Using about 4 million events recorded with the ALEPH detector at LEP I, distributions of various event-shape variables are measured as a function of the polar angle of the thrust axis with respect to the e+e- beam direction cos(theta). New calculations which include the event orientation are fitted to the data yielding a measurement of the strong coupling constant alpha_s. The result is compared to that obtained from unoriented event-shape distributions. Preliminary results on alpha_s and on the size of the QCD-corrections to the angular distribution are given

Stenzel, H

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Pa04-051Oriented Event Shapesin Hadronic Z DecaysThe ALEPH CollaborationAbstractUsing about 4 million events recorded with the ALEPH detector at LEP I, distributionsof various event-shape variables are measured as a function of the polar angle of the thrustaxis with respect to the e+e beam direction cos . New calculations which include the eventorientation are tted to the data yielding a measurement of the strong coupling constant s.The result is compared to that obtained from unoriented event-shape distributions. Preliminaryresults on sand on the size of the QCD corrections to the angular distribution are given.(Contribution to the International Conference on high Energy PhysicsWarsaw, Poland, 25-31 July 1996)OPEN-99-36925/07/19961 IntroductionThe analysis of event shapes in the process e+e ! hadrons has corroborated the theoryof strong interactions, QCD, and has provided accurate measurements of its strong couplingconstant s. Usually both experimental distributions and theoretical calculations are integratedover the scattering angle , the angle between the beam and the thrust axis. In this paper theevent orientation is retained and the distributions of the event-shape variables Y3, thrust, widejet broadening and heavy jet mass are measured as function of cos  and compared to next-to-leading order QCD predictions. Using this calculation the strong coupling constant is determinedand compared to a measurement based on event shapes integrated over the orientation. Notethat the measurements of spresented here suer from the missing resummed prediction in thetwo-jet region. Therefore it is used only for a check of the oriented calculations. Finally, afterintegration over the event shapes, the QCD eects on the event orientation itself are analyzed.Similar studies have been performed both at the Z resonance [1], [2] and at lower energies [3].Higher order QCD corrections are predicted to atten the distribution of cos  for Z ! qq, whichis proportional to 1 + cos2.2 Detector and Data SampleThe ALEPH detector is described in detail in [4] and its performance in [5]. Hadronic eventsare selected requiring at least 6 well measured tracks whose total energy exceeds 15 GeV. Atrack is dened as well measured when the angle to the beam axis is greater then 18:2, thereare at least four TPC points used for the t of the track, and it passes through a cylindercentered around the tted interaction point with a radius of 2 cm and a length of 10 cm. Thetotal visible energy of neutral and charged particles must exceed 45 GeV, and the thrust-axis isrequired to be well contained within the detector acceptance, i.e. cos   0:9. The thrust-axisdoes not distinguish between the forward and backward directions, so it is chosen with cos   0,and cos  is called the event orientation. The selection eciency is 87%, and the background isabout 0.2 % stemming from Z ! + and two-photon events. After these cuts, about 3:6106hadronic events recorded in 1991 to 1995 remain for further analysis.3 Analysis of Oriented Event Shapes3.1 Denition of Variables and Theoretical PredictionsThe event shape variable Y3is dened using the Durham [6] jet-nding algorithm, in whichthe distance between particles or clusters is given by yij= 2min(E2i; E2j)(1   cos ij)=E2vis,where E2visdenotes the visible energy and i, j are the indices of charged tracks and neutralobjects reconstructed by the ALEPH tracking system and the calorimeters. All particles areclustered until 3 jets remain, and Y3is the smallest yijbuilt with these jets. Thrust is denedby TR= maxPij~Pi:~nTj =Pij~Pijwhere ~nTis the thrust-axis. For the wide jet broadeningBWeach event is divided into two hemispheres Saccording to the thrust axis. The quantitiesB=Pi2Sj~Pi~nTj =2Pij~Pij are calculated and BW= max(B+; B ). The same denitionof hemispheres applies for the heavy jet mass MH. The invariant mass in each hemispheres iscalculated and the larger one is called MH.All numerical calculations are done at the level of partons using a Monte Carlo program EVENT2[7] based on the dipole formalism [8] for the integration of the O(2s) matrix elements. In thisframework the double dierential cross section can be written as follows:11d2(X; cos)dXd cos=s(2)2A(X; cos)+ s(2)2!2"A(X; cos)2b0ln 2s!