
x
z
B
z
y
rectangular symmetry
deflection in y - z plane
y
cylindrical symmetry
deflection in x - y ( ρ - φ ) plane
tracking detectors arranged in
parallel planes
measurement of curved trajectories
in y - z planes at fixed z
An Introduction to Charged Particles Tracking – Francesco Ragusa
4
Tracking Systems: ATLAS
Pixel Detector
3 barrels, 3+3 disks: 80×106 pixels
barrel radii: 4.7, 10.5, 13.5 cm
pixel size 50×400 µm
σrφ= 6-10 µm σz = 66 µm
SCT
4 barrels, disks: 6.3×106 strips
barrel radii:30, 37, 44 ,51 cm
strip pitch 80 µm
stereo angle ~40 mr
σrφ= 16 µm σz = 580µm
TRT
barrel: 55 cm < R < 105 cm
36 layers of straw tubes
σrφ= 170 µm
400.000 channels
An Introduction to Charged Particles Tracking – Francesco Ragusa
5
Tracking Systems: CMS
Pixel Detector
2 barrels, 2 disks: 40×106 pixels
barrel radii: 4.1, ~10. cm
σrφ= 10 µm σz = 10 µm
Internal Silicon Strip Tracker
Pixel
End cap –
TEC-
2,4
m
pixel size 100×150 µm
Outer
Barrel –TOBInner Barrel
–TIBInner Disks
–TID-
4 barrels, many disks: 2×106 strips
barrel radii:
strip pitch 80,120 µm
5.4
m
σrφ= 20 µm σz = 20 µm
volume 24.4 m3
External Silicon Strip Tracker
6 barrels, many disks: 7×106 strips
barrel radii: max 110 cm
strip pitch 80, 120 µm
σrφ= 30 µm σz = 30 µm
30
cm
An Introduction to Charged Particles Tracking – Francesco Ragusa
m
93 c
6
Tracking Systems: ATLAS & CMS
ATLAS
CMS
An Introduction to Charged Particles Tracking – Francesco Ragusa
7
Momentum Measurement
The momentum of the particle is
projected along two directions
PL
ρ
P
orders of magnitude
P⊥ = 1GeV
B = 2T
R = 1.67 m
P⊥ = 10GeV
B = 2T
R = 16.7 m
the sagitta s
λ
P⊥
s
z
in ρ - φ plane we measure the
transverse momentum P⊥
φ
P⊥ = P cos λ = 0.3BR
R
2α =
L
R
s = R ( 1 − cos α )
2α
α2
L2
s ≈R
=
2
8R
ρ
in the ρ - z plane we measure the
dip angle λ
ρ
λ
assume a track length of 1 m
P⊥ = 1GeV
s = 7.4 cm
P⊥ = 10GeV
s = 0.74 cm
z
An Introduction to Charged Particles Tracking – Francesco Ragusa
8
Momentum Measurement
Once we have measured the transverse
momentum and the dip angle the total
momentum is
P =
P⊥
0.3BR
=
cos λ
cos λ
the error on the momentum is easely
calculated
P
∂P
= ⊥
∂R
R
∂P
= −P⊥ tan λ
∂λ
 ∆P 2
 ∆R 2

= 
+ ( tan λ∆λ )2


 P 
 R 
We need to study
the error on the radius measured in
the bending plane ρ - φ
the error on the dip angle in the
ρ - z plane
We need to study also
contrubution of multiple scattering to
momentum resolution
Comment:
in an hadronic collider the main
emphasis is on transverse momentum
elementary processes among partons
that are not at rest in the
laboratory frame
use of momentum conservation only in
the transverse plane
An Introduction to Charged Particles Tracking – Francesco Ragusa
9
The Helix Equation
The helix is described in parametric form
(
)
(
)
R (m ) =
p⊥ (GeV )
0.3B (T )
hs cos λ


x ( s ) = xo + R  cos Φo +
− cos Φo 
R


PL
hs cos λ


y ( s ) = yo + R  sin Φo +
− sin Φo 
R


z ( s ) = zo + s sin λ
λ
B
P⊥
P
λ is the dip angle
h =±1 is the sense of rotation on the helix
The projection on th x-y plane is a circle
( xo , yo )
( x − xo + R cos Φo )2 + ( y − yo + R sin Φo )2 = R 2
xo and yo the coordinates at s = 0
Φo is also related to the slope of the
tangent to the circle at s = 0
Φo
R sin Φo
An Introduction to Charged Particles Tracking – Francesco Ragusa
R cos Φo
10
The Helix Equation
To reconstruct the trajectory we put position
measuring planes along the particle path
We will consider the track fit separately in
the the 2 planes
perpendicular to B (x,y): circle
containing B (x,z), (y,z) or (ρ,z) straight line
( x1, y1, z1 )
In the plane containing B (for example y - z plane)
the trajectory is a periodic function of z
(
)
hs cos λ


y ( s ) = yo + R  sin Φo +
− sin Φo 
R


(
s =
( x 2 , y2 , z 2 )
z − zo
sin λ
)
h ( z − zo )


y ( z ) = yo + R  sin Φo +
− sin Φo 


R tan λ
however, for large momenta, i.e. R tanλ >> ( z-zo ),
assuming for simplicity h = 1, Φo = 0
y ( z ) ≈ yo +
1
( z − zo )
tan λ
y ( z ) = yo + ctanλ ( z − zo )
An Introduction to Charged Particles Tracking – Francesco Ragusa
straight line
11
The Helix Equation
An Introduction to Charged Particles Tracking – Francesco Ragusa
12
Straight Line Fit
This is a well known problem
θ
b = tg θ = ctg λ
a reference frame
N+1 measuring detetectors at z0,…, zn, …,zN
a particle crossing the detectors
a
N+1 coordinate measurements y0,…, yn, …,yN
each measurement affected by uncorrelated
errors σ0,…, σn, …,σN
yo
yn
yN
zo
zn
zN
Find the best line y = a + b z that fit the track
The solution is found by minimizing the
χ2
N
2
χ =
∑
a = ( Sy Szz − Sz Szy ) / D
n =0
b = ( S1Szy − Sz Sy ) / D
N
the covariance matrix (at z = 0) is
 σa2 cab 
S

