Is Z^+(4430) a loosely bound molecular state?
aa r X i v : . [ h e p - ph ] N ov Is Z + (4430) a loosely bound molecular state? Xiang Liu , ∗ Yan-Rui Liu , † Wei-Zhen Deng , ‡ and Shi-Lin Zhu § Department of Physics, Peking University, Beijing 100871, China Institute of High Energy Physics, P.O. Box 918-4, Beijing 100049, China (Dated: October 26, 2018)Since Z + (4430) lies very close to the threshold of D ∗ ¯ D , we investigate whether Z + (4430) couldbe a loosely bound S-wave state of D ∗ ¯ D or D ∗ ¯ D ′ with J P = 0 − , − , − , i.e., a molecular statearising from the one-pion-exchange potential. The potential from the crossed diagram is much largerthan that from the diagonal scattering diagram. With various trial wave functions, we notice thatthe attraction from the one pion exchange potential alone is not strong enough to form a boundstate with realistic pionic coupling constants deduced from the decay widths of D and D ′ . PACS numbers: 12.39.Pn, 12.40.Yx, 13.75.Lb
I. INTRODUCTION
Recently Belle Collaboration observed a sharp peakin the π + ψ ′ invariant mass spectrum in the exclusive B → Kπ + ψ ′ decays with a statistical significance of 7 σ [1]. This resonance-like structure is named as Z + (4430).The fit with a Breit-Wigner form yields its mass m =4433 ± ± +17 − (stat) +30 − (syst) MeV. It is very interesting to notethat the width of Z + (4430) is roughly the same as thatof D .The product branching fraction is measured to be B ( B → KZ + (4430)) · B ( Z + (4430) → π + ψ ′ ) = (4 . ± . ± . × − [1]. For comparison, we listthe production rate of X (3872) and Y (4260) in B decays.From Ref. [2] we have B ( B − → K − X (3872)) · B ( X (3872) → J/ Ψ π + π − )= (1 . ± . × − , and from Ref. [3] B ( B − → K − X (3872)) · B ( X (3872) → J/ Ψ π + π − )= (10 . ± . ± . × − . For Y (4260), Babar Collaboration gave the upper limitof the branching fraction [3] B ( B − → K − Y (4260)) · B ( Y (4260) → J/ Ψ π + π − ) < . × − . It’s plausible that (1) B ( B → KZ + (4430)) is compa-rable to both B ( B − → K − X (3872)) and B ( B − → K − Y (4260)); (2) π + ψ ′ is one of the main decay modesof Z + (4430) if it is a resonance.The peak Z + (4430) inspired several theoretical specu-lations of its underlying structure. Rosner suggested that ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] Z + (4430) is a S-wave threshold effect because Z + (4430)lies close to the D ∗ (2010) ¯ D (2420) threshold [4]. Theproduction mechanism was speculated as follows. The b quark first decays into a strange quark and a pair of c ¯ c while a pair of light quarks are created from the vacuum.In other words, B meson first decays into a K and a pairof D mesons. Then the D meson pair re-scatters into π + ψ ′ . He also suggested other possible charged states.Maiani, Polosa and Riquer identified this signal as thefirst radial excitation of the tetraquark supermultipletto which X (3872) and X (3876) belong [5]. With theirassignment the quantum number of Z + (4430) is J P C =1 + − where the C-parity is for the neutral member withinthe same multiplet. Z + (4430) decays into π + ψ ′ via S-wave. Moreover their scheme requires the ground statewith J P C = 1 + − around 3880 MeV which decays into π + ψ and η c ρ + .With a QCD-string model, Gershtein, Likhoded andPronko argued that both X(3872) and Z + (4430) aretetraquark states [6]. They speculate that the two quarksand two anti-quarks sit on the four corners of a squarewhile any q ¯ q pair is a color-octet state. The decays oftetraquarks involve the reconnection of the color string.Cheung, Keung and Yuan discussed the bottom analogof Z + (4430) assuming it is a tetraquark bound state [7].According to their estimate, the doubly-charged Z bc statelies around 7.6 GeV while the bottomonium analog Z bb of Z + (4433) is about 10.7 GeV.Qiao suggested Z + (4433) be the first radial excitationof Λ c − Σ c bound state [8]. Within Qiao’s scheme, all ofthe recently observed states Y (4260), Y (4361), Z + (4430)and Y (4664) are accommodated in the extended heavybaryonium framework.Lee, Mihara, Navarra and Nielsen calculated the massof Z + (4430) m Z = (4 . ± .
