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CMS NoteFebruary 1, 2006

Vertex Fitting in the CMS Tracker

T. Speer, K. Prokofiev

University of Zurich, Switzerland

R. Fruhwirth, W. Waltenberger

Institute of High Energy Physics of the Austrian Academy of Sciences, Vienna, Austria

P. Vanlaer

Interuniversity Institute for High Energies, ULB, Brussels, Belgium

Abstract

Several algorithms for vertex fitting have been implemented and studied in the CMS recon-struction framework. In the present note, the most successful algorithms are compared tothe standard Kalman filter.

1 IntroductionThe task of Vertex Fitting is, given a set of tracks, to compute the best estimate of the vertex parameters(position, covariance matrix, constrained track parameters and their covariances) as well as indicators ofthe success of the fit (total �� , number of degrees of freedom, track weights).

Several algorithms have been implemented and studied, differing mainly by their sensitivity to outlyingtracks, either mismeasured tracks (type 1 outliers) or tracks from another vertex (type 2 outliers). Thealgorithms implemented can be divided into linear, or least-squares, and non-linear algorithms. In least-squares algorithms all tracks are used, with a unit weight. Non-linear algorithms are able to down-weightor discard tracks, and are thus less sensitive to outliers, since the weights of the tracks depend on theirdistance to the vertex. Since the algorithms are extensively described in the relevant references, only abrief description is given in the next section. The two non-linear algorithms presented here, the Adaptivefilter and the Trimmed Kalman filter, have been developped at CMS.

The goal of the present study is to evaluate the best performance achievable with the studied algorithms,and to understand their behaviour in different topologies with Monte Carlo simulated events. Studiesconducted on a simplified simulation, where both types of outliers contaminated the vertex, were alreadydescribed in a previous document [1]. To disentangle the features of the vertex fitters from the finders,only the tracks from the studied vertex, matched to the simulated tracks, will be used. In this way, onlymismeasured tracks contaminate the fitted vertex, without contamination of tracks from another vertex.The effects of the finders on the reconstruction will be studied in a forthcoming note. A further goal ofthis study is to gain an understanding of the effects of misalignment on the reconstructed vertices, andhow this affects the performance of the different algorithms.

2 AlgorithmsAll algorithms discussed here rely on the Kalman filter. Because of its sequential nature, the algorithmsproceed through several well-defined steps. If no initial position is provided, a fast, coarse approximationof the vertex position, called the linearization point, is computed. This point is the linear expansion pointof the measurement equation, which describes the dependence of the measured track parameters on thestate vector (vertex position and momentum of the tracks at the vertex). The track parameters will beestimated in the vicinity of this point. In addition, for robust algorithms, it is the point against which theinitial weights of the tracks are determined; a good initial estimate is thus crucial.

An iterative procedure is then applied. First, the track parameters are estimated and the measurementequation is linearized. For non-linear algorithms, the weights of the tracks are then computed accordingto their distance to the linearization point. Finally, the estimate of the vertex is obtained by a sequen-tial, track-by-track, update of the vertex parameters. These iterations are performed until the transversedistance between the vertex computed in the previous and the current iterations is less than the specifiedconvergence criterion, or until the maximum number of iterations is reached. For each new iteration, thevertex estimated in the previous iteration is used as the linearization point.

Finally, if required, the momenta of the tracks and the full track-to-track covariance matrix can be re-estimated with the vertex constraint, thus improving the track parameters at this point.

2.1 The linearization point finder

Among the several linearization point finder studied, the Fraction-of Sample Mode with Weights-estimator(FSMW) was found to have a good performance, in terms of resolution, robustness and computationalspeed [2]. It is now the default linearization point finder for all vertex fitters. This method is based on thecrossing points of the tracks. A crossing point is the algebraic mean of the two points of closest approachof two tracks. It is attached a weight which depends on the distance of the two tracks, such that closertracks have crossing points with larger weights.

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The FSMW finds the mode of the crossing points, in each of the three spatial coordinates separately.The one-dimensional mode finding starts by finding the shortest interval containing points with a sumof weights exceeding a fixed fraction of the sum of all weights (0.4 by default). The procedure is thenrepeated on the found interval, until at most two points remain. The mode is finally the average of theremaining points.

2.2 The track model

The often used perigee parameterization [3] has been chosen to describe a charged particle track in amagnetic field, where the trajectory of the particle is modelled by a helix. This parameterization isdefined with respect to a reference point (the linearization point in this case), and defines the track at itspoint of closest approach (in the transverse plane) to the reference point.

For a charged particle, the five parameters are:

�� : signed transverse distance of the point of closest approach. By convention, the sign is positiveif the reference point is at the left of the point of closest approach, looking along the trajectory (i.e.if the angle between the vector from the reference point to the point of closest approach and thetrajectory is �� ).

� � � : longitudinal distance of the point of closest approach.

�� : polar angle of the momentum vector.

�� : azimuthal angle of the momentum vector at the point of closest approach.

�� : signed transverse curvature, where the sign is the negative of the charge.

To describe neutral particles, a modified perigee parameterization is used, where the signed transversecurvature � is replaced by the inverse of the transverse momentum �� .

2.3 The Kalman Filter

The most often used algorithm for vertex reconstruction is the well-known Kalman Filter [4] (KVF).It is mathematically equivalent to a global least-squares minimization, which is the optimal estimatorwhen the model is linear, all random noise is Gaussian and there are no outlying measurements. In thatcase, the estimators are unbiased and have minimum variance, residuals and pulls of estimated quantitieshave Gaussian distributions and the value of the objective function (the function which is minimized)at the minimum obeys a �� distribution. For non-linear models or non-Gaussian noise, it is still theoptimal linear estimator. It is nevertheless very sensitive to outliers, and the bias in the fitted parametersis proportional to the bias in the outlying measurements.

2.4 The Adaptive filter

A more robust algorithm is the Adaptive filter [2, 5, 1] (AVF). It has been shown that this algorithm isvery stable with a high break-down point �� . It is an iterative re-weighted least-squares fit which down-weights tracks according to their standardized ( � � ) distance to the vertex. The weight-function is thesigmoidal function ��

� ��! ��"� #%$'&

()+*�,.-0/21 &((03 4 (1)

576The break-down point is defined as the fraction of outliers below which the vertex fit is not affected.

