Rapidity evolution of gluon TMD from low to moderate x
PPrepared for submission to JHEP
JLAB-THY-15-2040
Rapidity evolution of gluon TMD from low tomoderate x I. Balitsky ∗ and A. Tarasov † ∗ Physics Dept., Old Dominion University, Norfolk VA 23529,USA and Theory Group, JLAB,12000 Jefferson Ave, Newport News, VA 23606,USA † Theory Group, JLAB, 12000 Jefferson Ave, Newport News, VA 23606,USA
E-mail: [email protected] , [email protected] Abstract:
We study how the rapidity evolution of gluon transverse momentum dependentdistribution changes from nonlinear evolution at small x (cid:28) to linear evolution at moderate x ∼ . a r X i v : . [ h e p - ph ] S e p ontents
11 Appendix B: Propagators in the shock-wave background 60
A TMD factorization [1–3] generalizes the usual concept of parton density by allowing PDFsto depend on intrinsic transverse momenta in addition to the usual longitudinal momentumfraction variable. These transverse-momentum dependent parton distributions (also calledunintegrated parton distributions) are widely used in the analysis of semi-inclusive processeslike semi-inclusive deep inelastic scattering (SIDIS) or dijet production in hadron-hadroncollisions (for a review, see Ref. [3]). However, the analysis of TMD evolution in these casesis mostly restricted to the evolution of quark TMDs, whereas at high collider energies themajority of produced particles will be small- x gluons. In this case one has to understandthe transition between non-linear dynamics at small x and presumably linear evolution ofgluon TMDs at intermediate x .The study of the transition between the low- x and moderate- x TMDs is complexified bythe fact that there are two non-equivalent definitions of gluon TMDs in small- x and “medium x ” communities. In the small- x literature the Weizsacker-Williams (WW) unintegratedgluon distribution [5] is defined in terms of the matrix element (cid:88) X tr (cid:104) p | D i U U † ( z ⊥ ) | X (cid:105)(cid:104) X | D i U U † (0 ⊥ ) | p (cid:105) (1.1)between target states (typically protons). Here tr is a color trace in the fundamentalrepresentation, (cid:80) X denotes the sum over full set of hadronic states and U z is a Wilson-lineoperator - infinite gauge link ordered along the light-like line U ( z ⊥ ) = [ ∞ n + z ⊥ , −∞ n + z ⊥ ] , [ x, y ] ≡ P e ig (cid:82) du ( x − y ) µ A µ ( ux +(1 − u ) y ) (1.2)– 1 –nd D i U ( z ⊥ ) = ∂ i U ( z ⊥ ) − iA i ( ∞ n + z ⊥ ) U ( z ⊥ ) + iA i ( −∞ n + z ⊥ ) U ( z ⊥ ) . In the spirit ofrapidity factorization, Bjorken x enters this expression as a rapidity cutoff for Wilson-lineoperators. Roughly speaking, each gluon emitted by Wilson line has rapidity restrictedfrom above by ln x B .One can rewrite the above matrix element (up to some trivial factor) in the form α s D ( x B , z ⊥ ) = − π ( p · n ) x B (cid:90) du (cid:88) X (cid:104) p | ˜ F aξ ( z ⊥ + un ) | X (cid:105)(cid:104) X |F aξ (0) | p (cid:105) (1.3)where F aξ ( z ⊥ + un ) ≡ [ ∞ n + z ⊥ , un + z ⊥ ] am n µ gF mµξ ( un + z ⊥ )˜ F aξ ( z ⊥ + un ) ≡ n µ g ˜ F mµξ ( un + z ⊥ )[ un + z ⊥ , ∞ n + z ⊥ ] ma (1.4)and define the “WW unintegrated gluon distribution” D ( x B , k ⊥ ) = (cid:90) d z ⊥ e i ( k,z ) ⊥ D ( x B , z ⊥ ) (1.5)(Here ( k, z ) ⊥ denotes the scalar product in 2-dim transverse Euclidean space.) It shouldbe noted that since Wilson lines in Eq. (1.1) are renorm-invariant α s D ( x B , k ⊥ ) does notdepend on the renormalization scale µ .On the other hand, at moderate x B the unintegrated gluon distribution is defined as[6] D ( x B , k ⊥ , η ) = (cid:90) d z ⊥ e i ( k,z ) ⊥ D ( x B , z ⊥ , η ) , (1.6) α s D ( x B , z ⊥ , η ) = − x − B π ( p · n ) (cid:90) du e − ix B u ( pn ) (cid:88) X (cid:104) p | ˜ F aξ ( z ⊥ + un ) | X (cid:105)(cid:104) X |F aξ (0) | p (cid:105) where | p (cid:105) is an unpolarized target with momentum p (typically proton). There are moreinvolved definitions with Eq. (1.6) multiplied by some Wilson-line factors [3, 4] followingfrom CSS factorization [7] but we will discuss the “primordial” TMD (1.6). The Bjorken x is now introduced explicitly in the definition of gluon TMD. However, because light-likeWilson lines exhibit rapidity divergencies, we need a separate cutoff η (not necessarily equalto ln x B ) for the rapidity of the gluons emitted by Wilson lines. In addition, the matrixelements (1.6) may have double-logarithmic contributions of the type ( α s η ln x B ) n while theWW distribution (1.3) has only single-log terms ( α s ln x B ) n described by the BK evolution[8, 9].In the present paper we study the connection between rapidity evolution of WW TMD(1.3) at low x B and (1.6) at moderate x B ∼ . We will assume k ⊥ ≥ few GeV so thatwe can use perturbative QCD (but otherwise k ⊥ is arbitrary and can be of order of s asin the DGLAP evolution). In this kinematic region we will vary Bjorken x B and look hownon-linear evolution at small x transforms into linear evolution at moderate x B . It shouldbe noted that at least at moderate x B gluon TMDs mix with the quark ones. In this paperwe disregard this mixing leaving the calculation of full matrix for future publications. (Forthe study of quark TMDs in the low- x region see recent preprint [10].)– 2 –n addition, we will present the evolution equation for the fragmentation function D f ( β F , k ⊥ , η ) = (cid:90) d z ⊥ e − i ( k,z ) ⊥ D f ( β F , z ⊥ , η ) , (1.7) α s D f ( β F , z ⊥ , η ) = − β − F π ( p · n ) (cid:90) du e iβ F u ( pn ) (cid:88) X (cid:104) | ˜ F aξ ( z ⊥ + un ) | p + X (cid:105)(cid:104) p + X |F aξ (0) | (cid:105) where p is the momentum of the registered hadron. It turns out to be free of non-linearterms, at least in the leading log approximation.It should be emphasized that we consider gluon TMDs with Wilson links going to + ∞ in the longitudinal direction relevant for SIDIS [11]. Note that in the leading order SIDIS isdetermined solely by quark TMDs but beyond that the gluon TMDs should be taken intoaccount, especially for the description of various processes at future EIC collider (see e.g.the report [12]).It is worth noting that another gluon TMD with links going to −∞ arises in the studyof processes with exclusive particle production (like Drell-Yan or Higgs production), see forexample the discussion in Ref. [13]. We plan to study it in future publications.The paper is organized as follows. In Sec. 2 we remind the general logic of rapidityfactorization and rapidity evolution. In Sec. 3 we derive the evolution equation of gluonTMD in the light-cone (DGLAP) limit. In Sec. 4 we calculate the Lipatov vertex of thegluon production by the F ai operator and the so-called virtual corrections. The final TMDevolution equation for all x B and transverse momenta is presented in Sec. 5 and in Sec. 6we discuss the DGLAP, BK and Sudakov limits of our equation. In Sec. 7 we demonstratethat the linearized evolution equation for unintegrated gluon distribution interpolates be-tween BFKL and DGLAP equations. In Sec. 8 we present the evolution equations forfragmentation TMD and Sec. 9 contains conclusions and outlook. The necessary formulasfor propagators near the light cone and in the shock-wave background can be found inAppendices. In the spirit of high-energy OPE, the rapidity of the gluons is restricted from above bythe “rapidity divide” η separating the impact factor and the matrix element so the properdefinition of U x is U ηx = Pexp (cid:104) ig (cid:90) ∞−∞ du p µ A ηµ ( up + x ⊥ ) (cid:105) ,A ηµ ( x ) = (cid:90) d k π θ ( e η − | α | ) e − ik · x A µ ( k ) (2.1)where the Sudakov variable α is defined as usual, k = αp + βp + k ⊥ . We define the light-likevectors p and p such that p = n and p = p − m s n , where p is the momentum of the target Alternatively, with the leading-log accuracy one can take the Wilson line slightly off the light cone, seeRef. [3]. To pave the way for future NLO calculation we prefer the “rigid cutoff” Eq. (2.1) which was usedfor the NLO calculations in the low- x case [14]. – 3 –article of mass m . We use metric g µν = (1 , − , − , − so p · q = ( α p β q + α q β p ) s − ( p, q ) ⊥ .For the coordinates we use the notations x • ≡ x µ p µ and x ∗ ≡ x µ p µ related to the light-conecoordinates by x ∗ = (cid:112) s x + and x • = (cid:112) s x − . It is convenient to define Fourier transformof the operator F ai F aηi ( β B , k ⊥ ) = (cid:90) d z ⊥ e − i ( k,z ) ⊥ F aηi ( β B , z ⊥ ) , F aηi ( β B , z ⊥ ) ≡ s (cid:90) dz ∗ e iβ B z ∗ (cid:0) [ ∞ , z ∗ ] amz gF m • i ( z ∗ , z ⊥ )) η (2.2)where the index η denotes the rapidity cutoff (2.1) for all gluon fields in this operator.Here we introduced the “Bjorken β B ” to have similar formulas for the DIS and annihilationmatrix elements ( β B = x B in DIS and β B = β F = z F for fragmentation functions). Also,hereafter we use the notation [ ∞ , z ∗ ] z ≡ [ ∞ ∗ p + z ⊥ , s z ∗ p + z ⊥ ] where [ x, y ] stands for thestraight-line gauge link connecting points x and y as defined in Eq. (1.2). Our conventionis that the Latin Lorentz indices always correspond to transverse coordinates while GreekLorentz indices are four-dimensional.Similarly, we define ˜ F aηi ( β B , k ⊥ ) = (cid:90) d z ⊥ e i ( k,z ) ⊥ ˜ F aηi ( β B , z ⊥ ) , ˜ F aηi ( β B , z ⊥ ) ≡ s (cid:90) dz ∗ e − iβ B z ∗ g (cid:0) ˜ F m • i ( z ∗ , z ⊥ )[ z ∗ , ∞ ] maz (cid:1) η (2.3)in the complex-conjugate part of the amplitude.In this notations the unintegrated gluon TMD (1.6) can be represented as (cid:104) p | ˜ F aηi ( β B , z ⊥ ) F aiη ( β B , ⊥ ) | p + ξp (cid:105) ≡ (cid:88) X (cid:104) p | ˜ F aηi ( β B , z ⊥ ) | X (cid:105)(cid:104) X |F aiη ( β B , ⊥ ) | p + ξp (cid:105) = − π δ ( ξ ) β B g D ( β B , z ⊥ , η ) (2.4)Hereafter we use a short-hand notation (cid:104) p | ˜ O ... ˜ O m O ... O n | p (cid:48) (cid:105) ≡ (cid:88) X (cid:104) p | ˜ T { ˜ O ... ˜ O m }| X (cid:105)(cid:104) X | T {O ... O n }| p (cid:48) (cid:105) (2.5)where tilde on the operators in the l.h.s. of this formula stands as a reminder that theyshould be inverse time ordered as indicated by inverse-time ordering ˜ T in the r.h.s. of theabove equation.As discussed e.g. in Ref. [15], such martix element can be represented by a doublefunctional integral (cid:104) ˜ O ... ˜ O m O ... O n (cid:105) = (cid:90) D ˜ AD ˜¯ ψD ˜ ψ e − iS QCD ( ˜ A, ˜ ψ ) (cid:90) DAD ¯ ψDψ e iS QCD ( A,ψ ) ˜ O ... ˜ O m O ... O n (2.6)with the boundary condition ˜ A ( (cid:126)x, t = ∞ ) = A ( (cid:126)x, t = ∞ ) (and similarly for quark fields)reflecting the sum over all intermediate states X . Due to this condition, the matrix element– 4 –2.4) can be made gauge-invariant by connecting the endpoints of Wilson lines at infinitywith the gauge link (cid:104) p | ˜ F ai ( β B , x ⊥ ) F ai ( β (cid:48) B , y ⊥ ) | p (cid:48) (cid:105)→ (cid:104) p | ˜ F ai ( β B , x ⊥ )[ x ⊥ + ∞ p , y ⊥ + ∞ p ] F ai ( β (cid:48) B , y ⊥ ) | p (cid:48) (cid:105) (2.7)This gauge link is important if we use the light-like gauge p µ A µ = 0 for calculations [16],but in all other gauges it can be neglected. We will not write it down explicitly but willalways assume it in our formulas.We will study the rapidity evolution of the operator ˜ F aηi ( β B , x ⊥ ) F aηj ( β B , y ⊥ ) (2.8)Matrix elements of this operator between unpolarized hadrons can be parametrized as [6] (cid:90) d z ⊥ e i ( k,z ) ⊥ (cid:104) p | ˜ F aηi ( β B , z ⊥ ) F aηj ( β B , ⊥ ) | p + ξp (cid:105) = 2 π δ ( ξ ) β B g R ij ( β B , k ⊥ ; η ) R ij ( β B , k ⊥ ; η ) = − g ij D ( β B , k ⊥ , η ) + (cid:0) k i k j m + g ij k ⊥ m (cid:17) H ( β B , k ⊥ , η ) (2.9)where m is the mass of the target hadron (typically proton). The reason we study theevolution of the operator (2.8) with non-convoluted indices i and j is that, as we shall seebelow, the rapidity evolution mixes functions D and H . It should be also noted that ourfinal equation for the evolution of the operator (2.8) is applicable for polarized targets aswell.We shall also study the evolution of fragmentation functions defined by “fragmentationmatrix elements” (1.7) of the operator (2.8). If the polarization of the fragmentation hadronis not registered, this matrix element can be parametrized similarly to Eq. (2.9) (cf. Ref.[6]) (cid:90) d z ⊥ e − i ( k,z ) ⊥ (cid:88) X (cid:104) | ˜ F aηi ( − β F , z ⊥ ) | p + X (cid:105)(cid:104) p + ξp + X |F aηj ( − β F , ⊥ ) | (cid:105) = 2 π δ ( ξ ) β F g (cid:104) − g ij D f ( β F , k ⊥ , η ) + (cid:16) k i k j m + g ij k ⊥ m (cid:17) H f ( β F , k ⊥ , η ) (cid:105) (2.10)Note that β F should be greater than 1 in this equation, otherwise the cross section vanishes.As to matrix element (2.4), it can be defined with either sign of β B but the deep inelasticscattering corresponds to β B = x B > . In our calculations we will consider β B > forsimplicity and perform the trivial analytic continuation to negative β B in the final formula(5.2).In the spirit of rapidity factorization, in order to find the evolution of the operator (2.8)with respect to rapidity cutoff η (see Eq. (2.1)) one should integrate in the matrix element(2.4) over gluons and quarks with rapidities η > Y > η (cid:48) and temporarily “freeze” fieldswith Y < η (cid:48) to be integrated over later. (For a review, see Refs. [17, 18].) In this case, weobtain functional integral of Eq. (2.6) type over fields with η > Y > η (cid:48) in the “external” Similarly, this gauge link is implied in Eq. (1.1) which is Eq. (2.7) at β B = 0 . – 5 –elds with Y < η (cid:48) . In terms of Sudakov variables we integrate over gluons with α between σ = e η and σ (cid:48) = e η (cid:48) and, in the leading order, only the diagrams with gluon emissions arerelevant - the quark diagrams will enter as loops at the next-to-leading (NLO) level.To make connections with parton model we will have in mind the frame where target’svelocity is large and call the small α fields by the name “fast fields” and large α fields by“slow” fields. Of course, “fast” vs “slow” depends on frame but we will stick to namingfields as they appear in the projectile’s frame. (Note that in Ref. [8] the terminology isopposite, as appears in the target’s frame). As discussed in Ref. [8], the interaction of“slow” gluons of large α with “fast” fields of small α is described by eikonal gauge factorsand the integration over slow fields results in Feynman diagrams in the background of fastfields which form a thin shock wave due to Lorentz contraction. However, in Ref. [8] (aswell as in all small- x literature) it was assumed that the characteristic transverse momentaof fast and slow fields are of the same order of magnitude. For our present purposes weneed to relax this condition and consider cases where the transverse momenta of fast andslow fields do differ. In this case, we need to rethink the shock-wave approach.Let us figure out how the relative longitudinal size of fast and slow fields depends ontheir transverse momenta. The typical longitudinal size of fast fields is σ ∗ ∼ σ (cid:48) sl ⊥ where l ⊥ isthe characteristic scale of transverse momenta of fast fields. The typical distances traveledby slow gluons are ∼ σsk ⊥ where k ⊥ is the characteristic scale of transverse momenta of slowfields. Effectively, the large- α gluons propagate in the external field of the small- α shockwave, except the case l ⊥ (cid:28) k ⊥ which should be treated separately since the “shock wave”is not necessarily thin in this case. Fortunately, when l ⊥ (cid:28) k ⊥ one can use the light-cone expansion of slow fields and leave at the leading order only the light-ray operators ofthe leading twist. We will use the combination of shock-wave and light-cone expansionsand write the interpolating formulas which describe the leading-order contributions in bothcases. As we discussed above, we will obtain the evolution kernel in two separate cases: the “shockwave” case when the characteristic transverse momenta of the background gluon (or quark)fields l ⊥ are of the order of typical momentum of emitted gluon k ⊥ and the “light cone” casewhen l ⊥ (cid:28) k ⊥ . It is convenient to start with the light-cone situation and consider the one-loop evolution of the operator ˜ F aηi ( β B , x ⊥ ) F aiη ( β B , y ⊥ ) in the case when the backgroundfields are soft so we can use the expansion of propagators in external fields near the lightcone [19].In the leading order there is only one “quantum” gluon and we get the typical diagramsof Fig. 1 type. One sees that the evolution kernel consist of two parts: “real” part with theemission of a real gluon and a “virtual” part without such emission. The “real” productionpart of the kernel can be obtained as a square of a Lipatov vertex - the amplitude of the– 6 – a) (b) Figure 1 . Typical diagrams for production (a) and virtual (b) contributions to the evolution kernel.The dashed lines denote gauge links. emission of a real gluon by the Wilson-line operator F ai : (cid:104) ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:105) ln σ = − (cid:90) σσ (cid:48) d − α α d − k ⊥ (cid:0) (cid:104) lim k → k ˜ F ai ( β B , x ⊥ ) ˜ A mρ ( k ) (cid:105)(cid:104) lim k → k A mρ ( k ) F aj ( β B , y ⊥ ) (cid:105) (cid:1) ln σ (cid:48) (3.1)Hereafter we use the space-saving notation d − n p ≡ d n p (2 π ) n . As we mentioned, the production (“real”) part of the kernel corresponds to square of Lipatovvertex describing the emission of a gluon by the operator F ai . The Lipatov vertex is definedas L abµi ( k, y ⊥ , β B ) = i lim k → k (cid:104) T { A aµ ( k ) F bi ( β B , y ⊥ ) }(cid:105) (3.2)(To simplify our notations, we will often omit label η for the rapidity cutoff (2.1) but it willbe always assumed when not displayed).We will use the background-Feynman gauge. The three corresponding diagrams areshown in Fig. 2. k k k (a) (b) (c) Figure 2 . Lipatov vertex of gluon emission.
In accordance with general background-field formalism we separate the gluon field into the“classical” background part and “quantum” part A µ → A q µ + A cl µ – 7 –here the “classical” fields are fast ( α < σ (cid:48) ) and “quantum” fields are slow ( α > σ (cid:48) ) . It shouldbe emphasized that our “classical” field does not satisfy the equation D µ F µν = 0 ; rather, ( D µ F cl µν ) a = − g ¯ ψγ ν t a ψ , where ψ are the “classical” (i.e. fast) quark fields. In addition, inthis Section it is assumed that the slow fields are hard and the fast fields are soft so onecan use the light-cone expansion. We will perform calculations in the background-Feynmangauge, where the gluon propagator is (cid:0) P +2 iF (cid:1) µν , see Appendix A.The first-order term in the expansion of the operator [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) in quantumfields has the form [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) = s ∂∂y ∗ [ ∞ , y ∗ ] nmy A m q i ( y ∗ , y ⊥ ) (3.3) − [ ∞ , y ∗ ] nmy ∂ i A m q • ( y ∗ , y ⊥ ) + i (cid:90) ∞ y ∗ d s z (cid:48)∗ [ ∞ , z (cid:48)∗ ] y A q • ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (to save space, we omit the label cl from classical fields). The corresponding vertex of gluonemission is given by lim k → k (cid:104) A a q µ ( k ) (cid:0) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:1) (cid:105) (3.4) = lim k → k (cid:104) s ∂∂y ∗ [ ∞ , y ∗ ] nmy (cid:104) A a q µ ( k ) A m q i ( y ∗ , y ⊥ ) (cid:105) − [ ∞ , y ∗ ] nmy (cid:104) A a q µ ( k ) ∂ i A m q • ( y ∗ , y ⊥ ) (cid:105) + i s (cid:90) ∞ y ∗ dz ∗ [ ∞ , z ∗ ] y (cid:104) A a q µ ( k ) A q • ( z ∗ , y ⊥ ) (cid:105) [ z ∗ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) To calculate the r.h.s. we can use formulas (10.47)-(10.48) from Appendix A. As we men-tioned, we need contributions to production part of the kernel with the collinear twist up totwo. However, it is easy to see that the light-cone expansion of gluon emission vertex startswith the operators of twist one ( ∼ F • i ) since the gauge links in the first term in Eq. (10.20)cancel in Eq. (3.4) and the remaining background-free emission of gluon is proportional to δ ( β B + k ⊥ αs (cid:1) which vanishes for β B > . Thus, to get the contribution to the production partof the kernel of collinear twist up to two it is sufficient to use formula (10.20) for Feynmanamplitude and formula (10.23) for complex conjugate amplitude with twist-one (one F • i )accuracy. In this case the quark terms do not contribute and the gluon terms simplify to lim k → k (cid:104) A a q µ ( k ) A b q ν ( y ) (cid:105) = − ie i k ⊥ αs y ∗ − i ( k,y ) ⊥ O abµν ( ∞ , y ∗ , y ⊥ ; k ) , (3.5) O µν ( ∞ , y ∗ , y ⊥ ; k ) = g µν [ ∞ , y ∗ ] y + g (cid:90) ∞ y ∗ dz ∗ (cid:16) − iαs k j ( z − y ) ∗ g µν [ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y + 4 αs ( δ jµ p ν − δ jν p µ )[ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:17) – 8 –ith the help of this formula Eq. (3.4) reduces to lim k → k (cid:104) A a q µ ( k ) (cid:16) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:1) (cid:105) (3.6) = (cid:110)(cid:0) k ⊥ α g ⊥ µi − p µ k i (cid:1)(cid:16) − igk j αs (cid:90) ∞ y ∗ dz ∗ ( z − y ) ∗ [ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , ∞ ] y (cid:17) an + ig (cid:0) αk ⊥ p µ − αs p µ (cid:1)(cid:16) [ ∞ , y ∗ ] y F • i ( y ∗ , y ⊥ )[ y ∗ , ∞ ] y − ik ⊥ αs (cid:90) ∞ y ∗ dz ∗ [ ∞ , z ∗ ] y F • i ( z ∗ , y ⊥ )[ z ∗ , ∞ ] y (cid:17) an + 2 gαs ( g µi k j − δ jµ k i ) (cid:90) ∞ y ∗ dz ∗ ([ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , ∞ ] y ) an (cid:111) e i k ⊥ αs y ∗ − i ( k,y ) ⊥ Note that k µ × (cid:0) r . h . s . of Eq . (3 . (cid:1) µ = 0 as required by gauge invariance. Integrating ther.h.s. of Eq. (3.6) over y ∗ we obtain L abµi ( k, y ⊥ , β B ) = i lim k → k (cid:104) A a q µ ( k )( F bi ( β B , y ⊥ ) (cid:1) (cid:105) = 2 ge − i ( k,y ) ⊥ αβ B s + k ⊥ (3.7) × (cid:104) αβ B sk ⊥ (cid:0) k ⊥ αs p µ − αp µ (cid:1) δ li − δ lµ k i + αβ B sg µi k l k ⊥ + αβ B s + 2 αk i k l k ⊥ + αβ B s p µ (cid:105) F abl ( β B + k ⊥ αs , y ⊥ ) At this point it is convenient to switch to the light-like gauge p µ A µ = 0 . Since k µ × (cid:0) r . h . s . of Eq . (3 . (cid:1) µ = 0 it is sufficient to replace αp µ in the r.h.s. of Eq. (3.7) by αp µ − k µ = − k µ ⊥ − k ⊥ αs p µ . One obtains L abµi ( k, y ⊥ , β B ) light − like = 2 ge − i ( k,y ) ⊥ (3.8) × (cid:104) k ⊥ µ δ li k ⊥ − δ lµ k i + δ li k ⊥ µ − g µi k l αβ B s + k ⊥ − k ⊥ g µi k l + 2 k ⊥ µ k i k l ( αβ B s + k ⊥ ) (cid:105) F abl ( β B + k ⊥ αs , y ⊥ ) + O ( p µ ) We