- Email: [email protected]
Particle identification with a track fit χ2
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Particle identification with a track fit w2 Ferenc Sikle´r KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary
a r t i c l e in fo
abstract
Article history: Received 14 November 2009 Received in revised form 7 February 2010 Accepted 8 March 2010 Available online 16 March 2010
Tracker detectors can be used to identify charged particles based on their global w value obtained during track fitting with the Kalman filter. This approach builds upon the knowledge of detector material and local position resolution, using the known physics of multiple scattering and energy loss. The proposed method is independent of the traditional way of identification using deposited energy. The performance for present LHC experiments is demonstrated. & 2010 Elsevier B.V. All rights reserved.
Keywords: Particle identification Multiple scattering Energy loss Tracker detectors
1. Introduction The momentum of a charged particle can be measured by examining the small angle scatters of the trajectory during propagation through the detector medium or tracker layers. For a recent application see Ref. [1] where the root mean square of the scattering angle distribution is computed for each track and compared to the theoretical estimate which is proportional to 1=bp. By assuming particle type, or at high momentum ðb 1Þ, p can be estimated. This classical method underestimates momentum since the particle loses energy and its momentum decreases. The Kalman filter is widely used in present particle physics experiments for charged track and vertex fitting and provides a coherent framework to handle known physical effects and measurement uncertainties [2]. It is equivalent to a global linear least-squares fit which takes into account all correlations coming from process noise. It is the optimum solution since it minimizes the mean square estimation error. Recent studies show that this technique can be successfully used to improve momentum resolution of particles, even in experiments without magnetic field [3]. It is possible via the effects of multiple scattering. If the detector is in magnetic field, the momentum of charged particles can be obtained from the bending of the trajectory. Hence track fitting may provide additional information that could constrain the velocity of the particle, thus contributing to particle separation or identification. This article is organized as follows: Section 2 introduces the merit function of a track fit w and discusses its characteristics.
Tel.: + 36 1 392 2222; fax: + 36 1 395 9151.
E-mail address: [email protected] 0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.03.098
Section 3 deals with physical effects during track propagation, while in Section 4 the basic scaling properties of w are given. In Section 5 the details of the Monte Carlo simulation and the obtained performance are shown. This work ends with conclusions and it is supplemented by two Appendices.
2. The merit function of the fitted track There are various merit functions that can characterize the goodness of a track fit: sum of the squared and properly normalized predicted (P), filtered (F) or smoothed (S) residuals. It can be easily shown that for each hit w2P ¼ w2F . The filtered residuals are uncorrelated and in the Gaussian case independent. P P 2 wF is chi-square distributed with r ¼ ½ k dimðmk Þnp Hence degrees of freedom, where dim(mk) is the dimension of the k th hit on track and np is the number of track parameters. Tests with smoothed residuals (e.g. for outlier removal) appear to be more powerful [2], but the correlations of these residuals between the states have to be taken into account. Their global covariance matrix Rkl between smoothed states k and l can be calculated [4] with the recursion n n ¼ Ak1 Ck,l , Ck1,l
kr l
and n HlT Rkl ¼ Vk dkl Hk Ck,l
where C is the smoothed covariance matrix, A is the gain matrix, V is the covariance of measurement noise, H is the measurement projection matrix. Here we follow the notations of Refs. [2,4]. The vector of smoothed residuals is described by a multivariate Gaussian distribution with the global covariance R obtained
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
478
above. Since Kalman filtering consists of a series of linear transformations, the smoothed residuals can be obtained from the predicted ones by a linear transformation rS ¼BrP. Note that no translation is allowed, since the average of both residuals is zero. The global covariance matrix of predicted residuals is RP , the covariance for rS is RS ¼ BRP BT . Thus, the expression for the corrected sum of smoothed values is X 0 T T 1 w2S ¼ rST R1 ðBP rP Þ S rS ¼ ðBP rP Þ BP RP BP X T T T 1 1 w2P : ¼ ðrP BP Þ BP 1RP BP ðBP rP Þ ¼ rPT R1 P rP ¼ It is clear that the correlations are transformed out and we get P 2 P 2 wP ¼ wF ¼ back simply the predicted or filtered values: P 2 0 ð wS Þ . Hence the most straightforward quantity to calculate is P w2P using predicted residuals which will be used in the sum w2 the remaining part of this study. During track propagation the mass of the tracked particle has to be assumed. In collider experiments it is often set to the mass of the most abundantly produced particle, the pion, or that of the muon. The obtained merit function with mass assumption m0 is X w2 ðm0 Þ ¼ rkT R1 k rk k
where the index k runs for all the measurements and Rk is the local covariance matrix for the k th measurement. If the largest contributions to Rk are independent in rf and z directions, w2 can be written as Xxi m ðm0 Þ2 X si ðmÞ 2 xi m ðm0 Þ2 X i i w2 ðm0 Þ ¼ ¼ ai zi si ðm0 Þ si ðm0 Þ si ðmÞ i i i ð1Þ where i runs for all split measurements, xi is the measured coordinate, mi ðmÞ is the predicted coordinate with mass hypothesis m, and s2i lists the diagonal elements of the covariance matrices R. The resulted sum is a linear combination of noncentrally chi-square distributed independent random variables zi with weights ai. The distribution functions are fX ðzi ; 1,li Þ where si ðmÞ 2 mi ðmÞmi ðm0 Þ 2 ai ¼ , li ¼ : si ðm0 Þ si ðmÞ The sum in Eq. (1) can approximated by a single rescaled non2 central chi-squared distribution 1=a2 fX ðx=a2 ; r,l Þ such that P 2 P 2 X a a a2 ¼ Pi i , r ¼ Pi i2 np , l2 ¼ li a i i i ai i where np is the number of track parameters. For details see Appendix B. If m¼m0, we get ai ¼1, a ¼ 1, l ¼ 0, and the distribution is a chi-squared one. If the ratio of expected variances ai are similar for all i, namely if the variances are scattering dominated, we get * + si ðmÞ 2 ð2Þ a2 si ðm0 Þ and r is the number of split measurements decreased by the number of track parameters. pffiffiffiffiffiffi At the same time the use of the variable w w2 appears to be more practical. It is described by a scaled non-central chi-distribution 1=a f ðw=a; r,lÞ and well approximated by a Gaussian with parameters qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffi ð3Þ mw ¼ a r12 þ l2 , sw ¼ a 12: For detailed derivation see Appendix A. The value of w can be calculated for each track during the track fit with Kalman filter. For different type of particles it will have different distribution function, because the parameters mw and sw
(via a and r) depend on the ratio of expected hit deviations si ðmÞ=si ðm0 Þ which are mass dependent (see Section 3). This observation allows to use this quantity in particle identification. Using the Gaussian approximation of Eq. (3), the separation power rw of w between particles of mass m1 and m2 is 2½mw ðm1 Þmw ðm2 Þ : rw ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2w ðm1 Þ þ s2w ðm2 Þ
