Gauge-invariant TMD factorization for Drell-Yan hadronic tensor at small x
PPrepared for submission to JHEP
JLAB-THY-20-3294
Gauge-invariant TMD factorization for Drell-Yanhadronic tensor at small x I. Balitsky
Physics Department, Old Dominion University, Norfolk, VA 23529, USA and Thomas JeffersonNational Accelerator Facility, Newport News, VA 23606, USA
E-mail: [email protected]
Abstract:
The Drell-Yan hadronic tensor for electromagnetic (EM) current is calculatedin the Sudakov region s (cid:29) Q (cid:29) q ⊥ with Q accuracy, first at the tree level and then withthe double-log accuracy. It is demonstrated that in the leading order in N c the higher-twistquark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion.The resulting tensor for unpolarized hadrons is EM gauge-invariant and depends on twoleading-twist TMDs: f responsible for total DY cross section, and Boer-Mulders function h ⊥ . The order-of-magnitude estimates of angular distributions for DY process seem toagree with LHC results at corresponding kinematics. a r X i v : . [ h e p - ph ] D ec ontents q ⊥ (cid:28) Q . 84.3 Power expansion of classical quark fields 9 s (cid:29) Q (cid:29) q ⊥ Q terms from J µAB ( x ) J νBA (0) J µAB ( x ) J νBA (0) Ξ Ξ Ξ and Ξ g µν f h ⊥ ¯Ξ and Ξ f ¯ f h ⊥ ¯ h ⊥ J µA ( x ) J νB (0) terms 35 W L W ∆ W T W ∆∆ W i ( q ) at q ⊥ (cid:29) m W T W L W ∆ W ∆∆ W F terms 509.2 Matching for W H terms 509.2.1 Rapidity-only cutoff for TMD 52
10 Conclusions and outlook 5211 Appendix 53 σ -matrices 5311.1.3 Formulas with γ -matrices and one gluon field 5411.1.4 Formulas with γ -matrices and two gluon fields 5511.2 Parametrization of leading-twist matrix elements 5611.3 Matrix elements of quark-quark-gluon operators 5811.4 Parametrization of TMDs from section 7.1 6111.5 Gluon power corrections from J µA ( x ) J Aµ (0) terms 628 The Drell-Yan (DY) process of production of lepton pairs with large invariant mass inhadronic collisions [1] is one of the most important tools to study QCD. From experimentalviewpoint, it is a unique source of information about partonic structure of hadrons [2]. Onthe theoretical side, it serves as a testing ground for factorization approaches in variouskinematics regions, like the classical collinear factorization [3–8], TMD factorization [9–13],and SCET [14–17].The differential cross section of DY process is determined by the product of leptonictensor and hadronic tensor. The hadronic tensor W µν is defined as W µν ( p A , p B , q ) def = 1(2 π ) (cid:88) X (cid:90) d x e − iqx (cid:104) p A , p B | J µ ( x ) | X (cid:105)(cid:104) X | J ν (0) | p A , p B (cid:105) = 1(2 π ) (cid:90) d x e − iqx (cid:104) p A , p B | J µ ( x ) J ν (0) | p A , p B (cid:105) . (1.1)– 1 –here p A , p B are hadron momenta, q is the momentum of DY pair, (cid:80) X denotes the sumover full set of “out” states and J µ is either electromagnetic or Z -boson current. In thispaper I consider only the case of electromagnetic current, the Z-boson case will be studied ina separate publication. For unpolarized hadrons, the hadronic tensor W µν is parametrizedby 4 functions, for example in Collins-Soper frame [18] W µν = − (cid:0) g µν − q µ q ν q (cid:1) ( W T + W ∆∆ ) − X µ X ν W ∆∆ + Z µ Z ν ( W L − W T − W ∆∆ ) − ( X µ Z ν + X ν Z µ ) W ∆ (1.2)where X , Z are unit vectors orthogonal to q and to each other (their explicit form ispresented in Sect. 8.2).Conventionally, the analysis of hadronic tensor (1.1) in the Sudakov region q ≡ Q (cid:29) q ⊥ is performed by using TMD factorization. For example, functions W T and W ∆∆ can berepresented in a standard TMD-factorized way [9, 19] W i = (cid:88) flavors e f (cid:90) d k ⊥ D ( i ) f/A ( x A , k ⊥ ) D ( i ) f/B ( x B , q ⊥ − k ⊥ ) C i ( q, k ⊥ )+ power corrections + Y − terms (1.3)where D f/A ( x A , k ⊥ ) is the TMD density of a parton f in hadron A with fraction of momen-tum x A and transverse momentum k ⊥ , D f/B ( x B , q ⊥ − k ⊥ ) is a similar quantity for hadron B , and coefficient functions C i ( q, k ) are determined by the cross section σ ( f f → µ + µ − ) ofproduction of DY pair of invariant mass q in the scattering of two partons.There is, however, a problem with Eq. (1.3) for the functions W L and W ∆ . The reasonis that while W T and W ∆∆ are determined by leading-twist quark TMDs, W L and W ∆ startfrom terms q ⊥ Q and ∼ q ⊥ Q determined by quark-quark-gluon TMDs. The power corrections ∼ q ⊥ Q were found in Ref. [20] more than two decades ago but there was no calculation ofpower corrections ∼ q ⊥ Q until recently. Also, the leading-twist contribution is not gaugeinvariant. It is well known from DVCS studies that check of EM gauge invariancesometimes involves cancellation of contributions of different twists (see e.g. [21–27]) so thefact that we need power corrections to check q µ W µν = 0 should not come as a surprise.Still, the absence of gauge invariance may cause discomfort in practical applications of TMDfactorization.In a recent paper [28] A. Tarasov and the author calculated power corrections ∼ q ⊥ Q tototal DY cross section production which are determined by quark-quark-gluon operators.In this paper I present the result of calculation of symmetric part of W µν ( q ) for unpolarizedhadrons at large s (cid:29) Q (cid:29) q ⊥ relevant for DY experiments at LHC. The method ofcalculation is based on the rapidity factorization approach developed in Refs. [28, 29]. Thecalculations will be performed in the leading order in perturbation theory, first at the treelevel and then in the double-logarithmic approximation for coefficient functions C i ( q, k ) . Inthis paper I consider only the production of leptons by virtual photon and leave the case ofZ-boson production for future publication. Hereafter gauge invariance of hadronic tensor means electromagnetic (EM) gauge invariance, namelythat q µ W µν = 0 . – 2 –o find all functions in Eq. (1.2) we need to have gauge-invariant expression for W µν in terms of TMDs. As noted above, only W T and W ∆∆ come from leading-twist quark-antiquark TMD while two other structures come from higher-twist quark-antiquark-gluonTMDs. Fortunately, in the leading order in N c the latter are related to the former by QCDequations of motion ([28], see also Ref. [20]). Moreover, in the small- x region x A , x B (cid:28) all structures can be expressed by just two leading-twist TMDs - f ( x, k ⊥ ) (responsible forthe total cross section) and h ⊥ ( x, k ⊥ ) (the Boer-Mulders function [30]). The results for fourfunctions in Eq. (1.2), presented in next Section, are of the type of Eq. (1.3) with TMDs f ( x, k ⊥ ) and/or h ⊥ ( x, k ⊥ ) and tree-level coefficient functions constructed of q and k ⊥ .The paper is organized as follows. In section 2 I present the resulting gauge-invariantexpression for W µν up to Q terms which is calculated in the rest of the paper. In section3 the TMD factorization is derived from the rapidity factorization of the double functionalintegral for a cross section of particle production. In section 4 I explain the method of calcu-lation of power corrections based on approximate solution of classical Yang-Mills equations.Using this method, DY hadronic tensor for small x is calculated in Sections 5, 6, and 7.Section 8 contains results of calculations and order-of-magnitude estimate of angular coef-ficients of DY cross section. The matching of obtained TMDs and coefficient functions C i in the double-log approximation is discussed in Sect. 9 and the last section 10 is devotedto conclusions and outlook. The necessary technical details can be found in appendices. To set up the stage, in this Section I present the final result for tree-level DY hadronictensor. It is determined by two leading-twist TMDs: the function f f ( x, k ⊥ ) responsiblefor the total DY cross section and Boer-Mulders time-odd function h ⊥ ( x, k ⊥ ) (the explicitdefinition of these functions is presented in the Appendix 11.2). The result reads W µν ( q ) = 1 N c (cid:88) f e f (cid:90) d k ⊥ (cid:104) F ( q, k ⊥ ) W Fµν ( q, k ⊥ ) + H f ( q, k ⊥ ) W Hµν ( q, k ⊥ ) (cid:105) (2.1)where e f are electric charges of quarks, q = x A p A + x B p B + q ⊥ and F f ( q, k ⊥ ) = f f (cid:0) x A , k ⊥ (cid:1) ¯ f f (cid:0) x B , ( q − k ) ⊥ (cid:1) + f f ↔ ¯ f f H f ( q, k ⊥ ) = h ⊥ (cid:0) x A , k ⊥ (cid:1) ¯ h ⊥ (cid:0) x B , ( q − k ) ⊥ (cid:1) + h ⊥ f ↔ ¯ h ⊥ f (2.2)– 3 –he gauge-invariant structures W Fµν and W Hµν are given by W Fµν ( q, k ⊥ ) = − g ⊥ µν + 2( k, q − k ) ⊥ Q g (cid:107) µν + 2 Q (cid:2) x A p Aµ k ⊥ ν + x B p Bµ ( q − k ) ⊥ ν + µ ↔ ν (cid:3) + 4 x A p Aµ p Aν Q k ⊥ + 4 x B p Bµ p Bν Q ( q − k ) ⊥ ,m W Hµν ( q, k ⊥ ) = − (cid:2) k ⊥ µ ( q − k ) ⊥ ν + k ⊥ ν ( q − k ) ⊥ µ + g ⊥ µν ( k, q − k ) ⊥ (cid:3) − g (cid:107) µν Q k ⊥ ( q − k ) ⊥ − x A (cid:2) p Aµ ( q − k ) ⊥ ν + µ ↔ ν (cid:3) k ⊥ Q − x B (cid:2) p Bµ k ⊥ ν + µ ↔ ν (cid:3) ( q − k ) ⊥ Q − x A p Aµ p Aν Q k ⊥ ( k, q − k ) ⊥ − x B p Bµ p Bν Q ( q − k ) ⊥ ( k, q − k ) ⊥ (cid:111) (2.3)where g ⊥ µν and g (cid:107) µν are transverse and longitudinal parts of metric tensor (the explicitdefinition is given by Eq. (3.2)). It is easy to check that q µ W Fµν = 0 and q µ W Hµν = 0 .As we will see below, in some of the structures the corrections to Eq. (2.1) are of order O ( x A ) and O ( x B ) while in others on the top of that there are corrections ∼ O (cid:0) N c (cid:1) timessome other higher-twist TMDs discussed in Ref. [28]. It should be also noted that W F partcoincides with the result obtained in Refs. [31, 32] using parton Reggeization approach toDY process [33].In the rest of the paper I will derive the above equations and discuss their accuracy.Let me mention upfront that since the approximations made in Eq. (2.1) are x A , x B (cid:28) and q ⊥ (cid:28) Q (cid:39) x A x B s , I hope that the results of this paper can be used for studies ofDY process at LHC with Q ∼ Last but not least, the derivation of theabove equations is lengthly so the readers interested in final formulas for structures W i andthe discussion of approximations can go directly to Sect. 8. As was mentioned in the Introduction, to find the TMD formulas of Eq. (1.3) type I use therapidity factorization approach to developed in Refs. [28, 29]. Let me quickly summarizebasic ideas of this approach. The sum over full set of “out” states in Eq. (1.1) can berepresented by a double functional integral (2 π ) W µν ( p A , p B , q ) = (cid:88) X (cid:90) d x e − iqx (cid:104) p A , p B | J µ ( x ) | X (cid:105)(cid:104) X | J ν (0) | p A , p B (cid:105) (3.1) = t f →∞ lim t i →−∞ (cid:90) d x e − iqx (cid:90) ˜ A ( t f )= A ( t f ) D ˜ A µ DA µ (cid:90) ˜ ψ ( t f )= ψ ( t f ) D ˜¯ ψD ˜ ψD ¯ ψDψ Ψ ∗ p A ( (cid:126) ˜ A ( t i ) , ˜ ψ ( t i )) × Ψ ∗ p B ( (cid:126) ˜ A ( t i ) , ˜ ψ ( t i )) e − iS QCD ( ˜ A, ˜ ψ ) e iS QCD ( A,ψ ) ˜ J µ ( x ) J ν (0)Ψ p A ( (cid:126)A ( t i ) , ψ ( t i ))Ψ p B ( (cid:126)A ( t i ) , ψ ( t i )) . The reader should not be confused by using small- x approximation at LHC with Q ∼ GeV. Oneshould distinguish between small- x approximation and small- x resummation. In the kinematics discussedin this paper x A ∼ x B ∼ . so small- x resummation of α s ln x A ( B ) is unnecessary. On the other hand, if wesee an expression like f ( x A , k ⊥ ) + x A f ⊥ ( x A , k ⊥ ) we can safely neglect the second term, see the discussionin Appendix 11.3. – 4 –here J µ = (cid:80) flavors e f ¯ ψ f γ µ ψ f is the electromagnetic current. In this double functionalintegral the amplitude (cid:104) X | J µ (0) | p A , p B (cid:105) is given by the integral over ψ, A fields whereasthe complex conjugate amplitude (cid:104) p A , p B | J µ ( x ) | X (cid:105) is represented by the integral over ˜ ψ, ˜ A fields. Also, Ψ p ( (cid:126)A ( t i ) , ψ ( t i )) denotes the proton wave function at the initial time t i and theboundary conditions ˜ A ( t f ) = A ( t f ) and ˜ ψ ( t f ) = ψ ( t f ) reflect the sum over all states X , cf.Refs. [34–36].We use Sudakov variables p = αp + βp + p ⊥ , where p and p are light-like vectorsclose to p A and p B so that p A = p + m s p and p A = p + m s p with m being the protonmass. Also, we use the notations x • ≡ x µ p µ and x ∗ ≡ x µ p µ for the dimensionless light-conecoordinates ( x ∗ = (cid:112) s x + and x • = (cid:112) s x − ). Our metric is g µν = (1 , − , − , − whichwe will frequently rewrite as a sum of longitudinal part and transverse part: g µν = g µν (cid:107) + g µν ⊥ = 2 s (cid:0) p µ p ν + p µ p ν ) + g µν ⊥ (3.2)Consequently, p · q = ( α p β q + α q β p ) s − ( p, q ) ⊥ where ( p, q ) ⊥ ≡ − p i q i . Throughout thepaper, the sum over the Latin indices i , j , ... runs over two transverse components whilethe sum over Greek indices µ , ν , ... runs over four components as usual.Following Ref. [29] we separate quark and gluon fields in the functional integral (3.1)into three sectors (see figure 1): “projectile” fields A µ , ψ A with | β | < σ p , “target” fields B µ , ψ B with | α | < σ t and “central rapidity” fields C µ , ψ C with | α | > σ t and | β | > σ p , seeFig. 1. Our goal is to integrate over central fields and get the amplitude in the factorized “Central” fields
Figure 2 . Typical diagram for the classical field with projectile/target sources. The Green func-tions of central fields are given by retarded propagators. diagrams with retarded Green functions gives fields C µ and ψ C that vanish at t → −∞ .Thus, we are solving the usual classical YM equations D ν F aµν = g (cid:88) f ¯Ψ f t a γ µ Ψ f , / P Ψ f = 0 , (4.5)where A µ = C µ + A µ + B µ , Ψ f = ψ fC + ψ fA + ψ fB , P µ ≡ i∂ µ + C µ + A µ + B µ , F µν = ∂ µ A ν − µ ↔ ν − i [ A µ , A ν ] , (4.6) We take into account only u, d, s, c quarks and consider them massless. In principle, one can include“massless” b -quark for q ⊥ (cid:29) m b . – 8 –ith boundary conditions A µ ( x ) x ∗ →−∞ = A µ ( x • , x ⊥ ) , Ψ( x ) x ∗ →−∞ = ψ A ( x • , x ⊥ ) , A µ ( x ) x • →−∞ = B µ ( x ∗ , x ⊥ ) , Ψ( x ) x • →−∞ = ψ B ( x ∗ , x ⊥ ) (4.7)following from C µ , ψ C t →−∞ → . These boundary conditions reflect the fact that at t → −∞ we have only incoming hadrons with A and B fields.As discussed in Ref. [29], for our case of particle production with q ⊥ Q (cid:28) it is possibleto find the approximate solution of (4.5) as a series in this small parameter. One solvesEqs. (4.5) iteratively, order by order in perturbation theory, starting from the zero-orderapproximation in the form of the sum of projectile and target fields A [0] µ ( x ) = A µ ( x • , x ⊥ ) + B µ ( x ∗ , x ⊥ ) , Ψ [0] ( x ) = ψ A ( x • , x ⊥ ) + ψ B ( x ∗ , x ⊥ ) (4.8)and improving it by calculation of Feynman diagrams with retarded propagators in thebackground fields (4.8).Let me now explain how the parameter m ⊥ /s comes up in the rapidity-factorizationapproach (for details, see Ref. [29]). When we expand quark and gluon propagators inpowers of background fields, we get a set of diagrams shown in figure 2. The typical baregluon propagator in figure 2 is p + i(cid:15)p = 1 αβs − p ⊥ + i(cid:15) ( α + β ) . (4.9)In the tree approximation, the transverse momenta in tree diagrams are determined byfurther integration over projectile (“A”) and target (“B”) fields in eq. (3.1) which convergeon either q ⊥ or m N . On the other hand, the integrals over α converge on either α q or α ∼ and similarly the characteristic β ’s are either β q or β ∼ . Since α q β q s = Q (cid:107) (cid:29) q ⊥ , onecan expand gluon and quark propagators in powers of p ⊥ αβs p + i(cid:15)p = 1 s ( α + i(cid:15) )( β + i(cid:15) ) (cid:16) p ⊥ /s ( α + i(cid:15) )( β + i(cid:15) ) + ... (cid:17) , (4.10) /pp + i(cid:15)p = 1 s (cid:16) /p β + i(cid:15) + /p α + i(cid:15) + /p ⊥ ( α + i(cid:15) )( β + i(cid:15) ) (cid:17)(cid:16) p ⊥ /s ( α + i(cid:15) )( β + i(cid:15) ) + ... (cid:17) . After the expansion (4.10), the dynamics in the transverse space effectively becomes trivial:all background fields stand either at x or at . Note that in this statement is solely aconsequence of Q (cid:29) q ⊥ and does not rely on small- x approximation. Now we expand the classical quark fields in powers of p ⊥ p (cid:107) ∼ m ⊥ s (the corresponding ex-pansion of classical gluon fields is presented in Ref. [29], but we do not need it here). Asdemonstrated in Ref. [28], expanding it in powers of p ⊥ /p (cid:107) we obtain Ψ( x ) = Ψ ( x ) + Ψ ( x ) + . . . , (4.11)– 9 –here Ψ = ψ A + Ξ , Ξ = − /p s γ i B i α + i(cid:15) ψ A = is σ ∗ i B i α + i(cid:15) ψ A , ¯Ψ = ¯ ψ A + ¯Ξ , ¯Ξ = − (cid:0) ¯ ψ A α − i(cid:15) (cid:1) γ i B i /p s = − is (cid:0) ¯ ψ A α − i(cid:15) (cid:1) B i σ • i Ψ = ψ B + Ξ , Ξ = − /p s γ i A i β + i(cid:15) ψ B = is σ • i A i β + i(cid:15) ψ B , ¯Ψ = ¯ ψ B + ¯Ξ , ¯Ξ = − (cid:0) ¯ ψ B β − i(cid:15) (cid:1) γ i A i /p s = − is (cid:0) ¯ ψ B β − i(cid:15) (cid:1) A i σ • i (4.12)and dots stand for terms subleading in q ⊥ Q and/or α q , β q parameters (hereafter we assumethe small- x approximation α q , β q (cid:28) in all calculations). In this formula α + i(cid:15) ψ A ( x • , x ⊥ ) ≡ − i (cid:90) x • −∞ dx (cid:48)• ψ A ( x (cid:48)• , x ⊥ ) , (cid:16) ¯ ψ A α − i(cid:15) (cid:17) ( x • , x ⊥ ) ≡ i (cid:90) x • −∞ dx (cid:48)• ¯ ψ A ( x (cid:48)• , x ⊥ ) (4.13)and similarly for β ± i(cid:15) . For brevity, in what follows we denote (cid:0) ¯ ψ A α (cid:1) ( x ) ≡ (cid:0) ¯ ψ A α − i(cid:15) (cid:1) ( x ) and (cid:0) ¯ ψ B β (cid:1) ( x ) ≡ (cid:0) ¯ ψ B β − i(cid:15) (cid:1) ( x ) . Let us estimate the relative size of corrections Ξ in Eq.(4.12) at small x . As we will see, α and β transform to α q and β q in our TMDs so Ξ ∼ ψ A m ⊥ α q √ s ∼ ψ A q ⊥ Q , Ξ ∼ ψ B m ⊥ β q √ s ∼ ψ B q ⊥ Q (4.14)if α q ∼ β q ∼ Q √ s (recall that we assume that the DY pair is emitted in the central regionof rapidity). For example, the correction ∼ [ ¯ ψ A γ µ Ξ ][ ¯ ψ B γ ν Ξ ] will be of order of q ⊥ Q incomparison to leading-twist contribution [ ¯ ψ A γ µ ψ B ][ ¯ ψ B γ ν ψ A ] . s (cid:29) Q (cid:29) q ⊥ In general, our method is applicable for calculation of power corrections at any s, Q (cid:29) q ⊥ , m N . However, the expressions are greatly simplified in the physically interesting case s (cid:29) Q (cid:29) q ⊥ which is considered in this paper.As we noted above, we take into account only hadronic tensor due to electromagneticcurrents of u, d, s, c quarks and consider these quarks to be massless. It is convenient todefine coordinate-space hadronic tensor multiplied by N c s (and denoted by extra “check”mark) as follows ˇ W µν ( p A , p B , x ) ≡ N c s (cid:104) p A , p B | J µ ( x ) J ν (0) | p A , p B (cid:105) (5.1) W µν ( p A , p B , q ) = s/ π ) N c (cid:90) d x e − iqx ˇ W µν ( p A , p B , x ) . The reader may wonder why there are no corrections ∼ q ⊥ Q coming from next terms in the expansion(4.11) like [ ¯ ψ A ( x ) γ µ ψ B ( x )][ ¯ ψ B (0) γ ν γ i s /p β α γ j ∂ i B j Ψ A (0)] . The reason is that β between ¯ ψ B (0) and B j (0) does not transform to β q and remains ∼ O (1) , see the discussion in the Appendix 8.3.4 of Ref. [28]. – 10 –or future use, let us also define the hadronic tensor in mixed representation: in momentumlongitudinal space but in transverse coordinate space W µν ( p A , p B , q ) = (cid:90) d x ⊥ e i ( q,x ) ⊥ W µν ( α q , β q , x ⊥ ) , (5.2) W µν ( α q , β q , x ⊥ ) ≡ π ) s (cid:90) dx • dx ∗ e − iα q x • − iβ q x ∗ (cid:104) p A , p B | J µ ( x • , x ∗ , x ⊥ ) J ν (0) | p A , p B (cid:105) . After integration over central fields in the tree approximation we obtain ˇ W µν ( p A , p B , x ) ≡ N c s (cid:104) A, B | J µ ( x • , x ∗ , x ⊥ ) J ν (0) | A, B (cid:105) (5.3)where J µ = J µA + J µB + J µAB + J µBA ,J µA = (cid:88) f e f ¯Ψ f γ µ Ψ f , J µAB = (cid:88) f e f ¯Ψ f γ µ Ψ f (5.4)and similarly for J µB and J µBA . Here (cid:104) A, B |O ( ψ A , A µ , ψ B , B µ ) | A, B (cid:105) denotes double func-tional integral over A and B fields which gives matrix elements between projectile andtarget states of Eq. (3.8) type.The leading-twist contribution to W µν ( q ) comes only from product J µAB ( x ) J νBA (0) (or J µBA ( x ) J νAB (0) ), while power corrections may come also from other terms like J µA ( x ) J νB (0) .We will consider all terms in turn. Q terms from J µAB ( x ) J νBA (0) Power expansion of J µAB ( x ) J νBA (0) reads ¯Ψ ( x ) γ µ Ψ ( x ) ¯Ψ (0) γ ν Ψ (0) + ... (5.5)where quark fields are given by Eq. (4.12). As we mentioned above, in Ref. [28] it isdemonstrated that terms neglected in the r.h.s. lead to power corrections ∼ q ⊥ α q s or ∼ q ⊥ β q s which are much smaller than q ⊥ α q β q s = q ⊥ Q (cid:107) ∼ q ⊥ Q (if DY pair is emitted in the central regionof rapidity). Note that since we want to calculate the leading power corrections, we cansubstitute Q (cid:107) with Q . In the limit s (cid:29) Q (cid:29) q ⊥ this change of variables can only lead toerrors of the order of subleading power terms. As to terms ∼ ¯Ψ ( x ) γ µ Ψ ( x ) ¯Ψ (0) γ ν Ψ (0) , they can be decomposed using eq. (4.12)as follows: (cid:2)(cid:0) ¯ ψ A + ¯Ξ (cid:1) ( x ) γ µ (cid:0) ψ B + Ξ (cid:1) ( x ) (cid:3) [ (cid:0) ¯ ψ B + ¯Ξ (cid:1) (0) γ ν (cid:0) ψ A + Ξ (cid:1) (0) (cid:3) + x ↔