do not write down the terms ∼ p µ since they do not contribute to the productionkernel ( ∼ square of the expression in the r.h.s. of Eq. (3.8)).For the complex conjugate amplitude one obtains from Eq. (10.49) − i lim k → k (cid:104) ˜ A aµ ( x ) ˜ A bν ( k ) (cid:105) = e − i k ⊥ αs x ∗ + i ( k,x ) ⊥ ˜ O abµν ( x ∗ , ∞ , x ⊥ ; k ) , ˜ O µν ( x ∗ , ∞ , x ⊥ ; k ) = g µν [ x ∗ , ∞ ] x + g (cid:90) ∞ x ∗ dz ∗ [ x ∗ , z ∗ ] x (cid:16) iαs ( z − x ) ∗ g µν ˜ F • j ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x k j − αs ( δ jµ p ν − δ jν p µ ) ˜ F • j ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x (cid:17) (3.9)where ˜ O µν is obtained from the Eq. (10.23) with twist-two accuracy (as we mentioned,quark operators start from twist two and therefore do not contribute to the productionkernel).Repeating steps which lead us to Eq. (3.8) we obtain ˜ L abiµ ( k, x ⊥ , β B ) light − like = − i lim k → k (cid:104) (cid:0) ˜ F ai ( β B , x ⊥ ) (cid:1) ˜ A b q µ ( k ) (cid:105) light − like = 2 ge i ( k,x ) ⊥ (3.10) × (cid:104) k ⊥ µ δ ki k ⊥ − δ kµ k i + k ⊥ µ δ ki − g µi k k αβ B s + k ⊥ − k ⊥ g µi k k + 2 k ⊥ µ k i k k ( αβ B s + k ⊥ ) (cid:105) ˜ F abk (cid:0) β B + k ⊥ αs , x ⊥ (cid:1) + O ( p µ ) – 9 –he product of Lipatov vertices (3.8) and (3.10) integrated according to Eq. (3.1) gives theproduction part of the evolution kernel in the light-cone limit. To get the full kernel, weneed to add the virtual contribution coming from diagrams of Fig. 1b type. To get the virtual part coming from diagrams of Fig. 1b type we need to expand theoperator F up to the second order in quantum field (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) = (3.11) = − g s (cid:90) ∞ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ([ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y A q • ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , y ∗ ] y (cid:105) ) nm F m • i ( y ∗ , y ⊥ )+ ig ∂∂y ∗ (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm A m q i ( y ∗ , y ⊥ ) (cid:105)− igs (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm ∂ i A m q • ( y ∗ , y ⊥ ) (cid:105) As we mentioned above, we are interested in operators up to (collinear) twist one. Lookingat the explicit expressions for propagators in Appendix A it is easy to see that the onlycontribution of twist one comes from (cid:104) A q • ( z ∗ , y ⊥ ) A q i ( y ∗ , y ⊥ ) (cid:105) propagator, which is given byEq. (10.45) with G • i ( z ∗ , y ∗ ; p ⊥ ) = − gαs (cid:90) z ∗ y ∗ dz (cid:48)∗ [ z ∗ , z (cid:48)∗ ] F • i ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] (3.12)We obtain (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) = g N c (cid:90) ∞ d − αα (cid:104) − ( y ⊥ | p ⊥ | y ⊥ )[ ∞ , y ∗ ] nm F m • i ( y ∗ , y ⊥ )+ iαs (cid:90) ∞ y ∗ dz ∗ ( y ⊥ | e − i p ⊥ αs ( z ∗ − y ∗ ) | y ⊥ )[ ∞ , z ∗ ] nm F m • i ( z ∗ , y ⊥ ) (cid:105) (3.13)where we used Schwinger’s notations ( x ⊥ | f ( p ⊥ ) | y ⊥ ) ≡ (cid:90) d − p ⊥ e i ( p,x − y ) ⊥ f ( p ) , ( x ⊥ | p ⊥ ) = e i ( p,x ) ⊥ (3.14)For the operator F ( β B , y ⊥ ) the Eq. (3.13) gives (cid:104)F ni ( β B , y ⊥ ) (cid:105) = − g N c (cid:90) ∞ d − αα ( y ⊥ | αβ B sp ⊥ ( αβ B s + p ⊥ ) | y ⊥ ) F ni ( β B , y ⊥ ) (3.15)For the complex conjugate amplitude (cid:104) ˜ F m • i ( x ∗ , x ⊥ )[ x ∗ , ∞ ] mnx (cid:105) = (3.16) = − g s (cid:90) ∞ x ∗ dz ∗ (cid:90) ∞ z ∗ dz (cid:48)∗ ˜ F m • i ( x ∗ , x ⊥ ) (cid:0) [ x ∗ , z ∗ ] x (cid:104) ˜ A q • ( z ∗ , x ⊥ )[ z ∗ , z (cid:48)∗ ] x ˜ A q • ( z (cid:48)∗ , x ⊥ ) (cid:105) [ z (cid:48)∗ , ∞ ] x (cid:1) mn − ig ∂∂x ∗ (cid:90) ∞ x ∗ dz ∗ (cid:104) ˜ A m q i ( x ∗ , x ⊥ ) (cid:0) [ x ∗ , z ∗ ] x ˜ A q • ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x (cid:1) mn (cid:105) + 2 is g (cid:90) ∞ x ∗ dz ∗ (cid:104) ∂ i ˜ A m q • (cid:0) [ x ∗ , z ∗ ] x ˜ A q • ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x (cid:1) mn (cid:105) – 10 –gain, the only contribution of twist one comes from (cid:104) ˜ A q i ( x ∗ , x ⊥ ) ˜ A q • ( z ∗ , x ⊥ ) (cid:105) given by Eq.(10.46) with ˜ G kli • ( x ∗ , z ∗ ; p ⊥ ) = − gαs (cid:90) z ∗ x ∗ dz (cid:48)∗ ([ x ∗ , z (cid:48)∗ ] ˜ F • i ( z (cid:48)∗ )[ z (cid:48)∗ , z ∗ ]) kl (3.17)(see Eq. (10.23)) so the virtual correction in the complex conjugate amplitude is propor-tional to (cid:104) ˜ F ni ( β B , x ⊥ ) (cid:105) = − g N c (cid:90) ∞ d − αα ( x ⊥ | αβ B sp ⊥ ( αβ B s + p ⊥ ) | x ⊥ ) ˜ F ni ( β B , x ⊥ ) (3.18)The total virtual correction is (cid:104) ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:105) virt = − g N c ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:90) ∞ d − αα (cid:90) d − p ⊥ αβ B sp ⊥ ( αβ B s + p ⊥ ) (3.19)Note that with our rapidity cutoff in α (Eq. (2.1)) the contribution (3.19) coming from thediagram in Fig. 1b is UV finite. Indeed, regularizing the IR divergence with a small gluonmass m we obtain (cid:90) σ dαα (cid:90) d p ⊥ αβ B s ( p ⊥ + m )( αβ B s + p ⊥ + m ) (cid:39) π σβ B s + m m (3.20)which is finite without any UV regulator (the IR divergence is canceled with the corre-sponding term in the real correction, see Eq. (3.24) below). This feature - simultaneousregularization of UV and rapidity divergence - is a consequence of our specific choice ofcutoff in rapidity. For a different rapidity cutoff we may have the UV divergence in theremaining integrals which has to be regulated with suitable UV cutoff (for example, seeRefs. [20, 21]). Let us illustrate this using the example of the Fig. 1b diagram calculatedabove. Technically, we calculated the loop integral in this diagram (cid:90) d − αd − βd − β (cid:48) d − p ⊥ − β B s ( β − i(cid:15) )( β (cid:48) + β B − i(cid:15) )( αβs − p ⊥ − m + i(cid:15) )( αβ (cid:48) s − p ⊥ − m + i(cid:15) ) (3.21)by taking residues in the integrals over Sudakov variables β and β (cid:48) and cutting the obtainedintegral over α from above by the cutoff (2.1). Instead, let us take the residue over α : iβ B (cid:90) d − p ⊥ m + p ⊥ (cid:90) d − βd − β (cid:48) θ ( β ) θ ( − β (cid:48) ) − θ ( − β ) θ ( β (cid:48) )( β (cid:48) + β B − i(cid:15) )( β − i(cid:15) )( β (cid:48) − β ) (3.22) = iβ B (cid:90) d − p ⊥ m + p ⊥ (cid:90) d − βd − β (cid:48) β (cid:48) + β B − i(cid:15) (cid:104) θ ( β ) θ ( − β (cid:48) ) − θ ( − β ) θ ( β (cid:48) )( β − i(cid:15) )( β (cid:48) − β ) + θ ( β (cid:48) )( β − i(cid:15) )( β (cid:48) − β + i(cid:15) ) (cid:105) = iβ B (cid:90) d − p ⊥ m + p ⊥ (cid:90) d − βd − β (cid:48) β (cid:48) + β B − i(cid:15) θ ( β )( β − i(cid:15) )( β (cid:48) − β + i(cid:15) ) = β B (cid:90) d − p ⊥ m + p ⊥ (cid:90) ∞ d − ββ ( β + β B ) which is integral (3.20) with the replacement of variable β = p ⊥ αs .– 11 – conventional way of rewriting this integral in the framework of collinear factorizationapproach is β B (cid:90) d − p ⊥ m + p ⊥ (cid:90) ∞ dββ ( β + β B ) = (cid:90) d − p ⊥ m + p ⊥ (cid:90) dz − z (3.23)where z = β B β B + β is a fraction of momentum ( β B + β ) p of “incoming gluon” (described by F i in our formalism) carried by the emitted “particle” with fraction β B p , see the discussion ofthe DGLAP kernel in the next Section. Now, if we cut the rapidity of the emitted gluon bycutoff in fraction of momentum z , we would still have the UV divergent expression whichmust be regulated by a suitable UV cutoff. Summing the product of Lipatov vertices (3.8) and (3.10) (integrated according to Eq. (3.1))and the virtual correction (3.19) we obtain the one-loop evolution kernel in the light-coneapproximation (cid:0) ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:1) ln σ (3.24) = 2 g N c (cid:90) σσ (cid:48) d − αα d − k ⊥ (cid:110) e i ( k,x − y ) ⊥ ˜ F ak (cid:0) β B + k ⊥ αs , x ⊥ (cid:1) F al (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) × (cid:104) δ ki δ lj k ⊥ − δ ki δ lj αβ B s + k ⊥ + k ⊥ δ ki δ lj + δ kj k i k l + δ li k j k k − δ lj k i k k − δ ki k j k l − g kl k i k j − g ij k k k l ( αβ B s + k ⊥ ) + k ⊥ g ij k k k l + δ ki k j k l + δ lj k i k k − δ kj k i k l − δ li k j k k ( αβ B s + k ⊥ ) − k ⊥ g ij k k k l ( αβ B s + k ⊥ ) (cid:105) − αβ B sk ⊥ ( αβ B s + k ⊥ ) ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:111) ln σ (cid:48) where rapidities of gluons in the operators in the r.h.s. are restricted from above by ln σ (cid:48) .Let us write down now the evolution equation for gluon TMDs defined by the matrixelement (2.9). If we define β B as a fraction of the momentum p of the original hadron wehave β B < . Moreover, in the production part of the amplitude we have a kinematicalrestriction that the sum of β B and the fraction carried by emitted gluon k ⊥ αs should be lessthan one. This leads to the upper cutoff in the k ⊥ integral k ⊥ ≤ α (1 − β B ) s and we getthe equation dd ln σ (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) (3.25) = g N c π (cid:90) d − k ⊥ (cid:110) e i ( k,x − y ) ⊥ (cid:104) p | ˜ F ak (cid:0) β B + k ⊥ σs , x ⊥ (cid:1) F al (cid:0) β B + k ⊥ σs , y ⊥ (cid:1) | p (cid:105)× (cid:104) δ ki δ lj k ⊥ − δ ki δ lj σβ B s + k ⊥ + k ⊥ δ ki δ lj + δ kj k i k l + δ li k j k k − δ lj k i k k − δ ki k j k l − g kl k i k j − g ij k k k l ( σβ B s + k ⊥ ) + k ⊥ g ij k k k l + δ ki k j k l + δ lj k i k k − δ kj k i k l − δ li k j k k ( σβ B s + k ⊥ ) − k ⊥ g ij k k k l ( σβ B s + k ⊥ ) (cid:105) θ (cid:0) − β B − k ⊥ σs (cid:1) − σβ B sk ⊥ ( σβ B s + k ⊥ ) (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) (cid:111) – 12 –there is obviously no restriction on k ⊥ in the virtual diagram).If the target hadron is unpolarized one can use the parametrization (2.9) (cid:104) p | ˜ F ai ( β B , z ⊥ ) F aj ( β B , ⊥ ) | p + ξp (cid:105) η = 2 π δ ( ξ ) β B g (cid:104) − g ij D ( β B , z ⊥ , η ) − m (2 ∂ i ∂ j + g ij ∂ ⊥ ) H ( β B , z ⊥ , η ) (cid:105) = 2 π δ ( ξ ) β B g (cid:104) − g ij D ( β B , z ⊥ , η ) − m (2 z i z j + g ij z ⊥ ) H (cid:48)(cid:48) ( β B , z ⊥ , η ) (cid:105) (3.26)where η ≡ ln σ , H ( β B , z ⊥ , η ) ≡ (cid:82) d − k ⊥ e − i ( k,z ) ⊥ H ( β B , k ⊥ , η ) and H (cid:48)(cid:48) ( β B , z ⊥ , η ) ≡ (cid:0) ∂∂z (cid:1) H ( β B , z ⊥ , η ) . Rewriting Eq. (3.25) in terms of variable β ≡ k ⊥ σs oneobtains ddη (cid:104) g ij α s D ( β B , z ⊥ , η ) + 4 m (2 z i z j + g ij z ⊥ ) α s H (cid:48)(cid:48) ( β B , z ⊥ , η ) (cid:105) (3.27) = α s N c π (cid:90) − β B dβ (cid:110) g ij J (cid:0) | z ⊥ | (cid:112) σβs (cid:1) α s D ( β B + β, z ⊥ , η ) × (cid:104) β B + βββ B − β B + 3 ββ B ( β B + β ) − β β B ( β B + β ) + β β B ( β B + β ) (cid:105) + J ( | z ⊥ | (cid:112) σβs ) (cid:0) z i z j z ⊥ + g ij (cid:1) α s D ( β B + β, z ⊥ , η ) ββ B ( β B + β )+ 4 m J (cid:0) | z ⊥ | (cid:112) σβs (cid:1) (2 z i z j + g ij z ⊥ ) α s H (cid:48)(cid:48) ( β B + β, z ⊥ , η ) (cid:104) β B + ββ B β − β B + ββ B ( β B + β ) (cid:105) + 4 g ij m z ⊥ J (cid:0) | z ⊥ | (cid:112) σβs (cid:1) α s H (cid:48)(cid:48) ( β B + β, z ⊥ , η ) × (cid:104) ββ B ( β B + β ) − β β B ( β B + β ) + β β B ( β B + β ) (cid:105)(cid:111) − α s N c π (cid:90) ∞ dβ β B β ( β B + β ) (cid:104) g ij α s D ( β B , z ⊥ , η ) + 4 m (2 z i z j + g ij z ⊥ ) α s H (cid:48)(cid:48) ( β B , z ⊥ , η ) (cid:105) where we used the formula π (cid:90) dθe i ( k,z ) ⊥ [2 k i k j + k ⊥ g ij ] = − J ( kz ) k ⊥ (cid:16) z ⊥ z i z j + g ij (cid:17) (3.28)The evolution equation (3.27) can be rewritten as a system of evolution equations for D – 13 –nd H (cid:48)(cid:48) functions ( z (cid:48) ≡ β B β + β B ): ddη α s D ( β B , z ⊥ , η ) (3.29) = α s N c π (cid:90) β B dz (cid:48) z (cid:48) (cid:110) J (cid:16) | z ⊥ | (cid:114) σsβ B − z (cid:48) z (cid:48) (cid:17)(cid:104)(cid:0) − z (cid:48) (cid:1) + + 1 z (cid:48) − z (cid:48) (1 − z (cid:48) ) (cid:105) α s D (cid:0) β B z (cid:48) , z ⊥ , η (cid:1) + 4 m (1 − z (cid:48) ) z (cid:48) z ⊥ J (cid:16) | z ⊥ | (cid:114) σsβ B − z (cid:48) z (cid:48) (cid:17) α s H (cid:48)(cid:48) ( β B z (cid:48) , z ⊥ , η ) (cid:111) ,ddη α s H (cid:48)(cid:48) ( β B , z ⊥ , η )= α s N c π (cid:90) β B dz (cid:48) z (cid:48) (cid:110) J (cid:16) | z ⊥ | (cid:114) σsβ B − z (cid:48) z (cid:48) (cid:17)(cid:104)(cid:0) − z (cid:48) (cid:1) + − (cid:105) α s H (cid:48)(cid:48) (cid:0) β B z (cid:48) , z ⊥ , η (cid:1) + m z ⊥ − z (cid:48) z (cid:48) J (cid:16) | z ⊥ | (cid:114) σsβ B − z (cid:48) z (cid:48) (cid:17) α s D (cid:0) β B z (cid:48) , z ⊥ , η (cid:1)(cid:111) where (cid:82) x dzf ( z ) g ( z ) + = (cid:82) x dzf ( z ) g ( z ) − (cid:82) dzf (1) g ( z ) . The above equation is our finalresult for the rapidity evolution of gluon TMDs in the near-light-cone case.It is instructive to check that the evolution equation (3.29) agrees with the (one-loop)DGLAP kernel. If we take the light-cone limit x ⊥ = y ⊥ ( ⇔ z ⊥ = 0 ) we get ddη α s D ( β B , ⊥ , η ) = α s π N c (cid:90) β B dz (cid:48) z (cid:48) (cid:104)(cid:0) − z (cid:48) (cid:1) + + 1 z (cid:48) − z (cid:48) (1 − z (cid:48) ) (cid:105) α s D (cid:0) β B z (cid:48) , ⊥ , η (cid:1) (3.30)One immediately recognizes the expression in the square brackets as gluon-gluon DGLAPkernel (the term δ (1 − z (cid:48) ) is absent since we consider the gluon light-ray operator mul-tiplied by an extra α s ). It should be mentioned, however, that Eq. (3.30) is not a properDGLAP equation since the latter is formulated for the gluon parton density on the lightcone defined by d g ( x B , ln µ ) = − π α s ( p · n ) x B (cid:90) du e − ix B u ( pn ) (cid:104) p | ˜ F ai ( un ) F ai (0) | p (cid:105) µ (3.31)where the light-ray gluon operator F ai ( un )[ un, F ai (0) is regularized with counterterms atnormalization point µ (recall that on the light ray T-product of operators coincide withthe usual product).Comparing Eqs. (2.4) and (3.31) we see that d g ( x B ) = D ( x B , z ⊥ = 0) modulo dif-ferent cutoffs: by counterterms for d g ( x B , µ ) and by “brute force” rapidity cutoff Y < η in D ( x B , z ⊥ = 0 , η = ln σ ) . However, with the leading-log accuracy subtracting the coun-terterms is equivalent to imposing a cutoff in transverse momenta of the emitted gluons k ⊥ < µ . If we would calculate the leading-order renorm-group equation for the light-rayoperator ˜ F ai ( β B , x ⊥ ) F aj ( β B , x ⊥ ) we would cut the integral over k ⊥ from above by µ and– 14 –eave the integration over rapidity ( α ) unrestricted. Thus, we would obtain (cid:104) p | ˜ F ni ( β B , x ⊥ ) F ni ( β B , x ⊥ ) | p (cid:105) µ (3.32) = α s π N c (cid:90) ∞ dαα (cid:90) µ αsµ (cid:48) αs dβ (cid:110) θ (1 − β B − β ) (cid:104) β − β B ( β B + β ) + β B ( β B + β ) − β B ( β B + β ) (cid:105) × (cid:104) p | ˜ F ni ( β B + β, x ⊥ ) F ni ( β B + β, x ⊥ ) | p (cid:105) µ (cid:48) − β B β ( β B + β ) (cid:104) p | ˜ F ni ( β B , x ⊥ ) F ni ( β B , x ⊥ ) | p (cid:105) µ (cid:48) (cid:111) = α s π N c (cid:90) ∞ dβ (cid:90) µ βsµ (cid:48) βs dαα (cid:110) θ (1 − β B − β ) (cid:104) β − β B ( β B + β ) + β B ( β B + β ) − β B ( β B + β ) (cid:105) × (cid:104) p | ˜ F ni ( β B + β, x ⊥ ) F ni ( β B + β, x ⊥ ) | p (cid:105) µ (cid:48) − β B β ( β B + β ) (cid:104) p | ˜ F ni ( β B , x ⊥ ) F ni ( β B , x ⊥ ) | p (cid:105) µ (cid:48) (cid:111) which should be compared to Eq. (3.24) with x ⊥ = y ⊥ (cid:104) p | ˜ F ni ( β B , x ⊥ ) F in ( β B , x ⊥ ) | p (cid:105) ln σ = α s π N c (cid:90) σσ (cid:48) dαα (cid:90) ∞ dβ (cid:110) θ (1 − β B − β ) (cid:104) β − β B ( β B + β ) + β B ( β B + β ) − β B ( β B + β ) (cid:105) × (cid:104) p | ˜ F ni ( β B + β, x ⊥ ) F ni ( β B + β, x ⊥ ) | p (cid:105) ln σ (cid:48) − β B β − β B + β (cid:104) p | ˜ F ni ( β B , x ⊥ ) F in ( β B , x ⊥ ) | p (cid:105) ln σ (cid:48) (cid:111) In the leading log approximation β ∼ β B = x B so one can replace the cutoff µ βs in Eq. (3.32)by the cutoff µ x B s = σ and hence d g ( x B , ln µ ) = D ( x B , z ⊥ = 0 , ln σs ) with the leading-logaccuracy. The equation (3.32) can be rewritten as an evolution equation µ ddµ (cid:104) p | ˜ F ni ( β B , x ⊥ ) F in ( β B , x ⊥ ) | p (cid:105) µ (3.33) = α s ( µ ) π N c (cid:90) ∞ dβ (cid:110) θ (1 − β B − β ) (cid:104) β − β B ( β B + β ) + β B ( β B + β ) − β B ( β B + β ) (cid:105) × (cid:104) p | ˜ F ni ( β B + β, x ⊥ ) F ni ( β B + β, x ⊥ ) | p (cid:105) µ − β B β ( β B + β ) (cid:104) p | ˜ F ni ( β B , x ⊥ ) F in ( β B , x ⊥ ) | p (cid:105) µ (cid:111) which can be transformed to the standard DGLAP form [22] µ ddµ α s ( µ ) d g ( x B , ln µ ) (3.34) = α s ( µ ) π N c (cid:90) β B dz (cid:48) z (cid:48) (cid:104)(cid:0) − z (cid:48) (cid:1) + + 1 z (cid:48) − z (cid:48) (1 − z (cid:48) ) (cid:105) α s ( µ ) d g (cid:0) β B z (cid:48) , ln µ (cid:1) There is a subtle point in comparison of our rapidity evolution of light-ray operators to theconventional µ evolution described by renorm-group equations: the self-energy diagramsare not regulated by our rapidity cutoff so the δ -function terms in our version of the DGLAPequations are absent. Indeed, in our analysis we do not change the UV treatment of thetheory, we just define the Wilson-line (or light-ray) operators by the requirement that gluons For Eq (3.34) the absence of these terms is accidental, due to an extra α s in the definition (2.4). – 15 –mitted by those operators have rapidity cutoff (2.1). The UV divergences in self-energyand other internal loop diagrams appearing in higher-order calculations are absorbed inthe usual Z -factors. So, in a way, we will have two evolution equations for our operators:the trivial µ evolution described by anomalous dimensions of corresponding gluon (orquark) fields and the rapidity evolution. Combined together, the two should describe the Q evolution of DIS structure functions. Presumably, the argument of coupling constant inLO equation (3.30) (which is µ by default) will be replaced by σβ B s in accordance withcommon lore that this argument is determined by characteristic transverse momenta. Weplan to return to this point in the future NLO analysis.
In this section we will find the leading-order rapidity evolution of gluon operator (2.8) (cid:0) ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:1) ln σ with the rapidity cutoff Y < η = ln σ for all emitted gluons. As we mentioned in theIntroduction, in order to find the evolution kernel we need to integrate over slow gluons with σ > α > σ (cid:48) and temporarily freeze fast fields with α < σ (cid:48) to be integrated over later. To thisend we need the one-loop diagrams in the fast background fields with arbitrary transversemomenta. In the previous section we have found the evolution kernel in background fieldswith transverse momenta l ⊥ (cid:28) p ⊥ where p ⊥ is a characteristic momentum of our quantumslow fields. In this section at first we will find the Lipatov vertex and virtual correctionfor the case l ⊥ ∼ p ⊥ and then write down general formulas which are correct in the wholeregion of the transverse momentum.The key observation is that for transverse momenta of quantum and background fieldof the same order we can use the shock-wave approximation developed for small- x physics.To find the evolution kernel we consider the operator (2.8) in the background of externalfield A • ( x ∗ , x ⊥ ) (the absence of x • in the argument corresponds to α = 0 ). Moreover, weassume that the background field A • ( x ∗ , x ⊥ ) has a narrow support and vanishes outsidethe [ − σ ∗ , σ ∗ ] interval. This is obviously not the most general form of the external field, butit turns out that after obtaining the kernel of the evolution equation it is easy to restorethe result for any background field by insertion of gauge links at ±∞ p , see the discussionafter Eq. (5.4).Since the typical β ’s of the external field are β fast ∼ l ⊥ α fast s the support of the shockwave σ ∗ is of order of β fast ∼ σ (cid:48) sl ⊥ . This is to be compared to the typical scale of slow fields β slow ∼ αsp ⊥ (cid:29) σ ∗ so we see that the fast background field can be approximated by a narrowshock wave. In the “pure” low-x case β B = 0 one can assume that the support of this shockwave is infinitely narrow. As we shall see below, in our case of arbitrary β B we need to Note that while in the usual renorm-group DGLAP the argument of coupling constant is a part of LOequation, with our cutoff this argument can be determined only at the NLO level, same as in the case ofNLO BK equation at low x [14]. This is not surprising since we use the rapidity cutoff borrowed from theNLO BK analysis. – 16 –ook inside the shock wave so we will separate all integrals over longitudinal distances z ∗ inparts “inside the shock wave” | z ∗ | < σ ∗ and “outside the shock wave" | z ∗ | > σ ∗ , calculatethem separately and check that the sum of “inside” and “outside” contributions does notdepend on σ ∗ with our accuracy. In the leading order there is only one extra gluon and we get the typical diagrams of Fig.3 type. The production part of the kernel can be obtained as a square of a Lipatov vertex (a) (b)
Figure 3 . Typical diagrams for production (a) and virtual (b) contributions to the evolution kernel.The shaded area denotes shock wave of background fast fields. - the amplitude of the emission of a real gluon by the operator F ai (see Eq. (3.1)) (cid:104) ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:105) ln σ = − (cid:90) σσ (cid:48) d − α α d − k ⊥ (cid:0) ˜ L ba ; µi ( k, x ⊥ , β B ) L abµj ( k, y ⊥ , β B ) (cid:1) ln σ (cid:48) (4.1)where the Lipatov vertices of gluon emission are defined as L abµi ( k, y ⊥ , β B ) = i lim k → k (cid:104) A aµ ( k ) F bi ( β B , y ⊥ ) (cid:105) ˜ L baiµ ( k, x ⊥ , β B ) = − i lim k → k (cid:104) ˜ F bi ( β B , x ⊥ ) ˜ A aµ ( k ) (cid:105) (4.2)(cf. Eqs. (3.2) and (3.10)). Hereafter (cid:104)O(cid:105) means the average of operator O in the shock-wave background. As we discussed above, we calculate the diagrams in the background of a shock wave ofwidth ∼ σ (cid:48) sl ⊥ where l ⊥ is the characteristic transverse momentum of the external shock-wavefield. Note that the factor in the exponent in the definition of F ( β B ) is ∼ β B σ (cid:48) sl ⊥ which isnot necessarily small at various β B and l ⊥ and therefore we need to take into account thediagram in Fig. 4c with emission point inside the shock wave. We will do this in a followingway: we assume that all of the shock wave is contained within σ ∗ > z ∗ > − σ ∗ , calculatediagrams in Fig. 4a-d and check that the dependence on σ ∗ cancels in the final result forthe sum of these diagrams. – 17 – a) (b) (c) (d) Figure 4 . Lipatov vertex of gluon emission.
