Revisit assignments of the new excited $\Omega_c$ states with QCD sum rules

In this article, we distinguish the contributions of the positive parity and negative parity $\Omega_c$ states, study the masses and pole residues of the 1S, 1P, 2S and 2P $\Omega_c$ states with the spin $J=\frac{1}{2}$ and $\frac{3}{2}$ using the QCD sum rules in a consistent way, and revisit the assignments of the new narrow excited $\Omega_c^0$ states. The predictions support assigning the $\Omega_c(3000)$ to be the 1P $\Omega_c$ state with $J^P={\frac{1}{2}}^-$, assigning the $\Omega_c(3090)$ to be the 1P $\Omega_c$ state with $J^P={\frac{3}{2}}^-$ or the 2S $\Omega_c$ state with $J^P={\frac{1}{2}}^+$, and assigning $\Omega_c(3119)$ to be the 2S $\Omega_c$ state with $J^P={\frac{3}{2}}^+$.

and take the full c-quark propagator C ij (x) in the momentum space [24], q = u, d, s, t n = λ n 2 , the λ n is the Gell-Mann matrix. In Eq. (12), we add the term s j σ µν s i originates from the Fierz re-ordering of the s isj to absorb the gluons emitted from other quark lines to form s j g s G a αβ t a mn σ µν s i to extract the mixed condensate sg s σGs . The term − 1 8 s j σ µν s i σ µν was introduced in Ref. [29]. We compute the integrals both in the coordinate space and momentum space to obtain the correlation functions Π j (p 2 ), then obtain the QCD spectral densities through dispersion relation, ImΠ j (s) π = p ρ 1 j,QCD (s) + ρ 0 j,QCD (s) , where j = 1 2 , 3 2 , the explicit expressions of the QCD spectral densities ρ 1 j,QCD (s) and ρ 0 j,QCD (s) can be rewritten in a concise form after multiplying the weight function exp − s We take the quark-hadron duality, introduce the continuum thresholds s 0 and the weight function exp − s T 2 to obtain the QCD sum rules: where j = 1 2 , 3 2 , s . The QCD sum rules can be written more explicitly, The contributions of the positive parity and negative parity Ω c states are separated explicitly. Firstly, we choose low continuum threshold parameters s 0 so as not to include the contributions of the 2S and 2P Ω c states (Ω ′ c ), and obtain the QCD sum rules for the masses of the 1S and 1P Ω c states, then obtain the pole residues λ + j and λ − j . Now we take the masses and pole residues of the 1S and 1P Ω c states as input parameters, and postpone the continuum threshold parameters s 0 to larger values to include the contributions of the 2S and 2P Ω c states, and obtain the QCD sum rules for the masses of the 2S and 2P Ω c states, then obtain the pole residues λ ′+ j and λ ′− j .

Numerical results and discussions
The input parameters are taken to be the standard values qq = −(0.24 ± 0.01 GeV) 3 , ss = (0.8 ± 0.1) qq , sg s σGs = m 2 0 ss , m 2 0 = (0.8 ± 0.1) GeV 2 , αsGG π = (0.33 GeV) 4 at the energy scale µ = 1 GeV [23,24,30,31], m c (m c ) = (1.275 ± 0.025) GeV and m s (µ = 2 GeV) = (0.095±0.005) GeV from the Particle Data Group [32]. The updated values from the Particle Data Group in version 2016 [32] are slightly different from the corresponding ones in version 2014, we take the old values to make consistent predictions with the same parameters and criteria chosen in previous works. If we choose the updated values m c (m c ) = (1.28 ± 0.03) GeV and m s (µ = 2 GeV) = 0.096 +0.008 −0.004 GeV [32], the central value of the predicted mass of the Ω c (1S) is 2.6991 GeV rather than 2.6983 GeV, the predicted mass presented in Table 2 survives, so the old values are OK. The values of the m 2 0 , ss / qq and sg s σGs / qg s σGq vary in rather large ranges from different theoretical determinations, for example, in Ref. [33], sg s σGs / qg s σGq = 0.95 ± 0.15, which differs from the standard value sg s σGs / qg s σGq = ss / qq = 0.8 ± 0.1 remarkably [30]. In this article, we take the standard values or the old values still accepted now [30,31].
