$\lambda$ and $\rho$ Regge trajectories for hidden bottom and charm tetraquarks $(Qq)(\bar{Q}{\bar{q}}^{\prime })$

Abstract

We propose the Regge trajectory relations for the heavy tetraquarks $(Qq)(\bar{Q}{\bar{q}}^{\prime })$ $(Q=b,c;q,q^{\prime }=u,d,s)$ with hidden bottom and charm. By employing the new relations, both the $\lambda$-trajectories and the $\rho$-trajectories for the tetraquarks $(Qq)(\bar{Q}{\bar{q}}^{\prime })$ can be discussed. The masses of the $\lambda$-mode excited states and the $\rho$-mode excited states are estimated, and they agree with other theoretical predictions. We show that the behaviors of the $\rho$-trajectories are different from those of the $\lambda$-trajectories. The $\rho$-trajectories behave as $M\sim x_{\rho }^{1/2}$ $(x_{\rho }=n_r,l)$ while the $\lambda$-trajectories behave as $M\sim x_{\lambda }^{2/3}$ $(x_{\lambda }=N_r,L)$. Moreover, the Regge trajectory behaviors for other types of tetraquarks are investigated based on the spinless Salpeter equation. We show that both the $\lambda$-trajectories and the $\rho$-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.

Comments: 18 pages,7 figures,15 tables