A systematic way to constructing optimized interpolating operators strongly coupled to QCD two-particle states is developed, which is achieved by incorporating inter-hadron spatial wavefunctions. To efficiently implement these operators in lattice QCD, a novel quark smearing technique utilizing noise vectors is proposed. Applied to the $\Omega_{ccc}\Omega_{ccc}$ system, the optimized operators resolve distinct eigenstates separated by only $\sim 5$ MeV near the threshold $2m_{\Omega_{ccc}} \simeq 9700$ MeV. This exceptional resolving power opens new possibilities for studies of a wide range of hadronic systems in QCD.

A systematic way to constructing optimized interpolating operators for two-hadron systems is developed by incorporating inter-hadron spatial wavefunctions. The wavefunctions can be obtained from an iterative process with an appropriate initial guess. To implement these operators, a novel quark smearing technique utilizing $Z_3$ noise vectors is proposed, which allows for effectively incorporating inter-hadron spatial wavefunctions at the source without using all-to-all quark propagators. Proof-of-principle application to the $\Omega_{ccc}\Omega_{ccc}$ system using physical-point lattice configurations with a large size $La\simeq8.1$~fm shows that optimized operators enables clear identification of states around $2m_{\Omega_{ccc}}\simeq 9700$ MeV with the energy gap as narrow as $\sim 5$ MeV. A comparison on correlation functions, effective energies, and HAL QCD potentials between unoptimized operators and optimized operators is given, with a special emphasis on the effects from nearby elastic scattering states. Potential applicability of the optimized operator to various two-hadron systems and its relation to the variational method are also discussed.

We present the first lattice QCD determination of the $\Lambda \to N$ vector and axial-vector form factors, which are essential inputs for studying the semileptonic decay $\Lambda \to p \ell \bar{\nu}_\ell$. This channel provides a clean, theoretically controlled avenue for extracting the CKM matrix element $|V_{us}|$ from the baryon sector. Our analysis uses a gauge ensemble with physical light, strange, and charm quark masses and yields the most precise determination to date of the full set of transition form factors -- including second-class contributions -- as well as the associated couplings, radii, and the ratio of muon-to-electron decay rates, an observable sensitive to possible non-standard scalar and tensor interactions. We compare our non-perturbative results with next-to-next-to-leading order expansions in the small parameter $\delta = (m_\Lambda - m_N)/m_\Lambda \approx 0.16$. We find that the common phenomenological approximation of neglecting the $q^2$-dependence of the form factors leads to a $\sim 4\%$ deviation in the decay rate. This underscores the critical importance of precise, fully non-perturbative form factor inputs for achieving the sub-percent precision targets of upcoming experimental programs.

We study the two-point field-strength correlation $g^2 \langle G_{\mu\nu}^a(s)G^b_{\alpha\beta}(s') \rangle$ in the Landau gauge in SU(2) and SU(3) quenched lattice QCD, as well as the gluon propagator $g^2 \langle A_\mu^a (s)A_\nu^b(s') \rangle$. The Landau-gauge gluon propagator $g^2 \langle A_\mu^a (s)A_\mu^a(s') \rangle$ is well described by the Yukawa-type function $e^{-mr}/r$ with $r\equiv |s-s'|$ for $r=0.1-1.0~{\rm fm}$ in both SU(2) and SU(3) QCD. Next, motivated by color-magnetic instabilities in the QCD vacuum, we investigate the perpendicular-type color-magnetic correlation, $C_{\perp}(r) \equiv g^2\langle H_z^a(s)H_z^a(s + r \hat \perp)) \rangle$ ($\hat \perp$: unit vector on the $xy$-plane), and the parallel-type one, $C_{\parallel}(r) \equiv g^2 \langle H_z^a(s)H_z^a(s + r \hat \parallel) \rangle$ ($\hat \parallel$: unit vector on the $tz$-plane). These two quantities reproduce all the correlation of $g^2\langle G^a_{\mu\nu}(s)G^b_{\alpha\beta}(s')\rangle$, due to the Lorentz and global SU($N_c$) color symmetries in the Landau gauge. Curiously, the perpendicular-type color-magnetic correlation $C_{\perp}(r)$ is found to be always negative for arbitrary $r$, except for the same-point correlation. In contrast, the parallel-type color-magnetic correlation $C_{\parallel}(r)$ is always positive. In the infrared region of $r \gtrsim 0.4~{\rm fm}$, $C_{\perp}(r)$ and $C_{\parallel}(r)$ strongly cancel each other, which leads to a significant cancellation in the sum of the field-strength correlations as $\sum_{\mu, \nu} g^2\langle G^a_{\mu\nu}(s)G^a_{\mu\nu}(s')\rangle \propto C_{\perp}(|s-s'|)+ C_{\parallel}(|s-s'|) \simeq 0$. Finally, we decompose the field-strength correlation into quadratic, cubic and quartic terms of the gluon field $A_\mu$ in the Landau gauge.

