Momentum-transfer dependence of the quark and gluon gravitational form factors of the nucleon: gluon multipole shape, large-$|t|$ behavior, and microscopic origin

The quark and gluon gravitational form factors (GFFs) of the nucleon encode how mass, momentum, and mechanical pressure are spatially distributed among quarks and gluons. A PDF-grounded extraction (Selyugin and Teryaev, 2024) found a dipole-like quark GFF but a tripole-like ($n\approx3$) gluon GFF with a gluon gravitational radius close to the proton electromagnetic radius, conjecturally caused by an extra gluonic propagator in quark–gluon Fock components of the nucleon wave function, and reliable only for momentum transfers $|t|\lesssim2~\mathrm{GeV}^2$. The open problem is to determine the $t$-dependence of $A_q(t)$ and $A_g(t)$ — the precise multipole power and scale of the gluon GFF, its behavior at larger $|t|$, and the QCD mechanism behind the differing quark and gluon shapes — with controlled systematics from lattice QCD, near-threshold quarkonium photoproduction, and other reactions. Later work (2023–2025) leaves all three elements open: analyses still fix the multipole power by assumption, lattice results stop at $|t|=2~\mathrm{GeV}^2$, radius determinations conflict, and rival frameworks predict either stronger or identical gluon versus quark $t$-dependence.

Background

Generalized parton distributions (GPDs) are functions $F(x,\xi,t)$ that describe how partons (quarks and gluons) with longitudinal momentum fraction $x$ are distributed in transverse space in a hadron, with $t$ the square of the momentum transfer and $\xi$ the skewness (longitudinal momentum asymmetry between initial and final hadron). Their second Mellin moments in $x$ at $\xi=0$ give the matrix elements of the QCD energy–momentum tensor (the conserved current coupling to gravity), parameterized by the gravitational form factors (GFFs) $A(t)$, $B(t)$, and $D(t)$ (X. Ji, Phys. Rev. D 55, 7114 (1997)). $A(0)$ obeys the momentum sum rule — quark plus gluon contributions $A_q(0)+A_g(0)=1$ — so the slope of $A_{q,g}(t)$ at $t=0$ defines quark and gluon 'gravitational radii' of the mass–energy distribution, while $D(t)$ encodes internal pressure and shear forces (the 'mechanical' structure). Because gravity is far too weak to probe hadrons directly, GFFs must be accessed indirectly. The prime experimental route is near-threshold heavy-quarkonium photoproduction, e.g. $\gamma p\to J/\psi\, p$: at threshold the process is gluon-dominated, so the cross section constrains the gluon GFFs; this was first exploited in the Jefferson Lab J/psi-007 extraction (Duran et al., Nature 615, 813 (2023)), which parameterized the gluon GFFs $A_g(t)$ and $C(t)=-D(t)/(2A(t))$ in a tripole (multipole power $n=3$) form, and in the updated analysis of Guo, Ji, Liu, and Yang (Phys. Rev. D 108, 034003 (2023)). On the theory side, lattice QCD (discrete-space-time numerical simulation of the QCD path integral) can in principle determine the flavor-separated GFFs from first principles. A convenient phenomenological shape is the multipole ansatz $A(t)=[1-t/L^2]^{-n}$: $n=1$ monopole, $n=2$ dipole, $n=3$ tripole; larger $n$ means a harder (more compact) large-$|t|$ falloff. Perturbative quark-counting rules (Brodsky–Farrar/Matveev–Muradyan–Tavkhelidze scaling) predict power-law falloffs of form factors at very large momentum transfer and motivate connecting observed powers to the number of propagators in the underlying partonic process. Selyugin and Teryaev (arXiv:2311.12988, Phys. At. Nucl. 87, 518 (2024)) built GPD $t$-profiles anchored to parton distribution functions (PDFs, the forward limit $x$-distributions measured in deep-inelastic scattering) and validated against the 912-point electromagnetic form factor database; taking the limit of zero skewness they obtained $A_q(t)$ in dipole form with $L^2=1.6~\mathrm{GeV}^2$, while the gluon GFF $A_g(t)$, propagated from several modern gluon PDFs, was best described by a multipole with power $n$ closest to 3 and $L^2\approx0.9~\mathrm{GeV}^2$, with a gluon gravitational radius of $0.85$–$0.87~\mathrm{fm}$, approximately equal to the proton electromagnetic charge radius. They conjectured that the stronger gluonic $t$-dependence arises from an additional gluonic propagator associated with quark–gluon Fock components (higher particle-number components) of the nucleon wave function, in line with counting rules, and stated that their profile is validated only for $|t|\lesssim2~\mathrm{GeV}^2$ and that 'various experimental, theoretical and model tests of this picture are highly desirable'.

Problem Statement

Determine the momentum-transfer dependence of the quark and gluon gravitational form factors $A_q(t)$ and $A_g(t)$ of the nucleon, so that the following linked elements are settled with controlled systematics: (i) the shape of the gluon GFF — whether it indeed follows a multipole form with power $n\approx3$ (tripole) and scale $L^2\approx0.9~\mathrm{GeV}^2$ while the quark GFF is dipole-like ($n=2$, $L^2\approx1.6~\mathrm{GeV}^2$), i.e., a determination of the exponent $n$ and scale $L^2$ from data and/or lattice QCD with quantified rather than assumed values; (ii) the behavior of both form factors at momentum transfers beyond the region $|t|\lesssim2~\mathrm{GeV}^2$ where PDF-grounded GPD $t$-profiles are validated and lattice data exist, including whether the dipole-versus-tripole distinction persists or is an artifact of the fitted window; and (iii) the microscopic QCD origin of the relative quark/gluon $t$-dependence — in particular whether the stronger gluonic falloff is caused by the extra gluonic propagator associated with quark-gluon Fock components of the nucleon wave function, as suggested by quark-counting arguments — together with the resulting quark and gluon gravitational radii. Answers may come from any sound route — combined analyses of near-threshold quarkonium photoproduction and other exclusive reactions, lattice QCD with controlled continuum/chiral/renormalization systematics, holographic or Dyson-Schwinger frameworks, or direct QCD derivations — but must confront, not assume, the multipole shape and must state the $|t|$ domain over which the claimed behavior is established.

References

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