In the Euclidean view one must first require that positivity not be violated, and from this modest demand, together with locality, a great deal follows: starting from a reflection-positive lattice formulation of pure SU(N) Yang-Mills theory we obtain a transfer operator with a uniform gap, while large Wilson loops already show an area law by means of convergent character (polymer) expansions; a finite-range, gauge-covariant multiscale analysis then carries these features from one scale to the next with interlaced inequalities whose small defects can be summed, so that exponential clustering and a strictly positive string tension endure in the continuum; the Osterwalder-Schrader reconstruction turns these Euclidean facts into a Minkowski theory with a self-adjoint Hamiltonian, the spectral gap lying above the vacuum and the linear potential for static charges appearing, which gives a concrete picture of confinement; the construction depends on no special regulator, for a single-scale Lipschitz control and a telescoping argument bind all admissible reflection-positive slicings into a unique limiting measure and thus secure universality; moreover, the same framework admits entry from weak coupling, so that the continuum reached from strong coupling meets the one approached along an asymptotically free trajectory, yielding one and the same theory; in my view this is how mathematical clarity and physical insight cooperate: positivity, locality, and renormalization working together so that the mass gap and confinement are not marvels to be assumed, but natural properties of the non-Abelian vacuum.

This work develops a framework for transforming between smeared spectral functions computed using different smearing kernels. The kernel-transformation problem naturally arises when information is available for one family of energy-smeared observables, while phenomenology or comparison with other calculations require a different smearing. For exact transformations, analytic conditions are established for the maps to exist and converge without arbitrary regularization. Explicit expressions are provided for several kernel classes of interest, including Cauchy-to-Gaussian transformations and Gaussian-to-Cauchy width mixtures. When exact transformations are unavailable, the inverse problem is tackled through regulated maps paired with bounds on the associated systematic error, directly computable from the given input data. Errors on the input smeared spectral functions, either statistical or in the form of pointwise rigorous bounds, are then propagated to the target observables. Enforcing spectral positivity can be used to tighten the bounds.

We study the scalar diquark mass and the quark--diquark potential by applying a HAL QCD-inspired potential method to a baryonic system composed of a scalar diquark and a static quark. The diquark mass is determined self-consistently by requiring that the p-wave baryonic spectrum obtained from two-point correlators be reproduced within the potential framework. Numerical calculations are performed using $2+1$ flavor QCD gauge configurations generated by the PACS-CS Collaboration on a $L^{3} \times T = 32^{3} \times 64$ lattice with $a^{-1} \approx 2.176$ GeV and the pion mass, $m_π \approx 702$ MeV. From the analysis, we obtain a scalar diquark mass which is close to the na\"ıve constituent quark estimate $ (2/3)m_{N}$, together with a quark--diquark potential of the Cornell type (Coulomb + linear). The string tension extracted from the quark--diquark potential agrees within approximately 5% with that obtained from the static quark--antiquark potential (Wilson Loop).

We propose a finite-volume scheme to perform controlled continuum extrapolations of the lattice step-scaling function, a key ingredient for determining the running coupling in a Hamiltonian lattice gauge theory in small volumes. As a testbed, we employ a dual Hamiltonian formulation of pure U(1) gauge theory in (2+1) dimensions and an operator basis that remains efficient toward weak coupling. We describe the implementation of static external charges on the spatial lattice and study, using matrix product states, the resulting confining string, from which we extract the static potential and a force-based renormalized coupling. Using the proposed finite-volume scheme, we demonstrate a stable continuum limit of the step-scaling function on the lattice sizes accessible to present Hamiltonian simulations. The method is readily extendable to other gauge groups and dimensions, providing a pathway toward Hamiltonian step-scaling studies in other theories.

Motivated by the process of reconstructing real-time spectral functions from Euclidean correlators in lattice QCD, we derive explicit expressions for the fermionic determinant and the propagator of the four-dimensional clover-improved Wilson-Dirac fermions on anisotropic lattices in terms of the three-dimensional Wilson-Dirac Hamiltonian operator. We derive an effective Hamiltonian that governs Euclidean time evolution at finite temporal lattice spacing, and demonstrate its hermiticity and particle-anti-particle symmetry. Our results allow to quantify lattice artifacts of the numerical spectral reconstruction based on Euclidean fermionic correlators at finite temporal lattice spacing.

