Hexaquark bound states and Regge trajectories in QCD via Bethe–Salpeter formalism

Abstract Hexaquarks, as exotic hadronic states composed of six quarks, represent a fascinating frontier in quantum chromodynamics (QCD). While not yet conclusively observed experimentally, these particles provide crucial insights into the structure of multi-quark systems and serve as important testing grounds for theoretical models. In this work, we investigate the ground-state masses and binding energies of hexaquark configurations formed by the binding of two baryons. We employ the Bethe–Salpeter equation (Salpeter and Bethe in Phys Rev 84:1232, 1951) framework with a phenomenological potential that incorporates color-Coulomb, linear confining, and spin-spin interaction terms. The resulting equation is solved using the Ansatz (Tazimi and Ghasempour in Adv High Energy Phys 2020:2541837, 2020) to obtain precise energy eigenvalues. Our calculations yield mass predictions for various hexaquark systems, including $$\Lambda _c\Lambda _c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> , $$\Lambda _b\Lambda _b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> </mml:math> , $$\Sigma _c\Sigma _c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> , $$\Sigma _b\Sigma _b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> </mml:math> , $$\Sigma _c^{}\Sigma _c^{}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mi>Σ</mml:mi> <mml:mi>c</mml:mi> <mml:mrow/> </mml:mmultiscripts> <mml:mmultiscripts> <mml:mi>Σ</mml:mi> <mml:mi>c</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:mrow> </mml:math> , and $$\Sigma _b^{}\Sigma _b^{}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mi>Σ</mml:mi> <mml:mi>b</mml:mi> <mml:mrow/> </mml:mmultiscripts> <mml:mmultiscripts> <mml:mi>Σ</mml:mi> <mml:mi>b</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:mrow> </mml:math> , which show good agreement with existing theoretical studies. Furthermore, we construct Regge trajectories for these hexaquark states, revealing linear behavior in both $$(J,M^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$(n_r,M^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> planes that aligns with established patterns in conventional hadronic spectroscopy. The calculated binding energies indicate the stability of these configurations, supporting their possible existence as resonant states. This comprehensive analysis demonstrates the efficacy of our approach in describing exotic multi-quark systems and contributes to the ongoing search for hexaquarks in experimental facilities.