This thesis develops lattice-QCD methods for studying hadron structure, scattering, and decays, with particular emphasis on multi-hadron resonances. The \texttt{Mathematica} package \texttt{OpTion} is developed to automate the construction of lattice multi-hadron operators. Finite-volume spectra are related to infinite-volume amplitudes through quantization conditions. For coupled-channel $D\pi$ scattering, the pion-mass dependence of the pole trajectories and the scattering length are determined, and the results support a two-pole structure for the $D_0^*(2300)$. The thesis then studies the three-pion decay of the $\omega(782)$. Two- and three-body spectra are calculated, and three-body quantization conditions and effective field theory descriptions are developed. The extracted resonance pole, extrapolated to the physical point, gives a mass and width consistent with experiment and establishes a practical first-principles procedure for interacting three-body systems. These methods are further applied to the $\pi(1300)$, providing, for the first time, first-principles predictions of its resonance pole at several pion masses and revealing significant three-body effects. Finally, radiative and semileptonic decays of charmed mesons are investigated, and a model-independent method is developed to extract transition form factors with improved precision. These results advance the first-principles study of coupled-channel and three-body hadron resonances and provide new insight into their structure and decays.

We show that a diffusion sampler for lattice-field configurations can be trained without preparing training data by an external Monte Carlo calculation. Starting from exactly sampled configurations at $\beta=0$, we construct a self-bootstrap sampler, SLDiffusion, in which periodic Gaussian proposals with a fixed learned score are Metropolis--Hastings corrected, at each $\beta$, against the same physical target at every noise level, and only replay configurations from the resulting chain are used to train the score at the next stage. In the two-dimensional compact XY model, self-training proceeds from $\beta=0.30$ to $0.50$ at $L=4$. At $\beta=0.5$, the energy and vortex densities for $L=4,6,8,12$ agree with independent Hybrid Monte Carlo calculations within $1.35\sigma$. Volume-native retraining at $L=8$ and $12$ improves both the proposal displacement and autocorrelation. The integrated autocorrelation times of the energy and vortex densities remain below two for all volumes studied. These results demonstrate that a Metropolis-corrected diffusion sampler can be self-trained without configurations drawn in advance from the target coupling.

Inclusive hadronic decay widths remain a long-standing challenge for first-principles QCD. We present a framework combining lattice QCD with strongly-coupled potential nonrelativistic QCD (pNRQCD) to compute inclusive P-wave heavy quarkonium decays to light hadrons. At leading order in the velocity expansion, all nonperturbative effects, apart from the square of the derivative of the wavefunction at the origin, are encoded in a single universal moment of the two-point chromoelectric correlator, which we determine for the first time from a quenched lattice QCD calculation matched to $\overline{\mathrm{MS}}$ via the gradient flow. Combined with perturbative short-distance coefficients and the square of the derivative of the wavefunction at the origin, our result reproduces the observed $\chi_{cJ}(1P)$ widths and, at the same time, provides predictions for the unmeasured $\chi_{bJ}(nP)$ widths. The framework extends naturally to inclusive decays and production of ordinary and exotic hadrons.

We analyze the low lying spin-$\frac{3}{2}$ $\Xi$ spectrum by means of the two point QCD sum rule approach. The analysis is motivated by the BESIII observation of the $\Xi(1720)$ resonance, reported with mass $1721.0\pm5.2_{\mathrm{stat.}}\pm3.4_{\mathrm{syst.}}~\mathrm{MeV}$ and favored quantum numbers $J^P=\frac{3}{2}^{+}$. In this framework, we include the $1S$, $1P$, $2S$ and $2P$ configurations as possible low lying spin-$\frac{3}{2}$ cascade states. The extracted masses are $1527.53\pm111.38~\mathrm{MeV}$, $1615.30\pm50.98~\mathrm{MeV}$, $1727.52\pm42.39~\mathrm{MeV}$ and $1803.71\pm64.50~\mathrm{MeV}$ for the $1S$, $1P$, $2S$ and $2P$ states, respectively. The mass obtained for the $2S$ configuration agrees with the BESIII value within uncertainties and favors the assignment of $\Xi(1720)$ as the first radial spin-$\frac{3}{2}$ excitation of the $\Xi$ baryon.

