Analysis of the double-spin asymmetry A_LT in inelastic nucleon-nucleon collisions
aa r X i v : . [ h e p - ph ] D ec Analysis of the double-spin asymmetry A LT ininelastic nucleon-nucleon collisions A. Metz , D. Pitonyak , A. Sch¨afer , J. Zhou Department of Physics, Barton Hall, Temple University, Philadelphia, PA 19122, USA Institute for Theoretical Physics, Regensburg University,Universit¨atsstraße 31, D-93053 Regensburg, Germany
September 4, 2018
Abstract
Within the collinear twist-3 framework, we analyze the double-spin asymmetry in collisions betweenlongitudinally polarized nucleons and transversely polarized nucleons with focus on hadron andjet production. As was the case in direct photon production, the double-spin dependent crosssection for hadron and jet production has the advantage of involving a complete set of collineartwist-3 functions for a transversely polarized nucleon. In addition, we outline further benefits ofthis observable for a potential future measurement at RHIC, which includes insight on the gluonhelicity distribution as well as information on the Efremov-Teryaev-Qiu-Sterman function T F ( x, x )that plays a crucial role in single-spin asymmetries. Spin asymmetries in hard scattering processes have been an interesting subject of research for severaldecades. Starting in the mid-1970s, the large single-spin asymmetries (SSAs) observed in inclusivehadron production [1–7] were initially an obstacle for perturbative QCD. Within the collinear partonmodel, these asymmetries should be on the order of α s m q /P h ⊥ [8,9], where m q is the mass of the quark,and P h ⊥ is the transverse momentum of the detected hadron. However, research pioneered in the early1980s [10] that went beyond the simplistic parton model showed that these SSAs could be generatedwithin a framework that involved collinear twist-3 parton correlators. This formalism, which is validwhen a process contains one large scale Q (with Λ QCD ≪ Q ), has also been extensively investigatedfor SSAs in various observables — see [11–19] for some specific examples. (We also mention that othermechanisms have been proposed to explain large SSAs [20–22].)Similarly, extensive work has been done on the longitudinal double-spin asymmetry (DSA) A LL inprocesses such as polarized lepton-nucleon collisions and polarized nucleon-nucleon collisions [23, 24].This differs from the derivations of SSAs in that A LL is a leading twist (twist-2) effect that gives accessto the helicity distributions of partons in the nucleon — see [25] for a recent global extraction of thesefunctions. The main goal of this research has been to understand how the spin of the nucleon can beexplained in terms of the partons that compose it. A real surprise occurred when it was determinedby EMC [26] (and later confirmed at SLAC [27, 28]) that the spins of the quarks contribute anunexpectedly small fraction to the spin of the nucleon. Clearly, the remaining percentage must comefrom the orbital angular momentum of the partons and the spin of the gluons. Much research hasbeen done on this front to determine exactly what contribution each of these pieces make — see,e.g., [29–31] for recent reviews on the subject. 1hile the areas of hadronic spin physics outlined in the previous two paragraphs have for the mostpart operated independently of each other, one observable, namely, the longitudinal-transverse DSA A LT , offers insight into both domains. More specifically, A LT (in processes with one large scale) is acollinear twist-3 effect that is also sensitive to parton helicities. The classic process that necessitatesthis formalism is A LT for inclusive deep-inelastic lepton-nucleon scattering (DIS). In that case, one canstudy the collinear twist-3 function g T . In addition, A LT has been analyzed in the Drell-Yan processinvolving two incoming polarized hadrons [32–35]. More recently, A LT was calculated in inclusivelepton production from W -boson decay in proton-proton scattering [36], for jet production in lepton-nucleon scattering [37], and for direct photon production in nucleon-nucleon collisions [38]. However,it was only in [38] that for the first time a spin dependent cross section was considered that required acomplete set of collinear twist-3 functions for a transversely polarized nucleon in order to fully describethe observable. We will see this same characteristic holds for hadron and jet production. (Note that aterm containing chiral-odd correlation functions was not computed in [38]. We will also neglect thesecontributions in the present work — see the discussion below Eq. (2).) These higher-twist functions donot have a probability interpretation and are lesser known than the collinear ones relevant at leadingtwist (namely, the unpolarized distribution f , helicity distribution g , and transversity distribution h [32, 39, 40]), but nevertheless they provide important insight into the spin structure of the nucleon.In the present work, we analyze the double-spin asymmetry A LT in nucleon-nucleon collisionsfor the case of hadron and jet production as well as review the results for direct photon productionfound in [38]. These results collectively can be considered the DSA analog to the SSAs derived inthe same processes [11, 12, 14, 15]. Furthermore, we briefly discuss plans for a future numerical studyand highlight the prospects for this observable to provide insight on important areas of hadronic spinphysics. These include information on the gluon helicity distribution and the Efremov-Teryaev-Qiu-Sterman (ETQS) function T F ( x, x ) that enters into SSAs in hadronic processes.The paper is organized as follows: in Sec. 2, we review the collinear twist-3 formalism includingthe relevant non-perturbative functions that enter into the calculation. In Sec. 3, we derive thedouble-spin dependent cross section for hadron and jet production, providing a few