A comprehensive analysis of weak transition form factors for doubly heavy baryons in the light front approach

The transition form factors for doubly heavy baryons into a spin-$1/2$ or spin-$3/2$ ground-state baryon induced by both the charged current and the flavor changing neutral current are systematically studied within the light-front quark model. In the transition the two spectator quarks have two spin configurations and both are considered in this calculation. We use an updated vertex functions, and inspired by the flavor SU(3) symmetry, we also provide a new approach to derive the flavor-spin factors. With the obtained transition form factors, we perform a phenomenological study of the corresponding semi-leptonic decays of doubly heavy baryons induced by the $c\to d/s \ell^+\nu$, $b\to c/u\ell^-\bar \nu$ and $b\to d/s\ell^+\ell^-$. Results for partial decay widths, branching ratios and the polarization ratios $\Gamma_{L}/\Gamma_{T}$s are given. We find that most branching ratios for the semi-leptonic decays induced by the $c\to d,s$ transitions are at the order of $10^{-3}\sim10^{-2}$, which might be useful for the search of other doubly-heavy baryons. Uncertainties in form factors, the flavor SU(3) symmetry and sources of symmetry breaking effects are discussed. We find that the SU(3) symmetry breaking effects could be sizable in charmed baryon decays while in the bottomed case, the SU(3) symmetry breaking effects are less significant. Our results can be examined at the experimental facilities in the future.


I. INTRODUCTION
In hadron physics, quark model has become a well-established tool for the classification of various hadronic states. Most predictions of the quark model have already been experimentally confirmed, but the quest for doubly heavy baryons, baryonic states made of two heavy charm/bottom quarks, has been conducted for a long time. These baryonic states had never been observed in experiments until 2017, when the LHCb collaboration announced the discovery of Ξ ++ cc via Ξ ++ cc → Λ + c K − π + π + [1] with the decay mode suggested in Ref. [2]. This discovery is subsequently confirmed in 2018 in the Ξ ++ cc → Ξ + c π + decay [3], and meanwhile triggered a series of experimental investigations [4][5][6][7]. Now studies of doubly-heavy baryons now open a window to study the hadron spectroscopy and strong interactions in a baryonic system in the presence of two heavy constituent quarks.
Among various properties on doubly heavy baryons, weak decays are of special importance. In the experimental searches for new type of particles the firstly-discovered ones are usually the ground states, which can only be reconstructed via weak decay final states. Thus theoretical analysis of their weak decays can greatly help to optimize the experimental resources. Meanwhile, there exists rich dynamics in weak decay processes and currently only few theoretical approaches are available, which makes them a wonderland full of challenges and opportunities.
On the theoretical side an ingredient in the weak decay is the transition matrix element of the parent particle to a daughter particle, which can be parameterized as form factors. Fortunately, there are various available methods for this part of transition on the market. Thus the decays of a doubly heavy baryon to a singly heavy baryon transition are studied intensively [2,. In particular, some of the form factors are studied under the light front quark model [8,13,22,25], QCD sum rules [26] and light cone sum rules [27,28].
This paper is organized as follows. In Sec. II, we will present the framework of the light-front approach under the diquark picture, and then the flavor-spin wave functions will be discussed. In the appendix, we will provide a new approach to derive the flavor-spin factors. Numerical results of various transition form factors are shown in this section. In Sec. III, phenomenological applications of the doubly heavy baryon decays will be carried out, including numerical results of the decay widths, branching ratios and Γ L /Γ T s of the semileptonic weak decays of doubly heavy baryons. The SU(3) symmetry breaking effect and error estimations will be also discussed in Sec. III. A brief summary is given in the last section. The appendix also contains some brief description of the flavor-spin wave functions, and helicity amplitudes.

II. THEORETICAL FRAMEWORK
The theoretical framework for the charged current and FCNC induced baryonic transitions will be briefly introduced in this section, including the definitions of the states for spin-1/2 and 3/2 baryons, and the extraction of the transition form factors. More details can be found in Refs. [50,54]. Flavor-spin wave functions will be given in the second subsection, while a new derivation is given in the appendices.
A. Light-front quark model FIG. 3: Feynman diagram for doubly heavy baryons B into a spin-1/2 and spin-3/2 ground-state baryons B ′ with two spectator quarks as a diquark. Here P and P ′ are the momentum of the initial and final baryons, respectively. In quark level, the transition is one heavy quark Q1 with momentum p1 decays into a lighter quark q1 with momentum p ′ 1 , and the diquark with momentum p2. The black ball means the weak interaction vertex.