ð4Þ
3. Physical effects When a stable charged particle propagates through material the most important effects which alter its momentum vector are multiple scattering (ms) and energy loss (el). In the following the expected spatial shift d and deviation s will be calculated. They are to be compared with the resolution of the local position measurement spos of the tracker layers. The distribution of multiple Coulomb scattering is roughly Gaussian [5], the standard deviation of the planar scattering angle is
y0 ¼
13:6 MeV pffiffiffiffiffiffiffiffiffiffi z x=X0 ½1 þ 0:038lnðx=X0 Þ bcp
ð5Þ
where p, bc, and z are the momentum, velocity, and charge of the particle in electron charge units, and x=X0 is the thickness of the scattering material in radiation lengths. While the expected shift is dms ¼ 0, the average deviation on the next tracker plane after a flight path l, in case of normal incidence, is
sms ly0 :
ð6Þ
Momentum and energy is lost during traversal of sensitive detector layers and support structures. To a good approximation the most probable energy loss D p , and the full width of the energy loss distribution at half maximum GD [6] are " # 2 2mc2 b g2 x 2 ð7Þ D p ¼ x ln þ0:2000 b d I2
GD ¼ 4:018x
ð8Þ
where
x¼
K 2Z x z r 2 A b2
is the Landau parameter; K ¼ 4pNA re2 me c2 ; Z, A and r are the mass number, atomic number and the density of the material, respectively [5]. Since this study deals with momenta below 2 GeV=c, the density correction d was neglected. In most cases tracker detectors are placed in magnetic field (B). Given the radius of the trajectory r and the length of the arc l, the central angle is j ¼ l=r. If the radius is changed by dr, the angle changes by dj ¼ l=r2 dr and the position shift of the trajectory after l path is
del ldj=2 ¼ l2 =2dr=r2 : At the same time p ¼0.3Br, E dE¼ p dp. Hence 2
del
0:3Bl /DS : 2 bp2
Similarly, the expected deviation is 2
sel
0:3Bl sD : 2 bp 2
The contributions to deviations and shifts of the predicted hit in a B ¼3.8 T magnetic field, after crossing x=X0 ¼ 2% silicon and further l¼ 5 cm propagation before reaching the next layer, are shown in Fig. 1. Standard deviations are dominated by multiple scattering, although at very low momentum the energy loss, at
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
K p e
0.4 0.3 0.2 0.1 0
0.5 Shift (w.r.t.apion) [mm]
Standard deviation [mm]
0.5
K p e
0.4 0.3 0.2 0.1 0
Multiple scattering -0.1
479
Energy loss -0.1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p [GeV/c]
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p [GeV/c]
Fig. 1. The contributions to deviations and shifts of the predicted hit in a B¼ 3.8 T magnetic field, after crossing x=X0 ¼ 2% silicon and further l ¼ 5 cm propagation before reaching the next layer, as a function of particle momentum. Left: expected standard deviations due multiple scattering. Right: expected shifts, compared to an average propagation with p mass assumption, due to energy loss. The curves give the limits of the lower and upper 7 1s confidence intervals for several particle types. For comparison lines corresponding to a local position resolution of 25 mm are drawn.
very high momentum the local position measurement also plays a role. Shifts from energy loss are only relevant at very low momentum, but they are still very small compared to standard deviations.
5 K p e
4
4. Properties of v
3
which can be further simplified, if spos 5 sms , to " !# bðm0 Þ z2 b2 ðmÞ 1 1 2 a bðmÞ 2 b ðm0 Þ
ð9Þ
Shifts come entirely from differences in energy loss, hence contributions to l are only substantial at low momentum: ! pffiffiffiffiffi pffiffiffi 0:3Bl2 /DðmÞDðm0 ÞS l rx 1 1 : l r p p 2bp2 ly0 b2 ðmÞ b2 ðm0 Þ The average shift /lS in a B ¼3.8 T magnetic field, with layer thicknesses of x=X0 ¼ 2% silicon, an average propagation length of l ¼5 cm, in case of r ¼16 number of degrees of freedom, is shown in Fig. 2. If l, z 51, the separation power rw between particles m and m0 is pffiffiffiffiffiffiffiffiffiffiffiffi
1bðmÞ=bðm Þ
0 : rw 2 2r1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1
0 0.2
0.4
0.6
0.8
1 1.2 p [GeV/c]
1.4
1.6
1.8
2
Fig. 2. The average shift for several particle types, in a B¼ 3.8 T magnetic field, with layer thicknesses of x=X0 ¼ 2% silicon, an average propagation length of l ¼5 cm, in case of r ¼ 16 number of degrees of freedom, as a function of the particle momentum.
where the sensitivity is defined as z ¼ spos =sms ðm0 Þ, it is proportional to 1=bp (Eqs. (5) and (6)). If the local position resolution can be neglected ðz 5 1Þ we get
bðm0 Þ : a bðmÞ
r = 16
It is important to study the sensitivity of the measured w distribution at a given total momentum p. The parameters which govern the distribution (Eq. (3)) are the rescaler a, the average shift l and the number of degrees of freedom r. In this section we estimate them, as well as the separation power rw listed in Eq. (4), based on physical effects. Since the deviations are dominated by multiple scattering and local position measurement, a in Eq. (2) can approximated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2ms ðmÞ þ s2pos a s2ms ðm0 Þ þ s2pos
ð10Þ
1 þ ½bðmÞ=bðm0 Þ2
Hence if the momentum is not very low and the local position resolution is small compared to deviations from multiple scattering, neither the rescaler a nor the separation power rw depends on the details of the experimental setup, such as
magnetic field, radii of tracker layers, value of local position resolution and material thickness. In this respect the only decisive parameter is the number of split measurements which enters the above expressions by the number of degrees of freedom r. The mean and variance of the corresponding Gaussians are fully determined by the momentum and mass of the particles via b. Although at low momentum the prediction of the means is more difficult due to the increasing l, the variances still stay the same. The measured w distribution of tracks in a phase space bin can be fit with a linear combination of Gaussians, giving the relative yields of the different particle species. The obtained separation power allows for a many-parameter fit.
4.1. Applications The measured value of w is sensitive to the proper spatial alignment of the detector layers and to the correct estimate of the variation of the predicted local position. If the alignment precision is sufficient, the latter is mostly determined by the contribution pffiffiffiffiffiffiffiffiffiffi from multiple scattering which is closely proportional to x=X0 .
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
480
Table 1 Important characteristics of the inner barrel detectors of the studied experimental setup. B (T)
Subdetector
Radius of layers (cm)
srf ðmmÞ
sz ðmmÞ
x=X0 (%)
zrf
zz
Split meas.
Exp A
2
Pixels (barrel) Strips (SCT)s Straw (TRT)
5.0, 8.8, 12.2 29.9, 37.1, 44.3, 51.4 56.3–106.6 (r 36 hits)
10 17 130
115 580 –
4 4 0.5
0.1 0.1 10
1 3 –
50
Exp B
0.4
Pixels (SPD) Drifts (SDD) Strips (SSD)s [gas (TPC)
3.9, 7.6 14.9, 23.8 38.5, 43.6 84.5 – 246.6 ( r 159 hits)
12 35 15 900
100 23 730 900
1 1 1 10 3
0.2 0.3 0.1
2 0.2 7 103–104]
12
Exp C
3.8
Pixels (PXB) Strips (TIB)s
4.4, 7.3, 10.2 25.5, 33.9
15
15 230
3 4
0.2 0.1
0.2 0.8
20
Strips (TIB) Strips (TOB)s
41.8, 49.8 60.8, 69.2
– 530
2 4
0.2 0.1
– 2
Strips (TOB)
78.0, 86.8, 96.5, 108.0
–
2
0.2
–
pffiffiffi 23= 2 35 pffiffiffi 53= 2 53, 35
For details see text at the beginning of Section 5.
