arXiv:1706.10025v2 [hep-ph] 9 Aug 2017 tde eoe ivsr-rcadLadisace sta- searched Leandri and ble Silvestre-Brac 32]. before. [31, studied cases massive dibaryon the of H in In the bound study that becomes show . the calculations heavy lattice in recent containing interest ex- fact, states These the ap- 30]. to molecular multiquark [29, led the model findings [22–24], perimental the rules and sum [25–28] approaches, QCD proach theoretical the many as using such studied were states pnitrcinwe oelgtqak a neatwith interact color- can the quarks light in more attraction when large interaction spin a from arise configurations [40]. large mass sufficiently quark heavy be the to if taken decay was strong found against also stable were be quarks to 39]. heavy interaction[38, two spin with flavor states and 37] [36, models spin quark color in studied with were pen- quarks Heavy heavy with taquarks [33–35]. Hamiltonian chromomagnetic pure nteΛ the in the in hidden-charm states observed recently has collaboration Furthermore, LHCb 20]. collaboration[19, WASA-at-COSY the 1–8.As,i h iaynsco,arsnnestruc- resonance a the in sector, observed finally dibaryon was the ture in Also, experiments other [16–18]. several by confirmed subsequently was el olbrto in Collaboration Belle excitement initial Θ the of of and finding existence dibaryon, the H the about bound out ex- deeply their ruled Unfortunately, a predicted search [1–3]. Jaffe experimental model since tensive bag interest the using of existence theme a became ujc rgee ytedsoeyo the of discovery the by triggered subject [5–14]. it confirm not could study experimental ‡ † ∗ lcrncades [email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic utqakcngrtoswt ev urswere quarks heavy with configurations Multiquark ihntecnttetqakmdl tbemultiquark stable model, quark constituent the Within utqakhdosmd fmr hntrequarks three than more of made Multiquark nteohrhn,teei eee neeti the in interest renewed a is there hand, other the On q 6 , etqak ihtohayatqak nasml chromomag simple a in antiquarks heavy two with Heptaquarks 1 q eateto hsc n nttt fPyisadApidP Applied and Physics of Institute and Physics of Department b 0 5 Q → gis h togdcyit n aynadtomsn sdis is cas two that breaking and find SU(3) We one the into model. in decay strong principle the exclusion constru against We Pauli the interaction. from color-spin using antiquarks heavy ASnmes 14.40.Rt,24.10.Pa,25.75.Dw numbers: PACS ol epoe yrcntutn h + Λ the reconstructing by probed be could and J/ψK eivsiaetesmer rpryadtesaiiyo th of stability the and property symmetry the investigate We .INTRODUCTION I. q 4 − QQ p + rcs 2] usqety these Subsequently, [21]. process B ′ 14)4 ae wya further as away faded (1540)[4] + ytmi h rmwr fa of framework the in system J/ψp → ao Park, Aaron I K nain asspectrum mass invariant ( J ± q 2 P π s 0(3 = ) + 3 s ¯ π 2 + with J/ψ X 1, + 37)b the by (3872) ∗ hne by channel ) I 1] which [15], osn Park, Woosung 0 = Dtd a ,2018) 9, May (Dated: S , φ = + 5 2 φ stems tbehpaur ofiuainthat configuration heptaquark stable most the is nain mass. invariant eato nTbeI hc a eotie from obtained be can in- which spin color I, matrix Table the in in the teraction appearing with invariant together the exclusion for them Pauli elements represent the We satisfying states principle. four are there tions, h aial niymti ofiuaini em of terms in configuration color antisymmetric is configuration maximally con- the attractive diquark most separated The con- two diquark has figurations. attractive later the most because while three system figuration for ΛΛ allows separate two former the the attrac- from more that is than H-dibaryon tive a example, with in For interaction configuration interaction. color-spin the the color-spin inspect attractive to most has the first one hadrons, in probed color-spin be with model to interaction. quark configurations constituent a the using configura- of work two this such part with Hence, also state are [40]. tetraquark tions antiquarks stable or quark a quarks constituent be heavy a will within there fact, the In model, within state stable model. heptaquark a soliton stable light chiral two a is and be there will quarks as there heavy [42– two quarks, long researchers of as composed few state that a suggested only Ref.[44] by studied 44]. been has quarks two with heptaquarks consider antiquarks. will heavy multiquark another con- we yet probe in configuration, further quarks To heavy with hadrons. figurations correlation[41], effec- separated diquark be to additional can compared as which large, understood in- quarks tively color-spin light enhanced the among keeping teraction while can small energy made kinetic be additional the quarks heavy, com- additional are the antiquarks If energy or hadrons. kinetic isolated having additional to pared generate con- the compact will at into However, quarks figuration additional configuration. bringing compact time, same a in other each − = 1, osuytepsil xsec fcmatexotic compact of existence possible the study To anti two and quarks five of composed Heptaquark X † i

