Track Parametrization
YUKIYOSHI OHNISHI
KEK, National Laboratory for high Energy Physics,
Tsukuba, 305, Japan
Abstract
This is a note1 for track parameters[1] and their introductive manipulations that
will be utilized in track reconstruction. This shows how to represent trajectory of
both helical and straight tracks and to implement multiple scattering and energy
loss eects. A right-handed coordinate system is used in this track parametrization
in which the magnetic eld is along the z-axis.
1 The Reference Point
Track ts should be done at dierent t points to nd the best representation of the track
in the vicinity of that point. Each t uses the dierent pivotal point x0 = (x0 ; y0; z0 )T
as the reference point. This pivotal point makes the beginning point of track segment
explicit and how corrections to the track parameters have been done clear. In the CDC
ts the wire position of the most inner hit is chosen as the pivotal point(pivot).
2 Fit Parameters
There are ve parameters a = (d; 0; ; dz ; tan )T to be derived by track tting. These
helix parameters;
d is the signed distance of the helix from the pivot in x-y plane,
0 is the azimuthal angle to specify the pivot with respect to the helix center( The
range of 0 takes 0 to 2 in BELLE ),
is 1=Pt (reciprocal of the transverse momentum) and the sign of represents the
charge of the track assigned by the track tting,
dz is the signed distance of the helix from the pivot in the z direction,
tan is the slope of the track, tangent of the dip angle.
1 Revised:
17 June, 1997
1
y
y
(b) Positive Track
(a) Negative Track
(x, y, z)
φ0
(x 0 ,y 0 ,z 0 )
w φ
v
φ0
φ
v
(x 0 ,y 0 ,z 0 )
(x, y, z)
w
x
x
Figure 1: Schematic representation of the helix parametrization for (a) negative and (b)
positive charged track. The vectors in the gure are dened by ~x = ~x 0 +(d + ) w~ ? ~v;
where w~ = (cos 0; sin 0)T and ~v = (cos(0 + ); sin(0 + ))T .
3 Charged Track in Uniform Magnetic Field
Charged particle in a uniform magnetic eld is represented by a helical trajectory. The
position along the helix is given by
8
>
< x = x0 + d cos 0 + fcos 0 ? cos(0 + )g
y = y + d sin + fsin ? sin(0 + )g
(1)
>
: z = z00 + dz 0 ? tan 0 ;
where alpha is the magnetic-eld-constant, = 1=cB = 10000=2:9979258=B [cm
(GeV/c)?1] at the strength of magnetic eld B [K Gauss], is the turning angle, that
is an internal parameter with a sign, (for instance, has a negative sign for out-going
positive tracks from the pivot.) and determines the location. The helix parameters
a = (d; 0; ; dz ; tan1 )T and error matrix are dened at the pivot x0 = (x0; y0; z0)T .
The helix center in x-y plane is
cos 0
yc = y0 + d + sin 0 ;
xc = x0 + d +
and the signed radius of the circle is
(3)
= :
(4)
t
(5)
In terms of these parameters, the track momentum is given by:
1
P =
1
(2)
jj
0 is also dened by 0 = atan2((d + =)(yc ? y0 ); (d + =)(xc ? x0 )):
2
y
Straight Track
(x 0 ,y 0 ,z 0 )
(x, y, z)
dρ
φ0
v
w
x
Figure 2: Schematic representation of the track parametrization for straight track. The
vectors in the gure are dened by ~x = ~x0 + d w~ + t ~v; where w~ = (cos 0 ; sin 0)T and
~v = (? sin 0 ; cos 0 )T .
or at the deection angle ,
0 1
1
0
px
?
sin(0 + )
1
B@ py CA =
B@ cos(0 + ) CA :
jj
pz
tan (6)
4 Neutral Track and the case of Non-magnetic Field
Neutral track and charged particles in the non-magnetic eld are represented by a straight
line using four parameters a = (d; 0; dz ; tan )T .
The straight line is given by a helix with innite radius:
8
>
< x = x0 + d cos 0 ?t sin 0
y = y + d sin +t cos >
: z = z00 + dz 0 +t tan ;0
(7)
where t is the projected path length onto x-y plane corresponds to in the helix case.
5 Pivot Transformation
It is very useful to change a pivot because an appropriately chosen pivot simplies calculations, for example, in energy loss or multiple scattering corrections. The pivot change
x0 = (x0; y0; z0)T ! x00 = (x00; y00 ; z00 )T :
(8)
includes the following change in the helix parameters
a = (d; 0; ; dz ; tan )T ! a0 = (d0; 00; 0; d0z ; tan 0)T
3
(9)
and the error matrix
!