Exp A
8000
Exp B
K p e
Exp C
6000
=0 pT = 0.4GeV/c
4000 2000 0 0
5
10
15
20 0
5
10
15
20 0
5
10
15
20 K p e
8000 6000
=0 pT = 0.8GeV/c
4000 2000 0 0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
Fig. 3. Distributions of w for several particle species. The relative yield of particles was set to p : K : p : e ¼ 70 : 10 : 18 : 2. Results are shown for Z ¼ 0, pT ¼ 0:4 GeV=c (upper row) and Z ¼ 0, pT ¼ 0:8 GeV=c (lower row) with setups Exp A, B and C. Individual fits with chi distributions are indicated by thin solid lines.
While p and r are well measured, the amount of material in the detector can be
Understood: The unfolding of the w distribution in a phase
space bin enables the measurement of yields of different particle species. Poorly known: The unfolding of the w distribution in a phase space bin may provide corrections to the material thickness. They can be extracted by fitting the w distribution with an additional rescaler. Note that the measurement of yields of different particle species is still possible, although with lower confidence.
5. Simulation The proposed method was verified by a Monte Carlo simulation. As examples from LHC, the performance of simplified models for the inner detectors of the following experiments were studied:
ATLAS (Exp A): Three layers of silicon pixels, five layers of double-sided silicon strips, up to 36 layers of straw tubes [7].
ALICE (Exp B): Two layers of silicon pixels, two layers of silicon
drifts and two layers of double-sided silicon strips [8,9]. Due to the large z value of the gas detector (TPC) its measurements were not included. CMS (Exp C): Three layers of silicon pixels, 10 layers of silicon strips (four of them double-sided) [10].
Some relevant details of the experimental setups are given in Table 1. For simplicity a homogeneous longitudinal magnetic field was used, and detector layers were assumed to be concentric cylinders around the beam-line. Pixels, double-sided strips (superscript s), drift layers and gas provide measurements in two dimensions (rf and z), while one-sided strips and straw tubes give only measurement in one direction ðr fÞ. While double-sided strips consists of two slightly rotated layers separated by a small gap in reality, in this simulation they were treated as single twodimensional layers with very different position resolution in r f and z directions. x=X0 values are given per layer and they are rounded to integers where possible. Sensitivity values zrf and zz are shown for pions at p ¼ 1 GeV=c, normal incidence, rounded to
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
6
6 Exp A
6 Exp B
5
2f it
Exp C
5
4
4
4
3
3
3
2
2
2
1
1
1
0 50
0 10
0 20
40
8
30
6
20
4
K p e
15 10
r
5
481
10
2
0 6
0 6
0 6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0 12
0 12
0 12
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0 0.5
1 1.5 p [GeV/c]
2
K p e
5
K p e
–K –p –e
0 0.5
1 1.5 p [GeV/c]
2
0.5
1 1.5 p [GeV/c]
2
Fig. 4. Performance of w measurement for particle identification, with setups Exp A, B and C. For details see text in Section 5.1.
one significant digit. The number of split measurements are also indicated. The initial state vector was estimated by fitting a helix to the first three hits. (These hits are two-dimensional in all three examined experimental setups.) The starting values of the track parameters were extracted at the closest approach to the beam line. The track fitting was performed by a classical Kalman filter [2] with pion mass assumption. The state vector x ¼ ðk,y,c,rf,zÞ is five dimensional, where
The propagation from layer to layer was calculated analytically using a helix model. Multiple scattering and energy loss in tracker layers was implemented with their Gaussian approximations shown in Eqs. (6)–(8). The propagation matrix F ¼ @f =@x was obtained by numerical derivation. The measurement vector m ¼ ðr f,zÞ is two-dimensional, the measurement operator is 0 0 0 1 0 H¼ : 0 0 0 0 1
k ¼ q=p ðsigned inverse momentumÞ
The covariance of the process noise Q is
y ¼ yð~ p Þ ðlocal polar angleÞ
Q ¼ ðFk FkT Þs2k þðFy FyT Þs2y þ ðFc FcT Þs2c
c ¼ fð~ p Þ ðlocal azimuthal angleÞ rÞ rf ¼ r fð~ z ¼ rL
ðglobal azimuthal positionÞ
ðglobal longitudinal positionÞ:
where sk ¼ ksD =b, sy ¼ sc ¼ y0 and Fa ¼ @f =@xa is a vector. The covariance of measurement noise V is ! s2rf 0 : V¼ 0 s2z
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
Note that multiple scattering contributes equally to the variation of y and c, while energy loss affects only k. A trajectory may contain outliers, hits that do not belong to the particle track. They can be correlated with the track (d’ s) or uncorrelated (detector noise, hits from another track). The w2 distribution of the outlier hits is imposed by pattern recognition and selection during track finding. In order to simulate this effect 1% of the hits were randomly flagged as outliers and the covariance of the measurement noise was multiplied by a factor 4 during hit creation in the detector. The outliers can be suppressed after the track fit. All hits are examined and those with smoothed w2 greater than 50 are removed and the fit is redone [2]. Note that this cut somewhat reduces the separation power of the global track w value at very low momentum, since it may remove some outlying measurements if the generated particle was a kaon or a proton.
5.1. Results
10 Exp A Exp B Exp C
8
6
482
4
2
0 0.5
In order study the performance of w, charged pions, kaons, protons and electrons with random azimuthal angle were generated and emitted normal to the line of the colliding beams ðZ ¼ 0Þ and run through the above outlined reconstruction. Distributions of w using 105 particle tracks are shown in Fig. 3 for pT ¼0.4 and 0:8 GeV=c. For a realistic particle composition the relative yields were set to p : K : p : e ¼ 70 : 10 : 18 : 2. At pT ¼ 0:4 GeV=c, in case of Exp A, the protons are detached, but there is a good p2p separation for Exp B and C, as well. For Exp A and C the p2K separation allows for yield estimation. Even at pT ¼ 0:8 GeV=c the observed resolution is enough to extract the protons. When fitting the histograms a sum of chi distributions was employed (thin solid lines), but a sum of Gaussians may also be sufficient. For a complete picture charged pions, kaons, protons and electrons with transverse momenta pT ¼ 0:2,0:4,0:6, . . . ,2:0 GeV=c and Z ¼ 0 were used, amounting to 104 particles per pT setting for each particle type and experimental setup. The performance of w as function of p for all three setups is shown in Fig. 4. The subsequent rows give the dependence of the measured rescaler a, the fitted number of degrees of freedom r, the merit function of the histogram fit with sum of chi distributions w2fig and the separation power rw . This latter was calculated by using the measured a and r values with help of Eqs. (3) and (4). The measured values are shown by the symbols. In case of a the line gives the plain bðm0 Þ=bðmÞ scaling (Eq. (9)) that works well for all three setups and for all particle types. The deviations seen at very low momentum ðp ¼ 0:2 GeV=cÞ are due to the w2 cut in the outlier removal discussed in Section 5. For r the horizontal lines show the number of split measurements for a given pT, decreased by the number of track parameters np. While these predictions are closely followed by the measured values in case of Exp C, there are substantial deviations with the other two setups. It can be traced back to low sensitivity measurements, whose contribution to the global w2 is very small and the measurement does not increase the number of degrees of freedom: large number of straw tubes with zrf ¼ 10 (Exp A), and two strip layers with zz ¼ 7 (Exp B). Also outliers and their removal procedure introduces shifts in the fitted value of r. In case of the separation power rw the lines show the approximation based on the predicted number of degrees of freedom and the ratio bðmÞ=bðm0 Þ, calculated with help of Eq. (10). The steps are due to the changing number of crossed detector layers with varying p. The approximation works well for Exp C, but strongly overestimates the measured value for Exp A. It is again due to the large number of low sensitivity measurements.