10) GeV in the frameworkof QCD sum rules, assuming it is a 0 − molecular stateof D ∗ (2010) ¯ D (2420) [9]. They predicted the analogousmesons Z s at m Z s = (4 . ± .
06) GeV, which is above the D ∗ s D threshold and Z bb around m Z bb = (10 . ± . Z + (4430) is a threshold cusp aris-ing from the deexcitation of the D ∗ (2010) D (2420) pairinto lower mass D states [10]. The imaginary part Imf ( s )of the elastic S-wave amplitude f ( s ) is a step functionnear the threshold. From the dispersion relation, thereal part of the amplitude near the threshold looks like Ref ( s ) ∼ R s θ ( s ) s ′ − s ds ′ ∼ ln( s − s ). Therefore | f ( s ) | con-tains a sharp cusp near threshold.However, none of the above schemes explains why Z + (4430) does NOT decay into π + ψ . One notes thatthe momentum of D and ¯ D is small and close to eachother in the rest frame of the parent B meson, especiallywhen one (or two) of the D meson pair is an excited state.Therefore, there is plenty of time for the D meson pairto move together and re-scatter into π + ψ ′ . However, themost puzzling issue of the re-scattering mechanism is theabsence of any signal in the π + J/ψ channel. One maywonder whether the mismatch of the Q-values of the ini-tial and final states plays an important role. If so, oneshould also expect a signal in the π + ψ (3 S ) channel sincethere is nearly no mismatch of the Q-value now. Anotherpotential scapegoat is the specific nodal structure in thewave functions of the final states. Detailed calculationsalong the above two directions are highly desirable to in-vestigate the origin of the non-observation of Z + (4430)in the π + ψ mode.In the heavy quark limit, the angular momentum of thelight quark j l = ~l + ~S q is a good quantum number where l is the orbital angular momentum and S q is the lightquark spin. For the P-wave heavy mesons, j l = or ,which correspond to the two doublets with J P = (1 + , + )and (0 + , + ) respectively. The ground states D and D ∗ belong to the J P = (0 − , − ) doublet with j l = . Sincethe 1 + state in the (1 + , + ) doublet decays into D ∗ π viaD-wave, it’s very narrow and denoted as D (2420) [11].The 1 + state in the (0 + , + ) doublet decays into D ∗ π viaS-wave. Hence it’s very broad and denoted as D ′ (2430)[11].Assuming Z + (4430) is a D D ∗ (or D ′ D ∗ ) S-wave reso-nance, Meng and Chao found that the open-charm decaymode D ∗ D ∗ π is dominant and the re-scattering effectsare significant in D D ∗ channel but not in D ′ D ∗ chan-nel since D ′ is very broad [12]. For the J P = 1 − can-didate, the ratio Γ( Z + → ψ ′ π + ) / Γ( Z + → J/ψπ + ) mayreach 5 . Z + (4430) is difficult to be found in J/ψπ + .Despite so many theoretical speculations proposedabove, a dynamical study of the Z + (4430) signal is stillmissing. In this work we will explore whether Z + (4430)could be a S-wave molecular state of D ∗ and ¯ D ′ (or ¯ D ),which is loosely bound by the long-range pion exchangepotential. We want to find out whether there exists theattractive force between D ∗ and ¯ D ′ ( ¯ D ) in differentchannels.This paper is organized as follows. We discuss thepossible quantum numbers of Z + (4430) and its possiblepartner states in Section II. We collect the effective La-grangians and various coupling constants in Section III.We derive the one-pion-exchange potential (OPEP) insection IV. Then we present our numerical result and a short discussion in Section V. II. QUANTUM NUMBER OF Z + (4430) ANDOTHER POSSIBLE STATES
