Almost all visible mass resides in nucleons, and in quantum chromodynamics (QCD) it is generated dynamically rather than by the Higgs-scale quark masses. How the nucleon mass should be partitioned into quark and gluon contributions is contested: several exact sum rules - Ji's four-term rest-frame decomposition, the variants studied by Lorcé and by Metz, Pasquini and Rodini, the Hatta-Rajan-Tanaka decomposition that splits the QCD trace anomaly into quark and gluon pieces, and the two-term trace decomposition - partition the same mass differently, and their individual terms are generally renormalization-scheme and scale dependent. The literature through August 2026 shows the problem open and actively contested: a 2026 first-principles lattice study verified several sum rules simultaneously yet obtained materially different partitions; a 2025 NNLO analysis quantified the scale dependence of each term and added yet another variant; and an August 2026 paper re-opened the operator-renormalization dispute over whether the trace anomaly can be split into quark and gluon parts at all. The open core is whether any decomposition possesses scheme-independent physical content - in particular the status of the gluonic trace anomaly and of the proposed quantum anomalous energy - and which observables could discriminate empirically between the competing decompositions.

Background

Nucleons (protons and neutrons) account for nearly all of the visible mass in the universe, and their masses far exceed the sum of the current masses of their valence quarks: the bulk of the mass is generated by the dynamics of quantum chromodynamics (QCD), the gauge theory of quarks and gluons. In QCD the nucleon mass MM is the rest-frame expectation value of the energy-momentum tensor (EMT) TμνT^{\mu\nu}, M=⟨P∣∫d3x T00(x)∣P⟩M=\langle P|\int d^3x\, T^{00}(x)|P\rangle at zero momentum, and the question of the origin of the mass is formalized as decomposing MM into forward (zero-momentum-transfer) matrix elements of renormalized quark and gluon operators. Those forward EMT matrix elements are conventionally parameterized by gravitational form factors (GFFs), whose dominant terms are fixed by the momentum fractions ⟨x⟩q,⟨x⟩g\langle x\rangle_q,\langle x\rangle_g carried by quarks and gluons - the same second moments of the parton distribution functions measured in deep-inelastic scattering.

