Contribution of Twist-3 Multi-Gluon Correlation Functions to Single Spin Asymmetry in Semi-Inclusive Deep Inelastic Scattering
aa r X i v : . [ h e p - ph ] A ug Contribution of Twist-3 Multi-Gluon CorrelationFunctions to Single Spin Asymmetryin Semi-Inclusive Deep Inelastic Scattering
Hiroo Beppu , Yuji Koike , Kazuhiro Tanaka and Shinsuke Yoshida Graduate School of Science and Technology, Niigata University, Ikarashi, Niigata950-2181, Japan Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan Department of Physics, Juntendo University, Inzai, Chiba 270-1695, Japan
Abstract
As a possible source of the single transverse spin asymmetry, we study the contribution frompurely gluonic correlation represented by the twist-3 “three-gluon correlation” functions inthe transversely polarized nucleon. We first define a complete set of the relevant three-gluoncorrelation functions, and then derive its contribution to the twist-3 single-spin-dependentcross section for the D -meson production in semi-inclusive deep inelastic scattering, whichis relevant to determine the three-gluon correlations. Our cross-section formula differs fromthe corresponding result in the literature, and the origin of the discrepancy is clarified.1 Introduction
Clarifying the origin of the large single spin asymmetries (SSAs) observed in various high-energy semi-inclusive processes [1]-[7] has been a big challenge during the past decades. (See[8] for a review.) They are now understood as direct consequences of the orbital motion ofquarks and gluons and/or the multi-parton correlations inside the hadrons, and thus providea new opportunity for revealing the QCD dynamics and hadron structures, which do notappear in the parton models and perturbative QCD at the conventional twist-2 level. Satis-factory formulation of these effects in QCD, however, requires sophisticated framework forwhich lots of technical developments are involved. Up to now, the SSAs have been formu-lated based on the (naively) “T-odd” distribution/fragmentation functions [9]-[18] withinthe transverse-momentum-dependent (TMD) factorization approach [19, 20, 21], and on thetwist-3 multi-parton correlation functions in the collinear factorization approach [22]-[38].These two mechanisms, in principle, cover different kinematic regions, such that the formerapproach is suitable for treating SSAs in the region of the small transverse momentum of aparticle observed in the final-state, while the latter is designed for systematic description ofSSAs at large values of the corresponding transverse momentum. On the other hand, in theintermediate region of the corresponding transverse momentum where both approaches arevalid, it’s been shown for some structure functions in semi-inclusive deep inelastic scattering(SIDIS) and in the Drell-Yan cross section that the two approaches provide an equivalentdescription of SSAs [39, 40, 41, 42]. Therefore, in practice, the two approaches, tied togetherat the intermediate transverse-momentum region, can be regarded as providing a uniqueQCD effect leading to SSAs over the entire kinematic regions. Although our understandingon SSAs has made a great progress armed with these mechanisms, further studies are yetto be done for a complete clarification of all the effects responsible for SSAs. Among sucheffects, in particular, the role of purely gluonic effects has not been widely studied in theliterature. Since the gluons are ample in the nucleon, they are potentially an importantsource of SSAs.In this paper, we study a purely gluonic effect as an origin of SSAs in the framework ofthe collinear factorization. To this end, we work on SSAs for a heavy meson production inSIDIS, in particular, the D -meson production, ep → eDX . Since this process is induceddominantly by the c ¯ c -pair creation through photon-gluon fusion, this is the most relevantprocess to probe the gluonic effects for SSAs, together with D -meson production in pp collisions [43, 33] ongoing at RHIC [44]. In the collinear factorization framework, SSA is atwist-3 observable and thus the gluonic effects responsible for SSAs are represented by thetwist-3 gluon correlation functions in the polarized nucleon, which were first introduced inthe most general form by Ji [24]. Recently the authors of [32] studied the SSAs in SIDIS, ep ↑ → eDX , applying the twist-3 mechanism. They derived a formula, in the leadingorder with respect to the QCD coupling constant, for the contribution to the single-spin-dependent cross section from a “three-gluon” correlation function. In our opinion, however,the starting formula used to derive the cross section in [32] is incomplete and cannot leadto the correct result. So we revisit the same issue in this paper.As is well-known, a twist-2 gluon distribution inside the nucleon is defined in terms2f the gauge-invariant lightcone correlation function of the gluon field-strength tensors,schematically written as h F α + a F β + a i . Likewise, the twist-3 “three-gluon distribution” func-tions are defined through the gauge-invariant correlation functions h C abc ± F α + a F β + b F γ + c i , withthe structure constants of the color SU(3) group, C abc + = if abc and C abc − = d abc . On the otherhand, when we derive the three-gluon contribution to the cross section for ep ↑ → eDX withthe large transverse momentum of the D meson, we start our analysis with a cut forward-amplitude for the cross section, in which the correlation functions of the gluon fields appearin the form ∼ h A αa A βb A γc i . Extraction of the twist-3 effect relevant for SSAs from the cor-responding amplitude, converting eventually the associated nucleon matrix elements from h A αa A βb A γc i into the gauge-invariant forms h C abc ± F α + a F β + b F γ + c i , is a highly nontrivial issue, un-like straightforward calculations of the twist-2 cross sections. In this connection, it is worthmentioning that, for the case where the twist-3 quark-gluon correlation functions partici-pate, the necessary formulation was achieved in [27]; there, Ward identities played a crucialrole to prove the factorization property and gauge invariance of the corresponding twist-3 cross section. Similarly, we will show that, owing to the constraints from the tree-levelWard identities satisfied by the relevant partonic hard parts, the twist-3 contribution to thecross section, associated with the three-gluon correlation functions h A αa A βb A γc i , can be recastinto the factorized expression in terms of the gauge-invariant functions h C abc ± F α + a F β + b F γ + c i .With this formalism we will derive the complete single-spin-dependent cross section arisingfrom the twist-3 three-gluon correlation functions of the nucleon. We will also clarify thedifference of our result from that of [32].The remainder of this paper is organized as follows: In section 2, we first define acomplete set of the three-gluon correlation functions in the transversely polarized nucleon.We show that, actually, the three-gluon correlation functions defined in [24] are not allindependent, i.e., some of them are redundant. So we newly define a genuine complete set ofthe three-gluon correlation functions. In section 3, we present our formalism for calculatingthe twist-3 single-spin-dependent cross section