II. WHY HEAVY HEPTAQUARK? TABLE I: The classification of two quarks and quark- antiquark color-spin interaction, with λi,σi respectively rep- resenting the color and spin matrix of the i quark. Two quarks In this work, we investigate the stability of the hep- state is determined to satisfy Pauli exclusion principle. We taquark using hyperfine potential. Even if the hyperfine denote antisymmetric and symmetric state as A and S respec- potential is attractive for a given heptaquark configura- tively. In the parenthesis, multiplet state is represented. tion, it cannot form a compact stable state if the addi- tional repulsion from kinetic energy is large. Hence, we qq need to consider which flavor state makes the additional Color A(3)¯ S(6) A(3)¯ S(6) kinetic energy lower. In the remaining part of this paper, Flavor A(3)¯ A(3)¯ S(6) S(6) we consider only hyperfine potential, but we can simply estimate the additional kinetic energy using the following Spin A(1) S(3) S(3) A(1) coordinate system and simple Gaussian spatial function. − · − − 4 8 λiλj σi σj 8 3 3 4 If we label the five light quarks as (i =1 5) and the qq¯ two heavy antiquarks as (i = 6, 7), then we∼ can choose Color (1) (8) (1) (8) the Jacobi coordinate system as follows. Spin A(1) A(1) S(3) S(3) 7 6 − · − 16 − 2 1 1 1 λiλj σi σj 16 2 3 3 m r˙ 2 Mr˙ 2 = M x˙ 2 2 i i − 2 CM 2 i i i=1 i=1 X X 7 where M = mi,M1 = M2 = mu, i=1 X 1 2 2mumQ 5mu(mu + mQ) where N is the total number of quarks, Cc = 4 λ the M3 = M4 = ,M5 = color Casimir operator, S the spin and I the isospin of mu + mQ 2(4mu + mQ) 4 10 the system. Using Cc = 3 , 3 for color anti-triplet and 7(mu + mQ)(4mu + mQ) M6 = sextet respectively, one notes, as given in the table, that 10(5mu +2mQ) while the most attractive channel has spin 0, the spin 1 7 state also has an attractive combination. For two anti- 1 rCM = miri (2) quarks, we can use the same table simply by replacing 3¯ M i=1 and 6 with 3 and 6¯ for color and flavor state, respectively. X For the quark antiquark configuration, we can construct a similar table as given in the lower part of Table I. Typ- 1 x1 = (r1 r2) ically, the color-spin interaction is inversely proportional √2 − to the two constituent quark masses involved 1/(m m ). i j 2 1 1 Hence, in forming a multiquark configurations, if the ad- x2 = ( r1 + r2 r3) dition involves a light quark and a light antiquark, the r3 2 2 − addition will just fall apart into a meson state. On the 1 x3 = (r4 r6) other hand, if the addition is composed of a light quark √2 − and a heavy antiquark, it could become energetically fa- 1 x4 = (r5 r7) vorable to be in a compact configuration. To probe such √2 − a possibility systematically, we investigate the symmetry 6 1 1 1 mu mQ property and the stability of the heptaquark containing x5 = ( r1 + r2 + r3 r4 r6) two identical heavy antiquarks in a simple chromomag- r5 3 3 3 − mu + mQ − mu + mQ netic model. 10 1 x = (m r + m r + m r + m r 6 7 {4m + m u 1 u 2 u 3 u 4 This paper is organized as follows. We first explain r u Q 1 why kinetic energy favors compact heptaquarks contain- + mQr6) (mur5 + mQr7) . (3) ing heavy flavors in Sec. II. In Sec. III, we represent − mu + mQ } the color and spin basis functions of the heptaquark configuration. In Sec. IV, we construct the flavor This coordinate system can describe the decay mode of color spin part of the wave function of the heptaquark⊗ the heptaquark consisting of one baryon and two mesons. in order⊗ to satisfy the Pauli exclusion principle. In In this coordinate system, x1, x2 describe the baryon sys- Sec. V, we represent the color-spin interaction part of tem, while x3, x4 represent the relative quark distances the Hamiltonian. In Sec. VI, we calculate the binding for the two mesons, respectively. If we chose a simple potential of the heptaquark and plot the results as a Gaussian form as the spatial function, then we can cal- function of the light/heavy quark mass ratio parameter culate the kinetic energy of the heptaquark as follows. η. Finally, we summarize our results in Sec. VII. x2 x2 x2 x2 x2 x2 R = e−a1 1−a2 2−a3 3−a4 4−a5 5−a6 6 , (4) 3

6 p2 6 3¯h2 resentation as follows. T = i = a . (5) 2M 2M i i=1 i i=1 i X X From the above expression, we can find that the addi- 2 2 3¯h 3¯h tional kinetic energy of the heptaquark is 2M5 a5 + 2M6 a6, corresponding to the additional kinetic energy from bringing in the two meson type of quark antiquark pair into a compact configuration. In the heavy quark limit, (3 3 3) (3 3¯) (3 3¯) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ M6 making one of the additional kinetic terms =(1 8 8 10) (1 8) (1 8) zero,→ hence ∞ there is no penalty in the kinetic energy, ⊕ ⊕ ⊕ ⊗ ⊕ ⊗ ⊕ =(1 8 8 10) while the extra light quark might contribute attractively ⊕ ⊕ ⊕ to the pentaquark configuration. On the other hand, if (1 1 8 8 8 8 10 10¯ 27) we chose only one antiquark to be heavy, then the re- ⊗ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (6) duced masses become M = m and M = 7mu(mu+mQ) , 5 u 6 2(6mu+mQ) so that both of the additional terms survive in the heavy quark mass limit. Therefore, we can conclude that if we want to make the additional kinetic energy of the hep- taquark sufficiently small, one needs to include at least two heavy quarks. However, one still needs to weigh in the attraction from the color spin interaction to deter- Among the above states, two (1 1), eight (8 8) and one mine which combination generates the most attractive (10 10¯ ) states can form the color⊗ singlet state.⊗ There- heptaquark configuration, which is the subject of this fore,⊗ there are eleven color basis functions for heptaquark. work. It should be noted that if we want to make both However, there is a more efficient way to represent the additional terms zero, then we have to replace one more color basis of heptaquark using the Young-Yamanouchi quark with heavy quark; such configurations with three basis. heavy quarks however is experimentally quite difficult to produce and will not be considered here.

