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ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742
Contents lists available at ScienceDirect
Nuclear Instruments and Methods inPhysics Research A
0168-90
doi:10.1
� Corr
E-m
journal homepage: www.elsevier.com/locate/nima
Generalization of the Gluckstern formulas II: Multiple scattering andnon-zero dip angles
M. Valentan �, M. Regler, R. Fruhwirth
Institute of High Energy Physics of the Austrian Academy of Sciences, Vienna, Austria
a r t i c l e i n f o
Article history:
Received 4 November 2008
Received in revised form
8 May 2009
Accepted 8 May 2009Available online 18 May 2009
Keywords:
Detector design
Track resolution
Momentum resolution
Covariance matrix
Gluckstern formulas
Multiple scattering
Dip angle
02/$ - see front matter & 2009 Elsevier B.V. A
016/j.nima.2009.05.024
esponding author.
ail address: [email protected] (M
a b s t r a c t
The first rules-of-thumb for the uncertainties in track momentum and direction of tracking detectors
under inclusion of multiple scattering, as developed by Gluckstern [Nucl. Instr. and Meth. 24 (1963)
381] in the times of the bubble chamber, were limited to tracks with low curvature and equidistant
measurement points with equal accuracy. The extension to strongly curved tracks with nonvanishing
incident angle, arbitrary detector configuration and accuracy was published recently [M. Regler, R.
Fruhwirth, Nucl. Instr. and Meth. A 589 (2008) 109]. However, this extension is restricted to the
(symmetry) plane with tracks with zero dip angle, perpendicular to the magnetic field, and does not
treat multiple scattering. The present study extends the analytical approximate formulas for the
calculation of uncertainties in track momentum and direction of ‘‘barrel’’ detectors to non-zero dip
angles, including multiple scattering. The dip angle dependence of all terms of the error matrix is
calculated. The results of a comparison with a linear least-squares fit are presented, showing excellent
agreement. An open source implementation of the exact covariance matrices is described.
& 2009 Elsevier B.V. All rights reserved.
1. Introduction
The normalized relative transverse momentum resolutionsðDpT=p2
TÞ is very often used to characterize the performance ofa tracking detector. For a subsequent kinematics fit, however, therelative transverse momentum resolution sðDpT=pTÞ is also ofinterest. In the following simple approximate formulas for thesequantities will be derived.
Typically multiple scattering dominates the momentumresolution over a wide range of pT up to several tens of GeV=c
[1,2]. That is why an exact treatment of multiple scattering hasbeen introduced at the first pp collider at CERN (SFM experimentat the Intersecting Storage Rings [3]). With the advent of the SLHC(Super Large Hadron Collider) and ILC (International LinearCollider) projects, detector performance optimizations grow inimportance, and popular rules-of-thumb (such as the formulasgiven by Gluckstern [4]) are applied surprisingly often. Frequentlythese rules are stressed far beyond their limits. Extensions ofGluckstern’s formulas for the general cases of significant curva-ture k and inclination b ¼ arctanj, arbitrary distances of thedetector layers, and arbitrary detector resolutions are available inRefs. [5,6] for the case dominated by detector resolution (theasymptotic region of sðDpT=p2
TÞ), but little is available for the casedominated by multiple scattering [7].
ll rights reserved.
. Valentan).
The covariance matrix describing the detector resolution andmultiple scattering, as used in the global track fit, has the form [8]
V ¼ Vdet þ Vms.
Vdet is (block-)diagonal; Vms, the covariance matrix of thedeviations of the disturbed track w.r.t. the extrapolation of thetrue track, has considerable correlations and depends strongly onthe dip angle l. If one of the matrices dominates the other one, therelative contribution of the two covariance matrices on the finalmomentum resolution can be evaluated quite independently. This,however, has to be proven in practice. Note that Vms by itself issingular.
Up to now, all rules-of-thumb were restricted to the planeperpendicular to the magnetic field (see Fig. 1). The aim of thisstudy is to consider the effect of multiple scattering and deriverules-of-thumb with validity for tracks leaving this plane (withlarge dip angle l) and down to low pT.
The method described below calculates the momentum anddirection uncertainties of tracks measured in a barrel detector. It isbased on the covariance matrices of only two tracks, one at low pT
and the other one at high pT, i.e. in the asymptotic region ofsðDpT=p2
TÞ. These matrices can be determined using tracks atl ¼ 0� and at a certain initial angle j. The covariance matrices canbe taken from either a simulation or rules-of-thumb. The methodis furthermore independent of the detector, i.e. of the radii, theprecision and the material budget of the layers, under the onlyrestriction of z-independent detector properties. The method isapplicable under the condition that the low energetic track’sradius is at least twice as large as the radius of the largest detector
ARTICLE IN PRESS
Fig. 1. Two tracks passing a detector layer. Up to now, rules-of-thumb were only
available for tracks in the plane perpendicular to the magnetic field (i.e. l ¼ 0, left
track). The present study extends the existing rules-of-thumb for tracks with la0
(right track).
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742 729
cylinder, in order to maintain close to perpendicular intersectionsof the particles with the detector layers.
The result of the calculations will be a couple of simpleformulas to determine the uncertainties of the dip angle l, theazimuth angle j, and the curvature k. From these the relativetransverse momentum resolution sðDpT=pTÞ and the normalizedrelative transverse momentum resolution sðDpT=p2
TÞ can becomputed. Moreover, a short discussion of the projected impactparameter � and of the z coordinate z0 in the vertex region is given.Finally, the influence of the uncertainty of l on sðDp=pÞ will bediscussed.
The formulas given in this study have been validated with theVienna fast simulation and reconstruction tool ‘‘LiC Detector Toy’’(LDT) [9]. LDT is written in MATLAB [10], but an Octave [11]version is available, too. The estimation of the track parameters isdone by a linear least-squares (LS) fit, implemented as a Kalmanfilter [12]. LDT can handle inefficiencies, measurement errors andmultiple scattering; in this study inefficiencies were switched off.LDT is presently used for optimization studies of both the barreland the forward region of a detector at the ILC, see Ref. [13]. Oursimulations used a sample silicon detector with equidistantcylinders; as mentioned above, the rules-of-thumb derived canbe applied to any other barrel detector.
The exact calculation of the covariance matrix of the trackparameters by a linear LS fit has been implemented in a MATLABfunction called Compute_C_MS (see Section 6). This implementa-tion is fully compatible with Octave 3.0.1 [11]. It has also beenimplemented in Java, as part of the JDOT Java Detector Optimiza-tion Tool [18].
2. Combining multiple scattering and measurement errors
We now show how to take into account the effect of multiplescattering in an approximate way. The algorithm can be applied toany barrel detector.
2.1. Matrix formalism for multiple scattering (global method)
2.1.1. The global coordinate system
The detector setup is assumed to be azimuthally symmetricw.r.t. the z-axis, and a-priori invariant w.r.t. translations along thez-axis. The axes ðx; y; zÞ define a right-handed orthogonal frame. Byconvention, the x-axis is chosen to be horizontal w.r.t. the ground,and the y-axis to point upwards. Detector surfaces are cylinders ofradius R and borders zlower � z � zupper in the ‘‘barrel region’’.
Besides Cartesian coordinates, cylinder coordinates and sphe-rical polar coordinates are defined for space points and/ormomenta.
�
Space points ~x: ½x; y; z�cart or ½R;F; z�cyl, withx ¼ R � cosF; R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2
qy ¼ R � sinF; F ¼ arctanðy=xÞ azimuth angle 0 � Fo2p.