1 p [GeV/c]
1.5
2
Fig. 5. The p2p separation power of the w measurement for the experimental setups, as a function of momentum. The lines are drawn to guide the eye.
Comparison of the p2p separation power of the w measurement for several experimental setups as a function of momentum is shown in Fig. 5. While Exp A clearly performs better for p o0:6 GeV=c, Exp C has better resolution for the more critical higher momentum region. With the most sensitive setups (Exp A and C) protons are 1s apart if p o1:4 GeV=c, while 2s separation is reached if p o 1 GeV=c. For kaons these numbers are p o 0:9 and 0:5 GeV=c, respectively.
6. Conclusions It was shown that tracker detectors can employed to identify charged particles based on their global w obtained during track fitting with the Kalman filter. Since the w value of the filter is equivalent to that of a global fit, the method is suitable for any minimum w2 track fit that properly models energy-loss and scattering effects. This approach builds upon the knowledge of detector material and local position resolution, using the known physics of multiple scattering and energy loss. The study using simplified models of present LHC experiment shows that p2K and p2p unfolding is possible at low momentum. The separation is better than 1s for p o 0:9 and 1:4 GeV=c, respectively. In general, the performance of an experiment is determined by the number of good sensitivity split measurements. It is also a strong function of particle momentum.
Acknowledgements The author wishes to thank to Krisztia´n Krajcza´r for helpful discussions. This work was supported by the Hungarian Scientific Research Fund and the National Office for Research and Technology (K 48898, K 81614, H07-B 74296).
Appendix A. Properties of some distributions In this section the definitions of some used distributions are listed along with their calculated or approximated values for the mean m and variance s2 .
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
Note that with l ¼ 0 we get back the mean of the w distribution (Eqs. (12) and (14), while the variances are the same in the central and non-central case (Eqs. (13) and (15))).
A.1. w2 distribution The distribution, mean and variance are Pðx; rÞ ¼
483
xr=21 ex=2 r G 2r=2 2
Appendix B. Sum of non-central chi-squared distributed independent variables
m ¼ r, s2 ¼ 2r:
The goal is to approximate the sum A.2. Non-central w2 distribution y¼ The distribution, mean and variance are pffiffiffi pffiffiffiffiffi eðx þ lÞ=2 xðr1Þ=2 l Ir=21 ð lxÞ Pðx; r,lÞ ¼ r=4 2ðlxÞ
where zi are non-central chi-squared distributed independent random variables with one degree of freedom and density function fX ðzi ; 1,li Þ. Although an explicit expression for the distribution of y exists, it is difficult to evaluate in practice [14]. Here this function is approximated by a rescaled non-central chi-squared distribution 1=a fX ðx=a; r,lÞ by requiring that the first two moments be the same. The means and variances are additive, thus the equations two solve are X /yS ¼ ai ð1 þ li Þ ¼ aðr þ lÞ
where In(x) is the modified Bessel function of the first kind. A.3. w distribution The distribution and mean are
i
2 =2
Pðx; rÞ ¼
G
ð11Þ
2
pffiffiffi rþ1 rffiffiffiffiffiffiffiffiffiffi 2G pffiffiffi 1 1 1 2 r m¼ ¼ r 1 þ O 2 r 4r 2 r G 2
X /ð y/ySÞ2 S ¼ 2 a2i ð1 þ 2li Þ ¼ 2a2 ðr þ 2lÞ: i
By assuming li 5 1 we get ð12Þ
P
a2i , i ai
a ¼ Pi
where Ref. [11] for r b 1 was used. The variance is
s2 ¼ rm2 12:
ai zi
i¼1
m ¼ r þ l, s2 ¼ 2ðr þ 2lÞ
21r=2 xr1 ex r
n X
ð13Þ
P 2 ai r ¼ Pi 2 , i ai
l¼
X
li
i 2
with relative corrections of the order Oðl =r 2 Þ. If the values of ai are similar some of the above expressions can be approximated by
a /ai S, r n:
A.4. Non-central w distribution The distribution and mean are 2
Pðx; r,lÞ ¼ rffiffiffiffi
m¼
p 2
eðx
xl
ðlxÞ
r=2
2
ðr=21Þ
L1=2
References
þ l2 Þ=2 r
l 2
Ir=21 ðlxÞ
!
where L(a) n (x) is the generalized Laguerre function. For r b 1, with Kummer’s second formula [12] r þ1 ! rffiffiffiffi G 2 p 1 r l 2 m¼ F , , 3 1 1 2 r 2 2 2 G G 2 2 where 1 F1 ða,b,zÞ is the confluent hypergeometric function of the 2 first kind. With help of Eq. (12) and Ref. [13], assuming l 5r " #
pffiffiffi 1 1 l2 1 1þ þO 2 m ¼ r 1 þ O 2 2r 4r r r " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffi 12l 1 1 2 r þ l : þO 2 r 1 ð14Þ 4r 2 r For the variance
s ¼ rm2 þ l2 12: 2
ð15Þ
[1] A. Grossheim, K. Zuber, Nucl. Instr. and Meth. A 533 (2004) 532, doi:10.1016/ j.nima.2004.06.146. [2] R. Fruhwirth, Nucl. Instr. and Meth. A 262 (1987) 444, doi:10.1016/01689002(87)90887-4. [3] A. Ankowski, et al., Eur. Phys. J. C 48 (2006) 667, doi:10.1140/epjc/s10052006-0051-3. [4] W. Hulsbergen, Nucl. Instr. and Meth. A 600 (2009) 471, doi:10.1016/ j.nima.2008.11.094. [5] C. Amsler, et al., Phys. Lett. B 667 (2008) 1, doi:10.1016/j.physletb.2008. 07.018. [6] H. Bichsel, Rev. Mod. Phys. 60 (1988) 663, doi:10.1103/RevModPhys.60.663. [7] G. Aad, et al., Expected performance of the atlas experiment—detector, Trigger and Physics, 2009, arXiv:0901.0512. [8] F. Antinori, Nucl. Instr. and Meth. A 511 (2003) 215, doi:10.1016/S01689002(03)01796-0. [9] D. Antonczyk, et al., Nucl. Instr. and Meth. A 565 (2006) 551, doi:10.1016/ j.nima.2006.06.022. [10] R. Adolphi, et al., JINST 3 (2008) S08004, doi:10.1088/1748-0221/3/08/ S08004. [11] R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, second ed., Addison-Wesley, Reading, MA, 1994 (answer to Problem 9.60). [12] W. Koepf, Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities, Vieweg, Braunschweig, Germany, 1998. [13] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972. [14] S. Gabler, C. Wolff, Statist. Papers 28 (1) (1987) 317, doi:10.1007/ BF02932611.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Particle identification with a track fit w2 Ferenc Sikle´r KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary
a r t i c l e in fo
abstract
Article history: Received 14 November 2009 Received in revised form 7 February 2010 Accepted 8 March 2010 Available online 16 March 2010
Tracker detectors can be used to identify charged particles based on their global w value obtained during track fitting with the Kalman filter. This approach builds upon the knowledge of detector material and local position resolution, using the known physics of multiple scattering and energy loss. The proposed method is independent of the traditional way of identification using deposited energy. The performance for present LHC experiments is demonstrated. & 2010 Elsevier B.V. All rights reserved.