0= [ ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) (5.6) + [¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + [ ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + [ ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + [¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + [¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + [¯Ξ ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + [ ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν Ξ (0) (cid:3) + x ↔ . Except for the leading-twist term where the difference between Q (cid:107) and Q matters. – 11 –here the square brackets mean trace over Lorentz and color indices.First, let us consider the leading-twist term coming from the first term in the r.h.s. ofthis equation. As we mentioned, the leading-twist term comes from J µAB ( x ) J νBA (0) and J µBA ( x ) J νAB (0) . Using Fierz transformation (11.1) one obtains N c s (cid:0)(cid:2) ¯ ψ A ( x • , x ⊥ ) γ µ ψ B ( x ∗ , x ⊥ ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ ↔ ν (cid:1) + x ↔ g µν s (cid:2) − ( ¯ ψ A ψ A )( ¯ ψ B ψ B ) + ( ¯ ψ A γ ψ A )( ¯ ψ B γ ψ B ) + ( ψ A γ α ψ A )( ¯ ψ B γ α ψ B )+ ( ψ A γ α γ ψ A )( ¯ ψ B γ α γ ψ B ) −
12 ( ψ A σ αβ ψ A )( ¯ ψ B σ αβ ψ B ) (cid:3) − s [( ψ A γ µ ψ A )( ¯ ψ B γ ν ψ B ) + µ ↔ ν ] − s ( ψ A γ µ γ ψ A )( ¯ ψ B γ ν γ ψ B ) + µ ↔ ν ]+ 12 s [( ψ A σ να ψ A )( ¯ ψ B σ µα ψ B ) + ( ψ A σ µα ψ A )( ¯ ψ B σ να ψ B )] + x ↔ (5.7)where all parentheses in the r.h.s. are color singlet. As usual, after integration over back-ground fields A and B we promote A , ψ A and B , ψ B to operators ˆ A , ˆ ψ . A subtle point isthat our operators are not under T-product ordering so one should be careful while chang-ing the order of operators in formulas like Fierz transformation. Fortunately, all operatorsin the r.h.s of Eq. (5.7) are separated either by space-like intervals or light-like intervals sothey commute with each other.From parametrization of two-quark operators in section 11.2, it is clear that the leading-twist contribution to W µν ( q ) comes from ˇ W lt µν = 12 s ( g µν g αβ − δ αµ δ βν − δ αν δ βµ ) (cid:104) ˆ¯ ψ ( x • , x ⊥ ) γ α ˆ ψ (0) (cid:105) A (cid:104) ˆ¯ ψ (0) γ β ˆ ψ ( x ∗ , x ⊥ ) (cid:105) B (5.8) + 12 s (cid:0) δ αµ δ βν + δ αν δ βµ − g µν g αβ (cid:1) (cid:104) ˆ¯ ψ ( x • , x ⊥ ) σ αξ ˆ ψ (0) (cid:105) A (cid:104) ˆ¯ ψ (0) σ ξβ ˆ ψ ( x ∗ , x ⊥ ) (cid:105) B + x ↔ Hereafter we use notations (cid:104)O(cid:105) A ≡ (cid:104) p A |O| p A (cid:105) and (cid:104)O(cid:105) B ≡ (cid:104) p B |O| p B (cid:105) for brevity . Thecorresponding leading-twist contribution to to W µν ( q ) has the form [40] W lt µν ( α q , β q , q ⊥ ) = 116 π N c (cid:90) dx • dx ∗ d x ⊥ e − iα q x • − iβ q x ∗ + i ( q,x ) ⊥ ˇ W lt µν ( x )= (cid:88) f e f N c (cid:90) d k ⊥ (cid:16) − g ⊥ µν (cid:2) f f ( α q , k ⊥ ) ¯ f f ( β q , q ⊥ − k ⊥ ) + ¯ f f ( α q , k ⊥ ) f f ( β q , q ⊥ − k ⊥ ) (cid:3) − (cid:2) k ⊥ µ ( q − k ) ⊥ ν + k ⊥ ν ( q − k ) ⊥ µ + g ⊥ µν ( k, q − k ) ⊥ (cid:3) × (cid:2) h ⊥ f ( α q , k ⊥ )¯ h ⊥ f ( β q , q ⊥ − k ⊥ ) + ¯ h ⊥ f ( α q , k ⊥ ) h ⊥ f ( β q , q ⊥ − k ⊥ ) (cid:3)(cid:17) (5.10) In a general gauge for projectile and target fields these matrix elements read (cid:104) p A | ˆ ψ f ( x ) γ µ ˆ ψ f (0) | p A (cid:105) = (cid:104) p A | ˆ ψ f ( x • , x ⊥ ) γ µ [ x • , −∞ • ] x [ x ⊥ , ⊥ ] −∞ • [ −∞ • , • ] ˆ ψ f (0) | p A (cid:105) , (cid:104) p B | ˆ ψ f ( x ) γ µ ˆ ψ f (0) | p B (cid:105) = (cid:104) p B | ˆ ψ f ( x ∗ , x ⊥ ) γ µ [ x ∗ , −∞ ∗ ] x [ x ⊥ , ⊥ ] −∞ ∗ [ −∞ ∗ , ∗ ] ˆ ψ f (0) | p B (cid:105) (5.9)and similarly for other operators. – 12 –et us discuss other terms proportional to different TMDs in parametrizations in Sect.11.2. To this end, we write down terms from Eq. (2.3) that we are looking for in Sudakovvariables: g µν ⊥ (cid:104) q ⊥ α q β q s (cid:105) , q µ ⊥ q ν ⊥ q ⊥ (cid:104) q ⊥ α q β q s (cid:105) , g µν (cid:107) (cid:104) q ⊥ α q β q s (cid:105) , α q s (cid:0) p µ q ν ⊥ + µ ↔ ν (cid:1) , β q s (cid:0) p µ q ν ⊥ + µ ↔ ν (cid:1) , p µ p ν β q s , p µ p ν α q s (5.11)Here zero in the third term means that the contribution of order one is actually absent. Asdiscussed in Sect. 11.2, all TMDs considered here can have only logarithmic dependenceon Bjorken x ( ≡ α q or β q ) but not the power dependence x . It is easy to see that otherquark-antiquark TMDs give contributions to W µ ( q ) which look like terms in Eq. (5.11) butwithout extra α q and/or β q so they are power suppressed in low-x regime s (cid:29) Q .Let us also specify the terms which we do not calculate. Roughly speaking, theycorrespond to terms in Eq. (5.11) multiplied by m ⊥ Q or by either α q or β q . Our strategy inthe next sections is to compare a certain term in ˇ W µν to terms in Eq. (5.11), and, if it issmaller, neglect, if it is of the same size, calculate. J µAB ( x ) J νBA (0) We separate terms in Eq. (5.6) according to number of gluon fields (contained in Ξ ’s ). ˇ W µν sym µ,ν = ˇ W lt µν + ˇ W (1) µν + ˇ W (2 a ) µν + ˇ W (2 b ) µν + ˇ W (2 c ) µν (6.1)where leading-twist terms without gluons (quark-antiquark TMDs) were considered in pre-vious Section, and ˇ W (1) µν ( x ) = N c s (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + (cid:2) ¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + (cid:2) ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + (cid:2) ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (6.2) ˇ W (2 a ) µν ( x ) = N c s (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + [¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (6.3) ˇ W (2 b ) µν ( x ) = N c s (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + (cid:2) ¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (6.4)and ˇ W (2 c ) µν ( x ) = N c s (cid:104) A, B | (cid:2) ¯Ξ ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + (cid:2) ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν Ξ (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (6.5)The corresponding contributions to W µν ( q ) will be denoted W (1) µν , W (2) aµν , W (2) bµν , and W (2) cµν ,respectively. We will consider these contributions in turn.– 13 – .1 Terms with one quark-quark-gluon operator In this section we consider terms in Eq. (6.2) which will lead to β q s p µ q ν ⊥ + µ ↔ ν and α q s p µ q ν ⊥ + µ ↔ ν contributions to W µν ( q ) . Ξ Let us start with the last term in Eq. (6.2). The Fierz transformation (11.1) yields
12 [ ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + µ ↔ ν = g µν (cid:8)(cid:2) ¯ ψ mA ( x ) (cid:54) p s γ i α ψ kA (0) (cid:3)(cid:2) ¯ ψ nB ¯ B nki (0) ψ mB ( x ) (cid:3) − ( ψ kA ⊗ ψ nB ↔ γ ψ kA ⊗ γ ψ nB (cid:9) + 14 ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) × (cid:8)(cid:2) ¯ ψ mA ( x ) γ α (cid:54) p s γ i α ψ kA (0) (cid:3)(cid:2) ¯ ψ nB ¯ B nki (0) γ β ψ mB ( x ) (cid:3) + ( γ α ⊗ γ β ↔ γ α γ ⊗ γ β γ ) (cid:9) −
14 ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) (cid:2) ¯ ψ mA ( x ) σ αξ (cid:54) p s γ i α ψ kA (0) (cid:3)(cid:2) ¯ ψ nB ¯ B nki (0) σ ξβ ψ mB ( x ) (cid:3) (6.6)where we used Eq. (4.12) Ξ (0) = − (cid:54) p s γ i ¯ B i α ψ A (0) . Note that all colors are in the funda-mental representation so e.g. B mn ( x ) ≡ ( t a ) mn B a ( x ) .Promoting A and B fields to operators and sorting out the color-singlet contributionswe get ˇ W (1)1 µν ( x ) = N c s (cid:104) A, B | [ ˆ¯ ψ A ( x ) γ µ ˆ ψ B ( x ) (cid:3)(cid:2) ˆ¯ ψ B (0) γ ν Ξ (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ g µν s (cid:8) (cid:104) ˆ¯ ψ ( x ) (cid:54) p γ i α ˆ ψ (0) (cid:105) A (cid:104) ˆ¯ ψ ¯ B i (0) ˆ ψ ( x )] (cid:105) B − ( ˆ ψ (0) ⊗ ˆ ψ ( x ) ↔ γ ˆ ψ (0) ⊗ γ ˆ ψ ( x ) (cid:9) + 12 s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) (cid:8) (cid:104) ˆ¯ ψ ( x ) γ α (cid:54) p γ i α ˆ ψ (0) (cid:105) A (cid:104) ˆ¯ ψB i (0) γ β ˆ ψ ( x ) (cid:105) B + ( ˆ ψ (0) ⊗ ˆ ψ ( x ) ↔ γ ˆ ψ (0) ⊗ γ ˆ ψ ( x ) (cid:9) (6.7) − s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) (cid:104) ˆ¯ ψ ( x ) σ αξ (cid:54) p γ i α ˆ ψ (0) (cid:105) A (cid:104) ˆ¯ ψB i (0) σ ξβ ˆ ψ ( x ) (cid:105) B + x ↔ It is convenient to treat terms ∼ g µν separately so we define ˇ W (1)1 µν ( x ) = ˇ W (1 a ) µν ( x ) +ˇ W (1 b )1 µν ( x ) where ˇ W (1 a ) µν ( x )= g µν s (cid:16)(cid:8) (cid:104) ¯ ψ ( x ) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ ¯ B i (0) ψ ( x )] (cid:105) B − ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) − (cid:8) (cid:104) ¯ ψ ( x ) γ α (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) γ α ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) + 12 (cid:104) ¯ ψ ( x ) σ αβ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ αβ ψ ( x ) (cid:105) B + x ↔ (6.8)Hereafter we omit “hat” notation from from operators: (cid:104)O(cid:105) A,B ≡ (cid:104) ˆ O(cid:105)
A,B for brevity. We will keep different notations A i and B i for the projectile and target fields because of the relations(11.10) and (11.14) – 14 –et us now estimate this contribution to ˇ W µν . First, recall that B i is of order of m ⊥ (more accurately, it will be ∼ q i after the Fourier transformation, see e.g. Eq. (11.42) orEq. (11.48)). Next, as demonstrated in Sect. 11.3 (see Eqs. (11.30), (11.31)), α in thetarget matrix element turns to ± α q after Fourier transformation. Due to this fact we willreplace α by α q in our estimates, even in the coordinate space. Similarly, for the estimateof the target matrix elements we will replaces operator β by β q wherever appropriate.Now we will demonstrate that three terms in the r.h.s. of Eq. (6.8) are small incomparison to terms listed in Eq. (5.11). The projectile matrix element in the first termin the r.h.s. of Eq. (6.8) brings factor s (see Eq. (11.29)) but the target matrix elementcan produce only factor x i so the first term is ∼ g µν m ⊥ α q s which is smaller than g µν q ⊥ Q that wehave in Eq. (5.11) (and will calculate in the next Section). As to the second term in ther.h.s. of Eq. (6.7), it can be rewritten as − (cid:8) (cid:104) ¯ ψ ( x ) γ j (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) γ j ψ ( x ) (cid:105) B + 2 s (cid:104) ¯ ψ ( x ) (cid:54) p (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) (cid:54) p ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) g µν s (6.9)The projectile matrix element in the first term in the r.h.s. of this equation brings factor s but, as we discussed above, the target matrix element cannot produce factor s so thisterm is again ∼ g µν m ⊥ α q s (cid:28) g µν q ⊥ Q . As to the second term, converting three γ -matricesin the projectile matrix element to a combination of γ ’s and γγ ’s and looking at theparametrization of Sect. 11.2, we see that s (cid:104) ¯ ψ ( x ) (cid:54) p (cid:54) p γ i α ψ (0) (cid:105) A is not proportional to s .In addition, as discuss in Sect. 11.2, the target matrix element (cid:104) ¯ ψB i (0) γ µ ψ ( x ) (cid:105) B knowsabout p only via the direction of Wilson lines so it can be proportional only to p µ p · p thatdoes not change at rescaling of p . Thus, (cid:104) ¯ ψB i (0) (cid:54) p ψ ( x ) (cid:105) B is ∼ O (1) and therefore thesecond term in Eq. (6.9) is even smaller than the first one. Finally, let us discuss the thirdterm in the r.h.s. of Eq. (6.8). If both α and β are transverse g µν s (cid:104) ¯ ψ ( x ) σ αβ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ αβ ψ ( x ) (cid:105) B ∼ g µν m ⊥ α q s (6.10)similarly to the first term in Eq. (6.9). If both indices are longitudinal, we get g µν s (cid:104) ¯ ψ ( x ) σ ∗• (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ •∗ ψ ( x ) (cid:105) B = g µν s (cid:104) ¯ ψ ( x ) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ •∗ ψ ( x ) (cid:105) B (6.11)The projectile matrix element brings a factor s , but the target one is ∼ O (1) due to thereason discussed above, so this contribution is negligible. Finally, let us consider the casewhen index α is longitudinal and β is transverse g µν s (cid:104) ¯ ψ ( x ) σ • j (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ j ∗ ψ ( x ) (cid:105) B (6.12)Again, the target matrix element is ∼ O (1) while the projectile one can bring one factorof s as can be seen from parametrization (11.29) by reducing the number of γ -matrices totwo. Thus, the contribution (6.12) is negligible and so is the total contribution (6.8).– 15 –e get ˇ W (1) µν ( x ) (cid:39) ˇ W (1 b ) µν ( x ) == 12 s (cid:8) (cid:104) ¯ ψ ( x ) γ µ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) γ ν ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) − s (cid:104) ¯ ψ ( x ) σ µξ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ξν ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ (6.13)Let us start with the case when both of the indices µ and ν are transverse. It is easy to seethat the power counting for the first term in the r.h.s. of Eq. (6.13) is the same as for Eq.(6.9) so it is small. Also, the estimate of the second term in Eq. (6.13) is similar either tothe estimate of Eq. (6.10) or (6.12) so it can be neglected.Next, let us consider both µ and ν longitudinal. It is easy to see that multiplication ofthe r.h.s. of Eq. (6.13) by p µ p ν gives zero so there is no term proportional to p µ p ν . Theterm proportional to p µ p ν has the form s p µ p ν (cid:8) (cid:104) ¯ ψ ( x ) (cid:54) p (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) (cid:54) p ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) − s p µ p ν (cid:104) ¯ ψ ( x ) σ • ξ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ξ • ψ ( x ) (cid:105) B (6.14)It is easy to see that both projectile and target matrix elements are proportional to thefirst power of s so the resulting estimate is p µ p ν α q s m ⊥ which is ∼ O (cid:0) m ⊥ s (cid:1) in comparison tothe corrseponding term in Eq. (5.11). If one index is p and the other p we get g (cid:107) µν s (cid:104) (cid:104) ¯ ψ ( x ) (cid:54) p (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) (cid:54) p ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (6.15) − (cid:104) ¯ ψ ( x ) σ • j (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ j ∗ ψ ( x ) (cid:105) B − s (cid:104) ¯ ψ ( x ) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ •∗ ψ ( x ) (cid:105) B (cid:105) It is easy to see that in all terms the projectile matrix element is ∼ s but the target one is ∼ O (1) so the corresponding contribution ∼ g µν m ⊥ s is negligible.Finally, let us consider the case when one of the indices in Eq. (6.13) is longitudinal andone transverse. For example, let µ be longitudinal and ν transverse, the opposite case willdiffer by replacement µ ↔ ν . Using the decomposition of g µν in longitudinal and transversepart (3.2) we get (cid:16) p µ p µ (cid:48) s + µ ↔ µ (cid:48) (cid:17) ˇ W (1 b ) µν ( x ) = (cid:16) p µ p µ (cid:48) s + µ ↔ µ (cid:48) (cid:17) × (cid:104)(cid:8) (cid:104) ¯ ψ ( x ) γ µ (cid:48) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) γ ν ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) − (cid:104) ¯ ψ ( x ) σ µ (cid:48) ξ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ξν ψ ( x ) (cid:105) B + µ (cid:48) ↔ ν (cid:105) + x ↔ (6.16)– 16 –he term proportional to p µ in the r.h.s. can be expressed using Eq. (11.13) as follows p µ s (cid:110)(cid:104) (cid:104) ¯ ψ ( x ) γ ν ⊥ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) (cid:54) p ψ ( x ) (cid:105) B + (cid:104) ¯ ψ ( x ) (cid:54) p (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) γ ν ⊥ ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:105) (6.17) − (cid:104) ¯ ψ ( x ) σ • ξ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ξν ⊥ ψ ( x ) (cid:105) B − (cid:104) ¯ ψ ( x ) σ ν ⊥ ξ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ξ • ψ ( x ) (cid:105) B (cid:111) = p µ s (cid:110)(cid:104) − (cid:104) ¯ ψ ( x ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p ˘ B ν ⊥ (0) ψ ( x ) (cid:105) B + s (cid:104) ¯ ψ ( x ) γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) γ ν ⊥ ˘ B i (0) ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x )) (cid:105) + i (cid:104) ¯ ψ ( x ) σ • j σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ jν ⊥ ψ ( x ) (cid:105) B + i (cid:104) ¯ ψ ( x ) σ ν ⊥ j σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ j • ψ ( x ) (cid:105) B − (cid:104) ¯ ψ ( x ) σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ • ν ⊥ ψ ( x ) (cid:105) B + 2 is (cid:104) ¯ ψ ( x ) σ • ν ⊥ σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ∗• ψ ( x ) (cid:105) B (cid:111) Hereafter we use notation ˘ B i ≡ B i − i ˜ B i γ .Let us at evaluate two the most important contributions. The first is − p µ s (cid:104) ¯ ψ ( x ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p ˘ B ν ⊥ (0) ψ ( x ) (cid:105) B = p µ s (cid:104) ¯ ψ ( x ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) B (0) (cid:54) p γ ν ⊥ ψ ( x ) (cid:105) B (6.18)As we shall see below, due to QCD equations of motion (cid:54) B in the r.h.s. of this equation canbe replaced by transverse momentum of the target TMD k ⊥ . Also, α will be replaced by α q so from the parametrizations (11.24) and (11.27) we see that p µ s (cid:104) ¯ ψ ( x ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) B (0) (cid:54) p γ ν ⊥ ψ ( x ) (cid:105) B ∼ p µ α q s k ν f ¯ f (6.19)which is of order of fourth term in Eq. (5.11). The second relevant term is ip µ s (cid:104) ¯ ψ ( x ) σ ν ⊥ j σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ j • ψ ( x ) (cid:105) B − p µ s (cid:104) ¯ ψ ( x ) σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ • ν ⊥ ψ ( x ) (cid:105) B = p µ s (cid:104) ¯ ψ ( x ) σ j ∗ α ψ (0) (cid:105) A (cid:104) ¯ ψ (0)[ B ν (0) σ • j − ν ↔ j ] ψ ( x ) (cid:105) B − p µ s (cid:104) ¯ ψ ( x ) σ ∗ ν ⊥ α ψ (0) (cid:105) A × (cid:104) ¯ ψB j (0) σ • j ψ ( x ) (cid:105) B = ip µ s (cid:104) ¯ ψ ( x ) σ ∗ ν ⊥ α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) B (0) (cid:54) p ψ ( x ) (cid:105) B (6.20)where we used formula (11.4) and the fact that for unpolarized protons (cid:104) p | ¯ ψ (0)[ A i (0) σ • j − i ↔ j ] ψ ( x ) | p (cid:105) = 0 (6.21)from parity conservation. Again, α will turn to α q and (cid:54) B can be replaced by (cid:54) k ⊥ for thetarget, so (6.11) is of order of ip µ s (cid:104) ¯ ψ ( x ) σ ∗ ν ⊥ α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) B (0) (cid:54) p ψ ( x ) (cid:105) B ∼ p µ α q s k ν h ¯ h (6.22) A rigorous argument goes like that: the matrix element (6.21) can be rewritten as (cid:15) ν ⊥ j (cid:15) kl (cid:104) ¯ ψ (0)[ A k (0) σ • l ψ ( x ) (cid:105) = (cid:15) jν ⊥ (cid:104) ¯ ψ (0) (cid:54) A (0) (cid:54) p γ ψ ( x ) (cid:105) . As demonstrated in Sect. 11.3, (cid:54) A in this for-mula can be replaced by (cid:54) k ⊥ so the contribution is proportional to matrix element k i (cid:104) ¯ ψ (0) iσ • i γ ψ ( x ) (cid:105) = k i (cid:15) ij (cid:104) ¯ ψ (0) σ • j ψ ( x ) (cid:105) which vanishes as seen from the parametrization (11.29). – 17 –et us demonstrate that the remaining terms in the r.h.s. of Eq. (6.17) are negligible.First, term coming from replacement ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) in Eq. (6.18) van-ishes since (cid:104) ¯ ψ ( x ) (cid:54) p γ ψ (0) (cid:105) A = 0 for unpolarized hadrons, see Eq. (11.28). Next, term p µ s (cid:104) ¯ ψ ( x ) γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) γ ν ⊥ ˘ B i (0) ψ ( x ) (cid:105) B is small because neither projectile no target ma-trix elements can bring factor s . Last, using Eq. (11.4) we get ip µ s (cid:2) (cid:104) ¯ ψ ( x ) σ • j σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ jν ⊥ ψ ( x ) (cid:105) B + 2 s (cid:104) ¯ ψ ( x ) σ • ν ⊥ σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ∗• ψ ( x ) (cid:105) B (cid:3) = ip µ s (cid:104) ¯ ψ ( x ) (cid:2) g ij + i(cid:15) ij γ + i ( σ ij − s g ij σ •∗ ) (cid:3) ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ jν ⊥ ψ ( x ) (cid:105) B + is (cid:104) ¯ ψ ( x ) (cid:2) g iν ⊥ + i(cid:15) iν ⊥ γ + i ( σ iν ⊥ − s g iν ⊥ σ •∗ ) (cid:3) ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ∗• ψ ( x ) (cid:105) B = ip µ s (cid:104) ¯ ψ ( x ) (cid:2) − s σ •∗ (cid:3) ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:2) B j (0) σ jν ⊥ + 2 s B ν ⊥ (0) σ ∗• (cid:3) ψ ( x ) (cid:105) B (6.23)It is easy to see that neither the projectile nor the target matrix element in the r.h.s. of thisequation gives s so these terms can be neglected in comparison to Eqs. (6.19) and (6.22).Thus, the two non-negligible terms in Eq. (6.17) give ˇ W (1)1 µν ( x ) = N c s (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ p µ s (cid:2) (cid:104) ¯ ψ ( x • , x ⊥ ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p γ ⊥ ν ψ ( x ∗ , x ⊥ ) (cid:105) B + i (cid:104) ¯ ψ ( x • , x ⊥ ) σ ∗ ν α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0)ˆ p ψ ( x ∗ , x ⊥ ) (cid:105) B (cid:3) + µ ↔ ν + x ↔ (6.24)Using formulas (11.30), (11.31), (11.33), (11.36), (11.41), and (11.43) for quark-antiquark-gluon operators and parametrizations from Sect. 11.2 we get the contribution to W µν inthe form W (1)1 µν ( q ) = 116 π s (cid:90) dx • dx ∗ d x ⊥ e − iαx • − iβx ∗ + i ( q,x ) ⊥ (6.25) × (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + x ↔ | A, B (cid:105) + µ ↔ ν = 164 π N c p µ s (cid:90) d k ⊥ (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:90) dx ∗ d x (cid:48)⊥ e − iβx ∗ + i ( q − k,x (cid:48) ) ⊥ × (cid:2) (cid:104) ¯ ψ ( x • , x ⊥ ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p γ ⊥ ν ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B + i (cid:104) ¯ ψ ( x • , x ⊥ ) σ ∗ ν α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0)ˆ p ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B + x ↔ (cid:3) + µ ↔ ν = p µ α q sN c (cid:90) d k ⊥ (cid:110) ( q − k ) ν (cid:2) f f ( α q , k ⊥ ) ¯ f f ⊥ ( β q , ( q − k ) ⊥ ) + ¯ f f ( α q , k ⊥ ) f f ⊥ ( β q , ( q − k ) ⊥ ) (cid:3) − k ν ( q − k ) ⊥ m (cid:2) h ⊥ f ( α q , k ⊥ )¯ h f ⊥ ( β q , ( q − k ) ⊥ ) + ¯ h ⊥ f ( α q , k ⊥ ) h f ⊥ ( β q , ( q − k ) ⊥ ) (cid:3)(cid:111) + µ ↔ ν where terms with replacement f f ↔ ¯ f f and h ⊥ f ↔ ¯ h ⊥ f come from x ↔ contribution.