Some topics about Color Glass Condensate

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

2. K. Itakura (KEK) June 19th, 2006 at BNL 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

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

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

5. 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 sourcer Stochastic Yang-Mills equation  weight functionWx[r] K. Itakura (KEK) June 19th, 2006 at BNL Theoretical framework for the CGC (1/2) Effective theory of a fast moving hadron McLerran-Venugopalan picture Large x partons (mostly quarks) Large longitudinal mom. p+  small LC energy ELC  long life time ~ 1/ ELC Small x partons (mostly gluons) Small longitudinal mom. p+  large LC energy ELC  short life time ~ 1/ ELC

6. Higher energy r Integrate x1 A+ rapidity K. Itakura (KEK) June 19th, 2006 at BNL Theoretical framework for the CGC (2/2) Renormalization group equation 1 r Only weight func. gets modified at ln (1/x) accuracy  RG for W x0 A+ 0 x0 > x1 x Wx0[r] Wx1[r] weight function for random source The JIMWLK equation(Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner)

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

8. K. Itakura (KEK) June 19th, 2006 at BNL 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 equation random gauge field

9. K. Itakura (KEK) June 19th, 2006 at BNL The Balitsky-Kovchegov equation Mean-field picture (2/4) Consequences • BFKL + non-linear term  saturates (unitarizes) at fixed b = (x+y)/2 : • Saturation scale Qs(Y )  typical size of gluons when area is occupied: 1/Qs(Y )  increases with rapidity Y : Qs2(Y ) ~ elY • Geometric Scaling  amplitude is a function of (x-y)Qs(Y ) • Approximate scaling persists even outside of the CGC regime

10. K. Itakura (KEK) June 19th, 2006 at BNL Emerging picture Mean-field picture (3/4) QS2(x) ~ 1/xl: grows as x  0 Extended scaling regime CGC Higher energies  QS4(x)/LQCD2 Non-perturbative (Regge) Parton gas BFKL BK 1/x in log scale DGLAP LQCD2 Fine transverse resolution  Q2 in log scale

11. 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 t t’ > t u=0:unstable K. Itakura (KEK) June 19th, 2006 at BNL The Reaction-Diffusion dynamics Mean-field picture (4/4) 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 ~ t, x ~ ln k2 and u(t, x) ~ Nt(k).

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

13. 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. 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”) K. Itakura (KEK) June 19th, 2006 at BNL Lessons from the reaction-diffusion dynamics Beyond mean-field picture (1/5)

14. K. Itakura (KEK) June 19th, 2006 at BNL Reaction-diffusion dynamics: exact implementation Beyond mean-field picture (2/5) Master equation ni(t) = number of particles at site i , {ni} = (n0 , n1, n2 …..)configuration P({ni}; t ) : probability to have a particular configuration {ni} at time t Change of P({ni}; t ) 1) Diffusion (hopping to the right ii+1, and to the left ii –1) at rate D/h2 2) Splitting (AAA) at each site i at rate ls 3) Merging (AAA) at each site i at rate lm h gain loss gain loss

15. Second quantization methodDoi ’76 , Peliti ’85, Cardy & Tauber ’98 a useful technique for a system with creation and annihilation of particles Introduce bosonic operators ai+creates a particle at site i. [ai , aj+] = dij [ai , aj] = [ai+, aj+] = 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 f is a coherent state eigen-value of annihilation operator, gs=ls, gm=lmh/2 cf) Reggeon field theory (but diffusion is in the impact parameter space) K. Itakura (KEK) June 19th, 2006 at BNL Field-theory representation Beyond mean-field picture (3/5)

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

17. K. Itakura (KEK) June 19th, 2006 at BNL Back to saturation physics Beyond mean-field picture (5/5) • Difference btw Balitsky and BK eqs. becomes significant • when gluon (dipole) number is small (high transverse momentum). • 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. K. Itakura (KEK) June 19th, 2006 at BNL 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.

19. K. Itakura (KEK) June 19th, 2006 at BNL 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 RAA and RdA , 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!