 Longitudinal double-spin asymmetries in semi-inclusive deep-inelastic scattering of electrons and positrons by protons and deuterons 

A comprehensive collection of results on longitudinal double-spin asymmetries is presented for charged pions and kaons produced in semi-inclusive deep-inelastic scattering of electrons and positrons on the proton and deuteron, based on the full HERMES data set. The dependence of the asymmetries on hadron transverse momentum and azimuthal angle extends the sensitivity to the flavor structure of the nucleon beyond the distribution functions accessible in the collinear framework. No strong dependence on those variables is observed. In addition, the hadron charge-difference asymmetry is presented, which under certain model assumptions provides access to the helicity distributions of valence quarks.

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Introduction
============

Measurements of deep-inelastic scattering (DIS) with both beam and target longitudinally polarized have provided access to the polarization-dependent structure of the nucleon (e.g., Table I of Ref. @Aidala:2012mv). Semi-inclusive DIS, in which an identified final-state hadron is observed in conjunction with the scattered lepton, have provided enhanced sensitivity through the fragmentation process to quark flavor and hence to individual parton distributions @Adeva:1997qz [@Ackerstaff:1999ey; @Airapetian:2003ct; @Airapetian:2004zf; @Alekseev:2007vi; @Airapetian:2008qf; @Alekseev:2009ac; @Alekseev:2010ub; @Avakian:2010ae; @Jawalkar:2017ube]. Until recently (e.g., Refs. @deFlorian:2008mr [@Leader:2010rb]), interpretation of these measurements was largely carried out within a collinear approximation, one for which the effects of transverse components of parton motion are assumed to be negligible. While yielding substantial knowledge on the longitudinal momentum and polarization structure of the nucleon, such interpretation excludes the rich phenomenology of transverse-momentum dependent (TMD) parton distribution and fragmentation functions @Mulders:1995dh [@Bacchetta07]. In particular, in the limit of small hadron transverse momentum semi-inclusive DIS is sensitive to intrinsic transverse momentum @Bacchetta:2008xw. A detailed theoretical picture has been developed, providing a framework for which semi-inclusive DIS measurements in any configuration of beam and target polarization are related to various combinations of distribution and fragmentation functions @Mulders:1995dh [@Bacchetta07].

If terms that depend on transverse nucleon polarization are neglected, the complete model-independent decomposition of the semi-inclusive DIS cross section in the one-photon–exchange approximation can be expressed in terms of moments of azimuthal modulations @Bacchetta07, $$\begin{aligned}
\frac{{{\rm\,d}}\sigma^{h}}{{{\rm\,d}}x {{\rm\,d}}y {{\rm\,d}}z {{\rm\,d}}P^2_{h\perp} {{\rm\,d}}{\phi}} = \frac{2\pi\alpha^2}{xyQ^2} \frac{y^2}{2(1-\epsilon)} \bigg( 1 + \frac{\gamma^2}{2x}\bigg) &  \nonumber \\
\bigg\{ F_{UU,T}^{h} + \epsilon F_{UU,L}^{h}  + \lambda \Lambda \sqrt{1-\epsilon^2} F_{LL}^{h}~~~~~~~~~~~~~~~~& \nonumber \\
+ \sqrt{2\epsilon} \, \Big[   \lambda \sqrt{1-\epsilon} \, F^{h,\sin \phi}_{LU}  + \Lambda \sqrt{1+\epsilon} \, F^{h,\sin\phi}_{UL}\Big]  & \sin\phi \nonumber \\ 
+ \sqrt{2\epsilon} \, \Big[ \lambda \Lambda \sqrt{1-\epsilon} \, F^{h, \cos \phi}_{LL} +  \sqrt{1+\epsilon} \, F^{h,\cos\phi}_{UU} \Big]  & \cos\phi \nonumber \\
+ \Lambda\epsilon \, F^{h,\sin2\phi}_{UL}   \sin2\phi + \epsilon  \, F^{h,\cos2\phi}_{UU}  \cos2\phi~\bigg\}&.
\label{eqn:big-cross}\end{aligned}$$ The variables $Q^2$, $y$, and $x$ are the negative squared four-momentum of the virtual photon, the fraction of beam energy carried by the virtual photon in the target rest frame, and the Bjorken scaling variable, respectively. Here, $x=Q^2/(2M\nu)$ with $M$ the mass of the proton and $\nu$ the energy of the virtual photon in the target rest frame. These variables are determined from the momentum and angle of the scattered lepton. The angle $\phi$ is the azimuthal angle of the hadron momentum vector **P**$_{h}$ about the virtual-photon direction with respect to the lepton-scattering plane as depicted in Fig. \[fig:paramsDefined\] and defined, e.g., in Ref. @Bacchetta04. The $F^{h,\text{mod}}_{XY,Z}$ of Eq.  represent structure functions whose subscripts denote the polarization of the beam, of the target (with respect to the virtual-photon direction), and—if applicable—of the virtual photon. The superscript indicates the dependence on the hadron type and the azimuthal modulation parametrized. Each of these structure functions is a function of $x$, $Q^2$, $z$, and $P_{h\perp}$, where $z$ is the fraction of the virtual-photon energy carried by the observed final-state hadron (in the target rest frame), while $P_{h\perp}$ is the magnitude of the hadron momentum component transverse to the virtual-photon direction. The helicity of the nucleon in the center-of-mass system of the virtual photon and the nucleon is denoted as $\Lambda$, while $\lambda$ represents the helicity of the beam lepton. Furthermore, the "photon polarization parameter" $\epsilon = \frac{1-y-\frac{1}{4}~\gamma^{2}~y^2}{1-y+\frac{1}{4}~y^{2}~(\gamma^2+2)}$ is the ratio of longitudinal-to-transverse photon flux, where $\gamma= Q/\nu$, and $\alpha$ is the fine-structure constant.

