DØ Note: XXXX
DØ Note: XXXX
V0.2
Inclusive μ + jet Cross-Section
and
Inclusive b-jet Cross-Section
Don Lincoln, Yevgeny Galyaev, Hong Luo, Neal Cason, Paul Bierdz, Phil Buksa, Markus Wobisch
ABSTRACT
In this Note, we describe the details of a measurement of the μ + jet cross-section at a (s of 1.96 TeV. An attempt is made to extract the inclusive b jet cross-section for the central region | y | < 0.5.
Introduction
While the Standard Model of particle physics does a brilliant job of explaining the bulk of measurements made by modern experimenters, it is well known that there exist a number of questions that can be easily asked but not as easily answered. One of these questions is the nature and cause of three particle generations, each similar in gross properties, yet somehow different among other things, including increasing mass. In a simple argument by analogy, one might compare this situation with the chemical Periodic Table, in which a similar phenomenon was successfully explained by atomic structure. Consequently a possible explanation of the problem of particle generations is the idea of quark compositeness. If quarks are composite (taking the quarks with charge –1/3), then one might envision that a down quark is in the ground state, while the strange and bottom quarks are in low-lying excited states. A parallel structure would describe the +2/3 charge quarks.
By this reasoning, it might be true that a study of the quarks of the third generation might exhibit deviation from point-like behavior. Thus one might choose to study the process
pp ( X ( qq, where the final state quarks would be either bottom or top. Given the resources at hand, it was decided to study X ( bb. A more compelling discussion of the theoretical concerns of such a search will be given in a forthcoming note however for the moment, one can restrict oneself to a study of the relevant concerns and performances of b-tagging in very high Pt jets and high mass di-b jets. If a quality job of high Pt b-tagging is impossible, then theoretical concerns are moot. If successful, an intermediate result of this study will naturally yield an inclusive jet cross section for jets originated by b quarks. Subsequent work will explore the dijet question.
In addition to the goal of this measurement, study of high Pt b jets will provide useful feedback to the Higgs group. It is also true that there exist possible physics mechanisms which might generate enhanced b-quark production at high Pt (i.e. any hypothetical heavy-mass object preferentially decaying into quarks of the third generation.) Conventional inclusive jet cross-section measurements would not reveal this behavior, being over shadowed by the more common light-quark and gluon behavior.
Analysis Techniques
The analysis technique used in this study is based heavily on Ariel Schwartzman’s d0root package [1]. This is a root-based package, in which one can quickly change the nature and details of the cuts, without the inconvenience involved in recompiling RECO and framework-based analyses.
The basic premise of the analysis utilizes the strength of DØ’s calorimeter. First jets are found using the conventional (η,φ ) 0.5 cone based jet finder. Objects that are relevant to heavy flavor identification are then appended to the jet. For example, RECO muons and secondary vertices that are collinear to the jet are attached, as are (when relevant) the initiating parton as specified by leading order Monte Carlo. In addition, Monte Carlo truth muons, b and c quark carrying hadrons are associated.
Within the context of the d0root package, this heavy flavor identification, calorimeter jet based construct is called a “D0JetInfo”. The details of each object, as well as the cuts that must be fulfilled to associate each object to the calorimeter jet are given below:
Calorimeter Jet:
Standard (η,φ ) jet, with cone of 0.5 [corrJCCB]. This includes JES and jet quality cuts.
Track-based Jet:
A track based jet finding algorithm was run. Tracks that can be used to create a track jet are required to have
|Aspect |Value |
|Minimum track Pt |1.0 GeV |
|Minimum # of SMT hits on track |2 |
|Track (x,y) DCA |< 0.15 cm |
|Track z DCA |< 0.4 cm |
|DCA significance |< 3.0 |
|χ2 |< 10 |
In addition, via btags_cert, I run a prefilter of 2PPV_NoV0 which removes tracks that come from Vo’s (Ko, Λo and γ conversions). As this prefiltering comes before track jet finding, this will slightly reduce the number of track jets, but will not affect the desired vertices.
With these “Vo-free” tracks, the track jet finder is run, requiring the following cuts.
|Aspect |Value |
|(η,φ) cone size |0.5 |
|ΔZ between tracks (unused) |< 2.0 cm |
|Minimum track Pt |0.5 GeV [overridden by track criteria] |
|Minimum Jet Pt |0.0 GeV |
|Minimum Pt of highest Pt track |1.0 GeV |
|Minimum # of SMT hits on track |1 [overridden by track criteria] |
|Track (x,y) DCA |< 0.2 cm |
|Track z DCA |< 0.4 cm |
A track jet is associated with a calorimeter jet if they match in (η,φ) space with a ΔR of less than 0.7 and a ΔZ of less than 1.5 cm.