We study the confining flux tube in the reconfined phase of trace deformed SU(2) Yang-Mills theory in (2+1) dimensions. Using lattice simulations above the standard deconfinement temperature, we analyze Polyakov-loop correlators and extract the ground state energy of the effective string. We show that the usual Nambu-Goto effective string description, including its standard higher-order corrections, fails to reproduce the data as the trace deformation is increased. Remarkably, deep in the reconfined regime the results are instead accurately described by the Polchinski-Yang rigid-string solution, corresponding to an effective string dominated by an extrinsic-curvature term. We further investigate the transverse profile of the chromo-electric flux tube and find significant deviations from the standard Yang-Mills behavior, including a substantial modification of the intrinsic width. Finally, we present an exploratory study of the phase diagram, finding evidence for a transition from a continuous to a first order reconfinement line as the deformation parameter increases. These results suggest that the reconfined phase realizes a qualitatively different effective-string regime from ordinary confinement.

We initiate a bootstrap study of pion form factors in large $N$ QCD. We consider the mixed system of the vector-current two-point function, the pion vector form factor, and the pion scattering amplitude in the chiral limit. At large $N$ these observables are meromorphic, with spectral data constrained by unitarity, crossing symmetry, and Regge boundedness. We obtain bounds of two kinds. The first are rigorous and universal: from analyticity, unitarity and the asymptotic Brodsky-Farrar scaling, we constrain low-energy form-factor coefficients. The second are more phenomenological, of the Shifman-Vainshtein-Zakharov type: feeding in the perturbative ultraviolet behavior at a finite scale lets us bound the pion decay constant, convert a large $N$ lattice measurement into a lower bound on the scale at which asymptotic freedom sets in, and constrain the pion charge radius. Combining these inputs, the space of allowed chiral Lagrangians shrinks toward the region where large $N$ QCD is expected to sit. Our results illustrate how local gauge-invariant probes provide a canonical bridge between the hadronic bootstrap and the microscopic QCD Lagrangian.

The (2+1)D U(1) pure gauge theory always exists in the confining phase, with strings of non-zero string tension giving a characteristic linear potential between static charges. This makes it a useful testing ground for quantum computing methods designed to study string dynamics of confining gauge theories. Here we implement a minimal U(1) quantum link model on a quantum computer with qubit degrees of freedom representing the dual height variables of the model. This facilitates an efficient realization of plaquette interactions and enables effective calculations of real-time dynamics that are inaccessible to traditional quantum Monte Carlo. A specifically tailored lattice geometry is chosen to match the heavy-hexagonal geometry of the IBM quantum hardware used here, minimizing non-adjacent qubit interactions. By performing quantum quenches from a simple initial string state, we probe the transverse quantum fluctuations of the string before it thermalizes. Our experimental results from digital quantum simulations, with up to 112 qubits, show good agreement with reference tensor-network calculations at short times and with thermal averages at long times. Near the phase transition, the quench dynamics exhibit large fluctuations of the initial string that extend across both spatial dimensions of the lattice. Nonetheless, our error-mitigated estimators from the quantum hardware also give accurate predictions in that regime, with noise-induced violations of local gauge symmetries comparable to finite-bond-dimension tensor-network results.

We assess the tension between theoretical predictions and the values quoted by the Particle Data Group (PDG) for the partial decay width and branching fraction associated with the radiative charmonium decay $J/ψ\toγη_c$. A profile scan over the most recent PDG data depending on the branching fraction $\mathcal{B}(J/ψ\toγη_c)$ suggests that the correlation between measured branching fractions is compatible with lattice QCD determinations of the partial decay widths $Γ(J/ψ\toγη_c)$ and $Γ(η_c\toγγ)$. We propose a theoretically grounded photon line shape for the radiative decay spectrum and a prescription for the extraction of (product) branching fractions involving the magnetic dipole (M1) transition $J/ψ\toγη_c$. This approach obviates the need to modify the photon energy spectrum line shape using empirical damping functions, as done in the most recent experimental extractions of $\mathcal{B}(J/ψ\toγη_c)$ from the photon line shape, thereby eliminating an inherent ambiguity in the determination of the derived observables.