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Figure 1: Example of weight function of the Adaptive Vertex Fitter, with � �� .

where � �� is the distance where the weight function drops to 0.5 and � is the (global) temperaturewhich controls the sharpness of the drop (Figure 1). To prevent the algorithm from falling into a localminimum, a geometric annealing schedule is introduced, where, starting from a pre-defined value, thetemperature is lowered at each iteration according to a fixed scheme. At each temperature value, theKalman filter is used in order to fit the vertex, accounting for the track weights computed at the previousiteration. The Adaptive filter is nevertheless not tied to a particular fitter, and it has been shown to givegood results with the Gaussian-sum filter as well [7]. The most often used annealing schedule is ageometric annealing, where, at every iteration, the temperature is multiplied by a factor ( �� ) untilit reaches a value of 1. This factor is called the annealing ratio.

This algorithm has the advantage that the weights can be fractional (soft assignment) and that they varyfrom one iteration to the next, until the fit converges. No prior estimate of the weights of the tracks or ofthe fraction of outliers is thus needed.

2.5 The Trimmed Kalman filter

The Trimmed Kalman filter (TKF), is a conventional robustification of the Kalman vertex fitter, wheretracks incompatible with the vertex are removed one by one from the vertex [6]. It is a hard-assignment,iterative fit.

First, all input tracks are fitted to a vertex. The track least compatible with that vertex is removed, andthe vertex is refitted. This procedure is repeated, until the compatibility of all tracks is below a giventhreshold. Once a track is rejected, it will not be included again in the vertex.

The compatibility of each track to the vertex is computed from the standardized distance to the vertex.For correctly assigned tracks, provided that the track parameter errors are correctly estimated at the trackfitting stage, this distance is distributed according to a � � distribution with 2 degrees of freedom (1 degreeof freedom for a 2-track vertex). If the above-mentioned threshold is chosen as the ( �� ) quantile ofthe � � distribution and if the vertex has no incorrectly assigned tracks, then the probability of rejectinga good track is equal to � . The probabilistic meaning of the compatibility cut renders the tuning of thealgorithm straightforward. However, if there were incorrectly assigned tracks, the probability of rejectinga good track may be well above � .

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Sample Nbr. SimTracks Nbr. RecTracks Mean � � [ �� ]�� (SV) 4 4 3.5�� (PV) 28.9 11.6 1.6�-jet 51.9 12.6 2.8��

(GF) 52.1 23.2 2.0��(IVB) 55.1 23.7 2.8�� jets 66.5 33.3 2.1� ��

69.21 36.7 3.0Drell-Yan 50.6 22.7 7.2� ��

79.3 44.3 4.0

Table 1: Mean number of simulated and associated reconstructed tracks per vertex and average trans-verse momentum of the reconstructed tracks for the different samples.

3 Performance

3.1 Event samples and algorithms

The performance of the algorithms has been assessed on primary and secondary vertices in the followingMonte Carlo generated samples:

� primary vertices:� ��

,�!"�#�

(produced either through �$� fusion (GF) or %&% fusion (IVB)),Drell-Yan,

�'�� ,� ��

, ��(�� jets.

� secondary vertices:�)�* � �� .

All samples are simulated with low luminosity pile-up ( + -, ��.0/1/32�4 $ �65 $ � ). The distributions oftrack multiplicities, transverse momenta and pseudo-rapidities of the simulated and matched recon-structed tracks for the different samples are shown in Figures 2, 3 and 4. Average multiplicities ofsimulated and reconstructed tracks, as well as average transverse momentum of the reconstructed tracks,are given in Table 1. As can be seen, a large number of tracks are not reconstructed, since a significantfraction of the tracks is outside the acceptance of the tracker (with 7 897#: (;=< ), and hadrons have a lowerreconstruction efficiency due to nuclear interactions in the tracker.

Only tracks matched to simulated tracks which were produced in the selected process are used in the fit,without cuts. Therefore, only mismeasured tracks will contaminate the fitted vertex, without contami-nation of tracks from another vertex. The reconstructed tracks are associated with the simulated tracksthrough their hits, by requiring that more than half of the reconstructed hits are associated to simulatedhits. No cut is placed on the transverse momenta of the tracks � � . For the three algorithms, the defaultvalue is used for all parameters. For the Adaptive Vertex Fitter, a geometric annealing is used, with acutoff � �� , an initial temperature of 256 and an annealing ratio of 0.25. For the Trimmed Kalmanfilter, the track-compatibility threshold is set to 5%.

3.2 Performance estimators

To compare the algorithms, the main properties of the fits are summarized in tables (e.g. Tables 2 and 3).To characterize the residual distributions, the mean and the standard deviation (called the resolution)of a Gaussian fitted to the distribution are reported. To estimate the amount of tails, the half-widthof the symmetric intervals covering 95% of the distribution (called 95% coverage) is also quoted. Incase of a Gaussian distribution, the 95% coverage corresponds to twice the standard deviation. For

> 6It should be noted that in the ?A@CBEDGF�HJI sample, a cut on the transverse momentum of the muons is placed bythe Monte Carlo generator. Therefore only decays for which both muons have a transverse momentum above2 GeV are retained.

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the standardized residuals (or pulls) distribution as well, a Gaussian distribution is fitted and the standarddeviation taken as pull value. The distribution of pulls is well described by a single Gaussian distribution.

The tables also indicate the average normalized- � � and � � -probability. For non-linear algorithms, a“chi-square” can be defined in analogy with the Kalman filter. This pseudo-chi-square is not chi-squaredistributed. The distributions of the pseudo- � � -probability do not feature a peak at low values anymore,and usually show a dip at 0, which is all the more prominent with an increasing number of tracks. Since,by construction, the tracks with a large � � -distance to the vertex, and hence large contribution to thepseudo- � � , are down-weighted, the distribution of the pseudo- � � -probability will be shifted to highervalues.

