SU(3) analysis of fully-light tetraquarks in heavy meson weak decays

We perform a SU(3) analysis for both semi-leptonic and non-leptonic heavy meson weak decays into a pseudoscalar meson and a fully-light tetraquark in 10 or 27 representation. A reduction of the SU(3) representation tensor for the fully-light tetraquarks is produced and all the flavor components for each representation tensor are listed. The decay channels we analysis include $B/D \to U/T~P~l\nu$, $B/D \to U/T~P $ and $B_c \to U/T~P/D$, with $U/T$ represents a fully-light tetraquark in 10 or 27 representation and $P$ is a pseudoscalar meson. Finally, among these results we list all the golden decay channels which are expected to have more possibilities to be observed in experiments.

with its quarks and antiquarks clustering into diquark-anti-diquark pairs is called a tetraquark which is combined by the color force, while the one with meson-molecule structure is combined by the electroweak force. In this paper, we will concentrate on the production of light four-quark states in the two-body weak decays of heavy mesons B, B c and D. Due to the rich yield of heavy mesons in the Heavy Flavor Factories, a considerable production of light four-quark states in these decay channels is expected.
Recently, most of the theoretical studied on the exotic states are relied on effective theories and models, for example the studies of mass spectrum within the simple quark model [27], Iachello mass formula [28] and QCD sum rules [29]. However, the light quark flavor SU(3) symmetry is also a useful tool to analyze the decays of hadrons, which has been successfully applied for the ordinary meson or baryon case [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. One advantage of SU(3) analysis is that it is irrelevant with the details of the hadron structure, particularly whether the four-quark state is a bounded by diquark and anti-diquark or is a meson-molecule, as well as its explicit quantum number J P C . For this reason we will simply call the four-quark state in this work tetraquark. In this paper, we will choose the SU(3) symmetry analysis to study the production of fully-light tetraquarks in the both semileptonic and non-leptonic B, B c and D decays. According to the reduction of SU(3) representation, a fully-light tetraquark can belong to a 27 representation, a 10 representation, a 10 representation, four 8 representations or two singlets. In this work, we will focus on the fully-light tetraquark in 27 or 10 representation, and study their production in the decays B/D → U/T P lν, B/D → U/T P and B c → U/T P/D, with U/T represents the 10 or 27 states and P is a pseudoscalar meson.
According to the SU(3) symmetry the relations among these decay channels can be obtained, which can be examined by the future experiments. This analysis is helpful to identify the decay modes that will be mostly useful to discover the fully-light tetraquark states.
The rest of this paper is organized as follows. In Sec. II, we give the representation of fullylight tetraquarks under the SU(3) symmetry. In Sec III we give the SU (3)

II. SU(3) IRREDUCIBLE REPRESENTATION OF FULLY-LIGHT TETRAQUARKS
In general, according to the SU(3) flavor symmetry, the fully-light tetraquarks q 1 q 2q3q4 are described by the inner-product representation 3 ⊗ 3 ⊗3 ⊗3, which can be further reduced into 9 irreducible representaions 3 ⊗ 3 ⊗3 ⊗3 = 27 ⊕ 10 ⊕ 10 ⊕ 8 ⊕ 8 ⊕ 8 ⊕ 8 ⊕ 1 ⊕ 1. (1) Explicitly, in the language of tensor reduction, an fully-light tetraquark can be represented by a rank (2,2) tensor H ij kl , and the reduction reads as where A ijn klm = δ i k δ j m δ n l + δ j k δ i m δ n l + δ i l δ j m δ n k + δ j l δ i m δ n k , B ijn klm = δ i k δ j m δ n l + δ j k δ i m δ n l − δ i l δ j m δ n k − δ j l δ i m δ n k , C ijn klm = δ i k δ j m δ n l − δ j k δ i m δ n l + δ i l δ j m δ n k − δ j l δ i m δ n k .
Here the coefficient of each term in Eq. (2) can always be rescaled by redefining the corresponding irreducible representation tensors. Explicitly, these tensors read as By writing down all the components of these tensors, one can find out the flavor structure of each tetraquark. For example, a tetraquark with flavor ssūd belongs to the 10 or 27 representation.
The corresponding components are (T 10 ) 333 = ssūd − ssdū , (T 27 ) 33 12 = Note that in the (T 10 ) 333 , the u, d quarks are anti-symmetric in the flavor space, which means that without angular momentum, they form a spin-0 structure. On the other hand, in the (T 27 ) 33 12 u, d are symmetric and thus form a spin-1 structure. All the components of the tensors in 27, 10 For the 27 representation, the notation of all the independent components are listed in Table I, where the (T 27 ) i3 k3 components are absent because they are related with other components as The electroweak effective Hamiltonian for semi-leptonic b or c decays in the SM is where q = d, s and q ′ = u. In the SU(3) representation the H b eff corresponds to a triplet operator H 3 with the components as (H 3 ) 1 = V ub and (H 3 ) 2,3 = 0. H c eff also corresponds to a triplet operator

