On the cos 2 phi asymmetry in unpolarized leptoproduction

We investigate the origin of the cos 2 phi azimuthal asymmetry in unpolarized semiinclusive DIS. The contributions to this asymmetry arising from the intrinsic transverse motion of quarks are explicitly evaluated, and predictions for the HERMES and COMPASS kinematic regimes are presented. We show that the effect of the leading-twist Boer-Mulders function h_1^perp(x, k_T^2), which describes a correlation between the transverse momentum and the transverse spin of quarks, is quite significant and may also account for a part of the cos 2 phi asymmetry measured by ZEUS in the perturbative domain.


Introduction
The importance of the transverse-momentum distributions of quarks for a full understanding of the structure of hadrons has been widely recognized in the last decade [1,2,3,4].In semi-inclusive deep inelastic scattering (SIDIS), the k T -dependent distributions give rise to various azimuthal and/or single-spin asymmetries, which are currently under direct experimental scrutiny [5,6].Two leading-twist distributions of great relevance for their phenomenological implications are the Sivers function f ⊥ 1T (x, k 2 T ) [7] and its chirally-odd partner h ⊥ 1 (x, k 2 T ), the so-called Boer-Mulders function [4].These two distributions describe time-reversal odd correlations between the intrinsic momenta of quarks and transverse spin vectors [8].In particular, f ⊥ 1T represents an azimuthal asymmetry of unpolarized quarks inside a transversely polarized hadron, whereas h ⊥ 1 represents a transverse-polarization asymmetry of quarks inside an unpolarized hadron.Recently, it has been proven by a direct calculation [9] that f ⊥ 1T and h ⊥ 1 are non-vanishing: interference diagrams with a gluon exchanged between the struck quark and the target remnant generate non-zero asymmetries.The presence of a quark transverse momentum smaller than Q ensures that these asymmetries are proportional to M/k T , rather than to M/Q, and therefore are leading-twist quantities.Moreover, a careful consideration of the Wilson-line structure of k T -dependent parton densities shows that f ⊥ 1T and h ⊥ 1 are not forbidden by time-reversal invariance [10,11] (for a possible chiral origin of these distributions, see [12]).

The Sivers function f ⊥
1T is known to be responsible for a sin(φ − φ S ) singlespin asymmetry in transversely polarized SIDIS [5,6,13].The Boer-Mulders function h ⊥ 1 produces azimuthal asymmetries in unpolarized reactions.Boer [14] argued that it can account for the observed cos 2φ asymmetries in unpolarized πN Drell-Yan processes [15,16].This was quantitatively confirmed in [17,18], where h ⊥ 1 was calculated in a simple quark-spectator model and shown to explain the Drell-Yan data fairly well.
A similar cos 2φ asymmetry occurs in unpolarized leptoproduction.As we shall see, there are three possible mechanisms generating this asymmetry: 1) non-collinear kinematics at order k 2 T /Q 2 [19]; 2) the leading-twist Boer-Mulders function [4] coupling to a specular fragmentation function, the socalled Collins function [20], which describes the fragmentation of transversely polarized quarks into unpolarized hadrons; 3) perturbative gluon radiation [21,22,23,24].The purpose of this paper is to study the first two sources of the cos 2φ asymmetry, both related to the intrinsic transverse motion of quarks.They are especially relevant in the HERMES kinematic regime ( Q 2 ∼ 2 GeV 2 ), but the Boer-Mulders contribution, being leading twist, can also survive at higher Q 2 and partly account for the asymmetry measured by ZEUS in this domain [25].
In recent years, the cos 2φ asymmetry in leptoproduction was phenomenologically studied by some authors [26,27].In [26] only the O(k 2 T /Q 2 ) term and the perturbative contribution were included, whereas the Boer-Mulders effect was not considered.Our calculation is more similar to that presented in [27], the main differences being that we use a model for h ⊥ 1 adjusted on the Drell-Yan data [18], and compute the asymmetry according to its experimental definition (which incorporates a cutoff on the transverse momentum of the final hadron). 1. Lepton and hadron planes in semi-inclusive deep inelastic scattering.

