Probing the three-gluon correlation functions by the single spin asymmetry in p^\uparrow p\to DX
aa r X i v : . [ h e p - ph ] J u l Probing the three-gluon correlation functionsby the single spin asymmetry in p ↑ p → DX Yuji Koike and Shinsuke Yoshida Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan Graduate School of Science and Technology, Niigata University, Ikarashi, Niigata950-2181, Japan
Abstract
We study the single transverse-spin asymmetry for the inclusive open-charm productionin the pp -collision, p ↑ p → DX , induced by the three-gluon correlation functions in thepolarized nucleon. We derive the corresponding twist-3 cross section formula in the leadingorder with respect to the QCD coupling constant. As in the case of the semi-inclusive deepinelastic scattering, ep ↑ → eDX , our result differs from the previous result in the literature.We also derive a “master formula” which expresses the twist-3 cross section in terms of the gg → c ¯ c hard scattering cross section. We present a model calculation of the asymmetryat the RHIC energy, demonstrating the sensitivity of the asymmetry on the form of thethree-gluon correlation functions. 1 Introduction
Open charm production in inclusive hard processes, such as semi-inclusive deep inelasticscattering (SIDIS), ep → eDX , and the D -meson production in pp collision, pp → DX ,is an ideal tool to investigate the gluon distributions in the nucleon. Similarly the singlespin asymmetry (SSA) in those processes plays a crucial role to reveal the multi-gluoncorrelations in the nucleon [1, 2, 3, 4, 5, 6]. When the transverse momentum of the final D -meson can be regarded as hard ( P T ≫ Λ QCD ), one can analyze the processes in theframework of the collinear factorization [7, 8, 9]. In this framework, SSA appears as a twist-3 observable and can be represented in terms of the multi-parton correlation functions.The purely gluonic correlations responsible for SSAs in the open charm production arerepresented by the “three-gluon” correlation functions. For the quark-gluon correlationfunctions in the nucleon, there have been many studies in the literature in connection withSSAs for the light hadron productions [7]-[23], and our understanding on the mechanism ofSSA has made a great progress.In our recent paper [4] we studied the contribution from the three-gluon correlationfunctions to SIDIS, ep ↑ → eDX . In that study, we have identified the complete set of thethree-gluon correlation functions and established the formalism for calculating the twist-3single-spin-dependent cross section induced by those functions. The gauge invariance andthe factorization property of the cross section have been shown explicitly in the leadingorder with respect to the QCD coupling constant. The result of that study differs fromthe previous study in the literature [2], and we clarified the origin of the discrepancy. Inour another paper [6], we have developed a novel “master formula” for the three-gluoncontribution to ep ↑ → eDX which expresses the corresponding twist-3 cross section in termsof the twist-2 γ ∗ g → c ¯ c scattering cross section, extending the similar formula known forthe soft-gluon-pole (SGP) contribution to SSA from the quark-gluon correlation functionsin the nucleon [16, 17]. This formula simplifies the actual calculation of the twist-3 crosssection and makes its structure transparent, and may be useful to include higher ordercorrections to the asymmetry.The purpose of this paper is to extend these studies to the contribution of the three-gluon correlation functions to the SSA in the pp collision, p ↑ ( p, S ⊥ ) + p ( p ′ ) → D ( P h ) + X, (1)where S ⊥ is the transverse spin vector of the polarized nucleon and P h is the momentum ofthe D -meson satisfying P h = m h with the D -meson mass m h . The initial nucleons’ momenta p and p ′ are in the collinear configuration and can be regarded as massless ( p = p ′ = 0)in the twist-3 accuracy. We will derive a corresponding formula for the twist-3 single-spin-dependent cross section in the leading order with respect to the QCD coupling constant.We will further derive a master formula for (1) which connects the twist-3 cross section tothe cross section for the gg → c ¯ c scattering. We will also present a model calculation for In the framework of the transverse-momentum-dependent factorization, which is useful to describethe low- P T hadron production, the corresponding gluonic effect is represented as the k ⊥ -dependent gluondistribution functions [24, 25]. A N = ∆ σ/σ ≡ ( σ ↑ − σ ↓ ) / ( σ ↑ + σ ↓ ), where σ ↑ ( ↓ ) represents the cross sectionfor (1) with the initial nucleon polarized along S ⊥ ( − S ⊥ ), and will obtain a constraint onthe three-gluon correlation functions, using the recent RHIC data on A N for the D -mesonproduction [26].The remainder of this paper is organized as follows: In section 2, we recall the completeset of the three-gluon correlation functions in the transversely polarized nucleon which arerelevant for our study. In section 3, we derive the twist-2 unpolarized cross section for theprocess pp → DX induced by the gluon-density in the nucleon. In section 4, we derive thetwist-3 single-spin-dependent cross section induced by the three-gluon correlation functionsfor the process (1), applying the formalism in [4]. In section 5, we derive the master formulawhich expresses the twist-3 cross section for (1) induced by the three-gluon correlationfunctions in terms of the cross section for gg → c ¯ c scattering in the twist-2 level. In section6, we present a model calculation of A N for the D -meson production at the RHIC energy,and demonstrate a sensitivity of the asymmetry on the form of the three-gluon correlationfunctions. Section 7 is devoted to a brief summary. In the appendix, we discuss sometechnical aspects in the derivation of the contribution from the initial-state-interactiondiagrams to the twist-3 cross section. The twist-3 three-gluon