⚛️ phenomenology

$\lambda$ - $\rho$ mode mixing and radial Regge trajectories for triply heavy baryons $\Omega_{ccc}$ and $\Omega_{bbb}$ in the diquark picture

This paper investigates permutation-enforced $\lambda$ - $\rho$ mode mixing in triply heavy baryons $\Omega_{ccc}$ and $\Omega_{bbb}$ within the diquark picture, demonstrating that the Pauli exclusion principle necessitates mixed-mode eigenstates that replace pure Regge trajectories with novel sprouting-branch-like effective radial trajectories exhibiting $M\sim x^{2/3}$ scaling.

Original authors: Jiao-Kai Chen, Xin-Ru Liu, Qi Liu

Published 2026-10-07

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CC BY 4.0

Original authors: Jiao-Kai Chen, Xin-Ru Liu, Qi Liu

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

Deep within the heart of matter, protons and neutrons are not solid, indivisible spheres but rather bustling cities built from smaller, more fundamental particles called quarks. These quarks are bound together by a powerful force known as the strong interaction, which behaves somewhat like a stretchy rubber band that refuses to let the particles drift apart. Physicists have long used a framework called the "diquark picture" to make sense of how these particles organize themselves. In this view, two quarks can pair up tightly to form a small, stable unit—a diquark—which then orbits around a third quark, much like a moon orbiting a planet. This model helps scientists predict the properties of heavy particles made of three quarks, known as baryons. While we have observed many variations of these particles, some of the most extreme versions have remained invisible to our detectors. These are the triply heavy baryons, specifically those made of three identical heavy quarks: either three charm quarks or three bottom quarks. Because they have never been seen in a laboratory, they serve as a perfect theoretical testing ground to see if our understanding of how matter holds together is truly complete.

A new study by researchers at Shanxi Normal University in China takes a fresh look at these elusive particles, focusing on a subtle but critical rule that governs the quantum world: the Pauli exclusion principle. This principle dictates that identical particles, like the three quarks in these heavy baryons, cannot occupy the exact same state at the same time. In simpler terms, if you have three identical twins in a room, they must arrange themselves in a way that keeps them distinct from one another. For particles made of three different types of quarks, this arrangement is straightforward. However, when all three quarks are identical, the rules become much stricter. The researchers found that the standard way of describing these particles—treating them as a simple pair orbiting a single third particle—fails to respect this strict symmetry. If you try to describe a particle made of three identical quarks using the old, simple model, you end up with a description that violates the fundamental laws of physics.

To solve this, the team developed a new approach that forces the mathematical description of these particles to obey the rules of symmetry from the very beginning. They realized that the internal vibrations of the particle, which were previously thought to happen in two separate, independent ways, must actually mix together. Imagine the particle as having two types of internal movement: one where the two quarks in the pair wiggle against each other, and another where the entire pair wiggles against the third quark. In particles with different types of quarks, these two movements stay separate. But in the case of three identical quarks, the laws of physics force these two movements to blend into a single, complex pattern. The researchers constructed a new set of formulas to describe these blended states, effectively creating a map of how the mass of these particles should change as they become more excited.

Using this new framework, the team calculated the expected masses for the ground states and the first few excited states of the triply heavy baryons made of charm and bottom quarks. Their calculations predicted that the charm-based particle, known as $\Omega_{ccc}$ , should have a mass in its lowest energy state, while the bottom-based particle, $\Omega_{bbb}$ , should be much heavier. These numbers align well with other theoretical predictions, suggesting that the new method is on the right track. More importantly, the study revealed a unique and surprising pattern in how these particles behave as they gain more energy. Instead of following a single, smooth line of increasing mass, the energy levels of these particles branch out like a tree sprouting new limbs. The lowest energy family starts from the very bottom, but higher energy families do not start from the same point; they "sprout" from higher thresholds and then diverge, creating a distinct, branching structure.

This branching behavior is a direct consequence of the forced mixing between the two types of internal movements. It is a feature that does not exist in particles made of different quarks, where the movements remain separate and the energy levels follow a simpler, more predictable path. The researchers emphasize that this is an approximation based on the simplest possible mixing, and they acknowledge that more complex interactions could alter the details for highly excited states. Nevertheless, the study provides a crucial correction to how we view these heavy particles. By accounting for the internal structure and the strict symmetry requirements of identical particles, the team has shown that the simple, separate paths of the past are no longer sufficient. The reality of these triply heavy baryons is a more intricate, intertwined dance of internal forces, and their mass spectrum reflects this complexity in a way that is both novel and physically necessary.

Technical Summary: $\lambda$ - $\rho$ Mode Mixing and Radial Regge Trajectories for Triply Heavy Baryons

Problem Statement
The paper addresses the theoretical description of triply heavy baryons composed of three identical quarks, specifically $\Omega_{ccc}$ and $\Omega_{bbb}$ . While the diquark picture successfully describes baryons with non-identical constituents (e.g., $\Omega_{ccb}$ and $\Omega_{cbb}$ ) by decoupling $\rho$ -mode (internal diquark) and $\lambda$ -mode (diquark-quark relative) excitations, this separation fails for systems with three identical fermions. The Pauli exclusion principle imposes strict permutation symmetry constraints on the total wave function of $\Omega_{ccc}$ and $\Omega_{bbb}$ . Consequently, diquark-quark basis states lacking full symmetrization violate exchange symmetry, necessitating that physical eigenstates be superpositions of $\rho$ - and $\lambda$ -mode components. The authors note that existing literature largely focuses on pure $\lambda$ -trajectories or treats $\lambda$ and $\rho$ separately, with no prior works investigating mixed-mode Regge trajectories arising from this permutation-enforced mixing.

