FLAG Review 2019 (1902.08191v3)

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 $D$- and $B$-meson decay constants, form factors, and mixing parameters. 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 $\alpha_s$. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal.

• The paper consolidates lattice QCD results to provide precise averages for light and heavy-quark masses using advanced simulation techniques.
• The paper demonstrates reliable determinations of mesonic decay constants and form factors that underpin robust CKM matrix element tests.
• The paper details improved extraction of low-energy constants, strengthening Standard Model validations and guiding future high-energy experiments.

Overview of Lattice QCD Results for Flavor Physics: A Review

The paper under review, produced by the Flavour Lattice Averaging Group (FLAG), presents a comprehensive synthesis of lattice QCD calculations pertinent to particle and nuclear physics. The work predominantly focuses on meson, nucleon, and quark quantities crucial for flavor physics, with the objective of consolidating lattice results to provide accessible values for these parameters to guide the particle and nuclear physics communities. This review serves as a pivotal resource by systematically presenting lattice determinations of light and heavy-quark masses, decay constants, form factors, CKM matrix elements, and the QCD coupling constant, integrating and averaging results from numerous collaborations.

Light and Heavy-Quark Masses

One of the primary achievements highlighted is the precision achieved in determining the light-quark masses $m_u$, $m_d$, and $m_s$ across different numbers of flavors ($N_f$). These calculations incorporate sophisticated techniques to correct for isospin-breaking and electromagnetic effects, although there remains some reliance on phenomenological inputs for these corrections. The review provides averages for both the $\overline{\text{MS}}$ scheme at 2 GeV and renormalization group invariant (RGI) quark masses for both $N_f=2+1$ and $N_f=2+1+1$ simulations.

Furthermore, the FLAG review discusses substantial agreement with theoretical expectations and experimental data for the charm and bottom-quark masses, showcasing results from multiple collaborations that demonstrate the computational strides in using lattice QCD to resolve these heavy masses. The heavy-bottom mass remains challenging, yet significant progress has been illustrated by groups utilizing NRQCD and other methods optimized for treating heavy quarks on the lattice.

Decay Constants and Form Factors

The lattice determination of mesonic decay constants, such as $f_K$ and $f_\pi$, and the form factor $f_+(0)$ are pivotal for understanding Standard Model (SM) processes. The paper substantiates the reliability of these calculations, highlighting their role in deriving CKM matrix elements like $|V_{us}|$ and $|V_{ud}|$, integral for evaluating unitarity in the first row of the CKM matrix—a key test for the SM's validity. The strong numerical consensus from different collaborations bolsters confidence in these SM tests, showcasing the precision of lattice techniques across varying masses and methodologies.

Low-Energy Constants and Chiral Perturbation Theory

Particular emphasis is placed on the extraction of low-energy constants (LECs) from lattice simulations, which parameterize the quark mass dependence of various observables within chiral perturbation theory (χPT). These constants are essential for phenomenological applications and inherently test SU(2) and SU(3) symmetries. The review delineates extensive determinations of LECs such as $F$, $\Sigma$, and other NLO parameters, primarily emphasizing their consistency and agreement with phenomenological estimates, albeit with some reliance on ongoing explorations to resolve scale setting and systematic extrapolation issues.

Practical and Theoretical Implications

Practically, the lattice results serve as benchmarks for particle detectors and future experiments, offering refined parameters critical for precise SM predictions. Theoretically, the FLAG review underscores the importance of lattice QCD as both a tool for confirming established physics and providing avenues for probing beyond the SM phenomena. The discussion on systematic errors and averaging methodologies further illustrates the complex landscape of lattice calculations and points towards areas where improvements are necessary, especially concerning charm-heavy formulations and the inclusion of isospin effects.

Future Developments

Looking ahead, continued improvements in computational techniques and resources promise further reductions in uncertainties associated with lattice-QCD-generated quantities. The integration of QCD and QED in lattice simulations represents a significant frontier, promising insights into electromagnetic contributions that impact light-quark masses and decay constants, directly influencing precision physics programs in ongoing and future high-energy physics experiments.

In summary, this FLAG review crystallizes a detailed picture of the current status in lattice QCD calculations pertinent to flavor physics, providing a well-curated body of results crucial for both applied and theoretical endeavors in contemporary particle and nuclear physics. This continued synthesis enhances the robustness of SM tests and lays the groundwork for novel explorations in high-energy physics.