QCD Phenomenology and Heavy Ion Physics Yuri Kovchegov

QCD Phenomenology and Heavy Ion Physics Yuri Kovchegov The Ohio State University

Outline We’ll describe application of Saturation/Color Glass Condensate physics to Heavy Ion Collisions, concentrating on: n Multiplicity vs. Centrality and vs. Energy, d. N/dη vs. rapidity η n Hadron production in p(d)A collisions: going from mid- to forward rapidity at RHIC, transition from Cronin enhancement to suppression. n Two-particle correlations, back-to-back jets.

Multiplicity

Particle Multiplicity In Saturation/Color Glass Physics one has Ø only one scale in the problem – the saturation scale QS. Ø the leading fields are classical: The resulting gluon multiplicity is given by such that since d 2 b ~ S ~ p R 2 , with R the nuclear radius.

Particle Multiplicity vs. Centrality Since and we get : which is not a constant due to running of the coupling: Thus

Particle Multiplicity vs. Centrality This simple reasoning allowed D. Kharzeev and E. Levin to fit multiplicity as a function of centrality. (from nucl-th/0108006)

Particle Multiplicity vs. Energy Let’s try to use the same simple formula to check the energy dependence of multiplicity. Start with From saturation models of HERA DIS data we know that with Therefore we write obtaining Kharzeev, Levin ‘ 01

Particle Multiplicity vs. Energy Using the known multiplicity at 130 Ge. V Kharzeev and Levin predicted multiplicity at 200 Ge. V using the above model: The result agreed nicely with the data: (PHOBOS) Ø Energy dependence works too!

d. N/dη To understand the rapidity dependence one has to make a few more steps. Starting with factorization assumption inspired by the production diagram, and assuming a saturation/CGC form of the unintegrated gluon distribution f:

d. N/dη Kharzeev and Levin obtained a successfull fit of the pseudo-rapidity distribution of charged particles in AA: The value of the saturation scale turned out to be (see also Kharzeev & Nardi ’ 00, Kharzeev, Levin, Nardi ’ 01)

d. N/dη in d. Au The same approach works for pseudo-rapidity distribution of total charged multiplicity in d. Au collisions: (from Kharzeev, Levin, Nardi, hep-ph/0212316)

Thermalization: Bottom-Up Scenario Baier, Mueller, Schiff, Son ‘ 00 ØIncludes 2 → 3 and 3 → 2 rescattering processes with the LPM effect due to interactions with CGC medium. ØDoes not introduce any new scale, one still has QS only, with ØCan fit the multiplicity data assuming that less particles were produced initially (smaller QS) but their numbers increased during thermalization. Baier, Mueller, Schiff, Son ‘ 02

Bottom-Up Scenario: Questions Ø Instabilities!!! Evolution of the system may develop instabilities. (Mrowczynski, Arnold, Lenaghan, Moore, Romatschke, Strickland, Yaffe) However, it is not clear whether instabilities would speed up thermalization process and how to interpret them diagrammatically. ØAnother problem is that since and It appears that Stronger than classical field? Stronger than any QCD gluon field?

Hadron Spectra Let’s consider gluon production, it will have all the essential features, and quark production could be done by analogy.

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

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

Gluon Production in p. A: Mc. Lerran-Venugopalan model Classical gluon production: we need to resum only the multiple rescatterings of the gluon on nucleons. Here’s one of the graphs considered. Yu. K. , A. H. Mueller, hep-ph/9802440 Resulting inclusive gluon production cross section is given by With the gluon-gluon dipole-nucleus forward scattering amplitude

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

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

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

Including Quantum Evolution To understand the energy dependence of particle production in p. A one needs to include quantum evolution resumming graphs like this one. This resums powers of a ln 1/x = a Y. This has been done in Yu. K. , K. Tuchin, hep-ph/0111362. The rules accomplishing the inclusion of quantum corrections are Proton’s BFKL and LO wave function where the dipole-nucleus amplitude N is to be found from (Balitsky, Yu. K. )

Including Quantum Evolution Amazingly enough, gluon production cross section reduces to k. T –factorization expression (Yu. K. , Tuchin, ‘ 01): with the proton and nucleus “unintegrated distributions” defined by with NGp, A the amplitude of a GG dipole on a p or A.

Our Prediction Our analysis shows that as energy/rapidity increases the height of the Cronin peak decreases. Cronin maximum gets progressively lower and eventually disappears. • Corresponding Rp. A levels off at roughly at Rp. A Toy Model! energy / rapidity increases (Kharzeev, Levin, Mc. Lerran, ’ 02) 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 v 2. ) k / QS At high energy / rapidity Rp. A at the Cronin peak becomes a decreasing function of both energy and centrality.

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

Rd. Au at different rapidities Rd. Au RCP – central to peripheral ratio Most recent data from BRAHMS Collaboration nucl-ex/0403005 Our prediction of suppression was confirmed!

