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In 2017, the LHCb collaboration observed five narrow excited
Ωc states, i.e.,Ωc(3000) ,Ωc(3050) ,Ωc(3066) ,Ωc(3090) , andΩc(3119) , in theΞ+cK− mass spectrum [1]. Recently, they reported four excitedΩb states in theΞ0bK− mass spectrum [2]:Ωb(6316):m=6315.64±0.31±0.07±0.50MeV,Γ<2.8MeV,Ωb(6330):m=6330.30±0.28±0.07±0.50MeV,Γ<3.1MeV,Ωb(6340):m=6339.71±0.26±0.05±0.50MeV,Γ<1.5MeV,Ωb(6350):m=6349.88±0.35±0.05±0.50MeV,Γ=1.4+1.0−0.8±0.1MeV. (1) Following this experimental progresses, there have been many theoretical works concerning various properties of these excited
ΩQ (Q=b,c ) states [3-26] and other excited heavy baryons [27-36].Two kinds of excitations, the ρ-mode and λ-mode, exist in these excited
ΩQ states. The ρ-mode excitation is the excitation between two strange quarks, while the λ-mode one is the excitation between the strange diquark and bottom (charm) quark. In Ref. [37], the authors systematically considered all possible baryon currents with a derivative for internal ρ- and λ-mode excitations and studied the P-wave charmed baryons using the QCD sum rule method in the framework of heavy quark effective theory. In Refs. [12, 22], the authors studied these excited states using the QCD sum rule method in the framework of QCD.In this paper, we construct the full QCD counterparts of the interpolating currents considered in Ref. [37] and study P-wave
Ωb excited states using the QCD sum rule method [38, 39]. The basic idea of the QCD sum rule method is that the correlation function of interpolating currents of hadrons can be represented in terms of hadronic parameters (the so-called hadronic side) and calculated at quark-gluon level by operator product expansion (OPE) (the so-called QCD side); then, by matching the two expressions, we can extract the physical quantities of the considered hadron.The rest of the paper is organized as follows. In Sec. II, we construct the interpolating currents and derive the required sum rules. Sec. III is devoted to numerical analysis, and a short summary is given in Sec. IV. In Appendix B, OPE results are shown.
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Following Ref. [37], we introduce the symbols [
Ωb,jl,sl,ρ/λ ] andJα1α2⋯αj−12j,P,Ωb,jl,sl,ρ/λ to denote the P-waveΩb multiplets and the interpolating currents, respectively, where j is the total angular momentum; P is the parity;jl andsl are the total angular momentum and spin angular momentum of the light components, respectively; and ρ(λ) denotes the ρ(λ)-mode excitations. The general interpolating currents ofΩb baryons can be written asJ(x)∼ϵabc[saT(x)CΓ1sb(x)]Γ2bc(x),
(2) where a, b, and c are color indices,
