This chapter provides a pedagogical introduction to theoretical studies of hadrons based on the fundamental theory of strong interactions - Quantum ChromoDynamics. A perturbative expansion in the strong coupling is not applicable at hadronic energy scales. Lattice Quantum Chromodynamics is the formulation of the fundamental theory on a discrete space-time grid, which enables first-principles, systematically improvable, numerical simulations of strong interaction physics. This chapter explains how the masses of strongly stable and strongly decaying hadrons are determined. The strongly decaying hadrons have to be inferred from the corresponding scattering processes. Therefore, one of the main aims is to describe how the scattering amplitudes are extracted from a lattice simulation. The examples of spectra, widths, and scattering amplitudes are shown for conventional as well as exotic hadrons.

Determining the spectrum of photons emitted by the quark-gluon plasma non-perturbatively remains an open computational challenge. In this letter we calculate two moments of that spectrum at a temperature $T\approx 254\,$MeV, employing lattice QCD with two flavors of $\mathrm{O}(a)$-improved Wilson fermions, without facing an inverse problem. Our central value for the difference of these two moments, which is sensitive to photon energies $\omega\gtrsim \pi T$, is lower than, but compatible with that obtained by integrating the leading-order weak-coupling photon spectrum. This study informs the $\textit{direct photon puzzle}$ in heavy-ion collision phenomenology, where it has proved difficult to simultaneously explain the yield and azimuthal anisotropy of photons not originating from final-state hadronic decays.

We study the behaviour of \SU{2} Yang-Mills fields on a $T_2\times R^2$ geometry where the two-torus is equipped with twisted boundary conditions. We monitor the evolution of the dynamics of the system as a function of the torus size $l_s$. For small sizes the behaviour of the system is well understood in terms of semiclassical predictions. In our case, the long distance structure is that of a two-dimensional gas of vortex-like fractional instantons with size and density growing with $l_s$. Our lattice Monte Carlo simulations confirm the semiclassical predictions and allow the determination of the relevant scale signalling the transition to the non-dilute situation. At low densities the string tension takes the standard value of a 2D center-vortex gas, growing with the density and approaching the value measured at infinite volume. Our work includes preliminary studies of the extension to \SU{N} and to the region of large sizes in which boundary conditions are irrelevant, and all physical scales are determined by the $\Lambda$ parameter.

The liquid-vapor transition is a classic example of a discontinuous (first-order) phase transition. Such transitions underlie many phenomena in cosmology, nuclear and particle physics, and condensed-matter physics. They give rise to long-lived metastable states, whose decay can be driven by either thermal or quantum fluctuations. Yet, direct experimental observations of how these states collapse into a stable phase remain elusive in the quantum regime. Here, we use a trapped-ion quantum simulator to observe the real-time dynamics of ``bubble nucleation'' induced by quantum fluctuations. Bubbles are localized domains of the stable phase which spontaneously form, or nucleate, and expand as the system is driven across a discontinuous quantum phase transition. Implementing a mixed-field Ising spin model with tunable and time-dependent interactions, we track the microscopic evolution of the metastable state as the Hamiltonian parameters are varied in time with various speeds, bringing the system out of equilibrium. Site-resolved measurements reveal the emergence and evolution of finite-size quantum bubbles, providing direct insight into the mechanism by which the metastable phase decays. We also identify nonequilibrium scaling behavior near the transition, consistent with a generalized Kibble-Zurek mechanism. Our results demonstrate the power of quantum simulators to probe out-of-equilibrium many-body physics, including quantum bubble nucleation, a key feature of discontinuous quantum phase transitions, with application to studies of matter formation in the early universe.

We introduce Scale Factorized-Quantum Field Theory (SF-QFT), a framework that performs path-integral factorization of ultraviolet (UV) and infrared (IR) momentum modes at a physical scale $Q^*$ before perturbative expansion. This approach yields a UV-finite effective action whose Wilson coefficients $C_i(Q)$ and coupling $a_{\mathrm{eff}}(Q)$ are fixed by matching to experiment. Because the two-loop $\beta$-function is universal in massless QCD, $a_{\mathrm{eff}}(Q)$ evolves with a scheme-independent equation, with higher-order $\beta$-coefficients absorbed into the $C_i$. Applying SF-QFT to the inclusive ratio $R_{e^{+}e^{-}}$ gives $R^{\mathrm{SF-QFT}}(31.6\,\mathrm{GeV}) = 1.04911 \pm 0.00084$, in excellent agreement with experiment ($R^{\mathrm{exp}}(31.6\,\mathrm{GeV})= 1.0527 \pm 0.005$) while requiring orders of magnitude fewer calculations than a conventional four-loop $\overline{\mathrm{MS}}$ approach. We find universal algebraic recursion relations that generate all higher-order contributions without additional Feynman diagrams, yielding scheme-invariant predictions with remarkable convergence. SF-QFT provides a rigorous proof for the existence of a positive mass gap in Yang-Mills theory, resolving one of the Millennium Prize Problems by demonstrating how non-perturbative effects emerge naturally from the path-integral factorization. For QED, the same formalism integrates out high-energy modes above $Q^*$, producing scheme-independent predictions for the electron anomalous magnetic moment with unprecedented precision ($a_e^{\text{theory}} = 0.001~159~652~183~56(76)$). SF-QFT heralds a paradigm shift in quantum field theory, replacing the pursuit of ever-higher loop orders with a unified framework that handles both perturbative and non-perturbative physics while maintaining manifest gauge invariance and eliminating renormalization ambiguities.

The relations between the window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment $a^{\text{HVP}}_{\mu}$ in spacelike and timelike domains are studied. Two types of window functions (abrupt and smooth) as well as two kinds of kinematic intervals (symmetric and asymmetric with respect to the spacelike/timelike flip) are addressed. It is shown that the evaluations of the window quantities for $a^{\text{HVP}}_{\mu}$ represented in terms of the hadronic vacuum polarization function, the Adler function, and the $R$-ratio of electron-positron annihilation into hadrons are mutually equivalent only if the additional contributions due to the window edge effects are properly taken into account and the explicit expressions for such contributions are derived. The obtained results enable one to accurately compare the window quantities for $a^{\text{HVP}}_{\mu}$ based, e.g., on MUonE or lattice data with the ones based on $R$-ratio data, even if the window function covers different kinematic ranges in spacelike and timelike domains.

Sampling from high-dimensional and structured probability distributions is a fundamental challenge in computational physics, particularly in the context of lattice field theory (LFT), where generating field configurations efficiently is critical, yet computationally intensive. In this work, we apply a previously developed hybrid quantum-classical normalizing flow model to explore quantum-enhanced sampling in such regimes. Our approach embeds parameterized quantum circuits within a classical normalizing flow architecture, leveraging amplitude encoding and quantum entanglement to enhance expressivity in the generative process. The quantum circuit serves as a trainable transformation within the flow, while classical networks provide adaptive coupling and compensate for quantum hardware imperfections. This design enables efficient density estimation and sample generation, potentially reducing the resources required compared to purely classical methods. While LFT provides a representative and physically meaningful application for benchmarking, our focus is on improving the sampling efficiency of generative models through quantum components. This work contributes toward the development of quantum-enhanced generative modeling frameworks that address the sampling bottlenecks encountered in physics and beyond.