Exclusive $c\to s,d$ semileptonic decays of ground-state spin-1/2 doubly charmed baryons

We evaluate exclusive semileptonic decays of ground-state spin-1/2 doubly heavy charmed baryons driven by a $c\to s,d$ transition at the quark level. Our results for the form factors are consistent with heavy quark spin symmetry constraints which are valid in the limit of an infinitely massive charm quark and near zero recoil. Only a few exclusive semileptonic decay channels have been theoretically analyzed before. For those cases we find that our results are in a reasonable agreement with previous calculations.

. Quantum numbers of double-c and single-c heavy baryons involved in this study.J π and I are the spin-parity and isospin of the baryon, while S π is the spin-parity of the two heavy or the two light quark subsystem.n denotes a u or d quark.

I. INTRODUCTION
Doubly heavy baryons offer a unique opportunity to study QCD in the presence of heavy quarks as well as providing, through their decays, information on the weak sector of the Standard Model.From the experimental point of view the SELEX Collaboration claimed evidence for the Ξ + cc baryon, in the Λ + c K − π + [1] and pD + K − [2] decay modes.The combined analysis gave a mass of M Ξ + cc = 3518.7 ± 1.7 MeV/c 2 .However, other experimental collaborations like FOCUS [3], BABAR [4] and BELLE [5] found no evidence for doubly charmed baryons and the Ξ + cc has only been assigned a one star status by the Particle Data Group (PDG) [6].Furthermore, no evidence for the Ω + cc has been reported so far.Nevertheless, being the lightest among the doubly heavy baryons, one expects doubly charmed baryons masses and decay properties to be measured in the near future.
While there are many different theoretical determinations of the doubly charmed baryon masses , that range from non-relativistic quark model calculations to unquenched lattice QCD, there are just a few studies of their decays.
Total decay widths were evaluated in Refs.[29][30][31][32], and total semileptonic and non-leptonic decay rates were predicted in Ref. [30].Some exclusive non-leptonic as well as semileptonic decay rates of the Ξ cc baryon were calculated in [31].Finally the decay Ξ cc → Ξ ′ c e + ν e was analyzed in Ref. [33] 1 .To our knowledge, there is not exist any systematic study of the exclusive semileptonic c → s and c → d decay channels of the Ξ cc and Ω cc baryons.This is the purpose of this work, where we shall concentrate in transitions to the lowest-lying, 1/2 + or 3/2 + , single-c baryons in the final state.Besides, we will pay a special attention to possible violations of heavy quark spin symmetry relations among the relevant form factors, which one might expect to be sizable at the charm mass scale.
In Table I, we show the quantum numbers of the baryons involved in our calculation.Quark model masses have been taken from our previous works in Refs.[25,34], where they were obtained using the AL1 potential of Refs.[35,36].Experimental masses are the ones quoted by the PDG and in the table we quote the average over the different charge states.With the exception of the Ξ cc , the agreement is fairly good.For the actual calculation of the decays we shall use experimental masses except for the Ξ cc , which is not well established, and for the Ω cc due to the absence of experimental data.In those two cases, we take our model predictions in Table I which are in agreement with different lattice estimates [13,17,26].
The paper is organized as follows: In Sec.II we give general formulae for the semileptonic decay width and the form factor decomposition of the hadronic matrix elements of the weak current.In Sec.III we will find out heavy quark spin symmetry relations between different form factors. Finally in Sec.IV we present the results.The paper contains also two appendices: In Appendix A we give a brief description of the baryon states within the model and the expressions for the wave functions of the different baryons and in Appendix B we relate the form factors to weak matrix elements and show how the latter ones are evaluated in the model.