for ep ↑ → eDX . We show that only a polecontribution of an internal propagator in the hard part leads to a real quantity relevantto the cross section, and that the tree-level Ward identities satisfied by the correspondingpole contributions play an essential role to give the factorized expression for the single-spin-dependent cross section, in terms of a complete set of the gauge-invariant correlationfunctions defined in section 2. In section 4, we present the final form of the single-spin-dependent cross section for ep ↑ → eDX and discuss its characteristic features. Section 5is devoted to a brief summary of our result. In Appendix A, we summarize the relevantsymmetry properties of the gluon correlation functions, which are used in section 3. As a straightforward extension of the quark-gluon correlation functions discussed in, e.g.,[26, 27], purely gluonic correlation functions can be defined as nucleon matrix elements ofthe three gluon fields on the lightcone [24]. Due to the different ways for contracting the3olor indices of the three gluon fields to obtain the color-singlet operators, one can definethe two-types of correlation functions as O αβγ ( x , x ) = − gi Z dλ π Z dµ π e iλx e iµ ( x − x ) h pS | d bca F βnb (0) F γnc ( µn ) F αna ( λn ) | pS i , (1) N αβγ ( x , x ) = − gi Z dλ π Z dµ π e iλx e iµ ( x − x ) h pS | if bca F βnb (0) F γnc ( µn ) F αna ( λn ) | pS i , (2)where F αna ≡ F αβa n β with F αβa = ∂ α A βa − ∂ β A αa + gf abc A αb A βc being the gluon field strengthtensor, d bca and f bca are, respectively, the symmetric and anti-symmetric structure con-stants of the color SU(3) group, and we have suppressed the gauge-link operators whichappropriately connect the field strength tensors so as to ensure the gauge invariance. p is the nucleon momentum, and S is the transverse spin vector of the nucleon normalizedas S = −
1. We obtain the twist-3 contributions of (1) and (2), when we regard all thefree Lorentz indices α , β , and γ to be transverse, and, in this twist-3 accuracy, p can beregarded as lightlike ( p = 0). n is another lightlike vector satisfying p · n = 1, and, to bespecific, we assume p µ = ( p + , , ⊥ ) and n µ = (0 , n − , ⊥ ); then, we have S µ = (0 , , S ⊥ ).Taking into account the constraints from hermiticity, invariance under the parity andtime-reversal transformations, and the permutation symmetry among the participatingthree gluon-fields, Ji decomposed the twist-3 contribution of (1) in terms of the real,Lorentz-scalar functions O and e O associated with the six tensor structures as [24]2 iM N h O ( x , x ) g αβ ǫ γpnS + O ( x , x − x ) g βγ ǫ αpnS + O ( x , x − x ) g γα ǫ βpnS + e O ( x , x ) ǫ αβpn S γ + e O ( x , x − x ) ǫ βγpn S α − e O ( x , x − x ) ǫ γαpn S β i , (3)where we have introduced the nucleon mass M N in order to define O and e O as dimen-sionless, and the similar decomposition of (2) was also introduced as [24]2 iM N h N ( x , x ) g αβ ǫ γpnS − N ( x , x − x ) g βγ ǫ αpnS − N ( x , x − x ) g γα ǫ βpnS + f N ( x , x ) ǫ αβpn S γ − f N ( x , x − x ) ǫ βγpn S α + f N ( x , x − x ) ǫ γαpn S β i , (4)with the other dimensionless, real functions N and f N . In [24], the above four functions O , e O , N and f N were treated as independent twist-3 three-gluon correlation functions.However, the six tensor structures in (3) and (4) are not all independent and thus thesedecompositions actually define redundant correlation functions. To see this, we recall theidentity, g µν ǫ αβρδ = g µα ǫ νβρδ + g µβ ǫ ανρδ + g µρ ǫ αβνδ + g µδ ǫ αβρν , (5)and contract both sides of this identity with the tensor g γµ S ν p ρ n δ . When the indices α , β and γ are associated with the transverse components, one obtains the relation, ǫ αβpn S γ = − g γα ǫ βpnS + g βγ ǫ αpnS , (6) This point was also noticed and briefly mentioned in [54]. e O and f N are reexpressed by thoseassociated with O and N , respectively, in (3) and (4). Accordingly, (1) and (2) should bedecomposed into the three tensor structures. This can be formally achieved by omitting theterms associated with e O and f N in (3) and (4), respectively, and we newly define the twotwist-3 functions O ( x , x ) and N ( x , x ) as independent three-gluon correlation functionsto represent (1) and (2) as O αβγ ( x , x )= 2 iM N h O ( x , x ) g αβ ǫ γpnS + O ( x , x − x ) g βγ ǫ αpnS + O ( x , x − x ) g γα ǫ βpnS i , (7) N αβγ ( x , x )= 2 iM N h N ( x , x ) g αβ ǫ γpnS − N ( x , x − x ) g βγ ǫ αpnS − N ( x , x − x ) g γα ǫ βpnS i . (8)In this paper, we use these O ( x , x ) and N ( x , x ) as constituting a genuine complete set toexpress the contribution of the three-gluon correlations to the twist-3 single-spin-dependentcross section for ep ↑ → eDX . Similarly to the decomposition (3) and (4) considered in [24],the independent tensor structures in (7) and (8) are parameterized by the common real func-tions O and N , respectively, as consequences of the constraints from hermiticity, invarianceunder the parity and time-reversal transformations, and the permutation symmetry amongthe participating three gluon-fields. In particular, these constraints imply the followingsymmetry relations for O ( x , x ) and N ( x , x ), which are associated with the C -odd and C -even combinations of the three gluon operators in (1) and (2), respectively: O ( x , x ) = O ( x , x ) , O ( x , x ) = O ( − x , − x ) , (9) N ( x , x ) = N ( x , x ) , N ( x , x ) = − N ( − x , − x ) . (10)The authors of [45] also pointed out that there are only two independent three-gluon correla-tion functions at twist-3, in the study of the evolution equations for the twist-3 distributionsin the transversely polarized nucleon. We also mention that the C -even three-gluon corre-lation function in (8) contributes to the twist-3 transverse-spin structure function g ( x, Q )in the inclusive DIS with polarized beam and target [46], and the twist-3 local operatorsassociated with the double moments of (2) have been analyzed in the framework of theoperator product expansion [47].As we will see in the next section, a contribution to the single-spin-dependent crosssection based on the corresponding factorization formula at the leading order in QCDperturbation theory is generated from a pole arising in the propagators in the hard partonicsubprocesses, and such a pole contribution fixes the momentum fractions in the correlationfunctions O αβγ ( x , x ) and N αβγ ( x , x ) at x = x ≡ x . This represents the situation whereone of the three gluon-lines participating in the partonic subprocesses has zero momentum(see (1), (2)), and thus the corresponding contributions relevant to SSA can be referred to asthe soft-gluon-pole (SGP) contributions. Combined with the above decompositions (7) and58), the single-spin-dependent cross section proves to receive contributions associated with O ( x, x ), O ( x, N ( x, x ) and N ( x, O ( x, x ) is different from the hard part associatedwith O ( x, N ( x, x ) is different from thatfor N ( x, ep ↑ → eDX . The three-gluon correlation functionsintroduced by the authors of [32] read T ( ± ) G ( x, x ) = Z dy − dy − π e ixp + y − xp + g βα ǫ Sγnp h pS | C bca ± F β + b (0) F γ + c ( y − ) F α + a ( y − ) | pS i , (11)with C bca + = if bca , C bca − = d bca , (12)and it was claimed that the corresponding twist-3 single-spin-dependent cross section atthe leading order in QCD perturbation theory can be expressed entirely in terms of thesetwo functions of x (see (60) below and the discussion following this formula). However, T ( ± ) G ( x, x ) in (11) can be obtained by the contraction of O αβγ ( x, x ) and N αβγ ( x, x ) in (1)and (2) with the particular tensor g βα ǫ Sγnp , and thus can be written, using (7) and (8), as xg π T (+) G ( x, x ) = − M N ( N ( x, x ) − N ( x, , (13) xg π T ( − ) G ( x, x ) = − M N ( O ( x, x ) + O ( x, . (14)Using these relations, the twist-3 single-spin-dependent cross section obtained in [32] impliesthe same partonic hard parts for O ( x, x ) and O ( x, N ( x, x ) and N ( x, O ( x, x ) and O ( x,