 = 1  zz


2
cab σb  D  −Sz
−Sz 


S1 
depends only
on σ, zn and N
( yn − a − bz n )2
1
S1 = ∑ 2
n = 0 σn
σn2
N
Sy =
N
z
Sz = ∑ n2
n = 0 σn
yn
∑ 2
n = 0 σn
N
Syz =
yn z n
∑ 2
n = 0 σn
N
Szz
z n2
= ∑ 2
n = 0 σn
An Introduction to Charged Particles Tracking – Francesco Ragusa
D = S1Szz − Sz Sz
13
Straight Line Fit: Matrix Formalism
The χ2 can be written as
It is useful to restate the problem using
a matrix formalism [4:Avery 1991]
χ2 = ( Y − Ap )T W ( Y − Ap )
This is useful because:
it is more compact
it is easely extensible to other linear
problems
it is more useful to formulate an
iterative procedure
the linear model is given by f = Ap
( yi − yi ) ( y j − y j
( V )ij = σi2 δij
if uncorrelated
AT WY
−1

 1 z 

0  a

  


 =  1 ...  b  = Ap

 


 1 z N 
( V )ij =
−1
p = ( AT WA )
VP = ( AT V−1A )
measurements and errors are
 y 
 0 

Y =  ... 
 
 yN 
The minimum χ2 is obtained by
The covariance matrix of the parameters
is obtained from the measurements
covariance matrix V
With the same assumption as before
 f   a + bz
 0  
0
  
f =  ...  = 
  
 fN   a + bz N
W = V −1
)
please notice (N+1 measurements, M
parameters)
dimensions A
= (N+1) × M
dimensions V
= (N+1) × (N+1)
T
dimensions A WA = M × M
dimensions ATW = M × (N+1)
dimensions Vp
=M ×M
An Introduction to Charged Particles Tracking – Francesco Ragusa
14
Straight Line Fit
Let’s consider the case of equal spacing
between zo and zN with equal errors on
coordinates σn = σ
L
The errors on the intercept and the
slope are
σa2

N zc2  σ 2


= 1 + 12

N + 2 L2  N + 1
σb2
σ2
12N
=
( N + 1 ) L2 ( N + 2 )
a
zc
important features:
zN
zo
L = z N − zo
zc =
both errors are linearly dependent
on the measurement error σ
z N + zo
2
both errors decrease as 1/ N + 1
The S are easily computed in finite form
N +1
z
Sz = ( N + 1 ) c2
2
σ
σ
2

N + 1  (N + 2 ) L
2

=
+
z
c

12
σ 2  N
S1 =
Szz
D =
L2 ( N + 1 )2 ( N + 2 )
N
12σ 4
the error on the slope decrease as
the inverse of the lever arm L
the error on the intercept increases
if zc increase
S and errors: all computed at z = 0
An Introduction to Charged Particles Tracking – Francesco Ragusa
15
Errors At The Center Of The Track
We can choose the origin at the center
of the track: z = zc
L
ip
zc
zN
zo
It is easely seen that for uniform
spacing and equal errors
N +1
N + 1  ( N + 2 ) L2 


