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## Track Fitting under the Influence of Multiple Scattering

When the track is affected by multiple scattering during its coordinate measurements the error matrix (E0 in Eq.3.2.15) should take that into account. To see how to implement multiple scattering, let us start with the modification of the track equation (Eq.3.2.7): where we have set (x0,y0) = (0,0) and .Multiple scattering modifies by
 (58)
which necessitates the modification of the track equation to
 (59)
This means
 (60)
where the suffix attached to the track track parameters remines us that they are the track parameters at the starting point before any multiple scattering. In the pivot convention taken here (x0,y0) = (0,0), is the x-location of the track. Therefore, the probability of observing (n+1) hit points, and , is
 (61)
where N is a normalization factor and .We must, however, take into account the probability of obtaining the multiple scattering :
 (62)
where use has been made of Eq.3.2.11. Since we do not know , we should integrate the probability over :
 (63)
which now gives the probability of observing when the track has the track parameter at y=0.

Now, we fit the (n+1) hits to our track model (Eq.3.1.1) by minimizing
 (64)
where is given by Eq.3.1.1 with x0=0. Since this problem is linear, we can easily obtain
 (65)
with
 (66)
where EM-1 is the inverse of the error matrix due to corrdinate measurement errors, for which only the upper triangle is explicitly shown. Of course, the true error matrix is subsect to the correction for multiple scattering. Let us now derive the true error matrix below. We first note that the displacement of induces the displacement of through Eq.3.2.34:
 (67)

Defining A and by
 (68)
we can rewrite Eq.3.2.36 as
 (69)
This equation implies the full error matrix including multiple scattering to be given by
 (70)
Notice that in the absence of multiple scattering, we have and, therefore, as it should be, where in the last line, we have used
 (71)
which can be readily verified.

Our problem is now reduces to how to evaluate when the multiple scattering correlates, for instance, to .In order to calculate the correlation matrix, we first note that
 (72)
Substituting this in Eq.3.2.32, we obtain the following formula for the probability of getting the measured points for a track whose track parameter vector at y=y0 is :
 (73)
which leads us to The integral is readily performed to yield
 (74)
The second term, which was induced by multiple scattering, has a familiar path integral form in field theories. After some straightforward manipulations such as to divide the path into small segments and to replace integrals by summentions and differentiations by differences, we arrive at
 (75)
for which is enough since B is apparently symmetric. Combining Eqs.3.2.39, 3.2.46, and 3.2.47, we obtain the full error matrix as the sum of the part due to coordinate measurement errors EM and the part due to the multiple scattering in the tracking volume EMS:
 (76)
with
 (77)
The calculation of EMS is tedious but doable. We will, however, restrict ourselves to the case where the (n+1) samples are equally spaced in y and measured with an equal accuracy .We fill further assume that n is large, since the final expression otherwise becomes too complicated to show here. The large n limit then gives
 (78)
and
 (79)
which, for instance, result in The error on due to multiple scattering is thus obtained to be where use has been made of Eq.3.2.11 and The X is the material thickness through which the track passes. To see the dependence on the magnetic field explicitly, we introduce a new constant defined by When we assume and |Q|=1 (unit charge), we finally get a familiar result:
 (80)
with
 (81)

Next: Track Fitting in r-z Up: Track Fitting in r- Previous: Inclusion of Multiple Scattering
Keisuke Fujii
12/4/1998