Compositeness for Resonance: Structural Diagnostics

• The paper defines resonance compositeness as a quantitative measure of hadronic two-body contributions derived from the scattering amplitude’s pole properties.
• Methodologies include scattering theory, operator-based definitions, and finite-volume normalization to separate explicit two-body channels from missing elementarity.
• Results indicate that near-threshold states afford model-independent compositeness estimates, while broad resonances require prescribed treatments for probabilistic interpretation.

Compositeness for resonance is the attempt to quantify how much of an unstable hadronic state is accounted for by explicit two-body hadronic degrees of freedom, as opposed to “elementary,” bare, compact, or otherwise missing components. In the modern literature, the relevant quantity is usually defined from the resonance pole of the scattering amplitude, from the norm of a two-body wave function, or from equivalent operator constructions, and is complemented by an elementariness parameter $Z$ such that $X+Z=1$ in the corresponding formulation. For bound states this interpretation can be probabilistic, while for resonances it is generically complicated by complex pole positions, open channels, energy-dependent interactions, and model-space dependence (Sekihara et al., 2014, Hyodo, 2015).

1. Formal definition in scattering theory

A standard formulation starts from the pole structure of the two-body scattering amplitude. Near a bound-state or resonance pole at $E_{\text{pole}}$, the partial-wave amplitude behaves as

$$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$

where the residue functions $\gamma_j(q)$ encode the channel couplings and the two-body wave function. In this framework,

$$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$

so the channel compositeness is the norm of the two-body component,

$$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$

which, for separable interactions, reduces to

$$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$

The complementary contribution is the elementariness,

$Z=1-\sum_j X_j,$

and for energy-dependent interactions it can be written as a derivative of the effective interaction with respect to energy (Sekihara et al., 2017).

The same structure appears in relativistic coupled-channel formulations, with $s$ replacing $X+Z=1$0 as the analytic variable. If $X+Z=1$1 near the pole, then

$X+Z=1$2

together with the sum rule

$X+Z=1$3

For resonances, this extension is implemented with the Gamow vector, so $X+Z=1$4 and $X+Z=1$5 are generally complex (Sekihara et al., 2014).

An operator-based definition gives an equivalent interpretation. In Quantum Field Theory, one may define compositeness as the expectation value of the free-particle number operator,

$X+Z=1$6

or, more generally, by summing the relevant asymptotic particle numbers. This makes explicit that compositeness measures the weight of continuum states in the physical state vector (Oller, 2022).

2. Energy dependence, missing channels, and sum rules

A central structural distinction is whether the interaction is energy-independent or energy-dependent. For energy-independent potentials, the total compositeness in the explicit two-body model space is unity: the state is fully represented by the included continuum channels. For energy-dependent interactions, the total compositeness deviates from unity, and the deviation is interpreted as a missing-channel contribution or elementariness. In the separable formalism this appears directly through

$X+Z=1$7

so the energy dependence of $X+Z=1$8 is not a technical detail but part of the structural interpretation (Sekihara et al., 2017).

This feature underlies much of the model dependence discussed in the literature. The division between explicit hadronic channels and implicit degrees of freedom is not unique, so $X+Z=1$9 and $E_{\text{pole}}$0 depend on the adopted model space, the regulator, and the renormalization scheme unless one is in a universal near-threshold regime. Hyodo’s review formulates this succinctly: compositeness is model-dependent in general, but the structure of near-threshold bound states and resonances can be determined model-independently in the weak-binding limit (Hyodo, 2015).

The number-operator approach sharpens this point from a different angle. For any finite-range, energy-independent potential, $E_{\text{pole}}$1 for bound states and likewise for resonances with respect to the asymptotic basis. The departure from $E_{\text{pole}}$2 therefore signals either explicit energy dependence, integrated-out channels, or a different choice of basis rather than a failure of the formalism itself (Oller, 2017).

3. Resonances, complex compositeness, and probabilistic prescriptions

The main conceptual difficulty for resonances is that $E_{\text{pole}}$3 and $E_{\text{pole}}$4 are generally complex. This follows from the pole being located on an unphysical Riemann sheet, from the decay character of the state, and from the bi-orthogonal normalization implicit in the Gamow-state treatment. The immediate probabilistic interpretation available for stable bound states is then lost (Sekihara et al., 2014).

Several prescriptions have been proposed to recover an interpretable quantity when the complex ambiguities are mild. One commonly used construction defines

$E_{\text{pole}}$5

so that $E_{\text{pole}}$6 and $E_{\text{pole}}$7 are real, nonnegative, and sum to unity. The condition $E_{\text{pole}}$8 is then taken as the criterion for a probabilistic reading (Sekihara et al., 2015). A related prescription writes

$E_{\text{pole}}$9

again with an uncertainty parameter $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$0, and again only regards the result as probabilistically meaningful when $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$1 is small (Hyodo, 2015).

