PHYSICAL REVIEW D 102, 074023 (2020)

Hexaquark picture for dð2380Þ

Hungchong Kim ,1,2,* K. S. Kim,3 and Makoto Oka 4,5 1Research Institute of Basic Science, Korea Aerospace University, Goyang 412-791, Korea 2Center for Extreme Nuclear , Korea University, Seoul 02841, Korea 3School of Liberal Arts and Science, Korea Aerospace University, Goyang 412-791, Korea 4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan 5Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan

(Received 25 August 2020; accepted 24 September 2020; published 27 October 2020)

Hexaquark wave function with the quantum numbers IðJPÞ¼0ð3þÞ, which might be relevant for dð2380Þ, is constructed under an assumption that this is composed only of u, d in an S wave. ¯ By combining three diquarks of either type, (3c;I ¼ 1)or(6c;I ¼ 0), we demonstrate that there are five possible configurations for the six- state. The fully antisymmetric wave function is constructed by linearly combining the five configurations on an equal footing. We then take this wave function as well as the five configurations to calculate the hexaquark mass using the contact type effective potential consisting of the color spin, color electric, and constant shift. The mass is found to be the same regardless of the configurations being used including the fully antisymmetric one. This result can be traced to the fact that the hexaquark system has a freedom in choosing three diquarks in the construction of its wave function. The calculated hexaquark mass using the empirical parameters independently fixed from the spectroscopy is found to be around 2342 MeV, which is indeed very close to the experimental mass of dð2380Þ. Therefore, the hexaquark picture is promising for dð2380Þ as far as the mass is concerned.

DOI: 10.1103/PhysRevD.102.074023

I. INTRODUCTION invariant mass of ΔΔ. In this sense, dð2380Þ could be the dibaryon state predicted long ago by Ref. [9] with The dð2380Þ resonance with the quantum numbers I JP 0 3þ the mass 2350 MeV in the isospin-spin channel of ð Þ¼ ð Þ has been reported recently by the WASA- I;J 0; 3 at-COSY Collaboration [1] from the exclusive reaction ð Þ¼ð Þ. The dynamics that leads to this resonance pn → dπ0π0 could be the attractive force between two Δs in the channel channel, . Its existence is also supported by I;J 0; 3 later experiments [2–6]. This resonance, as a state of ð Þ¼ð Þ [10]. double-pionic fusion to , may provide a plausible One problem in this view is the small decay width of d 2380 explanation for the substantial enhancement seen long ago ð Þ, which is about 70 MeV [1]. Given the fact that 3 the Δ decay width, ΓΔ ≈ 115 MeV, it is not easy to in the He missing mass spectrum from the inclusive 3 understand the small width if d ð2380Þ is viewed as a reaction, pd → HeX [7,8]. of two Δ s. Alternatively, as advocated in Based on the reaction channels that lead to its observa- Refs. [11,12], dð2380Þ might be a hexaquark state that is tion, dð2380Þ is expected to be a six-quark state composed dominated by the hidden-color component. Indeed, accord- of u, d quarks only. Then, as is often encountered in the ing to Refs. [13,14], the six-quark state with all the quarks multiquark studies, one important issue to discuss is in an S wave is found to have more probability to stay in whether dð2380Þ is a molecular state of two color-singlet the hidden-color configuration rather than in the ΔΔ objects, namely the anticipated dibaryon state, or a hex- configuration. In this regard, the hexaquark picture needs aquark state that has the hidden-color component in to be investigated more concretely because, after all, it is addition. The dð2380Þ mass, which is measured to be more extensive in that this picture can accommodate the around M ≈ 2380 MeV, is about 80 MeV less than the molecular picture as its component. To investigate the hexaquark possibility, Ref. [15] has *[email protected] performed a calculation based on a variation method and concluded that the hexaquark picture is not realistic for Published by the American Physical Society under the terms of dð2380Þ because the calculated mass from this picture is the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to too large. This conclusion is questioned later by Ref. [16], the author(s) and the published article’s title, journal citation, which claims that the important medium-range interaction and DOI. Funded by SCOAP3. was missing in that calculation. More recently, Shi et al. [17]