We take into account the energy-scale dependence of the input parameters from the renormalization group equation, , Λ = 213 MeV, 296 MeV and 339 MeV for the flavors n f = 5, 4 and 3, respectively [32], and evolve all the input parameters to the optimal energy scales µ to extract the masses of the Ω c states. The energy scale dependence of the quark masses and quark condensates is known beyond the leading order, the energy scale dependence of the mixed quark condensates is only known in the leading order [34,35]. In this article, we take the leading order approximation in a consistent way, and take the energy scale dependence of the mixed condensates presented in Refs. [34,35], while a quite different energy scale dependence of the mixed condensates is presented in Refs. [33,36]. It is interesting to take the energy scale dependence presented in Refs. [33,36], this may be our next work. For the heavy degrees of freedom, we take the favors n f = 4, the power in the m c (µ) is 12 25 . For the light degrees of freedom, we take the flavors n f = 3, the powers in the ss (µ), sg s σGs (µ) and m s (µ) are 4 9 (or 12 27 ), 2 27 and 4 9 , respectively. If we take the favors n f = 4, the powers in the ss (µ), sg s σGs (µ) and m s (µ) are 12 25 , 2 25 and 12 25 , respectively, in fact, the induced tiny difference in numerical calculations can be neglected. As far as the fine constant α s (µ) is concerned, we choose the next-to-next-to-leading order approximation, which is consistent with the values determined experimentally [32].
In Fig.1, we plot the correlation functions Π j,+ and Π j,− with variations of the energy scales µ and the Borel parameters T 2 , From the figure, we can see that the Π j,+ and Π j,− increase remarkably with increase of the energy scale µ at the region T 2 > 4.0 GeV 2 , while at the region T 2 < 3.0 GeV 2 , the Π j,+ and Π j,− increase slowly with increase of the energy scale µ. All in all, we cannot obtain energy scale independent QCD sum rules, some constraints are needed to determine the energy scales of the QCD spectral densities in a consistent way. Now we take a short digression to discuss how to choose the optimal energy scales. In the heavy quark limit, the heavy quark Q serves as a static well potential and combines with a light quark q to form a heavy diquark in color antitriplet, or combines with a light diquark in color antitriplet to form a heavy baryon in color singlet. The heavy antiquark Q serves as another static well potential and combines with a light antiquarkq ′ to form a heavy antidiquark in color triplet, or combines with a light antidiquark in color triplet to form a heavy antibaryon in color singlet. Then the heavy diquark and heavy antidiquark combine together to form a hidden-charm or hidden-bottom tetraquark state. The heavy baryons B and tetraquark states X/Y /Z are characterized by the effective heavy quark masses M Q (or constituent quark masses) and the virtuality 2 (or bound energy not as robust). The diquark-quark type baryon states and diquark-antidiquark type tetraquark states are expected to have the same effective Q-quark masses M Q , which embody the net effects of the complex dynamics [29,37]. In Refs. [29,38], we study the acceptable energy scales of the QCD spectral densities for the hidden-charm (hidden-bottom) tetraquark states and molecular states in the QCD sum rules in details for the first time, and suggest an energy scale formula µ = M 2 X/Y /Z − (2M Q ) 2 by setting µ = V to determine the optimal energy scales with the effective heavy quark masses M Q .
We fit the effective c-quark mass M c to reproduce the experimental value of the mass of the Z ± c (3900) in the scenario of tetraquark state [29]. In this article, we use the empirical energy scale formula µ = M 2 Ωc − M 2 c to determine the optimal energy scales of the QCD spectral densities, and take the updated value of the effective c-quark mass M c = 1.82 GeV [39]. For detailed discussions about the energy scale formula µ = M 2 Ωc − M 2 c , one can consult Ref. [37]. According to the energy scale formula µ = M 2 Ωc − M 2 c , we extract the masses of the ground states (see Eqs. (25)(26)) and the first radial excited states (see Eqs. (27)(28)) at different energy scales.
In Fig.2, we plot the masses and pole residues of the Ω c (1S, 1 2 ), Ω c (1S, 3 2 ), Ω c (1P, 1 2 ) and Ω c (1P, 3 2 ) with variations of the energy scale µ for the central values of the Borel parameters and threshold parameters shown in Table 1. From the figure, we can see that the predicted masses decrease monotonously but mildly with increase of the energy scale µ, the constraint µ = M 2 Ωc − M 2 c is not difficult to satisfy. On the other hand, the pole residues increase monotonously and mildly with increase of the energy scale µ, which is consistent with Fig.1, as the Borel parameters are chosen as T 2 < 3.0 GeV 2 . At the vicinities of the energy scales presented in Table 1, the uncertainties induced by the uncertainties of the energy scales are tiny.
For the Z c (3900), the uncertainty of the energy scale of the QCD spectral density is about δµ = 0.1 GeV, the uncertainty of the effective c-quark mass M c can be estimated as δM c = µ 0 4Mc δµ = 0.02 GeV from the equation, where the µ 0 is the central value. The uncertainties δµ in this article can be estimated as δµ = Mc µ 0 δM c < 0.02 GeV from the equation, The predicted masses and pole residues are not sensitive to variations of the energy scales, the small uncertainty δM c = 0.02 GeV or δµ < 0.02 GeV can be neglected safely. We search for the ideal Borel parameters T 2 and continuum threshold parameters s 0 according to the four criteria: 1 · Pole dominance at the hadron side, the pole contributions are about (40 − 70)%; 2 · Convergence of the operator product expansion, the dominant contributions come from the perturbative terms; 3 · Appearance of the Borel platforms; 4 · Satisfying the energy scale formula µ = M 2 Ωc − M 2 c , by try and error, and present the optimal energy scales µ, ideal Borel parameters T 2 ,    Table 1, where the A, B, C and D correspond to the Ω c (1S, 1 2 ), Ω c (1S, 3 2 ), Ω c (1P, 1 2 ) and Ω c (1P, 3 2 ), respectively.
