Pivot Transformation

The parameter vector for a r-z track has only two components (see Eq.3.1.1): ${\bf a} \equiv ( \zeta, a )^T$ which transforms as

$$\begin{array} {lll} \nonumber d{\bf a} = - \pmatrix{1 \cr 0} dz_0 + T \cdot {\bf a} dy_0 ,\end{array}$$

where

$$\begin{array} {lll} T \equiv \pmatrix{ 0 & 1 \cr 0 & 0 }.\end{array}$$ (82)

As we did in the r-$\phi$ case, we set dz₀=0. Then, we have, in vacuum,

$$\begin{array} {lll} \frac{ d{\bf a} }{dy_0} = T \cdot {\bf a}.\end{array}$$ (83)

The equation to transform the error matrix is then

$$\begin{array} {lll} dE = \left( T \cdot E + E \cdot T^T \ri... ...atrix{ 0 & 0 \cr 0 & 1 } \left\vert dy_0 \right\vert\end{array}$$ (84)

with  

$$\begin{array} {lll} \sigma_a^2 & = & (1+\tan^2\lambda)^2 \s... ...ht)^2 \frac{ (1+a^2)^{1/2} \vert dy_0\vert }{X_0}.\end{array}$$ (85)

The solution to this equation can be written in the form:

$$\begin{array} {lll} E(\Delta y_0) = E_G(\Delta y_0) + E_S(\Delta y_0)\end{array}$$ (86)

where E_G is the general solution for K'=0 and E_S is a special solution:

$$\begin{array} {lll} E_G = \left( \frac{\partial {\bf a}}{\p... ...rac{\partial {\bf a}}{\partial {\bf a}_0 } \right)^T\end{array}$$ (87)

with

$$\begin{array} {lll} \left(\frac{\partial {\bf a}}{\partial ... ...a}_0}\right) = \pmatrix{ 1 & \Delta y_0 \cr 0 & 1 }\end{array}$$ (88)

and

$$\begin{array} {lll} E_S = \pmatrix{ \frac{K'}{3} \vert\Delt... ... \vert\Delta y_0\vert^2 & K' \vert\Delta y_0\vert },\end{array}$$ (89)

where E₀ is the error matrix at $\Delta y_0 = 0$.Notice that the second term, which is due to multiple scattering, must be dropped when one moves the pivot in the sensitive volume of the tracker which gave E₀, since the effect of multiple scattering is presumably taken into account in E₀. For instance, if one wants to extrapolate the track in the increasing y direction, one should first move the pivot to the outermost hit with K'=0 and then use to above formulae.