�
Momenta ~p: ½px; py; pz�cart, ½p; l;j�sph or ½pT;j; pz�cyl, withpx ¼ p � cos l � cosj; p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
x þ p2y þ p2
z
q; p ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
T þ p2z
qpy ¼ p � cos l � sinj; l ¼ arcsinðpz=pÞ dip angle�
p2� l �
p2
pT ¼ p � cos l; W ¼p2� l polar angle 0 � W � p
pz ¼ p � sin l; j ¼ arctanðpy=pxÞ azimuth angle 0 � jo2p.
The magnetic field is assumed to be homogeneous and alignedparallel or anti-parallel to the z-axis. It is defined by the fluxdensity ~B ¼ ½0;0;Bz�cart. This implies a helix track model, with thehelix axis being parallel to z.
The following units are used: length in mm, angle in rad,momentum in GeV=c, B field in T (Tesla), and charge in e
(elementary charge). For a particle with momentum p and chargeq, the radius of the helix and its signed inverse is
rH ¼1
K�
p � cosljq � Bzj
; k ¼ �signðq � BzÞ �1
rH
with the unit-dependent constant
K ¼ 10�15c ¼ 0:000299792458GeV=c
T �mm.
Our sign convention corresponds to signðkÞ ¼ signðdj=dsÞ, whichis the sense of rotation in the ðx; yÞ-projection. Note that in theabsence of matter p and l are constants of motion.
The helix equations for a starting point ½xS; yS; zS� and a startingazimuthal direction angle jS, as functions of the runningparameter j, are
xðjÞ ¼ xS þ1
k� ðsinj� sinjSÞ; yðjÞ ¼ yS �
1
k� ðcosj� cosjSÞ
zðjÞ ¼ zS þ1
k � tan l � ðj�jSÞ; sðjÞ ¼ 1
k � ðj�jSÞ= cos l
where sðjÞ is the path length along the helix.
2.1.2. The track parameters
The helix is described by five track parameters, which aredefined at a certain reference surface with radius Rref . These trackparameters are
½RF; z; l;j;k� at R ¼ Rref
where RF is the arc on the reference surface Rref �F, z is the z
position, l is the dip angle, j is the azimuth angle (in the globalcoordinate system), and k is the curvature (see Fig. 2). It iscommon use in collider experiments to define the trackparameters for a first single track fit at a reference surface near
ARTICLE IN PRESS
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742730
the vertex. This could be, e.g. the inner side of the beam tube or ofthe innermost detector layer.
In this paper we only consider the kinematic parameters l, jand k. In addition, in Section 3.1 a short discussion of the perigeeparameter � (the distance of closest approach) is given. It isfollowed by a brief treatment of the z-coordinate z0 in the vertexregion, as far as the scope of this work allows it.
2.1.3. The moving orthonormal frame for multiple scattering
Multiple scattering by the material budget of detector layer r
can be accounted for by adding terms to the covariance matrixdescribing the detector resolution. Two independent scatteringangles in two orthogonal planes with the track tangent asintersection, Dlr and Dfr , allow to evaluate the correspondingterms in the error matrix [3,8].
Fig. 3(a) shows a partially cut open detector surface and aparticle coming from the vertex, following a helical track. Forbetter visualization, the helix track is embedded in the partiallyshown helix cylinder. The moving orthonormal frame for multiplescattering is defined at the intersection point of the helix and thedetector surface. It is built by the track tangent ~e1, a vector ~e2
perpendicular to ~e1 in the tangential plane of the helix cylinder,and a vector ~e3 perpendicular to both ~e1 and ~e2. Note that thismoving orthonormal frame is the Frenet–Serret frame, knownfrom differential geometry, rotated around ~e1 by the angle p. Therotation changes the orientation of ~e2 and ~e3.
When traversing a scattering layer, say layer r, the direction ofthe momentum is slightly changed by an angle Yr in space. Itsabsolute value does not change, as the scattering processpreserves the particle momentum. The scattering process isdescribed by the two independent scattering angles Dlr andDfr . The scattering angle Dlr is defined in the plane spanned by~e1 and~e2, while Dfr can be found in the plane spanned by~e1 and~e3. Note that there are more appropriate choices of ~e2 and ~e3 for
Fig. 2. The coordinates and track parameters.
Fig. 3. Definition of the moving orthonormal frame for mu
different magnetic field configurations. However, with thisdefinition of the scattering planes, the scattering angle Dlr
directly adds to the dip angle of the global coordinate system:
l0r ¼ lr þDlr .
Before adding multiple scattering to the azimuth angle jr , thescattering angle Dfr has to be projected to the plane perpendi-cular to the z-axis, where jr is defined. This results in anadditional factor 1= cos lr:
Djr ¼1
cos lr� Dfr ; j0r ¼ jr þDjr ¼ jr þ
1
cos lr� Dfr .
2.1.4. Matrix formalism
After projecting the scattering angle in space Yr into themoving orthonormal frame, the variance of the projected scatter-ing angle Yp;r can be computed according to several approximateformulas. Although a more sophisticated approach has beenpublished recently [14], the most common in use is Highland’sformula [15]:
ffiffiffiffiffiffiffiffiffiffiffiffiffihY2
p;ri
q¼
0:0136 GeV
bcp
ffiffiffiffiffiffiffilr
L0;r
s1þ 0:038 ln
lrL0;r
� �� �(1)
where b ¼ v=c is the relative velocity of the scattered particle, p itsmomentum in GeV=c, c the velocity of light, lr the effectivethickness of detector r, and L0;r the radiation length of its material.The subscript ‘‘p’’ denotes the projection into the movingorthonormal frame, whereas the subscript ‘‘T’’ denotes theprojection into the transverse plane perpendicular to ~B, forexample the transverse momentum pT. The effective thickness lrtakes into account the crossing angle of the particle and readslr ¼ dr=ðcoslr cosðjr �FrÞÞ, where dr is the nominal thickness ofdetector r. Note that lr and L0;r have to be in the same units.
With the special choice of the orthonormal frame described inSection 2.1.3, the variances of the scattering angles read
varðlrÞ ¼ hDl2r i ¼ hY
2p;ri; varðjrÞ ¼ hDj
2r i ¼
1
cos2l� hDf2
r i
¼1
cos2l� hY2
p;ri.
Thus, the covariance matrix of the effect of the two scatteringangles on the track angles in the global coordinate system isdiagonal and reads
covðl;jÞr ¼hY2
p;ri 0
01
cos2l� hY2
p;ri
0B@
1CA.
This assumes the scattering angles to be so small that sinðDlÞ �Dl and sinðDfÞ � Df holds.
ltiple scattering: (a) total view and (b) zoomed view.
ARTICLE IN PRESS
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742 731
Multiple scattering causes a physical change of the track,which changes the intersection points of the track with thesubsequent detector surfaces, i.e. the observations of the track.The matrix describing the impact of multiple scattering on theseobservations contains off-diagonal terms. Without loss of general-ity we assume that both RF and z are measured in each detectorlayer. With the notation m1
i ¼ ðRFÞi and m2i ¼ zi, the general form
of the covariance of two observations mki and ml
j is [8,16]
covðmki ;m
ljÞ ¼
Xminði�1;j�1Þ
r¼0
hY2p;ri
@mki
@Dlr�@ml
j
@Dlrþ@mk
i
@Dfr
�@ml
j
@Dfr
!
¼Xminði�1;j�1Þ
r¼0
hY2p;ri
@mki
@Dlr�@ml
j
@Dlrþ
1
cos2l@mk
i
@Djr
�@ml
j
@Djr
!.
(2)
The assumption that multiple scattering takes place in twoorthogonal planes leads to a block diagonal form of the multiplescattering covariance matrix:
covðmki ;m
ljÞ �
varðRF�!Þ 0
0 varð~zÞ
!.
