Weak Decays of Stable Doubly Heavy Tetraquark States

In the light of the recent discovery of the $\Xi^{++}_{cc}$ by LHCb collaboration, we study the stable doubly heavy tetraquarks. These states are compact exotic hadrons which can be approximated as the diquark-anti-diquark correlations. In the flavor SU(3) symmetry, they form an SU(3) triplet or anti-sextet. The spectra of the stable doubly heavy tetraquark states are predicted by Sakharov-Zeldovich formula. We find that the $T^+_{cc\bar{u}\bar{d}}({\bf 3})$ is about 16MeV below the $DD^*$ threshold, while $T^{-}_{bb\bar{u}\bar{d}}({\bf 3})$ is about 73MeV below the $BB^*$ threshold. We then study the semileptonic and nonleptonic weak decays of the stable doubly heavy tetraquark states. The decay amplitudes are parametrized in terms of SU(3)-irreducible amplitudes. Ratios between decay widths of different channels are also derived. At the end, we collect the Cabibbo allowed two-body and three-body decay channels, which are most promising to search for the stable doubly heavy tetraquark states at LHCb and Belle II experiments.


I. INTRODUCTION
Up to now most hadrons found by experiments can be well established as quark-antiquark pair and triquarks configurations [1]. Based on the principle of color confining, the multiquark color singlet states such as qqqq (tetraquarks) and qqqqq (pentaquarks) can also exist. On the experimental aspect, many multiquark candidates have been observed even though their physical figures are still not established. The most aged of these exotic resonances is the neutral X(3872) discovered in B ± → K ± X(X → J/ψπ + π − ) by Belle in 2003 [2]. Four years later, the Belle Collaboration observed a charged hidden charm tetraquark candidate, i.e. Z + (4430) [3]. In 2013, the BESIII Collaboration discovered Z + c (3900) through the channel Y (4260) → π − π + J/ψ [4], which directly hadronic decays into J/ψπ + , and then implies that it shall be a meson with quark contents ccud. In 2015, the LHCb Collaboration discovered that two exotic baryons P c (4380) and P c (4450) hadronic decaying into J/ψp, which are candidates for pentaquark states and shall be a baryon with quark contents ccuud.
The LHCb Collaboration have recently observed the doubly charmed baryon Ξ ++ cc in the Λ + c K − π + π + invariant mass spectrum, whose mass is measured to be 3621.40 ± 0.72(stat.) ± 0.27(syst.) ± 0.14(Λ + c ) MeV [5]. This discovery has attracted wide attentions from both the theoretical and experimental sides in high energy physics. From the diquark-based model, the doubly heavy quarks can provide a static color source as the attractive diquark in the color 3 representation. The attractive heavy diquark and the light quark in the color 3 representation then form a color singlet hadron. Thus it is natural to conceive the doubly heavy tetraquark states with attractive heavy diquark and attractive light diquark. From the basic principles of QCD, the long-distance interactions among light quarks and gluons has a characteristic scale of the order of 300MeV. When the two heavy quarks attract each other and their separation is smaller enough than the separation to the light quark, the two heavy quarks interact with a perturbative one-gluon-exchange Coulomb-like potential. When the two heavy quarks have a large separation, the four quarks will form two weakly interacting mesons. It is an important issue to be discussed about whether or not the stable doubly heavy tetraquark states exist. When they steadily exist, it is another important issue on how to detect them.
On the theoretical aspect, the mass spectra of the doubly heavy tetraquark states have been studied in many literatures [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Most of them supported the existence of the doubly heavy tetraquark states, however, they predicted differently the spectra of the doubly heavy tetraquark states in these works. The structures of the doubly heavy tetraquark states were also different in their descriptions. Unlike the QqQq system which can be classified into four kinds of four quark structures [21], the structures of the QQqq system are relatively simple. Take the bbqq for example, the four quark structures may be classified into two groups: one is treated as a bound state made of a loosely bound BB meson pair or two far separated and essentially weak interacting B mesons; the other one is treated as a bound state made of a heavy diquark with color anti-triplet and a light anti-diquark with color triplet. Theoretical description of doubly heavy tetraquark states decays is few in current studies. Whether or not the QCD factorization is valid in the doubly heavy tetraquark states decays is an open question. An alternative and model-independent approach is to employ the flavor SU(3) symmetry, which has been successfully applied into the B meson and the heavy baryon decays [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. In this paper, we will investigate the amplitudes and decay widths of doubly heavy tetraquark states under the flavor SU(3) symmetry. The paper will be presented as follows. In Sec. II, we classify the doubly heavy tetraquark states into an SU(3) triplet or anti-sextet according to the decomposition of3 ⊗3 = 3 ⊕6. Other related baryons and mesons are also listed in SU(3) flavor symmetry. In Sec. III, we give the spectra of the doubly heavy tetraquark states. Their stability properties are essential for the discovery and will be discussed. In Sec. IV and V, we mainly study the semileptonic and nonleptonic weak decays of the stable doubly heavy tetraquarks. The decay amplitudes are explored with the SU(3) flavor symmetry. The ratios between the decay widthes of different decay channels are predicted. We summarize and conclude in the end.

