Author: Denis Avetisyan
New research explores the potential formation of complex, three-body states involving charmed and bottomed mesons, challenging conventional understandings of strong force interactions.
This study investigates the existence of $DD^ar{K}^$ and $BB^K^$ three-body molecular states and their connection to the Zc(3900) virtual state.
The precise nature of exotic hadronic states remains a central challenge in understanding the strong force, particularly regarding multi-body interactions. This work, ‘Existence of the $DD^\bar{K}^$ and $BB^K^$ three-body molecular states’, investigates the potential formation of three-body molecular states in the D^<i>K^</i> and B^<i>K^</i> systems within a one-boson-exchange model, demonstrating a strong correlation between their existence and the pole position of the Z_c(3900) virtual state. Our analysis reveals conditions for both bound states and resonances, finding more relaxed criteria for the bottom analog system. Could future experimental measurements of Z_c(3900) or the observation of three-body states in D\bar{D}K\pi\pi channels ultimately resolve the underlying structure of these complex hadronic systems?
The Strong Force: Unveiling the Hadronic Landscape
The strong force, fundamentally described by the theory of Quantum Chromodynamics (QCD), governs the interactions between quarks and gluons, the constituents of all hadrons. This force isn’t simply attractive; its unique properties, including asymptotic freedom and color confinement, dictate how quarks bind together to form composite particles like mesons. Mesons, comprised of a quark and an antiquark, represent a crucial testing ground for QCD, as their mass spectrum and decay patterns are directly influenced by the strong interaction’s dynamics. Accurately modeling this interaction is therefore paramount; understanding the intricacies of the strong force allows physicists to predict the properties of not only well-known mesons like the B and D mesons, but also to explore the exotic hadronic states continually being discovered, ultimately refining the broader understanding of matter itself.
Mesons, fleeting composite particles born from the union of a quark and its antiquark counterpart, represent a crucial tier in the standard model of particle physics. Examples such as the B, D, and K* mesons aren’t merely exotic curiosities; they act as sensitive probes of the strong force, providing insights into the very fabric of matter. Their internal dynamics, governed by Quantum Chromodynamics, are extraordinarily complex, necessitating sophisticated theoretical frameworks to accurately predict properties like mass, decay rates, and interaction cross-sections. A precise understanding of these particles is vital, not only for validating the standard model but also for interpreting experimental results from facilities like the Large Hadron Collider and for furthering research into the behavior of matter under extreme conditions, such as those found in neutron stars.
The inherent complexity of the strong force presents a significant challenge to accurately modeling systems containing multiple hadrons. Traditional computational techniques, often relying on perturbative approaches, falter when confronted with the non-linear nature of Quantum Chromodynamics at the energy scales relevant to hadron interactions. These methods struggle to account for the constant creation and annihilation of virtual particles – gluons and quark-antiquark pairs – that mediate the strong force and dramatically influence the behavior of multi-hadron systems. Consequently, predicting the properties and decay patterns of these complex arrangements – crucial for understanding phenomena ranging from the quark-gluon plasma to the structure of neutron stars – remains a substantial undertaking, demanding innovative theoretical frameworks and computational strategies to overcome these limitations.
Modeling Hadronic Interactions: The One-Boson-Exchange Approach
The One-Boson-Exchange (OBE) model describes the strong nuclear force, specifically the interactions between hadrons, as resulting from the exchange of virtual mesons. These mesons, such as pions, rho mesons, and omega mesons, act as force carriers, mediating the interaction between constituent quarks within the hadrons. The strength and range of the interaction are determined by the mass and coupling constants of the exchanged mesons; heavier mesons contribute to shorter-range interactions, while lighter mesons mediate longer-range forces. This framework posits that the strong force is not a fundamental interaction, but rather an effective theory arising from the exchange of these composite particles, offering a means to calculate hadron-hadron scattering cross-sections and binding energies.
The One-Boson-Exchange model employs an Effective Lagrangian formalism to derive the interaction potential between hadrons. This Lagrangian includes terms representing the exchange of vector mesons, such as ρ and ω, which mediate short-range repulsive interactions and contribute to the tensor force. Simultaneously, the Lagrangian incorporates the exchange of scalar mesons, most notably the sigma meson (σ), responsible for the long-range attractive force and contributing to the central potential. The complete potential is constructed as a sum of these vector and scalar exchange contributions, allowing for a description of both attractive and repulsive aspects of the hadron-hadron interaction and ultimately, the binding of hadronic systems.
The One-Boson-Exchange model specifically targets the residual strong force – the attractive potential that binds hadrons such as protons and neutrons into composite structures like nuclei. This binding arises from the exchange of virtual mesons, which mediate the interaction. The strength and range of this attractive force are determined by the mass and coupling constants of the exchanged mesons, with lighter mesons generally contributing to longer-range interactions. Capturing the characteristics of these exchanged mesons within the Effective Lagrangian allows for the calculation of nuclear potentials, ultimately aiming to reproduce observed hadron binding energies and scattering data. The model’s success relies on accurately representing the contribution of different meson exchanges – including π-mesons, ρ-mesons, and ω-mesons – to the overall attractive potential.
