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Nucleon mass: trace anomaly and $σ$-terms

Martin Hoferichter, Jacobo Ruiz de Elvira

TL;DR

The paper provides a pedagogical framework to understand nucleon mass origin through the trace anomaly and $\sigma$-terms in QCD. It develops the formalism of the trace of the energy-momentum tensor, including Abelian and non-Abelian cases, and defines $\sigma$-terms via scalar matrix elements and the Feynman–Hellmann theorem. It reviews the Cheng–Dashen low-energy theorem and Roy–Steiner analyses for $\sigma_{\pi N}$, presents lattice QCD results for $\sigma_{\pi N}$, $\sigma_s$, and heavy-quark terms, and discusses perturbative predictions for heavy flavors. It then provides a quantitative decomposition of $m_N$ showing gluonic contributions dominate (about two-thirds), with substantial heavy-quark content and smaller light-quark contributions, while highlighting uncertainties and open questions for cross-checking methods and improving precision.

Abstract

We give a pedagogical introduction to the origin of the mass of the nucleon. We first review the trace anomaly of the energy-momentum tensor, which generates most of the nucleon mass via the gluon fields and thus contributes even in the case of vanishing quark masses. We then discuss the contributions to the nucleon mass that do originate from the Higgs mechanism via the quark masses, reviewing the current status of nucleon $σ$-terms that encode the corresponding matrix elements.

Nucleon mass: trace anomaly and $σ$-terms

TL;DR

The paper provides a pedagogical framework to understand nucleon mass origin through the trace anomaly and -terms in QCD. It develops the formalism of the trace of the energy-momentum tensor, including Abelian and non-Abelian cases, and defines -terms via scalar matrix elements and the Feynman–Hellmann theorem. It reviews the Cheng–Dashen low-energy theorem and Roy–Steiner analyses for , presents lattice QCD results for , , and heavy-quark terms, and discusses perturbative predictions for heavy flavors. It then provides a quantitative decomposition of showing gluonic contributions dominate (about two-thirds), with substantial heavy-quark content and smaller light-quark contributions, while highlighting uncertainties and open questions for cross-checking methods and improving precision.

Abstract

We give a pedagogical introduction to the origin of the mass of the nucleon. We first review the trace anomaly of the energy-momentum tensor, which generates most of the nucleon mass via the gluon fields and thus contributes even in the case of vanishing quark masses. We then discuss the contributions to the nucleon mass that do originate from the Higgs mechanism via the quark masses, reviewing the current status of nucleon -terms that encode the corresponding matrix elements.

Paper Structure

This paper contains 15 sections, 35 equations, 1 figure, 1 table.

Table of Contents

  1. Nucleon mass: trace anomaly and $\boldsymbol{\sigma}$-terms
  2. Introduction
  3. Trace anomaly of the energy-momentum tensor
  4. General considerations
  5. Abelian case
  6. Non-Abelian case
  7. Quark-mass contributions and scheme definitions
  8. Nucleon $\boldsymbol{\sigma}$-terms
  9. Phenomenological determinations of $\boldsymbol{\sigma_{\pi N}}$
  10. Isospin conventions and chiral limit
  11. SU(3) relations for $\boldsymbol{\sigma_s}$
  12. Lattice QCD
  13. Heavy quarks
  14. Summary of the nucleon mass decomposition
  15. Conclusions

Figures (1)

  • Figure 1: Pie chart for the decomposition of the nucleon mass into contributions arising from the light quark masses $m_q$ with $q=u,d$ and $q=s$, the heavy quarks $Q=c,b,t$, and the gluon field strength $F_{\mu\nu}^a F^{\mu\nu}_a$. The numerical values, taken from Eq. \ref{['mN_numerics']}, in some cases still carry substantial uncertainties, as discussed in this chapter, while the general hierarchy among the different contributions is robust.