Landau equation and QCD sum rules for the tetraquark molecular states
aa r X i v : . [ h e p - ph ] M a r Landau equation and QCD sum rules for the tetraquark molecularstates
Zhi-Gang Wang Department of Physics, North China Electric Power University, Baoding 071003, P. R. China
Abstract
The quarks and gluons are confined objects, they cannot be put on the mass-shell, it isquestionable to apply the Landau equation to study the Feynman diagrams in the QCD sumrules. Furthermore, we carry out the operator product expansion in the deep Euclidean region p → −∞ , where the Landau singularities cannot exist. The Landau equation servers asa kinematical equation in the momentum space, and is independent on the factorizable andnonfactorizable properties of the Feynman diagrams in the color space. The meson-mesonscattering state and tetraquark molecular state both have four valence quarks, which formtwo color-neutral clusters, we cannot distinguish the contributions based on the two color-neutral clusters in the factorizable Feynman diagrams. Lucha, Melikhov and Sazdjian assertthat the contributions at the order O ( α ks ) with k ≤ O ( α s ). Such an assertion is questionable, we refute the assertion in details, andchoose an axialvector current and a tensor current to examine the outcome of the assertion.After detailed analysis, we observe that the meson-meson scattering states cannot saturate theQCD sum rules, while the tetraquark molecular states can saturate the QCD sum rules. TheLandau equation is of no use to study the Feynman diagrams in the QCD sum rules for thetetraquark molecular states, the tetraquark molecular states begin to receive contributions atthe order O ( α s /α s ) rather than at the order O ( α s ). PACS number: 12.39.Mk, 12.38.LgKey words: Molecular state, QCD sum rules
In 2003, the Belle collaboration observed a narrow charmonium-like structure X (3872) in the π + π − J/ψ invariant mass spectrum in the exclusive B -decays [1], which cannot be accommodatedin the traditional or normal quark-antiquark model. Thereafter, more than twenty charmonium-likeexotic states were observed by the BaBar, Belle, BESIII, CDF, CMS, D0, LHCb collaborations[2], some exotic states are still needed confirmation and their quantum numbers have not beenestablished yet. There have seen several possible interpretations for those X , Y and Z states,such as the tetraquark states, tetraquark (or hadronic) molecular states, dynamically generatedresonances, hadroquarkonium, kinematical effects, cusp effects, etc [3, 4].Among those possible interpretations, the tetraquark states and tetraquark molecular statesare outstanding and attract much attention as the exotic X , Y and Z states lie near the thresholdsof two charmed mesons. In 2006, R. D. Matheus et al assigned the X (3872) to be the J P C = 1 ++ diquark-antidiquark type tetraquark state, and studied its mass with the QCD sum rules [5]. Itis the first time to apply the QCD sum rules to study the exotic X , Y and Z states. Thereafterthe QCD sum rules become a powerful theoretical approach in studying the masses and widthsof the exotic X , Y and Z states, irrespective of assigning them as the hidden-charm (or hidden-bottom) tetraquark states or tetraquark (or hadronic) molecular states, and have given manysuccessful descriptions of the hadron properties [4, 5, 6, 7, 8, 9, 10, 11, 12]. In the QCD sumrules for the tetraquark states and tetraquark molecular states, we choose the diquark-antidiquarktype currents or meson-meson type (more precisely, the color-singlet-color-singlet type currents), E-mail: [email protected]. J µ = ε ijk ε imn √ n u Tj Cγ c k ¯ d m γ µ C ¯ c Tn − u Tj Cγ µ c k ¯ d m γ C ¯ c Tn o , = 12 √ n i ¯ ciγ c ¯ dγ µ u − i ¯ cγ µ c ¯ diγ u + ¯ cu ¯ dγ µ γ c − ¯ cγ µ γ u ¯ dc − i ¯ cγ ν γ c ¯ dσ µν u + i ¯ cσ µν c ¯ dγ ν γ u − i ¯ cσ µν γ u ¯ dγ ν c + i ¯ cγ ν u ¯ dσ µν γ c o , (1)where the i , j , k , m , n are color indices.In the correlation functions for the color-singlet-color-singlet type currents, Lucha, Melikhovand Sazdjian assert that the Feynman diagrams can be divided into or separated into factorizablediagrams and nonfactorizable diagrams in the color space in the operator product expansion, thecontributions at the order O ( α ks ) with k ≤
1, which are factorizable in the color space, are exactlycanceled out by the meson-meson scattering states at the hadron side, the nonfactorizable diagrams,if have a Landau singularity, begin to make contributions to the tetraquark (molecular) states, thetetraquark (molecular) states begin to receive contributions at the order O ( α s ) (according to theFierz rearrangements, see Eq.(1)) [13].About ten years before the work of Lucha, Melikhov and Sazdjian, Lee and Kochelev studiedthe two-pion contributions in the QCD sum rules for the scalar meson f (600) (or f (500) namedby the Particle Data Group now [2]) as the tetraquark state, and observed that the contributionsof the order O ( α ks ) with k ≤ In the following, we write down the two-point correlation function Π µν ( p ) in the QCD sum rulesas an example, Π µν ( p ) = i Z d xe ip · x h | T n J µ ( x ) J † ν (0) o | i , (2)where J µ ( x ) = 1 √ h ¯ u ( x ) iγ c ( x )¯ c ( x ) γ µ d ( x ) + ¯ u ( x ) γ µ c ( x )¯ c ( x ) iγ d ( x ) i . (3)The color-singlet-color-singlet type current J µ ( x ) has the quantum numbers J P C = 1 + − , at thehadron side, the quantum field theory allows non-vanishing couplings to the D ¯ D ∗ + D ∗ ¯ D scatteringstates or tetraquark molecular states with the J P C = 1 + − .At the QCD side, when we carry out the operator product expansion, Lucha, Melikhov andSazdjian assert that the Feynman diagrams can be divided into or separated into factorizable di-agrams and nonfactorizable diagrams, the Feynman diagrams of the orders O ( α s ) and O ( α s ) arefactorizable, the factorizable diagrams are exactly canceled out by the meson-meson scatteringstates, while the nonfactorizable Feynman diagrams, which are of the order O ( α s ), if have a Lan-dau singularity, begin to make contributions to the tetraquark (molecular) states, the tetraquark2igure 1: The nonfactorizable Feynman diagrams of the order O ( α s ) for the color-singlet-color-singlet type currents, other diagrams obtained by interchanging of the heavy quark lines (dashedlines) and light quark lines (solid lines) are implied.(molecular) states begin to receive contributions at the order O ( α s ) [13], see the Feynman diagramsshown Fig.1. In fact, such an assertion is questionable. Firstly , we cannot assert that the factorizable Feynman diagrams in color space are exactlycanceled out by the meson-meson scattering states, because the meson-meson scattering state andtetraquark molecular state both have four valence quarks, which can be divided into or separatedinto two color-neutral clusters. We cannot distinguish which Feynman diagrams contribute to themeson-meson scattering state or tetraquark molecular state based on the two color-neutral clusters.
Secondly , the quarks and gluons are confined objects, they cannot be put on the mass-shell, itis questionable to assert that the Landau equation is applicable in the nonperturbative calculationsdealing with the quark-gluon bound states [15].If we insist on applying the Landau equation to study the Feynman diagrams in the QCD sumrules, we should choose the pole masses rather than the
M S masses to warrant that there existsa mass pole which corresponds to the mass-shell in pure perturbative calculations, just like in thequantum electrodynamics, where the electron, muon and tau can be put on the mass-shell.According to the assertion of Lucha, Melikhov and Sazdjian, the tetraquark (molecular) statesbegin to receive contributions at the order O ( α s ) [13], it is reasonable to take the pole masses ˆ m Q as, ˆ m Q = m Q ( m Q ) " α s ( m Q ) π + f (cid:18) α s ( m Q ) π (cid:19) + g (cid:18) α s ( m Q ) π (cid:19) , (4)to put the heavy quark lines on the mass-shell, the explicit expressions of the coefficients f and g can be found in Refs.[2, 16]. It is straightforward to obtain ˆ m b = m b ( m b ) (1 + 0 .
