Single-Transverse Spin Asymmetry in Semi-Inclusive Deep Inelastic Scattering

We study the single-transverse spin asymmetry in semi-inclusive hadron production in deep inelastic scattering. We derive the leading contribution to the asymmetry at moderate transverse momentum $P_{h\perp}$ of the produced hadron in terms of twist-three quark-gluon correlation functions, and compare with the approach based on the factorization at fixed transverse momentum involving the asymmetric transverse-momentum and spin-dependent quark distribution. We verify that the two approaches yield identical results in this regime. By a comparison with our earlier calculations for the single-spin asymmetry in the Drell-Yan process we recover the well-established process-dependence of the time-reversal-odd transverse-momentum-dependent quark distributions that generate single-spin phenomena.


1.
The study of single-transverse spin asymmetries (SSAs) has been at the forefront of experimental and theoretical research in strong interaction physics ever since the first observation of strikingly large asymmetries in hadronic scattering in the 1970s [1]. The size of the asymmetries posed a significant challenge for QCD. With the advent of new experimental information from lepton scattering [2] and from RHIC [3], and with major recent theory advances, we are now beginning to obtain a much clearer picture of the possible origins of SSAs in QCD [4].
In particular, two mechanisms for generating SSAs had been identified in the literature: asymmetric transverse-momentum-dependent parton distribution in a transversely polarized proton (the so-called Sivers functions) [5], and twist-three transverse-spin-dependent quarkgluon correlation functions (the so-called Efremov-Teryaev-Qiu-Sterman (ETQS) mechanism) [6,7]. For a long time, despite a wide-spread belief that these two mechanisms were not completely unrelated, the precise connection between them remained obscure. Early efforts to link the two were made in [8,9,10]. In two recent publications [11], we have demonstrated that the two mechanisms each have their own domain of validity, and that they consistently describe the same physics in kinematic regime where they both apply.
We have shown this in [11] for the case of the SSA for Drell-Yan production of dilepton pairs with invariant mass Q and transverse momentum q ⊥ . At large q ⊥ ∼ Q, the ETQS mechanism applies, and the resulting SSA is of twist-three nature. At small q ⊥ ≪ Q, a factorization in terms of transverse-momentum dependent (TMD) parton distribution applies [12,13,14,15], involving in case of the SSA the Sivers functions. If q ⊥ is much larger than Λ QCD , the dependence of these functions on transverse momentum may be computed using QCD perturbation theory. At the same time, the result obtained within the ETQS formalism may also be extrapolated into the regime Λ QCD ≪ q ⊥ ≪ Q, and we demonstrated in Ref. [11] that the result of this extrapolation is identical to that obtained using the TMD approach. In this sense, we have unified the two mechanisms widely held responsible for the observed SSAs.
In the present letter, we extend our previous calculations to the case of semi-inclusive hadron production in lepton-hadron deep inelastic scattering (SIDIS) [16], e(ℓ) + p(P ) → e(ℓ ′ ) + h(P h ) + X, which proceeds through exchange of a virtual photon with momentum q µ = ℓ µ − ℓ ′ µ and invariant mass Q 2 = −q 2 . A similar calculation has also been reported in [17] in the quark-gluon correlation approach, where the twist-three effects in the parton distribution [7] as well as in the fragmentation function [18] were considered, and they focused on the SSAs at large transverse momenta, P h⊥ ∼ Q, with P h⊥ the transverse momentum of the final-state hadron in the "hadron frame" defined below. Our present calculation extends the work of [17]. In particular, only the so-called "derivative" contributions were taken into account in [17]. These may or may not dominate the spin-dependent cross section at large P h⊥ ∼ Q. When P h⊥ ≪ Q, there are definitely other equally important contributions, which we will calculate. More importantly, we are interested in hadron production at intermediate transverse momenta, Λ QCD ≪ P h⊥ ≪ Q, where we will compare the predictions from the two mechanisms. At first sight, this additional verification of the consistency of the two mechanisms might appear to be of rather limited interest. However, there are several reasons why we believe that this is a valuable addition. Foremost, the SIDIS process is of greater current interest experimentally than Drell-Yan, with several experiments producing data for SSAs in lepton scattering [2]. We stress that, apart from clarifying the theoretical description of SSAs, our work also provides a detailed scheme for the practical analysis of single-spin asymmetries, since it addresses the asymmetries over the whole kinematic regime of transverse momentum. Secondly, as is well-known by now, the peculiar gauge-dependence properties of the Sivers functions [19,20,21] predict a sign change of the functions when going from the Drell-Yan process to SIDIS. It is important to verify this sign change in an explicit calculation of a physical process, and our way of doing this is to confront our earlier Drell-Yan calculation with that for SIDIS. This provides a test of the QCD factorization and of the (non-)universality of spin-dependent TMD parton distributions.