In order to probe the polarization-dependent structure of the nucleon with minimal experimental systematic uncertainties, spin asymmetries are typically measured instead of cross sections. Ideally, cross sections are compared in all combinations of 100% polarized beams (with respect to beam direction) and targets (with respect to virtual-photon direction) to form @Diehl:2005 $$A_{LL}^{h}\equiv 
    \frac{\sigma^{h}_{+-}-\sigma^{h}_{++}+\sigma^{h}_{-+}-\sigma^{h}_{--}}{\sigma^{h}_{+-}+\sigma^{h}_{++}+\sigma^{h}_{-+}+\sigma^{h}_{--}}.
\label{eqn:ALL}$$ Here, $\sigma^{h}_{\lambda\Lambda}$ denotes the cross section in a given configuration of equal and opposite beam and target helicities. In a typical experimental situation of incomplete polarizations of beam and target, the degrees of polarization of the beam and target must be divided out.

The $P_{h\perp}$ dependence of semi-inclusive asymmetries is sensitive to the transverse-momentum contributions from both the partonic structure of the nucleon and the fragmentation process through which final-state hadrons are produced. Transverse-momentum distributions have in recent years become topics of great interest. The focus has been primarily on their relationship to transverse-spin asymmetries. However, even for unpolarized or longindually polarized beam/target, the $P_{h\perp}$ dependence has been shown to be sensitive to various sources of transverse momentum in the nucleon @Anselmino:2006yc [@Avakian:2007xa; @Musch:2010ka].

In the limit of small hadron transverse momentum ($P_{h\perp}\ll zQ$), the various contributions to $A^{h}_{LL}(x,Q^2,z,P_{h\perp},\phi)$ can be expressed in terms of convolutions of TMD distribution with fragmentation functions. The azimuthally uniform $A^{h}_{LL}(x,Q^2,z,P_{h\perp})$ enters with a single leading-twist contribution: $$F^{h}_{LL} \propto \sum_q e_q^2 \bigg[~g_{1L}^q (x,p_T^2) \otimes_{\mathcal{W}_1} D_1^{q\to h}(z,k_T^2)~\bigg].$$ Here, "$\otimes_{\mathcal{W}_1}$" represents a convolution of the distribution and fragmentation functions over the intrinsic transverse momentum $p_T$ of the parton $q$ (with fractional charge $e_q$) and the transverse-momentum contribution $k_T$ from the fragmentation process with a kinematic "weight" $\mathcal{W}_1$. The function $\mathcal{W}_1$ \[and $\mathcal{W}_2$ from Eq. \] is given explicitly, e.g., in Ref. @Bacchetta07. In the collinear limit, $F^{h}_{LL}$ reduces to the well-known product of the collinear helicity distribution $g^{q}_{1}(x)$ and the collinear fragmentation function $D_{1}^{q\to h}(z)$.[^4]

While there are no possible azimuthal moments at leading twist, cosine modulations are potentially present at twist-three level, i.e., suppressed by a single power of $M/Q$. Taking the Wandzura–Wilczek approximation (neglecting interaction-dependent terms, which depend on quark-gluon-quark correlators, and neglecting terms linear in quark masses) @Wandzura:1977qf, the following expression remains: $$\begin{aligned}
F_{LL}^{h,\cos\phi} \propto \frac{M}{Q}\sum_q e_q^2 \bigg[~g_{1 L}^{q} (x,p_T^2) & \otimes_{\mathcal{W}_2} D_1^{q\to h} (z,k_T^2)~\bigg]
\label{eq:subleading-TMD}.\end{aligned}$$ This combination of distribution and fragmentation functions was studied, e.g., in Ref. @Oganessyan:2002pc, and is sometimes referred to as the "polarized Cahn effect", which combines transverse momentum of longitudinally polarized partons inside the target nucleon with transverse momentum produced in the fragmentation process.

The unpolarized denominator of Eq.  has been extensively studied. Kinematic dependences of its azimuthal modulations, which include contributions arising from, e.g., the <span>*Boer–Mulders*</span> @Boer:1997nt and <span>*Cahn*</span> @Ravndal73 [@Kingsley74; @NCahn1978269] effects, have been explored thoroughly in Refs. @Airapetian:2012yg [@Adolph:2014pwc; @Yan:2016ods].

In general, the use of Eq.  to extract information on the nucleon spin structure in terms of parton distributions requires knowledge of the hadronization process. The advantage of such information is a more detailed sensitivity to the various quark flavors than that of purely inclusive DIS.

The hadron charge-difference double-spin asymmetry provides additional spin-structure information and is not trivially constructible from the simple semi-inclusive asymmetries. Under certain symmetry assumptions for fragmentation functions (cf. Sec. \[sec:charge-difference\]) charge-difference asymmetries provide a direct extraction of valence-quark polarizations @Frankfurt:1989wq.

It is the primary goal of this article to present the kinematic dependences of hadron-tagged longitudinal double-spin asymmetries as completely as possible with the available data. In comparison to the analysis of the HERMES unpolarized data presented in Ref. @Airapetian:2012yg, the size of the data presented here did not allow for a complete five-dimensional kinematic unfolding of the data. Decisions were made about the best possible kinematic projections of these data, within the constraints of the theoretical framework described above and with the goal of providing the maximum possible access to physics of interest.

*For the purpose of discussion and comparison* some assumptions will be made in the analysis, but the lepton-nucleon asymmetry $A^{h}_{\parallel}$ \[cf. Eq. \] should be taken to be the primary model-independent observable and is provided in the data tables in all cases. This asymmetry differs from $A^{h}_{LL}$ only in the direction in which the nucleon polarization is measured, either with respect to the beam for the former, or with respect to the virtual photon for the latter. Because of this, $A^{h}_{\parallel}$ contains a relatively small, but non-vanishing component of $A^{h}_{LT}$ @Diehl:2005.[^5] More significantly, in order to relate $A^{h}_{\parallel}$ to the virtual-photon–nucleon asymmetry $A^{h}_1$ \[cf. Eq. \], a parameterization of the longitudinal-to-transverse photoabsorption cross-section ratio $R=\sigma_L/\sigma_T$ must be assumed. To date, this quantity has only been measured in inclusive DIS. However, in semi-inclusive DIS this ratio might depend strongly on the hadron kinematics, in particular on $P_{h\perp}$.

