The decay of a charmonium vector meson $c\bar c$ into the $\Lambda\bar\Sigma^0 + \mathrm{c.c.}$ state can be considered purely electromagnetic under the assumption of isospin conservation. The first-order interaction is therefore mediated by a virtual photon, similar to the non-resonant process $e^+e^- \to \Lambda\bar\Sigma^0 + \mathrm{c.c.}$ in the Born approximation. Given the nature of...
We study the renormalization and mixing of the $\Delta F=2$ Four-Fermion Operators (FFO), starting from the problem of the computation of the perturbative running for $N_\mathrm{f}=3$ and showing how this problem is solved through the use of the Poincaré-Dulac theorem. We then illustrate the procedure to evaluate the non-perturbative running from the hadronic scale ($O(\Lambda_{QCD})$) to the...
The inclusive hadronic decays of the $\tau$ lepton offer an alternative method for extracting the CKM matrix elements $V_{\mathrm{ud}}$ and $V_{\mathrm{us}}$. In this talk, I will discuss recent results from the ETM Collaboration on the inclusive hadronic decay rate of the $\tau$, obtained in $N_f=2+1+1$ QCD using the novel HLT method. This approach circumvents the well-known inverse Laplace...
We present our preliminary determinations of the $\Lambda$--parameter of $\mathrm{SU}(N=3,5,8)$ Yang--Mills theories and of its large--$N$ limit in the Twisted Gradient Flow renormalization scheme on the lattice. First, we determine $\Lambda$ in units of a low-energy renormalization scale $\mu_{\mathrm{had}}$ using the step-scaling method. Then, to express $\Lambda$ in units of the...
The fundamental theory of strong interactions, known as Quantum ChromoDynamics (QCD), exhibits rich symmetry properties that underlie the behavior of hadronic matter. In the limit of $N_f$ light quark masses, QCD possesses an approximate chiral symmetry $U(1)_V\otimes U(1)_A\otimes SU(N_f)_V\otimes SU(N_f)_A$. The special unitary part of this symmetry group is spontaneously broken to its...
The (in)famous $X(3872)$ was the first exotic particle discovered in 2003, compatible with a tetraquark interpretation. More than twenty years have passed since then, yet its internal dynamic remain an open question. Currently, the most established models describe the $X(3872)$ either as a compact tetraquark, where quarks interact via color forces, or as a $\bar{D}^0D^{0}$ mesonic molecule,...
