Possible existence of a dibaryon candidate $N\Delta$ ($D_{21}$)

Inspired by the experimental report by WASA-at-COSY Collaboration, we investigate the possibile existence of the dibaryon candidate $N\Delta$ with quantum numbers $IJ^P=21^+$ ($D_{21}$). The dynamical calculation shows that we cannot obtain the bound $D_{21}$ state in the models which can obtain the experimental $d^{*}$. %although the $D_{21}$ state can bound in the range of parameters in quark models. The low-energy scattering phase shifts of the $N-\Delta$ scattering give the same conclusion. Besides, the mass calculation by using the Gursey-Radicati mass formula and the analysis of the matrix elements of the color magnetic interaction show that the mass of $D_{21}$ is larger than that of $D_{12}$ ($N\Delta$ with $IJ^P=12^+$), which indicate that it is less possible for the $D_{21}$ than the $D_{12}$ to form bound state.


I. INTRODUCTION
Very recently, an isotensor dibaryon N ∆ with quantum numbers IJ P = 21 + (D 21 ) with a mass M = 2140 (10) MeV and a width Γ = 110 (10) MeV was reported by WASA-at-COSY [1]. In their measurements of the quasifree pp → ppπ + π − reaction by means of pd collisions at T p = 1.2 GeV, total and differential cross sections have been extracted covering the energy region T p = 1.08−1.36 GeV, which is the region of N * (1440) and ∆∆ resonance excitations. Calculations describing these excitations by t-channel meson exchange are contradictory with the measured differential cross sections and underpredict substantially the experimental total cross section. And the new dibaryon D 21 can be used to overcome these deficiencies. This state reported by experiment is in good agreement with the prediction of Dyson and Xuong [2]. Both mass and width are just slightly smaller than the results of Faddeev equation calculation by Gal and Garcilazo [3]. This report invokes our interest to the dibaryon resonance of the N ∆ system.
For the N ∆ system, the early analyses of p−p and n−p scattering data by Arndt et al provided the evidence for the existence of D 12 in the I = 1 1 D 2 and 3 F 3 nucleonnucleon channels [25]. Some quark models calculation found that D 12 was almost 200 MeV too high [7,8], but the subsequent chiral quark model calculations showed the mass of it was about 2170 MeV, slightly below the threshold of N ∆ [23]. The recent Faddeev equation calculation also supported the existence of N ∆ (D 12 ) [3,18]. Moreover, in Ref. [18], another N ∆ state D 21 was also found slightly below threshold by solving πN N Faddeev equations. However, this N ∆(D 21 ) was unbound in the early chiral quark model calculation [23].
In our previous work of dibaryon system, we have showed that the d * (D 03 ) was a tightly bound six-quark system rather than a loosely bound nucleus-like system of two ∆ [11,17,[26][27][28]. The ∆∆ (D 30 ) was another dibaryon candidate with smaller binding energy and larger width [29]. The N ∆ (D 12 ) state could be a resonance state in the N N D−wave scattering process only in one quark model calculation, which gave a lower mass of d * , while in other quark model calculations it was unbound [17]. Even though a resonance appeared only in one quark model calculation, the mass was very close to N ∆ threshold. Moreover, the large ∆ decay width when included would cause the state to straddle the N ∆ threshold in Ref. [17]. We therefore considered a N ∆ (D 12 ) resonance near the N ∆ threshold to be possible in quark model calculations in Ref. [17]. The N ∆ (D 21 ) state, which has a mirrored quantum numbers for spin and isospin with D 12 , was unbound in our initial calculation [27]. This situation calls for a more quantitative study of N ∆ (D 21 ).
In the various methods of investigating the baryonbaryon interaction, QCD-inspired quark models are still the main approach, because the direct use of quantum chromodynamics (QCD) in nucleon-nucleon interaction is still out of reach of the present techniques, although the lattice QCD has made considerable progress recently [30]. In our previous study of the dibaryon system, two quark models with different intermediate-range attraction mechanisms were used: one is the chiral quark model (ChQM) [23], in which the σ meson is indispensable to provide the intermediate-range attraction; the other is the quark delocalization color screening model (QDCSM) [26], in which the intermediate-range attraction is achieved by the quark delocalization, and the color screening is needed to make the quark delocalization feasible and it might be an effective description of the hidden color channel coupling [31]. Both QDCSM and ChQM give a good description of the N N scattering phase shifts and the properties of deuteron despite the different mechanisms used in models [32]. Besides, both models give d * (D 03 ) resonances reasonable well. Therefore, we use these two models to study the existence of N ∆ (D 21 ) resonance in this work. The hidden color channels are added to the ChQM to check their effect in the N ∆ system. The structure of this paper is as follows. A brief introduction of two quark models is given in Sec. II. Section III is devoted to the numerical results and discussions. The last section is a summary.

