Coupled-channel approach to isotensor $πππ$ scattering from lattice QCD

TL;DR

This study extends finite-volume unitarity methods to a coupled-channel three-body system in QCD by analyzing the $I=2$ $ ext{πππ}$ sector with a $ ho(770)$ sub-channel, using lattice QCD spectra at two unphysical pion masses. The three-body amplitude is constructed in an isobar-spectator framework, with IAM two-body inputs for the $ ext{πρ}$ and $ ext{πG}$ channels, and mapped to infinite volume via a generalized QC; two subtraction schemes are developed to control cutoff dependence. Fits to the finite-volume spectrum reveal a predominantly repulsive three-body interaction driven by a short-range three-body force, with a smaller influence from $ ext{πρ}$ exchange and an even smaller contribution from $ ext{πG}$ exchange. In the infinite-volume limit, the resulting three-body amplitude and production process show moderate regulator sensitivity, and a narrow-$ ho$ analysis yields negative, repulsive $ ext{πρ}$ phase shifts, broadly consistent with leading-order effective Lagrangian predictions. The work demonstrates the feasibility of extracting three-body dynamics from lattice data in coupled-channel settings and highlights the need for more volumes and boosted frames to sharpen the results and chase chiral trajectories.

Abstract

The quest to understand three-body dynamics from first-principle QCD includes the study of non-resonant and resonant systems. The isospin $I=2$ system is of particular interest having no three-body resonance but featuring a resonance in a sub-channel, while also being a coupled-channel problem. In this study, we calculate the finite-volume spectrum from lattice QC at two different pion masses, map the amplitude to the infinite volume through a generalized FVU three-body quantization condition, investigate the limit of a narrow $ρ$, and compare with an effective Lagrangian prediction at leading order. Chiral extrapolations between different pion masses are performed.

Figures (13)

• Figure 1: Kinematical coverage of the considered heavy (left panel) and light pion mass (right panel) finite-volume setups. Notation: $\bm{p}$ -- three-momentum of the spectator; Full horizontal line -- physical two-body threshold of the two-body cluster; Thick vertical lines -- $\pi\pi\pi$ non-interacting levels; Dotted vertical lines -- $\pi\rho$ pseudo-non-interacting levels.
• Figure 2: Cutoff dependence test between method 1 (left panel) and method 2 (right panel). In both figures the ground state level is predicted with no three-body force using a hard cutoff $i_{max}\in\{2,3,4\}$. Solid lines linearly connect the first and second values to guide to the eye.
• Figure 3: Summary of the lattice results for ensembles in \ref{['table:gwu_lattice']}. Left panel: Overview of the finite-volume lattice spectrum with respect to the relevant thresholds and pion mass values. Error bars represent a combination of statistical and systematic error from model averaging. Right panel: Covariance matrix of the pertinent results.
• Figure 4: Example of the level determination through the FVU approach. Top panel: Fit 2 (see \ref{['tab:all-fits']} using method 2 \ref{['eq:QC-breve-method2']}) to the heavy pion mass lattice QCD spectrum (open symbols) for $\sqrt{s}<5\,m_\pi$ and using. Energy eigenvalues correspond to the zeros (vertical orange lines) of $T_{T_{1g}}^{-1}$ (green/blue lines). Thin green/blue lines denote the same quantity but for turned off inter-channel coupling. Lower panel: Prediction of the light-pion mass spectrum using fit 2 parameters. In both plots the thick solid (dashed) vertical lines denote non-interacting levels of the $3\pi(\pi\rho)$ system.
• Figure 5: Residua of the determined finite-volume energy eigenvalue states for the heavy pion mass setup (fit 2). Residua are compared between the $\pi\rho\to\pi\rho$ state of zero momenta $(\pi\rho(0))$ with the $\pi G\to \pi G$ state with back-to-back momenta of magnitude $1$ in lattice units $(\pi G(1))$.