Inverse problems play a central role in current areas of research in particle phenomenology. In these lectures we focus on two examples, the extraction of Parton Distribution Functions (PDFs) from experimental data (or, equivalently, from pseudo- and quasi-PDFs computed in lattice QCD), and the extraction of spectral functions from lattice Euclidean time correlators. We investigate in detail three different approaches, namely Backus-Gilbert, Gaussian Processes and fits based on Neural Network parametrizations.

The Monte Carlo method, applied to lattice quantum field theory, gives access to Euclidean correlation functions with well-understood error bars. Recovering the observables one cares about, such as the spectral density, requires solving an ill-posed inverse problem, usually tackled with heuristics that lose rigorous control of the error. Instead of trying to find the ``best'' spectral density $\rho(\omega)$, we ask how small or large linear functionals $\int_{\mathbb{R}^+} G(\omega) \rho(\omega) \mathrm{d} \omega$ of it can be, given the Monte Carlo data and the reflection positivity of the lattice action. This is a convex but infinite-dimensional problem. We show how its dual can be rigorously relaxed into a hierarchy of finite semidefinite programs, solvable with standard solvers and enjoying strong convergence guarantees. The resulting bounds are rigorous even when the relaxation is not tight, and converge quickly to the regime where the error is entirely dominated by Monte Carlo statistics. The method also flags implausible Monte Carlo data, for instance underestimated error bars, through an infeasibility certificate. We demonstrate it on lattice $\phi^4$ theory in two dimensions.

The quark mass dependence of the pseudoscalar meson decay consntants, $f_\pi, f_K$ and $f_\eta$, are determined from three-flavor N$^2$LO ChPT till pion masses around $780$ MeV, near the SU(3) limit. This is done by conducting an analysis of recent LQCD data using the LASSO method, a machine-learning technique which allows to pin down the relevant low-energy-constants with high precision. Since the pion decay constant is a fundamental quantity which usually appears in relevant phenomenological lagrangians or Effective Field Theories based on QCD at low energies, this analysis can be used as input to evaluate the quark mass dependence of hadronic states. As an example, we predict the masses of the octect baryons in the SU(3) limit within covariant Baryon Chiral Perturbation Theory.

The lattice QCD calculation of generalized form factors are exploited to determine the subtraction constants through all order dispersion relations of Deeply Virtual Compton Scattering (DVCS). The leading order relation is found to constrain significantly the real part of the Compton Form Factors (CFFs), and the higher order one reduces considerably both the real and imaginary part of CFFs in a global analysis of proton data. This is realized by a synthesis of the DVCS data and LQCD calculations within a neural network framework, whose architecture is specifically designed for a reliable extrapolation to unmeasured kinematic regime. By leveraging dispersion relations beyond leading order, our framework allows for adding higher moments of generalized parton distributions (GPDs) from LQCD into the extraction of CFFs from DVCS data.

We present the first systematic calculations of $\phi$-, $\eta_c$-, and $J/\psi$-nucleus ground states up to mass number $A{=}12$ based on the HAL QCD meson-nucleon potentials at near-physical point. The $(A{+}1)$-body Schrödinger equation is solved with a neural-network variational Monte Carlo framework, generalized to incorporate mesonic degrees of freedom. Benchmarking on light nuclei from $^2$H to $^{12}$C yields ground-state energies consistent with experiment. Meson-nucleus bound states emerge at $A\ge2$ for $\phi$, $A\ge4$ for $J/\psi$, and $A\ge6$ for $\eta_c$. The $\phi$-nucleus systems exhibit the strongest binding, with binding energies reaching tens of MeV. The $J/\psi$-nucleus and $\eta_c$-nucleus systems are weakly bound at the few-MeV and sub-MeV scale, respectively. The binding energy per nucleon deepens nearly linearly with $A$ for charmonium systems, whereas the $\phi$-nucleus system exhibits a non-monotonic behavior peaking at $^4$He -- a distinctive hallmark of the short-range and strongly attractive $\phi N$ interaction. The meson compresses the nucleon distribution relative to the parent nucleus, and evolves from a halo configuration to one embedded inside the nucleus with increasing $A$. Our results provide predictions for future experimental searches, and establish a quantitative bridge between lattice QCD meson-nucleon interactions and the emergent many-body phenomena in meson-nucleus bound states.