II. PARTICLE MULTIPLETS
Following the flavor SU(3) group, the doubly heavy tetraquark states and their decay products can be grouped into the particle multiplets.
In principle, doubly heavy tetraquark states with the QQqq are similar to theQQqq. We are focusing on the QQqq states for simplication. The doubly heavy tetraquark (QQqq) can form an SU(3) triplet or anti-sextet by the decomposition of3 ⊗3 = 3 ⊕6. The triplet has the expression While the doubly heavy tetraquark in anti-sextet can usually strong decay into the triplets and are not stable, then we will not consider them here. When we study the weak decays of the doubly heavy tetraquarks under the flavor SU(3) symmetry, we should classify the products. The charmed bottom baryons can form a SU(3) triplet with F bc = (Ξ + bc (bcu), Ξ 0 bc (bcd), Ω 0 bc (bcs)). The charmed anti-baryons are classified into a triplet and an anti-sextet Similarly, the singly charmed baryons F c3 and F c6 can be described in the same way, whose explicit expressions can be found in Refs. [34,37]. The anti-baryons with light quarks can be classified into an octet and an anti-decuplet. We write the octet as while the anti-decuplet can be written as The light pseudo-scalar mesons belong to an octet and a singlet. The explicit expression of the octet can be found in Refs. [34,37]. The flavor singlet η 1 will not be considered here for simplicity.