II. DECAY WIDTH AND FORM FACTOR DECOMPOSITION OF THE HADRONIC CURRENT
The total decay width for semileptonic c → l transitions, with l = s, d, is given by where |V cl | is the modulus of the corresponding Cabibbo-Kobayashi-Maskawa (CKM) matrix element for a c → l quark transition, for which we shall use |V cs | = 0.97345 and |V cd | = 0.2252 taken from Ref. [6].G F = 1.16637(1) × 10 −11 MeV −2 [6] is the Fermi decay constant, P, M (P ′ , M ′ ) are the four-momentum and mass of the initial (final) baryon, q = P − P ′ and w is the product of the baryons four-velocities . In the decay, w ranges from w = 1, corresponding to zero recoil of the final baryon, to a maximum value given, neglecting the neutrino mass, by , which depends on the transition and where m is the final charged lepton mass.Finally L αβ (q) is the leptonic tensor after integrating in the lepton momenta and H αβ (P, P ′ ) is the hadronic tensor.
The leptonic tensor is given by where with The hadronic tensor reads with J the initial baryon spin, B, r P B ′ , r ′ P ′ the initial (final) baryon state with three-momentum P ( P ′ ) and spin third component r (r ′ ) in its center of mass frame.J µ cl (0) is the charged weak current for a c → l quark transition Baryonic states are normalized such that with E the baryon energy for three-momentum P .
A. Form factors for 1/2 → 1/2 and 1/2 → 3/2 transitions Hadronic matrix elements can be parameterized in terms of form factors.For 1/2 → 1/2 transitions the commonly used form factor decomposition reads The u r are Dirac spinors normalized as (u r ′ ) † u r = 2E δ rr ′ .v µ , v ′µ are the four velocities of the initial and final baryons.The three vector F 1 , F 2 , F 3 and three axial G 1 , G 2 , G 3 form factors are functions of w or equivalently of For 1/2 → 3/2 transitions we follow Llewellyn Smith [37] to write Here u B ′ λ r ′ is the Rarita-Schwinger spinor of the final spin 3/2 baryon normalized such that (u In appendix B we give the expressions that relate the form factors to weak current matrix elements and show how the latter ones are evaluated within the model.

III. HEAVY QUARK SPIN SYMMETRY
In hadrons with a single heavy quark the dynamics of the light degrees of freedom becomes independent of the heavy quark flavour and spin when the mass of the heavy quark is much larger than Λ QCD and the masses and momenta of the light quarks.This is the essence of heavy quark symmetry (HQS) [38][39][40][41].However, HQS can not be directly applied to hadrons containing two heavy quarks.The static theory for a system with two heavy quarks has infra-red divergences which can be regulated by the kinetic energy term hQ (D2 /2m Q )h Q .This term breaks the heavy quark flavour symmetry, but not the spin symmetry for each heavy quark flavour [42].This is known as heavy quark spin symmetry (HQSS).HQSS implies that all baryons listed in Table I with the same flavour wave-function are degenerate.The invariance of the effective Lagrangian under arbitrary spin rotations of the c quark leads to relations, near the zero recoil point (w = 1 ↔ q 2 = (M − M ′ ) 2 ↔ | q | = 0), between the form factors for vector and axial-vector currents between the Ξ cc and Ω cc baryons and the single charmed baryons listed in Table I.These decays are induced by the semileptonic weak decay of the c quark to a d or a s quark.The consequences of spin symmetry for weak matrix elements can be derived using the "trace formalism" [43,44].To represent the lowest-lying S-wave ccl baryons we will use wave-functions comprising tensor products of Dirac matrices and spinors, namely [45] 2 : where we have indicated Dirac indices α, β and γ explicitly on the right-hand side and r is a helicity label for the baryon.Under a Lorentz transformation, Λ, and a c quark spin transformation S c , this wave-function of the form Γ αβ u γ transforms as: The state in Eq. ( 10) is normalized3 to (−ūu = −2M ), with M the mass of the state.On the other hand, the Λ c , Σ c and Σ * c final baryons are represented by the following spinor wave functions [44] Λ For the Σ * c , u λ γ (v ′ , r ′ ) is a Rarita-Schwinger spinor.For Σ c , we have taken into account that the light quarks are coupled to total spin 1 that gives a total spin 1/2 for the baryon when the spin of the light subsystem is summed with the spin of the charm quark.Under a Lorentz transformation , Λ, and a c quark spin transformation S c , the above spinor wave functions transform like S(Λ) U and S c U, respectively, with U (= u, 1 √ 3 (v ′λ + γ λ )γ 5 u, u λ ) each of the spinors appearing in Eqs. ( 12)- (14).States are normalized to ūu = 2M ′ , (−ūu = −2M ′ ) and ūλ u λ = −2M ′ for the Λ c , Σ c and Σ * c , respectively.We can now construct amplitudes for semileptonic Ξ cc → Λ c , Σ c , Σ * c decays, determined by matrix elements of the weak current J µ = dγ µ (1 − γ 5 )c.To that end, we write the most general form for the matrix element respecting the heavy quark spin symmetry, taking into account that under a c quark spin transformation J µ → J µ S † c .We should distinguish two situations depending on whether the total spin of the two light quarks in the final baryon is S = 0 or S = 1.In the first (second) case, the spinor wave-function U that represents the final baryon does not have (has) a Lorentz index.With all these considerations, we have + ūλ Note that near the zero recoil point, where the spin symmetry should work best, HQSS considerably reduces the number of independent form factors, and it relates those that correspond to transitions where the spin of the two light quarks in the final baryon is S = 1.Indeed, we find at w = 1 where the total spin of the two light quarks in the final baryon is S = 0: In the equal mass transition case one would find that η 0 is normalized as η 0 (w = 1) = 3 2 .
• Total spin of the two light quarks in the final baryon is In the equal mass transition case one would have that η 1 (w = 1) = 1 √ 2 when the two light quarks in the final state are different and η 1 (w = 1) = 1 when they are equal (Ω c and Ω * c ).
Relations (20), ( 21) and ( 22) are exactly satisfied in the quark model when the heavy quark mass is made arbitrarily large, and thus the calculation is consistent with HQSS constraints.
Comparison of F1 + F2 + F3 (solid) and 3G1 (dashed) for the specified transitions.The two light quarks in the final baryon have total spin S = 0.In the limit in which the heavy quark mass is made arbitrarily large one has that, near zero recoil (w = 1), F1 + F2 + F3 = 3G1.

