aa r X i v : . [ nu c l - t h ] S e p Possible molecular states from N ∆ interaction Zhi-Tao Lu, Han-Yu Jiang, Jun He ∗ Department of Physics and Institute of Theoretical Physics,Nanjing Normal University, Nanjing 210097, People’s Republic of China (Dated: September 4, 2020)Recently, a hint for dibaryon N ∆ ( D ) was observed at WASA-AT-COSY with a mass about 30 ±
10 MeVbelow the N ∆ threshold. It has a relatively small binding energy compared with the d ∗ (2380) and a width closeto the width of the ∆ baryon, which suggests that it may be a dibaryon in a molecular state picture. In this work,we study the possible S-wave molecular states from the N ∆ interaction within the quasipotential Bethe-Salpeterequation approach. The interaction is described by exchanging π , ρ , and ω mesons. With reasonable parameter,a D bound state can be produced from the interaction. The results also suggest that there may exist two morepossible D and D states with smaller binding energies. The π exchange is found to play a most importantrole to bind two baryons to form the molecular states. Experimental search for possible N ∆ ( D ) and N ∆ ( D )states will be helpful to understanding the hint of the dibaryon N ∆ ( D ). I. INTRODUCTION
In the past two decades, the study of exotic hadronsbecomes one of the most important topics in the communityof hadron physics. The core issue of the hadron physics isto understand how quarks combine into a hadron. In theconventional quark model, a hadron is composed of q ¯ q as ameson or qqq as a baryon. It is natural to expect the existenceof hadrons composed of more quarks, which are called exoticstates. The deuteron can be also seen as a hadron, which is aquark system with six quarks, though we called it a nucleus.The existence of the nucleus and the hypernucleus inspiresus to search for molecular states as loosely bound states ofhadrons. Such a picture has been widely applied to interpretthe experimentally observed XYZ particles and the hidden-charm pentaquarks P c [1–14]. More and more structuresobserved near thresholds of two hadrons give people moreconfidence about the existence of molecular states. If weturn back to the deuteron, which is a molecular state, it isinteresting to study possible molecular states composed of twonucleons and / or its resonances, such as systems N ∆ and ∆∆ .The hadron carrying baryon number B = ∆∆ ( D , ) and of dibaryon N ∆ ( D , )were predicted as 2376 and 2176 MeV, respectively. Afterobserving an experimental hint in 1977 [16], Kamae andFujita made a calculation in the one-boson-exchange modelat hadronic level to reproduce an anomaly at 2380 MeV inthe process γ d → pn [17]. The existence of the ∆∆ ( D )was supported by many theoretical calculations especially theconstituent quark model [18–21]. The N ∆ ( D ) was alsopredicted in the literature [22–24]. The existence of N ∆ ( D )state was favored by some early analyses of experimental data, ∗ Electronic address: Corresponding author: [email protected] such as partial-wave analysis of the reaction π + d → pp [25],an analyses of pp and np scatterings by the SAID group [26],and a study of the phase shifts for the N ∆ scattering witha nearby S-matrix pole based on the data of process pp → np π + [27]. However, the N ∆ ( D ) was not supported by theearly calculation in the constituent quark model [24, 28, 29].In Ref. [30], without the dibaryon the experimental data werealso reproduced.After the e ff orts of more than half a century in boththeoretical and experimental sides[16–29, 31–35], a candidateof dibaryon with I ( J P ) = + ) carrying a mass of about2370 MeV and a width of about 70 MeV was observed inthe process pp → d π π at WASA-at-COSY [36], denotedas d ∗ (2380). Later, a series of measurements confirmed theexistence of this state [37–39]. Such state was also confirmedby a recent measurement within the Crystal Ball at MAMI,where the photoproduction process was performed [40].The observation of the d ∗ (2380) attracts much attention oftheorists, and a large amount of interpretations were proposedto understand its properties and internal