High Energy QCD and Pomeron Loops

Transcription

1 High Energy QCD and Pomeron Loops Dionysis Triantafyllopoulos Saclay (CEA/SPhT) Based on : E. Iancu, D.N.T., Nucl. Phys. A 756 (2005) 419, Phys. Lett. B 610 (2005) 253 J.-P. Blaizot, E. Iancu, K. Itakura, D.N.T., Phys. Lett. B 615 (2005) 221 Y. Hatta, E. Iancu, L. McLerran, A. Stasto, D.N.T., hep-ph/ D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 1/35

2 Outline Approaches to High Energy QCD (s, Λ 2 QCD Q2 s) Color Dipole Picture Color Glass Condensate (CGC)- JIMWLK equation QCD - Statistical Physics correspondence Effective Action, Pomeron Vertices, Reggeized Gluons,... Outline: The BFKL Pomeron Pomeron Mergings, Saturation and the CGC The Saturation Momentum Pomeron Splittings and Fluctuations Pomeron Loops, Evolution Equations at High Energy Duality, Effective Hamiltonian The Saturation Momentum Revisited D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 2/35

3 The BFKL Pomeron Probe gluon distribution of generic hadron with small color dipole Dipole size : r 2 = (x y) 2 Λ 2 QCD, Gluon momentum : Q2 4/r 2 Lowest order in pqcd D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 3/35

4 The BFKL Pomeron Probe gluon distribution of generic hadron with small color dipole Dipole size : r 2 = (x y) 2 Λ 2 QCD, Gluon momentum : Q2 4/r 2 Y = ln(1/x) = ln(p + /k + ) One soft gluon: α s Y D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 3/35

5 The BFKL Pomeron Probe gluon distribution of generic hadron with small color dipole Dipole size : r 2 = (x y) 2 Λ 2 QCD, Gluon momentum : Q2 4/r 2 Two soft gluons: (α s Y ) 2 D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 3/35

6 The BFKL Pomeron Probe gluon distribution of generic hadron with small color dipole Dipole size : r 2 = (x y) 2 Λ 2 QCD, Gluon momentum : Q2 4/r 2 n soft gluons: (α s Y ) n Resum all (α s Y ) n terms when α s Y 1 Gluon ladder : BFKL Pomeron D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 3/35

7 The BFKL Equation (1/3) Equivalent to diagram resummation Write evolution equation for scattering amplitude T View soft gluon emission in projectile At large-n c : Gluon Quark-Antiquark pair Either daughter dipole can scatter off target ᾱ s Y + BFKL Equation (coordinate space) dt xy dy = ᾱs 2π d 2 z M(x, y, z) }{{} (x y) 2 (x z) 2 (z y) 2 [T xz + T zy T xy ] K T xy Kernel : dipole splitting differential probability D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 4/35

8 The BFKL Equation (2/3) Linear evolution Solve eigenvalue problem The easy problem: Integrate over impact parameter K r 2γ = [2ψ(1) ψ(γ) ψ(1 γ)] r 2γ = χ 0 (γ) r 2γ The hard problem: Fixed impact parameter Solution: Superposition of (evolved) eigenfunctions Both cases: High energy, fixed r 2 same eigenvalue dominates Energy dependence in asymptotics T α s ϕ α 2 s n α 2 s exp[ω P Y ] ω P = 4ᾱ s ln2 = ᾱ s χ 0 (1/2) = hard pomeron intercept Exponential increase of gluon distribution ϕ, dipole density n in target D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 5/35

9 The BFKL Equation (3/3) Pathologies of BFKL Equation Violation of Unitarity: Amplitude must satisfy T(r, b) 1 Maximal allowed gluon density is ϕ a a A 2 1/g 2 1/α s Sensitivity to non-perturbative physics: Transverse coordinates ( momenta) not strongly ordered Non-local in transverse coordinates ( momentum) kernel Random-walk in lnr 2 Diffusion to infrared: r 2 1/Λ 2 QCD BFKL evolution is not self-consistent Next to leading BFKL: resum α s (α s Y ) n terms Will not save from difficulties Simply adds O(α 2 s) correction to ω P D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 6/35

10 Saturation - Unitarity (1/2) ᾱ s Y O(α s ϕ) ᾱ s Y O(α 2 sϕ 2 ) Second diagram small in perturbation theory ϕ 1/α s Equally important at high density ϕ 1/α s T (2) T : Allow both dipoles to scatter D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 7/35

11 Saturation - Unitarity (1/2) ᾱ s Y O(α s ϕ) ᾱ s Y O(α 2 sϕ 2 ) Second diagram small in perturbation theory ϕ 1/α s Equally important at high density ϕ 1/α s T (2) T : Allow both dipoles to scatter Third diagram (equiv to second): target evolution Merging of two pomerons Gluon recombination D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 7/35