!
@ a0
@ a0 T
Ea ! Ea =
Ea
@a
@a
The new helix parameters a0 = (d0; 00; 0; d0z ; tan 0 )T are
0
0
d = x0 ? x0 + d + cos 0 cos 00
0
+ y0 ? y0 + d + sin 0 sin 00 ? 0
0
0
y0 ? y0 + d + sin 0 ;
0 = atan2 d +
: d0 +
x0 ? x00 + d + cos 0
0 = 0
0
dz = z0 ? z0 + dz ?
(00 ? 0) tan tan 0 = tan :
0
(10)
(11)
(12)
(13)
(14)
(15)
6 Track Extrapolation
6.1 Energy Loss Correction
The pivot transformation is useful to correct energy loss eect because only one of track
parameters, can be aected by energy loss eect at the point to be corrected. The
procedure is rst to move the pivot to the energy loss point x00 as
x0 = (x0; y0; z0)T ! x00 = (x00; y00 ; z00 )T
(16)
and also track parameter vector and its error matrix change as discussed in the previous
section.
a = (d; 0; ; dz ; tan )T ! a0 = (d0; 00; 0; d0z ; tan 0)T
Ea
Then correct 0 as
!
!
!
@ a0
@ a0 T
Ea =
E
@a a @a
0
0corr: = 0 + dE
;
dx
(17)
(18)
where dEdx is calculated from the average energy loss in the material.
The error matrix dose not change by energy loss correction since eq.(18) implies
!
@ a0corr:
= 1:
@ a0
(19)
Unless the pivot is chosen at the energy loss point, all track parameters except for
tan are aected by the energy loss. This is one of the advantages of having the freedom
to arbitrarily choose the pivot in this helix parametrization.
4
6.2 Multiple Scattering Correction (material of thin layer)
This section shows how to take the eect of the multiple scattering on the error matrix
into account. The rst procedure is the same as the energy loss correction. Move the
pivot to the material, the track parameters and its error matrix change simultaneously.
Then the error matrix for the track extended through the material becomes
Ea corr: = Ea + EMS ;
0
0
(20)
where the second term on the right-hand side represents the correction to the error matrix
due to the multiple scattering:
2
(EMS )22 = (1 + tan2 ) MS
2
(EMS )33 = 2 tan2 MS
(EMS )35 = (EMS )53
2
= tan (1 + tan2 ) MS
2
(EMS )55 = (1 + tan2 )2 MS
;
(21)
(22)
(23)
(24)
(25)
with all the other components are zero. The MS is given by
s
L 0
:0136 L MS =
1 + 0:038 ln X ;
p(GeV) X0
0
(26)
where , p, and L=X0 are the velocity in units of light velocity, the momentum, and the
thickness of the material in unit of its radiation length.
When the error matrix for extrapolated track is calculated, it is easy to estimate the
position error at a deection angle in terms of
@x
Ex () =
@ a0
!
Ea corr: 0
!
@x T
:
@ a0
(27)
References
[1] This follows the parametrization used by the TOPAZ Collaboration in Nucl. Inst.
Meth. A264 (1988) 297.
5
A Propagation of Error
Propagation of error matrix is written by
Ea
!
!
!
@ a0 T
@ a0
E
;
Ea =
@a a @a
(28)
0
where a and a0 indicates original helix parameter vector and new helix parameter vector
respectively. Jacobian is
0
BB
BB
!
@ a0
BB
=
BB
@a
B@
where
@d0
@d
@d0
@0
@d0
@
@d0
@dz
@00
@d
@00
@0
@00
@
@00
@dz
@0
@
@0
@d
@d0z
@d
@d0z
@0
@d
@d
@0
@d
@
@d
@dz
@d
@ tan @d
@d
@0
@0
@0
@
@0
@dz
@0
@ tan @0
0
0
0
0
0
0
0
0
0
0
0
0
@ tan @0
@ tan @
@ tan @dz
@ tan @ tan @ tan 0
0
0
0
@d
0
0
0
@d
@dz
@0
@dz
@
@dz
@dz
@dz
@ tan @dz
0
0
0
0
@d
@
@0
@
@
@
@dz
@
@ tan @
0
0
1
CC
CC
CC ;
CC
CA
(29)
= cos 00 cos 0 + sin 00 sin 0
(30)
=
(31)
=
d +
(sin 00cos0 ? cos 00 sin 0 )
f
1 ? (cos 00 cos 0 + sin 00 sin 0)g
0
2
@d
= @ tan = 0
= ? 0 1 (sin 00 cos 0 ? cos 00 sin 0)
d + !
