Rapidity evolution of Wilson lines

Transcription

1 Rapidity evolution of Wilson lines I. Balitsky JLAB & ODU QCD evolution May 014 QCD evolution May /

2 Outline 1 High-energy scattering and Wilson lines High-energy scattering and Wilson lines. Evolution equation for color dipoles. Leading order: BK equation. NLO high-energy amplitudes Conformal composite dipoles NLO BK kernel in QCD. NLO hierarchy of Wilson-lines evolution. Evolution of 3-Wilson-line baryon operator Conclusions QCD evolution May 014 /

3 DIS at high energy At high energies, particles move along straight lines the amplitude of γ A γ A scattering reduces to the matrix element of a two-wilson-line operator (color dipole): QCD evolution May /

4 DIS at high energy At high energies, particles move along straight lines the amplitude of γ A γ A scattering reduces to the matrix element of a two-wilson-line operator (color dipole): Formally, A(s) = d k 4π IA (k ) B Tr{U(k )U ( k )} B means the operator expansion in Wilson lines QCD evolution May /

5 Four steps of an OPE Factorize an amplitude into a product of coefficient functions and matrix elements of relevant operators. Find the evolution equations of the operators with respect to factorization scale. Solve these evolution equations. Convolute the solution with the initial conditions for the evolution and get the amplitude QCD evolution May /

6 DIS at high energy: relevant operators At high energies, particles move along straight lines the amplitude of γ A γ A scattering reduces to the matrix element of a two-wilson-line operator (color dipole): d k A(s) = 4π IA (k ) B Tr{U(k )U ( k )} B [ ] U(x ) = Pexp ig du n µ A µ (un + x ) Wilson line QCD evolution May /

7 DIS at high energy: relevant operators At high energies, particles move along straight lines the amplitude of γ A γ A scattering reduces to the matrix element of a two-wilson-line operator (color dipole): Formally, d k A(s) = 4π IA (k ) B Tr{U(k )U ( k )} B [ ] U(x ) = Pexp ig du n µ A µ (un + x ) means the operator expansion in Wilson lines Wilson line QCD evolution May /

8 Rapidity factorization Y > Y < η - rapidity factorization scale Rapidity Y > η - coefficient function ( impact factor ) Rapidity Y < η - matrix elements of (light-like) Wilson lines with rapidity divergence cut by η [ ] = Pexp ig dx + A η + (x +, x ) U η x A η µ(x) = d 4 k (π) 4 θ(eη α k )e ik x A µ (k) QCD evolution May /

9 Spectator frame: propagation in the shock-wave background. Boosted Field Each path is weighted with the gauge factor Pe ig dx µa µ. Quarks and gluons do not have time to deviate in the transverse space we can replace the gauge factor along the actual path with the one along the straight-line path. z z x y Wilson Line [ x z: free propagation] [U ab (z ) - instantaneous interaction with the η < η shock wave] [ z y: free propagation ] QCD evolution May /

10 High-energy expansion in color dipoles Y > Y < The high-energy operator expansion is T{ĵ µ (x)ĵ ν (y)} = d z 1 d z Iµν LO (z 1, z, x, y)tr{ûz η 1 Û z η } + NLO contribution QCD evolution May /

11 High-energy expansion in color dipoles Y > Y < η - rapidity factorization scale Evolution equation for color dipoles d dη tr{uη x U η y } = α s π d (x y) z (x z) (y z) [tr{uη x U y η }tr{ux η U η N c tr{u η x U η y }] + α s K NLO tr{u η x U η y } + O(α s ) y } (Linear part of K NLO = K NLO BFKL ) QCD evolution May /

12 Evolution equation for color dipoles To get the evolution equation, consider the dipole with the rapidies up to η 1 and integrate over the gluons with rapidities η 1 > η > η. This integral gives the kernel of the evolution equation (multiplied by the dipole(s) with rapidities up to η ). α s (η 1 η )K evol QCD evolution May /

13 Evolution equation in the leading order d dη Tr{Û x Û y} = K LO Tr{Û x Û y} +... d dη Tr{Û x Û y} shockwave = K LO Tr{Û x Û y} shockwave x a x * x x x x * * x a x * x b a a b y b (a) (b) (c) (d) b U ab z = Tr{t a U z t b U z } (U x U y) η 1 (U x U y) η 1 + α s (η 1 η )(U x U z U z U y) η Evolution equation is non-linear QCD evolution May /

