A Mixing Coupling Scheme for Spectra of Singly Heavy Baryons with Spin-1 Diquarks in P-waves

A new scheme of state classification is proposed and applied to analyze masses of the heavy baryons $\Omega_{Q}$, $\Sigma_{Q}$ and $\Xi_{Q}^{\prime}$ in P-waves. The results confirm all excited $\Omega_{c}$ and $\Omega_{b}$ baryons reported recently by LHCb to be bound states of a P-wave $ss$-diquark and a respective charm or bottom quark, and thereby predict Regge trajectories for more excited $\Omega_{c}$ and $\Omega_{b}$ baryons. We suggest one excited {$J^{P}=5/2^{-}$} $\Omega_{b}$ state to be unseen by LHCb around $6352$ MeV, and predict P-wave masses of all spin-partners of the odd-parity baryons $\Sigma_{c}(2800)/\Xi_{c}^{\prime}(2942)$ and $\Sigma_{b}(6097)$/$\Xi_{b}^{\prime }(6227)$. A computation is further given in a relativized potential quark models to explain matched values of spin couplings of all considered baryons, by which a scaling law for these spin couplings is discussed.

For charmed baryons, spectroscopy is known to be intricate, with many excited states expected [4-8, 12, 14, 15]. The common practice is to utilize the heavy quark symmetry(HQS) [34], which becomes exact in the limit of heavy quark m Q → ∞. Assuming HQS by which spin S Q of heavy quark Q is conserved, one can classify heavy hadrons by their total spin J and the angular momentum j = J − S Q of light degrees of freedom(named j j coupling). This is analogous to the hydrogen-like atoms whose quantum states are well labeled by quantum numbers of outermost electron. While this picture works well in describing the normal charmed and bottom baryons [5][6][7] with strangeness |S | < 2, its applicability to the Ω c,b baryons remains to be explored.
The purpose of this work is to explore quantum numbers and inner structures of the odd-parity bound systems of a heavy Q-quark (Q = c, b) and a light spin-1 qq-diquark(qq = nn, ns, n = u, d) and corresponding main feature of relative magnitudes of the spin coupling parameters for the systems. We propose a new scheme of state classification (named the Jls mixing coupling) to perform systematic mass analysis for the newly LHCb-reported resonances of the excited Ω c and Ω b , which interprets all of these resonances to be the P-wave negative-parity baryons, and enables us to predict Regge trajectories of the mean-spin masses of their excited states. We update earlier computations for the negative-parity baryons Σ Q and Ξ ′ Q , and explain the matched spin-couplings of these baryons in a relativized potential quark model, To achieve that, we determine the mean-spin massM and the spin coupling strengths (a 1 , a 2 , b, c) as effective parameters by matching computed and measured masses of heavy baryons considered, which implies that a central potential plus spin-interactions describe the main features of most established hadrons. Due to their connections with the interquark potential, these knowledges about spin-couplings are useful and forms a basis for further understanding QCD interaction within hadrons.
We apply our proposed scheme of the Jls mixing coupling to mass analysis of the excited Ω c baryons in Sect. II, where preferred spin-parity are assigned for them. We examine other plausible spin-parity arrangements in Sect. III. Similar mass analysis is given for excited Ω b baryons in Sect. IV. In Sect. V, the Jls mixing coupling is applied to the less strange baryons Σ Q and Ξ ′ Q . A computation is given in the relativized quark model to explain the matched spin-couplings in Sect. VI. We end with conclusions and discussions in Sect. VII.  [2] and Ω b [3], and excited baryons Σ Q and Ξ ′ Q [1], with the proposed values of spin-parity J P shown also.

II. INNER STRUCTURES OF THE c(ss) SYSTEMS
In heavy quark-diquark picture, two strange quarks in a c(ss) system form a S-wave anti-color triplet(3 c ) diquark (ss), with spin one(S ss = 1) due to the spatial symmetry under exchange of two fermions. The diquark spin(= 1) can couple with spin S c = 1/2 of the charm quark c to form a total spin S = 1 ± 1/2 = 1/2, 3/2. Let us consider the relative P-wave excitations of diquark ss with respect to the charm quark c(the relative orbital angular momentum L = 1). Coupling of L = 1 with the spin S = 1/2 gives states with total spin J = 1/2, 3/2, while coupling with S = 3/2 leads to states with J = 1/2, 3/2 and 5/2. Symbolically, they are written as, in terms of adding of angular momentum, In total, one has five P-wave states with J = 1/2, 3/2, 1/2 ′ , 3/2 ′ and 5/2 with negative parity P = −1 which implies 5! = 120 a priori possible assignments for their J P quantum numbers. For most heavy baryons with less strangeness, the j j coupling has been commonly used to classify their states in terms of hadron eigenstates |J, j , with J the eigenvalues of J and j of j respectively. In the case of doubly strange Qss baryons with the ss-diquark comparable with the heavy quark Q in mass, the finite mass effect of the heavy quark may become important and makes it appropriate to go beyond the j j coupling.