Vertexing:
Primary vertexing was done within the context of btags_cert. The objects returned were the two-pass primary vertices. If multiple primary vertices were returned, the first one was used as the event primary vertex.
Secondary vertices were found using SVKalmanBtagger. First trackjets were found using the above-described algorithm. Then the algorithm tries to create secondary vertices using a “build up” method. Tracks are added to a secondary vertex if they pass the following cuts and a vertex candidate is retained if the cuts listed below are satisfied.
|Aspect |Value |
|Track addition to VTX χ2 |< 15. |
|Total VTX χ2 |< 100 |
|# Tracks |> 1 |
|Zdca |< 0.4 cm |
|DCA Significance |> 0 |
|VTX Collinearity |> 0.9 |
When secondary vertices were found, they were inspected to find the “best” secondary vertex. A best secondary vertex satisfied the following cuts. The dot product between the vector starting at the primary vertex and ending on the secondary vertex and the vector of the momentum of the secondary vertex must be positive. This essentially means that the secondary vertex is on the same side of the primary vertex as the secondary vertex’ direction of motion. The two dimensional decay length separating the primary and secondary vertex must be less than 2.5 cm (to suppress Ks, Λ’s and other strange quark carrying hadron decays). Finally the secondary vertex with the greatest three-dimensional decay length significance (defined to be the decay length divided by the combined uncertainty of the two vertices) was chosen. This cut uses the 3D decay length significance, rather than 2D, for historical reasons.
RECO Muons:
RECO muons were associated to the JetInfo object if they were of MEDIUM quality as specified by the Muon ID group. In addition, the Pt of the muon was required to exceed 5.0 GeV, the reasons for which will be mentioned in the text below. A RECO muon is associated with a calorimeter jet if they match in (η,φ ) space with a ΔR of less than 0.5 and a ΔZ of less than 1.5 cm.
D0JetInfo Kinematics:
In addition to the kinematics of the calorimeter jet and the associated muon, the standard heavy flavor JES (v5.3, no T42) is applied, resulting in an approximate energy for the parent parton. This is done separately to allow for cuts on both the calorimeter jet and the “right” (i.e. jet + μ) energy.
Run and Event number:
Each jet had the run and event number associated with it.
Leading Order Primary Parton:
In Monte Carlo, it is frequently convenient to know the partonic parentage of a particular jet. Within the context of the Pythia Monte Carlo, two leading order partons are generated which are subsequently modified by parton shower models. There was not available when the study began a method to unambiguously determine the original partons, so a simple algorithm was devised. The program loops over all partons in the event (including ones from parton showers). It finds the parton with the largest Pt and declares this a leading-order parton. It then loops again over the remaining partons and finds the highest Pt parton which also has a Δφ from the hot parton of more than (π – 0.7). This parton is declared to be the second leading order parton.
These two potential leading order partons are then compared to the two leading jets. The partons and jets are associated if they match in (η,φ ) space with a ΔR of less than 1.0 and a ΔZ of less than 100 cm. This matching was intentionally left loose to enhance efficiency and, since there are only two leading order partons allowed, this can be done without much effect on fake matches.
Monte Carlo Secondary Vertices:
Monte Carlo secondary vertices are those particles in the event record containing either a bottom or charm quark and decay weakly. The parent heavy flavor hadron is recorded, as are all daughters with no attention paid to the stability or charge state of the daughters. The correspondence between the parent b-hadron and its daughter c-hadron is lost.
The MC secondary vertices and calorimeter jets are associated if they match in (η,φ ) space with a ΔR of less than 0.5 and a ΔZ of less than 3 cm.
Monte Carlo Muons:
Monte Carlo muons are those muons recorded in the truth table. They include heavy flavor as well as light meson decay. While in principle a source of muons might be Hadronic showers in the calorimeter or from light meson decay even in the magnet or calorimeter, the MC generated for this study restricted the radius of the decay vertex of the particle decaying inclusively into muons to be less than 53 cm. The MC muons and calorimeter jets are associated if they match in (η,φ ) space with a ΔR of less than 0.5 and a ΔZ of less than 3 cm.
