Analysis of the X 0 (2900) as the scalar tetraquark state via the QCD sum rules
aa r X i v : . [ h e p - ph ] O c t Analysis of the X (2900) as the scalar tetraquark state via theQCD sum rules Zhi-Gang Wang Department of Physics, North China Electric Power University, Baoding 071003, P. R. China
Abstract
In this article, we study the axialvector-diquark-axialvector-antidiquark ( AA ) type andscalar-diquark-scalar-antidiquark ( SS ) type fully open flavor cs ¯ u ¯ d tetraquark states with thespin-parity J P = 0 + via the QCD sum rules. The predicted masses M AA = 2 . ± .
12 GeVand M SS = 3 . ± .
10 GeV support assigning the X (2900) to be the AA -type scalar cs ¯ u ¯ d tetraquark state. PACS number: 12.39.Mk, 12.38.LgKey words: Tetraquark state, QCD sum rules
Recently, the LHCb collaboration reported a narrow peak in the D − K + invariant mass spectrumin the decays B ± → D + D − K ± with the statistical significance much greater than 5 σ [1, 2]. Thepeak has been parameterized in terms of two Breit-Wigner resonances: X (2900) : J P = 0 + , M = 2866 ± , Γ = 57 ±
13 MeV ; (1) X (2900) : J P = 1 − , M = 2904 ± , Γ = 110 ±
12 MeV . (2)This is the first exotic hadron with fully open flavor, the valence quarks or the constituent quarks are cs ¯ u ¯ d [1, 2]. In Ref.[3], Karliner and Rosner assign the narrow peak to be the scalar-diquark-scalar-antidiquark type tetraquark state with the spin-parity J P = 0 + . Subsequently, other assignmentsare proposed, such as the D ∗ ¯ K ∗ molecular state [4, 5, 6], the radial excited tetraquark state ororbitally excited tetraquark state [7], the triangle singularity [8], the scalar tetraquark state [9],non scalar tetraquark state [10].In 2019, the BESIII collaboration explored the process J/ψ → φηη ′ and observed a structure X in the φη ′ mass spectrum [11]. The fitted mass and width are M X = (2002 . ± . ± .
0) MeVand Γ X = (129 ± ±
7) MeV respectively with the assignment J P = 1 − , while the fitted massand width are M X = (2062 . ± . ± .
2) MeV and Γ X = (177 ± ±
20) MeV respectively withthe assignment J P = 1 + . In Ref.[12], we study the axialvector-diquark-axialvector-antidiquarktype scalar, axialvector, tensor and vector ss ¯ s ¯ s/qq ¯ q ¯ q tetraquark states with the QCD sum rulesin a systematic way. The predicted mass M X = 2 . ± .
12 GeV for the axialvector tetraquarkstate supports assigning the new structure X (2060) from the BESIII collaboration to be a ss ¯ s ¯ s tetraquark state with the spin-parity-charge-conjugation J P C = 1 + − . In Ref.[13], we constructvarious scalar, axialvector and tensor tetraquark currents to study the mass spectrum of the groundstate hidden-charm tetraquark states with the QCD sum rules in a comprehensive way, and revisitthe assignments of the X , Y , Z states, such as X (3860), X (3872), X (3915), X (3940), X (4160), Z c (3900), Z c (4020), Z c (4050), Z c (4055), Z c (4100), Z c (4200), Z c (4250), Z c (4430), Z c (4600), etcin a consistent way. For the axialvector-diquark-axialvector-antidiquark type ( AA -type) scalartetraquark states, we obtain the masses [12, 13], M qq ¯ q ¯ q = 1 . ± .
11 GeV ,M ss ¯ s ¯ s = 2 . ± .
13 GeV ,M cq ¯ c ¯ q = 3 . ± .
09 GeV . (3) E-mail: [email protected]. AA -type cs ¯ u ¯ d tetraquark state crudely, M cs ¯ u ¯ d = M qq ¯ q ¯ q + M ss ¯ s ¯ s + 2 M cq ¯ c ¯ q . ± .
