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 $V_{ud}$. 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 review the determinations of nucleon charges from the matrix elements of both isovector and flavour-diagonal axial, scalar and tensor local quark bilinears, and momentum fraction, helicity moment and the transversity moment from one-link quark bilinears. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model.

Recent developments in non-perturbative renormalization for lattice QCD are reviewed with a particular emphasis on RI/MOM scheme and its variants, RI/SMOM schemes. Summary of recent developments in Schroedinger functional scheme, as well as the summary of related topics are presented. Comparison of strong coupling constant and the strange quark mass from various methods are made.

We report the first lattice QCD calculation of pion valence quark distribution with next-to-next-to-leading order perturbative matching correction, which is done using two fine lattices with spacings $a=0.04$ fm and $0.06$ fm and valence pion mass $m_\pi=300$ MeV, at boost momentum as large as $2.42$ GeV. As a crucial step to control the systematics, we renormalize the pion valence quasi distribution in the recently proposed hybrid scheme, which features a Wilson-line mass subtraction at large distances in coordinate space, and develop a procedure to match it to the $\overline{\rm MS}$ scheme. We demonstrate that the renormalization and the perturbative matching in Bjorken-$x$ space yield a reliable determination of the valence quark distribution for $0.03\lesssim x \lesssim 0.80$ with 5-20\% uncertainties.

We extend the Rome-Southampton regularization independent momentum-subtraction renormalization scheme(RI/MOM) for bilinear operators to one with a nonexceptional, symmetric subtraction point. Two-point Green's functions with the insertion of quark bilinear operators are computed with scalar, pseudoscalar, vector, axial-vector and tensor operators at one-loop order in perturbative QCD. We call this new scheme RI/SMOM, where the S stands for "symmetric". Conversion factors are derived, which connect the RI/SMOM scheme and the MSbar scheme and can be used to convert results obtained in lattice calculations into the MSbar scheme. Such a symmetric subtraction point involves nonexceptional momenta implying a lattice calculation with substantially suppressed contamination from infrared effects. Further, we find that the size of the one-loop corrections for these infrared improved kinematics is substantially decreased in the case of the pseudoscalar and scalar operator, suggesting a much better behaved perturbative series. Therefore it should allow us to reduce the error in the determination of the quark mass appreciably.

We present a calculation of the renormalization coefficients of the quark bilinear operators and the K-Kbar mixing parameter B_K. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed non-perturbatively and then matched perturbatively to the MSbar scheme. The coefficients are calculated on the RBC/UKQCD 2+1 flavor dynamical lattice configurations. Specifically we use a 16^3 x 32 lattice volume, the Iwasaki gauge action at beta=2.13 and domain wall fermions with L_s=16.

We revisit the calculation of the next-to-leading order (NLO) corrections to dijet production in electron-ion collisions at small $x$. We focus on the back-to-back configuration where the relative transverse momentum $P_\perp$ of the measured jets is much larger than both their momentum imbalance $K_\perp$ and the target saturation momentum $Q_s(x,A)$. In this regime, we present for the first time a complete calculation of the real NLO corrections at leading power in $1/P_\perp$. Our result exhibits TMD factorisation, with the same hard factor as at tree-level and a NLO correction to the Weiszäcker-Williams (WW) gluon transverse momentum dependent (TMD) distribution which involves four Wilson-line operators. By studying different kinematical regimes for $K_\perp$ and for the radiated gluon, we recover all the quantum evolutions that were previously identified for this process at NLO: the B-JIMWLK high-energy evolution and the CSS evolution of the gluon WW TMD, and the DGLAP evolution of the gluon PDF. When both $K_\perp$ and the transverse momentum transferred by the target are large compared to $Q_s$, all the Wilson-line operators boil down to the unintegrated gluon distribution and our NLO result for the gluon TMD can be used to isolate the transverse-momentum dependent gluon splitting function.

The PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC) has measured electrons from heavy flavor (charm and bottom) decays for 0.3 < p_T < 9 GeV/c at midrapidity (|y| < 0.35) in Au+Au collisions at sqrt(s_NN) = 200 GeV. The nuclear modification factor R_AA relative to p+p collisions shows a strong suppression in central Au+Au collisions, indicating substantial energy loss of heavy quarks in the medium produced at RHIC. A large azimuthal anisotropy, v_2, with respect to the reaction plane is observed for 0.5 < p_T < 5 GeV/c indicating non-zero heavy flavor elliptic flow. Both R_AA and v_2 show a p_T dependence different from those of neutral pions. A comparison to transport models which simultaneously describe R_AA(p_T) and v_2(p_T) suggests that the viscosity to entropy density ratio is close to the conjectured quantum lower bound, i.e., near a perfect fluid.

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.

