We use the (1+1)-dimensional Schwinger model to investigate the nonequilibrium dynamics of a finite lattice system under a constant external electric field. The lattice Hamiltonian is constructed under open boundary conditions. The vacuum state is prepared using the variational quantum eigensolver (VQE). Scans over the external field strength show the flip of the vacuum state at several field strengths. The critical field strengths agree with theoretical predictions. We further investigate the real-time evolution of the zero-field vacuum under an external electric field using a second-order Trotter-Suzuki decomposition. By comparison with exact diagonalization (ED), we verify that the quantum-simulation protocol reproduces the main features of field-induced boundary charge separation, decay of the vacuum-state fidelity, and quasiperiodic energy redistribution between the electric-field energy term and the fermionic sector. Our results indicate that combining VQE-based state preparation with digital real-time evolution provides a useful approach for studying nonequilibrium dynamics in strong-field lattice gauge theories.

We present numerical results of staggered fermions with a taste splitting mass term on dynamical configurations. The rise of gluonic counterterms from rotational symmetry breaking is studied for a single taste operator and the pion propagator is computed. Preliminary numerical results are given for lattice sizes up to 16^4.

In this work, we propose a novel supervised machine-learning-based strategy for extracting smeared spectral functions from Euclidean correlation functions. The strategy revisits the numerically ill-posed spectral reconstruction problem within the framework of Operator Learning through the use of DeepONet-like architectures. To illustrate the method, we construct an ensemble of neural networks trained on mock data generated from a specific class of functions. This ensemble is then employed to estimate the systematic uncertainty associated with the fact that a neural network provides only an approximation to the target operator. The procedure is fully validated on previously unseen noisy mock data. To demonstrate the potential of the method for phenomenological applications, we reconstruct the inclusive rate above the four-particle threshold and up to high energies in the $1+1$-dimensional O(3) non-linear $\sigma$ model, starting from correlation functions computed in Monte Carlo simulations. The final result is consistent with the known analytic spectral density and, compared with the state-of-the-art Hansen-Lupo-Tantalo algorithm, exhibits a significant reduction in the total uncertainty. While the extent to which this improvement persists in the absence of prior physical knowledge remains to be quantified, the proposed strategy can be naturally extended to more phenomenologically relevant observables and, more generally, to other operations commonly encountered in lattice QCD.

We present a high-precision calculation of the hadronic running of electroweak gauge couplings from first principles. Employing lattice QCD in the low-energy regime, we achieve permille precision for virtualities $Q^2 \lesssim 12\;\mathrm{GeV}^2$. At $Q^2 \simeq 1\;\mathrm{GeV}^2$, our determination deviates by up to $7\sigma$ from estimates based on $e^+e^-$ measurements. Combining lattice QCD with perturbative QCD via the Euclidean split technique, we obtain for the electromagnetic coupling $\Delta\alpha^{(5)}_{\mathrm{had}}(M_Z^2) = 0.027821(34)_{\mathrm{lat}}(35)_{\mathrm{pQCD}}$, which is more than twice as precise as recent phenomenological determinations. We assess improvement scenarios by which the precision target for next-generation electroweak measurements could be reached.

Nucleon Mellin moments of parton distribution are computed up to the fourth order in lattice QCD. The computation is performed using one ensemble of twisted mass fermions at the physical pion mass point. We employ boosted frames to access the higher-order Mellin moments of generalized parton distributions. We also extract the forward-limit Mellin moments $\langle x^{n-1}\rangle$ for $n=2,3,4$. These Mellin moments are used to construct unpolarized parton distribution functions and compare to phenomenological extractions.

Trace estimation is central in many lattice QCD computations, but the accuracy of the standard, stochastic Hutchinson method improves only with the square root of the sample size, making precise results expensive. We investigate two complementary variance reduction strategies. First, multigrid multilevel Monte Carlo uses a multigrid hierarchy to construct an unbiased multilevel estimator via recursive coarse grid corrections available from the multigrid hierarchy of the solver. Second, stochastic probing uses distance-$d$ graph colorings; we propose a torus based coloring that requires substantially fewer colors than hierarchical probing at the same distance. We test these approaches on two representative problems: the connected pseudoscalar correlator and disconnected fermion loops. For the connected pseudoscalar two-point function, the multilevel decomposition yields a variance reduction of up to $\mathcal{O}(10^5)$ at large time separations and translates into a clear cost reduction at fixed accuracy, thus confirming earlier results of arXiv:2412.06347. For the disconnected loops, in contrast, the multilevel decomposition provides only moderate gains, whereas probing combined with dilution delivers a substantial cost reduction that improves as the number of probing vectors is increased. Overall, the results highlight a pronounced complementarity: deflation schemes are most effective for observables dominated by long distance propagation, while probing is most effective for localized quantities.