10 + 0 .
05 + 0 .
03) =4 . ± .
06 GeV [2].If the Landau equation is applicable in the QCD sum rules for the tetraquark states andtetraquark molecular states, it is certainly applicable in the QCD sum rules for the traditional ornormal charmonium and bottomonium states. In the case of the c -quark, the pole mass ˆ m c =1 . ± .
07 GeV from the Particle Data Group [2], the Landau singularity appears at the s -channel √ s = p p = 2 ˆ m c = 3 . ± .
14 GeV > m η c and m J/ψ . While in the case of the b -quark, the polemass ˆ m b = 4 . ± .
06 GeV from the Particle Data Group [2], the Landau singularity appears atthe s -channel √ s = p p = 2 ˆ m b = 9 . ± .
12 GeV > m η b and m Υ . It is odd or unreliable that themasses of the charmonium (bottomonium) states lie below the threshold 2 ˆ m c (2 ˆ m b ) in the QCDsum rules for the η c and J/ψ ( η b and Υ), as the integrals of the forms Z s m c δ (cid:16) s − m η c /J/ψ (cid:17) exp (cid:16) − sT (cid:17) ds , Z s m b δ (cid:16) s − m η b / Υ (cid:17) exp (cid:16) − sT (cid:17) ds , (5)3igure 2: The nonfactorizable Feynman diagrams contribute to the vacuum condensates h ¯ qg s σGq i for the color-singlet-color-singlet type currents, where the solid lines and dashed linesdenote the light quarks and heavy quarks, respectively.at the hadron side are meaningless, where the T is the Borel parameter. The tiny widths of the η c , J/ψ , η b and Υ valuate the zero-width approximation, the hadronic spectral densities are of theform δ (cid:16) s − m η c /J/ψ/η b / Υ (cid:17) . Thirdly , the nonfactorizable Feynman diagrams which have the Landau singularities beginto appear at the order O ( α s /α s ) rather than at the order O ( α s ), and make contributions to thetetraquark molecular states, if the assertion (the nonfactorizable Feynman diagrams which haveLandau singularities make contributions to the tetraquark molecular states) of Lucha, Melikhovand Sazdjian is right.The nonperturbative contributions play an important role and serve as a hallmark for thenonperturbative nature of the QCD sum rules, the nonfactorizable contributions appear at theorder O ( α s ) due to the operators ¯ qg s Gq ¯ qg s Gq , which come from the Feynman diagrams shown inFig.2. Such Feynman diagrams can be taken as annihilation diagrams, which play an importantrole in the tetraquark molecular states [17]. If we insist on applying the landau equation to studythe Feynman diagrams shown in Fig.2 and choose the pole mass of the c -quark, we obtain asub-leading Landau singularity at the s -channel s = p = ( ˆ m c + ˆ m c ) , which indicates that itcontributes to the tetraquark molecular states. From the operators ¯ qg s Gq ¯ qg s Gq , we can obtainthe vacuum condensate h ¯ qg s σGq i , where the g s = 4 πα s is absorbed into the vacuum condensate,so the Feynman diagrams in Fig.2 can be counted as of the order O ( α s ). The nonfactorizableFeynman diagrams appear at the order O ( α s ) or O ( α s ) (based on how to account for the g s inthe vacuum condensates), not at the order O ( α s ) asserted in Ref.[13]. Fourthly , the Landau equation servers as a kinematical equation in the momentum space, andis independent on the factorizable and nonfactorizable properties of the Feynman diagrams in thecolor space. Without taking it for granted that the factorizable Feynman diagrams in the colorspace only make contributions to the two-meson scattering states, the Landau equation cannotexclude the factorizable Feynman diagrams in the color space, those diagrams can also have theLandau singularities.In the leading order, the factorizable Feynman diagrams shown in Fig.3 can be divided intoor separated into two color-neutral clusters, each cluster corresponds to a trace both in the colorspace and in the Dirac spinor space. However, in the momentum space, they are nonfactorizablediagrams, the basic integrals are of the form, Z d qd kd l p + q − k + l ) − m c q − m q k − m q l − m c . (6)If we choose the pole masses, there exists a Landau singularity or an s -channel singularity at s = p = ( ˆ m u + ˆ m d + ˆ m c + ˆ m c ) , which is just a signal of a four-quark intermediate state.We cannot assert that it is a signal of a meson-meson scattering state or a tetraquark molecular4igure 3: The Feynman diagrams for the lowest order contributions, where the solid lines anddashed lines represent the light quarks and heavy quarks, respectively.state, because the meson-meson scattering state and tetraquark molecular state both have fourvalence quarks, q , ¯ q , c and ¯ c , which form two color-neutral clusters. The Landau singularity isjust a kinematical singularity, not a dynamical singularity [18], it is useless in distinguishing thecontributions to the meson-meson scattering state and tetraquark molecular state. If we switch offthe assertion that the factorizable Feynman diagrams shown in Fig.3 make contributions to themeson-meson scattering states alone, the s -channel singularity at s = p = ( ˆ m u + ˆ m d + ˆ m c + ˆ m c ) supports that they also contribute to the tetraquark molecular states. Fifthly , only formal QCD sum rules for the tetraquark states or tetraquark molecular statesare obtained based on the assertion of Lucha, Melikhov and Simula in Ref.[13], no feasible QCDsum rules with predictions can be confronted to the experimental data are obtained up to now.