0) are associated with the different partonic hard parts, and similarly for N ( x, x ) and N ( x, T ( ± ) G ( x, x ) only. Inthis connection, we also note that (7) and (8) can be converted to give O ( x, x ), O ( x, N ( x, x ) and N ( x,
0) as O ( x, x ) = i M N (3 g αβ ǫ γpnS − g αγ ǫ βpnS ) O αβγ ( x, x ) , (15) O ( x,
0) = − i M N ( g αβ ǫ γpnS − g αγ ǫ βpnS ) O αβγ ( x, x ) , (16) N ( x, x ) = i M N (3 g αβ ǫ γpnS − g αγ ǫ βpnS ) N αβγ ( x, x ) , (17) N ( x,
0) = i M N ( g αβ ǫ γpnS − g αγ ǫ βpnS ) N αβγ ( x, x ) . (18)6rom these relations, we see that the contributions of certain types of the twist-3 compo-nents in the three-gluon correlation functions, obtained as the contractions of O αβγ ( x, x ), N αβγ ( x, x ) with the tensor g αγ ǫ βpnS , were not taken into account in [32]. Taking into ac-count the components contracted with g αγ ǫ βpnS , as well as those contracted with g αβ ǫ γpnS ,is required on general grounds by the Bose statistics of the gluon, and, in (15)-(18), thereis no reason to anticipate that the latter types of components are more important than theformer. ep ↑ → eDX Here we summarize the kinematics for the SIDIS process, e ( ℓ ) + p ↑ ( p, S ) → e ( ℓ ′ ) + D ( P h ) + X. (19)As noted in the last section, one can assume that the initial nucleon’s momentum p islightlike, p = 0, in the twist-3 accuracy. But, we keep the mass m h of the final charmedhadron ( D -meson) as P h = m h . The corresponding results for the case of the light-mesonproduction can be obtained by the replacement m h → S ep = ( p + ℓ ) , x bj = Q p · q , Q = − q = − ( ℓ − ℓ ′ ) , z f = p · P h p · q , q T = q − q t . (20)Here, q t is the “transverse” component of q defined as q µt = q µ + m h p · q ( p · P h ) − P h · qp · P h ! p µ − p · qp · P h P µh , (21)satisfying q t · p = q t · P h = 0. In the actual calculation we work in the “hadron frame” [48]where the virtual photon and the initial nucleon are collinear, i.e., both move along the z -axis. In this frame, specifically, their momenta q and p are given as q µ = ( q , ~q ) = (0 , , , − Q ) , (22)and, similarly, p µ = Q x bj , , , Q x bj ! , (23)and the outgoing D -meson is assumed to reside in the xz plane: P µh = z f Q q T Q + m h z f Q , q T Q , , − q T Q + m h z f Q ! . (24)7he transverse momentum of the D -meson in this frame is given by P hT = z f q T , which istrue in any frame where the 3-momenta ~q and ~p are collinear. The lepton momentum inthis frame can be parameterized as ℓ µ = Q ψ, sinh ψ cos φ, sinh ψ sin φ, − ,ℓ ′ µ = Q ψ, sinh ψ cos φ, sinh ψ sin φ, , (25)where cosh ψ = 2 x bj S ep Q − . (26)We parameterize the transverse spin vector of the initial nucleon S µ as S µ = (0 , cos Φ S , sin Φ S , , (27)where Φ S represents the azimuthal angle of ~S measured from the hadron plane. With theabove definition, the cross section for ep ↑ → eDX can be expressed in terms of S ep , x bj , Q , z f , q T , φ and Φ S in the hadron frame. Note that φ and Φ S are invariant under boostsin the ~q -direction, so that the cross section presented below is the same in any frame where ~q and ~p are collinear. The differential cross section for ep ↑ → eDX can be obtained as d ∆ σ = 12 S ep d ~P h (2 π ) P h d ~ℓ ′ (2 π ) ℓ ′ e q L µν ( ℓ, ℓ ′ ) W µν ( p, q, P h ) , (28)where L µν ( ℓ, ℓ ′ ) = 2( ℓ µ ℓ ′ ν + ℓ ν ℓ ′ µ ) − Q g µν is the leptonic tensor for the unpolarized elec-tron, and W µν ( p, q, P h ) is the hadronic tensor. In the present study we are interested inthe contribution to W µν ( p, q, P h ) from the three-gluon correlation functions for the initialnucleon, in which c and ¯ c are created through the photon-gluon fusion process and oneof them fragments into a D ( ¯ D ) meson. The fragmentation function D ( z ) for a c -quarkto become the D -meson with momentum P h is defined from the corresponding lightconecorrelation function as X X Z dλ π e − iλ/z h | ψ i (0) | D ( P h ) X ih D ( P h ) X | ¯ ψ j ( λw ) | i = (/ p c + m c ) ij D ( z ) + · · · , (29)where the ellipses denote the terms associated with the gamma matrix structures which areirrelevant for the present purpose. In the left-hand side, we have suppressed the gauge-link8perators to be connected to the quark fields, as well as the trace over the color indices.Here, z is the relevant momentum fraction, w is the lightlike vector of O (1 /Q ) satisfying P h · w = 1, and p c is the momentum of the c (or ¯ c ) quark with mass m c , such that p µc = P µh /z + rw µ , with r = ( m c z − m h /z ) / p c = m c . At the leading twist-2accuracy for the quark-fragmentation process, we set w µ = p µ / ( P h · p ), and, in the hadronframe, p c is expressed as p µc = b zQ q T Q + m c b z Q , q T Q , , − q T Q + m c b z Q ! , (30)where b z = z f z . Note that, when we make the replacement, P X | D ( P h ) X ih D ( P h ) X | →| p ′ c ih p ′ c | in the left-hand side of (29), with | p ′ c i the state with a c -quark having the momentum p ′ c ≡ p c | z → z ′ , the ellipses in the right-hand side vanish and D ( z ) → δ (1 /z ′ − /z ). Thefragmentation function D ( z ) of (29) is factorized from W µν as W µν ( p, q, P h ) = Z dzz D ( z ) w µν ( p, q, p c ) , (31)where the summation over the c and ¯ c quark contributions is implicit. To extract thetwist-3 effect in w µν , one needs to analyze the diagrams of the type shown in Fig. 1.However, as shown in Appendix A, in the leading order with respect to the QCD couplingconstant for the partonic hard scattering parts, the contribution of Fig. 1(a) can not giverise to the single-spin-dependent cross section, and only Fig. 1(b) contributes to SSA, dueto the symmetry properties of the correlation functions of two and three gluon fields inthe polarized nucleon. So we shall focus on the analysis of Fig. 1(b) below. The goal ofour analysis is to show that all the contributions from Fig. 1(b) in the twist-3 accuracycan be expressed in terms of the gauge-invariant correlation functions O αβγ ( x , x ) and N αβγ ( x , x ) defined as (1) and (2) in the previous section. For this purpose, we work inFeynman gauge and apply the collinear expansion to the hard scattering part in Fig. 1(b),keeping all the terms contributing in the twist-3 accuracy [27]. For simplicity in the notation,we shall henceforth omit the Lorentz indices µ and ν for the virtual photon in w µν ( p, q, p c )of (31) and write it as w ( p, q, p c ).The contribution from Fig. 1(b) to w ( p, q, p c ) can be written as w ( p, q, p c ) = Z d k (2 π ) Z d k (2 π ) S abcµνλ ( k , k , q, p c ) M µνλabc ( k , k ) , (32)where S abcµνλ ( k , k , q, p