Strange baryons with two heavy quarks

The LHCb Experiment at CERN has observed a doubly-charmed baryon $\Xi_{cc}^{++} =ccu$ with a mass of $3621.40 \pm 0.78$ MeV, consistent with many predictions. We use the same methods that led us to predict $M(\Xi_{cc},~J^P{=}1/2^+) = 3627 \pm 12$ MeV and $M(\Xi_{cc}^*,~J^P{=}3/2^+) = 3690 \pm 12$ MeV to predict $M(\Omega_{cc}^+,J^P{=}1/2^+) = 3692 \pm 16$ MeV and $M(\Omega_{cc}^*,~J^P{=} 3/2^+) = 3756 \pm 16$ MeV. Production and decay are discussed briefly, and predictions for $M(\Omega_{bc})$ and $M(\Omega_{bb})$ are included.


I INTRODUCTION
The LHCb Experiment at CERN has observed a doubly-charmed baryon Ξ ++ cc = ccu with a mass of 3621.40±0.78 MeV [1]. This value is consistent with several predictions, including our value of 3627 ± 12 MeV [2,3]. It is more than 100 MeV above a candidate Ξ + cc for an isospin partner claimed by the SELEX Collaboration [4], but not seen by others. Here we use similar methods to those in Ref. [2] and earlier works [5] to predict the mass of the ground-state ccs state with spin-parity J P = 1/2 + M (Ω + cc ) = 3692 ± 16 MeV and its hyperfine partner with J P = 3/2 + , M (Ω * cc ) = 3756 ± 16 MeV. Binding effects lead the difference between the strange and nonstrange doubly charmed baryon masses to be less than half the constituent-quark mass difference between the strange and nonstrange light quarks. These results were obtained using constituent-quark masses appropriate for baryons. Use of quark masses universal for baryons and mesons leads to M (Ω cc ) and M (Ω * cc ) about 40 MeV higher, with similar systematic variations expected for Ω bc and Ω bb , due mostly to uncertainty in how strongly a strange quark binds to a heavy diquark.
In Section II we list contributions to M (Ω + cc ) that are straightforward extrapolations of the calculation of M (Ξ cc ). The pair of charmed quarks is treated as a (cc) diquark antisymmetric (a 3 * ) in color and hence symmetric in spin (S = 1). The difference in binding between a  [2] with corresponding contributions to the mass of Ω cc .

II EXTRAPOLATIONS FROM Ξ cc PREDICTION
We compare the contributions to the Ξ cc mass studied in Ref. [2] to similar contributions to the Ω cc mass in Table I. We take quarks in a baryon to have effective masses m b q = 363 MeV(q = u or d), m b s = 538 MeV, m b c = 1710.5 MeV, and m b b = 5043.5 MeV. We ignore isospin splitting, treated in [6] and references therein.
The effect of the spin-spin interaction between the q quark and the (cc) diquark is parametrized by a term −4a/m b q m b c , while that between s and (cc) is parametrized by −4a /m b s m b c with a = a m b s /m b q taken so that the two terms have the same strength. This is motivated by comparing the spin-spin interaction in the cs and Our calculations of the masses of light hadrons, based on the ideas of Ref. [7], use constituent-quark masses and do not require separate binding energies. However, for systems without q involving heavy quarks one must take into account additional binding. For example, when calculating the mass of the S-wave cs system, it was found necessary to include a supplemental binding energy of 69.9 MeV, while a binding energy of 258 MeV was needed to describe S-wave charmonium [2]. Hence the last energy in Table I represents a subtotal; we estimate the binding energy of s with the diquark (cc) in the next section.

III DIQUARK-LIGHT QUARK BINDING
We shall interpolate between thecs andcc binding energies to find that between (cc) and s. All three cases involve the interaction of a color antitriplet with a color triplet. We compare the masses m 1 , m 2 of the constituents and reduced mass µ ≡ m 1 m 2 /(m 1 +m 2 ) of the composite  Table II For use in subsequent discussion of the masses of Ω bc ≡ bcs and Ω bb ≡ bss, we include the binding energies between the diquark (bc) and s and between the diquark (bb) and s. The mass of the bc diquark is calculated to be m where the error reflects uncertainty in the bc binding energy. As the mass eigenstates are of indefinite bc spin (rather, they are approximately states of definite cs spin), we ignore small hyperfine effects. The mass of the bb diquark is calculated to be 2m The reduced mass of the (cc)s system lies between those ofcs andcc. Assuming a powerlaw dependence on µ, B = Aµ p , gives p = 1.636 and B((cc)s) = 98. 9 MeV. An alternate method makes use of the Feynman-Hellmann theorem [8], which relates the derivative of an energy expectation value with respect to a parameter µ to the expectation value of the derivative of the Hamiltonian: In the present case, the right-hand side is −(1/µ) T , where T is the kinetic energy. Let us now assume T is independent of the reduced mass. This is indeed the case for a logarithmic potential [9,10], which has been shown to suitably interpolate between charmonium and bottomonium. We shall assume T is constant also for our interpolation. Then the shift in binding energy between a system with reduced mass µ 1 and one with µ 2 is The binding energy increases with increased reduced mass, as expected. One can determine T = 235.
The average of the two determinations is 109.4 ± 10.5 MeV, where we take the error to be half of their difference. Similar methods apply to the estimates of B((bc)s) and B((bb)s) quoted in Table II, where the averages are those of the power-law (lesser value) and Feynman-Hellmann (greater value) methods of interpolation. Subtracting this from the subtotal in Table I, whose error was assumed to be the same as in the calculations of M (Ξ cc ), and adding the error of ±10.5 MeV in quadrature, we find M (Ω cc ) = 3692 ± 16 MeV, M (Ω * cc ) = 3756 ± 16 MeV.