S1 =
S
0
S
=
=
z
zz
12 
σ2
σ 2  N
L2 ( N + 1 )2 ( N + 2 )
D =
N
12σ 4
S1 and D do not change
Sz and Szz change
The errors are now
σb2
σ2
12N
=
( N + 1 ) L2 ( N + 2 )
σa2
σ2
=
N +1
unchanged
changed
The intercept is different from the
previous case (origin at z = 0)
Obviously, taking properly into account
error propagation ve get the same result
for the impact parameter ip
ip = f (−zc ) = a − bzc
2
σip
= σa2 + zc2σb2
2
σip
σa and σb uncorrelated if origin at zc
σ2
σ 2 12N zc2
=
+
N + 1 N + 1 N + 2 L2
An Introduction to Charged Particles Tracking – Francesco Ragusa
16
Vertex Detectors
The previous result shows clearly the need
for vertex detectors to achieve a precise
measurements of the impact parameter
L
2
σip
σ2
σ2
12N zc2
=
+
N + 1 N + 1 N + 2 L2
We should have
small measurement errors σ
large lever arm L
place first plane as near as possible
to the production point: small zc
zc
zo
zc
zN
The “amplitude” of the error is determined
by the error on the slope σ/L
Increasing the number of points also
improves but only as N + 1
The technology used is silicon detectors
with resolution of the order of σ ~ 10 µm
by the distance of the point to which
we extrapolate zc/L
expensive
by the size of the measurement error
small L
small N
An Introduction to Charged Particles Tracking – Francesco Ragusa
17
Vertex Detectors
Summarizing:the error on the impact
parameter is
σip
L
r =
σ
= Z ( r, N )
N +1
zc
L
ip
zc
for the ATLAS pixel detector
z0 = 4.1 cm, z2 = 13.5 cm
Z ( r, N ) =
L = 9.4, zc=6.8, r = 0.72
σip = 12 µm
zN
zo
N+1 = 3, σ = 10 µm
r
N
1
2
5
9
19
∞
1 + 12
N
r2
N +2
5.0
3.0
2.0
1.5
1.0 0.75 0.60 0.50
10.1
12.3
14.7
15.7
16.5
17.3
6.08
7.42
8.84
9.45
9.94
10.4
4.12
5.00
5.94
6.35
6.67
7.00
3.16
3.81
4.50
4.81
5.04
5.29
2.24
2.65
3.09
3.29
3.44
3.61
An Introduction to Charged Particles Tracking – Francesco Ragusa
1.80
2.09
2.41
2.55
2.67
2.78
1.56
1.78
2.02
2.13
2.22
2.31
1.41
1.58
1.77
1.86
1.93
2.00
18
Vertex Detector + Central Detector
We have seen that the error on the
impact parameter is
2
σip
= σa2 + zc2σb2
The second term depends on the error on
the slope and is limited by the small
lever arm L typical of vertex detectors
(~ 10 cm)
It is usually very expensive to increase
this lever arm
L
A solution is a bigger detector (Central
Detector) less precise (usually less
expensive) but with a much bigger lever
arm L
ip
zc
zo
zN
The first term:
depend only on the precision of the
vertex detector
it is equivalent to a very precise
measurement (σa ~ 5 µm) very near
to the primary vertex (zc)
The error on the slope then become
smaller
The error on the extrapolation
become smaller
This is the arrangement usually adopted
by experiments who want to measure the
impact parameter
An Introduction to Charged Particles Tracking – Francesco Ragusa
19
Momentum Measurement: Sagitta
To introduce the problem of momentum
measurement let’s go back to the sagitta
a particle moving in a plane perpendicular
to a uniform magnetic field B
R=
p
0.3B
δp
δR
=
p
R
the trajectory of the particle is an arc of
radius R of length L
sagitta
y3 s
y1
y2
R
L
2α =
R
2α
y1 + y2
2
L2 δR
δs =
∼ δy
8R R
L2 δ p
= δy
8R p
δp
8R
= 2 δy
p
L
δp
8p
=
δy
p
0.3BL2
δp
8δy
=
p2
0.3BL2
important features
the percentage error on the momentum
is proportional to the momentum itself
s = R ( 1 − cos α )
the error on the momentum is inversely proportional to B
α2
L2
s ≈R
=
2
8R
the error on the momentum is inversely proportional to 1/L2
assume we have 3 measurements: y1, y2, y3
s = y3 −
the error on the radius is related to the
sagitta error by
δs =
3
δy ∼ δy
2
the error on the momentum is proportional to coordinate measurement error
An Introduction to Charged Particles Tracking – Francesco Ragusa
20
Tracking In Magnetic Field
The previous example showed the basic
features of momentum measurement
Let’s now turn to a more complete treatement of the measurement of the charged particle trajectory
We have already seen that for an homogeneus magnetic field the trajectory
projected on a plane perpendicular to the
magnetic field is a circle
2
2
( y − yo ) + ( x − xo ) = R
b=
a = y (0)
1
c=−
2R
dy
dx
2
for not too low momenta we can use a
linear approximation
y = yo + R2 − ( x − xo )2
R2 >> ( x - xo )2
2

( x − xo ) 
y ≈ yo + R  1 −


2R2 
x
x
1 2
y = yo + R − o 2 + o x −
x
R
2R
2R
(
)
we are led to the parabolic approximation
of the trajectory
y = a + bx + cx 2
x =0
let’s stress that as far as the track parameters is concerned the dependence is
linear
The parameters a,b,c are
intercept at the origin
slope at the origin
radius of curvature (momentum)
An Introduction to Charged Particles Tracking – Francesco Ragusa
21
Quadratic Fit
Assume N detectors measuring the y
coordinate [Gluckstern 63]
yo
yn
yN
xo
xn
xN
However we can use the matrix formalism
developed for the straight line:
 yo 
 

Y =  ... 
 
 yN 
a 
 

p =  b 
 
 c 
−1
p = ( AT WA )
A track crossing the detectors
gives the measurements y0, …,yn, …, yN
Each measurement has an error σn
Using the parabola approximation, the track
parameters are found by minimizing the χ2
χ2 =
∑
n =0
( yn − a

0 


... ..0 

1 

0
σN2 

0
let’s recall the solution
The detectors are placed at positions
xo, …, xn, …, xN
N
 1


2 
 σ 2
x0 
 0


...  W =  0



2 

x N 
 0


 1 x 0

A =  ... ...

