ported by BMf`WF, project no. OWP-50). OCR Output”Permanent address: Dept. of Nuclear Physics. Comenius University, Bratislava, Slovak Republic (sup

the LEP collider.

to the Kalman filter. The implementation is shown to work also on real data from

samples of charm and bottom decays. It is shown that the !ll—estimator is superior

filter and its extension to an M—estimator, both estimators are studied on simulated

ing and estimating secondary vertices. After a brief review ofthe standard KalmanThis paper presents the application of a robust estimator to the problem of find

Abstract

(Submitted to Comp.Phys.Comm.)

ge 2ENikolsdorfer Gasse 18, A—lO5O Wien, Austria §

{DOsterreichischen Akzmdemie der Wissenschaften

Institut fur Hochenergiephysik der E(3 i

R. Friihwirth, P. Kubinec*, W. Mitaroff and M. Regler

the LEP Collider Using Robust Algorithms

Vertex Reconstruction and Track Bundling at

26 March 1996

HEPHY-PUB—598/94 (rcv.2)Fachbereich ASE

at the LEP collider (section 4). OCR Output

proposed by the authors works well with real data collected by the DELPHI experimentparticles. The results are presented in section 3. We also demonstrate that the algorithmof simulated events containing secondary decay vertices of short—lived charm and bottom

Wfe have made a detailed comparison of least—squares and 1lf—estimators on samplesestimator. A brief review of the Kalman filter and of the f\~{—estimator is given in section 2.is the obvious candidate. as it can be implemented as an iterated weighted least-squaresthem are not suitable to the analysis of a large number of events. The 1W—estimator [4]the use of robust estimators. These tend to be computationally expensive, so many ofIt therefore seems worthwile to investigate whether this situation can be remedied byto a certain degree. As a. consequence, the probability of recognizing them is impaired.at a common point in space. The fitted vertex position is distorted by secondary tracksconsidered as outliers, since they violate the assumption that all tracks in the fit originatetails of the error distributions. In a fit of the primary vertex, secondary tracks have to beto the model used in the regression. or if their errors are unusually large due to excessiveing to serious distortions. Measurements are called outliers either if they do not conformthe objective function, outlying measurements have a large influence on the estimate, leaderties, the most prominent of which is a lack of robustness. Due to the quadratic form of

Beside these nice features, least—squares estimators also have some less desirable propestimate is efficient, i.e. has the smallest possible variance among all unbiased estimates.variance among all linear estimates. In an exact linear model with Gaussian errors thelinear least—squares estimators: estimates are unbiased, consistent, and have minumum

The Kalman filter and the smoother possess all the well—known optimal properties ofused to discriminate between primary and secondary tracks.shown how the Xistatistics associated with each step of the filter and smoother can bethat this algorithm is a special case of the Kalman filter and the smoother It also wasgeneral regression problem, O(n.3) operations would be needed. Finally it was recognizedcovariance matrix of the estimates in O(n.2) steps, n. being the track multiplicity In athe estimates of the vertex position and the track parameters in O(n) steps, and the fullregression problem Later it was shown that there exists an algorithm which computesadvantage of the particular structure of the vertex Ht when it is formulated as a linearized

It had been recognized already in 1973 by one of the a.uthors (MR-) that one can takevertex reconstruction have to be adapted accordingly.be and will grow to truly enormous figures in colliders like LHC. The algorithms used forby automatic pattern recognition. In addition, track multiplicity is larger than it used tois now a problem which is no longer amenable to visual inspection, but has to be solvedscattering. The identification of secondary tracks and subsequently of secondary verticesPrecision is gained by high-resolution vertex detectors and a correct treatment of multiplebe extrapolated back in order to reconstruct the primary and secondary decay vertices.vertex is inside the beam tube, wl1erea.s tracks are reconstructed outside and have to

Today, the situation has changed enormously. In a collider experiment, the primaryidentification. Secondary vertices and their associated tracks could be recognized visually.prerequisite to a complete kinematics fit. which was one of the primary means of particleof bubble chamber experiments. In those days the geometric vertex fit was a necessaryThe reconstruction of vertices by least-squares methods has a long tradition since the time

1 Introduction

cov{_;ck.q,€) : Ek : —CkAk’ Gk,BkWk.. OCR Output

f`OV(qk) Z Dk Z Wk —f- WkBk’ GkAkCkAk’ GkBkWk Z Wk ·l* EkC§Ek,1l

wvtrirt) I Ct Z (Oki, + AtGkAr).fB_l

{It I WrBt’ Gttpt _ Ctr ·· Atitt

it = C'tlC';1¤ik—i + AtGr(Pk —€t.t)l»T'B

is - l tracks:

I The Italmau filter describes how to add a new track lr to a vertex already fitted with(mes qk,5)‘with Ak : (Ohk/U20]? and Bk : (Oh),/Oqklk. being the (5 >< Jl) matrices of derivatives at

ht(=¤t<1r)~ hr<¤¤€»qr_J + Arte — me) + Bttqr — qt.?) = Ctrl + Aw + Btqc

expansion at some point (acmqkf):A linear regression model is obtained by approximating hk by a first order Taylor

included as additional noise in Vk.

lf there is multiple scattering between the vertex and the reference surface. it has to be

pr = hk(<¤»<1r)+ er, <r<>v(e».> = Vr

which in general is non-linea.r:the equation of motion (track model). and is described by the measurement equation,The functional dependence of the track parameters on the state vector is determined by

mk Z $k_1 Z SD.

system equation is simply the identity:the state vector is augmented by the 3·momentum vector qk. Without process noise, theno such information is available, :1:0 is arbitrary and CS1 is set to zero. For each track.of the prior information about the vertex position, mo. and its covariance matrix CU. lfestimated are called state vector in filter language. Initially, the state vector consists onlyterminology and notation used below is the same as in [3. 6]. The parameters to beFor an introduction to filter theory the reader is referred to a textbook (e.g. The

2.1 The Kalman Filter Method

this vertex) and "bad” tracks (belonging to another vertex or to background).pattern recognition problem of discriminating between "good" tracks (originating frommentum vectors qk of all tracks at the common vertex. ln addition. it can be used for the

The aim of the vertex fit is the estimation of the vertex position :1: and of the mocovariance matrix of all pk is (5 >< 5) bloclediagonal.the vertex fit. lt is assumed that the tracks a.re uncorrelated. i.e. that the joint (512 >< 571)detectors, and are now in turn considered as (virtual) measurements for the purpose offitted track parameters contain the full information from measurements by the trackingthe corresponding covariance matrix Vk : Gk_l, defined at some reference surface. Thecommon vertex. For each reconstructed track, we have 5 fitted track parameters pk andSuppose that there are rn, tracks of particles in a magnetic field which are to be fitted to a

2 Review of Efficient Vertex Evaluation Techniques

all other formulas can be computed by Ott:.) operations. OCR Output(Yomput ation ofthe full covariance matrix requires Ot ng) arithmetic operations. whereas

elements of tracks not taking part in the kineinatic fit need not be computed.

needed onlv if the vertex fit is followed bv a kinematic fit. Even in this case. matrix

where are the smoothed residuals of track Lv. Normally the full covariance matrix is

rt = pt — bt Pt = ct. + At-dn,. + Btét

k:l

T -12 ~ — V ’1‘ Yu I (wu - wt) C0 (ws — $1.)+ Z rt Gtrt.