We start the calculation with the expansion of the gluon fields in F ( β B , z ⊥ ) in the firstorder in slow “quantum” field: F ni ( β B , y ⊥ ) st = g (cid:90) d s y ∗ e iβ B y ∗ (cid:104) − i s β B [ ∞ , y ∗ ] nmy A q mi ( y ∗ , y ⊥ ) − [ ∞ , y ∗ ] nmy ∂ i A q m • ( y ∗ , y ⊥ ) + 2 igs (cid:90) ∞ y ∗ dz ∗ [ ∞ , z ∗ ] y A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) (4.3)where the gauge links and F m • i are made of fast “external” fields. The corresponding vertexof gluon emission is given by lim k → k (cid:104) A a q µ ( k ) F ni ( β B , y ⊥ ) (cid:105) = lim k → k g (cid:90) dy ∗ e iβ B y ∗ (cid:104) A a q µ ( k ) (cid:8) − iβ B [ ∞ , y ∗ ] nmy A m q i ( y ∗ , y ⊥ ) − s [ ∞ , y ∗ ] nmy ∂ i A m q • ( y ∗ , y ⊥ ) + 4 igs (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] (cid:1) nm F m • i ( y ∗ , y ⊥ ) (cid:9) (cid:105) (4.4)The diagrams in Fig. 4a, 4b, and 4(c-d) correspond to different regions of integration over y ∗ in Eq. (4.3): y ∗ > σ ∗ , − σ ∗ > y ∗ , and σ ∗ > y ∗ > − σ ∗ , respectively.The trivial calculation of Fig. 4a contribution yields i lim k → k (cid:104) A aqµ ( k ) F ni ( β B , y ⊥ ) (cid:105) Fig . (4.5) = ig (cid:90) ∞ σ ∗ dy ∗ e iβ B y ∗ [ ∞ , y ∗ ] nm lim k → k (cid:104) A aµ ( k ) (cid:2) − iβ B A mi ( y ∗ , y ⊥ ) − s ∂ i A m • ( y ∗ , y ⊥ ) (cid:3) (cid:105) = gδ an g µi αβ B s + 2 αk i p µ αβ B s + k ⊥ e i ( β B + k ⊥ αs ) σ ∗ − i ( k,y ) ⊥ Next step is the calculation of Fig. 4b contribution. Using the vertex of gluon emissionfrom the shock wave (11.30) one obtains ig lim k → (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ k (cid:104) A a q µ ( k )[ ∞ , y ∗ ] nmy (cid:8) − iβ B A m q i ( y ∗ , y ⊥ ) − s ∂ i A m q • ( y ∗ , y ⊥ ) (cid:9) (cid:105) = g (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:90) d z ⊥ e − i ( k,z ) ⊥ (cid:8) − iβ B O µi ( ∞ , y ∗ , z ⊥ ; k )( z ⊥ | e i p ⊥ αs y ∗ | y ⊥ ) − is O µ • ( ∞ , y ∗ , z ⊥ ; k )( z ⊥ | p i e i p ⊥ αs y ∗ | y ⊥ ) (cid:9) am ( U † y ) mn (4.6)– 18 –here O is given by Eqs. (11.31): O abµν ( ∞ , y ∗ , z ⊥ ; k ) y ∗ < − σ ∗ = g µν U abz + g (cid:90) ∞−∞ dz ∗ (cid:0) [ ∞ , z ∗ ] z (cid:2) − iz ∗ αs g µν (2 k j − iD j ) + 4 αs ( δ jµ p ν − δ jν p µ ) (cid:3) F • j ( z ∗ , z ⊥ )[ z ∗ , −∞ ] z (cid:1) ab + 4 gα s (cid:90) ∞−∞ dz ∗ (cid:0) [ ∞ , z ∗ ] z (cid:8) − ip µ p ν D j F • j ( z ∗ , z ⊥ )[ z ∗ , −∞ ] z + g (cid:90) z ∗ −∞ dz (cid:48)∗ (cid:2) iαg µν z (cid:48)∗ − s p µ p ν (cid:3) F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , −∞ ] z (cid:9)(cid:1) ab + g αs (cid:110)(cid:90) ∞−∞ dz ∗ (cid:90) z ∗ −∞ dz (cid:48)∗ ¯ ψ ( z ∗ , z ⊥ )[ z ∗ , ∞ ] z t a U z t b [ −∞ , z (cid:48)∗ ] z γ ⊥ µ (cid:54) p γ ⊥ ν ψ ( z (cid:48)∗ , z ⊥ ) − (cid:90) ∞−∞ dz ∗ (cid:90) ∞ z ∗ dz (cid:48)∗ ¯ ψ ( z ∗ , z ⊥ ) γ ⊥ ν (cid:54) p γ ⊥ µ [ z ∗ , −∞ ] z t b U † z t a [ ∞ , z (cid:48)∗ ] z ψ ( z (cid:48)∗ , z ⊥ ) (cid:111) (4.7)where we replaced y ∗ by −∞ since we assumed that there is no gauge field outside the [ − σ ∗ , σ ∗ ] interval.Let us compare relative size of terms in the r.h.s. of this equation. The leading g µν term is ∼ U z ∼ and it is clear that all other g µν terms are small. Indeed, the first termin the second line is ∼ gαs (cid:82) dz ∗ z ∗ (2 k j − iD j ) F • j ∼ σ ∗ αs k j ∂ j U ∼ k ⊥ αs σ ∗ (cid:28) since the widthof the shock wave is ∼ sσ (cid:48) k ⊥ and α (cid:29) σ (cid:48) (recall that in this Section l ⊥ ∼ k ⊥ ). Similarly, thefirst term in the fourth line is ∼ g αs (cid:82) dz ∗ dz (cid:48)∗ z (cid:48)∗ F • j ( z ∗ ) F j • ( z (cid:48)∗ ) ∼ σ ∗ αs ∂ j U ∂ j U ∼ σ ∗ k ⊥ αs (cid:28) .Next, let us find out the relative size of quark terms in Eq. (4.7). The “power counting”for external quark fields in comparison to gluon ones is g s (cid:82) dz ∗ ¯ ψ (cid:54) p ψ ( z ∗ ) ∼ gs (cid:82) dz ∗ D i F • i ( z ∗ ) ∼ k ⊥ U ∼ k ⊥ and each extra integration inside theshock wave brings extra σ ∗ . Thus, the two last terms in Eq. (4.7) are ∼ g ⊥ µν σ ∗ αs k ⊥ (cid:28) .After omitting small terms the expression (4.7) reduces to O µν ( ∞ , −∞ , z ⊥ ; k ) = g µν U z + 4 gαs ( δ jµ p ν − δ jν p µ ) (cid:90) ∞−∞ dz ∗ [ ∞ , z ∗ ] z F • j ( z ∗ , z ⊥ )[ z ∗ , −∞ ] z + 4 gα s p µ p ν (cid:90) ∞−∞ dz ∗ [ ∞ , z ∗ ] z (cid:110) − iD j F • j ( z ∗ , z ⊥ )[ z ∗ , −∞ ] z − gs (cid:90) z ∗ −∞ dz (cid:48)∗ F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , −∞ ] z (cid:111) = g µν U z + 2 iαs ( δ jµ p ν − δ jν p µ ) ∂ j U z − p µ p ν α s ∂ ⊥ U z (4.8)where we used the formula ∂ ⊥ U z = g (cid:90) ∞−∞ dz ∗ [ ∞ , z ∗ ] z (4.9) × (cid:16) is D j F • j ( z ∗ , z ⊥ )[ z ∗ , −∞ ] z + 8 gs (cid:90) z ∗ −∞ dz (cid:48)∗ F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , −∞ ] z (cid:17) Note, however, that the quark term ∼ gα s p µ p ν (cid:82) dz ∗ D j F • j ( z ∗ ) ∼ p µ p ν k ⊥ α s U is of the same orderof magnitude as the gluon term ∼ g α s p µ p ν (cid:82) dz ∗ dz (cid:48)∗ F • j ( z ∗ ) F j • ( z (cid:48)∗ ) . – 19 –sing Eq. (4.8) one obtains for the r.h.s. of Eq. (4.6) i lim k → k (cid:104) A aqµ ( k ) F ni ( β B , y ⊥ ) (cid:105) Fig . (4.10) = g (cid:90) d z ⊥ e − i ( k,z ) ⊥ (cid:8) − O µi ( ∞ , −∞ , z ⊥ ; k )( z ⊥ | αβ B sαβ B s + p ⊥ e − i ( β B + p ⊥ αs ) σ ∗ | y ⊥ ) − s O µ • ( ∞ , −∞ , z ⊥ ; k )( z ⊥ | p i αsαβ B s + p ⊥ e − i ( β B + p ⊥ αs ) σ ∗ | y ⊥ ) (cid:9) am ( U † y ) mn = − ge − iβ B σ ∗ ( k ⊥ | (cid:2) g µi U − ip µ αs ∂ i U + 2 β B s (cid:0) p µ U + iα ∂ ⊥ µ U − p µ α s ∂ ⊥ U (cid:1) p i (cid:3) αβ B sαβ B s + p ⊥ U † | y ⊥ ) an where we used the fact that p ⊥ αs σ ∗ (cid:28) when all the transverse momenta are of the sameorder. Next step is the calculation of Fig.4 c,d contributions. Using the vertex of gluon emissionfrom the shock wave (10.47) and Eqs. (11.6), (11.7) one obtains lim k → k ig (cid:90) σ ∗ − σ ∗ dy ∗ e iβ B y ∗ (cid:104) A a q µ ( k ) (cid:8) − iβ B [ ∞ , y ∗ ] nmy A m q i ( y ∗ , y ⊥ ) (4.11) − s [ ∞ , y ∗ ] nmy ∂ i A m q • ( y ∗ , y ⊥ ) + 4 igs (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm F m • i ( y ∗ , y ⊥ ) (cid:9) (cid:105) = g (cid:90) σ ∗ − σ ∗ dy ∗ (cid:8) e i ( β B + k ⊥ αs ) y ∗ (cid:2) − iβ B O amµi ( ∞ , y ∗ , y ⊥ ; k ) − s (cid:0) ik i + ∂∂y i (cid:1) O amµ • ( ∞ , y ∗ , y ⊥ ; k ) (cid:3) + 4 igs (cid:90) ∞ y ∗ dz ∗ e iβ B y ∗ + i k ⊥ αs z ∗ (cid:0) O µ • ( ∞ , z ∗ , y ⊥ ; k )[ z ∗ , y ∗ ] y F • i ( y ∗ , y ⊥ ) (cid:1) am (cid:9) [ y ∗ , ∞ ] mny e − i ( k,y ) ⊥ = ge − i ( k,y ) ⊥ (cid:90) σ ∗ − σ ∗ dy ∗ (cid:104) e iβ B y ∗ (cid:0) − i { β B O µi ( ∞ , y ∗ , y ⊥ ; k ) + 2 s k i O µ • ( ∞ , y ∗ , y ⊥ ; k ) } [ y ∗ , ∞ ] y − s ∂∂y i (cid:0) O µ • ( ∞ , y ∗ , y ⊥ ; k )[ y ∗ , ∞ ] y (cid:1) + 4 igs (cid:90) ∞ y ∗ dz ∗ O µ • ( ∞ , y ∗ , y ⊥ ; k )[ y ∗ , z ∗ ] y F • i ( z ∗ , y ⊥ ) × [ z ∗ , ∞ ] y (cid:1) + 4 igs (cid:90) ∞ y ∗ dz ∗ e iβ B y ∗ + i k ⊥ αs z ∗ O µ • ( ∞ , z ∗ , y ⊥ ; k )[ z ∗ , y ∗ ] y F • i ( y ∗ , y ⊥ )[ y ∗ , ∞ ] y (cid:105) an where O µν = G µν + Q µν + ¯ Q µν and G , Q and ¯ Q are given by Eqs. (11.6) and (11.7). Aswe mentioned above, the contributions with extra ( z − σ ) ∗ are small and so are the quarkterms (except term ∼ D j F • j ). So, we have Q µi = ¯ Q µi = 0 and O µi ( ∞ , y ∗ , y ⊥ ; k )[ y ∗ , ∞ ] y = g µi − gp µ αs (cid:90) ∞ y ∗ dz ∗ [ ∞ , z ∗ ] y F • i ( z ∗ , y ⊥ )[ z ∗ , ∞ ] y , O µ • ( ∞ , y ∗ , y ⊥ ; k )[ y ∗ , ∞ ] y = p µ + g (cid:90) ∞ y ∗ dz ∗ [ ∞ , z ∗ ] y (cid:8) αs F • µ ( z ∗ , y ⊥ )[ z ∗ , ∞ ] y (4.12) − ip µ α s D j F • j ( z ∗ , y ⊥ )[ z ∗ , ∞ ] y − gp µ α s (cid:90) z ∗ y ∗ dz (cid:48)∗ F • j ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y F j • ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , ∞ ] y (cid:9) – 20 –fter some algebra the r.h.s. of Eq. (4.11) reduces to i lim k → k (cid:104) A aqµ ( k ) F ni ( β B , y ⊥ ) (cid:105) Fig . = − g µi αβ B s + 2 αk i p µ αβ B s + k ⊥ (cid:2) e i ( β B + k ⊥ αs ) σ ∗ − e − i ( β B + k ⊥ αs ) σ ∗ (cid:3) e − i ( k,y ) ⊥ gδ an + ge − i ( k,y ) ⊥ (cid:110) αβ B s (cid:0) αp µ δ ji + β B p µ δ ji − k i δ jµ (cid:1)(cid:2) F j ( β B , y ⊥ ) − i∂ j U y U † y e − iβ B σ ∗ (cid:3) − αp µ k ⊥ F i ( β B , y ⊥ ) + 2 p µ k i α s β B [ V ( β B , y ⊥ ) − e − iβ B σ ∗ ∂ ⊥ U y U † y ]+ 2 iαβ B s ∂ yi (cid:8) F µ ( y ⊥ , β B ) − i∂ µ U y U † y e − iβ B σ ∗ − p µ αs [ V ( β B , y ⊥ ) − e − iβ B σ ∗ ∂ ⊥ U y U † y ] (cid:9) + 4 gβ B s (cid:90) dz ∗ dz (cid:48)∗ e iβ B min( z ∗ ,z (cid:48)∗ ) × [ ∞ , z ∗ ] (cid:8) αs F • µ ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y − ip µ α s D j F • j ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y (cid:9) F • i ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , ∞ ] y − g p µ α s β B (cid:90) dz ∗ dz (cid:48)∗ dz (cid:48)(cid:48)∗ θ ( z ∗ − z (cid:48)(cid:48)∗ ) e iβ B min( z (cid:48)∗ ,z (cid:48)(cid:48)∗ ) [ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)(cid:48)∗ ] y F j • ( z (cid:48)(cid:48)∗ , y ⊥ ) × [ z (cid:48)(cid:48)∗ , z (cid:48)∗ ] y F • i ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , ∞ ] y − αβ B s e − iβ B σ ∗ ( ∂ µ U y ∂ i U † y + ip µ αs ∂ ⊥ U y ∂ i U † y ) (cid:111) an (4.13)where V ( β B , y ⊥ ) ≡ g (cid:90) ∞−∞ dz ∗ e iβ B z ∗ (cid:16) is [ ∞ , z ∗ ] y D j F • j ( z ∗ , y ⊥ )+ 8 gs (cid:90) ∞ z ∗ dz (cid:48)∗ [ ∞ , z (cid:48)∗ ] y F • j ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , z ∗ ] y F j • ( z ∗ , y ⊥ ) (cid:17) [ z ∗ , ∞ ] y (4.14)– 21 – .2.3 Lipatov vertex The sum of Eqs. (4.5), (4.10), and (4.13) gives the Lipatov vertex of gluon emission in theform L abµi ( k, y ⊥ , β B ) = gδ ab g µi αβ B s + 2 αk i p µ αβ B s + k ⊥ e − iβ B σ ∗ − i ( k,y ) ⊥ − ge − iβ B σ ∗ ( k ⊥ | (cid:2) g µi U − ip µ αs ∂ i U + (cid:0) p µ U + iα ∂ ⊥ µ U − p µ α s ∂ ⊥ U (cid:1) p i β B s (cid:3) αβ B sαβ B s + p ⊥ U † | y ⊥ ) ab + ge − i ( k,y ) ⊥ (cid:110) αβ B s (cid:0) αp µ δ ji + β B p µ δ ji − k i δ jµ (cid:1)(cid:2) F j ( β B , y ⊥ ) − i∂ j U y U † y e − iβ B σ ∗ (cid:3) − αp µ k ⊥ F i ( β B , y ⊥ ) + 2 p µ k i α s β B [ V ( β B , y ⊥ ) − ∂ ⊥ U y U † y e − iβ B σ ∗ ]+ 2 iαβ B s ∂ yi (cid:8) F µ ( β B , y ⊥ ) − i∂ µ U y U † y e − iβ B σ ∗ − p µ αs [ V ( β B , y ⊥ ) − e − iβ B σ ∗ ∂ ⊥ U y U † y ] (cid:9) + 4 gβ B s (cid:90) dz ∗ dz (cid:48)∗ e iβ B min( z ∗ ,z (cid:48)∗ ) × [ ∞ , z ∗ ] (cid:8) αs F • µ ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y − ip µ α s D j F • j ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y (cid:9) F • i ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , ∞ ] y − g p µ α s β B (cid:90) dz ∗ dz (cid:48)∗ dz (cid:48)(cid:48)∗ θ ( z ∗ − z (cid:48)(cid:48)∗ ) e iβ B min( z (cid:48)∗ ,z (cid:48)(cid:48)∗ ) [ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)(cid:48)∗ ] y F j • ( z (cid:48)(cid:48)∗ , y ⊥ ) × [ z (cid:48)(cid:48)∗ , z (cid:48)∗ ] y F • i ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , ∞ ] y − αβ B s e − iβ B σ ∗ ( ∂ µ U y ∂ i U † y + ip µ αs ∂ ⊥ U y ∂ i U † y ) (cid:111) ab (4.15)This expression explicitly depends on the cutoff σ ∗ . However, we can set σ ∗ = 0 in ther.h.s. of Eq. (4.15) (and eliminate few terms as well). To demonstrate this, let us considertwo cases: β B (cid:28) σ ∗ and β B ≥ σ ∗ . In the first case L abµi ( k, y ⊥ , β B ) β B σ ∗ (cid:28) = − ge − i ( k,y ) ⊥ αp µ k ⊥ F abi ( β B , y ⊥ ) + g ( k ⊥ | g µi αβ B s + 2 αk i p µ αβ B s + k ⊥ (4.16) − (cid:2) g µi U − ip µ αs ∂ i U + 2 β B s (cid:0) p µ U + iα ∂ ⊥ µ U − p µ α s ∂ ⊥ U (cid:1) p i (cid:3) αβ B sαβ B s + p ⊥ U † | y ⊥ ) ab = g ( k ⊥ | g µi (cid:0) αβ B sαβ B s + p ⊥ − U αβ B sαβ B s + p ⊥ U † (cid:1) + 2 αp µ (cid:0) p i αβ B s + p ⊥ − U p i αβ B s + p ⊥ U † (cid:1) + (cid:2) ip µ αs ∂ i U − iαβ B s ∂ ⊥ µ U p i + 2 p µ α s β B ∂ ⊥ U p i (cid:3) αβ B sαβ B s + p ⊥ U † − αp µ p ⊥ i ( ∂ i U ) U † | y ⊥ ) ab and all other terms are small since they contain extra factors e iβ B z ∗ − e − iβ B σ ∗ (cid:39) ( z − σ ) ∗ (or z (cid:48)∗ − σ ∗ or z (cid:48)(cid:48)∗ − σ ∗ ) in the integrand.In the second case σ (cid:48) β B s ≥ p ⊥ so αβ B s (cid:29) p ⊥ and we get L abµi ( k, y ⊥ , β B )= e − i ( k,y ) ⊥ (cid:16) (cid:0) p µ αs − αp µ k ⊥ (cid:1) F i ( β B , y ⊥ ) − k i αβ B s F µ ( β B , y ⊥ ) + 2 iαβ B s ∂ yi F µ ( β B , y ⊥ )+ 8 g αβ B s (cid:90) dz ∗ dz (cid:48)∗ e iβ B min( z ∗ ,z (cid:48)∗ ) [ ∞ , z ∗ ] F • µ ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y F • i ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , ∞ ] y (cid:17) ab (4.17)– 22 –here we used the formula ( k ⊥ | g µi (cid:0) αβ B sαβ B s + p ⊥ − U αβ B sαβ B s + p ⊥ U † (cid:1) + 2 αp µ (cid:0) p i αβ B s + p ⊥ − U p i αβ B s + p ⊥ U † (cid:1) | y ⊥ )= ( k ⊥ | αβ B sg µi + 2 αp µ k i αβ B s + k ⊥ (2 ik j ∂ j U − ∂ ⊥ U ) 1 αβ B s + p ⊥ U † + 2 iαp µ ∂ i U αβ B s + p ⊥ U † | y ⊥ ) (4.18)Let us now compare the contributions of various terms in the r.h.s. of Eq. (4.15) to theproduction part of the evolution kernel defined by the square of Lipatov vertices (4.15). Itis clear that the square of the first term ∼ (cid:0) p µ αs − αp µ k ⊥ (cid:1) F i is proportional to ˜ F i k ⊥ F i andcontributions of all other terms are down by at least one power of p ⊥ αβ B s . Thus, with ouraccuracy L abµi ( k, y ⊥ , β B ) αβ B s (cid:29) p ⊥ = 2 e − i ( k,y ) ⊥ (cid:0) p µ αs − αp µ k ⊥ (cid:1) F abi ( β B , y ⊥ ) (4.19)We see that in both cases (4.16) and (4.19) one can replace σ ∗ by 0. Moreover, with ouraccuracy the Lipatov vertex (4.15) can be reduced to the “direct sum” of Eqs. (4.16) and(4.19): L abµi ( k, y ⊥ , β B ) = 2 ge − i ( k,y ) ⊥ (cid:0) p µ αs − αp µ k ⊥ (cid:1) [ F i ( β B , y ⊥ ) − U i ( y ⊥ )] ab (4.20) + g ( k ⊥ | g µi (cid:0) αβ B sαβ B s + p ⊥ − U αβ B sαβ B s + p ⊥ U † (cid:1) + 2 αp µ (cid:0) p i αβ B s + p ⊥ − U p i αβ B s + p ⊥ U † (cid:1) + (cid:2) iβ B p µ ∂ i U − i∂ ⊥ µ U p i + 2 p µ αs ∂ ⊥ U p i (cid:3) αβ B s + p ⊥ U † − αp µ p ⊥ U i | y ⊥ ) ab where we introduced the notation U i ≡ F i (0) = i ( ∂ i U ) U † . It is clear that at β B σ ∗ (cid:28) thefirst term in the r.h.s. of this equation disappears and we get the r.h.s. of Eq. (4.16). Onthe other hand, as we saw above, at β B σ ∗ ≥ all terms in the last two lines in the r.h.s.of Eq. (4.20) are small except ( k ⊥ | p µ αs U i − αp µ p ⊥ U i | y ⊥ ) ab which cancels the second termin the first line of Eq. (4.20) so we get the r.h.s. of Eq. (4.19). It is worth noting that at β B = 0 Eq. (4.20) agrees with the Lipatov vertex obtained in Ref. [23].It is instructive to check the Lipatov vertex property k µ L abµi ( k, y ⊥ , β B ) = 0 . Oneobtains k µ × (r . h . s . of Eq . (4 . µ (4.21) = g ( k ⊥ | k i (cid:0) αβ B sαβ B s + k ⊥ − U αβ B sαβ B s + p ⊥ U † (cid:1) + k ⊥ (cid:0) k i αβ B s + k ⊥ − U p i αβ B s + p ⊥ U † (cid:1) + ( αβ B s [ p i , U ] + [ p ⊥ + αβ B s, U ] p i ) 1 αβ B s + p ⊥ U † − U i | y ⊥ ) = 0 – 23 – .3 Lipatov vertex for arbitrary transverse momenta Let us demonstrate that for arbitrary transverse momenta the Lipatov vertex of gluonemission is given by the following “interpolating formula” L abµi ( k, y ⊥ , β B )= g ( k ⊥ | αβ B sg µi + 2 αp µ k i αβ B s + k ⊥ (2 ik j ∂ j U − ∂ ⊥ U ) 1 αβ B s + p ⊥ U † + 2 iαp µ ∂ i U αβ B s + p ⊥ U † + 2 iαs p µ ∂ i U αβ B sαβ B s + p ⊥ U † − (cid:2) i∂ µ U − p µ αs ∂ ⊥ U (cid:3) p i αβ B s + p ⊥ U † − αp µ p ⊥ i ( ∂ i U ) U † | y ⊥ ) ab + 2 ge − i ( k,y ) ⊥ αβ B s + k ⊥ (cid:2) − δ jµ k i + 2 αk i k j p µ αβ B s + k ⊥ + αβ B sg µi k j αβ B s + k ⊥ + β B p µ δ ji − αp µ αβ B sk ⊥ δ ji (cid:3) × (cid:2) F j (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) − U j ( y ⊥ ) (cid:3) ab (4.22)Let us consider at first the light-cone limit corresponding to the case when the char-acteristic transverse momenta of the external “fast” gluon fields are small in comparisonto the momenta of “slow” gluons which we integrated over. As we discussed above, thehigher-twist terms ∼ D j F • k or ∼ F • j F • k exceed our accuracy so we can eliminate terms ∼ ∂ ⊥ U and commute operators ∂ j U with p ⊥ + αβ B s resulting in r . h . s . of Eq . (4 . light − cone = 2 g ( k ⊥ | k j αβ B sg µi + 2 αp µ k i ( αβ B s + k ⊥ ) U j + αp µ αβ B s + k ⊥ U i + p µ β B αβ B s + k ⊥ U i − k i αβ B s + k ⊥ U µ ⊥ − αp µ k ⊥ U i | y ⊥ ) ab + 2 ge − i ( k,y ) ⊥ αβ B s + k ⊥ (cid:2) − δ jµ k i + 2 αk i k j p µ αβ B s + k ⊥ + αβ B sg µi k j αβ B s + k ⊥ + β B p µ δ ji − αp µ αβ B sk ⊥ δ ji (cid:3) × (cid:2) F j (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) − U j ( y ⊥ ) (cid:3) ab (4.23)It is clear now that the first two lines in the r.h.s. cancel the last term in the square bracketsin the last line so we recover the light-cone result (3.8).Next we consider the case when the transverse momenta of fast and slow fields arecomparable so the Lipatov vertex is given by Eq. (4.20) above. The difference between ther.h.s.’s of Eq. (4.22) and Eq. (4.20) is ge − i ( k,y ) ⊥ αβ B s + k ⊥ (cid:2) − δ jµ k i + 2 αk i k j p µ αβ B s + k ⊥ + αβ B sg µi k j αβ B s + k ⊥ (cid:3)(cid:2) F j (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) − U j ( y ⊥ ) (cid:3) ab (4.24)where we used Eq. (4.18). It is easy to see that the expression (4.24) is small in both β B (cid:28) σ ∗ and β B ≥ σ ∗ cases. Indeed, when β B (cid:28) σ ∗ the integral representing F j (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) − U j ( y ⊥ ) contains an extra factor e i ( β B + k ⊥ αs ) z ∗ − ∼ ( β B + k ⊥ αs (cid:1) σ ∗ (cid:28) in theintegrand and in the β B ≥ σ ∗ case the Eq. (4.24) is ∼ k ⊥ αβ B s (cid:28) in comparison to theleading term in this limit (4.19).As in the light-cone case, for calculation of the evolution kernel it is convenient to goto the light-like gauge p µ A µ = 0 . Since k µ × (cid:0) r . h . s . of Eq . (4 . (cid:1) µ = 0 (see Eq. (4.21))– 24 –t is sufficient to replace αp µ in the r.h.s. of Eq. (4.22) by αp µ − k µ = − k µ ⊥ − k ⊥ αs p µ . Oneobtains L abµi ( k, y ⊥ , β B ) light − like (4.25) = g ( k ⊥ | αβ B sg µi − k ⊥ µ k i αβ B s + k ⊥ (2 ik j ∂ j U − ∂ ⊥ U ) 1 αβ B s + p ⊥ U † − ik ⊥ µ ∂ i U αβ B s + p ⊥ U † − i∂ µ U p i αβ B s + p ⊥ U † + 2 k ⊥ µ k ⊥ U i | y ⊥ ) ab + 2 ge − i ( k,y ) ⊥ (cid:104) k ⊥ µ δ ji k ⊥ − δ jµ k i + δ ji k ⊥ µ − g µi k j αβ B s + k ⊥ − k ⊥ g µi k j + 2 k ⊥ µ k i k j ( αβ B s + k ⊥ ) (cid:105) × (cid:2) F j (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) − U j ( y ⊥ ) (cid:3) ab + O ( p µ ) As usual, we do not display the term ∼ p µ since it does not contribute to the evolutionkernel. Using [ p ⊥ , U ] = − ip j ∂ j U + ∂ ⊥ U one can rewrite this vertex as L abµi ( k, y ⊥ , β B ) light − like = i lim k → k (cid:104) A a q µ ( k ) F bi ( β B , y ⊥ ) (cid:105) light − like (4.26) = g ( k ⊥ | U p ⊥ g µi + 2 p ⊥ µ p i αβ B s + p ⊥ U † − k ⊥ g µi + 2 k ⊥ µ k i αβ B s + k ⊥ | y ⊥ ) ab + 2 gk ⊥ µ k ⊥ e − i ( k,y ) ⊥ F abi (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) − ge − i ( k,y ) ⊥ (cid:104) δ jµ k i + δ ji k ⊥ µ − g µi k j αβ B s + k ⊥ + g µi k ⊥ k j + 2 k ⊥ µ k i k j ( αβ B s + k ⊥ ) (cid:105) ˘ F abj (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) + O ( p µ ) where we introduced the notation ˘ F j (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) ≡ F j (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) − F j (0 , y ⊥ ) (4.27)(recall that F j (0 , y ⊥ ) = U j ( y ⊥ ) ≡ i∂ j U y U † y ).It should be emphasized that while we constructed the Lipatov vertex (4.22) as aformula which interpolates between the light-cone result (3.8) for small transverse momentaof background fields and shock-wave result (4.20) for comparable transverse momenta, wehave just demonstrated that with our leading-log accuracy our final expression (4.22) iscorrect in the whole range of the transverse momenta.It is convenient to rewrite the Lipatov vertex (4.26) in a different form without explicitsubtraction (4.27). Starting from Eq. (4.25) we get L abµi ( k, y ⊥ , β B ) light − like (4.28) = g ( k ⊥ |F j (cid:0) β B + k ⊥ αs (cid:1)(cid:110) αβ B sg µi − k ⊥ µ k i αβ B s + k ⊥ ( k j U + U p j ) 1 αβ B s + p ⊥ U † − k ⊥ µ U g ij αβ B s + p ⊥ U † − g µj U p i αβ B s + p ⊥ U † + 2 k ⊥ µ k ⊥ g ij (cid:111) | y ⊥ ) ab + O ( p µ ) where the operator F i ( β ) is defined as usual ( k ⊥ |F i ( β ) | y ⊥ ) ≡ s (cid:90) dy ∗ e iβy ∗ − i ( k,y ) ⊥ F i ( y ∗ , y ⊥ ) (4.29)– 25 –et us prove that Eq. (4.28) coincides with Eq. (4.26) with our accuracy. First, aswe discussed above, in the light-cone case ( l ⊥ (cid:28) p ⊥ ) we can drop higher-twist terms andcommute operators U with p i and p ⊥ + αβ B s which gives us Eq. (3.8). Second, considerthe “shock-wave” case l ⊥ ∼ p ⊥ . When β B (cid:28) σ ∗ the integral representing F j (cid:0) β B + k ⊥ αs (cid:1) contains an exponential factor e i ( β B + k ⊥ αs ) z ∗ ∼ e i ( β B + k ⊥ αs ) σ ∗ . This factor can be approximatedby one, since k ⊥ αs σ ∗ (cid:28) in the shock-wave case (see the discussion above), so we can replace F j (cid:0) β B + k ⊥ αs (cid:1) by U j and get L abµi ( k, y ⊥ , β B ) light − like (4.30) (cid:39) g ( k ⊥ | (cid:110) αβ B sg µi − k ⊥ µ k i αβ B s + k ⊥ ( k j i∂ j U + i∂ j U p j ) 1 αβ B s + p ⊥ U † − k ⊥ µ i∂ i U αβ B s + p ⊥ U † − i∂ µ U p i αβ B s + p ⊥ U † + 2 k ⊥ µ k ⊥ U i (cid:111) | y ⊥ ) ab + O ( p µ ) which gives the first two lines in the r.h.s. of Eq. (4.25). As it was shown above, the lasttwo lines in the r.h.s. of Eq. (4.25) are small at β B (cid:28) σ ∗ so Eq. (4.28) coincides withEq. (4.25) at β B (cid:28) σ ∗ with our accuracy. Finally, in the β B ≥ σ ∗ case αβ B s (cid:29) p ⊥ andtherefore the Eq. (4.28) reduces to L abµi ( k, y ⊥ , β B ) light − like (cid:39) k ⊥ µ k ⊥ e − i ( k,y ) ⊥ g F abi (cid:0) β B + k ⊥ αs , y ⊥ (cid:1) + O ( p µ ) which is the same as Eq. (4.25) in this limit.Similar calculation for complex-conjugate amplitude gives ˜ L baiµ ( k, x ⊥ , β B ) light − like = − i lim k → k (cid:104) ˜ F bi ( β B , x ⊥ ) ˜ A aqµ ( k ) (cid:105) light − like (4.31) = g ( x ⊥ | ˜ U p ⊥ g µi + 2 p ⊥ µ p i αβ B s + p ⊥ ˜ U † − p ⊥ g µi + 2 p ⊥ µ p i αβ B s + p ⊥ | k ⊥ ) ba + 2 ge i ( k,x ) ⊥ k ⊥ µ k ⊥ ˜ F bai (cid:0) β B + k ⊥ αs , x ⊥ (cid:1) − ge i ( k,x ) ⊥ (cid:104) δ jµ k i + k ⊥ µ δ ji − g µi k j αβ B s + k ⊥ + g µi k ⊥ k j + 2 k i k j k ⊥ µ ( αβ B s + k ⊥ ) (cid:105) ˘˜ F baj (cid:0) β B + k ⊥ αs , x ⊥ (cid:1) + O ( p µ ) where ˘˜ F j (cid:0) β B + k ⊥ αs , x ⊥ (cid:1) ≡ ˜ F j (cid:0) β B + k ⊥ αs , x ⊥ (cid:1) − ˜ F j (0 , x ⊥ ) (4.32)Similarly to Eq. (4.28) we can rewrite the above expression in the form without subtractions ˜ L baiµ ( k, x ⊥ , β B ) light − like (4.33) = g ( x ⊥ | (cid:110) ˜ U αβ B s + p ⊥ ( ˜ U † k j + p j ˜ U † ) αβ B sg µi − k ⊥ µ k i αβ B s + k ⊥ − k ⊥ µ g ij ˜ U αβ B s + p ⊥ ˜ U † − g µj ˜ U p i αβ B s + p ⊥ ˜ U † + 2 k ⊥ µ k ⊥ g ij (cid:111) ˜ F j (cid:0) β B + k ⊥ αs (cid:1) | k ⊥ ) ba + O ( p µ ) The production part of the evolution kernel is proportional to the cross section of gluonemission given by the product of Eqs. (4.26) and (4.31) integrated according to Eq. (3.1).To find the full kernel we should calculate the virtual part.– 26 – .4 Virtual correction
To get the virtual correction shown in Fig. 5 we should use the expansion (3.11) of theoperator F up to the second order in quantum field. From Eq. (3.11) one gets (cid:104)F ni ( β B , y ⊥ ) (cid:105) (4.34) = 2 g s (cid:90) dy ∗ e iβ B y ∗ (cid:104) β B (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm A m q i ( y ∗ , y ⊥ ) (cid:105)− is (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm ∂ i A m q • ( y ∗ , y ⊥ ) (cid:105)− gs (cid:90) ∞ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ([ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y A q • ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , y ∗ ] y (cid:105) ) nm F m • i ( y ∗ , y ⊥ ) (cid:105) As in the case of production kernel we will calculate the diagrams in Fig. 5a, 5b, and 5cseparately and then check that the final result does not depend on the size of the shockwave σ ∗ (it is easy to see that the diagram in Fig. 5d vanishes in Feynman gauge). (a) (b) (c) (d) Figure 5 . Virtual part of the evolution kernel.
