Partners of Z(4430) and Productions in B Decays
aa r X i v : . [ h e p - ph ] M a r Partners of Z (4430) and Productions in B Decays
Ying Li a , Cai-Dian L¨u b and Wei Wang b,c a Physics Department, Yantai University, Yantai, 264005, P.R. China b Institute of High Energy Physics, P.O. Box 918(4) Beijing, 100049, P.R. China c Graduate University of Chinese Academy of Sciences, Beijing, 100049, P.R. China
Recently, Belle Collaboration has reported a resonant state produced in B → Kπψ ′ ,which is called Z (4430). This state is charged, so it can not be interpreted as an ordinarycharmonium state. In this paper, we analyze the octet to which this particle belongs andpredict the masses of mesons in this octet. Utilizing flavor SU(3) symmetry, we studyproduction rates in several kinds of B decays. The ¯ B → Z − s π + → K − ψ ′ π + and B − → ¯ Z s π − → K S ψ ′ π − decay channels, favored by Cabibbo-Kobayashi-Maskawa matrix elements,can have branching ratios of O (10 − ). This large branching ratio could be observed at therunning B factories to detect Z s particles containing a strange quark. We also predictlarge branching ratios of the Z and Z c (¯ cc ¯ cD, D = u, d, s ) particle production rates in non-leptonic B c decays and radiative B decays. Measurements of these decays at the ongoing B factories and the forthcoming Large Hadron Collider-b experiments are helpful to clarify themysterious Z particles. I. INTRODUCTION
Recently, there are many exciting discoveries on new hadron states especially in the hidden-charm sector. Among these discoveries, the most intriguing one is the new relatively narrow peaknamed Z (4430) found by Belle Collaboration in the invariant mass spectrum of πψ ′ in the decaymode B → Kπψ ′ [1]. There is a large branching fraction for the following decay chain: BR ( ¯ B → K − Z + (4430)) × BR ( Z + (4430) → π + ψ ′ ) = (4 . ± . ± . × − . (1)Mass and width of this particle are measured as: m Z = (4433 ± ± Z = (44 +17+30 − − )MeV . (2)The most prominent characteristic is that it is electric charged, that’s to say, this new particle cannot be described as an ordinary charmonium state or a charmonium-like state such as ¯ ccg . Onthe other hand, this particle can decay to π + ψ ′ with a large rate through strong interactions, soit involves at least four quarks c ¯ cu ¯ d , though there is not any further detailed information on itsinner dynamics at present.In order to elucidate this particle, many theoretical studies [2, 3, 4, 5, 6, 7, 8] have been putforward. This meson could be viewed as a genuine tetraquark state with diquark anti-diquark[ cu ][¯ c ¯ d ] content which has a large rate to πψ ′ [3]. Moreover based on QCD-string, two differentfour-quark descriptions are proposed in Ref. [6]: one can be reduced to the ordinary diquark-diquarkpicture and the other one can not. Besides this kind of explanation, it has also been identified asthe resonance of D ( D ′ ) D ∗ [2, 4] as its mass is close to the thresholds of D ∗ (2010) D (2420) and D ∗ (2010) D ′ (2430). Within this picture, the authors in Ref. [4] explored the production of πψ and πψ ′ . Short distance contribution to Z → πψ ( ψ ′ ) is neglected and the main contribution isfrom long distance re-scattering effect via D ∗ D . With proper parameters, they can successfullyexplain the much larger production rate of πψ ′ than that of πψ . In Ref. [5], Bugg took this mesonas a D ∗ (2010) ¯ D (2420) threshold cusp. Recently, Qiao also tried to explain this meson with thebaryonium picture [7]. Using the technique of QCD sum rules, Lee et.al calculated the masses ofthis particle