– 18 –ext we consider the remaining ∼ p µ term in Eq. (6.13) which can be rewritten as p µ s N c (cid:104) A, B | [ ¯ ψ A ( x ) (cid:54) p ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) | A, B (cid:105) = p µ s (cid:8) (cid:104) ¯ ψ ( x ) γ ν ⊥ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) (cid:54) p ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x )] (cid:9) + p µ s (cid:104) ¯ ψ ( x ) σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ ∗ ν ⊥ ψ ( x ) (cid:105) B + ip µ s (cid:104) ¯ ψ ( x ) σ ν ⊥ j σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ j ∗ ψ ( x ) (cid:105) B = p µ s (cid:8) (cid:104) ¯ ψ ( x ) γ ν ⊥ (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) (cid:54) p ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x )] (cid:9) + p µ s (cid:104) ¯ ψ ( x ) σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (0)[ B i (0) σ ∗ ν ⊥ + i ↔ ν ⊥ ] ψ ( x ) (cid:105) B − p µ s (cid:104) ¯ ψ ( x ) σ ∗ ν ⊥ α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ i ∗ ψ ( x ) (cid:105) B (6.26)where again we used formula (11.4). Note that while the matrix elements between projectilestates give contributions ∼ sα q k ⊥ , the target matrix elements cannot give s . Indeed, thesetarget matrix elements know about ˆ p only through direction of Wilson lines so they shouldnot change under rescaling p → λp , see the discussion in Sect. 11.2. Thus, the r.h.s. ofEq. (6.26) is ∼ p µ α q s k ⊥ ν which means that the p µ term in Eq. (6.16) is p µ s N c (cid:104) A, B | [ ¯ ψ A ( x ) (cid:54) p ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) | A, B (cid:105) ∼ p µ q ⊥ ν α q s (6.27)so the corresponding contribution to W µν is ∼ p µ q ⊥ ν α q s which is O (cid:0) s (cid:1) in comparison to thatof Eq. (6.28).Thus, the contribution of the first term in Eq. (6.2) to W µν is W (1)1 µν ( q ) = 116 π s (cid:90) dx • dx ∗ d x ⊥ e − iαx • − iβx ∗ + i ( q,x ) ⊥ × (cid:104) A, B | (cid:2) ¯ ψ A (0) γ µ ψ B (0) (cid:3)(cid:2) ¯ ψ B ( x ) γ ν Ξ ( x ) (cid:3) + ( x ↔ | A, B (cid:105) + µ ↔ ν = p µ α q N c (cid:90) d k ⊥ (cid:104) ( q − k ) ⊥ ν F f ( q, k ⊥ ) − k ν ( q − k ) ⊥ m H f ( q, k ⊥ ) (cid:105) + µ ↔ ν (6.28)where F f ( q, k ⊥ ) and H f ( q, k ⊥ ) are give by expressions (2.2) with x A ≡ α q and x B ≡ β q F f ( q, k ⊥ ) = f f ( α q , k ⊥ ) ¯ f f ( β q , ( q − k ) ⊥ ) + f f ↔ ¯ f f H f ( q, k ⊥ ) = h ⊥ f ( α q , k ⊥ )¯ h ⊥ f ( β q , ( q − k ) ⊥ ) + h ⊥ f ↔ ¯ h ⊥ f (6.29)Let us consider now the second term in Eq. (6.2). The calculation repeats that of thefirst term so we will indicate here main steps and pay attention to non-negligible terms– 19 –nly. If one of the indices (say, µ ) is longitudinal and the other transverse, we get (cid:16) p µ p µ (cid:48) s + µ ↔ µ (cid:48) (cid:17) N c s (cid:104) A, B | (cid:2) ¯Ξ ( x ) γ µ (cid:48) ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ (cid:48) ↔ ν | A, B (cid:105) = (cid:104) A, B | p µ (cid:0)(cid:2) ¯Ξ ( x ) (cid:54) p ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + (cid:2) ¯Ξ ( x ) γ ν ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) (cid:54) p ψ A (0) (cid:3)(cid:1) + p µ (cid:2) ¯Ξ ( x ) γ ν ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) (cid:54) p ψ A (0) (cid:3) | A, B (cid:105) N c s (6.30)where we used ¯Ξ = − (cid:0) ¯ ψ A α (cid:1) γ i B i /p s . The most important terms are those proportional to p µ . Using Fierz transformation and separating color singlets, they can be rewritten as (cf.Eq. (6.17)) N c s (cid:104) A, B | (cid:2) ¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) = p µ s (cid:110)(cid:2) (cid:104) ¯ ψ α ( x ) γ i (cid:54) p (cid:54) p ψ (0) (cid:105) A (cid:104) ¯ ψ (0) γ ⊥ ν B i ( x ) ψ ( x ) (cid:105) B + (cid:104) ¯ ψ α ( x ) γ i (cid:54) p γ ⊥ ν ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p B i ( x ) ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:3) − (cid:104) ¯ ψ α ( x ) γ i (cid:54) p σ • ξ ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ ξν ⊥ B i ( x ) ψ ( x ) (cid:105) B − (cid:104) ¯ ψ α ( x ) γ i (cid:54) p σ ν ⊥ ξ ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ ξ • B i ( x ) ψ ( x ) (cid:105) B (cid:111) + µ ↔ ν (6.31)After some algebra with γ -matrices this can be transformed to W (1)2 µν ( x ) = N c s (cid:104) A, B | (cid:2) ¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ p µ s (cid:110) (cid:104) ¯ ψ α ( x ) (cid:54) p ψ (0) (cid:105) A (cid:104) ¯ ψ (0) γ ⊥ ν (cid:54) p (cid:54) B ( x ) ψ ( x ) (cid:105) B − i (cid:104) ¯ ψ α ( x ) σ ∗ ν ⊥ ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p (cid:54) B ( x ) ψ ( x ) (cid:105) B (cid:111) + µ ↔ ν + x ↔ (6.32)plus terms small in comparison to p µ q ⊥ ν α q s . Using Eq. (11.31) we can transform (cid:0) ¯ ψ α (cid:1) ( x ) to (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) (cid:0) ¯ ψ α (cid:1) ( x )Γ ψ (0) (cid:105) A = i (cid:90) dx • (cid:90) x • −∞ dx (cid:48)• d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) ¯ ψ ( x (cid:48)• , x ⊥ )Γ ψ (0) (cid:105) A = − α q (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) ¯ ψ ( x • , x ⊥ )Γ ψ (0) (cid:105) A (6.33)Using QCD equation of motion and other formulas from sections 11.2 and 11.3 one gets π s (cid:90) dx • dx ∗ d x ⊥ e − iαx • − iβx ∗ + i ( q,x ) ⊥ × (cid:104) A, B | (cid:2) ¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) = p µ α q sN c (cid:104) ( q − k ) ν f f ( α q , k ⊥ ) ¯ f f ⊥ ( β q , ( q − k ) ⊥ ) − k ν ( q − k ) ⊥ m h ⊥ f ( α q , k ⊥ )¯ h f ⊥ ( β q , ( q − k ) ⊥ ) (cid:105) + µ ↔ ν (6.34)– 20 –o the contribution of Eq. (6.28) is effectively doubled. Again, the term with x ↔ exchange leads to Eq. (6.34) with f ↔ ¯ f and h ⊥ ↔ ¯ h ⊥ replacement.Thus, the sum of first and second terms in Eq. (6.2) leads to twice Eq. (6.28) W (1) µν = 2 p µ α q s N c (cid:90) d k ⊥ (cid:104) ( q − k ) ν ⊥ F f ( q, k ⊥ ) − k ν ⊥ ( q − k ) ⊥ m H f ( q, k ⊥ ) (cid:105) + µ ↔ ν (6.35) Ξ In this Section we calculate the third term in Eq. (6.2). ˇ W (1)3 µν = N c s (cid:104) A, B | [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (6.36)Again, main contribution correspond to one index (e.g. µ ) being longitudinal and the othertransverse so we need (cid:16) p µ p µ (cid:48) s + µ ↔ µ (cid:48) (cid:17) N c s (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ⊥ ψ A (0) (cid:3) + µ (cid:48) ↔ ν | A, B (cid:105) = (cid:104) A, B | p µ (cid:0)(cid:2) ¯ ψ A ( x ) (cid:54) p Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ⊥ ψ A (0) (cid:3) + (cid:2) ¯ ψ A ( x ) γ ν ⊥ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) (cid:54) p ψ A (0) (cid:3)(cid:1) + p µ (cid:2) ¯ ψ A ( x ) γ ν ⊥ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) (cid:54) p ψ A (0) (cid:3) | A, B (cid:105) N c s (6.37)(recall that Ξ ( x ) = − (cid:54) p s γ i A i β ψ B ( x ) so (cid:54) p Ξ = 0 ).Let us consider first the term proportional to p µ . Performing Fierz transformation(11.1) and sorting out the color-singlet contributions we get (cf. Eq. (11.31)) p µ s p µ (cid:48) N c s (cid:104) A, B | [ ¯ ψ B (0) γ µ (cid:48) ψ A (0) (cid:3)(cid:2) ¯ ψ A ( x ) γ ν ⊥ Ξ ( x ) (cid:3) + µ (cid:48) ↔ ν | A, B (cid:105) = p µ s (cid:110) (cid:8) (cid:104) ¯ ψγ ν ˘ A i ( x ) ψ (0) (cid:105) A (cid:104) ¯ ψ (0) γ i β ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) − s (cid:104) ¯ ψ (cid:54) p ˘ A ν ( x ) ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p β ψ ( x ) (cid:105) B + ( ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:9) − s (cid:104) ¯ ψA j ( x ) σ ∗ ν ⊥ ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ • j β ψ ( x ) (cid:105) B − is (cid:104) ¯ ψA i ( x ) σ ∗• ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ ∗ ν ⊥ σ • i β ψ ( x ) (cid:105) B + is (cid:104) ¯ ψ (0) σ ∗ j σ • i β ψ ( x ) (cid:105) B (cid:104) ¯ ψA i ( x ) σ jν ψ (0) (cid:105) A + is (cid:104) ¯ ψ (0) σ ν ⊥ j σ • i β ψ ( x ) (cid:105) B (cid:104) ¯ ψA i ( x ) σ j ∗ ψ (0) (cid:105) A (cid:111) (6.38)It is clear that matrix elements in the first line in the r.h.s. can produce only transversefactors ∼ q ⊥ so the corresponding contribution ∼ p µ q ⊥ ν m ⊥ β q s can be neglected. Also, matrixelement (cid:104) ¯ ψ (0) (cid:54) p β ψ ( x ) (cid:105) B vanishes as seen from Eq. (11.28). The remaining terms can be– 21 –ewritten as p µ s p µ (cid:48) N c s (cid:104) A, B | [ ¯ ψ B (0) γ µ (cid:48) ψ A (0) (cid:3)(cid:2) ¯ ψ A ( x ) γ ν ⊥ Ξ ( x ) (cid:3) + µ (cid:48) ↔ ν | A, B (cid:105) = p µ s (cid:110) − (cid:104) ¯ ψ (cid:54) p ˘ A ν ( x ) ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p β ψ ( x ) (cid:105) B − (cid:104) ¯ ψA j ( x ) σ ∗ j ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ • ν ⊥ β ψ ( x ) (cid:105) B + (cid:104) ¯ ψ ( x ) (cid:2) A ν ⊥ ( x ) σ ∗ j − A j ( x ) σ ∗ ν ⊥ (cid:3) ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ j • β ψ ( x ) (cid:105) B + is (cid:104) ¯ ψA i ( x ) σ jν ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:2) g ij − i(cid:15) ij γ − iσ ji + 2 s g ij σ ∗• (cid:3) β ψ ( x ) (cid:105) B − i (cid:104) ¯ ψA i ( x ) σ ∗• ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:2) g iν − i(cid:15) iν ⊥ γ + iσ iν ⊥ + 2 s g iν σ ∗• (cid:3) β ψ ( x ) (cid:105) B (cid:111) (6.39)where we have used Eq. (11.4). It is clear that only the first line in the r.h.s. cangive the non-negligible contribution to W µν . Indeed, matrix element (cid:104) ¯ ψ ( x ) (cid:2) A ν ⊥ ( x ) σ ∗ j − A j ( x ) σ ∗ ν ⊥ (cid:3) ψ (0) (cid:105) A vanishes for unpolarized hadrons due to parity, see Eq. (6.21). In thethird line in r.h.s., neither matrix element can produce s so the corresponding contributionis again ∼ p µ q ⊥ ν m ⊥ α q s while contribution from the last line is even smaller, of order of p µ q ⊥ ν m ⊥ α q s .Thus, we get p µ s p µ (cid:48) N c s (cid:104) A, B | [ ¯ ψ B (0) γ µ (cid:48) ψ A (0) (cid:3)(cid:2) ¯ ψ A ( x ) γ ν ⊥ Ξ ( x ) (cid:3) + µ (cid:48) ↔ ν | A, B (cid:105) (6.40) = p µ s (cid:110) (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p γ ν ⊥ ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p β ψ ( x ) (cid:105) B + i (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ • ν ⊥ β ψ ( x ) (cid:105) B It remains to prove that the last term in Eq. (6.37) proportional to p µ is small. One canrewrite that term similarly to Eq. (6.26) with replacement p ↔ p and (projectile matrixelements) ↔ (target ones). After that, the proof repeats arguments after Eq. (6.26) andone obtains the estimate p µ s N c (cid:104) A, B (cid:2) ¯ ψ A ( x ) γ ν ⊥ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) (cid:54) p ψ A (0) (cid:3) | A, B (cid:105) ∼ p µ q ⊥ ν m ⊥ β q s (6.41)Similarly, by repeating arguments from Section 6.1.1 with replacement p ↔ p and projec-tile matrix elements ↔ target ones, one can demonstrate that terms in Eq. (6.36) with µ, ν both longitudinal or both transverse are small in comparison to terms listed in Eq. (5.11).Thus, ˇ W (1)3 µν ( x ) = N c s (cid:104) A, B | [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ p µ s (cid:2) (cid:104) ¯ ψ (cid:54) A ( x • , x ⊥ ) (cid:54) p γ ν ⊥ ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p β ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B + i (cid:104) ¯ ψ (cid:54) A ( x • , x ⊥ ) (cid:54) p ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ • ν ⊥ β ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B + µ ↔ ν (cid:105) + x ↔ (6.42)Using QCD equation of motion and formulas from Appendix, we obtain the corresponding– 22 –ontribution to W µν in the form π s (cid:90) dx • dx ∗ d x ⊥ e − iαx • − iβx ∗ + i ( q,x ) ⊥ × (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3) [ ¯ ψ B (0) γ ν ψ A (0) (cid:3) | A, B (cid:105) + µ ↔ ν = 164 π N c p µ s (cid:90) dx ∗ dx • (cid:90) d k ⊥ (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:90) dx ∗ d x (cid:48)⊥ e − iβx • + i ( q − k,x (cid:48) ) ⊥ × (cid:2) (cid:104) ¯ ψ (cid:54) A ( x • , x ⊥ ) (cid:54) p γ ν ⊥ ψ (0) (cid:105) A (cid:104) ¯ ψ (0) (cid:54) p β ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B + i (cid:104) ¯ ψ (cid:54) A ( x • , x ⊥ ) (cid:54) p ψ (0) (cid:105) A (cid:104) ¯ ψ (0) σ • ν ⊥ β ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B + µ ↔ ν (cid:105) = p µ β q sN c (cid:90) d k ⊥ (cid:104) k ν f f ( α q , k ⊥ ) ¯ f f ⊥ ( β q , ( q − k ) ⊥ ) − ( q − k ) ν k ⊥ m h ⊥ f ( α q , k ⊥ )¯ h f ⊥ ( β q , ( q − k ) ⊥ ) (cid:105) + µ ↔ ν (6.43)Same as in previous Section, the term with x ↔ exchange leads to Eq. (6.43) with f ↔ ¯ f and h ⊥ ↔ ¯ h ⊥ replacement so we get W (1)3 µν = p µ β q sN c (cid:90) d k ⊥ (cid:104) k ν F f ( q, k ⊥ ) − ( q − k ) ν k ⊥ m H f ( q, k ⊥ ) (cid:105) + µ ↔ ν (6.44)Repeating arguments from previous Section, it is possible to show that the contribution ofthe fourth term ˇ W (1)4 µν ( x ) = N c s (cid:104) A, B | [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ p µ s (cid:2) (cid:104) ¯ ψγ ν ⊥ (cid:54) p (cid:54) A (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) (cid:54) p ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B − i (cid:104) ¯ ψ ( x ∗ , x (cid:48)⊥ (cid:54) p (cid:54) A (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) σ • ν ⊥ ψ ( x ∗ , x (cid:48)⊥ ) (cid:105) B + µ ↔ ν (cid:105) + x ↔ (6.45)doubles that of the third term so we get the full contribution of the terms with one quark-antiquark-gluon operators in the form W (1) µν = 2 N c (cid:90) d k ⊥ (cid:104)(cid:16) p µ k ⊥ ν β q s + p µ ( q − k ) ⊥ ν α q s (cid:17) F f ( q, k ⊥ ) − (cid:16) p µ ( q − k ) ⊥ ν β q s k ⊥ m + p µ k ν ⊥ α q s ( q − k ) ⊥ m (cid:17) H f ( q, k ⊥ ) (cid:105) + µ ↔ ν (6.46)This result agrees with the corresponding /Q terms in Ref. [20]. Ξ and Ξ Let us start with the first term in the r.h.s. of Eq. (6.3). Performing Fierz transformation(11.1) we obtain N c s (cid:104) A, B | ( ¯ ψ mA ( x ) γ µ Ξ m ( x ))( ¯ ψ nB (0) γ ν Ξ n (0)) + µ ↔ ν | A, B (cid:105) + x ↔ g µν ˇ V + ˇ V µν + ˇ V µν (6.47)– 23 –here ˇ V = N c s (cid:104) A, B | − [ ¯ ψ nA ( x )Ξ m (0)][ ¯ ψ nB (0)Ξ m ( x )] + [ ¯ ψ mA ( x ) γ Ξ n (0)][ ¯ ψ nB (0) γ Ξ m ( x )] (6.48) + [ ¯ ψ mA ( x ) γ α Ξ m (0)][ ¯ ψ nB (0) γ α Ξ n ( x )] + [ ¯ ψ mA ( x ) γ α γ Ξ m (0)][ ¯ ψ nB (0) γ α γ Ξ n ( x )] | A, B (cid:105) + x ↔ , ˇ V µν = N c s (cid:104) A, B | − [ ¯ ψ mA ( x ) γ µ Ξ n (0)][ ¯ ψ nB (0) γ ν Ξ m ( x )] − [ ¯ ψ mA ( x ) γ µ γ Ξ n (0)][ ¯ ψ nB (0) γ ν γ Ξ m ( x )] + µ ↔ ν | A, B (cid:105) + x ↔ , (6.49)and ˇ V µν = N c s (cid:104) A, B | [( ¯ ψ mA ( x ) σ µα Ξ n (0)][ ¯ ψ nB (0) σ αν Ξ m ( x )] + µ ↔ ν − g µν [ ¯ ψ mA ( x ) σ αβ Ξ n (0)][ ¯ ψ nB (0) σ αβ Ξ m ( x ) | A, B (cid:105) + x ↔ (6.50)It is convenient to define ˇ V µν to be traceless. In next Sections, we will consider these termsin turn. g µν Using Ξ = − g/p s γ i B i α + i(cid:15) ψ A and Ξ = − g/p s γ i A i β + i(cid:15) ψ B from Eq. (4.12) and extractingcolor-singlet contributions one obtains ˇ V = 12 s × (cid:110) − (cid:104) (cid:104) ¯ ψA i ( x ) (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) (cid:54) p γ i β ψ ( x ) (cid:105) B − ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:105) + (cid:104) (cid:104) ¯ ψA i ( x ) γ k (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) γ k (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:105) + 2 s (cid:104) (cid:104) ¯ ψA i ( x ) (cid:54) p (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) (cid:54) p (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:105)(cid:111) + x ↔ (6.51)Let us start with the first term. Using Eq. (11.21) and the fact that (cid:104) ¯ ψ ( x ) (cid:2) A k σ ∗ j − A j ( x ) σ ∗ k (cid:3) ψ (0) (cid:105) A = 0 (see the footnote 11), we obtain − s (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p β ψ ( x ) (cid:105) B − s (cid:104) ¯ ψ (cid:54) A ( x ) γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) γ i β ψ ( x ) (cid:105) B (cid:105) − s [ (cid:104) ¯ ψA i ( x ) σ jk α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ jk β ψ ( x ) (cid:105) B = − s (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p β ψ ( x ) (cid:105) B (cid:104) O (cid:0) q ⊥ s (cid:1)(cid:105) (6.52)where we used the fact that projectile and target matrix elements in the two last terms inthe l.h.s. cannot produce factor of s .Next, consider second term in Eq. (6.51). Using Eqs. (11.17) and (11.21), one canrewrite is as s (cid:104) (cid:104) ¯ ψA i ( x ) γ k (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) γ k (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:105) = 1 s (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p γ i β ψ ( x ) (cid:105) B (6.53)– 24 –imilarly, from Eq. (11.21) we get the third term in the form s (cid:104) (cid:104) ¯ ψA i ( x ) (cid:54) p (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) (cid:54) p (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:105) = 14 s (cid:104) (cid:104) ¯ ψ ˘ A i ( x ) γ j α ψ (0) (cid:105) A (cid:104) ¯ ψ ˘ B j (0) γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:105) (6.54)Since both projectile and target matrix elements cannot give factor s this contribution is O (cid:0) q ⊥ s (cid:1) in comparison to that of the two first terms. Thus, we get ˇ V = 1 s (cid:16) (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p β ψ ( x ) (cid:105) B + (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p γ i β ψ ( x ) (cid:105) B (cid:17)(cid:104) O (cid:0) q ⊥ s (cid:1)(cid:105) + x ↔ (6.55)Next, using QCD equations of motion (11.40), (11.43) and formulas from Appendix 11.2,we obtain the contribution to W µν in the form g µν V ( q ) = g µν π N c (cid:90) dx • dx ∗ d x ⊥ e − iα q x • − iβ q x ∗ + i ( q,x ) ⊥ ˇ V ( x )= g µν α q β q sN c (cid:90) d k ⊥ (cid:104) ( k, q − k ) ⊥ F f ( q, k ⊥ ) − m k ⊥ ( q − k ) ⊥ H f ( q, k ⊥ ) (cid:105) (6.56)where replacements f f ↔ ¯ f f and h ⊥ f ↔ ¯ h ⊥ f come from x ↔ term. f Separating color-singlet contributions one can rewrite Eq. (6.49) as ˇ V µν = − s (cid:8) (cid:104) ¯ ψA i ( x ) γ µ (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) γ ν (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) + µ ↔ ν } + x ↔ (6.57)We need to consider three cases: both µ and ν are transverse, both of them are longitudinal,and µ is longitudinal and ν transverse (plus vice versa ).In the first case we can use formula (11.18) and get ˇ V µ ⊥ ν ⊥ = − s (cid:8) (cid:104) ¯ ψA i ( x ) γ µ ⊥ (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) γ ν ⊥ (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) + µ ↔ ν } + x ↔ − g ⊥ µν s (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p γ i β ψ ( x ) (cid:105) B + x ↔ (6.58)which gives the contribution to W µν in the form V µ ⊥ ν ⊥ = 116 π N c (cid:90) dx • dx ∗ d x ⊥ e − iα q x • − iβ q x ∗ + i ( q,x ) ⊥ ˇ V µ ⊥ ν ⊥ ( x )= − g ⊥ µν α q β q sN c (cid:90) d k ⊥ ( k, q − k ) ⊥ F f ( q, k ⊥ ) (6.59)– 25 –here we again used formulas from Appendices 11.2 and 11.3.Next, if both µ and ν are longitudinal, we get ˇ V µν = − s ( p µ p ν + µ ↔ ν ) (cid:8) (cid:104) ¯ ψA i ( x ) (cid:54) p (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) (cid:54) p (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) } + x ↔ (6.60)Using formula (11.20) we rewrite r.h.s. of Eq. (6.57) as follows ˇ V µν = − s ( p µ p ν + µ ↔ ν ) (cid:8) (cid:104) ¯ ψ ˘ A i ( x ) γ j α ψ (0) (cid:105) A (cid:104) ¯ ψ ˘ B j (0) γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) } + x ↔ (6.61)Since matrix elements in the r.h.s. cannot give factor s , the contribution of this term to W µν is ∼ q ⊥ s times that of Eq. (6.59).Finally, let us consider the case when one index is longitudinal and the other transverse.Using Eq. (11.23) we get ˇ V (cid:107)⊥ µν = − s (cid:8) p µ (cid:104) ¯ ψA i ( x ) (cid:54) p (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) γ ν ⊥ (cid:54) p γ i β ψ ( x ) (cid:105) B + p µ (cid:104) ¯ ψA i ( x ) γ ν ⊥ (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) (cid:54) p (cid:54) p γ i β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) + µ ↔ ν } + x ↔ − s (cid:8) p µ (cid:104) ¯ ψ ( x ) γ i A ν ( x ) 1 α ψ (0) (cid:105) A (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p γ i β ψ ( x ) (cid:105) B + p µ (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p γ i α ψ (0) (cid:105) A (cid:104) ¯ ψ (0) γ i B ν (0) 1 β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) + µ ↔ ν } + x ↔ (6.62)It is clear that (cid:104) ¯ ψ (cid:54) A ( x ) (cid:54) p γ i α ψ (0) (cid:105) A and (cid:104) ¯ ψ (cid:54) B (0) (cid:54) p γ i β ψ ( x ) (cid:105) B bring one factor s so ˇ V (cid:107)⊥ µν ∼ p µ q ⊥ ν + µ ↔ να q β q s m ⊥ or ∼ p µ q ⊥ ν + µ ↔ να q β q s m ⊥ (6.63)which is α q s or β q s correction in comparison to Eq. (6.46). Thus, the contribution to W µν is given by Eq. (6.59) V µν = − g ⊥ µν α q β q sN c (cid:90) d k ⊥ ( k, q − k ) ⊥ F f ( q, k ⊥ ) (6.64) h ⊥ Let us consider now ˇ V (cid:48) µν = N c s (cid:104) A, B | [( ¯ ψ mA ( x ) σ µα Ξ m (0)][ ¯ ψ nB (0) σ αν Ξ n ( x )] + µ ↔ ν + x ↔ (6.65)(the trace will be subtracted after the calculation). Separating color-singlet contributions,we get ˇ V (cid:48) µν = 12 s (cid:104) ¯ ψA i ( x ) σ µα (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ αν (cid:54) p γ i β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ (6.66)– 26 –irst case is when µ and ν are transverse ˇ V (cid:48) µ ⊥ ν ⊥ = − s (cid:104) ¯ ψA i ( x ) σ µ ⊥ k σ ∗ j α ψ )0) (cid:105) A (cid:104) ¯ ψB j (0) σ kν ⊥ σ • i β ψ ( x ) (cid:105) B − s (cid:104) ¯ ψA i ( x ) σ • µ ⊥ σ ∗ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ ∗ ν ⊥ σ • i β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ (6.67)With the help of Eq. (11.4) the first term in the r.h.s. turns to s (cid:104) ¯ ψA i ( x )[ g µj σ ∗ k − g jk σ ∗ µ ⊥ ] 1 α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0)[ g νi σ k • − δ ki σ • ν ⊥ ] 1 β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔
0= 12 s (cid:16) (cid:104) ¯ ψA ν ( x ) σ ∗ k α ψ (0) (cid:105) A (cid:104) ¯ ψB µ (0) σ k • β ψ ( x ) (cid:105) B − (cid:104) ¯ ψA ν ( x ) σ ∗ µ ⊥ α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ • j β ψ ( x ) (cid:105) B − (cid:104) ¯ ψA i ( x ) σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB µ (0) σ • ν ⊥ β ψ ( x ) (cid:105) B + (cid:104) ¯ ψA i ( x ) σ ∗ µ ⊥ α ψ (0) (cid:105) A (cid:104) ¯ ψB i (0) σ • ν ⊥ β ψ ( x ) (cid:105) B (cid:17) + µ ↔ ν + x ↔ (6.68)After some algebra, it can be rewritten as − g ⊥ µν s (cid:104) ¯ ψA i ( x ) σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ • j β ψ ( x ) (cid:105) B + 1 s (cid:8) (cid:104) ¯ ψ (cid:0) A k σ ∗ µ ⊥ − g µk σ ∗ j A j (cid:1) ( x ) 1 α ψ (0) (cid:105) A × (cid:104) ¯ ψ (cid:0) B k σ • ν ⊥ − δ kν σ • j B j (cid:1) (0) 1 β ψ ( x ) (cid:105) B + µ ↔ ν (cid:9) + x ↔ (6.69)where again we used property (6.21). Using QCD equations of motion (11.41), (11.43) andparametrization (11.47) one can write the corresponding contribution to W µν as V (cid:48) µ ⊥ ν ⊥ = g ⊥ µν α q β q sN c (cid:90) d k ⊥ k ⊥ ( q − k ) ⊥ m H f ( q, k ⊥ ) (6.70) + 1 α q β q sN c (cid:90) d k ⊥ m (cid:2) ( k ⊥ µ ( q − k ) ⊥ ν + µ ↔ ν )( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ µ ( q − k ) ⊥ ν − ( q − k ⊥ ) k ⊥ µ k ⊥ ν − g ⊥ µν k ⊥ ( q − k ⊥ ) (cid:3) H fA ( q, k ⊥ ) where we introduced the notation H fA ( q, k ⊥ ) ≡ h fA ( α q , k ⊥ )¯ h fA ( β q , ( q − k ) ⊥ ) + h fA ↔ ¯ h fA (6.71)The second term in Eq. (6.67) can be rewritten as ˇ V (cid:48) µ ⊥ ν ⊥ = − s (cid:104) ¯ ψA i ( x ) (cid:0) g ⊥ µj − i(cid:15) µ ⊥ j γ − iσ µ ⊥ j − is g ⊥ µj σ •∗ (cid:1) α ψ (0) (cid:105) A × (cid:104) ¯ ψB j (0) (cid:0) g ⊥ iν − i(cid:15) iν ⊥ γ + iσ iν ⊥ + 2 s g ⊥ iν σ ∗• (cid:1) β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ (6.72)where we used Eq. (11.4). It is clear that neither projectile no target matrix element inthe r.h.s. can bring factor s so ˇ V (cid:48) µ ⊥ ν ⊥ ∼ m ⊥ α q β q s (6.73)– 27 –hich is O (cid:0) m ⊥ s (cid:1) in comparison to Eq. (6.70).Next, consider the case when both µ and ν are longitudinal. The non-vanishing termsare ˇ V (cid:48) µν = 4 p µ p ν s s (cid:104) ¯ ψA i ( x ) σ ∗ α (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ α • (cid:54) p γ i β ψ ( x ) (cid:105) B + 4 p ν p µ s s (cid:104) ¯ ψA i ( x ) σ • α (cid:54) p γ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ α ∗ (cid:54) p γ i β ψ ( x ) (cid:105) B (6.74) − p µ p ν s s (cid:104) ¯ ψA i ( x ) σ • k σ ∗ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ k ∗ σ • i β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ The first two terms in the r.h.s. can be rewritten as ˇ V (cid:48) µν = g (cid:107) µν s (cid:104) ¯ ψA i ( x ) σ ∗ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ • i β ψ ( x ) (cid:105) B + x ↔ g (cid:107) µν s (cid:104) ¯ ψA i ( x ) σ ∗ i α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ • j β ψ ( x ) (cid:105) B + g (cid:107) µν s (cid:104) ¯ ψ (cid:0) A i ( x ) σ ∗ j − g ij A k ( x ) σ ∗ k (cid:1) α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ • i β ψ ( x ) (cid:105) B + x ↔ (6.75)The corresponding contribution to W µν yields V (cid:48) µ (cid:107) ν (cid:107) = − g (cid:107) µν α q β q sN c (cid:90) d k ⊥ k ⊥ ( q − k ) ⊥ m H f ( q, k ⊥ ) − g (cid:107) µν α q β q sN c (cid:90) d k ⊥ m (cid:2) ( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ (cid:3) H fA ( q, k ⊥ ) (6.76)where again we used QCD equations of motion (11.41), (11.43) and parametrization (11.47).Next, it is easy to see that the third term in Eq. (6.74) is small in comparison to Eq.(6.76): − p µ p ν s s (cid:104) ¯ ψA i ( x ) σ • k σ ∗ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ k ∗ σ • i β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ − p µ p ν s (cid:104) ¯ ψA i ( x ) (cid:2) g jk + i(cid:15) jk γ + iσ jk − is g jk σ •∗ (cid:3) α ψ (0) (cid:105) A × (cid:104) ¯ ψB j (0) (cid:2) g ik − i(cid:15) ik γ + iσ ik + 2 s g ik σ ∗• (cid:3) β ψ ( x ) (cid:105) B ∼ q ⊥ s × g (cid:107) µν α q β q s (6.77)because neither projectile no target matrix element can bring factor s .Finally, take one of the indices (say, µ ) longitudinal and the other transverse. FromEq. (6.66) we get ˇ V µ (cid:107) ν ⊥ = − p µ s (cid:104) [ ¯ ψA i ( x ) σ • k σ ∗ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ kν ⊥ σ • i β ψ ( x ) (cid:105) B − ip µ s (cid:104) [ ¯ ψA i ( x ) σ • ν ⊥ σ ∗ j α ψ (0) (cid:105) A (cid:104) ¯ ψB j (0) σ • i β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ (6.78)– 28 –sing formulas (11.4) this can be rewritten as follows ˇ V µ (cid:107) ν ⊥ = ip µ s (cid:104) ¯ ψA i ( x ) (cid:2) g jk + i(cid:15) jk γ + iσ jk − is g jk σ •∗ (cid:3) α ψ (0) (cid:105) A × (cid:104) ¯ ψB j (0) (cid:2) g iν ⊥ σ • k − g ik σ • ν ⊥ (cid:3) β ψ ( x ) (cid:105) B − ip µ s (cid:104) ¯ ψA i ( x ) (cid:2) g jν ⊥ + i(cid:15) jν ⊥ γ + iσ jν ⊥ − is g jν ⊥ σ •∗ (cid:3) α ψ (0) (cid:105) A × (cid:104) ¯ ψB j (0) σ • i β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ (6.79)As we discussed above, projectile matrix elements in the r.h.s. like (cid:104) ¯ ψB j (0) σ • i β ψ ( x ) (cid:105) B canbring factor s but the target matrix elements cannot so the corresponding contribution to W µν is of order ˇ V µ (cid:107) ν ⊥ ∼ (cid:0) p µ q ⊥ ν + µ ↔ ν (cid:1) m ⊥ α q β q s (6.80)which is O (cid:0) q ⊥ s (cid:1) in comparison to Eq. (6.46).Next, the sum of Eqs. (6.70) and (6.76) is V (cid:48) µν = g ⊥ µν − g (cid:107) µν α q β q sN c (cid:90) d k ⊥ k ⊥ ( q − k ) ⊥ m H f ( q, k ⊥ ) (6.81) + 1 α q β q sN c (cid:90) d k ⊥ m (cid:8) [ k ⊥ µ ( q − k ) ⊥ ν + µ ↔ ν ]( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ µ ( q − k ) ⊥ ν − ( q − k ⊥ ) k ⊥ µ k ⊥ ν − g ⊥ µν k ⊥ ( q − k ⊥ ) − g (cid:107) µν (cid:2) ( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ (cid:3)(cid:9) H fA ( q, k ⊥ ) so subtracting trace we obtain V µν = V (cid:48) µν − g µν V (cid:48) ξξ (6.82) = g ⊥ µν − g (cid:107) µν α q β q sN c (cid:90) d k ⊥ m k ⊥ ( q − k ) ⊥ H f ( q, k ⊥ )+ 1 α q β q sN c (cid:90) d k ⊥ m (cid:8) [ k ⊥ µ ( q − k ) ⊥ ν + µ ↔ ν ]( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ µ ( q − k ) ⊥ ν − ( q − k ⊥ ) k ⊥ µ k ⊥ ν + g ⊥ µν ( k, q − k ) ⊥ − g ⊥ µν k ⊥ ( q − k ⊥ ) (cid:3)(cid:9) H fA ( q, k ⊥ ) As we will see in Sect. 8, cancellation of terms ∼ g (cid:107) µν proportional to h A in the r.h.s of thisequation is actually a consequence of (EM) gauge invariance.Let us now assemble the contribution of terms (6.47) to W µν . Summing Eqs. (6.56),– 29 –6.64), and (6.82) we get V µν ( q ) = 132 π (cid:90) dx • dx ∗ d x ⊥ e − iαx • − iβx ∗ + i ( q,x ) ⊥ (cid:2) (cid:104) A, B | ( ¯ ψ mA ( x ) γ µ Ξ m ( x ))( ¯ ψ nB (0) γ ν Ξ n (0)) + µ ↔ ν | A, B (cid:105) + x ↔ (cid:3) = g (cid:107) µν α q β q sN c (cid:90) d k ⊥ (cid:110) ( k, q − k ) ⊥ F f ( q, k ⊥ ) − m k ⊥ ( q − k ) ⊥ H f ( q, k ⊥ ) (cid:111) + 1 α q β q sN c (cid:90) d k ⊥ m (cid:8) [ k ⊥ µ ( q − k ) ⊥ ν + µ ↔ ν ]( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ µ ( q − k ) ⊥ ν − ( q − k ⊥ ) k ⊥ µ k ⊥ ν + g ⊥ µν ( k, q − k ) ⊥ − g ⊥ µν k ⊥ ( q − k ⊥ ) (cid:3)(cid:9) H fA ( q, k ⊥ ) (6.83)Finally, to get W (2 a ) µν ( q ) of Eq. (6.3) we need to add the contribution of the term [¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) . Similarly to the case of one quark-quark-gluon operatorconsidered in Sect. 6.1, it can be demonstrated that this contribution doubles the result(6.83) so we get W (2 a ) µν ( q ) = 132 π (cid:90) dx • dx ∗ d x ⊥ e − iαx • − iβx ∗ + i ( q,x ) ⊥ (cid:2) (cid:104) A, B | ( ¯ ψ mA ( x ) γ µ Ξ m ( x )) × ( ¯ ψ nB (0) γ ν Ξ n (0)) + [¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (cid:3) = 2 g (cid:107) µν Q (cid:107) N c (cid:90) d k ⊥ (cid:110) ( k, q − k ) ⊥ F f ( q, k ⊥ ) − m k ⊥ ( q − k ) ⊥ H f ( q, k ⊥ ) (cid:111) + 2 Q (cid:107) N c (cid:90) d k ⊥ m (cid:8) [ k ⊥ µ ( q − k ) ⊥ ν + µ ↔ ν ]( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ µ ( q − k ) ⊥ ν − ( q − k ⊥ ) k ⊥ µ k ⊥ ν + g ⊥ µν ( k, q − k ) ⊥ − g ⊥ µν k ⊥ ( q − k ⊥ ) (cid:3)(cid:9) H fA ( q, k ⊥ ) (6.84)where Q (cid:107) ≡ α q β q s ¯Ξ and Ξ Let us start with the first term in Eq. (6.4). ˇ W (2 b )1 µν = N c s (cid:104) A, B | (cid:2) ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (6.85)After Fierz transformation (11.1) we obtain ˇ W (2 b )1 µν = − N c s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) (cid:104) A, B | (cid:8) [ ¯ ψ mA ( x ) γ α ψ nA (0)][¯Ξ n (0) γ β Ξ m ( x )] (6.86) + γ α ⊗ γ β ↔ γ α γ ⊗ γ β γ (cid:9) | A, B (cid:105) + N c s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) (cid:104) A, B | [ ¯ ψ mA ( x ) σ αξ ψ nA (0)][¯Ξ n (0) σ ξβ Ξ m ( x )] | A, B (cid:105) + x ↔ (note that ¯Ξ Ξ = ¯Ξ γ Ξ = 0 ). Using explicit expressions (4.12) for quark fields andseparating color-singlet terms we get ˇ W (2 b )1 µν = ˇ V µν + ˇ V µν (6.87)– 30 –here ˇ V µν = − s ( δ αµ p ν + δ αν p µ − g µν p α ) (cid:16) (cid:104) ¯ ψ ( x ) A j ( x ) γ α A i (0) ψ (0) (cid:105) A × (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i (cid:54) p γ j β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:17) + x ↔ (6.88)and ˇ V µν = 1 s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) × (cid:110) − p β (cid:104) ¯ ψ ( x ) A j ( x ) σ αk A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ k • γ j β ψ ( x ) (cid:105) B + (cid:104) ¯ ψ ( x ) A j ( x ) σ α • A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ • β ⊥ γ j β ψ ( x ) (cid:105) B (cid:111) + x ↔ (6.89)We will consider them in turn. f ¯ f Let us start with g µν term in Eq. (6.88). g µν s (cid:104) ¯ ψ ( x ) A j ( x ) (cid:54) p A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i (cid:54) p γ j β ψ ( x ) (cid:105) B + (cid:54) p ⊗ γ j ↔ (cid:54) p γ ⊗ γ j γ (6.90)It is obvious that the target matrix element can bring factor s . On the contrary, as wediscussed above, the projectile matrix element cannot produce s since (cid:104) ¯ ψ ( x ) A j ( x ) γ α A i (0) ψ (0) (cid:105) A ∼ p α p · p × [ g ij φ ( x ⊥ ) + x i x j ξ ( x ⊥ )] + ... (6.91)Indeed, since projectile matrix elements know about p only through the direction of Wilsonlines, the l.h.s. can be proportional only to factor p α p · p that does not change under rescalingof p . Also, due to Eq. (11.31) (cid:104) (cid:0) ¯ ψ β (cid:1) (0) ⊗ β ψ ( x ) (cid:105) B can be replaced by − β q (cid:104) ¯ ψ (0) ⊗ ψ ( x ) (cid:105) B .Consequently, the r.h.s. of Eq. (6.90) is ∼ g µν m ⊥ β q s g µν s (cid:104) ¯ ψ ( x ) A j ( x ) (cid:54) p A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i (cid:54) p γ j β ψ ( x ) (cid:105) B + (cid:54) p ⊗ γ j ↔ (cid:54) p γ ⊗ γ j γ ∼ g µν m ⊥ β q s (6.92)which is O (cid:0) m ⊥ α q β q s (cid:1) in comparison to Eq. (6.84).We get ˇ V µν = − p µ s (cid:16) (cid:104) ¯ ψ ( x ) A j ( x ) γ ν A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i (cid:54) p γ j β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:17) + µ ↔ ν + x ↔ (6.93)If the index ν is transverse, the contribution of this equation to W µν is of order of ˇ V µν ∼ p µ q ⊥ ν m ⊥ β q s (6.94)– 31 –hich is O (cid:0) m ⊥ β q s (cid:1) in comparison to Eq. (6.46).For the longitudinal indices µ and ν we get ˇ V µν = − p µ p ν s (cid:16) (cid:104) ¯ ψ ( x ) A j ( x ) (cid:54) p A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i (cid:54) p γ j β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:17) − g (cid:107) µν s (cid:16) (cid:104) ¯ ψ ( x ) A j ( x ) (cid:54) p A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i (cid:54) p γ j β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:17) + x ↔ (6.95)Similarly to Eq. (6.92), the contribution of the second term to W µν is ∼ g (cid:107) µν m ⊥ β q s = O (cid:16) α q m ⊥ β q s (cid:17) × (cid:2) r . h . s . of Eq . (6 . (cid:3) (6.96)so we are left with the first term in the r.h.s. of Eq. (6.95). Using Eq. (11.8) it can berewritten as ˇ V µν = − p µ p ν s (cid:16) (cid:104) ¯ ψ ( x ) (cid:54) A ( x ) (cid:54) p (cid:54) A (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) (cid:54) p β ψ ( x ) (cid:105) B + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:17) + x ↔ (6.97)The corresponding contribution to W µν is obtained from QCD equation of motion (11.44)and formula (11.31) from Appendix 11.3: V µν ( q ) = 4 p µ p ν β q s N c (cid:90) d k ⊥ k ⊥ F f ( q, k ⊥ ) (6.98) h ⊥ ¯ h ⊥ Let us start with g µν term in Eq. (6.89). g µν s (cid:104) ¯ ψ ( x ) A j ( x ) σ • k A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ k • γ j β ψ ( x ) (cid:105) B + x ↔ (6.99)The target matrix element is proportional to s while the projectile one cannot bring s dueto Eq. (6.91), so the contribution of the r.h.s of Eq. (6.99) to W µν is of order ∼ g µν β q s m ⊥ = O (cid:0) m ⊥ α q sβ q (cid:17) × (cid:2) r . h . s . of Eq . (6 . (cid:3) (6.100)similarly to Eq. (6.96). We get ˇ V µν = 1 s (cid:110) − p µ (cid:104) ¯ ψ ( x ) A j ( x ) σ νk A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ k • γ j β ψ ( x ) (cid:105) B + (cid:104) ¯ ψ ( x ) A j ( x ) σ µ • A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ • ν ⊥ γ j β ψ ( x ) (cid:105) B (cid:111) + µ ↔ ν + x ↔ (6.101)– 32 –et us at first consider the second term in this formula: s (cid:104) ¯ ψ ( x ) A j ( x ) σ µ ⊥ • A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ • ν ⊥ γ j β ψ ( x ) (cid:105) B + 2 p µ s (cid:104) ¯ ψ ( x ) A j ( x ) σ ∗• A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ • ν ⊥ γ j β ψ ( x ) (cid:105) B (cid:111) + µ ↔ ν (6.102)Similarly to Eq. (6.91), projectile matrix elements cannot give factor s so the correspondingcontribution to W µν is of order of ( aq ⊥ µ q ⊥ ν + bq ⊥ g ⊥ µν ) m ⊥ β q s or 2 p µ s m ⊥ β q s (6.103)that are O (cid:0) α q m ⊥ β q s (cid:1) in comparison to Eqs. (6.46) and (6.84), respectively.We are left with ˇ V µν = − p µ s (cid:104) ¯ ψ ( x ) A j ( x ) σ νk A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ k • γ j β ψ ( x ) (cid:105) B + µ ↔ ν + x ↔ − p µ p ν s (cid:104) ¯ ψ ( x ) A j ( x ) σ ∗ k A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ k • γ j β ψ ( x ) (cid:105) B − g (cid:107) µν s (cid:104) ¯ ψ ( x ) A j ( x ) σ • k A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ k • γ j β ψ ( x ) (cid:105) B − (cid:16) p µ s (cid:104) ¯ ψ ( x ) A j ( x ) σ νk A i (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) γ i σ k • γ j β ψ ( x ) (cid:105) B + µ ↔ ν (cid:17) + x ↔ (6.104)First, note that the two last terms are small, of order of Eq. (6.103), for the same reasonas Eq. (6.101) above. As to the first term in r.h.s. of Eq. (6.104), using Eq. (11.7) it canbe rewritten as ˇ V µν = − p µ p ν s (cid:104) ¯ ψ ( x ) (cid:54) A ( x ) σ ∗ k (cid:54) A (0) ψ (0) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) σ k • β ψ ( x ) (cid:105) B + x ↔ (6.105)so the corresponding contribution to W µν takes the form V µν = − p µ p ν β q s N c (cid:90) d k ⊥ m k ⊥ ( k, q − k ) ⊥ H f ( q, k ⊥ ) (6.106)where we used Eqs. (11.31) and (11.45).The full result for W (2 b ) µν is given by the sum of Eqs. (6.98) and (6.106) W (2 b )1 µν = 4 p µ p ν β q s N c (cid:90) d k ⊥ (cid:104) k ⊥ F f ( q, k ⊥ ) − m k ⊥ ( k, q − k ) ⊥ H f ( q, k ⊥ ) (cid:105) (6.107) Let us start now consider the second term in Eq. (6.4). ˇ W (2 b )2 µν = N c s (cid:104) A, B | (cid:2) ¯Ξ ( x ) γ µ ψ B ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + µ ↔ ν | A, B (cid:105) + x ↔ (6.108)– 33 –fter Fierz transformation (11.1) we obtain ˇ W (2 b )2 µν = − N c s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) (cid:104) A, B | (cid:8) [¯Ξ m ( x ) γ α Ξ n (0)][ ¯ ψ nB (0) γ β ψ mB ( x )] (6.109) + γ α ⊗ γ β ↔ γ α γ ⊗ γ β γ (cid:9) | A, B (cid:105) + N c s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) (cid:104) A, B | [¯Ξ m ( x ) σ αξ Ξ n (0)][ ψ n B (0) σ ξβ ψ m B ( x )] | A, B (cid:105) + x ↔ Sorting out color-singlet terms, we get similarly to sum of Eqs. (6.88) and (6.89) ˇ W (2 b )2 µν = − s ( δ αµ p ν + δ αν p µ − g µν p α ) (cid:16) (cid:104) ¯ ψ ( x ) B j ( x ) γ α B i (0) ψ (0) (cid:105) B × (cid:104) (cid:0) ¯ ψ α (cid:1) (0) γ i (cid:54) p γ j α ψ ( x ) (cid:105) A + ψ (0) ⊗ ψ ( x ) ↔ γ ψ (0) ⊗ γ ψ ( x ) (cid:17) + x ↔