The cos 2φ asymmetry in unpolarized SIDIS
The process we are interested in is unpolarized SIDIS: The SIDIS cross section is expressed in terms of the invariants where q = ℓ − ℓ ′ and Q 2 ≡ −q 2 .We adopt a reference frame such that the virtual photon and the target proton are collinear and directed along the z axis, with the photon moving in the positive z direction (Fig. 1).We denote by k T the transverse momentum of the quark inside the proton, and by P T the transverse momentum of the hadron h.The transverse momentum of h with respect to the direction of the fragmenting quark will be called p T .All azimuthal angles are referred to the lepton scattering plane (we call φ the azimuthal angle of the hadron h, see Fig. 1).
Taking the intrinsic motion of quarks into account, the SIDIS cross section reads at leading order dσ dx dy dz d where f a 1 (x, k 2 T ) is the unintegrated number density of quarks of flavor a and D a 1 (z, p 2 T ) is the transverse-momentum dependent fragmentation function of quark a into the final hadron.We recall that the non-collinear factorization theorem for SIDIS has been recently proven by Ji, Ma and Yuan [28] for P T ≪ Q.
As shown long time ago by Cahn [19], the transverse-momentum kinematics generates a cos 2φ contribution to the unpolarized SIDIS cross section, which has the form dσ (HT )  dx dy dz where h ≡ P T /P T .Notice that this contribution is of order k 2 T /Q 2 , hence it is a (kinematic) higher twist effect.
The second k T -dependent source of the cos 2φ asymmetry involves the Boer-Mulders distribution h ⊥ 1 coupled to the Collins fragmentation function H ⊥ 1 of the produced hadron.The explicit expression of this contribution to the cross section is [4] dσ (LT )  dx dy dz It should be noticed that this is a leading-twist contribution, not suppressed by inverse powers of Q.
The asymmetry measured in experiments is defined as where the integrations are performed over the measured ranges of x, y, z and with a lower cutoff P c on P T , which is the minimum value of P T of the detected charged particles.Using Eqs. ( 3) and ( 5), cos 2φ h is given by where and (χ is the angle between P T and k T )

