High energy QCD at NLO: from light-cone wave function to JIMWLK evolution

Journal of High Energy Physics, May 2017

Soft components of the light cone wave-function of a fast moving projectile hadron is computed in perturbation theory to the third order in QCD coupling constant. At this order, the Fock space of the soft modes consists of one-gluon, two-gluon, and a quark-antiquark states. The hard component of the wave-function acts as a non-Abelian background field for the soft modes and is represented by a valence charge distribution that accounts for non-linear density effects in the projectile. When scattered off a dense target, the diagonal element of the S-matrix reveals the Hamiltonian of high energy evolution, the JIMWLK Hamiltonian. This way we provide a new direct derivation of the JIMWLK Hamiltonian at the Next-to-Leading Order.

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High energy QCD at NLO: from light-cone wave function to JIMWLK evolution

Received: October High energy QCD at NLO: from light-cone wave function to JIMWLK evolution Michael Lublinsky 0 1 2 Yair Mulian 0 2 Open Access 0 c The Authors. 0 0 Beer-Sheva 84105 , Israel 1 Physics Department, University of Connecticut 2 Department of Physics, Ben-Gurion University of the Negev Soft components of the light cone wave-function of a fast moving projectile hadron is computed in perturbation theory to the third order in QCD coupling constant. At this order, the Fock space of the soft modes consists of one-gluon, two-gluon, and a quark-antiquark states. The hard component of the wave-function acts as a non-Abelian background eld for the soft modes and is represented by a valence charge distribution that accounts for non-linear density e ects in the projectile. When scattered o a dense target, the diagonal element of the S-matrix reveals the Hamiltonian of high energy evolution, the JIMWLK Hamiltonian. This way we provide a new direct derivation of the JIMWLK Hamiltonian at the Next-to-Leading Order. NLO Computations; QCD Phenomenology Contents 1 Introduction and summary Basics of JIMWLK Light cone QCD Hamiltonian Field quantisation Light cone wave-function of a fast hadron Eigenstates of free Hamiltonian Eikonal approximation for QCD Hamiltonian Eikonal scattering LO JIMWLK Hamiltonian NLO JIMWLK Hamiltonian The light cone wave function at NLO Third order perturbation theory Matrix elements Technical aspects of the calculation Computation of the NLO wave function Quark anti-quark state Two gluon state One gluon state The nal result Computation of qq Computation of Computation of Computation of Computation of Computation of JJSSJJ and JJJSJ Virtual contributions The NLO JIMWLK Hamiltonian assembled Reduction of the (LO)2 contribution C Integrals and Fourier transformations D Properties of the NLO JIMWLK kernels The phase of the wave function G Supplementary for section 3 G.1 