A Novel Technique for Studying the Z Boson Transverse Momentum Distribution at Hadron Colliders
aa r X i v : . [ h e p - e x ] J u l A Novel Technique for Studying the Z BosonTransverse Momentum Distribution atHadron Colliders
M. Vesterinen, T. R. Wyatt.
Particle Physics Group, School of Physics and Astronomy, University ofManchester, UK.
Abstract
We present a novel method for studying the shape of the Z boson transverse mo-mentum distribution, Q T , at hadron colliders in p ¯ p/pp → Z/γ ∗ → l + l − . The Q T is decomposed into two orthogonal components; one transverse and the other par-allel to the di-lepton thrust axis. We show that the transverse component is al-most insensitive to the momentum resolution of the individual leptons and is thusmore precisely determined on an event-by-event basis than the Q T . Furthermore, wedemonstrate that a measurement of the distribution of this transverse componentis substantially less sensitive to the dominant experimental systematics (resolutionunfolding and Q T dependence of event selection efficiencies) reported in previousmeasurements of the Q T distribution. Key words:PACS:
The shape of the Z boson momentum distribution transverse to the beamdirection ( Q T ) at a hadron collider tests the predictions of quantum chromodynamics (QCD), since non-zero Q T is generated through radiation from theinitial state partons. A good understanding of electroweak vector boson pro-duction is important in precision measurements (e.g. top quark and W bosonmass) and in Higgs boson and new phenomena searches at hadron collider ex-periments. In many of these searches the signal and/or background processesinvolve electroweak vector bosons produced in association with jets. Preprint submitted to Elsevier July 30, 2008 t low Q T ( Q T ≪ Q ), where the emission of multiple soft gluons is important,calculations in fixed order perturbative QCD diverge. There exist resummationtechniques, in which singular contributions from all orders of α s are resummedto give a finite result. Resummation was first applied to the Drell-Yan processby Collins, Soper and Sterman (CSS) [1]. The resummation is carried out inimpact parameter ( b ) space and includes a non-perturbative (NP) form factorthat needs to be determined from data. Brock, Landry, Nadolsky and Yuan(BLNY) proposed the following parameterisation [2]: S NP ( b, Q ) = " g + g ln Q Q ! + g g ln(100 x i x j ) b (1)where x i and x j are the fractions of the hadron momenta carried by the initialstate partons, Q is an arbitrary scale set to 1 . g i need to be fitted from data.Using the BLNY NP form factor, the CSS formalism was able to describe si-multaneously Tevatron Run I Z data and lower Q Drell-Yan data in a globalfit of the parameters g i [2]. The Q T distribution at the Tevatron ( Q ∼ M Z )is sensitive to g and insensitive to g and g . A larger value of g correspondsto a harder Q T distribution. The CSS formalism is implemented in the nextto leading order (NLO) event generator ResBos [3]. In Run II the DØ Collab-oration reported a Q T measurement in the di-electron channel with 1 fb − ofdata [4]. For low Q T ( Q T <
30 GeV), the DØ data is, within the measurementuncertainties, well described by the CSS/BLNY formalism.At low Q T the overall uncertainties were dominated by the parton distributionfunctions (PDFs) and the following experimental systematics [4]: • Unfolding the Q T measurement to account for the resolution in the mea-surement of the E T of the electrons. • Correcting for the Q T dependence of the overall event selection efficiency.As a result of these substantial experimental systematics, the low Q T regionwas not much better measured in this 1 fb − Run II analysis than in the100 pb − Run I analysis [5]. In both analyses, a measurement was made of g . The measurements: 0 . ± .