 1 x

N
2
− bx n − cx n2
σn2
)
−1
( A WA )
T
F
 0

=  F1

 F
 2
 M 
 0 


AT WY =  M 1 


 M 
 2 
An Introduction to Charged Particles Tracking – Francesco Ragusa
F1
F2
F3
AT WY
−1
F2 

F3 

F4 
N
x nk
Fk = ∑ 2
n = 0 σn
N
Mk =
yn x nk
∑ 2
n = 0 σn
22
Quadratic Fit
The result is [4: Avery 1991, BlumRolandi 1993 p.204, Gluckstern 63]
∑ ynGn
a =
∑Gn
∑ ynGn
b=
∑ xnGn
∑ ynGn
c=
∑ xn2Gn
The result can be found in [Blum-Rolandi,
p. 206]
To get some idea of the covariance
matrix let’s first compute it by setting
the origin at the center of the track
Gn = F 11 − x n F 21 + x n2 F 31
N
xc =
Pn = −F 12 + x n F 22 − x n2 F 32
n =0
with this choice one can “easely” find
Qn = F 13 − x n F 23 + x n2 F 33
The quantities Fij are the determinants
of the 2x2 matrices obtained from the
3x3 matrix F by removing row i, column j
The covariance matrix
−1
Vp = ( AT WA )
F
 0

=  F1

 F
 2
F1
F2
F3
x
∑ N +n 1
−1
F2 

F3 

F4 
F1 = F3 = 0
N +1
σ2
L2 ( N + 1 )( N + 2 )
F2 = 2
12N
σ
2
L4 ( N + 1 )( N + 2 )( 3N + 6N − 4 )
F4 = 2
σ
240N 3
L4 ( N − 1 )( N + 1 )2 ( N + 2 )( N + 3 )
S = F0F4 − F22 = 4
σ
180N 3
F0 =
An Introduction to Charged Particles Tracking – Francesco Ragusa
23
Momentum Resolution
The covariance matrix is
 F4


1
 0
Vp =
F0F4 − F2F2 

 −F2
0
F0F4 − F2F2
F2
0
−F2 


0 


F0 
we are mostly interested on the error on
the curvature
σc2
CN
F0
σ2
=
= 4 CN
F0F4 − F2F2
L
180N 3
=
( N − 1 )( N + 1 )( N + 2 )( N + 3 )
it can be shown that the error on the
curvature do not depend on the position of
the origin along the track
Let’s recall from the discussion on the
sagitta
R=
δp
δR
=
p
R
p
0.3B
also recall that
c =
1
2R
σc =
1
δR
2R 2
and finally the momentum error
δp
σ
=
p2
0.3BL2
4C N
the formula shows the same basic features
we noticed in the sagitta discussion
we have also found the dependence on the
number of measurements (weak)
An Introduction to Charged Particles Tracking – Francesco Ragusa
24
Momentum Resolution
We stress again that a good momentum
resolution call for a long track
δp
1
∼
p2
L2
any trick that can extend the track length
can produce significant improvements on
the momentum resolution
the use of the vertex can also improve
momentum resolution:
the common vertex from which all the
tracks originate can be fitted
the point found can be added to every
track to extend the track length at
Rmin → 0
the position of the beam spot can also be
used as constraint
Extending Rmax can be very expensive
An Introduction to Charged Particles Tracking – Francesco Ragusa
25
Momentum Resolution
We can now give a rough estimate of the
momentum resolution of the ATLAS
tracking systems
There are different systems: some
simplifying guess
TRT: 36 point with σ = 170 µm
from 55 cm to 105 cm: as a single
point with σ = 28 µm at Rmax = 80
δp
(%)
p
p⊥ = 500 GeV
cm
Rmin = 4.7 cm, L = 75 cm
N+1= 3 + 4 + 1 = 8
σ = 12, 16, 28 ~ 20 µm
C N ≈ 12
4C N ≈ 7
δp
−4
−1
∼
4
×
10
GeV
p2
At 500 GeV
η
δp
= 20 × 10−2
p
An Introduction to Charged Particles Tracking – Francesco Ragusa
26
Momentum Resolution
An Introduction to Charged Particles Tracking – Francesco Ragusa
27
Slope And Intercept Resolution
For completeness we give also the errors
on the slope and intercept
The error on the slope is given by
σ2
1
σb2 =
= BN 2
F2
L
BN =
12N
( N + 1 )( N + 2 )
The only off diagonal element of the
covariance matrix different from 0 is
between intercept and curvature and we
have
σac
F2
σ2
=−
= −DN 2
F0F4 − F2F2
L
DN = −
15N 2
( N − 1 )( N + 1 )( N + 3 )
We find the same qualitative behaviour
we had for the straight line fit
The error on the intercept is
σa2 =
F4
= AN σ 2
F0F4 − F2F2
3 ( 3N 2 + 6N − 4 )
AN =
4 ( N − 1 )( N + 1 )( N + 3 )
An Introduction to Charged Particles Tracking – Francesco Ragusa
28
Extrapolation To Vertex
L
We want now compute the extrapolation
to vertex and compare the behaviour of
the results of the fit:
L = 8.8 cm
xc = 9.1 cm
r = xc/L ~ 1
inside magnetic field
no magnetic field
having measured the parameters a,b,c at
the center of the track, the intercept is
yip = a + bx v +
cx v2
Propagation of the errors gives
2
σip
= σa2 + x v2 σb2 + x v4 σc2 + 2x v2 σac
The calculation gives [Blum-Rolandi 1993]
σip =
σ
Baa ( r , N )
N +1
where Baa(r,N) is analogous to Z(r,N)
defined for the straight line fit (see
next slide for a table of values)
xv
xc
xc
let’s compare the error assuming the
geometry of the ATLAS pixel detector:
Rmin = 4.7, Rmax = 13.5, N+1 = 3
we have r = 1 and from the 2 tables we get
Baa(1,2) = 7.63
Z(1,2) = 2.65
We see that the error is degraded by a
factor ~ 2.9
The reason is that the error on momentum
cause an additional contribution to the error
in the extrapolation
a central tracking detector is needed
An Introduction to Charged Particles Tracking – Francesco Ragusa
29
Extrapolation To Vertex
σip =
σ
Baa ( r , N )
N +1
Baa ( r , N )
r
N
2
3
5
10
19
∞
5.0 3.0
2.0
1.5
1.0 0.75 0.60 0.50
211
224
250
282
304
335
32.9
35.2
39.4
44.6
48.1
53.0
18.1
19.5
21.9
24.8
26.8
29.5
7.63
8.29
9.39
10.7
11.6
12.8
75/3
80.2
89.5
101
109
120
4.10
4.48
5.10
5.84
6.32
7.00
2.65
2.84
3.20
3.65
3.95
4.37
An Introduction to Charged Particles Tracking – Francesco Ragusa
2.05
2.07
2.26
2.54
2.72
3.00
30
Parameters Propagation
We have seen that changing the origin of
the reference frame
the track parameters change
the covariance matrix changes
It is often useful to propagate the
parameters describing the track from one
“origin” (point 0) to “another” (point 1)
For linear models this is very easely
expressed in matrix form
pi = fi ( pk ) = Di pk
Di =
∂fi
∂pk
To better understand the above formulas
let’s apply them to the straight line and
to the parabola
straight line
 a 
p0 =  
 b 
 a + bz1 

