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

Abstract

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 Lüscher'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.

arXiv comment66 pages including 3 appendices, 20 figures, 18 tables. Full list of operators included as ancillary file. v2: References added, minor corrections that do not affect the results. v3: Matches published version

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