Supplement for computation of G.2 Supplement for computation of G.3 Supplement for computation of G.4 Supplement for computation of G.5 Supplement for computation of G.6 Supplement for computation of H Supplement for section 4 H.1 Supplement for computation of qq H.2 Supplement for computation of H.3 Supplement for computation of H.4 Supplement for computation of JJSSJJ Colour Glass Condensate (CGC). functional equation of the form Introduction and summary dO = HJIMWLK O : Here the rapidity Y JIMWLK was argued to be incomplete). to the lowest order (LO) in recent progress in s corrections to the triple Pomeron vertex. framework. rst part, the light coordinate). Soft gluons with momenta smaller than the cuto do not participate in scattering. at NLO schematically has the form j i = (1 ) j no soft gluonsi + ) j one soft gluoni ) j two soft gluonsi + gs2 6 j quark antiquarki : rather than being sourced by any topology. in which s corrections enhanced by the density were resummed. the cuto space. In fact, is a rapidity evolution operator = e Y HJIMWLK Y HJIMWLK + Y 2 HJ2IMWLK : : : de ned as HJIMWLK d Y j Y =0 our direct calculation. check on our calculation. devoted to the calculation of Basics of JIMWLK tonian formalism. Light cone QCD Hamiltonian nates, four-vectors are x = (x+; x ; x), where x+ 0 + x3 and x x3 stand metric [86]. The light-cone gauge: Aa+ = Aa0 + Aa3 = 0: sketched in appendix A: HLC QCD dx d2x 1 a(x ; x) a(x ; x) + gf abcAibAjc: i 2 (1; 2) is a transverse component index; denotes Dirac's 4-component quark spinor,1 with free Hamiltonian H0 given by: (@iAja)2 + i +y @@i+@i The interaction Hamiltonian Hint reads gf abcAibAjc@iAja + f abcf adeAibAjcAidAje @+ (Aib@+Aic) @1+ (Ajd@+Ae) j @+ (Aib@+Aic) @1+ ig2 +ytatb iAia @1+ ( +yta +) + 2g2 1 gauge group. Field quantisation gauge elds: Aia(x) = Z 1 dk+ Z aia(k+; k)e ik x + aiay(k+; k)eik x : p+) (2)(k Transforming to coordinate space, aia(k+; k) = e ik z aia(k+; z) ; aiay(k+; k) = eik z aiay(k+; z); 1The avour index is suppressed in this section. p+) (2)(x b (k+; k)e ik x + d y(k+; k)eik x 1 = 2 = 2 1 1 2 : b 1 (k+; k); b y(p+; p)o = n 2 d 1 (k+; k); d y(p+; p)o 2 = (2 ) b 1 (k+; x); b y(p+; y)o = n 2 d 1 (k+; x); d y(p+; y)o 2 = 2 H0 = Z 1 dk+ Z (2 )2 2k+ aiay(k+; k) aia(k+; k) + X hb y(k+; k) b (k+; k) d (k+; k) d y(k+; k) the commutation relation becomes: For the quark elds: +(x) = = 12 The polarisation vectors are: Z 1 dk+ Z + 12 = 2 = The anti-commutation relations: Light cone wave-function of a fast hadron , which is implicitly related are not are also referred as valence modes. valence modes, and A for the soft gluon eld. of partons in the LCWF get shifted up ( gure 1): j iY0 = j i : j iY = j i get lifted above Ignoring the modes below momenta. be to Eigenstates of free Hamiltonian aia(k+; k) j0i = 0; bi (k+; k) j0i = 0; di (k+; k) j0i = 0; for any One gluon state. jgia(k)i (2 )3=2 j0i