correlation functions in the transversely polarized nucleon werefirst introduced in [1]. Then, as was clarified in [27, 28, 4], there are two independentthree-gluon correlation functions due to the difference in the contraction of color indices.Following the notation in [4], we call those functions O ( x , x ) and N ( x , x ), which aredefined from the lightcone correlation functions of the three field-strengths of the gluon inthe polarized nucleon as O αβγ ( x , x ) = − g ( i ) 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 = 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 , (2) N αβγ ( x , x ) = − g ( i ) 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 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 . (3)where F αβa ≡ ∂ α A βa − ∂ β A αa + gf abc A αb A βc is the gluon’s field strength, and we used thenotation F αna ≡ F αβa n β and ǫ αpnS ⊥ ≡ ǫ αµνλ p µ n ν S ⊥ λ with the convention ǫ = 1. d bca and f bca are the symmetric and anti-symmetric structure constants of the color SU(3) group,and we have suppressed the gauge-link operators which ensure the gauge invariance. p isthe nucleon momentum, and S ⊥ is the transverse spin vector of the nucleon normalized3s S ⊥ = −
1. In the twist-3 accuracy, p can be regarded as lightlike ( p = 0), and n isanother lightlike vector satisfying p · n = 1. To be specific, we set p µ = ( p + , , ⊥ ), n µ =(0 , n − , ⊥ ), and S µ ⊥ = (0 , , S ⊥ ). The nucleon mass M N is introduced to define O ( x , x )and N ( x , x ) dimensionless. The decomposition (2) and (3) takes into account all theconstraints from hermiticity, invariance under the parity- and time-reversal transformationsand the permutation symmetry among the participating three gluon-fields. The functions O ( x , x ) and N ( x , x ) are real and have the following symmetry properties, O ( x , x ) = O ( x , x ) , O ( x , x ) = O ( − x , − x ) , (4) N ( x , x ) = N ( x , x ) , N ( x , x ) = − N ( − x , − x ) . (5) pp → DX fromgluon-fusion We first recall the twist-2 unpolarized cross section for the process (1) which is the denom-inator of A N . It receives main contribution from the c ¯ c -creation due to the gluon-fusionwith the subsequent fragmentation of the c (or ¯ c ) quark into the D (or ¯ D ) meson (Fig. 1).The corresponding unpolarized cross section can be written as P h dσd P h = α s S X f = c, ¯ c Z dx ′ x ′ G ( x ′ ) Z dzz D f ( z ) Z dxx G ( x )ˆ σ Ugg → c δ (˜ s + ˜ t + ˜ u ) (6)where S = ( p + p ′ ) is the center-of-mass energy squared and α s = g / (4 π ) is the strong cou-pling constant. The c → D (or ¯ c → ¯ D ) fragmentation function D f ( z ) and the unpolarizedgluon distribution in the nucleon G ( x ) are, respectively, defined as X X N 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 f ( z ) + · · · , (7)1 x Z dλ π e iλx h p | F µna (0) F νna ( λn ) | p i = − G ( x ) g µν ⊥ + · · · , (8)where N = 3 is the number of colors, p c is the momentum of the c (or ¯ c ) quark fragmentinginto the D ( ¯ D )-meson with p c = m c and w is another lightlike vector of O (1 /p + ) satisfying P h · w = 1. Thus p c is related to P h as p µc = P µh /z + ( m c z − m h /z ) / w µ . g µν ⊥ is definedas g µν ⊥ ≡ g µν − p µ n ν − p ν n µ . The symbol · · · denotes higher-twist contributions which areirrelevant here. The partonic hard cross section ˆ σ Ugg → c can be obtained from the 9 diagramsshown in Fig. 2 in the leading order (LO) with respect to the QCD coupling constant, andis given by ˆ σ Ugg → c = 12 N (cid:18) t ˜ u − NC F s (cid:19) ˜ t + ˜ u + 4 m c ˜ s − m c ˜ s ˜ t ˜ u ! , (9)4 Figure 1: Generic diagrams for the twist-2 cross section for pp → DX induced by the gluondensities in the initial nucleons. x0p0xp p x0p0 xp Figure 2: Leading order diagrams for the twist-2 unpolarized hard cross section for ˆ σ Ugg → c appearing in (6).where the invariants for gg → c ¯ c scattering are defined as˜ s = ( xp + x ′ p ′ ) ˜ t = ( p c − xp ) − m c ˜ u = ( p c − x ′ p ′ ) − m c , (10)and C F = N − N . p ↑ p → DX induced by thethree-gluon correlation functions The twist-3 single-spin-dependent cross section for p ↑ p → DX induced by the three-gluoncorrelation functions can be obtained by applying the formalism developed for ep ↑ → eDX [4]. The twist-3 cross section occurs from the diagrams of the type shown in Fig.5 (a) (b)Figure 3: Generic diagrams which give the twist-3 cross section for p ↑ p → DX inducedby the purely gluonic effect in the polarized nucleon (lower blob) convoluted with theunpolarized gluon density (upper blob) and the twist-2 fragmentation function for the D -meson (middle blob). A pair of circles in each figure represent gg → c ¯ c hard scatteringamplitudes. Figures (a) and (b) represent, respectively, the contributions from the final-state-interaction (FSI) and the initial-state-interaction (ISI). The mirror diagrams alsocontribute.3, where the extra coherent gluon is exchanged between the hard scattering part and thenucleon matrix element. In Fig. 3, the gluon density G ( x ′ ) in the unpolarized cross section(upper blob) and the fragmentation function D f ( z ) for the D -meson (middle blob) are al-ready factorized. Owing to the symmetry property of the correlation functions (lower blobsof Figs. 3(a) and (b)) defined by M µνλabc ( k , k ) = g Z d ξ Z d ηe ik ξ e i ( k − k ) η h pS | A νb (0) A λc ( η ) A µa ( ξ ) | pS i , (11)the SSA occurs only from a pole part of an internal propagator in the corresponding hardpart S abcµνλ ( k , k , x ′ p ′ , p c ) , (12)where a , b , c are color indices and the momenta k and k are assigned as shown in Fig.3. Here and below we employ the convention that the QCD coupling constant g associatedwith the attachment of the coherent gluon into the hard part is included in the matrixelement (11) consistently with the definition of the three-gluon correlation functions in (2)and (3). The hard part of the diagrams in Fig. 3 gives rise to the pole contributions at x = x . (See discussions