Weak Decays of Doubly Heavy Baryons: the $1/2\to 1/2$ case

Very recently, the LHCb collaboration has observed in the final state $\Lambda_c^+ K^-\pi^+\pi^+$ a resonant structure that is identified as the doubly-charmed baryon $\Xi_{cc}^{++}$. Inspired by this observation, we investigate the weak decays of doubly heavy baryons $\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$, $\Omega_{cc}^{+}$, $\Xi_{bc}^{(\prime)+}$, $\Xi_{bc}^{(\prime)0}$, $\Omega_{bc}^{(\prime)0}$, $\Xi_{bb}^{0}$, $\Xi_{bb}^{-}$ and $\Omega_{bb}^{-}$ and focus on the decays into spin $1/2$ baryons in this paper. At the quark level these decay processes are induced by the $c\to d/s$ or $b\to u/c$ transitions, and the two spectator quarks can be viewed as a scalar or axial vector diquark. We first derive the hadronic form factors for these transitions in the light-front approach and then apply them to predict the partial widths for the semi-leptonic and non-leptonic decays of doubly heavy baryons. We find that a number of decay channels are sizable and can be examined in future measurements at experimental facilities like LHC, Belle II and CEPC.


I. INTRODUCTION
The constituent quark model predicts the existence of multiplets of baryon and meson states [1].
Including the heavy charm and bottom quark, baryons made of three quarks are in a big family of hadron spectroscopy. When considering u, d, s and c, the baryon ground states, those with no orbital or radial excitations, consist of a 20-plet with J P = 1/2 + and a 20-plet with J P = 3/2 + .
For the five flavors, u, d, s, c, b, the ground states are then composed of a 40-plet with J P = 1/2 + and 35-plet with J P = 3/2 + . All these ground states with zero or one heavy quark have been well established on experimental side [2].
The search for doubly heavy baryons is a long-standing problem in the last decade. The only evidence in the past from the experimental side was found for Ξ + cc by the SELEX collaboration [3,4]. However this evidence has not been confirmed by any other experiments [5][6][7][8]. Very recently, the LHCb collaboration has observed the doubly charmed baryon Ξ ++ cc with the mass given as [9] m Ξ ++ cc = (3621.40 ± 0.72 ± 0.27 ± 0.14)MeV.
It is anticipated that this observation will have a great impact on the hadron spectroscopy and with no doubt it will trigger much more interests in this research field. On the other hand, after the observation in the Ξ ++ cc → Λ + c K − π + π + decay mode, we also believe that experimental investigations should be conducted in a number of other decay channels. Thus from this viewpoint theoretical studies on weak decays of doubly heavy baryons, not only Ξ ++ cc , will be of great importance and are highly demanded. Some attempts have been made in Refs. [10][11][12][13][14][15][16][17][18][19][20][21], but a comprehensive study is not available in the recent literature, and the aim of this work and the forthcoming ones is to fill this gap. To do so, we will calculate the transition form factors and use these results to study the weak decays of bottom quark and charm quark.
The quantum numbers of the doubly heavy baryons are given in Table I. Among various doubly charmed baryons, three of them can decay only through weak interactions, a Ξ cc isodoublet ccu, ccd, and an Ω cc isosinglet ccs. There are three doubly bottom baryons similarly. For the bottom-charm baryons, the ones with two different heavy flavors, there are two sets of SU(3) triplets, Ξ bc , Ω bc and Ξ ′ bc , Ω ′ bc . These two triplets have different total spin for the heavy quark system bc, but in reality they will probably mix with each other. Only the lighter ones can weak decay with sizable branching fractions. The mixing scheme between the two triplets is unknown yet, and we will consider both types in this work. All these baryons that can weak decay have spin 1/2. The ones with spin 3/2 can radiatively decay into the lowest-lying ones if the mass splitting is not large enough, or decay into the lowest-lying ones with the emission of a light pion when they are heavy enough. The decay final state of the Ξ cc and Ω cc contains the baryons with one charm quark. These baryons form an anti-triplets and sextets of charmed baryons, as shown in Fig. 1. This is also similar for baryons with one bottom quark. The total spin of the baryons in Fig. 1 is 1/2, while another sextet has the spin 3/2. In this work, we shall focus on the 1/2 → 1/2 transition, and leave the 1/2 → 3/2 transition in a forthcoming publication.
