Analytical properties of NLL-BK equation

Transcription

1 Guillaume Beuf Brookhaven National Laboratory Institute for Nuclear Theory, Seattle, September 28, 2010

2 Outline 1 Introduction 2 Traveling wave solutions of BK with fixed coupling 3 Asymptotic behavior of the solutions of the BK equation with running coupling and NLL terms Shape of the solutions and scaling properties The saturation scale and its evolution G.B., arxiv: G.B., in preparation. 4 Conclusion 5 Outlook: How to really do NLL gluon saturation phenomenology.

3 Introduction DIS at low x and saturation The starting point for most studies in DIS at low x is the dipole factorization (see previous talks), involving the dipole-target amplitude N(r, b, Y ).

4 Introduction DIS at low x and saturation The starting point for most studies in DIS at low x is the dipole factorization (see previous talks), involving the dipole-target amplitude N(r, b, Y ). N(r,b,Y ) determine the structure functions F 2 and F L, and diffraction, DVCS, exclusive vector meson production.

5 Introduction DIS at low x and saturation The starting point for most studies in DIS at low x is the dipole factorization (see previous talks), involving the dipole-target amplitude N(r, b, Y ). N(r,b,Y ) determine the structure functions F 2 and F L, and diffraction, DVCS, exclusive vector meson production. And its Fourier transform appears in several observables in pp and pa collisions.

6 Introduction DIS at low x and saturation The starting point for most studies in DIS at low x is the dipole factorization (see previous talks), involving the dipole-target amplitude N(r, b, Y ). N(r,b,Y ) determine the structure functions F 2 and F L, and diffraction, DVCS, exclusive vector meson production. And its Fourier transform appears in several observables in pp and pa collisions. N(r,b,Y ) is a more useful quantity than the usual gluon distribution in the saturation regime.

7 Introduction The dipole-target amplitude Two main methods to obtain N(r,b,Y) in practice: 1 Phenomenological models Problem: not under control from the theory.

8 Introduction The dipole-target amplitude Two main methods to obtain N(r,b,Y) in practice: 1 Phenomenological models Problem: not under control from the theory. 2 Numerical simulations of the BK equation Problem: So far fails to give a correct impact parameter dependance (see Tuomas talk and tomorrow s discussion).

9 Introduction The dipole-target amplitude Two main methods to obtain N(r,b,Y) in practice: 1 Phenomenological models Problem: not under control from the theory. 2 Numerical simulations of the BK equation Problem: So far fails to give a correct impact parameter dependance (see Tuomas talk and tomorrow s discussion). Status of the numerics: Strict LL BK equation: too fast evolution of the saturation scale. fails to reproduce the data.

10 Introduction The dipole-target amplitude Two main methods to obtain N(r,b,Y) in practice: 1 Phenomenological models Problem: not under control from the theory. 2 Numerical simulations of the BK equation Problem: So far fails to give a correct impact parameter dependance (see Tuomas talk and tomorrow s discussion). Status of the numerics: Strict LL BK equation: too fast evolution of the saturation scale. fails to reproduce the data. LL BK equation with running coupling: very good agreement with the data. Albacete, Armesto, Milhano, Salgado (2009) Can we understand why running coupling effects are so important?

11 Introduction The dipole-target amplitude Two main methods to obtain N(r,b,Y) in practice: 1 Phenomenological models Problem: not under control from the theory. 2 Numerical simulations of the BK equation Problem: So far fails to give a correct impact parameter dependance (see Tuomas talk and tomorrow s discussion). Status of the numerics: Strict LL BK equation: too fast evolution of the saturation scale. fails to reproduce the data. LL BK equation with running coupling: very good agreement with the data. Albacete, Armesto, Milhano, Salgado (2009) Can we understand why running coupling effects are so important? Full NLL BK equation: known for a few years Balitsky, Chirilli (2007)

12 Introduction Traveling waves A part from the numerical simulations, one of the most powerful tools to study N(r,b,Y ) is the traveling wave method. In general: Ebert, van Saarloos (2000) For the QCD case: Munier, Peschanski ( ).

13 Introduction Traveling waves A part from the numerical simulations, one of the most powerful tools to study N(r,b,Y ) is the traveling wave method. In general: Ebert, van Saarloos (2000) For the QCD case: Munier, Peschanski ( ). It allows to Obtain the exact (too?) large Y behavior of the solutions of BK.