Standard routes to a light composite Higgs either rely on tuning a single channel near criticality or protect a pseudo-Nambu--Goldstone coordinate of a coset. We introduce a third mechanism class in which the protected object is an \emph{eigenvalue} of the renormalized multi-operator scalar kernel of the strong sector. Two basis-invariant diagnostics, a sector participation number $\Psec$ and a sector gap ratio $\Rcoll$, identify collective lightness, but they cannot distinguish an accidental small determinant from a protected zero mode; the missing discriminator is the microscopic sensitivity $\Delta_g=\left|\partial\ln|m_H^2|/\partial\ln g\right|$. A protected collective Higgs is defined by $\Rcoll\gg1$, $\Psec>1$, and $\Delta_g=\mathcal{O}(1)$. We prove that this class is nonempty. A rank-one TC--DTC locking invariant forbids tree-level aligned curvature, while universal vectorlike DQCD bridge fermions, massless in the microscopic Lagrangian but acquiring a common DQCD constituent mass, obey $\partial_h^2\sum_A\Tr\mathcal{M}_A^2\big|_0=0$. The aligned scalar is therefore lifted only at joint two-spurion order, $m^2_{\mathrm{br}}=-(d_X/2\pi^2)g_T^2g_D^2(f_B^4/f_H^2)L_X$, giving $\Delta_{g_T}=\Delta_{g_D}=2$ at leading logarithmic order. The same DTC topology admits a collective top completion and a vector-decoupling route to reducing the positive technicolor contribution to $S$. The mechanism is falsifiable by sector-restricted lattice spectroscopy and coupling-response scans.

Quasi-PDFs provide a connection between Euclidean spatial correlations in lattice QCD and lightcone parton distributions. Their perturbative expressions contain both the infrared divergence required for matching and the scheme-dependent contributions associated with the renormalization prescriptions. The separation of these two ingredients is not always transparent. In this work we use a transverse-momentum cutoff as a simple setting in which these ingredients can be systematically decomposed into a scheme-dependent sector and a remainder for the nonsinglet quark quasi-PDF at one loop. We choose the minimal transverse-momentum-cutoff scheme, where the scheme-dependent sector is identified by its explicit cutoff dependence, while the remainder contains the full collinear infrared divergence and the finite contribution relevant for matching to the lightcone PDF. After expressing the quasi-PDF in terms of distributions, we show how to deal with the linear divergence and the logarithmic terms in the counterterm, and discuss the dependence of the renormalization-group behavior on the renormalization prescriptions. This organization clarifies how scheme dependence enters the quasi-PDF before the final matching is performed, and provides a benchmark for examining analogous separations in other renormalization schemes.

We investigate the spectroscopic properties of the double strange hybrid baryon with quark content $ssqg$ within the framework of QCD sum rule. Using an interpolating current with explicit gluonic degrees of freedom, the two-point correlation function is analyzed in terms of two independent Lorentz structures, $\slashed{q}$ and $I$. The operator product expansion is carried out by including vacuum condensates up to dimension ten, and the corresponding sum rules are derived for both structures. By taking the average of the results obtained from the two Lorentz structures, we extract the masses and pole residues of the ground and first excited states. For the ground state, we obtain a mass of $\widetilde{M } = (1593.44\pm 130.29)~ \mathrm{MeV}$ and a residue of $\widetilde{\lambda } = (2.57\pm 0.40) \times10^{-3} ~\mathrm{GeV}^5$. For the first excited state, the corresponding values are $M = (1897.47\pm 124.44)~ \mathrm{MeV}$ and $\lambda = (2.88\pm 0.62) \times10^{-3} ~\mathrm{GeV}^5$. The obtained results provide theoretical predictions for the double strange hybrid baryon spectrum and may be useful for future experimental searches as well as further nonperturbative studies of hybrid hadrons.