11 GeV , (4)which is consistent with the mass of the X (2900) within uncertainties.In this article, we construct the scalar-diquark-scalar-antidiquark type ( SS -type) and axialvector-diquark-axialvector-antidiquark type ( AA -type) scalar currents to study the masses of the cs ¯ u ¯ d tetraquark states with the QCD sum rules in details and explore the possible assignment of the X (2900) as the scalar tetraquark state.The article is arranged as follows: we obtain the QCD sum rules for the masses and poleresidues of the scalar tetraquark states in Sect.2; in Sect.3, we present the numerical results anddiscussions; and Sect.4 is reserved for our conclusion. Firstly, we write down the two-point correlation functions Π( p ) in the QCD sum rules,Π( p ) = i Z d xe ip · x h | T (cid:8) J ( x ) ¯ J (0) (cid:9) | i , (5)where J ( x ) = J AA ( x ), J SS ( x ), J AA ( x ) = ε ijk ε imn s Tj ( x ) Cγ α c k ( x ) ¯ u m ( x ) γ α C ¯ d Tn ( x ) ,J SS ( x ) = ε ijk ε imn s Tj ( x ) Cγ c k ( x ) ¯ u m ( x ) γ C ¯ d Tn ( x ) , (6)the i , j , k , m and n are color indexes, the C is the charge conjugation matrix. The attractiveinteractions of one-gluon exchange favor formation of the diquarks in color antitriplet [14, 15]. TheQCD sum rules calculations indicate that the favored quark-quark configurations are the scalarand axialvector diquark states [16, 17, 18, 19, 20].At the hadron side, we insert a complete set of scalar tetraquark states with the same quantumnumbers as the current operators J ( x ) into the correlation functions Π( p ) to obtain the hadronicrepresentation [21, 22, 23]. After isolating the pole terms of the lowest cs ¯ u ¯ d tetraquark states X ,we obtain the result: Π( p ) = λ X M X − p + · · · , (7)where the pole residues λ X are defined by h | J (0) | X ( p ) i = λ X .Now, we briefly outline the operator product expansion for the correlation functions Π( p ) inperturbative QCD. Firstly, we contract the u , d , s and c quark fields in the correlation functionsΠ( p ) with Wick theorem, and obtain the result:Π AA ( p ) = i ε ijk ε imn ε i ′ j ′ k ′ ε i ′ m ′ n ′ Z d x e ip · x Tr (cid:2) γ µ C kk ′ ( x ) γ ν CS Tjj ′ ( x ) C (cid:3) Tr (cid:2) γ ν U m ′ m ( − x ) γ µ CD Tn ′ n ( − x ) C (cid:3) , (8)Π SS ( p ) = i ε ijk ε imn ε i ′ j ′ k ′ ε i ′ m ′ n ′ Z d x e ip · x Tr (cid:2) γ C kk ′ ( x ) γ CS Tjj ′ ( x ) C (cid:3) Tr (cid:2) γ U m ′ m ( − x ) γ CD Tn ′ n ( − x ) C (cid:3) , (9)where the U ij ( x ), D ij ( x ), S ij and C ij ( x ) are the full u , d , s and c quark propagators, respectively, U/D ij ( x ) = iδ ij x π x − δ ij h ¯ qq i − δ ij x h ¯ qg s σGq i − ig s G aαβ t aij ( xσ αβ + σ αβ x )32 π x − h ¯ q j σ µν q i i σ µν + · · · , (10)2 ij ( x ) = iδ ij x π x − δ ij m s π x − δ ij h ¯ ss i
12 + iδ ij xm s h ¯ ss i − δ ij x h ¯ sg s σGs i
192 + iδ ij x xm s h ¯ sg s σGs i − ig s G aαβ t aij ( xσ αβ + σ αβ x )32 π x − iδ ij x xg s h ¯ ss i − δ ij x h ¯ ss ih g s GG i − h ¯ s j σ µν s i i σ µν + · · · , (11) C ij ( x ) = i (2 π ) Z d ke − ik · x (cid:26) δ ij k − m c − g s G nαβ t nij σ αβ ( k + m c ) + ( k + m c ) σ αβ ( k − m c ) − g s ( t a t b ) ij G aαβ G bµν ( f αβµν + f αµβν + f αµνβ )4( k − m c ) + · · · ) ,f αβµν = ( k + m c ) γ α ( k + m c ) γ β ( k + m c ) γ µ ( k + m c ) γ ν ( k + m c ) , (12)and t n = λ n , the λ n is the Gell-Mann matrix [23, 24, 25]. We retain the terms h ¯ q j σ µν q i i and h ¯ s j σ µν s i i come from Fierz re-ordering of the h q i ¯ q j i and h s i ¯ s j i to absorb the gluons emitted fromother quark lines to extract the mixed condensate h ¯ qg s σGq i and h ¯ sg s σGs i , respectively [25]. Thenwe compute the integrals both in the coordinate space and momentum space to obtain the cor-relation functions Π( p ). Finally, we obtain the QCD spectral densities ρ ( s ) at the quark levelthrough dispersion relation, ρ ( s ) = lim ǫ → ImΠ( s + iǫ ) π . (13)In this article, we carry out the operator product expansion up to the vacuum condensates ofdimension-11, and assume vacuum saturation for the higher dimensional vacuum condensates.There are three light quark propagators and one heavy quark propagator in the correlation func-tions Π( p ), if the heavy quark line emits a gluon and each light quark line contributes a quark-antiquark pair, we obtain a quark-gluon operator g s G µν ¯ qq ¯ qq ¯ ss , which is of dimension 11, and canlead to the vacuum condensates h ¯ qq i h ¯ sg s σGs i and h ¯ qq ih ¯ ss ih ¯ qg s σGq i , we should take into accountthe vacuum condensate up to dimension 11 in a consistent way. As the vacuum condensates are thevacuum expectations of the quark-gluon operators, we take into account the quark-gluon operatorsof the orders O ( α ks ) with k ≤ s and perform theBorel transform to obtain the QCD sum rules: λ X exp (cid:18) − M X T (cid:19) = Z s m c ds ρ ( s ) exp (cid:16) − sT (cid:17) , (14)where the T is the Borel parameter, ρ ( s ) = ρ AA ( s ), ρ SS ( s ), we neglect the explicit expressionsfor simplicity.We differentiate Eq.(14) with respect to τ = T , then eliminate the pole residues λ X and obtainthe QCD sum rules for the masses of the scalar cs ¯ u ¯ d tetraquark states, M X = − ddτ R s m c ds ρ ( s ) exp ( − sτ ) R s m c ds ρ ( s ) exp ( − sτ ) . (15) We take the standard values of the vacuum condensates h ¯ qq i = − (0 . ± .
01 GeV) , h ¯ ss i = (0 . ± . h ¯ qq i , h ¯ qg s σGq i = m h ¯ qq i , h ¯ sg s σGs i = m h ¯ ss i , m = (0 . ± .
1) GeV , h α s GGπ i = (0 .
33 GeV) atthe energy scale µ = 1 GeV [21, 22, 23, 28], and take the M S masses m c ( m c ) = (1 . ± . m s ( µ = 2 GeV) = (0 . ± . M S masses according to the renormalization group equation [30], h ¯ qq i ( µ ) = h ¯ qq i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) − nf , h ¯ ss i ( µ ) = h ¯ ss i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) − nf , h ¯ qg s σGq i ( µ ) = h ¯ qg s σGq i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) − nf , h ¯ sg s σGs i ( µ ) = h ¯ sg s σGs i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) − nf ,m c ( µ ) = m c ( m c ) (cid:20) α s ( µ ) α s ( m c ) (cid:21) − nf ,m s ( µ ) = m s (2GeV) (cid:20) α s ( µ ) α s (2GeV) (cid:21) − nf ,α s ( µ ) = 1 b t (cid:20) − b b log tt + b (log t − log t −
1) + b b b t (cid:21) , (16)where t = log µ Λ , b = − n f π , b = − n f π , b = − n f + n f π , Λ = 213 MeV, 296 MeVand 339 MeV for the flavors n f = 5, 4 and 3, respectively [29]. For the fully open flavor cs ¯ u ¯ d tetraquark states, we choose the flavor numbers n f = 4, and the typical energy scale µ = 1 GeV.Let us choose the continuum threshold parameters as √ s = M X + (0 . ∼ .
