This report reviews the improved method of primary vertex off-line reconstruction. The new method for primary vertex seed finding as well as modified procedure for vertex fit have been briefly described. Results of the primary vertex reconstruction performance are presented and compared to the previous version of the package. Improvement in the primary vertex reconstruction efficiencies is observed. Small but statistically significant bias, coming mostly from the long-living tracks as well as from a bias of the Velo measurements present in the DC06 simulation, affects the determination of the primary vertex position along the beam direction

Krasowski, M

Kucharczyk, M

Männer, W

Polok, G

Witek, M

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LHCb/2007-011OFFLINESeptember 19, 2007Primary vertex reconstructionM. Krasowski, M. Kucharczyk, W. Manner, G. Polok, M. WitekInstitute of Nuclear Physics PAN, Krakow, PolandAbstractThis report reviews the improved method of primary vertex off-line recon-struction. The new method for primary vertex seed finding as well as modifiedprocedure for vertex fit have been briefly described. Results of the primaryvertex reconstruction performance are presented and compared to the previ-ous version of the package. Improvement in the primary vertex reconstructionefficiencies is observed. Small but statistically significant bias, coming mostlyfrom the long-living tracks as well as from a bias of the Velo measurementspresent in the DC06 simulation, affects the determination of the primary ver-tex position along the beam direction.Contents1 Introduction 22 Primary vertex seeding 23 Fitting procedure 34 Performance of primary vertex reconstruction 34.1 Reconstruction efficiency and the rate of false PV’s . . . . . . . . . . 34.2 Primary vertex resolution . . . . . . . . . . . . . . . . . . . . . . . . 64.3 Study of bias on primary vertex z position . . . . . . . . . . . . . . . 65 Conclusions 911 IntroductionThere are three kinds of primary vertex (PV) finding procedures, two of them atthe trigger level [1, 2] and one for the off-line analysis. In the present note weconcentrate on the off-line primary vertex reconstruction.There are two main steps of the primary vertex reconstruction. The first one isto search for PV seeds providing the set of z coordinates of PV candidates. In thesecond step PV reconstruction is performed relying on the iterative fit using tracksextrapolated to the z-coordinate of PV seed.The primary vertex algorithms have beed adapted for the new MC data release(DC06) and new track event model. New matrix algebra from the MathCore packagehas been applied instead of CLHEP package.The results shown in the present note have been prepared using a sample ofabout 5.0× 104 inclusive bb¯ events reconstructed by Brunel v30r14 for DC06. Thecomparison to DC04 performance was based on the data reconstructed by Brunelv29r11.2 Primary vertex seedingThe initial estimate of the primary vertex z coordinate is based on the seedingprocedure employing the method of analytical clusterization1. It turned out thatmore emphasis should be put on the tracks with large angles θ with respect tothe z axis as the error σz along the beam is proportional to σd0/sinθ, where d0stands for impact parameter and σd0 denotes its error. The cluster is defined by thezclu coordinate and its uncertainty σcluz . The procedure starts with initial clustersdetermined by the closest approaches of the tracks with respect to the z axis as wellas the errors assigned to them. The general principle of the method is as follows.In each iteration a list of all pairs of clusters are considered, and a single pair withminimum distance is selected. The distance itself is defined as:Dpair = |zclu1 − zclu2 |/√(σclu1z )2 + (σclu2z )2. (1)Selected pair is merged into one cluster under the condition that Dpair < 5. Thezclu as well as σcluz of the merged cluster are taken as a weighted mean of the zcoordinates of two initial clusters, i.e.zclu =1w· (w1 · zclu1 + w2 · zclu2), (2)σcluz = 1/√w, (3)where wi = (1/σcluiz )2 (i = 1, 2) and w = w1 + w2. Then the next iteration isperformed. The procedure stops if no more pair of clusters remains to be merged.The final clusters are the candidates for primary vertex seeds. In the final step the1The previous version of the primary vertex reconstruction package relied on the histogrammingand peak finding method.2quality of such candidates is checked. Clusters with the multiplicity (number oftracks the cluster consists of) below 6 are removed. Additional quality conditionsare also applied to reduce the rate of false clusters. Main source of false seeds comesfrom clusters with relatively low multiplicities, which are formed around tracks withsmall errors pointing close to the high-multiplicity primary vertex. Stable workingpoint has been found, balancing between high efficiency and the rate of false seeds.3 Fitting procedureFor each PV seed the fitting procedure is performed in decreasing order of the seedmultiplicities. The Long, Upstream as well as Velo-only tracks are used. They areextrapolated to the position of the seed. Long and Upsteram tracks are extrapolatedusing full transport service while Velo-only tracks are extrapolated as straight lines.In the present note only main idea of the fitting procedure is described. More detailscan be found in [3]. The position of the primary vertex is determined by the leastsquare method minimizing following χ2:χ2PV =ntracks∑i=1d20iσ2d0i, (4)where d0 denotes the impact parameter of a track while σd0 its error. Estimationof the PV position is determined iteratively. Initial selection of the tracks is basedon a z coordinate of their closest approach to the z axis. Relatively wide window±30 mm around z of the seed is used.In each iteration the new position of PV is determined. The tracks are extrapo-lated to the z coordinate of new PV position and the impact parameter significance,IPS = d0/σd0 , is estimated. The track with the highest IPS is removed from the setof tracks with IPS > 4 and then next iteration is performed. The procedure stopsif there is no more track to discard. In the final phase the quality conditions areimposed. PV candidates with multiplicity (number of reconstructed tracks comingfrom the vertex) below 6 are removed.The whole procedure is started for the next seed excluding tracks already usedby any previous PV.4 Performance of primary vertex reconstruction4.1 Reconstruction efficiency and the rate of false PV’sPrimary vertex reconstruction efficiency is defined as a ratio of the number of re-constructed PV’s and the number of reconstructible PV’s. Reconstructible primaryvertex is the vertex which contains at least 6 reconstructed Velo tracks correspond-ing to the Monte Carlo particle coming from a given primary vertex. It is usefulto divide PV’s into two subclasses: isolated reconstructible PV’s and close recon-structible PV’s as well as to measure the rate of false reconstructed PV’s. Thecorresponding definitions are the following:3• Reconstructible PV is called isolated if its distance to the closest reconstructiblePV |∆z| > 10 mm.• Reconstructible PV is called close if its distance to the closest reconstructiblePV |∆z| < 10 mm.• Reconstructed PV is defined as false if its distance to any reconstructible PV|∆z| > 5σz, where σz denotes the estimated position error along z axis. Largefraction of such vertices represent reconstructed PV’s corresponding to realMonte Carlo PV’s, which contains reconstructed tracks, but do not fulfil theconditions to be reconstructible.Reconstruction efficiencies as a function of PV multiplicity are shown in Fig.2.Better efficiency for the lower PV multiplicities is clearly seen for DC06. Since theshapes of track multiplicity distributions for new (DC06) and old (DC04) MonteCarlo samples used in this study are not significantly different (see Fig.1), one cancompare the overall efficiencies. The results are listed in Table 1.Number of tracks per event0 20 40 60 80 100 120 140 160 180 20000.010.020.030.040.05Figure 1: Comparison of track multipilcity distributions for