and its strange partner Z s in Ref. [8].Whether or not these scenarios describe the true dynamics of Z (4430), this strange meson indeedplays an important role in the charmonium spectroscopy. In the present paper, we do not intendto give an explanation of this meson’s structure, but we want to analyze its partners within SU(3)symmetry: the octet to which the meson Z (4430) belongs and the corresponding singlet meson.Up to now, these is no experimental information on these mesons except Z ± . The decays of B meson provide a firm potential in searching for these exotic mesons [9, 10], just like the observeddecay channel B → KZ → Kπψ ′ . We will investigate the possibilities to detect these Z mesonsin B q ( q = u, d, s, c ) decays. In doing this, we will analyze decay amplitudes with the assumptionof SU(3) flavor symmetry: to construct effective Hamiltonian using flavor SU(3) meson matrixes.The decay amplitudes can also be studied by using Feynmann diagrams. In the discussion of Z production with the graphic technique, we only consider short-distance contributions and neglectsoft final state interactions. Specifically, the considered decays are divided into three categories:Cabibbo-Kobayashi-Maskawa (CKM) allowed non-leptonic decays; CKM suppressed non-leptonicdecays; radiative decays. The first kind of decays have similar branching ratios with the observed¯ B → K − π + ψ ′ , while the second type of decays is suppressed by about one order in magnitudeand we will show that the running B factories could hardly detect this kind of decays. Radiative B decay is a natural filter to exclude the 0-spin mesons and furthermore this kind of process maygo through with a sizable branching ratio.In the next section, we will analyze the octet of Z meson within flavor SU(3) symmetry and tryto estimate their masses. We will construct the effective Hamiltonian using meson matrices andthen use them to study the production rates of Z mesons in B decays. In Sec.III, we will introduce Z c meson which consists of three charm quarks, together with a brief discussion on its productionin B c decays. We will summarize this note in the last section. II. THE OCTET AND THE SINGLET
Just as stated above, Z (4430) involves at least four quarks in constituent quark model, andthere is an octet which Z (4430) belongs to in flavor SU(3) symmetry. Generally, we can deducethe particles in this octet using group theory: these particles, under the name Z ± , Z , Z ± s , Z s , Z s and Z , are shown in Fig. 1. Besides, there exists one singlet meson called Z . In constituentquark model, quark contents of these mesons are listed by: Z + = c ¯ cu ¯ d ; Z = 1 √ c ¯ c ( u ¯ u − d ¯ d ); Z − = c ¯ cd ¯ u ; Z + s = c ¯ cu ¯ s ; Z − s = c ¯ cs ¯ u ; Z s = c ¯ cd ¯ s ; Z s = c ¯ cs ¯ d ; Z = 1 √ c ¯ c ( u ¯ u + d ¯ d − s ¯ s ); Z = 1 √ c ¯ c ( u ¯ u + d ¯ d + s ¯ s ) . (3)In reality, s quark is slightly heavier than u, d quark which is one of the origins for SU(3) symmetrybreaking. Accordingly, the singlet Z can mix with eighth component of the octet Z , in analogywith η and η ′ . Physical particles, named Z α and Z β , are mixtures of them and can be expressedas: Z α Z β = cos θ sin θ − sin θ cos θ Z Z . (4)The mixing angle θ can be determined through measuring decays of these two particles in future.For simplicity, we will assume the mixing is ideal, i.e. θ = 54 . ◦ . In this case, the quark contentsare: Z α = 1 √ c ¯ c ( u ¯ u + d ¯ d ) , Z β = c ¯ cs ¯ s. (5)All together, one can use the following meson matrix to describe these mesons: Z = Z √ + Z √ Z + Z + s Z − − Z √ + Z √ Z s Z − s ¯ Z