Double-spin cos(phi) Asymmetry in Semi-inclusive Electroproduction

We consider the double-spin cos(phi) asymmetry for pion electroproduction in semi-inclusive deep inelastic scattering of longitudinally polarized leptons off longitudinally polarized protons. We estimate the size of the asymmetry in the approximation where all twist-3 interaction-dependent distribution and fragmentation functions are set to zero. In that approximation at HERMES kinematics a sizable negative cos(phi) double-spin asymmetry for pi+ electroproduction is predicted.


I. INTRODUCTION
Semi-inclusive deep inelastic scattering (SIDIS) of leptons off a nucleon is a suitable process to extract information on the quark-gluon structure -correlations between the spins of hadron or quark/gluon and the momentum of the quark/gluon with respect to that of the hadron. The parton intrinsic transverse momentum allows in the SIDIS cross section particular non-perturbative correlations, which can be probed in measurements of azimuthal asymmetries. The complete tree-level result for the SIDIS cross section in terms of distribution (DF) and fragmentation (FF) functions at leading and subleading order in 1/Q has been given in Ref. [1]. In particular the combination with T-odd fragmentation functions leads to single-spin asymmetries. The HERMES collaboration has recently reported on the measurement of such single target-spin asymmetries in the distribution of the azimuthal angle φ of produced pions relative to the lepton scattering plane, in semi-inclusive charged and neutral pion production on a longitudinally polarized hydrogen target [2,3].
Other consequences of non-zero intrinsic transverse momentum of partons are the spin-independent cos φ and cos 2φ asymmetries [4] and double-spin azimuthal asymmetries [1,5]. In this paper we investigate the specific cos φ azimuthal asymmetry at order 1/Q in semiinclusive π + production with longitudinally polarized electrons on longitudinally polarized protons.
The kinematics of SIDIS is illustrated in Fig. 1: k 1 (k 2 ) is the 4-momentum of the incoming (outgoing) charged lepton, Q 2 = −q 2 , where q = k 1 − k 2 , is the 4-momentum of the virtual photon. The momentum P (P h ) is the momentum of the target (observed) hadron. The scaling variables are x = Q 2 /2(P · q), y = (P · q)/(P · k 1 ), and z = (P · P h )/(P · q). The momentum k 1T is the incoming lepton transverse momentum with respect to the virtual photon momentum direction, and φ is the azimuthal angle between P h⊥ and k 1T . We will consider the case of a polarized beam, the helicity being denoted by λ e . Note that for the specific case in which the target is polarized parallel (anti-parallel) to the beam a transverse spin in the virtual photon frame arises which only can have azimuthal angle 0 (π). The value of this transverse spin component is [6] where θ γ is the virtual photon emission angle and S is target polarization parallel/antiparallel to the incoming lepton momentum. The quantity sin θ γ is of order 1/Q and given by where M is the nucleon mass. We will distinguish the situations by referring to (LL) lab which will produce both LL and LT polarization in the virtual photon frame.

II. THE SEMI-INCLUSIVE CROSS SECTION
The cross section for one-particle inclusive deep inelastic scattering is given by The quantity L µν is the well-known lepton tensor. The full expression for the symmetric and antisymmetric parts of the hadronic tensor W µν at leading 1/Q order are given by Eqs. (77), (78) of Ref. [1]. In order to investigate the cosφ azimuthal asymmetry we keep only the terms producing contributions in the cross section 1 that are φ-independent or proportional to cos φ where {µ ν} indicates symmetrization of indices and [µ ν] indicates anti-symmetrization. In the above expression we have used the shorthand notation g 1s and similarly for h ⊥ 1s . The contraction of leptonic and hadronic tensors leads to the cross section with the following terms where Here by k T x (p T x ) we denote the x component of the final (initial) parton transverse momentum vector.

III. THE CORRELATION FUNCTIONS
In the asymmetries considered in this paper a number of functions appear beyond the well-known leading twist DF's f q 1 , g q 1 and h q 1 and the FF D q 1 . Note that we do not consider polarization in the fragmentation process. These additional functions are • The DF's g q 1T and h ⊥ q 1L , interpreted as distributions of longitudinally and transversely polarized quarks (of flavor q) in transversely and longitudinally polarized nucleons, respectively. The most interesting p T -integrated functions in these case are the transverse moments • The DF's f ⊥ q and g q L appear at subleading (1/Q) order in the above expression.
• The FF's D ⊥ q and E q appear at subleading (1/Q) order.
For all of these functions the relevant transverse moments can be expressed into the leading twist functions f 1 , g 1 , and h 1 and interaction-dependent functions, indicated with a tilde. The relations are of the same type as the Wandzura-Wilczek relation [7] for the subleading function g q T measured in inclusive leptoproduction with a transversely polarized target.
The relations needed in our case and details on them are found in Refs. [1,8]. For DF one needs h ⊥(1) while for DF we need The approximation that we will use below consists in setting all interaction dependent (tilde) functions to zero. There is in fact no justification for this, except the observation that the same approximation for g q T (the Wandzura-Wilczek approximation (WW)) seems to work well [9]. We want to add another point concerning potential contributions in the cross section that have already been neglected, namely those proportional to α s (Q 2 ). Contributions proportional to α s (Q 2 ) . . . f 1 . . . will likely appear at the same point where the function (M/Q) . . . f ⊥ appears [10], while contributions proportional to α s (Q 2 ) . . . g 1 . . . will likely appear at the same point where the function (M/Q) . . . g ⊥ L appears.