ϵabc is the totally antisymmetric tensor, C is the charge conjugation operator, T denotes the matrix transpose on the Dirac spinor indices, ands(x) andb(x) are the strange and bottom quark fields, respectively. The state function corresponding to the diquarkϵabc[saT(x)CΓ1sb(x)] can be written as|color⟩⊗ |flavor,spin,space⟩ and should be antisymmetric under the interchange of the two strange quarks. Now, the color part and flavor part are antisymmetric and symmetric, respectively. The spin part is antisymmetric for the scalar diquarkϵabc[saT(x)Cγ5sb(x)] and symmetric for the axial-vector diquarkϵabc[saT(x)Cγμsb(x)] . The spatial wave function is antisymmetric and symmetric corresponding to the the ρ-mode and λ-mode excitation, respectively. For example, if the spin angular momentum of the diquark is 0, the excitation in theΩb state should be the ρ-mode, and we have the baryon-multiplet [Ωb , 1, 0, ρ]. Consequently, the P-waveΩb states can be classified into four multiplets, i.e., [Ωb , 1, 0, ρ], [Ωb , 0, 1, λ], [Ωb , 1, 1, λ], and [Ωb , 2, 1, λ], and the corresponding interpolating currents are● [
Ωb , 1, 0, ρ]:J1/2,−,Ωb,1,0,ρ(x)=iϵabc{[DμsT(x)]aCγ5sb(x)−sT(x)Cγ5[Dμs(x)]b}γμγ5bc(x),Jα3/2,−,Ωb,1,0,ρ(x)=iϵabc{[DμsT(x)]aCγ5sb(x)−sT(x)Cγ5[Dμs(x)]b}Γαμbc(x),
(3) with
Γαμ=gαμ−14γαγμ ,● [
Ωb , 0, 1, λ]:J1/2,−,Ωb,0,1,λ(x)=iϵabc{[DμsT(x)]aCγμsb(x)+sT(x)Cγμ[Dμs(x)]b}bc(x),
(4) ● [
Ωb , 1, 1, λ]:J1/2,−,Ωb,1,1,λ(x)=iϵabc{[DμsT(x)]aCγνsb(x)+sT(x)Cγν[Dμs(x)]b}σμνbc(x),Jα3/2,−,Ωb,1,1,λ(x)=iϵabc{[DμsT(x)]aCγνsb(x)+sT(x)Cγν[Dμs(x)]b}Γαμν1bc(x),
(5) with
Γαμν1=(gαμγν−gανγμ−14γαγμγν+14γαγνγμ)γ5 ,● [
Ωb , 2, 1, λ]:Jα3/2,−,Ωb,2,1,λ(x)=iϵabc{[DμsT(x)]aCγνsb(x)+sT(x)Cγν[Dμs(x)]b}Γαμν2bc(x),Jα1α25/2,−,Ωb,2,1,λ(x)=iϵabc{[DμsT(x)]aCγνsb(x)+sT(x)Cγν[Dμs(x)]b}Γα1α2μνbc(x),
(6) where
Γαμν2=(gαμγν+gανγμ−12gμνγα)γ5,
(7) Γα1α2μν=gα1μgα2ν+gα1νgα2μ−13gα1α2gμν−16gα1μγα2γν−16gα1νγα2γμ−16gα2νγα1γμ−16gα2μγα1γν.
(8) In the above equations,
Dμ(x)=∂μ−igsAμ(x) is the gauge-covariant derivative; a, b, and c are color indices; C is the charge conjugation operator; T denotes the matrix transpose on the Dirac spinor indices; ands(x) andb(x) are the strange and bottom quark fields, respectively. -
To obtain the mass sum rules for the P-wave excited
Ωb states, we begin with the following two-point correlation function of the interpolating currents constructed in the previous subsection,Πα1α2⋯αj−12β1β2⋯βj−12(p)=i∫dx4eipx⟨0∣T[Jα1α2⋯αj−12j,P,Ωb,jl,sl,ρ/λ(x)×ˉJβ1β2⋯βj−12j,P,Ωb,jl,sl,ρ/λ(0)]∣0⟩.
(9) First, we need to phenomenologically represent the two-point correlation function (9) in terms of hadronic parameters. To this end, we insert a complete set of states with the same quantum numbers as the interpolating field, perform the integral over space-time coordinates, and finally obtain
Π(Phy)α1⋯αj−12β1⋯βj−12(p)=1m2j,P,Ωb,jl,sl,ρ/λ−p2×⟨0|Jα1⋯αj−12j,P,Ωb,jl,sl,ρ/λ|j,P,Ωb,jl,sl,ρ/λ,p⟩×⟨j,P,Ωb,jl,sl,ρ/λ,p|ˉJβ1⋯βj−12j,P,Ωb,jl,sl,ρ/λ|0⟩+higherresonances.
(10) We parameterize the matrix element
⟨0|Jα1α2⋯αj−12j,P,Ωb,jl,sl,ρ/λ |j,P,Ωb,jl,sl,ρ/λ,p⟩ in terms of the current-hadron coupling constant (pole residue)fj,P,Ωb,jl,sl,ρ/λ and spinoruα1α2⋯αj−12(p) ,⟨0|Jα1α2⋯αj−12j,P,Ωb,jl,sl,ρ/λ|j,P,Ωb,jl,sl,ρ/λ,p⟩=fj,P,Ωb,jl,sl,ρ/λuα1α2⋯αj−12(p).
