We study the profile of the flux tube in the SU(2) gauge model in (2 + 1) dimensions, with a particular attention to the so called "intrinsic width" which drives the exponential decay of the flux density at large transverse distances and represents a new physical scale of the model. This quantity is directly related to the confining mechanism which generates the flux tube and can be used to test its properties. We study a wide range of different values of lattice spacing, temperature and flux tube lengths and show that our data are precise enough to distinguish between different confining models. In particular we show that at high temperatures (just below the deconfinement transition) the data are perfectly described by an Ising-like effective model based on the Svetitsky-Yaffe mapping. At lower temperatures this approximation does not hold anymore. In this regime (which is the most interesting one from a physical point of view) we test several alternative proposals and show that the dual superconductor model is the one which better fits the data. However, this proposal is not fully satisfactory, because the values of the Ginzburg-Landau parameter extracted from the fits increase with the length of the flux tube, which is not a feature predicted by the model. This suggests that a more sophisticated model is needed to explain confinement in non-abelian gauge theories and, at the same time, that our data on the intrinsic width may be a powerful tool to benchmark these candidates.

We demonstrate that gauge equivariant diffusion models can accurately model the physics of non-Abelian lattice gauge theory using the Metropolis-adjusted annealed Langevin algorithm (MAALA), as exemplified by computations in two-dimensional U(2) and SU(2) gauge theories. Our network architecture is based on lattice gauge equivariant convolutional neural networks (L-CNNs), which respect local and global symmetries on the lattice. Models are trained on a single ensemble generated using a traditional Monte Carlo method. By studying Wilson loops of various size as well as the topological susceptibility, we find that the diffusion approach generalizes remarkably well to larger inverse couplings and lattice sizes with negligible loss of accuracy while retaining moderately high acceptance rates.

We present the complete one-loop matching of the baryon- and lepton-number-conserving low-energy effective field theory (LEFT) to the QCD gradient flow. Using Euclidean conventions and the background-field formulation of the gradient flow, we derive the short-flow-time expansion for the full LEFT operator basis up to mass dimension six. The matching is performed in dimensional regularization in the algebraically consistent 't Hooft-Veltman scheme, including a systematic treatment of evanescent operators and the finite counterterms required to restore chiral symmetry in the spurion sense. Keeping fully generic flavor structures, we verify the cancellation of spurious chiral-symmetry-violating terms with the known finite symmetry-restoring counterterms. This demonstrates that the gradient flow as a gauge-invariant ultraviolet regulator enables an efficient extraction of both divergent and finite counterterms in addition to the matching contributions. We provide the matching coefficients both before and after field redefinitions that remove redundant operators, as well as power-divergent mixings into lower-dimensional operators. Our results establish a consistent perturbative link between continuum LEFT calculations and gradient-flow-based lattice-QCD matrix elements, enabling precision low-energy phenomenology beyond leading-logarithmic accuracy.

Motivated by the necessity of developing theoretical models for studying the electroweak structure of baryons in a nuclear medium, we apply a covariant quark model to study interactions of baryons with nuclear matter. The electromagnetic and axial form factors of the octet baryons are determined by combining a covariant quark model that takes into account the meson cloud dressing of the baryon cores, developed for free space, with the quark-meson coupling model in the extension to the nuclear medium. We discuss the medium modifications on the electroweak form factors of octet baryons for the range of densities from $\rho=0$ up to $\rho=2 \rho_0$, where $\rho_0= 0.15$ fm$^{-3}$ is the normal nuclear matter density. We also study how the shape of the form factors is modified in finite nuclei due to the profile of the nuclear density distributions compared with calculations using the average density of the nucleus

We show that small perturbations in the boost-invariant color fields of the glasma exhibit an exponential growth with the square root of time. We interpret this growth rate as a Lyapunov exponent, related to entropy production and the thermalization timescale in the earliest stage of heavy-ion collisions. Working in a regime that is linear in this perturbation, we extract the time dependence of this mode as $\sim \exp(0.4\sqrt{g^2\mu\tau})$ for SU($2$), where $g^2\mu$ is proportional to the saturation scale and the square-root dependence is caused by the boost-invariant expansion of the system. We show that the growth rate of this mode is, unlike its amplitude, remarkably insensitive to the details of how the perturbations are initialized. In particular, we show that the unstable mode couples to all momentum scales present in the initial perturbation.

We formulate and solve a Euclidean Bethe-Salpeter equation for the lightest scalar glueball (0++) in pure Yang-Mills theory, using the refined Gribov-Zwanziger gluon tree-level propagator as an infrared-complete input. In a minimal ladder truncation with an effective constant kernel strength g_C^2 and the dominant s-wave component, we extract scalar glueball masses in the range 1.7-2.3 GeV for representative values of g_C^2, with a preferred value around 1.9 GeV near g_C^2 = 0.54. The result is consistent with RGZ correlator-based infrared moment analyses and with lattice expectations, providing a cross-check of RGZ-inspired infrared gluodynamics from a bound-state viewpoint.