A different line of development emphasizes phase freedom in the $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$2-matrix. Because the phases of the residues can be changed by diagonal phase rotations, the modulus $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$3 is invariant under such transformations and has been argued to be the physically relevant quantity for resonances (Oller, 2022). Related work derives a positive compositeness relation from a rank-1 projection operator, provided that the Laurent expansion around the resonance pole converges in a finite region of the physical axis near $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$4; under this analytic condition the transformed coefficients become real and non-negative (Guo et al., 2015).

More recently, a probabilistic decomposition into three quantities,

$$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$5

has been proposed for near-threshold $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$6-wave resonances. Here $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$7 denotes certainly composite content, $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$8 certainly elementary content, and $$T_{L,jk}(E;q',q)=\frac{\gamma_j(q')\gamma_k(q)}{E-E_{\text{pole}}}+\text{regular part},$$9 intrinsically uncertain content. In that scheme, resonances with unphysically large decay widths are excluded from interpretation, and interpretable near-threshold resonances above threshold are found to have small composite fraction (Kinugawa et al., 2024). The coexistence of these prescriptions is itself part of the subject: resonance compositeness is not a single universally agreed probability assignment, but a family of closely related diagnostics with different analytic assumptions.

4. Near-threshold universality and its limits

The most model-independent statements arise for shallow $\gamma_j(q)$0-wave bound states. Weinberg’s weak-binding relation expresses the scattering length $\gamma_j(q)$1 and effective range $\gamma_j(q)$2 in terms of the compositeness $\gamma_j(q)$3: $\gamma_j(q)$4 with $\gamma_j(q)$5. When $\gamma_j(q)$6, the compositeness is determined by observables and is insensitive to ultraviolet model details (Hyodo, 2015).

This universality can be generalized to unstable near-threshold states. For quasi-bound states the scattering length and pole energy become complex, and the generalized relation contains additional correction scales. The formal conclusion is still that sufficiently near-threshold states permit a model-independent structural diagnosis, but $\gamma_j(q)$7 becomes complex and must be interpreted with care (Hyodo, 2015). Hyodo’s review places this result in a broader perspective: near-threshold structure can be model-independently determined, whereas states farther from threshold or involving several coupled channels are more model-dependent (Hyodo, 2015).

At threshold itself, an $\gamma_j(q)$8-wave zero-energy resonance satisfies $\gamma_j(q)$9, reflecting the diverging size of the wave function. This statement does not extend to higher partial waves (Hyodo, 2015). By contrast, the 2024 study of near-threshold $$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$0-wave resonances above threshold concludes, within its new probabilistic scheme, that such resonances have small composite fraction, in sharp contrast to shallow bound states below threshold (Kinugawa et al., 2024). Taken together, these results indicate that “near threshold” does not by itself fix the structural classification; the location relative to threshold and the interpretive scheme both matter.

Charged systems require further modification because Coulomb and short-range interactions coexist. In this setting the Coulomb-modified effective range expansion leads to

$$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$1

with $$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$2 determined by the eigenmomentum and Coulomb functions, and an “interpretable” compositeness $$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$3 can be constructed so that $$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$4 even for bound, virtual, or resonance states (Kinugawa et al., 24 Apr 2026). For repulsive Coulomb plus short-range interactions, the near-threshold state is fully characterized by the Coulomb scattering length, the Coulomb effective range, and the Bohr radius; in this case a shallow bound state turns directly into a resonance, bypassing a virtual state, and when $$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$5 both shallow bound states and near-threshold resonances can exhibit large compositeness as a remnant of low-energy universality (Kinugawa et al., 30 Jul 2025).

5. Extraction from finite volume and from decay observables

Because resonance wave functions are not square-integrable in infinite volume, finite volume provides an alternative route with an explicitly probabilistic normalization. In a periodic box of size $$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$6, the Hamiltonian eigenstates are normalizable, and one defines

$$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$7

with

$$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$8

For resonances, one associates the infinite-volume state with the finite-volume level that remains in the resonance energy region over a window of $$\langle \mathbf q_j|\Psi\rangle=\frac{\gamma_j(q)}{E_{\text{pole}}-\mathcal E_j(q)},$$9, and then averages $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$0 over that window. This yields a real probabilistic compositeness for unstable states (Tsuchida et al., 2017).