Our Model Rd. Au RCP p. T from D. Kharzeev, Yu. K. , K. Tuchin, hep-ph/0405045, where we construct a model based on above physics + add valence quark contribution

Our Model We can even make a prediction for LHC: Dashed line is for mid-rapidity p. A run at LHC, the solid line is for h=3. 2 d. Au at RHIC. Rd(p)Au p. T from D. Kharzeev, Yu. K. , K. Tuchin, hep-ph/0405045

Two-Particle Correlations

Back-to-back Correlations Saturation and small-x evolution effects may also deplete back-to-back correlations of jets. Kharzeev, Levin and Mc. Lerran came up with the model shown below (see also Yu. K. , Tuchin ’ 02) : which leads to suppression of B 2 B jets at mid-rapidity d. Au (vs pp):

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

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: The evolution between the jets makes the correlations disappear: from Kharzeev, Levin, Mc. Lerran, hep-ph/0403271

Back-to-back Correlations Ø Disappearance of back-to-back correlations in d. Au collisions predicted by KLM seems to be observed in preliminary STAR data. (from the contribution of Ogawa to DIS 2004 proceedings)

Back-to-back Correlations Ø The observed data shows much less correlations for d. Au than predicted by models like HIJING:

Back-to-back Correlations Ø However, KLM calculations are just a model. An exact calculation of two-particle inclusive cross section in p(d)+A (or DIS) has been performed in J. Jalilian-Marian and Yu. K. , ’ 04. Ø The resulting expression for the cross section is so horrible that no sane person would show it in a talk. It won’t fit in the Power. Point format anyway. Nevertheless it exists and can be used to make numerical predictions, though after a lot of work. (One has to solve 6 integral equations to get the answer. )

Conclusions • Particle multiplicity in Au. Au and d. Au collisions varies as a function of energy, centrality and rapidity in apparent agreement with saturation/CGC predictions. • New RHIC d. Au data at forward rapidity seem to confirm expectations of Saturation / CGC physics: at mid-rapidity we see Cronin enhancement, while at forward rapidity we see suppression arising from the small-x evolution. • Back-to-back correlations seem to disappear in a certain transverse momentum region in d. Au, in agreement with preliminary CGC expectations. • Implications for AA collisions need to be understood.

Backup Slides

Extended Geometric Scaling A general solution to BFKL equation can be written as where It turns out that the full solution of nonlinear evolution equation N(z, y) is a function of a single variable, N=N(z QS(y)), with (geometric scaling): (i) Inside the saturation region, , where nonlinear (ii) evolution dominates (Levin, Tuchin ‘ 99 ) (ii) In the extended geometric scaling region, where g≈1/2: (Iancu, Itakura, Mc. Lerran ‘ 02)

Geometric Scaling in DIS Geometric scaling has been observed in DIS data by Stasto, Golec-Biernat, Kwiecinski in ’ 00. Here they plot the total DIS cross section, which is a function of 2 variables - Q 2 and x, as a function of just one variable:

“Phase Diagram” of High Energy QCD III II QS I High Energy or Rapidity kgeom = QS 2 / QS 0 QS Cronin effect and low-p. T suppression Moderate Energy or Rapidity p T 2

Region I: Double Logarithmic Approximation At very high momenta, p. T >> kgeom , the gluon production is given by the double logarithmic approximation, resumming powers of Resulting produced particle multiplicity scales as with where y=ln(1/x) is rapidity and QS 0 ~ A 1/6 is the saturation scale of Mc. Lerran-Venugopalan model. For pp collisions QS 0 is replaced by L leading to Kharzeev Yu. K. Tuchin ‘ 03 as QS 0 >> L. Rp. A < 1 in Region I There is suppression in DLA region!

Region II: Anomalous Dimension At somewhat lower but still large momenta, QS < k. T < kgeom , the BFKL evolution introduces anomalous dimension for gluon distributions: with BFKL g=1/2 (DLA g=1) The resulting gluon production cross section scales as (we loose one power of QS) such that Kharzeev, Levin, Mc. Lerran, hep-ph/0210332 For large enough nucleus Rp. A << 1 – high p. T suppression! How does energy dependence come into the game?

Region II: Anomalous Dimension A more detailed analysis gives the following ratio in the extended geometric scaling region – our region II: Rp. A is also a decreasing function of energy, leveling off to a constant Rp. A ~ A-1/6 at very high energy. Rp. A is a decreasing function of both energy and centrality at high energy / rapidity. (D. Kharzeev, Yu. K. , K. Tuchin, hep-ph/0307037)

Region III: What Happens to Cronin Peak? ü The position of Cronin peak is given by saturation scale QS , such that the height of the peak is given by Rp. A (k. T = QS (y), y). ü It appears that to find out what happens to Cronin maximum we need to know the gluon distribution function of the nucleus at the saturation scale – f. A (k. T = QS, y). For that we would have to solve nonlinear BK evolution equation – a very difficult task. ü Instead we can use the scaling property of the solution of BK equation Levin, Tuchin ’ 99 Iancu, Itakura, Mc. Lerran, ‘ 02 which leads to We do not need to know f. A to determine how Cronin peak scales with energy and centrality! (The constant carries no dynamical information. )