Natively, the smaller its angular momentum, the lowerthe mass of Z + (4430). Since Z + (4430) lies very close tothe D ∗ (2010) ¯ D (2420) threshold, we consider only thepossibility of Z + (4430) being the loosely bound S-wavestate of D ∗ and ¯ D ′ (or ¯ D ). Therefore, its possible an-gular momentum and parity are J P = 0 − , − , − . More-over, Z + (4430) was observed in the ψ ′ π + channel. So itis an isovector state with positive G -parity, i.e., I G = 1 + .For a charged member within a molecular isovectormultiplet, one can construct its flavor wave function withdefinite G parity in the following way. Suppose | A i is onecomponent of its flavor wave function. Then the G = +state reads: | + i = 1 √ (cid:16) | A i + ˆ G | A i (cid:17) (1)where ˆ G = e iI y π ˆ C is the G -parity operator. Similarly,the G = − state reads: |−i = 1 √ (cid:16) | A i − ˆ G | A i (cid:17) (2)In the present case, the flavor wave function of Z + (4430) is | Z + i = 1 √ (cid:2) | A ′ i + | B ′ i (cid:3) (3)with | A ′ i = | ¯ D ′ D ∗ + i and | B ′ i = | D ′ +1 ¯ D ∗ i , where D ′ and D ′ +1 belong to the (0 + , + ) doublet in the heavyquark effective field theory. Or | Z + i = 1 √ (cid:2) | A i + | B i (cid:3) (4)with | A i = | ¯ D D ∗ + i and | B i = | D +1 ¯ D ∗ i , where D and D +1 belong to the (1 + , + ) doublet.The flavor wave functions of the partner states of Z + can be derived in the following way. Charmed mesonsbelong to the fundamental representation of flavor SU (3).Therefore, the system with a charmed meson and an anti-charmed meson belongs to × ¯3 = + . We list the wavefunctions of these hidden charm states, whose names arelisted in Fig. 1. Here we use the system of D ∗ ( s ) and D ( s )1 as an illustration.The flavor wave functions of these states are | Z + i = 1 √ (cid:16) | ¯ D D ∗ + i − c | ¯ D ∗ D +1 i (cid:17) , ¯ U U − Z + Z − U U + V ,V ,Z FIG. 1: The multiplets composed with charmed mesons andanticharmed mesons. | Z i = 12 h(cid:16) | D − D ∗ + i − c | D ∗− D +1 i (cid:17) − (cid:16) | ¯ D D ∗ i − c | ¯ D ∗ D i (cid:17)i , | Z − i = − √ (cid:16) | D − D ∗ i − c | D ∗− D i (cid:17) , | U + i = − √ (cid:16) | ¯ D D ∗ + s − c | ¯ D ∗ D + s ii (cid:17) , | U i = − √ (cid:16) | D − D ∗ + s i − c | D ∗− D + s i (cid:17) , | U − i = − √ (cid:16) | D − s D ∗ i − c | D ∗− s D i (cid:17) , | ¯ U i = 1 √ (cid:16) | D − s D ∗ + i − c | D ∗− s D +1 i (cid:17) , | V i = 12 √ h(cid:16) | D − D ∗ + i − c | D ∗− D +1 i (cid:17) + (cid:16) | ¯ D D ∗ i − c | ¯ D ∗ D i (cid:17) − (cid:16) | D − s D ∗ + s i − c | D ∗− s D + s i (cid:17)i , | V i = 1 √ h(cid:16) | D − D ∗ + i − c | D ∗− D +1 i (cid:17) + (cid:16) | ¯ D D ∗ i − c | ¯ D ∗ D i (cid:17) + (cid:16) | D − s D ∗ + s i − c | D ∗− s D + s i (cid:17)i . We use Z in the J = 0 case as an example to illustratehow to determine c . At the quark level, this molecularstate may be written as J Z = 12 [ J − cJ − ( J − cJ )] (5)where J = (¯ c a γ µ γ d a )( ¯ d e γ µ c e ) ,J = ( ¯ d a γ µ γ c a )(¯ c e γ µ d e ) ,J = (¯ c a γ µ γ u a )(¯ u e γ µ c e ) ,J = (¯ u a γ µ γ c a )(¯ c e γ µ u e ) . In the above equation, a and e are the color indices. Un- der charge conjugate transformation, we haveˆ CJ ˆ C − = − ( ¯ d a γ µ γ c a )(¯ c e γ µ d e ) = − J , ˆ CJ ˆ C − = − (¯ c a γ µ γ d a )( ¯ d e γ µ c e ) = − J , ˆ CJ ˆ C − = − (¯ u a γ µ γ c a )(¯ c e γ µ u e ) = − J , ˆ CJ ˆ C − = − (¯ c a γ µ γ u a )(¯ u e γ µ c e ) = − J . Therefore, we getˆ CJ Z ˆ C − = 12 [ − J + cJ − ( − J + cJ )] . (6)In other words, the C -parity of Z is C = ± for c = ± Z + (4430) is a state