Measurement of observables sensitive to coherence effects in hadronic Z decays with the OPAL detector at LEP
Nadine Fischer, Stefan Gieseke, Stefan Kluth, Simon Plätzer, Peter Skands, OPAL collaboration
OOPAL PR435KA-TP-09-2015COEPP-MN-15-2MPP-2015-98
Measurement of observables sensitive to coherence effects in hadronic Z decays with the OPAL detector at LEP N. Fischer , , S. Gieseke , S. Kluth , S. Pl¨atzer , , P. Skands , , andThe OPAL collaboration [1] : Institute for Theoretical Physics, Karlsruhe Institute of Technology, Karlsruhe, Germany : School of Physics and Astronomy, Monash University, Melbourne, Australia : Max-Planck-Institute for Physics, Munich, Germany : Institute for Particle Physics Phenomenology, Durham University, Durham, UnitedKingdom : School of Physics and Astronomy, University of Manchester, Manchester, UnitedKingdom : Theoretical Physics, CERN, Geneva, Switzerland Abstract
A study of QCD coherence is presented based on a sample of about 397 000 e + e − hadronic an-nihilation events collected at √ s = 91 GeV with the OPAL detector at LEP. The study is basedon four recently proposed observables that are sensitive to coherence effects in the perturbativeregime. The measurement of these observables is presented, along with a comparison with thepredictions of different parton shower models. The models include both conventional partonshower models and dipole antenna models. Different ordering variables are used to investigatetheir influence on the predictions.
Processes involving the strong interaction, described in the standard model (SM) by quantum chro-modynamics (QCD), dominate in high energy particle collisions. It is therefore important to accountfor QCD effects and to model them accurately. Colour coherence, the destructive interference effectbetween colour-connected partons, is an important aspect of high energy collisions and QCD partoncascades. Coherence is itself a subject of considerable interest, and QCD offers a situation in whichcoherence effects in a perturbative framework can be studied in a uniquely precise way. Furthermore,by testing different theoretical schemes for coherence, QCD Monte Carlo (MC) event generators (seeRefs. [2–5] for recent reviews) can be modified to better describe the results of experiments. Forexample, in new-physics searches at the CERN LHC, QCD multijet events often represent the mostdifficult SM background to characterize. Improvements in the reliability of QCD event generatorsmay help to better constrain this background. 1 a r X i v : . [ h e p - e x ] M a y he e + e − annihilation process offers a favorable environment to study colour coherence, becausethe lack of strong interactions in the initial state allows simple and conclusive comparisons betweenexperiment and theory. Previous studies of coherence in e + e − annihilation events are presented, forexample, in Refs. [6,7]. Within the context of a QCD shower, coherence implies an ordering condition,such as a requirement that each subsequent emission angle in the shower be smaller than the previousangle [8, 9]. However, there are many ambiguities in the definition of the ordering variable and in itsimplementation. In this study, we present the first experimental tests of recently proposed [10] observ-ables designed to discriminate between coherence schemes. The data were collected with the OPALdetector at the CERN LEP collider at a centre-of-mass energy of √ s = 91 GeV. The observablesexamined here are based on four-jet e + e − annihilation configurations in which a soft gluon is emittedin the context of a three-jet topology, with two of the three jets approximately collinear. This eventconfiguration has been shown to be favorable for the manifestation of coherence [11] and sensitive tothe choice of the ordering variable in the shower [12].We examine six different models for coherence, which are implemented in currently availableQCD MC event generator programs. Specifically, we compare the default ˜ q parton shower of H ER - WIG ++ [13] with angular-ordering, the p ⊥ dip - and q -ordered dipole showers of H ERWIG ++ [11],the default p ⊥ evol -ordered shower of P YTHIA p ⊥ ant - and m -ordered showers ofV INCIA [15], a plugin to the P
YTHIA
YTHIA
We consider hadronic events from e + e − annihilation at the Z boson peak and use the Durham k T clustering algorithm [16] to cluster all particles of an event into jets, keeping track of the clusteringscales along the way. The algorithm begins by assigning all particles in an event to a list. Each entryin the list is called a jet. The algorithm then computes, for all pairs of four-momenta i and j in theevent, the distance measure y ij = 2min( E i , E j )(1 − cos θ ij ) /s , (1)where E i and E j are the corresponding energies and θ ij is the angle between objects i and j . Thecenter-of-mass energy-squared is denoted by s . The pair of objects with the smallest y ij is combinedby summing their four-momenta and the sum is added to the list while the original four-momenta areremoved. This procedure is iterated until one entry is left in the list. To obtain an inclusive four-jet event sample in the perturbative regime we impose an explicit requirement on the value of theclustering scale at which the event goes from having four to having three jets. Denoting this scale(given by the value of min( y ij ) evaluated at the stage when the event has been clustered to four jets)by y → , we require y → > . (corresponding to ln( y → ) > − . ), as in Ref. [10]. Thisvalue originates from a compromise; on the one hand a smaller value results in a data sample withgreater statistical precision, and on the other hand a larger value provides a more direct representationof the shower properties. 