The color state of two antiquarks is a triplet or an anti- III. COLOR AND SPIN BASIS FUNCTIONS sextet. Therefore, in order to construct the color singlet heptaquark state, the color state of five quarks should be A. Color basis function an anti-triplet or a sextet. For five quarks, the number of the Young-Yamanouchi basis of anti-triplets and sextets We can represent the color state of the heptaquark us- are five and six, respectively. Therefore, we can represent ing color decomposition in the SU(3) fundamental rep- eleven color basis functions for heptaquark as follows.

1 2 1 3 1 2 1 3 1 4 6¯ 6¯ 6¯ 6¯ 6¯ C1 = 3 4 , , C2 = 2 4 , , C3 = 3 5 , , C4 = 2 5 , , C5 = 2 5 , , | i  7¯  | i  7¯  | i  7¯  | i  7¯  | i  7¯  5 5 4 4 3                     1 2 3 1 2 4 1 3 4 1 2 5 C6 = 4 , 6¯ 7¯ , C7 = 3 , 6¯ 7¯ , C8 = 2 , 6¯ 7¯ , C9 = 3 , 6¯ 7¯ , | i   | i   | i   | i   5 5 5 4                 1 3 5 1 4 5 C10 = 2 , 6¯ 7¯ , C11 = 2 , 6¯ 7¯ . (7) | i   | i   4 3        

In the appendix, we present the color basis of the hep- B. Spin basis function taquark using the tensor notation. The expectation value of all the color operators for the heptaquarks can be ob- 7 5 3 1 Seven quark system can have spin 2 , 2 , 2 and 2 . We tained using this color basis. represent the spin basis functions in terms of Young- 4

Yamanouchi basis for each spin values. IV. WAVE FUNCTION S = 7 : one basis function with Young tableau [7] • 2 There are two ways of constructing the wave function of the heptaquark. First, we can consider the flavor state 7 2 of five quarks in SU(3) flavor symmetry. In our previous S = 1 2 3 4 5 6 7 (8) | 1 i work [46], we have already classified all the possible flavor state for five light quarks. There are five possible flavor 5 S = : six basis functions with Young tableau [6,1] states for five quarks as follows: • 2

5 1 2 3 4 5 6 5 1 2 3 4 5 7 S 2 = , S 2 = , [3]¯ = , [6] = , [15]¯ = , | 1 i 7 | 2 i 6 F F F

5 5 2 1 2 3 4 6 7 2 1 2 3 5 6 7 S3 = , S4 = , | i 5 | i 4 [24]F = , [21]F = . (12) 5 1 2 4 5 6 7 5 1 3 4 5 6 7 S 2 = , S 2 = . (9) | 5 i 3 | 6 i 2 For a given isospin and spin, we can choose the possible flavor states and construct the remaining part of the wave function using color and spin symmetry. In this descrip- S = 3 : fourteen basis functions with Young 2 tion, the wave function should be antisymmetric for five • tableau [5,2] light quarks and for two heavy antiquarks, respectively. Second, we can calculate the wave function of the hep- 3 3 3 taquark in the SU(3) breaking case, fixing the position of 2 1 2 3 4 5 2 1 2 3 4 6 2 1 2 3 5 6 S1 = , S2 = , S3 = , strange quarks. In the SU(3) flavor symmetry breaking | i 6 7 | i 5 7 | i 4 7 case, we have to construct the wave function to be an- 3 1 2 4 5 6 3 1 3 4 5 6 3 1 2 3 4 7 tisymmetric separately for the u, d quarks, s quarks and S 2 = , S 2 = , S 2 = , | 4 i 3 7 | 5 i 2 7 | 6 i 5 6 two heavy antiquarks, due to the Pauli exclusion princi- ple. 3 3 3 2 1 2 3 5 7 2 1 2 4 5 7 2 1 3 4 5 7 Both approaches can be shown to give the same re- S7 = , S8 = , S9 = , | i 4 6 | i 3 6 | i 2 6 sult [47]. In this work, we follow the second method

3 3 3 for convenience of calculation. To do this, we assume 2 1 2 3 6 7 2 1 2 4 6 7 2 1 3 4 6 7 S10 = , S11 = , S12 = , the spatial function to be symmetric such that the rest | i 4 5 | i 3 5 | i 2 5 of the wave function represented by flavor color ⊗ ⊗ 3 3 spin should be antisymmetric. Using color spin coupling 2 1 2 5 6 7 2 1 3 5 6 7 S13 = , S14 = . (10) scheme [45, 46], we represent the wave function of the | i 3 4 | i 2 4 heptaquark by flavor color-spin coupling basis. ⊗ S = 1 : fourteen basis functions with Young • 2 tableau [4,3] A. q5Q¯2 : {12345}{67}

1 1 2 3 4 1 1 2 3 5 1 1 2 4 5 Here, we calculate the wave function of the heptaquark S 2 = , S 2 = , S 2 = , | 1 i 5 6 7 | 2 i 4 6 7 | 3 i 3 6 7 in the flavor SU(3) breaking case and fix the position of each quarks on q(1)q(2)q(3)q(4)q(5)Q¯(6)Q¯(7). In this 1 1 1 2 1 3 4 5 2 1 2 3 6 2 1 2 4 6 case, the flavor color spin wave function should sat- S = , S = , S = , ⊗ ⊗ | 4 i 2 6 7 | 5 i 4 5 7 | 6 i 3 5 7 isfy the following symmetry : 12345 67 { }{ } 1 1 3 4 6 1 1 2 5 6 1 1 3 5 6 S 2 = , S 2 = , S 2 = , 1 | 7 i 2 5 7 | 8 i 3 4 7 | 9 i 2 4 7 2 ¯ 1 1 1  6  2 1 2 3 7 2 1 2 4 7 2 1 3 4 7 ψFCS = 3 , (13) S10 = , S11 = , S12 = ,  7¯  | i 4 5 6 | i 3 5 6 | i 2 5 6  4    1 1  5  2 1 2 5 7 2 1 3 5 7   FCS S13 = , S14 = . (11)   | i 3 4 6 | i 2 4 6 5

