Covariant three-dimensional account of the proton spin: QCD scale evolution of the composition functions $u(\epsilon,Q^2)$, $v(\epsilon,Q^2)$ and experimental access to $V(Q^2)$

The proton spin puzzle — the empirical finding that quark spins carry only $\Delta\Sigma\lesssim1/3$ of the proton's spin — is revisited by a covariant, three-dimensional quark-parton formulation in which the quark spin contribution is fixed by angular-momentum composition through two scalar functions $\mathrm{u}(\epsilon,Q^2)$ and $\mathrm{v}(\epsilon,Q^2)$ of the quark energy in the proton rest frame, with the moment $V(Q^2)=\int\mathrm{v}\,d^3\mathbf{p}$ signaling higher-angular-momentum admixtures in the proton wave function. Whether the $Q^2$ evolution of $\mathrm{u},\mathrm{v}$ is calculable in perturbative QCD, and how to measure $V(Q^2)$ , are both open. Gluon degrees of freedom are deliberately neglected at leading order in the source. The problem is to complete — or definitively refute — this frame-independent account of the small quark-spin fraction, whose status remains formulation-dependent.

Background

One of the long-standing puzzles of nucleon structure is the proton spin problem. In polarized deep-inelastic scattering (DIS, lepton scattering that resolves the proton's quark content) one measures the spin structure functions $g_1$ and $g_2$ , whose first moments encode how the proton's spin- $1/2$ is shared among its constituents. The total quark helicity contribution

\Delta\Sigma=\sum_q\int_0^1\left(q^+(x)-q^-(x)\right)dx

where $q^\pm(x)$ count quarks of flavor $q$ with spin parallel/antiparallel to the proton spin at momentum fraction $x$ , was expected in the naive quark model to carry the whole proton spin; polarized DIS experiments since the late 1980s instead find $\Delta\Sigma\approx0.25$ – $0.35$ in standard renormalization schemes (the source paper quotes $\Delta\Sigma=0.32\pm0.03$ at $Q^2=3\,\mathrm{GeV}^2$ ). The remainder must reside in quark orbital angular momentum (OAM), gluon spin, and gluon OAM, but how the split arises dynamically — and in which theoretical formulation it is even frame-independently meaningful — is unsettled.

The conventional parton model treats partons as free and collinear with the proton in an infinite-momentum frame; in that one-dimensional kinematics, quark transverse motion, OAM and the structure function $g_2$ are difficult to treat covariantly, while the scale ( $Q^2$ ) dependence of collinear parton distribution functions (PDFs) is governed by DGLAP evolution — the perturbative-QCD description of how a parton's momentum distribution changes with resolution scale. Zavada and coworkers developed instead a covariant quark-parton model (CQM), formulated with fully relativistic, Lorentz-covariant kinematics in the proton rest frame, in which relativistic invariance and rotational symmetry relate PDFs, transverse-momentum-dependent distributions (TMDs, which keep the quark transverse momentum) and the rest-frame quark momentum distribution $\rho(\mathbf{p})$ through exact model relations. Within the CQM the collective quark spin vector $\mathbf{w}$ — the contribution of the quark spins to the proton spin — is fixed by the general rules of angular-momentum composition to the form

\mathbf{w}(\mathbf{\omega},\epsilon)=\left(\mathrm{u}(\epsilon)-\mathrm{v}(\epsilon)\right)\mathbf{S}+2\,\mathrm{v}(\epsilon)(\mathbf{n}\cdot\mathbf{S})\mathbf{n}

where $\epsilon=p\cdot P/M$ is the quark energy in the proton rest frame ( $P$ the proton four-momentum, $M$ the proton mass) and two scalar composition functions $\mathrm{u}(\epsilon,Q^2)$ , $\mathrm{v}(\epsilon,Q^2)$ encode the angular-momentum structure; their integrals $\mathrm{U}=\int\mathrm{u}\,d^3\mathbf{p}$ and $V(Q^2)=\int\mathrm{v}\,d^3\mathbf{p}$ carry the composition information. A nonzero $V$ would signal admixtures of single-quark states with total angular momentum $j>1/2$ or orbital quantum number $l_p\ge1$ in the $J=1/2$ proton. The ratio $\tilde\mu=\langle m/\epsilon\rangle$ of effective quark mass to energy controls a relativistic contraction of the quark-spin contribution compensated by quark OAM, $\langle L_z\rangle/\langle S_z\rangle=(2-2\tilde\mu)/(1+2\tilde\mu)$ , leading to the composition constraint $\Delta\Sigma=\frac{1}{3}(1-2J^g)(1+2\tilde\mu)$ with $J^g$ the gluon total angular momentum. The covariant kinematics then make $\Delta\Sigma\lesssim1/3$ natural — but $\mathrm{u},\mathrm{v}$ (and $\tilde\mu$ , $J^g$ ) are model inputs with unknown scale dependence, so the agreement with data is a compatibility, not a QCD prediction.