2 ) b) θ c) d) θ * e) f) M H2 M L2 M L M H2 Figure 1: a) The event topologies resulting from four-jet events with requirements on the anglesbetween the jets: θ > π/ , θ > π/ , and θ < π/ . b) Illustration of the observable θ , theangle between the first and fourth jet in the latter events. c) The event topologies resulting from four-jet events with requirements θ > π/ , θ > π/ , θ < π/ , and θ < π/ . d) Illustration ofthe observable θ ∗ = θ − θ , the difference in opening angles. e) The sketch shows event topologieswhere the third and fourth clusterings occur within the same jet and hence the mass ratio M L /M H issmall. f) Events with large mass ratios, where the third and fourth clusterings occur on opposite sidesof the event.We investigate four different observables, where for the first three we consider the event clusteredinto four jets, and order the jets in energy. To be sensitive to coherence, the angles between the jets areconstrained such that the first (hardest) jet lies back-to-back to a nearly collinear jet pair, formed bythe second and third jet: θ > π/ , θ > π/ , and θ < π/ . The event topology resulting fromthese requirements is shown in Fig. 1 a) . To investigate QCD colour coherence effects we examinethe following observables: • θ , proposed in Ref. [12]:The emission angle of the soft fourth jet with respect to the first jet; a sketch can be found inFig. 1 b) . • θ ∗ , proposed in Ref. [11]:A restriction on the angle between the second and fourth jet, θ < π/ , is imposed in orderto require the fourth jet to be close in angle to the nearly collinear (23) jet pair, see Fig. 1 c) .The observable is the difference in opening angles, θ ∗ = θ − θ , and is sensitive to coherentemission from the (23) jet system. A sketch of this observable is shown in Fig. 1 d) . • C (1 / , proposed in Ref. [17]: 3n general we have the freedom to chose the exponent β of the 2-point energy correlation doubleratio C ( β )2 . With the choice β = 1 / and a nearly collinear (23) jet pair we can express the 2-point double ratio as C (1 / ≈ E θ / θ /
523 4 E vis / ( E E θ /
51 23 ) , with the total visible energy E vis in the event. θ
23 4 denotes the angle between the softest jet and the (23) jet pair and analogouslyfor θ . Therefore the 2-point double ratio is mainly sensitive to the relative energy of thefourth jet.Strong ordering in the parton shower refers to strong ordering of the clustering scales, y → (cid:29) y → (cid:29) . . . , with y ( n +1) → n the value of the jet distance parameter in the Durham algorithm for whichthe configuration changes from n + 1 to n jets. In contrast, events with, e.g., y → ∼ y → are moresensitive to the ordering condition and to situations where the same parton participates in two splittingprocesses, hence to effective → splittings. For the final observable, we cluster events into twojets and apply the restriction y → > . y → . This forces events into a compressed hierarchy, i.e., ahierarchy without strong ordering. The investigated observable is: • ρ = M L /M H , proposed in Ref. [12]:The ratio of the invariant masses-squared of the jets at the end of the clustering process, orderedsuch that M L ≤ M H . For “same-side” events, where one → splitting occurs, the mass ratiois close or equal to zero, whereas for “opposite-side” events with → ⊗ → splittings,the mass ratio is larger. In Fig. 1 e) and f) we illustrate examples of these event topologies. Forreferences to heavy jet masses see, e.g. Refs. [18, 19].To exhibit the differences between the theory models more clearly, we introduce the asymmetryfor a given observable x , N left N right = (cid:80) i with x ( i )
8) or multiple partons (H
ERWIG ++).In the following we briefly describe the main differences between the theory models used in thispaper, mostly concentrated on the aspects described above. H
ERWIG ++ ˜ q [13], a parton showermodel based on DGLAP splitting kernels, uses global recoils. The evolution is ordered in a variableproportional to energy times angle, ˜ q = Q I M IK Q K ( M IK − Q I − Q K ) . (4)The shower includes a matrix-element correction for the first emission and uses two-loop running of α s . The QCD coherence properties are respected due to the angular ordering of the parton branchingcascade. The second shower model in the H ERWIG ++ event generator is H
ERWIG ++ p ⊥ dip [11],which is based on partitioned CS dipoles with local recoils within dipoles. The ordering variable isthe relative transverse momentum of the splitting pair, p ⊥ dip = Q I Q K ( M IK − Q I − Q K )( M IK − Q I ) . (5)We do not apply matching or matrix-element corrections and use one-loop running of α s . The dipoleshower with ordering in transverse momentum respects QCD coherence. As an alternative we use thesame shower model, but with a different ordering variable. H ERWIG ++ q [11] orders the showercascade in virtuality of the splitting pair, q = Q I . (6)As before we do not apply matching or matrix-element corrections and use one-loop running of α s .V INCIA p ⊥ ant [15] is a shower model based on antenna functions with local recoils within antennae.The ordering variable is the transverse momentum of the antenna, p ⊥ ant = Q I Q K M IK . (7)Matrix-element corrections at LO [28] and NLO [29] are switched off and we use one-loop running of α s . Colour coherence is respected, since it is an intrinsic property of the antenna functions. Transversemomentum as the evolution variable is the preferred choice in V INCIA , as has been shown in Ref. [29].However, we also use V
INCIA m [15] as an alternative to the transverse momentum ordering, whichorders the shower evolution in antenna mass, defined as m = min( Q I , Q K ) . (8)The last shower model is P YTHIA p ⊥ evol [14], a parton shower based on DGLAP splitting kernelsand ordered in transverse momentum, defined as p ⊥ evol = Q I ( M IK − Q K )( Q I + Q K )( M IK + Q I ) . (9)In contrast to the angular ordered H ERWIG ++ shower, local recoils within dipoles are applied. Amatrix-element correction for the first emission is included and we use one-loop running of α s . Toobtain QCD coherence properties, the shower applies angular vetoes.6esides the shower models used in this paper, there are several other models: A RIADNE [30],based on antenna functions, which is very similar to V
INCIA ; the CS dipole shower models ofWeinzierl et al. [31], and S
HERPA [32], which are similar to the H
ERWIG ++ dipole shower; the de-ductor by Nagy and Soper [33], which is not interfaced with a hadronization model; and the virtuality-ordered final-state showers of P
YTHIA [24, 34], N
LLJET [35] and H
ERWIRI [36].To compare the models on as equal a footing as possible, the shower and hadronization parametershave been readjusted with the P
ROFESSOR [37] tuning system, utilizing LEP data available throughR
IVET [38]. The tuning procedure is described in Ref. [10], where the resulting parameter values canalso be found.