IV. RESULTS AND DISCUSSION
We start by checking that our calculation respects the constraints on the form factors deduced from HQSS.In Figs. 1  and 2, we show to what extent the relations of ( 20), ( 21) and ( 22) deduced above are satisfied for the actual m c value.In all cases we see moderate deviations, that stem from 1/m c corrections, at the level of about 10% near zero recoil, though larger than those found in [46] for the b → c transitions of the Ξ bc and Ξ bb baryons.These discrepancies tend to disappear when the mass of the heavy quark is made arbitrarily large.This is illustrated in Fig. 3 where we show, for w = 1 and for three different heavy quark masses, the form factor ratio 3G1 F1+F2+F3 from the Ξ ++ cc → Ξ + c transition, the form factor ratio 3/5G1 F1+F2+F3 for the Ω + cc → Ω 0 c transition and the ratio 3,4,5 form factors from the Ω + cc → Ω * 0 c transition and the F 1,2,3 from the Ω + cc → Ω 0 c one.The ratios are shown as a function of the corresponding pseudoscalar P heavy-light meson mass.As the pseudoscalar meson mass increases (the heavy quark mass increases) the ratios tend to one as expected.Similar results are obtained in the other cases.Even though we are not in the infinite heavy quark mass limit, HQSS turns out to be a useful tool to understand the dynamics of the c → s, d Ξ cc and Ω cc decays near zero recoil.One also sees that at w = 1 our results for η 0 (w = 1), η 1 (w = 1) are systematically smaller than would be expected for an equal mass transition.This is a reflection of the mismatch in the wave functions due to the different initial (c) and final (d or s) quark masses in the c → d, s decays.Now we discuss the results for the decay widths.Those are shown in Table II for the dominant (c → s) and sub-dominant (c → d) exclusive semileptonic decays of the Ξ cc and Ω cc to ground state, 1/2 + or 3/2 + , single charmed baryons and with a positron in the final state 5 .For the Ω + cc baryon, semileptonic decays driven by a s → u transition at the quark level are also possible.However, in this latter case phase space is very limited and we find the decay widths are orders of magnitude smaller than the ones shown.To our knowledge there are just a few previous theoretical evaluations of the Ξ cc semileptonic decays.In Ref. [33] the authors use the relativistic three-quark model to evaluate the Ξ cc → Ξ ′ c e + ν e decay, while in Ref. [31], using heavy quark effective theory and non-relativistic QCD sum rules, they give both the lifetime of the Ξ cc baryon and the branching ratio for the combined decay Ξ cc → Ξ c e + ν e + Ξ ′ c e + ν e + Ξ * c e + ν e from which we have evaluated the semileptonic decay widths shown in the table.We find a fair agreement of our predictions with both calculations.In Ref. [30], using the optical theorem and the operator product expansion, the authors evaluated the total semileptonic decay rate finding it to be 0.151 ps −1 for Ξ ++ cc and 0.166 ps −1 for Ξ + cc .These values are roughly a factor of two smaller than the sum of our partial decay widths or the results in Ref. [31].For the Ω + cc a total semileptonic decay width of 0.454 ps −1 is given in Ref. [30].In this case this is in better agreement with the sum of our partial semileptonic decay widths which add up to 0.353 ps −1 .
An estimate of part of the uncertainties in our model can be done by evaluating the decay widths using wave functions produced with different interquark interactions.We have done this by using the AP1 [35,36] and Bhaduri [47] interquark potentials finding changes in the decay widths to be at the level of 10 %.Another source of uncertainties may come from the contribution from intermediate heavy-light vector meson (D * and D * s ) exchanges [48].They are neither considered in this work nor in the previous quark model calculation of Ref. [33] 6 .We expect such exchanges FIG. 2. Solid (dashed): F1 + F2 + F3 (3G1/5) for the specified transitions.Dotted: the combination for the transition with the corresponding 3/2 baryon (Σ * c , Ξ * c or Ω * c ) in the final