_ , — ~ __ 1 1 _ _ · co\(qQ.qé;L) - WkBt GtAkC.,,AJ GJBJWJ - EQ CH E3'. ityéj.T —l

smoothing:It required. the full covariance matrix and the total chi—square can be computed after

(`()V(1i,l,qZ) Z 2 —CnAk’ GkBtWk.

~ , cov(qQ) = : Wk + WkBkfGkAkCnAkf GkBtWk : Wk + CnT _1

QZ I WkBk1 Gtfpt — Cm — At€i37t)~

mentum vectors and covariance matrices with the final estimate of the vertex position:As there is no process noise, the smoother requires no more than recomputing the mo

kzl

cov(5:,L) : on Z (05 + Z A,JG,fAk)" ,. ’·*

fc:]

~ -1 wu = Gn[C0 wo + Z At Gt (pt — ctsllt’1‘ B

can actually be computed in one go ("global fit" ):Due to the simple form of the system equation the hnal estimate of the vertex position

the filter can be recomputed until convergence has been reached.errors ek. lf necessary. the function hk can be re—expanded at the new point (ick, qk) andapproximation error of the linear expansion should be small compared to the measurement

The choice of the expansion point (zzzmqkx) is in principle arbitrary; however, thedegrees of freedom otherwise.lt has 2k degrees of freedom if there is prior information on the vertex position, and 2/{ — 3

_ ,\t — Xt-1 + \t,F·2 2 2

(which are independent random variables):The total chi—square of the fit is equal to the sum of the chi—squares vi} of all filter steps

Tk : Pt _ pp pk Z Ck; + Akxk `l’ Bkqt

2 _ — ~ ~ ~ 1 xt; — (wt — wei) Gtntrvt - wth) + rt Gtrt.T -1

The clii—square of the filter step has two degrees of freedom. lt is given by:

Wt : (BkGtBt).T'`1

with:

the least—squares or L2-estimator. A choice of t;·(t) : sgn(t) yields the Lyestimator. OCR OutputClearly, there is a large variety of possible M-estimators. lf we choose u~(f.) : t we obtain

where \lf(t) is a suitablefunction, usually specified in terms of its derivative rjitt) : di}/dt.

rzl

Nita?) Z Z ‘I’(7`¢/Us},

of a:. The ilI—estimator is based on a generalized objective function:Nlinimization of this objective function with respect to :1: yields the least—squares estimate

.1=\

TL Z yi " L(Li_}JiJ·

with the residuals:

i=l

·’J(¤=> = ZW/¤i)2~

function C can be written as:

We assume for the time being that V is diagonal with elements (af, . . . ,02). The objective

y = Aa: + 6. cov(e) : V.

problem with m parameters :1:, a {11 >< m) model matrix A and 72 measurements y:the measurements with large residuals. Let us consider a homogeneous linear regressionsuch that outliers have less influence on the estimate. This is achieved by downweightingestimator The basic idea of the M—estima.tor is a modification ofthe objective functionless sensitive to outliers by using a robustified version of the Kalman filter, based on the iWAs first suggested by one of the authors (RT.), the estimation of the vertex can be made

2.2 The M-estimator Method

outliers, and the power of the test decreases. This will be demonstrated in sect. 3.4.there are several such tracks. ln that case, the estimate iz" is biased by the remainingparticular secondary tracks. lts usefulness is, however, rather limited, particularly whenthe chi—square of the smoother can be used as a. test criterium for outlying tracks, inIn contrast to the filter, the smoothed chi-squares @5 are no longer independent. Clearly,

fl _ L iH*)(C)(” i "IL* 7],1 G Tl. \t,.s — n it lv {Bn ‘”1t)+7't k""t·T7l*—1

The smoothed chi-square of track lc can now be computed from rf and dz)?

cov(:i:Z*) = CQ" = (C? — A;,G;,A;,)fB`1

er Z <>z*tC.:1¢. — Aifeftpt — wi.

Ixialrnan, filter using —Gk instead of Gr:estimate 5:2* which results from removing track lc from the fitted vertex by an zinrersrof the smoother, which has two degrees of freedom as well. To this end we compute theHtted with tracks l to lc — l. .·\ symmetric test can be constructed by using the chi—squarethe fitted vertex. This test is, however. not symmetric: track is is tested against a vertex

The chi—square of the Hlter can be used to test whether a track is compatible with

to one of three criteria; OCR Output

The test of compatibility of a track with the fitted vertex can bc computed according

reader is referred to the book of leluber

the ;lI—estimator is consistent and asymptotically normal. For mathematical details theresults available. lt can be shown that under certain regularity and uniqueness conditionsthe small sample properties of the iU—estimator. There are, however, some asymptoticresponding analysis of the linear least-squares estimator; in fact. little can be said about

A statistical analysis of the iU—estimator is considerably more difficult than the corposition of the vertex wa.nted — e.g. the beam interaction profile for a primary vertex.stable behaviour. This can be avoided by including prior information (:1:0, C0) about the

ln case of several equally well measured vertices, the ill-estimator may exhibit a multiapplied again.

above, yielding a new weight matrix Gk/’. The global least—squares formalism is thenthe preceding subsection; in subsequent iterations the weights are modified as describedThe first iteration of the i\J—estin1ato1· is just the global least—squares fit as described in

pk} I Ukpk, Ak, I UkAg, Bk, Z UkBk, AY 2 T, .... 72.

of derivatives have to be transformed accordingly:containing the (positive) eigenvalues of Gk. The virtual measurements and the matriceswhere the rows of Uk are orthonormal eigenvectors of Gk, and Gkf is a diagonal matrix

Gkl:UkGkUk1, f£ZT,...,’IZ,

an orthogonal transformation Uk (because they are symmetric):be transformed to diagonal form. This can be done, separately for each track k, by usingthe virtual measurements (track parameters), which are in general non-diagonal, have to

If the iw-estimator is to be applied to the vertex tit, the weight matrices Gk : Vk" offunction having several local minima.some choices of the function iii the il’I—estimator is not necessarily unique, the objectivechange in the minimum value of the objective function JV!. lt should be noted that forthe weights for iteration k. The iteration is stopped as soon as there is no significantestimate of iteration In — 1 is used to compute residuals which are then used to computeestimate is computed by ordinary least—squares, corresponding to wi = 1 for all i. Theunknown residuals ri. Therefore one has to resort to an iterative procedure. The initialA/I. The estimate cannot be computed explicitly, however, as the weights depend on thewi/afi, thus downweighting the contribution of large residuals to the objective functionThe ilI—estimator is formally identical to a least-squares estimator with modified weights

r,/ai R0)/|n], [ml > Ha,.wl S RU;wi : t(’("z/U1) :{ l, '

with:

i:1

Z E(7“i/O';) · (aijwi/ai) 1-‘ Z 1,. . . ,771.,

function A/l is minimized with respect to cc by setting the derivatives equal to O:R is called the constant of r0b2.:.sfnes.<. lt is usually chosen between l and 3. The ob_jective