c ) is the partonic hard scattering part represented by the middle blobof Fig. 1(b) and M µνλabc ( k , k ) is the corresponding nucleon matrix element (lower blob)defined as M µνλabc ( k , k ) = g Z d ξ Z d η e ik ξ e i ( k − k ) η h pS | A νb (0) A λc ( η ) A µa ( ξ ) | pS i . (33)Note that, for later convenience, we include one QCD coupling constant in the definitionof this nucleon matrix element. In (32), a real contribution relevant to the cross section for9 Php k kM(cid:22)(cid:23)(2)S(2)(cid:22)(cid:23)p q Php k1 k2k2 (cid:0) k1 Sab (cid:22)(cid:23)(cid:21)M(cid:22)(cid:23)(cid:21)ab p (a) (b)Figure 1: Generic diagrams for the hadronic tensor of ep ↑ → eDX induced by the gluoniceffect in the nucleon. Each one is decomposed into the nucleon matrix element (lower blob), D -meson matrix element (upper blob), and the partonic hard scattering part by the virtualphoton (middle blob). In the expansion by the number of gluon lines connecting the middleand lower blobs, the first two terms, (a) and (b), are relevant to the twist-3 effect inducedby the gluons in the nucleon.SSA occurs from an imaginary part of the color-projected hard part C bca ± S abcµνλ ( k , k , q, p c )with (12), since C bca ± M µνλabc ( k , k ) are pure imaginary quantities as shown in Appendix A.This means that only the pole contribution produced by an internal propagator in the hardpart can give rise to SSA.In the leading order with respect to the QCD coupling constant, we find that fourtopologically distinct diagrams shown in Fig. 2, together with their mirror diagrams, giverise to the “surviving” pole contributions; here, a short bar indicates the quark propagatorthat produces the corresponding pole contribution. The other pole contributions turn out tocancel among themselves after summing the contributions of all the leading-order diagramsfor (32). With the assignment of the momenta k and k of gluons as shown in Fig. 2,the condition for those poles is given by ( p c − k + k ) − m c = 0. After we perform thecollinear expansion and reach the collinear limit, k i → x i p ( i = 1 , x , x and x − x representing the longitudinal momentum fractions of the relevant three gluons, thiscondition reduces to x = x and hence a pole of such type is referred to as the soft-gluonpole (SGP). In the following, we assume that S abcµνλ ( k , k , q, p c ) in (32) represents the sum ofthe contributions of the diagrams in Fig. 2 and their mirror diagrams, in which the barredpropagator is replaced by its pole contribution. Also, for simplicity of notation, we suppressthe color indices a, b, c and the momenta q and p c in the hard part S abcµνλ ( k , k , q, p c ), writingit simply as S µνλ ( k , k ), and correspondingly, we write M µνλabc ( k , k ) as M µνλ ( k , k ).To perform the collinear expansion, we decompose the relevant gluon momenta k i ( i =10 Figure 2: Feynman diagrams for the partonic hard part in Fig. 1(b), representing thephoton-gluon fusion subprocesses that give rise to the “surviving” pole contribution for ep ↑ → eDX in the leading order with respect to the QCD coupling constant. The short baron the internal c -quark line indicates that the pole part is to be taken from that propagator.In the text, momenta are assigned as shown in the upper-left diagram, where p c denotesthe momentum of the c -quark fragmenting into the D -meson in the final state. The mirrordiagrams also contribute.1 ,
2) as k µi = ( k i · n ) p µ + ( k i · p ) n µ + k µ ⊥ ≡ x i p µ + ω µν k νi , (34)where x i = k i · n and ω µν ≡ g µν − p µ n ν . Since p µ ∼ g µ + Q in the hadron frame with (23)and thus the component along p µ gives the leading contribution in (34) with respect to thehard scale Q , we expand S µνλ ( k , k ) around k i = x i p . Expressing also the gluon field A α in the Feynman gauge as A α = ( p α n κ + ω ακ ) A κ = p α n · A + ω ακ A κ , (35)we note that, in the matrix element M µνλ ( k , k ) of (33), the components associated with thesecond term of (35) give rise to the contributions suppressed by ∼ /Q or more, compared11ith the corresponding contribution due to the first term, p α n · A (see, e.g., [27]). Accordingto the decomposition (35), the integrand of (32) can be expressed as S µνλ ( k , k ) M µνλ ( k , k )= S µνλ ( k , k ) ( p µ n κ + ω µκ )( p ν n τ + ω ντ )( p λ n σ + ω λσ ) M κτσ ( k , k )= S ppp ( k , k ) M nnn ( k , k ) + S αpp ( k , k ) ω ακ M κnn ( k , k ) + S pαp ( k , k ) ω ακ M nκn ( k , k )+ · · · + S pαβ ( k , k ) ω ακ ω βτ M nκτ ( k , k ) + S αβγ ( k , k ) ω ακ ω βτ ω γσ M κτσ ( k , k ) , (36)where S ppp ( k , k ) ≡ S µνλ ( k , k ) p µ p ν p λ , M nnn ( k , k ) ≡ M µνλ ( k , k ) n µ n ν n λ , etc. By per-forming the collinear expansion of the hard part of each term in the right-hand side of thisformula, we can organize the integrand of (32) based on the order counting with (34), (35),keeping the terms necessary in the twist-3 accuracy. For the first term in the right-handside of (36), the Taylor expansion about k i = x i p gives, S ppp ( k , k )= S ppp ( x , x ) + ω ακ k κ ∂S ppp ( k , k ) ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + ω ακ k κ ∂S ppp ( k , k ) ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + 12 ω ακ k κ ω βτ k τ ∂ S ppp ( k , k ) ∂k α ∂k β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + 12 ω ακ k κ ω βτ k τ ∂ S ppp ( k , k ) ∂k α ∂k β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + ω ακ k κ ω βτ k τ ∂ S ppp ( k , k ) ∂k α ∂k β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + 16 ω ακ k κ ω βτ k τ ω γσ k σ ∂ S ppp ( k , k ) ∂k α ∂k β ∂k γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + 16 ω ακ k κ ω βτ k τ ω γσ k σ ∂ S ppp ( k , k ) ∂k α ∂k β ∂k γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + 12 ω ακ k κ ω βτ k τ ω γσ k σ ∂ S ppp ( k , k ) ∂k α ∂k β ∂k γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + 12 ω ακ k κ ω βτ k τ ω γσ k σ ∂ S ppp ( k , k ) ∂k α ∂k β ∂k γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + · · · . (37)Here and below, we use the notation S µνλ ( x , x ) ≡ S µνλ ( x p, x p ) for simplicity. Wehave written down the expansion explicitly up to the third-order terms. This is because,according to the above notice with the decompositions (34), (35), the third-order terms in(37) behave as the same order as the first term in S αβγ ( k , k ) ω ακ ω βτ ω γσ M κτσ ( k , k ) = [ S αβγ ( x , x ) + · · · ] ω ακ ω βτ ω γσ M κτσ ( k , k ) , (38)which is obtained by the Taylor expansion of the last term of (36). If we substitute (38)directly into the integrand of (32) and perform the integrals over k and k , the first term12 = 0+ (a) (b)Figure 3: Ward identities used for the pole contribution. The dotted line represents ascalar-polarized gluon, and the quark lines marked by a bar are on-shell.in the right-hand side produces the contribution, which is associated with the hard scat-tering between the three physical gluons from the nucleon and behaves as the same or-der as the formal convolution of S αβγ ( x , x ) with the twist-3 correlation functions of (1)and (2). Actually, this corresponds to a quantity of twist-3. Compared to this, the el-lipses in (38) give rise to the terms suppressed by 1 /Q or more corresponding to twist-4and higher, and thus are irrelevant here. We can write down the collinear