p1 = 
b


zo
z1
 1 z1 
∂f1
∂p1

=
= 


∂p0
∂p 0
 0 1 
parabola
xo
x1
 a 
 

p0 =  b 
 
 c 
a + bx1 + cx12 




p1 = b + 2cx1



c

1 x
x12 

1

∂f1
∂p1

=
=  0 1 2x1 


∂p0
∂p0
 0 0
1 

Using the matrix D we can also propagate
the covariance matrix of the parameters
−1
V1 = ( DT V0−1D )
An Introduction to Charged Particles Tracking – Francesco Ragusa
31
Multiple Scattering
Particles moving through the detector
material suffer innumerable EM collisions
which alter the trajectory in a random
fashion (stochastic process)
Few examples:
Argon
Xo = 110 m
Silicon
Xo = 9.4 cm
consider a 10 GeV pion
X
θp
εp
θp
θp2 = K
X
X0
1 X 2
εp2 = K
X
3 X0
1 X
εp θp = K
X
2 X0
Argon:
1 m 0.10 × 10-3 80
εp
µm
Silicon: 300 µm 0.08 × 10-3 0.01 µm
The effect goes as 1/p: for a pion of
1 GeV the effect is 10 times larger
 0.0136 2
K = z 
 pβ 
The lateral displacement is proportional
to the thickness of the detector: usually
can be neglected for thin detectors
strongly correlated
In what follow we will consider only thin
detectors
2
ρ=
εp θp
εp2
θp2
= 0.87
For thick detectors ( for example large
volume gas detectors) see [Gluckstern 63
Blum-Rolandi 93, Block et al. 90]
An Introduction to Charged Particles Tracking – Francesco Ragusa
32
Multiple Scattering
The scattering angle has a distribution that is almost gaussian
P ( θp ) =
1
sin 4
θp2 = K
θp
2


1
2
exp  −
θ
 2 θp2 p 


1
2π θp2
X
X0
 0.0136 2
K = z 
 pβ 
2
At large angles deviations from gaussian distributions appear
that manifest as a long tail going as sin-4θ/2
In thick detectors the distribution of the lateral
displacement should also be considered
The joint distribution of the scattering angle
and the lateral displacement is
P ( εp , θp ) =
3
π θp2

3εp2  
2  2 3εp θp

exp − 2  θp −
+ 2  
2


X
X
X  
 θp 
An Introduction to Charged Particles Tracking – Francesco Ragusa
33
Multiple Scattering
Multiple scattering is a cumulative effect
and introduces correlation among the
coordinate measurements
The treatment of multiple scattering is
different for:
discrete detectors
Here we consider only the simplest case
of discrete and thin detectors
For continous detectors see for example
[5: Avery 1991]
Let’s consider 3 thin detectors
yi
zi
δθi
zj
yk
The 3 coordinate have measurement
errors σi, σj, σk due to the detector
resolution
yi = yi
on plane i
yj
= yj
yk
= yk
yi = y i
Because of multiple scattering on plane i
the actual trajectory cross plane j at
Because of multiple scattering on planes
i,j the actual trajectory cross plane k at
δθ j
yj
yj
y j = y j + ( z j − z i ) δθi
yk
yi
yi
They also have mean value
continous detectors
yj
A track cross the 3 planes at positions
yk
zk
yk = yk + ( z k − z i ) δθi + ( z k − z j ) δθj
Since
δθ = 0
yi = y i
An Introduction to Charged Particles Tracking – Francesco Ragusa
yj = yj
yk = yk
34
Multiple Scattering
we can now compute the covariance matrix of the coordinate
measurements including multiple scattering
Vnm = ( ym − ym )( yn − yn )
yk
First the diagonal elements
2
Vii = ( yi − yi )
yi
2
( yi − yi )
=
yi
=
(yj
δθi
zi
Vii = σi2
Vjj =
yj
2
− yj )
2
(yj − yj )
=
(yj
yj
zj
yk
zk
2
− y j + ( z j − z i ) δθi )
2
+ ( z j − zi )
δθi2 +2 ( z j − z i ) ( y j − y j ) δθi
2
δθi2
2
δθi2 + ( z k − z j )
Vjj = σ 2j + ( z j − z i )
δθ j
it easy to verify that
Vkk = σk2 + ( z k − z i )
2
δθj2
An Introduction to Charged Particles Tracking – Francesco Ragusa
35
Multiple Scattering
yk
Vnm = ( ym − ym )( yn − yn )
yj
yi
yi
The off-diagonal elements are
δθi
zi
δθ j
zj
Vij = ( yi − yi ) ( y j − y j ) =
( yi − yi )( y j − y j + ( z j − zi ) δθi )
Vik = ( yi − yi ) ( yk − yk )
uncorrelated: <>=0
=
Vjk =
(yj
− y j ) ( yk − y k )
( y j − y j + ( z j − zi ) δθi )( yk
= ( z j − z i ) δθi ( z k − z i ) δθi
= ( z j − z i ) ( z k − z i ) δθi2
=
Vjk
( yi − yi )( yk − yk + ( zk − zi ) δθi + ( zk − z j ) δθj )
yk
yj
zk
Vij = 0
Vik = 0
uncorrelated: <>=0
− yk + ( z k − z i ) δθi + ( zk − z j ) δθj )
An Introduction to Charged Particles Tracking – Francesco Ragusa
36
Multiple Scattering
Summarizing, the covariance matrix is
 σ2
 i
V =  0

