Extraction of transverse-momentum-dependent helicity distributions of the nucleon

Introduction

The three-dimensional internal structure of the nucleon remains a central focus of quantum chromodynamics (QCD) and hadronic physics. Considerable progress has been made in determining the longitudinal momentum distributions of quarks and gluons through collinear parton distribution functions (PDFs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. These distributions do not resolve the transverse motion of partons. Transverse-momentum-dependent distributions (TMDs) extend this description by resolving correlations between partonic transverse momentum and spin, providing insight into spin-orbit dynamics and their connections to partonic orbital angular momentum [14, 15, 16, 17, 18, 19].

The TMD helicity distribution $g_{1L}(x,k_T^2)$ describes, at longitudinal momentum fraction $x$ and transverse momentum $k_T$, the difference between the number densities of quarks polarized parallel and antiparallel to the longitudinal polarization of the parent nucleon [20, 21]. Its transverse-momentum dependence complements the collinear helicity PDF, while the ratio $g_{1L}/f_1$ characterizes the quark polarization relative to the unpolarized density. A qualitative motivation for studying this dependence comes from relativistic spin kinematics: in light-front quark models, Wigner rotations associated with boosts can suppress the longitudinal polarization of quarks with nonzero transverse momentum [22, 23, 24, 25]. Experimentally, helicity TMDs can be accessed through the transverse-momentum dependence of the longitudinal double-spin asymmetry $A_{LL}$ in semi-inclusive deep-inelastic scattering (SIDIS), where $g_{1L}$ enters the polarized structure function through a convolution with an unpolarized TMD fragmentation function (FF). Measurements with different targets and identified hadrons therefore provide information on both the transverse-momentum dependence and the flavor structure of quark polarization.

Global TMD analyses have advanced substantially over the past decade. Unpolarized TMD PDFs and FFs have been determined through fits to Drell-Yan and SIDIS data [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. The transverse-polarization sector, including the Sivers function, transversity, and the Collins effect, has also been studied intensively [38, 39, 40, 41, 42]. Lattice QCD calculations are providing complementary first-principles information on TMD structure and evolution [43, 44, 45, 46]. In the longitudinal-polarization sector, two independent phenomenological extractions of helicity TMDs from SIDIS data have recently been reported [47, 48].

In our preceding study [47], we performed the first global extraction of TMD helicity distributions using the HERMES [49] and CLAS [50] SIDIS measurements. The extracted polarization rates exhibited a characteristic transverse-momentum dependence: the magnitude of quark polarization decreased with increasing $k_T$ in the valence-dominated region, qualitatively consistent with the Wigner-rotation expectation, whereas a contrasting trend was favored at smaller $x$. The latter behavior motivates further investigation of the dynamics involving sea quarks and gluons, whose contributions remain poorly constrained by existing measurements [49, 50, 51]. These findings call for a refined determination of what the present data establish, comparisons with complementary observables, and an assessment of the improvements achievable with future measurements.

In this work, we reanalyze the existing world SIDIS $A_{LL}$ data within an improved phenomenological framework. We incorporate kinematic corrections associated with the target and detected-hadron masses, upgrade the small-$b_T$ operator-product-expansion (OPE) matching for $f_1$, $g_{1L}$, and $D_1$ to next-to-next-to-leading order (NNLO), and retain next-to-next-to-leading logarithmic (NNLL) TMD evolution. We also refine the treatment of correlated systematic uncertainties by combining a penalty-term $\chi^2$ with the $T_0$ prescription to mitigate bias from multiplicative uncertainties. These refinements provide the basis for the updated extraction and the subsequent predictions and impact study. The present data constrain the up- and down-quark helicity TMDs, whereas the sea-quark and gluon distributions remain weakly constrained. The up- and down-quark distributions integrated over transverse momenta up to the hard scale agree with the corresponding collinear helicity distributions within uncertainties.