7) GeV = 2 . . ∼ .
7) GeV tentatively according to the mass gap m ψ ′ − m J/ψ = 0 .
59 GeV [29], and vary theparameters √ s to obtain the best Borel parameters T to satisfy pole dominance at the hadronside and convergence of the operator product expansion at the QCD side.After trial and error, we obtain the ideal Borel parameters or Borel windows T and contin-uum threshold parameters s , therefore the pole contributions of the ground state scalar cs ¯ u ¯ d tetraquark states and the convergent behaviors of the operator product expansion, see Table 1. Inthe Borel windows, the pole contributions are about (38 − | D (11) | are about (2 − − AA -typeand SS -type tetraquark states, respectively. The operator product expansion is well convergent.At the beginning, we assume the ground states of the scalar tetraquark states cs ¯ u ¯ d have themasses about 2 . X (2900), and choose the continuum threshold parameters √ s = 2 . . ∼ .
7) GeV tentatively to search for the optimal values via trial and error tosatisfy the constraint √ s = M X + (0 . ∼ .
7) GeV besides the two basic criteria of the QCDsum rules. From Table 1, we can see that for the AA -type scalar tetraquark state, the contin-uum threshold parameter √ s = 2 . . ∼ .
7) GeV happens to coincide with the optimalvalue 3 . ± . SS -type scalar tetraquark state, the continuum threshold pa-rameter √ s = 2 . . ∼ .
7) GeV is slightly smaller than the optimal value 3 . ± . √ s , for example, √ s = 2 . . ∼ .
7) GeV as the initial point, and obtain the optimal values shown Table 1.Now we take into account all uncertainties of the input parameters, and obtain the values ofthe masses and pole residues of the fully open flavor cs ¯ u ¯ d tetraquark states, which are shownexplicitly in Table 1 and Fig.1. In Fig.1, we plot the masses of the AA -type and SS -type scalar cs ¯ u ¯ d tetraquark states with variations of the Borel parameters T in much larger ranges than theBorel windows. From the figure, we can see that there appear platforms in the Borel windows, itis reliable to extract the tetraquark masses. 4he predicted mass M AA = 2 . ± .
12 GeV is consistent with the experimental value 2866 ± X (2900) to be the AA -type cs ¯ u ¯ d tetraquark state with the spin-parity J P = 0 + . While the predicted mass M SS =3 . ± .