Let us start with the diagram in Fig. 5a. Using Eq. (3.11) and (11.27) we get s (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy gF m • i ( y ∗ , y ⊥ ) (cid:105) Fig . = (4.35) = 2 g s (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:104) β B (cid:90) ∞ σ ∗ dz ∗ (cid:0) (cid:104) A q • ( z ∗ , y ⊥ ) U y (cid:1) nm A m q i ( y ∗ , y ⊥ ) (cid:105)− is (cid:90) ∞ σ ∗ dz ∗ (cid:0) (cid:104) A q • ( z ∗ , y ⊥ ) U y (cid:1) nm ∂ i A m q • ( y ∗ , y ⊥ ) (cid:105) (cid:105) = − ig s f nkl (cid:90) ∞ d − αα (cid:90) − σ ∗ −∞ dy ∗ (cid:90) ∞ σ ∗ dz ∗ ( y ⊥ | e − i p ⊥ αs z ∗ (cid:8) β B G • i ( ∞ , −∞ ; p ⊥ )+ 2 s [ G •• ( ∞ , −∞ ; p ⊥ ) + Q •• ( ∞ , −∞ ; p ⊥ )] p i (cid:9) e i ( β B + p ⊥ αs ) y ∗ U † | y ⊥ ) kl = ig f nkl (cid:90) ∞ d − α ( y ⊥ | p ⊥ e − i p ⊥ αs σ ∗ (cid:8) αβ B s G • i ( ∞ , −∞ ; p ⊥ )+ 2 α [ G •• ( ∞ , −∞ ; p ⊥ ) + Q •• ( ∞ , −∞ ; p ⊥ )] p i (cid:9) αβ B s + p ⊥ e − i ( β B + p ⊥ αs ) σ ∗ U † | y ⊥ ) kl – 27 –as usual we assume that there are no external fields outside [ σ ∗ , − σ ∗ ] interval). Moreover,from Eq. (11.25) we see that G • i ( ∞ , −∞ ; p ⊥ ) = − iα ∂ i U and from Eqs. (11.25), (11.7)and (4.9) that G •• ( ∞ , −∞ ; p ⊥ ) + Q •• ( ∞ , −∞ ; p ⊥ ) = − α ∂ ⊥ U so we obtain s g (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . (4.36) = g f nkl (cid:90) ∞ d − αα ( y ⊥ | p ⊥ e − i p ⊥ αs σ ∗ [ αβ B s∂ i U − i∂ ⊥ U p i ] 1 αβ B s + p ⊥ e − i ( β B + p ⊥ αs ) σ ∗ U † | y ⊥ ) kl = g f nkl e − iβ B σ ∗ (cid:90) ∞ d − αα ( y ⊥ | p ⊥ [ αβ B s∂ i U − i∂ ⊥ U p i ] 1 αβ B s + p ⊥ U † | y ⊥ ) kl (recall that p ⊥ αs σ ∗ (cid:28) if the transverse momenta in the loop are of order of transversemomenta of external fields). To get the contribution of the diagram in Fig. 5b we need the gluon propagator with onepoint in the shock wave (11.8), which we will rewrite as follows (cid:104) A aµ ( z ∗ , z ⊥ ) A bν ( y ∗ , y ⊥ ) (cid:105) = (cid:90) −∞ d − α α (cid:8) ( y ⊥ | e − i p ⊥ αs ( y − z ) ∗ (4.37) × (cid:2) G baνµ ( y ∗ , z ∗ ; p ⊥ ) + Q baνµ ( y ∗ , z ∗ ; p ⊥ ) (cid:3) | z ⊥ ) + ( z ⊥ | ¯ Q baνµ ( y ∗ , z ∗ ; p ⊥ ) e − i p ⊥ αs ( y − z ) ∗ | y ⊥ ) (cid:9) with G and Q given by Eqs. (11.6) and (11.7) G bai • ( y ∗ , z ∗ ; p ⊥ ) = 2 gαs (cid:90) z ∗ y ∗ dz (cid:48)∗ ([ z ∗ , z (cid:48)∗ ] F • i ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ]) ab , Q i • ( y ∗ , z ∗ ; p ⊥ ) = ¯ Q i • ( y ∗ , z ∗ ; p ⊥ ) = 0 , G ba •• ( y ∗ , z ∗ ; p ⊥ ) = − g α s (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:90) z (cid:48)∗ y ∗ dz (cid:48)(cid:48)∗ ([ z ∗ , z (cid:48)∗ ] F • j ( z (cid:48)∗ )[ z (cid:48)∗ , z (cid:48)(cid:48)∗ ] F j • ( z (cid:48)(cid:48)∗ )[ z (cid:48)(cid:48)∗ , y ∗ ]) ab (4.38) Q ba •• ( y ∗ , z ∗ ; p ⊥ ) = − igα s (cid:90) z ∗ y ∗ dz (cid:48)∗ ([ z ∗ , z (cid:48)∗ ] D j F • j ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ]) ab , ¯ Q •• ( y ∗ , z ∗ ; p ⊥ ) = 0 and therefore from Eq. (4.34) we get s (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy gF m • i ( y ∗ , y ⊥ ) (cid:105) Fig . = (4.39) = 2 g s (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:104) β B (cid:90) σ ∗ − σ ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm A m q i ( y ∗ , y ⊥ ) (cid:105)− is (cid:90) σ ∗ − σ ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm ∂ i A m q • ( y ∗ , y ⊥ ) (cid:105) (cid:105) First, let us show that the second term in the r.h.s. of this equation vanishes. From Eq.(4.37) we see that (cid:104) A a q • ( z ∗ , y ⊥ ) ∂ i A b q • ( y ∗ , y ⊥ ) (cid:105) (4.40) = − i (cid:90) −∞ d − α α ( y ⊥ | p i e − i p ⊥ αs ( y − z ) ∗ (cid:2) G ba •• ( y ∗ , z ∗ ; p ⊥ ) + Q ba •• ( y ∗ , z ∗ ; p ⊥ ) (cid:3) | y ⊥ ) = 0 – 28 –ecause operators in Eq. (4.37) do not contain p and ( y ⊥ | p i e − i p ⊥ αs ( y − z ) ∗ | y ⊥ ) = 0 .Now consider the first term in the r.h.s. of Eq. (4.39). From Eq. (4.37) we get (cid:104) A a • ( z ∗ , y ⊥ ) A bi ( y ∗ , y ⊥ ) (cid:105) = gs (cid:90) ∞ d − αα ( y ⊥ | e − i p ⊥ αs ( z − y ) ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ [ z ∗ , z (cid:48)∗ ] F • i ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] | y ⊥ ) ab (4.41)and therefore gs (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . = 2 g s if nkl (cid:90) σ ∗ − σ ∗ dz (cid:48)∗ F kli ( z (cid:48)∗ , y ⊥ ) × (cid:90) ∞ d − αα ( y ⊥ | (cid:2) e − i p ⊥ αs σ ∗ − e − i p ⊥ αs z (cid:48)∗ (cid:3) αβ B sp ⊥ ( αβ B s + p ⊥ ) e − i ( β B + p ⊥ αs ) σ ∗ | y ⊥ ) (4.42)where F kli ( z (cid:48)∗ , y ⊥ ) ≡ g ([ ∞ , z (cid:48)∗ ] y F • i ( z (cid:48)∗ , y ⊥ )[ z (cid:48)∗ , ∞ ] y ) kl , see Eq. (1.4). Since p ⊥ αs σ ∗ (cid:28) ther.h.s. of this equation can be simplified to gs (cid:90) − σ ∗ −∞ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . = (4.43) = − g s f nkl e − iβ B σ ∗ (cid:90) ∞ d − αα ( y ⊥ | β B αβ B s + p ⊥ | y ⊥ ) (cid:90) σ ∗ − σ ∗ dz (cid:48)∗ ( z (cid:48) − σ ) ∗ F kli ( z (cid:48)∗ , y ⊥ ) (cid:39) because it is O ( p ⊥ αs σ ∗ ) in comparison to Eq. (4.36). As in previous Sections, we start from rewriting Eq. (3.11) gs (cid:90) σ ∗ − σ ∗ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . (4.44) = 2 g s (cid:90) σ ∗ − σ ∗ dy ∗ e iβ B y ∗ (cid:104) − gs (cid:90) ∞ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ([ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y A q • ( z (cid:48)∗ , y ⊥ ) (cid:105)× [ z (cid:48)∗ , y ∗ ] y ) nm F m • i ( y ∗ , y ⊥ ) + β B (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm A m q i ( y ∗ , y ⊥ ) (cid:105)− is (cid:90) ∞ y ∗ dz ∗ (cid:0) [ ∞ , z ∗ ] y (cid:104) A q • ( z ∗ , y ⊥ )[ z ∗ , y ∗ ] y (cid:1) nm ∂ i A m q • ( y ∗ , y ⊥ ) (cid:105) (cid:105) – 29 –sing the propagator (11.8) with point y inside the shock wave (and point z anywhere) we obtain (hereafter ∂ i ( C ) ≡ − i [ p i , C ] ) gs (cid:90) σ ∗ − σ ∗ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . (4.45) = g s if nkl (cid:90) σ ∗ − σ ∗ dy ∗ e iβ B y ∗ (cid:90) ∞ y ∗ dz ∗ (cid:90) ∞ d − αα (cid:110) ( y ⊥ | e − i p ⊥ αs ( z − y ) ∗ (cid:2) − β B ([ ∞ , z ∗ ] × G • i ( z ∗ , y ∗ ; p ⊥ )[ y ∗ , ∞ ]) kl + 2 is (cid:8) ∂ i (cid:0) [ ∞ , z ∗ ] O •• ( z ∗ , y ∗ ; p ⊥ )[ y ∗ , ∞ ] (cid:1) − ∂ i ([ ∞ , z ∗ ]) O •• ( z ∗ , y ∗ ; p ⊥ )[ y ∗ , ∞ ] − [ ∞ , z ∗ ] O •• ( z ∗ , y ∗ ; p ⊥ ) ∂ i ([ y ∗ , ∞ ]) (cid:9) kl (cid:3) | y ⊥ )+ 4 gs (cid:90) z ∗ y ∗ dz (cid:48)∗ ( y ⊥ | e − i p ⊥ αs ( z − z (cid:48) ) ∗ (cid:0) [ ∞ , z ∗ ] O •• ( z ∗ , z (cid:48)∗ ; p ⊥ )[ z (cid:48)∗ , y ∗ ] F • i ( y ∗ )[ y ∗ , ∞ ] (cid:1) kl | y ⊥ ) (cid:111) where G • i is given by Eq. (11.6) G • i ( x ∗ , y ∗ ; p ⊥ ) = − gαs (cid:90) x ∗ y ∗ dz ∗ [ x ∗ , z ∗ ] F • i ( z ∗ )[ z ∗ , y ∗ ] (4.46)and O •• ≡ G •• + Q •• by Eqs. (11.6), (11.7) O •• ( x ∗ , y ∗ ; p ⊥ ) = gα s (cid:90) x ∗ y ∗ dz ∗ [ x ∗ , z ∗ ] (cid:8) − iD j F • j ( z ∗ )[ z ∗ , y ∗ ] − gs (cid:90) z ∗ y ∗ dz (cid:48)∗ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] (cid:9) (4.47)Using these expressions, one obtains after some algebra gs (cid:90) σ ∗ − σ ∗ dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . (4.48) = g f nkl (cid:90) ∞ d − αα ( y ⊥ | p ⊥ ( αβ B s + p ⊥ ) | y ⊥ ) (cid:110)(cid:90) σ ∗ − σ ∗ dw ∗ (cid:0) e iβ B w ∗ − e − iβ B σ ∗ (cid:1) × (cid:16) − iαβ B F i ( w ∗ , y ⊥ ) − s ∂ yi V ( w ∗ , y ⊥ ) + 2 is V ( w ∗ , y ⊥ ) (cid:90) σ ∗ w ∗ dz (cid:48)∗ F i ( z (cid:48)∗ , y ⊥ ) (cid:17) + 2 igs (cid:90) σ ∗ − σ ∗ dz (cid:48)∗ (cid:90) σ ∗ z (cid:48)∗ dw ∗ (cid:16) igs [ ∞ , w ∗ ] y D j F • j ( w ∗ , y ⊥ )[ w ∗ , ∞ ] y (cid:2) e iβ B z (cid:48)∗ − e − iβ B σ ∗ (cid:3) + 8 g s (cid:90) w ∗ z (cid:48) dw (cid:48)∗ [ ∞ , w ∗ ] y F • j ( w ∗ , y ⊥ )[ w ∗ , w (cid:48)∗ ] y F j • ( w (cid:48)∗ )[ w (cid:48)∗ , ∞ ] y × (cid:2) e iβ B z (cid:48) − e − iβ B σ ∗ (cid:3)(cid:17) F i ( z (cid:48)∗ , y ⊥ ) (cid:111) kl where, as usual, F kli = g ([ ∞ , w ∗ ] y F • i ( w ∗ , y ⊥ )[ w ∗ , ∞ ] y ) kl and V ( w ∗ , y ⊥ ) ≡ ig [ ∞ , w ∗ ] y D j F • j ( w ∗ , y ⊥ )[ w ∗ , ∞ ] y + 8 g s (cid:90) ∞ w ∗ dw (cid:48)∗ [ ∞ , w (cid:48)∗ ] y F • j ( w (cid:48)∗ , y ⊥ )[ w (cid:48)∗ , w ∗ ] y F j • ( w ∗ , y ⊥ )[ w ∗ , ∞ ] y (4.49)(cf. Eq. (4.14)). Strictly speaking, one should depict Eq. (4.44) as several diagrams with points z (and z (cid:48) ) inside andoutside the shock wave. – 30 – .4.4 The sum of diagrams in Fig. 5 The total virtual correction coming from Fig. 5 is given by the sum of Eqs. (4.36) and(4.48) s g (cid:90) dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . (4.50) = g f nkl e − iβ B σ ∗ (cid:90) ∞ d − αα ( y ⊥ | p ⊥ [ αβ B s∂ i U − i∂ ⊥ U p i ] 1 αβ B s + p ⊥ U † | y ⊥ ) kl + g f nkl (cid:90) ∞ d − αα ( y ⊥ | p ⊥ ( αβ B s + p ⊥ ) | y ⊥ ) (cid:110)(cid:90) σ ∗ − σ ∗ dw ∗ (cid:0) e iβ B w ∗ − e − iβ B σ ∗ (cid:1) × (cid:16) − iαβ B F i ( w ∗ , y ⊥ ) − s ∂ yi V ( w ∗ , y ⊥ ) + 2 is V ( w ∗ , y ⊥ ) (cid:90) σ ∗ w ∗ dz (cid:48)∗ F i ( z (cid:48)∗ , y ⊥ ) (cid:17) + 2 igs (cid:90) σ ∗ − σ ∗ dz (cid:48)∗ (cid:90) σ ∗ z (cid:48)∗ dw ∗ (cid:16) igs [ ∞ , w ∗ ] y D j F • j ( w ∗ , y ⊥ )[ w ∗ , ∞ ] y (cid:2) e iβ B z (cid:48)∗ − e − iβ B σ ∗ (cid:3) + 8 g s (cid:90) w ∗ z (cid:48) dw (cid:48)∗ [ ∞ , w ∗ ] y F • j ( w ∗ , y ⊥ )[ w ∗ , w (cid:48)∗ ] y F j • ( w (cid:48)∗ )[ w (cid:48)∗ , ∞ ] y × (cid:2) e iβ B z (cid:48)∗ − e − iβ B σ ∗ (cid:3)(cid:17) F i ( z (cid:48)∗ , y ⊥ ) (cid:111) kl Let us prove that with our accuracy it can be approximated as s g (cid:90) dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . (4.51) = g f nkl (cid:90) ∞ d − αα ( y ⊥ | p ⊥ [ αβ B s∂ i U − i∂ ⊥ U p i ] 1 αβ B s + p ⊥ U † | y ⊥ ) kl − ig f nkl (cid:90) ∞ d − αα ( y ⊥ | αβ B sp ⊥ ( αβ B s + p ⊥ ) | y ⊥ )[ F i ( β B , y ⊥ ) − i∂ i U y U † y ] kl To this end we compare the size of different terms in the r.h.s. of equations (4.50) and(4.51) at β B σ ∗ (cid:28) and β B σ ∗ ≥ . In the first case (at β B σ ∗ (cid:28) ) the only surviving termsin the r.h.s.’s of these equations are the first terms and they are obviously equal.In the second case let us start from Eq. (4.51). Since β B σ ∗ ∼ β B σ (cid:48) sp ⊥ ≥ we have αβ B s (cid:29) p ⊥ so r . h . s . of Eq . (4 .
51) = − ig f nkl (cid:90) ∞ d − αα ( y ⊥ | p ⊥ | y ⊥ ) F kli ( β B , y ⊥ ) (4.52)Let us now compare the size of different terms in the r.h.s. of Eq. (4.50). Since s (cid:82) dw ∗ V ( w ∗ ) ∼ ∂ ⊥ U U † the first term in the fourth line ∼ (cid:82) dw ∗ αβ B F • i ( w ∗ , y ⊥ ) ∼ αβ B s∂ i U y U † y ismuch greater than the second term ∼ s (cid:82) dw ∗ ∂ i V ( w ∗ , y ⊥ ) ∼ ∂ i ∂ ⊥ U y U † y or the third term ∼ s (cid:82) dw ∗ ∂ i V ( w ∗ , y ⊥ ) (cid:82) dw (cid:48)∗ F • i ( w (cid:48)∗ ) ∼ ∂ ⊥ U y ∂ i U † y . Moreover, it is easy to see that the termsin the last three lines in Eq. (4.50) are of the same order as the terms ∼ V in the fourthline so they are again small in comparison to the term ∼ F i . Thus, we get r . h . s . of Eq . (4 .
50) = g f nkl e − iβ B σ ∗ (cid:90) ∞ d − αα ( y ⊥ | p ⊥ (cid:2) ∂ i U − iαβ B s ∂ ⊥ U p i (cid:3) U † | y ⊥ ) kl − ig f nkl (cid:90) ∞ d − αα ( y ⊥ | p ⊥ | y ⊥ ) 2 s (cid:90) σ ∗ − σ ∗ dw ∗ (cid:0) e iβ B w ∗ − e − iβ B σ ∗ (cid:1) F kli ( w ∗ , y ⊥ ) (4.53)– 31 –hich coincides with the r.h.s of Eq. (4.52).Last but not least, let us prove that one can use the formula (4.51) in the light-conelimit l ⊥ (cid:28) p ⊥ where it coincides with Eq. (3.15). First we notice that the term ∼ ∂ ⊥ U hastwist two and so exceeds our twist-one light-cone accuracy. Next, since the commutator [ p ⊥ , ∂ i U ] consists of operators of collinear twist two (or higher), one can rewrite the firstterm in the r.h.s of Eq. (4.51) in the form ( y ⊥ | p ⊥ αβ B s∂ i U αβ B s + p ⊥ U † | y ⊥ ) (cid:39) ( y ⊥ | αβ B sp ⊥ ( αβ B s + p ⊥ ) | y ⊥ ) ∂ i U y U † y (4.54)so it cancels with last term in the r.h.s of Eq. (4.51) and we obtain s g (cid:90) dy ∗ e iβ B y ∗ (cid:104) [ ∞ , y ∗ ] nmy F m • i ( y ∗ , y ⊥ ) (cid:105) Fig . (4.55) = − ig f nkl (cid:90) ∞ d − αα ( y ⊥ | αβ B sp ⊥ ( αβ B s + p ⊥ ) | y ⊥ ) F kli ( β B , y ⊥ ) which is the light-cone result Eq. (3.15).Thus, the final result for the sum of diagrams in Fig. 5 is Eq. (4.51) (cid:104)F ni ( β B , y ⊥ ) (cid:105) Fig . = − ig f nkl (cid:90) σσ (cid:48) d − αα (cid:110) ( y ⊥ | p ⊥ [ αβ B si∂ i U + ∂ ⊥ U p i ] 1 αβ B s + p ⊥ U † | y ⊥ ) kl + ( y ⊥ | αβ B sp ⊥ ( αβ B s + p ⊥ ) | y ⊥ )[ F i ( β B , y ⊥ ) − U i ( y ⊥ )] kl (cid:111) (4.56) = − ig f nkl (cid:90) σσ (cid:48) d − αα ( y ⊥ | p j p ⊥ (2 ∂ i ∂ j U + g ij ∂ ⊥ U ) 1 αβ B s + p ⊥ U † + αβ B sp − ⊥ αβ B s + p ⊥ F i ( β B ) | y ⊥ ) kl where we imposed our cutoff σ > α > σ (cid:48) . Again, let us note that the above expression isvalid with our accuracy in the whole range of transverse momenta.Similarly to Eq. (4.28) we can rewrite this formula in the form without subtractions (cid:104)F ni ( β B , y ⊥ ) (cid:105) Fig . = − ig f nkl (cid:90) σσ (cid:48) d − αα ( y ⊥ | − p j p ⊥ F k ( β B )( i ← ∂ l + U l ) (4.57) × (2 δ kj δ li − g ij g kl ) U αβ B s + p ⊥ U † + F i ( β B ) αβ B sp ⊥ ( αβ B s + p ⊥ ) | y ⊥ ) kl where F k ← ∂ l ≡ ∂ l F k = − i [ p l , F k ] . Indeed, in the light-cone case l ⊥ (cid:28) p ⊥ one can neglectthe operators with high collinear twist so both equations (4.56) and (4.57) reduce to thelast terms in the r.h.s’s which are the same. Also, as we discussed above, in the shock-wavecase ( l ⊥ ∼ p ⊥ ) and β B small one can replace F i ( β B ) by U i so the r.h.s’s of Eq. (4.56) andEq. (4.57) coincide after some trivial algebra. Finally, if l ⊥ ∼ p ⊥ and β B ≥ σ ∗ we have αβ B s (cid:29) p ⊥ so again the equations (4.56) and (4.57) reduce to the last terms in the r.h.s’s. The calculation of the virtual correction in the complex conjugate amplitude is very similarso we will only outline it. As in the previous Section, we start with the formula (3.16)– 32 –hich can be rewritten as s (cid:90) dx ∗ e − iβ B x ∗ (cid:104) ˜ F m • i ( x ∗ , x ⊥ )[ x ∗ , ∞ ] mnx (cid:105) = (4.58) = 2 s (cid:90) dx ∗ e − iβ B x ∗ (cid:110) β B (cid:90) ∞ x ∗ dz ∗ ˜ A m q i ( x ∗ , x ⊥ ) (cid:0) [ x ∗ , z ∗ ] x ˜ A q • ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x (cid:1) mn (cid:105) + 2 is (cid:90) ∞ x ∗ dz ∗ (cid:104) ∂ i ˜ A m q • (cid:0) [ x ∗ , z ∗ ] x ˜ A q • ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x (cid:1) mn (cid:105)− s (cid:90) ∞ x ∗ dz ∗ (cid:90) ∞ z ∗ dz (cid:48)∗ ˜ F m • i ( x ∗ , x ⊥ ) (cid:0) [ x ∗ , z ∗ ] x ˜ A q • ( z ∗ , x ⊥ )[ z ∗ , z (cid:48)∗ ] x ˜ A q • ( z (cid:48)∗ , x ⊥ )[ z (cid:48)∗ , ∞ ] x (cid:1) mn (cid:111) Using Eq. (11.28) we get s (cid:90) dx ∗ e − iβ B x ∗ (cid:104) ˜ F m • i ( x ∗ , x ⊥ )[ x ∗ , ∞ ] nmx (cid:105) Fig . = (4.59) = − ig s f nkl (cid:90) ∞ d − αα (cid:90) − σ ∗ −∞ dx ∗ (cid:90) ∞ σ ∗ dz ∗ ( x ⊥ | ˜ U e − i ( β B + p ⊥ αs ) x ∗ (cid:8) β B ˜ G i • ( −∞ , ∞ ; p ⊥ )+ 2 s p i [ ˜ G •• ( −∞ , ∞ ; p ⊥ ) + ˜ Q •• ( −∞ , ∞ ; p ⊥ )] (cid:9) e i p ⊥ αs z ∗ | x ⊥ ) kl + g s if nkl (cid:90) σ ∗ − σ ∗ dx ∗ e − iβ B x ∗ (cid:90) ∞ x ∗ dz ∗ (cid:90) ∞ d − αα (cid:110) ( x ⊥ | (cid:2) − β B ([ ∞ , x ∗ ] ˜ G i • ( x ∗ , z ∗ ; p ⊥ )[ z ∗ , ∞ ]) kl − is (cid:8) ∂ i (cid:0) [ ∞ , x ∗ ] ˜ O •• ( x ∗ , z ∗ ; p ⊥ )[ z ∗ , ∞ ] (cid:1) − ∂ i ([ ∞ , x ∗ ]) ˜ O •• ( x ∗ , z ∗ ; p ⊥ )[ z ∗ , ∞ ] − [ ∞ , x ∗ ] ˜ O •• ( x ∗ , z ∗ ; p ⊥ ) ∂ i ([ z ∗ , ∞ ]) (cid:9) kl (cid:3) e − i p ⊥ αs ( x − z ) ∗ | x ⊥ )+ 4 s (cid:90) ∞ z ∗ dz (cid:48)∗ ( x ⊥ | (cid:0) [ ∞ , x ∗ ] ˜ F • i ( x ∗ )[ x ∗ , z ∗ ] ˜ O •• ( z ∗ , z (cid:48)∗ ; p ⊥ )[ z (cid:48)∗ , ∞ ] (cid:1) kl e − i p ⊥ αs ( z − z (cid:48) ) ∗ | x ⊥ ) (cid:111) Similarly to Eq. (4.50) it is possible to demonstrate that the last three lines in the r.h.s.of this equation exceed our accuracy, and moreover, one can neglect factors e − iβ B σ ∗ . Usingformulas (11.29) for ˜ G µν and (11.10) for ˜ Q µν we obtain the virtual correction in the complexconjugate amplitude in the form (cid:104) ˜ F ni ( β B , x ⊥ ) (cid:105) σ = 2 gs (cid:90) dx ∗ e − iβ B x ∗ (cid:104) ˜ F m • i ( x ∗ , x ⊥ )[ x ∗ , ∞ ] mnx (cid:105) σ = − ig f nkl (cid:90) σσ (cid:48) d − αα (cid:110) ( x ⊥ | ˜ U αβ B s + p ⊥ (cid:8) − iαβ B s∂ i ˜ U † + p i ∂ ⊥ ˜ U † (cid:9) p ⊥ | x ⊥ ) kl + (cid:0) ˜ F i ( β B , x ⊥ ) − i∂ i ˜ U x ˜ U † x (cid:1) kl ( x ⊥ | αβ B sp ⊥ ( αβ B s + p ⊥ ) | x ⊥ ) (cid:111) (4.60)where we have imposed our cutoffs in α and used the formula ∂ ⊥ U † z = g (cid:90) ∞−∞ dz ∗ [ −∞ , z ∗ ] z × (cid:16) − is D j F • j ( z ∗ , z ⊥ )[ z ∗ , ∞ ] z + 8 gs (cid:90) ∞ z ∗ dz (cid:48)∗ F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , ∞ ] z (cid:17) Similarly to Eq. (4.51) this expression is also valid in the light-cone case l ⊥ (cid:28) p ⊥ where itcoincides with Eq. (3.18). – 33 –lternatively, one can use the expression without subtractions (cf. Eq. (4.57)) (cid:104) ˜ F ni ( β B , x ⊥ ) (cid:105) σ = − ig f nkl (cid:90) σσ (cid:48) d − αα ( x ⊥ | ˜ U αβ B s + p ⊥ ˜ U † (4.61) × (2 δ ki δ lj − g ij g kl )( i∂ k − ˜ U k ) ˜ F l ( β B ) p j p ⊥ + ˜ F i ( β B ) αβ B sp ⊥ ( αβ B s + p ⊥ ) | x ⊥ ) kl Now we are in a position to assemble all leading-order contributions to the rapidity evolutionof gluon TMD. Adding the production part (3.1) with Lipatov vertices (4.28) and (4.33)and the virtual parts from previous Section (4.57) and (4.61) we obtain (cid:0) ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (cid:1) ln σ (5.1) = − g (cid:90) σσ (cid:48) d − α α (cid:90) d − k ⊥ Tr { ˜ L µi ( k, x ⊥ , β B ) light − like L µj ( k, y ⊥ , β B ) light − like }− g (cid:90) σσ (cid:48) d − αα Tr (cid:110) ˜ F i ( β B , x ⊥ )( y ⊥ | − p m p ⊥ F k ( β B )( i ← ∂ l + U l )(2 δ km δ lj − g jm g kl ) U αβ B s + p ⊥ U † + F j ( β B ) αβ B sp ⊥ ( αβ B s + p ⊥ ) | y ⊥ )+ ( x ⊥ | ˜ U αβ B s + p ⊥ ˜ U † (2 δ ki δ lm − g im g kl )( i∂ k − ˜ U k ) ˜ F l ( β B ) p m p ⊥ + ˜ F i ( β B ) αβ B sp ⊥ ( αβ B s + p ⊥ ) | x ⊥ ) F j (cid:0) β B , y ⊥ (cid:1)(cid:111) + O ( α s ) where Tr is a trace in the adjoint representation. In the explicit form the evolution equationreads dd ln σ ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (5.2) = − α s Tr (cid:110)(cid:90) d − k ⊥ ( x ⊥ | (cid:110) ˜ U σβ B s + p ⊥ ( ˜ U † k k + p k ˜ U † ) σβ B sg µi − k ⊥ µ k i σβ B s + k ⊥ − k ⊥ µ g ik ˜ U σβ B s + p ⊥ ˜ U † − g µk ˜ U p i σβ B s + p ⊥ ˜ U † + 2 k ⊥ µ k ⊥ g ik (cid:111) ˜ F k (cid:0) β B + k ⊥ σs (cid:1) | k ⊥ ) × ( k ⊥ |F l (cid:0) β B + k ⊥ σs (cid:1)(cid:110) σβ B sδ µj − k µ ⊥ k j σβ B s + k ⊥ ( k l U + U p l ) 1 σβ B s + p ⊥ U † − k µ ⊥ g jl U σβ B s + p ⊥ U † − δ µl U p j σβ B s + p ⊥ U † + 2 g jl k µ ⊥ k ⊥ (cid:111) | y ⊥ )+ 2 ˜ F i ( β B , x ⊥ )( y ⊥ | − p m p ⊥ F k ( β B )( i ← ∂ l + U l )(2 δ km δ lj − g jm g kl ) U σβ B s + p ⊥ U † + F j ( β B ) σβ B sp ⊥ ( σβ B s + p ⊥ ) | y ⊥ )+ 