Measurement and analysis
========================

HERMES experiment and analysis formalism
----------------------------------------

The data were collected using the HERMES spectrometer @Ackerstaff:1998av at the HERA storage ring during the 1996–2000 running period. A longitudinally polarized lepton (electron or positron) beam with a momentum of 27.6 GeV was scattered off a longitudinally polarized atomic hydrogen or deuterium gas target. The sign of the target polarization was randomly chosen each 60 s for hydrogen and 90 s for deuterium, providing yields in both spin states while controlling systematic uncertainties. The experimental configurations by year are summarized in Table \[tab:configByYear\]. Typical values for the beam (target) polarization are around 53% (84%).

  ------ -------- ---------- ---------------------- -----------------
           Beam     Target           Hadron          Hadron Momentum
   Year    Type      Gas              Type                $P_h$
   1996   e$^+$       H           $\pi^{\pm}$          4–13.8 GeV
   1997   e$^+$       H           $\pi^{\pm}$          4–13.8 GeV
   1998   e$^-$       D       $\pi^{\pm}, K^{\pm}$      2–15 GeV
   1999   e$^+$       D       $\pi^{\pm}, K^{\pm}$      2–15 GeV
   2000   e$^+$       D       $\pi^{\pm}, K^{\pm}$      2–15 GeV
  ------ -------- ---------- ---------------------- -----------------

  :  Experimental configurations by year of longitudinally polarized beam and target data taking. The varieties of hadrons identified and the hadron-momentum range are determined by the particle-identification systems available at the time. A threshold Cherenkov counter was used during the hydrogen data-taking period and a ring-imaging Cherenkov detector was used throughout the deuterium running period. <span data-label="tab:configByYear"></span>

The asymmetries are computed using basically the same data set and procedure presented in prior HERMES publications on longitudinal double-spin asymmetries @Ackerstaff:1999ey [@Airapetian:2003ct; @Airapetian:2004zf; @Airapetian:2007mh]; differences from previous analyses are discussed below. The lepton-nucleon asymmetry is $$A^{h}_{\parallel}  \equiv \frac{C^{h}_{\phi}}{f_D}\left[\frac{L_{\rightrightarrows}N^{h}_{\rightleftarrows} - 
        L_{\rightleftarrows}N^{h}_{\rightrightarrows}}{L_{P,\rightrightarrows} 
        N^{h}_{\rightleftarrows} + 
        L_{P,\rightleftarrows}N^{h}_{\rightrightarrows}}\right]_\text{B}  .
    \label{eq:a-parallel-born}$$ Here, $N^{h}_{\rightrightarrows(\rightleftarrows)}$ represents the hadron yield containing events that meet the kinematic requirements summarized in Table \[tab:cuts\], and $L_{\rightrightarrows(\rightleftarrows)}$ and $L_{P,\rightrightarrows(\rightleftarrows)}$ represent the luminosity and polarization-weighted luminosity in the parallel (antiparallel) experimental beam/target helicity configuration.[^6] The square brackets, $\left[~\right]_\text{B}$, indicate that the enclosed quantity is corrected to Born level, i.e., unfolded for radiative and detector smearing, using Born and smeared Monte Carlo simulations according to the essentially model-independent procedure described in Ref. @Airapetian:2004zf. The unfolding is carried out in the same dimension used to present the data (see also Section \[sec:results\] and Table \[tab:binning\]). The factor $f_D$ represents the dilution of the polarization of the nucleon with respect to that of the nucleus and is explained in Section \[sec:fD\]. Finally, $C^{h}_{\phi}$ is a correction that compensates for any distortion caused by the convolution of the azimuthal moments of the polarization-independent cross section with the non-uniform detector acceptance, which is described in more detail in Section \[sec:azimcorr\].

   Kinematic Requirements
  ------------------------
   $Q^2  > 1.0$ GeV$^{2}$
     $W^2 > 10$ GeV$^2$
         $y < 0.85$
     (0.1) $0.2<z<0.8$
       $x_{F} > 0.1$

  :  Inclusive and semi-inclusive kinematic requirements (value in parentheses is the limit for the <span>*extended*</span> range discussed in Section \[sec:min-z\]). Here, Feynman-$x$ ($x_{F} $) is defined as the ratio of the hadron's longitudinal momentum component in the virtual-photon–nucleon center-of-mass system to its maximal possible value. <span data-label="tab:cuts"></span>

The virtual-photon–nucleon asymmetry $A_1^{h}$ is defined as $$\begin{aligned}
        A_1^h&\equiv 
        \frac{\sigma^{h}_{1/2}-\sigma^{h}_{3/2}}{\sigma_{1/2}^{h}+\sigma_{ 
        3/2}^{h}}, 
        \label{eq:A1} 
    \end{aligned}$$ where $\sigma^{h}_{1/2}$ ($\sigma^{h}_{3/2}$) is the photoabsorption cross section for photons for which the spin is antiparallel (parallel) to the target-nucleon spin. $A_1^{h}$ is computed from $A^{h}_{\parallel}$ as $$\begin{aligned}
        A_1^h &= \frac{1}{D(1+\eta\gamma)}A_{\parallel}^{h},
        \label{eq:A1exp} 
    \end{aligned}$$ where the contributions from the spin structure function $g_{2}$ and, in case of a deuterium target, from the tensor structure function $b_{1}$ are negligible @Airapetian:2005cb. Furthermore, $$\eta = 
    \frac{\epsilon\gamma y}{1-\left(1-y\right)\epsilon }\label{eq:eta}$$ is a kinematic factor, and $$D = \frac{1-(1-y)\epsilon}{1 + \epsilon R}$$ accounts for the limited degree of polarization transfer at the electron–virtual-photon vertex, including the ratio $R$ of longitudinal-to-transverse cross sections. In this analysis, $R$ was taken from the R1999 parameterization @collaboration-1999-452 for all calculations of $A^{h}_1$, which—strictly speaking—is valid only for inclusive DIS measurements as pointed out above.

