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Numerical study of the three-boson bound-state problem in partial-wave and vector-variable formulations

This paper presents a systematic benchmark of three-boson bound-state calculations in momentum space by comparing one-dimensional, two-dimensional, and three-dimensional formulations, demonstrating high-precision agreement in binding energies and spatial observables across different interaction types and equation forms.

Original authors: Wolfgang Schadow, Mohammadreza R. Hadizadeh

Published 2026-09-08

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CC BY 4.0

Original authors: Wolfgang Schadow, Mohammadreza R. Hadizadeh

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of atomic nuclei, nature often plays a game of three. When three particles, such as protons or neutrons, bind together, they form a system that is far more complex than the simple pairing of two. While physicists have long mastered the mathematics of how two particles stick together, predicting the behavior of three interacting particles has remained a stubborn challenge. The difficulty lies not in the laws of physics themselves, which are well understood, but in the sheer computational complexity of solving the equations that describe how these three bodies move and interact simultaneously. To understand the structure of matter, from the simplest atoms to the cores of stars, scientists must be able to calculate these three-body systems with extreme precision. If the tools used to make these calculations are flawed, the resulting picture of the physical world will be distorted.

A team of researchers has now performed a rigorous test of the different mathematical tools used to solve this three-body problem. They focused on a system of three identical, spinless particles, often called bosons, which act as a clean, simplified model for more complex nuclear systems. The team compared three distinct ways of setting up the calculation: a method that reduces the problem to a single line of data, a method that breaks the problem into two dimensions, and a newer, more direct approach that treats the particles' motion in full three-dimensional space without simplifying the angles. Their goal was not to discover a new force or a new particle, but to verify that these different mathematical routes lead to the exact same physical reality. By forcing all three methods to use the same underlying rules for how the particles attract one another, the researchers could isolate and measure the tiny errors introduced by the computer's numerical processing, such as rounding numbers or approximating curves.

The study confirmed that all three approaches, despite their different structures, converge on the same answer. When the researchers calculated the binding energy—the amount of energy required to hold the three particles together—they found that the results from the one-dimensional, two-dimensional, and three-dimensional methods agreed to within a few millionths of a millionth of an electron volt. This level of agreement is so precise that it effectively rules out any significant systematic error in the way the three-dimensional method handles the complex geometry of the particles' movement. The team also tested two different ways of driving the calculation: one that uses a pre-calculated "scattering" function and another that uses the raw force between the particles directly. Both methods produced identical results, validating that the complex geometric permutations required to swap the positions of the particles were being handled correctly by the computer code.

To ensure their findings were not just a mathematical trick, the researchers translated their results from the abstract language of momentum into the physical language of space. They calculated how far apart the particles typically sit from one another and how the probability of finding them is distributed in space. These spatial maps, derived from the three-dimensional vector approach, matched perfectly with the maps generated by the older, simpler methods. This cross-check provided a powerful confirmation that the new, more direct way of solving the problem does not lose any physical information, even though it avoids the traditional step of breaking the wave function into angular components.

The researchers also explored how these methods handle different types of forces. They tested the calculations against "separable" potentials, which are mathematically smooth and easy to handle, and "local" potentials, which are sharper and more difficult to compute because they involve strong, short-range repulsion. Even with the more difficult local forces, the three-dimensional method held its ground, matching the results of the established two-dimensional approach to within a few parts in a million. The study demonstrated that the newer, more computationally intensive three-dimensional vector method is not only accurate but also robust enough to handle the messy, high-energy details of real nuclear interactions.

This work serves as a critical benchmark for the future of few-body physics. By proving that the direct three-dimensional approach yields results indistinguishable from the highly trusted, traditional methods, the study removes a major barrier to using these more flexible techniques. Scientists can now employ the direct vector-variable formulation with confidence, knowing it captures the full angular dependence of the wave function without the need for complex approximations. This opens the door to tackling even more complicated systems, such as those involving four or more particles, where the traditional methods become too cumbersome to apply. The study does not claim to have solved the entire mystery of nuclear binding, but it has firmly established that the new tools are reliable, precise, and ready for the next generation of challenges in understanding the building blocks of matter.

Technical Summary: Numerical Study of the Three-Boson Bound-State Problem in Partial-Wave and Vector-Variable Formulations

Problem Statement
The quantitative description of quantum systems with three or more interacting particles remains a central challenge in few-body physics, requiring the solution of Faddeev or Faddeev–Yakubovsky equations. While partial-wave decomposition offers an efficient framework for bound states by reducing the problem to coupled one-dimensional (1D) or two-dimensional (2D) integral equations, formulations based directly on vector momentum variables provide a complementary approach that retains full angular dependence without explicit expansion. However, 3D vector-variable approaches introduce complex numerical structures, particularly regarding multidimensional integrals with shifted permutation arguments. This work addresses the need for rigorous internal consistency benchmarks to validate these different numerical realizations, specifically testing discretization, interpolation, and quadrature errors inherent in 3D vector-variable formulations against established partial-wave methods.