below.) For those contributions, following the same step as [4]in the collinear expansion to S abcµνλ ( k , k , x ′ p ′ , p c ), we eventually end up with the followingexpression for the LO twist-3 cross section induced by the three-gluon correlation function(see Appendix): P h d ∆ σd P h = α s S X f = c, ¯ c Z dx ′ x ′ G ( x ′ ) Z dzz D f ( z ) Z dx x Z dx x ∂S abcµνλ ( k , k , x ′ p ′ , p c ) p λ ∂k σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p pole ω µα ω νβ ω σγ M αβγF,abc ( x , x ) , (13)where ω µα = g µα − p µ n α , and M αβγF,abc ( x , x ) is the lightcone correlation function of thefield-strengths defined as M αβγF,abc ( x , x ) = − g ( i ) 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 = N d bca ( N − N − O αβγ ( x , x ) − if bca N ( N − N αβγ ( x , x ) (14)with O αβγ ( x , x ) and N αβγ ( x , x ) defined in (2) and (3), respectively. The symbol [ · · · ] pole indicates the pole contribution is to be taken from the hard part. We emphasize thateven though the analysis of Fig. 3 starts with the gauge-noninvariant correlation function(11) and the corresponding hard part S abcµνλ ( k , k , x ′ p ′ , p c ), gauge-noninvariant contributionsappearing in the collinear expansion either vanish or cancel and the total surviving twist-3contribution to the single-spin-dependent cross section can be expressed as in (13), usingthe gauge-invariant correlation functions (2) and (3).The pole contribution to the hard part (cid:20) ∂S abcµνλ ( k , k , x ′ p ′ , p c ) p λ /∂k σ (cid:12)(cid:12)(cid:12) k i = x i p (cid:21) pole occursfrom two types of diagrams shown in Figs. 3 (a) and 3(b), which are refered to as thefinal state interaction (FSI) and the initial state interaction (ISI), respectively, by theparton lines to which the coherent gluon is attached. Figs. 4 (a) and 4(b) show the LOdiagrams contributing to the hard part S abcµνλ ( k , k , x ′ p ′ , p c ) in Figs. 3(a) (FSI) and 3(b)(ISI), respectively. The mirror diagrams of Fig. 4 also contribute. The poles are producedfrom the bared propagator, and gives rise to the δ -function at x = x in the collinearlimit ( k i → x i p ), hence the poles are refered to as the soft-gluon-pole (SGP). Other polecontributions cancel among each other by taking the sum of the whole diagrams.By calculating (cid:20) ∂S abcµνλ ( k , k , x ′ p ′ , p c ) p λ /∂k σ (cid:12)(cid:12)(cid:12) k i = x i p (cid:21) pole from Fig. 4 contracted with thecoefficient tensors in the decomposition of (2) and (3), one obtains the twist-3 single-spin-dependent cross section as [5] P h d ∆ σd P h = α s M N πS ǫ P h pnS ⊥ X f = c ¯ c Z dx ′ x ′ G ( x ′ ) Z dzz D f ( z ) Z dxx δ (cid:0) ˜ s + ˜ t + ˜ u (cid:1) z ˜ u × (cid:20) δ f (cid:26)(cid:18) ddx O ( x, x ) − O ( x, x ) x (cid:19) ˆ σ O + (cid:18) ddx O ( x, − O ( x, x (cid:19) ˆ σ O + O ( x, x ) x ˆ σ O + O ( x, x ˆ σ O (cid:27) + (cid:26)(cid:18) ddx N ( x, x ) − N ( x, x ) x (cid:19) ˆ σ N + (cid:18) ddx N ( x, − N ( x, x (cid:19) ˆ σ N + N ( x, x ) x ˆ σ N + N ( x, x ˆ σ N (cid:27)(cid:21) . (15) k1 k2k2 (cid:0) k1x0p0 x0p0 (a) k1 k2x0p0x0p0k2 (cid:0) k1 p (b)Figure 4: The LO diagrams for the partonic hard part S abcµνλ ( k , k , x ′ p ′ , p c ) for the twist-3cross section. Diagrams in (a) represent the FSI contribution and those in (b) representthe ISI contribution.where δ c = 1 and δ ¯ c = −
1. The partonic hard cross sections are given by ˆ σ O = C F ˜ u − ˜ t ˜ s ˜ t ˜ u + 1 C F ˜ u ˜ s ˜ t − N C F ˜ s ˜ t ˜ u ! ˜ t + ˜ u + 4 m c ˜ s − m c ˜ s ˜ t ˜ u ! , ˆ σ O = C F ˜ u − ˜ t ˜ s ˜ t ˜ u + 1 C F ˜ u ˜ s ˜ t − N C F ˜ s ˜ t ˜ u ! ˜ t + ˜ u + 8 m c ˜ s − m c ˜ s ˜ t ˜ u ! , ˆ σ O = C F ˜ u − ˜ t ˜ t ˜ u + 1 C F t − N C F ˜ s ˜ t ˜ u ! (cid:16) m c ˜ s − m c ˜ t ˜ u (cid:17) , ˆ σ O = C F ˜ u − ˜ t ˜ t ˜ u + 1 C F t − N C F ˜ s ˜ t ˜ u ! (cid:16) m c ˜ s − m c ˜ t ˜ u (cid:17) , (16)8nd ˆ σ N = C F ˜ t + ˜ u ˜ s ˜ t ˜ u + 1 C F ˜ u ˜ s ˜ t − N C F ˜ s ˜ t ˜ u ! ˜ t + ˜ u + 4 m c ˜ s − m c ˜ s ˜ t ˜ u ! , ˆ σ N = − C F ˜ t + ˜ u ˜ s ˜ t ˜ u + 1 C F ˜ u ˜ s ˜ t − N C F ˜ s ˜ t ˜ u ! ˜ t + ˜ u + 8 m c ˜ s − m c ˜ s ˜ t ˜ u ! , ˆ σ N = C F ˜ t + ˜ u ˜ s ˜ t ˜ u + 1 C F t − N C F ˜ s ˜ t ˜ u ! (cid:16) m c ˜ s − m c ˜ t ˜ u (cid:17) , ˆ σ N = − C F ˜ t + ˜ u ˜ s ˜ t ˜ u + 1 C F t − N C F ˜ s ˜ t ˜ u ! (cid:16) m c ˜ s − m c ˜ t ˜ u (cid:17) , (17)The hard cross sections associated with the first term in the first parentheses in (16) and(17) come from ISI (Fig. 4(b)), and those associated with the second and the third termsin the same parentheses come from FSI (Fig. 4(a)). As in the case of ep ↑ → eDX , the crosssection in (15) receives the contribution from the four functions O ( x, x ), O ( x, N ( x, x )and N ( x, O and N functions. From (15), it is clear that the process p ↑ p → DX itself is notsufficient for the complete separation of the four functions. For the separation, the process ep ↑ → eDX serves greatly, since it has five structure functions with different dependenceson the azimuthal angles to which the four functions contribute differently [4]. For themassless quark fragmenting into a light hadron (i.e. m c → σ O ,O ,N ,N → σ O = σ O and σ N = − σ N . Therefore the three-gluon correlation functions appear in thecombination of x ( d/dx )( O ( x, x ) + O ( x, − O ( x, x ) + O ( x, x ( d/dx )( N ( x, x ) − N ( x, − N ( x, x ) − N ( x, m c = 0.Our result in (15) differs from a previous work [3]: The result in [3] is obtained from(15) by omitting the terms with ˆ σ O ,O and ˆ σ N ,N and by the replacement O ( x, x ) → O ( x, x ) + O ( x,
0) and N ( x, x ) → N ( x, x ) − N ( x, { O ( x, x ) , O ( x, , N ( x, x ) , N ( x, } is a consequence of thesymmetry property implied in the decomposition (2) and (3), in particular, the differentcoefficient tensors in front of O ( x, x ) and O ( x,