Differences from prior analyses {#sec:analysis-differences}
-------------------------------

Although the analysis has much in common with those in prior HERMES publications, several changes are made, which increase statistical precision and reduce the systematic uncertainties.

### Nucleon-polarization correction {#sec:fD}

The factor $f_D$ in Eq.  is the ratio of the polarization of the target nucleon to that of the host nucleus. This value is unity for protons, and 0.926 for deuterons due to the D-state admixture in the deuteron wave function @physRevC.60.035201. The application of this correction directly to the asymmetries differs from the analysis of Ref. @Airapetian:2004zf. In this prior publication, the nucleon polarization correction was applied in a calculation of quark polarizations but not to the asymmetries themselves.

### Minimum-$z$ requirement {#sec:min-z}

As in prior analyses, a constraint on the hadron energy-fraction of $z>0.2$ is applied. Only for the two-dimensional binning performed in $x$ and $z$, an additional low-$z$ bin (0.1$< z <$ 0.2) is added, which provides access to the kinematic behavior of the asymmetry outside the region that is typically used to separate current and target-remnant regions. This bin is omitted, however, from polynomial fits of Table \[tab:xz-chisq\] as the fit is intended as a check in a $z$ range commonly used in global analyses.

### Minimum-$P_h$ requirement

The hadron momentum range accepted is determined by the capabilities of the hadron identification apparatus. For the hydrogen sample, a threshold Cherenkov counter requires a minimum hadron momentum of 4 GeV in order to distinguish charged pions from heavier hadrons. For the deuterium sample the installation of a dual-radiator ring-imaging Cherenkov detector (RICH) @Akopov:2000qi enabled identification of hadrons with momentum larger than 2 GeV. For historical reasons, prior asymmetry analyses required the minimum hadron momentum to be the same for the two targets. This restriction has been relaxed on the deuterium sample as it unnecessarily removes low-momentum hadrons.

### Event-level RICH unfolding

In comparison to prior analyses, the RICH hadron identification algorithm was improved to reconstruct better multi-hadron events @Airapetian:2012yg. For each event, hit patterns are produced for each possible combination of hadron hypotheses so that the effect of all tracks is taken into account simultaneously. Previously each hadron track was identified individually, which increased the probability of misidentification for cases where Cherenkov rings overlapped.

### Multidimensional unfolding

Event migration due to radiative and detector smearing is corrected for in an unfolding procedure as in the previous HERMES analyses. The exploration of multidimensional dependences in this analysis required unfolding not only in $x$ but also in the other variables under study (s. below). However, unlike the case of unfolding of polarization-independent hadron yields in the measurement of hadron multiplicities @Airapetian:2012ki, which are strongly dependent on the hadron kinematics and which thus require also unfolding in those variables, the dependence of longitudinal double-spin asymmetries on hadron kinematics in this analysis is weak and the unfolding procedure is found to be robust against possible model dependence when performed in only those dimensions presented here, e.g., when the polarization-dependent yields are integrated over $z$ or $P_{h\perp}$.

### Azimuthal-acceptance correction {#sec:azimcorr}

The factor $C^{h}_{\phi}$ in Eq.  is a correction applied to the semi-inclusive asymmetries that compensates for the influence of the spectrometer acceptance in the implicit integration over kinematic variables in the semi-inclusive yields. It is primarily the integral over $\phi$, which combines a non-uniform detector acceptance with azimuthal modulations in the polarization-independent yield, produced, for example, by the Cahn effect @NCahn1978269, which distorts the semi-inclusive asymmetries. In practice, the *actual* asymmetry that is measured, $\tilde{A}_{\parallel}^h$, involves a convolution with an acceptance function $\xi(\phi)$, such that $$\begin{aligned}
        \tilde{A}_{\parallel}^h(x&, Q^2, z, P_{h\perp}) \nonumber \\
         &= \frac{\int 
        {{\rm\,d}}\phi~\sigma_{\parallel}^h\left(x,Q^2,z,P_{h\perp},\phi\right)~\xi(\phi) 
        }{\int 
        {{\rm\,d}}\phi~\sigma_{UU}^h\left(x,Q^2,z,P_{h\perp},\phi\right)~\xi(\phi)}. 
        \label{eq:acc} 
    \end{aligned}$$

In order to correct for this effect in the denominator of the asymmetry, a recent parameterization of the azimuthal moments of HERMES unpolarized data @Airapetian:2012yg was used. This parameterization was produced by unfolding unpolarized semi-inclusive yields in all five kinematic degrees of freedom simultaneously. The unfolding was conducted simultaneously in 10800 (5$x~\times$ 5$y~\times$ 6$z~\times$ 6$
    P_{h\perp}~\times$ 12$\phi$) bins, correcting the measured yields for acceptance and smearing effects.

The unpolarized correction factor, $$C_\phi^h =  \frac{\mathcal{A}_{\parallel}^h}{\tilde{\mathcal{A}}_{\parallel}^h},$$ is formed by taking the ratio of two Monte Carlo simulated asymmetries computed in acceptance: $\mathcal{A}_{\parallel}^h$, which is generated without azimuthal modulations (e.g., Cahn and Boer–Mulders effects), and $\tilde{\mathcal{A}}_{\parallel}^h$, which is weighted event-by-event by the parameterized azimuthal modulation of the polarization-independent cross section @Airapetian:2012yg, to reproduce the effect of the non-uniform azimuthal acceptance. By applying this ratio, the unpolarized denominator of the measured asymmetry is corrected for azimuthal acceptance effects. This correction is typically less than a few percent, reaching (and occasionally exceeding) 10% only in the kinematic region of large $x$.