We examine the predictive implications of the extraction through complementary spin observables. Using the extracted helicity TMDs, we calculate the $\cos\phi_h$ azimuthal modulation of the longitudinal double-spin asymmetry in SIDIS within the Wandzura-Wilczek (WW)-type approximation [52, 53, 54, 55]. The resulting predictions agree within uncertainties with the HERMES [49] and CLAS [50] measurements that were not included in the fit, providing a consistency check of the description within this approximation. We also predict the azimuthally averaged SIDIS asymmetry $A_{LL}$ in charged-pion production at CLAS12. Its large-$x$ spin-structure program [56] and multidimensional SIDIS measurements with a 10.6 GeV electron beam [57] provide opportunities to examine the extracted transverse-momentum dependence in the valence region.

A complementary process is $W^\pm$ production in longitudinally polarized proton-proton collisions. The STAR measurements at the Relativistic Heavy Ion Collider (RHIC), performed at $\sqrt{s}=500$ and 510 GeV, provide flavor-sensitive access to quark and antiquark helicities through leptonic $W$ decays [58, 59]. We use the SIDIS extraction to predict the charged-lepton-level single-spin asymmetries as functions of the $W$-boson transverse momentum at STAR kinematics, including the decay kinematics and lepton acceptance. These predictions extend the application of the extracted helicity TMDs to a different process at the electroweak scale and are compared with preliminary STAR measurements [60] that are not included in the fit.

More precise flavor separation, particularly in the sea-quark sector, requires measurements with higher statistics and broader kinematic coverage. The proposed Electron-Ion Collider in China (EicC) [61] is designed to collide polarized electron beams with polarized proton and ${}^{3}\mathrm{He}$ beams over center-of-mass energies of 15–20 GeV and high luminosity [62], with longitudinal polarization available for both beams [63]. Its coverage of intermediate $x$, extending down to $x \simeq0.005$, complements the large-$x$ program at Jefferson Lab and the small-$x$, high-$Q^2$ reach of the future Electron-Ion Collider (EIC) at Brookhaven National Laboratory [62, 64]. High luminosity and polarization, together with measurements of identified hadrons on different polarized targets, make the EicC well suited to precision TMD studies [39, 40, 65, 66, 67]. We quantify its potential impact by fitting the existing world data together with projected EicC SIDIS pseudo-data, using a more flexible flavor parametrization than in the world-data fit. Within the adopted framework, the projected data reduce the uncertainties of the quark and antiquark components, with substantial potential to improve constraints on the presently weakly determined sea-quark distributions.

The remainder of the paper is organized as follows. Section II presents the theoretical framework, and Sec. III describes the updated global analysis. Section IV presents the WW-type comparison for the $\cos\phi_h$ modulation and the predictions for CLAS12 SIDIS and STAR $W^\pm$ production. Section V assesses the projected impact of EicC measurements. A summary is given in Sec. VI.

Theoretical Framework

SIDIS Kinematics and Cross Section

We consider the SIDIS process

$$\ell(l) + N(P) \to\ell(l') + h(P_h) + X(P_X), \tag*{(1)}$$

where the scattered lepton and a final-state hadron $h$ are detected, while $X$ denotes the unobserved final state. The symbols in parentheses specify the particle four-momenta. With $q=l-l'$, the kinematic variables are

$$Q^2 \equiv-q^2 = -(l-l')^2, \tag*{(2)}$$

$$x \equiv\frac{Q^2}{2P\cdot q}, \qquad y \equiv\frac{P\cdot q}{P\cdot l}, \qquad z \equiv\frac{P\cdot P_h}{P\cdot q}, \tag*{(3)}$$

$$\gamma\equiv\frac{2xM}{Q} = \frac{MQ}{P\cdot q}, \tag*{(4)}$$

where $M$ is the nucleon mass.