For this, we propose a new scheme of state classification(named the Jls mixing coupling) in which the eigenfunctions (bases) of spin multiplets of the Q(ss) system(Q = c, b) diagonalize all spin-orbit interactions between the heavy quark Q and the diquark ss. The spin-dependent interaction for the Q(ss) systems, in the heavy quark-diquark picture, is [5,35] where the first two terms are spin-orbit interactions, the third is the tensor energy, and the last the contact interaction between the heavy quark spin S Q and the diquark spin S ss . Here, L is the orbital angular momentum of the system. Though four strengths a 1,2 , b and c of spin-couplings will be treated, in a sense, as effective parameters in this work, they are closely related to interquark interactions within hadrons and assumed, in potential quark model [5-7, 14, 32], to be computable from QCD analogues of Breit-Fermi interaction in QED. If Q is very heavy, the first spin-orbit term L·S ss should dominate over the others in Eq.
(2) if one assumes spin-dependent interactions enter the color hyperfine interactions through the magnetic moment e i S i /m i of the quark i = Q or ss with mass m i , by analogy with the spin-relevant relativistic correction of the heavy quarkonium [36]. As such, the heavy quark spin S Q decouples with the light degree of freedom and is conserved in the heavy quark limit, making the basis eigenfunctions of L · S ss (the j j coupling) appropriate to classify the baryon states. Here, the baryon spin J and the diquark angular momentum j = J − S Q are both conserved, enabling baryon states to be labelled by the j j coupling states |J, j . In the case of the c(ss) or b(ss) baryons, in which the diquark mass m ss (near 1 GeV) is comparable to M Q (about 1.5 GeV), the terms like a 2 L · S Q in Eq. (2) may become important. Taking a 1 = a 2 , for instance, the spin-orbit interaction becomes proportional to L · [S ss + S Q ] = L · S and diagonal in the LS coupling, in which S Q first couples S ss to form the total quark spin S = S ss + S Q and then to L of the quark-diquark system to form J, as shown in Eq. (1).
In the Jls mixing coupling considered in this work, the bases |J, j LS = j ′ diagonalize the interaction a 1 L · S ss + a 2 L · S Q + bS 12 , instead of L · S ss solely, in Eq. (2). The Jls scheme reduces to the j j coupling when the ratio ǫ = a 2 /a 1 (expected to scale as m ss /M Q ) tends to zero in the heavy quark limit, where b is expected to be suppressed by 1/M Q [35,37]. Finding the Jls mixing eigenstates |J, j LS = j ′ can be done by solving the linear eigenstate equation of 2 × 2 matrices ∆M J of the mass shift interaction (2) in the spin subspace of J = 1/2 and J = 3/2 [35,37](see Ref. [38] also). In terms of the LS bases 2S +1 P J = { 2 P 1/2 , 4 P 1/2 , 2 P 3/2 , 4 P 3/2 , 4 P 5/2 }, the matrix forms of these mass shift interactions are(see appendix A of Ref. [17]) Given J P assignments of the five Ω c states(two states of J P = 1/2 − , two of J P = 3/2 − , and one of J P = 5/2 − ) there should, in principle, exist one unique solution for the four parameters a 1 , a 2 , b and c. As we shall find below, there is one solution in which all states are P-waves with reasonable values of parameters and the mass pattern as reported by LHCb for the excited Ω c states in Table  I, where our preferred J P assignments are shown.
Diagonalizing the mass shift operator L · S ss + ǫL · S Q + b 1 S 12 , with b 1 ≡ b/a 1 , one can compute the mass shifts ∆M, the eigenvalues of Eqs. (4)-(6), by treating the contact term cS ss · S Q as a perturbation(c 1 = c/a 1 is expected to be small for the relative 1P-wave between diquark and Q). The lowest order perturbation theory gives(see appendix A) where six functions ∆ 1,2 (ǫ, x), ∆ ± 3 (ǫ, x) and ∆ ± 4 (ǫ, x) are defined by with ∆ − 3,4 (ǫ, x) obtained from ∆ + 3,4 (ǫ, x) by merely replacing ∆ 1,2 → − ∆ 1,2 . This expresses the mass shifts in terms of four parameters (a 1 , a 2 , b, c) nonlinearly. Expanding Eqs. (7) to the leading order of ǫ, b 1 and c 1 , it reduces to the linear mass shift in Ref. [17] in j j coupling. The spinweighted sum of these mass shifts in Eqs. (4)-(6) vanishes: J T r J (∆M J )(2J + 1) = 0, and the same holds for the eigenvalues ∆M(J, j ′ ) in Eqs. (7). Note that the sums of eigenvalues of ∆M 1/2 and ∆M 3/2 are equal to the traces of the respective matrices in Eq. (7). Note that an alternative expression similar to the mass-shifts(7) is obtained recently in Ref. [39].
