Calculating Efficiencies

We want to determine the efficiency of the road finder for finding accepted GEANT tracks. An accepted GEANT track is defined (for now) as a primary GEANT track⁵ that made both horizontal and vertical raw hits in the second and third MuID planes. The road-finding efficiencies thus exclude effects due to the geometrical coverage of the detectors and due to detection inefficiencies. The basic idea of calculating efficiencies is to determine what fraction of these accepted tracks match roads found in the pattern recognition.

Roads are matched to GEANT tracks through a dominant-contributor method. Specifically, for any combination of a road i and GEANT track j, we calculate the contributor ratio

$$f_{ij} = \frac{C_{ij}}{C_i} = \frac{C_{ij}}{\sum_{all\ j} C_{ij}}$$ (1)

where C_ij is the number of raw hits produced by GEANT track j and attached to road i, and C_i is the number of raw hits attached to road i (produced by any GEANT track). Any GEANT track that has an f_ij greater than some threshold value (we currently use 60%) is defined as the dominant contributor for that road.

We can now define the road-finder efficiency as the ratio of the total number N_dom of accepted GEANT tracks that were dominant contributors to at least one road to the total number N_acc of accepted GEANT tracks.

$$\epsilon_{road} = \frac{N_{dom}}{N_{acc}}$$ (2)

The following tables show some preliminary efficiencies obtained using this method.

Table 2: Preliminary road-finding efficiencies, as a function of two-pack detection efficiency, for muons from decays of $J/\psi$ (Z=0 cm vertex position) with no background. The data sample was 1300 events.

Arm	Two-pack	Number of	Number of	Average
	efficiency (%)	accepted tracks	found roads	efficiency (%)
S	40.0	42	32	76.2
S	50.0	77	68	88.3
S	60.0	155	140	90.3
S	70.0	262	243	92.7
S	80.0	402	375	93.3
S	90.0	624	589	94.4
S	100.0	919	872	94.9

Table 3: Preliminary road-finding efficiencies, as a function of the vertex cut parameter, for muons from decays of $J/\psi$ mixed with central HIJING Au+Au events (Z=0 cm vertex position). The data sample was 50 events. The South Muon Arm had no additional shielding.

Arm	Vertex cut (cm)	Number of	Number of	Average
		accepted tracks	found roads	efficiency (%)
S	100.0	66	38	57.6
S	150.0	66	44	66.7
S	180.0	66	48	72.7
S	200.0	66	48	72.7
S	400.0	66	51	77.3
N	180.0	80	71	88.8