(11) As a result, we have
● for spin-
12 baryon:Π(Phy)(p)=f21/2m21/2−p2(⧸p+m1/2)+higherresonances,
(12) ● for spin-
32 baryon:Π(Phy)α1β1(p)=f23/2m23/2−p2(⧸p+m3/2)(−gα1β1+γα1γβ13+2pα1pβ13m23/2−pα1γβ1−pβ1γα13m3/2)+higherresonances,
(13) ● for spin-
52 baryon:Π(Phy)α1α2β1β2(p)=f25/2m25/2−p2(⧸p+m5/2)[˜gα1β1˜gα2β2+˜gα1β2˜gα2β12−˜gα1α2˜gβ1β25−110(γα1γβ1+γα1pβ1−γβ1pα1m5/2−pα1pβ1m25/2)˜gα2β2−110(γα2γβ1+γα2pβ1−γβ1pα2m5/2−pα2pβ1m25/2)˜gα1β2−110(γα1γβ2+γα1pβ2−γβ2pα1m5/2−pα1pβ2m25/2)˜gα2β1−110(γα2γβ2+γα2pβ2−γβ2pα2m5/2−pα2pβ2m25/2)˜gα1β1]+higherresonances,
(14) where we have used the following formulas
∑su(p,s)ˉu(p,s)=⧸p+m1/2,
(15) ∑suα1(p,s)ˉuβ1(p,s)=(⧸p+m3/2)(−gα1β1+γα1γβ13+2pα1pβ13m23/2−pα1γβ1−pβ1γα13m3/2),
(16) ∑suα1α2(p,s)ˉuβ1β2(p,s)=(⧸p+m5/2)[˜gα1β1˜gα2β2+˜gα1β2˜gα2β12−˜gα1α2˜gβ1β25−110(γα1γβ1+γα1pβ1−γβ1pα1m5/2−pα1pβ1m25/2)˜gα2β2−110(γα2γβ1+γα2pβ1−γβ1pα2m5/2−pα2pβ1m25/2)˜gα1β2−110(γα1γβ2+γα1pβ2−γβ2pα1m5/2−pα1pβ2m25/2)˜gα2β1−110(γα2γβ2+γα2pβ2−γβ2pα2m5/2−pα2pβ2m25/2)˜gα1β1],
(17) with
˜gμν=gμν−pμpνp2 .Conversely, the correlation function (9) can be calculated theoretically via the OPE method at the quark-gluon level. We take the current
J1/2,−,Ωb,1,0,ρ(x) as an example to illustrate the involved technologies. Inserting the interpolating currentJ1/2,−,Ωb,1,0,ρ(x) (3) into the correlation function (9) and contracting the relevant quark fields using Wick's theorem, we findΠ(OPE)(p)=−4iϵabcϵa′b′c′∫d4xeipxγμγ5S(b)cc′(x)γμ′γ5×{Tr[γ5S(s)bb′(x)γ5C∂μx∂μ′yS(s)Taa′(x−y)C]−Tr[γ5∂μxS(s)bb′(x)γ5C∂μ′yS(s)Taa′(x−y)C]}y=0+4ϵabcϵa′b′c′∫d4xeipxgsAμad(x)γμγ5S(b)cc′(x)γμ′γ5×{Tr[γ5∂μ′yS(s)bb′(x−y)γ5CS(s)Tda′(x)C]−Tr[γ5S(s)bb′(x)γ5C∂μ′yS(s)Tda′(x−y)C]}y=0+⟨0|gsˉsσ⋅Gs|0⟩96ϵabcϵa′b′c′∫d4xeipxgsγμγ5S(b)cc′(x)γμ′γ5×(λn2)ad{(λn2)da′xνTr[γ5∂μ′yS(s)bb′(x−y)γ5σμν]−(λn2)bb′xνTr[γ5∂μ′yS(s)da′(x−y)γ5σμν]}y=0, (18) where a, b,
⋯ are color indices,λn,n=1,2,⋯,8 are the Gell-Mann matrix,Aμad(x)=Anμ(x)(λn2)ad is the gluon field,gs is the strong interaction constant, andS(b)(x) andS(s)(x) are the full bottom- and strange-quark propagators, respectively, whose expressions are given in Appendix A. Inserting the expressions for the full quark propagators into (18) and performing the involved integrals, we haveΠ(OPE)(p)=⧸p(∫∞(mb+2ms)2dsρ(s)s−p2+m2s⟨0|ˉss|0⟩212(m2b−p2))+otherLorentzstructures,
(19) where
ρ(s) is the QCD spectral densityρ(s)=−364π4∫1aminda(1−a)3a2(m2b−as)3+3m2s16π4∫1aminda(1−a)2a(m2b−as)2−3ms⟨0|ˉss|0⟩4π2∫1aminda(1−a)(m2b−as)−m2b⟨0|g2sGG|0⟩256π4∫1aminda(1−a)3a2−5⟨0|g2sGG|0⟩256π4∫1aminda(1−a)(m2b−as)−m2s⟨0|g2sGG|0⟩192π4(1−amin)2−ms⟨0|ˉss|0⟩⟨0|g2sGG|0⟩96π2M2B(1−amin),
(20) with
amin=m2b/s ; here,ms is the mass of the strange quark,mb is the mass of the bottom quark. andM2B is the Borel parameter, introduced to make the Borel transform in the next step.Finally, we match the phenomenological side (12) and the QCD representation (19) for the Lorentz structure
⧸p ,f21/2,−,Ωb,1,0,ρm21/2,−,Ωb,1,0,ρ−p2+higherresonances=∫∞(mb+2ms)2dsρ(s)s−p2+m2s⟨0|ˉss|0⟩212(m2b−p2),
(21) According to the quark-hadron duality, the higher resonances can be approximated by the QCD spectral density above some effective threshold
s1/2,−,Ωb,1,0,ρ0 ,f21/2,−,Ωb,1,0,ρm21/2,−,Ωb,1,0,ρ−p2+∫∞s1/2,−,Ωb,1,0,ρ0dsρ(s)s−p2=∫∞(mb+2ms)2dsρ(s)s−p2+m2s⟨0|ˉss|0⟩212(m2b−p2).