Adding spin-independent massM, which equals to the spin-averaged mass of the five excited css systems, the baryon mass becomes M(J, j LS = j ′ ) =M + ∆M(J, j ′ ), with ∆M(J, j ′ ) given by Eqs. (7). Confronting M(J, j LS = j ′ ) with the observed masses in Table I leads to the values of a 1 , a 2 (= ǫa 1 ), b(= a 1 b 1 ) and c(= a 1 c 1 ), with the help of the following criteria: (i) The parameter a 1 should be positive but smaller than 119 MeV= M(Ω c , 3119)−M(Ω c , 3000).
(ii) The parameter a 2 is of same order with a 1 but no more than a 1 roughly as a 2 /a 1 = ǫ scales as m ss /M c .
(iii) The parameter b should be smaller than a 1 and a 2 as b scales like 1/(m ss M c ). The parameter c should be smallest, less than b as it scales as P-wave wave function near the origin.
We carry out mass analysis for all 5! = 120 a priori possible assignments of P-wave Ω c states and list the preferred assignments in Table I with ǫ = 0.96, b 1 = 0.50, c 1 = 0.15. For our preferred assignment, the inner structures of the five excited Ω c baryons are 3050.2 3065.6 3090.2 3119.1 4 P 1/2 (97%) 2 P 1/2 (97%) 4 P 3/2 (98%) 2 P 3/2 (98%) 4 where the third line gives the main component in terms of the normal LS coupling ( 2S +1 P J , see below). One sees that the mass degeneration within the J = 1/2 and J = 3/2 multiplets removed by the flip of the charm quark spin S c . Within the same multiplet, the lower state consists mainly of S = 3/2 configuration c ↑ (s ↑ s ↑ ), while the higher consists mainly of that with S c flipped, c ↓ (s ↑ s ↑ ) with S = 1/2. Expressing in terms of the LS eigenstates | 2S +1 P J , with 1/2 ≤ J ≤ 5/2 and S = 0, 1, the Jls states in Eq. (13) become (see Appendix B), Dominately, they are | 4 P 1/2 , | 2 P 1/2 , | 4 P 3/2 , | 2 P 3/2 and | 4 P 5/2 with the monatomically increasing mass, respectively, as shown in Eq. (13). This is in contrast with the normal state classification of heavy baryon systems via the j j coupling(see [35], for instance), mixing significantly between the states with J = 1/2 and 3/2. It is of interest to express the mixing states (14) in terms of the j j bases, giving(see Appendix B) which mix almost equally between the j j bases 1 2 , j = 0, 1 . Eq. (16) implies that the angular momentum j = J − S c of the diquark in our mixing states with J = 1/2 may be observed to take values either j = 0 or j = 1, with almost equal probabilities (0.51 v.s. 0.49), while those for the J = 3/2 are mainly the j j eigenstates |3/2, j = 1, 2 (at 0.92), slightly mixing the respective |3/2, j = 2, 1 (at 0.08). Hence, the J = 1/2 states do not conserve j = J − S c of the diquark and thereby S c . This differs the css states from the nonstrange charmed baryons, all of which can be well classified by the j j eigenstates [5-7, 17, 35].
One alternative solution to assignment of the five Ω c = css states involves identification of the five masses to be that at 3000, 3066, 3050, 3090 and 3119 MeV, respectively. This gives rise to the parameters and the mean mass in Ref. [17] This assignment is disfavored by unreasonable relative ratio (1 : 2) between a 1 and a 2 in the sense of the criterion (ii). Using Eq. (7) further, we performed systematic search for all remaining permutations among all 5! possibilities and find no acceptable solution for considered permutations: the parameters obtained either have negative (unacceptable) signs of a 1 and a 2 or unreasonable values in the light of the criteria (i)-(iii).  (13) and the alternative arrangement of four LHCbobserved masses of the excited Ω c systems. All 2 * 5 = 10 possible choices of mass inputs out of the five measured masses are considered. The predicted mass at presumably unseen state is marked in square bracket. All parameters except the Regge slope a in MeV. We also made a "recovery" test by fitting four out of the five observed Ω c masses for the J P assignments (13) and the alternative solution with M(1/2, 1 ′ ) and M(3/2, 1 ′ ) interchanged, corresponding to the parameters (17), to see if the remaining measured mass, unused in fitting inputs, can be reproduced at the site of given spin-states(marked by square bracket). The results for the "recovery" test are shown in Table II for all 2 * 5 = 10 possibilities of state arrangements for chosen inputs and indicate that the test successes for both, whereas the parameter values clearly favor the assignment (13) in the light of the criterion (ii): a 2 is notably large over a 1 , nearly two times, for the alternative solution.
One can also use the mass scaling of the spin-interaction couplings, from D s mesons to the css baryons, to explain the values (12) for our preferred assignment. Using the scaling relation [35,37], one finds which are close to the values (12), but away from that in (17). Here, a 1,2 (cs) are the parameters of  Table I(cf. the discussion following (21) in Sect. III).

