NUC In the literature: contested

Pion-nucleon $\sigma$ term: lattice QCD versus pion-nucleon scattering

In plain words

The $\sigma$ term measures how much of the proton mass comes from the small masses of the up and down quarks. Extractions from pion-proton scattering give about 59 MeV, while several lattice QCD calculations (QCD solved numerically on a space-time grid) give about 40 to 45 MeV.

Precise statement

sigma_piN = m_hat <N| u-bar u + d-bar d |N> with $m_{\mathrm{hat}} = (m_{u} + m_{d})/2$. Roy-Steiner analysis of $\pi N$ scattering gives $59.1 \pm 3.5\,\mathrm{MeV}$ (Hoferichter et al., PRL 115, 092301, 2015); lattice QCD with standard excited-state treatment gives about $40 \text{ to } 45\,\mathrm{MeV}$ (for example $43.7 \pm 3.6\,\mathrm{MeV}$, Agadjanov et al., arXiv 2303.08741), while including $N\pi$ and $N\pi\pi$ excited states raises it to $59.6 \pm 7.4\,\mathrm{MeV}$ (Gupta et al., PRL 127, 242002, 2021). Answer: sigma_piN with uncertainty below $3\,\mathrm{MeV}$ agreed between physical-point lattice QCD and the dispersive value, with the cause of the discrepancy identified.

What would settle it

Lattice QCD at physical quark masses with variational nucleon, N pi and N pi pi operator bases and continuum extrapolation, compared with the Roy-Steiner value.

Status in the literature

Unverified note

A two-loop chiral analysis of lattice data gave $55.9(2.5)\,\mathrm{MeV}$ (Liang et al., arXiv 2508.11435, 2025), and a variational lattice study at $m_{\pi} = 429\,\mathrm{MeV}$ showed reduced excited-state contamination with multi-hadron operators (Barca, Bali, Collins, arXiv 2412.13138, 2024); no physical-point lattice consensus existed as of 2026.