The ππ scattering amplitude at large Nc

A bstract We study the scaling of meson-meson scattering amplitudes with the number of colors, N c . We use lattice calculations in a theory with N f = 4 degenerate flavors, with N c = 3 – 6 and pion mass M π ≈ 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 $$ {T}_{cs0}^0(2900) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>cs</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mfenced> <mml:mn>2900</mml:mn> </mml:mfenced> </mml:math> , $$ {T}_{c\overline{s}0}^{++}(2900) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mover> <mml:mi>s</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mn>2900</mml:mn> </mml:mfenced> </mml:math> , $$ {T}_{c\overline{s}0}^0(2900) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mover> <mml:mi>s</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mfenced> <mml:mn>2900</mml:mn> </mml:mfenced> </mml:math> and $$ {T}_{cs1}^0(2900) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>cs</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mfenced> <mml:mn>2900</mml:mn> </mml:mfenced> </mml:math> 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 c scaling and are sensitive to subleading N c corrections. In addition, we constrain the scaling of different combinations of low-energy constants from matching to large N c ChPT. The results for the channel corresponding to a $$ \left({\pi}^{+}{D}_s^{+}-{K}^{+}{D}^{+}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> state show evidence of a virtual bound state with energy E virtual = 1.63(10) M π for N c = 3, while this pole disappears at N c &gt; 3. This may be connected to the exotic states found in experiment.