The presentation of this paper will very closely follow our previous work. We will start by calculating the SSA for SIDIS at large transverse momentum of the produced hadron, P h⊥ ∼ Q. We will then expand the obtained result for P h⊥ ≪ Q, in order to make contact with the expression provided by TMD factorization [14], and we will verify that also for SIDIS both approaches contain the same physics in the region Λ QCD ≪ P h⊥ ≪ Q. For this to hold true, the sign change mentioned above is vital. In this paper, we focus entirely on the single-spin asymmetries coming from the quark-gluon correlation function and/or the Sivers functions in the polarized proton (referred to as the "Sivers-type" SSA in the following) in SIDIS. There are also other contributions to the SSA, for example associated with the quark-gluon correlation in the fragmentation functions and the so-called Collins effect [17,22]. We reserve the study of these for future work.

2.
We start by calculating the single-transverse-spin dependent differential cross section for SIDIS at large transverse momentum P h⊥ of the produced hadron (defined below in the "hadron frame"), where the ETQS formalism is relevant. The differential cross section may be calculated from the formula where α em is the electromagnetic coupling and x B ≡ Q 2 /2P ·q, z h ≡ P ·P h /P ·q, y ≡ P ·q/P ·ℓ.
We also introduce S ep = (P + ℓ) 2 , the electron-proton center of mass energy squared. L µν and W µν are the leptonic and hadronic tensors, respectively. The latter depends on the transverse proton spin vector, S ⊥ . We consider scattering of unpolarized leptons by virtualphoton exchange, in which case the leptonic tensor is given by The hadronic tensor has the following expression in QCD: where J µ is the quark electromagnetic current and X represents all other final-state hadrons other than the observed particle h.
It is convenient to write the momentum of the virtual photon in terms of the incoming and outgoing hadron momenta in SIDIS, with q µ t transverse to the momenta of the initial and final hadrons, q µ t P µ = q µ t P hµ = 0. q t is a space-like vector; we define The hadronic tensor W µν in Eq. (3) can be decomposed in terms of five parity and current conserving tensors V µν i [16]: where the W i are structure functions which may be projected out from W µν by W i = W αβṼ αβ i , with the corresponding inverse tensorsṼ i [16]. Both V i andṼ i can be constructed from four orthonormal basis vectors [16]: with q ⊥ ≡ q 2 ⊥ and normalizations In the following, we will only consider the contributions associated with the tensor V 1 . While the tensor V 5 does not contribute when contracted with a symmetric L µν , the other three tensors make contributions that may also be relevant at large transverse momentum P h⊥ ∼ Q, and which need to be included in phenomenological analyses [17]. However, as we discussed in the Introduction, we are primarily interested in this paper in hadron production in an intermediate transverse momentum region, Λ QCD ≪ P h⊥ ≪ Q, and in the connection between the ETQS mechanism and the TMD factorization approaches, and we therefore want to investigate the limit P h⊥ ≪ Q of the ETQS result. In that limit, V 1 alone provides the leading behavior. This is known from the literature [16] for the unpolarized cross section, and we have verified it by explicit calculation for the (Sivers-type) single-transverse-spin dependent polarized cross section. The tensors V 1 andṼ 1 are given by [16] V µν The definitions (7) for the coordinate vectors still leave freedom to associate the axes with specific momentum directions. In the following, we will perform our calculations in the so-called hadron frame, where the virtual photon and target proton are taken to have a spatial component only in the z-direction [16]: where the light-cone momenta are defined as P ± = (P 0 ± P 3 )/ √ 2, and p µ = (1 + , 0 − , 0 ⊥ ), n µ = (0 + , 1 − , 0 ⊥ ) are two light-like vectors with p · n = 1. Usually one chooses the photon to have a vanishing energy component, corresponding to P + = Q/ √ 2x B . In the hadron frame, the final state hadron will have the momentum where z h has been defined above. Using the expression for q µ in (4), one can show that in this frame q ⊥ = P h⊥ /z h with P h⊥ = P 2 h⊥ . The differential unpolarized and single-transversespin dependent cross sections will be calculated in terms of q ⊥ , which will immediately give their dependence on P h⊥ . In the following, we will use both q ⊥ and P h⊥ when discussing the transverse momentum in SIDIS, keeping in mind that they are essentially the same in the hadron frame.
