Topological vertex search in collider experiments

Nuclear Instruments and Methods in Physics Research A 337 (1993) 66-94 North-Holland

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A

Topological vertex search in collider experiments Gerhard Lutz

Max-Planck-Institut für Physik, München, Germany Received 21 September 1992 and in revised from 18 May 1993

A general method for vertex topology analysis suitable for charged and neutral tracks inside of a homogeneous magnetic field is described . It is thus applicable to most collider experiments which usually apply a homogeneous magnetic field oriented parallel to the beam axis for momentum measurement of charged particles . Two vertex fitting procedures have been developed, a complete fit constraining tracks to a common vertex and a fast fit which gives only the vertex but not the constrained tracks. The momentum sum, mass and incident track parameters of secondary tracks belonging to a common vertex are calculated . Some suggestions for applying the analysis methods to specific tasks are given .

1. Introduction The growing interest in heavy flavour physics together with the possibility of very precise particle trajectory measurement with the help of position sensitive semiconductor detectors has led to the desire of analysing complicated short range decay topologies . As an ideal one would like to uniquely assign particle trajectories to the in general multi-step decay vertices. Due to the short lifetime of the heavy flavour particles and the limited precision of the backwards extrapolated tracks many ambiguities in the assignment will remain . Selection of only one assignment by e.g. chi-square criteria will often lead to wrong results . Therefore it was the aim of this method to retain all possible track-vertex assignments and to select those being compatible with a specific physics channel only afterwards. Analysing the vertex topology in such a general way requires the investigation of a large number of track combinations . Therefore a rather fast method for fitting a vertex to a given set of tracks has been developed as well as a efficient strategy for handling the combinatorial system . Furthermore for tackling the physics analysis a set of modular tools have been developed which may be combined in an appropriate manner so as to select a specific channel . The problem of track and vertex fitting is a rather old one, dating back to the analysis of bubble chamber events . However published literature directly applicable to the specific questions of heavy flavour analysis in collider experiments is rather scarce . General considerations of this problem may be found in refs. [1,2] and references quoted therein . Contrary to other published vertex fitting procedures [3,4] no experiment dependent approximations are done in this paper. The formalism is exact when starting with proper track parameters and errors . The strongly experiment dependent multiple scattering properties are assumed to have been taken into account in the track parameters. The methods described in this paper have originally been developed for the NA32 fixed target experiment at CERN #` where the charm decay topology was investigated in a field free region . Later on they have been modified and developed much further for application in the ALEPH experiment at LEP #2 . This development contains the adaptation of the formalism to the case of a homogeneous magnetic field in which charged tracks are represented as helices as well as the extension of the formalism to neutral tracks . Furthermore kinematic fits have been developed which give as result e.g. the invariant mass of particles combined to a vertex and the incident track parameter as well as all error #t ACCMOR : Amsterdam - Bristol - CERN - Cracow - Munich - Rutherford - Valencia Collaboration . #2 ALEPH Collaboration . 0168-9002/93/$06.00 © 1993 - Elsevier Science Publishers B .V. All rights reserved

G. Lutz / Topological vertex search in collider experiments

67

correlations . These modular tools can easily be adapted to select and to investigate specific physics channels of interest . In the main part of this paper the choice representation of physical quantities (chapter 2) and the basic fitting methods (chapters 3-7) are described. The following chapter contains suggestions on the application of the program. 2. Representation of physical quantities The specific track and vertex representations have been chosen in view of the cylindrical symmetry of colliding beam experiments in which the interaction point is close to the origin of a cartesian coordinate system with the z-axis parallel to the homogeneous magnetic field. The charged and neutral track representations are closely related in order to keep the mathematics rather similar. 2.1. Charged particle track representation Charged particles trajectories inside a homogeneous magnetic field are helices. Their representation is chosen similar to the one used in the TASSO experiment [3] however with an important change in the sign convention which allows a unique mathematical treatment for all sign combinations of helix parameters . All helix parameters are defined in the vicinity of the coordinate origin where highest numerical precision for vertexing is required . The helix is described by five parameters (compare fig. 1): - p = l1r signed positive for counterclockwise bending, negative for clockwise bending. r is the radius of the helix projection into the x-y plane. - r = 8z/8s vertical slope. - 0 0 (0 < 0 0 < 2 ,rr) projected particle direction at closest approach of projected circle to origin . - D o projected distance of projected circle from origin, signed positive if particle has a positive angular momentum around the origin at this point, negative otherwise. - zo z-coordinate at point of closest approach of projected circle to the origin x = y = 0. A point P on the helix is defined by these parameters and the projected distance s along this circle measured from the point of closest approach in the projection from the origin . The sign convention illustrated in fig. 2 leads to unique mathematical formulations valid for all four possible sign combinations of p and Do .

Fig . 1 . Helix representation : projection of the helix onto the x-y plane .

68

G. Lutz / Topological oertex search in collider experiments

p>O D,,<0 Fig. 2. Helix sign convention .

2.2. Neutral track representation The parameters describing the straight trajectories of neutral particles have been chosen as close as possible to those of helices in order to unify the formalism as much as possible: - p momentum signed positive. - r = 8z/8s vertical slope. - (P O (0 < 0 tß < 2 ,rr) projected direction . - D o projected distance of projected track from origin, signed positive if particle has a positive angular momentum around the origin, negative otherwise . - z, z-coordinate at point of closest approach of projected track to origin . A point P on the neutral track is defined by these parameters and the projected distance s along the track measured from the point of closest approach in the x-y projection. 2.3. Notation Throughout the paper we will use vector respectively matrix notation as far as possible . The following notation of basic quantities will be used Helix h = (h l , . . ., h 5 ) T, 0 0 , D o , z ) helix parameters,

Do
i

Fig. 3. Neutral track representation and sign convention .

G . Lutz / Topological vertex search in collider experiments

69

H complete variance matrix Hi,i = (Ohj Ah; ), d transverse distance of a point to the helix d = (dl, d2) = (d d,), D transverse distance variance matrix Di,j = (Adi A dj ), n h number of helices . Neutral track n = (n,_ . ' n5 ) = (p, -r, (P O , Do, zo), N complete variance matrix N ,j = (OnjAnj), d transverse distance of a point to the helix d = (dl , d2) _ (dr , d,), D transverse variance matrix Di,j _ ( i1di0 dj ), n number of neutral tracks . Vertex u = (u ,, u2, U 3 ) = (L' X , uy , vZ) vertex coordinates, V variance matrix V,,i = (Dui Dud ) . Momentum p = (pl, p2, p 3) = (P X , p, pZ) momentum components, P momentum variance matrix Pi,j = (0 pj Api ) . Mass m mass of particle. In many instances the distinction between charged and neutral tracks will be avoided . Then the following notation will be used: Tracks t = (t, . 0 track parameters, T corresponding complete 5 x 5 variance matrix Ti,j = (i1 ti0 t; ), d transverse distance of a point to the track d = (dl , d2) = (d d,), D transverse distance variance matrix Di,j _ (OdiAdi), n t = n h + n  number of tracks . In some cases we will have input (measured) and output (fitted) track or vertex parameters of the same notation simultaneously . Then the input parameters will be marked with a tilde (~) . 3. Constraint fit of tracks to a common vertex The basic task in doing a topological vertex search is the identification of tracks belonging to a specific decay . This requires the fit of tracks to a common vertex . Furthermore one usually wants to know the invariant mass of the decay products as well as the incoming track parameters and all error correlations . Due to the many tracks involved an extremely large number of track combinations will have to be tested. Most of these combinations will fail and it is not necessary to have complete information on most tries. Therefore two methods of vertex fitting have been developed . One which makes a refit of all tracks to a common vertex and provides complete information on the decay while the second fast version only looks for a point which is close to all tracks considered simultaneously. While for the complete fit the computer time rises with the square of the number of tracks involved, the fast fitting method shows only a linear increase with nt. In this chapter the general fit is described while the fast fitting method is defered to the following chapter . The track parameters are varied according to their measurement precision so as to be constrained to a common vertex . In order to treat this problem in more generality, the task will be expanded to a common fit of charged and neutral tracks as well as other vertices - which might have been found previously by combining a subset of tracks - to a new common vertex . Thus the formalism may also be applied to some other common tasks as e.g. checking if a track may be attached to a vertex or if two vertices are really, separated . There are 3n v + 5n h + 5n input parameters for nv vertices, nh helices, and n  neutral tracks . Two constraints for every fitted track requiring passage to the fitted vertex reduce the number of independent