The correlation coefficients rððRFÞi; zjÞ do not exceed 10�2, whichhas been checked for i ¼ j and for i ¼ jþ 1 using the derivativesgiven by exact helix extrapolation. Thus, assuming a blockdiagonal form enables us to decompose this matrix into twoindependent sub-matrices:
covððRFÞi; ðRFÞjÞ ¼Xminði�1;j�1Þ
r
hY2p;ri
@ðRFÞi@Dlr
�@ðRFÞj@Dlr
�
þ1
cos2l@ðRFÞi@Djr
�@ðRFÞj@Djr
�(3)
covðzi; zjÞ ¼Xminði�1;j�1Þ
r
hY2p;ri
@zi
@Dlr�@zj
@Dlr
�
þ1
cos2l@zi
@Djr
�@zj
@Djr
�. (4)
2.2. Global track fit formalism
In Ref. [4], the approximate formulas for the calculation ofmomentum and direction uncertainties are valid only in the planeperpendicular to the magnetic field, i.e. for tracks with l ¼ 0. Wenow derive rules that describe how the variances and covariancesof the track parameters can be evaluated at la0 if they are knownat l ¼ 0, regardless of whether they are taken from a precise LSestimation or from approximate formulas.
We assume that the detector consists of n concentric, cylindricdetector surfaces parallel to the direction of the magnetic field.The detector layers measure the position z and the azimuth RF ofthe particles crossing them. The momentum and direction part ofthe track is represented by the dip angle l, the azimuthal angle jand by the curvature k.
First, only l and k are considered; j will be treated later inSection 2.2.3. Note that first only kinematic variables areconsidered. If one intends to do a subsequent vertex fit, oneshould make use of LDT or of the program in Section 6.
In a global LS fit the covariance matrix of the fitted trackparameters is calculated according to
C ¼ ðDTV�1totDÞ�1
where the total covariance matrix V tot ¼ Vms þ Vdet is composedof the multiple scattering covariance matrix Vms and of thecovariance matrix due to detector errors Vdet. The dimension ofV tot is 2n 2n. D is the matrix of derivatives of the observations
w.r.t. l and k; it is of dimension 2n 2. The superposition ofmultiple scattering and measurement errors is reflected by addingthe respective covariance matrices for multiple scattering andmeasurement errors to get the total covariance matrix:
V tot ¼ Vms þ Vdet �VmsðRF
�!Þ 0
0 Vmsð~zÞ
!þ
VdetðRF�!Þ 0
0 Vdetð~zÞ
!.
Vms ¼ covðmki ;m
ljÞ, which is given by Eq. (2), is block-diagonal. At
the moment we assume multiple scattering to dominate, so thatthe contribution of Vdet to V tot can be neglected.
2.2.1. Consideration of multiple scattering
When describing multiple scattering using the orthonormalframe defined in Section 2.1.3, the covariance matrix for multiplescattering is given by Eq. (2), where hY2
p;ri is calculated accordingto Eq. (1). For the remainder of this study, the term containing thelogarithm in Eq. (1) will be neglected. Furthermore, we make theassumption that the mass of the particle is negligible incomparison to its momentum (i.e. b � 1), so we use the formula:
hY2p;ri /
lrL0;r�
1
p2. (5)
hY2p;ri is the projected multiple scattering variance.The effective thickness of the scatterer behaves like
lr ¼ lT;r= cos l, where lT;r ¼ dr= cosðjr �FrÞ is the effective thick-ness of the scatterer in the transverse plane perpendicular to ~B.The difference jr �Fr is often called br , and is the angle betweenthe track’s direction and the surface normal of a detector surfacein the (x; y)-projection. Note that the effect of the increasedthickness in the transverse plane due to a non-perpendicularintersection of the track with the detector layer has already to beconsidered by the reference track at l ¼ 0�, while the approximateformulas to be derived in this work only care about the effect ofleaving this very plane, i.e. the transition to la0�.
The absolute momentum behaves like p ¼ pT= cos l / 1=ðk cos lÞ, and so the l and the k dependencies of the projectedmultiple scattering variance read
hY2p;ri / k2 cos l. (6)
One of the two independent scattering angles changes the dipangle l, whereas the other one changes the azimuthal angle j.Therefore, in the following, we will be talking about multiplescattering in l and in j.
2.2.1.1. Multiple scattering in l. If we assume multiple scattering inl only, submatrix (4) of the general covariance matrix (2) takesthe simple form:
ðVmsðlÞÞi;j ¼ covðzi; zjÞmsðlÞ ¼Xminði�1;j�1Þ
r
hY2p;ri �
@zi
@Dlr�@zj
@Dlr
which yields the lower block of the block diagonal multiplescattering covariance matrix. The other derivatives are@zi=@Djr ¼ 0, because we switch off multiple scattering in j. Tocalculate the remaining derivative we consider the relation:
yðz; l;kÞ ¼ 1
k � sinðkz cot lÞ (7)
which describes the sine form of the track in the ðy; zÞ-projection.Eq. (7) can be derived from the helix equations quoted in Section2.1.1, assuming jS ¼ p=2 and xS ¼ yS ¼ zS ¼ 0 without loss ofgenerality (see Fig. 4). We have to evaluate this equation at the y
coordinate of the intersection point y ¼ y0;i, where the helixcrosses the layer i with radius Ri. This point can be shown to be
ARTICLE IN PRESS
x
y
Φi
Ri
RH = 1/κ
y0, i
Fig. 4. A track projected to the plane perpendicular to the magnetic field.
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742732
(see Fig. 4)
y0;i ¼ Ri �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1
4ðkRiÞ2
q. (8)
We thus can express the z coordinate of a track’s intersection withlayer i as a function of l and k:
ziðl;kÞ ¼1
k � tanl � arcsin kRi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�
1
4ðkRiÞ
2
r !. (9)
Expanding the square root and the arcsine function in Eq. (9) inpower series w.r.t. kRi and neglecting terms of order OððkRiÞ
4Þ and
greater, we can express ziðl;kÞ as
ziðl;kÞ ¼1
k � tanl � kRi � 1þ1
24ðkRiÞ
2þ OððkRiÞ
4Þ
� �.
So, under the condition ðkRiÞ2¼ ðRi=RHÞ
2524 we can use the
approximation:
ziðl;kÞ � Ri � tan l. (10)
Note that this condition does not only apply here locally, but willoccur again below. Later applications of this approximation willrestrict us to ðkRiÞ
251, so that the condition ðkRiÞ
2524 needed
here is satisfied in a natural way.Differentiating Eq. (10) w.r.t. l yields
@zi
@l�
Ri
cos2 l.
Using this and relation (6) we determine the l dependence of thecovariance matrix due to multiple scattering in l to
VmsðlÞ /k2
cos3 l. (11)
To derive the variance of l from Eq. (11) according to the LSmethod, we calculate
varðlÞmsðlÞ ¼
@z1
@l
..
.
0@
1A
T
� V�1msðlÞ �
@z1
@l
..
.
0@
1A
0B@
1CA�1
/1
cos2l�
R1
..
.
0@
1AT
�k2
cos3 l
� ��1
�1
cos2l�
R1
..
.
0@
1A
0B@
1CA�1
/ k2 cos l.
Thus, the variance of l due to multiple scattering in l shows thefunctional dependency:
varðlÞmsðlÞ / k2 cos l
¼)varðlÞmsðlÞ ¼ a1;1 � r�2 � cos l
where we have introduced the relative transverse momentumr ¼ pT=pref
T ¼ kref=k. The coefficient a1;1 ¼ varðlÞjl¼0;pT¼prefT
is thevariance of l for tracks in the transverse plane perpendicular to ~B
with a reference momentum prefT . Note that varðlÞjl¼0;pT¼pref
T
already contains the effect of increased material due to particles
traversing a scattering layer i at a nonzero incident angle
bi ¼ ji �Fi.