Keywords: Particle identification Multiple scattering Energy loss Tracker detectors
1. Introduction The momentum of a charged particle can be measured by examining the small angle scatters of the trajectory during propagation through the detector medium or tracker layers. For a recent application see Ref. [1] where the root mean square of the scattering angle distribution is computed for each track and compared to the theoretical estimate which is proportional to 1=bp. By assuming particle type, or at high momentum ðb 1Þ, p can be estimated. This classical method underestimates momentum since the particle loses energy and its momentum decreases. The Kalman filter is widely used in present particle physics experiments for charged track and vertex fitting and provides a coherent framework to handle known physical effects and measurement uncertainties [2]. It is equivalent to a global linear least-squares fit which takes into account all correlations coming from process noise. It is the optimum solution since it minimizes the mean square estimation error. Recent studies show that this technique can be successfully used to improve momentum resolution of particles, even in experiments without magnetic field [3]. It is possible via the effects of multiple scattering. If the detector is in magnetic field, the momentum of charged particles can be obtained from the bending of the trajectory. Hence track fitting may provide additional information that could constrain the velocity of the particle, thus contributing to particle separation or identification. This article is organized as follows: Section 2 introduces the merit function of a track fit w and discusses its characteristics.
Tel.: + 36 1 392 2222; fax: + 36 1 395 9151.
E-mail address: [email protected] 0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.03.098
Section 3 deals with physical effects during track propagation, while in Section 4 the basic scaling properties of w are given. In Section 5 the details of the Monte Carlo simulation and the obtained performance are shown. This work ends with conclusions and it is supplemented by two Appendices.
2. The merit function of the fitted track There are various merit functions that can characterize the goodness of a track fit: sum of the squared and properly normalized predicted (P), filtered (F) or smoothed (S) residuals. It can be easily shown that for each hit w2P ¼ w2F . The filtered residuals are uncorrelated and in the Gaussian case independent. P P 2 wF is chi-square distributed with r ¼ ½ k dimðmk Þnp Hence degrees of freedom, where dim(mk) is the dimension of the k th hit on track and np is the number of track parameters. Tests with smoothed residuals (e.g. for outlier removal) appear to be more powerful [2], but the correlations of these residuals between the states have to be taken into account. Their global covariance matrix Rkl between smoothed states k and l can be calculated [4] with the recursion n n ¼ Ak1 Ck,l , Ck1,l
kr l
and n HlT Rkl ¼ Vk dkl Hk Ck,l
where C is the smoothed covariance matrix, A is the gain matrix, V is the covariance of measurement noise, H is the measurement projection matrix. Here we follow the notations of Refs. [2,4]. The vector of smoothed residuals is described by a multivariate Gaussian distribution with the global covariance R obtained
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
478
above. Since Kalman filtering consists of a series of linear transformations, the smoothed residuals can be obtained from the predicted ones by a linear transformation rS ¼BrP. Note that no translation is allowed, since the average of both residuals is zero. The global covariance matrix of predicted residuals is RP , the covariance for rS is RS ¼ BRP BT . Thus, the expression for the corrected sum of smoothed values is X 0 T T 1 w2S ¼ rST R1 ðBP rP Þ S rS ¼ ðBP rP Þ BP RP BP X T T T 1 1 w2P : ¼ ðrP BP Þ BP 1RP BP ðBP rP Þ ¼ rPT R1 P rP ¼ It is clear that the correlations are transformed out and we get P 2 P 2 wP ¼ wF ¼ back simply the predicted or filtered values: P 2 0 ð wS Þ . Hence the most straightforward quantity to calculate is P w2P using predicted residuals which will be used in the sum w2 the remaining part of this study. During track propagation the mass of the tracked particle has to be assumed. In collider experiments it is often set to the mass of the most abundantly produced particle, the pion, or that of the muon. The obtained merit function with mass assumption m0 is X w2 ðm0 Þ ¼ rkT R1 k rk k
where the index k runs for all the measurements and Rk is the local covariance matrix for the k th measurement. If the largest contributions to Rk are independent in rf and z directions, w2 can be written as Xxi m ðm0 Þ2 X si ðmÞ 2 xi m ðm0 Þ2 X i i w2 ðm0 Þ ¼ ¼ ai zi si ðm0 Þ si ðm0 Þ si ðmÞ i i i ð1Þ where i runs for all split measurements, xi is the measured coordinate, mi ðmÞ is the predicted coordinate with mass hypothesis m, and s2i lists the diagonal elements of the covariance matrices R. The resulted sum is a linear combination of noncentrally chi-square distributed independent random variables zi with weights ai. The distribution functions are fX ðzi ; 1,li Þ where si ðmÞ 2 mi ðmÞmi ðm0 Þ 2 ai ¼ , li ¼ : si ðm0 Þ si ðmÞ The sum in Eq. (1) can approximated by a single rescaled non2 central chi-squared distribution 1=a2 fX ðx=a2 ; r,l Þ such that P 2 P 2 X a a a2 ¼ Pi i , r ¼ Pi i2 np , l2 ¼ li a i i i ai i where np is the number of track parameters. For details see Appendix B. If m¼m0, we get ai ¼1, a ¼ 1, l ¼ 0, and the distribution is a chi-squared one. If the ratio of expected variances ai are similar for all i, namely if the variances are scattering dominated, we get * + si ðmÞ 2 ð2Þ a2 si ðm0 Þ and r is the number of split measurements decreased by the number of track parameters. pffiffiffiffiffiffi At the same time the use of the variable w w2 appears to be more practical. It is described by a scaled non-central chi-distribution 1=a f ðw=a; r,lÞ and well approximated by a Gaussian with parameters qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffi ð3Þ mw ¼ a r12 þ l2 , sw ¼ a 12: For detailed derivation see Appendix A. The value of w can be calculated for each track during the track fit with Kalman filter. For different type of particles it will have different distribution function, because the parameters mw and sw
(via a and r) depend on the ratio of expected hit deviations si ðmÞ=si ðm0 Þ which are mass dependent (see Section 3). This observation allows to use this quantity in particle identification. Using the Gaussian approximation of Eq. (3), the separation power rw of w between particles of mass m1 and m2 is 2½mw ðm1 Þmw ðm2 Þ : rw ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2w ðm1 Þ þ s2w ðm2 Þ
ð4Þ
3. Physical effects When a stable charged particle propagates through material the most important effects which alter its momentum vector are multiple scattering (ms) and energy loss (el). In the following the expected spatial shift d and deviation s will be calculated. They are to be compared with the resolution of the local position measurement spos of the tracker layers. The distribution of multiple Coulomb scattering is roughly Gaussian [5], the standard deviation of the planar scattering angle is