s − q Z + Z √ . (6) Z − Z + Z Z s ¯ Z s Z Z − s Z + s FIG. 1: Weight diagram for Z meson octet. With the quark contents given in the above, we are ready to estimate masses of these parti-cles. Isospin analysis predicts the equal masses for the four mesons with neither open nor hiddenstrangeness: Z ± , Z , Z α . For the mesons with a strange quark, the mass differences between thelighter u, d quarks and the heavier s quark are required. One can compare masses of D ∗ and D ∗ s to get some information: the mass of D ∗ s is 100 MeV larger than that of D ∗ . In heavy quarklimit m c → ∞ , the light system will not be affected by different heavy quark systems, thus we cansimply assume a similar difference for Z mesons which predicts the mass of Z s around 4533 MeV.Because the mass of newly observed Z meson is not far from the threshold of D ∗ (2010) D (2420), Z − meson is regarded as the resonance of D ∗ D (2420) [2]. Under this mechanism, we could givemore precise predictions on the masses for other Z mesons using experimental results for the D ∗ and D mesons. Our results are listed in Tab. I and uncertainties in this table are from that ofmasses of the charmed mesons. In the heavy quark limit, mesons with the same light system can berelated to each other. But if the Z particles are viewed as tetra-quark states, the effective strangequark mass in Z could be different from that in the usual mesons as the light systems in the twokinds of particles are different. If Z mesons are described as molecules, probably they would notbelong to a full SU(3) nonet and the predicted masses may not be suitable. Currently, there is nobetter solution and we will use this assumption in the present study. The recent QCD sum rulestudy predicts the mass by [8]: m Z s = (4 . ± . , (7)which is above the D ∗ s D and D ∗ D s threshold by about 160 MeV. More experimental studies arerequired to test this description.Experimentalists have observed the Z particle through the B → ZK with Z → πψ ′ . AssumingS-wave decay for Z meson, the quantum numbers can be determined as J P C = 1 + − [3]. In order TABLE I: Z meson and its massMeson Constituent Meson Mass(MeV) Decay Mode Z + , Z − , Z , Z α D ∗ (2010) ¯ D (2420) 4432 . ± . ψ ′ π/η ( η ′ ) , η c (2 S ) ρ/ωZ + s , Z − s , Z s , ¯ Z s D ∗ s (2112) ¯ D (2420) /D ∗ (2010) ¯ D s (2536) 4534 . ± . / . ± . ψ ′ K, η c (2 S ) K ∗ Z β D ∗ s (2112) ¯ D s (2536) 4647 . ± . ψ ′ η ( η ′ ) , η c (2 S ) φ to detect the other Z mesons, experimentalists will choose the proper final states to re-constructthem, thus the predictions on Z ’s strong decays are required. Using the flavor SU(3) symmetryand Z → πψ ′ , we also list the strong decays of other Z mesons in Tab. I. With the assumption J P C = 1 + − , another kind of possible decay modes is Z → η c (2 S ) V [3], where V denotes a lightvector meson.In order to explore the production in B decays, one can construct the effective Hamiltonianat hadron level using meson matrices [11]. In the following, to construct the related effectiveHamiltonian, we will assume the flavor SU(3) symmetry. In B u,d,s decays, the initial state B =( B − , ¯ B , ¯ B s ) forms an SU(3) anti-triplet. The transition at quark level is either b → c ¯ cs or b → c ¯ cd , which is described by the effective electro-weak Hamiltonian: H = G F √ V cb V ∗ cD (cid:2) C (¯ c α b β ) V − A ( ¯ D β c α ) V − A + C (¯ c α b α ) V − A ( ¯ D β c β ) V − A i + H.c. , (8)where D = d, s . α and β are color indices. The transition b → c ¯ cs is CKM favored: V cb V ∗ cs ∼ b → c ¯ cd transition is suppressed by | V ∗ cd /V ∗ cs | = λ = 0 .