The formulation question this program targets is real: the standard decompositions of the proton spin (for example Ji's kinetic decomposition, which splits the proton momentum and angular momentum into quark and gluon pieces through gauge-invariant kinetic operators, versus canonical, Chen-type splittings of the gluon field into pure-gauge and physical components) differ in their OAM terms and boost behavior, and the review of Leader and Lorcé concluded that the choice among gauge-invariant decompositions is essentially a matter of taste and convenience. The CQM's promise is a three-dimensional, Lorentz-invariant formulation in which the spin composition is fixed by angular-momentum algebra rather than by frame or gauge choices. Whether that promise can be completed — whether the composition functions possess a QCD-consistent scale evolution and observable consequences — is the open problem stated by the source paper.

Problem Statement

Within the Lorentz-covariant, three-dimensional quark-parton formulation of the proton spin (Zavada, Phys. Rev. D 89, 014012 (2014)), deliver a complete, QCD-consistent and experimentally testable account of the proton spin composition, by which the observed small quark-spin fraction $\Delta\Sigma\lesssim1/3$ is predicted frame-independently rather than fitted. This single objective — a closed covariant composition whose ingredients are scale-evolved and whose nonstandard moment $V(Q^2)$ is experimentally accessible — requires, as jointly necessary and source-listed ingredients:

(1) Scale evolution. Establish the $Q^2$ dependence of the composition functions $\mathrm{u}(\epsilon,Q^2)$ and $\mathrm{v}(\epsilon,Q^2)$ that define the covariant quark spin vector: either derive a practical evolution algorithm from perturbative QCD that is consistent with the model's exact PDF↔TMD↔ $\rho(\mathbf{p})$ relations and with rotational symmetry in the proton rest frame and reduces to standard DGLAP evolution of the collinear PDFs upon integration, or demonstrate that no such perturbative evolution exists within the model's assumptions, in which case specify the non-perturbative input that fixes the $Q^2$ dependence.

(2) Experimental access. Provide a method to determine the moments $\mathrm{U}(Q^2)$ and especially $V(Q^2)=\int\mathrm{v}(\epsilon,Q^2)\,d^3\mathbf{p}$ from observables — in particular to decide whether $V(Q^2)\ne0$ , i.e. whether the $J=1/2$ proton contains admixtures of single-quark states with $j>1/2$ or $l_p\ge1$ .

The covariant three-dimensional composition may likewise be refuted by an internal inconsistency with QCD. The two ingredients are linked by the common goal of a closed, frame-independent, QCD-evolved composition.

References

P. Zavada, 'Proton spin in leading order of the covariant approach', Phys. Rev. D 89, 014012 (2014). DOI: 10.1103/PhysRevD.89.014012; arXiv:1307.0699, https://arxiv.org/abs/1307.0699
P. Zavada, 'Kinematics of deep inelastic scattering in leading order of the covariant approach', Phys. Rev. D 85, 037501 (2012). DOI: 10.1103/PhysRevD.85.037501; arXiv:1106.5607, https://arxiv.org/abs/1106.5607
A. V. Efremov, O. V. Teryaev, P. Zavada, 'The QCD evolution of TMD in the covariant approach', J. Phys. Conf. Ser. 678, 012001 (2016). DOI: 10.1088/1742-6596/678/1/012001; arXiv:1511.01164, https://arxiv.org/abs/1511.01164
S. Bastami, A. V. Efremov, P. Schweitzer, O. V. Teryaev, P. Zavada, 'Structure of the nucleon at leading and subleading twist in the covariant parton model', Phys. Rev. D 103, 014024 (2021). DOI: 10.1103/PhysRevD.103.014024; arXiv:2011.06203, https://arxiv.org/abs/2011.06203
P. Zavada, 'Role of gluons and the quark sea in the proton spin', Phys. Lett. B 751, 525 (2015). DOI: 10.1016/j.physletb.2015.11.019; arXiv:1503.07924, https://arxiv.org/abs/1503.07924
P. Zavada, 'On the role of quark orbital angular momentum in the proton spin', arXiv:1410.5637 (2014), https://arxiv.org/abs/1410.5637 (unpublished companion version of the source paper)
E. Leader, C. Lorcé, 'The angular momentum controversy: What's it all about and does it matter?', Phys. Rep. 541, 163–248 (2014). DOI: 10.1016/j.physrep.2014.02.010; arXiv:1309.4235, https://arxiv.org/abs/1309.4235
K.-F. Liu, 'Status on lattice calculations of the proton spin decomposition', AAPPS Bulletin 32, 8 (2022). DOI: 10.1007/s43673-022-00037-4; arXiv:2112.08416, https://arxiv.org/abs/2112.08416
M. M. Yazdanpanah, A. Mirjalili, A. Behjat Ramezani, 'Mass and energy dependence of the transverse momentum densities in covariant parton model', Mod. Phys. Lett. A 30, 1550133 (2015). DOI: 10.1142/S0217732315501333, https://doi.org/10.1142/S0217732315501333

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Covariant three-dimensional account of the proton spin: QCD scale evolution of the composition functions $u(\epsilon,Q^2)$ , $v(\epsilon,Q^2)$ and experimental access to $V(Q^2)$

Background

Problem Statement

References