gia(k+; z) (2 )1=2 j0i ; These is a normalised state with the normalisation Two gluon state. Dgjb(p) jgia(k) E = ab ij (2)(k p) (k+ gia(k+; z) gjb(p+; z0)E gia(k) gjb(p) aiay(k) ajby(p) aiay(k+; z) ajby(p+; z0) b y(k) d y(p) b y(k+; z) d y(p+; z0) with the energy Egg(k; p) Quark anti-quark state. q 2 (p+; z0) q 1 (k+; z) elds with the bar contain modes with k+ > e Y this decomposition into Hint is done in appendix B. +(x) = +(x) where the underlined elds contain (soft modes) while (valence modes). Substitution of +). The action of @1+ is employ the eikonal approximation. The result is The rst eight contributions de ned by (B.5) (B.12), account for soft-soft and soft-valence a background for the soft elds, and is de ned by: ( p) + qaq( p); if abc Z 1 dk+ Z a Z 1 dk+ Z d2k p) b (k+; k) + d (k+; k) d y(k+; k where a(x) is a Fourier transform of a(p): a(x) = ( k) = d2x e ik x a(x) : (B.9) in terms of further details and write the nal results, Hg = Hg qq = 1; 2= 12 dk+ Z dk+ dp+ 2jk+j3=2 haiay(k+; k) a( k) + aia(k+; k) a(k)i ; (2 )2 (2 )2 2(p+ dp+ dq+ Z hb y(p+; p) d y(q+; q) + h:c:i p+ + q+ p+ + q+ dk+ dp+ (2 )2 (2 )2 igf abc 2p2k+p+q+ i aia(k)ajby(p)ajcy(q) (3)( k + p + q) i aiay(k)ajby(p)ajc(q) (3)(k + p q); + h:c: dp+ dq+ Z d2q ig2f abc (2 )2 (2 )2 pp+q+ 2(p+ + q+)q2) aiby(p+; p) aicy(q+; q) ( p (2 )2 (2 )2 (p+ + q+)2 Hg gg = Hgg inst = Hqq inst = functional Lie derivative: on the state jSvi as a S^ ga(x) j i = JL; adj (x) S^ E a Eikonal scattering S^ bi y(x+; x) j0i = S (x) di y(x+; x) j0i : (2.36) SAab(x) and S SAab(x) = (x) = dx+ T cA c(x+; x) dx+ tcA c(x+; x) tr SA(x)T a SAy(x) [SA(x) JR(x)]a = tr T aSA(x) SAy(x) S (x) S (x) taSy(x) ; Sy(x)ta : In a similar manner we can also express qq: JLa; F (x) ST (x) ST (x) S(x)ta taS(x) hJLa(x); J Lb(y)i = if abcJLc (x) (x y); (2.45) hJLc (x); SAab(z)i = if cadSAdb(z) (x z): (2.46) LO JIMWLK Hamiltonian where N LO is determined by the normalisation condition, = 1: = N The emission vertex is de ned by vanishing matrix-element is that of Hg, (2.29): Therefore, (2.47) becomes: = N hgia(k) j Hg j 0i = 2 3=2pk+k2 using the relations (2.44), we arrive at: X Y h JLa(x) JLa(y) + JRa (x) JRa (y) 2JLa(x) SAab(z) J Rb(y) i : LO . The coordinate densities a = N dk+ Z x; z 2 3=2X2 a(x) gia(k+; z) ; where Xi zi. From (2.48): where Y i zi. The scattered wave function reads: LO = 1 2 Z e Y dk+ Z X Y a(y) a(x) + O(g4); 2 Z e Y dk+ Z x; z 2 3=2X2 S^ a(x) gia(k+; z) : function, while the gluon from the valence current. HJLIOMWLK = X Y h JLa(x) JLa(y) + JRa (x) JRa (y) YLO , which gives (see 2JLa(x) SAab(z) J Rb(y) : transformation, S ! Sy together with JL ! and the charge conjugation symmetry S ! S . JR, which in [84] was identi ed as signature, NLO JIMWLK Hamiltonian x; y;z;z0 x; y; z; z0 w ;x; y; z; z0 w ;x; y; z HJNILMOWLK = KJSJ(x; y; z) hJLa(x)JLa(y) + JRa (x)JRa (y) 2JLa(x)SAab(z)J Rb(y) KJSSJ(x; y; z; z0) hf abcf def