Weak Decays of Stable Open-bottom Tetraquark by SU(3) Symmetry Analysis

The exotic state $X(5568)$ which was observed by D0 Collaboration is very likely to be a tetraquark state with four different valence quark flavors, though the existence was not confirmed by other collaborations. The possibility of such state still generate lots of interests in theory. In the paper, we will study the properties of the state under the SU(3) flavor symmetry. This four quark state with a heavy bottom quark and three light quarks(anti-quark) can form a $6$ or $\overline {15}$ representation. The weak decays can be dominant and should be discussed carefully while such state is stable against the strong interaction. Therefor we will study the multi-body semileptonic and nonleptonic weak decays systematically. With the help of SU(3) flavor symmetry, we can give the Hamiltonian in the hadronic level, then obtain the parameterized irreducible amplitudes and the relations of different channels. At the end of the article, we collect some Cabibbo allowed two-body and three-body weak decay channels which can be used to reconstruct $X_{b6}$ states at the branching fraction up to be $10^{-5}$.

The studies on stable open-flavor tetraquarks or pentaquarks are of even higher interests in literatures, especially on the doubly heavy flavor tetraquarks [30][31][32][33][34][35][36][37][38][39][40], Theoretical calculations on the masses of bbq 1q2 with q i = u, d or s always imply these states are stable against strong decays. However, the productions of such doubly heavy-flavor tetraquark states are too rare in experiments, especially at the current stage [30,31]. To search for stable open-flavor tetraquarks, one of the present authors proposed the possible tetraquark composed of bsūd in Ref. [27], which is supported by the quark delocalization color screening model [29]. Such stable tetraquark state has a unique advantage in the experimental searches. It has a lifetime as large as ordinary B mesons, so that it decays at a secondary vertex at the proton-proton colliders which rejects most of the backgrounds from the preliminary vertex. Besides, its production is much larger than the bb-tetraquarks. Therefore the searches of such stable singly heavy-flavor tetraquarks would be more promising.
To hunting for the possible stable open-bottom tetraquarks, a systematical analysis of decay modes is helpful. A successful example is the discovery of the Ξ ++ cc via the final state of Λ + c K − π + π + [41], which was first pointed out as the most favorable mode in [42]. Although the theoretical calculation of such multi-body decay modes has to deal with the nonperturbative strongly-coupled gauge dynamics, some dynamics-screening approaches always reduce the difficulties. In this paper, we will adopt the light quark SU(3) flavor symmetry to deal with the weak decays of the openbottom tetraquarks. SU(3) flavor symmetry analysis has been applied into the studies of B meson and heavy baryon decays  and it provides a general insight for different decay modes.
The open-bottom four quark multiplets Qq iqjqk can form a3, 6 or 15 representation in the SU(3) flavor symmetry. It will be seen in the next section that the fifteen-fold states are the excited tetraquark states which can hadronic decay into the the sextet states, and the anti-triplet can usually electromagnetic decay into B meson. Therefore we will study the possible weak decays of sextet states (denoted as X b6 ) on the main body of the paper. If the X b6 multiplets can be stable against the strong and electromagnetic decays, the weak decay analysis of such states will be the key tool for searching these exotic states in experiment. By constructing the weak decay Hamiltonian in hadronic level and parameterizing the amplitudes into some irreducible parts, we will give the weak decay amplitude expressions for the X b6 states and obtain the relations among different decay channels.
The paper is organized as followed. We give the multiplets of the open-bottom tetraquarks and the related hadrons in the SU(3) flavor symmetry in Sec. II. We will discuss the energy thresholds of sextet open-bottom tetraquarks, and list the possible masses from corresponding literatures in Sec. III. From Sec. IV to Sec. V, we mainly study the semi-leptonic and non-leptonic weak decays of the X b6 states. In Sec. VI, we will select some possible golden channels which may be performed in future experiments. We summarize and conclude in the end.
[jk]} (antisymmetric tensor ǫ 123 = −1 and ǫ 123 = 1). While the 15 states usually strong decay into the sextets and we will not consider them here. On the other hand, the3 tetraquarks have a quark-antiquark pair and can electromagnetic decay into B meson, we will only focus on the sextet tetraquark states. The traceless sextet (X b6 ) i j,k satisfies the condition that the flavor components are antisymmetric under the exchange of j and k [24], which can be written explicitly The light pseudo-scalars form an octet plus a singlet, and the octet mesons can be written as For the baryon states, the light anti-baryons form an octet and an anti-decuplet representation. 1 Throughout the paper, the charge conjugation is assumed and one can get the corresponding properties forbqiqj q k under the charge conjugation.
The octet can be written as For the anti-charmed anti-baryons, they form a triplet and an anti-sextet It is easily to get the components for the singly charmed baryons which form an anti-triplet T c3 and a sextet T c6 . Their explicit expressions can be found in Refs. [40,55,57,59].