Sixthly , in the QCD sum rules, we carry out the operator product expansion in the deepEuclidean space, − p → ∞ , then obtain the physical spectral densities at the quark-gluon levelthrough dispersion relation [19, 20, 21], ρ QCD ( s ) = 1 π Im Π( s + iǫ ) | ǫ → , (7)where the Π( s ) denotes the correlation functions. The Landau singularities require that the squaredmomentum p = ( ˆ m u + ˆ m d + ˆ m c + ˆ m c ) in the Feynman diagrams, see Fig.3 and Eq.(6), it isquestionable to perform the operator product expansion. Now let us assume that the assertion of Lucha, Melikhov and Sazdjian is right, the tetraquarkmolecular states begin to receive contributions at the order O ( α s ), the contributions at the order O ( α ks ) with k ≤ µν ( p ) and Π µναβ ( p ) in the5CD sum rules, Π µν ( p ) = i Z d xe ip · x h | T n J µ ( x ) J † ν (0) o | i , (8)Π µναβ ( p ) = i Z d xe ip · x h | T n J µν ( x ) J † αβ (0) o | i , (9)where J µ ( x ) = 1 √ h ¯ u ( x ) iγ c ( x )¯ c ( x ) γ µ d ( x ) + ¯ u ( x ) γ µ c ( x )¯ c ( x ) iγ d ( x ) i , (10) J µν ( x ) = 1 √ h ¯ s ( x ) γ µ c ( x )¯ c ( x ) γ ν γ s ( x ) − ¯ s ( x ) γ ν γ c ( x )¯ c ( x ) γ µ s ( x ) i . (11)The current J µ ( x ) has the quantum numbers J P C = 1 + − , while the current J µν ( x ) has definitecharge conjugation, the components J i ( x ) and J ij ( x ) have positive-parity and negative-parity,respectively, where the space indexes i , j = 1, 2, 3. The charged current J µ ( x ) couples potentiallyto the D ¯ D ∗ + D ∗ ¯ D scattering state or tetraquark molecular state with the J P C = 1 + − , while theneutral current J µν ( x ) couples potentially to the D ∗ s ¯ D s − D s ¯ D ∗ s meson-meson scattering states ortetraquark molecular states with the J P C = 1 ++ and 1 − + . Thereafter, we will denote the charged D ¯ D ∗ + D ∗ ¯ D tetraquark molecular state with the J P C = 1 + − as the Z c , and denote the neutral D ∗ s ¯ D s − D s ¯ D ∗ s tetraquark molecular states with the J P C = 1 ++ and 1 − + as the X + c and X − c ,respectively, where the superscripts ± on the X ± c denote the positive-parity and negative-parity,respectively.In the following, we write down the possible current-hadron couplings explicitly, h | J µ (0) | D ( q ) ¯ D ∗ ( p − q ) i = 1 √ f D m D m c f D ∗ m D ∗ ε µ ( p − q ) , h | J µ (0) | D ∗ ( q ) ¯ D ( p − q ) i = 1 √ f D m D m c f D ∗ m D ∗ ε µ ( q ) , h | J µ (0) | D ( q ) ¯ D ( p − q ) i = 1 √ f D m D m c f D ( p − q ) µ , h | J µ (0) | D ( q ) ¯ D ( p − q ) i = 1 √ f D m D m c f D q µ , (12) h | J µ (0) | Z c ( p ) i = λ Z ε µ ( p ) , (13) h | J µν (0) | D ∗ s ( q ) ¯ D s ( p − q ) i = 1 √ f D ∗ s m D ∗ s f D s m D s ε µ ( q ) ε ν ( p − q ) , h | J µν (0) | D s ( q ) ¯ D ∗ s ( p − q ) i = − √ f D s m D s f D ∗ s m D ∗ s ε ν ( q ) ε µ ( p − q ) , h | J µν (0) | D ∗ s ( q ) ¯ D s ( p − q ) i = i √ f D ∗ s m D ∗ s f D s ε µ ( q )( p − q ) ν , h | J µν (0) | D s ( q ) ¯ D ∗ s ( p − q ) i = − i √ f D s f D ∗ s m D ∗ s q ν ε µ ( p − q ) , h | J µν (0) | D s ( q ) ¯ D s ( p − q ) i = i √ f D s f D s q µ ( p − q ) ν , h | J µν (0) | D s ( q ) ¯ D s ( p − q ) i = − i √ f D s f D s q ν ( p − q ) µ , (14)6 | J µν (0) | X − c ( p ) i = λ X − M X − ε µναβ ε α ( p ) p β , h | J µν (0) | X + c ( p ) i = λ X + M X + [ ε µ ( p ) p ν − ε ν ( p ) p µ ] , (15)the ε µ are the polarization vectors of the vector and axialvector mesons or tetraquark molecularstates, the f D , f D s , f D ∗ , f D ∗ s , f D , f D s and f D s are the decay constants of the traditional ornormal heavy mesons, the λ Z and λ X ± are the pole residues of the tetraquark molecular states.The charged D ¯ D ∗ + D ∗ ¯ D tetraquark molecular state Z c with the J P C = 1 + − and the neutral D ∗ s ¯ D s − D s ¯ D ∗ s tetraquark molecular state X − c with the J P C = 1 − + differ from the traditionalmesons significantly, and are good subjects to study the exotic states.Now we take a short digression to give some explanations for the definitions of the current-hadron couplings in Eq.(12) and Eq.(14). Firstly, let us write down the standard definitions forthe decay constants of the traditional or normal heavy mesons, h | ¯ q (0) iγ c (0) | D ( q ) i = f D m D m c , h | ¯ q (0) γ µ c (0) | D ∗ ( q ) i = f D ∗ m D ∗ ε µ ( q ) , h | ¯ s (0) γ µ c (0) | D ∗ s ( q ) i = f D ∗ s m D ∗ s ε µ ( q ) , h | ¯ q (0) γ µ c (0) | D ( q ) i = f D q µ , h | ¯ s (0) γ µ c (0) | D s ( q ) i = f D s q µ , h | ¯ s (0) γ µ γ c (0) | D s ( q ) i = f D s m D s ε µ ( q ) , h | ¯ s (0) γ µ γ c (0) | D s ( q ) i = if D s q µ , (16)based on the properties of the vector currents and axialvector currents and their conservationfeatures, where q = u , d . On the other hand, the heavy meson fields have the properties, h | D ( s/ /s (0) | D ( s/ /s ( q ) i = 1 , h | D ∗ ( s ) µ (0) | D ∗ ( s ) ( q ) i = ε µ ( q ) , h | D s ,µ (0) | D s ( q ) i = ε µ ( q ) , (17)which imply that ¯ q ( x ) iγ c ( x ) = f D m D m c D ( x ) + · · · , etc, at the hadron degrees of freedom. FromEq.(13) and Eqs.(15)-(17), we can express the four-quark currents J µ ( x ) and J µν ( x ) in terms ofthe heavy meson fields (in other words, the Eq.(13) and Eqs.(15)-(17) imply that), J µ ( x ) = 1 √ f D m D m c f D ∗ m D ∗ (cid:2) D ( x ) D ∗− µ ( x ) + D ∗ µ ( x ) D − ( x ) (cid:3) + 1 √ f D m D m c f D (cid:2) D ( x ) i∂ µ D − ( x ) + i∂ µ D ( x ) D − ( x ) (cid:3) + λ Z Z c,µ ( x ) + · · · , (18) J µν ( x ) = 1 √ f D ∗ s m D ∗ s f D s m D s (cid:2) D ∗ + s,µ ( x ) D − s ,ν ( x ) − D + s ,ν ( x ) D ∗− s,µ ( x ) (cid:3) − √ f D ∗ s m D ∗ s f D s (cid:2) D ∗ + s,µ ( x ) ∂ ν D − s ( x ) − ∂ ν D + s ( x ) D ∗− s,µ ( x ) (cid:3) + 1 √ f D s f D s m D s (cid:2) i∂ µ D + s ( x ) D − s ,ν ( x ) − D + s ,ν ( x ) i∂ µ D − s ( x ) (cid:3) − √ f D s f D s (cid:2) i∂ µ D + s ( x ) ∂ ν D − s ( x ) − ∂ ν D + s ( x ) i∂ µ D − s ( x ) (cid:3) − λ X − M X − ε µναβ i∂ α X − βc ( x ) − λ X + M X + (cid:2) i∂ µ X + c,ν ( x ) − i∂ ν X + c,µ ( x ) (cid:3) + · · · , (19)7ccording to the assumption of current-hadron duality. It is straightforward to obtain the current-hadron couplings in Eq.(12) and Eq.(14).At the hadron side, we insert a complete set of intermediate hadronic states with the samequantum numbers as the current operators J µ ( x ) and J µν ( x ) into the correlation functions Π µν ( p )and Π µναβ ( p ) to obtain the hadronic representation [19, 20]. We isolate the contributions of themeson-meson scattering states and the lowest axialvector and vector tetraquark states accordingto Eqs.