I = 5 • 2

1 2  6¯  ψFCS = 1 2 3 4 5 , 6¯ 7¯ 3 , = F1 CS1 (14) F ⊗  7¯  ⊗    4     5   CS  

I = 3 • 2

1 5 1 4 1 1 2 3 4 2 6¯ 1 2 3 5 2 6¯ ψFCS = , 6¯ 7¯  ,  , 6¯ 7¯  ,  2 5 ! ⊗ 3 7¯ − 4 ! ⊗ 3 7¯  F   F    4   5   CS  CS     1 3 1 2 1 2 4 5 2 6¯ 1 3 4 5 3 6¯ + , 6¯ 7¯  ,  , 6¯ 7¯  ,  3 ! ⊗ 4 7¯ − 2 ! ⊗ 4 7¯ F   F     5   5   CS  CS 1     = (F CS F CS + F CS F CS ) (15) 2 1 ⊗ 4 − 2 ⊗ 3 3 ⊗ 2 − 4 ⊗ 1

I = 1 • 2

1 4 1 3 1 2 1 ¯ ¯ ¯ 1 2 3 ¯ ¯ 6 1 2 4 ¯ ¯ 6 1 3 4 ¯ ¯ 6 ψFCS = , 6 7  2 5 ,  , 6 7  2 5 ,  + , 6 7  3 5 ,  √5 4 5 ! ⊗ 7¯ − 3 5 ! ⊗ 7¯ 2 5 ! ⊗ 7¯  F 3 F 4 F 4  CS  CS  CS       1 3 ¯ 1 2 ¯ 1 2 5 ¯ ¯ 6 1 3 5 ¯ ¯ 6 + , 6 7  2 4 ,  , 6 7  3 4 ,  3 4 ! ⊗ 7¯ − 2 4 ! ⊗ 7¯ F 5 F 5   CS  CS 1     = (F1 CS5 F2 CS4 + F3 CS3 + F4 CS2 F5 CS1) (16) √5 ⊗ − ⊗ ⊗ ⊗ − ⊗

B. q4sQ¯2 : {1234}5{67}

We fix the position of each quarks on q(1)q(2)q(3)q(4)s(5)Q¯(6)Q¯(7). In this case, the fla- vor color spin wave function should satisfy the following⊗ symmetry⊗ : 1234 5 67 because there is no symmetry between u,d{ quarks} { and} s quark in SU(3) breaking case.

1 2 6¯ ψFCS =  , 5 ,  (17) 3 7¯    4    FCS   6

I =2 • 1 2 6¯ ψFCS = 1 2 3 4 , 5 , 6¯ 7¯  , 5 ,  = F1 CS1 (18) F ⊗ 3 7¯ ⊗      4   CS   I =1 • 1 4 1 3 1 ¯ ¯ 1 2 3 ¯ ¯ 6 1 2 4 ¯ ¯ 6 ψFCS = , 5 , 6 7  2 , 5 ,  , 5 , 6 7  2 , 5 ,  √3 4 ! ⊗ 7¯ − 3 ! ⊗ 7¯  F 3 F 4  CS  CS     1 2 ¯ 1 3 4 ¯ ¯ 6 + , 5 , 6 7  3 , 5 ,  2 ! ⊗ 7¯ F 4   CS 1   = (F1 CS3 F2 CS2 + F3 CS1) (19) √3 ⊗ − ⊗ ⊗ I =0 • 1 1 2 1 3 6¯ 1 3 1 2 6¯ ψFCS = , 5 , 6¯ 7¯ , 5 , , 5 , 6¯ 7¯ , 5 , √2 3 4 ⊗ 2 4 7¯ − 2 4 ⊗ 3 4 7¯  !F !CS !F !CS  1 = (F1 CS2 F2 CS1) (20) √2 ⊗ − ⊗

C. q3s2Q¯2 : {123}{45}{67}

We fix the position of each quarks on q(1)q(2)q(3)s(4)s(5)Q¯(6)Q¯(7). In this case, the fla- vor color spin wave function should satisfy the following⊗ symmetry⊗ : 123 45 67 { }{ }{ } 1 4 6¯ ψFCS = 2 , , (21)  5 7¯  3   FCS  

I = 3 • 2 1 4 6¯ ψFCS = 1 2 3 , 4 5 , 6¯ 7¯ 2 , , = F1 CS1 (22) F ⊗  5 7¯  ⊗   3  CS   I = 1 • 2 1 1 2 1 3 4 6¯ 1 3 1 2 4 6¯ ψFCS = , 4 5 , 6¯ 7¯ , , , 4 5 , 6¯ 7¯ , , √2 3 ⊗ 2 5 7¯ − 2 ⊗ 3 5 7¯  !F !CS !F !CS  1 = (F1 CS2 F2 CS1) (23) √2 ⊗ − ⊗