0+ 1 s ( δ αµ δ βν + δ αν δ βµ − g µν g αβ ) × (cid:110) − p α (cid:104) (cid:0) ¯ ψ α (cid:1) (0) γ i σ k ∗ γ j α ψ ( x ) (cid:105) A (cid:104) ¯ ψ ( x ) B j ( x ) σ βk B i (0) ψ (0) (cid:105) B + (cid:104) (cid:0) ¯ ψ α (cid:1) (0) γ i σ ∗ α ⊥ γ j α ψ ( x ) (cid:105) A (cid:104) ¯ ψ ( x ) B j ( x ) σ β ∗ B i (0) ψ (0) (cid:105) B (cid:111) + x ↔ (6.110)Starting from this point, all calculations repeat those of Sections 6.3.1 and 6.3.2 withreplacements of p ↔ p , α q ↔ β q and exchange of projectile matrix elements and thetarget ones. The result is Eq. (6.111) with these replacements so we finally get W (2 b ) µν = 4 p µ p ν β q s N c (cid:90) d k ⊥ (cid:104) k ⊥ (cid:2) f f ( α q , k ⊥ ) F f ( q, k ⊥ ) − m k ⊥ ( k, q − k ) ⊥ H f ( q, k ⊥ ) (cid:105) + 4 p µ p ν α q s N c (cid:90) d k ⊥ (cid:104) ( q − k ) ⊥ F f ( q, k ⊥ ) − m ( q − k ) ⊥ ( k, q − k ) ⊥ H f ( q, k ⊥ ) (cid:105) (6.111) Let us consider the first term in the r.h.s. of Eq. (6.5). After Fierz transformation it turnsto
12 [(¯Ξ m ( x ) γ µ Ξ m ( x ))( ¯ ψ nB (0) γ ν ψ nA (0)) + µ ↔ ν ]= − g µν m ( x ) ψ nA (0))( ¯ ψ nB (0)Ξ m ( x )) + g µν m ( x ) γ ψ nA (0))( ¯ ψ nB (0) γ Ξ ( x ))+ g µν m ( x ) γ α ψ nA (0))( ¯ ψ B (0) γ α Ξ m ( x )) + g µν m ( x ) γ α γ ψ A (0))( ¯ ψ nB (0) γ α γ Ξ ( x )) −
14 [(¯Ξ m ( x ) γ µ ψ nA (0))( ¯ ψ nB (0) γ ν Ξ m ( x )) + µ ↔ ν ] −
14 (¯Ξ m ( x ) γ µ γ ψ nA (0))( ¯ ψ nB (0) γ ν γ Ξ m ( x )) + µ ↔ ν ]+ 14 [(¯Ξ m ( x ) σ να ψ nA (0))( ¯ ψ nB (0) σ µα Ξ m ( x )) + µ ↔ ν ] − g µν m ( x ) σ αβ ψ nA (0))( ¯ ψ nB (0) σ αβ Ξ m ( x )) (6.112)Let us demonstrate that after sorting out color-singlet matrix elements the contribution W (2 c ) µν is O (cid:0) N c (cid:1) in comparison to W (2 a ) µν (and W (2 b ) µν ). Consider a typical term in the r.h.s.– 34 –f Eq. (6.112) N c s (cid:104) A, B | ¯Ξ m ( x )Γ ψ nA (0))( ¯ ψ nB (0)Γ Ξ m ( x )) | A, B (cid:105) (6.113) = N c s (cid:104) A, B | (cid:0) ¯ ψ kA α (cid:1) ( x ) A mlj ( x ) γ i /p s Γ ψ nA (0))( ¯ ψ nB (0)Γ /p s γ j B kmi ( x ) 1 β ψ lB ( x )) | A, B (cid:105)
After separation of color singlet contributions (cid:104)
A, B | ( ¯ ψ kA ( A i ) ml ψ nA )( ¯ ψ nB ( B j ) km ψ lB ) | A, B (cid:105) = (cid:104) ¯ ψ kA ( A i ) km ψ nA (cid:105) A (cid:104) ¯ ψ nB ( B j ) ml ψ lB ) (cid:105) B − if abc (cid:104) ¯ ψ kA t ckl A ai ψ nA (cid:105) A (cid:104) ¯ ψ nB B bj ψ lB (cid:105) B = 1 N c (cid:104) ¯ ψ A A i ψ A (cid:105) A (cid:104) ¯ ψ B B j ψ B (cid:105) B − if abc (cid:104) A ai ( ¯ ψ A t c t d ψ A ) (cid:105) A (cid:104) B bj ( ¯ ψ B t d ψ B ) (cid:105) B − N c f abc (cid:104) A ai ( ¯ ψ A t c ψ A ) (cid:105) A (cid:104) B bj ( ¯ ψ B ψ B ) (cid:105) B = 1 N c (cid:104) ¯ ψ A A i ψ A (cid:105) A (cid:104) ¯ ψ B B j ψ B (cid:105) B − i f abc N c − (cid:104) A ai ( ¯ ψ A t c t b ψ A ) (cid:105) A (cid:104) ¯ ψ B B j ψ B (cid:105) B = − N c ( N c − (cid:104) ¯ ψ A A i ψ A (cid:105) A (cid:104) ¯ ψ B B j ψ B (cid:105) B (6.114)we get N c s (cid:104) A, B | ¯Ξ m ( x )Γ ψ nA (0))( ¯ ψ nB (0)Γ Ξ m ( x )) | A, B (cid:105) (6.115) = − N c − s (cid:104) (cid:0) ¯ ψ A α (cid:1) A j ( x ) γ i /p Γ ψ (0) (cid:105) A (cid:104) ¯ ψ (0)Γ /p γ j B i (0) 1 β ψ B ( x )) (cid:105) B Since projectile and target matrix elements can bring s each and α and β convert to α q and β q , the typical contribution of (6.115) to W µν is ∼ N c × (cid:18) g µν ⊥ q ⊥ α q β q s , q µ ⊥ q ν ⊥ α q β q s , g µν (cid:107) q ⊥ α q β q s (cid:19) (6.116)In Ref. [28] we calculated the sum of these structures corresponding to convolution of µ and ν . In principle, one can repeat that calculation and find contribution to thesestructures separately. However, since the corresponding matrix elements of quark-quark-gluon operators are virtually unknown, in this paper we we will disregard such N c terms.Thus, the contribution of Eq. (5.6) to W µν ( q ) is given in the leading order in N c bythe sum of equations (6.46), (6.84), and (6.111). J µA ( x ) J νB (0) terms Power corrections of the second type come from the terms ¯Ψ ( x ) γ µ Ψ ( x ) ¯Ψ (0) γ ν Ψ (0) + x ↔ (7.1)– 35 –here Ψ and Ψ are given by Eq. (4.12). We get (cid:2)(cid:0) ¯ ψ A + ¯Ξ (cid:1) ( x ) γ µ (cid:0) ψ A + Ξ (cid:1) ( x ) (cid:3) [ (cid:0) ¯ ψ B + ¯Ξ (cid:1) (0) γ ν (cid:0) ψ B + Ξ (cid:1) (0) (cid:3) + x ↔
0= [ ¯ ψ A ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ B (0) (cid:3) (7.2) + [¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ B (0) (cid:3) + [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ B (0) (cid:3) + [ ¯ ψ A ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ B (0) (cid:3) + [ ¯ ψ A ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + [¯Ξ ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ B (0) (cid:3) + [ ¯ ψ A ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν Ξ (0) (cid:3) + [¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ B (0) (cid:3) + [¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ B (0) (cid:3) + [ ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) + x ↔ . First, let us demonstrate that contributions to W µν from the second to fifth lines in eq. (7.2)vanish. Obviously, matrix element of the operator in the second line vanishes. Formally, (cid:90) dx • e − iα q x • (cid:104) p A | ˆ ψ ( x • , x ⊥ ) γ µ ˆ ψ ( x • , x ⊥ ) | p A (cid:105) = δ ( α q ) (cid:104) p A | ˆ ψ (0) γ µ ˆ ψ (0) | p A (cid:105) , (cid:90) dx ∗ e − iβ q x ∗ (cid:104) p B | ˆ ψ (0) γ ν ˆ ψ (0) | p B (cid:105) = δ ( β q ) (cid:104) p B | ˆ ψ (0) γ ν ˆ ψ (0) | p B (cid:105) (7.3)and, non-formally, one hadron cannot produce DY pair on his own.It is easy to see that contributions to ˇ W µν from the third and the fourth lines in Eq.(7.2) vanish due to the absence of color-singlet structure. Indeed, let us consider for examplethe term [¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ B (0) (cid:3) = − (cid:2)(cid:0) ¯ ψ mA α (cid:1) ( x ) γ i B mni /p s γ µ ψ nA ( x ) (cid:3)(cid:2) ¯ ψ lB (0) γ ν ψ lB (0) (cid:3) (7.4)The corresponding term in ˇ W µν is − N c s (cid:104) (cid:0) ¯ ψ m α (cid:1) ( x ) γ i /p s γ µ ψ n ( x ) (cid:105) A (cid:104) ¯ ψ l (0) B mni (0) γ ν ψ l (0) (cid:105) B + µ ↔ ν (7.5)which obviously does not have color-singlet contribution. Similarly, other three terms inthe third and fourth lines in Eq. (7.2) vanish.Next, let us demonstrate that the contribution of the fifth line in Eq. (7.2) vanishes forthe same reason as in Eq. (7.3). Let is consider for example the first term in the fifth line [¯Ξ ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ B (0) (cid:3) = 2 p µ s (cid:2)(cid:0) ¯ ψ A α (cid:1) ( x ) γ i B i ( x ) /p γ j B j ( x ) 1 α ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν ψ B (0) (cid:3) = p µ N c s (cid:2)(cid:0) ¯ ψ mA α (cid:1) ( x ) γ i /p γ j α ψ mA ( x ) (cid:3)(cid:2) B ai B aj ( x ) ¯ ψ nB (0) γ ν ψ nB (0) (cid:3) (7.6) In the appendix 8.3.2 to [28] it is demonstrated that higher-order terms in the expansion Eq. (4.11)(denoted by dots) are small in our kinematical region s (cid:29) Q (cid:29) q ⊥ . – 36 –here we separated color-singlet contribution in the last line. The corresponding term in W µν is π N c p µ s (cid:90) d k ⊥ (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:90) dx ∗ d x (cid:48)⊥ e − iβx • + i ( q − k,x (cid:48) ) ⊥ (7.7) × (cid:104) (cid:0) ¯ ψ α (cid:1) ( x • , x ⊥ ) γ i /p γ j α ψ ( x • , x ⊥ ) (cid:105) A (cid:104) B ai ( x ∗ , x (cid:48)⊥ ) B aj ( x ∗ , x (cid:48)⊥ ) ¯ ψ (0) γ ν ψ (0) (cid:105) B = δ ( α q ) 132 π N c p µ s (cid:104) (cid:0) ¯ ψ α (cid:1) (0) γ i /p γ j α ψ (0) (cid:105) A × (cid:90) dx ∗ d x (cid:48)⊥ e − iβx • + i ( q,x (cid:48) ) ⊥ (cid:104) B ai ( x ∗ , x (cid:48)⊥ ) B aj ( x ∗ , x (cid:48)⊥ ) ¯ ψ (0) γ ν ψ (0) (cid:105) B = 0 Similarly, the contribution of the second term in the fifth line of Eq. (7.2) will be propor-tional to δ ( β q ) and hence vanish.Let us now discuss the non-vanishing contributions coming from last two lines in Eq.(7.2). For example, the first term in the sixth line is [¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ ν Ξ (0) (cid:3) = (cid:0) ¯ ψ mA α (cid:1) γ i B mni /p s γ µ ψ nA ( x ) ¯ ψ kB γ ν /p s γ j A klj β ψ lB (0)= (cid:2)(cid:0) ¯ ψ mA α (cid:1) γ i A klj (0) /p s γ µ ψ nA ( x ) (cid:3)(cid:2) ¯ ψ kB γ ν /p s γ j B mni ( x ) 1 β ψ lB (0) (cid:3) (7.8)Separating color-singlet contributions with the help of the formula (cid:104) ¯ ψ m A ai ψ n (cid:105) = 2 t amn N c − (cid:104) ¯ ψA i ψ (cid:105) (7.9)we get the corresponding term in ˇ W µν in the form N c N c − s (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) A j (0) γ i /p γ µ ψ ( x ) (cid:105) A (cid:104) ¯ ψ (0) γ ν /p γ j B i ( x ) 1 β ψ (0) (cid:105) B (7.10)which is similar to Eq. (6.57) with exception of extra color factor N c N c − (cid:39) N c . Consequently,as discussed in Sect. 6.2.2, non-negligible contributions come from transverse µ and ν only.We calculate them in next Section. In this section we calculate the traceless part of sixth and seventh lines Eq. (7.2). Since weconsider only transverse µ and ν , to simplify notations we will call them m and n in thisSection. Using eq. (4.12) and separating color-singlet matrix elements with the help of Eq.– 37 –7.9), we rewrite the traceless part of sixth and seventh lines in Eq. (7.2) as (cid:0) [¯Ξ ( x ) γ m ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ n Ξ (0) (cid:3) + [ ¯ ψ A ( x ) γ n Ξ ( x ) (cid:3)(cid:2) ¯Ξ (0) γ n ψ A (0) (cid:3) +[¯Ξ ( x ) γ m ψ A ( x ) (cid:3)(cid:2) ¯Ξ (0) γ n ψ B (0) (cid:3) + [ ¯ ψ A ( x ) γ m Ξ ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ n Ξ (0) (cid:3) + m ↔ n (cid:1) = 12( N c − s (cid:16)(cid:2)(cid:0) ¯Ψ A α (cid:1) ( x ) γ j /p γ m A k (0) ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ n /p γ k B j ( x ) 1 β ψ B (0) (cid:3) + (cid:2) ¯Ψ A ( x ) γ n /p γ j A k (0) 1 α ψ A ( x ) (cid:3)(cid:2)(cid:0) ¯ ψ B β (cid:1) (0) γ k /p γ m B j ( x ) ψ B (0) (cid:3) + (cid:2)(cid:0) ¯Ψ A α (cid:1) ( x ) γ j /p γ m A k (0) ψ A ( x ) (cid:3)(cid:2)(cid:0) ¯ ψ B β (cid:1) (0) γ k /p γ n B j ( x ) ψ B (0) (cid:3) + (cid:2) ¯Ψ A ( x ) γ m /p γ j A k (0) 1 α ψ A ( x ) (cid:3)(cid:2) ¯ ψ B (0) γ n /p γ k B j ( x ) 1 β ψ B (0) (cid:3) + m ↔ n (cid:17) (7.11)To save space, hereafter we do not display subtraction of trace with respect to m, n indices but it is always assumed. Using formulas (11.22) we can write down the contributionto ˇ W µν from sixth and seventh lines in Eq. (7.2) in the form ˇ W mn = N c ( N c − s (cid:16) (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) /p ˘ A m (0) ψ ( x ) (cid:105) A (cid:104) ¯ ψ (0) ˘ B n ( x ) /p β ψ (0) (cid:105) B + (cid:104) ¯ ψ ( x ) ˘ A m (0) /p α ψ ( x ) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) /p ˘ B n ( x ) ψ (0) (cid:105) B + (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) /p ˘ A m (0) ψ ( x ) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) /p ˘ B n ( x ) ψ (0) (cid:105) B + (cid:104) ¯ ψ ( x ) ˘ A m (0) /p (cid:0) α ψ (cid:1) ( x ) (cid:105) A (cid:104) ¯ ψ (0) ˘ B n /p (cid:0) β ψ (cid:1) (0) (cid:105) B + m ↔ n (cid:17) + x ↔ , = N c ( N c − s (cid:16) (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) /p ˘ A m (0) ψ ( x ) + ¯ ψ ( x ) ˘ A m (0) /p α ψ ( x ) (cid:105) A × (cid:104) (cid:0) ¯ ψ β (cid:1) (0) /p ˘ B n ( x ) ψ (0) (cid:105) B + ¯ ψ (0) ˘ B n ( x ) /p β ψ (0) (cid:105) B + m ↔ n (cid:17) + x ↔ (7.12)– 38 –et us now consider corresponding matrix elements. It is easy to see that π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:104) ¯ ψ ( x • , x ⊥ ) ˘ A i (0) /p α ψ ( x • , x ⊥ ) (cid:105) A = 1 α π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) x • −∞ dx (cid:48)• (cid:104) ¯ ψ ( x • , x ⊥ ) /p [ F ∗ i + iγ ˜ F ∗ i ](0) ψ ( x (cid:48)• , x ⊥ ) (cid:105) A = k i α j ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:104) (cid:0) ¯ ψ α (cid:1) ( x • , x ⊥ ) /p ˘ A i (0) ψ ( x • , x ⊥ ) (cid:105) A = − α π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) x • −∞ dx (cid:48)• (cid:104) ¯ ψ ( x (cid:48)• , x ⊥ ) /p [ F ∗ i − iγ ˜ F ∗ i ](0) ψ ( x • , x ⊥ ) (cid:105) A , = − k i α ¯ j ( α, k ⊥ ) , π s (cid:90) dx • e − iαx • (cid:104) ¯ ψ (0) ˘ A i ( x • , x ⊥ ) /p α ψ (0) (cid:105) A = − α π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) −∞ dx (cid:48)• (cid:104) ¯ ψ (0) /p [ F ∗ i + iγ ˜ F ∗ i ]( x • , x ⊥ ) ψ ( x (cid:48)• , ⊥ ) (cid:105) A = − k i α ¯ j (cid:63) ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:104) (cid:0) ¯ ψ α (cid:1) (0) /p ˘ A i ( x • , x ⊥ ) ψ (0) (cid:105) A = 1 α π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) x • −∞ dx (cid:48)• (cid:104) ¯ ψ ( x (cid:48)• , ⊥ ) /p [ F ∗ i − iγ ˜ F ∗ i ]( x • , x ⊥ ) ψ (0) (cid:105) A , = k i α j (cid:63) ( α, k ⊥ ) , (7.13)where we used parametrization (11.49). For the target matrix elements, we obtain π s (cid:90) d x ⊥ dx ∗ e − iβx ∗ + i ( k,x ) ⊥ (cid:104) ¯ ψ ( x • , x ⊥ ) ˘ B i (0) /p β ψ ( x ∗ , x ⊥ ) (cid:105) B = k i β j ( β, k ⊥ ) , π s (cid:90) d x ⊥ dx ∗ e − iαx ∗ + i ( k,x ) ⊥ (cid:104) (cid:0) ¯ ψ β (cid:1) ( x ∗ , x ⊥ ) /p ˘ B i (0) ψ ( x ∗ , x ⊥ ) (cid:105) A = − k i β ¯ j ( β, k ⊥ ) , π s (cid:90) d x ⊥ dx ∗ e − iβx ∗ + i ( k,x ) ⊥ (cid:104) ¯ ψ (0) ˘ B i ( x ∗ , x ⊥ ) /p β ψ (0) (cid:105) A = − k i β ¯ j (cid:63) ( β, k ⊥ ) , π s (cid:90) d x ⊥ dx ∗ e − iβx ∗ + i ( k,x ) ⊥ (cid:104) (cid:0) ¯ ψ β (cid:1) (0) /p ˘ B i ( x ∗ , x ⊥ ) ψ (0) (cid:105) A = k i β j (cid:63) ( β, k ⊥ ) , (7.14)The corresponding contribution to (traceless) W ( α q , β q , x ⊥ ) takes the form W mn ( q ) − trace = s/ π ) N c (cid:90) d x e − iqx (cid:0) ˇ W mn ( x ⊥ ) − trace (cid:1) , = 1( N c − α q β q s (cid:90) d k ⊥ (cid:16) [( j − ¯ j )( α q , k ⊥ )( j (cid:63) − ¯ j (cid:63) )( β q , ( q − k ) ⊥ ) + c . c . ] × [ k m ( q − k ) n + m ↔ n + g mn ( k, q − k ) ⊥ ] (7.15)where we have recovered the subtraction of trace.– 39 –he trace part can be obtained in a similar way. Using Eq. (11.21) one gets g mn ˇ W mn = 2 N c ( N c − s (cid:16) (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) ˘ A m (0) /p ψ ( x ) (cid:105) A (cid:104) ¯ ψ (0) /p ˘ B m ( x ) 1 β ψ (0) (cid:105) B + (cid:104) ¯ ψ ( x ) /p ˘ A m (0) 1 α ψ ( x ) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) ˘ B m ( x ) /p ψ (0) (cid:105) B + (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) ˘ A m (0) /p ψ ( x ) (cid:105) A (cid:104) (cid:0) ¯ ψ β (cid:1) (0) ˘ B m ( x ) /p ψ (0) (cid:105) B + (cid:104) ¯ ψ ( x ) /p ˘ A m (0) (cid:0) α ψ (cid:1) ( x ) (cid:105) A (cid:104) ¯ ψ (0) /p ˘ B m (cid:0) β ψ (cid:1) (0) (cid:105) B + m ↔ n (cid:17) + x ↔ , = 2 N c ( N c − s (cid:16) (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) ˘ A m (0) /p ψ ( x ) + ¯ ψ ( x ) /p ˘ A m (0) 1 α ψ ( x ) (cid:105) A × (cid:104) (cid:0) ¯ ψ β (cid:1) (0) ˘ B m ( x ) /p ψ (0) (cid:105) B + ¯ ψ (0) /p ˘ B m ( x ) 1 β ψ (0) (cid:105) B (cid:17) + x ↔ (7.16)The corresponding contribution to trace part of W ( α q , β q , x ⊥ ) takes the form g mn W mn ( q ) = s/ π ) N c (cid:90) d x e − iqx g mn (cid:0) ˇ W mn ( x ⊥ ) − trace (cid:1) , (7.17) = − N c − α q β q s (cid:90) d k ⊥ ( k, q − k ) ⊥ [( j − ¯ j )( α q , k ⊥ )( j (cid:63) − ¯ j (cid:63) )( β q , ( q − k ) ⊥ ) + c . c . ] which agrees with Eq. (6.2) from Ref. [28] after replacements j = j tw32 − i ˜ j tw32 and ¯ j = j tw31 + i ˜ j tw31 . It should be noted that the difference between j and j in traceless vs trace part is due to difference in formulas (11.22) and (11.21).Thus, the result is the sum of Eqs. (7.15) and (7.17) W µν ( q ) = s/ π ) N c (cid:90) d x e − iqx ˇ W mn ( x ⊥ )= 1( N c − α q β q s (cid:90) d k ⊥ (cid:16) [( j − ¯ j )( α q , k ⊥ )( j (cid:63) − ¯ j (cid:63) )( β q , ( q − k ) ⊥ ) + c . c . ] × [ k µ ( q − k ) ν + µ ↔ ν + g ⊥ µν ( k, q − k ) ⊥ ] − g ⊥ µν ( k, q − k ) ⊥ [( j − ¯ j )( α q , k ⊥ )( j (cid:63) − ¯ j (cid:63) )( β q , ( q − k ) ⊥ ) + c . c . ] (7.18)As we mentioned in the Introduction, in this paper we we will take into account onlyleading and sub-leading terms in N c and leave the N c corrections discussed above for futurepublications.Finally, as proved in Appendix 11.5, we can neglect contributions proportional to theproduct of quark and gluon TMDs. Assembling Eqs. (5.10), (6.46), (6.84), (6.111), and (7.18) we get the result for W µν ( q ) thatconsists of two parts: W µν ( q ) = W µν ( q ) + W µν ( q ) (8.1)– 40 –he first, gauge-invariant, part is given by W µν ( q ) = W Fµν ( q ) + W Hµν ( q ) ,W Fµν ( q ) = (cid:88) f e f W fFµν ( q ) , W fFµν ( q ) = 1 N c (cid:90) d k ⊥ F f ( q, k ⊥ ) W Fµν ( q, k ⊥ ) ,W Hµν ( q ) = (cid:88) f e f W fHµν ( q ) , W fHµν ( q ) = 1 N c (cid:90) d k ⊥ H f ( q, k ⊥ ) W Hµν ( q, k ⊥ ) (8.2)where F f and H f are given by Eq. (6.29) and W Fµν ( q, k ⊥ ) = − g ⊥ µν + 1 Q (cid:107) ( q (cid:107) µ q ⊥ ν + q (cid:107) ν q ⊥ µ ) + q ⊥ Q (cid:107) q (cid:107) µ q (cid:107) ν + ˜ q µ ˜ q ν Q (cid:107) [ q ⊥ − k, q − k ) ⊥ ] − (cid:104) ˜ q µ Q (cid:107) (cid:16) g ⊥ νi − q (cid:107) ν q i Q (cid:107) (cid:17) ( q − k ) i ⊥ + µ ↔ ν (cid:105) (8.3) m W Hµν ( q, k ⊥ ) (8.4) = − k ⊥ µ ( q − k ) ⊥ ν − k ⊥ ν ( q − k ) ⊥ µ − g ⊥ µν ( k, q − k ) ⊥ + 2 ˜ q µ ˜ q ν − q (cid:107) µ q (cid:107) ν Q (cid:107) k ⊥ ( q − k ) ⊥ − (cid:16) q (cid:107) µ Q (cid:107) (cid:2) k ⊥ ( q − k ) ⊥ ν + k ⊥ ν ( q − k ) ⊥ (cid:3) + ˜ q µ Q (cid:107) (cid:2) k ⊥ ( q − k ) ⊥ ν − k ⊥ ν ( q − k ) ⊥ (cid:3) + µ ↔ ν (cid:17) − ˜ q µ ˜ q ν + q (cid:107) µ q (cid:107) ν Q (cid:107) (cid:2) q ⊥ − k, q − k ) ⊥ (cid:3) ( k, q − k ) ⊥ − q (cid:107) µ ˜ q ν + ˜ q µ q (cid:107) ν Q (cid:107) (2 k − q, q ) ⊥ ( k, q − k ) ⊥ where q (cid:107) µ ≡ α q p + β q p and ˜ q µ ≡ α q p − β q p . These are the same expressions as in Eq.(2.3) if one identifies x A with α q and x B with β q and neglects O (cid:0) m s (cid:1) terms in p A and p B and O (cid:0) q ⊥ Q (cid:1) corrections due to difference between Q and Q (cid:107) . It is easy to see that q µ W Fµν = 0 and q µ W Hµν = 0 . Note that q µ W Fµν and q µ W Hµν are exactly zero withoutany q ⊥ Q corrections. This is similar to usual “forward” DIS, but different from off-forwardDVCS where the cancellations of right-hand sides of Ward identities involve infinite towersof twists [41–43]The second part is W µν ( q ) = 1 N c (cid:88) f e f Q (cid:90) d k ⊥ (cid:20) m (cid:8) [ k ⊥ µ ( q − k ) ⊥ ν + µ ↔ ν ]( k, q − k ) ⊥ − k ⊥ ( q − k ) ⊥ µ ( q − k ) ⊥ ν − ( q − k ⊥ ) k ⊥ µ k ⊥ ν + g ⊥ µν ( k, q − k ) ⊥ − g ⊥ µν k ⊥ ( q − k ⊥ ) (cid:3)(cid:9) H fA ( q, k ⊥ )+ N c N c − (cid:16) [ k ⊥ µ ( q − k ) ⊥ ν + µ ↔ ν + g ⊥ µν ( k, q − k ) ⊥ ] J f ( q, k ⊥ ) − g ⊥ µν ( k, q − k ) ⊥ J f ( q, k ⊥ ) (cid:17) + O (cid:0) N c (cid:1)(cid:21) + O (cid:0) Q ⊥ Q (cid:1) (8.5)where H A is given by Eq. (6.71) and J f ( q, k ⊥ ) = ( j − ¯ j )( α q , k ⊥ )( j (cid:63) − ¯ j (cid:63) )( β q , ( q − k ) ⊥ ) + c . c .J f ( q, k ⊥ ) = ( j − ¯ j )( α q , k ⊥ )( j (cid:63) − ¯ j (cid:63) )( β q , ( q − k ) ⊥ ) + c . c . (8.6)– 41 –hese terms are not gauge invariant: q µ W µν ( q ) (cid:54) = 0 . The reason is that gauge invarianceis restored after adding terms like m ⊥ Q × Eq . (5 . which we do not calculate in this paper.Indeed, for example, q µ W µν ( q ) ∼ q ⊥ ν q ⊥ α q β q s and q µ × p µ q ν ⊥ q ⊥ α q β q s = q ⊥ ν q ⊥ α q β q s (8.7)They are of the same order so one should expect that gauge invariance is restored aftercalculation of the terms ∼ p µ q ν ⊥ q ⊥ α q β q s which are beyond the scope of this paper. For the samereason we see that all structures in Eq. (5.11) except g ⊥ µν q ⊥ α q β q s and q ⊥ µ q ⊥ ν α q β q s are determined byleading-twist TMDs f and h ⊥ .Sometimes it is convenient to represent hadronic tensor in transverse coordinate space.Introducing f ( α, b ⊥ )¯ f ( α, b ⊥ ) (cid:27) = (cid:90) d k ⊥ π e i ( k,b ) ⊥ (cid:26) f ( α, k ⊥ )¯ f ( α, k ⊥ ) (8.8)(and similarly for target TMDs) we get W Fµν ( α q , β q , b ⊥ ) (8.9) = 4 π (cid:88) f e f (cid:110) − g ⊥ µν + iQ (cid:107) ( q (cid:107) µ ∂ ⊥ ν + q (cid:107) ν ∂ ⊥ µ ) − q (cid:107) µ q (cid:107) ν + ˜ q µ ˜ q ν Q (cid:107) ∂ ⊥ (cid:105) f ¯ f − q µ ˜ q ν Q (cid:107) ( ∂ i f f )( ∂ i ¯ f f ) − (cid:104) ˜ q µ Q (cid:107) (cid:16) δ iν − q (cid:107) ν Q (cid:107) i∂ i (cid:17) ( f ∂ i ¯ f f − ¯ f ∂ i f f ) + µ ↔ ν (cid:105) + f ↔ ¯ f (cid:111) + O ( α s ) where f = f ( α q , b ⊥ ) , ¯ f ≡ ¯ f ( β q , b ⊥ ) everywhere except f ↔ ¯ f terms where it is opposite(the question about rapidity cutoffs for TMDs will be addressed in Sect. 9).Similarly, we can write down W H contribution in coordinate space. For future use,however, it is convenient to define Fourier transform in a slightly different way. Introduce h i ( k ⊥ ) ≡ k i h ⊥ ( k ) , ¯ h i ( k ⊥ ) ≡ k i ¯ h ⊥ ( k ) and h i ( α, b ⊥ )¯ h i ( α, b ⊥ ) (cid:27) = (cid:90) d k ⊥ π e i ( k,b ) ⊥ (cid:26) h i ( α, k ⊥ )¯ h i ( α, k ⊥ ) (8.10)then W Hµν can be represented as m W Hµν ( α q , β q , b ⊥ ) = 4 π (cid:88) f e f (cid:16) g ⊥ µν h fj ¯ h fj − h fµ ¯ h fν − h ν ¯ h fµ + 2 q (cid:107) µ q (cid:107) ν − ˜ q µ ˜ q ν Q (cid:107) ∂ i h fi ∂ j ¯ h fj + q (cid:107) µ Q (cid:107) (cid:2) ( i∂ i h fi )¯ h fν + h fν i∂ i ¯ h fi (cid:3) + ˜ q µ Q (cid:107) (cid:2) ( i∂ i h fi )¯ h fν − h fν i∂ i ¯ h fi (cid:3) + µ ↔ ν (cid:17) − ˜ q µ ˜ q ν + q (cid:107) µ q (cid:107) ν Q (cid:107) × (cid:2) ∂ ⊥ ( h fi ¯ h if ) + ∂ i h fj ∂ i ¯ h fj (cid:3) − q (cid:107) µ ˜ q ν + ˜ q µ q (cid:107) ν Q (cid:107) ∂ i [( h fj ∂ ¯ h jf − ¯ h fj ∂h jf )] + h ↔ ¯ h (8.11)where h i = h i ( α q , k ⊥ ) and ¯ h i ≡ ¯ h i ( β q , ( q − k ) ⊥ ) everywhere except h ↔ ¯ h terms where it isopposite, cf. Eq. (8.9). – 42 – .2 Four Lorentz structures of hadronic tensor The four Lorentz structures of hadronic tensor in Collins-Soper frame are given by Eq. (1.2)where ( Q ⊥ ≡ | q ⊥ | ) Z = ˜ qQ (cid:107) = 1 Q (cid:107) ( α q p − β q p ) , X = (cid:104) Q ⊥ Q (cid:107) Q ( α q p + β q p ) + Q (cid:107) Q ⊥ Q q ⊥ (cid:105) (8.12)such that q · X = q · Z = X · Z = 0 and X = Z = − .First, let us check the structure corresponding to the total cross section of DY pairproduction. From Eq. (8.1) we get W µµ ( q ) = − N c (cid:88) f e f (cid:90) d k ⊥ (cid:110)(cid:2) − k, q − k ) ⊥ Q (cid:3) F f ( q, k ⊥ ) (8.13) + 2 k ⊥ ( q − k ) ⊥ m N Q H f ( q, k ⊥ ) + N c N c − k, q − k ) ⊥ J f ( q, k ⊥ ) (cid:111)(cid:104) O (cid:0) N c (cid:1)(cid:105) + O (cid:0) q ⊥ Q (cid:1) which agrees with Eq. (6.2) from Ref. [28]. This equation gives the sum of structures W µµ = − (2 W T + W L ) . W L The easiest structure to get is W L . Multiplying Eq. (8.1) by Z µ Z ν and comparing to Eq.