The OPAL experiment at LEP operated between August 1989 and November 2000. The detectorcomponents were arranged around the beam pipe, in a layered structure. A detailed description canbe found in Refs. [39–41]. The tracking system consisted of a silicon microvertex detector, an innervertex chamber, a jet chamber, and chambers outside the jet chambers to improve the precision of the z -coordinate measurement. The jet chamber was approximately long and had an outer radiusof about .
85 m . This device had sectors each containing sense wires spaced by 1 cm. Alltracking systems were located inside a solenoidal magnet, which provided a uniform axial magneticfield of .
435 T along the beam axis. The magnet was surrounded by a lead glass electromagneticcalorimeter and a sampling hadron calorimeter. The electromagnetic calorimeter consisted of lead glass blocks, divided into barrel and endcap sections, covering of the solid angle. Outsidethe hadron calorimeter, the detector was surrounded by a system of muon chambers. Similar layers ofinstrumentation were located in the endcap regions.Since the energy resolution of the electromagnetic calorimeter is better then that of the hadroncalorimeter, the resolution of jet directions and energies is not significantly improved by incorporat-ing hadron calorimeter information. Thus, our analysis relies exclusively on charged particle infor-mation recorded in the tracking detectors and on clusters of energy deposited in the electromagneticcalorimeter.
In the first phase of LEP operation, denoted LEP1 (1989 to 1995), the e + e − center-of-mass energy waschosen to lie at or near the mass of the Z boson, √ s ≈
91 GeV . During the second phase of operation,denoted LEP2 (1995-2000), the center-of-mass energy was increased in successive steps from to
209 GeV . Interspersed at various times during the LEP2 operation, calibration runs were collected atthe Z boson peak. In this analysis, we utilize data collected at √ s = 91 . GeV during the the LEP2calibration runs. This allows us to exploit conditions when the detector was operating in its final,most advanced configuration. In addition, this will facilitate possible future comparisons with datacollected under essentially identical conditions at higher energies. We use a sample corresponding toan integrated luminosity of . − . This sample is of sufficient size that systematic uncertaintiesdominate the statistical terms. To correct the data in order to account for experimental acceptanceand efficiency, simulated event samples produced with MC event generators are used. The process OPAL uses the right-handed coordinate system defined with the x -axis pointing towards the center of the LEP ring, thepositive z points along the direction of the e − beam and the y -axis upwards. r is the coordinate normal to the beam axisand the polar angle θ and the azimuthal angle ϕ are defined with respect to x and z . + e − → q ¯ q is simulated using P YTHIA √ s = 91 . GeV. Corresponding samples usingH
ERWIG
300 ps decay. In contrast,“detector level” refers to MC events that are processed through the G
EANT
OPAL [46], and that have been reconstructed using the same softwareprocedures that are applied to the data. The MC events generated for the detector-level samples are thesame as the hadron-level samples except that K S mesons and weakly decaying hyperons are declaredto be stable, as these particles can interact with detector material before decaying, and so their decaysare handled within the G EANT framework.In addition, for comparisons with the corrected data, large samples of hadron-level MC eventsare employed, using the event generators H
ERWIG ++ 2.7.0 [47], P
YTHIA IN - CIA
YTHIA
The same criteria for the selection of charged tracks and electromagnetic clusters are applied as de-scribed in Ref. [49]. Charged tracks are required to have transverse momentum relative to the beamaxis larger than .
