Chiral Unitary Approach in Hadronic Interactions

• Chiral Unitary Approach is a nonperturbative framework that uses chiral low-energy interactions and coupled-channel resummation to dynamically generate resonances and bound states.
• It employs on-shell Bethe-Salpeter and N/D-type methods with energy-dependent kernels and dimensional regularization of loop functions to ensure exact two-body unitarity.
• Its applications span meson-baryon, meson-meson, and heavy-hadron sectors, providing insights into structures like Λ(1405), Ξ(1620), and hidden-charm pentaquarks.

The chiral unitary approach is a nonperturbative framework in which low-energy chiral dynamics supplies the interaction kernel and unitarization resums two-body scattering so that resonances, bound states, and threshold structures emerge from the amplitude itself rather than from explicit bare poles. In the papers considered here, it appears in meson-baryon, meson-meson, finite-volume, few-body, and heavy-hadron settings, but its defining structure is the same: a chiral interaction $V$, a loop function $G$, and a resummed amplitude $T$ whose poles encode the spectrum (Miyahara et al., 2015, Oset et al., 2011, Suntharawirat et al., 23 Jun 2026).

1. Core construction

At the formal level, the approach is usually formulated as an on-shell factorized Bethe-Salpeter or $N/D$-type resummation. A representative coupled-channel equation is

$$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$

equivalently

$$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$

or, in the notation used in several papers,

$T=[1-VG]^{-1}V.$

Here $V_{ij}$ is the interaction kernel from chiral perturbation theory or from an equivalent chiral effective interaction, $G_i$ is the two-body loop function, and the matrix structure implements coupled-channel unitarity (Miyahara et al., 2015, Nishibuchi et al., 2023).

The framework is used in several channel sectors. In $s$-wave meson-baryon scattering it is applied to $G$0, $G$1, $G$2, $G$3, $G$4, and $G$5 systems; in meson-meson scattering it is applied to $G$6, $G$7, and related scalar channels; and in heavy-hadron applications it is combined with heavy-quark spin symmetry and local hidden gauge dynamics (Xu et al., 2015, Guo et al., 2012, Suntharawirat et al., 23 Jun 2026).

The chiral unitary approach is therefore not a single model with fixed field content. It is a construction principle: start from the chiral low-energy interaction, preserve exact two-body unitarity through resummation, and analyze the resulting analytic structure in the complex energy plane. This is why the same formalism can generate light scalar mesons, the $G$8, near-threshold $G$9 states, and hidden-charm pentaquark poles within different channel spaces (Oset et al., 2011, Miyahara et al., 2015, Nishibuchi et al., 25 Jul 2025).

2. Interaction kernels, loop functions, and renormalization

The most common driving term is the Weinberg-Tomozawa interaction. In the single-channel bound-state analysis, the kernel is written as

$T$0

while in decuplet-baryon–pseudoscalar-meson scattering it takes the form

$T$1

These expressions make explicit two standard features of the approach: the interaction is fixed by chiral symmetry at leading order, and it is generally energy dependent (Hyodo et al., 2010, Xu et al., 2015).

The loop function $T$2 encodes the unitarity cut and is typically regularized dimensionally. Its finite part depends on a subtraction constant $T$3, and several papers treat the subtraction constants as the main phenomenological parameters. In the $T$4-$T$5 analysis, for example, the scale dependence is absorbed by the subtraction constant according to

$T$6

so that the amplitude is scale independent once this is taken into account (Xu et al., 2015).

The subtraction constants are not merely technical. In the compositeness analysis, the dependence of the pole residue on $T$7 is traced directly to the energy dependence of the chiral interaction. For a bound state generated from the Weinberg-Tomozawa kernel, the coupling extracted from the residue depends on both $T$8 and $T$9, and therefore so does the compositeness. This is the basis of the connection between subtraction constants, natural renormalization, and hidden CDD-pole-like contributions (Hyodo et al., 2010).

Several works extend the minimal kernel. The $N/D$0 study adds vector-baryon channels and an explicit singlet meson-baryon interaction because the $N/D$1 is mostly singlet and the standard lowest-order interaction is suppressed. The $N/D$2 unitary chiral perturbation theory work includes explicit scalar, vector, and pseudoscalar resonance fields and then unitarizes one-loop amplitudes and form factors. In both cases, the chiral unitary approach remains recognizable because the extended kernel is still inserted into a nonperturbative unitarization scheme (Oset et al., 2010, Guo et al., 2012).

3. Analytic structure, poles, and resonance interpretation

The physical content of the approach lies in the analytic continuation of $N/D$3 into the complex energy plane. Near a pole,

$N/D$4

and the pole position $N/D$5 identifies a resonance, bound state, quasibound state, or virtual-state analogue depending on the Riemann sheet and threshold location (Xu et al., 2015, Nishibuchi et al., 2023).