(1.1) we get W L ( q ) = Z µ Z ν W µν ( q ) = (cid:88) f e f Q N c (cid:90) dk ⊥ (cid:110) ( q − k ) ⊥ F f ( q, k ⊥ ) (8.14) + 1 m (cid:0) k ⊥ ( q − k ) ⊥ − [ k ⊥ + ( q − k ) ⊥ ]( k, q − k ) ⊥ (cid:1) H f ( q, k ⊥ ) (cid:111)(cid:104) O (cid:0) q ⊥ Q (cid:1) + O (cid:0) N c (cid:1)(cid:105) Thus, one may say that W L is known at LHC energies at q ⊥ (cid:28) Q as far as f and h ⊥ areknown. W ∆ Using formula q µ ⊥ Z ν W µν = ( X · q ) ⊥ W ∆ = − Q (cid:107) Q ⊥ Q W ∆ we get W ∆ = QQ (cid:107) Q ⊥ N c (cid:88) f e f (cid:90) d k ⊥ (cid:110) ( q, q − k ) ⊥ F f ( q, k ⊥ ) − ( q, q − k ) ( k, q − k ) ⊥ m H f ( q, k ⊥ ) (cid:111)(cid:104) O (cid:0) q ⊥ Q (cid:1) + O (cid:0) N c (cid:1)(cid:105) (8.15)Again, we see that W ∆ is expressed via f and h ⊥ with great accuracy.– 43 – .2.3 W T Next, from Eqs. (8.13), (8.14) and W µµ = − (2 W T + W L ) one easily obtains W T ( q ) = 1 N c (cid:88) f e f (cid:90) d k ⊥ (cid:110)(cid:2) − q ⊥ Q (cid:107) (cid:3) F f ( q, k ⊥ )+ 12 m Q (cid:107) (cid:0) k ⊥ ( q − k ) ⊥ + [ k ⊥ + ( q − k ) ⊥ ]( k, q − k ) ⊥ (cid:1) H f ( q, k ⊥ )+ N c N c − k, q − k ) ⊥ J f ( q, k ⊥ ) (cid:111)(cid:104) O (cid:0) N c (cid:1)(cid:105) + O (cid:0) q ⊥ Q (cid:1) (8.16) W ∆∆ Finally, the easiest way to pick out W ∆∆ is to multiply W µν by q ⊥ µ q ⊥ ν /q ⊥ . One obtains fromEq. (1.1) q ⊥ µ q ⊥ ν q ⊥ W µν ( q ) = Q (cid:107) Q ( W T − W ∆∆ ) . On the other hand, from Eqs. (8.3) and (8.4)one gets q µ ⊥ q ν ⊥ q ⊥ W µν ( q )= 1 N c (cid:88) f e f (cid:90) d k ⊥ (cid:110) F f ( q, k ⊥ ) + (cid:104) ( k, q − k ) ⊥ m − q, k ) ⊥ ( q, q − k ) ⊥ m q ⊥ (cid:105) H f ( q, k ⊥ ) (cid:111) (8.17)and from Eq. (8.5) q µ ⊥ q ν ⊥ q ⊥ W µν ( q ) = 1 N c (cid:88) f e f Q (cid:107) q ⊥ (cid:90) d k ⊥ (cid:110) m (cid:8) q, k ) ⊥ ( q, q − k ) ⊥ ( k, q − k ) ⊥ (8.18) − k ⊥ ( q, q − k ) ⊥ − ( q − k ⊥ ) ( q, k ) ⊥ − q ⊥ ( k, q − k ) ⊥ + q ⊥ k ⊥ ( q − k ⊥ ) (cid:3)(cid:9) H fA ( q, k ⊥ )+ N c N c − (cid:16) [2( q, k ) ⊥ ( q, q − k ) ⊥ − q ⊥ ( k, q − k ) ⊥ ] J f ( q, k ⊥ ) + q ⊥ ( k, q − k ) ⊥ J f ( q, k ⊥ ) (cid:17)(cid:111) . Thus, we get W ∆∆ = W T − Q Q (cid:107) q ⊥ µ q ν ⊥ q ⊥ W µν = 1 N c (cid:88) f e f (cid:90) d k ⊥ (cid:110) q ⊥ Q (cid:107) F f ( q, k ⊥ ) (8.19) + (cid:16) q, k ) ⊥ ( q, q − k ) ⊥ q ⊥ − ( k, q − k ) ⊥ + 1 Q (cid:8) k ⊥ ( q − k ) ⊥ + 12 ( k, q − k ) ⊥ [ k ⊥ + ( q − k ) ⊥ ] + q ⊥ ( k, q − k ) ⊥ − q, k )( q, q − k ) ⊥ (cid:9)(cid:17) m H f ( q, k ⊥ )+ 1 m Q (cid:107) (cid:16) k ⊥ q ⊥ ( q, q − k ) ⊥ + ( q − k ⊥ ) q ⊥ ( q, k ) ⊥ − k ⊥ ( q − k ⊥ ) + ( k, q − k ) ⊥ − q, k ) ⊥ ( q, q − k ) ⊥ ( k, q − k ) ⊥ q ⊥ (cid:17) H fA ( q, k ⊥ )+ N c N c − Q (cid:107) (cid:104) ( k, q − k ) ⊥ − q, k ) ⊥ ( q, q − k ) ⊥ q ⊥ (cid:105) J f ( q, k ⊥ ) + O (cid:0) N c (cid:1)(cid:111) + O (cid:0) q ⊥ Q (cid:1) – 44 –his is the only function which has a O (cid:0) Q (cid:1) , leading- N c contribution proportional to twist-three TMD H A not related to leading-twist TMDs by equations of motion. The functions W T , W L , and W ∆ do not have such contributions (although they have such contributionsat the N c level). W i ( q ) at q ⊥ (cid:29) m Following the analysis in Ref. [28], let us estimate the relative strength of Lorentz structures W i at q ⊥ (cid:29) m . First, we assume that N c is a good parameter and leave only terms leadingin N c . Second, at q ⊥ (cid:29) m we probe the perturbative tails of TMD’s f ∼ k ⊥ and h ⊥ ∼ k ⊥ [44]. So, as long as Q (cid:29) q ⊥ (cid:29) m we can approximate f ( α z , k ⊥ ) (cid:39) f ( α q ) k ⊥ , h ⊥ ( α q , k ⊥ ) (cid:39) m N h ( α q ) k ⊥ , ¯ f (cid:39) ¯ f ( α q ) k ⊥ , ¯ h ⊥ (cid:39) m N ¯ h ( α q ) k ⊥ (8.20)(up to logarithmic corrections). Similarly, for the target we can use the estimate f ( β z , k ⊥ ) (cid:39) f ( β z ) k ⊥ , h ⊥ ( β z , k ⊥ ) (cid:39) m N h ( β z ) k ⊥ , ¯ f (cid:39) ¯ f ( β z ) k ⊥ , ¯ h ⊥ (cid:39) m N ¯ h ( β z ) k ⊥ (8.21)as long as k ⊥ (cid:28) Q . Thus, we get an estimate F f ( q, k ⊥ ) (cid:39) F f ( α q , β q ) k ⊥ ( q − k ) ⊥ , F f ( α q , β q ) ≡ f f ( α q ) ¯ f f ( β q ) + f f ↔ ¯ f f ,H f ( q, k ⊥ ) (cid:39) m H f ( α q , β q ) k ⊥ ( q − k ) ⊥ , H f ( α q , β q ) ≡ h f ( α q )¯ h f ( β q ) + h f ↔ ¯ h f (8.22)Note that due to the “positivity constraint” [45] h ⊥ ( x, k ⊥ ) ≤ m | k ⊥ | f ⊥ ( x, k ⊥ ) (8.23)we can safely assume that the functions f ( x ) and h ( x ) defined in Eqs. (8.20) and (8.21)are of the same order of magnitude. Moreover, both theoretical [46] and phenomenological[47, 48] analysis indicate that h ⊥ is several times smaller than f so in numerical estimateswe will disregard the contribution of h ⊥ . Substituting the above approximations to Eq. (8.13) we get the following estimate of thestrength of power corrections for total DY cross section [28] W µµ ( q ) = − N c (cid:88) e f (cid:90) d k ⊥ (cid:110)(cid:104) − k, q − k ) ⊥ Q (cid:105) F f ( q, k ⊥ ) + 2 k ⊥ ( q − k ) ⊥ m N Q H f ( q, k ⊥ ) (cid:111) (cid:39) − N c (cid:88) e f (cid:90) d k ⊥ (cid:110)(cid:104) − k, q − k ) ⊥ Q (cid:105) F f ( α q , β q ) k ⊥ ( q − k ) ⊥ + 2 m Q H f ( α q , β q ) k ⊥ ( q − k ) ⊥ (cid:111) (cid:39) − N c (cid:88) e f (cid:90) d k ⊥ (cid:104) − k, q − k ) ⊥ Q (cid:105) F f ( α q , β q ) k ⊥ ( q − k ) ⊥ (8.24)– 45 –here we used estimates (8.22) and the fact that ( k, q − k ) ⊥ ∼ q ⊥ (cid:29) m . Thus, the relativeweight of the leading term and power correction is determined by the factor − ( k,q − k ) ⊥ Q .Due to Eqs. (8.20) and (8.21), the integrals over k ⊥ are logarithmic and should be cut frombelow by m N and from above by Q so we get an estimate (cid:90) d k ⊥ k ⊥ ( q − k ) ⊥ (cid:39) πq ⊥ ln q ⊥ m , (cid:90) d k ⊥ ( k, q − k ) ⊥ k ⊥ ( q − k ) ⊥ (cid:39) − π ln Q q ⊥ (8.25)where we assumed that the first integral is determined by the logarithmical region q ⊥ (cid:29) k ⊥ (cid:29) m N and the second by Q (cid:29) k ⊥ (cid:29) q ⊥ . Taking these integrals to Eq. (8.24) oneobtains W µµ ( q ) = − πN c (cid:88) e f (cid:104) q ⊥ ln q ⊥ m N + 1 Q ln Q q ⊥ (cid:105) F f ( α q , β q ) (8.26)By this estimate, the power correction reaches the level of few percent at q ⊥ ∼ Q . W T Let us now consider estimates described in Sect. 8.3.1 for W T given by Eq. (8.16). At large N c , we can omit the third line so W T ( q ) = 1 N c (cid:88) f e f (cid:90) d − k ⊥ (cid:104) − q ⊥ Q (cid:105) F f ( q, k ⊥ ) (8.27) + 12 m Q (cid:0) k ⊥ ( q − k ) ⊥ + [ k ⊥ + ( q − k ) ⊥ ]( k, q − k ) ⊥ (cid:1) H f ( q, k ⊥ ) (cid:39) N c (cid:88) f e f (cid:90) d − k ⊥ (cid:110)(cid:104) − q ⊥ Q (cid:105) F f ( α q , β q ) k ⊥ ( q − k ) ⊥ + m Q (cid:16) k ⊥ + ( q − k ) ⊥ ] ( k, q − k ) ⊥ k ⊥ ( q − k ) ⊥ (cid:17) H f ( α q , β q ) k ⊥ ( q − k ) ⊥ (cid:111) (8.28)Again, due to q ⊥ (cid:29) m the second term in braces can be neglected and we get W T ( q ) (cid:39) πN c (cid:104) q ⊥ − Q (cid:105) ln q ⊥ m (cid:88) e f F f ( α q , β q ) (8.29)Thus, for W T the power correction reaches 10% level at q ⊥ ∼ Q . W L Again, using estimates from Sect. 8.3.1 one obtains W L ( q ) = (cid:88) f e f Q N c (cid:90) dk ⊥ k ⊥ ( q − k ) ⊥ (cid:110) ( q − k ) ⊥ F f ( α q , β q )+ m (cid:16) − [ k ⊥ + ( q − k ) ⊥ ] ( k, q − k ) ⊥ k ⊥ ( q − k ) ⊥ (cid:17) H f ( q, k ⊥ ) (cid:111) (8.30)which gives approximately W L ( q ) (cid:39) πQ N c (cid:104) ln q ⊥ m N + 2 ln Q q ⊥ (cid:105) (cid:88) f e f F f ( α q , β q ) (8.31)– 46 –n agreement with Eqs. (8.26) and (8.29). The estimate of the ratio of W L /W T is W L ( q ) W T ( q ) (cid:39) q ⊥ Q (cid:104) Q /q ⊥ ln q ⊥ /m (cid:105) (8.32) W ∆ It is easy to see that W ∆ vanishes if one uses the estimates (8.20) and (8.21). Indeed, withthese formulas F ( q, k ⊥ ) and H ( q, k ⊥ ) are symmetric under replacement k ⊥ ↔ ( q − k ) ⊥ whereas ( q, q − k ) ⊥ in the integrand in Eq. (8.15) is antisymmetric. Moreover, this vanish-ing of W ∆ will occur for any factorizable model of TMDs f and h ⊥ : if f ( α, k ⊥ ) = f ( α ) φ ( k ⊥ ) and h ⊥ ( α, k ⊥ ) = h ( α ) ψ ( k ⊥ ) the integral (8.15) vanishes. On the other hand, W ∆ is only ∼ Q ⊥ Q W T so without better knowledge of TMDs it is impossible to tell whether W ∆ issmaller or bigger than, say, W L . Also, if the parameter α q QQ ⊥ is not negligible, to compare W ∆ and W L one needs to take into account O ( α q ) corrections to W ∆ defined by TMDsother than f and h ⊥ . W ∆∆ Let us consider the relative weight of the terms in the r.h.s. of Eq. (8.19). As we mentioned,we assume that N c is a valid small parameter so we can omit the last J term. Also, itis natural to assume that H fA ( q, k ⊥ ) is of the same order of magnitude as H f ( q, k ⊥ ) and,since the term with H A is a power correction, it is not unreasonable to neglect this term inthe first approximation. Using now estimates (8.22) and the integrals (cid:90) d k k i ( q − k ) j k ⊥ ( q − k ) ⊥ θ ( k ⊥ − m ) θ (( q − k ) ⊥ − m ) (cid:39) π q ⊥ (cid:16) g ⊥ ik + 2 q i q k q ⊥ (cid:17) ln q ⊥ m (cid:90) d k k i ( q − k ) j k ⊥ ( q − k ) ⊥ θ ( k ⊥ − m ) θ (( q − k ) ⊥ − m ) (cid:39) πq ⊥ (cid:16) g ⊥ ik + 4 q i q k q ⊥ (cid:17) ln q ⊥ m (8.33)one gets an estimate W ∆∆ (cid:39) N c (cid:88) f e f (cid:90) d k ⊥ (cid:110) q ⊥ Q (cid:107) F f ( q, k ⊥ )+ (cid:16) q, k ) ⊥ ( q, q − k ) ⊥ q ⊥ − ( k, q − k ) ⊥ + 1 Q (cid:2) k ⊥ ( q − k ) ⊥ + 12 ( k, q − k ) ⊥ [ k ⊥ + ( q − k ) ⊥ ] + q ⊥ ( k, q − k ) ⊥ − q, k )( q, q − k ) ⊥ (cid:3)(cid:17) m H f ( q, k ⊥ ) (cid:39) πQ N c ln q ⊥ m (cid:88) f e f (cid:104) F f ( α q , β q ) + 4 m Q q ⊥ (cid:16) − q ⊥ Q (cid:17) H f ( α q , β q ) (cid:105) (8.34) It is easy to see that if one neglects H in Eq. (8.14) the ratio of W L and W ∆∆ is approxi-mately W L W ∆∆ (cid:39) Q /q ⊥ ln q ⊥ /m (8.35)It seems like the Lam-Tung relation works better if we move closer to the domain of collinearfactorization Q ∼ Q ⊥ (cid:29) m . – 47 – .3.8 Estimates of asymmetries The differential cross section of DY process is parametrized as (cid:16) dσd q (cid:17) − dσd Ω d q = 34 π ( λ + 3) (cid:0) λ cos θ + µ sin 2 θ cos φ + ν θ cos 2 φ (cid:1) (8.36)where Ω is the solid angle of the lepton in terms of its polar and azimuthal angles inthe center-of-mass system of the lepton pair. The angular coefficients λ , µ , and ν can beexpressed in terms of the hadronic tensor: λ = W T − W L W T + W L , µ = W ∆ W T + W L , ν = 2 W ∆∆ W T + W L (8.37)For an estimate, let us take s =8 TeV and Q =90 GeV so that x A ∼ x B ∼ . in central regionof rapidity. Although we did not include the contribution of Z -boson, we can compare ourorder-of-magnitude estimates with experimental data at this kinematics [49, 50]. Let ustake Q ⊥ = 20 GeV so the power corrections ∼ Q ⊥ Q are small but sizable, of order of few percent. At this kinematics, we obtain − λ = 2 W L W T + W L (cid:39) ln Q /q ⊥ ln q ⊥ /m Q q ⊥ − + 2 ln Q /q ⊥ ln q ⊥ /m (cid:39) . (8.38)from Eq. (8.31) which agrees with estimates in Ref. [51]. Next, in our kinematics theexpression in square brackets in the r.h.s. in Eq. (8.34) is approximately F + 0 . H . Sincethe Boer-Mulders function seem to be of order of few percent of f (see the discussion inSect. 8.3.1), the term with H can be safely neglected and we get ν = 2 W ∆∆ W T + W L (cid:39) Q q ⊥ − + 2 ln Q /q ⊥ ln q ⊥ /m (cid:39) . (8.39)As to µ coefficient, as we mentioned, we cannot estimate it since with factorization hy-pothesis for TMDs it vanishes. Reversing the argument, if µ will be measured to be muchsmaller than ν , it will be an argument in favor of factorization hypothesis for TMDs f and h ⊥ . Actually, there are experiments at much lower q ⊥ and Q ∼ few GeV which indicatethat µ is very small [52].Last but not least, let us estimate Lam-Tung relation. With our approximation in theabove kinematics we get W L W ∆∆ (cid:39) Q /q ⊥ ln q ⊥ /m (cid:39) . (8.40)so it seems to be violated at this kinematics. Again, these order-of-magnitude estimates donot include the contribution to DY cross section mediated by the Z -boson.– 48 – Coefficient functions and matching of rapidity cutoffs
The result (8.2) is a tree-level formula and to fully understand Eq. (1.3) we should specifythe rapidity cutoffs for f ’s and h ⊥ ’s. As we discussed in section 3, the rapidity cutoff forlongitudinal momenta in f ( α q , k ⊥ ) is β ≤ σ p and for f ( β q , k ⊥ ) α ≤ σ t , where σ p and σ t are rapidity bounds for central fields. To avoid double counting, the region where both α < σ t and β < σ p should give only small power corrections. This is achieved if one takes σ p , σ t ∼ Q ⊥ √ s so power corrections from double counting are Q ⊥ Q . In this case, the region α q > α > σ t , β q > β > σ p gives Sudakov double-log factor C (cid:16) q, k, α q σ t , β q σ p (cid:17) ∼ e − αscFπ ln αqσt ln βqσp (9.1)where the coefficient α s c F π is two times γ cusp for quarks. A more precise formula can beobtained from the requirement that the product of two TMDs and the coefficient function(9.1) does not depend on the “rapidity divides” σ p and σ t . For simplicity, let us start withthe leading-twist term ∼ g ⊥ µν F . Rapidity evolution of the function f ( α q , b ⊥ ; ς p ) was foundin Ref. [53] dd ln ς p f ( α q , b ⊥ ; ς p ) = α s C F π (cid:2) − ln α q ς p −
12 ln b ⊥ s − γ E + O ( α s ) (cid:3) f ( α q , b ⊥ ; ς p ) dd ln ς t f ( β q , b ⊥ ; ς t ) = α s C F π (cid:2) − ln β q ς t −
12 ln b ⊥ s − γ E + O ( α s ) (cid:3) f ( β q , b ⊥ ; ς t ) (9.2)where ς p = σ p b ⊥ √ s , ς t = σ t b ⊥ √ s are b ⊥ -dependent cutoffs providing conformal invarianceof the leading-order TMD rapidity evolution (in the coordinate space) and γ E is Euler’sconstant. Similar equation holds true for ¯ f since it is obtained from the evolution of thesame operator.Looking at Eqs. (9.1) and (9.2) one can guess that the coefficient function ∼ g µν ⊥ timestwo TMDs f in the coordinate space has the form W g ⊥ ( α q , β q , b ⊥ ) ∼ M ( α q , β q , b ⊥ ; ς p , ς t ) (cid:2) f ( α q , b ⊥ ; ς p ) ¯ f ( β q , b ⊥ ; ς t ) + f ↔ ¯ f (cid:3) (9.3)where M ( α q , β q , b ⊥ ; ς p , ς t ) = e − αscFπ ln (cid:0) αqb ⊥√ sςt e γE (cid:1) ln (cid:0) βqb ⊥√ sςp e γE (cid:1) + αscF π ln ς p ς t (cid:2) O ( α s ) (cid:3) (9.4)It is easy to check that with M given by Eq. (9.4) we have ddς p (r.h.s. of Eq. (9.3)) = 0 and ddς t (r.h.s. of Eq. (9.3)) = 0 so our guess (9.4) for the coefficient function is correct up to O ( α s ) terms. To write precise matching for other parts of W µν is a more complicated task. Let usstart with W Fµν terms considered in the next Section. As noted in Ref. [53], the factor ∼ γ E depends on the exact way to cut integrals over α and β . Herethe factor − γ E corresponds to “smooth” cutoffs e − ασt and e − βσp , see the discussion in Ref. [53] The Eq. (9.4) is obtained in the leading order in α s so the argument of coupling constant is leftundetermined. One should expect Sudakov formula with running coupling constant [11, 54] at the NLOlevel. – 49 – .1 Matching for W F terms We need to multiply Eq. (8.9) in coordinate space by M ( α q , β q , b ⊥ ; ς p , ς t ) . First, recall that M ( α q , β q , b ⊥ ; ς p , ς t )[ f ( α q , b ⊥ ; ς p ) ¯ f ( β q , b ⊥ ; ς t ) + f ↔ ¯ f (cid:3) (9.5)does not actually depend on the “rapidity divides” ς p and ς t . However, the differentiation ∂∂b i affects evolution equations (9.2). In this case we modify the derivative with respect to b i as follows ˜ ∂ i f ( α q , b ⊥ ; ς p ) ≡ (cid:0) ∂ i − α s c F π b i b ⊥ ln ς p (cid:1) f ( α q , b ⊥ ; ς p ) , ˜ ∂ i f ( β q , b ⊥ ; ς t ) ≡ (cid:0) ∂ i − α s c F π b i b ⊥ ln ς t (cid:1) f ( β q , b ⊥ ; ς t ) (9.6)(and similarly for ¯ f ’s) so that the l.h.s.’ of these equations satisfy Eqs. (9.2). Notealso that ∂ i ( M f ¯ f ) = M ( f ˜ ∂ i ¯ f + ¯ f ˜ ∂ i f ) . With this definitions, one can write W Fµν in thedouble-log approximation in the form W F ( α q , β q , b ⊥ ) = (cid:88) flavors e f W fF ( α q , β q , b ⊥ ) (9.7) W fF ( α q , β q , b ⊥ ) == 4 π (cid:110) − g ⊥ µν + iQ (cid:107) ( q (cid:107) µ ∂ ⊥ ν + q (cid:107) ν ∂ ⊥ µ ) − q (cid:107) µ q (cid:107) ν + ˜ q µ ˜ q ν Q (cid:107) ∂ ⊥ (cid:105) M f ¯ f − q µ ˜ q ν Q (cid:107) M ( ˜ ∂ i f f )( ˜ ∂ i ¯ f f ) − (cid:104) ˜ q µ Q (cid:107) (cid:16) δ iν − q (cid:107) ν Q (cid:107) i∂ i (cid:17) M ( f ˜ ∂ i ¯ f f − ¯ f ˜ ∂ i f f ) + µ ↔ ν (cid:105) + f ↔ ¯ f (cid:111) + O ( α s ) where M = M ( α q , β q , b ⊥ ; ς p , ς t ) and f = f ( α q , b ⊥ , ς p ) , ¯ f ≡ ¯ f ( β q , b ⊥ , ς t ) everywhereexcept f ↔ ¯ f terms where it is opposite.It is easy to check gauge invariance: ( α q p µ + β q p µ + i∂ µ ⊥ ) W Fµν ( α q , β q , b ⊥ ) = 0 . W H terms First, with our definitions (8.10) the Eq. (11.29) reads π s (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) σ i • ψ f (0) | A (cid:105) = 1 m N h fi ( α, k ⊥ )18 π s (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) σ i • ψ f ( x • , x ⊥ ) | A (cid:105) = − m N ¯ h fi ( α, k ⊥ ) (9.8)and similarly for the target matrix elements. With such definition, the evolution equationfor h i ( α, b ⊥ ) ≡ π (cid:82) d k ⊥ e i ( k,x ) ⊥ h i ( α, k ⊥ ) is the same as Eq. (9.2) dd ln ς p h i ( α q , b ⊥ ; ς p ) = α s C F π (cid:2) − ln α q ς p − ln | b ⊥ |√ s − γ E (cid:3) h i ( α q , b ⊥ ; ς p ) dd ln ς t h i ( β q , b ⊥ ; ς t ) = α s C F π (cid:2) − ln β q ς t −
12 ln b ⊥ s − γ E (cid:3) h i ( β q , b ⊥ ; ς t ) (9.9) Strictly speaking, the difference between ˜ ∂ i f and ∂ i f is ∼ O ( α s ) but since our matching is correct atsingle-log limit (see Eq. (9.4)) we keep ˜ ∂ i f to avoid corrections ∼ O ( α s ln ς ) in our matching formulas (8.9)(and (8.11) below). – 50 –nd similarly for ¯ h i . The reason is that one-loop rapidity evolution for ˆ¯ ψ ( x • , x ⊥ )Γ ˆ ψ (0) inthe Sudakov region is the same for all matrices Γ between ˆ¯ ψ ( x • , x ⊥ ) and ˆ ψ ( due to thefact that the “handbag” diagram in Fig. 3c is small and in two other diagrams (as well asself-energy corrections) the matrix Γ between ˆ¯ ψ ( x • , x ⊥ ) and ˆ ψ (0) just multiplies the resultof calculation. + perm.(a) (b) (c) x 0 x 0x 0 Figure 3 . Typical diagrams for the rapidity evolution of quark TMD in the Sudakov regime.
