Experimental tests of QCD predictions for event shape distributions combining
contributions from hard and soft processes are discussed. The hard processes
are predicted by perturbative QCD calculations. The soft processes cannot be
calculated directly using perturbative QCD, they are treated by a power
correction model based on the analysis of infrared renormalons. Furthermore, an
analysis of the gauge structure of QCD is presented using fits of the colour
factors within the same combined QCD predictions.Comment: 6 pages, 7 figures, contribution to Siena 2001, Seventh Topical
  Seminar on The legacy of LEP and SLC, Siena, Italy, 8 to 11 October 2001,
  proceedings to appear in Nucl. Phys. B (Proc. Suppl.

Abbaneo

Abreu

Acciarri

Barate

Beneke

Bethke

Catani

Dokshitzer

Ellis

Fernández

Kluth

Kunszt

S Kluth

English

arXiv

Experimental tests of QCD predictions for event shape distributions combining contributions from hard and soft processes are discussed. The hard processes are predicted by perturbative QCD calculations. The soft processes cannot be calculated directly using perturbative QCD, they are treated by a power correction model based on the analysis of infrared renormalons. Furthermore, an analysis of the gauge structure of QCD is presented using fits of the colour factors within the same combined QCD predictions

Kluth, S

CERN Document Server

1Tests of hard and soft QCD with e+e− Annihilation DataS. KluthaMPI]Max-Planck-Institut fu¨r Physik,Fo¨hringer Ring 6, 80805 Munich, Germanya[Experimental tests of QCD predictions for event shape distributions combining contributions from hard and softprocesses are discussed. The hard processes are predicted by perturbative QCD calculations. The soft processescannot be calculated directly using perturbative QCD, they are treated by a power correction model based onthe analysis of infrared renormalons. Furthermore, an analysis of the gauge structure of QCD is presented usingfits of the colour factors within the same combined QCD predictions.1. INTRODUCTIONHadron production in e+e− annihilation is awell-suited process for the study of the interplaybetween hard and soft QCD. The lack of inter-ference with the initial state improves the under-standing of the hadronic nal state in terms ofQCD. The availability of data sets at many en-ergy points with centre of mass (cms) energiespsbetween 14 and 209 GeV allows to separate thehard and soft contributions.The hard QCD processes, i.e. the radiationof gluons from the quarks produced at the elec-troweak vertex and subsequent higher order pro-cesses are well understood in perturbative QCD(pQCD) [1]. The production of multi-jet eventsis clear evidence for the existence of gluons andis successfully described by pQCD.The soft contributions to hadron production ine+e− annihilation are connected with the transi-tion from the partons (quarks and gluons) gener-ated by hard processes to the hadrons observedin the experiment. This transition is referredto as hadronisation and takes place at energyscales of approximately a light hadron mass, i.e.O(100) MeV, where pQCD calculations are notreliable due to the rapid growth of the strongcoupling αS. The soft contributions are knownto scale like an inverse power of the cms energy;for most observables the scaling is 1/ps.2. POWER CORRECTIONS AND IN-FRARED RENORMALONSAt energy scales Q = QCD perturbative evolu-tion of αS(Q) breaks down completely due to theLandau pole at QCD. It is therefore impossibleto attempt pQCD calculations of soft processeswithout further assumptions.In order to clarify which assumptions areneeded a phenomenological model of hadronisa-tion is considered. The longitudinal phase spacemodel or tube model goes back to Feynman [2].One considers a qq system produced e.g. in e+e−annihilation in the cms system. The two primarypartons move apart with velocity v/c ’ 1. Insuch a situation the production of soft gluons willbe approximately independent of their (pseudo-)rapidity η0 = − log(tan(i/2)), see gure 1.The change to e.g. the observable Thrust 1−Tdue to the production of a soft gluon at angle iwith transverse momentum kt,i is (1 − T )i ’kt,i/Qe−jη′ij. This observation is generalised towrite the soft or non-perturbative contribution tothe mean value of the 