Low x evolution equation for quadrupole operator A.V. Grabovsky - PowerPoint PPT Presentation

Low x evolution equation for quadrupole operator A.V. Grabovsky Budker Inst. of Nuclear Physics and Novosibirsk University Photon 2015, BINP Novosibirsk, 18.06.2015 A.V. Grabovsky Low x evolution equation for quadrupole operator

Outline Definitions Introduction Shockwave formallism Results for qadrupole operator Summary A.V. Grabovsky Low x evolution equation for quadrupole operator

Definitions Introduce the light cone vectors n 1 and n 2 n 2 = 1 n + 1 = n − n 1 = ( 1 , 0 , 0 , 1 ) , 2 ( 1 , 0 , 0 , − 1 ) , 2 = n 1 n 2 = 1 For any p define p ± p + = pn 2 = 1 � p 0 + p 3 � p − = pn 1 = p 0 − p 3 , , 2 p 2 = 2 p + p − − � p 2 ; The scalar products: ( p k ) = p µ k µ = p + k − + p − k + − � p = p + n 1 + p − n 2 + p ⊥ , p � k . A.V. Grabovsky Low x evolution equation for quadrupole operator

Definitions Wilson line describing interaction with external field b − η made of slow gluons with p + < e η d 4 p � + ∞ � −∞ dz + b − η ( z + ,� z ) , ( 2 π ) 4 e − ipz b − ( p ) θ ( e η − p + ) . U η z = Pe ig b − η = � A.V. Grabovsky Low x evolution equation for quadrupole operator

Introduction Dipole picture s ≫ Q 2 ≫ Λ 2 QCD γ p 1 � � d 2 r | Ψ γ ∗ ( r , Q 2 ) | 2 σ dip ( r , s ) , σ γ ∗ ( s , Q 2 ) = σ dip ( r , s ) = 2 d b ( 1 − F ( b , r , s )) N c r = r 1 − r 2 — dipole size, b = 1 2 ( r 1 + r 2 ) — impact parameter, F = tr ( U 1 U † 2 ) , — dipole Green function, U i = U η — Wilson lines, describing fast moving quarks interacting with the target. i η — rapidity divide, gluons with p + > e η belong to photon wavefunction, gluons with p + < e η belong to Wilson lines, describing the field of the target. A.V. Grabovsky Low x evolution equation for quadrupole operator

Introduction tr ( U 1 U † 2 ) obeys the LO Balitsky-Kovchegov evolution equation ∂ tr ( U 1 U † � r 2 2 ) � = α s � � 12 tr ( U 1 U † 4 ) tr ( U 4 U † 2 ) − N c tr ( U 1 U † d � r 4 2 ) . 2 π 2 � r 2 14 � r 2 ∂η 42 LO equation was obtained in 1996-99, NLO — in 2007-2010 (Balitsky and Chirilli). A.V. Grabovsky Low x evolution equation for quadrupole operator

Shock wave For a fast moving particle with the velocity − β and the field strength tensor F ( x + , x − ,� x ) in its rest frame, in the observer’s frame the field will look like = λ F − i ( λ y + , 1 F i ( � F − i � y + , y − ,� λ y − ,� y + � � � y y ) → δ y ) , F − i ≫ F ... � 1 + β in the Regge limit λ → + ∞ , λ = 1 − β . Therefore the natural choice for the gauge is b i , + = 0 , b − is the solution of the equations ∂ b − ∂ y i = δ ( y + ) F i ( � y ) , i . e . b µ ( y ) = δ ( y + ) B ( � y ) n µ 2 It is the shock-wave field. A.V. Grabovsky Low x evolution equation for quadrupole operator

Propagator in the shock wave background Choose the gluon field A in the gauge A n 2 = 0 as a sum of external classical b and quantum A . b µ ( x ) = δ ( x + ) B ( � x ) n µ A = A + b , 2 . The A - b interaction lagrangian has only one vertex ← − → � � L i = g ∂ 2 f acb ( b − ) c g αβ A a ∂ x − A b . α β ⊥ The free propagator G µν 0 ( x + , p + ,� p ) = = − d µν 0 ( p + , p ⊥ ) p 2 x + e − i � + n µ θ ( x + ) θ ( p + ) − θ ( − x + ) θ ( − p + ) 2 n ν 2 p + � � 2 . . . , 2 p + ⊥ − p µ ⊥ n µ − n µ ⊥ n ν 2 + p ν 2 n ν p 2 2 � d µν 0 ( p ) = g µν 2 ( p + ) 2 . p + A.V. Grabovsky Low x evolution equation for quadrupole operator

