Diquark Study in Quark Model

Abstract

To investigate diquark correlation in baryons, the baryon spectra with different light–heavy quark combinations are calculated using Gaussian expansion method within both the naive quark model and the chiral quark model. By computing the diquark energies and separations between any two quarks in baryons, we analyze the diquark effect in the $ud$-q/Q, $us$-Q, $ss$-q/Q, and $QQ$-q/Q systems (where $q=u,d$, or s; $Q=c,b$). The results show that diquark correlations exist in baryons. In particular, for $qq$-Q and $QQ$-q systems, the same type of diquark exhibits nearly identical energy and size across different baryons. In the orbital ground states of baryons, scalar–isoscalar diquarks have lower energy and a smaller size compared to vector–isovector diquark, which qualifies them as “good diquarks”. In $QQ$-q systems, a larger mass of Q leads to a smaller diquark separation and a more pronounced diquark effect. In $qq$-Q systems, the separation between the two light quarks remains larger than that between a light and a heavy quark, indicating that the internal structure of such diquarks must be taken into account. A comparison between the naive quark model and the chiral quark model reveals that the introduction of meson exchange slightly increases the diquark size in most systems.

1. Introduction

In 1964, Gell-Mann and Zweig independently proposed the quark model; the diquark was introduced during this period as an important component for explaining hadron structure [1,2]. In Quantum Chromodynamics (QCD) based on $S{U}_3$-color, the diquark carries color charge. Due to the confinement of the strong interactions, diquark cannot be observed experimentally and can only serve as internal components of hadrons. Understanding the structure of hadrons is a key issue in hadron physics. In early research, diquarks were considered as effective constituents of hadrons. Baryons can be regarded as combinations of a quark and a diquark; diquarks were introduced to simplify the structural analysis of baryons by reducing the three-body problem to a two-body one. In this context, the diquark was treated as a point-like particle. An important motivation for this treatment is to address the problem of missing states [3]; the number of baryon states predicted by quark model is much higher than those observed experimentally. Further theoretical studies indicated that diquarks possess spatial extension and cannot be simply regarded as point-like particles. Therefore, modern studies of diquarks focus on the quark–quark (diquark) correlations and emphasize the dynamical nature of diquarks [3,4,5,6,7,8,9]. Lattice QCD simulations supported the existence of diquark correlation [7,10]. Approaches based on Dyson–Schwinger equations and Bethe–Salpeter equations have calculated the masses of mesons and diquarks, arguing that two systems have similar behaviors. A comparative study of the ground and excited states of light octet and decuplet baryons within a three-body Faddeev framework and quark–diquark approximation showed that the two approaches yield consistent results [11]. Experiments have also found evidence for diquark correlations in the flavor separation of the proton’s electromagnetic form factors [12]. However, whether diquarks should be understood only as mathematical tools or as genuine physics degrees of freedom in the hadrons remains debated and under study. For more detailed information, readers may refer to comprehensive review articles [3,5].

In the present work, a powerful method for few-body systems, the Gaussian expansion method (GEM) [13], is employed to investigate the masses of the three-body systems, baryons, within the framework of quark models. After obtaining the wave functions of the systems, the separations between any two quarks and the masses of diquarks are calculated. By analyzing the separations and the masses of diquark, the diquark correlation are examined.

This paper is organized as follows. In Section 2 and Section 3, the model Hamiltonian, the wave functions, and the calculation method are described. The results are presented in Section 4 and a brief summary is given in Section 5.

2. Quark Model and Wave Functions

To check the model dependence of diquark correlations, calculations are performed using two types of quark models. One is the naive quark model, which includes only gluon exchange interactions. The other is the chiral quark model, which incorporates, in addition to the gluon exchange potentials, exchange potentials of Goldstone bosons and corresponding scalar mesons.

2.1. The Naive Quark Model (NQM)

The constituent quark model has been successfully applied to describe hadron properties and baryon–baryon interactions. The naive quark model is a relatively simple model among the constituent quark models. In this model, the phenomenological Hamiltonian takes the form of kinetic energy term (T), confinement potential ($V^{CON}$), and one gluon exchange potential ($V^{OGE}$). The confinement potential reflects the long-range behavior of QCD, while the short-range behavior of QCD is asymptotically free, which is represented by one-gluon exchange (OGE) interaction potential [14,15].

$$\begin{array}{ccc}\hfill H& =& T+V_{ij}^{CON}+V_{ij}^{OGE},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill T& =& \sum _{i=1}^3\left(m_i+{\displaystyle \frac{p_i^2}{2m_i}}\right)-T_{CM}\hfill \end{array}$$

(2)

$$V_{ij}^{CON}=-a_c{\mathit{\lambda }}_i^c·{\mathit{\lambda }}_j^c\left(r_{ij}^2-V_0\right)$$

(3)

$$V_{ij}^{OGE}={\displaystyle \frac{1}{4}}{\alpha }_s{\mathit{\lambda }}_i^c·{\mathit{\lambda }}_j^c\{{\displaystyle \frac{1}{r_{ij}}}-{\displaystyle \frac{\pi }{2}}\delta \left({\mathbf{r}}_{ij}\right)({\displaystyle \frac{1}{m_i^2}}+{\displaystyle \frac{1}{m_j^2}}+{\displaystyle \frac{4{\mathit{\sigma }}_i·{\mathit{\sigma }}_j}{3m_i{m}_j}})\}$$