expansionsfor the hard parts associated with the other terms in (36), similarly as (37): We expand S αpp ( k , k ), S pαp ( k , k ) and S ppα ( k , k ) through the terms involving the second deriva-tives, and S αβp ( k , k ), S αpβ ( k , k ) and S pαβ ( k , k ) through the terms involving the firstderivatives. Thus, the collinear expansion of S µνλ ( k , k ) in (32) produces lots of terms asabove, and each of those terms is not gauge invariant. At first sight, it looks hopeless toreorganize those into a form of the convolution with only the gauge-invariant correlationfunctions O αβγ ( x , x ) and N αβγ ( x , x ) of (1) and (2) used.However, as was the case for the pion production associated with the twist-3 quark-gluoncorrelation functions [27], great simplification occurs due to Ward identities satisfied by thecorresponding partonic hard-scattering function, S µνλ ( k , k ). To derive the Ward identities,we note that the contribution to S µνλ ( k , k ) from each diagram in Fig. 2 all contains thetwo delta functions, δ (( k + q − p c ) − m c ) and δ (( p c + k − k ) − m c ), representing theon-shell conditions associated, respectively, with the final-state cut on the unobserved ¯ c -quark line and with the unpinched pole contribution of the barred propagator. Also, in eachdiagram in Fig. 2, the c -quark line fragmenting into the final-state D -meson is on-shell, see(30). Due to these on-shell conditions for the diagrams in Fig. 2 and the similar conditionsfor their mirror diagrams, S µνλ ( k , k ) satisfies the tree-level Ward identities, k µ S µνλ ( k , k ) = 0 ,k ν S µνλ ( k , k ) = 0 , ( k − k ) λ S µνλ ( k , k ) = 0 . (39)Here, the last identity and the first two identities are represented diagrammatically in13igs. 3(a) and 3(b), respectively. In the collinear limit, k i → x i p ( i = 1 , δ (( p c + k − k ) − m c ) → (1 / p c · p ) δ ( x − x ), while the other delta function associatedwith the diagrams in Fig. 2 implies x > x bj (see (59) below); the similar relations hold alsofor the corresponding mirror diagrams. Thus, the collinear expansion of Ward identities of(39) produces a series of relations in the collinear limit: S pνλ ( x , x ) = 0 , (40) S µpλ ( x , x ) = 0 , (41)( x − x ) S µνp ( x , x ) = 0 , (42) ∂S µpλ ( k , k ) ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = 0 , ∂S pνλ ( k , k ) ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = 0 , (43) S ανλ ( x , x ) + x ∂S pνλ ( k , k ) ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = 0 , (44) S µβλ ( x , x ) + x ∂S µpλ ( k , k ) ∂k β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = 0 , (45) ∂ S µpλ ( k , k ) ∂k α ∂k β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = ∂ S pνλ ( k , k ) ∂k α ∂k β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = 0 , (46) x ∂ S µpλ ( k , k ) ∂k α ∂k β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p + ∂S µβλ ( k , k ) ∂k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = 0 , (47)and so on, where α = +, β = +. These can be used to reorganize various terms obtainedby the collinear expansion of (36), and one obtains the following results for the contributionin each order:(i) By the relations (40), (41) and (43), the first three terms in (37), and all terms arisingin (36) up to the order in 1 /Q of those three terms, vanish.(ii) By the relations (42), (44)-(47), the sum of the contributions of the next higher orderin (36), which behave as the same order as the terms involving the second derivativein (37), vanish (see the discussion below (52)).(iii) The remaining contributions in (36), behaving as the same order as the first term in(38), eventually yield the gauge-invariant twist-3 contribution to w ( p, q, p c ) in (57)below.Instead of describing in detail those rather lengthy calculations in the framework ofthe standard collinear expansion, we may employ a somewhat different approach: Among14he relevant relations (40)-(47), in particular, (44), (45) and (47) play a role to connectthe two terms that are generated from the different terms in the right-hand side of (36)through the collinear expansion. Namely, to obtain the gauge-invariant result, we have tocombine the contributions from different terms in (36), and, furthermore, those terms areassociated with different numbers of derivatives. These facts suggest an approach to applydirectly Ward identities of (39) to the second line of (36), before the collinear expansion,i.e., without the expansion implied by the second equality in (36), nor the Taylor expansionabout k i = x i p . Indeed, substituting the decomposition (34) into the first two identities in(39), we obtain S pνλ ( k , k ) = − x ω µα k α S µνλ ( k , k ) ,S µpλ ( k , k ) = − x ω νβ k β S µνλ ( k , k ) , (48)and we apply these relations to the second line of (36), yielding S µνλ ( k , k ) M µνλ ( k , k )= S µνλ ( k , k ) 1 x ω µα ( − k α n κ + k · ng ακ ) × x ω νβ (cid:16) − k β n τ + k · ng βτ (cid:17) ( p λ n σ + ω λσ ) M κτσ ( k , k ) , (49)where some factors combined with the matrix element (33) in the right-hand side can bereexpressed as, restoring the color indices a, b and c ,( − k α n κ + k · ng ακ ) (cid:16) − k β n τ + k · ng βτ (cid:17) M κτσabc ( k , k ) ≡ M αβσA,abc ( k , k )= g Z d ξ Z d η e ik ξ e i ( k − k ) η h pS | F βnb (0) A σc ( η ) F αna ( ξ ) | pS i , (50)up to the correction terms beyond the present lowest-order calculation in QCD perturbationtheory, so that S µνλ ( k , k ) M µνλ ( k , k ) = S µνλ ( k , k ) ω µα ω νβ x x h p λ M αβnA ( k , k ) + ω λσ M αβσA ( k , k ) i . (51)As the next step, we perform the collinear expansion of the hard-scattering function.Based on the definition (50) and the decomposition (35), we note that the first and secondterms in the parentheses in (51) correspond, respectively, to the “second” and “third” ordersin the order counting relevant to the collinear expansion like (37). Thus, the collinearexpansion of (51), up to the desired order, reads (see the discussion below (38)) S µνλ ( k , k ) M µνλ ( k , k ) = ω µα ω νβ x x S µνp ( x , x ) + ω λκ k κ ∂S µνp ( k , k ) ∂k λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p ω λκ k κ ∂S µνp ( k , k ) ∂k λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p M αβnA ( k , k ) + S µνλ ( x , x ) ω λσ M αβσA ( k , k ) . (52)First, we consider the first term in the right-hand side, which is proportional to S µνp ( x , x ) = S µνλ ( x , x ) p λ and corresponds to the second-order term in the above-mentioned ordercounting. One can show that, by the direct diagrammatic calculation of S µνp ( x , x ), thecorresponding SGP contributions from the diagrams in Fig. 2 cancel with those from theirmirror diagrams (i.e., S µνp ( x , x ) arising in (42) equals zero even for x = x ), and thusthe first term in the parentheses ( · · · ) in the right-hand side of (52) vanishes. It is worthnoting that the similar vanishing property of the SGP contributions was used also for thecase of the pion production associated with the twist-3 quark-gluon correlation functions(see, e.g., [27]). On the other hand, S µνλ ( x , x ) ω λσ , arising in the last term in (52), doesnot vanish, but this can be recast using the relation, S µνλ ( x , x ) ω λσ = ( x − x ) ω λσ ∂S µνp ( k , k ) ∂k λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p , (53)which is