The spin structure of the nucleon

The electromagnetic interaction of a lepton with a hadronic or nuclear target proceeds by the exchange of a virtual photon. The first-order amplitude, known as the first Born approximation, corresponds to a single photon exchange, see figure 1. In the case of electron scattering, where the lepton mass is small, higher orders in perturbative quantum electrodynamics (QED) are needed to account for bremsstrahlung (real photons emitted by the incident or the scattered electron), vertex corrections (virtual photons emitted by the incident electron and re-absorbed by the scattered electron) and 'vacuum polarization' diagrams (the exchanged photon temporarily turning into pairs of charged particles). In some cases, such as high-Z nuclear targets, it is also necessary to account for the cases where the interaction between the electron and the target is transmitted by the exchange of multiple photons (see e.g. [16]). This correction will be negligible for the reactions and kinematics discussed here. Perturbative techniques can be applied to the electromagnetic probe, since the QED coupling $\alpha \approx 1/137$, but not to the target structure whose reaction to the absorption of the photon is governed by the strong force at large distances where the QCD coupling $\alpha_s$ can be large.

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In inclusive reactions the final state system X is not detected. In the case of an 'elastic' reaction, the target particle emerges without structure modification. Alternatively, the target nucleon or nucleus can emerge as excited states which promptly decay by emitting new particles (the resonance region), or the target can fragment, with additional particles produced in the final state as in DIS.

We first consider measurements in the laboratory frame where the nucleon or nuclear target is at rest (figures 1 and 2). The laboratory energy of the virtual photon is $\newcommand{\e}{{\rm e}} \nu\equiv E-E'$. The direction of the momentum $\newcommand{\e}{{\rm e}} \overrightarrow{q}\equiv \overrightarrow{k}-\overrightarrow{k'}$ of the virtual photon defines the $\overrightarrow{z}$ axis, while $\overrightarrow{x}$ is in the $(\overrightarrow{k},\overrightarrow{k'})$ plane. $\overrightarrow{S}$ is the target spin, with $\theta^{*}$ and $\phi^{*}$ its polar and azimuthal angles, respectively. In inclusive reactions, two variables suffice to characterize the kinematics; in the elastic case, they are related, and one variable is enough.

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During an experiment, the transfered energy $\nu$ and the scattering angle $\theta$ are typically varied. Two of the following relativistic invariants are used to characterize the kinematics:

• The exchanged 4-momentum squared $\newcommand{\e}{{\rm e}} Q^2 \equiv -(k-k'){}^2 \,=$$\newcommand{\e}{{\rm e}} 4EE'\sin^2\frac{\theta}{2}$ for ultra-relativistic leptons. For a real photon, Q²  =  0.
• The invariant mass squared $\newcommand{\e}{{\rm e}} W^2\equiv (\,p+q){}^2\,=$ $\newcommand{\e}{{\rm e}} M_t^2+2M_t\nu-Q^2$, where M_t is the mass of the target nucleus. W is the mass of the system formed after the lepton-nucleus collision; e.g. a nuclear excited state.
• The Bjorken variable $\newcommand{\e}{{\rm e}} x_{Bj} \equiv \frac{Q^2}{2p.q} = \frac{Q^2}{2M_t\nu}$. This variable was introduced by Bjorken in the context of scale invariance in DIS; see section 3.1.2. One has $0<x_{Bj}<M_t/M$, where M the nucleon mass, since $W \geqslant M_t$, Q²  >  0 and $\nu>0$.
• The laboratory energy transfer relative to the incoming lepton energy $y=\nu/E$.

Depending on the values of Q² and $\nu$, the target can emerge in different excited states. It is advantageous to study the excitation spectrum in terms of W since each excited state corresponds to specific a value of W rather than $\nu$, see figure 3.

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In what follow, 'hadron' can refer to either a nucleon or a nucleus. The reaction cross section is obtained from the scattering amplitude T_fi for an initial state i and final state f . T_fi is computed from the photon propagator and the leptonic current contracted with the electromagnetic current of the hadron for the exclusive reaction $\newcommand{\e}{{\rm e}} \ell H \to \ell^\prime H^\prime$, or a tensor in the case of an incompletely known final state. These quantities are conserved at the leptonic and hadronic vertices (gauge invariance).

In the first Born approximation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{fi}=\left\langle k'\right|j^{\mu}(0)\left|k\right\rangle \frac{1}{Q^2}\left\langle P_X\right|J_{\mu}(0)\left|P\right\rangle, \label{eq:amplitran} \nonumber \end{align} \tag{ 1 }$$

where the leptonic current is $j^{\mu}=e\overline{\psi_l}\gamma^{\mu}\psi_l$ with $\psi_l$ the lepton spinor, e its electric charge and $J^\mu$ the quark current. The exact expression of the hadron's current matrix element $\left\langle P_X\right|J_{\mu}(0)\left|P\right\rangle$ is unknown because of our ignorance of the nonperturbative hadronic structure and, for non-exclusive experiments, that of the final state. However, symmetries (parity, time reversal, hermiticity, and current conservation) constrain the matrix elements of $J^\mu$ to a generic form written in terms of the vectors and tensors pertinent to the reaction. Our ignorance of the hadronic structure is thus parameterized by functions which can be either measured, computed numerically, or modeled. These are called either 'form factors' (elastic scattering, see section 3.3), 'response functions' (quasi-elastic reaction, see section 3.3.2) or 'structure functions' (DIS case, see section 3.1). A significant advance of the late 1990s and early 2000s is the unification of form factors and structure functions under the concept of GPDs. The differential cross-section ${\rm d}\sigma$ is obtained from the absolute square of the amplitude (1) times the lepton flux and a phase space factor, given e.g. in [18].

The leptonic tensor $\newcommand{\e}{{\rm e}} \eta^{\mu\nu}$ and the hadronic tensor $W^{\mu\nu}$ are defined such that $\newcommand{\e}{{\rm e}} {\rm d}\sigma\propto\left|T_{fi}\right|^2=\eta^{\mu\nu}\frac{1}{Q^{4}}W_{\mu\nu}$. That is, $\newcommand{\e}{{\rm e}} \eta^{\mu\nu}\equiv \frac{1}{2}\sum j^{\mu *} j^{\nu}$, where all the possible final states of the lepton have been summed over (e.g. all of the lepton final spin states for the unpolarized experiments), and the tensor

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle W^{\mu\nu}=\frac{1}{4\pi} \int {\rm d}^4 \xi \, {\rm e}^{{\rm i}q_\alpha \xi^\alpha} \left\langle P \right|\left[J^{\mu \dagger} (\xi),J^\nu (0)\right] \left| P \right\rangle, \label{hadronic tensor} \nonumber \end{align} \tag{ 2 }$$

follows from the optical theorem by computing the forward matrix element of a product of currents in the proton state. The contribution to $W^{\mu\nu}$ which is symmetric in ${\mu , \nu}$—thus constructed from the hadronic vector current—contributes to the unpolarized cross-section, whereas its antisymmetric part—constructed from the pseudo-vector (axial) current—yields the spin-dependent contribution.

In the unpolarized case; i.e. with summation over all spin states, the cross-section can be parameterized with six photoabsorption terms. Three terms originate from the three possible polarization states of the virtual photon. (The photon spin is a 4-vector but for a virtual photon, only three components are independent because of the constraint from gauge invariance. The unphysical fourth component is called a ghost photon.) The other three terms stem from the multiplication of the two tensors. They depend in particular on the azimuthal scattering angle, which is integrated over for inclusive experiments. Thus, these three terms disappear and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left|T_{fi}\right|^2=\frac{e^2}{Q^2(1-\epsilon)}\left[(w_{RR}+w_{LL})+2\epsilon w_{ll}\right], \label{eq:sep} \nonumber \end{align} \tag{ 3 }$$

where R, L and l label the photon helicity state (they are not Lorentz indices) and $\newcommand{\e}{{\rm e}} \epsilon\equiv 1/\left[1+2\left(\nu^2/Q^2+1\right)\tan^2 (\theta/2)\right]$ is the virtual photon degree of polarization in the m_e  =  0 approximation. The right and left helicity terms are w_RR and w_LL, respectively. The longitudinal term w_ll is non-zero only for virtual photons. It can be isolated by varying $\newcommand{\e}{{\rm e}} \epsilon$ [19], but w_RR and w_LL cannot be separated. Thus, writing $w_T=w_{RR}+w_{LL}$ and $w_L=w_{ll}$, the cross-section takes the form:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm d}\sigma\propto\left|T_{fi}\right|^2=\frac{e^2}{Q^2(1-\epsilon)}\left[w_T+2\epsilon w_{L}\right]. \label{eq:tensor} \nonumber \end{align} \tag{ 4 }$$

The total unpolarized inclusive cross-section is expressed in terms of two photoabsorption partial cross-sections, $\sigma_{L}$ and $\sigma_T$. The parameterization in term of virtual photoabsorption quantities is convenient because the leptons create the virtual photon flux probing the target. For doubly-polarized inclusive inelastic scattering, where both the beam and target are polarized, two additional parameters are required: $\sigma_{TT}$ and $\sigma_{LT}'$. (The reason for the prime $'$ is explained below). The $\sigma_{TT}$ term stems from the interference of the amplitude involving one of the two possible transverse photon helicities with the amplitude involving the other transverse photon helicity. Likewise, $\sigma_{LT}'$ originates from the imaginary part of the longitudinal–transverse interference amplitude. The real part, which produces $\sigma_{LT}$, disappears in inclusive experiments because all angles defined by variables describing the hadrons produced during the reaction are averaged over. This term, however, appears in exclusive or semi-exclusive reactions, see e.g. the review [20].

The basic observable for studying nucleon spin structure in doubly polarized lepton scattering is the cross-section asymmetry with respect to the lepton and nucleon spin directions. Asymmetries can be absolute: $A=\sigma^{\downarrow\Uparrow}-\sigma^{\uparrow\Uparrow}$, or relative: $A=(\sigma^{\downarrow\Uparrow}-\sigma^{\uparrow\Uparrow})/(\sigma^{\downarrow\Uparrow}+\sigma^{\uparrow\Uparrow})$. The $\downarrow$ and $\uparrow$ represent the leptonic beam helicity in the laboratory frame whereas $\Downarrow$ and $\Uparrow$ define the direction of the target polarization (here, along the beam direction). Relative asymmetries convey less information, the absolute magnitude of the process being lost in the ratio, but are easier to measure than absolute asymmetries or cross-sections since the absolute normalization (e.g. detector acceptance, target density, or inefficiencies) cancels in the ratio. Measurements of absolute asymmetries can also be advantageous, since the contribution from any unpolarized material present in the target cancels out. The optimal choice between relative and absolute asymmetries thus depends on the experimental conditions; see section 3.1.7.

One can readily understand why the asymmetries appear physically, and why they are related to the spin distributions of the quarks in the nucleon. Helicity is defined as the projection of spin in the direction of motion. In the Breit frame where the massless lepton and the quark flip their spins after the interaction, the polarization of the incident relativistic leptons sets the polarization of the probing photons because of angular momentum conservation; i.e. these photons must be transversally polarized and have helicities $\pm 1$. Helicity conservation requires that a photon of a given helicity couples only to quarks of opposite helicity, thereby probing the quark helicity (spin) distributions in the nucleon. Thus the difference of scattering probability between leptons of $\pm 1$ helicities (asymmetry) is proportional to the difference of the population of quarks of different helicities. This is the basic physics of quark longitudinal polarization as characterized by the target hadron's longitudinal spin structure function. Note also that virtual photons can also be longitudinally polarized, i.e. with helicity 0, which will also contribute to the lepton asymmetry at finite Q².

A lepton pair detected in the final state corresponds to the Drell–Yan process, see figure 4, panel A. In the high-energy limit, this process is described as the annihilation of a quark from a proton with an antiquark from the other (anti)proton, the resulting timelike photon then converts into a lepton-antilepton pair. Hence, the process is sensitive to the convolution of the quark and antiquark polarized PDFs $\Delta q(x_{Bj})$ and $\Delta \overline{q}(x_{Bj})$. (They will be properly defined by equation (25).) Another process that leads to the same final state is lepton-antilepton pair creation from a virtual photon emitted by a single quark. However, this process requires large virtuality to produce a high energy lepton–anti-lepton pair, and it is thus kinematically suppressed compared to the panel A case.

An important complication is that the Drell–Yan process is sensitive to double initial-state corrections, where both the quark and antiquark before annihilation interact with the spectator quarks of the other projectile. Such corrections are 'leading twist'; i.e. they are not power-suppressed at high lepton pair virtuality. They induce strong modifications of the lepton-pair angular distribution and violate the Lam-Tung relation [21].

A fundamental QCD prediction is that a naively time-reversal-odd distribution function, measured via Drell–Yan should change sign compared to a SIDIS measurement [22–25]. An example is the Sivers function [26], a transverse-momentum dependent distribution function sensitive to spin–orbit effects inside the polarized proton.

Inclusive diphoton production $\overrightarrow{p}\overrightarrow{p} \rightarrow \gamma \gamma+X$ is another process sensitive to $\Delta q(x_{Bj})$ and $\Delta \overline{q}(x_{Bj})$. The underlying leading order (LO) diagram is shown on panel B of figure 4.

The structure functions probed in lepton scattering involve the quark charge squared (see equations (21) and (23)): they are thus only sensitive to $\Delta q + \Delta \overline{q}$. W^+/− production is sensitive to $\Delta q(x_{Bj})$ and $\Delta \overline{q}(x_{Bj})$ separately. Panel (C) in figure 4 shows how W^+/− production allows the measurement of both mixed $\Delta u \Delta \overline{d}$ and $\Delta d \Delta \overline{u}$ combinations; thus combining W^+/− production data and data providing $\Delta q + \Delta \overline{q}$ (e.g. from lepton scattering) permits individual quark and antiquark contributions to be separated. The produced W is typically identified via its leptonic decay to $\nu l$, with the $\nu$ escaping detection.

These processes are $\overrightarrow{p}\overrightarrow{p} \rightarrow \gamma+X$, $\overrightarrow{p}\overrightarrow{p} \rightarrow \pi+X$, $\overrightarrow{p}\overrightarrow{p} \rightarrow jet(s)+X$ and $\overrightarrow{p}\overrightarrow{p} \rightarrow \gamma+jet+X$. At high momenta, such reactions are dominated by either gluon fusion or gluon–quark Compton scattering with a gluon or photon in the final state; see panel (D) in figure 4. These processes are sensitive to the polarized gluon distribution $\Delta g(x,Q^2)$.

Another process which is sensitive to $\Delta g(x,Q^2)$ is D or B heavy meson production via gluon fusion $\overrightarrow{p}\overrightarrow{p} \rightarrow D+X$ or $\overrightarrow{p}\overrightarrow{p} \rightarrow B+X$. See panel E in figure 4. The heavy mesons subsequently decay into charged leptons which are detected.

The kinematic domain of DIS where leading-twist Bjorken scaling is valid requires $W \gtrsim 2$ GeV and $Q^2 \gtrsim 1$ GeV². Due to asymptotic freedom, QCD can be treated perturbatively in this domain, and standard gauge theory calculations are possible. In the Bjorken limit where $\nu\to\infty$ and $Q^2\to\infty$, with $x_{Bj}=Q^2/(2M\nu)$ fixed, DIS can be represented in the first approximation by a lepton scattering elastically off a fundamental quark or antiquark constituent of the target nucleon, as in Feynman's parton model. The momentum distributions of the quarks (and gluons) in the nucleon, which determine the DIS cross section, reflect its nonperturbative bound-state structure. The ability to separate, at high lepton momentum transfer, perturbative photon-quark interactions from the nonperturbative nucleon structure is known as the factorization theorem [27]—a direct consequence of asymptotic freedom. It is an important ingredient in establishing the validity of QCD as a description of the strong interactions.

