Regge trajectory relations for the universal description of the heavy-light systems: diquarks, mesons, baryons and tetraquarks

The spinless Salpeter equation (SSE) Godfrey:1985xj; Ferretti:2019zyh; Bedolla:2019zwg; Durand:1981my; Durand:1983bg; Lichtenberg:1982jp; Jacobs:1986gv reads as

where $M_{0}=\omega_{1}+\omega_{2}$. $M$ is the bound state mass (diquark, meson, baryon and tetraquark). $\Psi_{d,m,b,t}({{\bf r}})$ are the diquark wave function, the meson wave function, the baryon wave function, and the tetraquark wave function, respectively. $V_{d,m,b,t}$ are the diquark potential, the meson potential, the baryon potential, and the tetraquark potential, respectively. $\omega_{1}$ is the relativistic energy of constituent $1$, and $\omega_{2}$ is of constituent $2$,

$m_{1}$ and $m_{2}$ are the effective masses of heavy constituent $1$ and light constituent $2$, respectively.

In the present work, the Cornell-like potential is considered Eichten:1974af; Ferretti:2019zyh; Bedolla:2019zwg; Ferretti:2011zz,

where $V_{c}\,{\propto}\,1/r$ is the color Coulomb potential or a Coulomb-like interaction for different hadrons Ferretti:2019zyh; Ferretti:2011zz. The second term is the linear confining potential and $\sigma$ is the string tension. $C$ is a fundamental parameter Gromes:1981cb; Lucha:1991vn. ${\bf{F}}_{i}\cdot{\bf{F}}_{j}$ is the color-Casmir,

From Eqs. (1) and (3), we see that the heavy-light diquarks, the heavy-light mesons, the heavy-light baryons, and the heavy-light tetraquarks are described in an unified approach Ferretti:2019zyh; Bedolla:2019zwg. Therefore, it is expected that these heavy-light systems can be described universally by the Regge trajectory approach.

The mass of the light constituent is assumed to approach zero, $m_{2}\to 0$ in Refs. Veseli:1996gy; Chen:2017fcs or is considered by correction term in Refs. Selem:2006nd; Chen:2014nyo; Afonin:2014nya; Sonnenschein:2018fph; Jakhad:2023mni. In the limit $m_{1}\to\infty$ and $m_{2}\to 0$, Eq. (1) is reduced to be

By employing the Bohr-Sommerfeld quantization approach brau:04bs; brsom, we have from Eq. (5)

where

Using Eq. (6), the parameterized formula can be written as Chen:2023cws

The parameter in Eq. (8) reads as Chen:2021kfw

The constants $c_{x}$ [$c_{n_{r}}$ and $c_{l}$] and $c_{c}$ are

Both $c_{fl}$ and $c_{fn_{r}}$ are theoretically equal to one and are fitted in practice. For the heavy-light mesons, the common choice of $m_{R}$ is Selem:2006nd; Chen:2021kfw; Jakhad:2023mni; Chen:2014nyo; Chen:2017fcs; Veseli:1996gy; Jia:2018vwl

$m_{R}$, $c_{x}$ and $\sigma$ are universal for the heavy-light systems. $c_{fx}$ and $c_{0x}$, which vary with different Regge trajectories, are determined by fitting the given Regge trajectory.

The usual Regge trajectory Eq. (8) with (11), which is obtained in the limit $m_{1}\to\infty$ and $m_{2}\to 0$, cannot give agreeable results of the heavy-light diquarks. Corresponding to different ways to include the light constituent’s mass, two modified formulas are proposed in Ref. Chen:2023cws, which can describe universally both the heavy-light mesons and the heavy-light diquarks. One is Eq. (8) with

where

Another reads

if $m_{2}{\ll}M$, where

where $\beta_{x}$ is in (9). (8) with (12) is an extension of $M=m_{1}+m_{2}+\sqrt{a(n_{r}+{\alpha}l+b)}$ Afonin:2014nya, while (14) with (15) is based on the results in Selem:2006nd; Sonnenschein:2018fph. As $m_{2}=0$, these two modified formulas, formulas (8) with (12) and (14) with (15), become identical. As $m_{2}=0$ and $C$ is neglected, these two modified formulas reduce to the usual Regge trajectory formula for the heavy-light mesons, i.e., (8) with (11). In this work, we apply these two modified formulas to the heavy-light systems: diquarks, mesons, baryons, tetraquarks.

The parameters are Ebert:2007rn; Ebert:2011kk; Ebert:2009ua

where $\{\;\}$ denotes the axial-vector diquark and $[\;]$ the scalar diquark. $c_{0x}$ and $c_{fx}$ vary with different Regge trajectories.

The values in Eq. (III.1) are used to calculate the masses of tetraquarks Ebert:2007rn, baryons Ebert:2011kk and mesons Ebert:2009ua. In Ref. Chen:2017fcs, the values are used to give an universal description of the heavy-light mesons and baryons. The values are also used to discuss diquarks Feng:2023txx; Chen:2023cws.

Diquarks have been discussed by various approaches, for example, the QCD sum rules Dosch:1988hu; Zhang:2006xp; Jamin:1989hh; deOliveira:2023hma; Kleiv:2013dta; Wang:2011ab; Esau:2019hqw; Wang:2010sh. In Ref. Chen:2023cws, we show that the Regge trajectories for the heavy-light diquarks can be well described by (8) with (12) and (14) with (15). The spectra of the heavy-light diquarks $(cu)$, $(cs)$, $(bu)$ and $(bs)$ obtained by using the Regge trajectory approach agree with other theoretical predictions. See Chen:2023cws for more details.

In case of Eq. (8) with (12), the heavy-light diquarks satisfy the universal description irrespective of both mass of the light constituents and mass of the heavy constituent. In case of Eq. (14) with (15), the heavy-light diquarks satisfy the universal description irrespective of mass of the heavy constituents.