Since in a scattering process the absolute value of themomentum remains unchanged, the estimated track parametersl and k are strictly correlated, i.e. with a correlation coefficient ofrðl;kÞ ¼ 1. Therefore we can derive the variance of k and thecovariance between l and k by straightforward error propagation.
Exploiting the condition p ¼ constant one can derive therelations:
k ¼ C
pT
¼C
p � cos l@k@l¼
C
p
sin lcos2 l
¼ k sin lcos l
(12)
where C is a constant (see Section 2.1.1). Using Eq. (12) for errorpropagation yields
varðkÞmsðlÞ ¼@k@l
� �2
� varðlÞmsðlÞ ¼ k2 sin2 lcos2 l
� a1;1 � r�2 � cos l
¼ a1;1 � k2ref � r
�4 �sin2 lcosl
covðl;kÞmsðlÞ ¼@k@l
� �� varðlÞmsðlÞ ¼ k sin l
cos l� a1;1 � r
�2 � cos l
¼ a1;1 � kref � r�3 � sin l.
The covariance matrix of the fitted parameters for multiplescattering in l only is therefore
Cðl;kÞmsðlÞ ¼
a1;1 � r�2 � cos l a1;2 � r
�3 � sin l
a1;2 � r�3 � sin l a2;2 � r
�4 �sin2 lcos l
0B@
1CA
with
a1;1 ¼ varðlÞ��l¼0�
pT¼prefT
; a1;2 ¼ a1;1 � kref
a2;2 ¼ a1;1 � k2ref ; kref ¼
KjqBzj
prefT
and the relative transverse momentum r ¼ pT=prefT ¼ kref=k.
ARTICLE IN PRESS
0 0.2 0.4 0.6 0.8 1 1.2 1.40
1
2
3
4
5
6
7
8x 10−10
Dip angle λ [Rad]
Var
ianc
e of
κ [1
/mm
2 ]
Variance of κ
λ scatteringφ scatteringboth
Fig. 5. Comparison of the magnitude of the two contributions to varðkÞ.
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742 733
2.2.1.2. Multiple scattering in j. If we assume multiple scatteringin j only, submatrix (3) of the general covariance matrix (2) takesthe form
ðVmsðjÞÞi;j ¼ covððRFÞi; ðRFÞjÞmsðjÞ
¼Xminði�1;j�1Þ
r
1
cos2 lhY2
p;ri@ðRFÞi@Djr
@ðRFÞj@Djr
which yields the upper block of the block diagonal multiplescattering covariance matrix. The other derivatives are@ðRFÞi=@Dlr ¼ 0, because we switch off multiple scattering in l.
Due to the condition that the helix radius RH is larger than thelargest detector radius, the derivatives @ðRFÞi=@Djr can beassumed not to depend on k. A detailed examination leads tothe condition ðkRiÞ
2¼ ðRi=RHÞ
251, which is the strongest assump-
tion one has to make. For ðkRiÞ2� 0:25 the change of @ðRFÞi=@Djr
with k stays below 10%. Thus, the helix radius RH has to be twiceas large as the largest detector radius, which translates into acondition on the crossing angle between the track and thedetector cylinder of jbij ¼ jji �Fij � 14:5�.
Furthermore, the derivatives @ðRFÞi=@Djr do not show any ldependence, because they are calculated in the transverse plane,i.e. at l ¼ 0�. However, the projection of the scattering angle to thetransverse plane yields an additional factor 1=cos2 l. Therefore thel and k dependence of the covariance matrix due to multiplescattering in j is
VmsðjÞ /k2
cos l.
Because measurements of RF cannot carry any information aboutl, we can only calculate the contribution of multiple scattering inj to the variance of k. To evaluate the derivative @ðRFÞi=@k neededfor the LS method, we first calculate the RF coordinate of thetrack’s intersection with layer i (see Fig. 4) and Eq. (8):
ðRFÞi ¼ Ri �Fi ¼ Ri � arcsiny0;i
Ri
� �¼ Ri � arcsin
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�
1
4ðkRiÞ
2
r !
¼)@ðRFÞi@k ¼ �
1
2
R2iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 14 ðkRiÞ
2q .
Expanding the square root in a power series w.r.t. kRi andneglecting terms of order OððkRiÞ
4Þ and greater, we can express
@ðRFÞi=@k as
@ðRFÞi@k
¼ �R2
i
2� 1þ
1
8ðkRiÞ
2þ OððkRiÞ
4Þ
� �.
So, under the condition ðkRiÞ258 (which is satisfied in a natural
way due to the condition taken in the approximation of@ðRFÞi=@Djr) we can use the approximation
@ðRFÞi@k
� �R2
i
2. (13)
Derivative (13) is independent of l and k, and consequently thevariance of k shows the same dependencies as VmsðjÞ:
varðkÞmsðjÞ /k2
cos l.
Using again the relative transverse momentum r ¼ pT=prefT , the
contribution of multiple scattering in j to the variance of k reads
varðkÞmsðjÞ ¼ b2;2 � r�2 �
1
cos l
where the coefficient b2;2 ¼ varðkÞjl¼0;pT¼prefT
is the variance of k fortracks in the transverse plane perpendicular to ~B with a referencemomentum pref
T .
2.2.1.3. Total multiple scattering covariance matrix. As scattering inl is uncorrelated with scattering in j, the respective contributionsto the variance of k can simply be added to obtain the total cov-ariance matrix of multiple scattering:
Cðl;kÞms ¼varðlÞ covðl;kÞ
covðl;kÞ varðkÞ
!
with
varðlÞ ¼ varðlÞmsðlÞ; covðl;kÞ ¼ covðl;kÞmsðlÞ
varðkÞ ¼ varðkÞmsðlÞ þ varðkÞmsðjÞ ¼ a2;2 � r�4 �
sin2lcos l
þ b2;2 � r�2 �
1
cosl.
To be able to quote a simple covariance matrix in closed form, wenow compare the magnitude of the different contributions to thevariance of k by a simulation study, using the covariance matrix ofa single track at pT ¼ 1 GeV=c and l ¼ 0�;5�; . . . ;60�, fitted by anexact Kalman filter. The results can be found in Fig. 5. Here, thesimulated values of varðkÞ are plotted versus the dip angle l. Thecurves with markers show the results when simulating withmultiple scattering in both planes, while the ones with � markersand the ones with n markers show the results when simulatingwith l scattering or j scattering only, respectively.
As can be seen in Fig. 5, varðkÞ () is dominated by j scatteringðnÞ, i.e. a2;2ob2;2. But the difference is only about one order ofmagnitude, which results in significant deviations for higher dipangles l. As long as the contribution of l scattering is sufficientlysuppressed by the term sin2 l, approximating the total variancevarðkÞ by the variance of j scattering varðkÞmsðjÞ is acceptable.Therefore the approximation varðkÞ � varðkÞmsðjÞ ¼ b2;2 � r
�2 �
1= cos l is applicable for lo45�. Note that a2;2 can becomedominant in cases where varðkÞ is measured very precisely bymany RF measurements without much degradation by multiplescattering in j, as is the case in a TPC, while other detector layersdegrade this information by multiple scattering in l, as siliconlayers do.
Summing up, the total covariance matrix Cðl;kÞms can beapproximated as follows:
Cðl;kÞms �
a1;1 � r�2 � cosl a1;2 � r
�3 � sin l
a1;2 � r�3 � sin l b2;2 � r
�2 �1
cos l
0B@
1CA. (14)
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Fig. 6. Geometry sketch for the calculation of varðlÞdet.
