Precise Measurements of Beam Spin Asymmetries in Semi-Inclusive $\pi^0$ production

We present studies of single-spin asymmetries for neutral pion electroproduction in semi-inclusive deep-inelastic scattering of 5.776 GeV polarized electrons from an unpolarized hydrogen target, using the CEBAF Large Acceptance Spectrometer (CLAS) at the Thomas Jefferson National Accelerator Facility. A substantial $\sin \phi_h$ amplitude has been measured in the distribution of the cross section asymmetry as a function of the azimuthal angle $\phi_h$ of the produced neutral pion. The dependence of this amplitude on Bjorken $x$ and on the pion transverse momentum is extracted with significantly higher precision than previous data and is compared to model calculations.

We present studies of single-spin asymmetries for neutral pion electroproduction in semi-inclusive deepinelastic scattering of 5.776 GeV polarized electrons from an unpolarized hydrogen target, using the CEBAF Large Acceptance Spectrometer (CLAS) at the Thomas Jefferson National Accelerator Facility. A substantial sin φ h amplitude has been measured in the distribution of the cross section asymmetry as a function of the azimuthal angle φ h of the produced neutral pion. The dependence of this amplitude on Bjorken x and on the pion transverse momentum is extracted with significantly higher precision than previous data and is compared to model calculations.
In recent years it has become clear that understanding the orbital motion of partons is crucial for achieving a more complete picture of the nucleon in terms of elementary quarks and gluons. Parton distribution functions have been generalized to contain information not only on the longitudinal momentum but also on the transverse momentum distributions of partons in a fast moving hadron. Intense theoretical investigations of Transverse Momentum Dependent (TMD) distributions of partons and the first unambiguous experimental signals of TMDs indicate that QCD-dynamics inside hadrons is much richer than what can be learned from collinear parton distributions.
TMDs were first suggested to explain the large transverse single-spin asymmetries observed in polarized hadron-hadron collisions. Since then, two fundamental mechanisms involving transverse momentum dependent distributions and/or fragmentation functions have been identified, which lead to single-spin asymmetries (SSAs) in hard processes: a) internal quark motion as represented by, e.g., the Sivers mechanism [1][2][3][4][5], which generates an asymmetric distribution of quarks in a nucleon that is transversely polarized and b) the Collins mechanism [4,6], which correlates the transverse spin of the struck quark with the transverse momentum of the observed hadron. The 'Sivers-type' mechanism requires non-zero orbital angular momentum of the struck parton together with initial-or final-state interactions via soft-gluon exchange [3][4][5]. This mechanism involves TMD distributions which describe the correlations between the transverse motion of the parton and its own transverse spin or the spin of the initial-or final-state hadron, thereby providing unprecedented information about spin-orbit correlations. * Corresponding author.
Semi-inclusive deep-inelastic scattering (SIDIS) has emerged as a powerful tool to probe nucleon structure and to provide access to TMDs through measurements of spin and azimuthal asymmetries. A rigorous basis for such studies of TMDs in SIDIS is provided by TMD factorization in QCD, which has been established in Refs. [7][8][9] for leading twist 1 single hadron production with transverse momenta being much smaller than the hard scattering scale. In this kinematic domain, the SIDIS cross section can be expressed in terms of structure functions [6,10,11] which are certain convolutions of transverse momentum dependent distribution and fragmentation functions. The analysis of TMDs thus strongly depends on the knowledge of fragmentation functions [12][13][14][15][16].
This Letter reports measurements of single-spin asymmetries in the production of neutral pions by longitudinally polarized electrons scattered off unpolarized protons. The helicity-dependent part (σ LU ) arises from the anti-symmetric part of the hadronic tensor [11]: 1 Each twist increment above leading twist (twist-2) contributes an extra suppression factor of 1/Q . with the structure function: The subscripts LU specify the beam and target polarizations (L stands for longitudinally polarized and U for unpolarized), α is the fine structure constant and φ h is the azimuthal angle between the leptonic and the hadronic planes defined according to the Trento convention [38]. The kinematic variables x, y, and z are defined as: receives contributions from the convolution of twist-2 and twist-3 distribution and fragmentation functions, such as the twist-2 Boer-Mulders DF h ⊥ 1 [39,40], the Collins FF H ⊥ 1 , and the twist-3 DFs e and g ⊥ . The Boer-Mulders DF h ⊥ 1 describes the correlation between the transverse motion of a quark and its own transverse spin, while g ⊥ can be interpreted as a higher twist analog of the Sivers function. Both functions represent spin-orbit correla- interaction-dependent parts of the higher-twist FFs G ⊥ and E, respectively, in which m q is the quark mass. The quantities f 1 and D 1 are the usual unpolarized twist-2 DF and FF, respectively.