obtained by the collinear expansion of the last Ward identity of (39). Furthermore,for the second term in the right-hand side of (52), we may use another relation, ∂S µνp ( k , k ) ∂k λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p = − ∂S µνp ( k , k ) ∂k λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p , (54)which can be derived for λ = ⊥ by direct inspection of the diagrams in Fig. 2 and their mirrordiagrams. We remind that a similar relation also holds for the hard part corresponding tothe quark-gluon correlation functions, as discussed for the case of the pion production [27].Thus, (52) yields, at the twist-3 accuracy, S µνλ ( k , k ) M µνλ ( k , k ) = ω µα ω νβ x x ω λκ ∂S µνp ( k , k ) ∂k λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p × [ − ( k κ − k κ ) n σ + ( k − k ) · ng κσ ] M αβσA ( k , k ) , (55)where the second line gives, similarly as in (50),[ − ( k κ − k κ ) n σ + ( k − k ) · ng κσ ] M αβσA,abc ( k , k )= ig Z d ξ Z d η e ik ξ e i ( k − k ) η h pS | F βnb (0) F κnc ( η ) F αna ( ξ ) | pS i , (56)up to the higher-order corrections beyond the present accuracy. Substituting these resultsinto (32), we obtain the final form for the relevant twist-3 contribution to the hadronictensor in SIDIS, as the factorization formula in terms of the gauge-invariant three-gluoncorrelation functions, w ( p, q, p c ) = Z dx x Z dx x ∂S abcµνλ ( k , k , q, p c ) p λ ∂k σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p ω µα ω νβ ω σγ M αβγF,abc ( x , x ) , (57)16here we have restored the color indices a, b, c as well as momentum variables q, p c , whichwere associated with the hard-scattering function in (32), and M αβγF,abc ( x , x ) denote thethree-gluon lightcone correlation functions, obtained by integrating (56) over k − i , k i ⊥ ( i =1 , M αβγF,abc ( x , x ) = − gi Z dλ π Z dµ π e iλx e iµ ( x − x ) h pS | F βnb (0) F γnc ( µn ) F αna ( λn ) | pS i = 340 d abc O αβγ ( x , x ) − i f abc N αβγ ( x , x ) , (58)with O αβγ ( x , x ) and N αβγ ( x , x ) in (1) and (2). As we demonstrated above in deriv-ing these results, all the gauge-noninvariant terms that could potentially contribute to w ( p, q, p c ) vanished or canceled among themselves, and the total twist-3 contribution to w ( p, q, p c ), relevant to SSA, proves to be expressed solely in terms of the gauge-invariantthree-gluon correlation functions O ( x , x ) and N ( x , x ) defined as (7) and (8).When one calculates the relevant hard part, ∂S abcµνλ ( k , k , q, p c ) p λ /∂k σ (cid:12)(cid:12)(cid:12) k i = x i p , arising in(57), one should note that the derivative with respect to k σ can hit the delta functions, δ (( k + q − p c ) − m c ) and δ (( p c + k − k ) − m c ), which are involved in S abcµνλ ( k , k , q, p c ) p λ as mentioned above (39). Such derivatives of these delta functions can be reexpressed bythe derivatives with respect to x , and then be treated by integration by parts, giving riseto the derivative of the three-gluon correlation functions of (7) and (8). After such manip-ulations, the former of the above delta functions, which represents the on-shell conditionfor the final-state cut on the unobserved ¯ c -quark line in the diagrams of Fig. 2, becomes δ (cid:16) ( k + q − p c ) − m c (cid:17)(cid:12)(cid:12)(cid:12) k = xp = 1 b zQ δ q T Q − (cid:18) b x − (cid:19) (cid:18) b z − (cid:19) + m c b z Q ! , (59)with b x = x bj x , and this factor appears in the final expression for our cross section based on(57).Here, we make a brief comment on the calculation presented in [32]. Using the notationin the present paper, the authors of [32] calculated w ( p, q, p c ) with the following formula,in place of the right-hand side of (57): Z dx x Z dx x ∂S abcµνλ ( k , k , q, p c ) p λ g µν ⊥ ∂k σ ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p ω σγ g αβ M αβγF,abc ( x , x ) , (60)where g µν ⊥ = g µν − p µ n ν − p ν n µ = − S µ S ν − ǫ µpnS ǫ νpnS . (61)In (60), we can make the replacement ω σγ → g σ ⊥ γ , up to the irrelevant corrections of twist-4and higher, and, using (61) and the property S γ g αβ M αβγF,abc ( x , x ) = 0, implied by (58) with(7) and (8), we see that the three-gluon correlation functions involved in (60) are indeedexpressed by the two types of functions of (11) after evaluating the SGP at x = x explicitly.17learly, (60) used in [32] leads to a result different from the result based on our completeformula (57). It is straightforward to see that, if the tensor structure of the three-gluoncorrelation functions M αβγF,abc ( x , x ) of (58) were assumed to be given by only one structure, g αβ ǫ γpnS , our formula (57) would reduce to the formula (60), up to the corrections of twist-4and higher. However, such assumption contradicts with the permutation symmetry requiredby the Bose statistics of the gluon, as emphasized in section 2 and represented in (7) and(8). L µν W µν Using the kinematic variables defined in section 3.1, the differential cross section (28) canbe expressed as d ∆ σdx bj dQ dz f dq T dφ = α em π x bj S ep Q z f L µν ( ℓ, ℓ ′ ) W µν ( p, q, P h ) , (62)where α em = e / (4 π ) is the QED coupling constant. We restore the implicit “free” Lorentzindices in (57) for the virtual photon, corresponding to µ and ν in w µν ( p, q, p c ) of (31)(see the discussion above (32)), and, substituting the result into (31), we obtain thethree-gluon-correlation contribution to W µν ( p, q, P h ) in (62). To calculate the contraction L µν ( ℓ, ℓ ′ ) W µν ( p, q, P h ) arising in (62), we introduce the following four vectors which areorthogonal to each other: T µ = 1 Q ( q µ + 2 x bj p µ ) ,X µ = 1 q T ( P µh z f − q µ − q T + m h /z f Q ! x bj p µ ) ,Y µ = ǫ µνρσ Z ν X ρ T σ ,Z µ = − q µ Q . (63)These are the extension of four basis vectors introduced for the massless case ( m h = 0)in [48] to the case of massive meson with m h in the final state [32]. These vectors become T µ = (1 , , , X µ = (0 , , , Y µ = (0 , , , Z µ = (0 , , ,
1) in the hadron frame. Inthe present case, we find that W µν can be expanded in terms of the following six independenttensors [29]: V µν = X µ X ν + Y µ Y ν , V µν = g µν + Z µ Z ν , V µν = T µ X ν + X µ T ν , V µν = X µ X ν − Y µ Y ν , V µν = T µ Y ν + Y µ T ν , V µν = X µ Y ν + Y µ X ν , (64)18here we have followed the notation in [48] but have not shown the explicit form of thethree tensors V , , among nine basis tensors because these three tensors are irrelevant forthe expansion of our W µν . We also introduce the inverse tensors e V µνk for the above V µνk : e V µν = 12 (2 T µ T ν + X µ X ν + Y µ Y ν ) , e V µν = T µ T ν , e V µν = −
12 ( T µ X ν + X µ T ν ) , e V µν = 12 ( X µ X ν − Y µ Y ν ) , e V µν = −