details of thecalculation. In addition, we briefly outline a future numerical study and emphasize potential benefitsfor a measurement of A LT at RHIC. In Sec. 4, we conclude the paper and summarize our work. To start, let us make explicit the process under consideration, namely, A ( P, ~S ⊥ ) + B ( P ′ , Λ) → C ( l ) + X, (1)where the 4-momenta and polarizations of the incoming nucleons A , B and outgoing particle (or jet) C are indicated. The Mandelstam variables for the process are defined as S = ( P + P ′ ) , T = ( P − l ) ,and U = ( P ′ − l ) , which on the partonic level give ˆ s = xx ′ S , ˆ t = xT /z , and ˆ u = x ′ U/z . Thelongitudinal momentum fraction x ( x ′ ) is associated with partons in the transversely (longitudinally)polarized nucleon.The first non-vanishing contribution to the cross section is given by terms of twist-3 accuracy andreads dσ ( ~l ⊥ , ~S ⊥ , Λ) = H ⊗ f a/A (3) ⊗ f b/B (2) ⊗ D C/c (2) + H ′ ⊗ f a/A (2) ⊗ f b/B (3) ⊗ D C/c (2) + H ′′ ⊗ f a/A (2) ⊗ f b/B (2) ⊗ D C/c (3) , (2)2 xP ( a ) ( b )( c ) ( d ) PxP + k ⊥ Px P + k ⊥ xP + k ⊥ x PP xP
Figure 1: Feynman diagrams for the twist-3 matrix elements that give contributions to f a/A (3) . Seethe text for more details.where a sum over partonic channels and parton flavors in each channel is understood. In Eq. (2), f a/A ( t ) denotes the twist- t distribution function associated with parton a in hadron A (and likewisefor f b/B ( t ) ), while D C/c ( t ) represents the twist- t fragmentation function associated with particle C in parton c . The factors H , H ′ , and H ′′ indicate the hard parts corresponding to each term, whilethe tensor product denotes convolutions in the appropriate momentum fractions. For the case ofthe SSA A UT (where B is now unpolarized), it has already been shown that the second term in (2),which involves chiral-odd twist-3 unpolarized distributions, is negligible because of the smallness ofthe hard scattering coefficients [41]. We believe a similar statement will hold for the A LT case, whichinvolves chiral-odd twist-3 helicity distributions, since the hard factors will be similar to the ones forthe unpolarized case. Arguments have been made that the first term in (2) for A UT (so-called Siversterm) is dominant [11, 12, 14]. However, recent work has shown that one cannot rule out significantcontributions from the third term (so-called Collins term) [42–44]. For this current work on A LT , wewill focus on the first term in (2) but cannot exclude that the third term could also play a critical role.Therefore, for the situation we consider, f b/B (2) = g b and D C/c (2) = D C/c , where g and D are thestandard twist-2 helicity distribution function and unpolarized fragmentation function, respectively.We then must determine what contributions are possible for f a/A (3) .A detailed discussion of collinear twist-3 functions and, in particular, those relevant for a trans-versely polarized nucleon, is given in Ref. [17]. Here we simply review the main aspects needed for thiswork. The twist-3 matrix elements that we must consider are given by the diagrams in Fig. 1. Notethat we have neglected matrix elements involving tri-gluon correlators. In the lightcone ( A + = 0)gauge, these graphs lead to the three matrix elements [17] h ¯ ψψ i , h ¯ ψ∂ ⊥ ψ i , h ¯ ψA ⊥ ψ i , (3)which result from Figs. 1(a), (b), and (d), respectively. We do not have to consider Fig. 1(c) becauseone does not need to simultaneously take into account k ⊥ expansion and A ⊥ gluon attachments (whichwould give rise to twist-4 contributions).Now that we have determined the relevant twist-3 matrix elements, we must parameterize themin terms of twist-3 functions that will eventually be involved in our final result. We first focus on the3uark-gluon-quark ( qgq ) matrix element h ¯ ψA ⊥ ψ i . One notices that this matrix element is not gaugeinvariant. This can be resolved in two ways: rewrite the gluon field A ⊥ in terms of the field strengthtensor F + µ ⊥ = ∂ + A µ ⊥ or rewrite it in terms of the covariant derivative D µ ⊥ = ∂ µ ⊥ − igA µ ⊥ . The formerleads to the matrix element being written in terms of the so-called “F-type” functions, while the lattergives the so-called “D-type” functions [11]. Respectively, we have Z dξ − π dζ − π e ix P + ξ − e i ( x − x ) P + ζ − h P, S ⊥ | ¯ ψ β (0) gF + µ ⊥ ( ζ − ) ψ α ( ξ − ) | P, S ⊥ i = M h F F T ( x, x ) ǫ µν ⊥ S ⊥ ν /n − G F T ( x, x ) iS µ ⊥ γ /n i αβ , (4)and Z dξ − π dζ − π e ix P + ξ − e i ( x − x ) P + ζ − h P, S ⊥ | ¯ ψ β (0) iD µ ⊥ ( ζ − ) ψ α ( ξ − ) | P, S ⊥ i = M P + h F DT ( x, x ) iǫ µν ⊥ S ⊥ ν /n + G DT ( x, x ) S µ ⊥ γ /n i αβ . (5)In Eqs. (4), (5), we have suppressed Wilson lines and have indicated the nucleon mass by M . We havealso introduced the lightcone vector n = (1 + , − ,~ ⊥ ), whose conjugate vector is ¯ n = (0 + , − ,~ ⊥ ). Notethat we have defined the F-type and D-type functions as in Ref. [45], which differs from those usedin [38]. These functions satisfy certain symmetry properties under the interchange of their arguments: F F T ( x, x ) = F F T ( x , x ) and G F T ( x, x ) = − G F T ( x , x ) , (6)while F DT ( x, x ) = − F DT ( x , x ) and G DT ( x, x ) = G DT ( x , x ) . (7)Moreover, it turns out the F-type and D-type functions are not independent of each other. Onecan establish the following relations between these functions [13]: F DT ( x, x ) = P V x − x F F T ( x, x ) , (8) G DT ( x, x ) = P V x − x G F T ( x, x ) + δ ( x − x ) ˜ g ( x ) , (9)where P V denotes the principal value. In order to derive these expressions, notice that we mustintroduce an additional twist-3 function ˜ g ( x ), whose definition is given by Z dξ − π e ixP + ξ − h P, S ⊥ | ¯ ψ β (0) (cid:18) iD µ ⊥ ( ξ − ) + g Z ∞ ξ − dζ − F + µ ⊥ ( ζ − ) (cid:19) ψ α ( ξ − ) | P, S ⊥ i = M h ˜ g ( x ) S µ ⊥ γ /n i αβ . (10)This function is associated with the quark-quark ( qq ) matrix element h ¯ ψ∂ ⊥ ψ i . We also mention that˜ g ( x ) is equivalent to the first k ⊥ -moment of the TMD g T ( x, ~k ⊥ ) for a longitudinally polarized quarkin a transversely polarized nucleon [17]:˜ g ( x ) = Z d ~k ⊥ ~k ⊥ M g T ( x, ~k ⊥ ) . (11)4 , S T P ′ , Λ l ˜ g, g T D g g D F F T , G