70

C . Lutz / Topological vertex search in collider experiments

Fig . 4 . Dependent and independent helix parameters .

parameters for each helix or neutral track to three . With the three common vertex coordinates one thus obtains 3 + 3n h + 3n  = 3 + 3n t free parameters to fit. We choose the following independent parameters: The common vertex: u = (v  U2 , v 3 ) = (L'x , v y , vZ ) vertex coordinates, For each helix h q = (tj = r - D o , T, (P O ), For each neutral track n q = (p, T, 0 o ), This set of o f = 3(1 + n h + n n ) = 3(1 + n t ) independent parameters leads to simple relations between standard track parameters and independent parameters which may be immediately read from the geometrical picture shown in fig . 4: Helix x, = -vl sin 0,, r=

Y '~ = rl COS

sign(

(L'1-XC)2+(1'2-YC)2

sin e= (v,

-

vl),

D o =r-'n,

Cos'e=

x c )/r,

S=r( ~P -00),

00 ,

-(v2-Y,)/r

ZO =v3-ST .

Neutral tracks Do = U 1

sin ~P, 1 - v 2 Cos (Po, ep + v 2 sin eO ,

S = v 1 COS

Z O = v 3 - ST .

Fitting is done by minimizing a chi-square which is composed of terms describing the difference between fitted and measured parameters (vertices, helices and neutral tracks). n nv h ii 2

n

n V,

y =1

a=1

= E axv + E R iih + E yXn a=1

ß=1

y aiiv +

n,

y

/c=1

KXt

The splitting up of the total chi-square in this form is possible due to the absence of error correlations between different tracks. In the rightmost formulation we have used a uniform notation for charged and neutral tracks . This praxis will be followed whenever possible, but keep in mind that the track parameters have a different meaning for these two species . Explicitly written in matrix form the contribution of input vertices and tracks to chi-square have the

G. Lutz / Topological vertex search in collider experiments

71

following form : a Ta V_1 a~,2 _=a j (il v V) (V

) T RH _1 ~ R h-Rh) RXh= (oh-Oh YXn =

(In -Yn)TVN_i(rn -Yn),

KXI =(Kt-KL)TKT_1(Kt-Kt) .

Oh'

1 h and Kr are the input vertices and tracks, v, Oh, Yn and xt the fitted vertex and tracks . In order to minimize the total chi-square we have to calculate the first and second derivatives with respect to the independent fit parameters . This requires the knowledge of the derivatives of the helix and parameters . Fortunately most of the derivatives neutral track parameters with respect to the in will be zero, we therefore write down only the non-zero ones - again omitting the track index. For helices one finds (omitting track index and helix subscript for easier readability)

"v,

a P __ _ au,

ah, au, ah, aqi ah 2 aq 2

=

ap D71_ aT

= aT

P 2 sin (P, -P2 cos(o - 00),

ah, av 2

aP _ -P2 av_

2

cos 0,

ah,a P _ _ P2 q = azo a q3

= 1,

ah a a0 o -= a(ho aq3 ah 4 aD o

= 1,

ah 4 aD o ar av 2 av2 av 2 au, av, au,

ar

ah 4

aD o

ar

aq,

a7j

a?,

ah 4 aq3

=

ah s

aD o

av,

av, az o

ah s

av 2

=

ar

sin 0, -

a"2

ah s

az o

av 3

av 3

ah s

az o

ah s

az o

-coS 0,

-1=cos(O-(Po)-1,

a 0o a 0o az o

sin(e - 0o),

=vl

sin(O-eo),

= -r[sin (P(~-eo) +cos 0], =T[cos 0(0-0o) -sin 0],

-=-=1 , aq, = a?) = - T[(O

a q2 ah 5 aq3

=

aT az o

-0o) cos(O -0 o) - sin( (h -0 o) ],

= -s,

--171[(e -0 o) sin(O - eo) +cos(O - 0o)] - r} . ao _

72

G. Lutz / Topological vertex search in collider experiments

For the neutral tracks one obtains similarly an,

ap

an 2

aq,

ap

aq 2

-=-=1,

a7

=-=1,

an 3

a(P o

aq 3

a(P0

-=

aT

=1,

an an 4 aD o 4 au,

aUl

a

aD o

an 4

az 0

-_

Du 1

au=

an s

cos

= U1

q3 = a0o

an s

aZ ()

aU3 aU 3 az o

aq 2

aT

an s

aZ ()

aq3

a-Po

tP o + U 2

cos Po,

-T

1

an s

8D O -COs eo, = aL2 aU 2 -

=sin 0o,

sin

= s,

-P o

an s

az o

aU 2

aL_ '2

_ -T

sin iP.,

1, cos

= -[VI

=4'[U 1

sin

00+U2

00-U2

sin

-s,

eo] =

cos 00]

=TDO .

For the derivatives of the chi-square terms with respect to the standard vertex and track parameters one obtains in the unified notation for charged and neutral tracks aaX 2 aUi a2 a

3

2.

=

j=1

2

Xv

a

a

V j 1( Uj

- a U-j ) ,

-1

=2 V,j

av i avj

a2 KX~

=2K

aKtiaKtj

Tt,j

1.

First and second derivatives with respect to the independent parameters À = (À1, ii2, . .-Anf) 1 1 1 'q3 2 , "'~hnh g3 ,ng1 , "' , nn q3) _ (L1 , U2, U 3 , hgl , hg2 , h~hq1

L1, U 2 , U3, ql~

1

21

'q

3,

2 ql, . . ., ntg3)

are obtained from these eqs. (8) and the derivatives of eqs. (6) and (7) of the standard track parameters with respect to the independent parameters : a

nv

2

X aak

=

E

a= 1

aa

2 Xv

aUk 3

- 2

a=1 j=1

nt

+

5

~~

K-1 =1

aKX2 aKti a

_ _ _ Vk,j ~ Uj aUj)

K

t i au k t

+2

5

5_

K=1i=1j=1

T,jl

(K

tj

_

_aK ti KT j) ôvk ,

G. Lutz / Topological vertex search in collider experiments a2X 2

n,,

avkavl

=l

a=1

_ ~ 5 a Kx' i=1

5

a2X 2

avk

aKt i a , tj

au,

1: 1:

aK ti _ 1/Ktl KT ,1 1 \ - K tj) aK '1k

5

aKta2 Kxt

aKt

j

i=1 j~--J,1 aKgk aKtiaKtj a K gl 5

5

aKti

2 i=1 j=1 a2 X

aK tj

a Kgk

5

aKgk aKgl

a 2KXi

a K ti

aKt i

5

+

Vk,l

a Kt .