The structure function F sin φ h LU in Eq. (2) is higher-twist by nature. Thus, related observables such as beam-spin asymmetries in single-pion production off an unpolarized target can only be accessed at moderate values of Q 2 . Such higher-twist observables are a key for understanding long-range quark-gluon dynamics. They have also been interpreted in terms of average transverse forces acting on a quark at the instant after absorbing the virtual photon [41].
Different contributions to the structure function in Eq. (2) have been calculated, related to both internal quark motion and the Collins mechanisms. Sizable beam SSAs were predicted for pion production [42] with spin-orbit correlations as the dynamical origin. Within this framework, the asymmetry generated at the distribution level is given by either the convolution of the T-odd Boer-Mulders DF h ⊥ 1 with the twist-3 FF E [43], or the convolution of the twist-3 T-odd DF g ⊥ with the unpolarized FF D 1 [44].
In contrast, calculations based on the Collins mechanism, eH ⊥ 1 , predict vanishing beam SSAs for neutral pions [45][46][47]. The surprising characteristic that favored and unfavored Collins FFs are roughly equal in magnitude but opposite in sign, as indicated by the latest measurements from HERMES [22], COMPASS [23] and Belle [37], put the π 0 in a unique position in SSA studies since the π 0 FF is the average of π + and π − FFs. Contributions to the beam SSA related to spin-orbit correlations could thus be studied without a significant background from the Collins mechanism.  Measurements of beam-spin asymmetries in the electroproduction of neutral pions in deep-inelastic scattering are presented from the E01-113 CLAS data set using a 5.776 GeV electron beam and the CEBAF Large Acceptance Spectrometer (CLAS) [48] at Jefferson Laboratory. Longitudinally polarized electrons were scattered off an unpolarized liquid-hydrogen target. The beam polarization was frequently measured with a Møller polarimeter and the beam helicity was flipped every 30 ms to minimize systematic instrumental effects. Scattered electrons were detected in CLAS. Electron candidates were selected by a hardware trigger using a coincidence of the gas Cherenkov counters and the lead-scintillator electromagnetic calorimeters (EC).
Neutral pions were identified by calculating the invariant mass of two photons detected with the CLAS EC and the Inner Calorimeter (IC) [49]. For events with more than two photons, the pair-wise combination of all photons was used. In each kinematic bin, π 0 events were selected by a Gaussian plus linear polynomial fit to the two-photon invariant mass distribution (see Fig. 1). In each φ h bin and for each beam helicity, the combinatorial background was subtracted using the linear component of the fit, and π 0 events were selected within the invariant mass region defined by the mean of the Gaussian ±3σ , as indicated by the vertical lines in Fig. 1.
Deep-inelastic scattering events were selected by requiring  uncertainties. An additional 3% scaling uncertainty arises from the beam polarization measurement and another 3% relative uncertainty from radiative effects which are not included in the band. for the eπ 0 system that are smaller than 1.5 GeV (M x (eπ 0 ) < 1.5 GeV) were discarded to exclude contributions from exclusive processes. A minimum value for the π 0 transverse momentum, P T > 0.05 GeV, ensures that the azimuthal angle φ h is well-defined. The total number of selected eπ 0 coincidences was ≈ 3.0 × 10 6 for the presented z range, 0.4 < z < 0.7, which selects the semi-inclusive region [28].
The beam-spin asymmetry A LU (φ h ) has been calculated for each kinematic bin as: where P = 0.794 ± 0.024 is the absolute beam polarization for this data set and N + π 0 and N −  Fig. 2 for a representative kinematic bin.
