Momentum Distribution of a mass-imbalanced three-body system across the Efimov-Unatomic transition

arXiv:2607.13769 · cond-mat.quant-gas, physics.atom-ph · Submitted 2026-07-15 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Momentum Distribution of a mass-imbalanced three-body system across the Efimov-Unatomic transition".

Mira: Here's the gist of the paper. The gist The asymptotic momentum distribution at Dc is governed by a distinct logarithmic scaling structure, which differs fundamentally from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime.

Kai: Emergence of Scaling Regimes The study investigates how dimensionality acts as an effective control parameter driving the evolution from discrete to continuous scale invariance in a mass-imbalanced three-body system.

Mira: The analysis identifies a narrow interval separating these regimes, referred to as an intermediate scaling regime (SIR), whose extent is strongly controlled by the mass ratio.

Kai: Momentum Space Structure at Transition The momentum distribution at the critical dimension Dc is governed by a distinct scaling structure characterized by a logarithmic dependence that differs fundamentally from both the log-periodic oscillations of the Efimov regime and the power-law scaling of the unatomic regime.

Paper summary: Kai: So, to recap what we’ve seen, this paper investigates the single-particle momentum distribution and contact parameters for massimbalanced three-body systems at D c, where the transition between discrete and continuous scale invariance happens when tuning dimension between three and two.

Mira: They show that at this critical dimension D c, the asymptotic momentum distribution follows a distinct logarithmic scaling structure, which is fundamentally different from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime.

Lev: The main claim is that this specific structure requires introducing an additional three-body contact parameter associated with a quadratic logarithmic contribution.

Kai: This C three'' term leads to a finite and well-defined description of the momentum tail at the transition, which was incomplete in previous analyses where the three-body contribution became formally divergent <ref:2607.13769#pg1>.

Mira: They point out that this coefficient C three'' depends sensitively on the mass imbalance, changing sign depending on mass configurations and vanishing for identical particles <ref:2607.13769#pg1>.

Lev: For heavy-light-light systems, they found that the subleading three-body term vanishes at a characteristic momentum, leaving the leading two-body contribution at the order retained in the asymptotic expansion.

Kai: They also analyze a narrow intermediate region connecting those Efimov and unatomic regimes, identifying it as an intermediate scaling regime whose extent is strongly controlled by the mass ratio.

Mira: Essentially, this work establishes that D c is a regime with emergent scaling properties and provides quantitative momentum-space signatures for probing the transition between discrete and continuous scale invariance in few-body quantum systems.

Lev: It sets up a clear picture of how dimensionality acts as an effective control parameter driving the system across this boundary.

Conclusion: Kai: Looking at this whole thing, the title itself points to something central: how momentum distribution behaves when you cross that boundary between discrete and continuous scaling in a mass-imbalanced three-body system.

Mira: What’s important for the broader context is that they found a new way to characterize that transition using these specific contact parameters.

Lev: So, if you think about this for experimentalists, it means they can look at the high-momentum tail of a momentum distribution and see how the mass imbalance affects its shape right near D c.

Kai: It connects the abstract mathematical concept of scale invariance directly to measurable features in momentum space that aren't obvious from just looking at energy spectra.

Mira: The implication is that by controlling the mass ratio or tuning the effective spatial dimension, we can move systems across this phase transition and observe fundamentally different scaling behaviors.

Lev: It suggests that probing these regimes might reveal underlying conformal symmetries in few-body interactions in ways that are currently hard to access.

Instituto Tecnol´ogico de Aeron´autica · Universit´e Paris-Saclay · Instituto de F´ısica Teórica, Universidade Estadual Paulista

cond-mat.quant-gas, physics.atom-ph

Submitted: 2026-07-15

Updated: 2026-10-20

DOI: 10.1103/ngg6-1mk4

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist The asymptotic momentum distribution at Dc is governed by a distinct logarithmic scaling structure, which differs fundamentally from both the log-periodic behavior of Efimov states and the

Key concepts

Critical Dimension ($D_c$)
This is the specific spatial dimension where the physics of three-body systems undergoes a fundamental change, transitioning from one type of scale invariance (discrete) to another (continuous). Tuning the system's dimension across this point reveals emergent scaling behaviors not seen in standard regimes.
Intermediate Scaling Regime (SIR)
This is a narrow region between the Efimov regime and the unatomic regime where neither description fully applies. The properties and size of this intermediate region are strongly influenced by the mass ratio of the particles, acting as a bridge between different physical behaviors.
Quadratic-Logarithmic Contribution ($C_3''$)
This term is an extra component in the asymptotic momentum distribution scaling at $D_c$. Its coefficient depends heavily on how imbalanced the masses are and can change sign. It dictates how three-body contributions cancel at a specific momentum, leaving the two-body interaction dominant.
Scale Invariance Transition
This refers to the shift in how physical properties scale with changes in system parameters. The paper shows that tuning dimensionality acts as a control parameter that drives this transition, moving the system from discrete scaling (like Efimov states) to continuous scaling (like power laws).

