Transverse single spin asymmetry in direct photon production in polarized pA collisions
aa r X i v : . [ h e p - ph ] F e b Transverse single spin asymmetry in direct photon production inpolarized pA collisions
Andreas Sch¨afer and Jian Zhou
Institut f¨ur Theoretische Physik,Universit¨at Regensburg, Regensburg, Germany
September 16, 2018
Abstract
We study the transverse single spin asymmetry in direct photon production in pA collisions withincoming protons being transversely polarized. To facilitate the calculation, we formulate a hybridapproach in which the nucleus is treated in the Color Glass Condensate (CGC) framework while thecollinear twist-3 formalism is applied on the proton side. It has been found that an additional termwhich arises from color entanglement shows up in the spin dependent differential cross section. Thefact that this additional term is perturbatively calculable allows us to quantitatively study colorentanglement effects.
The phenomenology of transverse single spin asymmetries (SSAs) in high energy scattering has at-tracted a lot of attention and has been under intense investigation during the past few decades. Thelarge size of the observed SSAs for single inclusive hadron production came as a big surprise and cannot be understood in the naive parton model [1, 2]. It signals that QCD phenomena are in generalmuch richer than in the well studied collinear limit. This opens up many possibilities. For example,SSAs could be especially sensitive to saturation phenomena. However, QCD beyond the collinear limitis usually more difficult and less under theoretical control, such that presently one is still searchingfor the most adequate theoretical frameworks to treat them. This contribution constitutes one moreattempt along these lines.Various recent studies show that it is necessary to take into account initial/final state gluon re-scattering interactions in order to generate large SSAs. Two ways of incorporating these initial/finalstate interactions are based on the transverse momentum dependent(TMD) factorization [3, 4] andthe collinear twist-3 factorization [5–10], respectively. In TMD factorization, the naive time reversalodd TMD distributions and fragmentation function, known as the quark/gluon Sivers functions [3]and the Collins fragmentation function [4] can account for the large SSAs, while in the collinear twist-3 approach, SSAs can arise from a twist-3 quark gluon correlator, the so-called Efremov-Teryaev-Qiu-Sterman(ETQS) function [5, 6], a tri-gluon correlation functions [8, 9], and a twist-3 collinearfragmentation functions [10]. It has been established that the k ⊥ moment of the Sivers function andthe Collins function can be related to the ETQS function and the corresponding twist-3 collinearfragmentation functions, respectively [10, 11].SSAs observed in various processes like pion production in single polarized pp collisions p ↑ p → πX or in SIDIS ep ↑ → πX receive contributions from both sources: the Sivers mechanism and the Collinsmechanism. A very recent study has shown that the Collins effect described within the collinear twist-3 framework might be the dominant contribution to the spin asymmetry in polarized pp collisions [12].1evertheless, it would be crucial to unambiguously pin down the Sivers mechanism in polarized ppcollisions, as the so-called ”sign mismatch” problem [13] (for a different version of this problem, seealso [14]) is still unsolved. It is thus desirable to investigate SSAs for cases of particle production inpolarized pp collisions for which the Collins effect is absent. Possible options are the SSA in directphoton production [7, 15, 18], jet production [16–18], or heavy quarkonium production in polarized ppcollisions [19, 20]. The SSA for direct photon production in polarized pp collisions has been calculatedin the collinear twist-3 approach in [7, 15, 18]. In the present work, we extend this analysis to singlepolarized p ↑ A collisions.Though most work in this field focuses on SSAs in ep ↑ or pp ↑ collisions, there exist a few exploratoryinvestigations devoted to the study of SSAs in p ↑ A collisions. The authors of Ref. [21] investigatedthe SSA for inclusive pion production at forward rapidities in p ↑ p collisions using a hybrid approachin which the target proton is treated in the CGC framework [22] while the spin-transverse momentumcorrelation in the projectile proton is described by the Sivers distribution. Their analysis can bestraightforwardly applied to p ↑ A collisions. Following the same line of reasoning, the SSA of Drell-Yan lepton pairs produced in p ↑ A collisions was computed in [23]. On the other hand, the SSA forinclusive pion production caused by the Collins mechanism after the transversely polarized quark fromthe projectiles is scattered off the background gluon field of the nucleus was investigated in Ref. [24].Furthermore, a recent GCG calculation suggests that SSAs also can be generated by the interactionof the spin-dependent light-cone wave function of the projectile with the target gluon field via C-oddodderon exchange [25].The purpose of the present work is to study SSA for direct photon production in p ↑ A collisions andto decide whether it provides a sensitive tool to establish and study saturation effects. First one cannote that the contribution from fragmentation to the spin asymmetry for direct photon production isfound to be negligible [17]. Next one observes that TMD factorization can not be applied on protonside for lack of an additional hard scale. Moreover, it has been shown that the odderon exchange doesnot give rise to a SSA for direct photon production [25]. Therefore, the only possible source for asizeable SSA is the Sivers effect which can only be described within the collinear twist-3 approach forthe process under consideration. To do so, we formulate a novel hybrid approach in which the nucleusis treated in the Color Glass Condensate (CGC) framework while the collinear twist-3 formalism isapplied on