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Reducing systematic errors in the selection of signals from two-photon mass spectra
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473 www.elsevier.com/locate/nima
Reducing systematic errors in the selection of signals from two-photon mass spectra A. Beddall, A. Beddall, A. Bingu¨l Department of Engineering Physics, University of Gaziantep, Gaziantep 27310, Turkey Received 31 May 2005; received in revised form 4 August 2005; accepted 10 August 2005 Available online 30 August 2005
Abstract When selecting signals from two-photon invariant mass spectra the signal significance can be optimised by selecting signal candidates from a narrow mass window around the signal peak. However, such tight cuts can lead to large systematic errors. In this study systematic errors due to inaccuracies in the Monte Carlo simulation of the peak width are studied for three selection methods. It is shown that in the presence of such inaccuracies a wider selection window is required to reduce systematic errors. The Ranking selection method is found to be much less sensitive to systematic errors of this type. r 2005 Elsevier B.V. All rights reserved. PACS: 29.85.+c Keywords: Eta; Signal significance; Systematic; Ranking; LHC
1. Introduction In the reconstruction of mesons that decay to two photons, for example p0 ! gg or Z ! gg, signal candidates are selected from two-photon invariant mass spectra. The signal in such mass spectra usually lies on a large combinatorial background formed by the incorrect combination of photon pairs. Signal significance is an important Corresponding author. Tel.: +90 342 360 1200x2202; fax:
+90 342 360 1100. E-mail address: [email protected] (A. Beddall).
factor, for example when using selected signal candidates to reconstruct decays such as r ! p p0 , and a 0 ! p Z. Most of the background can be rejected by selecting signal candidates from within a mass window. An example of this is given in Fig. 1 where an invariant mass spectrum is shown for photon pairs from three million pp events where the decay B0s ! J=c Z has been tagged; detector effects are simulated by passing all photons through a detector simulation. In this example the Z signal is seen clearly above the background which is dominantly from wrong combinations of
0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.08.050
ARTICLE IN PRESS 470
A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473
2. Event and detector simulation
Fig. 1. Invariant mass of photon pairs from three million pp events where the decay B0s ! J=c Z has been tagged. The Z signal is seen clearly above the background. Most of the background is rejected by selecting Z candidates from a mass window around the Z signal peak.
photon pairs (combinatorial background), also Z’s not originating from the B0s decay are considered here as background. Most of the background is rejected by selecting Z candidates from a window of width 2sM centred about the position of the reconstructed signal peak M P . A skew is seen in the Z signal shape, this is due to the increase in the reconstructed Z peak mass as the energy of the Z increases (see Section 3). Here sM represents the width of the peak which in this study has been obtained from a Gaussian fit to the Z signal, the value of M P is also extracted from the fit. A width of two standard deviations (m ¼ 2) is a typical value that gives both good selection purity and efficiency. The value of m can be optimised such that the signal significance of the selected Z candidates is maximised. The photon lower transverse momentum cut pcut T is also an important factor in the signal significance and so optimisation in m and pcut T space can be performed. Statistical considerations are not the only concern in signal selection. Inaccuracies in Monte Carlo simulations of electromagnetic calorimeters can lead to significant systematic errors in efficiency corrections. The aim of this study is to illustrate how signal selection procedures can be modified so that they are less sensitive to such systematic errors while still being well optimised statistically.