Summary

The gist The asymptotic momentum distribution at Dc is governed by a distinct logarithmic scaling structure, which differs fundamentally from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime.

Emergence of Scaling Regimes

The study investigates how dimensionality acts as an effective control parameter driving the evolution from discrete to continuous scale invariance in a mass-imbalanced three-body system. The analysis identifies a narrow interval separating these regimes, referred to as an intermediate scaling regime (SIR), whose extent is strongly controlled by the mass ratio.

Momentum Space Structure at Transition

The momentum distribution at the critical dimension Dc is governed by a distinct scaling structure characterized by a logarithmic dependence that differs fundamentally from both the log-periodic oscillations of the Efimov regime and the power-law scaling of the unatomic regime. The asymptotic expansion reveals a logarithmic hierarchy involving constant, linear, and quadratic logarithmic terms. This structure requires the introduction of an additional three-body contact parameter C''3 associated with a quadratic logarithmic contribution.

Contact Parameters and Mass Imbalance

The coefficient C''3 depends sensitively on the mass imbalance and changes sign across different mass configurations. This sign inversion plays a crucial role in determining the structure of the high-momentum tail at the critical dimension. For systems where C''3 > 0, three-body contributions can cancel at a finite momentum scale, leaving the asymptotic tail entirely determined by the two-body contact C2 0, the three-body contributions can cancel at a finite momentum scale">.

Experimental Signatures and Connections

The mass dependence of the contact parameters provides a direct and accessible signature of the interplay between mass imbalance and dimensionality. In ultracold atomic systems, mixtures such as 133Cs–6Li2 naturally realize the HLL configuration and can be tuned close to unitarity via Feshbach resonances. This cancellation point could be probed through measurements of the high-momentum tail of the momentum distribution. These results establish a concrete bridge between ultracold atoms and nuclear few-body systems.

Role of Intrinsic Scales

In nuclear halo systems, intrinsic physical scales such as finite effective range and centrifugal barriers can play a role analogous to external confinement, potentially driving the system toward the boundary between discrete and continuous scaling. These effects may play a role analogous to geometric confinement in atomic systems, effectively driving the system toward the boundary between discrete and continuous scale invariance.

Conclusion on Dimensionality Control

Tuning dimensionality provides a powerful route to explore the interplay between few-body correlations and scale invariance. The emergence of a distinct logarithmic structure at the critical dimension, together with its signatures in momentum-space observables, opens new directions for experimental and theoretical investigations in universal few-body physics.

How it works

  1. Dimensionality acts as an effective control parameter driving the evolution from discrete to continuous scale invariance. The Efimov regime, characterized by log-periodic behavior and an imaginary scaling exponent, evolves continuously toward a regime where the scaling exponent becomes real and the system exhibits power-law behavior.

  2. The critical dimension Dc is where the Efimov parameter vanishes and the scaling structure is qualitatively modified. This transition point cannot be described within standard scaling frameworks and requires a refined treatment of underlying three-body correlations.

  3. The asymptotic momentum distribution at Dc exhibits a logarithmic hierarchy involving constant, linear, and quadratic logarithmic terms. This structure requires an additional three-body contact parameter C''3 associated with a quadratic logarithmic contribution.

  4. The coefficient C''3 depends on the mass imbalance and changes sign across different mass configurations. This sign inversion plays a crucial role in determining the structure of the high-momentum tail at the critical dimension.

  5. The cancellation of three-body contributions can occur for HLL systems where C''3 > 0, leaving the asymptotic tail entirely determined by the two-body contact C2 0, the three-body contributions can cancel at a finite momentum scale">. This cancellation defines a momentum scale beyond which the asymptotic behavior is effectively governed solely by the two-body contact C2.

  6. The intermediate scaling regime (SIR) exists between Dc and D, where neither Efimov nor unatomic descriptions are sufficient. The extent of this regime is strongly controlled by the mass ratio.

  7. The mapping between effective dimension and external confinement aspect ratio allows probing this regime experimentally.

Improvements for AI systems

  1. Bold header: Momentum Distribution Analysis at Critical Dimension

This improved system can accurately characterize the asymptotic behavior at Dc, which is described as a distinct logarithmic scaling structure that differs fundamentally from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime.

  1. Bold header: Extraction of Quadratic Logarithmic Parameter

The AI can now reliably identify and quantify the additional three-body contact parameter associated with a quadratic logarithmic contribution, which is essential because it is absent in both the Efimov and unatomic limits and emerges uniquely at the critical dimension.

  1. Bold header: Mass Imbalance Dependent Tail Suppression

The system will be able to predict how mass imbalance affects the asymptotic tail, specifically noting that for heavy-light-light systems, the three-body contributions can cancel at a finite momentum scale, leaving the tail entirely determined by the two-body contact.

  1. Bold header: Intermediate Scaling Regime Mapping

The AI can map experimental parameters to physical regimes by using the relation that allows one to translate variations in D into experimentally tunable confinement parameters, thereby enabling probing of the narrow intermediate region connecting the Efimov and unatomic regimes.

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