The nucleon mass in N_f=2 lattice QCD: finite size effects from chiral perturbation theory

TL;DR

This work uses relativistic SU(2)_f baryon chiral perturbation theory to compute finite-volume corrections to the nucleon mass up to O(p^4) in lattice QCD with N_f=2, identifying contributions from pions traveling around the box and expressing the total shift as m_N(L) − m_N(∞) = Δ_a(L) + Δ_b(L) with explicit, parameter-free formulas once low-energy constants are fixed from large-volume data. The O(p^3) term Δ_a(L) and the O(p^4) term Δ_b(L) yield leading and subleading finite-size effects that agree remarkably with UKQCD/QCDSF and CP-PACS/JLQCD data across several volumes and pion masses, indicating robust applicability of chiral EFT in finite volumes. Fits using phenomenologically consistent parameters reproduce the large-volume data and produce finite-size predictions that match the measured masses without extra free parameters, underscoring the predictive power of the approach and its potential to guide lattice extrapolations. The study also connects the finite-volume expressions to Lüscher-type dispersive forms, highlighting the significance of multi-winding pion effects in finite boxes and suggesting directions for ratio-based analyses to reduce scale-setting uncertainties.

Abstract

In the framework of relativistic SU(2)_f baryon chiral perturbation theory we calculate the volume dependence of the nucleon mass up to and including O(p^4). Since the parameters in the resulting finite size formulae are fixed from the pion mass dependence of the large volume nucleon masses and from phenomenology, we obtain a parameter-free prediction of the finite size effects. We present mass data from the recent N_f=2 simulations of the UKQCD and QCDSF collaborations and compare these data as well as published mass values from the dynamical simulations of the CP-PACS and JLQCD collaborations with the theoretical expectations. Remarkable agreement between the lattice data and the predictions of chiral perturbation theory in a finite volume is found.

Figures (5)

• Figure 1: One-loop graphs of NLO (a) and NNLO (b, c) contributing to the nucleon mass shift. The solid circle denotes a vertex from the leading order Lagrangian, the diamond a vertex from the $O(p^2)$ Lagrangian.
• Figure 2: Nucleon mass data on (relatively) large and fine lattices. The star indicates the physical point. The curve corresponds to Fit 1 in Table \ref{['tab:fitresults']}.
• Figure 3: Volume dependence of the nucleon mass for $m_\pi = 545 \, \hbox{MeV}$ (data points 31, 36, 41). The dotted curve shows the contribution of the $p^3$ term, while the solid curve includes also the $p^4$ correction, with the parameters taken from Fit 1.
• Figure 4: Volume dependence of the nucleon mass for $m_\pi = 717 \, \hbox{MeV}$ (data points 12, 13, 14). The dotted curve shows the contribution of the $p^3$ term, while the solid curve includes also the $p^4$ correction, with the parameters taken from Fit 1.
• Figure 5: Volume dependence of the nucleon mass for $m_\pi = 732 \, \hbox{MeV}$ (data points 30, 35, 40). The dotted curve shows the contribution of the $p^3$ term, while the solid curve includes also the $p^4$ correction, with the parameters taken from Fit 1.