the proton side. In this hybrid approach, we take into account one extra gluon exchangefrom the proton side and sum gluon re-scattering to all orders on the nucleus side.The resulting spin dependent differential cross section computed in this hybrid approach is pro-portional to a convolution of the ETQS function and various Wilson lines. These Wilson lines can befurther related to two different types of k ⊥ dependent gluon distributions, one of which is the dipoletype gluon TMD. The other arises from a color entanglement effect which is due to the non-trivialinterplay of gluons from both the nucleon and nucleus [26]. This effect seems to be a unique feature ofnon-abelian theories, i.e. it is linked to one of the most fundamental aspects of QCD. The fact that thisadditional term can be perturbatively calculated in the Mclerran-Venugopalan (MV) model [22] allowsus to quantitatively study the color entanglement effect and to test the MV model. A measurement ofthis observable would also provide a hint to the size of generalized TMD factorization breaking effect.Let us note that it was argued that such a color entanglement effect could also manifest itself throughazimuthal asymmetries in the Drell-Yan process [27].In a more general context, the present work is part of the effort to address the interplay betweenspin physics and saturation physics. Apart from the studies mentioned above, early work in this veryactive field includes the study of small x evolution of spin dependent structure function g [28] andof the quark Boer-Mulders distribution and the linearly polarized gluon distribution inside a largenucleus, see Refs. [29]. The small x evolution equations for the linearly polarized gluon distributionswere derived in Ref. [30]. Several ways of accessing the linearly polarized gluon distributions inside2 large nucleus have been discussed in [29–32]. Furthermore, the asymptotic behavior of transversesingle spin asymmetries at small x was discussed in Ref. [33, 34]. It has been shown that SSAs atsmall x are generated by polarized odderon exchange whose size is determined by the anomalousmagnetic moment [34]. The quark Sivers function was computed in the quasi-classical Glauber-Mueller/MV approximation [35]. More recently, the authors of the paper [36] have investigated thespin asymmetries in pA collisions by going beyond the Eikonal approximation.The paper is organized as follows. In section II, we briefly review the existing calculations for directphoton production, including the collinear twist-3 calculation for direct photon production in polarizedp ↑ p collisions and the CGC calculation for direct photon production in unpolarized pA collisions. Insection III, we develop the hybrid approach and explain all technical steps in details. We focus on thederivative term contribution and identify a term arising from the color entanglement effect. It is shownthat the spin dependent differential cross section derived in collinear factorization can be recoveredfrom our result without color entanglement effect being incorporated in the kinematical limit wherethe produced photon transverse momentum is much larger than the saturation scale. The paper issummarized in section IV. In this section, we review how the calculation of the SSA for direct photon production is formulatedwithin the collinear twist-3 approach in p ↑ p collisions, following by a brief reminder of the applicationof the CGC framework to direct photon production in unpolarized pA collisions. ↑ p collisions The dominant production mechanism for prompt photons in high energy collisions is Compton scat-tering gq → γq . We start by introducing the relevant kinematical variables and assign 4-momenta tothe particles according to g ( x ′ g ¯ P ) + q ( xP ) −→ γ ( l γ ) + q ( l q ) (1)where ¯ P µ = ¯ P − n µ and P µ = P + p µ with n µ and p µ being the commonly defined light cone vectors,normalized according to p · n = 1. The Mandelstam variables are defined as: S = ( P + ¯ P ) , T = ( P − l q ) and U = ( P − l γ ) . The corresponding unpolarized Born cross section reads, d σd l γ ⊥ dz = α s α em N c z [1 + (1 − z ) ] l γ ⊥ X q e q Z x min dx f q ( x ) x ′ g G ( x ′ g ) (2)where z ≡ l γ · n/ ( xP · n ) is the fraction of the incoming quark momentum xP carried by the outgoingphoton, and l γ ⊥ is the photon transverse momentum. The meaning of the other coefficients should beself-evident. Note that x ′ g = − xTxS + U is a function of x ; and x min is given by x min = − US + T . In the aboveformula, f q ( x ) and G ( x ′ g ) are the usual integrated quark and gluon distributions, respectively.To generate the spin asymmetry, one additional gluon must be exchanged between the activepartons and the remanent part of the polarized proton projectile. The hard part, if an additional gluonis attached, can be calculated perturbatively, while the non-perturbative part describes the relevantthree parton correlations. The strong interaction phase factor necessary for having a non-vanishingspin asymmetry arises from the interference between an imaginary part of the partonic scatteringamplitude with an extra gluon and the real scattering amplitude without a gluon attachment, asshown in Fig.1. The imaginary part is due to the pole of the parton propagator associated with theintegration over the gluon momentum fraction x g . This effectively implies that one of the internal3 gP + p?xgP + p? (x + xg)P + p?x0g (cid:22)P x0g (cid:22)PxP (x + xg)P + p?x0g (cid:22)P x0g (cid:22)PxPl(cid:13) l(cid:13) (x + xg)P + p?