The polarization-dependent numerator of Eq.  is also subject to possible azimuthal modulations, which can enter the cross section at subleading twist @Bacchetta:2008xw. Sizable subleading-twist effects have in fact been observed in unpolarized-beam, longitudinally polarized target asymmetries @Airapetian:2005jc, which underscores the need to proceed with some caution. As the unbinned yields are limited in the dataset for a complete five-parameter kinematic unfolding, a full parameterization of polarization-dependent modulations is not possible, preventing a correction similar to that described for the unpolarized azimuthal acceptance. In order to address this, but also to access these additional degrees of freedom in the polarization-dependent cross section, $A^{h}_{\parallel}$ was unfolded simultaneously in $x$ and $\phi$. The $ \cos\phi $ and $ \cos2\phi $ moments of $A^{h}_{\parallel}$, which will be presented in Section \[sec:azimresults\], were found to be consistent with zero.

### Analytic fits

The two-dimensionally ($x$-$P_{h\perp}$ and $x$-$z$) binned virtual-photon–nucleon asymmetries $A^{h}_{1}$ are simultaneously fit with polynomial functions in both dimensions. This has two significant benefits. First, as kinematic variables are correlated to some degree, fitting provides a means of separating the underlying kinematic dependences of the asymmetries from kinematic correlations. Different hypotheses for kinematic dependence can easily be compared on the basis of their goodness-of-fit. As an example, a weak though non-vanishing dependence on $P_{h\perp}$ of $A^{h}_{1}$ has been suggested by lattice-QCD calculations @Musch:2010ka, which would have to be disentangled from the much stronger dependence on $x$. Second, such fits present a more intuitive picture of the statistical significance of data for which there are large covariances between bins. As is the case when a model-independent radiative and detector-smearing unfolding procedure is applied, some inflation of the on-diagonal error matrix elements occurs. While this causes the uncertainties to appear larger, the effect is compensated for by the statistical correlations between bins @Airapetian:2004zf [@Airapetian:2007mh]. By presenting a fit in addition to the data points with their single-bin uncertainties, the statistical power of the data to constrain models is also conveyed.

Results {#sec:results}
=======

One- and two-dimensional projections of $A_{\parallel}^h$
---------------------------------------------------------

The leading contribution to the longitudinal double-spin asymmetry  is the azimuthally uniform $A_{\parallel}^{h}(x,z,P_{h\perp})$. Traditionally, its collinear version, i.e., integrated over $P_{h\perp}$, has been presented as a function of $x$ only as the dependence on $z$ through the spin-independent fragmentation functions in the numerator and denominator largely cancels. Nevertheless, further information on the underlying interplay of parton distribution and fragmentation functions can be obtained by analyzing the multi-dimensional dependences. They are, in addition, less prone to potential detector effects that arise from integration of the numerator and denominator of Eq.  separately over the larger region of phase space on which the detector acceptance and physics observable might depend.

In this analysis the polarization-dependent experimental hadron yields were corrected for radiative and detector smearing by an unfolding procedure as described in Ref. @Airapetian:2004zf. As pointed out already, in contrast to the earlier analysis, unfolding was performed not only in one dimension, $x$, but also in two or three dimensions. Due to limited yields, the binning in the kinematic variables differs for each of the projections chosen, with the highest resolution in $x$ for the one- and three-dimensional presentations.

<span>lc</span>\
\
\
$x$: & 0.023 – 0.055 – 0.1 – 0.6\
$z$: & 0.1 – 0.2 – 0.3 – 0.4 – 0.5 – 0.6 – 0.7 – 0.8\
\
$x$: & 0.023 – 0.055 – 0.1 – 0.6\
\
\
$x$:  & 0.023 – 0.04 – 0.055 – 0.75 – 0.1 – 0.14 – 0.2 – 0.3 – 0.4 – 0.6\
\
$z$: & 0.2 – 0.35 – 0.5 – 0.8\

The dominating systematic uncertainty stems from the knowledge of both beam and target polarization, and amounts to an average relative uncertainty of 6.6% for the hydrogen and 5.7% for the deuterium data. Contributions to the systematic uncertainties from the RICH as well as acceptance and smearing unfolding were found to be substantially smaller than those. Contributions from the azimuthal-acceptance correction amount up to about 3% at large $x$ while becoming negligible at small $x$. The total systematic uncertainty, quoted in the data tables, is the quadratic sum of all contributions. In the figures they are added in quadrature to the statistical uncertainty.

In order to produce asymmetries in a fine binning in $x$, yields were binned in two dimensions: $x$ and two ranges in $Q^2$. The low-$Q^2$ bin was added, spanning 0.5 to 1 $\text{GeV}^2$, to allow for a better control of migration of events in the unfolding procedure. Likewise, the $x$ region of interest was subdivided into nine bins (see Table \[tab:binning\]), with again additional "padding" bins at low $x$. This quasi two-dimensional binning made it possible to perform kinematic unfolding (as described above) in $x$ and $Q^2$ simultaneously, which compensated for events that migrated from one joint $x$-$Q^2$ bin to another due to QED radiative corrections or detector smearing.

The resulting $x$ dependence of the asymmetries is presented for hydrogen and deuterium targets in Fig. \[fig:lowQ2d\]. The asymmetries extracted were found to be essentially identical to those in prior HERMES analyses @Airapetian:2004zf.