(22) Subtracting the contributions of the excited and continuum states, we obtain
f21/2,−,Ωb,1,0,ρm21/2,−,Ωb,1,0,ρ−p2=∫s1/2,−,Ωb,1,0,ρ0(mb+2ms)2dsρ(s)s−p2+m2s⟨0|ˉss|0⟩212(m2b−p2),
(23) To improve the convergence of the OPE series and suppress the contributions from the excited and continuum states, it is necessary to make a Borel transform. As a result, we have
f21/2,−,Ωb,1,0,ρe−m21/2,−,Ωb,1,0,ρ/M2B=∫s1/2,−,Ωb,1,0,ρ0(mb+2ms)2dsρ(s)e−s/M2B+m2s⟨0|ˉss|0⟩212e−m2b/M2B,
(24) where
M2B is the Borel parameter. Applying the operator−dd(1/M2B) to (24) and dividing the resulting equation with (24), we obtain the mass sum rulem21/2,−,Ωb,1,0,ρ=−dd(1/M2B)(∫s1/2,−,Ωb,1,0,ρ0(mb+2ms)2dsρ(s)e−s/M2B+m2s⟨0|ˉss|0⟩212e−m2b/M2B)∫s1/2,−,Ωb,1,0,ρ0(mb+2ms)2dsρ(s)e−s/M2B+m2s⟨0|ˉss|0⟩212e−m2b/M2B.
(25) In Sec. III, we will numerically analyze (25) and (24) and estimate the values of the mass
m1/2,−,Ωb,1,0,ρ and the pole residuef1/2,−,Ωb,1,0,ρ .For other interpolating currents, we do the same analysis, and the corresponding OPE results are given in Appendix B.
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The sum rule (25) contains some parameters, various condensates, and quark masses, whose values are presented in Table 1. The values of
mb andms are the¯MS values. In addition to these parameters, we need to determine the working intervals of the threshold parametersj,P,Ωb,jl,sl,ρ/λ0 and the Borel massM2B in which the masses and pole residues are stable. We take the continuum threshold to be approximatelymj,P,Ωb,jl,sl,ρ/λ+(0.7± 0.1)GeV , while the Borel parameter is determined by demanding that both the contributions of the higher states and continuum are sufficiently suppressed and the contributions coming from higher dimensional operators are small.Table 1. Input parameters required for calculations.
We define two quantities: the ratio of the pole contribution to the total contribution (Pole Contribution, abbreviated PC) and the ratio of the highest dimensional term in the OPE series to the total OPE series (Convergence, abbreviated CVG), as follows,
PC≡∫sj,P,Ωb,jl,sl,ρ/λ0(mb+2ms)2dsρ(s)e−sM2B∫∞(mb+2ms)2dsρ(s)e−sM2B,CVG≡∫sj,P,Ωb,jl,sl,ρ/λ0(mb+2ms)2dsρ(d=7)(s)e−sM2B∫sj,P,Ωb,jl,sl,ρ/λ0(mb+2ms)2dsρ(s)e−sM2B,
(26) where
ρ(d=7)(s) are the terms proportional to⟨0|ˉss|0⟩⟨0|g2sGG|0⟩ in the spectral density.For the current
J1/2,−,Ωb,1,0,ρ(x) , the numerical results are shown in Fig. 1. In Fig. 1(a), we compare the various condensate contributions as functions ofM2B withs1/2,−,Ωb,1,0,ρ0=6.952GeV2 . From the figure, it is clear that the OPE has good convergence. Fig. 1(b) shows PC and CVG varying withM2B ats1/2,−,Ωb,1,0,ρ0=6.952GeV2 . The figure shows that the requirementPC⩾50 % givesM2B⩽5.5GeV2 . The dependences of the massm1/2,−,Ωb,1,0,ρ and the pole residuef1/2,−,Ωb,1,0,ρ on the Borel parameterM2B are depicted in Fig. 1(c) and (d) at three different values ofs1/2,−,Ωb,1,0,ρ0 , respectively. It is obvious that the mass and the pole residue are stable in the interval4.5GeV2⩽M2B⩽5.5GeV2 . The mass and the pole residue are estimated to bem1/2,−,Ωb,1,0,ρ=(6.28+0.11−0.10)GeV andf1/2,−,Ωb,1,0,ρ=(0.35±0.06)GeV4 , respectively.