II. TWO QUARK MODELS
In this work, we use two constituent quark models: ChQM and QDCSM, which have been used in our previous work to study the dibaryon system [17]. The Salamanca version of ChQM is chosen as the representative of the chiral quark models. It has been successfully applied to hadron spectroscopy and N N interaction. The details of two models can be found in Ref. [17,23,26,27]. In the following, only the Hamiltonians and parameters of two models are given.

A. Chiral quark model
The ChQM Hamiltonian in the nonstrange dibaryon system is Where S ij is quark tensor operator, Y (x), H(x) and G(x) are standard Yukawa functions, T c is the kinetic energy of the center of mass. All other symbols have their usual meanings.

B. Quark delocalization color screening model
The QDCSM and its extension were discussed in detail in Ref. [26,27]. Its Hamiltonian has the same form as Eq.(1) but without σ meson exchange. Besides, a phenomenological color screening confinement potential is used in QDCSM.
Here, µ is the color screening constant to be determined by fitting the deuteron mass in this model. The quark delocalization in QDCSM is realized by replacing the left-(right-) centered single Gaussian functions, the singleparticle orbital wave function in the usual quark cluster model, with delocalized ones, The mixing parameter ǫ( S i ) is not an adjusted one but determined variationally by the dynamics of the multiquark system itself. This assumption allows the multiquark system to choose its favorable configuration in a larger Hilbert space. So the ansatze for the wave functions (Eq. 5) is a generalization of the usual quark cluster model ones which enlarges the variational space for the variational calculation. It has been used to explain the cross-over transition between hadron phase and quarkgluon plasma phase [33].
Since both of these two models give good descriptions of the deuteron, the nucleon-nucleon scattering phase shifts, and the dibaryon resonance d * in our previous work [17], the same models and parameters are used in this work. All parameters are listed in Table I. Here, the same values of parameters: b, α s , α ch , m u , m π , Λ are used for these two models, which are labeled as ChQM and QDCSM1 in Table I. Thus, these two models have exactly the same contributions from one-gluon-exchange and π exchange. The only difference of the two models comes from the short and intermediate-range part, σ exchange for ChQM, quark delocalization and color screening for QDCSM. By doing this, we can compare the intermediate-range attraction mechanism of these two models. Moreover, another set of parameters in QDCSM (labeled as QDCSM2) is used to test the sensitivity of the model parameters.