12 Saturation - Unitarity (2/2) First Balitsky Equation dt xy dy = K BFKL T xy ᾱs 2π d 2 zm(x, y, z)t (2) xz;zy Mean field approximation: closed equation (Kovchegov) Fixed points T = 0 unstable T = 1 stable Pathologies are cured T (2) (xz; zy) T(x, z)t(z, y) Amplitude satisfies unitarity bound Non-linear term cuts diffusion to the infrared Saturation line Q 2 s Λ 2 exp(λ s Y ) where T(r 2/Q s ) = O(1) Justifies weak coupling approximation: α s (Q s ) 1 D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 8/35

13 The Color Glass Condensate Fast moving partons with momentum p + Large lifetime x + 1/p = 2p + /p 2 Time scale separation: Frozen sources for slow partons with momenta k + = xp + p + Small-x gluons color field radiated by fast partons Solve Classical Yang-Mills equation A(ρ) for given source ρ (D ν F νµ ) a (x) = δ µ+ ρ a (x, x) Non linear Calculate observable O(A) = O(ρ) O[ρ] Y = Dρ W Y [ρ]o[ρ] W Y [ρ] = probability distribution of color sources at rapidity Y D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 9/35

14 The JIMWLK Equation (RGE) (1/2) Increase rapidity Y Y + Y Previously slow modes now become fast Integrate to include them in source Obtain change of W Y [ρ] Leading order in ᾱ s ln(1/x) All orders in classical fiels A[ρ] Resum ᾱ s Y terms in presence of strong color field Still a classical theory at Y + Y D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 10/35

15 The JIMWLK Equation (RGE) (2/2) Renormalization Group Evolution Equation (JIMWLK) Y W Y [ρ] = H [ ρ, d ] W Y [ρ] dρ Hamiltonian better expressed in terms of (covariant gauge) color field α(x, x) A + (x, x) H = 1 16π 3 uvz M(u, v, z) [ ] ab 1 + Ṽ uṽv Ṽ uṽz Ṽ z δ Ṽv δα (u) a δ δα b (v) x 1/k + : Action takes place in last layer of longitudinal extent Wilson lines arise from propagator of integrated modes Ṽ x[α] = P exp i g dx α a (x, x) T a D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 11/35

16 The Observables Hilbert space : Gauge invariant operators built from Wilson lines O[α] = tr ( V x 1 V x2 V x 3 V x4... ) tr ( V y 1 V y2... )... Indeed: Left moving quark with eikonal trajectory ψ(x ) γ A + (x ) ψ(x ) δ (2) (x x) δ(x + ) A + (x ) S-matrix Wilson line in fundamental represenation Scatter single dipole off target S(x, y) = 1 N c tr ( V xv y ) = 1 T(x, y) = 1 g2 4N c [α a (x) α a (y)] 2 + O(g 3 ) 1 T 0 (x, y) + O(g 3 ) D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 12/35

17 The Balitsky Equations Evolution of observables O Y = Dα W Y [α]h O = H O First Balitsky equation S xy Y = ᾱs 2π z M(x, y, z) [ S xz S xz S xy ] Second Balitsky Equation S xz S zy Y = Sxz Y S zy + S zy S xz Y + O ( ) tr(6v ) N 3 c Projectile evolution One dipole two dipoles three dipoles + non-dipolar state Infinite hierarchy, factorization not justified, but consistent at large-n c D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 13/35

18 The Saturation Momentum (1/3) Y T Line-2 Saturation 2 1 γ 0 log 1 α s Saturation Line T = const O(1) Critical Line BFKL DGLAP T = const O(α 2 s ) log(1/α 2 s ) ω P Line-1 T 0 ρ = log(q 2 /µ 2 ) Increase momentum, increase rapidity so that T = const Line-1: DGLAP γ 0 Line-2: Hard Pomeron Intercept γ = 1/2 Saturation-Line: 0 < γ s < 1/2 D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 14/35

19 The Saturation Momentum (2/3) Can we use BFKL dynamics? Yes, but put absorptive boundary Cuts diffusive paths to saturation, mimics non-linear term Require constant T and saddle point χ 0 (γ s ) + (1 γ s )χ 0(γ s ) = 0 γ s = ( ) Q 2 1 γs ) T = s (ln Q2 Q 2 + c Scaling form Q 2 s Exact eigenfunction, valid up to diffusion radius Y towards UV λ s d lnq2 s dy = ᾱ s χ 0 (γ s ) 1 γ s 3 2(1 γ s ) 1 Y = 4.88ᾱ s 2.39 Y Confirmed by rigorous analysis of non-linear equations (Calculated 1/Y 3/2 term) Full JIMWLK on lattice: Almost same Factorization (at large-n c ) D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 15/35