2470-0010=2020=102(7)=074023(10) 074023-1 Published by the American Physical Society HUNGCHONG KIM, K. S. KIM, and MAKOTO OKA PHYS. REV. D 102, 074023 (2020) investigated the hexaquark possibility for dð2380Þ in a fully antisymmetric wave function. The resulting hexa- model where the spin-1 with the color structure of quark wave function will be tested by calculating its mass ¯ 3c has been adopted in the construction of the hexaquark using a semiempirical effective potential of the contact type wave function. Even though this spin-1 diquark is known to composed of color spin, color electric, and some constant be the bad diquark [18], their calculation seems to reproduce shift, the same potential type that has been used in the the experimental mass and decay width of dð2380Þ previous studies of [20–24]. The parameters relatively well. This result, however, was disputed later by appearing in this potential will be independently fixed from Ref. [19], which pointed out some problems in calculating the baryon octet and decuplet. Thus, our approach in this the decay width and an unrealistic nature inherited in the work is different from Ref. [15] in that we use this contact calculated mass. type potential that freezes the spatial dependence of the The diquark approach, which normally relies on a potential by fitting the effective parameters from the baryon compact diquark with a hope to generate an optimal spectroscopy handled in the same footing. multiquark configuration, has conceptual problems espe- This paper is organized as follows. In Sec. II, we introduce cially when it applies to the hexaquark system of qqqqqq an effective potential that will be used in our study. In (q ¼ u, d). First of all, such a hexaquark, as it is built from Sec. III, all the possible hexaquark configurations are one diquark type only, cannot satisfy the fully antisym- constructed in color, isospin space separately and also in metric condition under exchange of any two quark among the combined color-isospin space. We also present explicit the six quarks [15,19]. Second, the resulting hexaquark expression for the fully antisymmetric wave function in configuration does not have a privilege over other possible Sec. III D. In Sec. IV, we make a few remarks on the configurations constructed from other diquark types as far interesting aspect from the expectation value of the effective as in generating the ground state configuration. The second potential when it is calculated with respect to the constructed statement is closely related to a freedom in dividing the six hexaquark wave functions. The hexaquark mass will be quarks into three diquarks. There are various ways to divide presented in Sec. V. We provide a summary in Sec. VI. the six quarks into three diquarks1 and, in principle, all of them must be equivalent to describe the hexaquark system II. EFFECTIVE POTENTIAL as it is completely arbitrary to choose any division especially in the qqqqqq system with q ¼ u, d only. It We begin with an effective potential that will be used to turns out that, if all the possible diquarks are considered in calculate the hexaquark mass. A in the constituent the hexaquark construction, there are five configurations in quark picture can be described by the Hamiltonian com- each division and, from the freedom mentioned above, it is posed of two terms, the quark mass term and the interaction possible to show that all the five configurations have the term among the participating quarks. The interaction can same expectation value for the potential (see Sec. IV). As a have two different sources, one- exchange potential result, all of the five configurations are equally important, and instanton-induced potential [25,26]. One way to para- and none of them are in fact better suited for the hexaquark metrize the effective potentials is to write them down in the description. In other words, there is no compact diquark that contact form composed of the three parts, color spin (VCS), can generate the compact hexaquark configuration. Instead, color electric (VCE), and constant shift [27], the five configurations can participate in constructing the V V V fully antisymmetric wave function as its components. eff ¼ CS þ CE þ constant X v X v Therefore, even though the diquark approach is conceptually 0 λ λ J J 1 λ λ v : ¼ m m i · j i · j þ m m i · j þ 2 ð1Þ problematic, it still provides one convenient basis to con- i

In this work, we investigate the hexaquark possibility for Here, λi denotes the Gell-Mann matrix for the color, Ji the d 2380 ð Þ. The hexaquark wave functions will be con- spin, and mi the quark mass. This interaction is a semi- S structed under an assumption that all the quarks are in an empirical type which acts on two quarks in one spatial wave. In our construction, we take all the possible diquarks V point. So the spatial dependence of eff is frozen in an as a convenient tool for describing the hexaquark wave average sense by fixing the empirical parameters, v0, v1, v2, function so the diquark types are not necessarily limited to from some baryon masses used as inputs. the compact one as in the usual diquark models. In this Hadron mass can be written formally by the mass sense, our approach is different from other studies that rely formula on one compact diquark type only. In fact, there are five X possible configurations that can be combined to form a M ≃ m V ; H i þh effi ð2Þ i 1This is in contrast to the system qqq¯q¯, which has only one division qq − q¯q¯ in a diquark model. The other like where the expectation value needs to be evaluated with qq¯ − qq¯ is not a division based on a diquark model. respect to an appropriate wave function constructed for the