The momentum distributions of quarks and gluons are parameterized by the structure functions: these distributions are universal; i.e. they are properties of the hadrons themselves, and thus should be independent of the particular high-energy reaction used to probe the nucleon. In fact, all of the interactions within the nucleon which occur before the lepton-quark interaction, including the dynamics, are contained in the frame-independent light-front (LF) wave functions (LFWF) of the nucleon—the eigenstates of the QCD LF Hamiltonian. They thus reflect the nonperturbative underlying confinement dynamics of QCD; we discuss how this is assessed in models and confining theories such as Light Front Holographic QCD (LFHQCD) in section 4.4. Final-state interactions—processes happening after the lepton interacts with the struck quark—also exist. They lead to novel phenomena such as diffractive DIS (DDIS), $\newcommand{\e}{{\rm e}} \ell p \to \ell^\prime p^\prime X$, or the pseudo-T-odd Sivers single-spin asymmetry $\vec {S}_p \cdot \vec {q} \times \vec {p}_q$ which is observed in polarized SIDIS. These processes also contribute at 'leading twist'; i.e. they contribute to the Bjorken-scaling DIS cross-section.

DIS is effectively represented by the elastic scattering of leptons on the pointlike quark constituents of the nucleon in the Bjorken limit. Bjorken predicted that the hadron structure functions would depend only on the dimensionless ratio x_Bj, and that the structure functions reflect conformal invariance; i.e. they will be Q²-invariant. This is in fact the prediction of 'conformal' theory—a quantum field theory of pointlike quarks with no fundamental mass scale. Bjorken's expectation was verified by the first measurements at SLAC [28] in the domain $x_{Bj} \sim 0.25$. However, in a gauge theory such as QCD, Bjorken scaling is broken by logarithmic corrections from pQCD processes, such as gluon radiation—see section 3.1.9. One also predicts deviations from Bjorken scaling due to power-suppressed $M^2/Q^2$ corrections called higher-twist processes. They reflect finite mass corrections and hard scattering involving two or more quarks. The effects become particularly evident at low Q² ($\lesssim$1 GeV²), see section 4.1. The underlying conformal features of chiral QCD (the massless quark limit) also has important consequence for color confinement and hadron dynamics at low Q². This perspective will be discussed in section 4.4.

An essential point of DIS is that the lepton interacts via the exchange of a virtual photon with the quarks of the proton—not at the same instant time t (the ' instant form' as defined by Dirac), but at the time along the LF, in analogy to a flash photograph. In effect DIS provides a measurement of hadron structure at fixed LF time $\tau = x^+ = t + z/c$.

The LF coordinate system in position space is based on the LF variables $x^{\pm} = (t \pm z)$. The choice of the $\hat z = {\hat x}^3$ direction is arbitrary. The two other orthogonal vectors defining the LF coordinate system are written as $x_{\bot}=(x,y)$. They are perpendicular to the $(x^+,x^-)$ plane. Thus $x^2 = x^+ x^- - x_\perp^2$. Similar definitions are applicable to momentum space: $p^{\pm} = (\,p^0 \pm p^3)$, $p_{\bot}=(\,p_1,p_2)$. The product of two vectors $a^{\mu}$ and $b^{\mu}$ in LF coordinates is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle a^{\mu} b_{\mu} = \frac{1}{2} (a^+b^- + a^-b^+) - a_{\bot} b_{\bot}. \label{lc vector product} \nonumber \end{align} \tag{ 5 }$$

The relation between covariant and contravariant vectors is $a^+=a_-$, $a^- = a_+$ and the relevant metric is:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left( \begin{array}{@{}cccc@{}} 0 & 1 & 0 & 0 \nonumber \\ 1 & 0 & 0 & 0 \nonumber \\ 0 & 0 & -1 & 0 \nonumber \\ 0 & 0 & 0 & -1 \end{array} \right). \nonumber \end{align*}$$

Dirac matrices $\gamma^\mu$ adapted to the LF coordinates can also be defined [29].

The LF coordinates provide the natural coordinate system for DIS and other hard reactions. The LF formalism, called the 'Front Form' by Dirac, is Poincaré invariant (independent of the observer's Lorentz frame) and 'causal' (correlated information is only possible as allowed by the finite speed of light). The momentum and spin distributions of the quarks which are probed in DIS experiments are in fact determined by the LFWFs of the target hadron—the eigenstates of the QCD LF Hamiltonian $H_{\rm LF}$ with the Hamiltonian defined at fixed $\tau$. $H_{\rm LF}$ can be computed directly from the QCD Lagrangian. This explains why quantum field theory quantized at fixed $\tau$ (LF quantization) is the natural formalism underlying DIS experiments. The LFWFs being independent of the proton momentum, one obtains the same predictions for DIS at an electron–proton collider as for a fixed target experiment where the struck proton is at rest.

Since important nucleon spin structure information is derived from DIS experiments, it is relevant to outline the basic elements of the LF formalism here. The evolution operator in LF time is $P^- = P^0 - P^3$, while $P^+ = P^0 + P^3$ and $P_\perp$ are kinematical. This leads to the definition of the Lorentz invariant LF Hamiltonian $H_{\rm LF} = P^\mu P_\mu = P^- P^+ - P^2_\perp$. The LF Heisenberg equation derived from the QCD LF Hamiltonian is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{\rm LF} |\Psi_H \rangle = M^2_H |\Psi_H \rangle, \nonumber \end{align} \tag{ 6 }$$

where the eigenvalues $M_H^2$ are the squares of the masses of the hadronic eigenstates. The eigensolutions $|\Psi_H \rangle$ projected on the free parton eigenstates $|n\rangle$ (the Fock expansion) are the boost-invariant hadronic LFWFs, $\langle n | \Psi_H \rangle = \Psi_n(x_i, \vec {k}_{\perp i}, {\lambda_i})$, which underly the DIS structure functions. Here $x_i= k^+_i/P^+$, with $\sum_i x_i = 1$, are the LF momentum fractions of the quark and gluon constituents of the hadron eigenstate in the n-particle Fock state, the $\vec {k}_{\perp i}$ are the transverse momenta of the n constituents where $\sum_i\vec {k}_{\perp i} = 0_\perp$; the variable $\lambda_i$ is the spin projection of constituent i in the $\hat z$ direction.

A critical point is that LF quantization provides the LFWFs describing relativistic bound systems, independent of the observer's Lorentz frame; i.e. they are boost invariant. In fact, the LF provides an exact and rigorous framework to study nucleon structure in both the perturbative and nonperturbative domains of QCD [30].

Just as the energy P⁰ is the conjugate of the standard time x⁰ in the instant form, the conjugate to the LF time x⁺ is the operator $P^- = {\rm i} \frac{{\rm d}}{{\rm d} x^+}$. It represents the LF time evolution operator

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P^- \Psi =\frac{(M^2+P_{\bot}^2)}{2P^+} \Psi, \label{LF Hamiltonian} \nonumber \end{align} \tag{ 7 }$$

and generates the translations normal to the LF.

The structure functions measured in DIS are computed from integrals of the square of the LFWFs, while the hadron form factors measured in elastic lepton–hadron scattering are given by the overlap of LFWFs. The power-law fall-off of the form factors at high-Q² are predicted from first principles by simple counting rules which reflect the composition of the hadron [31, 32]. One also can predict observables such as the DIS spin asymmetries for polarized targets [33].

LF quantization differs from the traditional equal-time quantization at fixed t [34] in that eigensolutions of the Hamiltonian defined at a fixed time t depend on the hadron's momentum $\vec {P}$. The boost of the instant form wave function is then a complicated dynamical problem; even the Fock state structure depends on $P^\mu$. Also, interactions of the lepton with quark pairs (connected time-ordered diagrams) created from the instant form vacuum must be accounted for. Such complications are absent in the LF formalism. The LF vacuum is defined as the state with zero P⁻; i.e. invariant mass zero and thus $P^\mu=0$. Vacuum loops do not appear in the LF vacuum since P⁺ is conserved at every vertex; one thus cannot create particles with $k^+ \geqslant 0$ from the LF vacuum.

It is sometimes useful to simulate LF quantization by using instant time in a Lorentz frame where the observer has 'infinite momentum' $P^z \to - \infty.$ However, it should be stressed that the LF formalism is frame-independent; it is valid in any frame, including the hadron rest frame. It reduces to standard nonrelativistic Schrödinger theory if one takes $c \to \infty$. The LF quantization is thus the natural, physical, formalism for QCD.

As we shall discuss below, the study of dynamics with the LF holograpic approach which incorporates the exact conformal symmetry of the classical QCD Lagrangian in the chiral limit, provides a successful description of color confinement and nucleon structure at low Q² [35]. An example is given in section 3.3.1 where nucleon form factors emerge naturally from the LF framework and are computed in LFHQCD.

Light-cone gauge.

Light-cone dominance.

Using unitarity, the hadronic tensor $W^{\mu \nu}$, equation (2), can be computed from the imaginary part of the forward virtual Compton scattering amplitude $\gamma^*(q) N(\,p) \to \gamma^* (q) N(\,p)$, see figure 6. At large Q², the quark propagator which connects the two currents in the DVCS amplitude goes far-off shell; as a result, the invariant spatial separation $x^2 = x_\mu x^\mu$ between the currents $J^\mu(x)$ and $J^\nu (0)$ acting on the quark line vanishes as $x^2 \propto {1\over Q^2}$. Since $x^2 = x^+ x^- - x^2_\perp \to 0$, this domain is referred to as 'light-cone dominance'. The interactions of gluons with this quark propagator are referred to as the Wilson line. It represents the final-state interactions between the struck quark and the target spectators ('final-state', since the imaginary part of the amplitude in figure 6 is related by the Optical Theorem to the DIS cross-section with the Wilson line connecting the outgoing quark to the nucleon remnants). Those can contribute to leading- twist—e.g. the Sivers effect [26] or DDIS, or can generate higher-twists. In QED such final-state interactions are related to the 'Coulomb phase'.

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More explicitly, one can choose coordinates such that $q^+ = -M x_{Bj}$ and $q^- = (2\nu + M x_{Bj})$ with $q_{\bot}=0$. Then $\newcommand{\bi}{\boldsymbol} q^\mu \xi_\mu = \big[ (2\nu+M x_{Bj})\xi^+ -M x_{Bj} \xi^- \big]$, with $\xi$ the integration variable in equation (2). In the Bjorken limit, $\nu \to \infty$ and x_Bj is finite. One verifies then that the cross-section is dominated by $\xi^+ \to 0$, $\xi^- \propto 1/(M x_{Bj})$ in the Bjorken limit, that is $\xi^+ \xi^- \approx 0$, and the reaction happens on the LC specified by $\xi^+ \xi^- = \xi^2 = 0$. Excursions out of the LC generate $M^2/Q^2$ twist-4 and higher corrections ($M^{2n}/Q^{2n}$ power corrections), see section 4.1.

It can be shown that LC kinematics also dominates Drell–Yan lepton-pair reactions (section 2.2) and inclusive hadron production in $e^+ ~ e^-$ annihilation (section 2.3).

Light-front quantization.

Two structure functions are measured in unpolarized DIS: $F_1(Q^2,\nu)$ and $F_2(Q^2,\nu)$⁴, where F₁ is proportional to the photoabsorption cross-section of a transversely polarized virtual photon, i.e. $F_1 \propto \sigma_T$. Alternatively, instead of F₁ or F₂, one can define $F_L = F_2 /(2x_{Bj}) - F_1$, a structure functions proportional to the photabsorption of a purely longitudinal virtual photon. Each of these structure functions can be related to the imaginary part of the corresponding forward double virtual Compton scattering amplitude $\gamma^* p \to \gamma^* p$ through the Optical Theorem.

The inclusive DIS cross-section for the scattering of polarized leptons off of a polarized nucleon requires four structure functions (see section 2.1.4). The additional two polarized structure functions are denoted by $g_1(Q^2,\nu)$ and $g_2(Q^2,\nu)$: the function g₁ is proportional to the transverse photon scattering asymmetry. Its first moment in the Bjorken scaling limit is related to the nucleon axial-vector current $\langle P | \overline{\psi}(0) \gamma^{\nu}\gamma_5 \psi(\xi) | P \rangle$, which provides a direct probe of the nucleon's spin content (see equation (2) and below). The second function, g₂, has no simple interpretation, but $g_t=g_1+g_2$ is proportional to the scattering amplitude of a virtual photon which has transverse polarization in its initial state and longitudinal polarization in its final state [37]. If one considers all possible Lorentz invariant combinations formed with the available vectors and tensors, three spin structure functions emerge after applying the usual symmetries (see section 2.1.3). One (g₁, twist-2) is associated with the P⁺ LC vector. Another one (g₃, twist-4, see equation (64)) is associated with the P⁻ direction. The third one, (g_t, twist-3) is associated with the transverse direction; i.e. it represents effects arising from the nucleon spin polarized transversally to the LC. Only g₁ and g₂ are typically considered in polarized DIS analyses because g_t and g₃ are suppressed as 1/Q and 1/Q², respectively.

The DIS cross-section involves the contraction of the hadronic and leptonic tensors. If the target is polarized in the beam direction one has [69]:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \left(\frac{{\rm d}^2\sigma}{{\rm d}\Omega {\rm d}E'}\right)_{\parallel} = \sigma_{\rm Mott} \bigg\{\frac{F_1(Q^2,\nu)}{E'} \tan^2\frac{\theta}{2} + \frac{{2E'F}_2(Q^2,\nu)}{M\nu} \nonumber \\ \pm \frac{4}{M}\tan^2\frac{\theta}{2} \bigg[ \frac{E+E'\cos\theta}{\nu}g_1(Q^2,\nu) - \gamma^2 g_2(Q^2,\nu)\bigg]\bigg\} , \label{eq:sigmapar} \nonumber \end{align} \tag{ 8 }$$

where $\pm$ indicates that the initial lepton is polarized parallel versus antiparallel to the beam direction. Here $\newcommand{\e}{{\rm e}} \gamma^2\equiv Q^2/\nu^2$. At fixed $x_{Bj} = Q^2/(2M\nu)$, the contribution from g₂ is suppressed as $\approx$1/E in the target rest frame.

It is useful to define $\sigma_{\rm Mott}$, the photoabsorption cross-section for a point-like, infinitely heavy, target in its rest frame:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma_{\rm Mott} \equiv \frac{\alpha^2\cos^2(\theta/2)}{4E^2\sin^{4}(\theta/2)}. \label{eq:mott} \nonumber \end{align} \tag{ 9 }$$

The $\sigma_{\rm Mott}$ factorization thus isolates the effects of the hadron structure.