14 Non linear evolution equation Û(x, y) 1 1 N c Tr{Û(x )Û (y )} BK equation d dη Û(x, y) = α sn c π d z (x y) } {Û(x, z) + (x z) (y z) Û(z, y) Û(x, y) Û(x, z)û(z, y) I. B. (1996), Yu. Kovchegov (1999) Alternative approach: JIMWLK ( ) QCD evolution May /

15 Non-linear evolution equation Û(x, y) 1 1 N c Tr{Û(x )Û (y )} BK equation d dη Û(x, y) = α sn c π d z (x y) } {Û(x, z) + (x z) (y z) Û(z, y) Û(x, y) Û(x, z)û(z, y) I. B. (1996), Yu. Kovchegov (1999) Alternative approach: JIMWLK ( ) LLA for DIS in pqcd BFKL (LLA: α s 1, α s η 1) QCD evolution May /

16 Non-linear evolution equation Û(x, y) 1 1 N c Tr{Û(x )Û (y )} BK equation d dη Û(x, y) = α sn c π d z (x y) } {Û(x, z) + (x z) (y z) Û(z, y) Û(x, y) Û(x, z)û(z, y) I. B. (1996), Yu. Kovchegov (1999) Alternative approach: JIMWLK ( ) LLA for DIS in pqcd BFKL (LLA: α s 1, α s η 1) LLA for DIS in sqcd BK eqn (LLA: α s 1, α s η 1, α s A 1/3 1) (s for semiclassical) QCD evolution May /

17 Why NLO correction? To check that high-energy OPE works at the NLO level. To check conformal invariance of the NLO BK equation(in N =4 SYM) To determine the argument of the coupling constant of the BK equation(in QCD). To get the region of application of the leading order evolution equation. QCD evolution May /

18 Hign-energy OPE In the next-to-leading order Y > Y < DIS structure function F (x): photon impact factor + evolution of color dipoles+ initial conditions for the small-x evolution Photon impact factor in the LO (x y) 4 T{ ˆψ(x)γ µ ν d ˆψ(x) ˆψ(y)γ ˆψ(y)} z 1 d z = z 4 Iµν LO (z 1, z )tr{ûz η 1 Û z η } 1 Iµν LO κ 4 [ (z 1, z ) = (κ ζ1 π 6 (κ ζ 1 ) 3 (κ ζ ) 3 x µ y ν )(κ ζ ) 1 κ (ζ 1 ζ ) ]. κ 1 sx + (p 1 s x p + x ) x y, ζ i ( p 1 s + z i p + z i ) NLO impact factor is calculated recently (G.A. Chirilli and I.B, 013) QCD evolution May /

19 NLO impact factor and conformal Möbius invariance z z 1 z z z 1 z y z 3 x y z 3 x z z (a) I NLO (x, y; z 1, z, z 3 ; η) = I LO λ z [ 13 π z ln eη s 1 z 3 4 Z 3 iπ ] + C The NLO impact factor is not Möbius invariant the color dipole with the cutoff η is not invariant However, if we define a composite operator (a - analog of µ for usual OPE) (b) [Tr{Ûz η 1 Û z η } ] conf + λ π = Tr{Û η z 1 Û η z } d z 1 z 3 z [Tr{T n Û 13 z z η 1 Û z η 3 T n Ûz η 3 Û z η } N c Tr{Ûz η 1 Û z η }] ln az 1 3 z 13 z 3 + O(λ ) the impact factor becomes conformal in the NLO. QCD evolution May /

20 Operator expansion in conformal dipoles T{Ô(x)Ô(y)} = + d z 1 d z I LO (z 1, z )Tr{Û η z 1 Û η z } conf d z 1 d z d z 3 I NLO (z 1, z, z 3 )[ 1 N c Tr{T n Û η z 1 Û η z 3 T n Û η z 3 Û η z } Tr{Û η z 1 Û η z }] I NLO = I LO λ π z [ 1 dz 3 z 13 z 3 ln z 1 eη as ] z Z 13 z 3 iπ + C 3 The new NLO impact factor is conformally invariant Tr{Û η z 1 Û η z } conf is Möbius invariant (in N = 4 SYM) We think that one can construct the composite conformal (in N = 4 SYM) dipole operator order by order in perturbation theory. Analogy: when the UV cutoff does not respect the symmetry of a local operator, the composite local renormalized operator in must be corrected by finite counterterms order by order in perturbaton theory. QCD evolution May /

21 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) QCD evolution May /

22 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) QCD evolution May /

23 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) We calculate the matrix element of the r.h.s. in the shock-wave background α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) QCD evolution May /

