Is the Z^+(4430) a radially excited state of D_s?
aa r X i v : . [ h e p - ph ] O c t Is the Z + (4430) a radially excited state of D s ? Takayuki Matsuki ∗ Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, JAPAN
Toshiyuki Morii † Graduate School of Human Development and Environment,Kobe University, 3-11 Tsurukabuto, Nada, Kobe 657-8501, JAPAN
Kazutaka Sudoh ‡ Nishogakusha University, 6-16 Sanbancho, Chiyoda, Tokyo 102-8336, JAPAN (Dated: August 15, 2008)We present the interpretation that the recently discovered Z + (4430) by the Belle Collaborationcan be a radial excitation of the c ¯ s state, being consistent with an observed value of the product ofbranching ratios, B ( B → K ∓ Z ± (4430)) × B ( Z ± (4430) → π ± ψ ′ ) ∼ − . We give an explicit c ¯ s candidate for this state by calculating the mass value in our semirelativistic quark potential modeland also give a natural understanding for the facts that the decay mode Z → J/ψπ + has not yetbeen seen while Z → ψ ′ π can be seen. PACS numbers: 12.39.Hg, 12.39.Pn, 12.40.Yx, 14.40.Lb, 14.40.NdKeywords: potential model; spectroscopy; heavy mesons
A series of exotic X , Y , and Z charmonium-like mesons have been discovered by the B factories, among whichthe recent discovery of Z + (4430) by the Belle Collaboration [1] draws attention of many physicists because of thefollowing reasons; Z + (4430) is charged and hence cannot be a c ¯ c charmonium state or a c ¯ cg hybrid meson and it mightbe the first charged tetraquark state because it is too heavy to be a charged Q ¯ q meson and furthermore, strangelyenough, only the decay mode Z + → ψ ′ π + is found while the mode Z + → J/ψπ + has not yet been seen. The productof branching ratios is determined to be B ( B → K ∓ Z ± (4430)) × B ( Z ± (4430) → π ± ψ ′ ) = (4 . ± . ± . × − . (1)Mass and width of this particle are measured as m Z = (4433 ± ±
2) MeV , Γ Z = (cid:18)
45 +18 + 30 − − (cid:19) MeV . (2)Looking at the above data, many people regard this particle Z + (4430) as a strong and plausible candidate of atetraquark state. According to such a point of view, the second factor of branching ratios given by Eq. (1) is in the O (1), because the decay width of Z + (4430) is 45 MeV as given by Eq. (2) and the decay Z ± (4430) → π ± ψ ′ occursthrough strong interaction, and then the first factor becomes the O (10 − ) from Eq. (1), whose suppression is due toa very small recombination probability for making four quarks ( c ¯ cq ¯ q ) into a tetraquark state. There already appearseveral papers [2]-[10] on this state whether a tetraquark or molecular state interpretation is possible or not. However,all models presented so far cannot give a convincing explanation on why the mode Z + → J/ψπ + has not been seen,though some people [4] partly account for the suppression of Z + → J/ψπ + based on the D D ∗ or D ′ D ∗ resonancemodel of Z + (4430). Furthermore, it should be noted that the decay width given by Eq. (1) is not a partial decaywidth for the process Z + → J/ψπ + but a total decay width of Z + (4430), because it is obtained from fitting theinvariant mass distribution of π + + ψ ′ to the Breit-Wigner resonance formula.In this letter, we would like to propose a different interpretation from these papers, i.e., this state can be a higherradial excitation of D s state. In this model, the first factor, B ( B → K ∓ D ± s (4430)), of the branching ratios of Eq. (1)replacing Z ± (4430) with D ± s (4430) is of the order O (1) because the numerator contains a gluon interaction as shownin Fig.1 below, while the second factor, B ( D ± s (4430) → π ± ψ ′ ), is of the order O (10 − ) because an excited D + s (4430)can decay to D K + /D + K and also to their excited particles D ( ∗ ) K ( ∗ ) via strong interactions with the decay widthof tens of MeV because of its large mass value 4430 MeV, in addition to the decay D + s (4430) → ψ ′ + π + occurring via ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]