with isospin 1 and G = +, onerequires c = − Z are I G ( J P C ) = 1 + (0 , , −− .We want to emphasize that the presence of both thecharm and anti-charm quark (and the light quark andanti-quark) in the expression of J i ensures there is noarbitrary phase factor under charge conjugate transfor-mation.There is one intuitive and natural way to interpretthe above results if we consider the flavor SU(4) sym-metry, which is of course broken badly in reality. If wenaively assume the flavor SU(4) symmetry, then stateswithin the same multiplet should carry the same coef-ficient under charge conjugate transformation. For ex-ample, ˆ C | D ∗− i = C ( D ∗− ) | D ∗ + i where the coefficient C ( D ∗− ) takes the same value as either C ( ρ ) or C ( J/ψ ).I.e., C ( D ∗− ) = −
1. Similarly, C ( D − ) = +1. In otherwords, the c = − Z state with nega-tive C -parity.In addition, one obtains the flavor wave functions ofthese states with opposite G -parity if we take c = + inthe above equations. For example, we will also discusswhether e Z + could be a molecular state: | e Z + i = 1 √ (cid:16) | ¯ D D ∗ + i − | ¯ D ∗ D +1 i (cid:17) . If ˜ Z + exists, this state may be discovered in either J/ψπ + π or ψ ′ π + π channel.If we replace one of the c (or ¯ c ) by b (or ¯ b ), we canget molecular states such as ( b ¯ q ) − (¯ cq ). With the dualreplacement c → b, ¯ c → ¯ b , we get the hidden bottommolecular states ( b ¯ q ) − (¯ bq ). III. EFFECTIVE LAGRANGIANS ANDCOUPLING CONSTANTS
We collect the effective chiral Lagrangian used in thederivation of the OPEP in this section. In the chiral andheavy quark dual limits, the Lagrangian relevant to ourcalculation reads [13, 14] L = ig Tr[ H b A/ ba γ ¯ H a ] + ig ′ Tr[ S b A/ ba γ ¯ S a ]+ ig ′′ Tr[ T µb A/ ba γ ¯ T µa ]+[ ih Tr[ S b A/ ba γ ¯ H a ] + h.c. ]+ { i h Λ χ Tr[ T µb ( D µ A/ ) ba γ ¯ H a ] + h.c. } + { i h Λ χ Tr[ T µb ( D/A µ ) ba γ ¯ H a ] + h.c. } , (7)where H a = 1+ v P ∗ µa − P a γ ] , (8) S a = 1+ v P ′ µ a γ µ γ − P ∗ a ] , (9) T µa = 1+ v n P ∗ µν a γ ν − r P ν a γ [ g µν − γ ν ( γ µ − v µ )] o (10)and the axial vector field A µab is defined as A µab = 12 ( ξ † ∂ µ ξ − ξ∂ µ ξ † ) ab = if π ∂ µ M + · · · with ξ = exp( i M /f π ), f π = 132 MeV and M = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − η √ . (11)After expanding Eq. (7) to the leading order of thepion field, we further obtain L D ∗ + D ∗ + π = g D ∗ + D ∗ + π ǫ αβµν D ∗ + α ( ∂ µ π )( ∂ ν D ∗− β ) + h.c., (12) L D ′ D ′ π = g D ′ D ′ π ǫ αβµν D ′ α ( ∂ µ π )( ∂ ν ¯ D ′ β ) + h.c., (13) L D D π = g D D π ǫ αβµν ( D α )( ∂ µ π )( ∂ ν ¯ D β ) + h.c., (14) L D ∗ D ′ π = g D ∗ D ′ π (cid:2) − ( ∂ α D ∗ β )( ∂ β π ) D ′ α + D ∗ β ( ∂ α π )( ∂ β D ′ α ) + ( ∂ µ D ∗ α )( ∂ µ π ) D ′ α (cid:3) + h.c. (15) L D ∗ D π = g D ∗ D π (cid:2) D ∗ β D ν g λν ( ∂ β ∂ λ π ) − D ∗ β D ν g βν ( ∂ λ ∂ λ π ) + 2 D ∗ β D ν g βλ ( ∂ ν ∂ λ π )+ 1 m D ∗ m D ( ∂ λ D ∗ ν )( ∂ α ∂ λ π )( ∂ α D ν ) (cid:3) , (16)where D ′ denotes the P-wave axial-vector state in the(0 + , + ) doublet while D is the 1 + state in the (1 + , + ) doublet. The coupling constants g D ∗ + D ∗ + π , g D ′ D ′ π , g D D π and g D ∗ D ( ′ )1 π are g D ∗ + D ∗ + π = − √ gf π , g D ′ D ′ π = √ g ′ f π ,g D D π = − g ′′ √ f π ,g D ∗ + D ′ +1 π = g D ∗ D ′ π = − i √ hf π ,g D ∗ D π = − g D ∗ + D +1 π = − √ m D ∗ m D √ f π Λ χ ( h + h ) . The coupling constant g was studied in many theoret-ical approaches such as QCD sum rules [15, 16, 17, 18]and quark model [13]. In this work, we use the value g = 0 . ± . ± .