The $N/D$6 is the canonical example. In the $N/D$7 chiral unitary description used for local-potential construction, the $N/D$8 amplitude has two poles induced by attractive interactions in both $N/D$9 and $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$0: a higher-energy $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$1 pole and a lower-energy $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$2 pole. In the specific example quoted there, the original poles are

$$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$3

The paper’s central technical point is that a local potential must reproduce not only the real-axis amplitude but also this complex-plane structure, otherwise the subthreshold dynamics relevant for $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$4 and kaonic nuclei are distorted (Miyahara et al., 2015).

Near-threshold $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$5 studies illustrate the same logic in a different sector. One chiral unitary model realizes a pole

$$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$6

as a quasibound state below the $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$7 threshold, whereas a model constrained to the ALICE $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$8 scattering length yields no pole on the physically relevant sheet and instead produces a cusp at threshold, with a quasivirtual-state pole on the $$T_{ij}(\sqrt{s}) = V_{ij}(\sqrt{s}) + V_{ik}(\sqrt{s}) G_k(\sqrt{s}) T_{kj}(\sqrt{s}),$$9 sheet. The point is not merely spectroscopic: a threshold enhancement can come from a quasibound pole, from a quasivirtual pole, or from cusp kinematics, and the line shape alone does not uniquely distinguish them (Nishibuchi et al., 2023, Nishibuchi et al., 2023).

The same theme recurs in the interpolation study of the $$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$0-related spectrum. Two models, one Belle-constrained and one ALICE-constrained, produce poles on different sheets,

$$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$1

and pole tracking under interpolation of subtraction constants shows that they are not continuously connected. Distinct analytic structures can therefore produce superficially similar low-energy enhancements (Nishibuchi et al., 25 Jul 2025).

There is also disagreement in the literature over how robust some pole patterns are. A variant calculation with off-shell corrections to the dimensionally regularized loop function reports that only one visible $$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$2-like state remains dynamically generated in the $$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$3, $$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$4 sector, in contrast with the standard two-pole picture used in other chiral unitary analyses (Dong et al., 2016). This establishes that analytic structure in the approach is sensitive to the treatment of the loop function and to the chosen renormalization prescription.

4. Compositeness and internal structure

One of the most developed structural interpretations within the chiral unitary approach is the compositeness program based on the field renormalization constant

$$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$5

with $$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$6 interpreted as the composite component of the physical bound state. In the nonrelativistic formulation,

$$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$7

and in the weak-binding limit it reduces to a Weinberg-type relation proportional to the bound-state coupling squared (Hyodo et al., 2010).

Applied to the chiral unitary amplitude, this construction shows that “dynamically generated” does not automatically mean “purely composite.” In the single-channel analysis, an energy-independent interaction gives a purely composite bound state, $$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$8, whereas the energy-dependent Weinberg-Tomozawa interaction introduces an elementary component that depends on the subtraction constant. The natural renormalization scheme is then interpreted as the choice that excludes CDD-pole contamination from the loop and yields a bound state dominated by the composite structure when the binding energy is small (Hyodo et al., 2011).

This structural language is directly connected to phenomenology in the $$T(\sqrt{s}) = \bigl[V(\sqrt{s})^{-1}-G(\sqrt{s})\bigr]^{-1},$$9 study based on the SIDDHARTA-constrained local $T=[1-VG]^{-1}V.$0 potential. Solving the Schrödinger equation with the constructed $T=[1-VG]^{-1}V.$1 potential gives the root-mean-squared $T=[1-VG]^{-1}V.$2 distance

$T=[1-VG]^{-1}V.$3

This is compared with the quoted proton and $T=[1-VG]^{-1}V.$4 charge radii, $T=[1-VG]^{-1}V.$5 fm and $T=[1-VG]^{-1}V.$6 fm, and is taken to indicate a meson-baryon molecular structure rather than a compact three-quark state (Miyahara et al., 2015).

A recurrent implication is that compositeness depends both on dynamics and on scheme. Large deviations of the subtraction constant from the natural value reduce $T=[1-VG]^{-1}V.$7, which the compositeness papers interpret as an increasing elementary or hidden-pole component. A plausible implication is that structural statements in the chiral unitary approach are sharpest for shallow bound states near threshold and less literal for broad resonances, virtual states, or models with substantial renormalization ambiguity (Hyodo et al., 2010, Hyodo et al., 2011).

5. Computational realizations and phenomenological applications

A major reason for the importance of the chiral unitary approach is that it can be embedded into practical calculations beyond elastic scattering. In finite volume, the infinite-volume loop $T=[1-VG]^{-1}V.$8 is replaced by a discrete sum $T=[1-VG]^{-1}V.$9, and the finite-volume amplitude becomes

$V_{ij}$0

For two channels, the energy levels satisfy

$V_{ij}$1

This construction is used to show that synthetic lattice-QCD spectra can be inverted to recover the pole positions of $V_{ij}$2, $V_{ij}$3, and $V_{ij}$4, while also demonstrating that threshold-induced level flattening should not be confused with resonance evidence (Oset et al., 2011).

