Early Thermalization in the CGC and a Couple of Other Crazy Ideas - PowerPoint PPT Presentation

Early ’Thermalization’ in the CGC and a Couple of Other Crazy Ideas Eugene Levin, Tel Aviv University WS:“High Density QCD” GGI, Jan 15, 2007 - Mar 9, 2007 D. Kharzeev and E.L: ( in preparation ) D. Kharzeev, E.L. and K. Tuchin: hep-ph/0602063 D. Kharzeev and K. Tuchin: hep-ph/0501234 Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 1

Thermalization: p T When? Why? How? p dn/dy = Const L Exp[ − p2 T / Q2 dn/d p2 T −> s ] t=0 (cylindrical phase space) t=tth (isotropic phase space) dn/d pLd p2 −> Exp[− E/T] T Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 2

Colour Glass Condensate: Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 3

Structure of parton cascade in CGC Space -time picture: Y Low, Nussinov (1975); BFKL(1976−1986); x+ k − y 1 GLR (1981); Mueller & Qiu (1986); Mueller (1994); i i McLerran &Venugopalan (1994);Balitsky (1995) y Kovchegov(1999); Bartels(1992 − 2000);Braun(2002); x − k + 1 rapidity x t Qs 1 0 τ 2 0 ∆ LLA of pQCD: E/m time t x s(y) α s (yi+1 1/Q Y − yi ) t 1 x+ d ρ /d y = K ( ρ − ρ 2 ) rapidity x − ρ (y, Qs ) = 1 x x+ − R (1/x) λ 2 π R2 x G(x, Qs ) Qs Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 4

Classical fields: Since all partons with rapidity larger than y live longer than parton with rapidity y, for a dense system as a nucleus they can be considered as the source of the classical field that emits a gluon with rapidity y. ( L. McLerran & R. Venugopalan, 1994) Duality principle: Quantum emission in each stage of the process should give the same result as the emission of the classical field JIMWLK= J.Jalilian-Marian, E. Iancu, L. McLerran, H. Weigert, A. Leonidov and A. Kovner 1999 Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 5

� � � � � � � � � � ✁ � � � ✁ ✁ ✁ � � ✁ ✁ ✁ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ − + + e γ + − e n + m( e e ) A question: ? S >> m or JIMWLK approach: Fields Fields Classical Quantum t=t’ 0 Quantum Fields Classical Fields t=t0 t E/m2 : L( ρ ) + j µ .A µ +L(A) At t=t0 ( t’ , z’ 1 ) (t 1 , z 1 ) 1 At t=t 0 : L( ρ ) + j µ A µ +L(A) . ’ (t , z ) (t’ 2 , z’ 2 ) 2 2 (t i , z ) ( t’ , z’ ) i i i (t , z ) ( t’ , z’ i+1 i+1 ) i+1 i+1 time(z) interaction time − t’ >> ... >> t i − t’ i >> t i+1 − t’ t 1 − t’ 1 >> t 2 i+1 2 Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 6

Fields of CGC Lienard-Weichart potential for a charge moving with v ≡ v z √ A − = g 2 1 � ; A + = 0 ; A ⊥ = 0 ; � 4 π 2 x 2 − + (1 − v 2 ) x 2 ⊥ Fields: (1 − v 2 ) � g x ⊥ � � v × � E z = 0 ; E ⊥ = ⊥ ) 3 / 2 ; H = � E ; (2 x 2 − + (1 − v 2 ) x 2 4 π Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 7

Lienard-Weichart potential for a parton in the parton cascade √ A − = g 2 Θ( x + − x − ) � ; A + = 0 ; A ⊥ = 0 ; � 4 π 2 x 2 − + (1 − v 2 ) x 2 ⊥ Fields: E z ∝ δ ( x + − x − ) ; (1 − v 2 ) � g x ⊥ � � v × � E ⊥ = ⊥ ) 3 / 2 Θ( x + − x − ) ; H = � E ; (2 x 2 − + (1 − v 2 ) x 2 4 π | � E ⊥ | = | � H ⊥ | ≪ E z Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 8

x − x i xt x − A + exists only for z ≥ 0 and the initial conditions depend on x − and x + . √ „ « ( x ⊥− x ⊥ )2 2 ρ x − , x ⊥ , x + + x − − d 3 x dx − d 2 x 2( x −− x − ) ρ ( � x, t − | � x − � x | ) Z Z A = = ( x ⊥− x ⊥ )2 4 π | � x − � x | 4 π x − − x − + 2( x −− x − ) √ „ « ( x ⊥ )2 2 ρ x − , x ⊥ , x + − dx − d 2 x 2 x − Z − → x 2 4 π ⊥ x − + 2 x − | � E ⊥ | = | � At high energies H ⊥ | ≪ E z Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 9

To replace complicated process of particle Main production by the production in the idea: background fields A − = A − ( x − ) • At t=0: E z = E 0 = Const and A − ( x − ) = − E 0 x − • At large t: x − >>R In pQCD: A − ( x − ) − → 1 /x − x − (Say A − ( x − ) = − E 0 − ); 1 + ω 2 x 2 x − >>R e − ωx − ; In n-pQCD: A − ( x − ) − → Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 10

Model for A + ( x − ) πp 2 • A + ( x − ) = E 0 � ⊥ ω (1 − e − ω x − ) = ⇒ Im S = gE 0 (1 + γ ) σ − − ( x ′ ⊥ ) 2 � � x ′ − , x ′ ⊥ , x + + x ′ • ρ = 2 x − � − − ω − 1 � x ′ d 2 k ⊥ e i� k ⊥ · � ⊥ δ x ′ k 2 ⊥ − Q 2 � � � c δ , s d 2 x ′ ⊥ d x ′ 2 x − δ ( x ′ − − ω − 1 ) J 0 ( x ′ ⊥ Q s ) = c ′ x − K 0 ( x − Q s ) • A + ( x − ) = c ′ � − 2 x 2 − + x ′ 2 4 π ⊥ Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 11

