Helix Parametrization

A charged particle in a uniform magnetic field follows a helical trajectory. In what follows we set our reference frame so that it has its z axis in the direction of the magnetic field. There are many ways to parametrize this helix. The following choice, however, greatly facilitates various operations we are going to make on tracks, such as track extrapolation and linking:  

$$\begin{array} {lll} \left\{ \begin{array} {lllllll} x & =... ...ppa} \tan \lambda \cdot \phi , \end{array}\right.\end{array}$$ (1)

where ${\bf x}_0 = (x_0,y_0,z_0)^T$ specifies an arbitrarily chosen pivotal point or simply pivot: we usually take as a pivot a hit or wire position of the tracking device or a position of track linking. Once the pivotal point is fixed, the helix is determined by a 5-component parameter vector ${\bf a} = (d_\rho,\phi_0,\kappa,d_z,\tan \lambda)^T$, where $d_\rho$ is the distance of the helix from the pivotal point in the xy plane, $\phi_0$ is the azimuthal angle to specifies the pivotal point with respect to the helix center, $\kappa$ is the signed reciprocal transverse momentum so that

$$\begin{array} {lll} \nonumber \kappa & = & Q/P_T \\ \rho & = & \alpha/\kappa \end{array}$$

with Q being the charge, $\rho$ being the signed radius of the helix, and $\alpha \equiv 1/c B$ being a magnetic-field-dependent constant, d_z is the distrance of the helix from the pivotal point in the z direction, and $\tan\lambda$ is the dip angle. The deflection angle $\phi$ is measured from the pivotal point and specifies the position of the charged particle on the helical track. The meanings of these parameters are depicted in Figs.2.1-a) for negatively charged and -b) for positively charged particles.

Notice that a negatively charged particle travels in the increasing $\phi$ direction, while a positively charged particle travels in the decreasing $\phi$ direction.

 

Figure 2.1:   A graphical explanation of helix parameters for (a) negatively charged and (b) positively charged tracks. Notice that the meaning of $\phi_0$ changes discretely by $\pi$, depending on the charge.

Taking this into account it is easy to get the following formula which relates the momentum of a charged particle to its helix parameters:

$$\begin{array} {lll} {\bf p} = - \left(\frac{Q}{\alpha}\righ... ...r ~~~ \cos (\phi_0+\phi) \cr ~~~ \tan \lambda } .\end{array}$$ (2)