(x + xg)P + p?xgP + p?x0g (cid:22)P x0g (cid:22)PxP l(cid:13) xgP + p?x0g (cid:22)P x0g (cid:22)PxP l(cid:13)(a)( ) (b)(d) Figure 1: Diagrams contributing to the single spin asymmetry for direct photon production in p ↑ pcollisions. Grey circles indicate all possible photon line attachments. The mirror diagrams are notshown here. The contributions from diagrams (c) and (d) to the spin asymmetry cancel.parton lines goes on shell. To isolate the imaginary part of such poles, the identity of distributions: x g ± iǫ = PV x g ∓ iπδ ( x g ) was used. Depending on which propagator’s pole contributes, the amplitudemay get contributions from x g = 0 (“soft-pole”) and x g = 0 (“hard-pole” ).It is convenient to carry out the calculation in the covariant gauge, in which the leading contributionof the exchanged gluon is the ”plus” component A + . The gluon’s momentum is given by p g = x g P + p ⊥ ,where x g is the longitudinal momentum fraction with respect to the polarized proton. In order tocalculate consistently with twist-3 accuracy, one has to expand the hard part in the gluon transversemomentum, H ( x g P + p ⊥ , l γ ) = H ( x g P, l γ ) + ∂H ( x g P + p ⊥ , l γ ) ∂p ρ ⊥ | p ⊥ =0 p ρ ⊥ + ... (3)In the above formula, the first term only contributes to the unpolarized Born cross section. We thushave to keep the linear term in p ⊥ at twist-3 level. In the second term, the p ⊥ factor can be combinedwith A + to yield ∂ ⊥ A + , which is an element of the field strength tensor F ∂ + . The above expansionallows us to integrate over three of the four components of each of the loop momenta p g . The four-dimensional integral is reduced to a convolution in the light-cone momentum fractions of the initialpartons. At this step, the relevant three parton correlation can be cast into the form of the ETQSfunction defined as [5, 6], T F,q ( x , x ) = Z dy − dy − π e ix P + y − + i ( x − x ) P + y − ×h P, S ⊥ | ¯ ψ q (0) γ + gǫ S ⊥ σnp F + σ ( y − ) ψ q ( y − ) | P, S ⊥ i (4)where we have suppressed Wilson lines. S ⊥ denotes the proton transverse spin vector. Note that ourdefinition of the ETQS functions differs by a factor g from the convention used in Ref. [15]. ThisETQS function plays an important role in describing SSA phenomenology.4 (cid:13) xPxP l(cid:13)(b)(a) ( )x0g (cid:22)P + k?x0g (cid:22)P + k? x0g1 (cid:22)P + k1? (x0g (cid:0) x0g1) (cid:22)P + k? (cid:0) k1?xP l(cid:13) Figure 2: Diagrams contributing to the unpolarized cross section for direct photon production in pAcollisions. The gluon line terminated by a cross surrounded with a circle denotes a classical field A A insertion. The contribution from diagram(c) vanishes because two poles are lying on the same halfplane.Making use of the ingredients described above, the calculation is straightforward. The spin depen-dent cross section has been calculated and given in [7, 15], d ∆ σd l γ ⊥ dz = α s α em N c N c − z [1 + (1 − z ) ] l γ ⊥ ( z − ǫ l γ S ⊥ np l γ ⊥ ! × X q e q Z x min dx x ′ g G ( x ′ g ) (cid:20) T F,q ( x, x ) − x (cid:18) ddx T F,q ( x, x ) (cid:19)(cid:21) (5)where we have omitted the soft fermion pole contribution [18]. In the next section, we will show thatthe above spin dependent cross section can be recovered from the proposed hybrid approach in thekinematical limit where the saturation scale is much smaller than the produced photon transversemomentum after neglecting terms arising from color entanglement effect. We now move on to review the CGC calculation for direct photon production in unpolarized pAcollisions which has been done in Ref. [37]. Roughly speaking, the CGC calculation for this processdiffers from the collinear factorization calculation in two ways. Firstly, in the small x region, transversemomenta carried by gluons are not necessarily much smaller than their longitudinal momenta. Onethus should keep gluon transverse momenta when computing the hard part. We fix kinematicalvariables accordingly, g ( x ′ g ¯ P + k ⊥ ) + q ( xP ) −→ γ ( l γ ) + q ( l q ) . (6)Secondly, due to the high gluon number density at small x, it is necessary to resum gluon re-scatteringto all orders.The multiple scattering between quark and the classical color field of the nucleus can be readilyresummed to all orders [38, 39]. This gives rise to a path-ordered gauge factor along the straight linethat extends in x + from minus infinity to plus infinity. More precisely, for a quark with incomingmomentum l and outgoing momentum l + k , the path-ordered gauge factor reads,2 πδ ( k + ) n µ [ U ( k ⊥ ) − (2 π ) δ ( k ⊥ )] , (7)with U ( k ⊥ ) = Z d x ⊥ e ik ⊥ · x ⊥ U ( x ⊥ ) , (8)5nd U ( x ⊥ ) = P e ig R + ∞−∞ dx + A − A ( x + , x ⊥ ) · t , (9)where t is the generators in the fundamental representation. We use this as building block to computethe amplitude for direct photon production in high energy pA collisions. It is straightforward to obtainthe production amplitude for diagram (a) illustrated in Fig. 2, M a = ¯ u ( l q )( ie ) ε/S F ( xP + x ′ g ¯ P + k ⊥ ) n/u ( xP ) h U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i (10)where a delta function is suppressed. In the above formula, ε µ is the polarization vector of the producedphoton, and S F ( xP + x ′ g ¯ P + k ⊥ ) = i xP/ + x ′ g ¯ P/ + k ⊥ / ( xP + x ′ g ¯ P + k ⊥ ) + iǫ is the quark propagator. The contribution ofdiagram (b) in Fig. 2 to the amplitude is similarly given by, M b = ¯ u ( l q ) n/S F ( xP − l γ )( ie ) ε/u ( xP ) h U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i (11)The contribution of diagram (c) in Fig. 2 vanishes. This is so because both x ′ g poles lie below thereal axis, such that one can close the integration contour above the real axis and get a vanishingcontribution. The total amplitude is thus given by M = M a + M b . By squaring the amplitude,one obtains the cross section [37], dσd l γ ⊥ dz = α em α s N c l γ ⊥ − z ) z X q e q Z x min dx Z d k ⊥ ( k ⊥ − l γ ⊥ /z ) x ′ g G DP ( x ′ g , k ⊥ ) f q ( x ) (12)where x ′ g G DP ( x ′ g , k ⊥ ) is the dipole type gluon TMD, defined as x ′ g G DP ( x ′ g , k ⊥ ) = k ⊥ N c π α s Z d x ⊥ d y ⊥ (2 π ) e ik ⊥ · ( y ⊥ − x ⊥ ) N c h Tr h U ( x ⊥ ) U † ( y ⊥ ) i i x ′ g (13)The Wilson lines appearing in the above formula can be explicitly evaluated in the MV model [22].The resulting dipole gluon distribution reads x ′ g G DP ( x ′ g , k ⊥ ) = k ⊥ N c π α s πR Z d r ⊥ (2 π ) e ik ⊥ · r ⊥ e − r ⊥ Q sq (14)where Q sq = α s C F µ ln r ⊥ Λ QCD is the quark saturation momentum with µ being the transverse colorsource density for a nucleus. Here R is the radius of nucleus.In the large transverse momentum region l γ ⊥ ≫ Q sq ∼ k ⊥ , the denominator in Eq. (12) can beapproximated as: 1 / ( k ⊥ − l γ ⊥ /z ) ≈ z /l γ ⊥ . After making this approximation and using the relation Z d k ⊥ x ′ g G DP ( x ′ g , k ⊥ ) = x ′ g G ( x ′ g ) , (15)one is able to reproduce Eq. (2) which was obtained from collinear factorization. ↑ A collisions
