1. Kazunori Itakura IPNS, KEK At the RBRC Alumni workshop 19 th June, 2006 Some topics about Color Glass Condensate

2. Plan Introduction Theoretical framework for the CGC Mean-field picture -- The Balitsky-Kovchegov equation -- Reaction-diffusion dynamics (F-KPP equation) Beyond mean-field picture -- Reaction-diffusion dynamics: exact implementation -- Fluctuation & Stochastic F-KPP equation -- Back to the saturation physics Summary K. Itakura (KEK)June 19 th, 2006 at BNL

3. Introduction higher energy Dilute gas Color Glass Condensate (CGC)!! High-energy limit of QCD is the Color Glass Condensate (CGC)!! Named by Iancu, Leonidov, & McLerran (2000) Color : Color : A matter made of gluons with colors. Glass: Glass: Almost “frozen” random color source creates gluon fields Condensate: Condensate: High density. Occupation number ~ O ( 1 /  s ) Experimental supports: Enhancement of gluon distribution in a proton at small x Geometric scaling at DIS and eA Suppression of R dAu at forward rapidity in deuteron-Au collisions, etc,... CGC: high density gluons K. Itakura (KEK)June 19 th, 2006 at BNL

4. Theoretical framework for the CGC K. Itakura (KEK)June 19 th, 2006 at BNL

5. Large x partons (mostly quarks) Large longitudinal mom. p +  small LC energy E LC  long life time ~ 1/ E LC Small x partons (mostly gluons) Small longitudinal mom. p +  large LC energy E LC  short life time ~ 1/ E LC Small-x gluons can be treated as classical radiation field created by static random color source on the transverse plane. need average over random color source   Stochastic Yang-Mills equation  weight function W x [  ] Theoretical framework for the CGC (1/2) Effective theory of a fast moving hadron McLerran-Venugopalan picture K. Itakura (KEK)June 19 th, 2006 at BNL

6. Theoretical framework for the CGC (2/2) The JIMWLK equation The JIMWLK equation (Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner) rapidity Renormalization group equation 0 1 x0x0  AA x1x1  AA Integrate Higher energy Wx0[]Wx0[] Wx1[]Wx1[] weight function for random source Only weight func. gets modified at ln (1/x) accuracy  RG for W x 0 > x 1 x K. Itakura (KEK)June 19 th, 2006 at BNL

7. Mean-field picture K. Itakura (KEK)June 19 th, 2006 at BNL

8. Mean-field picture (1/4) Gluon number ~ 2pt function of Wilson lines Evolution equation for 2pt operator contains 4pt function  Mean-field approx. : necessary to obtain a closed equation The Balitsky-Kovchegov equationThe Balitsky-Kovchegov equation random gauge field K. Itakura (KEK)June 19 th, 2006 at BNL

9. Mean-field picture (2/4) BFKL + non-linear term  saturates (unitarizes) at fixed b = (x+y)/2 : Saturation scale Q s (Y )  typical size of gluons when area is occupied: 1/ Q s (Y )  increases with rapidity Y : Q s 2 (Y ) ~ e Y Geometric Scaling  amplitude is a function of (x-y)Q s (Y ) Approximate scaling persists even outside of the CGC regime The Balitsky-Kovchegov equation Consequences K. Itakura (KEK)June 19 th, 2006 at BNL

10. Mean-field picture (3/4) Emerging picture Non-perturbative (Regge) 1/x in log scale Q 2 in log scale Parton gas Extended scaling regime CGC Higher energies  Fine transverse resolution  BFKL BK DGLAP  QCD 2 Q S 2 (x) ~ 1/x : grows as x  0 Q S 4 (x)/  QCD 2 K. Itakura (KEK)June 19 th, 2006 at BNL

11. Mean-field picture (4/4) The Reaction-Diffusion dynamics Munier & Peschanski (2003~) Within a reasonable approximation, the BK equation in momentum space is rewritten as the F-KPP equation (Fisher, Kolmogorov, Petrovsky, Piscounov) where t ~  x ~ ln k 2 and u(t, x) ~ N  (k). FKPP = “logistic” + “diffusion” Logistic : “reaction” part g  gg (increase) vs gg  g (recombination) rapid increase saturation Diffusion : expansion of stable region Traveling wave solution Wave front : x(t)=vt  saturation scale Translating solution: u(x-vt)  geometric scaling u=1 : stable u=0 :unstable tt’ > t K. Itakura (KEK)June 19 th, 2006 at BNL

