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As shown in Section 3, in the high momentum limit,
the r- and the r-z track fittings are decoupled so that the
error matrix or its inverse corresponding to
the coordinate measurement errors has a
blockwise diagonal form:
| |
(1) |
where the component matrices are given by
| |
(2) |
and
| |
(3) |
Notice that, in this limit, the error matrix is determined completely by
the y-locations
of the sampling points and the spatial resolutions
thereat.
If (n+1) sampling points are equally spaced and have common
resolutions, and , then the above equations
become
| |
(4) |
and
| |
(5) |
where we have defined the lever arm length L by
| |
(6) |
Notice that in the above equations, we have shown only upper triangles of
the matrices, since they are all symmetric.
By matrix inversions, we thus obtain
| |
(7) |
and
| |
(8) |
The above error matrices are defined with the pivotal
point (x0,y0,z0) chosen to be at the -th hit.
From the above formulae, we can estimate, for instance,
the transverse momentum resolution or as
| |
(9) |
where the last line is none other thatn the familiar
text book expression
for the momentum resolution valid in the large n limit.
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Keisuke Fujii
12/4/1998