Monte Carlo Truth Particles:
Monte Carlo truth particles are ALL truth particles collinear with the calorimeter jet, even if the particle subsequently decays into other particles that will also be listed as collinear with the calorimeter jet. Restricting the particles to the stable ones is done via JetInfo methods. The truth particles and calorimeter jets are associated if they match in (η,φ ) space with a ΔR of less than 0.5.
Monte Carlo Partons:
Monte Carlo truth partons are ALL truth partons collinear with the calorimeter jet, even if the parton subsequently showers into other partons that will also be listed as collinear with the calorimeter jet. The truth partons and calorimeter jets are associated if they match in (η,φ ) space with a ΔR of less than 0.5. Note this is distinct from the leading order parton, as it provides another way to look at b and c quark content.
Monte Carlo Samples
Initially, twenty thousand events were requested at three different Pt ranges. These Pt ranges were 080-160, 160-320 and 320-980 GeV. Separate requests were made for direct b-quark production, direct c-quark production and also light quark (standard jet production). Later, additional requests were made. The low-numbered requests are used for the following PtRel discussion. The high-numbered requests were used for all others. The following tables list the available statistics, as well as the original MC production job number and SAM dataset definition.
|MC Request |Request Type |# Events |SAM Dataset Definition |
|# | | | |
|13334 |bb 80-160 |20000 |req-id-13334-tmb-good |
|13383 |bb 80-160 |18490 |req-id-13383-tmb-good |
|13336 |bb 160-320 |18193 |req-id-13336-tmb-good |
|13385 |bb 160-320 |27000 |req-id-13385-tmb-good |
|13338 |bb 320-980 |18297 |req-id-13338-tmb-good |
|13387 |bb 320-980 |17259 |req-id-13387-tmb-good |
|13335 |cc 80-160 |20500 |req-id-13335-tmb-good |
|13384 |cc 80-160 |20397 |req-id-13384-tmb-good |
|13337 |cc 160-320 |21000 |req-id-13337-tmb-good |
|13386 |cc 160-320 |20000 |req-id-13386-tmb-good |
|13339 |cc 320-980 |20140 |req-id-13339-tmb-good |
|13388 |cc 320-980 |20750 |req-id-13388-tmb-good |
|13372 |qcd 80-160 |51000 |req-id-13372-tmb-good |
|13373 |qcd 80-160 |51000 |req-id-13373-tmb-good |
|13374 |qcd 80-160 |50500 |req-id-13374-tmb-good |
|13375 |qcd 80-160 |51000 |req-id-13375-tmb-good |
|13376 |qcd 160-320 |33500 |req-id-13376-tmb-good |
|13377 |qcd 160-320 |24108 |req-id-13377-tmb-good |
|13378 |qcd 320-980 |20000 |req-id-13378-tmb-good |
Data Selection: (skim, cuts, etc.)
The initial data processing was from the Common Samples group’s CSG_QCD skim. The following URL gives more details.
:
All data, spanning the run range 161973-193780 (15 August 2002 - 07 June 2004) was used. The total number of the events in the CSG QCD skim was 40,460,043 events and ALL events were present in our TMBTrees. We subsequently skimmed the data so as to have a more manageable data set. One skim, the Muon_skim, required that at least one of the two leading jets have a collinear MEDIUM RECO muon, with Pt > 4.0 GeV. This data set comprised 405,671 events. In addition, although not directly germane to this analysis, a 2VTX_skim was done, which required that at least one of the two leading jets contain a “Best Secondary Vertex” (see above). This data set comprised 1,538,291 events. For both of these skims, the offline data quality database was consulted and if any of the following fields were listed as bad, the event was rejected: SMT, CAL, MET, CFT, MUO & JET.
After reasonable (yet loose) jet thresholds were determined (discussed below), a “micro skim” was performed. The requirement for this skim was that for each QCD trigger, the jet containing the tag was required to be above a threshold, for which the trigger was 90% efficient. The thresholds are [JT25 – 70, JT45 – 90, JT65 – 120, JT95 – 165]. Also, the “tagged” jet was required to have a rapidity [| y | < 0.5]. When the muon-based micro skim was completed, 18,328 events remained. Microskims were also performed on the SVX data set (for both Pt and Mass thresholds), but these skims are not relevant to this analysis.