1−T distribution as follows:h1 − T iNP =∫ktQ(kt)dktkt∫e−jη′jdη0 (1)The function (kt)  αS(kt) is the distribu-tion of soft particles in kt. The rst integral inequation (1) is summarised as α0/Q independentof the observable and the second as a constantc1−T dependent on the observable. The quan-2ktq–qη’=-ln(tan(Θi/2))ΘiFigure 1. Sketch of a qq system in η0 − kt space.tity α0 ∫αS(kt)dkt can only exist when αS(kt)is identied with a non-perturbative strong cou-pling αNPS (kt) which is assumed to be nite at lowkt around and below the Landau pole.A more formal approach to the origin of softcontributions is the study of infrared renor-malons, i.e. the divergence of asymptotic pQCDpredictions due to the integration of low momentain quark loops in gluon lines [3]. The infraredrenormalon divergence of the O(αnS) term in apQCD prediction is factorial in n: rnαn+1S (2β0/p)nn!αn+1S . With Sterlings formula to re-place n! one nds rnαn+1S  (2β0αS/p)nnne−nαS.The convergence of the series is optimal for n =p/(2β0αS). Based on this relation one ndsrnαn+1S (QCDQ)p(2)where the rst order relation between αS andQCD has been used. The result shows that in-frared renormalon contributions to pQCD predic-tions scale like Q−p similar to the soft contribu-tion studied in the tube model, see equation (1).The power correction model of Dokshitzer,Marchesini and Webber (DMW) extracts thestructure of power correction terms from anal-ysis of infrared renormalon contributions [4].The model assumes that a non-perturbativestrong coupling exists around and below theLandau pole and that the quantity α0(µI) =1/µI∫ µI0 αNPS (kt)dkt can be dened. The value ofµI is chosen to be safely within the perturbativeregion, usually µI = 2 GeV.The main result for the eects of power cor-rections on distributions F (y) of the event shapeobservables 1− T , MH and C is that the pertur-bative prediction FPT(y) is shifted [5{7]:F (y) = FPT(y − cyP ) (3)where cy is an observable dependent constant andP  MµI/Q(α0(µI) − αS) is universal, i.e. in-dependent of the observable [6]. The factor Pcontains the 1/Q scaling and the so-called Milan-factor M which takes two-loop eects into ac-count. The non-perturbative parameter α0 isexplicitly matched with the perturbative strongcoupling αS. For the event shape observables BTand BW the predictions are more involved andthe shape of the pQCD prediction is modied inaddition to the shift [7]. For mean values of 1−T ,MH and C the prediction is:hyi = hyiPT + cyP (4)For hBTi and hBWi the predictions are also moreinvolved.3. EXPERIMENTAL TESTSDistributions and mean values of event shapeobservables measured in e+e− annihilation atmany cms energy points between 14 and morethan 200 GeV have been used recently by sev-eral groups to study power corrections [8{11].This report will present mainly results from [8].In all of these studies pQCD predictions inO(α2S)+NLLA [12{14] are used for event shapedistributions while O(α2S) [15] predictions are em-ployed for mean values. The pQCD predictionsare added together with the power correction pre-dictions. The resulting expressions are functionsof the strong coupling αS(MZ) and of the non-perturbative parameter α0. The complete pre-dictions are compared with data for event shapedistributions or mean values as published by theexperiments.3110 210 410 610 810 1010 1210 1410 1610 180 0.1 0.2 0.3TASSO 14 GeVTASSO 22 GeVHRS 29 GeVMARK2 29 GeVTASSO 35 GeVJADE 35 GeVTASSO 44 GeVJADE 44 GeVAMY 55 GeVSLD 91 GeVL3 91 GeVDELPHI 91 GeVALEPH 91 GeVOPAL 91 GeVL3 133 GeVDELPHI 133 GeVALEPH 133 GeVOPAL 133 GeVL3 161 GeVDELPHI 161 GeVOPAL 161 GeVL3 172 GeVDELPHI 172 GeVOPAL 172 GeVL3 183 GeVDELPHI 183 GeVOPAL 183 GeVL3 189 GeVOPAL 189 GeV1-T1/σ dσ/d(1-T)Figure 2. Scaled distributions of 1−T . The solidlines show the result of the t of the combinedpQCD and power correction prediction. The dot-ted lines represent an extrapolation of the re-sult [8].Figure 2 shows the results tting the datafor 1 − T distributions measured at cms ener-gies from 14 to 189 GeV. The data from exper-iments atps < MZ are corrected for the eectsof e+e− ! bb events [8]. The data in the ttedregions are well reproduced by the theory (solidlines). The dotted lines present extrapolationsoutside of the tted regions using the t resultsfor αS(MZ) and