To be more explicit, we will investigate the following decay modes of doubly-heavy baryons.
In the above, the quark components have been explicitly shown in the brackets, in which the first quarks denote the quarks participating in the weak decays.
To deal with the strong interaction in the transition, we will adopt the light front approach and calculate the decay form factors. This approach has been widely applied to various mesonic transitions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], and some analyses of baryonic transitions in this approach can be found in Refs. [40][41][42]. We will use diquark picture and only consider the ground states of baryons, thereby the two spectator quarks are considered to be a scalar diquark with J P = 0 + or an axial vector diquark with J P = 1 + . Both types of diquarks will contribute and their contributions are calculated respectively.
The rest of this paper is organized as follows. In Sec. II, we will give a very brief overview of the spectroscopy and lifetimes of the doubly heavy baryons. Sec. III is devoted to the calculation of transition form factors in the covariant light-front quark model. In Sec. IV and V, we apply our results to calculate the partial widths for semileptonic B → B ′ ℓν ℓ decays, and the nonleptonic decays, respectively. A brief summary and some discussions on the future improvements are given in the last section.
In Tab. II, we have neglected the isospin splittings, that is, we have used m Ξ ++ The isospin splittings for doubly heavy baryons have been studied in Ref. [59] with the results: As one can see from the above equation, the isospin splittings are at most a few MeV. We find that their impact on the form factors and decay widths of semi-leptonic and non-leptonic decays is small especially compared to hadronic uncertainties. This, however, will be improved when the experimental data for the masses are available.
For the baryons with a strange quark, one expects that they should be higher than the corresponding states with a u or d quark. Theoretical results from Ref. [58] did respect this expectation for the ccq and bbq baryons, however, the predicted mass for Ω bc is only 55 MeV higher than that for Ξ bc . It should be warned that using these results might introduce some theoretical uncertainties to form factors and decay widths.
The lifetime of the baryons is determined by the inclusive decays. Thus one can in principle use the optical theorem to obtain the total width (lifetime) of the heavy hadron by calculating the absorptive part of the forward-scattering amplitude. The lifetimes of the baryons have been studied in Refs. [46,[60][61][62][63][64][65], where the results differ significantly. For the instance, the lifetime of the Ξ ++ cc baryon is predicted in the range 200 fs to 700 fs. This large ambiguity will introduce dramatic uncertainties to the decay branching fractions, and we intend to improve the precision of the lifetime in the future. In this work, for the lifetime of Ξ cc we will use the results from Ref. [21], while other lifetimes are taken from Ref. [46,61].  [58] lifetimes 300 [21] 100 [21] 270 [61] 244 [46] 93 [46] 220 [61] 370 [46] 370 [46] 800 [61] In heavy quark limit, the interaction between the heavy quark and gluon is independent of the heavy quark spin. Thus the spin of heavy quarks is conserved and can be used for classification of hadrons. For the lowest lying bbq and ccq (q = u, d, s) system with L = 0, the bb and cc must have spin 1 due to the symmetry between the two heavy quarks. For the bcq baryon, the bc system can have spin 0, corresponding to 1/2 baryons, or spin 1, corresponding to the 1/2 or 3/2 baryons as shown in Tab. I. The physical hadrons, mass eigenstates, might be mixtures of the spin eigenstates of heavy quark subsystem: We expect the mixing effects are at the order Λ QCD /m Q , and in this case m Q is very probably the charm quark mass. But currently we are unable to determine the mixing angle in a reliable way, and thus in the following we will calculate the decays for both Ξ bc and Ξ ′ bc . It is necessary to point out that once the mixing scheme is determined, only one of the two sets of baryons have sizable branching fractions for weak decays. The ones with higher mass will radiatively decay into the lower one.