23. To construct the effectiveHamiltonian at hadron level, only the flavor structures needs to be concerned. The effectiveelectro-weak Hamiltonian given in Eq. (8) can also be written as an SU(3) triplet: H i (i=1 (u),2 (d), 3(s)), where the only non-zero elements are H = 1 for CKM favored decays b → c ¯ cs , and H = 1 for CKM suppressed channels b → c ¯ cd . The final state mesons can be described by twononet matrices: Z and M . The effective Hamiltonian at hadron level could be constructed as: H = A B i H i Z kl M lk + B B i H j Z il M lj + C B i H j Z kj M ik + D B i H j Z ij M ll + E B i H j Z ll M ij , (9)where the upper index labels rows and the lower labels columns.The above effective Hamiltonian can be related to Feynmann diagrams with the one-to-onecorrespondence and the lowest order diagrams are given in Fig. 2. The second term in eq.(9) If the ¯ cc quark pair is generated from the QCD vacuum rather than directly produced by the four-quark operator,this kind of contribution is expected to be suppressed by α s (2 m c ) since there is at least one hard gluon requiredto produce the ¯ cc quark pair. b bZ Z Z b Z bb Z FIG. 2: Typical Feynman diagrams: annihilation (first row), emission (second row) and gluonic diagrams(third row) of Z meson production in B decays. corresponds to the second diagram in Fig. 2 (called Z -recoiling diagram) in which the spectatorlight quark in B meson enters into the heavy Z meson. If the spectator quark goes to the lightmeson, we call this kind of diagram (the third one in Fig. 2) as the Z -emission diagram whichcorresponds to the third term in the effective Hamiltonian. In order to estimate relative sizes ofthese terms, we have to analyze diagrams at quark level. Final state mesons move very slowly andthus the gluon generating the q ¯ q quark pair is soft: α s ∼ O (1). Thus after integrating out highenergy scales, decay amplitudes can be expressed as matrix elements of a soft four-quark operatorbetween initial and final states. The first term in Eq. (9) corresponds to the annihilation diagram(the first one in Fig. 2), as flavor indices of B and H in this term are contracted with each other.This kind of diagram is expected to be suppressed in two-body non-leptonic B decays. But heresince the gluons are soft, decay amplitudes can also be expressed as time-ordered products of a softfour-quark operator and the O (1) interaction Hamiltonian which contains only soft fields, thus thiskind of contribution is comparable with contributions from the second and third terms in Eq. (9).For SU(3) flavor singlet mesons η and Z , there are additional contributions which are given bythe last two terms in Eq. (9). One kind of typical Feynmann diagram is also shown in Fig. 2 asthe last two diagrams and it is the contribution from the higher Fock states of η and Z . Even incharmless two-body B → K ( π ) η ( η ′ ) decays [12], this kind of gluonic contribution is sizable. Herewe do not have any implication and thus one can not neglect it with any a priori. TABLE II: SU(3) decomposition of ∆ S = 1 B u,d,s decays, whose decay amplitudes are proportional to V cb V ∗ cs Mode A B C D E B − → Z K − / √ B − → Z − ¯ K B − → Z − s π / √ B − → ¯ Z s π − B − → Z K − / √ − p / B − → Z K − / √ / √ √ B − → Z − s η − p / / √ B − → Z − s η / √ / √ √ B → Z + K − B → Z ¯ K − / √ B → Z − s π + B → ¯ Z s π − / √ B → Z ¯ K / √ − p / B → Z ¯ K / √ / √ √ B → ¯ Z s η − p / / √ B → ¯ Z s η / √ / √ √ B s → Z + s K − B s → Z s ¯ K B s → Z − s K + B s → ¯ Z s K B s → Z + π − B s → Z − π + B s → Z π B s → Z η −√ / −√ / −√ B s → Z η −√ / −√ / −√ B s → Z η / / B s → Z η / / With the effective Hamiltonian given in Eq. (9), we give the decay amplitudes for the first kindof non-leptonic B u,d,s decay channels in Table II. These decays are induced by the CKM allowedtransition b → c ¯ cs and go through with a large decay rate (typically the same order with theobserved ¯ B → K − π + ψ ′ ). The flavor SU(3) symmetry implies the following relations