JLa(x)SAbe(z)SAcf (z0)JRd (y) NcJLa(x)SAab(z)J Rb(y) Kqq(x; y; z; z0) h2 JLa(x) tr[Sy(z) ta S(z0)tb] J Rb(y) JLa(x) SAab(z) J Rb(y) JLa(w) SAcd(z) SAbe(z0) JRd (x) J Re(y) + (JLc (x) J Lb(y) JLa(w) J Rc(x) J Rb(y) JRa (w)) KJJSJ(w ; x; y; z) f bde hJLd(x) JLe (y) SAba(z) JRa (w) JLa(w) SAab(z) JRd (x) J Re(y) + (JLd(x) JLe (y) J Lb(w) JRd (x) J Re(y) J Rb(w)) : z; X0 KJJSJ(w; x; y; z) = KJSSJ(x; y; z; z0) = KJJSSJ(w; x; y; z; z0) = (Y 0)j Xi (Y 0)2X2 Y i(X0)j (X0)2Y 2 (W 0)j Zi W iZj W 2Z2 z0; Y X2W 2 W i(W 0)j W 2(W 0)2 Y 2W 2 z0; W (W 0)2 X2(Y 0)2 + (X0)2Y 2 Y )2Z2 Z4(X2(Y 0)2 (X0)2Y 2) X2(Y 0)2 (X0)2Y 2 Y 2(X0)2 X2(Y 0)2 Y 2(X0)2 X2(Y 0)2 (X0)2Y 2 2I(x; z; z0) 2I(y; z; z0) + Ke (x; y; z; z0); z ; W 0 I(x; z; z0) (X0)2 (X0)2 X2 + (X0)2 2(X0)2 X2 + (X0)2 (X0)2 (X0)2 (X0)2 (X0)2 Ke (x; y; z; z0) KJJSSJ(x; x; y; z; z0) KJJSSJ(y; x; y; z; z0) KJJSSJ(x; y; x; z; z0) + KJJSSJ(y; y; x; z; z0) : z0 ; Z ; (2.58) Explicitly: Ke (x; y; z; z0) = (X0)2Z2(Y 0)2 + (X0)2Z2Y 2 (X0)2X2(Y 0)2 X2(X0)2Y 2 Z2(Y 0)2 (X0)2 X2Z2Y 2 + (x $ y) : (2.62) The following useful equality holds: Ke (x; y; z; z0) = Ke (x; y; z; z0): The kernel KJSJ reads:2 KJSJ(x; y; z) = ln 2) + -function: KJSJ(x; y; z) X Y Kqq(x; y; z; z0) = The quark sector: An alternative representation for KJSJ: ln 2) + Ke (x; y; z; z0): If (x; z; z0) Z2(X2 (X0)2Y 2 + (Y 0)2X2 Y )2Z2 Z4 (X2(Y 0)2 (X0)2Y 2) X2(Y 0)2 (X0)2Y 2 If (x; z; z0) If (y; z; z0) (X0)2 relations among them can be found in appendix D. The light cone wave function at NLO 2The term 2b( there as is clear from [92]. { 16 { ! jii) and also normalising the state when dividing by its norm: = N Ei Ej Ei Ej Ek 2Ei2 Ej2 sation of the state: NLO is determined from normali= N LO is the LO contribution (see section 2.7): g d2k gki a( k) k+k2 jgia(k)i = ig a(x)Xi x;z 2 3=2X2 Order g2 states: NLO + h0 jHintj ii hi jHintj ji hj jHintj 0i 2Ei Ej h0 jHintj ii hi jHintj ji hj jHintj ki hk jHintj 0i 2Ei Ej Ek 3 jhj jHintj 0ij2 jhi jHintj 0ij2 : 8Ei2 Ej2 can be nevertheless uniquely cording to its soft component content: Our objective is to compute NLO . It is convenient to split the wave function ac Order g3 states: The following contributions vanish: Matrix elements expressions are computed based on (2.29) (2.34). At NLO there are two matrix elements Gluon splits into quark and anti-quark pair. 8 3=2pk+ Triple gluon interaction. igf abc (3)(k 8 3=2pk+p+q+ kn + q k+ + q+ k+ + p+ Dglc(q) gjb(p) jHgj gia(k) E = The denominator in the rst line of (E.3): (Ei(0))2 j6=0 j iN = j0i n(i)E n(i) jHintj 0 + n(i)E n(i) jHintj n(j) n(i)E n(i) jHintj n(j) n(j) jHintj n(k) E(0)Ej(0)E(0) i k (Ei(0))2Ej(0) E(0)E(0) The phase of the wave function NLO d(w) = 0: NLO (3.84) is inserted in (F.1). of the LCWF and the phase: d(w) d(w) NLO = h gg j d(w) j gg i + h gg j d(w) j g i + h0j d(w) i NLO j0i = 0 KJJSSJ(y; w; x; z; z0)] + h0j d(w) i NLO j0i = 0: in section 4.7. i NLO = v ;x; y; z; z0 