Spinning protons and gluons in the ${\eta }^{\prime }$

1. Introduction

Harald Fritzsch is renowned for his many creative and seminal contributions covering the origins and development of QCD, electroweak physics, ideas with possible unification of fundamental interactions and cosmology, all driven by a strong physical intuition. My first meeting with Harald was at the Rencontres de Blois conference in 1994. Then followed many stimulating discussions, primarily in Austria and in Munich. In this contribution I cover a selection of topics involving non-perturbative glue in QCD and the transition from current to constituent quarks. The emphasis will be on the spin structure of the proton and physics of the ${\eta }^{\prime }$ meson, topics where Harald made initial key contributions. The present status of this physics is reviewed along with some challenges for fresh and forthcoming experiments from high energy deep inelastic scattering through to searches for ${\eta }^{\prime }$ meson bound states in nuclei. The proton’s internal spin structure along with the ${\eta }^{\prime }$ brings together many aspects of QCD including confinement dynamics, dynamical chiral symmetry breaking, the QCD axial anomaly and both perturbative and topological properties of gluons as well as fresh ideas about quantum entanglement in QCD.

2. Gluons and Hadrons

QCD is our theory of strong interactions and the structure of hadrons. Historically, it developed from the Eightfold Way patterns observed in hadron spectroscopy with wavefunctions described in terms of SU(3) flavor, SU(2) spin and, inside baryons, antisymmetric in a new SU(3) color label plus the parton description of deep inelastic scattering. Then came the insight that color is a dynamical quantum number and the discovery of QCD as a non-abelian local gauge theory with colored gluons as the gauge bosons mediating interactions between quarks and gluons.^1,2 Asymptotic freedom^3,4 with the essential role of the non-abelian three gluon vertex provided the connection between high energy and low energy phenomenology with the realization that the QCD coupling ${\alpha }_s(Q^2)$ decreases logarithmically with increasing momentum transfer $Q^2$, with small interaction strength in the ultraviolet and strong interactions in the infrared. The glue that binds the proton plays an essential role in its mass and internal spin structure.

Low energy QCD is characterized by confinement and dynamical chiral symmetry breaking, DChSB. The gluonic confinement potential contributes most of the proton’s mass of 938MeV with the rest determined by small quark mass perturbations. The masses of the proton’s constituent two up quarks and one down quark are about 2.2MeV for each up quark and 4.7MeV for the down quark from the QCD Lagrangian. The color hyperfine one-gluon-exchange potential, OGE, between confined quarks generates the mass splitting between the spin $\frac{1}{2}$ nucleon and its spin $\frac{3}{2}$ lowest mass $\mathrm{\Delta }(1232)$ resonance excitation.⁵ Chiral symmetry is dynamically broken with formation of a vacuum quark condensate. The lightest mass pseudoscalar mesons, the pion and kaon, are would-be Nambu–Goldstone bosons associated with DChSB and special with their mass squared proportional to the masses of their valence quarks inside, $m_P^2\propto m_q$.⁶ Their isoscalar partners, the $\eta$ and ${\eta }^{\prime }$ mesons, are sensitive both to Goldstone dynamics and to non-perturbative gluon topology in the singlet channel which gives them extra mass and interaction — see Sect. 4 below.

Hadron masses are connected to gluonic matrix elements via the trace anomaly in the QCD energy–momentum tensor.^7,8 Whereas QCD with massless quarks is classically scale invariant, the proton mass is finite with infrared physics characterized by the infrared scale ${\mathrm{\Lambda }}_{\mathrm{\text{qcd}}}\approx 200$MeV associated with the running QCD coupling ${\alpha }_s$. Scale/conformal transformations are associated with the scale or dilation current $d^{\mu }=x_{\nu }{\theta }^{\mu \nu }$ with ${\theta }_{\mu \nu }$ the QCD energy–momentum tensor; $d_{\mu }$ satisfies the divergence equation ${\partial }_{\mu }{d}^{\mu }={\theta }_{\mu }^{\mu }$ with