(4)

where ${\lambda }_c$ and $\sigma$ are the $S{U}_3$ color and $S{U}_2$ spin matrices; $T_{CM}$ is the center-of-mass kinetic energy; and ${\alpha }_s$ is the quark–gluon coupling constant. However, in a non-relativistic quark model, the wide energy range covered to describe the systems with light, strange, and heavy quarks requires an effective scale-dependent strong coupling constant ${\alpha }_s$ that cannot be obtained from the usual one-loop expression of the running coupling constant because it diverges when $Q\to {\Lambda }_{QCD}$. So, we use an effective scale-dependent strong coupling constant explained by Ref. [16].

$${\alpha }_s={\displaystyle \frac{{\alpha }_0}{ln\left(\frac{{\mu }^2}{{\Lambda }_0^2}+\frac{{\mu }_0^2}{{\Lambda }_0^2}\right)}},$$

(5)

where $\mu$ is the reduced mass of two interacting quarks and ${\alpha }_0$, ${\mu }_0$ and ${\Lambda }_0$ are model parameters. For the confinement potential $V_{ij}^{CON}$, quadratic form is used in our calculations. The $\delta$ function, arising as a consequence of the non-relativistic reduction of the one-gluon exchange diagram between point-like particles, has to be regularized in order to perform exact calculations. It reads [17,18]

$$\delta \left({\mathbf{r}}_{ij}\right)={\displaystyle \frac{1}{{\beta }^3{\pi }^{3/2}}}{e}^{-r_{ij}^2/{\beta }^2},$$

(6)

where $\beta$ is a parameter.

2.2. The Chiral Quark Model (ChQM)

The Salamanca version of ChQM is chosen as a representative of chiral quark models [19,20]. It has been successfully applied to describe both hadron spectroscopy and hadron–hadron interactions. The model details can be found in Refs. [19,20]. Here, only the Hamiltonian in the baryon–baryon sector is given below.

$$\begin{array}{ccc}\hfill H& =& T+V_{ij}^{CON}+V_{ij}^{OGE}+V_{ij}^{\mathrm{OBE}}\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill T& =& \sum _{i=1}^3\left(m_i+{\displaystyle \frac{p_i^2}{2m_i}}\right)-T_{CM}\hfill \end{array}$$

(8)

The kinetic energy term (T) is same as the naive quark model. Compared to the confinement potential in the NQM, the ChQM employs a screened confinement, introducing an additional parameter ${\mu }_c$.

$$V_{ij}^{\mathrm{CON}}=\left[-a_c(1-e^{-{\mu }_c{r}_{ij}})+V_0\right]({\lambda }_i^c·{\lambda }_j^c),$$

(9)

$$V_{ij}^{\mathrm{OGE}}={\displaystyle \frac{1}{4}}{\alpha }_s({\lambda }_i^c·{\lambda }_j^c)\left[{\displaystyle \frac{1}{r_{ij}}}-{\displaystyle \frac{({\sigma }_i·{\sigma }_j)}{6m_i{m}_j}}{\displaystyle \frac{e^{-r_{ij}/r_0\left(\mu \right)}}{r_{ij}{r}_0^2\left(\mu \right)}}\right],$$

(10)

where the contact term has been regularized as

$$\delta \left({\mathbf{r}}_{ij}\right)\sim {\displaystyle \frac{1}{4\pi r_0^2}}{\displaystyle \frac{e^{-r_{ij}/r_0}}{r_{ij}}}.$$

(11)

The ChQM is based on the fact that a nearly massless current light quark acquires a dynamical, momentum-dependent mass; namely, the constituent quark mass, due to its interaction with the gluon medium. To preserve the chiral invariance of the QCD Lagrangian, new interaction terms, given by Goldstone-boson exchanges, should appear between constituent quarks. The partner of Goldstone boson, scalar mesons, also appear. Therefore, the chiral part of the quark–quark interaction can be expressed as follows:

$$\begin{array}{ccc}\hfill V_{ij}^{\mathrm{OBE}}& =& (v_{ij}^{\pi }+v_{ij}^{a_0})\sum _{a=1}^3{\lambda }_i^{f,a}{\lambda }_j^{f,a}+(v_{ij}^K+v_{ij}^{\kappa })\sum _{a=4}^7{\lambda }_i^{f,a}{\lambda }_j^{f,a}\hfill \\ & & +(v_{ij}^{\eta }cos{\theta }_P+v_{ij}^{f_0}){\lambda }_i^{f,8}{\lambda }_j^{f,8}+(-v_{ij}^{\eta }sin{\theta }_P+v_{ij}^{\mathit{\sigma }}){\lambda }_i^{f,0}{\lambda }_j^{f,0},\hfill \\ \hfill v^{\chi }\left({\mathbf{r}}_{ij}\right)& =& {\displaystyle \frac{g_{ch}^2}{4\pi }}{\displaystyle \frac{m_{\chi }^2}{12m_i{m}_j}}{\displaystyle \frac{{\Lambda }_{\chi }^2}{{\Lambda }_{\chi }^2-m_{\chi }^2}}{m}_{\chi }\left[Y\left(m_{\chi }{r}_{ij}\right)-{\displaystyle \frac{{\Lambda }_{\chi }^3}{m_{\chi }^3}}Y\left({\Lambda }_{\chi }{r}_{ij}\right)\right]({\mathit{\sigma }}_i·{\mathit{\sigma }}_j),\chi =\pi ,K,\eta ,\hfill \\ \hfill v^s\left({\mathbf{r}}_{ij}\right)& =& -{\displaystyle \frac{g_{ch}^2}{4\pi }}{\displaystyle \frac{{\Lambda }_s^2}{{\Lambda }_s^2-m_s^2}}{m}_s\left[Y\left(m_s{r}_{ij}\right)-{\displaystyle \frac{{\Lambda }_s}{m_s}}Y\left({\Lambda }_s{r}_{ij}\right)\right],s=\sigma ,a_0,\kappa ,f_0.\hfill \end{array}$$