06 GeV for Run I and 0 . ± .
06 GeV forRun II, have comparable uncertainties .An observable that is sensitive to the Q T , but less sensitive to these experi- The DØ Run I and Run II measurements cannot be directly compared since theyused different PDFs and the Run I measurement used the the Ladinsky-Yuan (LY)parameterisation [15] of the NP form factor as opposed to the BLNY parameteri-sation. Only the third term in g g is different and any shift in the fitted g is farsmaller than the uncertainties on these measurements. a T , are presented. The a T observable has previouslybeen used in the selection of l − l + ν ¯ ν final states at LEP by the OPAL collab-oration [6]. The UA2 Collaboration used a similar observable, p Zη , which werefer to as b T , in a Q T measurement [7]. The measured Q T is highly sensitive to the lepton p T resolution. Our goal is tobuild an observable that is less sensitive to this resolution, whilst still sensitiveto the Q T . We keep in mind the fact that collider detectors generally have farbetter angular resolution than calorimeter E T or track p T resolution. Fig. 1. A schematic representation in the transverse plane, of the construction of a T and a L in a typical leptonic Z decay. The hadronic recoil is expected to have equaland opposite transverse momentum to the Z . For events with di-lepton azimuthal separation, ∆ φ ll > π , the Q T is decom-posed into orthogonal components as follows (See figure 1): • The thrust axis is defined as: b t = ~p T (1) − ~p T (2) | ~p T (1) − ~p T (2) | where ~p T ( i ) is the trans-verse momentum vector of lepton i . The two leptons have equal momentumtransverse to this axis. • The transverse momentum vector of the di-lepton system, ~Q T = ~p T (1) + ~p T (2) ,is decomposed into components transverse to the axis, a T = | ~Q T × b t | , andaligned with the axis, a L = ~Q T · b t .For events with ∆ φ ll < π , a T is set equal to the Q T , while a L maintains thesame definition for all values of ∆ φ ll .At low Q T , ∆ φ ll ∼ π , hence the uncertainty on a T is approximately theuncertainty on the individual lepton p T ’s multiplied by the sine of a smallangle. In contrast, the uncertainty on a L (and thus also Q T ) is approximately3he uncertainty on the individual lepton p T ’s multiplied by the cosine of asmall angle.An alternative observable is discussed, b T , whose construction is identical tothat of a T except that the decomposition is performed relative to the di-leptonperpendicular bisector axis: b b = ~p T (1) − r ~p T (2) | ~p T (1) − r ~p T (2) | where r = | ~p T (1) | / | ~p T (2) | . Thetwo leptons have equal acoplanarity with respect to this axis. In an event inwhich the leptons have equal p T , the values of a T and b T are equal. No studyis presented of the component longitudinal to the axis, b L .As discussed below, the relative sensitivity to lepton p T mis-measurement of a T and b T depends on the correlation between the lepton p T and its resolution. Monte Carlo (MC) events are generated using pythia [8], which treats atleading order (LO) the process p ¯ p → Z/γ ∗ → µ + µ − plus up to one jet, at acentre of mass energy of 1.96 TeV, and mass between 60 and 130 GeV. pythia uses additional parton showering to simulate “soft” transverse momentumgeneration. Although this study is carried out with Z → µ + µ − events, theidea is applicable to studying the di-electron channel.In order to simulate the imperfect muon p T resolution of a detector, Gaussiansmearing of width 0.003 GeV − is applied in 1 /p T to both muons, which isapproximately the design muon p T resolution of the Run II DØ detector [9].The design tracking resolution of other current and future hadron colliderdetectors; CDF, ATLAS and CMS [10,11,12] vary between ≈ − and ≈ − GeV − depending on the track p T and pseudo-rapidity ( η ). We refer tothe constant δ (1 /p T ) form of the resolution p T dependence as muon-like . Analternative resolution dependence is studied; electron-like resolution with theform: δp T /p T = 0 . /p / , which would be the form expected for a calorimeter-based measurement appropriate for electrons. Gaussian smearing is appliedto the lepton azimuthal angle φ of width 0.0005 rad., which is the typicalresolution of a detector.In order to study the dependence of event selection efficiency on Q T , a T , a L and b T , the following simple cuts are applied to the event sample. These arerepresentative of the cuts typically applied to select Z decays at a hadroncollider. • The di-muon invariant mass must be between 70 and 110 GeV. • Kinematic cuts on both muons: p T >
15 GeV and | η | < An isolation cut on both muons: f iso < . f iso = P p T (∆ R < . R = q (∆ η ) + (∆ φ ) is the radius of a cone around the candidatemuon. The sum is over all particles excluding both the candidate muon andany neutrinos. pythia is a LO event generator and is not expected to give a good descriptionof the Q T distribution in real data. The events are re-weighted in Q T and Z rapidity ( y ) to match the prediction of ResBos using the default settings, whichis in good agreement with DØ Run II data [4] in the region of low Q T ( Q T <
40 GeV). Event samples are generated using ResBos, and the re-weightingprocedure is carried out as follows: • Two dimensional histograms in Q T and | y | are produced for both the pythia and ResBos event samples. • Dividing the ResBos histogram by the pythia histogram gives an event-weight histogram. • The pythia events are now given an event-weight based on their Q T and | y | such that the distributions of these variables are the same as the ResBosprediction.Hereafter, unless otherwise stated, MC refers to pythia re-weighted to Res-Bos. p T Mis-measurement (GeV)
T gen Q g e n X | / g e n - X d e t X m ea n | -1
10 1 T Q T a L a T b (a) muon-like (GeV) T gen Q g e n X | / g e n - X d e t X m ea n | -1
10 1 T Q T a L a T b (b) electron-like Fig. 2. Mean resolution, | X det − X gen | /X gen for each of the observables, Q T , a T , a L and b T as a function of Q T gen , with (a) muon-like resolution( δ (1 /p T ) = 0 .