We present the first lattice QCD calculation of the universal axial $\gamma W$-box contribution $\square_{\gamma W}^{VA}$ to both superallowed nuclear and neutron beta decays. This contribution emerges as a significant component within the theoretical uncertainties surrounding the extraction of $|V_{ud}|$ from superallowed decays. Our calculation is conducted using two domain wall fermion ensembles at the physical pion mass. To construct the nucleon 4-point correlation functions, we employ the random sparsening field technique. Furthermore, we incorporate long-distance contributions to the hadronic function using the infinite-volume reconstruction method. Upon performing the continuum extrapolation, we arrive at $\square_{\gamma W}^{VA}=3.65(8)_{\mathrm{lat}}(1)_{\mathrm{PT}}\times10^{-3}$. Consequently, this yields a slightly higher value of $|V_{ud}|=0.97386(11)_{\mathrm{exp.}}(9)_{\mathrm{RC}}(27)_{\mathrm{NS}}$, reducing the previous $2.1\sigma$ tension with the CKM unitarity to $1.8\sigma$. Additionally, we calculate the vector $\gamma W$-box contribution to the axial charge $g_A$, denoted as $\square_{\gamma W}^{VV}$, and explore its potential implications.

We use a method to calculate the hadron's charge radius without model-dependent momentum extrapolations. The method does not require the additional quark propagator inversions on the twisted boundary conditions or the computation of the momentum derivatives of quark propagators and thus is easy to implement. We apply this method to the calculation of pion charge radius $\langle r_\pi^2\rangle$. For comparison, we also determine $\langle r_\pi^2\rangle$ with the traditional approach of computing the slope of the form factors. The new method produces results consistent with those from the traditional method and with statistical errors 1.5-1.9 times smaller. For the four gauge ensembles at the physical pion masses, the statistical errors of $\langle r_\pi^2\rangle$ range from 2.1% to 4.6% by using $\lesssim50$ configurations. For the ensemble at $m_\pi\approx 340$ MeV, the statistical uncertainty is even reduced to a sub-percent level.

The agreement of hydrodynamic predictions of differential elliptic flow and radial flow patterns with Au+Au data is one of the main lines of evidence suggesting the nearly perfect fluid properties of the strongly coupled Quark Gluon Plasma, sQGP, produced at RHIC. We study the sensitivity of this conclusion to different hydrodynamic assumptions on chemical and thermal freezeout after the sQGP hadronizes. We show that if chemical freezeout occurs at the hadronization time, the differential elliptic flow for pions increase with proper time in the late hadronic phase until thermal freezeout and leads to a discrepancy with the data. In contrast, if both chemical and thermal equilibrium are maintained past the hadronization, then the mean pT per pion increases in a way that accidentally preserves v2(pT) from the sQGP phase in agreement with the data, but at the cost of the agreement with the hadronic yields. In order that all the data on hadronic ratios, radial flow, and differential elliptic flow be reproduced, the sQGP must expand with a minimal viscosity, \eta_Q, that is however even greater than the viscosity, \eta_H, of its hadronic corona. However, because of the large entropy density difference of the two phases of QCD matter, the larger viscosity in the sQGP phase leads to nearly perfect fluid flow while the smaller entropy density of the hadronic corona strongly hinders the applicability of Euler hydrodynamics. The ``perfect fluid'' property of the sQGP is thus not due to a sudden reduction of the viscosity at the critical temperature Tc, but to a sudden increase of the entropy density of QCD matter and is therefore an important signature of deconfinement.

We consider the $O(3)$ nonlinear sigma model with the $\theta$-term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $\theta$, and enjoys the periodicity in $\theta\rightarrow \theta+2\pi$. At $\theta=0,\pi$ it also has a charge-conjugation $C$-symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable $\mathbb RP^2$ manifold, after which the $2\pi$ periodicity of $\theta$ and the $C$ symmetry at $\theta=\pi$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation $C$, parity $P$, and time-reversal $T$ symmetries when $\theta=\pi$. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as $C$, $P$ and $T$ are all good symmetries of the theory. We make several consistency checks with various semi-classical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity and time reversal (but not involving the $SO(3)$-spin symmetry), requiring that the ground state is never trivially gapped, even if $SO(3)$ spin symmetry is explicitly and completely broken. We also consider generalizations to $\mathbb{C}P^{N-1}$ models and show that the $C$-$P$-$T$ anomaly exists for even $N$.