Substituting the tensors in Eq. (8) into (6) and into the formula (1) for the differential cross section, we obtain At large transverse momentum of the final state hadron, we may use collinear factorization and compute W 1 in terms of parton distribution functions, fragmentation functions for the produced hadron, and hard partonic cross sections that may be calculated using QCD perturbation theory. The lowest-order (LO) contributions to the latter arise from the processes γ * q → qg and γ * g → qq. Again, as we will eventually be interested in the extrapolation of our results to P h⊥ ≪ Q, we focus on the part that will dominate at small P h⊥ , namely the channel γ * q → qg. We then find for the unpolarized SIDIS cross section: where the contributionσ unp associated with the tensor structure V 1 has been given in the literature [16]:σ Here the variables ξ andξ are defined as ξ = x B /x andξ = z h /z, with x and z the initial-and final-state partonic momentum fractions, respectively. q(x) denotes the quark parton distribution function, andq(z) the fragmentation function for a quark going into the observed hadron. We have for simplicity suppressed their dependence on a factorization scale, and also a sum over all quark and anti-quark flavors in (12). Finally, α s denotes the strong coupling constant, and C F = 4/3.
The main objective of this paper is to calculate the single-transverse-spin dependent cross in the polarized proton. At large transverse momentum P h⊥ ≫ Λ QCD , the corresponding SSA is generated by the ETQS mechanism in terms of twist-three transverse-spin dependent quark-gluon correlation functions [7]. The difference between the physics of the unpolarized cross section and transverse-spin dependent one is that the latter involves an additional polarized gluon from the polarized proton, which interacts with partons in the hard part, in accordance with the twist-three nature of the observable. In Fig. 1, we show a generic Feynman diagram for such a contribution. The lower shaded oval of the diagram represents the transverse-spin-dependent quark-gluon correlation function for the polarized-proton target and is defined as [7]: P S|ψ (0) where the sums over color and spin indices are implicit, |P S denotes the proton state, ψ the quark field, and F + α the gluon field tensor. In Eq. (14), x 1 = k + q1 /P + and x 2 = k + q2 /P + are the fractions of the polarized proton's light-cone momentum carried by the initial quark lines in Fig. 1, while x g = k + g /P + = x 2 − x 1 is the fractional momentum carried by the gluon; L is the light-cone gauge link, L(ζ 2 , ζ 1 ) = exp −ig ζ 1 ζ 2 dξ − A + (ξ − ) , that makes the correlation operator gauge-invariant, and ǫ αβ ⊥ is the 2-dimensional Levi-Civita tensor with ǫ 12 ⊥ = 1.
