Three-particle scattering amplitudes from lattice QCD

TL;DR

This work surveys the progress in extracting three-hadron scattering amplitudes from lattice QCD through the three-particle quantization condition (QC3). It outlines the formalism, including the relation between finite-volume spectra and infinite-volume amplitudes via $\mathcal K_{\rm df,3}$ and the divergence-free framework, and describes end-to-end applications that determine $\mathcal M_3$ from lattice data across systems such as $3\pi$, $\pi\pi K$, $DD\pi$, and $N\pi\pi$, as well as exploratory studies of three-neutron interactions. Key results include the first physical-mass three-particle amplitude for pions and kaons (with $\mathcal K_{\rm df,3}$ consistent with zero for $3\pi^+$ and nonzero for $3K^+$), initial resonance determinations for $\pi(1300)$ at heavier masses, and novel approaches to treat left-hand cuts in $DD^*$ via a $DD\pi$ framework. The paper highlights remaining challenges, such as reducing model dependence in $\mathcal K_{\rm df,3}$, cross-validating different formalisms, and extending to more complex multi-particle channels, while outlining a practical path toward combining LQCD results with EFT and phenomenology to achieve quantitative predictions for resonances and many-body interactions.

Abstract

I review recent progress in calculating scattering amplitudes and resonance properties involving three particles using results from lattice QCD. The necessary input is the finite-volume spectrum, and the outputs -- via solutions of integral equations -- are scattering amplitudes that can be continued into the complex plane to search for resonance poles. I describe the outlook for future extensions and applications of this work.

Figures (7)

• Figure 1: Schematic form of the integral equations for $\mathcal{D}$ and $\mathcal{M}_{\rm df,3}$. Notation is explained in the text.
• Figure 2: Comparison of the leading two terms in $\mathcal{K}_{\rm df,3}$ for $3\pi^+$ scattering with ChPT (from Ref. Dawid:2025doq); "ETMC" refers to Ref. Fischer:2020jzp). NLO ChPT bands are obtained using standard values and errors of low-energy coefficients, see Ref. Dawid:2025doq. The lowest $M_\pi$ value corresponds to physical quark masses.
• Figure 3: Energy dependence of $\mathcal{M}_3$ with $J^P=0^-$ with incoming and outgoing particles in the equilateral configuration (from Ref. Dawid:2025doq). $E$ is the CM frame energy, $E_{\rm thr}$ is the CM energy at threshold. Left panel: comparing different processes at physical quark masses. Right panel: comparing $\mathcal{M}_{3\pi}$ at different quark masses with the NLO ChPT prediction.
• Figure 4: Results from Ref. Yan:2025mdm for the resonance pole in the $I=1$, $J^P=0^-$ channel. Those at $M_\pi=305\;$MeV are obtained from the three-particle formalism alone, while those at the physical pion mass involve model-based chiral extrapolations. The region in which the three-particle formalism is valid is indicated by the magenta arrow. Shaded areas correspond to $1\sigma$ regions from different fits; the AIC bars are from an average over fits using the Akaike information criterion.
• Figure 5: Left panel: $u$-channel pion exchange contribution to $DD^*$ scattering, showing three-particle cut. Right panel: sketch of how $D D^*$ scattering and the $u$-channel exchange emerge in a $DD\pi$ amplitude.