14 Introduction Traveling waves A part from the numerical simulations, one of the most powerful tools to study N(r,b,Y ) is the traveling wave method. In general: Ebert, van Saarloos (2000) For the QCD case: Munier, Peschanski ( ). It allows to Obtain the exact (too?) large Y behavior of the solutions of BK. Understand the outcome of numerical simulations.

15 Introduction Traveling waves A part from the numerical simulations, one of the most powerful tools to study N(r,b,Y ) is the traveling wave method. In general: Ebert, van Saarloos (2000) For the QCD case: Munier, Peschanski ( ). It allows to Obtain the exact (too?) large Y behavior of the solutions of BK. Understand the outcome of numerical simulations. Obtain suitable informations to be put in phenomenological models. Iancu, Itakura, Munier (2004).

16 Universal traveling waves solutions Saturation with fixed coupling Generic gluon saturation equation At leading logarithmic (LL) accuracy, the BK equation writes formally as { } Y N(L,Y ) = ᾱ χ( L )N(L,Y ) Nonlinear damping with the BFKL eigenvalue χ(γ) = 2Ψ(1) Ψ(γ) Ψ(1 γ), in position space for the dipole-target amplitude, with L log(r 2 ), in momentum space for the unintegrated gluon distribution, with L log(k 2 ).

17 Universal traveling waves solutions Saturation with fixed coupling Properties of saturation equations The BK equation has the following properties: 1 Exponential growth and diffusion in the dilute regime N(L,Y ) 1. 2 Nonlinear damping in the dense regime N(L,Y ) 1. 3 Admits a family of uniformly translated wave-front solutions N(L,Y ) = N γ (L vᾱy ), which have a tail in the UV of the type N γ (s) e γs a dispersion relation v(γ) = χ(γ)/γ

18 Universal traveling waves solutions Saturation with fixed coupling Properties of saturation equations The BK equation has the following properties: 1 Exponential growth and diffusion in the dilute regime N(L,Y ) 1. 2 Nonlinear damping in the dense regime N(L,Y ) 1. 3 Admits a family of uniformly translated wave-front solutions N(L,Y ) = N γ (L vᾱy ), which have a tail in the UV of the type N γ (s) e γs a dispersion relation v(γ) = χ(γ)/γ Theorem: For the equations with such properties, all solutions with steep enough initial conditions converge to the uniformly translated wave-front solution of minimal velocity v c = v(γ c ), in an universal way. Bramson (1983)

19 Universal traveling waves solutions Saturation with fixed coupling Universality of traveling wave solutions In QCD: color transparency steep enough initial conditions all physically relevant solutions have a universal asymptotic behavior. Munier, Peschanski ( )

20 Universal traveling waves solutions Saturation with fixed coupling Universality of traveling wave solutions In QCD: color transparency steep enough initial conditions all physically relevant solutions have a universal asymptotic behavior. Munier, Peschanski ( ) Allows to confirm, better justify and extend previous analytical results from: Gribov, Levin, Ryskin (1983) Iancu, Itakura, McLerran (2002) Mueller, Triantafyllopoulos (2002)

21 Universal traveling waves solutions Saturation with fixed coupling Universality of traveling wave solutions log N(L, Y ) 0 L γc vc ᾱ Y γ0 v0 ᾱ Y Three different regimes for the solutions: the saturated regime, the universal dilute regime (or geometric scaling window), and the initial condition dominated dilute regime, which is progressively invaded by the universal one.

22 Universal traveling waves solutions Saturation with fixed coupling Universality of traveling wave solutions General method to calculate the universal relaxation of the solutions: Ebert, van Saarloos (2000)

23 Universal traveling waves solutions Saturation with fixed coupling Universality of traveling wave solutions General method to calculate the universal relaxation of the solutions: Ebert, van Saarloos (2000) Outcome: position of the wave-front / saturation scale: ( ) Q s (Y ) 2 L s (Y ) log = v 2 c ᾱy 3 log(ᾱy ) + Const. Q 0 2γ c 3 γ 2 c 2π χ (γ c )ᾱy + O ( ) 1 ᾱy when translated into QCD notations: Munier, Peschanski (2004).

