$\lambda$ and $\rho$ Regge trajectories for bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$

Abstract

Using the newly proposed tetraquark Regge trajectory relations, we investigate three series of Regge trajectories for bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$ with $q,q'=u,d,s$: the $\rho_1$-, $\rho_2$-, and $\lambda$-trajectories. We provide rough estimates for the masses of the $\rho_1$-, $\rho_2$-, and $\lambda$-excited states. Except for the $\lambda$-trajectories, the complete forms of the other two series of Regge trajectories for bottom-charm tetraquarks are lengthy and cumbersome. We show that the $\rho_1$- and $\rho_2$-trajectories cannot be obtained by simply imitating meson Regge trajectories, because mesons have no substructures. To derive these trajectories, the tetraquarks' structure and substructure must be taken into consideration. Otherwise, the $\rho_1$- and $\rho_2$-trajectories would have to rely solely on fitting existing theoretical results or future experimental data. Consequently, the fundamental relationship between the slopes of the obtained trajectories and string tension would become unobvious, and the predictive power of the Regge trajectories would be compromised. Moreover, we show that the lengthy complete forms of the $\rho_1$- and $\rho_2$-trajectories can be well approximated by simple fitted formulas. For the bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$, $\rho_1$- and $\rho_2$-trajectories exhibit a behavior of $M{\sim}x^{1/2}$ $(x=n_{r_1},n_{r_2},l_1,l_2)$, whereas $\lambda$-trajectories exhibit a behavior of $M{\sim}x^{2/3}$ $(x=N_{r},L)$. All three series of trajectories display concave downward behavior in the $(M^2,\,x)$ plane when the confining potential is linear. This conclusion holds irrespective of whether light-quark masses are included, owing to the large masses of the heavy quarks.