For the purpose of the study a high event-byevent yield of Z mesons is obtained by selecting the decay B0s ! J=c Z. Events are generated using the Herwig event generator [1] with the process of QCD heavy b-quark production under LHC conditions (pp collisions at the centre of mass energy of 14 TeV). Events containing B0s mesons are selected forcing the decay B0s ! J=c Z, and forcing the decay J=c ! mþ m . The tagging of this event in LHC is simulated by requiring both muons to have a pseudo-rapidity jZjo2:4 and the transverse momentum of one muon pT 46 GeV=c and the transverse momentum of the other muon pT 43 GeV=c [2]. Three million events of this type are generated and passed through the detector simulation described next. As this is a general study, a detailed simulation of a specific LHC electromagnetic calorimeter (ECAL) is not necessary, instead a toy simulation of relevant detector effects is employed. Values of ECAL parameters (such as stochastic and constant terms in the photon energy resolution) are taken as typical values from LHC detectors [3,4]. The simulation of detector effects proceeds as follows:
Energy resolution is simulated by Gaussian smearing the energy of photons; the pffiffiffirelative ffi energy resolution has the form 10%= E 1% where E is measured in GeV. Spatial resolution is simulated by Gaussian smearing the angular position of photons; pffiffiffithe ffi angular resolution has the form 40 mrad= E . Reconstructed photons with a transverse momentum less than 1 GeV=c are rejected, this imposes a lower transverse momentum limit on ECAL cluster selection. Reconstructed photons with a pseudo-rapidity jZj42:4 are rejected, they lie outside the acceptance of the detector. Due to limited spatial resolving power of the detector, photons that are closer than five degrees are merged into a single cluster [2]. Such photons are assumed to be identified as neutral pions from their shower shape and are rejected as background. This feature has two important effects: (a) the two-photon invariant
ARTICLE IN PRESS A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473
mass spectra has very little neutral pion content; (b) a bias towards larger reconstructed angles (after spatial smearing) gives rise to significant shifting of the reconstructed mass of the signal, the bias increasing with energy.
471
variations originate from the simple features implemented in the toy detector-simulation; one can expect greater variations in the real world where detector and reconstruction algorithms are much more complex.
3. Calibration As expected, the values of M P and sM vary with energy and so a calibration procedure is required to position correctly the mass window around the signal peak and to make sure the width is related in some consistent way to the width of the signal. While the details of calibration are not intended to be discussed in this paper it is useful to see the results of calibration in the context of the scale of the variations in M P ðEÞ and sM ðEÞ that occur due to detector effects. Calibration results for the data used in this study are shown in Fig. 2. The values of M P and sM are seen to vary over a large range, for example the peak width is seen to vary between 60 and 120 MeV=c2 a span of 60 MeV=c2 . These
Fig. 2. Calibration of the peak width sM and peak mass M P for reconstructed Z mesons. The data points are the results for the standard deviation and mean of Gaussian fits to the Z signal separated into 1 GeV energy intervals, the inlay shows an example fit. The dashed line at 548 MeV=c2 indicates the position of the nominal Z mass.
4. Optimisation of signal significance Optimal values of m and pcut T can be chosen such that the product of efficiency and purity1 (e P) is maximum; this is equivalent to maximising the signal significance. With respect to this optimisation, three selection methods are compared: the Standard method, the Highest pT method, and the Ranking method. In the Standard method the invariant mass of photon-pairs is formed after the minimum transverse momentum cut pcut T is applied to the photons, then Z candidates are selected from within the mass window of width msM . The Highest pT method applies the same procedure, but only the two highest pT photons in the event are considered. The Ranking method is described in detail elsewhere [5] for the selection of neutral pions in the presence of high combinatorial background. In this method pion candidates are ranked according to a pion estimator (discriminant) based on the w2 from a mass constraint and the opening angle between the daughter photons (candidates with small w2 values and small opening angles are more likely to be true pions). Signal candidates sharing photons with other candidates are removed depending on their relative rank. The Ranking method in the present study is applied by subjecting the Ranking procedure to Z candidates that pass the Standard selection procedure. Here, it is found that the opening angle between the photons provides an effective Ranking estimator, Fig. 3, other parameters such as the Z energy and the w2 from a kinematic fit do not improve significantly the performance of the estimator. 1
For this study the Z selection efficiency is defined as e ¼ S=S0 , where S is the selected signal and S0 is the total signal in the data. The Z selection purity is defined as P ¼ S=ðS þ BÞ where B is the background (including p0 ’s, combinatorial background, and Z’s not originating from B0s decays).