We present a continuum-QCD determination of the leading-order hadronic vacuum-polarization contribution to the anomalous magnetic moment of the muon within the Dyson-Schwinger and Bethe-Salpeter equation framework. The calculation incorporates pion back-reaction, a fully dressed quark-photon vertex with a dynamically generated $\rho$-resonance structure in the timelike region, and both strong and electromagnetic isospin breaking treated self-consistently at the quark level. We obtain $a_\mu^{\mathrm{HVP,LO}}(u+d+s+c)|_{\mathrm{ISB}} = 709.7 \times 10^{-10}$, in good agreement with recent lattice-QCD determinations, and find an isospin-breaking shift of $\Delta a_\mu^{\mathrm{HVP,LO}} = 4.5 \times 10^{-10}$ ($0.6\%$), demonstrating that isospin-breaking effects, while quantitatively modest, are not negligible. Including the bottom-quark contribution and an indicative estimate of the systematic uncertainties, we obtain our final result, $a_\mu^{\mathrm{HVP,LO}}(u+d+s+c+b)|_{\mathrm{ISB}} = (710.0 \pm 14.5) \times 10^{-10}$.

We present the first lattice QCD calculation of single-channel $DD$ scattering with quantum numbers $I(J^P)=1(0^+)$ and $0(1^-)$. The calculation is performed on the $2+1$ flavor Wilson-Clover ensembles with a lattice spacing $a\simeq 0.077$ fm and two different pion masses, $m_{\pi}\simeq207$ and $305$ MeV. The scattering parameters are determined using the Lüscher's finite volume method. Our results indicate a weak repulsive interaction in the $1(0^+)$ channel and a slightly attractive interaction in the $0(1^-)$ channel. The $S$-wave isovector $DD$ scattering length and effective range, extrapolated to the physical pion mass, are $(-0.25\pm0.08\pm 0.12)$ fm and $(-5.7\pm4.5\pm 1.7)$ fm, respectively.

We formulate thermal quantum field theory on a finite spatial periodic volume incorporating finite rotations. Traditional compactifications at finite temperature without rotations typically involve ${\mathbb T}^4$ as the space-time manifold within a path integral formulation and also moving frames can be accommodated by shifted boundary conditions on the same space. We show that consistent descriptions of certain finite rotations are possible on space-time manifolds with topology different from ${\mathbb T}^4$ but still flat and without boundary and we classify all possible geometries. The non-trivial topology may be implemented by rotated boundary conditions allowing for a path integral formulation. The purely imaginary angular velocity in temperature units cannot be arbitrary but several discrete values are possible. We also discuss finite volume effects in detail.

We study the temperature dependence of meson screening masses in two-flavour lattice QCD using dynamical truncated overlap fermions (TOF), a type of lattice chiral fermions. The screening masses for the $\pi$, $\rho$, $a_1$, $a_0$, and the sigma $(f_0)$ mesons are extracted by computing spatial correlation functions. Above the pseudocritical temperature $T_{\rm pc}$, the $\pi$ and $f_0$ screening masses become degenerate, consistent with chiral restoration. The $(\pi,f_0)$ and $(\rho,a_1)$ pairs also show the expected degeneracy. Decomposition of the $f_0$ propagator reveals that the connected contribution dominates above $T_{\rm pc}$, while the disconnected part becomes significant below $T_{\rm pc}$, explaining the reduced statistical clarity observed at low $T$. These results demonstrate that dynamical TOF simulations can capture the qualitative thermal behaviour of the scalar sector.