This is non-vanishing for massless quarks with $\beta ({\alpha }_s)/4{\alpha }_s{G}_{\mu \nu }^a{G}_a^{\mu \nu }$ the trace anomaly term. Here ${\gamma }_m$ is the quark mass anomalous dimension, ${\mu }^2\frac{d}{d{\mu }^2}{m}_q={\gamma }_m{m}_q$ with ${\gamma }_m=-{\alpha }_s/\pi +\cdots$, $m_q$ is the renormalized quark mass and $\mu$ is the renormalization scale; $\beta ({\alpha }_s)$ is the QCD $\beta$-function The forward proton matrix element of ${\theta }_{\mu \nu }$ is $〈p,s|{\theta }_{\mu \nu }|p,s〉=p_{\mu }{p}_{\nu }$ with trace $〈p,s|{\theta }_{\mu }^{\mu }|p,s〉=M^2$ relating the proton mass M squared to the gluonic trace anomaly term.⁹ (Here $p_{\mu }$ denotes the proton’s momentum vector and $s_{\mu }$ its spin vector). In contrast, pions and kaons would be massless in the chiral limit with massless quarks. Here the gluonic trace anomaly term must vanish. Internal binding cancels against individual quark–antiquark terms as manifest in, e.g., the Nambu–Jona-Lasino model^10,11 with the massless pions and kaons emerging as Goldstone bosons. At low resolution the three valence quarks in the proton behave as massive constituent quark quasiparticles through interaction with the vacuum condensate produced by DChSB.

3. The Spin Structure of the Proton

The spin structure of the proton has brought many surprises and continues to inspire a vast global programme of research to understand QCD spin dynamics. Key experiments include inclusive and semi-inclusive polarized deep inelastic scattering and high energy polarized proton–proton collisions.

One finds that just about 30% of the proton’s spin is carried by the spin of the quarks inside. This result is deduced from measurements of the proton’s $g_1(x,Q^2)$ deep inelastic spin structure function; $Q^2$ denotes minus the four-momentum transfer squared in the deep inelastic process and x is the Bjorken variable. One evolves all data points with $Q^2>1$GeV² using next-to-leading order, NLO, QCD evolution to the same $Q^2$ and then takes the first moment :

Here $g_A^{(3)}$, $g_A^{(8)}$ and $g_A^{(0)}{|}_{\mathrm{\text{inv}}}$ are the isovector, SU(3) octet and scale-invariant flavor-singlet axial-charges respectively. The flavor non-singlet $c_{\mathrm{\text{NS}}\ell }$ and singlet $c_{\mathrm{\text{S}}\ell }$ Wilson coefficients are calculable in $\ell$-loop perturbative QCD. These perturbative QCD coefficients have been calculated to $O({\alpha }_s^3)$ precision.^12,13 For ${\alpha }_s=0.3$ typical of the deep inelastic experiments one finds $\{1+{\sum }_{\ell =1}^3{c}_{\mathrm{\text{NS}}\ell }{\alpha }_s^{\ell }(Q)\}=0.85$ and $\{1+{\sum }_{\ell =1}^3{c}_{\mathrm{\text{S}}\ell }{\alpha }_s^{\ell }(Q)\}=0.96$. The term ${\beta }_{\infty }$ represents a possible leading-twist subtraction constant from the circle at infinity when one closes the contour in the complex plane in the dispersion relation.¹⁴ The subtraction constant affects just the first moment and corresponds to a possible contribution at Bjorken x equal to zero.

In terms of the flavor-dependent axial-charges

the isovector, octet and singlet axial charges are The singlet axial-charge comes with a two-loop anomalous dimension and is often quoted as the value evolved to infinity using 3 flavor QCD evolution, coinciding with the scale invariant form $g_A^{(0)}{|}_{\mathrm{\text{inv}}}$ with the renormalization scale dependence parametrized by the factor $E({\alpha }_s)$ factored out.

In the parton model the quantities $\mathrm{\Delta }q$ are interpreted (before gluonic effects in the flavor-singlet channel) as the fraction of the proton’s spin carried by quarks and antiquarks of flavor q. The isovector axial-charge is measured independently in neutron $\beta$-decays ($g_A^{(3)}=1.275\pm 0.001$¹⁵) and the octet axial charge is commonly taken to be the value extracted from hyperon $\beta$-decays assuming a two-parameter SU(3) fit ($g_A^{(8)}=0.58\pm 0.03$¹⁶). The SU(3) symmetry assumption here may be strongly broken, e.g., by pion cloud effects — see below — and the error on $g_A^{(8)}$ could really be as large as 25%.⁹