III. SPECTRA OF THE DOUBLY HEAVY TETRAQUARKS
The wave function of a tetraquark consists of four parts: space-coordinate, flavor, color, and spin subspaces, where R(x i ), χ f (f i ), χ λ (λ i ), and χ s (s i ) denote the radial, flavor, color, and spin wave functions, respectively. The sub-labels 1, 2, 3, 4 in the above equation denote Q, Q ′ ,q,q ′ , respectively. For the two quark system, there are eight distinct diquark multiplets in flavor ⊗ color ⊗ spin space. According to the Pauli exclusion principle, the diquark-antidiquark configuration [QQ ′ ][qq ′ ] of doubly heavy tetraquark state only has four possible topologies 1 where the sub-label f, c, s denote the flavor, color, spin spaces, respectively. S and A denote the symmetric and antisymmetric properties. Each half bracket denotes the diquark configuration. For example, |1 f (S), 3 f (A) denotes the diquark [QQ ′ ] is the singlet in flavor space and thus is symmetric (S), while the diquark [qq ′ ] is the triplet in flavor space and thus is antisymmetric (A). Consider that the color sextet diquarks have larger color electrostatic energy thus is not a well-favored configuration, and the odd parity diquark operators will vanish in the single mode configuration. The diquarks |3 c (A) ⊗ |0 s (A) and |3 c (A) ⊗ |1 s (S) in Eqs. (6a) and (6c) are the "scalar" and "axial-vector" diquarks respectively, as the "good" and "bad" diquarks named by Jaffe. Other configurations of the diquark are "worse" diquarks. For simplication, we will not consider these "worse" diquarks in the prediction of the spectra. The constituent quark models have a robust power to predict hadron spectra, especially for the S-wave states. In these quark models, the hadrons are bound states composed of the constituent quarks. In Sakharov-Zeldovich formula, the hadron mass is given by [39,40] where the overall strength can be given as C ij = v ij δ(r ij ) with the coupling v ij and the strength of the radial wave function at zero separation δ(r ij ) which is dependent on the hadron constituent quark flavors. The λ i is the Gell-Mann matrix for color SU(3) group, and s i = σ i /2 is the quark spin operator with the Pauli matrix σ i . The parameters in Eq. (7) can be fitted by the hadron spectra [41], which have been given in Tab. II. In Tab. II, the overall factor C ij /(m i m j ) can be extracted from the hadron mass differences. The constituent quark masses are chosen as: m u,d = 305 MeV, m s = 490 MeV, m c = 1670 MeV, and m b = 5008 MeV for SET I [42][43][44][45]; m u,d = 330 MeV, m s = 500 MeV, m c = 1600 MeV, and m b = 4950 MeV for SET II.
The spectra of the triplet doubly charm tetraquark for SET I are determined as m(T + ccūs (3)) = m(T ++ ccds (3)) = 4.10GeV, In above, T + ccūd (3) lies the spectrum about 16MeV below the DD * threshold and about 120MeV above the DD threshold. However, the T + ccūd (3) is an axial-vector meson and can not directly hadronic decay to DD. Thus T + ccūd (3) with spin-parity 1 + is a stable tetraquark, which shall be tested in experiment. T + ccūs (3) and T ++ ccds (3) lie the spectrum about 124MeV above the D s D * threshold and about 118MeV above the D * s D threshold. These two states can hadronic decay thus are not stable.
The spectra of the triplet charm-beauty tetraquark for SET I are determined as m(T 0 bcūd (3)) = 7.20GeV, m(T 0 bcūs (3)) = m(T + bcds (3)) = 7.43GeV, In this kind, T 0 bcūd (3) lies the spectrum about 86MeV below the BD * threshold but about 5.8MeV above the B * D threshold. Thus T 0 bcūd (3) can hadronic decay to B * D and has a large decay width. T 0 bcūs (3) and T + bcds (3) lie the spectrum about 56MeV above the B s D * threshold and about 137MeV above the B * D s threshold. These two states are also not stable.
The spectra of the triplet doubly bottom tetraquark for SET I are determined as For bottom sector, T − bbūd (3) lies the spectrum about 73MeV below the BB * threshold. Thus T − bbūd (3) with spin-parity 1 + is also a stable tetraquark, which shall be tested in experiment. T − bbūs (3) and T 0 bbds (3) lie the spectrum about 78MeV above the B s B * threshold and about 75MeV above the B * s B threshold, which are not stable. Considering the uncertainties from the constituent quark masses, we adopt another input for these parameters. When we adopt the constituent quark quark with SET II, we can easily get the tetraquark mass differences and also predict the doubly heavy quark tetraquark spectra. The mass of T + ccūd (3) will been reduced by 90MeV, while T + ccūs (3) and T ++ ccds (3) will been reduced by 105MeV. The conclusions of the stability discussions of them are unchanged. For the triplet charm-beauty tetraquark with SET II, the mass of T 0 bcūd (3) will been reduced by 78MeV, while T 0 bcūs (3) and T + bcds (3) will been reduced by 93MeV. Then T 0 bcūd (3) is about 159MeV below the BD * threshold and about 67.2MeV below the B * D threshold, which indicates that T 0 bcūd (3) is a stable state. For the triplet doubly bottom tetraquark with SET II, the mass of T − bbūd (3) will been reduced by 66MeV, while T − bbūs (3) and T 0 bbds (3) will been reduced by 81MeV. Thus T − bbūd (3) becomes more stable. T − bbūs (3) and T 0 bbds (3) is about 3MeV below the B s B * threshold and about 6MeV below the B * s B threshold, which become stable. Future experiments shall tell us the answers to these different predictions between SET I and SET II.
To hunt for these possible doubly heavy tetraquarks, we will study their weak decays properties. Their semi-leptonic and nonleptonic decay amplitudes can be parametrized in terms of SU(3)-irreducible amplitudes. For completeness, we will investigate the weak two-body, three-body and four-body decays of the doubly heavy tetraquarks in flavor triplet.

Decays into mesons and ℓν ℓ
First we study the decays into mesons and ℓν ℓ , where the electro-weak Hamiltonian is with q ′ = u, c. The corresponding Feynman diagrams are given in Fig. 3. The transition of b → cℓ −ν ℓ belongs to a SU(3) singlet, while the b → uℓ −ν ℓ transition belongs to a SU(3) triplet and them can be described as H 3 which has the matrix elements (H 3 ) 1 = V ub and (H 3 ) 2(3) = 0 with the CKM matrix element V ub . Except the two three-body decay channels shown in Fig. (3c), i.e.
most modes involve four particles in the final states. The semileptonic T bb3 decays can be described through the hadron-level Hamiltonian which is written as Here the a i s are the nonperturbative model-independent parameters. The a 3 and a 5 will be present in Fig. (3a), and a 4 is related to the annihilation diagrams in Fig. (3b). Through the Hamiltonian, we can obtain the decay amplitudes for different decay modes, which are given in Tab. III. To satisfy the SU(3) flavor symmetry, one can ignore the effects of phase space when analyzing their decay widths. From Tab. III, all six decay channels into a B and a D meson have the same decay widths. Besides, we can get the relations for the decays into a B meson and a light meson:

Decays into a bottom baryon, a light anti-baryon and ℓν ℓ
As shown in the last two panels in Fig. 3, the T bb3 can transit into a bottom baryon and a light anti-baryon. Since the decuplet is anti-symmetric for light quarks in flavor space, the two spectator light quarks will not go into the decuplet. We write the Hamiltonian as: For convenience, we label the decay channels with different final states: class I for an octet baryon plus a heavy triplet baryon; class II for an octet baryon plus a heavy sextet baryon; class III for a decuplet light baryon plus a heavy sextet baryon. The last type of decays can occur only through the annihilation of bū shown in Fig. (3c). The explicit amplitudes can be found in Tab. IV. From them, the relations for class I decays can be found: The relations for class II decays become: The relations for class III decay widths are 3. Decays into a charmed and bottomed baryon plus a light anti-baryon and ℓν ℓ T bb3 can also decay into a light octet or anti-decuplet anti-baryon and a charmed and bottomed baryon for the b → c transition. After removing the forbidden constructions, the Hamiltonian becomes: The amplitudes are derived and given in Tab. V. From them, the relations of the related decay widths are:

II. Tccqq decays
The effective Hamiltonian from the charm semileptonic decays into a light quark is: cs is introduced for the heavy-to-light quark operators. We plotted the corresponding Feynman diagrams in Fig. (4). The triplet state T cc3 can decay to a charmed meson plus ℓ + ν : and their Feynman diagram are given in Fig. (4c). Thus we obtained: The effective Hamiltonian for decays into a charmed meson and a light meson is written as: The related Feynman diagrams are plotted in Fig. 4(a,b), and the related results for the decay width relations are given in Tab. VI. Thus we have

III. Semileptonic T bcqq decays
Both the bottom and charm quarks can decay in the semileptonic T bcqq decays. For the bottom decay in T bcqq , one can easily get the decay amplitude from those for T bbqq decays with T bbqq → T bcqq , B → D. For the charm decays in T bcqq , one can easily get them from those for T ccqq decays with the replacement of T ccqq → T bcqq , D → B. Thus we do not need to give the tedious results here.

V. NON-LEPTONIC T bbqq DECAYS
Next, we will study the non-leptonic decay amplitudes. For the bottom quark decay, there are four types: where q i with i = 1, 2, 3 denote the light flavors. We will discuss these decay modes one by one in the following.
b c The transition b → c ord/s →c can be signed as W-exchange topology, and we plotted the corresponding Feynman diagrams in Fig. 5. The effective Hamiltonian by this kind of transition is: We gave the decay amplitudes in Tab. VII, from which the relations of decay widths are:

Decays into an anti-charmed anti-baryon and a charmed bottom baryon
The transition b → ccd/s can lead to the process of an anti-baryon plus a baryon, where the anti-charmed antibaryons form a triplet or anti-sextet and the charmed bottom baryon form a SU(3) triplet. The effective Hamiltonian is described as: The related decay amplitudes are given in Tab.VIII, in which class I represents triplet anti-baryon plus the charmed bottom baryon in the final states, and class II denotes the anti-sextet anti-baryon plus the charmed bottom baryon. For class I, we have the relations: For class II, we have:

Decays into three mesons
The transition b → ccd/s leads to three body decays where the effective Hamiltonian becomes: The T bb3 decays amplitudes into J/ψ plus a bottom meson and a light meson are given in Tab. IX, from which we have: Decay amplitudes for T bb3 decays into B c meson plus a charmed meson and a light meson are given in Tab. X. The decay width relations are: The T bb3 decays amplitudes into a bottom meson plus a charmed meson and an anti-charmed meson are given in Tab. XI. And the relations of decay widths become:

Decays into a bottom meson and a charmed meson by W-exchange process
For the bottom quark decays to a charm quark, the effective Hamiltonian is given by:

hadron-level effective Hamiltonian
Decays amplitudes are collected in Tab. XII, from which the relations of decay widths become:

Decays into a light anti-baryon and a charmed bottom baryon
There are two kinds of multiplet for the final states, which lead to the Hamiltonian For class II, we have the relations: Decay amplitudes are collected in Tab. XIV, where no relation for decay widths is found.