ARTICLE IN PRESS 472
A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473 Table 1 Results from the optimisation of pcut T and m where the signal significance is maximised. Three selection methods are investigated: Standard, Highest pT , and the Ranking method; the method ‘all’ shows the case where all candidates are taken without cuts on pcut T or m. The seventh column sstat shows the relative statistical uncertainty in the number of selected Z’s. The final column ssyst shows the relative systematic errors due to a 10 MeV=c2 underestimation of the signal width by the Monte Carlo (see Section 5). The accuracy of the values is implied by the number of significant figures.
Fig. 3. Distributions of the angle between photons pairs f12 for the Z signal and its background. The data is taken from a 4sM mass window and for a 2 GeV=c pT cut. The probability of a candidate being signal increases for smaller angles.
Optimisation results for the three selection methods are shown in Table 1. The three methods agree for the optimal cuts pcut T 2:0 GeV=c and m ¼ 1:7 and all result in a statistical uncertainty in the number of selected Z’s of sstat 1:0% with the Ranking method performing slightly better than the other selection methods. The final column ssyst contains results from the systematic study discussed in Section 5. Differences in the three methods are seen in the details of the efficiency and purity values; the Highest pT method obtains a high purity while the Ranking method obtains a high efficiency. The results so far are not remarkable, though a preference towards high efficiency or high purity may be valuable to some analyses. Important differences, however, do arise when we consider the performance of each selection method in the presence of inaccuracies in the simulation of the peak width. This is considered next.
5. Systematic errors due to inaccuracies in the simulation of the peak width Now consider the case where a Monte Carlo detector simulation underestimates the width of the Z signal by 10 MeV=c2 . The selection of this value is arbitrary but one can imagine this could be a realistic value when one considers the scale of the
Method
m pcut T
e
P
e P sstat ð%Þ ssyst ð%Þ
All Standard Highest pT Ranking
1.0 2.1 2.0 1.9
1.000 0.376 0.276 0.427
0.006 0.287 0.343 0.267
0.006 0.108 0.095 0.114
1 1.7 1.7 1.7
4.30 0.98 1.05 0.96
0.0 6.1 6.1 5.9
calibration functions in Fig. 2; however, the value actually depends on the details of a specific experiment and detector simulations. The resultant relative systematic error in an efficiency correction is defined as ssyst ¼ ðeMC eRD Þ=eRD where eMC is the efficiency determined by the Monte Carlo, and eRD is the true efficiency in the real data. In this study we are only dealing with Monte Carlo data, the inaccuracy in the Monte Carlo is generated by simply shifting the calibrated width of Fig. 2 down by 10 MeV=c2 ; the real data (with the true efficiency) is represented by the Monte Carlo without the shift. The resultant systematic error ssyst is shown in the final column of Table 1, the systematic error for each method is 6%, much larger than the statistical uncertainty. To reduce such systematic errors one can choose a wider mass window. The choice in this study can be determined by re-optimisation the cuts pcut T and m such that the total error sstat ssyst is minimised. The results of the re-optimisation are given in Table 2. For the Standard and Highest pT methods optimisation chooses much wider mass windows, subsequently the systematic errors are reduced to the same order of the statistical uncertainties. The response of the Ranking method is noticeably different, while the statistical uncertainty is still smaller the systematic error almost vanishes even though the value of m is
ARTICLE IN PRESS A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473
473
Table 2 Results from re-optimisation of the cuts pcut T and m such that the total error sstat ssyst is minimised. The accuracy of the values is implied by the number of significant figures Method
pcut T
m
sstat (%)
ssyst (%)
Total (%)
Standard Highest pT Ranking
2.1 2.1 1.5
5.4 5.4 4.1
1.21 1.26 1.17
0.69 0.75 0.07
1.4 1.5 1.2
smaller than those of the other methods. Also a 2 much lower pcut T value is preferred. A further illustration of the performance of the Ranking method can be seen when the error in the signal width is varied, this is shown in Fig. 4 where the mass window is fixed at a reasonable wide value of 4sM . The results show that relative to other selection methods, Ranking is insensitive to errors in the signal width. This can be explained by the tendency of the Ranking method to select signals that are closer to the signal peak.
