vertex finding : fgfndvtxsci

Figure 3: Various type of vertices defined in the scifi tracker

1. The intersection of two(or more) tracks (Fig. 3(a),(b))
   1. Find the intersection of two or more 3D tracks
      1. For $N_{trk}$, i-th track : $x = a_i + b_i \cdot z$ / $y = c_i + d_i \cdot z$
      2. Vertex position($X_v$,$Y_v$,$Z_v$) as a fitting parameter
      3. $\chi^2$ is defined as
         
         $$\chi^2 = \sum_{i=1}^{N_{trk}} {(X_v-(a_i+b_i\cdot Z_v))^2 \over 1+a_i^2} + {(Y_v-(c_i+d_i\cdot Z_v))^2 \over 1+c_i^2}$$
         
         

      4. Find ($X_v$,$Y_v$,$Z_v$) to minimize $\chi^2$
      5. If ${\chi^2 \over 4\cdot N_{trk}-4} < 4$, ($X_v$,$Y_v$,$Z_v$) will be a vertex which has $N_{trk}$-associated tracks.
   2. Then, the fine fitting is performed to get the better vertex position($X_v$,$Y_v$,$Z_v$) and $\chi^2$. Here, the track parameter( $a_i,b_i,c_i,d_i$) as well as the vertex position($X_v$,$Y_v$,$Z_v$) will be considered as fitting parameter. [2]

2. For single track(which doesn't intersect with other tracks), the extrapolated point to the mid of the closest water tank of the track. (See Fig. 3(c))
   1. For the track of Fig. 3(c) which starts at 5th layer of SciFi : $x = a + b \cdot z\;\; / \;\; y = c + d \cdot z$
   2. The mid point of the water target( locates between 4th and 5th SciFi layer) : $Z_{tgt}$
   3. $Z_v = Z_{tgt}$, $X_v = a + b \cdot Z_{tgt}$, and $Y_v = c + d \cdot Z_{tgt}$.
   4. If vertex is located inside SciFi volume, $(X_v,Y_v,Z_v)$ will be a vertex which has one associated tracks.
3. For the entering track to the scifi volume, the entering point. (See Fig. 3(d))
   1. At the step '2 (c)', if vertex is located outside SciFi volume, Find the entering point of track.
   2. In the case of Trk1 of Fig. 3, the track enters from the front of SciFi detector and the vertex will be the track start point, i.e. this case, the 1st layer hit of track.
   3. In the case of Trk2 of Fig. 3, the track enters from the top of SciFi detector and the vertex will be on the top-side boundary of SciFi detector. So, $Y_v = 120 cm$ and $Z_v = (Y_v - c)/d$, $X_v = a + c\cdot Z_v$.