(wtf) = " 'j R — sgn(t), [tl > R.tl S R

kept and passed to the analysis program. OCR Output(track elements. TE). information on the simulated vertices (SP) and their tracks (ST) isas expected by a subsequent local pattern recognition in the individual detector modulesavoids the generation of detailed digitizings (raw data). lnstead, it simulates the results

The detector simulation is performed by the standard program FASTSIM [9] which(P.1},.p), with 0§1)§rr and 0§(<f>;,2) < 2vr.for space points in cylindrical coordinates is (R, (D, 2), and for vectors in polar coordinatesdirection of the e" beam. and the .r-axis pointing to the centre of the LEP ring. Notation

The global coordinate system has its origin at the centre of the detector, the z—axis infield is homogeneous with B : (0, 0, 1.2) T in the central tracking region.and in the "forward region": two Forward Drift, Chambers (li(`A and PCB). The magnetic(VD). lnner Detector (ID), Time Projection Chamber (TPC) and Outer Detector (OD):detector modules are, from the inside outwards, in the "barrel region”: Vertex Detector

This study is based on the DELPHI detector layout of 1990 The track—sensitivesimulation program (see below).ment into B or B" mesons only. Decays of the hadrons are generated by the detectorThe bottom sample consists of 200 selected events e`e+—>Z°—>bl1(g)—»hadr0ns which fragtion model. For the charm sample, 100 events e"e`l'—>Z°—>c6(g)—+hadr0ns are simulated.Both data samples are generated by the program JETSET [7) using the Lund fragmenta

3.1 Event Generation and Detector Simulation

detector and the reconstruction of tracks by the usual DELPH1 data ana.lysis.decay lengths L : Bqwcr with B7 = P/m. This is followed by a simulation of the DELPHIare r w 124, 140, 317 and 388 pm/0 for D0, Dj Di and B mesons, respectively, yieldingobtained by decays of charm or bottom events generated at the Z0 pole. Mean lifetimesthe M-estimator (sect. 2.2) methods. respectively. A clean but realistic environment istematically compared for simple impact parameter tests, the 1{ahnan filter (sect. 2.1) andln this section the efficiencies of testing track association to the correct vertex are sys

3 Monte Carlo Study of Charm and Bottom Events

assess, but should normally be small.bias the total chi-square in opposite directions, the net effect of which is difficult toerrors (loss of “good” tracks) and type ll errors (inclusion of “bad" tracks in the fit) willtice, however, outlier removal by one of the tests above is never 100 % efficient; type Ithe original weight matrices yielding a correct chi—square and covariance matrix. In prac

After removing outlying tracks, the vertex is refitted with ordinary least—squares andbe determined by simulation studies.

an outlier; its distribution is unknown. Therefore the critical values of the tests have to

actually (2-distributed even if there is no outlier. The product rrr is "small" if track is isThe “chi-squares” and \i_S' are "large” if track lu is an outlier: they are no longer

• The product '/Tk : HL, wwf of the final weight corrections of track ls.

• The smoothed chi—square xfs,. rising the hnal (downweighted) weight matrix Gt"

• The smoothed chi—square vis, using the original weight matrix Gif;

(1e\'"' OCR Outputrn1s(;kI’°l) 0.018 <:;(»y·* 1 0.0 10rr11s($.,:) 7.5 111rad 1 (5.1) mrad 1

rrr1s(Ar}) (1.1 mradmrad I 5.8pmrn’1s(A:) 1810 pm 1SSB

nmrr11s(AR<I>) 140 nm 1 138

Parameter cc sample ( BB sample

components of Ap = p —— p"“‘. the r.m.s. of which are given below:factor of V. The precision of tl1e track fit may be derived from the distribution of the 5where ‘n, is a (0,1) normally distribiited, uncorrelated random 5—vector. and S is a(1ho1cskyp p—» : p""‘ + Sn. SST : V,

track parameters p by parameters simulated in accordance with V:Tlierefore. we keep only the track fit°s covariance matrices V. a11d replace all fitted

X2 probability which shows a. pronounced U—like shape.coniponents wl1icl1 differ strongly from a. (0,1) normal distribution, and the track fit”s total

exhibited by plotting the normalized deviations (p —— pm"),/\/(V),, for the parametersurement errors and/or multiple scattering a11d iriefficiencies in the track search are clearly

lnconsistencies between FASTSIM and DELANA in the treatnient of detector n1ea—

the “true” track parameters 1J’"‘€.

tl1e original track (ST) from its true vertex position to the refererice surface yieldstrack is not the product of a secondary reaction in the detector. Then an extrapolation of

Pointers contained i11 the data help to identify corresponding Tl{X~ST pairs if thecovariance matrix is cov(p) : V.track”s charge and the z—component of the magnetic field, respectively. The correspondingP being tl1e track’s mon1e11tu1n. /1 : sgn(dt,¤/Os) : —sg11(QB;), a11d and BZ being theThe 5—dime11sional pa1·ameter vector is defined by p : (R<I>,:,1},cp. h/P) at RBEA, withco111mon reference surface at RHEA : T cm, i.e. 0.8 cn1 i11side the Al beam tube (TKX).the possibility of removing an “outlier” TE. The fitted tracks (TK) are extrapolated to aof a TS list, using a helix track model with correct treatment of multiple scattering and

The next stage performs the track fit by a Kalnian filter algorithm [2, 3] on all Tlis111easurements into the subsequent track fit. This is essential for our study.using a-priori information from simulation. thus makiiig sure to enter the ‘“correct" VI)

At this stage a modification is introduced for tl1e VD: its TE is linked to the lD`s TEperformed, linking the TEs into track strings (TS).of the individual detector niodules. First, a track search (global pattern recognition) ispattern recognition and starts witl1 the simulated Tl] data as "virtiial nieasu1‘e111e11ts"As explained above, the sta1nda1`d data analysis prograni DELANA [1Ul skips the local

3.2 Track Reconstruction

obtained by a st.1‘aigl1t-li11e "fit” ofthe HG) 11c1easure1i1e11ts at Rl aud R2.and var(:) : af; var(,:) : 20%,/(R2 — R])2 and the co1·1·elatio11 p(l?<D.,:) : -1//5 areTE consists il1e1·el`o1·c of 23 para111cte1·s p : (H<1>,;,,;) at I? : Rl with var(R®) : afm,layer is siinulatecl at Rl, measiiring also z with an a.cc11ra.cy 0: : ISO ;m1. The simulatedtracks at a cIista11cc Apt;) < 1DO pm. 111 a. 111odificatio11 111adc for this study, a double-sidedm€asu1‘<2 RQ with an accuracy am, : 5 pm for single tracks resp. aha, : 30 pru for two

The VD°s two cylixxdrical layers 0f Si—mic1‘0st1·ips at 1‘a dii H1 = 8.6 cm, R2 = 11.3 cm

original weight matrix Gt': OCR Output(b) the M-c.stirmit0r”s smoothed "probability`°. formally calculated with the

being tested:(a) the Ixrafmcm filter is smoothed \i_S probability, obtained by removal of the track

