Lecture II

3. Growth of the gluon distribution and unitarity violation

Solution to the BFKL equation Coordinate space representation  = ln 1/x Mellin transform + saddle point approximation Asymptotic solution at high energy dominant energy dep. is given by exp{  S  }  = 4 ln 2 =2.8 

Can BFKL explain the rise of F 2 ?

Actually, exponent is too large  = 4ln2 = 2.8  NLO analysis necessary!  But! The NLO correction is too large and the exponent becomes NEGATIVE!  Resummation tried  Marginally consistent with the data (but power behavior always has a problem cf: soft Pomeron) exponent

High energy behavior of the hadronic cross sections – Froissart bound Intuitive derivation of the Froissart bound ( by Heisenberg) BFKL solution violates the unitarity bound. Total energy Saturation is implicit

4. Color Glass Condensate

Saturation & Quantum Evolution - overview dilute Low energy BFKL eq. [Balitsky, Fadin,Kraev,Lipatov ‘78] N : scattering amp. ~ gluon number  : rapidity  = ln 1/x ~ ln s exponential growth of gluon number  violation of unitarity High energy dense, saturated, random Balitsky-Kovchegov eq. Gluon recombination  nonlinearity  saturation, unitarization, universality [Balitsky ‘96, Kovchegov ’99]

Population growth  Solution population explosion N : polulation density T.R.Malthus (1798) Growth rate is proportional to the population at that time. P.F.Verhulst (1838) Growth constant  decreases as N increases. (due to lack of food, limit of area, etc) 1. Exp-growth is tamed by nonlinear term  saturation !! (balanced) 2. Initial condition dependence disappears at late time dN/dt =0  universal ! 3. In QCD, N 2 is from the gluon recombination gg  g. Logistic equation linear regime non-linear exp growth saturation universal  Time (energy) -- ignoring transverse dynamics --

McLerran-Venugopalan model (Primitive) Effective theory of saturated gluons with high occupation number (sometimes called classical saturation model) Separation of degrees of freedom in a fast moving hadron Large x partons slowly moving in transverse plane  random source,  Gaussian weight function Small x partons classical gluon field induced by the source LC gauge (A + =0) Effective at fixed x, no energy dependence in  Result is the same as independent multiple interactions (Glauber).

Color Glass Condensate Color : gluons have “color” in QCD. Glass : the small x gluons are created by slowly moving valence-like partons (with large x ) which are distributed randomly over the transverse plane  almost frozen over the natural time scale of scattering This is very similar to the spin glass, where the spins are distributed randomly, and moves very slowly. Condensate: It’s a dense matter of gluons. Coherent state with high occupancy (~1/  s at saturation). Can be better described as a field rather than as a point particle.

CGC as quantum evolution of MV Include quantum evolution wrt  = ln 1/x into MV model - Higher energy  new distribution W  [  ] - Renormalization group equation is a linear functional differential equation for W  [  ], but nonlinear wrt . - Reproduces the Balitsky equation - Can be formulated for  (x) (gauge field) through the Yang-Mills eq. [D, F  ] =    x T        x T  x T    is a covariant gauge source   JIMWLK equation 

JIMWLK equation Evolution equation for W  [  ], wrt rapidity  = ln 1/x Wilson line in the adjoint representation  gluon propagator Evolution equation for an operator O

JIMWLK eq. as Fokker-Planck eq. The probability density P(x,t) to find a stochastic particle at point x at time t obeys the Fokker-Planck equation D is the diffusion coefficient, and F i (x) is the external force. When F i (x) =0, the equation is just a diffusion equation and its solution is given by the Gaussian: JIMWLK eq. has a similar expression, but in a functional form Gaussian (MV model) is a solution when the second term is absent.

DIS at small x : dipole formalism Life time of qq fluctuation is very long >> proton size This is a bare dipole (onium). _ 1/ M p x  1/(E qq -E  * )  Dipole factorization

DIS at small x : dipole formalism N: Scattering amplitude

S-matrix in DIS at small x Dipole-CGC scattering in eikonal approximation scattering of a dipole in one gauge configuration Quark propagation in a background gauge field average over the random gauge field should be taken in the weak field limit, this gives gluon distribution ~ (  (x)-  (y)) 2 stay at the same transverse positions

The Balitsky equation Take O=tr(V x + V y ) as the operator V x + is in the fundamental representation The Balitsky equation -- Originally derived by Balitsky (shock wave approximation in QCD) ’96 -- Two point function is coupled to 4 point function (product of 2pt fnc) Evolution of 4 pt fnc includes 6 pt fnc. -- In general, CGC generates infinite series of evolution equations. The Balitsky equation is the first lowest equation of this hierarchy.

The Balitsky-Kovchegov equation (I) The Balitsky equation The Balitsky-Kovchegov equation A closed equation for First derived by Kovchegov (99) by the independent multiple interaction Balitsky eq.  Balitsky-Kovchegov eq. (large Nc? Large A) N  (x,y) = 1 - (Nc -1 )  is the scattering amplitude

The Balitsky-Kovchegov equation (II) Evolution eq. for the onium (color dipole) scattering amplitude - evolution under the change of scattering energy s (not Q 2 ) resummation of (  s ln s) n  necessary at high energy - nonlinear differential equation resummation of strong gluonic field of the target - in the weak field limit reproduces the BFKL equation (linear) scattering amplitude becomes proportional to unintegrated gluon density of target  = ln 1/x ~ ln s is the rapidity

R Saturation scale - Boundary between CGC and non-saturated regimes - Similarity between HERA (x~10 -4, A=1) and RHIC (x~10 -2, A=200) Q S (HERA) ~ Q S (RHIC) - Energy and nuclear A dependences LO BFKL NLO BFKL [Gribov,Levin,Ryskin 83, Mueller 99,Iancu,Itakura,McLerran’02] [Triantafyllopoulos, ’03] A dependence is modified in running coupling case [Al Mueller ’03] 1/Q S (x) : transverse size of gluons when the transverse plane of a hadron/nucleus is filled by gluons

Geometric scaling Geometric scaling persists even outside of CGC!!  “Scaling window” [Iancu,Itakura,McLerran,’02] DIS cross section  x,Q) depends only on Qs(x)/Q at small x [Stasto,Golec-Biernat,Kwiecinski,’01] Once transverse area is filled with gluons, the only relevant variable is “number of covering times”.  Geometric scaling!! = Qs(x)/Q=1  Natural interpretation in CGC Qs(x)/Q=(1/Q)/(1/Qs) : number of overlapping 1/Q: gluon size times Scaling window = BFKL window consistent with theoretical results Saturation scale from the data

Summary for lecture II BFKL gives increasing gluon density at high energy, which however contradicts with the unitarity bound. CGC is an effective theory of QCD at high energy – describes evolution of the system under the change of energy -- very nonlinear (due to ) -- derives a nonlinear evolution equation for 2 point function which corresponds to the unintegrated gluon distribution in the weak field limit Geometric scaling can be naturally understood within CGC framework.