[Paper Review] A toy model for "elementariness"

Peter C. Bruns|arXiv (Cornell University)|Mar 31, 2022

Crystallography and Radiation Phenomena4 citations

TL;DR

This paper introduces a solvable non-relativistic toy model using a finite-range spherical well-with-wall potential to study the 'elementariness' and 'compositeness' of bound states. By tuning the potential's wall height and radius, it demonstrates a continuous interpolation between compact, elementary-like states and extended, composite-like states, showing that elementariness is quantified by the ratio of the derivative of the inverse K-matrix to the total derivative of the energy-dependent phase shift, with quantitative results confirming that the deuteron is highly composite when finite-range effects are included.

ABSTRACT

Motivated by recent efforts to analyze corrections to Weinberg's relations for the scattering length and effective range in the presence of a near-threshold bound state, we play around with an instructive toy model for non-relativistic scattering in a central potential. The model allows to interpolate between bound-state configurations of high "compositeness", where the wave function is spread over a wide region beyond the range of the interaction, and compact configurations of high "elementariness", where the wave function is confined to a small region around the center of the potential.

Motivation & Objective

• To develop a simple, analytically tractable model to study the concept of 'elementariness' in bound states, particularly in the context of near-threshold systems.
• To investigate how the interplay between interaction range and bound-state wave function structure affects the compositeness and elementariness of a state.
• To test the validity of the Weinberg relations and their linear approximations in a controlled, solvable potential model.
• To provide a quantitative framework for assessing whether a bound state like the deuteron can be considered elementary or composite based on its spatial wave function distribution.

Proposed method

• A central, finite-range potential is constructed as a spherical well of radius d with a surrounding wall of thickness δ and height W₀, with a negative well depth V₀.
• The s-wave Schrödinger equation is solved analytically in three regions: inside the well, within the wall, and outside the potential range, with matching conditions at r = d and r = d+δ.
• The bound-state wave function is normalized, and the scattering amplitude is derived from the K-matrix formalism, with poles corresponding to bound states.
• The compositeness $\mathcal{C}_B^0$ and elementariness $\mathcal{E}_B^0$ are computed using the derivative-based formula involving the K-matrix and the wave number derivative.
• The effective range expansion and Weinberg's linear approximations are applied to extract the scattering length $a_0$ and effective range $r_0$, and the consistency of $X_a$ and $X_r$ with $\mathcal{C}_B^0$ is tested.
• The model is extended to a harmonic well-with-wall potential to explore more realistic shapes, with solutions expressed in terms of confluent hypergeometric functions.

Experimental results

Research questions

• RQ1 How does the spatial extent of a bound-state wave function depend on the range and depth of a finite-range potential?
• RQ2 To what extent can the concept of 'elementariness' be quantified in a non-relativistic quantum mechanical model with a finite-range interaction?
• RQ3 How well do the linear approximations of the Weinberg relations (for scattering length and effective range) hold in a model with non-zero interaction range?
• RQ4 Can the deuteron be considered an elementary particle based on its spatial wave function distribution and the derived compositeness measure?
• RQ5 What is the role of the interaction range in determining the compositeness of a near-threshold bound state?

Key findings

• For a potential with zero wall height (W₀ = 0), the bound state is highly composite ($\mathcal{C}_B^0 = 1.042$), with 92.4% of the probability density outside the interaction range (r > d+δ).
• As the wall height W₀ increases, the state becomes more compact and elementary-like: at W₀ = 50, $\mathcal{C}_B^0 = 0.206$, and only 18.4% of the probability lies beyond r = d+δ.
• The quantities $X_a$ and $X_r$, derived from the effective range expansion, agree well with $\mathcal{C}_B^0$ for small binding energies, validating the linear approximation in the near-threshold limit.
• The model shows that the deuteron, with $X_a \approx 1.68$, cannot be described by a zero-range interaction, requiring a minimum interaction range of $R \gtrapprox 0.79\,M_\pi^{-1}$ to be consistent with its observed compositeness.
• Using a harmonic well-with-wall potential, the model reproduces the deuteron's physical parameters with $V_0 = -0.098\mu$, $d = 6.168\mu^{-1}$, yielding $\mathcal{C}_B^0 = 1.666$ and $P(r>d) = 0.502$, confirming its composite nature.
• The analogy of a couple seen from a distance illustrates that a state with significant amplitude beyond the interaction range is not elementary, as it can be resolved into constituent parts.