20 The Saturation Momentum (3/3) Next to leading BFKL (collinearly improved) + boundary a b All with running coupling a. brown : L BFKL with Y 0 =0 b. green : L BFKL c. blue : L BFKL + boundary d. magenta : L RG BFKL + boundary e. black : NL RG BFKL + boundary c d e Coupling decreases along saturation line Running is dominant effect Analytic expression (Line-c) λ s = 1.80 (Y + Y0 ) (Y + Y 0 ) 5/6 Full NLO result: Close to phenomenology λ s 0.3 D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 16/35

21 Deficiencies of Balitsky-JIMWLK Extreme sensitivity to the UV: Reconstructing solution in two (or more) steps by completeness Contributions from momenta up to ln(q 2 /Q 2 s) ᾱ s χ 0 (γ s)y Embarrassing: Some orders of magnitude in Q 2 Violation of Unitarity (!) O(1) T 1 αs 2 T 1 T 2 and for T 1 < αs 2 then T 2 > 1 Absence of Pomeron splittings: dα n dy = H JIMWLK α n α } α...α {{} 2 δ δα δ δα αm with m n Two ladders merge, but how could we have them in the first place? Nucleus target Many sources Many BFKL pomerons No more dynamics needed - Initial condition to be lost at high energy Pomeron Splittings D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 17/35

22 The Missing Diagram(s) Diagrammatic illustration of splitting ᾱ s Y O(αsϕ 2 2 ) ᾱ s Y O(αsϕ 3 3 ) ᾱ s Y O(αsϕ) 3 Third diagram not included in JIMWLK Important when ϕ α s T αs, 2 where JIMWLK has problems Low density region fluctuations Measure fluctuations: probe with two dipoles First Balitsky equation remains unchanged D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 18/35

23 The Color Dipole Picture (1/2) Evolution of dipole density n(x, y) Y = ᾱs 2π ᾱs 2π z z [ M(x, y, z) n(x, y) + M(x, z, y) n(x, z) + M(z, y, x) n(z, y) ] K xyz n(x, y) BFKL Equation for density D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 19/35

24 The Color Dipole Picture (2/2) Evolution of dipole-pair density n (2) (x 1, y 1 ; x 2, y 2 ) Y [ = ᾱs K x1 y 2π 1 z n (2) (x 1, y 1 ; x 2, y 2 ) z ] + M(x 1, y 2, x 2 ) n(x 1, y 2 ) δ (2) (x 2 y 1 ) Multi-dipole density equations not consistent with factorization At low density n (2) n, rather than n (2) n 2 D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 20/35

25 Pomeron Splittings Measure BOTH child dipoles Evolution equation for dipole-pair scattering at large-n c T (2) x 1 y 1 ;x 2 y 2 Y = split ( αs ) 2 ᾱ s 2π 2π uvz M(u, v, z) A 0 (x 1, y 1 u, z) A 0 (x 2, y 2 z, v) 2 }{{} u 2 vt uv }{{} Dip Dip Scatt Dip density Low density fluctuations are the seed for higher-point correlations Equivalent to Bartels 1 2 Vertex D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 21/35

26 The (Large-N c ) Equations At large-n c, only 1 2 process T (n) T (n+1) Structure of the Equations (adding large-n c Balitsky) dt dy = T T (2) dt (2) dy = T (2) T (3) + T dt (n) dy = T (n) }{{} BFKL T (n+1) }{{} merging + T (n 1) }{{} splitting Can be summarized in a Langevin Equation Certain approximations stochastic-fkpp equation dt dy = T T 2 + T ν with ν(y )ν(y ) = δ(y Y ) D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 22/35

27 Pomeron Loops Splittings + Mergings Loops This is a simple loop Pomeron Loops will be built through evolution On can construct effective pomeron vertices" But system is non-linear Need to solve hierarchy D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 23/35

28 Duality (1/3) D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 24/35

29 Duality (1/3) Merging in target splitting in projectile Pomeron Loop manifestly symmetric Effective theory with both splittings and mergings is self-dual High density Low density or Saturation Fluctuation Duality Splitting Hamiltonian H α α δ δ δα δα δ }{{ δα } 2 D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 24/35

30 Hamiltonian Approach A proposed splitting Hamiltonian at large-n c g2 H 1 2 = 16Nc 3 ᾱ s 2π uvz M(u, v, z) ρ a u ρ a v [ δ δρ b u δ ] 2 [ δ δρ b z δρ c z δ δρ c v ] 2 Two ρ s, four δ/δρ s 1 2 process Charge density is related to dipole density ρ a (x) ρ a (y) = g 2 N c n(x, y) Acting on n(x 1, y 1 ) n(x 2, y 2 ) Splitting term in evolution Assume two-gluon exchange in scattering Acting on T (2) 0 (x 1, y 1 ; x 2, y 2 ) T 0 (x 1, y 1 ) T 0 (x 2, y 2 ) Splitting term in evolution D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 25/35