III. THE RESULTS AND DISCUSSIONS
In this work, we study the possibility of the existence of the dibaryon state N ∆ (D 21 ). For the first step, we can estimate the mass of dibaryon states from the dynamical symmetry calculation method. The mass spectrum of baryons and dibaryons can be obtained by means of the Gursey and Radicati mass formula [34] where the term C  Table II and III, from which we can see that all ground baryons can be described well by using this mass formula. For the non-strange dibaryons, we can obtain the experimental mass of the deuteron (N N (D 01 )) and the d * resonance (∆∆ (D 03 )). Although the mass of ∆∆(D 30 ) is higher than that of ∆∆ (D 03 ), it is still under the threshold of two ∆s, which indicates that the D 30 is a bound state within this method. All these results are consistent with our quark model calculations [29]. Besides, we find that the mass of the N ∆ (D 12 ) is only 3 MeV lower than its threshold, and the mass of N ∆ (D 21 ) is 14 MeV larger than that of N ∆ (D 12 ). Therefore, we cannot obtain the bound N ∆ (D 21 ) state within this method.   Then, we do a dynamical calculation to investigate the existence of the dibaryon state N ∆ (D 21 ) within the quark models introduced in Section II. The resonating group method (RGM), described in more detail in Ref. [37], is used to calculate the mass of the dibaryon states. The channels involved in the calculation are listed in Table IV. Here the baryon symbol is used only to denote the isospin, the superscript denotes the spin, 2S + 1, and the subscript "8" means color-octet, so 2 ∆ 8 means the I, S = 3/2, 1/2 color-octet state. The hidden-color channel means that the dibaryon composed of two coloroctet baryons. We do four kinds of calculation in this work. The first one is the single-channel calculation of the N ∆ state, which is labeled as sc; the second one is the the S−wave channel-coupling calculation, which is labeled as cc1; the third one is the cc1 coupling with the D−wave channels, labeled as cc2; the last one is cc2 coupled with all hidden-color channels, which is labeled as cc3. Note that the cc3 is carried out only in ChQM, because the quark delocalization and color screening used in QDCSM were showed to be an effective description of the hidden-color channel-coupling in our previous work of dibaryons [17,31], so we do not need to couple the hidden-color channels again in QDCSM.
The mass for the N ∆(D 21 ) state is shown in Table V, from which we can see that the mass of the pure N ∆ (D 21 ) state is above the N ∆ threshold in all these quark models (ChQM, QDCSM1, and QDCSM2). This means that the pure D 21 is unbound. In various kinds of coupling, this state is still unbound, except in QDCSM2 by coupling the S− and D−wave channels (cc2). The result is similar to that of N ∆ (D 12 ) state in our previous calculation [17]. In Ref. [17], the D 12 was unbound except in QDCSM with the third set of parameters [17], which is labeled as QDCSM2 in the present work. In our previous work, although all these three models can give good description to the deuteron and the nucleon-nucleon scattering phase shifts [17], only ChQM and QDCSM1 fit the data of d * (∆∆ D 03 ) well, while the QDCSM2 give too low mass of the d * resonance. Therefore, we cannot obtain the bound D 21 state in the models which can obtain the experimental d * , although the D 21 state can bound in QDCSM2. To investigate the possibility of the existence of the D 21 state, more work in depth is needed.
To understand the above results, the effective potentials between N and ∆ with quantum numbers IJ P =  Fig. 1. The effective potential between two colorless clusters is defined as V (s) = E(s) − E(∞), where E(s) is the diagonal matrix element of the Hamiltonian of the system in the generating coordinate. The results show that the potentials are all attractive in three quark models, and the attraction in QDCSM2 is the largest one, followed by that in QDCSM1, and the lowest one is obtained from ChQM. However, the attraction in neither of the quark models is strong enough to form the bound state D 21 , that is why we cannot obtain the pure bound D 21 state in three quark model calculations. To compare the result to other possible dibaryons, we also calculate the effective potentials of dibaryons N N (D 01 ), N N (D 10 ), N ∆(D 12 ), ∆∆(D 03 ), and ∆∆ (D 30 ), which are illustrated in Fig. 2. To save space, we only show the results in QDCSM1 here, and the potentials in other quark models are similar to that in QD-CSM1. Although the potentials of all these states are attractive, only the attractions of ∆∆ (D 03 ) and ∆∆ (D 30 ) are strong enough to form bound states [29] and the N N (D 01 ) state can be bound by coupling D−wave channels [32]. Besides, the attractions of N N (D 01 ), N N (D 10 ) and N ∆(D 21 ) are almost same with each other, while that of the N ∆ (D 12 ) state is a little larger. It seems that the possibility of forming a N ∆ (D 21 ) bound state is smaller than forming a N ∆ (D 12 ) state. All these potentials of dibaryons indicate that the attraction between two decuplet baryons is larger than that between decuplet baryon and octet baryon, and the attraction between two octet baryons is the smallest one. This regularity has been proposed in our previous work of dibaryons [27].
Obviously, the states N ∆ (D 12 ) and N ∆ (D 21 ) have mirrored quantum numbers of spin and isospin with each other, so do the states N N (D 01 ) and N N (D 10 ), ∆∆ (D 03 ) and ∆∆ (D 30 ). However, in our previous study of D 03 and D 30 states, we found the naive expectation of the spin-isospin symmetry was broken by the effective one gluon exchange (OGE) between quarks, and the mass of D 30 state was larger than that of D 03 state [29].
So it is interesting to study the spin-isospin symmetry of the states N ∆ (D 12 ) and N ∆ (D 21 ) to see which one is more possible to form a bound state. In fact, the mass splitting part in OGE interaction is the color magnetic interaction (CMI). It contributes the attraction to the internal energy of an octet baryon, CM I N = −8C (see Eq.(6) below, where C is the orbital matrix element, the subscripts A and S denote antisymmetric and symmetric), because of the equal attractive and repulsive qq pairs within octet baryon. On the contrary there are only repulsive qq pairs within decuplet baryon, CM I ∆ = 8C (see Eq. (7)), which causes the decuplet baryon about 300 MeV heavier than the octet baryon. When two nucleons merge into an orbital totally symmetric color singlet sixquark state, the color-spin part of the matrix elements (M.E.) of CMI is 56/3C for deuteron N N (D 01 ) (see Eq. (14)), and 24C for N N (D 10 ) (see Eq. (15) (19)), and this also indicates that the mass of N ∆(D 21 ) state is larger than that of N ∆ (D 12 ) state. All these laws are in complete agreement with the results in Table III, which are obtained from the Gursey-Radicati mass formula.
Then we extract the scattering length a 0 and the effective range r 0 from the low-energy scattering phase shifts by using the formula where k is the momentum of relative motion with k = √ 2µE c.m. , and µ is the reduced mass of N and ∆. The results are listed in Table VI. From Table VI, we can see that in both ChQM and QDCSM1, the scattering lengths a 0 are negative, while in QDCSM2 the scattering