6. Summary and conclusion Signal candidates are selected from two-photon invariant mass spectra within a mass window around the signal peak. Analyses tend to choose a mass window with a width of 2sM or less. Although such tight cuts are statistically optimal this study has shown that they can give rise to large systematic errors when in the presence of inaccuracies in the Monte Carlo simulation of the peak width (or similarly the peak position). Such systematic errors can be reduced by increasing the width of the selection mass window though at the expense of a lower signal significance. The optimal size of the mass window depends on the level of
2 Reconstruction to a lower momentum can be important, for example in measurements of particle production where differential cross-sections need to be studied at low particle momenta.
Fig. 4. Results for varying the peak width error. The value of m is fixed at 4.
statistics and the size of the systematic error, the latter being very difficult to estimate. A width of 4sM or more maybe be preferable in analyses where systematic errors are suspected to be large. The Ranking method provides some relief to this dilemma as it is much less sensitive to systematic errors while at the same time giving better statistical performance.
References [1] HERWIG 6.5, G. Corcella, I.G. Knowles, G. Marchesini, S. Moretti, K. Odagiri, P. Richardson, M.H. Seymour, B.R. Webber, JHEP 0101 (2001) 010 [hep-ph/0011363]; hep-ph/ 0210213. hhttp://hepwww.rl.ac.uk/theory/seymour/herwig/i [2] F. Ohlsson-Malek, M. Melcher, ATLAS Calorimeter performance at low pT for g, p0 and Z particles identification in the B-Physics Context, ATLAS Internal Note, ATLCOM-PHYS-2001-021 10/09/2001, and ISN-01.75. [3] A. Golutvin, Nucl. Instr. and Meth. A 453 (2000) 192. [4] Peter Schacht, Nucl. Instr. and Meth. A 535 (2004) 446. [5] A. Beddall, et al., Nucl. Instr. and Meth. A 482 (2002) 520.
Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473 www.elsevier.com/locate/nima
Reducing systematic errors in the selection of signals from two-photon mass spectra A. Beddall, A. Beddall, A. Bingu¨l Department of Engineering Physics, University of Gaziantep, Gaziantep 27310, Turkey Received 31 May 2005; received in revised form 4 August 2005; accepted 10 August 2005 Available online 30 August 2005
Abstract When selecting signals from two-photon invariant mass spectra the signal significance can be optimised by selecting signal candidates from a narrow mass window around the signal peak. However, such tight cuts can lead to large systematic errors. In this study systematic errors due to inaccuracies in the Monte Carlo simulation of the peak width are studied for three selection methods. It is shown that in the presence of such inaccuracies a wider selection window is required to reduce systematic errors. The Ranking selection method is found to be much less sensitive to systematic errors of this type. r 2005 Elsevier B.V. All rights reserved. PACS: 29.85.+c Keywords: Eta; Signal significance; Systematic; Ranking; LHC
1. Introduction In the reconstruction of mesons that decay to two photons, for example p0 ! gg or Z ! gg, signal candidates are selected from two-photon invariant mass spectra. The signal in such mass spectra usually lies on a large combinatorial background formed by the incorrect combination of photon pairs. Signal significance is an important Corresponding author. Tel.: +90 342 360 1200x2202; fax:
+90 342 360 1100. E-mail address: [email protected] (A. Beddall).
factor, for example when using selected signal candidates to reconstruct decays such as r ! p p0 , and a 0 ! p Z. Most of the background can be rejected by selecting signal candidates from within a mass window. An example of this is given in Fig. 1 where an invariant mass spectrum is shown for photon pairs from three million pp events where the decay B0s ! J=c Z has been tagged; detector effects are simulated by passing all photons through a detector simulation. In this example the Z signal is seen clearly above the background which is dominantly from wrong combinations of
0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.08.050
ARTICLE IN PRESS 470
A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473
2. Event and detector simulation
Fig. 1. Invariant mass of photon pairs from three million pp events where the decay B0s ! J=c Z has been tagged. The Z signal is seen clearly above the background. Most of the background is rejected by selecting Z candidates from a mass window around the Z signal peak.
