Hyperfine mass splittings in ground and radially excited states of heavy-flavored QQQ¯Q¯ tetraquarks: PGM defect and fractional order effects

In this study, we investigate the masses and fall-apart decays of the radial excitations in $cc\overline{c}\overline{c}$ and $bb\overline{b}\overline{b}$ tetraquarks using a framework that incorporates topological defects and fractional-order derivatives via the generalized conformable fractional formalism. The interaction potential combines the Cornell potential with harmonic oscillator and inversely quadratic terms, adapted for diquark configurations. The Hamiltonian includes spin-independent and spin-dependent components, with the latter breaking degeneracies between spin singlets and triplets. The Schrödinger equation is solved analytically using the generalized fractional Nikiforov-Uvarov method. Our analysis focuses on axial-vector diquarks in the ground ($1^3{S}_1$) and first radially excited states ($2^3{S}_1$), ensuring compliance with symmetry principles. The computed diquark and tetraquark masses, for states with $J^{PC}=0^{++}$, $1^{+-}$ and $2^{++}$, align well with existing results, with deviations within reasonable ranges. Incremental differences observed in our models with varying δ values highlight the influence and adaptability of the fractional parameter, providing insights into the modeling approach. Heavy tetraquark states should be broad and difficult to detect experimentally. However, for some first radially excited states, we have calculated masses below their corresponding two-meson thresholds, indicating stability against fall-apart decays, while others with masses above these thresholds are likely to decay rapidly. The enhanced stability of lower-mass states increases their chances of experimental observation, as ongoing accelerator experiments search for exotic hadrons. Additionally, it is found that the $X\left(6400\right)$ and $X\left(6600\right)$ structures observed by ATLAS in the di-$J/\psi$ invariant mass spectrum can be attributed to the $T_{\left(4c\right)0^{++}}\left(6469\right)\left(2S\right)$ and $T_{\left(4c\right)2^{++}}\left(6643\right)\left(2S\right)$ states, respectively. These findings, consistent with prior studies, deepen our understanding of exotic hadrons and emphasize the significance of fractional dynamics and topological defects in quark-quark interactions, offering a solid foundation for future research on heavy tetraquark systems.