15 GeV , and photons to have energies larger than .
10 GeV ( .
25 GeV ) in the bar-rel (endcap) region of the electromagnetic calorimeter. The selection of hadronic annihilation eventsis the same as described in Ref. [50]. Briefly, a minimum of five charged tracks is required, and a con-tainment condition | cos θ T | < . is applied, where θ T is the polar angle of the thrust axis [51, 52]with respect to the beam axis, calculated using all accepted charged tracks and electromagnetic clus-ters. A total of
397 452 candidate hadronic annihilation events are selected, with a negligible expectedbackground.Since the energy loss due to initial-state radiation is highly suppressed at the Z peak, we do notapply a cut to that effect. However, radiative corrections are applied by requiring √ s − √ s (cid:48) < for the MC detector-level samples used to correct the data, where √ s (cid:48) is the effective center-of-massenergy after initial-state radiation. For each of the accepted events, the values of all observables described in Section 1 are computed.To avoid double-counting of energy between tracks and electromagnetic clusters, an energy-flow al-gorithm [53, 54] is applied, which matches the tracks and clusters and retains only those clusters thatare not associated with a track.Figure 2 shows a comparison of the uncorrected data with the detector-level predictions of P
YTHIA
ERWIG θ , ρ , θ ∗ , and C (1 / variables. The θ and θ ∗ variables are normalized by afactor of π . The simulations are seen to provide a generally adequate description of the measurements.To correct the data for detector and resolution effects, we implement an unfolding procedurebased on the R OO U NFOLD [55] framework. We use the iterative Bayes method, as proposed byD’Agostini [56], with four iterations, which is the recommendation from Ref. [55]. A necessary in-gredient for the unfolding is the response matrix of the MC event generator used for the correctionprocedure. For the standard analysis, P
YTHIA . . . . . . θ /π . . . . . σ − d σ / d ( θ / π ) a) Angle between 1st and 4th jet, θ /π OPAL data, det. levelH
ERWIG
OPAL P YTHIA
OPAL . . . . . . ρ . . . . . σ − d σ / d ρ b) Ratio of jet masses, ρ OPAL data, det. levelH
ERWIG
OPAL P YTHIA
OPAL . . . . . θ ∗ /π σ − d σ / d ( θ ∗ / π ) c) Difference in opening angles, θ ∗ = θ − θ OPAL data, det. levelH
ERWIG
OPAL P YTHIA
OPAL .
36 0 .
38 0 .
40 0 .
42 0 .
44 0 .
46 0 . C (1 / σ − d σ / d C ( / ) d) 2-point double ratio, C (1 / OPAL data, det. levelH
ERWIG
OPAL P YTHIA
OPAL
Figure 2: The uncorrected distributions of a) the emission angle θ , b) the mass ratio ρ = M L /M H , c) the difference in opening angles θ ∗ , and d) the 2-point double ratio C (1 / , in comparison with thepredictions of the H ERWIG
YTHIA .The corrected distributions are presented in Fig. 3 and Tables 2-3. Tables 2-3 include the co-variance matrices calculated with R OO U NFOLD . The statistical uncertainties are given by the squareroot of the corresponding diagonal element in the covariance matrices. Systematic uncertainties arediscussed in Section 5.3. Figure 3 includes the predictions of P
YTHIA
ERWIG θ and theasymmetry for the other observables, are listed in Table 4. The quantities are determined by summingand dividing the histogram entries. The statistical uncertainties are evaluated from propagation oferrors, while the systematic uncertainties are determined as described in Section 5.3. Systematic uncertainties are evaluated by repeating the analysis with different selection requirementsand with variations in the correction procedure. Specifically, we consider the following: • The requirement on the thrust angle direction is changed to | cos θ T | < . from the default | cos θ T | < . . • The minimum number of charged tracks is increased to seven from the default of five. • Variation of the reconstruction procedure: All tracks and clusters are taken into account. In thiscase the detector correction takes care of the double counting. • H ERWIG
YTHIA • We use the unfolding method with three and five instead of four iterations. • Instead of the iterative method, we use the singular value decomposition, as proposed by H¨ockerand Kartvelishvili [57] and implemented in R OO U NFOLD .We find the systematic variations that arise from these two checks to be smaller or comparable tothe variation observed when using H
ERWIG
YTHIA
6. Since adding all these effectstogether would likely double count the uncertainty associated with the unfolding procedure we do notadd the observed differences to the systematic uncertainty.
In this section, we present a comparison between the coherence schemes described in Section 2.2 andthe data. For this purpose, samples of × events are generated for each MC model, using the10 . . . . . . θ /π . . . . . σ − d σ / d ( θ / π ) a) Angle between 1st and 4th jet, θ /π OPAL dataH
ERWIG YTHIA . . . . . . ρ . . . . . σ − d σ / d ρ b) Ratio of jet masses, ρ OPAL dataH
ERWIG YTHIA . . . . . θ ∗ /π σ − d σ / d ( θ ∗ / π ) c) Difference in opening angles, θ ∗ = θ − θ OPAL dataH
ERWIG YTHIA .
36 0 .
38 0 .
40 0 .
42 0 .
44 0 .
46 0 . C (1 / σ − d σ / d C ( / ) d) 2-point double ratio, C (1 / OPAL dataH
ERWIG YTHIA Figure 3: The corrected distributions of a) the emission angle θ , b) the mass ratio ρ = M L /M H , c) the difference in opening angles θ ∗ , and d) the 2-point double ratio C (1 / , in comparison withthe predictions of the H ERWIG
YTHIA /π σ − d σ/ d( θ ∗ /π )0 . − .
15 0 . ± . ± . . − .
20 1 . ± . ± . . − .
25 1 . ± . ± . . − .
30 1 . ± . ± . . − .
35 1 . ± . ± . . − .
40 1 . ± . ± . . − .
45 1 . ± . ± . . − .
50 1 . ± . ± . . − .
55 1 . ± . ± . . − .
60 1 . ± . ± . . − .
65 1 . ± . ± . . − .