To calculate the SSA for direct photon production in polarized pA collisions, we have to take intoaccount one extra gluon exchange between the active partons and the remanent part of the polarizedproton, while gluon re-scattering inside the nucleus must be resummed to all orders. A typical diagramcontributing to this process is illustrated in Fig. 3. It is worthwhile to mention that gluons from the6 :: ::: ::::::
Figure 3: A typical diagram contributing to the SSA in direct photon production in polarized pAcollisions. The multiple re-scattering of the incoming partons (including the unpolarized quark andlongitudinally polarized gluon) from the proton off the classical gluon field of the nucleus needs to beresummed to all orders.nucleus could also interact with the color source inside the proton. Such an interaction is not shownin Fig. 3. In this section, we derive the spin dependent amplitude in the CGC framework. Wefurther calculate the derivative term contribution with the obtained amplitude, and also show thatthe full polarized cross section can be reduced to the one computed from the standard collinear twist-3approach at high photon transverse momentum provided that the 1 /N c suppressed color entanglementeffect has been neglected. As mentioned in the previous section, the multiple scattering between the incoming quark and theclassical color field of the nucleus can be resummed into a Wilson line. Similarly, this procedure alsoapplies to the case for which the incoming parton is a transversely polarized gluon. However, in theprocess under consideration, multiple scattering between incoming gluon and background gluon fieldof the nucleus can not be described by a simple Wilson line, since the incoming gluon from the protonside is longitudinally polarized.The formula for a longitudinally polarized gluon scattering off a nucleus has been worked out inRef. [40]. The expression for the gauge field created through the fusion of the incoming gluon fromthe proton and small x gluons from the nucleus contains both singular terms (proportional to δ ( x + ))and regular terms, A µ ( q ) = A µreg ( q ) + δ µ − A − sing ( q ) . (16)The regular terms A µreg are given by A µreg = A µp + igq + iq + ǫ Z d p ⊥ (2 π ) n C µU ( q, p ⊥ ) h ˜ U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i + C µV,reg ( q ) h ˜ V ( k ⊥ ) − (2 π ) δ ( k ⊥ ) io ρ p ( p ⊥ ) p ⊥ (17)7here ρ p ( p ⊥ ) is the color source distribution inside a proton, and A µp is the gauge field created by theproton alone. In the MV model, it is given by, A µp = 2 πgδ µ + δ ( q − ) ρ p ( q ⊥ ) q ⊥ , (18)In second term of the formula 17, p ⊥ is the momentum carried by the incoming gluon from the protonand k ⊥ defined as k ⊥ = q ⊥ − p ⊥ is the momentum coming from the nucleus. For the polarized case,there exists a correlation between the transverse momentum p ⊥ and the transverse proton spin vector S ⊥ . As shown below, such a correlation can be described by the ETQS function, and leads to a SSA fordirect photon production. The four vectors C µU ( q, p ⊥ ) and C µV,reg are given by the following relations C + U ( q, p ⊥ ) = − p ⊥ q − + iǫ , C − U ( q, p ⊥ ) = k ⊥ − q ⊥ q + + iǫ , C iU ( q, p ⊥ ) = − i ⊥ (19) C µV,reg ( q ) = 2 q µ − δ − µ q q + + iǫ (20)where the subscript ′ reg ′ indicates that the corresponding term of A µ does not contain any δ ( x + )when expressed in coordinate space. Here, we specified the q + pole structure according to the factthat this term arises from an initial state interaction. It is crucial to keep the imaginary part of thispole in order to generate the non-vanishing spin asymmetry. The notation p ⊥ is used to denote fourdimension vector with p ⊥ = − p ⊥ . ˜ U ( k ⊥ ) and ˜ V ( k ⊥ ) are the Fourier transform of Wilson lines in theadjoint representation,˜ U ( k ⊥ ) = Z d x ⊥ e ik ⊥ · x ⊥ ˜ U ( x ⊥ ) , ˜ V ( k ⊥ ) = Z d x ⊥ e ik ⊥ · x ⊥ ˜ V ( x ⊥ ) (21)with ˜ U ( x ⊥ ) = P exp (cid:20) ig Z + ∞−∞ dz + A − A ( z + , x ⊥ ) · T (cid:21) , (22)˜ V ( x ⊥ ) = P exp (cid:20) i g Z + ∞−∞ dz + A − A ( z + , x ⊥ ) · T (cid:21) (23)where the T are the generators of the adjoint representation. The singular terms reads, A − sing ( q ) = − igq + + iǫ Z d p ⊥ (2 π ) h ˜ V ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i ρ p ( p ⊥ ) p ⊥ (24)The peculiar Wilson line ˜ V differs from the normal one ˜ U by a factor 1 / V cancel in the unpolarized amplitudes for gluonproduction and quark pair production in pA collisions [40, 41]. It will be shown below that the ˜ V terms also drop out in the spin dependent amplitude for direct photon production.Following the method outlined in Ref. [41], we calculate the contributions from the regular termsand the singular terms separately. Let us begin with the regular terms which do not contain a deltafunction δ ( x + ). Their contributions are represented by the diagrams in Fig. 4 and Fig. 5. Theamplitude from Fig. (4a) reads, M a = [ igA µreg ( q )] [¯ u ( l q ) γ µ t a S F ( xP − l q )( ie ) ε/u ( xP )] (25)where the momentum carried by the gluon produced through the fusion of a longitudinally polarizedgluon from the proton and a small x gluons from the nucleus is given by q = x g P + p ⊥ + x ′ g ¯ P + k ⊥ . The8 PxP lq lqq = xgP + p? + x0g (cid:22)P + k?(a) Areg Aregq = xgP + p? + x0g (cid:22)P + k? l(cid:13)l(cid:13) (b)