01 extracted by fitting the experi-mental width of D ∗ [19]. Falk and Luke obtained an ap-proximate relation | g ′ | = | g | / | g ′′ | = | g | in quarkmodel [13]. However the phase between g ′ and g ′′ isnot fixed. With the available experimental information,Casalbuoni and collaborators extracted h = − . ± . h ′ = ( h + h ) / Λ χ = 0 .
55 GeV − [14]. If we re-place the meson field in the heavy quark limit in theabove equations by the fields in full QCD and scale thecoupling constants by a factor √ m D ∗ etc, we get the ef-fective Lagrangian in full QCD, which is used below inthe derivation of the potential. IV. DERIVATION OF THE ONE PIONEXCHANGE POTENTIAL
Study of the possible molecular states, especially thesystem of a pair of heavy mesons, started more thanthree decades ago. The presence of the heavy quarkslowers the kinetic energy while the interaction betweentwo light quarks could still provide strong attraction.Okun and Voloshin proposed possibilities of the molecu-lar states involving charmed mesons [20]. Rujula, Geogiand Glashow once suggested ψ (4040) as a D ∗ ¯ D ∗ molecu-lar state [21]. T¨ornqvist studied possible deuteronliketwo-meson bound states such as D ¯ D ∗ and D ∗ ¯ D ∗ us-ing a quark-pion interaction model [22]. Dubynskiy andVoloshin proposed that there exists a possible new reso-nance at the D ∗ ¯ D ∗ threshold [23, 24].Several groups suggested X (3872) could be a goodmolecular candidate [25, 26, 27, 28, 29]. However, Suzukiargued that X (3872) is not a molecule state of D ¯ D ∗ +¯ D D ∗ [30]. Instead, X (3872) may have a dominant c ¯ c component with some admixture of D ¯ D ∗ + ¯ D D ∗ [30, 31, 32].In this work we will explore whether Z + (4430) could bea S-wave molecular state of D ∗ and ¯ D ′ (or ¯ D ), which isloosely bound by the long-range pion exchange potential.We first derive the scattering matrix elements betweenthe pair of D mesons as shown in Fig. 2. With the Breitapproximation, we can get the one-pion exchange poten-tials. Since the flavor wave function of Z + contains twocomponents, we have to consider both the direct scatter-ing diagram Fig. 2 (a) and the crossed diagram Fig. 2(b). Note only the crossed diagram contributes to OPEPin the case of X (3872).Recall the flavor wave function of Z + (4430) reads | Z + i = 1 √ (cid:2) | A ′ i + | B ′ i (cid:3) (17)with | A ′ i = | ¯ D ′ D ∗ + i and | B ′ i = | D ′ +1 ¯ D ∗ i or | Z + i = 1 √ (cid:2) | A i + | B i (cid:3) (18)with | A i = | ¯ D D ∗ + i and | B i = | D +1 ¯ D ∗ i . The mass of Z + (4430) is expressed as M Z + = m D ∗ + m D ( ′ )1 + T + E + δ, (19)where T is the kinetic energy in the center ofmass frame, E = h D ∗ + ¯ D ( ′ )01 |H | D ∗ + ¯ D ( ′ )01 i and δ = h D ∗ + ¯ D ( ′ )01 |H | D ′ +1 ¯ D ∗ i . H and H correspond to theinteraction in Fig. 2 (a) and (b) respectively. For thepossible ˜ Z + state with negative G-parity, we can get itsmass through the replacement + δ → − δ in Eq. (19). ¯ D ( ′ )01 D ∗ + ¯ D ( ′ )01 D ∗ + π (a) ¯ D ( ′ )01 D ∗ + D ( ′ )+1 ¯ D ∗ π (b)FIG. 2: (a) The single pion exchange in the direct scatter-ing process D ∗ + ¯ D ( ′ )01 → D ∗ + ¯ D ( ′ )01 . (b) The crossed process D ∗ + ¯ D ( ′ )01 → D ( ′ )+1 ¯ D ∗ . A. Scattering amplitudes