2. Tests of frac/c association to this primary vertex. using the following criteria:

the influence of outlier tracks (e.g. those originating from secondary vertices).and convergence criterium IA/l(:Z:°"f) — A/l(:i:"“'*')I f 10, in order to downweight

(b) by the M—c.stimuIor method (sect. 2.2) with a constant of robustness H = 1.5

(a) by the Ifnlnzzm jilfer method (sect. 2.1), or

1. First appromimafe primary vertex fit, using all fitted tracks as input, either

module embedded within DELANA, performing the following steps:(VV.l\’1.), using the Kalman filter resp. 111-estimator algorithms of section 2. It is run as a

The program for vertex reconstruction (FV) has been developed by one of the authorsnot used as a measurement for the fit, so CSI : 0.tracking at the point of closest approa.ch in the :cy—projection. The interaction profile isaz,. : :::0 (the centre of the simulated interaction profile) and qkf as obtained from inwardThe expansion point chosen for the first approximate primary vertex Ht (see below) is

The parameters to be fitted are similarly defined as cc = (.r, y, z) and qk = (19,1,a, h/P).reversing the sign of the 5th parameter if BZ > 0.fit. The parameters a.re defined as above, except for a re—definition of fz —+ h : sgn(Q),RBEA inside the beam tube, are now regarded as “virtual measurements°° for the vertexAll parameter vectors pk and corresponding error matrices Vk (/r : 1,.. .,11). defined at

3.3 Vertex Evaluation

beam tube made of Be. and a closer third layer of the VD.)improvement is achieved since 1991 by a new DELPHI detector layout including a smallertube (1.12 % r.l.), blow up the corresponding covariance matrix elements. (However. anwith additional multiple scattering in the VD (0.48 % r.l. per layer) and in the Al beamsimulation of a layout with the ID removed [11]. Then inward error propagation, togetherthe VD°s position information, thus permitting a "zig-zag`° fit; this has been proved byand TPC, their direction and momentum information gets effectively decoupled fromimprove the resolution of ap and P`1: because of multiple scattering in the walls of IDof am, : 5 lum, might seem a paradox. In fact, this measurement can only marginally

The poor resolution in R<I> resp. the projected I/P, despite an accuracy of the VD

mean(I/P proj.) 348 nm I 320 pmmean(I/P space) 1435 inn I 1350 ,uniImpact parameter cc sample I BB sample

and their mean values are shown below:vertex position (SP), either in space or projected onto the .ry—plane. have been plotted,

For comparison, also the impact parameters of all tracks with respect t0 the true

value" is preferred in place of "test criterium" and "cut value". respectively. OCR OutputNote that in matlicmatical statistics the terminology "test statistic" and "critical

This is a. consequence of the .11-estimatorls robustifying properties.is a least-squares estimator e in (a). and that they depend much less on the event type.the theoretical value equal to rr : 0.1 than the cut value for the Kalman filter e which

It is remarkable that the cut values for the lll-estimator in (b) and (c) are closer to

I/P projected 1500 nm ) 1250 amI/P in space 1400 lam | *1000 am

0.78(d) | 0.82

0.107((2)) 0.13

0.063M-estimator (b) ) 0.087

0.018Kalman filter (a) ) 0.07

(see above) ) cc sample ) BB sampletest criterion I cut value for 0 : l0 % losses

test criteria follow from the PV plots (like Figs. 1 and 2, left-hand side):Fixing the first quantile at ci = 10 % losses, the corresponding cut values on different

be large, and L} should be low.contaminating with type 11 errors. Therefore, lat? is the power of the test, which shouldevaluate from the SV plots (like Figs. 1 and 2, right—hand side) the percentage IB of tracksthis value for each test criterium at a fixed quantile or for type I errors. we can directly(type I errors) and contamination with some “bad" SV tracks (type Il errors). Choosingprimary vertex fit will always be a compromise between loss of some “good” PV tracks

Whatever the test criterium, choosing a cut value to exclude SV tracks from the finalmade for the impact parameters with respect to the simulated PV position.

For comparison with usual track association tests, similar plots (not shown) can bein Fig. 1a-c and Fig. 2a—c for the cc and BB samples, respectively.test criteria are shown (left-hand side} for PV tracks and (rig/it-[rand side) for SV tracksidentified (for this purpose only) by simulation information. The plots for the first threeplotted separately for primary vertex (PV) and secondary vertex (SV) tracks, which aredistribution of the test criterium being investigated (step 2a resp. step 2b. c or d) can be

Starting with either the Kalman filter (step 1a) or the ;W—estimator (step lb), themeasure is needed. This may be obtained as follows;For a systematic comparison of the efficiency of different. test criteria, a quantitative

3.4 Results

steps 1 and 2 alone.are presented elsewhere [ll). Ilere. we present only results which have been obtained fromparameters and of the accuracy of reconstructing the simulated interaction profile, whichof the final primary and secondary vertex fits with respect to the simulated true vertexand secondary ircrfeirjfs by the Kalman filter method) are used for studies of the accuracy

Further steps of FV (Srcondrzry Mrlcr stare/2 by (`lll-S((1lEll`(‘ tests and Finn} prirnary

(d) the .)r[—esfimat0r`.s product rr). of the final weight corrections for one track.

final (downweighted) weight matrix Gt"(c) the 11»I—estimat0r’s smoothed vis, “probabi1ity”, formally calculated with the

10 OCR Output

complex decays of heavy-flavoured hadrons at LEP.association to the primary vertex and is particularly powerful for the identification of

ln conclusion, the 1W—estimator is shown to be the best method for testing track

the end of sect. 3.2.

improvements are to be expected with DELPHl°s new VD layout as mentioned atB f 45.1 % (16.2 %) with SV tracks for decay lengths 6 > 5 mm (10 mm). FurtherBB sample and accepting o : 10 % losses of PV tracks results in a contamination ofUsing the [VI-estimator°s smoothed Xis probability (b) as a test criterium for the

Thus, the /ll-estimator is now able to exhibit its superiority.