074023-2 HEXAQUARK PICTURE FOR Dð2380Þ PHYS. REV. D 102, 074023 (2020) hadron of concern. In the constituent quark picture, the spin-up state. Then, the rest color-isospin part must be quark mass used in Eq. (2) should be regarded as an antisymmetric. To achieve this, we start with all the possible effective mass that includes the kinetic energy of constitu- diquark types that obey the Pauli principle within the two ent quarks [17]. quarks and use them as a convenient tool to construct the The color-spin interaction VCS alone is often used to five possible configurations for qq − qq − qq in the color- investigate hadron masses [28–32] in the context of Eq. (2). isospin part. The resulting configurations therefore are One advantage of using VCS is that it reproduces quite antisymmetric only between the two quarks in each diquark. well the mass difference among with the same The fully antisymmetric wave function will be constructed flavor content [20,22,33–37] because the other terms in the by linearly combining all the five configurations. potential, the color electric and constant shift, are canceled away in the mass difference. But in general the other terms A. Color part are not negligible in calculating the mass itself. Indeed, the We start with the color part of the hexaquark wave mass formula, Eq. (2), with all the three terms kept in the function by combining three diquarks of all the possible potential, has been applied to the baryon system success- types. For an illustrative purpose, we label the six quarks by fully [27] using the three parameters, v0, v1, v2, fitted from q1q2q3q4q5q6 and divide them into three diquarks grouped the experimental masses of N, Δ, Λ. To make our as (12)(34)(56). The six-quark colors can be expressed by presentation self-contained, this fitting process has been the diquark colors in this division of (12)(34)(56) as explained in the Appendix with three different cases: (I) v0 ≠ 0;v1 ¼ v2 ¼ 0, (II) v0 ≠ 0, v1 ≠ 0, v2 ¼ 0, and f½3c ⊗ 3c ⊗ ½3c ⊗ 3c ⊗ ½3c ⊗ 3cg1 (III) v0 ≠ 0, v1 ≠ 0, v2 ≠ 0. We find that the third case with c the determined parameters 6 ⊕ 3¯ ⊗ 6 ⊕ 3¯ ⊗ 6 ⊕ 3¯ ; ¼f½ c c ½ c c ½ c cg1c ð5Þ v −199 6 3;v71 2 3; ¯ 0 ¼ð . MeVÞ 1 ¼ð . MeVÞ using the group multiplication of 3c ⊗ 3c ¼ 6c ⊕ 3c. 1 v2 ¼ 122.5 MeV; ð3Þ Here the subscript c of the total bracket denotes that the hexaquark is in a color singlet. reproduces the baryon masses very well as shown in Table I It is now easy to see that, among the various terms that in the Appendix. Eq. (5) can generate, only five color configurations can The effective potential, Eq. (1), can be simplified further form a color-singlet state totally. This coincides with the when it applies to the hexaquark system of our concern. As general statement that a six-quark state has five color mentioned already, we consider in this work the hexaquark configurations no how it is divided into subparts composed of u, d quarks only so we can set all the quark [38,39]. The five color configurations in this diquark masses to be equal, mi ¼ mj ≡ m. Its value is taken to be division can be written explicitly as 330 MeV as in the previous works on tetraquarks [20–23]. f½6c ⊗ 6c6¯ ⊗ 6cg ; ð6Þ Also, all the six quarks are in the spin-up state in order to c 1c make the total spin J ¼ 3. Hence, the spin-dependent part ¯ ¯ ¯ J J 1=4 f½3c ⊗ 3c3 ⊗ 3cg ; ð7Þ in Eq. (1) is trivially evaluated to be h i · ji¼ for any c 1c i, j. These two aspects simplify Eq. (1) further into the form ¯ ¯ f½6c ⊗ 3c3 ⊗ 3cg ; ð8Þ v v X c 1c V 0 1 λ λ v : eff ¼ 4m2 þ m2 i · j þ 2 ð4Þ ¯ ¯ i

074023-3 HUNGCHONG KIM, K. S. KIM, and MAKOTO OKA PHYS. REV. D 102, 074023 (2020)

