The precise value of the strong coupling $\alpha_s(m_{Z})$ at the $Z$-boson mass $m_{Z}$ is essential for high-energy phenomenology and precision tests of quantum chromodynamics (QCD). We present the status of a program targeting a $\sim 0.3\%$ determination of $\alpha_s(m_{Z})$ using the renormalization group $\beta$-function in the infinite volume gradient flow scheme based on lattice QCD simulations of degenerate four-flavor highly improved staggered quark (HISQ) ensembles. In particular, we analyze both tree-level cutoff effects and finite-mass effects. We also outline the next steps of the analysis, including the infinite-volume and continuum extrapolations required for a precise determination of $\alpha_s(m_Z)$.

We study QCD with three degenerate flavors of dynamical quarks using first-principles lattice simulations. For a specific choice of imaginary isospin chemical potential, this theory possesses an exact center symmetry, just like pure gauge theory. This exact symmetry is expected to be intact at low temperatures and spontaneously broken in the high-temperature regime. By analyzing the finite-size scaling of the Polyakov loop distribution, obtained with a dedicated multi-histogram approach, we demonstrate that there is a first-order deconfinement phase transition in between. Our results are obtained employing stout-smeared rooted staggered quarks at one lattice spacing. Using simulations at different quark masses we sketch the behavior of QCD in the mass-isospin chemical potential plane, shedding new light on this corner of the fundamental phase diagram of the strong interactions and the relationship between chiral symmetry breaking and deconfinement.

Strongly coupled theories are of phenomenological interest, for example as dark matter candidates. Theories that can undergo first order thermal phase transitions are particularly appealing as potential sources of a stochastic gravitational wave background. Determining the expected gravitational wave signal from a first order phase transition requires accurate information on the bubble nucleation rate, but thus far for strongly coupled models these have relied on semiclassical methods. As a first step towards determining the nucleation rate, in this paper we study the confinement-deconfinement phase transition in a 4D SU(8) pure gauge model, using multicanonical Monte Carlo. Resolving the critical bubble for the first time in a pure Yang-Mills model, we determine the critical bubble probability and compare it to results from thin wall calculations. We also compare the effectiveness of different lattice pseudo-order parameters at resolving the condensation transition between the metastable phase and critical bubble branch, and point out the choice of order parameter is crucial to accurately resolve the critical configurations.

Evidence from lattice and continuum studies supports the existence of a fully nonperturbative Higgs mechanism generating mass for gluons in the gauge sector of the strong interactions in linear covariant gauges. The broken charge is the Kugo-Ojima charge. The corresponding unphysical Goldstone boson is a bound-state superposition of two gluons, three gluons and a ghost-antighost pair. Mass generation occurs via the Schwinger mechanism, triggered by the formation of the Goldstone boson. Once corrected for symmetry breaking, the color charge operator is unbroken and confining.

We develop a strategy for computing the $B\to K\ell^+\ell^-$ and $\bar{B}_s\to\gamma\ell^+\ell^-$ decay amplitudes using lattice QCD (where $\ell^\pm$ are charged leptons). We focus on those terms which contain complex contributions to the amplitude, due to on-shell intermediate states propagating between the weak operator and electromagnetic current(s). Such terms, which are generally estimated using model calculations and represent significant uncertainties in the phenomenological predictions for these decays, cannot be computed using standard lattice QCD techniques. It has recently been shown that such contributions can be computed using spectral-density methods and our proposed strategy, which we discuss in detail, is built on this approach. The complex contributions include the ``charming penguins" (matrix elements of the current-current operators $O_1^{(c)}$ and $O_2^{(c)}$ defined in Eq. (6) below), in which the charm-quark loop can propagate long distances, particularly close to the region of charmonium resonances. They also include the contributions from the chromomagnetic operator ($O_8$ in standard notation, defined in Eq. (8) below). We discuss the renormalization of the ultra-violet divergences, and in particular those which arise due to ``contact" terms, and explain how those which appear as inverse powers of the lattice spacing can be subtracted non-perturbatively. We apply the spectral density methods in an instructive exploratory computation of the charming penguin diagram in $B\to K\ell^+\ell^-$ decays in which the virtual photon is emitted from the charm-quark loop (the diagram in Fig. 1(a) below) and discuss the prospects and strategies for the reliable determination of the amplitudes in future dedicated computations; computations which are however, beyond the scope of the present paper.

