Weak decays of doubly heavy baryons: the FCNC processes

The discovery of doubly heavy baryon provides us with a new platform for precisely testing Standard Model and searching for new physics. As a continuation of our previous works, we investigate the FCNC processes of doubly heavy baryons. Light-front approach is adopted to extract the form factors, in which the two spectator quarks are viewed as a diquark. Results for form factors are then used to predict some phenomenological observables, such as the decay width and the forward-backward asymmetry. We find that most of the branching ratios for $b\to s$ processes are $10^{-8}\sim10^{-7}$ and those for $b\to d$ processes are $10^{-9}\sim10^{-8}$. The flavor SU(3) symmetry and symmetry breaking effects are explored. Parametric uncertainties are also investigated.

Since then, great theoretical interests have been devoted to the study of doubly heavy baryons, some of them can be found in Refs. . Recently, some more new results were reported on Ξ ++ cc by LHCb collaboration, including the first measurement of the lifetime [23] and the first observation of the new decay mode Ξ ++ cc → Ξ + c π + [24]. After discovering Ξ ++ cc in the decay mode of Ξ ++ cc → Λ + c K − π + π + , LHCb collaboration is also continuing to search for the Ξ + cc and Ξ bc baryons [25]. Comprehensive theoretical studies on weak decays are highly demanded and our previous and forthcoming works aim to fill this gap. In our previous works [4,5], we have presented the calculations of 1/2 to 1/2 and 1/2 to 3/2 weak decays. As a continuation, we investigate the flavor-changing neutral current (FCNC) processes in this work.
FCNC processes are considered to be an ideal place to the precise test of Standard Model (SM) and the search for new physics (NP), while the discovery of the doubly heavy baryon provides us a new platform. b → d/s process in SM is induced by the loop effect, thus its decay width is small. NP effects manifest themselves in two different ways. One is to enhance the Wilson coefficients, and the other is to introduce new effective operators which are absent in the SM. The typical value of branching ratio for FCNC processes is ∼ 10 −6 for mesonic sector. However, the small branching ratio can be compensated by the high luminosity at the B factories. Also, with the accumulation of data, we are in an increasingly better position to study these semi-leptonic process. Baryonic rare decay modes, which are also induced by b → d/s l + l − at the quark level, are also important as its mesonic counterparts. Serious attention is deserved, both theoretically and experimentally.
A doubly heavy baryon is composed of two heavy quarks and one light quark. Light flavor SU(3) symmetry arranges them into the presentation 3. For 1/2 + doubly heavy baryons, we have Ξ ++,+ cc and Ω + cc in the cc sector, Ξ 0,− bb and Ω − bb in the bb sector, while there are two sets of baryons in the bc sector depending on the symmetric property under the interchange of b and c quarks. For the symmetric case, the set is denoted by Ξ +,0 bc and Ω 0 bc , while for the asymmetric case, the set is denoted by Ξ ′+,′0 bc and Ω ′0 bc . 1 In reality these two sets probably mix with each other, which will not be considered in this work.
In the above, the quark components of the baryons have been explicitly presented in the brackets, and the quarks that participate in weak decay are put in the first place. Taking the b → s process in bc sector as an example, the final baryons Ξ +,0 c belong to the presentation of3, while Ξ ′+,′0 c and Ω 0 c belong to the presentation of 6, as can be seen from Fig. 1.
Light front approach will be adopted to deal with the dynamics in the decay. This method has been widely used to study the mesonic decays [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. Its application to baryonic sector can be found in Refs. [45][46][47][48][49]. As in our previous works, diquark picture is once again adopted, i.e., the two spectator quarks are viewed as a whole system, as can be seen from Fig. 2. The two spectator quarks form a scalar diquark or an axial-vector diquark. Generally speaking, both types of diquarks contribute to the decay process and their contribution weights can be determined by the wave functions of the baryons in the initial and final states. SU(3) analyses for FCNC processes will also be conducted. A quantitative predictions of SU(3) symmetry breaking effects will be performed within the light-front approach.
The rest of the paper is arranged as follows. In Sec. II, we will present the effective Hamiltonian responsible for the b → d/s l + l − process. Then the framework of light-front approach under the diquark picture will be briefly introduced, then flavor-spin wave functions will also be discussed. Some phenomenological observables are collected in the last subsection of Sec. II. Numerical results are shown in Sec. III, including the results for form factors, decay widths, forward-backward asymmetry, the SU(3) symmetry breaking and the error estimates. A brief summary is given in the last section.
1 is the initial (final) quark momentum, p2 is the diquark momentum and the cross mark denotes the weak interaction.