Figure 4: The contribution from the regular terms to the spin dependent amplitude. A black dotdenotes a classical field A reg insertion.soft gluon pole contribution to the amplitude M a from the first term of Eq. (17) cancels between thediagram Fig. 4a and its mirror diagram in the same way as between the diagrams Fig. 1c and Fig. 1d.In addition, the contributions from the first part of C µV,reg cancel between Fig. 4a and Fig. 4b due tothe Ward identity. The spin dependent amplitude thus can be explicitly written as, M a = − ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ × ¯ u ( l q ) (cid:26) C U / ( q, p ⊥ ) q + iǫ t b S F ( xP − l q ) ε/ h ˜ U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i ba − n/x g P + iǫ t b S F ( xP − l q ) ε/ h ˜ V ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i ba ) u ( xP ) (26)For the diagram in Fig. 4b, it is easy to verify that the first term of the Eq. (17) gives rise to thevanishing contribution. One thus obtains for the amplitude from Fig. 4b, M b = − ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ × ¯ u ( l q ) (cid:26) ε/S F ( xP + q ) C U / ( q, p ⊥ ) q + iǫ t b h ˜ U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i ba − ε/S F ( xP + q ) n/x g P + iǫ t b h ˜ V ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i ba ) u ( xP ) (27)The incoming quark also can directly interact with the classical gluon field from the nucleus. Thisis illustrated in Fig. 5. The first term of A reg does not contribute to the spin dependent part of theamplitudes from the diagrams in Fig. 5. One further notices that all of the x ′ g poles in the amplitudesfrom the diagrams in Fig. 5d, Fig. 5e and Fig. 5f are lying in the same half plane. Therefore, aftercarrying out the x ′ g integration using the theorem of the residues, one has M d = M e = M f = 0 (28)We are left with the contributions from Fig. 5a, Fig. 5b and Fig. 5c. After carrying out the x ′ g integration, it becomes evident that the contributions from the first part of the C µV,reg term cancel9etween the diagrams in Fig. 5a, Fig. 5b and Fig. 5c. With these simplifications, the expression forthe amplitude of Fig. 5a is given by, M a = Z d k (2 π ) πδ ( k +1 )[ igA µreg ( q − k )] × ¯ u ( l q ) γ µ t a S F ( xP − l γ + k ) n/ h U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i S F ( xP − l γ ) ieε/u ( xP )= − ieg Z dk − d k ⊥ (2 π ) Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ ¯ u ( l q ) C U / ( q − k , p ⊥ )( q − k ) + iǫ t b S F ( xP − l γ + k ) n/ × h U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i S F ( xP − l γ ) ε/u ( xP ) h ˜ U ( k ⊥ − k ⊥ ) − (2 π ) δ ( k ⊥ − k ⊥ ) i ba + ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ Z d x ⊥ e ik ⊥ · x ⊥ ¯ u ( l q ) n/x g P + iǫ t b [ U ( x ⊥ ) − S F ( xP − l γ ) × ε/u ( xP ) h ˜ V ( x ⊥ ) − i ba (29)where we have applied the Eikonal approximation to the quark propagator S F ( xP − l γ + k ) whichappears in the hard part associated with the term containing h ˜ V ( x ⊥ ) − i ba . The k − and k ⊥ inte-grations have been carried out in the second term after making the Eikonal approximation. Followingthe similar procedure, we obtain the amplitude from the diagram in Fig. 5b, M b = − ieg Z dk − d k ⊥ (2 π ) Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ ¯ u ( l q ) ε/S F ( xP + q ) C U / ( q − k , p ⊥ )( q − k ) + iǫ t b S F ( xP + k ) × n/ h U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i u ( xP ) h ˜ U ( k ⊥ − k ⊥ ) − (2 π ) δ ( k ⊥ − k ⊥ ) i ba + ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ Z d x ⊥ e ik ⊥ · x ⊥ ¯ u ( l q ) ε/S F ( xP + q ) n/x g P + iǫ t b [ U ( x ⊥ ) − u ( xP ) × h ˜ V ( x ⊥ ) − i ba (30)The amplitude of the diagram in Fig. 5c does not receive any contribution from the second part ofthe C µV,reg term since both k − poles are lying in the same half plane. One thus obtains, M c = − ieg Z dk − d k ⊥ (2 π ) Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ ¯ u ( l q ) C U / ( q − k , p ⊥ )( q − k ) + iǫ t b S F ( xP − l γ + k ) ε/ × S F ( xP + k ) n/ h U ( k ⊥ ) − (2 π ) δ ( k ⊥ ) i u ( xP ) h ˜ U ( k ⊥ − k ⊥ ) − (2 π ) δ ( k ⊥ − k ⊥ ) i ba . (31)We now turn to discuss the contributions from the singular terms. As explained in Ref. [41], itis convenient to compute it in coordinate space. The expression for the singular term in coordinatespace is then given by [41], A − sing = − i g A − A ( x ) · T ] ˜ V ( x + , −∞ ; x ⊥ ) θ ( − x − ) 1 ∇ ⊥ ρ p ( x ⊥ ) (32)where the theta function θ ( − x − ) reflects the fact that this gluon field is created through an initialstate interaction. ˜ V ( x + , −∞ ; x ⊥ ) denotes an incomplete Wilson line:˜ V ( x + , −∞ ; x ⊥ ) = P exp " i g Z x + −∞ dz + A − A ( z + , x ⊥ ) · T (33)10 regAreg Areg(a) (b) ( )Areg Areg Areg(d) (e) (f) Figure 5: The contribution from the regular terms to the spin dependent amplitude. Black dotdenotes a classical field A reg insertion, while the gluon line terminated by a cross surrounded with acircle denotes a classical field A A insertion.In order to correctly compute the singular contributions, it is necessary to regularize δ ( x + ) by givingit a small width δ ( x + ) −→ δ ǫ ( x + ) (34)where δ ǫ ( x + ) is a regular function whose support is [0 , ǫ ], which becomes δ ( x + ) when ǫ goes to zero.The final result is independent of the precise choice of the regularization. The field A µsing is inserted onthe quark line at the ’times’ x + , the incoming quark then rescatters off the field A µA of the nucleus in theranges [0 , x + ] and [ x + , ǫ ]. The photon can only be emitted from the quark line either before multiplegluon re-scattering or after gluon re-scattering, because in the limit ǫ →