5 aKt 5_ aKti . l _KT_ 2 Yi,j alli K=l i=1 j=1 aUk nt

E

a Kgk

5

av k av l + K=1 i=1 j ~1

n

aX 2

5

nt

a2axv

73

5

2

all k a'gl

5

5

aKgk

aKt

i=1 j=1

K

,

- l aKt . T`,' aKgl

a 2KX1

al) k

a K tj

aKt;aKtj aKg l

a

5

aKt_ aKt . t KT, l t,j aKgl i-1 j=1 vk

2 y_

It may be worth mentioning that in calculating the second derivatives of chi-square terms of the form a2t .

J _1 l / T,! av k ar, l '

K t L - K-\/K t

i.e . those containing second derivatives of the fitted track parameters with respect to the independent parameters have been neglected. This is justified for a good fit where the measured and fitted track parameters are rather close to each other. It is also advisable in case of bad vertex candidates since inclusion of these terms may lead to non-positiveness of matrices and corresponding breakdown of the iterative fitting procedure. This may also be the case for good track combinations if the initial vertex is chosen far off the true value . First and second derivatives with respect to the independent fit parameters can be written in matrix form in the same order as the free parameters A . They then have the structure a ax e ax e , . . .a~ a~ 1 of

a 2x 2

av k av l

av k a 1g t

a x

2

avka 1g1

. ..

2 2 a x ~ aUk

an

tgl

e

- ( aU 1 '

a2x 2

2

G=

ax

2

a x

a' a ' ax e 2

. . .

ax

e 3

ax e a1gl

a2x 2

, . . .,

x

e

aKgk

, . . .,

x

e

), a n g3

1

av k antg l

2

al a lgk gl

0

0

...

o

o

0 0

a2x 2 ant gk an ~gl

i.e . cross terms between tracks and vertex but not between different tracks.

(I0)

G. Lutz / Topological vertex search in collider experiments

74

The deviation of the independent free parameters from the chi-square minimum is estimated as

sÀ = G - ' g .

(11)

A= iG -1

gives the variance matrix of the o f fit parameters A . It may be used to extract the estimated errors of the vertex as well as the standard helix and neutral track parameters, and their correlations, respectively . For the vertex this is especially simple as it has been chosen as the first three independent parameters. Correspondingly one has vk,l -

( OvkOL' [) - ZGk,I

k, 1 = 1, 2, 3 .

,

(12)

Formally this may also be written in matrix notation as

V=2FvG -1 Fv

F,,=

with

1 0 0 0

0 1 0 0

0 0 1 0

10

0

0)

(13)

The standard helix and neutral track parameters may be recalculated with the formulas given previously in eqs . (1) and (2) . The errors of the track parameters are obtained with the relations ~

K

3 tk -

i-1

aKt k

OKgi K a qi

+

3

aKt k

i=7

aUi

Ov i .

(14)

Correspondingly one finds the variance matrix for fitted helices and neutral tracks as 3 3 aKt k aKt _ t KTk,t - \~ Kt k Gm +I,IYL+j aK Ktll 2 aKq[ 1-1 j-1 qj

A

1

aKt aKt aKt k t + Gm+i,j av a Kq i avj

+

aKt k au I

Gi,m+j

t

+ K a qj

aKt k i

_

G "j1

aKtt j

(15)

with m=3 +(K - 1)3=3K . Similar expressions can easily be written down for error correlations between track parameters and vertex coordinates . Besides vertex and refitted track parameters one is usually interested in the mass and the track parameters of the incoming decaying particle. This question will be taken up again in chapter 5 after an excursion to an alternative fast fitting procedure described in the following chapter . 4. Fast vertex fit without refit of tracks For large numbers of tracks the vertex fitting method described in the previous chapter is rather computer-time consuming. It provides refitted parameters of all tracks, quantities which very often are of little interest, especially when one only tries to find out which tracks can be combined to originate from a common vertex . Therefore an alternative fast method has been developed to find the vertex only but leave the track parameters unchanged . There one tries to find a point whose weighted distance sum from the measured tracks is minimal . The weighting is done in such a way as to lead for the vertex to identical results as the elaborate procedure described in the previous chapter .

G . Lutz / Topological vertex search in collider experiments

75

0, Y<

Fig . 5 . Distance of point (A, B, C) from a helix.

In the following we first find the distance of a point from a helix or a neutral track before writing down the X Z to be minimized for finding the vertex . 4.1. Distance of a point from a helix The following important quantities may be calculated from the helix parameters and a point (A, B, C) as can be read off fig . 5: - (xo , y o) coordinates of closest approach in x-y production to the origin and (x c , yc) center of projected circle x o = D o sin 1~ o, x, =xo - r sin 0 O,

yo = -Do cos 0o, Y, =yo + r cos eo .

(16)

(B -

(17)

- Distance in x-y projection d, and in z-direction dZ of point (A, B, C) from helix, corresponding point at helix and several related quantities . 1= sign( r)V (A - XC)Z sin e= (A -x,,)11, x=x,+(A-x,,)r/l,

s=(e-tP o )r, d,=(l-r),

+

cos

YC)2 ,

e= -(B-yJ/1,

y=y,+(B-yJr/l,

O d Z =C-ZO -s-r . Z=Z +s7,

(1s)

The following derivatives are later on of interest : ad,

sin 4P, M = adz _ -T cos (P r/l, aA ai sin dP, aA = ax '~

-sin 0,

ad,

-cos 0,

aB ad, aB = --r sin (P r/l, ai aB

=

-cos

e,

- = cos (P, ay,

ad, ac = 0, adZ = 1, ac

(19)

76

G. Lutz / Topological vertex search in collider experiments

ad, ar

al axc al aye -+---1 ax, ar aye ar

= sin 0 sin 00 + cos 0 cos 00 - 1 = cos(O - 00 ) -

ad, a~o ad, aDo

al ax~ al ayC =(r-Do) sin(P-0o), ax ~ a~ o + a Y, a0o al ax~ al ayc - cos(Oax, aDo ay~ aDo

ad, ad, 0, a-r - 0 ' az o ad, 'd z 'd z as fr 0 r -s' lsin(P - Ou)], ~o aT az -1' ar -T ar -TL 0 adz r r-Do ad z r = _ Tlsin(e- ~P o) . cos(-~o)], aq5o -Tr [1 l Do 4.2.

Distance of a point from a neutral track

The following important quantities may be calculated from the neutral track parameters and a point (A, B, C) : - (x o , y o ) coordinates of closest approach in projection to the origin x o = Do sin eo,

Yo =

-Do cos

4P 0 .

(20)

- Distance in r-O projection S, and in z-direction S Z of point (A, B, C) from neutral track and corresponding point at track and several related quantities .

