The study of neutral $D$-meson mixing provides a unique probe of long-distance effects in the charm sector, where Standard Model contributions are dominated by nonperturbative effects. In this work, we investigate the feasibility of using spectral reconstruction techniques within lattice QCD to compute the long-distance contributions to $D^0- \bar{D}^0$ mixing. After outlining the general formalism describing neutral meson mixing in the charm sector, we focus on the determination of the mixing amplitudes and the dimensionless parameters $x = \Delta m_D / \Gamma_D$ and $y = \Delta \Gamma_D /(2 \Gamma_D)$, which respectively encode the mass and width differences between the $D$-meson mass eigenstates. We discuss in detail the required theoretical and computational framework, including the definition and renormalization of the four-quark operators entering the $\Delta C = 1$ weak Hamiltonian, and strategies for evaluating the relevant correlation functions employing variance-reduction techniques. To extract the mixing amplitudes, we explore methods for reconstructing the spectral density from lattice correlators, providing preliminary assessments of the data quality required to reach the scaling regime, where the smearing width is small enough to yield physically meaningful results. Our findings lay the groundwork for future precision determinations of long-distance contributions to $D$-meson mixing from first principles.

We present our progress on calculating leading-order QED corrections to meson masses and bare quark masses. As lattice QCD calculations become more precise, these QED corrections are becoming more important. However, one of the challenges in adding QED effects to QCD calculations is avoiding power-law suppressed finite-volume effects. By using the recently introduced infinite-volume reconstruction method for QED, we are able to avoid this problem and perform calculations with exponentially-suppressed finite-volume effects.

We present the results of our lattice QCD computation of the hadronic matrix elements relevant to the $h_{c}\to \eta_{c}\gamma$ and $h_{b}\to \eta_{b}\gamma$ decays by using the gauge configurations produced by the Extended Twisted Mass Collaboration with $N_{f}=2+1+1$ dynamical Wilson-Clover twisted mass fermions at five different lattice spacings with physical dynamical $u$ , $d$, $s$ and $c$ quark masses (except for the the coarsest lattice for which the lightest sea quark corresponds to a pion with $m_{\pi}\simeq 175~\mathrm{MeV}$). While the hadronic matrix element for $h_{c}\to \eta_{c}\gamma$ is obtained directly, the one relevant to $h_{b}\to\eta_{b}\gamma$ is reached by working with heavy quark masses $m^{(n)}_{H} = \lambda^{n-1} m_{c}$, with $\lambda \sim 1.24$ and $n=1,2, \ldots ,6$, and then extrapolated to $m_{b}$ by several judicious ans\"atze. In the continuum limit we obtain $\Gamma( h_{c}\to \eta_{c} \gamma ) = 0.604(24)~\mathrm{MeV}$, which is by a factor of $2.3$ more accurate than the previous lattice estimates, and in good agreement with the experimental measurement. In the $b$-quark case we obtain $\Gamma( h_{b}\to \eta_{b} \gamma) =46.0(4.8)~\mathrm{keV}$.

We present the first lattice QCD determination of coupled $DD_s^*$ and $D^*D_s$ scattering amplitudes in the $J^{P}=1^{+}$ channel and elastic $DD_s$ scattering amplitude in the $J^{P}=0^{+}$ channel.The aim is to investigate whether tetraquarks with flavor $cc\bar u\bar s$ exist in the region near threshold. Lattice QCD ensembles from the CLS consortium with $m_{\pi} \sim 280$ MeV, $a\sim0.09$ fm and $L/a = 24, 32$ are utilized. Finite-volume spectra are determined via variational analysis of two-point correlation matrices, computed using large bases of operators resembling bilocal two-meson structures within the distillation framework. The scattering matrix for partial wave $l=0$ is determined using lattice eigenenergies from multiple inertial frames following L\"uscher's formalism as well as following the solutions of Lippmann-Schwinger Equation in the finite-volume on a plane-wave basis. We observe small nonzero energy shifts in the simulated spectra from the noninteracting scenario in both the channels studied, which points to rather weak nontrivial interactions between the mesons involved. Despite the nonzero energy shifts, the lattice-extracted $S$-wave amplitudes do not carry signatures of any hadron pole features in the physical amplitudes in the energy region near the threshold.