Methodology
The authors present a systematic benchmark comparing three distinct numerical formulations for the three-boson bound-state problem in momentum space:

1D Spectator-Amplitude Formulation: Applied to separable Yamaguchi-type potentials, reducing the problem to coupled 1D integral equations for spectator amplitudes.
2D Partial-Wave Formulation: Applied to both separable and local Malﬂiet–Tjon (MT) potentials, solving coupled 2D integral equations for Faddeev components dependent on subsystem and spectator momenta.
3D Vector-Variable Formulation: Solving integral equations directly in Jacobi vector momentum space, retaining continuous angular dependence.

To ensure a controlled comparison, the study embeds the same finite partial-wave interaction space into each formulation. This ensures that discrepancies between results reflect numerical errors (discretization, interpolation, quadrature) rather than differences in the underlying model space. The study employs two distinct driving mechanisms for the Faddeev equations in both 2D and 3D formulations:

$t$ -matrix-driven: Using the fully off-shell two-body transition operator.
Bare-potential-driven ( $V$ -driven): Using the bare interaction directly, avoiding the explicit pre-calculation of the off-shell $t$ -matrix.

Numerical implementation utilizes Gauss–Legendre quadrature on discretized momentum grids, with hybrid Filon-type quadrature employed for Fourier transforms to coordinate space to handle highly oscillatory integrands. The study utilizes Yamaguchi potentials (separable) and Malﬂiet–Tjon potentials (local, Yukawa-type) as interaction models.

Key Contributions

Unified Benchmark Framework: The paper establishes a rigorous cross-validation framework where the same interaction space is embedded across 1D, 2D, and 3D formulations, isolating numerical errors from model-space truncation effects.
Validation of $V$ -driven 3D Equations: It demonstrates that the potential-driven 3D vector-variable formulation, which avoids the explicit construction of the off-shell $t$ -matrix, yields results consistent with the standard $t$ -matrix-driven approach. This validates the complex permutation geometry and interpolation schemes required for the $V$ -driven kernel.
Coordinate-Space Consistency: The study implements a two-step Fourier transformation to coordinate space, comparing results obtained via partial-wave expansion (spherical Bessel transforms) against direct vector-variable transforms. Consistency between these methods provides an independent check on the momentum-space solutions.
Convergence Analysis: The work details the convergence behavior of binding energies and expectation values with respect to momentum cutoffs and grid density, highlighting the significantly higher computational demands required for expectation values compared to eigenvalue determination.

Results

Binding Energy Agreement: For separable Yamaguchi interactions, binding energies across 1D, 2D, and 3D formulations agree at the $10^{-6}$ MeV level. For local Malﬂiet–Tjon interactions, agreement is achieved at the few- $10^{-6}$ to $10^{-5}$ MeV level.
Internal Consistency: The residuals between the eigenvalue energy ( $E_3$ ) and the Hamiltonian expectation value ( $\langle H \rangle$ ) are consistently at the $10^{-6}$ to $10^{-5}$ MeV level across all formulations, validating the reconstructed wave functions.
$t$ -matrix vs. $V$ -driven: The agreement between $t$ -matrix-driven and $V$ -driven calculations in both 2D and 3D formulations confirms that the permutation geometry and interpolation schemes are correctly implemented, even when avoiding the off-shell $t$ -matrix.
Partial-Wave Content: Decomposition of the 3D vector-variable wave function into partial waves shows excellent agreement with 1D and 2D results, accurately reproducing the angular content of the state despite the absence of an explicit partial-wave basis in the 3D solver.
Local Potential Challenges: Local Malﬂiet–Tjon potentials exhibit slower partial-wave convergence and require denser grids and higher momentum cutoffs compared to separable potentials to stabilize expectation values, reflecting their stronger short-range structure.

Significance
The paper claims that its primary significance lies in providing a stringent cross-validation framework for 3D vector-variable calculations. By controlling the interaction representation, the authors demonstrate that 3D vector-variable approaches can achieve numerical accuracy comparable to established partial-wave methods. The successful validation of the $V$ -driven 3D formulation suggests that it is a viable alternative that may reduce computational costs by avoiding the explicit construction and interpolation of the fully off-shell two-body $t$ -matrix. Furthermore, the consistent results in coordinate space provide an independent verification of the momentum-space solutions, reinforcing the reliability of these numerical methods for few-body physics.

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