We present predictions for the elastic electromagnetic form factors of the $\Omega$ baryon in both space-like and time-like regions, computed within a confining, symmetry-preserving framework based on a vector$\,\otimes\,$vector contact interaction. The calculation is performed in the rainbow-ladder truncation of QCD's Dyson-Schwinger equations, combined with a Poincaré-covariant Faddeev equation for the three-quark bound state. The $\Omega$ baryon, composed solely of strange quarks, provides a particularly clean environment in which to investigate the role of quark mass and SU(3)-flavor symmetry in shaping baryon structure. Within this approach, the electromagnetic current is constructed consistently with the Ward-Takahashi identity, yielding four independent form factors associated with the electric monopole, magnetic dipole, electric quadrupole, and magnetic octupole moments. We compute these form factors over a broad kinematic domain and analyze their behavior in both space-like and time-like regions. The resulting static electromagnetic moments and multipole form factors of the $\Omega$ baryon exhibit a visible sensitivity to the dressed-quark anomalous magnetic moment, particularly in the magnetic and higher-order multipole sectors. The time-like form factors are obtained through asymptotic analytic continuation of the corresponding space-like solutions, allowing the construction of the effective form factor and its comparison with available experimental data and recent phenomenological analyses.

We propose an analog quantum simulation of chiral magnetic dynamics using ultracold atoms in an optical superlattice. The massive Schwinger model in the zero gauge coupling limit maps onto the Rice-Mele model, with the fermion mass and topological angle encoded in the superlattice parameters. We study the real-time dynamics of the vector current following two quench protocols that drive continuous chirality injection and chirality relaxation. Simulations with realistic superlattice parameters and experimental noise demonstrates clear mass dependence of the current dynamics in both protocols, robust against experimental imperfections. The vector current may be directly measurable via single-bond-resolved detection, establishing cold atom superlattices as a viable platform for probing non-equilibrium chiral phenomena.

Lattice simulations provide the thermodynamics of quantum chromodynamics (QCD) as a function of the temperature, at zero-to-moderate values of the baryonic chemical potential. However, the contribution of single hadronic species cannot be directly isolated from lattice calculations. In this work, we find linear combinations of up to fourth order susceptibilities which isolate the contribution of hadrons to the QCD pressure according to their baryon number $B$, electric charge $Q$ and strangeness $S$ content. These combinations are valid, provided that the thermodynamics of a strongly-interacting gas in the low-temperature regime can be modeled as a gas of non-interacting hadrons and their resonances. Finally, we test the validity of these linear combinations in the Hadron Resonance Gas (HRG) model and compare them to available lattice QCD results, using continuum-estimated susceptibilities.

We present a dispersive amplitude analysis of the low-energy $\pi^0\pi^0$ system in the radiative decay $J/\psi\to\gamma\pi^0\pi^0$, using the mass-independent BESIII $0^{++}$ and $2^{++}$ intensities, the total spectrum, and the measured $0^{++}-2^{++}$ $E1$ phase difference. The isoscalar $S$-wave is described by a coupled-channel $\pi\pi/K\bar K$ Muskhelishvili-Omnès representation, which implements the strong final-state interactions associated with the $f_0(500)$ and $f_0(980)$. The $D$-wave is treated with a single-channel $\pi\pi$ Muskhelishvili-Omnès representation, where we identify all kinematic constraints of helicity amplitudes before transforming them to the experimental $E1$, $M2$, and $E3$ multipoles. Smooth short-distance production of a pseudoscalar-meson pair is encoded in subtraction polynomials, while left-hand-cut effects are estimated and found to be numerically subleading. We identify the negative solution of the BESIII $0^{++}-2^{++}$ $E1$ phase ambiguity, after using the modulo-$\pi$ freedom of production amplitudes, as the phase solution compatible with unitarity constraints, showing that the measured phase information can be accommodated with the Omnès phase motion without requiring large additional phases. By normalizing the BESIII intensities with the extracted branching fraction, we fix the absolute scale of the fitted amplitudes, making them suitable as input for future dispersive studies of two-pion contributions to gravitational form factors.