for b → c ¯ cs decays:2 BR ( B − → Z K − ) = BR ( B − → Z − ¯ K ) = BR ( ¯ B → Z + K − ) = 2 BR ( ¯ B → Z ¯ K ) , (10)2 BR ( B − → Z − s π ) = BR ( B − → ¯ Z s π − ) = BR ( ¯ B → Z − s π + ) = 2 BR ( ¯ B → ¯ Z s π ) , (11) BR ( B − → Z α K − ) = BR ( ¯ B → Z α ¯ K ) , BR ( B − → Z β K − ) = BR ( ¯ B → Z β ¯ K ) , (12) BR ( B − → Z − s η ) = BR ( ¯ B → ¯ Z s η ) , (13) BR ( ¯ B s → Z + s K − ) = BR ( ¯ B s → Z s ¯ K ) , BR ( ¯ B s → Z − s K + ) = BR ( ¯ B s → ¯ Z s K ) , (14) BR ( ¯ B s → Z + π − ) = BR ( ¯ B s → Z − π + ) = BR ( ¯ B s → Z π ) , (15)where mass differences and lifetime differences of B mesons are neglected which can not producelarge corrections. Although all of these decays are expected to go through with large branchingfractions, decay rates may differ from each other for distinct coefficients. Two of the decays in thefirst line have been observed experimentally, while the possibility to observe the other two channelsis a little smaller as the daughter meson π from Z is relatively more difficult to measure. Thedecays in the second line is contributed from the third term of effective Hamiltonian given in Eq.(9), which should also have similar production rates. Among these four channels, ¯ B → Z − s π + and B − → ¯ Z s π − can have large branching ratios and the final states ( K − ψ ′ π + or K S ψ ′ π + ) areeasily to be measured on the experimental side. Thus measurements of the Kψ ′ invariant massdistribution in these two channels are helpful to detect the Z s particles and determine relative sizesof B and C . The other B decays are less possible to be measured in the running B factories aseither π or η is produced in the final state. The forthcoming LHC-b experiments and Super-Bfactories can measure these decays, together with the ¯ B s decays.For ¯ B → K − Z + , the heavy b quark decays into c ¯ cs , and q ¯ q is produced from vacuum. Sub-sequently, c ¯ c , q and the spectator ¯ u can be transferred into Z , and the quarks left form a kaon.The other Z states can also be produced by selecting a different quark pair q ¯ q or changing the s quark by d quark. We give the decay amplitudes for non-leptonic B u,d,s decay channels induced by b → c ¯ cd transition in Table III. These decays are suppressed by CKM matrix elements | V cd /V cs | . B − → Z − s K is one example of this kind of decays and the product branching ratio is: BR [ B − → Z − s K ] × BR [ Z − s → K − ψ ′ ]= | V cd V cs | × BR [ ¯ B → Z + (4430) K − ] × BR [ Z + (4430) → π + ψ ′ ]= (2 . ± . ± . × − , (16)where | V cd | = 0 .
23 and | V cs | = 0 .
957 [13]. The uncertainties are from the experimental results for BR [ ¯ B → Z + (4430) K − ] × BR [ Z + (4430) → π + ψ ′ ]. In the above calculation, mass differences and TABLE III: SU(3) decomposition of ∆ S = 0 B u,d,s decays, whose decay amplitudes are proportional to V cb V ∗ cd . Mode A B C D E B − → Z π − / √ − / √ B − → Z − π − / √ / √ B − → Z − s K B − → Z s K − B − → Z π − / √ / √ B − → Z π − / √ / √ √ B − → Z − η / √ / √ B − → Z − η / √ / √ √ B → Z + π − B → Z − π + B → Z π / / B → Z + s K − B → Z s ¯ K B → Z − s K + B → ¯ Z s K B → Z π − / √ − / √ B → Z π − / √ − / √ − p / B → Z η − / √ − / √ B → Z η − / √ − / √ − p / B → Z η / / B → Z η / √ / √ / √ B → Z η / √ / √ / √ B → Z η / / B s → Z + s π − B s → Z s π − / √ B s → Z − K + B s → Z K − / √ B s → Z K − p / / √ B s → Z K / √ / √ √ B s → Z s η / √ − p / B s → Z s η / √ / √ √ K S is moredifficult than K − , thus it could hardly be measured at the present two B factories. The relationsfor the b → c ¯ cd decay can be derived similarly using the effective Hamiltonian which are also usefulin searching for the Z mesons: BR ( B − → Z − s K ) = BR ( ¯ B s → Z + s π − ) = 2 BR ( ¯ B s → Z s π ) , (17) BR ( B − → Z s K − ) = BR ( ¯ B s → Z − K + ) = 2 BR ( ¯ B s → Z K ) , (18) BR ( B − → Z − π ) = BR ( B − → Z π − ) , (19) BR ( ¯ B → Z + π − ) = BR ( ¯ B → ¯ Z s K ) , BR ( ¯ B → Z − π + ) = BR ( ¯ B → Z s ¯ K ) , (20) BR ( B − → Z α,β π − ) = 2 BR ( ¯ B → ¯ Z α,β π ) , BR ( B − → Z − η ( η ′ )) = 2 BR ( ¯ B → Z η ( η ′ )) . (21) ZZb ¯ c ¯ cb Z ¯ cb Z b ¯ c FIG. 3: Leading order Feynman diagrams of Z meson production in B c decays. As pointed out in ref.[14], the study of charmonium like states production in B c decays is easier.Here we also consider the Z(4430) particle production in B c decays. In this case, the spectator isa ¯ c quark, thus the initial state is very simple: a singlet of flavor SU(3) group. But the effectiveelectro-weak Hamiltonian can form an octet: 3 ⊗ ¯3 = 8 ⊕