We calculate the magnetic dipole moments of doubly strange hidden-charm pentaquark states with spin-parity $J^P = 1/2^-$ and $3/2^-$ using QCD light-cone sum rules (LCSR), presenting the first systematic LCSR investigation of the electromagnetic multipole structure in the $S = -2$ sector. To assess the model dependence, we employ independent interpolating currents in diquark-diquark-antiquark form, which probe different assumptions about the internal color-spin correlations. For the spin-$3/2$ states, we also compute the electric quadrupole and magnetic octupole moments. The magnetic dipole moments exhibit a considerable spread across currents, ranging from $-2.15\,\mu_N$ to $5.74\,\mu_N$ for spin-$1/2$ pentaquarks and from $-4.25\,\mu_N$ to $-0.43\,\mu_N$ for spin-$3/2$ states, reflecting the sensitivity of magnetic moments to the internal wave function. A quark-level decomposition reveals that the charm quark dominates in most configurations, while strange quarks play a decisive role only in currents favoring axial-vector diquark structures. The electric quadrupole moments lie between $-2.01\times10^{-2}$ fm$^2$ and $5.55\times10^{-2}$ fm$^2$, and the magnetic octupole moments are typically an order of magnitude smaller. The current dependence of the magnetic dipole moments provides a quantitative measure of the theoretical uncertainty arising from the choice of interpolating operator. The pronounced isospin sensitivity of $J^3_\mu(x)$ across all three multipole moments arises from its axial-vector diquark structure, which isolates the light quark from spin averaging and allows the charge asymmetry $e_u/e_d = -2$ to propagate directly into the electromagnetic moments; the ratio $\mu_u/\mu_d = -2.00$ confirms this mechanism exactly. Our predictions offer benchmarks for future experiments and lattice QCD calculations, and may help discriminate among competing structural models.

The equation of state (EoS) of strongly interacting matter at low temperature and high baryon density is a central ingredient in the physics of compact stars, but it is still difficult to determine directly from first-principles QCD calculations because of the sign problem. Two-colour QCD (QC$_2$D) provides a useful and controllable laboratory: for an even number of fundamental flavours the fermion determinant is real and positive, while the theory shares the nonperturbative properties with three-colour QCD at least zero chemical-potential regime. In this short review we summarize recent lattice progress on dense QC$_2$D, with emphasis on the phase structure, the emergence of superfluidity, the Bose--Einstein-condensation (BEC) to Bardeen--Cooper--Schrieffer(BCS) crossover, and thermodynamic quantities. A particularly interesting outcome is that the sound velocity in the cold dense superfluid regime can exceed the conformal value $c_{\rm s}^2/c^2=1/3$. This behaviour, now seen in the simulations for several QCD-like theories, gives a stiff strongly interacting matter. It is expected to offer insight into at least some aspects of the physics realized inside neutron stars.

We introduce a dynamical lattice regulator for Euclidean quantum field theories on a fixed hypercubic graph $\Lambda\simeq\mathbb{Z}^d$, in which the embedding $x:\Lambda\to\mathbb{R}^d$ is promoted to a dynamical field and integrated over subject to shape regularity constraints. The total action is local on $\Lambda$, gauge invariant, and depends on $x$ only through Euclidean invariants built from edge vectors (local metrics, volumes, etc.), hence the partition function is exactly covariant under the global special Euclidean group SE(d) at any lattice spacing. The intended symmetry restoring mechanism is not rigid global zero modes but short-range local twisting of the embedding that mixes local orientations. Our universality discussion is conditioned on a short-range geometry hypothesis (SR): after quotienting the global SE(d) modes, connected correlators of local geometric observables have correlation length O(1) in lattice units. We prove Osterwalder-Schrader reflection positivity for the coupled system with embedding $x$ and generic gauge and matter fields $(U,\Phi)$ in finite volume by treating $x$ as an additional multiplet of scalar fields on $\Lambda$. Assuming (SR), integrating out $x$ at fixed cutoff yields a local Symanzik effective action in which geometry fluctuations generate only SO(d)-invariant irrelevant operators and finite renormalizations. For example, in d=4 we recover the standard one-loop $\beta$-function in a scalar $\phi^4$ test theory. Finally, we describe a practical local Monte Carlo update and report d=2 proof-of-concept simulations showing O(1)-scale geometry correlations, a direct SO(2)-connection diagnostic of short-range local twisting, and evidence for reduced axis-vs-diagonal cutoff artefacts relative to a fixed lattice at matched bare parameters.

We investigate a non-Abelian SU$(2)$ quantum link model in $2+1$ dimensions on a hexagonal lattice using tensor network methods. We determine the static quark potential for a wide range of bare coupling values and find that the theory is confining. We also probe the existence of a Lüscher term and find a clear signal with a $g^2$ dependent coefficient, in qualitative agreement with a strong coupling expansion. Correspondingly, the width of the strings scales logarithmically with the string length again for all $g^2$-values, providing evidence for a rough string, with no indication for a roughening transition.