With this input, the polarized deep inelastic scattering experiments are interpreted in terms of a small value for the flavor-singlet axial-charge. Using the SU(3) value for $g_A^{(8)}$ and assuming no leading twist subtraction constant COMPASS found¹⁷

or $g_A^{(0)}{|}_{\mathrm{\text{pDIS}},Q^2\to \infty }=0.31\pm 0.06$ taking into account QCD renormalization group evolution. (This deep inelastic quantity misses any contribution to $g_A^{(0)}{|}_{\mathrm{\text{inv}}}$ from a possible delta function at $x=0$). When combined with $g_A^{(8)}=0.58\pm 0.03$, the value of $g_A^{(0)}{|}_{\mathrm{\text{pDIS}}}$ in Eq. (5) corresponds to a negative strange-quark polarization $\mathrm{\Delta }{s}_{Q^2\to \infty }=\frac{1}{3}(g_A^{(0)}{|}_{\mathrm{\text{pDIS}},Q^2\to \infty }-g_A^{(8)})=-0.09\pm 0.02$ — that is, polarized in the opposite direction to the spin of the proton. With this $\mathrm{\Delta }s$, the following values for the up and down quark polarizations are obtained: $\mathrm{\Delta }{u}_{Q^2\to \infty }=0.84\pm 0.02$ and $\mathrm{\Delta }{d}_{Q^2\to \infty }=-0.44\pm 0.02$. As a consistency check, the Bjorken sum-rule relates the difference in the first moments of $g_1$ for proton and neutron targets to the isovector $g_A^{(3)}$ with extracted value $1.29\pm 0.05\pm 0.10$.¹⁷ This value agrees well with the number from neutron $\beta$-decays, so here the theory is working as it should.

The value in Eq. (5) compares with the estimates of about 0.6 from the simplest relativistic quark models like the MIT Bag, which associates the extra 40% with quark orbital angular momentum. The value 0.6 is also the value one would expect by taking the octet axial charge if extracted assuming good SU(3) flavor symmetry in the hyperon $\beta$-decays and assuming $\mathrm{\Delta }s=0$. The initial EMC measurement at CERN inspired considerable surprise with a first value of $g_A^{(0)}$ consistent with zero.¹⁸

Looking in the data the small value of $g_A^{(0)}$ measured in the experiments is associated with a collapse in the isoscalar deuteron spin structure function to something close to zero in the low x region between 0.004 and 0.05. The convergence of the first moment integrals is shown in Fig. 1. Here “Bjorken” denotes the theoretical expectation for the isovector sum-rule ${\int }_0^{1}dx{g}_1^{(p-n)}$ and “Ellis–Jaffe” denotes the expectation for the isosinglet combination ${\int }_0^{1}dx{g}_1^{(p+n)}$ if the strange term $\mathrm{\Delta }s$ were zero and if good SU(3) symmetry were working with the nucleon’s octet axial charge. One observes that the proton spin puzzle is associated with this collapse in the isosinglet structure function for $x\lesssim 0.05$ which needs to be understood. This is in contrast with the isovector spin structure function which continues to rise with decreasing x and with unpolarized deep inelastic scattering, where the low x structure function is dominated by large isoscalar gluonic pomeron exchanges.¹⁹

Key issues are the interpretation of the flavor-singlet axial-charge and possible SU(3) breaking in the octet term used to extract it. Theoretical QCD analysis based on the axial anomaly leads to the formula

(see Refs. 14, 21,22,23,24). Here $\mathrm{\Delta }{g}_{\mathrm{\text{partons}}}$ is the amount of spin carried by polarized gluon partons in the polarized proton with ${\alpha }_s\mathrm{\Delta }g\sim constant$ as $Q^2\to \infty$^21,22 and the growth in gluon polarization at large $Q^2$ compensated by similar growth with opposite sign in the gluon orbital angular momentum; $\mathrm{\Delta }{q}_{\mathrm{\text{partons}}}$ measures the spin carried by quarks and anti-quarks carrying “soft” transverse momentum $k_t^2\sim {𝒪}(P^2,m^2)$ where $P^2$ is a typical gluon virtuality in the nucleon and m is the light quark mass. The polarized gluon term is associated with events in polarized deep inelastic scattering where the hard photon strikes a quark or anti-quark generated from photon–gluon fusion and carrying $k_t^2\sim Q^2$.²³ It is associated with the QCD axial anomaly in perturbative QCD.

${{𝒞}}_{\infty }$ denotes a potential non-perturbative gluon topological contribution with support only at Bjorken $x=0$.¹⁴ It is associated with a possible subtraction constant in the dispersion relation for $g_1$ and, if finite, it would be associated with a $J=1$ Regge fixed pole with non-polynomial residue. If non-zero it would mean that ${lim}_{ϵ\to 0}{\int }_{ϵ}^{1}dx{g}_1$ will measure the difference of the singlet axial-charge and the subtraction constant contribution; that is, polarized deep inelastic scattering measures the combination $g_A^{(0)}{|}_{\mathrm{\text{pDIS}}}=g_A^{(0)}-{{𝒞}}_{\infty }$. Any finite ${{𝒞}}_{\infty }$ term would show up as a difference²⁵ in the value of $\mathrm{\Delta }s$ extracted from polarized deep inelastic scattering and from elastic neutrino proton scattering, which measures the combination ${(\mathrm{\Delta }u-\mathrm{\Delta }d-\mathrm{\Delta }s)}_{Q^2\to \infty }$ up to small heavy quark radiative corrections, evaluated to NLO accuracy in Ref. 26, and with any ${{𝒞}}_{\infty }$ term included in the full axial-current matrix elements. Leading twist subtraction constant corrections proportional to $\delta (x)$ are known in unpolarized deep inelastic scattering through the Schwinger term sum-rule for the longitudinal structure function $F_L(x,Q^2)$.²⁷

