The pion mass in finite volume

TL;DR

This work addresses finite-volume corrections to the pion mass in two-flavor QCD by merging NNLO CHPT forward scattering amplitudes with Lüscher's asymptotic formula. The authors derive a compact expression for the relative mass shift $R_M(M_\pi,L)$ as a xi-series with integrals $I_{2m}(\lambda)$ computed from the CHPT $\pi\pi$ amplitude, enabling a robust assessment of convergence and subleading effects. Numerical analysis shows good CHPT convergence for moderate to heavy pions and $L\gtrsim 2$ fm, while for very light pions or smaller volumes subleading exponentials and full two-loop finite-volume CHPT become important. The results provide practical guidance for lattice QCD extrapolations, quantifying when Lüscher's leading term suffices and when more complete finite-volume treatments are required.

Abstract

We determine the relative pion mass shift $M_π(L)/M_π-1$ due to the finite spatial extent $L$ of the box by means of two-flavor chiral perturbation theory and the one-particle Lüscher formula. We use as input the expression for the infinite volume $ππ$ forward scattering amplitude up to next-to-next-to-leading order and can therefore control the convergence of the chiral series. A comparison to the full leading order chiral expression for the pion mass in finite volume allows us to check the size of subleading terms in the large-$L$ expansion.

Figures (7)

• Figure 1: The mass shift due to quantized momenta in the self-energy corrections amounts to a finite-size effect from pion exchange "around the world" (left), depicted by a "thermal insertion" (cross) in diagrammatic language (right).
• Figure 2: Integration contour in the complex $\nu$ plane where $\nu$ is the crossing variable in the (Minkowski space) forward scattering amplitude. The cut is generated only at NLO in the chiral expansion.
• Figure 3: The integrals $B^{2k}$ and $R_i^k$ which are needed for the evaluation of the finite volume corrections to NNLO. The $B^{2k}$ are known analytically, the $R_i^k$ have been determined numerically.
• Figure 4: The plot on the left shows the $1\sigma$-band of the pion mass dependence of $F_\pi$. The plot on the right shows the dependence of $\xi$ on the pion mass, without including any uncertainty.
• Figure 5: The integrand $I(y)=e^{-\sqrt{M_{\pi}^2+y^2}\,L}\,F(\mathrm{i}y)$ with LO, NLO, NNLO input from CHPT. Note the change in scale in the lower panel.