III. STRONG DECAY THRESHOLDS FOR THE X b6 TETRAQUARKS
In this section, we will look at the strong and electromagnetic decay thresholds for the X b6 tetraquarks. One should note that the states of X sūd , X uds and X dsū are purely open flavor states and can not electromagnetic decay into B meson. For the other three states of Y (uū,dd)s , Y (uū,ss)d and Y (dd,ss)ū , even though they have a quark-antiquark pair, they will not electromagnetic decay into B meson because of the antisymmetric quark structure for X b6 . The X b6 tetraquarks may have different spin-parities, and the ground states of the X b6 states will have the spin-parity with J P = 0 + . Higher excited X b6 states will strong decay into the ground states. Thus we only focus on the j P = 0 + X b6 ground states.
The BK mass threshold at 5.77GeV will influence the decay properties if the mass of X sūd state is higher than 5.77GeV. The B s π mass threshold at 5.51GeV will influence the decay properties if the masses of X uds , X dsū and Y (uū,dd)s states are higher than 5.51GeV. While for the Y (uū,ss)d and Y (dd,ss)ū states, their strong decay threshold lies in the Bπ mass at 5.41GeV. Therefor, it can  [22,27], simple constituent quark model [27], QCD sum rules [18,26], quark delocalization color screening model [29]. The predictions for the masses of the Y (uū,dd)s , Y (uū,ss)d and Y (dd,ss)ū states are missing.
In literatures, there are lots of theoretical predictions for these open-bottom tetraquarks. Using the diquark-antidiquark model, the mass of X sūd state was predicted at 5.637 GeV in Ref. [27] by one of the present authors, while the states of X uds and X dsū were predicted with the mass at 5.7 GeV in Ref. [22] by another one of the present authors and the collaborator. Using the simple constituent quark model, the the mass of X sūd state was predicted at 6.119 GeV [27].
The masses of the lowest-lying open-flavor bottom tetraquarks have been explored in recent articles, so we listed them in Tab. I.
From the predictions of the mass spectra of X b6 states, one can see some predictions support the existence of the X b6 states blow the strong decay thresholds. The processes of weak decays are dominant once the stable tetraquarks are confirmed. It is very useful to study the weak decays by the SU(3) symmetry analysis without any assumptions of factorization or dynamic information.
The semi-leptonic and nonleptonic decay amplitudes have been parameterized in terms of SU (3) irreducible representations. For completeness, the weak two body and three body decays of open bottom tetraquarks will be studied in the following.