α0 which describe the data rea-sonably well. The agreement between theory andexperiment is similar for the other observables.Figure 3 (solid line) presents the result of a t00.020.040.060.080.10.120.140 20 40 60 80 100 120 140 160 180 200√−−s [GeV]ÆMH2/sæTASSOMark IIJADEL3AMYDELPHIOPALSLDALEPHFigure 3. The cms energy dependence of h1 − T iis shown. The solid lines show the result of the tof combined O(α2S) QCD calculations with powercorrections while the dashed lines indicate theperturbative contribution [8].to measurements of hM2Hi atps = 14 to 189 GeV.The data are well described by the tted theory.The dotted line in gure 3 shows the perturbativepart of the prediction; one observes that the soft(power correction) contributions increase at lowps.The combined results from the analysis ofdistributions are αS(MZ) = 0.1111+0.0048−0.0037 andα0(2 GeV) = 0.579+0.100−0.072. The relatively smallvalue of αS(MZ) compared e.g. to the world av-erage αS(MZ) = 0.1184  0.0034 [16] is due tosmall results from MH and BW. The individualand the combined results are shown as points withone-standard-deviation error ellipses in gure 4a).From the analysis of mean values the com-bined results are αS(MZ) = 0.1187+0.0031−0.0020 andα0(2 GeV) = 0.485+0.066−0.045 in reasonable agreementwith the results from distributions. Figure 4 b)presents the individual and combined results fromts to mean values as points with one-standard-deviation error ellipses. Combining both analysesyields the nal results: αS(MZ) = 0.1171+0.0032−0.002040.40.50.60.70.80.910.08 0.09 0.1 0.11 0.12 0.13average1-TMH, MH2BTBWCαS(MZ)α0a)Distributions0.30.40.50.60.70.80.1 0.11 0.12 0.13average<1-T><MH2><BT><BW><C>αS(MZ)α0b)Mean ValuesFigure 4. Results for αS(MZ) and α0(2 GeV)for distributions (top) and mean values (bottom).The error ellipses show one standard deviationuncertainties (38% CL) [8].and α0(2 GeV) = 0.513+0.066−0.045 .Figure 5 shows a summary of results forαS(MZ) and α0(2 GeV) obtained in compara-ble analyses by several groups [8{11]. One ob-serves reasonable agreement between the individ-ual measurements. The comparable results forMH and BW from [8] and [10] dier, becausein [10] hadron mass eects on power correctionsare considered [2]. The non-perturbative param-0.30.40.50.60.70.80.910.09 0.1 0.11 0.12 0.131-TMH2BTBWC EECL3 means (prelim.)PPTSL meansPPTSL dist.DELPHI means (prelim.)DELPHI dist. (prelim.)ALEPH dist. (prelim.)αS(MZ)α0Figure 5. The gure shows a summary of resultsfor αS(MZ) and α0(2 GeV) from power correctionstudies [8{11]. The results labelled PPTSL arefrom [8].eter α0 appears universal within about 20% asexpected [6] while the values for αS(MZ) are com-patible with the world average [16].4. QCD COLOUR FACTORSThe analysis of event shape distributions de-scribed above can be generalised to study thegauge structure of QCD [17]. The O(α2S)+NLLApQCD predictions can be decomposed into ad-ditive terms which are proportional to productsof the QCD colour factors. This analysis consti-tutes an experimental test of the gauge symme-try of QCD via radiative corrections. The valuesof the colour factors correspond to the relativesize of contributions from three fundamental pro-cesses i) gluon radiation from a quark (CF), ii)conversion of a gluon in a qq pair (TFnf) and iii)radiation of a gluon from a gluon (triple gluonvertex TGV) (CA). Higher order processes likethe quartic gluon vertex are not accessible withthe available pQCD calculations.The values of the colour factors are determinedin the theory when a particular gauge symmetry50.1 0.125 0.4 0.6 0 5 100.1 0.125 0.4 0.6 2 3 40.05 0.1 0 0.5 1 2 41-TCBTαS(MZ) α0 nf1-TCBTαS(MZ) α0 CA1-TCBTBWαS(MZ) α0 CF00.511.522.50 1 2 3 4 5 6U(1)3SU(1)SU(2)SU(4)SU(5)QCD fitSU(3) QCD predictionCACF68% C.L.95% C.L.Figure 6. The gure at the top shows results forαS(MZ), α0(2 GeV) and one of the colour factorsfor individual observables as indicated. The errorbars show total uncertainties and the dotted linesshow the expectation from SU(3) QCD. The g-ure at the bottom presents combined results forCA and CF from ts to αS(MZ), CA and CF with1 − T and C. The square and triangle symbolsindicate expectations for CA and CF for severalsymmetry groups [17].group is chosen under which the Lagrangian oftheory must be locally invariant. The