We present the third part of a systematic calculation of the two-loop anomalous dimensions for the low-energy effective field theory below the electroweak scale (LEFT): insertions of dimension-six operators that conserve baryon number. In line with our previous publications, we obtain the results in the algebraically consistent 't Hooft-Veltman scheme for $\gamma_5$, corrected for evanescent as well as chiral-symmetry-breaking effects through finite renormalizations. We compute the renormalization of the dimension-six four-fermion and three-gluon operators, as well as the power corrections to lower-dimension operators in the presence of masses, i.e., the down-mixing into dimension-five dipole operators, masses, gauge couplings, and theta terms. Our results are of interest for a broad range of low-energy precision searches for physics beyond the Standard Model.

We investigate defects in the two-dimensional transverse-field Ising ferromagnet on periodic $L\times L$ lattices after quantum annealing from high to vanishing field. With exact numerical solutions for $L \le 6$, we observe the expected critical Kibble-Zurek (KZ) time scale $\propto L^{z+1/\nu}$ (with $z=1$ and $1/\nu \approx 1.59$) at the quantum phase transition. We also observe KZ scaling of the ground-state fidelity at the end of the process. The excitations evolve by coarsening dynamics of confined defects, with a time scale $\propto L^2$, and interface fluctuations of system-spanning defects, with life time $\propto L^3$. We build on analogies with classical simulated annealing, where we characterize system-spanning defects in detail and find differences in the dynamic scales of domain walls with winding numbers $W=(1,0)/(0,1)$ (horizontal/vertical) and $W=(1,1)$ (diagonal). They decay on time scales $\propto L^3$ (which applies also to system-spanning domains in systems with open boundaries) and $\propto L^{3.4}$, respectively, when imposed in the ordered phase. As a consequence of $L^{3.4}$ exceeding the classical KZ scale $L^{z+1/\nu}=L^{3.17}$ the probability of $W=(1,1)$ domains in SA scales with the KZ exponent even in the final $T=0$ state. In QA, also the $W=(1,0)/(0,1)$ domains are controlled by the KZ time scale $L^{2.59}$. The $L^3$ scale can nevertheless be detected in the excited states, using a method that we develop that should also be applicable in QA experiments.

We demonstrate that the polaron theory from solid state physics can serve as an interesting analogue model for non-perturbative QCD, at least in the description of nucleons and related low-energy physics of strong interactions. By drawing explicit analogies between polaron physics, arising for an electron moving in an ionic crystal, and physics of pion-nucleon interactions, certain rules for the "polaron/QCD correspondence" are proposed. In polaron theory, the effective fermion mass as a function of the coupling constant is known both in the weak and strong coupling limits. The conjectured "polaron/QCD correspondence" translates these results into strong interactions. It is then shown how application of these rules leads to unexpectedly good quantitative predictions for the nucleon mass and the pion-nucleon sigma term. The polaron approach also predicts that the quark degrees of freedom in the form of the constituent quark account for one-third of the nucleon mass, consistent with lattice predictions. We discuss possible physical reasons underlying the observed quantitative similarity between polaron physics and non-perturbative QCD.