=A cos 0. + B sin ~P o, x=xo +s cos Oo, Y=Yo+s sin (Po, z=z o +s7, d,=A sin iPo-B cos e,-D ., dZ =C-z o -sr . s

Fig . 6 . Distance of point (A, B, C) from neutral track .

(21) (22)

G. Lutz / Topological oertex search in collider experiments

77

The following derivatives are later on of interest: ad,

ad, sin - -cos ~P o, 0o, aA = aB ad, ad z - T cos ~o, aB = -,r sin 0 . 1 M ad, =A cos fo + B sin ~o = s, azo ad,

adr

aDO

-1,

ad,

aT =

aC ad z

= 0,

a C =

ad,

0,

az o =

1,

0,

(23)

ad,

-S,

aT

ad,

azo = -1,

ad,

'dz

=T(A sin 0o-B cos I~ o), a-Po

aD 0 -0.

4.3. "Transverse" error matrix As mentioned previously we attempt to fit a vertex by taking the distance from the tracks as a measure . The transverse distance in x-y projection and the corresponding difference in z between the vertex and the "closest" (in projection) point on the track will be taken as the relevant quantities . We therefore have the task of finding the 2 X 2 variance matrix for these quantities from the 5 X 5 variance matrix of the tracks . This matrix of course is dependent on the position along the track . The 2 X 2 variance matrix D may be calculated from the complete 5 X 5 helix variance matrix H by with A the 2

=

X

5 matrix for the helix

ad,

=- P2~COS(P-00)-11,

ad, A1,3= a0= (r - Do) sin(O 0 A,,4=

adr ad

= aPZ

A 2,3

=

ad, ado

(25)

- cos((P-0o),

aDO0

A 2,1

-0 o),

=

Tr 2 ~~ - ~ 0

=

Tr ~ 1 -

-

r-D j

r lsin(O - (Po)],

l o cos(O --Po)Il

ad A2,2= aT

-S,

1

aD r ad A 2,4 = = -T - sin(0 - 0o) , A2,5= aDO0 ôzZo and all other terms being zero. Similarly one obtains for neutral tracks D = ANAT,

(26)

78

G. Lutz / Topological vertex search in collider experiments

with A the 2

x

5 matrix

ad r ad

r

a(Po -S,

-1,

aDo

ad z ad z ad -T(A -S' A2,2 = A2,a= aT a~ 0

sin 0 0 -B cos

eo),

(27)

Z A2 .5=

az

-1,

0

and all other terms being zero. 4.4. Vertex fit The vertex coordinates may be found by the standard procedure of minimizing a chi-square function which we write down as a function of three terms representing the input vertices, the helices and neutral tracks. h

nn X2 -

E

aX v +

a=l

n

E

RXh +

p=I

n

YXn =

Y =I

n,

E

aXv +

«=1

K=I

(28)

KXt .

The splitting up of the total chi-square in this form is possible due to the absence of error correlations between different tracks . Explicitly written in matrix form the individual terms of chi-square have the following form:

RXh =ßdh ßD-IRdh, Y

Xn

K

2=K

Xt

Y D_ I

=Ydn

TK dt

(29)

Ydn

D - IKd t ,

with d = (d r , d z ) given in eqs . (18) and (22) for charged and neutral tracks respectively . Following the standard procedure for minimizing the expression for chi-square with respect to the vertex coordinates v = (A, B, C) we calculate first and second derivatives of chi-square with respect to these three parameters : 2

a

n

aa

nh

2

Xv

= E

2

8, ôk - X + ~, aU k aR a=1 aU k 1i = 1 i-1 n°

aaxv

a1 G>

Gk,l

_ -

aUk

a 2X 2

n, +

_

2

a2«

2 aRd . Xh di

a

KXr

aKd i

a

di

au k

=1 i= n

aß

n°

aUk

2

a Y x 2 a Ydl

I j~

ayd i

aU k

(30)

,

XI

aU k aU l - a-1 avkaVI nh

2

2

. a R di aPd1

a 2Rx 2

~

E + )s = 1 i aOdiaßdj aU 3U ß j n,.

a1

a2aX V

n,

2

81)k 2

aVl

a2KX_

aU k aU 1 + =1 i-1 a"diaKdj j=

nn

2

2

a2Y

2 Xn

+ Y-1 i-1 j-1 ayd i aydj aKd i aKd j aUk

aUl

.

aYd k

i

a Yd . l

G. Lutz / Topological vertex search in collider experiments

79

The expressions for g and G may also be written in matrix form as n.

g=

aV-1( U - aU )

ZL a=1 ( n"

aV_1 +

G = 2I y La=1

n,

B=

I

ad,

av,

av2

av3

ad,

ad,

ad,

au,

av2

aV31

-Cos 0

sin (P COS

0

R - LT sin O

-cos ~P O B - ( sin 00 -T

(31)

K=1

ad,

R

KBTKD

K=1

ad,

- LT

n,

KBTKD_1KBI,

with

B=

+

cos ~P O

-T

0

sin iP

0 1

(32) for helices,

for neutral tracks,

as is found with the definition of d (eq. (18) for helices and eq . (22) for neutral tracks). The displacement of the assumed vertex from the position of the X Z minimum is obtained by the relation 0U = G -1g

(33)

It is used for calculation of the next iteration of the vertex The variance matrix for the fitted vertex is given by V = -fl G-1 .

(35)

4.5. Finding of the vertex starting value In order to save computer time it is advisable to have a starting value not too far off the true vertex. Being close to the true value is even more important in some special cases when one has more than one local minimum in the chi-square distribution and the result of the iterative procedure depends on the starting value. An obvious example for this situation is the vertex of two helices with similar vertical slope T . In this case the two crossing points of the projected circles are determined and the one giving smaller separation in z is selected . This selection procedure has been generalized for nt tracks by applying the same procedure to several pairs of tracks out of the sample and requiring consistency of the results. The method of obtaining the vertex starting value is also available for the general vertex fit described in chapter 3 . It is however advisable to use the result of the fast fit as input for the complete fit. 5. Momenta and invariant mass at vertex Having in chapter 3 performed a fit of charged and neutral tracks to a common vertex one may be interested in the invariant mass of the system . In order to get this information one has first to determine

80

G. Lutz / Topological uertex search in collider experiments

Fig. 7 . Relation between helix parameters and particle momentum .

the momenta of all particles at the vertex . For doing proper error treatment one furthermore needs the derivatives with respect to vertex coordinates and independent track parameters. 5.1. Momenta of refitted tracks The relation between helix parameters vertex and particle momentum can be read from fig . 7: PT = KIp = Kr,

pz = PTT = KrT, cos O = Kr cos -P, y =PT sin (P= Kr sin ~P , P x =PTP

(36)

where K is the proportionality constant relating momentum and radius of the helix r. It is negative for positive charge and magnetic field aligned along the positive z-direction . The radius r and angle 0 are functions of the vertex coordinates v = (v,, v,, v 3) and independent helix parameters hq = (7l, T, 00 ) as given in eqs . (1) . The derivatives with respect to vertex coordinates and independent helix parameters follow from simple algebra with the help of eqs . (1) . Only the non zero terms are listed below: ab, 2

apx

ahql

=

apx 377

= K cos 00 ,

apx ahgs

apx = -Krl sin 0,, aoo

av, ap y

ap y.

ahgi

arl

= K sin

apy __ apy = Kvl cos fi , a(po ahg3

apz ap z = KT sin e, = -KT cos e, au, ôv 2 apza ap z pz apz =KT cos( - ~o) , Kr = PT, =-a hq , a hg2 aT a77 apz _ apz =Krl sin((P-0o) . arho 303

(37)

G. Lutz / Topological vertex search in collider experiments

81

The equivalent relations for the neutral tracks obtained directly from the independent track parameters n q = (p, T, 0,,) are given below: P

PT =

Px

PZ = PTT

1+T2

(38)

P y = PT sin e0,

=

PT COS 00~

apx

apx

Px

apx

apx

angl

ap

P

aq2

aT

aPx

aPx

ang3

a00

= -P y

ap y

ap y

Py

ap y

ap y

a n gl

ap

a n g2

aT

apZ

_ apZ _

ang2

a-r

ap y

apy

=

ang3

a-po

apZ

apZ

= P ' = Px

P

x

,r

1+7

Py

+ ,r2 , 1-

(39)

,

PZ

ap a n gl =

-

P '

PT 1 + ,r 2 .