  ---------------------------------------------------------------------------------------------------- ----------------------- ----------------------- ----------------------- ----------------------- --------------------- ---------------------
                                                                                                        $ A^{\pi^{+}}_{1,p} $   $ A^{\pi^{-}}_{1,p} $   $ A^{\pi^{+}}_{1,d} $   $ A^{\pi^{-}}_{1,d} $   $ A^{K^{+}}_{1,d} $   $ A^{K^{-}}_{1,d} $
  \[.5mm\] $\chi^{2~(NDF=16)}_{~C_1^h+C_2^hx}$                                                                  12.6                    10.0                    13.4                     9.1                   10.7                  26.0
  \[3mm\] $\chi^{2~(NDF=15)}_{~C_1^h+C_2^hx+C_3^hz}$                                                            12.2                     6.3                     7.2                     7.2                   10.1                  24.8
  \[3mm\] $\chi^{2~(NDF=12)}_{~C_1^h+C_2^hx+C_3^hz+C_{4}^{h} x^{2} + C_{5}^{h} z^{2} +C_{6}^{h} xz}$            10.3                     4.5                     5.5                     4.8                    5.8                  16.1
  \[.5mm\]                                                                                                                                                                                                                   
  ---------------------------------------------------------------------------------------------------- ----------------------- ----------------------- ----------------------- ----------------------- --------------------- ---------------------

The $z$ dependence of fragmentation functions is in principle quark-flavor dependent. This can result in an additional dependence of $A_{\parallel}^{h}$ on $z$. Nevertheless, the $z$-dependence of longitudinal double-spin asymmetries is a largely unexplored degree of freedom. This is addressed in a two-dimensional analysis, in which the unfolding was performed with a fine $z$ but coarse $x$ binning (see Table \[tab:binning\]). The low-$z$ bin spans the range $0.1<z<0.2$, which is excluded from asymmetries that are integrated over $z$. The resulting $A_{\parallel}^h(z)$ is shown for the three $x$ slices in Fig. \[fig:2DXZ\]. No strong dependence on $z$ is visible, in agreement with results by the COMPASS collaboration for charged-hadron production from longitudinally polarized deuterons @Alekseev:2010dm [@Adolph:2016vou].

To better evaluate any potential $z$ dependence, and in order to avoid, e.g., possible influence of the $y$ dependence of $A^{h}_{\parallel}$ through its kinematic prefactors, $A^{h}_{1}$ was determined from $A^{h}_{\parallel}$ according to Eq. . A set of polynomial functions—one linear in $x$ only, one linear in both $x$ and $z$, and one second order in both variables—was then fit to all 18 data points with correlated uncertainties for each of the resulting $A^{h}_{1}$ asymmetries. It was found that within the precision of the asymmetries, the goodness-of-fit was not significantly improved by including a $z$ dependence. The $\chi^2$ values are given in Table \[tab:xz-chisq\].

![image](Figures/Fig3b-Apar-x-phperp-final){width="67.00000%"}

  ------------------------------------------------------------------------------------------------------------------------------- ----------------------- ----------------------- ----------------------- ----------------------- --------------------- ---------------------
                                                                                                                                   $ A^{\pi^{+}}_{1,p} $   $ A^{\pi^{-}}_{1,p} $   $ A^{\pi^{+}}_{1,d} $   $ A^{\pi^{-}}_{1,d} $   $ A^{K^{+}}_{1,d} $   $ A^{K^{-}}_{1,d} $
  \[.5mm\] $\chi^{2~(NDF=16)}_{~C_1^h+C_2^hx}$                                                                                             12.7                    14.0                    33.7                    22.9                   16.0                  24.4
  \[3mm\] $\chi^{2~(NDF=15)}_{~C_1^h+C_2^hx+C_3^hP_{h\perp}}$                                                                              12.7                    13.9                    31.9                    20.6                   16.0                  23.6
  \[3mm\] $\chi^{2~(NDF=12)}_{~C_1^h+C_2^hx+C_3^hP_{h\perp}+C_{4}^{h} x^{2} + C_{5}^{h} P_{h\perp}^{2} +C_{6}^{h} xP_{h\perp}}$             8.5                     5.1                    29.7                    12.0                   12.2                  18.7
  \[.5mm\]                                                                                                                                                                                                                                              
  ------------------------------------------------------------------------------------------------------------------------------- ----------------------- ----------------------- ----------------------- ----------------------- --------------------- ---------------------

The $x$-$P_{h\perp}$ dependence of $A^{h}_{\parallel}$ is obtained by binning and unfolding in both of these variables simultaneously (see Table \[tab:binning\]), as done for the $x$-$z$ projection of $A^{h}_{\parallel}$. A dependence on the transverse hadron momentum may arise from different average transverse momenta of quarks with their spin aligned to the nucleon spin compared to the case of the spins being anti-aligned. The asymmetries are presented in Fig. \[fig:A1xpt\] as a function of $P_{h\perp}$ for three disjoint $x$ ranges. No strong dependence on $P_{h\perp}$ is visible, consistent with the weak dependences reported by the CLAS @Avakian:2010ae and COMPASS @Alekseev:2010dm [@Adolph:2016vou] collaborations.

In order to evaluate in more detail any potential $P_{h\perp}$ dependence, each of the asymmetries was transformed into a corresponding $A^{h}_{1}$ asymmetry and then fit with a set of polynomial functions as was done for the $x$–$z$ dependence—one linear in $x$ only, one linear in both $x$ and $P_{h\perp}$, and second order in both variables. Again, the goodness-of-fit of these polynomial fit functions, given in Table \[tab:chisq\], shows no clear preference for any of the functional forms used. Figure \[fig:A1xpt-fitcol\] shows as an example $A_{1}^{\pi^+}(P_{h\perp})$ from deuterons in three $x$ ranges as given in the different panels. Uncertainty bands are overlaid for two fits. They are presented to provide a realistic indication of the model-constraining power of these data.

