Diquark Study in Quark Model

1. Introduction

In 1964, Gell-Mann and Zweig independently proposed the quark model, 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, diquark carries color charge. Due to the confinement of the strong interactions, diquark cannot 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 are considered as effective components of hadrons, baryons can be regarded as combinations of one quark and one diquark, diquarks were introduced to simplify the structural analysis of baryons, three-body problem is reduced to two-body one. In this case diquark was treated as a point-like particle. The important motivation for this treatment is to tackle the problem of missing states [3], the number of baryon states predicted by quark model is are much higher the states reported by experiments. The further theoretical studies indicated that diquarks possess spatial extension and cannot be simply regarded as point-like particles. So the modern studies of diquarks are focused on the quark+quark (diquark) correlations and emphasize the dynamical nature of diquarks [3,4,5,6,7,8]. Lattice QCD simulations supported the existence of diquark correlation [7,10]. Dyson-Schwinger equations and Bethe-Salpeter equations approach calculated the masses of mesons and diquarks, argued that two systems have similar behaviors. The comparative study of ground and excited states of light octet and decuplet baryons in three-body Faddeev framework and quark+diquark approximation showed that two approaches gave mutually consistent results [11]. Experiment also found signals for diquark correlations in the flavor-separation of the proton’s electromagnetic form factors [9]. However, whether diquarks should be understood only as mathematical tools or as “physics” degrees of freedom in the hadrons is still in debate and under study. For more detailed information, good review articles [3,5], can be referred.

In the present work, a powerful method for few-body systems, Gaussian expansion method (GEM) [12], is employed to investigate the masses of the three-body systems, baryons in the framework of quark models. After obtaining the wavefunctions 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 checked.

This paper is organized as follows. In Sec. 2 and Sec. 3, the model Hamiltonian, the wave functions and the calculation method are separately described. The results are given in Sec. 4 and a short summary is given in the last section.

2. Quark Model and Wave Functions

Two types of quark models are employed to do the calculations to check the model dependence of the diquark correlations. One is the naive quark model, where only gluon exchange are used. Another is the chiral quark model, in which the Goldstone bosons and corresponding scalar mesons exchange potentials are introduced.

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 [13,14].

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

(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)

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

(3)

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

(4)

where ${\lambda }_c$, $\sigma$ are the $S{U}_3$ color and $S{U}_2$ spin matrices; $T_{CM}$ is the center-of-mass kinetic energy; ${\alpha }_s$ 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. [15].

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

(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 [16,17]

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

(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 [18,19]. It has been successfully applied to describe both hadron spectroscopy and hadron-hadron interactions. The model details can be found in Refs. [18,19]. 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 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:

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

(12)

where ${\lambda }^{f,a}$ is 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 $\sqrt{2/3}$ which is according to the normalization property of Gell-Mann matrices. In fact, The different terms of the OBE potential contain central, tensor and spin-orbit contributions; only the central ones will be considered attending the goal of the present manuscript and for clarity in our discussion.

Table 1. 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	${\mu }_c$ (fm−1)	-	0.677
	$V_0$ (MeV)	21.38	64.57
	${\alpha }_0$	5.02	0.852
	${\Lambda }_0$ (fm−1)	0.1874	1.8445
OGE	${\mu }_0$ (MeV)	109.298	659.93
	$\beta$	0.485	-
	$r_0$ (MeV fm)	-	40.73
	$m_{\pi }$ (fm−1)	-	0.70
	$m_K$ (fm−1)	-	2.51
	$m_{\eta }$ (fm−1)	-	2.77
	${\Lambda }_{\pi }={\Lambda }_{\sigma }$ (fm−1)	-	4.20
	${\Lambda }_{\eta }$ (fm−1)	-	5.20
Goldstone boson	${\Lambda }_K$ (fm−1)	-	5.20
	${\theta }_P{(}^o)$	-	-15
	$g_{ch}^2/\left(4\pi \right)$	-	0.54
SU(3)	$m_{\sigma }$ (fm−1)	-	3.42
Scalar nonet	${\Lambda }_s$ (fm−1)	-	5.20
$s=\sigma ,a_0,\kappa ,f_0$	$m_s$ (fm−1)	-	4.97

