Nucleon mass and sigma term from lattice QCD with two light fermion flavors

TL;DR

The paper analyzes two-flavor lattice QCD data for the nucleon mass across pion masses down to 157 MeV and confronts it with covariant baryon chiral perturbation theory up to next-to-next-to-leading order. A novel aspect is the simultaneous fitting of $M_N(m_\pi)$ and the directly computed pion-nucleon sigma-term via the Feynman-Hellmann relation, with finite-volume corrections incorporated and the Sommer scale $r_0$ determined self-consistently. The study finds a physical sigma-term of $\sigma_{\rm phys}=37(8)(6)$ MeV and a Sommer scale $r_0=0.501(10)(11)$ fm, obtained from robust ${O}(p^4)$ fits within the range $m_\pi<435$ MeV. These results indicate reasonable agreement with other $N_f=2$ and $N_f=2+1$ determinations, while highlighting convergence challenges above $m_\pi\sim250$ MeV and the need for additional low-$m_\pi$ data to better constrain higher-order low-energy constants.

Abstract

We analyze Nf=2 nucleon mass data with respect to their dependence on the pion mass down to mpi = 157 MeV and compare it with predictions from covariant baryon chiral perturbation theory (BChPT). A novel feature of our approach is that we fit the nucleon mass data simultaneously with the directly obtained pion-nucleon sigma-term. Our lattice data below mpi = 435 MeV is well described by O(p^4) BChPT and we find sigma=37(8)(6) MeV for the sigma-term at the physical point. Using the nucleon mass to set the scale we obtain a Sommer parameter of r_0=0.501(10)(11) fm.

Figures (11)

• Figure 1: Lattice data for $r_0M_N$ versus $(r_0m_\pi)^2$ for different lattice spacings ($\beta=5.25$, $5.29$, $5.40$) and volumes. Open (filled) symbols refer to points where $L\le3r_0$ ($L>3r_0$). A black-framed (yellow) circle indicates the physical point assuming $r_0=0.5$ fm. The dotted lines connect points of the same $(\beta,\kappa)$ but different lattice size. They are meant to guide the eye and to illustrate finite-volume effects.
• Figure 2: Volume dependence of our pion mass data. Left top: $\beta=5.29$, $\kappa=0.13632$. Left bottom: $\beta=5.29$, $\kappa=0.13640$. Right top: $\beta=5.40$, $\kappa=0.13640$. Right bottom: $\beta=5.40$, $\kappa=0.13660$. Dashed-dotted lines represent finite-volume extrapolations, obtained from applying the method of Colangelo:2005gd to the point for the smallest $r_0/L$. Vertical dotted lines denote constant $m_\pi L$. Gray areas mark the region ($m_\pi L<3.5$) where finite-volume corrections are not under control.
• Figure 3: Fit (surface) to the nucleon mass data (full circles) for a range of $r_0/L$ and $(r_0m_\pi)^2$ values. Black circles lie above the surface, gray (or half gray) below (or on) the surface. The line and the red diamonds at $r_0/L=0$ mark the fitted infinite volume prediction. For the sake of simplicity, for each $(r_0m_\pi)^2$ only one extrapolated (red) point is shown at $r_0/L=0$.
• Figure 4: Simultaneous fits to the nucleon mass (left) and $\sigma$-term data (right) for three fitting windows with $(r_0m_\pi)_{\max}^2=3.0$, 1.6 and 1.3 (from top to bottom). These fits are labeled Soo3, Soo2 and Soo1 in Table \ref{['tab:fitparaS']}, where $c_2\equiv 3.3\,\mathrm{GeV}^{-1}$, $c_3\equiv -4.7\,\mathrm{GeV}^{-1}$ and $\bar{l}_3\equiv3.2$. Lines, error bands and (full red) points are shown for the limit $L\to\infty$ (cf. Fig. \ref{['fig:3dPlot']}). A black-framed circle marks the location of the physical point using---for each plot separately---the $r_0$-value for which the fit is self-consistent. Open points did not enter any fit. In the left plots, the overlap of red points at $(r_0m_\pi)^2=0.436$ and 0.538 indicates the quality of the (fitted) finite-volume corrections.
• Figure 5: Left panel: stand-alone fit to the nucleon mass data (see fit Noo2 in Table \ref{['tab:fitparaN']}). Right panel: comparison of the $\sigma$-term data (open symbols) for a $32^3\times 64$ and $40^3\times64$ lattice Bali:2011ks and the BChPT expression for $\sigma(m_\pi)$ with the parameters $M_0$, $c_1$ and $e_r^1$ taken from the nucleon mass fit shown in the left panel. The black-framed circle is the value for $r_0M_N$ ($r_0\sigma_{N}$) at the physical point for these parameters. In both panels, the one-sigma error band is shown in gray. As in Fig. \ref{['fig:Splots']}, the overlap of (red) points (left panel) at same $(r_0m_\pi)^2$ indicates the quality of the (fitted) finite-volume corrections.