5.2. Momentum sum In order to calculate the invariant mass of the tracks forming the (decay) vertex we will attempt to calculate the four momentum sum vector, respectively three momentum sum vector ps = (pX, Py, Pz) and energy ES 

nn

Ps =

ß=1

ES =

nh

13=1

ßPh +

y=1 n

OE h +

y=1

n, YPn -

K=1

KPt

(40)

n,

_' E. __

Y, KE, .

K=1

Forgetting for the moment the distinction between neutral and charged tracks, which only enter through the different relation between dependent and independent track parameters one may write down the same relation in component form n,

(41)

Pi _ 1: KPi K=1

(i = 1, 2, 3 representing the x, y and z components) and obtains for the corresponding error 3

Api

nc

r- r= k=1 K=1

aKP~ aVk

n DUk+

3

1, 1:

K=1

k=1

aKP i

AKgk " K a gk

(42)

As in chapter 3 we define the vector representing the independent fit parameters as  = ( 1 , À2, . . .,

Ànf)

2 nh 1 .' ng 1 1 1 _ (i)1, U2 , U3 , h gl , h g2 , h g3 , h gl, "' , h g3,n gl, . n 3) 1 1 2 n, g1, " . ., q3) . = (U1 I 02, U3, gl , g2 3, , 'q

43

82

G. Lutz / Topological vertex search in collider experiments

The equation relating the momentum sum error with the errors on the independent track parameters may then be written in matrix representation as Ap' =

A

(44)

A AFP,

with n,

n`

aKp1 , avt

K~ n,

K~ n,

aKp, aU 2

,

aU,

ape

api

aA 1

aA,

aA,

api

ap2

FP =

api aA n

=l

al

Pl

aU 2

K=1 aU3 a]

P2

P3

a'p,

a 1P2

a 1P3

a lg2

a'q2

a'q2

a 1P 1

a 1P 2

a 1 P3

a 'q3

a 'q3

a lg3

aKp 1

aKp2

aKp3

akq,

a kq,

a k q,

an,

ani p3

a n,P I antg3

3

K a p3

aU3

K=1

aui aK

a Kp2

K=1 aU3 api

n`

aU 2

1

aKp 3

K~ ,

aKp2

aKPl

al

n,

aKp2

P2

ant

an , g3

(45)

g3

The variance matrix for the momentum sum thus may be written as Ps

Pl sk =(OpsApk) i, k=1, 2, 3

with

='FpTG - 'FP

(46)

the variance between vertex position and momentum sum similarly may be written as C Vp

= iF'G - 'Fp ,

with Fv and \ C ~P)i,j

Fp

(47)

defined in eqs . (13) and (45). Writing out this equation in individual terms one finds

= (AUiApj

-

3

E

1

2

n,

~

k=1 K=1 3

nt

k=1 K=1

Gik

1

aa pj

aUk

1

nc

+ 2 Y_

3

Y_

K=1 k=1

aKP aU k

nt

1

Gi,3K+k 3

K=1 k=1

aKPi aKgk aKp . a qk

(48)

83

G. Lutz / Topological vertex search in collider experiments 5.3 . Invariant mass

The invariant mass may be calculated from the momentum sum four vector 3

psz mz - Esz _ ps 2 - Esz E

(49)

with nc p i'= Y- KPi, K=1

n,

Es =

3

K=1

i=1

KPz +Kmz

aEs a"Pi

"Pi "E

(50)

The error on the mass is found with the derivatives with respect to the independent parameters : aEs

api) OKPi -P~ a a Pi Pi K=1 i=1 aK nt y 3 EsK 3 aK P KP` + Ov k Pi Pi' ( -2 E ( a qk k=1 avk KE K=1 i=1

Omz = 2mOm =

3

c

2( ES

0"qk) .

(51)

Here we have made use of the fact that the standard track parameters "pi are only dependent on the vertex v k and the proper three independent track parameters "qk. With the definition Y. (E8 Kpi -p ;) K=1 i=1 nr

3

K =1i=1 3

F, -

a aPi

K a KPi P, Es K - PiS ) av2 E

ES KPi - p5 a"pi K ) av3 E K=1 i=1 3 a 1Pi 1Pi (E s - _P s l i 'E l l a q1 =1 l 3

=1

3 i=1

one may write

(52)

1 Pi _Ps a 1Pi Es-'E a 1qz

Esn`Pi n`E

_

s

an`Pi

P` ) an `g3

1

) (53) 2mZ Fi G -1 Fm (Omz

Similarly one obtains for the correlation between vertex and mass Crv,n -

1

2m

Fv G-'Fm

with F defined in eq . (13).

(54)

84

G. Lutz / Topological uertex search in collider experiments

5.4. Momentum sum and invariant mass for fast vertex fit As the complete vertex fit is time consuming - especially when refitting a large number of tracks to a common vertex - it may be desirable to get slightly less precise values and estimated errors but with substantially less demands on computing time. This can be achieved by performing the simplified vertex fit and taking the momenta from the track directions at the point of closest approach to the vertex in x-y projection. A certain difficulty arises in the error estimations as the errors of the fitted vertex are correlated with the track parameter errors . We will neglect this correlation and consider the relevant quantities as function of the 5 x n t track parameters and the three vertex coordinates . As has been checked by numerical comparison with the exact procedure described in the previous section, the approximation generally leads to very small changes . The relation between momentum and track parameters is given in eq. (36) for charged and in (38) for neutral tracks . p and T are now the measured track parameters (curvature and vertical slope), while (b is the projected track direction of the helix at point of closest approach to the vertex obtained from eqs . (16) and (17). Starting from eqs . (50) and (49) calculation of momentum sum and mass is straightforward and it remains to estimate the errors. In order to do so, the derivatives of the track momentum components with respect to the measured track parameters and to the vertex coordinates are needed . A straightforward calculation leads for helices (again omitting helix index and subscript for easier readability) to

apX

aqi apX

apX

= ap = -PTr { cos 6 + jr [cos ap,

Do

1- r [sin ~o + cos ~ sin(O - 0,)], aqs 1 _apx __ ap, = PTl [-cos 0 o +cos(b cos( -~o) ], aqa aD o 1 apX ap, _ - = -p,- sin 0, au, aA 1 =

_apX __ ap, av}; aB

= PT

cos 0 cos(O - 0o)] ,

1

-py l sin ~b,

apy apy r r sin + [sin (b, - sin cß cos( - 0o)] ~, -PTr! aq, = ap l apy apy D -r -= [-cos 0 o +sin sin(¢-~O )], = PT () a0 0 1 aq3 apy apy 1 = PTl [ - sin 0. + sin cß cos( - 00)], aqa 8DO 1 ap, ap, cos ~, =Px avX aA l ap, au, apz

1 ap, cos ~, = pyaB

= apz = -PTr7, aqi ap ap, ap, _ -PT, aT aqz

with all other derivatives being zero.