state.In all cases the two light quarks in the final baryon have total spin S = 1.In the limit in which the heavy quark mass is made arbitrarily large one has that, near zero recoil (w = 1), for the Ω + cc → Ω 0 c transition and the ratio to produce small effects7 in the integrated widths, specially for the decays considered in this work, for which the D * and D * s poles are located far from q 2 max .This is in sharp contrast with the situation for the B → π and D → π decays [48,49].The model could be also improved by considering two body operators, and going in this manner beyond the spectator approximation.However, two body current contributions are not straightforward to compute, Decay widths in units of ps −1 .We use |Vcs| = 0.97345 and |V cd | = 0.2252 taken from Ref. [6].Results with an ( * ), our estimates from the total decay widths and branching ratios in [31].Similar results are obtained for µ + νµ leptons in the final state.
and since we expect moderate effects8 , similar to the other uncertainties mentioned above, we will leave this issue for future research.Moreover, there exists a greater source of uncertainties affecting our results that comes from our limited knowledge on the masses of the initial double charmed baryons.As we pointed out in the introduction, for the Ξ cc and the Ω cc baryons, we have used our quark model predictions in Table I.If the SELEX Collaboration measured mass for the Ξ cc baryon is used instead, we would find significantly smaller decay widths by about 20%.This is just because of the reduction on the available phase-space for the decay.None of the theoretical works mentioned in Table II use the SELEX mass value.
To summarize this work, we would like to point out that we have carried out the first systematic study of all dominant and sub-dominant semi-leptonic transitions of the doubly charmed Ξ cc and Ω cc baryons to the lowest-lying, 1/2 + or 3/2 + , single-c baryons.To that end, we have employed a simple constituent quark model scheme, which benefits from the important simplifications [21,34] of the non-relativistic three body problem that stem from the application of HQSS.We have also derived, for the first time, HQSS relations among the relevant form factors that govern these decays near zero recoil, and have found the size of the deviations induced by the finite charm quark mass.
Predictions of this framework have been successfully tested in the past in the context of the Λ b and Ξ b semileptonic decays [50].There, we obtained results for partially integrated decay widths that nicely compared with lattice results [51], and from the experimental Λ b −semileptonic decay, we could also determine the V cb CKM matrix element in excellent agreement with the accepted values quoted in the PDG [6].
In terms of matrix elements, the different form factors for the spin 1/2-baryon to spin 1/2-baryon transitions can be evaluated as For the spin 1/2-baryon to spin 3/2-baryon case the relations between form factors and weak matrix elements are For this latter case, 1/2-baryon to 3/2-baryon transitions, the following restrictions are observed (B23) (B24) (B25) The vector matrix elements have the general structure Here we have a c → l transition at the quark level, while l ′ is the light quark originally present in the initial baryon.The V Similarly, for the axial matrix elements we have and we have four vector (C V 3,4,5,6 (w)) and four axial (C A 3,4,5,6 (w)) form factors.

1 FIG. 3 .
FIG. 3. Form factor ratio squares) constructed with the C A 3,4,5 form factors from the Ω + cc → Ω * 0 c transition and the F1,2,3 from the Ω + cc → Ω 0 c one.Ratios are shown as a function of the pseudoscalar P heavy-light meson mass for three different heavy quark masses and for w = 1.
SF depend on the flavour and spin structure of the baryons involved.Their values for the different transitions appear in Table III.When the final baryon has just one s quark then φ(B ′ ) should be interpreted as φ(B ′ ) sn or φ(B ′ ) ds , for the case of c → s or c → d transitions, respectively.