003 GeV − ) and (b) electron-like resolution ( δp T /p T = 0 . /p / ). detector level quantity, X det , andthe unsmeared or generator level quantity, X gen , where X corresponds to Q T , a T , a L or b T .Figures 2 (a) and 2 (b) show the mean resolution, | X det − X gen | /X gen as afunction of Q T gen for muon-like and electron-like resolution respectively. It canbe seen that for low to moderate values of Q T ( Q T gen <
50 GeV), a T and b T are significantly better measured than a L or Q T . Either a T or b T are thereforeparticularly well suited to studying Z production at low to moderate Q T . Inthe region of low Q T ( Q T gen <
15 GeV), a T and b T have similar resolutions foreither the (a) muon-like or (b) electron-like cases. At increasing Q T , a T is moresuited for (a), and b T more suited for (b). The construction of a T ensures thatthe acoplanarity angle to the thrust axis is smaller for the larger p T leptonwhich tends to have poorer resolution in the muon-like case. In the followingsections, for brevity, we discuss the case of muon-like smearing. Figure 3 shows the dependence of the event selection efficiency (separatelyfor the cuts on muon | η | , p T and isolation) on the generator level Q T , a T , a L and b T . The cuts are applied in the following order: | η | ; p T ; isolation. Theefficiency dependence for each cut is calculated having applied all previouscuts. For large Q T , the muons tend to be more central, increasing the η cutefficiency. The same correlation is apparent for a T and a L , although weakerthan for Q T .The muon p T cut dependence on a T is flat in the range considered. Converselylarge values of a L , generate an asymmetry in the p T ’s of the two muons andtend to push the lower p T muon below the cut threshold. The dependenceon the isolation cut is substantially flatter for a T than for a L . Large a L corre-sponds to a high p T hadronic recoil with a large fraction of its p T aligned alongthe thrust axis and thus possibly lying within the isolation cone of one of thetwo muons. There is no such dependence on a T , since a T is the component ofthe recoil p T transverse to the thrust axis.In summary, the efficiencies of the cuts on muon | η | , p T and isolation dependless strongly on a T than Q T . The dependence of b T on each of the cuts issimilar to that of a T . 6 (GeV) T a X ˛ T p ˛ iso ˛ h ˛ (GeV) T b X ˛ (GeV) L a X ˛ (GeV) T Q X ˛ Fig. 3. The dependence of event selection efficiencies on the generator level (a) a T ,(b) b T , (c) a L and (d) Q T in Z → µ + µ − MC events. The efficiencies ǫ η , ǫ p T and ǫ iso (evaluated relative to the previous cut in the order; | η | , p T , isolation) are shownseparately for cuts on muon | η | , p T and isolation respectively. Note that the rangeof Q T shown in this figure is √ × larger than for a T and a L . Figure 4 compares (separately for Q T , a T , a L and b T ) the generator level distributions without selection cuts and the detector level distributions with selection cuts. The detector level Q T and a L distributions are substantiallyaffected by the detector resolution and event selection efficiency. In contrastthe distributions in a T and b T are almost completely unaffected.In order to extract the underlying shape, the measured distributions wouldneed to be unfolded for the detector resolution and corrected for the eventselection efficiency with associated systematic uncertainties. These uncertain-7 ) - ( G e V T a / s d s / gen X (all cuts) det X (a) (GeV) T a g e n X ) / g e n - X d e t X ( -0.06-0.04-0.0200.020.040.06 ) - ( G