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742734
2.2.2. Covariance matrix at higher energy
Up to now we have considered only the multiple scatteringdominated case of very low momentum, where the covariancematrix of detector errors can be neglected (see Section 2.2.1). Werecall the covariance matrix of the fitted parameters in the linearLS formalism:
Cðl;kÞ ¼ ðDTV�1totDÞ�1 with V tot ¼ Vms þ Vdet. (15)
When moving to higher energies, the detector errors grow inimportance relative to multiple scattering and have to beincluded. To this end, the exact formula (15) is approximated by
Cðl;kÞ � ðDTV�1msDÞ�1
þ ðDTV�1detDÞ
�1¼ Cðl;kÞms þ Cðl;kÞdet.
This approximation allows to use the total multiple scatteringcovariance matrix derived above. In general, the approximationwould be exact if D and Vms were regular, or if Vms ¼ aVdet, witha 2 R. Obviously, both conditions are not met in our case.
The covariance matrix for detector errors Cðl;kÞdet is easily setup. With the help of Fig. 6 we calculate varðlÞdet as follows. Themeasurement of l behaves according to
sðlÞdet /sp
L
where sp is the uncertainty perpendicular to the track direction and L
denotes the lever arm. We gather from Fig. 6 that the measurement
Cðl;kÞms �
¼)Cðl;j;kÞms � r�2 �
error sz (horizontal) and the uncertainty projected to the planeperpendicular to the track direction sp are related according to
sp ¼ sz � cos l.
This results in a factor cos2 l as first contribution to varðlÞdet. Thecontribution of the lever arm can be derived as follows:
LðlÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRF2þ zðlÞ2
q¼ RF �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ tan2 l
p¼
RFcos l
which again contributes with a factor cos2 l to the variance of l.Thus, the variance varðlÞdet behaves like
varðlÞdet / cos4 l.
Because the dip angle l does not affect the measurement of thecurvature k, varðkÞdet can be assumed to be independent of l.
At high energies the sine form of the trajectory in the ðy; zÞ-projection degenerates to a straight line, which means that thetwo projections ðy; zÞ and ðx; yÞ are uncoupled. Therefore we arefree to set covðl;kÞdet ¼ 0 in good approximation.
All terms of the covariance matrix for detector errors can beshown to be independent of pT down to 5 GeV=c, where multiplescattering already dominates by orders of magnitude. Summingup, this covariance matrix reads
Cðl;kÞdet �c1;1 � cos4 l 0
0 c2;2
!.
The coefficients ci;j can easily be determined by simulating andreconstructing a track at high momentum, in the asymptoticregion of sðDpT=p2
TÞ, or by using Gluckstern’s formulas foruncertainties of momentum and track direction [4].
2.2.3. Inclusion of the azimuthal angle jWe now extend the previous track representation ðl;kÞ by the
azimuthal angle j to ðl;j;kÞ. Consequently, we have to extendthe covariance matrices by one line and one column to thedimension 3 3.
The azimuthal angle j can only be determined by RFmeasurements. Consequently, the only contribution to theuncertainty of j due to multiple scattering origins in multiplescattering in j. We recall the covariance matrix of the RFmeasurements due to multiple scattering in j:
ðVmsðjÞÞi;j ¼Xminði�1;j�1Þ
r
1
cos2 lhY2
p;ri@ðRFÞi@Djr
@ðRFÞj@Djr
/k2
cos l.
Since the derivatives @ðRFÞi=@j and @ðRFÞi=@k, which are used inthe LS method, depend neither on l nor on k, the global fit of theazimuthal angle j preserves the l and k dependence of VmsðjÞ,and the variance of j and the covariance between j and k due tomultiple scattering read
varðjÞms ¼ b3;3 � r�2 �
1
cos l
covðj;kÞms ¼ b2;3 � r�2 �
1
cos l.
Due to the assumption of two independent scattering angles thecovariance covðl;jÞ can be neglected. Therefore the total multiplescattering covariance matrix can be extended according to
(16)
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with the relative transverse momentum r ¼ pT=prefT . The corre-
sponding covariance matrix for detector errors reads
Cðl;kÞdet �
¼)Cðl;j;kÞdet � (17)
2.3. Summary of the computation
The covariance matrix Cðl;j;kÞ is computed according to thefollowing step-by-step description:
(1)
Determine the coefficients of Cðl;j;kÞms at l ¼ 0 and atpT ¼ prefT , either by calculation with e.g. Ref. [4], or bysimulation with e.g. Ref. [9].
(2)
Determine the coefficients of Cðl;j;kÞdet at l ¼ 0, either bycalculation with e.g. Ref. [6], or by simulation with e.g. Ref. [9].(3)
Use the given l dependencies in Eqs. (16) and (17) to leave theplane perpendicular to the magnetic field.(4)
Use the given pT dependencies in Eq. (16) to map the results tothe desired momentum.(5)
Add the two covariance matrices Cðl;j;kÞ ¼ Cðl;j;kÞmsþCðl;j;kÞdet.
For multiple scattering the extrapolation to la0� is almost exact,and the quality is mainly limited by the quality of the lowmomentum covariance matrix itself, used for determining thecoefficients ai;j and bi;j. Note that the combination of the respectivecontributions of multiple scattering and detector errors, however,mixes different l-dependencies. This ‘‘interpolation’’ is part of themethod derived here, and will be examined in greater detail inSection 4.
3. The impact parameters
3.1. The projected impact parameter �
The projected (or 2D) impact parameter, which is the perigeeparameter � [17], is defined as the distance of closest approachbetween the reconstructed track’s projection and the true track’sorigin in the plane perpendicular to the magnetic field ~B. It is afunction of the azimuthal position angle F, the azimuthaldirection angle j and the curvature k, all defined at a certainreference cylinder surface with radius RrefoRi. This is commonuse for single track fits in collider experiments. A short tracksegment between the vertex and the reference cylinder can beapproximated to be a parabola, thus we can make a parabolicansatz for the impact parameter:
� � Rref �F� Rref �jþk2� R2
ref .
Note that Rref �F is the arc length ðRFÞref on the reference cylinder.Because of ðkRref Þ
2¼ ðRref=RHÞ
251 we can neglect the term
k=2 � R2ref ¼ 1=ð2kÞ � ðkRref Þ
2. Thus, using basic error propagation,the variance of the projected impact parameter reads
varð�Þ � R2ref � ½varðFÞ þ varðjÞ � 2 covðF;jÞ�. (18)
With the incident angle b ¼ j�F, which is a common fitparameter in cylindrical detectors, we can state a basic formulafor the variance of the projected impact parameter:
varð�Þ � R2ref � varðbÞ.
According to the previous sections, the variance of the azimuthaldirection angle j can be expressed as
varðjÞ � varðjÞjms þ varðjÞjdet
� varðjÞj l¼0�
pT¼prefT
�r�2
cos lþ varðjÞj l¼0�
pT!1
.
In contrast, the variance of RF cannot be parametrized in such asimple way. For high momenta, varðRFÞ asymptotically reaches aplateau determined only by the degradation due to measurementerrors, which is independent of l. But unlike the variance of j,which grows like p�2
T when going to small momenta, the error ofRF reaches yet another asymptotic plateau, roughly determinedby the RF resolution of the innermost measurement as an upperbound. A simple interpolation formula for the region betweenthese plateaus has not yet been found. A detailed examination isbeyond the scope of this paper, so that in the following only thelimits of very small and very high pT are discussed.
The correlation coefficient rðF;jÞ reaches values of 0.4–0.5 forsome momenta. Nevertheless, the contribution of covðF;jÞ inEq. (18) can be neglected in cases where one of the variancesvarðFÞ or varðjÞ dominates strongly, which is the case for bothvery high and very low momentum. Thus, restricting ourselves tothese limiting cases, we neglect the covariant term and write
varð�Þ � R2ref � ½varðFÞ þ varðjÞ�.