D. s3q2Q¯2 : {123}{45}{67} calculation. In this case, the flavor color spin ⊗ ⊗ We fix the position of each quarks on s(1)s(2)s(3)q(4)q(5)Q¯(6)Q¯(7) for convenience in 7 wave function should satisfy the following symmetry : Isospin Spin M Isospin Spin M 5 ¯2 5 3 3 2 ¯2 3 7 123 45 67 q Q 2 2 1 q s Q 2 2 1 { }{ }{ } 1 5 2 1 2 3 1 3 5 3 ¯ 2 2 1 2 8 4 6 3 1 ψFCS = 2 , , (24) 3 7  5 7¯  2 2 3 1 3 1 7 1 FCS 2 2 2   1 7 5   2 2 1 2 5 I = 1 : The wave function is the same as in the 5 3 3 13 • 3 2 2 3 2 2 case of q s Q¯ with I = . 3 1 2 2 4 2 13 1 q2s3Q¯2 7 I =0 2 3 1 2 1 4 ¯2 5 5 • q sQ 2 2 1 2 3 3 3 1 2 4 2 8 4 6¯ 1 1 ¯ ¯ 2 4 2 7 ψFCS = 1 2 3 , , 6 7  2 , 4 5 ,  5 ! ⊗ 7¯ 7 5 F 3 1 2 1 0 2 3  CS 5 3   2 5 2 6 = F1 CS1 (25) 3 1 ⊗ 2 10 2 7 1 4 ¯2 1 5 2 10 qs Q 2 2 1 7 3 4 2 0 2 1 2 4 E. s qQ¯ : {1234}5{67} 5 1 2 3 2 4 3 5 ¯2 5 2 7 s Q 0 2 1 We fix the position of each quarks on 1 3 ¯ ¯ 2 6 2 3 s(1)s(2)s(3)s(4)q(5)Q(6)Q(7). In this case, the fla- 1 vor color spin wave function should satisfy the 2 3 following⊗ symmetry⊗ : 1234 5 67 { } { } TABLE II: All the possible heptaquark states containing two heavy antiquarks with the corresponding multiplicity. M rep- 1 resents the multiplicity of the color ⊗ flavor ⊗ spin state. 2 6¯ ψFCS =  , 5 ,  (26) 3 7¯    4    FCS V. COLOR-SPIN INTERACTION   I = 1 : The wave function is the same as in the • 2 case of q4sQ¯2 with I = 2. In this article, we investigate the stability of the heptaquark configurations using the hyperfine potential given as F. s5Q¯2 : {12345}{67} 1 H = A λcλcσ σ , (28) − m m i j i · j We fix the position of each quarks on i

VI. RESULTS 1200 S=1.5 Heptaquark can decay into one baryon and two 1000 S=0.5 mesons. The differences in the confining and coulomb B 800 potentials are proportional to the two body color force V λiλj . As far as the compact heptaquark, baryon and mesons are taken to occupy the same size, the differ- 600 ence in these energies between the heptaquark and the 400 sum of the baryon and two mesons are negligible. This 0.0 0.2 0.4 0.6 0.8 1.0 is so because if the heptaquark, baryon and mesons have the same size, the confining potential will just be propor- Η tional to a common value and the sum of all the two body V q5Q¯2 I 5 interaction λiλj , which are equal to -56/3, -8, -16/3 FIG. 1: B of with = 2 (unit: MeV). for the heptaquark, baryon and meson respectively. The main differenceP comes from the difference in the color- spin potential. Therefore, we define the binding potential 1000 S=0.5 of the heptaquark as the difference in the color-spin in- teraction between the heptaquark and the sum of baryon 800 and two mesons as follows: B 600 V S=1.5 VB = Hheptaquark Hbaryon Hmeson1 Hmeson2. (29) − − − S=2.5 400 To search for possible stable configurations, we plot the binding potential of the heptaquark as a function of the 200 heavy quark mass using a variable η defined as follows: 0.0 0.2 0.4 0.6 0.8 1.0 m Η η =1 u . (30) − mQ 5 ¯2 3 FIG. 2: VB of q Q with I = 2 (unit: MeV). Here, we fix the mass to 632 MeV, which comes from our previous work [46]. When the flavor of antiquarks is strange, charm, and bottom, η values are approximately 0.46, 0.82, 0.93, respectively. Decay chan- C. q3s2Q¯2 : {123}{45}{67} nels that give the lowest potential can change as η varies. Hence, in some figures, there are graphs with turning The heptaquark containing two strange quarks with points that have sudden change in the slope. 3 I = 2 in Fig. 7 shows no possibility of a stable hep- 3 2 ¯2 1 5 taquark. In the case of q s Q with I = 2 and S = 2 , 5 2 as shown in Fig. 8, there is a configuration with a slight A. q Q¯ : {12345}{67} negative binding potential when the antiquarks are light quarks. Furthermore, for I = 1 and S = 1 , 3 configu- As we can see in Fig 1-3, there is no possibility of stable 2 2 2 1 3 rations, the potential becomes attractive when the mass heptaquark except I = 2 and S = 2 when the antiquarks of antiquarks becomes very large. We represent the ex- areu ¯ or d¯ quarks. However, the absolute values of the pectation values of the hyperfine potential for the hep- binding potential is very small, so it cannot be compact when we consider the total Hamiltonian including the kinetic term. S=0.5 600

B. q4sQ¯2 : {1234}5{67} 400 B V

4 2 200 S=2.5 In the case of q sQ¯ with I = 2 in Fig. 4, there is no S=1.5 possibility of stable heptaquark. However, in the case S=3.5 5 0 with I = 1 and S = 2 in Fig. 5, there can be a stable heptaquark when the antiquarks are light quarks. But, 0.0 0.2 0.4 0.6 0.8 1.0 the absolute value of the binding potential is still small. In contrast, as can be seen in Fig. 6, the heptaquark Η configuration with I = 0 and S = 1 , 3 can be stable 2 2 5 ¯2 1 when the mass of antiquarks becomes very large. FIG. 3: VB of q Q with I = 2 (unit: MeV). 9

