Alain Blondel

1
Track fit conventions for HARP
The fit outputs five parameters w 1/R the
signed curvature, convention a positive
particle in a positive B field will have a
positive cuvature. (turns clockwise) d0 the
signed impact parameter. Convention a
positive impact parameter means that the origin
of axes is inside the track circle. j0 the
azimuthal angle of the track vector at the point
of closest approach (the oriented tangent to the
circle) z0 z- coordinate of the point of
closest approach of the track to the origin tan
l , where l is the dip-angle of the track
(latitude, not polar angle)
y
uj
ur
yo
j0
b0
a0
X
C
dy
x
M0
M
Case where Rgt0, d0gt0 .
N.B. The triedron ( ur, uj,Oz) is always direct
according to Lorentz law. d0 and R are the
coordinates of O and C on the axis M0ur N.B.B.
The choice of sign of R assumes a certain
direction of motion, parallel to uj ! dy will
decrease with time if Rgt0. One could describe the
same trajectory with the consistent change
R-gt -R, j0 ? j0 p , tan l ? - tan l
which would describe a particle of the other
charge advancing in the other direction
2
If we use cylindrical coordinates, we need the
distance D of the center of the circle to the
origin and its azimuth a0. We also need the same
for the point of closest approach d and b0.
Finally the points on the circle will be
described by their azimuth on the circle, y, y0
being that of the point of closest approach,
and dyy-y0.
y
uj
ur
yo
j0
b0
a0
y
X
C
uj
dy
x
M0
ur
yo
M
j0
X
Figure 1a) Case where Rgt0, d0gt0 . a0 j0 -
p/2 b0 j0 p/2 y0 j0 p/2
C
a0
dy
M0
b0
x
M
Figure 1b) Case where Rgt0, d0lt0 . a0 j0 -
p/2 b0 j0 - p/2 y0 j0 p/2

• figures 1 a,b,c,d illustrate the conventions and
  signs
• for the four cases
• Rgt0, d0gt0,
• Rgt0, d0lt0,
• Rlt0, d0gt0,
• Rlt0, d0lt0

3
y
yo
b0
a0
y
X
C
dy
x
M0
j0
ur
yo
M
uj
X
C
a0
dy
O
b0
M0
Figure 1c) Case where Rlt0, d0gt0 . a0 j0
p/2 b0 j0 - p/2 y0 j0 - p/2
x
j0
ur
M
uj
Figure 1d) Case where Rlt0, d0lt0 . a0 j0
p/2 b0 j0 p/2 y0 j0 -p/2
4
From these figures one can see that all angles
can be derived from j0, R and d0 with a single
formula a0 j0 - R/R p/2 b0 j0 R/R
d0/ d0 p/2 y0 j0 R/R p/2 One can see
also that the distance of the center of the
circle to the origin is given as D R-d0 The
Cartesian coordinates of the center of the circle
are xc (R-d0) cos a0 (R-d0) cos( j0 -
R/R p/2 ) yc (R-d0) sin a0 (R-d0) sin
( j0 - R/R p/2 ) and the point on the circle
can be written as x xc R cos (y) xc
R cos (y0 dy) xc R cos (j0 R/R p/2
dy) xc R cos (j0 p/2dy) xc - R sin
(j0 dy) y yc R sin (y) yc R sin
(y0 dy) yc R sin (j0 R/R p/2 dy)
yc R sin (j0 p/2dy) yc R cos (j0
dy) since cos (jp) - cos (j) sin (jp) -
sin (j) cos (jp/2) -sin(j) sin (jp/2)
cos(j) One can see that the distance to the
point of closest approach of a point on the
circle is ds - Rdy the convention on R means
that normally for a particle originating from the
center, dsgt0. The equation in z then becomes z
z0 tan l ds or z z0 - tan l R dy
5
SIGN of R? The fundamental C.T ambiguity.
y
y
yo
yo
j0
b0
a0
b0
a0
X
C
X
C
dy
x
dy
M0
x
M0
j0
M
M
y
?
In many cases the convention to orient the track
according to where the measurement points are
will give the right answer. and describe
properly where the particle comes from and where
it goes. (above) In other cases the TPC alone can
not tell COSMICS LOOPERS or even particles
from secondary decays. RPC or TOF or other
physics algorithms will be needed to validate
the choice or solve the C.T ambiguity in
ambiguous cases.
yo
b0
a0
X
C
dy
x
M0
j0
M
6
SUMMARY
With the conventions and notations described
above, a track can be described unambiguously by
the following parametric equation x (R-d0)
cos( j0 - R/R p/2 ) - R sin (j0 dy) y
(R-d0) sin ( j0 - R/R p/2 ) R cos (j0 dy)
z z0 - tan l R dy The parameter dy describes
the progression of the particle along its track.
At the level of tracking with TPC alone, the
choice of sign for R is made such that all -- or
the majority -- of the TPC points lie with ds
- R dy gt 0
One could describe the same track with the
consistent change R-gt -R, j0 ?
j0 p , tan l ? - tan l which would correspond
to a particle of the other charge advancing in
the other direction. This C.T ambiguity cannot be
solved by the TPC alone, and must be validated
with other detectors or physics algorithms.