At low momenta the resolution of the projected impact parameteris dominated by the angular error varðjÞ and the spatialresolution varðFÞ can be neglected, while the case of highmomenta is vice versa.
For the multiple scattering dominated region of smallmomenta varðRFÞ is bounded from above by the point resolutionof the innermost measurement and can be neglected. Thus, forsmall momenta the variance of � behaves like
varð�Þms � R2ref � varðjÞms � R2
ref � varðjÞj l¼0�
pT¼prefT
�r�2
cos l.
For the detector error dominated region of high momenta varðRFÞreaches a plateau which is considerably higher than the one ofR2
ref varðjÞ. This means that the contribution of the direction angleuncertainty can be neglected and the error of RF directlytranslates into the error of � when going from the referencesurface to the interaction region:
varð�Þdet � varðRFÞj l¼0�
pT!1
.
3.2. The resolution of the z coordinate in the vertex region
Because of ðkRref Þ251 a short track segment between the
reference surface and the vertex region can be assumed to be astraight line in the ðR; zÞ-projection. Thus, the z coordinate z0 inthe vertex region is a function of the z coordinate and the dipangle l on the reference surface:
z0 � z� Rref tanl.
Using basic error propagation, the variance of z0 reads
varðz0Þ � varðzÞ þR2
ref
cos4 lvarðlÞ � 2
Rref
cos2 lcovðz; lÞ.
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In contrast to varðlÞ, which can be approximated by
varðlÞ � varðlÞms þ varðlÞdet
� varðlÞj l¼0�
pT¼prefT
� r�2 � coslþ varðlÞj l¼0�
pT!1
� cos4 l
the variance of z cannot be parametrized in such a simple way.Similar to the case of RF, for high momenta varðzÞ asymptoticallyreaches a plateau determined only by the degradation due tomeasurement errors, and for low momenta it reaches yet anotherasymptotic plateau, roughly determined by the z resolution of theinnermost measurement. Again, an interpolation formula wouldbe needed to describe the region of intermediate momenta. In thefollowing only the limits of very low and very high pT arediscussed.
The correlation coefficient rðz; lÞ reaches values of 0.6–0.8 forsome momenta. Again, the contribution of covðz; lÞ can beneglected for momenta where one of the variances varðzÞ orvarðlÞ dominates strongly, which is the case for very highmomenta, or when the correlation coefficient itself is small,which is the case for very low momenta. Thus, restrictingourselves to very low and very high momenta, we neglect thecovariant term and write
varðz0Þ � varðzÞ þR2
ref
cos4 lvarðlÞ.
For the multiple scattering dominated region of small momentavarðzÞ cannot be neglected, although it is bounded from above bythe point resolution of the innermost measurement, which isindependent of the momentum and independent of l. Thus, forsmall momenta the variance of z0 behaves like
varðz0Þms � varðzÞms þR2
ref
cos4 l� varðlÞms
� varðzÞj l¼0�
pT¼prefT
þ r�2 �R2
ref
cos3 l� varðlÞj l¼0�
pT¼prefT
.
This case is shown in Fig. 7(a). The approximate error of z0, theerror of z and the term containing the error of l are plotted asfunctions of l, at pT ¼ 1 GeV=c.
For the detector error dominated region of high momentavarðzÞ reaches a plateau which is considerably higher than the one
0 10 20 30 400
5
10
15
20
25
30
35
40
Dip angle [deg]
(z0) m
s [µm
]
The contributions to (z0)ms as functions of at pT=1[GeV/c]
(z)ms
Rref/cos2 ( )ms
(z0)ms
(z0)ms=[ 2(z)ms+R2ref/cos4 2( )ms]
1/2
(z)ms
Rref/cos2 ( )ms
(z0)ms
(z)
[µm
]
Fig. 7. The contributions to sðz0Þ as functions of l, neglecting covðz; lÞ. �: sðz0Þ, : sðzpT ¼ 1000 GeV=c. Note that the y scale is a factor of 10 smaller than the one in (a).
of the terms containing varðlÞ. This means that the contribution ofthe dip angle uncertainty can be neglected and the error of z
directly translates into the error of z0 when going from thereference surface to the interaction region:
varðz0Þdet � varðzÞj l¼0�
pT!1
.
This case is shown in Fig. 7(a) for pT ¼ 1000 GeV=c.
4. Comparison of the rules-of-thumb for l, j and k with a linearleast-squares fit
In this section we compare the results of the approximateformulas with the results of a linear LS fit. To do this, we take thecovariance matrix of a fitted low momentum track at 3 GeV=c (inaccordance with the restriction ðkRiÞ
251) and l ¼ 0 and the
covariance matrix of a fitted high momentum track at 1000 GeV=c
and l ¼ 0. Using these covariance matrices, one can determine theconstants aij, bij and cij.
Simulation and track fit were carried out with the Vienna fastsimulation tool LiC Detector Toy 2.0 (LDT) [9]. The detector modelused consists of 11 cylindric silicon detector layers, equally spacedbetween 10 mm � Ri � 1010 mm, with a thickness of 1% radiationlength each, and a point resolution of sðRFÞ ¼ sðzÞ ¼ 5mm. Themagnetic field is assumed to be a homogeneous field of 4 T. Notethat LDT assumes the reference surface to be the inner side of thebeam tube, which is common use for single track fits in colliderexperiments. LDT performs exact helix tracking with kinks due tomultiple scattering in thin layers, and does not use theapproximations used by the method developed in this work.
The behaviour of this example silicon detector is determined attransverse momenta in the range 1 � pT � 1000 GeV=c, for twodifferent dip angles l ¼ 0� and 45�. Fig. 8 shows a directcomparison of the approximate formulas with an linear LS fit.The three plots of the left column are sðDpT=pTÞ, sðlÞ and sðjÞ as afunction of pT at l ¼ 0�, whereas the three ones of the rightcolumn show the same at l ¼ 45�. The dashed line denotes thelow energetic reference momentum pref
T used in the approximateformulas. Note that one can extrapolate the results of theapproximate formulas for momenta pTopref
T with sufficient
0 5 10 15 20 25 30 35 40 450
0.5
1
1.5
2
2.5
3
3.5
4
Dip angle [deg]
0de
t
The contributions to (z0)det as functions of at pT=1000[GeV/c]
(z)det
Rref/cos2 ( )det
(z0)det
(z0)det=[ 2(z)det+R2ref/cos4 2( )det]
1/2
(z)det
Rref/cos2 ( )det
(z0)det
Þ, n: contribution of sðlÞ. (a) Low-energy lim. pT ¼ 1 GeV=c. (b) High-energy lim.
ARTICLE IN PRESS
Fig. 8. Direct comparison of sðDpT=pTÞ (top), sðlÞ (middle) and sðjÞ (bottom) as obtained by the approximate formulas (�) and by an LS fit ðÞ; Left column: l ¼ 0� , right
column: l ¼ 45� . The dashed line denotes the low energetic reference momentum prefT .
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742 737
accuracy, although these low energetic tracks do not fulfill therestriction ðkRiÞ
251 anymore.
Since the constants used in the approximate formulas aretaken from LDT itself, these plots really show the deviation whenusing the approximate formulas instead of the exact1 linear LS fitin the LDT. This error appears to be very small. Moreover, the ldependence of the calculated resolutions seems to be very wellreproduced by the relations used in the approximate formulas.