b → qℓν ℓ : Semi-leptonic decays into mesons
In the following, we will focus on the bottom quark decays. For the b quark decay, the electroweak Hamiltonian can be expressed as with q ′ = u, c. Therein the b → c transition forms a singlet in SU(3) flavor symmetry, while Firstly, the effective Hamiltonian for X b6 decays to a light meson and ℓν ℓ in hadron level can be easily written as where and in the following the coefficient a i represents the nonperturbative parameters. The related Feynman diagram is plotted in Fig. 1 The effective Hamiltonian for X b6 semileptonic decays into two mesons can also be constructed as The corresponding Feynman diagrams are given in Fig. 1.(b,c). The full term l ν ℓ which are related to Fig. 1.(c) and Fig. 1.(b) respectively. It should be mentioned that the process of X uds → π + K + ℓ −ν is trivial because of the exchangeable antisymmetric antiquarksd ands in the initial state. One can obtain the decay amplitudes from the effective Hamiltonian. The results for a light meson plus a charmed meson in final states are given in Tab. III, while the ones for two light mesons in final states are given in Tab. IV. For the semi-leptonic decays into a charmed meson and a light meson, one has the relations as For the semi-leptonic decays into two light mesons, one has

b → qℓν ℓ : Semileptonic decays into a light baryon plus a light anti-baryon
The X b6 tetraquark can also decay into a light baryon which is from an octet or anti-decuplet plus a light anti-baryon. Thus there are four different combinations for final states. The irreducible Hamiltonian can be constructed as follows The four kinds of amplitudes are given in Tab. V and Tab. VI. We labelled the different final states as class I for an octet anti-baryon plus an octet baryon, class II for an octet anti-baryon plus a decuplet baryon, class III for an anti-decuplet anti-baryon plus an octet baryon, and class IV for an anti-decuplet anti-baryon plus a decuplet baryon. The decay amplitudes of X − uds in Class IV disappear. In topology level, the corresponding Feynman diagrams are shown in Fig. 1. (d,e). For class I, the corresponding results for the decay widths become The relations for class II are For class III, the relations are For class IV, the relations become The amplitudes for different channels are given in Tab. VII for a charmed anti-triplet baryon plus an octet anti-baryon and a charmed sextet baryon plus an octet anti-baryon. We plotted the Feynman diagram in The relations of decay widths for class I are u/c The relations of decay widths for class II are In quark-level transitions, the bottom quark non-leptonic weak decays can be divided into four different where q 1,2,3 represent the light quark. For these four kinds of transitions, the weak effective Hamiltonian H ef f are given separately , , , where O i is the four-quark effective operators and C i is the Wilson short-distance coefficient. O 1 , O 2 are tree level operators while O 3 − O 10 are called as penguin operators. In hadron level, it is easily to discuss the X b6 non-leptonic decay modes when writing down the effective Hamiltonian using the SU(3) flavor symmetry.

Two-body decays into mesons
The b → ccd/s transition leads to form a SU(3) triplet operator. For X b6 decays to two mesons, the effective Hamiltonian is constructed as The amplitudes for the decays to the J/ψ plus a light meson are given in Tab The relations for decay widths are given as respectively

Two-body decays into a charmed baryon and an anti-charmed anti-baryon
The transition b → ccd/s can also produce a charmed baryon and an anti-charmed anti-baryon. Charmed baryons can be an anti-triplet or a sextet, while the anti-charmed anti-baryons can form the triplet or antisextet baryons. The effective Hamiltonian in hadron level is constructed as We listed the decay amplitudes in Tab The relations of decay widths for class I are The relations of decay widths for class II are For class III, the relations of decay widths are For class IV, the relations of decay widths are

Three-body decays into mesons
The b → ccd/s transition can also lead to three-body meson decays in which the effective Hamiltonian in hadron level is The Feynman diagrams are shown in Fig. 2. (e-j). It should be noted that the diagram in Fig. 2.(e) can lead to the process X sūd → K 0 K − J/ψ. However, the total amplitude of the process vanishes for the antisymmetric lower indexes in X b6 . The decay amplitudes of X b6 decays to J/ψ and two light mesons are given in Tab. XII, while the amplitudes of X b6 decays to an anti-charmed meson plus a charmed meson and a light meson are given in Tab. XIII. From them, we obtain the results for these decay widths anti-baryon triplet for class I, a charmed baryon sextet and an anti-charmed anti-baryon triplet for class II. anti-baryon anti-sextet for class III, a charmed baryon sextet and an anti-charmed anti-baryon anti-sextet for class IV.
effective Hamiltonian is then written as The decays amplitudes are given in Tab. XIV, and the relations among different decay widths become