(12)-(15), and get the results,Π µν ( p ) = Π( p ) (cid:18) − g µν + p µ p ν p (cid:19) + · · · , (20)Π µναβ ( p ) = Π − ( p ) (cid:18) g µα g νβ − g µβ g να − g µα p ν p β p − g νβ p µ p α p + g µβ p ν p α p + g να p µ p β p (cid:19) +Π + ( p ) (cid:18) − g µα p ν p β p − g νβ p µ p α p + g µβ p ν p α p + g να p µ p β p (cid:19) , (21)where Π( p ) = λ Z M Z − p + Π T W ( p ) + · · · , Π − ( p ) = P µναβ − Π µναβ ( p ) = λ X − M X − − p + Π − T W ( p ) + · · · , Π + ( p ) = P µναβ + Π µναβ ( p ) = λ X + M X + − p + · · · , (22)we project out the components Π − ( p ) and Π + ( p ) by introducing the operators P µναβ − and P µναβ + respectively, P µναβ − = 16 (cid:18) g µα − p µ p α p (cid:19) (cid:18) g νβ − p ν p β p (cid:19) ,P µναβ + = 16 (cid:18) g µα − p µ p α p (cid:19) (cid:18) g νβ − p ν p β p (cid:19) − g µα g νβ , (23)Π T W ( p ) = λ DD ∗ π Z s ∆ ds s − p p λ ( s, m D , m D ∗ ) s (cid:20) λ ( s, m D , m D ∗ )12 sm D ∗ (cid:21) + λ DD π Z s ∆ ds s − p q λ ( s, m D , m D ) s λ ( s, m D , m D )12 s + · · · , (24)Π − T W ( p ) = λ D ∗ s D s π Z s ∆ ds s − p q λ ( s, m D ∗ s , m D s ) s " λ ( s, m D ∗ s , m D s )12 sm D ∗ s λ ( s, m D ∗ s , m D s )12 sm D s + λ D ∗ s D s π Z s ∆ ds s − p q λ ( s, m D ∗ s , m D s ) s λ ( s, m D ∗ s , m D s )12 s + λ D s D s π Z s ∆ ds s − p q λ ( s, m D s , m D s ) s λ ( s, m D s , m D s )12 s + · · · , (25)8igure 4: The Feynman diagrams for the two-meson intermediate states, where the dashed linerepresents the cut. λ DD ∗ = f D m D f D ∗ m D ∗ m c ,λ DD = f D m D f D m c ,λ D ∗ s D s = f D ∗ s m D ∗ s f D s m D s ,λ D ∗ s D s = f D ∗ s m D ∗ s f D s ,λ D s D s = f D s f D s m D s , (26)∆ = ( m D + m D ∗ ) , ∆ = ( m D + m D ) , ∆ = ( m D ∗ s + m D s ) , ∆ = ( m D ∗ s + m D s ) , ∆ = ( m D s + m D s ) , (27) λ ( a, b, c ) = a + b + c − ab − bc − ca . The components Π − ( p ) and Π + ( p ) receive contributionsfrom the D ∗ s ¯ D s − D s ¯ D ∗ s meson-meson scattering states or tetraquark molecular states with the J P C = 1 − + and 1 ++ , respectively. The conventional hidden-flavor mesons have the normal quan-tum numbers, J P C = 0 − + , 0 ++ , 1 −− , 1 + − , 1 ++ , 2 −− , 2 − + , 2 ++ , · · · . The component Π − ( p )receives contributions with the exotic quantum numbers J P C = 1 − + , while the component Π + ( p )receives contributions with the normal quantum numbers J P C = 1 ++ . In this article, we study thetetraquark molecular states (in other words, the exotic states), it is better to choose the componentΠ − ( p ) with the exotic quantum numbers J P C = 1 − + , so we discard the component Π + ( p ) withthe normal quantum numbers J P C = 1 ++ . Thereafter, we will neglect the superscript − in the X − c for simplicity.Now we give some explanations for the components Π T W ( p ) and Π − T W ( p ) in Eqs.(24)-(25). Wedraw up the Feynman diagrams for the two-meson scattering state contributions in the correlationfunctions Π µν ( p ) and Π µναβ ( p ), see Fig.4, and resort to the Cutkosky’s rule to calculate theimaginary parts Im Π T W ( p ) and Im Π − T W ( p ) with the simple replacements of the two heavy-9eson lines, 1 q − m A + iǫ → − πi δ (cid:0) q − m A (cid:1) , p − q ) − m B + iǫ → − πi δ (cid:0) ( p − q ) − m B (cid:1) , (28)where the m A and m B denote the masses of the two heavy mesons, respectively. Then it is straightforward to carry out the integral over the four-vector q α , and obtain the two-meson scattering statecontributions Π T W ( p ) and Π − T W ( p ) through dispersion relation.In this article, we carry out the operator product expansion to the vacuum condensates upto dimension-10, and take into account the vacuum condensates which are vacuum expecta-tions of the quark-gluon operators of the order O ( α ks ) with k ≤
1. In calculations, we assumevacuum saturation for the higher dimensional vacuum condensates. For the current J µ ( x ), wetake into account the vacuum condensates h ¯ qq i , h α s π GG i , h ¯ qg s σGq i , h ¯ qq i , g s h ¯ qq i , h ¯ qq ih α s π GG i , h ¯ qq ih ¯ qg s σGq i , h ¯ qq i h α s π GG i , h ¯ qg s σGq i . The four-quark condensate g s h ¯ qq i comes from the terms h ¯ qγ µ t a qg s D η G aλτ i , h ¯ q j D † µ D † ν D † α q i i and h ¯ q j D µ D ν D α q i i , rather than comes from the perturbativecorrections of the h ¯ qq i . The four-quark condensate g s h ¯ qq i plays an important role in choosingthe input parameters due to the relation g s = 4 πα s ( µ ), which introduces explicit energy scaledependence, on the other hand, it plays a minor important role in numerical calculations. Forthe current J µν ( x ), we take into account the vacuum condensates h ¯ ss i , h α s π GG i , h ¯ sg s σGs i , h ¯ ss i , h ¯ ss ih α s π GG i , h ¯ ss ih ¯ sg s σGs i , h ¯ ss i h α s π GG i , h ¯ sg s σGs i , and neglect the condensate g s h ¯ ss i . Aftercarrying out the operator product expansion, we obtain the analytical expressions of the correlationfunctions Π( p ) and Π − ( p ) at the quark-gluon level,Π( p ) = Z s m c ρ Z,QCD ( s ) s − p + · · · , Π − ( p ) = Z s m c ρ X,QCD ( s ) s − p + · · · , (29)where the ρ Z,QCD ( s ) and ρ X,QCD ( s ) are the QCD spectral densities, ρ Z,QCD ( s ) = 1 π ImΠ( s + iǫ ) | ǫ → ,ρ X,QCD ( s ) = 1 π ImΠ − ( s + iǫ ) | ǫ → . (30)According to the assertion of Lucha, Melikhov and Sazdjian [13], all the contributions of theorder O ( α ks ) with k ≤ p ) = Π T W ( p ) + · · · , Π − ( p ) = Π − T W ( p ) + · · · , (31)at the hadron side, as we carry out the operator product expansion by taking into account only thecontributions of the order O ( α ks ) with k ≤
1. Now let us take the quark-hadron duality below thecontinuum threshold s and saturate the hadron side of the correlation functions with the meson-meson scattering states, then perform Borel transform with respect to the variable P = − p toobtain the QCD sum rules:Π T W ( T ) = λ DD ∗ π Z s ∆ ds p λ ( s, m D , m D ∗ ) s (cid:20) λ ( s, m D , m D ∗ )12 sm D ∗ (cid:21) exp (cid:16) − sT (cid:17) + λ DD π Z s ∆ ds q λ ( s, m D , m D ) s λ ( s, m D , m D )12 s exp (cid:16) − sT (cid:17) = κ Z s m c ds ρ Z,QCD ( s ) exp (cid:16) − sT (cid:17) , (32)10 − T W ( T ) = λ D ∗ s D s π Z s ∆ ds q λ ( s, m D ∗ s , m D s ) s " λ ( s, m D ∗ s , m D s )12 sm D ∗ s λ ( s, m D ∗ s , m D s )12 sm D s exp (cid:16) − sT (cid:17) + λ D ∗ s D s π Z s ∆ ds q λ ( s, m D ∗ s , m D s ) s λ ( s, m D ∗ s , m D s )12 s exp (cid:16) − sT (cid:17) + λ D s D s π Z s ∆ ds q λ ( s, m D s , m D s ) s λ ( s, m D s , m D s )12 s exp (cid:16) − sT (cid:17) = κ Z s m c ds ρ X,QCD ( s ) exp (cid:16) − sT (cid:17) , (33)the explicit expressions of the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ) are given inthe Appendix, where we have rewritten the terms of the forms dds δ ( s − m c ), d ds δ ( s − m c ), · · · , dds δ ( s − e m c ), d ds δ ( s − e m c ), · · · in more concise forms. We saturate the QCD side of the correlationfunctions with the two-meson scattering states at the hadron side ”by hand” according to theassertion of Lucha, Melikhov and Sazdjian [13]. In Sect.2, we present detailed discussions toapprove that the assertion is questionable, we have to introduce some parameters to evaluate theassertion in practical calculations. In Eqs.(32)-(33), we introduce the parameter κ to measure thedeviations from 1, if κ ≈
