Hadron Resonances and Coupled-Channel Scattering

An unstable hadron is defined by a pole of the analytically continued scattering amplitude, together with the pole’s Riemann sheet and residues into every coupled channel. A real-axis bump, a Breit–Wigner fit parameter, or a finite-volume energy level may provide evidence for that pole, but none is the definition. This page develops the two-body coupled-channel construction and the checks needed before a resonance claim is meaningful.

Required background. Hadron quantum numbers and the QCD spectrum supplies the channel labels; resonance poles, widths, and unstable particles supplies the analytic $S$-matrix picture. Helpful background. Analyticity and crossing of amplitudes explains the origin of cuts and continuation domains.

Partial waves, channels, and unitarity

Fix all exact quantum numbers—total angular momentum and parity, conserved flavor charges, and any other channel labels—and choose two-body channels $i,j$ with thresholds $s_i^{\rm th}$. In one useful normalization,

$$S(s)=\mathbf 1+2i\,\rho^{1/2}(s)t(s)\rho^{1/2}(s), \qquad \rho_{ij}(s)=\delta_{ij}\frac{2k_i(s)}{\sqrt{s}}\,\theta(s-s_i^{\rm th}),$$

where $t_{ij}$ is a dimensionless partial-wave amplitude and $k_i$ is the center-of-mass momentum. Other conventions move factors of $2$, $8\pi$, or $\sqrt{s}$ between $t$ and $\rho$; pole positions are unchanged, while residues must always be quoted with the convention.

On the physical real axis, two-body unitarity $S^\dagger S=\mathbf 1$ gives

$$t-t^\dagger=2i,t^\dagger\rho t, \qquad \operatorname{Im}t^{-1}=-\rho.$$

The second identity follows by multiplying the first relation by $t^{\dagger-1}$ and $t^{-1}$. It isolates the universal right-hand-cut imaginary part. Defining a real symmetric matrix $K$ by $K^{-1}=\operatorname{Re}t^{-1}$ in the elastic region yields

$$t(s)=\bigl[K^{-1}(s)-i\rho(s)\bigr]^{-1}.$$

For one open channel, $S=e^{2i\delta}$ and hence

$$t(s)=\frac{1}{\rho(s)\,[\cot\delta(s)-i]}.$$

A phase shift passing through $\pi/2$ can signal resonance-like behavior, but a nearby threshold, a background zero, or strong coupling to another channel can prevent that simple pattern. The invariant definition therefore lies in the continued amplitude Briceño, Dudek, and Young 2018, § II.A, pp. 3–4.

Sheets, poles, and channel residues

Each channel momentum contains a square root,

$$k_i(s)=\frac{\lambda^{1/2}(s,m_{i1}^2,m_{i2}^2)}{2\sqrt{s}}, \qquad \lambda(x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz,$$

so every two-body threshold introduces a branch point. The physical sheet is fixed by the usual $+i0$ prescription; continuing across channel $i$’s right-hand cut reverses the sign choice of $k_i$. With $N$ channels there are $2^N$ local sign combinations. A pole report must therefore specify both $s_\star$ and the sign convention for its sheet.

Near an isolated simple pole, time-reversal-invariant coupled-channel amplitudes factorize as

$$t_{ij}(s)=\frac{g_i^{(\star)}g_j^{(\star)}}{s_\star-s}+t_{ij}^{\rm reg}(s).$$

The complex residues $g_i^{(\star)}g_j^{(\star)}$ encode how the pole couples to the chosen channels in the stated normalization. For a pole near the physical region one often writes

$$\sqrt{s_\star}=M_{\rm pole}-\frac{i}{2}\Gamma_{\rm pole},$$

but interpreting $-2\operatorname{Im}\sqrt{s_\star}$ as a total width requires an isolated pole and a sensible branch of the square root. Bound-state poles lie on the physical sheet below threshold; virtual-state and resonance poles occupy different unphysical sheets. Coupled channels can produce several nearby poles associated with one observed enhancement, so “the mass and width” are not sufficient identifiers Briceño, Dudek, and Young 2018, § II.B, pp. 4–5.

A representative two-channel analysis

Suppose channels $1$ and $2$ carry the same exact quantum numbers. A minimal real-axis parameterization might be

$$K_{ij}(s)=\frac{c_i c_j}{m_0^2-s}+\gamma_{ij}, \qquad \gamma_{ij}=\gamma_{ji}\in\mathbb R,$$

with real parameters over the fitted energy interval. In the normalization above, $K$ and $\gamma$ are dimensionless and $c_i$ has mass dimension one. The analysis proceeds as follows:

1. Specify channels and domain. Include every two-body channel whose threshold and coupling can matter in the energy window; state where omitted three-body or left-hand singularities are assumed negligible.
2. Fit real-axis information. Constrain $K$ using phase shifts, inelasticities, production data with a consistent final-state interaction model, or finite-volume spectra.
3. Continue the same amplitude. Replace the physical-sheet momenta by the chosen sheet signs in $\rho(s)$ and solve $$\det\!\bigl[K^{-1}(s)-i\rho^{(\text{sheet})}(s)\bigr]=0.$$
4. Extract residues. Expand the inverse matrix about each zero. For a simple pole, the residue matrix must be rank one up to numerical and parameterization uncertainty.
5. Vary admissible forms. Repeat with additional smooth $K$-matrix terms, alternative left-hand-cut treatments, fit windows, and channel content. Pole stability is more probative than stability of $m_0$.

