Geometric scaling violation:

Transcription

1 Geometric scaling violation: [1] DIS with nuclei [] running QCD coupling Eugene Levin, UTFSM / TAU Frontiers in QCD, October 17, 11, Seattle B. D. Saez and E. L., Violation of the geometric scaling behaviour of the amplitude for running QCD coupling in the saturation region, [arxiv: [hep-ph]] A. Kormilitzin, E. L., S. Tapia, Geometric scaling behavior of the scattering amplitude for DIS with nuclei, [arxiv: [hep-ph]]. Geometric scaling violation E. Levin 1

2 Why we believe in GS? 1. A natural consequence of one dimensional scale: Q s ;. For τ = r Q s 1 it follows from BFKL and GLAP evolution; 3. Deeply in the saturation region follows from B-K equation at at fixed α S ; 4. Two models with simplified BFKL kernel with fixed α S reproduce GS: kernel in diffusion approximation, B-K reduces to F-KPP equation with traveling wave solution; only leading twist contribution to the kernel, B-K reduces to wave equation with traveling wave solution; Geometric scaling violation E. Levin

3 Will GS survive if there is an additional dimensional scale? New scales: Running α S : Λ QCD ; DIS with nuclei: Q s at low energy initial condition; Intuition: Running α S : violation of GS since conformal symmetry of the kernel is broken; DIS with nuclei: no violation since we expect that deeply inside the saturation region the solution does not depend on initial conditions; Geometric scaling violation E. Levin 3

4 Simplified BFKL kernel N A r, Y ; b Y = C F α S π d r K N A r, Y ; b r; r {N A r, Y ; b 1 r r N A r, Y ; b 1 r r N A r r, Y ; b 1 } r For τ = r Q s < 1 d r K r, r = d r r r r r π r 1 Λ QCD r dr r 4 For τ = r Q s > 1 d r K r, r π r dr 1/Q s Y,b r + π r d r r 1/Q s Y,b r r Geometric scaling violation E. Levin 4

5 χ γ = n 1 for τ = r Q γ s 1 ; logs: α S ln1/r Λ QCD n 1 for τ = r Q 1 γ s 1 ; logs: α S lnr Q s B-K equation inside the saturation region: Ñ r, Y ; b Y ln r = ᾱ S { 1 Ñ r, Y ; b ln r Ñ r, Y ; b } Ñ r, Y ; t = = r dr N r, Y ; b /r Geometric scaling violation E. Levin 5

6 Solution where N = 1 e φξ,y ; φ Y e φ = ᾱ S Ñ e φ ; ξ s = ln φ Y ξ = ᾱ S 1 e φy ;ξ Q s Y /Q s Y = Y = 4ᾱ S Y Y, and ξ = ln φ ξ s ξ = 1 4 r Q s Y = Y 1 e φy ;ξ Geometric scaling violation E. Levin 6

7 φ z 1 φ x = 1 e φy ;ξ 4 with z = ξ s + ξ and x = ξ s ξ 1. Traveling wave solution: φ φ c + 1 λ κ dφ φ = κ x + λ z 1 + e φ. Self-similar solution: φ = φ ζ with ζ = ξ s ξ ζ d φ ζ dζ + dφ ζ dζ = e φζ A.D. Polyanin and V. F. Zaitsev, Handbook of nonlinear Partial Differential Equations,. Chapman & Hall/CRC, 4 Geometric scaling violation E. Levin 7

8 Initial conditions from τ = r Q s > 1: φ t z =, z = φ ; φ z t z =, z = 1 φ ; Finally: traveling wave solution with κ = s GS saturation region z = GS x = perturbative QCD Geometric scaling violation E. Levin 8

9 BFKL Kernel: K r; r 1, r = ᾱ S r { r For the region: Running α S r 1 r + 1 r 1 ᾱs r 1 ᾱ S r r } ᾱs r ᾱ S r1 1 r r r 1 1/Q s and r r 1 r 1/Q s d r K r, r π r 1/Q s Y,b ᾱ S r 1 dr 1 r 1 + π r 1/Q s Y,b ᾱ S r dr r Ñ r, Y ; b = r N r, Y ; b Y 1/Q s dr ᾱsr r N r, Y ; b = Ñ r, Y ; b 1 N r, Y ; b Geometric scaling violation E. Levin 9

