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A quantitative description of hadrons and nuclei in Quantum Chromodynamics (QCD) requires control over multi-body scattering amplitudes. Over the past two decades, lattice QCD combined with finite-volume formalisms has made two-body scattering a precision tool for hadron and nuclear spectroscopy. Many states of current interest, however, couple strongly to three-particle channels -- including light mesons, baryon excitations, and states relevant for nuclear and flavor physics. As lattice QCD calculations approach the physical pion mass and resolve increasingly excited spectra, incorporating three-body dynamics are no longer optional.

Finite-volume quantization conditions now provide a nonperturbative connection between the discrete spectra computed in lattice QCD and intermediate infinite-volume quantities, while integral equations reconstruct the physical scattering amplitudes from these quantities. This methodology, colloquially called the L\"uscher framework~\cite{Briceno:2017max}, is illustrated in Fig.~\ref{fig:methodology}. In this contribution, I summarize this broader program and focus on a remaining step that is essential for phenomenology: the construction of symmetrized three-body amplitudes. This step becomes particularly important when extending the formalism beyond the lowest partial waves, where multiple angular-momentum components and their interference must be treated consistently. I highlight recent lattice QCD applications. For further discussion on three-body dynamics from QCD, see the contributions by S.~Dawid and M.~Mai in this collection.

%
%
%
\begin{center}
\textbf{From lattice QCD spectra to three-body amplitudes}
\end{center}

The basic observables in lattice QCD are Euclidean correlation functions,
\begin{align}
  \Cc_{ab}(t) = \braket{\Oc_a(t)\,\Oc_b^\dag(0)} \, ,
\end{align}
where $\Oc_a$ and $\Oc_b$ are interpolating operators chosen to create the states of interest. In a finite cubic volume of extent $L^3$, the spectrum is discrete, and the correlator admits the spectral decomposition
\begin{align}
  \Cc_{ab}(t) = \sum_n Z_a^{(n)} Z_b^{(n)\,*}\, e^{-E_n t} \, ,
\end{align}
with $Z_a^{(n)} \equiv \bra{\mathrm{vac}}\!\Oc_a(0)\!\ket{n}$. The finite-volume energies $E_n$ are determined stochastically from gauge-field ensembles by evaluating the required Wick contractions and analyzing matrices of correlators built from large operator bases.

These energies are not themselves resonance observables. Resonances are encoded in infinite-volume scattering amplitudes, while lattice QCD is formulated in Euclidean spacetime and in a finite box, where asymptotic scattering states are absent. The essential theoretical development is therefore the finite-volume mapping between the discrete spectrum and the infinite-volume amplitude. For two-body systems, this relation is provided by the L\"uscher formalism and its extensions~\cite{Luscher:1986n2,Luscher:1991n1,Briceno:2014oea}. For three-body systems, the corresponding quantization conditions relate the finite-volume energies to an intermediate three-body $K$ matrix,
\begin{align}
	\label{eq:QC3_clean}
	\det\!\Big[ F_3^{-1}(E_n,L\,|\,\Kc_2) + \Kc_3(E_n) \Big] = 0 \, .
\end{align}
Here $F_3$ is a known finite-volume geometric quantity that depends on the box size and on the two-body dynamics encoded in $\Kc_2$, while $\Kc_3$ parameterizes short-range three-body interactions~\cite{Hansen:2014eka,Hansen:2015zga,Mai:2017bge,Jackura:2022gib}. The determinant is over spectator momenta and two-body partial waves. Although the explicit representation depends on the conventions used to derive the quantization condition, all consistent formulations are constrained by $S$ matrix unitarity.

This is the natural three-body generalization of the two-body program: lattice QCD determines a finite-volume spectrum, the quantization condition constrains an intermediate $K$ matrix, and the physical amplitude is then reconstructed in infinite volume. What makes the three-body case qualitatively richer is the increase in kinematic complexity. The amplitudes depend not only on the total energy but also on the momenta of the initial and final spectators, together with the quantum numbers of the interacting pair. As a result, both the finite-volume analysis and the infinite-volume reconstruction require a much more elaborate treatment than in two-body scattering.