F T ( b )( a ) Figure 2: Graphs showing factorization for contributions to A LT from (a) qq correlators and (b) qgq correlators.The other relevant qq matrix element h ¯ ψψ i leads to a contribution from the well-known twist-3 function g T ( x ), whose definition is given by2 MP + S µ ⊥ g T ( x ) = Z dy − π e ixP + y − h P, S ⊥ | ¯ ψ (0) γ µ γ ψ ( y − ) | P, S ⊥ i . (12)However, g T ( x ) can be related to the D-type functions (and, therefore, due to (8), (9), also the F-typefunctions) through the QCD equations of motion (EOM) [10, 32]: x g T ( x ) = Z dx [ G DT ( x, x ) − F DT ( x, x )] . (13)From the above discussion, we have identified six twist-3 functions relevant for a transverselypolarized nucleon: ˜ g, g T , F F T , G
F T , F DT , G DT . However, from the relations given in Eqs. (8), (9),(13), in the end one has only three independent collinear twist-3 functions relevant for a transverselypolarized nucleon. At the outset of a calculation, one can choose to work with either the F-typefunctions and ˜ g ( x ) or the D-type functions and ˜ g ( x ). One cannot simply use the F-type or D-typefunctions alone, but rather the function ˜ g ( x ) must also be included — see, e.g., Ref. [46]. The factorization of the process under consideration is shown in Fig. 2. This includes collinear factorsassociated with the longitudinally polarized nucleon (top gray blob), the outgoing particle (or jet)(middle gray blob), and the transversely polarized nucleon (bottom gray blob) as well as hard factors(white blobs). We choose to work with the F-type functions and ˜ g ( x ). For each partonic channel,the main task becomes calculating the hard scattering coefficients for each of these functions, whichthen allows us to write down the double-spin dependent cross section. We will denote each channel by ab → cd , where a ( b ) is the parton associated with the transversely (longitudinally) polarized nucleonand c is the parton that fragments into the detected particle (or jet).5 a ) ( b ) ( c )( d ) ( e ) Figure 3: Hard scattering diagrams for the qq ′ → qq ′ channel involving (a) qq correlators and (b)–(e) qgq correlators. Note that Hermitian conjugate diagrams for the qgq graphs are not shown.Here we will focus on the qq ′ → qq ′ channel in order to present a few details of the calculation. Therelevant hard scattering diagrams for this channel are shown in Fig. 3. First, we consider the graph inFig. 3(a). If we keep the transverse momentum of the initial state parton q (as in Fig. 1(b)), then wecan determine the hard scattering coefficient for ˜ g ( x ). In fact, this will lead to terms involving both˜ g ( x ) and its derivative, as was first detailed in [11,12]. On the other hand, if we neglect the transversemomentum of q in the initial state (as in Fig. 1(a)), then we obtain the hard part for g T ( x ). However,since we work with the F-type functions and ˜ g ( x ), we use Eq. (13) in conjunction with Eqs. (8), (9)to write g T ( x ) in terms of those functions. Lastly, we must attach gluons in all possible ways toFig. 3(a), which leads to Figs. 3(b)–(e) and their Hermitian conjugates (not shown). In these graphswe can neglect the transverse momenta of the initial state parton q and gluon (as in Fig. 1(d)). Thesediagrams allow us to find the qgq contributions to the hard factors for the F-type functions. Note thatwe can combine the graphs in Figs. 3(b)–(e) with their Hermitian conjugates by using the symmetryrelations in Eq. (6).We remark at this stage that in general the qgq diagrams are not always real but can acquire animaginary part whenever internal parton lines go on-shell. This requires the use of the distributionidentity 1 x ± iǫ = P V x ∓ iπδ ( x ) . (14)However, unlike the case of SSAs, the P V part survives when we combine the various cut diagrams,whereas the pole term vanishes — see also [38]. A related feature is that all of the qgq graphscontribute to the hard scattering coefficients for the F-type functions, unlike the situation for SSAswhen one considers the so-called soft gluon pole (SGP) term. For example, if one were calculating theSGP term to the SSA for AB → CX for the qq ′ → qq ′ channel, only Figs. 3(b), (e) (after includingthe Hermitian conjugate graphs) provide such a pole [14].Finally, collecting all the terms, we find for the qq ′ → qq ′ channel the following contribution to the6ouble-spin dependent cross section: l dσ qq ′ → qq ′ ( ~S ⊥ , Λ) d ~l = − α s MS ~l ⊥ · ~S ⊥ Λ X a, b, c Z z min dzz D C/c ( z ) Z x ′ min dx ′ x ′ x ′ S + T /z z ˆ u g b ( x ′ ) 1 x × (cid:26) ˜ g a ( x ) (cid:20) C F N c s − ˆ t ˆ t (cid:21) − x d ˜ g a ( x ) dx (cid:20) C F N c ˆ u − ˆ s ˆ t (cid:21) + Z dx P V x − x G aF T ( x, x ) (cid:20) (cid:18) u − ˆ s ) ξ ˆ t + ˆ s (ˆ s − ˆ u )(1 − ξ )ˆ t + ˆ u ˆ t (cid:19) + 12 N c (cid:18) s − ˆ u )ˆ t (cid:18) t − ˆ uξ ˆ t + 11 − ξ (cid:19) − ˆ u ˆ t (cid:19)(cid:21) − Z dx P V x − x F aF T ( x, x ) (cid:20) (cid:18) ˆ u ˆ t − ˆ s (ˆ s − ˆ u )(1 − ξ )ˆ t (cid:19) + 12 N c (cid:18) − s − ˆ u )ˆ t (1 − ξ ) − ˆ u ˆ t (cid:19)(cid:21)(cid:27) , (15)where x = − x ′ ( U/z ) / ( x ′ S + T /z ), x ′ min = − ( T /z ) / ( U/z + S ), and z min = − ( T + U ) /S . We haveintroduced ξ = x g /x , where x g = x − x , and understand 1 /ξ to mean P V (1 /ξ ). The SU (3) colorfactors depend on C F = 4 / N c = 3. We note that the coefficient of ( d/dx )˜ g ( x ) in Eq. (15)matches the hard factor for the qq ′ → qq ′ channel in the leading order (LO) calculation of A LL foundin [24]. This is to be expected given the Dirac projectors associated with g b ( x ) and ˜ g a ( x ) and thefact the “derivative term” at