We collect the scattering amplitudes in thedifferent channels below. For the process D ∗ + ( p , ǫ ) ¯ D ′ ( p , ǫ ) → D ∗ + ( p , ǫ ) ¯ D ′ ( p , ǫ ), theamplitude is i M [ D ∗ + ( p , ǫ ) ¯ D ′ ( p , ǫ ) → D ∗ + ( p , ǫ ) ¯ D ′ ( p , ǫ )]= 2 igg ′ f π q − m π ε αβµν ε α ′ β ′ µ ′ ν ′ q µ q µ ′ p ν p ν ′ × ( ǫ λ α ǫ λ α ′ )( ǫ λ ′ β ǫ λ ′ β ′ ) . (20)For D ∗ + ( p , ǫ ) ¯ D ( p , ǫ ) → D ∗ + ( p , ǫ ) ¯ D ( p , ǫ ), onegets i M [ D ∗ + ( p , ǫ ) ¯ D ( p , ǫ ) → D ∗ + ( p , ǫ ) ¯ D ( p , ǫ )]= − igg ′′ f π q − m π ε αβµν ε α ′ β ′ µ ′ ν ′ q µ q µ ′ p ν p ν ′ × ( ǫ λ α ǫ λ α ′ )( ǫ λ ′ β ǫ λ ′ β ′ ) . (21)For D ∗ + ( p , ǫ ) ¯ D ′ ( p , ǫ ) → D ′ +1 ( p , ǫ ) ¯ D ∗ ( p , ǫ ), theamplitude is i M [ D ∗ + ( p , ǫ ) ¯ D ′ ( p , ǫ ) → D ′ +1 ( p , ǫ ) ¯ D ∗ ( p , ǫ )]= 2 ih f π q − m π [ − p α q β − q α p β + ( p · q ) g αβ ] × [ − p α ′ q β ′ − q α ′ p β ′ + ( p · q ) g α ′ β ′ ]( ǫ λ β ǫ λ α ′ )( ǫ λ ′ α ǫ λ ′ β ′ ) . (22)The amplitude of the process D ∗ + ( p , ǫ ) ¯ D ( p , ǫ ) → D +1 ( p , ǫ ) ¯ D ∗ ( p , ǫ ) is i M [ D ∗ + ( p , ǫ ) ¯ D ( p , ǫ ) → D +1 ( p , ǫ ) ¯ D ∗ ( p , ǫ )]= − im D ∗ m D ( h + h ) f π Λ χ q − m π h q ν q β − g βν q + g βν m D ∗ m D ( p · q )( q · p ) ih q ν ′ q β ′ − g β ′ ν ′ q + g β ′ ν ′ m D ∗ m D ( p · q )( q · p ) i ( ǫ λ β ǫ λ ν ′ )( ǫ λ ′ ν ǫ λ ′ β ′ ) . (23)Here the polarization vector is defined as ǫ ± = √ (0 , ± , i,
0) and ǫ = (0 , , , − B. The one-pion-exchange potential
We impose the constraint on the scattering amplitudesthat initial states and final states should have the sameangular momentum. The molecular state | J, J z i com-posed of the 1 − and 1 + charm meson pair can be con-structed as | J, J z i = X λ ,λ h , λ ; 1 , λ | J, J z i| p , ǫ ; p , ǫ i (24)where h , λ ; 1 , λ | J, J z i is the Clebsch-Gordan coeffi-cient. Combining the equation with the scattering am-plitudes, one gets the matrix element i M ( J, Jz ).With the Breit approximation, the interaction poten-tial in the momentum space is related to i M ( J, Jz ) V ( q ) = − qQ i m i Q f m f M ( J, Jz ) (25)where m i and m f denote the masses of the initial andfinal states respectively. We collect the expressions of thepotential in Tables I and II. We have explicitly shownthe resulting potentials are the same for the different J z component. TABLE I: The one-pion-exchange potential between D ∗ ( ¯ D ∗ )and ¯ D ′ ( D ′ ). Here the expressions are for the J z = 0 compo-nents. A ′ = ¯ D ′ D ∗ + and B ′ = D ′ +1 ¯ D ∗ .State A ′ ( B ′ ) → A ′ ( B ′ ) A ′ ( B ′ ) → B ′ ( A ′ )0 − gg ′ f π q q + m π − h f π ( q ) ( q ) − q − m π − gg ′ f π q z q + m π − h f π ( q ) ( q ) − q − m π − − gg ′ f π q − q z q + m π − h f π ( q ) ( q ) − q − m π TABLE II: The one-pion-exchange potential between D ∗ ( ¯ D ∗ )and ¯ D ( D ). In this table, A = ¯ D D ∗ + and B = D +1 ¯ D ∗ .State A ( B ) → A ( B ) A ( B ) → B ( A )0 − − gg ′′ f π q q + m π h ′ f π ( q ) ( q ) − q − m π − − gg ′′ f π q z q + m π − h ′ f π q ) − q z q ( q ) − q − m π − gg ′′ f π q − q z q + m π h ′ f π ( q ) − q z q +9 q z ( q ) − q − m π TABLE III: The one pion exchange potential in the coordinatespace with A ′ = ¯ D ′ D ∗ + and B ′ = D ′ +1 ¯ D ∗ .State A ′ ( B ′ ) → A ′ ( B ′ ) A ′ ( B ′ ) → B ′ ( A ′ )0 − gg ′ f π [ δ ( r ) − m π πr e − m π r ] h ( q ) πf π cos( µr ) r − gg ′ f π [ δ ( r ) − m π πr e − m π r ] h ( q ) πf π cos( µr ) r − − gg ′ f π [ δ ( r ) − m π πr e − m π r ] h ( q ) πf π cos( µr ) r