The strong interaction phase necessary for having a non-vanishing SSA arises from the interference between an imaginary part of the partonic scattering amplitude with the extra gluon and the real scattering amplitude without a gluon in Fig. 1. The imaginary part is due to the pole of the parton propagator associated with the integration over the gluon momentum fraction x g . Depending on which propagator's pole contributes, ∆σ(S ⊥ ) may get contributions from x g = 0 ("soft-pole") [7] and x g = 0 ("hard-pole") [11,23,24]. When we calculate the partonic scattering amplitudes, we have to attach the polarized gluon to any propagator of the hard part represented by the light circles in the diagram of Fig. 1.
In particular, if the polarized gluon attaches to the outgoing quark in the final state, the on-shell propagation of the quark line will generate a soft gluonic pole. A hard pole arises when internal quark propagators go on-shell with nonzero x g . In Figs. 2 and 3 we show the relevant soft-and hard-pole partonic diagrams, respectively. There are a total of eight diagrams contributing to the soft-pole part, four of which we show in Fig. 2. The remaining four diagrams can be obtained by attaching the gluon on the right side of the cut. There are twelve diagrams for the hard-pole contributions, and again only half of them are shown in Fig. 3. We note that only diagrams with an s-channel quark propagator can have a hard pole. All diagrams in Figs. 2 and 3 are crossed versions of the ones needed for the SSA in the Drell-Yan process considered in [11].
The calculations of the soft-pole and hard-pole contributions follow the same procedure as we used for the Drell-Yan process [11]. We only give a brief outline here and refer the reader The contribution to the single-transverse-spin asymmetry arises from terms linear in k g⊥ in the expansion of the partonic scattering amplitudes. One important contribution to the k g⊥ expansion comes from the on-shell condition for the outgoing "unobserved" gluon, whose momentum depends on k g⊥ . This leads to a term involving the derivative of the correlation function T F . In addition, the soft and hard poles in the diagrams may also arise as double poles [7], which will lead to a derivative contribution as well. The hard-pole contributions by the individual diagrams in Fig. 3 also give derivative terms. However, the derivative contributions cancel out in their sum, similar to what we found for the Drell-Yan case in [11]. For example, the derivative contribution from Fig. 3(a) is canceled out by part of 3(b), 3(c) by another part of 3(b). The remaining contributions contain only non-derivative terms. We note that in order to obtain the correct result for the hard-pole contributions it is crucial to sum only over physical polarization states of the "unobserved" gluon in the Feynman diagrams.
Combining the contributions by all the diagrams, we find for the single-transverse-spin dependent cross section: whereσ unp has been defined in Eq. (13). Again, we have kept only the contribution associated with the tensor structure V 1 . All other terms have been neglected, because they are suppressed by q ⊥ /Q in the limit of q ⊥ ≪ Q. Similar to the Drell-Yan process, the hard part for the derivative term is proportional to the unpolarized cross section. The last term in the above equation comes from the hard-pole contributions. As one can see, these are characterized by a dependence on the quark-gluon correlation function T F (x, x B ), unlike the soft-pole ones which enter with two identical momentum fractions in T F . We note in passing that we have also performed all calculations in a frame where the initial proton and the produced hadron are collinear and move in the z-direction. We found identical results for both the soft-pole and the hard-pole contributions.
We point out that the derivative contribution in Eq. (15) agrees with that derived in [17].
Our non-derivative terms for the soft-pole and the hard-pole contributions are new, however.
We emphasize that even though the derivative contribution is expected to dominate in some kinematic situations [7], the non-derivative parts become of equal importance for q ⊥ ≪ Q, as we shall see shortly. Since it is our goal in this paper to match the result obtained within the ETQS formalism to the one based on TMD factorization, it is crucial that we keep the non-derivative parts. This is also to be seen in the context that the bulk of the SIDIS event rate in experiment is generally located at relatively modest q ⊥ .
We also note that the angular correlation between the observed hadron's transverse momentum P h⊥ and the target proton's polarization vector S ⊥ as shown in Eq. (15) is characteristic of the contribution from the quark-gluon correlation in the proton. Other contributions, like the twist-three quark-gluon correlation in the fragmentation function, will lead to a different angular correlation between these two [17]. With different angular dependence, these contributions can be easily disentangled experimentally (see, e.g., [2]).