0) (likewise for N ( x, x ) and N ( x, x = x = x lead to different hard cross sections for the above four functions. See [4] formore details. p ↑ p → DX gg → c ¯ c scattering To obtain the twist-3 cross section based on (13), one has to calculate the derivative of thehard part [ ∂S abcµνλ ( k , k , x ′ p ′ , p c ) p λ /∂k σ (cid:12)(cid:12)(cid:12) k i = x i p ] pole from Fig. 4 contracted with the coefficient9ensors in the decomposition of (2) and (3). This calculation produces lots of terms at theintermediate step and is extremely complicated. Alternatively, application of the “masterformula” developed for the contribution of the quark-gluon correlation functions [16, 17] andalso for the three-gluon correlations for ep ↑ → eDX [6] provides us with a more transparentand simpler method to calculate the cross section. To extend the method to the contributionof the three-gluon correlation functions for p ↑ p → DX , we first note that the diagrams inFig. 4 are obtained by attaching the extra gluon-line to the external-lines of the twist-2hard part in Fig. 2, and the pole contribution is given by the propagator next to thevertex to which this extra-gluon line attaches. Because of this structure, the derivativecan be performed by keeping the structure of the hard part corresponding to those in Fig.2 almost intact. Based on this observation, one can obtain the master formula also forthe three-gluon contribution. To be specific, we consider the case in which the c -quarkfragments into the D -meson below.To present the result, we first define the hard part for the unpolarized cross sectionshown in Fig. 2 as H U,abµν ( xp, x ′ p ′ , p c ), where µν and ab are, respectively, the Lorentz andthe color indices for the gluon line with the momentum xp . Those indices for the gluonline with the momentum x ′ p ′ are already contracted to factorize G ( x ′ ) in (6). With thisconvention the partonic hard cross section ˆ σ gg → c in (6) is related to H U,abµν asˆ σ Ugg → c (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) = 1( N − δ ab (cid:18) − g µν ⊥ (cid:19) H U,abµν ( xp, x ′ p ′ , p c ) . (18)As is shown below the hard part for the twist-3 cross section has a simple relation with this H U,abµν ( xp, x ′ p ′ , p c ).In [6], we have shown that the twist-3 hard cross section for ep ↑ → eDX inducedby the three-gluon correlation functions can be obtained from the Born cross section forthe γ ∗ g → c ¯ c scattering. There the SGP contribution occurs from the FSI. Accordingly,the FSI contribution for p ↑ p → DX shown in Fig. 4(a) can also be expressed in termsof the Born cross section for the gg → c ¯ c scattering. We write the FSI contribution to S abcµνλ ( k , k , x ′ p ′ , p c ) p λ in (13) as S F,abcµνλ ( k , k , x ′ p ′ , p c ) p λ . Then, following the same procedureas [6], one can show that the hard part for the FSI is given by ∂S F,abcµνλ ( k , k , x ′ p ′ , p c ) p λ ∂k σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p pole = (cid:20) x − x + iǫ (cid:21) pole ∂∂p σc − p cσ p λ p · p c ∂∂p λc ! H F,abcµν ( x p, x ′ p ′ , p c )= (cid:20) x − x + iǫ (cid:21) pole ddp σc H F,abcµν ( x p, x ′ p ′ , p c ) , (19)where H F,abcµν ( x p, x ′ p ′ , p c ) is obtained from H U,abµν ( x p, x ′ p ′ , p c ) simply by adding the extracolor matrix t c in the same place where the coherent gluon line is attached in Fig. 4(a).10ere one needs to be cautious in taking derivative with respect to p σc : In the expressionafter the first equality of (19), the on-shell limit p c = m c should be taken after performingthe derivative with respect to p σc . For the derivative in the expression after the secondequality of (19), the form p µc = (cid:18) p + c = m c + ~p c ⊥ p − , p − c , ~p c ⊥ (cid:19) should be used for p c , i.e., on-shellcondition for p c should be used by regarding p + c as a dependent variable of p − c and ~p c ⊥ .One can also derive the similar relation for the ISI contribution. We write the ISIcontribution to S abcµνλ ( k , k , x ′ p ′ , p c ) p λ in (13) as S I,abcµνλ ( k , k , x ′ p ′ , p c ) p λ . For the ISI diagramsin Fig. 4(b), the coherent gluon couples to the initial gluon-line of the diagrams in Fig. 2through the three-gluon coupling. One can still apply the same method as [6], and obtainsfor the ISI contribution as ∂S I,abcµνλ ( k , k , x ′ p ′ , p c ) p λ ∂k σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i = x i p pole = (cid:20) − x − x + iǫ (cid:21) pole ∂∂ ( x ′ p ′ σ ) − p ′ σ p λ p · p ′ ∂∂ ( x ′ p ′ λ ) ! H I,abcµν ( x p, x ′ p ′ , p c ) , = (cid:20) − x − x + iǫ (cid:21) pole dd ( x ′ p ′ σ ) H I,abcµν ( x p, x ′ p ′ , p c ) , (20)where H I,abcµν ( x p, x ′ p ′ , p c ) differs from H U,abµν ( x p, x ′ p ′ , p c ) only with its extra color index c associated with the attachment of the coherent gluon line in Fig. 4(b). As in (19) theon-shell limit p ′ → p ′ µ = (cid:18) p ′ + = ~p ′ ⊥ p ′− , p ′− , ~p ′⊥ (cid:19) should beused in the expression after the second equality of (20). In (20), we first make p ′ σ ⊥ = 0in taking the derivative and then take the p ′ σ ⊥ → p and p ′ are collinear. Inserting (19) and (20) into (13), one obtains thesingle-spin-dependent cross section as P h d ∆ σd P h = α s S Z dx ′ x ′ G ( x ′ ) Z dzz D c ( z ) Z dxx ( − iπ ) ω µα ω νβ ω σγ M αβγF,abc ( x, x ) × " ddp σc H F,abcµν ( xp, x ′ p ′ , p c ) − dd ( x ′ p ′ σ ) H I,abcµν ( xp, x ′ p ′ , p c ) . (21)The hard part H F,abcµν and H I,abcµν contain the factor δ (( xp + x ′ p ′ − p c ) − m c ) = δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) as an on-shell condition for the final unobserved ¯ c -quark. For convenience we separate this δ -function and introduce the following functions by taking the color contraction: N d bca ( N − N − H F,abcαβ ( xp, x ′ p ′ , p c ) ≡ H ( F,d ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) , − if bca N ( N − H F,abcαβ ( xp, x ′ p ′ , p c ) ≡ H ( F,f ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) , d bca ( N − N − H I,abcαβ ( xp, x ′ p ′ , p c ) ≡ H ( I,d ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) , − if bca N ( N − H I,abcαβ ( xp, x ′ p ′ , p c ) ≡ H ( I,f ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) . (22)Using these forms in (21), one can write the cross section as P h d ∆ σd P h = α s S Z dx ′ x ′ G ( x ′ ) Z dzz D c ( z ) Z dxx ( − iπ ) × " O αβγ ⊥ ( x, x ) ( ddp γc H ( F,d ) αβ ( xp, x ′ p ′ , p c ) − dd ( x ′ p ′ γ ) H ( I,d ) αβ ( xp, x ′ p ′ , p c ) ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) + N αβγ ⊥ ( x, x ) ( ddp γc H ( F,f ) αβ ( xp, x ′ p ′ , p c ) − dd ( x ′ p ′ γ ) H ( I,f ) αβ ( xp, x ′ p ′ , p c ) ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) , (23)where O αβγ ⊥ ( x, x ) and N αβγ ⊥ ( x, x ) are the functions obtained by setting x = x = x in (2)and (3): O αβγ ⊥ ( x, x ) = 2 iM N h O ( x, x ) g αβ ⊥ ǫ γpnS ⊥ + O ( x, g βγ ⊥ ǫ αpnS ⊥ + g γα ⊥ ǫ βpnS ⊥ ) i ,N αβγ ⊥ ( x, x ) = 2 iM N h N ( x, x ) g αβ ⊥ ǫ γpnS ⊥ − N ( x, g βγ ⊥ ǫ αpnS ⊥ + g γα ⊥ ǫ βpnS ⊥ ) i . (24)We remind the derivatives d/dp γc and d/d ( x ′ p ′ γ ) in (23) also hit δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) . The relation(23) shows that the partonic hard cross section for O ( x, x ) and O ( x, N ( x, x ) and N ( x, O ( x, x ) and N ( x, x ) We first consider the contribution in (23) occuring from O ( x, x ) and N ( x, x ) in (24). Usingthe functions (22), we write the corresponding hard part as H ( F,j ) αβ ( xp, x ′ p ′ , p c ) g αβ ⊥ ǫ γpnS ⊥ ≡ K ( F,j ) (˜ s, ˜ t, ˜ u, m c ) ǫ γpnS ⊥ ,H ( I,j ) αβ ( xp, x ′ p ′ , p c ) g αβ ⊥ ǫ γpnS ⊥ ≡ K ( I,j ) (˜ s, ˜ t, ˜ u, m c ) ǫ γpnS ⊥ , (25)for j = d, f , where we have used the fact that the scalar functions K ( F,j ) and K ( I,j ) ( j = d, f ) become the functions of ˜ s , ˜ t , ˜ u and m c . For the scalar functions K ( F,j ) , K ( I,j ) and δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) , one can perform the derivative with respect to p γc and x ′ p ′ γ in (23) throughthat with respect to ˜ u as ddp γc K ( F,j ) (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) = − p cγ (cid:18) ˜ s ˜ t (cid:19) ∂∂ ˜ u K ( F,j ) (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) , (26) dd ( x ′ p ′ γ ) K ( I,j ) (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) = − p cγ ∂∂ ˜ u K ( I,j ) (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) , (27)12or j = d, f , where we have used the fact that p + c and p ′ + are the dependent variables (asnoted after (19) and (20)), and have set p ′ γ ⊥ → Using (26),one obtains for the FSI contribution with O ( x, x ) in (23) as Z dxx O ( x, x ) ddp γc H ( F,d ) αβ (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) g αβ ⊥ ǫ γpnS ⊥ = − ǫ p c pnS ⊥ Z dxx O ( x, x ) (cid:18) ˜ s ˜ t (cid:19) ∂∂ ˜ u K ( F,d ) (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) = − ǫ p c pnS ⊥ Z dxx (cid:18) ˜ s ˜ t (cid:19) " ∂K ( F,d ) ∂ ˜ u + x ˜ u ∂K ( F,d ) ∂x − K ( F,d ) ˜ u ! O ( x, x ) + K ( F,d ) ˜ u x dO ( x, x ) dx × δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) = − ǫ p c pnS ⊥ Z dxx (cid:18) ˜ s ˜ t (cid:19) " ∂K ( F,d ) ∂ ˜ u + ˜ t ˜ u ∂K ( F,d ) ∂ ˜ t + ˜ s ˜ u ∂K ( F,d ) ∂ ˜ s − K ( F,d ) ˜ u ! O ( x, x )+ K ( F,d ) ˜ u x dO ( x, x ) dx δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) = − ǫ p c pnS ⊥ Z dxx (cid:18) ˜ s ˜ t ˜ u (cid:19) " − m c ∂K ( F,d ) ∂m c O ( x, x ) + K ( F,d ) x dO ( x, x ) dx − O ( x, x ) ! × δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) . (28)In the first equality of (28), we transformed the derivative hitting δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) into thederivative with respect to x and performed the partial integration. In the last equality wehave used the relation ˜ s ∂∂ ˜ s + ˜ t ∂∂ ˜ t + ˜ u ∂∂ ˜ u + m c ∂∂m c ! K ( B,j ) (˜ s, ˜ t, ˜ u, m c ) = 0 , (29)for B = F, I and j = d, f , resulting from the scale-invariance property for the dimensionlessfunction K ( B,j ) (˜ s, ˜ t, ˜ u, m c ) = K ( B,j ) ( λ ˜ s, λ ˜ t, λ ˜ u, λm c ). Similarly to (28), by using (27), oneobtains for the ISI contribution with O ( x, x ) in (23) as − Z dxx O ( x, x ) dd ( x ′ p ′ γ ) H ( I,d ) αβ (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) g αβ ⊥ ǫ γpnS ⊥ = 2 ǫ p c pnS ⊥ Z dxx O ( x, x ) ∂∂ ˜ u K ( I,d ) (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) For the FSI, one can set p ′ γ ⊥ = 0 from the beginning.
13 2 ǫ p c pnS ⊥ Z dxx u " − m c ∂K ( I,d ) ∂m c O ( x, x ) + K ( I,d ) x dO ( x, x ) dx − O ( x, x ) ! δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) . (30)By the replacement K ( B,d ) (˜ s, ˜ t, ˜ u, m c ) → K ( B,f ) (˜ s, ˜ t, ˜ u, m c ) ( B = F, I ) and O ( x, x ) → N ( x, x ) in (28) and (30), one obtains the formula for the N ( x, x ) contributions. From (28)and (30), one can make the following important observations for the O ( x, x ) and N ( x, x )contributions:(1) The partonic hard cross sections for x ddx O ( x, x ) and O ( x, x ) are connected by thesimple relation. In particular, in the m c → x ddx O ( x, x ) − O ( x, x ). The same relation holds for x ddx N ( x, x ) and N ( x, x ).(2) The partonic hard cross sections K ( B,j ) ( B = F, I , j = d, f ) defined in (25) areobtained from H F,abcµν and H I,abcµν by the same Lorentz contraction as the unpolarizedcross section in (18). Therefore the contribution to K ( B,j ) from each diagram differsfrom those for ˆ σ Ugg → c only in the color factors.These features are the extension of those obtained in [17] for the SGP contribution ofthe quark-gluon correlation function with massless partons to the case of the three-gluoncorrelation functions with massive partons in the final state. O ( x, and N ( x, Next we consider the contribution from O ( x,
0) and N ( x,
0) in (23), which arise from thesecond terms in (24). For this purpose, we introduce the two fixed vectors X µ = (0 , , , Y µ = (0 , , , g βγ ⊥ = − X β X γ − Y β Y γ . (31)Then the derivative d/dp γc hitting the FSI hard part in (23) for O ( x,
0) and N ( x,