 0
0
σ 2j
0


0
0
0   0

 

2
2
2

z
−
z
δθ
z
−
z
z
−
z
δθ
0  +  0
(
)
(
)
(
)

j
i
i
i
j
i
i
k

 


σk2   0 ( z − z )( z − z ) δθ 2 ( z − z )2 δθ 2 + ( z − z )2 δθ 2 
i
j
i
i
i
i
j
i
j 
k
k


The second matrix has
diagonal elements due to any previous material affecting
the trajectory impact point at the given plane
off diagonal elements: only presents if a previous material layer
affects at the same time the trajectory impact points for the 2 planes
the same scattering at plane i
affects the trajectory at plane j and plane k
yj
yi
yi
zi
An Introduction to Charged Particles Tracking – Francesco Ragusa
δθi
yk
δθ j
yk
yj
zj
zk
37
Track Fit With Multiple Scattering
The methods developed to fit a track to
the measured points can be used to
perform a fit taking into account M.S.
the covariance matrix is computed
the same fit procedure is applied
Let’s now try to understand qualitatively
the effect of multiple scattering on the
determination of tracks parameters:
the size of the effect goes as 1/p
then the effect is important for low
momentum track
Assume we are dominated by multiple
scattering
the momentum resolution is given by
δp
σ
=
p2
0.3BL2
4AN
we have then
δp
0.0136 X
1
∼
p
β
X 0 0.3BL
4AN
N
We conclude:
for low momentum the percentage
momentum resolution reach a almost
constant value (still dependent on β)
δp
→ constant
p
The momentum resolution only
improves as 1/L
The additional factor 1/N can help
but in this case uniform spacing is
essential
the coordinate error due to M.S. is
σ ∼
L
L 0.0136 X
δθ =
N
N pβ
Xo
An Introduction to Charged Particles Tracking – Francesco Ragusa
38
Momentum Resolution with M.S.
δp
p2
δp
p2
[TeV −1 ]
[TeV −1 ]
solenoidal field
no beam constraint
η
η
δp
p2
δp
p2
[TeV −1 ]
[TeV −1 ]
η
An Introduction to Charged Particles Tracking – Francesco Ragusa
solenoidal field
with beam constraint
uniform field
no beam constraint
η
39
Track Fit With Multiple Scattering
Same kind of considerations for the
error on the slope and on the intercept
the multiple scattering error is
σ ∼
L
L 0.0136 X
δθ =
N
N pβ
Xo
As far as the impact parameter
resolution
σip =
σip =
σ
Baa ( r , N )
N +1
Baa ( r , N ) L 0.0136
N + 1 N pβ
the error on the slope is
σb =
σb =
σb =
BN
σ
L
L 0.0136 X
BN
LN p β Xo
BN 0.0136
N
pβ
X
Xo
we cannot improve anymore the error on
the slope (direction) by increasing the
lever arm
the limit is set by the multiple
scattering angle itself
X
Xo
large lever arm degrade the impact
parameter resolution
for a given error on the slope set by
the multiple scattering angle the
error on the extrapolation goes as
the lever arm
Unfortunately both ATLAS and CMS have
a lot of material
silicon detectors for high precision
silicon for radiation hardness
silicon for rate capabilities
An Introduction to Charged Particles Tracking – Francesco Ragusa
40
Material in ATLAS
An Introduction to Charged Particles Tracking – Francesco Ragusa
41
Tracker Resolutions With M.S.
We have seen that for low momentum
track the momentum resolution and the
impact parameter resolution are dominated by multiple scattering
the momentum resolution tend to
kp
δp
→
p
p2
Kp
δp
→
p sin θ
p2
X
Xo
the impact parameter resolution tend
to
σip
kip
→
p
X
Xo
σip →
K ip
p sin θ
The amount of material actually traversed
by the particles depend on the polar angle
Since the multiple scattering error and
the measurement error are independent
to total error is sum in quadrature of the
2 term
For the ATLAS detector montecarlo
studies have shown that the resolutions
can be parametrised as
σip = 11 ⊕
73
p⊥ sin θ
[ µm ]
δ p⊥
0.013
=
⊕
0.00036
2
p⊥ sin θ
p⊥
σip
[GeV −1 ]
140
120
100
80
X
sin θ
p⊥ = 1 GeV
p⊥ = 20 GeV
p⊥ = 200 GeV
60
40
θ
20
0
0
An Introduction to Charged Particles Tracking – Francesco Ragusa
0.5
1
1.5
2
2.5
η
42
Sign Of The Charge
The sign of the charge is defined by the
sign of 1/R
Q = +1
1
>0
R
Q = −1
1
<0
R
We remember that in our exemple we
had
CN = 12, L = 75 cm, s = 20 µm
if we require a 3 σ identification
3σy
1
> 3σ 1 = 6σc = 2
R
L
R
This measurement becomes more and
more difficult as the momentum increases
Let’s find up to which momentum the
ATLAS tracker will be able to measure
the sign of a charged particle
We recall taht the error on the radius
as determined from the parabola fit is
σc2
σ2
= 4 CN
L
3σy
1
>
0.3BR
0.3BL2
4C N
4C N
0.3BL2
p<
3 4C N σy
inserting numerical values we find
p < 800 GeV
An Introduction to Charged Particles Tracking – Francesco Ragusa
43
Systematic Effects
Recall the formulas we found for the
parabola fit
∑ ynGn
a =
∑Gn
∑ ynGn
b=
∑ xnGn
∑ ynGn
c=
∑ xn2Gn
Using those formulas it is easy to evaluate systematic effects on the track
parameters due to systematic errors on
the position measurements
Inserting, for example, in the formula
for the curvature