1, we can get the conclusion tentatively that the meson-meson scatteringstates can saturate the QCD sum rules. Then we differentiate Eqs.(32)-(33) with respect to T ,and obtain two additional QCD sum rules, − d Π T W ( T ) d (1 /T ) = − κ dd (1 /T ) Z s m c ds ρ Z,QCD ( s ) exp (cid:16) − sT (cid:17) , (34) − d Π − T W ( T ) d (1 /T ) = − κ dd (1 /T ) Z s m c ds ρ X,QCD ( s ) exp (cid:16) − sT (cid:17) . (35)Thereafter, we will denote the QCD sum rules in Eqs.(34)-(35) as the QCDSR I, and the QCDsum rules in Eqs.(32)-(33) as the QCDSR II.On the other hand, if the meson-meson scattering states cannot saturate the QCD sum rules,we have to introduce the tetraquark molecular states to saturate the QCD sum rules, λ Z exp (cid:18) − M Z T (cid:19) = Z s m c ds ρ Z,QCD ( s ) exp (cid:16) − sT (cid:17) , (36) λ X exp (cid:18) − M X T (cid:19) = Z s m c ds ρ X,QCD ( s ) exp (cid:16) − sT (cid:17) . (37)We differentiate Eqs.(36)-(37) with respect to T , and obtain two QCD sum rules for the massesof the tetraquark molecular states, M Z = − dd (1 /T ) R s m c ds ρ Z,QCD ( s ) exp (cid:0) − sT (cid:1)R s m c ds ρ Z,QCD ( s ) exp (cid:0) − sT (cid:1) , (38) M X = − dd (1 /T ) R s m c ds ρ X,QCD ( s ) exp (cid:0) − sT (cid:1)R s m c ds ρ X,QCD ( s ) exp (cid:0) − sT (cid:1) . (39)11 Numerical results and discussions
At the QCD side, we choose 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) at the energy scale µ = 1 GeV [19, 20, 21], and choose the M S masses m c ( m c ) = (1 . ± . m s ( µ = 2 GeV) = (0 . ± . m u = m d = 0. Moreover, we take into account the energy-scale dependence ofthe input parameters, h ¯ qq i ( µ ) = h ¯ qq i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) , h ¯ ss i ( µ ) = h ¯ ss i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) , h ¯ qg s σGq i ( µ ) = h ¯ qg s σGq i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) , h ¯ sg s σGs i ( µ ) = h ¯ sg s σGs i (1GeV) (cid:20) α s (1GeV) α s ( µ ) (cid:21) ,m c ( µ ) = m c ( m c ) (cid:20) α s ( µ ) α s ( m c ) (cid:21) ,m s ( µ ) = m s (2GeV) (cid:20) α s ( µ ) α s (2GeV) (cid:21) ,α s ( µ ) = 1 b t (cid:20) − b b log tt + b (log t − log t −
1) + b b b t (cid:21) , (40)where t = log µ Λ , b = − n f π , b = − n f π , b = − n f + n f π , Λ = 210 MeV, 292 MeV and332 MeV for the flavors n f = 5, 4 and 3, respectively [2, 22], and evolve all the input parameters tothe ideal energy scales µ with n f = 4 to extract the tetraquark molecular masses or the parameters κ . The QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), and the thresholds 4 m c depend on theenergy scales µ , the values of the parameters κ , masses M Z/X and pole residues λ Z/X extractedfrom the QCD sum rules in Eqs.(32)-(39) vary with the energy scales µ , we should resort to somemethods to choose the ideal energy scales (or pertinent energy scales) µ to extract those quantitiesin a consistent way.At the hadron side, we take the hadronic parameters as m D = 1 . m D s = 1 . m D ∗ = 2 . m D ∗ s = 2 . m D = 2 . m D s = 2 . m D s =2 . f D = 0 .
208 GeV, f D s = 0 .
240 GeV, f D ∗ = 0 .
263 GeV, f D ∗ s = 0 .
308 GeV, f D = 0 .
373 GeV, f D s = 0 .
333 GeV [23], f D s = 0 .
364 GeV from the QCD sumrules.The D ¯ D ∗ + D ∗ ¯ D and D ∗ s ¯ D s − D s ¯ D ∗ s thresholds are m D + m D ∗ = 3 .
88 GeV and m D ∗ s + m D s =4 .
65 GeV, respectively. For the conventional heavy mesons, the mass-gaps between the groundstates and the first radial excited states are about 0 . − . √ s = 4 . ± .
10 GeV and 5 . ± .
10 GeV, respectively.We search for the acceptable Borel parameters T to warrant convergence of the operator prod-uct expansion and pole dominance via trial and error. Firstly, let us define the pole contributionsPC, PC = R s m c ds ρ Z/X,QCD ( s ) exp (cid:0) − sT (cid:1)R ∞ m c ds ρ Z/X,QCD ( s ) exp (cid:0) − sT (cid:1) , (41)12 .0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.00.10.20.30.40.50.60.70.80.91.0 P C (GeV) A B
Figure 5: The pole contributions with variations of the energy scales µ , where the A and B correspond to the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), respectively.and the contributions of the vacuum condensates D ( n ), D ( n ) = R s m c ds ρ Z/X,QCD ; n ( s ) exp (cid:0) − sT (cid:1)R s m c ds ρ Z/X,QCD ( s ) exp (cid:0) − sT (cid:1) , (42)where the subscript n in the QCD spectral densities ρ Z/X,QCD ; n ( s ) represents the vacuum con-densates of dimension n .In Fig.5, we plot the pole contributions with variations of the energy scales of the QCD spectraldensities with the parameters T Z = 2 . , p s Z = 4 .
40 GeV and T X = 3 . , p s X =5 .
15 GeV for the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), respectively. We choose thosetypical values because the continuum threshold parameters s and Borel parameters T have therelation s Z T Z = s X T X , the weight functions exp (cid:0) − sT (cid:1) have the same values. From Fig.5, we can seethat the pole contributions increase monotonically and considerably with the increase of the energyscales at the region µ < . µ = 1 . . ρ Z,QCD ( s ) and ρ X,QCD ( s ), respectively.In Fig.6, we plot the absolute values of the D (6) with variations of the energy scales µ of theQCD spectral densities with the parameters T Z = 2 . , p s Z = 4 .
40 GeV and T X = 3 . , p s X = 5 .
15 GeV for the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), respectively. Thecontributions of the vacuum condensates of dimension 6 play a very important role in the QCDsum rules for the hidden-charm or hidden-bottom tetraquark (molecular) states. From Fig.6, wecan see that the contributions | D (6) | decrease monotonically with the increase of the energy scales.A larger energy scale µ leads to a larger pole contribution, but a smaller contribution of the vacuumcondensate D (6). Too small contributions of the vacuum condensates will impair the stability ofthe QCD sum rules. We saturate the hadron side of the QCD sum rules with the meson-meson scattering states alone,and study the QCD sum rues shown in Eqs.(32)-(35). In this article, we choose the pole con-tributions as large as (40 − .0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.000.030.060.090.120.150.180.210.240.270.30 | D ( ) | (GeV) A B
Figure 6: The absolute values of the D (6) with variations of the energy scales µ , where the A and B correspond to the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), respectively. J P C T (GeV ) √ s (GeV) µ (GeV) pole κ I κ II + − (¯ uc ¯ cd ) 2 . − . . ± .