![Effective potential calculations for the <span class="katex-eq" data-katex-display="false">Z_c(3900)[D\bar{D}^*]^{C=-1}_{I=1}</span> system, considering ρ, ω, and σ meson exchanges (represented by dashed blue, dotted yellow, and dot-dashed green lines, respectively), demonstrate a total potential (solid red line) calculated with a cutoff of 1.10 GeV and parameter values <span class="katex-eq" data-katex-display="false">(R_\beta, R_\lambda, R_s) = (1.0, 1.0, 2.0)</span>.](https://arxiv.org/html/2602.12010v1/x2.png)
Numerical Solutions: Mapping the Three-Body Schrödinger Equation
The Gaussian Expansion Method (GEM) is employed to numerically solve the three-body Schrödinger equation for systems such as DDK and BBK. This method represents the wave function as a linear combination of Gaussian functions, allowing for an efficient expansion basis suitable for handling the short-range interactions typical of hadronic systems. By varying the parameters of these Gaussian functions – including their widths and centers – GEM constructs a basis set that effectively describes the spatial correlation between the three particles. The Schrödinger equation is then discretized and solved using this expanded wave function, yielding approximate solutions for the system’s energy and wave function. The accuracy of the solution is dependent on the number of Gaussian functions used in the expansion and the range of parameters explored during the calculation.
The Complex Scaling Method (CSM) is employed as a complementary technique to the Gaussian Expansion Method (GEM) for analyzing three-body systems. CSM involves a non-physical rotation of the spatial coordinates in the complex plane, effectively transforming the Schrödinger equation into one where bound states manifest as poles on the complex energy plane. This allows for the identification of both resonant and bound states that may not be readily apparent through real-space calculations. Specifically, the location of these poles reveals the energies and lifetimes of resonant states, while the existence of poles with negative imaginary parts indicates the presence of bound states; these bound states are characterized by their binding energy, determined from the pole positions. The method is particularly useful for systems with weakly bound or highly resonant states, providing crucial insights into their stability and decay properties.
Application of the Gaussian Expansion Method and Complex Scaling Method to hadronic systems modeled with the One-Boson-Exchange potential allows for the prediction of system properties, including the existence and energy levels of bound states. Calculations reveal the emergence of a bound state within these systems, characterized by a binding energy on the order of a few MeV. This binding energy is determined through numerical solution of the three-body Schrödinger equation using the specified computational techniques, providing quantitative insight into the stability of these complex multi-hadron configurations.
The Zc(3900) Meson: Constraining the Hadronic Interaction
The enigmatic virtual state of the Z_c(3900) meson serves as a critical regulator of the Sigma Exchange interaction, a force governing the binding of certain exotic hadrons. This interaction, responsible for mediating attraction between particles, isn’t a fixed quantity; its strength is demonstrably influenced by the properties of the Z_c(3900). Calculations reveal that the proximity of the Z_c(3900) pole to the particle threshold directly constrains how strongly particles can interact via Sigma Exchange. Effectively, the virtual state acts as a tuning parameter, dictating the feasibility of forming bound states like the DDK^<i> and BBK^</i> molecules, and providing a novel means of probing the underlying dynamics of hadron interactions.
Calculations concerning the exotic three-body systems, DDK and BBK, have been significantly improved by incorporating constraints derived from the virtual state of Zc(3900). Previous models often treated the interaction between these particles with broad assumptions; however, acknowledging the specific properties of Zc(3900) – particularly its influence on the strong force – allows for a more precise mapping of the potential energy surface. This refinement reveals how subtle changes in the interaction strength impact the formation of bound states, clarifying the delicate balance needed for these molecules to exist. The resulting models demonstrate a heightened ability to predict the characteristics of DDK and BBK, offering a more nuanced understanding of the forces governing these unusual hadronic systems and paving the way for further investigation into the nature of exotic matter.
Calculations reveal a compelling link between the existence of exotic molecular states and the characteristics of the Z_c(3900) virtual particle. Specifically, a bound state designated DDK emerges when the energy pole of the Z_c(3900) lies within approximately -10 MeV of the relevant threshold. A similar, though more stringent, condition governs the formation of the BBK bound state, requiring the Z_c(3900) pole to be situated within -25 MeV of its threshold. This establishes a quantifiable correlation: the closer the Z_c(3900) virtual state is to zero energy, the more likely these complex molecular configurations are to exist, suggesting the Z_c(3900) plays a crucial role in mediating the binding forces within these systems.
The pursuit of defining hadronic molecular states, as detailed in this investigation of $DD^ar{K}^$ and $BB^K^$ systems, echoes a commitment to discerning fundamental order within seeming complexity. The research meticulously dissects three-body interactions, seeking to establish the conditions for stable, yet exotic, particle formations. This aligns with Leonardo da Vinci’s observation: “Simplicity is the ultimate sophistication.” The study’s reliance on methods like the Complex Scaling Method and Heavy Quark Symmetry exemplifies a reductionist approach-stripping away extraneous variables to reveal underlying principles. The correlation found with the Zc(3900) virtual state underscores the idea that clarity emerges not from adding layers, but from identifying the essential connections.
Where This Leads
The correlation between three-body states and the Zc(3900) is notable, yet correlation is not causation. Further investigation must define if this pole represents a genuine dynamical origin, or merely a coincidental proximity. Abstractions age, principles don’t. The search for exotic hadrons requires precision, not proliferation of models.
Current methods, like complex scaling, provide glimpses, but lack the definitive power to fully resolve the internal dynamics of these systems. Three-body interactions are inherently difficult. Every complexity needs an alibi. Future work should focus on incorporating more realistic potentials, and exploring the role of hidden color configurations.
Heavy quark symmetry offers a guiding principle, but its limitations must be acknowledged. The extension of this research to other heavy-light combinations – and ultimately, to systems without heavy quarks – represents a crucial test. The goal isn’t to find more particles, but to understand the fundamental forces that bind them.
Original article: https://arxiv.org/pdf/2602.12010.pdf
Contact the author: https://www.linkedin.com/in/avetisyan/
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2026-02-15 15:19