1. Small-x Theory Overview Yuri Kovchegov The Ohio State University Columbus, OH

2. Outline A brief review of Saturation/Color Glass Physics A brief review of Saturation/Color Glass Physics Hadron production in p(d)A collisions: going from mid- to forward rapidity at RHIC, transition from Cronin enhancement to suppression. Hadron production in p(d)A collisions: going from mid- to forward rapidity at RHIC, transition from Cronin enhancement to suppression. Open questions: what needs to be measured? Open questions: what needs to be measured?  Di-leptons in p(d)A.  Back-to-back jet correlations.  Is suppression tied to fragmentation region?  Open charm production in p(d)A.

3. Imagine an UR nucleus or hadron with valence quarks and sea gluons and quarks. Define Bjorken (Feynman) x as Nuclear/Hadronic Wave Function p k

4. Rest frame of the hadron/nucleus Longitudinal coherence length of a gluon is such that for small enough x Bj we get with R the nuclear radius. (e.g. for x=10 -3 get l coh =100 fm)

5. Color Charge Density Small-x gluon “sees” the whole nucleus coherently in the longitudinal direction! It “sees” many color charges which form a net effective color charge Q = g (# charges) 1/2, such that Q 2 = g 2 #charges (random walk). Define color charge density such that for a large nucleus (A>>1) Nuclear small-x wave function is perturbative!!! McLerran Venugopalan ’93-’94

6. McLerran-Venugopalan Model  The density of partons in the nucleus (number of partons per unit transverse area) is given by the scale  2 ~ A  R 2   This scale is large,  >> Λ QCD, so that the strong coupling constant is small, α S (  ) << 1.  Leading gluon field is classical! To find the classical gluon field A μ of the nucleus one has to solve the Yang-Mills equations, with the nucleus as a source of color charge: Yu. K. ’96 J. Jalilian-Marian et al, ‘96

7. Shadowing Ratio Defining the shadowing ratio for the unintegrated gluon distributions we plot it for the distribution found Shadowing! (small k  small x) Enhancement (anti-shadowing), most glue are here  Gluons are redistributed from low to high p T.

8. DIS in the Classical Approximation The DIS process in the rest frame of the target is shown below. It factorizes into with rapidity Y=ln(1/x)

9. DIS in the Classical Approximation The dipole-nucleus amplitude in the classical approximation is A.H. Mueller, ‘90 1/Q S Color transparency Black disk limit,

10. Quantum Evolution As energy increases the higher Fock states including gluons on top of the quark-antiquark pair become important. They generate a cascade of gluons. These extra gluons bring in powers of  S ln s, such that when  S >1 this parameter is  S ln s ~ 1.

11. BFKL Equation The BFKL equation for the number of partons N reads: Balitsky, Fadin, Kuraev, Lipatov ‘78 Start with N particles in the proton’s wave function. As we increase the energy a new particle can be emitted by either one of the N particles. The number of newly emitted particles is proportional to N.

12. Nonlinear Equation Yu. K. ’99 (large N C QCD) I. Balitsky ’96 (effective lagrangian) At very high energy parton recombination becomes important. Partons not only split into more partons, but also recombine. Recombination reduces the number of partons in the wave function. Number of parton pairs ~

13. Nonlinear Equation: Saturation Gluon recombination tries to reduce the number of gluons in the wave function. At very high energy recombination begins to compensate gluon splitting. Gluon density reaches a limit and does not grow anymore. So do total DIS cross sections. Unitarity is restored!

14. Nonlinear Evolution at Work First partons are produced First partons are produced overlapping each other, all of them about the same size. When some critical density is When some critical density is reached no more partons of given size can fit in the wave function. The proton starts producing smaller partons to fit them in. The story repeats itself for smaller The story repeats itself for smaller partons. The picture is similar to Fermi statistics.This way some critical density is never exceeded. Color Glass Condensate Proton

15. Phase Diagram of High Energy QCD Saturation physics allows us to study regions of high parton density in the small coupling regime, where calculations are still under control! We are making real progress in understanding proton/nuclear structure! QSQSQSQS Transition to saturation region is characterized by the saturation scale (or p T 2 )

16. What Happens to Gluon Distributions? With the onset of evolution, as energy/rapidity increases, the shadowing ratio starts to decrease: Shadowing Anti-shadowing Suppression! A new phenomenon. (Toy model picture)

17. What Happens to Gluon Distributions? At very high energy “anti-shadowing” disappears: Suppression everywhere! (Toy model picture)

18. Why Suppression? Kharzeev, Levin, McLerran, hep-ph/0210332 In general one can write: Without quantum evolution  =1 and there is no suppression: Quantum evolution leads to  =1/2, such that leading to

19. Hadron Production Let’s start with gluon production, it will have all the essential features, and quark production will become clear after that.

