Lecture IVLecture IVRecent progress in

phenomenology (cont’d)

and theory

6. Recent progress in 6. Recent progress in phenomenologyphenomenology

HERA (Lecture III) RHIC Au-Au

RHIC deuteron-Au

RHIC physics:RHIC physics:Au-Au collisionAu-Au collision

CCGGCC at RHIC (Au-Au) at RHIC (Au-Au)

CCGGCC at RHIC (Au-Au) at RHIC (Au-Au)Most of the produced particles have small momenta less than 1 GeV ~

Effects of saturation may be visible in bulk quantities Qs(RHIC)

Multiplicity : pseudo-rapidity & centrality dependences

in good agreement with the data

[Kharzeev,Levin,’01]

RHIC physics:RHIC physics:deuteron-Au collisiondeuteron-Au collision

Experimental resultsExperimental results

Nuclear modification

factor

- Cronin peak at =0, suppression at =3.2

- More enhanced for central at =0, More suppressed for central at =3.2

BRAHMS data for deuteron-Au collisions at RHIC

Theoretical analyses based on CGCTheoretical analyses based on CGC

Numerical studies Balitsky-Kovchegov equation

Albacete, Armesto, Kovner,

Salgado, Wiedemann 03

from top to bottom: =0 to 10

Cronin peak exists at =0, but rapidly disappears after evolution.

For >1, the ratio monotonically increases as a function of pt.

Analytical studies based on the MV model ----- can be used for mid-rapidity (moderate energy)

Cronin peak

based on the BK equation -- can be used for forward rapidity (high energy)

High pT suppression 　　　　　　　　　　　　　　　

　　　　　　　　　　　　　　　　　　　　　　　　　　　　 Iancu, Itakura, Triantafyllopoulos, ’04

Cronin effect from MV modelCronin effect from MV model

Bremsstrahlung at high kt

Simplified, but shows essentially

the same behavior as RpA

At mid-rapidity, we assume that we can use the MV model.

Approximate solutions in each regime

1) CGC

2) BFKL anomalous dimension

absorptive, scaling, scaling violation

3) DLA (double log approximation)

Quantum evolutionQuantum evolution

High pt suppression (I)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)

BFKL(A)

DLA(p)

Y=0.1

Y=0.7

Y=s y

BFKL(A)BFKL(p)

Y=0.75

Y=1.95

Y=s y

High pt suppression (II)High pt suppression (II)One can show in the linear regime (both p and A) within the saddle po

int approximation that the ratio is ….1) a decreasing function of rapidity

2) an increasing function of kt

3) a decreasing function of A

where is the BFKL kernel in the Mellin space

and is the saddle point.

pAproton evolves faster than nucleus. Proton: far from saturation, fast evolution Nucleus: already close to saturation, slow evolution

General arguments

saturation DLA

More phenomenological analysisMore phenomenological analysisJalilian-Marian ’04 used the CGC parametrization data most forward rapidity y=3.2

hadronize via fragmentation function

Kharzeev, Kovchegov & Tuchin ’04 improved at high mom

CCGGCC at LHC at LHCObviously, CGC becomes more important in LHC with higher sc

attering energy.

√sNN = 14 TeV for pp, 5.5 TeV for PbPb

Rough estimate tells the saturation scale at LHC is increased by

a factor of 3 than that of RHIC.

Qs2(LHC) ~ 3 -- 10 GeV2

(mid) (forward)

Number of gluons in the saturation regime increases.

Effects of saturation can be more visible.