continuum threshold parameters s 0 , pole contributions and perturbative contributions in Table 1. From Table 1, we can see that the criteria 1 and 2 can be satisfied, the two basic criteria of the QCD sum rules can be satisfied, and we expect to make reliable predictions. We take into account all uncertainties of the input parameters, and obtain the masses and pole residues of the 1S, 1P, 2S and 2P Ω c states, which are shown explicitly in Table  2. From Table 2, we can see that the criterion 4 can be satisfied. In Figs.3-4, we plot the masses and pole residues of the 1S, 1P, 2S and 2P Ω c states with variations of the Borel parameters T 2 at much larger intervals than the Borel windows shown in Table  1. In the Borel windows, the uncertainties originate from the Borel parameters T 2 are very small, the Borel platforms exist, the criterion 3 can be satisfied. Now the four criteria are all satisfied, and we expect to make reliable predictions. In the Borel windows, the uncertainties of the predicted masses are about (3 − 5)%, as we obtain the masses from a ratio, see Eqs. (25)(26)(27)(28), the uncertainties originate from a special parameter in the numerator and denominator cancel out with each other, so the net uncertainties are very small. On the other hand, the uncertainties of the pole residues are about (10 − 16)%, which are much larger. The uncertainties δλ Ωc are compatible with the uncertainties of the decay constants f π = 127 ± 15 MeV and f ρ = 213 ± 20 MeV from the QCD sum rules [31].
In Table 2, we also present the experimental values [1,32]. The present predictions support assigning the Ω c (3000) to be the 1P Ω c state with J P = 1  Table 2: The masses and pole residues of the Ω c states, the masses are compared with the experimental data, the values of the Ω c (1P) with J P = 5 2 − are taken from Ref. [8]. respectively. In Ref. [12], Aliev, Bilmis and Savci use the same interpolating currents to study the Ω c states by taking into account the 1S and 1P states with J = 1 2 and 3 2 in the pole contributions, and assign the Ω c (3000) and Ω c (3066) to be the (1P, 1 2 − ) and (1P, 3 2 − ) states, respectively. In Refs. [2,5,12], the contributions of the Ω c states with positive parity and negative parity are not separated explicitly, there are some contaminations from the 2S or 1P states. In Ref. [8], we separate the contributions of the positive parity and negative parity Ω c states explicitly, and study the new excited Ω c states with the QCD sum rules by introducing an explicit P-wave involving the two s quarks. The predictions support assigning the Ω c (3050), Ω c (3066), Ω c (3090) and Ω c (3119) to be the P-wave Ω c states with J P = 1 2 − , 3 2 − , 3 2 − and 5 2 − , respectively. Compared with Refs. [2,5,12], the methods used in the present work and Ref. [8] have the advantage that the contributions of the Ω c states with positive parity and negative parity are separated explicitly, there are no contaminations from the 2S or 1P states.
In the diquark-quark models for the heavy baryon states, the angular momentum between the two light quarks is denoted by L ρ , while the angular momentum between the light diquark and the heavy quark is denoted by L λ . In Refs. [2,5,12] and present work, the currents with L ρ = L λ = 0 are chosen to explore the P-wave Ω c states, although the currents couple potentially to the P-wave Ω c states, we are unable to know the substructures of the P-wave Ω c states, and cannot distinguish whether they have L λ = 1 or L ρ = 1. In Ref. [8], we choose the currents with L λ = 1 to interpolate the Ω c states, and obtain the predicted masses Now we summarize the assignments based on the QCD sum rules in Table 3. From Table 3, we can see that all the calculations based on the QCD sum rules support assigning the Ω c (3000) to be the 1P 1 2 − state, while the assignments of the other Ω c states are under debate. We have to study the decay widths to make the assignments on more solid foundation. In Ref. [5], Agaev, Azizi and Sundu study the decays of the Ω c states to the Ξ + c K − by calculating the hadronic coupling constants g ΩcΞcK with the light-cone QCD sum rules, however, they use an over simplified hadronic representation and neglect the contributions of the excited Ξ c states.
Experimentally, we can search for those new excited Ω c states through strong decays and electromagnetic decays to the final states Ξ + , Ω c (2695)γ, Ω c (2770)γ, and measure the branching fractions precisely, which can shed light on the nature of those Ω c states. More theoretical works on the partial decay widths based on the QCD sum rules are still needed.