y0 ¼
13:6 MeV pffiffiffiffiffiffiffiffiffiffi z x=X0 ½1 þ 0:038lnðx=X0 Þ bcp
ð5Þ
where p, bc, and z are the momentum, velocity, and charge of the particle in electron charge units, and x=X0 is the thickness of the scattering material in radiation lengths. While the expected shift is dms ¼ 0, the average deviation on the next tracker plane after a flight path l, in case of normal incidence, is
sms ly0 :
ð6Þ
Momentum and energy is lost during traversal of sensitive detector layers and support structures. To a good approximation the most probable energy loss D p , and the full width of the energy loss distribution at half maximum GD [6] are " # 2 2mc2 b g2 x 2 ð7Þ D p ¼ x ln þ0:2000 b d I2
GD ¼ 4:018x
ð8Þ
where
x¼
K 2Z x z r 2 A b2
is the Landau parameter; K ¼ 4pNA re2 me c2 ; Z, A and r are the mass number, atomic number and the density of the material, respectively [5]. Since this study deals with momenta below 2 GeV=c, the density correction d was neglected. In most cases tracker detectors are placed in magnetic field (B). Given the radius of the trajectory r and the length of the arc l, the central angle is j ¼ l=r. If the radius is changed by dr, the angle changes by dj ¼ l=r2 dr and the position shift of the trajectory after l path is
del ldj=2 ¼ l2 =2dr=r2 : At the same time p ¼0.3Br, E dE¼ p dp. Hence 2
del
0:3Bl /DS : 2 bp2
Similarly, the expected deviation is 2
sel
0:3Bl sD : 2 bp 2
The contributions to deviations and shifts of the predicted hit in a B ¼3.8 T magnetic field, after crossing x=X0 ¼ 2% silicon and further l¼ 5 cm propagation before reaching the next layer, are shown in Fig. 1. Standard deviations are dominated by multiple scattering, although at very low momentum the energy loss, at
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
K p e
0.4 0.3 0.2 0.1 0
0.5 Shift (w.r.t.apion) [mm]
Standard deviation [mm]
0.5
K p e
0.4 0.3 0.2 0.1 0
Multiple scattering -0.1
479
Energy loss -0.1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p [GeV/c]
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p [GeV/c]
Fig. 1. The contributions to deviations and shifts of the predicted hit in a B¼ 3.8 T magnetic field, after crossing x=X0 ¼ 2% silicon and further l ¼ 5 cm propagation before reaching the next layer, as a function of particle momentum. Left: expected standard deviations due multiple scattering. Right: expected shifts, compared to an average propagation with p mass assumption, due to energy loss. The curves give the limits of the lower and upper 7 1s confidence intervals for several particle types. For comparison lines corresponding to a local position resolution of 25 mm are drawn.
very high momentum the local position measurement also plays a role. Shifts from energy loss are only relevant at very low momentum, but they are still very small compared to standard deviations.
5 K p e
4
4. Properties of v
3
which can be further simplified, if spos 5 sms , to " !# bðm0 Þ z2 b2 ðmÞ 1 1 2 a bðmÞ 2 b ðm0 Þ
ð9Þ
Shifts come entirely from differences in energy loss, hence contributions to l are only substantial at low momentum: ! pffiffiffiffiffi pffiffiffi 0:3Bl2 /DðmÞDðm0 ÞS l rx 1 1 : l r p p 2bp2 ly0 b2 ðmÞ b2 ðm0 Þ The average shift /lS in a B ¼3.8 T magnetic field, with layer thicknesses of x=X0 ¼ 2% silicon, an average propagation length of l ¼5 cm, in case of r ¼16 number of degrees of freedom, is shown in Fig. 2. If l, z 51, the separation power rw between particles m and m0 is pffiffiffiffiffiffiffiffiffiffiffiffi
1bðmÞ=bðm Þ
0 : rw 2 2r1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1
0 0.2
0.4
0.6
0.8
1 1.2 p [GeV/c]
1.4
1.6
1.8
2
Fig. 2. The average shift for several particle types, in a B¼ 3.8 T magnetic field, with layer thicknesses of x=X0 ¼ 2% silicon, an average propagation length of l ¼5 cm, in case of r ¼ 16 number of degrees of freedom, as a function of the particle momentum.
where the sensitivity is defined as z ¼ spos =sms ðm0 Þ, it is proportional to 1=bp (Eqs. (5) and (6)). If the local position resolution can be neglected ðz 5 1Þ we get
bðm0 Þ : a bðmÞ
r = 16
It is important to study the sensitivity of the measured w distribution at a given total momentum p. The parameters which govern the distribution (Eq. (3)) are the rescaler a, the average shift l and the number of degrees of freedom r. In this section we estimate them, as well as the separation power rw listed in Eq. (4), based on physical effects. Since the deviations are dominated by multiple scattering and local position measurement, a in Eq. (2) can approximated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2ms ðmÞ þ s2pos a s2ms ðm0 Þ þ s2pos
ð10Þ
1 þ ½bðmÞ=bðm0 Þ2
Hence if the momentum is not very low and the local position resolution is small compared to deviations from multiple scattering, neither the rescaler a nor the separation power rw depends on the details of the experimental setup, such as
magnetic field, radii of tracker layers, value of local position resolution and material thickness. In this respect the only decisive parameter is the number of split measurements which enters the above expressions by the number of degrees of freedom r. The mean and variance of the corresponding Gaussians are fully determined by the momentum and mass of the particles via b. Although at low momentum the prediction of the means is more difficult due to the increasing l, the variances still stay the same. The measured w distribution of tracks in a phase space bin can be fit with a linear combination of Gaussians, giving the relative yields of the different particle species. The obtained separation power allows for a many-parameter fit.
4.1. Applications The measured value of w is sensitive to the proper spatial alignment of the detector layers and to the correct estimate of the variation of the predicted local position. If the alignment precision is sufficient, the latter is mostly determined by the contribution pffiffiffiffiffiffiffiffiffiffi from multiple scattering which is closely proportional to x=X0 .
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
480
Table 1 Important characteristics of the inner barrel detectors of the studied experimental setup. B (T)
Subdetector
Radius of layers (cm)
srf ðmmÞ
sz ðmmÞ
x=X0 (%)
zrf
zz
Split meas.
Exp A
2
Pixels (barrel) Strips (SCT)s Straw (TRT)
5.0, 8.8, 12.2 29.9, 37.1, 44.3, 51.4 56.3–106.6 (r 36 hits)
10 17 130
115 580 –
4 4 0.5
0.1 0.1 10
1 3 –
50
Exp B
0.4
Pixels (SPD) Drifts (SDD) Strips (SSD)s [gas (TPC)
3.9, 7.6 14.9, 23.8 38.5, 43.6 84.5 – 246.6 ( r 159 hits)
12 35 15 900
100 23 730 900
1 1 1 10 3
0.2 0.3 0.1
2 0.2 7 103–104]
12
Exp C
3.8
Pixels (PXB) Strips (TIB)s
4.4, 7.3, 10.2 25.5, 33.9
15
15 230
3 4
0.2 0.1
0.2 0.8
20
Strips (TIB) Strips (TOB)s
41.8, 49.8 60.8, 69.2
– 530
2 4
0.2 0.1
– 2
Strips (TOB)
78.0, 86.8, 96.5, 108.0
–
2
0.2
–
pffiffiffi 23= 2 35 pffiffiffi 53= 2 53, 35
For details see text at the beginning of Section 5.
