In the NNLO QCD mass decomposition of the proton and pion, the interaction-generated ('parton-correlation') contributions are governed by the forward values of the twist-four gravitational form factors Cˉq,g(0,μ)\bar{C}_{q,g}(0,\mu), with Cˉq+Cˉg=0\bar{C}_{q}+\bar{C}_{g}=0. Tanaka (2025) derived exact force-type representations of these quantities — a QCD analogue of the virial theorem, obtained from the dilatation transformation — in which Cˉq(0,μ)\bar{C}_{q}(0,\mu) is the hadron matrix element of the first transverse moment of the three-parton quark–gluon-force operator ψˉγαgFαβψ\bar{\psi}\gamma_{\alpha}gF^{\alpha\beta}\psi and Cˉg(0,μ)\bar{C}_{g}(0,\mu) of the gluonic operator FαλDβFβλF^{\alpha\lambda}D^{\beta}F_{\beta\lambda}, describing the forces between quarks and gluons and among gluons; the paper explicitly defers the details 'to be discussed elsewhere'. As of August 2026 no such analysis has appeared. The open problem is to supply it: the complete derivation and renormalization of the force-type operators, and their dynamical interpretation as inter-parton forces in a form that withstands the current debate on mechanical interpretations of the energy-momentum tensor. The scheme dependence of the quark/gluon splitting of this mass component (MS‾\overline{\mathrm{MS}} versus the D2 scheme) and a direct quantitative determination from lattice QCD or phenomenology are closely related open directions.

Background

Almost all visible mass originates not from the Higgs mechanism but from QCD dynamics: the proton and neutron masses are generated by the interactions of quarks and gluons. The quantitative language for this is the decomposition of the hadron mass through the QCD energy-momentum tensor (EMT), Tαβ=Tqαβ+TgαβT^{\alpha\beta}=T_q^{\alpha\beta}+T_g^{\alpha\beta}, whose gauge-invariant quark and gluon parts are Tqαβ=iψˉγ(αD↔β)ψT_q^{\alpha\beta}=i\bar{\psi}\gamma^{(\alpha}\overleftrightarrow{D}^{\beta)}\psi and Tgαβ=−FαλF λβ+ηαβF2/4T_g^{\alpha\beta}=-F^{\alpha\lambda}F^{\beta}_{\ \lambda}+\eta^{\alpha\beta}F^{2}/4 (FαλF^{\alpha\lambda} is the gluon field-strength tensor, DαD^{\alpha} the covariant derivative, ηαβ\eta^{\alpha\beta} the metric tensor, and parentheses denote symmetrization of indices). Hadron matrix elements of the EMT are parameterized by gravitational form factors (GFFs); their forward values (zero momentum transfer) enter the mass sum rules. Two matter here: Aq,g(0,μ)A_{q,g}(0,\mu), the average momentum fractions carried by quarks and gluons, which are the second moments of the parton distribution functions (PDFs), and the twist-four form factor Cˉq,g(0,μ)\bar{C}_{q,g}(0,\mu), which obeys the sum rule Cˉq+Cˉg=0\bar{C}_{q}+\bar{C}_{g}=0. ('Twist' classifies EMT operators by mass dimension minus spin: the twist-two pieces carry the partons' forward momentum and motion, while the twist-four pieces are induced by interactions among partons; μ\mu is the renormalization scale of the composite operators. Quantitative results are stated in the MS‾\overline{\mathrm{MS}} minimal-subtraction scheme in dimensional regularization, with 'NNLO' denoting next-to-next-to-leading order in the strong coupling αs\alpha_s.)

The classical starting point is Ji's separation of the nucleon mass into quark energies, gluon energy, quark masses, and the trace anomaly — the anomalous quantum breaking of scale invariance, appearing in the trace of the EMT through the gluon term β(g)F2/2g\beta(g)F^{2}/2g and the quark-mass term (1+γm)mψˉψ(1+\gamma_{m})m\bar{\psi}\psi, where β\beta and γm\gamma_m are the QCD beta function and quark-mass anomalous dimension. Later work established that the trace anomaly itself decomposes unambiguously into quark and gluon parts (Hatta, Rajan and Tanaka at two loops; Tanaka at three loops), which yields model-independent NNLO predictions for the forward twist-four form factors: Cˉq(0,μ0)=−Cˉg(0,μ0)=−0.163(4)\bar{C}_{q}(0,\mu_0)=-\bar{C}_{g}(0,\mu_0)=-0.163(4) for the proton and −0.03(2)-0.03(2) for the pion at μ0=2\mu_0=2 GeV, the proton at a few-percent accuracy with σ\sigma-terms (nucleon matrix elements of the quark-mass operator) and PDF moments as inputs. The decomposition scheme matters, however: besides MS‾\overline{\mathrm{MS}}, part of the literature uses the 'D2 scheme', in which the split of the trace (twist-four) sector into quark and gluon pieces is defined at the level of the bare operators rather than the renormalized ones, and the two schemes assign different quark/gluon splittings to this sector. Whether that split has physical meaning is actively debated — the 'quantum anomalous energy' controversy between Ji, Liu and Schäfer on one side and Lorcé, Metz, Pasquini and Rodini on the other, and recent critiques of D2-type 'reducible-basis' renormalization by Yang.