Particle production in the background field Educated guess: k ⊥ factorization, for inclusive cross section: (Catani,Ciafaloni & Hautman;E.L., Ryskin,Shabelski & Shuvaev; Collins & Ellis ; 1991) ε dσ d 3 p = 4 πN c 1 dk 2 ⊥ α S ϕ P ( Y − y, k 2 ⊥ ) ϕ T ( y, ( p − k ) 2 � ⊥ ) N 2 p 2 c − 1 ⊥ Proven in CGC by Kovchegov & Tuchin (2002) only for DIS !? Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 12

Our formula: dσ dyd 2 p ⊥ = ϕ P ( Y − y, p ⊥ ) ϕ T ( y, p ⊥ ) with ϕ P ( Y − y, p ⊥ ) ∝ d 2 k ⊥ Im D ( Y − y, � p ⊥ − � � k ⊥ ) Im D ( Y − y, � p ⊥ ) Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 13

Sources of classical fields y = t y t’y t t’ 0 time Sources of classical fields φ P y p x t t’ y t y k t x x t t’ p t − kt Γ (G−>2G) φ T time Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 14

Equation of motion in the background field • G µ = A µ + W µ , A µ is a classical background field; ∂ 2 L [ A ] L [ A + W ] = L [ A ] + ∂ L [ A ] ∂A µ W µ + 1 • ∂A µ ∂A ν W µ W ν ; 2 − ( ∂ λ − igA λ ) 2 δ µν + 2 i g F µν [ A ] � � • W µ = 0 ;   0 0 0 E z 0 0 0 0   • F µν =  ,   0 0 0 0  − E z 0 0 0 • Introducing W ± = W 0 ± iW 3 we obtain: ( ∂ λ − igA λ ) 2 ± 2 gE z � � − W ± = 0 Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 15

Gluon propagator in the background field Schwinger (1951); Marinov & Popov (1972-1974);Kluger, Eisenberg, Svetitsky & . . . (1991-1994); Dunne (2004) Semiclassical solution: Solution in the form W σ = e − iS σ − ip ⊥ · x ⊥ with σ = ± 1 assuming • ∂ + S ∂ − S ≫ ∂ + ∂ − S ; • Introducing p − ( x − , x + ) = ∂S/∂x + and p + ( x − , x + ) = ∂S/∂x − , we obtain − 2 p + ( p − − gA − ( x − )) + p 2 ⊥ + 2 g σE z = 0 ; dx − • = − 2 ( p − − gA − ( x − )) ; dt dx + • = − 2 p + ; dt dS • = − 2 p + p − − 2 p + ( p − − gA − ( x − )) ; dt dp + = − 2 p − gA ′ − ( x − ) − 2 gσ E ′ ; • dt Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 16

Solution: ⊥ + 2 g σ E 0 f ′ ( τ ) d p 2 d τ p 2 ⊥ + 2 g σ E z ( x − ) 1 � � • S K = − x − = − p 0 − − g A − ( x − ) g E 0 γ + f ( τ ) A − ( x − ) = − E 0 • ω f ( τ = ω x − ) ; x - > p 0 x - x + ω p + + • Adiabaticity parameter: γ = g E ) ≈ k i, + ; > x x - + Im S σ = − π p 2 • t=0: γ ≪ 1 ; 2 g E 0 ; ⊥ x + t=0: Im D ( Y − y, p ⊥ ) ∝ e − 2 P σ Im S σ • 2 π p 2 ⊥ α S e − = S P s ( y ) = g E 0 g E 0 ; Q 2 2 π ; Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 17

Thermalization by a pulse of the chromelectric field ( Kluger, Eisenberg, Svetitsky & ... (1991 -1994) dk iz dε dt = g E z ∼ Q 2 ⋆ s ; dt =0; i ≃ ∆ k − ∆ k + ⋆ ∼ Q s ; i ⋆ ω = ⇒ Q s ; γ ≫ 1; ω e − τ and γ > 1 E 0 For A − = π p 2 2 g E 0 γ = 4 π p + ⊥ • Im S σ = ω √ dσ T = ω 2 d 3 p ∝ e − ǫ T ; • 4 π ; Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 18

Thermalization time ∆ p − = p − ( x − ) − p 0 • + = − ω ; τ = ω x − ≃ ln γ ; − gE 0 • ∆ p + = ω ; Therefore ln γ ln γ ≈ 0 . 6 1 � • ω = g E 0 ; x − ≃ = √ g E 0 √ Q s 2 π Q s Q s • • • T = ≈ 0 . 3 Q s ; • • • √ 4 π Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 19

Nuclear gluon distributions McLerran-Venugopalan formula: • dN MV 1 d 2 x ⊥ dy ∝ α S ¯ � s ( x )) � 4 x 2 ⊥ Q 2 s ( x ) ln( x 2 ⊥ Q 2 1 − e − 1 Our approach: • dN LLA d 2 x ⊥ dy ∝ ln(1 /x ) � s ( x )) � 1 − e − 1 4 x 2 ⊥ Q 2 s ( x ) ln( x 2 ⊥ Q 2 • dN LLA 1 � x 2 ⊥ Q 2 � d 2 x ⊥ dy ∝ α S ln s ( x ) ¯ � s ( x )) � 4 x 2 ⊥ Q 2 s ( x ) ln( x 2 ⊥ Q 2 1 − e − 1 Early ’Thermalization’ in CGC and Other Crazy Ideas E. Levin 20