Luminosity:
The luminosity was calculated according to the instructions from the CSG web page. While the full recorded luminosity for this data set was approximately 389 pb-1, when bad runs were removed, the actual luminosity for the unprescaled JT_125TT trigger was 293.8 pb-1. As each trigger was prescaled by an amount that varied with instantaneous luminosity, the luminosity database was used to determine the luminosity for each trigger. These were:
|Trigger |Luminosity (pb-1) |
|JT_25TT |1.81 |
|JT_45TT |28.48 |
|JT_65TT |142.46 |
|JT_95TT |292.8 |
|JT_125TT |293.8 |
The error on the luminosity is taken to be 6.5%.
V0 removal:
The technique of V0 removal is given in [2]. The essential point is that it is possible for strange quark-carrying hadrons to decay at a distance from the interaction point and thus be reconstructed as a secondary vertex. Prior to vertex finding, the tracks from each event are scanned and an attempt is made to reconstruct Kos, Λo or γ conversions in the silicon system. Tracks which are found to come from one of these occurrences are removed and not used for future secondary vertex finding. This was done within the btags_cert package.
Identification of Muons:
Conventional Wisdom[1] reports that the use of MEDIUM muons gives an optimum separation between muons and fakes. TIGHT muons do not markedly improve the separation, but do decrease the efficiency. This wisdom was generated during studies of isolated muons for Z and W searches.
Our initial analysis effort took this wisdom to heart. However, since this analysis uses muons inside jets (and therefore manifestly not isolated), studies were undertaken to see if conventional wisdom held. Figure 1 shows the φ distribution for μ-tagged jets using MEDIUM muons in our event sample.
[pic][pic]
Figure 1 Left: φ distribution of μ-tagged jets (in radians). Right: Same, but in rφ coordinates. The curve with higher statistics is for jets tagged with MEDIUM muons. The lower statistics curve is for jets containing muons which both satisfy the MEDIUM criteria, as well as the additional scintillator BC cut discussed in the text.
What was observed was the expected lack of muons in the bottom of the detector, where the muon system coverage is restricted. However, there were two regions (near -45º and -135º) for which the cuts for MEDIUM muons are loosened.
The definitions of muon quality criteria are given in [3]. Briefly, the definition of a MEDIUM muon changes dependent on the number of muon segments (nseg) and the octant number in the muon system. A medium nseg = 3 is defined to be a muon with both A and BC segments, nseg = 2 requires a BC segment matched to a central track, while a nseg = 1 muon requires an A segment matched to a central track. The following chart shows the criteria for MEDIUM muons for the various nseg definitions.
|Cut Variable |nseg |
| |3 |2 |1 |
|# A Wire Hit |( 2 |( |( 2 |
|# A Scint Hit |( 1 |( |( 1 |
|# BC Wire Hit |( 2 |( 2 |( |
|# BC Scint Hit |( 1 |( 1 |( |
|Octant |( |5,6 |5,6 |
|| η | |( |< 1.6 |< 1.6 |
Table 1 Cuts to define a MEDIUM muon. “(” indicates no restrictions.
When a muon is on the bottom of the detector, the criteria are loosened. The reason that we are seeing a large muon presence in the bottom of the detector is partly because we are looking inside jets. Even though D( has a deep calorimeter, there is often some amount of MIP punch-through which carries minimal energy. However, this MIP punch-through will fire the A layers of the muon system. Coupled by the fact that this activity occurs within a jet (which has a lot of tracks), it is often possible for hits in the muon A layer to be correlated with a track in the central tracker. Consequently there are many muon fakes inside jets for nseg = 1 MEDIUM muons. Requiring a BC scintillator hit greatly reduces the punch-through background. This point is illustrated in Figure 1.
In addition to this very important muon cut, there are some additional muon cuts necessary to clean up the sample. For instance, the Pt of the muon can be measured in three ways: in the local muon system, in the central tracking system and in via a combined fit (global). For a very tiny fraction of the events, the global and central Pt are in disagreement. It was found that imposing the extremely loose cut
|Pt(global) – Pt(central)| < 15 GeV
would substantially improve the cross-section measurement. Failure to impose this cut would give too many jets with improperly-reconstructed high Pt muons. As illustrated in Figure 2, this cut removes far fewer than 1% of the events.
[pic]
Figure 2 Linear and log histograms of the difference between global and central Pt measurements.
With the imposition of the requirement that there be a BC scintillator hit, the question of muon penetration arises. What Pt range of muons can penetrate the calorimeter and muon steel? Towards this end, we take central muon-tagged jets (|yjet| < 0.5, Pt(μ) > 4 GeV) and plot the muon Pt distribution both before and after the BC scintillator cut was imposed. To ensure only “better” muons, only reconstructed muons from the top half of the detector (0 < φ(μ) < π) were used. A typical result is shown in figure 3.