Two-body decays into a charmed baryon and an anti-baryon
Concerning the two-body decays into a charmed baryon and an anti-baryon, the efficient Hamiltonian which includes four different kinds of combinations in final states can be written as The related decay amplitudes are given in Tab. XV and Tab. XVI, which are for anti-triplet charmed baryon plus octet anti-baryon (class I), sextet charmed baryon plus octet anti-baryon class II, anti-triplet charmed baryon plus anti-decuplet anti-baryon class III, sextet charmed baryon plus anti-decuplet anti-baryon class IV, respectively. The relations of decay widths for class I are given as The relations of decay widths for class II are The relations of decay widths for class III become The relations of decay widths for class IV become

Three-body decays into mesons
For the X b6 decays into three mesons, one constructs the effective Hamiltonian as follows The decay amplitudes are given in Tab. XVII. The relations of different decay widths become (III). b → ucd/s transition

Two-body decays into mesons
The ( Class IV  Y 0 (uū,ss)d → D + s π 0 K − (−a 9 +a 11 +a 12 +a 15 )V * subscripts and replacing V cs by V cd at the same time, one gets the nonzero components for the b → ucd transition. The effective Hamiltonian for X b6 produce two mesons by b → ucd/s transition is Decay amplitudes for different channels are shown in Tab. XVIII, which leads to the following relations

Two-body decays into a baryon and an anti-baryon
The b → ucd/s transition can lead to the decays into an anti-charmed anti-baryon plus a light baryon.

The effective Hamiltonian in hadron level is then written as
The related decay amplitudes for different channels are given in Tab. XIX and Tab. XX. Therein class I corresponds with the processes with octet baryon plus triplet anti-baryon, class II corresponds with the processes with octet baryon plus anti-sextet anti-baryon, class III corresponds with the processes with decuplet baryon plus triplet anti-baryon, class IV corresponds with the processes with decuplet baryon plus anti-sextet anti-baryon. The relation of decay widths for class I are given as: The relations of decay widths for class II are given as The relations of decay widths for class III are given as The relations of decay widths for class IV are given as:

Three-body decays into mesons
For X b6 decays into three meson modes, the hadron-level effective Hamiltonian can be constructed as The related amplitudes for different channels are presented in Tab. XXI and Tab. XXII, from which one obtain relations for decay widths

Two-body decays into mesons
The charmless tree operator of (q 1 b)(q 2 q 3 ) can lead to a triple H 3 , a sextet For the ∆S = 1(b → s) transition, the nonzero entries can be obtained from Eq. (22) by the exchange 2 ↔ 3.
The relations of decay widths for Class III are given as The relations of decay widths for Class IV become       channel amplitude X + uds → π + π 0 K 0 X − sūd → π + K − K − √ 2 (c 3 + c 4 − c 8 + c 12 + c 13 + c 14 + c 15 ) Y 0 (uū,ss)d → π + π 0 K − −c8+c12+c13+c14+c15 √ 2   on semi-leptonic and non-leptonic weak decays of the ground states of sextet representations whose masses are below the thresholds of strong and electromagnetic decays. Their decay amplitudes were discussed by constructing the relevant Hamiltonian at the hadronic level and parameterizing the interactions into some constants (a i , b j , ...). It is easily to obtain the relations of different channels when we ignore the small effect of phase space. We have given the Cabibbo allowed two-body and three-body decay channels, which shall play an important role to hunting for the stable open-bottom tetraquark X b6 states.