2( x ⊥ | ˜ U σβ B s + p ⊥ ˜ U † (2 δ ki δ lm − g im g kl )( i∂ k − ˜ U k ) ˜ F l ( β B ) p m p ⊥ + ˜ F i ( β B ) σβ B sp ⊥ ( σβ B s + p ⊥ ) | x ⊥ ) F j ( β B , y ⊥ ) (cid:111) + O ( σ s ) – 34 –he operators ˜ F j ( β ) and F i ( β ) are defined as usual, see Eq. (4.29) ( x ⊥ | ˜ F i ( β ) | k ⊥ ) = 2 s (cid:90) dx ∗ ˜ F i ( x ∗ , x ⊥ ) e − iβx ∗ + i ( k,x ) ⊥ ( k ⊥ |F i ( β ) | y ⊥ ) = 2 s (cid:90) dy ∗ e iβy ∗ − i ( k,y ) ⊥ F i ( y ∗ , y ⊥ ) (5.3)The evolution equation (5.2) can be rewritten in the form where cancellation of IR andUV divergencies is evident dd ln σ ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) (5.4) = − α s Tr (cid:110)(cid:90) d − k ⊥ ( x ⊥ | (cid:110) ˜ U σβ B s + p ⊥ ( ˜ U † k k + p k ˜ U † ) σβ B sg µi − k ⊥ µ k i σβ B s + k ⊥ − k ⊥ µ g ik ˜ U σβ B s + p ⊥ ˜ U † − g µk ˜ U p i σβ B s + p ⊥ ˜ U † (cid:111) ˜ F k (cid:0) β B + k ⊥ σs (cid:1) | k ⊥ ) × ( k ⊥ |F l (cid:0) β B + k ⊥ σs (cid:1)(cid:110) σβ B sδ µj − k µ ⊥ k j σβ B s + k ⊥ ( k l U + U p l ) 1 σβ B s + p ⊥ U † − k µ ⊥ g jl U σβ B s + p ⊥ U † − δ µl U p j σβ B s + p ⊥ U † (cid:111) | y ⊥ ) + 2 (cid:90) d − k ⊥ ( x ⊥ | ˜ F i (cid:0) β B + k ⊥ σs (cid:1) | k ⊥ ) × ( k ⊥ |F l (cid:0) β B + k ⊥ σs (cid:1)(cid:110) k j k ⊥ σβ B s + 2 k ⊥ σβ B s + k ⊥ ( k l U + U p l ) 1 σβ B s + p ⊥ U † + 2 U g jl σβ B s + p ⊥ U † − k l k ⊥ U p j σβ B s + p ⊥ U † (cid:111) | y ⊥ )+ 2 (cid:90) d − k ⊥ ( x ⊥ | (cid:110) ˜ U σβ B s + p ⊥ ( ˜ U † k k + p k ˜ U † ) k i k ⊥ σβ B s + 2 k ⊥ σβ B s + k ⊥ + 2 ˜ U g ik σβ B s + p ⊥ ˜ U † − U p i σβ B s + p ⊥ ˜ U † k k k ⊥ (cid:111) ˜ F k (cid:0) β B + k ⊥ σs (cid:1) | k ⊥ )( k ⊥ |F j (cid:0) β B + k ⊥ σs (cid:1) | y ⊥ )+ 2 ˜ F i ( β B , x ⊥ )( y ⊥ | − p m p ⊥ F k ( β B )( i ← ∂ l + U l )(2 δ km δ lj − g jm g kl ) U σβ B s + p ⊥ U † | y ⊥ )+ 2( x ⊥ | ˜ U σβ B s + p ⊥ ˜ U † (2 δ ki δ lm − g im g kl )( i∂ k − ˜ U k ) ˜ F l ( β B ) p m p ⊥ | x ⊥ ) F j ( β B , y ⊥ ) − (cid:90) d − k ⊥ k ⊥ (cid:104) ˜ F i (cid:0) β B + k ⊥ σs , x ⊥ (cid:1) F j (cid:0) β B + k ⊥ σs , y ⊥ (cid:1) e i ( k,x − y ) ⊥ − σβ B sσβ B s + k ⊥ ˜ F i ( β B , x ⊥ ) F j ( β B , y ⊥ ) (cid:105)(cid:111) + O ( α s ) . The evolution equation (5.4) is one of the main results of this paper. It describes the rapidityevolution of the operator (2.8) at any Bjorken x B ≡ β B and any transverse momenta.Let us discuss the gauge invariance of this equation. The l.h.s. is gauge invariant aftertaking into account gauge link at + ∞ as shown in Eq. (2.7). As to the right side, it wasobtained by calculation in the background field and promoting the background fields tooperators in a usual way. However, we performed our calculations in a specific backgroundfield A • ( x ∗ , x ⊥ ) with a finite support in x ⊥ and we need to address the question how can– 35 –e restore the r.h.s. of Eq. (5.4) in a generic field A µ . It is easy to see how one canrestore the gauge-invariant form: just add gauge link at + ∞ p or −∞ p appropriately. Forexample, the terms U z ( z | σβs + p ⊥ | z (cid:48) ) U † z (cid:48) in r.h.s. of should be replaced by U z [ z ⊥ − ∞ p , z (cid:48)⊥ −∞ p ]( z | σβs + p ⊥ | z (cid:48) ) U † z (cid:48) . After performing these insertions we will have the result which is(i) gauge invariant and (ii) coincides with Eq. (5.4) for our choice of background field. Atthis step, the background fields in the r.h.s. of Eq. (5.4) can be promoted to operators.However, the explicit display of these gauge links at ±∞ will make the evolution equationmuch less readable so we will assume they are always in place rather than written explicitly.When we consider the evolution of gluon TMD (1.6) given by the matrix element (2.4) ofthe operator (2.8) we need to take into account the kinematical constraint k ⊥ ≤ α (1 − β B ) s in the production part of the amplitude. Indeed, as we discussed in Sect. 3.3, the initialhadron’s momentum is p (cid:39) p so the sum of the fraction β B p and the fraction k ⊥ αs p carriedby the emitted gluon should be smaller than p . We obtain ( η ≡ ln σ ) ddη (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) η (5.5) = − α s (cid:104) p | Tr (cid:110)(cid:90) d − k ⊥ θ (cid:0) − β B − k ⊥ σs (cid:1)(cid:104) ( x ⊥ | (cid:16) U σβ B s + p ⊥ ( U † k k + p k U † ) × σβ B sg µi − k ⊥ µ k i σβ B s + k ⊥ − k ⊥ µ g ik U σβ B s + p ⊥ U † − g µk U p i σβ B s + p ⊥ U † (cid:17) ˜ F k (cid:0) β B + k ⊥ σs (cid:1) | k ⊥ ) × ( k ⊥ |F l (cid:0) β B + k ⊥ σs (cid:1)(cid:16) σβ B sδ µj − k µ ⊥ k j σβ B s + k ⊥ ( k l U + U p l ) 1 σβ B s + p ⊥ U † − k µ ⊥ g jl U σβ B s + p ⊥ U † − δ µl U p j σβ B s + p ⊥ U † (cid:17) | y ⊥ )+ 2( x ⊥ | ˜ F i (cid:0) β B + k ⊥ σs (cid:1) | k ⊥ )( k ⊥ |F l (cid:0) β B + k ⊥ σs (cid:1)(cid:16) k j k ⊥ σβ B s + 2 k ⊥ σβ B s + k ⊥ ( k l U + U p l ) 1 σβ B s + p ⊥ U † + 2 U g jl σβ B s + p ⊥ U † − k l k ⊥ U p j σβ B s + p ⊥ U † (cid:17) | y ⊥ )+ 2( x ⊥ | (cid:16) U σβ B s + p ⊥ ( U † k k + p k U † ) k i k ⊥ σβ B s + 2 k ⊥ σβ B s + k ⊥ + 2 U g ik σβ B s + p ⊥ U † − U p i σβ B s + p ⊥ U † k k k ⊥ (cid:17) ˜ F k (cid:0) β B + k ⊥ σs (cid:1) | k ⊥ )( k ⊥ |F j (cid:0) β B + k ⊥ σs (cid:1) | y ⊥ ) (cid:105) + 2 ˜ F i ( β B , x ⊥ )( y ⊥ | − p m p ⊥ F k ( β B )( i ← ∂ l + U l )(2 δ km δ lj − g jm g kl ) U σβ B s + p ⊥ U † | y ⊥ )+ 2( x ⊥ | U σβ B s + p ⊥ U † (2 δ ki δ lm − g im g kl )( i∂ k − U k ) ˜ F l ( β B ) p m p ⊥ | x ⊥ ) F j ( β B , y ⊥ ) − (cid:90) d − k ⊥ k ⊥ (cid:104) θ (cid:0) − β B − k ⊥ σs (cid:1) ˜ F i (cid:0) β B + k ⊥ σs , x ⊥ (cid:1) F j (cid:0) β B + k ⊥ σs , y ⊥ (cid:1) e i ( k,x − y ) ⊥ − σβ B sσβ B s + k ⊥ ˜ F i ( β B , x ⊥ ) F j ( β B , y ⊥ ) (cid:105)(cid:111) | p (cid:105) η + O ( α s ) Strictly speaking, we need to consider matrix element (cid:104) p | ˜ F ai ( β B , x ⊥ ) F ai ( β B , y ⊥ ) | p + ξp (cid:105) proportionalto δ ( ξ ) , see Eq. (2.4) – 36 –ote that we erased tilde from Wilson lines since we have a sum over full set of statesand gluon operators at space-like (or light-like) intervals commute with each other. Thisequation describes the rapidity evolution of gluon TMD (1.6) with rapidity cutoff (2.1) inthe whole range of β B = x B and k ⊥ ( ∼ | x − y | − ⊥ ). In the next section we will considersome specific cases. First, let us consider the evolution of Weizsacker-Williams (WW) unintegrated gluon dis-tribution (1.1) which can be obtained from Eq. (5.5) by setting β B = 0 . Moreover, inthe small- x regime it is assumed that the energy is much higher than anything else so thecharacteristic transverse momenta p ⊥ ∼ ( x − y ) − ⊥ (cid:28) s and in the whole range of evolution( (cid:29) σ (cid:29) ( x − y ) − ⊥ s ) we have p ⊥ σs (cid:28) , hence the kinematical constraint θ (cid:0) − β B − k ⊥ αs (cid:1) in Eq. (5.5) can be omitted. Under these assumptions, all F i (cid:0) β B + p ⊥ σs (cid:1) and F i ( β B ) canbe replaced by i∂ i U U † and similarly for the complex conjugate amplitude. To simplifyalgebra, it is convenient to take the production part of the kernel in the form of product ofLipatov vertices (4.26) and (4.31) noting that the “subtraction terms” ˘˜ F i and ˘ F j vanish inthis limit. One obtains the rapidity evolution of the WW distribution in the form dd ln σ ˜ U ai ( x ⊥ ) U aj ( y ⊥ ) = − α s Tr (cid:110)(cid:0) x ⊥ (cid:12)(cid:12) ˜ U p i ˜ U † (cid:0) ˜ U p k p ⊥ ˜ U † − p k p ⊥ (cid:1)(cid:0) U p k p ⊥ U † − p k p ⊥ (cid:1) U p j U † | y ⊥ ) − (cid:104) ( x ⊥ | ˜ U p i p k p ⊥ ˜ U † p k p ⊥ | x ⊥ ) −
12 ( x ⊥ | p ⊥ | x ⊥ ) ˜ U i ( x ⊥ ) (cid:105) U j ( y ⊥ ) − ˜ U i ( x ⊥ ) (cid:104) ( y ⊥ | p k p ⊥ U p j p k p ⊥ U † | y ⊥ ) −
12 ( y ⊥ | p ⊥ | y ⊥ ) U j ( y ⊥ ) (cid:105)(cid:111) (6.1)where we used the formula − ( y ⊥ | p k p ⊥ (2 ∂ j ∂ k U + g jk ∂ ⊥ U ) 1 σβs + p ⊥ U † | y ⊥ )= ( y ⊥ | p k p ⊥ U p ⊥ g jk + 2 p j p k σβ B s + p ⊥ U † − σβ B s + p ⊥ U j | y ⊥ ) (6.2)In this form Eq. (6.1) agrees with the results of Ref. [17]. To see the relation to the BKequation it is convenient to rewrite Eq. (6.1) as follows [24] (cf. Ref. [25]): ddη ˜ U ai ( z ) U aj ( z ) (6.3) = − g π Tr (cid:8) ( − i∂ z i + ˜ U z i ) (cid:2)(cid:90) d z ( ˜ U z ˜ U † z − z z z ( U z U † z − (cid:3) ( i ← ∂ z j + U z j ) (cid:9) We have left ˜ F as a reminder of different signs in the exponents of Fourier transforms in the definitions(2.2) and (2.3). – 37 –here η ≡ ln σ as usual. In this equation all indices are 2-dimensional and Tr standsfor the trace in the adjoint representation. It is easy to see that the expression in thesquare brackets is actually the BK kernel for the double-functional integral for cross sections[17, 26]. Hereafter, to ensure gauge invariance, U i ( z ⊥ ) must be understood as U i ( z ⊥ ) ≡ F i (0 , z ⊥ ) = s (cid:82) dz ∗ [ ∞ , z ∗ ] z F • i ( z ∗ , z ⊥ )[ z ∗ , ∞ ] and gauge links at ∞ p must beinserted as discussed after Eq. (5.4).It is worth noting that Eq. (6.3) holds true also at small β B up to β B ∼ ( x − y ) − ⊥ s sincein the whole range of evolution (cid:29) σ (cid:29) ( x − y ) − ⊥ s one can neglect σβ B s in comparison to p ⊥ in Eq. (5.5). This effectively reduces β B to 0 so one reproduces Eq. (6.3). Now let us discuss the case when β B = x B ∼ and ( x − y ) − ⊥ ∼ s . At the start of theevolution (at σ ∼ ) the cutoff in p ⊥ in the integrals of Eq. (5.4) is ∼ ( x − y ) − ⊥ . However,as the evolution in rapidity ( ∼ ln σ ) progresses the characteristic p ⊥ becomes smaller due tothe kinematical constraint p ⊥ < σ (1 − β B ) s . Due to this kinematical constraint evolution in σ is correlated with the evolution in p ⊥ : if σ (cid:29) σ (cid:48) the corresponding transverse momenta ofbackground fields p (cid:48)⊥ are much smaller than p ⊥ in quantum loops. This means that duringthe evolution we are always in the light-cone case considered in Sect. 3 and therefore theevolution equation for β B = x B ∼ and ( x − y ) − ⊥ ∼ s is Eq. (3.25) which reduces to thesystem of evolution equations for gluon TMDs D ( β B , | z ⊥ | , ln σ ) and H ( β B , | z ⊥ | , ln σ ) in thecase of unpolarized hadron. Finally, let us consider the evolution of D ( x B , k ⊥ , η = ln σ ) in the region where x B ≡ β B ∼ and k ⊥ ∼ ( x − y ) − ⊥ ∼ few GeV . In this case the integrals over p ⊥ in the production partof the kernel (5.5) are ∼ ( x − y ) − ⊥ ∼ k ⊥ so that p ⊥ (cid:28) σβ B s for the whole range of evolution > σ > k ⊥ s . For the same reason, the kinematical constraint θ (cid:0) − β B − p ⊥ σs (cid:1) in the lastline of Eq. (5.5) can be omitted and we get dd ln σ (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) real (6.4) = 4 α s N c (cid:90) d − p ⊥ p ⊥ e i ( p,x − y ) ⊥ (cid:104) p | ˜ F ai (cid:0) β B + p ⊥ σs , x ⊥ (cid:1) F aj (cid:0) β B + p ⊥ σs , y ⊥ (cid:1) | p (cid:105) As to the virtual part dd ln σ (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) virtual (6.5) = 4 α s N c (cid:90) d − p ⊥ p ⊥ (cid:104) − σβ B sσβ B s + p ⊥ (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) (cid:105) − α s Tr (cid:104) p | ( x ⊥ | ˜ U σβ B s + p ⊥ ˜ U † (2 δ ki δ lm − g im g kl )( i∂ k − ˜ U k ) ˜ F l ( β B ) p m p ⊥ | x ⊥ ) F j ( β B , y ⊥ ) − ˜ F i ( β B , x ⊥ )( y ⊥ | p m p ⊥ F k ( β B )( i ← ∂ l + U l )(2 δ km δ lj − g jm g kl ) U σβ B s + p ⊥ U † | y ⊥ ) | p (cid:105) , – 38 –he two last lines can be omitted. Indeed, as we saw in the end of Sect. 4.4.4, these termsare non-vanishing only for the region of large p ⊥ ∼ σβ B s . In this region one can expand theoperator O ≡ F k ( β B )( i ← ∂ l + U l )(2 δ km δ lj − g jm g kl ) U as O ( z ⊥ ) = O ( y ⊥ )+( y − z ) i ∂ i O ( y ⊥ )+ ... and get ( y ⊥ | p m p ⊥ O σβ B s + p ⊥ | y ⊥ ) = O y ( y ⊥ | p m p ⊥ ( σβ B s + p ⊥ ) | y ⊥ ) + i∂ m O y πσβ B s + ... The first term in the r.h.s of this equation is obviously zero while the second is O (cid:0) m N σβ B s (cid:1) in comparison to the leading first term in the r.h.s. of Eq. (6.5) (the transverse momentainside the hadron target are ∼ m N ∼ GeV).Thus, we obtain the following rapidity evolution equation in the Sudakov region: dd ln σ (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) (6.6) = 4 α s N c (cid:90) d − p ⊥ p ⊥ (cid:104) e i ( p,x − y ) ⊥ (cid:104) p | ˜ F ai (cid:0) β B + p ⊥ σs , x ⊥ (cid:1) F aj (cid:0) β B + p ⊥ σs , y ⊥ (cid:1) | p (cid:105)− σβ B sσβ B s + p ⊥ (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) (cid:105) As we mentioned above, the integrals over p ⊥ in the production part of the kernel (6.6)are k ⊥ whereas in the virtual part the logarithmic integrals over p ⊥ are restricted fromabove by an extra p ⊥ + σβ B s leading to the double-log region where (cid:29) σ (cid:29) ( x − y ) − ⊥ s and σβ B s (cid:29) p ⊥ (cid:29) ( x − y ) − ⊥ . In that region only the first term in the r.h.s. of Eq. (6.6)survives so the evolution equation reduces to dd ln σ (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) η =ln σ (6.7) = − g N c π (cid:90) d − p ⊥ p ⊥ (cid:2) − e i ( p,x − y ) ⊥ (cid:3) (cid:104) p | ˜ F ai ( β B , x ⊥ ) F aj ( β B , y ⊥ ) | p (cid:105) η which can be rewritten for the TMD (1.6) as dd ln σ D ( x B , z ⊥ , ln σ ) = − α s N c π D ( x B , z ⊥ , ln σ ) (cid:90) d p ⊥ p ⊥ (cid:2) − e i ( p,z ) ⊥ (cid:3) (6.8)We see that the IR divergence at p ⊥ → cancels while the UV divergence in the virtualcorrection should be cut from above by the condition p ⊥ ≤ σs following from Eq. (6.6).With the double-log accuracy one obtains dd ln σ D ( x B , z ⊥ , ln σ ) = − α s N c π D ( x B , z ⊥ , ln σ ) ln σsz ⊥ + ... (6.9)where dots stand for the non-logarithmic contributions. This equation leads to the usualSudakov double-log result D ( x B , k ⊥ , ln σ ) ∼ exp (cid:8) − α s N c π ln σsk ⊥ (cid:9) D ( x B , k ⊥ , ln k ⊥ s ) (6.10)– 39 –t is worth noting that the coefficient in front of ln σsk ⊥ is determined by the cusp anomalousdimension of two light-like Wilson lines going from point y to ∞ p and ∞ p directions (withour cutoff α < σ ). Indeed, if one calculates the contribution of the diagram in Fig. 6 forWilson lines in the adjoint representation, one gets p p Figure 6 . Cusp anomalous dimension in the leading order. (cid:104) [ ∞ p , , ∞ p ] (cid:105) = − ig N c (cid:90) d − αd − βd − p ⊥ θ ( σ > | α | > σ (cid:48) ) α ( β − i(cid:15) )( αβs − p ⊥ + i(cid:15) )= − g N c (cid:90) σσ (cid:48) d − αα (cid:90) d − p ⊥ p ⊥ = − g π N c ln σσ (cid:48) (cid:90) dp ⊥ p ⊥ (6.11)which coincides with the coefficient in Eq. (6.8), cf Ref. [27]. It is instructive to present the evolution kernel (5.5) in the linear (two-gluon) approximation.Since in the r.h.s. of Eq. (5.5) we already have ˜ F k and F l (and each of them has at least– 40 –ne gluon) all factors U and ˜ U in the r.h.s. of Eq. (5.5) can be omitted and we get dd ln σ (cid:104) p | ˜ F ai ( β B , p ⊥ ) F aj ( β B , p (cid:48)⊥ ) | p (cid:105) (7.1) = − α s N c (cid:90) d − k ⊥ (cid:110) θ (cid:0) − β B − k ⊥ σs (cid:1)(cid:104)(cid:16) ( p + k ) k σβ B s + p ⊥ σβ B sg µi − k ⊥ µ k i σβ B s + k ⊥ − k ⊥ µ g ik + p i g µk σβ B s + p ⊥ (cid:17) × (cid:16) σβ B sδ µj − k µ ⊥ k j σβ B s + k ⊥ ( p (cid:48) + k ) l σβ B s + p (cid:48) ⊥ − k µ ⊥ g jl + δ µl p (cid:48) j σβ B s + p (cid:48) ⊥ (cid:17) + 2 g ik (cid:16) k j k ⊥ σβ B s + 2 k ⊥ σβ B s + k ⊥ ( p (cid:48) + k ) l σβ B s + p (cid:48) ⊥ + 2 g jl σβ B s + p (cid:48) ⊥ − p (cid:48) j k l k ⊥ ( σβ B s + p (cid:48) ⊥ ) (cid:17) + 2 g lj (cid:16) ( p + k ) k σβ B s + p ⊥ k i k ⊥ σβ B s + 2 k ⊥ σβ B s + k ⊥ + 2 g ik σβ B s + p ⊥ − p i k k k ⊥ ( σβ B s + p ⊥ ) (cid:17)(cid:105) × (cid:104) p | ˜ F k (cid:0) β B + k ⊥ σs , p ⊥ − k ⊥ (cid:1) F l (cid:0) β B + k ⊥ σs , p (cid:48)⊥ − k ⊥ (cid:1) | p (cid:105) + 2 k ⊥ (cid:104) (2 k l p (cid:48) j − k j p (cid:48) l ) δ ki σβ B s + ( p (cid:48) + k ) ⊥ + (2 p i k k − k i p k ) δ lj σβ B s + ( p + k ) ⊥ (cid:105) (cid:104) p | ˜ F ak ( β B , p ⊥ ) F al ( β B , p (cid:48)⊥ ) | p (cid:105)− k ⊥ (cid:104) p | (cid:104) θ (cid:0) − β B − k ⊥ σs (cid:1) ˜ F i (cid:0) β B + k ⊥ σs , p ⊥ − k ⊥ (cid:1) F j (cid:0) β B + k ⊥ σs , p (cid:48)⊥ − k ⊥ (cid:1) − σβ B sσβ B s + k ⊥ ˜ F ai ( β B , p ⊥ ) F aj ( β B , p (cid:48)⊥ ) (cid:105) | p (cid:105) (cid:111) where we performed Fourier transformation to the momentum space. Also, the forwardmatrix element (cid:104) p | ˜ F i ( β B , p ⊥ ) F j ( β B , p (cid:48)⊥ ) | p (cid:105) is proportional to δ (2) ( p ⊥ − p (cid:48)⊥ ) . Eliminatingthis factor and rewriting in terms of R ij (see Eq. (2.9)) we obtain ( η ≡ ln σ ) ddη R ij ( β B , p ⊥ ; η ) (7.2) = − α s N c (cid:90) d − k ⊥ (cid:110)(cid:104)(cid:16) (2 p − k ) k σβ B s + p ⊥ σβ B sg µi − p − k ) ⊥ µ ( p − k ) i σβ B s + ( p − k ) ⊥ − p − k ) ⊥ µ g ik + p i g µk σβ B s + p ⊥ (cid:17) × (cid:16) σβ B sδ µj − p − k ) µ ⊥ ( p − k ) j σβ B s + ( p − k ) ⊥ (2 p − k ) l σβ B s + p ⊥ − p − k ) µ ⊥ g jl + δ µl p j σβ B s + p ⊥ (cid:17) + 2 g ik (cid:16) ( p − k ) j (2 p − k ) l − p j ( p − k ) l ( p − k ) ⊥ ( σβ B s + p ⊥ ) + ( p − k ) j (2 p − k ) l ( σβ B s + ( p − k ) ⊥ )( σβ B s + p ⊥ ) + 2 g jl σβ B s + p ⊥ (cid:17) + 2 g lj (cid:16) ( p − k ) i (2 p − k ) k − p i ( p − k ) k ( p − k ) ⊥ ( σβ B s + p ⊥ ) + ( p − k ) i (2 p − k ) k ( σβ B s + ( p − k ) ⊥ )( σβ B s + p ⊥ ) + 2 g ik σβ B s + p ⊥ (cid:17)(cid:105) × θ (cid:0) − β B − ( p − k ) ⊥ σs (cid:1) R kl (cid:0) β B + ( p − k ) ⊥ σs , k ⊥ (cid:1) + 2 δ ki ( k j p l − k l p j ) + δ lj ( k i p k − p i k k ) k ⊥ [ σβ B s + ( p − k ) ⊥ ] R kl ( β B , p ⊥ ; η ) − (cid:104) θ (cid:0) − β B − ( p − k ) ⊥ σs (cid:1) ( p − k ) ⊥ R ij (cid:0) β B + ( p − k ) ⊥ σs , k ⊥ ; η (cid:1) − σβ B sk ⊥ ( σβ B s + k ⊥ ) R ij ( β B , p ⊥ ; η ) (cid:105)(cid:111) Let us demonstrate that Eq. (7.2) reduces to BFKL equation in the low- x limit. Indeed,in this limit R ij is proportional to the WW distribution (1.1):– 41 – ij (0 , k ⊥ ) ∼ (cid:82) d xe i ( k,x ) ⊥ (cid:104) p | tr { ˜ U i ( x ) U j (0) }| p (cid:105) . In the leading-order BFKL approximation(cf. Ref. [14]) (cid:104) p | tr { ˜ U i ( x ) U j ( y ) }| p (cid:105) (7.3) = α s π (cid:90) d q ⊥ q ⊥ q i q j e i ( q,x ) ⊥ − i ( q,y ) ⊥ (cid:90) d q (cid:48)⊥ q (cid:48) ⊥ Φ T ( q (cid:48) ) (cid:90) a + i ∞ a − i ∞ dω πi (cid:16) sqq (cid:48) (cid:17) ω G ω ( q, q (cid:48) ) Here Φ T ( q (cid:48) ) is the target impact factor and G ω ( q, q (cid:48) ) is the partial wave of the forwardreggeized gluon scattering amplitude satisfying the equation ωG ω ( q, q (cid:48) ) = δ (2) ( q − q (cid:48) ) + (cid:90) d pK BFKL ( q, p ) G ω ( p, q (cid:48) ) (7.4)with the forward BFKL kernel K BFKL ( q, p ) = α s N c π (cid:104) q − p ) ⊥ − δ ( q ⊥ − p ⊥ ) (cid:90) dp (cid:48)⊥ q ⊥ p (cid:48) ⊥ ( q − p (cid:48) ) ⊥ (cid:105) Thus, in