20. How to Calculate Observables Start by finding the classical field of the McLerran-Venugopalan model. Start by finding the classical field of the McLerran-Venugopalan model. Continue by including the quantum corrections of the nonlinear evolution equation. Continue by including the quantum corrections of the nonlinear evolution equation. Works for structure functions of hadrons and nuclei, as well as for gluon production in various hadronic collisions. Let us consider pA collisions first. Works for structure functions of hadrons and nuclei, as well as for gluon production in various hadronic collisions. Let us consider pA collisions first.

21. To find the gluon production cross section in pA one has to solve the same classical Yang-Mills equations for two sources – proton and nucleus. Gluon Production in Proton-Nucleus Collisions (pA): Classical Field This classical field has been found by Yu. K., A.H. Mueller in ‘98

22. Gluon Production in pA: McLerran-Venugopalan model Yu. K., A.H. Mueller, hep-ph/9802440 The diagrams one has to resum are shown here: they resum powers of

23. McLerran-Venugopalan model: Cronin Effect To understand how the gluon production in pA is different from independent superposition of A proton-proton (pp) collisions one constructs the quantity We can plot it for the quasi-classical cross section calculated before (Y.K., A. M. ‘98): (see also Kopeliovich et al, ’02; Baier et al, ’03; Accardi and Gyulassy, ‘03) Enhancement (Cronin Effect) Classical gluon production leads to Cronin effect! Classical gluon production leads to Cronin effect! Nucleus pushes gluons to higher transverse momentum! Kharzeev Yu. K. Tuchin ‘03

24. Understanding Cronin Effect It is easier to understand this enhancement if we note that, amazingly, one can rewrite gluon production cross section in a k T –factorized form in terms of gluon distributions: Since for nuclear unintegrated gluon distribution we had enhancement at high p T, it leads to enhancement in R pA.

25. Proof of Cronin Effect  Plotting a curve is not a proof of Cronin effect: one has to trust the plotting routine.  To prove that Cronin effect actually does take place one has to study the behavior of R pA at large k T (cf. Dumitru, Gelis, Jalilian-Marian, quark production, ’02-’03): R pA approaches 1 from above at high p T  there is an enhancement! Note the sign!

26. Cronin Effect Cronin Effect  The height and position of the Cronin maximum are increasing functions of centrality (A)! The position of the Cronin maximum is given by k T ~ Q S ~ A 1/6 as Q S 2 ~ A 1/3. Using the formula above we see that the height of the Cronin peak is R pA (k T =Q S ) ~ ln Q S ~ ln A.

27. Including Quantum Evolution To understand the energy dependence of particle production in pA one needs to include quantum evolution resumming graphs like this one. It resums powers of  ln 1/x =  Y. (Yu. K., K. Tuchin, ’01)

28. Including Quantum Evolution Amazingly enough, gluon production cross section reduces to k T –factorization expression (Yu. K. Tuchin, ‘01):

29. Our Prediction Toy Model! Our analysis shows that as energy/rapidity increases the height of the Cronin peak decreases. Cronin maximum gets progressively lower and eventually disappears. Corresponding R pA levels off at roughly at D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0307037; (see also numerical simulations by Albacete, Armesto, Kovner, Salgado, Wiedemann, hep-ph/0307179 and Baier, Kovner, Wiedemann hep-ph/0305265 v2.)  At high energy / rapidity R pA at the Cronin peak becomes a decreasing function of both energy and centrality. R pA k / Q S energy / rapidity increases (Kharzeev, Levin, McLerran, ’02)

30. Understanding Suppression Again, using the k T -factorization formula And remembering that the unintegrated nuclear gluon distribution at high (forward) rapidity y is suppressed at all p T, we conclude that R pA is also suppressed.

31. Other Predictions Color Glass Condensate / Saturation physics predictions are in sharp contrast with other models. The prediction presented here uses a Glauber-like model for dipole amplitude with energy dependence in the exponent. figure from I. Vitev, nucl-th/0302002, see also a review by M. Gyulassy, I. Vitev, X.-N. Wang, B.-W. Zhang, nucl-th/0302077

32. R dAu at different rapidities Most recent data from BRAHMS Collaboration nucl-ex/0403005 R dAu R CP – central to peripheral ratio Our prediction of suppression was confirmed!

33. Our Model from D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0405045, where we construct a model based on above physics + add valence quark contribution R dAu pTpT pTpT R CP

34. Our Model from D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0405045 We can even make a prediction for LHC: R d(p)Au pTpT Dashed line is for mid-rapidity pA run at LHC, the solid line is for  dAu at RHIC.