TABLE I: First order mass spectra of the radial excited D s ( n = 2 , ,
4) in 1 /m Q . Units are in MeV. n = 2 S (0 − ) S (1 − ) P (0 + ) ” P ”(1 + ) ” P ”(1 + ) P (2 + ) D (1 − ) ” D ”(2 − )observed – 2715 2856 – – – – –calculated 2563 2755 2837 3082 3094 3157 4449 1366 n = 3observed – – – – – – – –calculated 3214 3371 3488 3792 3670 3736 3356 3413 n = 4observed – – – – – – – –calculated 3763 3870 4074 4440 4333 4411 3906 3966TABLE II: Optimal values of parameters.Parameters α n =2 , , s a (GeV − ) b (GeV) m u,d (GeV) m s (GeV) m c (GeV) m b (GeV)0.344 1.939 0.0749 0.0112 0.0929 1.032 4.639 weak interactions. Based on our semirelativistic quark potential model [11]-[16], we give not only the numerical massvalue corresponding to Z + (4430) but also give the reason why the decay mode Z + → J/ψπ + has not yet been seen.The masses of radially excited D s ( c ¯ s ) states are calculated and presented in Table I by using our semirelativisticquark potential model, which succeeds in reproducing the mass levels of all existent heavy mesons including recentlydiscovered higher states of D, D s , B and B s [13, 14]. In Table I, the same physical parameters as in Refs. [13, 14, 15]are used and given in Table II except for α n = is where i = 1 , , , and 4 denote the principal quantum number. In thispaper we have adopted the same value α is for i = 3 and 4 as that for i = 2 [14]. From Table I, one may identifyone of two states, n = 4 P (1 + ) 4440 MeV, and n = 4 P (2 + ) 4411 MeV, as a candidate for Z + (4430). Here wehave discarded the last two columns of Table I because these states, D (1 − ) and ” D (2 − )” in each row, are largelyaffected by the off-diagonal elements of the interaction Hamiltonian among these states with the quantum number S +1 L J in the larger state space [11]. Furthermore, it is natural to consider that the Z + (4430) does not have a largeorbital angular momentum but that it is rather the S state or at most the P state, even if it has large n .The parity is not a good quantum number in the decay process D s (4430) → ψ ′ π + since this process goes via a weakinteraction as shown in Fig. 3(a) and furthermore the spin of the excited D s is 1 if the particles ψ ′ and π + originatedfrom the excited D s are in the relative S state, while it can be 2, 1 or 0 if they are in the relative P state. Thus,both of n = 4 ” P ”(1 + ) and n = 4 P (2 + ) can be a possible candidate. This should be compared with the case of Z + (4430) to be a tetraquark, in which a decay of tetraquark to ψ ′ + π occurs through a strong interaction and hencethe allowed spin-parity of a tetraquark should be 1 + for the relative S state of ψ ′ and π + or 2 − , − , and 0 − for therelative P state of ψ ′ and π + because the parity is conserved in a strong interaction process.Next, if the Z + (4430) is a radially excited state of D s , how can we explain the suppression of the mode Z + → J/ψπ ?The answer is as follows. Let us