$\pi \pi$ scattering at Large $N_{\text{c}}$

We study the Large $N_{\text{c}}$ scaling of pion-pion scattering amplitudes for $N_{\text{f}}=4$ degenerate quark flavors. We focus on the standard isospin-2 channel and the adjoint-antisymmetric (AA) representation. The latter only exists for $N_{\text{f}}\ge 4$ and a representative state is $\frac{1}{\sqrt{2}}(\mathrm{\mid }{D}_s^{+}{\pi }^{+}⟩-\mathrm{\mid }{D}^{+}{K}^{+}⟩)$. We compare the results obtained for two regularizations (Wilson and twisted-mass fermions) and three values of the lattice spacing, and observe significant discretization effects in the AA channel. Finally, we match our results to NLO SU(4) and NNLO U(4) Chiral Perturbation Theory and constrain the $N_{\text{c}}$ scaling of the relevant low-energy couplings.