this stage is obtained by neglecting transverse momentum everywhereexcept in the on-shell delta function [11, 12]. We have checked for all channels that at this point inthe calculation an agreement occurs between the derivative term and the LO A LL coefficients.We can rewrite (15) in terms of the D-type functions and ˜ g ( x ) by using Eqs. (8), (9). If one doesso, a nice simplification occurs involving ˜ g ( x ) and its derivative: l dσ qq ′ → qq ′ ( ~S ⊥ , Λ) d ~l = − α s MS ~l ⊥ · ~S ⊥ Λ X a, b, c Z z min dzz D C/c ( z ) Z x ′ min dx ′ x ′ x ′ S + T /z z ˆ u g b ( x ′ ) 1 x × (cid:26)(cid:18) ˜ g a ( x ) − x d ˜ g a ( x ) dx (cid:19) (cid:20) − N c (ˆ t − ˆ u )(ˆ s − ˆ u )ˆ t (cid:21) + Z dx G aDT ( x, x ) (cid:20) (cid:18) u − ˆ s ) ξ ˆ t + ˆ s (ˆ s − ˆ u )(1 − ξ )ˆ t + ˆ u ˆ t (cid:19) + 12 N c (cid:18) s − ˆ u )ˆ t (cid:18) t − ˆ uξ ˆ t + 11 − ξ (cid:19) − ˆ u ˆ t (cid:19)(cid:21) − Z dx F aDT ( x, x ) (cid:20) (cid:18) ˆ u ˆ t − ˆ s (ˆ s − ˆ u )(1 − ξ )ˆ t (cid:19) + 12 N c (cid:18) − s − ˆ u )ˆ t (1 − ξ ) − ˆ u ˆ t (cid:19)(cid:21)(cid:27) . (16)We will comment more on this “compact” form involving ˜ g ( x ) and its derivative as well as make othergeneral remarks on the analytical result in the next subsection.7 .2 Final analytical result Following for the remaining channels the outline given above for calculating hard factors, we find thecross section relevant for the DSA A LT in AB → CX is given by l dσ ( ~S ⊥ , Λ) d ~l = − α s MS ~l ⊥ · ~S ⊥ Λ X i X a, b, c Z z min dzz D C/c ( z ) Z x ′ min dx ′ x ′ x ′ S + T /z z ˆ m i g b ( x ′ ) 1 x × (cid:26)(cid:20) ˜ g a ( x ) − x d ˜ g a ( x ) dx (cid:21) H i ˜ g + Z dx (cid:2) G aDT ( x, x ) H iG DT − F aDT ( x, x ) H iF DT (cid:3)(cid:27) , (17)where i denotes the channel and ˆ m i the corresponding partonic Mandelstam variable for that channel(see Table 1 in Appendix A). The result in Eq. (17) is if the detected particle is a hadron, with the hardscattering coefficients H i given in Appendix A. However, one can also obtain the expression for thedouble-spin dependent cross section for jet production by setting D C/c ( z ) = δ (1 − z ). The hard partsin this case are again given in Appendix A, but now one can combine channels that differ by a crossingof the final state partons. Likewise, for direct photon production one must set D c ( z ) = δ (1 − z ) butalso must make the replacement α s → α em e a for one factor of α s , where e a is the charge of a quarkwith flavor a (in units of e ). In this case, the number of channels reduces significantly, and theirrespective hard parts, which first appeared in [38], are given in Appendix B. We note a correction inthe overall sign for the hard factors for the qg → γq channel.A few comments are in order on the analytical result. First, we again mention that this calculationof A LT is the analog to the calculation of A UT in the same processes [11, 12, 14, 15]. Second, as we sawwith the qq ′ → qq ′ channel, when we write the result using the D-type functions instead of the F-typefunctions, ˜ g ( x ) and its derivative combine in the same compact form as T F ( x, x ) did for SSAs in directphoton and inclusive pion production [11, 14, 15]. This form was also seen in the A LT calculationsdone in Refs. [36,37] (and, as mentioned above, Ref. [38]). Finally, we see Eq. (17) involves a completeset of collinear twist-3 functions for a transversely polarized nucleon. This is because the hard partsassociated with each contribution are not the same, and, therefore, we cannot combine them into asimpler function. Thus, in principle this process in conjunction with other reactions allows us to accessa complete set of collinear twist-3 functions for a transversely polarized nucleon.At this point, we would also like to make a few observations about the hard scattering coefficients H iF DT and H iG DT in Appendices A and B. We remark that these hard parts, as we have made explicitin the Appendices, can all be written in the form H i = H i + 11 − ξ H i + 1 ξ H i , (18)where H i , H i , and H i are independent of ξ , and we have dropped the F DT and G DT subscripts fromthe H ’s. First, notice that H i , F DT = H i , G DT , which means one could pull these hard factors out ofthe integral over x and, using (13), write a term involving g T ( x ). For some channels, like qq ′ → qq ′ ,this is a trivial statement because the only ξ -independent terms in H iF DT and H iG DT come from the g T ( x ) contribution — see the second paragraph of Sec. 3.1. However, for other channels, like qg → qg ,the qgq contributions to H iF DT and H iG DT also contain ξ -independent terms. We find it interestingthat these additional ξ -independent terms are always the same for H iF DT and H iG DT . Second, one seesthat H i , F DT = − H i , G DT and H i , F DT = 0. We leave the former as another intriguing observationon the structure of the result. For the latter, we remark that the pole contribution 1 /ξ comes frominitial/final state interactions and can be written as a kinematical factor times the Born cross section8 see, e.g., [11, 12, 14]. In this case, the Born diagram corresponding to F DT vanishes, which leads to H i , F DT = 0. Finally, one notices that H ab → cdF DT = − H ab → dcF DT (ˆ t ↔ ˆ u ), and similarly for H ab → cd ( dc ) G DT , where(ˆ t ↔ ˆ u ) means interchange of ˆ t and ˆ u . Since one might surmise these hard parts for ab → dc can beobtained from those for ab → cd by an interchange of ˆ t and ˆ u (because we neglect k ⊥ and k ⊥ ), thenegative sign might be a bit unexpected. This negative sign appears to be due to the sensitivity ofthe transversely polarized gluon attachments to the transverse momentum of the outgoing partons.When one interchanges final state partons, the transverse momenta of the outgoing partons changesigns, which is reflected in the crossed-channel hard parts. Also, the fact that H ab → cd ˜ g