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

TL;DR

This study uses lattice simulations to analyze pion-pion scattering across different numbers of colors, providing insights into the large $N_c$ limit and constraining low-energy couplings in Chiral Perturbation Theory.

Contribution

It is the first lattice investigation of $mbda$ scattering with varying $N_c$, connecting lattice results to large $N_c$ ChPT predictions.

Findings

Identified the nature of two scattering channels at different $N_c$ values.

Extracted scattering amplitudes near threshold using L"uscher's formalism.

Constrained the $N_c$ scaling of low-energy couplings in ChPT.

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 $N_\text{f} \geq 4$. A representative state is $\left( |D_s^+ \pi^+\rangle - |D^+ K^+\rangle\right)/\sqrt{2}$. Using L\"uscher'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.