I've written a routine to calculate a "first guess" set of
parameters for CDC tracking. In order to do real tracking,
a set of reasonable starting parameters will be needed. I
describe the method briefly here and show some plots generated
from HDGeant data. The short summary is that the following
resolutions were achieved for gamma p -> n pi+ pi+ pi-:
| Momentum resolution |
~5% |
| Momentum resolution in X-Y plane |
~5.5% |
| Momentum resolution in Z |
~4.8% |
| Phi resolution |
~1.2% |
| Theta resolution |
<1% |
To start with, I took x,y,z coordinates from:
hddm->physicsEvents->in[i].hitView->centralDC->rings->in[j].straws->in[k].cdcPoints->in[m].z
I also used the "track" field to sort the hits into tracks. The hits in
a given track were sorted by z.
The method used two assumptions:
- a uniform magnetic field (i.e. the track represents a segment
of a perfect circle in the X/Y plane
- the track passed through the beamline.
This second assumption was needed for the following reason:
The coordinate system of the xy hits used the beamline as the origin.
The center of the circle on which the track lies (in the X-Y plane) can
be given the coordinates Xo,Yo. Let the hits themselves be denoted by Xi, Yi;
To fall on a circle, the following is true for all hits:
(Xi-Xo)^2 + (Yi-Yo)^2 = Ro^2
Expanding this, the Xo^2, Yo^2, and Ro^2 terms all cancel if the
circle passes through the beamline (i.e. Xo^2 + Yo^2 = Ro^2)
This leaves a relation that is linear in Xo and Yo and can be
solved with the standard linear regression technique.
Here are a few pictures of events with the fitted circles drawn over the
tracks. These pictures show a view of looking directly upstream.
The charge of the particle is determined by whether it is rotating
clockwise or counter-clockwise as z-increases. The technique used
could probably be improved upon, but here's what I did:
The basic idea is that you calculate the 2-D cross product of two
points at different z values. The sign of the cross product
will be the sign of the charge.
Since it's possible to have a "bad hit", I had it sum cross products
between the first half of the hits and the second half. This should
be a little more robust. For now, I just assumed singly charged
particles.
The theta angle was determined by looking at the phi angle relation
to the z coordinate of the hits. The value dphi/dz was calculated
numerically from the hits. The value of theta is then given by
atan(Rdphi/dz).
The phi angle is just the phi angle of the point Xo,Yo shifted
by + or - pi/2 depending on whether the charge is poisitive or negative.
The momentum is directly proportional to both the radius of the circle
and the magnetic field. I estimated a value for the magnetic field
at first, but then used the offset of the thrown and calculated
momenta to correct it. The total momentum is calculated by dividing the
transverse momentum by sin(theta).
The accuracy of this method is shown in the following plots. Each plot
represents the value of (thrown-reconstructed)/thrown for the specified
parameter.
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