[Paper Review] Scale setting, sigma terms and the Feynman-Hellman theorem

Phiala E. Shanahan, A. W. Thomas+1 more|arXiv (Cornell University)|Jan 15, 2013

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper investigates the impact of lattice scale setting on the determination of nucleon sigma terms using the Feynman-Hellman theorem applied to chiral perturbation theory fits of PACS-CS octet baryon mass data. It finds that mass-dependent scale setting yields a precise strange sigma term of $\sigma_s = 21 \pm 6$ MeV, significantly smaller than earlier estimates, and highlights the critical role of scale setting in lattice QCD calculations of matrix elements relevant to dark matter searches.

ABSTRACT

The authors recently presented new values for the octet baryon sigma terms. These were extracted using the Feynman-Hellman theorem from a chiral perturbation theory fit to octet baryon mass data from the PACS-CS collaboration. Of particular interest is the precise determination of the strangeness sigma term $σ_s = 21 \pm 6$ MeV. In this work, we elaborate on the critical effect which the choice of scale setting has on this value. We discuss the prospect that the comparison of direct and 'spectrum' determinations of the sigma terms from the lattice can provide insight not only into scale setting on the lattice, but into QCD itself.

Motivation & Objective

• To assess how different lattice scale setting procedures affect the determination of baryon sigma terms.
• To improve the precision of the strange sigma term $\sigma_s$ using the Feynman-Hellman theorem and chiral perturbation theory fits to lattice data.
• To compare 'spectrum' method results (derived from mass splittings) with direct lattice calculations to probe scale setting ambiguities.
• To provide refined values for individual up, down, and strange quark sigma terms in the proton and neutron for use in supersymmetric dark matter searches.
• To investigate the consistency of $\sigma_s$ with QCD dynamics and the implications of scale setting for lattice QCD accuracy.

Proposed method

• Apply the Feynman-Hellman theorem to extract sigma terms as derivatives of baryon masses with respect to quark masses: $\sigma_{Bq} = m_q \partial M_B / \partial m_q$.
• Use chiral perturbation theory to fit PACS-CS lattice data for octet baryon masses, enabling extrapolation to physical quark masses.
• Implement two scale setting schemes: (1) fixed $\beta$ (mass-independent), and (2) $r_0$-based (mass-dependent), to assess their impact on sigma term results.
• Use the physical up-down quark mass ratio $R = m_u/m_d = 0.553 \pm 0.043$ to break isospin symmetry and extract individual $f_{Tq}^{(p)}$ and $f_{Tq}^{(n)}$ terms.
• Define and compute the ratio $z = (B_u - B_s)/(B_d - B_s)$ to quantify the relative contribution of strange quark density in nucleons.
• Propagate uncertainties from the quark mass ratio and fit parameters in quadrature to obtain final error estimates.

Experimental results

Research questions

• RQ1 How does the choice of lattice scale setting (mass-independent vs. mass-dependent) affect the calculated value of the strange sigma term?
• RQ2 What is the precise value of the nucleon strange sigma term $\sigma_s$ when derived from the Feynman-Hellman theorem and chiral fits to lattice data?
• RQ3 How do the individual up, down, and strange quark sigma terms in the proton and neutron compare with previous estimates, especially in the context of dark matter direct detection?
• RQ4 To what extent do 'spectrum' method results (derived from mass splittings) agree with direct lattice calculations of matrix elements?
• RQ5 Can the comparison of direct and spectrum-based sigma terms provide insight into the reliability of scale setting in lattice QCD?

Key findings

• The strange sigma term is determined to be $\sigma_s = 21 \pm 6$ MeV using mass-dependent scale setting, which is significantly smaller than earlier estimates of ~300 MeV.
• The individual proton strange sigma term is found to be $f_{Ts}^{(p)} = 0.023(7)$, and the neutron value is $f_{Ts}^{(n)} = 0.022(6)$, with uncertainties including correlated and phenomenological inputs.
• Using mass-independent scale setting (fixed $\beta$), the strange sigma term increases to $f_{Ts}^{(p)} = 0.063(7)$, indicating a strong dependence on scale setting procedure.
• The ratio $z = 1.27(3)$, which quantifies the relative contribution of strange quark density, is found to be consistent across scale-setting schemes, suggesting robustness in this observable.
• The up and down quark sigma terms in the proton are $f_{Tu}^{(p)} = 0.019(3)$ and $f_{Td}^{(p)} = 0.027(4)$, respectively, with the neutron values being $f_{Tu}^{(n)} = 0.015(2)$ and $f_{Td}^{(n)} = 0.033(5)$.
• The results suggest that mass-dependent scale setting is more consistent with direct lattice calculations and provides a more reliable determination of sigma terms for dark matter and QCD studies.