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Consider a particle passing through a thin layer, whose
track parameter vector changes from to
due to multiple scattering there.
We assume here that the track parameter vector of the particle
is measured after the multiple scattering to be with its error matrix ^{}.
In the Gaussian approximation valid for small angle multiple scattering,
the probability of getting when the true track parameter
vector before the multiple scattering is is given by

| |
(1) |

where *E*'_{MS} is the diagonalized error matrix corresponding to
multiple scattering^{}
*N* is a normalization factor depending only on
and *E*'_{MS}, and
with *B* being a matrix whose only nonzero component is
.Making use of the fact that *B* is nilpotent, i.e.
*B*^{2} = 0, we can easily eliminate from
the exponent of Eq.A.1:
| |
(3) |

where we have defined
| |
(4) |

and
| |
(5) |

Changing the integration variable from
to , Eq.A.1 now becomes
| |
(6) |

Notice that the last line is a constant independent of
and ,which proves that the error matrix including the multiple scattering
has to be
| |
(7) |

The derivation of the explicit form of *E*'_{MS}
is straightforward.
The multiple scattering changes the helix
parameter vector from to and, when the pivot is chosen to be the point of the multiple scattering,
the tangential vector thereat changes as

| |
(8) |

Then the direction change of the track in space is
given by
| |
(9) |

where and are defined as
components of .The above equation determines *E*'_{MS} through
| |
(10) |

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*Keisuke Fujii*

*12/4/1998*