Interpretation of excited $\Omega_b$ signals

Recently LHCb reported the discovery of four extremely narrow excited $\Omega_b$ baryons decaying into $\Xi_b^0 K^-$. We interpret these baryons as bound states of a $b$-quark and a $P$-wave $ss$-diquark. For such a system there are exactly five possible combinations of spin and orbital angular momentum. We predict two of spin 1/2, two of spin 3/2, and one of spin 5/2, all with negative parity. We favor identifying the observed states as those those with spins 1/2 and 3/2, and give a range of predicted masses for the one with spin 5/2. We update earlier predictions for these states based on the five narrow excited $\Omega_c$ states reported by LHCb. An alternative picture of the states in which one of $J=1/2$ is extremely wide and hence not seen by LHCb is discussed.

: Masses and widths of Ω b = bss candidates reported by the LHCb Collaboration [1]. The proposed values of spin-parity J P are ours.
(d) Can one understand the mass pattern? Yes; the favored pattern, based on contributions of spin-orbit, spin-spin, and tensor force interactions, was uniquely selected out of 5! = 120 possible permutations of the five states.
(e) Are there other similar states with different quark content, in particular very narrow excited Ω b baryons? LHCb has now observed four out of the five predicted 1P excitations [1], leaving a fifth to be predicted and observed.
The same questions can be asked for the four observed Ω b states. Which of the expected five Ω b states is missing, and what is its mass? Is the spin-weighted average of the 1P excitations consistent with expectation? In Sec. II we comment on P -wave bss baryons. We then analyze spin-dependent forces for the bss system in Sec. III, building upon similar results [3] obtained previously for the negative-parity Ω c states. We evaluate the energy cost for a P -wave bss excitation in Sec. IV, compare our present results with our earlier predictions for the Ω b system in Sec. V, discuss alternative interpretations of the spectrum in Sec. VI, and conclude in Sec. VII.
All five states have negative parity P . Those with J P = 1/2 − decay to Ξ 0 b K − in an S-wave, while those with J P = 3/2 − , 5/2 − decay to Ξ 0 b K − in a D-wave. The LHCb experiment sees only four of the predicted five P -wave excitations in the Ω b = bss system [1]. Only four of the five predicted Ω c states are seen by Belle in e + e − collisions [4]; the omitted state is the heaviest, Ω c (3119). This makes sense as kinematic suppression is greatest for the heaviest state. For an initial state with no heavy flavor, the minimum mass recoiling against a css state such as Ω c (3119) is M(Ω c ) = 2695.2 ± 1.7 MeV while typical e + e − c.m.s. energy is M(Υ(4S)) = 10579.4 ± 1.2 MeV [5]. In keeping with our identification of the Ω c (3119) as the state with J P = 5/2, we shall assume that it is the J P = 5/2 − Ω b which is missing, and focus on the mass range above M(Ω b (6350)) for it.

III Spin-dependence of masses
The masses of the P -wave excitations of the ss diquark with respect to b are split by spin-orbit forces, a tensor force, and hyperfine interactions, leading to a spin-dependent potential [3] States with the same J but different S mix with one another, so the mass shift operators ∆M 1/2,3/2 may be written as 2 × 2 matrices in bases labeled by S = 1/2, 3/2: The spin-weighted sum of these mass shifts is zero: Note that the sums of eigenvalues of ∆M 1/2 and ∆M 3/2 are equal to the traces of the corresponding matrices, making the verification of Eq. (6) simple. There are four measured masses and four independent parameters leading to four mass shifts with respect to a spin-weighted average for which one needs the fifth mass. Thus the determination of the constants a 1 , a 2 , b, c has one free parameter which we may take as M 5/2 . We identify the four known masses as shown in Table I.
The spin-weighted average mass M = J [(2J +1)M J ]/18 is linear in the unknown mass M 5/2 , with slope 1/3. Anticipating the optimal fit M 5/2 = 6358 MeV (cf. the discussion following Eq. (10)), M can be rewritten in terms of the deviation from this fit, The limited range of M will be of use when we study the P -wave excitation energy.
Units of χ 2 are in MeV 2 . The favored range of M 5/2 is in the left-hand figure.
We now determine the parameters a 1 , a 2 , b, c from the masses in Table I. The measured masses permit one to write two identities which are helpful in finding solutions. We denote the two eigenvalues of M 1/2 by M 1 and M 2 , and the two eigenvalues of M 3/2 by M 3 and M 4 . A shorthand for M 5/2 is M 5 . We find Varying M 5/2 above 6350 MeV, we find solutions for the ranges 6355.4 MeV < M 5/2 < 6382.5 MeV and 6379.9 MeV < M 5/2 < 6406.9 MeV, as shown in the left-hand and righthand panels of Fig. 1, respectively, with two branches for a 1 , b, and c, and a single branch for a 2 .
The constants in the figure may be compared with those favored in a fit to excited Ω c states [3]: It was argued in Ref. [3] that the hyperfine term c should be no larger for the Ω b system than for the excited Ω c states. In that case one selects the lower branch of the dashed (red) ellipse in the left-hand panel, which is correlated with the upper branch of the solid (black) upper ellipse, and favors values of M 5/2 in the range of 6356 to 6366 MeV, with the most probable (lowest) value of c ≃ 3 MeV for M 5/2 = 6358 MeV.