(12)

where ${\lambda }^{f,a}$ is the a-th Gell-Mann matrix of flavor $S{U}_3^f$. ${\lambda }^{f,0}$ is just the $3\times 3$ identity matrix multiplied by a factor of $\sqrt{2/3}$, according to the normalization property of Gell-Mann matrices, and $Y\left(x\right)$ is the Yukawa function defined as $Y\left(x\right)={\displaystyle \frac{e^{-x}}{x}}$.

In fact, the OGE and OBE potentials contain central, tensor, and spin–orbit interactions; only the central ones will be considered attending the goal of the present manuscript and for clarity in our discussion. For the ground-state baryon, all the orbital angular momenta are zero, and the contribution of spin–orbit interaction is zero. The tensor interaction can contribute but its contribution is small $\sim 1\%$ compared to that of central interactions and can be neglected.

3. Wave Functions

As for the baryon’s wave function, each quark has color($r,g,b$), spin($\alpha ,\beta$), flavor($u,d,$ $s,c,b$) and spatial degrees of freedom. According to the empirical fact that color sources have never seen as isolated particles, the color wave function of a baryon must be color singlet, which can be easily written as

$${\chi }^c={\displaystyle \frac{1}{\sqrt{6}}}(rgb-rbg+gbr-grb+brg-bgr).$$

(13)

The spin wave functions ${\chi }_{S{M}_S}^{\sigma }$ of a 3-quark system, taking into account all possible quantum number combinations, are as follows.

$${\chi }_{{\displaystyle \frac{3}{2}},{\displaystyle \frac{3}{2}}}^{\sigma }\left(3\right)=\alpha \alpha \alpha ,$$

(14)

$${\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{{\sigma }_1}\left(3\right)={\displaystyle \frac{1}{\sqrt{6}}}(2\alpha \alpha \beta -\alpha \beta \alpha -\beta \alpha \alpha ),$$

(15)

$${\chi }_{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}^{{\sigma }_2}\left(3\right)={\displaystyle \frac{1}{\sqrt{2}}}(\alpha \beta \alpha -\beta \alpha ),$$

(16)

The charm and bottom quarks are much heavier than the light ones: $u,d$ and s quark. Therefore, we investigate the baryon with quark content $u,d,s$ and c or b in the $SU\left(3\right)$-flavor case and the corresponding flavor wave functions ${\chi }_{I{M}_I}^f$ are given by

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{N_1}={\displaystyle \frac{1}{\sqrt{6}}}(2uud-udu-duu),$$

(17)

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{N_2}={\displaystyle \frac{1}{\sqrt{2}}}(ud-du)u,{\chi }^{\Delta }=uuu,$$

(18)

$${\chi }_{00}^{{\Lambda }_1}={\displaystyle \frac{1}{2}}(usd-dsu+sud-sdu),$$

(19)

$${\chi }_{00}^{{\Lambda }_2}={\displaystyle \frac{1}{\sqrt{12}}}(2uds-2dus+usd-dsu-sud+sdu),$$

(20)

$${\chi }_{10}^{{\Sigma }_1}={\displaystyle \frac{1}{\sqrt{12}}}(2uds+2dus-usd-dsu-sud-sdu),$$

(21)

$${\chi }_{10}^{{\Sigma }_2}={\displaystyle \frac{1}{2}}(usd-sud+dsu-sdu),$$

(22)

$${\chi }_{10}^{{\Sigma }^{*}}={\displaystyle \frac{1}{\sqrt{6}}}(uds+usd+dus+dsu+sud+sdu),$$

(23)

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{{\Xi }_1}={\displaystyle \frac{1}{\sqrt{6}}}(uss+sus-2ssu),$$

(24)

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{{\Xi }_2}={\displaystyle \frac{1}{\sqrt{2}}}(us-su)s,$$

(25)

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{{\Xi }^{*}}={\displaystyle \frac{1}{\sqrt{3}}}(uss+sus+ssu),$$

(26)

$${\chi }_{00}^{\Omega }=sss.$$

(27)

For the light–heavy and full-heavy baryons, where Q represents either c- or b-quark, the flavor wave functions are given by

$${\chi }_{00}^{{\Lambda }_Q}={\displaystyle \frac{1}{2}}(ud-du)Q,$$

(28)

$${\chi }_{10}^{{\Sigma }_Q}={\displaystyle \frac{1}{2}}(ud+du)Q,$$

(29)

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{{\Xi }_Q}={\displaystyle \frac{1}{2}}(us-su)Q,$$

(30)

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{{\Xi }_Q^{\prime }}={\displaystyle \frac{1}{2}}(us+su)Q,$$

(31)

$${\chi }_{00}^{{\Omega }_Q}=ssQ,$$

(32)

$${\chi }_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}^{{\Xi }_{QQ}}=uQQ,$$