V from eff using the parameter set given in Eq. (3). Thus, the The division (12)(34)(56) above is just one particular hexaquark configuration of Eq. (6) is constructed from the choice in representing the color configurations of the stable diquark 6c only while Eq. (7), which has been used in hexaquark. Since dividing the six quarks into three ¯ Ref. [17], is built from the less stable diquark 3c only. The diquarks is completely arbitrary, one can choose a rest of the three configurations, Eqs. (8), (9), (10) contain different division like (13)(24)(56) that can be obtained ¯ q ↔ q both types, 6c and 3c. from (12)(34)(56) by 2 3. The equation, like For a mathematical convenience, it is useful to represent Eq. (5), still holds in this new division and one can the five color configurations Eqs. (6)–(10) by a tensor get similar five configurations that differ from jCii only 2 ¯ notation. In this notation, 6c and 3c are expressed by by the numerical subscripts on the rhs. But it is individual quark color as clear that jCii should be related to these new configu- rations through some linear combinations or vice versa 1 because any diquark that is in a definite color state in the 6c∶ Sab ¼ pffiffiffi ½qaqb þ qbqa; ð11Þ ¯ 2 (12)(34)(56) division is in a mixture of 6c and 3c when it is viewed in the different division like (13)(24)(56). This 1 ffiffiffi ¯ a abc p abc aspect can be utilized to prove that there are no lowest 3c∶ T ¼ pffiffiffi ϵ ½qbqc − qcqb¼ 2ϵ qbqc: ð12Þ 2 configuration among the five. We will come back to this discussion later when we calculate the expectation value, V From this expression, one can explicitly see that Sab is h effi,inSec.IV. symmetric and Ta is antisymmetric under exchange of the quark colors. Their inner products are normalized as B. Isospin part Next we develop the isospin part of the wave function S ;S δ δ δ δ ; ð ab cdÞ¼ ac bd þ ad bc ð13Þ that needs to be combined into the color part, jCii,in Eqs. (16)–(20). The isospin part is crucial for constructing ðTa;TbÞ¼4δab; ð14Þ the fully antisymmetric wave function under exchange of any two quark among the six quarks but, since the effective V and they are orthogonal, potential eff is blind on isospin, the isospin part does not V practically participate in calculating h effi. c ðSab;T Þ¼0: ð15Þ In our construction, we first impose the antisymmetric constraint only on each diquark in the combined space of Now, it is straightforward to write Eqs. (6)–(10) in terms color isospin. Any diquark in the configurations jCii is c of Sab;Ta, namely, either symmetric (Sab) or antisymmetric (T ) in color. To make each diquark antisymmetric in the color-isospin 1 abc a0b0c0 space, the diquark isospin (Idi) is restricted to be Idi ¼ 0 jC1i¼ ϵ ϵ ðS12Þaa0 ðS34Þbb0 ðS56Þcc0 ; ð16Þ c 12 for the Sab diquark and Idi ¼ 1 for the T diquark. Then the total isospins are determined from the multiplication 1 a b c of three isospins of the diquarks. Specifically, C1 in jC2i¼ pffiffiffi ϵabcðT12Þ ðT34Þ ðT56Þ ; ð17Þ j i 8 6 Eq. (16) consists of three diquarks of the Sab type with the isospin Idi ¼ 0. Consequently, the total isospin of 1 a b jC1i is I ¼ 0 only. For jC2i of Eq. (17), each diquark is jC3i¼ pffiffiffi ðS12ÞabðT34Þ ðT56Þ ; ð18Þ 8 3 in the Idi ¼ 1 state, so possible isospins of jC2i are I ¼ 0,1,2,3.ForjC3i; jC4i; jC5i, possible isospins are 1 a b I ¼ 0,1,2.ThismeansthattheI ¼ 0 is the only jC4i¼ pffiffiffi ðT12Þ ðS34ÞabðT56Þ ; ð19Þ 8 3 common isospin state that exists in all the five configu- rations. As we shall see below, since all the five color 1 a b configurations are necessary in constructing the fully jC5i¼ pffiffiffi ðT12Þ ðT34Þ ðS56Þab: ð20Þ 8 3 antisymmetric wave function, the I ¼ 0 is the only possible isospin for the hexaquark and this is indeed On the right-hand side (rhs), we have added the numerical consistent with the isospin of dð2380Þ. This observation subscripts in our tensors in order to specify the division here provides an alternative explanation why there is (12)(34)(56) more clearly.3 These five color configurations only one isospin state, I ¼ 0, when the six-quark state is are orthonormal, hCijCji¼δij. in the spin state J ¼ 3 totally [9]. It is straightforward to derive the five isospin I 0 2For technical details on this tensor notation, see Ref. [40]. configurations, with the total isospin ¼ , that can be 3 Our expression for jCii is different from Eq. (21) in Ref. [15] multiplied to the corresponding five color configurations, in that ours are written in a diquark basis. Eqs. (16)–(20),respectively.Theyare

074023-4 HEXAQUARK PICTURE FOR Dð2380Þ PHYS. REV. D 102, 074023 (2020) jI1i¼½ud½ud½ud; ð21Þ S S ⊗ ud ð ½udÞaa0 ¼ð 56Þaa0 ½ 1 1 1 jI2i¼pffiffiffi ½ðuufudg − fudguuÞdd − ðuudd − dduuÞfudg ¼ ðuada0 þ ua0 daÞ − ðdaua0 þ da0 uaÞ; ð30Þ 6 2 2 þðfudgdd − ddfudgÞuu; ð22Þ and the other terms are similarly defined. Again, the 1 five color-isospin wave functions are orthonormal, jI3i¼pffiffiffið½uduudd−½udfudgfudgþ½uddduuÞ; ð23Þ hψ ijψ ji¼δij. 3 1 D. Fully antisymmetric color-isospin part jI4i¼pffiffiffiðuu½uddd−fudg½udfudgþdd½uduuÞ; ð24Þ 3 By construction, all the five color-isospin configurations jψ ii in Eq. (27) are antisymmetric only under exchange of 1 ψ jI5i¼pffiffiffiðuudd½ud−fudgfudg½udþdduu½udÞ; ð25Þ the two quarks in each diquark. None of j ii is fully 3 antisymmetric under exchange of any two quark among all the six quarks, and, therefore, none of jψ ii can be regarded where we have introduced the short-hand notations, as the physical state. But they constitute the full compo- 1 1 nents of the hexaquark wave function because one can ½ud ≡ pffiffiffi ðud − duÞ; fudg ≡ pffiffiffi ðud þ duÞ; ð26Þ construct the fully antisymmetric wave function jΨi by 2 2 linearly combining all the five configurations as to represent the antisymmetric (Idi ¼ 0) and symmetric X5 (Idi ¼ 1) combination, respectively. These five isospin Ψ ai ψ i : 31 configurations are also orthonormal, hIijIji¼δij. j i¼ j i ð Þ i¼1 C. Color-isospin part We determine the coefficients ai by two conditions, the The color-isospin part of the hexaquark wave function normalization of the full wave function and the antisym- can be constructed from the direct product, metric constraint imposed on any two quarks in the six quarks. This has been worked out explicitly and we find that jψ ii¼jCii ⊗ jIii; ði ¼ 1; 2; 3; 4; 5Þ: ð27Þ 1 As we have mentioned already, to make each diquark a1 ¼ a2 ¼ −a3 ¼ −a4 ¼ −a5 ¼ pffiffiffi : ð32Þ antisymmetric in the combined space of color-isospin, 5 the product here must act only between a diquark in color and the corresponding diquark in isospin; that is, the (12) Therefore, the fully antisymmetric wave function in the diquark in color must be combined only with the (12) color-isospin space is given as diquark in isospin and so on. To show what we meant ψ explicitly, j 5i, which is obtained by multiplying Eqs. (20) 1 and (25), takes the form jΨi¼pffiffiffi ½jψ 1iþjψ 2i − jψ 3i − jψ 4i − jψ 5i: ð33Þ 5 ψ ∝ T T S ab − T T S ab j 5i ð uuÞað ddÞbð ½udÞ ð fudgÞað fudgÞbð ½udÞ T T S ab: From this expression, we see that all the five color-isospin þð ddÞað uuÞbð ½udÞ ð28Þ configurations jψ ii are equally important in making the final ab jΨi fully antisymmetric. Here ðTuuÞa, ðS ud Þ in the first term are defined as ½ Finally, before closing this subsection, we present the pffiffiffi explicit expression of this fully antisymmetric wave func- T T ⊗ uu 2ϵ ubuc; ð uuÞa ¼ð 12Þa ¼ abc ð29Þ tion given as