III. ALTERNATIVE POSSIBILITIES EXAMINED
The possibility that must be checked is that not all of the Ω c states in Table I are relative P-wave excitations between ss diquark and the charmed quark [5,23,24,40,41]. For instance, two 2S wave candidates are predicted for the c(ss) at masses 3088 MeV (J P = 1/2 + ) and 3123 MeV(J P = 3/2 + ) in Ref. [5], not far from two higher masses(3090 and 3119 MeV) reported by LHCb. This leaves possibility that some of the lower states may be that out of the five states of P-waves, with other P-wave states unseen somehow, probably due to locating below the Ξ + c K − threshold (≃ 2962 MeV), or invisible due to the near degeneracies [5]. To disentangle this possibility, we extend the mass analysis in Sect. II by assuming one of the five measured masses of the Ω c states in Table I to be that of 2S state and comparing it with mass estimation of the 2S state with the help of Regge trajectory [42,43]. The results suggest that all measured masses of the excited Ω c states are too low to be a 2S state.
We use the mass shift formula (7) to perform mass analysis for all 5! permutations of the J P arrangement with one presumed 2S state removed from five mass inputs. Apart from the 10 arrangements already given in Table II, we list, in Table III, the other candidate solutions with the parameters that are not too far away from the criteria (i)-(iii), and the thereby predicted P-wave mass(presumably unseen) enclosed in square bracket. Though two arrangements in the first and second data rows recover approximately the observed masses absent in the inputs, all of these permutated arrangements are disfavored by the criteria (i)-(iii), having unreasonable values of the parameters listed: either a 2 or b is notably large over a 1 or negative, or c is abnormally large compared with one of a 2 and b, or both. III: Parameters and spin-averaged masses (MeV) for the selected permutations of J P assignments of the four states out of excited Ω c levels. Only the arrangements that leads to the parameters near to reasonable values are listed among 5 * 4! = 120 possible permutations of mass inputs considered, with the mass prediction at unseen level marked by square bracket. The shown also includes the ensuing prediction for the Regge slope a (see text, in GeV 2 ) and spin-averaged 2S-wave mass(in MeV) of the c(ss) system. m ss = 991 MeV. The mass estimation of the Ω c systems in 2S wave can be given utilizing a Regge-like relation for the spin-averaged massM of the charmed baryons [37,44] in which a is the Regge slope, m d is the light diquark mass involved, m barec = 1.275 GeV the bare mass of charm quark and L the orbital angular momentum of the systems. The charm quark mass M c = 1.44 GeV is determined by confronting the relation (21) with the charmed baryons Σ c /Ξ ′ c [37]. In Ref. [42], a trajectory slope ratio π : 2 for the radially and angular excitations is suggested in applying Eq.  By analogy of the c(ss) system with D s = cs mesons, we estimate the 2S-wave mass of the c(ss) state using Eq. (21) with a replaced by a(π/2) and L by n, the radial quantum number of the c(ss) system,M where m d = 0.991 GeV and the Regge slope a (listed in Table II and III) are solved from (21) Usage of the mean mass (11) in P-wave(L = 1) and the mass (23) in S-wave(L = 0) in Eq. (21) gives Ω c : m d = m ss = 0.991 GeV, a = 0.316 GeV 2 .
Given trajectory parameters, one can predict Regge trajectories of the excited baryons Ω c (FIG.  1) with radial quantum number n = 0 and 1. In Table IV in which n stands for n + 1 = 1, 2, we list corresponding spin-averaged masses of the excited Ω c 's predicted by Eqs. (21)and (22), which are quite helpful for further constructing whole families of the baryon Ω c 's.  Depending on state arrangements having different P-wave mean-mass, the above values vary slightly. The 2S-wave masses of the c(ss) system estimated via (22) are listed in Table II and III for given assignments. Evidently, the 2S-wave massM(2S ) ≥ 3220 MeV, and both of the higher mass states at 3090 MeV and 3119 MeV are too low to be a 2S state candidates. The same conclusion holds also for other low-lying masses, as shown in Table II and III. For other assignments searched, no solutions of the predicted unseen states, below the 2962 MeV threshold (of Ξ + c K − ) or near degeneracies with other masses, are found up to requirements (i)-(iii) in Sect. II.  (Table I) makes it timely to apply mass analysis of the Ω c = c(ss) states to the bottom systems b(ss) which consists of a bottom quark b and a spin-1 diquark ss. Heaviness of the b quark implies that Eqs. (7) should certainly be applicable for the P-wave Ω − b states. We use it to search the possible assignments with the following inputs: (i) The parameter a 1 is positive and no more than the maximum gap between the observed masses : a 1 < 2 * (6349.88 − 6315.64) = 68. 48 MeV. (ii) The parameter a 2 is of same order of a 1 roughly in magnitude, but should be smaller than a 1 . (iii) The tensor parameter b should be smaller than a 1 in magnitude. (iv) The hyperfine parameter c is set to zero.