The average weight is the average of the sum of the track weights divided by the number of initial tracksin each vertex. For the Kalman filter, the average weight is by definition always 1, since all tracks areused with a unit weight. For the Trimmed Kalman filter which discards tracks, the weight of a discardedtrack is simply zero, and the average weight is equivalent to the average fraction of tracks remaining inthe vertex.

The last entry is the failure rate. A fit is considered to have failed if either the fitter does not converge (andhence does not return a vertex) or the sum of the weights of all the tracks is smaller than 1. Furthermore,for each sample, all relevant distributions (the distributions of the residuals and pulls of the � and the � -coordinates, of the � � -probability, of the number of tracks which have been removed or down-weightedand of the computation time) are given at the end of the note.

3.3 Performance with a perfect detector

In the first series of tests, a perfectly aligned detector is simulated. For each sample, the main propertiesof the fits are summarized in Tables 2 and 3.

As expected, the resolution of the reconstructed vertex improves with the number of tracks and thetransverse momentum. In each case, both the residual and the pull distributions for the � -coordinatesfeature significantly more tails than the corresponding distribution for the � -coordinates. Comparing thedifferent algorithms, the robust algorithms are more effective than the Kalman filter, especially in hightrack-multiplicity scenarios, as the outliers can be better identified and the vertex better defined by theremaining tracks. Indeed, in the fit of � or

��decay vertices (Figures 5, 6), where three or four tracks

only are fitted, little improvement can be seen on the estimated positions with respect to those obtainedwith the Kalman filter, with only a slight improvement of the coverage. The estimated error is slightlymore precise, as is attested by the pull distributions, which show nearly no tails and are nearly perfectlyGaussian with a standard deviation very close to 1. In the high-multiplicity primary vertex of

� �� events

(Figures 11, 12), the improvement on the resolution is in the order of 30%, and the tails are significantlyreduced. The estimated error is also very much improved with the robust fitters. The pulls are nearlyperfectly Gaussian with a standard deviation very close to 1, which is not the case for the Kalman filter.

For the decay�� , by using the reconstructed primary and secondary vertices, the improve-

ment of the resolution of the primary vertex yields an improvement of the reconstructed proper decaylength ( �� ). This decay length can be measured either in two dimensions, in the transverse plane, orin three dimensions. The residuals and pulls of the proper decay length measured in two dimensions,in the transverse plane, are shown in Figure 9, and those of the proper decay length measured in threedimensions in Figure 10. As expected, the resolution and the pulls show some improvement when usingthe robust fitters, with the best estimate given by the Adaptive Vertex Fitter. For the Kalman filter, theresolution on the �� in three dimensions is

� .0;�� 4 and the coverage is � 4 , while for the AdaptiveVertex Fitter, the resolution is � ��; � 4 and the coverage is ��. .� 4 (Table 8).

For all samples, the main change can be seen in the � � -probability distribution. For vertices fitted with theKalman filter, a large number of fits have a low � � -probability, typically below 0.01, large values of � �

occurring more frequently than expected. These have indeed been shown to be due to the mismeasured

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tracks [7], specifically the non-Gaussian tails in the distribution of the track parameters errors. Whereasin the case of the decay

� �� !� �� , approximately 10% of the fits have a � � -probability below 0.01,this number can be as high as 80% for

��vertices.

The average weight and the average number of discarded tracks for the Adaptive filter and the TrimmedKalman filter are very similar, event though the former does a soft track-assignment and the latter ahard track-assignment. Since, on average, the same number of tracks remain in the vertex, the estimatedposition is nearly equal. The Trimmed Kalman filter features a somewhat higher failure rate, of the orderof 0.1%. For only 1 vertex, in a total of some 80’000, has the Kalman filter been seen not to converge,the maximum number of iterations allowed being 10 in these tests.

Finally, the comparison of the computation time between the three algorithms shows that, as expected,the Kalman filter is the fastest. The non-linear algorithms are obviously slower, since more iterationshave to be performed. The Trimmed Kalman filter is faster than the Adaptive Vertex Fitter for trackmultiplicities below 30, and slower for larger multiplicities, indicating different algorithm complexities.In the Adaptive filter, all tracks contribute in every iteration. For the Trimmed Kalman filter, discardedtracks do not contribute to the fit anymore and will not be reused in subsequent iterations. It shouldbe mentioned that the timing estimates can not be directly compared between the samples, as programswere run on different machines in different computing centers and different operating systems.

3.4 Robustness with respect to degraded track reconstruction

To gauge the robustness of the algorithms, the tests have been repeated on a detector with long or short-term alignment scenario. The main properties of the fits are summarized in Tables 4 and 5 for the short-term alignment and in Tables 6 and 7 for the long-term alignment. The short-term alignment scenario issupposed to be typical of the misalignment of the initial data taking period, while the long-term alignmentshould reflect the improvements achieved after the detector has been aligned [9]. The pixel has the samemisalignment in both scenarios. In the silicon-strip tracker (SST) the short-term alignment is ten timesworse than the long-term alignment. The effects of misalignment of the tracker on various aspects oftrack and vertex reconstruction have been extensively studied and reported in [10].

The misalignment significantly degrades the estimated positions, both in terms of resolution, coverageand bias. The resolution is degraded by the same order of magnitude (around 10-12 4 for the short-term alignment and 7-10 4 for the long-term alignment) in all samples, regardless of their hardness ortrack multiplicities. In the case of the Kalman filter, the degradation of the coverages indicates a largeincrease of the tails.

For the bias, different trends can be observed for the three scenarios. With a perfect alignment, the biasis, as expected, consistent with 0. For the long-term alignment, the bias is nearly identical for all samples,as it is dominated by the shifts of the pixel detector. The biases in the three coordinates are consistentwith the simulated shifts of the support structures of the pixel half-barrel

�� 4��

4� � � . Indeed, since

the innermost track hits are reconstructed at a position shifted by �� 4��

4� � � with respect to their

true position, and about half of the tracks cross each half-barrel, the observed biases are approximatelythe mean of the shifts of the two half-barrels. For the short-term alignment, the bias towards negativevalues, similar to those observed with the long-term alignment, is much larger for soft events than forhard events. Indeed, for soft tracks, because of multiple scattering, the hits in the SST have a smallercontribution in the track-fit. The misalignment of the pixel detector therefore dominates. For hardertracks, the misalignment of the SST will have a larger influence, which is more pronounced, and visible,with the short-term alignment, since the misalignment of the SST is ten times worse than in the long-termalignment.