III. TRANSITION FORM FACTORS IN THE LIGHT-FRONT APPROACH
A. Light-front approach for baryons Doubly heavy baryons are made of two heavy quarks and a light quark. For a very solid analysis from QCD of their weak decays, one has to take into account all three quarks, which is very complicated and far beyond our capability now. Starting from the initial doubly heavy baryons, it might be better to treat the two heavy quarks in the initial state as a diquark system. However, if one adopts the heavy-QQ-diquark picture for the doubly heavy baryons, the diquark system must be smashed in the weak transition of doubly heavy system to singly heavy system. An observation is, in the decay transition, one of the two heavy quarks decays, while the other heavy quark and the light quark will act as spectators. Thus as an approximation it might be plausible to treat the two spectators as a system. Here it should be stressed that this system is not tightly bounded as a usual diquark system. Only for brevity, we use the symbol di to denote the heavy-light system.
In the light-front approach, it is convenient to use the light-front decomposition of the momentum p = (p − , p + , p ⊥ ), with p ± = p 0 ± p 3 and p ⊥ = (p 1 , p 2 ) and thus p · p = p − p + − p 2 ⊥ . A baryon with total momentum P , spin S = 1 2 and a scalar/axial vector diquark can be expanded as where p 1 and p 2 are the momenta of the quark q 1 and the diquark [di], respectively. The convention is chosen as:p The baryon mass is denoted as M , and the quark q 1 mass and the diquark [di] mass are m 1 and m 2 respectively. The minus component of the momenta can be determined by their corresponding on-shell condition, One can introduce the momentum fraction x 1,2 of q 1 and [di] through It is often convenient to use x ≡ x 2 and hence x 1 = 1 − x. DenoteP ≡ p 1 + p 2 and M 2 0 ≡P 2 . In P rest frame, e 1,2 corresponds to the energy of q 1 and [di], respectively. The 3-momentum of [di] is k = (k ⊥ , k z ). Then M 0 can be expressed as a function of the internal variables x and k ⊥ : Using e 1 + e 2 = M 0 and the on-shell conditions of q 1 and [di], one can obtain: with Γ = 1 for the scalar diquark and Γ = − 1 √ 3 γ 5 / ǫ * (p 2 , λ 2 ) for the axial vector diquark [42]. The light-front wave function is given as where A = 1 for the scalar diquark and A = which implies that the light-front wave function φ(x, k ⊥ ) should satisfy the following constraint In the practical calculations, a Gaussian form function is widely used, with where the parameter β characterizes the momentum distributions between the constituents, and is usually obtained by fitting the data. With the relations of one can obtain form factors as [40] The final expressions for form factors are given: where x ′ = x, x ′ 1 = x 1 = 1 − x and k ′ ⊥ = k ⊥ + x 2 q ⊥ since we choose the coordinate system which satisfies q + = 0.

D. Mixing of transition form factors
It should be noted that, in the above calculation, what we have obtained is simply the transition denoting a scalar or axial vector diquark spectator, respectively. The hadronic transition matrix elements are actually linear combinations of the ones with the scalar or axial vector diquarks, where the coefficients c S,A are determined by the wave functions of the initial and final states.
For the doubly charmed baryons, the wave functions are with q = u, d or s for Ξ ++ cc , Ξ + cc or Ω + cc , respectively. The superscripts describe the symmetry between the two charm quarks. The ones of the doubly bottom baryons are replaced by c → b.
For the bottom-charm baryons, there are two sets of states, with bc as a scalar or an axial vector diquarks. The wave functions of bottom-charm baryons with axial vector bc diquark are while those with a scalar bc diquark are given as with q = u, d or s for Ξ bc , respectively. The wave functions of the anti-triplet singly charmed baryons are For the sextet of singly charmed baryons, the wave functions are The ones of the singly bottom baryons are similar with the replacement of c by b.
