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On the equivalence of left-hand-cut and three-body formalisms

This paper analytically and numerically demonstrates the equivalence between the three-body formalism and the two-body left-hand cut approach for resolving exchange singularities in lattice QCD, while highlighting that exponentially suppressed finite-volume effects neglected by both methods can significantly compromise amplitude extraction at small lattice volumes.

Original authors: André Baião-Raposo, Raul Briceño, Sebastian M. Dawid

Published 2026-09-03

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CC BY 4.0

Original authors: André Baião-Raposo, Raul Briceño, Sebastian M. Dawid

Original paper licensed under CC BY 4.0 (http://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 subatomic world, scientists use a technique called lattice quantum chromodynamics to study how particles interact. Imagine trying to understand the rules of a game by watching it played inside a small, enclosed room. In this case, the "room" is a computer simulation with a finite size, and the "game" is the behavior of quarks and gluons, the building blocks of matter. By calculating the energy levels of particles trapped in this digital box, researchers can deduce how these particles would scatter and interact in the vast, open universe. However, this method hits a snag when the particles involved are very close to forming a bound state, like a molecule made of two smaller particles. In these delicate situations, long-range forces create mathematical complications known as left-hand cuts. These are not physical cuts in space, but rather points in the mathematical description where the rules of standard analysis break down, making it difficult to extract the true, infinite-volume behavior of the system from the finite-box data.

For years, physicists have developed different ways to navigate this problem. One approach treats the system as a collection of three individual particles, while another treats it as a single particle bouncing off a pre-existing pair. Both methods aim to correct for the distortions caused by the small size of the simulation box, but they do so using very different mathematical tools. A new study by researchers at Ruhr-University Bochum, the University of California, Berkeley, and Indiana University has now rigorously tested whether these two distinct approaches are actually saying the same thing. They set out to prove that, despite their different starting points, both methods lead to the exact same physical reality, provided the simulation box is large enough.

The team focused on a specific, simplified model where three identical particles interact. In this scenario, two of the particles can stick together to form a stable pair, while the third particle bounces off them. This setup mimics real-world systems where a particle scatters off a bound state, a situation relevant to understanding certain exotic particles recently discovered in nature. The researchers applied both the three-particle method and the two-particle method to this model. They first showed analytically, using pure mathematical logic, that the equations governing the two approaches could be rearranged to look identical. They demonstrated that the complex interactions between the three individual particles could be reorganized to match the description of a single particle interacting with a pair, effectively proving that the two formalisms are equivalent in theory.

To ensure this equivalence held up in practice, the team then ran extensive numerical simulations. They generated energy levels for the system in boxes of various sizes and used both methods to extract the scattering amplitudes, which describe how likely the particles are to bounce off each other at different speeds. Above the energy threshold where the bound state can exist, the two methods agreed with each other to an extraordinary degree of precision, matching to within less than one percent. This confirmed that both approaches correctly capture the physics of the system when the particles are energetic enough.

However, the story became more nuanced when the researchers looked at lower energies, deep below the threshold where the bound state forms. In this region, the "left-hand cut" problem is most severe. Here, the two methods still agreed, but the match was slightly less perfect, differing by a few percent. The researchers traced this small discrepancy not to a flaw in either method, but to the inevitable limitations of the simulation itself. Because the digital boxes are finite, there are tiny, exponentially suppressed effects that the mathematical proofs had to ignore to keep the equations solvable. These effects, which depend on the size of the box and the mass of the particles, become noticeable when the box is small. The study found that as the box size decreased, these neglected effects grew larger, causing the slight divergence between the two methods.

This finding is significant because it validates the use of the simpler two-body approach for studying complex three-body systems, provided the simulation volume is sufficiently large. It confirms that the complicated machinery of treating every particle individually is not strictly necessary if one correctly accounts for the bound state and the long-range forces. At the same time, the study serves as a crucial warning: in the smallest simulation boxes currently accessible to scientists, these tiny, ignored effects can become large enough to compromise the accuracy of the results. The researchers conclude that while the two methods are fundamentally equivalent, extracting precise physical data from lattice QCD simulations requires careful attention to the size of the box, especially when dealing with systems that are close to forming bound states. This work provides a clear roadmap for future studies of exotic particles, ensuring that the tools used to decode the universe's building blocks are both reliable and understood.

Technical Summary: On the Equivalence of Left-Hand-Cut and Three-Body Formalisms

Problem Statement
In lattice Quantum Chromodynamics (QCD), extracting scattering amplitudes from finite-volume energy spectra relies on quantization conditions, such as the Lüscher formalism. However, these standard relations break down in the presence of "left-hand cuts" (LHC), which arise from long-range interactions like one-particle exchange (OPE). These singularities invalidate the assumptions of standard finite-volume methods, particularly for systems involving sub-threshold states (e.g., dinucleon systems or the $T_{cc}^+(3875)$ ). To address this, two distinct approaches have recently been proposed: a relativistic three-body formalism treating the bound state as an emergent pole, and a two-body left-hand-cut formalism that explicitly isolates OPE diagrams in the diagrammatic expansion. While both aim to resolve the LHC problem, their analytical equivalence had not been rigorously established, nor had the magnitude of discrepancies arising from neglected finite-volume effects been quantified.