Kalman filter’s low cut value of 1.8% for a loss of cr = 10% (see table above).the spike in the first bin of Fig. 2a (left—hand side), and is the cause of theleast—squares is significantly distorted by outlier tracks. This is clearly seen bycascade decays. In such events the first approximate primary vertex fit bylower momenta and wider decay angles than charm mesons. and have alsoBottom mesons, because of their higher masses, have secondary vertices with

PV tracks only (test option).almost no difference when including either all tracks (standard method) or truecomparing the total smoothed probability distributions for that fit, showingM-estimator cannot give substantial improvement. This has been proven byvertex fit by least—squares is only slightly distorted by SV tracks, and thewith jet—like 2- or 3-particle decays. Therefore, a first approximate primaryCharm events have a characteristic topology of two opposite secondary vertices

kinematical differences between the two samples:

(a) in the case of cc, but are clearly superior in the case of BB. This is due to theThe rll-estimator tests (b—d) do not signilicantly differ from the Kalman filter test.

impact parameter tests, which use only part of the information available.The Kalman filter and il/I—estimator tests (a~d) are clearly superior to the usual

together with the effects of multiple scattering in the YD and Al beam tube.is caused by the ltinematical differences of the two saniples, as explained below,efficient for ci events (Fig. Ba) than for BB events (Fig. Bb) for 6 > 5 mm. ThisAn overall coinparison between the two samples shows all six test criteria. being less

Main results are:

samples, respectively.those not shown in Figs. l and 2) are presented in Figs. lla and 3h for the cc and BBcentres of these intervals are 6,. The graphs /?(6,) for six different test criteria {includingseparately for five 6-intervals with limits at 0, 0.1, 0.25, 0.5, 1 and 5 cm (not shown); theand secondary vertex, the distribution ofthe test criteriuin for SV tracks can be plotted

Since B is expected to depend strongly 011 the true distance 6 between the primary

11 OCR Output

all fitted tracks as input.

criterium l.i\/l(:i:°"t) — A/t(:ic'""`)] 3 10 (yielding on average -1 iterations), using(ib) by the robust ill!-esfrniafor method (sect. 2.2) with R : 1.5 and the convergence

impact parameter (w.r. to the centre of the beam spot) § 5 cm; or(a] by the [ralnzan ji/fer method (sect. 2.1), using fitted tracks with a projected

l. First approrévnate primary ’U€‘I‘f€;lf fit, either

above. lt performs the following steps;Our vertex evaluation program (PV) is run stand-alone with the input data described

in case of real data, and (1.3, 1. 1, 1.3, 1) in case of simulated data.le : 1 ...72. As a result, the following 5 scaling factors are used: (1.55, 1.15, 1.3, 1.55, 1.2)t.o be (0,1) normally distributed, from a subsample of "good” primary vertex fits of tracks

P2 : cov([>Q) : AkC,,A;,f + A;,EQBkf + (AkEQBk)+ B;,D]_lBkTT

(pt—i>L‘)i/t/<Vr—PZ).». i=1 ··-· 5 .

prob(yi, ndf) distribution to be flat. and the reduced smoothed residuals (“pu11s”)are scaled by factors 2 1. These have been determined by tuning carefully the total

ln order to correct for underestimated errors, the covariance matrices V], : cov(pk)at REE,] : 5 cm. These data correspond to the TKX of section 3.2.eters from DST”s perigee coordinates to our standard coordinates p = (RG), Z,19,gO,f1./P)muon chambers using standard “1oose cuts”); and transformation of the track fit paramtion; selection of hadronic events with at least two associated ui tracks (fiagged by the

The following post-DST processing [17] was done: decoding of the particle identificaThis will, of course, influence the track reconstruction efficiency.ln the case of simulated data (see below), no use was made of any a.·priori information.included in a second stage by a special DELANA module [16], and the tracks are re-fitted.the detector-local digitizings and for the global track search. The VD measurements areprogram DELANA [10] version 92-C. Standard pattern recognition was used, both for

LEP raw data of the 1992 run at the Z0 peak were processed by the full data analysisalignment errors) for single-hits resp. with 0]% :2: 150 pm for multi-hits.layers are single-sided, measuring only R<1> with an accuracy of JR.; : 11 pm (includinga better Vertex Detector (VD) with an additional closer third layer at H0 : 6.3 cm; alltube at Rm = 5.3 cm, made of Be (0.4 % r.l.) in the central region < 28.5 cm, andThe DELPH1 detector°s layout of 1992 [15] is an improved one, notably by a smaller beam

4.1 Detector Layout and Event Reconstruction

e’e+——>ZO—>bb(g). followed by a colour-suppressed decay of one tn-hadron. B——>.]/»r[·X_,.2-prong decays J/11¤—>)1.+;r". At LEP energies, these are produced predominantly by

Here, we present a study similar to that of section 3, based upon reconstructedmesons into J/rb and tj*(2.$`). Physics results of those are published elsewhere [13, 1+1].of 1992, have been performed with the aim of reconstructing “S1l()l`f·l1\`(?(.1il decays of B“°1ong-1ived°` 2-prong decays of .\° and .\° [12]. 1`ill1`l,l1(3l` studies, using real event dataevent data, taken in 1990 a.nd 1991 with the DELPH1 detector at LEP, by reconstructingA first demonstration of our vertex evaluation algorithms has been performed with real

4 Study of Real Bottom Decays into J/z1>—>;j';.4

12 OCR Output

J/t" decay.

tone of them in the mass interval defined above) which do not originate from a "true°`structed. Looking at a·priori simulation information. we find only if secondary verticesln the mass interval 3.0 < in,ff(;i+;i") < 3.2 (1eV (gray band). 837 events were recon(Fig. 1). A Gaussian fit to the J/ew peak yields a mass resolution of 0,,, = (43 5; 1.8) MeV.shows up in the effective mass rn.ff(;1+,a‘) of reconstructed 2—prong secondary vertices

After vertex evaluation FV (bundling criteria see [11]), practically no backgroundcriteria [14] gives 2,291 simulated events with J/iti—>;i+,a' decays.[10] was the same as that used for real data. Applying the standard post—DST selectionfull detector simulation DELSINI [20] for the 1992 set—up. The full data analysis DELANA(Ya. 25,000 Monte Carlo events Z°—+bh. B—»J/1pX, were generated by JETSET [19] and

4.2 Analysis of Simulated Data

given below.

Monte Carlo data. by using a-priori information from the event generator. Details arerequires a clean sample of primary iuerter tracks, which can only be obtained from

On the other hand, determination of the cut values for a fixed or-quantile (see sect.only ;z—candidate tracks with inomenta P 2 3 GeV are used.Since an efficient part.icle ideritification by the muon chambers is essential for this analysis,ing from reconstructed 2—prong secondary vertices with effective mass in the J/tp region.

Our strategy is based on regarding as secondary vertex tracks only a+ and tf originat·Further processing is done interactively with the program PAW [18].a “‘beam spot data base” or is a-priori known, for real or simulated data, respectively.fits (steps 1 and rfa). The corresponding information mo, CU : cov(¤20) is either taken fromAs an option, the interaction prohle can be included in the first and final primary vertex

finally cutting on a total prob(yi. ndf) > 1 %.