1000 binding potential negative. S=0.5 As we mentioned in introduction, there is a possibility 800 of a stable heptaquark state as long as there is a stable

S=1.5 meson state composed of two heavy quarks and two light B 600 V quarks within the chiral soliton model [44]. It is well S=2.5 P + known that Tcc with J = 1 , I = 0 could be a sta- 400 ble tetraquark state[40, 41]. Therefore, taking the result in Ref. [44] to be valid, there should be a stable con- 200 figuration composed of five light quarks and two heavy 0.0 0.2 0.4 0.6 0.8 1.0 3 1 1 antiquarks with S = 2 or S = 2 and I = 2 . It should be Η 5 ¯2 1 noted that although our results for q Q with I = 2 does not support a stable heptaquark, the configuration with 4 2 3 1 1 FIG. 4: VB of q sQ¯ with I = 2(unit: MeV). 3 2 ¯2 q s Q with S = 2 , 2 and I = 2 indeed may be stable heptaquark states. As we can see in Table II, since these two states have large multiplicities compared to the other states, it may lead a low binding potential. 400 S=0.5

300 600 S=0.5 S=1.5 200 500 B

V 400 100 S=1.5 S=3.5 B 300 0 V S=2.5 S=2.5 200 -100 0.0 0.2 0.4 0.6 0.8 1.0 100 S=3.5 Η 0 0.0 0.2 0.4 0.6 0.8 1.0 4 2 FIG. 5: VB of q sQ¯ with I = 1(unit: MeV). Η

3 2 ¯2 3 FIG. 7: VB of q s Q with I = 2 (unit: MeV).

1 taquark configuration with S = 2 and the lowest de- cay mode in Table III. We take the mass to be 1930 MeV as extracted from fits to the heavy 150 S=1.5 baryon masses using variational method [46]. For the 3 2 2 1 1 100 q s s¯ (I = 2 ,S = 2 ) case, there is an additional inter- S=0.5 S=2.5 action between the u quarks as compared to the isolated S=3.5 B 50 baryon meson states. However, the strength of the inter- V action between u quark and s quark is reduced. When 0 the antiquarks are heavy quarks, there is an additional repulsion between the s quarks, while there is also an -50 additional attraction u quark and s quark, making the 0.0 0.2 0.4 0.6 0.8 1.0 Η

100 3 2 ¯2 1 S=0.5 FIG. 8: VB of q s Q with I = 2 (unit: MeV). S=1.5 50

B S=2.5 V 0 S=3.5 D. s3q2Q¯2 : {123}{45}{67} -50

0.0 0.2 0.4 0.6 0.8 1.0 In this study, the heptaquark with three strange quarks and two antiquarks leads to the most stable configura- Η tion. In the case of s3q2Q¯2 with I = 1 in Fig. 9, there is no possibility of stable heptaquark. As we can see in 4 ¯2 FIG. 6: VB of q sQ with I = 0(unit: MeV). Fig. 10, however, there is large negative binding energy 5 with I = 0 and S = 2 . In Table IV, we can see there is 10

ing from the color-spin interaction will be common to all TABLE III: The expectation values of the hyperfine potential 3 2 ¯2 1 1 models and hence the most attractive configurations will divided by constant factor A for q s Q (I = 2 ,S = 2 ) and the corresponding lowest decay mode. point to the possible stable heptaquark state. Another interesting possibility is that the string ten- sion in the compact heptaquark configuration will be Heptaquark The lowest decay mode 3 2 2 1 1 smaller than those in usual hadrons. Such possibility has q s s¯ (I = 2 ,S = 2 ) Ξ+ K + K been discussed in Ref.[52] in relation to a stable dibaryon. 2.51 34.22 5.98 128 8 − 2 − − 2 − 3 + 3 2 mu mums ms mums ms The nonperturbative gauge field configuration for gener- 3 2 2 1 1 q s c¯ (I = 2 ,S = 2 ) Λ+ D + Ds ating the confining potential may change in the presence 4.3 11.7 3.01 8 16 16 − 2 − + 2 − 2 − − of other color sources and lead to a smaller string ten- mu mums ms mu mumc msmc 15.04 16.99 3.07 sion in a heptaquark or dibaryon configuration. In such − − + 2 mumc msmc mc cases, even if the heptaquark, baryon and mesons have the same size, the contributions from the confining po- tential in the heptaquark will be smaller than the sum of the baryon and mesons. Furthermore, due to a smaller a considerable amount of additional attraction between string tension, the wave function of the heptaquark will 3 2 2 5 the u and s quarks for s q s¯ (I = 0,S = 2 ) compared be more extended leading to a smaller additional kinetic to the corresponding lowest decay mode. However, when energy. These two effects could lead to a more stable and the antiquarks are heavy quarks, there is an additional strongly bound heptaquark configuration. repulsion between s quarks, so it makes the binding po- tential smaller. 500 S=0.5 In Table V, we present the additional kinetic energy and binding potential of the heptaquark for two most 400 stable cases. Here, we calculate the additional kinetic en- −2 300 ergy in Eq. (4) with a5 = a6 = 2.5fm , which assumes B that the interquark distance of heptaquark is similar to V 200 S=2.5 that of a proton. As we can see in the table, when the 100 antiquark is a heavy quark, the additional kinetic energy S=3.5 S=1.5 is reduced for both cases. 0 3 2 ¯2 1 1 0.0 0.2 0.4 0.6 0.8 1.0 For the q s Q (I = 2 ,S = 2 ) case, when the anti- quarks are heavy quarks, the binding potential is also Η reduced. However, the additional kinetic energy is still 3 2 2 much larger than the absolute value of the binding po- FIG. 9: VB of s q Q¯ with I = 1(unit: MeV). tential. 2 3 ¯2 5 For q s Q (I = 0,S = 2 ) case, when the antiquarks are light quarks, the expectation value of the binding potential is largest and becomes smaller when the anti- 400 quarks are heavy quarks. This is so because the interac- S=1.5 tion between u, d quarks and antiquarks is reduced due 300 to the 1/m factor. As sizeable repulsion comes from the 200 interaction between the two strange quarks for this quan- B S=0.5 V 100 tum number, replacing the strange quark with the heavy 0 quark might lead to a stable heptaquark state. S=2.5 It should be noted however, that the numbers for the -100 additional kinetic energy shown in Table V are obtained assuming that one brings the additional quarks into a 0.0 0.2 0.4 0.6 0.8 1.0 2 1/2 −1/2 Η compact size of around r = a4 0.632 fm. Assuming that the size becomesh i larger by∼ a factor of 3 2 2 2, the additional kinetic energy would be reduced by a FIG. 10: VB of s q Q¯ with I = 0(unit: MeV). 3 2¯2 1 1 factor of 4. Then the q s b (I = 2 ,S = 2 ) and the 2 3 2 5 q s s¯ (I = 0,S = 2 ) configurations could become sta- ble. These states will have masses of around 11949 MeV and 3572 MeV, respectively within our model. In par- 4 2 2 3 2 5 E. s qQ¯ : {1234}5{67} ticular, the q s s¯ (I = 0,S = 2 ) state could decay into Λ+ φ + φ, which is easy to reconstruct. The exact value of the mass and the additional kinetic energy depends on The wave function of s4qQ¯2 is the same as q4sQ¯2 with the model employed as we explain the case for MIT bag I = 2. The only difference is the mass factor in the hy- model in the appendix. However, the attraction com- perfine potential. As we can see in Fig. 11, there is no 11

