Scheme-unambiguous first-principles decomposition of the proton mass into Higgs and QCD-dynamical contributions

Less than one percent of the proton's 938 MeV mass comes from the Higgs-generated masses of its valence quarks; the remainder is generated by quantum chromodynamics (QCD). Yet the quantitative decomposition of that mass into quark-mass (sigma-term), quark kinetic-and-potential energy, gluon energy, and trace-anomaly contributions is not unique: the split of the trace of the QCD energy-momentum tensor (EMT) between quark and gluon operators depends on finite renormalization parameters ($x$, $y$) that no symmetry fixes, and existing numerical evaluations combine inputs determined in different schemes. The source work (Metz, Pasquini, Rodini, Phys. Rev. D 102, 114042 (2020)) established this scheme dependence up to three loops, showed that the quark-mass (sigma-term) share of the mass ranges from below 10% to nearly 20% depending on the treatment of the charm sigma term, and noted that a direct calculation of the renormalized scalar gluon expectation value $\langle(F^{\alpha\beta}F_{\alpha\beta})_R\rangle$ could provide further information about the role played by the Higgs mechanism for the numerics of the decomposition; no physical principle selecting $x$ and $y$ has been identified. Later work has narrowed but not closed the problem: the interpretive debate over the four-term sum rule and the 'quantum anomalous energy' remains unresolved, first lattice calculations of the nucleon trace anomaly exist only at unphysical masses or single lattice spacings, and the first complete scheme-consistent lattice validation of the mass sum rules has been demonstrated only for charmonia. The open core is a quantitative account of the proton mass in which the convention for the quark/gluon split of the EMT trace is either fixed by a physical principle or shown to be irreducible with its scheme-independent content identified, and in which the terms entering the sum rules are determined from first principles in a common scheme — including the fraction generated by the Higgs mechanism (the sigma terms) and the distribution of the remainder among quark energy, gluon energy and the trace anomaly.

Background

The visible mass of the universe is essentially the mass of protons and neutrons, and only about one percent of the proton mass is accounted for by the Higgs-generated masses of its two up quarks and one down quark. The rest is produced by QCD, the theory of quarks and gluons. How this works quantitatively is discussed through mass sum rules: identities that split the proton mass $M$ into contributions from forward (zero-momentum-transfer) matrix elements of operators in the QCD energy-momentum tensor (EMT), the conserved current associated with spacetime-translation symmetry. The pioneering four-term sum rule of Ji (Phys. Rev. Lett. 74, 1071 (1995)) writes the mass as the sum of the quark kinetic and potential energy, a quark-mass term, the gluon energy, and a trace-anomaly term.

Two specialist notions enter centrally. First, the trace anomaly: in classical QCD with massless quarks the EMT is traceless (scale invariance), but quantum renormalization breaks this symmetry. The running of the coupling, encoded in the beta function $\beta(g)$, and the quark-mass anomalous dimension $\gamma_m$ (the logarithmic rate at which renormalized quark masses change with the energy scale) make the trace nonzero, with the forward proton matrix element schematically $2M^2=\frac{\beta}{2g}\langle(F^{\alpha\beta}F_{\alpha\beta})_R\rangle+\sum_q(1+\gamma_m)\langle(m_q\bar\psi_q\psi_q)_R\rangle$, where $F^{\alpha\beta}$ is the gluon field-strength tensor and the subscript $R$ denotes renormalized operators. This anomaly is the term most often identified as the dominant engine of mass generation. Second, the sigma terms $\sigma_q=m_q\,\partial M/\partial m_q$, measurable in pion–nucleon scattering and in lattice QCD (the first-principles numerical formulation of QCD on a discrete spacetime grid), which quantify the quark-mass contribution — the part of the mass tied to the Higgs mechanism.