DC06 (points) and DC04(solid line) samples.Lower values for DC04 sample presented in Table 1 come mostly from low multi-plicity PV’s. This difference is mainly caused by two effects: i) more efficient seedingprocedure for DC06 version and ii) additional requirement in the reconstruction al-gorithm for DC04 causing the stop of primary vertex search if less than 10% of alltracks remains not assigned to any PV vertex.The primary vertex reconstruction efficiencies for vertices with highest and sec-ond highest multiplicity are presented in Tables 2 and 3 respectively.4Number of tracks0 10 20 30 40 50 60Efficiency00.20.40.60.81(a)Number of tracks0 10 20 30 40 50 60Efficiency00.20.40.60.81(b)Number of tracks0 10 20 30 40 50 60Efficiency00.20.40.60.81(c)Figure 2: Primary vertex reconstruction efficiencies as a function of primary vertex mul-tiplicity for: a) all reconstructible; b) isolated and c) not isolated PV’s obtained for DC06(black line) and DC04 (red or ligther line) samples.Table 1: Reconstruction efficiencies for different categories of reconstructible PV’s obtainedfor DC06 and DC04 samples.Efficiency [%] DC04 DC06All reconstructible 90.0 97.5Isolated 92.0 98.4Close 69.7 85.6False 1.2 0.95Table 2: Reconstruction efficiencies for different categories of reconstructible PV’s withhighest multiplicity obtained for DC06 and DC04 samples.Efficiency [%] DC04 DC06All reconstructible 99.5 99.6Isolated 99.6 99.6Close 96.2 98.8Table 3: Reconstruction efficiencies for different categories of reconstructible PV’s withsecond highest multiplicity obtained for DC06 and DC04 samples.Efficiency [%] DC04 DC06All reconstructible 76.6 93.3Isolated 79.0 95.6Close 59.9 78.54.2 Primary vertex resolutionSpatial resolution of the primary vertex has been determined by looking at thedifference of positional coordinates between Monte Carlo primary vertex and recon-structed primary vertex.Resolutions for DC06 sample are shown in Fig.3. The values of σx, σy and σz arecompared in Table 4 for DC06 and DC042. One can see that the mean value of thedistribution for ∆z differs significantly from zero. Origin of this shift is discussed inthe next section.Table 4: Primary vertex resolutions σx, σy and σz for DC06 and DC04 samples. Thevalues denote fit results employing single Gaussian distribution.Resolution [µm] DC04 DC06σx 8 8σy 10 10σz 50 59One can notice that results for old and new data samples do not differ much, slightlyworse resolutions may be observed for DC06.4.3 Study of bias on primary vertex z positionThe mean value as well as standard deviation of ∆z as a function of primary vertexmultiplicity for DC06 and DC04 samples are presented in Fig.4. A statistically2As has been mentioned before track multiplicity distributions for new (DC06) and old (DC04)samples are similar, thus the comparison between overall resolutions for DC06 and DC04 is justified.6x [mm]∆Primary vertex -0.1 -0.05 0 0.05 0.1x)∆dN/d(02004006008001000120014001600Entries  19317Mean   0.0001755RMS    0.01355 / ndf 2χ   1096 / 83Constant  14.9±  1385 Mean      0.0000777± 0.0002002 Sigma     0.00008± 0.01049 (a)y [mm]∆Primary vertex -0.1 -0.05 0 0.05 0.1y)∆dN/d(02004006008001000120014001600Entries  19317Mean   0.0003386RMS    0.01332 / ndf 2χ   1057 / 87Constant  15.1±  1391 Mean      0.0000775± 0.0003069 Sigma     0.00008± 0.01046 (b)z [mm]∆Primary vertex -0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5z)∆dN/d(0200400600800100012001400Entries  19317Mean   -0.01242RMS    0.08277 / ndf 2χ   1474 / 96Constant  13.7±  1203 Mean      0.000443± -0.008908 Sigma     0.00050± 0.05887 (c)Figure 3: Primary vertex resolutions for DC06 sample. A fit to a single Gaussian is used.significant bias in the mean value of ∆z is clearly visible for both DC06 and DC04(Fig.4a). Its meaningful contribution results from the admixture of long-living tracksthat come close to the primary vertex and therefore cannot be removed by the cuton their impact parameter significance. The bias for DC06 consists of a secondcomponent caused by the Velo RZ cluster position bias.In Fig.4b