appears to haveno relation to H ab → dc ˜ g might seem a bit strange. However, one can see this will be the case, e.g., bynoticing the k ⊥ dependence changes when one interchanges final state partons.We conclude this section with a brief discussion of a future numerical study and of the key insightsa measurement of this observable at RHIC might provide. In order to estimate the size of A LT forhadron, jet, and photon production, we must determine the input for the twist-3 functions that enterinto (17). We can obtain information on the function ˜ g ( x ) through its relation (11) to g T ( x, ~k ⊥ ) —see [37] for a recent example as well as [47–49]. In addition, one can choose to pull the ξ -independentterms in the hard factors for H iF DT and H iG DT out of the integral over x and write a term involving g T ( x ) — see the discussion in the previous paragraph. We also have information on this function, e.g.,through the Wandzura-Wliczek approximation [50, 51]. The main obstacle then is the qgq correlatorcontributions. The off-diagonal contributions (i.e., x = x ) to F F T and G F T needed for this DSAobservable are not as well-determined as the diagonal pieces that enter into SSAs. In Ref. [52], aGaussian form was assumed for F F T ( x, x ) ( T q,F in their notation) that was a maximum for x = x and fell off for x = x . This study was done in the context of the evolution of F F T ( x, x ). In Ref. [53],an analysis of higher-twist functions was conducted using light-cone wave functions that include qqqg Fock states. In contrast to [52], this study found F F T ( x, x ) ( T qF q in their notation) takes on itsgreatest values when x = x and some of its lowest values when x = x . It is our plan to determinethe impact of the qgq correlators on the size of A LT and provide a complete estimate for the observable.Given this estimate, the importance of the measurement of A LT in pion production from proton-proton ( pp ) collisions at RHIC is threefold. First, through this observable one might be able to probethe gluon helicity ∆ g down to momentum fractions x ∼ − (or even lower), more than an order ofmagnitude below the sensitivity of all current measurements [54–62]. Given the recent debate as tothe size of ∆ g at smaller x — see [63] and references therein, A LT could offer valuable insight into thematter. Second, a measurement of A LT in this process would be a first step towards extracting (non-diagonal) information on the 3-parton correlators F F T and G F T . Information on these functions isbeneficial in its own right, but, as alluded to in the previous paragraph, one must know the off-diagonalcontributions to F F T and G F T in order to fully determine the evolution of F F T ( x, x ) [52, 64–67]. Thisevolution is a vital aspect if one wants to fully understand SSAs. Finally, the “sign mismatch” issuethat has arisen involving F F T ( x, x ) (or T F ( x, x )) and the Sivers function [68] has called into questionwhether the collinear twist-3 framework is the correct formalism to describe, e.g., the large SSAs seenin inclusive hadron production from pp collisions [1–7]. The study of DSAs may provide new insightson this point. For instance, should a significant discrepancy exist between a numerical estimate of A LT in pion production and a future measurement of this observable at RHIC, one may ask whetherthe collinear twist-3 approach taken in this paper is the appropriate mechanism to consider for bothSSAs and DSAs. Of course, one must keep in mind that knowledge of the relevant twist-3 functionsat present is rather limited [52, 53]. 9 Summary
In conclusion, we have calculated the double-spin dependent cross section for inclusive hadron andjet production for the longitudinal-transverse asymmetry A LT in nucleon-nucleon collisions within thecollinear twist-3 framework. We have also reviewed the results for A LT in direct photon productionfrom nucleon-nucleon scattering [38]. These derivations are the DSA analog to the SSAs calculated inthe same processes [11,12,14,15]. Furthermore, these reactions require a complete set of collinear twist-3 functions for a transversely polarized nucleon in order to fully describe the process. We emphasizeagain that we did not consider contributions involving chiral-odd correlation functions. We have foundthat the solution, when written in terms of D-type functions, allows for a “compact” form involving˜ g ( x ) and its derivative; similar forms have manifested themselves in other reactions [11, 14, 15, 36–38].We also made some intriguing observations on the structure of the hard factors, in particular for H iF DT and H iG DT .In addition, we have briefly outlined our plan for a future numerical study of this observable. Themain difficulty underlying such an analysis is how to handle the contributions from the 3-parton corre-lators F F T and G F T since, unlike the case for SSAs, A LT is sensitive to the off-diagonal contributionsto these functions. Such an undertaking is worthwhile, however, since a measurement of this effectat RHIC could provide insight on some important areas of research in hadronic spin physics. Theseinclude not only information on 3-parton correlators, which are important in their own right, butalso access to the gluon helicity distribution ∆ g at momentum fractions not yet explored ( x ∼ − ),information on the evolution of the ETQS function T F ( x, x ) that appears in SSAs, and a generalunderstanding of the mechanism that causes twist-3 spin asymmetries in nucleon-nucleon collisions. Acknowledgments:
We would like to thank C. Aidala, E. Aschenauer, and B. Surrow for helpfulexchanges on the possibility of measuring this observable at RHIC. We also appreciate useful discus-sions with V. Braun and A. Manashov with regards to Ref. [53]. This work has been supported bythe NSF under Grant No. PHY-1205942 and by the BMBF under Grant No. OR 06RY9191.