[pic]
Figure 3 Effect of imposing BC scintillator cut on muon. This is for JT_45TT trigger.
In this figure, each bin is 0.5 GeV wide and the first bin covers 4.0-4.5 GeV. While this is for a specific trigger, the basic message is the same for all. Imposition of the BC scintillator cut seems to deplete muons with Pt less than 5 GeV (as evidenced by the blue curve in the top plot). It also appears that for a Pt of the muon between 4 and 4.5 GeV, there are an excess of reconstructed muons (probably punch through). Thus we decided to restrict the Pt of the muon to exceed 5 GeV, independent of jet energy. In addition, an additional 10% inefficiency is introduced in muon reconstruction due to the BC scintillator cut.
PtRel Failure:
A standard method for tagging b-jets is to look for an associated soft lepton, often a muon. In order to determine the fraction of muon-bearing jets that came from a b-hadron decay, the relative Pt (PtRel) of the muon with respect to the combined muon+jet axis is determined. Because b-hadrons have a mass typically of order 4.5 Gev, larger than the 1.5 GeV of c-hadrons and the low mass of lighter hadrons, the PtRel for b-hadrons is usually larger. This technique is described in [4]. Since the jets that would be of interest to a New Phenomenon search would necessarily be of high Pt, the behavior of PtRel in high Pt jets needs to be determined. If the PtRel distributions between the three components that make up the mu-tagged jet population (b, c and light/gluon jets) are sufficiently different, it would be possible to fit the data to “templates” of the three sub-distributions to the same distribution in data mu-tagged jet events. The sub-distribution fractions (and subsequently purity) of the data sample could thus be determined.
In order to determine the templates of the various sub-distributions, Monte Carlo was used to determine the PtRel template of b-hadrons and c-hadrons. First the invariance of the PtRel with respect to variations in the jet Pt was established (for Pt > 100 GeV), allowing single Pt-independent templates to be used. For light quarks and gluon initiated jets (which we call loosely “light jets”), the templates were determined first in Monte Carlo, followed by data. In data, the PtRel of tracks within the jet was used instead of muons (which were used in the heavy flavor samples). Basically the idea for light quarks is that some of the particles in the jet (predominantly pions and other light mesons) can decay into muons. If the decay was as such that the muon and pion Pt were similar, the muon finding algorithm would find both the muon and the original decay pion in the tracking volume and connect the two. Thus the pion/track distribution is taken to be representative of the light quark fakes. One slight consequence of this approach is that during the determination of the light quark template, the PtRel of the track is determined with respect to the jet axis, rather than the jet+track axis (as the track/pion is presumably measured in the calorimeter, unlike the muon of the heavy flavor case.)
Carefully study has revealed that while the templates for the three sub-distributions are somewhat different, they are sufficiently similar that it is not possible to use them to reliably extract the b-quark initiated content. Systematic errors associated with the template shapes erase any measuring power, especially at high Pt.
[pic][pic][pic]
Figure 4 PtRel templates for b-quark and c-quark Monte Carlo and QCD, for three different Pt bins. We see that the three templates become more similar as the Pt increases.
MC Predictions of tagging rates:
Tables 2 and 3 summarize important Monte Carlo information. What is seen is that the fraction of cross section from b quark production is low…on the order of 1-2%. The amount of b content from light parent Leading Order partons with heavy flavor gluon radiation is similar in magnitude to the Leading Order b-quark production.
Table 4 shows the efficiency and purity for both muon-only tagged as well as muon plus secondary vertex tagging. It is evident that the b content can be strongly enhanced by these cuts. It also is true that using Monte Carlo reconstructed variable muon information gives a smaller b-jet purity than Monte Carlo truth studies.
|xsec Fraction |080-160 |160-320 |320-980 |
|udsg |92.45 ( 0.09 |94.02 ( 0.16 |93.41 ( 0.16 |
|c | 2.32 ( 0.05 | 1.89 ( 0.09 | 2.97 ( 0.11 |
|b | 1.50 ( 0.04 | 1.51 ( 0.08 | 2.41 ( 0.10 |
|unknown | 3.73 ( 0.06 | 2.57 ( 0.11 | 1.20 ( 0.07 |
Table 2 Fraction of cross-section associated to identified Leading Order partons. Approximately 1.5-2.0% is b quarks, 2.0-3.0% is c quark and 1.5-3.5% is unknown. See table 3, bottom “Truth” section for fraction of light-quark jets with heavy flavor content from heavy-flavor parton showers. This is typically 5%, with 1.5% being b-quark and 3.5% being c-quark.