structure [41–53].Because the ∆ signal can be found in the final states of itsdecay, one may guess that it is a ∆∆ bound state. However,such assumption leads to a binding energy about 80 MeVconsidering the mass of ∆ baryon is about 1232 MeV. Suchlarge binding energy prefers a compact hexaquark instead ofa bound state of two ∆ baryons. The conclusion is furthersupported by the relatively smaller width of 70 MeV of the d ∗ (2380), which is even smaller than the width of one ∆ baryon, about 120 MeV. Gal and Garcilazo proposed that the d ∗ (2380) is from a three-body N ∆ π system with a Faddeevequation calculation [41, 42]. In their study, the N ∆ ( D , )was also studied in the three-body NN π interaction and foundslightly below the N ∆ threshold.Recently, an isotensor dibaryon N ∆ with quantum numbers I J P = + ( D ) with a mass of 2140(10) MeV and a width of110(10) MeV was reported at WASA-at-COSY [54]. Its massis about 30 ±
10 MeV below the N ∆ threshold. Consideringthat the width of nucleon is zero (for proton) or very small(for neutron) and the ∆ baryon has a width of about 120MeV, the width of this N ∆ ( D ) state is almost the sum ofnucleon and ∆ baryon. Hence, compared with the d ∗ (2380),such state is obviously consistent with the molecular statepicture. In Ref. [55], the authors studied the N ∆ states in theconstituent quark model, and found that it is less likely for N ∆ ( D ) than N ∆ ( D ) to form a bound state. In this work,with the help of the e ff ective Lagrangians, we will constructthe interaction in the one-boson-exchange model, and insert itinto the quasipotential Bethe-Salpeter equation (qBSE) to findS-wave bound states from the N ∆ interaction.The paper is organized as follows. After the introduction,we present the e ff ective Lagrangians and relevant couplingconstants to describe the N ∆ interaction, with which wededuce the potential. And the qBSE is also briefly introduced.In Sec. III, we will present the numerical results, and thecontributions from di ff erent exchanges and diagrams are alsodiscussed. Finally, the article ends with a summary in Sec. IV. II. THEORETICAL FRAME
In the current work, we will describe the N ∆ interactionin the one-boson-exchange model, in which the interactionis usually mediated by the exchange of the light mesonsincluding pseudoscalar mesons ( π and η ), vector mesons ( ρ , ω and φ ) and scalar meson σ . The coupling of η meson andnucleon is small [56–63], and the coupling of the φ mesonand nucleon is suppressed according to the OZI rule. Besides,we do not consider the scalar meson exchange as done inRefs. [64, 65]. Hence, we only consider the exchanges of π , ρ ,and ω mesons in the calculation. There are two diagrams forthe N ∆ interaction as shown in Fig. 1. In the cross diagram,the ω exchange is forbidden due to the conservation of theisospin. N N ∆∆ V Σ i I di = ∆ ∆ π , ρ , ω N N + Σ j I cj N ∆ N π , ρ ∆ FIG. 1: The diagrams for the direct (left) and cross (right) potentials.The thin (brown) and thick (blue) lines are for N and ∆ mesons,respectively. The I ( d , c ) i is the flavor factor for direct or cross diagramwith i exchange, which is explained in text. We need the Lagrangians for the vertices of nucleon, ∆ baryon, and pseudoscalar meson π , which is written as [64,65] L NN π = − g NN π m π ¯ N γ γ µ τ · ∂ µ π N , (1) L ∆∆ π = g ∆∆ π m π ¯ ∆ µ γ γ ν T · ∂ ν π ∆ µ , (2) L N ∆ π = g N ∆ π m π ¯ ∆ µ S † · ∂ µ π N + H . c ., (3)where the N , ∆ , and π are nucleon, ∆ baryon, and pion mesonfields. The coupling constants g NN π / π = . g ∆∆ π = . g N ∆ π = − . ∆ baryon, and vector meson ρ/ω are written as [64, 65], L NN ρ = − g NN ρ ¯ N [ γ µ − κ ρ m N σ µν ∂ ν ] τ · ρ µ N , L NN ω = − g NN ω ¯ N [ γ µ − κ ω m N σ µν ∂ ν ] ω µ N , L ∆∆ ρ = − g ∆∆ ρ ¯ ∆ τ ( γ µ − κ ∆∆ ρ m ∆ σ µν ∂ ν ) ρ µ · T ∆ τ , L ∆∆ ω = − g ∆∆ ω ¯ ∆ τ ( γ µ − κ ∆∆ ω m ∆ σ µν ∂ ν ) ω µ ∆ τ , L N ∆ ρ = − i g N ∆ ρ m ρ ¯ ∆ µ γ γ ν S † · ρ µν N + H . c ., (4)where ρ µν = ∂ µ ρ ν − ∂ ν ρ µ , and ρ or ω denotes the ρ or ω meson field. The coupling constants g NN ρ = − . g ∆∆ ρ = . g N ∆ ρ = . κ ρ = . κ ω = κ ∆∆ ρ = .