0) can bewritten as ddp γc H ( F,j ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) (cid:16) g βγ ⊥ ǫ αpnS ⊥ + g αγ ⊥ ǫ βpnS ⊥ (cid:17) = − X µ ddp µc H ( F,j ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) (cid:16) X β ǫ αpnS ⊥ + X α ǫ βpnS ⊥ (cid:17) − Y µ ddp µc H ( F,j ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) (cid:16) Y β ǫ αpnS ⊥ + Y α ǫ βpnS ⊥ (cid:17) . (32)To perform the derivatives in this equation, we introduce the scalar functions J ( F,j )1 , (˜ s, ˜ t, ˜ u )( j = d, f ) by the decomposition: H ( F,j ) αβ ( xp, x ′ p ′ , p c ) (cid:16) X β ǫ αpnS ⊥ + X α ǫ βpnS ⊥ (cid:17) ≡ J ( F,j )1 (˜ s, ˜ t, ˜ u, m c ) ( p c · X ) ǫ p c pnS ⊥ + J ( F,j )2 (˜ s, ˜ t, ˜ u, m c ) ǫ XpnS ⊥ , (33)14here we used the kinematic condition p γ ⊥ = p ′ γ ⊥ = 0. One also obtains the similar decom-position for H ( F,j ) αβ ( xp, x ′ p ′ , p c ) (cid:16) Y β ǫ αpnS ⊥ + Y α ǫ βpnS ⊥ (cid:17) by the replacement X → Y in (33)with the same functions J ( F,j )1 , . Then the derivative in (32) can be performed with the helpof (26) as − X µ ddp µc h J ( F,j )1 (˜ s, ˜ t, ˜ u, m c )( p c · X ) ǫ p c pnS ⊥ δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17)i + ( X → Y )= 3 ǫ p c pnS ⊥ J ( F,j )1 (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) +2 ~p c ⊥ ǫ p c pnS ⊥ (cid:18) ˜ s ˜ t (cid:19) ∂∂ ˜ u (cid:16) J ( F,j )1 (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17)(cid:17) , (34) − X µ ddp µc h J ( F,j )2 (˜ s, ˜ t, ˜ u, m c ) ǫ XpnS ⊥ δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17)i + ( X → Y )= − ǫ p c pnS ⊥ ˜ s ˜ t ∂∂ ˜ u (cid:16) J ( F,j )2 (˜ s, ˜ t, ˜ u, m c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17)(cid:17) . (35)Using these results and the relation ~p c ⊥ = ˜ t ˜ u ˜ s − m c in (32), and following the same procedureleading to (28), one obtains the FSI contribution with O ( x,
0) and N ( x,
0) in (23) in termsof J ( F,j )1 , in (33) as Z dxx ddp γc H ( F,d ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) (cid:16) g βγ ⊥ ǫ αpnS ⊥ + g αγ ⊥ ǫ βpnS ⊥ (cid:17) O ( x, (cid:16) H ( F,d ) αβ → H ( F,f ) αβ , O ( x, → − N ( x, (cid:17) = ǫ p c pnS ⊥ Z dxx " J ( F,d )1 O ( x,
0) + 2˜ s ˜ t ˜ u ˜ t ˜ u ˜ s − m c ! × − J ( F,d )1 − m c ∂J ( F,d )1 ∂m c O ( x,
0) + J ( F,d )1 x dO ( x, dx − O ( x, ! − s ˜ t ˜ u − m c ∂J ( F,d )2 ∂m c O ( x,
0) + J ( F,d )2 x dO ( x, dx − O ( x, ! δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) + (cid:16) J ( F,d )1 , → J ( F,f )1 , , O ( x, → − N ( x, (cid:17) . (36)The ISI contribution in (23) with O ( x,
0) and N ( x,
0) can also be obtained followingthe same procedure as above. Similarly to (33), one can decompose the ISI hard part as H ( I,j ) αβ ( xp, x ′ p ′ , p c ) (cid:16) X β ǫ αpnS ⊥ + X α ǫ βpnS ⊥ (cid:17) J ( I,j )1 (˜ s, ˜ t, ˜ u, m c ) ( p c · X ) ǫ p c pnS ⊥ + J ( I,j )2 (˜ s, ˜ t, ˜ u, m c ) ǫ XpnS ⊥ + J ( I,j )3 (˜ s, ˜ t, ˜ u, m c ) ( x ′ p ′ · X ) ǫ p c pnS ⊥ + J ( I,j )4 (˜ s, ˜ t, ˜ u, m c ) ( p c · X ) x ′ ǫ p ′ pnS ⊥ (37)for j = d, f , where we ignored the terms which vanish in the limit p ′ γ ⊥ → d/d ( x ′ p ′ γ ). Similar relation can be written down for H ( I,j ) αβ ( xp, x ′ p ′ , p c ) (cid:16) Y β ǫ αpnS ⊥ + Y α ǫ βpnS ⊥ (cid:17) , using the same functions J ( I,j )1 , , , . Compared with(33), note the existence of the J ( I,j )3 , terms in (37), since one has to take the derivative withrespect to p ′ γ ⊥ before taking the p ′ γ ⊥ → J ( I,j )1 , , , , one eventually obtainsthe ISI contribution in (23) with O ( x,
0) and N ( x,
0) as − Z dxx dd ( x ′ p ′ γ ) H ( I,d ) αβ ( xp, x ′ p ′ , p c ) δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) (cid:16) g βγ ⊥ ǫ αpnS ⊥ + g αγ ⊥ ǫ βpnS ⊥ (cid:17) O ( x, (cid:16) H ( I,d ) αβ → H ( I,f ) αβ , O ( x, → − N ( x, (cid:17) = ǫ p c pnS ⊥ Z dxx u ˜ t ˜ u ˜ s − m c ! J ( I,d )1 + m c ∂J ( I,d )1 ∂m c O ( x, − J ( I,d )1 x dO ( x, dx − O ( x, !) + 2˜ u − m c ∂J ( I,d )2 ∂m c O ( x,
0) + J ( I,d )2 x dO ( x, dx − O ( x, ! − (2 J ( I,d )3 + J ( I,d )4 ) O ( x, i δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) + (cid:16) J ( I,d )1 , , , → J ( I,f )1 , , , , O ( x, → − N ( x, (cid:17) . (38) gg → c ¯ c scattering Using the results (28), (30), (36) and (38) in (23), one obtains the final result for thetotal twist-3 single-spin-dependent cross section for p ↑ p → DX induced by the three-gluoncorrelation functions as P h d ∆ σd P h = 2 πM N α s S ǫ P h pnS ⊥ Z dx ′ x ′ G ( x ′ ) Z dzz D c ( z ) Z dxx × "(cid:18) s ˜ t ˜ u (cid:19) ( m c ∂K ( F,d ) ∂m c O ( x, x ) − K ( F,d ) x dO ( x, x ) dx − O ( x, x ) !) + 2˜ u ( − m c ∂K ( I,d ) ∂m c O ( x, x ) + K ( I,d ) x dO ( x, x ) dx − O ( x, x ) !) (cid:16) O ( x, x ) → N ( x, x ) , K ( F,d ) → K ( F,f ) , K ( I,d ) → K ( I,f ) (cid:17) +3 J ( F,d )1 O ( x,
0) + 2˜ s ˜ t ˜ u ˜ t ˜ u ˜ s − m c ! × − J ( F,d )1 − m c ∂J ( F,d )1 ∂m c O ( x,
0) + J ( F,d )1 x dO ( x, dx − O ( x, ! − s ˜ t ˜ u − m c ∂J ( F,d )2 ∂m c O ( x,
0) + J ( F,d )2 x dO ( x, dx − O ( x, ! + (cid:16) O ( x, → − N ( x, , J ( F,d )1 , → J ( F,f )1 , (cid:17) + 2˜ u ˜ t ˜ u ˜ s − m c ! J ( I,d )1 + m c ∂J ( I,d )1 ∂m c O ( x, − J ( I,d )1 x dO ( x, dx − O ( x, ! + 2˜ u − m c ∂J ( I,d )2 ∂m c O ( x,
0) + J ( I,d )2 x dO ( x, dx − O ( x, ! − (2 J ( I,d )3 + J ( I,d )4 ) O ( x, (cid:16) O ( x, → − N ( x, , J ( I,d )1 , , , → J ( I,f )1 , , , (cid:17)i δ (cid:16) ˜ s + ˜ t + ˜ u (cid:17) . (39)where K ( B,j ) (˜ s, ˜ t, ˜ u, m c ) ( B = F, I , j = d, f ), J ( F,j )1 , (˜ s, ˜ t, ˜ u, m c ) ( j = d, f ) and J ( I,j )1 , , , (˜ s, ˜ t, ˜ u, m c )( j = d, f ) are the functions defined, respectively, in (25), (33) and (37) and can be calcu-lated from the twist-2 diagrams in Fig. 2. By the direct calculation of these functions, wefound that J ( F,j )1 and J ( I,j )1 , , ( j = d, f ) are of O ( m c ) and thus vanish in the m c → K ( B,j ) = J ( B,j )2 ( B = F, I , j = d, f ) at m c = 0, which is consistent with the result in(15). For the twist-3 cross section for the ¯ D -meson production, the sign of the contributionfrom O ( x, x ) and O ( x,
