We determine the zero eigenmode spectrum of Minimally Doubled Fermions (MDF), namely in Karsten-Wilczek (KW) and Borici-Creutz (BC) formulations on the 4-dimensional space-time lattice. We employ background gauge fields with integer valued topological charges. The Atiyah-Singer index theorem is verified in the presence of two different background gauge fields, namely Smit-Vink [1] and cooled down MILC asqtad ensembles with $N_f=2+1$ dynamical flavors of quarks [2]. Using flavored mass terms [3,4], we find that the spectral flow of the eigenvalues detects the topology of the background gauge field. With the use of the modified chirality operator, we obtain chiralities of the zero eigenmodes and the fermionic topological charge.

The pole structure of the $\Lambda(1405)$ has been a topic of debate for a long time. Chiral perturbation theory predicts that its experimental spectrum may be explained by a two pole structure originating in the $SU(3)$ chiral dynamics of the baryon-meson interaction. The $SU(3)$-symmetric flavor point is readily accessible in lattice QCD, in this work we study the baryon-meson states directly at this point. We construct interpolation operators that belong to the irreducible representations of $SU(3)$ that are attractive in the channel with the quantum numbers of the (singlet and two octets). The extracted energy levels can be used as input for chiral perturbation theory to find the poles associated with each representation. The relevant correlation functions are computed on $SU(3)$-symmetric ensembles with $M_{\pi}\approx 714$ MeV using the distillation technique.

We consider glueball dark matter (DM) in a Yang-Mills dark sector confined at $\Lambda_D$ scale and coupled to the Standard Model through electrically and dark-color charged vector-like fermion portals, with the mass scale $m_\psi$. In a simple case with two lightest mass-degenerate vector-like fermions with opposite electric charges the effective amplitudes with one $C$-odd glueball (oddball) and odd number of photons vanish, rendering the lightest $C$-odd spin-1 state with mass $m_\chi$ a viable DM candidate provided that $m_\psi\gtrsim 5.5 \Lambda_D$. We develop a controlled effective field theory framework with non-perturbative information supported by QCD phenomenology leading to a quantitative prediction for coherent elastic glueball scattering off nuclei. We find a steep scaling of the spin-independent cross section $\sigma_{\rm SI}\propto \Lambda_D^{2.15} m_\psi^{-8}$. This implies that the sensitivity of the current and next-generation xenon experiments in the range of $\sigma_{\rm SI} \sim 10^{-46} - 10^{-48}$ cm$^2$ corresponds to $m_\psi \simeq 3-30$ GeV, respectively, for $\Lambda_D\simeq 0.55-5.5$ GeV. We provide a minimal UV completion of the portal sector compatible with collider phenomenology. Our results pave a quantitative foundation for testing glueball DM in direct-detection experiments.

We develop a generative framework based on denoising diffusion for the model-independent reconstruction of hadronic form factors from sparse and noisy data. The generative prior is built from a large ensemble of synthetic curves drawn from ten distinct functional classes rooted in different theoretical approaches to hadron structure. Applied to the proton gravitational form factors $A(t)$, $J(t)$, and $D(t)$, the framework yields non-parametric reconstructions consistent with lattice QCD across the full kinematic range $0\le -t\le 2~\mathrm{GeV}^{2}$, remaining robust even when only one or two conditioning points are retained. The densely sampled output enables a direct extraction of the chiral low-energy constants $c_8=-4.6\pm 0.8$ and $c_9=-0.61\pm 0.19$. Using these values at the physical pion mass, we obtain $D(0)=-4.3\pm 0.8$ for the nucleon $D$-term.

Variational quantum circuits (VQCs) constitute a central building block of near-term quantum machine learning (QML), yet the principled design of expressive and trainable architectures remains a major open challenge. The VQC design space grows combinatorially with the number of qubits, layers, entanglement structures, and gate parameterizations, rendering manual circuit construction inefficient and often suboptimal. We introduce an autonomous agent-based framework for VQC architecture search that integrates high-level reasoning with a quantum simulation environment. The agent proposes candidate circuit architectures, evaluates them through fully automated training and validation pipelines, and iteratively improves its design strategy via performance-driven feedback. Empirically, we show that the agent autonomously evolves circuit architectures from simple initial ansätze toward increasingly expressive designs, progressively trying to improve task performance. This demonstrates that agentic AI can effectively navigate and refine the VQC design landscape with minimal human intervention, providing a scalable methodology for automated quantum model development in the Noisy Intermediate-Scale Quantum (NISQ) regime.