We argue that a field theory with a $\mathbb Z_N$ $p$-form symmetry generically admits, in addition to a Higgs phase and a 'confining' phase, a Coulomb phase in which the infrared theory contains Abelian $p$-form electrodynamics, similar to the behaviour of Yang-Mills theory coupled to adjoint or fundamental matter. We illustrate our claim with continuum and lattice examples.

Monte Carlo methods have led to profound insights into the strong-coupling behaviour of lattice gauge theories and produced remarkable results such as first-principles computations of hadron masses. Despite tremendous progress over the last four decades, fundamental challenges such as the sign problem and the inability to simulate real-time dynamics remain. Neural network quantum states have emerged as an alternative method that seeks to overcome these challenges. In this work, we use gauge-invariant neural network quantum states to accurately compute the ground state of $\mathbb{Z}_N$ lattice gauge theories in $2+1$ dimensions. Using transfer learning, we study the distinct topological phases and the confinement phase transition of these theories. For $\mathbb{Z}_2$, we identify a continuous transition and compute critical exponents, finding excellent agreement with existing numerics for the expected Ising universality class. In the $\mathbb{Z}_3$ case, we observe a weakly first-order transition and identify the critical coupling. Our findings suggest that neural network quantum states are a promising method for precise studies of lattice gauge theory.

The isoscalar axial-vector form factor of the nucleon plays a key role in understanding the electroweak interaction of nucleons. For the interpretation of the spin structure of the nucleon the non-singlet isoscalar axial charge is indispensable. Moreover, $G_A^{u+d-2s}(Q^2)$ together with the isovector and singlet isoscalar form factors are needed for the flavor-decomposition of the axial-vector form factor. Detailed knowledge of the flavor decomposition facilitates extractions of Standard Model (SM) parameters from low energy experiments such as the electroweak charge of the proton in the P2 experiment. Here we present a lattice determination of $G_A^{u+d-2s}(Q^2)$ on $N_f=2+1$ $\mathcal{O}(a)$ improved Wilson fermions, with a full error budget concerning extrapolations and interpolations to the continuum and infinite volume and physical quark masses.

In this study, we conduct a thorough examination of the spectrum of the open confining string in 3+1 dimensions, commonly referred to as the open flux-tube, across various gauge groups of $SU(N_c)$. Our primary objective is to explore its behaviour as we approach the large-$N_c$ limit and the identification of possible world-sheet axion states. Specifically, we undertake a detailed analysis of the associated spectrum for $N_c=3, 4, 5, 6$. This marks the first systematic investigation of the open flux-tube spectrum within the context of the large-$N_c$ limit. More specifically, we analyse the spectra of flux-tubes that form between a static quark-antiquark pair, considering a significant number of radial excitations and eight irreducible representations characterized by the quantum numbers of angular momentum $\Lambda$, charge conjugation and parity $\eta_{CP}$ and the reflection symmetry $\epsilon$ for $\Lambda=0$. To this purpose we employ a diverse set of suitable operators, an anisotropic action, smearing techniques, and solve the generalized eigenvalue problem. We compare our findings with predictions from the Nambu-Goto string model to assess potential tensions indicative of novel phenomena such as the existence of axion-like state along the flux-tube world-sheet. Notably, we provide undoubted evidence of the existence of a massive axion-like particle with the same mass as the corresponding axion extracted within the context of closed flux-tube. This strengthens the conjecture that the axion is a property of the world-sheet of the QCD string.

In this paper, we discuss the role of Roberge-Weiss periodicity in the thermodynamics of quantum chromodynamics at moderately high temperature, where the semi-quark-gluon plasma is expected. From the construction of the grand canonical partition function at zero and also at finite density via the canonical approach, we are able to discuss the relation between contributions of the Polyakov-loop phase and Roberge-Weiss periodicity. Then, we can conclude that the existence of Roberge-Weiss periodicity is a necessary condition to reproduce exact results at moderately high temperature.