10 Introducing l = r dr ᾱsr = 4N r c b with ξ = ln r Λ QCD ξ ln 1/ᾱ S r = 4N c b ln ξ We obtain: φr,y ;b = 1 e φr,y ;b Y l Equation for saturation scale: 3N c b Y = ξ Solutions: 1. Traveling wave solution contradicts initial conditions Geometric scaling violation E. Levin 1

11 . Self-similar solution?! φ 1 Equation: Obvious solution: φy,l Y l = 1 φ Y, l; b = Y l + F Y + G l = Y l l s 1 φ Main problem with IC; e l e l s 1 e l e l s + φ No indication on GS; However for ξ ξ s solution: ξ ξ s ξ s φ Y, l; b φ 3 4 φ ξ ξ s = φ + 1 φ z Solution leads to φ <, contradicts unitarity?! Geometric scaling violation E. Levin 11

12 3. Searching self-similar solution: φ ζ with ζ = Y l l s Equation: ζ d φ ζ dζ + φ ζ dζ = 1 e φζ IC: φ ζ = = φ ; dφ ζ = dζ = 1 φ /ξ s = 1 φ / 3N c b Y IC at large Y: φ ζ = = φ ; dφ ζ = dζ = 1 φ /ξ s = Good chance for self-similar solution at Y 1 Geometric scaling violation E. Levin 1

13 4. Searching self-similar solution: numerical solution. Φ Ζ l, l Φ Ζ l, l l 1 1 Ζ 8 l Ζ No SSS Approaching SSS Self- similar solution but at unreasonable large ζ Geometric scaling violation E. Levin 13

14 5. GS solution in vicinity of the saturation scale. In terms of z = 16N c b z equation looks as: 16Nc b z d φ z; b + d φ z; b Y d z d z = 1 e φ z;b GS for 16N c b z Y GS for α S Q s ln r Q s 1 Geometric scaling violation E. Levin 14

15 DIS with nuclei s IC GS saturation region GS =??? z = Y = 1/3ln A A perturbative QCD x = IC: McLerran-Venugopalan formula α S ln1/x 1; α S A1/3 1 N A r ; Y ; b = 1 exp r Q s A; Y = Y A; b Geometric scaling violation E. Levin 15

16 Φ Ξ 5 Solution: φ 1 4 φ ξs,ξ = 1 4 ; or Ξ φ t φ x = 1 4 ; No GS General solution: φ ξ s, ξ = 1 4 ξ s ξ + F 1 ξ s + F ξ Solution for t = z < x ξ < : φ 1 z = 1 8 z + φ z + φ Solution for t = z > x ξ > : φ z, ξ = z /8 ξ /8 + φ e ξ + 1 φ ξ s Geometric scaling violation E. Levin 16

17 s saturation region 1 s z = GS no GS x = IC GS GS saturation region no GS z =? Y = 1/3ln A A perturbative QCD perturbative QCD x = For CGC McL-V formula : GS for τ = r Q s > 1 and r < 1/Q Y = Y A no GS for τ = r Q s > 1 and r > 1/Q Y = Y A Geometric scaling violation E. Levin 17

18 BFKL Pomeron Calculus for DIS with nuclei Goal: discuss Y A > Y > 1 BF KL NA = Simplified B-K equation = T A b Assumption: b b b. d b BF KL NN r, Y ; b b T A b d BF KL b N r, Y ; b BF KL = NN r, Y ; t = T A b R dipole - nucleon = R + α P Y R A N T A b = + dz ρ b, z Geometric scaling violation E. Levin 18

19 N eff N r, Y ; t = Y = ᾱs π d r r r r r { N eff1 N r, Y ; t = N eff1 N r, Y ; t = T A b N eff1 N r, Y ; t = N eff1 N r r, Y ; t = } N A = G 3P + gtb eff N N = G T b 3P A + g Geometric scaling violation E. Levin 19

20 Equation for Y < Y A T A b; Y = +1/mx 1/mx dz ρ b, z = ρ R A A 1/3 for Y Y A ; e Y ρ/m for Y < Y A ; Y II geometric scaling behaviour of the a m p l i t u d e s Y A 1s I 1 violation of the geometric scaling behaviour of the a m p l i t u d e ξ s = 4ᾱ S Y Y A and ξ = ξ 1s = 1 + ᾱ S YA Y Geometric scaling violation E. Levin

21 Φ y, z N z, b.5 3 y 1 5 z. 1 5 b 4 z 6 φ for Y < Y A N A for Y > Y A saturation region s IC GS z = GS no GS Y = 1/3ln A A perturbative QCD x = Geometric scaling violation E. Levin 1