identical heavy antiquarks. We constructed the flavor TABLE IV: The expectation values of the hyperfine potential 3 2 ¯2 5 color spin wave function satisfying the Pauli divided by constant factor A for s q Q (I = 0,S = 2 ) and ⊗ ⊗ the corresponding lowest decay mode. principle in the flavor SU(3) breaking case. We then searched for the heptaquark configuration with the lowest color-spin interaction, and found that the s3q2s¯2 Heptaquark The lowest decay mode 5 3 2 2 5 with I =0,S = 2 configuration is the most stable state. s q s¯ (I = 0,S = 2 ) Λ+ φ + φ For this quantum number, when seven quarks form 5.97 12.16 9.28 8 32 − 2 − + 2 − 2 + 3 2 mu mums ms mu ms a compact configuration, the interaction between u-d 3 2 2 5 ∗ ∗ s q c¯ (I = 0,S = 2 ) Λ+ Ds + Ds quarks is reduced compared to that in the Λ, but the 6.83 4.71 8.25 8 32 450 − 2 − + 2 − 2 + 3 S=0.5 mu mums ms mu msmc 4.24 1.32 2.74 400 S=2.5 − − + 2 mumc msmc mc 350 300 S=1.5 B

V 250 TABLE V: The additional kinetic energy(∆K) and bind- 200 ing potential(VB ) of the heptaquark. The first table is for 150 3 2 ¯2 1 1 2 3 ¯2 q s Q (I = 2 ,S = 2 ) and the second one is for q s Q (I = 100 5 0,S = 2 ). In the third table, we represent the parameters 0.0 0.2 0.4 0.6 0.8 1.0 used to calculate the additional kinetic energy. The unit of ∆K and VB is MeV. Η

4 2 1 1 1 3 2 2 3 2 2 3 2 2 ¯ ¯ FIG. 11: VB of s qQ with I = 2 (unit: MeV). I = 2 ,S = 2 q s s¯ q s c¯ q s b 2 2 2 2 2 2 3¯h 3¯h 3¯h 3¯h 3¯h 3¯h a5 a6 a5 a6 a5 a6 ∆K 2M5 2M6 2M5 2M6 2M5 2M6 388.75 294.73 210.03 139.51 163.97 65.08

VB -9.84 -31.75 -40.69

5 2 3 2 2 3 2 2 3¯2 additional interaction between the light quarks and the I = 0,S = 2 q s s¯ q s c¯ q s b 2 2 2 2 2 2 3¯h 3¯h 3¯h 3¯h 3¯h 3¯h s quarks result in the additional attracting that could a5 a6 a5 a6 a5 a6 ∆K 2M5 2M6 2M5 2M6 2M5 2M6 make the heptaquark state stable. This state could 271.57 245.82 201.34 135.18 162.46 63.89 be probed by reconstructing the Λ + φ + φ invariant VB -98.84 -16 3.13 mass or by its weak decay products if it is strongly bound.

mu ms mc mb a5 a6 343 632 1930 5305 2.5 2.5 MeV MeV MeV MeV fm−2 fm−2

stable heptaquark with four strange quarks. Addition- acknowledgements 5 ¯2 5 ¯2 5 ally, the plot of s Q is the same as q Q with I = 2 .

This work was supported by the Korea Na- tional Research Foundation under the grant number VII. SUMMARY 2016R1D1A1B03930089. The work of W.S. Park was supported by the National Research Foundation of Ko- In this work, we investigated the symmetry property rea (NRF) grant funded by the Ministry of Education and the stability of the heptaquark containing two (No.2017R 1D 1A 1B 03028419). 12

Appendix A: Color basis of the heptaquark

Here, we present the color basis of the heptaquark using the tensor form. The expectation value of all the color operators for the heptaquarks can be obtained using this basis.