0) should be reversed in (39). The result calculated from (39), ofcourse, agrees with (15) which were obtained by the direct calculation of Fig. 4.The origin of the above master formula (39) is the relations in (19) and (20). Althoughthey were derived in the LO QCD, the derivation was based on the quite general structureof the diagrams for the SGP contribution at x = x shown in Fig. 3, i.e., the coherent-gluon line is attached to the external parton line of the twist-2 diagrams in Fig. 1. As longas this structure is kept after including higher-order corrections, the relations (19) and (20)hold. Therefore we expect that the formula (39) will become a powerful tool to includehigh-order corrections to the twist-3 SSA [6].17 Numerical calculation of the asymmetry
As is shown in (15), four nonperturbative functions O ( x, x ), N ( x, x ), O ( x,
0) and N ( x, A DN . Unlike twist-2 parton distributions, twist-3multiparton correlation functions do not have probability interpretation and thus cannot beconstrained by a certain positivity bound. They have to be determined by comparing thecalculated SSAs with experimental data, or by some nonperturbative techniques in QCD. Atpresent there is no information on these functions. Preliminary data on A DN by the PHENIXcollaboration [26] suggests | A DN | ≤ | x F | < . √ S = 200 GeV. Sincethe unpolarized cross section for the pion production at RHIC has been well describedby the next-to-leading order (NLO) calculation in the collinear factorization [30, 31, 32],comparison of the asymmetry calculated by our twist-3 cross section with the RHIC datawill be the first step for determining the magnitude of the three-gluon correlation functionsin the nucleon. Here we present a simple model calculation of the asymmetry at theRHIC energy taking into account of the preliminary RHIC data. P d σ || d P h ( nb ⋅ G e V - ⋅ c ) x F Figure 5: Unpolarized cross section for pp → DX by the gluon fusion process at the RHICenergy √ S = 200 GeV and P T = 2 GeV.To see the relative importance of each term appearing in (15), we assume the same formfor the four nonperturbative functions as O ( x, x ) = O ( x,
0) = N ( x, x ) = − N ( x, . (40)As a functional form of these functions, we employ the following ansatz:Model 1 : O ( x, x ) = K G x G ( x ) , (41) For a comprehensive review on the positivity bounds of parton distributions, see [29]. The size of the NLO correction to the twist-3 cross section in (15) could be different from that for thetwist-2 unpolarized cross section, which may lead to significantly different A DN . One thus should take thepresent calculation as only an estimate of the order-of-magnitude. Minus sign is introduced for N ( x, σ N has an opposite sign compared with ˆ σ O ,O ,N . O ( x, x ) = K ′ G √ x G ( x ) , (42)where G ( x ) is the twist-2 unpolarized gluon density, and K G and K ′ G are the constants whichwe determine so that the calculated asymmetry is consistent with the RHIC data. Since the -0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F ˆ σ O1 O(x,x)ˆ σ O2 O(x,0)ˆ σ N1 N(x,x)ˆ σ N2 N(x,0) -0.0006-0.0004-0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F ˆ σ O3 ˆ σ O4 ˆ σ N3 ˆ σ N4 (a) -0.1-0.05 0 0.05 0.1 0.15-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F ˆ σ O1 O(x,x)ˆ σ O2 O(x,0)ˆ σ N1 N(x,x)ˆ σ N2 N(x,0) -0.002-0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F ˆ σ O3 ˆ σ O4 ˆ σ N3 ˆ σ N4 (b)Figure 6: (a) Contribution to A DN from the 8 components proportional to σ O ,O ,N ,N (left)and σ O ,O ,N ,N (right) in (15) obtained by using the model 1 in (41) with K G = 0 . σ O ,O ,N ,N are also plotted bythin lines labeled by O ( x, x ), O ( x, N ( x, x ) and N ( x, K ′ G = 0 . A DN
19o the small- x behavior of the three-gluon correlation functions. For the numerical calcula-tion, we use GJR08 distribution [33] for G ( x ) and KKKS08 fragmentation function [34] for D f ( z ). We also assume the same scale dependence for O ( x, x ) etc as G ( x ) for simplicity.We calculate A N for the D and ¯ D mesons at the RHIC energy of √ S = 200 GeV and thetransverse momentum of the D -meson P T = 2 GeV with the parameter m c = 1 . µ = q P T + m c .For completeness, we first show in Fig. 5 the unpolarized cross section for pp → DX based on the gluon fusion process (6) at √ S = 200 GeV and P T = 2 GeV. This willconstitute the denominator of A DN in our calculation below. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F (a) -0.4-0.3-0.2-0.1 0 0.1 0.2-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F -0.4-0.3-0.2-0.1 0 0.1 0.2-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N x F (b)Figure 7: (a) A DN for the D (left) and ¯ D (right) mesons for Model 1 in (41) with K G =0 . A DN for the D (left) and ¯ D (right) mesons for Model 2 in (42) with K ′ G = 0 . A DN from each term in the twist-3 crosssection shown in (15) with K G = 0 .
002 for the model 1 (Fig. 6(a)) and K ′ G = 0 . σ O ,O ,N ,N is negligible compared to the contributions fromˆ σ O ,O ,N ,N . In the left figures of Figs. 6 (a) and (b), we have also plotted the contributionfrom the nonderivative terms proportional to ˆ σ O ,O ,N ,N , which shows that the derivativecontributions dominate in these terms at large x F ( > x F < σ O and ˆ σ O give rise to numerically very close asymmetries at theRHIC energy for the two models, and likewise for ˆ σ N and ˆ σ N . The asymmetries causedby ˆ σ O ,O and ˆ σ N ,N are also similar.Fig. 7 shows the result for A DN for the D and ¯ D mesons including all the contributionsin (15) together with the preliminary data by the PHENIX collaboration [26]. Because thesign of the contribution from { O ( x, x ) , O ( x, } changes between D and ¯ D as shown in(15), { O ( x, x ) , O ( x, } and { N ( x, x ) , N ( x, } contribute to the asymmetry constructively(destructively) for the D ( ¯ D ) meson, leading to a large (small) A DN for D ( ¯ D ). If one reversesthe relative sign between O and N from (40), the result for the D and ¯ D mesons will beinterchanged. The values K G = 0 .