24 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) We calculate the matrix element of the r.h.s. in the shock-wave background α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) Subtraction [ ] of the (LO) contribution (with the rigid rapidity cutoff) 1 v prescription in the integrals over Feynman parameter v Typical integral dv 1 [ 1 ] (k p) v + p (1 v) v = 1 + p ln (k p) p QCD evolution May /

25 Gluon part of the NLO BK kernel: diagrams (I) (II) (III) (IV) (V) (VI) (VII) (VIII) (IX) (X) (XI) (XII) (XIII) (XIV) (XV) QCD evolution May /

26 Diagrams for 1 3 dipoles transition (XVI) (XVII) (XVIII) (XIX) (XX) (XXI) (XXII) (XXIII) (XXIV) (XV) (XXVI) (XXVII) (XVIII) (XXIX) (XXX) QCD evolution May /

27 Diagrams for 1 3 dipoles transition (XXXI) (XXXII) (XXXIII) (XXXIV) QCD evolution May /

28 "Running coupling" diagrams x y (I) (II) (III) (IV) (V) (VI) (VII) (VIII) (IX) (X) QCD evolution May 014 /

29 1 dipole transition diagrams a x x a x x x a x x x x a * * a x * * * q q q q q b b b c c b b k k k k k k k k k k c d (a) c d d (b) (c) d c (d) (e) x x x x x * x x x x x * * * * a a a b q q q b a q a k q k k k d d d k d c k k k k c c k c c (f) (g) b (h) b (i) b (j) QCD evolution May /

30 NLO evolution of composite conformal dipoles in QCD I. B. and G. Chirilli a d da [tr{u z 1 U z }] comp a = α ( s π d z 3 [tr{u z1 U z 3 }tr{u z3 U z } N c tr{u z1 U z }] comp a z [ 1 z 1 + α sn c ( b ln z 13 z 3 4π 1 µ + b z 13 z 3 z ln z z 3 z π ) ] 3 + α s d z {[ 4 4π + z 3 z 3 + z 4 z 13 4z 1 z 34 (z 3 z 3 z 4 z 13 ) ln z 3 z ] 3 z 4 34 z 4 z 13 [tr{u z1 U z 3 }tr{u z3 U z 4 }{U z4 U z } tr{u z1 U z 3 U z4 U z U z3 U z 4 } (z 4 z 3 )] + z 1 z [ 34 z ln z 1 z ( 34 z 1 13 z 4 z 3 z z ) 34 3 z 13 z 4 z 3 z ln z 13 z ] 4 3 z 3 z 3 } [tr{u z1 U z 3 }tr{u z3 U z 4 }tr{u z4 U z } tr{u z1 U z 4 U z3 U z U z4 U z 3 } (z 4 z 3 )] b = 11 3 N c 3 n f K NLO BK = Running coupling part + Conformal "non-analytic" (in j) part + Conformal analytic (N = 4) part Linearized K NLO BK (for composite dipoles) reproduces the result for the forward NLO BFKL kernel for Green function of two reggeized gluons. QCD evolution May /

31 NLO hierarchy of evolution of Wilson lines (G.A.C. and I.B., 013) a) b) c) d) e) f) Figure : Typical NLO diagrams: self-interaction (a,b), pairwise interactions (c,d), and triple interaction (e,f) QCD evolution May /

32 Self-interaction (gluon reggeization) a) b) d dη (U 1) ij = α s d z 4 d z 5 8π 4 z 45 ([ I 1 4 ] z 45 { U4 dd (Uee 5 U4 ee ) f ade f bd e (t a U 1 t b ) ij + (z 14, z 15 ) z 14 z 15 + α s N c d z {[ π 3 z (U4 ab Uab 1 )(ta U 1 t b ) ij 14 3 ln z 14µ π 3 ln z [ 14 if ad e z ({t d, t e }U 1 t a ) ij if ade (t a ] )} U 1 {t d, t e }) ij 15 ] I 1 I(z 1, z 4, z 5 ) = ln z 14 /z [ 15 z 14 + z 15 z 14 z 15 z (z 14, z 15 ) 45 z (z 14, z 15 ) ] 14 z 15 QCD evolution May /

33 Pairwise interaction c) d) d dη (U 1) ij (U ) kl = α s 8π 4 d z 4 d z 5 (A 1 + A + A 3 ) + α s 8π 3 d z 4 (B 1 + N c B ) QCD evolution May /