IV UNIVERSAL QUARK MASSES
For many years it has been realized that fits to baryon masses require constituent quarks about 55 MeV heavier than those in fits to low-lying mesons [11,12]. An alternative, secondary, description [13] makes use of quark masses appropriate for both mesons and baryons, adding a term S = 165.1 MeV to characterize the extra mass in a baryon due to a string junction [14]. The contributions to M (Ω cc ), before accounting of binding between the (cc) diquark and the strange quark, are shown in Table III   a From Ref. [13] b Mass eigenstates of indefinite bc spin; small hyperfine terms ignored Interpolating via a power law with B = Aµ p one finds p = 1.721, B((cc)s) =75.2 MeV, while interpolating via the Feynman-Hellmann theorem (2) one finds T = 227.4 MeV and B((cc)s) = 87.9 MeV. Hence B((cc)s) = 81.6 ± 6.4 MeV, implying M (ccs, 1/2 + ) = 3732 ± 14 MeV. This is 40 MeV above the value we obtained with separate quark masses for meson and baryons. The uncertainty reflects in part the uncertainty in estimating the binding energy between a strange quark and the heavy diquark. A precise measurement of M (Ω cc ) could help distinguish between the two pictures compared here. We also quote the predicted value of M (Ξ cc ) = 3640 ± 12 MeV in the scheme with universal quark masses. This is not as close to the experimental value as that in Ref. [2], but still acceptable. For the Ω * cc (J P = 3/2 + ) we replace the term −4a /m s m c = −37.6 MeV in Table III by Table IV contains also the (bc)s and (bb)s binding energies, obtained in an analogous way. The latter are used in Sec. VI to predict the masses Ω bc and Ω bb .

V PRODUCTION AND DECAY
We can estimate the rate for production of Ω cc = ccs by reference to that for Ξ ++ cc = ccu. Imagine that some process gives rise to the (cc) diquark, which then fragments into Ξ cc by picking up a u quark. The corresponding process giving rise to Ω cc then involves (cc) picking up a s quark. What is the ratio of these two processes?
There is information on b quark fragmentation in hadronic collisions from the CDF Collaboration [15], which measures f s 0.3f u inpp collisions at √ s = 1.96 TeV. One could expect a similar ratio for (cc) to pick up a u or s quark. At 13 TeV, in a sample of pp collisions consisting of an integrated luminosity of 1.7 fb −1 , the LHCb experiment accumulated 313±33 Ξ ++ cc events [1]. One might then expect the same sample to contain about (100±10)R Ω + cc identifiable events, where R is the ratio of Ω cc to Ξ cc decays into identifiable branching fractions.
The Ξ cc was seen in the final state Λ c K − π + π + . One decay process depicted in Fig. 1 of Ref. [1] involves the initial u and one of the initial charmed quarks c in the Ξ cc = ccu ending up in the Λ c = ucd. If the initial baryon is Ω cc = ccs, an initial s and one of the initial charmed quarks will end up instead in a Ξ 0 c = scd. The detectability of the Ω cc will then depend on the relative efficiencies for reconstruction of Ξ 0 c and Λ c . Another potentially useful decay mode of Ξ ++ cc is into π + Ξ + c . Its visibility at LHCb will depend on relative efficiencies for reconstruction of Ξ + c and Λ c . The corresponding decay mode of Ω cc is into π + Ω c . The LHCb experiment has detected not only the Ω c but several excited states of it [16] in the final state Ξ + c K − , providing a test of ability to reconstruct Ξ + c .

VI MASSES OF Ω bc = bcs AND Ω bb = bbs
We have seen that much of the uncertainty in prediction of M (Ω cc ) lies in uncertainty of the binding energy between the (cc) diquark and the strange quark. The same is true when predicting the masses of Ω bc and Ω bb . Extrapolating our results for nonstrange states [2] to ones in which q = u, d is replaced with s, we take account of (1) the s − q mass difference, (2) differences in ((QQ )q) and ((QQ )s) binding, and (3) small differences in hyperfine splittings, to obtain the results in Table V. The use of universal quark masses raises the prediction of all Ω Q 1 Q 2 masses by about 40 MeV. Table V: Summary of predictions of Ω QQ masses, in MeV. "Separate" denotes separate quark masses for mesons and baryons; "universal" denotes universal quark masses for mesons and baryons.

VII RESULTS
Using the same methods used to obtain an accurate prediction of the mass of the recently discovered doubly-charmed baryons Ξ ++ cc , we predict the mass of its strange partner: M (Ω cc ) = 3692 ± 16 MeV. The hyperfine partner of this state, with J P = 3/2 + , is predicted to have a mass M (Ω * cc ) = 3756 ± 16 MeV. Predictions for the ground state masses of the bcs baryons Ω bc and Ω bc and the bbs baryon Ω bb are also presented. The use of universal quark masses with an added "string-junction" contribution for baryons raises these predictions by about 40 MeV.
Our predictions for M (Ω cc ) are compared with a number of others in Tables VI (nonlattice) and VII (lattice). The predictions based on lattice gauge theory are shown separately as they have less of a spread. The corresponding values are plotted in Figs. 1 and 2.
In the picture with separate quark masses for mesons and baryons, the prediction of a rather large value of B((cc)s) distinguishes our approach from a number of others [3]  The production cross section for Ω cc was estimated to be about 0.3 times that for Ξ ++ cc . Its detectability then depends on the relative efficiency for reconstructing Ξ 0 c and Λ c .