Decays into two mesons by W-exchange process
We write the effective Hamiltonian for the anti-charm quark production as According to the flavor SU (3) b u We get the effective Hamiltonian For a bottom meson and an anti-charmed meson produced, the amplitudes are given in Tab. XV. And we have: T bb3 decays amplitudes into a charmed bottom meson and a light meson for different channels are given in Tab. XVI. And we have

Decays into an anti-charmed anti-baryon and a bottom baryon
The effective Hamiltonian for decays into an anti-charmed anti-baryon and a bottom baryon is  class I amplitude(/V ub ) class IV amplitude(/V ub ) Different decay channel amplitudes are given in Tab. XVII, where class I corresponds with triplet anti-baryon plus antitriplet baryon; class II corresponds with triplet anti-baryon plus sextet baryon; class III corresponds with anti-sextet anti-baryon plus anti-triplet baryon; class IV corresponds with anti-sextet anti-baryon plus sextet baryon.
For class I, we obtain the relations of decay widths: For class II, we obtain the relations of decay widths: For class III, we obtain the relations of decay widths: For class IV, the results are:

Decays into three mesons
The effective Hamiltonian is written as The decay amplitudes into a bottom meson plus an anti-charmed meson and a light meson are given in Tab. XVIII.
Deriving the formulae, we get:   channel amplitude The related decay amplitudes of an anti-charmed bottom meson plus two light mesons are given in Tab. XIX. Thus we obtain the relations as follows: IV. Charmless b → q1q2q3 transition

Decays into a bottom meson and a light meson by W-exchange process
The bottom to light quark transition leads to the effective Hamiltonian: where O i is the weak four-fermion effective operators. The tree operators are described as a vector H 3 , a tensor H 6 , and a tensor H 15 . The penguin operators are described as another vector H 3 . The nonzero components of these operators are for the non-strange decays. After doing the exchange of 2 ↔ 3, we will get the formulae for the ∆S = 1(b → s) decays. We get the effective hadron-level Hamiltonian for decays into the bottom anti-triplet The amplitudes are given in Tab. XX for the ∆S = 0(b → d) decays and Tab. XXI for the ∆S = 1(b → s) decays.

Decays into a light anti-baryon and a bottom baryon
There are four kinds of different final states which are light octet or anti-decuplet anti-baryon plus anti-triplet or sextet baryon respectively. Thus the Hamiltonian become  class I amplitude Decay amplitudes are given in Tab. XXII for the transition b → d , Tab. XXIII for the transition b → s. We remove the similar contributions in the amplitudes, such as c 1 −2c 1 , 2c 2 −c 2 , c 3 −2c 3 +c 3 ′ , c 5 +2c 5 +c 5 ′ , 2d 2 +d 2 , d 1 −2d 1 +d 1 ′ , There is no relation of decay widths for class I. The relations of decay widths for class II become: The relations of decay widths for class III become:  class I amplitude The relations of decay widths for class IV become: There are no relation of decay widths for class I. The relations of decay widths for class II become: The relations of decay widths for class III are: The relations of decay widths for class IV become: .
The c 11 and c 11 terms give the same contribution which always contain the factor c 11 − c 11 . We remove the c 11 term in the expanded amplitudes. The amplitudes are given in Tab. XXIV for the transition b → d and Tab. XXV for the transition b → s. The relations are:

VI. NON-LEPTONIC T bcqq DECAYS
The charm decays or (and) bottom decays can be present in the decays of T bcqq . For the bottom decay, there is a new decay channel in which two heavy quarks of tetraquark interact by a virtual W-boson. The others can be obtained from those for T bbqq decays with B → D and B c → J/ψ. For the charm decays, the decay amplitudes can be obtained from those for T ccqq decays with the replacement of D → B and J/ψ → B c . Thus we do not present those results again.

I. Tccqq
For the T ccqq decays, we collected Cabibbo allowed decays in Tab. XXXIII. From the data of the D meson decays, we conclude that these Cabibbo allowed decay channels in Tab. XXXIII may lead to branching fractions at a few percent level. Note that to reconstruct the final charm meson, another factor of 10 −3 is required.

II. T bcqq
To reconstruct the T bcqq tetraquark, lists of possible modes are given in Tab. XXXIV. The decay width is dominant by the charm quark decay from the estimation of the magnitudes of CKM matrix elements. For the charm quark decay, the typical branching fractions in Tab. XXXIV are estimated to be a few percents. If the bottom quark decays, Three body decays (charm decays) Two body decays (bottom decays) We found that the stable T + ccūd (3) is below the DD * threshold, and T − bbūd (3) is below the BB * threshold. In order to hunting for these stable doubly heavy tetraquarks, we investigated systematically the semileptonic and nonleptonic weak decay amplitudes of the stable doubly heavy tetraquarks under the flavor SU(3) symmetry, which is a powerful tool to analyze the general decay properties. The ratios between decay widths of different channels were also given. We have given the Cabibbo allowed two-body and three-body decay channels of the stable doubly heavy tetraquarks, which have large branching ratios and shall be employed as the "discovery channels" in the reconstructions at LHCb and Bell II experiments.