The Review summarizes much of particle physics and cosmology. Using data from previous editions, plus 2,717 new measurements from 869 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. Particle properties and search limits are listed in Summary Tables. We give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Most of the 120 reviews are updated, including many that are heavily revised. The Review is divided into two volumes. Volume 1 includes the Summary Tables and 97 review articles. Volume 2 consists of the Particle Listings and contains also 23 reviews that address specific aspects of the data presented in the Listings. The complete Review (both volumes) is published online on the website of the Particle Data Group (pdg.lbl.gov) and in a journal. Volume 1 is available in print as the PDG Book. A Particle Physics Booklet with the Summary Tables and essential tables, figures, and equations from selected review articles is available in print, as a web version optimized for use on phones, and as an Android app.

I show comparisons of the pseudoscalar meson vector form factor from simulations of QCD with N_c = 3, 4 and 5 colors and N_f = 2 flavors of degenerate mass fermions at a common (matched) fermion mass, lattice spacing, and simulation volume. The dependence of the form factor on the momentum transfer is nearly independent of the number of colors, and is consistent with the expectations of vector meson dominance.

I show comparisons of the pion - pion scattering length in the isospin - two channel from simulations of of QCD with N_c = 3, 4 and 5 colors and N_f = 2 flavors of degenerate mass fermions. The scattering length varies as 1/N_c, as expected from large N_c counting arguments.

This White Paper presents an overview of the current status and future perspective of QCD research, based on the community inputs and scientific conclusions from the 2022 Hot and Cold QCD Town Meeting. We present the progress made in the last decade toward a deep understanding of both the fundamental structure of the sub-atomic matter of nucleon and nucleus in cold QCD, and the hot QCD matter in heavy ion collisions. We identify key questions of QCD research and plausible paths to obtaining answers to those questions in the near future, hence defining priorities of our research over the coming decades.

We present a lattice calculation of the low energy constants of QCD with $N_c=3$, 4 and 5 colors and $N_f=2$ flavors of degenerate mass fermions. We fit data for the pseudoscalar meson mass, the pseudoscalar decay constant, and the Axial Ward Identity fermion mass to formulas from next to next to leading order chiral perturbation theory. We extract the next to leading order low energy constants and study their behavior as a function of $N_c$. Pre-existing analyses of $N_c=3$ inform our fitting strategies.

I make some simple observations about the calculation of weighted averages over energy of Minkowski space spectral densities from weighted averages over time of Euclidean space correlation functions, measured in latice simulations. The correlator of two vector currents is used as an example, where it appears that a determination of a weighted average of the spectral function near the rho pole at the five per cent level is possible from lattice simulations.

Contribution from the USQCD Collaboration to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021).

It is well known that the deconfinement transition temperature for $SU(N_c)$ gauge theory is almost independent of $N_c$, and the transition is first order for $N_c \ge 3$. In the real world ($N_c=3$, light quarks) it is a crossover located far away from the pure gauge value. What happens to the transition temperature at fixed fermion mass if the number of fermion flavors is held constant ($N_f=2$) and $N_c$ is varied? There are multiple plausible stories, only one of which appears to be true when the systems are simulated on the lattice. I describe the physics issues which surround the question and my lattice - based answer to it.

We present preliminary results from our calculation of the low energy constants (LECs) of the chiral effective theory for 3, 4 and 5 color QCD with $N_f=2$ dynamical fermion flavors. We simulate with clover fermions over a range of lattice couplings and quark masses. We observe the expected $N_c$ scaling for the LECs appropriate to the condensate $B$ and pseudoscalar decay constant $F$. The range of quark masses over which leading order chiral perturbation theory describes the data grows as $N_c$ rises.

Lattice quantum chromodynamics has proven to be an indispensable method to determine nonperturbative strong contributions to weak decay processes. In this white paper for the Snowmass community planning process we highlight achievements and future avenues of research for lattice calculations of weak $b$ and $c$ quark decays, and point out how these calculations will help to address the anomalies currently in the spotlight of the particle physics community. With future increases in computational resources and algorithmic improvements, percent level (and below) lattice determinations will play a central role in constraining the standard model or identifying new physics.

I present a numerical study of the crossover between the low temperature chirally broken phase and the high temperature chirally restored phase in SU(Nc) gauge theory with Nc = 3− 5 colors and Nf = 2 degenerate fermion flavors. Fermion masses span a range of intermediate values (represented by the squared ratio of pseudoscalar to vector meson masses (mPS/mV ) 2 ∼ 0.25 to 0.63). Observables include the temperature dependent chiral condensate and screening masses. At each fermion mass these quantities show nearly identical temperature dependence across Nc. ∗Electronic address: thomas.degrand@colorado.edu

I present a numerical study of the crossover between the low temperature chirally broken phase and the high temperature chirally restored phase in SU ( N c ) gauge theory with N c = 3 − 5 colors and N f = 2 degenerate fermion flavors. Fermion masses span a range of intermediate values corresponding to pseudoscalar to vector meson masses ( m PS /m V ) 2 ∼ 0 . 25 to 0.63. Observables include the temperature dependent chiral condensate and screening masses. At each fermion mass these quantities show nearly identical temperature dependence across N c .