Comparing the three algorithms, similar features can be seen. In each sample, the bias is identical for thealgorithms, since all tracks used in the fit are shifted by approximately the same amount in each vertex.In a given sample, for each algorithm, the resolutions, tails and the pulls are degraded by the same

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proportion (in the order of 30%). The robust algorithms remain effective in removing the track-outliersin the presence of misalignment.

These features help explain the results for the proper decay length of the� �� decay (Table 8).

The misalignment significantly degrades the estimated proper decay length, with an absolute degrada-tion of the resolution in the order of � 4 for all fitters, and an increase of the tails. This effect beingsignificant compared to the intrinsic resolution, little can be gained from robust fitters. The tails are how-ever significantly reduced even in the misaligned scenarios. The Adaptive Vertex Fitter performs slightlybetter than the Trimmed Kalman filter in that respect. With a similar degradation of the resolution ofthe primary and secondary vertices in both misalignment scenarios, the resolution on �� remains indeedsimilar. The improvement of the coverage of �� between the short-term and the long-term alignmentreflects the reduction of the tails of the estimated vertices. Finally, no significant bias is observed, sincethe biases of the primary and secondary vertices are largely in the same direction, which would not affectthe proper decay length. Nevertheless, even in the short-term alignment, where the biases differ by a fewmicrons between the primary and secondary vertices, no significant bias on �� is to be expected, sincethe decays are isotropic in � and to some extent in 8 .

4 ConclusionIn this study, it is seen that the Kalman filter gives consistent results over a range of processes, yieldinga good resolution on the reconstructed vertex and a good performance with a low timing and no failures.It is nevertheless seen that a significant number of fits have a large � � due to non-Gaussian tails of trackparameter errors.

With their ability to downweigh or discard outlying track, both the Adaptive Vertex Fitter and theTrimmed Kalman filter improve the estimate of the position and the error of the reconstructed vertex.These algorithms are most effective for high-multiplicity vertices, where an improvement of resolutionof up to 30% is obtained, together with a significant reduction of the tails. For low-multiplicity vertices,little improvement should be expected on the position, although it is seen that the error estimate is moreprecise. The draw-back of these algorithms is of course the timing, since they are slower by a factorbetween two and four, which is nevertheless by no means prohibitive.

An improvement of the � � -probability distribution is also obtained, although it has to be kept in mindthat the pseudo- � � computed by the robust algorithms is not � � -distributed anymore. In an analysis,different variables would thus have to be used instead of this � � , such as the weights of the tracks.

The misalignment of the tracker as implemented in the two examples described results in a global rescal-ing of the residual distribution of the estimated vertex. The robust algorithms remain effective in remov-ing the outlying track in the presence of misalignment, improving thus the estimates of the vertex withrespect to the Kalman filter, but the degradation due to the misalignment can obviously not be recovered.

References[1] J. D’Hondt, P. Vanlaer, R. Fruhwirth and W. Waltenberger, IEEE Trans. Nucl. Sci. 51, 2037 (2004)

[2] W. Waltenberger, “Development of Vertex Finding and Vertex Fitting Algorithms for CMS”, Ph.D.Thesis, Technischen Universitat Wien.

[3] P. Billoir and S. Qian, Nucl. Instrum. and Methods A 311, 139 (1992)

[4] R. Fruhwirth, R. Kubinec, W. Mitaroff and M. Regler, Comp. Phys. Comm. 96, 189 (1996)

[5] R. Fruhwirth, K. Prokofiev, T. Speer, P. Vanlaer and W. Waltenberger, Nucl. Instrum. and MethodsA 502, 699 (2003)

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[6] The CMS collaboration, The Trigger and Data Acquisition Project, Volume II, Data Acquisitionand High-Level Trigger, Technical Design Report, CERN/LHCC 2002-26, CMS TDR 6.2, 15 De-cember 2002

[7] R. Fruhwirth and T. Speer, Nucl. Instrum. and Methods A 534, 217 (2004)T. Speer and R. Fruhwirth, “A Gaussian-sum Filter for Vertex Reconstruction”, submitted for pub-lication.

[8] W. Adam, R. Fruhwirth, A. Strandlie and T. Todorov, J. Phys. G 31, N9 (2005)

[9] O. Buchmuller et al., “ORCA Misalignment Simulation for CMS Tracking Devices”, CMS publicNote 2005/XXX (in preparation).

[10] P. Vanlaer, L. Barbone, N. De Filippis, T. Speer, O. Buchmueller, F.-P. Schilling, “Impact of Sili-con Tracker Misalignment on Track and Vertex Reconstruction”, CMS Anlaysis Note, CMS-AN-2005/014.

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Filter Average Average Average Average Failure� � ��

�� Weight Time Rate

[ms] [%]� � � �� - secondary vertex (Figures 7, 8)KVF 1.32 0.46 1 1.2 0AVF 0.97 0.52 0.927 3.8 0.025TKF 0.73 0.61 0.927 2.1 0.074�'�� - primary vertexKVF 1.29 0.47 1 3.3 0AVF 0.79 0.68 0.929 12 0.01TKF 0.68 0.77 0.934 7.7 0.1� ��#�

(GF) - primary vertexKVF 1.42 0.45 1 7.4 0AVF 0.74 0.8 0.929 32 0.01TKF 0.65 0.88 0.935 24 0.091� ��#�

(IVB) - primary vertexKVF 1.58 0.39 1 7.9 0AVF 0.76 0.79 0.919 33 0TKF 0.66 0.88 0.924 27 0.02�6�� jets - primary vertexKVF 1.9 0.27 1 8.4 0AVF 0.77 0.83 0.908 37 0TKF 0.67 0.92 0.913 37 0� �

- semi-leptonic decay - primary vertexKVF 1.85 0.28 1 8.7 0AVF 0.76 0.85 0.911 39 0.01TKF 0.67 0.93 0.917 42 0

Drell-Yan - primary vertexKVF 1.78 0.31 1 8 0.01AVF 0.83 0.72 0.91 33 0.04TKF 0.7 0.84 0.913 27 0.12� ��

4 �� ! � . �� - primary vertex (Figures 13, 14)

KVF 2.05 0.21 1 13 0AVF 0.77 0.87 0.905 54 0TKF 0.67 0.95 0.911 60 0.01

Table 2: Main statistical properties for the three fitters, for the different data samples, for a detector withperfect alignement.