Direct observable probes have also been developed. For $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$1, the radiative decay $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$2 is dominated by the $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$3 component because the $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$4 and $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$5 contributions strongly cancel, so a large decay width to $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$6 implies large $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$7 compositeness (Sekihara et al., 2013). For the $$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$8–$$X_j=\int \frac{d^3q}{(2\pi)^3}\langle \tilde\Psi|\mathbf q_j\rangle\langle \mathbf q_j|\Psi\rangle,$$9 system, the mixing intensity $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$0 is proportional to $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$1 and hence correlates with $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$2; the empirical bound $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$3 implies

$$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$4

so the two resonances cannot simultaneously be $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$5 molecular states (Sekihara et al., 2014).

A more general decay-based method uses both the production rate of a resonance and the near-threshold invariant-mass distribution of its constituents. Applied to $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$6, this strategy concludes that the compositeness $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$7 can be extracted with about $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$8 of uncertainty by comparing the resonance-production rate with the $$X_j=-\gamma_j^2\left.\frac{dG_j}{dE}\right|_{E=E_{\text{pole}}}.$$9 mass distribution close to threshold (Molina et al., 2016).

6. Representative results and the case of $Z=1-\sum_j X_j,$0

Applications to specific hadrons illustrate both the usefulness and the limitations of the concept. In a wave-function-based analysis of dynamically generated resonances, the $Z=1-\sum_j X_j,$1 has $Z=1-\sum_j X_j,$2, the higher pole of $Z=1-\sum_j X_j,$3 has $Z=1-\sum_j X_j,$4, and $Z=1-\sum_j X_j,$5 has $Z=1-\sum_j X_j,$6, all supporting dominant molecular components in the corresponding channels (Sekihara et al., 2017). A broader coupled-channel study finds that $Z=1-\sum_j X_j,$7 and $Z=1-\sum_j X_j,$8 are dominated by $Z=1-\sum_j X_j,$9 and $s$0 composite states, whereas $s$1 and $s$2 are elementary (Sekihara et al., 2014).

For baryons, the $s$3 exhibits a non-negligible $s$4 component. In a chiral unitary treatment with a constrained fit, the quoted value is $s$5, while a related probabilistic analysis gives $s$6–$s$7; by contrast, the considered $s$8, $s$9, $X+Z=1$00, and $X+Z=1$01 components of $X+Z=1$02 and $X+Z=1$03 are negligible, with dominant missing-channel contributions (Sekihara et al., 2015, Sekihara et al., 2015). For $X+Z=1$04, the generalized compositeness condition applied to the $X+Z=1$05, $X+Z=1$06, $X+Z=1$07, and $X+Z=1$08 channels gives a total meson-baryon fraction ranging from $X+Z=1$09 to $X+Z=1$10, with average $X+Z=1$11, leaving room for only about $X+Z=1$12 of other components (Aceti et al., 2014).

The $X+Z=1$13 provides a particularly clear example of why resonance compositeness is not automatically model-independent. In the study devoted to this state, the compositeness is evaluated in three coupled-channel unitary models with different channel content and regularization prescriptions. The result is that the inferred channel composition varies significantly with the renormalization scheme and with the number of channels retained; even the dominant component can switch among $X+Z=1$14, $X+Z=1$15, or a mixture, and the uncertainty parameter

$X+Z=1$16

can be sizable, cautioning against a direct probabilistic interpretation (Lu et al., 2016). In the same work, an exploratory large-$X+Z=1$17 analysis shows that for moderate $X+Z=1$18 the mass and width of $X+Z=1$19 deviate from those of a genuine $X+Z=1$20 baryon, implying the relevance of meson-baryon components in its wave function. In the strict $X+Z=1$21 limit, however, an SU(8) Weinberg–Tomozawa analysis hints at a possibly sub-dominant $X+Z=1$22 component that would become dominant when the number of colors gets sufficiently large (Lu et al., 2016).

Heavy-quark exotics extend the same diagnostic logic to threshold states in the charmonium and bottomonium sectors. The $X+Z=1$23 and $X+Z=1$24 are reported with compositeness $X+Z=1$25 and $X+Z=1$26, respectively, supporting predominantly molecular interpretations; $X+Z=1$27, $X+Z=1$28, and $X+Z=1$29 have $X+Z=1$30, indicating sizable but not dominant molecular content; $X+Z=1$31 has total compositeness around $X+Z=1$32–$X+Z=1$33, suggesting it is not predominantly molecular; and $X+Z=1$34 is found to be predominantly a $X+Z=1$35 molecule (Oller et al., 2022).

Across these examples, a consistent picture emerges. Compositeness for resonances is a quantitative structural diagnostic tied to pole residues, loop derivatives, and explicit channel spaces. It is most robust for weakly bound near-threshold states, increasingly prescription-dependent for broad or multichannel resonances, and often most informative when compared with complementary probes such as finite-volume spectra, decay observables, and large-$X+Z=1$36 behavior.