The semi-inclusive asymmetry binned in three dimensions {#sec:3dresults}
-------------------------------------------------------

![ $A_{\parallel, N}^h(x,z,P_{h\perp})$ as a function of $x$ in three different $z$ ranges and three different $P_{h\perp}$ ranges as labeled (see Table \[tab:binning\] for details), with $N=p,d$ denoting the target nucleus and $h=\pi^{\pm},K^{\pm}$ the final-state hadron detected. Data points for the second $P_{h\perp}$ slice are plotted at their average kinematics, while the ones for the remaining $P_{h\perp}$ slices are slightly shifted horizontally for better legibility. The inner error bars represent statistical uncertainties while the outer ones statistical and systematic uncertainties added in quadrature. <span data-label="fig:Apar-p-xzpt"></span>](Figures/Fig5-Apar-3D-prot-deuteron-final-b){width="48.00000%"}

The hadron-tagged longitudinal double-spin asymmetry binned simultaneously in $x$, $z$, and $P_{h\perp}$ as measured by HERMES for hydrogen and deuterium targets are presented in Figs. \[fig:Apar-p-xzpt\]. The asymmetry is binned in a grid with nine bins in $x$, three bins in $P_{h\perp}$, and three bins in $z$ (see Table \[tab:binning\]), and is plotted as a function of $x$ for those ranges in $z$ and $P_{h\perp}$. The binning was selected to populate the bins with statistics as uniformly as reasonable while maintaining a degree of kinematic uniformity across each bin. Within the precision of the measurements, the asymmetries display no obvious dependence on the hadron variables. There is possibly an indication that the non-vanishing asymmetry for $\pi^{-}$ from protons observed in the one-dimensional binning in $x$ (cf. Fig. \[fig:lowQ2d\]) is caused to a large extent by low-$z$ pions. This is in line with expectation considering that disfavored fragmentation, e.g., fragmentation of quark flavors that are not part of the valence structure of the hadron produced, is sizable in that region. As such, $\pi^{-}$ production from up quarks—which carry a large positive asymmetry—may still play a dominant role in that kinematic region compared to larger values of $z$, where disfavored fragmentation will be more and more suppressed.

These data as well as those of the other asymmetry results discussed are available as Supplemental Material @asymData. A statistical covariance matrix is also provided, which describes the uncertainties of the asymmetry in every kinematic bin as well as the degree of correlation between them, which comes about as a result of the unfolding process. This complete covariance information should be included in any derivative calculation as omitting it—that is using the single-bin uncertainties alone—underestimates the statistical significance of these data. These three-dimensionally binned asymmetries are the most complete, unintegrated, longitudinally polarized double-spin dataset to date.

Azimuthal asymmetries {#sec:azimresults}
---------------------

\[t\]

As described in the introduction, azimuthal moments of asymmetries are potentially sensitive to unique combinations of distribution and fragmentation functions, a number of which vanish when integrated over semi-inclusive kinematic parameters.

<span>cc</span> $x$ binning & $z$ binning\
0.023 – 0.1 – 0.6 & 0.2 – 0.32 – 0.44 – 0.56 –0.68 – 0.8\
\
$x$ binning & $P_{h\perp}$\[ GeV\] binning\
0.023 – 0.1 – 0.6 & 0 – 0.3 – 0.4 – 0.5 –0.6 – 2\
\
$z$ binning & $x$ binning\
0.2 – 0.4 – 0.6 & 0.023 – 0.04 – 0.055 – 0.075 – 0.14 – 0.6

For each hadron and target combination, the asymmetry is divided into 10 $\phi$ bins and fit with an azimuthally periodic function in each of either 2 $x$ $\times$ 5 $z$-bins, 2 $x$ $\times$ 5 $P_{h\perp}$-bins, or 2 $z$ $\times$ 5 $x$-bins as detailed in Table \[tab:ALLcosphiBinning\]. The functional form used included constant, $\cos\phi$, and $\cos 2\phi$ terms. Each of these cosine moments is found to be consistent with zero. (A similar result was obtained for unidentified hadrons for deuteron data from the COMPASS experiment @Alekseev:2010dm [@Adolph:2016vou].) The $P_{h\perp}$ projections of the $\cos \phi$ moments for charged pions for each target, as well as for charged kaons in case of a deuterium target are presented in Fig. \[fig:ALLpt\]. All other projections of the $\cos \phi$ moments are included in the data tables in @asymData, including the statistically more precise results for unidentified hadrons.[^7]

A vanishing $\cos 2\phi$ asymmetry as found here can be expected because in the one-photon-exchange approximation there is no $A_{LL}^{h,\cos 2\phi}$ contribution to the cross section \[cf. Eq. \] and thus a non-zero $A_{\parallel}^{h,\cos2\phi}$ can arise in this approximation only through the very small transverse component of the target-spin vector in a configuration where the target is polarized along the beam direction @Diehl:2005.

The hadron charge-difference asymmetry {#sec:charge-difference}
--------------------------------------

The hadron charge-difference asymmetry $$A_1^{h^+ - h^-}(x) \equiv 
        \frac{\left(\sigma_{1/2}^{h^+}-\sigma_{1/2}^{h^-}\right)-\left(\sigma_{3 
        /2}^{h^+}-\sigma_{3/2}^{h^-}\right)}{\left(\sigma_{1/2}^{h^+}-\sigma_{1/ 
        2}^{h^-}\right)+\left(\sigma_{3/2}^{h^+}-\sigma_{3/2}^{h^-}\right)}
        \label{eq:A1ChargeDiff}$$ provides additional spin-structure information and is not trivially constructible from the simple semi-inclusive asymmetries. The difference asymmetries for pions from the hydrogen target and pions, kaons, and undifferentiated hadrons from the deuterium target are shown in Fig. \[fig:diffAsy\], together with results from the COMPASS Collaboration for unidentified hadrons from a $^6$LiD target @Alekseev:2007vi. A feature that might be unexpected is that the uncertainties for the kaon asymmetry are considerably smaller than those on the pion asymmetry despite the smaller sample size. This is a result of the larger difference between yields of charged kaons compared to that of the charged pions (as $K^-$ shares no valence quarks with the target), which causes a significantly larger denominator of Eq. .