(33)

$${\chi }_{00}^{{\Omega }_{QQ}}=sQQ,$$

(34)

$${\chi }_{00}^{{\Omega }_{QQQ}}=QQQ,$$

(35)

The total wave functions of baryons are

$${\Psi }_{I{M}_{I}J{M}_J}=\mathcal{A}\left[{\left[{\psi }_L(\mathit{\rho },\mathit{\lambda }){\chi }_S^{\sigma }\right]}_{J{M}_J}{\chi }_{00}^c{\chi }_{I{M}_I}^f\right],$$

(36)

where ${\psi }_{L{M}_L}(\mathit{\rho },\mathit{\lambda })$ is the spatial wave function, and $\mathit{\rho },\mathit{\lambda }$ are Jacobi coordinates, which are defined as,

$$\mathit{\rho }={\mathbf{r}}_1-{\mathbf{r}}_2,\mathit{\lambda }={\mathbf{r}}_3-{\displaystyle \frac{m_1{\mathbf{r}}_1+m_2{\mathbf{r}}_2}{m_1+m_2}}.$$

(37)

$\mathcal{A}$ is the antisymmetrization operator, $\mathcal{A}=1-\left(13\right)-\left(23\right)$ is used for three identical particles, and $\mathcal{A}=1$ is used for other cases, because the permutation symmetry of the first two-particle has been considered by choosing the appropriate wave functions of color, spin, flavor, and spatial degrees of freedom.

Among the different methods to solve the three-body Schrödinger equation we use the Rayleigh–Ritz variational principle, which is one of the most extended tools to solve eigenvalue problems due to its simplicity and flexibility. However, it is of great importance how to choose the basis on which to expand the wave function. In this work, we choose a set of Gaussians to expand the radial part of the spatial wave function. So, the spatial wave function of a 3-quark system is written as follows:

$${\psi }_{L{M}_L}={\left[{\varphi }_{l_1}\left(\mathit{\rho }\right){\varphi }_{l_2}\left(\mathit{\lambda }\right)\right]}_{L{M}_L}.$$

(38)

$${\varphi }_{l_1{m}_1}\left(\mathit{\rho }\right)=\sum _{n_1=1}^{n_{max}}{c}_{n_1{l}_1}{N}_{n_1{l}_1}{\rho }^{l_1}{e}^{-{\nu }_{n_1}{\rho }^2}{Y}_{l_1{m}_1}\left(\widehat{\rho }\right),$$

(39)

$$N_{n_1{l}_1}={\left[{\displaystyle \frac{2^{l_1+2}{\left(2{\nu }_{n_1}\right)}^{l_1+\frac{3}{2}}}{\sqrt{\pi }(2l_1+1)!!}}\right]}^{{\displaystyle \frac{1}{2}}}.$$

(40)

This choice is convenient because, for a nonrelativistic system, the center-of-mass kinetic term $T_{CM}$ can be completely eliminated. To deal with the complicated case, the orbital angular momentum is not zero, and the infinitesimally-shifted Gaussians (ISG) can be employed [13],

$${\varphi }_{lm}\left(\mathit{\rho }\right)=\sum _{n=1}^{n_{max}}{c}_{nl}{N}_{nl}\underset{\epsilon \to 0}{lim}{\displaystyle \frac{1}{{\left({\nu }_n\epsilon \right)}^l}}\sum _{k=1}^{k_{max}}{C}_{lm,k}{e}^{-{\nu }_n{(r-\epsilon D_{lm,k})}^2},$$

(41)

where the limit $\epsilon \to 0$ must be carried out after the matrix elements have been calculated analytically. This new set of basis functions makes the calculation of three- and, in general, few-body matrix elements very easy without the laborious Racah algebra. Moreover, all the advantages of using Gaussians remain with the new basis functions. In order to make the calculation tractable, the sizes of the Gaussians are arranged in a geometric progression,

$${\nu }_n={\displaystyle \frac{1}{r_n^2}},r_n=r_1{a}^{n-1},a={\left({\displaystyle \frac{r_{n_{max}}}{r_1}}\right)}^{{\displaystyle \frac{1}{n_{max}-1}}}.$$

(42)

By using Rayleigh–Ritz variational principle, the three-body Schrödinger equation can be reduced to the following generalized eigen-equation,

$$\sum _{n=1}^{n_{max}}\left(H_{n^{\prime }n}-E{N}_{n^{\prime }n}\right)C_n=0,n=(n_1,n_2).$$

(43)