%
\begin{figure}[t!]
	\centering
	\includegraphics[width=0.9\textwidth]{methodology.pdf}
	\caption{Work flow for extracting the spectrum and couplings of hadrons from QCD. Lattice QCD gauge configurations are used to determine finite-volume spectra and matrix elements from multi-hadron correlation functions. Infinite-volume K-matrices are then constrained using non-perturbative finite-volume mappings. Finally, the spectral content is extracted from the reconstructed amplitudes.}
	\label{fig:methodology}
\end{figure}
%

\begin{center}
\textbf{Amplitude reconstruction and symmetrization}
\end{center}

Once $\Kc_3$ has been constrained from the finite-volume spectrum, it can be continued to continuous kinematics and projected to definite quantum numbers. In the partial-wave representation used in Refs.~\cite{Jackura:2023qtp,Briceno:2024ehy}, the three-body $K$ matrix becomes a function $\Kc_{3,pk}(E)$ of the total energy and the magnitudes of the initial and final spectator momenta. Near threshold, analyticity implies that $\Kc_3$ admits a multivariate expansion in these variables, and a convenient, model-agnostic parameterization can be written in factorized form, $\Kc_{3,pk}(E) = h_p(E)\,\widetilde{\Kc}_3(E)\,h_k(E)$, where the reduced matrix $\widetilde{\Kc}_3(E)$ captures the energy dependence and the functions $h_p$, $h_k$ encode spectator-momentum dependence. This provides a flexible representation suitable for fitting lattice spectra while maintaining the analytic structure expected near threshold~\cite{Briceno:2024ehy}.

The physical $3\to3$ amplitude is obtained by solving a coupled set of integral equations that implements $S$ matrix unitarity and resums rescattering effects~\cite{Hansen:2015zga,Mai:2017vot, Jackura:2018xnx,Jackura:2022gib}. However, the quantization condition Eq.~\eqref{eq:QC3_clean} and integral equations are defined in the pair-spectator basis, where the $K$ matrices and amplitudes are asymmetric objects $\Mc^{(j',j)}$ called , where $j$ and $j'$ label the initial- and final-state spectator choices. These amplitudes incorporate both short-range dynamics through $\Kc_3$ and long-range effects associated with one-particle exchange (OPE). The latter plays a central role in shaping the analytic structure of the amplitude, as discussed in Ref.~\cite{Dawid:2023jrj} and within these proceedings. In particular, the relativistic partial-wave projection of the OPE kernel yields logarithmic dependence of the form $\Gc_{pk}(E)\propto \log(\zeta_{pk})$~\cite{Jackura:2023qtp}, giving rise to a nontrivial network of branch points and an infinite set of Riemann sheets. These singularities encode rescattering effects and are expected to be especially important in systems where triangle mechanisms and exchange dynamics are prominent~\cite{Jackura:2018xnx}.

These amplitudes depend on the arbitrary choice of spectator and therefore do not yet correspond directly to physical observables. The final step is to restore the full particle-exchange symmetry of the system. This is straightforward in principle,
\begin{align}
  \Mc = \sum_{j,j'} \Mc^{(j',j)} \, ,
\end{align}
but becomes nontrivial in a partial-wave basis, where amplitudes defined in different pair-spectator channels must be related by recoupling transformations. The full workflow for symmetrizing relativistic three-body partial-wave amplitudes was discussed in Ref.~\cite{Jackura:2025wbw}, with illustrative applications to $3\pi$ systems. The resulting symmetrized amplitudes of definite quantum numbers can be written schematically as
\begin{align}
  \Mc^{J^P} = \sum_{j,j'} \Rc_{j'}\, \Mc^{(j',j)\,J^P}\, \Rc_j \, ,
\end{align}
where the recoupling coefficients $\Rc_j$ are fixed by kinematics and symmetry. In the angular-momentum sector they reduce to Wigner rotations, while with flavor symmetry they also incorporate isospin recoupling~\cite{Jackura:2025wbw}. This construction is what converts the amplitudes produced by the integral equations into physical amplitudes that can be interpreted and compared across channels.