1 √3 ijk i j k lmn m n 1 ijk i j k lmn m n C1 = ε q (1)q (2)q (3)ε q (4)q (5) + ε q (1)q (2)q (4)ε q (3)q (5) | i √6{−4√2 4√6 1 1 εijkqi(1)qj (2)qk(5)εlmnqm(3)qn(4) εijkqi(1)qj (3)qk(4)εlmnqm(2)qn(5) − 2√6 − 2√6 1 ijk i j k lmn m n + ε q (1)q (3)q (4)ε q (2)q (5) εlprq¯p(6)¯qr(7) √6 } 1 1 ijk i j k lmn m n 1 ijk i j k lmn m n C2 = ε q (1)q (2)q (3)ε q (4)q (5) ε q (1)q (2)q (4)ε q (3)q (5) | i √6{4√2 − 4√2 1 ijk i j k lmn m n + ε q (1)q (2)q (5)ε q (3)q (4) εlprq¯p(6)¯qr(7) 2√2 } 1 1 ijk i j k lmn m n 1 ijk i j k lmn m n C3 = ε q (1)q (2)q (3)ε q (4)q (5) ε q (1)q (2)q (4)ε q (3)q (5) | i √6{4√2 − 4√2 1 ijk i j k lmn m n + ε q (1)q (3)q (4)ε q (2)q (5) εlprq¯p(6)¯qr(7) 2√2 }

1 √3 ijk i j k lmn m n 1 ijk i j k lmn m n C4 = ε q (1)q (2)q (4)ε q (3)q (5) ε q (1)q (2)q (3)ε q (4)q (5) εlprq¯p(6)¯qr(7) | i √6{4√2 − 4√6 } 1 1 ijk i j k lmn m n C5 = ε q (1)q (2)q (3)ε q (4)q (5) εlprq¯p(6)¯qr(7) | i √6{2√3 } 1 1 ijk lmn m n 1 ijk lmn m n C6 = ε qi(1)qj (2)qk(4)d q (3)q (5) ε qi(1)qj (2)qk(5)d q (3)q (4) | i √6{√30 − √30 1 ijk lmn m n 1 ijk lmn m n + ε qi(1)qj (3)qk(4)d q (2)q (5) ε qi(1)qj (3)qk(5)d q (2)q (4) √30 − √30

√3 ijk lmn m n + ε qi(1)qj (4)qk(5)d q (2)q (3) dlprq¯p(6)¯qr(7) √10 }

1 √3 ijk lmn m n 1 ijk lmn m n C7 = ε qi(1)qj (2)qk(3)d q (4)q (5) + ε qi(1)qj (2)qk(4)d q (3)q (5) | i √6{4√5 4√15 1 ijk lmn m n 1 ijk lmn m n ε qi(1)qj (2)qk(5)d q (3)q (4) ε qi(1)qj (3)qk(4)d q (2)q (5) − √15 − 2√15 2 ijk lmn m n + ε qi(1)qj (3)qk(5)d q (2)q (4) dlprq¯p(6)¯qr(7) √15 } 1 1 ijk lmn m n 1 ijk lmn m n C8 = ε qi(1)qj (2)qk(3)d q (4)q (5) ε qi(1)qj (2)qk(4)d q (3)q (5) | i √6{−4√5 − 4√5 1 ijk lmn m n + ε qi(1)qj (2)qk(5)d q (3)q (4) dlprq¯p(6)¯qr(7) √5 } 1 1 ijk lmn m n 1 ijk lmn m n C9 = ε qi(1)qj (2)qk(3)d q (4)q (5) ε qi(1)qj (2)qk(4)d q (3)q (5) | i √6{4 − 4 1 + εijkq (1)q (3)q (4)dlmnqm(2)qn(5) d q¯ (6)¯q (7) 2 i j k } lpr p r

1 1 ijk lmn m n √3 ijk lmn m n C10 = ε qi(1)qj (2)qk(3)d q (4)q (5) + ε qi(1)qj (2)qk(4)d q (3)q (5) dlprq¯p(6)¯qr(7) | i √6{−4√3 4 } 1 1 ijk lmn m n C11 = ε qi(1)qj (2)qk(3)d q (4)q (5) dlprq¯p(6)¯qr(7) (A1) | i √6{√6 } 13

abc where the non-vanishing d and dabc constants are

111 222 333 d = d111 = d = d222 = d = d333 =1

412 421 523 532 613 631 1 d = d412 = d = d421 = d = d523 = d = d532 = d = d613 = d = d631 = . (A2) √2

Appendix B: Kinetic energy ical hadron decays into two color singlet hadrons of N1 and N2 quarks respectively, their masses will also follow Consider the simple MIT bag model mass formula for the same formula as Eq. (B1) after replacing the num- a Hadron composed of N = N1 + N2 quarks[48–50] in S ber of quarks to either N1 or N2. For each hadrons, the wave. bag radius R is determined by minimizing the mass with respect to R. However, comparing the mass of the multi- ω 4 Z E = N + B πR3 0 . (B1) quark composed of N quarks to the sum of two hadrons, N R 3 − R one notices that the multiquark state has one less factor of the center of mass term. This difference is the addi- Here, ω 2.04 and R,B is the bag radius and pressure, tional kinetic energy needed to bring the N + N quarks respectively.∼ The last term was originally introduced as 1 2 into a compact configuration compared to two isolated the zero point energy or Casimir energy effect but is un- hadrons. derstood to be taking care of the center of mass motion of the hadrons composed of N quarks[51]. If the hypothet-

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