Microscopic KKNN Potential Overview

• Microscopic KKNN potential is a rigorously derived nucleon–cluster interaction model using resonating-group methods for n–α scattering and chiral coupled-channels for K⁻NN absorption.
• It employs central, spin–orbit, and parity-dependent Gaussian terms fitted to elastic phase shifts and kaonic atom data, ensuring accurate low-energy predictions.
• Numerical methods like the variable-phase approach and iterative self-consistency are used to resolve energy-dependent phase shifts and density effects in nuclear media.

The term "microscopic KKNN potential" refers to rigorously constructed nucleon–cluster or meson–nucleon interaction potentials, where "KKNN" typically denotes the contributions or nomenclature associated with models by Kanada, Kaneko, Nagata, and Nomoto. In modern nuclear physics and hadronic atom studies, two distinct classes of "microscopic KKNN" potentials are prominent: the KKNN $n$–$α$ (neutron–alpha) potential developed for elastic $n$–$α$ scattering, and the $K^-NN$ (antikaon–two-nucleon) absorption potential that describes the non-mesonic channels in kaonic atom systems. Both types are "microscopic" in the sense that they are explicitly derived from many-body dynamical models with minimal recourse to phenomenology, and their parameters are ultimately fit to experimental scattering or absorption data.

1. Functional Formulations of Microscopic KKNN Potentials

In the $n$–$α$ context, the microscopic KKNN potential $V_{n\alpha}(r)$ comprises a sum of central, spin–orbit, and parity-dependent Gaussian terms, with all parameters rooted in rigorous resonating-group or cluster model calculations. No explicit tensor or exchange nonlocalities are retained in the published local form for low-energy elastic scattering. The full functional form is

$$V_{n\alpha}(r)=V_{\rm C}(r)+V_{\rm C}^{\rm PD}(r)+V_{LS}(r),$$

where \begin{align*} V_{\rm C}(r) &= \sum_{i=1}^2 V_i^C\,e^{-\mu_i^C r^2}, \ V_{\rm C}^{\rm PD}(r) &= (-1)^{\ell} \sum_{i=1}^3 V_{\ell i}^C\,e^{-\mu_{\ell i}^C r^2}, \ V_{LS}(r) &= (\boldsymbol\ell \cdot \mathbf s)\left{V^{LS}e^{-\mu^{LS} r^2} + [1+0.3(-1)^{\ell-1}]\sum_{i=1}^2 V_{\ell i}^{LS}e^{-\mu_{\ell i}^{LS} r^2}\right}. \end{align*} Empirical parameters for all strengths and ranges are fixed by fits to elastic phase shifts and resonance structure (Khachi, 14 Dec 2025).

In the case of $K^-NN$ absorption, the microscopic potential formalism is derived from SU(3) chiral coupled-channels models for $K^-N$ scattering, extended via diagrammatic two-nucleon ($K^-NN\to YN$) absorption processes computed from meson-baryon Lagrangians. The real and imaginary parts are tied to density-dependent amplitudes and quadratic in the local baryon density: $$V_{K^-NN}(r)\simeq -\frac{2\pi}{\mu_{K^-N}}F_{K^-NN}\bigl(\sqrt{s}(r),\rho(r)\bigr)\rho^2(r),$$ where $F_{K^-NN}$ is constructed from explicit intermediate-state summation, antisymmetrization, and low-momentum transfer approximations (Óbertová et al., 2022).

2. Microscopic Origin and Parameter Determination

The $n$–$α$ KKNN model arises from a resonating-group–model (RGM) treatment formulated in a harmonic-oscillator–Gaussian basis, ensuring automatic antisymmetrization between the neutron and the $\alpha$ core. The parameters $(V_i, \mu_i)$ are determined by fitting to low-energy $S_{1/2}$, $P_{3/2}$, and $P_{1/2}$ phase shifts up to $E_{\text{lab}} \approx 20~$MeV, the $P_{3/2}$ resonance near 1 MeV, and higher-energy cross-section shapes. Higher-order exchange effects (beyond folding) are retained via the parity-dependent term, reflecting the microscopic RGM kernel expansion (Khachi, 14 Dec 2025).

In the $K^-NN$ model, amplitudes derive from SU(3) chiral Lagrangian-based scattering matrices, including full coupled Lippmann–Schwinger or Bethe–Salpeter solutions for all relevant meson–baryon channels. In-medium modifications (Pauli blocking, multiple scattering, subthreshold kinematics) and explicit two-nucleon absorption diagrams control the overall normalization and density dependence. No fitted free parameters appear beyond those already fixed in $K^-N$ amplitude fits to SIDDHARTA and threshold data (Óbertová et al., 2022).

3. Numerical Implementation and Partial-Wave Structure

For $n\,$–$α$ scattering, the local KKNN potential is implemented in the radial Schrödinger equation for each $(\ell, j)$ partial wave as

$$V_{\ell j}(r)= V_{\rm C}(r)+(-1)^\ell V_{\rm C}^{(\ell)}(r)+ \langle\ell j | \boldsymbol\ell \cdot \mathbf{s}| \ell j\rangle V_{LS}(r)+\frac{\hbar^2\ell(\ell+1)}{2\mu r^2},$$

with $\ell=0,1$ and $j=1/2,3/2$ for the relevant channels. There are no off-diagonal partial-wave couplings.