For the J P = 1/2 + baryon states, their wave functions in light-front quark model can be written as here Q 1 = b, c is initial heavy quark, and "(di)" presents the diquark shown in Fig. 3. Ψ SSz is the momentum-space wave function and can be shown with the following equation, here Γ is the coupling vertex of the decay quark Q 1 and the diquark in the baryon state, and when the diquark is a scalar diquark, the coupling vertex is defined as Γ S = 1. In Ref. [55], when an axial-vector diquark is involved, the vertex should be In Eq. (2), φ is a Gaussian-type function: In analogy to the 1/2 + baryon case, a 3/2 + baryon state has a similar expression to Eq. (1) except a different coupling vertex: where With the help of Eqs. (1) and (2), the spin-1/2 to spin-1/2 transition matrix element with (V − A) and tensor current can be derived as With the help of Eqs. (1), (2) and (5), the spin-1/2 to spin-3/2 transition matrix element with (V − A) and tensor current can be derived as In Eqs. (2)- (3) and (5)- (10), and p 1 and p ′ 1 are the four-momentum of the initial and final quark, respectively. P and P ′ are the four-momentum of the initial baryons B and final baryon states B ′ , respectively. q 1 = u, d, s, c means the lighter quark in the final states shown in Fig. 3. When the diquark is a scalar diquark, the coupling vertex is defined as, and when an axial-vector diquark is involved, the vertex should bē Then the extraction of these form factors f 1,2,3,S(A) can be performed as Ref. [54]. (7) and Eq. (15), respectively. At the same time, the approximation P (′) →P (′) need to be taken for the integral. After summing the polarizations of the initial and final baryon states up, we can get the three linear equations as follows, with (Γ µ ) i = {γ µ ,P µ ,P ′µ } and (Γ µ ) i = {γ µ , P µ , P ′µ }. Using the above Eq. (17), we can get the specific expression of the form factors f 1,2,3,S(A) as follows: The form factors g 1,2,3,S(A) can be calculated using the similar process, Then tensor form factors f T 1,2,S(A) or g T 1,2,S(A) defined by Eq. (16) can also be extracted in the similar way with the form factors f 1,2,3,S(A) and g 1,2,3,S(A) , the differences are only ( with in the final state baryons can not be a scalar state, so the transition matrix element q 1 [Q 2 q] S |Γ µ |Q 1 [Q 2 q] S is zero and the physical form factor is given as In this subsection, via performing the inner product of the flavor-spin wave functions of the initial and final states, the overlapping factors c S and c A in Eqs. (51) and (53) can be calculated easily. For shortage of the paper, the detail calculation of the wave functions for the initial and final baryons is arranged in the Appendix A. The flavor spin wave functions for the doubly charmed SU(3) triplets Ξ ++ cc , Ξ + cc and Ω + cc are here the two charm quarks noted by c 1 and c 2 are symmetric. The flavor spin wave functions of the doubly bottomed SU(3) triplets Ξ 0 bb , Ξ − bb and Ω − bb can be obtained through the replacement c → b. While the bottom-charm baryons could form two sets of SU(3) triplets, (Ξ bc , Ω bc ) and (Ξ ′ bc , Ω ′ bc ). The flavor spin wave functions of bottom-charm baryons (Ξ bc , Ω bc ) can be given as while the flavor spin wave functions of bottom-charm baryons (Ξ ′ bc , Ω ′ bc ) are given as The flavor-spin wave functions of the anti-triplet singly charmed baryons can be shown as follows, while the flavor spin wave functions of the sextet of singly charmed baryons are demonstrated as Then we can get the wave functions of the singly bottom baryons by changing c in Eqs. (57)-(58) to b. While for the baryons B * with spin-3/2 in the final states, their flavor spin wave function are given as follows, with q (′) = u, d, s, and Q (′) = c, b.
With the above wave functions of doubly heavy baryons and singly heavy baryons, the overlapping factors c S,A for each transition can be got. The corresponding results of the overlapping factors c S,A for the 1/2 → 1/2 transitions induced by the charged current and the FCNC in Eq. (52) are collected in Tab. II. For the 1/2 → 3/2 transitions induced by the charged current and the FCNC, the numerical results of the overlapping factors c A are listed in Tab. III. Under SU(3) symmetry the doubly heavy baryons can be formed into triplets and the singly heavy baryons can be formed into an anti-triplet and a sextet. The overlapping factors c S,A can be calculated with SU(3) approach, and the detail calculation can be found in the Appendix C. Using the SU(3) approach, one gets the same numerical results of c S(A) as those listed in Tabs. II and III. Then for a spin 1/2 finial state with a scalar and an axial-vector diquark, the physical form factors are then obtained by where these form factors f i , g i , f T i or g T i are defined in Eqs. (22)-(23).