II. THEORETICAL FRAMEWORK
A. The effective Hamiltonian The effective Hamiltonian for b → sl + l − is given as Here the explicit forms of the four-quark and the penguin operators O i can be found in Ref. [50] and C i are their corresponding Wilson coefficients, which are presented in Table I in the leading logarithm approximation [50]. The transition amplitude for B → B ′ l + l − turns out to be Note that the sign before C eff 7 is different in literatures. Our result coincides with those in Refs. [51,52], but is different from that in Ref. [53]. In Eq. (3), C eff 7 and C eff 9 are defined by [54] C eff The auxiliary functions h are given by The effective Hamiltonian and transition amplitude for b → d process can be written down in a similar way. Light-front approach for 1/2 → 1/2 FCNC transition will be briefly introduced in this subsection, including the definitions of the states for spin-1/2 baryons, and the extraction of form factors. More details can be found in Ref. [45,49].
In the light-front approach, the wave function of 1/2 + baryon with a scalar or an axial-vector diquark are expressed as where q 1 stands for b/s quark in the initial/final state, and the diquark is denoted by (di), which is composed of one b quark and one light quark. The momentum-space wave function Ψ SSz is given as with A = 1 and Γ = 1 for the case of a scalar diquark involved, and A = 3(m1M0+p1·P ) 3m1M0+p1·P +2(p1·p2)(p2·P )/m 2 2 and Γ = − 1 √ 3 γ 5 / ǫ * (p 2 , λ 2 ) for the case of an axial-vector diquark involved. A Gaussian-type function is usually adopted for φ: Taking the V − A current of b → s process as an example, the transition matrix element can be derived as where In a similar way, one can obtain from the second equation of Eqs. (15) Taking the extraction of f i as an example, these form factors can be extracted as follows [49]. Multiplying the corresponding expressions in Eq. (9) and Eqs. (14) byū(P, S z )(Γ µ ) i u(P ′ , S ′ z ) with (Γ µ ) i = {γ µ , P µ , P ′µ } respectively, and taking the approximation P (′) →P (′) within the integral, and then summing over the polarizations in the initial and final states, one can arrive at with (Γ µ ) i = {γ µ ,P µ ,P ′µ }. Then f i can be determined by solving linear equations. The form factors g i can be obtained in a similar way. Only f T 2,3 or g T 2,3 can be extracted in this way with (Γ µ ) i = {γ µ , P µ }, f T 1 or g T 1 is then obtained by Eq. (17) or (18).

C. Flavor-spin wave functions
In fact, the flavor-spin wave functions are not taken into account in the last subsection. This problem will be fixed in this subsection. We consider first the initial states. For the doubly bottomed baryons, the wave functions are given as with q = u, d or s for Ξ 0 bb , Ξ − bb or Ω − bb , respectively. For the bottom-charm baryons, there are two sets of states, as discussed in Sec. I. The wave functions of bottom-charm baryons with an axial-vector bc diquark are given as while those with a scalar bc diquark are with q = u, d or s for Ξ bc , respectively. For the final states, the singly charmed baryon which belongs to anti-triplets are given as For the sextet of singly charmed baryons, the following wave functions are needed The final states of singly bottomed baryons can be written down in a similar way. Finally, the overlapping factors are determined by taking the inner product of the flavor-spin wave functions in the initial and final states. The corresponding results are collected in Table II for both b → s and b → d processes. The physical form factors are then obtained by where F S(A) denotes the form factor f i , g i , f T i or g T i with a scalar diquark (an axial-vector diquark) involved.