0, there is not sufficienttime for emitting a photon inside the nucleus. The eight diagrams contributing to the amplitude areillustrated in Fig. 6. Combining the contributions from diagram Fig. 6a, Fig. 6b, Fig. 6c and Fig. 6d,the resulting amplitude in coordinate space is, M a +6 b +6 c +6 d = Z d xe iq · x ¯ u ( l q ) U (+ ∞ , x + ; x ⊥ ) n/ h igt a A − asing ( x ) i S F ( xP − l γ ) × ieε/U ( x + , −∞ ; x ⊥ ) u ( xP )= − eg Z d xe iq · x ¯ u ( l q ) U (+ ∞ , x + ; x ⊥ ) n/ h t a A − asing ( x ) i S F ( xP − l γ ) × ε/U † (+ ∞ , x + ; x ⊥ ) U (+ ∞ , −∞ ; x ⊥ ) u ( xP ) (35)where the incomplete Wilson lines in the fundamental representation are defined as U (+ ∞ , x + ; x ⊥ ) = P exp (cid:20) ig Z + ∞ x + dz + A − A ( z + , x ⊥ ) · t (cid:21) (36) U ( x + , −∞ ; x ⊥ ) = P exp " ig Z x + −∞ dz + A − A ( z + , x ⊥ ) · t (37)11 sing Asing(b)(a) Asing( ) AsingAsing(e) Asing(f ) Asing(g) Asing(h)(d) Figure 6: The contribution from the singular terms to the spin dependent amplitude. A black dotdenotes a classical field A sing insertion, while the gluon line terminated by a cross surrounded with acircle denotes a classical field A A insertion.In order to simplify this expression, we use the algebraic identity, U (+ ∞ , x + ; x ⊥ ) t a U † (+ ∞ , x + ; x ⊥ ) = t b ˜ U ba (+ ∞ , x + ; x ⊥ ) (38)and also the formula first derived in Ref. [41], i g Z + ∞−∞ dx + ˜ U (+ ∞ , x + ; x ⊥ )[ A − A ( x ) · T ] ˜ V ( x + , −∞ ; x ⊥ ) = ˜ U ( x ⊥ ) − ˜ V ( x ⊥ ) (39)After carrying out the x − and x + integrations, the above expression is simplified to, M a +6 b +6 c +6 d = ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ Z d x ⊥ e ik ⊥ · x ⊥ × ¯ u ( l q ) n/t b S F ( xP − l γ ) ε/U ( x ⊥ ) u ( xP ) 1 x g P + iǫ h ˜ U ( x ⊥ ) − ˜ V ( x ⊥ ) i ba (40)Following the same procedure, it is straightforward to write down the amplitude from the diagramsin Fig. 6e, Fig. 6f, Fig. 6g and Fig. 6h M e +6 f +6 g +6 h = ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ Z d x ⊥ e ik ⊥ · x ⊥ × ¯ u ( l q ) ε/S F ( xP + q ) n/t b U ( x ⊥ ) u ( xP ) 1 x g P + iǫ h ˜ U ( x ⊥ ) − ˜ V ( x ⊥ ) i ba (41)Collecting all pieces together, the total amplitude reads, M = − ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ Z dk − d k ⊥ (2 π ) ¯ u ( l q ) × (cid:26) ε/S F ( xP + q ) C U / ( q − k , p ⊥ )( q − k ) + iǫ t b S F ( xP + k ) n/U ( k ⊥ )12 C U / ( q − k , p ⊥ )( q − k ) + iǫ t b S F ( xP − l γ + k ) n/U ( k ⊥ ) S F ( xP − l γ ) ε/ + C U / ( q − k , p ⊥ )( q − k ) + iǫ t b S F ( xP − l γ + k ) ε/S F ( xP + k ) n/U ( k ⊥ ) (cid:27) × u ( xP ) h ˜ U ( k ⊥ − k ⊥ ) − (2 π ) δ ( k ⊥ − k ⊥ ) i ba + ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ Z d x ⊥ e ik ⊥ · x ⊥ × ¯ u ( l q ) n/S F ( xP − l γ ) ε/ + ε/S F ( xP + q ) n/x g P + iǫ t b U ( x ⊥ ) u ( xP ) h ˜ U ( x ⊥ ) − i ba (42)where the ˜ V ( x ⊥ ) terms drop out as expected. The spin dependence of the total amplitude comesfrom the correlation between p ⊥ and the transverse spin vector of the proton S ⊥ . With the obtainedamplitude, we are ready to compute the twist-3 spin dependent cross section. However, since thecalculation of the full polarized cross section is quite involved, we restrict ourself here to the discussionof two results, namely the derivative term contribution in a dense medium and the cross section inthe large photon transverse momentum limit. We believe that it is sufficient to demonstrate themost interesting feature of the complete result, on the one hand, and to check the consistence of ourformalism by extrapolating the result to the high transverse momentum limit and comparing it withthe polarized cross section computed in the collinear twist-3 approach, on the other hand. One of the terms in the polarized cross section proportional to the derivative of the ETQS function isoften refereed to as the derivative term, which is usually considered to be the dominant contribution tothe spin asymmetry in the forward region. In this subsection, we sketch a few key steps when derivingthe expression for the derivative term. As mentioned in the previous section, the spin dependent hardpart is calculated from an interference of two partonic scattering amplitudes, as illustrated in Fig. 3.We thus proceed by defining a hard part H µνBorn ( p ⊥ , k ⊥ ) according to the following equation, X spins , color x M | derivative M ∗ δ ( l q ) = Z d p ⊥ (2 π ) x g P + iǫ H µνBorn ( p ⊥ , k ⊥ ) n µ n ν δ ( l q ) × Z d x ⊥ d y ⊥ e ik ⊥ · ( x ⊥ − y ⊥ ) Tr c (h U † ( y ⊥ ) − i t b U ( x ⊥ ) ρ p,a ( p ⊥ ) p ⊥ ) h ˜ U ( x ⊥ ) − i ba (43)where M | derivative represents the last term in Eq. (42). As we shall explain below, the C U termsin Eq. (42) do not give rise to a derivative term contribution. The next step is to expand the hardpart in terms of p ⊥ , H µνBorn ( p ⊥ , k ⊥ ) δ ( l q ) = H µνBorn ( p ⊥ , k ⊥ ) δ ( l q ) | p ⊥ =0 + ∂H µνBorn ( p ⊥ , k ⊥ ) δ ( l q ) ∂p ρ ⊥ | p ⊥ =0 p ρ ⊥ + ... (44)where the spin dependent part is the term linear in p ⊥ , in which we only keep the contributioncontaining the derivative of the delta function, leading to a derivative of the ETQS function by partialintegration. More precisely, the relevant contribution is given by, " ( k ρ ⊥ − l ργ ⊥ ) l q · P H µνBorn ( p ⊥ , k ⊥ ) ∂δ ( l q ) ∂x p ⊥ =0 p ⊥ ,ρ (45)13nce the collinear expansion has been carried out, p ⊥ is set to zero in the hard part as indicated inthe above formula. At this point, it becomes clear why the C U terms do not give rise to a derivativeterm contribution, namely simply because C U vanishes when p ⊥ = 0, C µU ( q − k , p ⊥ = 0) = 0 (46)We proceed by combining p ⊥ with the color source term to yield the gluon field strength operatorusing Eq. (18), ρ p,a ( p ⊥ ) p ⊥ p ⊥ ,ρ −→ F + ρ,a ( p ⊥ ) (47)Since the hard part is independent of p ⊥ after the collinear expansion, the p ⊥ integration can betrivially carried out. The quark gluon correlator can subsequently be parameterized through theETQS function, Z d p ⊥ (2 π ) h ¯ ψ ρ p,a ( p ⊥ ) p ⊥ p ⊥ ,ρ ψ i proton −→ ǫ ρS ⊥ np N c − π t a T F ( x, x + x g ) (48)where the correlation between the transverse spin vector of the proton and p ⊥ becomes manifest.The color structure associated with the ETQS function is fixed by following the argument made inRefs. [6, 7, 10]. Moreover, one needs to isolate the imaginary part of the soft gluon pole using theidentity x g P + iǫ = P x g P − iπδ ( x g P ). The contributions from its real part cancel out between mirrordiagrams. With all these calculation recipes, one can readily compute the contribution from thederivative term. The spin dependent cross section