(55)

G. Lutz / Topological vertex search in collider experiments

85

For neutral tracks one finds similarly aPx aqi

= a = Pp , ap x

x

p

apx

apx

aq2

a-r

apx

apx

aq3

a0o

ap y aql

=

ap y

apy

ap

=

_

apZ

=

ap Z

,

py

p

py 1

aT

aq 3

aq ,

ap y

1 + TZ

= -py,

apy

aq2

T

-Px

T +

(56) T2 ,

ap y px -= (~ a o

ap = p , apZ

pZ

ap Z

1 PT _f_ + 7.2 ,

aT aq 2 -

with all other derivatives being zero . Momentum sum and error are similarly to eqs.

dpi

K=1 n,

=

pc

K=1 ~ k=1 5 aa Kgk

OKgk

(41)

and

(42)

given as

(57)

+r

aUk k=1 3 aKp.

OU k ,

3 representing the x, y and z components) Assuming absence of error correlation between track parameters of different tracks and vertex one obtains (i = 1, 2,

nt s 5 aKp, aKp 3 3 a Kp. aKpi K K (DUk Av t i ~. (OpiApi) - ~ (O gkA gli + Y- 1: I: aKgk aU ai k t a Kgt ) k=1 l=1 K=1 ~ k=1 t=1

(58)

The invariant mass is given by eq. (49) and its error are obtained similarly to eq. (51) as Am'

nt 3 = 2mAm = y_ Y_

aE s 2 ES K a P~ K=1 i=1 ~

=2

nt

3

K=1 i=1

(

ESKPi KE

5 _ PS)[ Y k=1

api

- pt Kp a K a KPt a gk

)

OKPi

O Kgk +

3

aKp . ~ wk . k=1 aU k J

(59)

86

G. Lutz / Topological vertex search in collider experiments

With the previous assumption of absence of error correlation between track parameters and vertex one obtains 1

((Om)2 ) -

m

Z

n,

3

3

K=1 i=1 j=1

(

E'K P! K

E

-P! )

3 3 a Kp, a + ~ k=1 r=1

aU'k

(OL'kOUt)

5

~ k=1 1=1

-P . '

aL

5

1

a KP, a

K

q

qk

a

K

a

qt

E"P . K

(60)

E

6. Track parameters from momentum sum and vertex In the previous chapter the momentum sum from tracks originating from a common vertex has been found . Assuming that these particles stem from the decay of a long lived parent particle one may want to know its track parameters. This e.g. is the case for a K ° where the incoming particle has a straight trajectory or for a D + where it is a helix. 6.1. Neutral tracks The neutral (incoming) track parameters (n1 , n2 , n 3 , n4 , ns ) _ (P, from fig . 8a: IIPx + Pv Py sin 0 0 = -,

PT =

cos 00

PT

D0 = vx sin 4)0 - vy cos PZ T= - , PT

Do, zo) can be read directly

+PZ

P = VPT

1

T, 4~0,

=

PX -,

PT

s = vz cos (ß, + vy sin 00, z o =v Z

-ST .

0

Fig . 8 . Calculation of track parameters from particle momentum and vertex for (a) neutral and (b) charged tracks .

(61)

87

G. Lutz / Topological vertex search in collider experiments

In order to find the variance matrix of the track parameters we calculate the partial derivatives of the (incoming) neutral track parameters with respect to momentum sum and vertex aP = P y _ -P, apy 8 ,r 7- sin ~ , apy PT cos 0 azo o , PT apy aDo - -cos 00, avy aDo a(Po

aP Pl _ =_ P , apx aT _ _ r cos (b,) _ aPx

PT

_azo _ sin 0 0 , PT apx aDo = sin 0o, OUx aDo

azo =s-, apx apx azo - - T cos 00, avx azo

acß0 aT =Dar- - s ap, apx ap,

sin

PT

(62)

aDo

=0, apz azo = 1, avZ

0o,

azv acß0 aT -=D,T - -sapy apy apy

The partial derivatives of three track parameters (p, zero .

apz

=0 , apz aDo = 0, avZ

apy

-T

apz

Pz -P

aibo

=S-,

apy azo -= avy

ap

T

azv

s

ap.

PT

and 00) with respect to the vertex cordinates are

6.2. Charged tracks

The helix parameters of the (incoming) charged track (h h 2 , h 3 , h 4 , h s ) _ (p, similarly be found from fig. 8b: PT = Px + pv , Py sin (ß = - , PT

K r = -- PT sign(B), x,=vX -r sin tß, ,17 = xC +YC sign(r), xc sin cßo = - -, 77 S=r((ß-`bo),

T,

00, Do, zo ) can be

pz T= PT COS

p

=

Px (ß = - , 1

r,

PT

Y c =vy +r cos Do =r-77, COs

Y,

(ßQ = -, 77

ZO =Uz-ST .

01

(63)

88

G. Lutz / Topological vertex search in collider experiments

In order to find the variance matrix of the track parameters we calculate the partial derivatives

ar apx a.~

ar

PT

apv a (~

sin 0

= -

OP, ax,

PT ,

a y,

a y,

r

=-, apx PT

a x -_ c -- sin(~o), 77 aux r

-cos PT

ax,. aa, = 0,

-0,

PT

,

aPZ

0, a v, =

a y,

= 0,

a77

aPZ

apy

=

r

PT

= 0,

D77

y,

-=-= cos(0o), auy 'ri -

= 0,

ay,

= l'

apy

a7j

C~ o,

= 0, aPZ ao =0 , apz

cos cß PT ,

aPy ay, aUy

apx

=

-- -

au, = 0'

ar

PT

Uy

= 0,

a,q

r

-sin

OP, ax,

-1,

avx apx ay,

r

-cos

auz =0, aq

sin

apz

= 0,

and the partial derivatives of the helix parameters with respect to momentum sum and vertex

al/r

cos cß

apx a-r

'PT T cos

apx a

PT cos

au-,

7

'

r sin ~o

apx aDo

nPT

aux

= sin 0.

aDo

r

ap x

PT

azo aux az

(cos 0 - cos (b (,),

-= Px PT

apy ai-

MT

aPZ a-r

T sin (b

apy a.

