Three-Body Quantization and Decay Amplitudes

Three-particle levels constrain a three-body amplitude only through a declared formalism that resolves a spectator, interacting two-body subchannels, symmetrization, and a genuine three-body kernel. In the relativistic field-theory construction used here, the finite-volume determinant contains the physical two-body $K$ matrix and a divergence-free three-body quantity $\mathcal K_{\mathrm{df},3}$; the latter is regulator and scheme dependent and becomes a physical $3\to3$ amplitude only after solving infinite-volume integral equations. A spectrum formula alone does not supply a decay-amplitude normalization.

Required background. Coupled-Channel Quantization and Inference supplies determinant, threshold, identifiability, and sheet discipline.

Helpful background. Three-Body Renormalization and Universality supplies infinite-volume three-body dynamics. Complete Lattice Error Budgets supplies the multistage uncertainty contract.

One explicit relativistic three-body branch

To make every statement testable, adopt the baseline Hansen–Sharpe branch:

• three identical spin-zero particles of mass $m$ in a periodic cubic box;
• a $\mathbb Z_2$ symmetry separating odd- and even-particle sectors, so $2\leftrightarrow3$ transitions are absent;
• total center-of-momentum energy in a stated interval below the first omitted higher-particle threshold;
• no two-body $K$-matrix pole in the original baseline interval, with a smooth cutoff that restricts spectator kinematics; and
• all partial waves and spectator momenta retained up to declared cutoffs.

Later extensions relax several of these assumptions. They are not silently folded into this baseline; their present domains and equivalences belong to the Lattice and Hamiltonian Field Theory Research area. In particular, the coupled two- and three-particle spectrum relation is constructed in Briceño, Hansen, and Sharpe 2017, §§ II–V, but using that extension requires declaring its enlarged particle-number and two-body-pole hypotheses.

Three-body formalism boundary. The determinant below is the stated identical-scalar, particle-number-parity branch with a declared spectator regulator and truncation; its divergence-free kernel is scheme dependent until the associated infinite-volume integral equations are solved.

The detailed choices are:

Field	Choice used on this page
Spectator	$\mathbf k=2\pi\mathbf n/L$ labels the on-shell spectator; the nonspectator pair is expanded in partial waves $\ell m$ in its own center-of-momentum frame
Symmetry	Identical-boson symmetrization is included; the baseline has a $\mathbb Z_2$ particle-number parity and no $2\leftrightarrow3$ mixing
Two-body input	The same physical $\mathcal K_2$ and its subthreshold continuation are used in the pair subchannel and all validation limits
Three-body input	$\mathcal K_{\mathrm{df},3}$ is divergence free only by a declared subtraction and cutoff scheme; it is not called the scattering amplitude
Quantization sign	$\det[F_3^{-1}+\mathcal K_{\mathrm{df},3}]=0$ defines the sign and normalization of $F_3$ on this page
Observable	A physical $\mathcal M_3$ is obtained only after the matching integral equations; a decay bridge additionally requires a current-specific finite-volume residue relation

With compound indices $(\mathbf k;\ell m)$, the structural condition is

$$\det_{\mathbf k\ell m} \left[ F_3(E,\mathbf P,L)^{-1} +\mathcal K_{\mathrm{df},3}(E^*) \right]=0.$$

$F_3$ is not a purely geometric zeta function. It contains finite-volume sums, the two-body $K$ matrix for the nonspectator pair, exchange between spectator choices, cutoff functions, and symmetrization. The determinant therefore resums repeated two-body scattering as well as genuine three-body interactions. The relativistic condition and its original hypotheses are derived by Hansen and Sharpe 2014, §§ II–VI.

Why a divergence-free kernel appears

A connected $3\to3$ amplitude contains physical singularities from a particle spectating while the other two scatter, followed by a change of spectator. Schematically,

$$\mathcal M_3 =\mathcal D[\mathcal M_2] +\mathcal M_{3,\mathrm{df}},$$

where $\mathcal D$ is a known sequence of pairwise scatterings with on-shell exchange singularities. Subtracting $\mathcal D$ defines a smoother divergence-free object, but the split depends on the cutoff and subtraction prescription. The formalism parameterizes that short-distance information by $\mathcal K_{\mathrm{df},3}$.

The physical amplitude is recovered from coupled integral equations of the form

$$\mathcal M_3 =\mathcal D[\mathcal M_2] +\mathcal L\, \mathcal T[\mathcal K_{\mathrm{df},3}]\, \mathcal R,$$

where $\mathcal L$ and $\mathcal R$ attach the pairwise rescattering and $\mathcal T$ solves an infinite-volume kernel equation. The symbols are structural because their measures and cutoff factors are formalism-specific. The essential check is exact: changing the allowed cutoff or subtraction shifts $\mathcal K_{\mathrm{df},3}$ and the intermediate functions, but a consistently matched $\mathcal M_3$ and predicted spectrum must remain invariant. Hansen and Sharpe give the amplitude relation and demonstrate this logical separation in Hansen and Sharpe 2015, §§ II–V.

Calling a fitted constant $\mathcal K_{\mathrm{df},3}^{\rm iso}$ a “three-body force” can be useful within one isotropic truncation, but it is not a scheme-independent observable. Only its effect on the spectrum and the matched $\mathcal M_3$ can be compared invariantly.

Free and weak-coupling benchmarks

For three noninteracting particles, the exact levels are

$$E^{(0)}_{\mathbf n_1\mathbf n_2\mathbf n_3} =\sum_{i=1}^3 \sqrt{m^2+ \left(\frac{2\pi\mathbf n_i}{L}\right)^2}, \qquad \mathbf n_1+\mathbf n_2+\mathbf n_3=\mathbf d.$$

After symmetrizing momentum triples and projecting the little-group irrep, the quantization implementation must reproduce these energies and multiplicities as the interactions are removed. This checks spectator enumeration, double-counting factors, boosts into the pair frame, and permutation symmetry.

