Vertexing and Kinematic
Fitting, Part III:
Vertex Fitting
Lectures Given at SLAC
Aug. 5 – 7, 1998
Paul Avery
University of Florida
avery@phys.ufl.edu
http://www.phys.ufl.edu/~avery
Paul Avery
1
Vertex fitting
Overview of plan
• 3rd of several lectures on kinematic fitting
• Focus in this lecture on vertex fitting theory
• Plan of lectures
•
•
•
•
Lecture 1: Basic theory
Lecture 2: Introduction to the KWFIT fitting package
Lecture 3: Vertex fitting
Lecture 4: Building virtual particles
• References
• KWFIT
http://www.phys.ufl.edu/~avery/kwfit/ or
http://w4.lns.cornell.edu/~avery/kwfit/
• Several write-ups on fitting theory and constraints
http://www.phys.ufl.edu/~avery/fitting.html
Paul Avery
2
Vertex fitting
Theory
Equations of motion in solenoidal field
Written as function of arc length s, the path of the particle is a helix:
px = p0 x cos ρs − p0 y sin ρs
py = p0 y cos ρs + p0 x sin ρs
pz = p0 z
0
0
p0 y
p0 x
1 − cos ρs
sin ρs −
a
a
p0 y
p
sin ρs + 0 x 1 − cos ρs
y = y0 +
a
a
p
z = z0 + 0 z s
p
x = x0 +
5
5
where a = –0.299792458BQ and ρ = a / p.
Paul Avery
3
Vertex fitting
Creating the constraint equations
Suppose we want to force n tracks to come from a common space
point x. Assume further that the vertex has some “prior information”,
i.e., it is a beam spot at x 0 with covariance matrix Vx 0 . (If the vertex
is completely unknown, we can just set the diagonal elements to
large values.)
The condition that track i pass through the vertex generates 2
constraints: (1) r–φ and (2) z
a
0 = px i ∆yi − pyi ∆xi − i ∆xi2 + ∆yi2
2
pz i −1
2
0 = ∆zi −
sin ai px i ∆xi + pyi ∆yi / px2i + pyi
ai
3
3
where
8
83
8
ai = −0.299792458 BQi
Qi = charge
∆xi = x x − xi , etc.
For n tracks, there are 7n parameters and 2n constraints
Paul Avery
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Vertex fitting
The χ 2 includes contributions from the track parameters α, vertex
parameters x and the constraints H(α, x ) = 0:
0 5 V 0α − α 5
+0 x − x 5 V 0 x − x 5
+2 λ 0Dδα + Eδx + d5
χ2 = α − α0
T
T
0
−1
α0
Tracks
0
−1
x0
Vertex
0
T
Constraints
α has length 7n
x has length 3
λ has length 2n
δα = α − α A (expanded around α A)
δx = x − x A (expanded around x A )
D is 2n × 7n
(coefficient of track parameters Dij = ∂Hi / ∂α j )
E is 2n × 3
(coefficient of vertex parameters Eij = ∂Hi / ∂x j )
d is 2n × 1
(constant term di = Hi α A , x A )
Paul Avery
0
5
5
Vertex fitting
The crucial fact about the vertex constraint is that the constraints do
not mix tracks.
E E E= E D
D=
1
1
2
n
D2
D n
This fact vastly speeds up the calculation for the solution because the
matrices that need to be inverted can be reduced to block diagonal
form.
DV
8 =
1
3
VD = DVα 0 D
T
10 D1
D2V2 0 D2T
T −1
DnVn 0
D −1
T
n
2×2
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Vertex fitting
The solution for α and x can be written (see CBX 98–37 for a
detailed discussion)
δα = δα 0 − Vα 0 DT λ
δx = δx 0 − Vx 0 E T λ = Vx Vx−01δx 0 − Vx E T λ 0
λ = λ 0 + VDEδx
0
λ 0 = VD Dδα 0 + d
3
VD = DVα 0 DT
8
−1
5
where δα = α − α A and δx = x − x A are the deviations of the
parameters from their expansion points.
