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Lecture IV

Lecture IV. Recent progress in phenomenology (cont’d) and theory. 6. Recent progress in phenomenology HERA (Lecture III) RHIC Au-Au RHIC deuteron-Au. RHIC physics: Au-Au collision. C G C at RHIC (Au-Au). C G C at RHIC (Au-Au).

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Lecture IV

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  1. Lecture IV Recent progress in phenomenology (cont’d) and theory

  2. 6. Recent progress in phenomenologyHERA (Lecture III) RHIC Au-AuRHIC deuteron-Au

  3. RHIC physics:Au-Au collision

  4. CGC at RHIC (Au-Au)

  5. CGC at RHIC (Au-Au) Most of the produced particles have small momenta less than 1 GeV ~  Effects of saturationmay be visible in bulk quantitiesQs(RHIC) Multiplicity : pseudo-rapidity & centrality dependences  in good agreement with the data [Kharzeev,Levin,’01]

  6. RHIC physics:deuteron-Au collision

  7. Experimental results BRAHMS data for deuteron-Au 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

  8. Theoretical analyses based on CGC Numerical studies 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. 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

  9. Cronin effect from MV model At mid-rapidity, we assume that we can use the MV model. Simplified, but shows essentially the same behavior as RpA Bremsstrahlung at high kt

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

  11. 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)

  12. 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

  13. More phenomenological analysis Jalilian-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

  14. CGC at LHC Obviously, CGC becomes more important in LHC with higher scattering 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.

  15. Phase diagram with numbers (I) x From the CGC fit Qs2(x)=(10-4/x)0.3 CGC Extended Scaling ~BFKL 10-4 Parton gas HERA 10-2 Q2 100 103

  16. Phase diagram with numbers (II) Extended scaling regime x~10-3 x~10-2 forward rapidity mid-rapidity from Dima Kharzeev’s talk at NSAC Subcommittee on Relativistic Heavy Ions, June 2004

  17. 7. Recent progress in theory Geometric scaling as traveling wave Physics beyond the BK equation -- effects of fluctuation, Pomeron loop -- odderon

  18. Geometric 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

  19. Geometric scaling as traveling wave Very important because FKPP equation has been investigated over the many 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

  20. Physics beyond the BK equation WHY?? 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.

  21. The 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 (t = i e). • n(x) is the fluctuation which is given by white noise and generates random gauge field a(x). • This equation generates everything ! •  evolution equations of arbitrary gluonic operators.

  22. Quadratic w.r.t. fluctuation Linear w.r.t. fluctuation The Balitsky eq. from the Langevin eq. Blaizot, Iancu, Itakura, 04 Diagramatic derivation of the Balitsky equation for tr(U+(x)U(y)) Due to white noise <na(x)>=0, <na(x)nb(y)> ~ dab d(x-y). Quadratic correlation of the fluctuation gives the nonzero result.

  23. 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). This includes evolution for tr(U+xUz) tr(U+zUy) = Sxz Szy dipoles Non-dipoles Non-dipole operators are created by the linear-noise term of the Balitsky eq.

  24. Role of the fluctuation term (II) Non-dipole term represents dipole-dipole interaction ! (Dipole branching gives just the fan diagram of Pomeron.) Dipole branching Dipole branching Dipole-dipole interaction Evolution of non-dipole operators generates dipole operators again (but less number of dipoles).  Eventually generates Pomeron loops !

  25. 1 dipole 2 dipoles 3 dipoles 4 dipoles 2 dipoles + 1 sextupole 3 dipoles + 1 quadrupole 5 dipoles (< 7 dipoles) Dipole branching, Normal evolution Pomeron loops !?

  26. Perturbative QCD Odderon Iancu,Itakura,McLerran,Hatta,in progress • 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?

  27. C-odd operators • Charge conjugation • Fermions mesonic baryonic • Gauge fields any combination of 3 gluons with d-symbol is C-odd.  (+ even, -- odd)  (+ even, -- odd)

  28. Intuitive 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

  29. C-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 scattering amplitude T (S = 1 + iT) • Weak field expansion  leading order is 3 gluons - should be gauge invariant combination

  30. Evolution of the dipole odderon (I) BFKL • Non-linear evolution eq. for the odderon operator can be 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) if 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 !

  31. Evolution of the dipole odderon (II) • The presence of imaginary part (odderon) affects the evolution equation for the scattering amplitude N(x,y). Balitsky equation new contribution!

  32. Open 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

  33. Neutrino Nucleon Cross Sections Contribution of small x partons Figure from Gluck, Kretzer and Reya, Astropart. Phys.11 (1999) 327

  34. Summary -- 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!!!

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