Under the assumption of leading-order (LO), leading-twist (LT) QCD, and charge-conjugation symmetry of the fragmentation functions, i.e., $$D_1^{q\to h^+}=D_1^{\bar{q} \to h^-},
        \label{eq:symAssume}$$ the difference asymmetry on the deuteron may be equated to a certain combination of parton distributions @Frankfurt:1989wq: $$A_{1,d}^{h^+ - h^-}   \stackrel{\textrm{\tiny{LO LT}}}{=} 
        \frac{g_{1}^{u_v}+g_{1}^{d_v}}{f_{1}^{u_v}+f_{1}^{d_v}} .
        \label{eq:diff-deuteron}$$ Here, $f_{1}^{q_v} \equiv f_{1}^{q}-f_{1}^{\bar{q}} $ ($g_{1}^{q_v} \equiv g_{1}^{q}-g_{1}^{\bar{q}} $) is the polarization-averaged (helicity) valence-quark distribution of the proton, and "LO LT" is a reminder of the assumptions mentioned previously. This is equivalent to assuming a well differentiated current and target region; a scenario in which the struck quark has no memory of the hadron variety to which it previously belonged.

By further assuming isospin symmetry in fragmentation, that is $$D_1^{u \to \pi^+}=D_{1}^{d \to \pi^-}~~\text{and}~~D_1^{u \to \pi^-}=D_{1}^{d\to\pi^+} ,$$ a second valence-quark expression using charge-difference asymmetries from a hydrogen target is given by $$A_{1,p}^{h^+ - h^-}  \stackrel{\textrm{\tiny{LO LT}}}{=} 
        \frac{4 g_{1}^{u_v}-g_{1}^{d_v}}{4 f_{1}^{u_v}-f_{1}^{d_v}} \; .
        \label{eq:diff-proton}$$

It follows that the charge-difference asymmetries should be independent of the hadron type, a feature consistent with the results shown in Fig. \[fig:diffAsy\]. Valence-quark helicity densities computed using Eqs.  and are presented in Fig. \[fig:helicityDensities\] alongside the same quantities computed from the previous HERMES purity extraction @Airapetian:2004zf. The results are largely consistent using two methods that have very different and quite complementary model assumptions. Whereas the method presented here depends on leading-order and leading-twist assumptions to provide the clean factorization, which ensures that fragmentation can proceed without memory of the target configuration, the purity method depends on a fragmentation model subject to its own uncertainties related to the model tune and the believability of its phenomenologically motivated dynamics. The lack of dependence on hadron type of the charge-difference asymmetries and the consistency of the derived valence-quark helicity distributions with the results of the purity analysis suggest that there is no significant deviation from the factorization hypothesis.

Conclusion
==========

Several longitudinal double-spin asymmetries in semi-inclusive deep-inelastic scattering have been presented. They extend the analysis of the previous HERMES publications to include also transverse-momentum dependence. Within the precision of the measurements, the virtual-photon–nucleon asymmetries $A_{1}^h(x,z)$ and $A_{1}^h(x, P_{h\perp})$ display no significant dependence on the hadron variables. Azimuthal moments, $A_{\parallel}^{h,\cos\phi}$, are found to be consistent with zero. The hadron charge-difference asymmetry $A_1^{h^+-h^-}(x)$ yields valence-quark helicity densities consistent with the result of the prior HERMES purity extraction. A common thread among these results is that within the available statistical precision the longitudinal sector shows no deviation from the leading-order, leading-twist assumption. In addition to this interpretation, these data are expected to provide an essentially model-independent constraint for theory and parameterization as they provide the first ever longitudinal double-spin semi-inclusive dataset binned in as many as three kinematic variables simultaneously. They point the way to future precision tests of models of nucleon structure that go beyond a collinear framework.

We gratefully acknowledge the DESY management for its support, the staff at DESY and the collaborating institutions for their significant effort. This work was supported by the State Committee of Science of the Republic of Armenia, Grant No. 18T-1C180; the FWO-Flanders and IWT, Belgium; the Natural Sciences and Engineering Research Council of Canada; the National Natural Science Foundation of China; the Alexander von Humboldt Stiftung, the German Bundesministerium für Bildung und Forschung (BMBF), and the Deutsche Forschungsgemeinschaft (DFG); the Italian Istituto Nazionale di Fisica Nucleare (INFN); the MEXT, JSPS, and G-COE of Japan; the Dutch Foundation for Fundamenteel Onderzoek der Materie (FOM); the Russian Academy of Science and the Russian Federal Agency for Science and Innovations; the Basque Foundation for Science (IKERBASQUE), the Basque Government, Grant No. IT956-16, and MINECO (Juan de la Cierva), Spain; the U.K. Engineering and Physical Sciences Research Council, the Science and Technology Facilities Council, and the Scottish Universities Physics Alliance; as well as the U.S. Department of Energy (DOE) and the National Science Foundation (NSF).

[^1]: Deceased

[^2]: Deceased

[^3]: Deceased

[^4]: The additional but suppressed contribution related to the $g_{2}$ structure function is neglected here due to the smallness of $g_{2}$ (e.g., Ref. @Airapetian:2011wu).

[^5]: The polarization component transverse to the virtual-photon direction (see Fig. \[fig:paramsDefined\]) is proportional to $\sin\theta_{\gamma^{*}}$, where $\theta_{\gamma^{*}}$ is the angle between the incoming lepton momentum and the virtual-photon direction. This transverse component is 10%-15% of the target polarization in typical HERMES kinematics, but can reach 20% for the largest $x$ values covered in this experiment.

[^6]: Note that if experimental polarizations are not alternated so that the average polarization of both beam and target samples are zero, terms in Eq.  with a single "$U$" in the subscript do not vanish, a priori, from both the numerator and denominator of the ratio. In contrast, Eq. , i.e., the combination of all four target- and beam-helicity states, leaves only the sum of terms from Eq.  with the "$LL$" subscript divided by the sum of terms with the "$UU$" subscript.

[^7]: Note that here and in the later discussed hadron charge-difference asymmetry the momentum requirement for unidentified hadrons is relaxed to $P_{h}>0.5$ GeV.

{'source': 'arXiv_out/1810'}