074023-5 HUNGCHONG KIM, K. S. KIM, and MAKOTO OKA PHYS. REV. D 102, 074023 (2020)

1 aa0 bb0 cc0 jΨi¼ pffiffiffi ϵabcϵa0b0c0 ðS ud Þ ðS ud Þ ðS ud Þ 12 5 ½ ½ ½

1 a b c a b c a b c þ pffiffiffi ϵabc½ðTuuÞ ðT ud Þ ðTddÞ − ðT ud Þ ðTuuÞ ðTddÞ − ðTuuÞ ðTddÞ ðT ud Þ 48 5 f g f g f g T a T b T c T a T b T c − T a T b T c þð ddÞ ð uuÞ ð fudgÞ þð fudgÞ ð ddÞ ð uuÞ ð ddÞ ð fudgÞ ð uuÞ

1 ab ab ab − pffiffiffi ½ðS ud Þ ðTuuÞ ðTddÞ − ðS ud Þ ðT ud Þ ðT ud Þ þðS ud Þ ðTddÞ ðTuuÞ 24 5 ½ a b ½ f g a f g b ½ a b

1 ab ab ab − pffiffiffi ½ðTuuÞ ðS ud Þ ðTddÞ − ðT ud Þ ðS ud Þ ðT ud Þ þðTddÞ ðS ud Þ ðTuuÞ 24 5 a ½ b f g a ½ f g b a ½ b

1 ab ab ab − pffiffiffi ½ðTuuÞ ðTddÞ ðS ud Þ − ðT ud Þ ðT ud Þ ðS ud Þ þðTddÞ ðTuuÞ ðS ud Þ : ð34Þ 24 5 a b ½ f g a f g b ½ a b ½

The jC1i, as is shown in Eq. (16), is the hexaquark IV. A FEW REMARKS ON hVeffi configuration constructed from the stable diquark 6c only To compute the hexaquark mass through Eq. (2), first while the jC2i in Eq. (17) is the configuration built from the V ¯ we need to calculate the expectation value of eff [Eq. (4)] less stable diquark 3c. One might naïvely expect from a with respect to the fully antisymmetric wave function, diquark model that V1 is the lowest and V2 is the highest Ψ V Ψ V h j effj i. Since eff is blind on isospin, the orthonormal among Vi. The others, V3, V4, V5, are expected to lie condition hIijIji¼δij guarantees that between the two as they contain the two diquark types as their constituents. But, Eq. (37) shows that the jC1i ψ V ψ 0; i ≠ j ; h ij effj ji¼ ð Þ ð35Þ configuration is not guaranteed to be the lowest energy state in the hexaquark system. This is certainly different and, for i ¼ j, from the naive expectation from a diquark model. The main V reason behind this is that eff acts not only on the diquarks ψ V ψ V ≡ V : h ij effj ii¼hCij effjCii i ð36Þ but also on other quark pairs that can be formed from the six quarks. Similar situation occurs in the tetraquark system The last equation also defines Vi, the expectation values of [20–24] where the tetraquark with the spin-1 diquark V ψ eff with respect to j ii. configuration turns out to be more stable than the one V One important characteristic of h effi is that with the spin-0 configuration even though the spin-0 diquark is more compact than the spin-1 diquark. Ψ V Ψ V V V : h j effj i¼ 1 ¼ 2 ¼¼ 5 ð37Þ In fact, the result in Eq. (37) is not accidental. As we have discussed briefly in the last paragraph of Sec. III A,itisa This basically says that jΨi as well as jψ ii have the same natural consequence coming from a freedom in dividing the expectation value of the potential. This also shows that the six quarks into three diquarks in constructing the hexaquark Ψ V fully antisymmetric state j i is not a better configuration as system. Since eff acts on all the pairs among the six quarks, far as in reproducing the lowest energy. its expectation value must be the same regardless of how the To prove Eq. (37), one can establish by direct calculation six quarks are divided into three diquarks. that the color-color part in Eq. (4) has the same expectation To put it more explicitly, let us rewrite jC1i in Eq. (16), value, which was written in the (12)(34)(56) division, in terms of X the new division (13)(24)(56) by moving q2, q3 in Eq. (16). hCij λj · λkjCii¼−16; ð38Þ We find that j