By using Eqs. (8) with (12) and (14) with (15) and data in Table 1, the radial and orbital Regge trajectories for the heavy-light mesons $D^{\ast}$, $D_{s}^{\ast\pm}$, $B^{\ast}$ and $B^{\ast}_{s}$ are obtained, see Fig. 1. The fitted values are listed in Table 2. In Fig. 1, the Regge trajectories (the red dashed lines) obtained by employing (14) with (15) lie above the Regge trajectories (the black lines) obtained by applying (8) with (12). It is because $m_{R}$ for these two Regge trajectories are different, see Eqs. (12) and (15). The figures show that the heavy-light mesons satisfy both of these two Regge trajectory relations.

From Fig. 2 and the fitted values in Table 2, we can see that both of the radial and orbital Regge trajectories in the $((M-m_{1}-m_{2}-C)^{2},\,x)$ planes almost overlap with each other irrespective of both the light quark flavor and heavy quark flavor. It shows an universal description of these heavy-light mesons. For the heavy-light mesons, the Regge trajectories in the $((M-m_{1}-C)^{2},\,x)$ plane are universal only irrespective of heavy quark flavor.

As employing Eq. (8) with (12), the average values of $c_{fn_{r}}$ and $c_{fl}$ are $1.026$ and $1.026$ for the heavy-light mesons, respectively. When applying Eq. (14) with (15), the average values of $c_{fn_{r}}$ and $c_{fl}$ are $1.108$ and $1.114$, respectively.

From Table 2, we can see that the fitted values of $c_{fn_{r}}$ and $c_{fl}$ decrease in the majority of cases as the masses of the constituents increase when employing (8) with (12). However, the fitted values of $c_{fn_{r}}$ and $c_{fl}$ obtained by using (14) with (15) frequently decrease with the increase of the masses of the heavy constituent and rise with the increase of the masses of the light constituent. This trend aligns with the results in Chen:2017fcs.

In the diquark picture, the baryons consisting of one light quark and one heavy diquark or consisting of one heavy quark and one light diquark are denoted as the heavy-light baryons and belong to the heavy-light systems. Employing (8) with (12) and (14) with (15) to the heavy-light baryons, the Regge trajectories are fitted. The used data are listed in Table 3 and the fitted values are in Table 2. The fitted Regge trajectories are in Fig. 3.

The difference of $m_{R}$ for two Regge trajectory relation leads to that the Regge trajectories (the red dashed lines) obtained by employing (14) with (15) lie above the Regge trajectories (the black lines) obtained by applying (8) with (12), see Fig. 3. From Fig. 4 and the fitted values in Table 2, we can see that the heavy-light baryons satisfy both of these two Regge trajectory relations. We also notice that some fitted Regge trajectories by employing (8) with (12) are better than the fitted Regge trajectories by applying (8) with (12), for example, the Regge trajectory for $\Omega_{c}$, see 3.

Similar to the meson case, from Fig. 4 and the fitted values in Table 2, we can see that both of the radial and orbital Regge trajectories for the heavy-light baryons plotted in the $((M-m_{1}-m_{2}-C)^{2},\,x)$ planes almost overlap with each other irrespective of both the light constituent and the heavy constituent. This shows an universal description of these heavy-light baryons. For the heavy-light baryons, the Regge trajectories in the $((M-m_{1}-C)^{2},\,x)$ plane are universal only irrespective of heavy constituent.

As employing Eq. (8) with (12), the average values of $c_{fn_{r}}$ and $c_{fl}$ are $0.794$ and $0.749$ for the heavy-light baryons, respectively. When applying Eq. (14) with (15), the average values of $c_{fn_{r}}$ and $c_{fl}$ are $0.896$ and $0.855$, respectively. The fitted $c_{fn_{r}}$ and $c_{fl}$ for the heavy-light baryons are smaller than that for the heavy-light mesons. The fitted values for the doubly heavy baryons approximate the fitted values for the heavy-light mesons because the doubly heavy baryons strongly resemble the heavy-light mesons.

The heavy-light tetraquarks denote tetraquarks consisting of one heavy diquark and one light antidiquark or consisting of one light diquark and one heavy antidiquark. The tetraquarks composed of a diquark and antidiquark in color $\bar{3}$ and $3$ configurations are considered.

Applying Eqs. (8) with (12) and (14) with (15) and using data in Table 5, the radial and orbital Regge trajectories for the heavy-light tetraquarks are obtained, see Fig. 5. The fitted values are listed in Table 2. Similar to the heavy-light mesons and the heavy-light baryons, the Regge trajectories (the red dashed lines) obtained by employing (14) with (15) lie above the Regge trajectories (the black lines) obtained by applying (8) with (12), see Fig. 5.

From Fig. 6 and the fitted values in Table 2, we can see that the heavy-light tetraquarks satisfy both of these two Regge trajectory relations. Moreover, we can see that the universal description of the heavy-light tetraquarks approximately holds. We notice that the fitted results are not reliable enough because the experimental data is lack and the theoretical data on the $\lambda$-excited states are scarce. [The theoretical data on the $\lambda$-excited states of the tetraquarks are scarce because the masses of these states lie above the corresponding thresholds.]

As employing Eq. (8) with (12), the average values of $c_{fn_{r}}$ and $c_{fl}$ are $0.553$ and $0.579$ for the heavy-light tetraquarks, respectively. When applying Eq. (14) with (15), the mean values of $c_{fn_{r}}$ and $c_{fl}$ are $0.647$ and $0.676$, respectively. The fitted $c_{fn_{r}}$ and $c_{fl}$ for the heavy-light tetraquarks are smaller than that for the heavy-light mesons and the heavy-light baryons.