Tanaka (2025) assembled these ingredients into an NNLO update of the proton and pion mass decompositions, including a new four-term formula that strictly separates, for each of the quark and gluon EMT parts, the twist-two contributions (partonic motion, which behaves similarly in the proton and the pion) from the twist-four contributions (parton correlations, which behave very differently, reflecting the pion's nature as the (pseudo-)Nambu–Goldstone boson of chiral symmetry breaking — i.e. the light collective excitation tied to the spontaneous breaking of the approximate chiral (left–right) symmetry of the light-quark sector). The physical basis of this twist-four sector is an exact force-type representation — a QCD analogue of the virial theorem (the relation, in a bound system, between interaction energy and internal forces), here derived by extending the field-theoretic virial-theorem treatment of Lorcé, Metz, Pasquini and Rodini via the dilatation (scale) transformation:

Cˉq(0,μ)=13M⟨∫d3x xβ[ψˉ(x)γα gFαβ(x)ψ(x)]⟩,\bar{C}_{q}(0,\mu)=\frac{1}{3M}\Bigl\langle\int d^{3}x\,x_{\beta}\left[\bar{\psi}(x)\gamma_{\alpha}\,gF^{\alpha\beta}(x)\psi(x)\right]\Bigr\rangle,

Cˉg(0,μ)=13M⟨∫d3x xα[Faαλ(x)DabβFβλb(x)]⟩,\bar{C}_{g}(0,\mu)=\frac{1}{3M}\Bigl\langle\int d^{3}x\,x_{\alpha}\left[F^{\alpha\lambda}_{a}(x)D^{\beta}_{ab}F_{\beta\lambda}^{b}(x)\right]\Bigr\rangle,

where ⟨⋯ ⟩\langle\cdots\rangle denotes the matrix element in a hadron of mass MM, [O][\mathcal{O}] the renormalized operator, and gg the strong coupling. The first relation exhibits Cˉq(0,μ)\bar{C}_{q}(0,\mu) as the matrix element of the first moment of the gauge-invariant three-parton (quark–gluon–quark) force operator ψˉγαgFαβψ\bar{\psi}\gamma_{\alpha}gF^{\alpha\beta}\psi, and the second exhibits Cˉg(0,μ)\bar{C}_{g}(0,\mu) through the purely gluonic operator FαλDβFβλF^{\alpha\lambda}D^{\beta}F_{\beta\lambda}. The source describes them as 'corresponding to the forces between the quarks and gluons, and to those among the gluons, the details of which will be discussed elsewhere'. That deferral, unmatched by any publication as of August 2026, defines the open problem: the quantitative mass decompositions exist, but the force-type operators behind their interaction sector have never been derived in detail, renormalized, interpreted, or independently determined.

Problem Statement

The problem, anchored in the source's explicit deferral — 'the details of which will be discussed elsewhere' — is to supply the deferred analysis of the force-type representation of the twist-four gravitational form factors for the proton and the pion. With the exact relations

Cˉq(0,μ)=13M⟨∫d3x xβ[ψˉ(x)γα gFαβ(x)ψ(x)]⟩,\bar{C}_{q}(0,\mu)=\frac{1}{3M}\Bigl\langle\int d^{3}x\,x_{\beta}\left[\bar{\psi}(x)\gamma_{\alpha}\,gF^{\alpha\beta}(x)\psi(x)\right]\Bigr\rangle,

Cˉg(0,μ)=13M⟨∫d3x xα[Faαλ(x)DabβFβλb(x)]⟩,\bar{C}_{g}(0,\mu)=\frac{1}{3M}\Bigl\langle\int d^{3}x\,x_{\alpha}\left[F^{\alpha\lambda}_{a}(x)D^{\beta}_{ab}F_{\beta\lambda}^{b}(x)\right]\Bigr\rangle,

where ⟨⋯ ⟩\langle\cdots\rangle denotes the matrix element in a hadron of mass MM and [O][\mathcal{O}] the MS‾\overline{\mathrm{MS}}-renormalized operator, the objective is one coherent goal — establishing the interaction-generated ('parton-correlation') sector of the hadron mass decomposition on the same rigorous footing as the twist-two sector — through the detailed treatment the source defers:

  1. the full derivation of these exact relations from the dilatation transformation (the QCD virial theorem), together with the operator-level bookkeeping: the renormalization and mixing of the gauge-invariant three-parton force operator ψˉγαgFαβψ\bar{\psi}\gamma_{\alpha}gF^{\alpha\beta}\psi and the gluonic operator FαλDβFβλF^{\alpha\lambda}D^{\beta}F_{\beta\lambda}, and their scale dependence;

  2. the dynamical interpretation of these matrix elements as the forces between quarks and gluons, and among gluons, inside the hadron — made precise enough to be confronted with the current debate on mechanical interpretations of the EMT (virial-theorem versus mechanical-equilibrium frameworks, and the status of 'quantum anomalous energy'), and consistent with translational invariance and the sum rule Cˉq+Cˉg=0\bar{C}_{q}+\bar{C}_{g}=0.

Together, the derivation and renormalization of the force-type operators and their dynamical interpretation as inter-parton forces supply the 'details' whose discussion the source defers. The scheme dependence of the quark/gluon splitting of the twist-four sector (a finite MS‾→\overline{\mathrm{MS}}\toD2 transformation) and a direct quantitative determination of these force-type matrix elements for the proton and the pion — from lattice QCD or from phenomenological and experimental constraints, consistent with the known NNLO forward values and their uncertainties — are closely related open directions that such a treatment will naturally engage, but the source's deferral does not enumerate them as conditions for discharging it.

References

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