1. OPEP from the direct scattering diagram
From Tables I and II, it’s very interesting to notethat the OPEP from the direct scattering diagram in the J P = 0 − and 1 − channel always has the same sign whileit’s opposite to that in the J P = 2 − channel. In otherwords, there must be attraction in one of three channelsno matter what sign gg ′ takes.After making the Fourier transformation, we get theone-pion-exchange potential in the configuration space.The final potentials are obtained with the following re-placements: x → r / x → r / x y → r /
15 etcsince we consider only the S-wave system. Alternatively,one may average the potential in the momentum spacefirst.In the derivation of the OPEP, we let q ≈ δ function, similarto OPEP in the nucleon-nucleon potential. TABLE IV: The one pion exchange potential in the coordinate space with A = ¯ D D ∗ + and B = D +1 ¯ D ∗ .State A ( B ) → A ( B ) A ( B ) → B ( A )0 − − gg ′′ f π [ δ ( r ) − m π πr e − m π r ] h ′ f π [ ∇ δ ( r ) − µ δ ( r ) − µ π cos µrr ]1 − − gg ′′ f π [ δ ( r ) − m π πr e − m π r ] − h ′ f π [ ∇ δ ( r ) − µ δ ( r ) − µ π cos µrr ]2 − gg ′′ f π [ δ ( r ) − m π πr e − m π r ] h ′ f π [ ∇ δ ( r ) − µ δ ( r ) − µ π cos µrr ]
2. OPEP from the crossed diagram
First we note from Table II that the OPEP from thecrossed diagram contains terms such as q in the numer-ator, which reflects the fact that D decays into D ∗ π via D-wave. But the approximation q ≈ q ≈ M D − M D ∗ ≈
410 MeV ≈ m π . In the following,we take J P = 0 − as an example to illustrate how to dealwith the crossed diagram. Since the D ′ meson decaysinto D ∗ π , the crossed diagram contains a small imagi-nary part, which is roughly of the order Γ D ′ . However @ GeV - D - - - V Dir @ GeV D (a) @ GeV - D - - V Cross @ GeV D (b)FIG. 3: (a) The solid, dotted and dashed lines correspondto the potentials from the direct scattering diagram A ( B ) → A ( B ) in the J P = 0 − , − , − channels respectively, where g = 0 . g ′ = g/ g ′′ = g . (b) The potential from thecrossed diagram A ( B ) → B ( A ). only the real part of the scattering amplitude contributesto the potential. Hence the principal integration is alwaysassumed throughout the Fourier transformation for thecrossed diagram. V ( r ) = Z ˆ P h − h f π ( q ) ( q ) − q − m π + iǫ e i q · r i d q (2 π ) = − h f π Z ˆ P h ( q ) µ − q + iǫ e i q · r i d q (2 π ) = h ( q ) πf π cos( µr ) r where µ = ( q ) − m π and q ≈ M D − M D ∗ . Theresulting potential is of very long range and oscillating.We list the expressions of the potential in the coordinatespace in Tables III and IV.For the ¯ D ∗ D ′ system, it’s clear from Table III that theOPEP from the crossed diagram has the same overall signin all three channels. Moreover, the sign is positive. Be-cause of the factor q ∼ m π , the OPEP from the crosseddiagram is numerically much larger than that from the @ GeV - D - - - V Dir @ GeV D (a) @ GeV - D - - V Cross @ GeV D (b)FIG. 4: (a) The solid, dotted and dashed lines correspond tothe potentials from the direct scattering diagram A ′ ( B ′ ) → A ( B ′ ) in the J P = 0 − , − , − channels respectively, where g = 0 . g ′ = g/ g ′′ = g . (b) The potential from thecrossed diagram A ′ ( B ′ ) → B ′ ( A ′ ). direct scattering diagram. For the ¯ D ∗ D case, we