KJJSSJ(v; x; y; z; z0)f abc b(y) a(x) c(v): x; y; z; z0 Supplementary for section 3 Supplement for computation of d2k d2p g3 a( k) tr[ta td] ki = I ; (k+)2 By using y = = 2 2peipej (k+)2 pi(kj pj ) + pj (ki pi)(kj pj ) + "il"jk(ki pi)(kj k+p+ "il"jk(kk p+(k+ 4 + 1)peipej + "il"jkpekpel : )2(k+)2 2"il"jkpekpel (k+)2 32 9=2 (1 )k4 ( (1 4 + 1)peipej + "il"jkpekpel k kipj + kj pi ki(kj pj ) + kj (ki pipj + "il"jkpkpl k+(k+ pk)pl "il"jk(kl pl)pk (p+)2 i j = i"ij 3 + ij I ; tr htatdi = "il"jk = ij kl { 58 { 2 Z e Y g1 = g1 = Integration over pe is done with the aid of (C.31): After expanding with (C.33) and taking ! 0 limit, the result becomes: (2 )d (2 )d k+k4 ( (1 2 Z e Y (4 )d=2 2 jgia(k)i : 4 + d) ki k+k2d Finally, integrating over according to (C.1): 64 7=2pk+k2 which we can write as in (3.50). Supplement for computation of d2k d2p g2f abcf dbc (k p)2 4 + 2) 64 7=2pk+k2 1 + 2 jgia(k)i : 4 3=2jk+j3=2 k+ + p+ k+ + p+ jgia(k)i : After some algebra: 2 Z e Y (2 )d (2 )d g2 = g3f abcf dbc a( k)ki 8(2 )3 3=2 (1 )k4 (k2 (1 By replacing the measure according to (3.28): 16 9=2k4 (k2 (1 g3Nc a( k)kipe2 g3Nc a( k)kipe2 MS g3Nc a( k)ki 16 7=2k2pk+ jgia(k)i : 2 Z e Y 2(4 )d=2 g3Nc a( k)ki )k2 d=2 1 After expanding with the aid of (C.33) taking the ! 0 limit and, the last result becomes: + 4 ln 2 g3Nc a( k)ki 32 7=2k2pk+ MS + 2 ln2 jgia(k)i : d2k d2p ig2f abc(p+ + k+) c( k + p) ji ! 2(2 )3pk+p+(k+ 4 3=2jp+j3=2 d2k d2p ig3f abc(1 + ) c( k + p) a( p)pi 16 9=2(1 operators according to (3.27), we notice that the contribution with one operator vanishes after integraion over p. The two part reads: d2k d2p ig3f abc(1 + )pi f c( k + p); a 32 9=2(1 The relevant integral for the integration over is (C.7), the result after the integration appears in (3.56). Supplement for computation of 1 Z e Y d2k d2p d2q ig2f abc(k+ 2(2 )3p(k+ 2q+) ij a( k) ! q+)q+(k+)2 gqj b(q) qi) c(k 4 3=2jk+ q+j3=2 After integration over q: g4 = d2k d2p ig3f abc(2p+ k+) c(k p) a( k)(ki 64 9=2 (k p)2 p+))2pp+ After changing variables according to (3.34): d2k d2p ig3f abc c(k 16 9=2 ( (k p) a( k)(ki pi)(2 p)2 + (1 )p2) p2(1 1) 3=2 )pk+ gib( k+; p)E : Now let us work out the case of matrix elements, (2.49), (3.23) and (3.22), g5 as de ned in (3.61). By inserting the relevant d2k d2p d2q igf abc 16 3=2pk+q+(k+ 4 3=2jk+j3=2 k+ + q+ gqj b(q) qk) c(k 4 3=2jk+ q+j3=2 By adding together g5 , we arrive at: d2k d2p ig3f abc c(k p) a( k) 3=2 16 9=2k2p2 ( (k p)2 + (1 )p2) (1 k2 + 2k p p arrive at: ig3f abc c(k p) a( k) 3=2 16 9=2k2p2 ( (k p)2 + (1 )p2) (1 2k p + 2 k2 (kj pj ) + 2kj + Equivalently, we can write the last result as: g5 = k p + ig3f abc c(k p) a( k) 3=2 16 9=2k2p2 ( (k p)2 + (1 )p2) (1 We rewrite the denominator of following de nitions: 4+5 by using Feynman parameter. We introduce the g x + x ; x + x and perform the shift k ! k + x p. Then we arrive at: d2k d2p 32 9=2pk+(1 g3Nc a( p) 3=2 )p2 (k2 + + 2 k + d2k d2p k2 + 2k p p g3Nc a( p) 3=2 32 9=2k2p2 ( (k p)2 + (1 )p2) (1 operators. Below, we will focus on the part which involve one operator only, which can be isolated via the prescription (3.27). 