70 0 . ± . ± . . − .
75 0 . ± . ± . . − .
80 0 . ± . ± . . − .
85 0 . ± . ± . . − .
00 0 . ± . ± . Correlation Matrix . .
252 1 . − .
037 0 .
105 1 . − . − .
062 0 .
103 1 . − . − . − .
108 0 .
190 1 . − . − . − . − .
082 0 .
141 1 . − . − . − . − . − .
106 0 .
164 1 . − . − . − . − . − . − .
105 0 .
153 1 . − . − . − . − . − . − . − .
123 0 .
273 1 . − . − . − . − . − . − . − . − .
100 0 .
292 1 . − . − . − . − . − . − . − . − . − .
090 0 .
181 1 . − . − . − . − . − . − . − . − . − . − .
097 0 .
290 1 . − . − . − . − . − . − . − . − . − . − . − .
110 0 .
273 1 . − . − . − . − . − . − . − . − . − . − . − . − .
108 0 .
259 1 . − .
075 0 . − . − . − . − . − . − . − . − . − . − . − .
079 0 .
179 1 . − .
072 0 . − . − . − . − . − . − . − . − . − . − . − .
092 0 .
055 0 .
190 1 . Table 2: The normalized corrected data and the correlation matrix at the hadron level for the emission angle θ . The first uncertainty is statisticaland the second systematic. σ − d σ/ d ρ . − .
06 1 . ± . ± . . − .
15 1 . ± . ± . . − .
38 0 . ± . ± . . − .
69 1 . ± . ± . . − .
00 1 . ± . ± . Correlation Matrix . .
435 1 . − . − .
007 1 . − . − .
530 0 .
088 1 . − . − . − .
490 0 .
387 1 . θ ∗ /π σ − d σ/ d( θ ∗ /π ) − . − .
04 0 . ± . ± . . − .
10 1 . ± . ± . . − .
16 2 . ± . ± . . − .
22 2 . ± . ± . . − .
28 3 . ± . ± . . − .
34 4 . ± . ± . . − .
43 1 . ± . ± . Correlation Matrix . − .
013 1 . .
358 0 .
367 1 . − .
188 0 .
010 0 .
232 1 . − . − . − .
092 0 .
278 1 . − . − . − . − .
295 0 .
245 1 . − . − . − . − . − .
168 0 .
304 1 . C (1 / σ − d σ/ d C (1 / . − .
402 4 . ± . ± . . − .
424 11 . ± . ± . . − .
446 11 . ± . ± . . − .
468 11 . ± . ± . . − .
495 1 . ± . ± . Correlation Matrix . .
314 1 . − .
385 0 .
218 1 . − . − .
493 0 .
187 1 . − . − . − .
225 0 .
299 1 . Table 3: The normalized corrected data and the correlation matrix at the hadron level for the mass ratio ρ = M L /M H , the difference in emissionangles θ ∗ , and the 2-point double ratio C (1 / . The first uncertainty is statistical and the second systematic. Central/Towards . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . θ ∗ N left /N right .
10 0 . ± . ± . .
16 0 . ± . ± . .
22 0 . ± . ± . .
28 1 . ± . ± . ρ N left /N right .
15 0 . ± . ± . .
38 0 . ± . ± . C (1 / , N left /N right .
42 0 . ± . ± . .
45 2 . ± . ± . Table 4: The corrected data for the derived distributions. The upper three tables list the results for θ asymmetry ratios defined in Eq. (3), withthe definitions of the towards, central, and away regions given in Table 1. The bottom three tables list the results for the asymmetries defined forthe other observables. The first uncertainty is statistical and the second systematic H ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q P YTHIA p ⊥ evol V INCIA p ⊥ ant V INCIA m θ /π . . × − ) 12 . . × − ) 16 . . × − ) 8 . . × − ) 8 . . × − ) 7 . . × − ) θ ∗ /π . . × − ) 2 . . × − ) 8 . . × − ) 2 . . × − ) 3 . . × − ) 5 . . × − ) C (1 / . . × − ) 6 . . × − ) 41 . . × − ) 4 . . × − ) 5 . . × − ) 2 . . × − ) ρ . . × − ) 10 . . × − ) 42 . . × − ) 2 . . × − ) 6 . . × − ) 3 . . × − ) (cid:104) p (cid:105) (4 . × − ) (4 . × − ) (1 . × − ) (7 . × − ) (7 . × − ) (6 . × − ) Table 5: The χ values (with p-values in parentheses) for the four observables considered in this analysis. The last line gives the average p-value, (cid:104) p (cid:105) , for the four observables. uned parameter sets mentioned in Section 2.2. We present the predictions for the different schemesin terms of the observables defined in Section 1. As a measure of the level of agreement with data, wecalculate the significance, defined as σ i = MC i − D i σ D i , (10)where MC i and D i represent the predicted and observed values in bin i of a distribution, with σ D i thecorresponding uncertainty in D i . In the following, we present a distribution of the significance in aplot below the distribution of the variables. In addition we calculate the χ values for the distributionas χ = ( MC − D ) T V − ( MC − D ) = N bins (cid:88) i,j =1 ( MC − D ) i ( V − ) ij ( MC − D ) j , (11)where V is the full covariance matrix, representing statistical terms, with systematic uncertaintiesadded to the diagonal elements. We present the χ results and corresponding p-values in Table 5. Thelatter are calculated with the R OOT [58] program and give the probability that the deviations of theMC predictions from the data are consistent with the evaluated uncertainties. θ In Figs. 4 a) and b) we show the normalized distribution of the emission angle of the soft fourth jetfrom the nearly collinear three-jet system, θ . All models are found to provide adequate descriptionsof the data, except that the H ERWIG ++ p ⊥ dip model lies about three standard deviations above themeasurements for a narrow region around θ ≈ . π . Even for this model, the p-value is . (Table 5), implying overall compatibility with the experimental results.We show the ratio C/T of the central-to-towards regions, which gives the relative amount of wide-angle to collinear emissions, in Figs. 5 a) and b) . For the p ⊥ dip -ordered dipole shower of H ERWIG ++and the parton shower of P
YTHIA