How should we understand the parton model formula, Eq. (6), in the context of the operator product expansion description of polarized deep inelastic scattering? The connection is subtle. There is no spin-one, local, gauge invariant gluonic operator with the quantum numbers of the flavor-singlet axial vector current. However, gluonic input does enter through the QCD axial anomaly. The gauge invariantly renormalized flavor-singlet axial-vector current $J_{\mu 5}={\sum }_q{\bar{\psi }}_q{\gamma }_{\mu }{\gamma }_5{\psi }_q$ satisfies the anomalous divergence equation

Here $G_{\mu \nu }^a$ is the gluon field tensor and ${\tilde{G}}_{\mu \nu }^a$ the corresponding dual tensor. The gauge invariant current can be written as the sum $J_{\mu 5}=J_{\mu 5}^{\mathrm{\text{con}}}+2f{K}_{\mu }$ with ${\partial }^{\mu }{K}_{\mu }=\frac{{\alpha }_s}{8\pi }G.\tilde{G}$ being a total divergence involving the gauge dependent Chern–Simons current $K^{\mu }=\frac{g^2}{32{\pi }^2}{ϵ}^{\mu \alpha \beta \gamma }{A}_{\alpha }^a(G_{\beta \gamma }^a-\frac{1}{3}g{c}^{abc}{A}_{\beta }^b{A}_{\gamma }^c)$ with ${\alpha }_s=g^2/4\pi$ and ${\partial }^{\mu }{J}_{\mu 5}^{\mathrm{\text{con}}}={\sum }_{q}2m_q{\bar{\psi }}_{q}i{\gamma }_5{\psi }_q$; $f=3$ is the number of active light flavors. The partially conserved axial-vector current $J_{\mu 5}^{\mathrm{\text{con}}}$ is not gauge invariant.

The QCD parton model is formulated in the light-front $A_{+}=0$ gauge with the leading twist term measured in the forward matrix element of the “$+$ component” of the flavor-singlet current, viz. $J_{+5}$. While $K_{\mu }$ is gauge dependent, the forward matrix elements of $K_{+}$ are actually invariant under residual gauge degrees of freedom in the light-front gauge $A_{+}=0$. Subject to these constraints, it is coincident with the “$+$ component” of the gauge invariant gluon spin operator (up to a possible surface term).^28,29 In more general gauges, one runs into the issue of invariance under large gauge transformations⁹ which connect to non-perturbative gluon topology beyond perturbation theory and involve shuffling a non-local zero mode around between “quark” and “gluonic” terms.²⁵ If the net zero mode term is finite, then it corresponds to a $x=0$ term in the spin dependent parton distributions and a subtraction at infinity in the $g_1$ dispersion relation.¹⁴

One can re-label the quark and gluon contributions to the spin structure functions through different choices of factorization “scheme” (or jet definitions) and include all the glue terms in Eq. (6) as absorbed in the “quark” piece, e.g. in the $\overline{\mathrm{\text{MS}}}$ scheme. For polarized deep inelastic scattering the decomposition in Eq. (6) is “physical” in the sense that the different terms correspond to measurable different jet processes with different $k_t$. The polarized gluon contribution $\mathrm{\Delta }g$ enters $g_1$ at small x after convolution with the gluon coefficient or hard part of the polarized ${\gamma }^{*}g$ cross section $C^g$, viz. $\sim {\int }_x^{1}dx\mathrm{\Delta }g(z)C^g(\frac{x}{z},{\alpha }_s)$. Within the parton picture the axial anomaly term corresponds to a term $-\frac{{\alpha }_s}{\pi }(1-z)$ in $C^g$ with z the Bjorken variable for the ${\gamma }^{*}g$ collision.³⁰

There is evidence in the data from polarized proton–proton collisions at RHIC for modest polarized gluon spin in the proton. Semi-inclusive measurements of fast kaon production in polarized deep inelastic scattering reveal no strong evidence for polarized strangeness.²⁰