Table 1. 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	${\mu }_c$ (fm−1)	-	0.677
	$V_0$ (MeV)	21.38	64.57
	${\alpha }_0$	5.02	0.852
	${\Lambda }_0$ (fm−1)	0.1874	1.8445
OGE	${\mu }_0$ (MeV)	109.298	659.93
	$\beta$	0.485	-
	$r_0$ (MeV fm)	-	40.73
	$m_{\pi }$ (fm−1)	-	0.70
	$m_K$ (fm−1)	-	2.51
	$m_{\eta }$ (fm−1)	-	2.77
	${\Lambda }_{\pi }={\Lambda }_{\sigma }$ (fm−1)	-	4.20
	${\Lambda }_{\eta }$ (fm−1)	-	5.20
Goldstone boson	${\Lambda }_K$ (fm−1)	-	5.20
	${\theta }_P{(}^o)$	-	-15
	$g_{ch}^2/\left(4\pi \right)$	-	0.54
SU(3)	$m_{\sigma }$ (fm−1)	-	3.42
Scalar nonet	${\Lambda }_s$ (fm−1)	-	5.20
$s=\sigma ,a_0,\kappa ,f_0$	$m_s$ (fm−1)	-	4.97

3. Wave Functions

As for the baryon’s wave function, each quark has color, spin, flavor 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 combination are as below.

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

(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 $SU\left(3\right)$-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

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

(36)

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

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

(37)

$\mathcal{A}$ is the antisymmetrization operators, $\mathcal{A}=1-\left(13\right)-\left(23\right)$ for three identical particles, $\mathcal{A}=1$ 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:

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

(38)

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

(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 complicate case, orbital angular momentum is not zero, the infinitesimally-shifted Gaussians (ISG) can be employed [20],

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

(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 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)

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

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

(44)

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

(45)

After obtaining the eigen-energie 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. The Results and Discussions

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 is spin scalar with flavor antisymmetric, another is spin vector with flavor symmetric. In the constituent quark model, the confinement potential is responsible for confining the quarks in a baryon, it is proportional to operator $-{\lambda }_i·{\lambda }_j$. Applying to 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 ${\lambda }_i·{\lambda }_j$, the factor $-{\displaystyle \frac{8}{3}}$ makes the attraction of color-Coulomb term. The second term is color magnetic interaction (CMI), it has color-spin operator $-{\lambda }_i·{\lambda }_j{\sigma }_i·{\sigma }_j$, it gives $-(-{\displaystyle \frac{8}{3}})\times (-3)=-8$ for scalar diquark, and $-(-{\displaystyle \frac{8}{3}})\times 1={\displaystyle \frac{8}{3}}$ for 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 complicate. 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 2. 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 analyse, 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 [17] and ChQM [20], are used to do the numerical calculations. The model parameters are fixed by fitting orbital ground state baryons and are listed in Table 1. 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, Table 4, Table 5 and Table 6. 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 $N\left(uud\right)$, $\Lambda \left(uds\right)$, ${\Lambda }_c\left(udc\right)$, ${\Lambda }_b\left(udb\right)$. For ${\Lambda }_c\left(udc\right)$ and ${\Lambda }_b\left(udb\right)$, the energies of diquark are almost 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 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 scalar diquark. The differences can be explained by CMI and Goldstone-boson-exchange, which have larger contribution to the energy in the vector diquark than that in the scalar diquark. Our results are 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 vector and scalar $us$ diquark is smaller than that of $ud$ diquark. The separation has the 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 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 a rough one.

Table 6 gives the mass spectra and the distances between two quarks of $QQ$-q/Q system. Same as the system $ss$-$q/Q$, there is only one type of orbital ground state diquark, vector diquark. The values of energy $E_{12}$ of $cc$ ($bb$), ∼ 3610 (10200) MeV and the separation $r_{12}$, ∼ 0.2 (0.1) fm are similar for different baryons. diquark correlation is clear. Only for the heavy diquark, the separations between two heavy quarks are smaller than that of heavy-light quarks.

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 diquark almost has the same energy and the same size, which means that diquark correlation really exists in baryons. When the $ud$, $us$ orbital is in the ground state, 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 diquark with the same flavor, $uu$, $ss$, and $QQ$, there is only one type of orbital ground state diquark, which the spin is 1. They have higher energy and larger separation that that of scalar diquark, which spin is 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 smaller the separation, the heavier the quark mass. In most case, 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.