and (13). While for each form factors are functions of q 2 , in order to obtain the dependence of form factors on the momentum q 2 , we take the following parametrization scheme for b → u, d, s, c processes, here F (0) is the numerical result of form factor at q 2 = 0, m fit and δ are two parameters waiting for fitting from numerical result of form factor at different q 2 values. When the fitting result of m fit is an imaginary result using the above parametrization scheme, we need to take the modified parametrization scheme as follows, In the tables, we mark the imaginary results with superscripts " * ". While for c quark decay process, the single pole structure is assumed for c → u, d, s decays, m pole s are respectively 1.87, 1.87, 1.97 GeV.
• For the charged current transition 1/2 → 1/2, the results for form factors with a scalar diquark or an axial-vector diquark spectator are shown in Tabs. V, III and III. As shown in Eq. (15), the numerical results of the form factors can be used to calculate the physical hadronic transition matrix elements. We take Ξ 0 bb → Σ + b as an example to show the q 2 -dependence of form factors in Fig. 4. There is no singular point for the form factors f 1,2,3 and g 1,2,3 in the integration interval shown in Fig. 4.
• For the FCNC transition 1/2 → 1/2, the results for form factors with a scalar diquark or an axial-vector diquark spectator are shown in Tabs. III, III and III. With the help of the results of the form factors and Eqs. (15)-(16), one can calculate the physical hadronic transition matrix elements. Ξ 0 bb → Λ 0 b is taken as an example to show the q 2 -dependence of these form factors which are depicted in Fig. 5. As one can see, these form factors are stable and no divergence exists in the integration interval.
• For the charged current transition 1/2 → 3/2, the results for form factors with an axial-vector diquark spectator are shown in Tabs. III and III. As shown in Eq. (22), the numerical results of the form factors can be used to calculate the physical hadronic transition matrix elements. In Fig. 6 we use Ξ 0 bb → Σ * 0 b as an example to show the q 2 -dependence of form factors. As shown in Fig. 6 these form factors are stable, which indicates our fitting result in the Tabs. III and III are reliable.
bc Ω * 0 bc masses 5.832 5.833 5.835 5.949 5.955 6.085 6.985 6.985 7.059 • For the FCNC transition 1/2 → 3/2, the results for form factors with an axial-vector diquark are shown in Tabs. III, XIV and XV. As shown in Eqs. (22) and (23), the numerical results of the form factors can be used to calculate the physical hadronic transition matrix elements. To describe the dependence of form factors on q 2 , we take the transition Ω bb → Ξ ′ * − b as an example shown in Fig. 7. The curves are all approaching to zero at large q 2 stably.

IV. SEMI-LEPTONIC WEAK DECAYS
For the charged current process c → d, s l + ν l , the effective Hamiltonian is and for b → u, c l −ν l , the effective Hamiltonian has been given as While for the FCNC process b → sl + l − , the effective Hamiltonian can be given as In Eqs. (68), (69) and (70), the Fermi constant G F and the CKM matrix elements are taken from Ref. [62]: The reader interested in the explicit forms of operators O i in Eq. (70) can consult Ref. [63]. Wilson coefficients C i for each operators O i are calculated in the leading logarithmic approximation, with m W = 80.4 GeV and µ = m b,pole [63] and can be given as follows, The numerical result of the parameters F (0), δ and m f it are shown in Tab. III. For the FCNC process B b → B ′ s l + l − , the amplitude can be obtained in following form, i,S(A) of doubly bottom baryon B bb decay with b → u, c processes. The parametrization scheme in Eq. (66) is introduced for these form factors with asterisk, and Eq. (65) for all the other ones. In Refs. [64][65][66], the signs before C eff 7 are different. In this paper, we take the same sign with the ones in Refs. [64,65], which is different from the one in Ref. [66]. In the above Eq. (73), C eff 7 and C eff 9 are obtained as [67] The auxiliary functions h have been given as While for the FCNC process b → dl + l − , the corresponding effective Hamiltonian and amplitude can be got by taking a replacement s → d similarly.     67) is introduced for these form factors, and the value of the singly pole m pole is taken as 1.87 GeV for c → u.        For the charged current induced transition, the kinematics are shown in Fig. 8, and the helicity amplitudes are defined by here ǫ µ and q µ are the polarization vector and four-momentum of the virtual propagator W, and λ W means the polarization of the virtual propagator W. λ and λ ′ are the helicities of the baryon in the initial and final baryon states, respectively. The detail derivation process of helicity amplitudes can be found in Appendix B. These helicity amplitudes are related to the form factors by the following expressions.  • The transition B i (λ) → B f (λ ′ ) matrix elements are parameterized as shown in Eq. (15), and the helicity amplitudes of 1/2 → 1/2 charged current transition can be expressed with following equations, Here f Then the total helicity amplitudes for (V-A) current can be shown as follows,   The polarized differential decay widths can be given as    , HV here f Then we can get the total helicity amplitudes, The polarized differential decay widths can be given as

the FCNC transition
For the FCNC induced transition, we adopt the helicity amplitudes as follows, and here ǫ µ and q µ are the polarization vector and four-momentum of the virtual vector propagator V, and λ V means the polarization of the virtual vector propagator V. λ and λ ′ are the helicities of the baryon in the initial and final baryon states, respectively. In the following, the superscripts "V l " and "A l " denote the corresponding leptonic counterparts lγ µ l andlγ µ γ 5 l, respectively.