+ B(X; cos )#; (1)whereb0=33  2nf12; X = Y3; TR; BW;MH:Here,  is the renormalization scale, and the coecients A and B have been computed [7].3.2 Correction ProcedureSince the theoretical prediction is valid only at the level of partons, corrections are needed forthe eects of initial state radiation, hadronization, geometrical acceptance, detector eciencyand resolution. Corrections are applied to the theoretical distributions by means of bin-by-binmultiplicative factors. The correction factors are obtained from various Monte Carlo models.Hadronization corrections have been computed with the generators Jetset [9] using both theparton shower (PS) and the O(2s) matrix element option (ME), with Herwig (HW) [10]and with Ariadne (AR) [11]. The important parameters of these models have been tunedto reproduce globally measured quantities. Acceptance and resolution corrections are estimatedwith a Monte Carlo detector simulation. Finally, theory and experiment are compared at thelevel of the detector, all corrections being applied to the theoretical expectation.3.3 Simultaneous Analysis of Event Shapes and OrientationOriented event-shape distributions are measured in nine bins of cos  between 0.1 and 0.9.The theoretical prediction of the formula (1) is tted to the data by means of a least squaresminimization with s(M2Z) as free parameter. The renormalization scale  is set to MZ=2. Inorder to obtain a good description of the data, the two-jet region has to be excluded from thet, which is based on second order calculations only. The t range has been put in the centralregion of the distribution where the correction factors are close to unity.In a rst step, sis determined in each bin of cos . The result is shown in Fig. 1. No systematicdependence on cos  is found.Subsequently the t is repeated for all bins of cos  simultaneously. The total 2of thesimultaneous t increases less than 10 % with respect to the 2of the t with individualvalues of sin each bin of cos . The two-dimensional distribution can be described by a uniquevalue of sand the results for each variable are given in Table 1. As an example the thrustdistribution is shown in Fig. 2, together with the result of the t, for dierent bins of cos .In order to check the reliability of the method, the event-shape distributions have been integratedover cos , and shas been determined using the integrated O(2s) prediction. An example ofthe t is shown in Fig. 3 and the results are given in Table 3. The central values of sare in goodagreement with those obtained from oriented event shapes, and the systematic uncertainties, tobe described in the following section, are also similar.3.4 Systematic StudiesThe uncertainty on the hadronization model used to compute corrections has been estimatedby the four models above. The results given in Table 1 are the mean values of s. Half of themaximum discrepancy between any of the models is given as systematic uncertainty.The main uncertainty related to xed order perturbation theory is due to the choice of therenormalization scale. The latter one is put by default to MZ=2 and has been varied from themass of the b-quark to the mass of the Z-boson.2Possible impacts of Bose-Einstein correlations on event shapes are conservatively derived byswitching on in Jetset a simple simulation of this eect (without changing other parameters).In this analysis both neutral objects and charged tracks have been used to compute the eventshapes. The stability of the corrections for detector eects has been checked by repeating theanalysis with charged tracks only and by recomputing the corrections. The shift in shas beentaken as a measure of the uncertainty. Finally the t range has been extended (reduced) atboth sides by three bins, and the largest deviation from the default range taken as error. Allsystematic errors are summarized in Table 2.3.5 Event OrientationThe integration over event shape variables is from a theoretical point of view more delicate,because in NLO framework used here, infrared divergiencies are present in the two-jet region.Therefore some cuts has to be applied in order to use the numerical prediction. This can also beseen as an advantage, since the impact on the event orientation is strongest for hard qqg-events.As illustrated in Fig. 4, the distribution becomes signicantly atter for events with low thrust.This can be seen clearly in Fig.5, where the ratio of the lowest order expectation (which isindependent of s) and the NLO prediction is shown together with the data.Another method relies on an analytic calculation in NNLO QCD. In contrast to the numericalcalculations with EVENT2, this next-to-next-to leading order prediction contains the lowest orderprocess Z ! qq, and also higher order corrections up to terms in 2s. In general the eventorientation can be written as follows :1dd cos =341 + cos2U+321  cos2L(2)where Uis the transverse unpolarized, Lthe longitudinally polarized cross section and  isthe sum of both. Both Uand Ldepend on sin the following way [12]:L= 2(8 ln32  3) R 0sCF(1 + ls) (3)where 0= 42=3s and R contains the vector couplings and axial-vector couplings of theZ-boson to the fermions and the propagator. The unpolarized cross section Uis obtained bysubtraction of Lfrom the total cross section [13]:U=    L(4) = 0Nc Ra"1 + d1s+ d2s2#+ Rv"1 + c1s+ c2s2#!Here, Rvcontains only the vector couplings and Rathe axial couplings. Note that both Uand Ldepend on electroweak parameters via the couplings and the coecients c and d. Theformula (2) has been tted to the data and sor equivalently Lare determined, although withlarge errors, since the QCD eects on the event orientation are small. The result is :s(M2Z) = 