PT sin

a(ßo

r cos 0,

apy aDO

77PT

aDO ap y

1

'

' '

- -COs 01, =

r PT

(sin (b - sin (b a),

rr azo - _ --sin 0o, au y 77

77

(sin

al/r

aUy

rr -cos cßo,

rT

sin (b

ay

'

a(b o

al Ir

r

--sin ~o), 71

r -r az r -- --(cos ¢--cos 4b1)), py PT 77

aPZ a avZ

1 PT = 0,

a'b t) aPZ aDO auz aDO,

0, (64)

= 0, = 0,

apz az(, = 1, av, az

a° pZ - -

r (~ PT

O, )

~

,

G . Lutz / Topological vertex search in collider experiments

89

Defining now the matrices at,

at,

au x

av y

au,

aux

av y

avZ

at3

at3

ai),

ai)y

avZ

at4

at4

at4

aux

ai)y

avZ

at5

at5

at5

avx

ai)y

avZ

ar l

at,

at,

apx ate

apy ate

apz at e

apx at3

ap y at3

apz at3

apx

apy at4

apz at4

apx at5

apy at5

apz at5

apx

ap y

apz

ate

SP =

at,

at4

ate

ate at3

(65)

(66)

with (vx, vy , vZ) the vertex, (px, py, p,) the momentum sum and the partial derivatives of the track parameters given by eqs . (62) and (64) for neutral or charged (incoming) track respectively, one may find the variance matrix for the (momentum sum) track parameters as T = SVVSV + SPPSSP + 2S vCv,pSP .

(67)

The variances V, Ps of vertex, momentum sum and the correlation term C ,P are given in eqs . (13), (46) and (47). 7. Adding of constraints to the vertex fit The desire to implement constraints to the vertex fit occurs rather frequently. A very common example is the reconstruction of K° mesons in which the constraint of the invariant mass to the nominal mass improves the precision of vertex and incoming track parameters. A more interesting example out of heavy flavour physics in which the method described below has been successfully employed [5] is the decay of the BS meson into qj' and (~ mesons with sucessive prompt decays into w+ w - and K + K-. Here one may want to constrain the w+ [,- mass to the nominal %~' and the K+ K- to the nominal 4 mass. The standard Lagrangian multiplier method is difficult to implement in our formalism in which we need analytical expressions for first and second derivatives of chi-square, the second derivative being

90

G. Lutz / Topological vertex search in collider experiments

positive definite in order to find the chi-square minimum in an iterative procedure. A simple trick allows an easy way around this problem . A penalty function is added in order to enforce the correct correlations between the "independent" fit parameters. The chi-square used for checking the quality of the fit after convergence is calculated without the penalty function while the estimated errors of the fit parameters respectively other dependent quantities are calculated from the variance matrix obtained with the penalty term. In order to illustrate the method we add a mass constraint of a specified subset of tracks belonging to the vertex :

to the chi-square expression defined in eq. (3) . m 2 is the invariant mass square given in eq. (49), m o the nominal mass and 0-mz an arbitrarily small constant which may be considered to represent the natural width of the resonance or simply as a constant determining the severity with which the constraint is enforced . Too small a choice for this quantity may lead to numerical problems due to limited computer precision . First and second derivatives of the penalty term Xm are easily calculated with eqs . (49)-(51) with the implicit assumption of skipping over all tracks not contained in the subset: 3 s" 2) n, m2 - M a "Pi m0) ~ ~ ( E Pi S =4 ( av k Q z "E -p ` l avk "=1i=1 Il (m2 -mô) 3 ES"p i a"Pi aXm S -4 ( " Pi ) a " , Q22 E aa K i q 2

aXm

a 2Xm2

_

au k au l

8

~ ~ E_

n,

Om2

3

n,

8 __ _

3

0 z i=1

a"gkaÀqi

"E

1 i-1

3 8 a 2Xm aua k a aq1 - z 0 _m 2 "-1 i-1 ~

a 2 Xm

"Pi

E

ES

s"

"

p

S

-P,

"P,

E

"E

- Pl., s

_Pi

a

n,

a "pi

a Uk x-l a "P, 3 ~ avk

"p

a"qk

,

(68)

r

'

E Àpj AE

3_

j-1

E SÀ

E (

j-1

E

3 _ ~ Es~pt. j-1

aE

P~

_

S

P'

~ ô aPj

a ÀPt

au,

S ~ - pj) a qi '

_PS

,

(69)

a~Pj a Àgi .

That this method gives correct results is demonstrated in fig . 9 where the result of a Monte Carlo simulation of events of the type e + e - - BS X

, LI_L+A_

L>K+K

and consequent reconstruction with the standard ALEPH software are given . Introduction of the mass constraint for the 0' - w + p, - decay reduces the width of the reconstructed BS mass by a factor three, in agreement with the expectation from the error calculation . Constraining in addition the (D mass gives no additional improvement, as the measurement precision of the (D is better than the natural width. 8. Vertex topology reconstruction Interest in vertex topology construction has centered on the field of heavy flavour analysis. In this type of physics several closely spaced vertices - partially from cascade decays - should be recognized and

G. Lutz / Topological vertex search in collider experiments

91

20 Ez

w w

10

40

20

5.00

5 .25

5 .50

MASS [GeV]

^ 5.75 I

I 6.00

Fig. 9. Invariant mass of the B, meson calculated from the reconstructed decay tracks w+ W - K + K- (top). Constraining the W+ w - invariant mass to the nominal t~' mass (bottom) reduces the width of the reconstructed Bs mass from 27 to 8.5 MeV.

separated from each other . Due to the limited measuring precision this in many cases will not be possible and for a large fraction of tracks usually there will not exist a unique assignment to single vertex from topological criteria alone . Therefore one usually will have to apply additional criteria originating from physics prejudice as e .g . assigning to a secondary vertex tracks identified as leptons or to sorting tracks according to some jet algorithm . The software based on topological criteria only has been written in the spirit of allowing easy introduction of these additional physics criteria while keeping all remaining ambiguities . In order to remain efficient in this task of considering a large number of combinations an elaborate system of flag handling tools has been introduced . Each track has assigned to it three markers describing : - The possible crossing with all other tracks . - The compatibility with possible particle assignment . - The identity of a possible parent particle . These flags are extensively used in most tasks some of which are sketched below.

8.1 . Particle identity assignment Particle identification usually involves a combination of information coming from different parts of the detector. In general only probability statements can be made and ambiguities will remain . In order to be able to handle these ambiguities, a marker word has been assigned to each track . For each possible particle identity assignment with a probability above a set minimum value one bit in the marker word is being set . 8.2 . Track selection In order to reduce the combinatoric task it is useful to restrict the tracks to be considered by rejecting right from the start uninteresting tracks . These may be e .g . badly measured tracks, sections of a spiralling track or, depending on the physics interest, low momentum tracks. The track indices of a subset of all reconstructed tracks are determined in this task .