We study the phase structure of the (3+1)-dimensional cold and dense QCD with the Kogut--Susskind quark in the strong coupling limit using the tensor renormalization group method. The chiral and nuclear transitions are investigated by calculating the chiral condensate and the quark number density as a function of the chemical potential. For a fixed temporal extent $N_τ=8$, we determine the critical quark masses $m_c^χ$ and $m_c^{n}$ for the chiral condensate and the quark number density, respectively, at which the first-order phase transition terminates with the vanishing discontinuity in thermodynamic quantities. We find that both quantities at the same quark mass exhibit a discontinuity at the same chemical potential, and the resulting critical quark masses are consistent with each other. We also compare our results for the critical quark masses with those obtained from the Monte Carlo simulation in the dual formulation and from the mean-field analysis. We further confirm the first-order phase transition at finite quark mass on a $1024^4$ lattice, which is essentially in the thermodynamic limit at zero temperature, as expected from the mean-field analysis.

A recent non-perturbative formulation of $2n$ dimensional chiral gauge theories relies on realizing chiral fermions on the $2n$ dimensional boundary of a $2n+1$ dimensional disk manifold. It also requires extending boundary gauge configurations into the interior of the disk using some flow prescription that preserves 2n dimensional gauge invariance. In this paper we propose a concrete realization of the equation of motion flow with the disk embedded on a square lattice. In addition, we couple the flow gauge field to fermions and demonstrate the mechanism of anomaly inflow and anmaly cancellation at work on the lattice.

The proposal to formulate chiral gauge theories using domain wall fermions on $2n+1$ dimensional Euclidean lattice with a slab geometry involves $2n$ dimensional dynamical gauge fields residing on one of the domain walls. The gauge fields are extended into the extra dimension using gradient flow decoupling the mirror fermions on the anti-wall. We implement this construction on the lattice for $n=1$ in the presence of $2n$ dimensional background gauge fields. We also formulate and implement an additional gauge field flow proposal, where the gauge fields satisfy $2n+1$ dimensional equation of motion away from the domain wall, known as the EOM (equation of motion) flow. In both cases, we couple the gauge fields to fermions and demonstrate how current conservation and anomaly inflow work on the lattice.

We present an investigation into the semileptonic and nonleptonic weak decays of bottom baryons $Ω^{*}_{b}$ and $Ω_{b}$ within the framework of three-point QCD sum rules. In the semileptonic sector, the $Ω^{*}_b\rightarrowΩ_c\ell\barν_{\ell}$ and $Ω_b\rightarrowΩ^*_c\ell\barν_{\ell}$ transitions are specifically considered. Utilizing the operator product expansion up to dimension six, the responsible form factors of these decays are obtained. The acquired form factors enable us to determine the decay widths in three leptonic channels. Branching ratios related to the $Ω_{b}$ baryon semileptonic decays are also presented. These invariant form factors are subsequently employed as inputs to determine the nonleptonic weak decay widths in various modes with emitting a pseudoscalar or vector meson. An extensive investigation into all possible decay channels of bottom baryons provides valuable information for future experiments to examine the SM predictions, explores the new physics effects in heavy baryonic decays, and advances the understanding of the internal structure of heavy baryons.

The spectrum of vector charmonium(-like) states in the 4.1\dash4.6~GeV energy region exhibits a long-standing tension between inclusive and exclusive measurements. While the inclusive $R$-value indicates only conventional vector charmonia such as $ψ(4160)$ and $ψ(4415)$, exclusive $e^+e^-$ cross sections reveal additional structures whose parameters strongly depend on the observed final states when fitted with Breit--Wigner functions. This puzzling pattern suggests that coupled-channel and threshold effects play an essential role. In this work, we develop a unified coupled-channel framework for the $1^{--}$ resonances in this energy region. The framework incorporates the $S$-wave open-charm channels $D\bar{D}_1$, $D^*\bar{D}_1$, and $D^*\bar{D}_2^*$ constrained by heavy-quark spin symmetry, optional bare poles associated with $ψ(4160)$ and $ψ(4415)$, and final-state interactions in the $Z_c$ channels. We perform simultaneous fits to the BESIII cross sections for $e^+e^-\to J/ψπ^+π^-$, $h_cπ^+π^-$, $D\bar{D}^*π$, $D^*\bar{D}^*π$, $J/ψη$, and $χ_{c0}ω$, together with invariant-mass distributions exhibiting the $Z_c(3900)$ and $Z_c(4020)$ structures. The benchmark models differ in the number of bare seed states and the fitting strategy. We show that even the purely dynamical scheme without bare charmonia captures the gross features of the analyzed distributions. The inclusion of bare compact states improves the fit quality but does not change the conclusion that the measured line shapes can be understood in terms of strong coupled-channel effects with dynamically generated poles. We also discuss possible heavy-quark spin partners of the exotic $1^{--}$ states.