If the target polarization is perpendicular to both the beam direction and the lepton scattering plane, then:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \left(\frac{{\rm d}^2\sigma}{{\rm d}\Omega {\rm d}E'}\right)_{\perp} = \sigma_{\rm Mott} \bigg\{\frac{F_1(Q^2,\nu)}{E'}\tan^2\frac{\theta}{2}+\frac{{2E'F}_2(Q^2,\nu)}{M\nu} \nonumber \\ \pm\frac{4}{M}\tan^2\frac{\theta}{2}E'\sin\theta \bigg[\frac{1}{\nu}g_1(Q^2,\nu)+\frac{2E}{\nu^2}g_2(Q^2,\nu) \bigg] \bigg\}. \label{eq:sigmaperp} \nonumber \end{align} \tag{ 10 }$$

In this case g₂ is not suppressed compared to g₁, since typically $\nu \approx E$ in DIS in the nucleon target rest frame. The unpolarized contribution is evidently identical in equations (8) and (10). Combining them provides the cross-section for any target polarization direction within the plane of the lepton scattering. The general formula for any polarization direction, including nucleon spin normal to the lepton plane, is given in [70].

From equations (8) and (10), the cross-section relative asymmetries are:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A_{\Vert}\equiv \frac{\sigma^{\downarrow\Uparrow}-\sigma^{\uparrow\Uparrow}}{\sigma^{\downarrow\Uparrow}+\sigma^{\uparrow\Uparrow}}=\frac{4\tan^2\frac{\theta}{2}\left[\frac{E+E'\cos\theta}{\nu}g_1(Q^2,\nu)- \gamma^2 g_2(Q^2,\nu)\right]}{M\left[\frac{F_1(Q^2,\nu)}{E'}\tan^2\frac{\theta}{2}+\frac{{2E'F}_2(Q^2,\nu)}{M\nu}\right]}, \label{Apar} \nonumber \end{align} \tag{ 11 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle A_{\bot}\equiv \frac{\sigma^{\downarrow\Rightarrow}-\sigma^{\uparrow\Rightarrow}}{\sigma^{\downarrow\Rightarrow}+\sigma^{\uparrow\Rightarrow}}=\frac{4\tan^2\frac{\theta}{2}E'\sin\theta\big[\frac{1}{\nu}g_1(Q^2,\nu)+\frac{2E}{\nu^2}g_2(Q^2,\nu)\big]}{M\left[\frac{F_1(Q^2,\nu)}{E'}\tan^2\frac{\theta}{2}+\frac{{2E'F}_2(Q^2,\nu)}{M\nu}\right]}. \label{Aperp} \nonumber \end{align} \tag{ 12 }$$

The beam and target must both be polarized to produce non-zero asymmetries in an inclusive cross-section. The derivation of these asymmetries typically assumes the 'first Born approximation', a purely electromagnetic interaction, and the standard symmetries—in particular C, P and T invariances. In contrast, single-spin asymmetries (SSA) arise when one of these assumptions is invalidated; e.g. in SIDIS by the selection of a particular direction corresponding to the 3-momentum of a produced hadron. Note that T-invariance should be distinguished from 'pseudo T-odd' asymmetries. For example, the final-state interaction in single-spin SIDIS $\newcommand{\e}{{\rm e}} \ell p_\updownarrow \to \ell' H X$ with a polarized proton target produces correlations such as $i \vec {S}_p \cdot \vec {q} \times \vec {p}_H$. Here $\vec {S}_p$ is the proton spin vector and $\vec {p}_H$ is the 3-vector of the tagged final-state hadron. This triple product changes sign under time reversal $T \to -T$; however, the factor i, which arises from the struck quark FSI on-shell cut diagram, provides a signal which retains time-reversal invariance.

The single-spin asymmetry measured in SIDIS thus can access effects beyond the naive parton model described in section 3.1.8 [71] such as rescattering or 'lensing' corrections [22]. Measurements of SSA have in fact become a vigorous research area of QCD called 'Transversity'.

The observation of parity violating (PV) SSA in DIS can test fundamental symmetries of the standard model [72]. When one allows for Z⁰ exchange, the PV effects are enhanced by the interference between the Z⁰ and virtual photon interactions. Parity-violating interactions in the elastic and resonance region of DIS can also reveal novel aspects of nucleon structure [73].

Other SSA phenomena; e.g. correlations arising via two-photon exchange, have been investigated both theoretically [74] and experimentally [75]. In the inclusive quasi-elastic experiment reported in [75], for which the target was polarized vertically (i.e. perpendicular to the scattering plane), the SSA is sensitive to departures from the single photon time-reversal conserving contribution.

In electromagnetic photo-absorption reactions, the probe is the photon. Thus, instead of lepton asymmetries, $A_{\Vert}$ and $A_{\bot}$, one can also consider the physics of photoabsorption with polarized photons. The effect of polarized photons can be deduced from combining $A_{\Vert}$ and $A_{\bot}$ (equation (19) below). The photo-absorption cross-section is related to the imaginary part of the forward virtual Compton scattering amplitude by the optical theorem. Of the ten angular momentum-conserving Compton amplitudes, only four are independent because of parity and time-reversal symmetries. The following 'partial cross-sections' are typically used [69]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma_{T,3/2} = \frac{4\pi^2\alpha} {M \kappa_{\gamma^*}} \left[F_1(Q^2,\nu) - g_1(Q^2, \nu) + \gamma^2 g_2(Q^2,\nu) \right] , \label{eq:sigma3/2} \nonumber \end{align} \tag{ 13 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma_{T,1/2}=\frac{4\pi^2\alpha}{M\kappa_{\gamma^*}}\left[F_1(Q^2,\nu)+g_1(Q^2,\nu)-\gamma^2g_2(Q^2,\nu)\right], \label{eq:sigma1/2} \nonumber \end{align} \tag{ 14 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma_{L,1/2}=\frac{4\pi^2\alpha}{M\kappa_{\gamma^*}}\left[-F_1(Q^2,\nu)+\frac{M}{\nu}(1+\frac{1}{\gamma^2})F_2(Q^2,\nu)\right], \label{eq:sigma1/2L} \nonumber \end{align} \tag{ 15 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\sigma}_{LT,3/2}'=\frac{4\pi^2\alpha}{\kappa_{\gamma^*}}\frac{\gamma}{\nu}\left[g_1(Q^2,\nu)+g_2(Q^2,\nu)\right], \label{eq:sigma3/2TL} \nonumber \end{align} \tag{ 16 }$$

where ${{{\it T},1/2} }$ and ${{{\it T},3/2}}$ refer to the absorption of a photon with its spin antiparallel or parallel, respectively, to that of the spin of the longitudinally polarized target. As a result, 1/2 and 3/2 are the total spins in the direction of the photon momentum. The notation ${{\it L}}$ refers to longitudinal virtual photon absorption and ${{\it LT}}$ defines the contribution from the transverse-longitudinal interference. The effective cross-sections can be negative and depend on the convention chosen for flux factor of the virtual photon, which is proportional to the 'equivalent energy of the virtual photon' $\kappa_{\gamma^*}$. (Thus, the nomenclature of 'cross-section' can be misleading.) The expression for $\kappa_{\gamma^*}$ is arbitrary but must match the real photon energy $\kappa_{\gamma}=\nu$ when $Q^2\to0$. In the Gilman convention, $\kappa_{\gamma^*}=\sqrt{\nu^2+Q^2}$ [76]. The Hand convention [77] $\kappa_{\gamma^*}=\nu-Q^2/(2M)$ has also been widely used. Partial cross-sections must be normalized by $\kappa_{\gamma^*}$ since the total cross-section, which is proportional to the virtual photon flux times a sum of partial cross-sections is an observable and thus convention-independent. We define:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sigma_T\equiv \frac{\sigma_{T,1/2}+\sigma_{T,3/2}}{2}=\frac{8\pi^2\alpha}{M\kappa_{\gamma^*}}F_1 , \ ~~~\sigma_{L}\equiv \sigma_{L,1/2}, \nonumber \end{align*}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma_{TT} \equiv \frac{\sigma_{T,1/2} - \sigma_{T,3/2}}{2} \equiv -\sigma_{TT}' = \frac{4\pi^2\alpha}{M\kappa_{\gamma^*}}(g_1-\gamma^2g_2), \label{sigmaTT} \ ~~ \sigma_{LT}'\equiv \sigma_{LT,3/2}' , \nonumber \end{align} \tag{ 17 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R\equiv \frac{\sigma^{L}}{\sigma^T}=\frac{1+\gamma^2}{2x}\frac{F_2}{F_1}-1, \label{R(Q2)} \nonumber \end{align} \tag{ 18 }$$

as well as the two asymmetries $\newcommand{\e}{{\rm e}} A_1\equiv \sigma^{TT} / \sigma^T, A_2\equiv \sigma^{LT} / \left(\sqrt{2}\sigma^T\right),$ with $\left|A_2\right|\leqslant R$, since $\left|\sigma^{LT}\right|<\sqrt{\sigma^T\sigma^{L}}$. A tighter constraint can also be derived: the 'Soffer bound' [78] which is also based on positivity constraints. These constraints can be used to improve PDF determinations [79]. Positivity also constrains the other structure functions and their moments, e.g. $\left|g_1\right|\leqslant F_1$. This is readily understood when structure functions are interpreted in terms of PDFs, as discussed in the next section. The A₁ and A₂ asymmetries are related to those defined by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A_{\Vert}=D(A_1+\eta A_2), \label{eq:asyp} \ \ ~~~A_{\bot}=d(A_2-\zeta A_1), \nonumber \end{align} \tag{ 19 }$$

where $\newcommand{\e}{{\rm e}} D\equiv \frac{1-\epsilon E'/E}{1+\epsilon R}$, $\newcommand{\e}{{\rm e}} d\equiv D\sqrt{\frac{2\epsilon}{1+\epsilon}}$, $\newcommand{\e}{{\rm e}} \eta\equiv \frac{\epsilon\sqrt{Q^2}}{E-\epsilon E'}$, $\newcommand{\e}{{\rm e}} \zeta \equiv \eta\frac{1+\epsilon}{2\epsilon}$, and $\newcommand{\e}{{\rm e}} \epsilon$ is given below equation (3).

One can use the relative asymmetries A₁ and A₂, or the cross-section differences $\Delta \sigma_{\parallel}$ and $\Delta \sigma_{\perp}$ in order to extract g₁ and g₂, The SLAC, CERN and DESY experiments used the asymmetry method, whereas the JLab experiments have used both techniques.

Extraction using relative asymmetries.

This is the simplest method: only relative measurements are necessary and normalization factors (detector acceptance and inefficiencies, incident lepton flux, target density, and data acquisition inefficiency) cancel out with high accuracy. Systematic uncertainties are therefore minimized. However, measurements of the unpolarized structure functions F₁ and F₂ (or equivalently F₁ and their ratio R, equation (18)) must be used as input. In addition, the measurements must be corrected for any unpolarized materials present in and around the target. These two contributions increase the total systematic uncertainty. Equations (11), (12) and (19) yield

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A_1=\frac{g_1-\gamma^2g_2}{F_1}, \label{eq:A1} \ \ ~~~A_2=\frac{\gamma(g_1+g_2)}{F_1}, \nonumber \end{align} \tag{ 20 }$$

and thus

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle g_1=\frac{F_1}{1+\gamma^2}\left[A_1+\gamma A_2\right] = \frac{y(1+\epsilon R)F_1}{(1-\epsilon)(2-y)}\left[A_{\parallel}+\tan(\theta/2) A_{\perp}\right], \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle g_2=\frac{F_1}{1+\gamma^{3}}\left[A_2-\gamma A_1\right]=\frac{y^2(1+\epsilon R)F_1}{2(1-\epsilon)(2-y)}\left[\frac{E+E'\cos\theta}{E'\sin\theta}A_{\perp}-A_{\parallel}\right]. \nonumber \end{align*}$$

Extraction from cross-section differences.

The advantage of this method is that it eliminates all unpolarized material contributions. In addition, measurements of F₁ and F₂ are not needed. However, measuring absolute quantities is usually more involved, which may lead to a larger systematic error. According to equations (8) and (10),

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Delta\sigma_{\parallel}\equiv \frac{{\rm d}^2\sigma^{\downarrow\Uparrow}}{{\rm d}E'{\rm d}\Omega}-\frac{{\rm d}^2\sigma^{\uparrow\Uparrow}}{{\rm d}E'{\rm d}\Omega}=\frac{4\alpha^2}{MQ^2}\frac{E'}{E\nu}\left[g_1(E+E'\cos\theta)-Q^2\frac{g_2}{\nu}\right], \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Delta\sigma_{\perp}\equiv \frac{{\rm d}^2\sigma^{\downarrow\Rightarrow}}{{\rm d}E'{\rm d}\Omega}-\frac{{\rm d}^2\sigma^{\uparrow\Rightarrow}}{{\rm d}E'{\rm d}\Omega}=\frac{4\alpha^2}{MQ^2}\frac{E'^2}{E\nu}\sin\theta\left[g_1+2E\frac{g_2}{\nu}\right], \nonumber \end{align*}$$

which yields

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle g_1=\frac{2ME\nu Q^2}{8\alpha^2E'(E+E')}\left[\Delta \sigma_{\parallel}+\tan(\theta/2)\Delta \sigma_{\perp}\right], \nonumber \\ ~ g_2=\frac{M\nu^2Q^2}{8\alpha^2E'(E+E')}\left[\frac{E+E'\cos\theta}{E'\sin\theta}\Delta \sigma_{\perp}-\Delta \sigma_{\parallel}\right]. \nonumber \end{align*}$$

DIS in the Bjorken limit.

The moving nucleon in the Bjorken limit is effectively described as bound states of nearly collinear partons. The underlying dynamics manifests itself by the fact that partons have both position and momentum distributions. The partons are assumed to be loosely bound, and the lepton scatters incoherently only on the point-like quark or antiquark constituents since gluons are electrically neutral. In this simplified description the hadronic tensor takes a form similar to that of the leptonic tensor. This simplified model, the 'parton model', was introduced by Feynman [80] and applied to DIS by Bjorken and Paschos [81]. Color confinement, quark and nucleon masses, transverse momenta and transverse quark spins are neglected and Bjorken scaling is satisfied. Thus, in this approximation, studying the spin structure of the nucleon is reduced to studying its helicity structure. It is a valid description only in the IMF [34], or equivalently, the frame-independent Fock state picture of the LF. After integration over the quark momenta and the summation over quark flavors, the measured hadronic tensor can be matched to the hadronic tensor parameterized by the structure functions to obtain:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F_1(Q^2,\nu) \to F_1(x)=\sum_i\frac{e_i^2}{2}\left[q_i^\uparrow(x)+q_i^\downarrow(x)+\overline{q}_{i}^\uparrow(x)+\overline{q}_i^\downarrow(x)\right], \label{eq:eqf1parton} \nonumber \end{align} \tag{ 21 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F_{_2}(Q^2,\nu) \to F_2(x)=2xF_1(x), \label{eq:Callan-gross} \nonumber \end{align} \tag{ 22 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_1(Q^2,\nu) \to g_1(x)=\sum_i\frac{e_i^2}{2}\left[q_i^\uparrow(x)-q_i^\downarrow(x)+\overline{q}_i^\uparrow(x)-\overline{q}_i^\downarrow(x)\right], \label{eq:eqg1parton} \nonumber \end{align} \tag{ 23 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_2(Q^2,\nu) \to g_2(x)=0, \nonumber \end{align} \tag{ 24 }$$

where i is the quark flavor, e_i its charge and $q^\uparrow(x)$ ($q^\downarrow(x)$) the probability that its spin is aligned (antialigned) with the nucleon spin at a given x. Electric charges are squared in equations (21) and (23), thus the inclusive DIS cross-section in the parton model is unable to distinguish antiquarks from quarks.