074023-6 HEXAQUARK PICTURE FOR Dð2380Þ PHYS. REV. D 102, 074023 (2020) pffiffiffi 1 3 jψ ii,Eq.(27). Because of this, Shi et al. [17] could have 0 − 0 : jC1i¼2 jC1i 2 jC5i ð40Þ gotten the same hexaquark mass if they took our effective potential, Eq. (4), as well as the mass formula, Eq. (2),in V Using this configuration in evaluating h effi, we find that their calculation that facilitates jψ 2i only. Therefore, in our approach, finding the fully antisymmetric wave function, 1 3 0 0 even though it is needed as a physical state, is not so crucial V1 ¼ V þ V ; ð41Þ 4 1 4 5 in determining the hexaquark mass. Our result is very different from Ref. [15] which V0 ;V0 V where 1 5 are the expectation values of eff with respect performed a variational calculation that takes into account 0 ; 0 4 ; 0 to jC1i jC5i, respectively. But, since jC1i jC1i differ only the spatial dependence of the effective potentials, and by the labeling either in (12)(34)(56) or in (13)(24)(56), obtained a much larger mass of 2630 or 2809 MeV V both must have the same expectation value for eff. The depending on the interaction type adopted in the calcu- ; 0 same thing applies also to jC5i jC5i. As a result, we must lation. The calculated mass there is too large to be a V V0 ;V V0 have 1 ¼ 1 5 ¼ 5. We emphasize that this is a dð2380Þ mass and thus excludes the hexaquark possibility consequence from the arbitrariness in labeling our hex- for dð2380Þ. On the other hand, in our approach, we rely V aquark system. Utilizing this in Eq. (41), we arrive at on a simplified effective potential of the contact type, eff in Eq. (4), and, as a result, the five configurations jψ ii have V1 ¼ V5 ð42Þ the same expectation value as the full wave function does. No variation is necessary in our simplified approach as all as anticipated from Eq. (37). Other relations in Eq. (37) can the six quarks are assumed to be in one spatial point in an be derived also by taking similar steps. Note that the average sense. equation like Eq. (37) may not be satisfied if the strange Nevertheless, it is interesting to compare our result with quarks are involved in the hexaquark system. the mass of the two-baryon state calculated long ago by Dyson and Xuong [9]. Purely from the SU(6) classification, V. HEXAQUARK MASS they predicted that the two-baryon mass in the ðI;JÞ¼ 0; 3 We now present and discuss the hexaquark mass calcu- ð Þ channel is around 2350 MeV, which is also very close d 2380 lated from our wave function. Plugging Eq. (38) into to the ð Þ mass. This old prediction therefore can d 2380 Eq. (4), we obtain the formula for hV i as be used to advocate a different picture for ð Þ, the eff molecular-type resonance composed of two . In this v v calculation, there are only two inputs, one is the deuteron V −16 0 1 v : h effi¼ 4m2 þ m2 þ 2 ð43Þ mass and the other is a parameter related to the mass formula for the baryon multiplet. Their approach, similarly Using the constituent quark mass as m ¼ 330 MeV and to ours in sprit, relies also on a simplified picture without explicit spatial dependence of the potential and so on. Their the effective parameters (v0, v1, v2) determined from the baryon spectroscopy in Eq. (3), we find that mass is only 10 MeV higher than our mass in Eq. (45) calculated based on the hexaquark picture with constituent hV i¼361.5 MeV: ð44Þ quarks. Therefore, as far as the mass is concerned, the two eff pictures can give a reasonable description for the dð2380Þ This value is positive suggesting that this hexaquark is a although they seem conflicting each other for its internal resonance state like the Δ baryon whose effective potential structure. Maybe one possible way of reconciling the two is also positive (see Table I). pictures can be sought from the fact that the hexaquark By putting Eq. (44) in the mass formula of Eq. (2),we picture is more extensive. In other words, the hexaquark finally arrive at our prediction for the hexaquark mass, picture can accommodate the two-baryon picture because it includes the two-baryon state as its component in addition to the hidden-color component. The resulting similar mass MH ¼ 2341.5 MeV: ð45Þ can be understood in our terminology as having similar V This is indeed very close to the experimental mass of value for h effi whether it is calculated with the two-baryon dð2380Þ, which is only 40 MeV below. Hence, as far as the component or with the hidden-color component. Deducing mass is concerned, the hexaquark picture may not be ruled from Eq. (37), and also as advocated in Sec. IV,itis out from a possible structure for dð2380Þ. In addition, as certainly possible that the different configurations have the V explained in the previous section, this mass is the same similar value for h effi. whether it is calculated with the fully antisymmetric wave function jΨi, Eq. (33), or with any of the five configurations VI. SUMMARY In summary, we have constructed in this work the 4 0 V 0 The mixing terms like hC1j eff jC5i are found to be zero. hexaquark wave function that might be relevant for the