002 and K ′ G = 0 . A N for the D -meson (left figures inFigs. 7(a) and (b)) under the assumption (40). By comparing the results for the models 1and 2 in Fig. 7, one sees that the behavior of the asymmetry at x F < x behavior of the three gluon correlation functions. Therefore A DN at x F < x behavior of the three-gluon correlation functions.As we saw in the left figures of Fig. 6, we found the relations ˆ σ O ≃ ˆ σ O and ˆ σ N ≃ ˆ σ N .This means that the combinations O ( x, x ) + O ( x,
0) and N ( x, x ) − N ( x,
0) can be taken asgood effective three-gluon correlation functions determining A DN ’s at RHIC energies. Fromthe left figures in Figs. 7 (a) and (b), if the ansatz (41) or (42) is a reasonable assumptionfor the x -dependence of the three-gluon correlation functions, K G = 0 .
002 or K ′ G = 0 . | O ( x, x ) + O ( x, | ≤ . x G ( x ) , | N ( x, x ) − N ( x, | ≤ . x G ( x ) , (43)and | O ( x, x ) + O ( x, | ≤ . √ x G ( x ) , | N ( x, x ) − N ( x, | ≤ . √ x G ( x ) , (44)although extraction of the separate constraint on the four functions is not possible. Weremark that even though the RHIC data suggests small A DN at | x F | < .
1, it can be muchlarger at | x F | > . x regions as shown in Fig. 7. In [5], we extracted a stronger constraint for the upper bound of | O ( x, x ) + O ( x, | and | N ( x, x ) − N ( x, | , since we assumed | A DN | < x F , while the data showing | A DN | < | x F | < . Summary
In this paper we have studied the SSA for the open-charm production in the pp collision, p ↑ p → DX , based on the twist-3 mechanism in the collinear factorization. Since the three-gluon correlation functions in the transversely polarized nucleon play a dominant role ingiving rise to SSA for this process, we have derived the corresponding twist-3 single-spin-dependent cross section in the leading order QCD. As in the case of our previous studyon ep ↑ → eDX , our result differs from the existing result in the literature. We have alsoderived the master formula which shows that the corresponding twist-3 cross section can beobtained from the hard part for the gg → c ¯ c scattering in the twist-2 level. The use of thisformula simplifies the actual calculation and is useful to make the structure of the twist-3cross section transparent. We expect that this master formula is useful for the inclusionof the higher-order corrections to the cross section. We also presented a model calculationof the asymmetry A DN in comparison to the preliminary data obtained at RHIC. We haveshown that A DN at x F < x behavior of the three-gluon correlationfunctions, and have given a modest upper limit on those functions. Acknowledgments
We thank D. Boer, Z.-B. Kang, K. Tanaka, M. Liu, J.-W. Qiu and F. Yuan for usefuldiscussions, and the authors of Ref. [34] for providing us with the Fortran code of their D -meson fragmentation function. The work of S. Y. is supported by the Grand-in-Aid forScientific Research (No. 22.6032) from the Japan Society of Promotion of Science. A Ward identity for the initial state interaction
To derive the cross section for p ↑ p → DX , one has to analyze Z d k (2 π ) Z d k (2 π ) S abcµνλ ( k , k , x ′ p ′ , p c ) M µνλabc ( k , k ) , (45)where M µνλabc ( k , k ) is defined in (11) and S abcµνλ ( k , k , x ′ p ′ , p c ) is the corresponding hard partas shown in Fig. 3. As was shown in [4], in order to be able to obtain the cross sectionfrom (13), it is essential that S abcµνλ ( k , k , x ′ p ′ , p c ) satisfies the Ward identities k µ S abcµνλ ( k , k ) = 0 , k ν S abcµνλ ( k , k ) = 0 , ( k − k ) λ S abcµνλ ( k , k ) = 0 . (46)It is easy to see that the hard part for the sum of the FSI diagrams shown in Fig. 4(a)satisfies (46) due to the on-shell condition for the bared quark-lines. However, the hardpart for the ISI diagrams in Fig. 4(b) does not satisfy (46). This is because the polarizationtensor for the gluon line producing the SGP (bared gluon line in Fig. 4(b)) is taken to be − g στ in the Feynman gauge, which contains the contribution from unphysical polarizations.Nevertheless, one can calculate the hard cross section for the ISI from (13). To show this,22e define the new hard part for the ISI, e S I,abcµνλ ( k , k , x ′ p ′ , p c ), which is obtained from theoriginal hard part for ISI, S I,abcµνλ ( k , k , x ′ p ′ , p c ), by replacing the polarization tensor for thebared gluon-propagator in Fig. 4(b) as − g στ → − g στ + q σ p τ + q τ p σ q · p , (47)where q is the momentum carried by the bared gluon-line. One can show that in the twist-3accuracy this e S I,abcµνλ ( k , k , x ′ p ′ , p c ) satisfies the relation Z d k (2 π ) Z d k (2 π ) S I,abcµνλ ( k , k , x ′ p ′ , p c ) M µνλabc ( k , k )= Z d k (2 π ) Z d k (2 π ) e S I,abcµνλ ( k , k , x ′ p ′ , p c ) M µνλabc ( k , k ) , (48)where M µνλabc ( k , k ) is defined in (11). The extra terms in e S I,abcµνλ ( k , k , x ′ p ′ , p c ) compared to S I,abcµνλ ( k , k , x ′ p ′ , p c ) which occur due to the replacement (47) can be shown to vanish withthe help of the Feynman gauge condition for the matrix element: k µ M µνλabc ( k , k ) = 0 , k ν M µνλabc ( k , k ) = 0 , ( k − k ) λ M µνλabc ( k , k ) = 0 . (49)From the above relation, one can use e S I,abcµνλ ( k , k , x ′ p ′ , p c ) for the calculation of the twist-3cross section. Since e S I,abcµνλ ( k , k ) satisfies k µ e S I,abcµνλ ( k , k ) = 0 , k ν e S I,abcµνλ ( k , k ) = 0 , ( k − k ) λ e S I,abcµνλ ( k , k ) = 0 , (50)the resulting twist-3 cross section from the ISI takes the form of (13) with S abcµνλ ( k , k , x ′ p ′ , p c )replaced by e S I,abcµνλ ( k , k , x ′ p ′ , p c ). Finally, one can show the relation ∂ e S I,abcµνλ ( k , k , x ′ p ′ , p c ) p λ ∂k σ (cid:12)(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 )= ∂S I,abcµνλ ( k , k , x ′ p ′ , p c ) p λ ∂k σ (cid:12)(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 ) , (51)where the right-hand-side is calculated with the original hard part for the ISI. This wayone can obtain the contribution of the three-gluon correlation functions to the twist-3 crosssection from (13). References [1] X. Ji, Phys. Lett.
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