consider the decay process D + s (4430) → ψ ′ π or D + s (4430) → J/ψπ as ( c ¯ s ) → ( c ¯ c ) + π by treating the π as an elementary Nambu-Goldstone boson. Then the decay amplitude is proportional to theoverlapping integral of the wave functions of ( c ¯ s ) and ( c ¯ c ) states. We notice that if the node of an initial c ¯ s statewave function is the same as that of a final c ¯ c state wave function, then the decay amplitude is expected to be large.On the other hand, if they are different, the magnitude of the decay amplitude is small. Here, we assume that theinitial state is a radially excited D s state with higher node. Then, if the final state is J/ψ whose node is zero, themagnitude of the decay amplitude is small or negligible. To see how it works, let us assume the trial wave functionsfor D s , J/ψ , and ψ ′ being expressed by the Hermite polynomials, Ψ X ( x, y, z ), asΨ X ( x, y, z ) = 1 N H m ( m X x ) H m ( m X y ) H m ( m X z ) e − m X ( x + y + z ) / , (3)where H m ( x ) is the 2 m -th Hermite polynomial with a node m and we have confined the wave function in the range0 ≤ x < ∞ , 0 ≤ y < ∞ , and 0 ≤ z < ∞ . N is a normalization and m X is mass of the particle X , which is introducedto make the arguments of the Hermite polynomials dimensionless. The ratio of decay rates of D s (4430) → J/ψπ to D s (4430) → ψ ′ π is proportional to | R d x Ψ Xm Ψ X | / | R d x Ψ Xm Ψ X | where m , 0, and 1 are nodes of the D s (4430), ¯ db ¯ us W ¯ cc W ¯ du ¯ sc ¯ B K − π + ψ ′ D s FIG. 1: Decay process through D s : ¯ B → Z + (4430) K − → ψ ′ π + K − . ¯ db s ¯ u W ¯ cc ¯ du ¯ B K − π + ψ ′ tetraquark FIG. 2: Decay process through tetraquark: ¯ B → Z + (4430) K − → ψ ′ π + K − . J/ψ , and ψ ′ , respectively. The calculated result is roughly given byΓ( D s (4430) → J/ψπ )Γ( D s (4430) → ψ ′ π ) = . × − ( n = 2 for D s )8 . × − ( n = 3)6 . × − ( n = 4) , (4)where n is the principal quantum number of the initial D s (4430) state and its node number is given by n −
1. Weknow that the node of
J/ψ is zero and that of ψ ′ is one, hence if we assume the node of D s (4430) is three with n = 4,then, the ratio of the production rate for the mode Z → J/ψπ to Z → ψ ′ π becomes 6 . × − , which is negligiblysmall.Some comments are in order for distinguishing the cases of Z = tetraquark and Z = radially excited D s . The chaindecay process ¯ B → Z + (4430) K − → ψ ′ π + K − for Z + (4430) being a radially excited D s or a tetraquark goes throughthe Feynman diagrams shown in Fig. 1 and Fig. 2, respectively. As mentioned early, the excited D s (4430) decays viastrong interactions as D s (4430) → DK ( D + s (4430) → D + k /D K + ). In addition to this mode, D s (4430) → D ( ∗ ) K ( ∗ ) are also possible where D ( ∗ ) and K ( ∗ ) are excited states of D and K , respectively, if they are kinematically allowed.These channels will be identified with D + K production accompanying one or more pions and they are also signalchannels for the Z + (4430). However, among these channels, D s (4430) → DK will be dominant because of the phasespace effect.1. Figure 1 in the case of Z = radially excited D s involves two weak bosons, hence this is the second order processin the weak interaction. On the other hand, Fig. 2 in the case of Z = tetraquark involves only one weak bosonand thus this is the first order weak interaction process. However the latter case needs to be multiplied with arecombination probability to form a