Possible assignments of the $$X(3872)$$ X ( 3872 ) , Zc(3900) Z c ( 3900 ) , and Zb(10610) Z b ( 10610 ) as axial-vector molecular states

In this article, we construct both the color singlet–singlet type and the octet–octet type currents to interpolate $$X(3872)$$ , Zc(3900) , and Zb(10610) , and we calculate the vacuum condensates up to dimension-10 in the operator product expansion. Then we study the axial-vector hidden charm and hidden bottom molecular states with the QCD sum rules, explore the energy-scale dependence of the QCD sum rules for the heavy molecular states in details, and we use the formula μ =√M²X/Y/Z-(2MQ)² with the effective masses MQ to determine the energy scales. The numerical results support assigning $$X(3872)$$ , Zc(3900) , and Zb(10610) as the color singlet–singlet type molecular states with JPC=1⁺⁺ , 1⁺⁻ , 1⁺⁻ , respectively, more theoretical and experimental work is still needed to distinguish the molecule and tetraquark assignments; while there are no candidates for the color octet–octet type molecular states.