local gaugesymmetry for QCD is SU(3) with CF = 4/3,CA = 3 and TF = 1/2. The number of ac-tive quark flavours is nf = 5 at the cms ener-gies of the currently available data. The per-turbative evolution (running) of the strong cou-pling is also sensitive to the colour factors sincequark loops and gluon loops have the oppositeeect. In O(αS) the evolution of the strongcoupling between two scales Q and µ is givenby αS(Q) = αS(µ)/(1 + 2β0αS(µ) ln(Q/µ)) withβ0 = (11CA − 2nf)/(12pi).The power correction calculations are also func-tions of the QCD colour factors. In the analysisthe colour factors in the pQCD predictions as wellas in the power corrections are considered as addi-tional free parameters to be determined from thedata. In this way the dependence on the hadroni-sation model is reduced compared to traditionalanalyses relying on Monte Carlo models of hadro-nisation, because in the Monte Carlo models thecolour factors are xed [18{20].Figure 6 (top) shows the results of simultane-ous ts of αS(MZ), α0(2 GeV) and one colourfactor where the other colour factors have beenkept xed at their SU(3) values. The ts are sta-ble with the observables 1 − T and C. Some tsare not stable with the observables BT and BW.In alternative ts the non-perturbative parame-ter α0 was kept xed at a previously measuredvalue and is varied by its total error as a system-atic uncertainty. The results and the uncertain-ties are consistent between the two types of t.The variation of α0 in the ts avoids a possiblebias which might be present when α0 is xed to avalue measured with colour factors xed to theirstandard values. Combined results for the colourfactors measured individually are nf = 5.641.35,CA = 2.880.27 and CF = 1.450.27. The corre-sponding results for αS(MZ) and α0 are consistentwith previous measurements.Figure 6 (bottom) presents the combined re-sults of simultaneous ts to 1 − T and C ofαS(MZ), CA and CF with α0 xed to previouslymeasured value. Such ts with α0 as a free pa-rameter turn out to be unstable mainly due tolimited precision of the data. The results areCA = 2.84  0.24 and CF = 1.29  0.18 witha consistent value of αS(MZ) = 0.1190.010 andconsistent with the expectation from SU(3). Sev-6eral other possible gauge groups are excluded.5. SUMMARYThe analyses of power corrections discussed insection 3 show that it is possible to describe eventshape distributions and mean values in e+e− an-nihilation data with combined pQCD and powercorrection predictions without the need for MonteCarlo based hadronisation corrections. Globalts of the combined theory with only αS(MZ)and α0(2 GeV) as free parameters to event shapedata measured atps = 14 GeV up to the highestLEP 2 energies lead to a satisfactory agreementwith the data. The non-perturbative parameterα0(2 GeV) is found to be universal, i.e. inde-pendent of the observable, within the expectedtheoretical uncertainty of about 20%.A generalisation of the power correction anal-ysis to also vary the QCD colour factors pro-vides a measurement of the colour factors basedon the QCD radiative corrections in the pQCDand the power correction predictions. The re-sults for the colour factors are consistent with theexpectations from SU(3) as the gauge group ofQCD while the total uncertainties are competi-tive with analyses of angular correlations in 4-jetnal states in e+e− annihilation.One of the legacies of LEP, SLC and theirpredecessors TRISTAN, PETRA and PEP isthe wealth of measurements using hadronic nalstates in e+e− annihilation. These data madeis possible to test and verify combined predic-tions for hard (pQCD) and soft (power correc-tions) processes. Based on these tests our under-standing of the hard and soft processes and theirinterplay has improved signicantly.The author would like to take the opportunityto thank the organisers for a stimulating and en-tertaining meeting.REFERENCES1. R.K. Ellis, W.J. Stirling, B.R. Webber: QCDand Collider Physics. Vol. 8 of CambridgeMonographs on Particle Physics, NuclearPhysics and Cosmology, Cambridge Univer-sity Press (1996)2. G.P. Salam, D. Wicke: JHEP 05 (2001) 0613. M. Beneke, V.M. 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Tests of hard and soft QCD with e+e- Annihilation Data

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