Assuming one of P-wave states is unseen experimentally the search is performed systematically via matching the observed masses of the Ω b states in Table I with that in Eq. (7) for all 4! = 24 a priori possible J P assignments of P-wave states and 5 possible permutations of the unseen P-wave state that is expected to exist somewhere around 6.
which lead, by Eqs. (7), to the following J P assignment, In terms of the LS basis, their inner structures are |1/2, 0 ′ = −0.190| 2 P 1/2 + 0.982| 4 P 1/2 , at 6316, which are mainly the P-wave states | 4 P 1/2 , | 2 P 1/2 , two mixings of | 4 P 3/2 and | 2 P 3/2 , and | 4 P 5/2 , respectively. The experimental missing of the J = 5/2 state in Table I at M(5/2, 2 ′ ) = 6351.5, which we predict, is most likely due to its degeneracy with the nearby state at M(3/2, 2 ′ ) = 6350 MeV. If this is the case, it may be hidden in the observed peak around 6350 MeV, which, though appears consistent with a single resonance, is actually composed of two. This is compared to the recent assignment in Ref. [33], where the mass for the unseen J    MeV) for the excited Ω b when one of four observed masses was removed from the mass inputs. The removed one (enclosed by square bracket) is reproduced by Eq. (7) approximately.  VI: Parameters and spin-averaged masses(in MeV) for the selected permutations of J P assignments of the four excited Ω b levels, with predicted mass at unseen level(marked by square bracket). All arrangements leading to highly unreasonable or unacceptable parameters are not listed. The shown includes also ensuing prediction for the Regge slope a (see text) and spin-averaged css masses in 2S wave. m ss = 991 MeV. We denoted the five states | 1 In addition, we carry out "recovery" test (Table V, all in MeV) for the excited Ω b 's, in which one of four observed masses is reproduced approximately by Eq. (7) at the site enclosed by square bracket when it was removed from four mass inputs. The uncertainty is about 2 at most. This further confirms our prediction M(5/2, 2 ′ ) = 6351.5 MeV for the unseen (excited) Ω b state.
Could it be possible that some higher states of these Ω b peaks are the 2S excitations? Our answer to this question is negative. We employ Eq. (7) to perform mass fitting for the four mass inputs of the Ω b baryons in Table I to find the three parameters a 1,2 , b andM(1P), and thereby estimate the 2S-wave mass of the Ω b baryons, shown in Table VI which disfavors identifying newly LHCb-reported Ω b to be a 2S excitation. During calculating, the ss diquark mass, m d=ss = 1.001 GeV, is determined from Eq. (21) applied to spin-averaged mass M(Ω b , 1S ) = 6061.4 MeV of the two lowest 1S states, the Ω b (1/2 + ) at 6046.1 ± 1.7 MeV and its partner Ω * b (3/2 + ) with mass 6046.1+ ∆E(Ω b , 1S ). Here, ∆E(Ω b , 1S ) stands for the 1S level-splitting between the Ω * b (3/2 + ) and Ω b (1/2 + ), and can be estimated from the corresponding 1S level-splitting of the ground states Ω c via the scaling [35], which yields the spin-averaged mass in 1S -wave: Hence, Eq. (31) disfavors the 2S state assignment for any of four observed Ω b states, as the 2S state is too high in levels to accommodate all Ω b states in Table I. Similar analysis giving Eq. (24) applies to the P-wave bottom baryons Ω b with the mean mass in Eq. (25) and gives a = 0.318 GeV 2 . This leads to the mean masses in Table IV with c(css) = 4.04 MeV given in Eq. (12), to be three initial inputs to give a b-dependent Pwave mass for each baryon using the nonlinear relation (7), and then find optimal fit of b as the initial inputs via matching b-dependent mass with the measured one for each baryons (Table I).
Finally, one evolves the obtained initial inputs via the relation (7) Table III of Ref. [37], which are extrapolated from Regge-trajectory of the Λ c /Ξ c spectra.
For the P-wave bottom baryons Σ b /Ξ ′ b , corresponding initial inputs (Table VII) follow from the following estimations: the initial parameter c is estimated, by scaling from the Ω c to the Ω b and then to the bottom baryon bqq, to be about with M c = 1.44 GeV and M b = 4.48 GeV [37]. The mass scaling a 2 (bss) ≃ a 2 (css)(M c /M b ) is used to estimate a 2 while the HQS relation a 1 (bqq) = a 1 (cqq) in Ref. [35] needs to be corrected, as indicated by a 1 (bss) = 8.98 < 26.96 = a 1 (css) and also by failure to find reasonable parameters if setting a 1 (Σ b ) = 39.4 MeV= a 1 (Σ c ) or a 1 (Ξ ′ b ) = 33.6 MeV= a 1 (Ξ ′ c ). We find that choosing initial values of a 1 (bqq) to be about one third of a 1 (cqq), as a 1 (bss) ≃ a 1 (css)/3 indicated, yields reasonable and stable values of the parameters during iteration of Eq. (7). In the Table VIII(  denotes mass observed), we show the results of iteration starting with the initial inputs in Table  VII, with reasonable values of the parameters achieved. Note that the obtained values of b is about one half of that of a 1 .  (7) for the less strange heavy baryons Σ c /Ξ ′ c and Σ b /Ξ ′ b in P-wave.