IV. SUMMARY
In the present work, we investigate the possible existence of the dibaryon state N ∆ (D 21 ). The dynamical calculation shows that although the potentials are all attractive in three quark models, and the attraction is not strong enough to form the bound state N ∆ (D 21 ) in the single channel calculation. By various kinds of coupling, this state is still unbound, except in QDCSM2 by coupling the S− and D−wave channels. But the QDCSM2 gives too large binding energy for the d * (D 03 ) resonance. Therefore, we cannot obtain the bound N ∆ (D 21 ) state in the models which can obtain the experimental d * . We also study the low-energy scattering phase shifts of the N ∆ (D 21 ) state and the same conclusion is obtained.
Both the mass calculation by using the Gursey-Radicati mass formula and the analysis of the color-spin part of the matrix elements of the color magnetic interac-tion show that the mass of N N (D 10 ) is larger than that of N N (D 01 ), the mass of N ∆ (D 21 ) larger than that of N ∆ (D 12 ), and the mass of ∆∆ (D 30 ) larger than that of ∆∆ (D 03 ). All these results indicate that it is less possible for the N ∆ (D 21 ) than the N ∆(D 12 ) to form bound state. Besides, the naive expectation of the spin-isospin symmetry is broken by the effective one gluon exchange between quarks. The non-strange dibaryon states searching will be another check of this gluon exchange mechanism and the Goldstone boson exchange model.
In our previous study of N N and ∆∆ systems, both ChQM and QDCSM1 can obtain similar results. Here, in the work of N ∆ system, the similar results are obtained again. This show once more that the σ-meson exchange used in the chiral quark model can be replaced by quark delocalization and color screening mechanism.