Figure 1. (color online) For the interpolating current
J1/2,−,Ωb,1,0,ρ(x) : (a) denotes the various condensate contributions as functions ofM2B withs1/2,−,Ωb,1,0,ρ0=6.952GeV2 ; (b) represents PC and CVG varying withM2B ats1/2,−,Ωb,1,0,ρ0=6.952GeV2 ; (c) and (d) depict the dependence of the mass and the pole residue onM2B with three different values ofs1/2,−,Ωb,1,0,ρ0 , respectively.For other interpolating currents, the same analysis can be performed. We summarize our results in Table 2 and compare the obtained masses with the results in Ref. [20] estimated using the QCD sum rule method in the framework of heavy quark effective theory. It is clear that they are in agreement with each other within the inherent uncertainties of the QCD sum rule method, except for the multiplet [
Ωb , 0, 1, λ]. We should provide some arguments regarding the result of the interpolating currentJ1/2,−,Ωb,0,1,λ(x) shown in Fig. 2. From Eqs. (B3) and (B4), we can see that all terms of the OPE series are proportional to the strange quark massms orm2s , except for the second term in (B4). As a result, the gluon-condensate term is much larger than the other terms, and OPE is invalid in this case. Moreover, the corresponding mass and pole residue are much lower than the others. All in all, our model can not give reasonable results in this case.Multiples Baryons ( jP )
Masses/ GeV 
Pole residues/ GeV4 
This work Ref. [20] [ Ωb , 1, 0, ρ]
Ωb (
12− )
6.28+0.11−0.10 
6.32+0.12−0.10 
0.35±0.06 
Ωb (
32− )
6.31+0.10−0.11 
6.32+0.12−0.10 
0.19±0.03 
[ Ωb , 0, 1, λ]
Ωb (
12− )
5.75+0.05−0.02 
6.34±0.11 
0.0183+0.0013−0.0007 
[ Ωb , 1, 1, λ]
Ωb (
12− )
6.33+0.10−0.11 
6.34+0.09−0.08 
0.62±0.10 
Ωb (
32− )
6.37+0.10−0.11 
6.34+0.09−0.08 
0.36+0.06−0.05 
[ Ωb , 2, 1, λ]
Ωb (
32− )
6.34+0.09−0.10 
6.35+0.13−0.11 
0.71±0.11 
Ωb (
52− )
6.54+0.07−0.08 
6.36+0.13−0.11 
0.15±0.02 
Table 2. Masses and pole residues of the P-wave excited
Ωb states.
Figure 2. (color online) For the interpolating current
J1/2,−,Ωb,0,1,λ(x) : (a) denotes the various condensate contributions as functions ofM2B withs1/2,−,Ωb,0,1,λ0=6.52GeV2 ; (b) representsRP andRH varying withM2B ats1/2,−,Ωb,0,1,λ0=6.52GeV2 ; (c) and (d) depict the dependence of the mass and the pole residue onM2B with three different values ofs1/2,−,Ωb,0,1,λ0 , respectively. -
In this paper, we consider all P-wave
Ωb states represented by interpolating currents with a derivative and calculate the corresponding masses and pole residues using the QCD sum rule method. The results are summarized in Table 2. Because of the large uncertainties in our calculation compared with the small difference in the masses of the excitedΩb states observed by the LHCb collaboration, it is necessary to study other properties of the P-waveΩb states represented by the interpolating currents investigated in the present work to gain a better understanding of the four excitedΩb states observed by the LHCb collaboration. For example, we could study their decay widths. Our results in this paper are necessary input parameters when studying their decay widths using the QCD sum rule method or light-cone sum rule method. -
One of the authors, Yong-Jiang Xu, thanks Hua-Xing Chen for useful discussion on the construction of interpolating currents.