24 Universal traveling waves solutions Saturation with fixed coupling Universality of traveling wave solutions General method to calculate the universal relaxation of the solutions: Ebert, van Saarloos (2000) Outcome: position of the wave-front / saturation scale: ( ) Q s (Y ) 2 L s (Y ) log = v 2 c ᾱy 3 log(ᾱy ) + Const. Q 0 2γ c 3 γ 2 c 2π χ (γ c )ᾱy + O ( ) 1 ᾱy when translated into QCD notations: Munier, Peschanski (2004). Remark: At fixed coupling the nuclear enhancement Qs,A 2 = A1/3 Qs,p 2 is introduced by the initial condition and left intact by the evolution, in the Const. term.

25 From fixed to running coupling Y N(L,Y ) = { } ᾱ χ( L )N(L,Y ) Nonlinear damping

26 From fixed to running coupling Y N(L,Y ) = { } ᾱ χ( L )N(L,Y ) Nonlinear damping Parent dipole/gluon prescriptions for the running coupling: ᾱ 1 bl, with b = 11N c 2N f 12N c ( 2 ) ( k r 2 2) Λ QCD and L = log 2 or L = log Λ QCD 4

27 From fixed to running coupling Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl Parent dipole/gluon prescriptions for the running coupling: ᾱ 1 bl, with b = 11N c 2N f 12N c ( 2 ) ( k r 2 2) Λ QCD and L = log or L = log 2 Λ QCD 4

28 Higher order corrections Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl

29 Higher order corrections Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping One can include the full NLL BFKL/BK corrections in that equation

30 Higher order corrections Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping +... One can include the full NLL BFKL/BK corrections in that equation, and formally NNLL and higher order contributions.

31 Balitsky s running coupling prescription Balitsky s prescription: LL BK equation + resummation of a part of the terms at each logarithmic order into the running coupling scale.

32 Balitsky s running coupling prescription Balitsky s prescription: LL BK equation + resummation of a part of the terms at each logarithmic order into the running coupling scale. expanded version: same form as the all order equation Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping +... but with [χ(γ) 2 χ (γ) 4γ χ(γ) ] χ NLL (γ) = b 2

33 Universal traveling waves with running coupling Properties of the equations with running coupling. Many features of the fixed coupling equations are still present in the running coupling case:

34 Universal traveling waves with running coupling Properties of the equations with running coupling. Many features of the fixed coupling equations are still present in the running coupling case: Instability of the dilute regime

35 Universal traveling waves with running coupling Properties of the equations with running coupling. Many features of the fixed coupling equations are still present in the running coupling case: Instability of the dilute regime Diffusion

36 Universal traveling waves with running coupling Properties of the equations with running coupling. Many features of the fixed coupling equations are still present in the running coupling case: Instability of the dilute regime Diffusion Nonlinear damping

37 Universal traveling waves with running coupling Properties of the equations with running coupling. Many features of the fixed coupling equations are still present in the running coupling case: Instability of the dilute regime Diffusion Nonlinear damping Color transparency steepness of the relevant initial conditions.

38 Universal traveling waves with running coupling Properties of the equations with running coupling. Many features of the fixed coupling equations are still present in the running coupling case: Instability of the dilute regime Diffusion Nonlinear damping Color transparency steepness of the relevant initial conditions. The universality of the traveling wave solutions at fixed coupling is due on these properties and to the existence of a family of exact scaling solutions.

39 Universal traveling waves with running coupling Properties of the equations with running coupling. Many features of the fixed coupling equations are still present in the running coupling case: Instability of the dilute regime Diffusion Nonlinear damping Color transparency steepness of the relevant initial conditions. The universality of the traveling wave solutions at fixed coupling is due on these properties and to the existence of a family of exact scaling solutions. What about exact scaling solutions in the running coupling case?

40 Universal traveling waves with running coupling Scaling properties and running coupling Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping +...

41 Universal traveling waves with running coupling Scaling properties and running coupling Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping +... No exact scaling solutions!

42 Universal traveling waves with running coupling Scaling properties and running coupling Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping +... No exact scaling solutions! But infinitely many families of approximate scaling variables s(l,y ) in the relevant regime L,Y with s(l,y )/L 0.

43 Universal traveling waves with running coupling Scaling properties and running coupling Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping +... No exact scaling solutions! But infinitely many families of approximate scaling variables s(l,y ) in the relevant regime L,Y with s(l,y )/L 0. Dispersion relation, and thus minimal velocity v c independent of the chosen type of approximate scaling variable.