$$\begin{array}{ccc}\hfill H_{n^{\prime }n}& =& 〈N_{n_1^{\prime }{l}_1^{\prime }}{N}_{n_2^{\prime }{l}_2^{\prime }}{\rho }^{l_1^{\prime }}{\lambda }^{l_2^{\prime }}{e}^{-{\nu }_{n_1^{\prime }}{\rho }^2}{e}^{-{\nu }_{n_2^{\prime }}{\lambda }^2}{\left[{\left[Y_{l_1^{\prime }}\left(\widehat{\rho }\right)Y_{l_2^{\prime }}\left(\widehat{\lambda }\right)\right]}_{L^{\prime }}{\chi }_S^{\sigma }\right]}_{J{M}_J}{\chi }^c{\chi }_{I{M}_I}^f\hfill \\ & & \left|H\right|N_{n_1{l}_1}{N}_{n_2{l}_2}{\rho }^{l_1}{\lambda }^{l_2}{e}^{-{\nu }_{n_1}{\rho }^2}{e}^{-{\nu }_{n_2}{\lambda }^2}{\left[{\left[Y_{l_1}\left(\widehat{\rho }\right)Y_{l_2}\left(\widehat{\lambda }\right)\right]}_L{\chi }_S^{\sigma }\right]}_{J{M}_J}{\chi }^c{\chi }_{I{M}_I}^f〉,\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill N_{n^{\prime }n}& =& 〈N_{n_1^{\prime }{l}_1^{\prime }}{N}_{n_2^{\prime }{l}_2^{\prime }}{\rho }^{l_1^{\prime }}{\lambda }^{l_2^{\prime }}{e}^{-{\nu }_{n_1^{\prime }}{\rho }^2}{e}^{-{\nu }_{n_2^{\prime }}{\lambda }^2}{\left[{\left[Y_{l_1^{\prime }}\left(\widehat{\rho }\right)Y_{l_2^{\prime }}\left(\widehat{\lambda }\right)\right]}_{L^{\prime }}{\chi }_S^{\sigma }\right]}_{J{M}_J}{\chi }^c{\chi }_{I{M}_I}^f\hfill \\ & & \left|N_{n_1{l}_1}{N}_{n_2{l}_2}{\rho }^{l_1}{\lambda }^{l_2}{e}^{-{\nu }_{n_1}{\rho }^2}{e}^{-{\nu }_{n_2}{\lambda }^2}{\left[{\left[Y_{l_1}\left(\widehat{\rho }\right)Y_{l_2}\left(\widehat{\lambda }\right)\right]}_L{\chi }_S^{\sigma }\right]}_{J{M}_J}{\chi }^c{\chi }_{I{M}_I}^f\right.〉.\hfill \end{array}$$

(45)

After obtaining the eigen-energy E and eigen-function ${\Psi }^E$ of a baryon, the energy and the size of diquark can be calculated as

$$E_{12}=\langle {\Psi }^E|H_{12}|{\Psi }^E\rangle ,$$

(46)

$$\sqrt{\overline{r_{12}^2}}=\sqrt{\langle {\Psi }^E\left|r_{12}^2\right|{\Psi }^E\rangle }.$$

(47)

4. Results and Discussion

Before the numerical calculation, we discuss the properties of diquark in a baryon analytically. To simplify the discussion, the orbital angular momentum between two quarks is set to 0, the ground-state diquark. Because of the requirement of color singlet, only symmetric flavor–spin diquarks are allowed in a baryon. There are two types of diquark, one has scalar spin with zac flavor, another has vector spin with symmetric flavor. In the constituent quark model, the confinement potential is responsible for confining the quarks in a baryon, it is proportional to the operator $-{\mathit{\lambda }}_i·{\mathit{\lambda }}_j$. Applying this to a color–antisymmetric quark pair, the operator gives ${\displaystyle \frac{8}{3}}$. The contribution of confinement potential to the energy of the diquark increases with the increasing separation between two quarks. It has the effect of confinement. For the one-gluon exchange potential, the first term is color–Coulomb with the color operator ${\mathit{\lambda }}_i·{\mathit{\lambda }}_j$, and the factor $-{\displaystyle \frac{8}{3}}$ results in the attraction of the color–Coulomb term. The second term is color magnetic interaction (CMI); it has a color–spin operator $-{\mathit{\lambda }}_i·{\mathit{\lambda }}_j{\mathit{\sigma }}_i·{\mathit{\sigma }}_j$, it gives $-(-{\displaystyle \frac{8}{3}})\times (-3)=-8$ for a scalar diquark, and $-(-{\displaystyle \frac{8}{3}})\times 1={\displaystyle \frac{8}{3}}$ for a vector diquark. So, CMI lowers the energies of scalar diquarks and lifts the energies of vector diquarks.

For the one-boson exchange potential, the situation is complicated. The spatial part of the Goldstone-boson exchange interaction is $Y\left(m_{\chi }r\right)-{\left({\displaystyle \frac{{\Lambda }_{\chi }}{m_{\chi }}}\right)}^{3}Y\left({\Lambda }_{\chi }r\right)$ with ${\Lambda }_{\chi }>m_{\chi }$, it is negative for the small separation ($r<r_0={\displaystyle \frac{2(ln{\Lambda }_{\chi }-ln{m}_{\chi })}{{\Lambda }_{\chi }-m_{\chi }}}$, for $\pi$, $r_0=1.02$ fm, for K, $r_0=0.54$ fm) and positive for the large separation. The matrix elements of flavor operators on light diquarks are shown in

Table 1. Matrix elements of flavor operators on light diquarks.