2 Z e Y (2 )d (2 )d g3Nc a( p) 3=2pj ) 2p2 (k2 + (1 + )(1 x) x(1 2 1 + 1 x 1 + g3Nc a( p) 3=2pj { 63 { After taking the x + x x 1 + x + x 2 1 + x + x )2 x 1 + (4 )d=2 (4 )d=2 2 x + x MS 2 !#! x + x k+ + p+ g(kn pn) c( k + p) 4 3=2jk+ p+j3=2 x + x x + x 2) ln ( ) + ( + 1) ln (1 Supplement for computation of we arrive at: dp+ dq+ d2k d2p d2q igf bcd (k p)2 4 3=2jq+j3=2 k + p) After simpli cations the last result becomes: d2k d2p ig3f bad b( p) a( k + p) 16 9=2 (k p)2 ((1 )p2 + (k { 64 { k2 + p and two via the prescription in (3.27). operator: d2k d2p )k p + 32 9=2 g3f badf bac c( k) k)2 + p)2 + We introduce again the variable according to p k we arrive at: d2k d2p )k (p + p)2 + 32 9=2 (p2 + jgla(k)i : last result as: k2 pl + 2) + 2 2(1 )2(1 + ) ddk ddp Z 1 k+ (2 )d (2 )d 2)(d + 2) + 2d(1 )(1 + ) )2(1 + ) (2 + d) 2( k+ (p2 + )k2)2 k2 2) + 4 (1 )(1 + ) (1 + )d jgla(k)i : (4 )d=2 { 65 { 2)d + 2 ( After taking the (1 + )d jgla(k)i : 2) + 2 2(1 32 7=2 k2pk+ )2(1 + ) )(1 + ) jgla(k)i : (1 + ) 2(4 )d=2 )(1 + ) (1 + ) d2k d2p )k p + After integration over x: g3Nc a( k)kl 32 7=2 (1 2) ln ( ) (1 + ) ln (1 (1 + ) 2 g3Nc a( k)ki 32 7=2k2pk+ + 2 ln2 jgia(k)i : This part can be deduced directly from (3.68): p and dividing by 2, we can write equivalently as: ig3f bad ( k + p) 32 9=2 (k p)2 ((1 )p2 + (k p)2 + d2k d2p ig3f bad 64 9=2 (1 ( k + p) )p2(k MS 2 !#! jgla(k)i : After integration over ( k + p) 32 9=2p2(k k pkl + Notice that under the change p ! k p the second summand inside the rectangled obtained after this change in (3.71). Supplement for computation of 1 Z e Y dk+ dp+ dq+ dr+ Z d2k d2p d2q d2r 4 3=2jp+j3=2 grj b(r) gqk c(q) 4 3=2jq+j3=2 4 3=2jr+j3=2 g7 = d2k d2p g3p2ki b(p) 64 9=2 jgia(k)i : By using the algebra of g7 = d2k d2p 2g3ki b(p) 64 9=2 jgia(k)i : { 67 { g3Nc a( k)ki (4 )d=2 g3Nc a( k)ki " 16 7=2pk+ k2 Expanding with the aid of (C.33) taking the ! 0 limit and, the last result becomes: After integration over MS + ln2 jgia(k)i : jgia(k)i : 2 !# MS jgia(k)i : 2 Y k+ according to (C.9): 2 g3Nc a( k)ki 32 7=2pk+k2 we obtain: Supplement for section 4 Supplement for computation of g7 = d2k d2p 2 Z e Y ddk ddp Integration over p by using (C.31) yields: We can now isolate the contribution to one via the prescription in (3.27). operator reads: k2 + p2 g3ki b(p) 8 9=2pk+ k2 jgia(k)i ; g4Nf JLa(x) T r Sy(z)taS(z0)tb J Rb(y) 512 10 ( (1 d2k d2pe d2u d2ve )k2 + pe2) ( (1 pej vei kiuj k2u2 simplify the denominators using: )Z2 + (X0 Z)2 = (1 )(X0)2 + X2: x;y;z;z0 0 g4Nf JLa(x) T r Sy(z)taS(z0)tb J Rb(y) 32 6Z4 ((1 )(X0)2 + X2) ((1 )(Y 0)2 + Y 2) (X0)2 + (Y 0)2 x;y;z;z0 0 g4Nf JLa(x) T r Sy(z)taS(z0)tb J Rb(y) 64 6Z4 ((1 )(X0)2 + X2) ((1 )(Y 0)2 + Y 2) )(X0)2 + X2 + (1 )(Y 0)2 + Y 2 +2 (1 It is possible to integrate over Supplement for computation of JSSJ Then, we arrive at: Rewriting the scalar products: X0 Z = X0 Y 0 = we arrive at: is shown in (4.15). to (4.19), we get: JNSLSOJ = (Y 0)2Y 2((Y 0)2 k+ f abcf def JLa(x) SAbe(z) SAcf (z0) JRd (y) (x; y; z; z0) (X0)2X2((X0)2 X2)2((X0)2Y 2 (y; x; z; z0) Y 2)2((X0)2Y 2 (Y 0)2 (X0)2 x; y; z; z0 (x; z; z0) T (y; z; z0) (X0)2(Y 0)2 k+ f abcf def JLa(x) SAbe(z) SAcf (z0) JRd (y) (x; z0; z) T (y; z0; z) ! X2Y 2 The following de nitions were used: jl(x; z; z0) = (x; y; z; z0) Xl(X0)j jl(y; z; z0) X2 + lj(y; z0; z) (X0)2 (X2 (X0)2) jl(y; z; z0) X2(X0)2 jl(x; z; z0) X2 + lj(x; z0; z) (X0)2 (X2 (X0)2) jl(x; z; z0) X2(X0)2 : from which we deduce: X2(X0)2(X2 (x; y; z; z0) X2(X0)2 (X0)2Z2 (X0)2Y 2 + X2(Y 0)2 X2(X0)2 2Z2(X2 (X0)2) X2 Z2 X0 Y 0 X2 X0 Z (X0)2 X Z + X X0 (X0)2 Y 0 Z X Z + X0 Y 0 X Z X2Z2 X2 X0 Y 0 Y Z (X0)2 X Y Y 0 Z X Z = X X0 = 2 1 (X0)2 X2 Z2 ; 12 X2 + (X0)2 Z2 : Y Z = 1 (Y 0)2 Y 2 Z2 ; (x; y; z; z0) X2(X0)2(X2 (X0)2)2 = 18 X2(Y 0)2 (X0)2Y 2 2 X2(Y 0)2 + (X0)2Y 2 4(X Y )2Z2 Z4(X2(Y 0)2 (X0)2Y 2) (X0)2Y 2 (X2 (X0)2)Z2 X2(Y 0)2 + Y 2(X0)2 + X2 + (X0)2 + 2(X0)2 + Z2 Z2 X2 Z2 (X0)2 X2(Y 0)2 Y 2(X0)2 +( X(Y0)02)Z4X2Y2 2 + Z2X2(Y 0)2 Y 4(X0)2 (X0)2(X X2Z2 Z2(Y 0)2 (X0)2Y 2 (X0)2 X2(X X2(Y 0)2 (x; y; z; z0) (X0)2X2((X0)2 X2)2((X0)2Y 2 (y; x; z; z0) Y 2)2((X0)2Y 2 (Y 0)2 (X0)2 Supplement for computation of JJSJ introduced in (4.28). Let us start with JJSJ g4f bde JLa(w) SAab(z) JRd (x) J Re(y) d2k d2p e ik X+ip (X Y ) p2(k k2(k k2p2 Ke (x; y; z; z0) = 16 4 ((X0)2Y 2 X2(Y 0)2) Z2(Y 0)2 X2(X Z2(Y 0)2 (Y 0)2(X (X0)2Y 2 (X0)2Z2 (X0)2 (X0)2 X2(Y 0)2 X2(Y 0)2 + (x $ y) ; we arrive at the conclusion that: (Y 0)4X2 (X0)2Z2Y 2 Y 4(X0)2 Z2X2(Y 0)2 (X0)2(X (Y 0)2(X X2Z2 (X0)2 following way: KJSSJ(x; y; z; z0): JNJLSOJ and After the change p ! k p, we nd: JNJLSOJ = g4f bde We can now use integral (C.27), and write: JNJLSOJ = ig4f bde hJLd(x) JLe (y) SAba(z) JRa (w) JLa(w) SAab(z) JRd (x) J Re(y) : JLa(w) SAab(z) JRd (x) J Re(y) p2(k k2(k d2k d2p e ik Y ip (X Y ) JJSJ g4f bde d2k d2p e ik X+ip (X Y ) p (p k2p2(k k) ki p2(k JLa(w) SAab(z) JRd (x) J Re(y) JJSJ = ig4f bde 2X0 Z Y (X0)2Z2Y 2W 2 w; x; y; z; z0 JLa(w) SAab(z) JRd (x) J Re(y) : Supplement for computation of JJSSJJ From (4.39) we deduce that: 2X0 Y 0 Y (X0)2(Y 0)2Y 2W 2 + X0 Y 0 Z (X0)2(Y 0)2Z2W 2 (X $ Y ) hJLd(x) JLe (y) SAba(z) JRa (w) JJSSJJ = w; v; x; y; z; z0 128 6 (1 X Y W 0 V 0 X Y W 0 V 0 X Y W 0 V 0 X Y W 0 V 0 Integrating over and reordering the J operators: JJSSJJ = 4X Y W 0 V 0 x; y; z; z0 X2Y 2(X0)2(Y 0)2 f bar f cdeJLr (x) SAbc(z0) SAad(z)J Re(y) : Open Access. { 72 { (1983) 1 [INSPIRE]. 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Michael Lublinsky, Yair Mulian. High energy QCD at NLO: from light-cone wave function to JIMWLK evolution, Journal of High Energy Physics, 2017, 97, DOI: 10.1007/JHEP05(2017)097