The parameter $m_0$ is a bare parameter of this chosen $K$ matrix, not the physical pole mass. Indeed, even in one channel the pole solves $m_0^2-s-i c^2\rho(s)=0$ only after continuation; the solution is shifted and generally complex.

Why a Breit–Wigner line shape can fail

For one narrow, isolated resonance with slowly varying phase space and background, the pole term can reduce locally to

$$t(s)\simeq\frac{g^2}{M^2-s-iM\Gamma(s)}.$$

This is an approximation on or near the real axis. It becomes unreliable when a threshold lies nearby, more than one channel is important, the background varies rapidly, or poles and zeros interfere. A measured production intensity has the form $|\mathcal A_i(s)|^2$ times phase space; its numerator and interference phases depend on the production process. Consequently, two processes may display different line shapes while sharing the same final-state pole.

The defensible reporting hierarchy is therefore:

Reported object	What it establishes	What must accompany it
Event enhancement	excess intensity in a specified process	resolution, acceptance, background, and channel definition
Phase shift or inelasticity	physical-axis scattering information	amplitude normalization and partial-wave convention
Breit–Wigner parameters	parameters of a stated real-axis model	fit window, background, thresholds, and model variation
Pole and residues	process-independent singularity of the amplitude and its channel couplings	sheet, amplitude convention, analytic continuation, and uncertainty provenance
Finite-volume energies	discrete QCD spectrum in a box	volume, irreducible representation, operator basis, and quantization condition

Finite-volume spectra as amplitude constraints

Finite volume replaces the scattering continuum by discrete energies. In a common schematic convention, those energies obey a condition of the form

$$\det\!\left[K^{-1}(E)+F(E,\mathbf P,L)\right]=0,$$

where the known geometric matrix $F$ mixes partial waves allowed by the finite-volume symmetry. The precise signs and kinematic factors depend on how $K$ and $F$ are defined. Several volumes, total momenta, and irreducible representations constrain a parameterization of $K$; only after fitting and analytic continuation does one obtain a pole Briceño, Dudek, and Young 2018, §§ IV.A–IV.C, pp. 16–25.

This inverse problem is performed in detail on scattering amplitudes and resonance poles from finite-volume spectra. The handoff must include the finite-volume energy covariance, volumes and boosts, channel masses, irreducible representations, partial-wave truncation, and the family of amplitude forms used.

Independent checks and failure modes

• Unitarity: on every fitted physical-axis point above threshold, verify $\operatorname{Im}t^{-1}=-\rho$ in the declared normalization. A generic sum of complex poles does not guarantee this identity.
• Real analyticity: away from cuts, the continued amplitude should obey $t(s^*)=t(s)^*$ with the corresponding sheet mapping. Failure usually signals an inconsistent branch choice.
• Decoupling: when off-diagonal couplings vanish, the matrix solution must reduce to independent single-channel amplitudes.
• Residue structure: a simple pole produces a rank-one residue matrix. A higher-rank numerical residue indicates unresolved poles, a defective fit, or an extraction error.
• Dimensions: with dimensionless $t$ and $\rho$, $K$ is dimensionless and the pole residue in $s$ has dimension mass squared. Translate residues before comparing papers using other normalizations.
• Model dependence: continuation amplifies incomplete real-axis information. Vary channels, left-hand-cut approximations, truncations, fit windows, and parameterizations rather than quoting only the statistical covariance of one model.
• Domain: the two-body relation above is insufficient once important three-body channels open, and long-range forces require modified analytic and finite-volume treatments.

Common pitfalls

Equating a fitted $K$-matrix pole with a resonance. The $K$-matrix parameter is representation dependent. Continue the full $t$ matrix and quote its pole and residues instead.

Naming a sheet only as “second.” In a multichannel problem that label is ambiguous. State which channel momenta have changed sign relative to the physical sheet.

Assigning branching fractions from residue magnitudes alone. Near thresholds and for broad states, phase space and analytic continuation prevent a universal probability interpretation. Report convention-defined residues and any derived partial-width approximation separately.

Informal self-check

For a one-channel amplitude $t^{-1}(s)=K^{-1}(s)-i\rho(s)$ with real $K$ on the physical axis, show that it satisfies elastic unitarity and identify the continuation needed to search for a resonance pole.

Answer

Because $K^{-1}$ is real, $t^{-1}-t^{\dagger-1}=-2i\rho$. Multiplying on the left by $t^\dagger$ and on the right by $t$ gives $t-t^\dagger=2it^\dagger\rho t$, the elastic unitarity relation. The physical right-hand cut comes from the square root in $k(s)$ and hence in $\rho(s)$. Searching for a resonance requires crossing that cut, equivalently reversing the channel-momentum sign in the analytically continued $\rho$, and solving $K^{-1}-i\rho^{(\mathrm{unphysical})}=0$. The sheet convention must be stated with the solution.

Handoffs

• Send pole-free stable external states and a specified local current to the form-factor route.
• Send finite-volume energy levels, covariance, channel basis, and truncations to the finite-volume amplitude method.
• Send short-distance production coefficients and partonic initial-state information to Perturbative QCD and Partons, keeping the final-state pole definition fixed.

References

• Briceño, Raúl A., Jozef J. Dudek, and Ross D. Young. “Scattering Processes and Resonances from Lattice QCD.” Reviews of Modern Physics 90 (2018): 025001. DOI · Open PDF