Appendices
A Hard scattering coefficients for hadron production
Here we give the hard scattering coefficients H i for hadron production. Table 1 lists all the channels i ( ab → cd ) and gives the corresponding partonic Mandelstam variable ˆ m i for that channel. Note that(ˆ t ↔ ˆ u ) means interchange of ˆ t and ˆ u . We define ξ = x g /x , where x g = x − x , and understand 1 /ξ to mean P V (1 /ξ ). We also mention that the SU (3) color factors depend on N c = 3. The double-spin dependent cross section for jet production takes on the same form as Eq. (17) but now with D C/c ( z ) = δ (1 − z ). This allows hard factors to be combined for channels that differ by an interchangeof the final state partons. 10 m i ˆ s ˆ t ˆ ui ( ab → cd ) q ¯ q → gg qg → gq qg → qgqq → qq qq ′ → q ′ q qq ′ → qq ′ q ¯ q → ¯ qq q ¯ q → q ¯ qq ¯ q ′ → ¯ q ′ q q ¯ q ′ → q ¯ q ′ q ¯ q → ¯ q ′ q q ¯ q → q ′ ¯ q ′ Table 1: Mandelstam variable ˆ m i for each channel i ( ab → cd ). qg → qg channel H ˜ g = 12 (cid:20) (ˆ s − ˆ u )ˆ u ˆ s ˆ t (cid:21) + 12 N c (cid:20) ˆ s − ˆ u ˆ u (cid:21) + 12( N c − (cid:20) (ˆ s − ˆ u ) ˆ t (cid:21) (19) H G DT = 12 (cid:20) ˆ s (ˆ s − ˆ t ˆ u )ˆ t ˆ u − ˆ u (ˆ t − ˆ u )ˆ s ˆ t − (ˆ s + ˆ u )(ˆ t − s ˆ u )(1 − ξ )ˆ s ˆ t ˆ u + 2ˆ s (ˆ s − ˆ u ) ξ ˆ t ˆ u (cid:21) + 12 N c (cid:20) − ξ − ˆ s + 2ˆ u ˆ s ˆ u + 2(ˆ s − ˆ u ) ξ ˆ s (cid:21) − N c − (cid:20) (ˆ s − ˆ u ) ˆ t (cid:18) − − ξ − ξ (cid:19)(cid:21) (20) H F DT = 12 (cid:20) ˆ s (ˆ s − ˆ t ˆ u )ˆ t ˆ u − ˆ u (ˆ t − ˆ u )ˆ s ˆ t + (ˆ s + ˆ u )(ˆ t − s ˆ u )(1 − ξ )ˆ s ˆ t ˆ u (cid:21) + 12 N c (cid:20) − − ξ − ˆ s + 2ˆ u ˆ s ˆ u (cid:21) − N c − (cid:20) (ˆ s − ˆ u ) (1 − ξ )ˆ t (cid:21) (21) qg → gq channel H ˜ g = 12 (cid:20) (ˆ s − ˆ t ) (ˆ t ˆ u − ˆ s )ˆ s ˆ t ˆ u (cid:21) + 12( N c − (cid:20) (ˆ s − ˆ t ) ˆ u (cid:21) (22) H G DT = − H qg → qgG DT (ˆ t ↔ ˆ u ) (23) H F DT = − H qg → qgF DT (ˆ t ↔ ˆ u ) (24)11 ¯ q → gg channel H ˜ g = − N c (cid:20) (ˆ t + ˆ u )(ˆ s + 2ˆ t ˆ u )ˆ s ˆ t ˆ u (cid:21) + N c (cid:20) ˆ s − t ˆ u ˆ t (cid:21) − N c (cid:20) ˆ s (ˆ t + ˆ u )ˆ u ˆ t (cid:21) (25) H G DT = 12 N c (cid:20) ˆ s (ˆ t − ˆ u )(ˆ s − ˆ t ˆ u )(1 − ξ )ˆ t ˆ u + 2ˆ s (ˆ t − ˆ u )(ˆ t + ˆ u ) ξ ˆ t ˆ u − (ˆ t − ˆ u )ˆ s (cid:18) s ˆ t ˆ u + (ˆ t − ˆ u ) ˆ s ˆ t ˆ u (cid:19)(cid:21) − N c (cid:20) (ˆ u − ˆ t )(ˆ t + ˆ u )(ˆ s − ˆ t ˆ u )ˆ s ˆ t ˆ u (cid:18) − − ξ − ξ (cid:19) − (ˆ t − ˆ u )(ˆ t + ˆ u )(ˆ s + 2ˆ t ˆ u )ˆ s ˆ t ˆ u (cid:21) + 12 N c (cid:20) − (ˆ t − ˆ u )ˆ s (1 − ξ )ˆ t ˆ u + ˆ s (ˆ t − ˆ u )(ˆ s + ˆ u ˆ t )ˆ t ˆ u (cid:21) (26) H F DT = 12 N c (cid:20) − ˆ s (ˆ t − ˆ u )(ˆ s − ˆ t ˆ u )(1 − ξ )ˆ t ˆ u − (ˆ t − ˆ u )ˆ s (cid:18) s ˆ t ˆ u + (ˆ t − ˆ u ) ˆ s ˆ t ˆ u (cid:19)(cid:21) − N c (cid:20) (ˆ u − ˆ t )(ˆ t + ˆ u )(ˆ s − ˆ t ˆ u )(1 − ξ )ˆ s ˆ t ˆ u − (ˆ t − ˆ u )(ˆ t + ˆ u )(ˆ s + 2ˆ t ˆ u )ˆ s ˆ t ˆ u (cid:21) + 12 N c (cid:20) (ˆ t − ˆ u )ˆ s (1 − ξ )ˆ t ˆ u + ˆ s (ˆ t − ˆ u )(ˆ s + ˆ u ˆ t )ˆ t ˆ u (cid:21) (27) qq ′ → qq ′ channel H ˜ g = − N c (cid:20) (ˆ t − ˆ u )(ˆ s − ˆ u )ˆ t (cid:21) (28) H G DT = − (cid:20) s − ˆ u ) ξ ˆ t − ˆ s (ˆ s − ˆ u )(1 − ξ )ˆ t − ˆ u ˆ t (cid:21) + 12 N c (cid:20) s − ˆ u )ˆ t (cid:18) t − ˆ uξ ˆ t + 11 − ξ (cid:19) − ˆ u ˆ t (cid:21) (29) H F DT = − (cid:20) ˆ s (ˆ s − ˆ u )(1 − ξ )ˆ t − ˆ u ˆ t (cid:21) + 12 N c (cid:20) − s − ˆ u )ˆ t (1 − ξ ) − ˆ u ˆ t (cid:21) (30) qq ′ → q ′ q channel H ˜ g = − (cid:20) ˆ s (ˆ t − ˆ s )ˆ u (cid:21) + 12 N c (cid:20) s − ˆ t )ˆ u (cid:21) (31) H G DT = − H qq ′ → qq ′ G DT (ˆ t ↔ ˆ u ) (32) H F DT = − H qq ′ → qq ′ F DT (ˆ t ↔ ˆ u ) (33)12 q → qq channel H ˜ g = 12 (cid:20) ˆ s (ˆ s − ˆ t )ˆ t ˆ u (cid:21) − N c (cid:20) s ˆ u − ˆ s (ˆ s − ˆ u )ˆ t (cid:21) − N c (cid:20) s ˆ u ˆ t (cid:21) + 12 N c (cid:20) s ˆ u ˆ t (cid:21) (34) H G DT = 12 (cid:20) s (ˆ u − ˆ t )(1 − ξ )ˆ t ˆ u + 2ˆ s (ˆ u − ˆ t ) ξ ˆ t ˆ u + ˆ s (ˆ u − ˆ t )ˆ t ˆ u (cid:21) − N c (cid:20) s (ˆ t − ˆ u )(2ˆ s − t ˆ u ) ξ ˆ t ˆ u − ˆ s (ˆ t − ˆ u )ˆ t ˆ u (cid:18) − ξ (cid:19)(cid:21) − N c (cid:20) ˆ s (ˆ u − ˆ t )ˆ t ˆ u (cid:18) − ξ (cid:19)(cid:21) + 12 N c (cid:20) ˆ s (ˆ t − ˆ u )ˆ t ˆ u (cid:18) − − ξ + 4 ξ (cid:19)(cid:21) (35) H F DT = 12 (cid:20) − s (ˆ u − ˆ t )(1 − ξ )ˆ t ˆ u + ˆ s (ˆ u − ˆ t )ˆ t ˆ u (cid:21) − N c (cid:20) − ˆ s (ˆ t − ˆ u )ˆ t ˆ u (cid:18) − − ξ (cid:19)(cid:21) − N c (cid:20) ˆ s (ˆ u − ˆ t )ˆ t ˆ u (cid:18) − − ξ (cid:19)(cid:21) + 12 N c (cid:20) ˆ s (ˆ t − ˆ u )ˆ t ˆ u (cid:18) − − − ξ (cid:19)(cid:21) (36) q ¯ q → q ¯ q channel H ˜ g = − (cid:20) ˆ u (ˆ s − t )ˆ s ˆ t (cid:21) − N c (cid:20) ˆ u ˆ s ˆ t (cid:18) s + 9ˆ t ˆ u ˆ t + ˆ s − u ˆ s (cid:19)(cid:21) − N c (cid:20) u (ˆ s − ˆ t )ˆ s ˆ t (cid:21) (37) H G DT = 12 (cid:20) ˆ u (3ˆ t + ˆ u )ˆ s ˆ t + 2ˆ u (ˆ t + ˆ u ) ξ ˆ s ˆ t − u (ˆ s + ˆ u ˆ t )(1 − ξ )ˆ s ˆ t (cid:21) − N c (cid:20) ˆ u (3ˆ t + ˆ u )ˆ s ˆ t − u (ˆ s − t )(1 − ξ )ˆ s ˆ t + 2ˆ u (2ˆ s − ˆ t (ˆ s − t )) ξ ˆ s ˆ t (cid:21) − N c (cid:20) − u ˆ s ˆ t (cid:18) − ξ (cid:19) − ˆ u ˆ s ˆ t (cid:18) − ξ (cid:19) − u ξ ˆ s ˆ t (cid:21) + 12 N c (cid:20) − ˆ u (ˆ t − ˆ s )ˆ s ˆ t (cid:18) − − ξ (cid:19) − u ˆ s ˆ t − u ξ ˆ t (cid:21) (38) H F DT = 12 (cid:20) ˆ u (3ˆ t + ˆ u )ˆ s ˆ t + 2ˆ u (ˆ s + ˆ u ˆ t )(1 − ξ )ˆ s ˆ t (cid:21) − N c (cid:20) ˆ u (3ˆ t + ˆ u )ˆ s ˆ t + 2ˆ u (ˆ s − t )(1 − ξ )ˆ s ˆ t (cid:21) − N c (cid:20) − u ˆ s ˆ t (cid:18) − − ξ (cid:19) − ˆ u ˆ s ˆ t (cid:18) − − ξ (cid:19)(cid:21) + 12 N c (cid:20) − ˆ u (ˆ t − ˆ s )ˆ s ˆ t (cid:18) − ξ (cid:19) − u ˆ s ˆ t (cid:21) (39)13 ¯ q → ¯ qq channel H ˜ g = 12 (cid:20) ˆ t (ˆ t − ˆ s )ˆ u (cid:21) + 12 N c (cid:20) t ˆ u − t (ˆ s − ˆ u )ˆ s (cid:21) − N c (cid:20) t ˆ u (cid:21) + 12 N c (cid:20) t ˆ s ˆ u (cid:21) (40) H G DT = − H