| |LO Parton |
| |80-160 |160-320 |320-980 |
|Parton |Efficiency |Purity |Efficiency |Purity |Efficiency |Purity |
|b |11.68 |
| |80-160 |160-320 |320-980 |
|Parton |Efficiency |Purity |Efficiency |Purity |Efficiency |Purity |
|b |7.61 |
|JT_25TT_NG |70 (Guessed) |
|JT_45TT |100 |
|JT_65TT |130 |
|JT_95TT |190 |
Table 5 Thresholds on jets + muons Pt for the various QCD triggers.
A threshold for JT_25TT_NG and the various lower Pt triggers was not determined (except by the guess listed above).
[pic]
Figure 6 Example plots that show how the efficient trigger points were determined.
JES Concerns
The jet energy scale used was v5.3. Unfortunately the JES in p14 TMB’s was applied improperly. In the D( data format, there are nominally JCCB (all 0.5 jets), corrJCCB (0.5 jets with Hadronic-only JES applied, plus only jets that pass the quality cuts retained) and corr_muJCCB (as corrJCCB, but with muon-included JES as appropriate). As it happens corrJCCB and corr_muJCCB were identical, e.g. the muon was never appropriately included. Consequently in making the D0JetInfo objects, JES 5.3 was applied properly.
In order to test that the jet energy scale did a good job, some simple tests were made. The first was to test the Pt balancing of dijet events. The cuts were that only two jets existed in corrJCCB (& JCCB, see below). The two jets needed to be separated by Δφ > 2.84. Both jets needed to be central |yjet| < 0.5. One jet was required to have a muon (as specified earlier in the paper) while the other jet was required to NOT have a muon.
In addition another cut turned out to be very important. It was also necessary to insist that there only be 2 JCCB jets. Without this additional cut, what occasionally occurred was that there were three JCCB jets, two normal and balancing and one tiny, low Pt jet. One of the normal jets failed the jet quality cuts, giving only two corrJCCB jets, with highly asymmetric Pt (and further, since one of the two leading jets was required to have a muon, the “good” jet tended to be the muon one). This effect gave a nasty asymmetry in the jet resolution.
With these events, the idea was simple: “hadronic-only” jets were considered to be “right”. An asymmetry variable was defined
[pic]
This asymmetry was plotted against the average jet Pt (Pt1 + Pt2)/2. This asymmetry is plotted in Figure 7.
[pic][pic]
Figure 8 Left: plot of asymmetry (y) as a function of average Jet Pt (x). Note the 4% offset. Right: Same plot after jets with muons have their Pt scaled by 1/1.038.
What was observed was a 3.8% offset in Pt for jets with muons. Each jet containing a muon was scaled by the inverse of this factor. When this scaling was done, the jets with and without muons were balanced. The 3.8% factor was independent above Pt of 50 GeV. Below that there was a significant difference. However in the data set, jets below a Pt of about 70 GeV were not fully efficient and therefore suspicious. No attempt was made to deal with the Pt scale below 50 GeV.
Jet Resolution
Jet resolution was dealt with in the usual way. The same cuts used for the JES verification (derived above) were used. Plots similar to figure 8(right) were used, except the RMS rather than the error on the mean were used. One can plot the RMS as a function of average Jet Pt and one observes:
[pic][pic]
Figure 9 LEFT: Top: Profile plot with error bars = RMS. Bottom: Plot of RMS as a function of average jet Pt with fit overlaid. RIGHT: Overlay of combined., Hadronic-only and best-estimate of mu+jet.
One can fit the resultant curve via the following form
[pic]
In order to get the resolution for the jets with muons, one must subtract the effect of the jet that does not contain a muon. This is done in quadrature.
[pic]
The relevant numbers are given in the below table. Note the “C” term for Extracted. This term is poorly determined, as the no-muon case is lower than the muon case. Measured means the result displayed in Figure 9, red curve. JES, no muons is that extracted by the JES group for JES 5.3, and Extracted (dashed black) is the result in which we are interested. The JES resolution for jets without muons is given in [5].
|Term |Measured |JES, no muons |Jet + μ, Extracted |
| |Mean |Error |Mean |Error |Mean |Error |
|N |7.68 |2.99 |0.000 |2.748 |7.68 |4.07 |
|S |2.14 |0.13 |0.950 |0.023 |1.92 |0.13 |
|C |0.000 |0.08 |0.07 |0.0027 |0.00 |0.08 |
Unsmearing
Unsmearing is done in the traditional way. There was an original spectrum (particle level truth), which is smeared by the detector jet resolution. Because the jet Pt resolution for jets containing muons is considerably larger than for those without muons, this is expected to be a considerable effect.