The unpolarized and polarized PDFs are respectively

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle q_i(x)\equiv q_i^\uparrow(x)+q_i^\downarrow(x), \ \ ~~~\Delta q_i(x)\equiv q_i^\uparrow(x)-q_i^\downarrow(x). \label{eq:pdf def.} \nonumber \end{align} \tag{ 25 }$$

These distributions can be extracted from inclusive DIS (see e.g. figure 7). The gluon distribution, also shown in figure 7, can be inferred from sum rules and global fits of the DIS data. However, the identification of the specific contribution of quark and gluon OAM to the nucleon spin (figure 8) is beyond the parton model analysis. Note that equation (25) imposes the constraint $\left|\Delta q_i(x)\right| \leqslant q_i(x)$, which together with equations (21) and (23) yields the positivity constraint $\left|g_1\right|\leqslant F_1$.

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Equations (21) and (23) are derived assuming that there is no interference of amplitudes for the lepton scattering at high momentum transfer on one type of quark or another; the final states in the parton model are distinguishable and depend on which quark participates in the scattering and is ejected from the nucleon target; likewise, the derivation of equations (21) and (23) assumes that quantum-mechanical coherence is not possible for different quark scattering amplitudes since the quarks are assumed to be quasi-free. Such interference and coherence effects can arise at lower momentum transfer where quarks can coalesce into specific hadrons and thus participate together in the scattering amplitude. In such a case, the specific quark which scatters cannot be identified as the struck quark. This resonance regime is discussed in sections 3.2 and 3.3.

The parton model naturally predicts (1) Bjorken scaling: the structure functions depend only on x  =  x_Bj; (2) the Callan-Gross relation [84], $F_2=2xF_1$, reflecting the spin-1/2 nature of quarks; i.e. F_L  =  0 (no absorption of longitudinal photons in DIS due to helicity conservation); (3) the interpretation of x_Bj as the momentum fraction carried by the struck quark in the IMF [34], or equivalently, the quarks' LF momentum fraction $x= k^+ / P^+$; and (4) a probabilistic interpretation of the structure functions: they are the square of the parton wave functions and can be constructed from individual quark distributions and polarizations in momentum space. The parton model interpretations of x_Bj and of structure functions is only valid in the DIS limit and at LO in $\alpha_s$. For example, unpolarized PDFs extracted at NLO may be negative [82, 85], see also [86].

In the parton model, only two structure functions are needed to describe the nucleon. The vanishing of g₂ in the parton model does not mean it is zero in pQCD. In fact, pQCD predicts a non-zero value for g₂, see equation (60). The structure function g₂ appears when Q² is finite due to (1) quark interactions, and (2) transverse momenta and spins (see e.g. [15]). It also should be noted that the parton model cannot account for DDIS events $\newcommand{\e}{{\rm e}} \ell p \to \ell' p X$, where the proton remains intact in the final state. Such events contribute to roughly 10% of the total DIS rate.

DIS experiments are typically performed at beam energies for which at most the three or four lightest quark flavors can appear in the final state. Thus, for the proton and the neutron, with three active quark flavors:

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle F_1^{\,p}(x) &= \frac{1}{2} \bigg(\frac{4}{9} \big(u(x)+ \overline{u} (x)\big)+ \frac{1}{9}\big(d(x)+\overline{d}(x)\big)+\frac{1}{9}\big(s(x)+\overline{s}(x)\big)\bigg), \nonumber \\ g_1^{\,p}(x) &=\frac{1}{2}\bigg(\frac{4}{9}\big(\Delta u(x)+\Delta\overline{u}(x)\big)+ \frac{1}{9}\big(\Delta d(x)+\Delta\overline{d}(x)\big) \nonumber \\ & \quad +\frac{1}{9}\big(\Delta s(x)+\Delta\overline{s}(x)\big)\bigg), \nonumber \\ F_1^n(x) &= \frac{1}{2}\bigg(\frac{1}{9}\big(u(x)+\overline{u}(x)\big)+ \frac{4}{9}\big(d(x)+\overline{d}(x)\big)\nonumber \\ & \quad +\frac{1}{9}\big(s(x)+\overline{s}(x)\big)\bigg), \nonumber \\ g_1^n(x) &= \frac{1}{2}\bigg(\frac{1}{9}\big(\Delta u(x)\Delta\overline{u}(x)\big)+ \frac{4}{9}\big(\Delta d(x)+\Delta\overline{d}(x)\big)\nonumber \\ & \quad ++\frac{1}{9}\big(\Delta s(x)+\Delta\overline{s}(x)\big)\bigg), \nonumber \end{align*}$$

where the PDFs $q(x), ~\overline q(x), ~ \Delta q(x)$, and $\Delta \overline q(x)$ correspond to the longitudinal light-front momentum fraction distributions of the quarks inside the nucleon. This analysis assumes SU(2)_f charge symmetry, which typically is believed to hold at the 1% level [87, 88].

In the Bjorken limit, this description provides spin information in terms of x (or x and Q² at lower energies, as discussed below). The spatial spin distribution is also accessible, via the nucleon axial form factors. This is analogous to the fact that the nucleon's electric charge and current distributions are accessible through the electromagnetic form factors measured in elastic lepton-nucleon scattering (see section 3.3). Form factors and particle distributions functions are linked by GPDs and Wigner Functions, which correlate both the spatial and longitudinal momentum information [89], including that of OAM [90].

In pQCD, the struck quarks in DIS can radiate gluons; the simplicity of Bjorken scaling is then broken by computable logarithmic corrections. The lowest-order $\alpha_s$ corrections arise from (1) vertex correction, where a gluon links the incoming and outgoing quark lines; (2) gluon bremsstrahlung on either the incoming and outgoing quark lines; (3) q-$\overline{q}$ pair creation or annihilation. This latter leads to the axial anomaly and makes gluons to contribute to the nucleon spin (see section 5.5). These corrections introduce a power of $\alpha_s$ at each order, which leads to logarithmic dependence in Q², corresponding to the behavior of the strong coupling $\alpha_s(Q^2)$ at high Q² [91].

Amplitude calculations, including gluon radiation, exist up to next-to-next-to leading order (NNLO) in $\alpha_s$ [92]. In some particular cases, calculations or assessments exist up to fourth order e.g. for the Bjorken sum rule, see section 5.5. These gluonic corrections are similar to the effects derived from photon emissions (radiative corrections) in QED; they are therefore called pQCD radiative corrections. As in QED, canceling infrared and ultraviolet divergences appear and calculations must be regularized and then renormalized. Dimensional regularization is often used for pQCD (minimal subtraction scheme, $\overline{MS}$) [93], although several other schemes are also commonly used. The pQCD radiative corrections are described to first approximation by the DGLAP evolution equations [56]. This formalism correctly predicts the Q²-dependence of structure functions in DIS. The pQCD radiative corrections are renormalization scheme-independent at any order if one applies the BLM/PMC [94, 95] scale-setting procedure.

The small-x_Bj power-law Regge behavior of structure functions can be related to the exchange of the Pomeron trajectory using the BFKL equations [57]. Similarly the t-channel exchange of the isospin I  =  1 Reggeon trajectory with $\alpha_R = 1/2$ in DVCS can explain the observed behavior $F_{2p}(x_{Bj},Q^2) - F_{2n}(x_{Bj},Q^2) \propto \sqrt {x_{Bj}}$, as shown by Kuti and Weisskopf [96]. This small-x Regge behavior is incorporated in the LFHQCD structure for the t-vector meson exchange [97]. A general discussion of the application of Regge dynamics to DIS structure functions is given in [98]. The evolution of $g_1(x_{Bj},Q^2)$ at low-x_Bj has been investigated by Kirschner and Lipatov, and Blumlein and Vogt [99], by Bartels, Ermolaev and Ryskin [100]; and more recently by Kovchegov, Pitonyak and Sievert [101]; See [10] for a summary of small-x_Bj behavior of the PDFs. The distribution and evolution at low-x_Bj of the gluon spin contributions $\Delta g(x_{Bj})$ and ${{\rm L}}_g(x_{Bj})$ is discussed in [102], with the suggestion that in this domain, ${{\rm L}}_g(x_{Bj}) \approx -\Delta g(x_{Bj})$. In addition to structure functions, the evolution of the distribution amplitudes in $\ln (Q^2)$ defined from the valence LF Fock state is also known and given by the ERBL equations [58].

Although the evolution of the g₁ structure function is known to NNLO [103], we will focus here on the leading order (LO) analysis in order to demonstrate the general formalism. At leading-twist one finds

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_1(x_{Bj},Q^2)=\frac{1}{2} \sum_i e_i^2 \Delta q_i(x_{Bj},Q^2), \label{g_1 LT evol} \nonumber \end{align} \tag{ 26 }$$

where the polarized quark distribution functions $\Delta q$ obey the evolution equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial \Delta q_i(x,t)}{\partial t}= \frac{\alpha_s(t)}{2\pi} \int_{x}^1 \frac{{\rm d}y}{y} \left[ \Delta q_i(y,t) P_{qq}\left(\frac{x}{y}\right) + \Delta g(y,t) P_{qg}\left(\frac{x}{y}\right) \right] , \label{quark LO evol} \nonumber \end{align} \tag{ 27 }$$

with $t=\ln(Q^2/\mu^2)$. Likewise, the evolution equation for the polarized gluon distribution function $\Delta g$ is

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial\Delta g(x,t)}{\partial t}= \frac{\alpha_s(t)}{2\pi} \int_{x}^1 \frac{{\rm d}y}{y} \bigg[\sum_{i=1}^{2f} \Delta q_i(y,t) P_{gq}\left(\frac{x}{y}\right) + \Delta g(y,t) P_{gg}\left(\frac{x}{y}\right) \bigg]. \label{gluon LO evol} \nonumber \end{align} \tag{ 28 }$$

At LO the splitting functions $P_{\alpha \beta}$ appearing in equations (27) and (28) are given by

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle P_{qq}(z) &= \frac{4}{3}\frac{1+z^2}{1-z}+2\delta(z-1), \nonumber \\ P_{qg}(z) &= \frac{1}{2}\big(z^2-(1-z)^2 \big), \nonumber \\ P_{gq}(z) &= \frac{4}{3}\frac{1-(1-z)^2}{z}, \nonumber \\ P_{gg}(z) &= 3\bigg[\big(1+z^4\big)\bigg(\frac{1}{z}+\frac{1}{1+z} \bigg) - \frac{(1-z)^3}{z}\bigg]+\bigg[\frac{11}{2}-\frac{f}{3} \bigg]\delta(z-1). \nonumber \end{align*}$$

These functions are related to Wilson coefficients defined in the operator product expansion (OPE), see section 4.1. They can be interpreted as the probability that:

• P_qq: a quark emits a gluon and retains z  =  x_Bj/y  of its initial momentum;
• P_qg: a gluon splits into q-$\overline{q}$, with the quark having a fraction z of the gluon momentum;
• P_gq: a quark emits a gluon with a fraction z of initial quark momentum;
• P_gg: a gluon splits in two gluons, with one having the fraction z of the initial momentum.

The presence of P_qg allows inclusive polarized DIS to access the polarized gluon distribution $\Delta g(x_{Bj},Q^2)$, and thus its moment $\newcommand{\e}{{\rm e}} \Delta G \equiv \int_0^1 \Delta g~{\rm d}x$, albeit with limited accuracy. The evolution of g₂ at LO in $\alpha_s$ is obtained from the above equations applied to the Wandzura–Wilczek relation, equation (60).

In general, pQCD can predict Q²-dependence, but not the x_Bj-dependence of the parton distributions which is derived from nonperturbative dynamics (see section 3.1). The high-x_Bj domain is an exception (see section 6.3). The intuitive DGLAP results are recovered more formally using the OPE, see section 4.1.

The success of modeling the nucleon with quasi-free valence quarks and with constituent quark models (see section 3.2.1) suggests that only quarks contribute to the nucleon spin:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J= \frac{1}{2}\Delta\Sigma+{{\rm L}}_q =\frac{1}{2}, \nonumber \end{align} \tag{ 29 }$$

where $\Delta \Sigma$ is the quark spin contribution to the nucleon spin J;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta \Sigma = \sum_q \int_0^1 {\rm d}x \, \Delta q(x), \nonumber \end{align} \tag{ 30 }$$

and ${{\rm L}}_q$ is the quark OAM contribution. Extracted polarized PDFs and modeled quark OAM distributions are shown in figures 7 and 8. It should be emphasized that the existence of the proton's anomalous magnetic moment requires nonzero quark OAM [63]. For instance, in the Skyrme model, chiral symmetry implies a dominant nonperturbative contribution to the proton spin from quark OAM [106]. It is interesting to quote the conclusion from [107]: 'Nearly 40% of the angular momentum of a polarized proton arises from the orbital motion of its constituents. In the geometrical picture of hadron structure, this implies that a polarized proton possesses a significant amount of rotation contribution to S_z and ${{\rm L}}_z$ comes from the valence quarks'. (Emphasis by the author). QCD radiative effects introduce corrections to the spin dynamics from gluon emission and absorption which evolve in $\ln Q^2$. It was generally expected that the radiated gluons would contribute to the nucleon spin, but only as a small correction (beside their effect of introducing a Q²-dependence to the different contributions to the nucleon spin). The speculation that polarized gluons contribute significantly to nucleon spin, whereas their sources—the quarks—do not, is unintuitive, although it is a scenario that was (and still is by some) considered (see e.g. the bottom left panel of figure 18 on page 46). A small contribution to the nucleon spin from gluons would also imply a small role of the sea quarks, so that $\Delta \Sigma$ and the quark OAM would then be understood as coming mostly from valence quarks. In this framework, it was determined that the quark OAM contributes to about 20% [107, 108] based on the values for F and D, the weak hyperon decay constants (see section 5.5), SU(3)_f flavor symmetry and $\Delta s=0$ [109–111]. This prediction was made in 1974 and predates the first spin structure measurements by SLAC E80 [112], E130 [113] and CERN EMC [114].

The origin of the quark OAM was later understood as due to relativistic kinematics [110, 111], whereas $\Delta \Sigma$ comes from the quark axial currents (see discussion below equation (2)). For a nonrelativistic quark, the lower component of the Dirac spinor is negligible; only the upper component contributes to the axial current. In hadrons, however, quarks are confined in a small volume and are thus relativistic. The lower component, which is in a p -wave, with its spin anti-aligned to that of the nucleon, contributes and reduces $\Delta \Sigma$. At that time, it seemed reasonable to neglect gluons, thus predicting a nonzero contribution to J from the quark OAM. The result was the initial expectation $\Delta\Sigma\approx 0.65$ and thus the quark OAM was about 18%. Since this review is also concerned with spin composition of the nucleon at low energy, it is interesting to remark that a large quark OAM contribution would essentially be a confinement effect.

The first high-energy measurements of $g_1(x_{Bj},Q^2)$ was performed at SLAC in the E80 [112] and E130 [113] experiments. The data covered a limited x_Bj range and agreed with the naive model described above. However, the later EMC experiment at CERN [114] measured $g_1(x_{Bj},Q^2)$ over a range of x_Bj sufficiently large to evaluate moments. It showed the conclusions based on the SLAC measurements to be incorrect. The EMC measurement suggests instead that $\Delta \Sigma \approx 0$, with large uncertainty. This contradiction with the naive model became known as the 'spin crisis'.