• The transition B i (λ) → B f (λ ′ ) matrix elements are parameterized with Eqs. (15)- (16), and the helicity amplitudes of 1/2 → 1/2 induced by FCNC transition can be obtained with following expressions, and where the "HV" and "HA" are corresponding to the Γ µ and Γ µ γ 5 parts in Eq. (90), respectively. The total helicity amplitude can be given as The specific expressions of H A l ,λ λ ′ ,λV are similar with the ones of H V,λ λ ′ ,λV , except Furthermore, the timelike polarizations of the virtual vector propagator V for the helicity amplitudes, H A l t are necessary for FCNC induced transitions, and In the above Eq. (92-97), the following notations have been introduced: and Here f 1 2 → 1 2 ,(T ) i (q 2 ) and g 1 2 → 1 2 ,(T ) i (q 2 ) are the physical form factors illuminated by Eq. (61). The longitudinally and transversely polarized differential decay widths read, with V CKM = V tb V * ts for b → s processes, V CKM = V tb V * td for b → d processes and | p 1 | = 1 2 q 2 − 4m 2 l .
• The transition 1/2 → 3/2 matrix elements are parameterized with Eqs. (22)- (23), and the helicity amplitudes induced by FCNC can be given by the following expressions, and where the "HV" and "HA" are corresponding to the Γ µ and Γ µ γ 5 parts in Eq. (90), respectively. Then we can get the total helicity amplitudes, The specific expressions of H A l ,λ λ ′ ,λV are similar with the ones of H V,λ λ ′ ,λV , except Furthermore, the timelike polarizations of the virtual vector propagator V for the helicity amplitudes, H A l t are necessary for FCNC induced transitions, and In Eqs. (102-110), the following notations are introduced.
Here f 1 2 → 3 2 ,(T ) i (q 2 ) and g 1 2 → 3 2 ,(T ) i (q 2 ) are physics form factors illuminated by Eq. (62). The longitudinally and transversely polarized differential decay widths read In the end, the total differential decay width can be written as then we can calculate total width using the following integral, where q 2 min = 0 for these decays with charged current, while q 2 min = 4m 2 l for other decays with FCNC. At the same time, the ratio of the longitudinal to transverse decay rates Γ L /Γ T can be calculated.

B. Results for semi-leptonic decays
• For the transition 1/2 → 1/2 with V − A current, the integrated partial decay widths, the relevant branching ratios and Γ L /Γ T s are shown in Tab. XVI. The dependence of q 2 of the differential decay widths can be shown in Fig. 9.
• For the transition 1/2 → 1/2 induced by FCNC, the integrated partial decay widths, the relevant branching ratios and Γ L /Γ T s are shown in Tab. XVII. The dependence of q 2 of the differential decay widths can be shown in Fig. 10.
• For the transition 1/2 → 3/2 with V − A current , the integrated partial decay widths, the relevant branching ratios and Γ L /Γ T s are shown in Tab. XVIII. The dependence of q 2 of the differential decay widths can be shown in Fig. 11.
• For the transition 1/2 → 3/2 with FCNC, the integrated partial decay widths, the relevant branching ratios and Γ L /Γ T s are shown in Tab. XIX. The dependence of q 2 of the differential decay widths can be shown with Fig. 12.
Some comments on the results for phenomenological observables are given as follows.
• It can be seen in Tabs. XVI-XIX that the decay widths for the four cases have the following hierarchical difference.