44 Universal traveling waves with running coupling Scaling properties and running coupling Y N(L,Y ) = 1 { } χ( L )N(L,Y ) Nonlinear damping bl + 1 { } (bl) 2 χ NLL ( L )N(L,Y ) Nonlinear damping +... No exact scaling solutions! But infinitely many families of approximate scaling variables s(l,y ) in the relevant regime L,Y with s(l,y )/L 0. Dispersion relation, and thus minimal velocity v c independent of the chosen type of approximate scaling variable. Universal wave-front formation a priori possible.

45 Universal traveling waves with running coupling Scaling variables and running coupling Examples: there is one family of approximate scaling variables s(l,y ) = L 2δ [ 1 ( ) ] 2vY δ bl 2 for each δ > 0.

46 Universal traveling waves with running coupling Scaling variables and running coupling Examples: there is one family of approximate scaling variables s(l,y ) = L 2δ [ 1 ( ) ] 2vY δ bl 2 for each δ > 0. For generic δ: Critical solution with same parameters v = v c and γ = γ c as as at fixed coupling.

47 Universal traveling waves with running coupling Scaling variables and running coupling Examples: there is one family of approximate scaling variables s(l,y ) = L 2δ [ 1 ( ) ] 2vY δ bl 2 for each δ > 0. For generic δ: Critical solution with same parameters v = v c and γ = γ c as as at fixed coupling. Particular case δ = 1/2: geometric scaling variable: ( 2vY r 2 Qs 2 s(l,y ) = L log (Y ) ) b 4 ( ) with Qs 2 (Y ) Λ QCD 2 exp 2vY /b

48 Universal traveling waves with running coupling Scaling variables and running coupling Approximate scaling variables for generic solutions with steep initial conditions: has to include relaxation to the scaling variable with minimal velocity v c, via diffusion effects. s(l,y ) = L 2δ [ 1 ( 2vc Y bl 2 ) δ ] 3ξ 1 4 (DL)1/3 +η + µ(dl) 1/3 + ν(dl) 2/3 + O ( L 1)

49 Universal traveling waves with running coupling Scaling variables and running coupling Approximate scaling variables for generic solutions with steep initial conditions: has to include relaxation to the scaling variable with minimal velocity v c, via diffusion effects. s(l,y ) = L 2δ [ 1 ( 2vc Y bl 2 ) δ ] 3ξ 1 4 (DL)1/3 +η + µ(dl) 1/3 + ν(dl) 2/3 + O ( L 1) First diffusive correction: already known. ξ 1 : zero of Airy function. D = χ (γ c) 2 χ(γ. c) Mueller, Triantafyllopoulos (2002); Munier, Peschanski (2004)

50 Universal traveling waves with running coupling Scaling variables and running coupling Approximate scaling variables for generic solutions with steep initial conditions: has to include relaxation to the scaling variable with minimal velocity v c, via diffusion effects. s(l,y ) = L 2δ [ 1 ( 2vc Y bl 2 ) δ ] 3ξ 1 4 (DL)1/3 +η + µ(dl) 1/3 + ν(dl) 2/3 + O ( L 1) First diffusive correction: already known. ξ 1 : zero of Airy function. D = χ (γ c) 2 χ(γ. c) Mueller, Triantafyllopoulos (2002); Munier, Peschanski (2004) Other coefficients η, µ and ν: new results.

51 Universal traveling waves with running coupling Calculation of the universal asymptotics First step: Solve the evolution equation in the regime L and s = O(1), roughly parallel to lines of constant density, assuming a low enough density to linearize the equation: N(L,Y ) = A e [s+ γcs 1 ( +(DL) 2/3 φ 2 (s)+(dl) 1 φ 3 (s)+o L 4/3)] γ c with polynomial functions φ 2 (s), φ 3 (s),....

52 Universal traveling waves with running coupling Calculation of the universal asymptotics Second step: Solve the evolution equation in the regime L and s = O(L 1/3 ), with z = s + ξ (DL) 1/3 1 = O(1), were one follow the spreading of the universal solution into the dilute tail : [ N(L,Y ) = A e γcs 1/3 Ai(z) (DL) Ai (ξ 1 ) + G 0(z) + (DL) 1/3 G 1 (z) +(DL) 2/3 G 2 (z) + O ( ] L 1) where the G n (z) are Airy functions times polynomials.