tetraquark, which must be O (10 − ) as mentioned early. Thus, it is expectedthat these two cases would be in the same order of magnitude for a chain decay rate. Therefore, we cannotdistinguish a tetraquark and a radially excited D s as long as we look at the product of branching ratios alonegiven by Eq. (1).2. As one can easily notice from Fig. 2, three charged states, ± and 0, of Z (4430) must be available if Z is atetraquark state because [ c ¯ cu ¯ d ], [ c ¯ cd ¯ u ], [ c ¯ c ( u ¯ u − d ¯ d ) / √ c ¯ c ( u ¯ u + d ¯ d ) / √
2] are possible. On the other hand,if Z is a radial excitation of D s , then only the ± charged states are possible because only c ¯ s and ¯ cs are possible c ¯ s c ¯ c ¯ d uW D + s ψ ′ π + (a) c ¯ s c ¯ d ¯ c uW D + s D + ¯ D (b) c ¯ s c ¯ u ¯ s u D + s D K + (c) c ¯ s c ¯ d ¯ s d D + s D + K (d) FIG. 3: D s Decay c ¯ cu ¯ d c ¯ cu ¯ d T + ψ ′ π + (a) c ¯ du ¯ c c ¯ du ¯ c T + D + ¯ D (b) c ¯ du ¯ c c ¯ uu ¯ sW T + D K + (c) c ¯ du ¯ c c ¯ dd ¯ sW T + D + K (d) FIG. 4: Tetraquark Decay states. Namely if Z is a tetraquark state, its isospin is one and/or singlet, while if Z is a radially excited stateof D s , then its isospin is 1 / D s , let usconsider the diagrams, Figs. 3 and 4, for Z = D s (4430) and Z = T (4430) ( T means a tetraquark), respectively. Ifwe assume Z = D s (4430), then the process D + s (4430) → ψ ′ π + (Fig. 3(a)) goes through one weak boson while theprocesses D + s (4430) → D K + (Fig. 3(c)) /D + K (Fig. 3(d)) are strong decays. The process D + s (4430) → ¯ D D + (Fig. 3(b)) needs to go through one weak boson exchange, hence the following ratios are expected to be obtained.Γ (cid:0) D + s (4430) → D + K /D K + (cid:1) Γ (cid:0) D + s (4430) → ψ ′ π + (cid:1) ≫ , Γ (cid:0) D + s (4430) → ¯ D D + (cid:1) Γ (cid:0) D + s (4430) → ψ ′ π + (cid:1) ≈ . (5)On the other hand, if we assume Z = T (4430), either the process T + (4430) → D K + (Fig. 4(c)) or T + (4430) → D + K (Fig. 4(d)) goes through one weak boson exchange. The process T + (4430) → ¯ D D + (Fig. 4(b)) is nota weak interaction but a strong interaction process. Hence the following ratios between these processes and theprocess T + (4430) → ψ ′ π + (Fig. 4(a)) are expected to be obtained.Γ (cid:0) T + (4430) → D + K /D K + (cid:1) Γ ( T + (4430) → ψ ′ π + ) ≪ , Γ (cid:0) T + (4430) → ¯ D D + (cid:1) Γ ( T + (4430) → ψ ′ π + ) ≈ . (6)Therefore if one measures the process Z → DK and its ratios to Z → ψ ′ π , i.e., Eqs. (5) and (6), then one candistinguish these two possibilities whether Z is D s (4430) or T (4430).4. Since a radially excited D s decays to ψ ′ + π via weak interactions, the partial decay width Γ( D + s (4430) → ψ ′ π )would be very small, contrary to the one for Z + (4430) = T + (4430), Γ( T + (4430) → ψ ′ π ), which is via a stronginteraction decay. However, as shown in Eq. (5), D + s (4430) will dominantly decay to D + K /D K + and hencethe total decay width of D + s (4430) becomes an order of a strong decay. As described in the beginning, in theBelle experiment only the total decay width of Eq. (2) being in the order of strong interactions, is obtainedby fitting the invariant mass of ψ ′ + π to the S -wave Breit-Wigner resonance formula. Therefore, the presentexperiment does not rule out the possibility of Z + (4430) being the D + s (4430).The Belle Collaboration discovered the rather narrow resonance Z + (4430) in the invariant mass distribution of ψ ′ + π + in the decay mode ¯ B → K − ψ ′ π + . However, in the ¯ B decay there are other decay modes, ¯ B → K − K D + /K − K + D or ¯ B → K − ¯ D D + , in addition to this special mode ¯ B → K − Z + (4430) → K − ψ ′ π + . Ifthe Z + (4430) is a radially excited D s state, one can find Z + (4430) more frequently in the invariant distribution of K D + /K + D in the mode ¯ B → K − K D + /K − K + D . On the other hand, if the Z + (4430) is a tetraquark, itmust be very difficult to find it in the mode ¯ B → K − K D + /K − K + D . Therefore, this type of decay mode is veryimportant to determine whether Z + (4430) is a tetraquark or a radially excited D s .Recently the CDF Collaboration reported the observation of the decay mode B ± c → J/ψπ ± [17]. This must becomea very interesting example to test our model: in this decay mode the heavy mesons in the initial and final states haveboth zero nodes and therefore, this decay rate should be very large compared with the rate for the process B ± c → ψ ′ π ± with the node of ψ ′ to be 1, which might be strongly suppressed or not be observed.In this letter, we have discussed the possible interpretation of Z + (4430) as a radially excited D s state. We presentedthe mass levels of radially excited D s states and gave the reasonable explanation on why the mode Z → J/ψπ wasnot observed in the Belle experiment. As shown in Table I, we can see a wealth of excited D s states. We wouldlike to stress that the experimental search for resonances in DK and D ( ∗ ) K ( ∗ ) invariant mass distributions is veryinteresting not only for observing Z + (4430) but also for discovery of n = 2 and n = 3 excited D s states. Related tothis, it is remarkable that the n = 2 S (1 − ) 2715 MeV and n = 2 P (0 + ) 2856 MeV were already observed andreproduced well by our semirelativistic model, as shown in Table 1.We would like to urge the analysis on the invariant mass distribution of K D + /K + D in the decay mode ¯ B → K − K D + /K − K + D which must contain a fruitful physics of excited D s states.AcknowledgmentsWe would like to thank S. Aoki for helping us in understanding the experimental data. [1] Belle Collaboration, S.-K. Choi, et al. , Phys. Rev. Lett. (2008), 142001.[2] J.L. Rosner, Phys. Rev. D (2007), 114002.[3] L. Maiani, A.D. Plosa, and V. Riquer, arXiv:0708.3997.[4] C. Meng and K.T. Chao, arXiv:0708.4222.[5] D.V. Bugg, arXiv:0709.1254.[6] S. S. Gershtein, A. K. Likhoded, and G. P. Pronko, arXiv:0709.2058.[7] C. F. Qiao, arXiv:0709.4066. [8] S. H. Lee, A. Mihara, F. S. Navarra, and M. Nielsen, arXiv:0710.1029.[9] X. Liu, Y.-R. Liu, W.-Z. Deng, and S.-L. Zhu, Phys. Rev. D (2008), 034003.[10] Y. Li, C.-D. Lu, and W. Wang, Phys. Rev. D (2008), 054001.[11] T. Matsuki and T. Morii, Phys. Rev. D (1997), 5646. The first order calculations in this paper are enough to compareour results with the experiments, which predicted D sJ observed by BaBar and CLEO in 2003.[12] T. Matsuki, K. Mawatari, T. Morii, and K. Sudoh, Phys. Lett. B (2005), 329, hep-ph/0411034.[13] T. Matsuki, T. Morii, and K. Sudoh, Prog. Theor. Phys. (2007), 1077, hep-ph/0605019.[14] T. Matsuki, T. Morii, and K. Sudoh, Eur. Phys. J. A (2007), 701, hep-ph/0610186.[15] T. Matsuki, T. Morii, and K. Sudoh, Phys. Lett. B (2007), 593, hep-ph/0712.1288.[16] T. Matsuki and K. Seo, Prog. Theor. Phys. (2007), 1087, hep-ph/0703158.[17] CDF Collaboration, A. Abulencia et al. , Phys. Rev. Lett. (2006), 082002; T. Aaltonen et al. , Phys. Rev. Lett.100