12 ( T µ Y ν + Y µ T ν ) , e V µν = 12 ( X µ Y ν + Y µ X ν ) . (65)Then, one obtains L µν W µν = X k =1 , ··· , , , [ L µν V µνk ] h W ρσ e V ρσk i ≡ Q X k =1 , ··· , , , A k h W ρσ e V ρσk i , (66)where A k ≡ L µν V µνk /Q is given by A = 1 + cosh ψ, A = − , A = − cos φ sinh 2 ψ, A = cos 2 φ sinh ψ, A = − sin φ sinh 2 ψ, A = sin 2 φ sinh ψ. (67)By the expansion (66), the cross section for ep ↑ → eDX consists of the five structurefunctions associated with A , , A , A , A and A , respectively, which have different de-pendences on the azimuthal angle φ .In closing this section, we summarize the prescription established for calculating thetwist-3 single-spin-dependent cross section that is generated by the three-gluon correlationfunctions of the nucleon: The corresponding differential cross section is given by (62) withthe expansion in the right-hand side of (66), in which W ρσ is given as (31) using ourfactorization formula (57) for w µν ( p, q, p c ). ep ↑ → eDX Using the formalism presented above, we now obtain the leading-order QCD formula for thesingle-spin-dependent cross section in the SIDIS, ep ↑ → eDX , generated from the twist-319hree-gluon correlation functions O ( x , x ) and N ( x , x ) of (7) and (8), as d ∆ σdx bj dQ dz f dq T dφ = α em α s e c M N πx bj S ep Q (cid:18) − π (cid:19) X k =1 , ··· , , , A k S k Z x min dxx Z z min dzz δ q T Q − (cid:18) − x (cid:19) (cid:18) − z (cid:19) + m c ˆ z Q ! × X a = c, ¯ c D a ( z ) " δ a ( ddx O ( x, x ) − O ( x, x ) x ! ∆ˆ σ k + ddx O ( x, − O ( x, x ! ∆ˆ σ k + O ( x, x ) x ∆ˆ σ k + O ( x, x ∆ˆ σ k ) + ( ddx N ( x, x ) − N ( x, x ) x ! ∆ˆ σ k − ddx N ( x, − N ( x, x ! ∆ˆ σ k + N ( x, x ) x ∆ˆ σ k − N ( x, x ∆ˆ σ k ) , (68)where the subscript k runs over 1 , , , , , A k defined in (67) and S k defined as S k = sin Φ S for k = 1 , , , S k = cos Φ S for k = 8 ,
9. The quark-flavor index a can, inprinciple, be c and ¯ c , with δ c = 1 and δ ¯ c = −
1, so that the cross section for the ¯ D -mesonproduction ep ↑ → e ¯ DX can be obtained by a simple replacement of the fragmentationfunction to that for the ¯ D meson, D a ( z ) → ¯ D a ( z ). α s = g / (4 π ) is the strong couplingconstant, and e c = 2 / c -quark. The lower limits ofthe integrals are given by z min = z f (1 − x bj ) Q x bj m c − vuut − x bj m c (1 − x bj ) Q " x bj q T (1 − x bj ) Q , (69)and x min = x bj (cid:20) z f q T + m c z f (1 − z f ) Q (cid:21) for z f r q T m c ! > ,x bj " m c Q r q T m c ! for z f r q T m c ! ≤ . (70)20artonic hard cross sections ∆ˆ σ ik ( i = 1 , · · · ,
4) are obtained as follows: ∆ˆ σ = q T ˆ xQ (1 − ˆ z ) ˆ z { Q ˆ z (1 − ˆ z )(1 − z + 2ˆ z − x + 2ˆ x + 12ˆ x ˆ z (1 − ˆ x )(1 − ˆ z ))+2 m c Q ˆ x (2ˆ z (1 − ˆ z ) + ˆ x (1 − z + 8ˆ z )) − m c ˆ x } , ∆ˆ σ = q T ˆ x Q (1 − ˆ z ) ˆ z { Q ˆ z (1 − ˆ x )(1 − ˆ z ) − m c ˆ x } , ∆ˆ σ = xQ (1 − ˆ z ) ˆ z (1 − z ) { Q ˆ z (1 − ˆ x )(1 − ˆ z ) − m c ˆ x } { Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x } , ∆ˆ σ = q T ˆ x Q (1 − ˆ z ) ˆ z { Q ˆ z (1 − ˆ x )(1 − ˆ z ) − m c ˆ x } { Q ˆ z (1 − ˆ z ) + m c } , ∆ˆ σ = ∆ˆ σ = 0 , (71) ∆ˆ σ = q T ˆ xQ (1 − ˆ z ) ˆ z { Q ˆ z (1 − ˆ z )(1 − z + 2ˆ z − x + 4ˆ x + 24ˆ x ˆ z (1 − ˆ x )(1 − ˆ z ))+4 m c Q ˆ x (2ˆ z (1 − ˆ z ) + ˆ x (1 − z + 8ˆ z )) − m c ˆ x } , ∆ˆ σ = 2∆ˆ σ , ∆ˆ σ = 2∆ˆ σ , ∆ˆ σ = − q T ˆ xQ (1 − ˆ z ) ˆ z ( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , ∆ˆ σ = xQ (1 − ˆ z ) ˆ z (1 − z )( Q ˆ z (1 − ˆ x )(1 − ˆ z ) − m c ˆ x ) , ∆ˆ σ = − q T ˆ xQ (1 − ˆ z ) ˆ z ( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , (72) ∆ˆ σ = q T ˆ x Q (1 − ˆ z ) ˆ z ( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x )( Q (1 − z + 6ˆ z ) − m c ) , ∆ˆ σ = − q T ˆ x Q (1 − ˆ z ) ˆ z ( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , ∆ˆ σ = − xQ (1 − ˆ z ) ˆ z (1 − z ) { Q ˆ z (1 − ˆ z ) (1 − x + 8ˆ x ) − m c Q ˆ x ˆ z (1 − x )(1 − ˆ z ) + 8 m c ˆ x } , ∆ˆ σ = − q T ˆ x Q (1 − ˆ z ) ˆ z ( Q ˆ z (1 − ˆ z ) + m c )( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , ∆ˆ σ = − xQ (1 − ˆ z ) ˆ z (1 − z )( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , ∆ˆ σ = − q T ˆ x Q (1 − ˆ z ) ˆ z ( Q ˆ z (1 − ˆ z ) + m c ) , (73)21 ∆ˆ σ = q T ˆ x Q (1 − ˆ z ) ˆ z ( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x )( Q (1 − z + 6ˆ z ) − m c ) , ∆ˆ σ = − q T ˆ x Q (1 − ˆ z ) ˆ z ( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , ∆ˆ σ = xQ (1 − ˆ z ) ˆ z (1 − z ) { Q ˆ z (1 − ˆ z ) (1 + 12ˆ x − x )+4 m c Q ˆ x ˆ z (3 − x )(1 − ˆ z ) − m c ˆ x } , ∆ˆ σ = − q T ˆ xQ (1 − ˆ z ) ˆ z ( Q ˆ z (1 − ˆ z ) (1 + ˆ x − x ) + m c Q ˆ x ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , ∆ˆ σ = xQ (1 − ˆ z ) ˆ z (1 − z )( Q ˆ z (1 + 2ˆ x )(1 − ˆ z ) + 2 m c ˆ x ) , ∆ˆ σ = − q T ˆ xQ (1 − ˆ z ) ˆ z ( Q ˆ z (1 − ˆ z )(1 + ˆ x ) + m c ˆ x ) , (74)where b x = x bj x , b z = z f z . (75)The single-spin-dependent cross section (68) can be decomposed into the five structurefunctions, based on the different dependences on the azimuthal angles Φ S and φ throughthe above-mentioned explicit forms of A k and S k , as d ∆ σdx bj dQ dz f dq T dφ = sin Φ S ( F + F cos φ + F cos 2 φ )+ cos Φ S ( F sin φ + F sin 2 φ ) . (76)The five independent azimuthal structures of this type have been observed also in thetwist-3 single-spin-dependent cross section for ep ↑ → eπX , generated from the quark-gluoncorrelation functions, as presented in [29, 35, 36]. Introducing the azimuthal angles φ h and φ S of the hadron plane and the nucleon’s spin vector ~S , respectively, as measured from the lepton plane , they are connected to the above Φ S and φ as Φ S = φ h − φ S , φ = φ h . One canrecast (76) into the superposition of five sine modulations with these new azimuthal anglesas d ∆ σdx bj dQ dz f dq T dφ h = sin( φ h − φ S ) F sin( φ h − φ S ) + sin(2 φ h − φ S ) F sin(2 φ h − φ S ) + sin φ S F sin φ S + sin(3 φ h − φ S ) F sin(3 φ h − φ S ) + sin( φ h + φ S ) F sin( φ h + φ S ) , (77)with the relations: F sin( φ h − φ S ) = F , F sin(2 φ h − φ S ) = F + F , F sin φ S = −F + F ,F sin(3 φ h − φ S ) = F + F , F sin( φ h + φ S ) = −F + F . (78)22he azimuthal-angle dependence of the single-spin-dependent cross section derived in theTMD approach [18] was presented in a form similar to (77), so that the decomposition (77)into five structure functions is convenient to make connection with the TMD approach inthe small- q T region.For completeness, we consider the unpolarized cross section for the SIDIS, ep → eDX ,and list the corresponding twist-2 contribution at the leading order in QCD perturbationtheory, which gives an extension of the study in [49, 48, 50] to the case with massive-hadronproduction in the final state. The result is d σ unpol dx bj dQ dz f dq T dφ = α em α s e c πx bj S ep Q X k =1 A k Z x min dxx Z z min dzz X a = c, ¯ c D a ( z ) G ( x )ˆ σ Uk × δ q T Q − (cid:18) − x (cid:19) (cid:18) − z (cid:19) + m c ˆ z Q ! , (79)with (67), (69), and (70). This is generated from the unpolarized gluon-density distribution G ( x ) for the nucleon, G ( x ) = − g βα x Z dλ π e iλx h p | F βna (0) F αna ( λn ) | p i , (80)and the partonic hard cross sections are obtained as, utilizing the formalism discussed insection 3.3, ˆ σ U = Q (1 − ˆ z ) ˆ z { Q ˆ z (1 − ˆ z )(1 − z + 2ˆ z − x + 2ˆ x + 12ˆ x ˆ z (1 − ˆ x )(1 − ˆ z ))+2 m c Q ˆ x (2ˆ z (1 − ˆ z ) + ˆ x (1 − z + 8ˆ z )) − m c ˆ x } , ˆ σ U = xQ (1 − ˆ z )ˆ z ( Q ˆ z (1 − ˆ x )(1 − ˆ z ) − m c ˆ x ) , ˆ σ U = q T ˆ xQ (1 − ˆ z ) ˆ z (1 − z )( Q ˆ z (1 − x )(1 − ˆ z ) − m c ˆ x ) , ˆ σ U = xQ (1 − ˆ z ) ˆ z ( Q ˆ z (1 − ˆ x )(1 − ˆ z ) − m c ˆ x )( Q ˆ z (1 − ˆ z ) + m c ) . (81)We note that these partonic hard cross sections coincide with the corresponding resultspresented in [32], except for ˆ σ U . Comparing (81) for ˆ σ Uk with the above result (71) for∆ˆ σ k , which represents the partonic hard cross sections associated with the derivatives, dO ( x, x ) /dx and dN ( x, x ) /dx , of the three-gluon correlation functions in (68), one findsthe following relations between the partonic hard cross sections at the twist-3 level andthose at the twist-2 level: ∆ˆ σ k = 2 q T b xQ (1 − b z ) ˆ σ Uk . (82)The other partonic cross sections at the twist-3 level, ∆ˆ σ ik ( i = 2 , ,