53 Universal traveling waves with running coupling Calculation of the universal asymptotics Third step: Match the two asymptotic expansions. Impose the causality requirements G n (z) 0 for z. Impose that the N(L,Y ) starts to saturate at a constant height. At the order considered, all parameters and integration constants are determined, except the normalization constant A of N(L,Y ), which is nevertheless independent of initial conditions..

54 Universal traveling waves with running coupling Dependance on the chosen scaling variable Asymptotic solution of the BK equation with running coupling (no NLL kernel). Y = 10,30,50,70. Blue: δ = 1/4. Black: δ = 1/2. Red: δ = 1. Left: with G 0 (z) = G 1 (z) = G 2 (z) = η = µ = ν = 0. Right: with all the orders calculated.

55 Universal traveling waves with running coupling Dependance on the chosen scaling variable Dependance on δ weaker and weaker when increasing the accuracy of the calculations. Exact solution independent on the choice of scaling variable s(l,y ).

56 Universal traveling waves with running coupling Dependance on the chosen scaling variable In any case, the exact solution scales approximately with the running coupling geometric scaling variable (δ = 1/2). Shows without bias the validity of (approximate) geometric scaling in the running coupling case.

57 Saturation scale with running coupling Extraction of the saturation scale The saturation scale Q s (Y ) is defined by a level line: From the result N(L s (Y ),Y ) = κ, with L s (Y ) log Q2 s (Y ) Λ 2 QCD N(L,Y ) = A e γcs [s + 1 γ c + (DL) 2/3 φ 2 (s) + O ( L 1) ], one gets s(l s (Y ),Y ) = Σ 1 + (D L s (Y )) 2/3 φ 2(Σ 1/γ c ), γ c γ c Σ 1 where Σ 1 ( W 1 γ c κ ), γ c A e and W 1 (x) is the 1 branch of the Lambert function.

58 Saturation scale with running coupling Notations Recall: γ c : solution of χ(γ c ) = γ c χ (γ c ) v c = χ(γ c) γ c and D = χ (γ c ) 2χ(γ c ) Higher derivatives of the LL BFKL eigenvalue parameterized by: K n = 2χ(n) (γ c ) n!χ (γ c ) for n 3 NLL contributions parameterized by: N n = χ(n) NLL (γ c) n!χ(γ c )b for n 0

59 Saturation scale with running coupling Universal part of the saturation scale Finally, one obtains the expansion L s (Y ) = log Q2 s (Y ) = Λ 2 QCD ( 2vc Y b ) 1/2 + 3ξ 1 4 ( +c 0 + c 1/6 D 2 2v cy b +O ( Y 1/2), ( D 2 2v cy b ) 1/6 ) 1/6 + c 1/3 ( D 2 2v ) 1/3 cy b with 3 new universal coefficients c 0, c 1/6 and c 1/3.

60 Saturation scale with running coupling Universal part of the saturation scale c 0 = Σ 1 γ c K N 0 c 1/6 = ξ 1 2 [ 1 72γ 2 c D 32 + K 3 3 4γ c 8 K ] 10 K 4 { Σ 2 [ K3 c 1/3 = ξ ] 6γ c 2γ c 3γc 2 Σ γc 3 + D 3γ c [ +K 3 2 3γc 2 + 3K 3 11K 2] ] 3 + K 4 [ 1γc +6K 3 3K 5 4γ c 4 } + 1 2γc 2 [1+3γ c K 3 ][N 0 γ c N 1 ] + N 2 DN 0

61 Saturation scale with running coupling Remarks: higher order contributions Dominant NLL effect at large rapidity: in the constant term c 0. Reduction of Q s (Y ) by a constant factor, and does not affects its evolution. Compatible with the numerics. Albacete, Kovchegov (2007)

62 Saturation scale with running coupling Remarks: higher order contributions Dominant NLL effect at large rapidity: in the constant term c 0. Reduction of Q s (Y ) by a constant factor, and does not affects its evolution. Compatible with the numerics. Albacete, Kovchegov (2007) First NLL contributions affecting the evolution of Q s (Y ): delayed to the order Y 1/3.