it may be observed that in overall primary vertex multiplicity rangethe resolution for DC06 is worse than for DC04.In order to exclude any potential distortion of the vertex reconstruction alone,the correctness of the fitting procedure was examined in two independent ways. Inorder to check the algebra of the fit the tracks used for primary vertex reconstructionwere shifted artificialy to an ideal position of MC truth vertex. The result of thiscross-check is shown in Fig. 5, where no significant bias in ∆z distribution may beobserved. Moreover, stability of this procedure was confirmed by varying seed z7coordinates around its nominal value.Number of tracks in vertex10 20 30 40 50 60z mean value [mm]∆-0.05-0.04-0.03-0.02-0.0100.010.02(a)Number of tracks in vertex10 20 30 40 50 60 [mm]zσ00.020.040.060.080.10.120.140.160.180.2 (b)Figure 4: (a) Comparison between mean values of ∆z as a function of primary vertexmultiplicity for DC06 (points) and DC04 (solid line) samples; (b) Resolution of primaryvertex z coordinate as a function of primary vertex multiplicity for DC06 (points) andDC04 (solid line) samples.| [mm]fit - zMC z = |z∆-0.04 -0.02 0 0.02 0.04 z)∆dN/d(020040060080010001200Entries  14421Mean   -2.884e-05RMS    0.007938 / ndf 2χ 175.5 / 92p0        15.2± 927.7 p1        5.562e-05± -1.561e-06 p2        0.000088± -0.004281 p3        12.5± 143.4 p4        2.000e-04± -9.548e-05 p5        0.00036± 0.01181 Figure 5: The result of the fit (solid line) to ∆z distribution, where the tracks used forprimary vertex reconstruction were coming directly from an ideal position of MC truthvertex.8Number of tracks in vertex10 20 30 40 50 60z mean value [mm]∆-0.05-0.04-0.03-0.02-0.0100.010.02(a)Figure 6: Comparison between mean value of ∆z as a function of primary vertex multi-plicity for DC06 with (points) and without (solid line) long-living tracks.The next check was to remove the bias coming from long-living tracks by requir-ing that primary vertex is fitted using only reconstructed tracks positively matchedto the Monte Carlo particles coming directly from the MC primary vertex. Theresult of this exercise is shown in Fig.6. A statistically significant shift of about 3-4µ is clearly visible for DC06, resulting from the Velo RZ cluster position bias, whilein the case of DC04 such effect is not present.5 ConclusionsThe efficiencies as well as spatial resolution of primary vertex reconstruction werecompared for two different Monte Carlo data samples: DC06 and DC04. For DC06the primary vertex algorithms have beed adapted to new track event model as wellas to new matrix algebra. New effective seeding procedure based on analyticalclusterization helped considerably with improving primary vertex reconstructionefficiencies, in particular for low multiplicity PV’s. Rate of the false reconstructedPV’s is below 1%. Slightly worse z coordinate resolution may be observed for DC06.About 10 µm bias may be observed for both DC04 and DC06. Its main contri-bution comes from long-living particles. This cannot be removed by employing asimple cut on track impact parameter significance. For DC06 an addition source ofthe shift due to bias of the Velo measurements present in the DC06 simulation wasdetermined to be about 4 µm.PV reconstruction procedure described above is planned to be further developed.In particular, the functionality of refitting PV’s excluding long-living particles will beexamined. Furthermore, the PV reconstruction software as well as Velo simulationpackage are being improved. Performance of the PV reconstruction will be then9updated within the subsequent note based on appropriate new data production.10References[1] The LHCb Level-1 Trigger: Architecture, Prototype, Simulation and Algorithm,V. Lindenstruth et al., Note LHCb-2003-064, Chapter 5.[2] LHCb Trigger System Technical Design Report, The LHCb Collaboration,R.A Nobrega et al., CERN/LHCC 2003-031, LHCb TDR 10, 2003.[3] The LHCb Level-2 Trigger, F. Teubert, I.R. Tomalin, J. Holt , Note LHCb-98-47.11

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