This Chapter introduces QCD at finite temperature and density. We first present the formulation of the thermal theory in the Euclidean path integral formalism. We then describe how the strong dynamics at high temperature can be inspected through thermal effective field theories. As a concrete example of thermodynamic quantity, we discuss the Equation of State, which characterises the equilibrium properties of the QCD plasma. We finally conclude with an overview of the phase diagram of strongly interacting matter.

The year 2024 marked the 50th anniversary of the discovery of the $J/\psi$ particle, which unveiled the charm quark and the charmonium spectrum, instigating the "November Revolution" in particle physics. This discovery catalyzed the development of quenched potential models, most notably the Cornell model, which provided a foundational quantitative description of the hadronic spectrum. However, the landscape of hadron spectroscopy has been profoundly transformed since the turn of the 21st century with the observation of numerous charmonium-like states, such as $X(3872)$, which exhibit properties starkly at odds with quenched model predictions. These discrepancies, exemplified by the "$X(3872)$ low-mass puzzle" and the "$Y$ problem" associated with vector states like $Y(4260)$, underscore the critical limitations of the quenched approximation and signal the necessity for a new theoretical paradigm. This review synthesizes recent advances in hadronic spectroscopy, arguing that the unquenched picture, which incorporates coupled-channel effects such as hadronic loops, is essential for a unified description of these new states and associated anomalies. We demonstrate how unquenched effects provide compelling solutions to long-standing puzzles in charmonium decays (e.g., the "$\rho\pi$ puzzle" and anomalous dipion transitions), predict and explain the existence of exotic charged states like $Z_c(3900)$ and $Z_b(10610)$ via mechanisms such as Initial Single Pion Emission, and offer a framework for understanding interactions between charmonia and with nucleons. Furthermore, we emphasize the universality of unquenched effects, extending their application to bottomonium and light-flavor sectors. With improving precision, we advocate systematic development of unquenched hadronic spectroscopy.

We apply score-based diffusion models to two-dimensional SU(2) lattice pure gauge theory with the Wilson action, extending recent work on U(1) gauge theories. The SU(2) manifold structure is handled through a quaternion parameterization. The model is trained on 10,000 configurations generated via Hybrid Monte Carlo at a fixed coupling $\beta_0= 2.0$ on an $8\times 8$ lattice, augmented to 20,000 samples via random gauge transformations. Through physics-conditioned sampling exploiting the linear $\beta$-dependence of the score function, we generate configurations at different values of the coupling without retraining; through the fully convolutional U-Net architecture with periodic boundary conditions, we generate configurations on lattices of different spatial extents. We validate our approach by comparing the average plaquette and Wilson action density against exact analytical predictions. At the training lattice size ($8\times 8$), the model reproduces the exact plaquette with biases $|\Delta| \leq 0.001$ for $\beta \in [1.5, 2.5]$ and $|\Delta| < 0.06$ across $\beta \in [1, 4]$. For lattices sharing the training extent $L=8$ in at least one direction, biases remain below $\sim 0.003$ for $\beta \in [1.5, 2.5]$, with larger deviations at higher couplings. This work demonstrates that diffusion models are a promising tool for non-Abelian gauge field generation and motivates further investigation toward higher-dimensional theories.

We introduce a renormalization-group invariant observable, the symmetry strength parameter $\kappa_{AB}$, for the quantitative characterization of symmetry breaking in QCD. As a first application, we employ $\kappa_{AB}$ to investigate the relative strength of $SU(2)_L \times SU(2)_R$ chiral symmetry and $U(1)_A$ axial symmetry breaking in $N_f=2+1+1$ lattice QCD using optimal domain-wall fermions at the physical point. Our study covers three lattice spacings and twelve temperatures in the range 164-385~MeV. We examine three independent symmetry-breaking channels in the nonsinglet sector with connected correlators: the $U(1)_A$-sensitive scalar-pseudoscalar channel ($\kappa_{PS}$), probing the $\pi$-$\delta$ system; the $SU(2)_L \times SU(2)_R$-sensitive vector--axial-vector channel ($\kappa_{VA}$), probing the $\rho$-$a_1$ system; and an additional $U(1)_A$-sensitive tensor--axial-tensor channel ($\kappa_{TX}$), probing the $\rho_T$-$b_1$ system. At finite lattice spacing, we observe a clear hierarchy $\kappa_{PS} > \kappa_{TX} \sim \kappa_{VA}$. A controlled continuum extrapolation reveals that this hierarchy collapses, with all three symmetry-breaking strengths becoming statistically indistinguishable within our precision. This result provides a new, model-independent benchmark from a chirally symmetric lattice action. Our findings indicate that the effective restoration scales for $SU(2)_L \times SU(2)_R$ and $U(1)_A$ in the nonsinglet sector converge closely near the chiral crossover, placing stringent quantitative constraints on the temperature window for chiral and axial symmetry manifestation in connected channels. These results support a two-stage restoration scenario, in which full symmetry restoration -- including the singlet sector -- occurs only at significantly higher temperatures once topological fluctuations are sufficiently suppressed.