1, cited fromRefs. [64–66]. The coupling constants for ω meson can berelated to these for ρ meson with SU(3) symmetry as g NN ω = g NN ρ , g ∆∆ ω = / g ∆∆ ρ , and κ ∆∆ ω = κ ∆∆ ρ . Besides, the T andthe S matrices are provided as follows, T · ϕ = r ϕ q ϕ + q ϕ − ϕ √ ϕ + √ ϕ − − ϕ q ϕ + q ϕ − − ϕ , (5) S · ϕ = − ϕ − q ϕ q ϕ + − q ϕ − q ϕ ϕ + , (6)where φ = π or ρ , and the + , − , and 0 denote the charges ofthe mesons.Using the Lagrangians above, the potential of the N ∆ interaction can be constructed as, i V d π = I d π g NN π g ∆∆ π m π ¯ u ( k ′ ) γ γ µ q µ u ( k ) · ¯ u α ( k ′ ) γ γ ν q ν u α ( k ) iP π ( q ) , (7) i V d ρ = − I d ρ g NN ρ g ∆∆ ρ ¯ u ( k ′ ) γ µ − κ ρ m N i σ µα q α ! u ( k ) · ¯ u κ ( k ′ ) γ µ + κ ∆∆ ρ m ∆ i σ νβ q β ! u κ ( k ) iP µνρ ( q ) , (8) i V d ω = − I d ω g NN ω g ∆∆ ω ¯ u ( k ′ ) γ µ − κ ω m N i σ µα q α ! u ( k ) · ¯ u κ ( k ′ ) γ µ + κ ∆∆ ω m ∆ i σ νβ q β ! u κ ( k ) iP µνω ( q ) , (9) i V c π = I c π − g N ∆ π m π ¯ u µ ( k ′ ) q µ u ( k ) ¯ u ( k ′ ) q ν u ν ( k ) iP π ( q ) , (10) i V c ρ = − I c ρ g N ∆ ρ m ρ ¯ u α ( k ′ ) γ ( γ µ q α − g µα / q ) u ( k ) · ¯ u ( k ′ ) (cid:16) γ ν q β − g νβ / q (cid:17) γ u β ( k ) iP µν , (11)where the u and u α are the spinor for nucleon and the Rarita-Schwinger vector-spinor for ∆ baryon, respectively. And q , k (1 , , and k ′ (1 , are the momenta of the exchange meson, initialand final nucleons or ∆ baryons. The flavor factors I ( d , c ) i forcertain meson exchange and total isospin are presented inTable I. TABLE I: The flavor factors I ( d , c ) i for certain meson exchange andtotal isospin. I d π I d ρ I d ω I c π I c ρ I = − √ / − √ / − / − / I = √ / √ / The propagator of exchanged mesons are of usualforms as P e ( q ) = i f i ( q ) / ( q − m e ) and P µν e ( q ) = i f i ( q )( − g µν + q µ q ν / m e ) / ( q − m e ) where m e is the mass ofthe exchanged meson. The form factor f i ( q ) is used tocompensate o ff -shell e ff ect of exchanged meson. In this work,we introduce four types of form factors to check the e ff ect ofthe form factor on the results, which have the forms as [67], f ( q ) = Λ e − m e Λ e − q , (12) f ( q ) = Λ e ( m e − q ) + Λ e , (13) f ( q ) = e − ( m e − q ) / Λ e , (14) f ( q ) = Λ e + ( q t − m e ) / q − ( q t + m e ) / + Λ e , (15)where the q t denotes the value of q at the kinematicalthreshold. The cuto ff is parametrized as a form of Λ e = m + α e .