IV. WEIGHTED CROSS SECTION
We will consider the differential cross section integrated over the transverse momentum of the produced hadron with different weights and denote them by [11,12] where W = W (P h⊥ , φ, φ S ). With the subscripts AB we denote the polarization of lepton and target hadron, respectively. We use U for unpolarized, L for longitudinally polarized and T for transversely polarized particles. From Eq. (7) we then obtain a number of asymmetries.
For each of them we have indicated the results after setting all interaction dependent functions equal to zero, i.e. only keeping the twist-2 functions. The results are The particular cos φ moment in the SIDIS cross section for which we will give an estimate is the following weighted integral of a cross section asymmetry, Here σ ++ , σ −− (σ +− , σ −+ ) denote the cross section with antiparallel (parallel) polarization of the beam and target respectively 2 . They are given by dσ 3 LT with φ S = π(0) for σ ++ and σ −+ (σ −− and σ +− ), respectively. The quantity M h is the mass of the final hadron. Using the Eqs. (16)-(23) and assuming 100% beam and target polarization one obtains For the experimentally measured cross sections that is determined without weighing with the transverse momentum of the produced hadron we use a further approximation, For the numerical estimate of A cos φ (LL) lab asymmetry we use the approximation, where only the twist-2 distribution and fragmentation functions are used, i.e. the interaction-dependent twist-3 parts are set to zero. It is important to point out that in this approximation the cos φ asymmetry reduces to a kinematical effect conditioned by intrinsic transverse momentum of partons similar to the cos φ asymmetry in unpolarized SIDIS [4]. Assuming a Gaussian parameterization for the distribution of the initial parton's intrinsic transverse momentum, p T , in the helicity distribution function g 1 (z, p 2 T ) one can get It is worth to note that these approximations lead to similar results obtained in the simple quark-gluon model with non-zero intrinsic transverse momentum in polarized SIDIS [5]. To estimate the transverse asymmetry contribution dσ 3 LT into the A cos φ (LL) lab , we proceed in the same way as in the Ref. [13].
In Fig. 2, the asymmetry A cos φ (LL) lab (x) of Eq.(26) for π + production on a proton target is presented as a function of x-Bjorken. The curves are calculated by integrating over the HERMES kinematic ranges corresponding to 1 GeV 2 ≤ Q 2 ≤ 15 GeV 2 , 4.5 GeV ≤ E π ≤ 13.5 GeV, 0.2 ≤ z ≤ 0.7, 0.2 ≤ y ≤ 0.8, and taking P h⊥ = 0.365 GeV as input. The latter value is obtained in this kinematic region assuming a Gaussian parameterization of the distribution and fragmentation functions with p 2 T = (0.44) 2 GeV 2 [14]. For the sake of simplicity, Q 2 -independent parameterizations were chosen for the distribution, g 1 (x) [15], and fragmentation, D 1 (z) [16], functions.
From Fig. 2 one can see that the approximation where all twist-3 DF's and FF's are set to zero gives the large negative double-spin cos φ asymmetry at HERMES energies. The 'kinematic' contribution to A cos φ (LL) lab (x) coming from the transverse component of the target polarization is small (up to 25% at large x-Bjorken).

V. CONCLUSION
The cos φ double-spin asymmetry of SIDIS of longitudinally polarized electrons off longitudinally polarized protons was investigated. We only kept the (1/Q)-order contribution to the spin asymmetry that arises from intrinsic transverse momentum effects related to twist-two DF and FF similar to the cos φ asymmetry in unpolarized SIDIS. With that approximation, a sizable negative cos φ asymmetry is found for HERMES kinematics. It is shown that the 'kinematical' contribution from target transverse component (S T ) is small. The approximation used to estimate the double-spin cos φ asymmetry is not complete in 1/Q order: it contains only 1/Q 'kinematical' twist-behavior of azimuthal distributions needs the inclusion of higher-twist and pQCD contributions. Nevertheless, if one consider the kinematics with P h⊥ < 1GeV and z < 0.8, the estimate shows the non-perturbative effects from the intrinsic transverse momentum of the partons in the nucleon. The double-spin cos φ asymmetry is a good observable to investigate the importance of leading and sub-leading effects at moderate Q 2 .