10 1 . − . ± .
40 1 . ± . − + (¯ sc ¯ cs ) 3 . − . . ± .
10 2 . − . ± .
09 0 . ± . κ for the QCDSR I and II, where we show the quarkconstituents of the meson-meson scattering states in the brackets.continuum threshold parameters, energy scales of the QCD spectral densities and pole contri-butions are shown explicitly in Table 1. In the Borel windows, the contributions of the higherdimensional vacuum condensates are | D (8) | = (3 − | D (10) | ≪
1% and D (8) = (1 − D (10) ≪
1% for the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), respectively. The operatorproduct expansion converges very very good.We take into account all uncertainties of the input parameters at the QCD side, and obtain thevalues of the κ from the QCDSR I and II directly, which are shown in Table 1. In calculations, weadd an uncertainty δµ = ± . µ . From Table 1, we can see that the values κ I = 1 . ± .
40 and κ II = 1 . ± .
40 overestimate the contributions of the ¯ uc ¯ cd meson-mesonscattering states with the J P C = 1 + − , while the values κ I = 0 . ± .
09 and κ II = 0 . ± . sc ¯ cs meson-meson scattering states with the J P C = 1 − + .In the two cases, the values of the κ from the QCDSR I and II deviate from 1 significantly, thetwo-meson scattering sates cannot saturate the QCD sum rules.In Fig.7, we plot the values of the κ with variations of the Borel parameters T with thecontinuum threshold parameters √ s = 4 .
40 GeV and 5 .
15 GeV for the ¯ uc ¯ cd and ¯ sc ¯ cs meson-meson scattering states, respectively, where we normalize the values of the κ to be 1 at the points T = 1 . and 2 . for the ¯ uc ¯ cd and ¯ sc ¯ cs meson-meson scattering states, respectively. Inthis way, we can see the variation trends of the κ with the changes of the Borel parameters moreexplicitly. From Fig.7, we can see that the values of the κ increase monotonically and quickly withthe increase of the Borel parameters T , no platform appears, which indicates that the QCD sumrules in Eqs.(32)-(33) obtained according to the assertion of Lucha, Melikhov and Sazdjian areunreasonable. Reasonable QCD sum rules lead to platforms flat enough or not flat enough, rather14 .5 1.8 2.1 2.4 2.7 3.0 3.3 3.6 3.9 4.2 4.50.81.01.21.41.61.82.02.22.42.62.8 A T (GeV ) (I) (II) B T (GeV ) (I) (II) Figure 7: The κ with variations of the Borel parameters T , where the A and B correspond to the¯ uc ¯ cd and ¯ sc ¯ cs meson-meson scattering states, respectively, the (I) and (II) correspond to QCDSRI and II, respectively, the κ values are normalized to be 1 for the Borel parameters T = 1 . and 2 . , respectively. J P C T (GeV ) √ s (GeV) µ (GeV) pole M (GeV) λ (10 − GeV )1 + − (¯ uc ¯ cd ) 2 . − . . ± .
10 1 . − . ± .
09 1 . ± . − + (¯ sc ¯ cs ) 3 . − . . ± .
10 2 . − . ± .
08 6 . ± . uc ¯ cd and ¯ sc ¯ cs tetraquark molecularstates.than no evidence of platforms.Now we can obtain the conclusion tentatively that the meson-meson scattering states cannotsaturate the QCD sum rules at the hadron side. We saturate the hadron side of the QCD sum rules with the tetraquark molecular states alone,and study the QCD sum rues shown in Eqs.(36)-(39).In Fig.8, we plot the masses with variations of the energy scales of the QCD spectral densitieswith the parameters T Z = 2 . , p s Z = 4 .
40 GeV and T X = 3 . , p s X = 5 .
15 GeV forthe ¯ uc ¯ cd and ¯ sc ¯ cs tetraquark molecular states, respectively. From Fig.8, we can see that the valuesof the masses decrease monotonically and slowly with the increase of the energy scales µ . Now weencounter the problem how to choose the pertinent energy scales of the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ).We describe the heavy tetraquark system Q ¯ Qq ¯ q (or the exotic X , Y , Z states) by a double-well potential with the two light quarks q and ¯ q lying in the two potential wells, respectively.In the heavy quark limit, the Q -quark serves as an static well potential, and attracts the lightquark q to form a diquark in the color antitriplet channel or attracts the light antiquark ¯ q toform a meson in the color singlet channel. Then the heavy tetraquark (molecular) states arecharacterized by the effective heavy quark mass M Q (or constituent quark mass) and the virtuality V = q M X/Y/Z − (2 M Q ) . It is natural to choose the energy scales of the QCD spectral densities15 .0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.03.03.33.63.94.24.54.85.15.45.76.0 M ( G e V ) (GeV) A B ESF
Figure 8: The masses with variations of the energy scales µ , where the A and B correspond to the¯ uc ¯ cd and ¯ sc ¯ cs tetraquark molecular states, respectively, the ESF denotes the energy scale formula M X/Z = p µ + 4 × (1 .
84 GeV) .as, µ = V = M X/Y/Z − (2 M Q ) . (43)Analysis of the J/ψ and Υ with the famous Coulomb-plus-linear potential or Cornell potentialleads to the constituent quark masses m c = 1 .
84 GeV and m b = 5 .
17 GeV [24]. If we set M c = m c = 1 .
84 GeV, we can obtain the dash-dotted line M X/Z = p µ + 4 × (1 .
84 GeV) in Fig.8,which intersects with the lines of the masses of the ¯ uc ¯ cd and ¯ sc ¯ cs tetraquark molecular states atthe energy scales about µ = 1 . . .
89 GeV and 4 .
67 GeV, which happen to coincide with the D ¯ D ∗ + D ∗ ¯ D and D ∗ s ¯ D s − D s ¯ D ∗ s thresholds 3 .
88 GeV and 4 .
65 GeV, respectively. The old value M c = 1 .
84 GeV and updated value M c = 1 .
85 GeV fitted in the QCD sum rules for the hidden-charm tetraquark molecular states areall consistent with the constituent quark mass m c = 1 .
84 GeV [11, 12]. We can set the value of theeffective c -quark mass as M c = 1 . ± .
01 GeV. In this article, we use the energy scale formula µ = q M X/Z − × (1 .
84 GeV) as the constraints to choose the best energy scales of the QCDspectral densities.Again, we choose the pole contributions as large as (40 − | D (8) | = (3 − | D (10) | ≪
1% and D (8) = (1 − D (10) ≪
1% for the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), respectively.Now let us take into account all uncertainties of the input parameters, and obtain the values ofthe masses and pole residues of the tetraquark molecular states, which are shown in Table 2 andFigs.9-10. In calculations, we add an uncertainty δµ = ± . µ accordingthe uncertainty in the effective c -quark mass M c = 1 . ± .