I present a numerical study of the crossover between the low temperature chirally broken phase and the high temperature chirally restored phase in $SU(N_c)$ gauge theory with $N_c=3-5$ colors and $N_f=2$ degenerate fermion flavors. Fermion masses span a range of intermediate values (represented by the squared ratio of pseudoscalar to vector meson masses $(m_{PS}/m_V)^2\sim 0.25$ to 0.63). Observables include the temperature dependent chiral condensate and screening masses. At each fermion mass these quantities show nearly identical temperature dependence across $N_c$.

We have calculated quantities of interest to a theory of compositeness. The lattice model, approximating the candidate theory, is the SU(4) gauge theory coupled to fermions in two color representations. For the composite Higgs, a current correlator gives one of the ingredients of the effective Higgs potential. For the partially composite top quark, we have hyperbaryon matrix elements that govern mixing of the fundamental quark with its heavy composite partner. The matrix elements turn out to be so small that the theory is disfavored as a source of a realistic top mass.

A number of proposed extensions of the standard model include new strongly interacting dynamics, in the form of SU(N) gauge fields coupled to various numbers of fermions. Often, these extensions allow N = 3 as a plausible choice, or even require N = 3, such as in twin Higgs models, where the new dynamics is a "copy" of QCD. However, the fermion masses in such a sector are typically different from (often heavier than) the ones of real-world QCD, relative to the confinement scale. Many of the strong interaction masses and matrix elements for SU(3) at heavy fermion masses have already been computed on the lattice, typically as a by-product of the approach to the physical point of real QCD. We provide a summary of these relevant results for the phenomenological community.

I present a calculation of the topological susceptibility $\chi_T$ in $SU(N_c)$ gauge theory with $N_c=3-5$ colors and $N_f=2$ degenerate flavors of fermions. The results lie on a common curve when expressed in terms of the combination $N_c m_{PS}^2 t_0$ where $m_{PS}$ is the pseudoscalar meson mass and $t_0$ is the flow parameter. $\chi_T$ approaches its quenched value as the pseudoscalar mass becomes large. The lattice simulations use clover fermions. They are done at a single lattice spacing, roughly matched across $N_c$, and over a restricted range of fermion masses.

Partial compositeness is a mechanism for the generation of fermion masses which replaces a direct Higgs coupling to the fermions by a linear mixing with heavy composite partners. We present the first calculation of the relevant matrix element in a lattice model which is very close to a candidate theory containing a composite Higgs boson and a partially composite top quark. Specifically, our model is an SU(4) gauge theory coupled to dynamical fermions in the fundamental and two-index antisymmetric (sextet) representations. The matrix element we obtain is small and hence our result disfavors the scenario of obtaining a realistic top mass in this model.

We present a progress report on our investigation of the thermodynamics of QCD with $N_f=2$ flavors of dynamical Wilson fermions in the limit of a large number of colors $N_c$. To date, studies of the thermodynamics of QCD at large $N_c$ have been limited to the quenched approximation, i.e., to the behavior of pure $\mathrm{SU}(N_c)$ gauge theory at large $N_c$. This is the first study of thermodynamics at large $N_c$ using dynamical fermions, and thus the first study able to test whether the quenched approximation is a valid way to investigate large $N_c$ thermodynamics. After reviewing 't Hooft's large $N_c$ limit, we discuss the automation we use to make this study feasible, and finally compare our preliminary physics results for $\mathrm{SU}(3-5)$ with large $N_c$ expectations.

These lectures about lattice field theory were written for, and given at, TASI 2019, ``The many dimensions of quantum field theory.'' The students at this TASI were mostly interested in formal things, and so these are slightly unusual lattice lectures: I wanted to give the physical motivation behind lattice calculations rather than describe all the technical details. A quick outline: (1) The really big picture: lattice basics, lattice confinement, getting rid of the lattice. (2) A walk through the parts of a lattice calculation -- an overview, to show what's involved. (3) Chiral fermions on the lattice. (This part might be interesting to lattice people.) (4) Case studies: the three dimensional Ising model, and QCD.

We investigate the phase structure of SU(4) gauge theory with the gauge field simultaneously coupled to two flavors of fermion in the fundamental representation and two flavors of fermion in the two-index antisymmetric representation. We find that the theory has only two phases, a low-temperature phase with both species of fermion confined and chirally broken, and a high-temperature phase with both species of fermion deconfined and chirally restored. The single phase transition in the theory appears to be first order, in agreement with theoretical predictions.