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G. Lutz / Topological vertex search in collider experiments

8.3. Reconstruction ofphoton conversions All pairs of oppositely charged tracks with track markers having the electron bit set are investigated for: - Tighter requirements on the electron probability . - Formation of a vertex . The fast vertex fitting procedure is used for this purpose - The vertex being outside the beam vacuum - The invariant mass of the pair is required to be below a set maximum value . For this purpose the complete vertex fitting procedure is done again, providing also the mass and the incident photon track . - The incident photon track is required to be compatible with the primary vertex (respectively the beam crossing point in the collider) . This check is done by performing a fast vertex fit using one vertex (beam crossing) and one neutral track (photon) as input . - Rejection of vertices with too many additional tracks . This check is done in order to reject secondary interactions . A loop over all other tracks is performed, calling the fast fitting routine with one input vertex and one additional track . If any of the checks is unsuccessful the specific track pair is abandoned from further consideration . If all checks are successful the photon is added to the available neutral tracks and the bit corresponding to a photon is set in the particle origin marker of the two electrons . 8.4. Reconstruction of Ko 's and A's All pairs of oppositely charged tracks with track markers compatible with Tr + rr -, pTr - or rr + p assignment are investigated for: - Formation of a vertex . The fast vertex fitting procedure is used for this purpose - The vertex being separated from the primary vertex or beam crossing. The fast vertex fitting procedure with two input vertices is used for this purpose . - The invariant mass of the pair is required to be within a given number of standard deviations from the nominal value. Only mass assignments corresponding to K0 , A and A, compatible with the decay particle identification markers, are tested . For this purpose the complete vertex fitting procedure is done again, providing also the invariant mass values and the incident Vo track . - The incident Vo track is required to be not too far of the primary vertex (respectively the beam crossing point in the collider) . This check is done by trying a fast vertex fit using one vertex (beam crossing) and one neutral track as input. - Rejection of vertices with too many additional tracks . This check is done in order to reject secondary interactions . A loop over all other tracks is performed, calling the fast fitting routine with one input vertex and one additional track . If all invariant mass checks or any of the other checks are unsuccessful the specific track pair is abandoned from further consideration . If at least one invariant mass value is accepted and all other checks are successful the incoming track is added to the available neutral tracks and bits corresponding to all successful assignments are set in the particle origin marker of the two decay tracks . 8.5. General vertex search In the case of photon conversion and Vo reconstruction the multiplicity of the vertex has been known beforehand . Thus one was able to apply rather straightforward the vertex fitting procedures described earlier in the text. In the more general case this is not so easy and a procedure has been developed with the aim of finding all track combinations which may be coming from a common vertex . The strategy of this procedure is as follows : - For the given set of tracks a vertex fit for all pairs of tracks is tried and a marker is generated for each track which contains for all successful fits a bit corresponding to the second track . Simultaneously the number of "crossings" with other tracks is saved for each track .

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All tracks which have no crossing (respectively have too few crossings if a minimum multiplicity was specified) are removed from the set and the markers are modified correspondingly - Try to find a vertex with the highest multiplicity. The multiplicity starting value is determined from the highest "crossing" multiplicity found in sufficient number of tracks (and from the maximum multiplicity if specified by the user) . All combinations of tracks with the required multiplicity are investigated. One may obtain more than one good vertex perhaps even with some tracks in common. - Repeat the procedure with multiplicity reduced by one. Track combinations which are contained already in a higher multiplicity vertex are skipped over. This step is repeated until multiplicity two (or a minimum specified by the user) is reached . Provisions have been taken to allow the introduction of some physics prejudices into the task of general vertex finding . One may require : - The inclusion of specific tracks or a minimum number of tracks of one set (respectively minimum numbers of several sets). This task is performed by generation of markers for the sets and requiring a minimum of coinciding bits with the marker containing the track combination to be tested for formation of a vertex. - Exclusion of specific track combinations -

8.6. Primary vertex reconstruction The primary vertex can be reconstructed from: - Beam crossing as given by the machine parameters of the collider. - Reconstructed charged tracks . - Neutral tracks for which the track position in the vicinity of the primary vertex is reasonably well determined as is the case for kaons lambdas and converted photons . In the most simple approach a common vertex of all these objects is searched for. This in general will be unsuccessful as there will be badly reconstructed tracks present and also genuinely missing tracks from short range secondary decays. The first problem can easily be dealt with by using the general fitting procedure described in the last chapter and selecting the highest multiplicity vertex with acceptable chi-square, respectively in case of ambiguities selecting the lowest chi-square vertex. This method however will often lead to distorted results in case of heavy flavour decays as quite frequently the multiplicity of tracks originating from the primary vertex is lower than those of the heavy flavour decays. In addition these tracks often have low momentum and due to multiple scattering give little position information when traced back to the primary vertex . An alternative method suggested for this case would be a separation of available tracks into jets, identification of tracks which are incompatible with coming from the beam crossing region and trying to produce a vertex out of the tracks belonging to the jet . The incident track of the jet may then be included into the primary vertex search. 8.7 Search for secondary vertices This topic is the most interesting and simultaneously most difficult one. The strategy one has to apply depends very strongly on the physics interest (aim) . If one wants to select a single or few decays of some heavy flavour particle as e.g. for a DS - 4)-rr ± decay a suitable strategy would be: - Loop over all pairs of oppositely charged tracks which are compatible with kaon signature. Require small vertex chi-square and (D mass. - Combine the incident q) track with all charged tracks which are compatible with pion assignment . Require small vertex chi-square, separation of vertex from primary vertex and look at invariant mass. All these tasks can be easily done with the elementary building blocks described in chapters 3-5. If one wants to perform a general search for new resonances with a priory undefined decay topology another strategy may be appropriate . One may want to select events which are likely to contain heavy

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flavour decays and save preliminary event topology information which can be used later on for a fast preselection when looking for specific channels . One strategy which selects event with secondary decays is described below: - Selection of tracks with momentum above a minimum value which are displaced from the primary vertex by a minimum chi-square distance . - Search for possible secondary vertices using the set of tracks selected in the previous step. The highest multiplicity vertex is selected . In case of ambiguities the one with lowest chi-square is chosen . Tracks belonging to the chosen vertex are removed from the sample and the procedure is repeated with the reduced samples until no further vertices can be formed . - Search for vertices of left over single displaced tracks with other (non displaced) tracks . - Investigation of ambiguities in track to vertex assignment . One tries to attach each track (above a minimum momentum) to each of the vertices and tables containing the results including the ambiguities are generated . 9. Summary A general tool for vertex topology analysis has been developed . It is extensively being used in the analysis of heavy flavour physics in to ALEPH experiment and has become part of the standard software of this experiment. Emphasis in this description has been given to the mathematical formalisms of fitting procedures which have been presented to some detail so as to allow easy transfer to other users. Complete or fast vertex fitting (returning only partial information) can be chosen . No experiment specific or other hidden simplifications have been made . Therefore the formalism is exact for any location of the vertex. The strategies of physics analysis have only been indicated . They strongly vary with the physics under consideration . However the tools have been written in sufficiently modular form so as to be easily adaptable to specific physics channels . Acknowledgements I have profited from many discussions with colleagues from the NA32 and ALEPH experiments . Special thanks belong to Martine Bosman who not only adapted the software to the ALEPH programming standard but also did the tedious task of implementing the program package into the general ALEPH software system, thereby correcting numerous mistakes and weaknesses . I am also indebted to J . Lauber, W. Manner, H.G. Moser, V. Sharma and F. Weber, the initial users in ALEPH who contributed with suggestions, constructive criticism and some programming effort to the topology analysis program package, and to G. Wolf who spotted a mathematical mistake . References [1]

P. Billoir, R. Friihwirth and M. Regler, Nucl . Instr. and Meth . A241 (1985) 115. [2] R.K . Bock, H. Grote, D. Notz and M. Regler, Data Analysis Techniques for High-Energy Physics Experiments (Cambridge University Press, 1990). [3] D.H. Saxon, Nucl . Instr. and Meth . A 234 (1985) 258. [4] E. Calligarich, R. Dolfini, M. Genoni and A. Rotondi, Nucl . Instr. and Meth . A 311 (1992) 251 . [5] D. Buskulic et al ., Phys . Lett . B 311 (1993) 425.