With the wave functions given above, the mixing coefficients for the transition matrix elements in (27) are given in Tables III, IV, V and VI.  TABLE VI: Same as Table III but for the bottom-charm baryons with b decays.     with the decay constants as [66][67][68] The values of the β parameters are then listed in Table VII. Other β's are taken directly from [39].
The masses of singly heavy flavor baryons are also collected in Table VII [2,69].
With the above inputs, the form factors with the scalar or axial vector diquarks in Eqs. (25) and (26) can be obtained. To access the q 2 distribution of the form factors, we adopt the following parametrized form where the F (0) is the form factor at q 2 = 0. The m fit and δ are two parameters to be fitted from numerical results. For the form factor g 2 , the above formula may lead to an imaginary result for The formula of (36) is adopted for the ones with asterisk, and that of (35) for all the others.
The results for form factors with scalar diquark spectators are given in Tables VIII, X, XII where the G F and V cs,cd,ub,cb are Fermi constant and Cabibbo-Kobayashi-Maskawa (CKM) matrix element, respectively. Leptonic parts can be computed in perturbation theory while hadronic contributions are paraemtrized in terms of form factors.
The B → B ′ form factors are parametrized in Eq. (19), and the helicity amplitudes of the vector current are related to these form factors through the following expressions: where . The M and M ′ is the mass for the initial and final baryon. The negative helicity amplitudes are derived as The helicity amplitudes for the left-handed current are obtained as The differential decay width for the B → B ′ lν decay is written as where p = M ′ √ ω 2 − 1 is the three-momentum of B ′ in the B rest frame. Integrating over the parameter ω, we obtain the total decay width One can also study the ratio of the longitudinal to transverse decay rates Γ L /Γ T .
It is also possible to express the differential decay width as  where the q 2 is the lepton pair invariant mass. The polarized decay widths are given as The total width is derived as:

B. Numerical results
For the numerical calculation, we will use the results for Fermi constant and CKM matrix elements from particle data group [2]: The integrated partial decay widths and the relevant branching ratios and Γ L /Γ T 's are given in Tables XVI, XVII  decay widths, branching ratios and Γ L /Γ T 's for semi-leptonic decays, with lepton mass neglected.
A few remarks are given in order.
• When presenting the numerical result for branching fractions, we have used the lifetimes as given in Table II, but as we have pointed out that there exist large uncertainties in the lifetimes. So we have also presented the results for decay widths.
• The Ξ ++ cc → Ξ + c l + ν l and Ξ ++ cc → Ξ ′+ c l + ν l are induced by the c → s transition. Their branching fractions, at a few percent level, are comparable to those of D → Ke + ν e .
• The branching ratio for Ξ ++ cc → Λ + c l + ν l is suppressed due to the CKM matrix element V cd , which is also comparable to those of the D → πe + ν e mode.
• The branching ratios for b → c transitions are typically at the order 10 −2 to 10 −3 , while the b → u transition is highly suppressed due to the smallness of |V ub |.
• In the calculation carried above, we have neglected the form factors f 3 (q 2 ) and g 3 (q 2 ). For semi-leptonic decays, the contribution from f 3 or g 3 is proportional to m l , thus it is safe to drop them if l = e, µ. As for l = τ , the transverse decay width dΓ T /dq 2 in Eq. (43) remains unchanged while the longitudinal decay width dΓ L /dq 2 in Eq. (42) should be re-calculated: wherem l ≡ m l / q 2 and H ± 1 2 ,t are given by

C. SU(3) analysis
Recently, an analysis of weak decays of doubly-heavy baryons based on flavor symmetry is available in Ref. [70]. In the SU(3) symmetry limit, there exist the a number of relations among these semileptonic decay widths, which we are going to examine in the following.
• cc sector • bb sector • bc sector with the c quark decay • bc sector with the b quark decay Comparing the above equations predicted by SU(3) symmetry with the corresponding results in this work, we have the following remarks.