Propagator in the shock-wave background Sum the diagrams b b b b does not depend on x − , hence the conservation of p + , p 2 ( x + 1 − x + � 2 ) b ∼ δ ( x + ) , hence e − i 2 p + → 1 in every internal vertex, g µν ⊥ d 0 νρ g ρσ ⊥ = g µσ ⊥ , hence no dependence on � p = ⇒ conservation of � x in every internal vertex Propagator in the shock-wave background: d 4 z δ ( z + ) F + i ( z ) U � � F + i ( z ) A ν ( y ) . z G µν ( x , y ) | x + > 0 > y + = 2 iA µ ( x ) − − → ∂ ∂ z − where the interaction with b is through Wilson line � x + y + dz + b − ( z + ,� z ) . z = Pe ig U � A.V. Grabovsky Low x evolution equation for quadrupole operator

Dipole picture A ∼ δ ( x + ) γ Color field of a fast moving particle A − ∼ δ ( z + ) A η ( z ⊥ ) A η ( z ⊥ ) contains slow components with rapidities < η Quark propagator in such an external field G ( x , y ) ∼ U η ( z ⊥ ) DIS matrix element contains a Wilson loop = color dipole operator U η 12 = tr ( U η ( z 1 ⊥ ) U η † ( z 2 ⊥ )) . Balitsky 1996 A.V. Grabovsky Low x evolution equation for quadrupole operator

Balitsky derivation of the BK equation To derive the evolution equation we have to change η → η + ∆ η and integrate over the fields with the rapidities in the strip ∆ η 12 + � 0 | T ( U ∆ η � 12 e i L ( z ) dz ) | 0 � U η +∆ η = U η . 12 � � 0 | T ( e i L ( z ) dz ) | 0 � z 1 b a b c d z 2 a f e ∂ U η z 2 � = α s � U η 14 U η 42 − N c U η 12 12 � � d � z 4 . 12 2 π 2 z 2 z 2 ∂η � 14 � 42 A.V. Grabovsky Low x evolution equation for quadrupole operator

Motivation γ γ p p Dipole picture, Evolution equation for BK equation for dipole quadrupole operator tr ( U 1 U † tr ( U 1 U † 2 U 3 U † 2 ) 4 ) A.V. Grabovsky Low x evolution equation for quadrupole operator

LO Evolution equation for quadrupole † ... U k U l † ) ≡ U ij † ... kl † , tr ( U i U j Jalilian-Marian Kovchegov Dumitru 2004, 2010 Dominguez Mueller Munier Xiao 2011 r 142 � ∂ U 12 † 34 † = α s � � d � r 0 r 402 ( U 10 † U 02 † 34 † + U 4 † 0 U 12 † 30 † − ( 0 → 1 )) 4 π 2 � r 102 � ∂η r 122 � + r 202 ( U 10 † U 02 † 34 † + U 2 † 0 U 10 † 34 † − ( 0 → 1 )) � r 102 � r 242 � − r 402 ( U 10 † U 02 † 34 † + U 30 † U 04 † 12 † − ( 0 → 4 )) 2 � r 202 � r 132 � − r 302 ( U 4 † 0 U 12 † 30 † + U 2 † 0 U 34 † 10 † − ( 0 → 1 )) 2 � r 102 � +( 1 ↔ 3 , 2 ↔ 4 ) } . A.V. Grabovsky Low x evolution equation for quadrupole operator

NLO corrections NLO evolution of 1 and 2 Wilson lines with open indices from Balitsky and Chirilli 2013 NLO evolution of 3 Wilson lines Grabovsky 2013 � K NLO ⊗ U 12 † 34 † � = α 2 r 5 ( G s + G a )+ α 2 � � s s d � r 0 d � d � r 0 ( G β + G ) , 8 π 4 8 π 3 A.V. Grabovsky Low x evolution equation for quadrupole operator

