A lattice study of $\pi \pi$ scattering at large $N_{\text{c}}$

Abstract

We present the first lattice study of pion-pion scattering with varying number of colors, $N_{\text{c}}$. We use lattice simulations with four degenerate quark flavors, $N_{\text{f}}=4$, and $N_{\text{c}}=3-6$. We focus on two scattering channels that do not involve vacuum diagrams. These correspond to two irreducible representations of the SU(4) flavor group: the fully symmetric one, $SS$, and the fully antisymmetric one, $AA$. The former is a repulsive channel equivalent to the isospin-2 channel of SU(2). By contrast, the latter is attractive and only exists for $N_{\text{f}}\ge 4$. A representative state is $\left(|D_s^{+}{\pi }^{+}⟩-|D^{+}{K}^{+}⟩\right)/\sqrt{2}$. Using Lüscher's formalism, we extract the near-threshold scattering amplitude and we match our results to Chiral Perturbation Theory (ChPT) at large $N_{\text{c}}$. For this, we compute the analytical U$(N_{\text{f}})$ ChPT prediction for two-pion scattering, and use the lattice results to constrain the $N_{\text{c}}$ scaling of the relevant low-energy couplings.

Comments: 40 pages, 15 figures, 8 tables. v2: references added and minor corrections. Matches published version