...particles.

The vectors in Fig.2.1 are defined by

$$\begin{array} {lll} \nonumber \vec{X} & = & \vec{X}_0 + ... ... & = & ( \cos(\phi_0+\phi), \sin(\phi_0+\phi) )^T. \end{array}$$

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...$\chi^2$

The terms containing the second derivatives of $\Delta_i$'s are proportional to $\Delta_i$ and are hence small and usually negligible near the $\chi^2$ minimum.

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...space

This assumption is valid for a thin material. If it is not, we can always slice the material into thin enough sublayers and apply the method explained here repeatedly. For a high momentum track, a different treatment is possible as described later.

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...material

We assume that the fluctuation of the energy loss is negligible in what follows.

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...process

Notice that the helix parameter vector so obtained is the one for the original track before energy loss and multiple scattering, which can then be used for vertexing. It should be also noted that the track segments combined here may belong to detector regions with different magnetic fields, as long as the fields can be regarded as constant in each region.

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...linking

Since any straight line can be regarded as the zero field limit of some helix, the results shown below is a special case of the results in the last subsection

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...follows

The *'s in Eq.2.5.34 represent $2 \times 2$ matrices.

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...HREF="node11.html#EQtrackav">2.5.31

It is easy to generalize the results to the cases in which two or more regions of no magnetic field are present.

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...by

In Subection 2.4.c, we have Taylor-expand Eq.2.1.1 to the lowest order to get a straight line track. The track model here, on the other hand, corresponds to the next order approximation.

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...0

This does not spoil the generality of our treatment, since a nonzero $\Delta x_0$ can always recovered as a simple shift in $\xi$.

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...$(\xi, \eta, \kappa)$

We will ignore the small effects on $\kappa \equiv Q/P_T$due to the change in the dip angle $a \equiv \tan\lambda$:

$$\begin{array} {lll} \nonumber \Delta \kappa = \frac{ a ~ \Delta a}{1+a^2} \kappa \simeq 0, \end{array}$$

which approximation is justifiable when $a \hbox{ \raise3pt\hbox to 0pt{$<$}\raise-3pt\hbox{$\sim$} }1$.

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...equation

In practice K is usually region-dependent. We should, therefore, integrate this equation segment by segment.

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...HREF="node21.html#EQdematerial">3.2.12

$\Delta y_0 \equiv y'_0 - y_0$ is now a finite shift of the y coordinate of the pivot.

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...(K=0)

In fact, we have

$$\begin{array} {lll} \nonumber \frac{d}{dy_0} E_G & = & \l... ...ft( \frac{dA}{dy_0} \right) \cdot A^{-1} \right]^T ,\end{array}$$

while we know

$$\begin{array} {lll} \nonumber A(\Delta y_0 + dy_0) \cdot A^... ...~~ \left( \frac{dA}{dy_0} \right) \cdot A^{-1} = - T\end{array}$$

from Eq.3.2.8.

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...sum

If the new pivot is within the sensitive volume of the tracker which measured the track in question, we should drop the E_S term, since the multiple scattering effects in the tracker must presumably be included in the original error matrix.

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...equation

The functional form of $\phi(y)$ is unknown so that it must be integrated out eventually.

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...shown

Notice that, in the high momentum limit where the track can be approximated by a parabola, E_M is determined solely by the configuration of the tracking detector in question: the y-locations of sampling points and the spatial resolutions thereat.

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...here

Exact formulae for E_M which are valid for any n are given in Appendix B.

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...instead

This method is applicable also to thin-layer multiple scattering.

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...$E_{{\bf a}_M}$

As a matter of fact, whether the particle track is measured before or after the multiple scattering does not make any difference in the result.

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...scattering

The multiple scattering only changes $\phi_0$,$\tan\lambda$, and, through the change in $\tan\lambda$,$\kappa$:

$$\begin{array} {lll} \Delta \kappa = \kappa \frac{\tan\lambda}{1+\tan^2\lambda} \Delta \tan\lambda.\end{array}$$ (2)

Notice that the independent variables here are thus $\Delta \phi_0$ and $\Delta \tan\lambda$, E'_MS in the above expression is diagonal when we use ${\bf a}''$ which is ${\bf a'}$ with its $\kappa$ component set equal to that of ${\bf a}$. This implies that the components of E'_MS^-1 corresponding to all but $\phi_0$ and $\tan\lambda$are infinity or in other words those of E'_MS are zero. Although E'_MS^-1 is ill-defined in this sense, we can treat the infinite components as if they are finite no matter how large and at the end of calculations let them go to infinity.

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...y-locations

In our coordinate system, y corresponds to r.

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