4) of (72)-(74), are alsorelated to the partonic cross sections (81) at the twist-2 level, although, unlike (82), thecorresponding relations cannot be manifested by direct comparison between the formulae of2372)-(74) and those of (81). These remarkable relations, as well as a single “master formula”behind them, will be presented elsewhere [51]. We mention that the similar master formula,which allows us to relate the 3 → → In this paper, we have investigated the single spin asymmetry for the D -meson productionin SIDIS, generated from the twist-3 three-gluon correlation functions for the nucleon. Wefirst showed, correcting the previous study in [24], that there are only two independentthree-gluon correlation functions of twist-3, O ( x , x ) and N ( x , x ), which correspond totwo possible ways to construct color-singlet combination composed of three active gluons.Then, we have formulated the method for calculating the twist-3 single-spin-dependentcross section generated from the three-gluon correlations. Our formulation is based ona systematic analysis of the relevant diagrams in the Feynman gauge and gives all thecontribution to the cross section at the twist-3 level in the leading order in perturbativeQCD, guaranteeing the gauge invariance of the result. As in the twist-3 mechanism for SSAgenerated from the quark-gluon correlation functions for the nucleon, the cross section inthe present case occurs as the pole contribution of an internal propagator in the partonichard-scattering subprocess, and the corresponding contribution leads to the cross sectionexpressed in terms of the four types of functions of the relevant momentum fraction x : O ( x, x ), O ( x, N ( x, x ) and N ( x, x contribute to the final form of the cross section. Thesefeatures discussed for the SSA in ep ↑ → eDX also apply to the case of A N in p ↑ p → hX ( h = π, K, D , etc.) [52].These new results have also revealed that the previous studies [32, 33] missed importantcontributions. In particular, the factorization formula for the cross section used in [32, 33]was written down in an ad-hoc way, and, compared to our factorization formula derivedin this paper, involved additional projection for the Lorentz structure onto a particularcomponent of the hard-scattering part as well as of the three-gluon correlation functions,which would have lead to the result incompatible with symmetry requirements in QCD. Acknowledgments
We thank Andreas Metz for bringing our attention to Ref. [54]. The work of S. Y. issupported by the Grand-in-Aid for Scientific Research (No. 22.6032) from the Japan Societyof Promotion of Science. 24
Symmetry constraints on the gluon correlation func-tions
In this Appendix, we consider the correlation functions of the gluon fields, A µa ( ξ ), in thenucleon. Such correlation functions of the two- or three-gluon fields, with non-lightlikeseparations between those fields, arise in the intermediate step of the analysis of the relevantFeynman diagrams for the SIDIS, ep ↑ → eDX , to the twist-3 accuracy, as discussed insection 3.2. We discuss the symmetry properties of those correlation functions and theirimplication on SSA: We show that Fig. 1(a) does not contribute to SSA, and that Fig. 2(b)can give rise to SSA.Similarly to (32), the contribution from Fig. 1(a) to w µν ( p, q, p c ) in (31) can be writtenas, Z d k (2 π ) S (2) µν ( k ) M µν (2) ( k, p, S ) , (83)where the Lorentz indices for the virtual photon in the partonic hard-scattering part S (2) µν ( k )are suppressed for simplicity, and the corresponding nucleon matrix element M µν (2) ( k, p, S )is defined as a correlation function of the type mentioned above: M µν (2) ( k, p, S ) = Z d ξe ik · ξ h pS | A νa (0) A µa ( ξ ) | pS i . (84)Note that the color indices of the gluon fields in this formula are summed over, since onlythe color-singlet combination is relevant as a matrix element in the color-singlet hadron.Also, the corresponding color projection is taken for S (2) µν ( k ) in (83). Now, invariance underthe parity ( P ) and time-reversal ( T ) transformations implies M µν (2) ( k, p, S ) ∗ = M µν (2) ( k, p, − S ) , (85)and, combined with the fact that the matrix element M µν (2) ( k, p, S ) depends on the spinvector S µ linearly , we find that the spin-dependent part of M µν (2) ( k, p, S ) is a pure imaginaryquantity. For the process ep ↑ → eDX , the leptonic tensor L µν of (28) is real. Accordingly,an imaginary contribution from the hard part S (2) µν ( k ) is necessary to give the real contribu-tion to the spin-dependent cross section. However, this is impossible for the leading-orderdiagrams contributing to the middle blob in Fig. 1(a). Therefore, (83) representing thecontribution from Fig. 1(a) does not give rise to SSA.Next, using the similar logic as above, we consider the contribution of Fig. 1(b) to (32).By projecting M µνλabc ( k , k ) in (33) into the color-singlet components, we define the twotypes of the contractions of the corresponding color indices as M µνλ ± ( k , k , p, S ) ≡ C bca ± M µνλabc ( k , k )= Z dξ Z dη e ik ξ e i ( k − k ) η h pS | C bca ± A νb (0) A λc ( η ) A µa ( ξ ) | pS i , (86)25ith C bca ± of (12). P T -invariance implies M µνλ ± ( k , k , p, S ) ∗ = M µνλ ± ( k , k , p, − S ) , (87)which shows that the spin-dependent parts of M µνλ ± ( k , k , p, S ) are pure imaginary quan-tities. Accordingly, only an imaginary contribution from the corresponding color-projectedhard part C bca ± S abcµνλ ( k , k , q, p c ) can give rise to SSA with (32) in ep ↑ → eDX . For thediagrams shown in Fig. 1(b), the pole contribution of an internal propagator in the hardpart can give rise to such imaginary contribution. The corresponding unpinched poles areallowed in Fig. 1(b), owing to the presence of an extra gluon line connecting the hard andsoft parts, compared with Fig. 1(a).At this point one might recall that the k ⊥ -dependent gluon distribution function [53](“gluon-Sivers function”) can give rise to SSA in the framework of the TMD factorizationand this fact may contradict with the above statement that (84) does not contribute toSSA. However, the gluon Sivers function can become well-defined only after supplyingan appropriate gauge-link operator between the gluon-fields, and the gauge-link operatoractually resums the effect of the extra gluon-lines connecting the hard and soft parts,i.e., the gluon-Sivers function represents some effect of the contribution from Fig. 1(b).Therefore, there is no contradiction between the two facts. The gluon-Sivers function mayrather represent the same effect as the three-gluon correlation functions in the region of theintermediate transverse momentum, similarly to the case of the quark-Sivers function andthe twist-3 quark-gluon correlation functions as shown in [39, 40, 41]. References [1] G. Bunce et al. , Phys. Rev. Lett. , 1113 (1976);A. M. Smith et al. , Phys. Lett. B185 , 209 (1987);B. Lundberg et al. , Phys. Rev.
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