01 GeV. From Figs.9-10, we can seethat there appear Borel platforms in the Borel windows indeed. The tetraquark molecular statesalone can satisfy the QCD sum rules. 16 .5 1.8 2.1 2.4 2.7 3.0 3.3 3.6 3.9 4.2 4.53.03.23.43.63.84.04.24.44.64.85.0 A M ( G e V ) T (GeV ) Central value Error bounds B M ( G e V ) T (GeV ) Central value Error bounds
Figure 9: The masses with variations of the Borel parameters T , where the A and B correspondto the ¯ uc ¯ cd and ¯ sc ¯ cs tetraquark molecular states, respectively, the regions between the two verticallines are the Borel windows. A ( - G e V ) T (GeV ) Central value Error bounds B ( - G e V ) T (GeV ) Central value Error bounds
Figure 10: The pole residues with variations of the Borel parameters T , where the A and B correspond to the ¯ uc ¯ cd and ¯ sc ¯ cs tetraquark molecular states, respectively, the regions between thetwo vertical lines are the Borel windows. 17 .3 Taking into account the meson-meson scattering states besides thetetraquark molecular states From the previous two subsections, we observe that the meson-meson scattering scattering statesalone cannot saturate the QCD sum rules, while the tetraquark molecular states alone can saturatethe QCD sum rules. However, the quantum field theory does not forbid the couplings between thefour-quark currents and meson-meson scattering states if they have the same quantum numbers.We should take into account both the tetraquark molecular states and the meson-meson scatteringstates at the hadron side.Now we study the contributions of the intermediate meson-meson scattering states D ¯ D ∗ , J/ψπ , J/ψρ , etc besides the tetraquark molecular state Z c to the correlation function Π µν ( p ) as anexample,Π µν ( p ) = − b λ Z p − c M Z − Σ DD ∗ ( p ) − Σ J/ψπ ( p ) − Σ J/ψρ ( p ) + · · · e g µν ( p ) + · · · , (44)where e g µν ( p ) = − g µν + p µ p ν p . We choose the bare quantities b λ Z and c M Z to absorb the divergencesin the self-energies Σ D ¯ D ∗ ( p ), Σ J/ψπ ( p ), Σ J/ψρ ( p ), etc. The renormalized energies satisfy therelation p − M Z − Σ DD ∗ ( p ) − Σ J/ψπ ( p ) − Σ J/ψρ ( p ) + · · · = 0, where the overlines above theself-energies denote that the divergent terms have been subtracted. As the tetraquark molecularstate Z c is unstable, the relation should be modified, p − M Z − ReΣ DD ∗ ( p ) − ReΣ
J/ψπ ( p ) − ReΣ
J/ψρ ( p ) + · · · = 0, and − ImΣ DD ∗ ( p ) − ImΣ
J/ψπ ( p ) − ImΣ
J/ψρ ( p ) + · · · = p p Γ( p ). Thenegative sign in front of the self-energies come from the special definitions in this article, if weredefine the self-energies, Σ( p ) → − Σ( p ) in Eq.(44), the negative sign can be removed. Therenormalized self-energies contribute a finite imaginary part to modify the dispersion relation,Π µν ( p ) = − λ Z p − M Z + i p p Γ( p ) e g µν ( p ) + · · · . (45)If we assign the Z c (3900) to be the D ¯ D ∗ + D ∗ ¯ D tetraquark molecular state with the J P C = 1 + − [11], the physical width Γ Z c (3900) ( M Z ) = (28 . ± .
6) MeV from the Particle Data Group [2].We can take into account the finite width effect by the following simple replacement of thehadronic spectral density, λ Z δ (cid:0) s − M Z (cid:1) → λ Z π M Z Γ Z ( s )( s − M Z ) + M Z Γ Z ( s ) , (46)where Γ Z ( s ) = Γ Z M Z √ s s s − ( M D + M D ∗ ) M Z − ( M D + M D ∗ ) . (47)Then the hadron sides of the QCD sum rules in Eq.(36) and Eq.(38) undergo the following changes, λ Z exp (cid:18) − M Z T (cid:19) → λ Z Z s ( m D + m D ∗ ) ds π M Z Γ Z ( s )( s − M Z ) + M Z Γ Z ( s ) exp (cid:16) − sT (cid:17) , = (0 . ∼ . λ Z exp (cid:18) − M Z T (cid:19) , (48) λ Z M Z exp (cid:18) − M Z T (cid:19) → λ Z Z s ( m D + m D ∗ ) ds s π M Z Γ Z ( s )( s − M Z ) + M Z Γ Z ( s ) exp (cid:16) − sT (cid:17) , = (0 . ∼ . λ Z M Z exp (cid:18) − M Z T (cid:19) , (49)18ith the value √ s = 4 .
40 GeV. We can absorb the numerical factors 0 . ∼ .
79 and 0 . ∼ . λ Z → . λ Z safely, the intermediate meson-loops cannot affect the mass M Z significantly, but affect the pole residue remarkably, which areconsistent with the fact that we obtain the masses of the tetraquark molecular states from afraction, see Eqs.(38)-(39). If we only take into account the tetraquark molecular states at thehadron side, we can obtain reasonable molecule masses but overestimate the pole residues. The quarks and gluons are confined objects, they cannot be put on the mass-shell, it is questionableto use the Landau equation to study the quark-gluon bound states. Furthermore, we carry outthe operator product expansion in the deep Euclidean region p → −∞ in the QCD sum rules,where the Landau singularities cannot exist. If we insist on applying the Landau equation to studythe Feynman diagrams in the QCD sum rules, we should choose the pole masses rather than the M S masses, which lead to obvious problems in the QCD sum rules for the traditional or normalcharmonium and bottomonium states.Lucha, Melikhov and Sazdjian assert that the contributions at the order O ( α ks ) with k ≤ O ( α s ). In fact, the nonfactorizableFeynman diagrams begin to appear at the order O ( α s /α s ) rather than at the order O ( α s ), andmake contributions to the tetraquark molecular states. Furthermore, the Landau singularities ob-tained by Lucha, Melikhov and Sazdjian are questionable, as the Landau singularities appear atthe region p ≥ ( ˆ m u/s + ˆ m d/s + ˆ m c + ˆ m c ) rather than at the deep Euclidean region p → −∞ .The meson-meson scattering state and tetraquark molecular state both have four valencequarks, which form two color-neutral clusters, we cannot distinguish which Feynman diagramscontribute to the meson-meson scattering state or tetraquark molecular state based on the twocolor-neutral clusters in the factorizable Feynman diagrams. The Landau equation servers as akinematical equation in the momentum space, and is independent on the factorizable and nonfac-torizable properties of the Feynman diagrams in the color space.We choose the axialvector current J µ ( x ) and tensor current J µν ( x ) to examine the outcomeif