e V T b / s d s / gen X (all cuts) det X (b) (GeV) T b g e n X ) / g e n - X d e t X ( -0.06-0.04-0.0200.020.040.06 ) - ( G e V L a / s d s / gen X (all cuts) det X (c) (GeV) L a g e n X ) / g e n - X d e t X ( -0.6-0.4-0.200.20.40.6 ) - ( G e V T Q / s d s / gen X (all cuts) det X (d) (GeV) T Q g e n X ) / g e n - X d e t X ( -0.6-0.4-0.200.20.40.6 Fig. 4. Detector and generator level distributions for (a) a T , (b) b T , (c) a L and(d) Q T . The detector level distributions are for Gaussian smearing in 1 /p T of width0.003 GeV − , and all selection cuts are applied. The generator level distributionsdo not include selection cuts. The lower halves of each plot show the fractionaldifferences. ties would be substantially smaller for a T or b T than for Q T or a L . In fact, forany realistic detector resolution, the measured a T or b T distribution describesthe underlying distribution within a few percent without any unfolding or ef-ficiency correction at all. The statistical sensitivity of a T or b T to the shapeof the underlying Q T distribution is also enhanced by the lack of resolutionsmearing compared to Q T . 8 (GeV) T * a g Z/ ) - ( G e V T / d a s d s / = 0.54 GeV g = 0.82 GeV g Fig. 5. The generator level prediction of the a T distribution of pythia (re-weightedto ResBos) for two different g values. In a “toy” MC measurement of the BLNY parameter g , we study both thesensitivity to systematic uncertainties and the statistical sensitivity of a T , a L , Q T and b T . Figure 5 shows the pythia (re-weighted to ResBos) prediction ofthe normalized a T distribution for two different values of g . The a T distribu-tion is clearly sensitive to the value of g .Event samples are generated using ResBos, for fifteen g values from 0.54 to0.82 (distributed around the world average, g = 0.68 +0 . − . GeV ) [2]. Using there-weighting procedure described earlier, pythia “MC samples” are producedcorresponding to each of the g values. An independent sample of 200k pythia events is re-weighted to the central g value and represents the experimental“pseudo-data” sample.For each of the 15 MC templates, the pseudo-data vs MC χ is calculated. Thefunction: y = a ( x − b ) + c is fitted to the χ as a function of g , and a best fit g is determined as b ± a − / where the uncertainty is statistical (∆ χ = ± g serves as an example analysis at an experiment.Free parameters in any NP model which affect the Q T could in principle bemeasured using this strategy. 9 .1 Systematic Uncertainties We study the systematic uncertainties on a fit to g . The following systematicvariations are carried out. These are typical of the experimental uncertaintiesexpected at a hadron collider although the size of the variations are chosen togive reasonable shifts in the fitted g : • In the MC, the smearing constant ( δ (1 /p T )) is varied by a factor ± − . • As a test of sensitivity to mis-measurement of the event selection efficiencydependencies on, | η | , p T and isolation, events in MC are given ±
10% moreweight for each muon with; 1 < | η | <
2, 15 < p T <
20 GeV, or 1 < f iso < . • In the MC, the φ smearing constant is varied by ± • As a test of sensitivity to mis-modeling of final state radiation (FSR), eventsin MC are given ±
10% more weight if the difference between the generatorlevel Z mass and the generator level di-lepton mass is greater than 1 GeV. Table 1The shifts (%) in the fitted g for systematic variations in the MC. a T a L Q T b T p T smearing ± − . ∓ ∓ . ∓ . p T ± ∓ . ± . ± . ∓ . f iso ± ∓ . ∓ . ∓ . ∓ . | η | ± ± . ± ± . ± . φ smearing ± − .