Although more recent measurements at COMPASS, HERMES and Jlab are consistent with a value of $\Delta \Sigma \approx 0.3$, the EMC indication still stands that gluons and/or gluon and quark OAM are more important than had been foreseen; see e.g. [115]. Since gluons are possibly important, J must obey the total angular momentum conservation law known as the 'nucleon spin sum rule'

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J = \frac{1}{2}\Delta\Sigma(Q^2)+{{\rm L}}_q(Q^2) + \Delta G(Q^2) + {{\rm L}}_g(Q^2) = \frac{1}{2}, \label{eq:spin SR} \nonumber \end{align} \tag{ 31 }$$

at any scale Q. The gluon spin $\Delta G$ represents with ${{\rm L}}_g$ a single term, $\Delta G+{{\rm L}}_g$, since the individual $\Delta G$ and ${{\rm L}}_g$ contributions are not separately gauge-invariant. (This is discussed in more detail in the next section.) The terms in equation (31) are obtained by utilizing LF-quantization or the IMF and the LC gauge, writing the hadronic angular momentum tensor in terms of the quark and gluon fields [110]. In the gauge and frame-dependent partonic formulation, in which $\Delta G$ and ${{\rm L}}_g$ can be separated, equation (31) is referred to as the Jaffe-Manohar decomposition. An alternative formulation is given by Ji's decomposition. It is gauge/frame independent, but its partonic interpretation is not as direct as for the Jaffe-Manohar decomposition [116].

The quantities in equation (31) are integrated over x_Bj. They have been determined at a moderate value of Q², typically 3 or 5 GeV². Equation (31) does not separate sea and valence quark contributions. Although DIS observables do not distinguish them, separating them is an important task. In fact, recent data and theoretical developments indicate that the valence quarks are dominant contributors to $\Delta \Sigma$. We also note that the strange and anti-strange sea quarks can contribute differently to the nucleon spin [117]. Finally, a separate analysis of spin-parallel and antiparallel PDFs is clearly valuable since they have different nonperturbative inputs.

A transverse spin sum rule similar to equation (31) has also been derived [118, 119]. Likewise, transverse versions of the Ji sum rule (see next section) exist [120, 121], together with debates on which version is correct. Transverse spin not being the focus of this review, we will not discuss this issue further.

The Q²-evolution of quark and gluon spins discussed in section 3.1.9 provides the Q²-evolution of $\Delta\Sigma$ and $\Delta G$. The evolution equations are known to at least NNLO and are discussed in section 5.5. The evolution of the quark and gluon OAM is known to NLO [122–126]. The evolution of the nucleon spin sum rule components at LO is given in [122]:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Delta\Sigma(Q^2) = {{\rm constant}}, \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle {{\rm L}}_q(Q^2) = &~\frac{-\Delta \Sigma(Q^2)}{2} + \frac{3n_f}{32+6n_f} \nonumber \\ & + \bigg({{\rm L}}_q(Q_0^2) + \frac{-\Delta \Sigma(Q^2_0)}{2} - \frac{3n_f}{32+6n_f}\bigg)(t/t_0)^{-\frac{32+6n_f}{9\beta_0}}, \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \Delta G(Q^2) = \frac{-4\Delta \Sigma(Q^2)}{\beta_0} + \bigg(\Delta G(Q_0^2)+ \frac{4\Delta \Sigma(Q^2_0)}{\beta_0}\bigg)\frac{t}{t_0}, \nonumber \end{align*}$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle {{\rm L}}_g(Q^2) =&~ -\Delta G(Q^2) + \frac{8}{16+3n_f} \nonumber \\ & + \bigg({{\rm L}}_q(Q_0^2) + \Delta G(Q_0^2) - \frac{8}{16+3n_f} \bigg)(t/t_0)^{-\frac{32+6n_f}{9\beta_0}} \label{eq:LO evol. of spin SR} \nonumber \end{align} \tag{ 32 }$$

with $t=\ln(Q^2/\Lambda_s^2)$ and $Q_0^2$ the starting scale of the evolution. The QCD $\beta$-series is defined here such that $\beta_0=11-\frac{2}{3}n_f$. The NLO equations can be found in [126].

Values for the components of equation (31) obtained from experiments, Lattice Gauge Theory or models are given in section 6.11 and in the appendix. It is important to recall that these values are convention-dependent for several reasons. One is that the axial anomaly shifts contributions between $\Delta \Sigma$ and $\Delta G$, depending on the choice of renormalization scheme, even at arbitrary high Q² (see section 5.5). This effect was suggested as a cause for the smallness of $\Delta \Sigma$ compared to the naive quark model expectation: a large value $\Delta G \approx 2.5$ would increase the measured $\Delta \Sigma$ to about 0.65. Such large value of $\Delta G$ is nowadays excluded. Furthermore, it is unintuitive to use a specific renormalization scheme in which the axial anomaly contributes, to match quark models that do not need renormalization. Another reason is that the definitions of $\Delta G$, ${{\rm L}}_q$, ${{\rm L}}_g$ are also conventional. This was known before the spin crisis [110] but the discussion on what the best operators are has been renewed by the derivation of the Ji sum rule [127]:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle J_{q,g}= \frac{1}{2} \int_{-1}^{1} x \big[E_{q,g}(x,0,0)+H_{q,g}(x,0,0)\big] {\rm d}x, \label{equation-Ji SR} \nonumber \end{align} \tag{ 33 }$$

with $\sum_q J^q + J^g = \frac{1}{2}$ being frame and gauge invariant and J_q,g and the GPDs E_q,g and H_q,g stand either for quarks or gluons. For quarks, $\newcommand{\e}{{\rm e}} J_q \equiv \Delta \Sigma/2 + {{\rm L}}_q$. For gluons, J_g cannot be separated into spin and OAM parts in a frame or gauge invariant way. (However, it can be separated in the IMF, with an additional 'potential' angular momentum term [67].)

Importantly, the Ji sum rule provides a model-independent access to ${{\rm L}}_q$, whose measurability had been until then uncertain. Except for Lattice Gauge Theory (see section 4.2.2) the theoretical assessments of the quark OAM are model-dependent. We mentioned the relativistic quark model that predicted about 20% even before the occurrence of the spin crisis. More recently, investigation within an unquenched quark model suggested that the unpolarized sea asymmetry $\overline{u} - \overline{d}$ is proportional to the nucleon OAM:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle {{\rm L}}(Q^2) \equiv {{\rm L}}_q(Q^2)+{{\rm L}}_g(Q^2) \propto \big(\overline{u}(Q^2) - \overline{d}(Q^2)\big), \label{equation-L propto sea} \nonumber \end{align} \tag{ 34 }$$

where $\overline{q}(Q^2)=\int_0^1 \overline{q}(x,Q^2){\rm d}x$. The non-zero $\overline{u} - \overline{d}$ distribution is well measured [128] and causes the violation of the Gottfried sum rule [129, 130]. The initial derivation of equation (34) by Garvey [131] indicates a strict equality, $L = (\overline{u} - \overline{d}) = 0.147 \pm 0.027$, while a derivation in a chiral quark model [132] suggests $L= 1.5(\overline{u} - \overline{d}) = 0.221 \pm 0.041$. The lack of precise polarized PDFs at low-x_Bj does not allow yet to verify this remarkable prediction [133]. Another quark OAM prediction is from LFHQCD: ${{\rm L}}_q(Q^2 \leqslant Q_0^2)=1$ in the strong regime of QCD, evolving to ${{\rm L}}_q=0.35 \pm 0.05$ at Q²  =  5 GeV², see section 4.4.

Beside equation (33) and possibly equation (34), the quark OAM can also be accessed from the two-parton twist-3 GPD G₂ [66]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\rm L}}_q=-\int\ G^q_2(x,0,0) {\rm d}x, \label{equation-quark OAM from twist-3} \nonumber \end{align} \tag{ 35 }$$

or generalized TMD (GTMD) [44, 53, 134]. TMD allow to infer ${{\rm L}}_q$ model-dependently [49].

Jaffe and Manohar set the original convention to define the angular momenta [110]. They expressed equation (31) using the canonical angular momentum and momentum tensors. This choice is natural since it follows from Noether's theorem [119]. For angular momenta, the relevant symmetry is the rotational invariance of QCD's Lagrangian. The ensuing conserved quantity (i.e. that commutes with the Hamiltonian) is the generator of the rotations. This definition provides the four angular momenta of the longitudinal spin sum rule, equation (31). A similar transverse spin sum rule was also derived [118, 119]. A caveat of the canonical definition is that in equation (31), only J and $\Delta\Sigma$ are gauge invariant, i.e. are measurable. In the light-cone gauge, however, the gluon spin term coincides with the measured observable $\Delta G$. (This is true also in the A⁰  =  0 gauge [15].) The fundamental reason for the gauge dependence of the other components of equation (31) is their derivation in the IMF.

What triggered the re-inspection of the Jaffe-Manohar decomposition and subsequent discussions was that Ji proposed another decomposition using the Belinfante–Rosenfeld energy-momentum tensor [135], which lead to the Ji sum rule [127], equation (33). The Belinfante–Rosenfeld tensor originates from General Relativity in which the canonical momentum tensor is modified so that it becomes symmetric and conserved (commuting with the Hamiltonian): in a world without angular momentum, the canonical momentum tensor would be symmetric. However, adding spins breaks its symmetry. An appropriate combination of canonical momentum tensor and spin tensor yields the Belinfante–Rosenfeld tensor, which is symmetric and thus natural for General Relativity where it identifies to its field source (i.e. the Hilbert tensor). The advantages of such definition are (1) its naturalness even in presence of spin; (2) that it leads to a longitudinal spin sun rule in which all individual terms are gauge invariant; and (3) that there is a known method to measure ${{\rm L}}_q$ (equation (33)), or to compute it using Lattice Gauge Theory (see section 4.2.2). Its caveat is that the nucleon spin decomposition contains only three terms: $\Delta \Sigma$, ${{\rm L}}_q$ and a global gluon term, thus without a clear interpretation of the experimentally measured $\Delta G$. While $\Delta \Sigma$ in the Ji and Jaffe-Manohar decompositions are identical, the ${{\rm L}}_q$ terms are different. That several definitions of ${{\rm L}}_q$ are possible comes from gauge invariance. To satisfy it, quarks do not suffice; gluons must be included, which allows for choices in the separation of ${{\rm L}}_q$ and ${{\rm L}}_g$ [136, 137]. The general physical meaning of ${{\rm L}}_q$ is that it is the torque acting on a quark during the polarized DIS process [39, 138]: Ji's ${{\rm L}}_q$ is the OAM before the probing photon is absorbed by the quark, while the Jaffe-Manohar ${{\rm L}}_q$ is the OAM after the photon absorption, with the absorbing quark kicked out to infinity. These two definitions of ${{\rm L}}_q$ have been investigated with several models, e.g. [137, 139], whose results are shown in section 6.11.2.

Other definitions of angular moments and gluon fields have been proposed to eliminate the gauge-dependence problem [140], leading to a spin decomposition equation (31) with four gauge-invariant terms. The complication is that the corresponding operators use non-local fields, viz fields depending on several space-time variables or, more generally, a field A for which $A(x) \neq {\rm e}^{-{\rm i}px} A(0) {\rm e}^{{\rm i}px}$.

Recent reviews on angular momentum definition and separation are given in [136]. It remains to be added that in practice, to obtain ${{\rm L}}_q$ in a leading-twist (twist 2) analysis, $\Delta \Sigma/2$ must be subtracted, see equation (33). Thus, since $\Delta \Sigma$ is renormalization scheme dependent due to the axial anomaly, ${{\rm ~L}}_q$ is too (but not their sum J_q). A higher-twist analysis of the nucleon spin sum rule allows to separate quark and gluon spin contributions (twist 2 PDFs/GPDs) from their OAM (twist 3 GPD G₂) [66, 67, 121, 134, 141]. It is expected that OAM are twist-3 quantities since they involve the parton's transverse motions. However, the quark OAM, as defined in equation (33) can be related to twist-2 GPDs. Beside GPDs, OAM can also be accessed with GTMDs [49, 53, 68, 142]. It is now traditional to call the Jaffe-Manohar OAM the canonical expression and denote it by l_z, the Ji OAM is called kinematical and denoted by L_z. We will use this convention for the rest of the review.

In summary, the components of equation (31) are scheme and definition (or gauge) dependent. Thus, when discussing the origin of the nucleon spin, schemes and definitions must be specified. This is not a setback since, as emphasized in the preamble, the main object of spin physics is not to provide the pie chart of the nucleon spin but rather to use it to verify QCD's consistency and understand complex mechanisms involving it, e.g. confinement. That can be done consistently in fixed schemes and definitions. This leads us to the next section where such complex mechanisms start to arise.

The basic classification of the hadron mass spectra was motivated by the development of constituent quark models obeying an SU(6) $\supset$ SU$(3)_{\rm flavor} \otimes$ SU$(2)_{\rm spin}$ internal symmetry [109, 143]. Baryons are modeled as composites of three constituent quarks of mass M/3 (modulo binding energy corrections which depend on the specific model) which provides the $J^{\rm PC}$ quantum numbers. The constituent quark model predates QCD but is now interpreted and developed in its framework. Constituent quarks differ from valence quarks—which also determine the correct quantum numbers of hadrons—in that they are not physical (their mass is larger) and are understood as valence quarks dressed by virtual partons. The large constituent quark masses explicitly break both the conformal and chiral symmetries that are nearly exact for QCD at the classical level; see sections 4.3. Constituent quarks are assumed to be bound at LO by phenomenological potentials such as the Cornell potential [144], an approach which was interpreted after the advent of QCD as gluonic flux tubes acting between quarks. The LO spin-independent potential is supplemented by a spin-dependent potential, e.g. by adding exchange of mesons [145], instantons or by including the interaction of a spin-1 gluon exchanged between the quarks ('hyperfine correction' [146, 147]). 'Constituent gluons' have also been used to characterize mesons that may exhibit explicit gluonic degrees of freedom ('hybrid mesons'). The constituent quark models, which have been built to explain hadron mass spectroscopy, can reproduce it well. In particular, they historically lead to the discovery of color charge. Of particular interest to this review, such an approach can also account for baryon magnetic moments which can be distinguished from the constituent quark pointlike (i.e. Dirac) magnetic moments. Another feature of these models relevant to this review is that the physical mechanisms that account for hyperfine corrections are also needed to explain polarized PDFs at large-x_Bj, see section 6.3.1. Hyperfine corrections can effectively transfer some of the quark spin contribution to quark OAM [149], consistent with the need for non-zero quark OAM in order to describe the PDFs within pQCD [150].

In non-relativistic constituent quark models, the quark OAM is zero and there are no gluons: the nucleon spin comes from the quark spins. SU(6) symmetry and requiring that the non-color part of the proton wavefunction is symmetric yield [146, 151]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{SU(6)\,p wavefunction} \left|p\uparrow\right\rangle = \frac{1}{\sqrt{2}}\left|u\uparrow(ud)_{\mathbf s=0,s=0}\right\rangle + \frac{1}{\sqrt{18}}\left|u\uparrow(ud)_{\mathbf s=1,s=0}\right\rangle \nonumber \\ \nonumber -\frac{1}{3}\left(\left|u\downarrow(ud)_{\mathbf s=1,s=1}\right\rangle - \left|d\uparrow(uu)_{\mathbf s=1,s=0}\right\rangle + \sqrt{2}\left|d\downarrow(uu)_{\mathbf s=1,s=1}\right\rangle \right), \nonumber \end{align} \tag{ 36 }$$

where the arrows indicate the projection of the 1/2 spins along the quantization axis, while the subscripts $\mathbf s$ and s denote the total and projected spins of the diquark system, respectively. The neutron wavefunction is obtained from the proton wavefunction via isospin $u \leftrightarrow d$ interchange. The spectroscopy of the excited states varies between models, depending in detail on the choice of the quark potential.