For further analysis of the approximation, we now calculate therelative deviation in percents of the results of the approximateformulas from those of the linear LS fit in the LDT. For the relative
1 In this context, the linear least-squares fit is called ‘‘exact’’, because LDT
assumes multiple scattering to take place in thin layers (the scattering angles are
so small that they can be assumed to be uncorrelated), and the reconstruction uses
the same assumptions as the simulation does.
momentum resolution the relative deviation is calculated by
D ¼sðDpT=pTÞappr � sðDpT=pTÞLDT
sðDpT=pTÞLDT
� 100. (19)
In the upper row of Fig. 9 this relative deviation is plotted as afunction of pT, for the two different dip angles l mentioned above.There are two plots for the detector resolutions sðRFÞ ¼ sðzÞ ¼20mm and sðRFÞ ¼ sðzÞ ¼ 5mm. One easily observes that therelative deviation has a (negative) maximum at a certainmomentum, where the approximation becomes worst. Thelower row of Fig. 9 shows the variance of k due to multiplescattering varðkÞms (falling lines) and the variance of k due todetector errors varðkÞdet (constant lines) in a double logarithmicscale. In the present approximation, these variances simply areadded instead of being inverted, added, and reinverted. Wetherefore expect the largest error to occur near the momentum
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where varðkÞms and varðkÞdet are comparable. This momentum canbe seen in the lower diagrams at the intersection point of the twogroups of curves. When varying the detector resolution, varðkÞdet
is changed and so is the intersection point. The plots in the upperrow of Fig. 9 clearly show that the position of the maximal relativedeviation follows the position of the intersection point. In the caseof the relative momentum resolution the maximal deviation issmaller than 10%.
The relative deviations of sðlÞ and sðjÞ are calculated in thesame manner as in Eq. (19). They are shown in Fig. 10. They arelarger than the one of sðDpT=pTÞ; the relative deviation of sðlÞeven reaches a value of almost 40%.
Fig. 9. Top: relative deviation of sðDpT=pTÞappr in percent; bottom: varðkÞms (falling lines
the momentum where varðkÞms and varðkÞdet are comparable.
Fig. 10. Top: relative deviation in percent of sðlÞappr (top left); sðjÞappr (top right
5. The relative momentum resolution
5.1. The uncertainty of Dp=p
When optimizing a detector, one of the central quantities tocharacterize detector performance is the relative momentumresolution Dp=p. In detectors with a solenoid magnetic field, it isquite common to examine separately the two projections of thehelix: a sine curve in ðx; zÞ and a circle in ðx; yÞ. Since the transversemomentum is obtained from the measurement of the track’scurvature in the ðx; yÞ-projection, it is more natural to workwith the transverse (ðx; yÞ-projected) momentum component
) and varðkÞdet (constant lines) separately. The largest relative deviation occurs near
); varðlÞms and varðlÞdet (bottom left); varðjÞms and varðjÞdet (bottom right).
ARTICLE IN PRESS
Table 1Results of the global formula (24) for simulation at 1 GeV=c.
l (deg) sðlÞsim s DpT
pT
� �sim
s Dp
p
� �sim
s Dp
p
� �appr
0 2:920 10�3 3:010 10�3 3:010 10�3 3:010 10�3
15 2:874 10�3 3:156 10�3 3:068 10�3 3:061 10�3
30 2:734 10�3 3:596 10�3 3:256 10�3 3:231 10�3
45 2:492 10�3 4:364 10�3 3:638 10�3 3:583 10�3
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742 739
pT ¼ p � cos l, or better with the curvature k / 1=pT, which has nosingularity at large pT. While the momentum p is conserved bymultiple scattering, the transverse momentum pT is directlyaffected (see below). With the definition of the relative momen-tum resolution:
Dp
p¼
p� pt
pt
where p is the fitted and pt is the true momentum, andpT ¼ p cos l, one gets
Dp
p¼
pT
cos l�
ptT
cos lt
ptT
coslt
¼
pTcos lt
cos l� pt
T
ptT
. (20)
Comparing this relation with DpT=pT ¼ ðpT � ptTÞ=pt
T, one can easilysee that the difference between DpT=pT and Dp=p depends only onthe term coslt= cos l. In the following, the influence of this termwill be investigated.
Assuming the difference Dl ¼ l� lt between the fitted andthe true angle l to be small w.r.t. lt, one can neglect terms of orderOðDl2
Þ, yielding
cos l ¼ cosðltþ DlÞ ¼ cos lt
� cosDl� sin lt� sinDl
� cos lt� sin lt
� Dl
and accordingly
cos lt
cos l�
coslt
coslt� sin lt
�Dl¼
1
1� tan lt�Dl� 1þ tanlt
� Dl.
(21)
In the last step we have used the first-order approximation ð1þxÞ�1� ð1� xÞ for small x. Inserting Eq. (21) into Eq. (20), one
obtains
Dp
p¼
p� pt
pt�
pT � ð1þ tanlt� DlÞ � pt
T
ptT
¼pT � pt
T
ptT
þpT
ptT
� tanlt� Dl �
DpT
pTþ tan lt
�Dl
assuming that DpT ¼ pT � ptT is small w.r.t. pt
T, i.e. pT=ptT � 1. Thus,
Dp=p is equal to DpT=pT plus a correction term dependent on l.Calculating the variance of Dp=p, one obtains
varDp
p
� �� var
DpT
pT
� �þ tan2 l � varðlÞ þ 2 tan l � cov l;
DpT
pT
� �(22)
where we have used varðDlÞ ¼ varðlÞ and covðDl;DpT=pTÞ ¼
covðl;DpT=pTÞ. The superscript ‘‘t’’ for the true dip angle lt hasbeen suppressed. As long as multiple scattering is not dominant,both varðlÞ and covðl;DpT=pTÞ are small, so that we do not have tocare about the difference between DpT=pT and Dp=p and it indeedmakes sense to use DpT=pT as a characteristic value for themomentum resolution. But as soon as multiple scattering gainsstronger influence, the polar angle error becomes comparable ordominant w.r.t. the error of pT, and the correlation between themomentum and the angle cannot be neglected anymore.
Since usually k is one of the track parameters in the fit, we getcovðl;kÞ from the covariance matrix rather than covðl;DpT=pTÞ.Using the relation k / 1=pT, one can prove the validity of theidentity:
cov l;DpT
pT
� �¼ �
1
k � covðl;kÞ.
With the formulas derived in Section 2.2.1.1 we can expresscovðl;kÞ in terms of varðlÞms:
varðlÞms ¼ a1;1 � r�2 � cosl
covðl;kÞ ¼ a1;1 � kref � r�3 � sin l ¼ varðlÞms � kref � r
�1 � tanl
with varðlÞms � varðlÞ � varðlÞdet. This allows us to write
cov l;DpT
pT
� �¼ � tanl � ðvarðlÞ � varðlÞdetÞ.
Here, we have used the relation r ¼ kref=k. Inserting this inEq. (22) yields a global formula for the relative momentumresolution:
s2 Dp
p
� �¼ s2 DpT
pT
� �þ tan2 l � 2s2ðlÞdet � s2ðlÞ
. (23)
In practise, sðlÞdet5sðlÞ as long as multiple scattering dominates,whereas in the detector error dominated case sðlÞ �sðlÞdet5sðDpT=pTÞ holds, so that we are free to use the simpleform of Eq. (23), which reads
s2 Dp
p
� �¼ s2 DpT
pT
� �� tan2 l � s2ðlÞ. (24)
Since each one of the quantities involved is the result of a correctfit, and therefore contains both the influence of detector errorsand the influence of multiple scattering, the global formula (23)holds for the detector error dominated case, the multiplescattering dominated case, as well as for the region in between.However, usually it is sufficient to use the simple form (24),because in well behaving detector setups sðlÞdet does notdominate the dip angle resolution down to the momentumregion, where the transverse momentum resolution sðDpT=pTÞ
becomes comparable to it. If multiple scattering can be neglected,i.e. in the asymptotic case of large momenta, the correction termcontaining sðlÞ becomes small and can be neglected.
