1. Event shape distributions at LEP Marek Taševský (Physics Institute Prague) for all LEP collaborations 21 April 2006 KEK-Tsukuba, Japan

2. Outline - Data samples and Event selection - Definitions & Properties of Event shape observables - Event shape observables at LEP1 and LEP2 energies The LEP alpha S measurement itself covered by T.Wengler ALEPH: EPJC35 (2004) 457 DELPHI: EPJC37 (2004) 1 L3: Phys.Rep.399 (2004) 71 OPAL: EPJC40 (2005) 287, PN519 (Preliminary)

3. Data samples and Event selection Typical numbers (ALEPH, 1994-2000) Main background: ISR for √s > 91 GeV - reduced by requiring -√s - √s, < 10 GeV WW,ZZ->4 fermions for √s > 2m W (2m Z ) E cm LumiN ev BG [GeV][pb -1 ][%] 9141> 10 6 < 1 13312806 < 1 16111319 5 17210257 10 183571319 12 1891743578 13 200 206 208 216 3528 3590 15

4. Correction procedure 1.Select hadronic event candidates 2.Construct distributions from tracks and clusters (avoid double counting) 3.Subtract bin-by-bin residual 4f- bg using grc4f and KORALW 4.Correct data bin-by-bin for effects of detector acceptance, resolutions and residual ISR using MC models. Justified by a good description of data and correlation between hadron and det.levels

5. Properties of event shape observables To make exper.tests of pert.QCD and to measure alpha S, we define physical observables that are sensitive to the HE pert.process but little sensitive to subsequent non-pert. hadronisation and decays. INCLUSIVE: - characterize geometry of event (2-jet or pencil-like, 3-jet or planar, 4+ -jet or spherical) - non-identified particles - only p and E needed to know pQCD ME diverge for process involving soft or collinear gluon emission Hence pQCD applicable only for quantities that are INFRA-RED SAFE: – not affected by soft gluon emission COLLINEAR SAFE: - not affected by replacing a parton by collinear partons with the same total 4-momentum

6. Properties of event shape observables 3-jet observables: sensitive to non-collinear emission of single hard gluon 4-jet observables: vanish in 3-jet limit All quantities approach 0 in the 2-jet limit. In experiment, pure 0 is never reached due to hadronisation. Measurement of alpha S : Based on fits of pQCD predictions to the corrected distributions of event shape observables. Standard set of observables is {1-T, M H, C, B T, B W, y 23 }. But let’s look at more of them. As theory predictions exist at parton level, they need to be corrected to hadron level by applying hadronisation corrections. For details about the alpha S measurement, see talk by T.Wengler

7. Thrust Thrust axis n T chosen to maximise the expression 1-T=0: 2-jet event 1-T=1/2: spherical event

8. Thrust major Thrust major axis n chosen to maximise the expression and to be orthogonal to n T T maj = 0: 2-jet event T maj =1/2: spherical event

9. Thrust minor T min =0: 2-jet event T min =0: 3-jet event T min =1/2: spherical event - 4-jet observable

10. Oblateness O=0: 2-jet and spherical event O=T maj for 3-jet events

11. Sphericity Quadratic momentum tensor: has three eigenvalues ordered such that λ 1 < λ 2 < λ 3. Being quadratic in p α,β, S αβ is not IR safe. Sphericity cannot be predicted reliably in pQCD S=0: 2-jet event S=1: spherical event

12. Aplanarity Sphericity tensor has three eigenvalues ordered such that λ 1 < λ 2 < λ 3. Being quadratic in p α,β, S αβ is not IR safe. Aplanarity cannot be predicted reliably in pQCD A=0: 2-jet and 3-jet event - 4-jet observable

13. C- parameter Linearised momentum tensor -linear in p α,β => it is IR safe. -has three eigenvalues ordered such that λ 1 < λ 2 < λ 3. M has unit trace => λ 1 + λ 2 + λ 3 = 1. We can thus form two indep. combinat.: 2 nd Fox-Wolfram moment C low: planar event (one of λ=0) C=1: isotropic event (λ 1 =λ 2 =λ 3 =1/3)

14. D- parameter Linearised momentum tensor -linear in p α,β => it is IR safe. -has three eigenvalues ordered such that λ 1 < λ 2 < λ 3. M has unit trace => λ 1 + λ 2 + λ 3 = 1. We can thus form two indep. combinat.: D=0: 2-jet and 3-jet event D=1: isotropic event (λ 1 =λ 2 =λ 3 =1/3) - 4-jet observable

15. Hemisphere observables So far, the variables have been constructed as global sums over all particles in the event. From now, let’s split the event into two hemispheres H 1 and H 2, divided by a plane orthogonal to the thrust axis. Invariant mass: Jet broadening:

16. Heavy jet mass - never zero due to finite masses of individual particles

17. Light jet mass M L =0: 2-jet and 3-jet events - 4-jet observable - never zero due to finite masses of individual particles

18. Wide jet broadening B W =0: 2-jet events to O(alpha S ): B W =B T =1/2T maj =1/2O Spherical event: B W =B N = π /16

19. Total jet broadening B T =0: 2-jet events to O(alpha S ): B W =B T =1/2T maj =1/2O Spherical event: B T = π /8

20. Jets The aim of jet algorithms is to group particles together such that the directions and momenta of partons are reconstructed. The jet algos include at least one free resolution parameter and N jets depends on its chosen value. Durham (or k T ) algo defines “scaled transverse momentum” for every pair of particles:. The pair with the smallest y ij is then replaced by a pseudoparticle with p ij =pi+pj and E ij =E i +E j (E-recomb.scheme; two other exist: P-scheme: E ij =|p i +p j | and E0-scheme: |p ij |=E i +E j ). This is repeated until all pairs have y ij >y cut (fixed value). Remaining pseudoparticles represent jets. [small y cut => many jets; large y cut ->1.0 => 1 jet]

21. y 23 – 2 to 3 jet transition Measure of how ‘3-jetlike’ event is. Y 23 : the highest y cut value for which the event is resolved into 3 jets. Events with N jet ≥3 have large y 23 values (max. y 23 =1/3 for 3 identical jets 120° apart), while 2-jet events at LEP have y 23 < 10 -3.

22. Event shapes in radiative hadronic events Measure event shape observables for a boosted qq system after final-state photon radiation. √s=91 GeV reduces to 20-80 GeV. Bg from non-rad. events: 5% (√s=78GeV) - 15% (√s=24GeV) - alpha S from radiative events measured by L3 and OPAL – results consistent with that from non-rad. events

23. Moments Another way to study the event structure – through moments: Y max is the max.kinematic. allowed value of observable Moments always sample all of available phase space: Lower moments are dominated by 2- and 3-jet events Higher moments are dominated by multi-jet events

24. Summary All LEP collaborations presented final measurements of event shape observables and their moments for all available data (√s = 91-209 GeV). Satisfactory description of data by Pythia, Herwig and Ariadne achieved. Discrepancies observed for LEP1 data in the extreme 2-jet region and for observables sensitive to 4+ -jet production.