FLAG Review 2021

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 $$f_+(0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> arising in the semileptonic $$K \rightarrow \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>→</mml:mo> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> transition at zero momentum transfer, as well as the decay constant ratio $$f_K/f_\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>π</mml:mi> </mml:msub> </mml:mrow> </mml:math> and its consequences for the CKM matrix elements $$V_{us}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>us</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $$V_{ud}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>ud</mml:mi> </mml:mrow> </mml:msub> </mml:math> . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $$SU(2)_L\times SU(2)_R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> </mml:math> and $$SU(3)_L\times SU(3)_R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> </mml:math> Chiral Perturbation Theory. We review the determination of the $$B_K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> 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 $$m_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> and $$m_b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> 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 $$\alpha _s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> . 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.