35. Future Experimental Tests

36. Di-lepton Production  To calculate hadron production one always needs to convolute quark and gluon production cross sections with the fragmentation functions. Since fragmentation functions are impossible to calculate and are poorly known in general, they introduce a big theoretical uncertainty.  Di-lepton production involves no fragmentation functions. It is, therefore, a much cleaner probe of the collision dynamics.  Theoretical calculation for di-lepton production in dAu is pretty straightforward.

37. Di-lepton Production from J. Jalilian-Marian, hep-ph/0402014  The photon does not interact (while everything else is just like for gluons): theoretical calculations are simpler! They were first performed by Kopeliovich, Schafer, and Tarasov in ’98. M 2 is the photon’s invariant mass, k T and q T are total and relative transverse momenta of the lepton pair.

38. Di-lepton Production from J. Jalilian-Marian, hep-ph/0402014 Again one should get suppression at forward rapidities at RHIC. Here we plot R p(d)A integrated over k T and q T, for both pA and dA, for y=2.2 as a function of M.

39. Di-lepton Production The suppression at forward rapidities at RHIC can also be seen as a function of k T : from Baier, Mueller, Schiff, hep-ph/0403201 R pA as a function of k T for M=2GeV for y=1.5 (short-dashed) and y=3 (dashed), as well as for M=4GeV and y=3 (lower solid line).

40. Di-lepton Production and as a function of centrality: from Baier, Mueller, Schiff, hep-ph/0403201 R pA as a function of A for k T =5 GeV, M=2GeV and y=3 for the saturation model (solid curve) and for analytical estimate R pA ~ A -0.124.

41. Back-to-back Correlations Saturation and small-x evolution effects may also deplete back-to-back correlations of jets. Kharzeev, Levin and McLerran came up with the model shown below (see also Yu.K., Tuchin ’02) : which leads to suppression of B2B jets at mid-rapidity dAu (vs pp):

42. Back-to-back Correlations and at forward rapidity: from Kharzeev, Levin, McLerran, hep-ph/0403271 Warning: only a model, for exact analytical calculations see J. Jalilian-Marian and Yu.K., ’04.

43. Back-to-back Correlations An interesting process to look at is when one jet is at forward rapidity, while the other one is at mid-rapidity: from Kharzeev, Levin, McLerran, hep-ph/0403271 The evolution between the jets makes the correlations disappear:

44. Back-to-back Correlations  Disappearance of back-to-back correlations in dAu collisions predicted by KLM seems to be observed in preliminary STAR data. (from the contribution of Ogawa to DIS2004 proceedings)

45. Back-to-back Correlations  The observed data shows much less correlations for dAu than predicted by models like HIJING:

46. Is Suppression due to Proximity to Fragmentation Region? Kopeliovich et al suggest that observed suppression may not be due to small-x evolution in the nuclear wave function, but is due to large-x effects in the deuteron. To resolve this issue one would ideally want to have d+Au collisions at higher energy, where we could test the same small-x region of the nuclear wave function for which x of the deuteron is not that large anymore – CGC would expect suppression, Kopeliovich no suppression. Alternatively one can run dAu at lower energy: large-x effects would stay the same, nuclear x would increase: CGC would expect less suppression, Kopeliovich would expect the same amount of suppression.

47. Open Charm Production Kharzeev and Tuchin model open charm production in the saturation/Color Glass formalism by the following model: It results in the spectra which fall off slower than PYTHIA: Caution: only a model, for exact calculation see Tuchin ’04 and Blaizot et al, ’04.

48. Open Charm Production Similar suppression applies to heavy flavors, in particular to open charm. The figure below demonstrates that in going from mid-rapidity to forward rapidity open charm production should start scaling slower than N coll, which indicates suppression. from D. Kharzeev and K. Tuchin, hep-ph\0310358 N coll Charmed Meson Yield

49. Summary A lot of interesting new information about saturation/Color Glass dynamics can be found by studying d+Au collisions at RHIC II. A lot of interesting new information about saturation/Color Glass dynamics can be found by studying d+Au collisions at RHIC II. Things to look at are: Things to look at are: R pA of hadrons in forward direction (suppression?). R pA of hadrons in forward direction (suppression?). R pA of open charm and J/y (forward suppression?). R pA of open charm and J/y (forward suppression?). Di-leptons: cleaner signal, expect suppression in forward direction as well. Di-leptons: cleaner signal, expect suppression in forward direction as well. Various back-to-back correlations: broadening, disappearance... Various back-to-back correlations: broadening, disappearance...

50. Summary Understanding nuclear and hadronic wave functions reveals interesting physics of strong gluonic fields and complicated interactions. Understanding nuclear and hadronic wave functions reveals interesting physics of strong gluonic fields and complicated interactions.