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Filter � -coordinate � -coordinateRes. Std. Dev. Res. Mean 95% Cov. Pull Res. Std. Dev. Res. Mean 95% Cov. Pull

[ 4 ] [ 4 ] [ 4 ] [ 4 ] [ 4 ] [ 4 ]� � � �� - secondary vertex (Figures 5, 6)KVF 54.4 0.2 164 1.08 72.9 -0.7 471 1.08AVF 53.1 -0.09 155 1.02 72.3 -0.5 440 1.02TKF 53.6 0.2 174 1.04 74.2 -0.5 502 1.05�'�� - primary vertexKVF 43.9 0.6 176 1.11 54 0.6 224 1.07AVF 38.2 0.3 94.9 0.94 48.3 0.08 140 0.94TKF 39 0.2 98.7 0.97 49 -0.06 144 0.95� ��#�

(GF) - primary vertexKVF 27.9 0.4 124 1.11 33.9 -0.1 152 1.06AVF 21.9 -0.3 73.7 0.9 29.1 -0.4 106 0.9TKF 22.8 -0.1 74.9 0.93 29.4 -0.6 111 0.92� ��#�

(IVB) - primary vertexKVF 23.8 0.4 150 1.2 34.1 0.005 211 1.15AVF 18.8 0.09 61.6 0.94 27.4 0.2 113 0.93TKF 19.1 0.07 62.3 0.96 28 0.2 117 0.95�6�� jets - primary vertexKVF 19 0.3 120 1.45 23.7 -0.1 140 1.32AVF 13.2 0.05 30.5 0.97 18.3 -0.1 46.9 0.96TKF 13.5 -0.04 32.1 0.99 18.8 -0.1 48.7 0.98� �

- semi-leptonic decay - primary vertexKVF 16.6 -0.04 102 1.41 20 -0.4 117 1.31AVF 12.1 -0.2 28 0.97 16.1 -0.3 41.8 0.97TKF 12.3 -0.3 28.8 0.98 16.5 -0.2 43 1

Drell-Yan - primary vertexKVF 15.3 0.06 77.1 1.51 26.4 -0.5 119 1.48AVF 12.6 -0.08 39.2 1.21 22.3 -0.2 60.4 1.18TKF 13.3 -0.06 39.6 1.21 22.8 -0.02 62.5 1.18� ��

4 �� ! � . �� - primary vertex (Figures 11, 12)

KVF 13.8 -0.3 118 1.51 17.7 0.04 122 1.46AVF 9.38 -0.2 21.1 0.99 12.9 -0.2 30.3 1TKF 9.72 -0.1 21.7 1.01 13.2 -0.3 31.7 1.02

Table 3: Resolution, bias, 95% coverage and pull of the � and � -coordinates of the reconstructed vertexfor the three fitters, for the different data samples, for a detector with perfect alignement.

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[ms] [%]� � � �� - secondary vertexKVF 1.2 0.48 1 1.2 0AVF 0.94 0.53 0.935 3.4 0.008TKF 0.72 0.62 0.935 1.9 0.008�'�� - primary vertexKVF 1.38 0.44 1 4.1 0AVF 0.81 0.68 0.925 14 0TKF 0.69 0.77 0.93 9.2 0.13� ��#�



- semi-leptonic decay - primary vertexKVF 1.93 0.24 1 9 0AVF 0.81 0.8 0.908 38 0TKF 0.7 0.91 0.913 43 0

Drell-Yan - primary vertexKVF 1.84 0.28 1 8.3 0AVF 0.86 0.69 0.908 30 0.04TKF 0.72 0.82 0.91 26 0.25� ��

4 �� ' � . �� - primary vertex

KVF 2.09 0.18 1 13 0AVF 0.81 0.81 0.903 51 0TKF 0.71 0.93 0.908 59 0

Table 4: Main statistical properties for the three fitters, for the different data samples, for a misaligneddetector (short-term alignment).

12


[ 4 ] [ 4 ] [ 4 ] [ 4 ] [ 4 ] [ 4 ]� � � �� - secondary vertexKVF 66.7 -2.6 189 1.12 83.7 2.4 520 1.09AVF 66.2 -3.2 173 1.07 82.9 2.3 491 1.04TKF 68.2 -3.1 200 1.11 84.3 2.8 568 1.07�'�� - primary vertexKVF 49.5 -8.1 233 1.16 57.7 -2.3 281 1.07AVF 42.6 -8.7 103 0.97 52.1 -2.6 146 0.94TKF 44.2 -8.6 108 1 53.2 -2.6 153 0.96� ��#�

(GF) - primary vertexKVF 35.7 -6.3 158 1.26 38.9 -1.9 192 1.11AVF 29.3 -6.6 81.3 1.04 33.8 -1.9 111 0.96TKF 30.4 -6.5 84 1.07 34.2 -1.7 114 0.98� ��#�

(IVB) - primary vertexKVF 35.4 -4 207 1.46 40.3 -0.4 288 1.25AVF 28.4 -4.7 75.2 1.19 34.3 -0.7 127 1.02TKF 29.1 -4.9 77.3 1.22 35.5 -0.9 129 1.04�6�� jets - primary vertexKVF 29.2 -2.3 212 1.84 32.9 1.4 252 1.5AVF 22.6 -2.4 48.7 1.35 26.5 0.8 63.7 1.15TKF 23.4 -2.5 50.5 1.38 27.1 0.8 64.6 1.17� �

- semi-leptonic decay - primary vertexKVF 27.1 -1.7 173 1.78 28 0.9 210 1.45AVF 20.6 -1.9 42.9 1.3 22.4 1 53 1.11TKF 21.4 -1.9 44.4 1.32 23.1 1 55.7 1.13

Drell-Yan - primary vertexKVF 29.1 3.5 131 1.74 35.8 2.4 199 1.48AVF 23.9 3.2 56.4 1.37 29.6 2.4 78.3 1.18TKF 25.5 2.9 58.1 1.4 30.6 2.6 78.8 1.19� ��


KVF 24.5 0.8 197 1.99 24.3 2.7 230 1.59AVF 18.2 0.4 37.1 1.4 19.2 2.1 41.7 1.16TKF 18.9 0.4 38.2 1.43 19.8 2.1 44.4 1.21

Table 5: Resolution, bias, 95% coverage and pull of the � and � -coordinates of the reconstructed vertexfor the three fitters, for the different data samples, for a misaligned detector (short-term alignment).

13



[ms] [%]� � � �� - secondary vertexKVF 1.18 0.49 1 1.2 0AVF 0.92 0.54 0.937 3.7 0TKF 0.72 0.62 0.939 2 0.025�'�� - primary vertexKVF 1.27 0.47 1 4.4 0AVF 0.8 0.68 0.931 14 0TKF 0.69 0.77 0.936 9.5 0.091� ��#�



- semi-leptonic decay - primary vertexKVF 1.79 0.29 1 11 0AVF 0.79 0.81 0.914 46 0TKF 0.7 0.91 0.919 46 0

Drell-Yan - primary vertexKVF 1.73 0.31 1 7.8 0AVF 0.85 0.7 0.914 29 0.02TKF 0.72 0.83 0.916 24 0.21� ��


KVF 1.96 0.23 1 12 0AVF 0.8 0.83 0.909 53 0TKF 0.7 0.93 0.914 58 0.01

Table 6: Main statistical properties for the three fitters, for the different data samples, for a misaligneddetector (long-term alignment).

14


[ 4 ] [ 4 ] [ 4 ] [ 4 ] [ 4 ] [ 4 ]� � � �� - secondary vertexKVF 63.7 -11 177 1.09 81.5 -3.8 500 1.07AVF 63.3 -11.2 170 1.05 80.2 -3.5 495 1.03TKF 64.7 -11.1 191 1.08 81.6 -3.8 555 1.06�'�� - primary vertexKVF 47.9 -10.8 187 1.13 57.2 -4.9 233 1.06AVF 42.3 -11.1 104 0.97 51.8 -5.5 145 0.94TKF 43.5 -11 109 1 53.2 -5.4 153 0.97� ��#�

(GF) - primary vertexKVF 33.8 -10.8 134 1.23 37.5 -5.2 157 1.09AVF 28.3 -11 82.2 1.03 33.2 -5.5 113 0.95TKF 29.3 -11.1 84.3 1.06 34 -5.5 116 0.97� ��#�

(IVB) - primary vertexKVF 31.8 -11.2 148 1.42 38.9 -5.2 215 1.21AVF 26.4 -11.1 75.7 1.17 33.2 -4.8 126 1.01TKF 27 -11.2 75.9 1.19 33.8 -4.6 128 1.03�6�� jets - primary vertexKVF 26.4 -11.3 127 1.73 30.5 -4.7 148 1.45AVF 20.9 -11.1 49.2 1.31 25.7 -4.9 62.9 1.16TKF 21.5 -11.1 50.7 1.33 26.2 -4.9 64.6 1.18� �

- semi-leptonic decay - primary vertexKVF 23.4 -11.2 107 1.65 25.7 -5.1 121 1.4AVF 18.8 -11.1 44.6 1.25 21.5 -4.9 53.6 1.1TKF 19.4 -10.8 45.9 1.28 22.2 -4.9 55.6 1.13

Drell-Yan - primary vertexKVF 22.7 -10.7 84.7 1.66 32.3 -4.6 132 1.47AVF 19.9 -10.6 53.2 1.38 27.8 -4.5 72.8 1.22TKF 21.8 -10.1 55.9 1.41 28.8 -4.3 75.5 1.21� ��


KVF 20.2 -11.4 102 1.8 22.2 -3.7 110 1.5AVF 16 -11.1 39.1 1.34 18 -4.1 42.4 1.15TKF 16.9 -10.9 40.1 1.37 18.9 -4.1 43.3 1.19

Table 7: Resolution, bias, 95% coverage and pull of the � and � -coordinates of the reconstructed vertexfor the three fitters, for the different data samples, for a misaligned detector (long-term alignment).

15

Filter �� /�� Res. Std. Dev. 95% Cov. Pull Res. Std. Dev. 95% Cov. Pull

[ 4 ] [ 4 ] [ 4 ] [ 4 ]Perfect alignment

KVF 40.6 129 1.36 47.3 164 1.25AVF 37.4 100 1.21 42.7 118 1.1TKF 38.2 117 1.25 43.7 135 1.13

short-term alignmentKVF 46.9 153 1.38 54.6 194 1.27AVF 44 105 1.23 49.8 127 1.11TKF 45.9 122 1.28 51.2 146 1.16

long-term alignmentKVF 45.3 131 1.34 52.1 168 1.24AVF 42.9 104 1.21 48.5 126 1.1TKF 44.4 118 1.27 50.1 140 1.15

Table 8: Resolution, 95% coverage and pull of the proper decay length �� of the�-�� , measured

in three dimensions ( �� /�� ) or in two dimensions ( �� ), for the three alignment scenarios.