involving the derivative term takes the followingform, d ∆ σd l γ ⊥ dz ∝ Z d k ⊥ dx ′ g dx h ǫ l γ S ⊥ np − ǫ k ⊥ S ⊥ np i l q · P " H µνBorn ( p ⊥ , k ⊥ ) ∂δ ( l q ) ∂x p ⊥ =0 X q e q T F,q ( x, x ) × Z d x ⊥ d y ⊥ e ik ⊥ · ( x ⊥ − y ⊥ ) h Tr c h(cid:16) U † ( y ⊥ ) − (cid:17) t b U ( x ⊥ ) t a i h ˜ U ( x ⊥ ) − i ba i x ′ g (49)The expression for the soft part from the nucleus side in the above formula can be further simplified.Using Eq. (38), one obtains,Tr c h(cid:16) U † ( y ⊥ ) − (cid:17) t b U ( x ⊥ ) t a i h ˜ U ( x ⊥ ) − i ba = C F Tr c h U † ( y ⊥ ) U ( x ⊥ ) i − Tr c h U † ( y ⊥ ) t a U ( x ⊥ ) t a i (50)Employing the Fierz identity, t aij t akl = 12 δ il δ kj − N c δ ij δ kl (51)the last term in Eq. (50) is rewritten as,Tr c h U † ( y ⊥ ) t a U ( x ⊥ ) t a i = 12 Tr c h U † ( y ⊥ ) i Tr c [ U ( x ⊥ )] − N c Tr c h U † ( y ⊥ ) U ( x ⊥ ) i (52)Inserting the above decomposition into Eq. (50), we obtain,Tr c h U † ( y ⊥ ) t b U ( x ⊥ ) t a i h ˜ U ( x ⊥ ) − i ba = N c c h U † ( y ⊥ ) U ( x ⊥ ) i −
12 Tr c h U † ( y ⊥ ) i Tr c [ U ( x ⊥ )] (53)14here the first term can be related to the dipole type gluon distribution, while the second term is a newcontribution that arises from the color entanglement effect. To arrive at the final expression for thepolarized cross section, we need to explicitly evaluate the hard part and carry out the x ′ g integrationusing the delta function δ ( l q ) which originates from the on shell condition. After combining thecontributions from the left and right cut diagrams, one ends up with, d ∆ σd l γ ⊥ dz = α s α em N c N c − − z ) zl γ ⊥ ( z − Z x min dx Z d k ⊥ h ǫ l γ S ⊥ np − ǫ k ⊥ S ⊥ np i ( k ⊥ − l γ ⊥ /z ) ( k ⊥ − l γ ⊥ ) × X q e q (cid:20) − x ddx T F,q ( x, x ) (cid:21) h x ′ g G DP ( x ′ g , k ⊥ ) − x ′ g G ( x ′ g , k ⊥ ) i (54)which is the main result of this section. Here we introduce a new gluon distribution G ( x ′ g , k ⊥ ). It isdefined as, x ′ g G ( x ′ g , k ⊥ ) = k ⊥ N c π α s Z d x ⊥ d y ⊥ (2 π ) e ik ⊥ · ( x ⊥ − y ⊥ ) N c h Tr c [ U ( x ⊥ )]Tr c [ U † ( y ⊥ )] i x ′ g (55)This new gluon distribution only shows up in the spin dependent cross section and is absent in theunpolarized cross section. The extra gluon exchange between the remnant of the proton and activepartons plays a crucial role in yielding the nontrivial Wilson line structure in the Eq. (56). Theadditional term associated with G ( x ′ g , k ⊥ ) thus essentially arises from the color entanglement effect.More interestingly, the gluon distribution G ( x ′ g , k ⊥ ) can be calculated in the MV model througha recursion procedure systemically developed in Ref. [41]. In the MV model, to our surprise, it issimply given by, x ′ g G ( x ′ g , k ⊥ ) = k ⊥ N c π α s πR Z d r ⊥ (2 π ) e ik ⊥ · r ⊥ N c e − r ⊥ Q sq = 1 N c G DP ( x ′ g , k ⊥ ) (56)We thus conclude that the novel gluon distribution G is sizable, though it is suppressed in the large N c limit as compared to the dipole type gluon distribution.We now make some observations on the polarized cross section in the different kinematic limits.At small photon transverse momentum Λ QCD ≪ l γ ⊥ ≪ Q sq ∼ k ⊥ , the denominator in the Eq. (54)can be approximated as, 1( k ⊥ − l γ ⊥ /z ) ( k ⊥ − l γ ⊥ ) ≈ zz k ⊥ · l γ ⊥ k ⊥ ! k ⊥ (57)Once adopting such approximation in the both unpolarized cross section and polarized cross section,it is easy to see that the spin asymmetry computed in the hybrid approach scales as l ⊥ at lowtransverse momentum. As a comparison, in the standard collinear twist-3 framework, the predicatedspin asymmetry is proportional to 1 /l ⊥ . For l γ ⊥ ≫ Q sq ∼ k ⊥ , the denominator can be approximatedby 1 / ( k ⊥ − l γ ⊥ /z ) ≈ z /l γ ⊥ . Using Eq. 15 and Eq. 56, the polarized cross section is correspondinglyreduced to, d ∆ σd l γ ⊥ dz = α s α em N c N c − ǫ l γ S ⊥ np z [1 + (1 − z ) ] l γ ⊥ ( z − × X q e q Z x min dx (cid:20) − x ddx T F,q ( x, x ) (cid:21) (cid:26) x ′ g G ( x ′ g ) − N c x ′ g G ( x ′ g ) (cid:27) (58)15hich recovers the result for the derivative term contribution computed in the collinear approachif one ignores the second term of the soft parts that arises from color entanglement effect. In thenext subsection, we show that the non-derivative term contribution also can be reproduced in ourhybrid formalism in the kinematical limit where l γ ⊥ ≫ Q sq ∼ k ⊥ provided that the G contributionis neglected.We now close this subsection with a few further remarks. • The color entanglement effect discovered for double spin asymmetries (DSA) leads to a violationof generalized TMD factorization [26]. In contrast, the process we study is factorizable thoughthere exists an additional term arises from the color entanglement effect. The reason is thatcollinear factorization is applied on the proton side and the basic building block of the soft parton the nucleus side, namely the Wilson line, is a universal object. • Apparently, the gluon distribution G ( x ′ g , k ⊥ ) vanishes in the single gluon exchange approxima-tion, thus requireing at least two gluon exchange. This is in line with the argument that twoextra gluon attachments from an unpolarized target are required to generate a non-trivial colorentanglement effect [26]. • Since G ( x ′ g , k ⊥ ) can be explicitly evaluated in the MV model, one can test it by measuringSSAs for photon production in p ↑ A collisions. • In general color entanglement plays a less important role for p ↑ p collisions than for p ↑ A collisionsas the existence of G ( x ′ g , k ⊥ ) requires at least two gluons exchange. However, the SSA for photonproduction in p ↑ p collisions might receive the significant contribution from the additional termproportional to the distribution G ( x ′ g , k ⊥ ) in the very forward region. • If we apply collinear factorization on both the proton and nucleus sides, according to the modelcalculation result Eq. 56, the color entanglement effect would survive. This might indicate thatthe collinear higher-twist factorization breaks down in the process we study.