Random matrix theory (RMT) provides a powerful framework for analyzing universal features of strongly coupled physical systems. In quantum chromodynamics (QCD), cold quark matter at asymptotically high density is expected to exhibit color superconductivity (CSC), the analogue of superconductivity in condensed-matter systems. Although CSC phases have been studied within RMT primarily in the macroscopic large-$N$ limit, where $N$ denotes the matrix size, it has remained unclear whether an RMT exists that realizes CSC in the microscopic large-$N$ limit. Here we answer this question in the affirmative by introducing a novel non-Hermitian chiral random matrix model. For three quark flavors, we show that the model exhibits spontaneous breaking of color $\mathrm{SU}(3)$ and flavor $\mathrm{SU}(3)$ symmetries down to the diagonal $\mathrm{SU}(3)$ subgroup, thereby reproducing color-flavor locking in dense QCD. For two flavors, we find that color $\mathrm{SU}(3)$ is spontaneously broken to $\mathrm{SU}(2)$ while the chiral symmetry $\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{SU}(2)_{\mathrm{R}}$ remains unbroken, consistent with the two-flavor color-superconducting phase.

We present results from co-designed quantum simulations of the neutrinoless double-beta decay of a simple nucleus in 1+1D quantum chromodynamics using IonQ's Forte-generation trapped-ion quantum computers. Electrons, neutrinos, and up and down quarks are distributed across two lattice sites and mapped to 32 qubits, with an additional 4 qubits used for flag-based error mitigation. A four-fermion interaction is used to implement weak interactions, and lepton-number violation is induced by a neutrino Majorana mass. Quantum circuits that prepare the initial nucleus and time evolve with the Hamiltonian containing the strong and weak interactions are executed on IonQ Forte Enterprise. Enabled by tuned model parameters, lepton-number violation is observed in real time, providing a clear signal of neutrinoless double-beta decay. This was made possible by co-designing the simulation to maximally utilize the all-to-all connectivity and native gate-set available on IonQ's quantum computers. Quantum circuit compilation techniques and co-designed error-mitigation methods, informed from executing benchmarking circuits with up to 2,356 two-qubit gates, enabled observables to be extracted with high precision. We discuss the potential of future quantum simulations to provide yocto-second resolution of the reaction pathways in these, and other, nuclear processes.

The tensor form factors of the $\Delta^+$ baryon are defined through the matrix element of the tensor current and describe its internal structure and spin distribution. We present the full Lorentz decomposition for the $\Delta^+ \rightarrow \Delta^+$ tensor current matrix element, including all independent structures consistent with Lorentz covariance, the Rarita-Schwinger constraints, and the discrete symmetries of Hermiticity, time-reversal, and parity invariance. By investigating the tensor form factors corresponding to both the isovector and isoscalar tensor currents, we observe differences that reflect the distinct contributions of up and down quark components in the $\Delta^+$ baryon.

Poincaré-covariant Bethe-Salpeter wave functions are used to calculate light-front wave functions (LFWFs) of the pion, $\pi$, and an analogue state, $\pi_{s\bar s}$. The current masses of the degenerate valence constituents in the $\pi_{s\bar s}$ are around $25$-times larger than those of the pion's valence constituents. Both valence spin-antialigned ($\mathcal L=0$) and valence spin-aligned ($\mathcal L=1$) components are obtained and combined to produce the complete LFWF for each system. Comparing predictions delivered by two distinct Bethe-Salpeter kernels, the impact of nonperturbative dynamical effects contained in the more sophisticated (bRL) kernel are seen to be significant; and contrasts between $\pi$, $\pi_{s \bar s}$ results reveal the interplay between emergent hadron mass and mass effects owing to Higgs-boson couplings. Amongst the results, one finds that for $\pi$, $\pi_{s\bar s}$, the LFWFs can be approximated by a separable form, with that representation being pointwise reliable in the bRL cases. Moreover, the $\mathcal L=1$ component is important; so a LFWF obtained after omission of this piece is typically a poor representation of the system. These features are naturally expressed in $\pi$, $\pi_{s\bar s}$ transverse momentum dependent parton distribution functions (TMDs). In this connection, it is found that a Gaussian \textit{Ansatz} can only provide a rough guide to TMD pointwise behaviour: magnitude deviations between \textit{Ansatz} and prediction exceed a factor of two on $k_\perp^2 \gtrsim 0.55\,$GeV$^2$. One should therefore be cautious in interpreting conclusions drawn from phenomenological analyses based upon Gaussian \textit{Ansätze}.