Phase diagram with numbers (I)Phase diagram with numbers (I)

100

10-4

x

Q2

10-2

103

From the CGC fitQs2(x)=(10-4/x)0.3

CGC Extended Scaling~BFKL

Parton gas

HERA

Phase diagram with numbers (II)Phase diagram with numbers (II)

from Dima Kharzeev’s talk at NSAC Subcommittee on Relativistic Heavy Ions, June 2004

x~10-2

x~10-3

Extended scaling regime

forward rapidity

mid-rapidity

7. Recent progress in theory7. Recent progress in theoryGeometric scaling as traveling wavePhysics beyond the BK equation -- effects of fluctuation, Pomeron loop -- odderon

Geometric scaling as traveling waveGeometric scaling as traveling wave • Munier & Peschanski ’03

The Balitsky-Kovchegov eq. with reasonable approximation (expansion around BFKL saddle point) is equivalent to the F-KPP equation.(Fisher, Kolmogolov, Petrovsky, and Piscounov)

change of variables

F-KPP equation

Logistic equation + spatial derivative

Geometric scaling as traveling waveGeometric scaling as traveling wave Very important because FKPP equation has been investigated over the man

y years and understood very well.

- This equation allows a traveling wave solution, which connects the unstable (u=0) and stable (u=1) fixed points.

u(x,t) = f (x-vt) : “geometric scaling”

- And the velocity of the wave front corresponds to the saturation scale!!!!!

velocity is essentially determined by the linear part (BFKL)

precise information about the saturation scale available

Physics Physics beyondbeyond the BK equation the BK equationWHY??

1. We have been looking at only the first part of Balitsky’s infinite hierarchy,

and even its simplified version. The Balitsky equation

along the path of quark

Assume Factorization + take large Nc limit Only the dipole operator which is given as the solution to the BK eq. is relevant.

- Balitsky-Kovchegov eq. = physics of independent dipoles

2. How to justify the factorization <NN> <N><N>?

Effects of fluctuation? Dipole-dipole correlations?

3. n-gluon exchange? (n Reggeon dynamics a la BKP or Korchemsky)

4. Role of non-dipole operator?? tr(Ux+UwUy

+UwUz+Uw)

5. Imaginary part of the dipole scattering amplitude?

So far N(x,y) has been always assumed to be real.

The Langevin approach for the CGCThe Langevin approach for the CGC

The Langevin approach is the simplest and most sophisticated method for the CGC. JIMWLK eq. = Fokker-Planck eq.

After one step of evolution, the gauge field which the dipole feels is given by

The index i is the rapidity step ( = i ). (x) is the fluctuation which is given by white noise and generates random gauge

field (x). This equation generates everything ! evolution equations of arbitrary gluonic operators.

The Balitsky eq. from the Langevin eThe Balitsky eq. from the Langevin eq.q.

Diagramatic derivation of the Balitsky equation for tr(U+(x)U(y))

Quadratic w.r.t. fluctuation

Linear w.r.t. fluctuation

Due to white noise <a(x)>=0, <a(x)b(y)> ~ ab (x-y).

Quadratic correlation of the fluctuation gives the nonzero result.

Blaizot, Iancu, Itakura, 04

Role of the fluctuation term (I)Role of the fluctuation term (I)

Before taking the average, the Balitsky equation has a term linear wrt noise.

This vanishes after taking the average, but is important for the evolution equations of dipole operators.

Consider one more step of evolution for a single dipole operator tr(U+xUy). T

his includes evolution for tr(U+xUz) tr(U+

zUy) = Sxz Szy

Non-dipole operators are created by the linear-noise term of the Balitsky eq.

dipoles

Non-dipoles

Role of the fluctuation term (II)Role of the fluctuation term (II)

Non-dipole term represents

dipole-dipole interaction !

Evolution of non-dipole operators generates dipole operators again (but less number of dipoles). Eventually generates Pomeron loops !

Dipole branching

Dipole branching Dipole-dipole interaction

(Dipole branching gives just the fan diagram of Pomeron.)

Pomeron loops !?Pomeron loops !?

1 dipole

2 dipoles

3 dipoles

4 dipoles

2 dipoles + 1 sextupole

3 dipoles + 1 quadrupole

5 dipoles (< 7 dipoles)

Dipole branching,Normal evolution

Perturbative QCD OdderonPerturbative QCD Odderon

• In QCD, the odderon is a three Reggeized gluon exchange which is odd under the charge conjugation

cf) BFKL Pomeron

= 2 gluon exchange, C-even

• What is the relevant operator for the odderon?