These lecture notes provide a self-contained, graduate-level introduction to the strong $CP$ problem and QCD axion physics from an effective field theory (EFT) viewpoint. We review the construction of the chiral EFT of QCD yielding a $\theta$-dependent potential, from which vacuum alignment, $\theta$ periodicity and branch structure follow. We further show how the framework leads to the Witten-Veneziano relation highlighting the role of the pure-glue topological susceptibility in organizing $\theta$-dependent hadronic observables. Using these tools, we show how to extract representative $CP$-odd mesonic and baryonic amplitudes, including the chiral estimate underlying the neutron EDM bound, and how to generalize the effective framework to confining SU(N) theories with fermions in arbitrary representations. We further show how to employ the Veneziano-Yankielowicz effective Lagrangian for N=1 supersymmetric Yang-Mills theory to extract salient information on the $\theta$-dependent physics of one-flavour QCD via orientifold planar equivalence. We also revisit a recent no strong $CP$ claim based on an ordering of limits in the sum over topological sectors and show, in the EFT language, that it amounts to introducing an extra non-propagating axion-like degree of freedom not required by QCD. We then present the standard dynamical resolution to the strong $CP$ problem, i.e. the Peccei-Quinn mechanism, the resulting axion potential and mass from chiral EFT and briefly review associated time-honored UV completions, and the axion quality problem from gravitational corrections.

Extracting continuum properties of quantum field theories from discretized spacetime is challenging due to lattice artifacts. Renormalization-group (RG)-improved lattice actions can preserve continuum properties, but are in general difficult to parameterize. Machine learning (ML) with gauge-equivariant convolutional neural networks provides a way to efficiently describe such actions. We test a machine-learned RG-improved lattice gauge action, the classically perfect fixed-point (FP) action, for four-dimensional SU(3) gauge theory through Monte Carlo simulations. We establish that the gradient flow of the FP action is free of tree-level discretization effects to all orders in the lattice spacing, making it classically perfect. This allows us to test the quality of improvement of the FP action, without introducing additional artifacts. We find that discretization effects in gradient-flow observables are highly suppressed and less than 1% up to lattice spacings of 0.14 fm, allowing continuum physics to be extracted from coarse lattices. The quality of improvement achieved motivates the use of the FP action in future gauge theory studies. The advantages of ML-based parameterizations also highlight the possibility of realizing quantum perfect actions in lattice gauge theory.

The hypothesis that dark matter could be a bound state of a strongly coupled non-Abelian gauge theory is theoretically appealing and has a variety of interesting phenomenological implications. In particular, an interpretation of dark matter as the lightest glueball state in the spectrum of a dark Yang-Mills theory, possibly coupled to the visible sector only through gravitational interactions, has been discussed quite extensively in the literature, but most of previous work has been focused on dark SU(N) gauge theories. In this article, we consider an alternative model, based on a symplectic gauge group, which has a first-order confinement/deconfinement phase transition at a finite critical temperature. We first determine the equation of state of this theory, focusing on temperatures close to the transition, and evaluating the associated latent heat. Then we discuss the evolution of this dark-matter model in the early universe, commenting on the mechanisms by which it could indirectly interact with the visible sector, on the spectrum of gravitational waves it could produce, and on the relic abundances it would lead to. Our discussion includes an extensive review of relevant literature, a number of comments on similarities and differences between our model and dark SU(N) gauge theories, as well as some possible future extensions of the present study.

Numerical studies of phase transitions in statistical and quantum lattice models provide crucial insights into the corresponding Conformal Field Theories (CFTs). In higher dimensions, comparing finite-volume numerical results to infinite-volume CFT data is facilitated by choosing the sphere $S^{d-1}$ as the spatial manifold. Recently, the fuzzy sphere regulator in Ref. [Zhu et al, Phys. Rev. X 13 021009 (2023)] has enabled such studies with exact rotational invariance, yielding impressive agreement with known 3D Ising CFT predictions, as well as new results. However, systematic improvements and a deeper understanding of finite-size corrections remain essential. In this work, we revisit the fuzzy sphere regulator, focusing on the original Ising model, with two main goals. First, we assess the robustness of this approach using Conformal Perturbation Theory (CPT), to which we provide a detailed guidebook. We demonstrate how CPT provides a unified framework for determining the critical point, the speed of light, and residual deviations from CFT predictions. Applying this framework, we study finite-size corrections and clarify the role of tuning the model in minimizing these effects. Second, we develop a novel method for extracting Operator Product Expansion (OPE) coefficients from fuzzy sphere data. This method leverages the sensitivity of energy levels to detuning from criticality, providing new insights into level mixing and avoided crossings in finite systems. Our work also includes validation of CPT in a 1+1D Ising model away from the integrable limit.