Nucleon mass: from lattice QCD to the chiral limit

TL;DR

The paper tackles the challenge of extrapolating lattice QCD results for the nucleon mass to the physical, light-quark regime using covariant baryon chiral perturbation theory with infrared regularization. It introduces and benchmarks an expansion up to ${\cal O}(p^4)$, studies the impact of explicit $\Delta(1232)$ degrees of freedom via the small-scale expansion, and performs a detailed error analysis including finite-volume effects. The results demonstrate stable extrapolations for $m_\pi$ up to about 0.6 GeV, show consistency of the extracted low-energy constants with pion–nucleon and nucleon–nucleon data when delta-dominance is accounted (notably for the $c_3$ term), and reveal that finite-volume data provide valuable additional constraints. The work argues for simultaneous, multi-observable fits with a common set of low-energy constants to further sharpen lattice-to-physical-mass extrapolations.

Abstract

Previous extrapolations of lattice QCD results for the nucleon mass to the physically relevant region of small quark masses, using chiral effective field theory, are extended and expanded in several directions. A detailed error analysis is performed. An approach with explicit delta(1232) degrees of freedom is compared to a calculation with only pion and nucleon degrees of freedom. The role of the delta(1232) for the low-energy constants of the latter theory is elucidated. The consistency with the chiral perturbation theory analysis of pion-nucleon scattering data is examined. It is demonstrated that this consistency can indeed be achieved if the delta(1232) dominance of the P-wave pion-nucleon low-energy constant c3 is accounted for. Introduction of the delta(1232) as an explicit propagating degree of freedom is not crucial in order to describe the quark-mass dependence of the nucleon mass, in contrast to the situation with spin observables of the nucleon. The dependence on finite lattice volume is shown to yield valuable additional constraints. What emerges is a consistent and stable extrapolation scheme for pion masses below 0.6 GeV.

Figures (11)

• Figure 1: Nucleon mass as function of $m_\pi^2$. Shown is the best fit interpolation between lattice results and the physical point (star), performed at chiral order $p^4$ using eq. (\ref{['massp4']}) (solid curve), with input given in column (a) of table \ref{['tab-paramsb']}. The physical point is included. The lattice data points in the grey region have not been used as input. Also shown are intermediate steps at orders $p^2$ and $p^3$ according to eqs. (\ref{['massp2']}), (\ref{['massp3']}). The parameter $e_1^{(3)}(\lambda)$ has been fitted to lattice data.
• Figure 2: Error band at 68% confidence level ("statistical error") and envelope of bands encoding input parameter uncertainties ("systematic envelope").
• Figure 3: Leading-one-loop diagram for the nucleon self-energy with an intermediate $\Delta\,(1232)$.
• Figure 4: Best fit curves based on the formula at order $\epsilon^3$ in SSE. The short-dashed curve refers to "fit delta I", while the long-dashed curve corresponds to "fit delta II" in table \ref{['tabledelta']}. For comparison, we plot the 68% statistical error band of the ${\cal O}(p^4)$ B$\chi$PT result shown in fig. \ref{['fig-band']}.
• Figure 5: Comparison between our ${\cal O}(\epsilon^3)$ "fit delta II" (dashed curve) and a band of ${\cal O}(\epsilon^4)$ fits using natural size assumptions for higher-order couplings.