B. Track Parameterization

Several track parameterizations are used in DELPHI. During the track reconstruction, TER and TKR forms are used (stored in the TE and TK TANAGRA banks respectively). For track extrapolation and fitting, the EXX form is used. The perigee parameters define the track at its point of closest approach to a reference point (usually the origin). Because the reference point is implicit, only five parameters are required. The other parameterizations also define the distance from the origin (in either $ R$ or $ z$) at which they apply. It is the perigee parameters and their weight matrix (inverse covariance matrix) that are stored on the DST and used in most of the subsequent analysis.

TER $ (R, \ensuremathbox{R\Phi}, z, \theta, \phi, -q/p_{xy})$ or $ (x, y, z, \theta, \phi, -q/p_{xy})$
TKR $ (R, \ensuremathbox{R\Phi}, z, \theta, \phi, -q/p)$ or $ (x, y, z, \theta, \phi, -q/p)$
EXX $ (R, \Phi, z, \theta, \beta, \kappa)$ or $ (z, x, y, \theta, \phi, \kappa)$     [sic]
Perigee $ (\epsilon,z,\theta,\phi,\kappa)$    

The parameters $ (R, \Phi, z)$ are the cylindrical position coordinates and $ (x,y,z)$ are the Cartesian position coordinates (in cm). The cylindrical forms are used mostly in the barrel and the Cartesians mostly in the endcaps. $ \theta$ and $ \phi$ are, respectively, the polar and azimuthal track directions at the specified point. $ \beta$ is the azimuthal angle, relative to a radial track ( $ \beta \equiv \phi - \Phi$). All angles are in radians.

The particle's momentum components are given by

$\displaystyle p_x$ $\displaystyle = p \sin\theta \cos\phi$     $\displaystyle (0 \le \phi$ $\displaystyle < 2\pi)$    
$\displaystyle p_y$ $\displaystyle = p \sin\theta \sin\phi$     $\displaystyle (0 \le \theta$ $\displaystyle < \pi)$ (B.1)
$\displaystyle p_z$ $\displaystyle = p \cos\theta$ $\displaystyle p_{xy}$ $\displaystyle = p \sin\theta$        

$ p$ is the momentum (in $ \ensuremathbox{\mathrm{GeV}/c}$), $ q$ the charge (relative to that of the proton), and $ \kappa$ the track curvature (in $ \ensuremathbox{\mathrm{cm}}^{-1}$), i.e. in the central region, where the magnetic field is parallel to the $ z$-axis,

$\displaystyle \kappa \equiv 1/\rho = -q B / p_{xy}\,,$ (B.2)

where $ \vert\rho\vert$ is the track's radius of curvature in the $ x$$ y$ plane, and $ B$ is the magnetic field ( $ 1.2\ \ensuremathbox{\mathrm{tesla}} = 0.0036\ \ensuremathbox{\mathrm{GeV}/c}/\ensuremathbox{\mathrm{cm}}$).

$ \epsilon$ is the geometric impact parameter, whose magnitude gives the distance from the origin (or other reference point) to the perigee, the point of the track's closest approach in the $ x$$ y$ plane. The sign of $ \epsilon$ is positive if the track passes to the right of the origin, when looking along $ -z$ (see figure B.1).B.1

Figure B.1: Illustration of the track impact parameter. In this case, the impact parameter, $ \epsilon$, and curvature, $ \kappa$, are positive. Since DELPHI's magnetic field is parallel to the $ z$-axis (pointing out of the paper), $ \kappa > 0$ corresponds to a negatively charged particle.

$\displaystyle \epsilon = (\mathbf{x_0} \times \ensuremathbox{\hat{\mathbf{p}}}) \cdot \ensuremathbox{\hat{\mathbf{z}}}$ (B.3)

where $ \mathbf{x_0}$ is the vector from the origin to the perigee, $ \ensuremathbox{\hat{\mathbf{p}}}$ is a unit vector along the track direction there, and $ \ensuremathbox{\hat{\mathbf{z}}}$ is a unit vector along the $ z$-axis. The position of the perigee is thus given by

$\displaystyle (x_0,y_0,z_0) = (\epsilon \sin\phi, -\epsilon\cos\phi, 0)$ (B.4)

For lifetime studies, it is often convenient to use another signing convention: the lifetime-signed impact parameter. In this case, a positive sign is assigned if the track intersects the direction vector (often estimated from the thrust axis) of a presumed decaying particle in front of the reference point (usually the beamspot or reconstructed primary vertex). For a large boost, the average lifetime-signed impact parameter is proportional to the decay lifetime [154]. Since we can reconstruct the $ \ensuremathbox{\mathrm{J/\psi}}$ vertex directly, this method and the lifetime signing convention are not used in this thesis.

Tim Adye 2002-11-06