the assertion of Lucha, Melikhov and Sazdjian is right. The axialvector current J µ ( x ) couplespotentially to the charged D ¯ D ∗ + D ∗ ¯ D meson-meson scattering states or tetraquark molecularstates with the J P C = 1 + − , while the tensor current J µν ( x ) couples potentially to the neutral D ∗ s ¯ D s − D s ¯ D ∗ s meson-meson scattering states or tetraquark molecular states with the J P C = 1 − + .The quantum numbers of the D ¯ D ∗ + D ∗ ¯ D and D ∗ s ¯ D s − D s ¯ D ∗ s differ from the traditional or normalmesons significantly, and are good subjects to study the exotic states. After detailed analysis, weobserve that the meson-meson scattering states cannot saturate the QCD sum rules, while thetetraquark molecular states can saturate the QCD sum rules. We can take into account themeson-meson scattering states reasonably by adding a finite width to the tetraquark molecularstates.The Landau equation is useless to study the Feynman diagrams in the QCD sum rules for thetetraquark molecular states, the tetraquark molecular states begin to receive contributions at theorder O ( α s /α s ) rather than at the order O ( α s ).19 ppendix The explicit expressions of the QCD spectral densities ρ Z,QCD ( s ) and ρ X,QCD ( s ), ρ Z,QCD ( s ) = ρ Z ( s ) + ρ Z ( s ) + ρ Z ( s ) + ρ Z ( s ) + ρ Z ( s ) + ρ Z ( s ) + ρ Z ( s ) + ρ Z ( s ) ,ρ X,QCD ( s ) = ρ X ( s ) + ρ X ( s ) + ρ X ( s ) + ρ X ( s ) + ρ X ( s ) + ρ X ( s ) + ρ X ( s ) + ρ X ( s ) , (50) ρ Z ( s ) = 14096 π Z dydz yz (1 − y − z ) (cid:0) s − m c (cid:1) (cid:0) s − sm c + 3 m c (cid:1) , (51) ρ Z ( s ) = − m c h ¯ qq i π Z dydz ( y + z ) (1 − y − z ) (cid:0) s − m c (cid:1) (cid:0) s − m c (cid:1) , (52) ρ Z ( s ) = − m c π h α s GGπ i Z dydz (cid:18) zy + yz (cid:19) (1 − y − z ) (cid:2) s − m c + s δ (cid:0) s − m c (cid:1)(cid:3) + 11024 π h α s GGπ i Z dydz ( y + z ) (1 − y − z ) s (cid:0) s − m c (cid:1) , (53) ρ Z ( s ) = 3 m c h ¯ qg s σGq i π Z dydz ( y + z ) (cid:0) s − m c (cid:1) − m c h ¯ qg s σGq i π Z dydz (cid:18) yz + zy (cid:19) (1 − y − z ) (cid:0) s − m c (cid:1) , (54) ρ Z ( s ) = m c h ¯ qq i π Z dy + g s h ¯ qq i π Z dydz yz (cid:2) s − m c + s δ (cid:0) s − m c (cid:1)(cid:3) − g s h ¯ qq i π Z dydz (cid:18) zy + yz (cid:19) (1 − y − z ) (cid:0) s − m c (cid:1) − m c g s h ¯ qq i π Z dydz (cid:18) zy + yz (cid:19) (1 − y − z ) (cid:2) s δ (cid:0) s − m c (cid:1)(cid:3) + g s h ¯ qq i π Z dydz ( y + z ) (1 − y − z ) (cid:0) s − m c (cid:1) , (55) ρ Z ( s ) = m c h ¯ qq i π h α s GGπ i Z dydz ( y + z ) (cid:18) z + 1 y (cid:19) (1 − y − z ) (cid:18) sT (cid:19) δ (cid:0) s − m c (cid:1) − m c h ¯ qq i π h α s GGπ i Z dydz (cid:18) yz + zy (cid:19) (1 − y − z ) (cid:20) s δ (cid:0) s − m c (cid:1)(cid:21) − m c h ¯ qq i π h α s GGπ i Z dydz (cid:20) s δ (cid:0) s − m c (cid:1)(cid:21) − m c h ¯ qq i π h α s GGπ i Z dy (cid:20) s δ (cid:0) s − e m c (cid:1)(cid:21) , (56) ρ Z ( s ) = − m c h ¯ qq ih ¯ qg s σGq i π Z dy (cid:16) sT (cid:17) δ (cid:0) s − e m c (cid:1) + h ¯ qq ih ¯ qg s σGq i π Z dy s δ (cid:0) s − e m c (cid:1) , (57)20 Z ( s ) = − h ¯ qg s σGq i π T (cid:18) − m c T (cid:19) Z dy s δ (cid:0) s − e m c (cid:1) − m c h ¯ qq i T h α s GGπ i Z dy " y + 1(1 − y ) δ (cid:0) s − e m c (cid:1) + m c h ¯ qq i T h α s GGπ i Z dy " y + 1(1 − y ) δ (cid:0) s − e m c (cid:1) + h ¯ qg s σGq i π Z dy (cid:18) sT (cid:19) δ (cid:0) s − e m c (cid:1) + m c h ¯ qq i T h α s GGπ i Z dys δ (cid:0) s − e m c (cid:1) , (58) ρ X ( s ) = 12048 π Z dydz yz (1 − y − z ) (cid:0) s − m c (cid:1) (cid:0) s − m c (cid:1) + 18192 π Z dydz yz (1 − y − z ) (cid:0) s − m c (cid:1) (cid:0) s − sm c + m c (cid:1) , (59) ρ X ( s ) = m s h ¯ ss i π Z dydz yz (cid:0) s − m c (cid:1) (cid:0) s − m c (cid:1) + m s h ¯ ss i π Z dydz yz (1 − y − z ) (cid:0) s − sm c + 3 m c (cid:1) + 9 m s m c h ¯ ss i π Z dydz (cid:0) s − m c (cid:1) , (60) ρ X ( s ) = − m c π h α s GGπ i Z dydz (cid:18) zy + yz (cid:19) (1 − y − z ) (cid:0) s − m c (cid:1) − m c π h α s GGπ i Z dydz (cid:18) zy + yz (cid:19) (1 − y − z ) (cid:20) s − m c + 4 s δ (cid:0) s − m c (cid:1)(cid:21) − π h α s GGπ i Z dydz ( y + z ) (1 − y − z ) (cid:0) s − m c (cid:1) (cid:0) s − m c (cid:1) + 112288 π h α s GGπ i Z dydz ( y + z ) (1 − y − z ) (cid:0) s − sm c + 3 m c (cid:1) , (61) ρ X ( s ) = − m s h ¯ sg s σGs i π Z dy y (1 − y ) (cid:0) s − e m c (cid:1) − m s h ¯ sg s σGs i π Z dydz yz (cid:20) s − m c + 4 s δ (cid:0) s − m c (cid:1)(cid:21) − m s m c h ¯ sg s σGs i π Z dy , (62) ρ X ( s ) = − m c h ¯ ss i π Z dy , (63)21 X ( s ) = m s m c h ¯ ss i π h α s GGπ i Z dydz (cid:18) zy + yz (cid:19) (cid:18) − sT (cid:19) δ (cid:0) s − m c (cid:1) + m s m c h ¯ ss i π h α s GGπ i Z dydz (cid:18) zy + yz (cid:19) (1 − y − z ) (cid:18)
14 + sT − s T (cid:19) δ (cid:0) s − m c (cid:1) + m s m c h ¯ ss i π h α s GGπ i Z dydz y (cid:16) − y − ysT (cid:17) δ (cid:0) s − m c (cid:1) − m s h ¯ ss i π h α s GGπ i Z dy (cid:2)
14 + 13 sδ (cid:0) s − e m c (cid:1)(cid:3) + m s h ¯ ss i π h α s GGπ i Z dydz ( y + z ) (cid:20)
34 + (cid:18) s + s T (cid:19) δ (cid:0) s − m c (cid:1)(cid:21) + m s m c h ¯ ss i π h α s GGπ i Z dy (cid:16) sT (cid:17) δ (cid:0) s − e m c (cid:1) , (64) ρ X ( s ) = 3 m c h ¯ ss ih ¯ sg s σGs i π Z dy (cid:16) sT (cid:17) δ (cid:0) s − e m c (cid:1) , (65) ρ X ( s ) = − m c h ¯ sg s σGs i π T Z dy s δ (cid:0) s − e m c (cid:1) + m c h ¯ ss i T h α s GGπ i Z dy y (cid:16) − ys T (cid:17) δ (cid:0) s − e m c (cid:1) + h ¯ sg s σGs i π T Z dy s δ (cid:0) s − e m c (cid:1) − m c h ¯ ss i T h α s GGπ i Z dy s δ (cid:0) s − e m c (cid:1) , (66)where R dydz = R y f y i dy R − yz i dz , R dy = R y f y i dy , y f = √ − m c /s , y i = − √ − m c /s , z i = ym c ys − m c , m c = ( y + z ) m c yz , e m c = m c y (1 − y ) , R y f y i dy → R dy , R − yz i dz → R − y dz when the δ functions δ (cid:0) s − m c (cid:1) and δ (cid:0) s − e m c (cid:1) appear. Acknowledgements
This work is supported by National Natural Science Foundation, Grant Number 11775079.
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