IV Energy cost of a P -wave excitation
We estimated the P -wave excitation energy for a ss diquark bound to a b quark with relative orbital angular momentum L = 1 in Section V of Ref. [3]. A crude value of 300 MeV was obtained. It was necessary to anticipate the hyperfine splitting in the S-wave bss ground state, as only the Ω b (1/2 + ) has been seen, with mass M(Ω b ) = (6046.1 ± 1.7) MeV. With an estimated hyperfine splitting between Ω b (1/2 + ) and Ω * b (3/2 + ) of 24 MeV, the spin-weighted average of the 1/2 + and 3/2 + S-wave bss states was estimated to be 6062 MeV. Subsequently we noted [7] that P -wave excitation energies obeyed an approximate linear relation where ΣB is the binding energy of the (ss) diquark and b quark, estimated to be 83.6 MeV, and µ R is the reduced mass of the ss-b system: With these inputs one finds ∆E P −S = 308 MeV, implying M = 6370 MeV. As we see above and below, under various assumptions the LHCb data imply values a bit lower than this.

V Evaluation of predictions for Ω b = b(ss) states
In addition to the predictions of Ref. [3] for the hyperfine parameter c and the S-P splitting, we predicted other parameters for the Ω b = bss states based on rescaling the fitted Ω c values quoted in Eq. (10). (c) The tensor force parameter b is favored to be within the limited range of ±20 MeV around zero, as in Ref. [6].
(c) State with j = 0 predicted to be very broad and not seen by LHCb.
(d) Experimental masses; proposed J P assignments.
2 P 3/2 , 4 P 1/2 , 4 P 3/2 , 4 P 5/2 . Typical errors in predictions are 10 to 20 MeV, except for the QCD sum rule calculation [39], whose errors are of order 100 MeV. One should pay more attention to splitting among levels than their absolute values. A general pattern emerges from these calculations. In the j − j coupling scheme, the single state with j = 0 is deemed to be very wide (see Table III), and hence not observable in the current data set of LHCb [1]. The two states with j = 1 and the two with j = 2 are expected to be narrow, for the most part within the experimental resolution. This behavior is not seen in the case of the Ω c states, where candidates for all five involving the spin-one ss pair in its ground state are seen [2].
Even if one assumes the reason for seeing four rather than five excited Ω b states is that the one with j = 0 is very broad, there is no unanimity on the order of the observed states. That is the question we address in considering the effects of spin-orbit, tensor force, and spin-spin couplings. We have found a consistent solution in which it is the state with J P = 5/2 − that is missing in the data.
We now repeat the exercise in which the states are described by j − j coupling and it is the one with j = 0 whose mass we vary in order to determine the parameters a 1 , a 2 , b, c.
In order to determine mass splittings in the linearized j − j coupling basis, we use lowest-order perturbation theory in the inverse of m b [3]: The independent parameters are a 1 , a 2 + c, and b, so the five mass splittings obey two sum where the first number refers to J and the second to j. We are assuming that the unseen state is the one with mass M(J = 1/2, j = 0). Eliminating ∆M(1/2, 0) from the above two equations and recalling that each ∆M ≡ M − M, we find an expression for M in terms of the four observed masses, whose value depends on the permutation of the masses assigned to each J P level: We can now obtain ∆M(1/2, 0) from Eq. (18), and find The spin-dependent coefficients are Table IV lists all 24 permutations of the masses of the observed four levels with (J, j) = (1/2, 1), (3/2, 1), (3/2, 2), (5/2, 2), obtaining parameters for each permutation. We denote the order in which the observed masses are monotonically increasing by the permutation 1 2 3 4. We then compare each set with values estimated in Ref. [3] by extrapolation from the excited Ω c spectrum. Some notable features are the following: • The values of a 1 are half or less that estimated by extrapolating from charm to bottom. It probably pays to choose the largest possible (positive) a 1 .
• The value of a 2 was estimated in [3] to be 8.72 MeV, while c was estimated to be small, less than a few MeV.
• Ref. [3] considered values of b lying within the range −20 < b < 20 MeV, satisfied by most sets in Table IV.
• The value of M varies within a narrow range around 6335 MeV, to be compared with the crude estimate of 6362 MeV in Ref. [3], the central value of 6344 MeV obtained in Sec. III, and the value of 6370 MeV found in Sec. IV.

VII Conclusions
We have interpreted the four narrow peaks seen by LHCb in the Ξ 0 c K − mass distribution [1] as P -wave excitations of a spin-1 ss diquark with respect to a spin-1/2 b quark. While such a system is expected to have five states -two of spin 1/2, two of spin 3/2, and one of spin 5/2 -, we advance arguments in favor of the spin-5/2 state being missed. When the four observed levels are assigned J P = 1/2 − , 1/2 − , 3/2 − , 3/2 − in order of ascending mass, solutions for spin-dependent parameters are obtained for 6355.4 MeV < M 5/2 < 6382.5 MeV and 6379.9 MeV < M 5/2 < 6406.9 MeV, with the lowest ∼ 10 MeV of this range favored by consideration of the derived spin-dependent parameters.
An alternative explanation of the missing state offered by several authors [36,37,39,40] envisions the states as approximately diagonal in the (J, j) basis, where J is the total angular momentum and j is the ss-diquark's total (spin plus L) angular momentum. In this basis the (1/2, 0) state is predicted to be very wide and hence not seen by LHCb. In this case the most plausible set of spin-dependent parameters is obtained when the four observed levels are assigned (J, j) = (1/2, 1); (3/2, 1); (3/2, 2); (5/2, 2) in order of ascending mass. Angular distributions of decay products should be able to distinguish between this scenario and the (favored) one in the preceding paragraph.