01 0.00 0.00 − . ± ∓ . ∓ . ∓ . ∓ . Table 1 shows the shifts in the fitted g for the +ve and -ve systematic vari-ations. For each of the variations, the shift in the fitted g is substantiallysmaller using a T or b T than Q T , and larger using a L . In fact the only variationto which a T or b T are more sensitive is the φ smearing, but for any realisticdetector φ resolution, the shift in g is negligible. Simply discarding information from a L is not optimal in terms of the statisticalsensitivity to the shape of the Q T distribution (and thus also the value of g ). As well as the basic observables, Q T , a T and a L , the following ideas are10roposed as possible optimised combinations of a T and a L : • A weighted quadrature sum of a T and a L with more weight ( w ) given to a T : Q ∗ T ( w ) = w q ( w · a T ) + a L . • A 2D fit to d σda T da L . Table 2The binning and 1 σ statistical uncertainties for each of the g fits, for a range of δ (1 /p T ). Q T a T a L Q ∗ T ( w = 5) d σda T da L b T nbins 20 20 20 20 20( a T ) × a L ) 20range (GeV) 0-30 0-20 0-20 0-30 0-20( a T ), 0-20( a L ) 0-20 δ (1 /p T ) (GeV − ) 1 σ statistical uncertainty (%)0.0000 1.4 2.2 2.4 1.9 1.4 2.20.0003 1.4 2.2 2.5 1.9 1.4 2.20.0010 1.8 2.2 3.6 1.9 1.6 2.20.0030 3.1 2.3 8.9 2.2 2.1 2.2 The binning and 1 σ statistical uncertainties for each of the fits is presentedin Table 2. Since the relative statistical sensitivity of each of the observablesdepends on the detector resolution, the fit is performed for different widths ofGaussian smearing. All of the fits use equal width bins. For “perfect” resolu-tion, a fit to Q T and the 2D fit have comparable statistical uncertainties. Afit to a T or a L alone is less statistically sensitive, as each contains informationon only one component of the Q T .At larger δ (1 /p T ), the sensitivity of a L is completely washed out by the smear-ing. For δ (1 /p T ) = 0 .
003 GeV − , a T alone gives better statistical precision than Q T . Over the resolution range covered, Q ∗ T ( w = 5) gives slightly better statis-tical precision than a T alone. Note that no attempt has been made to optimizethe weight in Q ∗ T ( w = 5) as a function of resolution, and w = 5 is a somewhatarbitrary choice. Clearly, for “perfect” resolution, w = 1 is the only sensiblechoice. Once experimental systematic uncertainties are taken into account, a T will need to be given more weight in any optimal combination of a T and a L . The NP form factor (Eq. 1) required an alteration, to describe deep inelasticscattering (DIS) data involving initial state partons with x < − [13]. Berge et al. [14] suggested that if this (so-called small- x broadening) was observed11t the Tevatron in an exclusive high | y | sample of Z bosons, a broader Higgs(and W, Z ) boson transverse momentum distribution could be expected at theLHC. The DØ Run II data on Q T [4] disfavoured this modification, althoughgiven the large uncertainties, the data was not particularly sensitive to suchbroadening. Even without taking into account the reduced systematic uncer-tainties, the optimised fits ( Q ∗ T , d σ/da T da L ) described earlier would be moresensitive to such effects. Once systematic uncertainties are taken into account,the sensitivity is further enhanced relative to the Q T . The example g measurement presented would be sensitive to the descrip-tion by the MC event generator(s) of the azimuthal correlation between thehadronic recoil and the leptonic decay. A measurement of d σ/da T da L couldbe used to study this correlation. Using MC simulations we demonstate the potential benefits of decomposingthe Q T into two orthogonal components, a T and a L . A measurement of the a T distribution would be substantially less sensitive to two of the dominantexperimental systematic uncertainties reported in previous Q T measurements:lepton p T or E T mis-measurement, and the Q T dependence of the event selec-tion efficiencies. A slightly different variable b T is demonstrated to be similarlyinsensitive to these uncertainties. An optimal combination of a T and a L , givingmore weight to a T gives, for any realistic detector resolution, better statisticalsensitivity to the shape of the region of low Q T . Two possibilities are proposed:a weighted quadrature sum of a T and a L or a fit to d σ/da T da L .A measurement of d σ/da T da L could potentially probe the azimuthal corre-lation between the Z boson decay axis and the hadronic recoil. The partialdifferential cross sections; d σ/da T dQ , d σ/da T dy and d σ/da T dQdy couldalso be measured, taking advantage of the reduced systematic uncertaintieson a T compared with similar measurements of differential cross sections withrespect to Q T . Such distributions would be sensitive to small- x broadeningeffects. 12 eferences [1] J. Collins, D. Soper, G. Sterman, Nucl. Phys. B 250, 199-224 (1985).[2] F. Landry et. al. , Phys. Rev. D 67, 073016 (2003).[3] C. Balazs, C.P. Yuan, Phys. Rev. D 56, 5558-5583 (1997).[4] DØ Collaboration, Phys. Rev. L. 100, 102002 (2008).[5] DØ Collaboration, B. Abbott et. al. , Phys. Rev. D 61, 032004 (2000); DØCollaboration, B. Abbott et. al. , Phys. Rev. L. 84, 2792 (2000).[6] OPAL collaboration, K. Ackerstaff et. al. Eur. Phys. J C4 47 (1998).[7] UA2 Collaboration, J. Alitti et. al.
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