16

Track/event0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Eve

nts

per

0.2

5

0

2000

4000

6000

8000

10000

12000Mean: 4.00

(a)

Track/event0 10 20 30 40 50 60

Eve

nts

per

3.0

0

0200

400600

80010001200

140016001800

200022002400 Mean: 11.65

(b)

Track/event0 10 20 30 40 50 60 70 80 90 100

Eve

nts

per

5.0

0

0

500

1000

1500

2000

2500

3000 Mean: 12.57

(c)

Track/event0 20 40 60 80 100 120 140

Eve

nts

per

7.0

0

0

200

400

600

800

1000

1200

1400

1600

1800

2000 Mean: 23.20

(d)

Track/event0 20 40 60 80 100 120 140

Eve

nts

per

7.0

0

0

200

400

600

800

1000

1200

1400

1600

1800

2000Mean: 23.67

(e)

Track/event0 20 40 60 80 100 120 140 160

Eve

nts

per

8.0

0

0

500

1000

1500

2000

2500Mean: 33.31

(f)

Track/event0 20 40 60 80 100 120 140 160

Eve

nts

per

8.0

0

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200 Mean: 36.67

(g)

Track/event0 20 40 60 80 100 120 140

Eve

nts

per

7.0

0

0

200

400

600

800

1000

1200

1400

1600

1800

2000Mean: 22.72

(h)

Track/event0 20 40 60 80 100 120 140 160 180

Eve

nts

per

9.0

0

0200400

600800

10001200

1400160018002000

22002400 Mean: 44.31

(i)

Figure 2: Distributions of the simulated (full line) and associated reconstructed (dashed line) track mul-tiplicities for the tested samples (from left to right, top to bottom: (a)

� � � ��A� (Secondary vertex),(b)

� � � ��A� (Primary vertex), (c)�-jet, (d)

� �#�(GF), (e)

� �#�(IVB), (f) �6�� jets, (g)

��,

(h) Drell-Yan and (i)� ��

).

17

[GeV/c]T

p0 20 40 60 80 100 120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410 (a)

[GeV/c]T

p0 20 40 60 80 100120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510 (b)

[GeV/c]T

p0 20 40 60 80 100120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510(c)

[GeV/c]T

p0 20 40 60 80 100 120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510 (d)

[GeV/c]T

p0 20 40 60 80 100120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510 (e)

[GeV/c]T

p0 20 40 60 80 100120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510(f)

[GeV/c]T

p0 20 40 60 80 100 120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510(g)

[GeV/c]T

p0 20 40 60 80 100120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510 (h)

[GeV/c]T

p0 20 40 60 80 100120140160180200

Tra

cks

per

2 G

eV/c

1

10

210

310

410

510

610

(i)

Figure 3: Distributions of the transverse momenta of the reconstructed tracks for the tested samples(fromleft to right, top to bottom: (a)

�)� � ��A� (Secondary vertex), (b)�)�� (Primary vertex), (c)�

-jet, (d)� ��#�

(GF), (e)��

(IVB), (f) ��(�� jets, (g)� ��

, (h) Drell-Yan and (i)� ��

).

18

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

100

200

300

400

500

600

700

800(a)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

(b)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

500

1000

1500

2000

2500

3000

3500

4000

4500(c)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

500

1000

1500

2000

2500

3000

3500

4000(d)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

500

1000

1500

2000

2500

3000

3500

4000

(e)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

1000

2000

3000

4000

5000

6000

(f)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

1000

2000

3000

4000

5000

6000

7000

(g)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

500

1000

1500

2000

2500

3000

3500

4000(h)

η0 0.5 1 1.5 2 2.5 3

Tra

cks

per

0.03

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

(i)

Figure 4: Distributions of the pseudo-rapidities of the reconstructed tracks for the tested samples (fromleft to right, top to bottom: (a)

�)� � ��A� (Secondary vertex), (b)�)�� (Primary vertex), (c)�

-jet, (d)� ��#�

(GF), (e)��

(IVB), (f) ��(�� jets, (g)� ��

, (h) Drell-Yan and (i)� ��

).

19

m]: KVFµRes. x coord. [-400 -300 -200 -100 0 100 200 300 4000

100

200

300

400

500

600

700

800

900 Mean: 0.25: 54.40σ

C(95): 164.29

m]: AVFµRes. x coord. [-400 -300 -200 -100 0 100 200 300 4000

100

200

300

400

500

600

700

800

900Mean: -0.09

: 53.14σC(95): 154.72

m]: TKFµRes. x coord. [-400 -300 -200 -100 0 100 200 300 4000

100

200

300

400

500

600

700

800

Mean: 0.23: 53.56σ

C(95): 174.28

Pull x coord.: KVF-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000 Mean: 0.00: 1.08σ

Pull x coord.: AVF-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000Mean: 0.00

: 1.02σ

Pull x coord.: TKF-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000Mean: 0.01

: 1.04σ

Figure 5: Secondary (decay) vertex of� � "� ��A� events (perfect alignment): residual (top) and pull

(bottom) of the � -coordinate of the reconstructed vertex for each of the three tested fitters.

m]: KVFµRes. z coord. [-600 -400 -200 0 200 400 6000

100

200

300

400

500

600

700

800

900

Mean: -0.72: 72.92σ

C(95): 471.05

m]: AVFµRes. z coord. [-600 -400 -200 0 200 400 6000

200

400

600

800

1000 Mean: -0.54: 72.25σ

C(95): 439.51

m]: TKFµRes. z coord. [-600 -400 -200 0 200 400 6000

100

200

300

400

500

600

700

800

900

Mean: -0.53: 74.21σ

C(95): 502.07

Pull z coord.: KVF-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800

900

Mean: 0.01: 1.08σ

Pull z coord.: AVF-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000 Mean: 0.01: 1.02σ

Pull z coord.: TKF-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000 Mean: 0.01: 1.05σ

Figure 6: Secondary (decay) vertex of� � "� ��A� events (perfect alignment): residual (top) and pull

(bottom) of the � -coordinate of the reconstructed vertex for each of the three tested fitters.