22 GeV. In the current work, we will change the q in the cross diagram to −| q | to avoid the singularities asdone in Ref. [68].In this work, we will adopt the Bethe-Salpeter equationto obtain the N ∆ scattering amplitudes. With the spectatorquasipotential approximation [69–71], the Bethe-Salpeterequation was reduced into a 3-dimensional qBSE, which isfurther reduced into a 1-dimensional equation with fixed spin-parity J P after a partial-wave decomposition as [72–74], i M J P λ ′ λ (p ′ , p) = i V J P λ ′ ,λ (p ′ , p) + X λ ′′ Z p ′′ d p ′′ (2 π ) · i V J P λ ′ λ ′′ (p ′ , p ′′ ) G (p ′′ ) i M J P λ ′′ λ (p ′′ , p) , (16)where the sum extends only over nonnegative helicity λ ′′ .With the spectator approximation, the G (p ′′ ) is reduced from4-dimensional propagator G D ( p ′′ ), which can be writtendown in the center-of-mass frame with P = ( W , ) as, G D ( p ′′ ) = δ + ( p ′′ ∆ − m ∆ ) p ′′ N − m N = δ + ( p ′′ ∆ − E ∆ (p ′′ ))2 E ∆ (p ′′ )[( W − E ∆ (p ′′ )) − E N (p ′′ )] . (17) Obviously, the δ function will reduce the 4-dimensionalintegral equation to a 3-dimensional one, and the Eq. (16)can be obtained after partial-wave decomposition. Here, asrequired by the spectator approximation, the heavier ∆ baryonis on shell, which satisfies p ′′ ∆ = E ∆ (p ′′ ) = q m ∆ + p ′′ assuggested by the δ function in Eq. (17). The p ′′ N for the lighternucleon is then W − E ∆ (p ′′ ). Here and hereafter, a definitionp = | p | will be adopted.The partial-wave potential is defined with the potential ofthe interaction obtained in the above as V J P λ ′ λ (p ′ , p) = π Z d cos θ [ d J λλ ′ ( θ ) V λ ′ λ ( p ′ , p ) + η d J − λλ ′ ( θ ) V λ ′ − λ ( p ′ , p )] , (18)where η = PP P ( − J − J − J with P and J being parityand spin for system, nucleon or ∆ baryon. The initial andfinal relative momenta are chosen as p = (0 , , p) and p ′ = (p ′ sin θ, , p ′ cos θ ). The d J λλ ′ ( θ ) is the Wigner d-matrix.In our qBSE approach, the ∆ baryon is set on-shell whilethe nucleon is still possible to be o ff -shell. Hence, weintroduce a form factor into the propagator to reflect theo ff -shell e ff ect as an exponential regularization, G ( p ) → G ( p )[ e − ( k − m ) / Λ r ] , where the k and m are the momentumand the mass of the nucleon. With such regularization, theintegral equation is convergent even if we do not consider theform factor into the propagator of exchanged meson. Thecuto ff Λ r is parameterized as in the Λ e case, that is, Λ r = m e + α r .