ERWIG ++ ˜ q model and the V INCIA p ⊥ ant model lie below the data by up totwo standard deviations in some regions, while the V INCIA m model lies about two standard devi-ations above the data in all regions. The two Vincia models exhibit the expected behavior: When theantenna mass is used as the evolution variable, soft wide-angle emissions are preferred over collinearones, which leads to higher values for the relative level of wide-angle to collinear emissions. Thisdemonstrates the sensitivity of the θ variable to the choice of evolution scheme. The largest devia-tion from the data in Figs. 5 a) and b) is observed for the q -ordered dipole shower of H ERWIG ++,for which the predictions lie up to around three standard deviations below the data in some regions.Thus, this model predicts too many collinear emissions compared to wide-angle emissions.In Figs. 5 c) and d) we show a comparison of the MC predictions to the data for the ratio C/Aof the central-to-away regions. This ratio measures the relative amount of wide-angle emissions toemissions in a backwards direction, away from the leading jet and near to the collinear (23) jet pair. Forthe H ERWIG ++ ˜ q model and for V INCIA , we find a good agreement with the data and observe smalldifferences for the different evolution variables of V
INCIA . P
YTHIA p ⊥ dip -ordered dipoleshower of H ERWIG ++ lie below the data, by around one and two standard deviations, respectively, andthus predict too few wide-angle emissions compared to the backwards emissions. The H
ERWIG ++ q model lies around one standard deviation above the data and thus predicts relatively too manywide-angle emissions. The observations the ratios C/T and C/A are confirmed by the measurementsof the ratio T/A of the towards-to-away regions, presented in Figs. 5 e) and f) .15 . . . . . . σ − d σ / d ( θ / π ) a) Angle between 1st and 4th jet, θ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q . . . . . . θ /π − − ( M C - d a t a ) / E rr o r . . . . . . σ − d σ / d ( θ / π ) b) Angle between 1st and 4th jet, θ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol . . . . . . θ /π − − ( M C - d a t a ) / E rr o r Figure 4: The corrected distribution of the emission angle θ of the soft fourth jet in comparisonwith the predictions of a) H ERWIG ++ and b) P YTHIA
INCIA . The thin solid lines correspondto H
ERWIG ++ with angular-ordering ( ˜ q ), the thick solid lines to the dipole shower of H ERWIG ++with ordering in p ⊥ dip , and the dash-dotted lines to ordering in q . V INCIA with ordering in p ⊥ ant is shown with medium solid lines, ordering in m with dashed lines, and P YTHIA θ ∗ In Figs. 6 a) and b) we show the normalized distribution of the difference in opening angles betweenthe third and the fourth jet with respect to the second jet, θ ∗ = θ − θ . All models are seen toprovide an adequate description of the data, with the exception of the region around θ ∗ ≈ . π (second bin of Figs. 6 a) and b) ), where the models predict somewhat fewer events than are observed.The largest discrepancy in this region arises from the H ERWIG ++ q model.We show the asymmetry as a function of the dividing point θ ∗ in the Figs. 6 c) and d) . The largestdiscriminating power is found for θ ∗ = 0 . π , where the q -ordered dipole shower of H ERWIG ++generates a deviation of almost four standard deviations with respect to the data. The number ofevents with large differences in the opening angles of the third and fourth jets is overestimated bythis non-coherent shower model. The p ⊥ dip -ordered H ERWIG ++ shower, based on the same showerkernels, but respecting coherence due to the choice of evolution variable, gives a better description ofthe asymmetry. This emphasizes the need for coherence in order to describe the data properly.16 C / T a) Central/Towards for θ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q definition of region − ( M C - d a t a ) / E rr o r C / T b) Central/Towards for θ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol definition of region − ( M C - d a t a ) / E rr o r C / A c) Central/Away for θ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q definition of region − − ( M C - d a t a ) / E rr o r C / A d) Central/Away for θ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol definition of region − − ( M C - d a t a ) / E rr o r T / A e) Towards/Away for θ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q definition of region − ( M C - d a t a ) / E rr o r T / A f) Towards/Away for θ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol definition of region − ( M C - d a t a ) / E rr o r Figure 5: The corrected data for the derived distributions in comparison with the predictions of a) H ERWIG ++ and b) P YTHIA
INCIA . The thin solid lines correspond to H
ERWIG ++ withangular-ordering ( ˜ q ), the thick solid lines to the dipole shower of H ERWIG ++ with ordering in p ⊥ dip ,and the dash-dotted lines to ordering in q . V INCIA with ordering in p ⊥ ant is shown with mediumsolid lines, ordering in m with dashed lines, and P YTHIA σ − d σ / d ( θ ∗ / π ) a) Difference in opening angles, θ ∗ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q . . . . . θ ∗ /π − − − ( M C - d a t a ) / E rr o r σ − d σ / d ( θ ∗ / π ) b) Difference in opening angles, θ ∗ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol . . . . . θ ∗ /π − − − ( M C - d a t a ) / E rr o r − N l e f t / N r i g h t c) Asymmetry for θ ∗ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q .