D. Phenomenological observables
The hadronic helicity amplitudes can be defined by and where ǫ µ (q µ ) is the polarization vector (four-momentum) for an intermediate vector particle, λ V denotes its polarization, λ (′) is the polarization of the baryon in the initial (final) state. Hereafter the superscript "V " ("A") always means that its corresponding leptonic counterpart islγ µ l (lγ µ γ 5 l). It should not be confused with the notation of the vector current (axial-vector current) in the hadronic matrix element. Note that Eqs. (14) and (15) have the same parameterization, so it is convenient to introduce the following notations: and Then the Γ µ and Γ µ γ 5 parts in Eq. (26) are calculated respectively as: and The total hadronic helicity amplitude is then given by H A,λ λ ′ ,λV has the complete the same form as the corresponding H V,λ λ ′ ,λV but with the following replacements: In addition, the timelike polarizations for H A are also needed and Finally, the angular distribution is given by the following expression Here the squared amplitude is and The differential decay width is given as where the q 2 is the invariant mass of the dilepton and the longitudinally and transversely polarized decay widths are respectively The normalized differential forward-backward asymmetry is defined by Then one can obtain when substituting Eqs. (36) and (37) into Eq. (43).

A. Inputs
The constituent quark masses are given as (in units of GeV) [36][37][38][39][40][41][42][43][44] where q = u, d, s. The masses and lifetimes of the parent baryons are collected in Table III [26,55,56]. The masses of the daughter baryons are given in Table IV [57]. Fermi constant and CKM matrix elements are give as [57] G F = 1.166 × 10 −5 GeV −2 , To access the q 2 -distribution, the following single pole structure is assumed for form factors Here F (0) is the value of the form factor at q 2 = 0, and the numerical results for f (T ) i and g (T ) i predicted by the light-front approach are collected in Tables V to VIII for b → s process and Tables IX to XII for b → d process. m pole is taken as 5.37 GeV for b → s process and 5.28 GeV for b → d process, which, in practice, are taken as the masses of B s and B mesons, respectively. The discussion for the validity of this assumption can be found in our previous work [44].
The physical form factors can then be obtained by Eq. (25) and Eq. (48).

C. Results for phenomenological observables
The decay widths are shown in Tables XIII to XV for b → s process and Tables XVI to XVIII for b → d process. Some comments are given in order.
• Since there exist uncertainties in the lifetimes of the parent baryons, there may exist small fluctuations in the results for branching ratios.
• It can be seen from these tables that, the decay widths are very close to each other for l = e/µ cases, while it is roughly one order of magnitude smaller for l = τ case. This can be attributed to the much smaller phase space for l = τ case.     (17) and (18) respectively.          • Most of the branching ratios are 10 −8 ∼ 10 −7 for b → s process and 10 −9 ∼ 10 −8 for b → d process, which are roughly one order of magnitude smaller than the corresponding mesonic cases. This is because we believe that the lifetime of the doubly heavy baryon is roughly one order of magnitude smaller than that of B meson.
The differential decay widths for Ξ 0 bb → Ξ 0 b l + l − with l = e, µ, τ are plotted in Fig. 3, where the resonant contributions are not taken into account. It can be seen that the curves for l = e/µ almost coincide with each other and the much smaller phase space for l = τ case can be seen clearly. The curves of forward-backward asymmetry (FBA) for Ξ 0 bb → Ξ 0 b l + l − with l = e, µ, τ are plotted in Fig. 4. It can be seen from this figure that, the zero-crossing point is around q 2 ≈ 2 GeV 2 for l = e/µ cases. The zero-crossing points for other b → s processes and for b → d processes can be found in Tables XIX and XX respectively. It can be seen from these tables that these s 0 roughly range from 2 to 3 GeV 2 .
Following Ref. [52], we now analyse the zero-crossing point s 0 of FBA which satisfies or Re(C eff 9 (s 0 )) + 2 Here R is defined by with The meaning of R can be seen more clear in Λ b → Λ process with the help of the heavy quark symmetry. In the heavy quark symmetry limit, the matrix elements of all the hadronic currents can be parameterized by only two independent form factors [58] where Γ is the product of Dirac matrices, v µ ≡ p µ Λ b /m Λ b is the four velocity of Λ b . Under the heavy quark symmetry, and R is reduced to the following form where we have also neglected the m Λ /m Λ b term. If we further take into account the fact that F 2 ≪ F 1 for Λ b → Λ process [59][60][61], then The values of R for FCNC processes of doubly heavy baryons can be found in Tables XIX and XX. It can be seen from these tables that R roughly ranges from 0.3 to 0.4 for bb sector, while it lies in the interval of [0.6, 0.7] for bc sector.  bc → Ξ ′0 c τ + τ − 6.45 × 10 −20 9.12 × 10 −9 0.99 Ω 0 bc → Ω 0 c τ + τ − 9.12 × 10 −20 3.05 × 10 −8 0.99