22 GeV with m e being the mass of exchanged mesonand α r serving the same function as the parameter α e . The α e and α r play analogous roles in the calculation of the bindingenergy. Hence, we take these two parameter as a parameter α for simplification. III. NUMERICAL RESULTS
The scattering amplitude of the N ∆ interaction can beobtained by inserting the potential kernel in Eqs. (7-11) intothe qBSE in Eq. (16). The bound state can be searched asthe pole in real axis of the complex energy plane below thethreshold. In the current work, we will consider four S-wavestates from the N ∆ interaction, D , D , D , and D , withisospin spin I J =
11, 12, 21, and 22, respectively. The resultswith the variation of the parameter α are presented in Fig. 2.Among the four S-wave states considered, three boundstates are produced from the N ∆ interaction, that is, D , D , and D . The D state, which hint was observed atWASA-at-COSY, appears at an α of about 2, and its bindingenergy increases with the increase of α . The experimentalvalue of the binding energy can be reached at α of about 3 to3.5. In the figure, we present the experimental results of themass and corresponding uncertainty as a horizontal line anda grey band in the middle panel for reference. The values of α can be determined by comparing the theoretical result andexperiment. Here, we take the results with f as an example,the determined value of α and its uncertainty are shown as ared vertical line and a cyan band. For f , , , two other bound E B (MeV) D D a f f f f D FIG. 2: The variation of the binding energy E B = M th − W onparameter α with M th and W being the N ∆ threshold and positionof bound states. The triangle (purple), square (red), circle (blue), anddiamond (green) and the corresponding lines are for form factors oftypes in Eqs. (12-15). The horizontal line and the grey band in middlepanel are for the experimental mass and its uncertainties observed atWASA-at-COSY [54]. The red line and cyan band are for the α determined by the experiment with form factor f . states appears at larger α , 2.5 and 2.8, for the D and D states, respectively. For f , values of α about 0.5 larger areneeded to produce these two states. If we choose the value of α for f as shown in figure as red line, the binding energiesof the D and D states are about 2 and 8 MeV, respectively.After considering the uncertainties, the binding energies ofthese two states are several and ten MeV, respectively. Hence,the D and D states are bound much more shallowly thanthe D state. In the current work, we consider four types ofthe form factors as shown in the figure. As suggested by theresults, the di ff erent choices of the form factor do not a ff ectthe conclusion obtained above with f .In the above, we present the results for the N ∆ interaction.The NN scattering has been studied explicity in Refs. [68,69] by Gross and his collaborators with the same spectatorapproximation adopted in the current work. Because there aresome di ff erences in the explicit treatment between the current work and Refs. [68, 69], it is interesting to see if the deuteroncan be reproduced with current Lagrangians and theoreticalframes. The potential can be obtained easily by replacing ∆ by N , and the σ exchange is introduced by a Lagrangian L σ NN = g σ NN ¯ NN σ with a coupling constant g σ NN ≈ NN interaction withisospin I = J = f ( q ). It is found thatwith an α about 2 the bound state was produced from the NN ( D ) interaction, which can be related to the deuteron.Considering the NN and N ∆ interaction are di ff erent, onecan say that the α about 3.2 adopted in the N ∆ interactionis consistent with the α value to reproduce the deuteron, about2.7. With only one of π , ρ , and σ exchanges, the bound statecan be found in the range of the parameter considered here,while with only ω exchange no bound state can not produced.It suggests that the π , ρ , and σ exchanges provide attractiveforce. If we remove the contribution from ω exchange, abound state will appear below α =
2, which concludes thatthe ω exchange provides repulsive force. Without π exchange,the bound state will disappear, while if we remove the ρ or σ exchange, the bound state is still remained. It suggests that the π exchange is essential to cancel the repulsive ω exchange.Moreover, the result with π exchange only is very close tothat with the full model. Hence, the π exchange is crucial toreproduce the deuteron. Such results are consistent with usualconclusion of the OBE model of the nuclear force [76]. Wewould like to remind that compared with the works by Gross et al . [68, 69] the calculation here is very crude and we do notfit experimental data of NN scattering also. It is given only toshow that our approach can give the basic results of nuclearforce. E B ( M e V ) Full model with only with only without with only without
FIG. 3: The variation of the binding energy E B for the NN interactionwith isospin I = S = f ( q ) in Eq. (13). Thehorizontal line are for the experimental mass of the deuteron. In Fig. 2, we choose a value of parameter α = .