(b) secondary vertex? ft with the tracks as associated by FVBUN2,

prob(Xi_S,2) > 2.5 % (Kalman filter) or p1‘0lJ(\f_S’,2) > S % (ill-estimator),(a) jnal primary vertc.1: ji? with tracks not associated to the secondary vertex if

Final p·r1'mary and secondary vc·rte;1v fits by the Kalnian Hlter method:

authors (PK.) lor 2—prongs with (phil") ancl locatecl inside of RBEA [14].version 65), using the test criteriuni (c) from above, and optiinizecl by one oftheSecmzdctry ve1·te;1· Search by conibinatorial bunrl/ing of all tracks (subnioclule FVBUN2

((1) the M—csfmzat0r}v pr0<li1ct rrr, 0f the tinal weiglit c0r1·ecti01is lor one track.

final (clowiiweiglitecl) weight matrix Gig"(0) the M-estinmt01·’s snioothed \'Z_5' "pr0baihility". ibrnially calculated with the

original weight iieiziirix Gi":<*»’ the ;\[—r.<fir1mfr>1".< sniootiied Q5 "pi‘0|JnbiIil,y". i`<>i‘1i1ziI]y cailciilated with the

ditioii 0i` the ti‘a.ctk being tested;

(a.) the Kalmmi ji/fc·r`s siiioothcd @5 pi·0baibilit.y. 0|>tai11c¤<i by removal resp. ad

Tests 0f tmc}: as.s0c2`ati0n to this primary vertex, using as criteriai:

lil OCR Output

[type ll errors).

percentages J of secondary vertex tracks contaminating the primary vertex track samplea possible bias from the background (which is not known). this allows to calculate thesample of simulated primary vertex tracks (see sect. 4.2). are applied as before. Neglectingfive ¢$-intervals. The lower resp. upper cut values. which have been determined by a cleanand secondary vertex. As before. each of the six test criteria is plotted separately for the

The 6—interval is now determined by the distance 6 between reconstructed primaryspace) resp. f 0.l cm (projection). as in the case of simulated data.vertex. both in space and in the .r_r)—projection. are also used as test criteria. if f 2 cm (instep 2); in addition. the impact. pa.rameters with respect to the reconstructed final primarycriteria. (a) of the Kalman filter (removals only) and (b). . .(d) of the M—estimator (FVdecays is tested for association of the ni to the first approximate primary vertex by the

The sample of S7 reconstructed 2—prong secondary vertices of candidate J/Qi'->/l+/Iare found: the estimated background is calculated to be lt).5 events.In the mass interval 3.0 < m..H()¢+;1`) < 3.2 GeV (gray band), S7 .]/1;% event candidatesof 0,,,, : (—l7 ;l: 7.2) MeV. and the deviation from the PDG value [2ll is less than 0.3 am.in Pig. 6. A fit with Gaussian peak plus exponential background yields a mass resolution

The effective mass nz.jf(;i+;1`) of reconstructed 2—prong secondary vertices is showna.nd passed to our vertex evaluation program FV (bundling vers. 65) [bl].of which 28,866 events with a.t least t;wo associated ni are select.ed by st.andard criteria.The 1992 data contain (after DPLANA [l0]) 720,360 reconstruct.ed "ha.dronic" Z0 events,

4.3 Analysis of Real LEP Data

postponed to sect. 4.4.are repeated in Pigs. 5c and Bd. with the last two <$—intervals concatenated. Discussion isprofile included or not included. respectively. in the primary vertex fits). The same graphs

The graphs l3(6i) for the six test criteria are shown in Pigs. Ba and 5b (for the beamsecondary vertex tracks contaminating the primary vertex track sample (type II errors).cut values determined above are applied, thus allowing to calciilate the percentages d of

Each of the six test criteria. is now plotted separately for the five 6-intervals. Theby the Kalman filter. the M—estimator and the impact parameters, as above.at 6)). The sample of secondary vertex tracks ,4.1* is tested for primary vertex associationinto one of five intervals with limits at 0. 0.l. 0.25. 0.5. l and 5 cm (their centres beingthe true distance 6 between sinnilated primary and secondary vertex is determined to fall

Returning to the 837 reconstructed 2—prong secondary vertices <>f`.]/1p——>;¢+;F decays:o : l0 % losses (type I errors). These six cut values will be used further on.criteria (a). . .(d) or impact. para.inet.ers. respectively). which will result in a fixed quantilecan proceed as in sect. 3.4: determining a corresponding lower or upper cut. value (for

Plotting each test criterium for the basic sample of "good" primary vertex tracks. we2 cm (in space) resp. § 0.l cm (project ion).

primary vertex. both in space and in the .ry—projection. are also used as test criteria if §(PV step 2): in addition. the impact parameters with respect to the reconstructed finalthe criteria. (a) of the Kalman filter (removals only) and (b). . .(d) of the M-estimator

mation. This sample is tested for association to the first approximate primary vertex byabove. "true`” primary vertex tracks can be identified by using a-priori simulation infor

Regarding the reconstructed primary vertices of the sample of 837 events defined

14 OCR Output

vertex fits.

again: this is, surprisingly. also the case when including the beam profile into the primaryBut the gap between Kalman filter (a) and M-estimator (bed) has become very significantag. for the ill-estimator at g3 z 35 % (with beam spot) resp. #15 (Zi (without beam spot) .significantly worse. while showing some "saturation effect" for decay lengths 6 > 0.25 cm.of the test criteria. effect of inclusion of the beam profile, etc. All test criteria behave

The general features are nevertheless consistent with those of simulated data: ranking

have been concatenated now.

are definitely much bigger than before. Note that the first two bins and the last two binslot of background, and it has a much smaller statistics; therefore, the errors on the graphs

For real data (Fig. T), two points have to be kept in mind: the data sample contains a

additional improvement.file has a "robustifying” effect by itself. Nevertheless, the ilI—estimator can still give andistortions caused by multiple outlier tracks. In other words, inclusion of the beam pro

This last effect is exactly as expected: inclusion of the beam profile counterbalancesand the gap between the Kalman filter and the M-estimator has narrowed.look at Fig. 5c shows the splitting among the il{—estimator’s test criteria has disappeared;all test criteria have improved over the whole range of decay lengths, as expected. A closer

The effect of using the beam profile shows up in a comparison with Fig. 5a or Fig. 5c:range 0.1 < 6 < 1 cm.The splitting of the last two, which is a new feature, is significant for decay lengths in the(a) worse than rlJ—estimator”s product—of—weights (d) worse than il/I—estimator’s X2 (b—c).the last two bins (Fig. 5d), the ranking becomes more significant again: Kalman filterhas become smaller, due to the fact that both have improved. But when concatenatingworse than Kalman filter (a) worse than M-estimator (b—d); the gap beteen the latter two

The ranking of the test criteria has remained the same: projected impact parameter(see sect. 4.1).compensates a possibly worse track reconstruction efficiency caused by pattern recognition

This is due to the very clean sample of selected J / ip decays, the effect of which overnow significantly better, with lower B over the whole range of decay lengths.

rameter’s B is about the same. But the Kalman filter’s and M—estmator’s test criteria aresimulation of B—>.]/zL¤—>;4+p" without beam profile (Fig. 5b): the projected impact pa

Comparing the earlier simulation of BB (Fig. 3b) with the corresponding presenthas been made for the VD set-up of 1992 (only R<I>, no z measured).being double—sided with a z—accuracy of 03 = 30 pm, see sect. 3.1); no such modification

This is, of course, due to our virtual modification of the VD set—up of 1990 (inner layerpercentage of contamination )3 > 90 (Zi even for big decay lengths.exception: the impact parameter in space is no longer a good test criterium. giving aare also consistent with those of the earlier simulation study of sect. 3 (Fig. 3), with oneThe results of this study (Figs. 5, 7) are consistent with each other. Its general features

4.4 Discussion

statistics, the errors are much larger than before.two and last two 6—intervals concatenated. Note, however, that due to the rather smallprofile included or not included, respectively, in the primary vertex fits), with the first

The graphs ,B(6,) for the six test criteria are shown in Figs. Ya and 7b (for the beam

15 OCR Output

support from the Austrian Ministry of Science and Research.Analysis Team for the data used in sect. 4. One author (P.K.) gratefully acknowledgesThe authors wish to thank D. Liko for many useful discussions. and the DELPHI Physics

Acknowledgements

gram Library.

