FLAG Review 2021 (2111.09849v2)

Abstract: We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU(2)_L\times SU(2)_R$ and $SU(3)_L\times SU(3)_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $\alpha_s$. We consider nucleon matrix elements, and review the determinations of the axial, scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities.

• The paper presents a comprehensive consolidation of lattice QCD calculations to determine quark masses and CKM matrix elements.
• It employs rigorous methodologies, including quality rating systems and cross-validation across lattice configurations, to ensure result reliability.
• The findings have significant implications for both theoretical Standard Model precision and practical high-energy physics simulations.

Overview of Lattice QCD Calculations in Flavour Physics

The paper under review is an extensive review conducted by the Flavour Lattice Averaging Group (FLAG), focusing on lattice Quantum Chromodynamics (QCD) calculations related to flavour physics. Published under the designations CERN-TH-2021-191 and JLAB-THY-21-3528, this comprehensive review discusses key results pertaining to pion, kaon, D-meson, B-meson, and nucleon physics. The primary objectives of this review are to provide a consolidated, accessible repository of lattice determinations for various physical quantities and to enhance the precision in our understanding of fundamental parameters within the Standard Model and its potential extensions.

Focus and Methodology

The FLAG review intricately analyzes a variety of crucial physical parameters. These include light-quark masses, CKM matrix elements, and the strong coupling constant. Critical evaluations are extended to semileptonic and leptonic decays of hadrons, offering insights into extraction of the CKM matrix elements such as $V_{us}$, $V_{ud}$, and $V_{ub}$.

The computational methods employed are rooted in lattice QCD, facilitating first-principle calculations of strong interaction effects. The FLAG collaboration delineates and synthesizes results from multiple lattice calculations, adopting stringent quality criteria and a rating system to ensure the robustness and reliability of the data included in their averages and estimates.

Numerical Results and Comparisons

The review articulates several pivotal findings. The average value for the strange-quark mass in the $2+1+1$ setup is calibrated at $\approx 93.40 \ \text{MeV}$, and for the $2+1$ setup at $\approx 92.03 \ \text{MeV}$. In heavy-quark sector analysis, the value for the charm quark mass is quoted with fine precision across different lattice setups: $1.278 \ \text{GeV}$ for $2+1+1$, reflecting a small divergence when compared to the $2+1$ framework, which cites $1.275 \ \text{GeV}$.

For leptonic and semileptonic decays, precision is enhanced by utilizing lattice inputs alongside experimental decay rates to draw comparisons and examine consistency across Standard Model predictions. The review finds $f_{+}(0)$ and $f_K/f_{\pi}$ ratios to reveal critical implications on CKM matrix unitarity, suggesting tensions leading as high as $3\sigma$ deviations.

Theoretical and Practical Implications

Beyond numerical analyses, the review explores the theoretical implications of these determinations. It addresses the role of lattice QCD in pushing the precision frontier, revealing subtle discrepancies or consistencies that could spotlight physics beyond the Standard Model. These efforts in improving computational techniques, alongside enhancing statistical methods, are crucial in validating the Standard Model’s accuracy and highlighting potential new physics.

On the practical front, these lattice QCD results have substantive implications on high-energy experiments and phenomenological models. Enhanced determinations of quark masses refine input parameters for hadron collider simulations, crucially influencing the interpretation of so-called “anomalies” in B-physics.

Looking Forward: Advancements and Challenges

FLAG emphasizes the ongoing need for incorporating QED corrections and isospin-breaking effects directly in lattice simulations to curb systematic uncertainties. Furthermore, it highlights the importance of cross-checking with results from different fermion actions and configurations, thereby reinforcing the robustness of conclusions drawn from lattice QCD.

A notable challenge remains to be the resolution of topological freezing, which might impact lattice simulations at extremely fine lattice spacings. Innovative strategies like open boundary conditions show promise and reflect the active pursuit of methodological advancements within the lattice community.

Conclusion

The FLAG 2021 review serves as a pivotal resource for academics and researchers in high-energy physics. It not only consolidates and rigorously evaluates critical lattice QCD results but also sets the stage for ongoing advancements and future research directions. As lattice methods continue to advance, they promise to sharpen our quantitative understanding of the Standard Model while probing the confines at which new physics might emerge.