As mentioned in section 3.1.10, the disagreement between the EMC experimental results [114], and the naive $\Delta \Sigma =1$ expectation from the simplest constituent quark models has led to the 'spin crisis'. Myhrer, Bass, and Thomas have interpreted the 'spin crisis' in the constituent quark model framework as a pion cloud effect [115, 152], which together with relativistic corrections and one-gluon exchange, can transfer part of $\Delta \Sigma$ to the quark OAM (mostly to $l^u_q$) [153]. Once these corrections have been applied, the constituent quark picture—which has had success in describing other aspects of the strong force—also becomes consistent with the spin structure data. Relativistic effects, one-gluon exchange and the pion cloud reduce the naive $\Delta \Sigma=1$ expectation by 35%, 25% and 20%, respectively. The quark spin contribution is transferred to quark OAM, resulting in $\Delta \Sigma /2 \approx 0.2$ and $l_q \approx +0.3$. These predictions apply at the low momentum scale where DGLAP evolution starts, estimated to be $Q_0^2 \approx 0.16$ GeV² [154], which could be relevant to the constituent quark degrees of freedom. Evolving these numbers from $Q_0^2$ to the typical DIS scale of 4 GeV² using equations (32) decreases the quark OAM to 0 ($l_q^d \approx - l_q^u \approx 0.1$), transferring it to $\Delta G +{{\rm L}}_g$. Thus, the Myhrer-Bass-Thomas model yields $\Delta \Sigma/2 \approx 0.18$, ${l_q} \approx 0$ and $\Delta G +{{\rm L}}_g \approx 0.32$, with strange and heavier quarks not directly contributing to J.

This result is not supported by those of [126, 155] which assessed the value of L_q at low scales by evolving down large scale LGT estimates of the spin sum rule components. A cause of the disagreement might be that [126, 155] use LGT input, i.e. with the quark OAM kinematical definition, while it is unclear which definition applies to the quark OAM in constituent quark models, such as that used in [154]. Furthermore, the high scale L_q input of [126, 155] stems from early LGT calculations which do not include disconnected diagrams. Those are now known to contribute importantly to the quark OAM, which makes the L_q input of [126, 155] questionnable. Finally, the scale evolutions are preformed in [126, 154, 155] at leading twist, which is known to be insufficient for scales below $Q_0\approx 1$ GeV [156, 157]. (We remark that some higher-twists are effectively included when a non-perturbative $\alpha_s$ is employed). The limitation of these evolutions in the very low scale region characterizing bag models (0.1–0.3 GeV²) is in particular studied in [126]. The authors improved the cloudy bag model calculation of [154] by using the gauge-invariant (kinematical) definition of the spin contributions. It yields $Q_0^2 \approx 0.2$ GeV², $\Delta \Sigma/2=0.23\pm 0.01$, $L_q=0.53\pm 0.09$ and $\Delta G+{{\rm L}}_g=-0.26\pm 0.10$. The importance of the pion cloud to J has also been discussed in [133, 158].

The first nucleon excited state is the P₃₃, also called the $\Delta(1232)~3/2^+$ ($M_{\Delta}=1232$ MeV) in which the three constituent quark spins are aligned while in an S-wave. Thus, the $\Delta(1232)~3/2^+$ has spin $J= 3/2,$ and its isospin is 3/2. The $\Delta(1232)~3/2^+$ resonance is the only one clearly identifiable in an inclusive reaction spectrum. It has the largest cross-section and thus contributes dominantly to sum rules (section 5) and moments of spin structure functions at moderate Q². The nucleon-to-$\Delta$ transition is thus, in this SU(6)-based view, a spin (and isospin) flip; i.e. a magnetic dipole transition quantified by the M₁₊ multipole amplitude. Experiments have shown that there is also a small electric quadrupole component E₁₊ ($E_{1+}/M_{1+} < 0.01$ at Q²  =  0) which violates SU(6) isospin-spin symmetry. This effect can be interpreted as the deformation of the $\Delta(1232)~3/2^+$ charge and current distributions in comparison to a spherical distribution. The nomenclature for multipole longitudinal (also called scalar) amplitudes $S_{l\pm}$, as well as the transverse $E_{l\pm}$ and $M_{l\pm}$ amplitudes is given in [20]. The small E₁₊ and S₁₊ components are predicted by constituent quark models improved with a M₁ dipole-type one-gluon exchange (see section 6.3).

Due to their similar masses and short lifetimes (i.e. large widths in excitation energy W), the higher mass resonances overlap, and thus cannot be readily isolated as distinct contributions to inclusive cross-sections. Their contributions can be grouped into four regions whose shapes and mean-W vary with Q², due to the different Q²-behavior of the amplitudes of the individual resonances. The second resonance region (the first is the $\Delta(1232)~3/2^+$) is located at $W\approx1.5$ GeV and contains the N(1440) 1/2⁺ P₁₁ (Roper resonance), the N(1520) 3/2⁻ D₁₃ and the N(1535) 1/2⁻ S₁₁ which usually dominates over the first two. The third region, at $W\approx1.7$ GeV, includes the $\Delta$(1600) 3/2⁺ P₃₃, N(1680) 5/2⁺ F₁₅, N(1710) 1/2⁻ P₁₁, N(1720) 3/2⁺ P₁₃, $\Delta$(1620) 1/2⁻ S₃₁, N(1675) 5/2⁻ D₁₅, $\Delta$(1700) 3/2⁻ D₃₃, and N(1650) 1/2⁻ S₁₁. The fourth region is located around $W\approx1.9$ GeV and contains the $\Delta$(1905) 5/2⁺ F₃₅, $\Delta$(1920) 3/2⁺ P₃₃, $\Delta$(1910) 1/2⁺ P₃₁, $\Delta$(1930) 5/2⁺ D₃₅ and $\Delta$(1950) 7/2⁺ F₃₇. Other resonances have been identified beyond W  =  2 GeV [18], but their structure cannot be distinguished in an inclusive experiment not only because of the overlap of their widths, but also because of the dominance of the 'non-resonant background'—incoherent scattering similar to DIS at higher Q². Its presence is necessary to satisfy the unitarity of the S matrix in the resonance region.

The DIS cross-section formulae remain valid in the resonance domain. Although the intepretation of structure functions as PDFs cannot be applied, the DIS cross-sections can nevertheless be related to overlaps of LFWFs, as shall be discussed below.

Bloom and Gilman observed [159] that the unpolarized structure function $F_2(x_{Bj},Q^2)$ measured in DIS matches $F_2(x_{Bj},Q^2)$ measured in the resonance domain if the resonance peaks are suitably smoothed and if the Q²-dependence of F₂—due to pQCD radiations and the non-zero nucleon mass—is corrected for. This correspondence is known as hadron–parton duality. It implies that Bjorken scaling, corrected for DGLAP evolution and non-zero mass terms (kinematic twists, see section 4.1), is effectively valid in the resonance region if the resonant structures can be averaged over. This indicates that the effect of the third source of Q²-dependence, the parton correlations (dynamical twists, see section 4.1), can be neglected. Thus the resonance region can be described in dual languages—either hadronic or partonic [160]. The understanding of hadron–parton duality for spin structure functions has also progressed and is discussed in section 6.10.

The doubly polarized elastic cross-section is:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}\sigma}{{\rm d}\Omega}=&~\frac{\sigma_{\rm Mott} E' Z^2}{E}\bigg[\bigg(\frac{Q^2}{\overrightarrow{q}^2}\bigg)^2R_{L}(Q^2,\nu)+\big(\tan^2(\theta/2)\nonumber \\ & -\frac{1}{2}\frac{Q^2}{\overrightarrow{q}^2}\big)R_T(Q^2,\nu)\pm\Delta(\theta^{*},\phi^{*},E,\theta,Q^2)\bigg] , \label{eq:unpocross} \nonumber \end{align} \tag{ 37 }$$

where Z is the target atomic number and the angles are defined in figure 2. R_L and R_T are the longitudinal and transverse response functions associated with the corresponding polarizations of the virtual photon. The cross-section asymmetry $\Delta$, where $\pm$ refers to the beam helicity sign [162], is:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Delta=&~-\left(\tan\frac{\theta}{2}\sqrt{\frac{Q^2}{\overrightarrow{q}^2}+\tan^2\frac{\theta}{2}}R_{T'}(Q^2)\cos\theta^{*}\right. \nonumber \\ &\left. -\frac{\sqrt{2}Q^2}{\overrightarrow{q}^2}\tan\frac{\theta}{2}R_{TL'}(Q^2)\sin\theta^{*}\cos\phi^{*}\right). \nonumber \end{align*}$$

Cross-sections for the targets used in nucleon spin structure experiments are given below:

Nucleon case.

The cross-section for scattering on a longitudinally polarized nucleon is:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}\sigma}{{\rm d}\Omega} =&~ \sigma_{\rm Mott}\frac{E'}{E}\left(W_2+2W_1\tan^2(\theta/2)\right)\label{eq:hadroncross}\nonumber \\ &\times \left(1\pm\sqrt{\frac{\tau_r W_1}{(1+\tau_r)W_2-\tau_r W_1}} \right. \nonumber \\ & \left. \frac{\frac{2M}{\nu}+\sqrt{\frac{W_1}{\tau_r\left((1+\tau_r)W_2-\tau_r W_1\right)}} \frac{2\tau_r M}{\nu}+2(1+\tau_r)\tan^2(\theta/2)}{1+\tau_r\frac{W_1}{\tau_r\left((1+\tau_r)W_2-\tau_r W_1\right)}\left(1+2(1+\tau_r)\tan^2(\theta/2)\right)}\right), \nonumber \end{align} \tag{ 38 }$$

with the recoil term $\newcommand{\e}{{\rm e}} \tau_r\equiv Q^2/(4M^2)$. The hadronic current is usually parameterized by the Sachs form factors, $G_E(Q^2)$ and $G_M(Q^2)$, rather than W₁ and W₂:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle W_1(Q^2)=\tau_r G_M(Q^2)^2, \ \ ~~~W_2(Q^2)=\frac{G_E(Q^2)^2+\tau_r G_M(Q^2)^2}{1+\tau_r}. \nonumber \end{align*}$$

In the nonrelativistic domain the form factors G_E and G_M can be thought of as Fourier transforms of the nucleon charge and magnetization spatial densities, respectively. A rigorous interpretation in term of LF charge densities is given in [163] (nucleon) and [164] (deuteron, see next section). The Dirac and Pauli form factors $F_1(Q^2)$ and $F_2(Q^2)$ can also be used (not to be confused with the DIS structure functions in section 3.1):

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle G_E(Q^2)=F_1(Q^2)-\tau_r\kappa_n F_2(Q^2), \ \ ~~~G_M(Q^2)=F_1(Q^2)+\kappa_n F_2(Q^2), \nonumber \end{align*}$$

where $\kappa_n$ is the nucleon anomalous magnetic moment. The helicity conserving current matrix element generates $F_1(Q^2)$. $F_2(Q^2)$ stems from the helicity-flip matrix element.

LF quantization of QCD provides an interpretation of $F_1(Q^2)$ and $F_2(Q^2)$ which can then be modeled using the structural forms for arbitrary twist inherent to the LFHQCD formalism [165], see section 4.4. In LF QCD, form factors are obtained from the Drell–Yan–West formula [166, 167] as the overlap of the hadronic LFWFs solutions of LF Hamiltonian P⁻, equation (7) [63]. In particular, $F_2(Q^2)$ stems from the overlap of L  =  0 and L  =  1 LFWFs. For a ground state system, the leading-twist of a reaction, that is, its power behavior in Q² (or in the LF impact parameter $\zeta$, see section 4.1), reflects the leading-twist $\tau$ of the target wavefunction, which is equal to the number of constituents in the LF valence Fock state with zero internal orbital angular momentum. This result is intuitively clear, since in order to keep the target intact after elastic scattering, a number $\tau-1$ of gluons of virtuality $\propto$Q² must be exchanged between the $\tau$ constituents. For example, at high-Q², all nucleon components are resolved and the twist is $\tau=3$. Higher Fock states including additional $q \overline{q}$, $q \overline{q} q \overline{q},\ldots$ components generated by gluons are responsible for the higher-twists corrections. These constraints are inherent to LFHQCD which can be used to model the LFWFs and thus obtain predictions for the form factors. Alternatively, one can parameterize the general form expected from the twist analysis in terms of weights reflecting the ratio of the higher Fock state probabilities with respect to the leading Fock state wavefunction. These weights provide the probabilities of finding the nucleon in a higher Fock state, computed from the square of the higher Fock state LFWFs. Two parameters suffice to describe the world data for the four spacelike nucleon form factors [165].

Deuteron case.

The deuteron is a spin-1 nucleus. Three elastic form factors are necessary to describe doubly polarized elastic cross-sections:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}\sigma}{{\rm d}\Omega} = \sigma_{M}\frac{E'}{E}\big(A(Q^2)+B(Q^2)\tan^2(\theta/2)\big)\big(1+A^V + A^T \big), \label{deuteron pol XS} \nonumber \end{align} \tag{ 39 }$$

where $A^V$ and A^T, the asymmetries stemming respectively from the vector and tensor polarizations of the deuteron, are

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle A^V=\frac{3P_b P_z}{\sqrt2} \bigg( \frac{1}{\sqrt2}\cos \theta^* T_{10} - \sin \theta^*T_{11}\bigg), \nonumber \end{align*}$$

where P_b is the beam polarization and P_z the deuteron vector polarization, $P_z = (n_+ - n_-)/n_{\rm tot}$. The n_i are the populations for the spin values i, and $n_{\rm tot}=n_+ + n_- + n_0$,

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle A^T=&~\frac{P_{zz}}{\sqrt{2}}\bigg(\frac{3\cos^2 \theta^{*} -1}{2}T_{20}-\sqrt{\frac{3}{2}}\sin(2\theta^{*})\cos \phi^{*} T_{21} \nonumber \\ & +\sqrt{\frac{3}{2}}\sin^2 \theta^{*} \cos(2\phi^{*})T_{22}\bigg), \nonumber \end{align*}$$

with the deuteron tensor polarization $P_{zz}=(n_+ + n_- - 2n_0)/$ $n_{\rm tot}$.

