1. Fragmentation in e + -e  Collisions Dave Kettler Correlations and Fluctuations Firenze July 7-9, 2006 p hadron ee e+e+ , Z 0 LEP PETRA color dipole s = Q 2

2. Dave Kettler2 QCD Issues for Nuclear Collisions How do partons scatter at low Q 2 (invariant mass squared) How do low-Q 2 partons fragment to hadrons What happens to low-Q 2 partons in heavy ion collisions to understand the second point we examine the systematics of parton fragmentation in p-p and especially e  -e  collisions we observe that a large fraction of RHIC C/F are due to minijets = low-Q 2 parton fragments

3. Dave Kettler3 Two-component p-p Spectrum Model ‘hard’ component: parton scattering and fragmentation p t and y t spectra S0S0 hard component vs n ch H0H0 200 GeV p-p minijets are isolated in single-particle spectra subtract S 0 replot

4. Dave Kettler4 Parton Scattering and Fragmentation e e´e´ , Z 0 x Q2Q2 ee e+e+ Z0Z0 s HERA LEP RHIC p proton Q2Q2 x1x1 x2x2 fragmentation – two issues: how distributed on momentum p how distributed on angle(s) e-p e-e p-p s, Q 2 parton  hadron jet (x,Q 2 )  i (p t,  ) i p proton

5. Dave Kettler5 p hadron Parton Fragmentation in e + - e  ee e+e+ , Z 0 LEP PETRA color dipole how are parton fragments (hadrons) distributed on momentum s = Q 2 LEP, PETRA fragmentation data: 1985-2000 color dipole radiation: ln(p hadron ) LEP PETRA e-e ln(p parton ) gluon coherence: a QCD triumph s, Q 2 ? an equilibration process

6. Dave Kettler6 Conventional Fragmentation Studies eeee CCOR  s = 63 GeV jet reconstruction leading particle xExE ? A.L.S. Angelis et al., NPB 209 (1982) x trigger vs associated fragmentation function ‘leading-particle’ strategy jet is not reconstructed, estimate parton with high-p t leading particle

7. Dave Kettler7 ln(p)rapidity y Fragment Distributions on Momentum  p = ln(1/x p ) fragmentation functions on logarithmic variables alternative: fragmentation functions on rapidity y conventional: fragment momentum relative to parton momentum D(  p,s) D(y,y max ) D(ln(p),s) D(x,s) x p = p hadron /p parton fragmentation function D(x,s)  D(y, y max ) LEP PETRA e-e non-pQCD physics! scaling violations pQCD

8. Dave Kettler8 Precision Analysis of Fragmentation fragmentation functions well described by simple model function g(u,y max ) = beta distribution on normalized rapidity u precisely models fragmentation functions (normalized) y min normalized rapidity redundant 7 46 GeV = Q/2 22 a form of equilibration p-pp-p - FNAL e-e - LEP STAR D(y,y max ) dijet multiplicity

9. Dave Kettler9 Why a Beta Distribution? Maximum Entropy Distributions GaussianExponential Beta Distribution Maximize Shannon Entropy with constraints Bounded interval Constraints reflect parton splitting and gluon coherence

10. Dave Kettler10 Identified Hadrons and Partons identified hadron fragmentsidentified partons the flavor/color chemistry of fragmentation pions kaons increasing meson mass bottom quark is anomalous quark and gluon shapes are different heavier fragments stay close to parent parton udsc gluon b

11. Dave Kettler11 y max ~ ln( Q/  ) Fragmentation Energy Systematics fits to quark and gluon jet multiplicities  (p,q) fits to fragmentation functions with beta distribution  (p,q) fragmentation functions represented over a broad energy range to few %! parton rapidity fragment rapidity 400 GeV fit parameters extrapolation to non-perturbative regime energy systematics (p,q) non-pQCD pQCD fit (p,q) 2n(y max ) g(u,y max ) energy sum rule

12. Dave Kettler12 Scaling Violations – Conventional Q/2 (GeV) G. Abbiendi et al. (OPAL Collab.), Eur. Phys. J. C 37, 25 (2004) excellent agreement with recent measurements data at right g-g

13. Dave Kettler13 Scaling Violations – Logarithmic near uniformity to right of dotted line C A /C F =2.25 ratio 2.25 P. Abreu et al. (DELPHI Collab.), Eur. Phys. J. C 13, 573 (2000) related to anomalous dimensions of QCD for x E  1 and s large ratio 2.25

14. Dave Kettler14 Comparisons with pQCD conventional FF description form factor on u MLLA gaussians peak modes 8% 10 GeV

15. Dave Kettler15 Summary Low-Q 2 partons play a dominant role in HI collisions Little was known about low-Q 2 parton scattering and fragmentation prior to this work We have described all measured e  -e  fragmentation functions with a precise (few %) model function The model function (beta distribution) allows us to extrapolate fragmentation trends to low Q 2 That system can then be used to describe low-Q 2 parton fragmentation in p-p and HI collisions ‘theoretical’ basis for minijet correlations in nuclear collisions