NLO corrections: symmetric part G s = G s 1 + n f G q + G s 2 + ( 1 ↔ 3 , 2 ↔ 4 ) . G s 1 = ( { U 0 † 34 † 15 † 02 † 5 − U 5 † 0 U 2 † 5 U 0 † 34 † 1 − ( 5 → 0 ) } + ( 5 ↔ 0 )) ( L 12 + L 32 − L 13 ) + ( { U 0 † 15 † 02 † 34 † 5 − U 0 † 5 U 5 † 1 U 2 † 34 † 0 − ( 5 → 0 ) } + ( 5 ↔ 0 )) ( L 12 + L 14 − L 42 ) , G q =( { U 0 † 34 † 12 † 5 + U 2 † 34 † 15 † 0 − U 0 † 5 U 2 † 34 † 1 − U 2 † 5 U 0 † 34 † 1 − ( 5 → 0 ) } + ( 5 ↔ 0 )) N 2 N c c × 1 + 1 L q 12 + L q 32 − L q L q 12 + L q 14 − L q � � � � 13 42 2 2 × ( { U 0 † 12 † 34 † 5 + U 2 † 34 † 15 † 0 − U 0 † 5 U 2 † 34 † 1 − U 5 † 1 U 2 † 34 † 0 − ( 5 → 0 ) } + ( 5 ↔ 0 )) , N c N 2 c 2 G s 2 = ( U 0 † 15 † 02 † 34 † 5 − U 0 † 5 U 5 † 1 U 2 † 34 † 0 + ( 5 ↔ 0 )) ( M 14 2 + M 12 4 + ( 5 ↔ 0 )) + ( U 0 † 34 † 15 † 02 † 5 − U 5 † 0 U 2 † 5 U 0 † 34 † 1 + ( 5 ↔ 0 )) ( M 23 1 + M 21 3 + ( 5 ↔ 0 )) + ( U 0 † 34 † 52 † 05 † 1 − U 0 † 1 U 2 † 5 U 4 † 05 † 3 + ( 5 ↔ 0 )) ( M 34 1 − M 24 1 + M 43 2 − M 13 2 + ( 5 ↔ 0 )) + ( U 0 † 35 † 02 † 54 † 1 − U 0 † 3 U 2 † 5 U 4 † 15 † 0 + ( 5 ↔ 0 )) ( M 14 3 − M 24 3 + M 41 2 − M 31 2 + ( 5 ↔ 0 )) A.V. Grabovsky Low x evolution equation for quadrupole operator

NLO corrections: antisymmetric part G a = G a 1 + G a 2 + G a 3 . G a 1 = ( U 0 † 1 U 2 † 5 U 4 † 05 † 3 + U 0 † 34 † 52 † 05 † 1 − ( 5 ↔ 0 )) ( M 31 2 − M 34 2 − M 42 1 + M 43 1 ) + ( U 0 † 3 U 2 † 5 U 4 † 15 † 0 + U 0 † 35 † 02 † 54 † 1 − ( 5 ↔ 0 )) ( M 13 2 − M 14 2 − M 42 3 + M 41 3 ) +( 1 ↔ 3 , 2 ↔ 4 ) . G a 2 = 1 2 ( U 0 † 34 † 15 † 02 † 5 − ( 5 ↔ 0 )) (˜ L 13 + 2 M 21 − 2 M 23 − M 23 1 + M 21 3 − ( 5 ↔ 0 )) + 1 2 ( U 0 † 15 † 02 † 34 † 5 − ( 5 ↔ 0 )) (˜ L 42 − 2 M 12 + 2 M 14 + M 14 2 − M 12 4 − ( 5 ↔ 0 )) +( 1 ↔ 3 , 2 ↔ 4 ) . G a 3 = 1 2 ( U 0 † 5 U 5 † 1 U 2 † 34 † 0 − ( 5 ↔ 0 )) (˜ L 12 + ˜ L 14 − 2 M 24 + M 14 2 + M 12 4 − ( 5 ↔ 0 )) + 1 2 ( U 5 † 0 U 2 † 5 U 0 † 34 † 1 − ( 5 ↔ 0 )) (˜ L 21 + ˜ L 23 − 2 M 13 + M 23 1 + M 21 3 − ( 5 ↔ 0 )) +( 1 ↔ 3 , 2 ↔ 4 ) . A.V. Grabovsky Low x evolution equation for quadrupole operator