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

Abstract

We present the first lattice study of pion-pion scattering with varying number of colors, N c . We use lattice simulations with four degenerate quark flavors, N f = 4, and N 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 f ≥ 4. A representative state is |D + s π + − |D + K + / √ 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 c . For this, we compute the analytical U(N f ) ChPT prediction for two-pion scattering, and use the lattice results to constrain the N c scaling of the relevant low-energy couplings.

Bootstrapping Pions at Large $N$

“…There are a few lattice studies of mesons in large N QCD (see e.g. [27][28][29][30][31]), but to the best of our knowledge the four-derivative Wilson coefficients have not been determined. The best we can currently do is to compare with real world, see figure 8.…”

“…A very safe assumption, confirmed by lattice studies for increasing values of N , see e.g [27][28][29][30][31]…”