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The full quark propagators are
Sqij(x)=i⧸x2π2x4δij−mq4π2x2δij−⟨ˉqq⟩12δij+i⟨ˉqq⟩48mq⧸xδij−x2192⟨gsˉqσGq⟩δij+ix2⧸x1152mq⟨gsˉqσGq⟩δij−igstaijGaμν32π2x2(⧸xσμν+σμν⧸x)+⋯
for light quarks, and
SQij(x)=i∫d4k(2π)4e−ikx[⧸k+mQk2−m2Qδij−gstaijGaμν4σμν(⧸k+mQ)+(⧸k+mQ)σμν(k2−m2Q)2+⟨g2sGG⟩12δijmQk2+mQ⧸k(k2−m2Q)4+⋯]
for heavy quarks. In these expressions,
ta=λa2 andλa are the Gell-Mann matrices,gs is the strong interaction coupling constant, andi,j are color indices. -
We choose the Lorentz structures
⧸p ,⧸pgαβ , and⧸pgα1α2gβ1β2 to obtain the sum rules for spin-1/2 , spin-3/2 , and spin-5/2 baryons, respectively. In this appendix, we will give the corresponding OPE results.For the interpolating current
Jα3/2,−,Ωb,1,0,ρ(x) ,Π(OPE)αβ(p)=⧸pgαβ(∫∞(mb+2ms)2dsρ(s)s−p2−m2s⟨0|ˉss|0⟩224(m2b−p2))+otherLorentzstructures,
where
ρ(s) is the QCD spectral density,ρ(s)=1384π4∫amin
For the interpolating current
J_{1/2,-,\Omega_{b},0,1,\lambda}(x) ,\Pi^{(\rm OPE)}(p) = {\not{p}}\left(\int^{\infty}_{(m_{b}+2m_{s})^{2}}{\rm d}s\frac{\rho(s)}{s-p^2} +\frac{2m^{2}_{s}\langle0|\bar{s}s|0\rangle^{2}}{3(m^{2}_{b}-p^{2})}+\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{192\pi^{2}(m^{2}_{b}-p^{2})}\right)+{\rm{other\; Lorentz\; structures}}, \tag{B3}
where
\rho(s) is the QCD spectral density,\begin{aligned}[b] \rho(s) = &-\frac{3m^{2}_{s}}{32\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{2}}{a}(m^{2}_{b}-as)^{2}-\frac{\langle0|g^{2}_{s}GG|0\rangle}{128\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a(1-a)(m^{2}_{b}-as) \\&+\frac{m^{2}_{s}\langle0|g^{2}_{s}GG|0\rangle}{384\pi^{4}}(1-a_{\min})^{2} +\frac{3m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{16\pi^{2}}\int^{1}_{a_{\min}}{\rm d}aa. \end{aligned}\tag{B4}
For the interpolating current
J_{1/2,-,\Omega_{b},1,1,\lambda}(x) ,\Pi^{(\rm OPE)}(p) = {\not{p}}\left(\int^{\infty}_{(m_{b}+2m_{s})^{2}}{\rm d}s\frac{\rho(s)}{s-p^2} +\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{192\pi^{2}(m^{2}_{b}-p^{2})}\right)+{\rm{other\; Lorentz\; structures}}, \tag{B5}
where
\rho(s) is the QCD spectral density,\begin{aligned}[b] \rho(s) = &-\frac{1}{8\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}}{a^{2}}(m^{2}_{b}-as)^{3}+\frac{27m^{2}_{s}}{32\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{2}}{a}(m^{2}_{b}-as)^{2}+\frac{3m_{s}\langle0|\bar{s}s|0\rangle}{\pi^{2}}\int^{1}_{a_{\min}}{\rm d}a (1-a)(m^{2}_{b}-as)\\&-\frac{m^{2}_{b}\langle0|g^{2}_{s}GG|0\rangle}{96\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}}{a^{2}}-\frac{3\langle0|g^{2}_{s}GG|0\rangle}{128\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a\frac{(1-a)^{2}}{a}(m^{2}_{b}-as) +\frac{\langle0|g^{2}_{s}GG|0\rangle}{128\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a(1-a)(m^{2}_{b}-as)\\& -\frac{3m^{2}_{s}\langle0|g^{2}_{s}GG|0\rangle}{128\pi^{4}}(1-a_{\min})^{2}-\frac{m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{16\pi^{2}}\int^{1}_{a_{\min}}{\rm d}a(4-7a) +\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{24\pi^{2}M^{2}_{B}}(1-a_{\min}) \\&-\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{96\pi^{2}s}a_{\min}. \end{aligned} \tag{B6}