q ¯ q → q ¯ qG DT (ˆ t ↔ ˆ u ) (41) H F DT = − H q ¯ q → q ¯ qF DT (ˆ t ↔ ˆ u ) (42) q ¯ q → q ′ ¯ q ′ channel H ˜ g = 12 (cid:20) ˆ t + ˆ u ˆ s ˆ t (cid:21) + 12 N c (cid:20) (ˆ t − s )(ˆ t + ˆ u )ˆ s ˆ t (cid:21) (43) H G DT = 12 (cid:20) u (ˆ t + ˆ u ) ξ ˆ s ˆ t + 2ˆ u (ˆ t − ˆ u )ˆ s + ˆ u (ˆ t + ˆ u )(1 − ξ )ˆ s ˆ t (cid:21) + 12 N c (cid:20) t + ˆ u )ˆ s ˆ t (cid:18) − ξ + 2 ξ (cid:19) − u (ˆ t − ˆ u )ˆ s (cid:21) (44) H F DT = 12 (cid:20) u (ˆ t − ˆ u )ˆ s − ˆ u (ˆ t − ˆ u )(1 − ξ )ˆ s ˆ t (cid:21) + 12 N c (cid:20) − u (ˆ t − ˆ u )ˆ s − t + ˆ u )(1 − ξ )ˆ s ˆ t (cid:21) (45) q ¯ q → ¯ q ′ q ′ channel H ˜ g = 12 N c (cid:20) (ˆ s − ˆ u )(ˆ t + ˆ u )ˆ s ˆ u (cid:21) (46) H G DT = − H q ¯ q → q ′ ¯ q ′ G DT (ˆ t ↔ ˆ u ) (47) H F DT = − H q ¯ q → q ′ ¯ q ′ F DT (ˆ t ↔ ˆ u ) (48) q ¯ q ′ → q ¯ q ′ channel H ˜ g = 12 (cid:20) ˆ u − ˆ s ˆ t (cid:21) + 12 N c (cid:20) (ˆ s − t )(ˆ u − ˆ s )ˆ t (cid:21) (49) H G DT = 12 (cid:20) ˆ u (ˆ s − ˆ u )(1 − ξ )ˆ t − ˆ u ˆ t (cid:21) + 12 N c (cid:20) s − ˆ t )(ˆ s − ˆ u ) ξ ˆ t − s − ˆ u )(1 − ξ )ˆ t − ˆ u ˆ t (cid:21) (50) H F DT = 12 (cid:20) − ˆ u (ˆ s − ˆ u )(1 − ξ )ˆ t − ˆ u ˆ t (cid:21) + 12 N c (cid:20) s − ˆ u )(1 − ξ )ˆ t − ˆ u ˆ t (cid:21) (51)14 ¯ q ′ → ¯ q ′ q channel H ˜ g = 12 (cid:20) ˆ t (ˆ t − ˆ s )ˆ u (cid:21) − N c (cid:20) s − ˆ t )ˆ u (cid:21) (52) H G DT = − H q ¯ q ′ → q ¯ q ′ G DT (ˆ t ↔ ˆ u ) (53) H F DT = − H q ¯ q ′ → q ¯ q ′ F DT (ˆ t ↔ ˆ u ) (54) B Hard scattering coefficients for photon production
Here we give the hard scattering coefficients H i for direct photon production. We define ξ = x g /x ,where x g = x − x , and understand 1 /ξ to mean P V (1 /ξ ). We also note that the SU (3) color factorsdepend on C F = 4 / N c = 3. The double-spin dependent cross section has the same form asEq. (17), but now we set D C/c ( z ) = δ (1 − z ) and for one factor of α s make the replacement α s → α em e a ,where e a is the charge of a quark with flavor a (in units of e ). qg → γq channel H ˜ g = N c N c − (cid:20) ˆ t − ˆ s ˆ s ˆ t (cid:21) H G DT = N c N c − (cid:20) (ˆ t − ˆ s )ˆ s ˆ t (cid:18) − − ξ − ξ (cid:19) (cid:21) − N c (cid:20) ˆ u (ˆ s + 2ˆ t )ˆ s ˆ t + ˆ u ˆ t (cid:18) − − ξ − ξ (cid:19) + 2ˆ uξ ˆ s (cid:21) H F DT = N c N c − (cid:20) ˆ t − ˆ s (1 − ξ ) ˆ s ˆ t (cid:21) − N c (cid:20) ˆ u (ˆ s + 2ˆ t )ˆ s ˆ t + ˆ u (1 − ξ ) ˆ t (cid:21) (55) q ¯ q → γg channel H ˜ g = 1 N c (cid:20) ˆ t + ˆ u ˆ t ˆ u (cid:21) H G DT = (ˆ t + ˆ u )ˆ t ˆ u (cid:18) − − ξ − ξ (cid:19) + 2 C F N c (cid:20) ˆ s (ˆ t − ˆ u )ˆ t ˆ u − (ˆ t − ˆ u )ˆ t (cid:18) − − ξ (cid:19) (cid:21) H F DT = ˆ t + ˆ u (1 − ξ )ˆ t ˆ u + 2 C F N c (cid:20) ˆ s (ˆ t − ˆ u )ˆ t ˆ u − (ˆ t − ˆ u )ˆ t (cid:18) − − − ξ (cid:19) (cid:21) (56) References [1] G. Bunce et al. , Phys. Rev. Lett. , 1113 (1976).[2] D. 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