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

TL;DR

This paper develops Regge trajectory relations for heavy tetraquarks with hidden bottom and charm, estimating their excited state masses and analyzing the differing behaviors of $ ho$- and $ ho$-trajectories, supported by theoretical calculations.

Contribution

It introduces new Regge trajectory relations for heavy tetraquarks and compares the behaviors of different trajectory types, extending understanding of their mass spectra.

Findings

$ ho$-trajectories behave as $M \\sim x_{\\rho}^{1/2}$

$\\lambda$-trajectories behave as $M \\sim x_{\\lambda}^{2/3}$

Regge trajectories are concave downward in the $(M^2, x)$ plane.

Abstract

We propose the Regge trajectory relations for the heavy tetraquarks $(Qq)(\bar{Q}\bar{q}')$ $(Q=b,\,c;\,q,\,q'=u,\,d,\,s)$ with hidden bottom and charm. By employing the new relations, both the $\lambda$-trajectories and the $\rho$-trajectories for the tetraquarks $(Qq)(\bar{Q}\bar{q}')$ 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.