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

Jiao-Kai Chen, He Song, Xin-Ru Liu

TL;DR

This work extends Regge-trajectory analysis to bottom-charm tetraquarks, deriving three trajectory series ($\rho_1$, $\rho_2$, and $\lambda$) within a diquark–antidiquark framework and incorporating substructure effects. Using the spinless Salpeter equation with a Cornell-like potential and Bohr–Sommerfeld quantization, the authors derive mass relations and provide complete vs. simplified forms; they show $\rho$-trajectories require substructure-aware constructions and cannot be obtained from meson analogies. The results yield rough mass estimates for excited states and demonstrate that the $\rho$-trajectories scale as $M\sim x^{1/2}$ while $\lambda$-trajectories scale as $M\sim x^{2/3}$, with all trajectories concave downward in the $(M^2,x)$ plane under linear confinement. The fitted formulas accurately reproduce the full trajectories, offering predictive power and illustrating the necessity of tetraquark-specific Regge relations for reliable phenomenology.

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 $ρ_1$-, $ρ_2$-, and $λ$-trajectories. We provide rough estimates for the masses of the $ρ_1$-, $ρ_2$-, and $λ$-excited states. Except for the $λ$-trajectories, the complete forms of the other two series of Regge trajectories for bottom-charm tetraquarks are lengthy and cumbersome. We show that the $ρ_1$- and $ρ_2$-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 $ρ_1$- and $ρ_2$-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 $ρ_1$- and $ρ_2$-trajectories can be well approximated by simple fitted formulas. For the bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$, $ρ_1$- and $ρ_2$-trajectories exhibit a behavior of $M{\sim}x^{1/2}$ $(x=n_{r_1},n_{r_2},l_1,l_2)$, whereas $λ$-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.

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

TL;DR

This work extends Regge-trajectory analysis to bottom-charm tetraquarks, deriving three trajectory series (, , and ) within a diquark–antidiquark framework and incorporating substructure effects. Using the spinless Salpeter equation with a Cornell-like potential and Bohr–Sommerfeld quantization, the authors derive mass relations and provide complete vs. simplified forms; they show -trajectories require substructure-aware constructions and cannot be obtained from meson analogies. The results yield rough mass estimates for excited states and demonstrate that the -trajectories scale as while -trajectories scale as , with all trajectories concave downward in the plane under linear confinement. The fitted formulas accurately reproduce the full trajectories, offering predictive power and illustrating the necessity of tetraquark-specific Regge relations for reliable phenomenology.

Abstract

Using the newly proposed tetraquark Regge trajectory relations, we investigate three series of Regge trajectories for bottom-charm tetraquarks and with : the -, -, and -trajectories. We provide rough estimates for the masses of the -, -, and -excited states. Except for the -trajectories, the complete forms of the other two series of Regge trajectories for bottom-charm tetraquarks are lengthy and cumbersome. We show that the - and -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 - and -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 - and -trajectories can be well approximated by simple fitted formulas. For the bottom-charm tetraquarks and , - and -trajectories exhibit a behavior of , whereas -trajectories exhibit a behavior of . All three series of trajectories display concave downward behavior in the 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.

Paper Structure

This paper contains 13 sections, 30 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Regge trajectory relations for the bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$
  3. Preliminary
  4. Spinless Salpeter equation
  5. Regge trajectory relations
  6. Regge trajectory relations for the bottom-charm tetraquarks
  7. Regge trajectories for the bottom-charm tetraquarks
  8. Parameters
  9. ${\rho}$-trajectories
  10. $\lambda$-trajectories
  11. Discussions
  12. Conclusions
  13. List of the Regge trajectory relations

Figures (4)

  • Figure 1: Schematic diagram of a tetraquark in the diquark-antidiquark picture.
  • Figure 2: Orbital and radial $\rho_1$-trajectories for tetraquarks $(bu)(\bar{c}\bar{u})$, $(bu)(\bar{c}\bar{s})$, $(bs)(\bar{c}\bar{u})$, and $(bs)(\bar{c}\bar{s})$. $n_{r_1}$ and ${l_1}$ are the radial and orbital quantum numbers for the $\rho_1$-mode, respectively. Circles represent the predicted data listed in Table \ref{['tab:massrho']}. The black dashed lines correspond to the complete forms of the $\rho_1$-trajectories, obtained from Eqs. (\ref{['t2qx']}) and (\ref{['pa2qQx']}) or (\ref{['summf']}) and (\ref{['pa2qQx']}). The pink lines correspond to the fitted formulas, obtained by linearly fitting the calculated data in Table \ref{['tab:massrho']}; these formulas are listed in Table \ref{['tab:formulas']}. The blue dotted lines correspond to the main parts of the full forms, which are also listed in Table \ref{['tab:formulas']}. $n_1^1s_0$ and $n^3_1s_1$ denote radial Regge trajectories for spin-0 and spin-1 diquarks, respectively; $1^1l_{1l_1}$ and $1^3l_{1l_{1}+1}$ denote orbital Regge trajectories for spin-0 and spin-1 diquarks, respectively.
  • Figure 3: Orbital and radial $\rho_2$-trajectories for tetraquarks $(bu)(\bar{c}\bar{u})$, $(bu)(\bar{c}\bar{s})$, $(bs)(\bar{c}\bar{u})$, and $(bs)(\bar{c}\bar{s})$. $n_{r_2}$ and ${l_2}$ are the radial and orbital quantum numbers for the $\rho_2$-mode, respectively. Circles represent the predicted data listed in Table \ref{['tab:massrhob']}. The black dashed lines correspond to complete forms of the $\rho_2$-trajectories, obtained from Eqs. (\ref{['t2qx']}) and (\ref{['pa2qQx']}) or (\ref{['summf']}) and (\ref{['pa2qQx']}). The pink lines correspond to the fitted formulas, obtained by linearly fitting the calculated data in Table \ref{['tab:massrhob']}; these formulas are listed in Table \ref{['tab:formulas']}. The blue dotted lines are for the main parts of the full forms, which are also listed in Table \ref{['tab:formulas']}. $n_2^1s_0$ and $n^3_2s_1$ denote radial $\rho_2$-trajectories for spin-0 and spin-1 antidiquarks, respectively; $1^1l_{2l_2}$ and $1^3l_{2l_{2}+1}$ denote orbital $\rho_2$-trajectories for spin-0 and spin-1 antidiquarks, respectively.
  • Figure 4: Orbital and radial $\lambda$-trajectories for tetraquarks $(bu)(\bar{c}\bar{u})$, $(bu)(\bar{c}\bar{s})$, $(bs)(\bar{c}\bar{u})$, and $(bs)(\bar{c}\bar{s})$. $N_{r}$ and ${L}$ are the radial and orbital quantum numbers for the $\lambda$-mode, respectively. Circles represent the predicted data listed in Table \ref{['tab:masslambda']}. The black dashed lines correspond to the $\lambda$-trajectories for the complete forms, obtained from Eqs. (\ref{['t2qx']}) and (\ref{['pa2qQx']}) or (\ref{['summf']}) and (\ref{['pa2qQx']}). The pink lines correspond to the fitted formulas, obtained by linearly fitting the calculated data in Table \ref{['tab:masslambda']}; these formulas are listed in Table \ref{['tab:formulas']}. The blue dotted lines are for the main parts of the full forms, which are also listed in Table \ref{['tab:formulas']}.