B. Effective Hamiltonian for non-leptonic b decays
The electroweak effective Hamiltonian for non-leptonic b decays in the SM is [50][51][52]: where O i is the four-quark operator and C i is its Wilson coefficient. The explicit forms of the O i s read as where the highly suppressed penguin contributions have been neglected, and while the operators containing other light flavors can be obtained by replacing the d, s quark fields.
Similar to the b decays, the tree operators of c decays transform under the flavor SU(3) symmetry For the Cabibbo allowed c → sud transition, the amplitudes are proportional to V cs V * ud and the decay operators are For the doubly Cabbibo suppressed c → dus transition, the amplitudes are proportional to V cd V * us and the decay operators are For the singly Cabbibo suppressed decays proportional to V cs V * us , we have and For the singly Cabbibo suppressed decays proportional to V cd V * ud , we have Note that since V cd V * ud = −V cs V * us −V cb V * ub ≈ −V cs V * us (with 10 −3 deviation), the contributions from the3 representation vanish, and the nonzero components are only from 6 and 15 representations.

D. Hadron Multiplets
In this subsection we display the SU (3) representation of the heavy mesons B, D, B c and pseudoscalar P . The B c meson contains no light quark and it is a singlet. The heavy mesons containing one heavy quark are flavor SU(3) anti-triplets: The light pseudoscalar P mesons are mixture of octets and singlets, and its representation contains nine hadrons where η 8 and η 1 η and η ′ into η and η ′ with the mixing angle θ η = cos θη 8 + sin θη 1 , η ′ = − sin θη 8 + cos θη 1 .
Note that η 8 and η 1 are not physical states, practically one can choose a basis η q and η s for the mixing, which read as [46] with where The Feynman diagram corresponding to these two terms is shown in Fig. 2. The decay amplitudes for B → T 10 P lν and B → T 27 P lν are listed in Table II and Table III respectively.
The effective Hamiltonian for D → T 10 /T 27 P lν is where D [ij] = ǫ ijk D k . The corresponding Feynman diagram has exactly the same topology as that of B → T 10 /T 27 P lν. The decay amplitudes are listed in Table IV and Table V respectively.
We then consider the two-body non-leptonic decay B → T 10 P and B → T 27 P . The effective Hamiltonian for B → T 10 P is The effective Hamiltonian for B → T 27 P is For simplicity we have used the same set of notations a i , b i , c i · · · both for the B → T 10 P and B → T 27 P cases. However, It should be kept in mind that these two sets are in fact independent.
for B → T 10 P decay and for B → T 27 P decay.
The decay amplitudes for b → d and b → s transitions with a 10 representation tetraquark produced in the final state are listed in Table VI and Table VII respectively. Note that these amplitudes are not all independent, and some of them are proportional to each other. For the b → d transitions, the decay widths for these channels are related as For the b → s transitions, the decay widths for these channels are related as −a 3 +a 6 +b 6 +b 15 +2c 15 +e 15 The decay amplitudes for b → d and b → s transitions with a 27 representation tetraquark produced in the final state are listed in Table VIII and Table IX respectively. These decay channels are also not totally independent. The decay widths of b → d transitions are related as −a 3 +a 6 +b 6 +b 15 +2c 15 +e 15 The decay widths of b → s transitions are related as C. Non-leptonic D decays Next we consider the two-body non-leptonic decay D → T 10 P and D → T 27 P with the final state containing an fully-light tetraquark in 10 or 27 representation. The effective Hamiltonian for Fig. 4 shows the Feynman diagrams corresponding to these Hamiltonians. The correspondence between each effective Hamiltonian above and the Feynman diagrams are: for D → T 10 P decay and for D → T 27 P decay.
The corresponding decay channels are classified by the Cabbibo suppression. For the Cabbibo allowed, singly Cabbibo suppresed and doubly Cabbibo suppressed D → T 10 P results, they are listed in Table X, Table XI and Table XII. The relations between these channels are: For the Cabbibo allowed, singly Cabbibo suppresed and doubly Cabbibo suppressed D → T 27 P results, they are listed in Table XIII, Table XIV and Table XV. The relations between these channels are: In this subsection we consider the two-body non-leptonic decay of B c . Now there are two possible transitions, one of them is Fig. 5 (a) shows the Feynman diagram corresponding to these two Hamiltonians. For B c → T 27 P decay. The amplitudes are listed in Table XVI and Table XVII respectively, where the left two columns correspond to b → d, while the right two correspond to b → s.
The effective Hamiltonian of B c → T 10 /T 27 P is These two terms are described by the Feynman diagram of Fig. 5 (b). The amplitudes are listed in Table XVIII and Table XIX respectively. Since these amplitudes have very simple forms, so the relationships among them are obvious and we will not explicitly list them here.

