The $\pi\pi$ scattering amplitude at large $N_\text{c}$

We study the scaling of meson-meson scattering amplitudes with the number of colors, $N_\text{c}$. We use lattice calculations in a theory with $N_\text{f}=4$ degenerate flavors, with $N_\text{c}=3-6$ and pion mass $M_\pi\approx 560$ MeV. We focus on three different scattering channels, two of which have the same quantum numbers as some tetraquark candidates recently found at LHCb: the $T_{cs0}^0(2900)$, $T_{c\bar{s}0}^{++}(2900)$, $T_{c\bar{s}0}^0(2900)$ and $T_{c\bar{s}1}^0(2900)$ states. Finite-volume energies are extracted using a large set of operators, containing two-particle operators with the form of two pions or two vector mesons, and local tetraquark operators. The resulting energy spectra is used to constrain the infinite-volume scattering amplitude by means of L\"uscher's quantization condition. We consider polynomial parametrizations of the phase shift, as well as one-loop chiral perturbation theory (ChPT) predictions. We find that our lattice results follow the expected $N_\text{c}$ scaling and are sensitive to subleading $N_\text{c}$ corrections. In addition, we constrain the scaling of different combinations of low-energy constants from matching to large $N_\text{c}$ ChPT. The results for the channel corresponding to a $(\pi^+ D^+_s - K^+ D^+)$ state show evidence of a virtual bound state with energy $E_\text{virtual}=1.63(10)M_\pi$ for $N_\text{c}=3$, while this pole disappears at $N_\text{c}>3$. This may be connected to the exotic states found in experiment.