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P dð2380Þ with the quantum numbers of IðJ Þ¼0ð3þÞ. APPENDIX: DETERMINATION OF v0, v1, v2 Our assumption in this construction is that the hexaquark is FROM BARYON SPECTROSCOPY u d S composed only of , quarks that are all in an -wave state. The effective potential, V in Eq. (1), contains three Since the spatial and spin parts of the wave function are eff parameters, v0, v1, v2, which are related to the color spin, symmetric under any quark exchange, the rest color-isospin color-electric potential, and the constant shift. In this part must be antisymmetric. The color-isospin part of the Appendix, we determine these parameters from the mass hexaquark wave function is constructed first in a diquark formula Eq. (2) when it is applied to the baryon octet and basis where constituting diquarks are being prepared to be decuplet. The inputs in this analysis are the constituent quark antisymmetric in color isospin. Note that the diquarks in masses, which we set as mu ¼md ¼330MeV, ms ¼500MeV. our work have been adopted as a convenient tool for Using the effective potential Eq. (1) and the appropriate describing the hexaquarks so they are not necessarily wave functions constructed for the baryon octet and restricted to the compact types as in the normal diquark decuplet, it is straightforward to derive the mass formulas, models. There are five color-isospin configurations in this v v construction. We then construct a fully antisymmetric m 3m 2 0 − 8 1 v ; N ¼ u þ 2 2 þ 2 ðA1Þ color-isospin wave function by linearly combining the five mu mu configurations. It turns out that all the five configurations v v m 3m − 2 0 − 8 1 v ; are equally important to the total wave function. In Δ ¼ u m2 m2 þ 2 ðA2Þ particular, all the five configurations were found to give u u v the same hexaquark mass as the total wave function does. m 2m m 2 0 Λ ¼ u þ s þ 2 For the effective potential, we have used the contact type mu composed of the color-spin, color-electric and constant 8 1 2 − v v ; shift, the same type that has been used in the previous 3 1 m2 þ m m þ 2 ðA3Þ – u u s works on tetraquarks [20 23]. The empirical parameters associated with the potential are determined from the 8 1 1 m 2m m − v − baryon spectroscopy. Using these parameters, we have Σ ¼ u þ s 3 0 4m2 m m u u s calculated the hexaquark mass to be around 2340 MeV, 8 1 2 which is quite close to the dð2380Þ mass, only 40 MeV − v v ; 3 1 2 þ m m þ 2 ðA4Þ below. Therefore, we conclude that the hexaquark picture is mu u s still promising as a possible structure for dð2380Þ. 8 1 1 m 2m m − v In closing, we want to make two remarks. First is the Σ ¼ u þ s 3 0 4m2 þ 2m m advantageous aspect of using the diquark basis in com- u u s 3q 3q 8 1 2 parison with other basis like ð Þð Þ partition. In par- − v v ; 1 2 þ þ 2 ðA5Þ ticular, it provides a convenient way to construct the 3 mu mums hexaquark wave function that are totally antisymmetric. 8 1 1 Namely, the color part of wave functions are conveniently mΞ ¼ mu þ 2ms − v0 − þ – 3 m m 4m2 classified according to Eqs. (16) (20) and these can be u s s easily incorporated to the isospin parts of Eqs. (21)–(25). 8 2 1 − v v ; This then straightforwardly leads to an explicit expression 1 þ 2 þ 2 ðA6Þ 3 mums ms for the fully antisymmetric wave function as given in Eq. (34). Another thing to mention is the fact that the 8 1 1 m m 2m − v diquark approach [17], whose wave function is not fully Ξ ¼ u þ s 0 þ 2 3 2mums 4ms antisymmetric, still yields the same hexaquark mass that 8 2 1 can be obtained from the fully antisymmetric wave func- − v v 1 þ 2 þ 2 tion. As discussed in Sec. IV, this result is originated from a 3 mums ms v v freedom in dividing the six quark into three diquarks in m 3m − 2 0 − 8 1 v : Ω ¼ s 2 2 þ 2 ðA7Þ constructing the hexaquark system. ms ms v v v V ACKNOWLEDGMENTS The three parameters 0, 1, 2 appearing in eff will be fixed in three different scenarios. In the first scenario The work of H. K. and K. S. K. was supported by the (Case I), we consider the color-spin potential only by National Research Foundation of Korea (NRF) grant funded setting v0 ≠ 0;v1 ¼ v2 ¼ 0 in Eq. (1). It is well known that by the Korea government (MSIT) (Grants No. NRF- the color-spin potential can explain the mass differences 2018R1A2B6002432 and No. NRF-2018R1A5A1025563). between the baryons with the same flavor content, namely ΔM ≈ Δ V The work of M. O. is supported by a JSPS KAKENHI Grant exp h CSi (see Table IV of Ref. [41]). Only input in No. JP19H05159. this case is the mass difference of N, Δ. Taking the