• SU(3) symmetry is respected very well in most cases, except for the following ones These five relations are broken considerably: larger than 20% but still less than 50% using • Since the mass difference between the u and d quark has been neglected in this work, the isospin symmetry is well respected. But since the strange quark is much heavier, the SU (3) relations for the channels involving u, d quark and s quark can be sizably broken. All relations given in Eq. (53) are of this type.
• The first 4 relations in Eq. (53) involve the c quark decay but the last one involves the b quark decay. It indicates that the c quark decay modes tend to break SU(3) symmetry easily.
This can be understood since the phase space of the c quark decay is smaller, and thus the decay amplitude is more sensitive to the mass of the initial and final baryons.

D. The loosely bounded diquark approximation and shape parameter uncertainty
In the above calculations, we have made the assumption that the two spectators, a heavy and a light quark, are treated as a system. Then baryons are composed of this loosely bounded system and the quark involved in the weak transition. We approximated this system as a heavy quark, and obtained the β parameter in the light-front wave function by comparing the doubly heavy baryons with heavy quarkonia, and singly heavy baryons with heavy-light mesons.
From a phenomenological viewpoint, namely since the size of the loosely bounded system is mainly determined by the size of the light quark, it might also be applicable to derive the shape parameter β under the approximation that the heavy-light spectator is treated a light system. Using these new parameters β, we have found that the form factors for the doubly charmed baryons are not significantly changed since the charm quark mass is not very large.  From this table, we can  TABLE XXII: Form factors f 1,2 and g 1,2 for the Ξ 0 bb → Σ + b process. For the left columns, β Ξ 0 bb = 1.472 GeV and β Σ + b = 0.562 GeV have been used; while the results in the right columns, β Ξ 0 bb = 0.562 GeV and β Σ + b = 0.318 GeV have been adopted. The fit formulae of (35) and (36) are adopted to access the q 2 distribution.
Though a QCD analysis of Ξ 0 bbu → Σ + buu transition is not available, we may expect similar power suppressions for these form factors, and the form factors might be of the order 0.1. Thus under the approximation of the two spectators as a loosely connected system, it may be reasonable to use the parameters β Ξ 0 bbu = 1.472 GeV and β Σ + buu = 0.562 GeV as an effective parameter.

V. NON-LEPTONIC DECAYS
In the following, we study two-body non-leptonic decays of doubly heavy baryons, B → B ′ M with M as a pseudoscalar (P ), vector (V ) or axial vector (A) meson. As to a first systematic analysis, we only consider the tree current-current operators in the effective Hamiltonian, taking c where where ǫ µ is the polarization vector of the final vector or axial vector mesons. In the heavy hadron decays, it has manifested that the factorization hypothesis works well in the heavy quark limit [72][73][74][75][76][77][78][79][80]. The above decay amplitudes in the factorization approach are expressed as The decay width for the B → B ′ P is given as where p is the magnitude of the three-momentum of the final-state particles in the rest frame of initial state. For B → B ′ V (A) decay, the decay width is where E(E ′ ) is the energy of final-state meson (baryon), and S = −A 1 , The values of the CKM matrix elements and the masses of the relevant mesons and baryons are taken from [2]. The decay constants are [26,39,66] f π = 130.4MeV, f ρ = 216MeV, f a 1 = 238MeV, f K = 160MeV, f K * = 210MeV, The partial decay widths and branching ratios for the two-body non-leptonic modes of the doubly heavy flavor baryon decays are given in Tables XXIII, XXIV, XXV, XXVI, XXVII, XXVIII and XXIX.
There are some remarks in the non-leptonic modes:   Table II. • For the bottom-charm baryons with the scalar or axial vector bc diquarks, only the lowest-  lying states can decay weakly. As it is not clear which state is the lowest-lying one, we assume that the masses and lifetimes of the two sets of states are the same with each other, The only difference between B ′ i → B f and B i → B f is the mixing coefficients in Tables V and VI. Our results would be useful for the studies in the future if the lowest-lying states are determined.
