Review of lattice results concerning low energy particle physics

TL;DR

The review consolidates lattice QCD results for low-energy pion and kaon physics, including light-quark masses, semileptonic form factors, and decay constants, and uses predefined quality criteria to form conservative averages across $N_f=2$ and $N_f=2+1$ simulations. It demonstrates consistent lattice determinations of $m_{ud}$, $m_s$, and their ratio, provides precise lattice inputs for $f_+(0)$ and $f_K/f_π$, and confirms CKM unitarity within uncertainties when combined with experimental data. The work also synthesizes lattice findings on SU(2) and SU(3) low-energy constants, and summarizes nonperturbative determinations of the kaon $B$-parameter $B_K$, emphasizing the interplay between lattice methodology, χPT, and phenomenology. Overall, the FLAG framework advances reliable, cross-validated lattice inputs for testing the Standard Model and constraining new physics in the flavor sector, with implications for precision CKM phenomenology and hadronic structure at low energy.

Abstract

We review lattice results relevant for pion and kaon physics with the aim of making them easily accessible to the particle physics community. Specifically, we review the determination of the light-quark masses, the form factor f_+(0), relevant for the semileptonic K -> pi transition at zero momentum transfer as well as the ratio f_K/f_pi of decay constants and discuss the consequences for the elements V_{us} and V_{ud} of the CKM matrix. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)_LxSU(2)_R and SU(3)_LxSU(3)_R Chiral Perturbation Theory and review the determination of the B_K parameter of neutral kaon mixing. We introduce quality criteria and use these when forming averages. Although subjective and imperfect, these criteria may help the reader to judge different aspects of current lattice computations. Our main results are summarized in section 1.2, but we stress the importance of the detailed discussion that underlies these results and constitutes the bulk of the present review.

Figures (11)

• Figure 1: Mass of the strange quark (${\overline{\mathrm{MS}}}$ scheme, running scale 2 GeV). The upper part shows the lattice results listed in Tables \ref{['tab:masses2']} and \ref{['tab:masses3']}. Full and empty symbols correspond to simulations with $N_{\space f}=2+1$ and $N_{\space f}=2$, respectively. Diamonds represent results based on perturbative renormalization, while squares indicate that, in the relation between the lattice-regularized and renormalized ${\overline{\mathrm{MS}}}$ masses, non-perturbative effects are accounted for. The vertical bands indicate our estimates (\ref{['eq:quark masses Nf=2']}) and (\ref{['eq:nf3msmud']}). The lower part shows recent determinations obtained from the evaluation of sum rules, together with the earliest result based on this method, as well as the most recent estimate of the Particle Data Group.
• Figure 2: Mean mass of the two lightest quarks, $m_{ud}=\frac{1}{2}(m_u+m_d)$. The meaning of the various symbols is explained in the caption of Fig. \ref{['fig:ms']}.
• Figure 3: Results for the ratio $m_s/m_{ud}$, in which the renormalization factors cancel. The upper part of the figure shows the lattice results listed in Table \ref{['tab:ratio_msmud']} (full and empty symbols correspond to $N_{\space f}=2+1$ and $N_{\space f}=2$). The vertical bands indicate our estimates (\ref{['eq:quark masses Nf=2']}) and (\ref{['eq:msovmud']}). The lower part shows results obtained on the basis of $\chi$PT or from QCD sum rules, as well as the most recent estimate of the Particle Data Group.
• Figure 4: Comparison of lattice results (red squares) for $f_+(0)$ and $f_K/ f_\pi$ with various model estimates based on $\chi$PT (blue triangles). Full and empty squares represent simulations with $N_{\space f}=2+1$ and $N_{\space f}=2$, respectively. The vertical bands indicate our estimates.$^\star$
• Figure 5: The plot compares the information for $|V_{ud}|$, $|V_{us}|$ obtained on the lattice with the experimental result extracted from nuclear $\beta$ transitions. The dotted arc indicates the correlation between $|V_{ud}|$ and $|V_{us}|$ that follows if the three-flavour CKM-matrix is unitary.