Exp A
8000
Exp B
K p e
Exp C
6000
=0 pT = 0.4GeV/c
4000 2000 0 0
5
10
15
20 0
5
10
15
20 0
5
10
15
20 K p e
8000 6000
=0 pT = 0.8GeV/c
4000 2000 0 0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
Fig. 3. Distributions of w for several particle species. The relative yield of particles was set to p : K : p : e ¼ 70 : 10 : 18 : 2. Results are shown for Z ¼ 0, pT ¼ 0:4 GeV=c (upper row) and Z ¼ 0, pT ¼ 0:8 GeV=c (lower row) with setups Exp A, B and C. Individual fits with chi distributions are indicated by thin solid lines.
While p and r are well measured, the amount of material in the detector can be
Understood: The unfolding of the w distribution in a phase
space bin enables the measurement of yields of different particle species. Poorly known: The unfolding of the w distribution in a phase space bin may provide corrections to the material thickness. They can be extracted by fitting the w distribution with an additional rescaler. Note that the measurement of yields of different particle species is still possible, although with lower confidence.
5. Simulation The proposed method was verified by a Monte Carlo simulation. As examples from LHC, the performance of simplified models for the inner detectors of the following experiments were studied:
ATLAS (Exp A): Three layers of silicon pixels, five layers of double-sided silicon strips, up to 36 layers of straw tubes [7].
ALICE (Exp B): Two layers of silicon pixels, two layers of silicon
drifts and two layers of double-sided silicon strips [8,9]. Due to the large z value of the gas detector (TPC) its measurements were not included. CMS (Exp C): Three layers of silicon pixels, 10 layers of silicon strips (four of them double-sided) [10].
Some relevant details of the experimental setups are given in Table 1. For simplicity a homogeneous longitudinal magnetic field was used, and detector layers were assumed to be concentric cylinders around the beam-line. Pixels, double-sided strips (superscript s), drift layers and gas provide measurements in two dimensions (rf and z), while one-sided strips and straw tubes give only measurement in one direction ðr fÞ. While double-sided strips consists of two slightly rotated layers separated by a small gap in reality, in this simulation they were treated as single twodimensional layers with very different position resolution in r f and z directions. x=X0 values are given per layer and they are rounded to integers where possible. Sensitivity values zrf and zz are shown for pions at p ¼ 1 GeV=c, normal incidence, rounded to
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
6
6 Exp A
6 Exp B
5
2f it
Exp C
5
4
4
4
3
3
3
2
2
2
1
1
1
0 50
0 10
0 20
40
8
30
6
20
4
K p e
15 10
r
5
481
10
2
0 6
0 6
0 6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0 12
0 12
0 12
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0 0.5
1 1.5 p [GeV/c]
2
K p e
5
K p e
–K –p –e
0 0.5
1 1.5 p [GeV/c]
2
0.5
1 1.5 p [GeV/c]
2
Fig. 4. Performance of w measurement for particle identification, with setups Exp A, B and C. For details see text in Section 5.1.
one significant digit. The number of split measurements are also indicated. The initial state vector was estimated by fitting a helix to the first three hits. (These hits are two-dimensional in all three examined experimental setups.) The starting values of the track parameters were extracted at the closest approach to the beam line. The track fitting was performed by a classical Kalman filter [2] with pion mass assumption. The state vector x ¼ ðk,y,c,rf,zÞ is five dimensional, where
The propagation from layer to layer was calculated analytically using a helix model. Multiple scattering and energy loss in tracker layers was implemented with their Gaussian approximations shown in Eqs. (6)–(8). The propagation matrix F ¼ @f =@x was obtained by numerical derivation. The measurement vector m ¼ ðr f,zÞ is two-dimensional, the measurement operator is 0 0 0 1 0 H¼ : 0 0 0 0 1
k ¼ q=p ðsigned inverse momentumÞ
The covariance of the process noise Q is
y ¼ yð~ p Þ ðlocal polar angleÞ
Q ¼ ðFk FkT Þs2k þðFy FyT Þs2y þ ðFc FcT Þs2c
c ¼ fð~ p Þ ðlocal azimuthal angleÞ rÞ rf ¼ r fð~ z ¼ rL
ðglobal azimuthal positionÞ
ðglobal longitudinal positionÞ:
where sk ¼ ksD =b, sy ¼ sc ¼ y0 and Fa ¼ @f =@xa is a vector. The covariance of measurement noise V is ! s2rf 0 : V¼ 0 s2z
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
Note that multiple scattering contributes equally to the variation of y and c, while energy loss affects only k. A trajectory may contain outliers, hits that do not belong to the particle track. They can be correlated with the track (d’ s) or uncorrelated (detector noise, hits from another track). The w2 distribution of the outlier hits is imposed by pattern recognition and selection during track finding. In order to simulate this effect 1% of the hits were randomly flagged as outliers and the covariance of the measurement noise was multiplied by a factor 4 during hit creation in the detector. The outliers can be suppressed after the track fit. All hits are examined and those with smoothed w2 greater than 50 are removed and the fit is redone [2]. Note that this cut somewhat reduces the separation power of the global track w value at very low momentum, since it may remove some outlying measurements if the generated particle was a kaon or a proton.
5.1. Results
10 Exp A Exp B Exp C
8
6
482
4
2
0 0.5
In order study the performance of w, charged pions, kaons, protons and electrons with random azimuthal angle were generated and emitted normal to the line of the colliding beams ðZ ¼ 0Þ and run through the above outlined reconstruction. Distributions of w using 105 particle tracks are shown in Fig. 3 for pT ¼0.4 and 0:8 GeV=c. For a realistic particle composition the relative yields were set to p : K : p : e ¼ 70 : 10 : 18 : 2. At pT ¼ 0:4 GeV=c, in case of Exp A, the protons are detached, but there is a good p2p separation for Exp B and C, as well. For Exp A and C the p2K separation allows for yield estimation. Even at pT ¼ 0:8 GeV=c the observed resolution is enough to extract the protons. When fitting the histograms a sum of chi distributions was employed (thin solid lines), but a sum of Gaussians may also be sufficient. For a complete picture charged pions, kaons, protons and electrons with transverse momenta pT ¼ 0:2,0:4,0:6, . . . ,2:0 GeV=c and Z ¼ 0 were used, amounting to 104 particles per pT setting for each particle type and experimental setup. The performance of w as function of p for all three setups is shown in Fig. 4. The subsequent rows give the dependence of the measured rescaler a, the fitted number of degrees of freedom r, the merit function of the histogram fit with sum of chi distributions w2fig and the separation power rw . This latter was calculated by using the measured a and r values with help of Eqs. (3) and (4). The measured values are shown by the symbols. In case of a the line gives the plain bðm0 Þ=bðmÞ scaling (Eq. (9)) that works well for all three setups and for all particle types. The deviations seen at very low momentum ðp ¼ 0:2 GeV=cÞ are due to the w2 cut in the outlier removal discussed in Section 5. For r the horizontal lines show the number of split measurements for a given pT, decreased by the number of track parameters np. While these predictions are closely followed by the measured values in case of Exp C, there are substantial deviations with the other two setups. It can be traced back to low sensitivity measurements, whose contribution to the global w2 is very small and the measurement does not increase the number of degrees of freedom: large number of straw tubes with zrf ¼ 10 (Exp A), and two strip layers with zz ¼ 7 (Exp B). Also outliers and their removal procedure introduces shifts in the fitted value of r. In case of the separation power rw the lines show the approximation based on the predicted number of degrees of freedom and the ratio bðmÞ=bðm0 Þ, calculated with help of Eq. (10). The steps are due to the changing number of crossed detector layers with varying p. The approximation works well for Exp C, but strongly overestimates the measured value for Exp A. It is again due to the large number of low sensitivity measurements.