In Fig. 3, the extracted A sin φ LU moment is presented as a function of P T for different x ranges. The results are summarized in Table 1. Systematic uncertainties, represented by the bands at the bottom of each panel, include the uncertainties due to the background subtraction, the event selection and possible contributions of higher harmonics. The first two contributions were estimated as the difference between the asymmetry moment extracted from data sets obtained with or without background subtraction, and by selecting the π 0 from the combination of all photons in an event or from events with exactly two photons. The contribution of higher harmonics was estimated by employing the fit functions p 0 sin φ h or p 0 sin φ h /(1 + p 1 cos φ h ). The contributions from other harmonics such as sin 2φ h or cos 2φ h were also tested and found to be negligible. All the above contributions were added in quadrature.
An additional 3% scaling uncertainty due to the beam polarization measurements should be added to the above-mentioned systematic uncertainties. Radiative corrections have not been applied.  vs. x compared to that of π + from an earlier CLAS measurement [51]. Uncertainties are displayed as in Fig. 3. For both data sets P T ≈ 0.38 GeV and 0.4 < z < 0.7. The right-hatched and left-hatched bands are model calculations involving solely the contribution from the Collins-effect [47].
However they have been estimated to be negligible for the sin φ h modulation [28,50] with an overall relative accuracy of 3%.
The A sin φ h LU moment increases with increasing P T and reaches a maximum at P T ≈ 0.4 GeV. There is an indication, within the available uncertainties, that the expected decrease of A sin φ h LU at larger P T could start already at P T ≈ 0.7 GeV. As a function of x, A sin φ h LU appears to be flat in all P T ranges shown in Fig. 4. Note, however, that Q 2 varies with x (see Table 1).
The measured beam-spin asymmetry moment for π 0 appears to be comparable with the π + asymmetry from a former CLAS data set [51] both in magnitude and sign, as shown in Fig. 5. For both data sets the average P T is about 0.38 GeV. Also shown are model calculations of A sin φ h LU , as indicated in the figure (righthatched and left-hatched bands), which take only the contribution from Collins-effect eH ⊥ 1 into account [45][46][47]52], suggesting that contributions from the Collins mechanism cannot be the dominant ones. In contrast, preliminary calculations of A sin φ h LU for pions [53], based on the models from Refs. [14,54], demonstrate a non-zero contribution from g ⊥ . Because this DF can be interpreted as the higher-twist analog of the Sivers function, it underscores the potential of beam SSAs for studying spin-orbit correlations. factor Q / f (y) versus x from CLAS and HERMES [20]. The 0.4 GeV < P T < 0.6 GeV range of the CLAS data is used to compare with HERMES, because this yields average kinematics closest to HERMES.
Beam SSAs for charged and neutral pions were also measured by the HERMES Collaboration at a higher beam energy of 27.6 GeV [20]. After taking into account the kinematic factors in the expression of the beam-helicity-dependent and independent terms [11] CLAS and HERMES measurements are found to be consistent with each other as shown in Figs. 6 and 7, indicating that at energies as low as 4-6 GeV, the behavior of beam spin asymmetries is similar to higher energy measurements. For comparison, CLAS data in the range 0.4 GeV < P T < 0.6 GeV are used in Fig. 6 and in the range 0.1 < x < 0.2 in Fig. 7, because these ranges yield average kinematic values similar to HERMES. The CLAS data provide significant improvements in the precision of beam SSA measurements for the kinematic region where the two data sets overlap, and they extend the measurements to the large x region not accessible at HERMES.
In summary, we have presented measurements of the kinematic dependences of the beam-spin asymmetry in semi-inclusive π 0 electroproduction from the E01-113 CLAS data set. The sin φ h amplitude was extracted as a function of x and transverse pion momentum P T , for 0.4 < z < 0.7. The asymmetry moment shows no significant x dependence for fixed P T . Note, however, that Q 2 factor Q / f (y) versus P T from CLAS and HERMES [20] (the same as in Fig. 6). The 0.1 < x < 0.2 range of the CLAS data is used to compare with HERMES, as this yields average kinematics closest to HERMES.
varies with x (see Table 1). The observed asymmetry moment for π 0 suggests that the major contribution to the pion beam SSAs originate from spin-orbit correlations.
The results are compared with published HERMES data [20]. They provide a significant improvement in precision and an important input for studies of higher-twist effects. Measured beam SSAs are in good agreement, both in magnitude and kinematic dependences, with measurements at significantly higher energies [20,25].