Inclusion of Multiple Scattering

  When the pivot is taken at the point of multiple scattering, only $\eta$ is affected among the track parameters $(\xi, \eta, \kappa)$. The scattering angle in the r-$\phi$ plane $\Delta \eta \equiv \Delta \phi_0$ has a variance  

$$\begin{array} {lll} \sigma_{\Delta \phi_0}^2 & = & ( 1 + ... ...}{X_0} \equiv K ~ \left\vert \Delta y_0 \right\vert\end{array}$$ (47)

with $C = 0.0141 {\rm GeV}$, P and $\beta$ are the momentum and the velocity of the particle, and X₀ is the radiation length of the material. This modifies (E)₂₂ and Eq.3.2.9 now becomes  

$$\begin{array} {lll} d E = - \left( T \cdot E + E \cdot T^T ... ...& 0 \cr 0 & K & 0 \cr 0 & 0 & 0 } \vert dy_0\vert.\end{array}$$ (48)

Notice that $(E)_{33} \equiv \sigma_\kappa^2$ is not affected by the pivot transformation, which allows us to readily integrate the above equation. The solution of the above equation can be written in the form:

$$\begin{array} {lll} \nonumber E = E_G + E_S\end{array}$$

where E_G is the general solution for K = 0, while E_S is a special solution of Eq.3.2.12. It is easily verified that  

$$\begin{array} {lll} E_S = \pmatrix{ \frac{K}{3}\vert\Delt... ...ert^2 & K \vert\Delta y_0\vert & 0 \cr 0 & 0 & 0 }\end{array}$$ (49)

satisfies Eq.3.2.12. It is also easy to show that  

$$\begin{array} {lll} E_G = \left( \frac{\partial {\bf a}}{\p... ...rac{\partial {\bf a}}{\partial {\bf a}_0 } \right)^T\end{array}$$ (50)

with  

$$\begin{array} {lll} \left( \frac{\partial {\bf a}}{\partial... ...& 1 & - \frac{1}{\alpha} \Delta y_0 \cr 0 & 0 & 1 }\end{array}$$ (51)

is the general solution of the homogeneous equation (K=0).

In summary, the propagated error matrix at the point which is outside the area of coordinate measurements is the sum:  

$$\begin{array} {lll} E(\Delta y_0) = E_G(\Delta y_0) + E_S(\Delta y_0) ,\end{array}$$ (52)

where E_G is the propagated error matrix without taking multiple scattering into account (Eqs.3.2.15 and 3.2.16) and E_S is the contribution from multiple scattering (Eq.3.2.14).

Now that we have established the machinary to move the pivot to anywhere in the detector system, it is easy to implement the information from external tracking devices. Assume that we have an extra coordinate measurement $x = \overline{x}$ at $y = \overline{y}$. Then we take $(0,\bar{y})$ as our new pivot and minimize

$$\begin{array} {lll} \chi^2 = ({\bf a'} - \overline{{\bf a}}... ...eft( \frac{\xi-\bar{x}}{\sigma_{\bar{x}}} \right)^2,\end{array}$$ (53)

which is a simple matrix inversion. The corresponding error matrix can also be readily obtained from

$$\begin{array} {lll} E'^{-1} = \frac{1}{2} \frac{\partial^... ...\bar{x}}^2 & 0 & 0 \cr 0 & 0 & 0 \cr 0 & 0 & 0 } .\end{array}$$ (54)

If there are more than one extra measurements, we can retransform E' to the next and do the same.

If $\sigma_{\bar{x}}$ is negligibly small, we might even include this as a constraint on the track parameter. In this case, the track becomes

$$\begin{array} {lll} x = \bar{x} - \eta ( y - \bar{y} ) + ... ...{2} \left(\frac{\kappa}{\alpha}\right) (y-\bar{y})^2\end{array}$$ (55)

which has only two parameters:

$$\begin{array} {lll} \nonumber {\bf A} \equiv ( \eta, \kappa )^T.\end{array}$$

The error matrix for ${\bf A}$ is given by

$$\begin{array} {lll} E_{\bf A} = \left[ \left(\frac{\parti... ...tial {\bf a}}{\partial {\bf A}}\right) \right]^{-1}\end{array}$$ (56)

with

$$\begin{array} {lll} \left(\frac{\partial {\bf a}}{\partial ... ...A}}\right) = \pmatrix{ 0 & 0 \cr 1 & 0 \cr 0 & 1 }.\end{array}$$ (57)