Next, to write down the product of m W Hµν and the coefficient function we need modifiedderivatives of h i ’s of Eq. (9.6) type: ˜ ∂ i h j ( α q , b ⊥ ; ς p ) ≡ (cid:0) ∂ i − α s c F π b i b ⊥ ln ς p (cid:1) h j ( α q , b ⊥ ; ς p ) , ˜ ∂ i h j ( β q , b ⊥ ; ς t ) ≡ (cid:0) ∂ i − α s c F π b i b ⊥ ln ς t (cid:1) h j ( β q , b ⊥ ; ς t ) (9.10)(and similarly for ¯ h ’s) so that ˜ ∂ i h j will satisfy same evolution equations (9.9) as h j . Weget then W Hµν ( α q , β q , b ⊥ ) = (cid:88) flavors e f W fHµν ( α q , β q , b ⊥ ) ,m π W fHµν ( α q , β q , b ⊥ ) = M (cid:16) g ⊥ µν h fj ¯ h fj − h fµ ¯ h fν − h ν ¯ h fµ + 2 q (cid:107) µ q (cid:107) ν − ˜ q µ ˜ q ν Q (cid:107) ˜ ∂ i h fi ˜ ∂ j ¯ h fj + q (cid:107) µ Q (cid:107) (cid:2) ( i ˜ ∂ i h fi )¯ h fν + h fν i ˜ ∂ i ¯ h fi (cid:3) + ˜ q µ Q (cid:107) (cid:2) ( i ˜ ∂ i h fi )¯ h fν − h fν i ˜ ∂ i ¯ h fi (cid:3) + µ ↔ ν (cid:17) − ˜ q µ ˜ q ν + q (cid:107) µ q (cid:107) ν Q (cid:107) × (cid:2) ∂ ⊥ ( M h fi ¯ h if ) + M ˜ ∂ i h fj ˜ ∂ i ¯ h fj (cid:3) − q (cid:107) µ ˜ q ν + ˜ q µ q (cid:107) ν Q (cid:107) ∂ i [ M ( h fj ˜ ∂ ¯ h jf − ¯ h fj ˜ ∂h jf )] + h ↔ ¯ h (9.11)where M = M ( α q , β q ; ς p , ς t ) and h i = h i ( α q , k ⊥ , ς p ) , ¯ h i ≡ ¯ h i ( β q , ( q − k ) ⊥ , ς t ) everywhereexcept h ↔ ¯ h terms where it is opposite, cf. Eq. (8.9).Let us comment on the choice of “rapidity divides” ς p and ς t in the product M ( α q , β q , b ⊥ ; ς p , ς t ) f ( α q , b ⊥ ; ς p ) ¯ f ( β q , b ⊥ ; ς t ) (and in similar M h ¯ h product). As we men-tioned in the beginning of this Section, in order to avoid double counting one should writedown factorization of the amplitude in projectile, target and central fields at ς p , ς t ∼ .After that, as discussed in Ref. [53], one can use the double-log Sudakov evolution (9.2)– 51 –ntil ς p ≥ ˇ ς p , ˇ ς p ≡ α q b ⊥ √ s , ς t ≥ ˇ ς t , ˇ ς t ≡ β q b ⊥ √ s (9.12)At this point, the result of Sudakov evolution is M ( α q , β q , b ⊥ ; ˇ ς p , ˇ ς t ) (cid:2) f ( α q , b ⊥ ; ˇ ς p ) ¯ f ( β q , b ⊥ ; ˇ ς t ) + f ↔ ¯ f (cid:3) = e − αscF π ln Q b ⊥ (cid:2) f ( α q , b ⊥ ; ˇ ς p ) ¯ f ( β q , b ⊥ ; ˇ ς t ) + f ↔ ¯ f (cid:3) (9.13)so in the final result (8.9) one should take M f ¯ f → e − αsNc π ln Q b ⊥ (cid:104) O (cid:0) α s (cid:1)(cid:105) f ( α q , b ⊥ ; ˇ ς p ) ¯ f ( β q , b ⊥ ; ˇ ς t ) (9.14)Similarly, for W Hµν ( α q , β q , b ⊥ ) one should take M h i ¯ h j → e − αsNc π ln Q b ⊥ (cid:104) O (cid:0) α s (cid:1)(cid:105) h i ( α q , b ⊥ ; ˇ ς p )¯ h j ( β q , b ⊥ ; ˇ ς t ) (9.15)at the end of Sudakov evolution (9.9). It should be emphasized that since factor M isuniversal for (9.14) and (9.15), our estimates of asymmetries in Sect. 8.3.8 are not affectedby summation of Sudakov double logs. As discussed in Refs. [28, 29], from the rapidity factorization (3.8) we get TMDs withrapidity-only cutoff | α | < σ t or | β | < σ p (or with modifications (9.12)). Such cutoff,relevant for small- x physics, is different from the combination of UV and rapidity cutoffsfor TMDs used by moderate- x community, see the analysis in two [55–57] and three [58]loops. For the tree-level formulas of Sect. 8, this difference in cutoffs does not matter, butif one uses the formulas from Sect. 9 and integrates models for TMDs with Sudakov factor M of Eq. (9.4), one has to relate TMDs with rapidity-only cutoffs to the TMD models withconventional cutoffs. This requires calculations at the NLO level which are in progress.
10 Conclusions and outlook
Main result of this paper is Eq. (8.1) which gives the DY hadronic tensor for electromagneticcurrent at small x with gauge invariance at the Q level. The part (8.2), determined byleading-twist TMDs f and h ⊥ , is manifestly gauge invariant. The only non-gauge invariantterm at the Q level is Eq. (8.6) with transverse µ and ν which is ∼ q ⊥ µ q ⊥ ν Q times twist-3TMDs. Also, in the leading- N c approximation the only structure affected by those terms is W ∆∆ , all other structures are calculated up to O (cid:0) q ⊥ Q (cid:1) terms. It is interesting to note that Q terms necessary for gauge invariance are calculated more than than two decades afterthe calculation of Q corrections in Ref. [20].It should be mentioned that, as discussed above, our rapidity factorization is differentfrom the standard factorization scheme for particle production in hadron-hadron scattering,namely splitting the diagrams in collinear to projectile part, collinear to target part, hardfactor, and soft factor [9]. Here we factorize only in rapidity and the Q evolution arises– 52 –rom k ⊥ dependence of the rapidity evolution kernels, same as in the BK (and NLO BK[59]) equations. Also, since matrix elements of TMD operators with our rapidity cutoffsare UV-finite [60, 61], the only UV divergencies in our approach are usual UV divergenciesabsorbed in the QCD running coupling. For the tree-level result (8.1) this does not matter,but if one intends to use the result like (8.9) with Sudakov logarithms for conventionalTMDs with double UV and rapidity cutoffs, one needs to relate our TMDs with rapidity-only cutoff to conventional TMDs. Needless to say, the gauge-invariant tree-level result(8.2) should be correct for TMDs with any cutoffs.An obvious outlook is to extend these results to the “real” DY process involving Z -bosoncontributions which are relevant for our kinematics. The study is in progress.The author is grateful to V. Braun, A. Prokudin, A. Radyushkin, J. Qiu , and A.Vladimirov for valuable discussions. This work is supported by Jefferson Science Associates,LLC under the U.S. DOE contract
11 Appendix
First, let us write down Fierz transformation for symmetric hadronic tensor
12 [( ¯ ψγ µ χ )( ¯ χγ ν ψ ) + µ ↔ ν ] (11.1) = − (cid:0) δ αµ δ βν + δ αν δ βµ − g µν g αβ (cid:1)(cid:2) ( ¯ ψγ α ψ )( ¯ χγ β χ ) + ( ¯ ψγ α γ ψ )( ¯ χγ β γ χ ) (cid:3) + 14 (cid:0) δ αµ δ βν + δ αν δ βµ − g µν g αβ (cid:1) ( ¯ ψσ αξ ψ )( ¯ χσ ξβ χ ) − g µν ψψ )( ¯ χχ ) + g µν ψγ ψ )( ¯ χγ χ ) σ -matrices It is convenient to define (cid:15) ij ≡ s (cid:15) ∗• ij = 2 s p µ p ν (cid:15) µνij (11.2)such that (cid:15) = 1 and (cid:15) ij (cid:15) kl = g ik g jl − g il g jk . The frequently used formula is σ µν σ αβ = ( g µα g νβ − g µβ g να ) − i(cid:15) µναβ γ − i ( g µα σ νβ − g µβ σ να − g να σ µβ + g νβ σ µα ) (11.3)with variations s σ • i σ ∗ j = g ij − i(cid:15) ij γ − iσ ij − is g ij σ •∗ , s σ ∗ i σ • j = g ij + i(cid:15) ij γ − iσ ij + 2 s g ij σ ∗• ,σ ij σ • k = − σ • k σ ij = − ig ik σ • j + ig jk σ • i , σ ij σ ∗ k = − σ ∗ k σ ij = − ig ik σ ∗ j + ig jk σ ∗ i (11.4) We use conventions from
Bjorken & Drell where (cid:15) = − and γ µ γ ν γ λ = g µν γ λ + g νλ γ µ − g µλ γ ν − i(cid:15) µνλρ γ ρ γ . Also, with this convention ˜ σ µν ≡ (cid:15) µνλρ σ λρ = iσ µν γ . – 53 –e need also the following formulas with σ -matrices in different matrix elements ˜ σ µν ⊗ ˜ σ αβ = −
12 ( g µα g νβ − g να g µβ ) σ ξη ⊗ σ ξη + g µα σ βξ ⊗ σ ξν − g να σ βξ ⊗ σ ξµ − g µβ σ αξ ⊗ σ ξν + g νβ σ αξ ⊗ σ ξµ − σ αβ ⊗ σ µν (11.5)and ˜ σ µξ ⊗ ˜ σ ξν = − g µν σ ξη ⊗ σ ξη + σ νξ ⊗ σ ξµ , σ ξη ⊗ ˜ σ ξη = ˜ σ ξη ⊗ σ ξη (11.6) σ µξ γ ⊗ σ ξν γ + µ ↔ ν − g µν σ ξη γ ⊗ σ ξη γ = − [ σ µξ ⊗ σ ξν + µ ↔ ν − g µν σ ξη ⊗ σ ξη ] σ k ∗ ⊗ γ i σ • k γ j = ˆ p γ k ⊗ (cid:54) p γ i γ k γ j = ˆ p γ k ⊗ (cid:54) p ( g ik γ j + g jk γ i − g ij γ k )= ˆ p ( g ik γ j + g jk γ i − g ij γ k ) ⊗ (cid:54) p γ k = ( γ j σ k ∗ γ i ) ⊗ σ • k (11.7)We will need also (cid:54) p ⊗ γ i (cid:54) p γ j + (cid:54) p γ ⊗ γ i (cid:54) p γ j γ = γ j (cid:54) p γ i ⊗ (cid:54) p + γ j (cid:54) p γ i γ ⊗ (cid:54) p γ (11.8) γ -matrices and one gluon field In the gauge A • = 0 the field A i can be represented as A i ( x • , x ⊥ ) = 2 s (cid:90) x • −∞ dx (cid:48)• A ∗ i ( x (cid:48)• , x ⊥ ) (11.9)(see eq. (4.3)). We define “dual” fields by ˜ A i ( x • , x ⊥ ) = 2 s (cid:90) x • −∞ dx (cid:48)• ˜ A ∗ i ( x (cid:48)• , x ⊥ ) , ˜ B i ( x ∗ , x ⊥ ) = 2 s (cid:90) x ∗ −∞ dx (cid:48)∗ ˜ B • i ( x (cid:48)∗ , x ⊥ ) , (11.10)where ˜ F µν = (cid:15) µνλρ F λρ as usual. With this definition we have ˜ A i = − (cid:15) ij A j and ˜ B i = (cid:15) ij B j so (cid:54) p ˘ A i = − (cid:54) A (cid:54) p γ i , ˘ A i (cid:54) p = − γ i (cid:54) p (cid:54) A, (cid:54) p ˘ B i = − (cid:54) B (cid:54) p γ i , ˘ B i (cid:54) p = − γ i (cid:54) p (cid:54) B (11.11)where ˘ A i ≡ A i − i ˜ A i γ , ˘ B i ≡ B i − i ˜ B i γ (11.12)We also use A i (cid:54) p ⊗ γ n (cid:54) p γ i + A i (cid:54) p γ ⊗ γ n (cid:54) p γ i γ = − (cid:54) p ˘ A n ⊗ (cid:54) p − (cid:54) p ˘ A n γ ⊗ (cid:54) p γ A i (cid:54) p ⊗ γ i (cid:54) p γ n + A i (cid:54) p γ ⊗ γ i (cid:54) p γ n γ = − ˘ A n (cid:54) p ⊗ (cid:54) p − ˘ A n (cid:54) p γ ⊗ (cid:54) p γ γ n /p γ i ⊗ /p B i + γ n /p γ i γ ⊗ /p γ B i = − /p ⊗ /p ˘ B n − /p γ ⊗ /p ˘ B n γ γ i /p γ n ⊗ /p B i + γ i /p γ n γ ⊗ /p γ B i = − /p ⊗ ˘ B n /p − /p γ ⊗ ˘ B n /p γ (11.13)and s (cid:2) (cid:54) p (cid:54) p γ i ⊗ B i γ n + (cid:54) p (cid:54) p γ i γ ⊗ B i γ n γ (cid:3) = γ i ⊗ γ n ˘ B i + γ i γ ⊗ γ n ˘ B i γ s (cid:2) γ i (cid:54) p (cid:54) p ⊗ B i γ n + γ i (cid:54) p (cid:54) p γ ⊗ B i γ n γ (cid:3) = γ i ⊗ ˘ B i γ n + γ i γ ⊗ ˘ B i γ n γ – 54 – γ -matrices and two gluon fields With definition (11.10), we have the following formulas A i ⊗ ˜ B j = g ij ˜ A k ⊗ B k − ˜ A j ⊗ B i , ˜ A i ⊗ B j = g ij A k ⊗ ˜ B k − A j ⊗ ˜ B i (11.14) ˜ A i ⊗ ˜ B j = − g ij A k ⊗ B k + A j ⊗ B i , ⇒ ˜ A i ⊗ ˜ B i = − A i ⊗ B i , ˜ A i ⊗ B i = A i ⊗ ˜ B i Using these formulas, after some algebra one obtains γ m (cid:54) p γ j A i ⊗ γ n (cid:54) p γ i B j + γ m (cid:54) p γ j A i γ ⊗ γ n (cid:54) p γ i B j γ = (cid:54) p ˘ A n ⊗ (cid:54) p ˘ B m + (cid:54) p ˘ A n γ ⊗ (cid:54) p ˘ B m γ γ j (cid:54) p γ m A i ⊗ γ n (cid:54) p γ i B j + γ j (cid:54) p γ m A i γ ⊗ γ n (cid:54) p γ i B j γ = (cid:54) p ˘ A n ⊗ ˘ B m (cid:54) p + (cid:54) p ˘ A n γ ⊗ ˘ B m (cid:54) p γ γ m (cid:54) p γ j A i ⊗ γ i (cid:54) p γ n B j + γ m (cid:54) p γ j A i γ ⊗ γ i (cid:54) p γ n B j γ = ˘ A n (cid:54) p ⊗ (cid:54) p ˘ B m + ˘ A n (cid:54) p γ ⊗ (cid:54) p ˘ B m γ γ j (cid:54) p γ m A i ⊗ γ i (cid:54) p γ n B j + γ j (cid:54) p γ m A i γ ⊗ γ i (cid:54) p γ n B j γ = ˘ A n (cid:54) p ⊗ ˘ B m (cid:54) p + ˘ A n (cid:54) p γ ⊗ ˘ B m (cid:54) p γ (11.15)and (cid:54) p ˘ A m ⊗ (cid:54) p ˘ B n + (cid:54) p ˘ A n γ ⊗ (cid:54) p ˘ B m γ = g mn (cid:54) p ˘ A k ⊗ (cid:54) p ˘ B k (cid:54) p ˘ A m ⊗ ˘ B n (cid:54) p + (cid:54) p ˘ A n γ ⊗ γ ˘ B m (cid:54) p = g mn (cid:54) p ˘ A k ⊗ ˘ B k (cid:54) p ˘ A m (cid:54) p ⊗ (cid:54) p ˘ B n + γ ˘ A n (cid:54) p ⊗ (cid:54) p ˘ B m γ = g mn ˘ A k (cid:54) p ⊗ (cid:54) p ˘ B k ˘ A m (cid:54) p ⊗ ˘ B n (cid:54) p + γ ˘ A n (cid:54) p ⊗ γ ˘ B m (cid:54) p = g mn ˘ A k (cid:54) p ⊗ ˘ B k (cid:54) p (11.16)The corollary of Eq. (11.16) is (cid:54) p ˘ A k γ ⊗ (cid:54) p ˘ B k γ = (cid:54) p ˘ A k ⊗ (cid:54) p ˘ B k , (cid:54) p ˘ A k γ ⊗ γ ˘ B k (cid:54) p = (cid:54) p ˘ A k ⊗ ˘ B k (cid:54) p γ ˘ A k (cid:54) p ⊗ (cid:54) p ˘ B k γ = ˘ A k (cid:54) p ⊗ (cid:54) p ˘ B k , γ ˘ A k (cid:54) p ⊗ γ ˘ B k (cid:54) p = ˘ A k (cid:54) p ⊗ ˘ B k (cid:54) p (11.17)From Eqs. (11.15) and (11.16) one easily obtains γ m (cid:54) p γ j A i ⊗ γ n (cid:54) p γ i B j + γ m (cid:54) p γ j A i γ ⊗ γ n (cid:54) p γ i B j γ + m ↔ n = 2 g mn (cid:54) p ˘ A k ⊗(cid:54) p ˘ B k (11.18)and γ m (cid:54) p γ j A i ⊗ γ n (cid:54) p γ i B j + γ m (cid:54) p γ j A i γ ⊗ γ n (cid:54) p γ i B j γ − m ↔ n = 2 (cid:54) p ˘ A n ⊗ (cid:54) p ˘ B m − m ↔ n,γ j (cid:54) p γ m A i ⊗ γ i (cid:54) p γ n B j + γ j (cid:54) p γ m A i γ ⊗ γ i (cid:54) p γ n B j γ − m ↔ n = 2 ˘ A n (cid:54) p ⊗ ˘ B m (cid:54) p − m ↔ n (11.19)We need also formulas s A i (cid:54) p (cid:54) p γ j ⊗ B j (cid:54) p (cid:54) p γ i = A i γ j ⊗ B j γ i − iA i γ j γ ⊗ ˜ B j γ i + i ˜ A i γ j ⊗ B j γ i γ + ˜ A i γ j γ ⊗ ˜ B j γ i γ , s (cid:0) A i (cid:54) p (cid:54) p γ j ⊗ B j (cid:54) p (cid:54) p γ i + A i (cid:54) p (cid:54) p γ j γ ⊗ B j (cid:54) p (cid:54) p γ i γ (cid:1) = γ j ˘ A i ⊗ γ i ˘ B j + γ j ˘ A i γ ⊗ γ i ˘ B j γ ,γ i ˘ A j γ ⊗ γ j ˘ A i γ = γ i ˘ A j ⊗ γ i ˘ B j − γ i ˘ A i ⊗ γ j ˘ B j (11.20)– 55 –nd A k γ i /p γ j ⊗ B j γ i /p γ k = /p ˘ A i ⊗ /p ˘ B i = (cid:54) A (cid:54) p γ i ⊗ (cid:54) B (cid:54) p γ i ,A k γ j /p γ i ⊗ B j γ k /p γ i = ˘ A i /p ⊗ ˘ B i /p = γ i (cid:54) p (cid:54) A ⊗ γ i (cid:54) p (cid:54) B,A k γ i /p γ j ⊗ B j γ k /p γ i = /p ˘ A i ⊗ ˘ B i /p = (cid:54) A (cid:54) p γ i ⊗ γ i (cid:54) p (cid:54) B,A k γ j /p γ i ⊗ B j γ i /p γ k = ˘ A i /p ⊗ /p ˘ B i = γ i (cid:54) p (cid:54) A ⊗ (cid:54) B (cid:54) p γ i , (11.21) A k γ m /p γ j ⊗ B j γ n /p γ k + m ↔ n − g mn A k γ i /p γ j ⊗ B j γ i /p γ k = ˘ A m (cid:54) p ⊗ ˘ B n (cid:54) p + m ↔ n − g mn ˘ A k (cid:54) p ⊗ ˘ B k (cid:54) p ,A k γ j /p γ m ⊗ B j γ k /p γ n + m ↔ n − g mn A k γ j /p γ i ⊗ B j γ k /p γ i = (cid:54) p ˘ A m ⊗ (cid:54) p ˘ B n + m ↔ n − g mn (cid:54) p ˘ A k ⊗ (cid:54) p ˘ B k ,A k γ m /p γ j ⊗ B j γ k /p γ n + m ↔ n − g mn A k γ j /p γ i ⊗ B j γ k /p γ i = ˘ A m (cid:54) p ⊗ (cid:54) p ˘ B n + m ↔ n − g mn ˘ A k (cid:54) p ⊗ (cid:54) p ˘ B k ,A k γ j /p γ m ⊗ B j γ n /p γ k + m ↔ n − g mn A k γ j /p γ i ⊗ B j γ k /p γ i == (cid:54) p ˘ A m ⊗ ˘ B n (cid:54) p + m ↔ n − g mn (cid:54) p ˘ A k ⊗ ˘ B k (cid:54) p , (11.22) s (cid:2) A i (cid:54) p (cid:54) p γ j ⊗ B j γ n (cid:54) p γ i + A i (cid:54) p (cid:54) p γ j γ ⊗ B j γ ν ⊥ (cid:54) p γ i γ (cid:3) (11.23) = − γ i ˘ A n ⊗ (cid:54) p ˘ B i − γ i ˘ A n γ ⊗ (cid:54) p ˘ B i γ = γ i ˘ A n ⊗ (cid:54) B (cid:54) p γ i + γ i ˘ A n γ ⊗ (cid:54) B (cid:54) p γ i γ , s (cid:2) A i γ n (cid:54) p γ j ⊗ B j (cid:54) p (cid:54) p γ i + A i γ n (cid:54) p γ j γ ⊗ B j (cid:54) p (cid:54) p γ i γ (cid:3) = −(cid:54) p ˘ A i ⊗ γ i ˘ B n − (cid:54) p ˘ A i γ ⊗ γ i ˘ B n γ = (cid:54) A (cid:54) p γ i ⊗ γ i ˘ B n + (cid:54) A (cid:54) p γ i γ ⊗ γ i ˘ B n γ . Let us first consider matrix elements of operators without γ . The standard parametrizationof quark TMDs reads (see e.g. Ref. [62])) π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) γ µ ψ f (0) | A (cid:105) (11.24) = p µ f f ( α, k ⊥ ) + k µ ⊥ f f ⊥ ( α, k ⊥ ) + p µ m N s f f ( α, k ⊥ ) , π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) ψ f (0) | A (cid:105) = m N e f ( α, k ⊥ ) for quark distributions in the projectile and π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) γ µ ψ f ( x • , x ⊥ ) | A (cid:105) (11.25) = − p µ ¯ f f ( α, k ⊥ ) − k µ ⊥ ¯ f f ⊥ ( α, k ⊥ ) − p µ m N s ¯ f f ( α, k ⊥ ) , π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) ψ f ( x • , x ⊥ ) | A (cid:105) = m N ¯ e f ( α, k ⊥ ) – 56 –or the antiquark distributions. The corresponding matrix elements for the target are obtained by trivial replacements p ↔ p , x • ↔ x ∗ and α ↔ β : π (cid:90) dx ∗ d x ⊥ e − iβx ∗ + i ( k,x ) ⊥ (cid:104) B | ¯ ψ f ( x ∗ , x ⊥ ) γ µ ψ f (0) | B (cid:105) (11.26) = p µ f f ( β, k ⊥ ) + k µ ⊥ f f ⊥ ( β, k ⊥ ) + p µ m N s f f ( β, k ⊥ ) , π (cid:90) dx ∗ d x ⊥ e − iβx ∗ + i ( k,x ) ⊥ (cid:104) B | ¯ ψ f ( x ∗ , x ⊥ ) ψ f (0) | B (cid:105) = m N e f ( β, k ⊥ ) , and π (cid:90) dx ∗ d x ⊥ e − iβx ∗ + i ( k,x ) ⊥ (cid:104) B | ¯ ψ f (0) γ µ ψ f ( x ∗ , x ⊥ ) | B (cid:105) (11.27) = − p µ ¯ f f ( β, k ⊥ ) − k µ ⊥ ¯ f f ⊥ ( β, k ⊥ ) − p µ m N s ¯ f f ( β, k ⊥ ) , π (cid:90) dx ∗ d x ⊥ e − iβx ∗ + i ( k,x ) ⊥ (cid:104) B | ¯ ψ f (0) ψ f ( x ∗ , x ⊥ ) | B (cid:105) = m N ¯ e f ( β, k ⊥ ) . Matrix elements of operators with γ are parametrized as follows: π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) γ µ γ ψ f (0) | A (cid:105) = − i(cid:15) µ ⊥ i k i g ⊥ f ( α, k ⊥ ) , π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) γ µ γ ψ f ( x • , x ⊥ ) | A (cid:105) = − i(cid:15) µ ⊥ i k i ¯ g ⊥ f ( α, k ⊥ ) (11.28)The corresponding matrix elements for the target are obtained by trivial replacements p ↔ p , x • ↔ x ∗ and α ↔ β similarly to eq. (11.27).The parametrization of time-odd Boer-Mulders TMDs are π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) σ µν ψ f (0) | A (cid:105) = 1 m N ( k µ ⊥ p ν − µ ↔ ν ) h ⊥ f ( α, k ⊥ ) + 2 m N s ( p µ p ν − µ ↔ ν ) h f ( α, k ⊥ )+ 2 m N s ( k µ ⊥ p ν − µ ↔ ν ) h ⊥ f ( α, k ⊥ ) , π (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) σ µν ψ f ( x • , x ⊥ ) | A (cid:105) = − m N ( k µ ⊥ p ν − µ ↔ ν )¯ h ⊥ f ( α, k ⊥ ) − m N s ( p µ p ν − µ ↔ ν )¯ h f ( α, k ⊥ ) − m N s ( k µ ⊥ p ν − µ ↔ ν )¯ h ⊥ f ( α, k ⊥ ) (11.29)and similarly for the target with usual replacements p ↔ p , x • ↔ x ∗ and α ↔ β .Note that the coefficients in front of f , g ⊥ f , h and h ⊥ in eqs. (11.24), (11.26), (11.28),and (11.29) contain an extra s since p µ enters only through the direction of gauge link so In an arbitrary gauge, there are gauge links to −∞ as displayed in eq. (5.9). – 57 –he result should not depend on rescaling p → λp . For this reason, these functions do notcontribute to W ( q ) in our approximation.Last but not least, an important point in our analysis is that any f ( x, k ⊥ ) may haveonly logarithmic dependence on Bjorken x but not the power dependence ∼ x . Indeed,the low- x behavior of TMDs is determined by pomeron exchange with the nucleon. Theinteraction of TMD with BFKL pomeron is specified by so-called impact factor and it iseasy to check that the impact factors for all leading-twist TMDs are similar and do not giveextra x factors. The only x may had come from some unfortunate definition of TMD whichincludes factor s artificially, but from power counting (5.11) we see that all definitions ofleading-twist TMDs do not have such factors. In this section we will demonstrate that matrix elements of quark-antiquark-gluon operatorsfrom section 6 can be expressed in terms of leading-power matrix elements from section 11.2.First, let us note that operators α and β in Eqs. (4.13) are replaced by ± α q and ± β q in forward matrix elements. Indeed, (cid:90) dx • e − iα q x • (cid:104) ¯Φ( x • , x ⊥ )Γ 1 α + i(cid:15) ψ (0) (cid:105) A (11.30) = 1 i (cid:90) dx • (cid:90) −∞ dx (cid:48)• e − iα q x • (cid:104) ¯Φ( x • , x ⊥ )Γ ψ ( x (cid:48)• , ⊥ ) (cid:105) A = 1 α q (cid:90) dx • e − iαx • (cid:104) ¯Φ( x • , x ⊥ )Γ ψ (0) (cid:105) A where ¯Φ( x • , x ⊥ ) can be ¯ ψ ( x • , x ⊥ ) or ¯ ψ ( x • , x ⊥ ) A i ( x • , x ⊥ ) and Γ can be any γ -matrix. Sim-ilarly, (cid:90) dx • e − iα q x • (cid:104) (cid:0) ¯ ψ α − i(cid:15) (cid:1) ( x • , x ⊥ )ΓΦ(0) (cid:105) A = 1 α q (cid:90) dx • e − iαx • (cid:104) ¯ ψ ( x • , x ⊥ )ΓΦ(0) (cid:105) A (11.31) (cid:90) dx • e − iα q x • (cid:104) (cid:0) ¯ ψ α − i(cid:15) (cid:1) ( x • , x ⊥ )Γ 1 α + i(cid:15) ψ (0) (cid:105) A = 1 α q (cid:90) dx • e − iα q x • (cid:104) ¯ ψ ( x • , x ⊥ )Γ ψ (0) (cid:105) A where Φ( x • , x ⊥ ) can be ψ ( x • , x ⊥ ) or A i ( x • , x ⊥ ) ψ ( x • , x ⊥ ) . We need also (cid:90) dx • e − iα q x • (cid:104) (cid:0) ¯ ψ α − i(cid:15) (cid:1) (0)ΓΦ( x • , x ⊥ ) (cid:105) A = − α q (cid:90) dx • e − iαx • (cid:104) ¯ ψ (0)ΓΦ( x • , x ⊥ ) (cid:105) A (cid:90) dx • e − iα q x • (cid:104) ¯Φ(0)Γ 1 α + i(cid:15) ψ ( x • , x ⊥ ) (cid:105) A = − α q (cid:90) dx • e − iαx • (cid:104) ¯Φ(0)Γ ψ ( x • , x ⊥ ) (cid:105) A (11.32)The corresponding formulas for target matrix elements are obtained by substitution α ↔ β (and x • ↔ x ∗ ).Next, we will use QCD equation of motion to reduce quark-quark-gluon TMDs toleading-twist TMDs (see Ref. [20]). Let us start with matrix element (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) /p ˘ A i ( x • , x ⊥ ) ψ (0) | A (cid:105) (11.33) = − (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) (cid:54) A ( x • , x ⊥ ) /p γ i ψ (0) | A (cid:105) = (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ × (cid:2) (cid:104) A | ¯ ψ ( x • , x ⊥ ) /k ⊥ /p γ i ψ (0) | A (cid:105) + i (cid:104) A | ¯ ψ ( x • , x ⊥ ) ← (cid:54) D ⊥ γ j /p γ i ψ (0) | A (cid:105) (cid:3) . – 58 –sing QCD equations of motion (4.1) we can rewrite the r.h.s. of eq. (11.33) as (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:2) (cid:104) A | ¯ ψ ( x • , x ⊥ ) /k ⊥ /p γ i ψ (0) | A (cid:105) + α q (cid:104) A | ¯ ψ ( x • , x ⊥ ) /p /p γ i ψ (0) | A (cid:105) (cid:3) = (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) − k i (cid:104) A | ¯ ψ ( x • , x ⊥ ) /p ψ (0) | A (cid:105) + α q s (cid:104) A | ¯ ψ ( x • , x ⊥ ) γ i ψ (0) | A (cid:105)− i(cid:15) ij k j (cid:104) A | ¯ ψ ( x • , x ⊥ ) /p γ ψ (0) | A (cid:105) + i s α(cid:15) ij (cid:104) A | ¯ ψ ( x • , x ⊥ ) γ j γ ψ (0) | A (cid:105) (cid:105) = − k i π sf ( α q , k ⊥ ) + 8 π sα q k i (cid:2) f ⊥ ( α q , k ⊥ ) + g ⊥ ( α q , k ⊥ ) (cid:3) , (11.34)where we used parametrizations (11.24) and (11.28) for the leading power matrix elements.Now, the second term in eq. (11.34) contains extra α q with respect to the first term , so it should be neglected in our kinematical region s (cid:29) Q (cid:29) q ⊥ and we get π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) /p ˘ A i ( x • , x ⊥ ) ψ f (0) | A (cid:105) (11.35) = − π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (cid:54) A ( x • , x ⊥ ) /p γ i ψ f (0) | A (cid:105) = − k i f f ( α q , k ⊥ ) By complex conjugation π s (cid:90) dx ⊥ dx • e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) ˘ A i (0) /p ψ f (0) | A (cid:105) (11.36) = − π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) γ i /p (cid:54) Aψ f (0) | A (cid:105) = − k i f f ( α q , k ⊥ ) . For the corresponding antiquark distributions we get π s (cid:90) dx ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) ˘ A i ( x • , x ⊥ ) /p ψ f ( x • , x ⊥ ) | A (cid:105) = 18 π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) − (cid:104) A | ¯ ψ (0) γ i /p /k ⊥ ψ ( x • , x ⊥ ) | A (cid:105)− i (cid:104) A | ¯ ψ (0) γ i /p (cid:54) D ⊥ ψ ( x • , x ⊥ ) | A (cid:105) (cid:105) = − k i ¯ f f ( α q , k ⊥ ) (11.37)and π s (cid:90) dx ⊥ dx • e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) /p ˘ A i (0) ψ f ( x • , x ⊥ ) | A (cid:105) = − k i ¯ f f ( α q , k ⊥ ) . (11.38)The corresponding target matrix elements are obtained by trivial replacements x ∗ ↔ x • , α q ↔ β q and /p ↔ /p .Next, let us consider π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) /p (cid:54) A ( x • , x ⊥ ) ψ (0) | A (cid:105) (11.39) = 18 π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ × (cid:104) (cid:104) A | ¯ ψ ( x • , x ⊥ ) /k ⊥ /p ψ (0) | A (cid:105) + i (cid:104) A | ¯ ψ ( x • , x ⊥ ) ← (cid:54) D ⊥ /p ψ (0) | A (cid:105) (cid:105) . As discussed in the end of Sect. 11.2, all leading-twist TMDs can have only logarithmic dependence onBjorken x (which is here either α q for the projectile or β q for the target matrix elements). – 59 –sing QCD equation of motion and parametrization (11.29), one can rewrite the r.h.s. ofthis equation as π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) (cid:104) A | ¯ ψ ( x • , x ⊥ ) /k ⊥ /p ψ (0) | A (cid:105) + α q (cid:104) A | ¯ ψ ( x • , x ⊥ ) /p /p ψ (0) | A (cid:105) (cid:105) = i k ⊥ m N h ⊥ ( α q , k ⊥ ) + α q m N (cid:2) e ( α, k ⊥ ) + ih ( α, k ⊥ ) (cid:3) . (11.40)Again, only the first term contributes in our kinematical region so we finally get π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) /p (cid:54) A ( x • , x ⊥ ) ψ f (0) | A (cid:105) = i k ⊥ m h ⊥ f ( α q , k ⊥ ) . (11.41)By complex conjugation we obtain π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) /p (cid:54) A (0) ψ f (0) | A (cid:105) = i k ⊥ m h ⊥ f ( α q , k ⊥ ) . (11.42)For corresponding antiquark distributions one gets in a similar way π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) /p (cid:54) A ( x • , x ⊥ ) ψ f ( x • , x ⊥ ) | A (cid:105) = i k ⊥ m ¯ h ⊥ f ( α q , k ⊥ ) , π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0) /p (cid:54) A (0) ψ f ( x • , x ⊥ ) | A (cid:105) = i k ⊥ m ¯ h ⊥ f ( α q , k ⊥ ) . (11.43)The target matrix elements are obtained by usual replacements x ∗ ↔ x • , α q ↔ β q and /p ↔ /p .Finally, we need π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) (cid:54) A ( x • , x ⊥ ) /p (cid:54) A (0) ψ (0) | A (cid:105) = 18 π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) (cid:16) /k ⊥ + i ← (cid:54) D (cid:1) /p (cid:0) /k ⊥ − i (cid:54) D (cid:1) ψ (0) | A (cid:105) = k ⊥ π f ( α q , k ⊥ ) + O ( α q , β q ) (11.44)and similarly π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) (cid:54) A ( x • , x ⊥ ) σ ∗ i (cid:54) A (0) ψ (0) | A (cid:105) = 18 π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) (cid:16) /k ⊥ + i ← (cid:54) D (cid:1) σ ∗ i (cid:0) /k ⊥ − i (cid:54) D (cid:1) ψ (0) | A (cid:105) = 116 π k i k ⊥ m h ⊥ ( α q , k ⊥ ) + O ( α q , β q ) (11.45)– 60 –or corresponding antiquark distributions we get π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ (0) (cid:54) A (0) /p (cid:54) A ( x • , x ⊥ ) ψ ( x • , x ⊥ ) | A (cid:105) = − k ⊥ π ¯ f ( α q , k ⊥ ) + O ( α q , β q )18 π s (cid:90) dx • dx ⊥ e − iα q x • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ ( x • , x ⊥ ) (cid:54) A ( x • , x ⊥ ) σ ∗ i (cid:54) A (0) ψ (0) | A (cid:105) = − π k i k ⊥ m h ⊥ ( α q , k ⊥ ) + O ( α q , β q ) (11.46)Also, as we saw in Sect. 6.2.3, at the leading order in N c there is one quark-antiquark-gluon operator that does not reduce to twist-2 distributions. It can be parametrized asfollows (cf. Eq. (11.43)) π s (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f ( x • , x ⊥ ) (cid:2) A i ( x ) σ ∗ j − g ij A k σ ∗ k ( x ) (cid:3) ψ f (0) | A (cid:105) = − ( k i k j + 12 g ij k ⊥ ) 1 m h fA ( α, k ⊥ ) , π s (cid:90) dx • d x ⊥ e − iαx • + i ( k,x ) ⊥ (cid:104) A | ¯ ψ f (0)[ A i (0) σ • j − g ij A k σ • k (0)] ψ f ( x • , x ⊥ ) | A (cid:105) = − ( k i k j + 12 g ij k ⊥ ) 1 m ¯ h fA ( α, k ⊥ ) (11.47)and similarly for the target matrix elements. We parametrize TMDs from section 7.1 as follows π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) x • −∞ dx (cid:48)• (cid:104) A | ¯ ψ ( x • , x ⊥ ) 2 /p s (cid:2) F ∗ i (0) + iγ ˜ F ∗ i (0) (cid:3) ψ ( x (cid:48)• , x ⊥ ) | A (cid:105) = k i j ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) x • −∞ dx (cid:48)• (cid:104) A | ¯ ψ ( x • , x ⊥ ) 2 /p s (cid:2) F ∗ i (0) − iγ ˜ F ∗ i (0) (cid:3) ψ ( x (cid:48)• , x ⊥ ) | A (cid:105) = k i j ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) x • −∞ dx (cid:48)• (cid:104) A | ¯ ψ ( x (cid:48)• , x ⊥ ) 2 /p s (cid:2) F ∗ i (0) − iγ ˜ F ∗ i (0) (cid:3) ψ ( x • , x ⊥ ) | A (cid:105) = k i ¯ j ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) x • −∞ dx (cid:48)• (cid:104) A | ¯ ψ ( x (cid:48)• , x ⊥ ) 2 /p s (cid:2) F ∗ i (0) + iγ ˜ F ∗ i (0) (cid:3) ψ ( x • , x ⊥ ) | A (cid:105) = k i ¯ j ( α, k ⊥ ) (11.48)– 61 –y complex conjugation we get π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) −∞ dx (cid:48)• (cid:104) A | ¯ ψ ( x (cid:48)• , ⊥ ) 2 /p s [ F ∗ i ( x ) − iγ ˜ F ∗ i ( x )] ψ (0) | A (cid:105) , = k i j (cid:63) ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) −∞ dx (cid:48)• (cid:104) A | ¯ ψ ( x (cid:48)• , ⊥ ) 2 /p s (cid:2) F ∗ i ( x ) + iγ ˜ F ∗ i ( x ) (cid:3) ψ (0) | A (cid:105) = k i j (cid:63) ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) −∞ dx (cid:48)• (cid:104) A | ¯ ψ (0) 2 /p s [ F ∗ i ( x ) + iγ ˜ F ∗ i ( x )] ψ ( x (cid:48)• , ⊥ ) | A (cid:105) = k i ¯ j (cid:63) ( α, k ⊥ ) , π s (cid:90) d x ⊥ dx • e − iαx • + i ( k,x ) ⊥ (cid:90) −∞ dx (cid:48)• (cid:104) A | ¯ ψ (0) 2 /p s (cid:2) F ∗ i ( x ) − iγ ˜ F ∗ i ( x ) (cid:3) ψ ( x (cid:48)• , ⊥ ) | A (cid:105) = k i ¯ j (cid:63) ( α, k ⊥ ) . (11.49)Note that unlike two-quark matrix elements, quark-quark-gluon ones may have imaginaryparts.Target matrix elements are obtained by usual substitutions α ↔ β , /p ↔ /p , x • ↔ x ∗ ,and ˆ F ∗ i ↔ ˆ F • i .For completeness let us present the explicit form of the gauge links in an arbitrarygauge: ¯ ψ ( x (cid:48)• , x ⊥ ) F ∗ i (0) ψ ( x • , x ⊥ ) → ¯ ψ ( x (cid:48)• , x ⊥ )[ x (cid:48)• , −∞ • ] x [ x ⊥ , ⊥ ] −∞ • (11.50) × [ −∞ • , ⊥ F ∗ i (0)[0 , −∞ • ] ⊥ [0 ⊥ , x ⊥ ] −∞ • [ −∞ • , x • ] x ψ ( x • , x ⊥ ) . J µA ( x ) J Aµ (0) terms There is one more type of contributions proportional to the product of quark and gluonTMDs J µA ( x ) J νA (0) == (cid:88) flavors (cid:16)(cid:2) ¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ A (0) γ ν Ξ (0) (cid:3) + (cid:2) ¯ ψ A ( x ) γ µ Ξ ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + (cid:2) ¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯Ξ (0) γ ν ψ A (0) (cid:3) + (cid:2) ¯ ψ A ( x ) γ ν Ξ ( x ) (cid:3)(cid:2) ¯ ψ A (0) γ ν Ξ (0) (cid:3)(cid:17) , (11.51)where we neglected terms which cannot contribute to W due to the reason discussed aftereq. (7.3), i.e. that one hadron (“A” or “B”) cannot produce the DY pair on its own.Let us consider the first term in the r.h.s of this equation ˇ W µν ( x ) = (cid:104) A, B | (cid:2) ¯Ξ ( x ) γ µ ψ A ( x ) (cid:3)(cid:2) ¯ ψ A (0) γ ν Ξ (0) (cid:3) | A, B (cid:105) (11.52) = − g s ( N c − (cid:104) (cid:0) ¯ ψ α (cid:1) ( x ) γ i /p γ µ ψ ( x ) ¯ ψ (0) γ ν /p γ j α ψ (0) (cid:105) A (cid:104) A ai ( x ) A aj (0) (cid:105) B – 62 –o estimate the magnitude of this contribution, first note that (cid:90) dx ∗ e − iβ q x ∗ (cid:104) B | A ai ( x ) A aj (0) | B (cid:105) (11.53) = 4 s (cid:90) dx ∗ e − iβ q x ∗ (cid:90) x ∗ −∞ dx (cid:48)∗ (cid:90) −∞ dx (cid:48)(cid:48)∗ (cid:104) B | F a • i ( x (cid:48)∗ , x ⊥ ) F a • j ( x (cid:48)(cid:48)∗ , ⊥ ) | B (cid:105) = 4 β q s (cid:90) dx ∗ e − iβ q x ∗ (cid:104) B | F a • i ( x ∗ , x ⊥ ) F a • j (0) | B (cid:105) = − β q π α s (cid:104) D g ( β q , x ⊥ ) + 1 m (2 ∂ i ∂ j + g ij ∂ ⊥ H g ( β q , x ⊥ ) (cid:3) where we used parametrization (3.26) from Ref. 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