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Lecture II

Lecture II. 3. Growth of the gluon distribution and unitarity violation. Solution to the BFKL equation. Coordinate space representation t = ln 1/x Mellin transform + saddle point approximation Asymptotic solution at high energy dominant energy dep.

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Lecture II

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  1. Lecture II

  2. 3. Growth of the gluon distribution and unitarity violation

  3. Solution to the BFKL equation • Coordinate space representation t = ln 1/x • Mellin transform + saddle point approximation • Asymptotic solution at high energy dominant energy dep. is given by exp{ w aS t } w = 4 ln 2 =2.8 

  4. Can BFKL explain the rise of F2 ?

  5. Can BFKL explain the rise of F2 ? Actually, exponent is too largew = 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

  6. 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

  7. 4. Color Glass Condensate

  8. Low energy BFKL eq.[Balitsky, Fadin,Kraev,Lipatov ‘78] dilute N :scattering amp. ~ gluon number t : rapidity t = ln 1/x ~ ln s exponential growth of gluon number  violation of unitarity [Balitsky ‘96, Kovchegov ’99] Balitsky-Kovchegov eq. dense, saturated, random Gluon recombination  nonlinearity saturation, unitarization, universality High energy Saturation & Quantum Evolution - overview

  9. T.R.Malthus (1798) N:polulation density Growth rate is proportional to the population at that time.  Solutionpopulation explosion P.F.Verhulst (1838) Growth constant k decreases as N increases. (due to lack of food, limit of area, etc) Logistic equation -- ignoring transverse dynamics -- Population growth linear regime non-linear exp growth saturation universal 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, N2 is from the gluon recombination ggg.  Time (energy)

  10. 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 m Result is the same as independent multiple interactions (Glauber).

  11. 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/as at saturation). Can be better described as a field rather than as a point particle.

  12. CGC as quantum evolution of MV Include quantum evolution wrt t = ln 1/x into MV model - Higher energy  new distributionWt[r] - Renormalization group equation is a linear functional differential equation for Wt[r], but nonlinear wrt r. - Reproduces the Balitsky equation - Can be formulated for a(x) (gauge field) through the Yang-Mills eq. [Dn , Fnm] = dm+r (xT)  - T2 a(xT) = r(xT) (r is a covariant gauge source)  JIMWLK equation D

  13. JIMWLK equation Evolution equation for Wt [a], wrtrapidityt = ln 1/x Wilson line in the adjoint representation gluon propagator Evolution equation for an operator O

  14. 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 Fi(x) is the external force. When Fi(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.

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

  16. DIS at small x : dipole formalism N: Scattering amplitude

  17. 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 ~ (a(x)-a(y))2 stay at the same transverse positions

  18. The Balitsky equation Take O=tr(Vx+Vy) as the operator Vx+ 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.

  19. The Balitsky-Kovchegov equation (I) The Balitsky equation The Balitsky-Kovchegov equation A closed equationfor<tr(V+V)> First derived by Kovchegov (99) by the independent multiple interaction Balitsky eq.  Balitsky-Kovchegov eq. <tr(V+V) tr(V+V)> <tr(V+V)> <tr(V+V)> (large Nc? Large A) Nt(x,y) = 1 - (Nc-1)<tr(Vx+Vy)>t is the scattering amplitude

  20. The Balitsky-Kovchegov equation (II) Evolution eq. for the onium (color dipole) scattering amplitude - evolution under the change of scattering energys (not Q2) resummation of (as ln s)n necessary at high energy -nonlineardifferential 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 t = ln 1/x ~ ln s is the rapidity

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

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

  23. 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.

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