Specifically, with gluon polarization a NLO global fit to spin data including from RHIC polarized proton–proton collisions gives ${\int }_{0.05}^{1}dx\mathrm{\Delta }g(x)\approx 0.2\pm 0.05$ and $|{\int }_{0.001}^{1}dx\mathrm{\Delta }g(x)|\lesssim 0.8$, each at $Q^2=10$ GeV².³¹ This interesting result is however not sufficient to alone resolve the spin puzzle through the polarized gluon term $-3\frac{{\alpha }_s}{2\pi }\mathrm{\Delta }g$ (which would need a value $\mathrm{\Delta }g\sim 2$ with ${\alpha }_s\approx 0.3$ assuming with good SU(3) in the determination of $g_A^{(8)}$) though it is consistent with theoretical model estimates $|\mathrm{\Delta }g(m_c^2)|\lesssim 0.3$³² and $\mathrm{\Delta }g(1{\mathrm{\text{GeV}}}^2)\approx 0.5$.³³ Other more recent fits to $\mathrm{\Delta }g(x,Q^2)$ including sensitivity to the overall sign of $\mathrm{\Delta }g$ are reported in Refs. 34 and 35. Improved constraints will come from future data with the Electron Ion Collider, EIC.

If $\mathrm{\Delta }g$ is too small to explain the small value of $g_A^{(0)}$, then what about possible SU(3) breaking in the nucleon’s axial charges? SU(3) breaking is induced through virtual pion cloud effects. QCD inspired models include both the color hyperfine OGE potential responsible for the nucleon-$\mathrm{\Delta }$ mass splitting⁵ and the pion cloud induced by dynamical chiral symmetry breaking. Taken alone (before pion effects) OGE gives the SU(3) $F/D$ ratio and 0.58 value for the nucleon’s octet axial charge extracted assuming good SU(3) symmetry. When the pion cloud is also included along with small kaon loop corrections, re-evaluation of the nucleon’s axial-charges in the Cloudy Bag model led to the value $g_A^{(8)}=0.46\pm 0.05$.³⁶ In these calculations $g_A^{(3)}$ retains its physical value. If one instead uses this new octet number in the analysis of polarized deep inelastic scattering, then the corresponding values extracted from the experiments become $g_A^{(0)}{|}_{\mathrm{\text{pDIS}},Q^2\to \infty }$$0.33\pm 0.06$ with $\mathrm{\Delta }s\sim -0.04\pm 0.03$. A recent joint fit to spin dependent parton distributions and fragmentation functions from inclusive and semi-inclusive deep inelastic scattering as well as inclusive $e^{+}{e}^{-}$ data gives the octet term peaked close to 0.5,³⁷ with value $0.46\pm 0.21$, which is close to the Cloudy Bag preferred value though with large uncertainty. Recent lattice calculations^38,39 give values of $g_A^{(0)}\approx 0.40\pm 0.04$ with $\mathrm{\Delta }s$ close to $-0.04$.

Summarizing the present status of this phenomenology, the OGE potential plus pion cloud effects together with the modest polarized glue suggested by theory and by the RHIC spin experiments are sufficient to resolve the small value of $g_A^{(0)}$ within the present experimental and theoretical errors. New data will come from the future EIC which will push the measurements to smaller Bjorken x, down to $x\sim 10^{-4}$, and with improved precision. One issue is the behaviour of $g_1$ at very small x, below the $x_{\mathrm{\text{min}}}=0.004$ achieved by COMPASS, where perturbative QCD resummation calculations⁴⁰ predict more divergent small x behaviour for the singlet part of $g_1$. Predictions in EIC kinematics are given in Ref. 41.

In parallel to high energy spin experiments, a precise measurement of elastic neutrino proton scattering would be valuable as a complementary probe of $\mathrm{\Delta }s$ and to test gluon topology ideas. Here, the present most accurate measurement is from KamLAND,⁴²$\mathrm{\Delta }s=-0.14_{-0.26}^{+0.25}$.

Confinement induces a parton transverse momentum scale and hence finite quark and gluon orbital angular momentum in the proton. The quark and gluon total angular momentum are related to terms in the corresponding QCD energy–momentum tensor allowing one to deduce information about their asymptotic behaviour where the ratio of quark to gluon contributions becomes 16:3 f with f the number of quark flavors.⁴³ Beyond this observation, specific total and orbital angular momentum definitions are not unique^9,43 with different definitions being more suitable to the interpretation of different spin processes. There is a vigorous experimental programme to measure $k_t$ dependent processes and spin-momentum correlations involving transverse-momentum dependent and non-forward generalized parton distributions, TMDs and GPDs, to map out the tomography of the proton (for a review see Ref. 20).

