Level rearrangement in K- p system

Technical Summary: Level Rearrangement in the K−p System

Problem Statement
The paper addresses the theoretical description of energy level shifts in the antikaon-nucleon ($K^-p$) system, specifically focusing on the interplay between the long-range Coulomb potential and the short-range strong nuclear interaction. Historically, two distinct types of (quasi-)bound states in this system have been studied separately: the "atomic" states of kaonic hydrogen (dominated by Coulomb attraction with strong interaction as a perturbation) and the $\Lambda(1405)$ resonance (a quasi-bound state dominated by strong interaction).

A significant paradox exists regarding the $1s$ level shift of kaonic hydrogen. While the strong $\bar{K}N$ interaction is attractive enough to form the $\Lambda(1405)$ resonance, experimental extractions of the $K^-p$ scattering length suggested a "repulsive" interaction, leading to the expectation of a "repulsive" level shift. However, recent high-precision experiments measuring the $2p \to 1s$ gamma-ray transition reported an "attractive" shift, creating a contradiction with the scattering data interpretation. Furthermore, the impact of the Coulomb interaction on the mass and width of the $\Lambda(1405)$ resonance had not been systematically studied. The authors aim to resolve these contradictions by studying both types of states simultaneously within a unified framework.

Methodology
The authors employed a non-relativistic quantum mechanical approach solving the Lippmann-Schwinger equation for the $K^-p$ system. They utilized two distinct solution methods:

1. Exact Separable Potential Solution: For separable potentials with Yamaguchi form factors, they solved the homogeneous equation $|\Psi\rangle = G^{(C)}(E)V^{(s)}|\Psi\rangle$ analytically, utilizing the Coulomb Green's operator $G^{(C)}(E)$ expressed via Whittaker functions.
2. Coulomb-Sturmian (CS) Basis Representation: To accommodate future work on three-body systems ($K^-pp$), they solved the equation using a CS basis. This method utilizes exact forms of the Coulomb Green's matrix in terms of hypergeometric functions and is applicable to both local and non-local potentials.

The study utilized two models for the strong antikaon-nucleon interaction:

• Real Potential ($V_{K^-p}^{1A,real}$): A one-channel separable potential fitted to reproduce the real part of the $\Lambda(1405)$ resonance energy and scattering length.
• Complex Potential ($V_{K^-p}^{1A,cmplx}$): A similar potential with complex strength parameters to account for absorption (decay into $\pi\Sigma$ channels), reproducing both the real and imaginary parts of the resonance energy.

The authors analyzed the system by gradually varying the strength constant of the strong potential ($\lambda_x = x\lambda$) from zero (pure Coulomb) to the physical value, and in the complex case, along different trajectories in the complex plane to investigate the stability of the results against absorption.

Key Contributions and Results

• Discovery of Level Rearrangement: The central finding is that the $K^-p$ system undergoes a "level rearrangement" when the strong interaction is added to the Coulomb potential. As the strength of the attractive strong interaction increases, the energy levels do not simply shift; they reorder.

  • The lowest physical state (the blue circle in the authors' analysis) evolves continuously from the pure Coulomb $1s$ state but eventually aligns with the trajectory of the pure strong bound state ($\Lambda(1405)$).
  • Consequently, the state that was originally the $2s$ atomic level drops down to occupy the energy region near the original pure Coulomb $1s$ level.
  • This rearrangement is confirmed by analyzing the wave functions: the state near the original $1s$ energy acquires a node (characteristic of a $2s$ state), while the new ground state is nodeless.

• Resolution of the "Repulsive" vs. "Attractive" Shift Paradox: The authors argue that previous definitions of level shifts were incorrect because they compared the pure Coulomb $1s$ energy to the wrong physical state (the shifted $2s$ state, which appeared closest in energy).

  • When the shift is correctly defined as the difference between the pure Coulomb $1s$ level and the actual new ground state (the rearranged $1s$), the shift is found to be attractive (positive $\Delta E$).
  • Similarly, the shift of the $\Lambda(1405)$ resonance caused by the addition of the Coulomb potential is also attractive.
  • The "repulsive" values previously reported (or expected) correspond to the difference between the pure Coulomb $1s$ and the shifted $2s$ level, which is a difference between neighbor levels, not a true level shift.

• Impact of Absorption: The study demonstrates that level rearrangement persists even in the presence of strong absorption (complex potentials). However, the visibility of this effect depends on the trajectory taken in the complex energy plane when turning on the interaction. For certain trajectories, the pole moves to an unphysical Riemann sheet, obscuring the rearrangement, but for physically motivated trajectories, the rearrangement effect remains valid.

• Quantitative Shifts:

  • The correctly calculated attractive shift of the kaonic hydrogen $1s$ level due to the strong interaction is large, approximately $7-8$ MeV.
  • The shift of the $\Lambda(1405)$ resonance due to the Coulomb interaction is smaller ($\sim 1-2$ MeV) but significant relative to its binding energy (approx. 28–43%).
  • The shift of the $2s$ atomic level due to the strong interaction is small ($\sim 5-6$ keV).

Significance
The paper claims to resolve the long-standing "kaonic hydrogen puzzle" by redefining how level shifts are calculated in systems where strong and Coulomb interactions are of comparable importance in determining the spectrum. The authors assert that the experimental observation of an "attractive" shift is correct, but the theoretical interpretation was flawed due to the failure to account for level rearrangement.

Furthermore, the paper highlights that the measured gamma-ray transition, previously assumed to be $2p \to 1s$, is likely a $2p \to 2s$ transition. This is because the strong interaction lifts the degeneracy between the $2s$ and $2p$ states (or shifts the $2s$ significantly while leaving $2p$ largely unchanged), while the $1s$ state moves deep into the strong interaction regime. The authors conclude that the "repulsive" scattering length and the "attractive" level shift are not contradictory; rather, the apparent repulsion arises from comparing the wrong energy levels. The correct application of Eq. (4) in their framework yields consistent attractive shifts for all levels.