S tate : This way of parameter relaxation via the nonlinear mass relation overcome the difficulty in previous works [35,37] within the scheme of the j j coupling, that the masses of the baryons Σ c /Ξ ′ c and Σ b /Ξ ′ b remain undetermined due to unknown b: a simple comparison of the b-dependent mass predictions [35,37] with experimental masses yields unreasonable values of b except for the Σ c . In our computation of the P-wave masses of the Σ c /Ξ ′ c 's and Σ b /Ξ ′ b 's, using of appropriate eigenstates, the bases of the Jls coupling here, is crucial to find stable parameters which keep them within a narrow range of a few MeV around the initial inputs in Table VII. We await upcoming measurement(mass and spin-parity) by LHCb and Belle to verify the predictions in Table VIII. Similar procedure can apply to the excited Ω b , giving nearly same parameters and masses as Eqs. (25) and (26), only that the mass of the J = 5/2 state evolves to M(5/2, 2 ′ ) = 6352.2 MeV, 0.7 MeV above that in Eq. (26), and c becomes a non-vanishing but small value of c = 1.35, as it should be.

VI. SPIN-COUPLINGS IN RELATIVIZED QUARK MODEL
One may ask if the parameters of spin interaction (2) can be computed based on dynamics of quark interaction, as it is done for fine and hyperfine structures of the hydrogen-like atoms? In this section, we employ a QCD analogues of Breit-Fermi spin interaction(Appendix C), derived from the Lorentz-invariant quark-antiquark scattering amplitude at the tree level [36], to compute the spin-couplings (a 1 , a 2 , b, c) and compare them with the experimentally matched ones.
Including the spin-dependent forces(Breit-Fermi spin interaction), the quasi-static potential be-tween quark 1 and antiquark 2 is [36,46], where V and S are the respective vector and scalar potentials, and V ′ , S ′ and V ′′ their derivatives. One way to relativize the instant potential V quasi-static is to use the replacements [46] V(r) →Ṽ(r) = m 1 m 2 where the relativistic factors m i /E i = m i / m 2 i + |p| 2 = 1 − v 2 i (i = 1, 2) rise from the length contraction r → r(m/E) when quark i moves relativistically, and these factors tend to unity in the low-momentum (p/m i → 0) limit of the quarks. Here, ǫ V,S are small homogeneous factors to be determined. Applying Eqs. (35) and (36) to a heavy quark Q in color triplet(3 c ) and a diquark d = qq(q = u, d, s) in the anti-color triplet(3 c ), four spin-couplings become then in which V + S stands for the confining potential between Q and d, to be approximated by that for the heavy quarkonia QQ, and the quantum average is made over the P-wave wavefunction Ψ Qd of the Qd system. For the spin interaction in Eq. (2), the relativistic treatment (35) implies that the factor 1/m i in Eq. (36) becomes 1/E i = (m 2 i + p 2 ) −1/2 , which makes sure, for example, that the hyperfine interactions of the light quark i do not blow up like 1/m i in the chiral limit(m i → 0), but rather they should have a finite limit determined by p −1 , which is in turn controlled by the radius of confinement.
To find P-wave wavefunction Ψ Qd of singly heavy baryons(Qd) we consider the relativized quark model [46] with the color-coulomb plus linear potential V + S (treating the spin interaction H S D perturbatively) where k s = 4α s /3 and α s is the strong coupling. By introducing auxiliary fields µ d and ν, one can solve the model (37) using auxiliary field (AF) method [47](Appendix C). The inputs of m d , a, and α s are given in Table IX, where the values of other parameters ξ, ζ and ǫ V,S for computing the spin couplings are also listed. The solved P-wave radial wavefunction of the quark-diquark system are given in FIG. 3. TABLE IX: Inputs for the model parameters and diquark masses defined in Hamiltonian (C1) (Appendix C) for the P-wave heavy baryons listed. The heavy quark masses m Q = M Q = 1440MeV(Q = c) and 4480MeV(Q = b), and light diquark masses m d are taken from Ref. [37]. The confining potential parameter a is set by Regge trajectories of corresponding baryons, taken from Ref. [37] and Eq. (24) and Sect. IV.  Table IX. Upon using (36) to compute the spin couplings, we take into account non-point nature of the diquark d via adding a form factor F(r) ≡ 1 − e −ξr−ζr 2 [5] in the short-range potential V so that Putting the improved V in Eq. (38) into Eq. (36), in which the relativized potentials become now, by Eq.