	Flavor Operators
Diquarks	${\sum }_{\mathit{a}=\mathbf{1}}^{\mathbf{3}}{\mathit{\lambda }}_{\mathbf{i}}^{\mathit{f},\mathit{a}}{\mathit{\lambda }}_{\mathit{j}}^{\mathit{f},\mathit{a}}$	${\sum }_{\mathit{a}=\mathbf{4}}^{\mathbf{7}}{\mathit{\lambda }}_{\mathit{i}}^{\mathit{f},\mathit{a}}{\mathit{\lambda }}_{\mathit{j}}^{\mathit{f},\mathit{a}}$	${\mathit{\lambda }}_{\mathit{i}}^{\mathit{f},\mathbf{8}}{\mathit{\lambda }}_{\mathit{j}}^{\mathit{f},\mathbf{8}}$	${\mathit{\lambda }}_{\mathit{i}}^{\mathit{f},\mathbf{0}}{\mathit{\lambda }}_{\mathit{j}}^{\mathit{f},\mathbf{0}}$
$uu$	1	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{2}{3}}$
$\sqrt{{\displaystyle \frac{1}{2}}}(ud+du)$	1	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{2}{3}}$
$dd$	1	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{2}{3}}$
$\sqrt{{\displaystyle \frac{1}{2}}}(ud-du)$	$-3$	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{2}{3}}$
$\sqrt{{\displaystyle \frac{1}{2}}}(us+su)$	0	2	$-{\displaystyle \frac{2}{3}}$	${\displaystyle \frac{2}{3}}$
$\sqrt{{\displaystyle \frac{1}{2}}}(us-su)$	0	$-2$	$-{\displaystyle \frac{2}{3}}$	${\displaystyle \frac{2}{3}}$
$ss$	0	0	${\displaystyle \frac{4}{3}}$	${\displaystyle \frac{2}{3}}$

. Combining four degrees of freedom, one can see that the Goldstone-boson exchange potentials are negative for the small separation between two quarks, and are positive for large separation between two quarks. The contributions of Goldstone bosons are attractive or repulsive depending on the wave function of diquarks. For the scalar meson exchange, the spatial part is positive. So, the contributions of scalar nonet are universally attractive.

From the above analysis, one can see that the “best” diquark is the one with antisymmetric color, spin, and flavor, $(ud-du)/\sqrt{2}$, in which all the potentials are attractive.

In the following, two quark models, NQM [18] and ChQM [21], are used to do the numerical calculations. The model parameters are fixed by fitting orbital ground-state baryons and are listed in

Table 2. Model parameters.

Model		NQM	ChQM
	$m_u$ (MeV)	313	313
	$m_d$ (MeV)	313	313
Quark mass	$m_s$ (MeV)	589	555
	$m_c$ (MeV)	1860	1620
	$m_b$ (MeV)	5209	5030
	$a_c$ (MeV)	60.845	202.1
Confinement	${\mu }_c$ (${\mathrm{fm}}^{-1}$)	-	0.677
	$V_0$ (MeV)	21.38	64.57
	${\alpha }_0$	5.02	0.852
	${\Lambda }_0$ (${\mathrm{fm}}^{-1}$)	0.1874	1.8445
OGE	${\mu }_0$ (MeV)	109.298	659.93
	$\beta$	0.485	-
	$r_0$ (MeV fm)	-	40.73
	$m_{\pi }$ (${\mathrm{fm}}^{-1}$)	-	0.70
	$m_K$ (${\mathrm{fm}}^{-1}$)	-	2.51
	$m_{\eta }$ (${\mathrm{fm}}^{-1}$)	-	2.77
	${\Lambda }_{\pi }={\Lambda }_{\sigma }$ (${\mathrm{fm}}^{-1}$)	-	4.20
	${\Lambda }_{\eta }$ (${\mathrm{fm}}^{-1}$)	-	5.20
Goldstone boson	${\Lambda }_K$ (${\mathrm{fm}}^{-1}$)	-	5.20
	${\theta }_P$ (°)	-	−15
	$g_{ch}^2/\left(4\pi \right)$	-	0.54
SU(3)	$m_{\sigma }$ (${\mathrm{fm}}^{-1}$)	-	3.42
Scalar nonet	${\Lambda }_s$ (${\mathrm{fm}}^{-1}$)	-	5.20
$s=\sigma ,a_0,\kappa ,f_0$	$m_s$ (${\mathrm{fm}}^{-1}$)	-	4.97

. The GEM parameters are determined by requiring the convergence of the results, $r_1=0.1$ fm, $r_{max}=3$ fm, and $n_{max}=12$. The calculated results are shown in

Table 3. Mass spectra (unit: MeV) and the distances (unit: fm) between two quarks of $ud$-diquark in baryons.

Baryon I(${\mathit{J}}^{\mathit{P}}$)	Exp.	Theo.		${\mathit{E}}_{12}$		${\mathit{r}}_{12}$		${\mathit{r}}_{13}$
		NQM	ChQM	NQM	ChQM	NQM	ChQM	NQM	ChQM
$N\left(uud\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	939	939	939	765	824	0.729	0.674	0.729	0.674
$\Delta \left(uuu\right){\displaystyle \frac{3}{2}}\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	1232	1232	1232	826	812	0.939	1.203	0.939	1.203
$\Lambda \left(uds\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	1116	1150	1206	769	813	0.708	0.694	0.610	0.622
$\Sigma \left(uds\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	1193	1172	1302	778	831	0.719	0.779	0.609	0.672
${\Lambda }_c\left(udc\right)0\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	2286	2288	2246	675	674	0.592	0.598	0.482	0.594
${\Sigma }_c\left(udc\right)1\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	2455	2471	2416	846	823	0.812	1.034	0.545	0.736
${\Lambda }_b\left(udb\right)0\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	5620	5608	5616	678	677	0.585	0.577	0.442	0.515
${\Sigma }_b\left(udb\right)1\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	5811	5816	5811	845	821	0.817	0.976	0.521	0.634
${\Sigma }^{*}\left(uds\right){\displaystyle \frac{3}{2}}\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	1383	1363	1397	835	825	0.881	1.095	0.737	0.936
${\Sigma }_c^{*}\left(udc\right)1\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	2520	2523	2456	839	811	0.845	1.133	0.581	0.835
${\Sigma }_b^{*}\left(udb\right)1\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	5830	5835	5826	842	817	0.829	1.075	0.533	0.722

,

Table 4. Mass spectra (unit: MeV) and the distances (unit: fm) between two quarks of $us$-diquark in baryons.

