Continuation to Straight Line

  Let us first consider the extrapolation of a track in a uniform magnetic field ${\bf B}$ to an adjacent region of a different magnetic field ${\bf B'}$:

$$\begin{array} {lll} \nonumber \alpha({\bf B}) \rightarrow \alpha({\bf B'}).\end{array}$$

If the pivot is taken at the intersection of the track with the boundary of the two regions, the helix parameter vector is left unchanged so is the error matrix, except for the energy loss and multiple scattering correctoins at the boundary.

The zero field case corresponds to the infinite $\alpha'$ limit, which implies that $\phi$ goes to zero while $\alpha' \phi$ is kept finite. Our helix parametrization allows us to take this limit easily. We expand the helix equations (Eq.2.1.1) in terms of the deflection angle $\phi$ to the first order:

$$\begin{array} {lll} \left\{ \begin{array} {lllllll} x & =... ...t) \cdot (~~~ \tan \lambda ~). \end{array}\right.\end{array}$$ (17)

These become our straight line parametrization, when we make the replacement:

$$\begin{array} {lll} \left( - \frac{\alpha' \phi}{\kappa} \right) \rightarrow \mbox{const.} \equiv t,\end{array}$$ (18)

where the straight line parameter vector is given by

$$\begin{array} {lll} {\bf b} = ( d_\rho, \phi_0, d_z, \tan\lambda )^T\end{array}$$ (19)

and their error matrix is obtained by deleting the third row and the third column of the error matrix for the helix:

$$\begin{array} {lll} E_{\bf b} = \left( \frac{\partial {\bf ... ...\frac{\partial {\bf b}}{\partial {\bf a'}} \right)^T\end{array}$$ (20)

with

$$\begin{array} {lll} \left( \frac{\partial {\bf b}}{\partial... ...& 0 \cr 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 1 }.\end{array}$$ (21)