In the transition 1/2 → 1/2 and 1/2 → 3/2 with FCNC cases, the decay widths are very close to each other for l = e/µ cases, while it is about one order of magnitude smaller for l = τ case. This can be attributed to the much smaller phase space for l = τ case.
• A reasonable modification with momentum-space wave function Ψ SSz in the case of an axial-vector diquark involved is performed in this work in Eqs. (2) and (3). While, in Refs. [8,13,22], the momentum-space wave function Ψ SSz in the case of an axial-vector diquark involved is defined as In Ref. [8] the extraction approach is different from the one used in this work and in Refs. [13,22]. In order to find out the impact on the form factors and decay widths of these two factors: the extraction method of the form factors and the baryon wave function related to the axial vector diquark, we list numerical results of theses form factors of the three decay channels and the corresponding partial decay widths in Tab. XX. The corresponding numerical results in Refs. [8,13,22] are also given in Tab. XX. Firstly, comparing each first two lines for Ξ ++ cc → Λ + c l + ν l , Ξ 0 bb → Ξ 0 b e + e − and Ξ ++ cc → Σ * + c l + ν l , we could find that partial decay width differences coming from the different wave function with axial-vector diquark are small, but there are some differences among the form factors. Secondly, comparing the second line and third line of the channel Ξ ++ cc → Λ + c l + ν l , we could find the extracting approach of these form factors will bring in some effect in the form factors and decay widths; So the effect in the form factors and decay widths brought in by the extraction approach is much larger than that of the definition of the wave function Ψ SSz .   • Since there exist uncertainties in the lifetimes of the parent baryons, there may be some small fluctuations in the results for branching ratios. Form Tab. XVII, we may find that These channels may be firstly examined at experimental facilities like LHC or BelleII.
Using these form factors, we have also preformed the calculation of phenomenological observables of these corresponding semileptonic weak decays of doubly heavy baryons with the results shown in Tabs. XVI-XIX. The flavor SU(3) symmetry and sources of symmetry breaking are also discussed in great details. We find that, • most branching ratios for spin-1/2 to spin-1/2 with c → d, s processes are at the order 10 −3 ∼ 10 −2 , which might be examined at experimental facilities at LHC or Belle-II; • the different extraction approaches could give sizable differences to form factors; • the uncertainties of form factors and decay widths caused by model parameters are sizable; • the ratios Γ L /Γ T s have the rules shown in Eqs. (122)-(126), for these decay channels have the same decay in quark level and the same overlapping factors and form factors; • most of our results are comparable to the theoretical results in Refs. [8,13,22]; • since the mass difference between the u and d quark has been neglected and the strange quark is much heavier, the SU(3) relations shown in Eq. (127) for the channels involving u, d quark and s quark can be broken; • the SU(3) symmetry breaking is sizable in the charmed baryon decays, while for the bottomed case the SU (3) symmetry breaking is small.
This work completes the study of form factors in the traditional light-front quark model with the quark-diquark constituent viewpoint. We hope our phenomenology predictions for these semi-leptonic decays could be tested by LHCb and other experiments in the future.
where q = u, d, s, Q 1 = b and Q 2 = c.
Then the flavor-spin wave functions of the singly heavy baryons in the final states are given with the following functions.
Then the the hadronic helicity amplitude can be defined as Ξ + c (P ′ , λ ′ )|sγ µ (1 − γ 5 )c|Ξ ++ cc (P, λ) ε * W,µ (λ W ) = Ξ + c (P ′ , λ ′ )|sγ µ c|Ξ ++ cc (P, λ) ε * W,µ (λ W ) − Ξ + c (P ′ , λ ′ )|sγ µ γ 5 c|Ξ ++ cc (P, λ) ε * W,µ (λ W ) While the leptonic part amplitude can be calculated by the follow equation, In this work, the dynamics involved in the the hadronic amplitude are all in the rest frame of the initial states B i , B i : P µ = (M, 0, 0, 0), B f : P ′µ = (E ′ , 0, 0, −| P ′ |), W : q µ = (E W , 0, 0, | P ′ |), While the dynamics involved in the the leptonic part are all in the rest frame of the virtual vector particle W . The spinor expressions involved in this work are given as follows: here λ denotes the helicity of the spinor and (θ,φ) are the direction of the momentum of the initial and final particles. In this work, we take the direction of the momentum of virtual vector particle W as positive with (θ, φ) = (0, 0) and the direction of the momentum of the final baryons as negative direction with (θ, φ) = (π, π), which are shown in Fig.8. The spinor expressions of anti-particles in the leptonic part are In this work, the direction of the momentum of l + is (θ l , φ l ), and the direction of the momentum of ν is (π − θ l , φ l + π).