V. GOLDEN DECAY CHANNELS
In this section, with all the decay channels listed above, we can choose the golden decay channels from them. Compared with other channels, the golden decay channels should have greater chance to be observed in the experiments. Generally, the criterion for choosing the golden channels is based on three requirements. The first one is to see whether the decay channel offers a large enough phase space to produce the fully-light tetraquark. For B or B c decays this is not the problem, while for D decays only the channels with the pion in the final state will be considered. Secondly, the channels with π 0 , η q , η s in the final state will not be considered. Although these neutral particles are able to be detected by the current experimental techniques, their high background prevent us to choose them as final states appearing in the golden channels. On the other hand, the channels with K 0 , K 0 should be kept since their corresponding excited states K 0 * , K 0 * can decay to charged pions. Finally, for D decays we only consider the Cabbibo allowed channels. The golden channels for semi-leptonic B, D decays are listed in Table XX and Table XXI. The golden channels for non-leptonic B → T /U P decays are listed in Table XXII and Table XXIII. The golden channels for D decays are listed in Table XXIV, which only contain the Cabbibo allowed channels. For the B c → U/T D decays, there are no preferred channels among them.
Furthermore, one should also check the possible final states of the fully-light tetraquark strong decays. The most suitable case is to consider the two-body decays U/T → P P . For T 27 states, the two-body strong decay can be described by a simple effective Hamiltonian: while for T 10 states, there are no such two-body processes. Although the effective Hamiltonian for decays into three pseudoscalar mesons do exist, due to the limited phase space we will not consider such three-body decays for the T 10 states. We only choose the T → P P channels with the two final states being both charged, and being either double pions or one pion and one kaon for B decays but only double pions for D decays. The channels satisfying these requirements are In Table XXV we list the best channels for reconstructing the T 27 from B/D decays. There is only one suitable D decay channel D + s → π + π − π + and the resonant contribution can be from T ′0 0 → π + π − or T ++ 2 → π + π + . The BarBar collaboration presented a Dalitz plot analysis for , where the S-wave contribution in the π + π − channel is measured. The possible candidates for such scalar particles are a 0 (980) or f 0 (980), which can have 4-quark structure, and for example, be a tetraquark T ′0 0 in 27 state. On the other hand, the tetraquark T ++ 2 has the flavor structure uudd, from which we can enumerate its mass to be around 2m π . However, as shown in the Fig. 3(d) of Ref. [54], at this region there is a large peak produced by the interference between the S-wave and f 2 (1270). This implies that a more sensitive measurement in the 2m π mass region for the π + π + channel is required to detect the potentially existed T ++ 2 .

VI. CONCLUSIONS
In this work we performed a SU(3) analysis for both semi-leptonic and non-leptonic heavy meson weak decays into a pseudoscalar meson and a fully-light tetraquark. We firstly gave a tensor reduction for the SU(3) representation of fully-light tetraquark and point out that the ssūd tetraquark that we mostly interested in belongs to the 10 or 27 representation. Accordingly, we used   there is no D → (T Q Iz → P P )P lν channels with all the three final mesons being pions.