We present the first lattice QCD determination of the electromagnetic form factors of the exotic tetraquark $T_{bb} \ (bb \bar u \bar d)$ with quantum numbers $I( J^P ) = 0( 1^+ )$. The extracted form factors encode information about its internal structure, including the charge distribution and the magnetic dipole moments, determined separately for the light and heavy quarks. Our results provide evidence in favor of it being a bound state consisting of a compact heavy diquark $[bb]$ in a color-antitriplet with spin one, and a light antidiquark $[\bar u \bar d]$ in a color-triplet with spin zero. The charge radius of $T_{bb}$ is found to be significantly smaller than the combined charge radii of $B$ and $B^*$ mesons. These two comprise the lowest-lying threshold $BB^*$ in the channel we are considering, and their electric charge form factors are also determined. The computations were performed on a single CLS ensemble with $N_f = 2+1$ dynamical quarks and a lattice spacing of approximately $a \approx0.064 \ \mathrm{fm}$ at the pion mass $m_\pi \approx 290 \ \mathrm{MeV}$.

I review recent lattice results on strongly interacting matter under extreme conditions, with emphasis on the finite-temperature QCD transition at $\mu_B=0$, its approach toward the chiral limit and the fate of the $U_A(1)$ anomaly, as well as recent constraints on the QCD phase boundary and the possible critical endpoint at $\mu_B>0$. I also discuss selected advances in lattice methods and in QCD thermodynamics under external conditions, in particular strong magnetic fields, isospin chemical potential, rotation, acceleration, and quark spin polarization.

Recently, a new framework for studying generic $2 \to 3$ hard exclusive reactions, referred to as single-diffractive hard exclusive processes, has been introduced to provide a cleaner separation of the underlying physical mechanisms. In this work, we expand this formalism to the case of exclusive real-photon electroproduction off a nucleon, $e(\ell) + N(p) \to e(\ell') + N(p') + \gamma(q')$, which represents the classical channel for accessing generalized parton distributions (GPDs) in nucleons and nuclei. This extension enables a more systematic and physically transparent formulation of the reaction dynamics, paving the way for improved extractions of GPDs from experimental data as compared to existing approaches.

The kaon, the lightest hadron containing a strangeness quark, is very peculiar. It is a Nambu-Goldstone boson, but significantly heavier than the pion. As a result, its interaction with a matter particle, such as the nucleon or a heavy-light meson, such as the $D$ meson, is completely determined by chiral dynamics and much stronger than its pion cousin. The strong attractive interaction has brought us many surprises and is manifested in the peculiar nature of many particles, such as the mysterious $\Lambda(1405)$ and $D_{s0}^*(2317)$. These two particles can be understood as $\bar{K}N$ and $DK$ hadronic molecules, respectively. They also imply the existence of three-body hadronic molecules that await future discovery. In this talk, I review some recent developments in our understanding of hadronic interactions involving the kaon.

By employing the gradient flow exact renormalization group (GFERG), we study the renormalization group (RG) flow of a manifestly gauge or BRST invariant non-perturbative ansatz of the 1PI Wilson action in quantum electrodynamics. The gauge invariance of the Wilson action is \emph{exactly\/} preserved under the RG flow. We explicitly solve the GFERG equation in the leading and partially next-to-leading orders of the large $N_f$ approximation, where $N_f$ is the number of flavors. We obtain gauge invariant critical exponents and the gauge invariant 1PI Wilson action at an infrared (IR) fixed point for~$D<4$, where $D$ is the spacetime dimension.