We calculate the equation of state in 2+1 flavor QCD at finite temperature with physical strange quark mass and almost physical light quark masses using lattices with temporal extent Nt=8. Calculations have been performed with two different improved staggered fermion actions, the asqtad and p4 actions. Overall, we find good agreement between results obtained with these two O(a^2) improved staggered fermion discretization schemes. A comparison with earlier calculations on coarser lattices is performed to quantify systematic errors in current studies of the equation of state. We also present results for observables that are sensitive to deconfining and chiral aspects of the QCD transition on Nt=6 and 8 lattices. We find that deconfinement and chiral symmetry restoration happen in the same narrow temperature interval. In an Appendix we present a simple parametrization of the equation of state that can easily be used in hydrodynamic model calculations. In this parametrization we also incorporated an estimate of current uncertainties in the lattice calculations which arise from cutoff and quark mass effects. We estimate these systematic effects to be about 10 MeV

We investigate the transverse dynamics in Au+Au collisions at \sqrt{s_NN}=200 GeV by emphasis upon the interplay between soft and hard components through p_T dependences of particle spectra, ratios of yields, suppression factors, and elliptic flow for identified hadrons. From hydrodynamics combined with traversing minijets which go through jet quenching in the hot medium, we calculate interactions of hard jets with the soft hydrodynamic components. It is shown by the explicit dynamical calculations that the hydrodynamic radial flow and the jet quenching of hard jets are the keys to understand the differences among the hadron spectra for pions, kaons, and protons. This leads to the natural interpretation for N_p/N_\pi ~ 1, R_{AA} >~ 1 for protons, and v_2^p > v_2^\pi recently observed in the intermediate transverse momentum region at Relativistic Heavy Ion Collider (RHIC).

We study QCD-like theories with pseudoreal fermions at finite baryon density. Such theories include two-color QCD with quarks in the fundamental representation of the color group as well as any-color QCD with quarks in the adjoint color representation. In all such theories the lightest baryons are diquarks. At zero chemical potential $\mu$ they are, together with the pseudoscalar mesons, the Goldstone modes of a spontaneously broken enlarged chiral symmetry group. Using symmetry principles, we derive the low-energy effective Lagrangian for these particles. We find that a second order phase transition occurs at a value of $\mu$ equal to half the mass of the Goldstone modes. For values of $\mu$ beyond this point the scalar diquarks Bose condense and the diquark condensate is nonzero. We calculate the dependence of the chiral condensate, the diquark condensate, the baryon charge density, and the masses of the diquark and pseudoscalar excitations on $\mu$ at finite bare quark mass and scalar diquark source. The relevance of our results to lattice QCD calculations and to real three-color QCD at finite baryon density is discussed.

We investigate the $N_f=2$ Schwinger model with the massive staggered fermions in the presence of a $2\pi$ periodic $\theta$ term, using the Grassmann tensor renormalization group. Thanks to the Grassmann tensor network formulation, there is no difficulty in dealing with the massive staggered fermions. We study the $\theta$ dependence of the free energy in the thermodynamic limit. Our calculation provides consistent results with the analytical solution in the large mass limit. The results also suggest that the $N_f=2$ Schwinger model on a lattice has a different phase structure from that described by the continuum theory.

We extend our previous study [Phys. Lett. B643 (2006) 46] of the cross-over temperatures (T_c) of QCD. We improve our zero temperature analysis by using physical quark masses and finer lattices. In addition to the kaon decay constant used for scale setting we determine four quantities (masses of the \Omega baryon, K^*(892) and \phi(1020) mesons and the pion decay constant) which are found to agree with experiment. This implies that --independently of which of these quantities is used to set the overall scale-- the same results are obtained within a few percent. At finite temperature we use finer lattices down to a <= 0.1 fm (N_t=12 and N_t=16 at one point). Our new results confirm completely our previous findings. We compare the results with those of the 'hotQCD' collaboration.

Quantum chromodynamics (QCD) contains field configurations which can be characterized by a topological invariant, the winding number Q_w. Configurations with nonzero Q_w break the charge-parity CP symmetry of QCD. We consider a novel mechanism by which these configurations can separate charge in the presence of a background magnetic field - the "Chiral Magnetic Effect". We argue that sufficiently large magnetic fields are created in heavy ion collisions so that the Chiral Magnetic Effect causes preferential emission of charged particles along the direction of angular momentum. Since separation of charge is CP-odd, any observation of the Chiral Magnetic Effect could provide a clear demonstration of the topological nature of the QCD vacuum. We give an estimate of the effect and conclude that it might be observed experimentally.

We present our results on the electromagnetic form factor of pion over a wide range of $Q^2$ using lattice QCD simulations with Wilson-clover valence quarks and HISQ sea quarks. We study the form factor at the physical point with a lattice spacing $a=0.076$ fm. To study the lattice spacing and quark mass effects, we also present results for 300 MeV pion at two different lattice spacings $a=0.04$ and 0.06 fm. The lattice calculations at the physical quark mass appear to agree with the experimental results. Through fits to the form factor, we estimate the charge radius of pion for physical pion mass to be $\langle r_{\pi}^2 \rangle=0.42(2)~{\rm fm}^2$.