1. The effective Hamiltonian at hadronlevel can be written by: H = A BH ij Z jl M li + B BH ij Z ki M jk + C BH ij M ji Z kk + D BH ij Z ji M ll , (22)where the non-zero elements of the transition Hamiltonian are H = 1 for CKM allowed channels b → c ¯ ud and H = 1 for CKM suppressed channels b → c ¯ ud with a factor V ∗ us /V ∗ ud . The correspond-ing Feynmann diagrams are given in Fig. 3. The coefficients for distinct contributions are given inTab. IV. The CKM matrix element for the decay channels induced by b → c ¯ ud is V cb V ∗ ud , which1 TABLE IV: SU(3) decomposition of B c induced by b → c ¯ ud (the first part) and b → c ¯ us transitions (thesecond part) Mode A B C D B − c → Z s K − B − c → Z − s K B − c → Z π − / √ − / √ B − c → Z − π − / √ / √ B − c → Z π − / √ / √ B − c → Z π − / √ / √ √ B − c → Z − η / √ / √ B − c → Z − η / √ / √ √ A B C D B − c → Z K − / √ B − c → Z − ¯ K B − c → ¯ Z s π − B − c → Z − s π / √ B − c → Z K − / √ − p / B − c → Z K − / √ / √ √ B − c → Z − s η − p / / √ B − c → Z − s η / √ / √ √ is in the same order with that of B → KZ (4430): V cb V ∗ cs . Thus without any other suppressions,these B c decays also have similar branching ratios ( O (10 − )) with ¯ B → K − Z + → K − π + ψ ′ . Thedecays in the second part of Tab. IV are suppressed by ( V ∗ us /V ∗ ud ) , which are expected to havesmaller decay rates ( O (10 − )). Furthermore, the SU(3) symmetry implies the following relations: BR ( ¯ B − c → Z π − ) = BR ( ¯ B − c → Z − π ) , (23)2 BR ( ¯ B − c → Z K − ) = BR ( ¯ B − c → Z − ¯ K ) , (24)2 BR ( ¯ B − c → Z − s π ) = BR ( ¯ B − c → ¯ Z s π − ) . (25)Besides non-leptonic B decays, Z particles can also be produced in radiative decays, as Fig. 4shows. Two-body radiative decays can serve as a natural filter to exclude spin-0 candidates of Zparticles, as the photon can only be transversely polarized. But in order to predict the productionrates, one has to know B → Z transition form factors. In the second diagram of Fig. 4, in orderto generate the ¯ cc , we require at least one hard gluon which will suppress the contribution from2 Zb b ¯ ccB Z Bγ γ FIG. 4: Z meson production via radiative decays. The right diagram gives a smaller contribution as at leastone hard gluon is required. this diagram. Na¨ıvely thinking, the first diagram will also be suppressed by the off-shell c quarkpropagator, but since the emitted photon is not energetic (the total energy release is only about0 . B decays could gothrough with considerable rates: B − → Z − s γ, ¯ B → ¯ Z s γ, ¯ B s → Z ( Z ) γ, ¯ B c → Z − γ. (26) III. Z c PARTICLE
In the above, we have utilized the flavor SU(3) symmetry for light quarks in Z mesons. One canalso replace one or two heavy c quarks by heavier b quarks which can predict Z b and Z bb mesons[15]. Another attempt is to replace a light quark by a heavy c quark. With this replacement, weobtain three Z c states which contain three heavy quarks and a light quark: ¯ cc ¯ cu , ¯ cc ¯ cd and ¯ cc ¯ cs ,together with their charge conjugates. We have to confess that this replacement may change theinternal dynamics, but here we assume the same dynamics with Z . In this case, the masses can beobtained by using mass differences of c and u, d, s quarks deriving from masses of ψ and D ∗ ( D ∗ s ).The rough predictions for masses are around 5520 and 5630 MeV. If these mesons are viewed as theresonance of ψ and D ( D s ), their masses could be predicted as (5519 . ± .