63 Saturation scale with running coupling Remarks: higher order contributions Dominant NLL effect at large rapidity: in the constant term c 0. Reduction of Q s (Y ) by a constant factor, and does not affects its evolution. Compatible with the numerics. Albacete, Kovchegov (2007) First NLL contributions affecting the evolution of Q s (Y ): delayed to the order Y 1/3. NNLL effects on log Q s (Y ) delayed until the order Y 1/2, as expected. Peschanski, Sapeta (2006); G.B., Peschanski (2007)

64 Saturation scale with running coupling Remarks: higher order contributions Dominant NLL effect at large rapidity: in the constant term c 0. Reduction of Q s (Y ) by a constant factor, and does not affects its evolution. Compatible with the numerics. Albacete, Kovchegov (2007) First NLL contributions affecting the evolution of Q s (Y ): delayed to the order Y 1/3. NNLL effects on log Q s (Y ) delayed until the order Y 1/2, as expected. Peschanski, Sapeta (2006); G.B., Peschanski (2007) Renormalization scheme independant result for Q s (Y ).

65 Saturation scale with running coupling Remarks: suppression of initial condition dependance Constant term in log Qs 2 (Y ): universal in the running coupling case. And initial condition effects on log Q 2 s (Y ) delayed until the order Y 1/2.

66 Saturation scale with running coupling Remarks: suppression of initial condition dependance Constant term in log Qs 2 (Y ): universal in the running coupling case. And initial condition effects on log Q 2 s (Y ) delayed until the order Y 1/2. Confirmation by direct calculation of this effect, already predicted Mueller (2003) and observed numerically Rummukainen, Weigert (2004); Albacete et al. (2005).

67 Saturation scale with running coupling Remarks: suppression of initial condition dependance Constant term in log Qs 2 (Y ): universal in the running coupling case. And initial condition effects on log Q 2 s (Y ) delayed until the order Y 1/2. Confirmation by direct calculation of this effect, already predicted Mueller (2003) and observed numerically Rummukainen, Weigert (2004); Albacete et al. (2005). The nuclear enhancement Qs,A 2 = A1/3 Qs,p 2 should completely disappear at very large rapidity.

68 Saturation scale with running coupling Remarks: suppression of initial condition dependance Constant term in log Qs 2 (Y ): universal in the running coupling case. And initial condition effects on log Q 2 s (Y ) delayed until the order Y 1/2. Confirmation by direct calculation of this effect, already predicted Mueller (2003) and observed numerically Rummukainen, Weigert (2004); Albacete et al. (2005). The nuclear enhancement Qs,A 2 = A1/3 Qs,p 2 should completely disappear at very large rapidity. This effect might be a problem for the EIC. Need to clarify how fast this effect sets in, in practice.

69 Saturation scale with running coupling Q s (Y ): Parent dipole vs. Balitsky s prescription Solid line: with Balitsky s prescription. Dotted line: with the parent dipole prescription.

70 Saturation scale with running coupling Q s (Y ): Parent dipole vs. Balitsky s prescription Running coupling prescription: important for the normalization of Q s (Y ) but not for its evolution, at large Y.

71 Saturation scale with running coupling Q s (Y ): Parent dipole vs. Balitsky s prescription Black: universal terms only. Red: with a shift Y Y + 20 in the leading term only.

72 Saturation scale with running coupling Q s (Y ): Parent dipole vs. Balitsky s prescription Very large initial conditions effects seem required to reduce λ(y ) to at moderate Y. But leads to a too large Q s (Y ).

73 Saturation scale with running coupling Q s (Y ) with the full NLL BFKL kernel Solid line: with the full NLL BFKL kernel. Dotted line: with no NLL contribution besides running coupling with parent gluon k.

74 Saturation scale with running coupling Q s (Y ) with the full NLL BFKL kernel Full NLL effects stabilize λ(y ) in the phenomenologically preferred range

75 Saturation scale with running coupling Q s (Y ) with the full NLL BFKL kernel Black: universal terms only. Red: with a shift Y Y 2.5 in the leading term only.

76 Saturation scale with running coupling Q s (Y ) with the full NLL BFKL kernel Moderate initial conditions effects sufficient to bring both λ(y ) and Q s (Y ) to reasonable values at intermediate rapidities.

77 Conclusion Conclusion Running coupling effects must be taken into account for the high energy evolution with the gluon saturation (different universality class).

78 Conclusion Conclusion Running coupling effects must be taken into account for the high energy evolution with the gluon saturation (different universality class). Approximate geometric scaling proven in the running coupling case, with log Q 2 s Y.