VERA ("Vertex Evaluation by Robust Algoritluns"), to be submitted to the (]P(l Proby its author (VV.M.) with the aim of a <letector—independent general-purpose package,

The vertex reconstruction program FV (presented in sect. 3.3 and 4.1) is being recodednot to take advantage of the capabilities of the M—estimator.real LEP data (see sect. 4) is increased by less than 1U %. There is therefore no reasonthe event. As an example, the CPU time spent. by the vertex evaluation program PV forvertex fit proper is usually only a tiny proportion of the total time required to analyzethan tl1e Kalman filter. This is, however, not a serious problem. as the time spent in thewhich is a great benefit in case of many tracks. Thus. on average the M-estimator is slowercan be performed separately on the (5 >< 5) weight matrices Gt. Ar : l ...11. (see sect. 2.2),iterations required until convergence. At startup. the extra requirement of diagonalization

The computational load of the il]-estimator is directly proportional to the number ofpronounced for B B—events.

type; for instance, our simulation study shows that it is negligible for cc-events, but quitehowever, be noted that the improvement due to the il»I—estimator depends on the eventreal data. Tests based on impact parameters turn out not to be competitive. It should,the test statistic based on the least-squares estimator, both with simulated and withthe test statistics based on the 1\»I—estimator give consistently better performance thanfrom the fitted primary vertex. leading to some sort of \2—statistic. \Ve have shown that

The separation of secondary tracks proceeds by measuring the distance of each trackKalman filter or global least-squares estimator.least-squares estimator, thereby requiring only minimal modifications to the traditionalthe influence of outlying secondary tracks. lt can be implemented as an iterated weightedwe have investigated the rU—estima1or which achieves robustificatiou by downweightingwords, the estimation of the primary vertex position has to be robustified. In this notean estimate of the primary vertex position which has as little bias as possible. ln otherThe detection of secondary tracks and eventually of secondary decay vertices requires

5 Suinmary and Outlook

feasible.

the beam profile, however, adds to the power of the test and should be used wheneverthe primary vertex, even in the "dirty" environment of complex real data. Inclusion ofvalues, has again been proven to be the best criterium for testing track associa.tion to

In conclusion, the /1/I—estimator’s smoothed X2, although distorted towards smaller

16 OCR Output

DIQLPHI 92-118/PROG-189/Rev.3. Geneva 1994.\r'. Perevozcliikov and N. Smirnov; PIIDST Package Description Ilser°s Manuals.1171

gram ofthe Vertex Detector. DELPHI 89-49/PROG-139/TRACl{-52. Geneva 1989.C. Troncon. M. Caccia and A. Zalewska: VDANA — The Pattern Recognition Pro[16}

be submitted to Nuc1.Instr.Met.li. A.

P. Abreu et al. (DELPHI Collaboration): Performance of the DELPHI Detector. ToU5]

Thesis. Comenius University. Bratislava 1995.

P. Kubinec: The Search for Secondary Vertices in the DELPHI Experiment. PhDU4]

with J/tb and $(25) at LEP. IIEPHY—PUB-625/95. Vienna. 1995.P. Kubinec. D. Liko and Wl. Mitaroff: Exclusive Reconstruction of Secondary Vertices[13]

J. Urban), Kosice 1992.

and A0/AO. Proc. Int. Conf. Hadron Structure 92, Stara Lesna (ed. D. Bruncko andP. Kubinec. D. Liko and W. Mitarofl`: DELPHI Vertex Evaluation Studies Using Kg{12]

(HEPHY—PUB—555/91).Experiment am LEP—Speicherring. PhD Thesis, Univ. ofTeclmo1ogy, Vienna 1991VV. Mitarofl`: Zur Bestimmung der Zerfallsvertices schwerer Teilchen im DELPHI—1111

DELPH1 89-44/PROG—137. Geneva 1989.A. Baroncelli et al.: DELPHI Data Analysis Program (DELANA) User’s Guide.[10]

PHI 87-27/PROG—72/Rev, Geneva 1988.J. Cuevas et al.: Fast Simulation for DEI.PH1(PZA.STSIM) Reference Manual. DEL[9]

P. Aarnio et a.l. (DELPH1 Collaboration): Nuc1.Instr.Met.h. A303(1991)‘233.[8]

Program Library Pool VV5035 etc., Geneva 1987.T. Sjostrand et al.: JETSET Vers. 6.3 — The Lund Monte Carlo Programs. CERN[7]

1990 (ISBN 0306-43371-0).Islands), July 1988 (ed. T. Ferbel). NATO ASI Series B204. Plenum Press, New YorkStudy Inst. on Techniques and Concepts in High Energy Physics. St. Croix (VirginM. Regler and R. Friihwirtliz Reconstruction of Charged Tracks. Proc. Bm Advanced[6]

1970.

A. Jazwinski: Stochastic Processes and Filtering Theoijv. Academic Press, New York[5]

P.J. Huber: Robust Statistics. Wilev. New York 1981.[4]

R. Friihwirtli: Nncl.Instr.Metl1. A2G2(1987)14~1.[3]

P. Billoir. R. Eriiliwirth a.nd M. Regler: Nll('l.ll1S11`.l\1(°Il1. A2l1(l985)1l5.[2}

NICOLE. CERN 73-2. Geneva 1973.M. Metcalf. M. Regler and C. Brollz A Split Field Magnet. Geometry Fit Program[1}

References

IT OCR Output

D5U(1994)1lT3.

L. Nlontanet et al. (Particle Data Group): Review Ol Partirrle Properties. Phys.Rev.[21]

DELPHI 89-67/PROG—l=l2 and DELPHI 89-68/PROG-143. Geneva 1989.A. de Angelis et al.: DELSIM ee DELPHI Event Generation and Detvector Sinnilation.[20]

TH-7112/93/Rev and ("ERN Program Library P00l \V5035/\\'5U—‘l¤l, Geneva l99~1.T. Sjéstrand: PYTIIIA X/ers. 5. 7 and JETSET Vers. 7.4 ae Pl2_x·.<1<·s and Manual. CERN[L9]

station. CERN Program Library Long VVriteup (2121. Geneva 1995.R. Brun, O. Couet, C. Vandoni and P. Zanarini: PALV — Physics Analysis VVOrl<[ls]

18 OCR Output

Figure 1: Distribution of test. critcaria for primary and secondary vortex tracks (cc)

c) zl[—estimato1·, yi_S "probability” with Hnal Gt,

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

20 20

40

out cutl 40 l60 60

PV tracks SV tracks

80 80

b) A/Lcstimator, XQS “probability” with original G;]