12. Beyond mean-field picture K. Itakura (KEK)June 19 th, 2006 at BNL

13. 2. Fluctuation (discreteness) becomes important when the number of particles are few.  At the tail of a traveling wave : u(x,t) ~ 1/N << 1  large effect: Diffusion controls the propagation. The velocity of a traveling wave is reduced. (Linear growth does not work without “seeds”) 1. The FKPP equation is not complete: It is for an averaged number density in the continuum limit and is valid when allowed number of particles N is quite large. Beyond mean-field picture (1/5) K. Itakura (KEK)June 19 th, 2006 at BNL Lessons from the reaction-diffusion dynamics

14. Beyond mean-field picture (2/5) Reaction-diffusion dynamics: exact implementation K. Itakura (KEK)June 19 th, 2006 at BNL Master equation n i (t) = number of particles at site i, {n i } = ( n 0, n 1, n 2 …..) configuration P({n i }; t ) : probability to have a particular configuration {n i } at time t Change of P({n i }; t ) 1) Diffusion (hopping to the right i  i+1, and to the left i  i –1 ) at rate D/h 2 2) Splitting (A  AA) at each site i at rate s 3) Merging (AA  A) at each site i at rate m  loss gain h loss

15. Beyond mean-field picture (3/5) Field-theory representation K. Itakura (KEK)June 19 th, 2006 at BNL Second quantization method Doi ’76, Peliti ’85, Cardy & Tauber ’98 a useful technique for a system with creation and annihilation of particles Introduce bosonic operators a i + creates a particle at site i. [a i, a j + ] =  ij [a i, a j ] = [a i +, a j + ] = 0 “Hamiltonian” is defined for a “probability vector” For A  AA and AA  A, the master equation is exactly reproduced by Can construct a “field theory” by using coherent state path-integral where  is a coherent state eigen-value of annihilation operator, g s = s, g m = m h/2 cf) Reggeon field theory (but diffusion is in the impact parameter space)

16. Beyond mean-field picture (4/5) Fluctuation & stochastic F-KPP equation K. Itakura (KEK)June 19 th, 2006 at BNL 1.Introduce a Gaussian noise  ( x,t ) as an auxiliary field to resolve the last “fluctuation” term  stochastic F-KPP eq. (reduces to FKPP in the limit N =g s /g m  infinity) 3.Systematic analysis of fluctuation possible  Front position as a collective coordinate for the F-KPP solution (work in progress) 2. Can perform functional integral for the conjugate field (  ) Mean-field limit   (K-V) : F-KPP equation At the tail V ~ 0 (  ~ 0)   (K) : Diffusion eq. F-KPP equation with a cutoff  large effect _

17. Beyond mean-field picture (5/5) Back to saturation physics K. Itakura (KEK)June 19 th, 2006 at BNL 1.Difference btw Balitsky and BK eqs. becomes significant when gluon (dipole) number is small (high transverse momentum). 2.Inclusion of full fluctuation replaces the F-KPP equation by the stochastic F-KPP equation.  Even the Balitsky equation must be modified so that it contains dipole splitting.  Pomeron loop 3. Saturation scale becomes slowly increasing due to diffusion at the edge, stochastic variable due to fluctuation term in sFKPP eq. By Mueller, Shoshi, Iancu, Munier, Tryantafyllopoulos, Soyez, …..

18. Summary Theoretical description of the Color Glass Condensate is improving. Analogy with the reaction-diffusion dynamics is very useful. In particular, the effects of fluctuation beyond the mean- field BK picture have been recognized to be significant in dilute regime (at high transverse momentum) Slowly-growing and stochastic saturation scale is obtained. Deeper understanding of the reaction-diffusion system is necessary. The field-theory representation will be useful. K. Itakura (KEK)June 19 th, 2006 at BNL

19. Thanks Stayed in BNL from June 2000 to May 2003. The first collision at RHIC was just after I started. Naturally (blindly?) entered CGC physics guided by Larry. Spent a great time with nice people and good physics. Shared a great exciting moment with many experimental discoveries: geometric scaling, high pt suppression of R AA and R dA, etc, etc… BNL and RBRC are keeping the position as the center for the CGC physics in the world, still involving some younger Japanese people (Y. Hatta, K. Fukushima, and more?) Many thanks to all the people in BNL during and after my stay. Always feel at home when I visit here. Hope that the activity in RBRC continue in the future! K. Itakura (KEK)June 19 th, 2006 at BNL