10 GeV lies above the experimental value 2866 ± X (2900) → D ¯ K can take place with the fall-apart mechanism andare kinematically allowed, therefore it is Okubo-Zweig-Iizuka super-allowed. The current J AA ( x )also couples potentially to the two-meson scattering states D ¯ K , which leads to a finite width tothe X (2900). The experimental value Γ X = 57 ±
13 MeV is small enough, the finite width effectcan be neglected safely. Analogous decay widths are obtained for the charmed partners [ su ][¯ c ¯ d ]of the X (5568) [31]. In Ref.[32], we study the Z c (3900) with the QCD sum rules in details byincluding the two-particle scattering state contributions and nonlocal effects between the diquarkand antidiquark constituents. The two-particle scattering state contributions, such as the J/ψπ , η c ρ , D ¯ D ∗ + h.c. , etc, cannot saturate the QCD sum rules at the hadron side, the contribution ofthe Z c (3900) plays an un-substitutable role, we can saturate the QCD sum rules with or withoutthe two-particle scattering state contributions. The conclusion is applicable in the present case.The contributions of the intermediate two-meson scattering states D ¯ K , D ∗ ¯ K ∗ , etc besides thescalar tetraquark candidate X (2900) can be written as,Π AA ( p ) = − b λ X p − c M X + Σ D ¯ K ( p ) + Σ D ∗ ¯ K ∗ ( p ) + · · · + · · · . (17)We choose the bare mass and pole residue c M X and b λ X to absorb the divergences in the self-energies Σ D ¯ K ( p ), Σ D ∗ ¯ K ∗ ( p ), etc. The renormalized self-energies contribute a finite imaginarypart to modify the dispersion relation,Π AA ( p ) = − λ X p − M X + i p p Γ X ( p ) + · · · , (18)with the (central value of) physical width Γ X ( M X ) = 57 MeV from the LHCb collaboration [1, 2].We can take into account the finite width with the simple replacement of the hadronic spectraldensity, λ X δ (cid:0) s − M X (cid:1) → λ X π M X Γ X ( s )( s − M X ) + M X Γ X ( s ) , (19)where Γ X ( s ) = Γ X M X s s s − ( M D + M K ) M X − ( M D + M K ) . (20)Then the hadron sides of the QCD sum rules in Eqs.(14)-(15) undergo the replacements, λ X exp (cid:18) − M X T (cid:19) → λ X Z s ( m D + m K ) ds π M X Γ X ( s )( s − M X ) + M X Γ X ( s ) exp (cid:16) − sT (cid:17) , = (0 . ∼ . λ X exp (cid:18) − M X T (cid:19) , (21) λ X M X exp (cid:18) − M X T (cid:19) → λ X Z s ( m D + m K ) ds s π M X Γ X ( s )( s − M X ) + M X Γ X ( s ) exp (cid:16) − sT (cid:17) , = (0 . ∼ . λ X M X exp (cid:18) − M X T (cid:19) , (22)5 (GeV ) √ s (GeV) pole | D (11) | M (GeV) λ (GeV )[ cs ] A [¯ u ¯ d ] A . − . . ± . − − . ± .
12 (1 . ± . × − [ cs ] S [¯ u ¯ d ] S . − . . ± . − − . ± .
10 (1 . ± . × − Table 1: The Borel windows, continuum threshold parameters, pole contributions, contributionsof the vacuum condensates of dimension 11, masses and pole residues for the scalar tetraquarkstates. AA M ( G e V ) T (GeV ) Central value Error bounds SS M ( G e V ) T (GeV ) Central value Error bounds
Figure 1: The masses of the AA -type and SS -type tetraquark states with variations of the Borelparameters T .with the central value of the continuum threshold parameter √ s = 3 .
50 GeV. We can absorb thenumerical factors 0 . ∼ .
98 and 0 . ∼ .
96 into the pole residue with the simple replacement λ X → (0 . ∼ . λ X safely. It is indeed that the finite width effects cannot affect the mass M X and pole residue λ X remarkably. However, we should bear in mind that there are non-polecontributions from the two-meson scattering states besides modifying the dispersion relation, whichare expected to play a minor important role in the vicinity of the pole. In this article, we construct the axialvector-diquark-axialvector-antidiquark type and scalar-diquark-scalar-antidiquark type currents to study the fully open flavor cs ¯ u ¯ d tetraquark states with thespin-parity J P = 0 + via the QCD sum rules by carrying out the operator product expansionup to the vacuum condensates of dimension 11 in a consistent way. We obtain the predictions M AA = 2 . ± .
12 GeV and M SS = 3 . ± .
10 GeV, the predicted mass for the axialvector-diquark-axialvector-antidiquark type scalar tetraquark state is consistent with the experimentalvalue 2866 ± X (2900) to bethe axialvector-diquark-axialvector-antidiquark type cs ¯ u ¯ d tetraquark state with the spin-parity J P = 0 + . Acknowledgements
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