For the interpolating current
J^{\alpha}_{3/2,-,\Omega_{b},1,1,\lambda}(x) ,\Pi^{(\rm OPE)\alpha\beta}(p) = {\not{p}}g^{\alpha\beta}\left(\int^{\infty}_{(m_{b}+2m_{s})^{2}}{\rm d}s\frac{\rho(s)}{s-p^2} -\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{576\pi^{2}(m^{2}_{b}-p^{2})}\right)+{\rm{other\; Lorentz\; structures}}, \tag{B7}
where
\rho(s) is the QCD spectral density,\begin{aligned}[b] \rho(s) = &\frac{1}{96\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}(3+a)}{a^{2}}(m^{2}_{b}-as)^{3}-\frac{3m^{2}_{s}}{32\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{2}(2+a)}{a}(m^{2}_{b}-as)^{2}\\&-\frac{m_{s}\langle0|\bar{s}s|0\rangle}{2\pi^{2}}\int^{1}_{a_{\min}}{\rm d}a (1-a)(1+a)(m^{2}_{b}-as)+\frac{m^{2}_{b}\langle0|g^{2}_{s}GG|0\rangle}{1152\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}(3+a)}{a^{2}}\\&-\frac{\langle0|g^{2}_{s}GG|0\rangle}{768\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a\frac{(1-a)^{2}(4-a)}{a}(m^{2}_{b}-as) -\frac{\langle0|g^{2}_{s}GG|0\rangle}{768\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a(1-a)(1+a)(m^{2}_{b}-as)\\& +\frac{m^{2}_{s}\langle0|g^{2}_{s}GG|0\rangle}{384\pi^{4}}(1-a_{\min})^{2}(2+a_{\min})-\frac{m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{48\pi^{2}}\int^{1}_{a_{\min}}{\rm d}a(3-4a+4a^{2})\\& -\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{144\pi^{2}M^{2}_{B}}(1-a_{\min})(1+a_{\min}) -\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{288\pi^{2}s}a_{\min}(1-a_{\min}). \end{aligned}\tag{B8}
For the interpolating current
J^{\alpha}_{3/2,-,\Omega_{b},2,1,\lambda}(x) ,\Pi^{(\rm OPE)\alpha\beta}(p) = {\not{p}}g^{\alpha\beta}\left(\int^{\infty}_{(m_{b}+2m_{s})^{2}}{\rm d}s\frac{\rho(s)}{s-p^2} +\frac{5m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{576\pi^{2}(m^{2}_{b}-p^{2})}\right)+{\rm{other\; Lorentz\; structures}}, \tag{B9}
where
\rho(s) is the QCD spectral density,\begin{aligned}[b] \rho(s) = &\frac{1}{96\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}(7+13a)}{a^{2}}(m^{2}_{b}-as)^{3}-\frac{3m^{2}_{s}}{32\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{2}(6+a)}{a}(m^{2}_{b}-as)^{2}\\&-\frac{m_{s}\langle0|\bar{s}s|0\rangle}{2\pi^{2}}\int^{1}_{a_{\min}}{\rm d}a (1-a)(5-7a)(m^{2}_{b}-as)+\frac{m^{2}_{b}\langle0|g^{2}_{s}GG|0\rangle}{1152\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}(7+13a)}{a^{2}}\\&-\frac{\langle0|g^{2}_{s}GG|0\rangle}{384\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a(1-a)(2+a)(m^{2}_{b}-as) +\frac{m^{2}_{s}\langle0|g^{2}_{s}GG|0\rangle}{384\pi^{4}}(1-a_{\min})^{2}(6+a_{\min})\\&+\frac{m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{48\pi^{2}}\int^{1}_{a_{\min}}{\rm d}a(1-4a+18a^{2}) -\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{144\pi^{2}M^{2}_{B}}(1-a_{\min})(5-7a_{\min})\\& -\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{288\pi^{2}s}a_{\min}(1+3a_{\min}). \end{aligned}\tag{B10}
For the interpolating current