• The contributions from the form factors of f 3 (q 2 ) and g 3 (q 2 ) are neglected in the non-leptonic decays. For the modes with a pseudoscalar meson in the final state, the terms with f 3 or g 3 are proportional to m 2 P /M 2 in the heavy quark limit, with M as the mass of the initial-state baryon. No matter for the light pseudoscalar mesons π or K in charm or bottom decays, or for the charmed mesons D or D s in bottom decays, these contributions are small and negligible. As to the processes with a vector or an axial vector meson in the final state, there is no contribution from f 3 or g 3 which is proportional to q · ǫ * = 0.
• Very recently, LHCb has observed the Ξ ++ cc in the final state of Λ + c K − π + π + with the significance of more than 12σ [9]. The multi-body charmed hadron decays are usually dominated  by the resonant contributions, since there are many resonances below the charm scale. The external W -emission contributions for this four-body process are Ξ ++ cc decaying into (csu) states and a charged pion, followed by (csu) fragmented into Λ + c K − π + . However, such contributions cannot be large. Only very high excited states of Ξ c can decay into Λ + c K − π + . But the high excited states are more difficult to be produced. The internal W -emission amplitudes can contribute to the four-body decay of Ξ ++ cc → Λ + c K − π + π + [21], as seen in Fig. 4.
For the internal W -emission amplitudes, unlike the case in B meson decays where the nonfactorizable contributions are color-suppressed and neglected, the non-factorizable contributions in charm decays are significantly enhanced due to the final-state interacting effects.
Empirically, the effective Wilson coefficient in Eq. (57) is |a eff 2 (µ c )| ∼ 0.7 as in D meson decays and in Λ + c decays. With this value of |a eff 2 |, we can obtain the branching fraction of Ξ ++ cc → Σ ++ c (2455)K * 0 = 4.1%, which is large enough for the experimental measurements.

VI. CONCLUSIONS
In the past decades, heavy quark decays have played a very important role in extracting the CKM parameters in the standard model, understanding the mechanism for the CP violation, and in shaping our understanding of dynamics in strong interactions and factorization theorem. This, however, has only made use of weak decays of the ground state of the heavy mesons/baryons with a single heavy quark. Weak decays of the doubly-heavy baryons are expected to provide equally important information.
Very recently, the LHCb collaboration has observed in the final state Λ c K − π + π + a resonant structure that is identified as the doubly-charmed baryon Ξ ++ cc . Such an important observation will undoubtedly promote the research on the hadron spectroscopy and also on the weak decays of the doubly heavy baryons. Inspired by this observation, we have investigated the decay processes of doubly heavy baryons Ξ ++ cc , Ξ + cc , Ω + cc , Ξ We find that a number of decay channels are sizable and can be examined in future measurements at experimental facilities like LHC, Belle II and CEPC. These results are also useful for a cross-check of the Ξ ++ cc and the search for other baryons. This work can be regarded as a first step towards a comprehensive understanding of weak decays of doubly-heavy baryons. The potential generalizations and improvements are given as follows.
• The 1/2 → 3/2 transition: This work has focused on the 1/2 → 1/2 transition, either the diquark spectator is a scalar or an axial vector system. When the diquark spectator is an axial vector, the final baryon may have spin 3/2. Such transitions will be calculated in the future.
• penguin dominated processes: The analysis of nonleptonic decay modes in this work are mainly dominated by treeoperators. Decay modes induced by penguin operators may have sizable branching fractions as shown in the heavy meson decays.
• Non-factorizable contributions: To give more precise predictions on branching ratios and more importantly the CP asymmetries, one has to reliably estimate the non-factorizable contributions.
• A comprehensive analysis from QCD: As we have discussed at the beginning of Section III, it is a very crude approximation to adopt one heavy and one light quarks as a system to study the heavy to light transition, though this approximation greatly simplifies the calculation. Since the initial and final states contain heavy quark, soft and collinear degrees of freedom, a power counting analysis might be possible in the framework of soft-collinear effective theory.