In few-body and nuclear applications, a coupled-channel chiral amplitude is often converted into a coordinate-space interaction. The $V_{ij}$5 local-potential construction uses Feshbach reduction to eliminate channels other than $V_{ij}$6, obtaining an effective single-channel interaction $V_{ij}$7, and then writes the local potential as

$V_{ij}$8

with Gaussian profile

$V_{ij}$9

The explicit complex, energy-dependent correction $G_i$0 is introduced so that the local potential reproduces the original amplitude in the complex plane as well as on the real axis (Miyahara et al., 2015).

The same logic underlies three-body calculations. In the Faddeev-chiral unitary treatment of $G_i$1, the two-body $G_i$2 amplitudes are generated from coupled-channel chiral SU(3) dynamics and then embedded into a relativistic Faddeev-type formalism. Using SIDDHARTA-constrained two-body input, the resulting prediction is

$G_i$3

in the $G_i$4-model and

$G_i$5

in the $G_i$6-model (Mizutani et al., 2012).

Reaction theory provides another major application. In $G_i$7 photoproduction, the $G_i$8 amplitude is built from gauge-invariant production mechanisms plus meson-baryon rescattering, and the line shapes are described with the $G_i$9 generated dynamically in the final-state interaction. The analysis concludes that nonresonant background is not negligible and shifts the apparent $s$0 peak by several MeV, while the dominant sensitivity is to the higher chiral-unitary pole (Nakamura et al., 2013). In near-threshold $s$1 and $s$2, unitarized meson-baryon amplitudes enter both the production subprocesses and the final-state interactions, and the strong suppression of $s$3 production is explained through destructive interference between pion and kaon exchange together with channel-dependent rescattering (Chen et al., 2011).

The framework has also been extended into the heavy sector. In the radiative decay $s$4, both pentaquarks are treated as $s$5-wave hadronic molecules generated in a coupled-channel chiral unitary approach with heavy-quark spin symmetry and local hidden gauge interaction. The calculation uses the pole residues from

$s$6

and finds a central width of

$s$7

with a conservative range of about $s$8 to $s$9 (Suntharawirat et al., 23 Jun 2026).

6. Ambiguities, limitations, and contested points

The chiral unitary approach is intrinsically sensitive to the way loop functions, subtraction constants, and off-shell effects are handled. In some channel sectors this sensitivity is used constructively, as in the $G$00-$G$01 study where the subtraction constant is treated as the only true free parameter and tuned to obtain a pole at $G$02 MeV. In the absence of strong experimental constraints, however, this also means substantial model dependence: varying $G$03 from $G$04 to $G$05 moves the pole from about $G$06 MeV to $G$07 MeV (Xu et al., 2015).

This dependence is not limited to spectroscopy. In the $G$08 analyses, the Belle-motivated pole condition and the ALICE-motivated scattering-length condition lead to non-overlapping allowed regions in the $G$09 plane within the same leading-order chiral unitary framework. The formalism can therefore generate a shallow quasibound-state interpretation or a cusp/quasivirtual-state interpretation, depending on which empirical constraint is imposed (Nishibuchi et al., 2023, Nishibuchi et al., 2023).

Another limitation concerns the meaning of “dynamically generated.” The compositeness analyses show that a dynamically generated pole in a chiral unitary amplitude is not automatically or universally a purely hadronic molecule. Energy-dependent kernels and unnatural subtraction constants can hide elementary or CDD-pole-like components in the renormalization prescription, so the structural interpretation is model- and scheme-dependent except in especially controlled regimes such as small binding (Hyodo et al., 2011).

There are also broader scheme issues. In nuclear forces, the leading and subleading chiral $G$10-exchange NN potentials derived with the method of unitary transformation differ, for selected classes, from the older S-matrix-matching results. The point there is that static $G$11-exchange is already scheme dependent, so off-shell consistency with the chosen interaction framework matters (Springer et al., 4 May 2025). This suggests, by close analogy, that chiral unitary constructions should be read as amplitude frameworks with explicit scheme choices rather than as unique microscopic reconstructions.

Taken together, these results define the chiral unitary approach less as a single model than as a disciplined class of nonperturbative chiral amplitudes. Its characteristic strengths are exact two-body unitarity, explicit threshold and coupled-channel dynamics, and direct access to pole structure. Its characteristic cautions are subtraction-constant sensitivity, production-model dependence, and the fact that real-axis line shapes do not map one-to-one onto pole content.