The importance of this step grows rapidly beyond the lowest partial waves. In realistic applications with interfering resonances, several partial waves can contribute at once, and only the symmetrized amplitude has the correct transformation properties and interference pattern needed for a physical interpretation. In that sense, symmetrization is not merely a technical refinement; it is a necessary part of turning the finite-volume three-body program into a practical tool for hadron spectroscopy.

\begin{center}
\textbf{Applications and outlook}
\end{center}

\begin{figure*}[t]
	\centering
	\includegraphics[width=\textwidth]{dalitz_2_1p.pdf}
	\caption{Dalitz plots of an intensity distribution for the isotensor $\pi\pi\pi$ amplitude determined from lattice QCD for $m_\pi \approx 400\mathrm{MeV}$ in Ref.~\cite{Briceno:2025yuq}. Dark bands are associated with the $\rho$ resonance. The theoretical framework developed to compute such plots was derived in Ref.~\cite{Jackura:2025wbw}.}
	\label{fig:dalitz}
\end{figure*}


A useful observable built from the symmetrized amplitude is an intensity distribution,
\begin{align}
\Ic^{I(J^P)} \sim \sum_{\lambda,\lambda'}
\left|\Mc^{I(J^P)}_{\lambda'\lambda}\right|^2 \, ,
\end{align}
which provides a direct analogue of the Dalitz-plot distributions used in experimental analyses. Such quantities make it possible to visualize resonance bands, kinematic singularities, and interference among different partial waves in a way that is closely connected to phenomenology.

This framework has now been applied to isotensor three-pion scattering. In Ref.~\cite{Briceno:2025yuq}, the Hadron Spectrum Collaboration completed the first lattice-QCD determination of an elastic $3\pi$ amplitude in this channel for lattices corresponding to $m_\pi \approx 400\mathrm{MeV}$. The symmetrization formalism of Ref.~\cite{Jackura:2025wbw} made it possible to construct Dalitz-plot-like intensity distributions from the resulting amplitudes. Shown in Fig.~\ref{fig:dalitz} are the results of Ref.~\cite{Briceno:2025yuq}, Dalitz distributions for the $J^P = 1^+$ channel for three-body energies $E/m_\pi = 3.3, 3.4, 3.5$, where the dominant waves are given by the $[\pi\pi]_{I=1}\pi$ system in $S$ and $D$ wave, and the $[\pi\pi]_{I=2}\pi$ system in $P$ wave. These distributions exhibit clear imprints of the underlying $\pi\pi$ subchannel dynamics, namely the dark bands highlight the dominant $\rho$ resonance in $[\pi\pi]_{I=1}$. The same study also illustrates the role of higher partial waves, for example in the $J^P=1^+$ channel where $\rho\pi$ in $S$ and $D$ waves and $[\pi\pi]_{I=2}\pi$ in $P$ wave all contribute nontrivially.

Going forward, the central ingredients of the three-body program are well-established. Lattice QCD supplies finite-volume spectra, quantization conditions constrain intermediate three-body $K$ matrices, integral equations reconstruct the infinite-volume amplitudes, and symmetrization restores the physical observables. The next challenges are to extend these methods to more complicated systems, especially those involving spin, coupled channels, and resonant substructures in multiple two-body subsystems. Meeting those challenges is essential for a first-principles description of the many hadronic resonances that live in genuinely three-body environments.

Three-body dynamics have become a central pillar of modern lattice-QCD spectroscopy. The field has progressed from establishing formal quantization conditions, to constructing amplitudes, and now to extracting physical observables with increasing precision. As these analyses continue to mature, studies of bound and resonant systems will enable a more systematic exploration of the rich and intricate structure inherent in the three-body problem.

\begin{acknowledgments}
I thank the organizers for hosting the workshop ``Universality in strongly interacting systems: from QCD to atoms'' at ECT$^*$ in Trento, Italy. This work contributes to the goals of the USDOE ExoHad Topical Collaboration, contract DE-SC0023598.
\end{acknowledgments}

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