The Elastic Arms algorithm is designed to use the output from the Multi-Hough algorithm as a seed for finding the global event vertex. It is based on a method of the same name listed in [68] which is sometimes referred to in the literature as the “method of deformable templates.” This method is described briefly here. For more information on the method of Elastic Arms, see [69], [68], [70], and [71].
data type number of entries mean RMS underflow
Table 6.5: Dot product between the Hough lines and the best matched MC particle trajectory in the X-view. The numbers in the underflow bin represent unmatched lines.
6.5.1 The Algorithm
The basic template for a NOvA event is a vertex with one or more particle tracks emanating outwards from that vertex. In the Elastic Arms method, each particle track is approximated by an “arm” (a vector pointing away from the vertex) whose direction can be adjusted to match the event topology for an identified vertex seed. For the application of this method to NOvA data, the number of arms is taken to be the largest number of Hough lines found for the event in either the XZ or YZ views. To determine the location of the vertex, a list of vertex candidates must be generated and evaluated.
From this list of vertex candidates, the best vertex is chosen as the one that minimizes an energy cost function given by
Here Mia is a distance measure from detector hit i to arm a, Via is the strength of the association between hit i and arm a, and Dais the distance from the vertex to the first hit
on arm a. The parameters λ and λv control the strength of the second and third terms.
The last term in this equation is unique to the NOvA application of the Elastic Arms method and penalizes arms whose first hits are far from the vertex. This term was added to prevent the best vertex from being the one that was always the farthest away from all of the hits, which in the extreme will always minimize the energy cost function. The hit/arm association term Via is given by
Via= e−βMia e−βλ+PM
b=1e−βMib, (6.12)
where e−βλ represents the likelihood that the hit is not associated with any arm and β can be interpreted as a range over which hits are allowed to be associated with arm a.
The list of vertex candidates is generated from information about the Hough lines and the spatial distribution of the hits in the event. From the Hough lines, vertex candidates are formed from the intersection points of the major lines in each view. The hits are sorted by their Z coordinates and additional vertex candidates are formed from selected hits at fixed intervals (2%, 5%, ... 50%) in this list. Note that this introduces a bias to favor vertices at lower values of Z but that this is exactly what one expects for beam neutrino events in the NOvA detectors.
The next step is to set the directions of the arms for each vertex candidate. To determine the arm directions, a list of possible vectors is generated from the directions of the Hough lines (matched between views by their peak heights in the Hough map) plus a minimum bias sample of vectors formed from the vertices of a dodecahedron. The arms are then set one-by-one by choosing the direction from this list that minimizes equation 6.11 before moving on to the next arm. Care is taken to ensure that the same or very similar arms are not reused for each vertex.
With a list of vertex candidates, each with a set of carefully chosen arms, equation
6.11 can now be evaluated to determine which vertex candidate will be deemed the “best one.” Once this vertex has been chosen, a process of simulated annealing is applied to allow the vertex to settle into an optimal location. The annealing is accomplished by varying the parameter β in equation 6.12 from low values (representing high temperatures) to high values (representing low temperatures.) This process allows the vertex to smoothly seek out the global minimum of equation 6.11 while avoiding potential local minima within that function.
6.5.2 Results and Performance
Shown in figure 6.14 is an event display from a far detector data event with a selected neutrino candidate. The Hough lines are drawn in red and the selected Elastic Arms vertex is drawn as a blue “X”. The Hough lines clearly pick out the major event features and align with the major particle tracks, and the Elastic Arms vertex sits at or very close to what appears to be the global event vertex.
Figure 6.14: Reconstructed far detector neutrino candidate event showing the Hough lines and the Elastic Arms vertex.
Together, the Multi-Hough and Elastic Arms algorithms achieve average event vertex
resolutions of 11.6, 10.9, and 28.8 cm for νµ CC, νe CC, and NC events respectively. For both the νµ CC and νe CC events, 68% of the vertices are within 10 cm of the true vertex (38 cm for the NC events.) This puts the vertex for the CC events within approximately 2 cell widths of the true vertex the majority of the time. The distribution of vertex resolutions broken into these three categories is shown in figure 6.15.
Figure 6.15: 3D Event vertex resolution with the Elastic Arms algorithm.