We investigate many-body chaos and scrambling in the Hyperbolic Ising model, a mixed-field Ising model living in the background of AdS2. The effect of the curvature is captured by site-dependent couplings obtained from the AdS2 metric applied to a flat nearest-neighbor spin chain. Using a combination of out-of-time-ordered correlators (OTOCs), Krylov complexity, and spectral statistics, we present consistent evidence that this model exhibits faster scrambling behavior relative to its flat counterpart. In particular, we observe signatures consistent with fast scrambling dynamics emerging from purely local interactions. At the system sizes accessible to tensor network simulations, the OTOCs display short-lived exponential growth regimes, from which we extract effective Lyapunov exponents. These effective finite-size exponents exhibit a temperature dependence broadly compatible with the Maldacena-Shenker-Stanford (MSS) bound within numerical uncertainty. Our results indicate that increasing spatial curvature can significantly decrease scrambling time in systems with only nearest-neighbor interactions, providing a minimal and computationally accessible platform for studying quantum chaos. This makes the model a promising test-bed for exploring scrambling and operator growth in near-term quantum simulation architectures.

Recent measurements of charmed-baryon semileptonic decays signal large tensions between experiment and theory, including $SU(3)_F$ analyses and lattice-quantum chromodynamics simulations. Possible sources of the discrepancy include the experimental normalization mode $\Xi_c^0 \to \Xi^- \pi ^+$. Using lattice-QCD inputs in an $SU(3)_F$ analysis including first-order symmetry breaking, we predict ${\cal B}( \Xi_c^+ \to \Sigma^0 \ell^+ \nu_\ell ) / {\cal B}( \Xi_c^+ \to \Xi^0 \ell^+ \nu_\ell )=(2.6\pm0.3)\%, $ and ${\cal B}( \Xi_c^+ \to \Lambda \ell^+ \nu_\ell )/{\cal B}( \Xi_c^+ \to \Xi^0 \ell^+ \nu_\ell )=(1.1\pm0.1)\%, $ with $\ell^+=(e^+,\mu^+)$. These ratios provide normalization-independent tests and may help clarify the origin of the present tension.

We explore the application of the Variational Quantum Eigensolver (VQE) to investigate the ground state properties, particularly the entanglement entropy, of the Transverse Field Ising Model (TFIM) in one, two, and three dimensions, considering systems of up to 27 spins. By benchmarking VQE results against exact diagonalization and analyzing the entanglement properties across different system sizes, we assess the algorithm's effectiveness in capturing faithful ground state. Using results of TFIM, we also investigate how VQE's expressivity and optimization influence the simulation of highly entangled quantum states. We employ different ansätze: the hardware-efficient EfficientSU2 from Qiskit, the physics-inspired Hamiltonian Variational ansätz (HVA) and HVA with symmetry breaking, and benchmark their performance using energy variance, entanglement entropy, spin correlations, and magnetization. We further discuss the implications for scaling these methods to larger quantum systems.

In a spin-1 hadron, tensor-polarized parton distribution functions (PDFs) exist. The twist-2 function is $f_{1LL}$ and a twist-3 one is $f_{LT}$. Because an experiment is under preparation at the Thomas Jefferson National Accelerator Facility (JLab) to measure the cross section of electron-deuteron deep inelastic scattering with the tensor-polarized deuteron target, these PDFs need to be understood theoretically. Especially, measurements will be done in a relatively low-$Q^2$ region at JLab, so that twist-3 contributions could become sizable in the cross section. In a previous work, a twist-2 relation was derived for $f_{LT}$ in terms of $f_{1LL}$ by using a nonlocal operator, and it corresponds to the Wandzura-Wilczek (WW) relation between $g_1$ and $g_2$. In addition, another relation similar to the Burkhardt-Cottingham (BC) sum rule was obtained. It is known that a formal way to derive the WW relation and the BC sum rule is to use the operator product expansion (OPE) with local operators. In this work, the WW-like relation and the BC-like sum rule for $f_{LT}$ are derived by using the local OPE method as a reliable independent way to establish these relations.