1 p [GeV/c]
1.5
2
Fig. 5. The p2p separation power of the w measurement for the experimental setups, as a function of momentum. The lines are drawn to guide the eye.
Comparison of the p2p separation power of the w measurement for several experimental setups as a function of momentum is shown in Fig. 5. While Exp A clearly performs better for p o0:6 GeV=c, Exp C has better resolution for the more critical higher momentum region. With the most sensitive setups (Exp A and C) protons are 1s apart if p o1:4 GeV=c, while 2s separation is reached if p o 1 GeV=c. For kaons these numbers are p o 0:9 and 0:5 GeV=c, respectively.
6. Conclusions It was shown that tracker detectors can employed to identify charged particles based on their global w obtained during track fitting with the Kalman filter. Since the w value of the filter is equivalent to that of a global fit, the method is suitable for any minimum w2 track fit that properly models energy-loss and scattering effects. This approach builds upon the knowledge of detector material and local position resolution, using the known physics of multiple scattering and energy loss. The study using simplified models of present LHC experiment shows that p2K and p2p unfolding is possible at low momentum. The separation is better than 1s for p o 0:9 and 1:4 GeV=c, respectively. In general, the performance of an experiment is determined by the number of good sensitivity split measurements. It is also a strong function of particle momentum.
Acknowledgements The author wishes to thank to Krisztia´n Krajcza´r for helpful discussions. This work was supported by the Hungarian Scientific Research Fund and the National Office for Research and Technology (K 48898, K 81614, H07-B 74296).
Appendix A. Properties of some distributions In this section the definitions of some used distributions are listed along with their calculated or approximated values for the mean m and variance s2 .
ARTICLE IN PRESS F. Sikle´r / Nuclear Instruments and Methods in Physics Research A 620 (2010) 477–483
Note that with l ¼ 0 we get back the mean of the w distribution (Eqs. (12) and (14), while the variances are the same in the central and non-central case (Eqs. (13) and (15))).
A.1. w2 distribution The distribution, mean and variance are Pðx; rÞ ¼
483
xr=21 ex=2 r G 2r=2 2
Appendix B. Sum of non-central chi-squared distributed independent variables
m ¼ r, s2 ¼ 2r:
The goal is to approximate the sum A.2. Non-central w2 distribution y¼ The distribution, mean and variance are pffiffiffi pffiffiffiffiffi eðx þ lÞ=2 xðr1Þ=2 l Ir=21 ð lxÞ Pðx; r,lÞ ¼ r=4 2ðlxÞ
where zi are non-central chi-squared distributed independent random variables with one degree of freedom and density function fX ðzi ; 1,li Þ. Although an explicit expression for the distribution of y exists, it is difficult to evaluate in practice [14]. Here this function is approximated by a rescaled non-central chi-squared distribution 1=a fX ðx=a; r,lÞ by requiring that the first two moments be the same. The means and variances are additive, thus the equations two solve are X /yS ¼ ai ð1 þ li Þ ¼ aðr þ lÞ
where In(x) is the modified Bessel function of the first kind. A.3. w distribution The distribution and mean are
i
2 =2
Pðx; rÞ ¼
G
ð11Þ
2
pffiffiffi rþ1 rffiffiffiffiffiffiffiffiffiffi 2G pffiffiffi 1 1 1 2 r m¼ ¼ r 1 þ O 2 r 4r 2 r G 2
X /ð y/ySÞ2 S ¼ 2 a2i ð1 þ 2li Þ ¼ 2a2 ðr þ 2lÞ: i
By assuming li 5 1 we get ð12Þ
P
a2i , i ai
a ¼ Pi
where Ref. [11] for r b 1 was used. The variance is
s2 ¼ rm2 12:
ai zi
i¼1
m ¼ r þ l, s2 ¼ 2ðr þ 2lÞ
21r=2 xr1 ex r
n X
ð13Þ
P 2 ai r ¼ Pi 2 , i ai
l¼
X
li
i 2
with relative corrections of the order Oðl =r 2 Þ. If the values of ai are similar some of the above expressions can be approximated by
a /ai S, r n:
A.4. Non-central w distribution The distribution and mean are 2
Pðx; r,lÞ ¼ rffiffiffiffi
m¼
p 2
eðx
xl
ðlxÞ
r=2
2
ðr=21Þ
L1=2
References
þ l2 Þ=2 r
l 2
Ir=21 ðlxÞ
!
where L(a) n (x) is the generalized Laguerre function. For r b 1, with Kummer’s second formula [12] r þ1 ! rffiffiffiffi G 2 p 1 r l 2 m¼ F , , 3 1 1 2 r 2 2 2 G G 2 2 where 1 F1 ða,b,zÞ is the confluent hypergeometric function of the 2 first kind. With help of Eq. (12) and Ref. [13], assuming l 5r " #
pffiffiffi 1 1 l2 1 1þ þO 2 m ¼ r 1 þ O 2 2r 4r r r " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffi 12l 1 1 2 r þ l : þO 2 r 1 ð14Þ 4r 2 r For the variance
s ¼ rm2 þ l2 12: 2
ð15Þ
[1] A. Grossheim, K. Zuber, Nucl. Instr. and Meth. A 533 (2004) 532, doi:10.1016/ j.nima.2004.06.146. [2] R. Fruhwirth, Nucl. Instr. and Meth. A 262 (1987) 444, doi:10.1016/01689002(87)90887-4. [3] A. Ankowski, et al., Eur. Phys. J. C 48 (2006) 667, doi:10.1140/epjc/s10052006-0051-3. [4] W. Hulsbergen, Nucl. Instr. and Meth. A 600 (2009) 471, doi:10.1016/ j.nima.2008.11.094. [5] C. Amsler, et al., Phys. Lett. B 667 (2008) 1, doi:10.1016/j.physletb.2008. 07.018. [6] H. Bichsel, Rev. Mod. Phys. 60 (1988) 663, doi:10.1103/RevModPhys.60.663. [7] G. Aad, et al., Expected performance of the atlas experiment—detector, Trigger and Physics, 2009, arXiv:0901.0512. [8] F. Antinori, Nucl. Instr. and Meth. A 511 (2003) 215, doi:10.1016/S01689002(03)01796-0. [9] D. Antonczyk, et al., Nucl. Instr. and Meth. A 565 (2006) 551, doi:10.1016/ j.nima.2006.06.022. [10] R. Adolphi, et al., JINST 3 (2008) S08004, doi:10.1088/1748-0221/3/08/ S08004. [11] R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, second ed., Addison-Wesley, Reading, MA, 1994 (answer to Problem 9.60). [12] W. Koepf, Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities, Vieweg, Braunschweig, Germany, 1998. [13] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972. [14] S. Gabler, C. Wolff, Statist. Papers 28 (1) (1987) 317, doi:10.1007/ BF02932611.