For twenty years, a persistent discrepancy between experimental measurements and theoretical calculations of the muon anomalous magnetic moment have provided tantalising hints of new physics. In recent years, improvements to the experimental precision have appeared to make the tension stronger and stronger. However, at the same time, our lattice calculation overturned the theoretical consensus, completely eliminating the tension. I will present the latest results from the Budapest-Marseille-Wuppertal (BMW) and DMZ collaborations, with a hybrid determination of the hadronic vacuum polarisation contribution to a precision of 0.45%

We develop efficient algorithms for evaluating higher-order terms in the hopping-parameter expansion of $\textrm{Tr}\ln M$ on $SU(N_\textrm{c})$ gauge configurations. The resulting algorithms, which exploit a trie data structure for the computation of high-order terms, evaluate the $\kappa^8$, $\kappa^{10}$, and $\kappa^{12}$ terms at computational costs of approximately $20$, $460$, and $8900$ times that of a single staple evaluation, respectively. The correctness of the algorithms is verified by comparison with a computationally expensive but reliable reference calculation. We emphasize that collaboration between human researchers and AI coding agents was essential to the development of these algorithms.

Gauge theories in (1+1)D have attracted renewed attention partially due to their experimental realizations in quantum simulation platforms. In this work, we revisit the truncated lattice massive Schwinger model and the truncated lattice Abelian-Higgs model in (1+1)D, where to facilitate quantum simulation, the electric field eigenvalues are truncated to a finite subset while preserving the exact gauge and global symmetries. We uncover previously overlooked universal features in these models, including the emergence of a supersymmetric quantum critical point when the Maxwell term's coefficient changes sign. Our primary focus is the truncated lattice Schwinger model at $\theta=0$, a model not equivalent to familiar spin models. We find that upon reversing the sign of the Maxwell term, the second-order charge conjugation symmetry breaking transition (or confinement-deconfinement transition in a sense) can become first-order. Furthermore, the two types of transitions are connected by a supersymmetric critical point in the tricritical Ising universality class. In the case of truncated Abelian-Higgs model at $\theta=0$, which we find to be equivalent to the quantum Blume-Capel model, the very existence of a symmetry-breaking phase requires a negative-sign Maxwell term. Similarly, there is a tricritical Ising point separating first-order and second-order phase transitions.

Dynamical mean-field theory (DMFT) is one of the most standard theoretical frameworks for addressing strongly correlated electron systems. In this study, we explore a holographic renormalization-group (RG)-like structure inherent in DMFT, which we refer to as "AdS/DMFT", by focusing on the background Bethe-lattice network behind DMFT for electrons with a semicircle density of states. We formulate an RG transformation for the branch Green's function from the outer edge to the interior of the background Bethe-lattice network, and then find that its fixed point can be interpreted as a self-consistent solution of Green's function in DMFT. Moreover, we clarify that the scaling dimensions for the branch Green's function and the boundary correlation functions of electrons at the outer edge of the Bethe-lattice network are characterized by the fixed-point Green's function, analogous to the behavior of a scalar field in an effective two-dimensional anti-de Sitter space. We also perform DMFT computations for the Bethe-lattice Hubbard model, which illustrate that the scaling dimensions capture the Mott transition in the deep interior.

The profile of the pion valence quark distribution function (DF) remains controversial. Working from the concepts of QCD effective charges and generalised parton distributions, we show that since the pion elastic electromagnetic form factor is well approximated by a monopole, then, at large light-front momentum fraction, the pion valence quark DF is a convex function described by a large-$x$ power law that is practically consistent with expectations based on quantum chromodynamics.

We present the complete one-loop matching of the baryon- and lepton-number-conserving low-energy effective field theory (LEFT) to the QCD gradient flow. Using Euclidean conventions and the background-field formulation of the gradient flow, we derive the short-flow-time expansion for the full LEFT operator basis up to mass dimension six. The matching is performed in dimensional regularization in the algebraically consistent 't Hooft-Veltman scheme, including a systematic treatment of evanescent operators and the finite counterterms required to restore chiral symmetry in the spurion sense. Keeping fully generic flavor structures, we verify the cancellation of spurious chiral-symmetry-violating terms with the known finite symmetry-restoring counterterms. This demonstrates that the gradient flow as a gauge-invariant ultraviolet regulator enables an efficient extraction of both divergent and finite counterterms in addition to the matching contributions. We provide the matching coefficients both before and after field redefinitions that remove redundant operators, as well as power-divergent mixings into lower-dimensional operators. Our results establish a consistent perturbative link between continuum LEFT calculations and gradient-flow-based lattice-QCD matrix elements, enabling precision low-energy phenomenology beyond leading-logarithmic accuracy.