1. Hadronization of Partons by Recombination Rudolph C. Hwa University of Oregon Summer School on RHIC Physics Wuhan, China, June 2005

2. 2 Outline An overview of the recombination model Some questions and answers on the basics Shower partons initiated by hard partons Hadronization in heavy-ion collisions

3. 3 Parton Recombination First studied for low-p T production in pp collision Das & Hwa, Phys. Lett. 68B, 459 (1977) pp x H(x) Ochs observation: H(x) is very similar to the valence quark distribution in a proton.

4. 4 Valon model -- to get the proton wave function Hwa, PRD (1980a) Valon-recombination model -- better formulation of recombination Hwa, PRD (1980b)

5. 5 p+A collisions Hwa & CB Yang (2002a) We studied the centrality dependence (or the number of collisions) in the valon-recombination model good data from NA49 Hadronic collisions Hwa & CB Yang (2002b) h + p  h’ +X h h’ p    K+   + K 

6. 6 Hadron production at high p T pp collision: mainly by fragmentation AA collision: there were puzzles according to fragmentation Recombination solved those puzzles Hwa & Yang, PRC 67, 034902 (2003); 70, 024905 (2004) Greco, Ko, Levai, PRL 90, 202302 (2003); PRC 68, 034904 (2003) Fries, Muller, Nonaka, Bass, PRL 90,202303(03); PRC 68, 044902 (03) More recent developments -- 2004, 2005 Correlations in jets

7. 7 Closer examination of the recombination formulas Pion : Proton : Questions: 1.What is the two-parton distribution? 2. What are the recombination functions ? Especially in heavy-ion collisions

8. 8 3. What about the gluons? 4. Does entropy decrease? 5. What about the spatial considerations? 6. Isn’t the pion a Goldstone boson? 7. Recombination versus fragmentation: Which is more important? 8. What is wrong with string fragmentation? More questions : Answer in reverse order.

9. 9 recombination 8. String fragmentation String model may be relevant for pp collisions, String/fragmentation has no phenomenological support in heavy-ion collisions. but not for AA collisions.

10. 10 High p T physics in pp collisions is well understood. What was a discovery yesterday is now used for calibration today.

11. 11 7. Recombination versus Fragmentation Parton distribution (log scale) p p hadron momentum q 1 +q 2 (recombine) higher yield q (fragment) heavy penalty suppressed by power-law

12. 12 6. Pion is a Goldstone boson Is it a boson due to spontaneous symmetry breaking? Or a bound state of quark-antiquark? Both are aspects of the pion. No theory exists that can continuously transform one to the other. In Drell-Yan process in  -p collisions, the quark contents of pion and proton are probed.  p ++ --

13. 13 5. Spatial considerations We have formulated recombination in momentum space only so far. Shouldn’t the spatial coordinates be important also? Isn’t hadron size relevant? In heavy-ion collisions there are two sizes: 1. nuclear transverse size R A 2. hadron transverse size r h If partons are parallel, but far apart, they cannot recombine If parton trajectories intersect, they must cross at the same space-time region --- relative momentum suppress recombination.

14. 14  Soft partons are restricted to the small spatial spread around the point where hard parton emerges from the nuclear medium. Groups at Duke University and Texas A&M University have Monte Carlo codes to implement space & momentum constraints on recombination. Our approach:  We consider only collinear partons.  Hard parton defines the direction of the hadron. We do not use Monte Carlo code to generate the soft partons throughout the expanding medium. We infer from the soft pion spectrum at low p T what the soft parton distribution is. Momentum space consideration is sufficient, and that is where observation is made.

15. 15 4. Entropy color: 3 X 3 1 spin: 2 X 2 1 degrees of freedom decreased depends on wave functionmomentum conservation g Soft gluon radiation: mutates color & carries away spin without changing RM cannot account for low momentum partons Entropy: a global quantity that should take into account expanding volume.

16. 16 3. How do gluons hadronize? In pp collisions the parton distributions are Gluons carry ~1/2 momentum of proton but cannot hadronize directly. Sea quark dist. F q ~ c (1-x) 7 Saturated sea quark dist. F’ q ~ c’ (1-x) 7 Gluon conversion to q-qbar Recombination of with saturated sea gives pion distribution in agreement with data. x 2 u(x) x 2 g(x) x [log]

17. 17 2. Recombination functions It depends on the wave function. Consider the time-reversed process What are the distributions of the quarks in momentum fractions in the infinite momentum frame?

18. 18 Deep inelastic scattering e e p FqFq We need a model to relate to the wave function of the proton FqFq Valon model p U U D valons

19. 19 p U U D Basic assumptions valon distribution is independent of probe parton distribution in a valon is independent of the hadron valence quark distr in proton valon distr in proton, independent of Q valance quark distr in valon, in proton or in pion Moments by convolution theorem known from CTEQ param cancel in the ratio the ratio can be determined

20. 20 3-valon exclusive distribution Recombination function proton pion From  initiated Drell-Yan process valon model Single-valon inclusive distribution Hwa & CB Yang, PRC66(2002)

21. 21 1. Two-parton distributions pp collisions: low p T and large x F Heavy-ion collisions: Low p L (mid-rapidity), large p T That is high p T physics. Traditionally, hadronization at high p T is by fragmentation. However, fragmentation model has met some difficulties, most notably in p/  ratio at intermediate p T in nuclear collisions.

22. 22 Before describing what the two-parton distribution should be at high p T in heavy-ion collisions, we must first discuss why fragmentation does not work phenomenologically what are the shower partons in fragmentation? how does the nuclear medium affect hadronization? Which parton recombines which parton is the core problem in the recombination model.