the BFKL approximation R ij (0 , q ⊥ ; ln σ ) = q i q j R ( q ⊥ ; ln σ ) = α s q i q j π q ⊥ (cid:90) d q (cid:48) q (cid:48) Φ T ( q (cid:48) ) (cid:90) a + i ∞ a − i ∞ dω πi (cid:16) σsqq (cid:48) (cid:17) ω G ω ( p, q (cid:48) ) (7.5)and the equation for Rdd ln σ R σ ( q ⊥ ; ln σ ) = (cid:90) d p ⊥ p ⊥ q ⊥ K BFKL ( q, p ) R σ ( p ⊥ ; ln σ ) (7.6)is obtained by differentiation of Eq. (7.5) with respect to ln σ using Eq. (7.4).Now it is easy to see that our Eq. (7.2) reduces to Eq. (7.6) in the BFKL limit. As wediscussed above, in this limit one may set β B = 0 and neglect k ⊥ σs in the argument of R ij .Substituting R ij (0 , k ⊥ ) = k i k j R ( k ⊥ ) into Eq. (7.2) one obtains after some algebra dd ln σ R ( p ⊥ ; ln σ ) = 2 α s N c (cid:90) d − k ⊥ (cid:104) k ⊥ p ⊥ ( p − k ) ⊥ R ( k ⊥ ; ln σ ) − p ⊥ k ⊥ ( p − k ) ⊥ R ( p ⊥ ; ln σ ) (cid:105) which coincides with Eq. (7.6). We have also checked that Eq. (7.1) at p ⊥ (cid:54) = p (cid:48)⊥ reducesto the non-forward BFKL equation in the low- x limit.Let us check now that the evolution of D ( β B , ln σ ) = − (cid:90) d − p ⊥ R ii ( β B , p ⊥ ; ln σ ) (7.7)reduces to DGLAP equation. As we discussed above, in the light-cone limit one can neglect k ⊥ in comparison to p ⊥ . Indeed, the integral over p ⊥ converges at p ⊥ ∼ σβ B s . On theother hand, extra k i k j in the integral over k ⊥ leads to the operators of higher collineartwist, for example (cid:90) d k ⊥ k i k j R nn ( β B , k ⊥ ; ln σ ) ∼ (cid:104) p | ∂ k ˜ F an ( β B , ⊥ ) ∂ j F an ( β B , ⊥ ) | p (cid:105) η =ln σ ∼ m g ij (cid:104) p | ˜ F an ( β B , ⊥ ) F an ( β B , ⊥ ) | p (cid:105) ln σ ∼ m D ( β B , ln σ ) (7.8)– 42 –where m is the mass of the target) so k ⊥ p ⊥ ∼ k ⊥ σβ B s ∼ m σs (cid:28) .Neglecting k ⊥ in comparison to p ⊥ and integrating over angles one obtains dd ln σ (cid:90) d p ⊥ R ii ( β B , p ⊥ ; ln σ )= α s N c π (cid:90) d p ⊥ (cid:104) p ⊥ − σβ B s + p ⊥ + 3 p ⊥ ( σβ B s + p ⊥ ) − p ⊥ ( σβ B s + p ⊥ ) + p ⊥ ( σβ B s + p ⊥ ) (cid:105) × (cid:90) d k ⊥ R ii (cid:0) β B + p ⊥ σs , k ⊥ ; ln σ (cid:1) − (cid:90) d k ⊥ σβ B sk ⊥ ( σβ B s + k ⊥ ) (cid:90) d p ⊥ R ii (cid:0) β B , p ⊥ ; ln σ (cid:1) which coincides with DGLAP equation (3.33).It would be interesting to compare Eq. (7.2) to CCFM equation [28] which also ad-dresses the question of interplay of BFKL and DGLAP logarithms. In this section we will construct the evolution equation for fragmentation function (1.7). Westart from Eq. (5.2) which enables us to analytically continue to negative β B = − β F . In theoperator form, the equation (5.2) has imaginary parts at negative β B = − β F correspondingto poles of propagators ( σβ F s − p ⊥ ± i(cid:15) ) − but we will demonstrate now that for theevolution of a “fragmentation matrix element” (2.10) (cid:104) ˜ F ai ( − β F , x ⊥ ) F aj ( − β F , y ⊥ ) (cid:105) frag ≡ (cid:88) X (cid:104) | ˜ F ai ( − β F , x ⊥ ) | p + X (cid:105)(cid:104) p + X |F aj ( − β F , y ⊥ ) | (cid:105) (8.1)we have the kinematical restriction σ ( β F − s > p ⊥ in all the integrals in the productionpart of the kernel (5.2). As to virtual part of the kernel, we will see that the imaginaryparts there assemble to yield the principle-value prescription for integrals over p ⊥ . The Again, strictly speaking we should consider (cid:80) X (cid:104) | ˜ F ai ( − β F , x ⊥ ) | p + X (cid:105)(cid:104) ( p + ξp ) + X |F aj ( − β F , y ⊥ ) | (cid:105) ,see Eq. (2.10). – 43 –fragmentation matrix element” (2.10) of Eq. (5.4) has the form dd ln σ (cid:104) ˜ F ai ( − β F , x ⊥ ) F aj ( − β F , y ⊥ ) (cid:105) frag (8.2) = − α s (cid:104) Tr (cid:110)(cid:90) d − k ⊥ θ (cid:0) β F − − k ⊥ σs (cid:1)(cid:104) ( x ⊥ | (cid:16) ˜ U − σβ F s − p ⊥ ( ˜ U † k k + p k ˜ U † ) σβ F sg µi + 2 k ⊥ µ k i σβ F s − k ⊥ + 2 k ⊥ µ g ik ˜ U σβ F s − p ⊥ ˜ U † + 2 g µk ˜ U p i σβ F s − p ⊥ ˜ U † (cid:17) ˜ F k (cid:0) − β F + k ⊥ σs (cid:1) | k ⊥ ) × ( k ⊥ |F l (cid:0) − β F + k ⊥ σs (cid:1)(cid:16) σβ F sδ µj + 2 k µ ⊥ k j σβ F s − k ⊥ ( k l U + U p l ) − σβ F s + p ⊥ U † + 2 k µ ⊥ g jl U σβ F s − p ⊥ U † + 2 δ µl U p j σβ F s − p ⊥ U † (cid:17) | y ⊥ ) + 2( x ⊥ | ˜ F i (cid:0) − β F + k ⊥ σs (cid:1) | k ⊥ ) × ( k ⊥ |F l (cid:0) − β F + k ⊥ σs (cid:1)(cid:16) − k j k ⊥ σβ F s − k ⊥ σβ F s − k ⊥ ( k l U + U p l ) 1 σβ F s − p ⊥ U † − U g jl σβ F s − p ⊥ U † + 2 k l k ⊥ U p j σβ F s − p ⊥ U † (cid:17) | y ⊥ )+ 2( x ⊥ | (cid:16) − ˜ U σβ F s − p ⊥ ( ˜ U † k k + p k ˜ U † ) k i k ⊥ σβ F s − k ⊥ σβ F s − k ⊥ − U g ik σβ F s − p ⊥ ˜ U † + 2 ˜ U p i σβ F s − p ⊥ ˜ U † k k k ⊥ (cid:17) ˜ F k (cid:0) − β F + k ⊥ σs (cid:1) | k ⊥ )( k ⊥ |F j (cid:0) − β F + k ⊥ σs (cid:1) | y ⊥ ) (cid:105) + 2 ˜ F i ( − β F , x ⊥ )( y ⊥ | p m p ⊥ F k ( − β F )( i ← ∂ l + U l )(2 δ km δ lj − g jm g kl ) U σβ F s − p ⊥ + i(cid:15) U † | y ⊥ )+ 2( x ⊥ | ˜ U − σβ F s − p ⊥ − i(cid:15) ˜ U † (2 δ ki δ lm − g im g kl )( i∂ k − ˜ U k ) ˜ F l ( − β F ) p m p ⊥ | x ⊥ ) F j ( − β F , y ⊥ ) − (cid:90) d − k ⊥ k ⊥ (cid:104) θ (cid:0) β F − − k ⊥ σs (cid:1) ˜ F i (cid:0) − β F + k ⊥ σs , x ⊥ (cid:1) F j (cid:0) − β F + k ⊥ σs , y ⊥ (cid:1) e i ( k,x − y ) ⊥ − (cid:2) σβ F sσβ F s − k ⊥ − i(cid:15) + σβ F sσβ F s − k ⊥ + i(cid:15) (cid:3) ˜ F i ( − β F , x ⊥ ) F j ( − β F , y ⊥ ) (cid:105)(cid:111) (cid:105) frag + O ( α s ) where we have restored ± i(cid:15) in the virtual part in accordance with Feynman rules.Let us prove that all non-linear terms in Eq. (8.2) can be neglected with our accuracy.(Naïvely, they were important at small β B but small β F are not allowed due to kinematicalrestrictions). First, consider the “light-cone” case when the transverse momenta of fastfields l ⊥ are smaller than the characteristic transverse momenta in the gluon loop of slowfields p ⊥ ∼ k ⊥ . As we discussed above, in this case with the leading-twist accuracy we cancommute all U ’s with p ⊥ operators until they form U U † = 1 and disappear. In this limit– 44 –he (8.2) turns to dd ln σ (cid:104) ˜ F ai ( − β F , x ⊥ ) F aj ( − β F , y ⊥ ) (cid:105) frag = − α s Tr (cid:110) ( x ⊥ | θ ( β F − − p ⊥ σs ) (8.3) × (cid:104) δ ki p µ ⊥ p ⊥ + g µk p i + p µ ⊥ δ ki − δ µi p k σβ F s − p ⊥ − i(cid:15) − δ µi p ⊥ p k + 2 p i p k p µ ⊥ ( σβ F s − p ⊥ − i(cid:15) ) (cid:105)(cid:104) p ⊥ µ δ lj p ⊥ + δ lµ p j + δ lj p ⊥ µ − g µj p l σβ F s − p ⊥ + i(cid:15) − g µj p ⊥ p l + 2 p ⊥ µ p j p l ( σβ F s − p ⊥ + i(cid:15) ) (cid:105) | y ⊥ ) (cid:104) ˜ F k (cid:0) p ⊥ σs − β F , x ⊥ (cid:1) F l (cid:0) p ⊥ σs − β F , y ⊥ (cid:1) (cid:105) frag + 12 (cid:104) ( x ⊥ | σβ F sp ⊥ ( σβ F s − p ⊥ − i(cid:15) ) | x ⊥ ) + ( y ⊥ | σβ F sp ⊥ ( σβ F s − p ⊥ + i(cid:15) ) | y ⊥ ) (cid:105) × (cid:104) ˜ F i ( − β F , x ⊥ ) F j ( − β F , y ⊥ ) (cid:105) frag (cid:111) Now consider the shock-wave case when l ⊥ ∼ p ⊥ . There are two “subcases”: when β F σ ∗ ≥ and when β F σ ∗ (cid:28) (where σ ∗ ∼ σ (cid:48) sl ⊥ ). In the former case we have β F σ (cid:48) sl ⊥ ≥ so σβ F s (cid:29) p ⊥ and only two last lines in Eq. (8.2) survive. Moreover, in this case Eq. (8.3)also reduces to the last two lines so Eq. (8.2) is equivalent to Eq. (8.3) in this case as well.If β F σ ∗ (cid:28) , as we discussed above, one can replace F j ( − β F ) (and F j (cid:0) − β F + p ⊥ αs (cid:1) )by U j . We will prove now that after such replacement the r.h.s. of Eq. (8.2) vanishes, andso does the r.h.s of Eq. (8.3), and therefore Eqs. (8.2) and (8.3) are equivalent in the caseof large β F σ ∗ also.Let us now prove that if we replace all F j ( − β F ) and F j (cid:0) − β F + p ⊥ αs (cid:1) by U j the r.h.s.of Eq. (8.2) vanishes. Indeed, a typical term in Feynman part of the amplitude vanishes: (cid:104) p + X | U j ( z ) U z (cid:48) U † z (cid:48)(cid:48) | (cid:105) = 0 (8.4)To prove this, let us consider the shift of U operator on s a ∗ p . Since the shift in the p direction does not change the infinitely long U operator, we get (cid:104) p + X | U j ( z ) U z (cid:48) U † z (cid:48)(cid:48) | (cid:105) = (cid:104) p + X | e i s ˆ P • a ∗ U j ( z ) U z (cid:48) U † z (cid:48)(cid:48) e − i s ˆ P • a ∗ | (cid:105) = e i ( β p + β X ) a ∗ (cid:104) p + X | U j ( z ) U z (cid:48) U † z (cid:48)(cid:48) | (cid:105) which can be true only if Eq. (8.4) vanishes. It is clear that for the same reason all terms inthe r.h.s of Eq. (8.2) (and r.h.s. of Eq. (8.3) as well) vanish. Summarizing, in all regimesthe Eq. (8.2) can be reduced to the light-cone version (8.3).One can rewrite Eq. (8.3) in the form: dd ln σ (cid:104) ˜ F ai ( − β F , x ⊥ ) F aj ( − β F , y ⊥ ) (cid:105) η ≡ ln σ frag (8.5) = 4 α s N c (cid:90) d − p ⊥ (cid:110) θ (cid:0) β F − − p ⊥ σs (cid:1) e i ( p,x − y ) ⊥ (cid:104) ˜ F ak (cid:0) p ⊥ σs − β F , x ⊥ (cid:1) F al (cid:0) p ⊥ σs − β F , y ⊥ (cid:1) (cid:105) η frag × (cid:104) δ ki δ lj p ⊥ + 2 δ ki δ lj σβ F s − p ⊥ + p ⊥ δ ki δ lj + δ kj p i p l + δ li p j p k − δ lj p i p k − δ ki p j p l − g kl p i p j − g ij p k p l ( σβ F s − p ⊥ ) − p ⊥ g ij p k p l + δ ki p j p l + δ lj p i p k − δ kj p i p l − δ li p j p k ( σβ F s − p ⊥ ) − p ⊥ g ij p k p l ( σβ F s − p ⊥ ) (cid:105) − θ ( σβ F s − p ⊥ ) p ⊥ (cid:104) ˜ F ai ( − β F , x ⊥ ) F aj ( − β F , y ⊥ ) (cid:105) η frag (cid:111) – 45 –here we used the formula (cid:90) d − p ⊥ p ⊥ (cid:104) σβ F sσβ F s − p ⊥ + i(cid:15) + σβ F sσβ F s − p ⊥ − i(cid:15) (cid:105) = (cid:90) d − p ⊥ p ⊥ θ ( σβ F s − p ⊥ ) (8.6)The Eq. (8.5) is our final evolution equation for fragmentation functions valid for all ( x − y ) ⊥ (and all β F ).If polarizations of fragmentation hadron are not registered we can use the parametriza-tion (2.10) (cid:104) ˜ F ai ( − β F , z ⊥ ) F aj ( − β F , ⊥ ) (cid:105) η frag = 2 π δ ( ξ ) β F g (cid:104) − g ij D f ( β F , z ⊥ , η ) − m (2 z i z j + g ij z ⊥ ) H (cid:48)(cid:48) f ( β F , z ⊥ , η ) (cid:105) (8.7)where H f ( β F , z ⊥ , η ) ≡ (cid:82) d − k ⊥ e i ( k,z ) ⊥ H f ( β F , k ⊥ , η ) and H (cid:48)(cid:48) f ( β F , z ⊥ , η ) ≡ (cid:0) ∂∂z (cid:1) H f ( β F , z ⊥ , η ) , cf. Eq. (3.26). After integration over angles similarto Eq. (3.27) one obtains ddη (cid:104) g ij α s D f ( β F , z ⊥ , η ) + 4 m (2 z i z j + g ij z ⊥ ) α s H (cid:48)(cid:48) f ( β F , z ⊥ , η ) (cid:105) (8.8) = α s N c π (cid:90) β F − dβ (cid:110) g ij J (cid:0) | z ⊥ | (cid:112) σβs (cid:1) α s D f ( β F − β, z ⊥ , η ) × (cid:104) β F − βββ F + 2 β F + 3 ββ F ( β F − β ) + 2 β β F ( β F − β ) + β β F ( β F − β ) (cid:105) + J ( | z ⊥ | (cid:112) σβs ) (cid:0) z i z j z ⊥ + g ij (cid:1) α s D f ( β F − β, z ⊥ , η ) ββ F ( β F − β )+ 4 m J (cid:0) | z ⊥ | (cid:112) σβs (cid:1) (2 z i z j + g ij z ⊥ ) α s H (cid:48)(cid:48) f ( β F − β, z ⊥ , η ) (cid:104) β F − ββ F β + 2 β F + ββ F ( β F − β ) (cid:105) + 4 g ij m z ⊥ J (cid:0) | z ⊥ | (cid:112) σβs (cid:1) α s H (cid:48)(cid:48) f ( β F − β, z ⊥ , η ) × (cid:104) ββ F ( β F − β ) + 2 β β F ( β F − β ) + β β F ( β F − β ) (cid:105)(cid:111) − α s N c π (cid:90) β F dββ (cid:104) g ij α s D f ( β F , z ⊥ , η ) + 4 m (2 z i z j + g ij z ⊥ ) α s H (cid:48)(cid:48) f ( β F , z ⊥ , η ) (cid:105) where β ≡ p ⊥ σs (and σ ≡ e η as usual). – 46 –his evolution equation can be rewritten as a system (cf. Eq. (3.29) for DIS) ddη α s D f ( 1 z F , z ⊥ , η ) (8.9) = α s N c π (cid:90) z F dz (cid:48) z (cid:48) (cid:110) J (cid:16) | z ⊥ | (cid:114) σsz F (1 − z (cid:48) ) (cid:17)(cid:104)(cid:0) − z (cid:48) (cid:1) + + 1 z (cid:48) − z (cid:48) (1 − z (cid:48) ) (cid:105) α s D f (cid:0) z (cid:48) z F , z ⊥ , η (cid:1) + 4 m − z (cid:48) z (cid:48) z ⊥ J (cid:16) | z ⊥ | (cid:114) σsz F (1 − z (cid:48) ) (cid:17) α s H (cid:48)(cid:48) f ( z (cid:48) z F , z ⊥ , η ) (cid:111) ,ddη α s H (cid:48)(cid:48) f ( 1 z F , z ⊥ , η )= α s N c π (cid:90) z F dz (cid:48) z (cid:48) (cid:110) J (cid:16) | z ⊥ | (cid:114) σsz F (1 − z (cid:48) ) (cid:17)(cid:104)(cid:0) − z (cid:48) (cid:1) + − (cid:105) α s H (cid:48)(cid:48) f (cid:0) z (cid:48) z F , z ⊥ , η (cid:1) + m z ⊥ (1 − z (cid:48) ) z (cid:48) J (cid:16) | z ⊥ | (cid:114) σsz F (1 − z (cid:48) ) (cid:17) α s D f (cid:0) z (cid:48) z F , z ⊥ , η (cid:1)(cid:111) where z (cid:48) = 1 − p ⊥ σβ F s = 1 − βz F . Here we introduced the standard notation z F ≡ β F forthe fraction of the “initial gluon momentum” carried by the hadron. By construction, thisequation describes the evolution of fragmentation TMD at any z F and any k ⊥ ∼ | z ⊥ | − .Let us demonstrate that Eq. (8.9) agrees with the DGLAP equation for fragmentationfunctions in the light-cone limit x ⊥ → y ⊥ . In this limit dd ln σ α s D f ( 1 z F , ⊥ , ln σs ) = α s π N c (cid:110)(cid:90) z F dz (cid:48) z (cid:48) (cid:104) z (cid:48) (1 − z (cid:48) ) + z (cid:48) (1 − z (cid:48) ) − (cid:105) (8.10) × α s D f ( z (cid:48) z F , ⊥ , ln σs ) − α s D f ( 1 z F , ⊥ , ln σs ) (cid:90) dz (cid:48) − z (cid:48) (cid:111) As explained in Eq. (3.30), with leading-log accuracy we can trade the cutoff in α for cutoffin µ . In terms of the standard definition of fragmentation functions [3] d f g ( z F , ln µ ) = − z F π α s N c ( p · n ) (cid:90) du e i uzF ( pn ) (cid:88) X (cid:104) | ˜ F aξ ( un ) | p + X (cid:105)(cid:104) p + X |F aξ (0) | (cid:105) µ (8.11)we have in the leading log approximation d f g ( z F , ln µ ) = z F N c D f ( 1 z F , z ⊥ = 0 , ln σs ) + O ( α s ) (8.12)so we can rewrite Eq. (8.10) in the form dd ln µ α s ( µ ) d f g ( z F , ln µ ) = (8.13) = α s ( µ ) π N c (cid:90) z F dz (cid:48) z (cid:48) (cid:16)(cid:2) − z (cid:48) (cid:3) + + 1 z (cid:48) + z (cid:48) (1 − z (cid:48) ) − (cid:17) α s ( µ ) d f g (cid:0) z F z (cid:48) , ln µ (cid:1) easily recognizable as the DGLAP equation for fragmentation functions [22]. (Here againthe term proportional to β -function is absent since ˜ F ai F ai is defined with an extra α s .)– 47 –inally, let us describe what happens if z F (cid:28) and we evolve from σ ∼ to σ ∼ z F z ⊥ s .With double-log accuracy we have an equation ddη D f (cid:0) z F , z ⊥ , η (cid:1) = − α s N c π D f (cid:0) z F , z ⊥ , η (cid:1)(cid:90) d p ⊥ p ⊥ (cid:2) − e i ( p,z ) ⊥ (cid:3) (8.14)(cf. Eq. (6.8)) with the solution of the Sudakov type D f (cid:0) z F , z ⊥ , ln σ (cid:1) ∼ exp (cid:8) − α s N c π ln σz F sz ⊥ (cid:9) D f (cid:0) z F , z ⊥ , ln z F z ⊥ s (cid:1) (8.15)The evolution with the single-log accuracy should be determined from the full system (8.9). We have described the rapidity evolution of gluon TMD (1.6) with Wilson lines goingto + ∞ in the whole range of Bjorken x B and the whole range of transverse momentum k ⊥ . It should be emphasized that with our definition of rapidity cutoff (2.1) the leading-order matrix elements of TMD operators are UV-finite so the rapidity evolution is theonly evolution and it describes all the dynamics of gluon TMDs (1.6) in the leading-logapproximation.The evolution equation for the gluon TMD (1.6) with rapidity cutoff (2.1) is given by(5.5) and, in general, is non-linear. Nevertheless, for some specific cases the equation (5.5)linearizes. For example, let us consider the case when x B ∼ . If in addition k ⊥ ∼ s , thenon-linearity can be neglected for the whole range of evolution (cid:29) σ (cid:29) m N s and we getthe DGLAP-type system of equations (3.29). If k ⊥ is small ( ∼ few GeV) the evolution islinear and leads to usual Sudakov factors (6.10). If we consider now the intermediate case x B ∼ and s (cid:29) k ⊥ (cid:29) m N the evolution at (cid:29) σ (cid:29) k ⊥ s will be Sudakov-type (see Eq.(6.6)) but the evolution at k ⊥ s (cid:29) σ (cid:29) m N s will be described by the full master equation(5.5).For low-x region k ⊥ ∼ few GeV and x B ∼ k ⊥ s we get the non-linear evolution describedby the BK-type equation (6.3). If we now keep k ⊥ ∼ few GeV and take the intermediate (cid:29) x B ≡ β B (cid:29) k ⊥ s we get a mixture of linear and non-linear evolutions. If one evolves σ ( ↔ rapidity) from 1 to k ⊥ s first there will be Sudakov-type double-log evolution (6.8) from σ = 1 to σ = k ⊥ β B s , then the transitional region at σ ∼ k ⊥ β B s , and after that the non-linearevolution (6.3) at k ⊥ β B s (cid:29) σ (cid:29) k ⊥ s . The transition between the linear evolution (6.8) andthe non-linear one (6.3) should be described by the full equation (5.5).Another interesting case is x B ∼ m N s and s (cid:29) k ⊥ (cid:29) m N . In this case, if we evolve σ from 1 to m N s , first we have the BK evolution (6.3) up to σ ∼ k ⊥ s and then for the evolutionbetween σ ∼ k ⊥ s and σ ∼ m N s we need the Eq. (5.5) in full.In conclusion, let us again emphasize that the evolution of the fragmentation TMDs(2.10) is always linear and the corresponding equation (8.8) describes both the DGLAPregion k ⊥ ∼ s and Sudakov region k ⊥ ∼ few GeV .– 48 –s an outlook, it would be very interesting to obtain the NLO correction to the evolu-tion equation (5.5). The NLO corrections to the BFKL [29] and BK [14, 30, 31] equation areavailable but they suffer from the well-known problem that they lead to negative cross sec-tions. This difficulty can be overcome by the “collinear resummation” of double-logarithmiccontributions for the BFKL [32] and BK [33] equations and we hope that our Eq. (5.5) andespecially its future NLO version will help to solve the problem of negative cross sectionsof NLO amplitudes at high energies.The authors are grateful to G.A. Chirilli, J.C. Collins, Yu. Kovchegov, A. Prokudin,A.V. Radyushkin, T. Rogers, and F. Yuan for valuable discussions. This work was sup-ported by contract DE-AC05-06OR23177 under which the Jefferson Science Associates,LLC operate the Thomas Jefferson National Accelerator Facility, and by the grant DE-FG02-97ER41028.
10 Appendix A: light-cone expansion of propagators
In this section we consider the case when the transverse momenta of background fast fields l ⊥ are much smaller than the characteristic transverse momenta p ⊥ of “quantum” slowgluons. As we discussed in Sect. 2, in this case fast fields do not necessarily shrink toa shock wave and one should use the light-cone expansion of propagators instead. Theparameter of expansion is the twist of the operator and we will expand up to operatorsof leading collinear twist two. Such operators are built of two gluon operators ∼ F • i F • j or quark ones ¯ ψ (cid:54) p ψ and gauge links. To get coefficients in front of these operators itis sufficient to consider the external gluon field of the type A • ( z ∗ , z ⊥ ) and quark fields (cid:54) p ψ ( x ∗ , x ⊥ ) with all other components being zero. For simplicity, let us again start with the expansion of a scalar propagator.