Away from the forward direction the matrix elements of $K_{\mu }$ are not gauge invariant at all. This means that for non-forward, transverse momentum dependent processes like polarized deeply virtual Compton scattering^44,45 one should use the $\overline{\mathrm{\text{MS}}}$ scheme with the full gauge invariant axial vector current including the anomaly included in the spin dependent “quark” distribution. The decomposition in Eq. (6) with the polarized gluon term linked to the Chern–Simons current only makes sense in the forward direction.^14,46

QCD quantum entanglement effects might be important in semi-inclusive processes with spin and transverse momentum dependence.^47,48 In hadron scattering processes color Wilson lines might overlap between the incoming (or outgoing) hadrons leading to novel factorization breaking effects.⁴⁷ Improved experimental precision will be necessary to test these QCD entanglement ideas.

Beyond the parton model picture encompassed in Eq. (6) further attempts to understand the role of gluonic spin degrees of freedom in the transition from current to constituent quarks have been explored in Refs. 9, 14, 33, 49,50,51,52,53,54. The singlet axial charge $g_A^{(0)}$ can also be related to the proton matrix element of the topological charge density. Consider first the non-forward matrix element

where $l_{\mu }={(p-p^{\prime })}_{\mu }$ and ${\tilde{s}}_{\mu }={\overline{u}}_{(p,s)}{\gamma }_{\mu }{\gamma }_5{u}_{(p^{\prime },s^{\prime })}/2M$. Since the ${\eta }^{\prime }$ meson has finite mass even in the chiral limit, it follows that there is no massless pole in $G_P(l^2)$ even in the chiral limit. Next define the chiralities $q.\tilde{s}{\chi }^q(l^2)=〈p,s|2m_{q}i\bar{q}{\gamma }_{5}q|p^{\prime },s^{\prime }〉$ for each flavor q and the gluonic term $q.\tilde{s}{\chi }^g(l^2)=〈p,s|\frac{{\alpha }_s}{4\pi }{G}_{\mu \nu }^a{\tilde{G}}_a^{\mu \nu }|p^{\prime },s^{\prime }〉$. Contracting Eq. (8) with $l^{\mu }$ and taking the limit $l^2\to 0$ gives With exactly massless quarks this involves just the $\frac{{\alpha }_s}{4\pi }{G}_{\mu \nu }^a{\tilde{G}}_a^{\mu \nu }$ topological charge density. However, with even very small quark masses, the quark and gluon terms here come proportional to the ratio of the light up and down quark masses.⁵⁵ This quark mass dependence cancels in each of the isovector, octet and full singlet axial charges when one sums over all the contributing terms. If the “gluon” term is taken alone, ${\chi }^g(0)$ is sensitive to large isospin violation. If one generalizes this discussion to polarized deep inelastic scattering from a polarized real photon target, then the corresponding “gluonic term” ${\chi }^g(0){|}_{\gamma }$ for a polarized photon target evaluates⁵⁶ to $2\pi /{\alpha }_s\approx +30$ if we take the ratio of light quark masses $m_u/m_d=\frac{1}{2}$! The infrared quark mass ratio dependence here contrasts with the $-\frac{{\alpha }_s}{2\pi }\mathrm{\Delta }g$ term in Eq. (6) which corresponds to the hard perturbative QCD process of two-quark-jet production at large $k_t^2\sim Q^2$.

One can also write a flavor singlet Goldberger–Treiman relation connecting $g_A^{(0)}$ and the ${\eta }^{\prime }$-nucleon coupling constant⁵²

Here $g_{{\eta }^{\prime }NN}$ is the ${\eta }^{\prime }$ nucleon coupling constant, $g_{GNN}$ denotes the proper vertex for coupling of the gluonic topological charge density to the nucleon, and F a renormalization scale invariant singlet decay constant; $g_{GNN}$ carries the renormalization scale dependence of $g_A^{(0)}$.