We propose a forward-mode automatic differentiation (AD) framework for tensor renormalization group methods. In this approach, evaluating the derivatives of the partition function up to the order of $k$ increases the matrix-multiplication cost by a factor of $(k+1)(k+2)/2$ compared to computing the free energy alone, and the memory footprint is only $k+1$ times that of the original calculation. In the limit where the derivatives of the singular value decomposition are neglected, we establish a theoretical correspondence between our forward-mode AD and conventional impurity methods. Numerically, we find that the proposed AD algorithm can calculate internal energy and specific heat significantly higher accuracy than the impurity method at comparable computational cost. We also provide a practical procedure to extract critical exponents from derivatives of the renormalized tensor in tensor renormalization group calculations in both two and three dimensions. In addition, we discuss how to efficiently differentiate an arbitrary tensor network.

We study the localization properties of the low-lying Dirac eigenmodes in QCD near the crossover temperature, using stout-smeared staggered fermions and Symanzik-improved gauge action on the lattice. On $N_{\mathrm{t}}=8$ lattices we find that localized low modes, absent at low temperature, appear at a temperature $T_{\mathrm{loc}}$ in the range $150\,\mathrm{MeV}\le T_{\mathrm{loc}}\le 160\,\mathrm{MeV}$, well within the chiral crossover range as determined from the chiral condensate and from the light-quark susceptibility. Since with our choice of lattice action neither the chiral transition region nor the renormalized mobility edges change significantly above $N_{\mathrm{t}}=8$, our conclusion that $T_{\mathrm{loc}}$ is in the chiral crossover region is expected to remain valid in the continuum limit.

Symmetric mass generation (SMG) has attracted growing interest in both condensed matter theory and lattice-QCD communities. Here we formulate general criteria for SMG and examine their compatibility with lattice-QCD. We propose possible RG-flow scenarios near the SMG transition, and argue that meson mass ratio can serve as a probe of the SMG transition viewed as a UV fixed point. We further identify Goldstone tetraquark meson states as phenomenological signatures of the "type-II'' SMG phase.

Recent progress in combining Clifford circuits with tensor-network (TN) methods has shown that local Clifford disentanglers can reduce bipartite entanglement across TN bonds prior to tensor compression, thereby improving the efficiency of TN simulations. In this work, we embed local Clifford disentanglers in the Grassmann-tensor language to define a Clifford-augmented Grassmann matrix product state (CAGMPS) ansatz, and develop a density-matrix renormalization group (DMRG) framework based on this ansatz while preserving locality and fermion-parity structure. We benchmark the resulting CAGMPS--DMRG method on representative fermionic lattice systems, including the tight-binding, $t$-$V$, and $t$-$V$-$V'$ models. In all cases, Clifford augmentation systematically suppresses bipartite entanglement and improves the accuracy of the ground-state energy at a fixed bond dimension. We further show that the Grassmann-evenness condition, together with equivalence under entangling action, restricts the relevant two-site Clifford candidates to 12 inequivalent representatives, enabling a more economical disentangling search than approaches based on the standard two-qubit Clifford gate set. Our results suggest that the CAGMPS--DMRG method provides a scalable and efficient variational tool for strongly correlated fermionic systems.

In practical physical systems, modeling assumptions of Gaussianity and basis independence break down due to self-interactions. We study a specific instance of one-dimensional $\phi^4$ theory on a lattice, analyzing how the interaction strength and system size jointly affect the marginal and joint distributions of frequency-based representation of the field (i.e., Fourier modes). We find that models relying on Gaussian and independent Fourier modes fail primarily from structured dependencies rather than marginal non-Gaussianity, since individual modes become approximately Gaussian despite mode coupling growing with size. Based on this, we identify three distinct regimes that delineate where traditional methods remain effective and where more expressive models are needed. Our results provide a computationally simple diagnostic to establish when Gaussian models are insufficient, and establish a concrete design criterion that future nonlinear models must satisfy.