The seven factors in equation (39), A, B, T₁₀, T₁₁, T₂₀, T₂₁ and T₂₂, are combinations of three form factors (monopole G_C, quadrupole G_Q and magnetic dipole G_M):

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle A &= G_{C}^2+\frac{8}{9}\tau_r^2G_{Q}^2+\frac{2}{3}\tau_r G_M^2, \nonumber \\ B &= \frac{4}{3}\tau_r(1+\tau_r)G_M^2, \nonumber \\ T_{10} &= -\sqrt{\frac{2}{3}}\tau_r(1+\tau_r)\tan(\theta/2)\sqrt{\frac{1}{1+\tau_r}+\tan^2(\theta/2)G_M^2}, \nonumber \\ T_{11} &= \frac{2}{3}\sqrt{\tau_r(1+\tau_r)}\tan(\theta/2)G_M\big(G_C+\frac{\tau_r}{3}G_Q\big), \nonumber \\ T_{20} &= -\frac{1}{\sqrt{2}}\left[\frac{8}{3}\tau_r G_{C}G_{Q}+\frac{8}{9}\tau_r^2G_{Q}^2+\frac{1}{3}\tau_r\left[1+2(1+\tau_r)\tan^2(\theta/2)\right]G_M^2\right], \nonumber \\ T_{21} &= \frac{2}{\sqrt{3}\left[A(Q^2)+B(Q^2)\tan^2(\theta/2)\right]\cos(\theta/2)}\tau_r\left[\tau_r+\tau_r^2\sin^2(\theta/2)G_MG_{C}\right], \nonumber \\ T_{22} &= \frac{1}{\sqrt{3}\left[A(Q^2)+B(Q^2)\tan^2(\theta/2)\right]}\tau_r G_M^2. \nonumber \end{align} \tag{ 40 }$$

P_zz produces additional quantities in other reactions too: in DIS, it yields the $b_1(x_{Bj},Q^2)$ and $b_2(x_{Bj},Q^2)$ spin structure functions [168]. The first one,

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle b_1(x_{Bj},Q^2)= \sum_i \frac{e_i^2}{2} \big[2q^0_{\uparrow}(x_{Bj},Q^2)-\big(q^1_{\downarrow}(x_{Bj},Q^2)-q^{-1}_{\downarrow}(x_{Bj},Q^2) \big) \big], \label{eq:b1 SF} \nonumber \end{align} \tag{ 41 }$$

has been predicted to be small but measured to be significant by the HERMES experiment [169]. For the PDFs $q^{-1,0,1}_{\uparrow,\downarrow}$, the superscript 0 or $\pm1$ indicates the deuteron helicity and the arrow the quark polarization direction, all of them referring to the beam axis.

The six quarks of the deuteron eigenstate can be projected onto five different color-singlet Fock states, only one of which corresponds to a proton–neutron bound state. The other five 'hidden color' Fock states lead to new QCD phenomena at high Q² [170].

Helium 3 case.

The doubly polarized cross-section for elastic lepton-³He scattering is

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle & \frac{{\rm d}\sigma}{{\rm d}\Omega}=\sigma_{\rm Mott}\frac{E'}{E}\left(\frac{G_E^2+\tau_r G_M^2}{1+\tau_r}+2\tau_r G_M^2\tan^2(\theta_{}/2)\right)\nonumber \\& \bigg(1 \pm \frac{1}{\left(\frac{Q^2}{2M\nu+\nu^2}\right)^2(1+\tau_r)G_E^2+\big(\frac{Q^2}{2M\nu+\nu^2}+2\tan^2(\theta/2)\big)\tau_r G_M^2} \nonumber \\ &\times\bigg[2\tau_r G_M^2\cos \theta^{*} \tan(\theta/2)\sqrt{\tan^2(\theta/2)+\frac{Q^2}{2M\nu+\nu^2}} \nonumber \\& +2\sqrt{2\tau_r(1+\tau_r)}G_MG_E\sin \theta^{*} \cos \varphi^{*} \frac{Q^2}{\sqrt{2}\left(2M\nu+\nu^2\right)}\tan(\theta/2)\bigg]\bigg), \nonumber \end{align*}$$

where the form factors are normalized to the ³He electric charge. The magnetic and Coulomb form factors F_m and F_c are sometimes used [172]. They are related to the response functions of a nucleus (A, Z) by $F_c = Z G_E$ and $F_m = \mu_A G_M$ where $\mu_A$ is the nucleus magnetic moment.

If the target is a composite nucleus and the transferred energy $\nu$ is greater than the nuclear binding energy, but still small enough to not resolve the quarks or excite a nucleon, the scattering loses nuclear coherence. For example, the lepton may scatter elastically on one of the nucleons, and the target nucleus breaks. This is quasi-elastic scattering. Its threshold with respect to the elastic peak equals the nuclear binding energy (2.224 MeV for the deuteron, 5.49 MeV for the ³He two-body breakup and 7.72 MeV for its three-body breakup). Unlike elastic scattering, the nucleons are not at rest in the laboratory frame since they are restricted to the nuclear volume. This Fermi motion causes a Doppler-type broadening of the quasi-elastic peak around the breakup energy plus Q²/(2M), the energy transfer in elastic scattering off a free nucleon. The cross-section shape is nearly Gaussian with a width of about 115 MeV (deuteron) or 136 MeV (³He) [173]. This model where the nucleon is assumed to be virtually free (Fermi gas model) provides a qualitative description of the the cross-section, but it does not predict the transverse and longitudinal components of the cross-section, nor the distortions of its Gaussian shape. To account for this, the approximation of free nucleons is abandoned and a model for the nucleon–nucleon interaction is introduced. The simplest implementation is via the 'plane wave impulse approximation' (PWIA), where the initial and final particles (the lepton and nucleons) are described by plane waves in a mean field. In this approach, all nucleons are quasi-free and therefore on their mass-shell, including the nucleon absorbing the virtual photon whose momentum is not changed by the mean field. The other nucleons are passive spectators of the reaction. The nucleon momentum distribution is given by the spectral function $P(k,E)$. Thus, the PWIA hypothesis enables the nuclear tensor to be expressed from the hadronic ones. The PWIA model can be improved by accounting for (1) Coulomb corrections on the lepton lines which distort the lepton plane waves. This corrects for the long distance electromagnetic interactions between the lepton and the nucleus whose interaction is no longer approximated by a single hard photon exchange; (2) final state interactions between the nucleon absorbing the hard photon and the nuclear debris; (3) exchange of mesons between the nucleons (meson exchange currents) which is dominated by one pion exchange; and (4) intermediate excited nucleon configurations such as the Delta-isobar contribution.

An example of a non-local structure function is g₃, equation (64). It depends on the quark field $\psi$ evaluated at the 0 and $\lambda n$ loci. As discussed, the OPE provides a local operator basis. Calculable quantities involve currents such as the quark axial current $\overline{\psi}\gamma_{\mu}\gamma_{5}\psi$. These currents correspond to moments of structure functions. In order to obtain those, e.g. g₁, the moments $\newcommand{\e}{{\rm e}} \Gamma_1^n\equiv \int x^{n-1}g_1{\rm d}x$ can be calculated and Mellin-transformed from moment-space to x_Bj-space. However, the larger the value of n, the higher the degree of the derivatives in the moments (see e.g. equations (57) and (56)), which increases their non-locality. Thus, in practice, only moments up to n  =  3 have been calculated in LGT, which is insufficient to accurately obtain structure functions (see e.g. [190–193] for calculations of $\Gamma_{1,2}^n$ and discussions). The higher-twist terms discussed in section 4.1 have the same problem, with an additional one coming from the twist mixing discussed on page 24. The mixing brings additional 1/a² terms which diverge when $a\to0$. This problem can be alleviated in particular cases by using sum rules which relate a moment of a structure function, whatever its twist content, to a quantity calculable on the lattice.

A method to avoid LGT's non-locality difficulty and compute directly x-dependencies of parton distributions has recently been proposed by X. Ji [194]. A direct application of LGT in the IMF is impractical because the $P \rightarrow \infty$ limit using ordinary time implies that $a \to 0$. Since LFQCD is boost invariant (see section 3.1.3) calculating LC observables using LF quantization would fix this problem. However, direct LC calculations are not possible on the lattice since it is based on Euclidean—rather than real—instant time and because the LC gauge A⁺   =  0 cannot be implemented on the lattice.

To avoid these problems, an operator $O(P,a)$ related to the desired nonperturbative PDF is introduced and computed as usual using LGT; it is then evaluated at a large 3-momentum oriented, e.g. toward the x³ direction. The momentum-dependent result (in the 'instant front form', except that the time is Euclidean: ix⁰) is called a quasi-distribution, since it is not the usual PDF as defined on the LC or IMF. In particular, the range of x_Bj is not constrained by 0  <  x_Bj  <  1. The quasi-distribution computed on the lattice is then related to its LC counterpart $o(\mu)$ through a matching condition $O(P,a) = Z(\mu/P) o(\mu) + \sum_{2n} C_n/P^n$, where the sum represents higher-order power-law contributions. This matching is possible since the operators $O(P,a)$ and $o(\mu)$ encompass the same nonperturbative physics. The matching coefficient $Z(\mu/P)$ can be computed perturbatively [2, 195]. It contains the effects arising from: (1) the particular gauge choice made in the LGT calculation, although it cannot be the LC gauge A⁺   =  0; and (2) choosing a different frame and quantization time when computing quantities using LF quantization and Euclidean instant time quantization in the IMF.

A special lattice with finer spacing a along ix⁰ and x³ is needed in order to compensate for the Lorentz contraction at large P³. Each of the two transverse directions requires discretization enhanced by a factor $\gamma$ (the Lorentz factor of the boost), which becomes large for small-x_Bj physics. The computed PDFs, i.e. the leading twist structure functions, can be calculated for high and moderate x_Bj, as well as the kinematical and dynamical higher-twist contributions. How to compute $\Delta G$ and L_q with this method is discussed in [60, 61, 196], and [186] reviews the method and prospects. Improvements of Ji's method have been proposed, such as e.g. the use of pseudo-distributions [197] instead of quasi-distributions.

The quark OAM definition using either the Jaffe-Manohar or Ji decomposition, see section 3.1.11, corresponds to different choices of the gauge links [67, 68, 134, 138, 198]. Results of calculations related to nucleon spin structure are given in [199–202]. In particular, Ji's method was applied recently to computing $\Delta G$ [203]. Although the validity of the matching obtained in this first computation is not certain, these efforts represent an important new development in the nucleon spin structure studies. More generally, the PDFs, GPDs, TMDs and Wigner distributions are in principle calculable with the innovative approaches described here, which are designed to circumvent the inherent difficulties in the lattice computation of parton distributions.

The Lagrangian for a free spin 1/2 particle is $\mathcal{L}= \overline{\psi}(i\gamma_{\mu}\partial^{\mu}-m)\psi$. The left-hand Dirac spinor is defined as $P_{l}\psi=\psi_{l}$, with $P_{l}=(1-\gamma_{5})/2$ the left-hand helicity state projection operator. Likewise, $\psi_{r}$ is defined with $P_{r}=(1 + \gamma_{5})/2$. If m  =  0 then $\mathcal{L}\mathcal{=L}_l + \mathcal{L}_r$ where $\psi_{l}$ and $\psi_{r}$ are the eigenvectors of P_l and P_r, respectively: the resulting Lagrangian decouples to two independent contributions. Thus, two classes of symmetrical particles with right-handed or left-handed helicities can be distinguished.

Chiral symmetry is assumed to hold approximately for light quarks. If quarks were exactly massless, then $\mathcal{L}_{\rm QCD} = \mathcal{L}^l _{\rm quarks} + \mathcal{L}^r_{\rm quarks} + \mathcal{L}_{\rm int} + \mathcal{L}_{\rm gluons}$. Massless Goldstone bosons can be generated by spontaneous symmetry breaking. The pion spin-parity and mass, which is much smaller than that of other hadrons, allows the identification of the pion with the Goldstone boson. Non-zero quark masses—which explicitly break chiral symmetry—then lead to the non-zero pion mass. The $\chi$PT calculations can be extended to massive quarks by adding a perturbative term $\overline{\psi}m\psi$ which explicitly breaks the chiral symmetry. The much larger masses of other hadrons are assumed to come from spontaneous symmetry breaking caused by quantum effects; i.e. dynamical symmetry breaking. Calculations of observables at small Q² use an 'effective' Lagrangian expressed in terms of hadronic fields, which incorporates chiral symmetry. The resulting perturbative series is a function of $m_{\pi}/M_n$ and the momenta of the on-shell particles involved in the reaction.

Once the quark masses are neglected, the classical QCD Lagrangian $\mathcal{L}_{\rm QCD}$ has no apparent mass scale and is effectively conformal. Since there are no dimensionful parameters in $\mathcal{L}_{\rm QCD}$, QCD is apparently scaleless. This observation allows one to apply the AdS/CFT duality [216] to semi-classical QCD, which is the basis for LFHQCD discussed next. The strong force is effectively conformal at high-Q² (Bjorken scaling), and at low Q², one observes the freezing of $\alpha_s(Q^2)$ [91]. The observation of conformal symmetry at high-Q² is a key feature of QCD. More recently, studying the conformal symmetry of QCD at low Q² has provided new insights into hadron structure, as will be discussed in the next section. However, these signals for conformal scaling fail at intermediate Q² because of quantum corrections—the QCD coupling $\alpha_s(Q^2)$ varies strongly near the mass scale $\Lambda_s$ (the scale arising from quantum effects and the dimensional transmutation property arising from renormalization). The QCD mass scale can also be expressed as $\sigma_{\rm str}$ (the string tension appearing in heavy quark phenomenology) and as $\kappa$ (LFHQCD's universal scale) which controls the slope of Regge trajectories. The pion decay constant $f_\pi$, characterizing the dynamical breaking of chiral symmetry, can also be related to these mass scales [217]. Other characteristic mass scales exist, see [91].

Assuming that quarks are in a S state, i.e. they have no OAM, a quark carrying all the nucleon momentum ($x_{Bj}\to 1$) must carry the nucleon helicity [329]. This implies $A_1 \xrightarrow[x_{Bj} \to 1] {}1$. This is a rare example of absolute prediction from QCD: generally pQCD predicts only the Q²-dependance of observables, see sections 3.1.1 and 3.1.9. (Other examples are the processes involving the chiral anomaly, such as $\pi^0 \to \gamma \gamma$.) Furthermore, the valence quarks dominance makes known the nucleon wavefunction, see equation (36). The BBS [33] and LSS [304]) global fits include these two constraints. The x_Bj-range where the S-state dominates is the only significant assumption of these fits which have been improved to include the $\left|{{\rm L}}_z(x_{Bj})\right|=1$ wavefunction components [150]. The phenomenological predictions [33, 150, 304] for $A_1 (x_{Bj} \to 1)$ are thus based on solid premises. Model predictions also exist and are discussed next.

Modeling the nucleon as made of three constituent quarks is justified in the high-x_Bj DIS domain since there, valence quarks dominate. This finite number of partons and the SU(6) flavor-spin symmetry allow one to construct a simple nucleon wavefunction, see equation (36), leading to $A_1^{\,p}=5/9$ and $A_1^n=0$. However SU(6) is broken, as clearly indicated e.g. by the nucleon-$\Delta$ mass difference of 0.3 GeV or the failure of the SU(6) prediction that $F_2^n/F_2^{\,p}=2/3$, see figure 10. The one-gluon exchange (pQCD 'hyperfine interaction', see page 19) breaks SU(6) and can account for the nucleon-$\Delta$ mass difference. It predicts the same $x_{Bj}\to1$ limits as for pQCD: $A_1^{\,p}= A_1^n=1$. A prediction of the constituent quark model improved with the hyperfine interaction [331] is shown in figure 11.

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Another approach to refine the constituent quark model is using chiral constituent quark models [332]. Such models assume a $\approx$1 GeV scale for chiral symmetry breaking, significantly higher than $\Lambda_s$ (0.33 GeV in the $\overline{MS}$ scheme) and use an effective Lagrangian [333] with valence quarks interacting via Goldstone bosons as effective degrees of freedom. The models include sea quarks. x_Bj-dependence is included phenomenologically in recent models, e.g. in the prediction [334].

Augmenting quark models with meson clouds provides another possible SU(6) breaking mechanism [115, 149]. [335] compares A₁ predictions with this approach and that of the 'hyperfine' mechanism.