4. Glue in the ${\eta }^{\prime }$

While the pseudoscalar pions and kaons fit well as would-be Goldstone bosons with their masses satisfying $m_P^2\propto m_q$, the isosinglet $\eta$ and ${\eta }^{\prime }$ are too heavy by about 400MeV and 300MeV to satisfy this relation. They are exceptional mesons with their masses and interactions sensitive to additional non-perturbative gluon dynamics in the flavor-singlet channel^{57,58,59,60,61,62} associated with the QCD axial anomaly in the flavor-singlet axial-vector current. Within the approximation of a leading-order one-mixing-angle scheme, if we include a ${\tilde{m}}_{{\eta }_0}^2=0.73$GeV² singlet gluonic mass term, then the $\eta$ and ${\eta }^{\prime }$ masses each come out correct to within 10% accuracy with the mixing angle $-20$${}^{\circ }$. Without this glue term in the isoscalar mass matrix, the ${\eta }^{\prime }$ would come out as a strange quark state with mass $\sqrt{2m_K^2-m_{\pi }^2}$ and the $\eta$ would be a light quark state degenerate with the pion. The glue associated with ${\tilde{m}}_{{\eta }_0}^2$ is associated with gluon topology^58,59 and its effect incorporated in axial U(1) extended chiral Lagrangians.^60,61,62 The theory involves the interface of local anomalous Ward identities and non-local topological structure. The $\eta$ and ${\eta }^{\prime }$ masses then satisfy the Witten–Veneziano mass formula $m_{\eta }^2+m_{{\eta }^{\prime }}^2=2m_K^2+{\tilde{m}}_{{\eta }_0}^2$. Recent lattice calculations with dynamical quarks of the meson masses and the gluonic term ${\tilde{m}}_{{\eta }_0}^2$ further confirm this picture at 10% accuracy.⁶³ With the leading order mixing angle $-20$ degrees, the ${\eta }^{\prime }$ has the biggest singlet component and hence the bigger sensitivity to OZI violating couplings to other hadrons proceeding through gluonic intermediate states. One expects OZI violation in the coupling of ${\eta }^{\prime }$ to other hadrons in scattering processes over a broad range of energies.⁶⁴ This includes the low energy ${\eta }^{\prime }$-nucleon coupling constant $g_{{\eta }^{\prime }NN}$⁶⁵ through to large branching ratios observed in high energy $B\to {\eta }^{\prime }X$ decays.⁶⁶

Recent investigations have focussed on the properties of ${\eta }^{\prime }$ mesons in nuclei. The light up and down quark contributions to the ${\eta }^{\prime }$ wavefunction are induced by the gluonic term in the $\eta -{\eta }^{\prime }$ mass matrix, and it is these that couple to the $\sigma$ (correlated two-pion) mean field inside the nucleus. Working within the Quark Meson Coupling model, QMC,^67,68 a $-37$MeV shift was predicted for the ${\eta }^{\prime }$ mass for an ${\eta }^{\prime }$ in a nucleus at nuclear matter density ${\rho }_0$ when the $\eta -{\eta }^{\prime }$ mixing angle is taken as $\theta =-20$ degrees.^69,70 Without the anomalous glue component in the ${\eta }^{\prime }$ mass the ${\eta }^{\prime }$ would be a strange quark state with much reduced interaction with the $\sigma$ mean field in the nucleus. ${\eta }^{\prime }$ photoproduction experiments at ELSA in Bonn from Carbon and Niobium targets subsequently revealed an $\approx -40$MeV shift in the ${\eta }^{\prime }$ mass at ${\rho }_0$ with a small ${\eta }^{\prime }$ width in medium. The measured ${\eta }^{\prime }$-nucleus optical potential has real and imaginary parts $V+iW$ with

(see Refs. 71,72,73). In these photoproduction experiments the meson is produced with reduced mass in the nucleus meaning that the production cross-section goes up. When it emerges from the nucleus it returns to its free mass at expense of the kinetic energy. Data with a Carbon target shows that the nuclear medium is approximately transparent to ${\eta }^{\prime }$ propagation, in contrast to ${\pi }^0$ and $\eta$ propagation where there is large interaction with the nucleus.⁷⁴ The small ${\eta }^{\prime }$ width relative to the mass shift (or potential depth) observed in these experiments means that possible ${\eta }^{\prime }$ bound states in nuclei might be accessible in experiments. If observed, these bound states would be a new state of matter bound just by the strong interaction, in contrast to pionic and kaonic atoms involving electrically charged pions and kaons bound by QED interactions. The present ${\eta }^{\prime }$ bound state search experiments constrain the possible parameter range with deep potential depths of $-150$ MeV so far excluded.^75,76,77 Further experiments in Germany and Japan are running or in planning to push the these measurements towards potentials typical of those suggested by the ELSA result and by QMC theory.^73,78

5. The $\mathrm{\Delta }$ Excitation in Polarized Photoproduction from Light Nuclei

A second interesting nuclear effect involves the $\mathrm{\Delta }$ resonance excitation in polarized photoproduction from polarized light nuclei where medium modification effects also enter.