3
∑ Gn
+δ n = 0 2
c=
2
∑ xnGn ∑ xnGn
∑ ynGn
≡ ctrue + δc
a more sofistcated effect could be
the rotation of the vertex detector
Examples:
displacement of vertex detector with
respect to the central detector
y 0 → y 0 + δ0
y1 → y1 + δ1 y2 → y2 + δ2
3
y 0 → y 0 + δ y1 → y1 + δ y2 → y2 + δ
c=
∑ ynGn
∑ xn2Gn
An Introduction to Charged Particles Tracking – Francesco Ragusa
∑ δnGn
+ n =0 2
∑ xnGn
≡ ctrue + δc
44
Systematic Effects: Misalignment
Let’s assume that one measurement is
systematically displaced: misalignment or
distortion (systematic error)
y k → yk + δ
1
1
=
+
2R
2Rtrue
Gk
∑ xn2Gn
δ
∆
1
δ
∼2 2
R
L
the second term is the systematic effect
on the radius due to the systematic
error on the measurement
y
Since the coefficients Gk are know the
effect can be precisely estimated
Please notice
xo
xN
xk
Recall the formula for the radius from the
parabola fit
1
=
2R
∑ ynGn
∑ xn2Gn
the sign of the systematic error on
1/R is fixed by the sign of δ
Q = +1
1
R
1
>0
R
increases
Q = −1
1
R
1
<0
R
decreases
introducing the coordinate with error
1
=
2R
∑ ynGn
∑ xn2Gn
+
Gk
∑ xn2Gn
δ
An Introduction to Charged Particles Tracking – Francesco Ragusa
45
Systematic Effects: Misalignment
As an example consider a systematic
effect as seen in the ALEPH TPC
As the plot clearly show the error is
quite large
The resolution was studied using muon
pairs produced in e+ e- annihilation at the
Z0 peak
To account for this error δ ~ 1 mm !
The muon are produced back to back and
have exactly half the c.m. energy each
The plot show the momentum reconstructed separately for positive and negative
muons
Magnetic field distortion
A correction procedure is the essential
The following plot shows the same
distribution after proper magnetic
distortion corrections are applied
Ebeam/ptrack
An Introduction to Charged Particles Tracking – Francesco Ragusa
Ebeam/ptrack
46
Problems With The Fit Procedure
We have learned how to use linear
models to fit the projection of the
charged particle track
The fit procedure gives the track
parameters at a given surface or plane
The method could be extendend to non
linear problems (inhomogeneous magnetic
field) by linearization and iteration
The solution of the problem is given by
−1
p = ( AT WA )
AT WY
W = V -1
to solve the problem the matrix V
has to be inverted
easy if V diagonal ( time O(n) )
We have seen that multiple scattering
introduces correlation among measurements and makes V non diagonal
For large detectors the dimension of V
can be prohibitively large (time O(n3))
Often prediction of the track crossing
point at a different plane is needed
impact parameter
match with calorimeters
match with particle ID (RICH)
The fit procedure described is not
optimal for this problem:
multiple scattering makes prediction
(extrapolation) non optimal
The fit is normally used to rank track
candidates during pattern recognition
An Introduction to Charged Particles Tracking – Francesco Ragusa
47
Kalman Filter
filtered
predicted
measured
smoothed
to start the Kalman Filter we need a seed
The filtered trajectory
the position on the next plane is predicted
The smoothed trajectory
the measurement is considered
prediction and measurement are merged
(filtered)
filtered trajectory
then new
prediction … measurement …
predicted trajectory
filtering … prediction … measurement …
smoothed trajectory
filtering … prediction … measurement …
An Introduction to Charged Particles Tracking – Francesco Ragusa
48
Kalman Filter
The filtering is nothing but a weighted
average of the
new measurement yn
The effect of multiple scattering, or any
other stochastic effect, can be handled
in the prediction
The advantages of this procedure are
the prediction yp
1
1
y
+
yn
p
σp2
σn2
yf =
1
1
+
σ p2
σn2
σ p2
σn2
yf = 2
yp + 2
yn
σp + σn2
σ p + σn2
is an iterative procedure
not necessary to invert large
matrices
is a local procedure: at any step the
estimate at the given plane is the
best that make use of the prevoius
measurements
Clearly if the new measurement has a
very large error
σn → ∞
y f → yp
measurement ignored
If the prediction has a large error (for
example large multiple scattering)
σp → ∞
y f → yn
prediction ignored
An Introduction to Charged Particles Tracking – Francesco Ragusa
49
Kalman Filter
The consequence is that if you want the
optimal measurement at the origin you
have to start the filter from the end of
the track
After all the measurements have been
used (filtered) it is possible possible to
build a procedure that
uses the (stored) intermediate
results of the filter
gives the best parameter estimation
at any point
This is the smoother
production vertex
direction of flight Æ
production vertex
Å direction of filter
An Introduction to Charged Particles Tracking – Francesco Ragusa
50
Kalman Filter
