Nucleon mass, sigma term and lattice QCD

TL;DR

This study addresses how the nucleon mass $M_N$ depends on the light-quark mass by combining lattice QCD results with relativistic $SU(2)_f$ baryon chiral perturbation theory up to ${\cal O}(p^4)$. The authors develop analytic expressions for $M_N(m_\pi)$ using infrared regularization, and fit them to a carefully selected lattice data set, finding that a one-loop ${\cal O}(p^3)$ description already provides a good interpolation; ${\cal O}(p^4)$ corrections are small yet refine the extracted parameters. They obtain $M_0 \approx 0.88$ GeV in the chiral limit and $\sigma_N \approx 49$ MeV at the physical pion mass (with $\sigma_N$ consistent with phenomenology, e.g., GLS). The results demonstrate a robust path to bridge lattice results and physical observables via chiral EFT, while highlighting the roles of low-energy constants and resonance physics, and noting remaining uncertainties from finite-volume effects and convergence at higher pion masses.

Abstract

We investigate the quark mass dependence of the nucleon mass M_N. An interpolation of this observable, between a selected set of fully dynamical two-flavor lattice QCD data and its physical value, is studied using relativistic baryon chiral perturbation theory up to order p^4. In order to minimize uncertainties due to lattice discretization and finite volume effects our numerical analysis takes into account only simulations performed with lattice spacings a<0.15 fm and m_pi L>5. We have also restricted ourselves to data with m_pi<600 MeV and m_sea=m_val. A good interpolation function is found already at one-loop level and chiral order p^3. We show that the next-to-leading one-loop corrections are small. From the p^4 numerical analysis we deduce the nucleon mass in the chiral limit, M_0 approx 0.88 GeV, and the pion-nucleon sigma term sigma_N= (49 +/- 3) MeV at the physical value of the pion mass.

Figures (4)

• Figure 1: One-loop graphs of NLO (a) and NNLO (b, c) contributing to the nucleon mass shift. The solid dot denotes a vertex from ${\cal{L}}_N^{(1)}$, the diamond a vertex from ${\cal{L}}_N^{(2)}$.
• Figure 2: Solid/dotted line: best fit curve using the one-loop result at chiral order $p^3$ Eq.(\ref{['massa']}). Input: four lowest lattice data points with $m_\pi<600\,{\rm{MeV}}$ and physical nucleon mass (Fit I). The dotted extension of this curve for $m_\pi^2>0.4\,{\rm{GeV}^2}$ indicates the region where the application of baryon ChPT is usually believed to become unreliable. For illustration we also show a subset of lattice data up to $m_\pi \approx 0.8\,{\rm{GeV}}$, those compatible with the cuts in lattice spacing and volume as explained in the text. The solid dots are CP-PACS data, the boxes refer to JLQCD and the empty circles to QCDSF. The dot-dashed, dashed and long-dashed curves show, respectively, the contributions from the sum of the first three, four and five terms in Eq.(\ref{['massachir']}).
• Figure 3: Solid curve: best fit (Fit II) using the NNLO result, Eq.(\ref{['massap4']}), at chiral order $p^4$. Dashed curve: NLO result (chiral order $p^3$) from Eq.(\ref{['massa']}) using as parameters the central values of Fit II. $\hat{e}_1$ has been deduced setting $c_2=3.2\,{\rm GeV}^{-1}$. For details on the data points see Fig.\ref{['figp3']}.
• Figure 4: The pion-nucleon sigma term as a function of $m_\pi^2$ from Eq.(\ref{['sigmap4']}), using as input the central values from Fit II (see table \ref{['table1']}). The small $m_\pi$ region is magnified in the right panel and plotted together with the frequently quoted empirical $\sigma_N=45 \pm 8\, {\rm{MeV}}$GLS.