In general, the observed spectrum can be written as F(pt) while the particle-level truth spectrum can be denoted f(p’t). The smearing function G(p’t - pt, p’t) is usually taken to be a gaussian. The observed spectrum can then be written as
[pic] (1)
The particle level truth function can be parameterized in a number of ways. One popular way is to use a four parameter function
[pic] (2)
Thus one inserts this function into the smearing equation and minimizes the difference between the smeared equation and the data (after all efficiencies are imposed). A second ansatz tried was:
[pic] (3)
The data was then fit to the smeared ansatz via the following χ2 function.
[pic] (4)
Finally, one can correct the data by the ratio of f(pt)/F(pt) (shown in figure 11b) and determine the corrected and unsmeared result. In essence, this is the same as defining the measurement as that of f(pt) (assuming a good fit of F(pt) ), but with the statistical variation preserved. In figure 10a we see the efficiency corrected (but not unsmeared) cross section. In figure 10b, we see an overlay of the data (points), point-by-point fit to the smeared function (stars, *), the unsmeared parent function (black line). The figure denotes a fit to the ansatz of equation (3). A fit to equation (2) was performed successful, but yielded numerically similar results. The parameters of the unsmeared function are:
|Variable |Fit Value |Parabolic Error |
|N1 |7.94 |0.42 |
|k1 |15.96 |1.44 |
|N2 |3.56 |0.67 |
|k2 |33.98 |3.21 |
There are 19 data points, but the top one is not used in the fit, as it contains only one jet. There are 4 parameters and the χ2 is 18.45, yielding a χ2/dof = 18.45/(19 –1 – 4) = 1.318.
The correlation coefficients are
| |N1 |k1 |N2 |k2 |
|N1 |1.000 |-0.990 |0.877 |-0.842 |
|k1 |-0.990 |1.000 |-0.931 |0.901 |
|N2 |0.877 |-0.931 |1.000 |-0.995 |
|k2 |-0.842 |0.901 |-0.995 |1.000 |
The error matrix is
| |N1 |k1 |N2 |k2 |
|N1 |0.176 |-0.598 |0.245 |-1.133 |
|k1 |-0.598 |2.077 |-0.894 |4.169 |
|N2 |0.245 |-0.894 |0.440 |-2.129 |
|k2 |-1.133 |4.169 |-2.129 |10.31 |
[pic][pic]
Figure 10 Left figure: the efficiency corrected (but not unsmeared) cross section. Right figure: an overlay of the data (points), point-by-point fit to the smeared function (stars, *), the unsmeared parent function (black line), the dashed line is an MC prediction of the inclusive jet cross-section.
[pic][pic]
Figure 11 Left: the fractional difference between the data and the smeared fit. Top and bottom are the same plots, except for having an expanded vertical scale. The highest Pt bin is not included in the fit. Right: ratio of unsmeared/smeared fit, as discussed in the text.
A second fit using the ansatz of equation (4) (setting γ = 0) gave numerically similar results. Figure 12 shows the difference in the unsmearing function for both ansatzes.
[pic]
Figure 11 The percentage difference between the unsmearing factors from both basic ansatzes. The blue and pink curves are simply mirror images around zero. The pink curve is the difference. Essentially, stating a 5% systematic error due to unsmearing ansatz is conservative for all jets with a transverse momentum greater than 100 GeV.
Results
What follows is the data. Initially we attempt an inclusive cross section measurement for jets tagged by muons. Later, we attempt to extract the heavy-flavor jet cross-section. We have required that the |yjet| < 0.5. Jet quality and the primary vertex cuts described above were imposed. Each jet is required to have a muon with a minimum Pt of 5 GeV within a ΔR of 0.5.
We now come to the final effort. If one combines all of the information included above, one can determine the final, efficiency-corrected cross-section, which is indicated in Figure 10. One multiplies this data by the ratio of the unsmeared to smeared fit to the data for each bin to determine the final unsmeared cross-section. The data is tabulated in Table 6 with the unsmeared cross-section given in figure 13.