34 Pairwise interaction A 1 = [ (t a U 1 ) ij (U t b ) kl + (U 1 t b ) ij (t a ] U ) kl [ ( f ade f bd e U4 dd (Uee 5 U4 ee ) K 4 z 4 + I 1 45 z + I )] 45 z 45 K = NLO BK kernel for N = 4 SYM A = 4(U 4 U 1 ) dd (U 5 U ) ee { i [ f ad e (t d U 1 t a ) ij (t e U ) kl f ade (t a ] U 1 t d ) ij (U t e ) kl J145 ln z 14 z 15 + i [ f ad e (t d U 1 ) ij (t e U t a ) kl f ade (U 1 t d ) ij (t a ] U t e ) kl J154 ln z } 4 z 5 J 145 J(z 1, z, z 4, z 5 ) = (z 14, z 5 ) z (z 15, z 45 )(z 15, z 5 ) 14 z 5 z 45 z + (z 5, z 45 ) 14 z 15 z 5 z 45 z 14 z 5 z 45 QCD evolution May /

35 Pairwise interaction A 3 = U4 {i [ dd f ad e (U 1 t a ) ij (t d t e U ) kl f ade (t a ] U 1 ) ij (U t e t d ) kl + (J 145 J 154 ) ln z ] 4 z (U 5 U ) ee 5 [ J 145 ln z 14 z 15 + i [ f ad e (t d t e U 1 ) ij (U t a ) kl f ade (U 1 t e t d ) ij (t a ] U ) kl [ J 145 ln z 4 z 5 + (J 145 J 154 ) ln z 14 z 15 ](U 5 U 1 ) ee } J 145 J (z 1, z, z 4, z 5 ) = (z 4, z 5 ) z (z 4, z 45 )(z 15, z 5 ) 4 z 5 z 45 z + (z 5, z 45 )(z 14, z 4 ) 4 z 5 z 15 z 45 z (z 14, z 4 )(z 15, z 5 ) 14 z 4 z 5 z 45 z 14 z 15 z 4 z 5 QCD evolution May /

36 Pairwise interaction B 1 = ln z 14 z 1 ln z 4 z 1 { (U 4 U 1 ) ab i [ f bde (t a U 1 t d ) ij (U t e ) kl + f ade (t e U 1 t b ) ij (t d ] [ (z 14, z 4 ) U ) kl z 1 14 z 4 z 14 + (U 4 U ) ab i [ f bde (U 1 t e ) ij (t a U t d ) kl + f ade (t d U 1 ) ij (t e U t b ) kl ] [ (z 14, z 4 ) z 14 z 4 1 z 4 ] ]} B = [ U ab { (z14, z 4 ) z 14 z 4 4 Uab 1 Uab [11 3 ln z 1µ π 3 ] [(t a U 1 ) ij (U t b ) kl + (U 1 t b ) ij (t a U ) kl ] ] 11[ z ln z 4 14 z z 4 ln z ] } 14 z 1 QCD evolution May /

37 Triple interaction e) f) J 1345 J (z 1, z, z 3, z 4, z 5 ) = (z 14, z 34 )(z 5, z 35 ) z 14 z 5 z 34 z 35 (z 14, z 45 )(z 5, z 35 ) z 14 z 5 z 35 z 45 + (z 5, z 45 )(z 14, z 34 ) z 14 z 5 z 34 z 45 + (z 14, z 5 ) z 14 z 5 z 45 QCD evolution May /

38 Triple interaction d dη (U 1) ij (U ) kl (U 3 ) mn = i α s π 4 d z 4 d z 5 {J 1345 ln z 34 z 35 f cde[ (t a U 1 ) ij (t b U ) kl (U 3 t c ) mn (U 4 U 1 ) ad (U 5 U ) be (U 1 t a ) ij (U t b ) kl (t c U 3 ) mn (U 4 U 1 ) da (U 5 U ) eb] + J 3145 ln z 14 z 15 f ade[ (U 1 t a ) ij (t b U ) kl (t c U 3 ) mn (U 4 U 3 ) cd (U 5 U ) be (t a U 1 ) ij (U t b ) kl (U 3 t c ) mn (U dc 4 Udc 3 )(Ueb 5 Ueb )] + J 1345 ln z 4 z 5 f bde[ (t a U 1 ) ij (U t b ) kl (t c U 3 ) mn (U 4 U 1 ) ad (U 5 U 3 ) ce (U 1 t a ) ij (t b U ) kl (U 3 t c ) mn (U 4 U 1 ) da (U 5 U 3 ) ec] (1) QCD evolution May /

39 Rapidity evolution of the baryon tripole Baryon operator B 13 = ε i j h ε ijh U i i (r 1 )U j j (r 1 )U h h (r 3 ) U 1 U U 3, Evolution equation in the LO (A. Grabovsky, 013) d dη B 13 = α [ s3 r ( 8π d r 1 a r ) 4 r 41 r 4 ln 1 r 41 r 4 ( B ] 6 (B 144B 34 + B 44 B 314 B 344 B 14 )) + (1 3) + ( 3). QCD evolution May /