Baryon I(${\mathit{J}}^{\mathit{P}}$)	Exp.	Theo.		${\mathit{E}}_{12}$		${\mathit{r}}_{12}$		${\mathit{r}}_{13}$
		NQM	ChQM	NQM	ChQM	NQM	ChQM	NQM	ChQM
${\Xi }_c\left(usc\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	2470	2504	2511	953	966	0.516	0.597	0.574	0.474
${\Xi }_c^{\prime }\left(usc\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	2578	2601	2571	1052	1028	0.617	0.794	0.634	0.524
${\Xi }_c^{*}\left(usc\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	2645	2644	2613	1044	1015	0.640	0.874	0.558	0.794
${\Xi }_b\left(usb\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	5797	5817	5880	956	970	0.506	0.572	0.440	0.534
${\Xi }_b^{\prime }\left(usb\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	5935	5935	5963	1052	1026	0.615	0.796	0.496	0.649
${\Xi }_b^{*}\left(usb\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	-	5951	5979	1050	1031	0.624	0.826	0.506	0.682

,

Table 5. Mass spectra (unit: MeV) and the distances (unit: fm) between two quarks of $ss$-diquark in baryons.

Baryon I(${\mathit{J}}^{\mathit{P}}$)	Exp.	Theo.		${\mathit{E}}_{12}$		${\mathit{r}}_{12}$		${\mathit{r}}_{13}$
		NQM	ChQM	NQM	ChQM	NQM	ChQM	NQM	ChQM
$\Xi \left(ssu\right){\displaystyle \frac{1}{2}}\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	1315	1341	1438	1202	1229	0.486	0.528	0.577	0.608
${\Xi }^{*}\left(ssu\right){\displaystyle \frac{3}{2}}\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	1530	1502	1559	1222	1200	0.580	0.722	0.673	0.826
$\Omega \left(sss\right){\displaystyle \frac{3}{2}}\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	1672	1613	1684	1238	1214	0.492	0.605	0.492	0.605
${\Omega }_c\left(ssc\right)0\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	2695	2729	2713	1248	1218	0.438	0.568	0.330	0.450
${\Omega }_c^{*}\left(ssc\right)0\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	2766	2763	2757	1244	1204	0.450	0.625	0.344	0.519
${\Omega }_b\left(ssb\right)0\left({{\displaystyle \frac{1}{2}}}^{+}\right)$	6045	6052	6103	1252	1217	0.429	0.526	0.292	0.359
${\Omega }_b^{*}\left(ssb\right)0\left({{\displaystyle \frac{3}{2}}}^{+}\right)$	-	6064	6121	1250	1201	0.568	0.591	0.400	0.423

and

Table 6. Mass spectra (unit: MeV) and the distances (unit: fm) between two quarks of $cc$-diquark and $bb$-diquark in baryons.

Baryon   ${\mathit{J}}^{\mathit{P}}$	Exp.	Theo.		${\mathit{E}}_{12}$		${\mathit{r}}_{12}$		${\mathit{r}}_{13}$
		NQM	ChQM	NQM	ChQM	NQM	ChQM	NQM	ChQM
${\Xi }_{cc}\left(ccu\right){{\displaystyle \frac{1}{2}}}^{+}$	3622	3698	3574	3607	3235	0.196	0.321	0.448	0.644
${\Xi }_{cc}^{*}\left(ccu\right){{\displaystyle \frac{3}{2}}}^{+}$	-	3761	3624	3604	3258	0.205	0.353	0.490	0.757
${\Omega }_{cc}\left(ccs\right){{\displaystyle \frac{1}{2}}}^{+}$	-	3842	3728	3614	3243	0.179	0.295	0.293	0.432
${\Omega }_{cc}^{*}\left(ccs\right){{\displaystyle \frac{3}{2}}}^{+}$	-	3879	3777	3611	3261	0.184	0.328	0.308	0.511
${\Omega }_{ccb}\left(ccb\right){{\displaystyle \frac{1}{2}}}^{+}$	-	8243	8107	3647	3275	0.139	0.270	0.108	0.206
${\Omega }_{ccb}^{*}\left(ccb\right){{\displaystyle \frac{3}{2}}}^{+}$	-	8247	8118	3646	3272	0.140	0.280	0.109	0.216
${\Xi }_{bb}\left(bbu\right){{\displaystyle \frac{1}{2}}}^{+}$	-	10,268	10,313	10,183	10,013	0.075	0.143	0.396	0.563
${\Xi }_{bb}^{*}\left(bbu\right){{\displaystyle \frac{3}{2}}}^{+}$	-	10,292	10,233	10,183	9961	0.076	0.120	0.411	0.531
${\Omega }_{bb}\left(bbs\right){{\displaystyle \frac{1}{2}}}^{+}$	-	10,383	10,463	10,187	10,015	0.068	0.132	0.229	0.354
${\Omega }_{bb}^{*}\left(bbs\right){{\displaystyle \frac{3}{2}}}^{+}$	-	10,397	10,364	10,186	9964	0.069	0.112	0.234	0.326
${\Omega }_{bbc}\left(bbc\right){{\displaystyle \frac{1}{2}}}^{+}$	-	11,458	11,457	10,201	10,019	0.056	0.123	0.093	0.189
${\Omega }_{bbc}^{*}\left(bbc\right){{\displaystyle \frac{3}{2}}}^{+}$	-	11,463	11,470	10,201	10,017	0.056	0.128	0.093	0.201
${\Omega }_{ccc}\left(ccc\right){{\displaystyle \frac{3}{2}}}^{+}$	-	4980	4751	3636	3269	0.160	0.302	0.160	0.302
${\Omega }_{bbb}\left(bbb\right){{\displaystyle \frac{3}{2}}}^{+}$	-	14,640	14,819	10,229	10,023	0.047	0.119	0.047	0.119

. In the following, we discuss the results in detail.