0 0.2 0.4 0.6 0.8 1 O 0.2 0.4 0.6 0.8 1

out

X l30cut

m3060 60

90 90

PV tracks SV tracks

120 120

a) Kalman Hlter, smoothed Xfj probability

0 0.2 0.4 0.6 0.8 1 O 0.2 0.4 0.6 0.8 1

ut ut

r30% T3060 60

90 90

SV tracksPV tracks

120120

l9 OCR Output

Figure 2: Distribution of test criteria for primary and secondary vertex tracks (BB)

C) ;lI—estimato1‘. Xzj "p1‘obability” with final G1.,

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

100cut 100 lCut

200 200

SV tracksPV tracks

300 300

b) AI-estimator, xfs “probability” with original Gkl

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

100 100cut cut

200 200

300 300

PV tracks SV tracks

400 400

a) Kalman filter, smoothed XZ5 probability

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

100 100

200 200

cut300 300

400

SV tracksPV tracks ` 400cut

500 500

20 OCR Output

Figure 4: Simulated data — distribution of the effective mass (pin`) / GeV

2.2 2.4 2.6 2.R 35 2 J 4 3.6 J H 4 4.2

"lv_L_J_

100

I50

200

250

300 (groyi J/w moss region)

(a) cc sample, (b) BB sampleFigt11·e 3: Percentage ef secondary vertex tracks contaminating the primary vertex

True vertex crstance [cm] True vertex drstance [cm]

0.5 1 1 5 2 2.5 3 3.5 4 4.5 5 0 5 ¤ 1.5 2 2.5 3 3 5 4 4 5 5

ts Impact parameter (projection) ¢.~ Impact parameter (prcycctron)

0 tmgaci paramerer (space) 0 Impact parameter (space)WL F 10··I M-e$hmal0r(d) • M-es1rmal¤r(d)

V M-estimator (c) V M-estimator tc)

A M·esIima\or(b; A M-estimator (b)20%

C KaIman1Uter(a) O Kalman tilter (a)20;30*

v x \ x \ x x ¤40*

x ` xgx xw _\\ \ X \ \ •

50»· \ x§\\ x\ x50k

60+ t \ \ x

` V ° L x t~\ \ \\ \ \ \ \

TQT.70%\ \

c.: r \ ` . lr \ t5 P\ ~\

>\’ E E p X R »§ . \ \E _OE ao — 9 ‘~ \. E so —_ \S i ‘ \ ` a5 *\

h ei • WZ y x - I x5 9O4···--_ 5 90 ‘ M s Y { \ >

r \

aa W w °“ .>E * Q `

t `

v¤‘|0O .,, 100 ,(B7 Snmulated c·cbar » no beam pro used (¤) Stmulated B~Bbar - no beam pro. used

'Zl OCR Output

(simulated data) — (a,c) beam profile used, (b,d) not usedFigure 5: Percentage of secondary vertex tracks contaminating the primary vertex

True vertex drsiance (cm] True venex cttstance lcml

C U 5 \ 1 5 2 2.5 3 3 5 4 4.5 5 0 0.5 1 t 5 2 2 5 3 3 5 A 4 5 5O————·——-——·——-——·——·

1D— 10—

\

» IeZO— 20

f`~" r Qcr \ k \ \

307Y\ 30_ " \ "l l Ofw

dO—"\l

*‘—2 >\l_ L"\sow 5°#U

2 impact par, (pro;.) c Impact par. (pro;.)

0 Impact par. (space) 0 Impact par (space)e0l—‘ lll

I M·esumator (d) ,\\ `T `O M·esnmatcr (d)l1s_ l

V M-estimator (c)’°i1 V M-esumator (1:)7¤L`1l*A M-estimator (D) A M-estimator (b)l p?O Kalman Utter (a) O Kalman filter (a)

V'S BO*‘ p‘·’ 0011*E CQ r. 5. M E lE I. g .1:1 go?

' §9(,L~- F i 0F

j:1¤01 ;,100gs r ?

g 7g I `

.,,1$3——·——·———·— $110j1::) Snmulated J/w - beam pmhle used (d) Srrnulated .1/01 - no beam pro used

True vertex dxszanoe [cm] True vertex drstance {cm]

0 0 s 1 1.5 2 2 s a as 4 4.5 s 0 0.5 1 15 2 2.5 e :.5 1 45 5

10 %L xi 10 — !

20 r- 1 *20 }¢ \•\

Rr l \

i ‘*` r M· 30 L vw so k Y1

[ 1 , ~\ V l` l \~ L l —do g s` ` \ T v 1 AD (ll

lmP ` r X `¤\so V = ” i 1 50 (- ( ‘

t \ \ l `c- !mpaclpar.(pr0).) c rmpactpar (pr¤llrx \ l ` ‘

0 Impact par. (space) 0 Impact par. (space)1 \ \ l =~~ ` ~Q \t l 50 ‘

• Mestimator (d) • M-estimalcr(¤)

. V M-est1maI0r(c) V M-estimaloru:)70 H" lQ v 7gA

A M-estima10r(b) A M»estnmaIcr (b)

5 Kalman filter (a)O 59 » O Kalman filter (a)a » , 6 r 1 B0 Q

c • \gi * `é t *1

E T.

E su d E 90; _

.. 1*

F 5 4; wu L 100

s r5 r3 1V, 170 · *2 , u1 110 (

{a) $»mu\ated J/w · beam prohte used (b) Sxmufated J/V - no beam pro used

22 OCR Output

(real data) — (a) beam profile used, (b) not usedFigure T: Percentage of secondary vertex tracks contaminating the primary vertex

Rec v€’1e¤ 21$\a"iCE 'C"Pac wma; ¤ stems ‘cm*

O O 5 X T 5 2 2 5 3 3 5 4 4 5 5O 0 S 1 15 2 2 5 3 3 5 4 4 5 5

c Impacl par (pwlr}¢ lmpactpar (proj.}

6 impact par. (space)O Impact par (space]¤c>>70..

I M-eslimanor (G)• M—esumar¤r (d)

V M-estimator (c)V M-estimator (C)

A M»es\ima\¤r (D)zeA M·es&imat¤r (b)20

O Ka!manhl1er(a)C Kalman finer ta)

a0—

' ~'-—- { i~40~

tu;

[ Mr so? \SOL qi t’ ip { ~»\

iY`¤60·—¤· iim; m . tx;i\‘\i LiiisiNv .»w1 \\% \`

-. ·¤L\ L"\¤ `_ Lgw " vu ,2 aa L`i;*\&H

80-:, aé P B0r·\>FxF - \ §

3 so— "0gg¤‘# E"’

L r, {` `_ § `

2 `°‘—' W- i C? _"émontE700-ats FiE L,,,i1¤~ fg Au»110—°£ .

ll:) Beal daxa ph — no beam pro. useu(ai Real dare p'u` ~ beam profile usac

Figure 6: Reed data ~ distribution ofthe efiectivc mass {;a+;2") / GeV

z.2 2.4 2.6 2.8 1 3.2 :.4 3.6 3.s 4 4.2

(grcyl J/W moss region)