The error of the μ + jet measurement is discussed below, but it is on the order of 20% at 100 GeV, rising to (-50%, +80%) at 400 GeV, thus the prediction and measurements are broadly within expectations, although different enough to cause some concern.
|Pt Bin |Mean |N |Luminosity |σ(Eff) |Δσ(Eff) |Smear |UnSmear |
|90-100 |94.0273 |96 |1.81 |1.22E+01 |1.25E+00 |7.58E+00 |7.73E-01 |
|100-110 |104.536 |727 |28.5 |5.48E+00 |2.03E-01 |3.32E+00 |1.23E-01 |
|110-120 |114.588 |458 |28.5 |3.38E+00 |1.58E-01 |2.00E+00 |9.33E-02 |
|120-130 |124.726 |293 |28.5 |2.14E+00 |1.25E-01 |1.23E+00 |7.21E-02 |
|130-140 |134.593 |838 |142.5 |1.23E+00 |4.26E-02 |6.98E-01 |2.41E-02 |
|140-150 |144.516 |538 |142.5 |8.03E-01 |3.46E-02 |4.46E-01 |1.92E-02 |
|150-160 |154.609 |367 |142.5 |5.54E-01 |2.89E-02 |3.02E-01 |1.58E-02 |
|160-170 |164.331 |265 |142.5 |4.12E-01 |2.53E-02 |2.21E-01 |1.36E-02 |
|170-180 |174.772 |172 |142.5 |2.69E-01 |2.05E-02 |1.42E-01 |1.08E-02 |
|180-190 |184.603 |126 |142.5 |2.02E-01 |1.80E-02 |1.05E-01 |9.35E-03 |
|190-200 |194.741 |160 |292.8 |1.26E-01 |9.97E-03 |6.47E-02 |5.12E-03 |
|200-210 |204.997 |98 |292.8 |7.75E-02 |7.83E-03 |3.93E-02 |3.97E-03 |
|210-220 |214.294 |75 |292.8 |6.12E-02 |7.07E-03 |3.07E-02 |3.54E-03 |
|220-230 |223.921 |62 |292.8 |5.14E-02 |6.53E-03 |2.55E-02 |3.24E-03 |
|230-250 |238.96 |91 |292.8 |3.73E-02 |3.91E-03 |1.82E-02 |1.91E-03 |
|250-270 |258.965 |36 |292.8 |1.47E-02 |2.45E-03 |7.05E-03 |1.18E-03 |
|270-330 |293.469 |47 |292.8 |6.55E-03 |9.56E-04 |3.08E-03 |4.49E-04 |
|330-400 |369.895 |10 |292.8 |7.71E-04 |2.44E-04 |3.50E-04 |1.11E-04 |
|400-500 |416.204 |1 |292.8 |1.33E-04 |1.33E-04 |5.94E-05 |5.94E-05 |
Table 7 A repeat of some of the information of Table 6, with the last two columns now containing the heavy-flavor only cross-section.
Error Estimation
The systematic errors on the cross-section include errors on the efficiencies, the error from the jet energy scale, luminosity, errors due to unsmearing, as well as the usual statistical ones
The individual efficiencies are given in the above sections, but one can tabulate the efficiencies and their associated uncertainty.
|Efficiency |Estimate |Uncertainty |
|Primary Vertex, εPV |0.84 |0.01 (?) |
|Trigger, εT |1.00 |0.00 |
|Muon, εμ |0.38 |0.04 (?) |
|Jet Finding, εj |0.99 |0.01 |
|Total |0.31 |0.04 |
Thus we see an approximate error of 10% due to efficiency estimations. For p14, the luminosity error is also quoted as 6.5%.
To incorporate the errors due to jet energy scale (JES), I determined the JES correction for each jet and increased it or decreased it by one standard deviation. I then created two new raw cross-section measurements, one for each Pt bin, which is determined by the number of events for the standard JES, as well as the high and low JES. The effect due to the high and low JES increased or decreased the number of events in each bin. The JES systematic error was delineated by the cross-sections as determined by the one-sigma variation of the JES. In all cases, the unsmearing factor was taken to be the same as the central value. The effect of this uncertainty is presented in figure 16 and more clearly in figure 17. The error is relatively small at 100 GeV (~20%) rising to (-50%, +80%) at 400 GeV.
Another error was due to the unsmearing ansatz fits to both functional forms (equations 2 and 3). The difference in the unsmeared cross-section was small ( ................
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