The results for the contributions by the structure function W 1 to the unpolarized and the single-transverse-spin dependent cross sections given in Eqs. (12) and (15) are valid when both P h⊥ , Q ≫ Λ QCD . In order to make contact with the TMD factorization formalism, we shall now extrapolate our results into the region of Λ QCD ≪ P h⊥ ≪ Q. This is also the region exclusively dominated by the contributions associated with the tensor V 1 that we have considered. In doing the expansion, we only keep the terms leading in P h⊥ /Q, and neglect all higher powers. For small P h⊥ /Q, the delta function in Eqs. (12) and (15) can be expanded as [25] δ q 2 Inserting this expression into Eq. (12), we find for the small-P h⊥ behavior of the unpolarized differential cross section [25]: Similarly, for the single-transverse-spin dependent cross section, we have where We stress that both soft poles and hard poles contribute to this result. Note that the T F function for the hard-pole contribution reduces to T F (x, x) at ξ = 1. It turns out that this property is crucial for obtaining the correct structure of the small-P h⊥ limit of the cross section, consistent with the TMD factorization. Because the contributions from all tensor structures other than V 1 vanish in the limit of P h⊥ ≪ Q, the above results are the final results for the unpolarized and (Sivers-type) single-transverse-spin dependent cross sections in this kinematical regime.
Comparing the small-P h⊥ behavior in Eqs. (18), (20) to the one we obtained for the Drell-Yan process at low pair transverse momentum q ⊥ ≪ Q [11], we find that the hard partonic parts are the same, with however an opposite sign. This sign difference comes from the fact that in the Drell-Yan SSA the strong interaction phase arises from initial-state interactions, while in DIS it is due to final-state interactions. Of course, the real physical asymmetries will also depend on the size of the unpolarized quark distribution and fragmentation functions and will not just differ by a sign. We note that this universality (up to a sign) of the Drell-Yan and the SIDIS twist-three partonic cross sections only happens at low transverse momentum.
At q ⊥ ∼ Q, there is no connection between the two processes at all. The universality of the partonic hard parts at low transverse momentum is actually a manifestation of the TMD factorization at P h⊥ ≪ Q, and of the universality of the TMD quark distributions and fragmentation functions. We will discuss this in the following section.
3. When P h⊥ ≪ Q, we know that a transverse-momentum-dependent factorization applies [14]. Following this reference, the differential SIDIS cross section may be written as where σ 0 = 4πα 2 em S ep /Q 4 × (1 − y + y 2 /2)x B , and where φ S and φ h are the azimuthal angles of the proton's transverse polarization vector and of the transverse momentum vector of the final-state hadron, respectively. Again, we only keep the terms we are interested in: F U U corresponds to the unpolarized cross section, and F (1) U T to the Sivers function contribution to the single-transverse-spin asymmetry. Other contributions, for example those related to the Collins effect [22], may be incorporated similarly [14]. F U U and F (1) U T depend on the kinematical variables, x B , z h , Q 2 , y, and P h⊥ . According to the TMD factorization formalism, they can be factorized into TMD parton distributions and fragmentation functions, and soft and hard parts. For example, F (1) U U has the following factorized form [14]: where q andq denote the unpolarized TMD quark distributions and fragmentation functions, respectively. H is a hard factor and is entirely perturbative. It is a function of Q ≫ P h⊥ only. The soft-factor S is a vacuum matrix element of Wilson lines and captures the effects of soft gluon radiation. Since the soft-gluon contributions in the TMD distribution and fragmentation have not been subtracted, the soft factor enters with inverse power. We have not displayed the dependence of the TMD quark distribution (fragmentation) functions on the variable ζ 2 = (2v · P ) 2 /v 2 (ζ 2 = (2ṽ · P h ) 2 /ṽ 2 ), which serves to regulate their lightcone singularities. Here, v andṽ are vectors off the light-cone. We finally introduce the soft-gluon rapidity cut-off ρ = (2v ·ṽ) 2 /v 2ṽ2 , on which the soft factor depends. In a special coordinate frame, one may choose x 2 B ζ 2 =ζ 2 /z 2 h = ρQ 2 [14]. There is also explicit renormalization scale dependence of the various factors in the factorization formula which, too, has been omitted for simplicity.