The AF fields (µ, ν) are solvable from the nonlinear equation (C4) (Appendix C) given the inputs of the masses m Q = M Q and m d , and the linear potential parameter a extracted from Ref. [37], as listed in Table IX, Table X, which agree reasonably with the parameters (a 1 , a 2 , b) given in Eq. (12) and Eq. (25) as well as that in Table VIII. One sees from Table X that our prediction for the spin couplings a 2 is slightly overestimated compared to that matched ones, about 4 − 8MeV, and the predictions for c is anomaly larger than the matched ones, about 7 − 16MeV.
One can list the resulted ratios of the spin couplings r 1,2 = a 1,2 (bqq)/a 1,2 (cqq) and r b = b(bqq)/b(cqq) between the bottom and charmed baryons and compare them to corresponding ratios (denoted as [.] Match ) obtained from Eqs. (12), (25) and from One sees that the simple mass scaling law(∼ M b /M c = 1440/4480 = 0.321) holds approximately for a 2 and b. In the case of a 1 , the scaling law a 1 (bqq) = a 1 (cqq) [35] is broken somehow as r 1 ∼ 1/3 is less than unity. Further, the matched value r 2 = 0.16 for the baryons Ω b vs. Ω c is somehow smaller than that expected by M b /M c and also than our computed value r 2 = 0.379.
From Table XI, which shows the computed values of the relevant factors and the averages in Eq. (40), one sees that the minus term∼ − m d /m Q a/r in Eq. (40) is notably suppressed and this gives relatively larger predictions for a 2 . We remark three features of our computations: (i) As a leading approximation, the Breit-Fermi-like interaction (34) have not considered nonperturbative corrections that may enter at long-distance regime. Assuming the spin interaction dominates at short distance, one expects that the interaction (34) includes the main spin-spin and spin-orbital forces via adding the scalar confining potential S ∼ r to the vector potential V, and such a nonperturbative corrections to Eq. (34) is relatively small.
(ii) Static approximation of quark-diquark potential V + S , which is valid in principle only for heavy quarkonia(QQ), has ignored rotation of QCD string when L 0 [41,53,54]. In orbitallyexcited state, the rotation of QCD string can change the static quark-diquark potential V + S in Eq. (37), as shown in Refs. [53,54]. For instance, this corrects L-dependence of the Hamiltonian, [m 2 d + p 2 r + L 2 /r 2 ] 1/2 ∼ L into a Regge-like one, H ∼ √ πaL due to addition of the orbital angular moment of the string [53].
(iii) A simple exponential form factor F(r) may not be enough in the short-range to describe the three-body quark systems of Q(qq). Three-body interaction [51,52] inside baryons may make F(r) quite nontrivial as the effective running of the color-charge of a non-point diquark d in colorantitriplet (3 c ) seen by Q in color-triplet (3 c ) becomes highly distance-dependent when d is close to Q. See Ref. [51](Eq. (18)) for other possible forms of the form factor, like Er f [c 0 r] or its linear combinations. For simplicity, we used the R H nL (r) in Eq. (C11) (Appendix C) to estimate c in Eq. (40) and this may overestimate values for c govern by integral of −F ′′ |R H nL (r)| 2 at short-range. Note that R H nL (0) ≡ 0 for the P-waves. In addition, Eq. (40) seems to imply the scaling laws a 1 ∼ O(m 0 Q ), a 2 ∼ 1/m Q ∼ b (see Eq. (24) in Ref. [35])), as heavy quark symmetry expected. However, the small variation of the averages within the baryons, manly in 1 + 2m d /m Q and k s = 4α s /3 along with the size(∼ a B ) of baryons, is enlarged by m 2 d in denominator for a 1 in Eq. (40) and this makes a 1 changes notably between the bottom and charmed sectors. This kind of enlargement does not happen for other spin couplings due to their overall suppression factor 1/m d m Q . Note also that the factor 1+m d /2m Q for a 2 changes As seen in Tables X and XI, the scaling law for the spin couplings breaks moderately, especially for a 1 , due to the injured heavy quark symmetry indicated by Eq. (40).

VII. CONCLUSIONS AND DISCUSSIONS
Exploring and building data set of the mean-spin massM and the spin coupling strengths(a 1 , a 2 , b, c) is important as it can be a basis to explore how quarks interact within hadrons. Although baryon is a three-body system with subtle short-range interactions, it is useful to employ a simple heavy quark-diquark picture for the singly heavy baryons to understand the measured mass data of excited heavy-baryons. In this work, we have proposed a new scheme of state classification, the Jls mixing coupling, to unify analysis of masses and inner-structures of the excited baryons Ω Q and other less-strange heavy baryons with spin-1 light diquarks. Firstly, we interpret the five Ω c and four Ω b excited states reported by LHCb to be the relative P-wave excitations between spin-1 ss diquark and the heavy quark, the charm quark c for the Ω c and the bottom quark b for the Ω b , respectively. Further analysis via Regge phenomenology for the low-lying excited states excludes the possibility that some of higher states of the resonances Ω c and Ω b are the 2S states since the 2S candidates are all too heavy to accommodate both of the resonances. Secondly, we applied our formalism based on the Jls mixing coupling to update the earlier computation of all P-wave masses of the less strange baryons Σ c /Ξ ′ c and Σ b /Ξ ′ b , for which only single measured mass available for each of these baryons. The masses of all spin-partners of the baryons Σ c (2800)/Ξ ′ c (2942) and Σ b (6097)/Ξ ′ b (6227) are then predicted. Finally, we demonstrated that the matched spin-couplings of the singly heavy baryons considered can be understood in a relativized potential quark model, based on which the scaling law for their spin coupling parameters are discussed.