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TABLE I. Calculated baryon masses in three different cases as well as experimental masses are reported here. See the text for details in calculating the masses in each case. The last four columns show individual contribution to the baryon mass from the quark mass, color spin, color electric, and the constant shift in Case III. The constant shift is of course the same in all the channels. All the numbers are given in MeV unit.

M Each term in Case III theory P Baryon Mexpt Case I Case II Case III mq VCS VCE v2 N 940 844 940 (input) 940 (input) 990 −146.0 −26.5 Δ 1232 1136 1232 (input) 1232 (input) 990 146.0 −26.5 Λ 1116 1014 1088 1116 (input) 1160 −146.0 −20.5 Σ 1193 1080 1154 1182 1160 −79.8 −20.5 122.5 Σ 1385 1273 1279 1375 1160 112.9 −20.5 Ξ 1320 1223 1347 1330 1330 −107.3 −15.5 Ξ 1531 1415 1472 1522 1330 85.4 −15.5 Ω 1672 1564 1605 1675 1500 63.6 −11.5

difference between Eqs. (A1) and (A2), we find mΔ −mN ¼ which lead to 2 −4v0=mu, which fixes

3 3 v1 71.2 MeV ; v2 122.5 MeV: A12 v0 ¼ð−199.6 MeVÞ ; ðA8Þ ¼ð Þ ¼ ð Þ if we use the experimental masses mΔ;mN as inputs. The The baryon masses in this scenario, which are given in the baryon masses determined in this scenario are given in the fifth column in Table I, have an excellent agreement with third column in Table I. The calculated masses are about M within 10 MeV. This also shows that the effective 100 MeV less than the experimental masses. So the color- expt potential, Eq. (1), as well as the empirical parameters v0, spin interaction alone, even though it is successful in v1, v2 given in Eqs. (A8), (A12) are successful in describ- reproducing the mass differences, is not precise enough ing the baryon spectroscopy. to generate the experimental masses. We also examine the relative importance of each In the second scenario (Case II), we include the color- potential term in Case III by separating the calculated electric potential in addition to the color-spin potential mass into the quark mass term, color-spin term, color- (v0 ≠ 0, v1 ≠ 0, v2 ¼ 0). The additional parameter v1 is electric term, and the constant shift. They are listed in the fixed from Eq. (A1) by using mN ¼ 940 MeV as an input. last four columns of Table I. A common feature is that the That is, quark mass term is the biggest as it should be in v v the constituent quark picture. The color-spin potential −8 1 m − 3m − 2 0 96 2 ¼ N u 2 ¼ MeV is the second biggest for N, Δ, but its contribution mu mu becomes less important in the resonances with → v −109 3 3: 1 ¼ð . MeVÞ ðA9Þ mostly because the color-spin interaction is proportional to 1=mimj as in Eq. (1).Relatingtothisistherelative The baryon masses determined in this case are given in the contribution of the constant shift, which is slightly less fourth column in Table I. This result is much better than than the color-spin term in N, Δ, becomes the second Case I, although the calculated masses deviate from the biggest in most resonances with strangeness. But in all experimental masses maximum up to 100 MeV. resonances, the color-electric term contributes marginally In the third scenario (Case III), we include all the in generating the baryon masses. three potentials, color spin, color electric, and constant Another thing to mention is that the parameters in v ≠ 0 v ≠ 0 v ≠ 0 m shift, ( 0 , 1 , 2 ). In this case, we use Λ ¼ Eqs. (A8), (A12) are determined from a baryon system 1116 MeV as an additional input. Plugging the input values of qqq. This system is similar to the hexaquark system in a v ;m ;m of 0 N Λ in Eqs. (A1), (A3), we find these two sense that both are composed of quarks only without constraints, antiquarks. This is in contrast to the tetraquark system which is composed of quarks and antiquarks. Indeed, the v0 v1 − 8 v2 96 ; value determined from the tetraquark system is v0 ≈ m2 þ ¼ MeV ðA10Þ u ð−193Þ3 MeV3, slightly less than Eq. (A8). Even though   8 1 2 the difference is rather small, our parameters determined − v v 102 ; from the baryon system must have better justification when 1 2 þ þ 2 ¼ MeV ðA11Þ 3 mu mums they are applied to hexaquarks.

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