photon pairs (combinatorial background), also Z’s not originating from the B0s decay are considered here as background. Most of the background is rejected by selecting Z candidates from a window of width 2sM centred about the position of the reconstructed signal peak M P . A skew is seen in the Z signal shape, this is due to the increase in the reconstructed Z peak mass as the energy of the Z increases (see Section 3). Here sM represents the width of the peak which in this study has been obtained from a Gaussian fit to the Z signal, the value of M P is also extracted from the fit. A width of two standard deviations (m ¼ 2) is a typical value that gives both good selection purity and efficiency. The value of m can be optimised such that the signal significance of the selected Z candidates is maximised. The photon lower transverse momentum cut pcut T is also an important factor in the signal significance and so optimisation in m and pcut T space can be performed. Statistical considerations are not the only concern in signal selection. Inaccuracies in Monte Carlo simulations of electromagnetic calorimeters can lead to significant systematic errors in efficiency corrections. The aim of this study is to illustrate how signal selection procedures can be modified so that they are less sensitive to such systematic errors while still being well optimised statistically.
For the purpose of the study a high event-byevent yield of Z mesons is obtained by selecting the decay B0s ! J=c Z. Events are generated using the Herwig event generator [1] with the process of QCD heavy b-quark production under LHC conditions (pp collisions at the centre of mass energy of 14 TeV). Events containing B0s mesons are selected forcing the decay B0s ! J=c Z, and forcing the decay J=c ! mþ m . The tagging of this event in LHC is simulated by requiring both muons to have a pseudo-rapidity jZjo2:4 and the transverse momentum of one muon pT 46 GeV=c and the transverse momentum of the other muon pT 43 GeV=c [2]. Three million events of this type are generated and passed through the detector simulation described next. As this is a general study, a detailed simulation of a specific LHC electromagnetic calorimeter (ECAL) is not necessary, instead a toy simulation of relevant detector effects is employed. Values of ECAL parameters (such as stochastic and constant terms in the photon energy resolution) are taken as typical values from LHC detectors [3,4]. The simulation of detector effects proceeds as follows:
Energy resolution is simulated by Gaussian smearing the energy of photons; the pffiffiffirelative ffi energy resolution has the form 10%= E 1% where E is measured in GeV. Spatial resolution is simulated by Gaussian smearing the angular position of photons; pffiffiffithe ffi angular resolution has the form 40 mrad= E . Reconstructed photons with a transverse momentum less than 1 GeV=c are rejected, this imposes a lower transverse momentum limit on ECAL cluster selection. Reconstructed photons with a pseudo-rapidity jZj42:4 are rejected, they lie outside the acceptance of the detector. Due to limited spatial resolving power of the detector, photons that are closer than five degrees are merged into a single cluster [2]. Such photons are assumed to be identified as neutral pions from their shower shape and are rejected as background. This feature has two important effects: (a) the two-photon invariant
ARTICLE IN PRESS A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473
mass spectra has very little neutral pion content; (b) a bias towards larger reconstructed angles (after spatial smearing) gives rise to significant shifting of the reconstructed mass of the signal, the bias increasing with energy.
471
variations originate from the simple features implemented in the toy detector-simulation; one can expect greater variations in the real world where detector and reconstruction algorithms are much more complex.
3. Calibration As expected, the values of M P and sM vary with energy and so a calibration procedure is required to position correctly the mass window around the signal peak and to make sure the width is related in some consistent way to the width of the signal. While the details of calibration are not intended to be discussed in this paper it is useful to see the results of calibration in the context of the scale of the variations in M P ðEÞ and sM ðEÞ that occur due to detector effects. Calibration results for the data used in this study are shown in Fig. 2. The values of M P and sM are seen to vary over a large range, for example the peak width is seen to vary between 60 and 120 MeV=c2 a span of 60 MeV=c2 . These
Fig. 2. Calibration of the peak width sM and peak mass M P for reconstructed Z mesons. The data points are the results for the standard deviation and mean of Gaussian fits to the Z signal separated into 1 GeV energy intervals, the inlay shows an example fit. The dashed line at 548 MeV=c2 indicates the position of the nominal Z mass.