Several exact sum rules compete. Ji's 1995 rest-frame decomposition splits MM into four terms: the quark-mass contribution ∑q⟨mqqˉq⟩\sum_q\langle m_q\bar q q\rangle (the sigma terms), the quark kinetic-plus-potential energy and the gluon field energy (both from the traceless parts of the quark and gluon EMTs and expressible through ⟨x⟩q,g\langle x\rangle_{q,g}), and a trace-anomaly term Ma=⟨β(g)2gGμνaGaμν⟩M_a=\langle\frac{\beta(g)}{2g}G^a_{\mu\nu}G^{a\mu\nu}\rangle. The trace anomaly is the quantum breaking of classical scale invariance: even with massless quarks the renormalized QCD EMT has a nonzero trace T μμ=∑qmqqˉq+β(g)2gGμνaGaμνT^\mu_{\ \mu}=\sum_q m_q\bar q q+\frac{\beta(g)}{2g}G^a_{\mu\nu}G^{a\mu\nu}, where β(g)\beta(g) is the QCD beta function governing the running of the coupling. Taking the forward matrix element of the trace itself yields the two-term trace decomposition M=∑q⟨mqqˉq⟩+⟨β(g)2gG2⟩M=\sum_q\langle m_q\bar q q\rangle+\langle\frac{\beta(g)}{2g}G^2\rangle, which partitions the mass into sigma terms plus anomaly only. Lorcé (2018) argued that decompositions with more than two terms run into consistency problems of physical interpretation and that the quark-mass and trace-anomaly terms are purely conventional, while also accounting for internal pressure effects; Metz, Pasquini and Rodini (2020) systematized the sum rules, computed their terms through three loops, and showed that the individual terms (beyond the anomaly as a whole) depend on the renormalization scheme and scale even though every complete sum rule is an exact identity. Hatta, Rajan and Tanaka (2018) renormalized the quark and gluon EMTs in a reducible operator basis - mixing the Lorentz trace and traceless components rather than treating them as separate irreducible blocks - using the gradient-flow representation (a smearing procedure that renders composite operator products finite), and concluded that in the minimal-subtraction scheme the trace anomaly itself can be unambiguously decomposed into quark and gluon parts, a result extended to three loops by Tanaka (2019). Ji, Liu and Schäfer (2021) argued that scale-symmetry breaking generates a distinct quantum anomalous energy (QAE) contribution to the mass, related through the field-theoretic virial theorem to M/4M/4 in the chiral limit; Lorcé, Metz, Pasquini and Rodini (2021) reviewed the entire landscape and concluded from translation (Poincaré) symmetry that QAE is not a genuine separate contribution to the mass sum rule. On the empirical side, the forward GFFs entering the decompositions are constrained by parton-distribution moments and lattice QCD, and anomaly-related mass radii have been extracted from near-threshold J/ψJ/\psi photoproduction (photoproduction of the charmonium ground state just above threshold, which probes the gluonic scalar density of the nucleon), though with model dependence. As of 2026 the disagreement is unresolved: a first-principles lattice study by Bollweg, Ding, Gao, Luo and Mukherjee (2026) simultaneously verified several sum rules for charmonia (bound states of a charm quark and its antiquark, in their case ηc\eta_c and J/ψJ/\psi) in a common scheme yet found materially different partitions (the anomaly share is about 6% of the mass in Ji's sum rule while the gluonic component of the anomaly in the Hatta-Rajan-Tanaka sum rule is about 35%); Tanaka (2025) evaluated the decompositions at next-to-next-to-leading order and introduced a further traceless/trace-organized variant; and Yang (2026) argued that the reducible-basis renormalization underlying the anomaly split is itself ill-conceived because the anomaly is regulator-, scheme- and scale-independent as a whole and cannot be traced back to separate quark and gluon ultraviolet contributions.

Problem Statement

Establish the physical, renormalization-scheme-independent content of the quark-gluon decomposition of the nucleon mass in QCD. Concretely: (1) does there exist a decomposition of the nucleon mass into quark and gluon contributions (with or without a separate anomaly term) whose individual terms are invariant under changes of renormalization scheme and scale, consistent with QCD symmetries (translation/Poincaré invariance and the anomalous dilatation Ward identity, i.e. the EMT trace identity stated above, which fixes the scalar sector of any decomposition in terms of the quark masses and the beta function) - or does every term beyond the exact total carry conventional, scheme-dependent content only, as Lorcé (2018) argues? (2) In particular, is the trace-anomaly contribution uniquely attributable to the gluon sector, does it admit a scheme-independent quark/gluon split as constructed by Hatta, Rajan and Tanaka, or is no sector assignment of the trace part meaningful, as Yang (2026) argues? (3) Is the proposed quantum anomalous energy a genuine additional contribution to the mass sum rule, or is it not independent, as Lorcé, Metz, Pasquini and Rodini (2021) conclude? (4) Which observables - forward gravitational form factors, near-threshold heavy-quarkonium photoproduction, lattice energy-momentum-tensor matrix elements - can discriminate empirically between the competing decompositions, and what precisely would they have to measure? A complete answer must set out, by whatever route, the scheme- and scale-transformation behaviour of every candidate term, must engage the existing constructions (Ji's four-term sum rule, the Lorcé and Metz-Pasquini-Rodini variants, the Hatta-Rajan-Tanaka anomaly split, and the quantum anomalous energy) explicitly rather than sidestep them, and must settle each of (1)-(3); it thereby either establishes one decomposition (existing or new) as physically meaningful, stating explicitly whether its individual terms are in principle measurable, or proves that no scheme-independent quark/gluon partition of the mass exists beyond the exact sum-rule identities. Sub-question (4) must be answered on whichever branch of that determination holds, stating on the negative branch what the impossibility of a scheme-independent partition implies for experiment.

References

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