79 Conclusion Conclusion Running coupling effects must be taken into account for the high energy evolution with the gluon saturation (different universality class). Approximate geometric scaling proven in the running coupling case, with log Q 2 s Y. Suppression of the nuclear enhancement of Q s at asymptotic rapidity. Slower A-dependance at the lowest x bins at the EIC?

80 Conclusion Conclusion Running coupling effects must be taken into account for the high energy evolution with the gluon saturation (different universality class). Approximate geometric scaling proven in the running coupling case, with log Q 2 s Y. Suppression of the nuclear enhancement of Q s at asymptotic rapidity. Slower A-dependance at the lowest x bins at the EIC? Large NLL effects on Q s (Y ) at finite rapidity. Full NLL equations required for the next generation of gluon saturation phenomenology.

81 Outlook What about a stable and reliable NLL BK equation? As for the BFKL equation, NLL corrections to the BK equation are very large, in particular in the collinear and anti collinear regime.

82 Outlook What about a stable and reliable NLL BK equation? As for the BFKL equation, NLL corrections to the BK equation are very large, in particular in the collinear and anti collinear regime. Resummations are required for gluon saturation phenomenology in DIS at NLL accuracy, and presumably already to get stable numerical solution of the NLL BK equation.

83 Outlook What about a stable and reliable NLL BK equation? As for the BFKL equation, NLL corrections to the BK equation are very large, in particular in the collinear and anti collinear regime. Resummations are required for gluon saturation phenomenology in DIS at NLL accuracy, and presumably already to get stable numerical solution of the NLL BK equation. Similar resummations have been performed for the NLL BFKL equation in momentum space. Ciafaloni, Colferai, Salam, Stasto ( ); Altarelli, Ball, Forte ( ); Thorne, White (2007)

84 Outlook What about a stable and reliable NLL BK equation? Physical origin of the big contributions:

85 Outlook What about a stable and reliable NLL BK equation? Physical origin of the big contributions: Running coupling.

86 Outlook What about a stable and reliable NLL BK equation? Physical origin of the big contributions: Running coupling. Non singular part of the DGLAP singlet splitting functions.

87 Outlook What about a stable and reliable NLL BK equation? Physical origin of the big contributions: Running coupling. Non singular part of the DGLAP singlet splitting functions. Crude kinematics of the low-x formalisms: gluon/dipole cascade ordered in longitudinal momentum rather than in lifetime.

88 Outlook What about a stable and reliable NLL BK equation? Physical origin of the big contributions: Running coupling. Non singular part of the DGLAP singlet splitting functions. Crude kinematics of the low-x formalisms: gluon/dipole cascade ordered in longitudinal momentum rather than in lifetime. For the BFKL equation in momentum space, the contributions of the third type can be resummed by imposing the kinematical constraint. Andersson, Gustafson, Kharraziha, Samuelsson (1996); Kwiecinski, Martin, Sutton (1996)

89 Outlook What about a stable and reliable NLL BK equation? Status of the resummations for NLL BK:

90 Outlook What about a stable and reliable NLL BK equation? Status of the resummations for NLL BK: Running coupling: already done. Balitsky s running coupling prescription.

91 Outlook What about a stable and reliable NLL BK equation? Status of the resummations for NLL BK: Running coupling: already done. Balitsky s running coupling prescription. Non singular part of DGLAP splitting functions: to do. Quite technical if one requires exact (N)LO DGLAP limits, but should be simple if one just resum the contributions appearing in NLL BFKL/BK.

92 Outlook What about a stable and reliable NLL BK equation? Status of the resummations for NLL BK: Running coupling: already done. Balitsky s running coupling prescription. Non singular part of DGLAP splitting functions: to do. Quite technical if one requires exact (N)LO DGLAP limits, but should be simple if one just resum the contributions appearing in NLL BFKL/BK. The improvement of the kinematics and the kinematical constraint have been studied in position space in Mueller s dipole model. Motyka, Stasto (2009)

93 Outlook What about a stable and reliable NLL BK equation? Status of the resummations for NLL BK: Running coupling: already done. Balitsky s running coupling prescription. Non singular part of DGLAP splitting functions: to do. Quite technical if one requires exact (N)LO DGLAP limits, but should be simple if one just resum the contributions appearing in NLL BFKL/BK. The improvement of the kinematics and the kinematical constraint have been studied in position space in Mueller s dipole model. Motyka, Stasto (2009) That should be rather easy to generalize to BK. But there are subtleties: unitarization at each order, virtual corrections?