λ and ρ Regge trajectories for hidden bottom and charm tetraquarks ( Q q ) ( Q ¯ q ¯ ′ )

We propose the Regge trajectory relations for the heavy tetraquarks ( Q q ) ( Q ¯ q ¯ ′ ) ( Q = b , c ; q , q ′ = u , d , s ) with hidden bottom and charm. By employing the new relations, both the λ -trajectories and the ρ -trajectories for the tetraquarks ( Q q ) ( Q ¯ q ¯ ′ ) can be discussed. The masses of the λ -mode excited states and the ρ -mode excited states are estimated, and they agree with other theoretical predictions. We show that the behaviors of the ρ -trajectories are different from those of the λ -trajectories. The ρ -trajectories behave as M ∼ x ρ 1 / 2 ( x ρ = n r , l ) while the λ -trajectories behave as M ∼ x λ 2 / 3 ( x λ = N r , L ) . Moreover, the Regge trajectory behaviors for other types of tetraquarks are investigated based on the spinless Salpeter equation. We show that both the λ -trajectories and the ρ -trajectories are concave downward in the ( M 2 , x ) plane. The Regge trajectories for the tetraquarks containing the light diquark and/or the light antidiquark also are concave in the ( M 2 , x ) plane when the masses of the light constituents are included and the confining potential is linear.