Finite volume effects for meson masses and decay constants

TL;DR

The paper develops a resummed Lüscher-type framework to quantify finite-volume effects on pseudoscalar meson masses and decay constants, integrating it with chiral perturbation theory to express shifts in terms of infinite-volume amplitudes. It provides explicit formulas for $M_P(L)$ and $F_P(L)$ for $ ext{π}$, $K$, and $ ext{η}$, including up to two- or three-loop order inputs and a polynomially simplified near $ u=0$ representation for practical use. Numerical analyses show finite-volume effects are typically a few percent or less for $L\,\geq\,2 ext{ fm}$, with the relative size governed by $M_ ext{π}L$ and the chiral order of input; kaon and eta corrections are generally small. The work demonstrates two applications: correcting lattice results for precision determinations like $V_{us}$ from $F_K/F_π$, and a pathway to constrain low-energy constants from finite-volume dependence, thereby linking two- and four-point functions. Overall, the resummed asymptotic framework provides accurate, analytically tractable finite-volume corrections that can guide lattice volume choices and enable LEC extractions.

Abstract

We present a detailed numerical study of finite volume effects for masses and decay constants of the octet of pseudoscalar mesons. For this analysis we use chiral perturbation theory and asymptotic formulae a la Luscher and propose an extension of the latter beyond the leading exponential term. We argue that such a formula, which is exact at the one-loop level, gives the numerically dominant part at two loops and beyond. Finally, we discuss the possibility to determine low energy constants from the finite volume dependence of masses and decay constants.

Figures (7)

• Figure 1: $M_{\pi}$ dependence (in infinite volume) of $F_{\pi},F_K,M_K,M_\eta$. For the latter three quantities the strange quark mass has been fixed as to reproduce $M_K^\mathrm{phys}$ at $M_{\pi}^{}\!=\!M_{\pi}^\mathrm{phys}$.
• Figure 2: $R_{M_{\pi}}$ vs. $M_{\pi}$ for $L\!=\!2,3,4\,\mathrm{fm}$ (top) and vs. $L$ for $M_{\pi}\!=\!100,300,500\,\mathrm{MeV}$ (bottom). The result of the original ("$n\!=\!1$") Lüscher formula (\ref{['luscher_mpi_ori']}) with LO/NLO/NNLO chiral input is to be compared to the resummed ("all $n$") formula (\ref{['eq:Rmpin']}) which amounts to an approximate 1/2/3-loop ChPT calculation in finite volume. With NNLO input the low energy constants lead to a non-negligible error band; with NLO input the error is smaller (not shown), with LO input it is zero (see text). In the region above the $M_{\pi} L\!=\!2$ line one is not safely in the $p$-regime and our results should not be trusted.
• Figure 3: $-R_{F_{\pi}}$ vs. $M_{\pi}$ for $L\!=\!2,3,4\,\mathrm{fm}$ (left) and vs. $L$ for $M_{\pi}\!=\!100,300,500\,\mathrm{MeV}$ (right).
• Figure 4: $-R_{F_K}$ vs. $M_{\pi}$ for $L\!=\!2,3,4\,\mathrm{fm}$ (left) and vs. $L$ for $M_{\pi}\!=\!100,300,500\,\mathrm{MeV}$ (right).
• Figure 5: $R_{M_K}$ (left) and $R_{M_\eta}$ (right) vs. $M_{\pi}$ for $L\!=\!2,3,4\,\mathrm{fm}$.