2, whichis determined from the experimental mass. If we choosethe parameter α = . N ∆ interaction will changea little. The binding energy of the D state deduces to about10 MeV, and the D state has a very small binding energy,about 1 MeV. With such parameter, the D may disappear.In the current work, we consider three exchanges of the π , ρ , and ω mesons. In Fig. 4 we present the results with only oneexchange to discuss the role played by each exchange. Herewe only present the results for three states which are boundby the interaction. For the D state, the bound state can notbe produced only with the ω exchange. With the ρ exchange,the bound state still exists but appears at larger α of about3.0. It suggests that the attraction is very weak comparedwith the full model, and a larger value of α is needed tocompensate it. For the results with only the π exchange,one can find that the binding even becomes stronger than thatwith all three exchanges. Explicit analysis suggests that the ω exchange will weaken the attraction, which leads to thelarger α needed in the full model, which is also analogousto the deuteron case. Hence, for the D state, the mainattraction is from the π exchange as in the deuteron case.The ρ exchange provides marginal attraction while inclusionof the ω exchange weakens the attraction. For the D and D states, only with π exchange, the bound states can beproduced, but the α needed is smaller than for the full model.It suggests that the π exchange plays the most important rolein producing the bound states as for the D state. f f f f D D with w with p with r E B (MeV) D with r D with w a D with p D with p D with w D with r D FIG. 4: The binding energy E B with variation of the α with exchangeof only one meson. In our model, two diagrams are considered for theinteraction, that is, the direct and cross diagrams as shownin Fig. 1. Here, we present the results with only onediagram. For di ff erent states di ff erent diagrams are importantin producing the bound states. The attraction from the directdiagram is enough to produce the D state while no boundstates can be produced with the direct diagram for the D and D states. These two states are mainly produced from thecontributions from the cross diagram. Such result suggests that the cross diagram is important and can not be neglectedin the calculation. Direct D Cross D E B (MeV) a D Cross D Direct D Cross f f f f D Direct
FIG. 5: The binding energy E B with variation of the α for direct andcross diagrams. IV. SUMMARY AND DISCUSSION
Inspired by the experimental hint of the dibaryon N ∆ ( D )at WASA-at-COSY, we study the possible molecular statesfrom the N ∆ interaction. Within the one-boson-exchangemodel, the interaction is constructed with the help ofthe e ff ective Lagrangians, which coupling constants aredetermined by experiment and SU(3) symmetry. Afterinserting the potential to the qBSE, we search the bound statesfrom the S-wave N ∆ interaction.Among four states considered in the current work, threebound states, D , D , and D , can be found in the rangeof parameter α considered here. We also perform a crudecalculation about the deuteron within the current theoreticalframe for reference. The deuteron can be reproduced fromthe NN interaction with a parameter a little smaller than theone for D state determined by the experimental mass. Theresults suggest that the π exchange plays the most importantrole in producing these bound states. The ρ exchange providesa marginal contribution to produce the D state while the ω exchange will weaken the interaction. The binding of the D state is deepest among the three states. With valuesof the parameter α for which the experimental value of thebinding energy for D state is obtained, the other two statesare predicted with much small binding energy.Here, we would like to address the possible uncertainties inthe current work. In our models, the spectator approximationand the replacement of q by −| q | in the propagator forthe cross diagram will introduce model uncertainties. Suchuncertainties will be absorbed by the parameter α . Hence, the α can vary a little. Besides, as discussed in the above section.The deuteron is reproduced at α = . N ∆ ( D )at WASA-at-COSY. With such value, the binding energy of D state becomes very small, and the D even disappears.Considering the model uncertainties, these two states mayhave very small binding energies, even do not exist.The d ∗ (2380) is the second observed dibaryon besides thedeuteron. However, it seems to be a compact hexaquarkinstead of a molecular state like the deuteron. It is interestingto find more dibaryons to understand the internal structure ofthe dibaryons. The masses of the dibaryon N ∆ ( D ) suggestedby the WASA-at-COSY Collaboration is close the the N ∆ threshold, and it has a width very close to the sum of widthsof a nucleon and a ∆ baryon, which supports it as a molecularstate. However, the experimental hint of the state N ∆ ( D ) at WASA-at-COSY is very weak and not confirmed by otherexperiments. The existence of such state requires furthertheoretical and experimental studies. Based on our work, theexistence of N ∆ ( D ) suggests the possible existence of othertwo N ∆ molecular states, D and D . It is interesting tosearch for such states in the experiment. Acknowledgments
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