586
R. Frühwirth et al.
Due to the second term on the right-hand side, V (v) ≈ 0 in regions where f i (v)
is significant for only one track. The form of V (v) can be modified to fold in
known physics information about probable vertex locations. For instance, V (v) can
be augmented by a further function f 0 (v) describing the location and spread of the
interaction point. In addition, V (v) may be modified by a factor dependent on the
angular location of the point v.
Vertex finding amounts to finding local maxima of the function V (v). The search
starts at the calculated maxima of the products f i (v)f j (v) for all track pairs. For
each of these points the nearest maximum of V (v) is found. These maxima are
clustered together to form candidate vertex regions. The final association of the
tracks to the vertex candidates can be done on the basis of the respective χ 2
contributions or by an adaptive fit (see Sect. 13.2.3.3). In [119] the topological
vertex finder was augmented by a procedure based on the concept of the minimum
spanning tree of a graph.
13.2.2.3 Iterated Estimators
Vertex finding can also be accomplished by iterated vertex fits (see Sect. 13.2.3).
The procedure can be summarized in the following way:
1. Fit one vertex with all tracks
2. Discard all incompatible tracks
3. Repeat step 1 with all discarded tracks
The iteration stops when no vertex with at least two tracks can be successfully fitted.
Step 2 might itself be iterative, especially if the vertex fit is not robust, so that
the incompatible tracks have to be removed sequentially. Iterative vertex finders
based on a least-squares fit (Sect. 13.2.3.1) and an adaptive fit (Sect. 13.2.3.3) are
implemented in the RAVE toolbox [120, 121].
13.2.3 Vertex Fitting
The input to the vertex fit is a vertex candidate, i.e., a set of estimated track
parameters { ˜
q 1 , . . . , ˜
q n } located at one or more reference surfaces, along with
their covariance matrices {C 1 , . . . , C n }. For instance, in the primary vertex fit in
a collider experiment the reference surface may be the beam tube. If possible, the
reference surface(s) should be chosen such that multiple scattering between the
vertex and the location of the track parameters is negligible.
The parameters to be fitted are the vertex position v and the track momenta
p i at the vertex. The functional dependence of the track parameters on the vertex
parameters requires a track model, which depends on the shape of the magnetic field
in the vicinity of the vertex. If the field is homogeneous, the track model is a helix;
R. Frühwirth et al.
Due to the second term on the right-hand side, V (v) ≈ 0 in regions where f i (v)
is significant for only one track. The form of V (v) can be modified to fold in
known physics information about probable vertex locations. For instance, V (v) can
be augmented by a further function f 0 (v) describing the location and spread of the
interaction point. In addition, V (v) may be modified by a factor dependent on the
angular location of the point v.
Vertex finding amounts to finding local maxima of the function V (v). The search
starts at the calculated maxima of the products f i (v)f j (v) for all track pairs. For
each of these points the nearest maximum of V (v) is found. These maxima are
clustered together to form candidate vertex regions. The final association of the
tracks to the vertex candidates can be done on the basis of the respective χ 2
contributions or by an adaptive fit (see Sect. 13.2.3.3). In [119] the topological
vertex finder was augmented by a procedure based on the concept of the minimum
spanning tree of a graph.
13.2.2.3 Iterated Estimators
Vertex finding can also be accomplished by iterated vertex fits (see Sect. 13.2.3).
The procedure can be summarized in the following way:
1. Fit one vertex with all tracks
2. Discard all incompatible tracks
3. Repeat step 1 with all discarded tracks
The iteration stops when no vertex with at least two tracks can be successfully fitted.
Step 2 might itself be iterative, especially if the vertex fit is not robust, so that
the incompatible tracks have to be removed sequentially. Iterative vertex finders
based on a least-squares fit (Sect. 13.2.3.1) and an adaptive fit (Sect. 13.2.3.3) are
implemented in the RAVE toolbox [120, 121].
13.2.3 Vertex Fitting
The input to the vertex fit is a vertex candidate, i.e., a set of estimated track
parameters { ˜
q 1 , . . . , ˜
q n } located at one or more reference surfaces, along with
their covariance matrices {C 1 , . . . , C n }. For instance, in the primary vertex fit in
a collider experiment the reference surface may be the beam tube. If possible, the
reference surface(s) should be chosen such that multiple scattering between the
vertex and the location of the track parameters is negligible.
The parameters to be fitted are the vertex position v and the track momenta
p i at the vertex. The functional dependence of the track parameters on the vertex
parameters requires a track model, which depends on the shape of the magnetic field
in the vicinity of the vertex. If the field is homogeneous, the track model is a helix;