An Estimate of the Inclusive Branching Ratio to ${\bar B}_c$ in $\Xi_{bbq}$ Decay

We estimate the branching ratio for the inclusive decays $\Xi_{bbq} \rightarrow {\bar B}_c^{(*)}+X_{c,s,q}$ to be approximately 1%. Our estimate is performed using non-relativistic potential quark model methods that are appropriate if the bottom and charm quarks are heavy compared to the strong interaction scale. Here the superscript $(*)$ denotes that we are summing over spin zero ${\bar B}_c$ and spin one ${\bar B}_c^*$ mesons and the subscript $q$ denotes a light quark. Our approach treats the two bottom quarks in the baryon $\Xi_{bbq}$ as a small color anti-triplet. This estimate for the inclusive branching ratio to ${\bar B}_c$ and ${\bar B}^*_c$ mesons also holds for decays of the lowest lying $T_{bb{\bar q}{\bar q}}$ tetraquark states, provided they are stable against strong and electromagnetic decay.


I. INTRODUCTION
In 2017, the doubly charmed baryon Ξ ++ cc (or in the notation used in this paper Ξ ccu ) was discovered at LHCb [1]. It has been observed in the exclusive decay modes, Ξ ++ cc → Λ + c K − π + π + (the discovery mode) and Ξ ++ cc → Ξ + c π + [2]. There is considerable interest in the detection of the analogous baryons containing two heavy bottom quarks Ξ bbq , q = u, d, partly because it would be the first step to observing the tetraquark states, T bbqq . They are thought to be stable with respect to the strong and electromagnetic interactions with masses that are around 100-200 MeV below theB qBq threshold [3][4][5].
Recently, Gershon and Poluektov [6] proposed the inclusive decay mode Ξ bbq →B c +X c,s,q as a potential discovery channel for the doubly bottom baryon Ξ bbq at the LHC. They made the clever observation thatB c 's that do not point back to the collision interaction point can only arise from the weak decay of a hadron with two bottom quarks. They also note that the decay chainB c → J/ψπ − → µ + µ − π − can be used to detect theB c meson 1 . OrdinaryB mesons that do not point back to the collision point cannot be used for this purpose 2 because they can arise from the weak decay of a long livedB c meson (via the weak decay of the anti-charm quark). The branching ratio forB c decay to ordinaryB mesons is not expected to be small and furthermore there will be many moreB c 's produced at the interaction point by hadronization then there are baryons with two bottom quarks.
In this paper we make an estimate of the inclusive branching ratio, Br(Ξ bbq →B c + X c,s,q ). Here the subscript c, s, q denotes the flavor quantum numbers of the inclusive final state and the superscript ( * ) denotes that we are summing over final state spin zeroB c and spin oneB * c mesons. AB * c meson decays to aB c plus a photon, so decays to the spin one state always result in aB c in the final state.
Our method relies on treating both the bottom and charm quark as heavy compared to the scale of the non-perturbative strong interactions, Λ QCD ∼ 200MeV. In this limit, the two bottom quarks in the Ξ bbq form a small (compared with 1/Λ QCD ) color anti-triplet 1 See [7] for a recent calculation of the branching ratio forB c → J/ψπ − . Their results imply that c + X c,s,q ) is then modeled by Γ(Φ bb →B ( * ) c + c + s), with the light quark q treated as a spectator. This decay rate is easily converted into a branching ratio since the total decay rate of the Ξ bbq is approximately twice the b quark decay rate 3 .
Our computation of Γ(Ξ bbq →B ( * ) c + X c,s,q ) does not include decay products from an excited (radial or orbital)B c (orB * c ) mesons. We will calculate the decay rates to the first radially excitedB c andB * c mesons and show they are suppressed, and then argue that decays to the other excited states are suppressed as well.
Our calculation of the inclusive decay rate of a Ξ bbq baryon toB c andB * c mesons is similar to the calculation of the inclusive B meson decay rate to J/Ψ [9] . One important difference is that the baryon decay is not color suppressed. Another difference is that the baryon decay matrix element is proportional to an overlap of wave-functions while the meson decay matrix element is proportional to the J/ψ wave function at the origin.