The Gerasimov–Drell–Hearn, GDH, sum-rule for polarized photoproduction relates the inclusive spin cross-sections for polarized photon–proton scattering to the ratio of the target proton anomalous magnetic moment and mass. The GDH sum-rule reads^79,80

where ${\sigma }_p$ and ${\sigma }_a$ are the spin dependent photoabsorption cross-sections involving photons polarized parallel and antiparallel to the target’s spin. Here $s_{\gamma p}$ is the photon–proton center-of-mass energy squared with $\kappa$ the target’s anomalous magnetic moment; M is the target mass and $S=\frac{1}{2}$ is its spin. The GDH sum-rule is derived from the very general principles of causality, unitarity, Lorentz and electromagnetic gauge invariance together with the single assumption that ${\sigma }_p-{\sigma }_a$ satisfies an unsubtracted dispersion relation.

For free protons the GDH sum-rule predicts a value of $+205$$\mu$b for the integral in Eq. (12) with the proton’s anomalous magnetic moment $\kappa =1.79$. This result is in excellent agreement with the experimental number $+209\pm 13$$\mu$b. This value is extracted as follows. Polarized photoproduction experiments at MAMI in Mainz and ELSA in Bonn have measured ${\sigma }_p-{\sigma }_a$ with photon beam energies between 200 MeV and 2.9 GeV giving a measured sum-rule contribution $+253.5\pm 5\pm 12$$\mu$b.^81,82 Here the biggest contribution comes from the $\mathrm{\Delta }$ magnetic transition excitation, about 190$\mu$b, with smaller contributions from heavier resonances. One also has a near threshold contribution $\approx -30$$\mu$b estimated from multipole models of pion photoproduction.⁸³ A further $\approx 10$% part of the sum-rule comes from high energy Regge contributions from energies above the maximum ELSA beam energy, $-15\pm 2$$\mu$b,⁸⁴ estimated from Regge fits to low $Q^2$ leptoproduction data. The high energy part of the proton GDH sum-rule is essentially all in the isovector channel with negligible isoscalar contribution, similar to the situation observed with polarized deep inelastic scattering in the so far measured kinematics (see Fig. 1). An independent confirmation of the GDH sum-rule comes from JLab, $+204\pm 11$$\mu$b from extrapolation of low $Q^2$ electroproduction data to the real photon point.⁸⁵

How about possible changes of nucleon properties in polarized photoproduction from (light) nuclei? Extending these GDH experiments from protons to light nuclear targets, experiments have so far been performed also with polarized deuterons and ³He targets. The present data involves measurements with photon energies from 200MeV up to 1.8GeV (deuteron data)⁸⁶ and up to 500MeV (³He data).⁸⁷

As recently noticed,⁸⁸ these data suggest a small shift in the excitation energy of the $\mathrm{\Delta }$ peak, up to $\approx -20$MeV, in the spin difference cross-section ${\sigma }_p-{\sigma }_a$ (specifically in the spin parallel cross-section ${\sigma }_p$ and not observed in the spin average cross-section) with the effect most visible in the more precise deuteron data. This observed cross-sections is qualitatively different from what one expects from smearing due to Fermi motion of the bound nucleons in the light nuclei. Small, few percent, medium modifications of nucleon properties in light deuterons are also observed in experimental measurements of the EMC nuclear effect where parton distributions of bound nucleons in the deuteron are seen to be modified relative to free protons,⁸⁹ as well as with theoretical lattice calculations of the nucleon’s axial and tensor charges in light nuclei.⁹⁰

It would be interesting to investigate whether the $\mathrm{\Delta }$ peak mass shift effect persists and might be enhanced with a larger polarized nuclear target where medium effects might be more pronounced. Experimentally, one needs a target where the spin of nucleus is carried close to all by a single polarized nucleon, e.g. ⁷Li where the nucleus is not so big that any effect is washed out by a huge spin dilution factor in the total asymmetry from scattering on unpolarized spectator nucleons carrying close to no net spin. More generally, if one could measure the GDH sum-rule for a bound nucleon over the full energy range, then the right-hand side static part would become sensitive to medium modifications in the proton mass and anomalous magnetic moment. Model estimates of these quantities suggest a possible enhancement in the sum-rule up to about a factor of two for a polarized proton at nuclear matter density.^88,91

6. Conclusions

QCD continues to inspire new advances in theory and experiments in our quest to understand the deep structure of the proton. In the topics of proton spin dynamics and ${\eta }^{\prime }$ interactions new data will soon follow from near threshold production processes up to high energy deep inelastic scattering. The physics puzzles that inspired Harald Fritzsch continue to inspire us today.

ORCID

Steven D. Bass  https://orcid.org/0000-0002-2089-9661