At weak coupling, first set $\mathcal K_{\mathrm{df},3}=0$ while retaining a small two-body scattering length. The leading shift of the threshold level must agree with the independent large-$L$ expansion driven by pairwise interactions. Only then turn on a three-body kernel and verify the first order at which it can contribute. The original condition was checked against multiple nontrivial orders of the threshold expansion (Hansen and Sharpe 2014, § VII).

A three-body bound state supplies another branch. Its finite-volume shift is exponential in a binding or breakup momentum and has spectator-dependent image structure. It is not obtained by inserting the bound-state energy into an elastic two-body zeta function. A controlled check reproduces a known nonrelativistic or synthetic bound-state limit before interpreting an interacting QFT level.

From levels to a physical amplitude

A reproducible inference has two nested inverse problems:

1. fit $\mathcal K_2$ to two-particle levels in the same masses, volumes, frames, and conventions;
2. hold its covariance and parametrization alternatives while fitting $\mathcal K_{\mathrm{df},3}$ to three-particle levels;
3. vary spectator cutoffs, pair partial waves, and the three-body kernel basis;
4. solve the infinite-volume integral equations for $\mathcal M_3$ in every outer resample; and
5. predict held-out three-particle levels and a free, threshold, bound-state, or synthetic amplitude benchmark.

The two-body input is not exact auxiliary data. Its uncertainty and analytic continuation into subthreshold pair energies can dominate the inferred three-body quantity. Shared two- and three-particle correlators also make their covariance nonzero.

For a decay or current-induced $1\to3$ amplitude, the spectrum determinant is only the strong final-state input. One additionally needs a finite-volume matrix-element relation with the residue of the three-body quantization condition, a current-renormalization convention, and an infinite-volume transition-amplitude map. Applying the one-to-two Lellouch–Lüscher factor to a three-particle level is not a controlled approximation.

Adversarial failure. Two cutoff functions yield very different fitted $\mathcal K_{\mathrm{df},3}$ constants but identical spectra. Reporting their spread as a physical systematic is wrong if the matching integral equations have not been solved. After consistent matching, $\mathcal M_3$ should agree; failure of that cancellation diagnoses implementation or truncation error.

The branch-status map below makes the three-body claim ceiling explicit. Its scheme-dependent intermediate kernel, subchannel treatment, regulator, integral-equation map, and benchmarks must close on their own; two-body success supplies input, not certification.

Finite-volume branches have different claim ceilings. A fixed massless-field prescription and an exact power-law test are durable; mutable prescription comparisons may move to Research. Coupled-channel, current-insertion, and three-particle claims require separate dated status and validation. The map is schematic and not to scale.

Observable-level validation

Before accepting a three-body quantity, verify that:

• the declared formalism’s particle content, $2\leftrightarrow3$ rule, two-body pole treatment, energy domain, spectator cutoff, and partial-wave truncation all apply;
• free energies, symmetrized multiplicities, and irrep projections are exact;
• the same $\mathcal K_2$ reproduces the two-body spectrum and is propagated with covariance into the three-body fit;
• a weak-coupling, threshold, bound-state, or synthetic benchmark is reproduced before physical inference;
• cutoff and subtraction changes move $\mathcal K_{\mathrm{df},3}$ but cancel in held-out spectra and the matched $\mathcal M_3$; and
• a decay claim includes a separate current residue, normalization, and renormalization check rather than borrowing the two-body factor.

What you can now do

You can now (1) identify the spectator, pair subchannel, symmetrization, $\mathcal K_2$, $F_3$, and every scheme-dependent object in one relativistic three-body condition, and (2) demand a free, weak-coupling, threshold, bound-state, or synthetic closure plus scheme cancellation before inferring a physical three-body amplitude.

General three-body dynamics and universality remain with Gauge Theories and the Standard Model, and infinite-volume amplitude analyticity remains with Scattering. Framework equivalence and current application status belong to the Lattice and Hamiltonian Field Theory Research area.

Exercises

1. Scheme dependence. Two schemes give $\mathcal K_{\mathrm{df},3}^{(A)}\ne \mathcal K_{\mathrm{df},3}^{(B)}$. What equality should be tested?

Solution

After using each kernel with its own $F_3$, cutoff, subtraction, and matching integral equations, the predicted finite-volume energies and physical $\mathcal M_3$ must agree within truncation and numerical uncertainty. Equality of the intermediate kernels is neither required nor generally expected.

2. Count a free level. At $\mathbf P=0$, take the momentum triple $(\mathbf n,-\mathbf n,\mathbf0)$ with $\mathbf n\ne0$. What is its free energy?

Solution

$E^{(0)}=2\sqrt{m^2+(2\pi\mathbf n/L)^2}+m$. Permutations do not produce new states before symmetrization; the momentum orbit must then be decomposed into cubic irreps with the correct multiplicity.

References

• Briceño, Raúl A., Maxwell T. Hansen, and Stephen R. Sharpe. “Relating the Finite-Volume Spectrum and the Two-and-Three-Particle $S$ Matrix for Relativistic Systems of Identical Scalar Particles.” Physical Review D 95 (2017): 074510. DOI. Open PDF.
• Hansen, Maxwell T., and Stephen R. Sharpe. “A Relativistic, Model-Independent, Three-Particle Quantization Condition.” Physical Review D 90 (2014): 116003. DOI. Open PDF.
• Hansen, Maxwell T., and Stephen R. Sharpe. “Expressing the Three-Particle Finite-Volume Spectrum in Terms of the Three-to-Three Scattering Amplitude.” Physical Review D 92 (2015): 114509. DOI. Open PDF.