The covariance matrices are
3
Vx = Vx−01 + E T VDE
8
−1
Vα = Vα 0 − Vα 0 DT VDDVα 0 + Vα 0 DT VDEVx E T VDDVα 0
cov(α , x ) = − Vα 0 DT VDEVx
and the χ 2 is given by
3
8
0
χ 2 = λ T VD−1 + EVx−01E T λ = λ T Dδα 0 + Eδx 0 + d
Paul Avery
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5
Vertex fitting
This solution requires that we invert the following
3
= 3V
VD = DVα 0 D
Vx
8
T −1
+ E VD E
T
x0
8
n 2 × 2 matrices
−1
a 3 × 3 matrix
DV
8 =
1
3
VD = DVα 0 D
V
=
T
10 D1
D2V2 0 D2T
T −1
D1
VD 2
VDn
DnVn 0
D −1
T
n
So we have found an efficient solution.
Paul Avery
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Vertex fitting
Comments About Solution
Vertex covariance matrix
The vertex covariance matrix Vx is an average of original covariance
matrix and a sum over track info
Vx =
E E E= E 3
Vx−01
V
=
8
−1
D1
1
2
VD
n
Vx
+ E VDE
T
= V
−1
x0 +
VD 2
∑ EiT VDi
i
VDn
E
−1
i
Note that when the initial vertex is totally unknown, we make Vx 0 as
large as possible, so that Vx−01 → 0 . In that case
Vx
= ∑ E V E −1
T
i Di i
i
Paul Avery
9
Vertex fitting
Track correlations
Note the structure of the updated track covariance matrix
Vα = Vα 0 − Vα 0 DT VDDVα 0 + Vα 0 DT VDEVx E T VDDVα 0
Improvement due to fit
to fixed vertex point.
Worsening due to “wiggle”
of vertex point.
Term 2 does not cause track - track correlations, since Vα 0 , D and
VD are block diagonal. Equivalent to fitting each track separately
though the same fixed space point.
Term 3 causes track-track correlations only through the “tiny” 3× 3
Vx matrix. Track-track correlations can thus be calculated by saving
some of the intermediate matrices.
Paul Avery
10
Vertex fitting
Vertex Fitting as a Kalman
Problem
Track fitting problem
Want to find 5 helix parameters α and 5 × 5 covariance matrix Vα by
adding in the information from n measurements.
Method is to move to measurement point, then average the
measurement (usually a 1 dim. quantity) with the track to get
improved track parameters and covariance matrix. This is repeated
until all measurements are included.
Paul Avery
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Vertex fitting
• Let α0, Vα 0 be the old track parameters & covariance matrix.
• Let ∆y, Vy be the measurement deviation & its covariance matrix.
• Let A represent the derivative of the measurement with each of the
5 track parameters.
Then the new track parameters and covariance matrix are
δα = δα 0 + Vα AT Vy−1∆y
3
Vα = Vα−01 + AT Vy−1 A
3
8
−1
= Vα 0 − Vα 0 A Vy + AVα 0 A
T
8
T −1
AVα 0
Note that only a 1 × 1 matrix has to be inverted at each step.
If the track does not have to be moved or modified between
measurements, then all the measurements can be added at once, at the
cost of inverting an n × n matrix.
Paul Avery
12
Vertex fitting
Vertex fitting
Want to find 3 vertex parameters x and 3 × 3 covariance matrix Vx
by adding information from n tracks.
Each track “measurement” consists of 7 parameters α0 and a 7 × 7
covariance matrix Vα 0
Method is to average the vertex with the track to get improved vertex
parameters and covariance matrix. This is repeated until all tracks are
included.
The only difference from the track fitting problem is that a constraint
is used to average the track info with the vertex information.
Paul Avery
13
Vertex fitting
Let the index i refer to track i. Assume we have a vertex x 0 with
initial covariance Vx 0 to which we want to add track i. The following
equations update the vertex parameters, covariance matrix and χ 2 .
3
δx = Vx Vx−01δx 0 − EiT VDi β 0 i
3
= 3β
Vx = Vx−01 + EiT VDi Ei
∆χ i2
0i
8
T
8
8
−1
3
+ Eiδx VDi β 0 i + Eδx 0
8
where β 0 i = Diδα 0 i + di . Repeat until all tracks are added.
It is also possible to make a quick algorithm where tracks can be
included or discarded. Only 3 sums needed:
1. Vector of length 3
∑ EiT VDi β 0 i
i
2. 3 × 3 matrix
∑ EiT VDi Ei
i
3. Scalar
∑ ∆χ i2 as described above
i
A track can be removed easily from the sums if its ∆χ 2 contribution
is too large. This process is similar to that used for removing hits
from a Kalman fit.
Paul Avery
14
Vertex fitting