II. THE DECAY RATE
In this section, we outline the calculation of the Φ bb →B c +c+s invariant matrix element where greek letters denote the color quantum numbers. We perform the calculation in the rest frame of the decaying bottom diquark state Φ bb , which is a color anti-triplet and has spin one. We assume that the relative momentum of the bound states are non-relativistic. The state vectors are then where repeated indices are summed over and the state |B c (k, s, m s ) corresponds to aB c meson if s = 0 and aB * c meson if s = 1. The bound states have been normalized such that B c (k 1 , s 1 , m s 1 )|B c (k 2 , s 2 , m s 2 ) = 2E k 1 δ s 1 s 2 δ ms 1 ms 2 (2π) 3 δ 3 (k 1 − k 2 ) and similarly for the Φ bb state. The state vectors on the right hand side of (2.1) have no hidden normalization factors, and are just the appropriate creation operators acting on the vacuum. The functionsψB c (p) andψ Φ bb (p) are the wavefunctions for the relative momentum of the quarks in the bound states and, in the non-relativistic limit, have support when p is much less than the masses of the bound quarks.
The weak Hamiltonian that induces the decay is 4 where 3) The operators O 1,2 and coefficients C 1,2 are evaluated at a subtraction point equal to the b quark mass. The invariant matrix element for the decay is then In eq. (2.4) theψ Φ bb (p) wavefunction restricts p to be much less than m b , so we can set u (b) (p, s 2 ) = u (b) (0, s 2 ) and E b (p) = m b . In addition, theB c wave function restricts |p + m b m b +mc k| to be much less than the charm quark mass, which means we can make the replacement p + k → (m c /(m b + m c ))k in E c and v (c) . In the non-relativistic limit, the masses of the bound states are approximately equal to the sum of their constituent quark masses, which implies E c ((m c /(m b + m c ))k) = (m c /(m c + m b ))EB c (k). After making these replacements, eq. (2.4) becomes Note, the position space wavefunctions are normalized so that |ψB c/Φbb (r)| 2 d 3 r = 1. To determine the differential decay rate, we square the matrix element, average over initial spins and colors and sum over final spins and colors. The spin sum involving the final stateB c spins is performed using the completeness relation, s,ms C (s,ms) sa,s b C * (s,ms) sa,s b = δ sasa δ s bsb . For the spin average over the Φ bb spin magnetic quantum numbers we note that, Rotational invariance implies that the decay rate is independent of the magnetic quantum number for the total spin of Φ bb . This means we can replace the average over its initial magnetic quantum numbers in the decay rate with the average over just the m = −1 and m = 1 magnetic quantum numbers. After integrating over the strange and charm momenta, the differential decay rate then becomes where, as mentioned in the introduction, the superscript ( * ) denotes that we are summing over the spin one and spin zeroB c mesons.

III. NUMERICAL RESULTS
To evaluate the form factor I(k), we need to determine the wave functions ψ Φ bb (r) and ψB c (r). We do this by numerically solving the non-relativistic Schrodinger equation with the Cornell potentials, The relative factor of 1/2 between V Φ bb (r) and VB c (r) reflects the fact that the Φ bb is a color anti-triplet while theB c is a color singlet 5 . We took the string tension to be 0.2 GeV 2 which fits the bb spectrum of bound states [10]. In addition, we chose the strong fine structure constant to be 0.3 and 0.4 for V Φ bb (r) and VB c (r). The charm and bottom quark masses are taken to be 1.5 GeV and 4.5 GeV. The form factor I(k) computed using the numerical ground state wavefunctions is plotted in Fig. 1 over the range of k allowed in the decay, The numerical solutions to the Schrodinger equation implies the radii squared of the ground state wavefunctions are r 2 Φ bb = 3.2 GeV −2 and r 2 B c = 2.8 GeV −2 . It turns out that the Coulomb-like wave functions which is also plotted in Fig. 1.
In Fig. 2 we plot dΓ/dk obtained using the ground state numerical wavefunctions. Integrating (2.7) over (3.2), we find that the decay rate is Using a total lifetime for Ξ bbq of 0.5 ps [8], the branching ratio is This branching ratio leaves outB c 's that arise from the decay of radially and orbitally excitedB c andB * c mesons. We computed the decay rate to the first radially excitedB c state with zero orbital angular momentum and found the branching ratio to be 7.3 × 10 −4 , which is an order of magnitude smaller than the branching ratio to the ground state.
Decays to other radially excitedB c states will be suppressed as well. The full Hamiltonian for the Φ bb system, including the kinetic terms and potential from (3.1), is almost equal to half the full Hamiltonian for theB c one. This means the spatial wave functions for the energy eigenstates of the two Hamiltonians are almost the same, which implies I(0) 1 for the ground stateB c mesons. In addition, it implies that the overlap integral for decays to radially excitedB c andB * c mesons will satisfy I(0) 0, which suppresses the branching ratio to these states. Decays to orbitally excitedB c mesons are also suppressed.
A recent work on production rates for hadrons with two heavy quarks at the LHC [12] estimates that σ(pp → Φ bb + X) 15 nb. Assuming most of the Φ bb diquarks end up as Ξ bbq baryons 6 this implies that in an integrated luminosity of 10 fb −1 there are around 10 8 Ξ bbq baryons. Our work then implies that the decays of these baryons produce around 10 6 B c 's that do not point back to the interaction point. About 10 2 of them end up in the final state µ + µ − π − , with the µ + µ − arising from J/ψ decay.

IV. CONCLUDING REMARKS
We calculated the inclusive decay rate for Ξ bbq →B c + X c,s,q to be 1.5 × 10 10 s −1 (which implies Br(Ξ bbq →B ( * ) c (k) + X c,s,q ) 8 × 10 −3 ). The initial bb system was treated as a tightly bound color anti-triplet diquark Φ bb and we evaluated its decay rate toB c +c+s. The Schrodinger equation was solved numerically to determine the non-relativistic wavefunctions for the Φ bb andB c . In reality, the relative momentum of the quarks in theB c bound state is not truly non-relativistic. In addition, we neglected the fact that the diquark initially exists in a hadron and interactions between the finalB c , c and s states and the soft degrees of freedom in Ξ bbq . Despite these approximations, we expect our calculation of the decay rate to be correct at the factor of two level.