0:121 0:022stat 0:011systL= = (1:22 0:21stat 0:11syst) 10 2In Fig. 6 the result of the t compared to the lowest order form and the corresponding systematicuncertainties are listed in Table 4.34 ConclusionThe distribution of oriented event shapes has been measured and compared to next-to-leading order QCD predictions. Good agreement over the whole range of cos  is observed.Measurements of susing oriented event shapes are in good agreement with those obtained bya standard method with integrated event shapes. The preliminary result using the variable Y3is:s(M2Z) = 0:1161 0:0003stat 0:0051syst(oriented event shapes)s(M2Z) = 0:1156 0:0003stat 0:0055syst(integrated event shapes)The analysis of the event orientation has demonstrated the expected attening with respect tothe lowest order 1 + cos2 form. This eect is of the order of 1 % in total and increases up to10 % for events with TR< 0:8. From a t of the next-to-next-to leading order prediction to theinclusive event orientation, sand the relative contribution of the longitudinally polarized crosssection to the total cross section are determined with large uncertainties.AcknowledgmentsWe would like to acknowledge the helpful support of B. Lampe and M. Seymour in this work. Wewish to thank our colleagues from the CERN accelerator divisions for the successful operationof LEP. We are indebted to the engineers and technicians in all our institutions for theircontribution to the good performance of ALEPH. Those of us from non-member states thankCERN for its hospitality.References[1] B. Adeva et al., L3 Collaboration, Phys. Lett. B263 (1991) 551.[2] P. Abreu et al., DELPHI Collaboration, Phys. Lett. B274 (1992) 498.[3] W. Braunschweig et al., TASSO Collaboration, Zeit. Phys. C47 (1990) 181.[4] D. Decamp et al., ALEPH Collaboration, Nucl. Instr. and Meth. A294 (1990) 121. `[5] D. Buskulic et al., ALEPH Collaboration, Nucl. Instr. and Meth. A360 (1995) 481.[6] S. Catani et al., Phys. Lett. B269 (1991) 432; W. J. Stirling, Durham Workshop,J. Phys. G17 (1991) 1657.[7] program EVENT2, M. Seymour.[8] S. Catani and M. Seymour, CERN-TH-96/28.[9] T. Sjostrand, Comp. Phys. Comm. 39 (1986) 347; 43 (1987) 367; 82 (1994) 74.[10] G. Marchesini et al., Nucl. Phys. B310 (1988) 571; Comp. Phys. Comm. 67 (1992) 269.[11] L. Lonnblad, Comp. Phys. Comm. 71 (1992) 465.[12] B. Lampe, Phys. Lett. B301 (1993) 435.[13] J. H. Kuhn and P. M. Zerwas., in \Z Physics at LEP 1", edited by G. Altarelli, CERNYellow Report 89-08 (1989) 267.4Variable Result on sY30:1161 0:0003 0:0051TR0:1309 0:0004 0:0152BW0:1202 0:0003 0:0065MH0:1238 0:0004 0:0081Table 1: Results on sfrom oriented event shapes. The rst error is the statistical one (includingthe error from limited Monte Carlo statistics), the second the total systematic error.sourceof Y3TRBWMHuncertaintyHadronization 0:0011 0:0057 0:0009 0:0028Scale 0:0044 0:0137 0:0058 0:0067Bose-Einstein 0:0012 0:0019 0:0011 0:0019Detector resolution 0:0019 0:0019 0:0019 0:0019Fit range 0:0003 0:0021 0:0017 0:0024Total 0:0051 0:0151 0:0065 0:0081Table 2: Systematic uncertainties on sfrom oriented event shapes.5Variable Result on sY30:1156 0:0003 0:0055TR0:1305 0:0005 0:0147BW0:1199 0:0003 0:0070MH0:1235 0:0004 0:0087Table 3: Results on sfrom integrated event shapes with the same systematic studies as fororiented event shapes.sourceof (s)uncertaintyHadronization 0:010EW 0:002Bose-Einstein 0:002Detector resolution 0:001Fit range 0:003Total 0:011Table 4: Systematic uncertainties for sfrom the event orientation, using the NNLO prediction.6Figure 1: Measurement of sin single bins of cos  for four event shape variables. The xed-orderprediction with  = MZ=2, corrected for hadronization with Jetset PS has been used.7Figure 2: A t of the theoretical prediction to the double dierential cross section1=N (1  TR)d2N=dTRd cos . The result of the t is s= 0:1309 0:0004. The parton showermodel PS has been used for hadronization corrections. Note that the two curves shown arenormalized to the number of events in that particular bin of cos , i.e. the trivial 1+cos2 formhas been folded out.8Figure 3: Integrated event shapes, compared to the NLO QCD prediction.9Figure 4: The event orientation for dierent cuts on TR. In principal the prediction can be usedto t s, but since the number of events in a given range of TR(used for normalization) dependalso on s, a scan over shas to be done.10Figure 5: The ratio of the lowest order prediction (/ 1 + cos2, independent of s) and thenext-to-leading order t results shows the size of changes induced by higher order QCD processes.11Figure 6: The inclusive distribution of cos  (without any cuts on TR) is compared to next-to-next-to leading order calculation, which includes the lowest order process Z ! qq and higherorder corrections up to terms in 2s. Note that the total 2for the QCD t is 2=d:o:f = 49=44,which has to be compared with 2=d:o:f = 85=45 for the lowest order expectation. A similar 2is obtained when s= 0:25 is used in the QCD formula. The lower plot shows the ratio of the1 + cos2 form and both the QCD t and the data. It can be seen that the attening inducedby QCD leads to a change in the distribution of the order of one %.12

Oriented Event Shapes in Hadronic Z Decays

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