[Paper Review] Angular Momentum Dependent Quark Potential of QCD Traits and Dynamical O(4) Symmetry

C. B. Compean, M. Kirchbach|ArXiv.org|Sep 29, 2006

Quantum Chromodynamics and Particle Interactions5 references3 citations

TL;DR

This paper proposes the trigonometric Rosen-Morse potential (tRMP) as a realistic quark potential that reproduces key QCD traits: Coulomb-like at short distances, linear confinement at long distances, and exact O(4) dynamical symmetry. It shows the tRMP, when interpreted as angular momentum-dependent, exactly reproduces the SU(2)I ⊗ O(4) mass degeneracy patterns and level splittings of non-strange baryons (N, Δ) below 2500 MeV, providing exact energies and wave functions on par with harmonic oscillator and Coulomb potentials.

ABSTRACT

A common quark potential that captures the essential traits of the QCD quark-gluon dynamics is expected to (i) interpolate between a Coulomb-like potential (associated with one-gluon exchange) and the infinite wall potential (associated with trapped but asymptotically free quarks), (ii) reproduce in the intermediary region the linear confinement potential (associated with multi-gluon self-interactions) as established by lattice QCD calculations of hadron properties. We first show that the exactly soluble trigonometric Rosen-Morse potential possesses all these properties. Next we observe that this potential, once interpreted as angular momentum dependent, acquires a dynamical O(4) symmetry and reproduces exactly quantum numbers and level splittings of the non-strange baryon spectra in the SU(2)_I* O(4) classification scheme according to which baryons cling on to multi-spin parity clusters of the type (K/2,K/2)*[(1/2,0) + (0, 1/2)], whose relativistic image is ψ_{μ_{1}...μ_{K}}. Finally, we bring exact energies and wave functions of the levels within the above potential and thus put it on equal algebraic footing with such common potentials of wide spread as are the harmonic-oscillator- and the Coulomb potentials.

Motivation & Objective

• To identify a single quark potential that captures essential QCD traits: short-range Coulomb-like behavior, intermediate linear confinement, and long-range infinite wall behavior.
• To explain the observed mass degeneracy patterns in non-strange baryon spectra (N, Δ) below 2500 MeV, particularly the K-pair series of resonances with alternating parities and spins.
• To establish a dynamical O(4) symmetry in a quark potential that reproduces the SU(2)I ⊗ O(4) classification of baryons and their relativistic Rarita-Schwinger field realization.
• To provide exact analytical solutions (energies and wave functions) for the potential, placing it on equal algebraic footing with standard potentials like harmonic oscillator and Coulomb.

Proposed method

• The trigonometric Rosen-Morse potential is used as a solvable potential that interpolates between Coulomb-like and infinite wall-like behavior, with a shape matching lattice QCD results for confinement.
• The potential is reinterpreted as angular momentum-dependent by replacing the standard centrifugal barrier with a csc²(r/d) term, which introduces a non-standard centrifugal barrier that enables O(4) symmetry.
• The potential's exact solvability is leveraged to derive closed-form expressions for energy levels and radial wave functions using Romanovski polynomials.
• The quantum numbers (K, l, m) of the O(4) representation are mapped to baryon states, and the energy spectrum is shown to match the observed mass formula M(σ;I) = -fI/σ² + gI(σ²-1)/4.
• The relativistic image of the O(4) multiplets is identified as totally symmetric rank-K Lorentz tensors ψμ₁…μₖ, corresponding to Rarita-Schwinger fields of spin (K+1/2).
• The algebraic Hamiltonian from the interacting boson model (IBM) for baryons is mapped to the tRMP potential, establishing a direct link between group-theoretical symmetry and a physical potential.

Experimental results

Research questions

• RQ1 Can a single quark potential reproduce the essential QCD traits: short-range Coulomb behavior, intermediate linear confinement, and long-range infinite wall behavior?
• RQ2 Does an angular momentum-dependent potential with a csc²(r/d) term lead to a dynamical O(4) symmetry that explains the observed SU(2)I ⊗ O(4) degeneracy in non-strange baryon spectra?
• RQ3 Can the exact energy levels and wave functions of such a potential be derived analytically and shown to match the observed baryon mass splittings and quantum numbers?
• RQ4 Is the tRMP potential capable of providing an algebraic framework equivalent to the harmonic oscillator and Coulomb potentials, enabling exact calculations in baryon spectroscopy?
• RQ5 Can the relativistic realization of the O(4) multiplets as Rarita-Schwinger fields ψμ₁…μₖ be consistently derived from this potential model?

Key findings

• The trigonometric Rosen-Morse potential (tRMP) with a csc²(r/d) non-standard centrifugal barrier successfully interpolates between Coulomb-like and infinite wall-like behavior, matching the expected QCD quark-gluon dynamics.
• The tRMP potential, when treated as angular momentum-dependent, exhibits a dynamical O(4) symmetry that exactly reproduces the SU(2)I ⊗ O(4) quantum numbers and level splittings observed in the non-strange baryon spectra.
• The potential yields exact analytical solutions for energy levels and radial wave functions, with the first three levels (1s, 2s, 2p, 3s, 3p, 3d) explicitly derived using Romanovski polynomials.
• The energy spectrum matches the empirical mass formula M(σ;I) = -fI/σ² + gI(σ²-1)/4 with fN = fΔ = 600 MeV and gN = 70 MeV, gΔ = 40 MeV, confirming the observed increase in level splittings with σ = K+1.
• The O(4) multiplets are shown to have a relativistic image as totally symmetric rank-K Lorentz tensors ψμ₁…μₖ, corresponding to Rarita-Schwinger fields of spin (K+1/2), linking the non-relativistic symmetry to relativistic field theory.
• The tRMP potential provides an algebraically equivalent framework to the harmonic oscillator and Coulomb potentials, enabling exact calculations within a QCD-inspired quark model.