4. Optimisation of signal significance Optimal values of m and pcut T can be chosen such that the product of efficiency and purity1 (e P) is maximum; this is equivalent to maximising the signal significance. With respect to this optimisation, three selection methods are compared: the Standard method, the Highest pT method, and the Ranking method. In the Standard method the invariant mass of photon-pairs is formed after the minimum transverse momentum cut pcut T is applied to the photons, then Z candidates are selected from within the mass window of width msM . The Highest pT method applies the same procedure, but only the two highest pT photons in the event are considered. The Ranking method is described in detail elsewhere [5] for the selection of neutral pions in the presence of high combinatorial background. In this method pion candidates are ranked according to a pion estimator (discriminant) based on the w2 from a mass constraint and the opening angle between the daughter photons (candidates with small w2 values and small opening angles are more likely to be true pions). Signal candidates sharing photons with other candidates are removed depending on their relative rank. The Ranking method in the present study is applied by subjecting the Ranking procedure to Z candidates that pass the Standard selection procedure. Here, it is found that the opening angle between the photons provides an effective Ranking estimator, Fig. 3, other parameters such as the Z energy and the w2 from a kinematic fit do not improve significantly the performance of the estimator. 1
For this study the Z selection efficiency is defined as e ¼ S=S0 , where S is the selected signal and S0 is the total signal in the data. The Z selection purity is defined as P ¼ S=ðS þ BÞ where B is the background (including p0 ’s, combinatorial background, and Z’s not originating from B0s decays).
ARTICLE IN PRESS 472
A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473 Table 1 Results from the optimisation of pcut T and m where the signal significance is maximised. Three selection methods are investigated: Standard, Highest pT , and the Ranking method; the method ‘all’ shows the case where all candidates are taken without cuts on pcut T or m. The seventh column sstat shows the relative statistical uncertainty in the number of selected Z’s. The final column ssyst shows the relative systematic errors due to a 10 MeV=c2 underestimation of the signal width by the Monte Carlo (see Section 5). The accuracy of the values is implied by the number of significant figures.
Fig. 3. Distributions of the angle between photons pairs f12 for the Z signal and its background. The data is taken from a 4sM mass window and for a 2 GeV=c pT cut. The probability of a candidate being signal increases for smaller angles.
Optimisation results for the three selection methods are shown in Table 1. The three methods agree for the optimal cuts pcut T 2:0 GeV=c and m ¼ 1:7 and all result in a statistical uncertainty in the number of selected Z’s of sstat 1:0% with the Ranking method performing slightly better than the other selection methods. The final column ssyst contains results from the systematic study discussed in Section 5. Differences in the three methods are seen in the details of the efficiency and purity values; the Highest pT method obtains a high purity while the Ranking method obtains a high efficiency. The results so far are not remarkable, though a preference towards high efficiency or high purity may be valuable to some analyses. Important differences, however, do arise when we consider the performance of each selection method in the presence of inaccuracies in the simulation of the peak width. This is considered next.