notefrom Table IV that the overall sign in the 1 − channel isdifferent from that in both 0 − and 2 − channels.With the coupling constants g = 0 . g ′ = g/ g ′′ = g , the variation of the potential listed in Tables IIIand IV with r (in unit of GeV − ) is shown in Figs. 3 and4. We also note that the OPEP does not depend on themass of the charmed mesons. In other words, the samepotential may be used in the discussion of the hiddenbottom molecular states ( b ¯ q ) − (¯ bq ). V. RESULTS AND DISCUSSIONS
Besides those coupling constants in Section III, we alsoneed the following parameters in our numerical analysis: m D ∗ = 2007 MeV, m D ′ = 2430 MeV, m D = 2420 MeV, m B ∗ = 5325 MeV, m B ′ = 5732 MeV, f π = 132 MeV, m π = 135 MeV [11]; m B = 5725 MeV [33].With the potentials derived above, we use the vari-ational method to investigate whether there exists aloosely bound state. Our criteria of the formation ofa possible loosely bound molecular state is (1) the ra-dial wave function extend to 1 fm or beyond and (2) theminimum energy of the system is negative. Our trialwave functions include (a) ψ ( r ) = (1 + αr ) e − αr ; (b) ψ ( r ) = (1 + αr ) e − βr ; (c) ψ ( r ) = r (1 + αr ) e − βr .Unfortunately a solution satisfy the above criteria doesnot exist for the system of D ′ − D ∗ or D − D ∗ in all J P =0 − , − , − channels with the realistic coupling constants g = 0 . g ′ = g/ g ′′ = g deduced from the width of D ∗ , D and D ′ . Such a solution also does not exist if weswitch the sign of gg ′ or enlarge the absolute value of gg ′ by a factor 3. The same conclusion holds for the systemof B ′ − B ∗ , B − B ∗ and e Z + with negative G -parity.In short summary, we have performed a dynamicalstudy of the Z + (4430) signal to see whether it is a looselybound molecular state of D − D ∗ or D ′ − D ∗ . We findthat the interaction from the one pion exchange potentialalone is not strong enough to bind the pair of charmedmesons with realistic coupling constants. Other dynam-ics is necessary if Z + (4430) is further established as amolecular state by the future experiments.It’s interesting to note that the one pion exchangepotential alone does not bind the deuteron in nuclearphysics either. In fact, the strong attractive force inthe intermediate range is introduced in order to bindthe deuteron, which is sometimes modeled by the sigma meson exchange. One may wonder whether the simi-lar mechanism plays a role in the case of Z + (4430) and X (3872). Further work along this direction is in progress.If the Z + (4430) is really a J P = 1 − molecularstate, the quantum number of its neutral partner state Z (4430) is J P C = 1 −− . Such a state can be searched forin the e + e − annihilation processes. Babar and Belle col-laborations have observed several new charmonium (orcharmonium-like) states with J P C = 1 −− around thismass range including Y (4260), Y (4320), and Y (4664)with the initial state radiation (ISR) technique, althoughthese states do not appear as a peak in the R distribu-tion. We strongly urge Belle and Babar collaborations tosearch for the Z (4430) state in the π ψ ′ channel usingthe ISR technique. The absence of a signal will be anindication that the J P of Z + (4430) is not 1 − . Acknowledgments
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