1. Cronin effect and high pt suppression from Color Glass Condensate Kazunori Itakura (SPhT, CEA/Saclay) based on hep-ph/0403103 with E.Iancu and D.Triantafyllopoulos

2. Terminologies a function of rapidityh and transverse momentum pt measure deviation from “collection of A nucleons RpA = 1” RpA > 1 at middle pt ~ a few GeV, the largest point of RpA RpA < 1 at high pt Observed in AuAu and is attributed to high energy density matter (which will not be created in pA collisions) I will use the same terminologies for a simpler quantity Nuclear modification factor for p(d)A collision Cronin effect/enhancement, Cronin peak High pt suppression

3. Motivation (from experiments) BRAHMS data for dAu collisions at RHIC Nuclear modification factor -Cronin peak at h=0, suppression at h=3.2 - More enhanced for central at h=0 More suppressed for central at h=3.2 - Similar results in other experiments (PHENIX, STAR, PHOBOS)

4. Some theoretical analyses based on CGC Motivation (from theory) Analytic studies - Cronin effect from classicalsaturation (McL-V. model) Gelis-Jalilian-Marian03 Kharzeev-Levin-McLerran02, Kharzeev-Kovchegov-Tuchin03 - High pt suppression from quantum evolution (ColorGlassCondensate) Numerical studies - McL.-V model - Balitsky-Kovchegov equation Albacete, Armesto, Kovner, Salgado, Wiedemann 03 from top to bottom:h=0 to 10 Cronin peak exists at h=0, butrapidly disappears after evolution. For h>1, the ratio monotonically increases as a function of pt.  Need to understand physics, mechanism, and detailed picture

5. = - Shows qualitatively the same behaviors properties of are transmitted to those of RpA - Allows for analytic control for unintegrated gluon distr. also for running coupling case - Direct measure of collective effects in nuclear wavefnc conceptually important ! =1  Nucleus=A nucleons (No collective effect) Our framework (I) Nuclear modification factor Ratio btw p and A wavefncs

6. McL-V model Our framework (II) BK=Balitsky-Kovchegov, JIMWLK=Jalilian-Marian,Iancu,McLerran,Weigert,Leonidov,Kovner Rapidity h large in pp/pA collision small-x component of p or A with large gluon density. Transition to larger h nonlinear quantumevolution eq.(BK, JIMWLK eqs). We work in a perturbative regime Q > LQCD Initial condition (h=0) Classical saturation McL-V model Large rapidities Saturation with quantum evolution Color Glass Condensate

7. Cronin effect > 1at lower energy

8. McLerran-Venugopalan model Classical Saturation Incoherent multiple scattering for a large nucleus (A >> 1)  Glauber-Mueller type rescattering Nucleus = collection of uncorrelated(randomly distributed) valence quarks  color sources for a small x gluon Initial condition (h=0) for the quantum evolution Unintegrated gluon distribution m: color charge squared per unit transverse area

9. McLerran-Venugopalan model Saturation scale Bremsstrahlung at high kt Saturation and twist parts Bremsstrahlung Saturation

10. Cronin effect from MV model Use Bremsstrahlung for a proton There exists the Cronin peak. Its height and position RpA ~ 0.281 ln A zmax ~ 0.435 are essentially determined by the saturation part. Higher twist terms develop a peak but far away from the real position At large kt, the ratio goes to 1 from above(can be checked analytically via “sum-rule”) The Cronin effect is due to redistribution of small kt gluons 1/3

11. High pt suppression < 1at higher energy

12. Quantum evolution The BK equation for scattering amplitude Includes nonlinear effects due to high gluon density  ColorGlassCondensate as saturated and dense gluon states Energy dependence of saturation scale Three kinematical regimes for p and A 1) CGC 2) BFKL geometric scaling 3) DLA (double log approximation)

13. Quantum evolution Approximate solutions in each regime 1) CGC 2) BFKLanomalous dimension absorptive, scaling, scaling violation 3) DLA(double log approximation)

14. Y=0.75 BFKL(A) BFKL(p) Y=1.95 Y=as y BFKL(A) DLA(p) Y=0.1 Y=0.7 Y=as y High pt suppression (I) - Distinguish three kinematical regimes for proton/nucleus - Use the approximate solutions in each domain - Form the ratio as a function of Large difference btw saturation scales: Qs(p,y) << Qs(A,y)

15. High pt suppression (II) General arguments One can show in the linear regime (both p and A) within the saddle point approximation that the ratio is …. 1) a decreasing function of rapidity 2) an increasing function of kt 3) a decreasing function of A where c is the BFKL kernel in the Mellin space and g is the saddle point. c(gp) > c(gA) : proton evolves faster than nucleus. Proton: far from saturation, fast evolution Nucleus: already close to saturation, slow evolution saturation DLA

16. Flattening of the Cronin peak Compare MV(A,evolved)/DLA(p) (black) and MV(A,unevolved)/DLA(p) (red). Suppression is due to proton’s faster evolution, while the shape change, the flattening, is due to nucleus evolution

17. Running coupling

18. Fixed coupling Running coupling Running coupling MV model Running coupling Unintegrated gluon distribution Integration analytically doable Essentially the same results for the Cronin effect

19. Quantum evolution with running coupling Solution in linear regime with absorptive boundary  Airy function scaling regime is narrower than the BFKL regime Saturation scale A dependence goes away at high energy - Difference btw p and A becomes less important - Reach at the universal value at yinfty But of course it approaches to 1 at kt  infty

20. Summary Developed a detailed analysis of the ratio -Cronin effect (from MV model) is due to saturation part of the distribution. Its height and position are computable. -Mismatch between p’s and A’s evolutions leads to High pt suppression (p is faster than A), and flattening of the Cronin peak (due to A’s evolution). - Running coupling case formulated Nuclear A dependence of saturation scale goes away.  the ratio approaches to universal value