The variable-phase approach (VPA) is employed to solve the Schrödinger equation, parametrizing the regular solution as

$$u_{\ell j}(k,r) = A_{\ell j}(r) [ \cos\delta_{\ell j}(r)\,\hat j_{\ell}(kr) - \sin\delta_{\ell j}(r)\,\hat n_{\ell}(kr)],$$

with the phase-function ODE

$$\frac{d}{dr}\delta_{\ell j}(r) = -\frac{1}{k}\frac{2\mu}{\hbar^2} V_{\ell j}(r)[\cos\delta_{\ell j}(r)\hat j_{\ell}(kr) - \sin\delta_{\ell j}(r)\hat n_{\ell}(kr)]^2.$$

Initial conditions $\delta_{\ell j}(0)=0$, $A_{\ell j}(0)=1$ are imposed, and integration proceeds via high-order Runge–Kutta methods (e.g. Dormand–Prince, Butcher schemes) with step sizes $\Delta r \sim 0.01~$fm to ensure numerical convergence (Khachi, 14 Dec 2025).

In the kaonic atom context, the Klein–Gordon equation with the full optical potential (including both $K^-N$, $K^-NN$ and phenomenological multi-nucleon components) is solved iteratively. Subthreshold corrections to $\sqrt{s}$ and self-consistency loops ensure proper energy dependence.

4. Physical Interpretation and Sample Results

The $n$–$α$ KKNN potential yields accurate and physically interpretable energy-dependent phase shifts. The large $S_{1/2}$ phase shift at low energy reflects dominant central attraction, while the narrow $P_{3/2}$ resonance at $E_{\mathrm{lab}} \simeq 1$ MeV results from constructive interplay of the parity-dependent and spin–orbit components. Weak, negative $P_{1/2}$ phase shifts are attributed to negative spin–orbit splitting. Parametric fits to numerically computed $\delta_{\ell j}(E)$ are accurate to better than $1^\circ$ below 20 MeV laboratory energy: $$\begin{aligned} \delta_{S_{1/2}}(E) &\approx \arctan\left[\frac{5.0}{\sqrt{E(\mathrm{MeV})}}\right], \ \delta_{P_{3/2}}(E) &\approx \arctan\left[\frac{1.2(E-1.0)}{(E-1.0)^2+(0.2)^2}\right], \ \delta_{P_{1/2}}(E) &\approx -\arctan\left[\frac{0.8}{\sqrt{E}}\right]. \end{aligned}$$ Specific phase shift values at 1, 5, and 10 MeV are tabulated (Khachi, 14 Dec 2025).

In the $K^-NN$ system, inclusion of the microscopic two-nucleon absorption reduces the global $\chi^2$ for kaonic atom data by a factor of two (from $\sim800$ to $400$–$560$ depending on in-medium amplitude treatment), and reproduces both old bubble-chamber and recent AMADEUS data for two-nucleon branching ratios (20–25%). Residual discrepancy with experiment points to possible missing higher-order ($3N$, $4N$) effects and motivates further refinement (Óbertová et al., 2022).

5. Density Dependence, Self-Consistency, and Practical Algorithms

Both KKNN classes exhibit distinctive density dependences critical for their physical viability. The $n$–$α$ potential acts in free space; its radial dependence arises solely from the Gaussian functional forms and the centrifugal barrier. Conversely, the $K^-NN$ absorption potential is quadratic in the local baryon density $\rho(r)$, with energy dependence modeled via subthreshold corrections and self-consistent solutions of the Klein–Gordon (or Schrödinger) equation. The iterative algorithm at each $r$ involves:

1. Initial guess for $B_{K^-}$, $V_{K^-}(r)$, yielding $\sqrt{s}(r)$,
2. Calculation of in-medium amplitudes $F_{K^-N}, F_{K^-NN}$,
3. Construction of optical potentials and solution for the kaonic atom ground state,
4. Update of $\sqrt{s}(r)$ and repetition until convergence.

Density profiles are imported from relativistic mean field (TM2/TM1) calculations; nuclear Coulomb potential is included by minimal substitution (Óbertová et al., 2022).

6. Comparison with Phenomenological and Alternative Potentials

Microscopic KKNN potentials, both in $n$–$α$ and $K^-NN$ systems, offer systematic improvements over purely phenomenological parametrizations. For $K^-NN$, the need for a large phenomenological multi-nucleon term is sharply reduced; fitted strengths decrease from $B\approx-1.3+1.9i~$fm and $\alpha\approx1$ to $B\approx-0.9+0.7i~$fm and $\alpha\approx0.6$ when the microscopic contribution is included. The remaining phenomenological term has weak density dependence ($\alpha<2$), suggesting possible missing self-energy or genuine three-nucleon mechanisms.

Future work aims to resolve remaining discrepancies by refining in-medium self-energies, including explicit $3N$ absorption diagrams, and incorporating new experimental results, particularly in sulfur kaonic atoms. The extensible structure of microscopic KKNN potentials provides a foundation for systematic, theory-driven advances in few-body nuclear and hadronic physics (Khachi, 14 Dec 2025, Óbertová et al., 2022).