For simplicity we will first perform the calculation for “scalar propagator” ( x | P + i(cid:15) | y ) . Aswe mentioned above, we assume that the only nonzero component of the external field is A • and it does not depend on z • so the operator α = i ∂∂z • commutes with all backgroundfields. The propagator in the external field A • ( z ∗ , z ⊥ ) has the form ( x | P + i(cid:15) | y ) = (cid:104) − iθ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + iθ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) (10.1) × e − iα ( x − y ) • ( x ⊥ | Pexp (cid:8) − i (cid:90) x ∗ y ∗ dz ∗ (cid:2) p ⊥ αs − gs A • ( z ∗ ) (cid:3)(cid:9) | y ⊥ ) The z • dependence of the external fields can be omitted since due to the rapidity ordering α ’s of thefast fields are much less than α ’s of the slow ones. – 49 –he Pexp in the r.h.s. of Eq. (10.1) can be transformed to ( x ⊥ | Pexp (cid:8) − i (cid:90) x ∗ y ∗ dz ∗ (cid:2) p ⊥ αs − gs A • ( z ∗ ) (cid:3)(cid:9) | y ⊥ )= ( x ⊥ | e − i p ⊥ αs ( x ∗ − y ∗ ) Pexp (cid:110) igs (cid:90) x ∗ y ∗ dz ∗ e i p ⊥ αs ( z ∗ − y ∗ ) A • ( z ∗ ) e − i p ⊥ αs ( z ∗ − y ∗ ) (cid:111) | y ⊥ ) (10.2)Since the longitudinal distances z ∗ inside the shock wave are small we can expand e i p ⊥ αs ( z ∗ − y ∗ ) A • e − i p ⊥ αs ( z ∗ − y ∗ ) = A • − z ∗ − y ∗ αs { p i , F • i } − ( z ∗ − y ∗ ) α s { p j , { p i , D j F • i }} + ... = A • − z ∗ − y ∗ αs (2 p i F • i − iD i F • i ) − z ∗ − y ∗ ) α s ( p i p j − ip j D i ) D j F • i + ... (10.3)This is effectively expansion around the light ray y ⊥ + s y ∗ p with the parameter of theexpansion ∼ | l ⊥ || p ⊥ | (cid:28) . As we mentioned, we will expand up to the operator(s) with twisttwo.We obtain O ( x ∗ , y ∗ ; p ⊥ ) ≡ Pexp (cid:110) igs (cid:90) x ∗ y ∗ dz ∗ e i p ⊥ αs ( z ∗ − y ∗ ) A • ( z ∗ ) e − i p ⊥ αs ( z ∗ − y ∗ ) (cid:111) (10.4) = 1 + 2 igs (cid:90) x ∗ y ∗ dz ∗ (cid:2) A • − ( z − y ) ∗ αs (2 p i F • i − iD i F • i ) − z ∗ − y ∗ ) α s ( p i p j − ip j D i ) D j F • i (cid:3) − g s (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:2) A • − z − y ) ∗ αs p i F • i (cid:3)(cid:2) A • − z (cid:48) − y ) ∗ αs p j F • j (cid:3) + ... It is clear that the terms ∼ A • will combine to form gauge links so the r.h.s. of the aboveequation will turn to O ( x ∗ , y ∗ ; p ⊥ ) = [ x ∗ , y ∗ ] − igαs (cid:90) x ∗ y ∗ dz ∗ ( z − y ) ∗ (cid:16) p j [ x ∗ , z ∗ ] F • j ( z ∗ ) − i [ x ∗ , z ∗ ] D j F • j ( z ∗ )+ 2 ( z − y ) ∗ αs ( p j p k [ x ∗ , z ∗ ] − ip k [ x ∗ , z ∗ ] D j ) D k F • j (cid:17) [ z ∗ , y ∗ ]+ 8 g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ( z (cid:48) − y ) ∗ (cid:16) i [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F j • ( z (cid:48)∗ ) (10.5) − p j p k ( z − y ) ∗ αs [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F • k ( z (cid:48)∗ ) (cid:17) [ z (cid:48)∗ , y ∗ ] + ... where dots stand for the higher twists.Thus, the final expansion of the propagator (10.1) near the light cone y ⊥ + s y ∗ p takesthe form ( x | P + i(cid:15) | y ) = (cid:104) − iθ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + iθ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) (10.6) × e − iα ( x − y ) • ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ O ( x ∗ , y ∗ ; p ⊥ ) | y ⊥ ) Note that the transverse arguments of all background fields in Eq. (10.6) are effectively y ⊥ .– 50 – For calculations of the complex conjugate amplitude we need also the propagator ( x | P − i(cid:15) | y ) = (cid:104) iθ ( y ∗ − x ∗ ) (cid:90) ∞ d − α α − iθ ( x ∗ − y ∗ ) (cid:90) −∞ d − α α (cid:105) (10.7) × e − iα ( x − y ) • ( x ⊥ | Pexp (cid:8) − i (cid:90) x ∗ y ∗ dz ∗ (cid:2) p ⊥ αs − gs ˜ A • ( z ∗ ) (cid:3)(cid:9) | y ⊥ ) For the calculation of the square of Lipatov vertex we need to consider point x inside theshock wave and point y outside. In this case one should rewrite Eq. (10.2) as follows ( x ⊥ | Pexp (cid:8) − i (cid:90) x ∗ y ∗ dz ∗ (cid:2) p ⊥ αs − gs ˜ A • ( z ∗ ) (cid:3)(cid:9) | y ⊥ )= ( x ⊥ | Pexp (cid:110) igs (cid:90) x ∗ y ∗ dz ∗ e − i p ⊥ αs ( x ∗ − z ∗ ) ˜ A • ( z ∗ ) e i p ⊥ αs ( x ∗ − z ∗ ) (cid:111) e − i p ⊥ αs ( x ∗ − y ∗ ) | y ⊥ ) (10.8)The light-cone expansion around x ⊥ + s x ∗ p is given by Eq. (10.3) with y ∗ → x ∗ e i p ⊥ αs ( z ∗ − x ∗ ) ˜ A • e − i p ⊥ αs ( z ∗ − x ∗ ) (10.9) = ˜ A • − z ∗ − x ∗ αs (2 ˜ F • j p j + i ˜ D j ˜ F • j ) − z ∗ − x ∗ ) α s ( ˜ D j ˜ F • i p i p j + i ˜ D i ˜ D j ˜ F • i p j ) + ... (the only difference with the expansion (10.3) is that we should put the operators p j to theright) and therefore ˜ O ( x ∗ , y ∗ ; p ⊥ ) ≡ Pexp (cid:110) igs (cid:90) x ∗ y ∗ dz ∗ e − i p ⊥ αs ( x ∗ − z ∗ ) ˜ A • ( z ∗ ) e i p ⊥ αs ( x ∗ − z ∗ ) (cid:111) = 1+ 2 igs (cid:90) x ∗ y ∗ dz ∗ (cid:2) ˜ A • − z ∗ − x ∗ αs (2 ˜ F • j p j + i ˜ D j ˜ F • j ) − z ∗ − x ∗ ) α s ( ˜ D j ˜ F • i p i p j + i ˜ D i ˜ D j ˜ F • i p j ) (cid:3) − g s (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:2) ˜ A • − z − x ) ∗ αs ˜ F • i p i (cid:3)(cid:2) ˜ A • − z (cid:48) − x ) ∗ αs ˜ F • j p j (cid:3) + ... (10.10)which turns to ˜ O ( x ∗ , y ∗ ; p ⊥ ) = [ x ∗ , y ∗ ] + 2 igαs (cid:90) y ∗ x ∗ dz ∗ [ x ∗ , z ∗ ] (cid:8) F • j ( z ∗ )[ z ∗ , y ∗ ] p j + i ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ]+ 2 ( z − x ) ∗ αs (cid:0) ˜ D k ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] p j p k + i ˜ D j ˜ D k ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] p k (cid:1)(cid:9) ( z − x ) ∗ + 8 g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ( z − x ) ∗ [ x ∗ , z ∗ ] (cid:16) − i ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] ˜ F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] − z (cid:48) − x ) ∗ αs ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] ˜ F • k ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] p j p k (cid:17) + ... (10.11)and we get ( x | P − i(cid:15) | y ) = (cid:104) iθ ( y ∗ − x ∗ ) (cid:90) ∞ d − α α − iθ ( x ∗ − y ∗ ) (cid:90) −∞ d − α α (cid:105) (10.12) × e − iα ( x − y ) • ( x ⊥ | ˜ O ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs ( x − y ) ∗ | y ⊥ ) Here (in Eq. (10.11)) the transverse arguments of all background fields are effectively x ⊥ .– 51 – For the calculation of Lipatov vertex we need the propagator in mixed representation ( k | P + i(cid:15) | z ) in the limit k → where k = αp + k ⊥ αs p + k ⊥ : k ( k | P + i(cid:15) | y ) = 2 s (cid:90) dx ∗ dx • d x ⊥ e ikx (cid:2) − s ∂∂x • ∂∂x ∗ + ∂ ⊥ (cid:3) ( x | P + i(cid:15) | y ) (10.13)First, we perform the trivial integrations over x • and x ⊥ : lim k → k ( k | P + i(cid:15) | y ) (10.14) = (cid:90) dx ∗ d x ⊥ e i k ⊥ αs x ∗ − i ( k,x ) ⊥ (cid:2) ∂∂x ∗ − iαs ∂ ⊥ (cid:3) θ ( x − y ) ∗ ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ O ( x ∗ , y ∗ ; p ⊥ ) | y ⊥ ) e iαy • = (cid:90) dx ∗ d x ⊥ e i k ⊥ αs x ∗ − i ( k,x ) ⊥ ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ ∂∂x ∗ θ ( x − y ) ∗ O ( x ∗ , y ∗ ; p ⊥ ) | y ⊥ ) e iαy • = e iαy • e i k ⊥ αs y ∗ (cid:90) dx ∗ ∂∂x ∗ θ ( x − y ) ∗ ( k ⊥ |O ( x ∗ , y ∗ ; k ) | y ⊥ )= e iαy • e i k ⊥ αs y ∗ − i ( k,y ) ⊥ O ( ∞ , y ∗ , y ⊥ ; k ) where O ( ∞ , y ∗ , y ⊥ ; k ) ≡ e i ( k,y ) ⊥ ( k ⊥ |O ( ∞ , y ∗ ; k ) | y ⊥ ) . In the explicit form O ( ∞ , y ∗ ; y ⊥ ; k )= [ ∞ , y ∗ ] y − igαs (cid:90) ∞ y ∗ dz ∗ ( z − y ) ∗ (cid:16) k j [ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ ) − i [ ∞ , z ∗ ] y D j F • j ( z ∗ , y ⊥ )+ 2 ( z − y ) ∗ αs ( k j k l [ ∞ , z ∗ ] y − ik l [ ∞ , z ∗ ] y D j ) D l F • j ( z ∗ , y ⊥ ) (cid:17) [ z ∗ , y ∗ ] y + 8 g αs (cid:90) ∞ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ( z (cid:48) − y ) ∗ (cid:16) i [ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y F j • ( z (cid:48)∗ , y ⊥ ) − k j k l ( z − y ) ∗ αs [ ∞ , z ∗ ] y F • j ( z ∗ , y ⊥ )[ z ∗ , z (cid:48)∗ ] y F • l ( z (cid:48)∗ , y ⊥ ) (cid:17) [ z (cid:48)∗ , y ∗ ] y (10.15)where the transverse arguments of all fields are y ⊥ and p j is replaced by k j .Similarly, for the complex conjugate amplitude we get lim k → k ( x | P − i(cid:15) | k ) = e − ikx ˜ O ( x ∗ , ∞ , x ⊥ ; k ) (10.16)where ˜ O ( x ∗ , ∞ , x ⊥ ; k ) ≡ ( x ⊥ | ˜ O ( x ∗ , ∞ , p ⊥ ) | k ⊥ ) e − i ( k,x ) ⊥ , or, in the explicit form ˜ O ( x ∗ , ∞ , x ⊥ ; k )= [ x ∗ , ∞ ] x + 2 igαs (cid:90) ∞ x ∗ dz ∗ [ x ∗ , z ∗ ] x (cid:8) F • j ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x k j + i ˜ D j ˜ F • j ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x + 2 ( z − x ) ∗ αs (cid:0) ˜ D l ˜ F • j ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x k j k l + i ˜ D j ˜ D l ˜ F • j ( z ∗ , x ⊥ )[ z ∗ , ∞ ] x k l (cid:1)(cid:9) ( z − x ) ∗ + 8 g αs (cid:90) ∞ x ∗ dz ∗ (cid:90) ∞ z ∗ dz (cid:48)∗ ( z − x ) ∗ [ x ∗ , z ∗ ] x (cid:16) − i ˜ F • j ( z ∗ , x ⊥ )[ z ∗ , z (cid:48)∗ ] x ˜ F j • ( z (cid:48)∗ , x ⊥ )[ z (cid:48)∗ , ∞ ] x − z (cid:48) − x ) ∗ αs ˜ F • j ( z ∗ , x ⊥ )[ z ∗ , z (cid:48)∗ ] x ˜ F • l ( z (cid:48)∗ , x ⊥ )[ z (cid:48)∗ , ∞ ] x k j k l (cid:17) + ... (10.17)– 52 –n the complex conjugate amplitude we expand around the light cone x ⊥ + s x ∗ p so thetransverse arguments of all fields in Eq. (10.17) are x ⊥ . Note that the second terms in ther.h.s. of Eqs. (10.6) and (10.7) (proportional to (cid:82) −∞ d − α ) do not contribute since α > forthe emitted particle. As we saw in the previous Section, to get the emission vertex (10.13) it is sufficient to writedown the propagator at x ∗ > y ∗ . The gluon propagator in the bF gauge has the form i (cid:104) A aµ ( x ) A bν ( y ) (cid:105) = ( x | P + 2 igF + i(cid:15) | y ) abµν (10.18) x ∗ >y ∗ = − i (cid:90) ∞ d − α α e − iα ( x − y ) • ( x ⊥ | Pexp (cid:8) − i (cid:90) x ∗ y ∗ dz ∗ (cid:2) p ⊥ αs − gs A • ( z ∗ ) − igαs F ( z ∗ ) (cid:3)(cid:9) | y ⊥ ) abµν = − i (cid:90) ∞ d − α α e − iα ( x − y ) • × ( x ⊥ | e − i p ⊥ αs ( x ∗ − y ∗ ) Pexp (cid:110) ig (cid:90) x ∗ y ∗ d s z ∗ e i p ⊥ αs ( z ∗ − y ∗ ) (cid:0) A • + iα F (cid:1) ( z ∗ ) e − i p ⊥ αs ( z ∗ − y ∗ ) (cid:111) abµν | y ⊥ ) where powers of F are treated as usual, for example ( F A • F F ) µν ≡ F ξµ A • g ξη F ηλ F λν . Theexpansion (10.3) now looks like e i p ⊥ αs ( z ∗ − y ∗ ) (cid:0) A • g µν + iα F µν (cid:1) e − i p ⊥ αs ( z ∗ − y ∗ ) = A • g µν + iα F µν + i z ∗ − y ∗ αs (cid:2) p ⊥ , A • g µν + iα F µν (cid:3) − ( z − y ) ∗ α s (cid:2) p ⊥ , (cid:2) p ⊥ , A • g µν + iα F µν (cid:3)(cid:3) + ... = g µν (cid:104) A • − z ∗ − y ∗ αs (2 p j F • j − iD j F • j ) − z − y ) ∗ α s ( p j p k D j F • k − ip k D k D j F • j ) (cid:105) + iα F µν + 2 i z ∗ − y ∗ α s p j D j F µν + 2 i ( z − y ) ∗ α s p j p k D j D k F µν + ... (10.19)– 53 –o we get G µν ( x ∗ , y ∗ ; p ⊥ ) ≡ Pexp (cid:110) ig (cid:90) x ∗ y ∗ d s z ∗ e i p ⊥ αs ( z ∗ − y ∗ ) (cid:0) A • + iα F (cid:1) ( z ∗ ) e − i p ⊥ αs ( z ∗ − y ∗ ) (cid:111) µν (10.20) = g µν + ig (cid:90) x ∗ y ∗ d s z ∗ e i p ⊥ αs ( z ∗ − y ∗ ) (cid:0) A • + iα F (cid:1) µν ( z ∗ ) e − i p ⊥ αs ( z ∗ − y ∗ ) − g (cid:90) x ∗ y ∗ d s z ∗ e i p ⊥ αs ( z ∗ − y ∗ ) × (cid:0) A • + iα F (cid:1) µξ ( z ∗ ) e − i p ⊥ αs ( z ∗ − y ∗ ) (cid:90) z ∗ y ∗ d s z (cid:48)∗ e i p ⊥ αs ( z (cid:48)∗ − y ∗ ) (cid:0) A • + iα F (cid:1) ξν ( z (cid:48)∗ ) e − i p ⊥ αs ( z (cid:48)∗ − y ∗ ) = g µν + 2 igs (cid:90) x ∗ y ∗ dz ∗ (cid:110) g µν (cid:104) A • ( z ∗ ) − ( z − y ) ∗ αs (2 p i F • i − iD i F • i ) − z − y ) ∗ α s ( p j p k D j F • k − ip k D k D j F • j ) (cid:105) + iα F µν + 2 i z ∗ − y ∗ α s p j D j F µν + 2 i ( z − y ) ∗ α s p j p k D j D k F µν (cid:111) − g s (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:2) g µξ A • + iα F µξ − g µξ p j ( z − y ) ∗ αs F • j (cid:3) × (cid:2) δ ξν A • + iα F ξν − p k ( z (cid:48) − y ) ∗ αs F • k δ ξν (cid:3) + ... = g µν [ x ∗ , y ∗ ] + g (cid:90) x ∗ y ∗ dz ∗ (cid:16) − iαs ( z − y ) ∗ g µν (cid:8) p j [ x ∗ , z ∗ ] F • j ( z ∗ ) − i [ x ∗ , z ∗ ] D j F • j ( z ∗ )+ 2 ( z − y ) ∗ αs ( p j p k [ x ∗ , z ∗ ] D j F • k − ip k [ x ∗ , z ∗ ] D k D j F • j ) (cid:9) + 4 αs ( δ jµ p ν − δ jν p µ ) (cid:8) [ x ∗ , z ∗ ] F • j ( z ∗ ) + 2( z − y ) ∗ αs p k [ x ∗ , z ∗ ] D k F • j ( z ∗ )+ 2 ( z − y ) ∗ α s p k p l [ x ∗ , z ∗ ] D k D l F • j (cid:9)(cid:17) [ z ∗ , y ∗ ]+ 8 g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:16) (cid:2) ig µν ( z (cid:48) − y ) ∗ − αs p µ p ν (cid:3) [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F j • ( z (cid:48)∗ ) − g µν αs p j p k ( z − y ) ∗ ( z (cid:48) − y ) ∗ [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F • k ( z (cid:48)∗ ) (cid:17) [ z (cid:48)∗ , y ∗ ] Note that F µξ F ξη F ην and higher terms of the expansion in powers of F µν vanish since theonly non-vanishing field strength is F • i .Finally, ( x | P + 2 igF + i(cid:15) | y ) abµν = (cid:104) − iθ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + iθ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) (10.21) × e − iα ( x − y ) • ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ G abµν ( x ∗ , y ∗ ; p ⊥ ) | y ⊥ ) For the complex conjugate amplitude we obtain in a similar way ( x | P + 2 igF − i(cid:15) | y ) abµν = (cid:104) iθ ( y ∗ − x ∗ ) (cid:90) ∞ d − α α − iθ ( x ∗ − y ∗ ) (cid:90) −∞ d − α α (cid:105) (10.22) × e − iα ( x − y ) • ( x ⊥ | ˜ G abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs ( x − y ) ∗ | y ⊥ ) – 54 –here ˜ G µν ( x ∗ , y ∗ ; p ⊥ ) ≡ Pexp (cid:110) ig (cid:90) x ∗ y ∗ d s z ∗ e i p ⊥ αs ( z ∗ − x ∗ ) (cid:0) ˜ A • + iα ˜ F (cid:1) ( z ∗ ) e − i p ⊥ αs ( z ∗ − x ∗ ) (cid:111) µν (10.23) = g µν [ x ∗ , y ∗ ] + g (cid:90) y ∗ x ∗ dz ∗ [ x ∗ , z ∗ ] (cid:16) iαs ( z − x ) ∗ g µν (cid:8) F • j ( z ∗ )[ z ∗ , y ∗ ] p j + i ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ]+ 2 ( z − x ) ∗ αs ( ˜ D j ˜ F • k ( z ∗ )[ z ∗ , y ∗ ] p j p k + i ˜ D k ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] p k ) (cid:9) − αs ( δ jµ p ν − δ jν p µ ) (cid:8) ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] + 2( z − x ) ∗ αs ˜ D k ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] p k + 2 ( z − x ) ∗ α s ˜ D k ˜ D l ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] p k p l (cid:9)(cid:17) + 8 g αs (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ [ x ∗ , z ∗ ] (cid:16)(cid:2) − ig µν ( z − x ) ∗ − αs p µ p ν (cid:3) ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] ˜ F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] − g µν αs ( z − x ) ∗ ( z (cid:48) − x ) ∗ ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] ˜ F • k ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] p j p k (cid:17) We do not impose the condition D i F • i = 0 so our external field has quark sources D i F a • i = g ¯ ψt a (cid:54) p ψ which we need to take into consideration. The corresponding contribution to gluonpropagator comes from diagrams in Fig. 7 (a) (b)z z’ z z’ Figure 7 . Gluon propagator in an external quark field. (cid:104) A aµ ( x ) A bν ( y ) (cid:105) Fig . = g (cid:90) d zd z (cid:48) ( x | P + i(cid:15) | z ) ac ( z (cid:48) | P + i(cid:15) | y ) db (10.24) × (cid:2) ( z | ¯ ψt c γ µ (cid:54) P i (cid:54) P + i(cid:15) γ ν t d ψ | z (cid:48) ) + ( z (cid:48) | ¯ ψt d γ ν i (cid:54) P + i(cid:15) (cid:54) P γ µ t c ψ | z ) (cid:3) As we mentioned above, we can consider quark fields with + spin projection onto p direction which corresponds to ¯ ψ ( ... ) ψ operators of leading collinear twist. In this approx-imation (cid:54) p ψ = 0 so the only non-zero propagators are (cid:104) A • ( x ) A • ( y ) (cid:105) , (cid:104) A • ( x ) A i ( y ) (cid:105) and (cid:104) A i ( x ) A j ( y ) (cid:105) . In addition, we assume that the quark fields ψ ( z ) depend only on z ⊥ and z ∗ (same as gluon fields) so the operator ˆ α = s ˆ p ∗ commutes with all background-fieldoperators. We get (cid:104) A a • ( x ) A b • ( y ) (cid:105) Fig . = 2 ig (cid:90) d zd z (cid:48) ( x | P + i(cid:15) | z ) ac ( z (cid:48) | P + i(cid:15) | y ) db (10.25) × (cid:2) ( z | ¯ ψt c (cid:54) p P • (cid:54) P + i(cid:15) t d ψ | z (cid:48) ) + ( z (cid:48) | ¯ ψt d (cid:54) P + i(cid:15) P • (cid:54) p t c ψ | z ) (cid:3) – 55 –n our gluon field (cid:54) P = α (cid:54) p + 2 (cid:54) p s P • + (cid:54) p ⊥ so (cid:54) P = 2 αP • − p ⊥ + is (cid:54) p γ j F • j and one canrewrite P • i (cid:54) P + i(cid:15) as P • αP • − p ⊥ + i(cid:15) = (cid:104) p ⊥ α + 12 α (cid:0) αP • − p ⊥ (cid:1)(cid:105) αP • − p ⊥ + i(cid:15) = 12 α + p ⊥ α αP • − p ⊥ + i(cid:15) (10.26)(the term s (cid:54) p γ j F • j does not contribute due to (cid:54) p ψ = 0 ). Similarly, αP • − p ⊥ + i(cid:15) P • = 12 α + 12 αP • − p ⊥ + i(cid:15) p ⊥ α (10.27)so one can rewrite the propagator (10.24) as (cid:104) A a • ( x ) A b • ( y ) (cid:105) Fig . = − ig ( x | α ( P + i(cid:15) ) D j F • j P + i(cid:15) | y ) ab (10.28) + ig ( x | (cid:0) α ( P + i(cid:15) ) (cid:1) ac (cid:0) p ⊥ ¯ ψ + 2 ip j ∂ j ¯ ψ − ∂ ⊥ ¯ ψ (cid:1) t c P + i(cid:15) (cid:54) p t d ψ (cid:0) P + i(cid:15) (cid:1) db | y )+ ig ( y | (cid:0) α ( P + i(cid:15) ) (cid:1) bd ¯ ψt d P + i(cid:15) t c (cid:54) p (cid:0) ψp ⊥ − i∂ j ψp j − ∂ ⊥ ψ (cid:1)(cid:0) P + i(cid:15) (cid:1) ca | x ) where in the first line we have rewritten g ¯ ψ [ t c , t d ] (cid:54) p ψ as − g ( D j F • j ) cd .Similarly, we get (cid:104) A a • ( x ) A bi ( y ) (cid:105) Fig . (10.29) = − ig ( x | (cid:0) P + i(cid:15) (cid:1) ac ( p j ¯ ψ − i∂ j ¯ ψ ) γ j (cid:54) p γ i t c P + i(cid:15) t d ψ (cid:0) P + i(cid:15) (cid:1) db | y ) − ig ( y | (cid:0) P + i(cid:15) (cid:1) bd ¯ ψt d P + i(cid:15) t c γ i (cid:54) p γ j ( ψp j + i∂ j ψ ) (cid:0) P + i(cid:15) (cid:1) ca | x ) , (cid:104) A ai ( x ) A bj ( y ) (cid:105) Fig . = ig ( x | (cid:0) αP + i(cid:15) (cid:1) ac ¯ ψγ i (cid:54) p γ j t c P + i(cid:15) t d ψ (cid:0) P + i(cid:15) (cid:1) db | y )+ ig ( y | (cid:0) αP + i(cid:15) (cid:1) bd ¯ ψt d P + i(cid:15) t c γ j (cid:54) p γ i ψ (cid:0) P + i(cid:15) (cid:1) ca | x ) for the remaining propagators.If now the point y lies inside the shock wave we can expand the gluon and quarkpropagators around the light ray y ⊥ + s y ∗ p . It is easy to see that the expansion of thegluon fields A • given by Eq. (10.3) exceeds our twist-two accuracy so we need only expansionof quark fields which is e i p ⊥ αs ( z ∗ − y ∗ ) ψe − i p ⊥ αs ( z ∗ − y ∗ ) = ψ + 2 z ∗ − y ∗ αs p j D j ψ + ... (10.30)(and similarly for ¯ ψ and D i F • i ).It is convenient to parametrize quark contribution in the same way as the gluon one(10.21) (cid:104) A aµ ( x ) A bν ( y ) (cid:105) Fig . = (cid:104) − θ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + θ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • × (cid:104) ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ Q abµν ( x ∗ , y ∗ ; p ⊥ ) | y ⊥ ) + ( y ⊥ | ¯ Q abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs ( x − y ) ∗ | x ⊥ ) (cid:105) (10.31)– 56 –n the leading order we need only the first two terms of the expansion (10.30) which gives Q ab •• ( x ∗ , y ∗ ; p ⊥ )= − igα s (cid:90) x ∗ y ∗ dz ∗ (cid:0) [ x ∗ , z ∗ ] D j F • j ( z ∗ )[ z ∗ , y ∗ ] + 2( z − y ) ∗ αs p k [ x ∗ , z ∗ ] D k D j F • j [ z ∗ , y ∗ ] (cid:1) ab + g α s (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:104)(cid:0) p ⊥ ¯ ψ ( z ∗ ) + 2 ip j ¯ ψ ← D j ( z ∗ ) (cid:1) (cid:54) p [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ )+ 2 z ∗ − y ∗ αs p ⊥ p j ¯ ψ ← D j ( z ∗ ) (cid:54) p [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ )+ 2 z (cid:48)∗ − y ∗ αs p ⊥ p j ¯ ψ ( z ∗ ) (cid:54) p [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] D j ψ ( z (cid:48)∗ ) (cid:105) (10.32)and ¯ Q ab •• ( x ∗ , y ∗ ; p ⊥ ) == − g α s (cid:90) x ∗ y ∗ dz ∗ (cid:90) x ∗ z ∗ dz (cid:48)∗ (cid:104) ¯ ψ ( z ∗ ) (cid:54) p [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] (cid:0) ψ ( z (cid:48)∗ ) p ⊥ − iD j ψ ( z (cid:48)∗ ) p j (cid:1) − z ∗ − y ∗ αs ¯ ψ ← D j ( z ∗ ) (cid:54) p [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ ) p ⊥ p j − z (cid:48)∗ − y ∗ αs ¯ ψ ( z ∗ ) (cid:54) p [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] D j ψ ( z (cid:48)∗ ) p ⊥ p j (cid:105) (10.33)Similarly, for propagator (cid:104) A • ( x ) A i ( y ) (cid:105) one gets Q ab • i ( x ∗ , y ∗ ; p ⊥ )= − g α s (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:104) [ p j ¯ ψ ( z ∗ ) − i ¯ ψ ← D j ( z ∗ )] γ j (cid:54) p γ i [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ )+ 2( z − y ) ∗ αs p j p k ¯ ψ ← D k ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ )+ 2( z (cid:48) − y ) ∗ αs p j p k ¯ ψγ j (cid:54) p γ i [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] D k ψ ( z (cid:48)∗ ) (cid:105) (10.34) ¯ Q ab • i ( x ∗ , y ∗ ; p ⊥ )= − g α s (cid:90) x ∗ y ∗ dz ∗ (cid:90) x ∗ z ∗ dz (cid:48)∗ (cid:104) ¯ ψ ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ]( ψp j + iD j ψ )( z (cid:48)∗ ) − z ∗ − y ∗ αs ¯ ψ ← D k ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ ) p j p k − z (cid:48)∗ − y ∗ αs ¯ ψ ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] D k ψ ( z (cid:48)∗ ) p j p k (cid:105) (10.35)For the propagator (cid:104) A i ( x ) A • ( y ) (cid:105) the corresponding expressions Q abi • ( x ∗ , y ∗ ; p ⊥ ) and ¯ Q abi • ( x ∗ , y ∗ ; p ⊥ ) are obtained from Eqs. (10.34) and (10.35) by replacements in the r.h.s.’s γ j (cid:54) p γ i → γ i (cid:54) p γ j and γ i (cid:54) p γ j → γ j (cid:54) p γ i , respectively.– 57 –inally, for the propagator (cid:104) A i ( x ) A j ( y ) (cid:105) we obtain Q abij ( x ∗ , y ∗ ; p ⊥ ) = g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:104) ¯ ψ ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ )+ 2( z − y ) ∗ αs p k ¯ ψ ← D k ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ )+ 2( z (cid:48) − y ) ∗ αs p k ¯ ψγ i (cid:54) p γ j [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] D k ψ ( z (cid:48)∗ ) (cid:105) (10.36) ¯ Q abij ( x ∗ , y ∗ ; p ⊥ ) = − g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) x ∗ z ∗ dz (cid:48)∗ (cid:104) ¯ ψ ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ ) − z ∗ − y ∗ αs ¯ ψ ( z ∗ ) ← D k γ j (cid:54) p γ i [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ ) p k − z (cid:48)∗ − y ∗ αs ¯ ψ ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] D k ψ ( z (cid:48)∗ ) p k (cid:105) (10.37)For the complex conjugate amplitude we get in a similar way (cid:104) ˜ A aµ ( x ) ˜ A bν ( y ) (cid:105) Fig . = (cid:104) − θ ( y ∗ − x ∗ ) (cid:90) ∞ d − α α + θ ( x ∗ − y ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • × (cid:104) ( x ⊥ | ˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs ( x − y ) ∗ | y ⊥ ) + ( y ⊥ | e − i p ⊥ αs ( x − y ) ∗ ¯˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) | x ⊥ ) (cid:105) (10.38)where ˜ Q ab •• ( x ∗ , y ∗ ; p ⊥ )= − igα s (cid:90) x ∗ y ∗ dz ∗ (cid:0) [ x ∗ , z ∗ ] ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] − x − z ) ∗ αs [ x ∗ , z ∗ ] ˜ D k ˜ D j ˜ F • j [ z ∗ , y ∗ ] p k (cid:1) ab + g α s (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ (cid:104) ˜¯ ψ ( z ∗ ) (cid:54) p [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] (cid:0) ˜ ψ ( z (cid:48)∗ ) p ⊥ − i ˜ D j ˜ ψ ( z (cid:48)∗ ) p j (cid:1) + 2 z ∗ − x ∗ αs ˜¯ ψ ← ˜ D j ( z ∗ ) (cid:54) p [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) p ⊥ p j + 2 z (cid:48)∗ − x ∗ αs ˜¯ ψ ( z ∗ ) (cid:54) p [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ D j ˜ ψ ( z (cid:48)∗ ) p ⊥ p j (10.39) ¯˜ Q ab •• ( x ∗ , y ∗ ; p ⊥ )= − g α s (cid:90) y ∗ x ∗ dz ∗ (cid:90) z ∗ x ∗ dz (cid:48)∗ (cid:104)(cid:0) p ⊥ ˜¯ ψ ( z ∗ ) + 2 ip j ˜¯ ψ ← ˜ D j ( z ∗ ) (cid:1) (cid:54) p [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) − z ∗ − x ∗ αs p ⊥ p j ˜¯ ψ ← ˜ D j ( z ∗ ) (cid:54) p [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) − z (cid:48)∗ − x ∗ αs p ⊥ p j ˜¯ ψ ( z ∗ ) (cid:54) p [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ D j ˜ ψ ( z (cid:48)∗ ) (cid:105) (10.40)– 58 –nd ˜ Q ab • i ( x ∗ , y ∗ ; p ⊥ )= − g α s (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ (cid:104) ˜¯ ψ ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ]( ˜ ψp j + i ˜ D j ψ )+ 2 z ∗ − x ∗ αs ˜¯ ψ ← ˜ D k ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) p j p k + 2 z (cid:48)∗ − x ∗ αs ˜¯ ψ ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ D k ˜ ψ ( z (cid:48)∗ ) p j p k (cid:105) (10.41) ¯˜ Q ab • i ( x ∗ , y ∗ ; p ⊥ )= − g α s (cid:90) y ∗ x ∗ dz ∗ (cid:90) z ∗ x ∗ dz (cid:48)∗ (cid:104) ( p j ˜¯ ψ − i ˜ D j ˜¯ ψ ) γ i (cid:54) p γ j [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) − z ∗ − x ∗ αs p j p k ˜¯ ψ ← ˜ D k ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) − z (cid:48)∗ − x ∗ αs p j p k ˜¯ ψ ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ D k ˜ ψ ( z (cid:48)∗ ) (cid:105) (10.42)To get ˜ Q abi • one should again make the replacement γ j (cid:54) p γ i → γ i (cid:54) p γ j in Eq. (10.41) and toget ¯˜ Q abi • the replacement γ i (cid:54) p γ j → γ j (cid:54) p γ i in Eq. (10.42). Finally, similarly to Eq. (10.36)one obtains ˜ Q abij ( x ∗ , y ∗ ; p ⊥ ) = g αs (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ (cid:104) ˜¯ ψ ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ ψ + 2 z ∗ − x ∗ αs ˜¯ ψ ← ˜ D k ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) p k + 2 z (cid:48)∗ − x ∗ αs ˜¯ ψ ( z ∗ ) γ i (cid:54) p γ j [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ D k ˜ ψ ( z (cid:48)∗ ) p k (cid:105) (10.43) ¯˜ Q abij ( x ∗ , y ∗ ; p ⊥ ) = − g αs (cid:90) y ∗ x ∗ dz ∗ (cid:90) z ∗ x ∗ dz (cid:48)∗ (cid:104) ˜¯ ψγ j (cid:54) p γ i [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) − z ∗ − x ∗ αs p k ˜¯ ψ ← ˜ D k ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) − z (cid:48)∗ − x ∗ αs p k ˜¯ ψ ( z ∗ ) γ j (cid:54) p γ i [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ D k ˜ ψ ( z (cid:48)∗ ) (cid:105) (10.44) Assembling terms from two previous Sections we get the final result forbackground-Feynman gluon propagator in external field in the form (cid:104) A aµ ( x ) A bν ( y ) (cid:105) (10.45) = (cid:104) − θ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + θ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (cid:8) ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ × (cid:2) G abµν ( x ∗ , y ∗ ; p ⊥ ) + Q abµν ( x ∗ , y ∗ ; p ⊥ ) (cid:3) | y ⊥ ) + ( y ⊥ | ¯ Q abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs ( x − y ) ∗ | x ⊥ ) (cid:9) – 59 –or Feynman propagator and (cid:104) ˜ A aµ ( x ) ˜ A bν ( y ) (cid:105) (10.46) = (cid:104) − θ ( y ∗ − x ∗ ) (cid:90) ∞ d − α α + θ ( x ∗ − y ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (cid:8) ( x ⊥ | (cid:2) ˜ G abµν ( x ∗ , y ∗ ; p ⊥ )+ ˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) (cid:3) e − i p ⊥ αs ( x − y ) ∗ | y ⊥ ) + ( y ⊥ | e − i p ⊥ αs ( x − y ) ∗ ¯˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) | x ⊥ ) (cid:9) for the anti-Feynman propagator in the complex conjugate amplitude. Repeating the steps which lead us to Eq. (10.14) we obtain lim k → k (cid:104) A aµ ( k ) A bν ( y ) (cid:105) = − ie iky O abµν ( ∞ , y ∗ , y ⊥ ; k ) , (10.47) O abµν ( ∞ , y ∗ , y ⊥ ; k ) = G abµν ( ∞ , y ∗ , y ⊥ ; k ) + Q abµν ( ∞ , y ∗ , y ⊥ ; k ) + ¯ Q abµν ( ∞ , y ∗ , y ⊥ ; k ) where G abµν ( ∞ , y ∗ , y ⊥ ; k ) ≡ e i ( k,y ) ⊥ ( k ⊥ |G abµν ( ∞ , y ∗ ; p ⊥ ) | y ⊥ ) , Q abµν ( ∞ , y ∗ , y ⊥ ; k ) ≡ e i ( k,y ) ⊥ ( k ⊥ |Q abµν ( ∞ , y ∗ , p ⊥ ) | y ⊥ ) , ¯ Q abµν ( ∞ , y ∗ , y ⊥ ; k ) ≡ e − i ( k,y ) ⊥ ( y ⊥ | ¯ Q abµν ( ∞ , y ∗ ; p ⊥ ) | k ⊥ ) (10.48)The explicit expressions can be read from Eqs. (10.20) and (10.32) - (10.37) by taking thetransverse arguments of all fields to be y ⊥ and replacing the operators p j with k j similarlyto Eq. (10.15).Similarly, for the complex conjugate amplitude the emission vertex takes the form lim k → k (cid:104) ˜ A aµ ( x ) ˜ A bν ( k ) (cid:105) = ie − ikx ˜ O abµν ( x ∗ , ∞ , x ⊥ ; k ) , (10.49) ˜ O abµν ( x ∗ , ∞ , x ⊥ ; k ) = ˜ G abµν ( x ∗ , ∞ , x ⊥ ; k ) + ˜ Q abµν ( x ∗ , ∞ , x ⊥ ; k ) + ¯˜ Q abµν ( x ∗ , ∞ , x ⊥ ; k ) where ˜ G abµν ( x ∗ , ∞ , x ⊥ ; k ) ≡ e − i ( k,x ) ⊥ ( x ⊥ | ˜ G abµν ( x ∗ , ∞ ; p ⊥ ) | k ⊥ ) , ˜ Q abµν ( x ∗ , ∞ , x ⊥ ; k ) ≡ e − i ( k,x ) ⊥ ( x ⊥ | ˜ Q abµν ( x ∗ , ∞ ; p ⊥ ) | k ⊥ ) , ¯˜ Q abµν ( x ∗ , ∞ , x ⊥ ; k ) ≡ e i ( k,x ) ⊥ ( k ⊥ | ¯˜ Q abµν ( x ∗ , ∞ ; p ⊥ ) | x ⊥ ) (10.50)Again, the explicit expressions can be read from Eqs. (10.23) and (10.39) - (10.44) bytaking the transverse arguments of all fields to be x ⊥ (and replacing the operators p j with k j ) similarly to Eq. (10.17).
