Basics of geometrical constraint fit.

Here we consider to find the common vertex consisting of N charged tracks. Let their track parameters and error matrixes are ${\bf a}_i$ and ${\bf E}_i$. Track parameter ${\bf a}_i$consists of 5 elements: dr_i, ${\phi_0}_i$, $\kappa_i$, dz_i, and $\tan\lambda_i$. In the geometrical constraint fit, ${\bf a}_i$are fitted by vertex coordinate, ${\bf v}$, and fitted helix parameter, ${\bf b}_i$. ${\bf b}_i$ consists of three elements: $\tilde{\phi_0}_i$, $\tilde{\kappa}_i$, and $\tan\tilde{\lambda}_i$. Parameters corresponding to dr_iand dz_i are zero as we constraint ${\bf b}_i$ to go through the common vertex point ${\bf v}$.

A helix, in terms of a, is expressed as

$$x(\phi) {x_0}_i + d\rho_i\cos{\phi_0}_i + {\alpha\over\kappa_i} \left[\cos{\phi_0}_i - \cos(\phi+{\phi_0}_i)\right]$$ = (1)

$$y(\phi) {y_0}_i + d\rho_i\sin{\phi_0}_i + {\alpha\over\kappa_i} \left[\sin{\phi_0}_i - \sin(\phi+{\phi_0}_i)\right]$$ = (2)

$$z(\phi) {z_0}_i + dz_i - {\alpha\over\kappa_i}\phi \tan\lambda_i$$ = (3)

where, ( x_0i, y_0i, z_0i) is the pivot and $\phi$ is a deflection angle relative to the pivot.

The same helix can be expressed, in terms of $\bf b$ as,

$$x(\phi) \tilde{x}_v + {\alpha\over\tilde{\kappa}_i} \left[\cos\tilde{\phi_0}_i - \cos(\phi+\tilde{\phi_0}_i)\right]$$ = (4)

$$y(\phi) \tilde{y}_v + {\alpha\over\tilde{\kappa}_i} \left[\sin\tilde{\phi_0}_i - \sin(\phi+\tilde{\phi_0}_i)\right]$$ = (5)

$$z(\phi) \tilde{z}_v - {\alpha\over\tilde{\kappa}_i}\phi \tan\tilde{\lambda}_i$$ = (6)

Using b, the center of the circle is given by

$$(x_c,y_c) = \left( \tilde{x}_v + {\alpha\over\tilde{\kappa}_i... ..._v + {\alpha\over\tilde{\kappa}_i}\sin\tilde{\phi_0}_i \right)$$ (7)

Therefore, the deflection angle at the pivot of input helix is given by

$$\phi_0' = \left\{ \begin{array}{c l} \tan^{-1}\left({{y_0}_i-... ..._c-{x_0}_i}\right) & \; \; ( \kappa_i > 0 ) \end{array}\right.$$ (8)

Note that $\phi_0'$ is a deflection angle for the input helix estimated from the fitting parameter b. The expected position of the helix at the pivot of the input helix is

$$\tilde{x}_v + {\alpha\over\tilde{\kappa}_i}(\cos\tilde{\phi_0}_i - \cos\phi_0')$$ x' = (9)

$$\tilde{y}_v + {\alpha\over\tilde{\kappa}_i}(\sin\tilde{\phi_0}_i - \sin\phi_0')$$ y' = (10)

$$\tilde{z}_v - {\alpha\over\tilde{\kappa}_i}(\phi_0'-\tilde{\phi_0}_i)\tan\tilde{\lambda_i}$$ z' = (11)

Using these formula, the helix parameter is expressed as follows by the fitting parameter b:

 

$$\left[ x' - {x_0}_i \right]\cos\phi_0' + \left[ y' - {y_0}_i \right]\sin\phi_0'$$ dr' = (12)

$$\phi_0' \left\{ \begin{array}{c l} \tan^{-1}\left({{y_0}_i-y_c \over {x_0... ...c-{y_0}_i \over x_c-{x_0}_i}\right) & \; \; ( \kappa_i > 0 ) \end{array}\right.$$ = (13)

$$\kappa' \tilde{\kappa}_i$$ = (14)

dz' = z' - z_0i (15)

$$\tan\lambda' \tan\tilde{\lambda}_i$$ = (16)

12pt

What we do in the geometrical constraint is to minimize $\chi ^2$ defined as

$$\chi^2=\sum_i ({\bf a}_i-{\bf a'}_i)^t {\bf E}_i^{-1} ({\bf a}_i-{\bf a'}_i)$$ (17)

where ${\bf a'}_i$ is the helix parameter of i-th track derived from the fitting parameter b: dr', $\phi_0'$, $\kappa'$, dz', and $\tan\lambda'$. $\chi ^2$ is a function of b, and we find the parameter b by using the Newtonian method.

The principle of the Newtonian method is as follows.

$\chi ^2$ can be expanded with respect to b as,

$$\chi^2 = \chi_0^2 + { \partial \chi^2 \over \partial {\bf b} ... ... \chi^2 \over \partial {\bf b}^2 } \Delta {\bf b}^2 + \cdots$$ (18)

where $\Delta {\bf b}\equiv {\bf b}_{n+1} - {\bf b}_n$, ${\rm b}_n$ is the parameter in the current iteration loop and ${\rm b}_{n+1}$ is the parameter for the next iteration. If we use the terms up to second order, $\chi ^2$ can be expressed as

$$\chi^2 = {1\over 2} {\partial^2\chi^2 \over \partial {\bf b}^... ...-1} {\partial \chi^2 \over \partial {\bf b}} \right] + Const.$$ (19)

and $\Delta {\bf b}$ to minimize $\chi ^2$ is given by

$$\Delta {\bf b}= -\left( {1\over 2}{\partial^2\chi^2 \over \p... ... b}^2 }\right)^{-1} {\partial \chi^2 \over \partial {\bf b}}$$ (20)

Namely the parameter for next iteration is given by

 

$${\bf b}_{n+1} {\bf b_n} + \Delta {\bf b}_i$$ = (21)

$${\bf b_n} - \left( {1\over 2}{\partial^2\chi^2\over \partial {\bf b}^T \partial {\bf b} } \right)^{-1} {\partial\chi^2 \over \partial {\bf b} }$$ = (22)

Note that the second derivative in Eq. 22, ${1\over 2}{\partial^2\chi^2\over \partial {\bf b}^T \partial {\bf b} }$represents the local curvature of the $\chi ^2$ surface of b coordinate system. If we use the exact form of the second derivative in the minimization, it becomes sensitive to the local property of the surface and the fitting process becomes to slow or divergent. To avoid this problem, we multiply the diagonal part of the matrix ${1\over 2}{\partial^2\chi^2\over \partial {\bf b}^T \partial {\bf b} }$by a factor $1+\epsilon$.

In the class JSFVirtualFit, from which JSFGeoCFit is derived, the initial value of $\epsilon$ is 10^-12, but is is multiplied by 100 or divided by 100 depending on the change of $\chi ^2$. The fit converged if the change of $\chi ^2$ during the minimization loop becomes less than 10^-6. The loop is abandoned if the number of loop exceeds 50 or $\epsilon$ becomes larger than 10⁶⁰.

The derivatives and the second derivatives of $\chi ^2$ with respect to b is given in the appendix 1.