10 0 .
16 0 .
22 0 . θ ∗ /π − − − − ( M C - d a t a ) / E rr o r − N l e f t / N r i g h t d) Asymmetry for θ ∗ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol .
10 0 .
16 0 .
22 0 . θ ∗ /π − − − − ( M C - d a t a ) / E rr o r Figure 6: The distribution of the difference in opening angles θ ∗ for a) H ERWIG ++ and b) P YTHIA
INCIA . The asymmetry with respect to the dividing point θ ∗ is shown for c) H ERWIG ++ and d) P YTHIA
INCIA . The thin solid lines correspond to H
ERWIG ++ with angular-ordering ( ˜ q ),the thick solid lines to the dipole shower of H ERWIG ++ with ordering in p ⊥ dip , and the dash-dottedlines to ordering in q . V INCIA with ordering in p ⊥ ant is shown with medium solid lines, orderingin m with dashed lines and P YTHIA .3 2-Point double ratio: C (1 / For the normalized distribution of the 2-point double ratio, C (1 / , shown in Figs. 7 a) and b) , wefind rather large deviations between the data and the MC prediction for most of the shower models.We again find that the q -ordered H ERWIG ++ shower exhibits the largest discrepancies. Only twomodels, P
YTHIA
INCIA m model, yield p-values larger than (Table 5).In Figs. 7 c) and d) , we show the asymmetry in the C (1 / variable as a function of the dividingpoint C (1 / , . Since C (1 / is proportional to the energy E of the fourth jet, the asymmetry in C (1 / measures the relative number of events of soft versus hard fourth-jet emissions. We observe largedeviations from the data, at the level of four standard deviations, for the H ERWIG ++ q model,which underpredicts the relative fraction of events with a very soft fourth jet. A similar discrepancy,at the level of around . standard deviations, is observed for the H ERWIG ++ ˜ q model. The twoversions of V INCIA exhibit deviations of about one standard deviation in the opposite sense, i.e.,V
INCIA m somewhat overpredicts the level of hard fourth-jet emissions, whereas V INCIA p ⊥ ant predicts too few hard fourth-jet emissions. In contrast, the H ERWIG ++ p ⊥ dip and P YTHIA ρ = M L /M H The normalized distributions of the ρ = M L /M H variable are shown in Figs. 8 a) and b) . Forthe P YTHIA
INCIA models, we find reasonable overall agreement with the data, withdifferences on the level of two standard deviations or less. The H
ERWIG ++ models demonstrate largerdifferences, with discrepancies reaching the level of four standard deviations for the H
ERWIG ++ q model.In Figs. 8 c) and d) we show the asymmetry of the ρ variable as a function of the dividing point ρ .This asymmetry is sensitive to the relative number of same-side versus opposite-side events, whosedefinitions were given in Section 2.1. This asymmetry is seen to provide discrimination between mostof the shower models. The P YTHIA
ERWIG ++ ˜ q models yield predictions that lie within onestandard deviation of the data. However, the H ERWIG ++ q model predicts too small an asymmetryby about four standard deviations, meaning that there are too few same-side compared to opposite-side events. The two V INCIA models also predict too few same-side events, but only at the level ofaround one standard deviation. In contrast, the H
ERWIG ++ p ⊥ dip model predicts relatively too manysame-side events, at the level of two standard deviations. We have presented measurements of distributions in e + e − annihilations at √ s = 91 . that aresensitive to QCD colour coherence, the ordering parameter in parton showers, and to whether four-jetevents arise from two separate → splittings or from a → splitting. The data, correspondingto a sample of about
397 000 hadronic annihilation events, were collected with the OPAL detector atLEP. The event selection criteria are defined in a way to minimize the influence of non-perturbative(hadronization) effects. We compared the data with six different models for the parton shower, basedon the H
ERWIG ++, P
YTHIA
8, and V
INCIA
Monte Carlo event generator programs, which differ inthe choice of the radiation function, ordering variable, and recoil strategy. Each of the six models wasfound to be in general agreement with the data. However, interesting differences between the models19 σ − d σ / d C ( / ) a) 2-point double ratio, C (1 / , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q .
36 0 .
38 0 .
40 0 .
42 0 .
44 0 .
46 0 . C (1 / − − − ( M C - d a t a ) / E rr o r σ − d σ / d C ( / ) b) 2-point double ratio, C (1 / , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol .
36 0 .
38 0 .
40 0 .
42 0 .
44 0 .
46 0 . C (1 / − − − ( M C - d a t a ) / E rr o r N l e f t / N r i g h t c) Asymmetry for C (1 / , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q .