J^{\alpha_{1}\alpha_{2}}_{5/2,-,\Omega_{b},2,1,\lambda}(x) ,\Pi^{(\rm OPE)\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}}(p) = {\not{p}}g^{\alpha_{1}\alpha_{2}}g^{\beta_{1}\beta_{2}}\left(\int^{\infty}_{(m_{b}+2m_{s})^{2}}{\rm d}s\frac{\rho(s)}{s-p^2} +\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{1728\pi^{2}(m^{2}_{b}-p^{2})}\right) +{\rm{other\; Lorentz\; structures}}, \tag{B11}
where
\rho(s) is the QCD spectral density,\begin{aligned}[b] \rho(s) = &\frac{1}{288\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}(1+a)}{a}(m^{2}_{b}-as)^{3}-\frac{1}{288\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a\frac{(1-a)^{4}(1+2a)}{a}s(m^{2}_{b}-as)^{2}\\& -\frac{1}{144\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a(1-a)^{5}s^{2}(m^{2}_{b}-as)-\frac{m^{2}_{s}}{32\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a (1-a)^{2}(m^{2}_{b}-as)^{2}\\&+\frac{m^{2}_{s}}{48\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a (1-a)^{3}s(m^{2}_{b}-as)-\frac{m_{s}\langle0|\bar{s}s|0\rangle}{18\pi^{2}}\int^{1}_{a_{\min}}{\rm d}a a(1-a)^{2}(3m^{2}_{b}-(a+1)s)\\&+\frac{m^{2}_{b}m_{s}\langle0|\bar{s}s|0\rangle}{18\pi^{2}}a_{\min}(1-a_{\min})^{3} +\frac{m^{2}_{b}\langle0|g^{2}_{s}GG|0\rangle}{3456\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a \frac{(1-a)^{3}(1+a)}{a}\\&-\frac{\langle0|g^{2}_{s}GG|0\rangle}{1152\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a(1-a)(2-a^{2})(m^{2}_{b}-as) +\frac{\langle0|g^{2}_{s}GG|0\rangle}{6912\pi^{4}}\int^{1}_{a_{\min}}{\rm d}a(1-a)^{2}(3-4a^{2})s\\& +\frac{\langle0|g^{2}_{s}GG|0\rangle s}{3456\pi^{4}}(1-a_{\min})^{4}+\frac{m^{2}_{b}\langle0|g^{2}_{s}GG|0\rangle}{3456\pi^{4}}a_{\min}(1-a_{\min})^{3}\\& -\frac{\langle0|g^{2}_{s}GG|0\rangle s^{2}}{10368\pi^{4}M^{2}_{\rm B}}(1-a_{\min})^{5}-\frac{m^{2}_{s}\langle0|g^{2}_{s}GG|0\rangle}{3456\pi^{4}}(1-a_{\min})^{2}(1-4a_{\min})\\& +\frac{m^{2}_{s}\langle0|g^{2}_{s}GG|0\rangle s}{3456\pi^{4}M^{2}_{\rm B}}(1-a_{\min})^{3}-\frac{m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{72\pi^{2}}\int^{1}_{a_{\min}}{\rm d}aa^{2}(1-2a)\\&-\frac{m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{72\pi^{2}}\int^{1}_{a_{\min}}{\rm d}aa(1-a)-\frac{m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{432\pi^{2}}a_{\min}(1-a_{\min})(1-8a_{\min})\\&-\frac{m^{2}_{b}m_{s}\langle0|g_{s}\bar{s}\sigma\cdot Gs|0\rangle}{108\pi^{2}M^{2}_{\rm B}}a_{\min}(1-a_{\min})^{2} -\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{1296\pi^{2}M^{2}_{\rm B}}(1-a_{\min})^{2}(4+a_{\min})\\& +\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle s}{1296\pi^{2}M^{4}_{\rm B}}(1-a_{\min})^{2}(5-4a_{\min})-\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle s^{2}}{1296\pi^{2}M^{6}_{\rm B}}(1-a_{\min})^{3}\\& -\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{5184\pi^{2}s}a_{\min} +\frac{m_{s}\langle0|\bar{s}s|0\rangle\langle0|g^{2}_{s}GG|0\rangle}{5184\pi^{2}M^{2}_{\rm B}}a_{\min}. \end{aligned}\tag{B12}
In the above equations,
a_{\min} = m^{2}_{b}/s , andM^{2}_{\rm B} is the Borel parameter.
P-wave Ωb states: masses and pole residues
- Received Date: 2021-10-25
- Available Online: 2022-04-15
Abstract: In this study, we consider all P-wave
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