We review perturbative matching between the regularization-invariant symmetric MOM (RI/SMOM) and MS schemes at the symmetric subtraction point, which is relevant for lattice QCD simulations. For bilinear operators we summarize three-loop conversion factors for local quark currents and for the n=2,3 twist-two moments of structure functions. For three-quark operators we summarize two-loop RI/SMOM matching for N=0 baryonic operators and three-loop anomalous dimensions together with two-loop matching for the N=1 Mellin moment, enabling improved lattice studies of baryon distribution amplitudes. All numerical results quoted here are given in Landau gauge.

The Standard Model prediction for the rare $K_L\to\mu^+\mu^-$ decay depends critically on the long-distance contribution coming from the exchange of two photons. Such a contribution can be computed using lattice QCD and an effective three-flavor theory including only the $u$, $d$ and $s$ quarks, provided terms falling as the inverse square of the omitted charm quark mass, $1/m_c^2$ are neglected. Because of the missing Glashow-Iliopoulos-Maiani cancelation, this three-flavor theory contains additional low-energy constants that depend on $m_c$. Here we show how these constants can be determined from a practical four-flavor lattice QCD calculation performed on a small volume with $u$ and $d$ quark masses that are heavier than physical.

Operator equations (OEs) underpin quantitative modeling across science and engineering. Finite-element (FE) methods discretize continuous OEs into finite-dimensional algebraic systems, whereas tensor networks (TNs) provide flexible variational representations of correlated discrete systems. Here, we develop a framework that connects FE with TN for analytic OEs. The power of this method comes from its ability to convert highly non-linear partial differential equations into linear matrix equations. In particular, we show that FE discretization induces a hierarchy of multilinear interaction tensors, through which differential, integral, nonlinear, memory, and delay equations can be expressed within a common algebraic structure. The resulting systems are reformulated as weighted-residual optimization problems over TN degrees of freedom. Matrix-product-state calculations for one-dimensional linear and nonlinear diffusion reproduce conventional solutions with controlled error while preserving continuity and Neumann boundary conditions. The framework provides a common variational language for analytic OEs and establishes a direct connection between FE numerical formalism and TN variational algorithms, offering a general foundation for TN-based and quantum-inspired approaches to solving OEs.

We introduce half dualization as a general principle to construct non-invertible duality defects. The construction performs a duality transformation only in a subregion of spacetime, leaving an interface between the original theory and its dual. A non-invertible duality defect can be hosted on this interface. Half dualization is applicable whenever a local duality transformation is available, and does not depend on the spacetime dimension. We demonstrate the construction procedure in the (2+1)-dimensional Villainized charge-$n$ XY model on a cubic lattice. Half dualization yields a non-invertible particle-vortex duality defect whose fusion with its orientation reverse produces a (1+1)-dimensional $\mathbb{Z}_n$ gauge theory on the fusion surface. We then apply half dualization to the $\mathbb{Z}_n$-gauged XY model relevant to the 3D XY$^\ast$ transition. In the gauged model, the half dualization interface hosts a gauge covariant duality wall, rather than a genuine gauge invariant duality defect. It flows to an invertible duality defect in infrared when the $\mathbb{Z}_n$ gauge theory is confined or Higgsed.

We present a comprehensive theoretical assessment of the masses of possible baryonic configurations characterized by the presence of two heavy top quarks, including $\Xi_{ttu}$, $\Xi_{ttd}$, $\Omega_{tts}$, $\Omega_{ttc}$, and $\Omega_{ttb}$ systems. This analysis is rigorously executed within the specialized framework of two-point $\mathrm{QCD}$ sum rules, focusing on their predicted ground state masses. Our interest in these systems arises from recent CMS and ATLAS reports indicating a pseudoscalar excess close to $t\bar{t}$ threshold. Our evaluation incorporates both perturbative terms and nonperturbative effects, including condensate contributions up to dimension eight. Based on our results, the extracted central masses for all channels are slightly above the sum of the constituent quark masses, which is consistent with the inherent uncertainties of the method. These quantitative predictions provide a useful first-principle theoretical reference, which may help future experimental searches for such heavy configurations at the LHC and inform sensitivity studies at next-generation facilities such as the FCC.

We present a comprehensive theoretical study of the masses of possible baryonic and mesonic configurations containing a single top quark. Our analysis includes the baryons $\Lambda_t$, $\Xi_t$, $\Sigma_t$, $\Xi'_t$, $\Omega_t$, $\Omega_{tcc}$, and $\Omega_{tbb}$, together with the pseudoscalar and vector mesons $T_{t\bar n}^{\mathrm{Ps}}$, $T_{t\bar n}^{\mathrm{V}}$, $T_{t\bar s}^{\mathrm{Ps}}$, $T_{t\bar s}^{\mathrm{V}}$, $T_{t\bar c}^{\mathrm{Ps}}$, $T_{t\bar c}^{\mathrm{V}}$, $T_{t\bar b}^{\mathrm{Ps}}$, and $T_{t\bar b}^{\mathrm{V}}$. Motivated in part by recent experimental indications of a pseudoscalar enhancement near the $t\bar t$ threshold reported by the CMS and ATLAS collaborations, this study is carried out within the framework of two-point QCD sum rules to determine the corresponding ground-state masses by including perturbative contributions and nonperturbative condensates up to dimension eight. For several channels, including the $\Lambda_t$, $\Xi_t$, $\Sigma_t$, $T_{t\bar b}^{\mathrm{Ps}}$, and $T_{t\bar b}^{\mathrm{V}}$ states, the extracted central masses lie slightly below the corresponding sums of constituent quark masses, which may indicate nontrivial binding dynamics or near-threshold multiquark configurations within the uncertainties of the method. Moreover, when the full theoretical uncertainties are taken into account in a conservative manner, a larger subset of the investigated states exhibits a consistent tendency toward weak binding behavior, suggesting that the possibility of loosely bound configurations cannot be excluded for most of the considered baryonic and mesonic channels. These results provide useful first-principles theoretical benchmarks for possible top-containing hadronic systems, which may support future searches at the LHC, along with sensitivity analyses for next-generation facilities such as the FCC.