11 Appendix B: Propagators in the shock-wave background
In this section we consider propagators of slow fields in the background of fast fields in thecase when the characteristic transverse momenta of fast fields ( k ⊥ ) and slow fields ( l ⊥ ) are– 60 –omparable. In this case the usual rescaling of Ref. [8] applies and we can again considerthe external fields of the type A • ( x ∗ , x ⊥ ) with A i = A ∗ = 0 .Actually, since the typical longitudinal size of fast fields is σ ∗ ∼ σ (cid:48) sl ⊥ and the typicaldistances traveled by slow gluons are ∼ σsk ⊥ our formulas will remain correct if l ⊥ (cid:29) k ⊥ since the shock wave is even thinner in this case. As we discussed above, we assume thatthe support of the shock wave is thin but not infinitely thin. For our calculations we needgluon propagators with both points outside the shock wave and propagator with one pointinside and one outside. It is convenient to start from the latter case since all the necessaryformulas can be deduced from the light-cone expansion discussed in the previous Section.To illustrate this, let us again for simplicity consider scalar propagator. For simplicity we will again perform at first the calculation for “scalar propagator” ( x | P + i(cid:15) | y ) . As usual, we assume that the only nonzero component of the external field is A • and it does not depend on z • so the operator α = i ∂∂z • commutes with all backgroundfields. The propagator in the external field A • ( z ∗ , z ⊥ ) is given by Eq. (10.1) and (10.2)which can be rewritten as ( x | P + i(cid:15) | y ) = (cid:104) − iθ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + iθ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (11.1) × ( x ⊥ | e − i p ⊥ αs ( x ∗ − y ∗ ) Pexp (cid:110) igs (cid:90) x ∗ y ∗ dz ∗ e i p ⊥ αs ( z ∗ − y ∗ ) A • ( z ∗ ) e − i p ⊥ αs ( z ∗ − y ∗ ) (cid:111) | y ⊥ ) Suppose the point y lies inside the shock wave (the point x may be inside or outside of theshock wave). Since the longitudinal distances z ∗ inside the shock wave are small ( ∼ σ (cid:48) sl ⊥ )we can use the expansion (10.3) but the parameter of the expansion is now p ⊥ αs σ ∗ ∼ σ (cid:48) α (cid:28) rather than twist of the operator. Consequently, the last term in Eq. (10.3) can be neglectedsince it has an extra factor p ⊥ αs σ ∗ in comparison to the second term: e i p ⊥ αs ( z ∗ − y ∗ ) A • e − i p ⊥ αs ( z ∗ − y ∗ ) = A • − z ∗ − y ∗ αs (2 p i F • i − iD i F • i ) − z ∗ − y ∗ ) α s ( p i p j − ip j D i ) D j F • i + ... = A • − z ∗ − y ∗ αs (2 p i F • i − iD i F • i ) + ... (11.2)This is again the expansion around the light ray y ⊥ + s y ∗ p but now with the parameterof the expansion ∼ p ⊥ αs σ ∗ (cid:28) . However, we need to keep the second term of this expansionsince the first term forms gauge links (for example, it is absent in the A • = 0 gauge).Since there are no new terms in the expansion (11.2) in comparison to (10.3) we canlook at the final result (10.5) for O ( x ∗ , y ∗ ; p ⊥ ) and drop the terms which are small with– 61 –espect to our new power counting. This way the Eq. (10.5) reduces to O ( x ∗ , y ∗ ; p ⊥ )= [ x ∗ , y ∗ ] − igαs (cid:90) x ∗ y ∗ dz ∗ ( z − y ) ∗ (cid:8) p j [ x ∗ , z ∗ ] F • j ( z ∗ ) − i [ x ∗ , z ∗ ] D j F • j ( z ∗ ) (cid:9) [ z ∗ , y ∗ ]+ 8 ig αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ( z (cid:48) − y ) ∗ [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] + ... (11.3)and the propagator has the form (10.6) ( x | P + i(cid:15) | y ) = (cid:104) − iθ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + iθ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) (11.4) × e − iα ( x − y ) • ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ O ( x ∗ , y ∗ ; p ⊥ ) | y ⊥ ) As we mentioned, this formula is correct for the point y inside the shock wave and the point x inside or outside.Similarly, for the complex conjugate amplitude we obtain the propagator in the form(10.12) with ˜ O ( x ∗ , y ∗ ; p ⊥ )= [ x ∗ , y ∗ ] + 2 igαs (cid:90) y ∗ x ∗ dz ∗ ( z − x ) ∗ [ x ∗ , z ∗ ] (cid:8) F • j ( z ∗ )[ z ∗ , y ∗ ] p j + i ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] (cid:9) − ig αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ( z − x ) ∗ [ x ∗ , z ∗ ] ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] ˜ F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] + ... (11.5)which is the expansion (10.11) but with fewer number of terms. Again, the formula (10.12)with ˜ O ( x ∗ , y ∗ ; p ⊥ ) given by the above expression is correct for the point x inside the shockwave and the point y inside or outside.The expressions for particle production are the same as (10.14) and (10.16) with O ( ∞ , y ∗ , y ⊥ ; k ) and ˜ O ( x ∗ , ∞ , x ⊥ ; k ) changed to Eqs. (11.3) and (11.5), respectively. As we saw in previous Section, the gluon propagator with one point in the shock wave canbe obtained in the same way as the propagator near the light cone, only the parameter ofthe expansion is different: p ⊥ αs σ ∗ rather than the twist of the operator. Careful inspectionof the expansions (10.19) and (10.30) reveals that there is no leading or next-to-leadingterms with twist larger than four so we can recycle the final formulas (10.45) and (10.46)for gluon propagators. At p ⊥ αs σ ∗ (cid:28) the expression (10.20) for G µν ( x ∗ , y ∗ ; p ⊥ ) turns to G µν ( x ∗ , y ∗ ; p ⊥ ) = (11.6) = g µν [ x ∗ , y ∗ ] + g (cid:90) x ∗ y ∗ dz ∗ (cid:16) − iαs ( z − y ) ∗ g µν (cid:8) p j [ x ∗ , z ∗ ] F • j ( z ∗ ) − i [ x ∗ , z ∗ ] D j F • j ( z ∗ ) (cid:9) + 4 αs ( δ jµ p ν − δ jν p µ )[ x ∗ , z ∗ ] F • j ( z ∗ ) (cid:17) [ z ∗ , y ∗ ]+ 8 g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:2) ig µν ( z (cid:48) − y ) ∗ − αs p µ p ν (cid:3) [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] – 62 –ooking at quark formulas (10.32) - (10.37) we see that at p ⊥ αs σ ∗ (cid:28) the only survivingterms are the first terms in the r.h.s’s of these equations. Let us compare now the size ofthese terms to the gluon contribution (11.6). The “power counting” for external quark fieldsin comparison to gluon ones is g s (cid:82) dz ∗ ¯ ψ (cid:54) p ψ ( z ∗ ) ∼ gs (cid:82) dz ∗ D i F • i ( z ∗ ) ∼ l ⊥ U ∼ l ⊥ and eachextra integration inside the shock wave brings extra σ ∗ . The first lines in r.h.s.’s of Eqs.(10.34) and (10.35) are of order of g p i α s (cid:82) dz ∗ ¯ ψ (cid:54) p ψ ( z ∗ ) ∼ g p i l ⊥ α s σ ∗ so they can be neglectedin comparison to the corresponding term gαs (cid:82) dz ∗ F • i ( z ∗ ) ∼ g l i α in Eq. (11.6). As to theterms (10.36) and (10.37), they are of the same odrer of magnitude as next-to-leading terms ∼ g µν in Eq. (11.6) so we keep them for now. With these approximations we obtain Q abµν ( x ∗ , y ∗ ; p ⊥ ) = − igα s p µ p ν (cid:90) x ∗ y ∗ dz ∗ ([ x ∗ , z ∗ ] D j F • j ( z ∗ )[ z ∗ , y ∗ ]) ab (11.7) + g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ¯ ψ ( z ∗ )[ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] γ ⊥ µ (cid:54) p γ ⊥ ν ψ ( z (cid:48)∗ ) , ¯ Q abµν ( x ∗ , y ∗ ; p ⊥ ) = − g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) x ∗ z ∗ dz (cid:48)∗ ¯ ψγ ⊥ ν (cid:54) p γ ⊥ µ [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ψ ( z (cid:48)∗ ) and the gluon propagator is given by Eq. (10.45) with the above G µν , Q µν , and ¯ Q µν : (cid:104) A aµ ( x ) A bν ( y ) (cid:105) (11.8) = (cid:104) − θ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + θ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (cid:8) ( x ⊥ | e − i p ⊥ αs ( x − y ) ∗ × (cid:2) G abµν ( x ∗ , y ∗ ; p ⊥ ) + Q abµν ( x ∗ , y ∗ ; p ⊥ ) (cid:3) | y ⊥ ) + ( y ⊥ | ¯ Q abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs ( x − y ) ∗ | x ⊥ ) (cid:9) As in the scalar case, it is easy to see that Eq. (11.8) holds true if the point y is inside theshock wave and the point x anywhere.Similarly, in the complex conjugate amplitude the gluon propagator is given by Eq.(10.46) with ˜ G µν ( x ∗ , y ∗ ; p ⊥ ) (11.9) = g µν [ x ∗ , y ∗ ] + g (cid:90) y ∗ x ∗ dz ∗ [ x ∗ , z ∗ ] (cid:16) iαs ( z − x ) ∗ g µν (cid:8) F • j ( z ∗ )[ z ∗ , y ∗ ] p j + i ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] (cid:9) − αs ( δ jµ p ν − δ jν p µ ) ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] (cid:17) + 8 g αs (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ [ x ∗ , z ∗ ] (cid:16)(cid:2) − ig µν ( z − x ) ∗ − αs p µ p ν (cid:3) ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] ˜ F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] (cid:17) and ˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) = 4 igα s p µ p ν (cid:90) y ∗ x ∗ dz ∗ (cid:0) [ x ∗ , z ∗ ] ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] (cid:1) ab (11.10) + g αs (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ ˜¯ ψ ( z ∗ ) γ ⊥ µ (cid:54) p γ ⊥ ν [ z ∗ , x ∗ ] t a [ x ∗ , y ∗ ] t b [ y ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) , ¯˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) = − g αs (cid:90) y ∗ x ∗ dz ∗ (cid:90) z ∗ x ∗ dz (cid:48)∗ ˜¯ ψγ ⊥ ν (cid:54) p γ ⊥ µ [ z ∗ , y ∗ ] t b [ y ∗ , x ∗ ] t a [ x ∗ , z (cid:48)∗ ] ˜ ψ ( z (cid:48)∗ ) – 63 –he expressions (11.9) and (11.10) are valid for point x inside the shock wave (and point y inside or outside).The corresponding expressions for the Lipatov vertex of gluon production are given byEqs. (10.47) -(10.50) with G , Q , and ¯ Q changed accordingly. In this section we will find the propagators with both points outside the shock wave. Again,we assume that the characteristic shock-wave transverse momenta are of order of transversemomenta of “quantum” fields with α > σ (cid:48) . As discussed in Sect. 2, we consider the widthof the shock-wave to be small but finite, consequently we can not recycle formulas fromRef. [8] for the infinitely thin shock-wave.
As in the previous Section, for simplicity we start with the scalar propagator (10.1) ( x | P + i(cid:15) | y ) x ∗ >y ∗ = − i (cid:90) ∞ d − α α e − iα ( x − y ) • ( x ⊥ | Pexp (cid:8) − i (cid:90) x ∗ y ∗ dz ∗ (cid:2) p ⊥ αs − gs A • ( z ∗ ) (cid:3)(cid:9) | y ⊥ ) (11.11)The Pexp in the r.h.s. of Eq. (11.11) can be transformed to ( x ⊥ | e − i p ⊥ αs x ∗ Pexp (cid:110) ig (cid:90) x ∗ y ∗ d s z ∗ e i p ⊥ αs z ∗ A • ( z ∗ ) e − i p ⊥ αs z ∗ (cid:111) e i p ⊥ αs y ∗ | y ⊥ ) = (cid:90) d z ⊥ d z (cid:48)⊥ (11.12) × ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ )( z ⊥ | Pexp (cid:110) ig (cid:90) x ∗ y ∗ d s z ∗ e i p ⊥ αs z ∗ A • ( z ∗ ) e − i p ⊥ αs z ∗ (cid:111) | z (cid:48)⊥ )( z (cid:48)⊥ | e i p ⊥ αs y ∗ | y ⊥ ) Next, we use the expansion (11.2) at y ∗ = 0 e i p ⊥ αs z ∗ A • e − i p ⊥ αs z ∗ = A • − z ∗ αs (2 p i F • i − iD i F • i ) + ... (11.13)This is an expansion around the light cone z ⊥ + s z ∗ p with the parameter of the expansion ∼ p ⊥ αs σ ∗ (cid:28) . Note that similarly to Eq. (11.2) we need to keep the second term of thisexpansion since the first term forms gauge links.From Eqs. (10.4), and (10.5) we obtain (cf. Eq. (10.6)) ( x | P + i(cid:15) | y ) = (cid:104) − iθ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + iθ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (11.14) × (cid:90) d z ⊥ ( x ⊥ | e − i p ⊥ αs x ∗ O ( x ∗ , y ∗ ; p ⊥ ) | z ⊥ )( z ⊥ | e i p ⊥ αs y ∗ | y ⊥ ) where O ( x ∗ , y ∗ ; p ⊥ )= [ x ∗ , y ∗ ] − igαs (cid:90) x ∗ y ∗ dz ∗ z ∗ (cid:0) p j [ x ∗ , z ∗ ] F • j ( z ∗ ) − i [ x ∗ , z ∗ ] D j F • j ( z ∗ ) (cid:1) [ z ∗ , y ∗ ]+ 8 ig αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ z (cid:48)∗ [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] + ... (11.15)– 64 –ere the transverse arguments of all fields turn effectively to z ⊥ . Note that this expressionis equal to Eq. (10.5) at y ∗ = 0 . For the complex conjugate amplitude one obtains (cf. Eq.(10.12)) ( x | P − i(cid:15) | y ) = (cid:104) iθ ( y ∗ − x ∗ ) (cid:90) ∞ d − α α − iθ ( x ∗ − y ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (11.16) × (cid:90) d z ⊥ ( x ⊥ | e − i p ⊥ αs x ∗ ˜ O ( x ∗ , y ∗ ; p ⊥ ) | z ⊥ )( z ⊥ | e i p ⊥ αs y ∗ | y ⊥ ) where ˜ O ( x ∗ , y ∗ ; p ⊥ ) = [ x ∗ , y ∗ ] + 2 igαs (cid:90) y ∗ x ∗ dz ∗ z ∗ [ x ∗ , z ∗ ](2 ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] p j + i ˜ D j ˜ F • j ( z ∗ )[ z ∗ , y ∗ ]) − ig αs (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ z ∗ [ x ∗ , z ∗ ] ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] z ˜ F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] (11.17)Again, this expression can be obtained from Eq. (10.11) by taking x ∗ → in parentheses. Similarly to Eq. (10.14) we get lim k → k ( k | P + i(cid:15) | y ) = (cid:90) dx ∗ d x ⊥ e i k ⊥ αs x ∗ − i ( k,x ) ⊥ (11.18) × (cid:2) ∂∂x ∗ − iαs ∂ x ⊥ (cid:3) θ ( x − y ) ∗ (cid:90) d z ⊥ ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ )( z ⊥ |O ( x ∗ , y ∗ ; p ⊥ ) e i p ⊥ αs y ∗ | y ⊥ ) e iαy • = (cid:90) dx ∗ ∂∂x ∗ θ ( x − y ) ∗ ( k ⊥ |O ( x ∗ , y ∗ ; p ⊥ ) e i p ⊥ αs y ∗ | y ⊥ ) e iαy • = (cid:90) d z ⊥ e − i ( k,z ) ⊥ O ( ∞ , y ∗ ; z ⊥ ; k )( z ⊥ | e i p ⊥ αs y ∗ | y ⊥ ) e iαy • where O ( ∞ , y ∗ , z ⊥ ; k ) is obtained from Eq. (11.15) O ( ∞ , y ∗ , z ⊥ ; k ) ≡ e i ( k,z ) ⊥ ( k ⊥ |O ( ∞ , y ∗ ; p ⊥ ) | z ⊥ ) (11.19) = [ ∞ , y ∗ ] z − igαs (cid:90) ∞ y ∗ dz ∗ z ∗ [ ∞ , z ∗ ] z (2 k j F • j − iD j F • j )( z ∗ , z ⊥ )[ z ∗ , y ∗ ] z + 8 ig αs (cid:90) ∞ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ z (cid:48)∗ [ ∞ , z ∗ ] z F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , y ∗ ] z For the complex conjugate amplitude we get lim k → k ( x | P − i(cid:15) | k ) = (cid:90) dy ∗ d y ⊥ e − i k ⊥ αs y ∗ + i ( k,y ) ⊥ (11.20) × (cid:2) ∂∂y ∗ + iαs ∂ y ⊥ (cid:3) θ ( y − x ) ∗ (cid:90) d z ⊥ ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ )( z ⊥ | ˜ O ( x ∗ , y ∗ ; p ⊥ ) e i p ⊥ αs y ∗ | y ⊥ ) e − iαx • = (cid:90) dy ∗ ∂∂y ∗ θ ( y − x ) ∗ ( x ⊥ | e − i p ⊥ αs x ∗ ˜ O ( x ∗ , y ∗ ; p ⊥ ) | k ⊥ ) e − iαx • = (cid:90) d z ⊥ e i ( k,z ) ⊥ ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ ) ˜ O ( x ∗ , ∞ ; z ⊥ ; k ) e − iαx • – 65 –here ˜ O ( ∞ , y ∗ , z ⊥ ; k ) is obtained from Eq. (11.17) in a usual way ˜ O ( x ∗ , ∞ ; z ⊥ ; k ) ≡ e − i ( k,z ) ⊥ ( z ⊥ | ˜ O ( x ∗ , ∞ ; p ⊥ ) | k ⊥ ) (11.21) = [ x ∗ , ∞ ] z + 2 igαs (cid:90) ∞ x ∗ dz ∗ z ∗ [ x ∗ , z ∗ ] z (2 k j ˜ F • j + i ˜ D j ˜ F • j )( z ∗ , z ⊥ )[ z ∗ , ∞ ] z − ig αs (cid:90) ∞ x ∗ dz ∗ (cid:90) ∞ z ∗ dz (cid:48)∗ z ∗ [ x ∗ , z ∗ ] z ˜ F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z ˜ F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , ∞ ] z The gluon propagator in a background gluon field (10.18) can be rewritten as i (cid:104) A aµ ( x ) A bν ( y ) (cid:105) = ( x | P + 2 igF + i(cid:15) | y ) abµν x ∗ >y ∗ = − i (cid:90) ∞ d − α α e − iα ( x − y ) • (11.22) × ( x ⊥ | e − i p ⊥ αs x ∗ Pexp (cid:110) igs (cid:90) x ∗ y ∗ dz ∗ e i p ⊥ αs z ∗ (cid:0) A • + iα F (cid:1) ( z ∗ ) e − i p ⊥ αs z ∗ (cid:111) µν e i p ⊥ αs y ∗ | y ⊥ ) ab Using the expansion (10.19) at y ∗ = 0 we obtain with our accuracy e i p ⊥ αs z ∗ (cid:0) A • g µν + iα F µν (cid:1) e − i p ⊥ αs z ∗ (11.23) = g µν (cid:104) A • − z ∗ αs (2 p j F • j − iD j F • j ) (cid:105) + iα F µν + i z ∗ α s (cid:16) p j D j F µν − iD j D j F µν (cid:17) + ... Similarly to Eq. (10.21) we get ( x | P + 2 igF + i(cid:15) | y ) abµν = (cid:104) − iθ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + iθ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • × (cid:90) d z ⊥ ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ )( z ⊥ |G abµν ( x ∗ , y ∗ ; p ⊥ ) e i p ⊥ αs y ∗ | y ⊥ ) (11.24)where G µν ( x ∗ , y ∗ ; p ⊥ ) (11.25) = g µν [ x ∗ , y ∗ ] + g (cid:90) x ∗ y ∗ dz ∗ (cid:16) − iαs z ∗ g µν (cid:0) p j [ x ∗ , z ∗ ] F • j ( z ∗ ) − i [ x ∗ , z ∗ ] D j F • j ( z ∗ ) (cid:1) + 4 αs ( δ jµ p ν − δ jν p µ ) (cid:8) [ x ∗ , z ∗ ] F • j ( z ∗ ) + 2 iz ∗ αs p k [ x ∗ , z ∗ ] D k F • j ( z ∗ ) (cid:9)(cid:17) [ z ∗ , y ∗ ]+ 8 g αs (cid:90) x ∗ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:2) ig µν z (cid:48)∗ − αs p µ p ν (cid:3) [ x ∗ , z ∗ ] F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] + ... Let us consider now quark terms coming from Fig. 7. From Eq. (10.24) it is clear thatthis contribution can be parameterized similarly to Eq. (10.31): (cid:104) A aµ ( x ) A bν ( y ) (cid:105) Fig . (11.26) = (cid:104) − θ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + θ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (cid:90) d z ⊥ (cid:104) ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ ) × ( z ⊥ |Q abµν ( x ∗ , y ∗ ; p ⊥ ) e i p ⊥ αs y ∗ | y ⊥ ) + ( y ⊥ | e i p ⊥ αs y ∗ | z ⊥ )( z ⊥ | ¯ Q abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs x ∗ | x ⊥ ) (cid:105) – 66 –here Q abµν and Q abµν are given by expressions (10.32) - (10.37) with z ∗ − y ∗ → z ∗ (andsimilarly z (cid:48)∗ − y ∗ → z (cid:48)∗ ). With our accuracy only the first terms in these expressions surviveso Q abµν and ¯ Q abµν are given by Eq. (11.7) from previous Section.Adding gluon contribution (11.24) one obtains the final expression for gluon propagatorin a shock-wave background: (cid:104) A aµ ( x ) A bν ( y ) (cid:105) = (cid:104) − θ ( x ∗ − y ∗ ) (cid:90) ∞ d − α α + θ ( y ∗ − x ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (11.27) × (cid:90) d z ⊥ (cid:104) ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ )( z ⊥ | [ G abµν ( x ∗ , y ∗ ; p ⊥ ) + Q abµν ( x ∗ , y ∗ ; p ⊥ )] e i p ⊥ αs y ∗ | y ⊥ )+ ( y ⊥ | e i p ⊥ αs y ∗ | z ⊥ )( z ⊥ | ¯ Q abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs x ∗ | x ⊥ ) (cid:105) where G abµν is given by Eq. (11.25) and Q abµν , ¯ Q abµν are given by Eq. (11.7).Similarly, for the complex conjugate amplitude one obtains (cf. Eq. (10.46)) (cid:104) ˜ A aµ ( x ) ˜ A bν ( y ) (cid:105) = (cid:104) − θ ( y ∗ − x ∗ ) (cid:90) ∞ d − α α + θ ( x ∗ − y ∗ ) (cid:90) −∞ d − α α (cid:105) e − iα ( x − y ) • (11.28) × (cid:90) d z ⊥ (cid:104) ( x ⊥ | e − i p ⊥ αs x ∗ (cid:2) ˜ G abµν ( x ∗ , y ∗ ; p ⊥ ) + ˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) (cid:3) | z ⊥ )( z ⊥ | e i p ⊥ αs y ∗ | y ⊥ )+ ( y ⊥ | e i p ⊥ αs y ∗ | z ⊥ )( z ⊥ | ¯˜ Q abµν ( x ∗ , y ∗ ; p ⊥ ) e − i p ⊥ αs x ∗ | x ⊥ ) (cid:105) where ˜ G µν ( x ∗ , y ∗ ; p ⊥ ) (11.29) = g µν [ x ∗ , y ∗ ] + g (cid:90) y ∗ x ∗ dz ∗ [ x ∗ , z ∗ ] (cid:110) iz ∗ αs g µν (2 ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] p j + i ˜ D j ˜ F • j [ z ∗ , y ∗ ]) − αs ( δ jµ p ν − δ jν p µ ) (cid:0) ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] + 2 i z ∗ αs ˜ D l ˜ F • j ( z ∗ )[ z ∗ , y ∗ ] k l (cid:1)(cid:111) + 8 g α s (cid:90) y ∗ x ∗ dz ∗ (cid:90) y ∗ z ∗ dz (cid:48)∗ [ x ∗ , z ∗ ] (cid:2) − iαg µν z ∗ − s p µ p ν (cid:3) ˜ F • j ( z ∗ )[ z ∗ , z (cid:48)∗ ] ˜ F j • ( z (cid:48)∗ )[ z (cid:48)∗ , y ∗ ] and ˜ Q abµν , ¯˜ Q abµν are given by Eq. (11.10). Note that the transverse coordinates of all fieldsare effectively z ⊥ . Similarly to Eq. (11.18) one obtains from Eq. (11.27) lim k → k i (cid:104) A aµ ( k ) A bν ( y ) (cid:105) = (cid:90) d z ⊥ e − i ( k,z ) ⊥ O abµν ( ∞ , y ∗ , z ⊥ ; k )( z ⊥ | e i p ⊥ αs y ∗ | y ⊥ ) e iαy • (11.30)– 67 –here O abµν ( ∞ , y ∗ , z ⊥ ; k ) is given by Eqs. (10.47)-(10.48). With our accuracy we get O abµν ( ∞ , y ∗ , z ⊥ ; k ) = g µν [ ∞ , y ∗ ] abz + g (cid:90) ∞ y ∗ dz ∗ (cid:16) [ ∞ , z ∗ ] z (cid:2) − iz ∗ αs g µν (2 k j − iD j ) + 4 αs ( δ jµ p ν − δ jν p µ ) (cid:3) F • j ( z ∗ , z ⊥ )[ z ∗ , y ∗ ] z (cid:17) ab + 4 gα s (cid:90) ∞ y ∗ dz ∗ (cid:110) − ip µ p ν [ ∞ , z ∗ ] z D j F • j ( z ∗ , z ⊥ )[ z ∗ , y ∗ ] z + g (cid:90) z ∗ y ∗ dz (cid:48)∗ (cid:2) iαg µν z (cid:48)∗ − s p µ p ν (cid:3) [ ∞ , z ∗ ] z F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , y ∗ ] z (cid:111) ab + g αs (cid:110)(cid:90) ∞ y ∗ dz ∗ (cid:90) z ∗ y ∗ dz (cid:48)∗ ¯ ψ ( z ∗ , z ⊥ )[ z ∗ , ∞ ] z t a [ ∞ , y ∗ ] z t b [ y ∗ , z (cid:48)∗ ] z γ ⊥ µ (cid:54) p γ ⊥ ν ψ ( z (cid:48)∗ , z ⊥ ) − (cid:90) ∞ y ∗ dz ∗ (cid:90) ∞ z ∗ dz (cid:48)∗ ¯ ψ ( z ∗ , z ⊥ ) γ ⊥ ν (cid:54) p γ ⊥ µ [ z ∗ , y ∗ ] z t b [ y ∗ , ∞ ] z t a [ ∞ , z (cid:48)∗ ] z ψ ( z (cid:48)∗ , z ⊥ ) (cid:111) (11.31)For the gluon emission in the complex conjugate amplitude one obtains (cf. Eq. (11.20) − lim k → k i (cid:104) ˜ A aµ ( x ) ˜ A bν ( k ) (cid:105) = (cid:90) d z ⊥ e i ( k,z ) ⊥ ( x ⊥ | e − i p ⊥ αs x ∗ | z ⊥ ) ˜ O abµν ( x ∗ , ∞ , z ⊥ ; k ) e − iαx • (11.32)where ˜ O abµν ( ∞ , y ∗ , z ⊥ ; k ) is given by Eqs. (10.49)-(10.50). With our accuracy ˜ O abµν ( x ∗ , ∞ , z ⊥ ; k ) = g µν [ x ∗ , ∞ ] abz (11.33) + g (cid:90) ∞ x ∗ dz ∗ (cid:16) [ x ∗ , z ∗ ] z (cid:110) iz ∗ αs g µν (2 k j + i ˜ D j ) ˜ F • j ( z ∗ , z ⊥ )[ z ∗ , ∞ ] z − αs ( δ jµ p ν − δ jν p µ ) (cid:0) ik l z ∗ αs ˜ D l (cid:1) ˜ F • j ( z ∗ , z ⊥ )[ z ∗ , ∞ ] z (cid:111)(cid:17) ab + 4 gα s (cid:90) ∞ x ∗ dz ∗ (cid:16) [ x ∗ , z ∗ ] z (cid:110) ip µ p ν ˜ D j ˜ F • j ( z ∗ , z ⊥ )[ z ∗ , ∞ ] z + g (cid:90) ∞ z ∗ dz (cid:48)∗ (cid:2) − iαg µν z ∗ − s p µ p ν (cid:3) ˜ F • j ( z ∗ , z ⊥ )[ z ∗ , z (cid:48)∗ ] z ˜ F j • ( z (cid:48)∗ , z ⊥ )[ z (cid:48)∗ , ∞ ] z (cid:111)(cid:17) ab + g αs (cid:110)(cid:90) ∞ x ∗ dz ∗ (cid:90) ∞ z ∗ dz (cid:48)∗ ˜¯ ψ ( z ∗ , z ⊥ )[ z ∗ , x ∗ ] z t a [ x ∗ , ∞ ] z t b [ ∞ , z (cid:48)∗ ] z γ ⊥ µ (cid:54) p γ ⊥ ν ˜ ψ ( z (cid:48)∗ , z ⊥ ) − (cid:90) ∞ x ∗ dz ∗ (cid:90) z ∗ x ∗ dz (cid:48)∗ ˜¯ ψ ( z ∗ , z ⊥ ) γ ⊥ ν (cid:54) p γ ⊥ µ [ z ∗ , ∞ ] z t b [ ∞ , x ∗ ] z t a [ x ∗ , z (cid:48)∗ ] z ˜ ψ ( z (cid:48)∗ , z ⊥ ) (cid:111) . 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