We study the decay mode of the $a_1(1260)$ into a ${\pi }^{+}$ in $p$ wave and the $f_0(980)$ that decays into ${\pi }^{+}{\pi }^{-}$ in $s$ wave. The mechanism proceeds via a triangular mechanism where the $a_1(1260)$ decays into $K^{*}\overline{K}$, the $K^{*}$ decays to an external ${\pi }^{+}$ and an internal $K$ that fuses with the $\overline{K}$ producing the $f_0(980)$ resonance. The mechanism develops a singularity at a mass of the $a_1(1260)$ around 1420 MeV, producing a peak in the cross section of the $\pi p$ reaction, used to generate the mesonic final state, which provides a natural explanation of all the features observed in the COMPASS experiment, where a peak observed at this energy is tentatively associated to a new resonance called $a_1(1420)$. On the other hand, the triangular singularity studied here gives rise to a remarkable feature, where a peak is seen for a certain decay channel of a resonance at an energy about 200 MeV higher than its nominal mass.

We present an approach that allows one to obtain information on the compositeness of molecular states from combined information of the scattering length of the hadronic components, the effective range, and the binding energy. We consider explicitly the range of the interaction in the formalism and show it to be extremely important to improve on the formula of Weinberg obtained in the limit of very small binding and zero range interaction. The method allows obtaining good information also in cases where the binding is not small. We explicitly apply it to the case of the deuteron and the $$D^{*}_{s0}(2317)$$Ds0∗(2317) and $$D^{*}_{s1}(2460)$$Ds1∗(2460) states and determine simultaneously the value of the compositeness within a certain range, as well as get qualitative information on the range of the interaction.

We extend the theoretical framework used to describe the $T_{cc}$ state as a molecular state of $D^{*}D$ and make predictions for the $D^{*}{D}^{*}$ and $D_s^{*}{D}^{*}$ systems, finding that they lead to bound states only in the $J^P=1^{+}$ channel. Using input needed to describe the $T_{cc}$ state, basically one parameter to regularize the loops of the Bethe-Salpeter equation, we find bound states with bindings of the order of MeV and similar widths for the $D^{*}{D}^{*}$ system, while the $D_s^{*}{D}^{*}$ system develops a strong cusp around the threshold.

We study the interaction of the doubly bottom systems $BB$, $B^{*}B$, $B_{s}B$, $B_s^{*}B$, $B^{*}{B}^{*}$, $B^{*}{B}_s$, $B^{*}{B}_s^{*}$, $B_s{B}_s$, $B_s{B}_s^{*}$, $B_s^{*}{B}_s^{*}$ by means of vector meson exchange with Lagrangians from an extension of the local hidden gauge approach. The full s-wave scattering matrix is obtained implementing unitarity in coupled channels by means of the Bethe-Salpeter equation. We find poles below the channel thresholds for the attractively interacting channels $B^{*}B$ in $I=0$, $B_s^{*}B-B^{*}{B}_s$ in $I=\frac{1}{2}$, $B^{*}{B}^{*}$ in $I=0$, and $B_s^{*}{B}^{*}$ in $I=\frac{1}{2}$, all of them with $J^P=1^{+}$. For these cases the widths are evaluated identifying the dominant source of imaginary part. We find binding energies of the order of 10–20 MeV, and the widths vary much from one system to the other: of the order of 10–100 eV for the $B^{*}B$ system and $B_s^{*}B-B^{*}{B}_s$, about 6 MeV for the $B^{*}{B}^{*}$ system and of the order of 0.5 MeV for the $B_s^{*}{B}^{*}$ system.

We study the molecular probability of the $X(3872)$ in the $D^0{\overline{D}}^{*0}$ and $D^{+}{D}^{*-}$ channels in several scenarios. One of them assumes that the state is purely due to a genuine nonmolecular component. However, it gets unavoidably dressed by the meson components to the point that in the limit of zero binding of the $D^0{\overline{D}}^{*0}$ component becomes purely molecular. Yet, the small but finite binding allows for a nonmolecular state when the bare mass of the genuine state approaches the $D^0{\overline{D}}^{*0}$ threshold, but, in this case the system develops a small scattering length and a huge effective range for this channel in flagrant disagreement with present values of these magnitudes. Next we discuss the possibility to have hybrid states stemming from the combined effect of a genuine state and a reasonable direct interaction between the meson components, where we find cases in which the scattering length and effective range are still compatible with data, but even then the molecular probability is as big as 95%. Finally, we perform the calculations when the binding stems purely from the direct interaction between the meson-meson components. In summary we conclude, that while present data definitely rule out the possibility of a dominant nonmolecular component, the precise value of the molecular probability requires a more precise determination of the scattering length and effective range of the $D^0{\overline{D}}^{*0}$ channel, as well as the measurement of these magnitudes for the $D^{+}{D}^{*-}$ channel which have not been determined experimentally so far.

We perform a fit to the LHCb data on the $$T_{cc}(3875)$$T cc ( 3875 ) state in order to determine its nature. We use a general framework that allows to have the $$D^0 D^{*+},$$D 0 D ∗ + , $$D^+ D^{*0}$$D + D ∗ 0 components forming a molecular state, as well as a possible nonmolecular state or contributions from missing coupled channels. From the fits to the data we conclude that the state observed is clearly of molecular nature from the $$D^0 D^{*+},$$D 0 D ∗ + , $$D^+ D^{*0}$$D + D ∗ 0 components and the possible contribution of a nonmolecular state or missing channels is smaller than 3%, compatible with zero. We also determine that the state has isospin $$I=0$$I = 0 with a minor isospin breaking from the different masses of the channels involved, and the probabilities of the $$D^0 D^{*+},$$D 0 D ∗ + , $$D^+ D^{*0}$$D + D ∗ 0 channels are of the order of 69% and 29% with uncertainties of 1%. The differences between these probabilities should not be interpreted as a measure of the isospin violation. Due to the short range of the strong interaction where the isospin is manifested, the isospin nature is provided by the couplings of the state found to the $$D^0 D^{*+},$$D 0 D ∗ + , $$D^+ D^{*0}$$D + D ∗ 0 components, and our results for these couplings indicate that we have an $$I=0$$I = 0 state with a very small isospin breaking. We also find that the potential obtained provides a repulsive interaction in $$I=1,$$I = 1 , preventing the formation of an $$I=1$$I = 1 state, in agreement with what is observed in the experiment.