In this subsection, we show that not only the derivative term contribution but also the non-derivativeterm contribution computed in the collinear factorization framework can be recovered from our hy-brid approach in the limit l γ ⊥ ≫ Q sq after neglecting the additional contribution results from colorentanglement effect. To achieve this goal, our main strategy is to systematically neglect all termssuppressed by powers of Q sq /l γ ⊥ (or k ⊥ /l γ ⊥ ).The first step is to set k ⊥ = 0 in the hard part in Eq. (42). This is a well justified approximationin the kinematical limit that we consider because the typical transverse momentum carried by smallx gluons is of the order of Q sq and thus much smaller than the photon transverse momentum l γ ⊥ .We then can trivially carry out the k ⊥ and k − integrations in Eq. (42) after applying the Eikonalapproximation to the quark propagators S F ( xP + k ) and S F ( xP − l γ + k ). As a result, the polarizedamplitude simplifies to, M ≈ − ieg Z d p ⊥ (2 π ) ρ p,a ( p ⊥ ) p ⊥ Z d x ⊥ e ik ⊥ · x ⊥ ¯ u ( l q ) × C L / ( q, p ⊥ ) S F ( xP − l γ ) ε/ + ε/S F ( xP + q ) C L / ( q, p ⊥ ) q + iǫ t b U ( x ⊥ ) u ( xP ) h ˜ U ( x ⊥ ) − i ba (59)16here C L / is the well known effective Lipatov vertex for the production of a gluon via the fusion oftwo gluons. It is given by, C L / ( q, p ⊥ ) q + iǫ = C U / ( q, p ⊥ ) q + iǫ − n/q + + iǫ = − p ⊥ ( q + iǫ )( q − + iǫ ) p/ + k ⊥ ( q + iǫ )( q + + iǫ ) n/ − q − n/ + p ⊥ /q + iǫ (60)One notices that the first term in the above formula can be neglected since it is beyond the order in p ⊥ that we consider. The second term contains one soft gluon pole and one hard gluon pole,1 q + + iǫ −→ x g = 0 , q + iǫ −→ x g = ( k ⊥ + p ⊥ ) x ′ g P · ¯ P (61)When the photon transverse momentum is much larger than the saturation scale, the hard gluon pole x g = ( k ⊥ + p ⊥ ) / (2 x ′ g P · ¯ P ) ≈ − q − n/ + p ⊥ /q + iǫ −→ q + p/ + k ⊥ /q + iǫ (62)Applying the Ward identity to the right side of the cut diagrams, we may make the following replace-ment for the gluon polarization vector, n µ −→ − k µ ⊥ q − (63)With these replacements, the hard part can be written as, − q − δ ( q )( x g P µ + k ⊥ ,µ )k ⊥ ,ν H µνBorn ( p ⊥ , k ⊥ ) δ ( l q ) (64)where H µνBorn ( p ⊥ , k ⊥ ) has been defined in the previous subsection. To proceed further, we keep theleading term which is proportional to k ⊥ and neglect all higher order terms in k ⊥ . After averagingover the azimuthal angle of k ⊥ , the hard part reads,12 q − k ⊥ ( P µ p ⊥ ,ν x ′ g P · ¯ P + d ⊥ ,µν ) h H µνBorn ( p ⊥ , k ⊥ ) δ ( l q ) δ ( q ) i k ⊥ =0 (65)where the tensor d ⊥ ,µν is defined as d ⊥ ,µν = − g µν + ( p µ n ν + p ν n µ ) /p · n . k ⊥ in the above formula canbe combined with the soft part, namely the Wilson lines, and related to the transverse momentumdependent gluon distributions G DP ( x ′ g , k ⊥ ) and G ( x ′ g , k ⊥ ). Since there is no k ⊥ dependence in thehard part any longer, the k ⊥ integration can be trivially carried out using the relations presented inEq. (15) and Eq. (56). The resulting soft part is simply the ordinary integrated gluon distribution ifwe ignore the term generated from G .On the other hand, the spin dependent hard part from Fig. 1a calculated in the collinear approachis proportional to the following expression12 d σ ⊥ ,ν h H µνBorn ( p ⊥ , k ⊥ ) δ ( l q ) δ ( q ) i k ⊥ =0 Λ σρµ p ρ (66)17ith the three gluon vertex being defined as,Λ σρµ = g σρ ( x ′ g ¯ P − x g P − p ⊥ ) µ + g ρµ (2 x g P + 2p ⊥ + x ′ g ¯ P ) σ + g µσ ( − x ′ g ¯ P − x g P − p ⊥ ) ρ (67)After few algebra steps, one finds that the expressions (65) and (66) for the hard parts derived in thetwo different formalisms agree with each other up to some trivial pre-factor. Following the procedureoutlined above, one can show that both formalisms also yield the same hard parts for the mirrordiagrams. Therefore, we confirmed that our result without the G related contribution being includedreduces to that computed in the standard collinear approach in the kinematical region l γ ⊥ ≫ Q sq . Weconsider this as an important consistency check for the hybrid approach. We studied the SSA in direct photon production in polarized p ↑ A collisions. The calculation iscarried out using a hybrid approach in which the nucleus is treated in the CGC framework while thecollinear twist-3 formalism is applied on the proton side. We derived the part of the polarized crosssection containing the derivative term, with particular emphasis on the contribution caused by thecolor entanglement effect. This effect arises from the non-trivial interplay between one extra gluonexchange from the proton side and multiple gluon exchanges from the nucleus. The identified newgluon distribution G ( x ′ g , k ⊥ ) can be explicitly evaluated in the MV model. As a result, measuringthis observable would provide us with a unique chance to quantitatively study color entanglementeffects.We have further shown that the spin dependent cross section computed in the standard collinearapproach can be recovered from the hybrid approach in the kinematical region where the transversemomentum of the produced photon is much larger than the saturation momentum, provided thatthe contribution arises from color entanglement effect is not included in our result. However, at lowphoton transverse momentum, in sharp contrast to the predication from the standard collinear twist-3approach, the spin asymmetry is found to be proportional to the photon transverse momentum.A direct extension of this work is to investigate the impact of the color entanglement effect on theSSA in Drell-Yan lepton pair production in p ↑ A collisions. One can also use the hybrid formalism tocalculate the SSAs for pion production and di-jet and photon-jet [42] production in p ↑ A collisions. Aproposed p ↑ A program at RHIC [43] is thus extremely welcome. Finally, we would like to emphasizethat the hybrid approach in principal can also be applied to p ↑ p collisions, where it, however, onlyis valid in the very forward region at low transverse momentum Λ QCD ≪ l γ ⊥ ≤ Q psq (where Q psq isthe proton saturation momentum). In other words, the standard collinear twist-3 approach is notadequate to describe SSA phenomenology in p ↑ p or p ↑ A collisions in the mentioned kinematic regionwhere the CGC framework can apply on target nucleus or proton side.
Acknowledgments:
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