20

): KVF2χProb(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

200

400

600

800

1000

1200

1400 Mean: 0.46

): AVF2χProb(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

40

60

80

100

120

140

160Mean: 0.52

): TKF2χProb(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

220Mean: 0.61

Effective nbr. removed tracks: AVF0 5 10 15 200

1000

2000

3000

4000

5000

Mean: 0.29

Effective nbr. removed tracks: TKF0 5 10 15 200

1000

2000

3000

4000

5000

6000

7000

8000

9000 Mean: 0.29

Figure 7: Secondary (decay) vertex of� � � ��A� events (perfect alignment): �� -probability (top) and

sum of downweighted tracks (bottom) of the reconstructed vertex for each of the three tested fitters.

Time per fit [ms]: KVF0 1 2 3 4 5 6 7 8

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200 Mean: 1.21

Time per fit [ms]: AVF0 1 2 3 4 5 6 7 8

0

200

400

600

800

1000

1200

1400

1600

1800

2000Mean: 3.77

Time per fit [ms]: TKF0 1 2 3 4 5 6 7 8

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200Mean: 2.10

Figure 8: Secondary (decay) vertex of� �A � �� events (perfect alignment): CPU time of the vertex

fits for each of the three tested fitters.

21

m]: KVFµ (2D) [τRes. c-250-200-150-100-50 0 50 1001502002500

100

200

300

400

500

600 Mean: -1.78: 47.30σ

C(95): 164.31

m]: AVFµ (2D) [τRes. c-250-200-150-100-50 0 50 1001502002500

100

200

300

400

500

600

Mean: -1.91: 42.73σ

C(95): 118.25

m]: TKFµ (2D) [τRes. c-250-200-150-100-50 0 50 1001502002500

100

200

300

400

500

600Mean: -1.87

: 43.68σC(95): 135.26

(2D): KVFτPull c-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800Mean: -0.05

: 1.25σ

(2D): AVFτPull c-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800

900Mean: -0.04

: 1.10σ

(2D): TKFτPull c-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800

900Mean: -0.05

: 1.13σ

Figure 9: Residual (top) and pull (bottom) of the proper decay length of the� ��

meson measured in twodimensions, for each of the three tested fitters (perfect alignment).

m]: KVFµ (3D) [τRes. c-250-200-150-100-50 0 50 1001502002500

100

200

300

400

500

600

Mean: -2.17: 40.63σ

C(95): 129.29

m]: AVFµ (3D) [τRes. c-250-200-150-100-50 0 50 1001502002500

100

200

300

400

500

600

700Mean: -2.17

: 37.41σC(95): 99.95

m]: TKFµ (3D) [τRes. c-250-200-150-100-50 0 50 1001502002500

100

200

300

400

500

600

700 Mean: -2.22: 38.17σ

C(95): 117.01

(3D): KVFτPull c-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

Mean: -0.08: 1.36σ

(3D): AVFτPull c-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800

Mean: -0.07: 1.21σ

(3D): TKFτPull c-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800Mean: -0.07

: 1.25σ

Figure 10: Residual (top) and pull (bottom) of the proper decay length of the� ��

meson measured inthree dimensions, for each of the three tested fitters (perfect alignment).

22

m]: KVFµRes. x coord. [-150 -100 -50 0 50 100 1500

100

200

300

400

500

600

700

800

900Mean: -0.31

: 13.80σC(95): 117.96

m]: AVFµRes. x coord. [-150 -100 -50 0 50 100 1500

200

400

600

800

1000

1200

1400Mean: -0.16

: 9.38σC(95): 21.09

m]: TKFµRes. x coord. [-150 -100 -50 0 50 100 1500

200

400

600

800

1000

1200

Mean: -0.12: 9.72σ

C(95): 21.66

Pull x coord.: KVF-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

Mean: -0.02: 1.51σ

Pull x coord.: AVF-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800

Mean: -0.01: 0.99σ

Pull x coord.: TKF-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800Mean: -0.01

: 1.01σ

Figure 11: Primary vertex of� ��

events (perfect alignment): residual (top) and pull (bottom) of the� -coordinate of the reconstructed vertex for each of the three tested fitters.

m]: KVFµRes. z coord. [-150 -100 -50 0 50 100 1500

100

200

300

400

500

600

700Mean: 0.04

: 17.68σC(95): 122.10

m]: AVFµRes. z coord. [-150 -100 -50 0 50 100 1500

200

400

600

800

1000Mean: -0.24

: 12.85σC(95): 30.35

m]: TKFµRes. z coord. [-150 -100 -50 0 50 100 1500

100

200

300

400

500

600

700

800

900

Mean: -0.26: 13.22σ

C(95): 31.74

Pull z coord.: KVF-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

Mean: 0.00: 1.46σ

Pull z coord.: AVF-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800Mean: -0.01

: 1.00σ

Pull z coord.: TKF-10 -8 -6 -4 -2 0 2 4 6 8 100

100

200

300

400

500

600

700

800 Mean: -0.01: 1.02σ


events (perfect alignment): residual (top) and pull (bottom) of the� -coordinate of the reconstructed vertex for each of the three tested fitters.

23

): KVF2χProb(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1000

2000

3000

4000

5000

6000 Mean: 0.21

): AVF2χProb(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

1800

2000

2200Mean: 0.87

): TKF2χProb(0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1000

2000

3000

4000

5000 Mean: 0.95

Effective nbr. removed tracks: AVF0 5 10 15 200

50

100

150

200

250

300

350

400

Mean: 4.20

Effective nbr. removed tracks: TKF0 5 10 15 200

200

400

600

800

1000

1200

1400

1600

1800 Mean: 3.93


events (perfect alignment): � � -probability (top) and sum of down-weighted tracks (bottom) of the reconstructed vertex for each of the three tested fitters.

Time per fit [ms]: KVF0 20 40 60 80 100 120 140

0

200

400

600

800

1000

1200

1400

1600

1800 Mean: 12.89

Time per fit [ms]: AVF0 20 40 60 80 100 120 140

0

50

100

150

200

250

300

350 Mean: 53.93

Time per fit [ms]: TKF0 20 40 60 80 100 120 140

0

20

40

60

80

100

120

140

160

180Mean: 60.15


events (perfect alignment): CPU time of the vertex fits for each of thethree tested fitters.

24

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