d + = d0 + (cos 00 cos 0 + sin 00 sin 0)
1
= 2 d0 + (sin 00 cos 0 ? cos 00 sin 0 )
0
0
= @ @
tan = 0
= 1
(32)
(33)
(34)
(35)
(36)
(37)
@0 @0
@0
= @
= @d = @ tan
=0
0
z !
tan = d0 + (sin 00 cos 0 ? cos 00 sin 0)
!
!
d + 0
0
=
tan 1 ? 0 (cos 0 cos 0 + sin 0 sin 0)
d + 6
(38)
(39)
(40)
(41)
@d0z
@
@d0z
@dz
@d0z
@ tan @ tan 0
@d
@ tan 0
@ tan =
2
0
tan 0 ? 0 ?
sin 00 cos 0 ? cos 00 sin 0
d0 + !!
(42)
= 1
(43)
= ? (00 ? 0)
0 @ tan 0 @ tan 0
= @ tan
= @ = @d = 0
@
(44)
= 1:
(46)
(45)
z
0
In order to estimate position error using error matrix, the following jacobian is needed:
0
! B
@x
=B
@
@a
@x
@0
@y
@0
@z
@0
@x
@d
@y
@d
@z
@d
@x @x
@ @dz
@y @y
@ @dz
@z @z
@ @dz
@x
@ tan @y
@ tan @z
@ tan 1
CC
A;
(47)
where
@x
@d
@x
@0
@x
@
@x
@dz
@y
@d
@y
@0
@y
@
@y
@dz
@z
@d
@z
@
@z
@dz
@z
@ tan = cos 0
= ?d sin 0 +
(48)
f? sin 0 + sin(0 + )g
(49)
= ? 2 fcos 0 ? cos(0 + )g
@x
= @ tan
=0
(51)
= sin 0
(52)
= d cos 0 + fcos 0 ? cos(0 + )g
= ? 2 fsin 0 ? sin(0 + )g
@y
= @ tan
=0
@z
= @
=0
2
= 1
= ?
tan (53)
(54)
(55)
(56)
0
=
(50)
(57)
(58)
(59)
:
7
B Multiple Scattering After Passing Through Thin
Layer
The correction to error matrix due to multiple scattering eect is shown in Section 6.2:
Ea = Eameas: + EMS
(60)
The multiple scattering changes the helix parameter from a to a0 . If pivot is chosen at
the point of multiple scattering, tangential vector changes from
to
@ @ @
@ sin
0 ; ? cos 0 ; ? tan =
@ =0
@x @y @z
!
!
@ =
@0 =0
sin 00 @ ; ? 0 cos 00 @ ; ? 0 tan 0 @
0
@x @y @z
!
@ @ @
@
' @ + 0 cos 0 0 @x ; 0 sin 0 0 @y ; ? 0 tan @z ;
where 00 = 0 + 0, 0 = + , and tan 0 = tan + tan .
Therefore, the direction change of track, is given by
@ @ 2
^ ()2 ' (sin )2 = @@ 2 @@ 2
@ @ (61)
(62)
(63)
(64)
0
!
0
2 tan 1
2
= 1 + tan
2 1 ? 1 + tan2 ( tan ) (0 )
!
1
2
tan
+ (1 + tan2 )2 1 ? 1 + tan2 ( tan ) ( tan )2 (65)
( tan )2 :
0 )2
+
(66)
' 1 (
+ tan2 (1 + tan2 )2
0 and tan are dened as components of a = a ? a0. due to deviation of tan is derived from momentum conservation.
tan tan :
+ ' + (67)
1 + tan2 The correction of error matrix, EMS , is determined from
!
2 = aT E ?1 a
(68)
MS
0
0
0
B
2
B
0 (1 + tan2 )MS
B
0
0
EMS = B
B
B
0
@0
0
0
MS
a = (0; 0; ; 0; tan )T
0
0
2
2
tan2 MS
0
2
tan (1 + tan2 )MS
8
0
0
0
0
2
0 tan (1 + tan2 )MS
0
0
2
0
(1 + tan2 )2MS
(69)
1
CC
CC
CC :(70)
A