D. SU(3) analyses
According to the flavor SU(3) symmetry, there exist the following relations among these FCNC processes. These relations can be readily derived using the overlapping factors given in Table II. For b → s process, we have for bb sector, for bc sector and for bc ′ sector. For b → d process, we have      Also taking the process of Ξ 0 bb → Ξ 0 b as an example, the uncertainties caused by the model parameters and the single pole assumption will be given in this subsection. The error estimates for the form factors can be found in Table XXIV, in which the errors come from β i , β f and m di , respectively. The error estimates for the decay widths are listed below: Γ(Ξ 0 bb → Ξ 0 b e + e − ) = (1.98 ± 0.49 ± 1.21 ± 0.13 ± 0.26) × 10 −19 GeV, Γ(Ξ 0 bb → Ξ 0 b µ + µ − ) = (1.92 ± 0.48 ± 1.18 ± 0.14 ± 0.26) × 10 −19 GeV, Γ(Ξ 0 bb → Ξ 0 b τ + τ − ) = (3.72 ± 0.96 ± 2.52 ± 0.51 ± 1.28) × 10 −20 GeV, where these errors come from β i , β f , m di and m pole , respectively. The first three model parameters are all varied by 10%, while m pole , which is responsible for the single pole assumption, is varied by 5%. It can be seen from Table XXIV and Eqs. (63) that, the uncertainties caused by these parameters may be sizable.

IV. CONCLUSIONS
In our previous work, we have investigated the weak decays of doubly heavy baryons for 1/2 to 1/2 case and for 1/2 to 3/2 case. As a continuation, we investigate the FCNC processes in this work. Light-front approach under the diquark picture is once again adopted to extract the form factors. The same method was applied to study the singly heavy baryon decays and reasonable results were obtained [62]. The extracted form factors are then applied XXIV: Error estimates for the form factors, taking Ξ 0 bb → Ξ 0 b as an example. The first number is the central value, and the following 3 errors come from βi = β Ξ 0 bb , β f = β Ξ 0 b and m di = m (bu) , respectively. These parameters are all varied by 10%. to study some observables in these FCNC processes. We find that most of the branching ratios for b → s processes are 10 −8 ∼ 10 −7 , while those for b → d processes are 10 −9 ∼ 10 −8 , which are roughly one order of magnitude smaller than those in mesonic sector. This is because we believe that the lifetime of the doubly heavy baryon is roughly one order of magnitude smaller than that of B meson. SU(3) symmetry and sources of symmetry breaking are discussed.
The error estimates are also investigated.