3) MeV, (5519 . ± . . ± .
6) MeV.Because of the large mass of Z c , these particles only appear in B c meson decays. The corre-sponding effective Hamiltonian responsible for non-leptonic decays is H = A BH i Z i M kk + B BH i Z j M ji , (27)where the first contribution A comes from the gluonic diagram shown as the first one in Figure 5;while the second term B comes from the Z -recoiling diagram shown as the second one in Figure 5.3 b Z c ¯ c ¯ cc b Z c ¯ c ¯ ccs ( d ) s ( d ) FIG. 5: Leading order Feynman diagrams of Z c meson production in non-leptonic B c decays. These two contributions give the following amplitudes for 8 decays: A ( ¯ B c → Z (¯ cc ¯ cu ) K − ) = A ( ¯ B c → Z (¯ cc ¯ cd ) ¯ K ) = B , (28) A ( ¯ B c → Z (¯ cc ¯ cs ) η ) = − r B (29) A ( ¯ B c → Z (¯ cc ¯ cs ) η ) = √ A + 1 √ B , (30) A ( ¯ B c → Z (¯ cc ¯ cu ) π − ) = −√ A ( ¯ B c → Z (¯ cc ¯ cd ) π ) = A ( ¯ B c → Z (¯ cc ¯ cs ) K ) = B , (31) A ( ¯ B c → Z (¯ cc ¯ cd ) η ) = 1 √ B (32) A ( ¯ B c → Z (¯ cc ¯ cd ) η ) = √ A + 1 √ B . (33)The first 4 decay channels shown in eq.(28-30) are induced by b → c ¯ cs at quark level and thus havebranching ratios of order O (10 − ), while the other decays shown in eq.(31-33) are suppressed by | V ∗ cd /V ∗ cs | , which have smaller decay rates ( O (10 − )). IV. SUMMARY
Belle Collaboration has reported a resonance named Z (4430), which consists of at least fourquarks in constituent quark model. In this note, we analyze the octet to which this Z meson belongsand the corresponding singlet meson. Using the picture that the Z mesons are the resonances of D ∗ and D mesons, we estimate the masses of these mesons. We investigate the production in non-leptonic B u,d,s decays by constructing the effective Hamiltonian using flavor SU(3) meson matrices.The transition at quark level is either b → c ¯ cs or b → c ¯ cd , where the former one is CKM favored andthe latter is suppressed by | V ∗ cd /V ∗ cs | . Thus the considered non-leptonic decays have either similarbranching ratios with the observed decay B ± → Z ± ¯ K (10 − ) or smaller branching ratios (10 − )as shown in the text. Utilizing the SU(3) symmetry, we also obtain many relations for variousdecay channels. Measurements of the Kψ ′ invariant mass distribution in ¯ B → Z − s π + → K − ψ ′ π + and B − → ¯ Z s π − → K S ψ ′ π − are helpful to detect the Z s particles and determine the relative size4of B and C . We also study the production rates in non-leptonic B c decays and radiative B decaysin a similar way. Replacing a light u, d, s quark by a heavy c quark, we get three states which havethree heavy quarks. The masses and the production in B c decays are also discussed. Measurementsof all these decays at the present B factories and the forthcoming LHC-b experiments will help usto clarify the new Z particles. Acknowledgements
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