23. 23 Not possible in fragmentation model: R p/π R p/π  1 u p/  ratio

24. 24

25. 25 The black box of fragmentation  q A QCD process from quark to pion, not calculable in pQCD z 1 Momentum fraction z < 1 Phenomenological fragmentation function D  /q z 1

26. 26 Let’s look inside the black box of fragmentation.  q fragmentation z 1 gluon radiation quark pair creation Although not calculable in pQCD (especially when Q 2 gets low), gluon radiation and quark-pair creation and subsequent hadronization nevertheless take place to form pions and other hadrons.

27. 27 Description of fragmentation by recombination known from data (e+e-,  p, … ) known from recombination model can be determined hard parton meson fragmentation shower partons recombination

28. 28 Shower parton distributions u g s s d duvalence sea L L  D  Sea K NS L  D  V G G  D  G L L s  D K Sea G G s  D K G RR RKRK 5 SPDs are determined from 5 FFs. assume factorizable, but constrained kinematically.

29. 29 Shower Parton Distributions Hwa & CB Yang, PRC 70, 024904 (04)

30. 30 BKK fragmentation functions

31. 31 If our shower parton distributions are reliable, based on the dynamical independence of the shower partons except for kinematical constraints, then we should be able to calculate the quark fragmentation function into a proton. Data on D u  p (z) not well determined. KKP parametrization has an error. Nevertheless, there is only a discrepancy of less than a factor of 2 over 4 order of magnitude.

32. 32 Once the shower parton distributions are known, they can be applied to heavy-ion collisions. The recombination of thermal partons with shower partons becomes conceptually unavoidable. D(z) h q AA Conventional approach

33. 33 Once the shower parton distributions are known, they can be applied to heavy-ion collisions. The recombination of thermal partons with shower partons becomes conceptually unavoidable. h Now, a new component

34. 34 Pion formation:distribution thermal shower soft component soft semi-hard components usual fragmentation (by means of recombination) Proton formation: uud distribution

35. 35 Thermal distribution Fit low-p T data to determine C & T. Shower distribution in AuAu collisions hard parton momentum distribution of hard parton i in AuAu collisions Contains hydrodynamical properties, not included in our model.

36. 36 density of hard partons with p T = k Input: parton distributions CTEQ5L nuclear shadowing EKS98 hard scattering pQCD Srivastava, Gale, Fries, PRC 67, 034903 (2003) C, B,  are tabulated for i=u, d, s, u, d, gK=2.5

37. 37 Thermal distribution Fit low-p T data to determine C & T. Shower distribution in AuAu collisions hard parton momentum distribution of hard parton i in AuAu collisions SPD of parton j in shower of hard parton i fraction of hard partons that get out of medium to produce shower calculable Contains hydrodynamical properties, not included in our model.

38. 38 thermal fragmentation softhard TS Pion distribution (log scale) Transverse momentum TT SS Now, we go to REAL DATA, and real theoretical results.

39. 39  production in AuAu central collision at 200 GeV Hwa & CB Yang, PRC70, 024905 (2004) fragmentation thermal

40. 40 Proton production in AuAu collisions TTS+TSS TSS

41. 41 Proton/pion ratio

42. 42 All in recombination/ coalescence model Compilation of R p/  obtained by 3 groups

43. 43 Puzzle in pA or dA collisions k T broadening by multiple scattering in the initial state. Unchallenged for ~30 years. If the medium effect is before fragmentation, then  should be independent of h=  or p Cronin Effect Cronin et al, Phys.Rev.D (1975) p q h A STAR, PHENIX (2003) Cronin et al, Phys.Rev.D (1975)  p >  

44. 44 RHIC data from dAu collisions at 200 GeV per NN pair Ratio of central to peripheral collisions: R CP PHENIX and STAR experiments found (2002) Can’t be explained by fragmentation. (in fragmentation model)

45. 45 STAR

46. 46 d d central peripheral more  T  more TS less  T  less TS d+Au collisions (to study the Cronin Effect)

47. 47 d+Au collisions Pions Hwa & CB Yang, PRL 93, 082302 (2004) No p T broadening by multiple scattering in the initial state. Medium effect is due to thermal (soft)-shower recombination in the final state. soft-soft

48. 48 Proton Thermal-shower recombination is negligible. Hwa & Yang, PRC 70, 037901 (2004)

49. 49 Nuclear Modification Factor This is the most important result that validates parton recombination. 2q  , each quark has ~1/2 of  momentum 3q  p, each quark has ~1/3 of p momentum

50. 50 Molnar and Voloshin, PRL 91, 092301 (2003). Parton coalescence implies that v 2 (p T ) scales with the number of constituents STAR data Azimuthal anisotropy

51. 51 Forward-backward asymmetry in d+Au collisions Expects more forward particles at high p T than backward particles If initial transverse broadening of parton gives hadrons at high p T, then backward has no broadening forward has more transverse broadening

52. 52 Backward-forward asymmetry at intermediate p T in d+Au collisions

53. 53 More interesting behavior found in large p T and large p L region. It is natural for parton recombination to result in forward-backward asymmetry Less soft partons in forward (d) direction than backward (Au) direction. Less TS recombination in forward than in backward direction. Forward-backward asymmetry by recombination

54. 54 Hwa, Yang, Fries, PRC 71, 024902 (2005) Forward production in d+Au collisions Underlying physics for hadron production is not changed from backward to forward rapidity. BRAHMS data

55. 55 Summary We have discussed some basic issues about recombination application to intermediate and high p T physics in heavy-ion collisions resolved several puzzles on single-particle distributions in HIC We have not covered Correlation of hadrons in jets (Thursday)