Methodology
The authors investigate the equivalence of these two formalisms using a specific scalar model: a system of three identical scalar particles of mass $m$ , where two form an S-wave bound state ( $b$ ) of mass $M$ ( $m < M < 2m$ ). The interaction involves short-range contact terms and one-particle exchange.

Analytical Proof:
- The authors reorganize the three-body integral equations (specifically the amputated scattering amplitude $d_S$ ) to demonstrate that they can be cast into the form of the two-body left-hand-cut amplitude ( $M_{cut}$ ).
- They define an effective coupling $g_{eff}$ and introduce a rescaled three-body OPE amplitude. By separating the OPE kernel into singular and non-singular parts, they show that the differences between the three-body and two-body kernels (denoted as $\epsilon$ and $\delta_2$ ) can be absorbed into a momentum-dependent, non-singular kernel $K_0^{(t)}$ .
- They prove that this $K_0^{(t)}$ can be mapped to the standard, momentum-independent $K_0$ of the two-body formalism via a specific integral equation, establishing that the infinite-volume amplitudes are equivalent up to exponentially suppressed terms.
- A similar mapping is performed for the finite-volume quantization conditions. They demonstrate that the three-body quantization condition (involving the function $F_3^{iso}$ ) reduces to the two-body condition (involving $M_0^{-1} + C_L + F$ ) when the finite-volume corrections to the two-body propagator and the OPE kernel are accounted for, again up to exponentially suppressed terms.
Numerical Verification:
- The authors perform numerical tests using the scalar model described above.
- Path 1: They compute finite-volume energy levels using the three-body quantization condition and extract the $K_0$ matrix by inverting the two-body quantization condition.
- Path 2: They solve the infinite-volume three-body integral equations to obtain the physical amplitude $M_{\phi b}$ , then match this to the two-body form to extract $K_0$ independently.
- They compare the $K_0$ matrices and the resulting physical amplitudes from both paths across various scattering lengths ($ma$) and lattice volumes ( $L$ ). They also analyze the convergence of the finite-volume sums with respect to momentum cutoffs.

Key Contributions and Results

Analytical Equivalence: The paper establishes that the relativistic three-body formalism and the two-body left-hand-cut formalism are analytically equivalent in the kinematic region where a two-body bound state exists. The three-body approach, which treats the bound state as a pole in the sub-channel, can be reorganized to reproduce the two-body amplitude structure, with the non-singular off-shell contributions absorbed into the $K$ -matrix.
Numerical Agreement:
- Above Threshold: For energies above the particle-bound-state threshold, the two methods agree at the sub-percent level.
- Below Threshold (Left-Hand Cut): When analytically continuing to the left-hand cut region, the agreement remains strong but degrades to the few-percent level.
Source of Discrepancy: The authors identify that the residual differences (few percent) are consistent with exponentially suppressed finite-volume effects that are neglected in the derivation of the equivalence. These effects scale as $e^{-mL}$ (single particle), $e^{-\kappa L}$ (bound state size, where $\kappa$ is the binding momentum), and $e^{-2\kappa_{th}L/\sqrt{3}}$ (proximity to the three-body threshold).
Practical Implication: The study reveals that while the two formalisms are theoretically equivalent, the "exponentially suppressed" corrections neglected in both can be numerically significant at currently accessible lattice volumes, particularly for shallow bound states. This limits the precision of amplitude extraction using either formalism in small volumes, even if the formalism correctly encodes the left-hand cut singularity.

Significance
The paper provides a rigorous theoretical bridge between two competing frameworks for handling left-hand cuts in lattice QCD. By proving their equivalence, it validates the two-body left-hand-cut formalism (developed in Refs. [40, 44]) as a viable alternative to the more complex three-body approach for systems dominated by a bound state. This is significant because the two-body formalism is a natural extension of existing Lüscher-like methods used in many lattice analyses, potentially simplifying the computational workflow for systems like the $T_{cc}^+(3875)$ or dinucleon scattering.

However, the authors modestly conclude that the numerical tests highlight a practical limitation: the equivalence holds only up to exponentially suppressed terms that can be large in small volumes. Therefore, while the formalisms are equivalent in principle, the accuracy of extracting physical amplitudes in current lattice simulations may be constrained by these finite-volume effects rather than the choice of formalism itself. The work clarifies the relationship between different regularization schemes and parametrizations of the $K$ -matrix, aiding in the systematic error analysis of future lattice QCD studies.

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