Other predictions for A₁ at high-x_Bj exist and are shown in figure 11. They are:

• The statistical model of [336]. It describes the nucleon as fermionic and bosonic gases in equilibrium at an empirically determined temperature;
• The hadron–parton duality (section 6.10). It relates well-measured baryons form factors (elastic or $\Delta(1232)~3/2^+$ reactions, all at high-x_Bj) to DIS structure functions at the same x_Bj [337]. Predictions depend on the mechanism chosen to break SU(6), with two examples shown in figure 11;
• Dyson–Schwinger equations with contact or realistic interaction. They predict $A_1(1)$ values significantly smaller than pQCD [174];
• The bag model of Boros and Thomas, in which three free quarks are confined in a sphere of nucleon diameter. Confinement is provided by the boundary conditions requiring that the quark vector current cancels on the sphere surface [338].
• The quark model of Kochelev [339] in which the quark polarization is affected by instantons representing non-perturbative fluctuations of gluons.
• The chiral soliton models of Wakamatsu [340] and Weigel et al [341] in which the quark degrees of freedom explicitly generate the hadronic chiral soliton properties of the Skyrme nucleon model.
• The quark-diquark model of Cloet et al [342].

Experimental results on A₁ [257, 267, 279, 280, 282, 283, 288, 291] are shown in figure 11. They confirm that SU(6), whose prediction is shown by the flat lines in figure 11, is broken. The x_Bj-dependence of A₁ is well reproduced by the constituent quark model with 'hyperfine' corrections. The systematic shift for $A_1^n$ at x_Bj  <  0.4 may be a sea quark effect. The BBS/LSS fits to pre-JLab data disagrees with these data. The fits are constrained by pQCD but assume no quark OAM. Fits including it [150] agree with the data, which suggests the importance of the quark OAM. However, the relation between the effect of states $\left|{{\rm L}}_z(x_{Bj})\right|=1$ at high x_Bj and $\Delta L$ in equation (31) remains to be elucidated. To solve this issue, the nucleon wavefunction at low x_Bj must be known. While the data have excluded some of the models (bag model [338], or specific SU(6) breaking mechanisms in the duality approach), high-precision data at higher x_Bj are needed to test the remaining predictions. Such data will be taken at JLab in 2019 [343].

$\Gamma_1^{\,p}$ and $\Gamma_1^n$ moments: the measured $\Gamma_1(Q^2)$ is constructed by integrating g₁ from $x_{Bj,{\rm min}}$ up to the pion production threshold. $x_{Bj,{\rm min}}$, the minimum x_Bj reached, depends on the beam energy and minimum detected scattering angle for a given Q² point. Table 1 on page 33 provides these limits. When needed, contributions below x_Bj,min are estimated using low-x_Bj models [345, 346]. For the lowest Q², typically below the GeV² scale, the large-x_Bj contribution (excluding elastic) is also added when it is not measured. The data for $\Gamma_1$, shown in figure 12, are from SLAC [267–270], CERN [114, 271]—[321, 324, 347, 348]—[323], DESY [344] and JLab [275]—[280, 282, 285, 286]—[288, 349, 350].

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Bjorken sum $\Gamma_1^{\,p-n}$: the proton and neutron (or deuteron) data can be combined to form the isovector moment $\Gamma_1^{\,p -n}$. The Bjorken sum rule predicts that $\Gamma_1^{\,p -n} \xrightarrow[Q^2 \to \infty] {} g_A/6$ [235]. The prediction is generalized to finite Q² using OPE, resulting in a relatively simple leading–twist Q²-evolution in which only non-singlet coefficients remain, see equation (54). The sum rule has been experimentally validated, most precisely by E155 [257]: $\Gamma_1^{\,p-n}=0.176 \pm 0.003 \pm 0.007$ at Q²  =  5 GeV², while the sum rule prediction at the same Q² is $\Gamma_1^{\,p-n} = 0.183 \pm 0.002$. $\Gamma_1^{\,p-n}$ was first measured by SMC [271] and then E143 [268], E154 [256], E155 [257] and HERMES [344]. Its Q²-evolution was mapped at JLab [282, 349, 350]. The latest measurement (COMPASS) yields $\Gamma_1^{\,p-n}= 0.192 \pm 0.007$ (stat) $\pm 0.015$ (syst) [321–324, 348].

As an isovector quantity, $\Gamma_1^{\,p -n}$ has no $\Delta(1232)~3/2^+$ resonance contribution. This simplifies $\chi$PT calculations, which may remain valid to higher Q² than typical for $\chi$PT [351]. In addition, a non-singlet moment is simpler to calculate with LGT since the CPU-expensive disconnected diagrams (quark loops) do not contribute. (Yet, the axial charge g_A and the axial form factor $g_A(Q^2)$ remain a challenge for LGT [352] because of their strong dependence to the lattice volume. Although the calculations are improving [353], the LGT situation for g_A is still unsatisfactory.) Thus, $\Gamma_1^{\,p -n}$ is especially convenient to test the techniques discussed in section 4. As for all moments, a limitation is the impossibility to measure the $x_{Bj} \to 0$ contribution, which would require infinite beam energy. The Regge behavior $g_1^{\,p-n}(x_{Bj}) = (x_0/x_{Bj}){}^{0.22}$ may provide an adequate low-x_Bj extrapolation [346] (see also [99–101]).

At Q²  =  0, the GDH sum rule, equation (45), predicts $d \Gamma_1/d Q^2$ (see figure 12). At small Q², $\Gamma_1(Q^2)$ can be computed using $\chi$PT. The comparison between data and $\chi$PT results on moments is given in table 2 in which one sees that in most instances, tensions exist between data and claculations of $\Gamma_1$.

Table 2. Comparison between $\chi$PT results and data for moments. The bold symbols denote moments for which $\chi$PT was expected to provide robust predictions. 'A' means that data and calculations agree up to at least Q²  =  0.1 GeV², 'X' that they disagree and '—' that no calculation is available. The p   +  n superscript indicates either deuteron data without deuteron break-up channel, or proton  +  neutron moments added together with neutron information either from D or ³He.

Reference	$\Gamma_1^{\,p}$	$\Gamma_1^n$	$\pmb{\Gamma_1^{\,p-n}}$	$\Gamma_1^{\,p+n}$	$\gamma_0^{\,p}$	$\gamma_0^n$	$\pmb{\gamma_0^{\,p-n}}$	$\gamma_0^{\,p+n}$	$\pmb{\delta_{LT}^n}$	$d_2^n$
Ji 1999 [210, 211]	X	X	A	X	—	—	—	—	—	—
Bernard 2002 [208, 209]	X	X	A	X	X	A	X	X	X	X
Kao 2002 [212]	—	—	—	—	X	A	X	X	X	X
Bernard 2012 [213]	X	X	A	X	X	A	X	X	X	—
Lensky 2014 [214]	X	A	A	A	A	X	X	X	∼ A	A

The models of Soffer–Teryaev [354], Burkert–Ioffe [355] and Pasechnik et al [356] agree well with the data, as does the LFHQCD calculation [357]. The Soffer–Teryaev model uses the weak Q²-dependence of $\Gamma_T=\Gamma_1+\Gamma_2$ to robustly interpolate $\Gamma_T$ between its zero value and known derivative at Q²  =  0 and its known values at large Q². $\Gamma_1$ is obtained from $\Gamma_T$ using the BC sum rule, equation (50), where PQCD radiative corrections and higher-twists are accounted for. Pasechnik et al improved this model by using for the pQCD and higher-twist corrections a strong coupling $\alpha_s$ analytically continued at low-Q², which removes the unphysical Landau-pole divergence at $Q^2=\Lambda_s^2$, and minimizes higher-twist effects [91]. This extends pQCD calculations to lower Q² than typical. The improved $\Gamma_1$ is continued to Q²  =  0 by using $\Gamma_1(0)=0$ and ${\rm d}\Gamma_1(0)/{\rm d}Q^2$ from the GDH sum rule. The Burkert–Ioffe model is based on a parameterization of the resonant and non-resonant amplitudes [358], complemented with a DIS parameterization [236] based on vector dominance. In LFHQCD, the effective charge $\alpha_{g_1}$ (viz the coupling $\alpha_s$ that includes the pQCD gluon radiations and higher-twist effects of $\Gamma_1^{\,p-n}$ [91]) is computed and used in the leading order expression of the Bjorken sum to obtain $\Gamma_1^{\,p-n}$.

The leading-twist Q²-evolution is shown in figure 12 by the gray band. The values a₈  =  0.579, g_A  =  1.267 and $\Delta \Sigma^{\,p} = 0.15$ ($\Delta \Sigma^n = 0.35$) were used to anchor the $\Gamma_1^{\,p (n)}$ evolutions, see equation (52). For $\Gamma_1^{\,p-n}$, g_A suffices to fix the absolute scale. In all cases, leading-twist pQCD follows the data down to surprisingly low Q², exhibiting hadron–parton global duality i.e. an overall suppression of higher-twists, see sections 6.9 and 6.10.

Neutron results.

Proton results.

Conclusion.

The Efremov–Leader–Teryaev sum rule.

The $\gamma_0^n$ extracted either from ³He [275] or D [279] agree well with each other. The MAID phenomenological model [247] agrees with the $\gamma_0^n$ data, and so do the $\chi$PT results (table 2), except the recent Lensky et al calculation [214]. For $\gamma_0^{\,p}$, the situation is reversed: only [214] agrees well with the data, but not the others (including MAID). This problem motivated an isospin analysis of $\gamma_0$ [350] since, e.g. axial-vector meson exchanges in the t  −  channel (short-range interaction) that are not included in computations could be important for only one of the isospin components of $\gamma_0$. $\chi$PT calculations disagree with $\gamma_0^{\,p +n}$ but MAID agrees. Alhough the $\Delta(1232)~3/2^+$ is suppressed in $\gamma_0^{\,p-n}$, $\chi$PT disagrees with the data. Thus, the disagreement on $\gamma_0^{\,p}$ and $\gamma_0^n$ cannot be assigned to the $\Delta(1232)~3/2^+$. MAID also disagrees with $\gamma_0^{\,p-n}$.

Since the $\Delta(1232)~3/2^+$ is suppressed in $\delta_{LT}$, it was expected that its $\chi$PT calculation would be robust. However, the $\delta_{LT}^n$ data [275] disagreed with the then available $\chi$PT results. This discrepancy is known as the '$\delta_{LT}$ puzzle'. Like $\gamma_0$, an isospin analysis of $\delta_{LT}$ may help with this puzzle. The needed $\delta_{LT}^{\,p}$ data are becoming available [326]. The second generation of $\chi$PT calculations on $\delta_{LT}^n$ [213, 214] agrees better with the data. At larger Q² (5 GeV²), the E155x data [270] agree with a quenched LGT calculation [190, 191]. At large Q², generalized spin polarisabilities are expected to scale as 1/Q⁶, with the usual additional softer dependence from pQCD radiative corrections [205, 234]. Furthermore, the Wandzura–Wilczek relation, equation (60), relates $\delta_{LT}$ to $\gamma_0$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \delta_{LT}(Q^2)\to{\frac{1}{3}}\gamma_0(Q^2)\ \ \ {\rm if~} g_2 \approx g_2^{\rm WW}. \label{eq:relation gam0 deltaLT} \nonumber \end{align} \tag{ 67 }$$

The available data being mostly at Q²  <  1 GeV², this relation and the scaling law have not been tested yet. Furthermore, the signs of the $\gamma_0$ and $\delta_{LT}$ data disagree with equation (67). These facts are not worrisome: for $\Gamma_1$ and $\Gamma_2$, scaling is observed for $Q^2 \gtrsim1$ GeV², when the overall effect of higher-twists decreases. For higher moments, resonances contribute more, so scaling should begin at larger Q². The violation of equation (67) is consistent with the fact that $g_2 \neq g_2^{\rm WW}$ in the resonance domain, see section 6.9.3.

At moderate Q², $\overline{d_2}^n$ is positive and reaches a maximum at $Q^2 \gtrsim0.4$ GeV². Its sign is uncertain at large Q². At low Q² the comparison with $\chi$PT is summarized in table 2. MAID agrees with the data. That MAID and the RSS datum (both covering only the resonance region) match the DIS-only global fits and E155x datum suggests that hadron–parton duality is valid for $d_2^n$, albeit uncertainties are large. The LGT [190, 191, 360], Sum Rule approach [362], Center-of-Mass bag model [363] and Chiral Soliton model [364] all yield a small $d_2^n$ at Q²  >  1 GeV², which agrees with data. At these large Q², the data precision is still insufficient to discriminate between these predictions. The negative $d_2^n$ predicted with a LC model [44] disagrees with the data.

Proton data are scarce, with a datum from RSS [277] and one from E155x [270]. In figure 13, the RSS point was evolved to the E155x Q² assuming the 1/Q-dependence expected for a twist 3 dominated quantity (neglecting the weak log dependence from pQCD radiation). The E155x and RSS results agree although RSS measured only the resonance contribution. As for $d_2^n$, this suggests that hadron–parton duality is valid for $d_2^{\,p}$. However, this conclusion is at odds with the mismatch between the (DIS-only) JAM global PDF fit [361] and the (resonance-only) result from RSS.

Overall, $\overline{d_2}$ is small compared to the twist 2 term ($| \Gamma_1| \approx 0.1$ typically at Q²  =  1 GeV², see figure 12) or to the twist 4 term ($f_2 \approx 0.1$, see figure 14). This smallness was predicted by several models. The high-precision JLab experiments measured a clearly non-zero $\overline{d_2}$. More data for $\overline{d_2^{\,p}}$ are needed and will be provided shortly at low Q² [326] and in the DIS [289], see table 1. Then, the 12 GeV upgrade of JLAB will provide $\overline{d_2}$ in the DIS with refined precision, in particular with the SoLID detector [365].

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The higher-twist contribution to $\Gamma_1$ can be obtained by fitting its data with a function conforming to equations (51)–(52) and (54)–(55). The perturbative series is truncated to an order relevant to the data accuracy. Once $\mu_4$ is extracted, the pure twist 4 matrix element f ₂ is obtained by subtracting a₂ (twist 2) and d₂ (twist 3) from equation (55). For $\Gamma_1^{\,p,n}$, $\mu_2^{\,p,n}$ is set by fitting high-Q² data, e.g. $Q^2 \geqslant 5$ GeV², and assuming that higher-twists are negligible there. For $\Gamma_1^{\,p-n}$, $\mu_2^{\,p-n}$ is set by g_A  =  1.2723(23) [17]. The resulting $\mu_2^{\,p,n}$, together with a₈ from the hyperons $\beta$-decay, yield $\Delta \Sigma = 0.169 \pm 0.084$ for the proton and $\Delta \Sigma = 0.35 \pm 0.08$ for the neutron [282, 367, 368]. The discrepancy may come from the low-x_Bj part of $\Gamma_1^n$, which is still poorly constrained, as the COMPASS deuteron data [272] suggest. Specifically, it may be the low-x_Bj contribution to the isoscalar quantity $\Gamma_1^{n+p}$, since $\Gamma_1^{\,p-n}$ agrees well with the Bjorken sum rule. Another possibility is a SU(3)_f violation. The $\Delta \Sigma$ obtained from global analyses (see section 6.11) mix the proton and neutron data and agree with the averaged value of $\Delta \Sigma^{\,p}$ and $\Delta \Sigma^n$.