Table 1 shows a comparison between the results of the globalformula (24) and the results of a simulation with LDT in theexample silicon detector mentioned in Section 4. An additionalpassive layer of 5% radiation length has been added in front of theinnermost sensitive layer to increase the uncertainty of l, whichmakes the difference between sðDpT=pTÞ and sðDp=pÞ visible.
The values of sðlÞsim, sðDpT=pTÞsim and sðDp=pÞsim are com-puted from 1000 simulated tracks at pT ¼ 1 GeV=c for fourdifferent dip angles l. They are used to calculate sðDp=pÞappr
according to the simple form of the global formula (24). Columns4 and 5 of Table 1 show that the global formula (24) yields resultswith excellent agreement, even for the worst case, which is aprojected helix radius of 0.75 m, corresponding to pT ¼ 1 GeV=c
and jBj ¼ 4 T.Fig. 11 shows the values of sðlÞsim, sðDpT=pTÞsim, sðDp=pÞsim and
sðDp=pÞappr according to the simple form of the global formula(24) as function of the transverse momentum pT at l ¼ 45�. Atvery low momenta, sðlÞ is comparable to the relative transversemomentum resolution, and therefore sðDpT=pTÞ and sðDp=pÞ
differ. When going to higher momentum, sðlÞ decreases while
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100 101 102 10310−4
10−3
10−2
10−1
σ (Δ
p T/p
T),σ
(Δp/
p),σ
(λ) [
1]
σ(ΔpT/pT)sim
σ(Δp/p)sim
σ(λ)sim
σ(Δp/p)appr
Transverse momentum pT [GeV/c]
λ = 45 [deg], σ (z) = 5 [μm]
Fig. 11. Examining the simple form of the global formula (24).
100 101 102 10310−3
10−2
10−1
σ (Δ
p T/p
T),σ
(Δp/
p),σ
(λ) [
1]
σ(ΔpT/pT)sim
σ(Δp/p)sim
σ(λ)sim
σ(Δp/p)appr
λ = 45 [deg], σ (z) = 5000 [μm]
Transverse momentum pT [GeV/c]
Fig. 12. Examining the full form of the global formula (23).
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742740
sðDpT=pTÞ increases, so we are free to neglect the correction termfor higher momentum.
As soon as the variance of l due to detector errors grows verybig, one has to use the full form of the global formula (23). Toshow this, we increase the detector accuracy in z tosðzÞ ¼ 5000mm. Now the term containing sðlÞdet cannot beneglected anymore. Note, however, that in practise this case canbe ignored. Fig. 12 shows the values of sðlÞsim, sðDpT=pTÞsim,sðDp=pÞsim and sðDp=pÞappr according to the full form of the globalformula (23) as a function of the transverse momentum pT atl ¼ 45�, for a detector setup with sðRFÞ ¼ 5mm andsðzÞ ¼ 5000mm.
Here, sðlÞ is dominated by sðlÞdet down to low momenta,where it becomes comparable to sðDpT=pTÞ. Therefore, in thisregion, sðDp=pÞ is larger than sðDpT=pTÞ. At very low momentum,multiple scattering dominates the uncertainty of l and sðDp=pÞ
grows smaller, whereas at very high momentum the correctionterms containing the uncertainty of l are negligible w.r.t.sðDpT=pTÞ again.
5.2. The contribution of multiple scattering
At low momenta the momentum resolution is dominated bymultiple scattering, and we can calculate sðDp=pÞ according toEq. (24). Expressing varðDpT=pTÞ from Eq. (24) and multiplying itby k2 yields
k2 � varDpT
pT
� �¼ k2 � tan2 l � varðlÞ þ k2 � var
Dp
p
� �. (25)
From error propagation using DpT=pT ¼ �Dk=k ¼ 1� k=kt we canconclude
varDpT
pT
� �¼
@DpT
pT
@k
0BB@
1CCA
2
� varðkÞ ¼ 1
ðktÞ2� varðkÞ
¼)k2 � varDpT
pT
� �¼ varðkÞ
with k � kt. On the other hand, using the approximate formulasfrom Section 2.2.1.1 we can prove the identity:
k2 � tan2l � varðlÞ ¼ varðkÞmsðlÞ
where we assume multiple scattering to dominate, i.e. varðlÞ �varðlÞmsðlÞ. So, Eq. (25) takes the form
varðkÞ ¼ varðkÞmsðlÞ þ k2 � varDp
p
� �.
Comparing this with varðkÞ ¼ varðkÞmsðlÞ þ varðkÞmsðjÞ we identify
varDp
p
� �¼
1
k2� varðkÞmsðjÞ.
Thus, varðDp=pÞ is affected by multiple scattering in j only and isinvariant w.r.t. multiple scattering in l, because the latter changespT but not p. In contrast, varðDpT=pTÞ is affected by multiplescattering in both l and j. In that sense, the global formula (24) isable to separate the contributions of multiple scattering in l andin j to the momentum resolution. On the other hand, fromSection 2.2.1.2 we know that varðkÞmsðjÞ is proportional to k2. So,varðDp=pÞ does not depend on pT in the multiple scatteringdominated case of low momentum.
6. MATLAB/Octave and Java implementation
The exact calculation of the covariance matrix of the trackparameters by a linear LS fit has been implemented in a MATLABfunction called Compute_C_MS. This implementation is fullycompatible with Octave 3.0.1 [11]. It allows both cylindrical andplane detector surfaces and includes the track positions ðy; zÞ forplane detectors and ðRF; zÞ for cylindrical detectors. The userinterface is shown in Fig. 13. The function is available from one ofthe authors (RF).
The exact calculation has also been implemented in Java, aspart of the JDOT Java Detector Optimization Tool [18]. This tooloptimizes the detector configuration by minimizing variousobjective functions that are based on the covariance matrix ofthe track parameters.
7. Conclusion and outlook
An analytical method has been developed that allows todescribe the momentum and direction resolution of a barreldetector with only eight coefficients. Four of them are needed toset up the covariance matrix of multiple scattering, while anotherfour are used to build the covariance matrix of detector errors.
ARTICLE IN PRESS
Fig. 13. User interface of the MATLAB/Octave function Compute_C_MS.
M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742 741
These coefficients can be determined from the covariancematrices of only two single tracks in the plane perpendicular tothe magnetic field: one low energetic track yielding thecoefficients of the multiple scattering covariance matrix, andone high energetic track yielding the coefficients of the detectorerror covariance matrix. Both tracks can be assumed to start at thebeam axis with x ¼ 0 and y ¼ 0, thus implying independence ofthe azimuthal angle j.
The derived approximate formulas allow the detector optimi-zation to be carried out in the plane perpendicular to the magnetic
field only. This plane can be left using the quoted l dependencies,while the derived pT dependencies allow to determine thedetector’s behaviour at any desired momentum.
The method is applicable under the following conditions:
(1)
Rotational symmetry of the detector. (2) Invariance w.r.t. translations parallel to the magnetic field (e.g.no z dependent resolution).
(3) The helix radius RH has to be at least twice as large as theradius of the largest detector cylinder.
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M. Valentan et al. / Nuclear Instruments and Methods in Physics Research A 606 (2009) 728–742742
points at la0 to model the l dependence of the resolution.
Condition (2) can be overcome by considering several referenceNote that condition (3), i.e. RH 2 �maxðRiÞ, only applies to thelow momentum reference track with pT ¼ pref
T . The analyticalformulas can be used to extrapolate the results for pTopref
T to acertain extent.
The variance of k due to multiple scattering has beenapproximated neglecting the contribution of multiple scatteringin l. This, however, has to be proven in practice. The case of a TPCcombined with silicon layers shows that the contribution ofmultiple scattering in l can become important.
Acknowledgements
Thanks are due to Winfried Mitaroff for fruitful discussions.Moreover, we thank the reviewer for many detailed and mindfulcomments.
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