31 Duality (2/3) Scattering of two evolved dipoles in two gluon exchange S Y = [ Dα R Dα L W Y y [α R ] W y [α L ] exp i z ] ρ a L(z) αr(z) a } {{ } S S symmetric under R L ; use 2 α R/L = ρ R/L, integrate by parts Lorentz (boost) invariance requires d S Y dy = 0 H [ α, ] δ i δα [ ] δ = H i δρ, ρ Conceptual problem (with no answer): Splitting in R-wavefunction Merging in L-wavefunction This is large-n c, do we understand dipole recombination? D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 26/35

32 Duality (3/3) From duality condition H 2 1 = g2 16N 3 c ᾱ s 2π uvz M(u, v, z) [α a u α a z] 2 [α b u α b z] 2 δ δα c u δ δα c v Gives correct equations of motion with Hilbert space T 0 s H 2 1 T 0(x, y) = ᾱs 2π z M(x, y, z) T (2) 0 (x, z; z, y) Can show that BFKL part H 0 is self-dual Self-dual Hamiltonian generating correct evolution at large N c H = H 0 + H H 2 1 D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 27/35

33 Dual of JIMWLK JIMWLK : Expressed in terms of Wilson lines Involves n 1 processes with n arbitrary, finite N c High density limit of full Hamiltonian Low density limit of full H is dual of JIMWLK δ i δα ρ, α δ i δρ, x x + Wilson lines, 1 n processes, finite N c H = 1 M(u, v, z) ρ a 16π (u)ρ b (v) [ 1 + W 3 u W v W u W z W z W v ] ab uvz W x = P exp g dx + δ δρ a (x +, x) T a Full Hamiltonian: NOT any simple interpolation of high and low density D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 28/35

34 Full Hamiltonian (1/2) n 1 1 n m n Resume n m in ᾱ s ln(1/x) for arbitrary n, m Renormalization group in rapidity (from the beginning) H eff expected to involve both V and W H eff expected to be dual under V W D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 29/35

35 Full Hamiltonian (2/2) x + =x + τ t x =x τ x x + W V V W z H eff = 1 2πg 2 N c x Tr [ V ( i W )( i V )W ] + permutations Three independent Wilson lines: V W V W = 1 Expand W s to order g 2 JIMWLK Expand V s to order g 2 dual of JIMWLK D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 30/35

36 The Saturation Momentum Reloaded (1/4) Splittings in target in dilute region Merging in projectile UV boundary Confirmed by analogy to statistical physics Hierarchy Langevin sfkpp cuttof at α 2 s = 1/(1 γ s ) ln(1/α 2 s) = separation of boundaries Within, amplitude drops from O(1) to O(α 2 s) Look for a Y -independent BFKL solution ( [χ ) z ] λ s T = 0, z = ln(q 2 /Q 2 z s) Only real combination satisfying boundary conditions T exp[ (1 γ r ) z] sin πz, γ i = π D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 31/35

37 The Saturation Momentum Reloaded (2/4) Real part γ r uniquely fixed in terms of γ i or or α s λ s = χ 0(γ) 1 γ with Im(λ s ) = 0 For large separation of boundaries 1 α s 1 λ s = χ 0(γ s ) π2 (1 γ s )χ 0(γ s ) ᾱ s 1 γ s 2 ln 2 (αs) 2 = ln 2 (α 2 s) Correction: parametrically suppressed, but coefficient huge Denominator : Effective transverse space (same in single boundary) Boundaries in BFKL too sharp Full equation has well-defined solution for reasonable values of α s D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 32/35

38 The Saturation Momentum Reloaded (3/4) Γ i Π Γ r 0.5 Real part of anomalous dimension: no significant change Reduces to γ s = when D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 33/35

39 The Saturation Momentum Reloaded (4/4) Λ Α s Α s Speed of saturation momentum: significant change Positive for reasonable values of coupling: λ s (α s 0.3) > 0 But not really under control: Can change α s κ α s Reduces to λ s = 4.88 ᾱ s when D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 34/35

40 Conclusion-Perspectives Evolution equations at high-density and low-density (large-n c ) Pomeron Loops: Basic building block to reach unitarity Free of divergencies NO diffusion to IR, NO diffusion to UV (good for numerics) More important than Next to Leading-BFKL corrections Go beyond multicolor limit (done) Self-dual effective theory Arbitrary density (semi-done) Phenomenology may change even at qualitative level D.N. Triantafyllopoulos (Saclay) Cracow School of Theoretical Physics, XLV Course, Zakopane, Poland, June 2005 High Energy QCD and Pomeron Loops p. 35/35