- Pomeron = tr(Vx+ Vy) with strong field (saturation)

2 gluon operator {a(x)-a(y)}2 in weak field limit (a(x) is the minus component of the gauge field) - Gauge invariant combination of 3 gluons? How to construct them?

Iancu,Itakura,McLerran,Hatta,in progress

C-odd operatorsC-odd operators• Charge conjugation

• Fermions mesonic

baryonic

• Gauge fields

any combination of 3 gluons with d-symbol is C-odd.

(+ even, -- odd)

(+ even, -- odd)

Intuitive construction of S-matrixIntuitive construction of S-matrix

• Dipole-CGC scattering in eikonal approximation

scattering of a dipole in one gauge configuration

average over the random

gauge field should be taken

stay at the same transverse positions

C-odd S-matrixC-odd S-matrix (dipole-CGC scattering)

• Transition from C-even to C-odd dipole states

• Relevant operator

Odipole(x,y) = tr(Vx+ Vy) – tr(Vy

+ Vx) = 2i Im tr(Vx+ Vy)

- constructed from gauge fields, but has the same symmetry as for the fermionic dipole operator M(x,y)-M(y,x)

anti-symmetric under the exchange of x and y

Odipole(x,y) = - Odipole(y,x)

- imaginary part of the dipole operator tr(Vx+ Vy). Real part of the scatteri

ng amplitude T (S = 1 + iT)

• Weak field expansion leading order is 3 gluons

- should be gauge invariant combination

Evolution of the dipole odderon (I)Evolution of the dipole odderon (I)• Non-linear evolution eq. for the odderon operator can b

e easily obtained from the Balitsky eq. for tr(V+xVy).

- N(x,y) = 1- 1/Nc Re tr(V+xVy) is the usual “scatt. amplitude” (real)

- the whole equation is consistent with the symmetry

Odipole(x,y) = - Odipole(y,x) and N(x,y) = N(y,x) - becomes equivalent to Kovchegov-Symanowsky-Wallon (2004) i

f one assumes factorization <NO> <N><O>. - linear part = the BFKL eq. (but with different initial condition) reproduces the BKP solution with the largest intercept found by Bartels, Lipatov and Vacca (KSW,04) - intercept reduces due to saturation

As N(x,y) 1, Odipole(x,y) becomes decreasing !

BFKL

Evolution of the dipole odderon (IEvolution of the dipole odderon (II)I)

• The presence of imaginary part (odderon) affects the evolution equation for the scattering amplitude N(x,y).

Balitsky equation new contribution!

Open problemsOpen problems• Application to Ultra High Energy Cosmic Ray

ideal play ground for CGC : x ~ 10-9 – 10-10

• Non-equilibrium properties

Langevin equation, Fokker-Planck equation

• Fluctuations (Balitsky eq. vs BK eq. etc )

• Impact parameter dependence

• Phenomenological analysis (RHIC, HERA)

• BKP equation (n-point function)

• Exact solution to the BK equation?

Exact solution found in 2+1 dimensions

Neutrino Nucleon Cross SectionsNeutrino Nucleon Cross SectionsContribution of small x partons

Figure from Gluck, Kretzer and Reya, Astropart. Phys.11 (1999) 327

SummarySummary-- Some of the physics at RHIC are consistent with CGC.

Brahms data on RdA (Cronin effect and high pT suppression)

-- The BK equation is essentially the same as FKPP eq. Geometric scaling corresponds to the traveling wave solution and its velocity is the saturation scale.

-- Interesting and rich physics is there if one looks beyond the Balitsky-Kovchegov equation.

-- Non-dipole operator (sextupole operator with 6 U’s) in the evolution of 2 dipoles appears.

-- This contribution is important since this physically represents the dipole-dipole interaction, and eventually leads to dipole fusion, namely, creates effectively the Pomeron loop.

-- There are still many interesting open problems, and it’s time to join this activity!!!