Quantum chromodynamics (QCD) is the fundamental theory that explains the strong force. At elevated momentum transfer, where asymptotic freedom justifies the use of perturbation theory, it has undergone quantitative validation in numerous experiments. Hadrons are clearly quark-bound particles that are held together through interactions facilitated by gluons. However, we still have not achieved a quantitative and predictive theory of states confined by quarks and gluons. We assess our understanding of the low-energy behavior of quarks and gluons in a setup known as hadron spectroscopy. Peaks in the invariant mass distributions of their decay products were observed, offering proof of pentaquarks. These assertions, which precisely reflected the perspective of the particle physics community during that period, lacked backing from further data or more precise experiments until an investigation of specific decays by LHCb. The LHCb collaboration’s discovery of pentaquark states in 2015 signified a pivotal moment in hadronic spectroscopy [1]-[3]. Among these unusual states, P_c(4550)⁺ has gained significant interest because of its substantial mass, potential molecular configuration, and its closeness to thresholds related to charmed baryons and anti-charmed mesons [2]. It displays a contrasting parity compared to the P_c(4380)⁺ state. Theoretical explanations of P_c(4550)⁺ typically categorize into two primary types: Hadronic molecular models, which view the pentaquark as a loosely bound entity like D ¯ * Σ c , and compact models, in which the five quarks are closely linked within a color-bound core [4]-[6]. Different theoretical models have been formulated, spanning from compact diquark frameworks [7][8] to hadronic molecules held together by meson exchange [9]-[11]. The molecular explanation is backed by heavy-quark spin symmetry (HQSS) [12]-[14], which forecasts almost degenerate partner states. Nonetheless, the internal dynamics and decay characteristics continue to be a topic of discussion [15][16]. Nonetheless, the exact characteristics of P_c(4550)⁺ are still contested. Conventional molecular methods that focus solely on one primary channel fall short of explaining specific details, like the measured decay widths or possible interference phenomena among closely situated states. In this paper, we consider that the pentaquark P_c(4550)⁺ may be interpreted as a mixed molecular state, resulting from a dynamic coupling between two nearly-degenerate configurations via a residual spin-orbit interaction analogously to fine-structure splittings in nuclear and atomic systems [17]-[19] and we so explore whether such a 1coupling can explain spatial structure of P_c(4550)⁺, by solving the coupled-channel Schrodinger equation with a non-perturbative mixing term. This mixing is modeled through a coupled double-well potential, where each well represents a distinct hadronic configuration with similar energy. This type of coupling, influenced by spin-orbit phenomena familiar in nuclear physics and quantum chemistry, might produce intricate spectral features that standard models do not account for. The LHCb collaboration’s discovery of pentaquarks has generated considerable interest in hadron physics, especially concerning states like P_c(4450) and P_c(4457). These unusual baryons challenge conventional quark models and offer different interpretations about their composition. Current interpretations indicate that pentaquarks could be basic molecular states (for instance, D ¯ * Σ c ) or dense baryonlike structures. This research intends to suggest a mixed double well model that includes residual spin-orbit coupling, drawing influence from low-energy QCD. The main goal is to clarify the characteristics of the P_c(4550)⁺ state and its significance for experimental findings.

The examination of linked molecular states signifies an intriguing domain of quantum chemistry and particle physics. Interactions among heavy particles, like mesons, are especially fascinating because of their intricate nature and abundance. These states can be considered quantum systems that interact with each other, leading to considerable mixing effects. Coupled molecular states emerge from the active interplay of the quantum fields linked to each state. and we compute the system with two Schrödinger equations linked by a spin-orbit term. Every channel is characterized by a Yukawa-like potential inspired by meson exchange [19]-[21]. The mesons D ¯ * and D ¯ engage with the baryons Σ c and Σ c * via meson exchange [22][23]. This interaction is characterized by a paired double well potential, which represents the energy levels of the states concerning the distance between the particles. The interplay between these states can be illustrated using an energy level diagram. When the two states are near in energy, the likelihood of transition between them rises, resulting in a change in the energy levels. This dynamic is essential for comprehending decay processes and particle distributions. The physical rationale for a coupled double well originates from the interaction dynamics among these molecular states, in which each state can affect the other and result in mixing. We consider two interacting channels:

Channel 1: D ¯ * Σ c and Channel 2: D ¯ Σ c *

The radial wave functions ϕ 1 ( r ) and ϕ 2 ( r ) satisfy coupled Schrodinger equations:

where μ₁, μ₂ are the reduced masses, V₁, V₂ the diagonal potentials, and V₁₂ the offdiagonal spin-orbit induced coupling. To Yukawa’s fundamental hypothesis, which holds that massive-particle exchange mediates the nuclear force. Although the fundamental understanding of that concept has changed throughout time (from a more fundamental to a more effective approach), the major driving force has been constant. The qualities of particle interaction that have been observed empirically seem to be best described by massive particle exchange. The connection between the mass of the exchanged particle and the range of the associated force reflects this feature with respect to the finite range of the nuclear force. The fact that this subdivision is still physically most relevant today, especially in light of QCD inspired approaches to the nuclear force, shows how forward thinking this concept was. To determine the energies of bound states in the pentaquark system P_c(4550)⁺, we utilized the factor f_reg (regularization factor) that is essential in the regularization of short-range divergent potentials for computing regularized potentials and approximate wave functions. The regulatory factor freg is implemented to manage the behavior of the potentials at short ranges. Indeed, meson-exchange interactions may lead to unphysical or divergent contributions when r → 0. The role of the regulator is to suppress these effects while preserving the physical structure of the long-range interactions. The formulas used are:

And for ajust the interaction strength for all potentials, we use:

with: c₀ = −50 for V₁(r), −30 for V₂(r), +20 for V_c(r) and g = −10.

And for Yukawa potential originates from the exchange of a massive particle with massμ:

where: g is coupling constant, which determines the strength of the interaction.μ: Mass of the exchanged particle. r: Distance between the two particles. A regularization factor f_reg(r) is introduced such that:

Here, f_reg(0) = 0 and f_reg(r) → 1 for r ≫ ζ . where ζ denotes a cutoff parameter ( ζ = 1.7 GeV). This term acts as a filter by eliminating contributions associated with large momentum transfers, which correspond to ultrashort distances. which, after Fourier transformation into coordinate space, leads to a regularized Yukawa interaction. The regulator suppresses the overly strong 1/r attraction of the bare Yukawa potential, while at long distances it preserves the characteristic exponential decay exp(−μr)/r. Formulas used for Wave Functions:

which:

ϕ i ( r ) is the radial wave function of channel i, ℓ i is the orbital moment, N normalization constant and β> 0.

The energy associated with a normalized trial function ϕ ( r ) is obtained by the expectation value of the Hamiltonian:

For two channels the elements of the Hamiltonian matrix are written:

where:

And h ⌢ 1   2 ( r ) = V 1   2 ( r a d ) ( r ) is the radial part of the coupling.

The wave function is normalized as:

The corresponding probability densities are:

We will discuss the implications of our findings for future experiments aimed at probing the structure of pentaquarks and other exotic states.

This study proposes a comprehensive model for understanding the P_c(4550)⁺ pentaquark as a coupled molecular state influenced by spin-orbit interactions. By employing a two-channel approach and deriving the necessary coupled equations, we hope to bridge theoretical predictions with experimental observations, paving the way for advancements in hadron physics. And the coupled-channel model with Yukawa-type interactions successfully reproduces the observed doublet structure of P_c(4440)⁺ and P_c(4457)⁺. The inter-channel coupling is essential for the emergence of these molecular states. So study of exotic hadrons, which cannot be interpreted as ordinary three-quark baryons or quarkantiquark mesons, would offer us a good opportunity to investigate heigh-energy quark dynamics. Coupled states and double well potentials also play an important role within the framework of the Standard Model of particle physics. By understanding these interactions, we can better grasp the decay mechanisms of particles as well as production processes in high-energy collisions. These findings support the interpretation of the P_c(4550)⁺ as a dynamically generated hadronic molecule rather than a compact multiquark configuration. More generally, our study highlights the necessity of coupled-channel analyses with short-range corrections in the investigation of exotic hadrons. Future extensions of this work may include the study of decay widths, three-channel couplings, and comparisons with lattice QCD simulations to further test the molecular picture.