With the help of the relation (7), we find that four measured masses for the excited Ω b 's are enough to make a quantitative prediction M(5/2 − ) = 6352 MeV for one unseen state with J P = 5/2 − . The missing of this 5/2 − state in LHCb experiment can be due to the degeneracy with the nearby state |3/2 − , 2 ′ at 6350 MeV. The same J P = 5/2 − state is predicted as the unseen state in a recent assignment [33], with two higher mass ranges predicted, about 10 MeV away or more.
One very recent prediction for the odd-parity Ω b 's is that by the LS coupling [45]: for which the explanation of near degeneracy applies only to two J = 3/2 states and assuming one of them to be unseen is favored. In the viewpoint of this work, with given mixing weights in Eq. Our computation via a simple heavy quark-diquark picture indicates that the measured masses of the excited heavy-baryons considered can be understood via a relativized potential quark model and resulted Breit-Fermi spin-orbit and tensor forces, provided that a proper relativizing is made to these spin interactions(forces). By the way, we confirm based on the relativized expression of the Breit-Fermi formula (40) for the spin coupling parameters that a mass scaling law [35] for them holds approximately for all baryons considered. Being at the order of O(1), the scaling relation a 1 (bqq) = a 1 (cqq) for a 1 is broken most, in contrast with other parameters at the order of O(1/m Q ). Due to involved QCD dynamics of the heavy flavor baryons(see [5,34] for instance), our analysis for the spin-couplings is of quite approximative, with further quantitative corrections remained to be explored.

Appendix C
Using auxiliary field (AF) method [47], which re-expresses nonlinear operator √ O = min λ>0 [O/(2λ) + λ/2] in terms of the AF field λ, for which the minimization is achieved when λ = √ O, one can rewrite the model (37) as where p = |p| is the 3-momentum of the heavy quark Q and the light diquark d, and the minimization conditions (C2) for the auxiliary fields (µ d , ν) are assumed to hold at semi-classical level. In a sense, the AF method linearizes the Hamiltonian H in (37) in a manner similar to mean-field method for many-body systems. Next, we solve the P-wave radial wavefunction R nL of the Qd system defined by Eq. (37) numerically in two steps.
In the first step, we find two self-consistent AF fields (µ d , ν) via solving Eq. (C2). For this, we ignore the (color) Coulomb term −k s /r to write the Hamiltonian (C1) as that of three dimensional harmonic oscillator(HO) with reduced mass µ ≡ µ d /(1 + µ d /m Q ), the HO frequency ω = a/ √ µν, the inverse HO length α = √ µω = (a 2 µ/v) 1/4 . As such, the wavefunction Ψ Qd is given by that of the HO, ψ N = R N n r L (r)Y Lm (θ, ϕ) with quantum number N = 2n r + L, where Y Lm is the spherical function and which can be solved by numerical iteration. The solution for (µ, ν) and µ d ≡ µ/(1 − µ/m Q ) are shown in Table 1(Table IX in revision). The solution of the wavefunction Ψ Qd can be solved from the Hamiltonian (C1) of the two-body Qd, which is In the long-range where the color-Coulomb term −k s /r is ignorable, the radial eigenfunctions of the Hamiltonian (C5) can be given by that of the HO wavefunction, where R nL (r) are the radial wavefunctions of these systems. The results for the P-wave solutions R nL (r) are shown in FIG. 3. It is useful to write the radial wavefunction R H n r L (r) in the small-r range, which is given by solution to the eigen-equation of the Hamiltonian (C1) with the HO potential(∼ r 2 ) ignored. It is the solution to the hydrogen-like atoms with the effective Bohr radius a B = 1/(µ H k s ) and the reduced mass µ H = µ dH /(1 + µ dH /m Q ) with µ dH = with n = n r + L + 1. Using averaging in the hydrogen-like wavefunction ψ H with n r = 0, p 2 H = k s µ H /[a B (L+1) 2 ] = k 2 s µ 2 H /(L+1) 2 , one finds that the AF parameter µ dH ≃ m 2 d + p 2 H is subjected to the self-consistent equation, µ dH = m 2 d + 1 (L + 1) 2 (k s µ dH ) 2 (1 + µ dH /m Q ) 2 The numerical results for µ dH and thereby-resulted µ H are shown in Table V also.
In the short-range where the term −k s /r is dominate over the HO potential, the radial solution of wavefunction to the Hamiltonian (C5) is that of the hydrogen-like: with n = n r + L + 1.