Applications of Kalman Filter:
navigation
radar tracking
sonar ranging
satellite orbit computation
stock prize prediction
It is used in all sort of fields
We only observe some measurements
from sensors which are noisy:
radar tracking
charged particle tracking detectors
As an additional complication the state
evolve in time with is own uncertainties:
process stochastic noise
Eagle landed on the moon using KF
deviation from trajectory due to
random wind
Gyroscopes in airplanes use KF
multiple scattering
Usually the problem is to estimate a
state of some sort and its uncertainty
location and velocity of airplane
track parameters of charged
particles in HEP experiments
In case of tracking in HEP instead of
time we can consider the evolution of the
track parameter at the discrete layers
where the detectors perform the
measurement
However we do not observe the state
directly
An Introduction to Charged Particles Tracking – Francesco Ragusa
51
Kalman Filter
We give here the basics equations of the
Kalman Filter
Detailed discussion can be found in [2:
Bock et al 1990], [6: Avery 1992], [7:
Frühwirth 1987], [8: Billoir 1985]
Consider 2 planes of our system
pk
pk −1
The covariance matrix of pk-1 is Ck-1
The covariance matrix Ck of pk is
Ck = Fk Ck −1FkT + Mms
The matrix Mms accounts for the effect
of multiple scattering on the parameters
covariance matrix
On plane k we have some measurements
mk with a covariance matrix V
Using the track model (k means: origin at
plane k ! )
k −1
k
The measurements up to plane k - 1
allowed us to get an estimate of the
track parameters pk-1
We then propagate pk-1 to plane k
pk = Fk pk −1
Fk =
∂fk
∂pk −1
yk = Hk pk
we can obtain a second estimate of the
track parameter at plane k: q
k
T
χ2 = ( yk − mk ) V−1 ( yk − mk )
T
χ2 = ( Hk pk − mk ) V−1 ( Hk pk − mk )
minimizing χ2 gives the second estimate q k
An Introduction to Charged Particles Tracking – Francesco Ragusa
52
Kalman Filter
Summarising we have
the estimate propagated
pk
with its covariance matrix
The second estimate from the
measurement at plane k
qk
with its covariance matrix
We can obtain a proper weigthed average
of those 2 estimate
This is the filtered value at plane k
The advantage of this method are
it is clearly iterative
at each step the problem has low
dimensionality and no large matrix
has to be inverted
the computation time increases only
linearly with the number of
detectors
The estimated track parameters
closely follows the real path of the
particle
the linear approximation of the track
does not need to be valid over the
whole track length but only from one
detector to the next
Details and formulas can be found in the
cited references
An Introduction to Charged Particles Tracking – Francesco Ragusa
53
Conclusions
We have seen the basic aspects of tracking with and without magnetic fields
In both cases we have shown the use of powerful and easy linear models
We have discussed optimization of
momentum resulution: track length
impact parameter resolution: vertex detector + central detector
We have discussed the importance of multiple scattering at low momentum
We have introduced the Kalman Filter
a powerful iterative technique to optimally solve the tracking problem
I Hope It Is Rather A Beginning
I hope you found tracking interesting and
that soon some of you will work on the tracking
detector of his/her experiment
An Introduction to Charged Particles Tracking – Francesco Ragusa
55
References
1
Gluckstern R.L.
Uncertainties in track momentum and direction, due
to multiple scattering and measurement errors
NIM 24 p. 381 (1963)
2
Bock R. K., Grote H., Notz D., Regler M.
Data Analysis technique for high energy physics
experiments
Cambridge University Press 1990
3
Blum W., Rolandi L.
Particle detection with drift chambers
Springer-Verlag 1993
4
Avery P.
Applied fitting teory I: General Least Squares Theory
Cleo note CBX 91-72 (1991)
see: http//www.phys.ufl.edu/~avery/fitting.html
5
Avery P.
Applied fitting teory III: Non Optimal Least Squares
Fitting and Multiple Scattering
Cleo note CBX 91-74 (1991)
see: http//www.phys.ufl.edu/~avery/fitting.html
6
Avery P.
Applied fitting teory V: Track Fitting Using the
Kalman Filter
Cleo note CBX 92-39 (1992)
see: http//www.phys.ufl.edu/~avery/fitting.html
7
Frühwirth R.
Application of Kalman Filtering to Track and Vertex
Fitting
NIM A262 p. 444 (1987)
8
Billoir P.
Track Element Merging Strategy and Vertex Fitting
in Complex Modular Detectors
NIM A241 p. 115 (1985)
thank you for reading this lecture note:
I hope you found it useful.
If you find errors I will be grateful if
you send me an email at
francesco.ragusa@mi.infn.it
An Introduction to Charged Particles Tracking – Francesco Ragusa
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