We work in the one-photon-exchange approximation and neglect the lepton mass. For a longitudinally polarized lepton with helicity $\lambda_e$ scattering off longitudinally polarized targets, one can express the differential cross section as [68]

$$\frac{\mathrm{d}\sigma}{\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}\phi_h\,\mathrm{d}P_{h\perp}^2} = \frac{2\pi\alpha^2}{xyQ^2}\frac{y^2}{2(1-\varepsilon)} \left(1+\frac{\gamma^2}{2x}\right) \left[ F_{UU} +\lambda_e S_L\sqrt{1-\varepsilon^2}F_{LL} +\lambda_e S_L\sqrt{2\varepsilon(1-\varepsilon)} \cos\phi_h F_{LL}^{\cos\phi_h} \right]. \tag*{(5)}$$

Here $S_L$ denotes the longitudinal target-spin component; in the target rest frame, positive $S_L$ corresponds to spin antiparallel to the virtual-photon momentum. The labels U and L denote unpolarized and longitudinally polarized states, respectively, with the first label referring to the lepton and the second to the target. An integration over the overall azimuth of the lepton scattering plane about the beam axis is performed. We only retain the azimuthally independent unpolarized contribution $F_{UU}$ and the double-longitudinal terms $F_{LL}$ and $F_{LL}^{\cos\phi_h}$ that are relevant to this study. The electromagnetic fine-structure constant is denoted by $\alpha$. The virtual-photon polarization parameter, defined as the ratio of longitudinal to transverse photon fluxes, is

$$\varepsilon=\frac{1-y-\frac{1}{4}\gamma^2y^2}{1-y+\frac{1}{2}y^2+\frac{1}{4}\gamma^2y^2}. \tag*{(6)}$$

Following the Trento conventions [69], we define the measured hadron transverse momentum and azimuthal angle relative to the virtual-photon and target directions, as illustrated in Figure 1. In a frame where the spatial momenta of $P$ and $q$ are defined along the longitudinal axis, $P_{h\perp}$ is the magnitude of the hadron transverse momentum. The corresponding covariant definitions are

$$P_{h\perp}=\sqrt{-g_{\perp}^{\mu\nu}P_{h\mu}P_{h\nu}}, \tag*{(7)}$$

$$\cos\phi_h=-\frac{l_\mu P_{h\nu}g_{\perp}^{\mu\nu}}{l_\perp P_{h\perp}}, \qquad \sin\phi_h=-\frac{l_\mu P_{h\nu}\epsilon_{\perp}^{\mu\nu}}{l_\perp P_{h\perp}}, \tag*{(8)}$$

where $l_\perp=\sqrt{-g_{\perp}^{\mu\nu}l_\mu l_\nu}$. The transverse metric and antisymmetric tensor are

$$g_{\perp}^{\mu\nu} = g^{\mu\nu} -\frac{q^\mu P^\nu+P^\mu q^\nu}{P\cdot q(1+\gamma^2)} +\frac{\gamma^2}{1+\gamma^2} \left(\frac{q^\mu q^\nu}{Q^2}-\frac{P^\mu P^\nu}{M^2}\right), \tag*{(9)}$$

$$\epsilon_{\perp}^{\mu\nu} = \epsilon^{\mu\nu\rho\sigma} \frac{P_\rho q_\sigma}{P\cdot q\sqrt{1+\gamma^2}}. \tag*{(10)}$$

For TMD factorization, it is convenient to define the longitudinal axis by the target and detected-hadron momenta, with the target moving along the positive direction and the detected hadron along the negative direction. We introduce light-cone components $v^\pm=(v^0\pm v^3)/\sqrt{2}$ with respect to this axis. The TMD factorization applies in the low-transverse-momentum region, $\delta\equiv q_T/Q\ll1$, where the quantity $q_T$ is the magnitude of the virtual-photon momentum transverse to these hadronic directions. Following Ref. [26], the kinematic target- and detected-hadron-mass corrections are implemented through the relations:

$$x_S=-\frac{q^+}{P^+}=\frac{2x}{\gamma^2} \left[ 1-\sqrt{1+\gamma^2\left(1-\frac{q_T^2}{Q^2}\right)} \right], \tag*{(11)}$$

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