5. Systematic errors due to inaccuracies in the simulation of the peak width Now consider the case where a Monte Carlo detector simulation underestimates the width of the Z signal by 10 MeV=c2 . The selection of this value is arbitrary but one can imagine this could be a realistic value when one considers the scale of the
Method
m pcut T
e
P
e P sstat ð%Þ ssyst ð%Þ
All Standard Highest pT Ranking
1.0 2.1 2.0 1.9
1.000 0.376 0.276 0.427
0.006 0.287 0.343 0.267
0.006 0.108 0.095 0.114
1 1.7 1.7 1.7
4.30 0.98 1.05 0.96
0.0 6.1 6.1 5.9
calibration functions in Fig. 2; however, the value actually depends on the details of a specific experiment and detector simulations. The resultant relative systematic error in an efficiency correction is defined as ssyst ¼ ðeMC eRD Þ=eRD where eMC is the efficiency determined by the Monte Carlo, and eRD is the true efficiency in the real data. In this study we are only dealing with Monte Carlo data, the inaccuracy in the Monte Carlo is generated by simply shifting the calibrated width of Fig. 2 down by 10 MeV=c2 ; the real data (with the true efficiency) is represented by the Monte Carlo without the shift. The resultant systematic error ssyst is shown in the final column of Table 1, the systematic error for each method is 6%, much larger than the statistical uncertainty. To reduce such systematic errors one can choose a wider mass window. The choice in this study can be determined by re-optimisation the cuts pcut T and m such that the total error sstat ssyst is minimised. The results of the re-optimisation are given in Table 2. For the Standard and Highest pT methods optimisation chooses much wider mass windows, subsequently the systematic errors are reduced to the same order of the statistical uncertainties. The response of the Ranking method is noticeably different, while the statistical uncertainty is still smaller the systematic error almost vanishes even though the value of m is
ARTICLE IN PRESS A. Beddall et al. / Nuclear Instruments and Methods in Physics Research A 554 (2005) 469–473
473
Table 2 Results from re-optimisation of the cuts pcut T and m such that the total error sstat ssyst is minimised. The accuracy of the values is implied by the number of significant figures Method
pcut T
m
sstat (%)
ssyst (%)
Total (%)
Standard Highest pT Ranking
2.1 2.1 1.5
5.4 5.4 4.1
1.21 1.26 1.17
0.69 0.75 0.07
1.4 1.5 1.2
smaller than those of the other methods. Also a 2 much lower pcut T value is preferred. A further illustration of the performance of the Ranking method can be seen when the error in the signal width is varied, this is shown in Fig. 4 where the mass window is fixed at a reasonable wide value of 4sM . The results show that relative to other selection methods, Ranking is insensitive to errors in the signal width. This can be explained by the tendency of the Ranking method to select signals that are closer to the signal peak.
6. Summary and conclusion Signal candidates are selected from two-photon invariant mass spectra within a mass window around the signal peak. Analyses tend to choose a mass window with a width of 2sM or less. Although such tight cuts are statistically optimal this study has shown that they can give rise to large systematic errors when in the presence of inaccuracies in the Monte Carlo simulation of the peak width (or similarly the peak position). Such systematic errors can be reduced by increasing the width of the selection mass window though at the expense of a lower signal significance. The optimal size of the mass window depends on the level of
2 Reconstruction to a lower momentum can be important, for example in measurements of particle production where differential cross-sections need to be studied at low particle momenta.
Fig. 4. Results for varying the peak width error. The value of m is fixed at 4.
statistics and the size of the systematic error, the latter being very difficult to estimate. A width of 4sM or more maybe be preferable in analyses where systematic errors are suspected to be large. The Ranking method provides some relief to this dilemma as it is much less sensitive to systematic errors while at the same time giving better statistical performance.
References [1] HERWIG 6.5, G. Corcella, I.G. Knowles, G. Marchesini, S. Moretti, K. Odagiri, P. Richardson, M.H. Seymour, B.R. Webber, JHEP 0101 (2001) 010 [hep-ph/0011363]; hep-ph/ 0210213. hhttp://hepwww.rl.ac.uk/theory/seymour/herwig/i [2] F. Ohlsson-Malek, M. Melcher, ATLAS Calorimeter performance at low pT for g, p0 and Z particles identification in the B-Physics Context, ATLAS Internal Note, ATLCOM-PHYS-2001-021 10/09/2001, and ISN-01.75. [3] A. Golutvin, Nucl. Instr. and Meth. A 453 (2000) 192. [4] Peter Schacht, Nucl. Instr. and Meth. A 535 (2004) 446. [5] A. Beddall, et al., Nucl. Instr. and Meth. A 482 (2002) 520.