40 Composite conformal baryon operator in the NLO General prescription: O conf = O + 1 O η r mn r im r in r mn r im r in ( r ln mn a r im r in (cf. Conformal JIMWLK by Kovner and Lublinsky, 014) Composite conformal baryon operator B conf 13 = B 13 + α s3 8π [ r d r 1 4 r 41 r 4 ( a r ln 1 r 41 r 4 ( B ] 6 (B 144B 34 + B 44 B 314 B 344 B 14 )) + (1 3) + ( 3). ) ) QCD evolution May /

41 NLO evolution of baryon operator (A. Grabovsky and I.B., 014) db conf 13 +L C 1 dη +M C 1 = LO α s 8π 4 d r 0 d r 4 ({ L 1 C ) (U 1 U 0 U 4 U 3 + tr [(U 0 U 4 U ) ) ) (U 0 U 4 U (U 1 U 0 U 4 U 3 (U ) ( ) 0 U 4 U 1 U 0 U U 3 U 4 ] 3 4 [B 144B 34 + B 44 B 134 B 344 B 14 ] + 1 B 13 ) ) ) [(U 0 U 4 U 3 (U U 0 U 1 U 4 + (U 1 U 0 U +Z 1 B 355 B 15 + (all 5 permutations 1 3) ( [ ( α s 11 r ) ( 8π 3 d r 5 ln r 5 r 5 1 ) r 15 r 1 r 01 r 5 ( 3 (B 155B 35 + B 55 B 135 B 355 B 15 ) 9B 13 ) (U 3 U 4 U 0 } ] U 4 ) + (0 4) ( r )] ln 1 µ ) ) + (1 3) + ( 3). QCD evolution May /

42 NLO evolution kernels Here L1 C BK = KNLO 1 + L 1 C BK = KNLO 1 + ( r 1 4 r 15 r 45 r ln r5 ) ( r 14 r 1 4 r + 45 r 1 4 r 5 r 45 r ln r15 r 4 14 r 45 r 1 ( r 1 4 r 15 r 45 r ln r5 ) ( r 14 r 1 4 r 45 r 1 4 r 5 r 45 r ln r15 r 4 14 r 45 r 1 Z 1 = r 1 [( r35 8 r 15 r 5 r 45 r r ) ( 5 r5 ) r r ln 45 r 4 r 45 r 1 + r 15 ( r 45 r ln r5 ) r r + r 13 ( 35 r 4 r 14 r ln r35 )] r 1 34 r (1 3), 5 r 13 ), ), QCD evolution May /

43 NLO evolution kernels ( M1 C = r 1 16 r 5 r 45 r ln r15 r ) ( 5 r 4 34 r 1 14 r 4 35 r + 14 r 4 16 r 15 r 45 r ln r35 4 r 4 45 r 4 ) 1 r 4 4 r 15 r 6 5 r 14 r 4 34 ( r r 5 r 45 r ln r15 4 r 35 r 6 ) ( 4 r 34 r 3 34 r 5 r 4 45 r r r 35 r 45 r ln r5 r ) 35 r r 4 15 r 4 r 34 ( r r 35 r 45 r ln r5 4 r ) ( 14 r 34 r r 15 r + 35 r r 15 r 45 r ln r5 4 r ) 14 r r 15 r 35 r 4 4 r ( 35 r r 15 r 5 r 45 r ln r15 r ) 35 r 4 4 r ( 3 r 1 34 r 5 r 45 r + 1 r 34 8 r 15 r 5 r 4 r ln r5 r ) 1 r r 15 r 3 r 4 r ( 14 r r 15 r 45 r 4 r ln r15 r 45 r ) 3 r 4 34 r 4 5 r 14 r 34 QCD evolution May /

44 Conclusions High-energy operator expansion in color dipoles works at the NLO level. QCD evolution May 014 /

45 Conclusions High-energy operator expansion in color dipoles works at the NLO level. The NLO BK kernel in for the evolution of conformal composite dipoles in N = 4 SYM is Möbius invariant in the transverse plane. The NLO BK kernel agrees with NLO BFKL equation. The correlation function of four Z operators is calculated at the NLO order. It gives the anomalous dimensions of gluon light-ray operators at the BFKL point j 1 NLO photon impact factor is calculated. NLO hierarchy of Wilson-line evolution is derived. NLO evolution of baryon operator ( odderon contribution) is obtained. QCD evolution May 014 /