Similarly to Eq. (22), the contribution to the Sivers single-transverse-spin asymmetry can be factorized as whereˆ P h⊥ is a unit vector in direction of P h⊥ and q T is the Sivers TMD quark distribution.
The proton mass M P is used to normalize the Sivers function and the unpolarized TMD quark distribution to the same mass dimension. For the operator definition of the Sivers function, see for example [11].
In order to make contact with the result for the ETQS formalism of the previous section, we compute the various factors in the factorization formulas (22), (23) at large transverse momentum (P h⊥ ≫ Λ QCD ), where their dependence on P h⊥ is perturbative. The unpolarized quark distribution and fragmentation functions at large P h⊥ can be expressed in terms of their respective k ⊥ -integrated distributions, multiplied by perturbatively calculable coefficients. Their expressions are well known (see, for example, Ref. [14]). For the quark distribution function, one has: where q(x) is the integrated quark distribution and ξ = x B /x. Likewise, the TMD quark fragmentation function is given bŷ whereq(z) is the integrated quark fragmentation function andξ = z h /z.
Similarly, the Sivers function at large k ⊥ can also be calculated perturbatively. Because it is (naively) time-reversal-odd, the only contribution comes from the twist-three quark-gluon correlation function T F in Eq. (14). The calculation follows the same procedure as for our calculation for the Drell-Yan process in [11]. The Feynman diagrams are the same, the only difference being that the gauge-link propagators have an opposite sign for their imaginary parts. Carrying out the calculations accordingly, we find where A has been defined in Eq. (19) and where ξ = x B /x. Indeed, as expected [19,20,21], we find that the Sivers function in DIS is the same as that in the Drell-Yan process, but with an opposite sign. As is well-known now [19,20,21], this sign difference comes from the different directions of the gauge links for the two processes: in DIS the gauge link arises from final-state interactions and runs to positive light-cone infinity, while in Drell-Yan it is due to initial-state interactions and goes to −∞.
In order to calculate the explicit P h⊥ -dependence generated by the TMD factorization, we let one of the transverse momenta k ⊥ , p ⊥ , and λ ⊥ be of the order of P h⊥ and the others much smaller. When λ ⊥ is large, for example, we neglect k ⊥ and p ⊥ in the delta function, and the integrations over these momenta yield either the ordinary quark distribution, or a k ⊥ moment of the Sivers function. The latter is related to the twist-three correlation [8]: where the minus sign on the right-hand-side of the second equation is again due to the direction of the DIS gauge link. In case λ ⊥ is neglected in the delta function, one makes use of the relation [14] d 2 λ ⊥ S(λ ⊥ ) = 1. We then obtain the following results for the unpolarized and single-transverse-spin dependent cross sections: where A and B are defined as in Eqs. (19), (20). It is evident that the above results reproduce the differential cross sections in Eqs. (17), (18).

4.
In conclusion, we have demonstrated in this paper that the two mechanisms for the Sivers-type single-transverse-spin asymmetry in semi-inclusive deeply-inelastic scattering are consistent at moderate transverse momentum, Λ QCD ≪ P h⊥ ≪ Q. This provides an additional test of the unification of the mechanisms discussed in [11]. It will be important to carry out a relevant experimental test of this unification. Furthermore, our calculation also explicitly exemplifies the process-dependence of the functions generating singletransverse-spin asymmetries. We finally note that another interesting SSA phenomenon in semi-inclusive DIS processes is associated with the so-called Collins effect [22]. A similar connection between the twist-three quark-gluon correlation mechanism in fragmentation [18] and the Collins function should exist. An extension to this case would be very interesting.