424 0 . C (1 / , − − − − − ( M C - d a t a ) / E rr o r N l e f t / N r i g h t d) Asymmetry for C (1 / , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol .
424 0 . C (1 / , − − − − − ( M C - d a t a ) / E rr o r Figure 7: The distribution of the difference in opening angles C (1 / for a) H ERWIG ++ and b) P YTHIA
INCIA . The asymmetry with respect to the dividing point C (1 / , is shown for c) H ERWIG ++ and d) P YTHIA
INCIA . The thin solid lines correspond to H
ERWIG ++ withangular-ordering ( ˜ q ), the thick solid lines to the dipole shower of H ERWIG ++ with ordering in p ⊥ dip ,and the dash-dotted lines to ordering in q . V INCIA with ordering in p ⊥ ant is shown with mediumsolid lines, ordering in m with dashed lines and P YTHIA . . . . . σ − d σ / d ρ a) Ratio of jet masses, ρ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q . . . . . . ρ − − ( M C - d a t a ) / E rr o r . . . . . σ − d σ / d ρ b) Ratio of jet masses, ρ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol . . . . . . ρ − − ( M C - d a t a ) / E rr o r N l e f t / N r i g h t c) Asymmetry for ρ , H ERWIG ++ OPAL dataH
ERWIG ++ ˜ q H ERWIG ++ p ⊥ dip H ERWIG ++ q .
15 0 . ρ − − ( M C - d a t a ) / E rr o r N l e f t / N r i g h t d) Asymmetry for ρ , V INCIA , P
YTHIA OPAL dataV
INCIA p ⊥ ant V INCIA m P YTHIA p ⊥ evol .
15 0 . ρ − − ( M C - d a t a ) / E rr o r Figure 8: The distribution of the difference in opening angles ρ = M L /M H for a) H ERWIG ++and b) P YTHIA
INCIA . The asymmetry with respect to the dividing point ρ is shown for c) H ERWIG ++ and d) P YTHIA
INCIA . The thin solid lines correspond to H
ERWIG ++ withangular-ordering ( ˜ q ), the thick solid lines to the dipole shower of H ERWIG ++ with ordering in p ⊥ dip ,and the dash-dotted lines to ordering in q . V INCIA with ordering in p ⊥ ant is shown with mediumsolid lines, ordering in m with dashed lines and P YTHIA
YTHIA IN - CIA , or between the different variants of V
INCIA . Our study of the asymmetry of the ratio of squaredjet masses, shown in Fig. 8 d) , shows that V INCIA predicts somewhat too many opposite-side events(i.e., events with two → splittings) compared to same-side events (i.e., events with a → splitting), and that the data prefer P YTHIA
8. We find that the different variants of V
INCIA can bedistinguished using the central-to-towards (Fig. 5 b) ) and central-to-away (Fig. 5 d) ) ratios in the θ variable, which indicate that the V INCIA variant based on antenna mass-squared evolution predictssomewhat too many wide-angle emissions for the soft fourth jet, compared to collinear emissions.To summarize the results of our study, we show the average p-value for each model, calculatedfrom the four values of the single observables, in the bottom row of Table 5. The variant of H
ERWIG ++with a q -ordered dipole shower is found to provide the least satisfactory description of the data. Thismodel does not contain coherence; it has intentionally been introduced to confront it with coherentevolution. Thus our results emphasize the importance of incorporating coherence into the descriptionof the QCD multijet process. Since H ERWIG ++ uses the cluster [23] and P
YTHIA
INCIA usethe Lund string [59, 60] hadronization model, a direct comparison of the predictions from the twogroups of shower models is somewhat ambiguous. It would be interesting to perform a comparisonbased on use of the same hadronization model for all models. However, when comparing all showermodels together, we find P
YTHIA
INCIA with evolution in transverse momentum to give thebest description of the measurements presented here.
Acknowledgements
We would like to express our gratitude to the members of the editorial board (S. Bentvelssen, S.Bethke, J.W. Gary, K. Rabbertz) for their careful review of the analysis and the draft resulting in thispublication.NF would like to thank CERN for hospitality during the course of this work. This work wassupported by the FP7 Marie Curie Initial Training Network MCnetITN under contract number PITN-GA-2012-315877. SG and SP acknowledge support from the Helmholtz Alliance “Physics at theTerascale”. SP acknoweldges support through a Marie Curie Intra-European Fellowship under con-tract number PIEF-GA-2013-628739.We particularly wish to thank the SL Division for the efficient operation of the LEP acceleratorat all energies and for their close cooperation with our experimental group. In addition to the supportstaff at our own institutions we are pleased to acknowledge the Department of Energy, USA,National Science Foundation, USA,Particle Physics and Astronomy Research Council, UK,Natural Sciences and Engineering Research Council, Canada,Israel Science Foundation, administered by the Israel Academy of Science and Humanities,Benoziyo Center for High Energy Physics,Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and a grant underthe MEXT International Science Research Program,Japanese Society for the Promotion of Science (JSPS),German Israeli Bi-national Science Foundation (GIF),Bundesministerium f¨ur Bildung und Forschung, Germany,National Research Council of Canada,Hungarian Foundation for Scientific Research, OTKA T-038240, and T-042864,22he NWO/NATO Fund for Scientific Research, the Netherlands.
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