Methodology
The authors employ the diquark picture, treating the baryon as a system of a color-antitriplet diquark and a color-triplet quark.

Wave Function Construction: The total wave function is constructed as a tensor product of color, flavor, spin, and spatial components. For $\Omega_{ccc}$ and $\Omega_{bbb}$ , the color and flavor parts are antisymmetric and symmetric, respectively. To satisfy the overall antisymmetry required by the Pauli principle, the spin and spatial parts must be combined appropriately.
Symmetrization and Mixing: The authors explicitly construct permutation-symmetrized spatial wave functions for radially excited $S$ -wave states with $J^P = (3/2)^+$ . Under a "minimal-mixing" assumption, they consider the mixing between states $|n, N\rangle$ and $|N, n\rangle$ (where $n$ and $N$ are radial quantum numbers for $\rho$ and $\lambda$ modes, respectively). The physical spatial wave function is defined as $\phi_{nsNS+} = (\phi_{nsNS} + \phi_{NsnS})/\sqrt{2}$ for $n \neq N$ .
Mass Formulation: The spin-independent mass of these symmetrized states is derived as the average of the unmixed masses: $M_{nsNS+} = (M_{nsNS} + M_{NsnS})/2$ . The authors utilize a general Regge trajectory ansatz based on the spinless Salpeter equation and the Cornell potential, which yields a nonlinear behavior $M \sim x^{2/3}$ for heavy systems, where $x$ represents radial or orbital quantum numbers.
Parameterization: Parameters for quark masses, string tension, and potential constants are taken from previous works. Coefficients for the trajectory formulas are determined via empirical fitting relations dependent on the reduced mass of the system. Spin-dependent contributions are neglected as they are found to be small (order of tens of MeV) and do not alter the qualitative Regge behavior.

Key Contributions

Formulation of Mixed-Mode Trajectories: The paper derives explicit formulas for mixed-mode Regge trajectories for $\Omega_{ccc}$ and $\Omega_{bbb}$ that account for the permutation-enforced $\lambda$ - $\rho$ mixing. This contrasts with the fixed diquark picture for non-identical baryons where pure $\rho$ and $\lambda$ trajectories exist independently.
Symmetrized Mass Relations: The authors provide a specific prescription for calculating the masses of physical eigenstates by symmetrizing the basis states, resulting in the relation $M_{nrNr+} = M_{Nrnr+}$ .
Identification of Novel Trajectory Behavior: The study identifies a "sprouting-branch-like" behavior in the effective radial Regge trajectories. Unlike standard trajectories originating from a common ground state, the mixed-mode trajectories for higher excited families (e.g., $|2^3s_1, N^4S_{3/2}\rangle_+$ ) "sprout" at finite thresholds of the total radial quantum number ( $n_r + N_r = 1, 2, \dots$ ) rather than at zero.

Results

Mass Predictions: The authors calculate the masses of radially excited $J^P = (3/2)^+$ states for $\Omega_{ccc}$ and $\Omega_{bbb}$ . For $\Omega_{ccc}$ , the ground state is predicted at 4.65, with excited states reaching approximately 6.23. For $\Omega_{bbb}$ , the ground state is 14.27, with excited states up to 15.52.
Trajectory Fitting: The calculated masses are fitted to simple functional forms of the type $M = m_R + \beta x (x + c_0)^{2/3}$ . These fits confirm that the effective mixed-mode trajectories retain the $M \sim x^{2/3}$ scaling behavior characteristic of heavy-heavy confinement dynamics, despite the complex mixing.
Comparison: The predicted masses show reasonable agreement with existing theoretical predictions from various models (e.g., lattice QCD, potential models), as detailed in the paper's comparison tables.

Significance and Claims
The paper claims to be the first investigation of mixed-mode trajectories arising specifically from permutation-enforced $\lambda$ - $\rho$ mixing in the diquark picture. The authors emphasize that for baryons with three identical quarks, the concept of pure $\rho$ - or $\lambda$ -trajectories ceases to exist; only mixed-mode effective trajectories are physically meaningful. The "sprouting-branch" morphology is presented as a direct consequence of Pauli exclusion principles combined with the minimal-mixing assumption.

The authors assert that accounting for the internal structure and substructure of baryons is indispensable for constructing these trajectories. Without such structural considerations, trajectory parameters and functional forms could only be obtained through pure fitting against data. The work provides a phenomenological framework for estimating the masses of unobserved triply heavy baryons and highlights the necessity of permutation symmetrization in few-body heavy quark systems. The authors modestly note that their effective trajectories are approximate descriptions for low-lying states, as systematic deviations may occur for highly excited states where further mixing channels (involving different orbital angular momenta or total radial sums) become relevant.

Technical Summary: λ\lambdaλ-ρ\rhoρ Mode Mixing and Radial Regge Trajectories for Triply Heavy Baryons

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Technical Summary: $\lambda$ - $\rho$ Mode Mixing and Radial Regge Trajectories for Triply Heavy Baryons