Calculation and Results
In order to calculate cos 2φ one needs to know the k T -and p T -dependent distribution and fragmentation functions appearing in Eqs. ( 9)- (11).Independent information on the Boer-Mulders function h ⊥ 1 (x, k 2 T ) can be obtained from the study of the cos 2φ azimuthal asymmetry in unpolarized Drell-Yan processes, which has been measured in πN collisions [15,16].In [17,18] this asymmetry .The dotted curve is the leading-twist Boer-Mulders contribution, the dashed curve is the higher-twist term, the solid curve is the sum of the two contributions.
was estimated by computing the h ⊥ 1 distribution of the pion and of the nucleon in a quark spectator model [29,30].To compute the cos 2φ azimuthal asymmetry in SIDIS we adopt the same distributions h ⊥ 1 (x, k 2 T ) and f 1 (x, k 2 T ) used in [18].We assume that the observables are dominated by u quarks (i.e., we consider π + production).The set of the transverse-momentum dependent distribution functions is (for simplicity, we consider a spectator scalar diquark [30,18]) where N is a normalization constant, m is the constituent quark mass, and Here Λ is a cutoff appearing in the nucleon-quark-diquark vertex and M d is the mass of the scalar diquark.As it is typical of all model calculations of quark distribution functions, we expect that Eqs. ( 12) and ( 13) should be valid at low Q 2 values, of order of 1 GeV 2 .The average transverse momentum of quarks inside the target, as computed from (12), turns out to be k 2 T 1/2 ≃ 0.54 GeV.
Coming to the fragmentation functions, for H ⊥ 1 we adopt the simple parametrization suggested by Collins [20]  where M C is a free parameter.We assume a Gaussian dependence for the unintegrated unpolarized fragmentation function: Finally, the integrated unpolarized fragmentation function for pions D 1 (z) is taken from the Kretzer-Leader-Christova parametrization [31], valid at Q 2 = 2.5 GeV 2 .For the parameters in Eqs. ( 12) and ( 13) we choose the values M d = 0.8 GeV, m = 0.3 GeV, Λ = 0.6 GeV, α s = 0.3, which are the same as in [18].As for the parameters in Eqs. ( 15) and ( 16), we fix M C to 0.3 GeV and show results for two values of the average transverse momentum: p 2 T 1/2 = 0.5 GeV and 0.6 GeV (we checked that a variation of The HERMES kinematics is characterized by the following ranges: 0.02 < x < 0.4, 0.1 < y < 0.85, 0.2 < z < 1, Q 2 = 2 GeV 2 .Our predictions for the cos 2φ asymmetry in this regime are displayed in Fig. 2, where we show separately the higher-twist term and the leading-twist Boer-Mulders contribution.For a typical transverse momentum cutoff P c = 0.5 GeV, these two terms are comparable and the predicted asymmetry lies in the range cos 2φ = 0.02−0.04.The x-dependence (with z integrated over the accessible interval) and the z-dependence (with x integrated over the accessible interval) are shown in Figs. 3 and 4, respectively.As one can see, the asymmetry is larger at small x and large z.In Fig. 5 we plot our results for the x-dependent asymmetry (integrated over z) in the COMPASS kinematic domain.The correlation between x and Q 2 is such that the lowest x bin (x = 0.005) corresponds to Q 2 ≈ 1 GeV 2 , whereas the highest x bin in Fig. 5 (x = 0.25) corresponds to Q 2 ≈ 24 GeV 2 .Again, the asymmetry is of order of few percent and decreases with x.
There are available data on the cos 2φ asymmetry in SIDIS coming from the ZEUS experiment [25].The ZEUS kinematic ranges are: 0.01 < x < 0. GeV Fig. 6.The SIDIS cos 2φ azimuthal asymmetry as a function of the cutoff P c in the ZEUS domain.Data are from [25].
higher twist contribution is clearly irrelevant.Since only the Q 2 evolution of the k T moments of h ⊥ 1 is known [32], and not that of h ⊥ 1 itself, we assume for simplicity that the distributions ( 12) and ( 13) scale exactly, i.e. that they are valid for any Q 2 (one should recall however that Sudakov form factors arising from soft gluon contributions may reduce the Boer-Mulders asymmetry at very high Q 2 [33]).The result for the cos 2φ asymmetry in the ZEUS kinematic domain is shown in Fig. 6, where it is compared with the experimental data.The agreement is rather good for low values of the P T cutoff (up to 0.5 GeV).For larger P T values one expects of course a relevant perturbative contribution.Including this contribution is beyond the purpose of this paper, which is primarily devoted to predictions for the low-Q 2 domain.A more extended analysis of the cos 2φ asymmetries, taking into account also the perturbative term, is in progress and will be reported soon [34].
For completeness recall that long time ago the European Muon Collaboration at CERN measured cos 2φ for Q 2 > 4 GeV 2 [35].The EMC data, however, are affected by large uncertainties and do not allow drawing definite conclusions about the magnitude and the shape of the asymmetry.The comparison of our predictions with these data is shown in Fig. 7.
In conclusion, we predicted the cos 2φ asymmetry for semi-inclusive deep inelastic scattering in the kinematic regions of the HERMES and COMPASS experiments.We found that cos 2φ is of order of few percent and tends to be larger in the small-x and large-z region.The combined analysis of the future data on cos 2φ and of the previous ZEUS measurements in the high-Q 2 domain (where higher twist effects are irrelevant) will allow to get information on the Boer-Mulders function, shedding light on the correlations between transverse spin and transverse momenta of quarks.

Fig. 2 . 2 T 1 / 2
Fig.2.The SIDIS cos 2φ azimuthal asymmetry in the HERMES domain as a function of the cutoff P c , for two values of p 2 T 1/2 .The dotted curve is the leading-twist Boer-Mulders contribution, the dashed curve is the higher-twist term, the solid curve is the sum of the two contributions.

Fig. 3 .
Fig.3.The SIDIS cos 2φ azimuthal asymmetry in the HERMES domain, as a function of x with P c = 0.5 GeV.The dotted curve is the leading-twist Boer-Mulders contribution, the dashed curve is the higher-twist term, the solid curve is the sum of the two contributions.

Fig. 4 .Fig. 5 .
Fig.4.The SIDIS cos 2φ azimuthal asymmetry in the HERMES domain, as a function of z with P c = 0.5 GeV.The dotted curve is the leading-twist Boer-Mulders contribution, the dashed curve is the higher-twist term, the solid curve is the sum of the two contributions.