A seemingly technical issue controls all the numerics. Renormalizing the EMT in dimensional regularization (the standard loop regulator, which continues spacetime to $d=4-2\epsilon$) leaves the finite parts of the quark and gluon trace operators undetermined: one must choose two finite parameters, conventionally $x$ and $y$, which fix how the EMT trace is shared between quark and gluon operators. Hatta, Rajan and Tanaka (JHEP 12 (2018) 008) showed that minimal-subtraction schemes (MS and $\overline{\mathrm{MS}}$, in which only the divergent $1/\epsilon$ poles are subtracted) fix $x$ and $y$ order by order in perturbation theory; Metz, Pasquini and Rodini (the source paper) additionally studied two 'D-type' schemes, D1 ($x=0$, $y=\gamma_m$) and D2 ($x=y=0$), which make the trace split diagonal or attribute the entire anomaly to gluons. Different choices give numerically different attributions — individual quark and gluon terms move by tens of percent between schemes. The quark-mass (sigma-term) term is the exception: it does not depend on $x$ and $y$, and its spread, from below 10% of the proton mass (without a charm sigma term) to nearly 20% (with the lattice charm sigma term included), is driven by the input used for the heavy-quark sigma terms rather than by the scheme choice. No symmetry fixes $x$ and $y$: the decomposition of the EMT trace is a scheme convention, and the D-type schemes additionally lack the renormalization infrastructure needed to 'improve' them to higher perturbative order with consistently matched inputs. The source paper also stresses that its numerical evaluation combines inputs (phenomenological parton momentum fractions, sigma terms from chiral perturbation theory or lattice QCD) determined in schemes different from the one in which the sum rule is stated, and that a direct determination of the renormalized matrix elements $\langle(F^{\alpha\beta}F_{\alpha\beta})_R\rangle$ and $\langle(m_q\bar\psi_q\psi_q)_R\rangle$ — including heavy flavors — would clarify the numerics of the Higgs-related share.

Problem Statement

The problem is to make the quantitative decomposition of the proton mass unambiguous and complete. Two linked open questions stand between the present situation and that objective, and both must be resolved for it to be met — a first-principles account of how the proton mass arises, in which how much of it is attributable to the Higgs mechanism (quark masses, the sigma terms) and how much to QCD dynamics (quark energy, gluon energy and the trace anomaly) is settled rather than convention-dependent:

(1) Scheme dependence of the operator-level decomposition. The finite renormalization parameters $x$ and $y$ that split the trace of the QCD energy-momentum tensor into quark and gluon contributions are not fixed by any symmetry, so different renormalization schemes (MS-type implementations and the D-type schemes D1, D2) yield different operator-level attributions and numerics. Establish either a physical principle that singles out a unique such split — that is, unique quark and gluon parts of the renormalized EMT trace, as parameterized by $x$ and $y$, in the mass sum rules of the cited references (Ji's four-term rule [Ref. 2], Lorcé's two- and four-term rules [Refs. 4, 5], the Hatta–Rajan–Tanaka two-term rule [Ref. 3]) — or a proof that the ambiguity is irreducible, together with an explicit characterization of which combinations of sum-rule terms carry scheme-independent physical meaning beyond the content already established (the total mass $M$, and the sigma-term piece $M_m$, which is independent of $x$ and $y$); here a scheme-invariant combination means a combination of sum-rule terms whose forward proton matrix element is unchanged when $x$ and $y$ are varied among the admissible schemes. In either case the Higgs-versus-QCD-dynamical split is not itself what the convention decides: the sigma-term piece is free of $x$,$y$ dependence, while what the scheme choice moves is the distribution of the remaining, QCD-dynamical part among quark energy, gluon energy and the trace anomaly.

(2) First-principles determination. Provide a first-principles determination of the renormalized matrix elements $\langle(F^{\alpha\beta}F_{\alpha\beta})_R\rangle$ and $\langle(m_q\bar\psi_q\psi_q)_R\rangle$ for the nucleon, including heavy (charm) flavors, at the physical quark masses and in the continuum limit, with every hadronic input entering the mass sum rules renormalized and matched in that same scheme.

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