The Table 3 shows the mass spectra and the distances between two quarks of $ud$-q/Q system. When the $ud$ orbital is in the ground state, the scalar diquark with color, spin, and flavor wave functions being all antisymmetric is the “best” diquark, resulting in a lower energy $E_{12}$ for these systems such as $N\left(uud\right)$, $\Lambda \left(uds\right)$, ${\Lambda }_c\left(udc\right)$, and ${\Lambda }_b\left(udb\right)$. For ${\Lambda }_c\left(udc\right)$ and ${\Lambda }_b\left(udb\right)$, the energies of diquark are almost the same—675 MeV and 678 MeV in NQM, 674 MeV and 677 MeV in ChQM—and the separations have the same behavior, 0.592 fm and 0.585 fm in NQM, 0.598 fm and 0.577 fm in ChQM. However, the separations between two light quarks are still larger than the separations between light and heavy quarks. So the point-like approximation of diquark is not a good one, even for the “best” diquark. For baryons $N\left(uud\right)$ and $\Lambda \left(uds\right)$, the masses of diquarks are a little larger, due to the using of $S{U}_3^f$ symmetry, in which all three particles are identical. For the $uu$ or $ud$ diquark (vector diquark) in baryons $\Delta \left(uuu\right)$, $\Sigma \left(uds\right)$, ${\Sigma }_c\left(udc\right)$, ${\Sigma }_b\left(udb\right)$, ${\Sigma }^{*}\left(uds\right)$, ${\Sigma }_c^{*}\left(udc\right)$, ${\Sigma }_b^{*}\left(udb\right)$, the masses of diquark are in the range, 826∼ 846 MeV, about 170 MeV higher than the masses of “best” diquark. The separations between two quarks in vector diquarks are also larger compared to the scalar diquark. The differences can be explained by CMI and Goldstone-boson exchange, which have a larger contribution to the energy in the vector diquark than that in the scalar diquark. Our results also show that the heavier the Q, the smaller the diquark, and the more pronounced the diquark effect. Generally, the size of diquark in ChQM is a little larger than that in NQM; this effect may come from the different model parameters.

In Table 4, the mass spectra and the distances between two light quarks of $us$-q/Q system are listed. Similar to the above discussion, the scalar $us$ diquarks have lower energy, 953∼956 MeV in NQM, than that of vector diquark, 1044∼1052 MeV. The energy difference ∼100 MeV between the vector and scalar $us$ diquark is smaller than that of $ud$ diquark. The separation has similar behavior.

Table 5 shows the mass spectra and the distances between two quarks of $ss$-q/Q system. In this case, only one type of orbital ground-state diquark is allowed, so the energies $E_{12}$ and the separation between two s quarks $r_{12}$ are all similar for different baryons. The separation between two s quarks is still larger than the separation between s and heavy quarks, so the point-like particle approximation is still rough.

Table 6 gives the mass spectra and the distances between two quarks of the $QQ$-q/Q system. There is only one type of orbital ground-state diquark, vector diquark, same as the $ss$-$q/Q$ system. The values of energy $E_{12}$ of $cc$ ($bb$), ∼3610 (10,200) MeV and the separation $r_{12}$, ∼0.2 (0.1) fm are similar for different baryons. Diquark correlation is clear. The separations between two heavy quarks are only smaller than that of heavy–light quarks for the heavy diquark.

5. Summary

We use the Gaussian expansion method to dynamically calculate the baryon spectra and the distances between quarks in various heavy–light quark combinations. By analyzing the energy and separation of two-particle systems, we study the diquark effect in systems such as $ud$-q/Q, $us$-Q, $ss$-q/Q, and $QQ$–q/Q (where $q=u$, d, or s; $Q=c$ or b).

The results show that the same type of diquark has nearly identical energy and the same size, indicating that diquark correlation indeed exists in baryons. When the $ud$, $us$ orbital is in the ground state, and the color, spin, and flavor wave functions are all antisymmetric, leading to lower energy and smaller quark separations, making these systems good diquarks. For the diquarks with the same flavor, $uu$, $ss$, and $QQ$, there is only one type of orbital ground-state diquark, with a spin of 1. They have higher energy and larger separation that those of scalar diquark with a spin of 0. The diquark effect is more pronounced with larger Q values. However, the hierarchy of the separations between two quarks is the same as the hierarchy of quark mass; the heavier the quark mass, the smaller the separation. In most cases, the separations of diquark are not small enough to take the diquark as a point-like particle.

In baryon models, the structure of these diquarks must be considered. Comparing the naive quark model (NQM) and the chiral quark model (ChQM), we find that introducing meson exchange in ChQM generally increases the distance between quarks in most systems.